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

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Part A 4

100 Part A Mathematical Methods

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Part A 4

101

Perturbation T

5. Perturbation Theory

Perturbation theory (PT) represents one of

the bridges that takes us from a simpler, exactly

solvable (unperturbed) problem to a corresponding

real (perturbed) problem by expressing its solutions

as a series expansion in a suitably chosen “small”

parameter ε in such a way that the problem

reduces to the unperturbed problem when ε = 0.

It originated in classical mechanics and eventually

developed into an important branch of applied

mathematics enabling physicists and engineers to

obtain approximate solutions of various systems of

differential equations [5.1–4]. For the problems of

atomic and molecular structure and dynamics, the

perturbed problem is usually given by the time-

independent or time-dependent Schrödinger

equation [5.5–8].

5.1 Matrix Perturbation Theory (PT) ............ 101

5.1.1 Basic Concepts .......................... 101

5.1.2 Level-Shift Operators ................. 102

5.1.3 General Formalism .................... 102

5.1.4 Nondegenerate Case .................. 103

5.2 Time-Independent Perturbation Theory 103

5.2.1 General Formulation ................. 103

5.2.2 Brillouin–Wigner and

Rayleigh–Schrödinger PT (RSPT)... 104

5.2.3 Bracketing Theorem and RSPT ..... 104

5.3 Fermionic Many-Body Perturbation

Theory (MBPT)...................................... 105

5.3.1 Time Independent Wick’s

Theorem .................................. 105

5.3.2 Normal Product Form of PT ......... 105

5.3.3 Møller–Plesset

and Epstein–Nesbet PT .............. 106

5.3.4 Diagrammatic MBPT ................... 107

5.3.5 Vacuum and Wave Function

Diagrams ................................. 107

5.3.6 Hartree–Fock Diagrams .............. 108

5.3.7 Linked and Connected Cluster

Theorems ................................. 108

5.3.8 Coupled Cluster Theory............... 109

5.4 Time-Dependent Perturbation Theory ... 111

5.4.1 Evolution Operator

PT Expansion ............................ 111

5.4.2 Gell–Mann and Low Formula...... 111

5.4.3 Potential Scattering

and Quantum Dynamics ............. 111

5.4.4 Born Series ............................... 112

5.4.5 Variation of Constants Method .... 112

References .................................................. 113

5.1 Matrix Perturbation Theory (PT)

A prototype of a time-independent PT considers an

eigenvalue problem for the Hamiltonian H of the form

H = H

+ V , V =

∞



i=1

, (5.1)

acting in a (ﬁnite-dimensional) Hilbert space V

,as-

suming that the spectral resolution of the unperturbed

operator H

is known; i. e.,



, P

= δ



= I ,

(5.2)

where ω

are distinct eigenvalues of H

,theP

form

a complete orthonormal set of Hermitian idempotents

and I is the identity operator on V

.ThePT problem

for H can then be formulated within the Lie algebra A

(see Sect. 3.2) generated by H

and V [5.9, 10].

5.1.1 Basic Concepts

Deﬁne the diagonal part X of a general operator X ∈ A

X=



, (5.3)

and recall that the adjoint action of X ∈ A,

adX : A → A,isdeﬁnedby

ad X(Y ) =[X, Y ] ,(∀ Y ∈ A),

(5.4)

Part A 5

102 Part A Mathematical Methods

where the square bracket denotes the commutator. The

key problem of PT is the ‘inversion’ of this operation,

i. e., the solution of the equation [5.9–11]

adH

(X) ≡[H

, X]=Y . (5.5)

Assuming that Y=0, then

X = R(Y ) +A ,

(5.6)

where A ∈ A is arbitrary and

R(Y ) =



i= j

∆

−1

, (5.7)

with ∆

= ω

− ω

, represents the solution of (5.5) with

the vanishing diagonal part R(Y )=0.

5.1.2 Level-Shift Operators

To solve the PT problem for H,(5.1), we search for

a unitary level-shift transformation U [5.9, 10], U

†

U =

†

= I ,

UHU

†

= U(H

+ V )U

†

= H

+ E , (5.8)

where the level-shift operator E satisﬁes the condition

E =E . (5.9)

To guarantee the unitarity of U, we express it in the form

U = e

, G

†

=−G , G=0 . (5.10)

Using the Haussdorff formula

B e

−A

∞



k=0

(k!)

−1

( ad A)

B , (5.11)

and deﬁning the operator

F =[H

, G] , (5.12)

we ﬁnd from (5.8)that

E + F = V +

[G, V + E]

∞



k=2

(k!)

−1

( adG)

(V − E), (5.13)

where we used the identity [5.12]



∞



k=0



∞



k=0

(k + 1)!



= I ,

(5.14)

and B

designates the Bernoulli numbers [5.13]

= 1 , B

=−

, B

2k+1

= 0 (k ≥ 1),

, B

, etc.

(5.15)

5.1.3 General Formalism

Introducing the PT expansion for relevant operators,

X =

∞



i=1

, X = E, F, G ; F

=[H

, G

] ,

(5.16)

(5.13) leads to the following system of equations

+ F

= V

+ F

= V

, V

+ E

] ,

+ F

= V

, V

+ E

]

, V

+ E

]

, [G

, V

− E

]] ,

etc. ,

(5.17)

which can be solved recursively for E

and G

by tak-

ing their diagonal part and applying operator R,(5.7),

since

E

=E

, G

=F

=0 ,

= G

, RE

= 0 . (5.18)

We thus get

=V

 ,

=V

+

[RV

, V

] ,

=V

+[RV

, V

]

[RV

, [RV

, 2V

+ E

]] , etc. , (5.19)

and

= RV

R[RV

, V

+ E

] , etc. (5.20)

Since

R(X)=RX=0 , R(X)Y =−XR(Y ) ,

R(XY ) = R(X)Y, R(XY) =XR(Y ),

(5.21)

Part A 5.1

Perturbation Theory 5.2 Time-Independent Perturbation Theory 103

these relationships can be transformed to a more con-

ventional form

=V

−V

 ,

=V

−V



R(V

)R(V

)[2V

+V

]

−

R(V

)[2V

+V

]R(V

)

[2 V

+V

]R(V

)R(V

) , etc.

(5.22)

However, in this way certain nonphysical terms arise that

exactly cancel when the commutator form is employed

(Sect. 5.3.7).

5.1.4 Nondegenerate Case

In the nondegenerate case, when P

=|ii|, with |i

representing the eigenvector of H

associated with the

eigenvalue ω

, the level-shift operator is diagonal and

its explicit PT expansion (as well as that for the cor-

responding eigenvectors) is easily obtained from (5.19)

and (5.20). Writing x

for the matrix element i|X| j,

we get

)

= (v

)

= (v

)

−





)

∆

)

= (v

)

−





)

+ (v

)

∆



j,k



)

∆

− (v

)





)

∆

etc.,

(5.23)

the prime on the summation symbols indicating that the

terms with the vanishing denominator are to be deleted.

Note that in contrast to PT expansions which directly

expand the level-shift transformation U, U = 1 + εU

+··· , the above Lie algebraic formulation has the

advantage that U stays unitary in every order of PT.This

is particularly useful in spectroscopic applications, such

as line broadening.

5.2 Time-Independent Perturbation Theory

For stationary problems, particularly those arising in

atomic and molecular electronic structure studies relying

on ab initio model Hamiltonians, the PT of Sect. 5.1 can

be given a more explicit form which avoids a priori the

nonphysical, size inextensive terms [5.6–8, 14, 15].

5.2.1 General Formulation

We wish to ﬁnd the eigenvalues and eigenvectors of the

full (perturbed) problem

K|Ψ

≡(K

+ W )|Ψ

=k

|Ψ

 , (5.24)

assuming we know those of the unperturbed problem

|Φ

=κ

|Φ

, Φ

|Φ

=δ

. (5.25)

For simplicity, we restrict ourselves to the nondegenerate

case (κ

= κ

if i = j) and consider only the ﬁrst order

perturbation [see (5.1), εV

≡ W, V

= 0fori ≥ 2].

Of course, K and K

are Hermitian operators acting

in a Hilbert space which, in ab initio applications, is

ﬁnite-dimensional.

Using the intermediate normalization for |Ψ

,

Ψ

|Φ

=1 , (5.26)

the asymmetric energy formula gives

= κ

+Φ

|W |Ψ

 . (5.27)

The idempotent Hermitian projectors

=|Φ

Φ

|, Q

= P

⊥

= 1 − P



j(=i)

|Φ

Φ

| ,

(5.28)

commute with K

,sothat

(λ − K

|Ψ

=Q

(λ − k

+ W )|Ψ

 , (5.29)

λ being an arbitrary scalar (note that we write λI simply

as λ). Since the resolvent (λ − K

)

−1

of K

is nonsingu-

lar on the orthogonal complement of the ith eigenspace,

we get

|Ψ

=|Ψ

−|Φ

=R

(λ)(λ − k

+ W )|Ψ

 ,

(5.30)

Part A 5.2

104 Part A Mathematical Methods

where

≡ R

(λ) = (λ − K

)

−1

= Q

(λ − K

)

−1



j(=i)

|Φ

Φ

λ − κ

(5.31)

assuming (λ = κ

). Iterating this relationship, we get

prototypes of the desired PT expansion for |Ψ

,

|Ψ

=

∞



n=0

(λ − k

+ W )]

|Φ

 , (5.32)

and, from (5.27), for k

= κ

∞



n=0

Φ

|W [R

(λ − k

+ W )]

|Φ

 . (5.33)

5.2.2 Brillouin–Wigner

and Rayleigh–Schrödinger PT (RSPT)

So far, the parameter λ was arbitrary, as long as λ =

( j = i). The following two choices lead to the two

basic types of many-body perturbation theory (MBPT):

Brillouin–Wigner (BW) PT

Setting λ = k

gives

= κ

∞



n=0

Φ



(BW)



|Φ

 , (5.34)

|Ψ

=

∞



n=0



(BW)



|Φ

 , (5.35)

where

(BW)



j(=i)

|Φ

Φ

− κ

. (5.36)

Rayleigh–Schrödinger (RS) PT

Setting λ = κ

gives

= κ

∞



n=0

Φ



(RS)

(κ

− k

+ W )



|Φ

 ,

(5.37)

|Ψ

=

∞



n=0



(RS)

(κ

− k

+ W )



|Φ

 , (5.38)

where now

≡ R

(RS)



j(=i)

|Φ

Φ

− κ

. (5.39)

The main distinction between these two PTs lies

in the fact that the BW form has the exact eigenval-

ues appearing in the denominators, and thus leads to

polynomial expressions for k

. Although these are not

difﬁcult to solve numerically, since the eigenvalues are

separated, the resulting energies are never size exten-

sive and thus unusable for extended systems. They are

also unsuitable for ﬁnite systems when the particle num-

ber changes, as in various dissociation processes. From

now on, we thus investigate only the RSPT, which yields

a fully size-extensive theory.

5.2.3 Bracketing Theorem and RSPT

Expressions (5.37)and(5.38) are not explicit, since they

involve the exact eigenvalues k

on the right-hand side.

To achieve an order by order separation, set

≡ k =

∞



j=0

( j)

, |Ψ

≡|Ψ =

∞



j=0

|Ψ

( j)

 ,

(5.40)

where the superscript ( j) indicates the jth-order in the

perturbation W. We only consider the eigenvalue expres-

sions, since the corresponding eigenvectors are easily

recovered from them by removing the bra state and

the ﬁrst interaction W [see (5.37)and(5.38)]. We also

simplify the mean value notation writing for a general

operator X,

X≡Φ

|X|Φ

 . (5.41)

Substituting the ﬁrst expansion (5.40)into(5.37)and

collecting the terms of the same order in W,weget

(0)

= κ

(1)

=W ,

(2)

=WRW ,

(3)



W(RW )



−W





(4)



W(RW )



−W



WR(RW )





(WR)



+W





−WRW





, etc.

(5.42)

The general expression has the form

(n)



W(RW )

n−1



+ R

(n)

, (5.43)

Part A 5.2

Perturbation Theory 5.3 Fermionic Many-Body Perturbation Theory (MBPT) 105

the ﬁrst term on the right-hand side being referred to as

the principal nth-order term, while R

(n)

designates the

so-called renormalization terms that are obtained by the

bracketing theorem [5.14, 16] as follows:

1. Insert the bracketings ··· around the W, WRW,

... ,WR ···RW operator strings of the principal

term in all possible ways.

2. Bracketings involving the rightmost and/or the left-

most interaction vanish.

3. The sign of each bracketed term is given by (−1)

where n

is the number of bracketings.

4. Bracketings within bracketings are allowed, e.g.,

WRWRWRWRW=W





5. The total number of bracketings (including the prin-

cipal term) is (2n − 2)!/[n!(n − 1)!].

5.3 Fermionic Many-Body Perturbation Theory (MBPT)

5.3.1 Time Independent Wick’s Theorem

The development of an explicit MBPT formalism is

greatly facilitated by the exploitation of the time-

independent version of Wick’s theorem. This version of

the theorem expresses an arbitrary product of creation

†

) and annihilation (a

) operators (see Chapt. 6)as

a normal product (relative to |Φ

) and as normal

products with all possible contractions of these oper-

ators [5.14,15],

···x

= N[x

···x

]+ΣN[x

······x

] ,

= a

†

or x

= a

) (5.44)

where

†

= a

= 0 ,

†

= h(µ)δ

µν

, a

†

= p(µ)δ

µν

, (5.45)

and

h(µ) = 1 , p(µ) = 0if|µ is occupied in |Φ



(hole states),

h(µ) = 0 , p(µ) = 1if|µ is unoccupied in |Φ



(particle states) .

(5.46)

The N-product with contractions is deﬁned as a pro-

duct of individual contractions times the N-product of

uncontracted operators (deﬁning N[∅] ≡ 1 for an empty

set) with the sign given by the parity of the permutation

reordering the operators into their ﬁnal order.

Note that the Fermi vacuum mean v alue of an

N-product vanishes unless all operators are contracted.

Thus, x

···x

 is given by the sum over all possible

fully contracted terms (vacuum terms). Similar rules fol-

low for the expressions of the type (x

···x

)|Φ.

Moreover, if some operators on the left-hand side

of (5.44) are already in the N-product form, all the terms

involving contractions between these operators vanish.

5.3.2 Normal Product Form of PT

Consider the eigenvalue problem for a general ab initio

or semi-empirical electronic Hamiltonian H with one-

and two-body components Z and V , namely,

H|Ψ

=E

|Ψ

 ,

H = Z + V =



z(i) +



i< j

v(i, j), (5.47)

and a corresponding unperturbed problem

|Φ

=ε

|Φ

 , (5.48)

= Z +U , Φ

|Φ

=δ

with U representing some approximation to V .Inthe

case that U is also a one-electron operator, U = Σ

u(i),

the unperturbed problem (5.48) is separable and reduces

to a one-electron problem,

(z + u)|µ=ω

|µ , (5.49)

which is assumed to be solved. Choosing the or-

thonormal spin orbitals {|µ} as a basis of the second

quantization representation [Chapt. 6,(6.8)], the N-

electron solutions of (5.48) can be represented as

|Φ

=a

†

···a

†

|0 , (5.50)



j=1

, (5.51)

the state label i representing the occupied spin orbital

set {µ

,µ

,... ,µ

}, while the one- and two-body

operators take the form

X =



µ,ν

µ|x|νa

†

, X = Z, U; x = z, u ,

(5.52)

V =



µ,ν,σ,τ

µν|v|στa

†

. (5.53)

Part A 5.3

106 Part A Mathematical Methods

Considering, for simplicity, a nondegenerate ground

state |Φ≡|Φ

=a

†

···a

†

|0, referred to as a Fermi

vacuum, we deﬁne the normal product form of these

operators relative to |Φ

≡ X −X=



µ,ν

µ|x|νN



†



(5.54a)

(X = Z, U, G; x = z, u, g)

≡ V −V −G



µ,ν,σ,τ

µν|v|στN



†





µ,ν,σ,τ

µν|v|στ



†



(5.54b)

where

µ|g|ν=



σ=1

µσ|v|νσ

, (5.55)

µν|v|στ

=µν|v|στ−µν|v|τσ , (5.56)

X=Φ|X|Φ,andN[···] designates the nor-

mal product relative to |Φ [5.14, 15]. (Recall

that N[x

···x

]=±b

†

···b

†

i+1

···b

,where

= b

or b

†

are the annihilation and creation op-

erators of the particle-hole formalism relative to |Φ,

i. e., b

= a

†

for µ ≤ N and b

= a

for µ>N,the

sign being determined by the parity of the permutation

p : j → µ

Deﬁning

K = H −H, K

= H

−H

=H

− ε

(5.57)

we can return to (5.24)and(5.25), where now

= E

−H,κ

= ε

− ε



µ=1

, (5.58)

and

W = K − K

= V − U −V − U . (5.59)

With this choice, W=0, so that for the reference

state |Φ,(5.42) simplify to (we drop the subscript 0

for simplicity)

(0)

= 0, k

(1)

= 0 ,

(2)

=WRW ,

(3)

=WRWRW ,

(4)



W(RW )



−WRW





, etc.

(5.60)

Note that W is also in the N-product form,

W = W

+ W

, W

= G

− U

, W

= V

(5.61)

5.3.3 Møller–Plesset

and Epstein–Nesbet PT

Choosing U = G we have

= Z + G ≡ F , (5.62)

so that (5.49) represent Hartree–Fock (HF) equa-

tions, and ω

and |µ the canonical HF orbit-

al energies and spin orbitals, respectively. Since

H=



µ=1



µ|z|µ+

µ|g|µ



is the HF energy,

k = k

gives directly the ground state correlation en-

ergy. (Note, however, that the N-product form of PT

eliminates the ﬁrst-order contribution k

(1)

=W in any

basis, even when F is not diagonal.) With this choice,

= 0, W = V

, and the denominators in (5.39)are

given by the differences of HF orbital energies

− κ



i=1

(ω

− ω

) ≡ ∆{µ

}; {ν

} , (5.63)

assuming that |Φ

 is a λ-times excited conﬁgu-

ration relative to |Φ obtained through excitations

→ ν

, i = 1,... ,λ. Using the Slater rules (or the

second quantization algebra), we can express the

second-order contribution in terms of the two-electron

integrals and HF orbital energies as

(2)



a,b,r,s

ab|v|rs



rs|v|ab−rs|v|ba



+ ω

− ω

(5.64)

where the summations over a, b (r, s) extend over all oc-

cupied (unoccupied) spin orbitals in |Φ. Obtaining the

corresponding higher-order corrections becomes more

and more laborious and, beginning with the fourth-order,

important cancellations arise between the principal and

renormalization terms, even when the N-product form

is employed. These will be addressed in Sect. 5.3.7.

The above outlined PT with H

given by the HF op-

erator is often referred to as the Møller–Plesset PT [5.17]

and, when truncated to the n-th order, is designated

by the acronym MPn, n = 2, 3,.... In this version,

the two-electron integrals enter the denominators only

through the HF orbital energies. In an alternative, less

often employed variant, referred to as the Epstein–

Nesbet PT [5.18, 19], the whole diagonal part of H is

Part A 5.3

Perturbation Theory 5.3 Fermionic Many-Body Perturbation Theory (MBPT) 107

a) b) c) d)

Fig. 5.1a–d Diagrammatic representation of one- and two-

electron operators

used as the unperturbed Hamiltonian, i. e.,



Φ

|H|Φ

P

. (5.65)

With this choice, the denominators are given as dif-

ferences of the diagonal elements of the conﬁguration

interaction matrix.

5.3.4 Diagrammatic MBPT

To facilitate the evaluation of higher order terms,

and especially to derive the general properties and

characteristics of the MBPT,itisusefultoemployadia-

grammatic representation [5.6–8, 14, 15]. Representing

all the operators in (5.42)and(5.43)or(5.60)inthe

second quantized form, we have to deal with the refer-

ence state (i. e., the Fermi vacuum) mean values of the

strings of annihilation and creation operators (or with

these strings acting on the reference in the case of a wave

function). This is efﬁciently done using Wick’s theorem

and its diagrammatic representation via a special form of

Feynman diagrams. In this representation we associate

with various operators suitable vertices with incident

oriented lines representing the creation (outgoing lines)

and annihilation (ingoing lines) operators that are in-

volved in their second quantization form. A few typical

diagrams representing operators (−U), W

and V are

shown in Fig. 5.1a, Fig. 5.1b and Fig. 5.1c, Fig. 5.1d, re-

spectively. Using the N-product form of PT with HF

orbitals (Sect. 5.3.3), we only need the two-electron

operator V or V

, which can be represented using ei-

ther non-antisymmetrized vertices (Fig. 5.1c), leading

to the Goldstone diagrams [5.20], or antisymmetrized

vertices (Fig. 5.1d), associated with antisymmetrized

two-electron integrals (5.56) and yielding the Hugen-

holtz diagrams [5.21].

5.3.5 Vacuum and Wave Function Diagrams

Applying Wick’s theorem to the strings of operators in-

volved, we represent the individual contractions, (5.45),

by joining corresponding oriented lines. To obtain a non-

a) b) c)

Fig. 5.2a–c The second-order Goldstone (a), (b) and

Hugenholtz

vanishing contribution, only contractions preserving the

orientation need be considered [cf. (5.45)]. The resulting

internal lines have either the left–right orientation (hole

lines) or the right–left one (particle lines). Only fully

contracted terms, represented by the so-called vacuum

diagrams (having only internal lines), can contribute

to the energy, while those representing wave function

contributions have uncontracted or free lines extending

to the left. When the operators involved are in the N-

product form, no contractions of oriented lines issuing

from the same vertex are allowed. The projection-like

operators R,(5.39), or their powers, lead to the denomi-

nators, (5.63), given by the difference of hole and particle

orbital energies associated with, respectively, hole and

particle lines passing through the interval separating the

corresponding two neighboring vertices. Clearly, there

must always be at least one pair of such lines lest the de-

nominator vanish. Thus, for example, the second-order

contribution WRW is represented either by the two

Goldstone diagrams [5.20](Fig.5.2a,b) or by the single

Hugenholtz diagram [5.21](Fig.5.2c).Therulesforthe

energy (vacuum) diagram evaluation are as follows:

1. Associate appropriate matrix elements with all ver-

tices and form their product. The outgoing (ingoing)

lines on each vertex deﬁne the bra (ket) states of

a given matrix element, and for the Goldstone dia-

grams, the oriented lines attached to the same node

are associated with the same electron number, (e.g.,

for the leftmost vertex in diagram (a) of Fig. 5.2 we

have ab|

v|rs≡a(1)b(2)|v|r(1)s(2)).

2. Associate a denominator, (5.63), or its appropri-

ate power, with every neighboring pair of vertices

(and, for the wave function diagrams, also with the

free lines extending to the left of the leftmost ver-

tex; with each pair of such free lines associate also

the corresponding pair of particle creation and hole

annihilation operators).

3. Sum over all hole and particle labels.

4. Multiply each diagram contribution by the weight

factor given by the reciprocal value of the order of

Part A 5.3

108 Part A Mathematical Methods

Fig. 5.3 Hugenholtz diagrams for the third-order energy

contribution

the group of automorphisms of the diagram (stripped

of summation labels) and by the sign (−1)

h+

,where

h designates the number of internal hole lines and

 gives the number of closed loops of oriented lines

(for Hugenholtz diagrams, use any of its Goldstone

representatives to determine the correct phase).

Applying these rules to diagrams (a) and (b)

of Fig. 5.2 we clearly recover (5.64) or, using the Hugen-

holtz diagram of Fig. 5.2, the equivalent expression

(2)



a,b,r,s

ab|v|rs

rs|v|ab

∆

−1

(a, b; r, s).

(5.66)

The possible third-order Hugenholtz diagrams are

showninFig.5.3 with the central vertex involving

particle–particle, hole–hole, and particle–hole interac-

tion [5.14, 15].

5.3.6 Hartree–Fock Diagrams

In the general case (non-HF orbitals and/or not normal

product form of PT), the one-electron terms, as well

as the contractions between operators associated with

the same vertex, can occur (the latter are alw ays the

hole lines). Representing the W

and (−U) operators as

shown in Fig. 5.1, the one-body perturbation W

repre-

sents in fact the three diagrams as shown in Fig. 5.4.The

second-order contribution of this type is then represented

by the diagrams in Fig. 5.5, which in fact represents nine

diagrams which result when each W

vertex is replaced

by three vertices as shown in Fig. 5.4.

Using HF orbitals, all these terms mutually cancel

out as seen above. For this reason, the diagrams involv-

ing contractions of lines issuing from the same vertex

a) b) c) d)

=++

Fig. 5.4a–d Schematic representation of W

= G

− U

Fig. 5.5 The second-order one-particle contribution

are referred to as Hartree–Fock diagrams. Note, how-

ever, that even when not employing the canonical HF

orbitals, it is convenient to introduce W

vertices of the

normal product form PT and replace all nine HF-type

diagrams by a single diagram of Fig. 5.5 (clearly, this

feature provides even greater efﬁciency in higher orders

of PT).

5.3.7 Linked and Connected Cluster

Theorems

Using the N-product form of PT, the ﬁrst nonvan-

ishing renormalization term occurs in the fourth-order

[cf. (5.60)]. For a system consisting of N noninteract-

ing species, the energy given by this nonphysical term is

proportional to N

, and thus violates the size extensivity

of the theory. It was ﬁrst shown by Brueckner [5.22]that

in the fourth-order these terms are in fact exactly can-

celed by the corresponding contributions originating in

the principal term. A general proof of this cancellation

in an arbitrary order was then given by Goldstone [5.20]

using the time-dependent PT formalism (Sect. 5.4).

To comprehend this cancellation, consider the

fourth-order energy contribution arising from the so-

called unlinked diagrams (no such contribution can arise

in the second- or the third-order) shown in Fig. 5.6.

An unlinked diagram is deﬁned as a diagram contain-

ing a disconnected vacuum diagram (for the energy

diagrams, the terms unlinked and disconnected are

synonymous). The numerators associated with both

diagrams being identical, we only consider the denomi-

nators. Designating the denominator associated with the

Fig. 5.6 The fourth-order unlinked diagrams

Part A 5.3