Lecture Notes in Chemistry Edited by:

Prof. Dr. Gaston Berthier Universite de Paris

Prof. Dr. Michael J. S. Dewar The University of Texas

Prof. Dr. Hanns Fischer Universitat Ziirich

Prof. Dr. Kenichi Fukui Kyoto University

Prof. Dr. George G. Hall University of Nottingham

Prof. Dr. Jiirgen Hinze Universitat Bielefeld

Prof. Dr. Hans Jaffe University of Cincinnati

Prof. Dr. Joshua Jortner Tel-Aviv University

Prof. Dr. Werner Kutzelnigg Universitat Bochum

Prof. Dr. Klaus Ruedenberg Iowa State University

Prof Dr. Jacopo Tomasi Universita di Pisa

59

S. Fraga M. Klobukowski J. Muszynska E. San Fabian K. M. S. Saxena 1. A. Sordo T. L. Sordo

Research in Atomic Structure

Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest

Authors

S. Fraga M. Klobukowski 1. Muszynska E. San Fabian K. M. S. Saxena 1. A. Sordo T. L. Sordo Department of Chemistry, University of Alberta Edmonton, Alberta, Canada T6G 2G2

ISBN-13 :978-3-540-56237-5 e-ISBN-13:978-3-642~93532-9

DOl: 10.1007/978-3-642-93532-9

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law.

© Springer-Verlag Berlin Heidelberg 1993

Typesetting: Camera ready by author 52/3140 - 5 4 3 2 1 0 - Printed on acid-free paper

To

Harry E. Gunning

Preface

Impressive advances have been made in the study of atomic structures, at both the experimental and theoretical levels. And yet, the scarcity of information on atomic energy levels is evident

At the same time there exists a need for data, because of the developments in such diverse fields as astrophysics and plasma and laser research, all of them of fundamental importance as well as practical impact.

This project of research in atomic structure, consisting of three components (formulation, computer program, and numerical results), constitutes a basic and comprehensive work with a variety of uses. In its most practical application, it will yield a rather accurate prediction of the energy levels of any atomic system, of use per se or in the interpretation and confirmation of experimental results. On the other hand, it will also be of use in the comparative study of the appropriateness of the various levels of approximation and as a point of reference.

Work on this project has extended over several years and a number of researchers have collaborated, at different stages of its development, during their stays at this laboratory: M. Klobukowski organized the generation of the SL-functions, J. Muszynska developed most of the basic formulation for the matrix elements, K.M.S. Saxena contributed his expertise on numerical integration, and E. San Fabian, J.A. Sordo and T.L. Sordo participated in the study of hyperfine­structure interactions. In addition J.D. Climenhaga and P. Clark adapted the original program, written for IBM-type mainframes, for use on CDC and CRA Y machines, respectively, and J. Jorgensen has been in charge of the word-processing component for the complete project.

Serafm Fraga Edmonton, Alberta

Canada

Table of Contents

Introduction ............................................................................... . 1

Theoretical Foundation ................................................................. 7

1 Hamiltonian Operator and Eigenvalue Equations ..................... 9 1.1 Hamiltonian operator ....................................................... 9

1.1.1 Extended Breit Hamiltonian operator ............................ 9 1.1.2 Generalized Hamiltonian operator ............................. 12

1.2 Eigenvalue equations ..................................................... 16

Basic Theoretical Formulation ............... ...................................... 19

2 Angular Functions: Coupling of Angular Momenta ............... 21 2.1 One-electron functions ................................................... 21 2.2 SL-functions .............................................................. 24 2.3 JM]- and FMF-functions ................................................. 28 2.4 Selection of functions .................................................... 32

3 Tensor-Operator Formulation ............................................. 35 3.1 Tensor operators .......................................................... 35 3.2 Wigner-Eckart theorem .................................................. 36 3.3 Reduced matrix elements ................................................ 37 3.4 Matrix elements ........................................................... 42

Application of the Basic Formulation ........................................... 45

x

4 Transformation of Operators to Tensor Form ....................... 47 4.1 Basic operators ............................................................ 47

4.1.1 Operators s(1),,1(1) and C(k) .••••.....••••.....•.•.......•..... 47 4.1.2 Other common operators ........................................ 53

4.2 Transformation rules ..................................................... 54 4.3 Application ................................................................ 62 4.4 Summary .................................................................. 62

5 Matrix Elements ................................................................ 69 5. 1 General formulation ...................................................... 69 5.2 General expressions ...................................................... 71

5.2.1. SMsLML-coupling .............................................. 71 5.2.2. 1M]-coupling ....................... , ............................. 72 5.2.3. FMF-coupling .................................................... 74

5.3 Examples for specific interactions .......................... , ........... 75

6 Summary of Theoretical Results ......................................... 79 6.1 Electronic energy ......................................................... 81 6.2 Mass variation ............................................................. 82 6.3 Specific mass effect ...................................................... 83 6.4 One-electron Darwin correction ......................................... 84 6.5 Two-electron Darwin correction ........................................ 85 6.6 Electron spin-spin contact interaction .................................. 86 6.7 Orbit-orbit interaction .................................................... 87 6.8 Spin-orbit coupling ....................................................... 89 6.9 Spin-spin dipole interaction ............................................. 92 6.10 Magnetic dipole and Fermi contact interactions ....................... 93 6.11 Electric quadrupole coupling ............................................ 95 6.12 Magnetic octupole coupling ............................................. 96 6.13 Zeeman effect (low field) ................................................ 97 6.14 Zeeman effect (high field) .............................................. 103 6.15 Zeeman effect (very high field) ........................................ 104 6.16 Stark effect ............................................................... 104 6.17 Nuclear-mass dependent orbit-orbit interaction ...................... 106 6.18 Nuclear-mass dependent spin-orbit coupling (electron spin) ....... 107 6.19 Nuclear-mass dependent spin-orbit coupling (nuclear spin) ........ 108

Implementation ........................................................................ . 111

XI

7 Practical Details ............................................................... 113 7.1 Selection of configurations ............................................. 114 7.2 Detennination of radial functions ...................................... 114 7.3 Selection rules ............................................................ 115 7.4 Masscorrections ......................................................... 119

8 Numerical Examples ......................................................... 121 8.1 Accurate energies ........................................................ 121 8.2 SLJ energy levels ........................................................ 122 8.3 Hyperfine-structure splittings .......................................... 126 8.4 Nuclear-mass dependent corrections .................................. 129

Bibliography ............................................................................ 131 References ......................................................................... 131 Reference texts .................................................................... 134 Data sources ....................................................................... 135

Units and Constants .................................................................. 137 Constants .......................................................................... 137 Units ............................................................................. 138

Notation and Symbols ............................................................... 139

Acknowledgements

The authors acknowledge the kindness of the following organizations for the use of copyright material:

American Physical Society; John Wiley & Sons, Inc.; National Research Council (Canada); Springer Verlag; Argonne National Laboratory (managed by The University of Chicago for the U.S. Department of Energy under Contract No. W-31-109-Eng-38).

Specific mention must be made, in particular, of the following works from which copyright material has been taken:

S. Fraga and J. Karwowski. Relativistic Treatments for Bound-State Atomic Energies. Theoretica Chimica Acta (Berlin) 35,183-187 (1974).

S. Fraga, M. Klobukowski, 1. Muszynska, K.M.S. Saxena and J.A. Sordo. Matrix Elements of the Breit Hamiltonian. Physical Review A34, 23-28 (1986).

S. Fraga, E. San Fabian, J.A. Sordo, M. Campillo, J.D. Climenhaga and M. Klobukowski. Atomic Energy Levels from Configuration Interaction Calculations with Relativistic Corrections. International Journal of Quantum Chemistry XXXV, 325-330 (1989).

R.A. Hegstrom. Nuclear-mass and Anomalous-moment Corrections to the Hamiltonianfor an Atom in a Constant External Magnetic Field. Physical Review A7, 451-456 (1973).

M. Klobukowski and S. Fraga. Accurate Theoretical Prediction of the Experimental Ground-state Total Atomic Energies. Physical Review A38, 1593-1594 (1988).

W.C. Martin. 'Energy Levels of Highly Ionized Atoms'. In 'Proceedings of the Workshop on Foundations of the Relativistic Theory of Atomic Structure held at Argonne National Laboratory, December 4-5, 1980'. Argonne National Laboratory, Report ANL-80-126, March 1981.

E. San Fabian and S. Fraga. Hyperfine-structure Interactions: Preliminary Results. Canadian Journal of Physics 66,583-585 (1988).

T.L. Sordo, J.A. Sordo and S. Fraga. Nuclear-mass Dependent Spin-orbit Interaction. Canadian Journal of Physics 69,161-163 (1991).

Introduction

The ultimate goal of research in atomic structure is the complete and accurate determination of the energy levels of any given atomic system, whether neutral or charged.

As such an objective is to be reached through the practical implementation of the appropriate formulation in numerical calculations, progress in this field has been dependent on the availability of fast, large-scale computers.

In fact, although the basic theoretical developments (variational principle and perturbation theory), approximations (use of one-electron functions), methods (self­consistent field and configuration interaction formalisms), and techniques (tensor operator algebra and numerical integration) were developed at an early stage, the level of sophistication at which they have been used throughout the years has paralleled the developments in computer technology, as exemplified in the work carried out at this laboratory.

The first step consisted of the evaluation of expectation values of simple operators and some physical quantities from analytical Hartree-Fock functions, yielding the first comprehensive descriptions [Fraga and Malli (1968), Fraga et al. (1969]. This work was later expanded, using numerical Hartree-Fock functions, into a more sophisticated and exhaustive description of the ground states of all the neutral atoms of the Periodic System and of many of their positive ions [Fraga et al. (1971; 1972a, b; 1973; 1976)].

Equivalent calculations could have been carried out next for excited states, but such an enormous effort would have been wasted to a large extent, if executed at the monoconfigurationallevel. Multiconfiguration Hartree-Fock calculations, on the other hand, would have been feasible but simultaneous consideration of the relativistic corrections was beyond the computing capabilities of the time.

It was possible, however, to incorporate such a treatment into a semiempirical scheme, whereby the fitting of experimental data could yield a satisfactory prediction of undetected levels [Fraga et al. (1979)]. complemented with the independent evaluation of pertinent quantities [Fraga and Muszynska (1981)].

At present it is finally possible to tackle the problem at the appropriate level of sophistication [Fraga et al. (1986, 1987)]. The approach adopted at this laboratory, consisting of an a priori multi configuration treatment, with consideration of the most significant relativistic corrections, using numerical Hartree-Fock functions, may be

2

rationalized on the basis of the following considerations [as discussed by Fraga and Karwowski (1974a)].

The calculations are based on the Breit generalization of the Dirac equation. Within the framework of the Hartree-Fock approximation, two approaches exist for the determination of approximate values of the relativistic energies. On one hand, the 'relativistic' method consists of the determination of the relativistic functions by a Hartree-Fock scheme based on the Dirac-Breit equation, from which the magnetic and retardation terms of the Breit correction have been omitted, and whose contribution may then be evaluated using the relativistic functions. [The omission of the magnetic and retardation terms of the Breit correction from a variational treatment and the evaluation of their contribution as a ftrst-order correction was discussed by Bethe and Salpeter (1957). An approximate form of second-order perturbation theory has been used in intermediate-coupling calculations; see the discussions by Condon and Shonley (1964) and Ermolaev and Jones (1972).] On the other hand, the 'non-relativistic' method involves the determination of the corresponding non­relativistic Hartree-Fock functions, which are then used for the evaluation of the contributions of the relativistic corrections as given in the non-relativistic form of the Dirac-Breit equation [e.g., as developed by Hegstrom (1973)].

Although the general trend has been towards calculations within the framework of the 'relativistic' method [see, e.g., the work of Pyykko (1986)], both approaches have merit, as seen from inspection of Table 1, where the following designations have been used: Er - energy evaluated within the framework of the 'relativistic' method, with the

nuclear potential modifted in order to take account of the ftnite size of the nucleus <ErJn) or considering a point nucleus <Er,pn); it does not include the contributions of the magnetic and retardation terms of the Breit correction, the electron-nucleus interactions (other than the electrostatic interaction) or the speciftc mass effect.

Em- - energy evaluated within the framework of the 'non-relativistic' method, considering a point nucleus; it includes the contributions of the magnetic and retardation terms of the Breit correction, all the electron-nucleus interactions, and the speciftc mass effect.

ED - contributions from the magnetic and retardation terms of the Breit correction; in the non-relativistic limit, it corresponds to the orbit-orbit and the electron spin-spin contact interaction [Armstrong (1966)].

ESM - speciftc mass effect L1Er - lowering of the 'relativistic' energy with respect to the 'non-relativistic'

energy (not including the contributions of the magnetic and retardation terms of the Breit correction or the speciftc mass effect), i.e., ~r = [Eur - (ED + ESM)] - Er,pn

~n - contribution to the energy, by comparison with the value obtained within the point-nucleus approximation, resulting from proper consideration of the ftnite size of the nucleus in the nuclear potential expression, i.e., ~ = Er,fn - Er,pn

The values in Table 1 show that, for Z<102, it is ~n«(ED+ESM), which indicates that there is no justiftcation in considering the effect of the ftnite size of the

3

Table 1. Total energies (in absolute value) and relativistic corrections (in hartrees) [Fraga and Karwowski (1974a)].

Total Energy Corrections Element Z Er,fna Er,pn b Enl &:r EB+ESMc ~ He 2 2.8618 2.8618 2.8617 0.0 0.0001 0.0 Be 4 14.5759 14.5759 14.5752 0.0 0.0007 0.0 Ne 10 128.6919 128.6920 128.6751 0.0007 0.0162 0.0001 Mg 12 199.9353 199.9024 0.0017 0.0312 A 18 528.6837 528.6854 528.5356 0.0194 0.1304 0.0017 Ca 20 679.7128 679.4916 0.0328 0.1884 Zn 30 1794.6122 1794.6277 1793.4731 0.4144 0.7402 0.0155 Kr 36 2788.8597 2788.8961 2786.2073 1.3159 1.3729 0.0364 Sr 38 3178.1267 3174.9710 1.4771 1.6786 Cd 48 5593.3191 5593.4802 5582.8536 6.9198 3.7068 0.1611 Xe 54 7446.8985 7447.2186 7426.7089 14.9856 5.5241 0.3201 Ba 56 8136.0530 8110.7773 19.0291 6.2466 Yb 70 14067.7301 14069.3615 13973.565 82.6543 13.1122 l.6314 Hg 80 19649.0836 19653.870 19430.911 202.4843 20.4747 4.7864 Rn 86 23602.4542 23611.517 23253.331 332.l107 26.0753 9.0628 Ra 88 25039.773 24622.183 389.4318 28.1582 No 102 36743.3761 36793.382 35629.187 11l8.2363 45.9587 50.0059

aMann and Johnson (1971).

bMaly and Hussonnois (1973).

CFraga et al. (1971, 1972a, b, 1973, 1975).

[Reprinted with permission of Springer Verlag.]

r SPECTRUM

Z H.

5 I.

e 0

10 ~.

15

A ,

20 e. T;

25 e, I.

~ ;

30 Zn

G.

35 s. K,

S,

40 Z,

"'. 45 R.

Pd

Cd 50 Sn

To

55 x. I.

e. 60 ~d

65

f' 70 Yb

HI

75 W

o. p,

80 H. Pb

85 P.

Rn

R. 90 T.

u

95 p.

e .. Ik

CI

100 1m Eo

~. Md L,

4

• VERY COMPLETE TO FAIR

181 FAIR TO FRAGMENTARY

o TRANSITION ARRAY IDENTIFIED ­NO lEVELS

Figure 1. Schematic representation of the information available on atomic energy levels [Martin 1981)]. [Reprinted with permission of the United States Department of Energy (Argonne National Laboratory).]

5

nucleus if the contribution of the Breit tenn is not included. The values of (EB+EsM) and (L1Er+~n) can be used to estimate the order of magnitude of the corrections neglected in the 'relativistic' (with finite nucleus) and the 'non­relativisitic' calculations, respectively. As observed in the Table, up to Z=48, (L1Er+~)<(EB+EsM), which indicates that for the frrst half of the Periodic System more accurate energy values, at least for neutral atoms, will be obtained within the context of the 'non-relativistic' method.

In addition, it must be remembered [Fraga et aL (1976)] that for Z<14 the correlation energy is greater (in absolute value) than the relativistic corrections (in absolute value). But even for Z> 14, correction for the correlation energy is required for an accurate prediction of energy levels, because of the different configurations involved (as the correlation energy correction depends on the configuration).

In summary: It is for these reasons that in this research project the efforts have been directed towards inclusion of the correction for the correlation energy (via a configuration interaction treatment) and consideration of the relativistic corrections (as a perturbation correction), starting from accurate non-relativistic functions.

This work presents now the corresponding theoretical treatment, summarizing the fonnulation for the required energy matrix elements, within a computer-based approach [Fraga et al. (1987)]. [For simplicity in the presentation of the material, the description of the notation, rather complex of necessity, has been collected in an appendix, so that its ready availability will simplify the reading.]

The reader interested in the application of this fonnulation through the use of the corresponding program to actual calculations on specific atomic systems may obtain a perspective of the work that remains to be done from inspection of Fig. 1, which presents graphically the information available [Martin (1981)] on energy levels. For sources of experimental data, the reader is referred to the publications of the National Bureau of Standards, U.S.A. [e.g., Moore (1949, 1952, 1958, 1970), Hagan and Martin (1972), Hagan (1977), etc.], the Journal of Physical and Chemical Reference Data (published by the American Chemical Society and the American Institute of Physics for the National Bureau of Standards) as well as to the references compiled in the International Bulletin on Atomic and Molecular Data for Fusion (published by the International Atomic Energy Agency, Vienna, Austria). The work of Martinson (1982) may serve as reference on the modem sources of data on atomic structure.

Theoretical Foundation

The determination of the energy levels of an N -electron atomic system implies the (approximate) solution of the eigenvalue problem corresponding to the appropriate Hamiltonian operator and therefore the ftrst step is to identify the operator to be used.

In this work we have adopted the extended Breit Hamiltonian operator transformed to non-relativistic form plus the relativistic and magnetic terms, to fIrst order in 11M, lima, m/M and m/ma (where m, rna and M denote the electron, nucleus and total mass, respectively), and augmented by inclusion of the additional terms corresponding to the interactions with the higher moments of the nucleus as well as with external electric and magnetic ftelds.

Solution of the corresponding eigenvalue equation is not possible and therefore a decision must be made as to what approximation is to be adopted. Inspection of the Hamiltonian operator suggests a perturbation-variation approach, in the form of a conftguration interaction treatment for the complete Hamiltonian operator, using either mono- or multi-conftguration Hartree-Fock functions as starting point

1 Hamiltonian Operator and Eigenvalue Equation

1.1 Hamiltonian Operator

1.1.1 Extended Breit Hamiltonian Operator

The Dirac Hamiltonian operator [Dirac (1928a, b)] for a one-electron atomic system is

JiD = ca-p + c28 + V(r)

where c is the velocity of light, p is the momentum operator (p = -iV), and the operators a and 8 are dermed by

8 =( 1 0) o -I

I is the two-dimensional unit matrix and the (}'P denote the Pauli matrices

(}'p=(O 1) x 1 0

at = (0 -i) y i 0

(}'p=( 10) Z 0 -1

V(r) is the central field potential and r is the position vector of the electron.

The Breit Hamiltonian operator [Breit (1929, 1930, 1932)] may be written as

where HI and H2 are one-electron Dirac Hamiltonian operators of the form

10

where A(ri) and ~(rj) denote the vector and scalar potentials of any electromagnetic field acting on the electron, including the nuclear potential. The tenn B 12 constitutes the Breit correction, defined by

and which is of the order of (aZ)2, where a is the fine-structure constant and Z is the atomic number.

The Breit Hamiltonian operator, extended to include the anomalous-moment interactions may be written [using the notation of Hegstrom (1973)] as

H = I. Hi + I. Uij i i<j

with

and

Hi = H +~. ej[aj x rij]/Iij J""I

In these equations, 1ti is the mechanical moment of particle i, Ai is the magnetic vector potential at position ri in the laboratory frame, H is the constant external magnetic field, and Hi and Ei are the magnetic and electric fields experienced by particle i and Ki is the anomalous magnetic moment of particle i.

This Hamiltonian operator, after elimination of the dependence on the centre of mass coordinates and reduction to a non-relativistic form (SchrOdinger Hamiltonian operator) plus relativistic and magnetic terms, is expressed [Hegstrom (1973)] as

11

K=I.mi+I.Kn (withn=O.I •...• 7) i n

with (in natural units. ~ = c = 1)

KI = - I. (1tt/8~) i

K2 = - I. I. (1tej/mi)(gillQi - ei/2mi)~(rij) i j'#i

K3 = - ~ ~. (ej/mi)(giJ.4)i - ei/2mJriI(si-[rij x 1ti]) 1 J'#l

K4 = ~ ~ (ej/mj)giJ.4)ifiI(si-[rij x 1tj]) 1 J'#l

Ks = - I. gil1Qi(Si-H)(I-1tf/2mf) - I. (giJ.4)i - ei/mi)(1!2mf)(Si-(1tf 1 - 1ti-1tj)-H) i i

where the summations extend over all the particles (nucleus and electrons). with mass. charge (affected by the appropriate sign) and spin denoted by m. e and s. res~tively. The g factor is defined by the relationship

gllO = 2(e!2m + K)

where IlO stands for the Bohr (IlB. for electrons) or the nuclear (IlN. for the nucleus)

magneton and K denotes the anomalous magnetic moment. The mechanical moment is defined by

1ti = Pi - ei(Ai - AR) - (mi/M)~ ejAj J

with

12

Ai = (1/2)[H x ri] AR = (l/2)[H x R]

where R is the centre of mass coordinate. M is the sum of the rest masses of all the particles in the system.

The above Hamiltonian operator applies only to particles with spin 1/2; for nuclei with a different spin, correct results (to frrst order in the inverse of the nuclear mass) are obtained if the observed nuclear magnetic moment and spin are substituted for the corresponding Dirac values in the Hamiltonian operator [Bethe and Salpeter (1957)].

1.1.2. Generalized Hamiltonian Operator

Developing the above expressions and retaining only the terms to first order in 11M, lima, m/M and mlIIla (where the subindex a denotes the nucleus) one can write, for a system with total momentum P=O and R=O,

41 (H) :n.l = Kmv + ~v

41 (m) (H) (H) (H) (mH) :n.3 + K4 = Ksol + Kso2 + Kmdl + Kso + Ksol +Kso2 +K~~h + Ksol

K5=~)+K~

(m) (H) (mH) ~ = Koo + Koo +Koo +Koo

where the subscripts stand for electronic (el), specific mass (sm), kinetic (k), mass variation (mv), Darwin (0), spin-orbit (so), magnetic dipole (md), nucleus (n), electrons (e), orbit-orbit (00), Fermi contact (Fc), spin-spin contact (sse) and spin­spin dipole (ssd), while the superscripts indicate the dependence on the external magnetic field (H) and the nuclar mass (m).

The development of the expressions, as outlined above, is straightforward and without complications but some comments are appropriate regarding the unit systems as discussed below.

The significance of the Hegstrom Hamiltonian operator lies in the fact that it allows us to obtain in a single step all the interaction operators, except for those involving the higher moments of the nucleus and the interaction with an external

13

electric field. Many of those operators were previously known, but they had been proposed on the basis of a variety of arguments, including the need for a proper interpretation of experimental data. [See, e.g., the work of Fraga and Malli (1968) and Fraga et al. (1975).]

It must be mentioned, however, that the Hegstrom operator, given in natural units (11= c = 1), will lead to expressions not showing explicitly the dependence on cor, equivalently, on the fine-structure constant a. Such a dependence (as well as the

fact that the relativistic corrections are of the order of ( 2) is observed in the usual expressions employed in the calculations, within the context of the so-called atomic unit system, with II = Ille = lei = 110 = 1. [In fact, because of the appearance of the reduced mass Il when the motion of the centre of mass is eliminated, it becomes necessary to operate within the Il-unit system, with II = Il = lei = 110 = 1, introducing the appropriate relativistic mass correction, with a final transformation to the universal m-unit system on completion of the calculation, as discussed in Chapter 7.]

For these reasons a compromise has been adopted in this work, as follows. The expressions of the interaction operators, to be used in the calculations, are presented in both Chapter 6 (in the usual fashion, in terms of vector operators) and Chapter 4 (in terms of tensor operators, as required for the evaluation of the matrix elements using tensor algebra). For completeness, in order to show how those expressions are obtained directly from the Hegstrom operator, the corresponding terms are presented below. In this connection it must be mentioned that because their main use is for comparative purposes, those expressions have been obtained for simplicity with neglect of the anomalous moment of the electron; that is, with gsllB == e/Ille,

IlB = e/2Ille, gs == 2, which has been a traditional approximation in the past. In such a case the expressions of the interaction terms are

ttsm = (11m.) 1: (Ppa· Po.) p<o

14

Kso2 = (e2flm2) I. I. rpA(Spe[2rpox Paa - rpax Ppal) p Oitp

Kmci = KFe + Kmcil + Kmd2

= (81te/3m)gIJ,LN I. (SpeI)S(rpJ p

+ (e/m)gIJ.1N I. rpl(Ie{[rpa x Ppal- s p + 3(SperpJ(rpJr~J}) p

K~':) = - (ZeI2mma) I. {rp1(PpaePoJ +rpa(rp~paeppJePoa} p<O

~) = -gIJ,LN(leH)

K~) = (e/2m) I. (SpeH) - (eflm2) I. (SpeH)P~a p p

K~~) = -(e3flm2) I. (rpb(AaaePpa + Apaepo.> p<O

+ r~(rp&r paeppJe Aaa + r pae(rpAr paeApJePaa)

K~~ = (z.e2flm2) I. rpl(Spe[r pa X Apa1) p

K~ = (e3flm) I. I. rpA(Spe[2rpa x Aa~ - rpax Apal) p Oitp

15

K~mH) = - (e/rnJ I. (ApaePoJ + (e212mJI. I. (ApaeAaJ p<a p 0I'p

K~~) = ('a,2{2mmJ I. {rpl(Ac:raePpa - ApaePc:rJ p<a

+ rpae(rp1rpaeppJeAaa - rpae(rplrpaeApJepc:ra}

where the summations extend over all the electrons, Z and e are used in absolute value and

Apa = (112)[0 x rpal

The generalized Hamiltonian operator may then be written as

Kg = K + K(m) + K(H) + K(mH)

where

K = He}. + Ksm + Keel

includes the electronic Hamiltonian operator Hea. (consisting of the electron kinetic, nuclear attraction and electrostatic repulsion energy terms), the operator for the

specific mass effect Ksm and the relativistic correction Keel. consisting of the mass variation (Kmv), the Darwin corrections (KDl and KD2), the electron spin-spin

contact interaction (KssC>, the electron orbit-orbit interaction (Koo), the spin-orbit coupling (Hso1 and H so2), the electron spin-spin dipole coupling (KsS<J> and the electron-nucleus magnetic dipole interactions (Kmd). The terms included in K(m), K(H) and K(mH) are those that contain a dependence on ma, K and rna and H, respectively, their subindices indicating the terms ofK to which they are related.

The complete, generalized Hamiltonian operator is finally obtained by adding terms (as discussed later in the text) for the interactions with the nuclear electric quadrupole (HQ) and the nuclear magnetic octupole (Ha) as well as with an external electric field (Ks).

[The operators in K, K(m) and K(H) have been studied extensively. The reader is referred to the work of Fraga and Malli (1968), Fraga and Karwowski (1974 a, b), Fraga et al. (1976), Fraga and Muszynska (1981), Sordo et al. (1991). References to past work may be found there as well as in the work of Hegstrom (1973).]

16

1.2.Eigenvalue Equations

The basic fact in the development of a theoretical formulation for atomic calculations is that the existence of the electrostatic repulsion energy tenns as well as the tenns embodying all the additional corrections, discussed above, make it impossible to solve exactly the eigenvalue equation corresponding to the complete Hamiltonian operator.

Recourse must be made therefore to computational methods which will allow us to generate solutions, numerically as accurate as desired, of those eigenvalue equations. Those methods are formally based on the successive application of perturbation and variational formalisms, as summarized below.

The complete Hamiltonian operator may be written, for the purpose of this discussion, as

where Ki consists of the electron kinetic and nuclear attraction tenns, He denotes the electrostatic repulsion energy tenns and He includes all the other corrections mentioned above.

The operator Ki is separable

2 Hi = l: {-(112) V p - Z/rpaJ = l: J£p

p p

in tenns of the one-electron operators Hp corresponding to the N electrons in the system. All these operators Hp are identical, except for the label identifying the electron under consideration, to a hydrogenoid Hamiltonian operator, whose solutions are easily obtainable. Those solutions are the one-electron functions and energies, c!>i and £j, denoted respectively as atomic orbitals and orbital energies, where the subscript i identifies the set of quantum numbers (~li needed to completely label each orbital. These orbitals may be separated into the product of a radial and an angular part

~(r,O,c!» = Rnt(r) Y.a.m(O,c!»

where r, 0 and c!> are the polar coordinates and Y.a.m(O,c!» denotes a spherical hannonic (see Chapter 2).

Solutions for the operator Ki may then be constructed from those atomic orbitals. Spin-orbitals are first constructed as products of atomic orbitals and spin functions and possible electronic configurations are defined by application of the Aufbau principle. The antisymmetric N-electron function corresponding to a given

17

configuration is constructed as a (sum of) Slater detenmnant(s) built from the spin­orbitals defming the configuration.

These solutions of Hi represent the zero-th order approximation to the solutions

of the operator (Hi + He) and may be used to obtain the corresponding energy to first order. A variational treatment applied to the energy expectation value corresponding to the lowest state of the symmetry and multiplicity designation under consideration leads to a set of integro-differential equations which may be solved iteratively, until self-consistency has been reached. The self-consistent field (SCP) method yields the best atomic orbitals within the framework of first-order perturbation theory. In an analytical SCF procedure the results depend on the size and characteristics of the basis set used for the expansion of the atomic orbitals. In the limit the results will tend to those obtained in a numerical SCF calculation and they are denoted as Hartree-Fock orbitals. The difference between the Hartree-Fock energy and the exact eigenvalue of the operator Hi + He is denoted as correlation energy, which can be accounted for (within the context of the present approach) by a multiconfigurational approach.

An analytical SCF procedure yields as many orbitals as basis functions have been used for their expansion: the occupied orbitals are those involved in the electronic configuration under consideration while the remaining ones are denoted as virtual orbitals. Different electronic configurations may be defined, and corresponding functions may be constructed, using virtual as well as occupied orbitals, that is, by excitation of electrons from occupied to virtual orbitals.

Functions corresponding to different configurations (of the same symmetry and multiplicity designation) may then be used in a multiconfigurational approach. The simplest procedure at this level consists of the approximation of the eigenfunctions

of the operator Hi + He by a linear combination of functions corresponding to different configurations, with expansion coefficients to be detenmned variationally. The accuracy of the results depends on the number and characteristics of the configurations considered.

Such a configuration interaction (CI) treatment may be coupled with a SCF procedure in what is denoted as a multiconfigurational self-consistent field method (MC SCP) or, in a numerical approach, multiconfigurational Hartree-Fock method (MCHF).

The contribution due to the relativistic corrections may be incorporated in a similar fashion. That is, to first-order, one may perform a CI treatment for the complete Hamiltonian operator.

Complete details on all the methods and formulations mentioned above are available from various sources [Carbo and Klobukowski (1991), Fischer (1977) and Fraga (1992)] and therefore it will be assumed hereafter that the reader is acquainted with them.

Basic Theoretical Formulation

The direct solution of the eigenvalue equation of the complete Hamiltonian operator is not possible and a perturbation-variation approach must be adopted.

The perturbation-variation approach involves a multiconfiguration treatment using either mono- or multi-configuration Hartree-Fock functions. Both modalities are formally analogous, differing only in the order in which the successive corrections are introduced:

(a) The single-configuration HF (SCHF) functions represent approximate solutions, within the one-electron function (orbital) approximation ofKel. , and the multiconfiguration treatment will correct the results, to the level allowed by the number and character of the configurations considered, by inclusion of the correlation and relativistic corrections in one single step.

(b) The multiconfiguration HF (MCHF) functions, on the other hand, already account for the correlation correction, to the level allowed by the number and character of the configurations considered, for the state under consideration and the multiconfiguration treatment will correct the results by inclusion of the relativistic corrections (with an additional multiconfigurational correction).

The determination of HF functions is carried out at present in a routine fashion [Fischer (1977)] and therefore only the formulation for the matrix elements of K, using such functions, will be considered here. Racah algebra (using the coupling scheme appropriate for the interaction under consideration and coefficients of fractional parentage) could, in principle, have been used throughout for this purpose but in this work a computer-oriented approach has been adopted. In a general manner, with the exceptions noted below, the matrix representations have been obtained in the set of SL-functions for the configurations under consideration, with the matrix elements ultimately expressed in terms of SL-vector coupling coefficients and radial and angular factors evaluated from Slater determinants. Tensor algebra, which has been used in this task, is examined in Chapter 3, after having discussed the angular functions in Chapter 2.

2 Angular Functions: Coupling of Angular Momenta

The HF functions are constructed as combinations of Slater determinants built from spin-orbitals defined by

where Tlms is the spin function (either Cl or~) and the HF orbital eIIDlm1. may be

expressed as

where Rn.t.(r) denotes the radial function and Y1.m1. (S,eII) is a spherical harmonic. The quantum numbers ..R.,m1. characterize the electron angular momentum and ms corresponds to the electron spin angular momentum; n is the principal quantum number. In order to simplify the presentation of the formulas, the quantum numbers n ..R.m1.ms will be represented as n..R.m1l and the angular part of orbitals and spin­orbitals will be represented in ket notation by 1n..R.m) and 1n..R.m1l), respectively.

In the complete atomic system, those angular momenta may be coupled in various ways, depending on the interaction to be considered. Therefore it is necessary to examine first of all their characteristics and then their coupling, which will result in new sets of quantum numbers labelling the complete function.

2.1 One-Electron Functions

A vector operator (Le., a tensor operator of rank 1)

j = h + jy + jz = jxux + jyUy + jzuz

(where ux, Uy, Uz denote the unit vectors along the three axes) is said to constitute an angular momentum operator when its components satisfy the commutation relations

22

Ux,jy] = ijz Uy,jz1 = ijx Uz,jx] = ijy

The related angular momentum operator squared,p, is defined by

p = (j.j) = jx2 + jl + jz2

and satisfies the commutation relations

while the so-called step opemtors

satisfy the commutation relations

Uz,h] =h UzJ] = -j-

The angular momentum opemtor squared may be expressed

in tenns of the step opemtors andjz.

The functions, which are associated with the angular momentum operator j and are denoted by Ijm), are defmed by the eigenvalue relations

PUm) = Urn) UG+ 1)]

jzljm) = Ijm) m

where j and m are the corresponding total and z-component quantum numbers, with -j~j (i.e., m = -j, -j + 1, -j +2, ... , j - 2, j - 1, j). Under the step operators, these functions tmnsform as

Njm) = Ij(m+l» [(j-m)G+m+I)]1/2

jJjm) = Ij(m-l» [G+m)(j-m+l)]l/2

In the study of atomic structure, the two basic angular momentum vector opemtors are those associated with the electron orbital and spin angular momenta, respectively. The orbital angular momentum opemtor is defmed by

.R. = [r x p] = -i[r x V]

where r is the vector that specifies the position of the electron with respect to the nucleus, located at the center of coordinates, and p represents the linear momentum

23

of the electron, which is mathematically expressed in tenns of the gradient vector V. Its components (in Cartesian and polar spherical coordinates) are

1x = -i(yoz - zOy) = i sin cp 00 + i cot e cos cp o. 1y = -i(zOx - xOz) = -i cos cp 00 + i cot e sin cp O.

lz = -i(xoy - yoJ = -io. (where a denotes the partial derivative with respect to the coordinate given as a subscript). The functions associated with this operator are the spherical hannonics,Y m(t.), which satisfy the eigenvalue equations

.12 Y m(t.) = Ym(t.)[.1(.1+1)]

lz Y m(t.) = Y m(t.) m

where .1 and m are the total and z-component orbital angular momentum quantum numbers, with -~~. Related to these functions are the so-called modified spherical hannonics

CqCK) = [41r/(2k+l)]lfl YqCK)

which constitute the components of the tensor CCK) of rank k.

The electron spin angular momentum vector operator is expressed

in terms of the Pauli matrices

0=(0 1) x 1 0

o = (0 -i) y i 0 0=(10)

z 0-1 for which oi = o~ = o~ = 1. There are only two spin functions, a and p, defmed by the eigenvalue equations

s2a=~ sza=!.a 4 2

s2p = 113 szp = -113 4 2

from which one can see that the associated quantum numbers, s and ~, have the

1 1 1 1 . values 2 and 2 (for a) and 2 and -2 (for P), respectively.

24

Consequently, the angular momentum behaviour of an electron may be described by a so-called spin-orbital function, given as a product of an orbital and a spin angular momentum function. The spin-orbitals may be labelled, in short, by the corresponding quantum numbers associated with the spherical harmonics and the spin functions, although the quantum number s is usually omitted because of its constant value. In this work, in order to avoid an exaggerated use of subindices, spin-orbitals are denoted by LRmJl), where m and Jl represent the orbital and spin angular momentum z-component quantum numbers, respectively.

The complete notation for a one-electron function, defined as the product of a radial and a spin-orbital function, is then In.R.mJl), where n denotes the principal quantum number (associated with the radial function).

The set of one-electron functions, with the same n and l quantum numbers, constitute a shell and an electronic configuration of an N-electron atom may be specified in terms of the occupancies of the various shells, (nlll)r(n2l2)S----(nili)t, with N = r + s + ... + t.

2.2 SL-Functions

In a complex atom, the individual electron spins, on one hand, and the individual orbital angular momenta, on the other, may be coupled together. The corresponding operators of interest are S2 and Sz and L2 and Lz, respectively, defmed by

S2 = (8-8) 8 = 1: sp 8z = 1: szp

where the summations extend to all the electrons, p, in the atom.

A monomial product of one-electron functions, denoted by (n.R.mJlh (n.R.mJlh ... (n.R.mJl)N (where the subindices label the electrons), is an eigenfunction of the operators Sz and Lz, with corresponding z-component quantum numbers

(where the summations extend to all the electrons, p, in the atom). A Slater determinant, obtained from the above monomial term through operation with the antisymmetric normalized permutation operator, will behave in the same way; that is, it will be characterized by the same total angular momentum quantum numbers, Ms and ML. In this work, such a determinant will be represented by I(l} (IDJl}MsML)

25

where {.R.} and {mJ.1} stand for the sets of quantum numbers.R.b .R.2, ... , .R.N and (mJ.1h, (mJ.1h, ... , (mJ.1)N, respectively, the set {.R.} being the same for all the Slater determinants associated with a given configuration.

Neither the monomial tenns nor the associated Slater determinants built up from them are necessarily eigenfunctions of the total operators S2 and L2. Single­configuration, N-electron functions which are eigenfunctions, simultaneously, of S2, Sz, £2 and Lz may be obtained, however, by direct diagonalization of the matrix representations of S2 and L2 in the set of Slater determinants corresponding to the configuration under consideration.

The actual procedure involves three steps, which will be illustrated below for the configuration p2. [In actual calculations, the program developed by Nussbaumer (1969) may be used for this task.]

1. Characterization of all the non-vanishing Slater determinants.

For the configuration p2 there are 15 non-vanishing Slater determinants:

{mJ.1}

l{p2}{mJ.1h 02) (1112) (1 - 1(1.)

l{p2]{mJ.1h 11) (1 112) (0 112)

l{p2}{mJ.1l3 01) (1 112) (0 - 1(1.)

l{p2} (mJ.1)4 01) (1 - 112) (0 1(1.)

I {p2}{mJ.1Js -11) (1 - 112) (0 - 112)

1{p2}{mJ.1}6 1O) (1112) (-1 1(1.)

I {p2}{mJ.1h 00) (1112) (-1 - 112)

l{p2}{mJ.1}S 00) (1 - 112) (-1112)

l{p2}{mJ.1}900) (0 112) (0 - 1(1.)

l{p2}{mJ.1ho -10) (1 - 112) (-1 -112)

l{p2}{mJ.1}11 1-1) (-1112) (0112)

I {p2}{mJ.1h2 0-1) (-1 -1(1.) (0112)

l{p2}{mJ.1h30-1) (-11(1.) (0 - 112)

l{p2}{mJ.1h4 -1-1) (-1 - 112) (0 - 112)

26

l{p2} {m~} 150-2) (-1112) (-1 - 112)

2. Determination of the possible SL values

(a) The determinants are grouped in sets according to their MS, ML values (i.e., one set for each pair Ms, Md. For the p2 configuration there are 11 such sets, corresponding to the Ms ML values 02, 11,01(2), -11, 10,00(3), -10, 1-1,0-1(2), -1-1, and 0-2 (where the numbers in parentheses indicate the number of determinants, if greater than 1).

(b) The set with the largest dimension is chosen. This set will be the one grouping the determinants with MS = 0, ML = 0 (if the total spins are integers) or 1MSI = 112, ML = 0 (if the total spins are half-integers). The dimension of this set defines the number of possible SL-states, some of which may have the same SL values. In the present case there are 3 determinants with MS = 0 and ML = 0 and therefore there will be 3 SL-states.

(c) Taking into account that -S~MS~S, -L~L~, one of these states will be characterized by L = max(ML) and S = max(Ms) (compatible with the condition for L). Elimination of all the (2S+ 1)(2L+ 1) pairs Ms, ML belonging to that term yields a reduced set of Ms, ML values on which the same procedure may now be applied, until all the terms have been found. In the present case this procedure yields the three states ID(S = 0, MS = 0; L = 2, ML = 2, 1, 0, -1, -2), 3p(S = 1, Ms = 1, 0, -1; L = 1, ML = 1, 0, -1) and IS(S = 0, MS = 0; L = 0, ML = 0).

3. Determination of the SL-functions

(a) The set of determinants with the largest dimension is chosen and the matrix representation of L2 in this set is constructed. In the present case, the three determinants to be considered are

l{p2} (m~17 00) l{p2} (m~}8 00) l{p2} (m~}9 00)

corresponding to the {m~} sets

(1 112) (-1 - 1/2) (1 - 1/2) (-1 1/2) (0 112) (0 - 1/2)

With

L =il +i2

L2 = (LeL) = «il + i2) e (il + i2» = i12 + 2(il ei2) + i22

= i12 + il-ri2- + il-R.2+ + 2l1zi2z + i22

one obtains

27

L21{p2) (~l7 (0) = 21{p2) (mj.1l7 (0) + 21{p2) (mj.1)9 (0)

L21{p2) (~}8 (0) = 21{p2) (mj.1)8 (0) - 21{p2) (~}9 (0)

L21{p2) (~}9 (0) = 21{p2) (mj.1l7 (0) - 21{p2} (~}8 (0) + 41{p2} (mj.1}9 (0)

so that the matrix representation of L2 becomes

(2 0 2) o 2 -2 2 -2 4

(b) The matrix obtained in the previous step is diagonalized, yielding linear combinations of Slater determinants, which are eigenfunctions of Sz, L2 and L,.. In the present case one obtains, using the notation 1MsLML) to represent the linear combinations,

1010) = (YT!2){I{p2}{mj.117 (0) + 1 {p2}{mj.1} 8 OO)}

1000) = (Y3"/3){I{p2) (mj.117 (0) - l{p2} (mj.1}8 (0) - l{p2} (mj.1)9 OO)}

1020) = (1/Y'O){-I{p2}{~17 (0) + l{p2}{mj.1}8 (0) - 21{p2}{mj.1}9 OO)}

(c) The behaviour under 52 of the linear combinations obtained in the preceding step is determined. In the present case, with

S = Sl + S2

one obtains

521010) = 21010)

521000) = 0

52020) = 0

which shows that those functions belong to the multiplets 3p, IS and 1D, respectively. Hereafter they will be labelled as 11010), 10000) and 1(020), respectively, cOITesponding to the general notation ISMsLML).

(d) Operation with the step operators, S+, S_, L+, L, on the above functions will then generate the SL-functions for all the possible MS, ML values.

28

The SL-functions are therefore expressed as linear combinations of the determinantal functions I{].} {mJ.1} uMsML>. The corresponding expansion coefficients are usually denoted as SL-vector coupling coefficients and will be

represented in this work as ({].) (mJ.1}uMsMLI~SL). That is,

where the summation extends to all the sets {mJ.1}u (i.e., Slater determinants) with

the correct MS,ML values. C denotes the configuration and P includes all the additional details (e.g., ordering indexes for states with the same SL designation) needed in order to completely label the function. When no confusion will arise, the

SL-states will be labelled in short as I~SL).

The total angular momentum operators squared, S2 and L2, commute with the Hamiltonian operator, which includes only the electron kinetic, nuclear attraction, and electrostatic repulsion terms, and therefore the matrix representation of the latter in the set of SL-functions for a given configuration is diagonal. This is not true, however, when the so-called SL-non-splitting interactions are included in the Hamiltonian operator and/or the set of SL-functions is expanded to include more than one configuration, and therefore the corresponding SL-functions are to be determined by diagonalization of the corresponding matrix:

(a) Diagonalization of the matrix representation of the Hamiltonian operator, which includes the non-splitting terms, in the set of SL-functions (with same S, L values) of a single configuration, yields SL-functions said to include state interaction.

(b) Diagonalization of the matrix representation of the electronic Hamiltonian operator in the set of SL-functions (with same S, L values) corresponding to more than one configuration, yields SL-functions said to include configuration interaction.

(c) SL-functions, including both state and configuration interactions, are obtained by diagonalization of the matrix representation of the Hamiltonian operator, including the non-splitting interactions, in the set of SL-functions (with same S, L values) corresponding to two or more configurations.

2.3 JMJ- and FMF-Functions

When fine- and/or hyperfine-structure interactions are included, the Hamiltonian operator does not commute with the operators s2 and L2 and, consequently, S and L are not good quantum numbers any longer.

29

Inclusion of those interactions implies a coupling of angular momenta such that the associated vector operators commute with each other. In such a case the components of the resulting vector operator satisfy the general commutation relations for angular momentum vector operators and the general formulation, developed for such cases, may be applied.

This formulation, which will allow us to express JM]- and FMF-functions in terms of SL-functions, is summarized briefly below before discussing the two specific cases.

Let us consider two angular momentum vector operators (such as the spin and orbital angular momentum vector operators), which act on different parts of the system. We will denote as j 1 and h those two operators and by Ij 1 m Ihm2) the functions which are simultaneously eigenfunctions

h2lhmU2m2) = IhmU2m2) 0IGl+1)] h zlhmU2m2) = IhmU2m2) ml

h2lhmU2nn) = IjlmU2m2) 02G2+1)] nzljlmU2ID2) = UlmU2ID2) ID2

The functions, ULi~m) or in short Ijm), associated with the resulting angular momentum operator

may be expressed

in terms of the functions IjlmLi2m2). The summations extend over ml and m2 with the restriction that m = ml + m2. The possible values of j are limited by the triangular condition ~Gu2i),

Ijl - hi ~j ~h + h

The coupling coefficients, GlmU2m2IjLi2im) or in short GlmU2m2Ijm), are the 80-

called Clebsch-Gordan coefficients, related

( jl h j) = (-1~1- j2+ m (2j + 1)-1/2 Glmlhm2Um) ml m2-m

to the 3-j symbols.

The functions Ijm) are simultaneous eigenfunctions of the set of operators h 2, h 2,P. andjz. Taking into account that

30

jz=hz + hz

one can proceed as follows:

Let us consider fIrst the functions Ijm) == I(h+h)m), with -jd2~h+h. The function I(h+h)(h+h» can only be generated from the function IjUlhi2), for which

h 2IjUU2i2) = ljUU2i2) UIG1+1)]

h 2IjUU2i2) = ljUU2i2) U2G2+1)]

PljUU2i2) = IjUU2i2) UIG1+1)+2jU2 + hG2+1)] = ljUU1.i2) lGl+h)Gl+h+l)]

jzljUU2i2) = IjUU2i2) Gl+h)

That is, the function

is an eigenfunction of h 2,h2,j2 andjz with eigenvalues hGl+1), h(i2+1), Gl+h)Gl+h+l) and Gl+h), respectively. Acting on this function with

j-=h-+h-

one can generate all the remaining functions IGl+h)m). For example:

for which one obtains

h 2jJG1+h)Gl+h» =jJG1+h)Gl+h» UIG1+1)]

h 2jJGl+j2)Gl+h» = jJGl+h)Gl+h» U2G2+1)]

pjJGl+h)Gl+h» = jJG1+h)Gl+j2» lGl+h)Gl+h+l)]

jziJGl+h)Gl+h» = jJG1+h)Gl+h» Gl+h-l)

That is, the function jJG l+h)Gl+h» is an eigenfunction of the operatorsj}2,h2,f2 and jz with eigenvalues jiG 1 + 1), hG2+ 1), G 1 +h)G 1 +h+ 1) and G 1 +h-l) and one can write, with the appropriate normalization constant,

or, equivalently,

31

IGl+jVGl+h-1» = IjlGl-1)hi2) GIGl-1)hivIGl+jVGl+h-1»

+ IjUlhG2-1» GUU2G2-1)IGl+h)Gl+h-1»

We consider next the functions IG2+h-1)m), with -jl-h+1S;mS;j}+h-1. The function IGl+h-1)Gl+j2-1» can only be generated from the functions IhGl-1)h.i2) and IjUU2G2-1), which are precisely the ones appearing in the expansion of the function IGl+h)Gl+h-1», obtained above. The two functions, IGl+jVGl+h-1» and IGl+h-1)Gl+h-1», must be orthogonal and therefore one can immediately write

IGl+j2-1)Gl +h-1» = (2jl+2h)-lfl{ Ijl Gl-1)h.i2)(2jv1/2..ljuU2G2-1»(2jl)l/2)

= IjlGl-1)j2i2) GIGl-1)j2i2IGl+h-1)Gl+h-1»

+ IjUUiG2-l» GUU2G2-1)IGl+h-1)Gl+h-1»

which can then be used to generate all the remaining functions IG 1 +h-1 )m).

Repett'"tion of the procedure for all the remaining possible values of j will generate all the functions Ijm).

Application of this formulation to the case when the total electron orbital and spin angular momenta

J=S+L

are Coupled leads to the single-configuration JMJ-functions

in terms of the SL-functions for that configuration, where 'Y includes all the additional details needed in order to completely label the function and the summations extend to the values of MS, ML compatible with the restriction M] = Ms + ML.

The functions ICySLJM]), or in short IJM]) whenever no confusion will arise, are simultaneous eigenfunctions of the operators s'l, L2, fl and Jz. Computationally they may be obtained by diagonalization of the matrix representations of the operator fl in the set of SL-functions, with one matrix for each MJ = MS + ML value. For example, in the case of the p2_configuration, the nine components of the 3p state would be grouped in five matrices corresponding to MJ = 2, 1,0, -1, -2 (with dimensions 1,2,3,2 and 1, respectively); diagonalization of these matrices would yield the functions 1112MJ), with MJ = 2, 1,0, -1, -2, I111MJ), with MJ = 1,0, -1, and (1100).

32

The fine-structure interaction Hamiltonian operator commutes with the operators fl and }z. That is, J and MJ are good quantum-numbers and JMJ-functions are to be used whenever the fine-structure interaction is included in the Hamiltonian operator. Diagonalization of the matrix representations, one for each MJ value, of the Hamiltonian operator including the fine-structure terms, in a set of ICySLJMJ) functions, for one or more configurations will yield, respectively, the

monoconfigurational ICyJMJ) or the multiconfigurationallyJMJ) functions. These functions are no longer eigenfunctions of either S2 or L2.

When the total electron and the nuclear spin angular momenta are coupled

F=J+I

the single-configuration FMF-functions may be expressed

in terms of the JMJ-functions of that configuration, where e includes all the additional details needed in order to completely label the function and the summations extend to those values of MJ compatible with the restriction MF = MJ+MJ.

The functions ICeJIFMF), or in short IFMF) whenever no confusion will arise, are simultaneous eigenfunctions of fl, fl, F2 and Fz and they could be determined by diagonalization of the matrix representations, one for each MF = MJ+MJ value, of the operator Jil in the set of JMJ-functions.

The hyperfine-structure interaction Hamiltonian operator commutes with p2 and Fz. That is, F and MF are good quantum numbers and FMF-functions are to be used whenever the hyperfme-structure interaction is included in the Hamiltonian operator. Diagonalization of the matrix representations, one for each MF value, of the Hamiltonian operator including the hyperfine-structure interactions, in a set of

ICeJIFMF) functions, for one or more configurations will yield, respectively, the

monoconfigurationallCeFMF) or the multiconfigurationalleFMF) functions. These functions are no longer eigenfunctions of either fl or fl.

2.4 Selection of Functions

There are, depending on the interaction terms included in the Hamiltonian operator, four cases of calculations that may be performed:

33

(i) with the electronic Hamiltonian operator (containing only the electron kinetic, nuclear attraction and electrostatic repulsion tenDs);

(ii) with the above operator incremented with the SL-non-splitting terms;

(iii) with the above operator incremented with the fme-structure terms;

(iv) with the above operator incremented with the hyperfine-structure terms.

Taking into account that the ICclIFMp) functions may be expressed in terms of

ICySLJMJ) functions and that the latter are expressed in terms of ICPSMSLML) functions, a brute force approach could be adopted for the construction of the interaction matrix, using the ICpSMSLMd functions, independently of the Hamiltonian operator considered.

Such an approach, however, would result in a waste of computer time and it is more efficient to construct the interaction matrix using the appropriate functions:

(a) for the cases (i) and (ii) above, the matrix representation of the Hamiltonian operator is constructed in the set of functions

ICPSMSLML), for all the configurations under consideration, with one matrix for each set of values S, L (for given MS, ML values);

(b) for the case (iii) above, the matrix representation of the Hamiltonian operator is constructed in the set of functions ICySLJMJ), for all the configurations under consideration, with one matrix for each J value (for a given MJ value);

(c) for the case (iv) above, the matrix representation of the Hamiltonian operator is constructed in the set of functions IC£JIFMp), for all the configurations under consideration, with one matrix for each F value (for a given Mp value).

In all cases the matrix elements will ultimately be evaluated from Slater determinants and the formulation in Chapter 3 and in the next section (Application of the basic formulation) will show how to make use of symmetry considerations in order to simplify that task.

3 Tensor-Operator Formulation

In this work, the SL-, J- and F-functions will be generated by diagonalization of the matrix representations of the appropriate Hamiltonian operator in the set of single­

configuration IJ3SMsLMd functions for the configurations under consideration.

The evaluation of the matrix elements may be considerably simplified by application of tensor algebra techniques, through the use of the so-called reduced matrix elements, as discussed below.

3.1 Tensor Operators

An irreducible tensor operator T(k) of rank k is defined as a set of (2k+ 1) operators T q(k) which serve as a basis of the (2k+ 1 )-dimensional representation of the rotation group. In particular, a vector operator is a tensor operator of rank 1.

The tensor product, [A<k) x B(k')](K), of two irreducible tensor operators A(k)

and B(k') yields a set of irreducible tensor operators of ranks K, with values

K = k+k', k+k'-l, ... , Ik-k'i

Their components are defined by

with summation over q arid q' (with the restriction Q = q+q'). In the particular case when k=k' and K=O, one has

which is related

36

to their scalar product

All the terms of the Hamiltonian operator, considered in this work, may be expressed in tenns of the basic tensor operators s(1), J.<I) and C(k). The tensor operators s(1) and J'<I) are the vector operators discussed in the previous chapter and the tensor operator C(k) has as components the modified spherical harmonics Cq(k), also discussed in the preceding chapter.

3.2 Wigner-Eckart Theorem

The reduced (or double-barred) matrix element of an irreducible tensor operator T(k), which acts on functions I~jm), is defined from the Wigner-Eckart theorem

which states that the dependence of the matrix element on the projection quantum numbers m is contained entirely in the Oebsch-Gordan coefficients, or equivalently in the 3-j symbols. The left-hand side matrix element vanishes unless the triangular condition

Ij-j'1 S k S j+j'

is satisfied and q = mom', so that in particular

(~jmITo(k)I~'j'm') = a(m,m')(2j+ l)-I/2G'mkOljm)(~jIlT(k)IIP'j')

(~jmITo(O)Ip'j'm') = S(m,m')(2j+ 1 )-I/2G'mOOljm)(~jllT<O)II~'j')

= S(m,m')(2j+ 1 )-I/2(~jIlT(O)II~'j')

The evaluation of the reduced matrix elements may be illustrated in the case of the tensor operator C(k). Denoting by Lim) the spherical harmonic Ym().) one can write

37

= (-I)m[(2R.+l)(2R.'+I)]l/2 Ii k i') (i k i') \0 0 0 -mqm'

which, when compared with

(ik i') (0 .... 11"" (k)Ii'm') = (-I)l--m (illdk)IIi') -.~ -mqm'

yields

~l k i') (illdk)IIi') = (-I)L[(2R.+l)(2R.'+I)] 1/2 . 000

For the tensor operators s(1) and i(1), corresponding to the electron spin and orbital angular momentum, the corresponding reduced matrix operators are

(slls(1)lIs) = [s(s+I)(2s+1)]1/2 = (3/2)1(2

(illl(l)IIl') = S(l,i')[l(i+ 1)(2R.+ 1)]1(2

3.3 Reduced Matrix Elements

The usefulness of the reduced matrix elements is due to the fact that the terms of the Hamiltonian operator, although scalars, are constituted in many cases by products of tensor operators.

The reduced matrix element formulation will permit us, as shown below, to express the matrix elements of the composite operators in terms of the reduced matrix elements of the component operators. Two cases will be distinguished, depending on whether the component tensors act on the same or different parts of the system (say, the spin and the orbital part).

In the case when the two component tensors act on the same part of the function one can write directly

38

with the summations running over q and q' = Q-q. Expansion of Bq,(k')Ip'j'm')

Bq,(k')IWfm') = 1; Ipljlml)(pljlm"IBq,(k')lp'j'm')

in tenns of the complete set of functions leads to

(pjmITQ(K)lp'j'm') = 1; (kqk'q'IKQ)(pjmIAq(k)lplljllmll)(p"jllmIlIB~~')lp'j'm')

Application of the Wigner-Eckart theorem to each matrix element in this equation yields

(2j+ 1 )-ll2(j'm'KQijm)(pjllT(K)IIp'j')

= 1; (kqk'q'IKQ){ (2j+ l)-ll2(jlmlkqljm)(pjIlA(k)IIPlj"»)

{(2j"+ l)-ll2(j'm'k'q'ljlm")(p"j"IlB(k')IIP'j'»)

which reduces to

(j'm 'KQijm)(pj IIT(K)IIP'j')

or

= l;(2j"+l)-112 (kqk'q'IKQ) (j"m"kqljm) (j'm'k'q'lj"m") (pjIlA(K)lIplj")

(P"jII liB (k')11P'j')

,~ ~ ~) (PjIlT(K)IIp'j') = 1;(-1)K+k+k'+j+j'+j"+Q+m"(2K+l)112

(• ·11 k) (k' K k) (k' ·11 .,) _~ ~II q -q' Q -q q' _~II r1., (pjIlA(k)np"j") (P"j"IB(k')np'j')

Taking into account that

1; (_l)k'+j"+k+q'+m"-q (_~ j" kV k' K k X k' j" j'\ mil qA, -q' Q -q q' -m" m]

39

(with summations over q, q' and mil) one obtains finally

(PjIlT(K)IIPT) = 1: (-1).i+K+j'(2K+l)l/2 {~, j~ J'} (pjIlA(k)lIplj") (Plj"IlB(k')IIPT)

In the case when the two component tensors act on different parts of the system, i.e., A(k) acts on the functions la) and B(k') acts on the functions Ib), one proceeds as follows. If we write

GmITq(k)Ij'm') = 1: Gmlj"m") GlmIITq(k)lIjl'm'") Glllm"'Ij'm')

taking into account the orthonormality of the functions Ijm) and where the summations run over allj", mil, jill and mill, we can express Tq(k) in general by

where the summations run now over all j, m, j' and m'. Taking into account the Wigner-Eckart theorem one obtains

Tq(k) = 1: (2j+l)-1/2 G'm'kqljm) GIIT(k)lIj') {ljm)G'm'l}

which can be rewritten as

or, equivalently

where the new tensor operator W(k)Gj') is defined by its components

Wq(k)(ij') = 1: Ijm) {(2k+l)1/2(2j+l)-1/2 G'm'kqljm)} G'm'l

=1:ljm){(-1).i+m(2k+l)1I2(j k j',)}(j'm'l \-m q m

This transformation may be applied either to the composite tensor operator [A(k) x B(k,)](K) or to the component tensors A(k) and B(k'), yielding respectively

40

(with summations over j andj') and

[A(k) x B(k')](K) = [(E(2k+1)-1/2(aIlA(k)lIa')W(k)(aa')} x

(E(2k' + l)-l/2(bIlB(k')lIb') W(k')(bb') }](K)

= E[(2k+ 1)(2k'+1)]-1/2(aIlA(k)lIa')(bIlB(k')lIb')[W(k)(aa') x

W(k')(bb')](K)

(with summations over a, a', band b'). Taking into account that both tensor operators, W(K)(jj') and [W(k)(aa') x W(k')(bb')](K), are irreducible and behave as angular momentum functions, one can expand

[W(k)(aa') x W(k')(bb')](K) = E «ab)j(a'b')j'KI(aa')k(bb')k'K) W(K)(jj')

(with summations over j and j') or, in short,

[W(k)(aa') x W(k')(bb')](K) = E (jj'Klkk'K) W(K)Oj')

which relate the two modes of coupling of the four angular momenta a, a', b and b'. The coupling coefficients in the above expression are related to the 9-j symbols and one can rewrite it as

Jab j \ = E[(2j+l)(2j'+1)(2k+l)(2k'+1)]1/2 ,a' b' j' J W(K)(jj')

k k'K

Comparison of the two original expressions for [A(k) x B(k')](K), taking into account the preceding result, leads finally to

(abjll[A (k) x B(k')](K)lIa'b'j')

(a b j )

= [(2j+l)(2j'+1)(2K+1)]1/2 a' b' j' (aIlA(k)lIa')(bIlB(k')lIb') k k'K

41

which expresses the reduced matrix element of the composite tensor operator, in the coupling scheme abj, in terms of the reduced matrix elements of the component tensors.

Application of this result to the most usual cases yields the following results:

(a) tensor operator of rank 0

(abjll[A(k) x B(k)](O)lIa'bT) = [(2j+l)(2j'+I)]112 a' b' j' (aIlA(k)lIa')(bIlB(k)lIb') {a b j} k k 0

In the particular case when j=j', taking into account that

I a b j \ . a' b' j' I = (-I)a'+b+j+k[(2j+ 1)(2k+ 1)]-1/2 {a, b, J } k k' 0 b a k

one obtains

(abjll[A (k) x B(k)](O)lIa'b'j')

= (_l)a'+b+j+k(2j+l)I/2(2k+1)-1/2 {~,~, ~} (aIlA(k)lIa')(bIlB(k)lIb')a(ij')

(b) scalar product

(abjll(A(k).B(k»lIa'b'j') = (-I)k(2k+l)l/2(abjll[A(k) x B(k)](O)lIa'b'j')

= (_l)a'+b+j(2j+ 1)1/2 { ~, ~, ~ } (aIlA(k)lIa')(bIlB(k)lIb')a(jj')

In the particular case when k=O, a=a' and b=b', the above further reduces to

(abjll(A(O).B(O»lIabj) = [(2j+ 1)/(2a+ 1)(2b+ 1)]112 (aIlA(O)lIa)(bIlB(O)lIb)

(c) single tensor operator

One can write

(abjIlA(k)lIa'bT) == (abjll[A(k) x l(O)](k)lIa'b'j')

42

Jab j \ = [(2j+l)(2j'+1)(2k+1)]lfl \a' b' j' I (aIlA(k)lIa)(blll(O)lIb')

k 0 k

With

(bill (0) lib') = (2b+ 1 )1/2 S(bb')

one obtains

(abjIlA(k)lIa'b'j') = (_1)a+b+j'+k[(2j+ 1) (2j'+ 1)]1/2 {j~ ~, ~ } (aIlA(k)lIa')S(bb')

In a similar fashion one obtains

(abjIlB(k)lIa'b'j') = (-I)a+b'+j+k[(2j+l)(2j'+I)]1/2 {j~ ~, : } (bIlB(k)lIb')S(aa')

3.4 Matrix Elements

Simple substitution of these expressions, as appropriate, in those of the original matrix elements yields the desired fmal expressions:

(a) for component tensors acting on the same part of the system

(pjml[A(k) x B(k')]Q(K)Ip'j'm') =(2j+ l)-I/2U'm'KQljm)(pjll[A(k) x B(k')](K)IIp'j')

= 1: (-1~+K+j'(2j+ 1)-1/2(2K+ 1)I/2Um'KQljm)

{ ~, ~ j~ } (PjIlA(k)lIpo'j")(P"j"IIB(k')IIWj')

(b) for component tensors acting on different parts of the system

(abjml[A(k) x B(k)]o(O)la'b'j'm')

= (2j+1)-I!2U'm'OOljm)(abjll[A(k) x B(k)](O)la'b'j')

43

= (_l)a'+b+j+k(2k+l)-I/2(j'm'OOljm)

{b~ ~ ~} (aIlA(lc)lIa')(bIlB(lc)IIW)li(jj')

= (-I~'+b+j+k+j'+m[(2j+l)/(2k+l)]I/2

(~, ~ _~) {b~ ~ ~} (aIlA(lc)lIa')(bIlB(lc)lIb')li(jj')

= (-I~'+b+j+k(2k+l)-I/2 {:. ~ ~} (aIlA(lc)lIa')(bIlB(lc)lIb')li(jj')

(abjml(A (lc).B(lc»la'b'j'm') = (-l)k(2k+ 1 )1/2(abjml[A (k) x B(k)]o(O)la'b'j'm')

= (-1 )a'+b+j {:' ~ ~} (aliA (lc)lIa')(bIlB(k)lIb')li(jj')

(abjmIAq(lc)la'b'j'm') = (2j+ l)-I/2(j'm'kqljm)(abjIlA (k)lIa'b'j')

(. k "){ b '} = (_l)a+b+j+j'+k-m[(2j+l)(2j'+I)]I/2 \~ q ~, j, a' k (aliA (k)lIa')li(bb')

(abjmIBq(k)la'b'j'm') = (2j+ 1 )-I/2(j'm'kqljm)(abjIlB(k)lIa'b'j')

, .. (jkj'){ab j } = (-I)a+b+k-m[(2J+l)(2j'+I)]-1/2 \m q m' j' a' k

(bIlB(k)lIb')li(aa')

(c) An important case, not covered above, is that of the matrix elements of the product of two tensor operators, which act on different pans of the system not properly coupled. In such a case one must fIrst decompose the matrix element into the product of two corresponding elements, one for each part, which are then handled as above. That is,

44

= 1: (kqk'(Q-q)IKQ)(a(l )ma( 1 )b(2)mb(2)1~)( 1 )B~(2)la'( l)~ (1 )b'(2) m~(2»

= 1: (kqk'(Q-q)IKQ)(a(1)ma(1)[~k)la'(l)~ (1»(b(2)ffib(2)1 B~(2)lb'(2)m~(2»

where the indices 1 and 2 label the parts of the system and the summation runs over q. In the case of the scalar product one has

(a(1)ma(1)b(2)mb(2)I(A(k)(1).B(k)(2)la'(1)m~(1)b'(2)m~(2»

= (-1)k(2k+l)l/2

(a(1)ma(1)b(2)mb(2)I[A(k)(1) x B(k)(2)]o(O)la'(1)~(1)b'(2)m~(2»

= (-1)k(2k+l)l/l1: (kqk-qIOO) (a(1)ma(1)IA~k)(1)la'(1)~(l))

(b(2)ffib(2)IB~)(2)lb'(2)m~(2»

= 1: (-l)q (a(1)ma(1)IA~k)(1)la'(1)~(1» (b(2)mb(2)IB~)(2)lb'(2)m~(2»

(a( 1 )IIA (k)( 1 )lIa'( 1) )(b(2)IIB(k)(2)lIb'(2»

= (-1)q[(2a+l)(2b+l)]-1/2 (a'm~kqlama> (b'm~-qlbmb)

(aIlA(k)lIa')(bIlB(k)lIb') O(ma+mb,~+ m~)

The practical application of the formulation presented in this chapter requires the knowledge of the reduced matrix elements of the component operators. For such operators the expressions of the complete matrix elements are either known or may be easily derived, with the result that their reduced matrix elements may be determined by direct application of the Wigner-Eckart theorem.

Application of the Basic Formulation

All the terms of the HamUtonian operator, although scalars, may be expressed in terms of tensor products (of rank 0). Consequently, the general formulation presented in the preceding chapter for the evaluation of matrix elements in terms of the corresponding reduced matrix elements may be used. Two steps are needed in this connection.

First of all it is necessary to express all the terms in the Hamiltonian operator in terms of products of tensor operators. The general rules for this transformation are discussed in Chapter 4, together with a survey of the basic tensor operators and of useful relationships for their reduced matrix elements. This chapter is completed with an illustration of the application of the general formulation to some specific cases and a summary of all the terms of the Hamiltonian operator in terms of tensor operators.

In Chapter 5 the matrix elements are first recast with separation into radial and angular factors and then the formulation obtained in Chapter 3 is applied to the latter, yielding the general expressions of the matrix elements in the various coupling schemes. The use of these expressions is finally illustrated for specific examples.

The final, general expressions for the matrix elements, obtained in Chapter 5, have been applied to the most important terms of the Hamiltonian operator: electronic energy, mass variation, specific mass effect, one- and two-electron Darwin correction, electron spin-spin contact interaction, orbit-orbit interaction, spin-orbit coupling, spin-spin dipole interaction, magnetic dipole and Fermi contact interactions, electric quadrupole coupling, magnetic octupole coupling, Zeeman and Stark effects and nuclear-mass dependent orbit-orbit interaction and spin-orbit coupling.

In each case the matrix element is given in terms of radial and angular parts over the spin-orbitals. The angular parts are given explicitly in each case while the radial parts are expressed in terms of radial integrals. Because some of the radial integrals may be common to more than one interaction, they have been collected together at the beginning of the chapter, being given in terms of the radial functions P (see Notation and Symbols).

4 Transformation of Operators to Tensor Form

The evaluation of the ma~ elements of the Hamiltonian operator through the use of the reduced matrix elements fonnulation presented in the preceding Chapter requires that its tenns be transfonned into tensor fonn (see Section 4.2).

All the tenns of the Hamiltonian operator may be expressed in tenns of the tensor operators s(1), .JW) and C(k). The transfonnation is achieved by fIrst expressing the basic operators of the Hamiltonian operator and then using recoupling techniques.

4.1 Basic Operators

4.1.1 Operators s(1),.JL(1) and C(k)

The reduced matrix elements 9f these operators have been presented in the preceding Chapter and here only some useful relationships involving the tensor operators C(k)will be discussed.

The reduced matrix element of this operator, when both components act on the same part of the system, may be obtained from the general expression given in the preceding Chapter:

(.JLII[C(k) x C(k,)](K)II.JL')

= (-1)l.+KH:(2K+ 1)112 ~ { f. : 1'} (.JLIIC(k)II.JLI)(.JL"IIC(k')II.JL')

48

{k' K k} = (-1)L+K+1.'(2K+l)1/2 I: .i.i".i' (.illdk)II.i")(.i"IIC(k')II.i')

(with summation over .i"). Substitution of the reduced matrix elements on the rhs (see preceding Chapter) yields

(.iIl[C(k) X C(k')](K)II.R.') = (-1)L+K+1.'(2K+l)1/2

( .i k ,R.") (.i" k' .i') o 0 0 0 0 0

Taking into account that

{ k' K k}(.i" k'.i') _ ( l'l.'+k'+1."(.i' k' l')JR.' k'.i" } .i .i".i' 0 0 0 - - ,- 0 0 0 lk .i K

_ <-l)1.'+k(.i K.2 )(k K k')(k .i .i") - 000000000

one obtains

( .i K .i')(k K k\(k .i .i")2 o 0 0 0 0 0) 0 0 0

Finally, with

one has

49

or, equivalently,

The most usual cases of interest are

[C(1) x C(k)](k+l) == [C(k) x C(l)](k+l) = [(k+l)/(2k+l)]l/2C(k+l)

[C(1) x C(k)](k) == [C(k) x C(1)](k) = 0

[C(1) x C(k)](k-l) == [C(k) x C(1)](k-l) = -[k/(2k+ l)]l(lC(k-l)

(b)[C(k+l) x T(l)](k) and [C(k-l) x T(1)](k)

From the general expressions obtained in the preceding Chapter, for operators acting on the same part of the system, one can write

= (-1)L+k+l.'(2k+l)l/2l: {~ ~"~!I (illdk+l)lIi")(i"IIT(k)lIi')

(with summation over i"). Using the result obtained above for C(k+l), this expression transforms into

50

k ' {I k k+ll = (-1)1.+ +.R.(2k+l)I/2[(2k+l)/(k+l)]l/2:E 1 1" l'

(J.JI[C(k) x C(I)] (k+ 1) 1I1")(1"lrr(l)1I1' )

= (-1)1.'+ 1 (2k+ 1)[(2k+3)/(k+ 1)]1/2

(with summations over 1" and 1"'). Proceeding in a similar way one obtains

(1I1[C(k-l) x T(l)](k)1I1') = (-I)1.'(2k+l)[(2k-l)/k]l/2

These two expressions may now be combined as

[(2k+3)(k+ 1)] 1/2(111 [C(k+ 1) x T(l)](k)1I1') - [(2k-l)k]I/2(1I1[C(k-l) x T(l)](k)1I1')

,l..' ,i." {I k k+lH 1 k+l1"1 = (-1r(2k+l):E(-lr {(2k+3) 1 1" l' 1 1'" k +

{ I k k-l!{1 k-l 1111 (2k-l) 11"1' 11'" k )

(1IIC(k)1I1'") (1'" I IC(l)1I1")(1" Irr(l)Il.R: )

which, taking into account the orthogonality property of the 6-j symbols, may be transfonned into

[(2k+3)(k+l)]I/2(1I1[C(k+l) x T(l)](k)1I1') - [(2k-l)k]I/2(1I1[C(k-l) x T(1)](k)1I1')

= (-I).R.'[(2k+ 1)/(21'+ I)]:E( -1)1." (1I1C(k)1I1') (1'lIc(l)1I1")(1"I1T(l)1I1')

In the case when T(1) == 1(1), the rhs vanishes and one can write [Innes and Ufford (1958)]

51

[C(k+l) x .R.(1)](k) = [k(2k-l)/(k+l)(2k+3)]lf2[C(k-l) x .RW](k)

which vanishes for k = O.

This result may be obtained also from the expressions of the components of the related tensor products [[C(k) x C(l)](k') x T(1)](k). One can write

=[(2k+l)(2k+3)]1/1~(-I)q+q'+1 (q'!q : ~;!) (q,_lq" ~" k~~,)

Cq,,(k>cq'-q,,(1Trq-q.(l)

[[C(k) x C(l)](k) x T(l)]q(k)

=(2k+l)~(-I)q+q'+1 (q'!q: _~,) (q'_~" ~" _~, )

Cq.(k)Cq'-q..(lTrq-q,(l)

[[C(k) x C(l)](k-l) x T(l)]q(k)

= [(2k+l)(2k-l)]1/2 ~ (-I)q+q'+l ( ,1 k k-l) ( 1 q -q q -q' q'-q"

so that

k k-l ) q" -q'

(2k+3)1/2[[C(k) x C(1)](k+l) x T(l)]q(k) + [[C(k) x C(l)](k) x T(l)]q(k)

+ (2k-l)1/2[[C(k) x C(1)](k-l) x T(1)]q(k)

= (2k+ 1)112 ~ (-1)q+q'+1 Cq,,(k)Cq'-q,,(l>Tq-q,(l)

( 1 k k+ 1) (I k k+ 1 ) {(2k+3) q'_q q _q' q'_q" q" _q'

( lk k)(l kk) + (2k+l) q'-q q _q' q'-q" q" _q'

52

( 1 k k-l) (1 k + (2k-l) q'_q q _q' q'_q" q" k-l) -q' }

which vanishes because of the orthogonality property of the 3-j symbols. Taking

into account that [C(k) x C(l)](k) vanishes, one can rewrite the above expression as

+ (2k-l)l/2[[C(k) x C(l)](k-l) x T(l)](k) = 0

which yields

[(k+l)(2k+3)]l/2[C(k+l) x T(l)](k) - [k(2k-I)]l/2[C(k-l) x T(l)](k) = 0

As no restriction has been imposed on T(l), the above expression may be used for the three cases of possible interest:

[C(k+l) x C(l)](k) = [k(2k-I)/(k+I)(2k+3)]lf2[C(k-l) x C(l)](k)

[C(k+l) x .2.(l)](k) = [k(2k-I)/(k+I)(2k+3)]lf1[C(k-l) x.2.(l)](k)

[C(k+l) x s(l)](k) = [k(2k-l)/(k+l)(2k+3)]l/2[C(k-l) x s(l)](k)

(c) [C(k) x .2.(l)](k)

From the general expression one obtains again in this case

(.2.II[C (k) x .2.(l)](k)II.2.')

= (-I)l.+k+1.'(2k+ l)lf1 {~ ~, lJ (.2.IIc(k)II.2.') (.2.'1 1.2.(1)11.2.' )

= (112)[.2.(.2.+ I) - .2.' (.2.'+ I)-k(k+ I)][k(k+ 1)]-1/2(.2.1Ic(k)11.2.')

or, equivalently [Innes and Ufford (1958)]

[C(k) x .2.(1)](k) = (112)[.2.(.2.+ I) - .2.' (.2.'+ I)-k(k+ I)][k(k+ 1)]-l/2C(k)

53

4.1.2 Other Common Operators

Operators which appear repeatedly in the expression of the Hamiltonian operator

(see Chapter 1) are the electron gradient (V p), linear momentum (pp), orbital angular momentum (ip), position (rp) and relative position (rpcr) vector operators. Their expressions, relationships and related quantities are listed below, while the details of their transformation to tensor form is discussed in Section 4.2

(a) Vp, Pp and ~

Vp == V(1)(p) = - 1'1 rp-I[C(l)(p) x .~"<I)(p)](1) + C(1)(p) dp

Pp == p(l)(p) = -i V(l)(p)

~ = .R.<l)(p) = [rp x pp] = -i[rp xV p]

The operator V p, acting on rp-l and rpcr- l , yields

V p rp-l = -rp-3 rp

V pfpcr- l = -rpcr-3 rpcr

(b) position vector rp (denoted in preceding Chapters as rpa>

(c) relative position vector rpcr and related quantities

rpcr = rp - rcr = rpC(l)(p) - rcrC(I)(cr)

l/rpcr = l: (rJc/r>k+I)I1c(cos 0» = l: (r<k/r>k+I)(C(k)(p)·C(k)(cr»

= l: (-I)k(2k+l)l/2(r<k/r>k+I)[C(k)(p) x C(k)(cr)]o(O)

(where r< and r> denote the lesser and the greater of rp and rcr, the summation over k

runs from 0 to 00, I1c (cos 0» is a Legendre polynomial, and 0) denotes the angle between the position vectors of the two electrons)

l/rpcr3 = [l/(r>2-r<2)]l: (-I)k(2k+I)3/2(r<k/r>k+I)[C(k)(p) x C(k)(cr)]O(O)

l/rpcr5 = [l/3(r>2-r<2)3]I, (_l)k(2k+ 1)3/2(rJc/r>k+I)[(2k+3)r>2-(2k-l)r<2]

54

(which are obtained by successive differentiation of the expansion for l/rpo with

respect to cos m and using the properties of the Legendre polynomials).

4.2 Transformation Rules

The transformation is carried out taking into account simple relationships and recouplings. as discussed below.

(a) Transformation of vector operators to spherical form

In order to be able to apply the Wigner-Eckart theorem it is necessary that the tensor opertors T q~) transform under any rotation in an identical fashion as the modified spherical harmonics Cq~). This condition is satisfied if the commutation relations of both T q(k.) and Cq(k.) with J +. JO. and J_ are the same.

Only the electron orbital angular momentum operator does not commute with Cq(k.). Writing

.J4 = lx + ily = cos ell de + i sin ell de + i cot 9 cos ell dcjl- cot 9 sin ell dcjl

= eicjl (de + i cot 9 dell)

1_ = 1x - i1y = -cos ell de + i sin ell de + i cot 9 cos ell dcjl + cot 9 sin ell dcjl

= e-icjl( -de + i cot 9 dcjl)

one obtains

[J+. Cq(k.)] = [k(k+l)-q(q+l)]l/2 Cq+l(k.)

[Jo, Cq(k.)] = q Cq(k.)

[J_. Cq(k.)] = [k(k+ l)-q(q-l)]l/l Cq_l(k.)

and therefore the operators Tq(k) will possess the same transformation properties under rotation as the modified spherical harmonics Cq(k) if they satisfy the commutation relations [Racah (1942)]

55

[Jo, Tq(k)] = q Tq(k)

[J_, Tq(k)] = [k(k+ 1) - q(q-1)]l/2 Tq_1(k)

These commutation relations may be used in order to find the spherical components of Cartesian tensors. For example, for a vector A (with Cartesian components Ax, Ay, Az) one can use the above relations as well as the commutation relations [Brink and Satchler (1979)]

[Jx, Ax] =0

[Jy, Ax] = -i Az

[Jz, Ax] = i Ay

One can write

so that

Then one has

[Jy, Ay] = 0

[Jz, Ay] = -i Ax

~1(1) = (1N!) [J+, Ao(1)] = (1/f1) {[Jx, Az] + i[Jy, Azl}

= (l/f1){ -i Ay + i(i Ax)} = -(1/f1)(Ax + i Ay)

A_1(1) = (1/f1)[J_, AO(1)] = (1/f1){ [Jx, Az] - i[Jy, Azl}

= (1/f1){ -i Ay - i (i Ax)} = (1/f1)(Ax - i Ay)

or, conversely

[Jx, Azl = -i Ay

[Jy, Azl = i Ax

[Jz, Az] =0

Thus for the position vector r (with Cartesian components x, y, z) one obtains

r+1(1) = -(1ti!)(x + iy) ro(1) = z q(1) = (1ti!)(x - iy)

or, equivalently, r(1) = re(1) (as given above).

(b) Products of vector operators

For the scalar product of the two vector operatores, A and B, one has originally

56

while in tensor fonn it is

(A(l).B(1» = v'1"[A (1) x B(l)]O(O)

= -13" { -(1/Yj)[-A+l(1) Rl(l) + Ao(l) BO(1) - A.l(1)B+l(1)]}

= (l!2)(Ax + i Ay)(Bx - i By) + AzBz + (1!2)(Ax - i Ay)(Bx + i Ay)

That is:

In the case of the cross-product, V = [A x B], one can write

Vx = AyBz - AzBy = (i/v'!)(A.l(1)BO(l) - AO(1)Rl(l) + ~l(l)Bo(1) - Ao(l)B+l(l»

Vy = AzBx - AxBz = O!V!)(-A.l(l)Bo(l) + Ao(1)Rl(l) + ~l(l)Bo(l) - Ao(l)B+l(l»

Vz = AxBy - AyBx = i(A.l(l)B+l(l)· A+l(l)B-l(l»

and therefore

V+l(l) = -i(~l(l)BO(l). Ao(l)B+l(l»

Vo(l) = i(A.l(l)B+l(l) - A+l(l)B.l(l»

V.l(l) = i(A.l(l)Bo(1) - Ao(l)Rl(1»

On the other hand, the tensor fonnulation yields

[A<l) x B(l)]+l(l) = (1!V!)(~I(l)Bo(l) - Ao(l)B+I(I»

[A(l) x B(l)]O(I) = O!V!)(AI(l)RI(I) - A.I(l)B+I(1»

so that comparison of both sets of expressions yields

[A x B] = - i v'! [A(l) x B(l)](l)

This result may be used, as an example, in the transfonnation of the linear momentum vector operator to an expression in tenns of modified spherical harmonics. Taking into account the definition of l and the properties of vector products of vectors one can write

57

[.1 x r] = [[r x p] x r] = (rer)p - (per)r

so that

(rer)p = [.1 x r] + (per)r

With p = -i V and using the expressions for the detennination of the spherical components of a vector operator one obtains

P+l(l) = - (1/'fI) [sin 9 sin ~ cos ~]-1 ei' Or

PO(l) = [cos 9]-1 Or

P_l(l) = - (l#!)[sin 9 sin ~ cos ~]-1 e-i, Or

and with r<1) = re(l) one finally obtains

(rer) = r2(C(l)eC(l) = r2

(per) = r(p(l)eC(l) = -i ~ Or .

and one can write [Innes and Ufford (1958)]

where use has been made of the commutation property of tensors in a cross-product (see below).

(b) Commutation of operators in a tensor product

Comparison of the two expressions

[A(k) x B(k')]Q(K) = 1: (kqk'q'IKQ) Aq(k)Bq,(k')

[B(k') x A(k)]Q(K) = 1: (k'q'kqIKQ) Bq,(k')Aq(k)

(with the condition Q = q + q') and taking into account the symmetry properties of the Clebsch-Gordan coefficients (or, equivalently, of the related 3-j symbols), one obtains

or, equivalently

58

In the case of the scalar product, this expression reduces to

(c) Recoupling of products of tensor operators

In some cases it may be advantageous to change the order in which the tensor products are perfonned and consequently it is necessary to obtain the corresponding relationships. Thus, e.g., one may be interested in establishing the relationship between the products [[A (2) x B (2)](0) x U (0)](0) and [A(2) x [B(2) x U(0)](2)](0).

In a straightforward manner one can write

[[A(2) x B(2)](0) x U(O)]O(O) = (If{5) {~(-I)f1Aq(2)B-q(2)}Uo(0)

[A (2) x [B(2) x U(0)](2)]O(0) = (If{5) ~ (-I)q(2-q 0012-q)AcP)B-q(2)UO(0)

= (1f{5) ~ (-1 )fIAq(2)B-q(2)Uo(O)

That is:

The same result may be obtained in a more elegant way in tenns of recoupling. The two products considered are related to the coupling schemes IGl.i2)h2j3jm) and 1j},Gli3)j23jm), respectively, for which one can write

(with summation over j12), where the expansion coefficients are related

to 6-j symbols. Thus, in the case considered, one can write

= «2,2)0,0,0012,(2,0)2,00) [[A (2) x B(2)](O) x U(O)](O)

59

= v'5"{ ~ ~ ~ }[[A (2) x B(2)](O) x u(O)](O)

= [[A(2) x B(2)](O) x u(O)](O)

One can proceed similarly in the case of, e.g., [[A(l) x B(l)](k) x [u(l) x V(l)](k)]O(O) which is to be transformed into a product

of the products [A(1) x U(1)](K) and [B(1) x V(l)](K). Expansion of both products, in terms of the components of the tensors involved, yields

= (-1)k(2k+l)l/2 ~ (-l)q (~, q~q' _~ ) (~II _~_qll ~ )

Aq,(I) Bq-<l.~I) Uq,,(I) V -<1.-<1.,,(1)

(with summations over q, q' and q") and

=(-1)K(2K+l)l/2 ~ (-l)Q (h, Q~Q' -cf) (h" -6-Q" _~

AQ,(I) BQ,,(I) UQ-Q~I) V -Q-Q"(1)

(with summations over Q, Q' and Q"). This last equation may be rewritten as

= (_1)K(2K+l)l/2 ~ (-l)q'+q" (~, ~II _q~qll) (~_q' q_~11 q~II)

Aq.(l) Bq-<l.~I) Uq,,(1) V -<1.-<1.,,(1)

(with summations over q, q' and q"). One can then write, for a given k,

(with summation over K) or equivalently, taking into account the above expansions,

60

(-1)k+q(2k+1)lfl (11k) (11k) q' q-q' -q q" -q-q" q

K ' " 1/2 (11K ) (11K ) =1:(-1) +q+q aK(2K+1) q' q" _q'_q" q-q' q-q" q'+q"

(with summation over K). The above expression yields an equation for each set of values q', q-q', q" and -q-q", which can be solved for the parameters aK. Thus, for k=O, q=O one obtains the values ao = 1/3, al = Y'Jf3 and a2 = 5f3. In terms of recoupling the transformation considered implies a recoupling of the general form

l(jlh)j 13,(j2i4)j24jm)

= 1: l(j Ih)j 12,(j3.i4)h4jm) «j Ih)j 12,(j3.i4)j34jml(j Ih)h3,(j2i4)j24jm)

(with summations over j12 andj34), where the expansion coefficients are related

{ jl h j12}

= [(2jI2+1)(2j34+1)(2j13+1)(2h4+1]lfl ~3 j~ j~4

113 J24J

to 9-j symbols. In the present case one obtains

[[A(1) x B(1)](O) x [U(I) x V(1)](O)]o(O)

= 1: «(1,l)k,(1,l)k,OOI(1,l)O,(1,l)O,OO)[[A(1) x U(l)](k) x [B(1) x V(l)](k)]O(O)

{II k} = 1: (2k+1) 11k [[A(l) x U(l)](k) x [B(1) x V(l)](k)]O(O) 000

and similarly

61

=1:{[8-2k(k+ 1)]/8YJ}(2k+ 1)1/2[[A (1) x U(1)](k) x [B(I) x V(1)](k)]O(O)

[[A (1) x B(1)](2) x [U(1) x V(1)](2)]O(O)

= 1:{ [3 [4-k(k+ 1)][3-k(k+ 1)]-16]/12Y3"}(2k+ 1)1/2

[[A (1) x U(1)](k) x [B(I) x V(1)](k)]O(O)

(with summations over k), where the coefficients (squared) satisfy the nonnalization condition.

Therefore, if the tenn appearing in the Hamiltonian operator is of the form (A·B)(U·V) one can write

In another case of interest one will write similarly

[[A (1) x B(1)](2) x [U(k) x V(k)](O)]q(2)

= 1: «1,k)k',(1,k)k",2ql(1,1)2,(k,k)O,2q)

{ 1 1 2} = 1: (5(2k'+1)(2k"+1)]l/2 k k 0

k' k" 2

= 1:(-1)k"+k+l[(2k'+1)(2k"+1)/(2k+l)]1!2{r ~,,~}

[[A (1) x U(k)](k') x [B(1) x V(k)](k")]q(2)

(with summations over k' and k").

62

4.3 Application

The tranformation of the tenns of the Hamiltonian operator to tensor fonn may now be easily accomplished.

For example, in the case of the specific mass effect one can write

(Vp.Vo) == (V(1)(p).V(1)(a»

= «- ff rp -l[C(l)(p) x .Jl.(1)(p)](1) + C(l)(p)dp)

·(-ff ro-1[C(1)(a) x ],(1)(a)](1) + C(1)(a)do»

Using, for simplicity, the notation

one obtains

(V p.V 0) = 2rp -lro-1(R(l,l)(p).R(l,l)(a» - ffrp -l(R(l,l)(p).C(1)(a»)dp

- ffro-1(C(l)(p).R(l,l)(a»dp + (C(l)(p).C(l)(a»dpdo

4.4 Summary

The transfonnation to tensor fonn has been applied [Fraga et al. (1986)] to most of the tenns of the Hamiltonian operator, discussed in Chapter 1. The corresponding expressions (reprinted with permission of the American Physical Society) are

summarized below, where the summations over the electrons run over o>p = 1 to N

in all cases except for the Zeeman effect, in which case they run over (J;tp = 1 to N.

Electronic interaction

Mass variation

Specific mass effect

63

Ksm = (lIIlla) l: [(C(1)(p)-C(1)(cr»)dpaa - y!(C(1)(p)-R(1,l)(cr»ra-1ap

- Y!(R(l,l)(p)-C(1)(cr»rp-1aa + 2(R(l,l)(p)-R(l,l)(cr»rp-1ra-1]

One-electron Darwin correction

Kdl = (1Ca.2Z/2) l: B(rp) = (a.2'Zj8) l: rp-2B(rp)

Two-electron Darwin correction

Kd2 = -(a.2/4) l: (rp-2B(rpa) l: (2k+1)(C(k)(p)-C(k)(cr»]

Electron spin-spin contact interaction

Ksc = -(2a.2/3) l: (rp-2B(rpa)(s(1)(p)-s(1)(cr» l: (2k+1)(C(k)(p)-C(k)(cr»]

Orbit-orbit interaction

Koo = (a.'N2) l:I: h(oo)(k;p,cr)

h(oo)(k;p,cr) = (C(k)(p)-C(k)(cr»[[[k(k+ 1)/(2k-1)](r<k-l/r>k)

- [k(k+ 1)/(2k+ 3)](r~+ 1/r>k+2)]apaa

+ (k/2)[(rpk-l/r~+l)dp - (r~-l/rpk+l)ap]

+ [(k+ 1)/2][ -(rpk/r~+2)ap + (r~/rpk+2)aa]]

+ (C(k)(p )-R(k,k)( cr) ) [[[k(k+ 1)] 1/2/(2k+ 3)] [-k(rpk+ 1/r~+2) + (k+ 3)(r~+ 1/rpk+2)]

+ [[k(k+ 1)]1/2/(2k-1)][(k-2)(rpk-l/r~) - (k+ 1)(r~-1/rpk)]]]ra-lap

+(R(k,k)(p)-C(k)(cr»[[[k(k+1)]1/2/(2k+3)][(k+3)(rpk+l/r~+2) - k(r~+1jrpk+2)]

+ [[k(k+ 1)] 1/2/(2k-1)][(k+ 1)(rpk-l/r~) - (k-2)(r~-1/rpk)]]rp-laa

+ (R(k,k)(p)-R(k,k)(cr»)[[k(k+3)/(2k+3)](r~+I/r>k+2)

- [(k-2)(k+ 1)/(2k-1)](r~-I/r>k)]rp-Ira-1

- (R(k-l,k)(p )_R(k-l,k)( cr»)[[2(2k-1)/(k+ 1)](r<kjr>k+ I )rp -Ira-I]

- (R(k-I,k)(p)-C(k)(cr»[k(2k-1)!2]I/2[(rpk-2/r~+I) - (r~/rpk+3)]

64

- (Cp(k).R(k-l,k)(a))[k(2k-l)/2]1/2[(rpk/rd'+3) - (rd'-2/rpk+l)]

Spin-orbit coupling

Jtso = (Za2/2) l; rp-3(s(l)(p) • .1(1)(p»

+ (a2{2) :El; (_I)k+l (h~!~)(k;p.a).(s(1)(p)+2s(1)(a)))

h~~)(k;p.a) = [C(k)(p) x C(k)( a)](1)[k(k+ 1 )(2k+ 1 )/3] l/2[(rpk-Vrd'+ 1 )+(rd'/rpk+2)]ap

+ [R(k,k)(p) x C(k)(a)](1)[(2k+l)/3]l/2[k(rd'/rpk+3)-(k+l)(rpk-2/rd'+1)]

- [R(k-l,k)(p) x C(k-l)(a)](1)[(2k+ 1)/3]l12(2k-1)(r(Jk-l/rpk+2) .

+ [R(k+l,k)(p) X C(k+I)(a)](1)[(2k+ 1)13] 112(2k+3)(rpk-l/rd'+2)

Spin-spin dipole interaction

ftsd = (a2t5):El; (_I)k[k(k+l)(2k-l)(2k+l)(2k+3)]1/2

([s(1)(p) x s(1)( a)](2).T(2»

T(2) = (rpk-I/rd'+2)[C(k-l)(p) x C(k+l)(a)](2)

+ (rd'-1/rpk+2)[C(k+I)(p) x C(k-l)(a)](2)

Magnetic dipole and Fenni contact interactions [Annstrong (1971)]

Jto = gclJ.B l; (rp-3(.1(I)(p)_(10)1/2[s(1)(p) x C(2)(p)](1»

+ (2/3)rp -2S(rp)s(1)(p») ·N(1)

Electric quadrupole coupling [Annstrong (1971); see also Judd (1963)]

KQ = -[YO"QIlI(2I-l)] L rp-3(C(2)(p).I(2»

Magnetic octupole coupling [Annstrong (1971)]

Jto = -(5/3)I12JlB l; rp-5{[6-(Sn)rpS(rp)][s(1)(p) x C(2)(p)](3)

- 2[C(2)(p) x .1(I)(p)](3»).N(3)

Zeeman effect

65

Kz = HIlB{E [[1-(m/mp)]-a2Tp]1(p)(p)

+ E {~(1-a2Tp) + (Za2/3)rp-l]so(1)(p)

+ (5/18)l/2Za2 E [s(l)(p) x C(2)(p)]o(l)rp-l

+ (m/mp) ~ (Y![C(l)(p) x Ca(l)]O(lh-pda

- 2[C(l)(p) x R(l,l)(cr)]o(l)rpra-1)

+ a2 EE [h(z)(k;p,cr)x(s(l)(p)+2s(l)(cr))]~1) + a2 U ti~~)(k;p,cr)}

-HIDIlNIo(l)

(with the summations over k running from 0 to 00 and 1 to 00, respectively)

h(z)(k;p,cr)

= [C(k-2)(p) x C(k)(cr)](2)(rpk/rJc+l)( _1)k+l[(k-l)k(2k-3)(2k+ 1)/12(2k-l)]l/2

+ [C(k+2)(p) x C(k)(cr)](2)(rJc/rpk+l)( -1)k[(k+ 1)(k+2)(2k+ 1)(2k+5)/12(2k+3)] 1/2

+ [C(k)(p) x C(k)(cr)] (2)(rpk/rJc+l )(-l)k[k(k+ 1)(2k+ 1)(2k+3)n2(2k-l)] 1/2

+ [C(k)(p) x C(k)(cr)](2)(rJc/rpk+l)( -1)k+l[k(k+ 1)(2k-l)(2k+ l)n2(2k+3)] 1/2

+ [C(k)(p) x C(k)(cr)](l)(r<k/r>k+l)(-I)k[k(k+l)(2k+l)/24]l/2

+ [C(k)(p) x C(k)(cr)](O)(rpk/rJc+l)(-I)k+l(kl3)(2k+l)1/2

+ [C(k)(p) x C(k)(cr)](O)(rJc/rpk+l)(-I)k[(k+l)/3](2k+l)l/2

ti~~)(k;P,cr)

= [C(k)(p) x C(k)(cr)]o(1)(rp/ra)k+2da(-I)k[(k+3)/(2k+3)][k(k+ 1)(2k+ 1)/12]1/2

+ [C(k)(p) x C(k)(cr)]o(1)(ra/rp)k+lda(-I)k+l[kI(2k+3)][k(k+ 1)(2k+ 1)/12] III

+ [C(k)(p) x C(k)(cr)]o(l)(rplra)kda( _l)k+l[(k+ 1)/(2k-l)][k(k+ 1)(2k+ 1)/12]l1l

+ [C(k)(p) x C(k)(cr)]o(l)(ra/rp)k-lda( -1)k[(k-2)/(2k-l)][k(k+ 1)(2k+ 1)/12]l1l

+ [C(k)(p) x R(k,k)(cr)]o(1)[(rpk+2/rJc+3)

66

+ (rci-/rpk+1)](-I)k[k(k+3)/(2k+3)][(2k+l)/12]1/2

+ [C(k)(p) x R(k,k)(cr)]O(l)[(rpk/rci-+1) + (rci--2/rpk-l)](-I)k+l

[(k-2)(k+ 1)/(2k-l)][(2k+ 1)/12]1/2

+ [C(k)(p) x R(k-l,k)(cr)]O(l)[(rpk+l/rci-+2) + (rci--1/rpk)](-I)k

[(2k-l)(2k-l)/3(2k+ 1)]112

+ [C(k+l)(p) x R(k+l,k)(cr)]O(1)[(rpk+l/rci-+2) + (rci--1/rpk)](-I)k

[(2k+3)(2k+3)/3(2k+ 1)]1/2

Stark effect

Hs = -F l;rpCo(1)(p)

Nuclear mass dependent orbit-orbit coupling

Hoo(m) = -a,2(ZI2JIm1a) U h(oo)(m)

h(oo)(m) = -2rp -2ra-1(R(l,l)(p).R(l,l)( cr» + fI rp -2(R(l,l)(p )·C(l)(cr»da

+ fI rp -lra-1(C(1)(p)·R(l,l)( cr»dp-rp -l(C(l)(p)·C(l)(cr»dpda

-2rp -2ra -l(C(l)(p)· R(l,l)(p »(C(l)(p ).R(l,l)( cr»

+ fI rp -2(C(1)(p)· R(l,l)(p »(C(l)(p )·C(l)( cr»da

+ fI rp-1ra-1 (C(l)(p).C(l)(p»(C(l)(p).R(l ,l)(cr»dp

- rp -l(C(l)(p).C(l)(p »(C(l)(p).C(l)( cr»dpda

Nuclear-mass dependent spin-orbit coupling (electron spin)

Hsos(m) = a,2(Z/mma> l;l; h(sos)(m)

h(sos)(m) = -2 rp-2ra-1(s(1)(pNC(l)(p) x R(l,l)(cr)](l»

+ fI rp -2(s(l)(p).[C(l)(p) x C(l)(cr)](l»da

Nuclear-mass dependent spin-orbit coupling (nuclear spin)

HsoI(m) = -a,2(Zlmma> U h(soI)(m)

67

5 Matrix Elements

The fact that the one- and two-electron tenns of the Hamiltonian operator may all be written, as shown in the preceding Chapter, as f(rp)T(O)(p) and f(rp,ra)T(O)(p,a), respectively, where f(rp) and f(rp,ra) denote radial-dependent functions, may be used for the separation of the angular and radial parts of the matrix elements.

5.1 General Formulation

It will be assumed, first of all, that the spin-orbitals within the Slater detemrinants of the two functions involved in the matrix element have been reordered (introducing the required sign change, which is omitted here for simplicity) in order to bring them into maximum coincidence. As a result of this reordering, the summations over electrons and spin-orbitals, in the development below, become equivalent.

Let us consider first the matrix element of a one-electron term of the Hamiltonian operator. Proceeding, without loss of generality, within the J-coupling scheme, one can write

(CyJMJI1:f(r p)T(O)(p )IC'1 J'M]')

= 1: (SMSLMLIJMJ) (S'MS'L'ML'IJ'M]')

(SMsLMLI1:f(rp)T(O)(p )IS'MS 'L'ML')

with summations over MS and ML (compatible with MJ = MS + ML), MS' and ML' (compatible with M], = MS' + ML') and p. The matrix element on the rhs of the above equation may now be expanded as

(SMSLMLI1:f(rp)T(O)(p)IS'MS'L'ML')

= 1: ({.i) (IIlJ.LluMsMLISL) ({.i') (m'Il' lvMs'ML'IS'L')

70

with summations over u and v (see Chapter 2) and p. The new matrix element appearing on the rhs of the above equation may be now expanded in terms of matrix elements over spin-orbitals

({nR.} {mll}uMsMLIU(rp)T(O)(p)l{n'l'} {m'Il'}vMs'ML')

= ~ Ei«nili)(1)lf(rt)l(ni'li')(1» «limilli)(1)IT(O)(1)I(~'mi'lli')(1»

= ~ EiR(ni~;ni'~') «~milli)(1)IT(O)(1)I(~'mi'lli')(l»

with summation over the spin-orbitals and where Ei may take the values 1 (for all i,

when the two configurations are identical), ~p (if the two configurations differ only in the spin-orbitals in the p-th position) and 0 (if the configurations differ in two or more spin-orbitals). R(ni~; ni'~') denotes a radial integral involving the orbital part of the spin-orbitals i and i'.

Therefore, for the original matrix element one can now write

where the matrix element on the rhs may now be expressed in terms of reduced matrix elements. For example, for the case

one would obtain

= (_l)S'+L+J{~' t. i}(SIIS(K)(P)IS')(LIIL<K)(P)IIL')

With

(SIIS(K)(p)IIS') = (2S+1)l/2(S'Ms'KOISMs)(SMsISo(K)(p)IS'Ms')

(LlIL(K)(p )IIL') = (2L+ 1)l/2(L'ML'KOILML)(LMLILQ(K)(p )IL'ML')

finally the matrix element would be given in terms of the matrix elements (l/21lIS0(K)Il/2Il')(imlLo(K)Il'm'), which may be obtained from the corresponding reduced matrix elements.

71

Proceeding in a similar fashion, the matrix element for a two-electron term of the Hamiltonian operator may be expressed as

(C'YJMJI~f(rp,rer)T(O)(p,o)ICYJ'MJ')

= ~ coperPper'R(np~,nerlcr; np'~',ner'.icr')(C'YJMJIT(O)(p,o)ICYJ'MJ')

with Pper' = I-Tper', where Tper' denotes the transposition of the two spin-orbitals

np'lp'mp'Jlp' and ncr'.icr'mcr'Jlcr'. The parameter coper takes the values 1 (for any

p,o if the two configurations are identical), Scrq (if the two configurations differ in

the spin-orbitals at the q-th position), BppBerq (if the two configurations differ in the spin-orbitals at the p- and q-th positions) and 0 (if the two configurations differ in more than two spin-orbitals).

Similar expressions may be obtained, proceeding in the same manner as above, for the matrix elements within the SL- and F-coupling schemes, in terms of the corresponding angular-dependent matrix elements.

5.2 General Expressions

The angular-dependent matrix elements may now be developed for the general types of interaction operators present in the Hamiltonian operator, within the context of the three coupling schemes considered in this work [Fraga et al. (1986)].

The corresponding expressions (reprinted with permission of the American Physical Society) are:

S.2.1 SMSLML coupling

The tensor operators in the electronic and non-splitting terms may be expressed, in a general fashion, as

T(O)(p) = (S(O)(p). L<O)(p»

T(O)(p,o) = (S(nO)(p,o).L<kkO)(p,o»

where S and L denote appropriate spin and orbital angular momentum operators, such as, e.g.,

~(O)(p) = 1

L(O)(p) = 1

S(llO)(p,o) = (s(1)(p).s(1)(o»

L(kkO)(p,o) = (C(k)(p )·C(k)( 0» etc.

72

The matrix elements are then given by

(CPSMSLMLI:Ef(rp)(S(O)(p ).L(O)(p )IC'f3'S 'MS'L'MS ')

= a(SMsLML, S'Ms'L'ML')

(CPSMSLMLI:Ef(rp,ra)(S(lClCO)(p,o).L(kkO)(p,o)IC'P'S'ML'L'ML')

= a(SMsLML, S'Mg'L'ML')

:E {( {l} {m~}uMsMLIPSL)({l'} (m'~'}vMsMLIf3'SL)

:Emij Pij' R(ni~, nj.R.j; ni'~', nj'lj')

(1/2 ~i 1/2 ~jISO(1C1CO)11/2 ~i' 1/2 ~j')(~mi.R.jDljILo(kkO)~'mi'.R.j'mj')}

where the summations run over the electrons (p<o) or the spin-orbitals (i<j) and the Slater-determinants (u,v) and where the matrix elements are to be developed in terms of the appropriate reduced matrix elements for each particular case (see below).

5.2.2 JMJ-coupling

General terms of the form

appear in the fme-structure interaction terms, such as, e.g.,

73

(S(l)(p)eL(l)(p» = (s(l)(p)ei(l)(p»

(S(1l2)(p,cr)eL«k-l)(k+l)2)(p,cr» = ([s{l)(p) x s(l)(cr)](2)e[C(k-l)(p)eC(k+l)(cr)](2»

etc.

The operators in the electronic and non-splitting terms, given above, represent special cases (K::::O) of these operators and therefore the matrix elements may be given, in general, as

IS' L' J } (-I)S'+L+][(2S+1)(2L+l)]l/2[(L'MrJ(OILML)(S'MsKOISMs)]-l L S K

:E{ ({i) {m~)uMsMLIPSL)({i'} {m'~' JvMsMLIP'S'L')

IeiR(ni~; ni'~')(1/2 ~iISo(K)I1/2 ~i')(~milLo(K)~'mO }

(CySLJM]I:Ef(rp,ra)(S(lClC'K)(p,cr)eL(kk'K)(p,cr)IC"y'S'L'J'M]')

= S(MsMLJM],MS'ML'J'M]')

IS' L' J } (_I)S'+L+][(2S+1)(2L+l)]l/2 [(L'MLKOILML)(S'MgKOISMs)]-l L S K

:E{ ({i) {m~}uMgMLIPSL)({i'} (m'~'lvMsMLIP'S'L')

:ECOij Pij' R(ni~,njlj; ni'~', nj'lj')

(1/2 ~i 1/2 ~jISo(1C1C'K)I1/2 ~i' 1/2 ~j')(~miljmjILo(kk'K)~'mi'lj'mj')}

where the summations run over the electrons (p<cr) or the spin-orbitals (i<j) and the Slater determinants (u,v). The matrix elements are to be developed in terms of the appropriate reduced matrix elements for each particular case (see below).

74

5.2.3 FMF-coupling

In this case one must consider both the interaction operators, discussed above, for the electronic, SL-non-splitting and fine-structure interaction terms as well as the proper hyperfme-structure coupling terms.

The interaction terms in the hyperfine-structure coupling operators are of the

form (J(K)(p).N(K)(p», where J and N represent total electron and nuclear spin angular momentum tensor operators such as, e.g.,

(J(1)(p).N(l) == ([s(l)(p) x C(2)(p)](l).N(l»

(J(2)(p ). N(2» == (C(2)(p). 1(2»

etc. The electronic and SL-non-splitting terms may be expressed in an analogous form, with K=O and taking N(O) as a unit tensor. Therefore the matrix elements may be written, in general, as

(CEIJFMFI1:f(rp)(J(K)(p).N(K»)IC'E'IJ'F'MF') = o(FMF, F'MF')

(-1)I+J+F[(2I+1)(2J+1)]l/2 (IMIKOIIMI)-I(IMI1No(K)IIMI)H r ~} l£iR(ni~; ni'~')(SLJII[S(1C) x L<k)](K)IIS'L'J')

= 8(MSMLFMF' MS'ML'F'MF')

(-I)I+J+F[(2I+ 1)(2J+ 1)(2J'+ 1)(2L+ 1)(2S+ 1)(2K+ 1)] 1/2

[(IMIKOIIMI)(L'MLkOILML)(S 'MSKOISM) ]-1

which reduces as follows for the two extreme cases:

75

for 1C = 0 (i.e .• k = K)

(CeIJMFMFI~f(rp)(J(K)(p). N(K»IC'e'IJ'F'MF')

= o(SMSMLFMF.S'MS'ML'F'MF')

(_I)I+2J+S+L'+F+K[(2I+ 1)(21+ 1)(2J'+ 1)(2L+ 1)]1/2

for k = 0 (Le .• 1C = K)

(CeIJFMFI~f(rp)(J(K)(p).N(K»IC'e'IJ'F'MF') = o(MSLMLFMF. MS'L'ML'F'MF')

(_1)I+I+I'+S+L+F+K[(2I+ 1)(21+ 1)(21'+ 1)(2S+ 1)]1/2

[(IMIKOIIMI)(S'MSKOISMS)]-1 [IMI1NO(K)IIMI]{ } f ~H ~ ~' i} ~{( {i} {mJ.l.)uMsMLI~SL)({i'} (m'J.l.'}v'MSMLIWS'L)

where the summations run over the electrons (p<cr) or the spin-orbitals (i<j) and the Slater determinants (u.v). The matrix elements are to be developed in terms of the appropriate reduced matrix elements for each particular case (see below).

5.3 Examples For Specific Interactions

The general expressions. obtained above, may now be applied to the various terms of the Hamiltonian operator and some illustrative examples will be presented here before summarizing the complete results.

(a) For example. for the electrostatic interaction

76

(where the summations extend over the electrons p<a and over k) the general expression becomes

(C~SMSLMLIl:(r<k/r>k+ 1 )(C(k)(p )·C(k)( a»IC'WS'Ms'L'ML')

= B(SMsLML, S'MS'L'ML')

with

l:{ ({i) {m~}uMsMLI~SL)({i'} (m'~'}vMsMLIWSL)

l:CJlij Pij' Re2(k)(ni~,njij; ni'~', nj'ij')

Ae2(k)(iimi~i,ijmj~j; ii'mi'~i' ,lj'mj'~j')}

Ae2(k) = «~mi~ih (ijmj~jhl(C(k)(1).C(k)(2»I(~'mi '~i') 1 (.Q;'mj'~j'h)

= B(~i,~j')B(~j,~j')«iimih(ijmjhl(C(k)(I).C(k)(2»I(~'mi')I(ij'mj'h)

= B(~i,~j')B(~j,~j')B(mi+mj,mi'+mj')(-l)q[ (21i+ 1)(21j+ 1)]-1/2

(~'mi'k-qliimi)(ij'mj'kqlijmj)(ii"C(k)"ii')(ij"C(k)"ij')

= O(~j,~i')O(~j,~j')O(mi+mj,mi'+mj')( -l)q

[(~'+ 1)(21j'+ 1)/(~+ 1) (21j+ 1)] 1/2

{~'mi'k-q~mil {ij'mj'kqlljmj}

with q = mi-mi' = mj'-mj and where

{i'm'kqllm} = (i'm'kqlim)(i'OkOliO)

so that one can fmally write

(C~SMSLMLIl:(r<k/r>k+ 1 )(C(k)(p )·C(k)( a»IC'WS'MS'L'ML')

= B(SMsLML, S'Ms'L'ML')l:({i}{m~}uMsMLI~SL)

({i') (m'~'}vMsMLIWSL)

77

[(lli'+ 1)(llj'+ 1 )/(2~+ 1)(21j+ 1)] If}.

{~'mi'k-ql~mil (ij'mj'kqlijmj}Re2(k)(n~,njij;ni'~' ,nj'ij')}

(b) For the spin-own orbit interaction

the general expression becomes

(C'ySUMJI(Za2/2) l: rp -3(s(1)(p ).i(l)(p) )ICYS 'L'J'M]')

= O(MSMLJMJ,MS'ML'J'M],)(-I)S'+L+J[(2S+1)(2L+l)]If}.

[(L'MLIOILML)(S'MSlOISMS)]-I{ t' ~: ~ } l:{ ({i) (mJl }uMSMLI~SL)( {i'} {m'Jl'} vMSMLIWS 'L')

with

= O(Jli,Jli')(1,H2")(1!2 Jli lOl l/2 Jli)(1!2 IIs(1)II1/2)

(c) One can proceed in a similar fashion in the case of the matrix elements of the hyperfine-structure interaction and for that reason only the details concerning the nuclear moments will be discussed here.

The nuclear dipole and octupole magnetic moments

Jll = (lMlINO(l)IIMI)

n = -(lMlINO(3)IIMI)

are related (for Ml=I) to the tensor operators in the corresponding operators. Therefore, the nuclear-dependent tenns may be rewritten (for MFI) as follows:

magnetic dipole interaction

78

(21+1)l/2(UI01ll)-I(II1NO(l)lll) = [(21+ 1)(1+ 1)!I]lflllI

magnetic octupole interaction

(21+ 1)(ll30Im-1(II1No(3)1I1) = [(1+1)(1+2)(21+ 1)(21+3)/I(I-l)(21-1)]lflQ

(d) In the case of the electric quadupole coupling, not covered by the general expressions given above, one can write

(CeIJFMFIKQIC'e'U'F'MF')

= -[~QI2I(21-1)](C~UFMFll;rp-3(C(2)(p).1(2»IC'P'UFMF')

= -[voQ/21(21-1)] a(SMsMLFMF, S'Ms'ML'F'MF')

(-1 )I+S+L'+F[ (21 + 1 )(21+ 1 )(2J' + 1 )(2L+ 1)] Ifl [(IMI20IIMI)(L'MLlOILML)]-1

2 {I J' F} {J' L'S} (lMIIIo( )IIMI) J I 2 L J 2

which, for MI=I and with

10(2) = [1(1) x 1(1)]0(2) = -(112v'O)4L - (112~)L4 + (2f/O)1z2

(llIlo(2)lm = 1(21-1)f{O

reduces to

(CeIJFMpIKQIC'e'U'F'Mp') = -(QI2)a(SMSMLfMp,S'Ms'ML'F'MF')

(_1)I+S+L'+F[(I+ 1)(21+ 1)(21+3)(21+ 1)(21'+ 1)(2L+ 1)/1(21-1)] Ifl

(L'MLlOILML)-I{ I J' F}{J' L'S} J I 2 L J 2

l;{ ({i} {mll}uMsMLI~SL)({i'} (m'Il'lvMsMLIp"SL')

~iRQ(n~i;ni'.Q.j')a(mi,mj')a(Ilj,lli') {i'mi20~mi}[(~'+1)/(~+1)]lfl}

6 Summary of Theoretical Results

The complete formulas fOt: the matrix elements of all the terms of the Hamiltonian operator are presented below.

All the expressions are given (with appropriate coefficients) in terms of SL­vector coupling coefficients and radial and angular parts. The latter are given explicitly in each case while the former have been expressed in terms of radial integrals.

Some of these radial integrals are common to several interactions and for that reason their expressions are presented below, where a (and b) have been used for simplicity in the expressions instead of t& (whenever no confusion will arise):

One-electron integrals

T(t&;n'.i) = J drP(t&;r)TP(n'.i;r)

I(k)(t&;n '.i') = J rkrlfP(t&;r)P(n'.i';r)

Imv(t&;n'.i) = Jdr(TP(t&;r)(TP(t&;r»

~l(t&;n'.i) = R(t&;O)R(n'.i;O)

Isc(aa';bb') = J r 2drP(a;r)P(a';r)P(b;r)P(b';r)

Iz(n.i;n '.i') = J rdrP(r&;r)(arP(n'l';r)/r)

II(n.i;n'l') = J r2drP(t&;r)(arP(n'l';r)/r)

(with integration in all cases between 0 and 00).

Two-electron integrals

80

D(k)(aa';bb') = I qdrlI U(k)(q,r2)r2<ir2P(a;fl)(arP(a';rl)/q)P(b;r2)(arP(b';r2)/r2)

N(k)(aa';bb') = N>(k)(aa';bb') + N>(k)(bb';aa')

N>(k)(aa';bb') = I (l/qk+3)drll r2k<ir2P(a;q)P(a';q)P(b;r2)P(b';r2)

R(k)(aa';bb') = R>(k)(aa';bb') + R>(k)(bb';aa')

V(aa';bb') = I qdrlI r2- ldr2P(a;rl)(arP(a';rl)/q)P(b;r2)P(b';r2)

V(k)(aa';bb') = V>(k)(aa';bb') + V «k)(aa';bb')

V «k)(aa';bb') = I qk<irII (l/r2k+l)dr2P(a;q)P(a';r2)P(b;r2)(arP(b';r2)/Q)

V>(k)(aa';bb') = I (l/flk+l)drII r2kdr2P(a;q)P(a';q)P(b;r2)(arP(b';r2)/r2)

X(aa';bb') = I fl-2dfll r2- ldr2P(a;fl)P(a';fl)P(b;r2)P(b';r2)

Y(aa';bb') = I drll r2- ldr2P(a;q)(arP(a';rl)/fl)P(b;r2)P(b';r2)

with integration between 0 and 00 for the integral over q in all cases and between 0

and 00 for the integral over f2 in all cases except for the integrals N>(k), R>(k), V>(k),

81

and W>(k.) (with integration from 0 to rt) and for the integrals V <(k.) and W <(k.) (with integration from rl to 00), and where

forrt < 1"2

()rP(r)/r == «()(P(r)/r)f()r) = r 1«()P(r)/iJr)-r2P(r)

6.1 Electronic Energy

Operator

He .. = -(1/2) ~ L\p - Z ~ rp-l + ~ rpa-1

(with summations over the electrons, p and cr, with cr> p = 1 to N).

Matrix element

(CySMsLMrJMJIKe .. IC'y'S 'MS 'L 'ML' J'M]')

= B(SMsLML1MJ, S'Ms'L'ML'J'M]')(CPSMSLML1Ke .. IC'P'SMsLMd

(CPSMsLMLIKe .. IC'P'S'Ms'L'ML')

= B(SMSLML,S'MS'L'ML') B(P,/3') ~ £iRet(niii;ni'lUAet(Jl.jmiJli;~'mi'~i')

~ {({i) {m~}uMsMLIPSL)({i'} (m'~'}vMsMLI/3'SL)

~roijPij' (~Re2(k:)(niii,njlj;ni'~' ,n j 'ij')

Ae2(k.)(~mi~i,ijmj~j;ii'mi'~i' ,lj'mj'~j')]}

(with summations over the spin-orbitals i = 1 to N, the Slater determinants u and·v, the spin-orbitals j > i = 1 to N, and k, respectively).

Radial parts

Ret(i;i') = T(ni~;ni'ii) - ZI{-l)(ni~;ni'ii)

Re2(k.)(ij;iT) = R(k)(ii';jj')

82

Angular partS

Ael(i;i') = a(~,~') S(mj,mj')S(J.1i.J.1i'>

Ae2(k)(ij;i'j') = S(mi+mj,mi'+mj')S(J.1i,J.1i')S(J.1j,J.1j')

(-l)q[(~'+ 1)(21j'+ l)/(21i+ 1)(21j+ 1)] Ifl{~'mi'k-qllimi} {1j'mj'kq I1jmj}

with

Remarks

The condition S(P,P') only applies to matrix elements between states of the same symmetry designation of the same configuration.

6.2 Mass Variation

Operator

Kmv = -(a.2!S) 1: Vp4

(with summation over the electrons p = 1 to N).

Matrix element

= S(SMsLMrJMJ,S'MS'L'ML'J'M]')(CPSMSLMLIKmvIC'P'SMsLML)

(CPSMSLMLlKmvIC'P'S'Ms'L'ML')

= S(SMsLML,S'Ms'L'ML')S(P,P')l:£iRmv(ni~;ni'.iOAmv(~miJ.1i;~'mi'J.1i')

(with summation over the spin-orbitals i = 1 to N)

Radialpart

83

Angular part

Remarks

The condition l)(P,J3') only applies to matrix elements between states of the same symmetry designation of the same configuration.

6.3 Specific Mass Effect

Operator

(with summations over the electrons, 0" > P = 1 to N).

Matrix element

= l)(SMSLMLJMJ,S'MS'L'ML'J'Mj')(CPSMSLMLlKsmIC'P'SMSLML)

(CPSMSLMLlKsmIC'P'S'MS'L'ML')

= l)(SMSLML,S'MS'L'ML')

1: {({.1) {IllJ.1}uMSMLIPSL)({.1'} (m'~'}vMSMLIJ3'SL)

1: COjjPij'Rsm(ni.1ionj.1j;ni'~' ,nj'.1j')

Asm(~mi~i,.1jmj~j;~'mi'~i' ,.1j'mj'~j')}

(with summations over the Slater determinants, u and v, and the spin-orbitals, j > i = 1 to N).

Radial pan

Rsm(ij;ij')

= -(l/ma) (D(ii';jj') - (1!2)[.1j(.1j+ 1)-.1j' (.1j'+ 1)-2JV(ii';jj')

- (l!2)[~(~+ l)-~' (~'+ 1)-2JV(jj';ii')

84

Angular pan

Asm(ij;i'j') = O(mi+mj,mi'+mj')O(Ili>Ili')O(llj,Ilj')

(-l)q[(~'+ 1 ) (2ij' + l)/(2ii+ l)(2ij+ 1)]l/2{li'mi'!-qllimi) {lj'mj'lqll.imj}

with

Remarks

As the calculations are perfonned in the Il-unit system (see Chapter 7), the numerical factor, lima, in the operator must be changed to

wma = mma/(m+ma)ma = m/m+ma

If the value of this correction is to be reported independently, in cm- l , the conversion is achieved by means of the factor 2RM, with

RM = Roo[mJ(m+ma)] = Roo[lI(1+m/ma)] == Roo(1-m/ma>

6.4 One-Electron Darwin Correction

Operator

(with symmations over the electrons p = 1 to N).

Matrix Element

(with summation over the spin-orbitals i = 1 to N).

85

Radial part

~l(i;i') = (o.2!2)Z Idl(n~i;ni'~)

AnguJarpan

Remarks

This interaction exists only for s orbitals.

6.S Two-Electron Darwin Correction

Operator

(with summations over the electrons cr > P = 1 to N).

Matrix element

(CySMSLMLJM]IKd2IC'y'S'MS'L'ML'J'M]')

= o(SMsLML,S'Ms'L'ML')

I,r""P"'Rd2(n'l' n'l"n"o" n"n")Ad2(o'm' n'm"o"m" n"m") Ul1J 1J 1 h J l' 1 AJ" J AJ AJ, hAJ 1'AJ, 1 oAJ J

Radial part

Rd2(ij;i'j') = -(a2/4)IsC<ii';jj')

Angular part

with

Spin part

86

(-I)q{~2k+ 1)[(21i'+ 1)(2.R.j'+ 1)/(21i+ 1)(2.R.j+ 1)]1/2

{1i'mi'k-ql1imil U.j'mj'kqLijmj} }

6.6 Electron Spin-Spin Contact Interaction

Operator

(with summations over the electrons (J > P = 1 to N)

Matrix element

(CPSMSLML1KscIC'P'S'MS'L'ML')

~ {( {i} {m~}uMSMLIPSL)({i'} (m'~'lvMsMLIWSL)

2~(J)ijPij'Rd2(oi1i,njlj;oi'ii', OJ 'lj')

~(1imi.ijmj;1i'mi',lj'mj')SsC<~i,~j;~i',~j')

87

RadiaJpart

See 'Two-electron Darwin correction'

Angular part

See 'Two-electron Darwin correction'

Spin part

with

6.7 Orbit-Orbit Interaction

Operator

(with summations over the electrons (J > P = 1 to N).

Matrix element

(CySMSLMLJMJ1KooICyS'Ms'L'ML'J'M]')

= 5(SMsLMLJMJ,S'MS'L'ML'J'M]')(CPSMsLMLlKooIC'P'SMsLML)

(CPSMsLML1KooIC'P'S'Ms'L'ML')

= 5(SMsLML,S'Ms'L'ML')

:E IDijPij'[:ERoop(k)(ni~,njlj;ni'li',nj'lj')

Aoop(k)(limi,ljmj~'mi' ,lj'mj')J}

(with summations over the Slater determinants u and v, the spin-orbitals j>i = 1 to N, p = 1 to 4, and k, respectively). The limits for the summations over k are:

88

forp = 1: max.{I.R.i-.ii'I,I.it.ij'I} ~ k ~ min{~+~' ;'j+.ij'}

forp = 2: max.{I~-.ii'l+l,l.it.ij'I+l,l} ~ k ~ min{~+~'+l;'j+.ij'+l}

for p = 3: max.{I~-.ii'l+l,l.ij-.ij'I,l} ~ k ~ min{~+~'+ 1 ;'j+.ij'}

for p =4: max.{I~-.ii'I,I.ij-.ij'l+l,l} ~ k ~ min{~+~' ,.ij+.ij'+ I}

(in steps of two in all cases).

Radialparts

Rool (k)(ij;i'j') = (a2/2) (k(k+ 1)[(2k-l)-ID(k-l)(ii';jj')-(2k+3)-ID(k+ 1)(ii';jj')]

+ (l/2)[(2k+3)-1(-k V>(k+2)(jj';ii')+(k+3) V <(k)(jj';ii'»

+(2k-1)-I«k-2) V>(k)(jj';ii')-(k+ 1) V <(k-2)(jj';ii'»]

[~(.ij+ 1)-.ij' (.ij'+ l)-k(k+ 1)]

+ (l/2)[(2k+3)-I«k+3) V<(k)(ii';jj')-k V>(k+2)(ii';jj'»

-(2k-1)-I«k+ 1) V <(k-2)(ii';jj')-(k-2) V >(k)(ii';jj'»]

[~(.ii+ 1)-~' (~'+ l)-k(k+ 1)]

+ (l/2)[k(V>(k)(jj';ii') - V>(k)(ii';jj'» + (k+l)(-V>(k+l)(jj';ii')+ V>(k+l)(ii';jj'»]

+ (l/4)[«k+ 3)/(k+ 1)(2k+3»N(k)(ii';jj')-«k-2)1k(2k-l»N(k-2)(ii';jj')]

[~(.ii+ 1)-~' (~'+ l)-k(k+ l)][.ij(.ij+ 1)-.ij' (.ij'+ l)-k(k+ I)]}

Roo2(k)(ij;i'j') = -(a2/2)[2k(k+ 1 )]-1 N«k-l)(ii';jj')

[(.ii+~'+k+ l)(.ij+.ij'+k+ 1 )(~' -~+k)(.ij' -.ij+k)]l/2

[(.ii-~'+k)(.it.ij'+k)(~+~'-k+ l)(.ij+.ij'-k+ 1] 1/2

Roo3(k)(ij;i'j') = (a2/8)[-N>(k-2)Gj';ii') + N>(k)(ii';jj')]

[2(~+~'+k+ l)(.ii' -.ii+k)(~-~'+k)(~+~'-k+ 1)] 1/2

Roo4 (k)(ij;i 'j') = (a2/8)[ -N> (k)(jj';ii') + N> (k-2)(ii';jj')]

[2(.ij+.ij'+k+ 1)(.ij' -.ij+k)(.ij-.ij'+k) (.ij+.ij'-k+ 1)] 1/2

89

Angular parts

Aool (k)(ij;ij') = O(mi+mj,mi'+mj')o(llioIlOO(llj,llj')

(-I)q[(2~'+ 1 )(2lj'+ 1)/(2~+ 1)(2lj+ 1)]l/2{~'mi'k-ql~mil {lj'mj'kqlljmj}

Aoo2(k)(ij;ij') = o(mj+mj,mi'+mj')o(lli,lli')O(Ilj.Ilj')

(-I)q[(~'+ 1) (2lj' + 1)/(2~+ 1)(2lj+ 1)] 1(2

(~'mi'k-qllimi)(~'Ok-l0IljO)(lj'mj'kqlljmj)(lj'Ok-l0IljO)

Aoo3(k)(ij;i'j') = o(mj+mj,mj'+mj')o(llj,1l00(llj,Ilj')

(-I)q[(~'+ 1) (21j'+ l)/(~+ 1) (21j+ 1)]1(2

(~'mj'k-qlljmj)(~'Ok-lOl~O) {lj'mj'kqlljmj)

Ao04(k)(ij;i'j') = o(mj+mj,mj'+mj') O(llj,llj')O(llj,Ilj')

with

(-I)q[(2~'+ 1) (21j'+ 1)/(2~+ 1)(2lj+ 1)] 1(2

{li'mj'k -q Ilimi} (lj'mj'kq L2.jmj)(lj'Ok-l0IljO)

6.8 Spin-Orbit Coupling

Operator

(with summations over the electrons cr > P = 1 to N and where Tpa denotes

transposition of the subindices p and cr).

Matrix element

90

(CPSMSLMLlttsoIC'P'S'MS'L'ML')

+ I OOijPjj' II Rsop(k)(nj.Jtj,nj.Jtj;nj'J4',nj'.Jtj')

Asop(k)(.Jtjmj~j,.Jtjmj~j;J4 'mj '~j' J.j'mj'~j') }

(with summations over the Slater determinants u and v, the spin-orbitals i = 1 to N, the spin-orbitals j > i = 1 to N, P = 2 to 7, and k, respectively). The limits for the summations over k are:

forp = 2: max{iJ4-J4'I,I.Jl.j-lj'I}:S k:S min {J4+J4, J..i+lj'}

forp = 3: max{iJ4-.Jti'I,I.Jl.j-.Jtj'I}:S k:S min{J4+J4' J.j+.Jtj'}

for p =4: max{~-.Jti'l+ 1,l.Jl.j-.Jtj'I+ I} :S k :S min{J4+J4'+ 1 J.j+.Jtj' + I}

for p =5: max{iJ4-.Jti'l+ 1 ,1.Jtj-.Jtj'I+ I} :S k :S min {J4+J4'+ 1 J.j+.Jtj' + 1 }

for p =6: max{iJ4-.Jtj'l-l ,1.Jtj-.Jtj'I-l,l} :S k :S min {J4+J4' -1 J.j+.Jtj'-l }

for p =7: max{ 1J4-.Jti'l-l ,1.Jtj-.Jtj'I-l,l} :S k :S min{J4+J4'-l..ij+.Jtj'-1 }

(in steps of two in all cases).

Radial parts

Rsol(i;i') = (Za2/2)I(-3)(njJ4;ni'.Jti)

Rso2(k)(ij;i'j') = (a2/2)(-1)k+l[(2J4'+ 1)(2.Jl.j'+ 1)/(2J4+ 1)(2lj+ 1)]1/2

([k(k+ 1)(2k+ 1)/3]l/ly(k)(jj';ii')+ In [J4(J4+ 1)-.Jtj' (J4'+ 1)-k(k+ 1)]

[(2k+l)/3k(k+l)]l/1[kN>(k)(ii';.ij') - (k+l) N>(k-2)(jj';ii')]}

91

Rso3(k)(ij;i'j') = (a2/2)(-I)k+l[(2~'+ 1)(21/+ 1)/(21j+ 1)(.R.j+ 1)] If2

([k(k+ 1)(2k+ 1)/3] If2V(k)(ii';jj')+ (l/2)[.R.j(.R.j+ 1)-.R.j' (.R./+ l)-k(k+ 1)]

[(2k+ 1)!3k(k+ 1)1/2[k N>(k)(jj';ii') - (k+ 1) N>(k-2)(ii';jj')]}

Rs04(k)(ij;i'j') = (a2/4)(-I)kN>(k-l)(ii';jj')[(~'+ 1)(21/+ 1)/(21j+ 1)(21j+ 1)]1/2

[(2k-l)(2k+ 1)/3k]I/2[(k+.R.j+.R.j'+ 1)(k+~-.R.j')(k-~+~')(~+.R.j'-k+ 1)] 1/2

Rsos(k)(ij;i'j') = (a2/4)(-I)kN>(k-l)(jj';ii')[(21j'+ 1)(21/+ 1)/(21j+ 1)(21j+ 1)]1/2

[(2k-l)(2k+ 1)/3k]I/2[(k+.R.j+..R.j'+ 1) (k+.R.j-.R./)(k-.1j+.R./)(.R.j+.R.j'-k+ 1)] 1/2

Rs06(k)(ij;i'j') = (a2/4) ( -1)kN>(k-l)(jj';ii')[(~'+ 1)(21/+ 1)/(21j+ 1)(21j+ 1)]1/2

[(2k+ 1)(2k+3)/3(k+ 1)]1/2

[(k+~+ ~'+2)(k+~-~'+ 1 )(k-.R.j+.R.j'+ 1 ) (.R.j+..R.j'-k)] 1/2

Rso7(k)(ij;i'j') = (a2/4)( _l)kN>(k-l)(ii';jj')[(~'+ 1)(21/+ l)/(21j+ 1)(21j+ 1)] 1/2

[(2k+ 1)(2k+3)/3(k+ 1)]1/2

[(k+.R.j+.R./+2) (k+.1j-.R./+ 1 )(k-.R.j+.R./+ 1 )(.R.j+.R./-k)] 1/2

Angular pans

Asol (i;i') = O(~,..R.j')O(mj,mj')O(J.1j,J.1j')millj

Aso2(k)(ij;i'j') = KlAo

Aso3(k)(ij;i'j') = -K2AO

As04(k)(ij;i'j') = Kl(kmj-mj'k-lmj-m/110)(~'mi'kmj-mj'~mj)

(~'Ok-l01~O) (.R.j'mj'k-lmj-m/l.R.jmj)

Aso5(k)(ij;i'j') = K2(kmj-mj'k-lmj-mj'1l0)(..R.j'mj'kmj-mj'l..R.jmj)

(..R.j'Ok-lOl..R.jO){..R.j'mj'k-lmj-mi'~mil

As06(k) (ij;i 'j') = K 1 (kmj-mj'k+ 1 mj-mj'llO)(..R.j'mj 'kmj-mj'l~mj)

(~'Ok+ 101..R.jO) {..R.j'mj'k+ Imj-mfl..R.jmj}

Aso7(k)(ij;i'j') = K2(lcnlj-mj'k+ Imj-mj'llO)(.R.j'm/kmj-mj'l.R.jmj)

92

with

AO = (kmi-mi'kmj-mj'110){~'mi'kmi-mi'l~mil {ljmjkmj-mj'lljmj}

Kl = O(mi+mj,mi'+mj')O<l..Li,~i')O(~j,~j')(~i+2~j)

K2 = O(mi+mj,mi'+mj')O(~i,~i')O(~j,~j')(~j+2~i)

6.9 Spin-Spin Dipole Interaction

Operator

(with summations over the electrons (J> P = 1 to N).

Matrix element

(Cj3SMsLML1KsdIC'j3'S'Ms'L'ML')

(Cj3SMsLML1KsdlC'j3'S'Ms'L'ML')

{~roijPij' ~~ Rsdp(k)(ni~,njlj;ni'li' ,nj'lj')

Asdp(k)(~mi~i,ljmj~j;li'mi'~i',lj'mj'~j') }

(with summations over the Slater determinants u and v, the spin-orbitals j > i = 1 to N, p = 1,2, and k, respectively).

93

Radial parts

Rsdl(k)(ij;i'j') = (3a2/4y3)(-I)k[(~'+ 1)(2lj'+ 1)/(2li+ 1)(2lj+ 1)]1/1

[k(k+ 1)(2k-l)(2k+ 1)(2k+3)] IflN>(k-l)(jj';ii')

Rsd2(k)(ij;i'j') = (3a2/4y'3")(-I)k[(~'+ 1)(2lj'+ 1)/(2li+ 1)(2lj+ 1)]1/1

[k(k+ 1)(2k-l)(2k+ 1)(2k+3)]lflN>(k-l)(ii';jj')

Angular parts

Asdl(k)(ij;i'j') = K(k-lmi-mi'k+ Imj-mj'120)

{.1i'mi'k-lmi-mi'l~nii} {.1j'mj'k+ Imj-mj'l.1jmj}

Asd2(k)(ij;i'j') = K(k+lmi-mi'k-lmj-mj'120)

{.1i'mi'k+ Imi-mi'l~mil {.1j'mj'k-lmj-mj'l.1jmj}

with

K = ~(mi+mj,mi'+mj')~(J.li+J.lj.lli'+J.lj')

(lJ.li-J.li'lJ.ltJ.lj'120)(l/2 J.li'lJ.li-J.li'll/2 J.li)(l/2 J.lj' IJ.lj-J.lj'Il/2 J.lj)

6.10 Magnetic Dipole and Fermi Contact Interactions

Operator

HD = gsJ.lBgIJ.lN1:rp-3I-([rp x pp] - sp + 3(sp-rp)(rplrp2»

+ (81t/3)gIJ.lN 1: (sp-I)~(rp)

(with summations over the electrons p = 1 to N).

Matrix element

(CeSMSLMLJMJIMIFMFIHDIC'e'S'Ms'L'ML'J'MJ'IMI'F'MF')

= ~(MsMLMIFMF,Ms'ML'MI'F'MF')( _1)I+J+F

94

[(I+l)(2I+l)(2J+l)(2J'+1)fl]112{} r ~} 1:( {l]{~}uMSMLIPSL)({l'} (mJ.L}uMsMLIP'S'L')

{li(S,S')(-I)S+L'+J+l(2L+l)I12(L'MLI0ILML)-I{[ ~' ~}

l:eiROl(nili;ni'~')Aol(~miJli;~'mi'J.Li')

+[3(2S+1)(2L+l)]I12[(S'MSlOISMS)(L'ML20ILML)J-1 S' L' J' {s L J} 1 2 1

{J' SJ' L1} +li(L,L')( -1 )S+L+J'+ 1 (2S+ 1) 112(S 'MS 1OISMs)-1 S

(with summations over the Slater determinants u and the spin-orbitals i =1 to N).

Radial parts

RDl(i;i') = I(-3)(nili;ni'~)

R02(i;i') = _(10)I12I(-3)(ni~;ni'~')

RD3(i;i') = (81t/3)[(R(ni~;r)R(ni'~;r)1r = 0

Angular parts

A02(i;i') = li(mi,mi')li(J.Li,J.Li')J.Li[(~'+ 1)/(2li+ 1)] 112{~'mi20~mil

AD3(i;i') = li(~,li')li(mi,mi')li(J.Li,J.LOJ.Li

Remarks

The conversion factor for this interaction, with the radial integrals given in atomic units, is

95

(gsJ.1BJ.1NIh)ao-3 = 2(l.OI159652193)(927.40154-1O-26rr-1)(O.50507866-1o-26rr-1)

(6.6260755-10-34 Js)-I(O.529177249-1O-1O m)-3

= 96.517101 MHz = 14.668959-10-10 hartrees

6.11 Electric Quadrupole Coupling

Operator

(with the summations extending over the orbitals p = 1 to N).

Matrix element

(CeSMsLMLJM]IMIFMFIIHQIC'e'S'Ms'L'ML'J'Mj'IMI'F'MF')

= -O(SMsMLMIFMF.S'Ms'ML'MI'F'MF')( -1 )S+L'+I+F(QI2)

[(1+ 1)(21+ 1)(21+3)(2J+ 1)(2J'+ 1)(2L+ 0/1(21-1)] 1(2

(L'ML20ILML)-I{ J' L' S} {I J' F} LJ2 JI2

(with summations over the Slater determinants u and the spin-orbitals i = 1 to N).

RadiaJpan

RQ(i;i') = 1(-3)(nili;ni'~')

Angular pan

Remarks

96

The conversion factor for this interaction, with the nuclear electric quadrupole moment Q measured in bams (10-24 cm2) and the radial integrals given in atomic units, is

e2 = (a. M c)(l0-24 cm2) ao-3

= (7.29735308-1O-3)(1.05457266-1O-34Js)(2.99792458-1O-1O cm s-l)

(10-24 cm2)(0.529177249-1 0-10 m)-3

= 234.96472 MHz = 0.35710641-10-7 hartrees

6.12 Magnetic Octupole Coupling

Operator

Hmo = -(5/3)l/2J.1B L rp -5 ((6-(8n)rpO(rp»

[s(l)(p) X C(2)(p)](3)-2[C(2)(p) x i(1)(p)](3)}-N(3)

(with the summations extending over the electrons p = 1 to N).

Matrix element

[5(1+ 1)(1+2)(21+ 1)(21+3)(2J+ 1)(2J'+ 1)/31(1-1)(21-1)]1I2{} f f } L( {i} {mJ.1}uMsMLI~SL)({i'} (mJ.1}uMSMLIWS'L')

{s L J} ([7(2S+1)(2L+l)] l/2[(S'MS 1OISMS)(L'ML20ILML)]-1 S' L' J'

1 2 3

97

(with summations over the Slater determinants u and the spin-orbitals i = 1 to N).

Radial parts

ROl (i;i') = 6 I(-5)(nili; n(R.i')-(8n)(R(niJ4;r)R(ni'li';r)/r4)r = 0

Ro2(i;i') = 2I(-5)(niJ4;ni'J4')

Angular parts

AOl (i;i') = O(Ili.Ili')o(mi,mi')lli[(~' + 1 )/(2J4+ 1)] 1/2 (li'mi201J4mil

A02(i;i ') = O(lli,IlOo(mi,mi')(1/2)[ (21i' + 1 )/(2J4+ 1)] 1/2

[(li+li'+4 ) (li' -li+ 3)(J4-li'+ 3 )(li+li'-2)/35] 1/2

(J4'mi 301J4mi)(J4'020 lliO)

Remarks

The conversion factor for this interaction, with the nuclear magnetic octupole

moment n measured in nuclear magneton barn units and the radial integrals in atomic units, is

(IlBIlNfh)(1o-24 cm2) ao-2 ao-3

= (927.401540 10-26 rr-1)(0.505078660 1O-26 rr-1)(6.62607550 1O-34 Js)-1

(10-24 cm2)(0.5291772490 1O-1O m)-5

= 17.035880 10-7 MHz = 25.8916430 10-17 hartrees

6.13 Zeeman-Effect (low field)

Operator

98

-JlB(m/mp)( 1: Jl.p + 1: {r p x PaD

-JlB a2 [ 1: (Jl.p + gssp)T P - (ZI2) 1: [[sp x V p(rp -I)] x r p]

+ (1/2) 1: [[(sp + 2sa) x V p(rpa-1)] x rp]

+ (1/2) 1: (rpa-I[rp x Pa] + rpa-3 [rp x ra](rpaepa))]}

(with the summations extending over the electrons a ':f: p = I to N).

Matrix element

+C 1 (2) 1: COijPij'

[RzI (nili,njlj;ni'~' ,nj 'lj')AzI (~miJli,ljmjJlj;~'mi 'Jli' ,lj'mj'Jlj')

+Rzl(njlj,nili;nj'lj',ni'lOAzl(ljmjJlj,limiJli;lj'mj'Jlj'.li'mi'Jli)]

+1: Cp(2) 1: COijPij'

[ 1: Rzp(k)(ni~,njlj;ni'li' ,nj'lj')Azp(k)(~miJli,ljmjJlj;~'mi'Jli' ,lj'mj'Jlj')

+ 1: Rzp(k)(njlj,ni~;nj'lj',ni'~')Azp(k)(ljmjJlj,~miJli;lj'mj'Jlj' ,~'mi'JlO]}

with the summations extending over the Slater determinants u and v, p = 1 to 3, the spin-orbitals i = 1 to N, the spin orbitals j > i = 1 to N, P = 2 to 10, the spin-orbitals j > i = 1 to N, and k, respectively. The limits for the summations over k are as follows:

forp = 2: max{lli-~'1+2,llj"lj'l} S k S min {~+~'+2J.j+lj'}

max{lli-~'I,llj-lj'I+2} ~ k S min{~+li' ,lj+lj'+2}

forp = 3: max{lli-~'1-2,llj-lj'l} S k S min{~+~'-2J.j+lj'}

max{lli-~'I,llj-lj'I-2} S k S min{~+~' J.j+lj'-2}

99

forp = 4: max{I~-~'I,lJtt.R.j'I} ~ k ~ min{~+~' JLj+.R.j'}

max{~-~'I,lJtt.R.j'l} ~ k ~ min{~+~' JLj+.R.j'}

forp = 5: max{I~-~'I,lltlj'l} ~ k ~ min{~+~' JLj+lj'}

forp = 6: max{lli-~'I,lJttlj'l} ~ k ~ min{~+~' JLj+lj'}

max{I~-~'I,lJttlj'l} ~ k ~ min{~+~' JLj+lj'}

forp = 7: max{>O,~-li'l,lJtj-lj'l} ~ k ~ min{~+~' ,ij+lj'}

forp = 8: max{>o,I~-li'I,llj-lj'l} ~ k ~ min{~+~' ,ij+lj'}

for p = 9: max{~-~'1+1,lJttlj'I+1} ~ k ~ min {li+1i'+ 1 ,ij+lj' + 1 }

for p = 10: max{>O,~-li'I-1,llj-lj'I-1} ~ k ~ min{li+1i'-1 ,ij+lj'-1}

(with the restrictions that the second summation does not exist for p = 5 and p = 7 and that the limits of both summations are the same for p ~ 8).

Coefficients

Cl = 5(MF,MF')(-1)J+I+F[(2I+ 1)(I+l)(2F'+ 1)/1] 1I2(F'MFlOIFMF) { ~,r ~}

C2 = 5(MF,MF')(-1)1'+I+F+l[3(2S+ 1)(2L+ 1)(2J+ 1)(2J'+ 1)(2F'+ 1)]1(2

(F'MFIOIFMF){ ~, j. n

{s L J} C3(1) = [(S'MSlOISMS)(L'ML20ILML)]-1 S' L' l'

121

100

{s L I} C5(2) = [(S'MSlOISMS)(L'MLlOILML)]-l S' L' I'

1 1 1

Qf.2) = C2(1)

C7(2) = Cg(2) = OJ(2) 0= ClO(2) = Cl(l)

Radial parts

RZ1(i;i') = S(ni,ni')(1-m/mp) - a2T(ni~;ni'~)

Rz2(i;i') = gs[S(ni,nj') - a2T(ni~;ni'~] + (Za2!3)I(-1)(n~;nj'.R.i)

Rz3(i;i') = (5/18)1/2 Za2 1(-1) (ni~;ni'~')

Rzl (ij;i'j') = (m/mp)I(l)(ni~;ni'~') (1(2)(nj,.1j;nj'.1j')

+ (1/2)[.1j' (.1j' + 1 )-.1j(.1j+ 1)+2]I(-l)(nj.1j;nj'.1j') }

Rz2(k)(ij;i'j') = (-1)k+la2[(k-l)k(2k-3)(2k+ 1)/12(2k-l)]l/2R>(k)(jj';ii')

Rz3(k)(ij;i'j') = (-I)ka2[(k+ 1)(k+2)(2k+ 1)(2k+5)112(2k+3)] l/2R>(k)(ii';jj')

Rz4(k)(ij;i'j') = (-1)ka2[k(k+ 1)(2k-l)(2k+ 1)n2(2k+3)]l/2R>(k)(ii';jj')

+ (-I)k+la2[k(k+l)(2k+l)(2k+3)n2(2k-l)]lflR>(k)(jj';ii')

Rz5(k)(ij;i'j') = (-1)ka 2[k(k+ 1)(2k+ 1)/24]l/2R>(k)(ii';jj')

Rz6(k)(ij;i'j') = (k!3)a2R>(k)(jj';ii') - [(k+ 1)!3]a2R>(k)(ii';jj')

Rz7(k)(ij;i'j') = (-I)ka2[k(k+ 1)(2k+ 1)/12] Ifl

([(k+3)/(2k+3)][W«k+l)(ii';jj') - W<(k+l)(jj';ii')]

- [k/(2k+3)][W>(k+l)(ii';jj') - W>(k+l)(jj';ii')]

-[(k+ 1)/(2k-l)][W <(k-l)(ii';jj') - W <(k-l)(jj';ii')]

+ [(k-2)/(2k-l)][W>(k-I)(ii';jj') - W>(k-l)(jj';ii')]}

101

RzS(k)(ij;i'j') = (_I)k+la2 (l/2)[(2k+ 1)/12k(k+ 1)]II2[.1j'(.1j'+ 1)-.1j(.1j+ 1)+k(k+l)]

([k(k+3)/(2k+3)][R>(k+2)(jj'ii') + R>(k)(ii';jj')]

- [(k-2)(k+l)/(2f-1)][R>(k)(jj';ii') + R>(k)(ii';jj')]}

Rz9(k)(ij;i'j') = (-I)ka2 0/2)

[(2k-l)(.1j+.1j'+k+ 1)(.1j'-.1j+k)(.1j-.1j'+k)(.1j+.1j'-k+ 1)/12k(2k-l)] 1/2

[R>(k+l)(jj';ii') + R>(k-l)(ii';jj')]

RzlO(2)(ij;i'j') = (_I)k+la2

(1/2)[(2k+3)(.1j+.1j'+k+2)(.1j' -.1j+k+ 1)(.1j".1j'+k+ 1 )(.1j+.1j'-k)/12(k+ 1)(2k+ 1)] 112

[R>(k+l)(jj';ii') + R>(k-l)(ii';jj')]

AnguIarparts

Azl (i;i') = O(~,.1j')O(~j,~j')mj

Adi;i') = O(~,~')O(~j,~j')~j

Az3(i;i') = O(~j,~j')~j{~'mj20~mil

Azl (ij;i'j') = o(mj+mj,mj'+mj')O(lljllj,llj'~j')( -1)mj'-mj(mj'-mj)

[(~'+ 1)(llj'+ 1)/(llj+ 1)(llj+ 1)]112 {~'mj'lmj-mj'l1jmj}

{.1j'mj'lmj-mj'l1jmj}

Az2(k)(ij;i'j') = o(mj+mj,mj'+mj')o(~j~j,~j'llj')(~j+2~j)

[(llj'+ 1 ) (llj' + 1)/(llj+ 1)(llj+ 1)] 112

(k-2 mj-mj' k mj"mj'120){~'mj' k-2 mj-mj'l1jmj) {.1j'mj' k mj"mj'l1jmj}

Az3(k)(ij;i'j') = o(mj+mj,mj'+mj')o(~j~j,~j'~j')(~j+2~j)

[(llj'+ 1)(llj'+ 1)/(llj+ l)(llj+l)] 112

(k+2 mj-mj' k mj-mj'120){.1j'mj'k+2 mj-mj'l1jmj) (.1j'mj'kmj"mj'l1jmj)

AZ4(k)(ij;i'j') = o(mj+mj,mj'+mj')o(~j~j,~j'~j')(~j+2~j)

102

[(~'+ 1)(2~'+ 1)/(~+ 1)(2~+ 1)} 1(2

(k mj-mj' k mtmj'120)(~'mj'kmj-mj'~mil {.ij'mj'kmtmj'L1jmj}

AzS(k)(ij;i'j') = 5(mj+mj,mj'+mj')5(lljllj,llj'llj')(llj-llj)

[(2.ij'+ 1)(2.ij'+ 1)/(~+1)(~+ 1)]1(2

(k mj-mj' k mj-mj'110){~'mj' k mj-mj'~mi) {.ij'mj' k mtmj'L1jmj}

Az6(k)(ij;i'j') = 5(mj+mj,mj '+mj')5(lljllj,llj'Ilj')(llj+ 2Ilj)( -1) I-mj+mj'

[(~'+ 1)(~'+ 1)/(~+ 1)(2.i+ 1)1(2

{~'mj k mj-mj'l~mil {.ij'mj' k mj-mj'l.ijmj}

Az7(k)(ij;i'j') = 5(mj+mj,mj'+mj')5(lljllj,llj'Ilj')

[(~'+ 1)(2.ij'+ 1)/(~+ 1)(2.ij+ 1)]1(2

(k mj-mj' k mj-mj'110){~'mj' k mj-mj'~mil {.ij'mj' k mj-mj·l.ijmj}

Az8(k)(ij;i'j') = Az7(k)

Az9(k)(ij;i'j') = 5(mj+mj,mj'+mj')5(lljllj,llj'Ilj')

[(2.ij'+ 1)(2.ij'+ 1)/(~+ 1)(2.ij+ 1)]1(2

(k-1 mj-mj' k mj-mj'110){~'mj' k-1 mj-mj'~mj}{~'mj' k-1 mj-mj'l.ijmj)

AzlO(k)(ij;i'j') = 5(mj+mj,mj'+mj')5(lljllj,llj'Ilj')

[(2.i'+ 1)(2.ij'+ 1)/(2.ij+ 1) (2.ij+ 1)]1(2

(k+ 1 mj-mj' k mtmj'llO){~'mj' k+ 1 mj-mj'l~mil {.ij'mj' k+ 1mj"mj'L1jmj}

Restrictions

l:(.ij+.ij') = even (with summation over all the spin-orbitals i)

l:(mj-mj') = 0 (with summation overall the spin-orbitals i)

103

6.14 Zeeman Effect (high field)

Operator

See the description for 'Zeeman Effect (low field),

Matrix element

(CE1M]IMIlKzIC'£'J'M]'IMI')

+ Cl (2) 1: IDijPij'

[Rzl (ni.Rq, njlj;ni'.Rq' ,n j 'lj')Azl (.Rqmilli,ljmjllj;li'mi'lli' ,1j'mj 'Ilj')

+ Rzl (njlj,ni.Rq;nj'lj' ,ni '.Rq')Azl (ljmjllj,.Rqmilli,lj' mj'llj' .1i 'mi'lli')]

+ 1: Cp(2)1:IDijPij'

[1: Rzp(lc)(ni.Rq,njlj;ni'.Rq', nj 'lj')Azp(lc)(.Rqmilli,ljmjllj;.Rq'mi'lli' ,1j'mj'llj ')

+ 1: Rzp(lc)(njlj,ni.Rq;nj'lj', ni'.Rq')Azp(lc)(ljmjllj.1imilli;lj'mj'llj' ,.Rq'mi'lli')]

with the summations extending over the Slater determinants u and v, p = 1 to 3, the spin-orbitals i = 1 to N, the spin-orbitals j > i = 1 to N, p = 2 to 10, the spin-orbitals j > i = 1 to N, and k, respectively. The limits for the summations over k are the same as those given for the low field case.

Coefficients

Cl = O(M],M],)MIfI

C2 = o(M],M],)[3(2S+1)(2L+l)(2J'+1)]l/2(J'M]1OIJM])

All the remaining coefficients are identical to those for the low field case.

Radial and angular parts

See the low field case.

104

6.15 Zeeman Effect (very high field)

Operator

Kz = H{~B ~ [lo(l)(p) + gsso(l)(p)] - ~ m:Io(l»)

(with the summations over the electrons p = 1 to N).

Matrix element

(CPSMsLMLIMIlKzIC'P'S'Ms'L'ML'IMi')

= S(CPSMsLMLMI,C'p'S'MS'L'ML'MI')H[~B(ML + gsMS)-~N~I MIll]

Remllrks

The relativistic, diamagnetic, and isotopic corrections have not been included.

6.16 Stark Effect

Operator

Ks = - ~ (Ferp) = -F ~ rpCo(l)(p) = -F hS

(with the summations over the electrons p = 1 to N).

Energy levels

low field

E'(CeJIFMF) = E(CeJIF)

+ p2 ~ (1(CeJIFMF1hsIC'e'J'IF'MF')12/[E(CeJIF) - E(C'e']'IF')))

(with the summations over C', e', ]', and F')

105

[(2L+l)(2J+l)(2J'+I)(2F'+I)]I!2(F'MplOIFMp)(L'MLlOILML)-1

{r. t. ~} {~, f. ~} (C~SMSLMLlhSIC'WS'MS'L'ML') high field

E'(CySLJM]) = E(CySLJ)

+ F21: (1(CySLJM]lhSIC'y'S'L'J'M],)12/[E(CySLJ)-E(C'y'S'L'J')]}

(with summations over C', 1, S', L', and J')

(CySLJM]lhSIC'1S'L'J'M]') = o(M],M],)(-1)S+L'+]+I[(2L+ 1)(2J'+ 1)] 1/2

very high field

E'(C~SMSLML) = E(C~SL)

+ F21: (1(C~SMsLMLlhsIC'WS'Ms'L'ML')12/[E(C~SL)-E(C'WS'L')]}

(with summations over C', W, S', and L')

Matrix element

(with summations over the Slater detenninants u and v and the spin-orbitals i = 1 to N)

Radialpart

RS(i;i') = I(I)(ni~;ni'.R.i')

Angular part

106

Restrictions

The matrix element exists only if

~(li+ii')+ 1 = even (with summation over the spin-orbitals, i = 1 to N)

~(mi-mj') = 0 (with summation over the spin-orbitals, i = 1 to N)

Remarks

The above formulation is only valid for those cases where there is no appreciable mixing of states.

6.17 Nuclear-Mass Dependent Orbit-Orbit Interaction

Operator

(with summations over the electrons (J > P = 1 to N).

Matrix element

(CySMSLMLJM]IKoo(m)IC"y'S'MS'L'ML'J'M]')

= O(JM],J'Ml)(CPSMsLMLIKoo(m)lc'p'S'Ms'L'ML')

(CPSMsLMLlKoo(m)lc'poS'Ms'L'ML')

~{( {i} {mll}uMsMLIPSL)({i'} (m'Il'}yMsMLIPOSL)

~ (J)ij Pij'[~ Rmoop(ni~,njij;ni'~',nj'ij')]

Amoo(~milli,ijmjlli;li'mi'lli' ,ij'mj'lli')}

(with summations over the Slater determinants u and v, the spin-orbitals j > i = 1 to Nand p = 1 to 4).

Radial parts

107

Rmool(ij;iJ') = -a2(ZJ41JUlla}[~(~+I)-Jli'(~'+I)-2]

[1j(Jlj+ 1)-1j' (1j'+ 1)-2]X(ij;i'j')

Rmoo2(ij;iJ') = a2(ZJ2mma>[~(~+ 1)-~' (~'+ 1)-2]V(-I)(ij;iJ')

Rmoo3(ij;iJ') = a 2(ZJ2IJUlla}[1j(Jlj+ 1)-1j' (1j'+ 1)-2]Y(ij;i'j')

Rmoo4(ij;iJ') = -a2(ZJmma>Z(ij;iJ')

Angularpan

Amoo(ij;iJ') = S(mi+mj,mi'+mj')S{f.1hfli')S(flj,flj')

[(2Jli'+ 1)(2Jlj'+ 1)/(~+ 1)(2Jlj+ 1)]112

{~'mi'lmi-mi'~mi} {1j'mj'1 mj-mj'l1jmj}

6.18 Nuclear-Mass "Dependent Spin-Orbit Coupling (Electron Spin)

Operator

(with summations over the electrons (J > P = 1 to N).

Matrix element

(CySMSLMLJMJlKso(m)IC'yS'MS'L'ML'J'MJ')

= S(JMJ ,J'M],)( -1 )S'+L+J[(2S+ 1 )(2L+ 1)] 1(2

[(L'ML'lOILML)(S'MS'lOISMS)]-1 {~, ;. :}

(CPSMsLMLlKso(m)lc'p'S'Ms'L'ML')

(CPSMsLMLiHso(m)lc'p'S'Ms'L'ML')

= S(MsML,Ms'ML')

108

1: Cl)ij Pij' Rmsop (ni~,njlj;ni'.li' ,nj'lj')

Amsop(~millioljmjllj;~ 'mi 'Ili' ,1j'mj'llj')

Radial parts

Rmso1 (ij;i'j') = a2(Z/mma)X(ij;i'j')

Rmso2(ij;i'j') = a2(Z/mma)Y(ij;i'j')

Angular parts

Amso1(ij;i'j') = -(1/2)K[lj(lj+I)-lj' (lj'+I)-2]

Amso2(ij;i'j') = K

with

K = ~(mi+mj,mi'+mj')B(lli,lli')B(llj,Ilj')lli

[(2~'+ 1)(21j'+ I)/(~+ 1)(21j+ 1)]1/2

{~'mi' 1 mi-mi'!.Rqmj} {lj'mj' 1 mj-mj'l1jmj}

6.19 Nuclear Mass Dependent Spin-Orbit Coupling (Nuclear Spin)

Operator

(with summation over the electrons p = 1 to N).

Matrix element

(CeSMsLMLJM]IMIFMFIKsoI(m)lC'e'S'Ms'M'ML'J'M]'IMI'F'MF')

= B(SMsMLMIFMF,S'Ms'ML'MI'F'MF')

(_l)I+2J+S+L'+F+1[(2I+ 1)(2J+ 1)(2J'+ 1) (2L+ I)] 1/2

109

[(IMIlOIIMv(L'MLlOILML)J-l{ I J' F} { J' L' S } JIl LJl

Radial partS

Rmson(i;i') = -a2('ZJrnma)(fI fl)I(-4)(i;i')

RmsoI2(i;i') = a2('ZJrnma>fI lI(i;i')

Angular partS

Amsoll = [(~+l) - ~'(~'+1) -~]K

AmsoI2=K

with

Implementation

The fonnulation presente<! in the preceding chapters has been implemented in a computer program [Fraga et al. (1987)] and applied to some chosen examples in order to illustrate the characteristics of the calculations and the possible difficulties to be faced.

The program uses numerical integration but that does not imply any restriction on the functions used as starting point for the CI calculation to be perfonned. It is true that the orbitals must be expressed in numerical fonn but they might be obtained not only from numerical Hartree-Fock calculations but also from analytical SCF calculations or approximated in any way deemed appropriate (e.g., using modified Slater rules, etc.). The most important point in this connection is that the goal of the calculations dictates what functions should be used as starting point. These details, as well as a summary of selection rules, are discussed in Chapter 7.

Finally, Chapter 8 presents a summary of the results obtained for accurate energies of lowest states and in the prediction of J-Ievels and hyperfine-structure splittings. The characteristics of each type of calculation are discussed, showing how a very accurate description of atomic energy levels is within reach, given the necessary computing power.

7 Practical Details

The final formulation presented in the preceding Chapter was incorporated in a computer program [Fraga et al. (1987)] to be used in the configuration interaction study of atomic energy levels with consideration of the non-splitting, fine-structure, and hyperfine-structure interactions.

Before discussing specific details of the calculations it is convenient to summarize first their general organization:

(a) The configurations to be included are selected and the common set of atomic orbitals in those configurations are listed.

(b) Radial functions are determined for all the atomic orbitals listed above. In this connection it must be pointed out that the formulation presented in this work is only valid for orthonormal orbitals and therefore care must be exercised in the determination of the radial functions of the atoms belonging to the same symmetry designation (see below).

(c) The SL-coupling coefficients for the states under study are determined [e.g., using the program of Nussbaumer (1969)]. When only the electronic and the SL­non-splitting correction terms are included, the calculations may be carried out for any of the MS, ML terms of the multiplet for the state under consideration; that is, the SL-coupling coefficients may be those, for example, corresponding to the term with highest MS and ML. When the fine- and hyperfine-structure interactions are included, the MS, ML terms to be used depend on the values of S,S' and L,L' of the interacting states.

(d) The interaction energy matrix is then constructed for the chosen Hamiltonian operator in the set of states arising from the configurations under consideration, with one matrix for each symmetry designation; that is, one matrix for each SL­designation (when operating with the electronic Hamiltonian operator, with or without the SL-non-splitting terms), J value (when including the fine-structure coupling) or F value (when including the hyperfine-structure interactions). The matrix elements are evaluated according to the present formulation, with appropriate

114

factors in order to include the relativistic mass correction [Karwowski and Fraga (1974)].

(e) Each energy matrix is diagonalized, yielding the eigenvalues and eigenvectors (in terms of the SL-states of the various configurations) for each symmetry designation.

(f) The final energy level prediction is then obtained by referring all the energy values to the lowest one, taken as ground state, and correcting for the normal mass effect [Karwowski and Fraga (1974)].

The most pertinent details regarding the selection of the configurations, determination of radial functions, selection rules, and mass corrections are presented below.

7.1 Selection of Configurations

The calculations may be performed in a brute force fashion, choosing indiscriminately a set of configurations, without an initial check of whether they contribute to the symmetry designation under consideration. Such an approach is not recommended as it may result in an appreciable waste of computer time.

The recommended approach is as follows. First, a list of possible configurations is prepared and then each of them is analyzed in order to find out whether they will give rise to states of the symmetry designation under consideration. The final list of appropriate configurations is then obtained by removal of those which will not contribute.

The calculations could now be performed for this set of configurations, but this direct approach might also result in a waste of computer time. It is recommended that, in all cases, first a calculation be performed for the electronic Hamiltonian operator, with the purpose of ordering the states (and corresponding configUrations) according to their increasing energies. This ordering will help in eliminating those configurations that give rise to high-lying states which may not contribute noticeably. It goes without saying, naturally, that there is no harm in including them if the necessary computer time is available.

7.2 Determination of Radial Functions

Several approaches are possible. each with deficiencies and advantages, depending on the purpose of the calculations. The main point to be emphasized here is that the orbitals of the same symmetry designation must be mutually orthogonal and

115

therefore in some of the approaches it may be necessary to perform a reorthonormalization.

(a) In calculations in which the objective is to obtain an accurate energy value for the lowest state of the symmetry designation under consideration it is appropriate to perform first a MCHF calculation for that state. In this way, the starting point for the following CI calculation (with inclusion of the relativistic corrections) is already rather accurate and the improved value will be very satisfactory. This approach, however, is not appropriate for the determination of energy levels, as the MCHF imposes constraints on the excited orbitals, which will result in a poor description of the interactions between excited states.

(b) For the determination of the set of energy levels, two different approaches may be followed. On one hand, one could perform SCHF calculations for as many states as necessary in order to determine the radial functions of all the orbitals needed. Afterwards those radial functions must be reorthonormalized. The advantage of this approach is that the starting point (Le., the Hartree-Fock energy for the lowest state) will be satisfactory and the improvement obtained in the CI calculation and the inclusion of the relativistic corrections will lead to a satisfactory prediction of the lowest levels.

The subsequent reorthonormalization may not be necessary if the SCHF calculations are performed for states arising from configurations which include all the orbitals needed for the CI calculations. The difficulty in this approach arises from the fact that some of the required configurations may be rather unusual.

An alternate approach, simpler and faster, consists of adopting a set of Slater­type functions and transforming them to numerical form, followed by reorthonormalization. The starting energy for the lowest state may not be satisfactory but it may still be possible to obtain an appropriate description of the energy levels if a large enough set of states is included in the calculations.

7.3 Selection Rules

As mentioned above, when the fine- and hyperfine-structure terms are included in the interaction Hamiltonian operator, the MS,ML terms to be used depend on the values of S,s' and L,L' 'of the interacting states. The functions used in the evaluation of the fine- and hyperfine-structure interactions are listed in Table 7.2, using the order numbering presented in Table 7.1.

Those matrix elements are subject to certain restrictions, given in Tables 7.3 and 7.4, imposed by the 3-j and 6-j symbols which appear in their expressions. Independently of these selection rules, the fme- and hyperfme-structure interactions

116

Table 7.1 Details of the functions used for evaluation of fine- and hyperfine-structure interactions.

Function MS ML Function MS ML

0,1/2 0 6 1/2, 1 L

2 0,1/2 7 S 0

3 0,1/2 L 8 S

4 1/2, 1 0 9 S L

5 1GI1

Table 7.2 Functions used for evaluation of fme- and hxperfine-structure interactions.

M.. 0 2 3 &max ;to L\S

o 9 8 8 7 7

o 9 8 7 7

3 2

o 9 8 7 7

6 5 4 4

2 3 2

117

Table 7.3 Selection rules.-

Spin-orbit Spin-spin Magnetic Electric Magnetic dipole dipole quadrupole octupole

IL\Msl 0 0 0 0 0

I~SI 0,1 0,1,2 0 0 0,1

0,1 0

0,1

IL\MLI 0 0 0 0 0

IM.I 0,1 0,1,2 0,1 0,1,2 0,1,2

0,1,2 0,1,2,3

0

I~JI 0 0 0 0 0

IL\JI 0 0 0,1 0,1,2 0,1,2,3

IL\MFI 0 0 0

It\FI 0 0 0

-The restrictions for the various tenns in a given matrix element are given on different lines. See Table 7.4 for the restrictions on the values of the individual quantum numbers.

118

Table 7.4 Additional restrictions on the values of the quantum numbers.

Forbidden value Case Remarlcs

MS 0 IASI=O IASlmax =0

0 IASI = IASlmax-1 IASlmax > 0

> Smin lAS I > 0

sa 0 o ~IASI < IASlmax IASlmax > 0

ML 0 IALI=O !ALlmax = 0

0 IALI = IALlmax-l IALlmax>O

>Lmin !ALI> 0

La 0 o ~ !ALI < IALlmax IALlmax>O

Ja 0 o ~ IAlI < IAllmax IAllmax >0

aThese restrictions hold for Sand/or S'. Land/or L'. and J and/or J'. respectively.

119

in, and between, certain states may vanish because of additional restrictions, due to the orbital angular dependence, etc.

7.4 Mass Corrections

Most of the terms of the Hamiltonian operator contain (see Chapter 1) a mass dependence, either on the mass of the electron, the mass of the nucleus, or both. This dependence is straightforward in all cases except for the electron kinetic energy term, which is expressed in terms of the reduced mass Il = mmJ(m+ma> (where lIla is the nuclear mass).

Usually, the calculations are carried in the Il-unit system, i.e., with Il = 1. For simplicity, it is assumed that m == Il and therefore it is necessary to introduce the corresponding numerical correction [Bethe and Salpeter (1957), Karwowski and Fraga (1974)], denoted as relativistic mass correction. Taking into account that

11m = 1I1l- 1I1Ila

the correction will take the fonn of

ijl/m)t: I-tijl/ma> = I-t(m/1J.a>

where the value of t depends on the tenn being considered (e.g., t = 3 for the mass variation term, etc.)

Proceeding in this way, the matrix elements and therefore the final energy values are obtained in the Il-unit system. Because the Il-unit system is specific for the species being considered, the final energy values must be corrected once again, in order to have them in the universal m-unit system This correction, denoted as the Donnal mass effect [Bethe and Salpeter (1957), Karwowski and Fraga (1974)] is introduced when the energy values are transfonned from the Il-unit system to cm-I , using the factor 2RM, with

RM == Roo [mJ(m+ma>] = Roo [1I(1+m/ma>] == Roo(1-m/Illa)

(neglecting higher powers of m/ma>.

8 Numerical Results

The formulation presented in this work was implemented in a computer program [Fraga et al. (1987)], used in some test calculations. Some of the corresponding results are presented here in order to illustrate the difficulties to be faced and the quality of the results.

8.1 Accurate Energies

Calculations were carried [Klobukowski and Fraga (1988)] for the two-electron systems He I, Li II and Be III. First of all, MCHF calculations were performed using an available program [Fischer (1978)]. These calculations yielded, in each case, a set of 15 orbitals (with 0 ~.1 ~ 4, .1 + 1~ n ~ 5) obtained from a total function which was a straightforward extension of an existing function [Fisher (1977)]. Those orbitals were then used to generate 35 configurations (n.i) 1 (n'.1') 1 ,

with.1 + 1 ~ n ~ 5, n ~ n' ~ 5. These configurations give rise to 53 J = 0 levels.

The calculations, including all the states and with consideration of all the relativistic corrections (except for the hyperfine-structure terms), yielded the following results (with the experimental values given in parentheses): He I, 637131.9 (637219.6); Li II, 1597643.4 (1597739.1); Be II, 2997220.0 cm- 1

(2997279.1). The accuracy of these results is 99.986, 99.994 and 99.998%, respectively. [See the work of Klobukowski and Fraga (1988) for the atomic data and conversion factors used.]

As mentioned in the preceding Chapter, these results illustrate that very accurate values may be obtained for lowest states when starting from good M;CHF functions. That is, the MCHF functions already yield a very accurate non-relativistic energy, which is then improved by the additional CI treatment with the consideration of the relativistic corrections. These calculations were straightforward but, evidently, their complexity will increase with the atomic number.

122

The main conclusion is that accurate energies for lowest states may be obtained now for all the elements up to, say, Z = 50.

8.2 SLJ-Energy Levels

Calculations were perfonned [Fraga et al. (1989)] for the elements, He through F, of the first row of the Periodic Table, followed by a more extended treatment of He.

In this case, the accurate, analytical expansions obtained in SCF calculations [Clementi (1965)], with HF accuracy, were adopted for the orbitals appearing in the ground state configuration, while for the excited orbitals simple Slater-type orbitals were used. Then the radial functions of all the above orbitals were transfonned to numerical fonn and orthonormalized.

The details of the calculations and of the accuracy obtained are as follows:

Helium: Calculations were carried out for IS, 3S, 1pO, 3pO, 10, 30, 1pO, and 3pO states, including 42 levels in each case. The average (absolute) error for the first 33 levels, with configuration designations 1s1ns1 (2SnS6), 2s1np1 (2SnS6), 1s1nd 1 (3SnS6), and 1s1nf1 (4 S n S 6) is 0.94%.

Lithium: Calculations were carried out for 2S (67 levels), 2pO (78 levels), 20 (74 levels), and 2pO (78 levels) states using 42 configurations in all cases. The average (absolute) error for the first 16 levels, with configuration designations 1s2ns1 (2 S n S 6), 1s2np1 (2 S n S 6), 1s2nd1 (3 S n S 6), and 1s2nf1 (4 S n S 6) is 0.80%.

Beryllium: Calculations were carried out for IS (42 levels), 3S (42 levels), 1pO (41 levels), 3pO (41 levels), 1p (28 levels), 3p (36 levels), 10 (44 levels), and 30 (39 levels) states. The average (absolute) error for the first 34 levels, with configuration designations 2s1np1 (2 S n S 6), 2s1ns1 (3 S n S 6), 2s1nd 1 (3 S n S 6), 2s2nf1 (4 S n S 6), and 2p2 is 2.14%.

Boron: Calculations were carried out for 204 odd levels (110 2pO, 34 4pO, 40 400, and 20 2S0 states) with J = 1/2. The average (absolute) error for 9 levels for which experimental data are available is 2.11 %.

Carbon: Calculations were carried out for 276 even levels (148 3p and 128 IS states) with J = O. The average (absolute) error for 11 states is 4.84%.

Nitrogen: Calculations were carried out for 389 odd levels (126 2pO, 95 200, 38 4S0, 48 4pO, 54 40 0, 20 4pO, and 6 600 states) with J = 3/2. The average (absolute) error for 19 states is 5.44%.

123

Oxygen: Calculations were carried out for 326 even levels (80 10,89 3p, 71 3D, 45 3F, 25 5p, 5 50, and 11 5F states) with J = 2. The average (absolute) error for 22 states is 6.71 %.

Fluorine: Calculations were carried out for 43 odd levels with J = 1/2, 57 odd levels with J = 3/2, 35 odd levels with J = 5/2, 14 odd levels with J = 7/2, 57 even levels with J = 1/2, 77 even levels with J = 3/2, 70 even levels with J = 5/2, 35 even levels with J = 7/2, and 14 levels with J = 9/2. The average (absolute) error for the frrst 31 levels is 7.40%.

[Reprinted with pennission of John Wiley & Sons, Inc.]

The results, in such a limited CI treatment, are not satisfactory (with the accuracy decreasing with increasing Z), although the overall description is encouraging, as observed in Fig. 8.1, which presents the comparison of the experimental and theoretical results of some energy levels.

The main information obtained from these calculations concerns, however, the relative importance of the relativistic corrections and of the correction for the correlation energy as introduced by the CI treatment. On one hand it was observed that for ground states the contributions from the relativistic corrections do not change appreciably with the CI treatment, being in agreement both at the mono- and multiconfigurationallevel with the 'experimental' values obtained from the total experimental energies and the accurate, theoretical non-relativistic energies. If this observation may be extrapolated to excited states, one would conclude that the accuracy of the prediction of the energy levels of an atomic species will depend, as long as the relativistic corrections are included, on how well the CI treatment will account for the correlation energy.

Improving CI results is not a simple matter of just increasing the number of interacting configurations, as seen from the results obtained in an extended treatment for the J=O levels of He. First a calculation was carried out for 330 levels arising from configurations nslmsl (lSnS15, nSmS15) and nplmpl (2SnS15, nSmS15); then another calculation was performed for 234 levels arising from configurations nslms l (lSnS15, nSmS15), nplmpl (2SmS9, nSmS9) and ndlmdl (3SnS8, nSmS8). The corresponding results are presented in Table 8.1, together with those of the initial calculation described above. These results show clearly that in spite of the drastic reduction in the number of configurations considered in the second calculation, the results are just as satisfactory, due to the fact that angular correlation is now better accounted for thanks to the consideration of the configurations ndlmd1. The average (absolute) error is 0.83%, indicating th~t a successful prediction of the energy levels of an atomic species is simply a matter of a judicious selection of a large enough set of configurations and of the availability of the required computing time.

The I-levels splitting are generally well reproduced, as observed in Fig. 8.2, which presents a comparison of theoretical versus experimental results. The

124

20.---------------------------~

~ ,.... Ie o

b

15

:!:.. 10 ur <l

5

5 10 15 20

~Ee (104 cm-1)

Figure 8.1 Comparison of theoretical (~u versus experimental (~) values of some energy levels for He through F. The experimental values were taken from Moore (1949), Biemont and Grievese (1973), Odintzova and Striganov (1979) and Stricker and Bauer (1975). [Reprinted with permission of John Wiley & Sons, Inc.]

0 • • • •

~ -1 ,.... I e • 0 •

'& -2 • • • :!:.. ur <l

-3 • • •

-4

•

-4 -3 -2 -1 0

~Ee (102 cm-1)

Figure 8.2 Comparison of theoretical (~u versus experimental (~e) values of some J-Ievel splittings in F. The experimental values were taken from Moore (1949). [Reprinted with permission of John Wiley & Sons, Inc.]

125

Table 8.1 Energy levelsa (in em-I) of He I [Fraga et al. (1989)]

Theoretical

State Experimental 421evelsb 3301evelsc 2341eveIsC

IS 0.0 0.0 0.0 0.0

IS 166271.7 166805.6(0.32) 166167.9(0.06) 166129.0(0.()9)

IS 184859.1 184984.4(0.07) 184255.1(0.33) 184207.9(0.35)

IS 190934.5 190426.6(0.27) 189707.3(0.64) 189658.4(0.67)

IS 193657.8 192890.2(0.40) 192066.1(0.82) 192016.7(0.85)

IS 195109.2 193982.5(0.58) 193292.6(0.93) 193242.9(0.96)

IS 195973.2 193864.7(1.08) 193814.7(1.10)

IS 195529.0 194037.0(1.27) 193987.2(1.29)

IS 196907.1 194485.5(1.23) 194435.7(1.26)

IS 197176.4 194954.9(1.13) 194904.9(1.15)

IS 195154.4 195104.5

IS 197524.3 195352.1(1.10) 195302.1(1.13)

IS 195460.4(1.11) 195410.7(1.13)

IS 197189.9 197139.9

3p 481198.0 480140.3(0.22l 481012.7(0·02l

lIThe values in parentheses represent the percentage errors (in absolute value). bWithout fine-structure interactions. cIncluding fine-structure interactions (for J =0). See the text for details of the configurations used. [Reprinted with permission of John Wiley & Sons, Inc.]

126

goodness of the comparison must be judged taking into account that one is considering differences of, at most, 200 cm-I for levels up to 140,000 cm- I .

8.3 Hyperfine-Structure Splittings

The evaluation of hyperfine-structure splittings is further complicated by the differences in the magnitudes of the corrections introduced by the various relativistic tenns: cm-I for the fine-structure couplings versus MHz for the hyperfine-structure interactions. Consequently, the brute force approach, followed in the calculations reported above, is not appropriate, as the hyperfine-structure contributions would not be detected

The approach to be adopted is as follows. For each F-value, the complete interaction matrix is constructed, with consideration of all the relativistic corrections, including the hyperfine-structure interactions; simultaneously, a matrix with only the contributions of the hyperfine-structure interactions is also constructed. Diagonalization of the complete interaction matrix yields the eigenvectors, which are then used in the evaluation of the hyperfine-structure contributions to the F-states. The calculation is repeated for every possible F-state and then the hyperfine-structure splittings for the F-Ievels arising from each SU level are obtained.

Calculations were performed [San Fabian and Fraga (1988)] for the F-Iev~ls arising from the J levels of the SL ground states of B through F as well as for AI. As in the calculations of I-levels, described above, Slater-type orbitals were adopted for the excited orbitals while for the orbitals appearing in the ground state configuration the analytical expansions [Clementi (1965)] obtained in accurate SCF calculations were used. The calculations were started using only the ground state configuration and continued, with inclusion of additional excited configurations, until a satisfactory agreement with experimental data was obtained. For B, F and Al only one excited configuration (arising from the 2p ~ 3p excitation in the case of B and F and from the 3p ~ 4p excitation in the case of AI) was used. For C and N the configurations arising from the 2p ~ 3p and 2s ~ 3s excitations were included, while for 0 it was necessary to include the configurations arising from the 2p ~ 3p, 2p ~ 4p and 2p ~ 5p excitations.

The agreement between the theoretical and experimental results is satisfactory, as observed in Table 8.2, taking into account the reduced number of configurations, the small magnitude of the interactions and the difficulties which are even present in the assignment of the experimental results. The precision of the results (with respect to the experimental values) is 96.7, 104.1, 100.3 and 93.8% for B, 94.3, 94.2 and 91.8% for C, 92.6, 96.1, 87.6 and 92.3% for 0,96.1 and 98.0 for F, and 80.3, 99.8,90.5 and 78.9% for AI.

127

Table 8.2 H~rfine-strueture levels and ~littingsa,b [San Fabian and Fraga !1988ll.

Level (MHz) Splitting (MHz) Atom State J Level (em-I) F Theor. Expti. Theor. Exptl.

13B 2pO 1{l 0.0 -442.5 -457.6

708.0 732.2

2 265.5 274.6

3{l 21.4 0 -269.0 -271.7

73.6 70.7

-195.4 -201.0

144.5 144.0

2 -50.9 -57.0

209.4 222.7

3 158.5 165.7

lle 3p 0 0.0 3{l 0.0 20.9 5{l 0.4 -1.3

-2.7 0.1

3{l -2.3 -1.2

5.7 7.5 1{l 3.4 6.3

2 50.9 7{l -194.1 -205.9

229.3 243.1

5{l 35.2 37.2

157.7 167.4 3{l 192.9 204.6

92.1 98.0

1{l 285.0 302.6

l3e 3p 0 0.0 1{l 0.0 20.9 1{l -0.3 -2.8

0.4 4.2

3{l 0.1 1.4

2 50.9 3{l -205.2 -223.6

342.0 372.7

5{l 136.8 149.1

14N 4S0 3{l 0.0 1{l -7.2 -26.1

4.3 15.6

3{l -2.9 -10.5

7.2 26.2

5{l 4.3 15.7

128

170 3p 2 0.0 9/2 -1013.5 -1095.5 917.1 990.6

7/2 -96.4 -104.9 705.6 733.9

5/2 609.2 629.0 499.9 570.5

3/2 1109.1 1199.5 298.2 323.2

1/2 1407.3 1522.7

190.7 3/2 -27.7 -12.9 14.3 4.0

5/2 -13.4 -8.9 37.4 22.1

7/2 24.0 13.2 0 267.3 5/2 0.0

19F 2pO 3/2 0.0 -2414.7 -2512.5 3863.5 4020.0

2 1448.8 1507.5

1/2 439.4 0 -7532.2 -7683.2 10042.9 10244.3

2510.7 2561.1

27Al 2pO 1/2 0.0 2 -705.3 -878.6 1209.1 1506.1

3 503.8 627.5

3/2 98.6 -433.0 -481.7 173.1 173.4

2 -259.9 -308.3 248.2 274.4

3 -11.7 -33.9 309.5 392.2

4 297.8 358.3

aThe experimental values were taken from the work of Lew and Title (1960), Harvey et al. (1972),

Haberstroch et al. (1964), Wolker et al. (1970), Holloway et al. (1962), Wilmer Anderson et al.

(1959), Harvey (1965), and Radford et al. (1961).

bThe experimental energy levels, when not given explicitly in the experimental work, have been evaluated as Ep= AK/2 + B{[(3/4)K(K+l)-I(l+I)J(J+l)]/21(21-1)J(2J-l)}

where A and B are the dipole and quadrupole hyperfine-structure coupling constants and

K = F(F+ 1) - 1(1+ 1) - J(1+ 1) [Reprinted with permission of the National Research Council of Canada.]

129

8.4 Nuclear-Mass Dependent Corrections

Simple calculations of the nuclear-mass dependent spin-orbit coupling (electron spin) were performed [Sordo et al. (1991)] for the ground states of C I (3PO,},2, A = 11, M = 11.0114333 au) and 0 I (3P2,1,O, A = 17, M = 16.9991333 au).

The splittings obtained, using monoconfigurational HF functions, were:

Carbon

Oxygen

3Pl-3PO: 244.9 MHz

3Pl-3P2: 809.6 MHz

3P2-3PO: 489.3 MHz

3P03Pl: 1619.2 MHz

It was concluded, from these results, that the contributions of the nuclear-mass dependent spin-orbit coupling (electron spin) are comparable with, or even greater than, those of the hyperfine-structure interactions. That is: this interaction as well as the nuclear-mass dependent orbit-orbit interaction and the nuclear-mass dependent spin-orbit coupling (nuclear spin) should be included in accurate calculations whenever isotopic effects are to be studied.

Bibliography

9.1 References

L. Armstrong, Jr. Theory of the Hyperjine Structure of Free Atoms. Wiley­Interscience, New York (1971).

H.A. Bethe and F.E. Salpeter. Quantum Mechanics of One- and Two-Electron Atoms. Academic Press Inc., New York (1957).

E. Biemont and N. Grevesse. At. Data Nuclear Data Tabl.12, 217 (1973).

G. Breit. Phys. Rev. 34,553 (1929).

G. Breit. Phys. Rev. 36,383 (1930).

G. Breit. Phys. Rev. 39, 616 (1932).

D.M. Brink and G.R. Satchler. Angular Momentum. Clarendon Press, Oxford (1979).

E. Clementi. Tables of Atomic Functions. International Business Machines Corporation, White Plains (1965).

R. Carbo and M. Klobukowski (editors). Self-Consistent Field Theory. Theory and Applications. Elsevier Science Publishers, Amsterdam (1991).

E.U. Condon and G.H. Shonley. The Theory of Atomic Spectra. Cambridge University Press, Cambridge (1964).

P.A. Dirac. Proc. Roy. Soc. Al17, 610 (1928a).

P.A. Dirac. Proc. Roy. Soc. A118, 351 (1928b).

A.M. Ermolaev and M. Jones. 1. Phys. B5, L225 (1972).

132

A.M. Ermolaev and M. Jones. J. Phys. B6, 1 (1973).

D. Feller, C.M. Boyle and E.R. Davidson. J. Chem. Phys. 86, 3424 (1987).

C.F. Fischer. The Hartree-Fock Method for Atoms. John Wiley & Sons, New York (1977).

C.F. Fischer. Comput. Phys. Commun. 14,145 (1978).

S. Fraga (editor). Computational Chemistry: Structure, Interactions and Reactivity. Elsevier Science Publishers, Amsterdam (1992).

S. Fraga and J. Karwowski. Theoret. Chim. Acta 35, 183 (1974a).

S. Fraga and J. Karwowski. Can. J. Phys. 52, 1045 (1974b).

S. Fraga, J. Karwowski and K.M.S. Saxena. Atomic Data Nucl. Data Tab. 12, 467 (1973).

S. Fraga, J. Karwowski and K.M.S. Saxena. Handbook of Atomic Data. Elsevier Science Publishers, Amsterdam (1976; second printing 1979).

S. Fraga, J. Karwowski and K.M.S. Saxena. Atomic Energy Levels: Data for Parametric Calculations. Elsevier Science Publishers, Amsterdam (1979).

S. Fraga, M. Klobukowski, J. Muszynska, K.M.S. Saxena and J.A. Sordo. Phys. Rev. A34, 23 (1986).

S. Fraga, M. Klobukowski, J. Muszynska, K.M.S. Saxena, J.A. Sordo, J.D. Climenhaga and P. Clark. Comput. Phys. Commun. 47, 159 (1987).

S. Fraga and G. Malli. Many-Electron Systems: Properties and Interactions. W.B. Saunders Co., Philadelphia (1968).

S. Fraga and J. Muszynska. Atoms in External Fields. Elsevier Science Publishers, Amsterdam (1981).

S. Fraga, E. San Fabian, J.A. Sordo, M. Campillo, J.D. Climenhaga and M. Klobukowski. Int. J. Quantum Chem. 35,325 (1989).

S. Fraga and K.M.S. Saxena. Atomic Data 4,255 (1972a).

S. Fraga and K.M.S. Saxena. Atomic Data 4, 269 (1972b).

S. Fraga, K.M.S. Saxena and J. Karwowski. Can. J. Phys. 51, 2063 (1973).

S. Fraga, K.M.S. Saxena and B.W.N. Lo. Atomic Data 3,323 (1971).

133

S. Fraga, J. Thorhallsson and C. Fisk. Can. J. Phys. 47, 1415 (1969).

RA. Haberstroch, W.J. Kossler, O. Ames and D.R Hamilton. Phys. Rev. B136, 932 (1964).

J.S.M. Harvey. Proc. Roy. Soc. A285, 581 (1965).

J.S.M. Harvey, L. Evans and H. Lew. Can. J. Phys. 50, 1719 (1972).

RA. Hegstrom. Phys. Rev. A7, 451 (1973).

W.W. Holloway, Jr., E. LUscher and R Novick. Phys. Rev. 126,2101 (1962).

F.R Innes and C.W. Ufford. Phys. Rev. 111, 194 (1958).

B.R Judd. Operator Techniques in Atomic Spectroscopy. McGraw-Hill Book Company, Inc., New York (1963).

J. Karwowski and S. Fraga. Can. J. Phys. 52, 536 (1974).

M. Klobukowski and S. Fraga. Phys. Rev. A38, 1593 (1988).

H. Lew and RS. Title. Can. J. Phys. 38, 868 (1960).

J. Maly and M. Hussonnois. Theoret. Chim. Acta 28,363 (1973).

J.B. Mann and W.R Johnson. Phys. Rev. A4, 41 (1971).

W.C. Martin. In Proceedings of the Workshop on Foundations of the Relativistic Theory of Atomic Structure. Argonne National Laboratory, Report ANL-80-126 (1981).

I. Martinson. Comments At. Mol. Phys. 12, 19 (1982).

C.E. Moore. Atomic Energy Levels as Derived from the Analyses of Optical Spectra. National Bureau of Standards, Washington (1949,1952,1958).

H. Nussbaumer. Comput. Phys. Commun. 1, 191 (1969).

G.A. Odintzova and A.R. Striganov. J. Phys. Chern. Ref. Data 8,63 (1979).

P. Pyykko. Relativistic Theory of Atoms and Molecules - A Bibliography (1916-1985). Springer Verlag, Berlin (1986).

G. Racah. Phys. Rev. 62, 438 (1942).

H.E. Radford, V.W. Hughes and V. Beltran-Lopez. Phys. Rev. 123, 153 (1961).

134

F. San Fabian and S. Fraga. Can. J. Phys. 66, 583 (1988).

C. Schwartz. Phys. Rev. 97, 380 (1955).

T.L. Sordo, J.A. Sordo and S. Fraga. Can. J. Phys. 69, 161 (1991).

J. Stricker and S.H. Bauer. Chem. Phys. Lett. 30, 447 (1975).

L. Wilmer Anderson, F.M Pipkin and J.C. Blaird. Phys. Rev. 116, 87 (1959).

O. Wolber, H. Figger, R.A. Haberstroch and S. Penselin. Z. Phys. 236, 337 (1970).

9.2 Reference Texts

L. Armstrong, Jr. Theory of the Hyperjine Structure of Free Atoms. Wiley­Interscience, New York (1971).

H.A. Bethe and E.E. Salpeter. Quantum Mechanics of One- and Two-Electron Atoms. Academic Press Inc., New York (1957).

L.C. Biedeham and H. Vail Dam. Quantum Theory of Angular Momentum. Academic Press Inc., New York (1965).

D.M. Brink and O.R. Satchler. Angular Momentum. Clarendon Press, Oxford (1979).

E.U. Condon and H. Odabasi. Atomic Structure. Cambridge University Press, Cambridge (1980).

E.U. Condon and O.H. Shonley. The Theory of Atomic Spectra. Cambridge University Press, ~bridge (1964).

A.R. Edmonds. Angular Momentum in Quantum Mechanics. Princeton University Press, Princeton (1960).

C.F. Fischer. The Hartree-Fock Method for Atoms. John Wiley & Sons, New York (1977).

A.P. Jucys and A.J. Savukynas. Mathematical Foundations of the Atomic Theory. Academy of Sciences of the Lithuanian SSR, Vilnius (1973).

A.P. Jucys and A.A. Bandzaitis. Theory of Angular Momentum in Quantum Mechanics. Academy of Sciences of the Lithuanian SSR, Vilnius (1977).

B.R. Judd. Operator Techniques in Atomic Spectroscopy. McGraw-Hill Book Company, Inc., New York (1963).

135

M.E. Rose. Elementary Theory of Angular Momentum. John Wiley & Sons, Inc., New York (1957).

C. Sanchez del Rio. Introduccion ala Teor(a del Atomo. Editorial Alhambra, S.A., Madrid (1977).

B.G. Wybourne. Spectroscopic Properties of Rare Earths. Interscience Publishers, New York (1965).

9.3 Data Sources

S. Bashkin and J.O. Stoner. Atomic Energy Levels and Grotrian Diagrams. North­Holland, Amsterdam (1975, 1978, 1981).

G.W. Erickson. J. Phys. Chern. Ref. Data 6,831 (1977).

G.H. Fuller. J. Phys. Chern. Ref. Data 5,835 (1976).

L. Hagan. Bibliography on Atomic Energy Levels and Spectra. National Bureau of Standards, Washington (1977).

L. Hagan and W.C. Martin. Bibliography on Atomic Energy Levels and Spectra. National Bureau of Standards, Washington (1972).

W.C. Martin, L. Hagan, J. Reader and J. Sugar. 1. Phys. Chern. Ref. Data 3, 771 (1974).

C.E. Moore. Atomic Energy Levels As Derived From the Analyses of Optical Spectra. National Bureau of Standards, Washington (1949,1952,1958).

C.E. Moore. Ionization Potentials and Ionization Limits Derived From the Analyses of Optical Spectra. National Bureau of Standards, Washington (1970).

E. Richard Cohen and B.N. Taylor. Europhys. News 18,65 (1987).

J.G. Stevens and B.D. Dunlap. 1. Phys. Chern. Ref. Data 5, 1093 (1976).

A.H. Wapstra and N.B. Gove. Nucl. Data Tables 9, 267 (1971).

as well as

International Bulletin on Atomic and Molecular Data for Fusion. Atomic and Molecular Data Unit, International Atomic Energy Agency, Vienna.

Units and Constants

10.1 Constants

Designation

Speed of light

Charge of the electron

Mass of the electron

Mass of the proton

Bohr radius

Electron magnetic moment

Bohr magnetron

Nuclear magnetron

Fine-structure constant

Planck constant

Rydberg constant

Definition Value

c = 2.99792458-1010 cm s-1

e = 1.60217733-10-19 C

m = IDe = 9.1093897-10-31 kg

mp = 1.6726231-10-27 kg

ao = ~2/IDee2 ao = 0.529177249-10-10 m

J.le = 928.47701-10-26 JT-l

IlB = e~!lIDec IlB = 927.40154-10-26 J T-l

IlN = e~!2lllpC IlN = 0.50507866-10-26 J T-l

a. = e2/~c a. = 7.29735308-10-3

~ = h/21t ~ = 1.05457266-10-34 J s

h = 6.6260755-10-34 J s

Roo = IDeca.2/2h Roo = 109737.31534 cm-1

[Taken from E.R. Cohen and B.N. Taylor, 'Fundamental Constants. 1986 Adjustments', Europhysics News 18,65 (1987). See also E.R. Cohen and B.N. Taylor, J. Phys. Chern. Ref. Data 17, 1795 (1988).]

138

10.2 Units

The unit of energy of interest in this work is the Hartree, as well as the electton­volt, em-1 and MHz, related as follows:

1 Hartree = 4.35974812-10-18 J = 27.2113957 eV

= 219474.625 em-I = 0.657968374-1010 MHz

Notation and Symbols

A consistent notation has been used throughout this work and the most often used symbols and conventions are summarized below.

Subindices

a: Nucleus

ij: Particles (in the Hamiltonian operator) or spio-orbitals (in the expressions for the matrix elements).

u, v: Slater determinants

p,O': Electrons

Mathematical symbols

O(r): One-dimensional Dirac function, 5(r) = 4m20(r)

O(r): Three-dimensional Dirac delta function

O(a,b): Kronecker delta

5(abc ... , a'b'c' ... ) = 5(a,a')5(b,b')5(c,c') ...

ra: Position vector of nucleus

rp: Position vector of electron p

rpa = rp - ra

fpa = fp - ra

r<: The lesser of rp and ra

r>: The greater of rp and ra

140

(jlmU2m21j3m3): Oebsch-Gordan coefficient

{j 1 m U2ffi2lj3m3} = (j 1 m U2m2Ij3m3)(j 10j:zQlj30)

Tensor operator notation

T(K): Tensor operator of rank K

T dK): Q-component of tensor operator T(K)

[U(k) x V(k')](K): Cross-product of tensor operators

(U(k). V(k»: Scalar product of tensor operators

,R,(1)(p): Orbital angular momentum tensor operator (for electron p)

y(1.)(p): spherical harmonic tensor operator (for electron p)

C(k)(p): Modified spherical harmonic tensor operator (for electron p)

R(k,k')(p) = [C(k)(p) x .5l(1)(p)](k')

Electron data

m: Mass

J..l: Reduced mass, defmed as J..l = mM/(m+M)

e: Charge

J.le: Magnetic dipole moment (in J..lB units)

gs: g-factor

s(1)(p): Spin angular momentum tensor operator (for electron p)

Pp: Linear momentum vector (for electron p)

T: Kinetic energy operator

141

Atomic data

A: Mass number

M: Mass

ma: Nuclearmass

RM: Rydberg constant for fmite nuclear mass

Roo: Rydberg constant for infinite nuclear mass

Z: Nuclear charge

Pa: linear momentum vecter

1(1): Nuclear spin angular momentum tensor operator

Ill: Nuclear magnetic dipole moment (in IlN units)

gr: Nuclear g-factor

Q: Nuclear electric quadrupole moment

n: Nuclear magnetic octupole moment

N(k): Nuclear tensor operator. The nuclear magnetic dipole and octupole moments are defmed [Schwartz (1955), Armstrong (1971)] by

III = (llINo(I)Ill)

n = -(llINo(3)III)

Quantum numbers

s: Individual electron spin angular momentum total quantum number

ms: Individual electron spin angular momentum z-projection quantum number. When affected by a subindex, labelling the electron, the symbol Il has been used

i: Individual electron orbital angular momentum total quantum number

ml.: Individual electron orbital angular momentum z-projection quantum number. When affected by a subindex, labelling the electron, the symbol m has been used

142

S: Atom electronic spin angular momentum total quantum number

Ms: Atomic electronic spin angular momentum z-projection quantum number

L: Atom electronic orbital angular momentum total quantum number

ML: Atom electronic orbital angular momentum z-projection quantum number

J: Atom electronic angular momentum total quantum number

MJ: Atom electronic angular momentum z-projection quantum number

I: Nuclear spin angular momentum total quantum number

MI: Nuclear spin angular momentum z-projection quantum number

F: Atom angular momentum total quantum number

MF: Atom angular momentum z-projection quantum number

Configurations and Slater determinants

In abbreviated form, the symbol C will be used in order to denote a configuration. Its detailed specification will include the complete listing of the quantum numbers.1, ma. and ms for all the spin-orbitals: (ilmllll)(i2m21l2) .... (iNmNIlN). Additional symbols associated with· a configuration are:

{nil: Set of n; i quantum numbers corresponding to the occupied orbitals. Whenever the details of the principal quantum number n are not required, the notation may be shortened to {il

{mlJ.l: Set ofma., ms quantum numbers corresponding to the occupied spin-orbitals

{ni}{mllluMsML: Slater determinant constructed from the configuration characterized by the sets of quantum numbers {nil and {mlllu. The subindex u labels the various determinants that may be built from a given configuration

( {i 1 (mIJ. 1 uMSMLIPSL): SL-vector coupling coefficient associated with the Slater

determinant {i 1 {mIJ. luMsML in the expansion of the SL-state CPSMsLML

States

PSL: Multiconfiguration SL-state. The parameter P includes whatever additional information is needed for a complete identification of the state (such as, e.g.,

143

an ordering index to distinguish states with the same S,L values). When full identification is needed, the label P S M S L M L is used. For monoconfiguration SL-states (with or without state interaction), the designations CpSL and CpSMsLML are used

yI: Multiconfiguration I-state. The parameter y includes whatever additional information is needed for a complete identification of the state (such as, e.g., an ordering index to distinguish states with the same I values). When full identification is needed, the label yIMJ is used. For monoconfiguration I-states (with or without state interaction), the designations CyI and CyIMJ are used. The label CySMSLMLIMJ is used for the components of a given JMJ-state, indicating the full parentage

eF: Multiconfiguration F-state. The parameter e includes whatever additional information is needed for a complete identification of the state (such as, e.g., an ordering index to distinguish states with the same F values). When full identification is needed, the label eFMF is used. For monoconfiguration F-states (with or without state interaction), the designations CeF and CeFMF

are used. The labels CEIMJFMF andCeSMsLM0MJFMF are used for the components of a given F-state, indi~ating the full parentage

Functions

R(ni;r): Radial function of the orbital characterized by the quantum numbers n.1.

P(ni;r) = rR(ni;r)

YllD1.(9,ej»: spherical harmonic

y m(1.)(9,ej»: spherical harmonic

Matrix elements

(TtimITQ(K)ITl'j'm'): Mattix element between states Ttim and TlTm'

(TtiIlT(K)It'lT): Reduced matrix element between states Tti and ll'j' (independent of mandm')

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Vol. 48: e. Pisani, R. Dovesi, e. Roetti, Hartree-Fock Ab Initio Treatment of Crystalline Systems. V, 193 pages. 1988.

Vol. 49: E. Roduner, The Positive Muon as a Probe in Free Radical Chemistry. VII, 104 pages. 1988.

Vol. 50: D. Mukherjee (Ed.), Aspects of Many· Body Effects in Molecules and Extended Systems. VIII, 565 pages. 1989.

Vol. 51: J. Koca, M. Kratochvil, V. Kvasnicka, L. Matyska, J. Pospfchal. Synthon Model of Organic Chemistry and Synthesis Design. VI, 207 pages. 1989.

Vol. 52: U. Kaldor (Ed.), Many·Body Methods in Quan­tum Chemistry. V, 349 pages. 1989.

Vol. 53: G.A. Arteca, F.M. Fernandez, E.A. Castro, Large Order Perturbation Theory and Summation Methods in Quantum Mechanics. XI, 644 pages. 1990.

Vol. 54: S.J. Cyvin, I. Brunvoll, B.N. Cyvin, Theory of Coronoid Hydrocarbons. IX, 172 pages. 1991.

Vol. 55: L.T. Fan, D. Neogi, M. Yashima. Elementary Intro· duction to Spatial and Temporal Fractals. IX, 168 pages. 1991.

Vol. 56: D. Heidrich, W. Kliesch, W. Quapp, Properties of Chemically Interesting Potential Energy Surfaces. VIII, 183 pages. 1991. .

Vol. 57: P. Turq, I. Barthel, M. Chemla, Transport, Relaxation, and Kinetic Processes in Electrolyte Solutions. XIV, 206 pages. 1992.

Vol. 58: B. O. Roos (Ed.), Lecture Notes in Quantum Chemistry. VII, 421 pages. 1992.

Vol. 59: S. Fraga, M. Klobukowski, J. Muszynska, E. San Fabian, K.M.S. Saxena, I.A. Sordo, T.L. Sordo, Research in Atomic Structure. XII, 143 pages. 1993.