https://ntrs.nasa.gov/search.jsp?R=19700010180 2020-03-12T00:58:24+00:00Z -1. The V a r i a t i o n a l Method I1 - - P e r t u r b a t i o n p he or^" S a u l T . E p s t e i n T h e o r e t i c a l Chemis t ry I n s t i t u t e a n d P h y s i c s bepactmelt t U n i v e r s i t y o f Wiscons in , Madison, i d i s c o c s i n I WOULD GREATLY APPRECLATE YOlJR CRlTTCISMS , SOT@lEfu'T, AXD SUGGESTIONS CONCERNING THE CONTENT AaD ORCYNIZATLON OE THESE LECT3RFS -Jk R e s e a r c h s u p p o r t e d by N a t i o n a l A e r o n a u t i c s and Space A d m i n i s t r a t i o n G r a n t NGL 50-002-001. XII. The V a r i d c j on P r i n c i p l e and P e r t u r b a t i 3 n T h e a r y - --- ----+ --- T h e r e a r e m a n y connectL~or?s be tween the ~ : a r i a t ~ c l i ~ rtlet l ~ o d and p e r t u r b a t i o n t h e o r y , I n t h e s u c c e e d i n g s e c t i o n s we wi 1 l b e c o n c e r n e d i n t h e main with u s i n g t h e v a r i a t i o n a l method t a aipproximate s o l u t i o n s o f the p e r t u r b 3 t o equations, and, b e f o r e t f ia- L ? w i . t i i t.he c l o s e l y re- f,,. i: .i; ec 1 .- problem of tire perlr!.ir.bai:Foil :?nt:lysis o f optl-ma!. t r i s i f u n c t i o n s and e n e r g i e s .. K<;i+ever t h e v a r i a k i o n p r inc i~p l : . can :.lay ,stl-ier r o l e s , . , 8- .- FOP ~:-:.am~:!(+ ':t. <:~a of'tei! , . ; z T ~ T i _ ~ , ~ - L & ! i t i n t i : p ~ i . ~ : L " > ~ t ' , o r i '.' ~lli+O?:)r ,T1 ql-, ~b I . T!i:is . ef 12 "$ btti.12 p e s ~ u r ~ - : ~ ~ f o n ~ t ; r a m p . t e r , TkLen Ll 4 "": - I <: ,-j: !..* f -.... -. * 1 IL, * I d - d ~ c e ~ p w ; - ~ ; Y ~ *& Is an ei .geL?funct loz? ;:not n t - - c e r ~ s a r ~ l y or' \ tlie Loi~esir !;bai:p) :hen jt js: an i_mrnedi.ate cc;ns.t+c!uenc. of t i l e v a r i a t i o n i '<,' \)2& ,k~. i r e a t \ .~% : i.s of o r d e r a n d we have tkic wel.1. known r e s u l t t h a t t o c a l c u l a t e t h e e n e r g y t h r o u g h o r d e r 21 Zn+B one n e e d s t o m know t h e wa.ve f u n c t i o n o n l y t h r o u g h o r d e r 7) This t heo rem i s u s u a l l y $4agvce by c l e v e r r e a r r a n g e m e n t o f t h e p e r t u r b a t i o n e q u a t i o n s . I n S e c . X I V . we w i l l p r e s e n t a g e n e r a l i z a t i o n o f t h i s t heo rem, I n t h i s s e c t i o n we w i s h t o d i s c u s s y e t a n o t h e v k i n d o f c o n n e c t i o n ; o n e which , s o t o speak , goes from p e r t u r b a t i - o n t h e o r y t o a v a r i a t i o n a L p r i n c i p l e r a t h e r t h a n t h e o t h e r wa.y r o u n d , The p o i n t i s s i m p l y t h e f o l l o w i n g , The mark o f a v a r i a t i o n a l p r i n c i p l e i s t h a t i t Leads t o e r r o r s of second o r d e r i n t i l e q u a n t i t y t o b e c a l c u l a t e d (we have been c o n c e n t r a t i n g on ihe e n e r g y ) , T h i s t h e n s u g g e s t s t h a t i f u s i n g p e r - t u r b a t i o n t h e o r y vie can q fve a fo rmula f o r c a L c u l a t i n g a q u a n t i t y c o r r e c t I:hrougl-I f i r s t ardtzr , i , e , , s o t h a t t h e error i.s of seeorid o r d e r , t h e n 3 Lorn z n l s ? : J x r i l ~ i a we s t~o~ i i i i : ue a b l e i30 icier ;i v a r i a ~ i o n p r i n c i p l e f o r t h e qr r ; l r , t i ty 1 1 1 q u e s r i o n , f T r t i s a l s o s u g g t s r s t h a t s t a r t i n g from t h e r e s u l t s of h igher order pe r tu rba t ion theory one could i n f e r "super -var ia t ion pr inc ip les" . See f o r example Kfkuta, Prog. Theo. Phys. 1210 (1954)) 14, 457 (1955). See a l s o Koster and - - Brooks, J. Math. Phys. 5-, 169 (1954) .) Rather than d i scuss ing i n d e t a i l how t o c a r r y out t h i s program we w i l l simply i l l u s t r a t e t h e point by a few examples, o t h e r s w i l l occur i n l a t e r s e c t i o n s . (For a genera l d i scuss ion s e e Pomraning, J . Math. Phys. g, 149 (196?) and re ferences given there,) F i r s t l e t us cons ider t h e energy i t s e l f . Then given a zero order wave func t ion 9"' , a zero orde r energy E'*' and a zerd o rde r H a m i l f o n i a n ~ ~ ) one has t h e f a m i l i a r r e s u l t t h a t through f i r s t o rde r t h e energy f o r a Hamiltonian H i s given by which we can w r i t e a s I n t h i s l a s t form a l l e x p l i c i t r e f e rence t o &\v' has disappeared. Fur ther it i s a formula which is c o r r e c t through f i r s t o rder i n t he e r r o r i n # , whatever qy9) may be. Hence we can i n f e r t h a t w i l l y i e l d a v a r i a t i o n a l approximation t o t h e energy, and we a r e back where we s t a r t e d from. To g l v e a n o t h e r exampie, l e t US u s e p e r t u r b a t i o n t h e o r y t o der rve a v a r i a r i o u p r i n c i p l s f o r a G r e e n ' s f u n c t i o n , l i s iag an o p e r a t o x n o t a - t i o n o a e has 4 - e ) C - - \ o r , c o r r e c t t h r o u g h f l l s r _ o r d e r g the Green's f u n c t i o n i s g i v e n by (XII -3) F o l l o w i n g t h e p a t t e r n in t h e p r e v i o u s case, we now e l m i n a t e e x p i i c i t r e f e r e n c e t o MW To d i t h i s we use ( 2 ) r o w r i t e ( 3 ) a s r w j . a -.. F..i -.-- whence, s i c c e tc ,> iir.,- i i d r y j we i n f e r t h a t r- yk - t-- 3 ''= /Pruhieim: P roye i i . i i i n : r e and show thai- t h i s q u a n t i t y i ' q ~ i h l s Cy p 1 ~ s ts~rns of o r d e r ( n o t e t h a t ~ y ~ , ~ ~ - ~ * t = i j 1 \ F t n z i L y i , ~ give an example from sca.c"iering eheory , We cons~e fe r the s p a t t e r i n g - 0" .. a particle of mass H3 and energy , C -- by a c e n t r a l f i e l d and e o n f f n e a t t e n t : i o n t o a singEe parcia1 wave, Then tl~ere is a w e l l IC'noi.7n fmEiula which gives '.'-I times the t a n g e n t of t h e p f z e s ~ . s h i . 3 ~ \'v] 5x1 (b) by o f We can nov e? 1-~riinace e x p l l e - i t I-efelence ro -7 ( a+ b7+ 0-6) d i ""6 hence i n f e r Char ,, .; vill furnish a v a r l a r i a a : a p p r o a i m s t i c n LO v the he i n g variation e q u a l t o z e r o ; i ? . i ~ r e s ~ i a t ~ a f t e r an igt-gra~i~~; by pa rks , r a n b?: 1sr-iti.m in a form identicel f-0 t h a z 7,;til-jch x e found for bocud s t a t e s . namely X I I I , Pe r tu rba t ion Analysis o f t h e V a r i a t i o n a l Method Let us suppose t h a t t h e Hamiltonian takes t h e form wi th we a r ev' and ' independent of t h e r e a l parameter 3 , and t h a t . f s . ." f i /"- e i n t e r e s t e d i n e x h i b i t i n g 9 and a s powgr s e r i e s i n (we 1 w i l l assume wi thout f u r t h e r comment t h a t t h i s i s poss ib l e ) . Qne way t o d e r i v e such expansions i s - s imp ly t o c a r r y ou t t h e cornpeete c a t c u l a t i o n and then t s . e x p n d the r e s u l t s . The o t h e r approach, and t h e one which we wish to exp lore i n t h i s section, is &o t r y t o produce ;the succes s ive terms Ln t h e s e r i e s . by an i t e r a t i v e procedure. /- While t h e second approach i s t he one which has u sua l ly been followed i n say ca l c u l a f i n g the response t o e x t e r n a l f i e l d s , o r i n t h e 'Iq? expansion, i n r ecen t yea r s t h e r e has been a t r end t h e o t h e r way. Thus t o c a l c u l a t e the s t a c i c p o l a r i z i - b i l i t y ; i n s t ead of d 5 r e c t l y -ca lcu la t - ing t h e second order energy, one c a l c u l a t e s $he %,otal energy i n a s e r i e s of weak f i e l d s and f i t s : & r e s u l t s t o a polynomial. See f o r example papers by Hurst and co l l abo ra to r s , Phys. Rev. 167, 1 (1868), Cohen and Roothaan, J. Chem. Phys ., 43, 534 (19651, and Pople \ e t . a l . , J. Chem. Phys. 49, 2960 (1968). / Before t u r n i n g t o t h i s however, l e t us s ay a few words about t h e var- p o s s i b i l i t i e s f o r ittQ9 and J &*that one encounters i n a p p l i c a t i o n s . F i r s t of ell a'] w i l l u sua l ly be e i t h e r t h e "accurate" Hamiltonian f o r an i s o l a t e d system o r some approximation t o it. I n say ing t h i s w e q u a l i f i e d t h e word a c c u r a t e by quota t ion marks because i t i s a r e l a t i v e term. ~ G u s f o r an i s o l a t e d atom or a molecule, under many circumstances t h e f ixed nucleus, n o n - r e l a t i v i s t i c (no magnetic e f f e c t s ) Hamiltonian, i s the s tandard of accuracy, wi th c o r r e c t i o n s adequate ly taken c a r e of by f i r s t o r d e r pe r tu rba t ion theory. I n o t h e r circumstances, p a r t i c u l a r l y f o r heavy elements, t h i s w i l l n o t be t h e case and r e l a t i v i s t i c e f f e c t s must be t r e a t e d more accu ra t e ly . I f nw' i s an accu ra t e Hamiltonien then J*'.' w i l l u sua l ly r ep re sen t t h e e f f e c t of an e x t e r n a l f i e l d , (i .e. , t h e s o r t of tching we have elsewhere included i n . On t h e o t h e r hand i f e.G> i s approximate then gqW might r ep re sen t t he e f f e c t of an e x t e r n a l f i e l d b u t it might ahso be t h e d i f f e r e n c e between t h e "exacis" aild appsox.imate .Hamiltonians of the i o s l a t e d system, i .e . , t h e )su cf e a r l i e r d i scuss ions . Also i t might be a q u a n t i t y f o r which we would l i k e t o c a l c u l a t e an accu ra t e expec ta t ion va lue and which us ing an in te rchange theorem, we have introduced a s a fictitious pe r tu rba t ion (i .e. , again a quan t i t y of t h e M For complete g e n e r a l i t y then we r e a l l y should r e p l a c e \) H by a sum of terms ( inc luding terms of d i f f e r e n t o rde r s ) , however s i n c e t h e modi f ica t ion i n t h e formalism which t h i s in t roduces should be q u i t e obvious, a t l e a s t i n low orders , we w i l l forgo complicat ing the n o t a t i o n a t t h i s po in t . 6 & a Let us now r e t u r n t o our problem -- t h e expansion of 9 and E &.n b hdc ,.;\l;~-y PC We w i l l conf ine ou r a t t e n t i o n V t o s i t u a t i o n s i n whjch t h e s e pf trial t- func t ions ? i s independent of J s i n c e t h i s i s o f t e n t h e case i n - p r a c t i c e . I f we denote t h e c o l l e c t i o n of parameters and/or func t ions rJ which l a b e l t h e members of t h e space by CL , and i f we denote a t y p i c a l t r i a l func t ion by ), then t h i s means t h a t A w i l l depend on 4 o n l y because t h e depend on d 6" (As we have s e e n A t h i s a l s o means t h a t 9 w i l l s a t i s f y tlze Hellmann-E'eynmtrn theorem f u r W -=+-rd and, g e n e r a l i z i n g to t h e c a s e of s e v e r a l p e r t u r b a t i o n s , that i n t e r c h a n g e theorems w i l l be s a t i s f i e d , ) I n a c c o r d with o u r program t h e n we w r i t e (6% [+&%a d F u r t h e r we wi.1 i assu!ite t h a t Q*., can be expancied j.n a pc\,ir-r s e l - e s i n @;' T h i s i s s u f f i c i e n t Lo y i e l d ( 3 ) and ( 4 ) b u t n o t :lecissa~:', Hawever we assume i t because i t s i m p l i f i e s t h e d i s c u s s i o n , bJe will comment on more g e n e r a l p 0 s s i b i l i t i . e ~ a t t h e end of t h i s s e c t i o n , A k 4 From o u r assumpt ion t h a t &. depends on on ly because C?, does 7 it then f o l l o w t h a t e t c . Now a p o i n t o f n o t a t i o n , The formulae ( 6 ) - (8: a r e q u i t ? e x p l F c i t \-* /n (excep t f o r o u r u s e o f t h e c o l l e c t i v e symbol '8 ~ h u z if tl-r '- a r e a s e t of parameters then the r ,h.s. of (7) i s s h o r t hand f o r z i However f o r compactness we w i l l r ep l ace z,w h'Q) '@ ' e t c . which we w i l l then f u r t h e r abb rev ia re by a- /a& k\v7 e t c . Thus we have This more compact n o t a t i o n should cause no confusion. However, one must keep i n mind t h a t some members of t h e 3' may be zero. (Can- s i d e r f o r example the S t a r k e f f e c t foy a c l ~ s e d s h e l l atom and l e t 04e N 4 f t h e a measure t h e amount of s t a t e . ) Note a l s o that: if t h e r e i-4 # i s no d i s t i n c t i o n between and GL (c . f . t h e S - l fmi t i n Helium) Cs) then q, ' &L,I . I\ qw F i n a l l y w e need 6% . We now remqrk t h a t we can exp lo re t he Q f i *tin i n t h e neighborhood of Q simply by making a r b i t r a r y changes i n a a lone . Thus it w i l l be s u f f i c i e n t t o cons ider 4 J S with $2') a r b i t r a r y wi th in t h e s e t ( i . e . i f &L i s an a r b i t r a r y -\a number &a; i s an a r b i t r a r y number; i f g; ig an a r b i t r a r y func t ion 7 of c e r t a i n va r i ab l e s , then 8 is an a r b i t r a r y func t ion of those (r v a r i a b l e s ' ) . We now expand 6% i n powers of 7/ t o f ind, us ing (9). I f we now s u b s t i t u t e tlhese korniuias i n t o t h e b a s i c e q u a t r o n of t h e v a r i a t i o r r a l ~it,:r'rrsd, t l ' i ~ r e s u l r s w i l l be Ti: t h e f ~ i n l o r power sex ,t - 5-ii d e q u a l t o zero, S i n c e @ i s a;:bitrral-y t h i s rnearze co- eff iciel- i ts of each power o f 3 ni l r s i ; v a n i s h s e p a r a t e l y , >lore s p e c i f j - c:all;r J e t 1:s p.;ri"i ei;he eynairions ti; tj-:e &r.11~-aj F C ~ L . ~ ! (1.13--.5) , k - I f now w e jrtsert tlle expansions , 3 assume t h e forln From w h i c h w e x a i l l der ive t h e sequence o f e q u a t t o u s 6 f k-r" - A. 'r- and " ' - \ * j The f i r s r few ~ " i ~ e qivcn by (XI 11- 1s) e t c . &\*\ The equat ions (15) and (16) now enable us t o determine the A f i \*j /a and the E recurs ' ively. ~ h u s from 6Sa=0 3 we will J h i ) ) determine %*' and , Then ~ 5 ~ ' ) yie lds , using (18), (11) . . . and (10). 6\03 (\.\17) C C * , 6 ~ ) t-j!y?- &) p) +- L., L. /t klr) which toge the r w i th s ~ % ~ w i l l s e rve t o determine t h e 0. and C C j 5 e tc . , e t c . This kind of approach has been s y s t e m a t i c a l l y pursued by J. N. Pv Silverman and c o l l a b o r a t o r s f o r ca ses i n which t h e Ch are parameters .. ' (see Phys'i Rev'. 162, 1175 (1967), a n d ' f u r t h e r r e f e rences there..). How- ever, i t is ' imp l i c i t knmanjt,applications of pe r tu rba t ion Fetsyy w i t h i 0 t h e variatTona l method. , . i". i t shou id be n o t e d however t h a t t h e assumpt ion t h a t a" can be iu. expanded i n i n t e g r a l powers of does n o t r e q u i r e t h a t Ca can a l s o I r l be s o expanded. For example Or migirt i n v o l v e and an example of t h i s s o r t i s men t io r~cd i n t h e n e x t s e c t i o n . Thus o u r d i s c u s s i o n i s n o t complete 1y g e n e r a l , Hov~ever, t h e genera 1 p a t c e r n --- i s c o r r e c t -L'hub; 83- we w i l l have 8%' o f "Ltlc g e n e r a l form (11) and hence one will 15) h';r>", and (10) . However, t h e d e t a i l r d formulae f o r t h e + and t h e &'"'." , at, give81 b:i (9) arid (10) i , ~ i l l nclt a p p l y , / t J Probk~ .n*~: F o f ~ T G \> we cou ld e v i d e n t l y e q u a l l y w e l l v a i y - by isak~n,: zP-x,.rg~s o a l y i n gt"T s a y . Shoi; I i t i l ~ ~ L S ~ o f a -&b\ \ w i t h d0- aahbi t rary y i e l d s t h e same e q u a r i o n s a s above / I s o w v I n o u r d i s c u s s i o n we have assumed t h e s p a c e of t r i a l f u u c t i o n s t o be independent o f 3 . It i s a l s o oE i n t e r e s t t o d i s c u s s one pa r t i c t l l r t r s o r t o f s i t u a t i o n i n which t h i s 5.s n o t t h e c a s e . It a r i s e s n a t u r a l l y f o r example i n e x t e n d i u g r e s t r i c t e d Har t ree -Fock t o g ~ j t - 1 ~ L e x t e r n a l f i e l d s . P,kso ii; a r i s e s i n a more g e n e r a l c o n n e c t i m which we w i l l d i s c u s s i n S e c , XVL Namely suppose t h a t id where $J( i s a g iven f u n c t i o n ( u s u a l l y d e r i v e d from sclule s e t o f t r i a l . f u n c t i o n s v i a $$%'=D b u t n o t p e c e s s a r i l y ) and where tzhe CJt ' \ ~ , r j 3 c o n t a i n a r b i t r a r y parameters a n d / o r f u p c t i o n s whose p p t i n 1 ~ 1 va lues we r e q u i r e t o be independent or' (Note t h a t w h a t ~ v ~ r t h e soucee of - - -I_YI_-_I-4? - 9 1 A \ d a- v 9 i n n o t a t i o n i s c o i w i s t e n t s i n c e e v i d e n t l y f o r J Z O t i le o p t i m a l A 105 i n t h e p r e s e n t c o n t e x t i s i j , . ) - Varying t h e s e q u a n t i t j - e s we w i l l t h e n e v i d e n t l y be Led a g a i n t o A e q u a t i o n s o f t h e form (15) excep t t h a t now sp- --Q and f o r t h e ,$ \") we as ye2 have no formulae l ike (15) ; f o r th;. ii;oll;t.~tc 3" t h e y a r e s i m p l y t h e v a r i d t i o n s o E i and not l l ing nlore. It is nab 44- i m p o r t a n t t o n o t l c e t h a t t h e s e e q u a t i o ~ l s now irnpose res c-rict Lolls on a . t he kinds r,f $ 6 ~ i d b ~ ~ , 1 l d ~ ;;.araiilzters aaci/or L i ~ u r e ~ , n s XWF .a i tched i r a (.u 3 t h e qtd'j! Thr i i c o i r s i d t r ;k.e e q i i a c ~ o n T f y i e ? c i a 'I\ \<> r g q C Y - 0 jXIT1-21) \&A+* A. ,] r " k ~ 3 a n d v t e l l s cis r:l. L sq iirui;z b? Z R allowed !:vsxrstjon u f 'but a l s o C l l e a imbedded i n 4 Then A , &giQ$ Rt.8.5 T., \.~) K~-.J\ f L whence: s i n c e the and &re independent , S $ iYk.U i m p l i e s and r t c.> $9 13, 2 h%3 E q , (23) then r e q u r r e s tihat --.- be an 31lo~td 7ddr ia t Lon a1 b& E%l> \$ b 2 l"j while ( 2 3 ) ~ r ~ q 0,. \ - , + - , r ~ ~ - i ~ l + f i + ~ " , v ~ ~ J ~ ~ q v ~ f i r ~ > - . , A603 - (1) t h e r e f o r e (21) implies r e s t r i c t i o n s on tl/ . However i n a, - - . . 4e have taken t h e p o i n t of v i e w t h a t U/ i s given and t h a t some cases one might want t o consider t he qb). A s given, whence (21) y i e l d s requirements on '$la) . /' XIV. A Genera 1 . Theorem . 3 . - As we have discussed, i f one knows a n exac t e igenfunct ion tor- . . M r e c t l y through order 2/ , t hen one can c a l c u l a t e the exac t e igenvalpe , . - - ~ r n + \ c o r r e c t l y through order . We now want t o e x h i b i t a s i m i l a r theorem w i t h i n the con tex t of the v a r i a t i o n a l method. I- : , For g e n e r a l i t y we w i l l a l h w the t r i a l fdnc t ions t o depend e x p l i c i t l y . on . Thus . . :-. ., . . . - . . : i . . . . . . .. ,.:s:..: ' - . . : . i s , ': ............ :..-;:..;.i : . . ................ . . ; a . . , - , : . - . ' , &. - .: . _ cL, . . . . . . . - . . . . . . . . . . . . . . . . -+ =.;. +..[a v ) . . ....... . . . . ' C , . . . .- . . . . . . : . . - , . . . . . . . . .. ... * . . . . . . . . . . Now cons ider t h e func t ion d e f i n e d by . . . . , . . . . . . . . . . . . . . . . . . . . . . . . . . .., . a+t. f poss ib ly terms of order 9 . . . . . . . . . . . . . . (XIV - 2) (XIV - 3) Y r 4 Thus t h e i m p l i c i t t) dependence o f * d i f f e r s from t h a t of 9 by ~ . t \ terms of order Y , while , i t s - - e x p l i c i t 9 dependence must be thp - $.h+"" ia$e a t l e a s t t o order . . i-; . . . . . . . . . . . 1 . - .- ,., . ... ; . :. W. ., .. ..,... .1 a'-.- r........ . . (XIV -4) Then one r e a d i l y f i nds from That Now from (2) and (3) i t fol lows &ha t %, 2 - . . \ Thqn we no te t h a t I\ n * ;:>p &q-J (a- - . , % , , - $ MS\ ?s.?.(i) o f order. Y 'w-3 so % t h a t t h e Last term, i n (5) is of o rde r 9 A ,and (ii) 'is an allowed v a r i a t i o n of 'k' s o t h a t it makes no contrltbu- t i o n t o t h e sum of t h e f i r s t t w o terms i n ( 5 ) . Thus i t fol lows t h a t f i even i f (c)-C> + is of q rde r zero, s t i l l n Thus i f t h e a invo.lve only i n t e g r a l power of \) we have t he P- J\ r e s u l t t h a t i f we know IX c o r r e c t l y through o rde r Y and if we , , i nc lude any e x p l i c i t .d dependence c o r r e c t l y a t l e a s t through ~rcjer ?/ -+\ Ir ?..W+t , ' then ,we can c a l c u l a t e E c o r r e ~ ; tly through order \f . .In p a r t c c u l a r %hen - i f - t h e space of t r i a l funcf;ions is independent. of 3 we have a d i r e c t g e n e r a l i z a t i o n of the V &hP\ theorem. O f A 'X c o u r s e if CH- E) = OLv') where S> CJ ice can realien t h e addi.t:iona 1 requ i rement on the e x p l i c i t 'd dependence, Namely e * t o d e r i v e ( 7 ) w e need o n l y t h a t '63 ,. ? L " , , ; ~ ~ ~ - u , + : j ! : : ; = C J ~ ki41!2 the cheer em, - . .. , . $ k%+'b p r i',oli id y.25 rxp.c t thz t h e ~ r ~ ~ ilo ha\ie d s e i n l aons@.cjcerrc.e for calcuBat is ;h~s of r!-$e so - - - "a L exefibpa,Ef i e d by XIXI-207 dew about l iaear \i-ls.riational. ca lcu:a Lions i n which the b a s i s fznet t r rns depend on -3' ? "1 A 1 L k s&y i.c,voZve h a l f integral potaers 96- ( ~ 1 3 ~ w h 3 - I g i v e t P an example i n a m o n ~ e n t ) then ( 3 ) requires t h a t t.~e kaelw c o r r e c t l y 13s*# through order Koqdever, i n some c a s e s accbracy through ordeiL v'' will s r i l l s u ~ f i c r . Namely suppose we r e p l a c e ( 3 ) b y Then ias-cesd of (6 ' i.re w i g nave ; o *ina-"3d Now a s i t s t z n d s k"r l o o k s t o he of i l ' s d ~ r L) buc ~ f , 38 ue \* . 4i- P- assuaie 5 and % caiT be B=.P;IL?SB~ i n i n l e g r a i porlers af 9 , a% - fir. ): I [nust in f a c t he ot- order 'L' ,- . i h u s the l a s t rzril ir IS) i s 3gai t l n ..-of order \I . - UL'V , Turning t o t he f i r s t two terms, i s s t i l l an allowed v a r i a t i o n , so t h a t on ly t h e OL~'~'') p a r t of & c o n t r i b r 'L* 2. u t e s . -This c o n t r i b u t i o n w i l l then be of order V i f - - As.a case i n po in t cons..ider the ground s t a t e of ~ & l i u m wichv* '9 w --&ha. e l e c t r o n - e l e c t r o q i n t e r a c t i o n and t h a t l i n e a r com'binatf on of " \Z&&?SI -. a n d i d i d Z p \ ,which y i e l d s a s p i n s i ng l e t , the^ one f i n d s ( s ee BFbberk, Proc, Phys. Soc, a, 33 (1967) and r e f e r e n c e s t h e r e , See a l s o H i b b ~ r t , gr-oe, Phys, S O G . - 1, 51048 (1968) f o r a , : . . - , s i m i l a r method applZed t o L i . For some reason i n t hese papers the & method i s r e f e r r e d t.o a s URHP) !bat a l though can be expanded i n . 1. A Ca . , in , tegra l powers of dal , t h e , namely P and 3 involve @ . + f i *>* However % is exac t i n t h e l i m i t d - 4 D C ' d ) and hence (9) i s s a t i s f i e d , Thus i n t h i s ca se one does f i n d t h a t knowing the o r b i t a l s - A c o r r e c t l y through order * enables one t o c a l c u l a t e c o r r e c t l y %*+I : - . through order Y ". ! :. B /Problem: Give a n ana logous d i scuss ion f o r o ther frae tiona l powers Problem : Since f o r a l i n e a r space- (which may depend on dl, i s a n e x a c t e igenfunct ion of , i t fol lows t h a t i f What i s the connect ion between t h i s r e s u l t and t h a t farrnd \above? J 7 The Var i a t i 1 l e d M e a n s f o r Approximating P e r t u r b a t i o n ___r Theory ,~k As we have shown in S e c , LV, i f , i n the v a r i a t i o n a l method, 69 tk k c a n be c o m p l e t e l y a r b i t r a r y , t h e n and s e k i s f y the Schryedingen. c.0 64%bWY v o y u n i i u n . S i m i l a r l y we w i l l now show t h a t i f &$'%' can be a r n i t r a r y , r"- s.-,-) kY\) t h e n t h e kk. and h s a t i s f y the bas i c e q u a t i o w of p e r t u r h a c ~ s n t heo ry , The p roof r e q u i r e s l i t t i 2 more t h e n writing c u t the succes- r3- 7 >* C, Cjdj r ., (jLF,L 1 a- cLv--> +- [-\,<+ -I "1-1 I+ &\Q-, ) 1 , . - . %\8) ROW L E 03: ~zs C C S I R D ; ~ ~ ~ ~ ~ ~ a r b i t r a r y then_, (1) y i e l d s theory, t e l l i n g us t h a t p7 and 6'~' should be an eigenfunction, and eigenvalue respec t ive ly of etv) . fh.03 ' Turning now t o ( 2 ) and us ing ( 4 ) , $q completely a r b i t r a r y I the- f ami l i a r f i r L t order equation - - - - and - --, rhen (3) with (4), (5), and 6qW' completely a r b i t r a r y , y i e l d s e t c . , e t c . % ;-< ../' + . s - . . .. . ,.. . J -'. 3.L. . . . , ?sf:-. -i..Fu~;khed;'35i acco'rd p i t h the d i scuss ion i n See. X I L I , we get t h e .&\$>: . ?same e q & a t t a d i = * be a'+ 'and le t ~ 6 . ~ ' be completely arbi- . . . fi ,v?~* :: t r a r y . ' ~ h u s k t h - sy (I)' y ie lds no informatian, but (2 ) with = bpF@ - and .&-'). . compliithiy & b i t r a r y y ie lds (4). m e n using (41, i . (3 ) w i t h -&%g And .&?? ioatplete a r b i t r e r y , y ie lds (5) e t c . S imi la r ly we se: the same r e s u l t s i f we put 6% and &Gh' equal to ('WJ z e r o a n d . c l e t 6 9 be cawle te - f f -a rb i t r a ry , e ~ e & I . . ' : ~ a v i n ~ ow seen i n d e t a i l how. .the v a r i a t i a n a l ate thod conta ins . .. . - , . per turbat ion theory, the following possib$f i t y immediately presentg A i t s e l f : J u s t a s one uses 8 3'0 t o der ive approximations t o the so lu t ions of the S c h r ~ e d i n g e r equati,on,. sb one might try t o use, ehe , J &$I*, . . tt + , - t d f i i d approxiimatiqnp- to . t h e . . solu t ions of the perturba- t i o n equations. The next s e v e r a l sec t ions w i l l be devoted t o i n - v e s t i g a t i n g t h i s program. om Thus w e assume t h a t * and Cw) E areexact, i.e., that (4) ts fi\~l s a t i s f i e d . Also we w i l l assume f o r convenience tha t % fs w - normal ized T h i s be ing t h e case we w i l l t h e n deno te* h'm and L $ 'km' and F?" r e s p e c t t i v e i y , Fl i r ther we w i l l i m p l i c i t l y assume t h a t *b) 7s e s s e n t i a l l y u n i q u e , i . e . t h a r @) i s son-degenera te . i d i s c u s s i o n of d e g e n e r a t e p e r t u r b a t i o n t h e o r y can be found i n J , C h e m , Phys, 44, t-> :> -b-9 3266 (1960;, Then f rom 3 zFs it i o l l o u s , that, 3s I!;+ ex;;pir, c #i. now i u i r . T o t h e de t e rn lnac ion of q' , w h l ~ t i we ill B S S U . ~ ~ fi we c s n ' t de t e rn i i r i e exac t ly . S l n a t we have been "handed" %?" i r is hw) is n a t u r a l t o b a s e our d i s c u s s i o n on (XIII-20),' Then s ince 9 exace and hence a l l variaeioas are allowed, t h e r e a r e no r e s t r i c t i o n s C-L- imposed o n t h e a n d f r o m f X I I I - 2 % ) x ~ e R a v e a s t h e e q u a c i s n c o & * s t , de te rmine 0- ?'-BW> JP- 1%) The e q u a t i o n then de te rmines : -., c- XVI. The Hylieraas- Var r a t i o n a l Method . . From t h e i r s t r u c t u r e s and from the remark t h a t the re w e no \ a p r i o r i r e s t r i c t i o n s on &PI', it i s c l e a r t h a t (XV-9) and (W-10) - can be viewed a s descr ib ing a process q u i t e analogous t o t h a t of t& v a r i a t i o n a l method, Namely de f ine a t r i a l second order energy -47' :Z 1 r, 2 : . > * . where 3( IS a t r i a l f i r s t o rde iwave functFondrawnPrornsurrte set;, Then (XV-9) says t h a t v' is thak v) which m a b s \r. s t a t i o n a r y f o r var ia teons of wi th in the Bet, while (XV-10) says that the . , -.. - +w.) , et4 s t a t i o n a r y value of i s b This observat ion a$so serves as ,a .statement of the Bylleraas v a r i a t i o n method, and we w i l l c a l l t he func t iona l on the rph .s . of (1) the Hylleraas functional * 1q and denote it by 3d . Thus (XV-9) and (XV-10) can be w r i t t e n The (probably impatient) reader w i l l no doubt have noticed t+t: I I . . "9 we could have w r i t t e n down d i r e c t l y a s y ie ld ing the v a r i a t i o n a l & equivalent of eq. (IN-5), and thereby could have avoided a l l the d i s r - - cussion of the preceding sec t ion . With t h i s sentiment: we- agree; 01le - , . ? & only plea is t h a t we have thereby placed (2) i n a more general. see t f ng, 'A i f i - c a n a l s o be r e l a t e d . t o t h e v a r i a t i o n a l f u n c t i o n a l . \ f o r a ree en" f u n c t i o n g iven i n (X'LL-41, Thus c o n s i d e r t h e e q u a t i o n where t h e r e a l number %E , and t h e funcc ion % a r e g i v e n , The a p p r o ~ r i a t e v a r i a t i o n a l f u n c t j ona l ts then Now t h e solution c~ (3) rs (i;e assam-e laa at -&- i s noc equal t o ars i ; i z i j r ~ 3 L ~ r i c f k bi"'3 1 t-4 where & i s a Hermit ian I n t ~ g r a l ope rz ro r (& t r i a l Green 's f u n c t i o n ) , Then ( 4 ) becomes where gc) i s clie rzzc'; s e ~ ~ r i $ order e n e r g y - We w i l l d i s c u s s tlhe s i t u a t i o n foe exc i res r srar-es rn S e c , XLX, T i l l then we w i l l be concerned e x c l u s i v e l y w i t h t h e ground statti: =:nd hsnce x u . i . l l r ; o ~ k e e ~ 2 k - l ' mentioning th,$$ f a c t e x p ' i h i t l v Thus .- it 12cr avz.rlarrc-n;'"i The proof i s simple.' We wri te where JZ\) i s the exact f i r s t order function s a t i s fy ing - , L#Q~?- ~ i m ) @ ) + c + - ~ l - ~ b\) JZWI- t3 Then using (8 ) and ( 9 ) one r ead i l y f inds :..< ;.E\y2 where i s the &act-second order energy: 1 , , (? ". i . , % J 1 . . . .- Cv-?. ~ ~ * y %I"J . (xvr -11) ~*), For k' the y&wes t eiie*value of {fd)we then c l e a r l y have (7). The f a c t t h a t we have a minimal p r inc ip le m y nor: seent su rp r i@iqg , - a _. . - After I a l l the general va r i a t i ona l method y ie lds a minimum uader the s imi lar circumstances. However, it i s i n f a c t not t r i v i a l and is . .,,. , l i ~ l & d t o our assumption t h a t 4 is exact. Namely we khow tha t f i - f o r the ground s t a t e - g % and therefore for a l l $ Now i n the absence of fu r the r information, about a l l that w e can fib--,) concludg, about the indiyidua 1 E is what w e know anyway, namely t ha t &\Q which follows from ( 12) when 3-9 . On the other band with & % 7 ~ E\3 exact s o t h a t and kh'Z~%' , we f ind , d ividing by $1- +La* ,anti setting 9 % ~ , G I L ~ G a s before. f i Following the argument r'eaiether, i f r$." i s a l s o exsct t hen we P-a* [ .--j expect (and can show from 5 - 3 ' " j 3 t D ) t L i = a l s o be exac t , whence we infer .Cram (23) t h s r The r e s u l t (14) air' coe~rse a 1s; Zoslc~%s from a-he sa rbac iona l rcethod $ 3 5.4 under the saine coodi i i o c j . > P t E and 9 -\C ptlh ' $'b%"Y4U yiild~ hi- f i r s t we p u t sp'= f i - + ' * ' ; i . . iien o @ t v g information, namely zrs e q o a t i s n f o r 9 0% n!le f o ~ m (Prqbleala: F i l l i n t he d e t a i l s ) w where i s , not s u r p r i s i n g l y , j u s t the v a r i a c r o n a l func t iona l 8 t h a t one would assoc i a t e with the second order e q u a t f 01% That i s &A ,y-- r."ij Fu r the r i t folLows from d ,9 t h a t - ~ 3 From t h i s we can then d i r e c t l y in fe r ( 1 4 ) . Namely we know tha t =f and from (17) it follows tha t when i s s ta t ionary when - 4~~~ C p ) i t s numerical value i s - NOW i f we wr i t e "\*) thei . . the only poss ible source of 'second o rde r t e r m s ' i n Sfl i e f i r s tcterm on the r . h. s .V ( 16), whence we i n f e r t h a t , . - - wh-ich. proves- the point . . . * ?be gyi le raas var iaxional method, eqs, (11, (15), and t he i r ketteral'ikations t o higher even orders, has been widely used t o yield approximations t o per turbat ion theory, fo r example i n t he ca lcula- . ,tion of s t a t i c p o l a r i z i b i l i t i e s and s u s c e p t i b i l i t i e s , i n the calcula - t i o n of the e f f e c t of Coulomb repulsion ( Yg approximation), i n the , ca lcu la t ion of long range forces etc. , e tc . , and many of these appl ica t ions w i l l be discussed i n d e t a i l i n l a t e r chapters. (Note t ha t s-ince i s l e s s than zero, an upper bound f a r eG7 yie lqs a lower bound fo r p o l a r i i i b i l i t i e s etc.) "^'=*,e> -+ . .., . We .would emphasize however t h a t (7) holds oqly i f ? while ( l a ) holds only i f and p3\ts.) e t c . Boweven kc ._ *\@I " t.)grant ing t h a t '.f. - , ustially one then derives from (1) and then. goes on t o use i t i n (15) etc. , ( a l so & is usually c a l c y ~ " bl* l a t ed from the formulae one gets f r o m S --o assuming t h a t +-I .is exact) and one may wel l wonder about the legitimacy of such e prnce+lqre. xn, general: of course one can only hope t ha t one has aehieved a . suff ic+enf ly accurate approximation t o p' , etc., so thaL a l l is well numerically. 104 One should a l s o n o t e t h a t t h e formulae of t h e H y l l e r a a s v a r i a - t i o n a l method w i t h t h e symbols s u i t a b l y r e i n t e r p r e t e d a l s o app ly t o p e r t u r b a t i o n t h e o r y w i t h i n t h e v a r i a t i o n a l method a s d i s c u s s e d i r i Sec , X I 1 1 i f t h e space of t r i a l f u n c t i o n i s l i n e a r , One can see f> ld t h i s i.n s e v e r a l ways. F i r s t o f a l l i n a Linear spece $- ~jv =-O for &v,#.-4, \ Thus (XI,{\ -19) has t h e form (XVtg] and s i m i l z i r kv , higher [k %--= C f - o r d e r s . Another approach i s t o n o t e t h a t pf"* @here 'k is an ?-+ (c,.,* .- L,: 1 e i g e n f u n c t i o n o f &4 Thus % , ~ h . i 3 E S . ' _ C i c - V eLc, b; <:,y<;3 i L.\ -------J F u r t h e r i f i s independent o i t h e n 2.: k$ dqd !&kb and a l l h igher o rde r pertj~rliacfoki v a n i s h , Thus t d c way app ly a l l che above fo rvu l ae excepc that w t should p u t %ars sr~E$-'~< ' ~ " 9 and ** and t h e ,, Bowever we may then drrp r'be 11.r~ on $.)wS and ~3%) s ince t h e func txons a l l . l i e i n t h e space, Now a s a p r a c t i c a l m a t t e r , one can, as we have d i s c u s s e d b e f o r e , u s u a l l y comple te ly c a r r y through a c a l c u l a t i o n i n a linear space, --b> i . e . ' f i n d t h e &Sa , o n l y i f t h e space i s f i n i t e l y p a r a m e t r i z e d , kf 7' t h e space i s r e a l l y i n f i n i t e ( t h e S - l i m i t i n He wi th 4t8aW' the e l e c t r o n - e l e c t r o n i n t e r a c t i o n i s a n exatnplr 1 then one n ~ u s c r esor t t o approximation, I f one u s e s a f i n i t e l y parametrazed f ~ ~ n e a r . . Subspace, t h e n t h e H y l l e r a a s formulae of course ag-in ~ ~ 9 % w i ~ h f u r t h e r r e i n t e r p r e t a t i o n of the symbols, I f one uses n s n - l i n e a r pa ramete rs too , then i n any case, t h e a n a l y s i s of Sec, X Z l X a p p l i e s , I f however one does u s e che formulae of the HyLleraas method, then Ab> one'must hope t h a t 9 is a s u f f i c i e n t l y accasrcte a%:roxir r~at ion t o - tq ~k e t c . .f.. I n t h e p r e z e d i ~ g d i se i t s s ion w2 hhve ;ass::tn~d L l i u L +""" 4; is exace. and wor r ied abou t ere. However i f %i\V) ;q tii easr t R a i i i i l ~ o n i a n . 8 . 1 . . . ' . . *-. . of an i s o l a t e d system (with more than 1 e lec t ron) then one r a r e l y \d ktio*s'.& with any prec.isi,on. Never the l e s s , many workers have used (1) and (2)with +'" replaced by an approximate p' and have taken pG a s an approximation t o E ' ~ . WF w i l l discuss some p roper t i e s of t h i s approximation i n Sec. XXI. . . 't XVTE. -THe Hylleraas Va'riational Mefhod for the Second Order Enery37 .j- .' MCficellaneous 'Results . ' - ' (A) If t h e r e a r e M r e a l i t y restrictions on 8P' than w e cap r ep lace (XV-8) by , .. 3 . I t i - h-1 - = &;6i;: ~.'l0'52 y .+ CY~L v,$y, *'WE) -O (mu- 19 with i t s obvious i n t e r p r e t a t i o n i n terms of nioments. (XVXI~ 2.) G * Mu.s" +\l v i s an allowed v a r i a t i o n wieh 69 r e a l then (9), together with (10) , is $lkowedJ theti one find's , - . I.. - . . .$. ->:. . . . ; .... < : . . s : . . . . ' ~ .>;. . - - ':" - . . . . &) , c*G9 <*+??- ~ ~ - 5 ) YLn) 1 < ;. :. ,,*:.. . . . . . * . . . which i s q u i t e anqlogous t o (WIz10). (Problem: Derive (3) and ( 5 ) , ) . %at> (C) For some purposes i t i s u s e f u l t o r e p l a c e t h e f u n c t i o n a l $ Y 7% by another , equ iva l en t , f u n c t i o n a l which is der ived from J (r;) a s -1-4 follows : Suppose t h a t t h e o v e r a l l s c a l e of t h e 9 i s a r b i t r a r y . That i s , suppose t h a t hJ" I&;. t,g zp' where e i s a v a r i a t i o n a l parameter and $ i s independent s f t CV I f t h e r e a r e no p r i o r i r ea l i . cy r e s t r i c t i o n s on C , hen rbe opcfmrzl! # & C. fo r a gxuren (we will not in t roduce a special . symbol fur t h i s It G ) i s e v i d e n t l y determi,-iad 3y whence w e have r-J I f now we use t h i s va lue f o r Cj. i n t he Hyl le raas Funct iona l , we ge t a new f u n c t i o n a l which we w i l l c a l l the D i r i c h l e t Funqt iona l A 6 $? then y i e l d s equatioas t o determine $" lz i s t h e n t q i" k%> determined by use of ( 7 ) l and Jg and i n p a r t i c u l a r Sg y i e l d -%-a A"\.> upper bounds t o k- The 3 D ferrnished by the paq-ticrstar choice , c ~ 3 ' ' : .-- 7 - * - / ~ % ' h a yie.lds the so- -ca l IeT tfassk bauna t o en + One sometimes re@ t h a t rr, is a b e t t e r approximation t o than i s ~ $ 7 . I n view of our discussion this sounds puzzling. What is meant is 'the f ollowitng: Suppose t h a t i n P F w e i n s e r t & (6) with ( 7 ) , but simply Then the conclusion w i l l c l e q r l j follow s ince we have not, adjusted the o v e r a l l s c a l e of ' qM ' i n qn ~ t i m a l way. Tbps f o r the same t r i a l funcnon, i . ~ . , qw= I)-*' , we have for the ground s t a t e - 4 3 L-? -49 ~n::~a; t i c u l a r , qdte t h a t l&ke ke , gSg is ,defi=itely c:,. -,:a;. . . . . . . . . ?'* ... , . - . . . " ,..y-> + - '. -'I : .. . . r . . , . . . t' .- . . . .. ' : .... y. . .-" .. . . . ; ...+ .-.- .-.; _. - . . 4% .i. ~r obi& : ~9rb;ve -(8) d Z ~ e c t.ly by shrrwZng that -$ is ,. 'tY.0 kai.i.,tfie gk$una- &ate. . \ - - .d . . (2) If pnc.. t r i e s :.:ko prove 4irecLl.y that .: : .,: ? 5 :. = . . r . '-.. Bi ". .."" .. - . ' . ?.,.<<.f' :a . , . ... . . . 3 , . - . . - ,:.. ... .: * 8 i , * . 2 .--. 3 . rue : .. : . 2 > ,. e &b-l , ' by mi t i n g '?a. ..=\j2' , - a . e tc , , - one- f &nds. '&t . one . mus f invoke lhe -. e s. /. Schwartz inequa l i ty ( ~ r o b l k & ~ i l l " k * - t h e d e t a i l s ) . We will .ow , show t h a t i n genera l (9) can be considered as a'speEi&l'-c+&e'dPJt&e, Schwar t e inequal i ty . \ i c , . . To avoid spurious s i n g u l a r i t i e s , let us replace . %.,ir s l i g h t l y - l e s s Chad E~'.-SO that C , I @ is a s t r l c ~ t l y posttivs '. - - ,., -- . 3 r - z - -- - ..a --. , operator . A t the end of the c a l c u l a t i o n we can thep let %-++be. T - W i . t h t h i s understanding, .gW' c e e ' b e mitten ( I . - . ^, , : : f : . , :-a I , . . e , . ;$ .'" .. , . ! 'A,. ....... .:I . . .... 2. - 4;. ~ 2 . . . . . . . . , 2 . I'f which we w i ll...vr&ii- &'.- . . . - - . .. where C q n L E;nr) and C%) *fh Now from .our hypotheses n d G e x i u t and are Heroii t iun, r rhrneru "- ") follows from the Schwarte i n e q u a l i t y , i i i a i Lor any P- Cb-3 -Up_ , w.&& J: - b q p,6b.4""-gbb ~9 &-? 151g>3 C& F) c &'r;ByaFy & Q il , "26 1 I -L 9 -% -4~ whence, s i n c e the r.h.s. equa's f cT\g3, &'~r XJ, hsve> a ti, of r ea r r ang ing t h a t which of ,course . i s j u s t . Sroblem: General ize the r e su l t s (4) - (c) to higher orders , (E) I n many s i t u a t i o n s q ~ ~ b ' desc r ibes t h e i n t e r a c t i o n of t h c s y s tem wi th two f i e Id? IVY 2. 'Lv) and one i s i n t e r e s t e d i n the cross term i n y E , i .e., t& term propor t iona 1 t o q\dv (sh ie ld ing coef f i c i n t s , electro-magqe& e f f e c t s , e t c . ) Now from our d iscuss ion s o f a r we have g v a r i a t i o n a l upper bound to k* E'*! , and hence a var i q t iona 1 upper bound tm f I , @uL dr -43 ) w and a v a r i a t i o n a l upper bound t o E, ( p u t d l = - - ~ )J. Haplfqver, tSris '@I y i e l d s no hold on \& . Thus for exemp le, suppose that wg would have 6 % ~ Then from !S a r ~ ' ~ we can't conclude anything about: thg relative -w s i z e s of %'?+and ir, . In t h i s s i t u a t i o n s e v e r a l a l t e r n a t i v e s a r e open and have been used, ( i ) Determine a Vt-"EL' and take the term i n ir propor t ' ooa l r tr3 t o *\lid a s the "bestt ' approximation t o the E , . Note however fu.3 t h a t i f we def ine 3 by 9,-+'3 and 'tPdL*3> thqn i f 'k involves fit4 non-linear parameters, then usua l ly f3 w i l l not s k q l y be a . quadra t ic form i n 3 .. f*w . ( i i ) Put Jb* a d determine an 5 ) . Then use eke t o i* compute a "best" approxima t ion t o EtV according to where . . . . . - ~, ,<; ,. .. . .. ?*. , .. .. , , i. - - , . > - - . -. . . . . . . . . I . . l . _ . . . . . . - .- - 1 :, ' ; . - Rev. 172, 1788 (1968) Sec." I1 f o r some comments on t h i s p o i n t . ) - tb.9 However, we need no t use 3 , i n t h i s way. One could, f o r example, *\.) i n s e r t f o r t h e from ( i i ) and f o r qz t h e @: from ( i i i ) w i t h t h e hope t h a t (14) would then provide a b e t t e r approximation t o C*) then e i t h e r (11) or (12). \ 4roblem: Show t h a t i f i n t h e procedures ( i ) - ( i v ) one draws a l l t he t r i a l func t ions from a common l i n e a r 2 space, then a l l four methods y i e l d t h e Shha answer. .+/ ( v ) Find a v a r i a t i o n a l lower bound f o r c9 (and hence f o r E';' 9 and @: 1, That is , given the second order energy, c a l l it Et , f o r t%) and t h e second order energy, c a l l it e - , f o r then ev ident ly -1%) - -=\1? 5?+-k= \, - La3 Thus a n upper (lower) bound f o r E + combined wi th a lower (upper) -49 bound f o r $2 y i e l d s an upper (lower) bound f o r k \, We w i 11 d i scuss such lower bounds i n ~ h a p t e r - ~ (a"\\] I n p r a c t i c e one often t a k e s f o r t h e l i n e a r v a r i a t i o n a l forrr w J where t h e +G a r e given b a s i s s e t and t h e g$' a r e v a r i a t i o n a l pa ramete rs . The +? can of c o u r s e a l s o c o n t a i n pa ramete rs , however we w i l l c o n s i d e r them f ixed f o r t h e purposes of zhe present d i s - c u s s i o n s , Now c l e a r l y we can r e p l a c e t h e 8 by a se t of $\ f u i ~ c t i o n s yb w i t h t h e p r o p e r t i e s Ca A namely t h e 2& and % a r e j u s t t h e %P-- and k- which emerge from t h e l i n e a r v a r i a t i o n a l method of Sec , VL w i t h and t h e *L a g a i n t h e b a s i s s e t . Thus w i t h o u t l o s s of g e n e r a l i t y we may r e p l a c e (1) by lL bb$ S i n c e we w i l l impose no r e a l i t y r e s t r i c t i o n s on t h e \o+ w e c a n now u s e ( X V I I - 2 ) t o f i n d , v a r y i n g 'G; , and u s i n g ( 2 ) and (31, whence and from (XVII-5) we have These formulae of course have a c lose resemblance t o the "sum over 4,-7 s t a t e s " formulae fo r and 8. I n the above discussion we have imp l i c i t l y assumed that none \0 ) of the i s equal t o c) . I f we would include 4 i n the bas i s se t , say choose , +'@ then one of the +\L , c a l l it , would a l s o be*': and would equal Under these circumstances (5) holds only fo r (A) w h i l e can be chosen qui te a r b i t r a r i l y . I n pa r t i cu la r then we can take i t equal t o zero. Then s ince from (2), c'%&,+~! w i l l equal zero fo r k-k) we find which have ?in even scroyiger reaerr2~1,anee ~:ZJ the "s~nrra over states" @\-> which wi.1.L be an upper, - - - . s t a t e , -The $, i .so~~.ssioa T~I- 0.i ,- 4.osely p;_raLle l. that ~:'clFch we gave & earlier :if c ;ondit~~:,aw unde- l;$li,ch E ~iJc~r~:j .d he an x ~ , p p e ~ b~u .~ l ; i t o foz exs i ted s r a t e s . F i r s t & a l l suppose t h a t there is sarne syrralc.eLr:y p r o p e r t y which ' = - i s mai~tsined La Lhe presence oif the p e r b u r b a r i o r ~ , Then if t t h e lowest z e r a order efgenszalve asssciated w i t h chat- s ~ 7 m r c r y . a n d if > b-9 t h e 89"' a11 knave t h a t symmetry then it irnmediateky f o l l o r ; ~ s from (m-1-18) t h a t s i n c e if w e imagine expanded in terms ' of the ei ge i l f inc t ions of kig7 , it can involve o n l y s . ~ ; i t t s with energy gxester tiran or equa l 1x1. Sec, 11, w e a.l.so pointed orst t h a t more genera l ly .we wernl.r%! P", k have u i f were orthtsgonal t o a l l the lower s taces . However we di.snal.saecl tihis a s betrag of lzo p r a c t t c d linterest: in ck?,e absence of symmetry, s ince one wo*~.kd not [<,no% the lower s t a t e s exactly. However i n the pr eserrt contexr. the ana kogoess r e s v lt becomes rnur e i n t e r e s t i n g . The psiflt i s the t one mdy we14 know t h e exact z e r o order wave f e ~ ~ c t f . o n s f o r the Icwer states, dnd t h a r h c n ~ i n g this i t is then easy t o en fo r ce o r t h o g o n a l i t y t o the exact lower states through f i r s t order. We w i l l now show t h a t i f one has orthogonali ty A\%) -?*9 through f i r s t order then & 3 Suppose then there a r e zero order s t a t e s with eigenfunctions v' and eigenvalues ~2' such t h a t E~'o)cE'@ (more p rec i se ly i f the re i s symmetry involved then 1$ is the number of s t a t e s with J E P ~ E* and having the symmetry of 4' , however w e w i l l not f l continue t o s t r e s s t h i s point) . We now require t ha t* be orthogonal t o % through f i r s t order. Then, s ince they a r e orthogonal i n zero order, && WQf3-M mat (XIX- 1) But from thel'sum over s ta tes1 ' formulae whence from (1) we have w> A\-> C*a , I s - - - ~29\P'- c'") t@ k\ - k\3 which i n tu rn equals C* 1 . Thus i f (1) is s a t i s f i e d then A w i l l be orthogonal t o e l l p L g', and hence we w i l l have, a s announced, $-'- g4 . Now one way t o ensure t ha t (1) i s s a t i s f i e d i s simply t o use t r i a l functions of the form w where X i s orthogonal t o t h e q ' . (This is the approach suggested by Sinanoglu, Phys. Rev. 122, 491 (1961)). A r a t he r more f l ex ib l e T- pw where t k e aL ?-re c ~ r . h i ~ r s ~ y va r id r io i l a l parameters and where)' cdn * be a n y t h i n g eszegt t h a t i t zhuu ld no t involve the GL e x ? l i r : i ~ l y . C W To see that this works, n.ote t h a t if we ask f o r t-he o.ptf:c,ak Q-%, fa r w a. given 7 then we w i 3 . 1 have and hence, i n p a r t i c u l a r , ( 2 ) vi ll be s a t i s f i e d , which proves t h e point. Note, h o l ~ e v e r , t h a t in gene ra l u n l e s s )I Ls ort-hogsnsl to t he % F i n a l l y let: u s t t i rn t o t h e analog of t h e v e r i a t i o n a 1 rrrethod f o r e x c i t e d s f a t e s discv4sed i n S e c . L T . Let u s suppose t h a t we hdve used the l inea r v d : , r i a t f o n ~ . l method of Sec. W P 1 P and have chosen k!L s u f f i c i e n t l - y l a rge , i n d the +L s u f f i c i e n t l y sell, s o t h d t T of t h e 6k are 4 E'~' ., We w i l l noT), show t h a t under t hese c i rcun~s tances A E ~ ) ? t E ~ + To do this w e w i l l i m i t a t e thne pl-ocedwre i..f See, V I and add one m r c e f u c c t i o n t o our b a s i s s e t , Thwa w e cons ide r t r L a l f u n c t i o n s s f the form a..' r a. I . riw M Ad fat\) 9 '"- b@P f c,'" y& " C - + (XIX- 2) where without loss of general i ty we may assume tha t i s normalized and orthogonal t o a l l the rk . It is now a simple matter t o der ive A t e (Problem: Carry through the d e t a i l s ) the r e s u l t t ha t the new E , 'B I 4 c a l l i t [fi*]) i s given by where &!%b%) i s given by (XVIII-71, where i s non-negative, and where (XIX -5) We w i l l now show tha t a C em) ?(D whence we w i l l have 'h tm and hence, assuming convergence, E (w) ?E?) a s announced. To show t h a t %Cc@j i s pos i t ive we note t ha t from (VI-10) the roo t s of t c&' would jus t be the e ) ~ t h a t one would derive by using a t r i a l function of . the form (2) i n the l inear va r ia t iona l method a s applied t o ~ ~ s c ) ' " . Thus i n pa r t i cu l a r i t follows from the separation theorem (VI-11) t h a t I c2 7-t-\ 5. %+! Fur ther we c e r t a i n l y have -l ET+\ and from our i n i t i a l assumption we have That D@@) i s posi . t ive now fo l lows simply by looking a t a graph. Namely ~ ' T U i s given by t he app rop r i a t e i n t e r s e c t i o n of and the s t r a i g h t l i n e 6-tb) : whence we see from t h e graph t h a t which proves t h e p o i n t . Thus we have he re an a l t e r n a t i v e method f o r \d g e t t i n g upper bounds which does no t involve use of t he 4% . O f .I19 course, i n accord with the discussion a t the end of Sec. V I , one w i l l ge t a be t t e r r e s u l t i f one does requ i re (1). Problem: Prove t h i s . / I n t h i s discussion we have imp l i c i t l y assumed t h a t none of the CL was equal t o e). I f , changing the nota t ion s l i g h t l y from Sec. X V I I I ,3C T+ = tkw) (for a s excited s t a t e t h i s i s not t he only way of having an f ~ r E ~ ~ ) however it i s the most common one) then one f inds r e s u l t s l i k e ( 3 ) - ( 6 ) except t h a t the term %=T-tl i s excluded from the sums. (Problem: Carry through the de t a i l s . ) Comparing - with (VIg10) we see t ha t t h i s corresponds t o having, i n the nota t ion of t h a t section, one of the ~$'*:t)'~~&] equal t o zero. However, the qua l i t a t i ve r e s u l t i s the same, the appropriate graph baing \ (Problem : Carry through the de t a i 1s. ) 120 - * Srs;iel ~ a ; . o : r ~ f ! : ~ &at-.~;.f:, ---- - - Val-ha ti.cs~t3lb T,Ja?;s. F-.rzil: ki":r:ls ------- Our discuaaio.i!, h6x-e ,;?:j.12 h e ;.st%e~ned a f t e r t&at: f n Seca X. '6Je a r e i n t e r e s t & , ig , , . ST-L~; ei ia t $d,Z QRC n y w i l l s a t i s f y varf 033s thec3ren7.s tYi-rsz;& i'tr si: (07. 3ec:ond) o r d e r , The genera- l i z a t i o n of the.;e r e s t ~ l t . ~ 6 0 FL$&,ET b;irdars kea.ve as exerc-jze f o r the r e a d e r . (A ) -8J 1 j-~ z e & ~ g & ~ ~ ~ ~ ~ ~ ~ ~ y g + ; ~ ; ~ ~ J > ~ ~ ~ ~ ~ ~ ~ I f b o t h %'P' an.' , ooriratn the parsme~t . rCT-~ c l e a r l y 199 e, knowing only 4 zxct @ : I c , Since E\D) doesn ' t depend on eq. (2) i s t r i v i a l and s o i s Eq. ( 3 ) s ince and 4' is independent of (3- . (More accura te ly ,lb) could involve F only i n a numerical phase fac tor . ) Thus we a r e l e f t with the quest ion, when w i l l ( 4 ) be s a t i s f i e d ? Well, w e have % d'cp7 &pl3-@Qy 1 &'9 Now l e t us suppose t h a t :q bb6 is an allowed v a r i a t i o n of ('-'--I . Then i n c a l c u l a t i n g 4 (@-g4) 3e whence we have @v we may ignore the terms i n v o l v i n g 3 2 BcF I f 'Y/BV*o then t h i s i s the same a s ( 4 ) . I f a:r4d, then t o agree with ( 4 ) we a l s o must r e q u i r e t h a t the c o n d i t i o n s for which w e dfsct , , sssd i n Sec. m-IP-(B). ~ h u s i f we 8%- i=.p se lec th$ : f rom a set ef t r i a l fp-~nct ions which a r e independent of s o t h a t '9 so- is 2 , a l l t , , e d v z r i a t i o n . a d which have an a r b i t r a r y 'Br over a1.l. s c a l e s o t ha t (5) is s a t i s f i e d , then ( 4 ) w i l l be s a t i s f i e d a u t o m a t i c a i l y . (Again, i f i d U / o ~ - ~ then w e d o n ' t ~ e d ( 5 ) . ) c@d=mk @P- % d Pr sblem: Discuss t h e Genera l i zed Kellrnann-F'eynrnala theorem =-- f o r ehe e a s e r--, $ Now Pet u s c o n s i d e r t h e inore g e n e r a l c a s e i a tJZlich bo th H"' and v H ~ ' rnay i n v o l v e The Generai tzed Helimann-Feyilman rr1:eorem now r e a d s Expariding ( 6 ) i n power of d and eqlaating cerms of l i k e order we t h e n f i n d i n p a r t i c u l a r and Now ( 7 ) i s c e r t a i n l y s a t i s f i e d s i n c e 4'" i s ail e i g e n f u n c t i o n of F\" Thus our g o a l i s ~o f i n d s u f f i c i e n t c o n d i t i o n s s o t h a t (8) i s s a t i s f i e d , Before p roceed ing f u r t h e r i t w i l l be u s e f u l t o r e w r i t e (8) s o a s t o e l i m i n a t e t h e 2 Q&J@- - 3a- $ 9 Now from Ctf\d- 4 ' ~ 9 = i t follows t h a t whence we can rep lace (8) by \@ Now l e t us assume t h a t 3% a, 3 s &\.) is an allowed v a r i a t i o n of . Then from the Hylleraas v a r i a t i o n a l p r inc ip le we find where we have used ' ~'k"', qW7 3a . I f we i n s e r t t h i s i n t o (9) i .e . , an i d e n t i t y and hence eq. (8) i s s a t i s f i e d . Now how might one ensure t h a t &4\*>= - b s i s an allowed v a r i a t i o n ? The one 9 6 genera l way we know of i s simply t o use t r i a l functions of the form w f-%d where i s 3 v a r i a t i o n a l parameter and i s a n y t h i n g which does 4 n o t depend e x p l i c i t l y on A . ( B ) - The or ems For s i m p l i c i ~ y l e t .;i sssu~ne that) i s independent of Then expanding rhe h y p e u v i r i a l theorem t h e z e r o o r d e r e q u a t i o n i s of c o u r s e t z B v i a l l y s a t i s f f e d whence w e a r e l ed t o c o n s i d e r t h e f i r s t o r d e r e q u a t i o n One now r e a d i l y f i n d s t h a t i f ia w i t h $"1 a r e a l pa ramete r , i s a n a l lawed v a r i a t i o n , then ( 7 ) wil.1 be s a t i s i f e d . (w: Prove t h i s , ) A g e n e r a l way of e n s u r i n g t h a t (8) i s allowed i s t o i n t r o d u c e i t v i a a l i n e a r v a r i a t i o n a l parameter , i. e . , t o use Note t h a t Phis i s j u s t the f i x s t o rder v e r s i o n of (XI-14). Problem: Fw What theorem i s s a t i s f i e d i f no r e a l i t y c o ~ t d i t i o n s a r e inlposed on G 2' : B C ( U ~ ~ k r A J v ~ L p l , ~ ~ 4 = 4"'t Q A "! 1 k q Here cur q u e s t i o n i s when a r e t h e f u n c t i o n s % .b-d\k lw\ '$9 \w> and 3 +--q o r r h o g o r ~ a l through f i r s t o r d e r , g iven t h a t q" and 125 b7 yU a r e two eigenfunctions of belonging t o d i f f e r e n t eigen- values, and hence orthogonal. Thus the quest ion i s rl (XX- 12) Let us now assume t h a t (XVII-1) and i t s analog for apply. That i s , we have Now l e t us suppose t h a t and (XX- 13) (XX- 14) (XX- 15 ) 1.3 a r e allowed va r i a t ions of and Y respect ive ly . Then i t follows from (13) and the complex conjugate of (14) t h a t and d 9. Subtract ing and using etfw- € " ' )Y~=D , and ~ ,@ )so we then f ind (9). Note a l s o t h a t from (16) i t follows t h a t and s imi l a r ly A i . e . , r e f e r r i n g t o t h e d i s c u s s i o n i.n Sec , X I X , w e s e e t h a t is o r t h o g o n a l t o the e x a c t thruugti f i r s t o rder and l i k e w i s e % i s J or thogonal t o through first order . Thus, i t E~~~ i s the ground k ~ ~ l s t a t e and t h e f i r s t e x c i t e d s t a c e i t w i l l a l s u f a l l o w t h a t & F u r t h e r , a s was t h e c z s e i n Sec. XIX, t h e g e n e r a l & way t o ensuupe tha:t the var ia t io ras (15) a r e a.llov,ied i s i n t r o d u c e 9 ' i n t o ' L $!"l u s e of a l i n e a r pa ramete r . "i.7 The c o n d i t i o n s on t h e which w i l l ensure t h e gauge i n v a r i a n c e of t h e energy through second o r d e r can be i n f e r r e d d i r e c t l y from t h e d i s c u s s i o n i n Sec. X I - ( E ) . However i t ma.y be worthwhi le t o f u r n i s h some f u r t h e r d e t a i l s i n c e i n c a s e s o f i n t e r e s t \dl i s u s u a l l y nor of t h e simp Le form ~ t ~ k ~ )+'I wkiich we have been assuming. Thus c o n s i d e r a sys tem i n a magnetic f i e l d , Then t y p i c a l l y h a s t h e farm L -where vH*' i s l i n e a r i n t h e f i e l d andU \%'9 q u a d r z t i c . Thus iron1 t h e d i s c u s s i o n i n Sec. X I - ( E ) f t f o l l o w s , s i n c e a f t e r a gauge t r a n s - * Z.A &!? fo rmat ion LI z e 3 1 - \S\ol-+ \1*" +- a- M"' d- t ha t ( k i.e ,ap o+ 3 1 0 4 ,J\+"' c- and tha t the .higher order terms which w ~ u l d r e s u l t from the sxpanerion of eL stmply vanish. tD.b Prom (7) l t than i%":lon. that E r for any $? ~ t n c e thrqugh f , i re t ordero Then, i n accord v&th the f a c t that 9f M** I \ &fO !nd V?&s* &en 9 , one osn r e a d i l y n h a ( h o b l a : :Do t h i s ) that i f we define then 9 1 where 'j\$ de the Hylleraas Func t iqnal for kt''*@ H"' u s h g p a s :the t r i a l function. Now l e t ,us slupposle that the a e t of t r i d l 4 Punetiqns yhich we propose t o use f m both ' qnd H 18 c 1osed undar (19) for ce r t a in r , i.e., tha t i f % is i n the a e t ,then *lQ' so %is , (This i s cIe4rl;y jus t the f i r a t order version af our reguirernent i n Sec. L X X - ( ~ ) , ) Then from (20), s h o e given the gaugeol cpq, Htd ~ ~ v ) and Q'4 \v't+*'b.2vy 1 are ff&eJ,numbers independent of the (' 19 8 ~ i . 1 funetione, i t follows tha t if Y maker ~j625a.o then+"' L . *. w i l l maka sQ$'*o , whence, u s ing (20) again, we w i l l have But the quantities on seeh aide are the rota1 eseaad ardar anerglee (paramagwetie p%,us dZ,amagaetia) i n each gauge, 'IYbberwfore undsr t h e a a M % $ ' t he energy v ~ i ll be i .nvariact t:hcauy;h esaond ordaz, t o tahc 1"rsasfkwlrmatLo~ generated by d k p & y 9 , h d @ I ~ q C , A q* a f i ~ - ~ ~ deal with the folEwl,ng .~%tuat%csn, We are g&ven a (ne~rmal '9,~~d) fu t~eg i~on +'@@) which i n @om aense epproxilmetea the emct (normaliaed) eigen- k@? function $" of. en ieolared system, whose Hamtltortlan i s , and we would 1.ik.e t o i l n p ~ d v s on the approximation t o C%") \l'3%7gy of ferad @"' or we would Ilks t o impeova on a f f p o i . a r i ~ i k t Z L e y t ~ , br CY 1 or what a l l , where H\'' $8 some opera tor , men double per tugbation theory (or the interchange theorem) offerb the Bo%%owkag presaript%on 1 for a a l c u l a t i n g f i r e t order ao r rec t i ane : F%.nd a Ham%lttanian euch t h a t C~bfl_ c \ ~ ~ q ) %w"I "0 where p) i s a r e a l number (and such t h a t r") is non-degenerate). I@ 1) Then determine e func f3. en @-k from, where One t h e n s h o ~ i ~ s ( i n t e r c h a n g e theorem) t h a t where and AD^ 1 y i e l d s a f i r s t o rde r ( i n the parameter ) c o r r e c t i o n t o k , T h a t i s one shows t h a t where which c l e a r l y & t h e f i r s t o r d e r c o r r e e t i s n , it?\) The advantage of t h e fo rmula t ion (4) over ( 7 ) i s t h a t 9 may LW) be e a s i e r t o d e a l w i t h t h a n a&: , i , e , , h may be a s imple r per - t u r b a t i o n than V. En e i t h e r e a s e one can use v a r i a t i o n a l t e c h n i q u e s t o approximate t h e i r s o l u t i o n , and w e w i l l comment on t h i s l a t e r , The p o i n t we want t o make now i s t h a t t h i s approach, e i t h e r i n t h e form ( 4 ) or ( 7 ) hiages on choosing a n M ~ ~ ' ~ NOW i n soole s i r u a - t i o n s t h e r e rnay he a n a t u r a l choice, hor~~ever u s u a l l y t h e r e w i l l n o t be, and s i n c e t h e a n s x e r s c e r t a i n l y depend on t h e c h o i c e of bb) i~ i s no t r e a l l y clear V t n t t o do , We now n o t e t h a t t h e v % r i a t i o n a l methad s u g g e s t s a n o t h e r approach which seems t o avoid t h e problem of choosing R ~ ' . This approach h a s been most developed by Delves, (See P r a c , Pkys. Soc. 92, 55 (1967') f o r a r e c e n t d i s ~ u s s i s n and refereirces t o h i s e a r l i e r p a p e r s , S i m i l a r methods have been sugges ted lr9y Deal, e t , ax, 5 . Chem, Phys , &4, 3395 (1968). ) Consider t h e Hami l t o n i a n Then i t s e i g e n v a l u e % is t h e quantity of i n t e r e s t t o us s i n c e 1%) E \ ' ~ ~ c ~ ~ ~ ~ ~ ~ ~ + ~ ' ) E , y\e \k t h e p o l a r i z i b i l i t y e t c . Now suppose t h a t from someplace one produces a f u n c t i o n depending on b? d which i s such t h a t f o r , 43 becomes $wl i. e . , -- +!v~) Now whatever $ i s we know t h a t e d e f i n e d by (XXZ - 9 ) w i l l d i f f e r from by terms of the- o r d e r of t h e square of t h e e r r o r i n * I n g e n e r a l rill be i n e r r o r , i . e . d i f f e r from $ t h e h a h W B 0$_Wi9- 3 e x a c t e i g e n f u n c t i o n of b y terms of ' where 3 i s some measure pr %=- a f Thus & ~ w i l l d i f f e r from by terms of o r d e r V ,' S V P W ~ xT and >J But no2 tha . t i s such t h a t i f 1 were e q u a l t o z e r o then $ would be e x a c t . Under t h e s e c i rcumstances wi11 be i n e r r o r only by t e r m s of o r d e r 5 whence E--$. and heace t h e $"L &\--$ 9 /J -/ w i l l be on ly of o r d e r $ A c a s e i n p o i n t i s URHF f o r an atom. 131 Then ~ f t h e r e a r e no d e g e n e r a c i e s f a r z=+& we can i d e n t i f y 3 a s B P r 2 Fur thes it is e Pear that under t h e s e c i z curns Lances URHF i s e x a c t when if i s any orae e l e c t r o n o p e r a t o r . Thus we have r e d e r i v e d , i n a v e r y d i r e c t way, ehe r e s u l t we found i n Sec. X I - @ ) by use of ~ r i l l o u i n ' s theorem, Now i n double p e r t u r b a t f u n t h e o r y $ i s i d e n t i f l e d w i t h t h e parameter in t roduced i n (5) and one e n s u r e s e x a c t n e s s i n t h e )ay e0 3 l i t n i t s imply by choos ing $h\) where +,@I' s a t i s f i e s t h e exac t X-AB equa t ion , narrle l y --3 CD I \\VL i n p a r t i c u l a r w r i t i n g '.l- "= z g n ~ o ) r ) and ELD,x - 2 $M PI^') P t h e f i r s t o r d e r e q u a t i o n i s j u s t ( 2 ) aga in . F u r t h e r , whatever i s w e have from ( 9 ) , w r i t i n g qz 5yn4'%' and e - 2 dW&'"3 I t h a t e t e . and one r e a d i l y f i n d s ( : F i l l i n t h e d e t a i l s ) t h a t w i t h dip2 +(OKI & ~ " j - F!vb 9 - CI~) -3 k as g iven i n ( 4 ) , However now suppose t h a t i n s t e a d of ( 2 ) we c o n s i d e r t h e e q u a t i o n P r e s u w b l y we can't solve it e x ~ e t l y , however l e t us i g n o r e t h i s l i t t l e d i f f i c u l t y f o r t h e moment, Then she ; ~ ~ ~ l n r is t h a t e l e a r l y e q u a l @' i-$ $--0 , i n d ireoic. i f we wo:ild i n s e r r i n on ly by terms of order xd NOW the v i r t u e of (18) i s of cour ee t h a t ir, doesn't- involve d m hwi b u t a l s o , a s uie have nored, one c a n ' t so lve i t . However, e x c e p t i,n s p e c i a l cases he'/% expinsi~on or the s h i e l d i n g approxiins- t i o n f o r example) t h i s w i l l u s u e l l y also be t h e case w i t h ( 2 ) so t h i s i s n o t a s p e c i a l , drawback. of the proposed method, I n eFthe.1; case olle can use va r i a t i s r r a l t e c h n i q u e s , Fo r ( 2 ) t h i s T A ~ U P ~ i m p l y be t h e bsi'% H y l l e r a a s approach w i t h H ~ ' p l a y i n g the r o l e which b pl.ayed i n our e a r I f e r d i s c u s s i o n s and a l l t h o s e d i seussPans wi l E e p p l y , For 7 lv.7 (13) however t h e s i t u a t i o n i s no t q u i t e t h e same, F i r s t of a l l k n o t e q u a l t o This has the f o l l o w i n g consequence: c \+~LE@~>$~~]~ is of @$-&') .is sf o r d e r y4 w h i l e C\k. $ T h i s then rnesns t h ~ t i f i t 3 a e s n n t v a n i s h for r e a s o n s of synlmetry, w i l l he of order & buch a behav ior would seem u n d e s i r e a b l e , Hence we r e p l a c e (10) by - where 3 i s a nlrmber t o be chosen s o t hn r c$@), 4b g whence t h e r e r i l l b e n u p r o b l e m s a s t+B Since 3 a i i l d i f f e l from ~~~1 o n l y by term; of order 5 , (14) w i l l s t i l l be e x a c t i n the F ~ B l i m i t , +WOl A f u r t h e r difficulty is t h e fo%lowing, Suppose t h a t i s such a poor a p p r ~ x i m ~ t i o n to t h a t e"' l i e s i n the i ontinuurn 133 of !+&' (such a s i t u a t i o n i s not a t a l l unusual. See Dougherty, e t . a l . J Chem, Phys. 45, 1803 (1966)). Then the homogenous equation has a s o l u t i o n and the inhomogenity w i l l u sua l ly not be orthogonal t o it, s o t h a t one has an incons i s t en t equation. This d i f f i c u l t y can a l s o be avoided, only r a t h e r a r t i f i c i a l l y . I n p r a c t i c e t h i s d i f f i c u l t y usually doesn' t appear e x p l i c i t l y because one usual ly couples a poor *tm' with an equal ly poor "solution" t o (12). A s we have sa id , the n a t u r a l way t o a t t a c k (12) i s with a va r i a - t i o n a l approach. Evidently the appropr ia te funct ional i s (XXI - 15 ) f' 6 (91 where 3 i s t o be chosen s o t h a t c*"'~, @ 1 =a a One could of course a l s o simply r e s t r i c t the s e t of t r i a l functions t o functions or thogona 1 t o and replace (15) by (XXI - 16 ) To change the sub jec t somewhat for a moment, it i s now of i n t e r e s t t o note t h a t , a s we remarked a t the end of Sec. XVI , t he functional , fi -ton (15), though o f t en with 3 replaced by k , is frequently used t o A approximate t h e second order energy, i . e . 9 i s used a s an approxima- t i o n t o E"'. The following a r e some p roper t i e s of t h i s approximation. I n the main we s t a t e them without proof (see J. Chem. Phys. 42, 3630 (1965)). &I%? \ b > \ f-%, (1) With 2 9; t h e value of i s (XXT- 17) One can thea aho-h7 t h d t %-z3*' i s of f i ~ order i n 3 and can have eYr 0 e i t h e r s i g n , (If one uses tnstead s f 3 t h e n B-E'~' c a n even be of ze-pa o r d e r i n 1 8 ) w7 ( 2 ) A s no ted above, If 9 is s o poor t h a t ( 1 2 9 actually i s an i n c o n s i s t e n t e q u a t i o n , then u s u a l l y one is not aware of t h i s , and rcay h even f i n d a r e a s o n a b l e r e s u l t f u r 3 because one w i l l p robab ly u s e a. Limited and " reasonab le" s e t o f ( 3 ) Wri t i ng then c l .ear l y avl If now symmetry c o n s i d e r a t i o n s a p p l y , and i f is less t h a n t h e energy sf f i r s t ~ t a t e ~ a p p r s p r i a t e sy rmet ry t h e n evFdentEy 0-P- m $3) I f one does not: have synmetry, then one car1 s t i l l show &Elat i f % P= i s t h e ground s t a t e , t h e n 0 w i l l be an upper bound t o ?& i f % i s s m a l l enough, R e t u r n i n g now t o doub re p e r t ~ r b a t i o n r h e s r y, t h e approximat ion of t h e s o l u t i o n of eq. ( 2 ) [ e S , (8) v a r i a t i o n a l l y , followed by use of %a> cw a A w] eq. ( 4 ) w i t h pi) rep laced by 3 r e q . ( 7 ) wi rh % rep laced by Ct. a A - t o approximate EbdJ ( f o r an a p p l i c z t i o n s e e Chen and ba lgarno , Prcc. 135 i t 2 ~3 Phys. Sce. p3 855, 399 (1965). For &--" t h e y uSe t h e B ~ ~ & ~ X & ! M * C WarniEtonian) i s of course j a s t t he same s o r t of procedure which we d i s c u s s e d i n Sec . XVniiZ-(E) p a r t (ii) birt ( i i i ) ] i n a d i f f e r e n t 60tww - z c t i o n , Here, as t h s r e , t h e n , t h e r e i s a n o t h e r a l t e r n a t i v e which d e a l s w i t h Et19 d i r e c t l y , Namely use of t h e f u n c t i o n a l (YXJP-LZ), which in our prestl-zt n o t a t i o n besornes and we can s i m p l y a p p l y t h e dissuss icsn which we gave t h e r e t o t h e p r e s e n t s i t u a t i o n , pn A s we have no ted , t h e use of 13 t o approximate E'%' L) c a f a t o approximate p o b a r i z i b i 1 i t i e s , i n v o l v e s a f i r s t o r d e r e r r o r i n 3 One can obv ious ly however g e t a n approx imat ion which i n v o l v e s o n i y a COZI second o r d e r e r r o r e i t h e r through t h e u s e of % , i . i , e . , v i a double &'?) p e r t u r b a t i o n theory , o.r b y t h e use of a d e f i n e d by an e q u a t i o n l i k e and t h e i r a s s o c i a t e d v a r i a t i s n a i p r i n c i p l e s , e t c , We leave t h e d e t a i l s t o t h e i n t e r e s t e d r e a d e r , Having come s o f a r we must now p o i n t o u t t h a t i n f a c t t h e Delves method i s j u s t a s p e c i a l e a s e of double p e r t u r b a r f o n theory , i , e , i n v o l v e s a s c i a 1 ( i m p l i c i t ) c h o i c e of k k D Lei @ be t h e p ro - F= j e c t i o n o p e r a t o r o n t o i e , e , i n D i r a c n o t a t i o n Then e v i d e n t l y qQJDD' i s a n e ig t ; i fu l l i t ion of R"' a n d r h e e i g e ~ i v a l u e i s j u s t b e P v t i ) C o t i s i d e ~ no>? the a s s o c i a t e d \ [\@I Now suppose u:e r r s c r t r t t h e ?P" r o be f-h~gona 3 t o + 1 the v a l u e of c%@'j %'\mn) is undeterrninad i i i j i i r e le \ i i r t i . Then .$ "!"? we may d r o p t h e pj ternis i n (19 ) &rid fus-tilei s i n c e (J = I ; and ct- @--I ?lq7= Tj93 ve may r e p l a c e bCG i n ( 3 9 ) by . But t h e n PJ we s e e t h a t we have 0 t n t h e form (16) Fn a s l i g h t l y d i f f e r e n t n o t a - - bP i n s t e a d of Thus one may say t h a t t h e Delves t i o n , i o e o ' - k p rocedure , w i t h t h e orthogona E i t y requ i rement , F s e q u i v a l e n t c o double p e r t u r b a t i o n t h e o r y w i t h b e i n g t h e o p e r a t o r ( 6 8 ) , A n a t u r a l q u e s t i o n a t t h i s p o i n t i s t h q i s t h i s \+'DD' i n soole s e n s e a 1 b e s t c h o i c e ? We w f l l not pursue t h i s q u e s t i o n h e r e \ The c h o i c e (18) h a s been made by &4.ansf f and Percrzs Phys, Rev, 26&, 1255 (1968) who a l s o n o t e t h a t ( 1 4 ) can be re- p laced by (16) . (18) h a s a l s o been c o n s i d e r e d by A d a m , .J, Chew, Phys, 45 %4%2 (1966>, i n ano the r e o n n e c t i o a , E v i - d e n t l y t h i s qiw9 i s not 4 s imple oper: . tur , It1 p a r t i c u l a r F t i s n o t obvious whi t i r s o t h e r e a g e n f u n c t i c n s a r e , Now- e v e r , r e f e r r i n g t o our e a r l i e r d i s c u s s i o n , w e can now say f o r what i t i s w o r t h t h a t i f Q"' is t he Luwestl e i g e n v a l u e 4% of t h i s %@'7 t h e n we w i l l have D $1 The procedure which we have c a l l e d t h e Delves procedure i s on ly one of s e v e r a l t h a t he h a s proposed ( s e e t h e r e - f e r e n c e g iven e a r l i e r ) , However, i t seems t o be abou t t h e b e s t of t h e l o t , Some of h i s e a r l i e r e f f o r t s were e s s e n - t i a l l y based on t h e o b s e r v a t i o n t h a t ~5'~'- E'~') i s i n a somewhat d i f f e r e n t n o t a t i o n ( 9~*" i n s t e a d of gw' 4" i n s t e a d of qtrp ) > j u s t t h e yb7 t h a t one d e r i v e s from ( X I T I - P O ) . r% I n view of t h e r e s t r i c t i o n s on t h e 6,"> which we d i s c u s s e d t h e r e , i t i s n o t s u r p r i s i n g t h a t t r y i n g t o use 6$'""*a t o de te rmine a "bes t " ~$3~' d i d n o t prove v e r y s a t i s f a c t o r y .

The variational method 2 - Perturbation theory

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