Higher­order SCF response functions in a quasi­energy formulation

Higher­order SCF 

response functions in a  quasi­energy formulation

Andreas J. Thorvaldsen

A dissertation for the degree of Philosophiae Doctor U ^{NIVERSITY   OF}  T ^ROMSØ

Faculty of Sciences

Centre for Theoretical and Computational Chemistry Department of Chemistry 

August 8,  2008

response funtions in a

quasi-energy formulation

Andreas J. Thorvaldsen

A dissertation for the degree of Philosophiae Dotor

UNIVERSITY OF TROMSØ

Faulty of Siene

Centre for Theoretial and Computational Chemistry

Department of Chemistry

Title:

Higher-orderSCF response funtions ina quasi-energyformulation

Author:

AndreasJohan Thorvaldsen

Address:

Centre for Theoretialand ComputationalChemistry

Departmentof Chemistry

University of Tromsø

NO-9037TROMSØ

NORWAY

E-mail address:

Andreas.Thorvaldsenhem.uit.no

Key words:

omputationalhemistry,Kohn-ShamDFT,omputationalspetrosopy,

responsefuntions,Floquettheory,quasi-energyderivatives,perturbation-

First of all, I would like to express my gratitude to my supervisor Prof.

Kenneth Ruud, for his advie, enouragement and patiene throughout the

ourse of this work.

I alsowant to thank my o-authors Sonia Coriani,Kasper Kristensen, Poul

Jørgensen, Mihaª Jaszu«ski, Lara Ferrighi and Antonio Rizzo, for their ef-

fortsandpatieneinbringingmypreliminaryderivationstoaomprehensive,

appliable andpublishable form,aswellasfor manyinformativeand edua-

tional disussions.

Thanks to olleagues Alemayehu Mekonnen, and Dmitry Shherbin, with

whom I have shared oe spae, and many more. I have found the Depart-

mentofChemistrytobeaalmandomfortableenvironmentduringmytime

here. I would also like to thank my parents for their enouragement, om-

mentsand proofreadingof this thesis, and Kerstinand Jan-Olov Strömberg

for their hospitality duringmy stay inStokholm.

Finally, I want to thank the University of Tromsø for funding this work,

andtheNorwegianSuperomputingConsortium(NOTUR)forprovidingthe

needed omputing resoures.

Tromsø, August 8,2008

This thesis is onerned with omputer modelling of moleules interating

with eletromagneti radiation, for appliations in spetrosopy. Response

theory is used, in whih time-dependent perturbation theory applied to the

ground state permits the study of both ground and exited states. For the

lassofself-onsistenteld (SCF)eletronistruturemodels,whihinludes

Hartree-Fok- and all Kohn-Sham DFT models, a full hierahy of new for-

mulas for response funtions have been derived. Although there are several

equivalent formulas for a given response funtion, typially a spei one is

preferabledue toomputationalonsiderations.

Thederived formulasare expressed intermsof theatomiorbital(AO)den-

sitymatrix,and validalsowithtime-andperturbationdependent AOs, suh

as the magneti eld-dependent London or gauge-inluding AOs, whih are

employedtoobtainimprovedbasissetonvergene andgauge-originindepen-

dent results. The density matrix has an advantage over the more ommon

moleularorbital oeientmatrix (MO) parameterizationinthat itdeays

rapidlywiththedistanebetween atoms(exeptindiretionsofondution).

Forlarge moleules onemay thereforetrunate the density matrix and treat

it as sparse. Although this is not presently utilized in our implementation,

it isexpeted to lead togreat omputationalsavings.

To resolve any ambiguity in the denition of response theory, we formulate

it by applying perturbation theory to Floquet theory, whih is a quantum-

mehanial theory that inludesso-alled semi-lassial radiation,by whih

both stimulatedand spontaneous emissionandabsorption an be predited.

The entralquantity inFloquettheory isthe quasi-energy,and this isthere-

fore the 'quasi-energy formalism' of response theory.

The DALTON quantum hemistry program has sine long been the leading

software for omputing moleularproperties. Using the programstrutures

already present in the ode, suh as integrals and integral derivatives, in

addition to reently implemented SCF and SCF-response program modules

funtions, relevant to spetrosopies suh as Cotton-Mouton, oherent anti-

Stokes Raman sattering (CARS), and eletri-eld-gradient indued bire-

fringene(EFGB).

I A. J. Thorvaldsen, K. Ruud, K. Kristensen, P. Jørgensen and S. Co-

riani: Adensity matrix-based quasienergyformulation of Kohn-Sham

densityfuntionalresponsetheoryusingperturbation-andtime-dependent

basis sets. Journal of Chemial Physis(aepted).

II A. J. Thorvaldsen, L. Ferrighi, K. Ruud, H. Ågren, S. Coriani and P.

Jørgensen: Analytiabinitioalulationsof Coherent anti-StokesRa-

man Sattering (CARS). Submitted to Physial ChemistryChemial

Physis.

III A. J. Thorvaldsen, K. Ruud and M. Jaszu«ski: Analyti alulations

ofvibrationalhyperpolarizabilitiesintheatomiorbitalbasis. Journal

of PhysialChemistry A (aepted).

IV A. J. Thorvaldsen, K. Ruud, A. Rizzo and S. Coriani: Analyti al-

ulations of frequeny-dependent hypermagnetizabilities and Cotton

Moutononstants usingLondonatomi orbitals. JournalofChemial

Physis (aepted).

V D. Shherbin, A. J. Thorvaldsen, K. Ruud, A. Rizzo and S. Cori-

ani: Analyti alulations of nonlinear mixed eletri and magneti

frequeny-dependentmoleularpropertiesusingLondonatomiorbitals:

Bukingham birefringene. Submitted to Physial Chemistry Chemi-

Related papers not inluded in the thesis

•

^{Radovan Bast, Andreas J.} Thorvaldsen, Magnus Ringholm and Ken- nethRuud: Atomi orbital-basedubiresponse theoryfor one-,two-

and four-omponent relativisti self-onsistent eld models. Submit-

ted toChemial Physis.

•

^AndreasJ.Thorvaldsen,KennethRuud,MaximFedorovskyandWerner Hug: An atomiorbital-based sheme for analytialulations of Ra-

man OptialAtivity Spetra. Manusript.

•

^{T. Kjærgaard, P. Jørgensen, A. J.} Thorvaldsen and S. Coriani: A gauge-originindependentformulationandimplementationofMagneto-

optialAtivity within atomi-orbital-densitybased Hartree-Fok and

1 Introdution 11

2 Quantum mehanis 13

2.1 Shrödingerequation . . . 13

2.2 Born-Oppenheimer approximation . . . 15

2.3 Self-onsistent eld approximation . . . 16

2.3.1 Hartree-Fok . . . 17

2.3.2 Kohn-Sham DFT . . . 18

3 Properties and spetra 21 3.1 Propagationand Floquet theory . . . 22

3.2 Radiationpotential . . . 24

3.3 Response theory. . . 27

3.4 Resonane . . . 30

4 Summary and outlook 37

Introdution

It was unexplained observations in spetrosopy that led to the advent of

quantum mehanis in the mid1920s 1

. The radiation emitted by hot gases

showed sharp peaks at ertain wavelengths, whih ould not be predited

with existing theories. The two equivalent theories of quantum mehanis

proposedbyHeisenbergandShrödinger 2

explainedthepeaksasarisingwhen

the moleule jumps between two of its eigenstates, with the wavelength of

the peak determined by the dierene between the two eigenenergies, and

theintensityofthepeaksbythepopulationsoftheeigenstatestogether with

the transitiondipolemoment.

However, both the spetra themselves and the Shrödinger equation, whih

must be solved in order to preditspetra, are vastlyomplex, as expressed

by another pioneer, Dira 3

:

The fundamental laws neessary for the mathematial treatment

of a large part of physis and the whole of hemistry are thus

ompletely known, and the diulty lies only in the fat that ap-

pliation of these laws leads to equations that are too omplex to

be solved.

In theearlydays of omputational(theoretial)hemistry, alulationswere

1

W.Heisenberg: ÜberQuantentheoretisheUmdeutun KinematisherundMehanis-

herBeziehungen,ZeitshriftfürPhysik,vol. 33,p. 879-893(1925)

2

E. Shrödinger: An UndulatoryTheoryof theMehanisofAtoms andMoleules,

Phys. Rev. 28(6): 10491070(1926)

3

P.A.M. Dira: Quantum mehanis of many-eletron systems, Pro. Royal So.

arriedoutbyhand(penilandpaper) 4,5

orbymehanialalulators. With

theinvention ofthe digitalomputer,omputationalhemistrysoonbeame

one of its main tasks, and has ontinued to be so. But still today, after 80

years of knowing the theoretial foundation and many billion-fold inreases

in omputing power, there is still a onsiderable gap between the auray

deliveredbyomputation,andthatoftheexperimentsondutedinhemial

laboratories 6

. Thus, at present itseems Dira wasright.

Althoughomputationhas yettorepliateexperiment,italreadyserveswell

toomplement,estimateorpreviewexperiment,as,forinstane,inthephar-

maeutial industry. Most of the eorts of omputational (and theoretial)

hemists, and their omputers, are put into solving the time-independent

Shrödingerequation(SE): Moleulargeometries,reationenergies, reation

barriers,eletronanities, ionizationenergies, dissoiationenergies, et. All

these tasks onsist of nding either just one, or a few solutions of the SE.

The predition of eletromagneti spetra, however, requires the solutionof

thetime-dependent Shrödingerequation(TDSE). Fortunately,onlyaslight

adaptiationof themethodsused tosolvethe SEare needed inorder fortheir

appliationto the TDSE. Moreover, the errorinherited from the underlying

SE method willtypiallydominate those introdued by the approximations

totheTDSE. Therefore,omputationalspetrosopy, thetopiofthis thesis,

ismainly onernedwith ndingthe right adaptations fora spei lass of

SEmodels,andinterpretingtheomputedresultsinrelationtoexperimental

observations.

The rest of this thesis is organized as follows: Chapter 2 presents the fun-

damental equation whih governs moleular quantum mehanis, namely

the Shrödinger equation, together with the Born-Oppenheimer and self-

onsistent-eldapproximationsapplied to it. In Chapter 3,moleular prop-

ertiesand spetrosopy are presented ina quasi-lassialformulationknown

asFloquet theory, where the eletrons and nulei obey quantum mehanis,

whereas the externaleletromagnetield obeys the lassialMaxwellequa-

tions. Responsetheoryisthenformulatedbyapplying(Rayleigh-Shrödinger)

perturbation theory to Floquet theory. Finally, Chapter 4 summarizes the

results in this thesis, as well as gives some remarks on future developments

and appliations.

4

W. Heitler and F. London: Interation of Neutral Atoms and Homopolar Binding

AordingtotheQuantumMehanis,ZeitshriftfürPhysik,vol. 44,p. 455(1927)

5

D.R.HartreeandW.Hartree: Self-onsistenteld, withexhange,fornitrogenand

sodium,Pro. RoyalSo. London,vol. 193(1034),p. 299-304(1948),whereW.Hartree

(Hartree'sfather)didthealulations.

6

Quantum mehanis

2.1 Shrödinger equation

In quantum mehanis, a system (moleule) onsisting of

N

^{partiles (ele-}

trons and nulei) is desribed by a wavefuntion

ψ( r ₁ , r ₂ . . . r _N )

^{, a omplex-}

valued funtion of the set of partile oordinates

1

r ₁ , r ₂ . . . r _N

ψ( r ₁ , r ₂ . . . r _N ) ∈ C .

^(2.1)

In the so-alled'Copenhagen interpretation' of the wavefuntion, the proba-

bility

P

^{of'nding'allpartileswithintherange}

δ

^{ofthepositions}

t ₁ , t ₂ . . . t _N

isthe integralof the square absolutevalueof the wavefuntion overthe or-

responding

3N

-dimensionalvolume

P = Z

k r ₁ − t ₁ k <δ

Z

k r ₂ − t ₂ k <δ

. . . Z

k r N − t N k <δ

| ψ | ² d r ₁ d r ₂ . . . d r _N .

^(2.2)

Thus

| ψ | ² = ψ ^∗ ψ

^{is the} probability density of the positions of the partiles.

Sine allthe partiles must be somewhere inspae, the orresponding prob-

ability

P

^for

δ = ∞

^{must be1 (whih means 100%)}

1 =

Z Z . . .

Z

ψ ^∗ ψ d r ₁ d r ₂ . . . d r _N = h ψ | ψ i ,

^(2.3)

whih is allednormalization of the wavefuntion

ψ

^{. The 'bra-ket'}

h . . . | . . . i

is a short-hand notation for suh integrals over all oordinates 2

. Addition-

ally, the wavefuntion should fulllso-alled spin-statistis: When idential

1

Partileshaveanadditionalspin oordinatewhihis'hidden'in

r _p

here.

2

Morepreisely,ratherthananintegral,itisanaverage overthe'enter-of-massoor-

fermions (nulei with an odd number of nuleons and eletrons) are inter-

hanged (swap oordinates), the wavefuntion should hange sign. This is

the Pauli exlusion priniple. Moreover, when idential bosons (nulei with

aneven number of nuleons) are interhanged, the wavefuntion should not

hange.

The time evolution of the moleule (its wavefuntion) is determined by the

time-dependentShrödinger equation, whih isa linear dierentialequation

Hψ ˆ = i _dt ^d ψ,

^(2.4)

wherethedierentialoperator

H ˆ

^{isthemoleule's}Hamiltonian. TheHamil- tonian onsists of a kineti energy operator

T ˆ p

^{for eah partile, and a po-}

tential energy operator

V ˆ pq

^{for eah (distint) pair of partiles. Ignoring}

interations due topartilespin, the kineti- and potentialenergy operators

are given by the Laplae operator and Coulomb potential

H ˆ = X

p

T ˆ p + X

p>q

V ˆ pq ,

^(2.5)

T ˆ p = − 1

2m p ∇ ² p = − 1 2m p

∂ ²

∂x ² _p + ∂ ²

∂y _p ² + ∂ ²

∂z _p ²

(2.6)

V ˆ pq = q _p q _q r pq

= q _p q _q

k r _p − r _q k = q _p q _q

p (x p − x q ) ² + (y p − y q ) ² + (z p − z q ) ² ,

^(2.7)

where atomi units have been used, and

m p

^{are the partiles' masses and}

q p

^{the harges. Notethat the Coulombpotential between partilesof oppo-}

site harge is attrative (

q p q q

^{negative), while it is repulsive (}

q p q q

^positive)

between those of same harge. The kineti energy isalways positive.

If the wavefuntion

ψ

^{is an} eigenfuntion (eigenstate) of the Hamiltonian witheigenvalue

E

^(the eigenenergy),itisastationarystate, as

e ^{− iEt} ψ

^solves

the time-dependent Shrödingerequation

Hψ ˆ = Eψ ⇒ H(e ˆ ^{− iEt} ψ) = i _dt ^d (e ^{− iEt} ψ),

^(2.8)

andthephasefator

e ^{− iEt}

^{anelswhenomputingthesquare absolutevalue}

| e ^{− iEt} ψ | ²

^,leavingtheinterpretation(probabilities

P

^{above)ofthe wavefun-}

tion onstant in time (stationary). The eigenstates

ψ

^{fulll the variation}

priniple,whihstates thatexpetationvalue ofthe Hamiltonian

h ψ | H ˆ | ψ i

^is

stationarywith respet tovariationsin

ψ

^{. One may thereforesearhfor the}

ground state, the eigenstate with lowest

E

^{, by minimizing this expetation}

2.2 Born-Oppenheimer approximation

The nulei are the heaviest partilesin a moleule; the lightest nuleus, the

proton

1 H

^is

≈

^{1836 times as heavy as aneletron, while the most abundant}

arbonnuleus

12 C

^is

≈

^{21863timesasheavy. Sinetheselargemassesappear}

in the denominator in the kineti energy operator in Eq. 2.6, nulei will

have little kineti energy relative to eletrons. In the Born-Oppenheimer

approximation 3

,thenulearkinetienergyoperators

T ˆ _n

^{areatrstseparated}

fromthe eletroni Hamiltonian

H ˆ ^el

^{, whih thenonsistsof} zero-eletron, 1- eletron and 2-eletron parts (

n, m

^{denoting nulei,}

e, f

^eletrons)

H ˆ ^tot = X

n

T ˆ n + ˆ H ^el

^(2.9)

H ˆ ^el = X

n>m

V ˆ nm = h nuc ,

^(2.10)

+ X

e

T ˆ e + X

n

V ˆ en

= ˆ h,

^(2.11)

+ X

e>f

V ˆ ef = ˆ g,

^(2.12)

where the nulear oordinates

r _n

enter

h _nuc

^and

ˆ h

^as parameters. The ele- troni Shrödingerequation isthen solved for alleletroni states (

k

⁾

H ˆ ^el ψ _k ^el ( r _e ; r _n ) = E _k ^el ( r _n )ψ ^el _k ( r _e ; r _n ), k = 0, 1 . . . ∞ ,

^(2.13)

eah depending parametrially 4

on

r _n

. The solutions

E _k ^el ( r _n )

^{are alled 'po-}

tential energy surfaes' (PES), and the 'equilibrium geometry' is dened as

theongurationof

r _n

that givesthe lowest eletronienergyontheground- state PES.

Inaseondstep, theompleteShrödingerequationissolvedwithanexpan-

sion over the eletroni solutions

ψ ^el _k ψ ^tot ( r _n , r _e ) = X

k

ψ _k ^nuc ( r _n )ψ _k ^el ( r _e ; r _n ), H ˆ ^tot ψ ^tot = E ^tot ψ ^tot ,

^(2.14)

where the

ψ _k ^nuc

^{are the oeients and the}

ψ ^el _k

^{the basis of the expansion.}

Sinethe

ψ ^el _k

^are eigenstatesof

H ˆ ^el

^{andorthogonal(forall}

r _n

),Eq.2.14 leads 3

M.Bornand R.Oppenheimer: ZurQuantentheoriederMolekeln, Ann. derPhysik

84,20(1927)

4

Thisisadierentialequationin

r _e

,butanordinary(parametri)equationin

r _n

.

toaninnite set of oupled Shrödingerequations

X

n

T ˆ n + E _k ^el ( r _n )

ψ _k ^nuc ( r _n ) = E ^tot ψ ^nuc _k ( r _n )

^(2.15)

wherethe potential energy surfaes

E _k ^el ( r _n )

^{have the role of potentialopera-}

tors (henethe name).

Although the equation set Eq. 2.15 is no less ompliated than the original

Shrödingerequation Eq. 2.4, it an betrunated to agood approximation,

both inthe numberof PESsinluded, and inthe rangeand preisionofeah

PES. The approximations range from the simplest, whih is to 'lamp' the

nulei in the equilibrium geometry (one PES, one

r _n

); to the harmoni, in whih the groundstate PES is approximated toseond order about apoint

r _n

;to more ompliatedapproximationsof several PESs et.

2.3 Self-onsistent eld approximation

EvenintherudestBorn-Oppenheimerapproximation,theso-alledlamped

nuleusapproximation,inwhihonlyonegeometry

r _n

ononePESissought, an

N

^{-eletron Shrödinger equation isstill too diulttosolve. In the self-}

onsistenteld(SCF)approximation,thisistakledbywritingthewavefun-

tionasaSlaterdeterminant,ananti-symmetrizedprodutof

N

orthonormal orbitals

φ 1 , φ 2 . . . φ N

^(1-eletron wavefuntions)

ψ( r ₁ , r ₂ . . . r _N ) = 1

√ N !

φ 1 ( r ₁ ) φ 2 ( r ₁ ) · · · φ N ( r ₁ ) φ 1 ( r ₂ ) φ 2 ( r ₂ ) · · · φ N ( r ₂ )

.

.

.

.

.

. .

.

. .

.

.

φ ₁ ( r _N ) φ ₂ ( r _N ) · · · φ _N ( r _N )

.

^(2.16)

A matrix determinant is the sum of all possible produts of one term from

eahrowandolumn,withsign

+

^or

−

^{dependingonwhetheritisaneven or}

oddpermutation. Thisensures thatthewavefuntionswithessignwhentwo

eletronsareinterhanged,asrequiredbythePaulipriniple. Moreover,sine

the orbitals are orthonormal, all the

N !

^{terms in the} determinant are also orthonormal, and the fator

1 / √

N !

^{gives a normalized}

ψ

^{. The SCF lass of}

modelshaveinommonthattheyattempttosolvean

N

^{-eletronShrödinger}

equation(Eq. 2.8)by solving oupled 1-eletron Shrödingerequations. The

term 'self-onsistent' is derived from the oupling between the 1-eletron

Applying the variation prinipleto the Slater determinant, one obtains the

Hartree-Fok model 5

,whihisthe mostbasi SCFmodel. Althoughderived

on a dierent basis, Kohn-Sham density funtional theory 6

models are a

broad lass of SCF models, and thus share the main harateristia with

Hartree-Fok.

2.3.1 Hartree-Fok

Inserting the Slater determinantEq. 2.16 into the expression for the energy

expetation value

E = h ψ | H ˆ | ψ i

^{,it isredued to}

E = h nuc + X

k

D φ k

− ¹ 2 ∇ ² − X

n

q n

k r − r _n k φ k

E

(2.17)

+ X

j>k

Z Z

φ ^∗ _j ( r ₁ )φ ^∗ _k ( r ₂ ) 1 r 12

h φ j ( r ₁ )φ k ( r ₂ ) − φ k ( r ₁ )φ j ( r ₂ ) i

d r ₁ d r ₂ ,

where the nulear repulsion

h nuc

^{is given by Eq. 2.10. Sine the diagonal}

terms

j = k

^{in the seond summationwill anel,it an be rewritten as}

1

2 X

jk

Z Z

φ ^∗ _j ( r ₁ )φ ^∗ _k ( r ₂ ) 1 r 12

h φ _j ( r ₁ )φ _k ( r ₂ ) − φ _k ( r ₁ )φ _j ( r ₂ ) i

d r ₁ d r ₂ ,

^(2.18)

where the ontributions from the rst term in the braket are alled the

Coulombrepulsion,andthosefromtheseondtermtheexhangeinteration.

Expanding the orbitalsin abasis of atomi orbitals 7

(AOs)

χ _µ ( r ) φ k ( r ) = P

µ χ µ ( r ) C _µk ,

^(2.19)

the energy an be written in matrix form in terms of the orbital oeient

matrix

C

as

E = h nuc + Tr C ^† HC + ¹ ₂ Tr C ^† G ( CC ^† ) C

(2.20)

= h nuc + Tr HD + ¹ ₂ Tr G ( D ) D ,

where

Tr

^{denotesmatrixtrae,and}

. . . ^†

^theomplex-onjugatedmatrixtrans- pose. Usingthe invarianeofthe traeunderyli permutationsofamatrix

5

G. G. Hall: The Moleular Orbital Theory of Chemial Valeny & A Method of

Calulating Ionization Potentials, Pro. Royal So. London A, vol. 205, p. 541-552

(1951)

6

W.Kohn,L.J.Sham: Self-ConsistentEquationsInludingExhangeandCorrelation

Eets,Phys. Rev.,vol. 140(4A),p. A1133-A1138(1965)

7

Commonlynuleus-enteredGaussian-typefuntions:

x ^k y ^l z ^m e ^{− ζr 2}

produt, the density matrix

D = CC ^†

has been introdued. The 1- and 2- eletronintegralmatries

H

and

G ( D )

^{ontainstheintegralsoftheoperators}

ˆ h

^and

ˆ g

^, respetively, overthe AO basis

χ µ

H _µν = χ µ

ˆ h

χ ν

= D χ µ

− ¹ 2 ∇ ² − X

n

q n

k r − r _n k χ ν

E

,

^(2.21)

G _µν,ρσ = Z Z

χ ^∗ _µ ( r ₁ )χ ^∗ _ρ ( r ₂ ) 1 r 12

h χ _ν ( r ₁ )χ _σ ( r ₂ ) − χ _σ ( r ₁ )χ _ν ( r ₂ ) i

d r ₁ d r ₂ , G ( D ) _µν = P

ρσ G _µν,ρσ D _ρσ .

^(2.22)

The orbitals

φ _k

^{are required to be} orthonormal, whih translates into the following matrix equationsto be satisedby

C

and

D

h φ j | φ k i = δ jk ⇒ C ^† SC = 1 ⇒ DSD = D ,

^(2.23)

where

S _µν = h χ _µ | χ _ν i

^{is the overlap matrix for the AO basis}

χ _µ

^{. The latter}

equationisommonlyreferredtoastheidempoteny onditionforthedensity

matrix.

Sine

C

must satisfy the orthonormality relation, it is onstrained, and the Lagrange multipler method

8

an be used toderive the variationalondition

E( C , Λ ) = h nuc + Tr C ^† HC + ¹ ₂ Tr C ^† G ( CC ^† ) C

(2.24)

− Tr Λ C ^† SC − 1 ,

∂

∂ C ^† E( C , Λ ) = H + G ( CC ^† )

C − SCΛ = 0 ,

^(2.25)

where

Λ

isthe Lagrangemultipliermatrix forthe orthonormalityondition.

Introduingthe Fok matrix

F = H + G ( D )

^{, the} variationalondition anbe expressed in terms of

D

as

9

FDS = SDF ,

^(2.26)

whih is the SCF equationin terms of the density matrix.

2.3.2 Kohn-Sham DFT

Hohenberg and Kohn 10

showed that there is a one-to-one relation between

the potential funtions

v( r )

^{in the eletroni Shrödinger equation, and the}

8

J.-L.Lagrange: Théoriedesfontionsanalytiques,(1797,p. 198)

9

P. Pulay: ImprovedSCFonvergeneaeleration,J.Comp. Chem. 3(4), 556-560

(1982)

10

P.Hohenberg,W.Kohn: InhomogeneousEletronGasPhys. Rev. B,vol. 136(3B),

eletron density

ρ( r )

^{of the groundstate (solution)}

ρ( r ) = N

Z Z . . .

Z

| ψ( r , r ₂ . . . r _N ) | ² d r ₂ . . . d r _N .

^(2.27)

Fora moleule,

v( r )

^{is the sum of Coulomb attrations toeahnuleus}

v( r ) = − X

n

q _n

k r − r _n k ,

^(2.28)

and it enters the eletroni Hamiltoniantogether with the nulearrepulsion

h nuc

^{, the eletroni kineti energy, and eletron repulsion. Basially this}

means that two dierent eletroni Hamiltonians (dierenent

v( r )

^{) annot}

have the same ground state density

ρ( r )

^{. For moleules this is perhaps not}

surprisingthepeaksinthegroundstatedensity, andtheirheights,indiate

the positions and harges of the nulei, fromwhih

v( r )

^{an be}determined.

Under the additional assumption that the ground state is non-degenerate

(has multipliity 1), Hohenberg and Kohn also proved the existene of a

variational density funtional

E v [ρ]

^{11 for the energy, whih minimum}

ρ( r )

is the ground state density orresponding to

v( r )

^{, hene the name 'density}

funtional theory' (DFT). The nulear repulsion and nulear attration are

known ontributions to

E v [ρ]

^{,while the kinetienergy}

T

^{and eletron repul-}

sion

V

^{are unknowns}

E v [ρ] = h nuc + R

v( r )ρ( r )d r + (T +V )[ρ].

^(2.29)

The formulation of a density funtional for the kineti energy is a diult

task, as the ground state kineti energy an hange abruptly with small

hanges in the density. To aount for this, Kohn and Sham 12

proposed to

expand the density in terms of orthonormal orbitals, and use the Hartree-

Fok (or non-interating) kineti energy

T s

^{as the main} ontribution, with the remaining kineti energy expeted to vary more slowly. Analogously,

the Coulombontribution

J [ρ]

^{(see Eq. 2.18) is separated from the eletron}

repulsion,leavingthe'exhange-orrelationfuntional'

E xc [ρ]

^astheunknown

E v [ρ] = h nuc + R

v ( r )ρ( r )d r + T s [ρ] + J[ρ] + E xc [ρ],

^(2.30)

J[ρ] = ¹ ₂ R

ρ( r ₁ ) _r ¹

12 ρ( r ₂ )d r ₁ d r ₂ ,

^(2.31)

E xc [ρ] = T − T s + V − J

[ρ].

^(2.32)

11

It is ustomary to write a density funtional with square brakets

[. . .]

^{around its}

argumentinsteadof

(. . .)

12

W.Kohn,L.J.Sham: Self-ConsistentEquationsInludingExhangeandCorrelation

This isthe formof the basi Kohn-Shamdensity funtionaltheory. If

E xc [ρ]

isanintegral over afuntion

F (ρ( r ))

^{,it issaid tobea loaldensity approx-}

imation (LDA), whereas an integral over

F (ρ( r ), k∇ ρ( r ) k )

^{is a} generalized gradient approximation (GGA). If an additional 'exat exhange' ontribu-

tion (meaning Hartree-Fok exhange, see Eq. 2.18) is separated from

V

^{, it}

isa 'hybrid' funtional.

As was the ase with Hartree-Fok in the previous setion, expanding the

orbitals in terms of AOs results in a matrix expression for the Kohn-Sham

energy, and a variational ondition of the same form as the SCF equation

Properties and spetra

The basi 1

interpretation of spetrosopy is that it measures dierenes be-

tween the stationary states of (atoms and) moleules, the eigenstates of the

Hamiltonian and solutions of the time-independent Shrödinger equation.

The moleule absorbs radiation at frequenies

ω yx = E x − E y

^{, whih orre-}

spond to dierenes between two eigenenergies, ata rate

A

proportional to theintensityofinomingradiation

I(ω yx )

^{,thetransitiondipolemoment,and}

the population

| c _y | ²

^{of the lower state}

ψ _y A(ω _yx ) ∝ I (ω _yx )

ψ _x

µ ˆ ψ _y

2 | c _y | ² .

^(3.1)

The moleulealsoemitsradiationatthe sameset offrequenies, atarate

S

proportionaltothesametransitionmomentandthepopulationofthehigher

state

ψ x

S(ω yx ) ∝

ψ y

µ ˆ

ψ x

2 | c x | ² .

^(3.2)

Thesetwoproessesare linear absorption and spontaneousemission, respe-

tively. Thus, in the absene of any inoming radiation to be absorbed, a

moleule in a mixture of states will eventually deay to the ground state

by spontaneous emission of radiation. Moreover, a system of moleules in

thermodynami equilibrium(onstant populations) willemit radiation with

frequeniesandintensitiesreetingthepopulationsandtransitionmoments.

1

This setion is based on Robert C. Hilborn: Einstein oeients, ross setions, f

3.1 Propagation and Floquet theory

Inordertopreditabsorption andemissionspetra, weneed away todeter-

minethe expansionoeients

c g (t)

^and

c x (t)

^{of the}wavefuntion

2

ψ(t) ˜

^{, for}

agiven experimentalenvironment

ψ(t) = ˜ c g ψ g + X

x

c x ψ x .

^(3.3)

In general, this amounts to solving the time-dependent Shrödinger equa-

tion,inwhihthe inomingradiationgivesrise toatime-dependentexternal

potential

V ˆ ^t

^{(whihwillbepresented inthe next setion)}

H ˆ + θ(t) ˆ V ^t ψ ˜ = i _dt ^d ψ, ˜

^(3.4)

where

θ(t)

^{is some funtion 'swithing' the radiation on; either instanta-}

neously,suhaswiththe stepfuntion

θ(t<0)=0

^,

θ(t>0)=1

^{;orgradually,as}

with the error funtion

θ(t)=erf(εt)

^{; or}exponentially,

θ(t)= exp(εt)

^{. As ini-}

tialondition of the lineardierentialequationEq. 3.4, one may speify the

wavefuntion atsome time, forinstane

ψ( ˜ −∞ )=ψ g

^or

ψ(0)=ψ ˜ g

^{. Thispro-}

edure of settinganinitialonditionfollowed by solving the time-dependent

Shrödinger equation is alled propagation, and treated in propagator theo-

ries.

3

In this work we make the simplest possible hoie of swithing funtion,

namely

θ(t)=1

^{. Rather than speifyingan initialondition, we require that}

thewavefuntionistheprodutofaphasefator

e ^{− iQt}

^andaquasi-periodi 4

wave funtion

ψ(t)

ψ(t) = ˜ e ^{− iQt} ψ(t) = e ^{− iQt} X

ω ∈ Ω

e ^iωt ψ ω ,

^(3.5)

where the frequeny set

Ω(V ^t )

haraterizing quasi-periodiity onsists of allombinationsofintegermultiplesof frequeniesinthe externalpotential

V ^t

^{. This means that}

ψ

^{is a Fourier series in all frequenies appearing in}

V ^t

^{. That is, if}

V ^t

^is monohromati, as when the moleule is irradiated by

2

Thetilde isput on thiswavefuntion,to reserve

ψ(t)

^{forthephase isolatedFloquet}

statein Eq.3.5.

3

J. Oddershede and P. Jørgensen: Polarization propagator methods in atomi and

moleularalulations,ComputerPhysisReports2(2),33-92(1984)

4

D. A. Telnov and S.-I. Chu: Generalized Floquet formulation of time-dependent

density funtional theory for many-eletron systems in intense laser elds, AIP Conf.

a single laser,

ψ

^{is aFourier series in the laser frequeny, and thusperiodi.}

Analogously,intheaseoftwolasers,

ψ

^{isabi-variateFourierseries,whihis}

periodionlywhenthetwofrequenieshaveaommondivisor, butgenerally

quasi-periodi. Unless the frequenies in

V ^t

^{have a ommon divisor (are}

ommensurate), the set

Ω(V ^t )

^{isdense inthe real numbers.}

Inserting the quasi-periodi wave funtion Eq. 3.5 into the time-dependent

ShrödingerequationEq.3.4with

θ(t)=1

^{,expandingthetimederivativeand}

anelling the phase fator, the time-dependent Shrödinger equation takes

the form of aneigenvalue equation

H ˆ + ˆ V ^t

e ^{− iQt} ψ = i _dt ^d e ^{− iQt} ψ,

^(3.6)

e ^{− iQt} H ˆ + ˆ V ^t

ψ = e ^{− iQt} Q + i _dt ^d

ψ,

^(3.7)

H ˆ + ˆ V ^t − i _dt ^d

ψ = Qψ.

^(3.8)

This willbe referred to as the Floquet-Shrödinger equation. The operator

H ˆ + ˆ V ^t − i _dt ^d

^{isthe Floquet operator,and itseigenvalue}

Q

^the quasi-energy.

The eigenfuntions

ψ

^{will in the following be referred to as Floquet states.}

The operator

i _dt ^d

^{is Hermitianin the} time-averaged salar produt

ψ

i φ ˙ _t = { dt ^d h ψ | φ i} ^t − i ψ ˙

φ _t = 0 + i ψ ˙

φ _t ,

^(3.9)

where thetime-averageiswelldened for quasi-periodifuntions and leads

tothe time average of atime derivative being zero

{ . . . } ^t = lim

r,s →∞

1 r+s

R s

− r . . . dt, _d

dt . . . _t = 0.

^(3.10)

The Floquet operator is therefore Hermitian and the quasi-energies

Q

^real-

valued. ForeahFloquetstate

ψ

^with quasi-energy

Q

^{,thereisaninniteset}

of Floquet states

e ^iωt ψ

^with quasi-energies

Q − ω

^{for all frequenies}

ω

^taken

fromtheset

Ω(V ^t )

^{,asanbeseenbyinserting}

e ^iωt ψ

^{inEq.3.5. Floquetstates}

that inthis way only dierby a phase fator

e ^iωt

^{are said tobe} degenerate.

The non-degenerate Floquet states are orthogonal (at eah time

t

^{). This is}

seenbyexpandingthematrixelementoftheFloquetoperatorintwodierent

ways

0 = ψ a

H ˆ + ˆ V ^t − i _dt ^d ψ b

− ψ a

H ˆ + ˆ V ^t − i _dt ^d ψ b

= ψ a

Q b ψ b

− H ˆ + ˆ V ^t − i _dt ^d ψ a

ψ b

+ i _dt ^d h ψ a | ψ b i

= (Q b − Q a ) h ψ a | ψ b i + i _dt ^d h ψ a | ψ b i .

^(3.11)

As taught in introdutory mathematis ourses, the general solution of this

rst-order linear dierentialequation is

h ψ a | ψ b i = c e ^{i(Q b − Q a )t} ,

^(3.12)

where

c

^{is a omplex onstant. Sine}

ψ a

^and

ψ b

^are quasi-periodi,

h ψ a | ψ b i

must also be quasi-periodi, but the frequeny

Q _b − Q _a

^{does not in general}

belong to the quasi-periodi set

Ω( ˆ V ^t )

^(unless

ψ a

^and

ψ b

^{happen to be de-}

generate), hene

c

^{must bezero and the states are} orthogonal.

3.2 Radiation potential

Thepotential

5

V ˆ ^t

^{arisingfromastatiexternal(rstorder-)}inhomogeneous eletri eld and a stati external homogeneous magneti eld is given by

the expression

V ˆ ^t = − F · µ ˆ − G · Θ ˆ − B · m ˆ − ¹ 2 B · ξB ˆ ,

^(3.13)

where

F

^,

G

^and

B

^{are the eletri eld (at the origin of the oordinate}

system),theeletrieldgradientandthemagnetield,respetively,whih

multiplythe (negative)eletridipole operator

ˆ µ = P

p q p r _p ,

^(3.14)

eletriquadrupole operator (symmetri

3 × 3

^matrix)6

Θ ˆ = P

p q p

2 r _p r ^T

p ,

^(3.15)

magneti dipoleoperator

ˆ m = P

p q p

2m p

ˆ l _p = P

p − iq p

2m p

r _p ×∇ ^p ,

^(3.16)

and magneti suseptibilityoperator (symmetri

3 × 3

^matrix)

ξ ˆ = P

p q ² _p 4m p

r _p r ^T

p − ( r _p · r _p ) 1

,

^(3.17)

respetively. Theexternalpotentialisinthis asetime-independent(stati),

and the notation

V ˆ ^t

^perhaps misleading, but as will be shown below, the presene of radiationleads to time-dependent

F

^,

G

^and

B

^{(and hene}

V ˆ ^t

^).

Aneletromagneti wave, radiationwitha single frequeny and diretion, is

asimple solutionof Maxwell'sequations 7

∇· F = 4πρ, _dt ^d F = c ² ∇× B − 4πj,

^(3.18)

∇· B = 0, _dt ^d B = − ∇× F ,

^(3.19)

5

ThissetionisbasedonL.D. BarronandC.G.Gray: MultipoleinterationHamil-

tonianfortime-dependentelds, J.Phys. A6(1),59-61(1973)

6

Thereare several waysto dene

Θ ˆ

. Inthis denition,

Θ ˆ

is nottraeless andsaled sothatitmultiples theeletrieldgradient.

7

TheseareMaxwell'sequationsinthe'Lorentzfore'onvention,wheretheeletriand

magnetieldsdierinmagnitudebyafator

1

c

^{,asopposedtothe'Gaussian'onvention.}

whih for empty spae (harge

ρ

^{and urrent density}

j

^{zero) state that the}

eldsaredivergene-free(havenosoures), andtime-evolutionisdetermined

by theopposite eld'surl(rotation). Thespeed oflight

c

^is

≈

^137inatomi

units.

Aneletromagneti wave withfrequeny

ω

^, propagatinginthe (normalized) diretion

k

ison the form

F ( r ) = f e ^{− iωt} exp( ^iω _c k · r ) + c.c.,

^(3.20)

B( r ) = b e ^{− iωt} exp( ^iω _c k · r ) + c.c.,

^(3.21)

where

f

^{is the wave's Jones vetor8, a omplexvetor whihdetermines the}

wave's intensity,phase,and (eletri)polarization,andisperpendiularto

k

. Theorrespondingmagnetivetorisgivenby

b= ¹ _c k × f

^,andisperpendiular to both

k

and

f

^{, and diers from}

f

^{in magnitude by a fator}

¹ _c

^{. The terms}

'

c.c.

^{' in Eqs. 3.20 and 3.21 denote the omplex onjugate of the preeding}

expression, thus the elds are real-valued.

The polarization of the wave is linear if the real and imaginary parts of

f

are parallel (or either is zero), irular (right or left) if perpendiular, and

ellipti in other ases.

The eletromagneti eld

F ( r ), B( r )

^is 'translated'to an external potential operator

V ˆ ^t

^{through the salar-and vetor potentials}

φ( r )

^and

A ( r )

^{,by the}

relation

V ˆ ^t = P

p iq p

2m p ( ∇ ^p · A ( r _p ) + A ( r _p ) ·∇ ^p )

^(3.22)

+ P

k q ² _p 2m p

A ( r _p ) · A ( r _p ) + P

p q p φ( r _p ),

where the potentials

φ( r )

^and

A ( r )

^{are related tothe elds by}

F ( r ) = − _dt ^d A ( r ) − ∇ φ( r ), B( r ) = ∇× A ( r ).

^(3.23)

However, these relations leave a great deal of freedom in the hoie of

φ( r )

and

A ( r )

^{, alled gauge9. By requiring}

r · A ( r ) = 0

^{, whih is to adopt the}

multipolar gauge 10

(about the origin), the potentials are given as simple

integrals overthe elds

φ( r ) = − r · R 1

0 F (u r )du, A ( r ) = − r × R 1

0 uB(u r )du,

^(3.24)

8

R.C. Jones, Newalulusfor thetreatmentofoptial systems,J.Opt. So. Am.,

vol. 31,p. 488493(1941),orhttp://en.wikipedia.org/wiki/Jones_vetor

9

P.Shwerdtfeger(ed.): RelativistiEletroniStrutureTheory. Part1. Fundamen-

tals,Elsevier (2002)

10

A.M. Stewart: Wavemehaniswithoutgaugexing, J. Mol. Stru. (Theohem),

whih for the elds given by Eqs. 3.20 and 3.21 an bealulated expliitly

φ( r ) = − r ·

f e ^{− iωt} exp( ^iω _c k · r ) − 1

iω

c k · r + c.c.

,

^(3.25)

A ( r ) = − r ×

b e ^{− iωt} (1 − ^iω c k · r ) exp( ^iω _c k · r ) − 1 ( ^ω _c k · r ) ² + c.c.

.

^(3.26)

Thewavelengths

λ = 2πc/ω

^{used inspetrosopy are ingeneralseveraltimes}

the size of the moleules studied. Therefore, it is onvenient to trunate

the salar and vetor potentials to seond and rst order in

r

, respetively, so that

V ˆ ^t

^{in Eq. 3.22 beomes aurate to rst order in11}

¹ _c

^{(due to the}

dierenein magnitude between

f

^and

b

⁾

φ( r ) = − r · f e ^{− iωt} (1 + ^iω _2c k · r ) + c.c.

= − f e ^{− iωt} + c.c.

· r − ^iω _2c ( k f ^T +f k ^T )e ^{− iωt} + c.c.

· rr ^T

= − F · r − ¹ ₂ G · rr ^T ,

^(3.27)

A ( r ) = − r × b e ^{− iωt} ( ¹ ₂ ) + c.c.

= − ¹ 2 r × B ,

^(3.28)

wherethe eletri eldat theorigin, the eletrield gradient and themag-

neti eld have been introdued

F = f e ^{− iωt} + c.c.,

^(3.29)

G = ^iω _c k f ^T + f k ^T

e ^{− iωt} + c.c.,

^(3.30)

B = b e ^{− iωt} + c.c. = ¹ _c ( k × f )e ^{− iωt} + c.c.

^(3.31)

Inserting these expressions into Eq. 3.22, the external potential operator

beomes

V ˆ ^t = P

p q p

2m p i ∇ p · ( − ¹ ₂ r _p × B) + ( − ¹ ₂ r _p × B) · i ∇ p

(3.32)

+ P

p q ² _p

2m p ( − ¹ ₂ r _p × B) · ( − ¹ ₂ r _p × B) − P

p q p F · r _p − P

p q p

2 G · r _p r ^T

p ,

whih an be rearranged into

V ˆ ^t = − B · P

p q p

2m p ( r _p × i ∇ ^p )

+ ¹ ₂ B · P

p q _p ² 4m p ( r _p r ^T

p − ( r _p · r _p ) 1 B

− F · ( P

p q p r _p ) − G · P

p q p

2 r _p r ^T

p

= − F · µ ˆ − G · Θ ˆ − B · m ˆ − ¹ 2 B · ˆ ξB ,

^(3.33)

11

Calledthene-strutureonstant,andommonlydenoted by

α

whihisonthe same formasEq. 3.13, exeptthe elds are time-dependent.

This isthe eletriquadrupolemagnetidipoleapproximationto the radia-

tion potential. Fora eld onsisting of several waves of dierent frequenies

and diretions, there will be several time-dependent ontributions to

F , G

and

B

^.

3.3 Response theory

When exposed to radiation, the moleule starts utuating, rotating and

vibrating in various ways. This means the moleular wavefuntion is dis-

tributed over anumberof eigenstates of the Hamiltonian

ψ(t) = c _g (t)ψ _g + X

x

c _x (t)ψ _x ,

^(3.34)

whihmakestheexpliitsolutionoftheFloquet-ShrödingerequationEq.3.8

verydemanding,andonlyappliabletosmallatoms. Formoleules,theonly

optionis therefore toresortto approximate solutionsby meansof perturba-

tion theory.

Having solved the time-independentShrödingerequation (with

V ˆ ^t =0

^{), and}

thusfoundatime-independenteigenstate

ψ _g

^witheigenenergy

E _g

^,thequasi-

energy

Q

^{and Floquet state}

ψ

^{an be written as a} perturbation expansion (Taylor series) inthe eld parameters

f, b, g

^{et.,that enter}

V ˆ ^t

Q

f,b,g = E g + f Q ^f + f ^∗ Q ^{f ∗} + b Q ^b + b ^∗ Q ^{b ∗} + g Q ^g + g ^∗ Q ^{g ∗}

^(3.35)

+ ¹ ₂ f f Q ^{f f} + f f ^∗ Q ^{f f ∗} + ¹ ₂ f ^∗ f ^∗ Q ^{f ∗ f ∗} + f b Q ^{f b} + f ^∗ b Q ^{f b} + ¹ ₂ bb Q ^bb + f b ^∗ Q ^{f b ∗} + f ^∗ b ^∗ Q ^{f ∗ b ∗} + bb ^∗ Q ^{bb ∗} + ¹ ₂ b ^∗ b ^∗ Q ^{b ∗ b ∗} + f g Q ^{f g} + f ^∗ g Q ^{f ∗ g} + bg Q ^bg + b ^∗ g Q ^{b ∗ g} + ¹ ₂ gg Q ^gg + f g ^∗ Q ^{f g ∗} + f ^∗ g ^∗ Q ^{f ∗ g ∗} + bg ^∗ Q ^{bg ∗ 1} ₂ g ^∗ g ^∗ Q ^{g ∗ g ∗} + b ^∗ g ^∗ Q ^{b ∗ g ∗} + gg ^∗ Q ^{gg ∗} + ¹ ₆ f f f Q ^{f f f} + . . . ψ

f,b,g = ψ g + f ψ ^f + f ^∗ ψ ^{f ∗} + b ψ ^b + b ^∗ ψ ^{b ∗} + . . .

^(3.36)

wheresupersriptsareusedasshort-handnotationforderivatives,i.e.

Q ^{f ∗ g} = _df ^d ∗

d dg Q

^,

and the vetor-tensor produts are ontrated. Note that also derivatives

with respet to the omplex-onjugate elds

f ^∗ , b ^∗ , g ^∗

^{appear in the series.}

Wewillrefertothederivativesofthequasi-energy

Q ^{f b ∗}

^{et.,asresponses,and}

derivatives of the wave funtion

ψ ^b

^{et., as perturbed} wavefuntions. They

are determined by the orresponding derivativesof the Floquet-Shrödinger

equation,Eq. 3.8 and the normalizationondition Eq. 2.3

d

df H ˆ + ˆ V ^t − i _dt ^d − Q

ψ = 0, _df ^d h ψ | ψ i − 1

= 0,

^(3.37)

Expandingthe derivatives and insertingEq. 3.33, we get

H ˆ + ˆ V ^t − i _dt ^d − Q

ψ ^f = (e ^{− iωt} µ ˆ + Q ^b )ψ,

^(3.38)

ψ ^{f ∗}

ψ +

ψ ψ ^f

= 0.

^(3.39)

Applying

h ψ | . . . i

^{to the rst equation and} rearranging

ψ

H ˆ + ˆ V ^t − i _dt ^d − Q ψ ^f

= h ψ | e ^{− iωt} µ ˆ + Q ^f | ψ i ,

^(3.40)

H ˆ + ˆ V ^t − i _dt ^d − Q

ψ ψ ^f

− i _dt ^d h ψ | ψ ^f i

= e ^{− iωt} h ψ | µ ˆ | ψ i + Q ^f h ψ | ψ i .

^(3.41)

The rst term vanishes and the last term is 1 due to normalization. Tak-

ing the time average, the seond term also vanishes (the average of a time

derivativeis zero, see Eq. 3.10)

− i _dt ^d h ψ | ψ ^f i _t = { e ^{− iωt} h ψ | µ ˆ | ψ i} ^t + Q ^f ⇒ Q ^f = − µ _ω ,

^(3.42)

andthe derivative

Q ^f

^{of the}quasi-energywith respet toanosillatingele- tri eld, is found to be minus the

ω

^{-frequeny omponent of the eletri}

dipolemoment. This property is known as the (time-dependent) Hellmann-

Feynmantheorem 12,13

: Therstderivativeisgivenby theexpetationvalue

of the perturbing operator. Thus, no knowledge of

ψ ^f

^{isrequired to obtain}

Q ^f

^.

Dierentiating

Q ^{b ∗}

^{, whih aording to the previous disussion is given by}

−{ e ^iωt h ψ | m ˆ | ψ i} ^t

^{,with respet to}

f

^{, the linear response}

Q ^{b ∗ f}

^isobtained

Q ^{b ∗ f} = − df ^d { e ^iωt h ψ | m ˆ | ψ i} ^t = −

e ^iωt ψ ^{f ∗}

m ˆ ψ

+ ψ

m ˆ

ψ ^f _t .

^(3.43)

In this ase, however,

ψ ^f

^{an not be eliminated from the formula. Going}

bak to Eq. 3.38, and using that the unperturbed wavefuntion

ψ=ψ g

^is

time-independent,and thus

Q ^f = − µ _ω

^{is zero (unless}

ω=0

⁾

H ˆ − i _dt ^d − E g

ψ ^f = e ^{− iωt} µψ ˆ g .

^(3.44)

Sine only the phase fator

e ^{− iωt}

^is time-dependent on the right-hand side, and

ψ ^f

^{is the only} time-dependent fator on the right-hand side,

ψ ^f

^must

12

H.Hellmann: Einfürung indieQuantenhemie(Leipzig: Deutike)(1937)

13

arry the same phase fator:

ψ ^f (t) = e ^{− iωt} ψ ^f (0)

^{. This means that}

_dt ^d ψ ^f = − iωψ ^f

^{, and the equationbeomes}

H ˆ − ω − E _g

ψ ^f = e ^{− iωt} µψ ˆ _g .

^(3.45)

If

ψ g

^{together with all the other} eigenstates

ψ x

^of

H ˆ

^{form a omplete or-}

thonormal set, we an write

H ˆ

^as

H ˆ = E g | ψ g ih ψ g | + X

x

E x | ψ x ih ψ x | ,

^(3.46)

and the inverse of the operator on the left-hand side of Eq. 3.45 an be

writtenas

H ˆ − ω − E g

₋ 1

= 1

E _g − ω − E _g | ψ g ih ψ g | + 1

E _x − ω − E _g | ψ x ih ψ x | .

^(3.47)

The solution

ψ ^f

^{is therefore given by}

ψ ^f = e ^{− iωt}

− h ψ g | µ ˆ | ψ g i

ω ψ + X

x

h ψ x | µ ˆ | ψ g i E x − ω − E g

ψ x

,

^(3.48)

from whih

ψ ^{f ∗}

^{is obtained by hanging}

ω

^to

− ω

^{. Inserting for}

ψ ^{f ∗}

^and

ψ ^f

in the linear response in Eq. 3.43, the so-alled sum-over-states expression

for the linear response isobtained

d

df Q ^{b ∗ f} = − n e ^iωt D

e ^{− iωt}

− h ψ g | µ ˆ | ψ g i

− ω ψ _g + X

x

h ψ x | µ ˆ | ψ g i E x + ω − E g

ψ _x m ˆ

ψ _g E

+ e ^iωt D ψ _g

m ˆ

e ^{− iωt}

− h ψ g | µ ˆ | ψ g i

ω ψ _g + X

x

h ψ x | µ ˆ | ψ g i E x − ω − E g

ψ _x Eo

t

= − X

x

h ψ g | µ ˆ | ψ x ih ψ x | m ˆ | ψ g i E x + ω − E g − X

x

h ψ g | m ˆ | ψ x ih ψ x | µ ˆ | ψ g i E x − ω − E g

.

^(3.49)

Observe that the ontributions fromthe rst terms in

ψ ^f

^and

ψ ^{f ∗}

^{have an-}

eled. Thisisalsothe asefor thetime-dependentphase fators,makingthe

time average redundant.

As there in general willbe innitely many exited states

ψ x

^{, using Eq. 3.47}

is not a pratial way to solve the response equation Eq. 3.45. Rather, it

is preferrables tosolve Eq. 3.45 iteratively, using a preonditioner (approxi-

mation to Eq. 3.45) to improve onvergene. An iterative tehnique is also

preferrablefornding

ψ g

^{intherstplae,andthe twotehiques arerelated.}

Due to the time-independene of the referene state

ψ g

^{, and the frequeny}

dependeneoftheappliedelds,quasi-energyderivatives(responses)arenon-

zeroonly whenthe frequenies ofthe elds sumtozero. Thus, the following

seondderivatives are zero, for instane

Q ^{f f} = Q ^bb = Q ^{f ∗ f ∗} = Q ^{b ∗ b ∗} = 0,

^(3.50)

sinethe frequenies of the eldssum to

2ω, 2ω, − 2ω

^and

− 2ω

^,respetively.

Forhigher-orderresponses, involvingseveral dierent frequenies, itismore

onvenienttouseanotationwhihspeiesboththe eldsandtheirfrequen-

ies, suh as

Q ^{F F} _{− ω,ω} , Q ^{F B} _{− ω,ω} , Q ^{F GF} _{− 2ω,ω,ω} , Q ^{F F BB} _{− ν − ω,ν,ω,0} .

^(3.51)

where

F

^{is understood as the eletrield,}

B

^{the magneti eld, and}

G

^the

eletrieldgradient, respetively. A well-establishednotationisthe double

braket 14

hh µ; µ, m ii ^ω,ν = ( − 1) ³ Q ^{F F B} _{− ω − ν,ω,ν} , hh µ; Θ, m, µ ii ^ω,ν,γ = ( − 1) ⁴ Q ^{F GBF} _{− ω − ν − γ,ω,ν,γ}

whih lists the perturbing operators, the rst designated as the 'outgoing'

eldandtheothersas'inoming'elds,alongwiththe inomingfrequenies.

3.4 Resonane

Even if the singularity in

ψ ^f

^{Eq. 3.48 at}

ω=0

^{is absent from the linear}

response funtion Eq. 3.49, singularities remain at all exitation energies

ω=E _x − E _g

^{. Atrstglanethis mayseemasaproblemwith response theory.}

But as will be explained in this setion 15

, these are resonanes disonti-

nous 'jumps' in

ψ

^and

Q

^{as the eld is swithed on.}

Forsimpliity,wewillonsider atwo-state system,sothat the Floquetstate

ψ

^{an bewrittenasa linearombinationof the two}unperturbed eigenstates

ψ g

^and

ψ x

ψ = c g (t)ψ g + c x (t)ψ x , H ˆ = E g | ψ g ih ψ g | + E x | ψ x ih ψ x | ,

^(3.52)

14

J. Linderbergerand Y. Öhrn: Propagatorsin quantum hemistry, 2nd ed., Wiley

(2004)

15

This setion is based on S. H. Autler and C. H. Townes: Stark Eet in Rapidly