Physics Reports vol.298

DENSITY FUNCTIONAL. THEORY AND APPLICATION TO ATOMS AND MOLECULES A¨ . NAGY Institute of Theoretical Physics, Kossuth L...

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DENSITY FUNCTIONAL. THEORY AND APPLICATION TO ATOMS AND MOLECULES

A¨ . NAGY Institute of Theoretical Physics, Kossuth Lajos University, H-4010 Debrecen, Hungary

AMSTERDAM — LAUSANNE — NEW YORK — OXFORD — SHANNON — TOKYO

Physics Reports 298 (1998) 1—79

Density functional. Theory and application to atoms and molecules A¨. Nagy Institute of Theoretical Physics, Kossuth Lajos University, H-4010 Debrecen, Hungary Received August 1997; editor: J. Eichler

Contents 1. 2. 3. 4.

Introduction Hohenberg—Kohn theorems The method of constrained search The Kohn—Sham scheme 4.1. Non-interacting system 4.2. Spin density functional theory 5. Exact theorems, relations, inequalities 5.1. Long-range asymptotic form of the density and the potentials 5.2. Exchange-correlation hole 5.3. Virial theorems 5.4. Coordinate scaling 5.5. Hierarchy of equations for the energy functional 5.6. Functional expansions 5.7. Adiabatic connection and perturbation theory 6. Fundamental concepts based on density functional theory 6.1. Chemical potential and electronegativity 6.2. Hardness and softness 6.3. Fukui function and local softness

4 5 6 7 7 10 12 12 13 15 20 24 26 28 29 29 30 31

6.4. Density functional theory as thermodynamics 6.5. Work formalism 7. Optimized potential method 8. Potentials from electron density 9. Functionals 9.1. Local density approximation (LDA) 9.2. Approximations containing the gradient of the density 10. Applications 10.1. Atoms 10.2. Molecules 11. Extensions of the density functional theory 11.1. Finite-temperature density functional theory 11.2. Density functional theory for excited states 11.3. Current density functional theory 11.4. Time-dependent density functional theory 11.5. Relativistic density functional theory 12. Concluding remarks References

33 35 37 39 41 41 43 45 45 48 50 50 52 60 62 68 71 73

Abstract The density functional theory is one of the most efficient and promising methods of quantum physics and chemistry. It is a theory of electronic structure formulated in terms of the electron density as the basic unknown function instead of the electron wave function. According to the fundamental theorems of Hohenberg and Kohn, the electron density carries all the information one might need to determine any property of the electron system. However, the way of obtaining it, is not at all trivial. In this report, the recent advances are summarized. After a review of the Hohenberg—Kohn theorems, the 0370-1573/98/$19.00 Copyright ( 1998 Elsevier Science B.V. All rights reserved PII S 0 3 7 0 - 1 5 7 3 ( 9 7 ) 0 0 0 8 3 - 5

A! . Nagy / Physics Reports 298 (1998) 1—79

3

method of constrained search and the Kohn—Sham scheme, exact theorems, relations and inequalities are discussed. There are several important concepts of chemistry (e.g. electronegativity, hardness, softness) that have recently obtained a firm foundation in the density functional theory. The optimized potential method and the methods that generate the potential from the electron density are reviewed. The local and nonlocal approximate functionals are compared. Extensions of the ground-state density functional theory (excited states, time-dependent, relativistic and finite temperature) are summarized. A review of the applications to atoms and molecules is presented. ( 1998 Elsevier Science B.V. All rights reserved. PACS: 31.15.Ew Keywords: Density functional theory; Hohenberg—Kohn theorems; Constrained search; Kohn—Sham theory


4

1. Introduction The story of the density functional theory dates back to the pioneering work of Thomas and Fermi [1]. After the fundamental steps taken by Gomba´s [2], Dirac [3], Slater [4] and Ga´spa´r [5] the theory has been given a firm foundation by Hohenberg and Kohn [6] and Kohn and Sham [7]. Since then an enormous amount of work has been done in this field and this approach to the solution of the many-electron problem has become competitive in accuracy with modern quantum chemical methods. An impressive development has taken place in the formalism, the basic principles and, due to the construction of more and more reliable functionals, in the applications. Several monographs, textbooks, reviews [8—39] have been devoted to this novel approach. The growing number of reviews reflects the great interest of the community of physicists and chemists. Because of the vast number of papers in the subject, even a review could focus on only selected key points. In this survey the basic theorems, relations and fundamental concepts are pinpointed, applications are outlined only from the point of view of judging the progress in the theory. We begin with a short summary of the ground state density functional theory. The essence of the ground state density functional theory is that a knowledge of the ground state electron density is sufficient in principle to determine all molecular properties. This statement can be simply understood following Bright Wilson’s [38] argument: A well-known theorem of quantum mechanics, Kato’s theorem [40] states that

K

1 n(r) Z "! b 2n(r) r

,

(1)

r/Rb where the partial derivatives are taken at the nuclei b. So the cusps of the density tell us where the nuclei (R ) are and what the atomic numbers Z are. On the other hand, the integral of the density b b gives us the number of electrons:

P

N" n(r) dr .

(2)

Thus, from the density, the Hamiltonian can be readily obtained from which every property can be determined. Certainly, we do not follow this way leading to the traditional quantum mechanics. (We have to mention, however, that this argument can only be applied to the Coulomb potential, while the density functional theory is valid for any local external potential.) The basic theorem of the novel treatment of the many-body problem, the Hohenberg—Kohn theorem is discussed in Section 2. Section 3 summarizes the constrained search method, one of the most powerful techniques of the density functional theory. Section 4 presents the Kohn—Sham scheme. Section 5 covers the exact theorems, relations of the density functional theory, such as relations for the exchange-correlation hole, different forms of the virial theorem, coordinate scaling, hierarchy equations and functional expansions. Fundamental new concepts, based on the density functional theory, including the softness, hardness and the local temparature are reviewed in Section 6. Exchange can be treated exactly via the optimized potential method which is outlined in Section 7. Section 8 describes how the Kohn—Sham and the exchange-correlation potentials can be determined exactly if the electron density is known. Section 9 outlines model functionals, including the local density approximation and approximations containing the gradient of the electron


5

density. Some key points of applications to atoms and molecules are presented in Section 9. Section 10 summarizes extensions of the density functional theory to finite temperature, excited states, magnetic fields, time-dependent phenomena and the relativistic formalism.

2. Hohenberg—Kohn theorems The density functional theory is based on the theorems of Hohenberg and Kohn [6]. Consider a system of N electrons enclosed in a large box and moving under the influence of some time-independent local external potential v(r). In this section only nondegenerate ground-states are considered. The first Hohenberg—Kohn theorem states that v(r) is determined within a trivial additive constant by the knowledge of the electron density n(r). The proof proceeds by reductio ad absurdum. Suppose that there exists another potential v@(r) leading to the same density n(r) and vOv@#const. That means that we have two different ground-state wave functions W and W@ corresponding to the two external potentials v(r) and v@(r) and consequently two different Hamiltonians HK and HK @ with ground-state energies E and E@ . The Hamiltonian is 0 0 N HK "¹K #»K # + v(r ) , %% i i/1 where ¹K and »K are the kinetic and electron—electron repulsion operators: %% 1 N ¹K "! + + 2 , i 2 i/1 N 1 , »K " + %% r ij i:j N »K " + v(r ) . i i/1 Atomic units are used everywhere. From the Rayleigh—Ritz variational principle it follows that

(3)

(4)

(5)

(6)

E "SWDHDWT(SW@DHDW@T"SW@DH@DW@T#SW@DH!H@DW@T 0

P

"E@ # n(r)[v(r)!v@(r)] dr . 0

(7)

Similarly, using the variational principle for the Hamiltonian H@ with the trial function W, we have E@ "SW@DH@DW@T(SWDH@DWT"SWDHDWT#SWDH@!HDWT 0

P

"E # n(r)[v@(r)!v(r)] dr . 0

(8)


6

Addition of Eqs. (7) and (8) leads to contradiction and one concludes that the density determines the external potential, consequently the Hamiltonian and thus all electronic properties of the system. If we write the total energy as

P

E [n]" n(r)v(r) dr#F [n] , v HK

(9)

the functional F [n] is the sum of the kinetic and electron—electron repulsion energies. HK The second Hohenberg—Kohn theorem states that for any trial density nJ E 4E[nJ ] 0 if nJ (r)50 and N":nJ (r) dr. The proof is based on the variational principle as for any trial wave function WI

P

SWI DHK DWI T" nJ (r)v(r) dr#F[nJ ]"E [nJ ]5E [n] . v v

(10)

(11)

Equality stands only in the true ground-state. The variation of the total energy at constant number of electrons

G

d E [n]!k v

CP

DH

n(r) dr!N

"0

(12)

leads to the Euler equation dF [n] dE [n] , k" v "v(r)# HK dn dn

(13)

where the Lagrange multiplicator k is the chemical potential (or the negative of the electronegativity). The functional F [n] is defined only for those trial n(r) that are v-representable. A v-representaHK ble density is one that is associated with a ground-state wave function of some Hamiltonian with a local external potential. The conditions for a density to be v-representable are yet unknown. It was demonstrated [41—43], however, that there exists a proper universal variational functional, which delivers the sum of the kinetic and repulsion energies and which does not require the density to be v-representable. This theory of constrained search is discussed in the next section.

3. The method of constrained search The v-representability problem was solved by the constrained search method by Levy and Lieb [41—43]. A universal functional F[n] is defined as a sum of kinetic and repulsion energies: F[n]"Min SWD¹K #»K DWT . %% W ?n

(14)


7

F[n] searches all wave functions W which yield the fixed trial density n. n need not be vrepresentable. The ground-state energy is searched in two steps:

T G T G C

U

N E "Min WD¹K #»K # + v(r )DW %% i 0 W i/1 N "Min Min WD¹K #»K # + v(r )DW %% i W n i/1 ?n

P

UH

DH

"Min Min SWD¹K #»K DWT# v(r)n(r) dr . %% W ?n n Using the definition of F[n], Eq. (15) can also be written as

G

P

(15)

H

E "Min F[n]# v(r)n(r) dr 0 n "Min E[n] , n

(16)

where

P

E[n]"F[n]# v(r)n(r) dr .

(17)

For v-representable densities F[n]"F [n] . (18) HK The functional F[n] is universal because it is independent of the external potential v. The constrained search formulization eliminates the limitation of the Hohenberg—Kohn theorems that there be no degeneracy in the ground state. In the constrained search only one of a set of degenerate wave functions is selected, the one corresponding to the density n. The method of constrained search is frequently applied in the density functional theory (see Sections 4, 5.2, 8 and 11.2).

4. The Kohn—Sham scheme 4.1. Non-interacting system The ground-state electron density can be in principle determined by solving the Euler Eq. (13): dF[n] #v(r)"k . dn

(19)

However, we do not know the exact form of the functional F[n]: F[n]"¹[n]#» [n] . %% Slater [4], Ga´spa´r [5] and Kohn and Sham [7] proposed the following scheme:

(20)


8

A non-interacting system, in which the electrons move independently in a common local potential v is constructed. The Hamiltonian is 4 N 1 N HK " + ! + 2 # + v (r ) . (21) 4 4 i 2 i i/1 i/1 Substituting the non-interacting wave function

A

B

1 W" det Mu ,u ,2,u N , 4 JN! 1 2 N

(22)

into the Schro¨dinger equation of the non-interacting system HK W "E W 4 4 4 4 we obtain the one-electron equations

(23)

hK u "M!1+2#v (r )Nu . 4 i i 4 i 2 i The kinetic energy of the non-interacting system is

(24)

T K A

BK U

TK

KU

1 N 1 N ¹ " W + ! +2 W " + u ! +2u , 4 4 4 i 2 i 2 i i i/1 i/1 while the density of the non-interacting system

(25)

N n(r)" + Du (r)D2 (26) i i/1 is equal to that of the interacting one. The exact kinetic energy functional ¹ is unknown, so we simply take the kinetic energy functional ¹ of the non-interacting system instead of ¹. Substituting ¹ into Eq. (20) of F[n] an 4 4 extra term, the difference ¹ "¹!¹ appears: # 4 F[n]"¹ [n]#» [n]#¹ [n] , (27) 4 %% # » [n]#¹ [n]"J[n]#E [n] , (28) %% # 9# i.e. F[n]"¹ [n]#J[n]#E [n] . (29) 4 9# With the help of the exchange-correlation energy functional E [n] the total energy E[n] has the 9# form

P

E[n]" n(r)v(r) dr#¹ [n]#J[n]#E [n] . 4 9#

(30)

The variation of Eq. (30) leads to the Euler equation dE[n] d¹ [n] dJ[n] dE [n] k" "v(r)# 4 # # 9# . dn dn dn dn

(31)


9

It can also be written as d¹ [n] k"v (r)# 4 , KS dn

(32)

where v (r)"v(r)#vJ(r)#v (r) KS 9# is the Kohn—Sham potential consisting of the external v, the classical Coulomb

P

dJ[n] n(r@) v (r)" " dr@ J dn Dr!r@D

(33)

(34)

and the exchange-correlation dE [n] v (r)" 9# , 9# dn

(35)

potentials. Taking the variation of the total energy expressed with the one-electron orbitals u i N N 1 E[u ]" + v(r)Du (r)D2 dr# + u*(r) ! + 2 u (r) dr i i i i 2 i/1 i/1 N 1 N N Du (r)D2 Du (r@)D2 i j dr dr@#E n" + Du (r)D2 . # + + 9# i 2 Dr!r@D i/1 i/1 j/1 with the constraint of orthonormality of the orbitals u i

P

P C

P

P

u*(x)u (x) dx"d , i j ij

D

C

D

(36)

(37)

we obtain

C

D

1 N (38) hK KSu " ! + 2#vKS(r) u " + e u . i i ij j 2 i/1 Making use of the fact that the operator hK is Hermitian, a unitary transformation of the orbitals KS leads to Kohn—Sham equations [!1+ 2#v (r)]u "e u . (39) 2 KS i i i This is probably the most important equation of the density functional theory. It tells us that the motion of the interacting electrons can be treated exactly as a system of independent particles. The electrons can be considered as if they move in a common effective local potential v . All the KS interaction between the electrons can be merged exactly into a single local potential v . KS The Kohn—Sham equations can also be derived with constrained search. In the following subsection this technique will be applied to obtain the Kohn—Sham equations of the spin density functional theory.


10

4.2. Spin density functional theory Up to this point we studied systems with scalar external potential only. The density functional theory was extended to systems in external magnetic fields by von Barth and Hedin [44] and Pant and Rajagopal [45]. To characterize a system in the presence of magnetic field B(r) we need more information beyond the electron density. The new quantity is the magnetization or electron spin density Q(r)"n (r)!n (r) , (40) ¬ which is defined as the difference of the electron densities of electrons with spin up n and down n . ¬ The Hamiltonian has the form 1 N N 1 N N HK "! + + 2# + # + v(r )#2b + B(r) ) s , i i % i 2 r i:j ij i/1 i/1 i/1 where

(41)

e+ b" % 2mc

(42)

is the Bohr magneton and s is the spin vector of the ith electron. Here, the interaction of the i magnetic field with the electronic current is neglected. (The current density functional theory is outlined in Section 11.3.) Let us consider the interaction with the external field N N »K " + v(r )#2b + B(r) ) s (43) i % i i/1 i/1 in the case of z-direction magnetic field b(r). Then the expectation value of »K has the form

P

P

SWD»K DWT" v(r)n(r) dr! b(r) ) m(r) dr ,

(44)

where m(r)"b (n (r)!n (r)) % ¬ is the magnetization density. The constrained search leads to

T G G

U

N N E "Min WD¹K #»K # + v(r )#2b + b(r )s DW 0 %% i % i zi W i/1 i~1 "Min n,n¬

(45)

P

Min SWD¹K #»K DWT# [v(r)n(r)!b(r)m(r)] dr %% ?n,n¬

W

P

H H

"Min F[n ,n ]# dr [(v(r)#b b(r))n (r)#(v(r)!b b(r))n (r)] , ¬ % % ¬ n,n¬

(46)


11

where F[n ,n ]" Min SWD¹K #»K DWT %% ¬ W ?n,n¬ is a universal functional that can be written as

(47)

F[n ,n ]"¹ [n ,n ]#J[n #n ]#E [n ,n ] ¬ 4 ¬ ¬ 9# ¬ to obtain the Kohn—Sham equations. The non-interacting kinetic energy has the form

(48)

C

P

D

1 ¹ [n ,n ]"Min ! + f dr u* (r)+ 2u (r) ip ip ip 4 ¬ 2 ip where

(49)

n (r)"+ f Du (r)D2 . (50) p ip ip ip p denotes C or B. The occupation numbers f are chosen so that the lowest eigenstates are occupied ip ( f "1) and the rest are unoccupied ( f "0). The minimization of the total energy ip ip 1 (51) E[n ,n ]"+ f dr u*(r) ! + 2#v (r) u (r)#J[n #n ]#E [n ,n ] i ip i ¬ 9# ¬ ¬ ip 2 ip subject to normalization constraint

P

A

B

P

Du (r)D2 dr"1 ip

(52)

leads to the spin-polarized Kohn—Sham equations: [!1+ 2#vKSr(r)]u (r)"e u (r) , ip ip ip 2 where

P

n(r@) dr@#v (r) , v (r)"v (r)# 9#p KSp p Dr!r@D

(53)

(54)

v (r)"v(r)#b b(r) , (55) % v (r)"v(r)!b b(r) , (56) ¬ % dE [n , n ] v (r)" 9# ¬ . (57) 9#p dn p The spin density functional theory can be applied even in the absence of magnetic fields. In this case the results should reduce to the spin-compensated results provided that the exact Kohn—Sham potential is applied. However, because the exact form of the exchange-correlation functional is unknown and one should use approximate potentials the results are different and the spin density functional theory generally provides a better description of a system. This is surely the case for open-shell atoms or molecules.


12

5. Exact theorems, relations, inequalities 5.1. Long-range asymptotic form of the density and the potentials The asymptotic expression for the Kohn—Sham orbitals [46] u (r)"C / (r)[1#O(r~1)] , i i i

(58)

where the functions / (r)"rbi exp(!i r) i i

(59)

decay exponentially. i "(!2e )1@2 , i i

(60)

b "Z /i !1 , i %&& i

(61)

Z "Z!N#1 . %&&

(62)

Z"+Z and N are the total nuclear charge of the atom or molecule and the number of electrons, b respectively. It can also be proven [46] that the uppermost occupied Kohn—Sham orbital energy is equal to the negative of the ionization potential e "!I. Consequently, the density has the m asymptotic behaviour: n(r)"r2bm exp(!2i r)[1#O(r~1)] . m

(63)

Studying the large-r expansion of the external and the classical electrostatic potentials the asymptotic form of the exchange-correlation potential is given by [46]:

AB

1 1 , v (r)"! #O 9# rp r

(64)

where p"4 for a nondegenerate spin-unpolarized or for the spin-up potential of a spin-polarized ground-state, otherwise p"3. The long range behavior of the exchange potential can be written as [47—50]

AB

1 1 . v (r)"! #O 9 r2 r

(65)

As the leading term in both the exchange-correlation and the exchange potentials is the same the correlation potential does not play any role in the asymptotic region. Another important exact expression for the density is the cusp condition [40] that has already been mentioned in the introduction (Eq. (1)). The cusp of the exchange and exchange-correlation potentials [51] have also been studied. Another set of properties of considerable interest is obtained by imposing rigorous bounds on the density and expectation values [52].


13

5.2. Exchange-correlation hole The electron—electron interaction energy can be written [53] as

P

1 SUD»K DUT" C»K dr dr . %% %% 1 2 2

(66)

The two-particle density matrix C can be split into four terms: C 1 2, where p corresponds to spin i p ,p up or down [53]. The pair-correlation function h 1 2 is defined as p ,p C 1 2(r ,r )"(1#h 1 2(r ,r ))n 1(r )n 2(r ) . p ,p 1 2 p 1 p 2 p ,p 1 2

(67)

To obtain the exchange-correlation energy of the density functional theory the coupling constant [54—56] integrated pair-correlation function [54] (see Section 5.7) should be determined: hM

P

(r ,r )" p1, p2 1 2

1

0

hj1 2(r ,r ) dj . p ,p 1 2

(68)

hj1 2(r ,r ) is the pair-correlation function corresponding to the Hamiltonian p ,p 1 2 HK "¹K #j»K #»K . j %% j

(69)

The coupling constant integration is done in such a way that the density remains constant at any value of the coupling strength j. j"1 and j"0 give the fully interacting and the non-interacting cases, respectively. The exchange-correlation energy has the form 1 E [n ]" + 9# p 2 p1, p2

P

n 1(r )n 2(r ) p 1 p 2 hM (r , r ) dr dr . Dr !r D p1, p2 1 2 1 2 1 2

(70)

It can also be expressed with the exchange-correlation hole oN 1 2 [4,57,58]: 9# p ,p 1 E [n ]" + 9# p 2 p1, p2

P

n 1(r )oN 1 2(r , r ) p 1 9#p ,p 1 2 dr dr . 1 2 Dr !r D 1 2

(71)

Using the definition of the exchange and correlation energies one arrives at 1 E [n ]" + 9 p 2 p

P

1 E [n ]" + # p 2 p1, p2

n (r )oN (r , r ) p 1 9r 1 2 dr dr , 1 2 Dr !r D 1 2

(72)

P

(73)

n 1(r )oN 1 2(r , r ) p 1 #p ,p 1 2 dr dr . 1 2 Dr !r D 1 2

The exchange or Fermi hole can be expressed with the help of the one-particle density matrix c: oN (r , r )"!1Dc (r , r )D2/n (r ) . p 1 9p 1 2 2 p 1 2

(74)


14

The exchange-correlation, the exchange and the correlation holes satisfy the following sum rules:

P

+ oN 1 2(r , r ) dr "!1 , 9#p , p 1 2 2 p2

P

(75)

oN 1(r , r ) dr "!1 , 9p 1 2 2

(76)

P

(77) + oN 1 2(r , r ) dr "0 . #p , p 1 2 2 p2 These sum rules can serve important tests of approximations. Recently, there is considerable interest in the properties of the so-called on-top electron pair density [59] defined as the diagonal of the two-particle density matrix C(r , r )"n(r )[n(r )#o (r , r )] , (78) 1 2 1 2 9# 1 2 where o (r , r ) is the exchange-correlation hole surrounding an electron at r (without the 9# 1 2 1 coupling constant integration). [n(r )#o (r , r )] is the conditional probability to find an electron 2 9# 1 2 in dr at r , given that there is one at r . For a uniform electron gas Eq. (78) becomes 2 2 1 C6/*&(n ,n ;w)"n[n#o6/*&(n ,n ;w)] (79) ¬ 9# ¬ with r "r #w. Then the on-top pair density for an electron gas with uniform spin densities 2 1 n and n is given by ¬ CI (r, r)"C6/*&(n ,n ;w"0) . (80) ¬ Numerical tests [59] show that there is a satisfactory agreement between the exact on-top exchange-correlation hole density and its LSD and GGA approximations. Based on this finding an alternative interpretation of the spin-density-functional theory (LSD) has been put forward. It is possible to set up a formal density functional theory [59] that yields the exact energy, density n and on-top pair density C(r,r). Using the constrained search approach a new functional SWD¹K #»K DWT (81) Min %% ?n/n`n¬ ? (n,n¬§w/0) is introduced. This means that the search is over all antisymmetric wave functions that yield the density n and the on-top pair density CI (r, r) until the minimum expectation value is reached. The total energy is defined FI [n ,n ]" ¬

W CI C6/*&

A

P

B

(82) EI "Min FI [n ,n ]# dr v(r)[n (r)#n (r)] . ¬ ¬ ¬ n ,n From the Rayleigh—Ritz variational principle EI is larger or equal to the true ground-state energy. The standard spin-density-functional theory allows to predict the spin magnetization density n !n , but the reliability of this prediction is more questionable than that of CI (r,r). This ¬ alternative theory is exact in the fully spin-polarized and low density limits and more accurate than LSD or GGA and does not lead to a symmetry dilemma [59].


15

5.3. Virial theorems The virial theorem of quantum mechanics in the case of a Coulomb potential in equilibrium molecular geometry has the form ¹"!E ,

(83)

where ¹ and E are the kinetic and total energies, respectively. This theorem is often used to judge the quality of approximate wave functions. Naturally, this theorem holds also in the density functional theory. However, in the Kohn—Sham scheme of the density functional theory, it takes a somewhat different form [60]: ¹ #¹ "!E . (84) 4 # The non-interacting kinetic energy ¹ differs from the interacting kinetic energy ¹. As the 4 difference ¹ is positive # ¹ '0 (85) # we arrive at the inequality ¹ (!E . 4 Using the Kohn—Sham equations one can easily derive the Levy—Perdew relation [60]

P

¹ #E "! nr ) +v (r) dr . 9# # 9#

(86)

(87)

This equation combined with Eq. (26) leads to another form of the Levy—Perdew relation [60]:

P

F#¹"! nr ) +

dF dr . dn

(88)

In the exchange-only case the Levy—Perdew relation takes the form

P

E "! n(r)r ) +v (r) , 9 9 where the exchange potential v is the functional derivative of the exchange energy E : 9 9 dE v (r)" 9 . 9 dn

(89)

(90)

In this case the virial theorem (Eq. (84)) reduces to ¹ "!E , 4 because

(91)

¹ "0 . (92) # In the following subsections the differential, the local and the spin virial theorems are summarized. For a recent review of other forms of the virial theorem, e.g., the integral and the regional virial theorems, see Ref. [61].


16

5.3.1. Differential virial theorem The differential virial theorem has been derived by Holas and March [62]. The starting point of the derivation is the many-body Schro¨dinger equation: HK W"EW ,

(93)

that can be written as »K #»K !E"!(¹K WR%)/WR%"!(¹K WI.)WI. , (94) %% Differentiating Eq. (94) with respect to r , then multiplying with (WR%)2, then repeating the 1b procedure after replacing WR% by WI. and adding the two final equations and integrating, we get the differential virial theorem of Holas and March:

P

n(r)+v# dr@C(r, r@)+ w(r,r@)"1+ 2+n(r)!2 div pL HM , 3 4

(95)

where C(r, r@) is the diagonal of the two-particle density matrix and pL HM is the kinetic energy density tensor defined by pL HM(r)"1(+"+ @#+ @"+ )o(r, r@)]D . 4 r/r{ The differential virial theorem of Holas and March can be written in the following form:

+v"!f (r; [w, n, o, C]) ,

(96)

(97)

where

CP

f (r; [w, n, o, C])"

D

1 dr@C(r, r@)+ w(r, r@)! + 2+n(r)#2 div pL HM /n(r) . r 4

o(r, r@) is the one-electron density matrix and w(r, r@)"1/Dr!r@D .

(98)

(99)

Eq. (97) presents an exact expression from which the external potential can be readily determined by a line integral or from a knowledge of the external potential the exchange-correlation potential can in principle be obtained. In the Kohn—Sham scheme of the density functional theory, Eq. (97) has the form:

+v "!f (r;[n,o ]) , 4 4 4 where

(100)

(101) f (r; [n, o ])"[!1+ 2+n(r)#2 divpL HM]/n(r) . 4 4 4 4 (The subscript s refers everywhere to the non-interacting system.) For spherically symmetric systems the differential virial theorem of Holas and March Eq. (95) reduces to the differential virial theorem of Nagy and March [63] 1 1 q@ q q@"! . @@@! . v@ # ! , 8 2 KS r2 r3

(102)


17

where the radial kinetic energy density has the form

K

K

1 2. (r@,r) q(r)"! 2 r2

. (r) 1 # + l (l #1) k , r2 2 k k r{/r k

(103)

1 q" + l (l #1). . (104) k 2 k k k . , . and v are the radial electron density, the radial electron density of orbital k and the k KS Kohn—Sham potential, respectively. For particles having zero angular momentum i.e. for s electrons the differential virial theorem reduces to the special form of March and Young [64] (105) q@"!1. @@@!1. v@ . 2 KS 8 Recently, using the differential virial theorem for spherically symmetric system [63] (Eq. (102)) an exact relation for the kinetic energy of atoms (or ions) with one p and one or more s shells has been derived in terms of the total and the s electron densities [65]. 5.3.2. Local virial theorem To derive the local virial theorem [66] we take the gradient of the Euler equation Eq. (13) and multiply by the density: n+

d¹ 4#n+v #n+v*"0 , 9# dn

(106)

where v*"v#v

(107)

div pL !n+v*"0 ,

(108)

J is the total classical electrostatic potential. The force equation can be written as

which gives the condition of static equilibrium for the system. The stress tensor pL was introduced by Bartolotti and Parr [67]. Following Ghosh and Berkowitz [68] the stress tensor pL connected with 4 the non-interacting kinetic energy is defined by d¹ 4. div pL "!n+ 4 dn

(109)

Deb and Ghosh [69] determined the form of pL . As can be seen from Eqs. (108) and (109) the 4 definition of the stress tensor is not unique and the corresponding non-interacting kinetic energy can also have several forms. The pressure is defined as p"!1tr pL . 3 The pressure connected with the non-interacting kinetic energy takes the form

(110)

p "2 t . 4 3 4 With the definitions of the exchange-correlation stress tensor pL

(111)

div pL "!n+v 9# 9#

9# (112)


18

and the exchange-correlation pressure p "!1tr pL , 9# 3 9# using the total stress tensor

(113)

pL "pL #pL , 4 9# Eqs. (111) and (113) and Eq. (114) lead to the local virial theorem

(114)

2t "3(p!p ) . 4 9# If the exchange—correlation stress tensor has the form

(115)

pL "!p IK 9# 9# the local virial theorem takes the form

(116)

P

=

n(r@)+r v (r@) ) dr@ . { 9# r In the local density approximation the local virial theorem reads 2t "3p#3 4

de 2 p" t #n2 9# , dn 3 4

(117)

(118)

where e (n)"ne (n) is the exchange-correlation energy density. 9# 9# Ghosh et al. [70] introduced the concept of local temperature into the density functional theory. The local temperature ¹(r) was defined by the ideal gas expression for the kinetic energy (119) t "3nk¹ , 4 2 where k is the Boltzmann constant. With this definition the local virial theorem provides the virial equation of state p"nk¹#p

9# and in the local density scheme p"nk¹#n2

de 9# . dn

(120)

(121)

This equation shows that the deviation of the real system from the ideal system (for which p"nk¹) is due to the exchange-correlation effects which contain a kinetic contribution. It is interesting to note that for a microscopic system the virial equation of state has a closed form and unlike for the macroscopic system a virial expansion is not needed. Local virial relations have also been derived by Bader et al. [71] and Zietsche et al. [72]. (For a discussion see Ref. [73].) Several forms of the integral virial theorem [74] can be obtained by integrating the local virial theorem Eq. (115) on the whole space. The Levy—Perdew relation (Eq. (109)) leads to an integral form of the virial theorem which is the generalization of the theorem derived by Bartolotti and Parr [67]. (Another form of the virial theorem was given by Ghosh and Parr [75].) It was shown [74] that the orbital energy sum can be approximated by the integral of the pressure — :p dr. In recent years there has been considerable interest in the general properties of subdomains of systems.


19

Regional virial theorem has been derived within the wave function theory [76] and in the density functional theory [77,74]. The regional analogue of the Levy—Perdew relation has also been derived [74,61]. 5.3.3. Spin virial theorem The spin virial theorem connects the difference of the spin-up and -down kinetic and potential energies [78,79]. The starting point of the derivation is given by the spin-polarized Kohn—Sham equations (Eq. (53)). After some manipulation we get the spin virial theorem

P

2(¹!¹¬)" dr(n r ) +v !n r ) +v ) . 4 4 KS ¬ KS¬

(122)

¹p and 4 n "+ u* u (123) p ip ip i (p) are the electron density and the non-interacting kinetic energy for spin p, respectively. The spin virial theorem is independent from the virial theorem. Contrary to the virial theorem the spin virial theorem has no classical counterpart; it is a completely quantum mechanical theorem. The spin virial theorem for free atoms, ions and molecules has the form 2D¹ "!D» , 4 where

(124)

D¹ "¹!¹¬ , 4 4 4 D»"» !» "w !w #y !y #q #x !x . ¬ ¬ ¬¬ ¬ ¬ The first and second terms in Eq. (126)

(125)

P C

D

Z Z p #R ) + p dr w !w "Dw"! Q+ p pDr!R D ¬ Dr!R D p p p come from the electron—nuclear attraction, where

(126)

(127)

Q(r)"n (r)!n (r) (128) ¬ is the (spin) magnetization density. The third and fourth terms in Eq. (126) arise from the electron—electron repulsion

P P P

1 n (r )n (r ) 1 2 dr dr , y " 1 2 2 Dr !r D 1 2 1 n (r )n (r ) ¬ 1 ¬ 2 dr dr , y " ¬¬ 2 Dr !r D 1 2 1 2 n (r ) n (r ) q " 1 ¬ 2 dr dr . ¬ 1 2 Dr !r D 1 2

(129) (130) (131)


20

The difference between the exchange-correlation virials gives the last two terms of Eq. (126):

P

x "! n r ) +v dr . p 9#p p

(132)

Another form of the spin virial theorem can be given as

P

Z p dr , DE#D¹ "DE !q !Dx# Q+ R ) + 4 9# ¬ p pDr!R D p p where

(133)

DE"E !E , (134) ¬ DE "E !E . (135) 9 9# 9#¬ This form of the theorem connects the total and kinetic energy differences. The spin-up and spin-down total energies are defined as

P

P

Z 1 p dr# n (r)/(r) dr#E . E "¹p! n (r)+ p 4 p 9#p Dr!R D 2 p p p The exchange-correlation energy

P

1 n (r ) p 1 . (r ,r ) dr dr E " 9#p 2 r 9#p 1 2 1 2 12 is expressed with average exchange-correlation hole

(136)

(137)

.

(r ,r )"+ n (r )hM (r ,r ) , (138) 9#p 1 2 p{ 2 9#pp{ 1 2 p{ where hM (r ,r ) is the coupling-constant averaged pair-correlation function. The spin virial 9#pp{ 1 2 theorem can be derived in the presence of magnetic field, too [79]. The spin virial theorem has conceptual and practical significance. It can be used to check the accuracy of the approximate spin orbitals. As the spin virial theorem is independent of the virial theorem, this is a new way of checking the accuracy of spin orbitals. 5.4. Coordinate scaling The density functional theory constitutes an enormous simplification of the many-electron problem. However, the exact exchange-correlation functional must be approximated. Levy and coworkers [60] have shown that coordinate scaling provides a powerful tool for constructing approximate potentials. (For reviews see Refs. [80,81].) Let U(r ,2,r ) be an eigenfunction of the 1 N Hamiltonian HK . The (uniform) coordinate scaling means that the coordinates r are changed into i ar , where a is any real constant. The scaled wave function has the form: i U (r ,2,r )"a3N@2U(ar ,2,ar ) . (139) a 1 N 1 N


21

The factor a3N@2 is present to preserve normalization to N electrons. From the Rayleigh—Ritz variational principle d SU DHK DU TD "0 . a a/1 da a

(140)

One can easily check the scaling of the kinetic and potential energies: SU D¹K DU T"a2SUD¹K DUT , a a SU D»K DU T"aSUD»K DUT , a %% a %% SU D»K DU T"aSUD»K DUT , a a where the scaled density can be written as

(141) (142) (143)

n (r)"a3n(ar) . (144) a The Hohenberg—Kohn theorem implies that the ground-state wave function, the kinetic and the electron—electron potential energies are functionals of the electron density n ¹[n]"SU[n]D¹K DU[n]T ,

(145)

» [n]"SU[n]D»K DU[n]T . (146) %% %% To derive scaling relations for the functionals ¹[n] and » [n] we use the method of constrained %% search ¹[n ]#» [n ]"F[n ]"Min SWD¹K #»K DWT a %% a a %% W ?na "SW aD¹K #»K DW aT(SU aD¹K #»K DU aT , aO1 , n %% n n %% n where the wave function W a is a solution of the Schro¨dinger equation n (¹K #»K #»K )W a"E aW a . %% n n n From Eqs. (141)—(143) the inequalities

(147)

(148)

¹[n ]#» [n ](a2¹[n]#a» [n] , aO1 (149) a %% a %% can be obtained. The wave function U differs from the wave function W (if aO1) because U is a a a a solution of the following Schro¨dinger equation: EU (r ,2,r )"HK (ar ,2,ar ) [a3N@2U(ar ,2,ar )] a 1 N 1 N 1 N 1 1 1 " ¹K # »K # »K U (r ,2,r ) . %% a 1 N a a a2

A

B

(150)

Or (¹K #a»K #a»K )U "(a2E)U . %% a a

(151)

22


It means that the wave function U minimizes the functional SWD¹K #a»K DWT. It leads to the a %% inequality SU D¹K #a»K DU T(SW aD¹K #a»K DW aT , aO1 . a %% a n %% n Making use of this inequality and Eqs. (141)—(143) one obtains

(152)

a2(¹[n]#» [n])(¹[n ]#a» [n ] , aO1 . %% a %% a Eqs. (149)—(153) lead to the inequalities:

(153)

¹[n ](a2¹[n] , a'1 , (154) a ¹[n ]'a2¹[n] , a(1 , (155) a » [n ](a» [n] , a(1 , (156) %% a %% » [n ]'a» [n] , a'1 . (157) %% a %% On the other hand, the scaling relation for the non-interacting kinetic energy ¹ takes the form 4 ¹ [n ]"a2¹ [n] (158) 4 a 4 as for the case of » "0, U "W a. %% a n From the definition of the exchange and correlation energies E [n]"SU.*/D»K DU.*/T!J[n] , (159) 9 n %% n E [n]"SW.*/D¹K #»K DW.*/T!SU.*/D¹K #»K DU.*/T , (160) # n %% n n %% n where U.*/ is the wave function that yields the given density n and minimizes the kinetic energy n ¹ [n]"SU.*/D¹K DU.*/T , (161) 4 n n we arrive at E [n ]"aE [n] , 9 a 9 E [n ]OaE [n] . # a # The correlation energy satisfies the following inequalities: E [n ](aE [n] , a(1 , # a # E [n ]'aE [n] , a'1 # a # coming from Eqs. (156), (157) and (159). Eq. (160) leads to the correlation energy E [n ]"SW.*/ D¹K #»K DW.*/ T!SU.*/ D¹K #»K DU.*/ T # a na %% na na %% na "a2[SW.*/,jD¹K DW.*/,jT!SU.*/D¹K DU.*/T] n n n n # a[SW.*/,jD»K DW.*/,jT!SU.*/D»K DU.*/T] , n %% n n %% n

(162) (163)

(164) (165)

(166)

as W.*/ "a3N@2W.*/,j(ar ,2,ar ) , j"a~1 . na n 1 N

(167)


23

The wave function W.*/,j yields the density n and minimizes S¹K #j»K T. It follows that the wave na %% function a3N@2W.*/, j(ar ,2,ar ) yields the density n and minimizes S¹K #ja»K T (i.e. S¹#»K T if n 1 N j %% %% j"a~1). For small values of j substituting the perturbation expansion of the wave function W.*/,j n = W.*/,j(r ,2,r )"U.*/(r ,2,r )# + jkg (r ,2,r ) (168) n 1 N n 1 N k 1 k k/1 into Eq. (166) one obtains E [n ]"a[n]#b[n]a~1#c[n]a~2#2 . # a It leads to the inequality

(169)

lim E [n ]'!R . # a a?= For small values of a (aP0 ill. jPR) from Eq. (166) it follows

(170)

E [n ] lim # a "!g[n] , (171) j a?0 where g[n]'0, as SW.*/,jD»K DW.*/,jT!SU.*/D»K DU.*/T tends to a negative constant. n %% n n %% n From the Levy—Perdew relation (87) [60] and Eqs. (159) and (160) we arrive at other important expressions derived by Levy and Perdew [60]:

P

!jEj[n]!j dr n(r)r ) + #

P

Ej[n]# dr n(r)r ) + #

A

A

B

dEj[n] # "¹j[n] , # dn(r)

B

dEj[n] dEj[n] # "j # . dj dn(r)

(172) (173)

Eliminating the integral term between these equations, results in dEj[n] ¹j[n]"!j2 # . # dj

(174)

Making use of the definition of ¹j[n]: # ¹j[n]"SWjD¹K DWjT!SUj/0D¹K DUj/0T , # and Eqs. (172)—(175) we have

P

2¹j[n]# dr n(r)r ) + #

A

B

d¹j[n] d¹j[n] # "j # . dn(r) dj

(175)

(176)

The exact identities Eqs. (173) and (176), respectively, involve Ej[n] and ¹j[n]. # # A set of other exact relations can be derived [82]. These conditions are very important to help approximate the exchange-correlation energy functional. It is often quite easy to ascertain how a functional behaves upon coordinate scaling. E.g. GGA [83](PW92) obeys many of the known exact relationships including those obeyed by the LDA and several others that are violated by


24

LDA. Anologous coordinate scaling relations hold for the second-order density matrix, the pair correlation function and the exchange-correlation hole [84]. Non-uniform coordinate scaling imposes further conditions upon the functionals [85]. 5.5. Hierarchy of equations for the energy functional For development of approximations, exact relations and criteria that are fulfilled by the exact functionals are of great help in constructing new approximate functionals. Hierarchies of equations for the energy functional [86—89] and their Legendre transformations [90] provide such relations. Here, hierarchies for the sum of the kinetic and the electron—electron energies and for the exchange and exchange-correlation are reviewed. Hierarchies of equations have been derived in several fields of physics (e.g. Bogoliubov—Born— Green—Kirkwood—Yvon (BBGKY) hierarchy [91]) or hierarchy equations for reduced density matrices [92]. 5.5.1. Hierarchy of equations for the sum of the kinetic and electron—electron energies Taking the gradient of the Euler—Lagrange equation (Eq. (13)) we obtain

+u(r)"!+

dF[n] , dn(r)

(177)

where dF u(r)"v(r)!k"! . dn

(178)

The zeroth equation of the hierarchy is the universal virial relation of Levy and Perdew (Eq. (88)) [60]. Its functional differentiation with respect to n(r) leads to the first equation of the hierarchy

P

dF[n] d¹[n] dF[n] # "!r ) + ! dr n(r )r ) + g(r, r ; n) , 1 1 1 1 dn(r) dn(r) dn(r)

(179)

where d2 F[n] du(r) g(r, r ; n)" "! (180) 1 dn(r)dn(r ) dn(r ) 1 1 is the hardness kernel [93]. With another functional differentiation we arrive at the second equation of the hierarchy:

P

dg(r, r ; n) d2 ¹[n] 1 . g(r, r ; n)# "!(r ) +#r ) + )g(r, r ; n)! dr n(r )r ) + (181) 1 1 1 2 2 2 2 dn(r ) 1 dn(r)dn(r@) 2 Further differentiations will lead to higher-order equations. The hierarchy of equations links the nth functional derivatives to the (n#1)-th functional derivatives and the electron density.


25

5.5.2. Hierarchy of equations for the exchange-correlation and the exchange energies The starting point of the derivation is the Levy—Perdew relation (Eq. (87)). The first

P

dv (r; n) d¹ [n] v (r; n)# # "!r ) +v (r; n)! dr n(r )r ) + 9# , 9# 9# 1 1 1 1 dn(r ) dn(r) 1 and the second equations of the hierarchy

(182)

P

dv (r; n) 1 d2v (r; n) d2 ¹ [n] dv (r; n) 9# # 9# # "!(r ) +#r ) + ) 9# ! dr n(r )r ) + (183) 1 1 dn(r ) 2 2 2 2dn(r )dn(r ) dn(r)dn(r ) 2 dn(r ) 1 1 2 1 1 can be obtained by subsequent functional differentiation of the Levy—Perdew relation. The functional differentiation of the Levy—Perdew-relation (Eq. (89)) for exchange leads to the first

P

dv (r; n) v (r; n)"!r ) +v (r; n)! dr n(r )r ) + 9 , 9 9 1 1 1 1 dn(r ) 1 the second

P

(184)

dv (r; n) dv (r; n) 1 d2v (r; n) 9 9 "!(r ) +#r ) + ) 9 ! dr n(r )r ) + , (185) 1 1 dn(r ) 2 2 2 2 dn(r )dn(r ) dn(r ) 2 1 1 1 2 and higher order equations of the hierarchy. This hierarchy is self-contained. It means that it contains only the exchange energy functional and its functional derivatives. It is the consequence of the fact that the exchange energy functional scales homogeneously as it has been shown by Ou—Yang and Levy [94]. The hierarchies of equations can be used to check the quality of approximate functionals. One can use first the zeroth order equations of the hierarchies. Inserting the approximate expressions into both sides of the zeroth order equation, the values of two integrals, i.e., the two numbers can be compared and used for the checking. On the other hand, if the first order equations of the hierarchies are applied as criteria for the approximate functionals, the comparison of the two sides of these equations would involve a comparison of two functions of r instead of two numbers. So the first order equations of hierarchy give more information. Similarly, higher order equations of hierarchy provide even more information. Certainly, to apply these equations, higher order derivatives of the functionals are needed. It is interesting to note that the Legendre transforms of energy functionals as functionals of potentials are available. The electron density appears as a functional derivative of Legendre transforms and the first order equations of the hierarchies provide equations for the electron density. The Legendre transform of the exchange energy is studied in the weighted density and gradient approximations. The first order equation of the hierarchy gives useful differential equation for the electron density. This equation can be used in constructing or improving approximating functionals. The hierarchies of equations are exact. The exact functionals satisfy these equations. On the other hand, these equations do not generally hold for approximate functionals. Occasionally, even


26

approximate functionals are exact solutions of the hierarchy equations. For instance the Thomas—Fermi [1] and the Thomas—Fermi—Dirac [3,2] models can be derived from the hierarchy equations for the non-interacting kinetic and exchange energies. The derivation is based on the locality assumption. With the assumption that the functionals are local the exact solutions are the Thomas—Fermi and Thomas—Fermi—Dirac solutions. The truncation of the hierarchies of the kinetic and exchange energies results in rigorous lower bounds to the kinetic energy and upper bounds to the exchange energy in the plane-wave approximation. In both cases an additional assumption was done (locality or truncation). This assumption includes an approximation. (So we cannot obtain the exact solution.) In these cases the hierarchy of equations cannot be used to judge the quality of the approximation and the validity of the approximation can be studied by other methods. 5.6. Functional expansions There is a growing interest in the properties of energy functionals. A recent approach in this field includes functional expansions. Functional expansions have been derived while studying hierarchies of equations for the noninteracting kinetic [86], exchange and exchange-correlation [87] functionals and the Legendre transforms of different energy functionals [90]. Important identities have been obtained making use of functional expansions [86, 87, 90, 95, 96]. For any well-behaved functional F[n] [96] up to a constant

P

PP

1 F[n]" dr n(r)F@(r; n)# 2

dr dr n(r )n(r )F@@(r , r ; n)#2 , 1 2 1 2 1 2

(186)

where it is supposed that the successive functional derivatives (denoted by primes) exist and the series converges at least in a region near n. The functional expansions have recently been applied to generate a transition functional method [97], which can be considered the functional generalization of Slater’s transition state method [98]. This transition functional method can be used to calculate energy differences if the functional derivative is known at the transition density:

P

F[n ]!F[n ]" dr [n (r)!n (r)]F@ (r; n)# third and higher order terms . 2 1 0 2 1

(187)

The second order terms disappeared. The functional derivatives are evaluated at the transition density n (r)"1(n (r)#n (r)) . (188) 0 2 1 2 For example for iso-electronic ions the kinetic and exchange energy differences have the forms

P P

¹ [n ]!¹ [n ]+ dr [n (r)!n (r)]v0 (r) , 4 2 4 1 1 2 KS

(189)

E [n ]!E [n ]+ dr [n (r)!n (r)]v0(r) , 9 2 9 1 2 1 9

(190)


27

respectively, where both the Kohn—Sham v0 and the exchange v0(r) potentials are taken at the KS 9 transition density. The transition functional method is especially useful if the functional itself is unknown, and only the first functional derivative is available. A very important example is as follows: Nowadays there are several methods [99—105] to determine the exchange-correlation potential if the electron density is known (see Section 8). One can obtain, however, the exchangecorrelation potential only as a function of the radial distance and not as a functional of the electron density. So, the exchange-correlation energy cannot be calculated. Using the transition functional method one can determine exchange-correlation energy differences even if the exchange-correlation energies are not known. Several other identities have been derived for local functionals and functionals containing gradient terms. Combining the constrained-search formalism with Taylor series expansions general expansions of E [n] and ¹ [n] in terms of homogeneous functionals can be derived [106—108]: Making the # # general postulate that there exists an expansion of Ej in powers of j # = jk dkEj[n] = jk # Ej[n]" + " + A [n] (191) # k! k! k djk j/0 k/1 k/1 and inserting it into Eq. (173) one obtains

A

P

! dr n(r)r ) +

B

A

B

dA [n] k "(1!k)A [n] . k dn(r)

(192)

It means that the kth component of E [n] and consequently ¹ [n] is a homogeneous functional of # # degree (1!k) in coordinate scaling, i.e., A [n ]"a1~kA [n] , k a k where

k"1,2,2 ,

n "a3n(ar) . a With j"1, Eq. (191) becomes

(193)

(194)

= A [n] E [n]" + k . # k! k/1 Similarly, one can obtain that

(195)

= A [n] k ¹j[n]"! + . (196) # (k!1)! k/1 If one assumes that A [n] is a local functional then up to third order the correlation energy can be k expressed [106] as

P

P

E "C N#C dr n2@3# dr n1@3#C , # 1 2 4

(197)

where N is the total number of electrons and the constants C —C can be determined by fitting. 1 4


28

5.7. Adiabatic connection and perturbation theory The adiabatic connection [54—56] is a key concept in the density functional theory. It is not only supposed that the electron density is the same for both the interacting and non-interacting systems, but there exists a continuous path between them. The coupling constant path is defined by the Schro¨dinger equation [¹K #a»K #»K ]DW T"E DW T . (198) %% a a a a The density is supposed to be the same for any value of the coupling constant a. a"1 corresponds to the fully interacting case, while a"0 gives the Kohn—Sham system. Consider the functional F [n] j F [n]"min S¹K #j»K T . j %% W ?n Noticing that for the interacting system

(199)

F [n]"¹[n]#» [n] , 1 %% while for the non-interacting case

(200)

F [n]"¹ [n] , 0 4 the exchange-correlation energy has the form

(201)

E [n]"F [n]!F [n]!J[n] 9# 1 0 1 F [n] " dj j !J[n] . j 0 Using the Hellmann—Feynmann theorem

P

F [n] j "S»K T , %% j j

(202)

(203)

we obtain the adiabatic connection formula for the exchange-correlation energy:

P

1 dj (S»K T !J) . %% j 0 From the scaling relation for the Coulomb repulsion energy E [n]" 9#

J[n ]"a~1J[n] , 1@a another form of the adiabatic connection formula follows:

P

(204)

(205)

1 a(» [n ]!J[n ]) da . (206) %% 1@a 1@a 0 With the coupling-constant integration important expressions can be derived for the correlation energy [109]. E [n]" 9#


29

It has been shown [110] that the Kohn—Sham potential has the form

C

D

dE [n ] v (r)"v (r)!a v (r)#v (r)#a # j , j"1/a , a 0 J 9 dn(r)

(207)

where v (r) is the non-interacting Kohn—Sham potential, v (r) and v (r) are the classical Coulomb 0 J 9 and exchange potentials, respectively, while E is the correlation energy. The density functional # perturbation theory is based on the Taylor series of v (r) a = v (r)" + akv (r) a k k/0 and E a

(208)

= E " + akE . (209) a k k/0 The zeroth order term is equal to the Kohn—Sham potential, while the higher order terms are related to the Coulomb, exchange and correlation potentials. It was shown that potentials v (r) are k functionals of the Kohn—Sham orbitals, eigenvalues and the potentials v (r),2,v (r). Thus all the 1 k~1 potentials v (r) can be calculated exactly for any given Kohn—Sham potential. In practice, however, k the potentials only up to some finite order can be determined. Similarly, the terms in the series of E can be obtained a

P

E "F [Mu N,Me N,v (r)2v (r)]# n(r)v (r) . k k i i 1 k~1 k

(210)

Comparing the density functional perturbation theory with the conventional quantum chemical perturbation theory (e.g. Møller—Plesset perturbation theory) the important difference is in the fact that the self-consistent procedure leading to the Kohn—Sham wave function already depends on the perturbation theory, so the perturbation comes into play at an earlier stage. With the help of the density functional perturbation theory formally exact expressions can be gained for the exchangecorrelation energy and potential. It seems reasonable that approximate functionals can be directly constructed from the Kohn—Sham orbitals and eigenvalues. It is not necessary to know how these can be explicitly formulated as functionals of the density. It is believed that functionals based directly on the Kohn—Sham orbitals and eigenvalues provide much more freedom in the construction of approximate expressions.

6. Fundamental concepts based on density functional theory 6.1. Chemical potential and electronegativity In the density functional theory the total energy E[n] is a unique functional of the density n. The variational principle leads to the Euler equation (19). The Lagrange multiplier k is the chemical


30

potential that plays an important role in the density functional theory. Another expression for the chemical potential written as

A B

k"

E N

(211) 7 can be easily seen by an infinitesimal change of the given ground-state energy into another ground-state energy:

P

dE"k dN# n(r) dv(r) dr ,

(212)

where v is the external potential. As the chemical potential is constant, i.e., takes the same value in every point of the molecule or solid considered, its analogy with the macroscopic chemical potential is transparent. It is even more important that one can define the electronegativity [111] s with the chemical potential: s"!k .

(213)

Mulliken [112] long ago showed that the absolute electronegativity of a species can be sensibly defined as the average of its ionization potential I and electron affinity A s "1(I#A) . (214) M 2 This is just the finite difference approximation to Eq. (211). From the definition of the electronegativity, Sanderson’s principle [113] immediately follows: the electronegativity equalizes when two species unite to form a new species leading to a single electronegativity or chemical potential (the same way as in ordinary thermodynamics). 6.2. Hardness and softness The hardness g of an electronic system is defined [114] as

A B A B

1 2E 1 k " . (215) g" 2 N2 2 N 7 7 This is a global quantity, often called absolute hardness to emphasize the fact that its value for an atom in some environment can be different from its value in isolation. The global softness [115] is the inverse of global hardness:

A B

N 1 S" " k 2g

. (216) 7 From the convexity of the total energy E(N) follows that g50. It can be easily shown [22] that the global hardness can be approximated by g+1(I!A)+1(e !e ), (217) 2 2 LUMO HOMO where HOMO and LUMO stand for the highest occupied molecular orbital and lowest unoccupied molecular orbital, respectively.


31

The global softness, on the other hand, can be connected with the fluctuation of the particle number [115]. Considering a grand canonical ensemble with temperature ¹, volume » and chemical potential k, the equilibrium fluctuation of the particle number is given by a well-known expression of statistical physics:

A B

1 SNT [SN2T!SNT2]" k¹ k

.

(218)

T,V

Noticing the analogy between Eqs. (216) and (218) the softness is 1 [SN2T!SNT2]"S . k¹

(219)

The concepts of chemical hardness and softness are useful in studying acid—base reactions. It turned out that [115] hard (soft) acids generally have high (low) positive charge, low (high) polarizability and small (large) size. On the other hand, hard (soft) bases possess the property of high (low) electronegativity, low (high) polarizability and it is difficult (easy) to oxidize them. Studying the concepts of chemical hardness and softness, Pearson [116] concluded that there is a principle of maximum hardness, i.e. molecules arrange themselves so as to be as hard as possible. This principle is equivalent to the principle of minimum softness. It was formally proved [117] making use of the fluctuation—dissipation theory of statistical mechanics. Another very important theorem is the so-called HSAB principle. In acid—base reactions hard (soft) acids prefer to coordinate with hard (soft) bases. Two formal proofs [118] are available for this principle. Both the principle of maximum hardness and the HSAB principle proved to be very powerful in the theory of chemical reactions. 6.3. Fukui function and local softness To understand the local properties of molecules or solids, local quantities which vary from place to place are introduced. In order to measure the chemical reactivity of a particular site in a molecule different local variables are defined. The Fukui function [119] f (r) is defined as

C D

f (r)"

dk dv(r)

.

(220)

N

It measures how sensitively the chemical potential reacts to an external perturbation at a particular point [22]. With the invertion of the order of variation and differentiation

A B

A B

dE d E " N dv(r) dv(r) N

(221)

the Fukui function f (r) has the form

A B

f (r)"

. (r) N

v

.

(222)


32

f (r) is established as an index of considerable importance for understanding molecular behaviour. The Fukui function measures reactivity:

A B . A B . A B

. (r) ~ , electrophilic , (223) N v (r) ` f `(r)" , nucleophilic , (224) N v (r) 0 , radical (225) f 0(r)" N v reagents. This reactivity index can be easily approximated using the HOMO and LUMO orbital densities l and l , respectively: HOMO LUMO f ~(r)+l (r) , (226) HOMO f `(r)+l (r) (227) LUMO (r)#l (r)) . (228) f 0(r)+1(l LUMO 2 HOMO The density functional definition of the Fukui function provides a firm foundation of the frontierelectron theory [120]. The local softness [115] is defined as f ~(r)"

C D n(r) k

(229)

S" s(r) dr .

(230)

s(r)"

v(r) and it yields the global softness upon integration:

P

The relationship between local softness and the Fukui function, s(r)"Sf (r) ,

(231)

reflects that these quantities contain the same information about the relative site reactivity in a molecule. For metals, the local and the global softness [115] have the forms s(r)"g(e , r) , (232) F S"g(e ) , (233) F respectively, where g(e , r) and g(e ) are the local and global density of states at the Fermi level. F F Higher functional derivatives have also been derived [93]. The most important ones are the second functional derivatives, the hardness and softness kernels; they have a fundamental role in the hierarchy equations for energy functionals (see Section 5.4).


33

6.4. Density functional theory as thermodynamics 6.4.1. Transcription of ground-state density functional theory into a local thermodynamics Ghosh et al. [70] developed a local thermodynamical picture within the framework of groundstate density functional theory. They introduced concepts like local temperature, local entropy and local free energy density. Recently, this theory has been enlarged into an exact local thermodynamics [121]. The theory of Ghosh et al. provides a phase-space approach to the density functional theory. The density functionals are considered as averages in the phase-space. A distribution function f (r, p) in the phase-space is introduced with the following properties:

P P P

dp f (r, p)"n(r) ,

(234)

dr n(r)"N ,

(235)

dp 1p2f (r, p)"t (r) . 2 4

(236)

The entropy density s and the entropy S are defined as

P

s(r)"!k dp f (ln f!1) ,

P

S" dr s(r) ,

(237) (238)

where k is the Boltzmann constant. The most probable distribution function is obtained by imposing the criterion of maximum entropy subject to the constraints of correct density (Eq. (234)) and correct kinetic energy (Eq. (236)). Thus, f (r, p)"e~a(r)e~b(r)p2@2 ,

(239)

where a(r) and b(r) are r-dependent Lagrange multipliers. The local temperature ¹(r) is defined in terms of the kinetic energy density t (r)"3n(r)k¹(r) , 4 2 i.e. by the ideal gas expression. Eqs. (236), (239) and (240) lead to b(r)"1/k¹(r) .

(240)

(241)

For the entropy density we obtain (242) s(r)"!kn ln n#3 kn ln ¹#1 kn[5#3 ln(2pk)] , 2 2 the Sackur—Tetrode equation. The exchange energy can also be derived [122]. Assuming ¹ to be a function of the density, the Thomas—Fermi—Dirac model can be obtained with numerical values of the coefficients C and C which provide improvements of the original values. Predicted F 9 exchange energies [122] and predicted Compton profiles [123] are very good.


34

6.4.2. The exact thermodynamics This phase-space approach of Ghosh et al. is, however, not exact. It is possible to go beyond this theory by postulating a thermodynamic description without using any statistical method [121]. The electron system under consideration has the ground-state electronic energy functional E,

P

E[n]" e[n; r] dr ,

(243)

where the energy density e is a functional of n. According to the Hohenberg—Kohn theorem the functional E takes its minimum value E[n ]"E (244) 0 0 at the exact ground-state electron density n . The variation of Eq. (244) leads to the Euler— 0 Lagrange equation

K

dE dn

(245)

P

(246)

"k n/n0 using the constraint n dr"N .

The Lagrange multiplier k is the chemical potential of the system. Now, we postulate thermodynamics by requiring the existence of a new functional

P

E[n, s]" e[n, s; r] dr ,

(247)

where the energy density e is a functional of both the density n and the entropy density s. The extremum principle for the ground-state takes the form d

GP

H

(e[n, s; r]!¹(r)s!kJ (r)n) dr "0 .

(248)

The Lagrange multipliers, the local temperature ¹ and the local chemical potential kJ are functions of r. The Euler—Lagrange equations of thermodynamics are

K K

dEI "¹(r) , (249) ds n dEI "kJ (r) . (250) dn s To build up a thermodynamical formulation we require that the local form of the fundamental equation EI "¹S!p»#kJ N

(251)


35

holds, i.e., e"¹s!p#kJ n ,

(252)

where p is the local pressure. The Kohn—Sham kinetic energy density is written as t "3nk¹ . 4 2 Certainly, we have

(253)

¹"¹[n] ,

(254)

e[¹,n]"e[n] .

(255)

We emphasize that generally ¹ is a functional of the density n, not simply a function of n. In the local density approximation, all quantities are functions of n. E.g. ¹(r)"c(n(r)). The entropy contains a correction to the Sackur—Tetrode equation (Eq. (242)) resulting from the nonideality. The locality assumption causes ambiguity in the description as it was shown in Ref. [121]. Taking advantage of this ambiguity in the limiting case of the original density functional theory the model of Ghosh et al. is recovered. There is another way of using the ambiguity: the local chemical potential is taken to be a constant, i.e., equal to the chemical potential of the original density functional theory. In this case, using the scaling argument, the well-known Thomas—Fermi and Thomas—Fermi—Dirac models are recovered. 6.5. Work formalism Harbola and Sahni [124] proposed an interesting interpretation of the exchange-correlation potential. The electron-interaction potential that contains a kinetic energy component, can also be interpreted as a work done to bring an electron from infinity to its position at r against a field E(r):

P

d(E%%[n] ) r v%%(r)" "! E(r@) ) dr@ . dn = The field E(r) can be separated into

(256)

(257) E(r)"E (r)#E #(r) . %% t The field E (r) arises from the classical Coulomb and exchange-correlation terms, while the field %% E #(r) comes from the difference of the interacting and non-interacting kinetic energy tensors. The t potential v%%(r) can be written as a sum of the works: v%%(r)"¼ (r)#¼ #(r) , %% t where

P P

r

E (r@) ) dr@ , %% = r ¼ #(r)"! E #(r@) ) dr@ . t t =

¼ (r)"! %%

(258)

(259) (260)


36

This interpretation is based on the fact that

+v%%(r)"!E(r) .

(261)

The condition on the path-independence of the work is

+]E(r)"0 .

(262)

The sum of the works ¼ (r) and ¼ #(r) is path-independent and for systems of a certain symmetry %% t such as closed shell atoms, jellium metal clusters, etc., the works ¼ (r) and ¼ #(r) are separately %% t path-independent. Following Harbola and Sahni [124] we write the electric field E (r) as a sum of the classical %% Coulomb E (r) and the exchange correlation E (r) terms: J 9# E (r)"E (r)#E (r) , (263) %% J 9# where

P P

E (r)" J

E (r)" 9#

n(r@)(r!r@) dr@ , Dr!r@D3

(264)

o (r, r@)(r!r@) 9# dr@ . Dr!r@D3

(265)

They supposed that

+v (r)"!E (r) . (266) 9# 9# Making use of the exchange-correlation hole density o (r, r@) and the (coupling-constant averaged) 9# pair-correlation function hM (r, r@) (Eq. (68)) the exchange-correlation energy has the form (Eq. (70))

P

1 1 E " n(r)n(r@)hM (r, r@) dr dr@ . 9# 2 Dr!r@D

(267)

It can be rewritten as

P

P

P

o (r, r@)(r!r@) 9# dr@" dr n(r)r ) E (r) . 9# Dr!r@D3

E " dr rn(r) 9#

(268)

Using Eq. (268) with the Levy—Perdew relation (Eq. (87)) we obtain

P

¹ "! n(r)r ) [E (r)#+v (r)] dr . 9# 9# #

(269)

Comparing this relation with Eq. (266) we notice that the Harbola—Sahni conjecture on the exchange-correlation field (Eq. (265)) does not satisfy [125,126] the Levy—Perdew relation. The Harbola—Sahni expression for the exchange can be derived from the differential virial theorem of Holas and March (Eq. (98)) which can be rewritten as a path-integral expression for the exchange-correlation potential [62]:

P

v (r )"! 9# 0

r

0

=

f (r) ) dr . 9#

(270)


37

Neglecting the difference between the interacting and non-interacting density matrices in f , it has 9 the form

P

1 . f (r)"! dr@[C(r, r@)!n(r)n(r@)]/n(r)+ 9 rDr!r@D

(271)

Expressing the diagonal of the two-particle density matrix C(r,r@) with the pair-correlation function h(r, r@) (Eq. (67)) one obtains

P

f (r)" 9

o (r, r@)(r!r@) 9 dr@ . Dr!r@D3

(272)

We recognize f (r) to be identical with the force field E (r) (Eq. (265)) of Harbola and Sahni. 9 9# Finally, we note here that, following the work of Holas and March [62] their expression for v has been generalized by Levy and March [127] for electron—electron interaction je2/r , where 9# ij 0(j41. They thereby have exhibited a kinetic correction to the Harbola—Sahni exchange-only potential. The path dependence of this latter contribution is thereby annulled. The Harbola—Sahni formalism has the advantage, besides the appealing interpretation character, that the exchange-correlation potential in this model shows the exact asymptotic behaviour. Several calculations have been performed using the work formalism [128]. For instance, electron removal energies and electron affinities have been determined for atoms and ions. Even excited state calculations have been done with this method [129].

7. Optimized potential method Though the Kohn—Sham approach to the density functional theory is an exact scheme to treat the ground state properties of many-particle electronic systems, unfortunately, the exchangecorrelation part of this Kohn—Sham potential is not known exactly. The exchange potential, however, can be exactly determined by finding the optimized effective potential. The question of how to obtain the local potential whose eigenfunctions would minimize a given energy functional was first investigated by Sharp and Horton [130]. Their integral equation for the local potential was independently derived by Talman and Shadwick [47]. From calculations for atoms, they and Aashamar et al. [131] found that the calculated one-electron and total energies were very close to that of the Hartree—Fock method. Norman and Koelling [132] combined the optimized effective potential method with the technique of including self-interaction correction proposed by Perdew and Zunger [133]. The optimized effective potential method was generalized for the spin polarized case by Krieger et al. [49]. Krieger et al. [49] introduced an approximate OPM method. Recently, an alternative derivation to the KLI method was proposed [134]. The optimized potential method can be applied when the total energy is given as a functional of the one-electron orbitals u : i

P

E[u ]"¹ [u ]#J[u ]#E [u ]# dr v(r)n(r) , i 4 i i 9# i

(273)


38

where ¹ [u ], J[u ] and E [u ] are the non-interacting kinetic, the Coulomb and the exchange4 i i 9# i correlation energies, respectively. The one-electron orbitals u are eigenfunctions of a local effective i potential » hK u "(!1+ 2#»)u "e u , (274) i 2 i i i with » being determined by requiring that E[u ] is minimized for all u obtained from Eq. (274). i i This results in

P

dE dE du*(r@) i "+ dr@#c.c. "0 . (275) d» du*(r@) d»(r) i i The functional derivative of the one-electron orbitals u with respect to the local effective potential i » can be calculated with the help of Green’s function: du*(r@) i "!G (r@, r)u (r) , i i d»(r)

(276)

(hK !e )G (r@, r)"d(r!r@)!u (r)u*(r@) , (277) i i i i Using Eqs. (274)—(277) an integral equation for the effective exchange-correlation potential » follows: 9#

P

H(r, r@)» (r@) dr@"Q(r) , 9#

H(r, r@)"+ u*(r)G (r, r@)u (r@) , i i i i

P

(278) (279)

(280) Q(r)"+ dr@ u*(r)G (r, r@)vi (r@)u (r@) . i i 9# i i The orbital dependent potential vi is given by 9# dE [u ] vi (r)" 9# i . (281) 9# u du* i i The effective exchange-correlation potential » can be determined from the effective potential »: 9# » (r)"»!v!v . (282) 9# J If the total energy were known as a functional of the one-electron orbitals u Eq. (282) would i result in the exact exchange-correlation potential. If the method is applied to the Hartree—Fock energy functional the exact exchange-only Kohn—Sham potential can be obtained. It was found [131,49,104,50] that the results of OPM and the HF calculations are almost identical, the total energies obtained from OPM are slightly higher than that of the HF. It is very difficult to calculate the effective potential » because of various numerical problems. It has already been done only for spherically symmetric systems. Krieger et al. [49] proposed an


39

accurate approximate approach to OPM. They found that the analytic expression n (r) (283) »KLI(r)"»S(r)#+ i (»M KLI!vN ) , 9i 9 9 n(r) 9i i where n (r) v 9i »S(r)"+ i (284) 9 n(r) i is the Slater potential, is an extremely accurate approximation. »M and vN are the expectation 9i 9i values of the exchange potential »KLI and the Hartree—Fock exchange potentials v with respect to 9 9i orbital u . n (r) is ith one-electron density. i i 8. Potentials from electron density The exact functional form of the Kohn—Sham potential of the density functional theory is still unknown. However, if the density is known it is possible to calculate the Kohn—Sham, the exchange or exchange-correlation potentials. Several methods have been developed to obtain the potentials from the electron density. In this section, the most important methods are outlined [136]. In a density functional calculation, usually the Kohn—Sham equations (Eq. (39)) are solved self-consistently. In the so-called inverse problem, the density is known and from Eqs. (26), (33) and (39) e ,u and v are determined. If we have only two electrons the inverse problem can be trivially i i KS solved: n"2DuD2 ,

(285)

1 1 v "e# + 2Jn . KS 2Jn

(286)

This question has been studied by Almbladh and Pedroza [102], Davidson [135] and Umrigar and Gonze [137]. In this case the exchange potential

P

1 n(r@) v "! dr@ 9 2 Dr!r@D

(287)

corresponds to the self-exchange. The correlation potential takes the form

P

1 n(r@) 1 1 + 2Jn!v! dr . v "v !v "e# # 9# 9 2 Dr!r@D 2Jn

(288)

The nontrivial cases of two- and three-level spherically symmetric systems are detailed in Refs. [61,136]. Several methods have been worked out for systems with arbitrary number of electrons. Stott et al. [103] proposed the following method. The one-electron orbital with the lowest energy can be expressed as

A

B

1@2 N u " n! + u2 . 1 k k/2

(289)


40

The Kohn—Sham potential is given by 1 v " + 2u #e KS 2u 1 1 1 1 N 1@2 " + 2 n! + u2 #e . (290) k 1 2(n!+N u2)1@2 k/2 k k/2 Combining Eqs. (39) and (290) a set of coupled nonlinear second-order differential equations

A

C

B

A

B D

N 1 1@2 1 + 2 n! + u2 u "eJ u , k"2,3,2,N ! + 2u # k k k k k 2(n!+N u2)1@2 2 k/2 k k/2 are obtained, where

(291)

eJ "e !e . (292) k k 1 If the density is known the single-particle wave functions can be determined from Eq. (291). Then Eq. (290) gives the Kohn—Sham potential. Parr and collaborators [101] used the constraint 1 2

PP

[n(r)!n (r)][n(r@)!n (r@)] 0 0 dr dr@"0 Dr!r@D

(293)

to determine the orbitals corresponding to the input density n . The minimization of the kinetic 0 energy min SDD¹K DDT D?n0 using the constraint (293) leads to the one-electron equations

(294)

[!1+2#v(r)#vj(r)]uj(r)"ejuj(r) , 2 # i i i where

(295)

P

vj (r)"j 9#

n(r)!n (r) 0 dr . Dr!r@D

(296)

The Kohn—Sham orbitals are obtained by extrapolating the Lagrange multiplier j to R. To speedup the convergence the one-electron equations

C

A

B

D

1 1 ! + 2#v(r)# 1! vj(r)#vj (r) uj(r)"ejuj(r) 9# i i i 2 N J

(297)

are solved instead of Eq. (295), where the factor (1!(1/N)) in the Coulomb potential

P

n(r@) dr vj" J Dr!r@D

(298)

can be thought of as the Fermi—Amaldi self-interaction correction. The exchange-correlation potential is given by

A

B

1 vj! vj . v "lim 9# j?= # N J

(299)


41

Another method of calculating the Kohn—Sham potential has been proposed by Go¨rling [104]. The linear response

P

dn(r)" dr@ G(r, r@)dv (r@) KS

(300)

of the electron density to the changes of the effective potential v can be obtained from KS perturbation theory. The Green’s function is given by 0## 6/0## u*(r) u (r)u*(r@)u (r@) s s i # c.c. G(r,r@)"2 + + i e !e i 4 i s To determine the change in the effective potential

P

dv (r)" dr@ G~1(r, r@)dn(r@) KS

(301)

(302)

the inverse of the Green’s function should be calculated. Go¨rling proposed an iterative method to obtain the Kohn—Sham potential and one-electron orbitals and energies. The present author [99] has worked out a numerical, iterative method. Starting from an appropriate potential, the Kohn—Sham equations are solved self-consistently then using the Kohn—Sham potential v(1) and the electron-density n(1) obtained, another Kohn—Sham potential KS constructed and the Kohn—Sham equations are solved again. This process goes on until the density equal to the input density is reached. It has turned out that the potential of the (i#1)th iteration can be constructed as

C

D

n »(i`1)"»(i) 0 c#(1!c) , KS KS n(i)

(303)

where c is a damping factor applied to speedup the convergence. From the Kohn—Sham potential the exhange-correlation potential can be readily obtained. If the input density is the exact one, the exact exchange-correlation potential can be determined. However, if only the Hartree—Fock density is available as an input, the exchange-correlation potential obtained by the abovementioned process is very close to the exact exchange potential [104,50]. A method very similar to this has been worked out by Baerends and coauthors [138,105]. As we have seen if the exact density is known, the exchange-correlation potential can be determined exactly. However, it should be emphasized that the exchange-correlation potential can only be obtained as a function of the radial distance r. The form of the exchange-correlation potential as a functional of the density still remained unknown.

9. Functionals 9.1. Local density approximation (¸DA) The density functional theory is exact in principle, but the exact form of the energy functional is unknown, so approximations have to be used in calculations. Numerous approximate methods


42

have already been proposed. The simplest forms are the local expressions. We have to emphasize that the term “local” is used in several meanings: First, a potential may be local in multiplicative sense. E.g. the Kohn—Sham potential (Eq. (39)) is local, whereas the Hartree—Fock exchange potential is nonlocal. Secondly, a functional F is local, if

P

F[n]" dr f (n(r)) ,

(304)

i.e. the density f corresponding to F depends on r only through the electron density n. In this sense the term “local density approximation (LDA)” is used. Third, a potential is local if its value at a given point depends on the value of the density (and finite number of its derivatives) at that point only. In this sense the gradient functionals

P

F[n]" dr f (n(r), n(r), 2 n(r),2) k kl

(305)

are also local. In a great majority of the papers on density functional theory the term “local” if it refers to a functional is used in the second sense and functionals in Eq. (305) are called nonlocal. In this subsection local functionals (in the second sense) are reviewed. These are the simplest and most widely used approximations. Starting with the exchange we can easily arrive at this approximation by studying the homogeneous electron gas. In the lowest order perturbation theory for a homogeneous system one obtains the contribution to the exchange energy density e 9 3 3 1@3 (e (n)) "! e2n(r)4@3 . (306) 9 LDA 4p p

AB

It is worth mentioning that an exchange energy density proportional to the 4 power of the density 3 can be obtained in several other ways, too. It was first obtained by Dirac [3], Slater [4], Ga´spa´r [5], Kohn and Sham [7]. It can be derived, e.g., from dimensional considerations [23]. The first equation of the hierarchy of the exchange energy [87,90] with the locality assumption leads also to this expression. It can also be derived from the thermodynamical picture of the density functional theory [121]. The value of the constant in front of the 4 power of the density, was a crucial question 3 in the so-called Xa method [98] which is still in use in special calculations [139]. Considering higher order perturbations for a homogeneous system, one notices that all higher order terms contribute to the correlation energy density e . Recent local density functional # calculations are based on the results of accurate Green’s function Monte Carlo calculations of Ceperley and Alder for the homogeneous electron gas [140]. The most widely used parametrisations are due to Vosko, Wilk and Nusair [141] and to Perdew and Zunger [133]. The most recent one is [142]. From the genesis of the local density approximation, one would expect that this approximation is reasonably accurate, if the density varies slowly. The approximation gives, however, acceptable results for real electronic systems, where this condition is (strongly) violated. One reason for this is that the local density exchange-correlation-hole satisfies the exact sum rule. Nevertheless, the local density approximation leads to a number of deficiencies, the most serious being a rather imperfect cancellation of self-interaction effects, which leads to the incorrect asymptotic limit of the local density exchange-correlation potential. This leads, e.g., to the result that negative ions are poorly


43

described by the local density approximation. One of the most frequently used method that incorporates self-interaction correction was proposed by Perdew and Zunger [133]. Definitely improved results can be obtained by this method. For a review see [143]. Another method, that goes beyond the local density approximation is the weighted density approximation (WDA) [144]. The gist of this approximation, which leads to nonlocal energy functional, is a more careful transcription of the exchange-correlation hole of the homogeneous electron gas to the case of an inhomogeneous system. o%9!#5(r, r@)PoWDA(r, r@)"n(r@)[h)0.(n ,Dr!r@D)!1] 0 N r . (307) 9# 9# 0 n ?n( ) The weighted density exchange-correlation hole features the correct density prefactor and the density argument of the homogeneous pair correlation function is replaced by a weighted density nN (r), which is determined by requiring that the basic sum rule (Eq. (75)) be satisfied for each value of r. This leads to an improved but not perfect cancellation of self-interaction effects. From among the several other local approximations we note in passing the Gomba´s—Lie— Clementi [145] or local Wigner functional for correlation parametrized also by Wilson and Levy [146] and Su¨le and Nagy [147]. The same expression was proposed by Lee and Parr [148] for exchange-correlation energy. 9.2. Approximations containing the gradient of the density There are several approximations that go beyond the local density approximation. Second order gradient corrections to the exchange-correlation energy functional ELDA can be obtained by 9# consideration of the lowest order energy shift due to linear response of a homogeneous system. Linear response also yields some of the higher order gradient corrections, but for a complete calculation of all contributions higher order response functions have to be considered. The static linear response argument (see Ref. [26]) leads to the second order gradient correction to the exchange-correlation energy having the form

P

E*2+[n]" dr B (n(r))(+n(r))2 . 9# 2

(308)

The coefficient B (n) obtained at the RPA level using a high density expansion is found to be 2 proportional to 1/n4@3. A more careful evaluation of the gradient contribution in the ring approximation was carried out by Geldart and Rasolt [149], Langreth and Perdew [150], and Hu and Langreth [151]. Unforturnately, it turned out that the addition of second order gradient corrections to local exchange-correlation energy functional ELDA did not lead to a systematic improve9# ment. This failure was the starting point for the development of generalised gradient approximations (GGA). The first ones were introduced starting with an ansatz of the form

P

DEGGA" dr n(r)4@3f (n,s) , 9# 9#

(309)

where s is the dimensionless variable s"((+n(r))2/n(r)8@3)1@2 .

(310)


44

The function f is chosen so as to reproduce the limiting cases of high as well as low density 9# gradients correctly. For the exchange functional Becke (1992-version [152]) proposed the following form: s2 f (n, s)"!b . 9 1#6bs arcsinh(s)

(311)

For small s (i.e., low gradients) it reproduces the gradient expansion result

P

(+n)2 . DEBP !b d3r 9 n4@3 s?0 For the case of an exponential tail density one has s"ce1@3crJn~1@3

(312)

(313)

which diverges as rPR. In this limit the exchange energy density 1 n(r) , eB(r)"! 9 2 r

(314)

falls off correctly. The parameter b is fitted to reproduce the Hartree—Fock exchange energies of the noble gas atoms from He to Rn giving b"0.0042 a.u. We note in passing that it was shown by Engel et al. [48] that generalized gradient approximations for exchange cannot simultaneously reproduce the correct asymptotic forms of both the energy and the potential. They also pointed out that the correct asymptotic form of the exchange energy does not lead to significant improvement. On the other hand, Baerends [153] and coworkers argued that the correct asymptotic behaviour of the exchange potential has a crucial effect on the results. To investigate more closely the reasons for the failure of the gradient expansion, following Perdew and Wang [154], we turn to study the exchange-correlation hole. The exchange hole should be strictly negative and should satisfy the basic sum rule (Eq. (76)). On the other hand, the correlation hole satisfies the sum rule (Eq. (77)), being negative close to the electron and positive further away. It can be shown that while the holes obtained from the local density approximation satisfy these conditions, the holes calculated from the gradient expansion do not. In order to construct a generalized gradient approximation the following cut-off procedure was proposed: 1.

G

oGE oGGA" 9 9 0

if oGE40, 9 if oGE50; 9

2. oGGA"0 for Dr!r@D5R , 9 9 where R is chosen so that the sum rule is satisfied. Then the resulting numerical exchange energy 9 density is fitted to an appropriate form. A similar analysis can be applied to the correlation part. The cut-off is given by oGGA"0 for Dr!r@D5R , # #


45

where R is chosen so that the sum rule for correlation (Eq. (77)) is satisfied. Regarding the rather # involved parametrisation of the numerical results we refer to [155]. It was pointed out by Engel and Vosko [156] that the Perdew—Wang form for exchange does not reproduce the exact OPM exchange potential correctly. They then constructed a new exchange energy EGGA, which reproduces the OPM exchange potential much better. (On the other hand, the 9 new form reproduces the exchange energy worse than the Perdew—Wang functional). We note in passing the Wigner-like Wilson—Levy [146](WL) functional, which was constructed to satisfy certain coordinate scaling relations:

P

an#bD+nD/n1@3 EWL[n]" dr , (315) # c#dD+nD/(n/2)4@3#r 4 a, b, c, and d are parameters [146], and r is the Wigner—Seitz radius: 4 r "(3/4pn)1@3 . (316) 4 Another frequently used approximation for the correlation energy is the Lee—Yang—Parr (LYP) [157] formula which involves the Weizsacker kinetic energy term and is based on the Colle—Salvetti expression [158]. Correlational functional involving the Laplacian of the density has recently been proposed by Proynov et al. [159]. They supposed that the local temperature (see Section 6.2) plays the role of a Fermi-hole correlation length. They found that combined with the exchange functional of Becke [160] and Perdew [161] their correlation functional gives improved results for the binding energies and the geometries of molecules.

10. Applications 10.1. Atoms First, we have to emphasize that exchange can be treated exactly within the optimized potential method (see Section 7). For atoms and atomic ions this method can be applied. To simplify the numerically rather involved method Krieger et al. [49] worked out a very accurate approximation. Table 1 presents exchange energies for several closed shell atoms. The gradient exchange functional of Becke [160] provides dramatic improvement over the local density approximation. Table 2 shows total energies. As it is expected the Hartree—Fock total energy is somewhat lower than the one obtained from the exchange-only OPM [48,131], though the difference is extremely small. One finds that the local density approximation does not perform too badly. Its error is of the same order as the error in Hartree—Fock, but in the opposite direction. In the GGA (with results above the experimental values) the accuracy is improved considerably. Turning to the correlation, following Gross and coworkers [162], the conventional quantum chemical and the density functional correlation energies are compared. In quantum chemistry the correlation energy is traditionally defined as the difference between the exact (non-relativistic) energy and the total Hartree—Fock energy: EDC"E !E . # %9!#5 HF

(317)


46

Table 1 Exchange energies of atoms calculated by the Hartree—Fock [48], the OPM [48,131], the LDA [160] and the gradient-corrected functional of Becke [160](in a.u.) Atom

HF

OPM

LDA

B

He Be Ne Mg Ar

!1.026 !2.667 !12.108 !15.994 !30.185

!1.026 !2.666 !12.105 !15.988 !30.175

!0.884 !2.312 !11.03 !14.61 !27.86

!1.025 !2.658 !12.14 !16.00 !30.15

Table 2 Total energies of atoms calculated by the Hartree—Fock, the OPM, the LDA and the GGA methods (in a.u.) Atom

Exp.

HF

OPM

LDA

GGA

He C Ne Si Cl

!2.9037 !37.8450 !128.939 !289.383 !460.217

!2.8617 !37.6886 !128.547 !288.854 !459.482

!37.6865 !128.546 !288.850 !459.477

!2.975 !38.0522 !129.317 !289.912 !460.838

!2.8989 !37.8243 !128.945 !289.368 !460.162

On the other hand, in the density functional theory the correlation energy is obtained by inserting the exact ground-state density into the correlation functional EDFT # EDFT"E [n] , (318) # # which is given by the difference of the exact exchange-correlation and the exchange energies: EDFT"EDFT[n]!EDFT[n] . (319) # 9# 9 Keeping in mind that the density functional exchange energy can be given by the Hartree—Fock expression for exchange energy providing that the exact Kohn—Sham orbitals are inserted into it: EDFT"EHF[uKS] . 9 9 i The density functional correlation energy can be expressed as

(320)

EDFT"E[n]!EHF[uKS] , (321) # i because E[n]"E , while the quantum chemical correlation energy has the form %9!#5 EQC"E[n]!EHF[uHF] . (322) # i Since the Hartree—Fock orbitals are the ones that minimize the Hartree—Fock total energy, the following inequality holds EDFT4EQC . # #

(323)


47

However, the difference between the two correlation energies is very small [162], as can be seen from Table 3. D in the last column denotes the value of DEQC !EDFT D/EDFT in percent. #,%9!#5 #,%9!#5 #,%9!#5 Table 4 presents correlation energies calculated by various approximate functionals: the Wilson—Levy (WL) [146], the Lee—Yang—Parr (LYP) [157], the Perdew and Wang (GGA) [83] the local Wigner functionals (LW) [163,147] and the local correlation functional of Perdew and Wang (LDA) [164]. For purposes of comparison experimental results [165] are also shown. The correlation energy is overestimated by roughly a factor of 2 in the local density approximation, while the GGA yields an accuracy below 5% (with the exception of He). The fact that total energy in the local density agrees much better with empirical data than either exchange or correlation demonstrates the often quoted cancellation of errors. All the models considered except the local density approximation yield rather close correlation energies, which agree satisfactorily with the available empirical data. There is only some relative overestimation of correlation for heavier atoms in the Lee—Yang—Parr model. The least deviation is achieved in the Wilson—Levy model. For ionic systems the picture is less consistent. All functionals fail to reproduce the correlation energy for the F~ anion. Table 3 Density functional, conventional quantum chemical correlation energies (QC) and their difference [162] (in a.u.) DFT H~ He Be`2 Ne`8 Be Ne

!0.041 !0.042 !0.044 !0.045 !0.096 !0.394

D

QC 995 107 274 694 2

!0.039 !0.042 !0.044 !0.045 !0.094 !0.390

821 044 267 693 3

#0.002 #0.000 #0.000 #0.000 #0.001 #0.004

D% 174 063 007 001 9

5.2 0.2 0.02 0.002 2.0 1.0

Table 4 Correlation energies of atoms obtained by various approximate correlation energy functionals (in a.u.) [166]

He Be Ne Mg Ar Kr Xe Li` Be2` Ne6` B` Li~ F~

WL

LYP

GGA

LW

LDA

Exp

0.042 0.094 0.383 0.444 0.788 1.909 3.156 0.044 0.045 0.109 0.101 0.0805 0.368

0.043 0.094 0.383 0.459 0.750 1.748 2.742 0.047 0.049 0.129 0.106 0.0732 0.362

0.046 0.094 0.383 0.451 0.771 1.916 3.150 0.051 0.053 0.123 0.103 0.078 0.362

0.042 0.094 0.374 0.462 0.771 1.948 3.174 0.060 0.075 0.187 0.114 0.069 0.332

0.112 0.223 0.743 0.888 1.426 3.267 5.173 0.134 0.150 0.334 0.252 0.182 0.696

0.042 0.094 0.392 0.444 0.787

0.044 0.044 0.187 0.111 0.073 0.400

48


Recently, the exact correlation energy density has been calculated for the He atom and compared with the model functionals [166]. The correlation energy density obtained from the abovementioned model functionals have quite different local behaviour. The form of exchange, correlation and exchange-correlation potentials is of considerable importance. Using the methods described in Section 8 these potentials can be exactly determined provided that the electron density is known. They can be readily compared with model potentials. Neither the local nor the gradient-corrected approximations provide potentials having the correct asymptotic behaviour. Baerends and coworkers [153] pointed out that the correct asymptotic behaviour of the potentials is much more important than that of the corresponding energy density. Recently, there has been considerable progress in determining the difference between the interacting and non-interacting kinetic energy [60,109,167]. Several approximate expressions have been compared in Ref. [168]. 10.2. Molecules The density functional theory has developed into a cost-effective general method of calculating molecular properties. There is a vast literature on the application of the Kohn—Sham theory to chemistry. It seems impossible even to give a complete list of the reviews on molecular computations. So, only two recent reviews are referred [169,170] and the present survey deals only with a brief comparison of some frequently used functionals based on systematic studies of Pople and coworkers [171] and Handy and collaborators [172]. Table 5 [166] presents correlation energies for 21 closed shell molecules calculated with the Wilson—Levy [146], the Lee—Yang—Parr [157], the local Wigner [163,147] and the Perdew—Wang [83] functionals. For these molecules all these functionals yield a reasonable estimate of the experimental correlation energy [173]. It was, however, found [166] that there are considerable differences in the correlation energy density. Especially the Wilson—Levy model leads to a local behaviour quite different from the others. It is well known that the Hartree—Fock method predicts bond lengths to be too short, vibrational frequencies to be systematically large and binding energies to be too low. On the other hand, MP2 predictions of the bond length for single bonds are also too short, but multiple bonds can be either too short or too long. MP2 gives generally too large vibrational frequencies and too low binding energies. The local density approximation (see Section 9.1) leads to severe overbinding (see atomization energies in Table 6), while other quantities such as bond length, bond angles, vibrational frequencies tend to agree quite well with experiment. The bond length, e.g., presented in Table 7 for some first-row diatomic molecules, is overestimated (the mean deviation is about 0.01 A_ ). The Lee—Yang—Parr correlation functional [157] combined with Becke’s exchange functional [160] gives bond lengths which are somewhat long, vibrational frequencies that are often better than MP2 and quite satisfactory atomization energies. Simple hydrides with lone-pair electrons (e.g. NH ) tend to bind less, while molecules with multiple bonds, such as F , overbind at this level 3 2 of approximation. The Becke—Perdew approximation [160,161] is an improvement over the local density approximation, especially for single bonds. For certain XH bonds, however, the bond length is too long and consequently vibrational frequencies are too low. For atomization energies this functional gives high accuracy predictions.


49

Table 5 Correlation energies of molecules obtained by various model correlation energy functionals Molecule

WL

LYP

LW

PW

Exp

H 2 Li 2 Be 2 B 2 C 2 N 2 O 2 F 2 H O 2 NH 3 CH 4 HF LiH LiF HCN CO H O 2 2 C H 2 2 C H 2 6 C H 2 4 CO 2

0.049 0.136 0.231 0.336 0.446 0.532 0.621 0.683 0.386 0.376 0.369 0.377 0.088 0.417 0.525 0.516 0.690 0.504 0.678 0.593 0.865

0.038 0.133 0.200 0.289 0.384 0.483 0.583 0.675 0.340 0.318 0.294 0.363 0.089 0.418 0.464 0.484 0.638 0.443 0.551 0.497 0.791

0.029 0.134 0.193 0.265 0.344 0.435 0.533 0.633 0.314 0.268 0.241 0.335 0.083 0.343 0.410 0.440 0.569 0.386 0.426 0.417 0.720

0.046 0.137 0.205 0.296 0.391 0.490 0.588 0.671 0.347 0.338 0.320 0.367 0.092 0.415 0.478 0.488 0.652 0.466 0.577 0.529 0.807

0.041 0.122 0.205 0.330 0.514 0.546 0.657 0.746 0.367 0.338 0.293 0.387 0.083 0.447 0.527 0.550 0.691 0.476 0.553 0.528 0.829

Note: The notations of model functionals are the same as in Table 4. Exp denotes the experimental correlation energies [165]. For CO and C H the experimental correlation energies were estimated by using experimental atomization energies 2 4 on the basis of Ref. [173]. All the energies are in a.u. The calculations were performed using the large TZV#3D basis Table 6 Atomization energies (in kcal mol~1) of some molecules calculated with HF, MP2, LDA, BLYP (using 6-31G* basis [171]) and BP, BRP (using 6-31G basis [172])

H 2 LiH NH 3 C H 2 2 H CO 2 F 2

HF

MP2

LDA

BLYP

BP

BRP

Exp

75.9 30.4 170.2 271.9 237.8 !34.3

86.6 39.8 232.4 365.6 335.5 36.8

100.2 57.5 306.0 438.6 417.6 83.6

103.2 54.9 270.1 383.4 361.8 54.4

107.8 55.8 289.5 398.6 371.5 49.6

106.9 58.9 286.7 404.0 372.5 47.1

103.3 56.0 276.7 388.9 357.2 36.9

The Becke—Roussel [174] exchange functional combined with the correlation functional of Perdew [161] leads to improved geometries for a lot of molecules. The predicted atomization energies are as good as the Becke—Perdew approximation and in many cases better. It was found [171] that all the model functionals predict dipole moments that are often significantly in error. Inclusion of gradient terms does not lead to improvement.


50

Table 7 Bond length (in A_ ) for some first-row diatomic molecules calculated with HF, MP2, LDA, BLYP (using 6-31G* basis [171]) and BP, BRP (using 6-31G basis [172])

H 2 BeH LiF CO N 2 NO

HF

MP2

LDA

BLYP

BP

BRP

Exp

0.730 1.348 1.555 1.114 1.078 1.127

0.738 1.348 1.567 1.150 1.130 1.143

0.765 1.370 1.544 1.142 1.111 1.161

0.748 1.355 1.561 1.150 1.118 1.176

0.747 1.356 1.580 1.135 1.103 1.160

0.741 1.353 1.582 1.130 1.101 1.158

0.741 1.343 1.564 1.128 1.098 1.151

Recently, a new class of hybrid (Hartree—Fock and density functional) methods have been proposed; the simplest form of that contains an exchange-correlation energy: E "EDFT#a(E%9!#5!EDFT) . (324) 9# 9# 9 9 The paremeter a is determined by fitting to experimental thermochemical data. This expression has been rigorously founded by Go¨rling and Levy [175]. They showed that these hybrid schemes are based on model systems which are defined as the Slater determinant which yields the exact ground-state density and minimizes the expectation value of the operator ¹K #a»K with a being %% a parameter having a value between zero and one. (These hybrid schemes belong to the class of generalized Kohn—Sham schemes derived recently [176].) This kind of exchange mixing reduces average bond energy errors considerably (to about 2 kcal/mol). Reaction barrier heights are also improved. Much progress has been made over the years in solving the molecular Kohn—Sham equations. Very large molecules (e.g. of biological interest) can be treated in the density functional theory. While in the Hartree—Fock method the computational costs increase as N4 or N3, those of the Kohn—Sham scheme as N3. But, in principle, within the density functional theory, it is possible to work out methods that scale linearly [177]. The reason lies in the fact that the whole electronic structure is determined solely by the electron density. Such methods will soon become a very important tool for molecular modelling. We mention in passing that combining the density functional theory with molecular dynamics has led to an extremely fruitful method of Car and Parinello [178] making possible large-scale simulations of molecules and solids.

11. Extensions of the density functional theory 11.1. Finite-temperature density functional theory An extension of the density functional theory can be used to treat systems at finite temperatures [7,179,180]. (For reviews see [8,10].) The forerunner of the theory is the temperature-dependent Thomas—Fermi model [181,182]. Mermin’s generalization of the Hohenherg—Kohn theorem [179]


51

states that in a grand canonical ensemble at a given temperature h and chemical potential k, the density n uniquely determines the difference v(r)!k. For a given v and k there exists a functional of nJ (r), the grand potential,

P

X[nJ ]" nJ (r)(v(r)!k) dr#G[nJ ]

(325)

such that it is an absolute minimum for the correct density n(r). The functional G is a universal temperature dependent functional of the density only. The proof proceeds by reductio ad absurdum like in the original Hohenberg—Kohn theorem. The equilibrium grand potential can also be written as

C

P

D

X0"Min X[n]"Min ¹[n]#» [n]!hS[n]# n(r)(v(r)!k) dr , (326) %% n where S[n] is the entropy. In the finite temperature Kohn—Sham theory [7,180] a system of non-interacting electrons with density n at temperature h is considered, where n equals the density of the interacting system. A free kinetic energy is defined as A [n]"¹ [n]!hS [n] , (327) 4 4 4 where the subscript s refers to the non-interacting system. Thus the grand potential takes the form

P

X[n]"A [n]# n(r)(v(r)!k) dr#J[n]#A [n] , 9# 4 the exchange-correlation contribution to the free energy A"¹#» !hS is given by %% A [n]"» [n]!J[n]#¹[n]!hS[n]!¹ [n]#hS [n] . 9# %% 4 4 The grand potential of the non-interacting system reads as

P

X [n]"A [n]# n(r)(v(r)!k) dr . 4 4

(328)

(329)

(330)

The Kohn—Sham equations have the form [!1+ 2#v (r,¹)]u (r)"e u (r) , 2 KS i i i where the effective potential

P

dA [n] n(r@, h) v (r, h)"v(r)# dr@# 9# KS dn(r, h) Dr!r@D

(331)

(332)

includes the external potential v(r), the Coulomb and exchange-correlation potentials. The electron density reads n(r, h)"+ Du (r)D2f (e !k) , i i i

(333)


52

where f (e !k)"Mexp[b(e !k)]#1N~1 (334) i i is the Fermi function. b"1/kh and k is the Boltzmann constant. The non-interacting electrons occupy the single-particle eigenstates according to the Fermi—Dirac statistics. Eqs. (331)—(334) have to be solved self-consistently. To do this one needs approximate expressions for the exchangecorrelation potential. For local density approximations and calculations see [183—186] The adiabatic-connection expression for the exchange-correlation expression was given by Perdew [161]. 11.2. Density functional theory for excited states The density functional theory was originally formalized for the ground-state [6]. It was soon noticed [187] that the original theory can also be applied to the lowest excited states with different symmetries. To calculate excitation energies Slater [188,98] introduced the so-called transition state method. It proved to be a very efficient and simple approach and was used to solve a large variety of problems. The density functional theory was first rigorously generalized for excited states by Theophilou [189]. Formalisms for excited states have also been provided by Fritsche [190] and English et al. [191]. A more general treatment was given by Gross, Oliveira and Kohn [192]. The relativistic generalization of this formalism has also been done [193]. 11.2.1. Density functional theory for ensembles Here, only the most general treatment of Gross et al. [192] is reviewed. (The subspace theory of Theophilou [189] can be considered as a special case of the former.) The density functional theory for ensembles is based on the generalized Rayleigh—Ritz variational principle [192]. The eigenvalue problem of the Hamiltonian HK is given by HK W "E W (k"1,2,M) , k k k where

(335)

E 4E 42 (336) 1 2 are the energy eigenvalues. The generalized Rayleigh—Ritz variational principle [192] can be applied to the ensemble energy M E" + w E , k k k/1 where w 5w 525w 50. The weighting factors w are chosen as 1 2 M i w "w "2"w "(1!wg)/(M!g) , 1 2 M~g w "w "2"w "w , M~g`1 M~g`2 M 04w41/M , 14g4M!1 .

(337)

(338) (339) (340) (341)


53

The limit w"0 corresponds to the eigenensemble of M!g states (w "2"w "1/(M!g) 1 M~g and w "2"w "0). The case w"1/M leads to the eigenensemble of M states M~g`1 M (w "w "2"w "1/M). 1 2 M The generalized Hohenberg—Kohn-theorems read as follows: (i) The external potential v(r) is determined within a trivial additive constant, by the ensemble density n defined as M n" + w n . k k k/1 (ii) For a trial ensemble density n@(r) such that

(342)

n@(r)50 ,

(343)

P

n@(r) dr"N

(344)

E[n]4E[n@] .

(345)

The ensemble functional E takes its minimum at the correct ensemble density n. The proof of the theorem goes exactly in the same way as for the ground-state. Using the variation principle the Euler-equation can be obtained: dE "k . dn

(346)

Kohn—Sham equations for the ensemble can also be derived: [!1+ 2#v ]u (r)"e u (r) . 2 KS i i i The ensemble Kohn—Sham potential

P

n (r) v (r; n )"v(r)# w dr#v (r; w, n ) KS w 9# w Dr!r@D

(347)

(348)

is a functional of the ensemble density MI 1!wg MI~gI I + + j Du (r)D2#w + j Du (r)D2 , + nI (r)" mj j mj j w M I~1 m/1 j m/MI~gI`1 j g is the degeneracy of the Ith multiplet. I I M "+ g I i i/1 is the multiplicity of the ensemble and j mj

04w41/M . I are the occupation numbers. The density matrix is defined as M DK M, g(w)" + w DW TSW D . m m m m/1

(349)

(350)

(351)

(352)


54

The ensemble exchange-correlation potential v is the functional derivative of the ensemble 9# exchange-correlation energy functional E 9# dE [n, w] v (r; w, n)" 9# . (353) 9# dn(r) The excitation energies can be expressed with the one-electron energies e as j 1 dEI(w) I~1 1 dEI(w) EM I" #+ , g dw M dw I w/wI i/2 I w/wi where

K

K

(354)

K

N~1`MI g N~1`MI~1 EI dEI(w) " + e! I + e # 9# , (355) j j M w w dw I~1 j/N n j/N`MI~1 04w 41/M . (356) i I It is emphasized that the excitation energy cannot generally be calculated as a difference of the one-electron energies. There is an extra term E /wD w to be determined. 9# n The two-particle density matrix of the ensemble is the weighted sum of the two-particle density matrices of the ground and excited states: M CM,g,w(r , r ; r@ , r@ )" + w Cm(r , r ; r@ , r@ ) . 1 2 1 2 m 1 2 1 2 m/1 The total ensemble energy has the form

(357)

P

EM,g"trMDK M,g(w)HK N"trMDK M,g(¹K #»K )N#trMDK M,g(w)»K N"FM,g(w)# n(r)v(r) dr , w %%

(358)

where n(r) is the ensemble density. The ensemble exchange energy is given by EM,g[w,n]"FM,g[w,n]!¹M,g[w,n]!J[n] , 9# 4 where

P

1 n(r)n(r@) J[n]" dr dr@ 2 Dr!r@D

(359)

(360)

is the ensemble Coulomb energy and ¹M, g[w, n] is the noninteracting ensemble kinetic energy. 4 11.2.2. Coordinate scaling and adiabatic connection formula The constrained-search formulation has also been applied for the ensemble theory [192]. The minimum is searched over the density matrices that yield the ensemble density n (361) FM,g[w; n], min trMDK M,g(¹K #»K )N . %% M,g D (w)?n Define W.*/,a as the wavefunction which yields n and minimizes S¹#a»K T. As Levy and n %% Perdew [84] and later Levy et al. [194] and Levy [195] showed the scaled wavefunction


55

j3N@2W.*/,a(jr ,2,jr ) yields n and minimizes S¹K #ja»K T. Hence, j3N@2W.*/,a(jr ,2,jr ) yields n 1 N a %% n 1 N n and minimizes S¹K #»K T if j"1/a. So a %% W.*/ (r , ,r )"j3N@2W.*/,a (jr ,2,jr ) , j"1/a . (362) nj 1 2 N n 1 N To obtain scaling relations for the ensemble theory the density matrix has to be studied instead of the wavefunction. Consider the density matrix DM,g[w, n] which yields the ensemble density n and a minimizes S¹K #a»K T. DM,g[w, n] is the density matrix considered before, while DM,g[w, n] is the %% 1 0 non-interacting density matrix, i.e. DM,g[w, n] yields n and minimizes just S¹K T. The coordinate 0 scaling for the density matrix DM,g[w; n ](r ,2, r ; r@ ,2, r@ )"j3NDM,g[w; n](jr ,2,jr ; jr@ ,2, jr@ ) , j"a~1 (363) j 1 N 1 N a 1 N 1 N can be readily obtained from Eq. (352) and the scaling law for the wavefunction (Eq. (167)). From Eqs. (352) and (167) we see that j3NDM,g[w, n](jr ,2, jr ; jr@ ,2, jr@ ) yields n and minimizes a 1 N 1 N j S¹K #»K T if j"1/a. %% From Eqs. (357) and (167) we get scaling properties of the two-particle ensemble matrix: CM,g,w (r , r ; r@ , r@ )"a6CM,g,w(ar , ar ; ar@ , ar@ ) . (364) na 1 2 1 2 n 1 2 1 2 The well-known ground-state adiabatic connection formula of the exchange-correlation energy (see Section 5.6)

P

1 (»a [n]!J[. ]) da (365) %% 0 is valid for the ensemble exchange energy, too, provided that n corresponds to the ensemble density. To prove this let us consider the functional FM,g[w, n]: j E [n]" 9#

FM,g[w, n]" min trMDK M, g(w)(¹K #j»K )N . %% j DM,g(w)?n j"1 gives the interacting system with

(366)

FM,g[w; n]"FM,g[w; n]"¹M,g[w; n]#»M,g[w; n] , 1 %% while j"0 corresponds to the noninteracting case

(367)

FM,g[w; n]"¹M,g[w; n] . 0 4 The ensemble exchange-correlation energy is given by

(368)

EM,g[w; n]"»M,g[w; n]!JM,g[w; n]#¹M,g[w; n]!¹M,g[w; n] 9# %% 4 "FM,g[w; n]!FM,g[w; n]!JM,g[w, n] 1 0 1 FM,g[w; n] " dj j !JM,g[w, n] . (369) j 0 It can be shown that the Hellmann—Feynmann theorem is valid for an ensemble [196]:

P

FM,g[w; n] j "trMDK M,g(w)»K N"S»K T . j %% %% j j

(370)


56

Using this theorem the adiabatic connection formula for the ensemble exchange-correlation energy:

P

EM,g[w; n]" 9#

1

0

dj(S»K T !J) . %% j

(371)

can be derived. From the scaling properties of the electron—electron energy »a [n]"a»1 [n ] and the %% %% 1@a Coulomb repulsion energy J[n ]"a~1J[n], another form of the adiabatic connection formula 1@a for the ensemble exchange-correlation energy

P

EM,g[w; n]" 9#

1

0

a(» [n ]!J[n ]) da %% 1@a 1@a

(372)

follows. This is the same form as the ground-state expression. (Of course, it contains the ensemble quantities.) 11.2.3. Excitation energies To solve the Kohn—Sham equations (Eq. (347)) one needs the ensemble exchange-correlation potential. Several approximations have been proposed for the ensemble exchange-correlation potential. Gross et al. [192] calculated the excitation energies of He atom using the quasi-localdensity approximation of Kohn [197]. The first excitation energies of several atoms [198] have been calculated with parameter-free exchange potential of Ga´spa´r [199]. As this potential depends explicitly on the spin orbitals it is very flexible and can be successfully applied not only in ground-state but also in ensemble-state calculations. Higher excitation energies have also been studied [200]. Ga´spa´r’s parameter-free potential proved to be remarkably good in these calculations. Several ground-state local density functional approximations have been tested [204]. The Gunnarsson—Lundqvist—Wilkins [202], the von Barth—Hedin [203] and Ceperley—Alder [140] local density approximations parametrized by Perdew and Zunger [133] and Vosko et al. [141] are applied to calculate the first excitation energies of atoms. As ground-state exchange-correlation potentials were used the extra term in Eq. (355) does not appear. Spin-polarized calculations [204] lead to a definite improvement compared with the non-spin-polarized results, still, in most cases the calculated excitation energies are highly overestimated. The results provided by the different local density approximations are quite close to each other (Table 8). The best one seems to be the Gunnarson—Lundqvist—Wilkins approximation. (In the non-spin-polarized case the Perdew—Zunger parametrization gives results closest to the experimental data.) In these calculations the minimum (w"0) and the maximum possible values of the weighting factor were applied. The results obtained with different weighting factors w are different. However, any value of w satisfying inequality (Eq. (340)) is appropriate. If we knew the exact exchangecorrelation energy functional, any value of w satisfying condition (Eq. (340)) would lead to the same result. As the exact form of the exchange-correlation energy is, however, unknown and we have to use approximate functionals, different values of w provide different excitation energies. The effect of w on the excitation energies has also been studied [201,204]. In certain atoms it causes only a small change while in other cases there is a considerable change in the excitation energy. The change is


57

Table 8 Electron configurations of ensemble states and calculated (spin-polarized) density functional, Hartree—Fock and experimental first excitation energies of atoms (in Ry). The upper rows contain the results belonging to w"0, the lower rows contain the results belonging to w"1/(g #g ) 1 2 Atom B

Na

Mg

Al

P

Ensemble state

Xa

GLW

VBH

VWN

2s2 2p 2s 2s 13 2p 123 ¬

0.285

0.326

0.340

0.335

0.034

0.184

0.202

0.191

3s 3p 0 3s 14 3p 34

0.152

0.166

0.162

0.159

0.156

0.167

0.164

0.162

3s2 3p 0 3s 3s 101 3p 109

0.244

0.254

0.253

0.249

0.045

0.143

0.152

0.149

3p 4s 0 3p 34 4s 14

0.174

0.216

0.210

0.196

0.183

0.216

0.209

0.202

3p 3 4s 0 3p 214 4s 34

0.427

0.459

0.446

0.434

0.538

0.554

0.544

0.541

HF

Exp

0.157

0.262

0.145

0.155

0.136

0.199

0.291

0.230

0.605

0.512

monotonic in all the approximations studied. We mention that a special choice of w, in certain cases, corresponds to Slater’s transition-state method. Relativistic calculations have also been performed [193]. 11.2.4. Local ensemble exchange potential Unfortunately, the currently existing ground-state exchange-correlation potentials do not always perform well for ensemble-states. Recently, a simple local ensemble potential has been proposed [205] in the form:

A B

3 1@3 v (n ,w)"!3a(w) n . 9 w 8p w

(373)

The w-dependence is incorporated in the parameter a. The corresponding exchange energy has the form

A B

P

9 3 1@3 E [n ,w]"! a(w) n4@3 dr . 9 w w 4 8p

(374)

Using the experimental energies a(w) is determined so that the calculated ensemble energy be equal to the ensemble energy obtained from the experimental energies. Two typical cases were found for light atoms. The functions a for the atoms B, C, O, Mg, Si and P have almost the same form, i.e. slight almost monotonic dependence on w. For the atom S a is almost constant. On the other hand, the function a has a very shallow minimum for the atoms F, Cl and Na.

58


As we have already seen the excitation energy is not simply the difference of the one-electron energies. The extra term E /wD w is generally not zero. At certain values of w it gives a significant 9# n contribution. The function E /wD w for atoms B, C, O, Mg, Si, S, and P has an almost linear 9# n dependence on w. On the other hand, for the atoms F, Cl and Na there is a value w"w , where it 0 disappears. It means that at w the excitation energy can be simply given by the difference 0 e !e . The importance of the existence of this w lies in the fact that it is possible to determine N`1 N 0 the excitation energy simply as a difference of one-electron energies at a certain value of w. The value of w is 0.0144 for F, 0.113 for Na and 0.0178 for Cl. The corresponding values of a are 0 0.76100, 0.75198, 0.74256 for the atoms F, Na and Cl, respectively. Naturally, calculations can be performed at any value of w (satisfying the condition (Eq. (356))). One can select, e.g., the maximum possible value of w, (i.e. the one corresponding to the subspace theory of Theophilou [189]). Table 9 contains the values of a corresponding to w for selected .!9 atoms. 11.2.5. Ensemble exchange potential and energy for multiplets The multiplet structure has already been treated using the density functional theory. The most important approaches have been proposed by Bagus and Bennett [206], Ziegler et al. [207] and von Barth [208]. All these methods have the same feature of not being completely within the frame of the density functional theory. The method of fractionally occupied states can be used to treat the multiplet problem, too. The method of obtaining the potential from the density (see Section 8) [99] can be applied to ensemble states without any difficulty [209]. Starting out from the Hartree—Fock densities [210] the ensemble exchange potentials for multiplets have been calculated. Writing the ensemble exchange potentials in the form

A B

3 1@3 vM,g(w, n; r)"!3aM,g(w) , n 9# w 8p

(375)

the factors aM,g(w) as functions of the radial distance r show a shell structure. (For the ground state the shell structure has also been demonstrated [99].) Though the ensemble exchange potentials are Table 9

Ground- and excited state configurations and the value of a corresponding to the maximal weighting factor w Atoms

B C O F Na Mg Al Si P S Cl

a

Configurations Ground-state

Excited-state

2P1@2(2p) 3P (2p2) 0 3P (2p4) 2 3P3@2(2p5) 2S1@2(3s) 1S0(3s2) 2P1@2(3p) 3P (3p2) 0 4S (3p3) 3@2 3P2(3p4) 3P3@2(3p5)

4P1@2(2s2p2) 5S (2p3s) 0 5S (2p33s) 0 4P3@2(2p43s) 2P1@2(3p) 3P0(3s3p) 2S1@2(4s) 5S(3s3p3) 4P(3p24s) 5S(3p34s) 4P5@2(3p44s)

0.80210 0.79115 0.77350 0.76390 0.75198 0.75105 0.74810 0.74832 0.74623 0.74472 0.74295


59

different from the ground state one, the difference is not too much and the factors a show a very similar behaviour. The fact that the exact exchange potential has similar behaviour for the ensemble of multiplets suggests that approximations might also be similar. Probably, a small change in the ground-state exchange functionals might lead to good approximation for ensembles of multiplets. It was emphasized in the theory of Gross et al. [192] that the ensemble exchange potential depends on w. It has been explicitly demonstrated [209] that the ensemble exchange factor for multiplets is different for different values of w. The method described in Section 8 makes it possible to calculate the ensemble energy [209]. The ensemble exchange energies are very close to the Hartree—Fock ones, the latter being somewhat lower as it is expected. It has been recently shown that the ensemble exchange energy for multiplets is linear in w [211]. Excitation energies can be calculated within the time-dependent theory that is detailed in Section 11.4. A new way of treating excited states within the density functional theory recently proposed by Go¨rling [212] is outlined in the next subsection. 11.2.6. Density functional theory for excited states via adiabatic connection and perturbation theory The density functional theory can be extended to excited states via the adiabatic connection and making use of perturbation theory [212] (see Section 5.7). The adiabatic connection characterized by the Schro¨dinger equation HK aDWaT"EaDWaT , (376) k k k HK a"¹K #a»K #»K (377) %% a represents a continuous connection between a non-interacting system and the real system. Here not only the ground-state but also the kth eigenstate Wa of the coupling constant Hamiltonian is k considered. The additional assumption here is that the energetic order of eigenstates Wa of HK a of the k same symmetry is preserved along the adiabatic connection. So the coupling constant path establishes a continuous connection between the kth eigenstate of non-interacting and the interacting Hamiltonian. The energy of the kth eigenstate

P

Ea"SU [n ]D¹K DU [n ]T#J [n ]#E [n ]#Ea [n ]# va(r)n0(r) k k 0 k 0 k 0 9, k 0 #, k 0 k

(378)

is a functional of the ground-state density n which is kept fixed in the coupling constant path. The 0 exchange and correlation energy functionals are defined as E [n ]"SU [n ]D»K DU [n ]T!J [n ] , (379) 9, k 0 k 0 %% k 0 k 0 Ea [n ]"SWa[n ]DHK aDWa[n ]T!SU [n ]DHK aDU [n ]T , (380) #, k 0 k 0 k 0 k 0 k 0 where J [n ] is the classical Coulomb energy of the kth eigenstate. k 0 In order to treat excited states in the Kohn—Sham formalism, first, the ground-state Kohn—Sham equations have to be solved. I.e. the ground-state one-electron energies and orbitals have to be determined. To obtain the excited-state exchange and correlation energy functionals E [n ] and 9,k 0 Ea [n ], the density functional perturbation theory (see Section 5.7) can be applied. Go¨rling [212] #,k 0


60

has shown that the ground-state one-electron energies are not just auxiliary quantities without physical meaning; their difference provides zeroth-order approximation to excitation energies. 11.3. Current density functional theory In the spin density functional theory (Section 4.2) the interaction of the magnetic field with the electronic current was neglected. Inclusion of currents is required whenever a large magnetic field coexists with a strongly inhomogeneous electronic structure, e.g. charge-density waves, spindensity waves, atoms and molecules in strong magnetic fields. (For a review see [214].) Recently, Vignale and Rasolt constructed the non-relativistic current density functional theory [213]. The most delicate point in the theory is the following. The non-relativistic expression for the orbital current density 1 j(r)"j (r)# n(r)A(r) , 1 c

(381)

includes the paramagnetic current density j (r) and the diamagnetic current density 1n(r)A(r). Since 1 c in the variational process the vector potential A(r) is kept constant, the variation with respect to j is the same as the variation with respect to j , consequently j should be used as a basic variational 1 1 object. The paramagnetic current density, however, is not gauge-invariant. Vignale and Rasolt managed to construct a gauge-invariant theory in terms of non-gauge-invariant variables. To write the Hamiltonian the kinetic energy operator has to be replaced by

A

B

1 N 2 1 ¹K " + !i+# A(r) . 2 c i/1 Then the total energy has the form

P C

(382)

D P

1 1 E"E # n(r) v(r)# A2(r) # j (r)A(r) dr . 0 2c2 c 1

(383)

The Hohenberg—Kohn theorem states that the external scalar and vector potentials v(r) and A(r) are determined within a trivial additive constant by the knowledge of the ground-state electron and current densities n(r) and j (r) and there is a variational principle for the total energy. The proof 1 again proceeds by reductio ad absurdum and is omitted here. The variational principle should be applied by imposing the constraints

P

N" n(r) dr ,

(384)

1 + j (r)# + [n(r)A(r)]"0 1 c

(385)

which corresponds to the constancy of the total number of electrons and the continuity equation for the total current.


61

The Kohn—Sham equations can also be readily obtained in the usual way (i.e. by introducing the non-interacting kinetic energy functional and performing the variation):

CA

B

D

1 2 1 !i+# A (r) #v (r) u (r)"e u (r) . %&& %&& i i i 2 c

The densities n(r) and j (r) are related to the one-electron orbitals by 1 N n(r)" + Du (r)D2 , i i/1 1 N j (r)" + (u*(r)+u (r)!u (r)+u*(r)) . 1 i i i i 2i i/1 The effective potentials in Eq. (386) have the forms: 1 v (r)"v(r)#v (r)#v (r)# [A2(r)!A2 (r)] %&& J 9# %&& 2c2 A (r)"A(r)#A (r) , %&& 9# where v (r) is the classical Coulomb potential and J dE [n, j ] 1 , v (r)" 9# 9# dn dE [n, j ] 1 . A (r)" 9# 9# dj 1 The gauge-invariancy of the theory can be expressed in the following compact form: E [n, j ]"EM [n, m] , 9# 1 9# i.e., the exchange-correlation energy is a functional of the gauge-invariant combination

(386)

(387) (388)

(389) (390)

(391) (392)

(393)

m(r)"+][ j (r)/n(r)] . (394) 1 From Eq. (393) one can easily check that the effective potentials v (r) and A (r) can be expressed %&& %&& with n(r) and m(r) and the gauge-invariancy of the effective potentials and the Kohn—Sham equations can be readily proved. Certainly, one should approximate the exchange-correlation energy. The local density approximation can be extended to the current density functional theory. In the local approximation the exchange-correlation energy density e which is a function of the n(r) and m(r) is taken to be the 9# exchange-correlation energy density of an electron gas with uniform n(r) and m(r).

P

ELDA[n,m]" n(r)e(n(r),Dm(r)D) dr . 9#

(395)

In case of a uniform magnetic field if the electron gas is at rest in the laboratory frame the current density j(r) vanishes everywhere, consequently m(r) is uniform and has the form: m(r)"m"!B/c .

(396)


62

Different expressions of the exchange-correlation energy density have been proposed for weak [213,215—217], intermediate and strong magnetic fields [218]. Handy and coworkers [219] calculated magnetisabilities, polarizabilities and nuclear shielding constants using current density functional theory. It is possible to include the spin in the current density functional theory. Vignale and Rasolt [213] proposed two alternative ways, the total current and the spin-current formulations. Recently, a time-dependent generalization of the current density functional theory has been developed by Ng [220] for electronic systems in weak electromagnetic fields. An alternative of the current density functional theory has been proposed by Grayce and Harris [221]. In this magnetic field density functional theory the density is the only fundamental variable. The magnetic field B instead of the current j appears explicitly in the energy functional: 1 1 n(r, B)n(r@, B) E"G[n, B]# n(r, B)v(r)# dr dr@ . (397) Dr!r@D 2

P

P

The disadvantage of this theory is the non-universality, i.e., the dependence on the magnetic field. While G has the same form for all v, it depends explicitly on the magnetic field B. It is universal only with respect to the scalar potential, but not with respect to the vector potential. Several calculations have been done with this method, i.e. [222, 223]. 11.4. Time-dependent density functional theory The roots of the time-dependent density functional theory date back to the time-dependent Thomas—Fermi model proposed by Bloch [224]. The first time-dependent Kohn—Sham equations were obtained by Peuckert [225] and Zangwill and Soven [226]. The rigorous foundation of the time-dependent density functional theory was started by the work of Deb and Ghosh [227] and Bartolotti [228]. The general proofs of the fundamental theorems of the time-dependent density functional theory were given by Runge and Gross [229]. For a recent review of time-dependent density functional theory see [230]. 11.4.1. Runge—Gross theorem The ground-state density functional theory is based on the Rayleigh—Ritz variation principle. In the case of a time-dependent external potential, however, no minimum principle exists. Instead, there is a stationary-action principle. The starting point of studying time-dependent systems is the Schro¨dinger equation W(t) i "HK W(t) , t

(398)

where the Hamiltonian HK (t)"¹K #»K #»K (t) %% includes the kinetic ¹K , the electron—electron repulsion »K and the external potential %% N »K " + v(r ,t) i i/1

(399)

(400)


63

operators. The densities of the system evolving from a fixed initial state W(t )"W (401) 0 0 are considered. The initial state W is not supposed to be an eigenstate of the initial potential v(r,t ), 0 0 i.e. the case of sudden switching is assumed. The potentials are required to be expandable in a Taylor series about t . Then the following theorem holds: The densities n(r,t) and n@(r,t) evolving 0 from a common initial state W under the influence of two potentials v(r,t) and v@(r,t) are always 0 different provided that the potentials differ by more than a purely time-dependent function: v(r, t)Ov@(r, t)#c(t) .

(402)

The reader should consult Ref. [229] concerning the proof. As a consequence of this theorem the time-dependent wave function is a functional of the time-dependent density W(t)"e~*s(t)W[n](t) .

(403)

This functional is unique up to a time-dependent phase s(t). Concerning the expectation value of any quantum mechanical operator, the ambiguity of the phase cancels out and it is a unique functional of the density. E.g. the current density j(r,t)"SW(t)D jK (r)DW(t)T 1 is also a functional of the density. Here,

(404)

1 N (405) jK (r)" + (+rjd(r!r )#d(r!r )+rj) j j 1 2i j/1 is the paramagnetic current density operator. The time-dependent particle and current densities can also be calculated from the hydrodynamical equations: n(r, t)"!+j(r, t) , t

(406)

j(r, t)"P(r, t) , t

(407)

where P(r, t)"!iSW(t)D[ jK (r),HK (t)]DW(t)T . 1 According to the stationary-action principle the quantum mechanical action integral

P T K

(408)

K U

t1 dt W(t) i !HK (t) W(t) (409) t 0 t has a stationary point at the correct time-dependent density. Consequently, the solution of the Euler equation A"

dA[n] "0 dn(r, t)

(410)


64

leads to the correct density. The functional A[n] can be separated as

P P

t1 dt n(r, t)v(r, t) dr , t0 where the time-dependent functional B[n] A[n]"B[n]!

P T K

(411)

K U

dt W(t) i !¹K (t)!»K W(t) %% t 0

t1

(412) t is universal as it is independent of the external potential v(r,t). The approach presented above is valid only for v-representable densities. There are proposals [231,232] for a Levy—Lieb type extension of the theory. In the following v-representability is assumed. B"

11.4.2. Time-dependent Kohn—Sham scheme Just like in the time-independent case a non-interacting system, in which the electrons move independently in a common local potential, is constructed. The time-dependent Kohn—Sham equations have the form

C

D

1 ! + 2#v (r, t) u (r, t)"i u (r, t) . KS i 2 t i

(413)

The density of the non-interacting system N n(r, t)" + Du (r, t)D2 (414) i i/1 is equal to that of the interacting one. The current density built up from the non-interacting orbitals 1 N (415) j(r,t)" + (u*(r, t)+u (r, t)!u (r, t)+u*(r, t)) i i i i 2i i/1 is also identical with the true current density of the interacting system. The time-dependent Kohn—Sham potential v (r, t)"v(r, t)#v (r, t)#v (r, t) KS J 9# includes the external v, the classical Coulomb

P

n(r@,t) dr@ v (r, t)" J Dr!r@D

(416)

(417)

and the exchange-correlation dA [n] v (r, t)" 9# , 9# dn

(418)

potentials. A [n] is the exchange-correlation part of the action functional. There is an important 9# difference between the time-independent and the time-dependent schemes. Here, the functionals,


65

such as A or B implicitly depend on the initial state W . Consequently, the Kohn—Sham potential 0 also depends on the initial orbitals. In principle, infinitely many Slater determinants reproduce the same electron density [233]. Any of them can, of course, be used in the formalism. If, however, both the interacting and the non-interacting initial states are non-degenerate ground-states, there is no dependence on the initial state. The formalism presented above is valid if the motion of the atomic nuclei is neglected. It can be, however, extended to treat the nuclear motion either quantum mechanically or classically (for discussion and references see [230]). 11.4.3. Time-dependent spin and current density functional theory The theory can be extended to time-dependent electromagnetic fields. First the time-dependent spin density functional theory is reviewed. The time-independent formalism was outlined in Section 4.2. Following the notations of Section 4.2, the interaction with the time-dependent external field is given by N N (419) »K " + v(r )#2b + B(r) ) s . i i % i/1 i/1 Using this operator in the action functional Eq. (409) one obtains the time-dependent Kohn—Sham equations

C

D

1 ! + 2#v (r, t) u (r, t)"i u (r, t) . KSp ip 2 t ip

(420)

The spin-dependent Kohn—Sham potential and the density of electrons with spin p have the forms v (r, t)"v (r, t)#v (r, t)#v (r, t) , (421) KSp p J 9#p Np n (r, t)" + Du (r, t)D2 , (422) p ip i/1 respectively. The spin-dependent exchange-correlation potential v (r,t) is defined as functional 9#p derivative of the exchange-correlation action functional A [n ,n ]. 9# ¬ Turning to the problem of coupling to orbital currents, the kinetic energy ¹K in Eq. (399) has to be replaced with

A

B

N 1 1 2 ¹K (t)" + !i+ # A(r , t) , (423) i c i 2 i/1 where A(r, t) is the time-dependent vector potential. The gauge-invariant current density is given by 1 j(r, t)"SW(t)D jK (r)DW(t)T# n(r, t)A(r, t) , 1 c

(424)

where jK (r) is the paramagnetic current density operator defined by Eq. (405). Then the basic 1 theorem of the time-dependent current density functional theory reads: The current densities j(r, t) and j@(r, t) evolving from a common initial state W under the influence of the four-potentials 0 (v(r, t), A(r, t)) and (v@(r, t), A@(r, t)) which differ by more than a gauge transformation are always different provided that the potentials can be expanded in Taylor series around the initial time t . 0


66

The Kohn—Sham scheme can be derived in the usual way and the Kohn—Sham equations have the form:

CA

B

D

1 2 1 !i+# A(r, t) #v (r, t) u (r, t)"i u (r, t) , KS i 2 c t i

(425)

where

A

B

1 N 1 N (426) j(r, t)" + (u*(r, t)+u (r, t)!u (r, t)+u*(r, t))# + Du (r, t)D2 A(r, t) . i i i i i c 2i i/1 i/1 These equations are rather complicated arising from the fact that here n(r, t) and j(r, t) are the basic variables. For electrons in static electromagnetic fields, Vignale and Rasolt [213] have formulated a current density functional theory in terms of the density and the paramagnetic current density. The time-dependent formalism in terms of the density and the paramagnetic current density is not available yet. Several extensions of the time-dependent formalism have been presented, e.g. superconductors in time-dependent electromagnetic fields. (For references see [230].) 11.4.4. Time-dependent linear density response Consider an electron system subject to external potentials v(r,t)"v (r)#v (r,t), where v (r) is the 0 1 0 external potential of the unperturbed system and v (r,t) is the time-dependent perturbation. The 1 unperturbed state is supposed to be the ground-state corresponding to v (r). The external potential 0 is a functional of the time-dependent density and the density—density response function can be expressed as

K

dn[v](r, t) s(r, t; r@, t@)" dv(r@, t@)

, (427) v*n0+ where the functional derivative is to be evaluated at the external potential corresponding to the unperturbed ground-state density n . The linear density response to the perturbation v (r,t) reads 0 1

P P

n (r, t)" dt@ dr@s(r, t; r@, t@)v (r@, t@) . 1 1

(428)

Expression (427) is valid in the Kohn—Sham scheme, too, writing s (r, t; r@, t@) instead of s(r, t; r@, t@) KS and v (r, t) instead of v(r, t). The response function (427) can also be given by a Dyson-type KS equation

P PP P

C

s(r, t; r@, t@)"s (r, t; r@, t@)# dr@@ dq dr@@@ dq@s (r, t; r@@, t) KS KS

D

d(q!q@) Dr@@!r@@@D

#[f (r@@, q; r@@@, q@)]s(r@@@, q@; r@, t@) , (429) 9# that expresses the relation between the interacting and non-interacting response functions. The time-dependent exchange-correlation kernel f (r, t; r@, t@)"dv (r, t)/dn(r, t) comprises all dynamic 9# 9# exchange and correlation effects to linear order in the perturbing potential. Taking the Fourier


67

transform with respect to time the linear density response has the form

P

C

P A

B

D

1 n (r, u)" dr@s (r, r@; u) v (r@, u)# dr@@ #f (r@, r@@; u) n (r@@, u) . 1 KS 1 9# 1 Dr@!r@@D

(430)

The Kohn—Sham response function that can be expressed in terms of the unperturbed Kohn—Sham orbitals: u (r)u*(r)u*(r@)u (r@) j k s (r, r@; u)"+ (j !j ) j k (431) KS k j u!(e !e )#ig j k j, k has poles at the Kohn—Sham orbital energy differences. (j stands for the occupation number of the j orbital u .) The linear response formalism can be extended to systems at finite temperature [234] j and current density response theory for arbitrary time dependent electromagnetic fields has also been worked out [220]. Higher-order response theory has also been developed [230] since recently there is a growing interest in nonlinear phenomena. A hierarchy of equations for the time-dependent density response has been derived [89]. 11.4.5. Excitation energies Just like in the time-independent case one has to use approximations to the time-dependent exchange-correlation potential. The simplest possible approximation is the time-dependent or “adiabatic” local density approximation (ALDA). The functional form of the static LDA is used with the time-dependent density: vALDA(r, t)"v)0.(n(r, t)) , (432) 9# 9# where v)0. is the exchange-correlation potential of the homogeneous electron gas. This approxima9# tion leads to an exchange-correlation kernel f ALDA having no frequency-dependence at all. To 9# incorporate the frequency-dependence into f Gross and Kohn [235] proposed that 9# f LDA(r, r@; u)"f )0.(n (r), Dr!r@D; u) , (433) 9# 9# 0 i.e. the frequency-dependent exchange-correlation kernel of the homogeneous electron gas. (For a detailed discussion see Ref. [230].) The optimized potential method reviewed in Section 7 has been extended to the time-dependent case. To construct an optimised effective potential the action functional is written as in Eq. (411), where the exchange-correlation part of the action functional has the form:

P P

1N t1 u*(r@, t)u (r@, t)u (r,t)u*(r, t) j i j A [n]"! + d i j dr dr@ . (434) dt i 9# pp 2 Dr!r@D ~= i, j A procedure similar to the one described in Section 7 leads to the time-dependent optimised potential method [236]. From the several kinds of applications (such as photoresponse of finite and infinite systems, van der Waals interactions) (see Ref. [230]) we mention only that the time-dependent density functional theory can be efficiently applied to calculate excitation energies. Table 10 presents excitation energies of several atoms calculated by Gross et al. [230, 237] with local density approximation,


68

Table 10 The lowest 1SP1P excitation energies of several atoms calculated by OPM, LDA, OPM#ALDA methods (in a.u.) Atoms

OPM

LDA

OPM#ALDA

Exp

Be Mg Ca Zn Sr Cd

0.196 0.164 0.117 0.211 0.105 0.188

0.200 0.176 0.132 0.239 0.121 0.214

0.199 0.165 0.118 0.209 0.106 0.185

0.194 0.160 0.108 0.213 0.099 0.199

optimised potential method and a method, in which the optimised potential method was used for v and local density approximation (ALDA) for the exchange-correlation kernel f . The values 9# 9# obtained by the optimised potential method are clearly superior to the local density results. A similar method based on the one-particle density matrix has been recently proposed by Casida [238]. Bauernschmitt and Ahlrichs [239] calculated excitation energies of several molecules by various local, gradient-corrected and hybrid functionals and found considerable improvement over Hartree—Fock-based approaches requiring comparable numerical work. 11.5. Relativistic density functional theory The relativistic extension of the ground-state density functional theory has also been done. The forerunner was the relativistic Thomas—Fermi model [240]. The relativistic counterparts of the Hohenberg and Kohn theorem were formalized by Rajagopal and Callaway [241]. The Kohn—Sham equations were derived by Rajagopal [242] and independently by MacDonald and Vosko [243]. The retardation corrections to the Coulomb interaction were taken into consideration, however, the radiative corrections were essentially neglected. A field theoretical background was addressed by Engel and Dreizler [244] and is not covered here. (For a recent review of relativistic density functional theory see [245].) 11.5.1. Relativistic Hohenherg—Kohn theorem and Kohn—Sham equations The relativistic Hohenherg—Kohn theorem is proved by reductio ad absurdum. (Renormalised quantities should be used to obtain the proof.) It states that there exists a one-to-one correspondence between the classes of external potentials » just differing by gauge transformations and the l ground-state four current jl. Fixing the gauge once and for all one arrives at the statement that all ground-state observables are unique functionals of the four current. The energy functional E[ jl] contains all relativistic kinetic effects for both electrons and photons and all radiative effects. The variational principle takes the form

P

d E[ jl]!k j0"0 djl

(435)

with the constraint of charge conservation. (All quantities are supposed to be fully renormalised.)


69

In the presence of only a purely electrostatic external potential: »l"(»0, 0) the ground-state energy and all the observables are unique functionals of the zeroth component j0, i.e. the density n only. To derive the Kohn—Sham equations one has to introduce single-particle four spinors u . Then k the exact ground-state four current has the form: jl"jl #jl , V D where

C

1 jl " V 2

(436)

D

+ uN clu ! + uN clu #Djl,(0) k k k k ~m:ek eky~m

(437)

and jl " + uN clu , (438) D k k ~m:ekyeF where Djl,(0) is the counterterm and e represents the Fermi level below which all orbitals are F occupied. (For the derivation see Refs. [244,245].) The total energy can be decomposed as E[ jl]"¹ [ jl]#E [ jl]#J[ jl]#E [ jl] , (439) s %95 9# where ¹ , E , J and E are the non-interacting kinetic, external, Coulomb and exchange-correla4 %95 9# tion energy parts, respectively. The relativistic Kohn—Sham equations have the form: c0(!ic+#m#e». #v. #v. )u "e u , J 9# k k k where

P

(440)

jl(y) v (x)"e2 dy , J Dx!yD

(441)

d E [ jl] 9# vl " 9# djl

(442)

and

are the Coulomb and exchange potentials, respectively. The Kohn—Sham equations should be solved self-consistently, which is a very complicated problem, as it includes summation over all negative and positive energy solutions and renormalisation in each iterative step. The most important simplification of this tedious problem is the no-sea approximation, in which all radiative contributions to the four current and ¹ and the vacuum contribution in E are 4 9# neglected. Another simplification may be applied if the external potential is purely electrostatic. In this case, only the density n"j0 is the variational object and the spatial current j is a functional of the density. If the electron—electron interaction decomposed into longitudinal and transverse parts one arrives at another approximation. It is straightforward for the classical part: E [n]"EL[n]#ET[ j[n]] , J J J

(443)


70

where

P

e2 n(r)n(r@) EL[n]" dr dr@ , J 2 Dr!r@D

P

j(r) j(r@) e2 , ET[ j[n]]"! dr dr@ J Dr!r@D 2

(444) (445)

while EL is defined by neglecting the transverse interaction to all orders in the free-electron 9# propagator and the remainder is called ET . In the longitudinal approximation the transverse 9# contributions ET and ET are neglected in the self-consistent calculations and added perturbatively J 9# to the total energy. Then the Kohn—Sham equations have the form: [!ia ) +#bm#vL]u "e u , k k k where a and b are the Dirac matrices, while vL is the total relativistic potential

(446)

vL"v#vL#vL J 9# containing the external v, the longitudinal Coulomb

(447)

P

n(r@) vL[n(r)]"e2 dr@ J Dr!r@D

(448)

and the longitudinal exchange dEL [n] vL " 9# 9# dn

(449)

potentials. The electron density is given by n" + uù , k k ~m:ekyeF

(450)

11.5.2. Relativistic exchange-correlation functionals Just like in the non-relativistic case the exact form of the relativistic exchange-correlation functional is unknown. On the other hand, if the orbital representation of the exchange energy is known the optimised potential method can be applied (see Section 7). Relativistic extension of the optimised potential method was put forward by Talman and coworkers [246] in the longitudinal no-sea level and was recently applied to atoms by Engel et al. [247]. In the longitudinal no-pair approximation for a purely electrostatic potential the method is completely analogous to the non-relativistic case. Calculations obtained with the exchange-only optimised potential method result in ground-state energies being extremely close to the relativistic Hartree—Fock ones. In complete analogy to the non-relativistic case the relativistic local density approximation is based on the relativistic homogeneous electron gas:

P

ERLDA[n]" dr eRLDA(n(r)) . 9# 9#

(451)


71

The lowest order contribution gives the exchange energy that can be written as the non-relativistic exchange energy multiplied by the relativistic correction factor (for references see [245]). The longitudinal part has the form eRLDA,L(n)"eNRLDA(n)UL(b) , 9 9 x where (3p2n)1@3 b" . mc

(452)

(453)

There is a similar expression for the transverse part. It turned out that the longitudinal contribution dominates in the low density limit and depends only weekly on b. On the other hand, the transverse part shows a stronger dependence on b and dominates in the high density limit. Relativistic correlation contribution to the local density approximation has only been studied as a partial resummation of those terms in the perturbation expansion in e2 which are the most relevant in the high density limit. The relativistic analogue of the non-relativistic weighted density approximation has also been worked out. It has the advantage of insuring the satisfactory cancellation of self-interaction effects and consequently reproducing the asymptotic r~1 proportionality of the exchange potential though with an incorrect prefactor. The local density approximation is the most frequently applied approach in the non-relativistic calculations. Though it is far from being exact, because of the partial cancellation of errors between the exchange and correlation contributions, it often leads to acceptable results. In the relativistic case, however, the situation is somewhat different. The error in the total longitudinal exchange energy is about 10% for light atoms and reduces to about 5% for heavier atoms. Considering the relativistic correction to the exchange energy, one can conclude that longitudinal correction is underestimated by about 20%, while the transverse one is overestimated by about 50%. These errors do not cancel, therefore the local density approximation reproduces rather poorly the exchange energies. Studying the exchange potential Engel and Dreizler [245] found that the local errors are quite significant. Table 11 presents the relativistic exchange and correlation energies for Ne, Xe and Hg atoms [244]. For comparison Hartree—Fock and Møller—Plesset data are also included. Correlation energy was studied using the RPA for relativistic corrections. The results are not satisfactory.

12. Concluding remarks I have made no attempt here to survay all contributions to the density functional theory. My apology to the authors of the many important papers that are not referenced in the present review. I have not covered topics of considerable importance such as Thomas—Fermi type density functional theory, momentum-space density functional theory [248], formulation of the density functional theory using local scaling transformation [24,249], numerical methods and applications. My objective has been to review several relevant aspects encountered in the development of the theory.

72


Table 11 Relativistic exchange and correlation energies [244] in atomic units NROPM, ROPM, RHF, RLDA, LDA and MBPT2 denote the non-relativistic optimized potential method, the relativistic optimized potential method, the relativistic Hartree—Fock method, the relativistic local density approximation, the non-relativistic local density functional approximation and second-order Møller—Plesset method, respectively Longitudinal exchange energy (!EL) 9 NROPM ROPM RHF RLDA

RHF RLDA

MBPT2 LDA MBPT2 LDA

Transverse exchange energy (!ET) 9

Correlation energy !DEL # (longitudinal) !ET # (transverse)

Ne

Xe

Hg

12.105 12.120 12.123 10.944

179.062 184.083 184.120 174.102

345.240 365.203 365.277 347.612

0.017 0.035

5.711 9.089

22.168 34.201

0.20 0.38 1.87 0.32

37.57 64.73 108.75 39.27

203.23 200.87 282.74 113.08

I think that the future prospects of density functional theory are bright. It is already competitive with the conventional methods and prospective applications to problems in several fields, such as molecular biology are particularly promising. Still, there remain a lot of problems to solve, and there are several open questions. Though, very efficient exchange-correlation functionals are nowadays available, the search for better and better functionals will go on. It is generally believed that exact relations and theorems, quite exhaustively detailed here, are especially useful in constructing approximate functionals. The number of these constraints, however, is still growing and for the time being it is not quite clear which ones are the most important. It is expected that orbital-dependent potentials are possible candidates. The optimised potential method solves the exchange-only problem exactly. Inclusion of an appropriate orbital-dependent correlation term might lead to an even more accurate solution. In spite of several efforts and considerable developments the form of the kinetic energy fuctional is still unknown. A sufficiently accurate approximation to this functional would lead to a breakthrough in the density functional theory, because the solution of the Euler-equation would generally need considerably less numerical effort than would the Kohn—Sham equations. There are a multitude of problems to be solved in the extension of the density functional theory. Further important developments are expected in these fields. Undoubtedly, there is room for further extentions of the density functional theory.


73

Acknowledgements The author is most grateful to Professor G. Parr for valuble discussions, encouragement and generous hospitality. The grant “Szećhenyi” from the Hungarian Ministry of Culture and Education is gratefully acknowledged. This work was supported by the grant MTA-NSF No. 93, the grants OTKA No. T 16623 and F 16621 and FKFP 0314/1997.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32]

L.H. Thomas, Proc. Cambridge Phil. Soc. 23 (1926) 542; E. Fermi, Z. Phys. 48 (1928) 73. P. Gomba´s, Die Statistische Theorie des Atoms und Ihre Anwendungen, Springer, Wien, 1949. P.A.M. Dirac, Proc. Cambridge Phil. Soc. 26 (1930) 376. J.C. Slater, Phys. Rev. 81 (1951) 381. R. Ga´spa´r, Acta. Phys. Hung. 3 (1954) 263. P. Hohenberg, W. Kohn, Phys. Rev. B 136 (1964) 864. W. Kohn, L.J. Sham, Phys. Rev. 140A (1965) 1133. A.K. Rajagopal, Theory of inhomogeneous electron systems: spin-density-functional formalism, Adv. Chem. Phys. 41 (1980) 59. A.S. Bamzai, B.M. Deb, The role of single particle density in chemistry, Rev. Mod. Phys. 53 (1981) 95. U. Gupta, A.K. Rajagopal, Phys. Rep. 87 (1982) 259. R.V. Ramana, A.K. Rajagopal, Inhomogeneous relativistic electron systems: a density-functional formalism, Adv. Chem. Phys. 54 (1983) 231. J. Keller, J.L. Ga´zquez (Eds.), Density Functional Theory, Springer, Berlin, 1983. S. Lundgvist, N.H. March (Eds.), Theory of the Inhomogeneous Electron Gas, Plenum Press, New York, 1983. R.G. Parr, Annu. Rev. Phys. Chem. 34 (1983) 631. J. Callaway, N.H. March, Density functional methods: theory and applications, Solid State Phys. 38 (1984) 135. J.P. Dahl, J. Avery (Eds.), Local Density Approximation in Quantum Chemistry and Solid State Physics, Plenum Press, New York, 1984. R.M. Dreizler, J. de Providencia (Eds.), Density Functional Methods in Physics, Plenum Press, New York, 1985. N.H. March, B.M. Deb, Single Particle Density in Physics and Chemistry, Academic Press, New York, 1987. R. Erdahl, V.H. Smith Jr. (Eds.), Density Matrices and Density Functionals, Reidel, Dordrecht, 1987. K.D. Sen (Ed.), Electronegativity, Springer, Berlin, 1987. D.R. Salahub, Adv. Chem. Phys. 68 (1987) 447. R.G. Parr, W. Yang, Density Functional Theory of Atoms and Molecules, Oxford Univ. Press, New York, 1989. N.H. March, Electron Density Theory of Atoms and Molecules, Academic Press, New York, 1989. E.S. Kryachko, E.V. Ludena, Density Functional Theory of Many-Electron Systems, Academic Press, New York, 1989. R.O. Jones, O. Gunnarsson, Rev. Mod. Phys. 61 (1989) 689. R.M. Dreizler, E.K. Gross, Density Functional Theory, Springer, Berlin, 1990. S.B. Trickey (Ed.), Density-functional theory of many-fermion systems, Adv. Quantum Chem. 21 (1990). J.K. Labanowski, J.W. Andzelm (Eds.), Density-Functional Methods in Chemistry, Springer, New York, 1991. T. Ziegler, Chem. Rev. 91 (1991) 651. K.D. Sen (Ed.), Chemical Hardness, Springer, Berlin, 1993. D.E. Ellis (Ed.), Density-Functional Theory of Molecules, Clusters and Solids, Kluwer, Dordrecht, 1994. J.M. Seminario, P. Politzer (Eds.), Density Functional Theory, A Tool for Chemistry, Elsevier, Amsterdam, 1994.

74


[33] S.B. Trickey, in: E.S. Kryachko, J.L. Calais (Eds.), Conceptual Trends in Quantum Chemistry, Kluwer, Dordrecht, 1994, p. 87. [34] E.K.U. Gross, R.M. Dreizler (Eds.), Nato ASI Series Vol. B 337, Plenum, New York, 1995. [35] D.P. Chong (Ed.), Recent advances in the density functional methods, in: Recent Advances in Computational Chemistry, vol. 1, World Scientific, Singapore, 1996. [36] R.M. Dreizler, Acta Phys. Chem. Debr. 30 (2) (1995) 21. [37] R. Nalewajski (Ed.), Density Functional Theory, Topics in Current Chemistry, vol. 179—182, Springer, Berlin, 1996. [38] N.C. Handy, in: Quantum Mechanical Simulation Methods for Studying Biological Systems, D. Bicout, M. Field (Eds.), Springer, Heidelberg, 1996, p. 1. [39] A. St-Amant, in: D. Bicout, M. Field (Eds.), Quantum Mechanical Simulation Methods for Studying Biological Systems, Springer, Heidelberg, 1996, p. 37. [40] T. Kato, Commun. Pure Appl. Math. 10 (1957) 151; E. Steiner, J. Chem. Phys. 39 (1963) 2365. [41] M. Levy, Proc. Natl. Acad. Sci. USA 76 (1979) 6002. [42] M. Levy, Phys. Rev. A 26 (1982) 1200. [43] M. Lieb, Int. J. Quantum. Chem. 24 (1982) 243. [44] U. von Barth, L. Hedin, Phys. Rev. A 20 (1972) 1629. [45] M.M. Pant, A.K. Rajagopal, Solid State Commun. 10 (1972) 1157. [46] C. Almbladh, U. von Barth, Phys. Rev. B 21 (1985) 3231. [47] J.D. Talman, W.F. Shadwick, Phys. Rev. A 14 (1976) 36. [48] E. Engel, J.A. Chevary, L.D. Macdonald, S.H. Vosko, Z. Phys. D 23 (1992) 7. [49] J.B. Krieger, Y. Li, G.J. Iafrate, Phys. Lett. A 146 (1990) 256; Int. J. Quantum. Chem. 41 (1992) 489; Phys. Rev. A 45 (1992) 101; Phys. Rev. A 46 (1992) 5453; in: E.K.U. Gross, R.M. Dreizler (Ed.), Density Functional Theory, Plenum, New York, 1995. [50] A. Holas, N.H. March, in: R. Nalewajski (Ed.), Density Functional Theory, Topics in Current Chemistry, vol. 180, Springer, Heidelberg, 1996. [51] S. Liu, R.G. Parr, A¨. Nagy, Phys. Rev. A 21 (1995) 2645. [52] F.J. Ga´lvez, I. Porras, Phys. Rev. A 44 (1991) 144; Phys. Rev. A 46 (1992) 105; Int. J. Quantum. Chem. 56 (1995) 763; I.C. Angulo, J.S. Dehesa, F.J. Ga´lvez, Phys. Rev. A 42 (1990) 641. [53] R. McWeeney, Rev. Mod. Phys. 32 (1960) 335. [54] O. Gunnarsson, B.I. Lundqvist, Phys. Rev. B 13 (1976) 4274. [55] J. Harris, R.O. Jones, J. Phys. F 4 (1974) 1170; J. Harris, Phys. Rev. A 29 (1984) 1648. [56] D.C. Langreth, J.P. Perdew, Phys. Rev. B 15 (1977) 2884. [57] M.S. Gopinathan, M.A. Whitehead, R. Bogdanovic, Phys. Rev. A 14 (1976) 1. [58] J.L. Gazquez, J. Keller, Phys. Rev. A 16 (1977) 1358. [59] J.P. Perdew, A. Savin, K. Burke, Phys. Rev. A 51 (1995) 4531; A. Savin, in: D.P. Chong (Ed.), Recent Advances in the Density Functional Methods, Recent Advances in Computational Chemistry, vol. 1, World Scientific, Singapore, 1996, p. 129; M. Ernzenhof, J.P. Perdew, K. Burke, in: R. Nalewajski (Ed.), Density Functional Theory, Topics in Current Chemistry, vol. 180, Springer, Berlin, 1996, p. 1. [60] M. Levy, J.P. Perdew, Phys. Rev. A 32 (1985) 2010. [61] A¨. Nagy, in: D.P. Chong (Ed.), Recent Advances in the Density Functional Methods, Recent Advances in Computational Chemistry, vol. 1, World Scientific, Singapore, 1996, p. 1. [62] A. Holas, N.H. March, Phys. Rev. A 51 (1995) 2040. [63] A¨. Nagy, N.H. March, Phys. Rev. A 40 (1989) 554. [64] N.H. March, W.H. Young, Nucl. Phys. 12 (1959) 237. [65] A¨. Nagy, N.H. March, Chem. Phys. Lett. 181 (1991) 279; Phys. Chem. Liq. 32 (1996) 319. [66] A. Nagy, R.G. Parr, Phys. Rev. A 42 (1990) 201. [67] L.F. Bartolotti, R.G. Parr, J. Chem. Phys. 72 (1980) 1593. [68] S.K. Ghosh, M. Berkowitz, J. Chem. Phys. 83 (1985) 2979. [69] B.M. Deb, S.K. Ghosh, J. Phys. B 12 (1979) 3857. [70] S.K. Ghosh, M. Berkowitz, R.G. Parr, Proc. Natl. Acad. Sci. USA 81 (1984) 8028.


75

[71] R.F.W. Bader, Atoms in Molecules: A Quantum Theory, Clarendon Press, Oxford, 1990. [72] P. Ziesche, D. Lehmann, Phys. Stat. Sol. 13 (1987) 467. See also P. Ziesche, J. Grafenstein, O.H. Nielsen, Phys. Rev. B 37 (198) 8167; Yu.A. Uspenskii, P. Ziesche, J. Grafenstein, Z. Phys. B 76 (1989) 193. [73] A¨. Nagy, N.H. March, Mol. Phys. 91 (1997) 597. [74] A¨. Nagy, Proc. Indian Acad. Sci. (Chem. Sci.) 106 (1994) 251. [75] S.K. Ghosh, R.G. Parr, J. Chem. Phys. 82 (1985) 3307. [76] S. Srebenik, R.F.W. Bader, T.T. Nguyen-Dang, J. Chem. Phys. 68 (1978) 3667; R.F.W. Bader, Chem. Phys. 73 (1980) 287; R.F.W. Bader, H. Esseń, in: J. Dahl, J. Avery (Eds.), Local Density Approximations in Quantum Chemistry and Solid State Physics, Plenum, New York, 1984. [77] A¨. Nagy, Phys. Rev. A 46 (1992) 5417. [78] M. Ishiara, Bull. Chem. Soc. Japan 58 (1975) 2472. [79] A¨. Nagy, Int. J. Quantum. Chem. 49 (1994) 353. [80] M. Levy, in: N.H. March, B.M. Deb (Eds.), Single Particle Density in Physics and Chemistry, Academic Press, New York, 1987. [81] M. Levy, J.P. Perdew, in: E.K.U. Gross, R.M. Dreizler (Eds.), Nato ASI Series Vol. B 337, Plenum, New York, 1995, p. 11. [82] M. Levy, Int. J. Quantum. Chem. 23 (1989) 617; Phys. Rev. A 43 (1991) 4637; A. Go¨rling, M. Levy, Phys. Rev. A 45 (1992) 1509; M. Levy, J.P. Perdew, Phys. Rev. B 48 (1993) 11 638. [83] J.P. Perdew, in: Electronic Structure of Solids ’91, Academic Press, Berlin, 1991; J.P. Perdew, Y. Wang, Phys. Rev. B 45 (1992) 13 244. [84] M. Levy, J.P. Perdew, Int. J. Quantum. Chem. 49 (1994) 539. [85] H. Ou-Yang, M. Levy, Int. J. Quantum. Chem. 40 (1991) 379; Phys. Rev. A 42 (1990) 155; Phys. Rev. A 42 (1990) 651. [86] A.A. Kugler, Phys. Rev. A 41 (1990) 3489. [87] A. Nagy, Phys. Rev. A 47 (1993) 2715. [88] D.J. Joubert, Phys. Rev. A 50 (1994) 3527; J. Chem. Phys. 101 (1994) 9701. [89] S. Liu, Phys. Rev. A 54 (1996) 1328. [90] A¨. Nagy, Phys. Rev. A 52 (1995) 984. [91] N.N. Bogoliubov, J. Phys. USSR 10 (1946) 257, 265; M. Born, H.S. Green, Proc. R. Soc. (London) Ser. A 188 (1946) 10; J.G. Kirkwood, J. Chem. Phys. 3 (1935) 300; J. Yvon, Actualities Scientifiques et Industrielles, vol. 203, Herman et Cie, Paris, 1935. [92] L. Cohen, C. Frishberg, Phys. Rev. A 13 (1976) 927; J. Chem. Phys. 65 (1976) 4234. [93] M. Berkowitz, R.G. Parr, J. Chem. Phys. 83 (1988) 2553. [94] H. Ou-Yang, M. Levy, Phys. Rev. A 44 (1991) 54. [95] R.G. Parr, J.L. Ga´zquez, J. Chem. Phys. 97 (1993) 3939. [96] R.G. Parr, S. Liu, A.A. Kugler, A. Nagy, Phys. Rev. A 52 (1995) 969; S. Liu, R.G. Parr, Phys. Rev. A 55 (1997) 1792. [97] A¨. Nagy, Phys. Rev. A 53 (1996) 3660. [98] J.C. Slater, The Self-Consistent Field for Molecules and Solids, McGraw-Hill, New York, 1974. [99] A¨. Nagy, J. Phys. B 26 (1993) 43; Phil. Mag. B 69 (1994) 779. [100] A¨. Nagy, N.H. March, Phys. Rev. A 39 (1989) 5512; 40 (1989) 5544. [101] Q. Zhao, R.G. Parr, J. Chem. Phys. 98 (1993) 543; R.G. Parr, Phil. Mag. B 69 (1994) 737; Q. Zhao, R.C. Morrison, R.G. Parr, Phys. Rev. A 50 (1994) 2138; R.C. Morrison, Q. Zhao, Phys. Rev. A 51 (1995) 1980; R.G. Parr, S. Liu, Phys. Rev. A 51 (1995) 3564. [102] C.O. Almbladh, A.P. Pedroza, Phys. Rev. A 29 (1984) 2322. [103] F. Arysetiawan, M.J. Stott, Phys. Rev. B 38 (1988) 2974; J. Chen, M.J. Stott, Phys. Rev. A 44 (1991) 2816; J. Chen, R.O. Esquivel, M.J. Stott, Phil. Mag. B (1994) 69. [104] A. Go¨rling, Phys. Rev. A 46 (1992) 3753; A. Go¨rling, M. Ernzerhof 51 (1995) 4501. [105] R. van Leeuwen, E.J. Baerends, Phys. Rev. A 49 (1994) 2421. [106] S. Liu, R.G. Parr, Phys. Rev. A 53 (1996) 2211; S. Liu, Phys. Rev. A 54 (1996) 4863. [107] A. Go¨rling, M. Levy, Phys. Rev. A 47 (1993) 13 105. [108] S. Liu, P. Su¨le, R. Lo´pez-Boada, A. Nagy, Chem. Phys. Lett. A 257 (1996) 68.

76


[109] A. Savin, Phys. Rev. A 52 (1995) 1805. [110] A. Go¨rling, M. Levy, Phys. Rev. B 47 (1993) 13 105; Phys. Rev. A 50 (1994) 196; Int. J. Quantum. Chem. S. 29 (1995) 93. [111] R.G. Parr, R.A. Donnelly, M. Levy, W.E. Palke, J. Chem. Phys. 69 (1978) 4431; S. Liu, R.G. Parr, Phys. Rev. A 55 (1997) 1792. [112] R.S. Mulliken, J. Chem. Phys. 2 (1934) 782. [113] R.T. Sanderson, Science 114 (1951) 670; Chemical Bond and Bond Energy, Academic Press, New York, 1976. [114] R.G. Parr, R.G. Pearson, J. Am. Chem. Soc. 105 (1983) 7512. [115] W. Yang, R.G. Parr, Proc. Natl. Acad. Sci. USA 82 (1985) 6723. [116] R.G. Pearson, J. Chem. Educ. 64 (1987) 561. [117] R.G. Parr, P.K. Chattaraj, J. Am. Chem. Soc. 113 (1991) 1854. [118] P.K. Chattaraj, H. Lee, R.G. Parr, J. Am. Chem. Soc. 113 (1991) 1855. [119] R.G. Parr, W. Yang, J. Am. Chem. Soc. 106 (1984) 4029. [120] K. Fukui, Science 218 (1987) 1442. [121] A¨. Nagy, R.G. Parr, Proc. Indian Acad. Sci. (Chem. Sci.) 106 (1994) 117. [122] C. Lee, R.G. Parr, Phys. Rev. A 35 (1987) 2377. [123] R.G. Parr, K. Repnik, S.K. Ghosh, Phys. Rev. Lett. 56 (1986) 1555. [124] M.K. Harbola, V. Sahni, Phys. Rev. Lett. 62 (1989) 489; V. Sahni, in: Nato ASI Series, vol. B 337, Plenum, New York, 1995, p. 217; in: Topics in Current Chemistry, vol. 182, Springer, Berlin, 1996, p. 1. [125] A¨. Nagy, Phys. Rev. Lett. 65 (1990) 2608. [126] M.K. Harbola, V. Sahni, Phys. Rev. Lett. 65 (1990) 2609. [127] M. Levy, N.H. March, Phys. Rev. A 55 (1997) 1885. [128] A. Solomatin, V. Sahni, Int. J. Quantum. Chem. S. 29 (1995) 31; V. Sahni, Int. J. Quantum. Chem. S 56 (1995) 265; K.D. Sen, Phys. Rev. A 44 (1991) 756; M. Slamet, V. Sahni, M.K. Harbola, Phys. Rev. A 49 (1994) 809. [129] K.D. Sen, Chem. Phys. Lett. 188 (1992) 510. [130] R.T. Sharp, G.K. Horton, Phys. Rev. 30 (1953) 317. [131] K. Aashamar, T.M. Luke, J.D. Talman, At. Data Nucl. Data Tables 22 (1978) 443. [132] M.R. Norman, D.D. Koelling, Phys. Rev. B 30 (1984) 5530. [133] J.P. Perdew, A. Zunger, Phys. Rev. B 23 (1981) 5048. [134] A¨. Nagy, Phys. Rev. A 55 (1997) 3465. [135] E.R. Davidson, Int. J. Quantum. Chem. 37 (1980) 811. [136] A¨. Nagy, Acta Phys. Chem. Debr. 30 (2) (1995) 47. [137] C.J. Umrigar, X. Gonze, Phys. Rev. A 50 (1994) 3827. [138] R. Leeuwen, E.J. Baerends, Phys. Rev. A 49 (1994) 2421; R. Leeuwen, O.V. Gritsenko, E.J. Baerends, in: Topics in Current Chemistry, vol. 180, Springer, Berlin, 1996, p. 107. [139] L. Ko¨ve´r (Ed.), Electronic Structure of Clusters: Development and Applications of the DV-Xa Method, Adv. Quantum Chem., in press. [140] D.M. Ceperley, B.J. Alder, Phys. Rev. Lett. B 45 (1980) 566. [141] S.H. Vosko, L. Wilk, M. Nusair, J. Phys. 58 (1980) 1200. [142] A.G. Koures, F.E. Harris, Int. J. Quantum Chem. 59 (1996) 3. [143] R. Ga´spa´r, A¨. Nagy, Acta Phys. Chem. Debr. 26 (1989) 7. [144] J.A. Alonso, L.A. Girifalco, Solid State Commun. 24 (1977) 135; O. Gunnarsson, R.O. Jones, Phys. Scr. 21 (1980) 394. [145] P. Gomba´s, Pseudopotential, Springer, Berlin, 1967; G.C. Lee, E. Clementi, J. Chem. Phys. 60 (1974) 1275. [146] L.C. Wilson, M. Levy, Phys. Rev. B 41 (1990) 12 930. [147] P. Su¨le, A¨. Nagy, Acta Phys. Chem. Debr. 29 (1994) 1. [148] C. Lee, R.G. Parr, Phys. Rev. A 42 (1990) 193. [149] D.J.W. Geldart, M. Rasolt, Phys. Rev. B 13 (1976) 1477. [150] D.C. Langreth, J.P. Perdew, Phys. Rev. B 21 (1980) 5469. [151] C.D. Hu, D.C. Langreth, Phys. Rev. B 33 (1986) 943. [152] A.D. Becke, J. Chem. Phys. 96 (1992) 2155.


77

[153] O. Gritsenko, R. van Leeuwen, E. van Lenthe, E.J. Baerends, Phys. Rev. A 51 (1995) 1944; R. van Leeuwen, O. Gritsenko, E.J. Baerends, Z. Phys. D 33 (1995) 229. [154] J.P. Perdew, in: Nato ASI Series, vol. B 337, Plenum, New York, 1995, p. 51; M. Ernzerhof, J.P. Perdew, K. Burke, in: Topics in Current Chemistry, vol. 180, Springer, Berlin, 1996, p. 1. [155] J.P. Perdew, J.A. Chevary, S.H. Vosko, K.A. Jackson, M.R. Pederson, D.J. Singh, C. Fiolhais, Phys. Rev. B 46 (1992) 6671. [156] E. Engel, S.H. Vosko, Phys. Rev. B 47 (1993) 13 164. [157] C. Lee, W. Yang, R.G. Parr, Phys. Rev. B 37 (1988) 785. [158] R. Colle, O. Salvetti, Theor. Chim. Acta 37 (1975) 329. [159] E.I. Proynov, A. Vela, D.R. Salahub, Chem. Phys. Lett. 230 (1994) 419; E.I. Proynov, D.R. Salahub, Int. J. Quantum Chem. 49 (1994) 67. [160] A.D. Becke, Phys. Rev. A 38 (1988) 3098. [161] J.P. Perdew, Phys. Rev. B 33 (1986) 8823; 34 (1986) 7406; J.P. Perdew, Y. Wang, Phys. Rev. B 33 (1986) 8800. [162] E.K.U. Gross, M. Petersilka, T. Grabo, in: T. Ziegler (Ed.), Density Functional Methods, American Chemical Society, Washington DC, 1996. [163] E.P. Wigner, Trans. Faraday Soc. 34 (1938) 678. [164] Y. Wang, J.P. Perdew, Phys. Rev. B 44 (1991) 13 298. [165] A. Savin, H. Stoll, H. Preuss, Theor. Chim. Acta 70 (1986) 407; E.R. Davidson, S.A. Hagstrom, S.J. Chakravorty, Phys. Rev. A 44 (1991) 7071. [166] P. Su¨le, O.V. Gritsenko, A. Nagy, E.J. Baerends, J. Chem. Phys. 103 (1995) 10 085. [167] M. Levy, A. Go¨rling, Phys. Rev. A 52 (1990) 1808. [168] P. Su¨le, Chem. Phys. Lett. 259 (1996) 8524. [169] R.G. Parr, W. Yang, Annu. Rev. Phys. Chem. 46 (1995) 701. [170] W. Kohn, A.D. Becke, R.G. Parr, J. Phys. Chem. 100 (1996) 12 974. [171] B.G. Johnson, P.M.W. Gill, J.A. Pople, J. Chem. Phys. 97 (1992) 7846; 98 (1993) 5612. [172] R. Neumann, R.H. Nobes, N.C. Handy, Mol. Phys. 87 (1996) 1. [173] N.O. Oliphant, R.J. Bartlett, J. Chem. Phys. 100 (1994) 6550. [174] A.D. Becke, M.E. Roussel, Phys. Rev. A 39 (1989) 3761. [175] A. Go¨rling, M. Levy, J. Chem. Phys. 106 (1997) 2675. [176] A. Seidl, A. Go¨rling, J.A. Majewski, P. Vogl, M. Levy, Phys. Rev. B 53 (1996) 3764. [177] W. Yang, Phys. Rev. Lett. 66 (1991) 1438; Phys. Rev. A 44 (1991) 7823; P. Cortona, Phys. Rev. B 44 (1991) 8454; 46 (1992) 2008; W. Hierse, E.B. Stechel, Phys. Rev. B 50 (1994) 17 811. [178] R. Car, M. Parinello, Phys. Rev. Lett. 55 (1985) 2471. [179] N.D. Mermin, Phys. Rev. 137 (1965) A1441. [180] W. Kohn, P. Vashishta, in: S. Lundgvist, N.H. March (Eds.), Theory of the Inhomogeneous Electron Gas, Plenum Press, New York, 1983, p. 79. [181] R.P. Feynman, N. Netropolis, E. Teller, Phys. Rev. 75 (1949) 1561. [182] R.D. Cowan, J. Ashkin, Phys. Rev. 105 (1957) 144. [183] F. Perrot, M.W.C. Dharma-Wardana, Phys. Rev. A 30 (1984) 2619. [184] D.G. Kanhere, P.V. Panet, A.K. Rajagopal, J. Callaway, Phys. Rev. A 33 (1986) 490. [185] S. Phatisena, R.E. Amritkar, P.V. Panet, Phys. Rev. A 34 (1986) 5070. [186] A. Ghazali, P.L. Hugon, Phys. Rev. Lett. 41 (1978) 1569. [187] O. Gunnarsson, B.I. Lundqvist, Phys. Rev. B 13 (1976) 4274. [188] J.C. Slater, Adv. Quantum Chem. 6 (1972) 1. [189] A.K. Theophilou, J. Phys. C 12 (1978) 5419. [190] L. Fritsche, Phys. Rev. B 33 (1986) 3976; L. Fritsche, Int. J. Quantum Chem. S 21 (1987) 15. [191] H. English, H. Fieseler, A. Haufe, Phys. Rev. A 37 (1988) 4570. [192] L.N. Oliveira, E.K.U. Gross, W. Kohn, Phys. Rev. A 37 (1988) 2805, 2809, 2821. [193] A¨. Nagy, Phys. Rev. A 49 (1994) 3074. [194] M. Levy, W. Yang, R.G. Parr, J. Chem. Phys. 83 (1985) 2334. [195] M. Levy, Phys. Rev. A 43 (1991) 4637.

78 [196] [197] [198] [199] [200] [201] [202] [203] [204] [205] [206] [207] [208] [209] [210] [211] [212] [213] [214] [215] [216] [217] [218] [219] [220] [221] [222] [223] [224] [225] [226] [227] [228] [229] [230] [231] [232] [233] [234] [235] [236] [237] [238] [239] [240] [241] [242]

A! . Nagy / Physics Reports 298 (1998) 1—79 A¨. Nagy, Int. J. Quantum Chem. 56 (1995) 225. W. Kohn, Phys. Rev. A 34 (1986) 737. A¨. Nagy, Phys. Rev. A 42 (1990) 4388. R. Ga´spa´r, Acta Phys. Hung. 35 (1974) 213. A¨. Nagy, J. Phys. B 24 (1991) 4691. A¨. Nagy, I. Andrejkovics, J. Phys. B 27 (1994) 233. O. Gunnarsson, B.I. Lundgvist, J.W. Wilkins, Phys. Rev. B 10 (1974) 1319. U. von Barth, L. Hedin, J. Phys. C 5 (1972) 1629. I. Andrejkovics, A. Nagy, Acta Phys. et Chim. Debr. 29 (1994) 7. A¨. Nagy, J. Phys. B 29 (1996) 389. P.S. Bagus, B.I. Bennett, Int. J. Quantum Chem. 9 (1975) 143. T. Ziegler, A. Rauk, E.J. Baerends, Theor. Chim. Acta (Berlin) 43 (1977) 261. U. von Barth, Phys. Rev. A 20 (1979) 1693. A¨. Nagy, Int. J. Quantum. Chem. Symp. 29 (1995) 297. E. Clementi, A. Roetti, At Data Nucl. Data Tables 14 (1974) 177. A¨. Nagy, Adv. Quantum Chem., in press. A. Go¨rling, Phys. Rev. A 54 (1996) 3912. G. Vignale, M. Rasolt, Phys. Rev. Lett. 59 (1987) 2360; Phys. Rev. B 37 (1988) 10635. G. Vignale, M. Rasolt, D.J.W. Geldart, Adv. Quantum Chem. 21 (1990) 235; in: R. Nalewajski (Ed.), Density Functional Theory, Topics in Current Chemistry, vol. 179—182, Springer, Berlin, 1996. A.K. Rajagopal, K.P. Jain, Phys. Rev. A 5 (1972) 1475. G. Vignale, M. Rasolt, D.J.W. Geldart, Phys. Rev. B 37 (1988) 2502. S. Ma, K.A. Brueckner, Phys. Rev. 165 (1968) 18. R.W. Danz, M.L. Glasser, Phys. Rev. 4 (1971) 96. A.M. Lee, N.C. Handy, S.M. Colwell, Chem. Phys. Lett. 229 (1994) 225; J. Chem. Phys. 103 (1995) 10 095; Phys. Rev. A 53 (1996) 1316; S.M. Colwell, N.C. Handy, Chem. Phys. Lett. 217 (1994) 271. T.K. Ng, Phys. Rev. Lett. 62 (1989) 2417. C.J. Grayce, R.A. Harris, Phys. Rev. A 50 (1994) 3089. C.J. Grayce, R.A. Harris, Mol. Phys. 72 (1991) 523. V.G. Maikin, O.L. Malkina, D.R. Salahub, Chem. Phys. Lett. 204 (1993) 80; 204 (1993) 87. F. Bloch, Z. Physik 81 (1933) 363. V. Peuckert, J. Phys. C 11 (1978) 4945. A. Zangwill, P. Soven, Phys. Rev. A 21 (1980) 1561. B.M. Deb, S.K. Ghosh, J. Chem. Phys. 77 (1982) 342; Theor. Chim. Acta 62 (1983) 209; J. Mol. Struct. 103 (1983) 163. L.J. Bartolotti, Phys. Rev. A 24 (1981) 1661; Phys. Rev. A 26 (1982) 2243; J. Chem. Phys. 80 (1984) 5687; Phys. Rev. A 36 (1987) 4492. E. Runge, E.U.K. Gross, Phys. Rev. Lett. 52 (1984) 997. E.U.K. Gross, J.F. Dobson, M. Petersilka, in: Topics in Current Chemistry, vol. 181, Springer, Berlin, 1996, p. 81. H. Kohl, R.M. Dreizler, Phys. Rev. Lett. 56 (1986) 1993. S.K. Ghosh, A.K. Dhara, Phys. Rev. A 38 (1988) 1149. J.E. Harriman, Phys. Rev. A 24 (1981) 680; G. Zumbach, K. Maschke, Phys. Rev. B 28 (1983) 54; 29 (1984) 1585. T.K. Ng, K.S. Singwi, Phys. Rev. Lett. 59 (1987) 2627; W. Yang, Phys. Rev. A 38 (1988) 5512. E.U.K. Gross, W. Kohn, Phys. Rev. Lett. 55 (1985) 2850; 57 (1986) 923. C.A. Ullrich, U.J. Grossman, E.U.K. Gross, Phys. Rev. Lett. 74 (1995) 872. M. Petersilka, U.J. Grossman, E.U.K. Gross, Phys. Rev. Lett. 76 (1996) 1212. M.F. Casida, in: Advances in Computational Chemistry, vol. 1, World Scientific, Singapore, 1996, p. 155. R. Bauernschmitt, R. Ahlrichs, Chem. Phys. Lett. 256 (1996) 454. M.S. Vallarta, N. Rosen, Phys. Rev. 41 (1932) 708; H. Jensen, Z. Phys. 82 (1933) 794. A.K. Rajagopal, J. Callaway, Phys. Rev. B 7 (1973) 1912. A.K. Rajagopal, J. Phys. C 11 (1978) L943.

A! . Nagy / Physics Reports 298 (1998) 1—79 [243] [244] [245] [246] [247] [248] [249]

79

A.M. MacDonald, S.H. Vosko, J. Phys. C 12 (1979) 2977. E. Engel, R.M. Dreizler, in: Nato ASI Series, vol. B, 337, Plenum, New York 1995, p. 65. E. Engel, R.M. Dreizler, in: Topics in Current Chemistry vol. 181, Springer, Berlin, 1996, p. 1. B.A. Shadwick, J.D. Talman, M.R. Norman, Comput. Phys. Commun. 54 (1989) 95. E. Engel, S. Keller, A. Facco Bonetti, H. Miller, R.M. Dreizler, Phys. Rev. A 52 (1995) 2750. B.G. Englert, Phys. Rev. A 45 (1992) 127; M. Cinal, B.G. Englert, Phys. Rev. A 45 (1992) 135; 48 (1993) 1893. E.S. Kryachko, E.V. Ludena, New J. Chem. 16 (1992) 1089; E.S. Kryachko, T. Koga, Int. J. Quantum Chem. 42 (1992) 591; E.V. Ludena, R. Lopez-Boada, J.E. Maldonado, E. Valderrama, E.S. Kryachko, T. Koga, J. Hinze, Int. J. Quantum Chem. 56 (1995) 285; E.V. Ludena, R. Lopez-Boada, in: Topics in Current Chemistry, vol. 180, Springer, Berlin, 1996, p. 169.

DARK OPTICAL SOLITONS: PHYSICS AND APPLICATIONS

Yuri S. KIVSHAR!, Barry LUTHER-DAVIES" ! Australian Photonics Co-operative Research Centre, Optical Sciences Centre, Research School of Physical Sciences and Engineering, Australian National University, ACT 0200, Canberra, Australia " Australian Photonics Co-operative Research Centre, Laser Physics Centre, Research School of Physical Sciences and Engineering, Australian National University, ACT 0200, Canberra, Australia

AMSTERDAM — LAUSANNE — NEW YORK — OXFORD — SHANNON — TOKYO

Physics Reports 298 (1998) 81—197

Dark optical solitons: physics and applications Yuri S. Kivshar!, Barry Luther-Davies" ! Australian Photonics Co-operative Research Centre, Optical Sciences Centre, Research School of Physical Sciences and Engineering, Australian National University, ACT 0200, Canberra, Australia " Australian Photonics Co-operative Research Centre, Laser Physics Centre, Research School of Physical Sciences and Engineering, Australian National University, ACT 0200, Canberra, Australia Received April 1997; editor: A.A. Maradudin

Contents 1. Introduction 2. Dark vs. bright solitons 2.1. Optical fibers 2.2. Nonlinear waveguides 2.3. When the NLS equation fails 2.4. Modulational instability and solitons 2.5. Dark solitons: mathematical tools 2.6. Integrals of motion 2.7. Physical interpretation of dark solitons 3. Perturbation theory and applications 3.1. Equations for soliton parameters 3.2. Physical applications 3.3. Dark-soliton jitter 3.4. Effect of third-order dispersion 3.5. Background of finite extent 4. Instability-induced soliton dynamics 4.1. Stability of dark solitons 4.2. Asymptotic approach 4.3. Examples of non-Kerr dark solitons

84 87 87 88 92 93 96 100 103 105 105 108 114 116 119 121 121 123 128

5. Multi-component dark solitons 5.1. Mode interaction: general overview 5.2. Dark—bright solitons 5.3. Modes with opposite dispersions 5.4. Polarization instability and domain walls 5.5. Parametric dark solitons in s(2) media 6. Experimental verifications 6.1. Dark solitons in fibers 6.2. Spatial dark solitons 6.3. Coupled dark—bright solitons 7. Dark solitons in higher dimensions 7.1. Introductory remarks 7.2. Transverse instability of plane solitons 7.3. Vortex solitons: theory 7.4. Vortex solitons: experiments 7.5. Ring dark solitons 8. Conclusion and open problems References

137 137 139 142 145 148 153 153 160 167 168 168 170 172 179 185 188 190

Abstract We present a detailed overview of the physics and applications of optical dark solitons: localized nonlinear waves (or ‘holes’) existing on a stable continuous wave (or extended finite-width) background. Together with the traditional problems involving properties of dark solitons of the defocusing cubic nonlinear Schro¨dinger equation, we also describe recent theoretical results on optical vortex solitons; ring dark solitons; polarization domain walls; parametric dark solitons in a dispersive s(2) medium; vector dark solitons; coupled dark—bright soliton pairs, and we discuss the 0370-1573/98/$19.00 Copyright ( 1998 Elsevier Science B.V. All rights reserved PII S 0 3 7 0 - 1 5 7 3 ( 9 7 ) 0 0 0 7 3 - 2

Y.S. Kivshar, B. Luther-Davies / Physics Reports 298 (1998) 81—197

83

instability-induced dynamics of dark solitons in the models of generalized (i.e., non-Kerr) optical nonlinearities. Special attention is paid to the experimental demonstrations of temporal dark solitons in optical fibres and spatial dark solitons, especially dark-soliton stripes and vortex solitons, in a defocusing bulk medium. ( 1998 Elsevier Science B.V. All rights reserved. PACS: 42.65.!k

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1. Introduction Optical solitary waves, temporal and spatial solitons, have been the subject of intense theoretical and experimental studies in recent years. Solitons — localized pulses in time or bounded self-guided beams in space — evolve from a nonlinear change in the refractive index of a material induced by the light intensity distribution. When the combined effects of the refractive nonlinearity and the pulse dispersion (in the case of temporal solitons) or beam diffraction (in the case of spatial solitons) exactly compensate each other, the pulse or beam propagates without change in shape and is said to be self-trapped. Nonlinear effects responsible for soliton formation in optical fibers are, in general, weak and Kerr-like, i.e. they induce a local index change directly proportional to the light intensity. In this case the main nonlinear equation governing the pulse evolution is the famous cubic nonlinear Schro¨dinger (NLS) equation for the complex amplitude envelope of the electric field which, depending on the sign of the group-velocity dispersion, has two distinct types of localized solutions, bright or dark solitons. These two types of waves look like two members of a general family of localized solutions, and this idea manifests itself in the drawing of Marc Haelterman, see Fig. 1. However, as will be seen from the results presented below, these two types of solitary waves are in fact very different, they have completely different nature and result from quite different physics. In the case of temporal solitons observed in optical fibers [Hasegawa (1989); an extended overview and history can be also found in the book by Hasegawa and Kodama (1995)], the group velocity dispersion is known to vanish at a wavelength of about 1.3 lm and is positive at larger wavelengths and negative at shorter ones. As a result, since silica optical fibers have always a positive Kerr coefficient, the two different signs of group-velocity dispersion support two different types of solitons, dark, in the former case, and bright, in the latter case. A similar situation occurs for self-guided beams or spatial optical solitons [Chiao et al. (1964); see also Chiao et al. (1993)] observed in planar waveguides or in a bulk medium. Here diffraction plays a role analogous to dispersion in the temporal domain, but now the nonlinearity may be either positive, for the so-called self-focusing nonlinear medium, or negative, for self-defocusing medium. This again gives rise to two distinct types of solitons, bright and dark, respectively.

Fig. 1. Do these ‘animals’ belong to the same soliton family? (the drawing made by Marc Haelterman in 1989).


85

When the group-velocity dispersion in an optical fiber is anomalous (or, similarly, when the nonlinearity of a planar waveguide is self-focusing), a constant amplitude continuous wave is unstable due to the modulational instability [see, e.g., Hasegawa (1989)], and breaks down into a sequence of localized pulses (or beams for the spatial domain). These pulses are bright solitons. Propagation of bright solitons in optical fibers has been verified in a number of elegant experiments, see, e.g., the pioneering paper by Mollenauer et al. (1980), as well as more recent investigations of long-distance soliton transmission in periodically amplified fibers [e.g., Mollenauer et al. (1990)]. These results have been presented in several review papers and books [e.g., Hasegawa (1989), Agrawal (1989), Hasegawa and Kodama (1995) and Haus and Wong (1996)]. In the case of normal group-velocity dispersion in fibers (or a self-defocusing nonlineariy in waveguides), bright solitons do not exist, instead initial pulses (or spatially localized beams) undergo enhanced dispersion (or diffraction) induced broadening and chirping. In this case a constant amplitude wave is modulationally stable and localized pulses can appear only as “holes” on a continuous wave (cw) background, i.e., as dark solitons. Interest in the behaviour of such dark solitons has been motivated by several experimental observations of temporal dark solitons in optical fibers (Emplit et al., 1987; Kro¨kel et al., 1988; Weiner et al., 1988) and spatial dark solitons in bulk media and waveguides (Andersen et al., 1990; Swartzlander et al., 1991; Allan et al., 1991; Skinner et al., 1991; Luther-Davies and Yang, 1992a,b; Duree et al., 1995; Taya et al., 1996; Z. Chen et al., 1996b). Although there has now been many successful experiments in which dark solitons have been observed in optical systems because of the relative ease of producing high intensity light beams or short (ps or fs) optical pulses, it should be remembered that the basic physics behind dark-soliton propagation on a modulationally stable background wave is quite fundamental and it applies to nonlinear problems with a different physical context. By way of illustration we note here some other experimental results including the excitation of nonpropagating n-kink surface modes in a long channel of shallow liquid driven parametrically (Denardo et al., 1990), the observation of dark-soliton standing waves in a discrete mechanical system [Denardo et al. (1992); see also the theory of this phenomenon presented by Kivshar (1993a)], the observation of high-frequency dark solitons during pulse propagation in thin magnetic films (M. Chen et al., 1993), and so on. Optical dark solitons have been investigated in many theoretical and experimental papers and several years ago the early results in this field were summarized in two review papers (Weiner, 1992; Kivshar, 1993b). However, recent experimental achievements have increased interest in the potential applications of optical dark solitons. For example, it was demonstrated (Luther-Davies and Yang, 1992b) that various types of all-optical switches may be ‘written’ using structures created during propagation and interaction of dark spatial solitons. As was demonstrated earlier for bright solitons (Reynaud and Barthelemy, 1990), these induced structures can guide a weak probe beam of a different frequency or polarization thus acting as light induced structured waveguides. These kinds of devices have very interesting properties, e.g., they may conserve transverse velocity — the key characteristic used in dark spatial soliton switching — even in the presence of two-photon absorption (Yang et al., 1994); the effect which can have a dramatic destructive influence on bright spatial solitons (Silberberg, 1990a). The purpose of this review paper is to describe, in the framework of an unified approach, the basic physics of the dark-soliton propagation using the examples taken from nonlinear waveguide optics. We present a systematic analysis of the properties of scalar dark solitons, in the framework of the generalized NLS equations, and vector dark solitons and their generalizations such as

86


bright—dark solitons, in the framework of coupled NLS equations. Although throughout we relate the theory to possible applications in guided wave optics, the analytical results are rather general and are applicable to other fields. First, we discuss the physical origin and properties of dark solitons and similar localized structures in the nonlinear systems with no or small dissipation. We consider the most interesting examples, with applications in nonlinear guided wave optics, and discuss the effect of perturbations and the stability of dark solitons, and their waveguiding properties. As a generalization of the concept of dark solitons, we include a summary of results on dark solitons in quadratic or s(2) media; vector dark solitons; and dark—bright solitons; polarization domain walls; (2#1)dimensional dark solitons of circular symmetry; etc. One of the important parts of our review is a summary of the experimental results demonstrating the generation and propagation of (1#1)and (2#1)-dimensional dark solitons. We emphasize that in presenting the analytical results, we avoid the traditional restrictions associated with consideration of only the cases of integrable models (i.e., the cubic NLS equation, for scalar solitons, or the Manakov equation, for vector solitons), as has been done in many previous review papers and books on optical solitons. Instead, we concentrate on the physics of the underlying phenomena and more realistic (generally nonintegrable) physical models. This involves naturally discussions of soliton stability and instabilityinduced evolution of dark solitons, since in nonintegrable models solitary waves can become unstable. At the same time, we do not discuss here some phenomena which can be also related to the physics of dark solitons, such as vectorial dark solitons of the ¹M type [e.g., Y. Chen (1991a,b)], different types of envelope shock waves connecting background of nonequal intensities [e.g., Christodoulides (1991), Kivshar and Malomed (1993), Kivshar and Turitsyn (1993), Cai et al. (1997)], dark surface modes in slab waveguide structures with defocusing nonlinearity [e.g., Andersen and Skinner (1991a,b), Miranda et al. (1992) and Y. Chen and Atai (1992)], dark solitons and dark-profile modes in discrete lattices (Kivshar, 1993a; Kivshar et al., 1994a,c), dark gap solitons in the systems with periodically varying parameters such as optical waveguides with grating [e.g., Feng and Kneubu¨hl (1993), Kivshar (1995) and Kivshar et al. (1995)]. We believe these, and some other related topics still require further analysis and deeper insight into stability as well as experimental verifications. The structure of our review paper is as follows. In Section 2 we provide a kind of framework for the remaining chapters. First, we discuss the basic equations describing the physics of dark solitons in nonlinear optics. This includes nonlinear pulse propagation in optical fibers (temporal solitons), where weak nonlinearity can be described in the framework of the Kerr effect, and self-guided optical beams (spatial solitons) that require to introduce generalized phenomenological models of non-Kerr nonlinearities. Next, we describe the physical origin of dark solitons and discuss their difference from bright solitons by analyzing the results of modulational instability of continuous waves. As we demonstrate, this leads to a different choice of mathematical tools for analyzing these two types of solitons, and a specific role of the integrals of motion and, in the case of dark soliton, the soliton phase. To discuss solitary waves in more realistic physical models described by a perturbed cubic NLS equation (temporal solitons) or in media with non-Kerr optical response (spatial solitons), we present a summary of the results of the perturbation theory developed for dark solitons (Section 3) and also discuss the characteristic scenarios of the instability-induced dynamics dark solitons (Section 4). In particular, we analyse some effects important for optical applications,


87

e.g. dark-soliton jitter (Section 3.3); the effect of third-order dispersion on dark solitons (Section 3.4); and briefly discuss the role of a finite-width background in experimental observations of dark solitons (Section 3.5). In Section 5 we present several important multi-component generalizations of dark solitons for systems describing incoherent and parametric interactions between optical polarization modes or harmonics. Section 6 gives a summary of experimental results on dark solitons in optical fibers and the (1#1)-dimensional dark spatial solitons. Extensions of the concept of dark soliton to higher dimensions are presented in Section 7, where we discuss also the theory and experimental demonstrations of optical vortex solitons. Finally Section 8 concludes the review being served as a guide to some open, unsolved problems.

2. Dark vs. bright solitons 2.1. Optical fibers To discuss the physics of dark solitons in optical fibers, we should start from the basic dynamical equation for the complex envelope amplitude of the electric field which can be derived taking into account the weak nonlinearity arising from the Kerr effect in a silica glass. This derivation is well-known and it is exactly the same as in the case of bright solitons [see, e.g., Agrawal (1989), Hasegawa (1989) and Hasegawa and Kodama (1995)]. The derivation is based on the well-known Maxwell’s equations for a dielectric medium in which is assumed that the electric displacement vector D can be split up into two parts, linear and nonlinear ones. The nonlinearity arises from the Kerr effect alone, and the nonlinear part D may be presented in the form, D "n DED2E, where E is nl nl 2 the electric field, and n is the Kerr coefficient. 2 The next important step of the derivation is the use of the fact that wave envelope function E(z, t) is a slowly varying function in the propagation coordinate z and retarded time t, which can be expanded using the Fourier space variable Du"u!u . This represents a small frequency shift of 0 the side band from the carrier frequency u , which in turn induces a small shift pf the carrier wave 0 number, Dk"k!k . The expansion of the wave number k(u) around k can be therefore 0 0 presented as a standard Fourier expansion, k k!k " 0 u

K

1 2k (u!u )# 0 2 u2

K

(u!u )2#2 , 0

u/u0 u/u0 where the second-order derivative describes the wave dispersion. Expanding the field envelope E and taking into account simultaneously both temporal (or group-velocity) dispersion and weak Kerr nonlinearity, we arrive at the well-known cubic nonlinear Schro¨dinger (NLS) equation. In an appropriate system of normalized coordinates, this equation becomes i

u p 2u ! #DuD2u"0 , z 2 x2

(2.1)

where p"$1 stands for the sign of the second-order derivative of k(u). This sign corresponds to two distinct types of the fiber group-velocity dispersion, namely anomalous, when p"!1, or normal, when p"#1. The meaning of other values is the following: u is the normalized amplitude

88


of the electric field envelope E describing the pulse, z is the normalized distance along the fiber, and the time variable t is a retarded time measured in the reference frame moving along the fiber with the group velocity. The normalization units are well known and, as a matter of fact, they are the same as for bright solitons [see, e.g. Hasegawa (1989)], zPz/Z , x"(t!z/» )/t . 0 ' # The most frequently used normalized units are pc¹2 ¹ Z "0.322 , t" , 0 # 1.763 j2D 0 2p d2k dk D" , »~1"2 . ' j2 du2 du 0 u/u0 u/u0 Here the parameter ¹ represents the full width at half maximum (FWHM) of the pulse intensity, and the pulse propagation is considered in the reference frame moving with the group velocity » . '

G H

G H

2.2. Nonlinear waveguides 2.2.1. Why temporal and spatial solitons are different Usually, the stationary beam propagation in planar waveguides is considered as the phenomenon similar to the pulse propagation in fibers referring to the so-called spatio-temporal analogy in wave propagation [Akhmanov et al. (1967); see also Svelto (1974)]. This means that the propagation coordinate z is treated as the evolution variable and the spatial beam profile along the transverse direction, for the case of waveguides, is similar to the temporal pulse profile, for the case of fibers. This analogy has been developed by many researchers, and it is based on a simple notion that both beam and pulse propagation can be described by the cubic NLS equation [see, e.g., Boardman and Xie (1993), Chiao et al. (1993)]. However, contrary to the accepted opinion, there exists a crucial difference between these two phenomena. Indeed, in the case of the nonstationary pulse propagation in fibers, the operation wavelength is usually selected near the zero of the group-velocity dispersion. This means that the absolute value of the fiber dispersion is small enough to be compensated by a weak nonlinearity such as that produced by the (very weak) Kerr effect in optical fibers which leads to a nonlinearity-induced change in the refractive index by the order of 10~10. Therefore, nonlinearity in fibers is always weak and it is well modeled by the cubic NLS equation, which is known to be integrable by means of the inverse scattering transform [Zakharov and Shabat (1971, 1973); see also Zakharov et al. (1980)]. However, for very short (fs) pulses the cubic NLS equation should be corrected to include some additional (but still small) effects such as higher-order dispersion, induced Raman scattering, etc. (e.g., Hasegawa (1989) and Hasegawa and Kodama (1995)]. Thus, in fibers nonlinear effects are very weak and they become important on large distances (of order of hundred meters or even kilometers). Contrary to the pulse propagation in optical fibers, the physics underlying the stationary beam propagation in planar waveguides and a bulk medium is different. In this case the nonlinear change in the refractive index should compensate for the beam spreading caused by diffraction which is not a small effect. That is why to observe spatial solitons much larger nonlinearities are usually required, and very often such nonlinearities are not of the Kerr type (e.g. they saturate at higher


89

intensities). This leads to the models of generalized non-linearities (see Section 2.2.3 below) with the properties of solitary waves different from those described by the integrable cubic NLS equation. For example, unlike the solitons of the integrable cubic NLS equation, solitary waves of generalized nonlinearities may be unstable, they also show some interesting properties such as fusion due to collision, etc. Propagation distances usually involved into the phenomenon of the beam self-focusing and spatial soliton propagation are of order of millimeters or centimeters. Nevertheless, the physics of spatial solitary waves is very rich and it should be understood in the framework of nonintegrable models. 2.2.2. Basic equations First, we consider the propagation of a monochromatic scalar electric field E in a bulk optical medium with an intensity-dependent refractive index, n"n #n (I), where n is the linear 0 /0 refractive index, and n (I) describes the variation in the index due to the field with the intensity /I"DED2. The function n (I) is assumed to be dependent only on the light intensity, and it is usually /introduced phenomenologically. Solutions of the governing Maxwell’s equation can be presented in the form E(R , Z; t)"E(R , Z)e*b0Z~*ut#c.c. , (2.2) M M where c.c. denotes complex conjugate, u is the source frequency, and b "k n "2pn /j is the 0 0 0 0 plane-wave propagation constant for the uniform background medium, in terms of the source wavelength j"2pc/u, c being the free-space speed of light. Further, we assume a (2#1)dimensional model, so that the Z-axis is parallel to the direction of propagation, and the X- and ½-axis are two transverse directions. The function E(R , Z) describes the wave envelope which in the absence of nonlinear and M diffraction effects E would be a constant. If we substitute Eq. (2.2) into the two-dimensional, scalar wave equation, we obtain the generalized nonlinear parabolic equation,

A

B

E 2E 2E 2ik n # # #2n k2n (I)E"0 . 0 0 Z 0 0 /X2 ½2

(2.3)

In dimensionless variables, Eq. (2.3) becomes the well-known generalized NLS equation, where local nonlinearity is introduced by the function n (I). /For the case of the Kerr (or cubic) nonlinearity we have n (I)"n I, n being the coefficient of /2 2 the Kerr effect of an optical material. Now, introducing the dimensionless variables, i.e. measuring the field amplitude in the units of k Jn Dn D and the propagation distance in the units of k n , we 0 0 2 0 0 obtain the (2#1)-dimensional NLS equation in the standard form, i

A

B

u 1 2u 2u # # $DuD2u"0 , z 2 x2 y2

(2.4)

where the sign ($) is defined by the type on nonlinearity, self-defocusing (‘minus’, for n (0) or 2 self-focusing (‘plus’, for n '0). 2 For propagation in a slab waveguide, the field structure in one of the directions, say y, is defined by the linear guided mode of the waveguide. Then, the solution of the governing Maxwell’s

90


equation has the structure n z~*ut#2 , E(R , Z; t)"E(X, Z)A (½)e*b(0) M n

(2.5)

where the function A (½) describes the corresponding fundamental mode of the slab waveguide. n Similarly, substituting this ansatz into Maxwell’s equations and averaging over ½, we come again to the renormalized equation of the form of Eq. (2.4) with the ½-derivative omitted, which in the dimensionless form becomes the standard cubic NLS equation i

u 1 2u # $DuD2u"0 . z 2 x2

(2.6)

As has been discussed above, Eq. (2.6) coincides formally with Eq. (2.1) of Section 2.1, which has been derived in the theory of pulse propagation in dispersive nonlinear optical fibers. 2.2.3. Models of optical nonlinearities The generalized NLS Eq. (2.3) has been considered in many papers for analyzing the beam self-focusing and properties of spatial bright and dark solitons [see, e.g., Zakharov et al. (1971), Zakharov and Synakh (1975), Kaplan (1985a,b), Enns and Mulder (1989), Mulder and Enns (1989), Gatz and Herrmann (1991, 1992), Herrmann (1992), Bass et al. (1992), Snyder and Sheppard (1993), Kro´likowski and Luther-Davies (1992, 1993), Kro´likowski et al. (1993), Valley et al. (1994), Christodoulides and Carvalho (1995), Pelinovsky et al. (1996a,b), Micallef et al. (1996)]. All types of non-Kerr nonlinearities discussed in relation with the existence of solitary waves in nonlinear optics can be divided, generally speaking, into three main classes: (i) competing nonlinearities, e.g. focusing (defocusing) cubic and defocusing (focusing) quintic nonlinearity [see, e.g., Kaplan (1985a,b), Gatz and Herrmann (1991); Kro´likowski et al. (1993)] and also generalization to a power nonlinearity [e.g., Pelinovsky et al. (1996a) and Micallef et al. (1996)]; (ii) saturable nonlinearities [see, e.g., Snyder and Sheppard (1993), Kro´likowski and Luther-Davies (1992, 1993), Valley et al. (1994) and Christodoulides and Carvalho (1995)], and (iii) transiting nonlinearities [see, e.g., Kaplan (1985a,b), Enns and Mulder (1989) and Bass et al. (1992)]. Usually, the nonlinear refractive index of an optical material deviates from the linear (Kerr) dependence for larger light intensities. Nonideality of the nonlinear optical response is known for semiconductor (e.g., AlGaAs, CdS, CdS Se ) waveguides and semiconductor-doped glasses [see, 1~x x e.g., Roussignol et al. (1987), Acioli et al. (1990) and Lederer and Biehlig (1994)]. Larger deviation from the Kerr nonlinearity is observed for nonlinear polymers. For example, recently the measurements of a large nonresonant nonlinearity in single crystal PTS (p-toluene sulfonate) at 1600 nm (Lawrence et al., 1994a,b) revealed a variation of the nonlinear refractive index with the input intensity which can be modeled by competing, cubic-quintic nonlinearity, n (I)"n I#n I2 . /2 3

(2.7)

This model describes a competition between self-focusing (n '0), at smaller intensities, and 2 self-defocusing (n (0), at larger intensities. Similar models are usually employed to describe the 3 stabilization of wave collapse in the (2#1)-dimensional NLS equation [e.g., Josserand and Rica (1997) and references therein].


91

In a more general case, the models with competing nonlinearities can be described by power-law dependence on the beam intensity, n (I)"n Ip#n I2p , (2.8) /p 2p where p is a positive constant and usually n n (0. p 2p Models with saturable nonlinearities are the most typical ones in nonlinear optics. For higher powers saturation of nonlinearity has been measured in many materials and consequently the maximum refractive index change has been reported [see, e.g., Coutaz and Kull (1991)]. We do not linger on the physical mechanisms behind the saturation but merely note that it exists in many nonlinear media being usually described by phenomenological models introduced more than 25 years ago [see, e.g., Gustafson et al. (1968), Reichert and Wagner (1968) and Marburger and Dawes (1968)]. The effective generalized NLS equation with saturable nonlinearity is also the basic model to describe the recently discovered (1#1)-dimensional photovoltaic dark solitons in photovoltaic—photorefractive materials as LiNbO [see Valley et al. (1994)]. Unlike the phenomenological 3 models usually used to describe saturation of nonlinearity, for the case of photovoltaic solitons this model finds its rigorous justification (Valley et al., 1994; Christodoulides and Carvalho, 1995). There exist several models of the nonlinearity saturation. From a general point of view, the function n (I) describing the nonlinearity saturation should be characterized by three independent /parameters: the saturation intensity, I , the maximum change in the refractive index, n , and the 4!5 = Kerr coefficient n which appears for small I. In particular, the phenomenological model 2 1 n (I)"n 1! , (2.9) /= (1#I/I )p 4!5 satisfies these criteria, provided n "n p/I . In the particular case p"1, the model defined by 2 = 4!5 Eq. (2.9) reduces to the well-known expression derived from the two-level model, which is used most frequently. For the case p"2 the model Eq. (2.9) possesses localized solutions for bright and dark solitons in an explicit analytical form (Kro´likowski and Luther-Davies, 1992, 1993). At last, bistable solitons introduced by Kaplan (1985a,b) usually require a special type of the intensity-dependent refractive index which changes from one type to another one, e.g., it varies from one kind of the Kerr nonlinearity, for small intensities, to another kind with different value of n , for larger intensities. This type of nonlinearity is known to support bistable dark solitons (Enns 2 and Mulder, 1989; Mulder and Enns, 1989) as well. One of the simplest models of such transiting nonlinearities describes a change from one type of the Kerr dependence, to the other one, i.e.,

G

H

G

n(1)I I(I #3 n (I)" 2 (2.10) /n(2)I I'I . 2 #3 A smooth transition of this kind can be modeled by the function (Enns and Mulder, 1989) n (I)"n IM1#a tanh[c(I2!I2 )]N , (2.11) /2 #3 where for I;I , n (I)Kn(1)I, where n(1)"n [1!a tanh2(cI2 )], and for I R . (7.72) 0 (2p)d

P

Since the calculation of the loop(s) in the denominator of Fig. 11 involves integrals like :ddR G (R, N), we obtain two volume factors coming from two rings. In the numerator the extra 0 volume factor will come because of the translational invariance of the interaction potential. These considerations explain the presence of the volume factor in Fig. 11. Calculation of the numerator

328

A.L. Kholodenko, T.A. Vilgis / Physics Reports 298 (1998) 251—370

Fig. 11. Feynman diagram for the calculation of the second virial coefficient A2.

involves calculation of the following expression:

P P P P P P

N N dq dq@ dr dr dr@ dr@ G (r !r ,q) 1 2 1 2 0 1 2 0 0 ]G (r !r , N!q)»(Dr (q)!r@ (q@)D) 0 2 1 2 2 ]G (r@ !r@ ,q@)G (r@ !r@ , N!q@) , (7.73) 0 1 2 0 2 1 where »(Dr !r D) is the polymer—polymer interaction potential. Substitution of Eq. (7.72) into 1 2 Eq. (7.73) and account of translational invariance immediately produce the following result for I: I"

P

I"N2[G (0,N)]2»3 dR »(DRD) 0

(7.74)

so that the result for A follows: 2 A "N2»(k"0) , 2 where »(k"0) is obtained by noticing that

(7.75)

P

»(k)" dR e*k > R»(DRD) .

(7.76)

To account for the constraint given by Eq. (7.71) it is useful to recalculate I using a different method, Feynman (1972). For this purpose, we have to consider the calculation of the following auxiliary functional integral for the closed path:

P P

G P

H

1 N N dq dq@rR 2#ik ) r(q) . D[r(q@)] exp ! (7.77) 4 r 0 0 (0)/r(N) To calculate such an integral it is very useful to introduce the following Fourier decomposition of r(q): IK "

A B

= npq r(q)"a # + a cos . 0 n N n/1

(7.78)


329

Using this decomposition the action in the exponent of the path integral of Eq. (7.77) can be written as (Kholodenko and Quian, 1988)

A B

p2 = = npq S" . (7.79) + a2 n2!ik ) + a cos n n 8N N n/1 n/0 The integral over the zero mode a produces d(k) so that the second term in Eq. (7.79) (with 0 components a other than zero) vanishes upon k-integration. The first term in Eq. (7.79) has the n same structure as that calculated for the closed paths (Feynman, 1972), and, whence will produce the same result as Eq. (7.72) (for R"0). The presence of d(k) is important since if we represent the interaction potential as

P P

ddk ddk@ k r k r »(r , r )" e* > 1`* {> 2»(k, k@) 1 2 (2p)d (2p)d

(7.80)

so that »(k,k@)"»(k)d(k#k@), then evidently, zero modes coming from two rings will remove k and k@ integrations in Eq. (7.80) thus producing the volume factor d(0) coming from d(k#k@). Collecting all terms together, we arrive again at the result of Eq. (7.75) as required. Consider now the more complicated case which involves the constraint of Eq. (7.71). In this case, we have to substitute into the path integral measure the d-factor given by

P

C A

P

P

BD

d3K N N exp iK ) R!1/N dq r(q)#1/N dq@r(q@) . (2p)3 0 0 The presence of this factor changes the action in the exponent of Eq. (7.77) into d"

P

P

1 N 1 N dq@rR 2!ik ) r(q)#i K ) dq@ r(q@) . S" N 4 0 0 Use of the Fourier expansion, Eq. (7.78), converts the above action into

(7.81)

(7.82)

A B

p2 = = npq S@" #i K ) a . (7.83) + a2n2!ik ) + a cos 0 n n 8N N n/1 n/0 Integration of the zero mode produces now the d-constraint: d(k!K). By integrating over k (see, e.g. Eq. (7.80)) we are left with the following action:

C

A BD

p2 = npq S@" . (7.84) + a2 n2!i K ) a cos n n 8N N n/1 Performing the Gaussian integration over each of a modes we obtain now the following result n including both rings and discarding factors like d(0):

P

P P C A B

d3K K R N N º(DRD) " e* > » (K) dq dq@ a (2p)3 k ¹ 0 0 B 2NK2 = 1 npq npq@ cos2 #cos2 ]exp ! + p2 N N n2 n/1

G

A BDH

.

(7.85)

330


Following Feynman (1972), we can replace the summation in the exponent of Eq. (7.85) by the integration via the rule:

P

= N = + 2P dx2 . p 0 n/1

(7.86)

This produces after a little calculation (upon completion of q integrations and rescaling):

A B P

A B

º(DRD) 2 const (a) R 4~a = sin xy dx xa~5(1!e~x2@R2)2 " , p k ¹ xy JN B 0

(7.87)

where const(a) was defined after Eq. (7.38) and y~1"JN. Straightforward convergence analysis indicates that the obtained integral is convergent for 14a43 in complete agreement with the results of Section 7.3. In order to actualy use Eq. (7.87) we notice that for y;1 we can subdivide the domain of x-integration into two parts, e.g. from 0 to 1 and from 1 to R. We then can appropriately Taylor series expand the integrand in each subdomain of integration by taking into account that (in chosen system of units) R251. For a"1 and R;N we obtain, in view of Eq. (7.69), the result of Everaers and Kremer (1996) given by Eq. (7.65) while for a"2 we obtain the result of Helfand and Pearson (1983). Comparison between these results and Eq. (7.44) indicates that the exponent a in Eq. (7.87) is likely to be bounded by the inequality 14a42 for any kind of entanglement of a given polymer with other polymers (or with itself). This observation leads us to Eq. (2.21) where, accordingly, we obtain 24u43. Obtained results allow us to calculate several additional quantities. For example, in view of Eq. (7.64), one can calculate the topological second virial coefficient AT between two non-entangled 2 polymers. Following Vologodskii et al. (1975), we obtain

P C

A

BD

1 F (R) AT" d3r 1!exp ! 0 2 2 k ¹ B

,

(7.88)

where F (R) is related to P (R) via 0 0 F "!k ¹ ln P (R) . 0 B 0

(7.89)

Substitution of Eq. (7.89) into Eq. (7.88) produces, in view of Eqs. (7.64) and (2.21), the following result for AT (for aK1): 2 AT"4pR3 . 2 3 L

(7.90)

It is quite remarkable that this result was obtained with help of only qualitative arguments by Frisch and Wasserman (1961) as discussed in Section 2. The existence of non-negative AT causes 2 additional repulsion between the polymer rings (not to be confused with depression of H temperature discussed in Section 7.5) thus leading to the effective reduction of their sizes. More quantitative analysis of this phenomenon is provided in Section 8.


331

8. Polymer dynamics: an interplay between topology and geometry 8.1. Statistical mechanics of a melt of polymer rings Already in Section 2 we have noticed that in the melt of linear polymers of length N the ratio JSR2T/NP0 for NPR. I.e. dynamically, for times q(q the melt of polymer rings and linear 5 polymers should behave very similar which is indeed the case, see, e.g. McKena et al. (1989) (especially Fig. 20 of this reference). This experimental observation is very important for the development of the dynamics of polymer melts of both linear and ring polymers. The entanglements which are inevitably present in such melts in the form of (quasi) links are not only responsible for the formation of the effective tube which surrounds the given polymer chain (which is well documented experimentally, Straube et al., 1995) but affect also the stiffness of the trapped chain. We have discussed this fact, in part, in Section 6.3 from the geometrical point of view. Here we would like to discuss the same problem from the topological point of view. To this purpose, let us consider again the partition function given by Eq. (4.10). Following the work of Brereton and Vilgis (1995), we shall concentrate our attention on a single ring placed in a melt of other rings. The many body problem which involves different rings is going to be reduced effectively to the one-body problem for the ring which is being singled out. To this purpose, let us rewrite Eq. (4.10) in the following form:

T

U

n n Z(Mc N,Mm N)" < d(lk(a, b),m ) < d(lk(b, b@), m ) (8.1) a bb{ ab bb{ bEa b;b{Ea where d(x,y) is, as before, the Kroneker’s delta and S2T denotes the polymer averaging, e.g. like that given by Eq. (7.29), of all n!1 chains, except one, which we denote as a. Since the Kroneker’s delta can be written as

P

2p dg expMig(x!y)N , 2p 0 Eq. (8.1) can be equivalently rewritten according to Brereton and Vilgis (1995) as d(x,y)"

(8.2)

P

n 2p dg bb{Z(Mc N;Mg N) expM!ig m N (8.3) Z(Mc N;Mm N)" < a bb{ bb{ bb{ a bb{ 2p bb{/1 0 where (ua(q)]ub(!q)) ) q 1 Z(Mc N,Mgbb{N)" exp + (g #g ) a ab ba q2 X bEa (ub(q)]ub{(!q)) ) q 1 ]exp + g , (8.4) bb{ q2 X bb{Ea and X is the volume of the system. In arriving at the result given by Eq. (8.3) the linking number, defined by Eq. (4.11), has been transformed with help of identities

T G G

Q

dlad(ra!r) ,

ua(r)"

c 1 qk rk " d3q exp(iq ) r) . q2 DrD3 2p2i a

P

HU

H

(8.5) (8.6)

332


Such transformation allows to calculate the partition function, Eq. (8.1), in formally closed form given by

P

P

n 1 p 2p~abb{ ds expMi2s m N Z(Mc N;Mm N)" < da bb{ bb{ bb{ a bb{ bb{ p2 abb{ 0 bb{/1 1 ]exp ! + [A (q;Ms N)l(q;Mc N)!B (q;Ms N)/(q;Mc N)] aa bb b aa bb{ a X q

G

H

(8.7)

where the matrices

G C G C

D H D H

n C2 ~1 A (q;Ms N)" C2 1# aa bb{ c(q) q2 n C3 C2 ~1 B (q;Ms N)" 1# aa bb{ c(q) q2 q2

,

aa ,

with matrix C being given by ab C ,[C(q)] "(1/n)s c(q) ; ab ab ab while c(q)"o

QQ QQ QQ

(8.8)

aa

dl ) dl@SexpMiq ) (r!r@)NT ,

(8.9)

(8.10)

ca ca

l(q;Mc N)" a

/(q;Mc N)" a

dl · dl@

ca ca ca c

expMiq ) (r(l)!r(l@))N , q2

expMiq ) (r(l)!r(l@))N dl]dl@ ) q q2 a

(8.11) (8.12)

with o"Nn/X. Evidently, in order to calculate Eq. (8.7) explicitly, some approximations should be made. These are discussed in Brereton and Vilgis (1995). The results can be considerably simplified if all linking numbers m in Eq. (8.1) are being put equal to zero which corresponds to the description of the ab melt of unlinked rings. In this case, after some algebra, one arrives at the result Z(Mc N, M0N)"expMiplk(a, a)Nexp[!(1/l )E[Mc N]] (8.13) a %&& a with the self-linking number lk(a,a) being defined by Eq. (4.19) while the knot energy E(Mc N) is a being given by

QQ

dl · dl@ E[Mc N]"lim a Dr(l)!r(l@)D1è e?0 ca ca with l "l(6/ol3p2). In the original paper of Brereton and Vilgis (1995) a different terminology for %&& E[Mc N] is being used (they call it the “self-inductance”). This choice of terminology is due to a chronological reasons: the paper by Brereton and Vilgis had appeared in 1995 while Kholodenko


333

and Rolfsen’s had been published in 1996. Both terms in the exponents of Eq. (8.13) were already discussed earlier in this work. The first term is associated with the choice of framing, see, e.g. Section 4.2, and causes the polymer chain to be more stiff, Section 6.1. The presence of the second term is essential if the polymer chain is effectively knotted as discussed in Sections 3, 5 and 7. Implicitly, it can also be associated with the probability for a given chain not to be entangled with other chains as discussed in Section 7.6. When the r.h.s. of Eq. (8.13) is substituted into the path integral, e.g. Eq. (7.2), for the ring C , this leads to the delicate competition between the stiffening a and softening. In Kholodenko (1991) only the stiffening effect was taken into account which amounts to the assumption (also implicitly present in de Gennes (1971) and Doi and Edwards (1978) treatment of reptation) that the chain trapped into the tube is knotless. If the rigidity wins, then one can use the scaling analysis of Section 2.2 in order to arrive at famous result: q JN3.4 for 5 the viscosity. Since, however, according to Eqs. (3.5) and (5.95), for NPR the fraction of the unknotted rings is completely negligible, the presence of the second term in the exponent of Eq. (8.13) is quite natural and effectively counterbalances the stiffening leading to the noticeable contraction of the ring in the polymer melt (Mu¨ller et al. 1996), in qualitative accord with calculations of Brereton and Vilgis (1995). Since the topological effects alone are unable to make the trapped polymer backbone more stiff, the geometrical factors discussed in Section 6.3 should be taken into account. They are responsible for making the longitudinal part of the trapped polymer motion more stiff so that the scaling analysis of Section 2.2 could be used. The transversal part of this motion requires additional discussion since it is responsible for the transition from the Rouse to the reptation regime of the dynamics of polymer melts (Kholodenko, 1996b,c; Kholodenko and Vilgis, 1994). 8.2. Statistical mechanics of planar rings in an array of obstacles (the replica approach) The transversal motion of the trapped polymer is usually described by the oscillator-like Schro¨dinger equation, see, e.g. Doi and Edwards (1978) and Eq. (6.55). We have demonstrated in Kholodenko and Vilgis (1994), that this oscillator-like Schro¨dinger problem can be reinterpreted in terms of magnetic language. In this language we are dealing with the quantum Landau-diamagnetism-like problem about the planar “motion” of charged particles placed in the constant magnetic field. Such reinterpretation allows us to look at the whole problem of chain confinement from a much wider perspective. In Kholodenko (1996a,b) it was shown that the Landau diamagnetism problem is also isomorphic to the problem about the planar random walk which encloses the fixed prescribed area A, see, e.g. Section 6.4. Now, we want to introduce some complications into this problem. Specifically, let us assume that our closed planar walk takes place at the punctured plane where the punctures are meant to represent the cross sections of other chains, or tubes. In the case of chains the punctures have infinitely small radius while in the case of tubes they have a finite radius. Topologically, however, this fact makes no difference, see, e.g. Kholodenko (1996b,c) and Appendix A.1. Whence, we may want to calculate the probability of enclosing a given area A by the planar random walk of N steps in the presence of impurities with some prescribed surface density oL "n /A where n is the total number of cross sections (punctures). We would like to 5 5 impose an additional constraint that no impurities are allowed to be inside the contour which encloses the area A. The presence of randomly distributed impurities introduces some sort of quenched (or annealed) disorder into the problem which is normally being treated with the use of replicas. Use of

334


replicas can be bypassed based on topological considerations, see, e.g. Kholodenko (1996b,c) and below, but it is of interest to compare the observables which can be calculated in both ways. To this purpose, we would like to consider a closely related problem about the properties of the closed planar walk of N steps which is entangled with a random array of cross sections, e.g. twodimensional analogue of Eq. (4.10) where the constant c is now a random variable with prescribed probability distribution. This problem was considered by Tanaka (1984) and, more recently, by Otto and Vilgis (1996). Related results were earlier obtained by Nechaev and Rostiashvili (1993) and Rostiachvili et al. (1993) based on the fundamental earlier work by Brereton and Shah (1980). To develop our results, let us recall a useful identity (Fulton, 1995), dz dx#i dy x dx#y dy !y dx#x dy d ln z" " " #i z x#iy x2#y2 x2#y2 ,d ln r#i dw(h) ,

(8.14)

where, according to Eq. (5.2), :N dw(h)"w and we used polar coordinates: x"r cos h, y"r sin h in 0 the last of our equations. Evidently, we can consider as well a combination +n5 d ln (z!a ) which would place singulari/1 i ities (punctures) of the complex z-plane at points a . Obviously, the total winding number w5 can be i written now as (Fulton, 1995),

C P

D

N n5 w5" + Im dq d ln(z!a ) i 0 i/1 N , dq rR (q) ) A[r(q)] , (8.15) 0 where the vector potential A[r(q)] is given by A"(A ,A ) with x y n5 n5 A "! + (y!a i)r~1, A " + (x!a i)r~1 and r2"(x!a i)2#(y!a i)2 . x x i y y i i x y i/1 i/1 For a single closed polymer chain of length N which is entangled with punctures the partition function Z(c) (with account of the excluded volume effects) can be written as

P

P

AP B A P

Z(c)" D[r(q)]d

N

0

dq rR d c!

B

N dq rR ) A[r(q)] expM!S[r(q)]N , 0

(8.16)

where

P

P P

1 N a2 N N S[r(q)]" dq rR 2# dq dq@d(r(q)!r(q@)) l2 2 0 0 0 with a2 being the two-dimensional excluded volume parameter. Since both the locations a of punctures as well as the total winding number w5,c are i fluctuating variables it is necessary to perform some sort of averaging of Z(c) in order to calculate the obsevables (e.g. SR2T, etc.). It is assumed, that the disorder associated with the location of punctures could be considered as annealed while the disorder associated with w5 as quenched. To


335

perform the average for the case of quenched disorder normally requires the use of replicas. This can be accomplished in several steps. First, we can rewrite the annealed average of Z(c) as

P

P

AP B

SZ(g)T " D[A]d(+ ) A) D[r(q)]d A

G

P

P

N dq rR 0

H

N 1 ]exp ! d2x(+]A)2!ig dq rR ) A!S[r(q)] , (8.17) 2u 0 0 where the parameter u is related to the distribution of obstacles which is assumed to be 0 Gaussian-like. The function Z(g) is related to Z(c) via Fourier transform:

P

= dg expMigcNSZ(g)T . A 2p ~= Second, upon introduction of the “current” J via Z(c)"

(8.18)

P

N dq rR (q)d(r!r(q)) 0 use of the Hubbard—Stratonovich transformation in Eq. (8.17) allows us to eliminate the A-field. This produces the following result for SZ(g)T : A N u g2 SZ(g)T " D[r(q)]d dq rR exp !S[r(q)]! 0 A[r(q)] . (8.19) A 2 0 Here, following Cardy (1994), we have introduced an area J(r)"

P

AP B G

H

P P

A[r(q)]" d2r d2r@SA (r)A (r@)TJ (r)J (r@) k k k k

(8.20)

which has the same meaning as the expression introduced earlier, see, e.g. Eq. (6.58). Upon the substitution of an identity 1":dA d(A!A[r(q)]) inside the path integral, Eq. (8.19) it is possible to rearrange terms so that the result for SZ(g)T now looks like this A u g2 1 SZ(g)T " D[A]d(+ ) A) dA daJ exp ! d2x(+]A)2! 0 !iaJ A A 2 2

P

P P

]Z(e, A) ,

G P

A

BH

(8.21)

where Z(e, A) is defined by

P

G

P

H

Z(e, A)" D[r(q)] exp !S[r(q)]!ie d2r A[r(q)] ) J(r(q)) .

(8.22)

with e"J2iaJ . Following Nechaev and Rostiashvili (1993), the last expression can be rewritten with the help of replicas in terms of the n-component complex scalar field theory path integral:

P P

Z(e, A)"lim Du Du* expM!S[u,u*]N , n?0

(8.23)

336


where u"Mu ,2,u N and 1 n n l2 ¸a2 n S[u,u*]" + u* m2! (+!ieA)2 u # + Du D2Du D2 . i i i j 4 4 i/1 i.j/i In this expression the mass variable m2 is &N~1 while ¸ is the average size of the polymer in the direction perpendicular to the plane. Substitution of Eq. (8.23) into Eq. (8.21) and integration over the field A produces in the replica symmetric approximation, i.e. +nDu D2"nDuD2, the following final i i result:

A

B

G P

H

, SZ(e, A)T "exp !n d2r ¸ %&& A

(8.24)

where

A

A B

B

l2 l2 ¸a2 DuD2 ¸ "iaJ ! DuD2 ln # DuD2 #(m2!¸a2M2)DuD2# DuD4 %&& 4p 2p 4 M2 with M2 being an arbitrary mass which appears as a result of regularization of the one-loop corrections coming from A-integration. By introduction of the “free energy” f (aJ ) via

CP

1 f (aJ )" » n

d2r ¸

%&&

D

(8.25)

"¸

%&&

if it is possible to rewrite Eq. (8.21) as

P A

A B

B

l2 l2 DuD2 SZ(g)T " dA d A# »DuD2ln ! »DuD2 e~Vf(A,g) A 4p 2p M2

(8.26)

where »":d2r and f (A,g) is defined by f (A,g)"(u g2/2»)/A#(m2!¸a2M2)DuD2#(¸a2/4)DuD4 . (8.27) 0 Use of this result in Eq. (8.18) with account that c is Gaussianly distributed random variable allows to calculate the average Z(c). Actual calculations of this quantity can be only performed with help of the saddle point approximation which produces the following consistency conditions: o "(¸/a)2(l2/4p) (u /D )(1!c2/D ) , # 0 # 0 # 1/N "(¸a2/2)o ln[¸3o ] , ¸3o '1 , # # # #

(8.28) (8.29)

and (8.30) A "(»/2p)o (1!1ln[¸3o ]) # # # 2 with o"DuD2 and parameters D and c characterizing the average total winding number (c is the # 0 0 mean winding number) and D is the dispersion of the winding number while u is the mean density # 0 of obstacles. For the fixed value of parameters u ,c and * the above results determine the critical 0 0 # length N so that below the critical length the polymer acts as if it is still fully flexible (Gaussian# like) while above N it collapses to the conformational state of branched polymer. Indeed, #


337

according to Eq. (8.26), we have A"»(l2/2p)(o!(1/2)oln(o¸3)) . If now we take into account that o"N/», this result can be written as (for N'N ): # A"(l2/2p)N(1!1 ln N#2) 2 K(l2/2p) N1~1@2 .

(8.31)

(8.32)

Since A&SR2T we obtain immediately JSR2TJlN1@4 which is the scaling law for the branched polymer without excluded volume. The main conclusions of the calculations just presented could be summarized as follows: 1. The problem about conformational properties of a planar ring trapped (entangled) in an array of obstacles was actually reduced to the problem about the calculation of the effective area which such a ring encloses. 2. It was shown that the problem is well defined only above a certain threshold (in parameter space). 3. Below this threshold the ring polymer collapses and acquires the shape of the branched polymer (the last result is being independently used by Obukhov et al. (1994) to describe the dynamics of rings in gels as discussed in Section 2). Below we shall reproduce these results using completely different (topological) methods which do not rely on use of replicas. By doing this some new aspects of the “trapping problem” will be revealed. 8.3. Statistical mechanics of planar rings in an array of obstacles (the Riemann surface approach) In Section 6.4 we had discussed configurational statistics of the planar random walks restricted by the area constraint. Surprisingly, the problem about the random entanglements considered in Section 8.2 happens to be very closely related to this area problem. In this section we will try to clarify why, indeed, such connections exist. As it was already noticed in Section 6.4, the planar area constraint problem is essentially equivalent to the standard Landau diamagnetism problem (Landau, 1930), in case the plane is not punctured. In such a (standard) case, the problem lies in quantum (and statistical mechanics) treatment of motion of the electron in the presence of constant magnetic field H defined by the vector potential A, see, e.g. Eq. (6.66). In Kholodenko (1996a) full analysis of this problem is given for both nonrelativistic and relativistic electrons (since this problem happens to be isomorphic to the problem of statistical description of deformable planar droplets of arbitrary rigidity). We shall discuss only the nonrelativistic limit in this review. The relativistic effects are briefly discussed in Kholodenko (1996a). In the nonrelativistic limit the solution is reduced to that known for the quantum harmonic oscillator with frequency depending upon the strength of the magnetic field H. Whence, for arbitrary small H we still have an infinite tower of equidistant discrete energy levels. The situation changes dramatically if the motion of an electron is considered on the punctured plane. In this case we may have, depending upon the surface density oL of punctures, a finite number of bound states or even no bound states at all (Kholodenko, 1996b,c). Whence, we may anticipate, that there is some threshold oL so that above (below) oL there will (will not) be bound states. The above picture # #

338


Fig. 12. (a) Fusion of two punctured spheres produces a sphere again. (b) Fusion of two punctured tori produces a new surface of genus 2.

can be now recast into polymer language (Kholodenko and Vilgis, 1994; Kholodenko, 1996c). The transversal part of the diffusive motion, Eq. (6.55), is isomorphic to the Landau diamagnetism problem (Kholodenko and Vilgis, 1994). In the presence of planar punctures Eq. (6.55) should be modified. Upon such modification the tube existence and stability will be determined by the number of available bound states. The transition from zero to finite number of bound states is discontinuous. We formulate our results in such a way that the numerical predictions of our theory related to the onset of tube creation (destruction) associated with transition from the Rouse (no tubes) to the reptation (tube assisted) regime could be directly compared with experimental data (Kholodenko, 1996c), and demonstrate very satisfactory agreement with the experiment. Quantitative results obtained below are in qualitative accord with the results of Otto and Vilgis (1996) discussed in Section 8.2. Let us begin with the following auxiliary example. Following Arnold (1978) (see, e.g. Appendix A.1), let us consider the classical motion of a particle in a square with periodic boundary conditions (i.e. on the torus). We shall complicate matters by putting inside a square another circle (hole) so that our particle can elastically scatter out of this hole and the walls of the square. The classical motion in such billard takes place actually on a Riemann surface which is known as a double torus (i.e. sphere with two handles). The double torus is obtained by gluing two copies of the usual torus with a hole in it as depicted in Fig. 12. The gluing is done around the circumference of a hole. It is well known that the Riemann surfaces represent the case of surfaces of constant negative curvature. The classical motion on such surfaces is chaotic (Arnold, 1978). To bring this auxiliary problem closer to our original tube problem, let us consider, instead of just one hole, many (with some surface density oL introduced in Section 8.2). Then, it is intuitively clear that we will end up with the Riemann surface of genus g (sphere with g handles) where the genus g is determined by the density of obstacles (holes). All this can be made quite rigorous by considering homotopy of the paths on the punctured plane with periodic boundary conditions and by using the van Kampen theorem as explained, e.g., in Massey (1967), Gilbert and Porter (1994) or Fulton (1995). We deliberately would like to avoid all these mathematical complications unfamiliar to most of the readers trained in


339

polymer physics. Instead, we would like to use more intuitive examples (including that of Arnold) which have some physical appeal. But since the van Kampen theorem tells us that the punctured plane is effectively the Riemann surface (irrespective of the classical mechanics example discussed above), one can exploit this fact “quantum mechanically”. Let us recall that the conformational properties of flexible chains in the external random potential » are described with the help of the end-to-end distribution function G(r, r@; N) which obeys the “equation of motion” (in three dimensions)

A

B

l ! +r2#»(r) G(r, r@; N)"d(r!r@)d(N) . N 6

(8.33)

Upon the decomposition of this equation into longitudinal and transversal parts (as discussed in Section 6.3) we are left with effectively two independent Schro¨dinger-like equations. The transversal (planar) problem could be treated, in principle, with the help of the methods described in Section 8.2. Following the seminal work of de Gennes (1971) on reptation (see, e.g. his Eqs. (2.4) and (2.5)), the random environment can be modeled, however, with the help of a Smoluchovskitype equation for G given by G"D G!c G . r2 r N

(8.34)

The actual values of constants D and c depend on the microscopic model used to arrive at Eq. (8.34), For example, in Nechaev et al. (1987) the “motion”on the regular lattice is considered (see also Nechaev (1990)) while in Nechaev (1988) “motion” on the Bethe lattice is being considered. Eq. (8.34) appears to be universal (Helfand and Pearson, 1983; Rubinstein and Helfand, 1985; Mehta et al. 1991) and independent of the dimensionality of the embedding space. In case the “motion” takes place on the regular lattice D"2pq, c"q!p, p"z~1, q"1!p and z is the coordination number of the lattice. The above equation should be actually supplemented with initial and boundary conditions, e.g. G(r, N"0)"d(r) ,

D(G/r)D !cG(0, N)"0 . (8.35) r/0 As it was argued in Kholodenko (1996b), to obtain the general solution of Eq. (8.35) it is permissible initially to ignore the boundary conditions: once the general solution is obtained, it will be forced to satisfy the specific boundary conditions. So far, we have not made any connection(s) between Eq. (8.34) and the topological properties of the underlying two-dimensional punctured plane. To do so, we would like to pose the question: is it possible to rewrite Eq. (8.34) in the form of diffusion-type equation on some curved manifold? The answer to this question is “yes”. To prove this, let us first bring Eq. (8.34) to the dimensionless form. If one chooses a"D/c2 and b"D/c, then one obtains the dimensionless analogue of Eq. (8.34) given by 2 G(x,q)" G! G . q x2 x

(8.36)

Let us demonstrate that this equation can be rewritten in an equivalent form as (/q)G"(1/Jg) (gabJg )G a b

(8.37)

340


for some metric tensor g . If we choose: g "1, g "g "0, g "e~2x, then we can obtain: ab xx rx xr rr g"e~2x, grr"1, gxx"e2x so that we get

A

B

A

B

1 (g Jg 2)"ex ex 2 #ex ex 2 . b g a ab x x u u

(8.38)

If G(r,q) is u-independent, then use of Eq. (8.38) in Eq. (8.37) produces back Eq. (8.36) as required. Once we have obtained the metric tensor g of surface, we can find out what kind of surface it ab determines. The first fundamental form of the surface (the length) can be written now, based on the above results, as ds2"dx2#e~2xdu2 .

(8.39)

By introducing a new variable y"ex we obtain dy"ex dx and, therefore, Eq. (8.39) can be rewritten as ds2"dx2#e~2x du2"(dy2#du2)/y2 .

(8.40)

In mathematical literature the metric given by the last expression of Eq. (8.40) is known as the hyperbolic metric (Arnold, 1978; Stillwell, 1992). The Poincare´ model H consists of a subset of the complex plane C defined by H"Mz"u#iy3CDy'0N

(8.41)

supplemented with hyperbolic metrics given by Eq. (8.40) (Poincare, 1882; Buser, 1992). If we would use complex variables z and zN , then Eq. (8.40) could be rewritten as ds2"(dz)2/(Im z)2 .

(8.42)

For finite distances d between z and z@ in this model we could obtain with the help of Eq. (8.42) the following result: Dz!z@D2 cosh d(z, z@)"1# 2 Im z Im z@

(8.43)

to be compared with the usual Euclidean distance d (z, z@)"Dz!z@D . (8.44) E With the help of d just defined, the solution of Eq. (8.36) (without boundary effects) is known to be (Buser, 1992; Kholodenko, 1996b)

P

1 = dx x e~x2@4q G (z, z@; q)" e~q@4 . (8.45) H 2(2pq)1@2 d(z,z{)Jcosh x!cosh d(z, z@) Earlier, when we have discussed Arnold’s billiard, the claim was made that the actual motion takes place on the Riemann surface (i.e. sphere with g handles) instead of H-plane (also known as the Lobachevski plane). There is no contradiction, however, between the earlier claim and the results just obtained since the Lobachevski plane is the universal covering surface for the Riemann


341

Fig. 13. The planar representation of a torus is just a square with opposite sides properly identified. In this representation the homotopy of paths around a puncture inside a square is equivalent to the homotopy of paths around a puncture near one of the corners.

surface of any genus g'1, Stillwell (1992). The notion of the covering surface could be easily understood using the following example. Let C be a discrete translation in the complex plane C (or R2), then a given square S can be obtained as an image of some fundamental square SK upon q q translation, i.e. S "CSK . The torus can be obtained as a quotient R2/C so that R2 is the universal q q covering surface for the torus. Analogously, every Riemann surface can be constructed from some fundamental 4g-gon on the H-plane (g'1), see, e.g. Figs. 13 and 14 for g"2 surfaces. Whence, any Riemann surface is just a quotient H/C where C is some generator of discrete translations in H-plane (Buser, 1992). The genus g of the surface is directly connected with the number of punctures in the plane and this fact is completely independent of whether these punctures are frozen or not. Moreover, the hyperbolicity will remain even if we remove the restriction that the motion should be strictly planar: because tubes are entangled in three dimensions (Kholodenko and Vilgis, 1994), and form quasi-knotted configurations, Brownian motion in the presence of such quasiknots will remain hyperbolic (Thurston, 1979). To understand intuitively how this happens we refer the reader to the Appendix. For the moment, let us consider again the simplest Arnold’s billiard which is just a union of two punctured toruses glued along the circumference of the punctures, see, e.g. Fig. 12. To construct such a billiard we need two copies of the Riemann sphere each having three holes. We can glue together two holes on each sphere thus converting it into punctured torus and, then, we can glue the resulting objects together to make the final product. It can be shown (Buser, 1992) that every Riemann surface of genus g'1 is just a collection of thrice punctured spheres along with the gluing prescription, which is used for their assembly. Once we recognized this fact, we can construct a finite square lattice made of m2 copies of the Arnold square. By gluing these squares together it is possible to insert yet another set of k holes into this lattice (Buser, 1992), thus forming a surface of genus g"1#1(m2#k) , 2

(8.46)

342


Fig. 14. Two punctured tori in planar representation: (a) can be transformed into (b) and, then, glued together along c , 1 and c thus resulting in the octagon (c). 2

k"M0,2,2,2mN if m is even or k"M1,3,5,2,2m!1N if m is odd. For m"1 using Eq. (8.46) we obtain g"2 in agreement with (Arnold 1978). Let, as in Section 8.2, oL "n /A, then, the logic of the previous discussion suggests us to choose 5 n "m2#k (8.47) 5 so that the filling fraction l can be defined now as (Kholodenko and Vilgis, 1994), l"pa2oL , i.e. we can think of cross sections of tubes as a planar gas of disk of area pa2. Suppose that there is some sort of interaction between such disks. Then, by analogy with other models of statistical mechanics, it is natural to expect that the system of such disks can undergo a phase transition (e.g. solid—liquid-like) which is controlled by l so that for some critical value l"l* we would have l*"pa2(o*)oL (o*) ,

(8.48)

where o* is the critical monomer density (recall, that o&N/»). The explicit dependence of a and oL on o is unknown in general (but, since oL "n /A, it is expected that oL &o) and should be 5 dependent upon the details of the model which is used.


343

Since the punctures in our plane have finite sizes, and since Eq. (8.37) accounts for the curvature effects only, we need to complicate this equation so that it can account for the finite sizes of cross sections. To this purpose we begin with planar case discussed in Sections 6.4 and 8.2. In particular, using Eq. (6.55) we can formally define the averaged size of our tube cross section as Sq2#q2T"l/w,a2 . (8.49) 2 3 This definition is in accord with that given in Doi and Edwards (1978) (with S2T being the usual polymer average, see, e.g. Eq. (8.89) below). By continuously changing w we obtain continuous changes in a2. This will no longer be true if the above average is considered in the multiply connected plane which is effectively the Riemann surface. To demonstrate this, we would like to reobtain Eq. (8.49) in a more systematic way which was outlined in Section 6.4. Using Eq. (6.71) we obtain in the limit of small Dp the following result for SAT: SATK 1 Dp(Nl)2 . (8.50) 12 For a circle of circumference N the area is N2/4p. It is the maximal area which can be enclosed by the walk of length N. Whence, one can rewrite Eq. (8.50) in equivalent form as SAT Ka2 , Dpl2

(8.51)

where on the r.h.s. the area a2 is identified with that given in Eq. (8.50). Since the smallest area SAT of the circle cannot be smaller than Kl2, then, evidently, in this extreme case we would have 1/DpKa2 .

(8.52)

Since [Dp]"[A~1] while [w]"[l~1] we, indeed, have reobtained Eq. (8.49). Use of the area (or magnetic language) formalism to determine the tube cross section is more advantageous, as compared with Eq. (8.49), since it allows to consider problems related to random walks with the area constraint on the Riemann surfaces. To this purpose, the following key observation is helpful (Kholodenko, 1996a,b). The probability P(A, N) for a random walk to enclose an area A is given by the ratio P(A, N)"Z(A, N)/Z(0, N) ,

(8.53)

where Z(A, N) is given by

P

Z(A, N)" dr G(r "r "r, NDA) 1 2

(8.54)

with G(r "r "r, NDA) being given by the r.h.s. of Eq. (6.57) (divided by pNl) and Z(0, N) is the 1 2 same but with A"0. The r.h.s. of Eq. (8.54) is just the usual statistical mechanical partition function (Feynman, 1972). Whence, by analogy with Section 7.2, to obtain Z(A, N) we need to know only the eigenvalues (and their degeneracies) of the corresponding Schro¨dinger-like operators. The spectrum of such operators on the Riemann surfaces can be also obtained where the partition function Z(A, N) is known in mathematical literature as Selberg’s trace formula (Buser, 1992).

344


Let us consider this topic in some detail. Since in the flat case the spectrum of Landau “electron” e is known to be just that for the harmonic oscillator (Kholodenko, 1996b), n e "Dpl(n#1) (8.55) n 2 with degeneracy g "DpNl/4, the partition function is easily obtainable n x = , (8.56) Z(Dp, N)" + g e~Nen" n sin x n/0 where x"DpNl/2. The function Z(Dp, N) is related to Z(A, N) via

P

Z(Dp, N)"

N2@4p

dA eADpZ(A, N)

(8.57)

0 and is even more convenient since

(8.58) SAT lim (/Dp)ln Z(Dp,N) . D p?0 Generalization of the result of Eq. (8.56) to the Riemann surface of genus g can be now accomplished without any problems with the result g!1 Z(Dp, N)" + (2DpNlR2!2n!1) expM!e NN n 4R2 1 0yny@b@~2

(8.59)

with

C

A

BD

1 1 1 2 !b2! n# !b e" n 4R2 4 2

(8.60)

and b"DpNlR2, R2"s2/l2 with s being an average distance between the obstacles in the plane. As it was noticed in Kholodenko (1996b) the parameter R plays the role of a radius of curvature of the manifold: for R2PR (flat case) one obtains: DpNl Z(Dp, N)+ sin (DplN/2)

(8.61)

which effectively coincides with Eq. (8.56). But for finite R@s one has to require that the partition function Z(Dp, N) remains nonnegative and well defined. The nonnegativity of Z(Dp, N) requires 2DpNlR2!2n!150

(8.62)

while for the sum in Eq. (8.59) to be well defined we have to require as well (8.63) DbD!1!n50 . 2 Taking into account the definition of b given after Eq. (8.60) we conclude that both inequalities Eqs. (8.62) and (8.63) are equivalent. In particular, the inequality (8.63) implies that the reduced “magnetic field” DbD should exceed a certain threshold, in our case, DbD51 2

(8.64)


345

in order for the tube to exist. Moreover, the theory which produced the result given by Eq. (8.59) dictates, yet another set of constraints:

A

a JB2!4DcD "arctanh lR B

B

,

(8.65)

b!2R2Jc"n#1 , (8.66) 2 where B"b/R2. The constant DcD can be eliminated from these equations thus producing tanh

AB S

S

a n#1/2 2b!n!1/2 " . lR b b

(8.67)

If DcD"0 in Eq. (8.65) we obtain tanh (a/(lR))"1

(8.68)

which leads to the requirement aPR for fixed R. In this case the tube does not exist. Hence, for the tube to exist one must require DcDO0 and (a/lR)41. A crude estimate for b can now be obtained from the following self-consistent equation for b which follows from Eq. (8.67) (for n"0) (8.69) b tanh 1KJb!1 , 4 e.g. it is assumed that a+R (or the size of the tube is of the order of the distance between the obstacles). Numerical solution of Eq. (8.69) produces bK0.86(1$0.643). From the theory of the operators on Riemann surfaces (Kholodenko, 1996b), it is known that f bK , 2(g!1)

(8.70)

where f"0,$1,$2,2 and g!1 is given by Eqs. (8.46) and (8.47). By combining these equations we obtain f bK "0.86(1$0.643) . n 5 If A"fpa2, then oL "n /A can be rewritten as 5 poL a2Kn /f . 5 By combining this result with Eqs. (8.49) and (8.71) produces l*K0.708 .

(8.71)

(8.72)

(8.73)

This result is too high as compared to the estimate l*K0.0286 which was obtained in Kholodenko and Vilgis (1994) with help of other methods to be discussed below. To improve the above estimate we can, e.g., require, by analogy with the theory of coil—globule transitions (Kholodenko and Freed, 1984a), that in addition to n"0 (i.e. to the ground state) there is at least one more discrete state, e.g. n"1. Using Eq. (8.67) and repeating previous calculations, produces l*K0.341 .

(8.74)

346


This result is considerably in better agreement with the earlier obtained as we shall demonstrate now by direct comparison of this result with the experimental data. To do so, we need to recall some basic facts from chemistry. Let n be the number of moles, then the total number of “particles” (polymers) NI is given by NI "nN where N is Avogadro’s number. A A Let ¼ be the total weight (mass) of polymer(s) of molecular weight M, then n"¼/M while the density o is defined by o"¼/» where », as before, is the total volume. Let » "AK SR2T3@2 be the 0 ' volume occupied by a single polymer chain. Here AK is some unknown constant, of order unity according to Fetters et al. (1994), and SR2T is related to M via ' SR2T"cL M , (8.75) ' where SR2T"1SR2T and cL is some proportionality factor. In writing Eq. (8.75) it is being assumed ' 6 that the individual chains in the melt are effectively at h point conditions (Fetters et al., 1994) Consider now a combination » oN /M,NI /(»/» )"cJ . By construction, this combination is just 0 A 0 a fraction cJ of “lattice sites” (of total number »/» ) which are occupied by polymers. Evidently, 0 04cJ 41. If, following Fetters et al. (1994), we assume cJ +1 (i.e. polymer melt), then we obtain 1KAK 6~3@2M1@2cL 3@2N o . (8.76) A If M is the molecular weight of the segment of polymer chain between the entanglements, then % Eq. (8.76) produces M "o~263(AK cL 3@2)~2N~2 . % A Using Eq. (8.75) we can eliminate the constant cL from Eq. (8.77) thus producing

AT UB

M Ko~2AK ~2 %

R2 M

~3 63N~2. A

(8.77)

(8.78)

In view of Eq. (8.75), it is reasonable to assume that a2KcL M , which then produces % SR2T a2 " (8.79) M M % in agreement with Fetters et al. (1994). Alternative expression was found by He and Porter (1992), who obtain instead M SR2T"28pa2M. The numerical factor of 28p cannot be further checked % using the data from Fetters et al. (1994) and, whence, we shall use the result of Eq. (8.79) in order to produce the final numbers. By combining Eqs. (8.78) and (8.79) we can estimate a as

S

A B

62@3 SR2T M MK . (8.80) % AK oN SR2T M A The last result is in complete accord with Eq. (3.3) of Fetters et al. (1994). According to this reference, Eqs. (8.78) and (8.80) could be used for the independent measurements of M and % a provided that o and SR2T/M are known. Let us now have another look at these results in the light of the discussion presented earlier in this section. Using Eqs. (8.79) and (8.80) we obtain aK

A B

62@3 M % aK AK oN a2 A

(8.81)


347

Table 1 Molecular characteristics of polymers at ¹"413 K Polymer

o (g cm~3)

PE PEO PEB-11.7

0.784 1.064 0.793

PMMA PEE

1.13 0.807

a"d5/2 (A_ )

AK

828 1624 1815

16.4 18.75 21.3

10013 11084

33.5 39.3

5.865 5.672 5.802 1 5.77 5.543

M%

Table 2 Molecular characteristics of polymers at ¹"298 K Polymer

o (g cm3)

M%

a"d5/2 (A_ )

AK

PEB-14 HHPP PEE PMA 65-MYRC

0.860 0.878 0.866 1.11 0.891

1522 3347 9536 8801 24874

18.45 24.2 35.0 30.35 44.05

6.903 6.59 6.29 6.94 8.00

or, equivalently, N a3o 1 A " . (8.82) AK 63@2M % This result can be compared now against Eq. (8.48) which can now be equivalently rewritten as pa3(oL /a)"l .

(8.83)

Taking into account the definitions of oL and o we can now identify oN /63@2M with oL /a and, A % hence, l/p with AK ~1. The theoretically obtained l* given by Eq. (8.74) can be used now to obtain AK K9.21. This result can be compared against the experimental data of Fetters et al. (1994). Based on the data from Tables 1 and 2 of Fetters et al. the Tables 1 and 2 of Kholodenko (1996c) are reproduced here (in units and notations used by Fetters et al., 1994). In calculating AK with help of Eq. (8.82) the conversion factor coming from the combination N o/63@2 is estimated to be 0.04082, based on N "6]1023, 1 A_ "10~8 cm. Also, the tube A A diameter d "2a since a is the tube radius. The results of Tables 1 and 2 are in good agreement with 5 our theoretical estimate AK "9.21 based on Eq. (8.74). In addition, by combining Eqs. (8.75) and (8.80) we can also write ao"const.

(8.84)

This result is obtained theoretically in Kholodenko and Vilgis (1994) using the analogy with quantum Hall effect (QHE) formalism. Independently, the same result was obtained by Kavasalis and Noolandi (1988) based on the packing model of reptation. The result of Eq. (8.84) is supported

348


by numerical simulations of Wittmer and Binder (1992) and Kremer and Grest (1994). Eq. (8.84) is based on the assumption that the combination SR2T/M is constant. Theoretical calculations performed by Lachowski et al. (1988) and Hutnik et al. (1991) indicate that this should be the case and provide for the above ratio the result: SR2T/M+1.03—1.1. This result is only in qualitative agreement with the data from Tables 1 and 2 of Fetters et al. (1994) as is acknowledged by the authors. Accordingly, use of the data given in Tables 1 and 2 of this work for computation of the product oa leads to less satisfactory results for AK : the results for AK are uniformly smaller (roughly by a factor of 6) than those given in Tables 1 and 2. This is not too surprising since the result given by Eq. (8.82) was obtained without restrictions on the ratio to be fixed and universal. Since the independent numerical data of Wittmer and Binder (1992) and Kremer and Grest (1994) and the theory of Kholodenko and Vilgis (1994) strongly support Eq. (8.84), we would like to present these theoretical arguments in favor of Eq. (8.84) in Section 8.4. 8.4. Statistical mechanics of planar rings in an array of obstacles (QHE approach) In Appendix A.1 we have discussed complications which arise from considering the Brownian motion at the twice punctured plane as compared to the well understood one puncture case discussed in Section 5.1. Here, we would like to develop the results of Appendix A.1 in order to illuminate some additional physical aspects of the whole problem. In physics literature, study of path integrals in multiply connected spaces was initiated to our knowledge by Shulman (1971) and later developed by many authors, see, e.g. Levay et al. (1996) and references therein. In the mathematics literature, the same problem was studied by Pitman and Yor (1986, 1989) who use methods which are noticeably different from that used in physics literature. It would be interesting to make a detailed comparison of these approaches in the future. The most typical (hydrogen atom-like) problem which is well studied is the problem related to the quantum mechanics of the particle on a circle which we had discussed in Section 6.2. The key idea of solving the circular problem lies in realizing the fact that the universal covering space for a circle S1 is just a straight line R1. Since the path integral for R1 is well known, then the path integral for S1 can be obtained by some sequence of operations leading from S1 to R1 and back to S1 (Tanimura and Tsutsui, 1995) (very much in accord with the results of Appendix A.1). Now, if we have a hole in the plane, the closed paths around a hole are homotopic to S1 (Fulton, 1995). If we would have some interaction between the Brownian particle and the hole (which could be just a world line of another particle), then this would be equivalent to having fractional statistics (with the strength of interaction d interpolating between the Bose and the Fermi statistics as it was explained in Section 6.2). Let us now have two holes instead, then we have to consider instead of S1 a product S1]S1 as depicted in Fig. 15. The universal covering space for the “figure eight” is known to be (see, e.g. Dubrovin et al., 1985), a four-valent Bethe lattice as depicted in Fig. 16. This explains why, e.g. Nechaev et al. (1987) and others had used a Bethe lattice to study the reptation. The “figure eight” can be also obtained by considering paths on the once punctured torus (which was discussed earlier in connection with Arnold’s billiard) as depicted in Fig. 17. If we make a puncture in a sphere S2 and glue two copies of S2 together, the result will be S2 as depicted in Fig. 12a, but if we glue two punctured tori together, as depicted in Fig. 12b, we shall obtain a surface of genus 2. At the same time, if we think about the torus as a square with sides properly identified, then the punctured torus will look like that in


349

Fig. 15. Homotopy of paths on the twice punctured plane (“figure eight”).

Fig. 16. Universal covering space of the “figure eight”.

Fig. 13. If we glue together these polygons as depicted in Fig. 14 we shall obtain a double torus in the planar representation. There are four distinct paths on this torus as depicted in Fig. 18b so that if we make cuts along these paths we re-obtain Fig. 14c. At the same time, if we would think of homotopy of these paths, we would obtain a bouquet of four circles (instead of two as in Fig. 15). Two out of our four circles had originated from the

350


Fig. 17. Topological structure of the once punctured torus.

Fig. 18. If the sides of the octagon are glued together in an order shown in (a), then the resulting surface (b) coincides with that depicted in Fig. 12b. Alternatively, if the cuts are made along a , a and b , b on the double torus, we will 1 2 1 2 obtain again Fig. 14c.

periodic boundary conditions (see, e.g. Fig. 14a) and were left unaccounted in Fig. 15. Evidently, the Bethe lattice structure of Fig. 16 becomes more complicated when the surfaces of higher genus are being considered. But, in any case, the Bethe lattice calculations, e.g. like that discussed in Nechaev (1990), are effectively calculations on the universal covering surface for the Riemann surface of given genus (Stillwell, 1993), so that S1PR1PS1 calculational procedure for the circle is replaced now by the H/CP¹PH/C where ¹ is the corresponding Bethe lattice. The Bethe lattice calculations are not readily extendable to account for the “magnetic field” effects and, hence, the


351

Riemann surface approach described in the previous subsection is more advantageous. Moreover, the QHE picture of tube stability developed in Kholodenko and Vilgis (1994) is in complete accord with the Riemann surface approach as we are going to demonstrate shortly. The QHE model of tube stability is based on the following observations. According to Section 6.4 the Landau Hamiltonian HK for an “electron” of “mass” m placed in the “magnetic” field H"e]A is known to be (Sondheimer and Wilson, 1951), 1 1 A2 HK " + 2# A ) e # , r r 2m mi 2m

(8.85)

so that the Bloch equation for the density matrix o can be written as !(/b)o"HK o

(8.86)

provided that o(r, r@; bP0)"d(r!r@). Using results of Sections 6.3 and 6.4 the last equation can be equivalently rewritten as

A

A

B

B

1 H H2 ! + 2! x !y # (x2#y2) o"0 . r 2mi y Lx 8m b 2m

(8.87)

The corresponding polymer problem is obtained by the following replacements: b¢N,

1 l H2 w2 ¢ , ¢ , 2m 6 8m 6

and, if one considers only the states with the total angular monumentum zero, then the last equation coincides exactly with Eq. (6.55) while the partition function Z, given by

P

(8.88)

P

(8.89)

Z" dr o(r"r@"r; b)

coincides with that given by Eq. (8.54) (with obvious redefinitions of w or Dp). If W (r) is the n eigenfunction of the Schro¨dinger-like operator given by Eq. (8.87), then the size of the tube can be estimated according to Eq. (8.49) as a2" d2z [W (z, zN )]2DzD2 , 0

where use was made of the planarity of the magnetic Schro¨dinger problem which allows us to introduce complex variables z"x#iy and zN "x!iy so that, upon rewriting the whole problem in terms of z and zN , one obtains for the lowest Landau level wave function W (z, zN ) the following 0 result:

A

w W (z, zN )"NI exp ! DzD2 0 l

B

(8.90)

with NI being a normalization constant. The validity of the approximation for a2 rests on the assumption that for large N’s the density matrix o can be approximated by o(r, r@; N)Ke~e0N W* (r)W (r@) . 0 0

(8.91)

352


The presence of holes (other polymers) in the plane can be accounted for by introducing the mutual winding number h constraint into the corresponding path integrals. In the absence of the ij “magnetic” field the functional integral for an assembly of topological interacting planar Brownian walks is given by

P

G P C

DH

n5 N n5 1 U n5 d G" < D[r(q )]exp ! dq + rQ 2#i + h(r (q)) i l i 2p dq ij 0 i/1 i/1 i:j where

A B

y(q) , h(r (q))"tan~1 ij x(q)

,

(8.92)

(8.93)

n was defined e.g. in Eq. (8.47) and U is some constant which is related to the filling fraction l, 5 defined before Eq. (8.48) (for details, please, consult Kholodenko and Vilgis, 1994) as l"4p/U .

(8.94)

The result given by Eq. (8.92) should be compared now against earlier discussed Eq. (6.9). The presence of the “magnetic” field which describes the tube(s) cross section(s) can be accounted easily now by analogy with Eq. (6.67). At the same time, although at the classical level the interaction term in the exponent of Eq. (8.92) is the total “time” derivative and, hence, can be discarded, it cannot be ignored at the quantum level as we have explained in Section 6.1. For U"0 the total Hamiltonian HK is just a collection of single particle Hamiltonians, i.e.

A

B AG

HB

n5 1 1 HK " + HK r , ,H r , . i i r i i r i i i/1 Topological interactions change HK into

AG

(8.95)

HB

U r , ! + +rih(r !r ) (8.96) i r i j 4p i iEj so that the Schro¨dinger-like equation (with or without magnetic field) can be written now as HK "H 5

W(Mr N, t)"HK (Mr N, t)W . i 5 i t

(8.97)

Elementary examples of the above procedure were discussed in Section 6.1. Eq. (8.97) by design assumes that all “particles” are moving in the same “time” t (in case of polymers N). In the theory of Brownian motion there is no need, however, to make such an assumption (McKean, 1969). In case of polymers this was recognized by des Cloizeaux and Jannink (1990). The sychronized “time” is used in the theory of directed polymers (Kardar and Zhang, 1987), without explicitly acknowledging this fact (e.g. see also Blatter et al., 1994). Use of one “time” (instead of many) is equivalent of saying that all polymers are of the same length and are indistinguishable. des Cloizeaux and Jannink (1990) had carefully analyzed this issue for polymers and found that this assumption may sometimes lead to wrong results. The indistinguishability is also closely associated with statistics as we have demonstrated in Section 6.2. Extension of the “anionic philosophy” to the case of distinguishable particles was recently made by Liguori and Mintchev (1995) and Isakov et al.


353

(1995). The Riemann surface approach presented in Section 8.3 does not require these assumptions and, hence, is more consistent with the traditional language used in polymer physics, see, e.g. des Cloizeaux and Jannink (1990). Nevertheless, the time peculiarity just described is not a major stumbling block towards obtaining physically meaningful results in the present case. Indeed, if instead of the correct single valued function W in Eq. (8.97) we would use the multi-valued functions

G

H

U (8.98) WI (Mr N,t)"exp ! + h(r !r ) W(Mr N, t) i j i i 2p i:j then this equation will be replaced by an equivalent single-particle Schro¨dinger equation for WI WI "HK WI , t

(8.99)

where HK is given by Eq. (8.95). The reader is referred again to Section 6.1 for illustrative elementary example of such transformation. Use of complex variables z and zN and ground state dominance assumption, Eq. (8.91), allow us to write the many-body wave function WI for our Landau-like problem in the form

G

H

w n5 (8.100) WI (Mz N, MzN N)"NI < (z !z )U@2p exp ! + Dz D2 . i j i 0 i i l i:j i For U"0 we obtain back the product of Landau wavefunctions (see, e.g. Eq. (8.90)), while for nonzero U we obtain, instead of Eq. (8.89), the following result for a2:

P

a2" d2z DzD2oL (z, zN ) */5

(8.101)

with

P

n5 (8.102) oL (z, zN )" < d2z DWI (Mz N,MzN N)D2 . i 0 i i */5 i/2 The combined use of Eqs. (8.100), (8.101) and (8.102) reduces the problem of computation of a2 to the calculation of the classical statistical mechanical average

P

1 n5 a2" < d2z Dz D2 expM!HI [z, zN ]N , i 1 Z i/1 where, in view of Eq. (8.94), we have

(8.103)

2w !4 + ln Dz !z D# + Dz D2 (8.104) HI [z, zN ]" i j i l l i:j i and Z is a normalization constant (partition function). The Hamiltonian HI is known in the literature as describing the one-component plasma (OCP), (Caillol et al., 1982), while the wave function of Eq. (8.100) is known as Laughlin wave function (Laughlin, 1983), used in the theory of quantum Hall effect (QHE). For n

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