Moller Plesset Perturbaton Theory In Moller-Plesset (MP) perturbaton theory one taes the unperturbed Hamltonan for an atom or molecule as the sum of the one partcle Foc operators H F() where the egenfunctons of F are the occuped and vrtual Hartree-Foc orbtals of the system and the egenvalues the assocated one electron energes. Fϕ εϕ The Hartree-Foc wavefuncton (,,!, ) Âϕ ()ϕ ()!ϕ ( ) s an egenfuncton of Ĥ wth an egenvalue equal to the sum of the one electron energes of the occuped spn orbtals ε Ths s the essental observaton n MP perturbaton theory : all Slater determnants formed by exctng electrons form the occuped to the vrtual orbtals are also egenfunctons of Ĥ wth an egenvalue equal to the sum of the one electron energes of the occuped spn orbtals. So a determnant formed by exctng from the th spn orbtal th n the Hartree-Foc ground state nto the a vrtual spn orbtal a Âϕ ()ϕ ()ϕ ( )ϕ a ()ϕ + ( +)!ϕ ( ) has the egenvalue a + a ε ε Smlarly, the doubly excted determnant, has the egenvalue + a + b ε ε ε ε J. F. Harrson //7
and so on. Wth electrons we have ground state spn orbtals (,,!, ) whle the number of vrtual orbtals depends on the number of functons n the expanson bass. Lets say we have vrtual orbtals (a,,!, ). We then have sngle exctatons, double exctatons, 3 3 trples, etc. up to fold exctatons. The total number of excted determnants and therefore the total number of excted egenfunctons of Ĥ s total Knowng all of the egenvalues and egenfunctons of Ĥ we can use Raylegh- Schrodnger perturbaton theory to fnd the energes and egenfunctons of Ĥ. We wrte the perturbaton as the dfference between the perturbed and unperturbed Hamltonans. As usual H H H f () + g(, ) < The Foc operator has the form F() f () + () where the Hartree-Foc potental s gven by * τ ϕ ϕ () d () () g(, )( P ) () The one-electron operators, f n H & dfference resultng n the perturbaton H are dentcal and cancel n tang the g(, ) ( ) < whch s the dfference between the nstantaneous and average electron-electron nteracton. Ths perturbaton s sometmes called the fluctuaton potental as one magnes that t measures the devaton from the mean of the electron-electron nteracton. J. F. Harrson //7
The frst order correcton to the energy s the average of the perturbaton over the unpertubed wavefuncton. In ths context ths s gven by g(, ) ( ) () < From the Slater-Condon rules we have and resultng n g (, ) ϕϕ g(,)( P ) ϕϕ < < ϕϕ g(, )( P ) ϕϕ () ϕϕ g(, )( P) ϕϕ We note that the energy through frst order s smply the Hartree-Foc energy. () + ε ϕϕ (, )( ) ϕϕ H g P The second order correcton to the ground state energy depends on the frst order correcton to the wavefuncton. Ths n turn depends on matrx elements of the perturbaton between the unperturbed ground and excted states of Ĥ. In ths context ths s () ν µν ν S + D + T +!+ The sngle exctatons contrbute S a ε ε a a J. F. Harrson //7 3
Snce a g(, ) ± g(, )( P) a < ϕ ϕ ϕ ϕ and a ± g(, )( P) a ϕ ϕ ϕ ϕ so (, ) a g and < S. The double exctatons contrbute D < a< b a b Because s a one electron operator all matrx elements between & vansh and only g(,) contrbutes < and therefore g(, ) l ± l g(, )( P) a b < ϕ ϕ ϕ ϕ D ϕϕ g(,)( P ) ϕ ϕ a b < a< b a b All matrx elements of the perturbaton nvolvng trple or hgher exctatons vansh and so T + Q +!+ J. F. Harrson //7 4
and () D 4 ϕϕ g(, )( P ) ϕ ϕ a b,, a b We have rewrtten the summatons as unrestrcted sums and note that the & a b terms vansh. ote also that denomnator s always negatve so as requred. () <, J. F. Harrson //7 5