PROBLEM SET #4 SOLUTIONS by Robert A. DiStasio Jr.

Transcription

1 PROBLM ST # SOLUTIONS y Roert. DStso Jr. Q. Prove tht the MP eergy s sze-osstet for two ftely seprted losed shell frgmets. The MP orrelto eergy s gve the sp-ortl ss s: vrt vrt MP orr Δ. or two moleulr systems d B tht re ftely seprted, the qudruple summto e wrtte s vrt vrt B B vrt vrt B vrt vrt B se the uped spe d vrtul spe ow e prttoed to two seprte spes elogg to moleulr systems d B respetvely. Ths e doe se the moleulr systems re ftely seprted. xpltly rryg out the multplto of summtos yelds the followg 6 terms: vrt vrt B B B B vrt vrt vrt vrt vrt vrt vrt vrt B B B B B vrt vrt B vrt vrt B vrt vrt B vrt B vrt B B B B vrt B vrt vrt B vrt vrt B vrt vrt B vrt B B B B B vrt B vrt B vrt B vrt B vrt B vrt B. rom the umertor of, we see tht the MP orrelto eergy wll vsh f. So, let s ow osder whh of these tegrls wll survve d whh wll vsh. The frst lss of tegrls tht wll vsh wll e those tht lude exttos from the uped spe of oe moleulr system to the vrtul spe of the other moleulr system,.e., tegrls of the

2 form B where p s MO elogg to moleulr system. The seod lss of tegrls wll vsh due to the tegrl kerel, r, whh s r, wll use terms lke to go to zero. Wth the removl of these terms, smplfes to: B B vrt vrt B B vrt B vrt B. Therefore, we re left wth oly two terms to osder. The frst term s gve s MP orr, vrt vrt 5 whle the seod term s gve s MP orr, B B B vrt B vrt B B B BB. 6 Therefore the MP orrelto eergy of two systems, d B, t fte seprto wll e MP MP equl to the sum of d, therey provg sze-osstey for the MP orrelto eergy. orr, orr, B xpltly wrte out the MP eergy orreto to sgle determt of sportls whh does ot stsfy the Brllou odto. rom Rylegh-Shrödger perturto theory, we kow tht the seod-order orreto to the eergy s gve y: V. 7 Se we kow the omplete set of egefutos of Ĥ, we expd the frst-order orreto to the wvefuto the egespe of the upertured mlto s follows

3 V V V V 8 where termedte ormlzto ws ssumed d the results of mpultg the λ equto. I the se of MP theory, s the referee soluto d the set of re the egefutos of the ok opertor.e., exted Slter determts. The dom of, the dex of summto 8 s { S D T Q L}, ut se V s two-eletro opertor, oly the sgles S d doules D re kept. Therefore, we ould rewrte 8 s V S, D S S D D V S V D. 9 If the referee stsfes Brllou s theorem, the wll equl zero d 9 S smplfes eve further to the stdrd MP eergy expresso: D V D D vrt vrt Δ. owever, f Brllou s theorem s ot stsfed the 9 wll ot smplfy to, ut wll lso lude otrutos from the sgles. Se S, the V S, so 9 eomes usg for the stdrd doule otruto: vrt vrt whh s the MP eergy orreto to sgle determt of sp-ortls whh does ot stsfy the Brllou odto.

4 overt ths to sptl ortl expresso for determt of sp-ortls tht re sp-restrted d desrg losed shell system wth eve umer of eletros. The sp-ortl expresso for the MP eergy orreto to sgle determt of sportls whh does ot stsfy the Brllou odto s gve y s vrt vrt. To overt ths to the restrted sptl ortl form, we proeed s follows: frst, we overt the sp-ortls to urestrted sptl ortl form, d the efore the restrted odto tht. So, our frst step s to overt to urestrted sptl ortl form. osderg the frst term we hve vrt vrt vrt vrt vrt vrt vrt vrt vrt d for the seod term we hve f the tegrls over sp re doe orretly vrt vrt vrt vrt vrt vrt vrt vrt vrt vrt vrt vrt vrt vrt vrt vrt vrt.

5 To omplete the dervto, we ow set the lph sptl ortls equl to the et sptl ortls. Therefore, eomes vrt vrt N / K N / 5 d eomes vrt N / K N / N / K N / vrt vrt [ N / K N / ] vrt vrt. 6 Therefore, the sp-restrted expresso for the MP eergy orreto to sgle determt of sp-ortls whh does ot stsfy the Brllou odto s gve y.f. 5 d 6 s, R N / K N / N / K N / [ ]. 7 or referee, we lso hve the sp-urestrted expresso for the MP eergy orreto to sgle determt of sp-ortls whh does ot stsfy the Brllou odto s gve y mpulto of d.. Revst prolem umer from prolem set, d ompute the MP orreto to the pproxmte S soluto tht resulted from sgle step from the tl guess whh hd oth eletros o e. Se ths wvefuto s ot true soluto, you should use your expresso from prolem ove. The ok mtrx geerted fter sgle step PS Q s gve s

6 from whh the eletro d totl R eerges were omputed s -.8 d hrtrees, respetvely. rom 7, we hve two otrutos to the MP eergy for ths referee wvefuto se Brllou s theorem s ot stsfed. The frst otruto s from the N K sgles d ws foud s N. To evlute ths term, we eed the ortl eergy egevlues of ths ok mtrx, foud ove s d -.6. Therefore the sgles.5 otruto s gve y hrtrees. The seod otruto s from the doules d ws foud s [ ]. To evlute ths term we eed the tegrl from PS whh ws omputed for us s.77. Therefore the doules.77 otruto s gve y hrtrees. Wth these orretos, we ow hve hrtrees, whh e dded to the totl R eergy ove, hrtrees, to yeld the totl ludg uler-uler repulso eergy to orret to seod order s -.57 hrtrees.. tomzto eergy of wter usg wvefuto-sed methods. Use eletro struture progrm to perform the followg lultos. Do your ow lultos t the MP d SDT levels to ompute the tomzto eergy usg the -pvdz ss set freezg the s ore ortl. ompre your results gst ext soluto I whh s vlle for ths model prolem J. hem. Phys., Use ths omprso to ssess the performe of, MP, SD vlle s output from the SDT lulto reltve to SDT reoverg the tomzto eergy of wter. Report the frto of the orrelto eergy reovered y MP, SD d SDT. t the MP/-pVDZ level of theory: O hrtrees R/-pVDZ o the sglet stte

7 O hrtrees U/-pVDZ o the trplet stte hrtrees U/-pVDZ o the doulet stte O - O. hrtrees 9 kl/mol t the SDT/-pVDZ level of theory: O hrtrees R/-pVDZ o the sglet stte O hrtrees U/-pVDZ o the trplet stte hrtrees U/-pVDZ o the doulet stte O - O. hrtrees 8 kl/mol rom PS, the expermetl vlue for the tomzto eergy of wter ws gve s 9. kl/mol. lerly, oth MP d SDT the -pvdz ss re le to predt ths vlue wth eptle level of ury. rom J. hem. Phys., 996, the I eergy for O the -pvdz ss wth ll eletros orrelted s hrtrees. Se predts the eergy of O s hrtrees the sme O ss, the ss set orrelto eergy s omputed s. hrtrees 8 kl/mol. The orrelto eerges ths prolem usg the sme O ss set were foud s.5 hrtrees MP,.5 hrtrees SD, d.8 hrtrees SDT. Therefore the peret orrelto eergy reovery y these methods s 9.8 % MP, 97. % SD, d 98.6 % SDT. lthough the I referee ludes the ore eletros, ths does ot tke wy from the m pot,.e., the peret orrelto eergy reovered follows the tred: MP < SD < SDT. Repet wth the lrger 6-Gdf,pd ss d ompre gst your results from prt d gst the est expermetlly derved vlue of 9. kl/mol. Dsuss refly the role of the ss set. t the MP/6-Gdf,pd level of theory:

8 O hrtrees R/6-Gdf,pd o the sglet stte O hrtrees U/6-Gdf,pd o the trplet stte hrtrees U/6-Gdf,pd o the doulet stte O - O.6 hrtrees 6 kl/mol t the SDT/ 6-Gdf,pd level of theory: O hrtrees R/6-Gdf,pd o the sglet stte O hrtrees U/6-Gdf,pd o the trplet stte hrtrees U/6-Gdf,pd o the doulet stte O - O.56 hrtrees kl/mol Usg the lrger 6-Gdf,pd O ss set, our predtos for the tomzto eergy re etter. or MP, we oserve derese the peret error from.9% -pvdz to.% 6- Gdf,pd. or SDT, we oserve derese the peret error from 5.% -pvdz to.7% 6-Gdf,pd. s we rese the qulty of the O ss set, we re removg some of the error ssoted wth usg omplete O ss.. Wrte the proeto equtos for the method usg the sme pproh used to derve the SD equto letures.e. usg termedte ormlzto. Idte whh terms re resposle for the flure of the method to e sze-osstet, d how they dffer from the SD equtos. xpd the SD wvefuto through qudruple exttos to show how d eletro-susttutos re derved s produts of lower luster opertors. Dsuss how ths permts the SD method to e sze-osstet. The wvefuto e wrtte s follows Ψ 9 where s the referee wvefuto, vrt

9 d vrt. Of ourse, the eergy ssoted wth the wvefuto s ORR. Therefore we hve three sets of ukows: the orrelto eergy ORR, the sgles mpltudes }, d the doules mpltudes { }. To fd these ukows, we use the suspe { proeto method trodued lss, d efore the followg three odtos: Ψ Ψ Ψ. Usg 9,, d, we mpulte s follows: Ψ ORR where I hve used Brllou s theorem d the expresso for the totl eergy gve ove to smplfy ths expresso. rom, we ow hve the followg expresso for the orrelto eergy ORR vrt

10 terms of the doules mpltudes. I smlr mer, I mpulte to yeld: Ψ 5 Itrodug y oveto, we wrte 5 s Ψ ORR vrt ORR ORR 6 therey gvg us other expresso reltg the orrelto eergy, the sgles mpltudes, d the doule mpltudes, mely k vrt k k vrt ORR. 7 Note: We ould ot OV equtos of the form gve 7 y proetg o the left wth vrt therey provdg us wth eough equtos to solve for ll of the OV sgles mpltudes. logous mpulto of yelds

11 ORR 8 vrt k k k vrt kl d d kl d kl d oe g, oe ould proet o the left wth,, vrt to yeld eough equtos of the form gve 8 to solve for ll of the O V doules mpltudes. The orrelto eergy s ot sze-osstet euse the LS of 7 d 8 re qudrt the vetors euse the orrelto eergy expresso ludes the doules mpltudes whle the RS of 7 d 8 re ler the vetors. If oe osders two moleulr systems d B, s we dd ove, the the uped spe d vrtul spe e prttoed to two seprte spes elogg to moleulr systems d B respetvely. Oe g, ths oly e doe se the moleulr systems re ftely seprted. Now, f we thk out doule extto, the 8 wll eome ORR ORR ORR [ B] 9 therey mplyg tht the vlue of, doules mpltude o depeds o the orrelto eergy of B ORR B. Se there wll e o term o the RS of 9 to el ths term, sze-osstey wll follow. s ws doe lss for D, oe lso derve the SD equtos usg the suspe proeto method desred ove. The SD wvefuto s very smlr form to the wvefuto 9, wth expoetl stz,.e., Ψ SD exp T T where d T re defed smlr mer to d, mely T

12 t T vrt d vrt T t wth the SD totl eergy gve y SD ORR SD. The suspe proeto equtos - lso requre slght modfto,.e., Ψ SD SD Ψ SD SD Ψ. SD SD The expoetl opertor e expded the stdrd wy exp T T T T T TT T T TT T L whh I hve segregted terms sed o the overll extto.e., sgles, doules, trples, et.. Se the mlto ossts of oe- d two-eletro opertors, we trute ths sum t the qudruples whe delg wth the suspe proeto oto the sp of the doules. Let s eg y usg : SD Ψ SD SD T T T SD SD T SD T SD T 5 SD T T ORR SD T T whh dretly provdes us wth expresso for the SD orrelto eergy:

13 SD T. 6 ORR T T T logous mpulto o d yelds the followg oupled expressos for the SD sgles d doules mpltudes: ORR SD t T T TT T T T TT T 7 d t ORR SD T T T T T T T T T T T T T T. 8 The mportt pot to ote here s tht exttos of hgher order th the level of truto the luster opertor T eter the mpltude equtos 7 d 8 d therey ffetg the mpltudes d the SD orrelto eergy. By omprg the SD equtos gve y 6-8 wth the equtos gve y, 7, d 8, we ote severl key dfferees. The frst dfferee s the orrelto eergy expresso, whh the SD eergy expresso ludes dsoeted doules whh s produt of sgles. I the sgles mpltude equtos, SD ludes dsoeted trples v T T d T, whle the doules mpltude equtos, SD ludes oth dsoeted trples v T T d T s well s dsoeted qudruples v T d T. It s the luso of these produts of exttos tht use SD to e sze-osstet, euse oth sdes of the SD equtos re ow osstet wth respet to order of t. or ste, the SD sgles mpltude equto 7 the LS s ow thrd order wrt t rsg from the multplto of the sgles mpltude d the oeted doules mpltude t d t k T dsoeted doule mpltude t t k oted wth the SD orrelto eergy. The RS of

14 7 s lso thrd order wrt t rsg from two types of dsoeted trples mpltudes, mely, from T T d t t tk from T. t t k