Multireference Perturbation Theories for the accurate calculation of energy and molecular properties

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Fa oltà di S ienze Matemati he, Fisi he e Naturali

Dottorato di Ri er a in S ienze Chimi he

Ci lo XXI

Multireferen e Perturbation Theories for the

a urate al ulation of energy and mole ular

properties

Coordinator:

Prof. Renzo Cimiraglia

Candidate:

Dr. Maria hiara Pastore

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an he dopolane diquestaesperienza, vorrei ringraziare

il Professor Renzo Cimiraglia, supervisore attento e presente della

mia attività di ri er a. Lo ringrazio per la stima e la du ia he ho

sempre per epito e peravermi dato tutti glistrumenti, leopportunità e

l'autonomia per res ere, nonsolo preofessionalmente

Celestino, per essere stato un ami o, prima an ora he uno dei miei

puntidiriferimento professionali

Maurizioper hè fortunatamenteva adormire sempretardi

Stefano Evangelisti, Jean-Paul Malrieu, tutte le persone on ui ho

lavorato e il assoulet per aver reso indimenti abili i miei soggiorni a

Toulouse

Sasha, per tutto il lavoro fatto insieme e per quella birra in quella

squallida birreria dellaperiferia diSto arda

Un grazie, poi, ai miei genitori e a mio fratello, per il loro ostante

supporto

RingrazioMassimoperessermia antoneimomenti ompli atienelle

s elte importanti e per essere una fonte inesauribile di entusiasmo e di

energia

Grazie,inne,adimieiami ipiù ari. GrazieadEva,Martina,Silvia,

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Prefa e 1

1 Mathemati al tools and methods 3

1.1 Completeset expansions . . . 3

1.2 Antisymmetry: Slater's formalism. . . 4

1.2.1 Conguration Intera tion Approa h . . . 5

1.3 An alternative approa h: se ondquantization . . . 7

1.3.1 The Fo kspa e . . . 7

1.3.2 Creation and annihilationoperators . . . 7

1.3.3 Representation of one-and two-ele tron operators. . . 8

1.3.4 The spin-tra ed repla ement operators . . . 10

1.4 One-determinant approximation: Hartree-Fo ktheory . . . 10

1.4.1 Self-Consistent Field(SCF) theory . . . 11

1.4.2 Koopmans'Theorem . . . 13

1.5 The Ele tron orrelationproblem . . . 14

1.5.1 Ele tron distribution: densityfun tionsand densitymatri es 14 1.5.2 The one-determinant approximation ase. . . 17

1.5.3 Stati al and Dynami alCorrelation . . . 19

1.6 Handling the Stati al Correlation: MCSCF Theory . . . 19

1.6.1 Complete A tive Spa e(CAS). . . 20

2

N

-ele tron Valen e State Perturbation Theory 23 2.1 Rayleigh-S hrödinger Perturbation Theory . . . 24

2.2 Møller-Plesset Theory . . . 25

2.3 NEVPT2 philosophy . . . 26

2.4 Internally ontra ted approa h . . . 27

2.4.1 The Partially Contra ted NEVPT2 . . . 29

2.4.2 The Strongly ontra ted NEVPT2 . . . 30

2.5 Major NEVPT2 properties. . . 31

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2.5.2 Invarian e underorbital rotations . . . 32

2.5.3 Size onsisten e . . . 32

2.6 Quasidegenerate NEVPT2 . . . 33

2.7 Thirdorder NEVPTand InternallyContra ted CI . . . 35

2.8 A test ase: the

X

1 _Σ

+

g

and

B

′1

_Σ

+

g

statesof C

2

. . . 36

2.8.1 Method of al ulation . . . 37

2.8.2 Resultsand dis ussion . . . 37

I Ex ited state al ulations 43 3 The hetero y lopentadienes 45 3.1 Ioni valen e states . . . 46

3.2 Valen eRydberg intera tion . . . 48

3.3 Computational approa h . . . 49

3.3.1 A tive Spa es . . . 51

3.4 Pyrrole. . . 53

3.4.1 TheUV absorption spe trum . . . 53

3.4.2 Thesinglet valen e states . . . 55

3.4.3 The

π

type Rydbergstates . . . 58

3.4.4 The

σ

type Rydbergstates . . . 59

3.5 Furan . . . 60

3.5.1 TheUV absorption spe trum . . . 60

3.5.2 Valen eRydberg mixing . . . 61

3.5.3 SingletValen e States . . . 64

3.5.4 SingletRydbergstates . . . 66

3.6 Thiophene . . . 69

3.6.1 Valen eRydberg mixing . . . 70

3.6.2 TheVUV absorptionspe trum . . . 76

4 The verti al ele troni spe trum of FreeBase Porphin 83 4.1 TheUV spe trumof free-baseporphin . . . 83

4.2 Computational approa h . . . 84

4.3 NEVPT results . . . 85

II MixedValen e systems 91 5 Ele tron transfer in a model spiro system 93 5.1 Introdu tion . . . 93

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5.3 The modelSpiro system . . . 96

5.4 Computational details . . . 98

5.5 Se ondand third orderstandard MRPT . . . 98

5.6 Failure ofa standard MRPTapproa h . . . 101

5.6.1 A simple two-statemodel . . . 101

5.6.2 Se ond order orre tion . . . 102

5.6.3 Third order orre tion . . . 106

5.6.4 Con lusive remarks . . . 107

5.7 The useofstateaveraged orbitals . . . 109

5.7.1 The energy barrier . . . 109

5.7.2 The energy splitting . . . 112

5.7.3 Ex itation energy to the

2 _A

2 (2)

state . . . 113 A PC-NEVPT2

S

(k)

l

spa es 115 A.0.4 The

S

(0)

ij,rs

Spa e. . . 115 A.0.5 The

S

(−1)

i,rs

Spa e . . . 115 A.0.6 The

S

(1)

ij,r

Spa e . . . 116 A.0.7 The

S

(−2)

rs

Spa e . . . 117 A.0.8 The

S

(2)

ij

Spa e . . . 118 A.0.9 The

S

(0)

i,r

Spa e . . . 119 A.0.10 The

S

(−1)

r

Spa e . . . 119 A.0.11 The

S

(1)

i

Spa e . . . 119

B Matrix elements of PC-NEVPT3 121 B.0.12

V(0)V(0)

Class . . . 121 B.0.13

V(0)V(1)

Class . . . 122 B.0.14

V

(0)V(−1)

Class . . . 123 B.0.15

V(0)V(2)

Class . . . 125 B.0.16

V

(0)V(−2)

Class . . . 125 B.0.17

V(0)V(0

′

₎

Class . . . 126 B.0.18

V(0)V(1

′

₎

Class . . . 126 B.0.19

V

(0)V(−1

′

₎

Class . . . 127 B.0.20

V(1)V(1)

Class . . . 127 B.0.21

V

(1)V(−1)

Class . . . 130 B.0.22

V(1)V(2)

Class . . . 133 B.0.23

V(1)V(0

′

₎

Class . . . 134 B.0.24

V

(1)V(−1

′

₎

Class . . . 135 B.0.25

V(1)V(1

′

₎

Class . . . 136

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Quantum Chemistry has be ome an important and powerful tool to investigate a

great dealof hemi al andphysi al phenomena. Nowadays,therapid growth ofthe

omputational power along with the orresponding development of methodologies,

tailored to approa h large s ale systems,allows to treat problems of in reasing size

and omplexity.

A large domain of appli ation of rigorous quantum me hani s al ulations is

the a urate predi tion of ex itation energies and other spe tros opi parameters

valuable for the interpretation of the experimental measurements. The des ription

of ele troni ally ex ited states represents a severe task for approximated

theoreti- al approa hes, even inthe ase of small-sized mole ules. In su h ases, thesimple

one-determinantapproximation(thewell-knownHartree-Fo ktheory)turnouttobe

defe tiveandamultireferen ewavefun tion,a ountingforalltherelevantele troni

ongurations, should be used. An important eld of appli ations of the

Multiref-eren e Perturbation Theories (MRPTs) is just the al ulation of the ele troni ally

ex ited statesof mole ules, where thestrong dierential orrelation ee ts and the

possiblemultireferen e natureofthewavefun tions an be,inprin iple,su essfully

handled by avariationalplus perturbation s heme.

ThisPh.D.thesisdealswiththedevelopmentandtheappli ationsof

N

-Ele tron Valen eStatePerturbationTheory(NEVPT),anovelformofMRPTputforwardin

ollaboration between thetheoreti al hemistry groupsoftheuniversities ofFerrara

and Toulouse.

After a rst general overview on the basi mathemati al tools and theoreti al

methods (Chap. 1), in Chapter 2 we will introdu e the NEVPT philosophy and

present the major development eort a omplishedduringthePh.D: the

implemen-tationofthethirdorder orre tiontotheenergyintheso alledpartially ontra ted

s heme. Then, the large part devoted to the appli ations follows. Part I on erns

the al ulationofele troni allyex ited states. Dierentissueswillbeaddressed: on

the one hand thetreatment of small aromati mole ules, Pyrrole, Furan and

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low-lying Rydberg statesand bythe ioni nature of some valen e states, extremely

sensitive to the so- alled dynami al

σ − π

polarization; on the other hand the ase of alarge-sized aromati mole ule,Free-Base Porphin (Chap. 4),for whi hthe

ru- ial problemis the hoi e ofa balan ed variational spa e to a uratelydes ribe the

wavefun tions of the groundand of theex ited states. Finally, Chap. 5 is devoted

to the des ription, by means of MRPT, of the Ele tron Transfer (ET) pro ess in

Mixed-Valen esystems. Theinvestigation is arriedout onamodelspiro

π − σ − π

ompound,forwhi h theETrea tion issimulatedusing asimpliedone-mode

two-state model. The inadequa y of a standard se ond order MRPT approa h will be

shownandtheappli ationofanalternativeandee tive omputationalstrategywill

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Mathemati al tools and methods

1.1 Complete set expansions

Let

f (x)

be afun tion dened intheinterval

(a, b)

and let

Φ = {φ

1 , φ

2 , . . . , φ

n

}

be a setoffun tionsdened inthesame interval. One anexpressthefun tion

f (x)

as a linear ombination, withproperly hosen oe ients, of

φ

i

f (x) ≃ f

n

(x) =

n

X

i=1

c

i

φ

i

(1.1)

where the oe ients

c

i

are determined through minimization of the mean-square deviation of

f

n

(x)

from

f (x)

. The a ura yofsu hexpansion dependsonthe om-pleteness of theset ofbasisfun tionsandat thelimit ofaninniteset (

n → ∞

)we have

f

n

(x) → f(x)

.

It ispossible to generalize these onsiderations to fun tions ofseveral variables.

Tothispurposewe onsiderthe aseofawavefun tion dependingonthe oordinates

of

n

ele trons

Ψ

el

(x

1 , x

2 , . . . , x

n

)

. Givena ompletesetofone-ele tronspin-orbitals,

{ψ

1 , ψ

2 , . . . , ψ

n

. . .}

, if the oordinates of

n − 1

ele trons are onsidered xed, the resulting fun tion an beexpanded inthe form

Ψ

el

(x

1 , x

2 , ..., x

n

) =

+∞

X

i=1

c

i

(x

2 , x

3 , ..., x

n

)ψ

i

(x

1 )

(1.2)

where the oe ients

c

i

, being a tually fun tions themselves, hold the dependen e on the oordinatesof the remaining

n − 1

ele trons.

Again, onsideringxed

x

3 , x

4 , . . . , x

n

,the oe ients

c

i

,whi harenowfun tions of a singlevariable,

x

2

, an be expanded onthesame basisofspin-orbitals as

c

i

=

+∞

X

j=1

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Repeating su h pro edure for the oordinates of theremaining

n − 2

ele trons, one an have the exa t expansion of the ele troni wavefun tion over the given set of

spin-orbitals:

Ψ

el

(x

1 , x

2 , . . . , x

n

) =

X

i,j,...,p

c

i,j...,p

ψ

i

(x

1 )ψ

j

(x

2 ) . . . ψ

p

(x

n

),

(1.4)

wheretheindi es

i, j, . . . , p

runoverallpossible hoi esofthespin-orbitalsbelonging to the basisset.

1.2 Antisymmetry: Slater's formalism

As above stated, equation (1.4) gives theexa t expansion of a many-parti le

wave-fun tion over a omplete setofmonoele troni spin-orbitals;however, a natural law

imposes a severe restri tion to a fermioni wavefun tion: the antisymmetry

prop-erty. In other terms, for a

n

-ele tron wavefun tion

Ψ

el

(x

1 , x

2 , . . . , x

n

)

thefollowing relationship mustbesatised:

P Ψ

el

(x

1 , x

2 , . . . , x

n

) = σ

P

Ψ

el

(x

1 , x

2 , . . . , x

n

)

(1.5) where

P

performs any permutation of the spin- oordinates

x

1 , x

2 , . . . , x

n

and

σ

P

equals

±1

a ordingasthe permutationisgivenbyanevenoroddnumberof trans-positions.

To over ome the di ulties of building an antisymmetri many-ele tron

wave-fun tion,apossiblestrategyistoperformanexpansionoverasetofantisymmetrized

spin-orbital produ ts, theSlater determinants:

Ψ

el

(x

1 , x

2 , . . . , x

n

) =

X

I

C

I

Φ

I

(1.6) with

Φ

I

=

1 √

n!

ψ

i

1 (1)

ψ

i

2 (1) · · · ψ

i

n

(1)

ψ

i

1 (2)

ψ

i

2 (2) · · · ψ

i

n

(2)

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

ψ

i

1 (n) ψ

i

2 (n) · · · ψ

i

n

(n)

=

√

1 n!

det |ψ

i

1 (1)ψ

i

2 (2) . . . ψ

i

n

(n)|

(1.7)

Herethebasissethasbeen hosenasorthonormal(

hψ

i

|ψ

j

i = δ

ij

)and, onsequently, theresultingsetofSlater determinantsturnsoutto be orthogonal(

hΦ

K

|Φ

L

i = 0

for

K 6= L

).

The use of Slater determinants automati ally guarantees the antisymmetry of

thewavefun tion,sin e thesignofthedeterminant ofthematrix(1.7) hangesupon

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Furthermore, in the ase of a one-determinant approximation to the wavefun tion,

the quantum-me hani al form of thePauli's prin iple dire tlyfollows, sin e the

de-terminantineq. (1.7)vanisheswhentwo olumnshavethesamevalue(twoidenti al

spin-orbitals).

1.2.1 Conguration Intera tion Approa h

WithintheBorn-Oppenheimerapproximation(xednu leimodel),inwhi hthe

ele -troni and nu lear motions an be de oupled and two separate equations an be

solved,the ele troni time-independent S hrödinger equation hastheform

ˆ

H

el

Ψ

el

(X; Q) = E

el

(Q)Ψ

el

(X; Q)

(1.8)

where the ele troni wavefun tion possesesa parametri dependen e on thenu lear

oordinates

Q

. Substitution of(1.6) inequation (1.8)gives:

X

I

ˆ

H

el

Φ

I

c

I

= E

el

X

I

Φ

I

c

I

(1.9)

By appli ation of thebra ve tor

hΦ

J

|

toboth sidesofequation (1.6) one has

X

I

hΦ

J

| ˆ

H

el

|Φ

I

i c

I

= E

el

c

J

(1.10)

whi h an be put inmatrix form

H

= E

(1.11)

where the matrix H haselements

H

JI

= hΦ

J

| ˆ

H

el

|Φ

I

i

and the oe ients

c

J

have been olle ted inthe olumn ve tor .

We note that the problem of solving the ele troni S rhödinger equation has been

redu ed to apurely algebrai problem ofdiagonalizing theHamiltonianmatrix

H

. Expression(1.11)isknownasthefullConguration Intera tion(FCI)expansion

andprovidestheexa tsolutiontotheele troni S rhödingerequationwithinagiven

one-ele tron basis set. The number of determinants in a FCI expansion, obtained

distributing

n

ele trons into

N

orbitals,isgiven by

N

n

!

=

N !

n!(N − n)!

(1.12)

This fa torial dependen e of the number of Slater determinants on the number of

spin-orbitals and ele trons makes the FCI approa h pra ti ally appli able only to

verysmallmole ularsystems[1,2℄. However,inthose asesinwhi hFCI al ulations

anbe arriedout,theresultsserveasusefulben hmarksforevaluatingthea ura y

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Slater's rules

Here, resorting to thewell-knownSlater's rules for one- andtwo-ele tron operators,

we shallillustrate a fastwayto evaluate theHamiltonian matrix elements

H

JI

. Givenaone-ele tronoperator

ˆ

F =

P

n

_i=1

f (i)

ˆ

,onlytwo asesinwhi hthematrix elements give a nonzero resultarepossible:

•

ifthe two determinants areidenti al,

Φ

J

= Φ

I

,one has

H

II

=

n

X

j=1

ψ

i

j

ˆ

f

ψ

i

j

(1.13)

•

if the two determinants have a single spin-orbital dieren e (

Φ

J

6= Φ

I

, with

ψ

j

k

6= ψ

i

k

) theresultis

H

JI

= hψ

j

k

| ˆ

f |ψ

i

k

i

(1.14)

Clearly,all thematrix elementsbetweenSlater determinantsdieringfor morethan

one spin-orbital arezero. Ina similarway,for a two-ele tron operator

ˆ

G =

1

2 n

X

i6=j

ˆ

g(i, j)

(1.15)

thefollowing threepossibilitieso ur:

•

if

Φ

J

= Φ

I

one has

G

II

=

1

2 n

X

k,l=1

(hψ

i

k

ψ

i

l

| ˆg |ψ

i

k

ψ

i

l

i − hψ

i

k

ψ

i

l

| ˆg |ψ

i

l

ψ

i

k

i)

(1.16)

•

if

Φ

J

6= Φ

I

for asingle spin-orbitaldieren e (

ψ

j

k

6= ψ

i

k

)

G

JI

=

n

X

l=1

(hψ

j

k

ψ

i

l

| ˆg |ψ

i

k

ψ

i

l

i − hψ

i

k

ψ

i

l

| ˆg |ψ

i

l

ψ

i

k

i)

(1.17)

•

if

Φ

J

6= Φ

I

for twospin-orbital dieren es (

ψ

j

k

6= ψ

i

k

and

ψ

j

l

6= ψ

i

l

)

G

JI

= hψ

j

k

ψ

j

l

| ˆg |ψ

i

k

ψ

i

l

i − hψ

j

k

ψ

j

l

| ˆg |ψ

i

l

ψ

i

k

i

(1.18)

Weshould stressthat intheabove expressions we haveimpli itly assumed thatthe

equal spin-orbitalsappearinthesameorderinthetwo determinants; if,instead,the

orderisdierent,thepossible hange insigndueto thepermutationsmustbetaken

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1.3 An alternative approa h: se ond quantization

1.3.1 The Fo k spa e

Theformalismwepresentinthisse tionisknownasse ondquantization; itwasrst

developed inphysi s (eldtheory) andlater widelyusedalsoinquantum hemistry

(see Ref.[3℄).

In the se ond quantization language there is a one-to-one orresponden e

be-tween the ele troni wavefun tion

Ψ

el

(x

1 , x

2 , . . . , x

n

)

, in whi h the spin-orbitals

ψ

i

, ψ

j

, . . . , ψ

p

are o upied by ele trons and a state ve tor (ket)

|ki

, where only the o upation numbers(0 or 1) ofthe wholesetof spin-orbitalsaregiven,thatis

|ki = |k

1 , k

2 , . . . , k

N

i , k

i

=







1

if

ψ

i

iso upied

0

if

ψ

i

isuno upied

.

(1.19)

The linear ve tor spa e spanned by basis ve tors in luding all possible kets (1.19),

obtained distributing

n

ele trons in

N

spin-orbitals, is known as the Fo k Spa e. Thereby,ea hSlaterdeterminant hasits orrespondingo upation number ve torin

the Fo kspa e andvi e versa:

|ij . . . pi =

√

1 n!

ψ

i

(x

1 ) ψ

j

(x

1 ) . . . ψ

p

(x

1 )

ψ

i

(x

2 ) ψ

j

(x

2 ) . . . ψ

p

(x

2 )

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

ψ

i

(x

n

) ψ

j

(x

n

) . . . ψ

p

(x

n

)

Due tothe antisymmetry property, theorderinwhi h thespin-orbitalsappear(the

labelsinthe ketve tors) isimportant and one has

|ji . . . pi =

√

1 n!

ψ

j

(x

1 ) ψ

i

(x

1 ) . . . ψ

p

(x

1 )

ψ

j

(x

2 ) ψ

i

(x

2 ) . . . ψ

p

(x

2 )

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

ψ

j

(x

n

) ψ

i

(x

n

) . . . ψ

p

(x

n

)

= − |ij . . . pi

therefore, ea h ve tor ismultipliedby

σ

P

(

= ±1

) underlabelpermutation.

A parti ular ve tor ofthe Fo kspa e is the va uum ve tor, representing the

situ-ation inwhi h noparti lesarepresent

|vaci = |0

1 , 0

2 , . . . , 0

N

i .

(1.20)

1.3.2 Creation and annihilation operators

In orderto onne t ve torswithdierent numberof ele trons,wedene two

opera-tors, alled reation and annihilation operators. The reation operator,

a

+

(16)

that

a

+

_r

_{|ij . . . pi =}







|rij . . . pi

if

r 6∈ (ij . . . p)

0

if

r ∈ (ij . . . p).

(1.21)

Therefore, ifthe ketdoes not in lude theo upation number ofthe spin-orbital

ψ

r

thena parti le isadded andan (

n + 1

)-ele tron wavefun tion isobtained,

Φ(x

1 , x

2 , . . . , x

n

, x

n+1

)

;otherwise,if

r

isalreadyo upiedintheasso iateSlater de-terminant,uponappli ationof

a

+

r

itvanishes,asa onsequen eoftheantisymmetry requirement (two identi al olumns).

Similarly,one maydene the annihilationoperator,

a

r

,su h that

a

_r

_{|rij . . . pi =}







|ij . . . pi

if

r ∈ (ij . . . p)

0

if

r 6∈ (ij . . . p).

(1.22)

where the se ond ase expresses the impossibility of annihilating an ele tron in a

uno upied spin-orbital.

Con luding,wenotethatallstateve tors anbegeneratedbyappli ation ofthe

properstring of reationoperatorsto theva uumstate

a

+

_i

a

+

_j

. . . a

+

_p

_{|vaci = |ij . . . pi}

and thatthe antisymmetry propertyof the basisve torsisensured bythe

anti om-mutativeproperties of theseoperators:

a

+

_i

a

+

_j

+ a

+

_i

a

+

_j

=

h

a

+

_i

, a

+

_j

i

+

= 0

a

_i

a

_j

+ a

_i

a

_j

=

a

_i

, a

_j

+

= 0

a

_i

a

+

_j

+ a

+

_j

a

_i

=

h

a

_i

, a

+

_j

i

+

= δ

ij

1.3.3 Representation of one- and two-ele tron operators

The formof aone-ele tron operator inrst quantizationis

ˆ

F

f q

=

n

X

i=1

f (i)

(1.23)

where thesumrunsoverthenumberofele trons

n

of thesystem. Re allingSlater's rules,illustratedinse tion(1.2.1),thisoperatorgivesnullmatrixelementswhenthe

Slater determinants dier for more than one spin-orbital. The se ond quantization

analogue of (1.23) an be expressedas a linear ombination of produ ts of reation

and annihilationoperators:

ˆ

F

sq

=

X

r,s

(17)

where the indi es

r

and

s

run overthe wholesetof spin-orbitalsandthethematrix

F

is hermitianwith

f

rs

= f

∗

sr

. As anbeeasily proved(see forinstan eRef.[4℄), by omparison withSlater's rules for a one-ele tronoperator (se tion1.2.1), hoosing

f

rs

=

Z

ψ

∗

_r

(x

i

) ˆ

f (x

i

)ψ

s

(x

i

)dx

i

(1.25)

the rst quantization one-ele tron operator

F

ˆ

in (1.23) is equivalent to the se ond quantizationform in(1.24).

We shallnow onsiderthe aseof a two-ele tron operator, su h as,for instan e,

the interele troni repulsion term of the ele troni Hamiltonian; as known, in rst

quantizationit isexpressedas

ˆ

G

f q

=

1

2 n

X

i,j

′

_g(x

i

, x

j

).

(1.26)

We re all that for a two-ele tron operator the matrix elements between two Slater

determinantsarenonzeroonlyifthedeterminants ontainatleasttwoele tronsand

ifthey do notdier bymore than twospin-orbitals.

Analogously,inse ondquantizationa two-ele tron operatorhasthe following form:

ˆ

G

sq

=

1

2 X

rstu

g

rs,tu

a

+

r

a

+

s

a

u

a

t

(1.27)

where thematrix

G

ishermitian(

g

rs,tu

= g

∗

tu,rs

)andthesymmetryproperty

g

rs,tu

=

g

sr,ut

is imposed.

One an easily demonstrate thatthe rst (1.26) and se ond quantization (1.27)

forms be ome identi aliftheparameter

g

rs,tu

areproperly hosenas

g

rs,tu

=

Z Z

ψ

∗

_r

(x

1 )ψ

s

∗

(x

2 )g(x

1 , x

2 )ψ

t

(x

1 )ψ

u

(x

2 )dx

1 dx

2

(1.28)

Making useof the above presentedresults for generi two- andone-ele tron

op-erators, we may now get the se ond quantization representation of the ele troni

Hamiltonian within the Born-Oppenheimer approximation:

ˆ

H

el

=

X

r,s

hψ

r

| h |ψ

s

i a

+

r

a

s

+

1

2 X

rstu

ψ

r

ψ

s

1 r

12 ψ

t

ψ

u

a

+

_r

a

+

_s

a

_u

a

_t

(1.29)

Con luding,itisworthwhilesummarizingtherelevant hara teristi sofoperatorsin

rstandse ondquantizationformalisms. Therstimportantdieren ebetweenthe

tworepresentations on ernsthedependen eonthenumberofele trons: whereasthe

rst quantization operators(1.23) and (1.26) make expli it referen eto thenumber

of ele trons,their se ondquantizationanalogues(1.24) and (1.27) donot havesu h

(18)

thebasissetdependen e. Inparti ular,inrstquantizationthedeterminantsdepend

onthespin-orbitalbasis,whiletheoperatorsareinvariantwithrespe ttothe hoi e

of the basis. On the ontrary, in the se ond quantization representation, the state

ve torsdonothaveanyreferen etothespin-orbitalsandthisinformationis,instead,

ontained inthe operators throughthe

f

rs

(1.25) and

g

rs,tu

(1.28) parameters. 1.3.4 The spin-tra ed repla ement operators

Ausefulsimpli ationintheevaluationofthematrixelementsofone-andtwo

ele -tronoperators anbeobtainedthroughthedenitionofso- alledspin-freeoperators.

Given a set of spin-orbitals (

ψ

i

, ψ

j

, . . . ψ

p

), originated from the same set of spatial orbitals(

φ

i

, φ

j

. . . φ

p

)with

α

and

β

o upations,foraspinlessone-ele tronoperators one has

ˆ

F =

X

rs

hφ

r

| t |φ

s

i (a

+

rα

a

sα

+ a

+

rβ

a

sβ

)

(1.30)

werewenotethatthesummationrunsjustoverthe spatialorbitals. Thespin-tra ed

repla ement operator isdened as

E

rs

= a

+

rα

a

sα

+ a

+

rβ

a

sβ

(1.31)

The ommutation rulefor two spin-tra ed operatorsis

[E

rs

, E

tu

] = δ

st

E

ru

− δ

ru

E

ts

(1.32)

and an important property of su h operators is that they ommute with the total

spin momentum

S

2

andwithits

z

omponent,

S

z

.

Followingtheaboves hemeonearrivesatthedenitionofaspinlesstwo-ele tron

operator:

ˆ

G =

1

2 X

rstu

hφ

r

φ

s

| g |φ

t

φ

u

i (E

rs

E

tu

− δ

ts

E

ru

).

(1.33)

So, nally, using expressions (1.30), (1.31) and (1.33), the ele troni Hamiltonian

an be written as

ˆ

H =

X

rs

h

rs

E

rs

+

1

2 X

rstu

φ

r

φ

s

1 r

12 φ

t

φ

u

(E

rs

E

tu

− δ

ts

E

ru

).

(1.34)

1.4 One-determinant approximation: Hartree-Fo k

the-ory

Among the simplest approximations to the ele troni wavefun tion, one an quote

theHartree-Fo ktheory,where only one Slater determinant

(19)

is onsideredandwhere thespin-orbitals

ψ

i

areoptimized byminimizing the expe -tation value of the ele troni energy

D

Ψ

_ˆ

H

_Ψ

E

. The Hartree-Fo k method an be applied to thedes riptionof theground state aswellastothatof thelowest-energy

state of any given spatial or spin symmetry. This simple and apparently rough

ap-proximation is, however, able to provide, parti ularly in losed shell systems near

their equilibriumgeometry,ele troni energiesthatareinerrorbylessthan1%,and

a number of mole ular properties (dipole moments, for e onstants et ...) with a

reasonable a ura y. Due to its low omputational ost, the Hartree-Fo k method

is routinely used for qualitative studies of large mole ular systems. For a urate

quantitative studies, instead, theHartree-Fo k wavefun tion representsthestarting

point for more sophisti ated approa hes, like the perturbative Møller-Plesset (MP)

orre tions andthe oupled- luster (CC)method( [4,5℄).

1.4.1 Self-Consistent Field (SCF) theory

Given the one-determinant expansion oftheele troni wavefun tion

Ψ(x

1 , x

2 , . . . , x

n

) = (n)

−1/2

det|ψ

1 ψ

2 . . . ψ

n

|

(1.36)

the entral point oftheHartee-Fo ktheoryisto nd thebest spin-orbitals

(

ψ

1 , ψ

2 , . . . , ψ

n

) to use in the Slater determinant. As is well-known, these optimal spin-orbitals aretheeigenfun tionsof aone-ele tron eigenvalueequation

ˆ

F ψ = ǫψ

(1.37)

where

F

ˆ

, termed the Fo k operator, is an operator of a single ele tron whi h takes a ountof anee tiveeld due to the presen eofthenu lei and oftheremaining

n − 1

ele trons. The Hartee-Fo k method is a parti ular form of the independent-parti le model (IPM), where the ele troni intera tions are evaluated by means of

anee tivepotential throughtheFo koperatorandthewavefun tionisexpressed

asan antisymmetri produ tofone-ele tron fun tions.

In order to obtain equation (1.37), we start expressing the variational energy

approximation oftheone-determinant wavefun tion (1.36)

E =

D

Ψ

_ˆ

H

_Ψ

E

=

n

X

i

hψ

i

|h| ψ

i

i +

1

2 n

X

i,j

hψ

i

ψ

j

|| ψ

i

ψ

j

i

(1.38)

where wehave useda shorternotation, indi ating

hψ

i

ψ

j

|| ψ

i

ψ

j

i = hψ

i

ψ

j

|g| ψ

i

ψ

j

i − hψ

i

ψ

j

|g| ψ

j

ψ

i

(1.39)

Let we hoose an orthonormalized set of spin orbitals, su h that

hψ

i

|ψ

j

| =i δ

ij

. At the stationary point, for any innitesimal variation

ψ

i

= ψ

i

+ δψ

i

the ondition

(20)

δE = 0

mustbefullled. Su haninnitesimalvariationofthespinorbitalbasis an beobtained applyingthe unitaryoperator

U = e

ˆ

T

to thewavefun tion

Ψ

,where

T

ˆ

is anantihermitian operator, thatinse ond quantization an be expressedas

ˆ

T = − ˆ

T

+

=

X

r,s

t

rs

a

+

r

a

s

(1.40)

with

t

rs

= −t

sr

.

Upon opportunemanipulations, onearrivesat

δE =

n

X

i=1

X

a>n

t

ai

hΨ| ˆ

H |Ψ

a

i

i + c.c.

(1.41)

where . . indi ates the omplex onjugate of therst term andthe onventionof

indi atingwithindi es

i, j . . .

theo upiedspin-orbitalsandwith

a, b, . . .

thevirtual ones has been adopted The relation

t

ai

= −t

∗

ia

has been used and we also have introdu ed theshorternotation

|Ψ

a

i

to indi atetheSlaterdeterminant inwhi hthe spin-orbital

ψ

i

hasbeen repla edby

ψ

a

.

Equation(1.41) dire tlyleads to thewell-known form of theBrillouin Theorem

[6,7℄

hΨ

a

i

| ˆ

H |Ψi = 0

(1.42) whi hstatesthatthebest spin-orbitalstousearesu hthattheintera tionbetween

Ψ

and any singlyex ited determinant

Ψ

a

i

is zero.

ResortingtoSlater'srules(se tion1.2.1)andintrodu ingtwoauxiliaryoperators,

ˆ

J

(Coulomb operator)

hψ

r

| ˆ

J |ψ

s

i =

n

X

j=1

hψ

r

ψ

j

|

1 r

12 |ψ

s

ψ

j

i

and

K

ˆ

(Ex hange operator)

hψ

r

| ˆ

K |ψ

s

i =

n

X

j=1

hψ

r

ψ

j

|

1 r

12 |ψ

j

ψ

s

i

one an writethe generalized Hartree-Fo k equations

ˆ

F |ψ

i

i =

n

X

j=1

|ψ

j

i ǫ

ji

(1.43)

where we have dened the Fo k operator

ˆ

F = ˆ

h + ˆ

_{J − ˆ}

K

and

ǫ

ji

= hψ

j

| ˆ

F |ψ

i

i =

hψ

i

| ˆ

F |ψ

j

i

∗

.

We an exploit the hermiti ity of

ǫ

, onsidering the unitary transformation

U

(21)

whi h diagonalizes

ǫ

and noting that hanging the spin-orbitals a ording to the transformation

ψ

_i

′

=

n

X

j=1

ψ

j

U

ji

the Fo koperator remains invariant undersu htransformation. So fromthe

gener-alized equations(1.43) one arrivesat the anoni al Hartree-Fo k equations:

ˆ

F ψ

′

_i

= ǫ

i

ψ

′

i

(1.44)

We re all that, sin e

F

ˆ

depends on its eigenfun tions

ψ

i

, eq. (1.44) annot be solved in a single step. An iterative method must instead be used, starting from a

guess of spin-orbitals, buildingan approximated

F

ˆ

,diagonalizing it andpro eeding until onvergen e isrea hed (self onsisten y).

1.4.2 Koopmans' Theorem

The eigenvalues ofthe anoni alFo kequations(1.44) aretermedorbital energies

and havea dire tphysi al interpretation, sin e

−ǫ

i

represents arst approximation totheIonizationPotential(IP),namelytheenergyneededtoremoveanele tronfrom

thespin-orbital

ψ

i

. Analogously,

−ǫ

r

isarstapproximationtotheEle tronAnity (EA) of the neutral mole ule. This resultis known asKoopmans'Theorem [8℄and

an interesting dis ussion an befound inRef.[9℄.

Let us onsider the ionized system obtained by removing an ele tron from the

spin-orbital

ψ

i

inthe Hartree-Fo k determinant

Ψ

. The energy of the

n − 1

deter-minant is

E

_i

+

=

D

a

_i

Ψ

_ˆ

_H

_a

_i

Ψ

E

=

D

Ψ

_a

+

_i

_Ha

ˆ

_i

_Ψ

E

(1.45)

= E +

D

Ψ

_[a

+

_i

, ˆ

_H]a

_i

_Ψ

E

.

(1.46) Equation(1.46) anbeeasilymanipulatedexploitingthe ommutationrulesbetween

reation and annihilation operators (see se tion 1.3.2) and one promptly arrives at

the formulationof the Koopmans' Theorem fortheionization energy:

D

Ψ

_[a

+

_i

, ˆ

_H]a

_i

_Ψ

E

_{= −h}

ii

− (J

ii

− K

ii

) = −ǫ

i

(1.47) An analogous expression anbe derived fortheEle tron Anity

E − E

−

k

= −ǫ

k

Thisapproximationisbasedonasimplemodelfortheopen-shellionizedsystem,

where theioni wavefun tionisnot allowedto relaxupontheionization pro ess

(re-laxationenergy)butitisinsteadbuiltfromthefrozenMOsoftheneutralmole ule;

asa onsequen e,toolargeIPsandtoosmallEAsareattained. Inadditionto these

orbital relaxation ee ts, theHF method also negle ts the orrelation energy;

how-ever,whilefortheIPs,theKTapproximationyieldsreasonableresults,duetoasort

(22)

1.5 The Ele tron orrelation problem

1.5.1 Ele trondistribution: densityfun tions and densitymatri es

Inorder tobetter dis usstheproblemoftheele tron orrelationenergy,whi h

rep-resents one of the entral issues in the ele troni stru ture theory, here, we shall

introdu e the on epts ofdensity fun tionsand density matri es [10 13℄. Thegreat

advantageof using this fun tionsbasi ally arises from their relative simpli ity,

par-ti ularlywhen omparedto the omplexityofsophisti atedwavefun tions,and from

thepromptinsight theygiveabout thephysi al ontent oftheele trondistribution.

Let us onsider a

n

Ψ(x

1 , x

2 , . . . , x

n

)

, the probability of ndingele tron

1

in

x

1

and at the same timeele tron

2

in

x

2

et . is given by

dP (x

1 , x

1 + dx

1 ; . . . ; x

n

, x

n

+ dx

n

) = Ψ(x

1 , x

2 , . . . , x

n

)Ψ

∗

(x

1 , x

2 , . . . , x

n

)dx

1 dx

2 .

(1.48)

Then, theprobabilityon anyof

n

ele tronin

dx

1

is expressedas

dP (x

1 , x

1 + dx

1 ) = dx

1 Z

Ψ(x

1 , x

2 , . . . , x

n

)Ψ

∗

(x

1 , x

2 , . . . , x

n

) dx

2 dx

3 . . . dx

n

(1.49)

By multiplying eq.(1.49) by the number of ele trons,

n

, we obtain the amount of harge in volume

dx

1

. We write this probability as

ρ(x

1 )dx

1

where we have intro-du ed thedensity fun tion

ρ(x

1 )

dened as

ρ(x

1 ) = n

Z

Ψ(x

1 , x

2 , . . . , x

n

)Ψ

∗

(x

1 , x

2 , . . . , x

n

)dx

2 dx

3 . . . dx

n

(1.50)

Weshould stressthat

x

1

on the left of eq. (1.50) doesnot indi ate the oordinates of ele tron

1

but thepoint 1 ofthewholespa e inwhi h thedensityis evaluated. Byintegrationoverthe spin oordinates, itisthenpossibletoobtaintheprobability

of ndingan ele tronat point

1

regardlessof itsspin:

P (r

1 ) =

Z

ρ(dx

1 )ds

1 .

(1.51)

Su h denitions given for a single ele tron an be easily extended to two or more

parti les; so,inthe ase oftwo ele trons, thepair density fun tion be omes

ρ(x

1 , x

2 ) = n(n − 1)

Z

Ψ(x

1 , x

2 , . . . , x

n

)Ψ

∗

(x

1 , x

2 , . . . , x

n

)dx

3 dx

4 . . . dx

n

(1.52)

and its spinless ounterpart is

P (r

1 , r

2 ) =

Z

(23)

Let

F =

ˆ

P

n

i=1

f (x

i

)

bea one-ele tronmultipli ative operatorand

Ψ(x

1 , x

2 , . . . , x

n

)

a

n

-ele tron wavefun tion, theexpe tationvalueof

F

ˆ

is

hF i =

n

X

i=1

Z

Ψ

∗

(x

1 , x

2 , . . . , x

n

)f (x

i

)Ψ(x

1 , x

2 , . . . , x

n

) dx

1 dx

2 . . . dx

n

= n

Z

Ψ

∗

(x

1 , x

2 , . . . , x

n

)f (x

1 )Ψ(x

1 , x

2 , . . . , x

n

) dx

1 dx

2 . . . dx

n

.

(1.54)

Sin e

f (x

1 )

isjust amultiplier, expression(1.54) an be rearranged, using the de-nition of densityfun tion givenineq. (1.50),to obtain

hF i =

Z

f (x

1 )ρ(x

1 ) dx

1 .

(1.55)

We note thatin themore general aseof non-multipli ative operator

f (x

1 )

,eq. (1.54) annot be simply put in the form (1.55), sin e

Ψ

∗

_(x

1 , x

2 , . . . , x

n

)

annot be shifted to the right of the operator. However, a simple mathemati al tri k an be

used: sin e

f (x

1 )

worksonly on fun tions of

x

1

,

Ψ

∗

an be made exempt from the

a tion of

f (x

1 )

just hangingthenameofthevariablefrom

x

1

to

x

′

1

;then, uponthe a tion of

f (x

1 )

on

Ψ

we an hange ba k

x

′

1 → x

1

and pro eed to the integration. Pra ti ally, the expe tationvaluebe omes

hF i =

Z

x

′

1 =x

1 f (x

1 )ρ(x

1 , x

′

1 ) dx

1 .

(1.56)

where the thedensity matrix

ρ(x

1 ; x

′

1 ) = n

Z

Ψ(x

1 , x

2 , . . . , x

n

)Ψ

∗

(x

′

1 , x

2 , . . . , x

n

) dx

2 dx

3 . . . dx

n

(1.57)

hasbeen introdu ed.

For two-ele tron operators, thetwo-parti le density matrix an be dened

ρ(x

1 , x

2 ; x

′

1 , x

′

2 ) = n(n − 1)

Z

Ψ(x

1 , x

2 , . . . , x

n

)Ψ

∗

(x

′

1 , x

′

2 , . . . , x

n

) dx

3 dx

4 . . . dx

n

(1.58)

and hen etheexpe tationvalueof ageneri two-ele tron operator

ˆ

G =

1

2 n

X

i6=j=1

ˆ

g(x

i

, x

j

)

an be obtainedsimplyevaluating

hGi =

1 ₂

Z

x

′

1 = x

1 x

′

2 = x

2 ˆ

g(x

1 , x

2 ) ρ(x

1 , x

2 ; x

′

1 , x

′

2 ) dx

1 dx

2

(1.59)

(24)

Integratingoverthespin oordinates, thespinlessdensity matri es analogousof the

spinlessdensityfun tions(1.51) and(1.53) aredened:

ρ(r

1 ; r

′

1 ) =

Z

s

′

1 =s

1 ρ(x

1 ; x

′

1 ) ds

1

(1.60) and

ρ(r

1 , r

′

1 ; r

2 , r

′

2 ) =

Z

s

′

1 = s

1 s

′

2 = s

2 ρ(x

1 , x

′

1 ; x

2 , x

′

2 ) ds

1 ds

2

(1.61)

Obviously,followingthesameformalism,densitymatri esforthreeormoreparti les

an be dened.

Finally, itis worthwhile to point out that the densitymatrix

ρ(x

1 ; x

′

1 )

does not have ana tual physi al meaning initself but onlyits diagonal part

ρ(x

1 ; x

1 )

,whi h oin ides withthedensityfun tion

ρ(x

1 )

.

Then, given a omplete set oforthonormal basis fun tions

{ψ

1 , ψ

2 , . . .}

,we may expand the one andtwo-parti le densitymatri esintheforms

ρ(x

1 ; x

′

1 ) =

X

i,j

R

ij

ψ

i

(x

1 )ψ

∗

j

(x

′

1 )

(1.62) and

ρ(x

1 , x

2 ; x

′

1 , x

′

2 ) =

X

i,j,k,l

R

ij;kl

ψ

i

(x

1 )ψ

j

(x

2 )ψ

k

∗

(x

′

1 )ψ

l

∗

(x

′

2 )

(1.63)

where the oe ients

R

ij

and

R

ij;kl

arenumeri al fa tors.

Finally,the expe tation values of one- and two-ele tron operators an be evaluated

respe tively as

ˆ

F =

Z

x

′

1 =x

1 ˆ

f (x

1 )ρ(x

1 ; x

′

1 ) dx

1 =

X

i,j

R

ij

F

ji

(1.64) and

ˆ

G =

1

2 Z

x

′

1 = x

1 x

′

2 = x

2 ˆ

g(x

1 , x

2 )ρ(x

1 , x

′

1 ; x

2 , x

′

2 ) dx

1 dx

2 =

1

2 X

i,j,k,l

R

ij;kl

G

kl;ij

(1.65)

where thematri es

F

and

G

have elements

F

ji

= hψ

j

|f(x

1 )| ψ

i

(1.66)

and

(25)

1.5.2 The one-determinant approximation ase

In the aseof aone-determinant

n

Ψ(x

1 , . . . , x

n

) =

1 √

n!

kψ

1 ψ

2 . . . ψ

n

k

the forms of the one-and two-parti le density matri es an be obtained omparing

the above expressions (1.64) and (1.65) withtheexpe tation valueof theele troni

Hamiltonian inSlater's formalism (seese tion1.2.1)

E =

D

Ψ

_ˆ

H

_Ψ

E

=

X

i

hψ

i

|h| ψ

i

i +

1

2 X

ij

(hψ

i

ψ

j

|g| ψ

i

ψ

j

i − hψ

i

ψ

j

|g| ψ

j

ψ

i

i)

(1.68)

Forthe one-ele tronpart of

ˆ

H

we have thatthefollowing equalitymust besatised

R

ij

= δ

ij

→ ρ(x

1 ; x

′

1 ) =

n

X

i=1

ψ

i

(x

1 )ψ

∗

i

(x

′

1 )

(1.69)

withboth

i

and

j

o upied;forthe two-ele tron omponent,weobtaintherelations











R

ij;ij

= 1

R

ij;ji

= −1

R

ii;ii

= 0

again with

i

and

j

o upied and thus

ρ(x

1 , x

2 ; x

′

1 , x

′

2 ) =

n

X

i,j=1

(ψ

i

(x

1 )ψ

j

(x

2 )ψ

i

∗

(x

′

1 )ψ

j

∗

(x

′

2 ) − ψ

i

(x

1 )ψ

j

(x

2 )ψ

∗

j

(x

′

1 )ψ

i

∗

(x

′

2 )).

(1.70)

An important result is that eq. (1.70) an be expressed in terms of one-ele tron

density matrix

ρ(x

1 , x

2 ; x

′

1 , x

′

2 ) = ρ(x

1 ; x

′

1 )ρ(x

2 ; x

′

2 ) − ρ(x

2 ; x

′

1 )ρ(x

1 ; x

′

2 )

(1.71)

and, moregenerally, for anyn-ele tron densitymatrix itmay be shownthat

ρ

n

(x

1 , . . . , x

n

; x

′

1 , . . . , x

′

n

) =

ρ(x

1 ; x

′

1 ) ρ(x

1 ; x

′

2 ) · · · ρ(x

1 ; x

′

n

)

ρ(x

2 ; x

′

1 ) ρ(x

2 ; x

′

2 ) · · · ρ(x

2 ; x

′

n

)

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

ρ(x

n

; x

′

1 ) ρ(x

n

; x

′

2 ) · · · ρ(x

n

; x

′

n

)

(1.72)

Re alling the denition given of the spinless density matri es (1.60, 1.61) and

dierentiating the spin-orbitals a ording to their spin fa tor, for a losed-shell

(26)

ρ(x

1 ; x

′

1 ) =

n/2

X

i

α

=1

φ

i

(r

1 )φ

∗

i

(r

1 ′

)α(s

1 )α

∗

(s

′

1 ) +

n/2

X

i

β

=1

φ

i

(r

1 )φ

∗

i

(r

1 ′

)β(s

1 )β

∗

(s

′

1 )

= P

₁

αα

α(s

1 )α(s

′

1 ) + P

ββ

1 β(s

1 )β(s

′

1 )

(1.73)

and integrating overthe spin we obtainthespinlessdensity matrix

P

1 (r

1 ; r

1 ′

) = P

1 αα

+ P

1 ββ

(1.74) with

P

αα

1 = P

1 ββ

.

We now turnto thepair densitymatrix;as an be shown, for awavefun tion of

denite spin, it onsistsof six omponents (

αααα

,

ββββ

,

αβαβ

,

βαβα

,

αββα

and

βααβ

), whi hredu e to four afterintegration over thespin

P

2 (r

1 , r

2 ; r

1 ′

, r

′

2 ) = P

2 αααα

+ P

2 ββββ

+ P

αβαβ

2 + P

βαβα

2

(1.75)

Re allingthatintheone-determinant asethetwo-parti ledensitymatrix anbe

fa -torizedintermsoftheone-parti ledensitymatri es(1.71),thefollowingexpressions

areobtained for thepair fun tions(imposing

r

′

1 = r

1

and

r

′

2 = r

2

)

P

₂

αα

(r

1 , r

2 ) = P

1 α

(r

1 )P

1 α

(r

2 ) − P

1 α

(r

1 ; r

2 )P

1 α

(r

2 ; r

1 )

(1.76)

P

₂

ββ

(r

1 , r

2 ) = P

1 β

(r

1 )P

1 β

(r

2 ) − P

1 β

(r

1 ; r

2 )P

1 β

(r

2 ; r

1 )

(1.77)

P

₂

αβ

(r

1 , r

2 ) = P

1 α

(r

1 )P

1 β

(r

2 )

(1.78)

P

₂

βα

(r

1 , r

2 ) = P

1 β

(r

1 )P

1 α

(r

2 )

(1.79)

From these expressions, indi ating the probability of nding ele trons

simultane-ously at two point in spa e with a given spin onguration, we an get a prompt

understanding of the ele tron orrelation problem. As is apparent, the motion of

ele trons withthesame spin,

αα

(1.76)or

ββ

(1.77),isdes ribedby orrelated fun -tions and

P

αα

2 (r

1 , r

2 )

vanishesas

r

2 → r

1

. Thistypeof orrelation,knownasFermi orrelation, avoids ele trons of parallel spin being at the same point of spa e and

dire tlyarises fromtheantisymmetry propertyof afermioni wavefun tion. Onthe

ontrary,fromeqs. (1.78) and(1.79),weseethatthereisno orrelationbetween the

motion of ele trons with opposite spin, sin e the probability of nding them in

r

1

and

r

2

atthe same timeisgivenjustbytheprodu toftheprobabilities oftheea h of two independent events. Thisla kof orrelation(Coulomb orrelation) is learly

a serious defe t in the one-determinant model, sin e the mutual repulsion between

pairs of ele trons is not properly taken into a ount and the probability of nding

(27)

1.5.3 Stati al and Dynami al Correlation

Fromaquantitative pointofview,the orrelationenergyisdened(Löwdin,1955)

asthedieren ebetweentheexa t energy(pra ti allytheenergyofFCI

wavefun -tion) and the energy oftheHartree-Fo kwavefun tion

E

corr

= E

exact

− E

H−F

within agiven basissetapproximation. Althoughinitself itrepresents avery small

fra tionoftheele troni energy,itsa uratetreatmentisessentialwhendealingwith

energydieren eswhi hareofthesameorderofmagnitudeofthe orrelationenergy

( hemi al rea tivity,ex itation energieset .).

A tually, twodierent ee ts ofele troni orrelationexist:

•

the stati al orrelation,whi h isasso iated withtheproblems ofthe multi on-gurational hara terof thewavefun tion;

•

the dynami al orrelation,whi h is, instead,related to theee ts ofthe inter-ele troni intera tions.

Referring the theHartree-Fo kdes ription of the

H

2

mole ule disso iation, the dis-tin tion between the stati al and thedynami alee ts be omes lear. At the

equi-librium geometry, thewavefun tion is qualitatively well des ribed bythe losed-sell

Hartree-Fo kdeterminant and the orrelationenergy essentiallyarises from the

dy-nami al ee tsof the interele troni repulsions. Ontheother hand,at the

disso ia-tionlimit,wherethereisno oulombrepulsionbetween thetwoele trons,thefailure

of the one-determinant approximation is due to the need to take into a ount the

near-degenera y between the

σ

2 g

and

σ

2 u

ongurations.

1.6 Handling the Stati al Correlation: MCSCF Theory

Asabovestated, inmany hemi al andphysi al phenomena,su h astherupture or

formation of hemi al bonds, or thedes ription of ele troni ally ex ited states, the

one-determinantapproximationdramati allyfailsdueto theintrinsi multireferen e

nature of the problem. Thesestati al orrelation ee ts an be properly taken into

a ount resorting to a multideterminant expansion of the wavefun tion, inwhi h a

simultaneous variational optimization of spin-orbitals and expansion oe ients is

performed: su h strategy is alled Multi ongurational Self-Consistent Field

(MC-SCF) approa h.

Startingfrom atrun ated CIexpansion

Ψ =

N

X

K=1

(28)

inordertobuildaMCSCFwavefun tion weneedtoimposethattheenergyvariation

withrespe ttoaninnitesimalvariationofbothorbitals(

φ

′

_{= φ+δφ}

)and oe ients

(

C

′

K

= C

K

+ δC

K

) iszero.

The optimization an be performed resorting to both a single-step

Newton-Rahpson te hnique and a two-step approa h (Super CI),where rst the oe ients

C

K

andthenthe orbitals areiteratively optimized until self- onsisten yis rea hed. Following thepro edurepresentedinse tion(1.4.1)for theHartree-Fo ktheory,

theself- onsisten y ondition ishereexpressedas

X

rs

t

rs

hΨ| ˆ

H |E

rs

Ψi − hE

sr

Ψ| ˆ

H |Ψi

= 0

(1.81)

and itissatised by theExtended-Brillouin Theorem[14℄

hΨ| ˆ

H |(E

rs

− E

sr

)Ψi = 0

(1.82)

Inotherterms, whentheenergyisstationary,the ontra tedsingleex itations

Ψ

r

s

=

(E

rs

−E

sr

)Ψ

donotintera twiththeoptimizedMCSCFwavefun tion. TheSuperCI method ispra ti ally basedupon aniterative pro edure, whi h onsistsin building

an improved wavefun tion

Ψ

′

= Ψ +

X

r>s

c

rs

Ψ

r

s

(1.83)

diagonalizing the CI matrix and then using the oe ients of the single-ex ited

fun tions,

c

rs

, for onstru ting the matrix

T

, whi h operates the unitary orbital transformation (

U

= e

T

).

1.6.1 Complete A tive Spa e (CAS)

The key issue in the onstru tion of a redu ed CI spa e in whi h to expand the

MC wavefun tion is essentially how to sele t a limited number of ele troni

ong-urations able to properly take into a ount the stati al orrelation energy ee ts.

In the present work we shall adopt a parti ular and largely used type of MCSCF

wavefun tion, known as Complete A tive Spa e Self-Consistent Field (CASSCF)

wavefun tion [15℄. Aswe shallwidelydis ussinthenext hapter, this fun tion

rep-resentsthe zero orderwavefun tion,

Ψ

(0)

,inour perturbative approa h.

Theidea of A tive Spa e provides a useful pre ept to hoosethe relevant

on-gurations of the CI expansion (1.80). It is based upon the partitioning of the

spin-orbitalsinto three lasses:

1. ore(i,j,...),whi hhaveo upationnumberequalto

1

inallthedeterminants

(29)

2. a tive (a,b,...),withall thepossible o upation number from

0

to

1

; 3. virtual(r, s,...),whi h arenevero upied inanydeterminant

Φ

K

.

The CASSCF wavefun tion is built by performing a Full CI expansion within the

a tive orbitals subspa e and then optimizing oe ients and orbitals until

self- onsisten y. However, it is important to stress that the CASSCF approa h is not

a bla k-box method and there is not a re ipe to sele t the right a tive spa e.

However,itshouldbealways arefully hoseninordertoin ludealltheorbitalsthat

are thought to be involved in some measure in the hemi al and physi al pro ess

(30)

(31)

N

-ele tron Valen e State

Perturbation Theory

Multireferen eperturbationtheories(MRPT)representapowerfulandrelatively

in-expensive toolfor thetreatment ofele troni orrelation inmole ules. Asdis ussed

in the previous hapter (se tion 1.5.3), in many mole ular phenomena su h as the

breaking of a hemi al bond or the ele troni transition to an ex ited state, a

sin-gle referen e wavefun tion does not su e to provide a good approximation to the

solution of the time independent S hrödinger equation; many ele troni

ongura-tions anbeimportantandazeroorderdes riptionoftheele troni stru tureofthe

mole ule may not leave out of onsideration su h quasidegenerate ongurations.

The in lusion of the quasidegenerate ongurations a ounts for what is alled the

stati al orrelation (se tion 1.6);the dynami al omponent ouldbe dealt with

per-turbationally witha suitable MRPT. A keyissue inMRPT on erns the denition

of a proper zeroorder Hamiltonian

H

0

. Inthe earlytheories,whi h were developed at the beginning of the 1970's, su h as CIPSI [16℄,

H

0

was dened in terms of a oneele tron, Fo klike, operator and the zero order fun tions(perturbers), usedto

buildtherstorder orre tiontothewavefun tion,weresimpleSlater determinants.

The idea that

H

0

should be basedon a oneele tron operator still persists inmost modernMRPT's. Forinstan einCASPT2[17,18℄,one ofthemost su essfulforms

of MRPT,

H

0

isa proje ted generalized Fo koperator andtheperturbersarebuilt in terms of internally ontra ted ex itations (vide infra). Dyall [19℄ showed that

theusageof orre tionfun tionsderiving fromaoneele tron operator introdu es a

bias in the energy al ulation sin e the zero order referen e wavefun tion properly

takesinto onsiderationthebiele troni intera tionso urringamongthea tive

ele -trons whereas the orre tion fun tionsarenot allowed to do so. In orderto obviate

su hdi ultyDyallproposedtheuseofamodelHamiltonian,partiallybiele troni .