Stochastic Variational Approach to Quantum-Mechanical Few-Body Problems

Preface widely used techniques to solve the eigenvalue problems of quantum-mechanical Hamiltonians. Its popularity is ...

Author: Yasuyuki Suzuki | Kalman Varga

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Preface

widely used techniques to solve the eigenvalue problems of quantum-mechanical Hamiltonians. Its popularity is due to its simplicity and flexibility. The most crucial point in the variational approach is the choice of a variational trial function. One usually attempts to construct the trial function from some adequate basis functions which contain a number of nonlinear parameters. The direct method of diagonalizing the Hamiltonian matrix on such a basis set may not be feasible, except for simple systems, because of the large number of degrees of freedom involved in specifying the system. One thus faces a problem of selecting the most suitable basis set. It is by the stochastic variational method, that is, by a trial and error procedure with an admittance test that we give an answer to this problem. The stochastic variational method has been developed through the search for precise solutions of nuclear few-body problems. The variational method is

In this method

it enables

us

we

one

of the most

set up the basis element

to test many

parameters

to monitor the energy convergence. The aim of this book is to give

as

after the other because

one

fast

as

possible

and

moreover

unified and

reasonably simple problems with the use few-body recipe of the stochastic variational method and to present its application to various few-body problems which one encounters in atomic, molecular,

nuclear, subnuclear and solid

a

bound-state

for solutions of

state

physics.

quantum systems is in general extremely difficult and challenging, great advances have been made in recent years, especially for systems of a few particles and it has become possible to obtain accurate solutions for the eigenvalue prob-

Though

a

unified

approach

to the diverse

quantum-mechanical Hamiltonians. The main interest the few-body problems lies in, e.g., finding an accurate solution for

lem of various

in

the system

so as

to understand the

dynamics

of its

constituents,

test-

VI

Preface

ing the equation of motion and the conservation laws and symmetries, or looking for unknown interactions governing the system. Quantum mechanics plays a fundamental role in atomic and subatomic physics. It is via quantum mechanics that one can understand the binding mechanism of atoms, molecules and atomic nuclei, that is, the structure of the building blocks of matter. The interaction between the particles depends on the system: For example, the longrange Coulomb interaction dominates in atoms and molecules but the

very different mass ratio of the electrons and the atomic nucleus plays a key role as well. In contrast, the protons and neutrons in nuclei have

equal masses and the interaction between them is short-ranged. The variational foundation for the time-independent Schr5dinger equation provides a solid and arbitrarily improvable framework for the solution of diverse bound-state problems. As mentioned above, the most crucial point in the variational approach is the choice of the trial function. There are two widely applied strategies for this choice: One is to use the most appropriate functional form to describe the short-range as well as the long-range correlations among the particles. Such calculations, however, are fairly complex for systems of more than three particles, and the integration involved is performed by the Monte Carlo method. Another way is to approximate the solution as a combination of a number of simple basis states which facilitate the analytical calculation of matrix elements. We follow the latter course almost

in this book and show that the stochastic variational method selects

important basis set without any bias, keeps the dimension of and, most importantly, provides a very accurate solution. The book is conceptually divided into two parts. The first seven chapters present the basic concepts of the variational method and the formulation using Gaussian basis functions. The latter four chapters of the book cover applications of the formulation to various quantummechanical few-body bound-state problems. In Chap. 2 a general formulation is developed to express the physical operators which are needed to specify the Hamiltonian in terms of an arbitrary set of independent relative coordinates. The linear transformation of the relative coordinates induced by the permutation of identical particles is also established in this chapter. In Chap. 3 we review the basic principles of the variational method with particular emphasis on the case where the variational trial function is given as a linear combination of nonorthogonal basis functions. We introduce in Chap. 4 a key algothe most

the basis low

rithm used in this

book,

the stochastic variational

method,

and show

Preface

that its trial and

error

search

procedure

makes it

possible to

Vii

select the

important basis functions without any bias in the function space spanned by the basis functions. Some other methods to solve few-body most

briefly introduced in Chap. 5. Chapter 6 defines the type of variational trial functions used extensively in the book, the correlated Gaussians and the correlated Gaussian-type geminals. They are chosen because they enable us to evaluate matrix elements analytically and because they provide. us with precise solutions for most problems of real interest. A simple but powerful angular function is introduced to describe orbital motion with nonzero orbital angular momentum. To facilitate the systematic and unified evaluation of matrix elements,

problems

are

it is shown that the above Gaussian basis functions

function. In

7

are

all obtained

show that the

generatChap. deriving the matrix elements of the Gaussian basis functions for an N-body system of essentially any interaction. Explicit formulas axe given in this chapter for the simplest possible Gaussian basis functions, because they are already found to be very useful. The matrix elements for a, general case are detailed in the appendix. We show also in this chapter that the. method can be extended to evaluate the matrix elements of nonlocal potentials and the seniirelativistic kinetic energy as well. Chapters 8-11 present application of the stochastic variational method to various systems: small atoms and molecules (Chap. 8), baryon spectroscopy (Chap. 9), excitonic complexes and quantum dots in solid state physics (Chap. 10), and nuclear few-body problems (Chap. 11). We hope that this book will be found useful by students who want to understand and make use of the variational approach to quantummechanical few-body problems, while it may also be of interest to researchers who axe familiar with the subjects. It will be our pleasure if this book serves to bridge the gap between graduate lectures and the literature in scientific journals, as well as to give impetus to further development in the deeper understanding of quantum-mechanical fewbody systems. We assume that readers have taken courses on quantum mechanics and mathematical physics at an undergraduate level. No special knowledge is assumed of, e.g., atomic physics or nuclear physics. To help readers to understand the text, we have attempted to make the book self-contained, put as much emphasis as possible on clarity, and given several Complements of an explanatory nature. The Complements are intended to further develop or to reinforce the arguments and ideas presented in the text. We have collected the derivation

from

ing

a

generating plays a

function

vital role in

we

VIII

Preface

of formulas that may possibly be difficult for readers solutions at the end of the chapters.

as

exercises with

Depending on their interest, readers may adopt several reading strategies. A thorough-going reader is advised to read all of the text including the Complements. A reader who wants to understand only the basic formulation can omit the Complements. Anyone who is familiar with the variational method may skip Chaps. 2 and 3. Experts, or those readers who are only interested in the performance of the stochastic variational approach or the physical consequences of the results, may skip Chaps. 2-7. The Gaussian basis has long been used in many areas of physics. The correlated Gaussians were first introduced in quantum chemistry by S.F. Boys and K. Singer. The application of the Gaussian basis is one of the key elements of the success of the ab initio calculations in quantum chemistry. The stochastic variational method is actually very similar to the so-called "random tempering", that has been used to find the optimal parameters of the basis in quantum chemistry. In the random tempering pseudo-random parameters axe generated, and the best basis functions are selected by sorting out the states which improve the energy. There exists another method which is also similar to the stochastic variational method, called the stochastic diagonalization. This method, originally developed in solid state physics, attempts to find the lowest eigenvalue of huge eigenvalue problems by randomly testing the contributions of different states. The random selection of the parameters of a Gaussian basis, named the stochastic variational method, was first used by V.I. Kukulin and V.M. Krasnopol'sky in nuclear physics. We are grateful to Prof, V.I. Kukulin for his interest and encouragement during this work. We are indebted to our collaborators, K. Arai, Y. FujiNvara, L.Ya. Glozman, R.G. Lovas, J. Afitroy, C. Nakamoto, H. Nemura, Y. 0hbayasi, Z. Papp, W. Plessas, G.G. Ryzhikh, N. Tanaka, J. Usukura, and R.F. Wagenbrunn for useful discussions and for many of the calculations which press

our

hearty

are

included in this book. We wish to

thanks to Prof. K.T.

Hecht,

ex-

Prof. D.

Baye, Prof. reading of the Despite all these

M. Kamimura and Prof. R.G. Lovas for their careful

manuscript and for much advice and many comments.

efforts mistakes may still remain. Needless to say, these are our own responsibility. Suggestions and criticisms from our readers would be welcomed. We

are

grateful

for the

use

of the computer VPP500 of (RIKEN). One of the

the Institute of Physical and Chemical Research

Preface

for the

authors

(K.V.)

ment of

Niigata University and

is

grateful

would like to thank the

Research of the

hospitality

the Linac

funding agencies,

Iffinistry

of

(Hu gary),

of the

Ix

Physics Depart-

Laboratory

of RIKEN. We

Grants-in-Aid for Scientific

Education, Science and Culture (Japan),

Japan Society for the Promotion of Science (JSPS), the Science and Technology Agency (STA) of Japan, the U. S. Department of Energy, Nucleax Physics Division, and the Hungarian Academy of Sciences (HAS), for their support which has been vital for the completion of the work. Finally, without the support and patience of our families, this book would not have been possible. OTKA Grants

Niigata Argonne September

the

Y. Suzuki K.

1998

Varga

Table of Contents

I..

Introduction

2.

Quantum-mechanical few-body problems

........................................

...........

7

2.1

Hamiltonian ......................................

7

2.2

Relative coordinates

..............................

9

..................................

15

2.3

Symmetrization

2.4

Permutation of the Jacobi coordinates

Complements C2.1 An N-particle Hamiltonian in the heavy-particle center

..............

.........................................

Variational

3.2

The variance of local energy The virial theorem

3.3 4.

principles

......................

19

4.2

21

21

.......................

30

................................

33

.......................

39

................................

39

...............................

43

......................

44

.........................

47

..............................

47

....................................

50

Basis

optimization A practical example 4.2.1 Geometric progression 4.2.2 Random tempering 4.2.3

Random basis

4.2.4

Sorting

4.2.5

Trial and

error

search

.......................

Refining 4.2.7 Comparison of different optimizing strategies Optimization for excited states

4.2.6

4.3

................

.............................

Stochastic variational method 4.1

18

18

Introduction to variational methods 3.1

16

...........

coordinate set

C2.2 Canonical Jacobi coordinates 3.

1

...................................

.................

50 53

.

54

....

56

.

I

Complements

.........................................

61

XII

Table of Contents

C4.1 Minimization of energy 5.

versus

variance

..............

Other methods to solve

5.2

few-body problems Quantum Monte Carlo method: The imaginary-time evolution of a system Hyperspherical harmonics expansion method

5.3

Faddeev method

5.4

The generator coordinate method

5.1

.........

...........

6.

..................................

Variational trial functions 6.1

6.2

6.4

65

67 70

72

..........................

75

Correlated Gaussians

Gaussian-type gerninals Orbital functions with arbitrary angular Generating function The spin function

Complements

..............

.

82

87

................................

94

C6.2 Solid

spherical harmonics C6.3 Angular momentum recoupling C6.4 Separation of the center-of-mass =

as a

basis

.....

96 96 105

.....................

106

motion

.........................

1/2

......................

arbitrary spin arrangement

112

115

......

115

...........................

116

............................................

118

an

C6.7 Six electrons with S

Matrix elements for

=

0

spherical Gaussians, generating function

............

123

..........

123

..............................

125

7.1

Matrix elements of the

7.2


7.3 7.4

Correlated Gaussians in two-dimensional systems Correlated Gaussian-type gendnals

7.5

Nonlocal

7.6

Semirelativistic kinetic energy

Complements

potentials

.

129

.................

131

.

.

.

.

...............................

......................

.........................................

C7.1 Sherman-Morrison formula Exercises

.

..........................

from correlated Gaussians

C6.5 Three electrons with S

Exercises

momentum

.........................................

C6.6 Four electrons in

75

..............................

C6.1 Nodeless harmonicoscillator functions

7.

65

...................

and correlated

6.3

.........

61

134 137

143

........................

143

............................................

145

Table of Contents

8.

Small atoms and molecules

.........................

8.3

Coulombic systems Coulombic three-body systems Four or more particles

8.4

Small molecules

8.1

8.2

.......

149

.....................

150

...................................

.........................................

C8.1 The cusp condition for the Coulomb potential C8.2 The chemical bond: The H,+ ion C8.4 9.

Stability hydrogen-like Application of global vectors

169

.................

to muonic, molecules

174

................................

177

One-gluon exchange model Meson-exchange model

.....

178 178

............................

181

quark

...........

188

...................................

191

....................

196

method

a

magnetic

204

..........................

213

...

213

.............

216

...............................

223

.........................................

230

11.2 Few-nucleon

systems with central forces

potentials

C11.1 Correlations in few-nucleon systems C11.2 Convergence of partial-wave expansions Pauli effect in s-shell A

C11.4 The "C nucleus

Appendix

202

field. 204

few-body systems Introductory remark on nucleon-nucleon potentials

Quark

....

.........................................

11. Nuclear

C11.3

187

..............................

C10.1 Two-dimensional electron motion in

as a

...............

230

............

233

..........

239

.

242

hypernuclei system of three alpha-particles

...............................................

Matrix elements for

A.1

model

........................

Few-body problems in solid state physics 10.1 Excitonic complexes 10.2 Quantum dots 10.3 Quantum dots in magnetic field 10.4 Quantum dots in the generator coordinate

Complements

171

....

9.2

11.3 Realistic

167

....................

The trial function in the constituent

11. 1

165

167

9.1

Complements

154

........

molecules

Baryon spectroscoPy

9.3

10.

of

149

........................

............................

Complements

C8.3

XIII

general


A. 1. 1

Overlap

A.1.2

Kinetic energy

Gaussians

..................

.............................

of the basis functions

247 247

247

................

247

.............................

249

Table of Contents

XIV

A.1.3 A.1.4

......................

250

..................

256

Two-body interactions Density multipole operators

A.2

Correlated Gaussians with different coordinate sets

A-3

Correlated

matrix elements

Spin

A.5

Three-body problem and spin-orbit forces

Complements

.................

262 263

with

central,

tensor

..............................

265

.........................................

280

CA. 1 Matrix elements of central potentials CA.2 Matrix elements of density multipoles

CA.3

...............

280

..............

283

matrix elements of the correlated Gaussians

Overlap a three-particle system

for

Exercises

References Index

257

.............................

Gaussian-type geminals

AA

...

.........................

285

............................................

288

..............................................

...................................................

299

307

1. Introduction

There

are a

countless number of examples of quantum-mechanical few-

body systems:

constituent

quarks

in subnuclear

physics, few-nucleon

few-cluster systems in nuclear physics, small atoms and molecules physics or few-electron quantum dots in solid state physics, etc. The intricate feature of the few-body systems is that they develop or

in atomic

depending on the number of constituent particles. The mesons and baryons, the alpha-particle and the 'Li nucleus, or the He atom and the Be atom have very different physical properties. The most important causes of these differences are the correlated motion and the Pauli principle. This individuality requires specific methods for the solution of the few-body Schr6dinger equation. Approximate solutions which assume restricted model spaces, mean field, etc. fail to describe the behavior of the few-body systems. The goal of this book is to show how to find the energy and the wave function of any few-particle system in a simple, unified approach. The system will normally be in the minimum energy quantum state. As forewarned, however, to find this state, the ground state, is a complicated matter. The present stage of the development of computer technology, however, makes a very simple approach possible: Searching for the ground state by "gambling". Without any a priori information on the true ground state, completely random states are generated. Provided that the random states axe general enough, after a series of trials one finds the ground state in a good approximation. The reader individual characters

may find this

a

little

suspicious but there are indeed a number of fine error procedure which makes the whole idea

tricks in the trial and

really practicable. Before bombarding the

reader with

sophisticated details, let us example. Let us try to determine the energy of a Coulombic three-body system: a positron and two electrons, the positronium negative ion, Ps-. The example is simdemonstrate the random search with

Y. Suzuki and K. Varga: LNPm 54, pp. 1 - 6, 1998 © Springer-Verlag Berlin Heidelberg 1998

an

1. Introduction

2

ple enough so t1i at a graphic illustration of the wave function can be given, but its solution is far from being trivial. The ground state of this particular system can be calculated by different methods accurately. In the rest of this chapter we will give just the outline of the gambling method without paying much attention to the details. To look for the wave function of the ground state of the system, we generate random functions. The functions we assume some

continuous a.re

parameters

or

randomly

chosen. For

Tf (a, 0, x,

y)

=

=

are

random in the

depends

on some

sense

that

discrete

or

"quantum numbers" and these paxameters example, we assume a simple Gaussian form

e`2 -

3Y2

(1.1)

7

1r2 -7'31 denotes the distance between the two electrons and I (r2 +r3) /2 r 1 1 denotes the distance between the center-of-mass

where Y

functional form which

x

=

-

of the electrons and the

positron.

randomly chosen generated states are calculated and compared. The one among them giving the lowest value is selected to be a successful paxameter set. Figures 1.1(a)-(d) show examples of four random states and their energy expectation values which are obtained with such successful paxameters. Not surprisingly, the energies of the "configurations" appreciably depend on the shape of the functions, that is, on the random paxameters. The lowest energy, -0.11 in atomic units (a.u.) in Fig. 1.1(d), is higher than that of the exact ground state (-0.262). Here in the atomic units M, (the electron mass), e (the electron charge magnitude), and h axe chosen to be basic quantities of units, so the unit of length is Bohr radius h?/(Mee2) 5.29x1O-11 m, theunit of energyis me4/h? 27.2 ao 2.42 x 10-17 S. (Actually the and the unit of time is h3/(me4) eV, minimum energy calculated analytically with one term of Eq. (1.1) is -0.177, but it may not be known for a general case.) That means that the trial function is not general enough to describe the ground state. To improve the trial function, let us take a linear combination of two of the above functions of Eq. (1.1), where one of the two is fixed as the one already selected and another is newly selected after a number of random trials. Figures 1.2(a)-(d) show examples for random wave functions which are calculated by the two terms. The energy improved because the model space increased but we still miss a substantial amount of binding energy. Increasing the model space further A certain number of parameter sets a and and then the energy expectation values of the

=

axe

=

-

=

by adding one,

we

more

and

more

functions to the linear combination

reach the exact energy and the

wave

function

one

(Fig. 1.3)

by

with

1. Introduction

E--0.09

E--3.15

E=0.18

E--0.11

Examples of the energy expectation value and the (normalized) xyTf (a, 3, x, y) of Ps- for one random basis state. Figure 1(a) is in the upper left, 1(b) in the upper right, 1(c) in the lower left, and 1(d) in the lower right. In each section the x axis denotes the distance

Fig.

wave

1-1.

function

between the two electrons and the y axis denotes the distance between the center-of-mass of the two electrons and the positron. Atomic units are used.

1. Introduction

E-0.14 E=-0.18

0.1

0.0

B-0.12 E=-0.17

0.1 0.0

Fig.

1.2.

Examples

xyjcj-T1(aj, Oj

of the energy expectation value and the

wave

function,

y) +c2Tf(a2, 02 Xi Y) ji of Ps- for combinations of two random. basis states. Figure 2(a) is in the upper left, 2(b) in the upper right, 2(c) in the lower left, and 2(d) in the lower right. See also the caption of -,

Fig.

1. 1.

x,

1

1. Introduction

a

combination of about 150 functions of the form

(1.1).

The

5

conver-

gence of the energy versus the number of the functions in the linear combination is shown in Fig. 1.4. The energy gain is large in the first

few steps and then it

slowly approaches the

exact energy.

E=-0.262

X

Fig. and

1.3. The energy and the wave function of Ps- obtained selection of 150 basis states

by the trial

error

The alert reader may question the importance of the steps that are taken to increase the model space and can ask why not start say, a linear combination of 150 functions and then, by random of all the parameters, one may find the solution. The reason for trials

with,

increasing the number of functions in the combinations is to control "convergence" of the energy. By comparing the energy gain in the successive steps one can guess how far the exact ground-state energy the

is. There is

no

guarantee that the ad hoe 150 functions would be

1. Introduction

6

-0.250

-0.252 M 41

-0.254

-0-256

-0.258 Z

-0.260

-0.262 20

40

60

80

100

Dimension of the basis

Fig.

1.4. The energy convergence of Ps- as a function of dimension of basis are increased one by one with the trial and error selection. The

states that

solid, dashed and dotted

curves

correspond to

sufficient to reach the solution in

cases

three different random

paths.

where the exact energy is not

known. A

skeptic may say that one should, instead of the above gambling method, try a deterministic parameter search such as that furnished by the Newton or conjugate gradient method. While this may be true for small systems, the random trials the

axe more

successful in most of i

in the

I

a. by avoiding being trapped omnipresent local Moreover, one may have discrete paxameters or quantum numbers

cases

where the deterministic seaxch may not be suitable. One may also wonder if the solution we reach is the in

ground state energy. These the succeeding chapters.

and other

really (close to)

questions will be answered

Quantum-mechanical few-body problems 2.

first step toward the goal of giving a unified and reasonably simple recipe for solutions of few-body bound-state problems the basic

As

a

notations and concepts

The motion of the ton

operator

axe

introduced here.

few-body system

(Hamiltonian)

and described

governed by the Hamilby the eigenfunction of the

is

depends on the positions and other degrees of freedom of the particles. See, for example, [1, 21 for textbooks on quantum mechanics. One can define the positions of the particles in several different ways by using single-particle or relative coordinates. The single-particle coordinates are useful if the particles move ah-nost independently of each other. The relative coordinates are advantageous to emphasize correlations between the particles and Hamiltonian. This

to

wave

function

separate the center-of-mass motion. One

can

relate the different

coordinate systems to each other by linear transformations. The definitions and the properties of these transformations are elaborated in this

chapter.

indistinguishable particles. To comply with the Pauli principle the wave function has to be properly symmetrized. In the spatial paxt of the wave function the symmetrization induces permutations of the coordinates. It is shown that these permutations can be imposed by linear transformations in the relative coordinate One often deals with

space.

2.1 Hamiltonian isolated system of N particles with masses ml., and let 7,1,, ,,rN denote the position vectors of the particles. -, mAT All of the particles may be identical or different, or some group(s) of them may be identical. The Hamiltonian of the system, with the

Let

us

consider

an

*

center-of-mass kinetic energy Tcm

being subtracted,


reads

as


2.

8

N

N

2

H=E 2mi

-T,.

As the interaction

(2.1)

Vij.

+

unchanged by the displacement of the origin, the above Hamiltonian is translationaUy invariant and depends on the internal degrees of freedom only. One may encounter few-body, problems where the system is subjected to some external field or the particles move in a single-particle potential. The Hamiltonian then reads as IV

H

=

N

2

E2

remains

Vij

+

1: Ui + 1: i=1

i=1

N

(2.2)

Vij.

i>i=l

Even with the presence of the one-body potential Ui it may happen that one wants to remove the contribution from the center-of-mass motion. This will be considered later in

evaluating the matrix elements

of various operators. An extension of the nonrelativistic kinetic energy to the relativistic kinematics will be discussed in Sect. 7.6.

Three-body potentials can be treated in principle but axe suppressed for the sake of simplicity. The two-body potentials are assumed to be local (but nonlocal potentials will also be considered in Sect. 7.5) and can in general be expressed as

Vij

=

1: VP(ri

-

rj) 0?.

P

=

1: f VP (r) J(ri

-

rj

-

r) 0?. dr.

(2.3)

P

Here p is the short-hand notation to specify the component of the potential which is characterized by 0?. (e.g., central, spin-orbit, etc.

and

VP(r)

is the

corresponding form factor.

To

specify

the

particle

motion, several degrees of freedom are in general needed besides the spatial one. For example, the nucleon has spin and isospin degrees

freedom, and the quark has spin, flavor and color degrees of fre&dom. The specific potentials related to nuclear, subnuclear and other of

systems will be discussed in later sections.

2.2 Relative coordinates

9

2.2 RelatiVe coordinates The

separation of the center-of-mass motion

care

of most

is

important to describe intrinsic excitations of the system. The center-of-mass motion is taken mass

conveniently by introducing

coordinates;i

(XIi X2

relative and the center-of-

xN). The symbol- stands for a trans-

pose of a matrix. In particular, x is often used to stand for one-column matrix whose-ith element is xi, whereas ,'c a I row

an

x

N

IV

x

1

one-

matrix. Here XN is chosen to be the center-of-mass coordinate of

the system and the rest of the coordinates

relative coordinates.

independent single-paxtide coordinates

i

-=-

They

(rj,

-'

fXI,

axe

in

-

-

-,

X N-I

general

I

is

a

set of

related to the

TN) by a linear transformation:

N

Uijrj,

xi

ri

=

j=1

E (U-I)ijxj

(i

N).

(2.4)

j=1

We show two

examples

of relative coordinates

is the Jacobi coordinate set, which is defined

by

x.

See

Fig.

2. 1. One

the matrix

C

2

4 3

0

0

(a) Fig.

2.1.

Examples

(b)

of relative coordinates for the

four-particle systein. (a)

the Jacobi coordinate set, xi =rl -r2, X2:-- (MIrI +M2P2)/(MI +M2)'r3 7 X3 = (Tnlrl + M2r2 + M3 r3)/(Ml + M2 + M3) -P47 and (b) the heavyparticle center coordinate set, xi = rl r47 X2 = r2 r47 X3 = r3 r4-

-

-

2.

10

Quantum-mechanical few-body problems 1

-1

MI

M2

M12

M12

M1

M2

0

0

...

0

(2.5)

Ui M12

...

M12

N-I

...

N-I

M1

M2

Tal2---IV

Tnl2---.LV

MN ...

...

M12

...

N

given by

Its inverse matrix is

M2

M3

M12

M123

MIV ...

Tni

M3

M12

M123

M12

---

M

MN ...

(Uj)

M12

...

N

(2.6)

M 12

0

M123

-"M12

0

0

...

N-1

M12 ...N

where M12 i +mi and, especially, 7nl2---N is the total Tnl+Tn2+ the "heavy-particle center" coordinate is Another the of mass system. -

...

set defined

UC

by 1

0

0

0

1

0

-1 -1

...

(2.7)

-

0

0 M12

---

N

...

-1

...

MN

M2

Mj_

...

...

M12

...

given by

and its inverse matrix is

...

N

M12

...

M

N

(UC )-I

M12

...

M12

...

N

M1V-I

M2

MI ...

MN-I

M2

M, M12

M12

M12 ---N

M

N

M12

N

M2

M12

...

N

MN-I

M2

MI

M12

...

N

M12

...

N

M12

...

N

(2.8) The choice of the when the other

mass

particles.

heavy-particle

See

center coordinates may be natural

particle is heavier than the masses of the Complement 2.1. Note, however, that it is always

of the Nth

2.2 Relative coordinates

11

possible to transform from one coordinate set to another. The Jacobian corresponding to the coordinate transformation (2.4) is unity for both U matrices.

Corresponding to the transformation (2.4), the momentum pi is -ih ya conjugate to the expressed in terms of the momentum -xj axj =

coordinate xj: IV =

Pi

N

1: Ujilrp

Wi

=

1:(U-l)jipj

(i

=

1,

...,

N).

(2.9)

j=1

j=1 N

Note that -7rAT

=

Ej=1 p,

is the total momentum. The center-of-mass

kinetic energy is given by T,,n ir2,v/(2MI2 N). The kinetic energy with the center-of-mass kinetic energy subtracted is then operator =

...

expressed

as

P

(27ni 6ij

2mi

i=1

i=1

2m12

...

N)

Pi

Pj

j=1

M-1 N-I

Aij-xi

2 i=1

-

(2.10)

-7rj,

j=1

where N

1: UikUjk Mk

Aij

(i,j

=

11

....

N

-

1).

(2.11)

k=1

To evaluate the potential energy matrix elements for the tions that contain

no

dependence

on

the center-of-mass

wave

func-

coordinate,

it

is convenient to express the single-particle coordinate relative to the center-of-mass, ri xN, the interparticle-distance vector, ri -,rj, and -

difference, pi pp in terms of x and easily done by using Eqs. (2.4) and (2.9):

the momentum This is

-

-x,

respectively.

1V-1

ri

-

XN

=

1: (U-l)ikXk

-W X,

(2.12)

k=1 N-I

'ri

-

r3

1: k=1

((U-l)ik (U-I)jk) -

Xk

=

W(j) X,

(2.13)


2.

12

N-1

2

(Pi

Pj

E (Uki

2

-

(ij )Ir-

Ukj)7rk

(2.14)

k=1

Note that the contribution from the term

momentum,

(Mi

is

when

legitimate

-

Mj)/(2Ml2 N)7rN, ...

we are

proportional to the total omitted in Eq. (2.14). This

is

in the center-of-mass

(,7rlV

system

=

0)

and

interested in the intrinsic motion of the system. The Hamiltonian of type (2.1) is now expressible in terms of independent relative coordinates alone. In what follows

and

x

7r are

relative coordinates unless otherwise

(i.e. (N

introduced to

the notation.

by the

W(i) EN-I k Xkk=1

=

they are (N-1) x 1

stated,

-

1)

1 one-column

x

I)-dimensional vectors) 0), Oj),

simplify

which is formed

-W X

-

represent only the

that

so

Eqs. (2.12)-(2.14), (N

one-column matrices. In matrices

meant to

Wx

E.g.,

is

a

and

(W)

are

Caitesian vector

multiplication of iv(i) and x, i.e., that Oj) and (W) satisfy the following

usual matrix Note

equalities M-1

-((ij)

3 W(ij) (k k

W( ij

=

(-ipw(ij)

k=1 N

W(ij)(1(ij)

W

k

i>i=l

(ij)

F(ij)

i>i=l

kI

N

where the relations UNTj

for k

=7 -

For

N

are

an

(N

-

1) x (N

-

1) symmetric

scalar

products

E xi-(Ax)i with this

i=1

=

1,

EN

Uki

=

0

A,

let the

quadratic

of the Cartesian vectors:

xv)

j=1

convention, that

1V -

matrix

Aijxi-xj. i=1

show,

ai(ri

(2.15)

N-1 N-1

i=1

It is easy to

1),

?ni/?nl2---N, (U-1-)iN

=

N-I =

-

used.

form, bAx, represent

..bAx

N

R, I

Jk, 2

2

i=1

ajw(')J))

x,

(2.16)

2.2 Relative coordinates N

1V

I:

a

(ri _,rj)2

ij

1:

=.i

a

ajjw

-(ij)

(ij)

(2.17)

X.

i>i=l

i>i=l

As

13

special case of the above relations, we have the following equalities

for the moment of inertia around the center-of-mass: N-1

N

N

Mi(ri

-

XN

MzMj

)2

M12

where the reduced is defined

mass

...

N(ri

Tj)

-

corresponding to

ILi

2

(2.18)

the Jacobi coordinate xi

by ?ni+17nl2 M12

...

z

...

(2.19)

1).

N

i+1

We also obtain 2

N

N

(ri

-

T'j )2

N

=

E(ri XN)2

(,ri

_

-

(2.20)

xiv)

i=1

The term are

the

E , (ri

-

XN)

masses

of all the

particles

same.

The total orbital mass

vanishes when the

system

(-xv

=

angular

0)

momentum

be

can

operator L in the center-of-

expressed

in terms of the

operators

relevant to the relative coordinates: N-1

N

hL

(ri

((,ri

-

X

Pj)

Tj)

(XN

X

2

(Pi

X

-

(Xi

7N)

X

7ri),

(2.21)

Pj))

i>i=l N

or i

-

X

'r

Rij)

-X

i>i=l

+

((ri

-

W('j)X

rj) XMi

-

2M12

X

Mj 7rN ...

N

(('j)-7r) =NU. 2

(2.22)


2.

14

Eq. (2.22) use is made of Eq. (2.15) in the last step. Thus the equality of Eq. (2.22) holds in the case one considers the orbital angular 0 or treats a system of paxticles momentum in the system of 7rjV with equal masses. In atomic physics the trial function of Hylleraas type or correlated In

=

exponential type is often used with success. This function contains the exponential form expressed in terms of the interparticle-distance coordinates. T-nstead of the exponential function let us consider its Gaussian analogue IV

T

=

exp

E

2

aij (ri

-

rj )2

(2.23)

i>i=l

aij determine the falloff of the interparticle functions and may be considered variational parameters. Since not all of the N(N 1)/2 interparticle-distance vectors, ri rj, are Here

N(N

1)/2 parameters

-

-

-

(Irl T2) + (T2 -r3), it is convenient to T3 e.g., TI rewrite T in terms of the independent coordinates x. This is needed to independent,

calculate, e.g., the Eq. (2.23) as T

=

exp

aij

are

-1

(

(N

where the

=

-

-

2 -

related

1)

norm

-

of T.

Equation (2.17) enables

one

to rewrite

iAx), x

(N

-

(2.24)

1) symmetric matrix A

and the parameters

by N

N

-

(ij)

AkI

ajjWk

(ij) WI

=

I:

aij

(W

(ij) W(ij)

) kl'

i>i=l

i>i=l N-I N-1

aij

-

1: 1: UkjAkIU1j k=1

=-

-(CTAU)ij

(i

(2.25)

i=l

'where jLjN mimN/(Tni + Talv) is the reduced mass for the ith and Nth particles. The potential energy VjV (i 1, ..., N 1) is a function =

=

of ri -rN on

xi and

=

the ith

can

be considered the one-body potential Uj

On the other hand the

particle. lighter particles can

N-I

H=E i=1

Potential

energy

function of ri -,rj xi thus be reduced to the type of Eq. (2.2)

between the

HamiltoTdan

-

2AjN

=

a

-

xj. The

N-1

N-I

2

^7r'

is

acting Vij acting

+EUi+ 1:

(Vij+

MN

7ri-7rj).

(2-34)

j>i=i

=1

problem is thus reduced to that of an (N-1)-paxticle system with an external field Uj and an additional separable "two-body potential"

The

oo, jLjjv goes to mi and the 7ri .7rj term The effect of the finite nucleax mass in the helium atom

7ri -7rj. In the limit of m1V -

disappears. was

discussed in

-->

[3].

The above formulation

can

also be

applied

to

a

system of identi-

particles when some, or most, of them form an inert core. As an example let us take up the nuclear shell model, where the nucleons

cal

are

divided into two groups, the passive or inactive nucleons and a small number of active nucleons. The passive nucleons form

relatively

C2.2 Canonical Jacobi coordinates

19

by filling the lowest possible single-particle orbits and exert a potential field Ui on each of the active nucleons. The active nucleons may occupy several single-particle orbits belonging to several major shells outside the inert core. In this approximation it is desiran

inert

core

able to describe the motion of the active nucleons without inclusion

arising from the center-of-mass motion. It is clear can be excluded by taking the full Hamiltonian have the form (2.34) with the core as the heavy particle.

of any excitations

that such excitations to

2.2 Canonical Jacobi coordinates

As

was

2.2, the Jacobi coordinates

noted in Sect.

that the transformation matrix Ui of Eq.

is,

neither

following

UjUi-

nor

relations

UjA-'Uj

Uj-Uj

equal

(2.5)

have the property orthogonal, that

to the unit matrix.

Instead, the

fulfilled:

are

L-1,

=

is

x

is not

UiLUi

=

(2.35)

A,

with

Aij

Jij

Lij

(2.36)

Jij, Ai

Mi

N where pi (i 1, 1) is the reduced mass belonging to the ith Jacobi coordinate and is defined in Eq. (2.19), while 1LV is equal to the =

-

the transformation between

the singleUir and p UT-x, of paxticle and the Jacobi coordinate systems, x identities the lead above relations the to and following (2.9), Eqs. (2.4) total

mass

'MI-2

N.

...

By using

=

N

N

E Tnir?

=

z

N

E Nx

Alternatively,

N

pi2

E

-

Mi

one can

introduce

Ir2i

=

-(2.37)

Ai

"canonical" set of Jacobi

a

coor-

dinates: IV

Vijrj

with

V

=

A

2

UiL 2.

(2.38)

j=1

The square root matrices of the diagonal matrices L and A axe simply given by the square root of the elements. This system of Jacobi

coordinates

Vf"

=

belongs

f7v

and therefore

17

to

an

orthogonal

transformation:

(2.39)

20

Complements 1V

1V

EVji j

IV

E 2 Er2

and

(2.40)

=

i

i*

i=1

j=I

i=1

Let the momentum

Simfla,rly

to the

canonically conjugate to i be denoted qi. transformation of coordinates, we obtain the trans-

formation of momenta

follows:

as

N

71i

=

N

E ViiPi,

A

E Viini

=

j=I

(2.41)

j=I

The total Idnetic energy does not take to q and can be expressed as follows:

diagonal

a

form with respect

I?

E Mi

E E (VLfr) ij?7i -77j wLf7rl.

i=1

i=1

j=I

The two systems of the Jacobi coordinates ical Jacobi coordinates X

=

For

ujf '

a

=

=

M1

rT; 1'2

?T 1 M

M2M3

3

M123

12MI23

VMM1232:

1

2

particles with equal I

1

(2.43)

V

vf2-

I

I

v/6-

T6

matrix V reads

0

12

M12

M123

I

'1

73T

disadvantage

masses

mi

axe

M3

mass

2

V"6_

(2.45)

1

73

of the canonical Jacobi coordinates

is that if the

equal then the center-of-mass motion is not sepais not the center-of-mass coordinate).

not

easily ( N

(2.44)

V-Ml'23

-

v/3-

as

0

-

V/2-

rated

=

V(Uj)-Ix.

VMM1123

The

can

-VEM12

L2 M12

and for

Vr

Ujr and the canoneasily be related by x

three-paxticle system the transformation

V

(2.42)

=

3. Introduction to variational methods

The variational method is

popular approaches to tackle quantum-mechanical few-body problems. Though it gives only an approximate solution except for some special cases (the Ritz variational method, for example, gives only an upper bound of the energy), one can get a virtually exact solution with an appropriately chosen function space. The function space is defined by basis states and the wave function of the system is expanded in that, basis. In this chapter we briefly introduce the theorems requisite for obtaining a vaxiational one

of the most

solution.

3.1 Variational Let

principles

physical system whose Hamiltonian H is self-adjoint (Hermitian), bounded from below and time-independent. We are interested in finding the discrete eigenvalues of H and its corresponding (normalized) eigenstates: us

consider

H!P,, The

=

a

En(fi,

energies En

n

axe

=

(3.1)

1, 2,...

real and

are

ordered such that El :! E2
-

El,


(3.2)


22

where the

equality

if and only if Tf

holds

is

an

eigenstate of H

with the

eigenvalue El. Proof.

elementary and can be found in of quantum mechanics. See for example [1, 2]. The ftmetion expanded in terms of the energy eigenstates

The

textbooks T1 may be

proof

of this theorem is

00

(3-3)

ai(fii.

In this

cluded and the

integration

over

(Tf jHn Itp) (TrI Tf )

the

with continuous

eigenvalues are sum must be extended appropriately to include them. Then we can show that for an integer n

expansion

eigenstates

0

Faci=2 A Eln)Jai 12 00, jai 12 Ei= n Y

inan

7

-

En I

-

-

(3-4)

I Clearly the right-hand side of Eq. (3.4) is non-negative for n and if for > if i It vanishes 2. > 0 because Ei 0 for E, only ai i > 2, that is, T, is an edgenstate of H with eigenvalue El. This proves I was used to prove the Ritz the Ritz theorem. Only the case of n theorem but other cases will be needed later to derive Temple's bound. =

=

-

=

approximate deexploited as ground-state energy El, namely the minimization principle of the mean value of H. Suppose that we choose a family of functions Tf (a) which are characterized by a finite number of parameters denoted a. In the case of Eq. (1.1), there axe two parameters a and P. We calculate the mean value E(a) of the Hamiltonian H in this trial function, and minimize E(a) with respect to a. The minimal value obtained in this way is an approximation to tile ground-state energy El of the system. Clearly, whether this variational method produces a satisfactory result or not substantially depends on the choice of the trial functions Tf (a). The Ritz theorem can be generalized to excited states as well: The Ritz theorem is the basis for

a

method of

it is

termination of the

Theorem 3.2

(Generalizecl

of the Hamiltonian H discrete eigenvalues.

value

Proof.

Let

E(TfIT)

=

us

(TfjHjTf)

sumed to be linear term

calculate

an

is

Theorem).

stationary

in the

increment JE of the

The

expectation neighborhood of its

mean

value us' g

+JT-1, where JTf is aschanged of higher order than a terms Neglecting

when Tf is

infinitely small. in JT1, we obtain

Ritz

to Tf

3.1 Variational

(TfITf)JE

=

=

The

(JTf IH

value E is

mean

that

J((TIIHITI)) -

-

23

principIcs

EJ((TIITI))

EITI)

+

(TfIH

stationary if JE

(3.5)

EIJTf).

-

=

0 for any infinitesimal

JT,

if

is,

(,NTtIH

EITI)

-

+

(TIIH

If JTf is chosen to be

the above

number,

-

-(H

EIST)

-

E)Tf,

(3.6)

0.

=

where

-

is

that the

equation implies

an

infinitesimal real

norm

of the function

(H E)Tf is zero, and thus the function (H E)Tf itself must be a null function, namely HTf = ET. Therefore the mean value E is -

-

stationary if and only if the state Tf from which E is calculated is an eigenstate of H, and the stationary value E is the corresponding

eigenvalue

of the Hamiltonian.

generalized Ritz theorem allows an approximate determination of the eigenvalues of the Hamiltonian. If the E(a) has several extrema, they are the approximate values of some of the energies E,,. In most cases for practical applications the. trial function is given as a linear combination of a finite number of independent functions This

!P(a): K

(3.7)

ciTf (ai).

Tf

independence of the functions will be discussed soon later. T' (a K) axe mutually orthogWe do not always assume that Tf (a I) onal because the use of nonorthogonal. functions is in fact quite useful. They can, however, always be made orthogonal if necessary, e.g. by a Gratn-Schmidt orthogonalization procedure. The variational method then reduces to the eigenvalue problem of the Hamiltonian inside the state space VK spanned by the set JTf((YI),...,Tf(aK)Ji that is, the space containing all linear combinations of T (a,), Tf(aK). The mean value E is given by The linear

,

...

i

...

Ctlic E

(3.8)

=

CtBC' where and

-

c

is

a

IC-dimensional column vector whose ith element is

ct is the Hermitian conjugate of c. The K

overlap

matrices W and B

are

defined

by

x

ci

K Hamiltonian and


24

Rij

=

(Tf(ai)JHJT'(cvj)),

Bij

(TI ((Yi) I Tf (aj)).

=

(3.9)

The linear parameter ci can be determined by the generalized Ritz theorem. The condition that E is stationary with respect to an ar-

bitrary, infinitesimal change problem

of ci leads to the

generalized eigenvalue

K

Wc

E&,

=

E(Rij

i.e.,

-

EBij)cj

=

0

(i

=

I,-, K).

(3.10)

j=J

The restriction of the thus

can

eigenvalue problem

of H to the

subspace VK

the solution.

simplify

We discuss the linear Tf (a K). The linear c

except for

c

0

=

can

function. In other

and

only

for

has

E

c

K

if c

independence of the functions Tf(al),..., independence of the functions means that no vector make

words,

0. This is

=

a

linear combination ,

EK i=1 ciTf(ai),

the combination becomes

equivalent

unique solution of c ci!F(ai) 0 is equivalent a

=

=

0.

a

a

null

null function if

that the

equation Bc 0 (To understand this, we show that

to

to Bc

saying =

0. Whe'n

=

EK ci!P(ai)

=

0,

0 for j obviously have Ef I ci (TI(aj) JTf (ai)) I,-, K, which but for 0 nothing j I,-, K. Conversely, when (Bc)j (&)i 0 for j I,-, K, by multiplying cj* (the complex conjugate of cj) and we

=

is

=

=

=

=

=

summing

over

j,

we

obtain ctL3c

=(.EK CiT,(Cv,) I EK I

K

which leads

us

solution

0 for the

Fj-1 ciTf (ai)

1

ciTf (ai))

=

0,

As the existence of

a unique 0.) equation Bc 0 is possible only when detB --A 0, the linear independence of the functions is assured by the condition detB :A 0. Because the overlap matrix B is at least positivectBc > 0 for any vector c, all eigenvalues semidefinite, that is (Tf JTf)

c

=

to

=

=

=

p of B become

Tf (a K) ar e real, positive when the functions Tf (a,), linearly independent. The positiveness of M is understood as follows: For the eigenvector c :A 0 corresponding to the eigenvalue JL, we have the relation pctc As the basis functions Tf are (Tf Ifl. linearly (ai) independent, Tf is not identically zero because otherwise we can make !rf EK ciTf (ai) vanish identically with c =A 0, which contradicts the i= I assumption of the linear independence of the basis functions. Thus (TfITf) is positive and of course ctc is positive, so the eigenvalue y has to be positive. We can also state that if there exists a vanisl-iing eigenvalue of B then the basis set JTf(a1),..-,Tf(aK) is linearly dependent. The solution of the eigenvalue problem, pc(A), gives us an ....

=

=

BcG l

orthonormal set:

=

3.1 Variational

principles

1: C(ii.') Tf (C'j),

01.1

25

(3-11)

2

i=l

where

c(l')

c(") tc(/")

is assumed to be normalized to

=

1. In many

practical problems it may happen that B has one or several very small eigenvalues. Then the eigenstate corresponding to the small eigenvalue has very large expansion coefficients cil") If this occurs then a - Flj,. small error in the matrix elements of R or B can lead to a larger error in the solution of Eq. (110). When the ill condition mentioned above does not occur, the generalized eigenvalue problem (3.10) can be solved safely. The eigenvalues q

(i

est

1,

=

...'

K)

eigenvalue

are

arranged be

El may

a

in

increasing order el :! good approximation to

62

:

the

The low-

...

ground-state

energy El if the state space V_T

>

(Ek

(Ek -

-

E)2

E)',

jai 12.

(3.28)

which proves the theorem.

The Weinstein criterion guarantees that there is

an

eigenstate

whose energy is in the interval [E G-, E + o-] but does not indicate which one. In case E is sufficiently dose to the ground-state energy, -

the theorem E-

G-

i=l

N

IV

function of the form

r-

Or

)

OV,ij (r)

(r

+ r=ri

Or

)r=ri-rj (3-55)

spherically symmetric potential the operator r-ar- reduces simply r-4-. If the calculation of matrix elements of the. operator WA is not dr

For to

a

difficult,

the virial theorem

When the

potential

W

can

be used

depends

as

on a

parameter A,

(9

(!PIWAI(P) Here the

=

((filWA

+A

in the Coulombic we

case.

obtain

d

WI-P)

-

A

dA

(3.56)

E(A).

theorem is used to express the expecW in terms of the energy eigenvalue E(A). In a

Hellmann-Feynman

tation value of

A'&

molecular system X may be a set of parameters which stand for the positions of nuclei and if W consists of only the Coulomb potentials

-W simply reduces WA + A-bX plication

of this relation.

to -W. See

Complement

8.2 for

an

ap-

4. Stochastic variational method

approach to the variational solution of quantumproblems is to diagonalize the Hamiltonian in a state space spanned by some appropriate functions TV (a-,) Tf (a2) Tf (CeK). The applicability of this approach is, however, very limited, because the diagonalization may not be feasible if the dimension K of the state space is very large. This is typical in many-particle problems

The most direct

mechanical bound-state

7 ...I

I

such

as

the Hubbard model

or

the nuclear shell model. This method

approach", because one sets up a basis in a well-defined way, e.g., by using a complete set of states that contains no parameters, and then obtains the energy by a diagonalization. Another possibility is basis optimization, which is actually designed to avoid the problem of the huge basis dimension in the direct approach. In this case one specifically selects the basis states that are really essential to get the energy and the wave function of the syscan

be called

a

"direct

certain accuracy. It is obvious that this selection may be state-dependent: Some functions might be adequate to describe the

tem to

a

others would be

appropriate to approximate an excited state, particularly when the ground state and the excited state have different spatial extensions. The stochastic variational method uses this second route by selecting the most appropriate basis functions in a trial and error procedure. but

ground state,

4.1 Basis

some

more

optimization

It goes without saying that the quality of the variational approximation crucially depends on the choice of the basis functions. Our

primary aim here is-

choose basis functions 1.

They

can

be

IV-particle problems. For this that meet the following requirements:

to solve

easily generalized

for


an

N-body system.

we

will


40

2. Their matrix elements

3.

They

are

analytically calculable. easily adaptable to the permutational symmetry are

of the

system. 4.

They

are

flexible

enough

to

approximate

even

rapidly changing

functions.

important condition (3) is non-trivial because the permutational symmetry looks very complicated when expressed in terms of the relThe

ative coordinates to be used.

A

possible

choice for the basis functions

tions is the correlated Gaussian of

fillfilling the

above condi-

Eqs. (2.23)-(2.25):

-1 N-1

1

c

exp

Ax)

NE E Aij

exp

i=1

xi

-

xj

j=1

IV

expf -2

E

aij (ri

-

(4.1)

rj

i>i=l

equivalently, aij are nonlinear paxameters are spherically symmetric. To take into account non-spherical states, the basis function has to be multiplied by an appropriate orbital angulax function. In addition to the spatial degree of freedom, the particles may have other degrees of freedom, such as spin and flavor, and thus the function has also to be multiplied by suitable trial functions in these additional spaces. These functions may bring other parameters or sets of quantum numbers. These sets of quantum numbers (e.g., total and intermediate spins, orbital angular The matrix elements

Aij

or,

of the basis. These functions

momenta,

etc.)

be considered

can

often be referred to

as

trial function will be

as

channels in the

given

in

discrete parameters, and will following. (More details on the

Chap. 6.)

The actual form of the basis function is not very important at this stage. The above discussion serves to draw the reader's attention to the fact that the basis function

depends

on

many

and discrete parameters. The parameters define the

linear, nonlinear shape of the basis

function and determine how well the variational function space contains the true eigenfunction. To find the best possible solution, one has to

optimize the paxameters. To have a crude guess of how much optimization amounts to, let us consider an N-particle syswith the simplest basis function (4.1). This correlated Gaussian

work the tem

has ea,r

N(N

-

1)/2 parameters, and, by assuming

combination of K

functions,

-we

face

an

that

we

need

a

lin-

optimization problem of

4.1 Basis

K(N(N 1) /2) -

the number of

41

paxameters. Tb is number increases quadratically with

particles. By taking

N

=

4 and K

end up with 1200 parameters. The main problem of the minimization of

case,

optimization

=

200

as a

typical

we

nipresence of local minima. A local function reaches

a

minimum in

a

There

function is the

om-

point where the

finite interval of variables and the

number of such minima tends to increase

of the

a

minimum is the

exponentially

with the size

are plenty problem. optia function, and these optimizations can be divided into two categories: the deterministic and the stochastic optimizations. A deterministic optimization moves downwards on the slope of the function according to a certain well-defined strategy. There exist many elaborated algorithms (conjugate gradient, Powell (direction set), etc. [141) and they are deterministic in the sense that, starting from a given point, they always reach the same (local or global) minimum. The drawback of these techniques is that they are time consuming and tend to converge to whichever local minimum they first encounter. The solution in these cases may not be the global minimum but a local minimum. These methods are sensitive to the starting point and are

of different methods for the

mization of

unable to search further for

a

better solution after

a

local minimum is

reached.

Stochastic optimizations address the problem of finding the global large number of undesired local minima

minimum in the presence of a by making random steps [15,

161.

One can, for

example, start deterstarting points and then pick up the minimum of these. Numerous strategies have been developed in the last few years, e.g., simulated annealing [171, genetic algorithm [18] etc. The simplest (and actually not very economical) stochastic optimization is a random search where one picks up random points and tries to find the minimum. This may not sound very sophisticated, but the optimization with a random trial and error procedure seems to be the most efficient one. In some cases it would be nearly impossible to apply other strategies than simple random trials. To reduce the load of optimization, an alternative is to shift the burden of minimization from a large number of parameters to a smaller number of more sensitive, "tempering", parameters. Several such possibilities have been explored, e.g., geometric progression [19, 20], random tempering [21, 22, 231, and Chebyshev grid [241. The common property of these methods is that they use a "grW in the parameter space. The grid is given by some ad hoe rule and each point of ministic searches from several random


42

grids defines a basis function. The grid can be defined by some simple functions which may depend on some additional- parameters to be optimized. The number of parameters contained in the functions which define the grid is chosen to be much smaller than that of the original function to be optimized. For example, in the case of a three-particle system, each of the basis functions has three parameters: All, A12 A21 and A22 or a12, a13 and C923- Some COnVenient the

=

choices for basis parameters

are

shown in Table 4.1.

Table 4. 1. Choice of parameters. In the random tempering p is an index to a prime number that is used to generate pseudo-random numbers.

denote

See

Eq. (4.8) for the

Geometric

All

=

notation
.

progression -2

(aiqk-i) (a2q2-1) I

A22

=

A12

=

(k

mi)

(k

M2)

(k

I,- mi.)

(k

1:

(k

1

(k

17

mi)

(k

17

M2)

-2

k

0

Random tempering a12

=

OL13

=

a23

=

exp(d, < k,p > +d2 < kp + I > exp(d3 < k,p + 2 > +d4 < k,p + 3 > exp(d5 < k,p +4 > +d6 < k,p + 5 >

,

...

...

M2)

M3)

Chebyshev gTid All

=

A22

=

A12

=

( a2tan ( altan

0

7r

2k-1

2

2 mi

7r

2k-I

2

2M2

4.2 A

distribute

certain number of basis functions to

a

part of the

wave

function and then

the basis elements in

4.2 A

regions

one

43

practical example

might

approximate the bulk

increase the

density of

where finer resolution is needed.

practical example

point of the previous section, let us consider an example for basis optimization. The example will at the same time provide insight into some other aspects of the methods as well. We consider the system of a positron and two electrons, Ps-, used in Chap. 1. The energy of this system has been very accurately calculated by various To illustrate the

approaches and

it has been found to be -0.262005 in atomic units

given in a.u. throughout this chapter unless otherwise mentioned.) The quality of the different optimization strategies used in this section can be judged by comparing their results with this value. Let the positron be labelled I and the electrons labelled 2 and 3. With the electron spins coupled to S 0, the orbital of the trial function that the antisymmetry requires part ought to be symmetric with respect to the interchange of the electron coordinates. The wave function is thus expanded as

(a.u.). (Energy

and

length

are

=

K

1

Tf

=ECk(1+P23) exp (-2-;

,Akx),

(4.2)

k=1

exchange operator P23 is introduced to assure the symr2 and X2 rl metry requirement. The coordinates are x, matrix 2 with three 2 and x Ak is a symmetric (rI + r2)/2 r3, nonlinear parameters All, A12 and A22. The wave function Tf has 3K parameters to be optimized. The linear parameter Ck is determined by solving the generalized eigenvalue problem (3.10). To reach the accuracy required in atomic physics, the minimum basis size of a three-body problem is at least K 100, as will be shown later. That means that even in this simple example we would have 300 parameters. This is already almost beyond the capability of most of the computer codes for optimization and this "full" optimization is out of the question for larger systems. The fact that the optimization of a large number of parameters is not feasible is just a small part of the problem. To have an efficient optimization one needs fast function evaluation as many times as required. There are two steps which are where the

=

-

=

-

-


44

necessary to calculate the energy expectation value: The calculation of overlap and Hamiltonian matrix elements and the diagonalization.

the In

a

fall

required

optimization,

one

time increases

(a K3-process)

as

has to recalculate all matrix elements K2) and to

at each calculation of the energy. This is

load

for

(the

the Hamiltonian

rediagonalize

a

considerably

small system. heavy computational Instead, one can try a partial optimization. One can, for example, fix the parameters of all basis states but one. In this partial optimizaeven

a

only one row (column) of the Hamiltonian and the overlap -nn atrix (the required time is of order K). Let -us assume that the Hamiltonian has been diagonalized over the fixed basis states. Then, in the successive step, only one row (column) is changed. As has been shown in Theorem 3.5, after the N x N diagonalization there is no need for an extra diagonalization to solve the eigenvalue problem on the (N + I)-dimensional basis. Consequently, the computational time required by the partial optimization is only a small ftaction of that of the full optimization, and, moreover, only a small number of paxameters (three in the present example) has to be optimized. tion

has to be recalculated

4.2.1 Geometric

First let can

us

try

to

progression

use a

grid defined by

a

define three different sets of Jacobi

which two

particles

are

connected first:

Sects. 2.4 and 7.6 and in these systems to be

given

Fig. by x('), X(2)

and

us

X(3)

(23)1, (31)2,

and

(12)3.

See

denote the Jacobi coordinates The trial function is

IT

assumed

in the form K

3

(P)

qf

Ckl P=1 k=1

with the

2.2. Let

geometric progression. One coordinates, depending on

exp(--l (-P)A(P)x(P))0(11)00(x(P)), 2

(4.3)

1

angular function

0(11)00 (X)

=

[Y1 (XI) X Y1 (X2)100 I

-

(-1)1-M

(4.4)

-: 2= ,+1Y1Ta(X1)Y1-m(X2)-

Here p denotes the arrangement channel and Y1. (,r) is

a

solid spherical

harmonic

YIM M

=

7"YM (P),

(4.5)

4.2 A

where Yl,,, (i

)

polynomial

a

is

a

spherical harmonic.

practical example

The solid

45

spherical harmonic is (see Complement

of degree 1 in the Cartesian coordinate

A() is always chosen to be diagonal. Therefore, the k spherical part of Eq. (4.3) is considered a special case of Eq. (4.1). (See also Complement 8.4.) The trial function of type (4.3) is used in the so-called Coupled Rearrangement Channel Variational Method [20, 25, 26]. The basic idea of this approach is that, by taking into account 6.2).

The matrix

the different Jacobi sets, one can introduce various correlations. This method has been used with great success especially for Coulombic

three-body problems [201. Because of the symmetry of the trial

function,

(2) C

(3) =

k1

The 2

x

cki

we

A (2)

may A

-

requirement imposed

on

the orbital part

that

assume

(3)

(4-6)

"k

k

diagonal matrix A(P) 1, (k K) has two nonlinear k and they are taken as a geometric progression

2

=

...'

rameters

(Ak(')),,

(P)

ai(p) (qi ) In

principle

one can use

k-1

(i

=

1, 2).

pa-

(4.7)

different geometric progression parameters,

ai(P) and qi(p), corresponding to the different arrangement channels and the Jacobi

that

is, they would depend on p and i in order even depend on the angular momentum 1. For simplicity, in this example we use the same parameters, a and q, for all. arrangement channels and all sets of Jacobi coordinates. The main reason for using the geometric progression as parametrization is clear: The number of parameters of the basis is reduced to just 2 (a and q), so the optimization is simple. Another reason for the choice of this specific parametrization is that the overlap integral of the basis functions can be easily controlled by a choice of q and thus the danger of the linear dependence of the basis functions can be avoided. (See Sect. 3 for the danger arising from the linear to

coordinates,

get better convergence. They might

dependence of the basis functions.) The disadvantage of the use of these diagonal matrices is that, without having some polynomials like the scalar product of Eq. (4.4), the energy does not converge to its accurate value. As is shown in Table 4.2, one reproduces only the first two figures (E -0.2618530) =

of the

ground-state energy without the polynomial part, i.e., by taking 0 in the expansion. only I

=


46

Table 4.2. The energy of Ps- in different energy is -0.262005. E

arrangement channels. "Exact"

(a.u.)

Channel

Partial

(23)1 (23)1

1

1 =0,2

-0-2392726

(31)2+(12)3 (31)2+(12)3 (31)2+(12)3

1

=

0

-0.2609626

1

=

-0-2619622

1

=

0,1 0, 1,2

(23)1+(31)2+(12)3 (23)1+(31)2+(12)3 (23)1+(31)2+(12)3

At this moment

=

1

=

1

=

1

=

a

wave

-0.2068096

0

-0-2619717

0

-0-2618530

0, 1 0, 1, 2

-0.2619804 -0.2619816

comment is in order. As the

spherical harmonies

angular functions, complete may think that the partial-wave expansion using only one of the Jacobi coordinate sets form

set for

a

one

would be sufficient to represent the wave function. The convergence as a function of I is, however, very slow if only one set of the Jacobi slow convergence in the partial-wave expansion will also be studied in detail in Complement 11.2.) E.g., the energy calculated with the (23)1 channel using the partial waves of

(The

coordinates is used.

up to I

2 is

=

just -0.2392726, which

of the three-channel calculation

(E

higher than the energy -0.2618530) using only I 0.

is much

=

=

Since the lowest threshold of Ps- is the Ps+e- channel at the

en-

ergy of -0.25, this result indicates that this single-channel calculation cannot bind the system. Moreover, the expansion with I has for prac-

tical purposes to be truncated to low values because the calculation of the matrix elements of high I waves generally becomes more time

consuming. Table 4.2 shows that, by combining the expansion Jacobi coordinate sets, the energy converges faster and few terms of the partial-wave expansion are needed. By

in the

the first

only using

10

grid

points for A(') k be

a

600

=

(=

we have found the optimal values of the parameters to 2.6. The variational energy obtained with this 0.06 and q =

2

x

10

x

10

x

with

computational load 180300

alized

(=

x

601/2)

one

repeated

a

calculation is -0.2619816. The

particular basis

set is the calculation of

matrix elements and the solution of

eigenvalue problem

has to be can

600

3)-dimensional

a

gener-

of dimension 600. For the optimization this

few dozen times.

By increasing the basis

get closer and closer to the exact energy.

size

one

4.2 A

4.2.2 Random

practical example

47

tempering

Another

popular tempering method is random tempering [21, 22, 231approach involves the generation of (pseudo-) random numbers with the basis parameters defined by the following prescription: This

I

ak

1: dj < k, j >

exp

(k

=

1,

...,

K),

(4.8)

j=1

where
is a

pseudo-random numer, the fractional part of (k(k + 1)/2)V/P---(j) P(j) being a prime number in the sequence 2, 3, 5, 7,..., that is, P(I) 2, P(2) 3,... By using this tempering number I of one a formula, optimizes parameters, dj, instead of the original 3K parameters. A possible application of the formula for a three-particle case is given in Table 4.1. The origiTi of this formula is the following. One may assume that the ground-state wave function is an integral transform of some known with

=

function with

some

weight

=

function. The known function in

our case

is the correlated Gaussian and its nonlinear parameters are the integration variables. The simplest way to carry out the integral trans-

formation is to

the Monte Carlo method. The

quadrature points integration can be generated by the above formula. As this formula provides "good lattice points" for the integration, these nonlinear parameters can be thought to be adequate to represent the wave function in a variational approach. While the random tempering works in many examples in a superb way, one has to note that it has a serious problem: It often leads to (almost) lineaxly dependent bases. In our actual example, the best energy of Ps- with random tempering is -0.261872, and further hnprovement of this value was difficult due to the linear dependence. The parameters of the random tempering used are, in the notation of Table 4.1, ?nj 7, Tn3 5, di 4, d2 -8, d3 Tn2 4, d4

required

use

in this

=

-7, d5

=

-1, d6

The parameters

are

=

=

1, and the basis dimension nearly optimized.

-11,

di

=

p

=

=

is K

=

245.

4.2.3 Random basis

The above calculations have

suggested

that not all of the

grid points equally important grids give nearly the same energy. The explanation is simple: The basis functions are nonorthogonal to each other, none of them is indispensable; they are dense, that are

but the different

can


48

will compensate any of thein can be omitted because some others for the loss. This property of the basis functions and the success of

is,

tempering suggest the idea of a completely random distribution of the parameters. To illustrate this possibility, let us generate K 100 sets of basis parameters in the expansion (4.2). The eleraents of Ak are randomly chosen from a "physical" interval: the random

=

1

1
i=l ozij(ri -,rj

actually widely used in variational the completeness of Gaussians for

example [67, 351.

Of course, we have to keep in mind that the Gaussian is not economical in describing the asymptotic behavior of the wave function at

laxge

distances

(see Fig. 6-2). Moreover,

it does not

predict

a

correct

specific quantities such as the cusp ratio. See Comasymptotics and the cusp ratio could be well plement described with exponential functions. For example, welinow that the Rylleraas-type functions give very accurate results for Coulombic fewbody system (see, e.g. [68, 691). The correlated exponential functions are not, however, amenable to analytic evaluation of matrix elements for a system of more than three paxticles. This makes it difficult to use the exponential functions as a vahational trial function for a general N-particle system. value for

some

8.1. Both the

We extend the above argument further to define the correlated we consider separately two cases for two types

Gaussians. For this

of Hamiltonians. First in be

expressed

hn

terms of

case a

of

set of

Eq. (2.1) the Hamiltonian can independent relative coordinates

6.1 Correlated Gaussians and correlated

,7

x

(xi,

=

...

,

xN-1).

As

was

shown in


Eqs. (2.23)-(2.25),

77

it is then

con-

venient to express F in terms of x, instead of N(IV- 1) /2

interparticlefunction, a so-called

distance vectors, ri rj. An N-particle basis correlated Gaussian, then looks like -

I:

Type where A is

TI

(N

an

exp

=

-

1)

(_2 :TbAx)

x

(N

0 (x),

(6-1)

1) positive-definite, symmetric

-

matrix

of nonlinear parameters, specific to each basis element. As mentioned in Sect. 2.2, the matrix A with these properties can in general be written

6DG with the use of an orthogonal matrix G and a diagonal

as

matrix D with all

positive diagonal elements. The function O(x) is a generalization spherical harmonics Y of Eq. (6.56) to the many-particle case. More details will be given below. of the solid

The second each

is

is suited to the Hamiltonian

case

governed by both

(2.2).

The motion of

the

single-particle potential and the two-body Eq. (6.1) to include an independent motion of the ith paxticle around some point R, Here PI, is not a dynamical coordinate but just a parameter vector. In this particle

interaction. It is thus useful to extend

case we

but

do not need to

can use

the

use

the relative and center-of-mass coordinates

single-particle coordinates. Therefore we are led to

the

following type of correlated Gaussians which are often called correlated Gaussian-type geminals N

expf

XeXpf Here

r

-

aij (,ri

-

2

-

2

Y

R stands for

i(ri a

N

E j>i=, aj(,r, _,rj)2 A defined N

3=i+l

can

through Aij as

Type

11

:

Tf

=

x

Ri

)210(,r

be written

=

exp

-

T3)2

set of vectors

aij, the correlated

expressed

_

Aji

=

firl

R). RI,

-

compactly

-aij

(i

j=1 kij + 2q X:N is given by EN 1 UiXi With

kii

j=1

term

+2

=

v

Since the factors roles in lish the

-

1

xj2

=

(xi xj)

and

-

are

L, where the

pprop,riate.

a

vector

co

scalar, they play

v

j cient8

no

active

the rotational motion. Theorems 6.1 and 6.3 estab-

desci ibing equivalence

between the

angular

Eqs. (6.3) and angular momenta a-Te the parity oL the basis

functions of

(6.4)

under the condition that the intermediate

restricted

stated in Theorem 6.3 and that

as

L

given by (_ 1) through the angular momentum L. The basis function whose parity is given by (_l)L is called tohave a natural parityThe construction of a general angular function with unna ural parity must be based on the vector-coupled form of Eq. (6.3). Unfortunately there is no simple function analogous to Eq. (6.4) for the unnatural parity case. One way to construct the angulax function with unnatural parity is for L > 1 function is

OLM (X)

V2K [YL-1(V)

.

X

W]LM

with v

=

iix

and

Sij (xi

w

x

(6.16)

xj),

skew-symmetric matrix which satisfies Sij 0-, YL-I(v) must be replaced -Sji. For the special case of LI with YI(v). A slightly simpler angular function would be possible by introducing another vector v' as follows:

where S is

an

N

x

N

=

OLM(X)

=

V2K [YL(V)

X

VILM)

Vr

=

I?X-

(6-17)

Eq. (6.7) that in the case of K 0 both ki and k-2 are limited to zero and only the stretched coupling, namely 11 + 12 L, is allowed. See Exercise 6.1. With an increasing K value the possible values of partial waves 11 and 12 increase including the case of nonWe note in

=

=

stretched

coupling.

This

applies

to the

case

of many variables

as

well.

6.3

To increase K is thus

one

way to include

Generating ftmction

higher partial

waves

87

in the

calculation. The matrix A of the correlated Gaussian is often assumed to be

diagonal in order

In this

case

to reduce the number of nonlinear

parameters.

has to increase K when the contribution of

one

high

important. However, in the case where expected partial A is not diagonal, additional and important partial-wave contributions come from the cross terms of the exponential part of the correlated 0. E.g., the term exp(-Aijxi xj), when Gaussian even with K expanded into power series, contains many terms of the form (X,.X,)n, which can describe high partial waves associated with the coordinates relation xi and xj. This is easily understood by noting the following to be

is

waves

-

=

for

arbitrary

vectors

(a .,r)n

a

and b:

Bkja

2k

r2k

2k+l=n

E

Yj,,,(a)*Yj,,,(r) M=-1

Bkja 2kr 2k(_1)1,,

F21-+l[Yl(a)xYl(r)loo, (6.18)

2k+l=n

Eq. (6.45) and the addition theorem (6.54) for spherical harmonics. This implies that even the basis function with 0 is expected to be useful if a general matrix A is used in the K variational calculation. The calculation for the dtl-L molecule given in Complement 8.4 will clarify the point discussed in this paragraph.

which results from

=

6.3

Generating function

The calculation of the matrix elements becomes

generating

function for the correlated Gaussian.

simpler if one uses a In fact, the following

function g, which contains an auxiliary "vector" A (8 1, ..., SV), is found to generate the correlated Gaussians of both type I and type 11 =

conveniently:

g(s; A, x)

=

exp

To relate the use

the

following

(-2 ;Mx 9x).

(6.19)

+

generating

function g to the correlated Gaussian

formula

B kja2k+l r 2ky1M (,r)

=

fYjm (ol) (a

.

r

)2k+lda

we

6. Variational trial functions

88

f Y,.(

=

,

(

92k+l

Aa-r

dal

e

a)

OA2k+l

(6.20)

A=O

which is

easily proved by using Eq. (6.18). Then the vector-coupled product (6.3) can be generated from the factor egx of the function g as follows: By choosing si aiti with a unit vector ti we obtain =

N

-I bAx)

exp

YI,., (xi)

2

dii- Y ,,., (Fi)

B01i

Oai

g(a It; A, x)

(6.21) By a symbol alt we mean a "vector" such that given by aiti, where & (a, aiv) with ai =

I

(ti,

...,

tN)

is

I

a

The

x

N

-

- -

one-row

7

each component is a real number and

matrix of Cartesian vectors ti.

in the above

key point equation is Eq. (6.20) which relates the solid spherical harinonics to e ". Since Eq. (6.20) yields a more general term r2kyl., (,r) than just the solid spherical harmonics, it may also be useful to use a simpler relation, which generates just the solid spherical haxmonics. The relation (6.58) serves for this purpose. By choosing tj (1 T,j 2, i(I +,ri 2), -27-j) (j 1, N), we obtain =

=

...,

IV

exp

2

TbAx)

Y1,m, (xi)

N

a Ii-Mi

01i

I

H=1 Climi Oaili O-Fili-mi x

g(alt; A ,X)

1

t =(,-, 2,i(l+, j j j

U=11

---

2),-2-rj)

(6.22)

IN)

where

47r(I M) 1 (21 + 1) (1 + m)! -

Cim

=

(-2)111

-

(6.23)

Note that the vector ti satisfies ti-tj -2(-Fi-'F,j)2, particularly ti 2 0. This formula requires only differentiation in contrast to Eq. (6.21), =

6.3

where both differentiation and

integration

Generating

function

needed. To

are

89

couple

the

spherical harmonics to the desired function in Eq. (6.3), one has multiply Eq. (6.21) or (6.22) by q, and sum over x. To construct the correlated Gaussian-type geminals from g we note

solid to

2RBR

exp

=

exp

(6.5)

f

-2

1

iAx

-

2

BR; A + B, x)

---

(x

to

serves

+

-

R)B(x

-

Eq. (6.6).

f

-2 (x

Jc- Ax

-

2

-

R)

+

(x

-

R)j-

generate the function O(x

It is easy to show that

or

exp

9R) g(s

1 -

0('-R)

The factor

-

R)B(x

-

R)

-

(6.24)

R)

of

Eq.

I

IV X

IIIX,_Ri12ki y1irni (X,

-

Ri)

N

02ki+li

fj BkjIj f dtiYjj.j (ti) Oaj 2kj+1j

i=1

x

x

exp

(

-

-1 kBR

-

2

-J-tR)

g(alt + BR; A + B, x)

(6.25)

1

aj.=O,...,CXJV=O ItAr 1=1 it, 1=1-, ...,

or

fn (A, B, R, x) I

I

expf -2 j Ax 2(x -

-

R)B(x

-

R)j

xip i=1

N

3

anip

fj fj atipnip

i=1

x

-

Rip )nip

P=j

exp( 2kBR iR) -

-

P=I

g(t + BR; A + B,

x))

1

(6.26)

6. Variational trial fimetions

90

where

n

stands for the set of

In,,, n12,n13,...,nNj,nN2,nN31-

Tn the

Cartesian representation the parameter a plays no active role but the x,y, and z components of each vector ti, (tillti2lti.), serve to construct the function

O(x

-

R).

To construct the

vector-coupled product OLM(X) of Eq. (6.3), one has to sum over mi's with appropriate Clebsch-Gordan coefficients in Eqs. (6.21) or (6.22). Apparently this is a very tedious task particularly when the number of particles is large. The choice Of OLM(X) of Eq. (6.4) leads us to the following very simple equation which relates the correlated Gaussian to g. By choosAe with a unit vector e, we obtain for t2 tN ing t1 =

V

=

iiX

fKLM (u, A, x)

' =

BKL

f

=_

exp

YL M (' )

(_2I FcAx (

v

2KYLM(v)

d2K+L

g(Aeu; L WA-2_K+

A, x)

dL

(6.27) We

from

Eqs. (6.21), (6.22), and (6.25)-(6.27) that the are explicitly constructed from the generating function g. Depending on the choice of the vector s, g leads to different forms of the correlated Gaussians, when followed by suitable operations acting on s. These are surn-ma J ed in Table 6.2. The construction of fKLM (u, A, x) is simplest among others and it has a wide range of applications as will be shown in later chapters. The correlated Gaussian of Eq. (6.27) contains only the relative coordinates and the center-of-mass motion is dropped from the outset. Thus there is no problem arising from the coupling between the intrinsic motion and the center-of-mass motion. If one uses the singleparticle coordinatesr instead of x in Eq. (6.27), the coupling between them occurs in general and has to be taken caxe of appropriately in can see

correlated Gaussians

order to calculate the energy of the intrinsic motion. A suitable choice of A and u will, however, lead to the result that the center-of-mass motion in

separated from the intrinsic motion. This will be discussed Complement 6.4. Chap. 2 we discussed the linear transformation of the coordi-

can

more

In nates

x

be

detail in

induced

by tJae permutation

P. It is

important

to Imow the

6.3

Generating

function

91

Relationship of the two types of the correlated Gaussians to generating function g of Eq. (6.19). The symbol alt indicates a set of vectors f aiti amtN I. Bkj and Cl,,, are defined in Eqs. (6.9) and (6.23), respectively. Table 6.2.

the


-I.;v2

exp

Ax) rIN

i=1

(I rIN

1

P01-i f

-

i= 1.

2

i= I

i=1

x

Ylimi(Xi) ali-Mi

ali

I

rIN

jg(alt; A, x))

diiYiini(Z)

-!.,'cAx) IIN

exp

Y1 irni (Xi)

climi aaili ajli-mi

g(alt; A, x)

I

t =(I-, 2,i(l+, j j j

2),-2-rj)

i=O"ri=O (i=l,.. N) .,

ld Ax) IV12Ky f dgYLm( ,) (

exp 1 =

Correlated

Uixi)

i=1

,

x)

).X=O,e=je-j=3-


-UMx

-

(In,iv exp

.1 2

2

x

EN

=

d2K+L j,-X2K+L g(Aeu; A,

F3 K-L

exp

(V

J(V)

L IV

2

I

-

,

Bkjjj 2

(x

-

R)B (x

f dii-Yi

RBR

-

I fT7

R)

=,

Ixi _RZ12ki Y1 imi(Xi_ 14)

a2ki+li iM

JiR) g(a It

-

+

BR; A + B, x)

expf .1.7cAx I(x--RR)B(x R) I IIN JJ'=j(Xip =ffrff, lip-=, t-,77ri- jexp(-1:YZBR-!R) g(t+BR; A+B, x))

-j=O,jtjj=j

3

-

-

-

-

, i=-

2

2

3

a7"P

i=

2

p

P

x

ti=o

p

-

Rip )ni,

6. Vahational trial functions

92

effect of P

the correlated Gaussians. Since the correlated Gaus-

on

generated from the generating function, it suffices to examproperty of g due to P. By using the relation Tpx (see Eqs. (2.26) and (2.29)), we obtain a very simple result

sian is

ine the transformation

Px

=

Pg(s; A, x)

=

g(Tp_.9; Tp-ATp, x).

(6.28)

One

only needs to change the matrix A and the vector s appropriately. An important fact is that the generating function preserves its functional form under P. This is also true for a more general linear transformation T of the coordinates x, e.g., a transformation from one set of coordinates to another. Combining this fact with Eq. (6.27), we

obtain

a

very useful

property of the correlated Gaussian fKLM-

Namely for the transformation of Tx

TfKLM(u, A, x)

7--

=

Tx,

we

have

fKLM(TU TAT, x).

(6.29)

Thanks to this nice property one only needs to redefine the parameters A and u of the basis function to construct the transformed wave function. This

property plays

important role

an

in

evaluating

the matrix

elements. The

generating function (6.19) plays a key role in generating the and, moreover, facilitates the evaluation of the

correlated Gaussians matrix elements of

physical operators.

the formulation based

It is therefore desirable that

the

generating function In a many-body system as well. In extending the results to laxger systems of identical paxticles we need to cope with the symmetry adaptation one can use

of the

the

wave

function. Tn such

generating

function

were

a case

on

it would sometimes be useful if

expressed

in

an

"uncorrelated7 form of

the coordinates x, because the symmetry adaptation can then be shnplified by using the technique of Slater determinants or permanents. In fact to the

we can show that the generating function can be related product of the Gaussian wave-packets centered around -4 =

(R,,..., RN) through an integral transformation. Using the definition Eq. (6.51) of Complement 6.1, we can express the product of the

of

Gaussian

wave-packets

as

N

det-V

_'i

ORj (Xi)

)

4

1 exp

:iFx + 2

k_Vx

1 -

2

kFR) (6-30)

with

an

N

x

N

diagonal

matrix

6.3

-YJ

0

0

72

Generating ftmetion

93

0

...

(6.31)

0

'YN

A direct calculation using

1:Mx +

exp

2

2

x)

dx

-9A-18

exp

detA

following equation which relates product of the Gaussians of Eq. (6.30): proves the

2

g of

Eq. (6.19)

(6.32) to the

g(s; A, x)

(detr)3

4

expf 'g(r, Arisl -

-

(47r-) N (det(rT

2

N

g(_V(.V-A)-1s;A(F-A)-'FR)

X

'Y'

ORi (xi)

dR.

(6.33) Note that

A(r

From this

we see

A)-' r

r(r

A)

-1

r

F is

symmetric matrix. The function g in the right-hand side of the above equation depends on the integration variable R and serves as just a weight which is needed to convert the product of the Gaussian wave-packets to the generating function. Equation (6.32) is proved in Exercise 6.2. We have seen that the correlated Gaussians are all generated from the same generating function as summa ized in Table 6.2. Furthermore, the latter can be obtained by the integral transformation of Eq. (6.33) involving the product of single-particle Gaussian wave-packets. -

=

-

that the calculations

-

can

a

be reduced to the matrix

N-particle wave functions involving the product of the wave-packets. The width parameters -yi of the wave packets can be chosen arbitrarily. To choose uniform width parameter for all of them is most convenient if xi indicates the single-particle coordinate of the identical Particles, because then the permutational symmetry of the wave function is simply reduced to that of the "generator coordinates" R. Even when x denotes the set of relative coordinates, this elements of

Gaussian

6. Variational trial fimcdons

94

nice property

be used

by including the center-of-mass coordinate as [311. integral representation of the generating function has been successfully employed in accurate solutions of few-body problems [311 as well as in microscopic descriptions of nuclear systems can

shown in

The

in multicluster models

6.4 The

X.L M

[29, 30].

spin fanction

=

1

cosO -1 .1 (0),

2

Alternatively,

22

.1; 2

-IM) 2

.1 + sin# .1 22 (1),

may set up the

we

.1; 2

.1 2

M).

spin function with

(6.34) a

continuous

parameter such as that of Eq. (6.34) by an elementary method instead of using the successive coupling. Its merit is that the construction of the

spin function is simple, that the evaluation of spin matrix elements

is easy, and

V in

moreover

Eq. (6.34),

that

one can use

continuous parameters such

The construction is done

as

follows. The

Young diagram

spin function with spin S for the N-fermion system

[(N12)

+S

as

in the variational calculation.

(N12)

-

S].

The maximum

weight function

is

for the

[n+ n-I

with M

=

=

S,

XSS, must have n+ spin-up functions and n- spin-down functions. The number of terms distributing n+ spin-up functions among the N

particles terras:

is

NS

=

(,+). Therefore xss is expressed n+

as a sum over

these

6.4 The

spin fimcdon

95

NS

XSS (A)

=

E Ai ii)

(6.35)

-

i=1

characterizing the spin function, satisfy the condition, are not independent 0, where S+ is the spin-raising operator. Acting with S+Xss(A) S+ on XSS (A) yields n+ + 1 spin-up functions and n- 1 spin-down N functions in each term and thus leads to (n++,) independent terms.

ANs)

The coefficients

of. each other but must

=

-

Since the coefficient of each term must N

I

V (n+ 1) (nl+) +

1

=

vanish,

one

(2S + 1)N! + S + 1)! (IN (IN 2 2

has

-

S)!

(6-36)

independent parameters to specify the vector A completely. Here the last minus one of Eq. (6.36) comes from the normalization of the spin function. XSM is easily obtained from Xss(/\) with the use of the spin-lowering operator and thus denoted as XSM(A). The overlap of two spin functions is independent of M and sh-nply given by iV. The independent pa(XSS(A)IXSS(/X')) (Xsm(/\)IXSM(/\')) varied be /X can continuously in the variarameters needed to specify =

=

tional calculation. The ner.

isospin

exactly the same manthe spin and isospin parts in Eq. (6.35), leading to a

function is also constructed in

permutation P on reordering of the terms

The action of the

simply produces

a

linear transformation of the vector /X to another vector denoted

A(P).

96

Complements

Complements 6. 1 Nodeless harmonic-oscillator functions

as a

basis

In variational calculations for bound states it is very important to have a set of basis functions which can approximate square-integrable

functions to any desired accuracy. For a single-variable function one may use the well-known complete, orthonormal eigenfunctions of, e.g., the harmonic-oscillator

(HO)

Hamiltonian. Or

one may try to use functions which may not be complete mathematically able to cope with many problems flexibly and, from a practical

nonorthogonal but

are

point of view, accurately. The amine the

possibility

functions. We will

by studying

purpose of this

of nodeless HO functions the

see

performance

Complement

is to

ex-

set of such "basis"

as a

of the nodeless HO functions

how well

they approximate a given function f (r). The nodeless HO functions with angular momentum Im have

continuous

parameter

-Palm (T)

=

Nal

( V3)

14 I/r

exp

2

T

with

a

solid spherical harmonic

the normalization constant

(2 1+2 a"

a

a:

2)y7n(,VF

(6.37)

I/r

1

(see the following Complement). where

Nj

is

given by

2

2

Nal

Here

(6-38)

=

(21 + 1)!! is introduced to scale the

length. The Gaussians employed in Complement 8.1 to solve the ground state of the hydrogen atom are 0. The overlap of two nodeless HO functions is special cases with I simply given by v

=

(-Vaiml-Va-'im)

aa1

Na I IVa'I

2

2

(6.39)

_

=

.

+.,1)2

(a+a)2 2

2

We attempt to approximate a normalized function momentum ITn in terms of combinations of

fl,,, (r)

with

angular

K

Am (T) The ance

error

cirailm (T) of the

approximation

(6.40) is estimated

by calculating

the vaxl-

C6.1 Nodeless harmonic-oscillator functions

0-

2(f)

=

fI

cillil,,-,

,,

97

a

set of

Or)Idr.

parameters

(P.41)

jal, a2,..., aKI by

the trial and

may set them up

by SVM. Once they are selected, minimizing 0-2 to determine the ci's priate way

basis

2

fl (r)

One may choose or

as a

error

leads to

some

appro-

procedure a

linear

of the

equation

K

E(Failml-Vajim)cj =' (Faiiralfim)

(i

=

1,

...,

(6.42)

K),

j=1

andthena 2 is As

a

I

given by

test function

K

-Y:ij=iCi*Cj(-Vai1mIrajIm)-

fim(r)

we

first take up the HO

wave

functions

angular frequency w hy/m, where n is the number *,,Im (r) of radial nodes. We employ a phase convention such that the radial HO wave functions are all positive for r greater than the outermost nodal is needed to point. The overlap of the functions Fal (r) and calculate o-2. It can be obtained by using a generating function of the with the

=

..

HO functions

Ik

4

A(k,,r)

exp

2

+

Nf2-vk-r

2

1 VT

-

2

21 (6.43)

0,,jm(r)*P,,jm(k), n1m

polynomial of the complex Bargmann space variable k, is the Bargmann transform [76, 771 of the HO wave functions in a spherical basis and its explicit form is given in terms of the solid spherical harmonics of (4.5) [781

where

Pnjm(k),

a

:(2

P,,lm(k) where

Bn1 n -

1) n + 1)! (2n

(6-44)

(k k) ny1m (k), -

Bnj, defined in Eq. (6.9), is the coefficient needed to Legendre polynomials Pl(x):

express xm

in terms of the

xm

21+1

E

=

4r

(6.45)

BnIP1(

2n+l=m

Expanding

(6.44),

we

the

overlap (F ,lm (r) I A(k, r))

obtain

in terms of the

polynomials

Complements

98

(-PaimlOnlm) Obviously,

(2n + 21 + 1)!! (2n)!! (21 + 1)11.1

the

1

(1+ )n (

2n)+!!

-

approximation

in

-

a

Eq. (6.40)

2

2Va

a

1 +

(6.46)

a

becomes less trivial

the number of radial nodes of

as

1 f increases. For Onlm (r) wit"h n 2, the combination of merely a few (::5: 3) terms leads to very good 5 it is hard to make u 2 less than approximations, whereas for n 10-10 with a double precision calculation. This is so because all ai's tend to take a value of unity, reflecting the fact that such a HO function can be expressed by combinations of higher order derivatives of Falm(r) with respect to a at a 1, and the optimization procedure tends to construct such derivatives out of -Tal,,, of almost equal parameters. One thus has to deal with an almost singular overlap matrix ((Faiiml-Vajim)) to obtain the solution of ci's. In stead of approximating each of the 0,,,. (r) with different sets of =

or

=

=

1),

(j:IV n=O 0r2(0nIM))/(N-.L

minimized the average value, (U2 )N with a single set of ai's. The values of the

ai's,

we

follow

a

=

geometric progression and

cessive terms

were

determined

of ten nodeless HO functions The next test

example

k

ai's

were

assumed 11-o

its first term and the ratio of

by Powell's method [141.

yielded (a 2)8

=

0.6

x

suc-

A combination

10-10 for I

is the shifted Gaussian defined

=

0.

by

I

( 3)4 (_IV(,r2+S2))jI(I/'Sr)y V

(r)

=

Fj

exp

-

where the function

ii(x)

(6.47)

7 IM

2

7F

is the modified

spherical

Bessel function of

the first kind

ij(X)

=

F '7-x'I,+.! (X) 2

X2k+1

1: (2k)!! (2k + 21 + 1)!!'

(6.48)

k=O

and where the normalization constant Fj is 4e

2

'us

with

F,j

2

(6.49)

2

To get the normalization constant, 00

fo

x

e-ax21, (bx) 1, (ex) dx

which is valid for Rea

>

Gaussian

wave-packet

I =

2a

0, Rev

shifted Gaussian because it is

we

>

have used the formula 2

exp(b +c2),,'(bc)'

-

4a

1. The function

closely

centered around

2a

(6.47)

related to the s

(6.50) is named

single-particle

C6.1 Nodeless harmonic-oscillator functions 3

,ps' (r)

99

(6.51)

11

-

2

T

through

basis

1 exp,

-

as a

the relation

4V'7-r Os

(6.52)

1M

Fj 1ra

Here

use

eVr_5

is made of the

=

evrscos'o

=

equation

E(21 + 1)ij(vrs)Pj(cosO)

(6.53)

1

with the rem

angleO between r and spherical harmonics

s, and the well-known addition theo-

for the

I

4w-

Pi (CosO)

1:

=

21+1

yjm(i )Yjm( )*

ra=-l

=

4-x-(-I)' /2_71-+I

lyl(p)

X

(6.54)

YIM100.

8 The radial paxt of the shifted Gaussian is hence peaked at r shifted the be to therefore would It approximate challenging -

/_2, Iv.

Note that large s in terms of the combinations of A e-211 A of of in terms as Eq. (6.43) expressible (NF2'91 T). 0,v(r) The overlap between the functions F,,la(r) and 7p,jn(r) is

Gaussian with

1

2

is

(r,im 1,0SW

v/-2-

N,, 1 Fs

1(1 + a)'+' 2

P(_ 1+a

ex

V I/

(6.55)

Assuming again that the ai's follow a geometric progression, 0-2 was minimized by Powell's method for a given value of ;. For the case 2 0 it is possible to make o- < 10`0 with ten nodeless HO of I =

functions for the shifted Gaussians of up to :! 10. All ai then tend to take values close to each other for large,;, and this requires a very

equation (6.42). Increasing the number 2 of nodeless HO functions gives us even smaller o- values and widens precise solution of the

the range of maximum approximated well.

linear

(;

value in which the shifted Gaussian

can

be

examples strongly suggest that the nodeless HO functions can approximate square-integrable functions to any desired accuracy, though the number of nodeless HO functions needed depends The above

Complements

100

on

the

shape of the

test function. A remarkable

shows that the linear function

approximated to high

r

itself, defined in

accuracy in terms of a

in

example given

[66]

finite range, can be combination of Gaussians a

e-ar2 ) We stress two remarkable points of the nodeless HO functions. One a combination of a few nodeless HO functions can approximate

is that

the shifted Gaussian is easy to make

2 a

even

with

less than

nodeless HO functions. In

an

a

large

value of

10-1-0 with expansion

a

;.

For

example, it only 15 HO functions,

combination of

in terms of

would need many more terms to obtain the same accuracy. This indicates the flexibility of a nonorthogonal Gaussian "basis" compared one

to the

orthogonal basis such as the HO functions. Another point is that there are many, possibly an infinite number, of sets f a,,..., aKI which approximate a given function equally well, even though the number K is fixed. This is what we have experienced in the above examples. We display graphic illustrations of Gaussian expansions for different functions with I 0. The most appropriate nonlinear 0, m paxameters ai are determined by optimization, while the linear ones ci are given by the solution of the least square equation (6.42). To point out the dependence on the number of Gaussians, we use different numbers of terms in the expansion (K 5, 10 and 20). In the first example we approximate the HO function 05oo(r) by =

=

=

Gaussians. This function is smooth but oscillates and

asymptotically falls

off like

a

(it

has five

nodes)

Gaussian function. As shown in

10 and K 20 Gaussians give a perfect fit to the 6.1, K function, so these curves are practically indistinguishable. =

In

=

the second

to fit

Fig.

exact

an exponential function e-', the wave function of the ground state of the hydrogen atom. The asymptotics of this function is quite different from that of a Gaussian. To approximate the asymptotic part of this function one needs many terms of Gaussians as is illustrated in Fig. 6.2. By increasing the number of Gaussians one has better and better agreement in the asymptotics. After a certain distance, the Gaussians fall off much more rapidly than the exponential function. In many practical applications, especially for bound states, however, one can always use enough Gaussians to reach the required accuracy. We note that the Gaussian fit gives a poor value (zero) for the derivative at the origin (the exact value is -1) as will be discussed in Complement 8.1. In the next example we try to approximate the absolute value fimction f (r) 12.5-rl. The Gaussian expansion, again, does a pretty

f (r)

=

=

case

we

attempt

C6.1 Nodeless harmonic-osefflator functions

as a

basis

101

10

5

0

-5

-10 0

4

2

6

8

r

0, m 5, 1 0) and its Fig. 6.1. The harmonic-oscillator function (n approximations by Gaussian expansions. The solid curve is the (UM3.ormalized) harmonic-oscillator function f (r) V 500, and the dotted, dashed and 5-, 10- and 20-term Gaussian expansions. long-dashed curves are the K The 10- and 20-term. expansions are practically indistinguishable from the -

=

exact

curve.

100

IT

-C

20

10-40

--60

10

40

10

0

10

30

20

40

50

r

Fig. 6.2. The exponential function and its approximations by'Gaussian exp(-r), expansions. The solid curve is the exponential function f (r) and the dotted, dashed and long-dashed curves are the K 5-, 10- and 20-term Gaussian expansions. =

=

102

Complements

good job (Fig. 6.3), would expect

of further Gaussians

by the inclusion

and

an even

one

better fit.

3

2

V

1

0

0

1

2

I

I

.

3

4

5

r

Fig. 6.3. The absolute value function and its approximations by Gaussian expansions. The solid line is the absolute value function f (r) 12.5 -,rl, and the dotted, dashed and long-dashed curves are the K 5-, 10- and 20-term Gaussian expansions. =

=

The approximation for the step function, f (r) 1 if 7- < 2.5, 2 if r > 2.5, is less impressive (Fig. 6.4), but it goes without f (r) that it is not trivial to fit that function. One can improve the saying =

=

fit

by increasing

the number of

Gaussians,

be pretty slow. The last example shows that

adequate

in

some cases.

Let

us

with Gaussians. This function is it simulates the behavior of

a

try

but the convergence

might

Gaussian expansion may be into

approximate f (,r)

practically

a wave

zero near

function of

a

the

12 =

r

2

e-T

origin, and

system -with very

core. The Lenard-Jones or other hard-core potentials such a function. Figure 6.5 shows that the Gaussians produce may generally give a good fit to this function. By scrutinizing the inner

strong repulsive

part of the approximation

reality

(see Fig. 6.6)

the Gaussians do not

produce

one can

exact

see,

however, that

zero near

the

in

origin, but

C6.1 Nodeless harmonic-oscillator functions

as a

basis

103

3

2

C, -

0 0

Fig.

6.4. The

1

2

3

4

5

step function and its approximations by Gaussian expansions.

The solid line is the step function f (r) = I if r < 2.5, f (r) and the dotted, dashed and long-dashed curves are the K 20-term. Gaussian

2 if

r > 2.5, 5-, 10- and

expansions.

the

approximate function oscillates around the exact one even for 20 Gaussians. This oscillation may be very unpleasent: To tame the hard core potential, we need a wave function which is effectively zero near the origin. The oscillating function leads to numerical problems, which makes it rather difficult to obtain the solution of interaction has

a

very

It is

problems

where the

strong hard-core part.

to know the

important completeness of the basis functions in L (square integrable functions), H1 and H2 (first and second Sobolev) spaces because the bound state in quantum mechanics is traditionally 2

formulated in L 2, and the mathematical solution of the

equation is formulated in H' and H2

spaces.

(The

Schr8dinger

space HI is

a

set of

functions whose derivatives of up to the mth order are all square integrable. Note that the kinetic energy operator requires the second order derivative of

a

basis

variational method

ple

in

[791,

and the

function.) are

The convergence properties of the Ritz demonstrated in H1 and H2 spaces, for exam-

completeness

in L 2 is not sufficient to

guarantee

the convergence of the Ritz method. The proof of the completeness in the space of L 2 and in the spaces of H' and H2 is presented in

[801.

This work proves that any function

can

be

approximated

to any

Complements

104

200

150

100

50

0

-50

-j

-100 0

1

2

3

4

5

r

Fig.

expansions. dashed

The solid

curves are

10 and 20 term

curve

the K

=,r

and its

is the ftmction and the

=

expansions

12e_,r2

approximations by Gaussian dotted, dashed and long5-, 10- and 20-term Gaussian expansions. The

f (r)

6.5. The fimction

are

practically indistinguishable from

the exact

curve.

0.002

.

.

.

.

.

i

.

.

I

I

0.001

0.000

-0.001

-0.002 0.0

0.1

0.2

0.4

0.3

0.5

0.6

r

in Fig. 6.5, but the inner paxt is 5- and 10-term Gaussian expan i n scope and not drawn here.

Fig.

6.6. The

curves

same as

with the K

=

magnified. axe

The

out of the

C6.2 Solid

prescribed

accuracy with

spherical

linear combination of

a

that the number of Gaussians in the the parameters of the Gaussians

harmonics

105

Gaussians, provided

expansion is sufficiently large and

are

appropriately

chosen.

6.2 Solid

spherical harmonics The solid spherical harmonic Yl,(r) r1YI,(i6) of Eq. (4.5) is a 0. 0, of the Laplace equation, V2f (,r) solution, regulax at r The irregulax solution is given by r-1-'YI (f). The solid spherical harmonic is a homogeneous polynomial of degree I in the Caxtesian =

=

==

..

coordinate:

21+1 +

rlyl.(,P)

(1 + Tn)! (I

47r

-

m)!

p! q!

p

-

q)!

pq

(_X+i )p (X _iy)q Y

X

2

where p and q p + q

A+

Aiuigi,

(7.65)

Complements

144

where p is either I or 2. As B is equal to A + leads to the

following

X,

the

change B,

of A

as

given by Eq. (7.65)

modification of

P

B

---+

B+

(7.66)

Aiuigi.

Therefore the calculation of the matrix element for the above

change

of A results in the calculation of the inverse and determinant of the

special form of matrix, B + a, Aiuigi. When the modification is given by just one term of the form AuO, the Sherm an-Morrison formula can

be used to obtain

Aub)-'

(B

+

det

(B + Aub)

B-1

-

-AB-1u,6B-1,

1 + AbB-lu

(7.67)

and

(I + A,5B-1u)detB.

(7-68)

See Exercise 7.2 for the derivation of the Sherman-Morrison formula.

advantage of these formulas is appaxent: By knowing B-' and detB one can easily calculate the right-hand side of the equations, and the A dependence is given in a very simple form. To change A, therefore The

there is

no

need for the evaluation of inverses and determinants of the

modified matrix B

(which would require (N

get the desired results by

_

1)3 operations),

but

we

simple multiplication and division. When the modification of B is given by Eq. (7.66) in fiL11 generality, then the inverse and determinant can be calculated by using the Woodbury formula [141, which is the block-matrix version of the Sherman-Morrison formula. If one wants to change a few of the aij's or one-column (and one-row) elements of A at the same time, the summa ion in Eq. (7.66) has to be further extended appropriately. a

Exercises

145

Exercises

Eq. (7.12).

7. 1. Derive

Solution.The the

integral

case

as

of

a

=

0 is

1 00rnCar2-brdr

dn 1

(_I)n x

case

of

a

>

0,

we

may

get

dn

(-I)n

=

0

By putting

In the

simple.

follows:

=

dbn 2

bl(2,Va-),

1000e-ar2-brdr I-exp ( ) erfc( \,Fa

dbn

V

b

4a

a

the above

2

(7-69)

-

equation becomes

00

fo

rne-ar2-brdr

=

V,-(_I)n

(

1

2

- fa- )

n+1

n

nt

k=O

(n

-

k)! k!

A (X)

gn-k (X) ,

(7-70)

where

fk (X)

=

A (X)

=

(_X2) (eXp (X2))

eXp

(k)

eXP(X2) (erfC(X))(k).

(7.71)

fk (x) defined Eq. (7.13). Remembering that the

It is easy to show that

above is

in

Hermite

not

(n)

(-I)n eX2 (exp(_X2)) difficult to check that gk(x)

.(X)

=

7.2. Derive

and

can

(erfc(x))

be

equal to the one given polynomial is given by (1)

expressed

_X21 V/,-X-,it is -

=

as

-2e

in

Eq. (7.13).

Eqs. (7.67) and (7.68).

Solution. Let X represent u,&. Then the inverse as follows:

(B + AX)-1

may be

calculated

(B + AX)

-1 =

(I + I\B-'-X) -'B-1 CO

=

j:(-A)n(B-IX)nB-1.

(7-72)

n=O

special form of X, XB-'X reduces to cX, where the constant factor c. is given by OB-'u. Therefore repeated use of this relation leads to the follwoing result: Because of the

Exercises

146

(B -I.X)n B-1

B-'(XB-'XB-'X

=

d'-'B-'XB-1

=

By substituting

this result into

(B + AX)-' which is

nothing

=

......

(n

Eq. (7.72),

>

we

B-'XB-17

-

(7.74)

Eq. (7-67).

but

given by Bij

P(A)

obtain

I+Ac

+ Auivj, let

us

(B + AX)

=

-

=

ao +

aj.X +

-

-

+

-

(i, j)

with its

suppose that the determinant is a The function P(A) is a polynomial

P(A) det(B+AX). degree (N 1) and can be expanded

function of A:

of at most

(7.73)

1).

A

B-1

To calculate the determ In ant of the matrix

element

B-'X)B-1

as

follows:

aN-O:I F-1,

(7-75)

where the coefficient ak is calculated by k!p(k) (0). The rule of dif0 ferentiating determinants leads us to the conclusion that ak =

for k ao

=

2 because of the

>

P(O)

=

special

form of the matrix X. We have

detB. The coefficient a, is obtained

as

B1,

B12

B, N-1

V1

V2

VN-1

Biv-, I

BN-1 2

BN-1N-1

follows:

N-1

Ui

a,

N-1 N-1

IV-1 N-1

E Y uivj,6ij

E T ujvjdetB(B-')jj

i=1

i=1

j=1

j=1

(,DB-1u) detB.

(7.76)

Aij is the (i, j) cofactor of the matrix B and use is made of the detB (B-1)jj. Thus P(A) is eq ial to (I+ADB-'u) detB, relation Aij Here

=

which is what

we

7.3. Calculate the

want to derive.

one-body density matrix for the generating function

g: Pi W,

r)

(g(sf; X, x) IJ(,ri

-

xN

-,r'f)) (,5(,ri

-

xN

-

r) lg(,s; A, x)).

(7.77)

147

Exercises

(2.12) shows,

Solution. As Eq. W The

ri

-

can

xN

be

expressed

as

EN-I 1=1

argument made for the nonlocal potential in Sect. 7.5 in exactly the same way. The can, therefore, be applied to this case only necessary change is to replace w('j) with 0). The density matrix When the wave function pi (r, r) takes the same form as Eq. (7.46). has the proper symmetry for a system of identical particles, the onebody density matrix does not depend on the suffix i.

w, xj.

Reproduce the nonrelativistic kinetic energy formula (7-5) by using the formulation presented in Sect. 7.6. 7.4.

Solution.

By using Eq. (7.50), the matrix element of the nonrelativistic

kinetic energy operator in the center-of-mass system becomes IV

N

2

(FA'(-X)JE

IFA(X))

2Tnj

=

(FA, (x) 11:

(i)

(7rN-l)

IF",(X))

2m,

3h2

(7.78)

(FAI IFA)

2mici

3h2 / (2mx) for the nonrelativistic kinetic energy is f (x, m) obtained by replacing Vlfh-2q2 + M2 in Eq. (7.61) with h2q 2/ (2m). To perform the summation over i in Eq. (7.78), we need to know the special matrix element M9 N-1: k

where

=

N

VM

(U(N) )kI (TT(i)

-

us

vectors of

Uj(N)

TnI , M2 ,

....

(1, 2,

N)

can

0) i

recall that

...,

mv

can

Uj and

EE

(7-79)

i

kN-i

Let

-I

IN-I

be obtained same

time

with the

i times. Rom this construction it is easy to

be obtained from

constructed from

N)

Uj('

(N)

in

Uj'

but

the column vectors. Then

the N column

by rearranging the masses the operation cyclic permutation

at the

according to

by rearranging

exactly

the

see

that

same manner as

by rearranging the row obtain (see Eq. (2.6))

U(') i U

M is

vectors instead of

we Mi

for

1: k i

for

I

M12---N

UW

(7-80) I N-1

-(I

'i

-

) M12---N

=

i.

Exercises

148

Using

this result in

0 for k

Eq. (7.79) and noting the relation

N enables

VW siimma

IV

i

n

3h2 -

2mici

over

3eT 2

which

was

(

Eq. (7.78)

(A(A

+

X)

can

easily

j(kv)

U

as

(7.81)

(UJ)ki

be done to obtain

-'A!A)

being

(7.82)

defined

kinetic energy from the total kinetic energy. This exercise serves as an indirect evidence for the

formulation

-

by Eq. (2.11). This agrees with Eq. obtained by explicitly subtracting the center-of-mass

with the matrix A

(7.5),

i in

I

to obtain the desired matrix element

(U'(V))ki

-

kIV-1

The

us

E'V I=

given

in Sect. 7.6.

validity of the

8. Small atoms and molecules

examples for the application of the method to atomic and molecular systems. The interaction between the charged particles is the Coulomb force. The long-range character of this force makes the solution of the few-body problems difficult, especially in the case of scattering. We restrict our attention to bound states, where This

chapter

contains

many different methods have been elaborated in the

These calculations

provide an

excellent

past decades.

possibility to test the

of our method. Relativistic effects in atoms

perimental precision of today, which calls

are

efficiency

withi-n the reach of ex-

for very accurate theoretical

calculations.

8.1 Coulombic

systems

The systems of charged by the masses (MI jn2 ,

particles

can

be characterized and classified

mN) and the charges (qj,q2,...,qN) of distinguish systems formed by either equal (unit) or unequal charges, and depending on the masses of the particles one can classify the systems as adiabatic and nonadiabatic ones. ITI the unit charge systems of more than two particles the constituents form a molecule and the binding energy depends only on the mass ratio(s) of the particles. Atoms are good examples for systems with unequal charges. The distinction between the adiabatic and nonadiabatic cases is dictated by the possibility of a simplified treatment in the former case. In the adiabatic case the masses of a group of particles are considerably heavier than those of the rest. The classical examples are the molecular ion (see Complement 8.2). In these H2 molecule and the H+ 2 cases the electrons move faster, while the protonic frame may rotate and vibrate by moving considerably slower. This physical picture is expressed in mathematical form as the Born-Oppenheimer approximation, where the electronic motion is first calculated by assiimin the constituents. One

....

can



150

form

CkA

JeXP

2

i AkX) jVk 12K+L YL

M

(f,-I") X Slus

k

(8.1)

UkiXi

Vk

The operator A is introduced to impose proper symmetries on identical particles (antisymmetry for fermions and symmetry for bosons) of the system. In most calculations the index K is assumed to be zero or at most zero or one. In the case of the PS2 molecule of Sect. 8.3, K is set

equal

to

zero.

The details of the calculations

can

be found in

some of the system presented in this chapter. The vari[33, 84, 851 ational parameters included in each basis function are the elements of

for

the -matrix

Ak and the coefficients

Uki, which define the

vk. Since the Hamiltonian used in this

chapter

global

vector

commutes with the

spin operator, the variational trial function can be chosen to 'have a definite spin value S. The value of S influences the symmetry of the orbital part of identical particles. Therefore, possible S values 'have to be tested in general to obtain the ground state. For treating the adiabatic system of small molecules in Sect. 8.4, we use combinations of the generating function g itself as the variational trial function.

8.2 Coulombic

three-body systems

8.2 Coulombic

13-16

figures, that the solution of

lem is

one

151

three-body systems

the Coulombic

three-body prob-

of the most useful benchmark tests to compare different

methods. The accuracy pursued in Coulombic cases is not of purely academic interest, but highly motivated by the high precision of the For

experiments.

example,

the fine-structure

splitting of the

state of the Li atom has been measured with

(parts

per

million).

include relativistic

including

an

the

18 22p2pj

accuracy of 20 ppm numbers one has to

experimental and quantum electrodynamics (QED) corrections, To

explain

terms of the second- and third-order in the fine-structure

require very high accuracy. In addition to the accurate reproduction of the energies, another motivation is that in the variational calculations, even though the energy is good, other physical observables might be less accurately determined. The increased accuracy of the energy, as we will demonstrate, eventually will lead to constant, which

very accurate values of other observables

as

well.

three-body general (m+M+m-)-type stability C A B system with unit charges has been thoroughly explored [861, and the like requirement for stability can be phrased as an empirical rule: In masses" ac[861. charges have to be borne by equal or nearly equal cordance with this prediction, systems such as H- (pe-e-), H2+ (ppe-), Ps-(e+e-e-), HD+(pde-), HT+(pte-), or tdg- are all bound, while (ppe-) or (pe+e-) are most probably not. Here p, p, d, and t are proton, antiproton, deuteron (2 H) and triton (3H) respectively. In the latter case, the particles with opposite charges form an atom, which does not bind the third particle. Some systems, e.g., the muonic molecules such as (ttft-) or (tdft-), remain bound even for L =-;,k 0 orbital angular domain of

The

a

"

,

momenta.

by

A second group of the Coulombic three-body problems is formed systems where not all the particles carry unit charges. The rep-

resentatives of this category are the helium atom (ae-e-) and the helium-like ions, where a stands for the 'He nucleus. These systems often form bound states with L atom

(ape-)

angular

has been observed

momentum states

We have

challenged

(L

our

=

=,4

0

as

[871 30

-

well. The

antiprotonic helium

and studied in very

bigh orbital

40) [88, 891.

method to calculate the

energies of

some

of these systems. In the calculations to be presented the number of basis functions superposed is mostly limited to be modest because our

primary

purpose is to demonstrate the overall

performance of the

cor-

related Gaussians and not to compete with well-established methods sharpened for these systems. We will increase the basis dimension to


152

reach

high

accuracy

only

in

cases

where such calculations

sidered to be important. The results of the calculations

are con-

axe

listed in

Table 8.1. Table 8.2 presents the parameters used in the calculations. The results are compared with those of other (mainly Hylleraas-type or

correlated

hyperspherical haxmonics basis)

calculations. Our results

reasonable agreement with other calculations. In most cases a basis size of K 200 was used in the SVM. The precision of the axe

in

=

results

be

improved by increasing the basis size as can be seen on the selected examples of Table 8.1. In these cases we reach almost the same precision as the other methods. The calculation extends to nonzero orbital angulax momentum states as well, including an e,,c 31 state of the antiprotonic helium atom or the a,mple for the L 3pe bound of the H- ion. These nonzero orbital angustates slightly can

=

lar momentum states the

as

well

as

L

=

0 states have been

investigated

by using global representation. The recovery of the results of other calculations (which are based on several different represenvector

tations of the orbital part of the

usefulness of the

wave

function)

convinces

us

of the

global vector representation. See Complement 8.4 for tdl-t molecule with the global vector

calculation of the

compaxative representation. a

To illustrate the convergence of the energy and the expectation values of average separation distances, the results at different basis sizes

are

tabulated in Table 8.3 for Ps-.

increasing the itself. One

Actually

accuracy is the conventional

can

the limit of further

precision of the computer

notice that at the basis size of K

100 the energy the first four figures of =

is accurate up to six decimal digits, but only the separation distances can be precisely determined.

the basis size, the virial coefficient falls below

high

accuracy of the calculation that all the

calculation

are

By increasing

10-'0, showing by the digits

of the reference

recovered.

Quite a few very accurate methods have been developed to solve the three-body Coulomb problem. It is very difficult to go beyond their precision. This is especially true for the methods which have been elaborated for a given system only, incorporating as much physical intuition as possible into the trial function or into the solution. In contrast with these methods, we use the same trial function, which is of Gaussian nature and therefore it is not tailored to Coulomb problems at all. Still, as the examples prove, one can get a sufficiently good solution in a unified and automatic way knowledge about the systems to cope with.

in

without

a

priori built-

The real power of the

8.2 Coulombic

three-body systems

153

Energies of different Coulombic three-body systems in atomic basis dhnension. See Table 8.2 for the constants which are the K is units. used in the calculations denoted superscripts a, b and c. Table 8.1.

System

State

K

SVM

Other method

K

Ref.

Ps-

Ise

600

-0.262005070226

-0.2620050702328

1488

COI-i-

600

-0.527710163

-0.527751016523

850

200

-0.1252865

-0.1252865

90

200

-112.97300a

-112.9730179a

200

-110.26210a

-110.2621165

a

ttA

Ise 3pe Ise 1PO

200

-105.98292

b

UIL

1D'

-105.982930b

2250

ttA

IF'

200

-101.43131'

-101.43'

200

MIL MIL td[t

IS'

200

-111.36444a

-111.364511474a

[691 [90] [911 [23] [231 [921 [931 [231 [20] [20] [941 [951 [681 [68] [681 [68] [68] [681 [681 [681 [681 [891 [96]

H-

UP

b

b

500 500

1400

IP,

200

-108.17923

1D'

200

-103.40849a

-103-408481a

1566

'He

Ise

600

-2.9037243769

-2.903724376984

700

He

Ise 3se

200

-2.9037242

-2.903724372437

100

He He He

He 'He

-108.179385

2662

200

-2.1752291

-2.175293782367

700

IP0 3po

200

-2.1238423

-2.123843086498

700

200

-2.1331635

-2.133164190779

700

1-D' 3D'

200

-2.0556201

-2.055620732852

700

700

-2.055338993068

-2.055338993337

700

IF' 3F'

200

-2.03125504

-2.031255144382

700

He

200

-2.03125506

-2.031255168403

700

He

IGe

200

-2.02000069058

-2.020000710898

700

He

3Ge

200

-2.02000069062

-2.020000710925

700

VHe+

L=31

300

-3.50760

-3.50763486

1728

'Li+

Ise

300

-7.279913

-7.279913

He

Table 8.2. The constants used in the calculations. The of the electron

mass.

energy in eV.

Set

a

masses are

m,,=7294.2618241, mp=1836.1515. R.

Set b

Set

c

Mt

5496.918

5496.92158

5496.918

Md

3670.481

3670.483014

3670.481

MI,

206.7686

206-768262

206.769

2R,,

27.2113961

27.2113961

27.2116

is the

in units

Rydberg


154

Energy and different separation distances for the (e+e-e-) three-body system as a function of the basis dimension K. The virial ratio 71 is defined by q 11 + (V)/(2(T))I. See Eq. (3.50). Atornic Table 8.3. Coulomble

=

units

are

-E

(,r2+_) (7-2

1 2

used. SVM

SVM

SVM

Hylleraas

(K=100)

(K=200)

(K=600)

[691

0.26200465

0.2620050648 0.262005070226 0.2620050702328

5.489

5.48962

5.489633252

5.489633252

8.548

8.54856

8.548580655

8.548580655

6.958

6.95832

6.95837

6.95837

9.65284

9.65291

9.65291

1

9.652

0.46

77

approach will

x

10-4

be

0.34

more

number of particles is most

as

The

or more

expedition

nium molecule

to

10-6

0.54

x

10-10

x

10-10

where the

than three and the method still works al-

while the other methods need tedious efforts.

particles

larger Coulombic systems

(PS2).

0.23

highlighted in the following sections, more

easily as before,

8.3 Four

x

starts with the

positro-

This exotic molecule consists of two electrons

and two

positrons. The possibility that the PS2 molecule or in general electron-positron system consisting of p positrons and q electrons form a bound system was originally suggested by Wheeler [97], and this question has been extensively studied since then. The existence of the positronium. negative ion Ps- (p 1, q 2) has experimentally been observed [981. The binding energy Of PS2 was first calculated by Hylleraas and Ore [991. To date, it has not been observed yet due to the dffficult experimental circum tances, and this fact has intensified the theoretical interest in solving- this Coulombic four-body problem [65, 100, 69, 101-1041. Actually the positron-electron annibil i n limits the

=

=

the lifetime of Ps2 to few nanoseconds. I-n

obtaining the

solution for the

PS2 molecule, it is useful to note PS2 is invariant with respect to the charge permutation, that is, the exchange of positive and negative charges. that the Hamiltonian for

The trial function should therefore either remain

unchanged or change

sign under the charge permutation operation. The ground PS2 turns out to be even under the charge permutation. its

state of

8.3 Four

or more

particles

155

The convergence of the energy Of PS2 against the increase of the basis dimension is shown in Table 8.4. The fact that the best vari-

ational calculation

[103]

in the correlated Gaussian basis is

already

at the basis size of 400 illustrates the power of the random

surpassed trials.

Table 8.4. The total energies (in a.u.) of the ground state and the bound excited-state of the PS2 molecule in atomic units. K is the basis dimension.

PS2

Method

(K (K SVM (K SVM (K SVM (K SVM (K CG [1031

SVM SVM

=

=

=

=

=

=

100) 200) 400) 800) 1200) 1600)

(L

0)

=

PS2

(L

=

1)

-0.516000069

-0-334376975

-0.516003119

-0.334405047

-0.516003666

-0.334407561

-0.516003778

-0.334408177

-0.5160037869

-0.334408234

-0-516003789058

-0-3344082658

-0-5160024 -0.51601+-0.00001

QMC [1041

possible existence of bound excited-states of the PS2 molecule [84, 85] by taldng all possible combinations of states with L 0, 1, 2 spins. By 0, 11 2,3 orbital angular momenta and S a bound excited state we mean such a state that cannot decay to any dissociation channels. The results of the calculation were negative in I (with negative parity) and all but one case. In the case of L of a second bound-state the existence S 0, the calculation predicts We have examined the

=

=

=

=

of the PS2 molecule. This unique bound-state has been found to be odd under the charge permutation operation. The convergence of the excited-state energy is shown in Table 8.4. Figure 8.1 summa izes the energy spectra of the bound states made up of two positrons and two electrons

together with the

relevant thresholds.

One may ask the question of why the second bound-state cannot decay to two Ps atoms in spite of the fact that it is located above the threshold of Ps (1S) +Ps(IS). Since the total spin of the state is to zero, it

that

they

dissociate into two Ps

can

have

equal spins

(ground state)

and the relative orbital

atoms

coupled provided

angular momentum

1. (Recall that the Hamiltonian preserves spin, between them is L orbital angular momentum and parity.) However, this is apparently =

impossible

equal spins are bosons and their L. Consequently, the PS2 molecule

because two Ps atoms of

relative motion

can

only have

even

156


0

Ps(2P)+e++e-0.1

-0.2

-

Ps(lS)+e++e-'

Ps-+e+

Cd -0.3

IP0

-

Ps(IS)+Ps(2P)

>1 0)

-0.4

-

W

-0.5

-Ps(IS)+PS(IS)

-0.6

Se

PS2

-0.7

Fig. 8.1. The energy spectrum are given in atomic units.

with L

=

I and

of electron and positron systems.

Energies

negative paxity cannot decay into the ground states of

two Ps

atoms, that is, the lowest threshold of Ps(IS)+Ps(IS). Since the energy of this L 1 state is calculated to be E -0-3344 a.u. (see =

Table

=

8.4), (-0.3125 au.) of Ps(IS) + Ps(2P), this state is stable against autodissociation into this which is lower than the next threshold

channel. The

binding energy of this state is 0.5961 e-V from this second threshold, about 40% more tightly bound than that of the ground state Of PS2 whose binding energy is 0.4355 eV from the lowest threshold. are

The expectation values of vaxious listed in Table 8.5. In the

quantities for the PS2 molecule

PS2 molecule we deal with antiparticles, so the electronpositron pair can annihilate. The second bound-state may decay either by annihilation or by an electric dipole transition to the ground state [851. The most dominant annihilation is accompanied by the emission of two photons with energy of about 0.5 MeV each. To have an estimate for the decay width due to the annihilation we have substituted

8.3 Four

or more

particles

157

Properties of the ground and excited states of the PS2 molecule. positrons; are labelled I and 3 and the electrons are 2 and 4. Because (r14) of charge permutation symmetry, some equalities hold, e.g. (r12) Table 8.5. The

=

(r32)

=

(r,34).

Atomic units

PS2

(L

are

used.

0)

=

PS2

(L

=

1)

(r12) (r13)

4.4871530

7.56881891

6.0332070

8.8575844

(r212

29.112633

80.173836

46.374735

96.085514

2

r13 3

( r12) 3 (r 13 )

253.04611

1041.3251

443.85244

1226.7955

(412)

2807.2718

15612.112

4 13

5202.0371

17939-574

(r 21) (r.3' -2) r12 -2) r13

0.36839693

0.24082648

0.22079007

0.147244820

0.30310361

0.16081514

0.073444303

0.032230158

(1'12'TI3) ('r12 'T14) (5(rl2)) (5(rl3))

23.187368

48.042757

(r ) 12

5.9252651

32.131079

0.0221151

0.0112091

0.0006259

0.00014591

(V2) 1

-0.258001894

-0.16720401

(V1 V2)

0.1307732538

0.091656853

(VI'V3) 11 + (V) /(2(T))

-0.0035446132

-0-016109693

*

the

0.3

10-9

x

probability density

(J(r12))

,

of

into the formula

rannihi

=

47

x

10-6

electron at the

position of

a

positron,

[1021

(MC622 ) 2hc(TIjJ(rj-T2)jTf) 62

=

an

0.36

4ir-(hc)

4

hcaOI(5(rI2))i

(8.2)

equal to J0 (Tf 15(rl r2) ITf) with ao Roughly speaking, the Metime is inversely probeing portional to the probability. The Metime due to the annihilation is estimated to be 0.44 ns. This is twice that of the ground state (0.22

where

(,S(r12)), given

in a.u., is

-

the Bohr radius.

ns). dipole transition from the excited state emits one photon with energy of 4.94 eV. The decay width Idjp&,, for this transition The electric


158

is calculated

through the reduced transition probability B(EI) dipole operator D.:

electric

16v

Tdipole

':--

(E) 3B(EI;

I-

0+),

-->-

he

9

for tile

(8-3)

where I

B(EI; 1-

--+

0+)

=

)7 I (Oind I DI, ,

_

M) 12

(8.4)

1

with 4

qi 1ri

Djz

X4

-

I Ylp (ri

-

X4)

IIIere X4 is the center-of-mass of the

(4.94 eV)

tation energy

B(El)

dipole

P-92 molecule and E is the exci-

of the second bound-state. We calculated the

value and obtained

electric

(8-5)

-

B(EI)

=

0.87e2a 2. The lifetime due 0

transition has been found to be 2.1

ns.

The

to the

branching

dipole transition is thus about 17 % of the total decay rate. Therefore, both branches contribute to the decay of the excited state of the PS2 molecule. Its lifetime is finally estimated to be about of the electric

The excitation energy of 4.94 eV found for PS2 is different by 0.16 eV from the corresponding excitation energy (5.10 eV) of a Ps 0.37

ns.

atom. This difference

to be

large enough to detect its existence, e.g. in the photon absorption spectrum of the positronium gas. Before discussing the spatial distribution of the PS2 molecule, let us recall that the average distance (r+-) between the positron and the electron is 3 a.u. in the ground state of the Ps atom, while it is 10

a.u.

first excited state. The root-mean-square radius

in its

(ri

-

X'j

)2),

culated to be 3.61

surprising a

system of

of the

a.u.

5.66 a.u., 1.5 times

if

seems

larger

one assumes a

ground

The

rms

PS2 molecule

is cal-

radius of the second bound-state is

than that of the

ground

state. This is not

that the second bound-state is

Ps atom in its

(spatially extended)

state of the

(rms),

ground

state and

a

excited state. To check the

essentially

Ps atom in its first

validity

of this

as-

sumption, we have restricted the model space to include only this type of configurations. This can be achieved by a special choice of the uki parameters of that is, the

Eq. (8-1). The energy converged to -0.323 a.u., Ps(1S) +Ps(2P) system with zero relative orbital angular

momentum forms state of the

a

bound state with energy close to that of L

PS2 molecule, therefore this configuration

is

lik-ely

=

to

1

be

8.3 Four

the dominant

configuration in this molecule.

uration, the Ps- + e+

or

Ps+ + e- with L

=

or more

There is

particles

159

second

config-

a

I relative orbital

motion,

intuitively may look important because two oppositely charged particles attract each other, but it is barely bound (E -0.315 a.u).

which

=

1 state The average distances in Table 8.5 show that in the L the two atoms are well separated. In fact we can estimate the root=

mean-square distance d between the two atoms

by

2

d2

I'l +7'2

7'3 +7'4

2

2

(2(r12 ) 2

4

(,r213)

+

-

) (8-6)

2(rl2*rl4)

The symmetry properties of the PS2 wave function are used to obtain 6.93 a.u. the second equality. Using the values of Table 8.5 yields d =

for the L

=

I excited state and d

state. One cannot

give

a

direct

Arij

The correlation function defined

gives

(TfIJ(ri

=

more

-

-

(r,2

of the

(,rj)2

ground ground or

is

0

laxge.

by

(8.7) on a

system than just various average

quantity can be calculated for the correlated Gaussians

by using Eq. (A.30) =

=

=

r) ITf)

detailed information

distances. This

Tf with L

rj

for the L

a.u.

geometrical picture

excited state because the variance

C(r)

4.82

=

or

(A.136).

0, C(r) becomes

monopole density.

a

ground-state wave function of only r, which is called the

For the

function

For the excited-state

wave

function with L

=

1,

monopole and quadrupole densities. and the electron-positron electron-electron the 8.2 displays Figure correlation functions r 2C(r) for the ground-state of the PS2 molecule. The peak position of the electron-electron correlation function is shifted to a larger distance than that of the electron-positron correla-

C(r)

consists of the two terms of

tion function. The latter has much broader distribution and reaches

farther in distances

compared

to the

corresponding function

of

a

Ps

atom.

Figure

8.3

displays

the electron-electron and

electron-positron

cor-

relation functions for the second bowid-state of the PS2 molecule. As I state consists above, the correlation function for the L of the monopole and quadrupole densities and their shapes depend on the magnetic quantum number M of the wave function. Of course the M-dependence of the shapes is not independent of each other

mentioned

=


160

0.020

0.015

Cd 0.010

0.005

0.000 0

Fig.

4

8.2. The correlation functions

molecule. The solid dashed

the

2

curve

r

6

8

r

(a.u.)

2C(r)

for the

12

ground

14

state of the

PS2

denotes the electron-electron correlation and the the electron-positron correlation. For the sake of comparison, curve

electron-positron correlation function for

dotted

10

a

Ps atom is drawn

by

the

curve.

but is related

by the Clebsch-Gordan coefficient. See Eq. (A.136). quadrupole density is contributed from only the P wave of the electron-positron relative motion, while the monopole density is contributed by both S and P waves. Figure 8.3 plots the correlation funcThe

tions for both

(a)

M

=

0 and

(b)

M

=

1. As the correlation function

axiaUy symmetric around the z axis and has a reflection symmetry with respect to the xy plane, the correlation function sliced on the xz plane is drawn as a function of x (x > 0), z (z > 0). The electronelectron correlation function has a peak at the point corresponding to the average distance of 7.57 a.u. The electron-positron correlation function has two peaks reflecting the fact that the basic structure of the second bound-state is a weakly coupled system of a Ps atom in the L 0 state and another Ps atom in the L 1 spatially extended state. The peak located at a larger distance from the origin is due to the P-wave component of the PS2 molecule. The hydrogen and positronium molecules can be considered as members of the same family as both are quantum-mechanical fermio-nic is

=

=

four-body systems of two positively and two negatively charged identical particles. But they are at the opposite ends of the (M+M+m-m-)-

8.3 Four

x

x

(a.u.)

(a.u.)

x

z

(a.u.)

z

(a.u.)

0 x

or more

(a.u.)

161

particles

z

(a.u.) z

(a.u.)

(a.u.)

Fig. 8.3. The correlation fimctions rC(r) in atomic units, raultiplied by one thousand, for the second bound-state of the PS2 molecule. The z comI for 0 for (a) and M ponent of the orbital angular momentum is M (b). Drawn on the xz plane are the corresponding contour maps. =

=


162

type Coulombic systems called biexcitons biexciton and two

(or

biexciton

holes,

molecule),

is observed in

a

a

or

bound

variety

excitonic molecules. The

complex of

two electrons

of semiconductors

[105, 106].

See also Sect. 10.1. The biexciton molecule is characterized mass

ratio

o-

=

m/M. Apart

by the connection, however, their

from this

properties are radically different, e.g., H2 is an adiabatic but PS2 is a highly nonadiabatic system. Moreover, while in the case of the H2 molecule many bound excited-states have been observed experimentally and later studied theoretically, in the case of the PS2 molecule only the ground state and the unique excited state discussed above have so far been predicted theoretically. See also Complement 8.3 for the stability of the biexciton molecule. Figure 8.4 displays the dependence of the binding energy of the biexciton molecule on the mass ratio a m/M. The changes of the binding energies in the ground state (L 0) and the excited statues with L 1 and negative parity is si-milar. Both the ground and excited states become less bound by changing the mass ratio from H2 to PS21 though the binding of the excited state decreases to a somewhat lesser =

=

=

extent. The energy of the transition from the excited state to the

ground state is also shown in this figure. This transition may take place in an external field, for example. By increasing the mass M of the positively charged paxticles tbowa,rd infinity, one arrives at the energy of the C I H,, 2p-x state of the H2 molecule. This state is formed by an excited H-atom and a groundstate HI-atom. Consequently, a statement similar to the case of the PS2 molecule is valid for the biexcitonic molecule: The second bound-state of the biexciton molecule is of

a

ground-state

dominantly formed by

exciton and

an

L

=

an

interacting

pair

1 excited-state exciton.

The rule that the Pauli

principle forbids odd partial waves between identical bosons also applies to the biexciton with L I and negative The second bound-state of the biexciton molecule cannot decay parity. to two ground-state excitons. A somewhat similar situation exists in the 3p, state of the H- ion as well, where its second bound state cannot decay due to the parity conservation. By changing the mass ratio in that (M+m-m-) system, however, this kind of state disappears for oI and the Ps- ion is known to have only one bound state. =

=

Tables 8.6-8.8 show tem

our

results for various other Coulombic sys-

.

The tivated

investigation of the stability of positronic atoms has been nioby the use of positrons as a tool for spectroscopy (positron

8.3 Four

or more

163

particles

0.15

0.10 CU

0.05

0.00 0.0

0.4

0.2

0.6

0.8

to

M/M

Fig.

8.4. The

mass

ratio

o-

binding

=

m/M.

state, and the solid

energy of the biexciton molecule as a function of the curve is the binding energy of the ground

The dotted

curve

is that of the first excited state with L

Table 8.6.

curve

Energies of different Coulombic four-body systems

I and

=

is the energy difference, multiplied negative parity third, between the first excited state and the ground state.

The dashed

by one

in atomic

units. K is the basis dimension.

System

State

K

SVM

Other method

K

Ref.

PS2

Ise IP0

800

-0-516003778

-0.516002

400

[102] [107] [1071 [1071 [1031

800

-0.334408112

600

-7.478058

-7.47806032

1589

Li

Ise 1PO

1000

-7.410151

-7.410156521

1715

Li

'D'

1000

-7-335520

-7.335523540

1673

'HPS

Ise

1200

-0.7891964

-0.7891794

PS2 Li


164

Table 8.7.

Energies of different Coulombic five-body systems

in atomic

units. K is the basis dimension. The Li- energy with K = oo is the extrapolated one [1091, where E = -7.500577 is given by multiconfiguration

Hartree-Fock calculations with K

=

2997.

System

State

K

SVM

Other method

K

Ref.

Be

Ise Ise

500

-14.6673

-14.667355

1200

600

-7-50012

-7-50076

00

[1081 [1091

Li-

Ise Ise

(27r+, 3,-1-) Li +

e+

Table 8.8.

Energies

200

-0-5493

1000

-7.53218

of different Coulombic

six-body systems

in atomic

units. K is the basis dimension.

System

State

K

SVM

(37,-+,37r-)

IS' Ise Ise

300

-0.820

600

-7.73855

1000

-14.692

Li + Ps

Be+e+

annihilation

spectroscopy) is whether

in condensed matter

physics.

An

intrigu-

or a chemically stable system containing a ing question positron or a positronium could be formed in the various targets. This

not

be answered

only by a sophisticated calculation or experiment because the mechanism responsible for binding the positron to the neutral atom is the polarization potential present in the atom+e+ system. The boundness of the hydrogen positride (positronium hydride) HPs was predicted theoretically by Ore [99] in 1951 and it has recently been created and observed in collisions between positrons and methane [1101. The properties of HPs is discussed in [851. The use of the SVM proved for the first time that the positronic lithium (Li+e-r [1111 and the positronic beryllium (Be+e+) [1121 are stable. We see from the tables that the positron separation energy of the positronic lithbun is 0.054 a.u. Below the Li+e+ channel the Li++Ps channel is question

can

open and the

energy of the

positronic lithium is only 0.0022 a.u. against the dissociation into the Li++Ps channel. A calculation has to be accurate at least to 10-3 a.u. to answer the stability of the

binding

positronic litbium. Likewise, the positron separation energy of the

Be+e+ system

typically

is

only

about 0.01

0.025

a.u.

a.u.

one

Due to the

tiny binding energies of

has to. be able to reach

high

accuracy

8.4 Small molecules

165

6-particle systems. A naive picture of these systems is that the positron orbiting around the neutral atom slightly polarizes the negative electron cloud, and the positron is bound by the resulting in these

attraction.

(as a fall N-body solution) for the investigation of the stability of much larger systems (e.g. Sodium plus positron) To extend the method

question. One can, however, try to use a "frozen core approximation?'. In this approximation the positively charged core is considered to be passive (its polarization is neglected) and the problem is is out of

model space where the single-paxticle orbitals are orthogonal to the core orbitals. One has to solve the modified Schr6dinger

solved in

a

equation of the form

(H + AP)Tf

=

with

ETI-

P

(8-8)

Oi) (Oi iGoccupied

produces wave functions that are orthogonal to the core orbitals provided the positive constant A is large enough. The projection operator P is an example for the nonlocal potentials discussed in Sect. 7.5. See also Complement 11.4. One can validate this approximation by comparing it to the "exact" fuU N-body calculation for Li+e+. This approximation turns out to be very accurate, reproducing the first six digits of the result of the full calculation [1121. Assuming that the accuracy holds for larger systems, one seems to find the stability of positronic sodium (Na+e+) [1121. which

8.4 Small molecules As it case

was

mentioned at the

demands

a

special

beginning

of this

assumed to take combinations of the form 1

g(s; A, x) where

Aij

and

functions. The x.

exp

=

Note that

s

(_2

=

is

a

Mx +

f8li S21

"generator x

chapter, the molecular

treatment. The variational trial function is

-7

(6.19)

9x),

SN-11

coordinates"

(8.9) are s

parameters of the basis

are

chosen

conforming

to

set of relative coordinates. Our aim here is

to calculate the energy of the

system which

can

be

directly

com-

pared to experiment without recourse to the adiabatic treatment like the Born-Oppenheimer approximation. Each basis function includes !V(IV 1) /2 + 3 (IV 1) parameters to be optimized. These parameters -

-


166

describe various correlations. The matrix A describes the electronic

correlations and motions, while the generator coordinate s makes the function flexible and allows us to represent several "peaks" of

wave

the

distribution

density

the holes

when, for example

in the

well separated and the electrons them. around 0 the function By choosing s are

=

hydrogenic limit, "atomic orbits"

axe on

(8.9)

at the

'has its

maxhnum

origin and this limit is suitable around a1, when m/M the paxticles with nearly equal masses are moving equally fast. At the hydrogenic limit, when the motion of the heavy particles are very slow compared to the light ones, the density distribution has several peaks =

=

axound the attractive centers, and to represent these configurations need to shift the maximum of the trial functions out of the origin

we

by choosings appropriately. The usefulness of the generator coordinates in the basis function (8.6) can be understood by the following example. Let us try to calculate the energy of the IH+ this basis with and without 2 by using (that is by setting them to zero) the generator coordinates. The latter

form

corresponds

to the correlated Gaussians for L

of that system is -0.6026

basis of K

=

a.u..

300 Gaussians

0. The energy Without the generator coordinates a

give

-0.5999

a.u.

=

for this molecule. The

inclusion of the generator coordinate immediately h-nproves the con10 basis states vergence and one can get -0.6024 a.u. by using K =

only! Table 8.9 shows ions

consisting

examples

of calculations for the molecules and the

of protons and electrons..

Table 8.9. Ground state

energies of small molecules

in atomic units. K is

the basis dimension.

System

K

"OH+ 2 0OH2 "OH+ 3

SVM

Other method

K

Ref.

50

-0.602634429

-0.602634214

160

100

-1-17445

-1-174475714

1200

100

-1.34351

-1.343835624

600

[1131 [1131 [1141

167

Complements 8.1 The cusp condition for the Coulomb potential It is desirable that the trial function satisfies the proper asymptotic behavior or the special boundary condition as demanded by a given

special boundary condition, the cusp condition [115] known for the Coulomb potential, by using the hydrogen atom as an example. The local energy for the hydrogen atom is given by Hamiltonian. We discuss

h2 1

Ej"'c

a

192 Tf

-

-r-2

2m Tf

where hl is the

angular

h2

2 (9Tf + r

-5r- )

momentum

1

e2

2

(8-10)

1 IIf

+

.M r2 Tf

r

operator. As

was

discussed in

Sect. 3.2, the local energy for the exact wave function turns out to be a constant. The Coulomb potential in the local energy gives a singular 0. For the local energy to be a constant, this singular behavior at r =

behavior must be

for

compensated

by

the kinetic energy term. For

an S wave, where the wave function Tf has no angular dependence, 12Tf 0 and the constancy of El.,r requires that the second term in =

the bracket in

Eq. (8.10) cancel

the

singular

behavior of the Coulomb

potential: I

2

aTf)

Tf 9r

-me h2

r=o

ao

where ao is the Bohr radius. It is easy to see that an exponenexp(-r/a) for the radial part of the ground-state wave

tial form of

function leads to sen

a

constant local

density

if and

only

if

to be ao. The constant of the local energy is then

-0/(2ma2) drogen

=

_Me4/(2h2)

=

-El

as

expected,

atom ionization energy without the

a

is cho-

equal

where E, is the

to

hy-

proton recoil effect, that

is, the well-known energy of 13.6 eV.

angular dependence, it has to take care I/r singularities. Then Tf may be expressed I/r as a product of radial and angular parts: Tf r'R(r)Y(S?), where s 0. is a positive constant and R(r) is assumed to be nonzero at r Substituting Tf into Eq. (8.10), we obtain For

a

general

case

when Tf has 2

and

of both the

=

=

h?

Eloc 2m

+

1

a2 R

( R -5r-2

h2 1 2 __1 Y 2Tar2 Y

2(s + 1)

8(8+1)

1 M

+

+ r

R 9r

r2

2 -

_.

r

(8-12)

Complements

168

As R does not vanish at

r

0, 1IR gives

=

origin. The condition that the local following result: 1

I

( OR) 9r R

We know that the second

12y equation

no

singularity

at the

energy is constant leads to the

=

(s + 1)ao'

=o

rise to

(8.13)

S(S + 1)y

is satisfied if and

only

if

s

is

a

positive integer 1, and then the first equation determines the correct behavior of R near the origin as R(r) oc exp(-r/(l + I)ao). Equations known to be the cusp conditions. Let us attempt to solve the S-wave hydrogen atom variationally with Gaussian basis functions, exp[-(a/2)(r/a0)2J' where a is a vari-

(8.11)

and

(8.13)

are

ational parameter. We functions that lead to

use a

Gaussians

as

an

example of such basis

rather accurate solution but

are

poor in

satisfying the cusp condition. When a single basis function is used, the optimal value of a is 16/(9z), giving the minimum energy of -8/(37t-)EI -0.849EI. A combination of a few terms approximates the ground-state energy quite well. The parameter values of a are determined by the SVM. Table 8.10 shows sample results of such calculations obtained with the code given in [81]. The calculation with five Gaussians already reproduces the energy up to three digits. The wave function obtained with ten Gaussians can reproduce both the energy and the mean values of r and I/r fairly accurately. The over=

lap of the wave function with the exact wave function is very close to unity. We may conclude that the Gaussian basis can predict physical quantities to high accuracy. Of course, the solution does not satisfy the proper asymptotic behavior at large distances and, moreover, always gives zero for the cusp value of Eq. (8.11). The local energy displayed in Fig. 8.5 for the variational wave function indicates that with increasing K it tends to show smaller and smaller deviations from the exact wave function except for the singular points mentioned above. The local energy at large r deviates from the correct value because the Gaussian basis has the wrong asymptotic behavior. It is possible to generalize the above arguments for the cusp value in a system of particles interacting via Coulomb potentials. Evaluating the cusp value for a pair of particles with charges qj and qj, we obtain

(If 16(ri

-

2'j) alriarj I Of)

(T/IJ(ri

-

rj)ITI')

I-tijqiqj h2

(8-14)

where tzij is the reduced mass of the two particles. The left-hand side of Eq. (8.14) is expressed with the matrix elements involving

C8.2 The chemical bond: The H+ ion 2

J(r)

=

J(r)/(47rr2)

used to test the

The cusp values for a pair of paxticles quality of the variational solution at the .

are

169

often

particles'

coalescence.

Table 8.10. Variational solution for the

hydrogen

of Gaussian basis functions. The last

shows the exact values. Ei is the is the Bohr radius.

hydrogen K

atom ionization energy and ao

E

E,

((_L_)-2) ao

atom with

a

number K

row

ao

((_E_)2) ao

ao

Overlap

1

-0.8488264

1.131774

0.8488284

1.499996

2.650706

0.9568351

3

-0.9939585

1.903352

0.9939409

1.491519

2.922759

0.9987560

5

-0.9996191

1.986219

0.9995692

1.499147

2.991284

0.9999446

10

-0.9999998

1.999700

0.9999958

1.500004

3.000014

0.9999999

-1

2

1

1.5

3

1

K= 1

0-, IN.

-4-

K=3

K=10

1 %

0

5

10

15

20

rlao Fig. 8.5. hydrogen

The local energy

curve

plotted for the variational solution

of the

atom. K denotes the number of Gaussian basis functions.

8.2 The chemical bond. The

H+ 2

ion

Quantum mechanics enables us to understand the chemical bond, which is responsible for the formation of molecules from isolated atoms. The chemical bonding phenomenon involves the delocalization of electrons in an atom to gain attraction from the other nuclei when

Complements

170

the atoms

close to each other. We take up the simplest possiH2+ ion, to understand what an important role the

come

ble molecule, the

Hellmann-Feynman

and virial theorems

[1]

of the chemical bond. See

play for clarifying the origin

for detail.

ffilly quantuin-mechanical description of a molecule is a complex problem. This problem is usually simplified by using the BornOppenheimer approximation, where the electronic motion is separated from the nuclear motion, considering the fact that the electron mass is much smaller than that of the nuclei. One starts with determining the motion of the electrons for a fixed configuration R of the nuclei and The

ground state, of energy U(R), of the electronic system. Then one assumes that,when R varies, the electronic system always remains in the ground state corresponding to R, that is the electrons follow adiabatically the motion of the nuclei. The chemical bond is then determined by studying the nuclear motion in a potential energy V(R) which comprises the Coulomb repulsion between the nuclei and obtains the

U(R)Ht 2

Let R be the distance vector between the two protons of the ion and

v

be the

vector of the electron with

position

respect

to the

center-of-mass of the protons. The electron motion is determined the Hamiltonian

2[t

Note that R is

equal

is

the

m

to

e2

e2

p2

H,e( R)

IT

just

a

-

by

stage. The reduced mass jL where M is the mass of the proton and

parameter

2Mm/(2M + m),

(8-15)

IT + RI 2

Al 2

at this

of the electron. The electronic energy U(R) is the lowest of the Hamiltonian (8.15). It is clear that U(R) becomes

mass

eigenvalue

only. The Schr8dinger equation for the Hamiltonian completely separated in elliptical coordinates with respect (8.15) to the foci, R/2 and -R/2. We do not need its exact solution in the following discussion, but note that it is well approximated by the variational calculation using a trial function of the form a

function of R is

Tf

-01"

where at

s.

(Z' R) +'OL' (Z' R),

01, (Z, s) is the charge Z is

The

(8.16)

_

2

2

ls a

hydrogenic orbital

of radius

variational parameter and its

ao/Z

centered

optimal

value

is deternUned to miniraize the energy for each R. The optimal value 1 for R ---* co. At 0 to Z 2 for R of Z decreases froin Z --+

cc

the system will switch

=

=

=

R

over

to

a

configuration

of the

hydrogen

C8.3

atom and the

Stability

proton. Between these

of

hydrogen-like

molecules

extremes, Z is

two

function of R. The energy U(R) is -4E, (2M/(2M + and approaches -E, (MI (M + m)) for R oo.

a

m))

171

decreasing for R

=

0

By using the virial theorem (3-49) and Eq. (3.56) with A R, Ry7 R, we can show that the expectation values, (T) and (W), of the kinetic energy and potential energy of H,..(R) satisfy the relation d

2(T)

+

(W)

+R

dR

U(R)

=

(8.17)

0. '9

Here we have used the fact that WA+R. aR W R d dR to

=

-W and

Rr-aR -U(R)

U(R), enables us equation, together with (T) + (W) U(R). the protons, between of the potential express (T) and (W in terms This

V(R)

=

U(R)

+

=

(e 2IR),

as

follows:

d

d

(T)

=

-U(R)

-

R

dR

U(R)

=

U(R)

=

-V(R)

-

R dR

=

2U(R)

+R

dR

For the chemical bond to must have

(T)

,

clude from

occur

2V(R)

in the

+R

dR

V(R)

-(8.18)

-

R

H+2

system, the potential V(R) --+ oc) -Er at some point

V(R V(Ro:) < -E-r have to be met. Since oc (see Eq. (3.53)), we can con-2E, at R at equilibrium (R Eq. (8.18) that, RO), the electronic

a

deeper

minimum

that is, V(Ro) Er and (W)

Ro,

e2

d

d

(W)

V(R),

=

than

-

0 and

-- -

-

=

kinetic energy is increased and the electronic potential energy is decreased. The lowering of the electronic potential energy is large enough to cancel the

repulsion

for the chemical bond.

and

V(Ro)

-

V(R

-+

between the protons, and that is

According to

oo)

=

an

exact

responsible 2.00ao calculation, Ro -

-2.79 eV.

challenging to perform a nonadiabatic calculation in which no separation of the electron and nuclear motion is made. The validity of the Born-Oppenheimer approximation can be tested in such a calculation. Furthermore, the development of the nonadiabatic treatment for a smaU molecule is of importance in its own right because the adiabaticity may be questionable when the electron is replaced with heavier particles like the muon or the pion. The excellent results obIt is

tained in Sects. 8.2-8.4 indicate that realistic nonadiabatic, calculations are

in fact

possible

in the correlated Gaussian basis.

172

8.3

Complements molecules

Stability of hydrogen-like

The existence of bound states of systems composed of particles with unit charge attracts considerable attention. We discuss this problem here

by applying

of the

some

principles discussed

in

Chap. 3. always bound

The system of the hydrogen-like atom, (M+m-), is and its binding energy is equal to jL/2 in units of e

1, where p stability of a

Mm/(M+m)

=

is the reduced

-mass.

=

I and h

=

What about the

hydrogen-like molecule (M+M+m-m-)? This system is characterized by the mass ratio om/M. Two well-known examples include the hydrogen molecule (a < 1) and the positronium molecule PS2 (o1). Another example is the biexciton molecule [105, 1061. See =

=

Sect. 8.3. The value of 0
1,

is the

where

velocity

of

v

is the

light.

mean

velocity of the

184

9.

Baryon spectroscopy

18001700-

El

E----1

L-J

=

16001500M

[MeV]

14001300-

12001100-

N

1000-

A

900-

1

1+

1-

3+

3

2

2

Y

Y

5+ 5 Y Y

1-

3+ 3i -2

Fig. 9.1. Comparison of low-lying baryon spectra predicted by the mesonexchange potential model with experiment. The solid lines denote the calculated spectra, and the shaded boxes show the error bars.

experimental

masses

with

their

(Continued

on

the next

page.)

One may think that the idea of the meson-exchange interaction acting between quarks is unfamiliar though it has produced the good results. To

judge

its merits the present calculations have to be

tended into several directions

ex-

(prediction physical observables, e.g. factors, decay widths, etc.). The complexity of the NN interaction is basically due to the compositeness of the nucleon. As N, A or Z belong to the same octet family, it is natural to attempt a unified description of the NN, AN or EN interaction from the underlying qq potentials. This goal is at present too difficult to achieve directly on the basis of QCD. In so of

form

far

as

QCD as yet presents no results, the effective theory formua microscopic quark-cluster model [1271 seems to be justified.

lated in

9.3

(Continued

from

Fig

Meson-exchange

model

185

9. 1.)

E] 1800

1700 1600

-

-

F-7

17.71

t

H

r--.7.

-

r__,_T

p G=

r_.j

0

p"

-

-

1500M

[MeVj

1400-

13001200-

1100-

A

1000900

1

1

_2

1

2

_2

_2

1+

1-

3+

1

_2

1800

1700 1600

1500 M

[MeVJ

1400

1300 1200 1100

1000

900 3-

1

11

3+

_2

_2

1

_2

186

The

9.

Baryon spectroscopy

intermediate-range

attraction of the

cannot be accounted for

by

tribution between the colorless the

baryon-baryon interactions produces no con-

the OGEP because it

It is not clear yet whether introduced above leads to a realistic de-

baryons.

meson-exchange potential scription of the NN interaction or not. It is probable that the OGEP combined with the meson-exchange potential is a useful, effective vehicle to obtain a reasonable description of both baxyon spectra and baryon-baxyon interactions at the same time. See, for example, [128] for the attempt along this line.

in solid state

10.

Few-body problems physics

This

chapter is a collection of examples of the application of the SVM to few-body problems in the field of solid state physics. The examples include excitonic complexes, biexcitons, and quantum dots with and without magnetic field. The biexciton and two

(or excitonic molecules), the bound state of two holes

electrons, provides

an

interesting few-body problem,

since it

may be thought as a positronium. or a hydrogen molecule with variable electron and hole masses realized in semiconductors. The biexcitons

also often confined and

are

can

be modelled

as

two-dimensional

(2D)

systems. In recent yeaxs there has been much excitement in the possible applications of ultrasmall. systems with a length scale of 10-100

A,("nanostructures"). The technological motivation (in electronics and optoelectronics)

is that the smaller components

to manufacture very small

contain

very few electrons

only a

are

faster. It is

possible

nanostructures, often called "dots", which

(N

>

L, the

quantum dot is quasi 2D. The confinement is usually modelled by a harmonic-oscillator potential. This system is somewhat similar to an atom in

nature, where the "confining" potential is the Coulomb

traction of the electrons to

quantum dots

are

Throughout this chapter

we use an

tron. When the dielectric constant of

length =

a

effective

mass

me*

for

material is denoted

an

by

elec-

r., the

and the energy will be measured in "atomic units" with Bohr h2n/ (M*e2 ) and haxtree (two times the Rydberg energy) e-

radius a*

2R*

at-

nucleus in the atom, and therefore the often referred to as "artificial atoms". a

=

e2/(Ka*), respectively.


188

10.

Few-body problems

10.1 Excitonic

in solid state

physics

complexes

The excitonic

complexes have considerable importance in the development of semiconductor physics and spectroscopy. The bound state of a positively charged hole with an effective mass m and an electron with an effective mass m,* is called an exciton in semiconductor physics. The mass ratio ocan between 0 (hydrogenic limit cchange m*/m*h and oI (positronium limit) depending on the material and other factors. Like in the case of hydrogen, where not only the hydrogen atom but the hydrogen molecule (H2), or the H2+, H-, H,+ ions are bound, the system of N, electrons and Nh holes can also be bound. The latter system is called an excitonic complex. These excitonic complexes, including the charged excitons, have been subject of intensive experimental [105, 1291 and theoretical [99, 130-1351 investigation. The properties and structure of these systems strongly change with the mass ratio, and, by approaching the two limiting cases, one arrives at two completely different worlds. The interest in these systems has =

=

e

=

been intensified when the advance in semiconductor

technology has possible the fabrication of artificial nanostructures with diameters comparable to atomic distances. This restricted geometry has a prominent effect on the dynamics of the excitonic species. In the following we present the ground-state energy and some other properties of 2D excitonic complexes. The general Hamiltonian made

be written

can

IV.

H

as

N

pi,

i=1

2m,

i=IV'+I

N

j>i=N' +l x

TC.

r"Iri

-

e2

+

2Mh

3>z=l N,

e2

E where

Ne

pi,

+

=

N

e2

E E

rj I

i=1

j=N,+1

is the dielectric constant of the

material,

and the position

2D vectors in the xy plane. In atomic units introduced in this chapter, the Hamiltonian becomes dimensionless and the envectors

axe

depends only

(Note

that

do not need to

specify what dependence is hidden in the atomic units.) This is easily understood by introducing dimensionless coordinates and momenta by ri* ri/a* and pi* pil(hla*). Then the ergy

value of

K

on o-.

is used because the

r,

=

Hamiltonian

(10-1)

we

is reduced to

=

10.1 Excitonic N,

H

2

2P'

2

P

2(o-N,

i=N +I

+N

N

+ Z

i>i=l

3

-

N'-

E

+

j>i=N,+l

2

0-

*2

E

+

189

N

/

1

2R*

complexes

Jr

-

'r

7,

3

P'

N,) N

E E

I

i=1

j=N,+l

J'r j*

r 3

I'

At this point, a comment is due. What we consider here is a welldefined quantum-mechanical problem to be solved. This is, however,

experimental situation. In the realistic case there considered, for example the effect of other electronic bands, the possibility that the interaction is different from the pure Coulomb force due to the confinement in the z direction, etc. The discussion of the importance of these effects is beyond the scope only are

model of the

a

many other factors to be

of this book.

As the

problem here is practically the same as that discussed in the chapter of small atoms and molecules (Chap. 8), the same trial function is used (see Eq. (8.1)), except that it is tailored to the 2D case, that is, the position vectors have only x and y components. The results for case

o-

=

0 and

o-

=

1

are

and in Table 10.2 for the 2D

shown in Table 10.1 for the 3D

case.

between the 3D and 2D results is the

The most

large

striking difference binding

increase in the

0) in energy in the 2D case. E.g., the binding energy of R2 (with o2D is about 8 times of that in 3D. The increase of binding has been =

found

experimentally

Table 10. 1.

as

well.

Energies and binding energies (in a.u.) of 3D

of electrons and holes for two orbital

angular

cases

momentum and

of the

mass

ratio

o-.

exciton complexes

L and S

are

The asterisk refers to the states which

are

found to be unbound. The binding

energy is understood with respect to the nearest threshold.

E(o-i)

B(o-=O)

B(o-=I)

-0.500

-0.250

0.500

0.250

-0.528

-0.262

0.028

0.012

-0.602

-0.262

0.102

0.012

-1.174

-OM6

0.174

0.016

System

(L, S)

E(o-=O)

eh

(0,0) (0,1/2) (0,1/2) (0,0) (0,1/2) (0,1/2)

eeh ehh

eehh eeehh eehhh

*

-1.344

the total

spin of the exciton complex, respectively.

0.169

190

10.

Few-body problems

in solid state

physics

Table 10.2. Energies and binding energies (in a.u.) of 2D excito-n complexes of electrons and holes for two cases of the mass ratio o-. M is the z component of the total orbital angular momentum and S is the total spin. See also the caption of Table 10.1.

System

(M, S)

E

eh

(0,0) (0,1/2) (0,1/2) (0,0) (0,1/2) (0,1/2)

-2.00

-1.00

2.00

1.00

-2.24

-1.12

0.24

0.12

-2.82

-1.12

0.82

0.12

-5.33

-2.19

1.33

0.19

eeh ehh eehh eeehh

eehhh

(o-

=

0)

E

(o-

=

1)

B

(o-

*

-6-82

1.50

=

0)

B

(o-

=

1)

10.2

equality 2(r+-)

distances have to fullfil the

=

Quantum

dots

V(r--)2 + (r++)2.

191

Ta-

ble 10.3 shows that this is not satisfied. Moreover, the fact that the uncertainties

Arij

(,ri2j

-

(rij 2

are

large

means

that

no

such in-

terpretation is possible. There is only one case where the uncertainty is in very small, that is the distance between the (heavy) positive charges with accordance in that In can limit. case one the hydrogenic assume, the adiabatic approximation, that the positive charges are fixed at the distance of 0.37 a.u. The equilibrium distance in the H2 molecule in 3D is 1.4

a.u.

Table 10.3. ratio

mass

Average distances in 2D biexcitons (eehh) as a function of the E.g., (r--) is the mean distance between the two negative

o-.

charges. Atomic 0-

(r++) (,r+-) 2 (r _) (r 2+) 2 (r _)

=

0

units o-

are

0.4

=

used. a

=

0.7

0-

=

0.67

1.26

1.55

0.37

1.14

1.49

1.80

0.47

0.93

1.16

1.38

0.59

2.09

3.19

4.33

0.14

1.69

2.95

4.33

0.31

1.29

2.05

2.88

1.80

dots

10.2

Quantum

Rapid

advances in semiconductor

rication of nanostructures called

[136, 137, 1381. electrons,

are

In

I

quanturn dots

technology

have led to the fab-

quantum dots a

few

or

artificial atoms

electrons, typically

2 to 200

bound at semiconductor interfaces. These few-electron

(2D) electron gases (MOS) structures are

systems arise when homogeneous two-dimensional metal-oxide-semiconductor

of heterojunctions

or

laterally confined

to diameters

comparable

to the effective Bohr

ra-

dius of the host semiconductor. The interest in quantum dots arises not only from the prospect of new technological applications but also from the desire to understand the in

physics of a few interacting electrons

external field. Needless to say, we concentrate on the few-electron where the effect of electron-electron interaction and their corre-

an

case

lation

seems

to be very

important.

192

10.

Few-body problems

The axtificial atoms

in solid state

consider here

we

of N electrons confined in 2D

physics are

modelled

by

a

system

by harmonic-oscillator potential. Before starting with artificial atoms in 2D we present a calculation for the energies of 2D "natural" atoms where the "co-ofi-ohn ' potential is a

Coulombic. The results for few-electron atoms and ions Table 10.4. The much

binding energies

are

again, due

are

to the 2D

given

in

geometry,

than in the 3D

case. Otherwise the properties of these laxger systems axe rather similax to those of their 3D counterparts. For ex-

ample,

akin to

3D,

the

H, Li

or

Be atom

can

bind

an

electron,

extra

while the He atom cannot.

Table 10.4.

Energies

of atoms and ions confined in 2D. M is the

nent of the total orbital

units

are

(M, S)

H

(0,0) (0,1/2) (0,0) (0,1/2) (0,0)

He Li

Be

momentum and S is the total

z

compo-

spin. Atomic

used.

System

H-

angular

Energy -2.00000000 -2.24

-11-89 -29-87 -56.77

The

ex6mple of quantum dots considered here is a system of 2D electrons in a quadratic confiniD potential. The Hamiltonian reads as N

N

9

Pi

H

+

_M

2m*

(V__

+

ivy)

M

where v., and v.

by

N

I 2

e2

*WO2,r? +

(10.2)

exp( IiUr),

(10.3)

-

2

are

the

x

and y components of the 2D vector

v

defined

10.2

Quantum dots

193

N V

(10.4)

Uiri.

The parameters ui and A of the basis function SVMDue to the external field

are

determined

(the single-particle potential),

by the

the Hamil-

tonian (10.2) of the system is not translationally invariant. Correspondingly, the basis function (10.3) is written in terms of single-

particle coordinates instead of relative ones. This basis function depends on the center-of-mass coordinate, and the energy obtained will be the total energy of the system, unlike the previous cases where the energy of the center-of-mass motion was always subtracted. The case of the quadratic confinement is very special, because in that case the energy of the center-of-mass can be separated. In general, for an arbitrary single-particle potential, however, this cannot be done. 1 electron is trivial and its solution is a simple The case of N 2 an analytical solution exists for cerharmonic oscillator. For N I a.u., the ground state tain frequencies [139]. For example, for hwo which is 3 is precisely reproduced by the numerical cala.u., energy culation. Not surprisingly, as is shown in Table 10.5, if the quadratic =

=

=

confinement is strong, the spectrum is close to that of a harmonic oscillator. In the realistic cases (hwo < I a.u.), however, the eigen-

strongly deviate from those of the harmonic oscillator. This example illustrates the importance of these systems: by changing the confinement one can "tune" the properties of these artificial atoms.

values

Energies of N-electron quantum dots with the z component of angular momentum M and the total spin S. The numbers parentheses are the eigenvalues without Coulomb interaction. Atomic

Table 10.5.

the total orbital in

units

are

used.

N

(M, S)

hwo

2

(0,0) (0,1/2) (0,0)

3.000

212.198

7.220

525.12

3 4

Figure

=

I

hwo

(2.000) (5.000) 10.600 (6-000)

10.1 shows the

=

654.45

100

(200.00) (500.00) (600.00)

energies of the ground and excited

states

of the energy of a harinonic-oscillator quantum) of two electrons as a function of the inverse square root of the oscillator constant, -1/2 If there would be no Coulomb interaction between the elec-

(in units

(hwo)

.

10.

194

Few-body problems

in solid state

physics

trons, the energy divided by hwo would be given by lines parallel to the horizontal axis., The deviation from the

straight lines is entirely interesting feature is that states but the ground state

due to the Coulomb interaction. Another level

crossings

always

occur

remains M

between the excited

0.

12

10

8

6

4

2 0

2

4

6

8

10

(hcoo )-1/2 10.1.

Fig. potential

Energy

with

an

levels of two electrons confined

by

a

oscillator constant wo. Atomic units

harmonic-oscillator

are

-used.

Quantum dots

10.2

195

3.5 3.0 2.5

2.0 ;-4

t__I (14

1.5 1.0 0.5 0.0

1

Fig.

3

2

4

(a.u.)

r

10.2. Electron densities of two-electron

quantum dots

as a

function

origin. The solid, the dotted and the dashed 5, respectively. Atomic 0.2, hwo 2, and hwo

of the radial distance from the curves

units

correspond to hwo

are

used.

1.2

1.0

0.8

A

0.6

0.4

0.2

0.0 0

4

2

6

r

Fig.

8

10

(a.u.)

(dashed curve), four- (dotted curve), (solid curve) quantum dots. The oscillator frequency of the

10.3. Electron densities of two-

and six-electron

quadratic confining potential

is hwo

=

0.2. Atomic units

are

used.

196

Few-body problems

10.

in

solid state

physics

peak density inreases with the number of electrons, that is, the equilibrium configuration is realized on rings of expanding diameter. In the case of N=4 electrons, for example, the paxticles may move along a ring, while they are situated at the vertices of a square to TniniTni e the Coulomb repulsion.

10.3

dots in

Quantum

magnetic field

A dramatic feature axises when the quantum dots are subjected to magnetic fields: The ground states are stabilized into magic-number states. When the

state

Tn

from

jumps owing

to the combined effect of the electron correlation and

appear

the Pauli

agnitude of the magnetic field is vaxied, the ground magic number state to another. Magic numbers

one

principle [140].

The Hamiltonian of the system is N

N

I

I

H

=

E 2m* (pi + e-A,)2 + -

N

2

us assume

ward in the

z

2

Wd 'ri

e2

E

1' Iri

i>i=l

Let

2

,,

-M

-

C

+

given by

that

a

(10.5)

-,rj I*

homogeneous magnetic

field B is

direction. The above Hamiltonian

can

applied

down-

then be rewritten

in the form

H

P'

2

+

-m w

2

2m*

N

2-

r?-

1 -

2

hw,

Ii,

e2

rdri

-

rj I

where u),

eB/(m*c) is the cyclotron frequency and w

We

the

=

V/W--22. rl4+wo

ignored magnetic interaction energy due to the electron spin. See Complement 10.1 for details. It is shown there that the part of

the Hamiltonian

corresponding to the single-particle motion can easily

be diagonalized by the use of associated Laguerre polynomials. Depending on the strength of the magnetic field, both the single-particle

10.3

Quantum dots

in

motion and the correlation due to the Coulomb

sive roles in

determining

function is the

same as

197

repulsion play deci-

the structure of the quantum dots. The trial

in the

For the two-electron with the

magnetic field

previous

case our

solution

section.

calculation is in

perfect agreement

[1411. The spectrum of two elec0) is shown in Fig. 10.4. See also

in

given analytical spin-singlet state (S Complement 10.1. The harmonic-oscillator frequency wo is taken as 0.01 a.u. The most intriguing phenomenon seen is the level hwo crossing. Due to these level crossings, the ground state changes with the strength of the magnetic field. This is shown in a magnified scale on the right-hand side of Fig. 10.4. If there is no Coulomb interaction between the electrons, no level crossing occurs. trons in the

=

=

10.0 18 9.5

16

9.0 14 +It

+Z

8.5

12 8.0

10 7.5 8

7.0 0

1

2

3

4

0.0

0.5

1.0

1.5

2.0

2.5

(,)C/ ("0

(J)C/(00

10.4. Energy levels of a two-electron quantum dot in magnetic field function of the cyclotron frequency w,. The figure on the right-hand side magnifies the level crossing. The oscillator frequency of the quadratic 0. Atomic units are 0.01. The total spin is S confining potential is hwo

Fig. as a

=

=

used.

Figures

plot the energy levels of a three-electron total spin being S 3/2) against the orbital

10.5 and 10.6

quantum dot

(with

the

=

198

10.

in solid state

Few-body problems

physics

25

20

15

10

5 0

2

4

6

8

10

12

14

COC/ coo 1.0.5.

Fig. z

Energy

levels of

a

three-electron quantum dot

component of the total orbital angular

IVI

=

6

(dotted curve),

and M

=

9

momentum M

(dashed curve)

frequency of the quadratic confining potential S

=

3/2.

Atomic units

are

used,

is hwc)

belonging

=

3

to the

(solid c=e),

orbital motion. The =

1. The total

spin

is

10.3

Quantum

3

4

dots in

magnetic

field

199

15

14--1

13

12

44

10-

9

8

7

6 0

2

1

5

6

7

9

8

10

M

Energy levels as a function of the z component of the total angular momentum, M, of a three-electron quantum dot in a low 3/2. Atomic 1). The total spin is S 0.2, hwo magnetic field (hw,

Fig.

10.6.

orbital

=

units

are

used.

=

=

200

10.

Few-body problems

of these orbits

in solid state

physics

larger (see Eq. (10.16)). The energy of the dots interplay of the single-paxticle energies and the interaction energy. By increasing the strength of the magnetic field, similarly to the case of low magnetic field, the ground state does -not take all possible values of M. By changing the magnetic field the states with M 3,6,9,12,... "magic numbers' become ground states. is determined

axe

by

the

=

40-

35-

30-

25

-

201

10

0

1

2

3

4

5

6

7

8

9

M 1.0.7. The

Fig. hwo

=

1).

same as

Atomic units

in

are

Fig.

10.6 but for

high magnetic

field

(hw,-

=

6,

used.

change of the ground-state angular momentum quantum numbers as a function of the magnetic-field strength is illustrated In Fig 10.5 for a three-electron quantum dot with spin S 3/2. This figure shows the energies of the states with M 3 (solid curve), M 6 M and 9 orbital The calmomentum. angular (dotted) (dashed) culation shows that by changing the strength of the magnetic field the orbital angulax momentum of the ground state of this system is M 3 for w,lw < 5, it is M 6 for 5 < w,lw < 11, and then it The

=

=

=

=

=

switches

=

over

to M

=

9, reproducing the "magic number"

sequence

10.3

(M

3,6,9,..).

Quantum dots

in

magnetic field

201

States with other orbital

angular momentum never ground state and are not included in the figure. The previous examples dealt with spin polaxized electrons (S 3/2). Actually, spin polarized (S 3/2) and spin unpolarized (S 1/2) states compete to be the ground state as the magnetic field changes. This very interesting phenomenon is illustrated in Fig. 10.8. As the magnetic field increases, the ground state changes as (M, S) (1, 1/2) -+ (2,1/2) --+ (3,3/2) and so on. At low magnetic field the lowest energy state is spin unpolarized (S 1/2) and as the magnetic field increases the spin becomes polaxized. The orbital angular momentum (M) changes continuously (in steps of unity). =

become the

=

=

=

=

=

5

4

3

C5

2

0 0.0

0.5

1.0

1.5

2.0

2.5

(j)C/ (1)0 1-0.8. Spin (S) and z component of the total orbital angular momentum (M) quantum numbers of the ground state of a three-electron quantum dot

Fig. as

a

function of

denote M and

magnetic field strength. hwo S, respectively. Atomic units are

=

1. Solid and dotted lines

used.

202

10.

10.4

Few-body problems

Quantum

in solid state

physics

dots in the generator coordinate

method In Sects. 10.2 and 10.3 the correlated Gaussian basis of

Eq. (10.3)

is

to obtain the solution of the

employed not the only

quantum dots. However, th is is Gaussian basis function for the quantum dots. In fact, the

quantum dots offer

(8.9)

that

was

application of another basis function study of small molecules. That basis func-

very nice

a

used for the

by the matrix A which describes the correlated particles and also by the generator coordinates s which represent several peaks of the density distribution of the

tion is characterized motion of the

allow

us

to

system. In this section

consider the quantum dots in 3D and

we

field. The

state of the

quantum dot

apply belong

can ground 3angular momentum (for example, in the N electron case). The application of the basis function (8.9) gives a very simple way to obtain accurate energies of the dots even with nonzero orbital angular momentum, without the hassle of partial-wave expansion or angular momentum algebra. As this basis function has no definite orbital angular momentum, in principle one has to project out the component with good angular momentum quantum numbers. In practice, however, the converged wave function already belongs to the correct ground-state quantum numbers and the energy gain due to an orbital angular momentum projection is negligible. As the Hamil2 tonian and the L operator commute, the optimization of the wave 2 function filters out the eigenstate of L as well. As was mentioned in Sect. 6.1, this type of calculation is called a variation before projection. In the limit that the variational trial function is chosen to be flexible enough, the solution of this type of calculation would reach the 2 eigenstate of the conserved quantity like L to a good approximation.

no

to

magnetic

nonzero

orbital

=

In Table 10.6 the results of the SVM with the basis function of

Eq. (8.9)

is

compared

to that of

The shell model works

a

extremely

large

scale shell-model calculation.

well in this

case

because the

con-

fining potential is a harmonic-oscillator well. By choosing the singleparticle wave functions to be the eigenfunctions of this single-paxticle potential, the diagonalization of the Hamiltonian with the relatively weak Coulomb interaction between the electrons is simple and gives an accurate solution. The results of the two methods are in perfect agreement. This agreement shows that the optimization of the basis of

Eq. (8.9) finds the

correct

ground

state

automatically

even

for the

10.4

case

with

sis size is

Quantum

nonzero

dots in the generator coordinate method

orbital

angular

sufficiently large. An

momentum

orbital

angular

provided

203

that the ba-

momentum

projection

may lead to a faster convergence, but as it can only be carried out by a three-dimensional numerical integration, it would prohibitively slow

down the calculation.

Energies of few-electron quantum dots in 3D calculated by SVM and by a large scale shell-model (SM) basis'. L is the total orbital angular momentum, S the total spin, and -x is the parity. K is the basis 0.5. Atomic units are used. dimension. hwo Table 10.6.

=

N

(L, S, -ri)

(1,1/2,-) 4(0,0,+) 5 (1,1/2,-) 3

a

P.

(SM)

K

E

4.01324

100

4.01324

6.35025

300

6.3506

39 hwo 14 hwo

9.00331

500

9.0032

6 hwo

E

(SVM)

Navratil, private

communication.

SM space

(in hwo)

Complements

204

Complements 10.1 Two-dimensional electron motion in

a

magnetic field

vaxiational solution to the

have

presented a previous section we few-body problems of quantum dots. In this approach a correlated Gaussian basis has been used and the problem has been formulated In a relative coordinate framework. The examples from atomic physics (Chap. 8) convincingly prove that this is a very powerful -way to cope In the

with the correlation between the electrons in the field of

an

attractive

only possible approach. One can, alternatively, use a basis set made up from products of single-particle states to diagonalize the Hamiltonian. The properties of these singleparticle states axe discussed in this Complement. The Hamiltonian of interest is given by center. This is,

H

however,

-2m*

not the

(P eAi)2 +

+

9*ABB

c

-

2

aj

I

+

vori)

2

(10.7)

+ X

i 'i

1ri

-,rj I

where m* is the effective electron mass, AB eh/(2m,c) the Bohr magneton of the electron, g* the effective g-factor of the electron, =

and

r.

is the dielectric constant of the bulk material. The

magnetic

interaction energy due to the electron spin is included in the Hamilltonian. The magnetic field is directed downward in the z direction, i.e. B

B

-

with B

> 0. For high magnetic field of, e.g. T, the value of ILBB becomes 0.5788 meV. The vectors ri

10

and pi

(0, 0, -B)

=

axe

2D vectors, which have

x

and y components. The

one-

body potential V(r) is to confine the quantum dots and is chosen to be quadratic, m* W02T 2/2. A typical value of the confinement en5 meV. By taking a symmetric gauge vector potential, ergy is hwo -

A

-r x

=

(P

B/2 2

e

)

_Ji2' j +

A

+

where A

B(y/2, -x/2, 0),

c

_2B 2

we

e. hB

2

(10.8)

4C2 +

_ey2-

and I-, is the

obtain

c

z

component of the angular

mo-

mentum.

To express the Hamiltonian in terms of dimensionless introduce

vaxiables,

-we


e,B W,

W

=

WC2

=

M

a

magnetic field

205

hI + W02

and

p

.

=

(10.9)

W U)

Here wc is the cyclotron frequency and p is the that the magnetic length is expressed by p

magnetic length. (Note using Bohr and hartree 2R*.) Then the Hamiltonian is expressed by the =

radius a*

- -2R*-1hwa*

dimensionless variable ri: 1

H

=

hw

A,

+

2

1

lr,2 2

hw, 2

E lj ,

-

2

g*AB B

aj ,

i

e2

+XE 1ri

with -

j>i

Here the is

hwc

=

length is

rjI

2R*a*

(10.10)

-

=

P

rIP

in unit of the

(2ra,/m*)ABB.

X

magnetic length. The cyclotron energy

The ratio of m,/rn* is of order 15 for the 2D

electron system in GaAs-Ga,,All-,,As heterojunctions. The strength of the Coulomb potential is of the order of X, while the level spacing of the single-particle energy is determined by hw. Therefore the ratio

Xlhw

-

VI'2--R*-Ihw is

the

quantity which

charac-

terizes the electronic motion. If the ratio is much smaller than

the

independent

tion. As it

the

motion of the electrons becomes

a

unity, good approxima-

increases, the correlated motion of the electrons overwhelms

independent

motion.

Recall that the second and third terms of the above Hamiltonian are

consistent with the well-known fact that

mass

M and

charge

qe in the

magnetic

a

spin 1/2 particle with magnetic

ffied B has the

interaction energy -IPB, where the magnetic moment tL of the particle is in general expressed by tL g1l + gs in units of ehl(2Mc). Here g, and g, are the respective gyromagnetic ratios for the orbital and =

spin angular momenta. The gi value is equal to q, while the Dirac 2q for a point charged paxticle like the equation would give g, 2.0023 ILB for the The measurement electron. gives I g, I precision electron and the deviation from 2 can be very accurately computed by the higher order corrections of quantum electrodynamics. On the other hand, the experimental values for the nucleons are far from the 5.586 for the proton Dirac value expected for point paxticles: g, =

=

=

and g, AN

=

=

-3.826 for the neutron in units of the nuclear

eh/(2mpc).

nucleon.

magneton

This fact is ascribed to the internal structure of the

206

Complements

The eigenvalue of the single-particle Hamiltonian, Ho hw (-,A/2 +r2 /2) -(hw,12)1., -(g*p.BB12)o-,,, is obtained easily. Rewriting the spatial coordinate as a complex variable z (x iy)/-\/-2, we defme =

-

I

at

72= (

=

0),

*

Z

-

'0

I

--.,f2- (z- OZ* ),

bt=

(z

a=

OZ

=

[at, b]

Hamiltonian

Ho

=

=

[at, bt]

=

OZ*

(z*

b

0 +

OZ

[a, at]

with the commutation relations

[a, bt]

+

0. We

can

[b, bt]

=

=

[a, b]

1 and

then express the

single-particle

as

hw(ata + btb + 1)

Ihw,(ata

-

-

2

btb)

Ig*ttBBc-z-

-

2

(10.12)

The operator at creates a right circular quantum of counterclockwise rotation about z axis, and bt creates a left one. Let a non-negative

eigenvalue of the total oscillator quanta ata+btb at a b b. and m denote an eigenvalue of the angular momentum 1,, Then the eigenvalues of A and btb are given by (N + m)/2 and (N m)/2, respectively. Since they should be positive integers or zero, the possible values of m for a given N are m -N+ N, N-2, N-4, 2, -N. The single-particle energy becomes integer N denote

an

=

-

-

=

...,

I

E,,,,,,

=

hw(N+1)

hwm

-

-

2

g*ABBS,

I =

with Sz

quanta

hw(2n + Iml

being

either

N is set

1/2

For

of

hw(N + 1),

1) or

to N

equal given N, possible

n.

a

+

hwm

-

-

2

-1/2, =

2n +

values of

(10.13)

g*ABB$,

and where the number of total

Iml

with

n are

0,

1

a

non-negative integer [N12]. The energies

....

with N + I-fold

degeneracy, corresponding to the 2D are independent of m conforming to N. The singleparticle energy is shown in Fig. 10.9. In the case of no magnetic field or very weak magnetic field (w,lwo < 1) the single-particle level shows a shell structure with degeneracy of N + 1. The single-particle level is then in the order of haxmonic-oscillator

(n, m)

=

(0, 0); (0, 1), (0, -1); (0, 2), (1, 0), (0, -2);...

The level

cross-

V/-I/-2

ing of the low-lying single-particle levels occurs at W,/WO 0.71 and the level with (n, m) (0, 2) becomes lower than the =

=

with

(0, -1).

Z-'

one

CIO. 1 Two-dimensional electron motion in

a

magnetic field

207

If wo 0, that is, there is no confining potential present or the magnetic field is very high (wc.Iwo > 1), w is (nearly) equal to (112)w, and the energy becomes E,,,,, hw,(n+(Iml -m)/2+1/2) -9*ABBS,, de=

=

(Iml

pending only

on n +

dent of m for

positive

-

m.

m) /2. Clearly the energy becomes indepen-

This

gives

rise to

infinitely degenerate levels

called the Landau levels with the Landau level index The fact that the

levels with

an

single-electron degeneracy

infinite

motion is

quantized

number in the

+

n

case

(IM I

-

m) /2.

to the Landau

of

no

confining

potential reflects the fact that the energy is independent of the center of the cyclotron motion. In the presence of the confining potential, the

single-particle energy increases with m as seen from Fig. 10.9. By using the polar coordinates r and 6 the eigenfimetions single-particle Hamiltonian are expressed as

rp2

ni n!

(n-t TP 2(n+lml)!

(r)

1-1

L (I ' 1) n

P

r2

(p2

-

r

ey-P

of the

2

2p2

(10.14) where

Ln(c) (x)

is

an

associated

Laguerre polynomial defined by

d L,(,o I (X) =ex-' (e-XX n+a n! dXn n

E k! (n

'k=O

F(n+a+l) (-X),. k)! -V(k + a + 1)

(10.15)

-

parity operation changes V to V + 7, so the parity of the singleparticle eigenfunction is given by (-l)'. For the lowest Landau states (n 0, m > 0) the larger angular momentum state corresponds to a wave function which is more extended from the origin:

The

=

2

(0o,jr loom

(Iral

The Han-d1tonian

+

1)P2.

(10.10)

(10.16)

contains the Coulomb interaction be-

electrons, which leads to interesting effects on quantum interplay between the magnetic field and the Coulomb interaction is an essential ingredient for studying the dots. The relative importance of various pieces of the Hamiltonian depends on m*, K, g*,

tween the

dots. The

and B. For two electrons the Hamiltonian

can

be

separated in the relative By introducing

and center-of-mass coordinates of the two electrons. the coordinates

7 C)

Is 6

5

4

3

2

1

0 0

1

2

3

4

5

6

7

8

9

10

O)C /0)0

Fig. 10.9. The single-particle energy E,,,,, of a quantum dot as a function of magnetic field strength. The Zeeman spin splitting of the energy is not included.


I

R=

-\f2-

('rl +r2),

Ir

a

magnetic field

I-(Irl -r2),

=

209

(10.17)

A/2

the Hamiltonian is reduced to H

=

hwf

I

,AR

-

+

2

1R21

'&V,, IR ,

-

"A'r + 21,r2 2

+ hW

hWcl

,T Z

2

-

C

.

2

2

g*[tBB(Sj,,

+

S2 ,)

X 1 +

(10.18)

-

vF2 r

by exactly the same Hamiltonian as the single-particle Hamiltonian and its energy and eigenfunction are given by Eqs. (10.13) and (10.14). The eigenfunction b(T) of the relative motion Hamiltonian may be obtained in polar coordinates. By separating the angular part as b(r) u(r)e " (m and the the function radial eigenvalue E, are obu(r) 0, 1, 2, ...), tained from the following equation The center-of-mass motion is determined

=

d2u

1 du +

-

.-

r

where

u(r)

r2

+

hW

dr

has to

M2

V2-X 1 +

satisfy

=

2Er

W, 7n

U

-

=

0,

(10.19)

r

the condition that its norm,

fo' drru(r)2, is

finite.

analytical solutions are available provided Xlhw belongs to a certain enumarably infinite set of values [1411. For a general case we can obtain the eigenvalue by numerical integrations. The eigenvalue 0, 1, 2,... for each m. We may be labelled by a quantum number n note the symmetry property of the eigenfunctions. The interchange of the position coordinates ri and V2 leads to r -r, which is # + 7r. Therefore the eigenfunction equivalent to the change of 6 and labelled by n m receives a phase change e"' (-I)m, so that it is symmetric for even m and antisymmetric, for odd m. Figure 10.10 displays the energy E, of the relative motion as a function of the cyclotron frequency. The most striking feature is the level crossing in the lowest level (ground state). With increasing magnetic field the so (0, 0) to (0, 1), (0, 2), (0, 3), ground state changes from (n, m) that the angular momentum m of the ground state increases one by one. (This figure is basically the same as Fig. 10.4 which is obtained by the variational calculation, but included here to show the significant Paxticular

=

---).

--*

=

=

role of the Coulomb

Finally

we

electrons in

a

..

-,

interaction.)

importance of the correlated motion of strong magnetic field has already been recognized in

note that the

the quantum Hall effect

[1421.

The

broadening of the Landau levels

210

Complements 12-

11

10I f I I

9-

8 It

7

*11

IA 41/

*, -

3-

2

1

0

-

1

0

1

2

3

4

5 0) C

Fig. dot

6

7

8

9

10

/ (00

10.10. The energy E, of the relative motion of a two-electron quantum function of magnetic field strength. The Zeeman spin splitting of

as a

the energy is not included. The solid curves are for states with even m and the dashed curves for states with odd m. hwo = 1. Atomic units are used.

CIOA Two-dimensional electron motion in is caused

by

scatterers such

as

a

magnetic field

211

impurities and lattice defects. The repulsion between many electrons

correlations due to the Coulomb

lead to the Anderson localization and the fractional quantum Hall effect. Laughlin [1431 successfully used a correlated basis of Jastrow

type

to describe the electron motion.

11. Nuclear

few-body systems

originally developed to solve physics. In this chapter we collect a

The stochastic variational method

was

few-body problems in nuclear few possible applications of the SVM for nuclear systems. The nuclear force, due to the effect of the underlying quark structure and relativistic motion, is a very complicated interaction. h addition to the spinisospin dependent central part, the two-nucleon interaction contains spin-orbit and tensor forces, and L2- and (LS)2_dependent terms. The two-body interaction designed to fit the nucleon-nucleon phase shift and the properties of the deuteron fails to reproduce the binding energies of the three- and four-nucleon nuclei, and a three-body interaction has to be introduced to get the correct binding energy. As the nucleons are not structureless point-like entities, the spatial part of the nuclear force contains a strong repulsion at short distances. These circumstances altogether make the solution of the nuclear few-body problem a

formidable task.

application is restricted to central forces. These schematic forces are used in simple model calculations and serve for benchmark tests. Examples of more realistic problems are shown J-n the last secThe first

tion.

Introductory remark

11.1

on

nucleon-nucleon

potentials The nucleon

(N)

has

spin and isospin degrees of freedom

in addition

of Eq. (2.3) is a product of these spatial one. The operator 0?. V factors, one acting on each coordinate of the three types:

to the

0?. 73

=

(0,,ij (space) 0,.ij (spin) -

Oij (isospin),


214

11. Nuclear

where,

e.g.,

few-body systems

0,,,,j (space) (IL

=

n,

r.

-

-K)

1,

is

a

spherical tensor a scalar product

spatial part, and product (-I)"A/2__K+ I[T,,, x U,,Ioo, which (T,,,.U,,) is defined with Clebsch-Gordan coefficients by operator of rank

acting

K

in the

stands for the tensor

[Tr-1

X

Ur-2]r-393

=

E

(11.2)

rv1jL1K2A2jK3A3 >TK,jL1 Ur-2/L2'

0, E(I, ab)

=

2 2 2) -g (a 2 xj+b X2,

(- 1)

1

(11.32)

for ab < 0. Since the

Legendre polynomial PI(cosO) is proportional to the tensor product [YI(i-1) x Yj(i2-)joo (see Eq. (6.54)) and has non-vanishing matrix elements between the functions of different partial waves, we find that the various multipole components contained in the potential bring about the mixing of the higher partial waves. The relative strength of the multipoles thus determines the extent to which each partial wave is contained in the solution. For a long-ranged potential (small p) the magnitude Of V1 (X1 X2) is determined by the factor IL21/ (21 1) 11, the soluto contributions main low that so partial waves give only tion. However, this is not the case for a short-ranged potential. The -

7

is, the larger the contribution of thehigher in the extreme case of V(I ax, + bX2 1) E.g., inultipole components. shorter the

potential

range

-

Quark

C11.3

Pauli effect in s-shell A

hypernuclei

239

(fk2/7)3/2 and letting 1L oc in Eq. J(ax, + bX2), by putting Vo (11.32) We get VI(X1, X2) ((21+ 1)/(47))J(Ialxl jbjX2)1(jabjXIX2), ,

=

-

-

which indicates that the

multipole strength is equally strong for arbi-

trary 1. The result of Table 11.10 the fact that the ATS3

repulsive-core part. Clearly the matrix

be understood in this way from contains the strong short-ranged

can

potential

element for the central

potential

between the

neutron and the proton vanishes for the functions of diffrent I values if they are expressed in terms of the different Jacobi coordinate set where

the first coordinate is chosen to the relative coordinate between the neutron and the proton. This consideration leads us to the following ansatz for the solution

(X(I), X(1)) [Y,(X(,)) YJ(X(1)) 100

f

1

1

2

X

2

1

+

X

fl(3) (X (3), (3)) [yJ(X(3))

X

1

+

where

fl(2) (X (2), (2)) [yJ(X(2)) 1

x('), x(') (i 1

=

2

X

X

1

2

1

2

1

1

y1I

(X(2))j 2

Yj(X 2(3))

00

100

(11-33)

1, 2, 3) stands for the ith Jacobi set (see Sect. 2.4).

This type of functions is used in the Faddeev method and the CRCG [26] method. It is known that this expansion converges much faster than the one using one particular Jacobi set, and the inclusion of low

partial waves

is

be sufficient to get accurate solutions (see In contrast to this approach, the CG basis

expected to

also Sect. 4.2.1 and

[30]).

particular Jacobi coordinate but takes account of the contribution of the high partial waves by the cross term 1412XVX2 in the exponent of the trial function. This ensures accurate solutions in the correlated Gaussian approach. It is noted that the calculation of matrix elements with high I values is computer time consuming (see Appendix A-5), whereas in the CG basis no such PWE is employed, which makes it possible to calculate matrix elements very quickly. of

Eq. (11.26)

uses

only

one

Quark Pauli effect in s-shell A hypernuclei 4 3 and 5A Ije are Called Among a few known A hypernuclei AH, 4jA 1, AIIe, s-shell hypernuclei because in the simplest version of the shell model 0 all the constituents can reside in an s orbit. By applying the L 11-3

-

=

240

Complements

correlated Gaussians of

Eq. (7.3),

of s-shell A

in order to reveal the

hypernuclei quark substructure of baryons of

in

we

calculate the

resolving

the

binding energies significant role of the long-standing problem

'He. Since the

pioneering work of [1711, the S-shell A hypernuclei have intriguing problems, among others, the anomalously small binding of 5AHe. According to a recent survey of hypernu5 clear physics [1721, "The anomalously small binding of AHe remains model calculations an enigma. Simple based upon AN potentials, parametrized to account for the low-energy AN scattering data and the binding energy of the A 3, 4 A-hypernuclei, overbindS A5He by offered several

=

2-3 MeV."

The trial function is

given as

a

combination of the correlated Gaus-

sians:

CkAjFAk (X)XkJM?7kTMTli

TfJMTMT

(11.34)

k

where the operator A

antisymmetrizes the nucleon coordinates, XkjjU (Since L 0, the spin S is equal to the total angular momentum J), and 77kTMT is the isospin function. The spin and isospin functions are obtained by successive couplings, for examPle, XkJM= X(S12SI23 )JM [[[Xl/2 X XI/21SI2 X X1/2JSI23-IJTM with k representing a set of the intermediate spins S12, S1237 The optimal set was chosen for each matrix Ak. The matrix Ak containing is the

spin function

=

:::--:

...

....

nonlinear parameters

was

selected

by

the SVM.

The NEnnesota potential [1471 with u I was used for the NN interaction. The MA potential is the same as was used in the previous Complement. Table 11.11 lists the results of the calculation. The calculation reproduces the data well except for the case Of AHe, for which the theory overestimates the binding energy by about 1.9 MeV, =

consistent with what is mentioned above. Another is that

4 He A

is

bound than

4 H. A

thing to be noticed

strongly introducing a charge-dependent component as the IVA interaction, because then 4AHe would be less tightly bound because of the Coulomb repulsion. We have so far treated a baryon as a structureless particle and distinguished A from N. What happens if we take their quaxk substructure into consideration? Recently it has been shown in [173] that the anomalous binding problem of 5A He can be, at least partly, resolved more

It would be difficult to

understand this without

by considering

the

quark substructure

of the

baryons.

C11.3

Quark Pauli effect

in s-shell A

241

hypernuclei

Binding energies (in units of MeV) of s-shell hypernuclei. The separation energy BA of the hypernucleus AAX is defined by the difference of the binding energies, B(-A'1X)-B(-'1-1X). Table 11.1-1.

A

3

AH

A411

4AHe

5 He A

B(AX) A B(`1-1X)

2.38

10.62

9.91

34.93

2.20

8.38

7.71

29.95

BA(Cal.) BA(Exp.)

0.18

2.23

2.20

4.98

0.130.05

2.040.04

2.390.03

3.120.02

Let

us assume

for the sake of simplicity that

baryon is (u, d, d), A a

a

coinpos-

(u, d, s), (u, u, d), n particle of three quarks, p well. harmonic-oscUlator the in orbit in the Os and each quark moves with size parameter b. The spatial part of the three-quark baxyon is described by the function ite

=

=

OB

-`

(7rb2)

1

-1 4

exp

Z3

b2

( '3

(_ (P2 P2 P2)) (_ _b 2) O(int) 2b2

1

+

2

+

3

3

exp

-

where pi is the

=

2rI

(11.35)

B

quark's position vector,

rl

=

(P1

+ P2 +

P30

is the

is the spatial part of the intrinsic baryon's position vector, and 0(int) B of the function wave baryon depending on p, P27 (PI + P2)/2 P3* The value of b can be estimated to be about 0.86 fin by requiring that the function (11.35) reproduce the charge radius of the proton [1741. When the spatial part of the two-baryon wave function is described we interpret it as with the configuration of expf -3/(2b2)(r21 + r2)1, 2 the Os in all six orbit, exp(-p 2/(2b 2)), quarks move indicating that of the common harmonic-oscillator well. with size parameter b. By increasing the number of baryons, we thus expect that the manybaxyon wave function may receive a special constraint arising from the quark Pauli principle that any single-particle orbit can accommodate at most six quarks (three colors and up-down spins) for each flavor. It is easy to see that we have no apparent quark Pauli-forbidden 4 hypernuclei. However, this is not the case for states up to A 5 5He: Four nucleons of A He, when they are on top of one another, A have already six u-quarks and six d-quarks in the Os orbit, so neither u nor d quark of A can take the same Os orbit. This leads us to the conclusion that the five baryons cannot take a configuration of -

=

-

242

Complements

exp(-3/(2b 2) 1:5i=1_ r i2) forbidden state for

rated,

is

.

To be

5AHe,

more

specific, the (normalized) Paulibeing sepa-

with its center-of-mass motion

given by 5

TfPF (X)

=

JV

'(7b2)-3eXP(_ 4

-

3

3

_b2 Dr'

X5)2

(11.36)

i=1

where X5 is the center-of-mass coordinate of the five baryons. The normalization constant is given by JV (4M2 + M2)/(4MN + MA)2' =

N

A

where MN is the mass of N. Other Pauli-forbidden states sibly exist, but this is the simplest and most evident one. The solution TI obtained above for 5He A

Pauli-forbidden component of Eq. quence is very

was

might

pos-

found to contain the

(11.36) by only 0.44:'YD, but its conse-

If this Pauli-forbidden component is simply subtracted from the wave function Tf, the BA value would have been

from 4.98 MeV to 2.74

changed of

A5He!

The reduction is

pectation may be

significant:

a

principle

MeV,

anomalously

small

binding

because the reduction in the

calculation of the so-called variation

6.1),

no

due to the fact that the energy exvalue in TfF(x) is very large, i.e., 513.6 MeV. However, this premature conclusion from the viewpoint of the variational

mainly

binding energy is based on a before projection type (see Sect.

that

is, the variation has been done before the elimination of the Pauli-forbidden component is made from the trial function. To calculate the

binding energy more precisely by taldng into acquark effect, we have repeated the calculation by replacing the correlated Gaussian in Eq. (11.34) with the one that 'has count the

no

Pauli

Pauli-forbidden component:

FAk (X)

11.4 The

FAA: (X)

C nucleus

-

TfPF (X) (TfPF (X) IFAI, (X))

-

(11.37)

system of three alpha-particles example application of nonlocal potentials, we take up simple model for the 12 C nucleus, the 3a model. In this model 12C

As a

12

=

an

of the

as a

C11.4 The

12C

nucleus

as a

system of three alpha-particles

243

5

>4

3

Exp.

Pq

2

0.6

0.4

0.2

0.0

b

Fig.

[fml

separation energy BA of 5H A three-quark baryons

11.4. The A

parameter b of

containing

1.2

1.0

0.8

six protons and six neutrons is

as

a

function of the size

approximatedas

bosonic

a

system of three alpha-particles. The physical reason behind this picture is in the fact that the alpha-particle is a strongly bound, very stable system compared to light neighboring nuclei. One needs an energy of about 20 MeV to excite the of the

alpha-particle.

Because of this

nuclei tend to form

unique

consist-

feature, light subsystem ing of two protons and two neutrons, which is called an alpha-cluster. The alpha-cluster would never be identical to the alpha-particle but it may happen that a model descriptioii assun-Ang such subsystems can explain many properties of the light nuclei [60, 175, 59]. To calculate the binding energy of 12C in the 3a model, one needs to know the potential between two alpha-paxticles. It is hard to derive a potential which acts between composite particles through the underlying interactions of the particles composing the composite particles. Thus we use a phenomenological potential which successfully reproduces the alpha-alpha scattering phase shifts. The one employed here is an I-independent local potential consisting of both the nuclear some

paxt of Gaussian form and the Coulomb part

V(r)

-Vo

e

_P,

[1761:

4e2

2 =

a

erf (,3r),

+ r

(11-38)

244

Complements

where VO

122.6225

=

MeV, p 0.22:ftn-2, and scattering data very well

0.75 fin-'.

=

fits the

this

Though

to about Em

30 potential MeV and reproduces the 0+ resonance at 92 keV quite well, we have to note that it predicts "redundant bound states", two (-72.625 and

-25.617

I

=

2

in the I

MeV)

state, these bound states ftom the

0

=

Since the two

wave.

wave

and

one

(-21.999 MeV)

known to form

in the

bound

alpha-paxticles considered spurious and must be excluded are

no

axe

space. The existence of

configuration

=

system of two nuclei is due understood in the

to the Pauli

spurious

principle

and

states for

can

be

a

easily

version of the nuclear shell model. This is

simplest orthogonality condition model [1771 which succeeded to give a foundation to the deep local potential of type (11.38) from the microscopic theory of scattering between composite particles. Let us denote the spurious states as 0, 1 for I (r), (n 0 for I 0, and n 2). We require that the wave function TI, for the 3a system be free from the spurious components in the alpha-alpha pairs, that is, the basis of the

=

=

=

( Pnim(ri

-

rAff)

=

(11.39)

0-

Here ri is the position vector of ith alpha-particle. Unless this condition is imposed, the ground-state energy with the potential (11.38)

strongly overbound because

would be

the

alpha-clusters,

as

bosons,

would occupy the lowest possible states. An alternative, convenient approach to eliminate reasonably well the spurious components is to use

the nonlocal

Fini

1M ==

potential of projection operator type

I

(ri

rj)) (W,,,M (ri

-

and include it in the Hamilto-nian

-

as a

kind of

N

I

=

T

-

Vij(lri-rjl)+A

Here A is

a

wave

positive

E EI'i'!

(11.41)

j>i=l n1m

j>i=l

that if the

"pseudo potential" [178] N

E

Tn, +

(11.40)

rj) 1,

constant chosen to be very

large. The idea

is

function contains the spurious components, then the energy would be comparatively high. Therefore, the variational lowest solution would approach a state that has a negligible overlap with the

spurious components. The spurious bound

state

lar

O,am (r)

Cae- 2 a

can

be well

approximated

in the form

2

Y1. (r)

(11.42)

CIIA The

12

Then the nonlocal

C nucleus

potential

as a

245

system of three alpha-particles

summed

over m

takes the form

_Vnlra

f

ij

M

21+1

(rr') -47r

R

(cosv) E ca*ca, e-la 2

r2_ 1 af rf2

(11.43)

aa,

where V is the

between

angle

r

and

r.

The trial function for the

system was chosen as a linear combination of the correlated Gaussians, FA(x), of Sect. 7.2. The matrix element of the nonlocal potential (11.43) in this basis can be calculated with the use of Eq. (7.46). 3a

high sensitivity of the energy on A calls for numerical calculations with a high accuracy in order for the pseudo potential to play the role of the Pauli projector. It was found [1791 that the pro0 and 2 do not jection effect is so strong that the partial waves 1 The

=

contain bound states, and a contribution of more than 95 % to the ground-state energy is due to the partial wave I 4. Table 11.12 lists =

the

energies

of the

ground

state and the excited states obtained in

a

[1801 using the basis function of Eq. (11.33). h (X1 X2) is expanded as combinations of Gaussians 2 The calculation does not reproduce the experix11 x12 exp(-ax 1 _OX2) 2 mental energies very well. In [1801 the energies of the 2+1 and 41+ states

variational calculation The radial part

are

also

given. They

are

-3.77 and -2.25

MeV, respectively, which

compared to the experimental energies of -2.84 and 6.80 MeV. Here the discrepancy between theory and experiment is more serious: state is lower than that of the 0+1 state and the The energy of the 2+ , energy of the 4+1 state is much lower than the experimental energy. are

Note that there is

no

Pauli-forbidden state in the 1

makes the contribution of the

high partial

> 4 waves,

waves too much

which

important.

This calculation suggests that the 3a model with the local aa potential has only limited success and is not very realistic to reproduce the experimental energy spectra. This doe not, however, exclude the

Table 11.12. Energies of the 0+ states of 12C calculated in the 3a model [180]. The energy is measured from the threshold of the 3a breakup.

0+ 2

0+ 3

-3.38

-1.43

3.70

-7.27

0.38

3.03

0+ 1

(MeV) Exp. (MeV) Cal.

246

Complements

a

+

8Be,

a (

and

'Be(a,-y)12c.

Appendix

Matrix elements for

Gaussians

general

The matrix element for the correlated Gaussian with

arbitrary physically important potentials including central, tensor and spin-orbit components. A simple and straightforward method is presented to calculate the matrix element for the general correlated Gaussian-type geminals.

angular

momentum is obtained in

a

closed form for most

A.1 Correlated Gaussians We start with

Eq. (7.2)

to evaluate the matrix element for

a general generating functions g is given perform the operations prescribed in Eq.

The matrix element between the

case.

in Table 7.1 and

one

has to

(7.2). A.1.1

The

Overlap

overlap

of the basis functions

matrix element

can

be obtained

through

(fK" LM (u, A", X) 1 fKLM (u, A, x)) I

BKLBK-'L

ff

_ d9de'YLm(i;_)YLm(e')

d2K+L+2K+L

dA2K+LdAI2K'+L

3

X

detB

exp

[q A2 + q"A/2 + PAA/e. er]

where


\,=0

,

(A. 1)

Appendix

248

B=A+A',

q=

2

fiB-lu,

B-lu ,

q

2

p

=; B-lu.

(A.2) To

perform

the

operation prescribed

Eq. (A.1),

in

we use

the

ex-

pansion

[qA2 + q, A/2 + pAA/ e. el]

exp

CO

00

CO

E 1: E H(n, q).ff(i , q) H(m, P) Vn+m y2n+m

fn

n=O n'=O m=O

(A-3) where

H(n, x)

is introduced to Xn

10

H(n, x)

for

n!

simplify the

notation and defined

non-negative integer

n

by

(A.4)

otherwise.

Differentiating with respect to A and Y, followed by A 0, 2K + L and gives non-vanishing contribution only when 2n + m 2K' + L, while the integration over the angles of e and e' 2n + m becomes nonzero if m is equal to L + 2k with non-negative integer k (see Eq. (6.18)): =

=

if

d iYLM(e-)yLM(,

di

1

j)*(e.eI)2k+L

K k and n Rewriting n overlap matrix element =

-

=

K'

-

k,

=

we

BI.-L

(A.5)

obtain the result for the

(fKIILM(UI, A', X) IfKLM(u, A, x)) 3

(2K+L)!(2K'+L)i

(27)N-1-

BKLBKIL

detB

2

min(K,K') X

E

H(K-k,q)H(K-k,q)H(L+2k,p)BkL- (A.6)

k=O

The B,,I value is given in Eq. (6.9). Note that the values of K and K' usually be chosen to be small. in practical cases, and then the sum

can

over

of K

k is limited to =

K'

=

just

a

few terms. In

0 the above result

particular, simplifies to

for the

special

case

A. 1 Correlated Gaussians

249

(ALM(Uli A i X) IfOLM(Ui A X)) 3

(2L + 1)!!

(2v) N_I

(

4v

detB

2

)

L

(A.7)

P

A.1.2 Kinetic energy

The kinetic energy with the center-of-mass kinetic energy subtracted is given in Eqs. (2.10) and (2.11). It is simply written as FrAn-/2. The matrix element of the kinetic energy is then

Table 7.1

easily obtained by using

as

I

(fK"LM(UIjXjX)j ?rA7rjfKLM(UjAjX)) h2

(2r) N-I

2BKLBKIL

detB

(

d2K+L+2K+L

[R

X

dA2K+LdA/2K'+L

X

exp

3 2

+

if

de-- do

pA2

+

YLm (&) YLM (4 )

p/A/2

+

QAA'e-.e-f

[qA2 + q' A/2 + PAAte.e/

(A.8)

where R

M(AB-'A!A),

=

P'=

P

=

-i B-'AAAB-lu ,

-i!B-1A!AA!B-'u, Q

=

2

The matrix element of the kinetic energy manipulation similar to the overlap case:

B-'AAXB-lu. can

be obtained

I

rA7rjfKLM(u, A, x))

(fKfLM(Uf7A!jX)j 2

3

(2r)N-1

h2(2K+L)!(2K+L)!

x

detB

BKLBK"L

2

E f RH(K

k, q)H(K'

-

2

-

k, q)H(L + 2k, p)

k

+PH(K

-

k

-

1, q)H(K'

-

k, q')H(L + 2k, p)

(A.9) by using

a

250

Appendix

+P'H(K

k, q)H(K'

-

-

k

-

1, q)H(L -IF 2k, p)

+QH(K-k,q)H(K'-k,q')H(L+2k-l,p)lBkLThe

case

K'

with K

=

0 reduces to

a

(A.10)

simple result

very

(ALM(U17A x) I --iAw I fOLM (u, A, x)) 2 ,

h? 2

A.1.3 Next

(2,x)'V-1

(2L + 1)!! (R+LQp- )-

detB

4-x

3 2

)

L

(A.11)

P

Two-body interactions

we

derive the matrix element for the interaction of Eqs.

(2.3)

and

(11.1). To evaluate the matrix element of the operator expressed as a tensor product, it is convenient to make use of the famous WignerEckart theorem

spherical

[70, 71, 721, which states that a matrix element of a operator 0.-, between states with angular momenta

tensor

JM and XIW

coefficient and

be

can a

expressed

as

product of

a

reduced matrix element which is

z-components of the

(J'M'JO-AIJM)

angular

a

Clebsch-Gordan

independent of the

momenta:

(JMrILJYM-) =

,V2J'

V 110. 11 J).

(A.12)

+ I

Here the reduced matrix element is barred matrix element. The factor

expressed by the so-called double-

I/V -2-Y+I is factored out because

then the reduced matrix element is

symmetric

to the bra and ket

interchange

IEV 110. 11 X) provided of

OKIL

is

=

(-I)J+r.-J,(y 110. 11 J),

(A-13)

that the matrix element is real and the Hermitian

(0'1')

t

=

E(- 1)

"L

Or,, -i-L

with

a

phase

factor

E(E2

conjugate =

1).

By applying the Wigner-Eckart theorem we can express the matrix element for the orbital and spin angular momentum coupled wave function

as

follows:

(Tf(LI SI) JM IV(I'ri

-rj 1)

(Or ij (space) 0,,ij (spin)) ITf(LS) JM)

U(LnJS; L'S) V,f(2L' + 1) (2S + 1)

-

A.1 Correlated Gaussians

(L' 11 V(Iri

x

rj J)0,,jj (space)

-

251

11 L) (S' 11 0,,jj (spin) 11 S). (A.14)

potential V(r) is assumed to be a function of r only. Eq. (A.14) and its extension to a general coupled tensor operator is given in Exercise A.I. The reduced matrix element of the spatial part is obtained through Here the form factor of the

The derivation of

(LMn1ijL'M') (L' II V(I ri v/2--L'+I (L'IWIV(Iri

=

jV(r)(LM'jJ(rj case

uncoupled by

where

wave

-rj

(_1)r. 2S'

one uses

-

I I L)

-

r)O,,,,,j (space) ILM)dr.

the orbital and spin matrix element

(A.15)

angular momentum

(A.14)

can

be obtained

(0,,,j (space) 0,j (spin)) lTf(LS)JM)

ri 1)

-

U(LKJS'; LIS)

+ I

2S+1

x

-rj

function, the

R(L"S")JM IV(Iri

rj 1) 0,,i, (space)

(space) ILM)

=

In the

-

(LMLts1-tjLIML')(SMSrvjS,MS)

(TfL,MLS,ms, IV(Iri -rj J)0,st,,j (space)Oc,jj (spin jTfLmLsms) (A.16)

Since the spin matrix element is easily calculated as will be shown in Appendix AA, we will focus on the spatial matrix element and evaluate it for the most important components of the nuclear potential, that is, central, tensor and spin-orbit components.

(i) central

and tensor interactions

The operator

Or-A,j (space)

can

be I for the central

potential

and

Y2,,(r-i---r-j)

for the tensor operator. See Eq. (11.7). Thus both the central and tensor components can be treated by assuming the form

---rj) for 0,,,,,3. (space). Expressing ri

of Y,,,,, (,r-i

(N 7.1,

-

1)

we

x

1 column matrix

-

rj

as w (ij) x

Oj) defined iii Eq. (2.13)

and

with the

using Table

obtain

(fK'L'M'(UI74 x)IS(ri

-

rj

-

r)Y,,j,(rj

-

rj) IfKLM(U7 -47 X))

Appendix

252

'::::-

6)(fKLM(UIi W 7x)IJ(W(ij)X-r)jfKLM(u,A,x))

Yrg(

(27r)N-2C

1

2

1

e-fc7'Y-"(i )BKLBKL' X

if

de^- d,

detB

3 2

)

d 2K+L+2K'+L'

iYjLM(e,-) YJL, M,

dA2K+LdAI2K'+L'

XeXp[q,X2+ A/2+ Aye.e/+7Ae.,r+,Y/,XIe/.,r] (A-17) where C

w(ij)B-lw (ij)

c

I

^/2

q=q-

712

q'-

2c

w(ij) B-1u, =P-

2c

cw(ij)B-1U',

7

1'7YI.

(A.18)

C

All of these quantities

depend on i and j but we omit the labels i and j to simplify notations. The integration over the angles of e and e' can be performed by expanding exp (pAVe- e' + -/Ae- r + -l'.Ve,- r) in -

power series and

-

using the relation

(e.e )nj(e.T)n2(eI.,r)n3 n2 +n3

Lel=l

njn2n3

RL Lfn

[[YjL( )

X

Y

X

(A.19)

Y. 00

LL'r.

with Rnjn2n3 2 LIts

=

(_I)nl+n2+n3

Bnl-LI7 Bn2-12 12 Bn,3 -13 2

'1

2

2

3

111213

E(2L+:1:1 ) (2

V

r.

+

212+1

1)

C(1112; L)C(1113; L)C(1213; K)

U(IIIILK; L'12).

(A.20)

See Exercise A.2 for the derivation of this relation. In the

defining

I or 0, and 12 equation for R the sum over 11 is limited to ni, ni 2, and 13 have similar ranges, respectively. Possible values of L, L", and r, -

...'

A.1 Correlated Gaussians

for

a

given set of values of ni, n2, and

n3

limited

axe

253

by the conditions

that L takes the values nj +n2, nj +n2 2, ..., I or 0, and L' and r, take similar values given by ni, n3 and n2, n3, respectively. In addition, the -

is restricted to

sum over

L, L,

for

values of

given

and n2 + n3

-

would vanish.

K

r.

even

values of L + L+ K.

Conversely,

L, L', and K, all of the nj + n2 L, nj + n3 L, have to be non-negative and even, otherwise Rnin2n3 -

-

LL'r.

R has the symmetry: The reduced matrix element becomes

Clearly

(fK'L'(UIi -4 i X) 11 V(17'i

-

ri Dyt#

njn2n3

RLLII,,

pnjn3n2 "L'Ln

i -rj) 11 fKL (u, A, x)) 'R

Lf

(2K + L)! (2K'

+

L') 1

BKLBKILI

E

x

(

(27) N-2 detB

C.)2

H(ni, #)H(n2, -y)H(n3, 71) I(n2+n3+2,c)RnL,-n2n3 LIn

n,n2n3

2K + L

-

XH

nj

-

2K+L'-ni-n3

n2

2

2

(A.21) where the

I is defined in

Eq. (7. 10) potential for different pairs of particles be calculated with the above formula by changing only 01) in

integral

The matrix element of the can

Eq. (A. 18). We note

some

useful

applications

of

Eq. (A.21). For example,

calculate the matrix element of I ri-rj I' simply by putting r' and K 0. The two-paxticle correlation function can

one

V(r)

=

(fK-'Ll (u, A', x)

J (I ri

-

rj

a) Y,, (ri

-

rj)

fKL (u, A, x))

(A.22) easily calculated by taking J(r a) for V(r). K' We note again that the formula (A.21) simplifies for K 0, that is, the triple sum reduces to a single sum over, say ni (nl_ 0, min( L, L', (L + L' r,)/2)) and n2 and n3 have to satisfy L and nj + n3 L, respectively. In addition, in this nj + n2 njn2n3 calculated is R case simply by a term with 11 special nj, 12 n3 alone in Eq. (A.20). This simplicity will be used to obtain n2, 13 the matrix element of a density multipole operator. See Complement can

also be

-

=

=

=

-

=

=

=

=

A.2 for the details.

254

Appendix

(ii)spin-orbit The operator

angular

interaction

0,,,,,, (space)

momentum

((ri

rj)

-

1-

x

2h

and

and

(2.14)

trix element in Table 7.1 into

(fK"L"M"(Ufi Al7X)jV(jT'i

substituting

pj))

See

,.

x

and

Eq. (11. 9). -r,

with the

we

the

corresponding

ma-

i

obt

I'jj)1jzjjjfKLM(u, A, x)) (27)N-2

.1 Cr2

drV(r)e-2

-

is the orbital

1,

Eq. (7.2),

-

(pi

in terms of

x

Eqs. (2.13)

of

=

spin-orbit potential

Fij)x W)7r)

Expressing 1,_,,j use

1,1i,

for the

3 2

C) I

detB

BKLBK'Ll

d2K+L+2K'+L'

dA2K+LdA/2K'+LI q

x

exp

X

ifn*

+

X

qf A/2

r)t,

-

+

PAy e. ef + yXe.,r + -Y/Ale/. -

q'Af (e,

X

(A.23)

r),,

with 77

=

(MXB-lu,

Note that to derive

an

arbitrary

=

Eq. (A.23)

V(r) e-c('-a)2 (r for

q'

vector

x

a

a)dr

(MAB-luf. use

=

(A.24)

is made of the relation

(A.25)

0

provided V(r)

is

a

function of

r.

See

Eq.

(A-163). Before

performing the operation prescribed

that the matrix element vanishes in the has

parity

(_I)L

and because the

case

Eq. (A.23), we note 0 L. Because fKLM

in

of L

spin-orbit potential

does not

change

A. 1 Correlated Gaussians

parity, IL

-

L'I

has to be

even.

On the other

hand,

255

the tensorial char-

spin-orbit potential imposes the condition IL LI :5 1. Both the conditions are met only when L is equal to L'. This special result is entirely due to the unique feature Of fKLM and does not always hold for general wave functions. Combining Eq. (A.19) and the relation acter of the

i

fqX(e

-

r) 0

X

4vf2--x 3

-

77W (e'

rjqA[Yj(ii)

X

x

r)

Yj(i )Jj, -,qA[Yj(( )

x

YI(,P)JI, (A.26)

yields

the reduced matrix element

(fK,ILI (Ul A! X) I I V(I'ri I

I

as

-rj 1) Iij

I I fKL (u, A, x)) '3

4 V2--x

JLLI (-1)

3

x

E

q

L

(2K + L)!(2K' + L)!.

(27r)

( N-2C.)2 detB

BKLBKIL

l(n2+n3+3,c)H(nl,p)H(n2,,y)H(n3,-y')

n,n2n3

2K+L-nl-n2-1 H

xH 2

(2K'+

L

-

nj

-

n3,,,)

2

n., n2 ns

C(AI; L)U(LAII; 1L)RAL1

x

A

I(n2+n3+3,c)H(nj,#)H(n2, -y) H(n3 7f)

+77'

,

njn2n3

H

x

(

2K+L-nl 2

-n2

q)H(

2K+L-nj

njn2n3

C(AI; L)U(LA11; 1L)RLAI

-n3

-

1

2

(A.27)

straightforward and easy to follow. As a simple check of the above formula, the matrix element of the orbital angular momentum is calculated in Exercise A.3. It is again possible, however, to get a simpler formula by performing the'P integration first, as was done in Complement A. 1. This task is reserved for Exercise A.4. The above derivation is

256

Appendix

A.1.4

Density multipole operators

The matrix element of

density multipole operator plays a substantial role in investigating the properties of a system, e.g. the density, the deformation, the electromagnetic transition rates and the electron scattering form factors. The basic element of the Multipole operator a

takes the form

0,,,L, (space)

:---

f (Iri

xm

-

I)YI,

(A.28)

Note that the argument of the

density multipole operator is not ri but is correctly taken as ri xN, which is the single-particle coordinate measured from the center-of-mass coordinate. Comparison of Eqs. (2.12) and (2.13) immediately suggests that the matrix element of the density multipole operator ought to be calculated in exactly the same manner as that of the two-body potential. In fact the reduced -

matrix element

(fK"L-'(U/7 X7 X) 11 f (ITi

-

XNI)yn(Tii

-

N)

fKL (u, A, x))

XN

(A.29) be calculated

can

replacements

of

by Oj)

the --+

same

formula

as

0) inEq. (A.18)

Eq. (A.21) and of

with the trivial

V(r)

--+

f (r)

in

Eq.

(7.10). Just

as

the matrix element for the

lated from that of relation function

As

we

J(ri

can

xN

-

-

r),

and ri

be calcu-

the matrix element for the

also be derived ftom. that of

J(ri

-

rj

-

-

-

reduced to the

are

r)jfKLM(u,A,x)).

in fact obtained in the derivation

one

(fKfL'M" (U , X7 X) IJ(17VX (2K + L)! (2K'

+

BKLBK'L'

-

L)!

following Eq. (A.17),

r) IfKLM(u, A, x)) (2T,

)N-2C)

32

detB

E(LMr.M -MjLM)Y,,m, -m(i )* M

of type

This matrix element

the fi-nal result:

x

r).

-

(fK'L'Mf(U'jA!jx)jJ(iv-x give

cor-

-

already noted in the above derivations, both of ri xN rj are expressible in terms of the relative coordinates x as iv- being an appropriate 1 x (N 1) row matrix. There-

fore all of these matrix elements

we

can

have

Cvx with

was

density operator

L'

-1) -e -v/-2Ll + I

2

so

that

A.2 Correlated Gaussians with different coordinate sets

E

X

257

rn2+n3 Rnin2n3 H(nl, P) H(n2, -y) H(n8, -y') LLIn

nin2n3

2K+L X

H

-nI -n2

2K' + L'

-

H

2

n,

-

n3

2

(A.30) 7/,

given in exactly the same manner as in Eq. (A.18) with w(W being replaced by w. Here, r, takes the values

where c, 7,

p

are

L+L', L+L'-2,..., IL-L'I,

and it is of course restricted

by IAF-MI :!

The above matrix element becomes very simple for the special case K' of K 0 as was noticed in the previous subsection. Since the is.

=

=

applications, we show in Complement A.2 simplified to just a double sum.

matrix element has useful

that

Eq. (A.30)

can

be

A.2 Correlated Gaussians with different coordinate sets As the correlated Gaussians treated in the previous section have a very simple transformation property implied by Eq. (6.29), we as-

they are all expressed by a particular set of the relative coordinates x. However, the correlated Gaussians with the angular function OLM(X) given by Eq. (6.3) do not have such a nice property. Moreover, the use of different sets of the relative coordinates for these correlated Gaussians leads in general to a faster convergence because they allow us the possibility of describing naturally different types of correlations and asymptotics. The lineax transformation of the coordinate sets, however, leads to a formidable task even for a system of only four particles, because the function OLM(X) shows no simple transformation rule. See Appendix A.5 where the matrix elements for a three-body system are explicitly evaluated by transforming the corsumed that

related Gaussians from

one

Jacobi coordinate set to another. The aim

of this section is to outline ments for are

we

a

method of

calculating

the matrix ele-

the correlated Gaussians which

N-body system using expressed by different sets of the coordinates. By making use of Eq. (6.22) to generate the correlated Gaussian, have to evaluate the following matrix element for the operator 0: an

N-1

exp,

2

x-fXx)

11 Yij, j, (x ) j=1

Appendix

258

N-1

-1:Z.Ax)

1

0 exp

x

N-1

Yli.i (xi)

2

Climi,9ai

N

ali-Mi

01i

-

1

01j,i +Mi'

(-')Mj 01j,

3

C11 -mlj aal - ar'1j'+mi' j j j

97i j=1

(A.31)

(g(a'Jt';A!,x')J0Jg(aJt;A,x))

x

ri=O,-r =O 3

(-1)'YI-,,,(xi) to identity (Yl,,,(xi))* take care of the complex conjugation in the bra side. Note that the vector ti is defined by 2, i(l + i 2), -2-Fi) and likewise t!j by Here

(1

is made of the

use

T,

/2' i(I + T/2), -2,Tj j

complex conjugated.

transformation T trix element

x'

as

(A.31)

of g in the bra side should Assume that x' is related to x by a linear

The vector

j

not be

=

=

tj'

Tx. With the

between two

g's which

different sets of relative coordinates

Eq. (6.28) expressed in

of

use are

the

ma-

terms of

be reduced to

can

(g(a'lt'; X, x')J0Jg(aJt;A, x))

(g(Ta'lt';i XTx)101g(alt;A,x)), and

can

(A-32)

be obtained from Table 7.1 for most operators. 1, the matrix element

(A.32) takes, where v alt+ except for a trivial constant factor, the form e2 B' is matrix the and x given by (A+ TXT) (N 1) (N 1) Ta'It' For the unit operator 0

=

-IbB'V

=

,

-

-

To

simplify the

SN-1,

a.el

=

aNtIV

=

we

SN,

---,

rename

alt,

afN-,eN-l

=

expressed by a (2N 2) by 9Bs, where B is expressed in

6B'v

Then

notation

matrix B

can

be

-

aN-1tN-1 a2N-2t2N-2 82N-2-

-=

si,

x

(2N

-.-,

-

2) symmetric

terms of B' and T

as

follows

B

=

Thus the

(

B'

TB'

Bi TB'i

(A.33)

operation prescribed

in

Eq. (A.31)

can

be reduced to the

form N-

Oi",`

exp,

(2IBs

+ ibs

))

(A.34) .1=0'---'12M-2=0, 1=0,---,'2N-2=0

A.2 Correlated Gaussians with different coordinate sets

259

with

(A.35)

i9ail 0-Til-m Here the factor iv-s is included because it is needed in the

case

of the

potential energy matrix elements. See Table 7.1. It is worthwhile noting that the present formulation leads to a unified prescription of Eq. (A.34), which is independent of the choice of the relative coordinate sets but requires only a very simple calculation of the matrix B. It would be extremely tedious if one were to try to rewrite the anguN 1 lar function of fL'=I Yj . (x'.) in terms of those angular functions 3 conforming to the coordinates x. Using Leibniz's formula we can express Eq. (A.34) as 3

2

2

T

3

=1

Aj! (Ij

'XiAi

-

Ai)! (Ai

(1i Mj)! (1i

-

-

-

Mi)! mi

-

Ai

+

Ai)!

2N-2

Oj"

X

exp

(2 Os) -ri=O

2N-2

ai

X

ie,

(A.36) aj=O -ri=O

The last factor is

easily evaluated by using Eq. (6.58):

2N-2

11

ali-'Xi'mi-Ai e-vs i Ui=O -ri=O 2N-2

(A-37)

CIi-'XiMi-/-tiY1i-)LiMi-Ai(Wi)-

Possible values of Ai and [ii are determined by the conditions that Oi=l

where

and

mij's

li's

are

all

non-negative integer and

must

satisfy the

following relations K

K

mij

=

k,

mij +

j>i=l

=

(i

ni

K).

1,

(A. 172)

j=1

A.6. Derive

Eq. (A.93).

Solution. First x

E mji + li

(

2 x

we

note that

(Al) '+Ae-2IpX2_,5X.y YLM 41

M)

(,Axe- .j'8X2_jx.Y) X2v+Ay(,Xl) 2

LM

+

e-12 jax 2-,6X.y A.'X 21/+Ay(,\I) LM PC

+2

(Vxe-2.jpX2_jx`.Y) (VXX

M)

2v+,X

-

AI) YL(M P9M

The differentiation in the second term of the

above equation

(AxX

2v+,\

can

be

right-hand side

performed by using Eq. (6.59)

AO

I

The third term

can

be

reduced, by

means

to

(Vxe- .IpX2_jX.y 2

of the

as

AI) YL(M

2v(2z/ + 2A + I)x 2v+,X-2 YL(M PC M (A.42),

(A.173)

VXX

2v+A

AI) YL(M (ic'

of the

(A.174) gradient formula

Exercises

X2v+X-l e

F+' OX

1

oX2 -,6x.,y

(2v + 2A + 1)

1

-A-+I 2A+1

297

X

YX-1 ('' )4

X

Y' Ml LM

2v[[(- x-Jy)XYX+I( 3)11\Xyl( )ILM (A.175)

By using the angular momentum algebra the equation is reduced to

first term in the round

bracket of the above

[[(-OX

-

JY)

,rL3

X

Y)'-' POD'

X

Y' M] LM

OxC(1A-I;A)YL(A')(:t, ) M

+ Sy

A

U(A-1 I L 1; A r,) C(11; r)Y(X-l LM

2A+lt XYL

X1) M

Jy E U(A

r.)

( O,

-

11 L 1; A

K) C(11; r,) YLM

K

(A.176) where

use

Similarly

is made of

Eq. (A.156)

for

C(l A

-

1;

A)

C(A

-

11; A).

the second term becomes

11(- X-*X YA+'(" )IXX Y1MILM XY(,X1) L M PC M I

yE U(A+I ILI; Ais)C(11; r,)YL(A+1r')(1b, M

The contribution of the first term is

(A.177)

easily obtained by using

Exercises

298

2 e- .l#X2_,5X.y

1

+

02X2 + j2y2 + 2,3dx -,y) e-'2,8X

where the last term

involving

the scalar

2

product

-,5x.y

(A.178)

1

x

-

y

can

be reduced

to

X.

Y Y(Ai) LM

=

,

E

-v -3[x

x

y]oo0'1) LM

U(11 ; 0"0 Ry

X

[X

X

Ily

X

YA-1 4A A

X

Y' MI LM

_X

Ry

X

Y +'P

X

YI(MILM*

-V

3

Y

X

X

1WILM

Y

r.=Al

2,X + I

(A.179)

right-hand side of the above equation appear Eqs. (A.176) and (A.177), respectively. Combining these results we obtain the desired equality (A.93) The two terms of the

Ax

(X

2 Ll +,X

l,3X2_,6x.y

e-2

AI) YL(M

30X2 +,32X4 + j2X2Y2 + 2zj(2i/ + 2A +

2 -2Sx 2v+A-l y e-LPx

I

X

x

):+

TA

I

in

1)) 2

X

1 _

-

20(271 + A)X2

2v+,X-2

2

_,SX.y

LpX2_jX.y

e- 2

V/

YL(M

7F

3

(2y + 2A + 1_ OX2)

EU(A-IlLl;Aiz)C(11;K)YL(A-l)(. c, ) M

Is

'A+I -

2A +I

x

(2y

_

OX2)

I:U(A+IILI;Ar,)C(11;r,,)YL K

M

(A.180)

Stochastic Variational Approach to Quantum-Mechanical Few-Body Problems

Stochastic Variational Approach to QM Few-Body Problems

Stochastic variational approach to quantum-mechanical few-body problems

Variational Approach to Fracture

Lagrange multiplier approach to variational problems and applications

Convex variational problems

Variational problems in geometry

Branching solutions to one-dimensional variational problems

A Variational Approach to Structural Analysis

From discrete to continuum: A variational approach

A variational approach to structural analysis

Convex analysis and variational problems

Bayesian Approach to Inverse Problems

A Variational Approach to Structural Analysis

Bayesian Approach to Inverse Problems

Bayesian approach to inverse problems

Bayesian Approach to Inverse Problems

Variational problems in materials science

Convex Analysis and Variational Problems

Convex analysis and variational problems

Variational Problems in Materials Science

Applications of variational inequalities in stochastic control

Applications of Variational Inequalities in Stochastic Control

Stability Problems for Stochastic Models

Stability Problems for Stochastic Models

Stability Problems for Stochastic Models

Stability Problems for Stochastic Models

One-Dimensional Variational Problems: An Introduction

One-dimensional variational problems: An introduction

Finite-dimensional Variational Inequalities and Complementarity Problems

Topological Degree Approach to Bifurcation Problems

Stochastic Variational Approach to Quantum-Mechanical Few-Body Problems

Stochastic Variational Approach to QM Few-Body Problems