Moller Plesset Perturbton Theory In Moller-Plesset (MP) perturbton theory one tes the unperturbed Hmltonn for n tom or molecule s the sum of the one prtcle Foc opertors H F() where the egenfunctons of F re the occuped nd vrtul Hrtree-Foc orbtls of the system nd the egenvlues the ssocted one electron energes. Fϕ εϕ The Hrtree-Foc wvefuncton (,,, ) A ϕ () ϕ () ϕ ( ) s n egenfuncton of wth n egenvlue equl to the sum of the one electron energes of the occuped spn orbtls ε The essentl observton n MP perturbton theory s tht ll Slter determnnts formed by exctng electrons form the occuped to the vrtul orbtls re lso egenfunctons of wth n egenvlue equl to the sum of the one electron energes of the occuped spn orbtls. So determnnt formed by exctng from the ground stte nto the th vrtul spn orbtl A ϕ () ϕ () ϕ ( ) ϕ ( ) ϕ ( + ) ϕ( ) + hs the egenvlue + ε ε Smlrly, the doubly excted determnnt, th b hs the egenvlue spn orbtl n the Hrtree-Foc b + + b ε ε ε ε J. F. Hrrson /7/7
nd so on. Wth electrons we hve ground stte spn orbtls (,,, ) whle the number of vrtul orbtls depends on the number of functons n the expnson bss. Lets sy we hve V vrtul orbtls (,,, V ). We then hve V sngle excttons, V double excttons, V 3 3 trples, etc. up to V fold excttons. The totl number of excted determnnts nd therefore the totl number of excted egenfunctons of s totl V Knowng ll of the egenvlues nd egenfunctons of we cn use Rylegh- Schrodnger perturbton theory to fnd the energes nd egenfunctons of. We wrte the perturbton s the dfference between the perturbed nd unperturbed Hmltonns. As usul V H H H f () + g(, ) < The Foc opertor hs the form F() f () + V () where the Hrtree-Foc potentl s gven by * τ ϕ ϕ V () d () () g(, )( P ) () The one electron opertors, f n H & dfference resultng n the perturbton H re dentcl nd cncel n tng the V g(, ) V ( ) < whch s the dfference between the nstntneous nd verge electron-electron ntercton. Ths perturbton s sometmes clled the fluctuton potentl s one mgne tht t mesures the devton from the men of the electron-electron ntercton. J. F. Hrrson /7/7
The frst order correcton to the energy s the verge of the perturbton over the unpertubed wvefuncton. In ths context ths s gven by () V g(, ) V ( ) < From the Slter-Condon rules we hve nd resultng n g (, ) ϕϕ g(,)( P ) ϕ ϕ < < V ϕϕ g(, )( P ) ϕϕ () ϕ ϕ g(, )( P ) ϕ ϕ We note tht the energy through frst order s smply the Hrtree-Foc energy. () + ε ϕϕ (, )( ) ϕϕ H g P The second order correcton to the ground stte energy depends on the frst order correcton to the wvefuncton. Ths n turn depends on mtrx elements of the perturbton between the unperturbed ground nd excted sttes of ths s () Vµν S D T + ν ν + + + The sngle excttons contrbute. In ths context S V V ε ε Snce J. F. Hrrson /7/7 3
± P < ϕ nd g(, ) ϕ ϕ g(, )( ) ϕ V ± g(, )( P ) ϕ ϕ ϕ ϕ so (, ) g V nd < S. The double excttons contrbute D V V b < < b ε + ε ε ε b Becuse V s one electron opertor ll mtrx elements between & b vnsh nd only nd therefore < g(, ) contrbutes b g(, ) l ± ϕϕl g(, )( P ) ϕϕ b < D V ϕϕ g(, )( P ) ϕ ϕ b ε + ε ε ε < < b b All mtrx elements of the perturbton nvolvng trple or hgher excttons vnsh nd so nd T Q + + + J. F. Hrrson /7/7 4
() D 4 V ϕϕ g(, )( P ) ϕ ϕ b ε + ε ε ε,, b b We hve rewrtten the summtons s unrestrcted sums nd note tht the & b s requred. terms vnsh. ote lso tht denomntor s lwys negtve so () <, J. F. Hrrson /7/7 5