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

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306 Part B Atoms

Vol. 21, ed. by S. B. Trickey (Academic, New York

1990) pp. 69–95

20.133 M. Levy: In: Density Functional Methods in Chem-

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20.134 A. Görling, M. Levy: Phys. Rev. A 45, 1509 (1992)

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Part B 20

307

Atomic Structu

21. Atomic Structure:

Multiconﬁguration Hartree–Fock Theories

This chapter outlines variational methods for the

determination of wave functions either in non-

relativistic LS or relativistic LSJ theory. The emphasis

is on Hartree–Fock and multiconﬁguration

Hartree–Fock theory though conﬁguration

interaction methods are also mentioned. Some

results from the application of these methods to

a number of atomic properties are presented.

21.1 Hamiltonians:

Schrödinger and Breit–Pauli................. 307

21.2 Wave Functions: LS and LSJ Coupling ..... 308

21.3 Variational Principle ............................ 309

21.4 Hartree–Fock Theory ............................ 309

21.4.1 Diagonal Energy Parameters

and Koopmans’ Theorem ........... 311

21.4.2 The Fixed-Core Hartree–Fock

Approximation.......................... 311

21.4.3 Brillouin’s Theorem ................... 311

21.4.4 Properties of Hartree–Fock

Functions ................................. 312

21.5 Multiconﬁguration

Hartree–Fock Theory ............................ 313

21.5.1 Z-Dependent Theory ................. 313

21.5.2 The MCHF Approximation............ 314

21.5.3 Systematic Methods................... 315

21.5.4 Excited States ........................... 316

21.5.5 Autoionizing States ................... 316

21.6 Conﬁguration Interaction Methods........ 316

21.7 Atomic Properties ................................ 318

21.7.1 Isotope Effects .......................... 318

21.7.2 Hyperﬁne Effects....................... 319

21.7.3 Metastable States

and Lifetimes ........................... 320

21.7.4 Transition Probabilities .............. 321

21.7.5 Electron Afﬁnities...................... 321

21.8 Summary ............................................ 322

References .................................................. 322

21.1 Hamiltonians: Schrödinger and Breit–Pauli

The state of a many-electron system is described by

a wave function Ψ that is the solution of a partial

differential equation (called the wave equation),

(H −E)Ψ = 0 ,

(21.1)

where H is the Hamiltonian operator for the system

and E the total energy. The operator H depends on the

system (atomic, molecular, solid-state, etc.) as well as

the underlying quantum mechanical formalism (nonrela-

tivistic, Breit–Pauli, Dirac–Coulomb, or Dirac–Breit,

etc.). In atomic systems, the Hamiltonian of the non-

relativistic Schrödinger equation is (in atomic units)

=−



i=1



∇





i< j

. (21.2)

Here Z is the nuclear charge of the atom with N elec-

trons, r

is the distance of electron i from the nucleus,

and r

is the distance between electron i and elec-

tron j. This equation was derived under the assumption

of a point-nucleus of inﬁnite mass. The term 2Z/r rep-

resents the nuclear attraction and 1/r

the inter-electron

repulsion. The operator H

has both a discrete and con-

tinuous spectrum: for the former, Ψ(r

, r

,... ,r

) has

a probability interpretation and consequently must be

square integrable. In the Breit–Pauli approximation, the

Hamiltonian is extended to include relativistic correc-

tions up to relative order (αZ)

. It is convenient to write

the Breit–Pauli Hamiltonian as the sum [21.1]

= H

+ H

rel

, (21.3)

where H

rel

represents the relativistic contributions. The

latter may again be subdivided into nonﬁne-structure

(NFS) and ﬁne structure (FS) contributions:

rel

= H

NFS

+ H

. (21.4)

Part B 21

308 Part B Atoms

The NFS contributions

NFS

= H

mass

+ H

ssc

+ H

(21.5)

shift nonrelativistic energy levels without splitting the

levels. The mass-velocity term

mass

=−



∇

(21.6)

corrects for the variation of mass with velocity; the one-

and two-body Darwin terms

=−



∇

−1



i< j

∇

−1

(21.7)

are the corrections of the one-electron Dirac equa-

tion due to the retardation of the electromagnetic

ﬁeld produced by an electron; the spin–spin contact

term

ssc

=−

8πα



i< j

·s

)δ(r

) (21.8)

accounts for the interaction of the spin magnetic mo-

ments of two electrons occupying the same space; the

orbit–orbit interaction

=−



i< j



· p

) · p



(21.9)

accounts for the interaction of two orbital moments.

The FS contributions

= H

+ H

soo

+ H

, (21.10)

split the nonrelativistic energy levels into a series of

closely-spaced ﬁne structure levels. The nuclear spin–

orbit interaction



·s

), (21.11)

represents the interaction of the spin and angular mag-

netic moments of an electron in the ﬁeld of the nucleus.

The spin–other-orbit term

soo

=−



i=j



× p



·(s

+2s

), (21.12)

and the spin–spin term

= α



i< j



·s

−

·r

)(s

·r

)



(21.13)

arise from spin-dependent interactions with the other

electrons in the system.

21.2 Wave Functions: LS and LSJ Coupling

In the conﬁguration interaction model, the approximate

wave function Ψ for a many-electron system is expanded

in terms of conﬁguration state functions (CSF).

The assignment of nl quantum numbers to elec-

trons speciﬁes a conﬁguration, often written as

)

···(n

)

,whereq

is the occupa-

tion of subshell (n

). Associated with each subshell

are one-electron spin-orbitals

φ(r,θ,ϕ,σ)= (1/r)P

(r)Y

(θ, ϕ)χ

(σ) ,

where P

(r) is the radial function, Y

(θ, ϕ) a spherical

harmonic, and χ

(σ) a spinor. Each CSF is a linear

combination of products of one-electron spin-orbitals,

one for each electron in the system, such that the sum

is an eigenfunction of the total angular momenta op-

erators L

, L

and the total spin operators S

, S

.It

can be considered to be a product of radial factors, one

for each electron, an angular and a spin factor obtained

by vector coupling methods. It also is required to be

antisymmetric with respect to the interchange of any

pair of electron co-ordinates. Often, the speciﬁcation of

the conﬁguration and the ﬁnal LS quantum numbers is

sufﬁcient to deﬁne the conﬁguration state, but this is not

always the case. Additional information about the order

of coupling or the seniority of a subshell of equivalent

electrons may be needed. Let γ specify the conﬁguration

information and any additional information about coup-

ling to uniquely specify the conﬁguration state function

denoted by Φ(γLS).

The wave function for a many-electron system is

usually labeled in the same manner as a CSF and gener-

ally designates the largest component. Thus, in the LS

approximation,

Ψ(γLS) =



α=1

Φ(γ

LS).

Part B 21.2

Atomic Structure: Multiconﬁguration Hartree–Fock Theories 21.4 Hartree–Fock Theory 309

However, cases are known where the conﬁguration states

are so highly mixed that no dominant component can be

found. Then the assignment is made using other criteria.

Clearly no two states should have the same label.

In the LSJ scheme, the angular and spin momenta

are coupled to form an eigenstate of the total mo-

menta J

, J

. The label often still includes an LS

designation, as in 2p

, but only the subscript J is

a good quantum number. Thus,

Ψ(γLSJ ) =



α=1

Φ(γ

J ). (21.14)

21.3 Variational Principle

Variational theory shows the equivalence between so-

lutions of the wave equation, (H − E)ψ = 0, and

stationary solutions of a functional. For bound states

where approximate solutions Ψ are restricted to a square

integrable subspace, say

H , the best solutions are those

for which the energy functional

E (Ψ) =Ψ |H|Ψ /Ψ |Ψ 

(21.15)

is stationary. The condition δE (Ψ ) = 0 leads to

δΨ |H −E|Ψ =0, ∀ δΨ ∈

H , E = E (Ψ) .

(21.16)

Several results readily follow. The eigenvalues of H

are bounded from below. Let E

≤ E

≤···.Then

≤ E (Ψ), ∀ Ψ ∈

H . (21.17)

Consequently, for any approximate wave function,

the computed energy is an upper bound to the

exact lowest eigenvalue. By the Hylleraas–Undheim–

MacDonald theorem (see Sect. 11.3.1) the computed

excited states are also upper bounds to the exact eigen-

values, provided that the correct number of states lies

below.

21.4 Hartree–Fock Theory

In the Hartree–Fock (HF) approximation, the approxi-

mate wave function consists of only one conﬁguration

state function. The radial function of each spin–orbital

isassumedtodependonlyonthenl quantum numbers.

These are determined using the variational principle and

the nonrelativistic Schrödinger Hamiltonian.

The energy functional can be written as an energy ex-

pression for the matrix element Φ(γLS)|H|Φ(γLS).

Racah algebra may be used to evaluate the spin–

angular contributions, resulting in two types of radial

integrals:

One-Body

Let L be the differential operator

L =

−

( +1)

. (21.18)

Then,

I(nl, n



) =−

∞



P(nl;r)

×LP(n



;r) dr . (21.19)

Two-Body

The other integrals arise from the multipole expansion

of the two-electron part



k+1

(cos θ) , (21.20)

where θ is the angle between the vectors r

and r

,and

, r

are the lesser and greater of r

, r

, respectively.

In general, let a, b, c, d be four nl quantum numbers,

two from the left (bra) and two from the right (ket) CSF.

Then

(ab, cd) =

∞



∞



P(a;r

)P(b;r

)

k+1

P(c;r

)P(d;r

) dr

(21.21)

which is called a Slater integral. It has the symmetries

(ad, cb) ≡ R

(cb, ad ) ≡ R

(cd, ab)

≡ R

(ab, cd ).

Part B 21.4

310 Part B Atoms

In the Hartree–Fock approximation, the Slater integrals

that occur depend on only two sets of quantum numbers.

These special cases are denoted separately as

(a, b) ≡ R

(ab, ab) and

(a, b) ≡ R

(ab, ba). (21.22)

The former is the direct interaction between a pair of

orbitals whereas the latter arises from the exchange

operator.

The energy expression may be written as

E (γLS) =





I(n

, n

)

−1



k=0

(i, i)F

, n

)





j<i



2min(l

)



k=0

(i, i)F

, n

)



k=|l

−l

(i, j )G

, n

)



(21.23)

In general, the coefﬁcients f

(i, j ) and g

(i, j ) depend

not only on the conﬁguration, but also on the coupling.

An extremely useful concept, introduced by Slater, is

the “average energy of a conﬁguration”, E (av).Thisis

a weighted average of all possible LS terms, where the

weighting factor is (2L +1)(2S +1). In this case the

coefﬁcients have simple formulas that depend on the

conﬁguration:

(i, i) = 1, k =0 ,

=−







000



, k > 0

(i, j ) = 1, i = j and k =0 ,

= 0, i = j and k > 0 ,

(i, j ) =−







000



. (21.24)

Let ( P

) be an integral that depends on P

.Thenδ

is deﬁned as the ﬁrst-order term of (P

+δP

) −(P

To derive the ﬁrst-order variation of the F

and G

integrals (and R

in general) it is convenient to re-

place the variables (r

, r

) by (r, s) and introduce the

function

(ab;r) =r

∞



k+1

P(a;s)P(b;s) ds







P(a;s)P(b;s) ds

∞







k+1

P(a;s)P(b;s) ds (21.25)

so that

(a, b) =

∞



(a;r)





(bb;r) dr (21.26)

and

(a, b) =

∞



P(a;r)P(b;r)





(ab;r) dr .

(21.27)

Then the variations are

δI(a, b) =−

(1+δ

a,b

)

∞



δP(b;r)L P(b;r) dr ,

δF

(a, b) = 2(1 +δ

a,b

)

∞



δP(a;r)P(a;r)





(bb;r) dr ,

δG

(a, b) = 2

∞



δP(a;r)P(b;r)





(ab;r) dr .

(21.28)

The part of the expression that depends on P(n

;r),for

example, is the negative of the removal energy of the

entire (n

)

subshell, say −

E [(n

)

]. The stationary

condition for a Hartree–Fock solution applies to this

expression, but since the variations must be constrained

in order to satisfy orthonormality assumptions, Lagrange

multipliers λ

need to be introduced. The stationary

condition applies to the functional

F [P(n

)]=−



)





P(n

)|P(n

) .

(21.29)

Part B 21.4

Atomic Structure: Multiconﬁguration Hartree–Fock Theories 21.4 Hartree–Fock Theory 311

Applying the variational conditions to each of the inte-

grals, and dividing by −q

, we get the equation



[Z −Y(n

;r)]−

l(l +1)

−ε



× P(n

;r)

X(n

;r) +



j=i

P(n

;r), (21.30)

where

Y(n

;r) = (q

−1)



(i, i)Y

;r)



j=i

;r),

X(n

;r) =



j=i



(i, j )Y

;r)

× P(n

;r). (21.31)

21.4.1 Diagonal Energy Parameters

and Koopmans’ Theorem

The diagonal (ε

) and off-diagonal (ε

)energypa-

rameters are related to the Lagrange multipliers by

= 2λ

and ε

= λ

. In fact,



)



−(q

−1)



(i, i)F

, n

) (21.32)

where

E[(n

)

] is the Hartree–Fock value of the re-

moval energy functional

E [(n

)

]. In the special case

where q

= 1, ε

is twice the removal energy, or ion-

ization energy. This is often referred to as Koopmans’

Theorem; but, as discussed in Sect. 21.4.3, if a rotation

of the radial basis leaves the wave function unchanged

while transforming the matrix of energy parameters (ε

the removal energies are extreme values obtained by

setting the off-diagonal energy parameters to zero. For

multiply occupied shells, ε

/2 can be interpreted as an

average removal energy, with a correction arising from

the self-interaction.

21.4.2 The Fixed-Core Hartree–Fock

Approximation

The above derivation has assumed that the solution is

stationary with respect to all allowed variations. In prac-

tice, it may be convenient to assume that certain radial

functions are “ﬁxed” or “frozen”. In other words, these

radial functions are assumed to be given. Such approxi-

mations are often made for core orbitals and so, this is

called a ﬁxed-core HF approximation.

21.4.3 Brillouin’s Theorem

The Hartree–Fock approximation has some special

properties not possessed by other single conﬁguration

approximations. One such property is referred to as

satisfying Brillouin’s theorem, though, in complex sys-

tems with multiple open shells of the same symmetry,

Brillouin’s theorem is not always obeyed.

Let Φ

(γ LS) be a Hartree–Fock conﬁguration

state, where γ denotes the conﬁguration and coupling

scheme. With Φ

(γ LS) are associated the m Hartree–

Fock radial functions P

;r), P

;r),

... ,P

;r). These radial functions deﬁne the oc-

cupied orbitals. To this set may be added virtual orbitals

that maintain the necessary orthonormality conditions.

Let one of the radial functions (nl) be replaced by

another (n



l), either occupied or virtual, without any

change in the coupling of the spin–angular factor. Let

the resulting function be denoted by F(nl →n



l).

The perturbation of the Hartree–Fock radial func-

tion, P(nl ;r) → P

(nl;r) +P(n



l;r) induces a per-

turbation Φ

(γ LS) → Φ

(γ LS) +F(nl → n



l).

But the Hartree–Fock energy is stationary with respect

to such variations and so,

Φ

(γ LS) | H | F(nl → n



l)=0 . (21.33)

If the function F(nl → n



l) is a CSF for a conﬁgur-

ation γ

∗

, or proportional to one, then Brillouin’s theorem

is said to hold between the two conﬁguration states.

When n



l is a virtual orbital, it may happen that F(nl →



l) is a linear combination of CSFs, as in the 2p → 3p

replacement from 2p

P, yielding a linear combination

of {2p

(

S)3p, 2p

(

P)3p, 2p(

D)3p}, the linear com-

bination being determined by coefﬁcients of fractional

parentage. Thus, Brillouin’s theorem will not hold for

any of the three individual conﬁguration states in the

above equation, only for the linear combination.

When perturbations are constrained by orthogonality

conditions between occupied orbitals, the perturbation

is of the form of a rotation, where both are perturbed

simultaneously,

P(nl;r) → P(nl ;r) +P(n



l;r),

P(n



l;r) → P(n



l;r) −P(nl ;r). (21.34)

Then the perturbation has the form F(nl → n



l, n



l →

−nl). For 1s

S, the simultaneous perturbations,

Part B 21.4

312 Part B Atoms

F(1s → 2s, 2s →−1s), lead to a linear combination of

{1s2s

S, 1s

S}.TheCSF for the 1s

S is identically

zero by antisymmetry, and so Brillouin’s theorem holds

for the lithium-like ground state. In the 1s2s

S state,

neither the 1s → 2s nor the 2s → 1s substitutions are

allowed; in fact, it can be shown that for these states,

Brillouin’s theorem holds for all mono-excited conﬁg-

urations. The same is not true for 1s2s

Swherethe

simultaneous perturbations lead to the condition



(1s2s

S)|H|







−Φ





= 0 .

(21.35)

Thus, Brillouin’s theorem is not obeyed for either the





or Φ





CSF in an HF calculation for 1s2s.

The importance of Brillouin’s theorem lies in the fact

that certain interactions have already been included to

ﬁrst order. This has the consequence that certain classes

of diagrams can be omitted in many-body perturbation

theory [21.2].

21.4.4 Properties of Hartree–Fock Functions

Term Dependence

The radial distribution for a given nl orbital may depend

signiﬁcantly on the LS term. A well known exam-

ple is the 1s

2s2p conﬁguration in Be which may

couple to form either a

Por

Pterm.Theenergy

expression differs only in the exchange interaction,

±(1/3)G

(2s, 2p),wherethe+ refers to

Pandthe

− to

P. Clearly, the energies of these two terms dif-

fer. What is not quite as obvious is the extent to which

the P(2p) radial functions differ for the two states. The

most affected orbital is the one that is least tightly bound,

which in this case is the 2p orbital. Figure 21.1 shows

the two radial functions. The

P orbital is far more dif-

fuse (not as localized) as the one for

P. Such a change

in an orbital is called LS term dependence.

Orbital Collapse

Another phenomenon, called orbital collapse, occurs

when an orbital rapidly contracts as a function of

the energy. This could be an LS dependent effect,

but it can also occur along an isoelectronic se-

quence. This effect is most noticeable in the high-l

orbitals. In hydrogen, the mean radius of an orbital

is r=(1/2)[3n

−l(l +1)]a

. Thus, the higher-l or-

bitals are more contracted; but in neutral systems, the

high-l orbitals have a higher energy and are more diffuse.

This is due, in part, to the l(l +1)/r

angular momentum

barrier that appears in the deﬁnition of the L operator.

0.5

0.4

0.3

0.2

0.1

0 5 10 15 20

Radius (a.u.)

P(r)

Fig. 21.1 A comparison of the 2p Hartree–Fock radial func-

tions for the 1s2p

1,3

P states of Be

In the Hartree model, it is possible for V(r) +l(l +1)/r

to have two wells: an inner well and an outer shallow

well. As the lowest eigenfunction changes rapidly from

the outer well to the inner well, as Z changes, orbital

collapse is said to occur.

Quantum Defects and Rydberg Series

Spectra of atoms often exhibit phenomena associated

with a Rydberg series of states where one of the elec-

trons is in an nl orbital, with n assuming a sequence

of values. An example is the 1s

2snd

D series in Be,

n = 3, 4, 5,.... For such a series, an useful concept is

that of a quantum defect parameter δ. In hydrogen, the

ionization energy (I

) in atomic units is 1/(2n

).Incom-

plex neutral systems, the effective charge would be the

same at large r.Asn increases, the mean radius be-

comes larger and the probability of the electron being in

the hydrogen-like potential increases. Thus, one could

deﬁne an effective quantum number, n

∗

= n −δ,such

that

(nl) = (1/2)/(n −δ)

. (21.36)

Table 21.1 The effective quantum number and quantum

defect parameters of the 2snd Rydberg series in Be

n n

∗

δ(nl) n

∗

δ(nl)

3 2.968 0.032 3.014 −0.014

4 3.960 0.040 4.012 −0.012

5 4.957 0.043 5.013 −0.013

6 5.955 0.045 6.013 −0.013

Part B 21.4

Atomic Structure: Multiconﬁguration Hartree–Fock Theories 21.5 Multiconﬁguration Hartree–Fock Theory 313

For ionized systems, the equation must be modiﬁed to

(nl) = (1/2)[(Z −N +1)/(n −δ)]

Often, this parameter is deﬁned with respect to

observed data, but it can also be used to evaluate

Hartree–Fock energies, where I

= ε

nl,nl

/2, so that

nl,nl

=[(Z −N +1)/(n −δ)]

.Table21.1 shows the

effective quantum number and quantum defect for the

Hartree–Fock 2snd

Dand

D orbitals in Be as a func-

tion of n. For the triplet part the quantum defect is

positive whereas for the singlet it is negative. This is

the effect of exchange. Note that as n increases the

quantum defect becomes constant. This observation is

often the basis for determining ionization potentials from

observed data.

21.5 Multiconﬁguration Hartree–Fock Theory

The Hartree–Fock method predicts many atomic prop-

erties remarkably well; but when analyzed carefully,

systematic discrepancies can be observed. Consider the

ionization potentials tabulated in Table 21.2 compared

with the observed values. In these calculations, the en-

ergy of the ion was computed using the same radial

functions as for the atom. Thus, no “relaxation” effects

were included.

The observed data include other effects as well,

such as relativistic effects, ﬁnite mass and volume of

the nucleus, but these are small for light atoms. For

these systems, the largest source of discrepancy arises

from the fact that the Hartree–Fock solution is an in-

dependent particle approximation to the exact solution

of Schrödinger’s equation. Neglected entirely is the no-

tion of “correlation in the motion of the electrons”; each

electron is assumed to move independently in a ﬁeld de-

termined by the other electrons. For this reason, the error

in the energy was deﬁned by Löwdin [21.3], to be the

correlation energy, that is,

corr

= E

exact

−E

. (21.37)

In this deﬁnition, E

exact

is not the observed energy –

it is the exact solution of Schrödinger’s equation which

itself is based on a number of assumptions.

21.5.1 Z-Dependent Theory

An indication of the important correlation corrections

can be obtained from a perturbation theory study of

the exact wave function. In the following section, we

follow closely the approach taken by Layzer et al. [21.4]

in the study of the Z-dependent structure of the total

energy.

Let us introduce a new scaled length, ρ = Zr .Then

the Hamiltonian becomes

H = Z



+ Z

−1



, (21.38)

where

=−





∇



(21.39)

V =



i> j

, (21.40)

and Schrödinger’s equation becomes



+ Z

−1



ψ =



−2



ψ. (21.41)

With Z

−1

V regarded as a perturbation, the expansions

of ψ and E in the powers of Z

−1

are

ψ = ψ

+ Z

−1

+ Z

−2

+··· , (21.42)

in the ρ unit of length, and

E = Z



+ Z

−1

+ Z

−2

+ Z

−3

+···



(21.43)

The zero-order equation is

= E

. (21.44)

The solutions of this equation are products of hydrogenic

orbitals.

Table 21.2 Observed and Hartree–Fock ionization poten-

tials for the ground states of neutral atoms, in eV. (See also

Table 10.3.)

Atom Obs. HF Diff.

Li 5.39 5.34 0.05

Be 9.32 8.42 0.90

B 8.30 8.43 −0.13

C 11.26 11.79 −0.53

N 14.53 15.44 −0.91

O 13.62 14.45 −0.85

F 17.42 18.62 −1.20

Ne 21.56 23.14 −1.58

Part B 21.5

314 Part B Atoms

Let |(nl)νLS be a conﬁguration state function

constructed by vector coupling methods from prod-

ucts of hydrogenic orbitals. Here (nl) represents a set

of N quantum numbers (n

, n

,... ,n

) and ν

any additional quantum numbers such as the coupling

scheme or seniority needed to distinguish the different

conﬁguration states. Then,

|(nl)νLS=E

|(nl)νLS , (21.45)

=−



. (21.46)

Since E

is independent of the l

, differentconﬁgurations

may have the same E

;thatis,E

is degenerate. Ac-

cording to ﬁrst-order perturbation theory for degenerate

states, ψ

then is a linear combination of the degenerate

conﬁguration state functions |(nl



)ν



LS with the same

set of principal quantum numbers n

and parity π.The

coefﬁcients are components of an eigenvector of the in-

teraction matrix (nl



)ν



LS|V |(nl)νLS,andE

is the

corresponding eigenvalue. This is the set of conﬁgur-

ations referred to as the complex by Layzer [21.5]and

denoted by the quantum numbers (n)πLS.

The zero-order wave function ψ

describes the

many-electron system in a general way. It can be shown

that the square of the expansion coefﬁcients of ψ

over

the degenerate set of conﬁguration states can be inter-

preted as a probability that the many-electron system is

in that conﬁguration state, that ψ

is then a weighted lin-

ear combination of ﬁrst-order corrections to each such

conﬁguration state. Let us now assume the nondegen-

erate case where ψ

= Φ

(

γ LS

)

. The conﬁgurations

interacting with γ LS are of two types: those that dif-

fer by a single electron (single substitution S) and those

that differ by two electrons (double substitution D). The

former can be further subdivided into three categories:

(i) Those that differ from γ LS by one principal quan-

tum number but retain the same spin–angular coupling.

These conﬁguration states are part of radial correlation.

(ii) Those that differ by one principal quantum number

but differ in their coupling. If the only change is the

coupling of the spins, the conﬁguration states are part of

spin-polarization.

(iii) Those that differ in the angular momentum of one

electron and are accompanied by a change in angular

coupling of the conﬁguration state and possibly also the

spin coupling. The latter represent orbital-polarization.

The sums over intermediate states involve inﬁnite

sums. In practice, the set of orbitals is ﬁnite. In the

nondegenerate case, these orbitals can be divided into

occupied orbitals and unoccupied, or virtual, orbitals

depending on whether or not they occur in the refer-

ence conﬁguration that deﬁnes γ LS. Single and double

(SD) replacements of occupied orbitals by other occu-

pied or virtual orbitals generate the set of conﬁgurations

that interact with ψ

. Consider the 1s

2s ground state

of Li and the {1s, 2s, 3s, 4s, 2p, 3p, 4p, 3d, 4d, 4f} set

of orbitals. The 1s and 2s orbitals are occupied and

all the other orbitals are virtual orbitals, vl.Thesetof

replacements can then be classiﬁed as follows:

Replacement Conﬁguration Type of correlation

1s → 2s 1s2s

Radial and

spin-polarization

2s → vs 1s

vs Radial

1s → vs 1svs(

S)2s Radial

1svs(

S)2s Spin-polarization

1s2s → vlv



l 1svlv



l Core-polarization

→ vlv



l 2svlv



l Core

The above discussion has considered only the

Z-dependence of the wave function, but the notion can

readily be extended to other properties. For example,

in transition studies, the dipole transition matrix ele-

ment decreases as 1/Z, whereas the transition energy

increases linearly with Z for ∆n = 0 transitions, and

quadratically as Z

otherwise. A ﬁrst-order theory for

oscillator strengths (FOTOS) [21.6] is based on similar

concepts.

21.5.2 The MCHF Approximation

In the multiconﬁguration Hartree–Fock (MCHF)

method, the wave function is approximated by a lin-

ear combination of orthogonal conﬁguration states so

that

Ψ(γ LS) =



(

)

, (21.47)

where



= 1 .

Then the energy expression becomes



(

γ LS

)





(21.48)

where



(

)







. (21.49)

Because H

= H

, the sum over i, j may be limited to

the diagonals and the lower part of the matrix H =(H

Part B 21.5

Atomic Structure: Multiconﬁguration Hartree–Fock Theories 21.5 Multiconﬁguration Hartree–Fock Theory 315

called the interaction matrix. Let c = (c

) be a column

vector of the expansion coefﬁcients, also called mixing

coefﬁcients. Then the energy of the system is

E = c

Hc . (21.50)

Let P be the column vector of radial functions,

, P

,...)

. Since the interaction matrix elements de-

pend on the radial functions, it is clear that the energy

functional depends on both P and c.

In deriving the MCHF equations, the energy needs

to be expressed in terms of the radial functions and c.

From the theory of angular momenta, it follows that



I(a, b) +



abcd;k

(ab, cd),

(21.51)

where the sum over ab or abcd covers the occupied

orbitals of conﬁguration states i and j.

Substituting into the energy expression (21.48), and

interchanging the order of summation, we get

E (Ψ) =



I(a, b) +



abcd;k

(ab, cd),

(21.52)

where



and

abcd;k



abcd;k

In this form, the energy is expressed as a list of integrals

and their contribution to the energy – a form suitable for

the derivation of the MCHF radial equations.

As in the derivation of the Hartree–Fock equations,

the stationary principle must be applied to a functional

that includes Lagrange multipliers for all the constraints.

Thus,

F (P, c) = E (Ψ) +



a<b



,

a|b−E



(21.53)

In deriving the stationary conditions with respect to

variations in c

, the most convenient form for E (Ψ)

is (21.48), which leads to the secular equation

Hc = Ec .

(21.54)

Thus, the Lagrange multiplier E is the total energy of

the system.

The requirement of a stationary condition with re-

spect to variations in the radial functions leads to

a system of equations with exactly the same form as the

Hartree–Fock equations except: (i) the occupation num-

bers q

are not integers but rather expected occupation

numbers, and (ii) the function X(r) arises not only from

the exchange of electrons within a conﬁguration state,

but also from interactions between conﬁguration states.

A solution of the MCHF problem requires the si-

multaneous solution of the secular equation and the

variational radial equations. When the latter are assumed

to be given, then only the secular problem needs to be

solved and the problem is called a conﬁguration inter-

action (CI) problem. If any radial function is optimized,

the calculation is called a multi-conﬁguration Hartree–

Fock (MCHF) calculation. The iterative procedure for

its solution is the MCHF-SCF method, details of which

can be found in [21.7].

21.5.3 Systematic Methods

For MCHF calculations with only a few conﬁguration

states, the latter must be carefully chosen. Accord-

ing to ﬁrst-order perturbation theory, the expansion

coefﬁcient is c

=Φ

|H|Φ

/(E

−E

). Clearly, con-

ﬁgurations near in energy to the state under investigation

are candidates for a strong interaction, but it is also

important to remember the numerator in this expres-

sion. Generally, the latter is large if the electrons in the

two conﬁgurations occupy the same region of space.

Thus in the ground state of beryllium, the strongest

mixing is with 2p

S, even though the energy of

the latter is far removed from the energy of 2s

In highly ionized systems, the complex identiﬁes the

conﬁgurations that might interact strongly, but for neu-

tral systems, there are many exceptions, particularly

in atoms such as the transition metals where the 4s

and 4p subshells may be ﬁlled before the 3d sub-

shell.

For large scale computation, it is desirable to de-

ﬁne the conﬁguration states systematically in terms of

an active set (AS) of orbitals. Several notations are com-

monly used for this set. A very simple designation is

“the n = 4 active set” in that all orbitals up to and

including those with principal quantum number n = 4

are included. Another notation common in quantum

chemistry is one where the number of orbitals of each

symmetry is speciﬁed, as in 4s3p2d1f.

Closely associated with this set is the set of conﬁg-

uration states that can be generated. The latter is referred

to as the complete active space (CAS). Often the CAS

is deﬁned relative to a set of closed shells (or subshells)

common to all conﬁgurations. For example, the com-

Part B 21.5