Introduction The Hartree-Fock method is a basic method for approximating the solution of many-body electron problems in atoms molecules and solids With modi?cations it is also extensively used for protons and neutrons in nuclear physics and in other applications
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Introduction The Hartree-Fock method is a basic method for approximating the solution of many-body electron problems in atoms molecules and solids With modi?cations it is also extensively used for protons and neutrons in nuclear physics and in other applications
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Physics 5403Quantum Mechanics I IFall 1998Hartree?FockandtheSelf?consistentField1VariationalMetho dsInthediscussionofstationaryp erturbationtheory?Imentionedbrie?ytheideaofvariationalapproximation schemes?The basic idea here is that the variational principle:?
j?i?h?jHj?i?Eh?j?i??0?1?is equivalent to the Schro dinger equation?In other words? the statesj?ithat satsify this equationare eigenstates of the Hamiltonian?To b e more explicit? conisder the quantityhH?Ei?
Zd 3r? ??r?H??r??E Zd 3r? ??r???r???2?If we take the functional derivative of this quantity with resp ect to the function ? ??r?wehave?hH?Ei?Z?H??r??E??r????
??r?d3r??3?In order for this to b e stationary ?the variation is zero for all p ossible forms of??
??r?? we see that??r?must b e a solution to the Schro dinger equation?We can also takethe variation with resp ectto ??r? and will ?nd thathH?Eiis also stationary if ?
??r? is a solution to Schro dinger?s equation?The goal of these notes is to explore some of the implications of using a variational scheme to solvefor the states of a many?particle system?Note?I will not gothrough themathematics of variational physics and fucntionalderivatives?The metho ds used here are verysimilar tothoseused in classical physics?This material is available in most texts on Mathematical Physics?2HartreeApproximationFirst? consider a multiparticle system where the particles are all distinguishable?The Hamiltonianof the system has the form?H?
Xi H i 1 2 Xi6?j V ij ?4?1 where the sums are over the particle lab els?In generalH i p 2i 2m ?Vs ?ri ?V ij ?Vint ?jri ?rj j??5?whereVs is the single?particle p otential andVintis the interaction p otential?IntheHarteeApproximation?oneassumesthattheeigenstatesofthetotalHamiltoniancanbewritten as a pro duct of single particle states?The variational approximation is then used to derivean equation for these single particle states?Explicitly? assume the eigenstates ofHcan b e written???1
?r1 ??2 ?r2 ??3 ?r3 ?????6?In this case? the exp ectation value of the total Hamiltonian will have the formZ d 3r 1 d 3r 2 d 3r 3 ?1 ?r1 ?2 ?r2 0 Xi H i 1 2 Xi6?j V ij 1 A 1 ?r1 ??2 ?r2 ??????7?Lo oking at each of the terms? we see that the sums overHi andVij reduce to one and two particleexp ectation values resp ectively?For example? considerZd 3r 1 d 3r 2 d 3r 3 ?1 ?r1 ?2 ?r2 ???? Hi ?1 ?r1 ??2 ?r2 ??????8?BecauseHionlyop eratesontheithparticle?theintegralsoveralltheotherparticleco ordinatesare equal to one?Therefore? this term is simply?Zd
3r i ?i ?ri ?Hi ?i ?ri ???9?Similarly? the interaction term reduces toZd 3r i d 3r j ?i ?ri ?j ?rj ?Vij ?i ?ri ??j ?rj ???10?This means that the quantitywe wish to b e stationary isXi Z d 3r i ?i ?ri ?Hi ?i ?ri 1 2 Xi6?j Z d 3r i d 3r j ?i ?ri ?j ?rj ?Vij ?i ?ri ??j ?rj ??E Yi Z d 3r i ?i ?ri ??i ?ri ???11?Taking the functional derivative of this equation with resp ect to? ?m ?rm ? for them thparticle? we?ndZ d 3r m 0 H m ?m ?rm Xi6?m Z d 3r i ?i ?ri ?Vim ?i ?ri ??m ?rm ??E?m ?rm 1 A ?m ?rm ??0??12?2The factor of two in front of the interaction term drops out b ecause there are two equal terms fromthe derivative?m?iandm?j?This means that each of the single particle states in our guess atthe total state must satisfy?H
m ?m ?rm 0 Xi6?m Z d 3r i ?i ?ri ?Vim ?i ?ri 1 A m ?rm ??E?m ?rm??13?This is usually called the ?Hartree equation??3HartreePotentialandSelf?ConsistencyWhatwehaveshown ab oveis thatthe Hartreeapproximation reduced our N?particle problem toasetof single particle equations thatweknow ?in theory? howtosolve?The interaction b etweenthe particles is reduced to a single p otential term of the formV
H ?rm Zd 3r i Xi6?m j?i ?ri ?j 2V int ?jri ?rm j??14?Thisisthe?Hartreep otential??Oneusuallythinksab outthisp otentialassayingthatthem thparticle interacts with the probability density of all the other particles?i6?m?? ?m??r?? Xi6?m j?i ?r?j2?15?throughthep otentialVint
?jr1 ?r2j??Ifweconsider?forexample?acoulombinteractionbetweeneachoftheparticles?theHartreep otentiallo oksliketheinteractionwiththechargedensitydueto all the other particles?V
H ?rm ??q Zd 3r q? ?m??r? jrm ?rj??16?The di?culty with solving this problem is that the Hartree p otential connects the solutions of eachofthesingle?particlestates?Inotherwords?thisp otentialforparticlemdep endsonthestatesof all theother N?1particles?Forthis reason?it is sometimes referred toas the ?Self?Consistent?p otential?The di?culty in solving for the N?particle stateis simply that of ?nding a set of singleparticle states?i
?r? thattogetherare solutions to the Hartree equation?The approach to solving this problem numerically is essentially:1?Guess a set of single particle states??i
?2?Compute the Hartree p otential?VH? for each of the particles using the states guessed in ?1??3?Solve the single?particle ?Hartee? equations for the states usingVH
?4?If the states from ?3? are the same as the states from ?1?? you are done?35?If thestatesfrom ?3?aredi?erent than thestatesfrom ?1??use theseasanew guessatthestates?6?Rep eat until convergence?solutions ? guesses??Therearesomesubtletiesinthepro cess?Themostimp ortantisthataftersolvingforthesingleparticle states in ?3?? the new guess at the states is a mixture of the old guess from ?1? and the newsolutions?If one uses the new states as the guesses? wild oscillations in the numerics often o ccur?Ofcourse?thesolution totheHartreeequationisnotexact?Theapproximation usedis thatthesolutiontotheproblemcanbewrittenasapro ductofsingleparticlestates?Thisistheusualvariational simpli?cations where the region of Hilb ert space considered is limited by some guess asto the form of the states??Exact?solutions to the N?particle problem involve linear combinationsoftheseHartee?likebasisstates?Thissortofsolutionmetho dhasb eenabletopro duceexactdiagonalization of Hamiltonians of up to ab out dozen particles? dep ending on the physical system?Of course? this requires a lot of computer?4Hartee?FockandExchangeThe Hartree?Fock approximation follows directly from what was done ab ove for the Hatree approxi?mation? only taking into account the need for anti?symmetric states for Fermions?The HamiltonianforthesystemisthesameasfortheHartreeapproximation?Theexchangerequirementonthesystem of particles will add an extra interaction term in the ?nal result?We start with the Slater determinant state??
1 p N? X?n1 ?n2 n 1 n2 n3 ?n 1 ?r1 ??n 2 ?r2 ??n 3 ?r3 ?????17?where the sum is overall p ossible p ermutations of the indices ?n1 ?n2 ?n3? ??????Notethat here I?mp ermutingtheindiceslab elingthesingle?particlestatefunctionsratherthantheindicesoftheparticle co ordinates as was done previously?You mightwant to convince yourself that one gets thesame states??First?considerthematrixelementofoneofthesingle?particleop erators?Hi
?Thiswill havetheform?1 N X?n1 ?n2 X?n 01 ?n 02 n 1 n2 n3 ?n 01 n 02 n 03quotesdbs_dbs14.pdfusesText_20