Drake G.W.F. (editor) Handbook of Atomic, Molecular, and Optical Physics

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316 Part B Atoms

plex for the 3s

P ground state of Si, is a neon-like

core, 1s

, coupled to the {N = 4,π = even,

CAS of the active set {3s, 3p, 3d}.IftheAS is extended

to {3s, 3p, 3d, 4s, 4p, 4d, 4f}, then all single, double,

triple, quadruple (SDTQ) replacements from the outer

four electrons are generated. Of course, the number of

CSF increase rapidly, both with the size of the active

set and with the number of electrons N deﬁning the

CAS. For this reason, other models may be used, such as

multi-reference singles and doubles (MR–SD) modeled

on the results of Z-dependent perturbation theory. These

multi-reference functions may be extended to include all

important contributors to a wave function. Calculations

in which correlation is restricted to a few outer electrons

is called valence correlation.

A consequence of a CAS expansion is that the

MCHF problem is over determined: a rotation of ra-

dial functions of the same symmetry merely transforms

the CSFs and the expansion coefﬁcients. The wave func-

tion and energy do not change. When similar situations

occur in Hartree–Fock theory, Koopmans’ theorem re-

quires that the lagrange multiplier associated with the

rotation be set to zero. In MCHF calculations, the de-

gree of freedom is usually used to eliminate a CSF or,

more precisely, to determine that solution for which

a speciﬁc CSF has a zero expansion coefﬁcient. A gener-

alized Brillouin’s Theorem (GBT) then holds. In the case

of the helium ground state (or any ns

pair function),

applying GBT leads to the natural orbital expansion

of the form {1s

, 2s

, 3s

,... ,2p

, 3p

,... ,3d

,...}

in which all CSF differ by two electrons. For other

symmetries, such as 1s2p

P, a reduced form [21.8]

can also be deﬁned in which all CSF differ by two

electrons, but now involves different sets of orthonor-

mal radial functions for the different partial waves

as in {1s2p

, 2s3p

, 3s4p

,... ,2p

, 3p

,...}.

Such expansions yield the fastest rate of convergence,

but are difﬁcult to apply in complex systems. For a his-

tory of Brillouin’s theorem and its use in solving multi-

conﬁguration self-consistent ﬁeld problems see [21.9].

21.5.4 Excited States

For ground states or atomic states lowest in their

symmetry, the variational procedure is a minimization

procedure, and consequently any approximate energy is

an upper bound. For all others the energies are stationary.

Such calculations may be difﬁcult. One of the most dif-

ﬁcult has been the HF calculation for 1s2s

S and, once

obtained is disappointing since the energy is too low. Ex-

cited state calculations become minimization problems

through the use of the Hylleraas–Undheim–MacDonald

theorem (Sect. 11.3.1). Consider a CAS calculation

over the active set {1s, 2s} for which the CSFsare

{1s

, 1s2s, 2s

}. In determining orbitals, we have a de-

gree of freedom so one CSF may be removed. Selecting

has the consequence that the eigenvalue for the 1s2s

state to be determined is now the second eigenvalue, and

hence an upper bound to the exact.

21.5.5 Autoionizing States

The MCHF variational method may be applied to core

excited states imbedded in the continuum, provided that

certain CSFs with ﬁlled shells are omitted. An example

is 1s2s2p

P of Lithium. An MCHF calculation omit-

ting 1s

np conﬁguration states represents the localized

charge distribution of a state in the continuum. How-

ever, the resulting energy is not guaranteed to be an

upper bound to the exact energy.

The saddle-point variational method, as described in

Chapt. 25 is a minimax method for such states.

21.6 Conﬁguration Interaction Methods

Conﬁguration interaction (CI) methods differ from vari-

ational methods like MCHF in that the radial functions

are assumed to be ﬁxed, and hence known in advance.

There are several situations where these methods can be

used effectively:

MCHF with Breit–Pauli

Since the Breit–Pauli operators are valid only as

ﬁrst order perturbations, variational calculations for

the Breit–Pauli Hamiltonian are not justiﬁed. Instead,

MCHF calculations are performed to provide a basis for

LSJ wave functions of (21.14). The expansion coef-

ﬁcients are obtained as a CI calculation. Usually such

expansions are a concatenation of the expansions of all

the LS terms of a conﬁguration and possibly some close

lying conﬁgurations. Most Breit–Pauli codes require

a single orthonormal basis. In order to have an orbital

basis that simultaneously describes the correlation of

all these LS terms from a systematic procedure, the

MCHF derivation described earlier has been extended

to derive systems of coupled equations for linear combi-

nations of energy functionals, one for each LS term and

Part B 21.6

Atomic Structure: Multiconﬁguration Hartree–Fock Theories 21.6 Conﬁguration Interaction Methods 317

eigenstate. With this extension, it has been possible to

perform “spectrum” calculations in which all LSJ lev-

els up to a certain point in the spectrum are included.

This requires a balanced correlation approach so that

the energy differences with respect to the ground state is

in good agreement with the observed excitation energy.

Once wave functions have been obtained for each level,

transition probabilities can be computed. With all E1

transitions between these levels and some E2/M1 tran-

sitions, the lifetimes of levels can be obtained [21.10].

Full-Core Methods

The full-core plus correlation method (FCPC) [21.16]

is a conﬁguration interaction method. It is mostly used

for an atomic system in which a well deﬁned “core”

is present. For example, in a system such as the three-

elec tron 1s

nl or the four-electron 1s

nln



, the system

has a 1s1s core. In this case, the wave function for an

N-electron system is written as

Ψ(1, 2, 3,... ,N ) =

Aφ

1s1s

(1, 2)



(3, 4,... ,N )



(1, 2, 3,... ,N ),

where φ

1s1s

(1, 2), the wave function of the core, is used

as a single term, and A denotes antisymmetrization. The

relaxation of the core and other correlation effects are

accounted for by the last term in this equation. The cor-

relation effect of the 1s1s core is, in general, very strong

and it is difﬁcult to fully account for this effect in a con-

ventional N-electron wave function. If inaccurate results

are obtained from such a wave function, it may be difﬁ-

cult to distinguish between errors coming from the core

part or from the other parts. In the FCPC, the correlation

effect in φ

1s1s

(1, 2) can be precalculated to a desired ac-

curacy such that ψ

no longer contains the contribution

from the unperturbed core. In physical processes such as

ionization, optical transitions and others, the 1s1s core

wave function in the ﬁnal state does not change much

from that of the initial state. Using the same φ

1s1s

(1, 2)

for the core may minimize possible errors due to the in-

accuracy of the core wave function. The application of

this method is not limited to systems with 1s1s cores.

For example, for the lithium-like 1s2snp

,forn ≥4,

the 1s2s

S can also be considered as an appropriate

“core”.

From a computational view point, the FCPC wave

function has the advantage that it reduces the matrix size

of the secular equation substantially. This drastically

Table 21.3 Comparison of theoretical and experimental en-

ergies for Be 1s

S in hartrees. All theoretical values

include some form of extrapolation

Ref. Method −E

−E

rel

[21.11] FEM MCHF 14.667 37 14.669 67

[21.12] MCHF 14.667 315

[21.13] Full-core CI 14.667 3492 14.669 6774

[21.14] Semi-empirical 14.667 353

[21.15] Experiment 14.6696759

reduces the memory and CPU time needed on a com-

puter. This method has been quite successful in getting

accurate results for four-electron systems as shown in

Table 21.3. See [21.13] for more details.

Slater Type Orbitals

The essential characteristic of a radial function can be

well represented by the expansion

(r) =



jnl

(r),

where

jnl

(r) =



(2ζ

jnl

)

jnl

(2I

jnl

)



1/2

jnl

exp(−ζ

jnl

is a Slater type orbital (STO). Optimized sets of pa-

rameters have been tabulated for many Hartree–Fock

wave functions and these may be used to represent the

core [21.17]. Others may be added and selected orbitals

exponent optimized (only the exponent is varied) so as

to augment the basis. This method has been used effec-

tively by A. Hibbert as implemented in the Conﬁguration

Interaction Version 3 program [21.18].

Spline Basis

The analytic basis methods described in the previous

section have some similarities with MCHF in that linear

combinations of STOs ﬁrst represent orbitals which then

deﬁne the CSFs. These bases result in extensive cancel-

lation as shown by Hansen et al. [21.19]. An expansion

in B-spline basis functions, B

i,k

(r), which are a basis for

a piecewise polynomial subspace (see Chapt. 8), provide

a more ﬂexible basis with very little cancellation in this

representation.

By solving a radial equation with a well chosen po-

tential using the spline Galerkin method that leads to

a matrix eigenvalue problem, a complete set of orbitals

for a piecewise polynomial space can be obtained. The

resulting orthogonal orbital basis may then be used in

Part B 21.6

318 Part B Atoms

CI calculations. Such methods have been reviewed by

Hansen et al. [21.19]. Methods that deal directly with the

primitive B-spline basis have been applied to the study

of Rydberg series [21.20], though orthogonality to tar-

get orbitals was still required. More recently, by using

non-orthogonal theory, an Rmatrix-CI method has been

developed that leads to a generalized eigenvalue prob-

lem. As in the close-coupling approximation, the wave

function is expressed in terms of one or more “targets,”

each coupled to a Rydberg orbital, and an arbitrary num-

ber of pseudo-states. Each Rydberg orbital is expressed

as a linear combination of B-splines but there is no re-

quirement of orthogonality of the Rydberg orbital to the

orbitals deﬁning the target function [21.21]

21.7 Atomic Properties

The discussion so far has concentrated entirely on deter-

mining accurate wavefunctions based on expressions for

the energy. Energy levels are a by-product of such cal-

culations, but once a wave function is known a number

of atomic properties can be evaluated.

21.7.1 Isotope Effects

The isotope shift observed in atomic transitions can be

separated into a mass shift and a ﬁeld shift. The mass

shift is due to differences in the nuclear mass of the

isotopes, and is the dominant effect for light atoms. The

volume shift arises from the ﬁnite volume of the nuclear

charge distribution, and is important for heavy atoms.

From a physical point of view the ﬁeld shift is the more

interesting, since it yields information about differences

in the nuclear charge distribution between the isotopes.

The isotope shift in a transition is given as the dif-

ference between the shift for the upper and lower level.

The individual shifts are often large, but cancel, and

therefore it is necessary to calculate them very accu-

rately in order to get a reliable value for the difference.

The energyshifts are evaluated in ﬁrst-order perturbation

theory with wave functions obtained from the zero-order

Schrödinger Hamiltonian, H

Mass Shift

Up to this point, the nuclear mass has been taken to

be inﬁnite. If instead it has a ﬁnite value M,anN-

electron atom turns into an (N+1)-particle dynamical

system. A transformation to center of mass R plus rel-

ative coordinates r



= r

−r

nuc

yields the transformed

Hamiltonian [21.22]

H =

tot



i=1

2

2µ



i< j



· p









(21.55)

where M

tot

= M + Nm is the total mass and

µ = Mm/(M +m) is the reduced mass. The ﬁrst term is

the kinetic energy of the center of mass, which can be ne-

glected if Ris an ignorable coordinate. For the remaining

terms, introduce the scaled distances ρ



,where

= (m/µ)a

is the scaled Bohr radius. If the potential

is entirely Coulombic, then H assumes the form

H =−



i=1

∇

−



i< j

∇

·∇

+V(ρ

)

(21.56)

in units of e

. This can be written in the form H =

+ H

,whereH

includes the ﬁrst and last terms

of (21.56), and

=−



i< j

∇

·∇





(21.57)

is the additional mass polarization term.

Equation (21.56) gives rise to two kinds of mass

shifts. First, since H is identical to the inﬁnite mass

Hamiltonian, all its eigenvalues E

are multiplied by

= µ/m, resulting in the normal mass shift (nms)

∆E

nms

=−(µ/M)E

. (21.58)

Second, if H

is treated as a ﬁrst order perturbation,

then the resulting speciﬁc mass shift (sms) is

∆E

sms

=Ψ

| H

| Ψ

 (e

) (21.59)

where Ψ

is an eigenvector of H

. The speciﬁc mass

shift parameter S is deﬁned by

S =



|−



i< j

∇

·∇

| Ψ



(21.60)

Field Shift

The ﬁeld shift of an atomic energy level is due to the

extended nuclear charge distribution. The ﬁeld inside

the nucleus deviates from the Coulomb ﬁeld of a point

Part B 21.7

Atomic Structure: Multiconﬁguration Hartree–Fock Theories 21.7 Atomic Properties 319

charge, and this is reﬂected in the calculated levels. For

light atoms, the resulting energy correction to the level

is expressed in terms of the nonrelativistic electron

probability at the origin, |Ψ(0)|

2π





|Ψ(0)|

, (21.61)

where





is the mean square radius of the nucleus. For

heavier atoms (Z > 10) it becomes necessary to include

a relativistic correction factor. Numerical values of this

correction factor can be found in [21.23].

Table 21.4 shows the convergence of an MCHF cal-

culation for the speciﬁc mass shift parameter and the

electron density at the nucleus of the ground state of

Boron II as the active set increases. The calculated

B−

B isotope shift is 13.3 mÅ with an estimated

uncertainty of 1%. The size of the isotope shift is sim-

ilar to the limit of resolution of the Goddard High

Resolution Spectrograph aboard the Hubble Space Tele-

scope [21.24].

21.7.2 Hyperﬁne Effects

The magnetic hyperﬁne structure (hfs) is due to an in-

teraction between the magnetic ﬁeld generated by the

electrons and the nuclear magnetic dipole moment. The

interaction couples the nuclear and electronic angular

momenta to a total momentum F = I+ J, and the in-

teraction energy can be written as the expectation value

of a scalar product between an electronic and a nuclear

tensor operator [21.25] (see Chapt. 16)

(J ) =



IJFM

| T

(1)

·M

(1)

| γ

IJFM



(21.62)

The nuclear operator M

(1)

is related to the scalar mag-

netic dipole moment, µ

, according to



II | M

(1)

| γ



= µ

. (21.63)

Table 21.4 Speciﬁc mass shift parameter and electron den-

sity at the nucleus as a function of the active set

Active set S(a.u.) |ψ(0)|

HF 0.000 00 72.629

2s 1p −0.020 17 72.452

3s 2p 1d 0.625 18 72.490

4s 3p 2d 1f 0.624 81 72.497

5s 4p 3d 2f 1g 0.601 69 72.501

6s 5p 4d 3f 2g 1h 0.598 03 72.503

7s 6p 5d 4f 3g 2h 1i 0.597 09 72.504

The magnetic dipole moments are known quantities, ob-

tained with high accuracy from experiments. For a recent

tabulation see [21.26].

The electronic operator is

(1)



i=1



(1)

(i)r

−3

−g

√



(2)

(i) ×s

(1)

(i)



(1)

−3

πδ(r

(1)

(i)



, (21.64)

where g

= 2.002 3193 is the electron spin g-fac-

tor, δ(r) the three-dimensional delta function and

(k)

√

4π/(2k +1)Y

, with Y

being a normalized

spherical harmonic. The ﬁrst term of the electronic op-

erator represents the magnetic ﬁeld generated by the

orbiting electric charges and is called the orbital term.

The second term represents the ﬁeld generated by the

orbiting magnetic dipole moments, which are coupled

to the spin of the electrons. This is the spin-dipole term.

The last term represents the contact interaction between

the nuclear magnetic dipole moment and the electron

magnetic moment. It is called the Fermi contact term

and contributes only for s-electrons.

By recoupling I and J, the interaction energy can be

rewritten as

(J ) =(−1)

I+J−F

W(IJIJ; F1)

×γ

JT

(1)

γ

Jγ

IM

(1)

γ

I ,

(21.65)

where W(IJIJ; F1) is a W coefﬁcient of Racah. When

the magnetic dipole interaction constant

[J(J +1)(2J +1)]

γ

JT

(1)

γ

J

(21.66)

is introduced, the energy is given by

(J ) =

C , (21.67)

where C = F(F +1) − J(J +1) − I(I +1). In theoret-

ical studies, the A-factor is often given as a linear

combination of the hyperﬁne parameters

=γLSM



(1)

(i)r

−3

| γLSM

 ,

(21.68)

Part B 21.7

320 Part B Atoms

=γLSM



(2)

(i)s

(1)

(i)r

−3

| γLSM

 , (21.69)

=γLSM



(1)

(i)r

−2

δ(r

) | γLSM

 , (21.70)

where M

= L and M

= S. These parameters corre-

spond to the orbital, spin-dipole and Fermi contact term

of the electronic operator.

The electric hyperﬁne structure is due to the interac-

tion between the electric ﬁeld gradient produced by the

electrons and the nonspherical charge distribution of the

nucleus. The interaction energy is

(J ) =γ

IJFM

| T

(2)

·M

(2)

| γ

IJFM

 , (21.71)

where the nuclear operator M

(2)

is related to the scalar

electric quadrupole moment, Q, according to

γ

II | M

(2)

| γ

II=

(21.72)

The electronic operator is

(2)

=−



i=1

(2)

(i)r

−3

, (21.73)

and represents the electric ﬁeld gradient. By introducing

the electric quadrupole interaction constant B,

=2Q



J(2J −1)

(J +1)(2J +1)(2J +3)



×γ

JT

(2)

γ

J , (21.74)

the interaction energy can be written as

(J ) = B

C(C +1) − I(I +1)J(J +1)

2I(2I −1)J(2J −1)

(21.75)

In many cases the electric hyperﬁne interaction is

weaker than the magnetic and manifests itself as a small

deviation from the Landé interval rule for the magnetic

hfs. If the electronic part of the interaction can be cal-

culated accurately, a value of the electric quadrupole

moment Q, which is a difﬁcult quantity to measure

with direct nuclear techniques, can be deduced from

the measured B-factor. A recent tabulation of nuclear

quadrupole moments is given in [21.27].

Table 21.5 shows the convergence of an MCHF ac-

tive space calculation for two different isotopes for the

2s2p

P state of B II [21.24]. Some oscillations are

Table 21.5 MCHF Hyperﬁne constants (in MHz) for the

2s2p

P state of B II

Active set

HF 60.06 8.338 179.36 4.001

2s 1p 60.22 8.360 179.83 4.011

3s 2p 1d 60.98 8.193 182.11 3.932

4s 3p 2d 1f 60.05 7.677 179.34 3.684

5s 4p 3d 2f 1g 60.48 7.764 180.62 3.725

6s 5p 4d 3f 2g 1h 60.85 8.052 181.71 3.864

7s 6p 5d 4f 3g 2h 1i 60.81 8.002 181.60 3.840

observed since each new “layer” of orbitals may localize

in different regions of space.

21.7.3 Metastable States and Lifetimes

States above an ionization threshold may decay via

a radiationless transition to a continuum. When the inter-

action with the continuum is spin-forbidden the state is

metastable. The nsnp

P of negative ions, for example,

decay through Breit–Pauli interactions with nsnp

dou-

blets. The latter, in turn interact with continuum states,

thus opening a decay channel. Such metastable states

may be treated as bound states.

The foundation for the theory of autoionization was

laid down by Fano [21.28], where he developed a con-

ﬁguration interaction (CI) theory for autoionization. Let

(N;γLS) be a normalized, multiconﬁguration com-

ponent of a discrete perturbor for an N-electron system,

in which all orbitals are bound orbitals, decaying ex-

ponentially for large r.LetΨ

be an asymptotically

normalized continuum component of the wave func-

tion, also for an N-electron system, at energy E,of

the form,

(N; Eγ



LS) =|Ψ



N −1;β



· φ(kl)LS ,

(21.76)

where Ψ



N −1;β



is a bound MCHF wave function

for the (N −1)-electron target system. Then, in the LS-

coupling scheme, the width of the autoionizing state is

given by the “Golden Rule”

Γ = 2πV

, (21.77)

where

=Ψ

(N;γLS)|H −E

|Ψ

(N; E



LS) .

(21.78)

A similar formula can be derived for the LSJ-scheme

and the Breit–Pauli Hamiltonian. In the above equation,

Part B 21.7

Atomic Structure: Multiconﬁguration Hartree–Fock Theories 21.7 Atomic Properties 321

=Ψ

(N;γLS)|H|Ψ

(N;γLS). The energy of the

core, or target, is E

target





N −1;β



|H|Ψ



N −

1;β



,whereH in the latter equation is the Hamilto-

nian for an (N −1)-electron system.

The wave function for the continuum compo-

nent is assumed to have only one unknown, namely

φ(kl) = P

(r)|ls, the one-particle continuum function,

where |ls is the known spin–angular part. The radial

equation for P

(r) has exactly the same form as (21.26),

except that ε

nl,nl

=−k

,whereE

= E

target

/2.

The radial function can be obtained iteratively using an

SCF procedure from outward integration.

One of the more accurate calculations of a lifetime

is that for He

−

1s2s2p

5/2

by Miecznik et al. [21.29].

In this case, a lifetime of (345±10) µs was found

and compared with a recent experimental value of

(350±15) µs [21.30]. In this LSJ state, the

Pinter-

acts with the 1s

k f

F. It was found that correlation in

the target of the continuum orbital modiﬁed the life-

time. In calculations like these, it is always a question

whether orthogonality conditions should be applied [as

in projection operator formalism (see Chapt. 25)]. Some

theorems relating to this question have been published

by Brage et al. [21.31].

The position and widths of autoionizing resonances

can also be determined from the study of photoioniza-

tion or photodetachment using a spline Galerkin method

together with inverse iteration. No boundary condition

need be applied nor is there an inner and outer re-

gion. Resonance properties are obtained from a ﬁtting of

the cross-section [21.32]. Non-orthogonal, spline-based

R-matrix methods with an inner and outer region have

also been developed where there is no need of the “Buttle

correction” and, at the same time, the non-orthogonality

eliminates the need of certain pseudo-states [21.33].

For an extensive review of the application of splines

in atomic and molecular physics, see [21.34].

21.7.4 Transition Probabilities

The most fundamental quantity for the probability of

a transition from an initial state i to a ﬁnal state f is the

reduced matrix element related to the line strength by

1/2

=Ψ

||O||Ψ

 , (21.79)

where O is the transition operator. In the case of the

electric dipole transition, there are two frequently used

forms: the length form O =



, and the velocity

form, O =



∇

,whereE

= E

−E

.Forex-

act non-relativistic wave functions, the two forms are

equivalent, but for approximate wave functions, the ma-

trix elements in general differ. Thus, the computation of

the line strength and the oscillator strength, or f -value,

where

f = (2/3)E

S/[(2S

+1)(2L

+1)] ,

forms a critical test of the wave function in non-

relativistic theory and also the model describing

a many-electron system. The same operators are often

also used in Breit–Pauli calculations. In this case, the

velocity form of the operator has neglected some terms

of order (αZ)

and the length value is preferred.

Some well-known discrepancies between theory and

experiment existed for more than a decade for the res-

onance transitions of Li and Na. For the nonrelativistic

2s–2p transition in Li, a full-core CI [21.35] calcula-

tion produced f -values of (0.747 04, 0.747 04, 0.753 78)

for the length-, velocity-, and acceleration form, respec-

tively. When relativistic corrections were included, the

value changed to 0.747 15, in agreement with a num-

ber of theories, tabulated to only four decimal places.

A fast beam-laser experiment by Gaupp et al. [21.36],

yielded a value of 0.7416±0.0012 but this value was

revised in 1996 by a beam-gas-laser experiment in per-

fect agreement with theory [21.37]. In the case of the

resonance transition in Na, when theory included cor-

relation in the core as well as core-polarization and

some relativistic effects, results were in agreement with

an almost simultaneous cascade of new experimental

values [21.38].

For MCHF calculations of transition data, an impor-

tant consideration is that the matrix element is between

two different states. For independently optimized wave

functions, the orbitals of the initial and ﬁnal states are

not orthonormal, as assumed when Racah algebra tech-

niques are used to evaluate the transition matrix element.

Through the use of biorthogonal transformations, the

orbitals and the coefﬁcients of expansion of the wave

functions of the initial and ﬁnal state can be transformed

efﬁciently so that standard Racah algebra techniques

may be applied [21.39]. Table 21.6 shows the conver-

gence of an MCHF calculation for the ground state of

Boron from independently optimized wave functions.

21.7.5 Electron Afﬁnities

By deﬁnition, the electron afﬁnity is A

= E(A

−

) −

E(A). Thus it is the energy difference between Hamilto-

nians differing by one electron. Correlation plays a very

important role in the binding of the extra electron in

the negative ion. It has been known for a long time that

the alkali metals have a positive electron afﬁnity, but

Part B 21.7

322 Part B Atoms

only recently has it been found, theoretically [21.40]

and experimentally [21.41], that some of the alkaline

earths may also be able to bind an extra electron. The

d-electrons need to play a strong role, so Be and Mg,

do not have a positive A

, but according to the most

recent experimental measurement, the electron afﬁn-

ity for Ca is 18 meV [21.42]. A calculation based on

the spline methods and using a model potential with

adjustable parameters to describe the core, obtained

a value of 17.7 meV [21.43] in close agreement with

experiment.

Atomic systems such as Ca are often thought of as

two-electron systems and indeed, for qualitative descrip-

tions, many observations can be explained. A number of

physical effects need to be considered when predicting

such electron afﬁnities:

(i) Valence correlation is crucial for obtaining binding.

(ii) Intershell correlation between the valence electrons

and the core (core polarization) is also important. The

ﬁrst electron may polarize the core considerably, but

this is reduced by the second electron since the two

avoid each other dynamically and prefer to be on oppo-

site sides of the core.

(iii) Core rearrangement, which occurs because of the

presence of one or more outer electrons, and is particu-

larly large if any of these penetrate the core. In the case of

, the ﬁxed-core Hartree–Fock energy of 3d

D state

is 300 meV higher if computed in the ﬁxed potential the

ion compared with a fully variational calculation!

(iv) Intracore exclusion effects due to the presence of an

Table 21.6 Convergence of transition data for the

→ 1s

2s2p

D transition in Boron with in-

creasing active set

N gf

∆E (cm

−1

)

3 0.6876 0.8156 2.5534 53 197

4 0.2456 0.2696 0.9959 48 720

5 0.2625 0.2695 1.0705 48 440

6 0.2891 0.2866 1.1868 48 125

7 0.2928 0.2900 1.2036 48 051

Expt.

0.28(02) 47 857

[21.44]

extra valence electron which reduces the correlation of

the core.

(v) Relativistic shift effects, which are present in

observed levels and are particularly important for

s-electrons.

Model potential methods attempt to capture all but

valence correlation in a potential so that calculations

for Calcium, for example, can proceed as though for

a two-electron system. A review of various theoretical

approaches, many of which include different effects,

may be found in [21.45].

For small systems such as Li, the electron afﬁnity has

been computed [21.46] to experimental accuracy [21.47]

of (0.6176 ±0.0002) eV. In neutral oxygen, it has been

found that there is an isotope effect on the electron

afﬁnity [21.48].

21.8 Summary

More comprehensive treatments of atomic structure

may be found [21.49, 50]. An atomic structure pack-

age is available for many of the calculations described

here [21.51]. A review of the application of systematic

procedures to the prediction of atomic properties has

been published [21.52,53].

References

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(1978)

21.2 I. Lindgren, J. Morrison: Atomic Many-Body Theory

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21.3 P.-O. Löwdin: Phys. Rev. 97, 1509 (1955)

21.4 D. Layzer, Z. Horák, M. N. Lewis, D. P. Thompson:

Ann. Phys. (N. Y.) 29,101(1964)

21.5 D. Layzer: Ann. Phys. (N. Y.) 8, 271 (1959)

21.6 C. A. Nicolaides, D. R. Beck: Chem. Phys. Lett. 35,79

(1975)

21.7 C. Froese Fischer: Comput. Phys. Rep. 3,273(1986)

21.8 C. Froese Fischer: J. Comput. Phys. 13, 502 (1973)

21.9 M. R. Godefroid, J. Lievin, J.-Y. Metz: Int. J. Quan-

tum Chem. XL, 243 (1991)

21.10 G. Tachiev, C. Froese Fischer: J. Phys. B 32,5805

(1999)

21.11 S. A. Alexander, J. Olsen, P. Öster, H. M. Quiney,

S. Salomonson, D. Sundholm: Phys. Rev. A 43, 3355

(1991)

21.12 C. Froese Fischer: J. Phys. B 26, 855 (1993)

21.13 K. T. Chung, X.-W. Zhu, Z.-W. Wang: Phys. Rev. A

47, 1740 (1993)

Part B 21

Atomic Structure: Multiconﬁguration Hartree–Fock Theories References 323

21.14 E. Lindroth, H. Persson, S. Salomonson,

A.-M. Mårtensson-Pendrill: Phys. Rev. A 45,1493

(1992)

21.15 R. L. Kelly: J. Phys. Chem. Ref. Data 16, 1371–1678

(1987), Suppl. 1

21.16 K. T. Chung: Many-Body Theory of Atomic Struc-

ture and Photoionization, ed. by T. N. Chang (World

Scientiﬁc, Singapore 1993) p. 83

21.17 E. Clementi, C. Roetti: At. Data Nucl. Data Tables 14,

177 (1974)

21.18 A. Hibbert: Comput. Phys. Commun. 9,141(1975)

21.19 M. Bentley, H. W. van der Hart, M. Landtman,

G. M. S. Lister, Y.-T. Shen, N. Vaeck: Phys. Scr. T47,

7 (1993)

21.20 T. Brage, C. Froese Fischer: Phys. Scr. 49, 651 (1994)

21.21 O. Zatsarinny, C. Froese Fischer: J. Phys. B 35, 4669

(2002)

21.22 D. S. Hughes, C. Eckart: Phys. Rev. 36, 694 (1930)

21.23 W. H. King: Isotope Shifts in Atomic Spectra

(Plenum, New York 1984)

21.24 P. Jönsson, S. G. Johansson, C. Froese Fischer:

Astrophys. J. 429, L45 (1994)

21.25 I. Lindgren, A. Rosén: Case Stud. At. Phys. 1772 4,

93 (1974)

21.26 P. Raghavan: At. Data Nucl. Data Tables 42,189

(1989)

21.27 P. Pyykkö: Z. Naturforsch. 47a, 189 (1992)

21.28 U. Fano: Phys. Rev. 129, 1866 (1961)

21.29 G. Miecznik, T. Brage, C. Froese Fischer: Phys. Rev.

A 47, 3718 (1993)

21.30 T. Andersen, L. H. Andersen, P. Balling, H. K. Hau-

gen, P. Hvelplund, W. W. Smith, K. Taulbjerg: Phys.

Rev. A 47, 890 (1993)

21.31 T. Brage, C. Froese Fischer, N. Vaeck: J. Phys. B 26,

621 (1993)

21.32 J. Xi, C. Froese Fischer: Phys. Rev. A 53, 3169

(1996)

21.33 O. Zatsarinny, C. Froese Fischer: J. Phys. B 35, 4161

(2002)

21.34 H. Bachau, E. Cormier, J. E. Hansen, F. Martin: Rep.

Prog. Phys. 64, 1815 (2001)

21.35 K. T. Chung: Proceedings of the international con-

ference on highly charged ions, Manhattan Kansas,

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21.36 A. Gaupp, P. Kuske, H. J. Andrä: Phys. Rev. A 26,

3351 (1982)

21.37 U. Volz, H. Schmoranzer: Phys. Scripta T65,48

(1996)

21.38 P. Jönsson, A. Ynnerman, C. Froese Fischer,

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Rev. Lett. 59, 2263 (1987)

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21.49 I. I. Sobel’man: Introduction to the Theory of

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Bristol 1997)

Part B 21

325

Relativistic Ato

22. Relativistic Atomic Structure

Relativistic quantum mechanics is required for

the description of atoms and molecules whenever

their orbital electrons probe regions of space with

high potential energy near the atomic nuclei.

Primary effects of a relativistic description include

changes to spatial and momentum distributions;

spin–orbit interactions; quantum electrodynamic

corrections such as the Lamb shift; and vacuum

polarization. Secondary effects in many-electron

systems arise from shielding of the outer electrons

by the distributions of electrons in penetrating

orbitals; they change orbital binding energies and

dimensions and so modify the order in which

atomic shells are ﬁlled in the lower rows of the

Periodic Table.

Relativistic atomic and molecular structure

theory can be regarded as a simpliﬁcation of the

fundamental description provided by quantum

electrodynamics (QED). This treats the atom

or molecule as an assembly of electrons and

atomic nuclei interacting primarily by exchanging

photons. This model is far too difﬁcult and

general for most purposes, and simpliﬁcations

are required. The most important of these is the

representation of the nuclei as classical charge

distributions, or even as point particles. Since the

motion of the nuclei is usually slow relative to the

electrons, it is often adequate to treat the nuclear

motion nonrelativistically, or even to start from

nuclei in ﬁxed positions, correcting subsequently

for nuclear motion.

The emphasis in this chapter is on relativistic

methods for the calculation of atomic structure

for general many-electron atoms based on

an effective Hamiltonian derived from QED in

the manner sketched in Sect. 22.2 below. An

understanding of the Dirac equation, its solutions

and their numerical approximation, is essential

material for studying many-electron systems,

just as the corresponding properties of the

Schrödinger equation underpin Chapt. 21.We

shall use atomic units throughout. Where it aids

interpretation we shall, however, insert factors

of c, m

and . In these units, the velocity of

light, c, has the numerical value

−1

= 137.035 999 11(46),whereα is the ﬁne

structure constant.

22.1 Mathematical Preliminaries.................. 326

22.1.1 Relativistic Notation:

Minkowski Space-Time .............. 326

22.1.2 Lorentz Transformations............. 326

22.1.3 Classiﬁcation of Lorentz

Transformations........................ 326

22.1.4 Contravariant and Covariant

Vectors..................................... 327

22.1.5 Poincaré Transformations........... 327

22.2 Dirac’s Equation .................................. 328

22.2.1 Characterization of Dirac States... 328

22.2.2 The Charge-Current 4-Vector ...... 328

22.3 QED: Relativistic Atomic

and Molecular Structure ....................... 329

22.3.1 The QED Equations of Motion ...... 329

22.3.2 The Quantized Electron–Positron

Field........................................ 329

22.3.3 Quantized Electromagnetic Field . 330

22.3.4 QED Perturbation Theory ............ 331

22.3.5 Propagators.............................. 333

22.3.6 Effective Interaction of Electrons . 333

22.4 Many-Body Theory For Atoms ............... 334

22.4.1 Effective Hamiltonians ............... 335

22.4.2 Nonrelativistic Limit:

Breit–Pauli Hamiltonian ............ 335

22.4.3 Perturbation Theory:

Nondegenerate Case.................. 335

22.4.4 Perturbation Theory:

Open-Shell Case........................ 336

22.4.5 Perturbation Theory: Algorithms . 337

22.5 Spherical Symmetry ............................. 337

22.5.1 Eigenstates of Angular

Momentum .............................. 337

22.5.2 Eigenstates of Dirac Hamiltonian

in Spherical Coordinates ............ 338

22.5.3 Radial Amplitudes ..................... 340

22.5.4 Square Integrable Solutions........ 341

22.5.5 Hydrogenic Solutions ................. 342

22.5.6 The Free Electron Problem

in Spherical Coordinates ............ 343

Part B 22

326 Part B Atoms

22.6 Numerical Approximation

of Central Field Dirac Equations ............ 344

22.6.1 Finite Differences ...................... 344

22.6.2 Expansion Methods ................... 345

22.6.3 Catalogue of Basis Sets

for Atomic Calculations .............. 347

22.7 Many-Body Calculations ....................... 350

22.7.1 Atomic States............................ 350

22.7.2 Slater Determinants................... 350

22.7.3 Conﬁgurational States................ 350

22.7.4 CSF Expansion ........................... 350

22.7.5 Matrix Element Construction....... 350

22.7.6 Dirac–Hartree–Fock

and Other Theories .................... 351

22.7.7 Radiative Corrections ................. 353

22.7.8 Radiative Processes ................... 353

22.8 Recent Developments........................... 354

22.8.1 Technical Advances ................... 354

22.8.2 Software for Relativistic Atomic

Structure and Properties ............ 354

References .................................................. 355

22.1 Mathematical Preliminaries

22.1.1 Relativistic Notation:

Minkowski Space-Time

An event in Minkowski space-time is deﬁned by

a 4-vector x ={x

} (µ = 0, 1, 2, 3) where x

= ct is

the time coordinate and x

, x

are Cartesian coordi-

nates in 3-space. The bilinear form (The Einstein sufﬁx

convention, in which repeated pairs of Greek subscripts

are assumed to be summed over all values 0, 1, 2, 3, will

be used where necessary in this chapter.)

(x, y) = x

µν

, (22.1)

in which

g =



µν





µν









1000

0 −100

00−10

000−1







(22.2)

are called metric coefﬁcients, deﬁnes the metric of

Minkowski space.

22.1.2 Lorentz Transformations

Lorentz transformations are deﬁned as linear map-

pings Λ such that

(Λx,Λy) = (x, y)

(22.3)

so that

µν

= Λ

ρσ

. (22.4)

This furnishes 10 equations connecting the 16 compo-

nents of Λ; at most 6 components can be regarded as

independent parameters. The (inﬁnite) set of Λ matri-

ces forms a regular matrix group (with respect to matrix

multiplication) called the Lorentz group, L, designated

O(3,1) [22.1, 2].

22.1.3 Classiﬁcation of Lorentz

Transformations

Rotations

Lorentz transformations with matrices of the form

Λ =



1 0



0 R



(22.5)

where R ∈ SO(3) is an orthogonal 3 × 3 matrix with de-

terminant +1, and 0 is a null three dimensional column

vector, correspond to three-dimensional space rotations.

They form a group isomorphic to SO(3).

Boosts

Lorentz transformations with matrices of the form

Λ =



γ(v) γ(v)v



γ(v)v I

+(γ(v) −1)nn





(22.6)

with v = vn a three dimensional column vector, |n|=1,

v =|v| and γ(v) = (1 −v

)

−1/2

, are called boosts.

The matrix Λ describes an ‘active’ transformation from

an inertial frame in which a free classical particle is at

rest to another inertial frame in which its velocity is v.

Boosts form a submanifold of L though they do not

in general form a subgroup. However, the set of boosts

in a ﬁxed direction n form a one-parameter subgroup.

Discrete Transformations

The matrices

P =



1 0



0 −I



, T =



−1 0



0 I



with PT =−I

(22.7)

are called discrete Lorentz transformations and form

a subgroup of the Lorentz group along with the iden-

Part B 22.1