Many Electron Theory: Time dependent perturbations and propagation.

Transcription

1 Many Elecron Theory: Time dependen perurbaions and propagaion. Coninued noes for a workgroup Sepember-Ocober 00. Noes prepared by Jan Linderberg, Sepember 00

2 Hamilonians wih ime dependence. Evoluion in ime is governed by he hamilonian of he sysem and when here is an explici dependence upon ime in he hamilonian i is no longer possible o separae sae veco ino a produc of a ime dependen phase facor and a saionary eigensae of a ime independen hamilonian. Specroscopy is buil on he premise ha absorpion and emission occu beween such saes and ha a low order perurbaion reamen suffices for he descripion of such processes. Molecular scaering processes and reacion dynamics is no well suied for such a reamen and requires more deailed analysis and somewha differen approximaion mehods. Variaional formulaions are useful boh for specroscopic and dynamical problems. The Schrödinger equaion obains from he condiion, aribued o Frenkel, [ ] = δ 1 ih Ψ 1 ih H Ψ + Ψ Ψ Ψ () Ψ and is a fi order differenial equaion in ime. I needs a saring value for he sae and is used o propagae his sae owards laer imes, or, occasionally, o previous ones. This is differen from he ime independen variaional form which leads o a second order parial differenial equaion and cerain boundary condiions. 0 Many elecron heory in he perurbaion form for a ime independen hamilonian was developed from an adiabaic hypohesis ha he elecron ineracion could be considered ime dependen and vanishing infiniely long ago. Convergence of his procedure has been quesioned. Jan Linderberg

3 Dirac considered he ime evoluion of he densiy marix in he Harree- Fock approximaion, ha is he sae vecor should a any ime be of he Slaer deerminan form wih ime dependen spin orbials. Then one calculaes he value of he lagrangian L ()= 1 ih Ψ i H + 1 h () Ψ Ψ Ψ Ψ Ψ for a sae THF e i = Λ( ) pq n The las erm in he lagrangian comes ou as in he normal Harree-Fock calculaion wih consideraion ha he densiy marix now depends on ime. I holds ha i ih THF = h Λ Λ() e () pq n = hλ () THF so ha L ()= h λ() ρsr () h() ρsr () 1 [( pq ) ( ps rq) ] ρ () ρ () pq This expression simplifies when he canonical spin orbials are used and becomes L ()= h l λl() lε ()+ 1 l ll ( ll l ) l ll l l [ ( )] which has he value zero when he sum of he λ s is a linear funcion of ime wih he slope being he negaive of he Harree-Fock energy expecaion value. qp sr A small deformaion of he Harree-Fock sae should no change he value of he lagrangian when a saionary soluion is found. We use 3 Jan Linderberg

4 THF THF + iδλ() THF ih THF hλ () THF hδλ () THF ihλ () δλ() THF and obain δ THF ih THF = h THF δλ () THF ih THF [ δλ(), Λ () ] THF = h δλ () ρsr () ih pδλ() λ sp() ρ pr () ρsp() λ [ pr () ]= h δλ () ρsr () h δλ() ρ sr () The previous resul for he variaion of he Harree-Fock energy expecaion value gives δl ()= h δλ () ρ () δλ () h ρ ()+ i f () ρ () ρ () f () sr p sp pr sp pr { [ ]} and he condiion for arbirary variaions is h ρsr ()+ i p[ fsp() ρpr () ρsp() fpr () ]= 0 A marix form of his equaion is [ ] ih ρ()= f(), ρ() which shows ha he densiy marix ransforms inveely o he equaion of moion for a Heisenberg operaor. sr A proper Harree-Fock densiy marix is a projecion operaor ono he occupied se of spin orbials and he fi erm of he variaion of he lagrangian vanishes when he ime derivaive of he deformaion generaor has no componens in his space. The ime dependen Harree-Fock equaions have found use in collision problems, molecular as well as nuclear. Our emphasis will be on he Jan Linderberg 4

5 specroscopy of elecronic sysems. We consider he effec of an exernal field wih some dependence on ime, h()= h + v() and a densiy marix wih a small deviaion from a saionary sae, ρsr ()= δsrns + δρsr () The basis is aken as he canonical one for he saionary sae. The elemens of he Fock marix are hen [ ] () ()= + ()+ ( ) ( ) f δ ε v pq rq ps δρ s pq qp These forms are insered ino he equaion of moion for he densiy marix and erms of second and higher order in he perurbaions are discarded o obain he linearized ime-dependen Harree-Fock equaions: hδρ sr ()+ ivsr () [ nr ns ]+ i[ εs εr ] δρsr () + in [ r ns] pq[ ( sr pq) ( sq pr) ] δρqp()= 0 Were we o neglec he erms ha depend explicily on he elecron ineracions, hen a direc inegraion gives i h ( ) i [ h( )( r) ] δρsr ()= δρsr ( 0)+ ns nr d vsr exp εs ε [ ] 0 Resonance occu when he perurbaion has a frequency componen ha equals he spin orbial energy difference. Fourier analysis ransforms he differenial equaions above o algebraic ones. We inroduce he noaion l o indicae occupied spin orbials and k o refer o unoccupied ones. Thus 5 Jan Linderberg

6 [ ] ( ) [ kl l k ( kk l l) ] δρ k l ( ω) [ ( )] ( )= [ ] ( ) l [ lk l k ( lk l k) ] δρ k l ( ω) [ lk k l ( ll k k) ] δρ ( ω)= hωδρ kl( ω)+ v kl( ω)+ εk εl δρ kl ω + ( ) pq l l l l δρ pq k k k k l k ω 0 + ( ) hωδρ lk ( ω) v lk( ω) εk εl δρ lk ω ( ) k ( ) k l l k 0 The marix elemens ha occur in hese equaions are he same as hose considered in he sabiliy analysis of he Harree-Fock sae. I is convenien o order he elemens of he perurbed densiy marix as a onedimensional array. We inroduce o his end an index σ o indicae a pair (kl) and use σ for he pair (lk). Then i holds ha hωδρ σ + v σ + σσδρ σ σ σ σδρ ( A, + A, σ )= 0; hωδρ σ + v σ + σ ( A σσ, δρ σ + A σ, σ δρ σ )= 0 which is a linear equaion sysem for he perurbed densiy marix elemens. A perurbaion, which is consan in ime, gives a deformaion of he Harree-Fock sae from hese coupled equaions provided he sae is sable. Singulariies are o be anicipaed in he equaion sysem above for cerain frequencies ω. Such frequencies are inerpreed as exciaion frequencies of he sysem. We saw above ha neglec of elecron ineracions, beyond wha is included in he Fock marix, gives he Bohr frequency rules. A posiive definie marix A has a Cholesky decomposiion such ha A= LL ; A± σ, ± σ = τ L± στ, L± σ, τ Jan Linderberg 6

7 and we define new quaniies as The invee relaion is denoed ( ) τ σ σ, τ σ σ, τ σ δρ = L δρ + L δρ δρ ± σ τ τ ± σ τ = L, δρ. A somewha shorened noaion is used for he complex conjugaes of he elemens of he invee o he marix L. The ransformed elemens of he densiy marix saisfy he equaion sysem 1 1 δρτ + hω τ Θττ, δρτ + σ ( Lτσ, vσ + Lτ, σv σ)= 0 ( ) 1 ττ = σ τσ 1 Θ, L, Lτ, σ Lτ, σl τ, σ I follows from he consrucion ha Θ is an hermiean marix and i can be shown ha is eigenvalues are occurring in pai of opposie sign. Real spin orbials and marix elemens provide for some simplificaions in he acual calculaion of hese marices. The soluion of he equaion sysem offe he possibiliy o calculae he perurbed expecaion value of an operaor. Such an operaor may be he dipole operaor wih he form so ha D= dar a D = dρsr = l dll + σ dlkδρσ + dklδρ σ There should also be inegraion over he frequencies from he Fourier ransformaion. A perurbaion wih a simple harmonic ime dependence gives an induced dipole momen wih he same frequency and he proporionaliy beween he induce dipole and he field srengh is 7 Jan Linderberg s ( )

8 expressed in erms of a polarizabiliy ha can be calculaed from he equaion sysem. Almos every elecronic srucure calculaion involves a spin independen hamilonian and he Harree-Fock orbials are normally spin degenerae and may be characerized as eiher α or β orbials. A single ground sae admis single exciaions only o anoher single or o a hree-fold degenerae riple level. Accordingly one can separae he densiy marix disorion elemens ino single and riple componens. The marix problem separaes ino four independen ones, hree of which are idenical and refer o riple exciaions. Poin group symmery simplifies he soluion furher when presen. The perurbaion from an elecric field can be expressed eiher in he dipole lengh form or in he dipole velociy form. Time dependen linearized Harree-Fock mainains he equivalence beween he wo formulaions provided ha dipole velociy operaor represenaion saisfies he fundamenal relaion D i D, H = [ ] h This holds in a complee basis and may be reasonably accurae in a large basis for he relevan marix elemens. Some model hamilonians are consruced so as o offer his relaion. Jan Linderberg 8