Интерпретация метода хартри-фока как метода частичного суммирования атомных фейнмановских диаграмм

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  • Acta Physica Academiae Scientiarum Hungaricae, Tomus 27, pp. 351--371 (1969)

    HHTEPFIPETAI2I/I~t METO,~A XAPTPH-qbOKA KAK METO,~A qACTHqHOFO CYMMHPOBAHH~I ATOMHblX

    r BCKHX ,~HAFPAMM*

    B. B. TOYlMALIEB

    YHHBEPCHTET qHYIE, CAHTHAFO, qH.rlE**

    (FI0CTyn~z0 20. II. 1969)

    Ha MeTo~e XapTpH--tDoKa [I, 2, 3] 0CHOBb~BatOTC~I n0qTH Bce np0a0An~bte B HaCTo~Imee BpeMfl aTOMHble ti MO~eKyJ~~lpHb~e pacqeTu, H rI03TOMy eF0 HHTepnpeTaIDt~t KaK MeToJxa CyMMH- p0BaHH~I orlpe~eJIeHHOFO K.rlacca aTOMHblX qbe~HMaHOBCKHX ~HaFpaMM [4] ~IB.rIJ~eTC~I oqeHb BaX'

  • 352 B.B. TOMMAMEB

    FIpe>~~e Bcero B03bMeM HeNOT0p0e cqbepriqecNH CHMMeTpHqH0e ~0 I< no~m ~~pa- -Z / r BHem~ee note U(r), 0pTOHOpMHpOBaHHb~e pa~~a:ibHbte ~ym~- R~H KOTOpOro onpeRean~TC~ ri3 ypa~HeHHa

    ( 1 d ~ 1 d + l ( l+ l ) Z 4- U(r))R, ,(r)=%,R ro(r), (4) 2 dr 2 r dr 2r ~ r

    j':~r~drR~t(r)R~~(r)= 1 (5)

    (npeRnoJlaraeM, qT0 cnercrp ent He riMeeT cJlyqa~H0r0 Bblp0)K~eHH~t BOJ~0p0j~0rl0

    jX06H0r0 aToMa, TO eCTb era peaabuo 3aBnC~T 0T l, H cKa~eM nanpriMep e2v # e~) C00TBeTCTBeHH0 Mbl A0~)KHb~ BBeCTH HOBb~~ Hy~eB0~ raMnabTOHHaH ti raMnJm-

    TOHriaH ~8ariMo~e~CTBria

    1 ~, /lr~ - ~. __Z + ~, U(r~) (61 HU- - 2 " ( I~

  • HHTEPFIPETAL[H~ METO~A XAPTPH--d:~OI
  • 354 B .B . TO.rlMALIEB

    B xapTp~-~b0KOBCrr npHS~i4~eH~4a HaM He06x0A~tMO paccqnTaTb ~3~9 AE ~~naab~ aTO~HbtX a~arpa~ roabxo nepnoro nopa~xa, Tax KaK

    En~ ~ Eo + AEx. (15)

    :3TOT pacqeT 0qeHb np0CT [4], a ~ pe3ym,TaTe rabl noayqaer~ ~~ BX~a~on aT0~- H~x JmarpaM~ nepBoro nopn~txa 0KOHqaTe~bHble mapa~eHnn

    rta[t 2 .~ (2l~+l)(2l~+l)Ro(n~ll, nzl~; n2iz, n~la), nJ.El ., ntl~ f .

    -- 2 (2l~+l)(2lz-4-1).~[l~lzl]2Rl(n~l~'nzl~; nxll'n~l~) ' ndxE n,It~f , l

    Na[

    0 -- 2 ~ (21~+l) U(nxl~;nxlx). nxlt~f,

    TaK~M o6pa3oM A~I~ no~I~0fi xapTp~-~pOKOBC~:0ii aHeprn~ Mil ~~eeM c yqeTo~ (15), (13)

    EHF = .~ 2(2/1 + 1) I(nl l~) -4- nl/i~fo

    + 2 ~ (2l~+l)(21~+l)Ro(n~l.,,~12; ,,~.I~,,,14) -- nxhEy ., nt l tEf o

    - 2 (24+1)(24+1)2144q~R~(n~l,~~/~;~xl,~.t~). (16) nd,Ef o, n, ltEf , l

    EcaH BBeCTn 06~,tqHbIe o [2] ~za noaHoro q~c~a C0CT0am~ 3Jlek'TpOrtH0i o6oa0q~e g(nl) = 2( /+ 1) H R.rla npn~oro H o6~eHHoro paa~aab- max nHTerpaaos

    F~(nl/1, n2/2) ~ Rl(nllo n2/2; n212q n~l~), G,(nl la, n2 12) ~ R~(n111, ni 12 ; nx 11, n212),

    TO Bblpa~eHne (16) MO>~nO IlpeRCTaBI4Tb B IIpHBbIqHOM BI4Re

    1 Er4F = .~ q(rh l~) I(nr la) -4- .~ o ~ q(nx l~) (q(nx la) -- 1) Fo(n t lx, nt la) +

    nd~ 6.f, ntlt E " (17) + .~" q(nll~) q(nzl2) Fo(n~ Ix, n2/2) -- 2 2 -'41'I F~(nx l~, na I~) --

    nll~ ( f , , nsltE ", nxl, Efo l~eO (ndt > n,4)

    -- ~.~ 2 Bt1','G'(nl ll'nzl2) ' nlll E fo, n ,h E ro d

    (n~4 >1 n,lt)

    Acta Physica ,4eademiae Scientiarum Hungaricae 27, 1969

  • HHTEPr lPETAUH~[ METO,]~A XAPTPH- -~OKA 355

    npttqe~ A~,~ = (21~+1)2[1~l~I] ~ npn l=/=0, (18)

    B,,t,t = 2(2/1+1) (2/~.+ 1) [l x 121] ~ , (19)

    r~Ie CrI~BO~ [/1 Iz l] o3HaqaeT B3~TbIi~ rio MoRy~IO C00TSeTCTSymmnti 3j - - Ko3r q~mt~enT c HyJ]eBb[Mri npoe~urmMn ~oMem'o~ [4].

    r iy~bl (18), (19) nO3S0~~IOT paccqHTbIBaTb ne0fixo;m~bIe 3HaqeHri~ K09~I}HI~HeHTOB A, B ~, B qaCTH0CTH, n0JIyqHTb BamHble Ta6a~ilLU

    ./l lxl B h l:d

    /=2

    $ " 0

    6 P 5

    $$

    sp

    PP

    /=0

    2

    0

    /=1

    0

    2

    /=2

    0

    0

    12 5

    9 Bb[pa~eHrle (17) ~~~ EHF BMecTe C C00TBeTCTByIOIIIHMH Ta6JmtlaMrl 3HaqeHHi~ K03~){~HIIHeHTOB A, B coBnallamT c nonyqaeMblMH 06blqHHM CII0C060M B Te0pHH xapTpH-~b0KoBcKor0 npH6~m~eHHa [2]; HaCT0~mH~ Cn0C0 no~yqeHnfl aToro IIpH6JIHXeH~~I II03BOJI~teT He rlpOBO2itiTb BbIqI4cJleHI4~ B rca:~,aoM KOHKpeTHOM cayqae, H cpa3y ~IaeT o6rIme qbopMynbI (18), (19) aJm K03~pqbmmeHTOB Al~t npH 1~0 H Blg.1.

    BB0~fl MHO>KHTe.rIIJ J-IarpaHn

  • 356 B. B, TOd-IMALIEB

    rlpiiqeM, pa3yMeeTcfl,

    .i+o ,o r z dr R~,(r) R~,(r) = 1. (23)

    HOMH0:~a~ npaByto H aeBylO qacTb (22) Ha Rnt(r) ii iiHTerpHpya rio r OT 0 RO + ~, C yqeTOM (23) riozyqHM ycaoniie

    2 U(nl; ni) -- 4 ~ (2lx+l)Ro(ntl~,nl; nl, n~li) + nttt ~ ", (24)

    +2 ~ (2 l~+1) .~ [l~ll']~Rr(n~lx, nl;n~l~,nl) = O. ntl~ ~ f~ l"

    9TO Ba~KU~ ttHTepHpeTrlpOBaTb Ha ~HarpaMMax: Ha ~HarpaMMHOM ~3~Ke OH0 o3HaqaeT, qTO

    0= -~, , , , + ,*-~~-.. ~ + -P-r

    (HpHqeM 06e BHemHiie .IIHHHH HHKaI< He Hy>I

  • HHTEPr IPETAUH~I METO,~A XAPTPH---~OI
  • 3581 B.B. TO~MAqEB

    l'IOJIyqHM, B qaCTHOCTH, npn n

    I 0 ! 9 ~'(~) -- 1 / - - 2 ~, (2/~ + 1) l~(n~l i, n21z, n2l, n,l,) A- "~ni,n.. En~t - - En,t n,l,Ef,

    .~~" (2/1 A- 1) .~' [t~t2t] ~ o ' +

    H np~ n~~n z

    (32)

    C() = 0 (33) n2,n 2 1-I0~CTaB~~ pa3ao~eu~a (30), (31), (32), (33) B Bb~pa~eH~e (16). OT~eTaM

    npeABapnTeJIbHO, qT0

    ~(nt ) = " ; E. . , C'.,.n C'.,,n =

    - - En, + 2E.I r~(~) -4- .~ En'~ C#~)n c'(O" --

    Ho TaI< KaK B CHJ'Iy yCa0Br~~ 0pToH0pMI4p0BKI4

    C/(l) ~l(1) ~ C/n(1) 1 C/n(l~m 0 nl,n I "~- ~nl,n i ~ /I"

    HMeeM

    =- 2 ' !,,,:'(') ~~"~.,nOg-'/(2) Cln (! ~n, .n n"

    TO npH6~H>KeHHO

    I (n l ) En I -~- .~ (En , l Ti:. A ClO) CJn(!]n -~ = - - ~I~l] ~n ' , r l * " " IT:

    (3a)

    Bo BTOp0M nopfl~Ke H0~yqHM

    nd, El, n

    nd i Ef , , n, l t ~fo n

    4 ~ (211 A- 1) (2/2 ~ 1) .~ [/1 ! !19 "~ r R~(n111, n nl Ii, n~/2) nalzEf , ,na l . Efo I n ~. (35)

    I-IOACTaB~a~ (32), (33) B (38), M~ n0~yq~M ~0BOJIbH0 rpo~o3~Koe sbIpa- >KeHae, IKHO npeACTaBnTb B B~Ae CyMMbI BI

  • IdHTEPFIPETAIIH~ METO~A XAPTPH--~OI
  • 360 B.B. TOYIMAMEB

    HblX ~II4HI4~ MeHbIIH4M HJHI paBHb~M y~B0eHHOMy qHc~y 9JIeK'Tp0HOB B 0TKpbITOH 060~10qKe 2q.

    l-lpa~~na C0CTaB~IeltI491 BK~Ia~0B 0T Tal

  • PIHTEPFIPETAI~HH METO~A XAPTPI4--~Ob(A 361

    pa/p~

    ~S

    2p

    ~D

    g(p3 SL, pZ S' L')

    'S 3p ~D

    -~~ a o ~ 3 5

    -g- -2- -6- - - - - - -3 - - 3

    o ! 2 T

    nop~AKa, t~u no~yqtiM A~~ nopan~a B cnyqae ~/-----p.

    o, r~la i~

    naL~ n~la nal a

    (n~~o)~ S' L'

    flete ~a[a

    g(p4 SL, p2 S ' L ' )

    p4/p~ l S

    ~S

    ~p -T

    ~D i ~

    3

    4

    ID

    5 3 5 3 8 3

    (gp~ SL ; pe S' L')

    p~/p~ 1S ap 1D

    3

    OCTaBngg BKJIaA~I aTONHNX dpe~HMaHOBCKHX ~Harpa~u~, TOJIbK0 nepBoro BeJltiql414b! AE'HFAJlfl BKJIaAOB At ia rpa~ nepBoro

    2 ,~ (2/~ + 1) Ro(n , l~, n a lo; no lo, n, I~), nd, EIa

    - - ,~, (2/1-[-1)X'[lalxl]2R,(nlll, nala;nt l l , nola), nll, Ef, l

    (__ 1)L' (~8",1/2,1/2 2 Rl(na la, na la; na la, na la) X l

    X [l a lo/]2 (2la + 1) y.t., /

    - - u(na la; n,, la),

    r~La nata ! ii

    ti OKoHqaTe~bH0 A~~ no~Ho~ xapTpti-qb0KOBCK0fi 3HeprHti paccMaTptiBaeMoro C0CT0~tHti~l (hala) q SL

    EHe= ~ 2(2 / t+ l ) I(n~l~)4- n,l lEh

    +2 ~ (2/1+ 1) (2/2 + X) Ro(nll. n~Z~;~~12, nl l l ) - - nlli (fa, nll, ~f,

    - .2" (2J1 + 1) (% + 1) ~, [l~ & tl ~ R,(nl I. ,,~ l~; ,,~ 11, "~.l~) + n,l, E. *, n,l, Ef* l

    -4-f [2(2/a 4- 1)I(nalo) + !

    Acta Physica .4cademiae Scientiarum Hungarieae 27, 1969

  • 362 B.B. TO2]MAL1EB

    -f- 4 ~ ' (2/a A- 1) (2/1 -4- 1) Ro(nt 11, na la; na la, n111) -- nj, ~fo

    2 ~'' (2la + 1) (2ll + 1).~[lal l l]2Rl(nala. nlll;nala, n~l~)l+ n,, Efo l J

    -4-.~ .~'g(Iq SL. PaS'L ') ( - 1)L" t,.,.L'J X l S ' L '

    X (2/a -1- 1)2 [l a la/]2 Rl(n a la, na la; na la, na la), (44)

    r~e f=q/2(2la+l) , H 3Ta BennqnHaxapaKTepH3yeT CTeneHb 3ano~neHn~ 3nek'Tp0- HaMH paccMaTpnBaeMo~ OTl

  • IdHTEPFIPETAI.~PIYI METO.~A XAPTPPI---OOI-(A 363

    Bbipa~eHHe (44) M0)KH0 TaKme cpaBHnTb C npHBoabIMnM XapTpn [2], KOTOpbd~ HcrIOYIb3OBaSl npoRe~ypy ycpejl~eHHa caMocor~acoBaHuoro noaa no pa3aaqHbIM HanpaBneHnaM B npocTpaHCTBe. DTO BbIpa)KeHt~e HMeeT caeAyro~rff BH~

    1 EnF = .~ q(n 1 l~) I(n~ 11) .2~ nx/x~~f. -~- q(n111) (q(/'gl 11) -- 1) Fo(nt lx, ni/1) -4-

    ngl EA

    + .~ q(nxll)q(nzl,,)Fo(n~ll, n212) - ~.~ .a~Al~tFt(nlll, n~l~)-- ndt E h, n,l, Ef, n~la (f, 1~0 (nd, > nal,)

    - - .~ ~, B,~t2, G~(nl la, nz lz) + nd~ (f,,, n,l.E eo 1

    (ntlj. ~ nsl2)

    -b q(na la) I(na la) + q(na la) .~ q(n111) Fo(na la ;nxll) + nd~ (f,

    1 n + 2 q( a la) (q(na lo)-- 1) IZo(na la; no la) --

    q(na la) ~ ~" Bl,,h, Gl(n a la; nj 11) -- 2(2/a+1) n,t, ~I,

    - - .~ A,,,z F~(na 1,~; na la). (48) l

    CpaBHH~a~t (48) C (44), rloayqaeM flBHbIe sbIpa~eH~~t ~aa K03OdpHRI4eHTOB XapTpa

    Bt.kt ~ 2(2/a + 1) (2/1 -b 1) [lalll]2, (49) I \ L ' Jlalal ] A~.,--=-- .,~g(/q SL; q a S'L) (-- 11 t,~*.L'f

    S'L"

    (21 a + 1)~ [lz la l]Z. (50)

    Ho 9THM ~bop~y~aM MOCHO COCTaBHTb Ta 3HaqeHrI~ 9THX ~O3~3t~HRrleHTO/3, IIOJ1HOCTbIO conrta~arotune co 3HaqeHH~mI~ H3 Ta6aHI L nprIBO~~MblX B [2].

    f / l

    1S

    All

    0 2

    --1 2 5 1 --1 5 1 --1 25

    f / t

    All

    3p

    1D

    Atl

    0 2

    4S --3 3 5

    2p --3 0

    2D --3 6 25

    p5/l ~ 2

    ~p 3 -T-

    Al1

    p41~ o 2

    1s -6 o

    3p -6 3 5 9 1D --6 25

    4eta Physica Academiae Scientiarum Hungaricae 27, 1969

  • 364 B. B. TOSIMALIEB

    {lalaO } (__ 1)L' t~l~ l~,L" 14 qT0 OTMeTI4M, qT0 BB14Ry Toro, qT0 la la L' = 214 +---------i- '

    1 [l~ ~~ o] - - - 2la + 1

    , MbI HMeeM

    At. 0 -- q(q -- 1) (51) 2

    Ho~yqHM Tenepb xapTpH-qborKI~Te~IVi YlarpaH.a H Bapbtlpy~ Bb~pa.eHne

    . . . . . . r 2 dr R,~~~(r) R,,~,(r) - - FHF EHF X ' 2(2/1 + 1) eml ~ n~l~ E fe

    lo ~~ -- 2(21 a + 1)fen~ u r e dr Rn.t.(r ) Rn.,~(r) - -

    - - 4f ~~,-~'~fi,~ (2/~ + 1) (2/1 + 1) e.~t~ ' ~~t