Hyperspherical Harmonics and Generalized Sturmians 9780306469442, 0306469448

Table of contents :
front-matter.pdf
fulltext.pdf
fulltext(2).pdf
fulltext(3).pdf
fulltext(4).pdf
fulltext(5).pdf
fulltext(6).pdf
fulltext(7).pdf
fulltext(8).pdf
back-matter.pdf

Citation preview

+@ (TXDWLRQ  EHFRPHV KLJKO\ DFFXUDWH IRU ODUJH YDOXHV RI Z ZKLOH IRU Z  N WKH DSSUR[LPDWLRQ GHWHULRUDWHV DQG VKRXOG QRW EH XVHG

18

&+$37(50$1 38] (2.16) where



 DQG ZKHUH

C H A P T E R 2. MOMENTUM-SPACE W A V E FUNCTIONS LV D *HJHQEDXHU SRO\QRPLDO

 7KH UHDGHU PD\ YHULI\ XVLQJ WKH K\SHUVSKHULFDO KDUPRQLFV LQ 7DEOH  WKDW )RFN V H[SUHVVLRQ LQGHHG JLYHV WKH PRPHQWXPVSDFH K\GUR JHQOLNH RUELWDOV RI 7DEOH  ZKLFK ZH IRXQG E\ ODERULRXVO\ WDNLQJ WKH )RXULHU WUDQVIRUPV RI WKH GLUHFWVSDFH RUELWDOV  7 K H QXPEHU RI OLQHDUO\ LQGHSHQGHQW GLPHQVLRQDO K\SHUVSKHULFDO KDUPRQLFV FRUUHVSRQGLQJ WR D SDUWLFXODU YDOXH RI WKH SULQFLSOH TXDQWXP QXPEHU LV JLYHQ E\     DQG LI ZH OHW n   WKLV FRUUHVSRQGV H[DFWO\ WR WKH GHJHQ HUDF\ R I W K H K\GURJHQOLNH RUELWDOV 7KLV JLYHV XV DQ H[SODQDWLRQ RI WKH n  GHJHQHUDF\ RI WKH K\GURJHQOLNH RUELWDOV ZKLFK ZH ZRXOG H[SHFW WR EH RQO\ l   IROG GHJHQHUDWH RQ WKH EDVLV RI WKH VSKHULFDO V\PPHWU\ RI WKH SRWHQWLDO ,W LV VWULNLQJ W R VHH WKDW DSDUW IURP WKH XQLYHUVDO IDFWRU M  p DQG D QRUPDOL]DWLRQ IDFWRU WKH K\GURJHQOLNH ZDYH IXQFWLRQV DUH U UHSUHVHQWHG LQ PRPHQWXP VSDFH E\ H[WUHPHO\ VLPSOH IXQFWLRQV RI WKH RUV VKRZQ LQ HTXDWLRQ   7KH s RUELWDO FRUUHVSRQGV WR  WKH p p,  p 3 DQG s RUELWDOV FRUHVSRQG W R u u  , u DQG u   ZKLOH WKH p p DQG p RUELWDOV FRUUHVSRQG UHVSHFWLYHO\ WR u  u   u u  DQG uu :H ZLOO VHH LQ ODWHU FKDSWHUV W K D W PRPHQWXUQVSDFH K\GURJHQ OLNH RUELWDOV FDQ EH XVHG DV FRQYHQLHQWEDVLVIXQFWLRQV IRU VROYLQJ PDQ\ SUREOHPV LQ TXDQWXP FKHPLVWU\ 7KXV K\SHUVSKHULFDO KDUPRQLFV SOD\ DQ LPSRUWDQW UROH LQ PRUQHQWXUQVSDFH TXDQWXP FKHPLVWU\ DQG IRU WKLV UHDVRQ ZH VKDOO H[SORUH VRPH RI WKHLU SURSHUWLHV LQ &KDSWHU  ,QWHU HVWLQJO\ LW WXUQV RXW WKDW WKHUH LV D dGLPHQVLRQDO JHQHUDOL]DWLRQ IRU HDFK RI WKH WKHRUHPV ZKLFK FDQ EH GHULYHG IRU GLPHQVLRQDO VSKHULFDO KDUPRQLFV

27 7DEOH

28

&+$37(5020(178063$&(:$9()81&7,216

Table 2.2 Hydrogenlike atomic orbitals and their Fourier transforms W N—[, W N —U

 7DEOH  GLPHQVLRQDO K\SHUVSKHULFDO KDUPRQLFV

30

CHAPTER 2. MOMENTUM-SPACE WAVE FUNCTIONS

31 Table 2.4 Alternative 4-dimensional hyperspherical harmonics

32

CHAPTER 2. MOMENTUM-SPACE WAVE FUNCTIONS

([HUFLVHV 1. Calculate the integral in equation (2.6) and show that it yields the result shown (2.7). 2. Starting with J calculate the integrals J and J by differentiating with respect t o kµ , as shown in equations (2.8) and (2.9). 3. Starting with J, use the recursion relation of equation (2.11) t o generate J, J and J. 4. Use the integrals JVO, in Table 2.1 to evaluate the Fourier transform of' the direct-space hydrogenlike orbitals and Show that the transforms correspond to the solutions of Fock, equation (2.15).

&KDSWHU  + Q  @´ DQG Q  ... ZKHQ Q LV HYHQ RU Q Q Q±   Q ±   ZKHQ Q LV RGG DQG



& + $ 3 7 ( 5  +@ ,I VXFK DQ H[SDQVLRQ LV SRVVLEOH WKHQ ZH FDQ ZULWH  ZKHUH WKH IXQFWLRQV IQ [ DUH KRPRJHQHRXV SRO\QRPLDOV 7KHQ IURP  ZH KDYH 



& + $ 3 7 ( 5  +@



7R LOOXVWUDWH HTXDWLRQ   ZH FDQ FRQVLGHU WKH FDVH ZKHUH )  [ LV D GGLPHQVLRQDO SODQH ZDYH  7KHQ  VR WKDW  \LHOGV

 :KHQ G

 WKLV UHGXFHV WR 

ZKHUH MRNU LV D VSKHULFDO %HVVHO IXQFWLRQ RI RUGHU ]HUR 1RWLFH WKDW LQ WKLV H[DPSOH ZH GLG QRW DFWXDOO\ H[SDQG WKH IXQFWLRQ )  [ LQ WHUPV RI SRO\QRPLDOV LQ WKH FRRUGLQDWHV [ O [  [ G  DOWKRXJK LW ZDV QHFHVVDU\ WR DVVXPH WKDW VXFK DQ H[SDQVLRQ FRXOG EH PDGH



Hyperspherical harmonics 6XSSRVH WKDW ZH KDYH VHYHUDO OLQHDUO\ LQGHSHQGHQW KDUPRQLF SRO\QR PLDOV RI RUGHU :H ZLOO WKHQ QHHG DGGLWLRQDO ODEHOV EHVLGHV WR GLVWLQJXLVK EHWZHHQ WKHP DQG ZH FDQ FDOO WKLV VHW RI LQGLFHV ^µ` $OO RI WKH KDUPRQLF SRO\QRPLDOV LQ VXFK D VHW ZLOO EH HLJHQIXQFWLRQV RI WKH JUDQG DQJXODU PRPHQWXP RSHUDWRU $² FRUUHVSRQGLQJ WR WKH VDPH HLJHQYDOXH  :H FDQ FKRRVH WKH OLQHDUO\ LQGHSHQGHQW KDUPRQLF SRO\QRPLDOV LQ VXFK D ZD\ WKDW WKH\ ZLOO VDWLVI\ RUWKRQRUPDOLW\ UHODWLRQV RI WKH IRUP  7KHUH ZLOO WKHQ EH D VHW RI K\SHUVSKHULFDO KDUPRQLFV UHODWHG WR WKH KDUPRQLF SRO\QRPLDOV E\ HTXDWLRQ    DQG WKH K\SHUVSKHULFDO KDUPRQLFV ZLOO REH\ WKH RUWKRQRUPDOLW\ UHOD WLRQV  ,Q WKH FDVH RI WKH GLPHQVLRQDO K\SHUVSKHULFDO KDUPRQLFV VKRZQ LQ 7DEOH  WKH VHW RI LQGLFHV ^µ` FRUUHVSRQGV WR ^l, m` 7KHVH LQGLFHV DUF ODEHOV IRU WKH LUUHGXFLEOH UHSUHVHQWDWLRQV RI D FKDLQ RI VXEJURXSV RI WKH GLPHQVLRQDO URWDWLRQ JURXS SO    7KH VWDQGDUG FKDLQ RI VXEJURXSV XVHG LQ FRQVWUXFWLQJ RUWKRQRUPDO VHWV RI K\SHUVSKHULFDO KDUPRQLFV LQ SXUH PDWKHPDWLFV LV >@  EXW WKLV FKRLFH LV QRW QHFHVVDULO\ WKH PRVW FRQYHQLHQW RQH LQ SK\VLFDO DSSOLFDWLRQV ,Q D ODWHU FKDSWHU ZH ZLOO GLVFXVV WKH FKDLQV RI VXEJURXSV RI S O ( d ) ZKLFK DUH DSSURSULDWH IRU FRQVWUXFWLQJ RUWKRQRUPDO VHWV RI K\SHUVSKHULFDO KDUPRQLFV DGDSWHG WR WKH V\PPHWU\ RI YDULRXV TXDQWXP PHFKDQLFDO SUREOHPV

44

& + $ 3 7 ( 5  +@ KDYH VKRZQ WKDW WKH VHFRQGLWHUDWHG PDQ\FHQWHU VHFXODU HTXDWLRQ LV FDSDEOH RI JLYLQJ PRUH DFFXUDWH UHVXOWV WKDQ FDQ EH REWDLQHG ZLWK WKH ILUVWRUGHU VHFXODU HTXDWLRQ 7KLV KLJKHU DFFXUDF\ UHVXOWHG IURP WKH IDFW WKDW LQ HYDOXDWLQJ WKH PDWUL[ .ð .RJD DQG 0DW VXKDVKL PDGH XVH RI VXP UXOHV ZKLFK LPSOLFLWO\ LQYROYHG EDVLV IXQFWLRQV ZKLFK ZHUH QRW SUHVHQW LQ WKHLU WUXQFDWHG EDVLV VHW 7KH LPSUHVVLYHO\ DFFXUDWH UHVXOWV RI RI D FDOFXODWLRQ RQ E\ WKHVH DXWKRUV DUH VKRZQ LQ 7DEOH  :KHQ i =   EHFRPHV  (TXDWLRQ  FDQ DOVR EH H[SUHVVHG LQ WKH IRUP  DQG DQDORJRXVO\ WKH L

 VHFXODU HTXDWLRQ 



&+$37(5  ,7(5$7,21 2) 7+( :$9( (48$7,21

FDQ EH H[SUHVVHG LQ WKH IRUP

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

7KH PDQ\FHQWHU SUREOHP LQ GLUHFW VSDFH :H ZRXOG OLNH WR VROYH WKH RQHHOHFWURQ ZDYH HTXDWLRQ  ZKHUH Y[ LV WKH PDQ\FHQWHU QXFOHDU DWWUDFWLRQ SRWHQWLDO VKRZQ LQ HTXDWLRQ   7R GR WKLV ZH FDQ H[SDQG WKH ZDYH IXQFWLRQ LQ WHUPV RI K\GURJHQOLNH 6WXUPLDQ EDVLV IXQFWLRQV ORFDWHG RQ WKH YDULRXV FHQWHUV  ZKHUH

VWDQGV IRU WKH VHW RI LQGLFHV ^D Q OP` DQG ZKHUH 

7KH PRPHQWXPVSDFH EDVLV IXQFWLRQV VKRZQ LQ HTXDWLRQ  DUH WKH )RXULHU WUDQVIRUPV RI WKH GLUHFWVSDFH EDVLV IXQFWLRQV VKRZQ LQ  

 6XEVWLWXWLQJ WKH H[SDQVLRQ  LQWR WKH ZDYH HTXDWLRQ  ZH REWDLQ  1H[W ZH PXOWLSO\ IURP WKH OHIW E\ D FRPSOH[ FRQMXJDWH IXQFWLRQ IURP WKH EDVLV VHW DQG LQWHJUDWH RYHU WKH HOHFWURQ¶V FRRUGLQDWHV  6LQFH  LW IROORZV WKDW  7DNLQJ WKH )RXULHU WUDQVIRUP RI   ZH REWDLQ

 )LQDOO\ PDNLQJ XVH RI  DQG   ZH GHULYH WKH UHVXOW

 /HW XV QH[W FRQVLGHU WKH UHPDLQLQJ PDWUL[ HOHPHQW LQ HTXDWLRQ  





& + $ 3 7 ( 5  ,7(5$7,21 2) 7 + ( : $ 9 ( (48$7,21

/HWWLQJ VWDQG IRU WKH VHW RI LQGLFHV ^ Q  O  P `  DQG PDNLQJ XVH RI HTXDWLRQ   ZH KDYH

 ,QVHUWLQJ WKHVH H[SDQVLRQV LQWR  DQG PDNLQJ XVH RI WKH SRWHQWLDO ZHLJKWHG RUWKRQRUPDOLW\ UHODWLRQ   ZH KDYH

 )URP  DQG  ZH FDQ VHH WKDW WKH VHFXODU HTXDWLRQ  FDQ EH ZULWWHQ LQ WKH IRUP  ZKLFK LV LGHQWLFDO ZLWK   )URP   ZH FDQ DOVR VHH WKDW WKH PDWUL[ .ð FDQ EH LQWHUSUHWHG LQ WHUPV RI QXFOHDU DWWUDFWLRQ LQWHJUDOV :KHQHYHU PHWKRGV IRU FDOFXODWLQJ WKHVH QXFOHDU DWWUDFWLRQ LQWHJUDOV DUH DYDLODEOH LW LV SUHIHUDEOH WR XVH WKHP IRU HYDOXDWLQJ .ð UDWKHU WKDQ HYDOXDWLQJ .ð E\ VTXDULQJ . ZLWK D WUXQFDWHG EDVLV VHW

$Q LOOXVWUDWLYH H[DPSOH 7R LOOXVWUDWH WKH LQFUHDVHG DFFXUDF\ JLYHQ E\ HTXDWLRQ  LQ WKH FDVH RI D WUXQFDWHG EDVLV VHW ZH FDQ FRQVLGHU WKH WKH WZRFHQWHU FDVH ZKHUH 

 Then from (6.44), (4.53), and (6.46) and we have: (6.47) where R

X2

–

X1, and similarly, (6.48)

With the help of (6.44) and (5.74), we can also obtain the terms which are off-diagonal in the indices a1 and a:

(6.49) while from (5.14) and (5.15),

(6.50)

A curve representing =

as a function of R = s/kµ can be generated by substituting these values of K² and K into equation (6.32). Such a curve, with 3 basis functions on each center, is shown in Figure 6.1, compared with the nearly exact, results of Koga and Matsuhashi [190-192]. Comparison shows that equation (6.32) yields a more accurate result with 3 basis functions on each center than could be obtained from (6.30) with 15 basis functions on each center. The numerical values of the ground-state energies for obtained using equation (6.32) with 3 basis functions on each center are shown in column a of Table 6.1, compared with the extremely high-precision results of Koga and Matsuhashi (column b ) and the best available direct-space results (column c).



& + $ 3 7 ( 5  ,7(5$7,21 2) 7 + ( : $ 9 ( (48$7,21

)LJXUH  7KLV ILJXUH VKRZV WKH JURXQGVWDWH HQHUJ\ RI LQ +DUWUHHV DV D IXQFWLRQ RI WKH LQWHUQXFOHDU VHSDUDWLRQ LQ %RKUV  FRP SDUHG ZLWK WKH QHDUO\H[DFW UHVXOWV RI .RJD DQG 0DWVXKDVKL >@ GRWV  7KH ILUVW H[FLWHGVWDWH HQHUJLHV DUH VKRZQ LQ WKH XSSHU FXUYH 7KH WZR FXUYHV ZHUH JHQHUDWHG E\ XVLQJ HTXDWLRQ  ZLWK  EDVLV IXQFWLRQV RQ HDFK FHQWHU



7DEOH  1XFOHDU DWWUDFWLRQ LQWHJUDOV XVHG LQ HTXDWLRQV  DQG   7KH ]D[LV LV WDNHQ WR EH LQ WKH GLUHFWLRQ RI 5 DQG WKXV V  V



&+$37(5 ,7(5$7,212)7+(:$9((48$7,21

7DEOH  *URXQGVWDWH HOHFWURQLF HQHUJLHV RI LQ +DUWUHHV 7KH UHVXOWV RE WDLQHG XVLQJ HTXDWLRQ  ZLWK  EDVLV IXQFWLRQV RQ HDFK FHQWHU DUH VKRZQ LQ FROXPQ D 7KH QHDUO\H[DFW PRPHQWXPVSDFH UHVXOWV RI .RJD DQG 0DWVXKDVKL >@ DUH VKRZQ LQ FROXPQ E FRPSDUHG ZLWK WKH EHVW SRVLWLRQVSDFH UHVXOWV FROXPQ F 



([HUFLVHV  6WDUWLQJ ZLWK HTXDWLRQV  DQG   GHULYH HTXDWLRQV  DQG    6KRZ WKDW

7DEOH  ZKHUH V

N—5 DQG ZKHUH

:KDW LV WKH OLPLW RI WKH LQWHJUDO DV V

"

 &RQVLGHU WZR QXFOHL ZLWK FKDUJHV =  DQG =  :ULWH GRZQ WKH PDWULFHV DQG IRU WKH FDVH ZKHUH WKH EDVLV VHW FRQVLVWV RI D VLQJOH V DWRPLF RUELWDO ORFDOL]HG RQ HDFK FHQWHU :K\ LV WKH VTXDUH RI WKH ILUVW PDWUL[ QRW HTXDO WR WKH VHFRQG"

&KDSWHU 

02/(&8/$5 67850,$16 &RQVWUXFWLRQ RI PDQ\HOHFWURQ 6WXUPLDQV IRU PROHFXOHV ,Q &KDSWHU  ZH VDZ WKDW LI

 LV WKH H[WHUQDO SRWHQWLDO H[SHULHQFHG E\ D FROOHFWLRQ RI 1 HOHFWURQV WKHQ LW LV SRVVLEOH WR FRQVWUXFW D VHW RI PDQ\HOHFWURQ 6WXUPLDQ EDVLV IXQFWLRQV > @ VDWLVI\LQJ

 SURYLGHG WKDW ZH DUH DEOH W R VROYH WKH RQHHOHFWURQ HTXDWLRQ

 7KH SURGXFW

 



& + $ 3 7 ( 5  0 2 / ( & 8 / $ 5 67850,$16

ZLOO WKHQ EH D VROXWLRQ RI   SURYLGHG WKDW WKH VXEVLGLDU\ FRQGLWLRQV

 DQG

 DUH IXOILOOHG $Q DQWLV\PPHWUL]HG ZDYH IXQFWLRQ EXLOW XS RI SURGXFWV RI WKH IRUP VKRZQ LQ  ZLOO DOVR EH D VROXWLRQ RI   7KH IRUPDOLVP GLVFXVVHG LQ &KDSWHUV  DQG  SURYLGHV XV ZLWK D PHWKRG IRU VROYLQJ  LQ WKH FDVH ZKHUH 

LH WKH FDVH ZKHUH 9 [ UHSUHVHQWV WKH PDQ\FHQWHU &RXORPE DWWUDF WLRQ SRWHQWLDO SURGXFHG E\ D VHW RI QXFOHL ZLWK FKDUJHV =D ORFDWHG DW WKH SRVLWLRQV ;D  1RWLFH WKDW ZH KDYH UHLQWURGXFHG WKH LQGH[ M WR ODEHO WKH HOHFWURQV RI DQ 1HOHFWURQ V\VWHP :H QHJOHFWHG WR ZULWH WKLV LQGH[ LQ &KDSWHUV  DQG  ZKHUH ZH FRQFHQWUDWHG RQ WKH RQHHOHFWURQ PDQ\ FHQWHU SUREOHP :H KDYH DOVR UHLQWURGXFHG WKH ZHLJKWLQJ IDFWRU E—N— LQWR WKH RQHHOHFWURQ ZDYH HTXDWLRQ VLQFH WKH RQHHOHFWURQ RUELWDOV ZLOO QRZ EH XVHG WR EXLOG XS D PDQ\HOHFWURQ JHQHUDOL]HG 6WXUPLDQ EDVLV VHW :H OHW  ZKHUH

 DQG ZKHUH UHSUHVHQWV WKH VHW RI LQGLFHV ^ DQ OP ` 6XEVWLWXWLQJ  LQWR   ZH REWDLQ  0XOWLSO\LQJ  E\

DQG LQWHJUDWLQJ ZH REWDLQ



 7KHQ IURP HTXDWLRQV  DQG  ZH REWDLQ WKH VHFXODU HTXDWLRQ  ZKHUH  DQG

 (TXDWLRQ  LV LGHQWLFDO ZLWK HTXDWLRQ   H[FHSW WKDW N— KDV EHHQ UHSODFHG E\ +DYLQJ FRQVWUXFWHG D VHW RI PDQ\HOHFWURQ PROHF FKRVHQ LQ XODU 6WXUPLDQ EDVLV IXQFWLRQV ¡Y [ ZLWK WKH SDUDPHWHUV VXFK D ZD\ WKDW DOO IXQFWLRQV LQ WKH VHW FRUUHVSRQG WR WKH VDPH YDOXH RI ZH FDQ QRUPDOL]H WKHP DFFRUGLQJ WR WKH UHTXLUHPHQW  $OWHUQDWLYHO\ WKH PDQ\HOHFWURQ 6WXUPLDQ EDVLV VHW PD\ EH QRUPDOL]HG LQ PRPHQWXP VSDFH E\ UHTXLULQJ WKDW 

,I WKH IXQFWLRQV ¡Y [ DUH H[DFW VROXWLRQV WR HTXDWLRQ   WKHQ WKH UHTXLUHPHQWV  DQG  DUH LGHQWLFDO EXW LI WKH VROXWLRQV DUH RQO\ DSSUR[LPDWH WKH GLUHFWVSDFH DQG PRPHQWXPVSDFH QRUPDOL]DWLRQ UHTXLUHPHQWV DUH RQO\ DSSUR[LPDWHO\ LGHQWLFDO +DYLQJ FRQVWUXFWHG RXU EDVLV VHW ZH FDQ XVH LW WR VROYH WKH PDQ\HOHFWURQ 6FKU|GLQJHU HTXDWLRQ 



& + $ 3 7 ( 5  02/(&8/$5 67 850O$16

ZKHUH



,Q HTXDWLRQ   9 [ LV WKH PDQ\FHQWHU QXFOHDU DWWUDFWLRQ SRWHQWLDO GHILQHG E\ HTXDWLRQV  DQG   ZKLOH  LV WKH LQWHUHOHFWURQ UHSXOVLRQ SRWHQWLDO /HWWLQJ

 ZH REWDLQ WKH VHFXODU HTXDWLRQ  ZKHUH

 7KH IDFWRU ZKLFK DSSHDUV LQ HTXDWLRQ  FDQ EH UHZULWWHQ E\ PHDQV RI WKH VXEVLGLDU\ FRQGLWLRQV  DQG    7KXV WKH VHFXODU HTXDWLRQ   EHFRPHV  7KH WHUP UHSUHVHQWLQJ WKH QXFOHDU DWWUDFWLRQ SRWHQWLDO LV DOUHDG\ GLDJ RQDO LQ   DQG RQO\ WKH LQWHUHOHFWURQ UHSXOVLRQ WHUP  UHPDLQV WR EH GLDJRQDOL]HG



$Q LOOXVWUDWLYH H[DPSOH $V D VLPSOH H[DPSOH LOOXVWUDWLQJ WKH PHWKRG GLVFXVVHG DERYH ZH FDQ FRQVLGHU WKH PROHFXOH +  LQ WKH DSSUR[LPDWLRQ ZKHUH RXU HOHFWURQ 6WXUPLDQ EDVLV VHW FRQVLVWV RI WKH GHWHUPLQHQWLDO ZDYH IXQFWLRQV

 ZKHUH  DQG ZKHUH LV WKH VROXWLRQ WR  ZKLOH LV WKH VROXWLRQ :H EHJLQ E\ FRQVWUXFWLQJ WKH PDWULFHV .ð DQG . XVLQJ WKH PHWKRGV GLVFXVVHG LQ &KDSWHU  $V ZH VDZ WKH PDWUL[ HOHPHQWV DUH IXQFWLRQV RI WKH SDUDPHWHU V = N—5 ZKHUH 5 LV WKH LQWHUQXFOHDU VHSDUDWLRQ&KRRVLQJ D YDOXH RI V  ZH WKHQ VROYH HTXDWLRQ  DQG REWDLQ EJ DQG EX IRU WKH JHUXGH DQG XQJHUXGH VROXWLRQV DV ZHOO DV WKH FRHIILFLHQWV DQG ,I ZH FKRRVH WR QRUPDOL]H RXU EDVLV VHW LQ PRPHQWXP VSDFH WKHQ HTXDWLRQ  UHTXLUHV WKDW  )URP WKH VXEVLGLDU\ FRQGLWLRQ  DQG IURP WKH IDFW WKDW E— = N— = ZH KDYH

DQG 

8VLQJ  DQG WKH IDFW WKDW  ZH FDQ UHZULWH  LQ WKH IRUP 



& + $ 3 7 ( 5  02/(&8/$5 67850,$16

7KH ILUVW LQWHJUDO LQ  LV HDV\ WR HYDOXDWH VLQFH  &RPELQLQJ  DQG   ZH REWDLQ  ZKHUH WKH PDWUL[ & LV DVVXPHG WR EH XQLWDU\ 6LPLODUO\  ZKHUH

 ,Q HTXDWLRQ  ZH KDYH PDGH XVH RI WKH UHODWLRQVKLS  /LNH WKH 6KLEX\D:XOIPDQ LQWHJUDOV WKH 6WXUPLDQ RYHUODS LQWHJUDOV VKRZQ LQ HTXDWLRQ  FDQ EH HYDOXDWHG E\ PHDQV RI HTXDWLRQV  DQG   VLQFH  ZKHUH  7KXV ZH DUH DEOH WR REWDLQ WKH HOHPHQWV RI WKH PDWUL[ 0 DV IXQFWLRQV RI WKH SDUDPHWHU V N—5 DQG WKH QRUPDOL]DWLRQ FRQVWDQWV FDQ EH IRXQG E\ PHDQV RI WKH UHODWLRQ 

 .QRZLQJ WKH EDVLV IXQFWLRQV ZH FDQ VROYH WKH 6WXUPLDQ VHFXODU HTXD WLRQV

 ZKHUH LV WKH PDWUL[ RI LQWHUHOHFWURQ UHSXOVLRQ LQWHJUDOV -XVW DV LQ WKH FDVH RI DWRPV WKLV PDWUL[ WXUQV RXW WR EH LQGHSHQGHQW RI 3  DQG LQ IDFW LW GHSHQGV RQO\ RQ V +DYLQJ FKRVHQ D YDOXH RI V ZH FDQ ILQG WKH FRUUHVSRQGLQJ YDOXH RI S E\ VROYLQJ   7KH YDOXHV RI HQHUJ\ DQG LQWHUQXFOHDU VHSDUDWLRQ FDQ EH IRXQG IURP WKH UHODWLRQVKLSV  DQG 5 VN— %\ UHSHDWLQJ WKLV SURFHGXUH IRU PDQ\ YDOXHV RI V  ZH FDQ ILQG WKH HQHUJ\ DQG ZDYH IXQFWLRQ RI WKH PROHFXOH DV IXQFWLRQV RI 5



& + $ 3 7 ( 5  02/(&8/$5 67850,$16 7DEOH  6WXUPLDQ RYHUODS LQWHJUDOV



7DEOH 

& + $ 3 7 ( 5  02/(&8/$5 67850,$16



Exercises  &DOFXODWH GHILQLWLRQ

XVLQJ 7DEOH  DQG WKH

DQG

 8VH WKH UHVXOWV RI ([HUFLVH  W R FDOFXODWH WKH 6WXUPLDQ RYHUODS LQWHJUDO

 &DOFXODWH WKH LQWHJUDO , RI ([HUFLVH  XVLQJ WKH HOOLSVRLGDO FRRU GLQDWHV U D DQG ¡ ZKHUH [M [M DQG [M  R  [M  5 DQG ZKHUH ¡ KDV LWV XVXDO PHDQLQJ ,Q HOOLSVRLGDO FRRUGLQDWHV WKH YROXPH HOHPHQW LV JLYHQ E\ –

&RPSDUH \RXU DQVZHU ZLWK WKH UHVXOWV RI ([HUFLVH  &RXOG HOOLSVRLGDO FRRUGLQDWHV EH XVHG WR FDOFXODWH 6KLEX\D:XOIPDQ LQ WHJUDOV"

Chapter 8

RELATIVISTIC EFFECTS Relativistic hydrogenlike Sturmians 7KH 'LUDF HTXDWLRQ IRU DQ HOHFWURQ PRYLQJ LQ WKH &RXORPE ILHOG RI D QXFOHXV ZLWK FKDUJH E— LV JLYHQ LQ WKH FODPSHG QXFOHXV DSSUR[LPDWLRQ E\ >@

 ZKHUH  DQG  ,Q HTXDWLRQ   PDWULFHV

LV D YHFWRU ZKRVH FRPSRQHQWV DUH WKH 3DXOL VSLQ

 ZKLOH , LV D  x  XQLW PDWUL[ ,I ZH OHW ^E—` EH D VHW RI SDUDPHWHUV HVSHFLDOO\ FKRVHQ VR WKDW DOO RI WKH IXQFWLRQV LQ WKH VHW FRUUHVSRQG WR WKH VDPH YDOXH RI WKH UHODWLYLVWLF HQHUJ\ >@ WKHQ WKH VHW RI IXQFWLRQV [—[ PLJKW EH FDOOHG D UHODWLYLVWLF K\GURJHQOLNH 6WXUPLDQ EDVLV VHW 

&+$37(5  5(/$7,9,67,& ())(&76



7KH FRPSRQHQW VROXWLRQV RI  FDQ EH ZULWWHQ LQ WKH IRUP

 ZKHUH

M

–

 DQG ZKHUH WKH IXQFWLRQV >@



DUH VSKHULFDO VSLQRUV EXLOW XS IURP VSKHULFDO KDUPRQLFV DQG FRPSRQHQW VSLQRUV FRPELQHG ZLWK DSSURSULDWH &OHEVFK*RUGDQ FRHIILFLHQWV 7KH VSKHULFDO VSLQRUV DUH VLPXOWDQHRXV HLJHQIXQFWLRQV RI -ð /ð DQG -] ZKHUH

 DQG ZKHUH M O DQG 0 DUH WKH TXDQWXP QXPEHUV ODEHOLQJ WKH HLJHQIXQF WLRQV RI WKH WKUHH RSHUDWRUV :KHQ M O 



ZKLOH ZKHQ M = O

–

1



 7KH UDGLDO IXQFWLRQV FRUUHVSRQGLQJ WR WKH ODUJH DQG VPDOO FRPSRQHQWV RI DUH

 DQG

 ZKHUH 

 

  DQG ZKHUH

LV D FRQIOXHQW K\SHUJHRPHWULF IXQFWLRQ

 7KH HQHUJLHV RI WKH UHODWLYLVWLF K\GURJHQOLNH 6WXUPLDQV DUH UHODWHG WR WKH HIIHFWLYH QXFOHDU FKDUJHV E— E\ 



&+$37(5  5(/$7,9,67,& ())(&76

/LNH WKH QRQUHODWLYLVWLF K\GURJHQOLNH 6WXUPLDQV WKH UHODWLYLVWLF IXQF WLRQV VKRZQ DERYH REH\ D SRWHQWLDOZHLJKWHG RUWKRQRUPDOLW\ UHODWLRQ :H FDQ VHH WKLV LQ WKH IROORZLQJ ZD\ 0DNLQJ XVH RI WKH VHOIDGMRLQWQHVV RI WKH RSHUDWRU ' HTXDWLRQ  ZH FDQ ZULWH

 ZKHUH [ — [ DQG DUH WZR VROXWLRQV WR   %\ PXOWLSO\LQJ E\ WKH DSSURSULDWH IXQFWLRQV LQWHJUDWLQJ RYHU FRRUGLQDWHV DQG VXEWUDFWLQJ WKH VHFRQG HTXDWLRQ LQ  IURP WKH ILUVW ZH FDQ VKRZ WKDW >@  DQG WKHUHIRUH LI E—



,I ZH FRPELQH WKLV ZHLJKWHG RUWKRJRQDOLW\ SURSHUW\ ZLWK UHVSHFW WR WKH SULQFLSDO TXDQWXP QXPEHUV ZLWK WKH RUWKRQRUPDOLW\ SURSHUWLHV RI WKH VSKHULFDO VSLQRUV ZH FDQ ZULWH 

ZKHUH ZH KDYHQRUPDOL]HG RXU EDVLV VHW LQ VXFK D ZD\ WKDW  7KH UHODWLYLVWLF K\GURJHQOLNH 6WXUPLDQ EDVLV IXQFWLRQV FDQ EH XVHG WR EXLOG XS VROXWLRQV WR WKH 'LUDF HTXDWLRQ LQ D QRQ&RXORPE SRWHQWLDO ,I ZH ZLVK WR VROYH WKH HTXDWLRQ 

 ZH FDQ UHSUHVHQW WKH ZDYH IXQFWLRQ DV D OLQHDU FRPELQDWLRQ RI LVRHQHU JHWLF VROXWLRQV WR HTXDWLRQ  

 6XEVWLWXWLQJ  LQWR   ZH REWDLQ

 ,I ZH QRZ PDNH XVH RI   ZH FDQ UHZULWH  LQ WKH IRUP  7KH QH[W VWHS LV WR PXOWLSO\  IURP WKH OHIW E\ DQ DGMRLQW IXQFWLRQ IURP RXU UHODWLYLVWLF 6WXUPLDQ EDVLV VHW DQG WR LQWHJUDWH RYHU WKH FRRU GLQDWHV PDNLQJ XVH RI WKH SRWHQWLDOZHLJKWHG RUWKRQRUPDOLW\ UHODWLRQ   7KLV JLYHV XV D VHFXODU HTXDWLRQ RI WKH IRUP  ZKHUH

 :H FDQ VROYH WKH 6WXUPLDQ VHFXODU HTXDWLRQ LQ WKH IROORZLQJ ZD\ )LUVW ZH SLFN D YDOXH RI :H WKHQ FDOFXODWH YDOXHV RI E—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ð= DQG





&+$37(5  5(/$7,9,67,& ())(&76

Figure 8.1:7KLV ILJXUH VKRZV WKH JURXQG VWDWH DQG WKH ILUVW IHZ H[FLWHG VWDWHV ZLWK O  DQG M  IRU DQ HOHFWURQ PRYLQJ LQ WKH VFUHHQHG &RXORPE SRWHQWLDO VKRZQ LQ HTXDWLRQ   7KH UHODWLYLVWLF HQHUJLHV IURP ZKLFK WKH HOHFWURQ¶V UHVW HQHUJ\ KDV EHHQ VXEWUDFWHG DUH H[SUHVVHG LQ +DUWUHHV DQG WKH\ DUH VKRZQ DV IXQFWLRQV RI = 7KH ILJXUH LV WDNHQ IURP $YHU\ DQG $QWRQVHQ UHIHUHQFH >@



Relativistic many-electron Sturmians ,I ZH ZLVK WR FRQVWUXFW UHODWLYLVWLF PDQ\HOHFWURQ 6WXUPLDQ EDVLV IXQF WLRQV ZH PXVW UHLQWURGXFH DQ LQGH[ M WR ODEHO WKH HOHFWURQV LQ DQ 1  HOHFWURQ V\VWHP 7KXV  DQG  EHFRPH UHVSHFWLYHO\  ZKHUH ZH KDYH OHW E—

DQG 

DQG ZKHUH

LV JLYHQ E\   ,I ZH OHW 

DQG  WKHQ WKH IXQFWLRQ  ZLOO VDWLVI\  SURYLGHG WKDW  EHFDXVH





&+$37(5  5(/$7,9,67,& ())(&76

$Q DQWLV\PPHWUL]HG IXQFWLRQ WR VDWLVI\   ,I ZH OHW

=

FDQ DOVR EH VHHQ

 WKHQ WKH 'LUDF&RXORPE HTXDWLRQ IRU DQ 1HOHFWURQ DWRP ZLWK QXFOHDU FKDUJH = FDQ EH ZULWWHQ LQ WKH IRUP

 ,Q RUGHU WR VROYH   ZH H[SDQG WKH ZDYH IXQFWLRQ LQ WHUPV RI RXU 1HOHFWURQ 6WXUPLDQ EDVLV VHW

 6XEVWLWXWLQJ  LQWR   PDNLQJ XVH RI   DQG WDNLQJ WKH VFDODU SURGXFW ZLWK DQ DGMRLQW IXQFWLRQ LQ RXU EDVLV VHW ZH REWDLQ

 :H QRZ OHW

 DQG ZH QRUPDOL]H RXU EDVLV VHW LQ VXFK D ZD\ WKDW



$ SRWHQWLDOZHLJKWHG RUWKRQRUPDOLW\ UHODWLRQ FDQ WKHQ EH HVWDEOLVKHG IURP WKH VHOIDGMRLQWQHVV RI WKH RSHUDWRU > ' ± ( @ LQ D PDQQHU VLPLODU WR HTXDWLRQV   :

 7KXV ZH REWDLQ WKH VHFXODU HTXDWLRQ



 ,Q RUGHU W R VROYH WKHVH HTXDWLRQV ZH EHJLQ E\ SLFNLQJ D YDOXH RI ( DQG D VHW RI FRQILJXUDWLRQV Y ^——ï  —ïï`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

The relativistic many-center one-electron problem ,Q &KDSWHU  ZH GLVFXVVHG WKH XVH RI PDQ\FHQWHU PDQ\HOHFWURQ 6WXU PLDQ EDVLV VHWV ,W LV LQWHUHVWLQJ WR DVN ZKHWKHU DQDORJRXV PHWKRGV FDQ EH XVHG IRU UHODWLYLVWLF FDOFXODWLRQV RQ PROHFXOHV 7R DQVZHU WKLV TXHV WLRQ OHW XV EHJLQ E\ FRQVLGHULQJ WKH PDQ\FHQWHU RQHHOHFWURQ 'LUDF HTXDWLRQ  ZKHUH ' M LV GHILQHG E\   DQG  :H FDQ WU\ WR EXLOG XS WKH PROHFXODU RUELWDO RI WKH IRUP

IURP EDVLV IXQFWLRQV

 ZKHUH WLRQV

VWDQGV IRU WKH VHW RI LQGLFHV ^D Q M , 0` DQG ZKHUH WKH IXQF  DUH VROXWLRQV WR





&+$37(5  5(/$7,9,67,& ())(&76

7KH FRQVWDQWV DUH FKRVHQ LQ VXFK D ZD\ WKDW DOO WKH IXQFWLRQV [Q M O 0 LQ WKH RQHHOHFWURQ EDVLV VHW FRUUHVSRQG WR WKH VDPH YDOXH RI )URP  DQG  LW IROORZV WKDW

 ,I ZH OHW

 WKHQ ZLWK WKH KHOS RI   HTXDWLRQ  FDQ EH UHZULWWHQ LQ WKH IRUP

 0XOWLSO\LQJ IURP WKH OHIW E\ DQ DGMRLQW IXQFWLRQ IURP WKH EDVLV VHW DQG LQWHJUDWLQJ ZH REWDLQ WKH VHFXODU HTXDWLRQ



,I ZH KDYH D PHDQV RI HYDOXDWLQJ WKH PDQ\FHQWHU QXFOHDU DWWUDFWLRQ LQWHJUDOV ZKLFK DSSHDU LQ   WKHQ ZH FDQ SURFHHG DV IROORZV :H EHJLQ E\ FKRRVLQJ D YDOXH RI 7KH FRQVWDQWV EQMO DUH WKHQ GHWHUPLQHG E\ LQYHUVLRQ RI WKH UHODWLRQVKLS 

6ROXWLRQ RI  WKHQ GHWHUPLQHV WKH YDOXH RI ZKLFK FRUUHVSRQGV WR WKH FKRVHQ YDOXH RI IRU WKH JURXQG VWDWH 2WKHU YDOXHV RI FRUUHVSRQG WR WKH FKRVHQ IRU WKH H[FLWHG VWDWHV %\ UHSHDWLQJ WKLV SURFHGXUH ZH DUH DEOH WR REWDLQ DV D IXQFWLRQ RI DV ZHOO DV WKH FRUUHVSRQGLQJ PROHFXODU RUELWDOV ERWK IRU WKH JURXQG VWDWH DQG IRU WKH H[FLWHG VWDWHV



Relativistic many-electron molecular Sturmians 6LQFH ZH FDQ VROYH  E\ WKH PHWKRG MXVW GHVFULEHG LW IROORZV WKDW LI ZH OHW  DQG  WKHQ IRU HDFK FRQILJXUDWLRQ Y ^— —ï —ïï ` ZH FDQ ILQG ( DV D IXQF WLRQ RI 7KH SURGXFW VKRZQ LQ  LV D VROXWLRQ WR

 ZKHUH ' LV GHILQHG E\  DQG  :H FDQ VHH WKDW LW LV D VROXWLRQ EHFDXVH IURP  DQG   ZH KDYH

 $Q DQWLV\PPHWUL]HG SURGXFW IXQFWLRQ ZLOO DOVR VDWLVI\   7KXV ZH DUH DEOH WR FRQVWUXFW EDVLV VHWV ZKLFK DUH WKH UHODWLYLVWLF DQDORJV RI WKH PDQ\HOHFWURQ PROHFXODU 6WXUPLDQV GLVFXVVHG LQ &KDSWHU  &KRRVLQJ D VHW RI IXQFWLRQV ^¡Yx `DOO RI ZKRVH PHPEHUV FRUUHVSRQG WR WKH VDPH YDOXH RI ( ZH FDQ XVH WKHP WR EXLOG XS VROXWLRQV WR WKH PDQ\FHQWHU 'LUDF&RXORPE HTXDWLRQ  ZKHUH 9ï [ LV WKH LQWHUHOHFWURQ UHSXOVLRQ SRWHQWLDO VKRZQ LQ HTXDWLRQ   :LWK WKH GHILQLWLRQV JLYHQ LQ HTXDWLRQV   WKH VHFXODU HTXDWLRQ EHFRPHV 



&+$37(5  5(/$7,9,67,& ())(&76

)RU HDFK YDOXH RI (  VROXWLRQ RI  JLYHV XV YDOXHV RI EHORQJLQJ WR WKH JURXQG VWDWH DQG WR H[FLWHG VWDWHV RI WKH V\VWHP %\ UHSHDWLQJ WKH SURFHGXUH DQG E\ LQWHUSRODWLQJ ZH FDQ ILQG WKH VROXWLRQV FRUUHVSRQGLQJ  ERWK IRU WKH JURXQG VWDWH RI WKH V\VWHP DQG IRU WKH H[FLWHG WR VWDWHV

 Table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¶V +DUWUHH)RFN YDOXHV >@ –

&+$37(5  5(/$7,9,67,& ())(&76



Exercises  6KRZ WKDW IRU M    DQG 0     WKH FRPSRQHQW VROXWLRQ WR WKH K\GURJHQOLNH 'LUDF HTXDWLRQ FDQ EH ZULWWHQ LQ WKH IRUP

:KDW LV WKH IRUP RI WKH VROXWLRQ FRUUHVSRQGLQJ W R M DQG 0 ±O"  /HWWLQJ E—  ILQG WKH YDOXHV RI N QU 1 DQG VROXWLRQV WR  LQ WKH Q  DQG Q  VKHOOV

 



IRU WKH

 7KH HQHUJLHV FDOFXODWHG LQ ([HUFLVH  LQFOXGH WKH HOHFWURQ UHVW HQHUJ\ PFð DQG DUH H[SUHVVHG LQ XQLWV RI PFð 6XEWUDFW WKH UHVW HQHUJ\ IURP WKH FDOFXODWHG YDOXHV DQG H[SUHVV WKH UHVXOWV LQ +DUWUHHV



Appendix A: Generalized Slater-Condon rules $ VOLJKW FRPSOLFDWLRQ LQ WKH XVH RI PDQ\HOHFWURQ 6WXUPLDQ EDVLV IXQF WLRQV FRPHV IURP WKH IDFW WKDW RUWKRJRQDOLW\ EHWZHHQ RQHHOHFWURQ RU ELWDOV FDQQRW EH DVVXPHG ZKHQ RQH LV HYDOXDWLQJ PDWUL[ HOHPHQWV EH WZHHQ FRQILJXUDWLRQV FRUUHVSRQGLQJ WR GLIIHUHQW YDOXHV RI 2QH PXVW WKHQ XVH WKH JHQHUDOL]HG 6ODWHU&RQGRQ UXOHV /HW XV FRQVLGHU WZR IXQF WLRQV ¡Y DQG ¡Yï LQ RXU VHW RI PDQ\HOHFWURQ 6WXUPLDQ EDVLV IXQFWLRQV 7KHVH PLJKW IRU H[DPSOH EH WKH 6ODWHU GHWHUPLQDQWV

$  DQG

$  ,I WKHQ ZH FDQQRW DVVXPH WKDW WKH WZR 6ODWHU GHWHUPLQDQWV ¡Y DQG ¡Yï DUH EXLOW XS IURP PXWXDOO\ RUWKRQRUPDO RQHHOHFWURQ VSLQ RUELWDOV DQG ZH PXVW PDNH XVH RI WKH IRUPDOLVP RI JHQHUDOL]HG 6ODWHU &RQGRQ UXOHV LQ HYDOXDWLQJ LQWHUFRQILJXUDWLRQDO PDWUL[ HOHPHQWV 7KLV IRUPDOLVP ZDV GHYHORSHG E\ /|ZGLQ $PRV +DOO DQG RWKHUV >@ ZKR VKRZHG WKDW LW LV SRVVLEOH WR VLPSOLI\ WKH HYDOXDWLRQ RI LQWHU FRQILJXUDWLRQDO PDWUL[ HOHPHQWV E\ PHDQV RI WZR VHSDUDWH XQLWDU\ WUDQV IRUPDWLRQV RQH PL[LQJ WKH RUELWDOV RI WKH FRQILJXUDWLRQ Y DQG WKH RWKHU PL[LQJ WKH RUELWDOV RI WKH FRQILJXUDWLRQ Yï :KHQ WKH PDWUL[

 $  LV LQ JHQHUDO QRW GLDJRQDO +RZHYHU ZH QRZ PDNH WKH XQLWDU\ WUDQV IRUPDWLRQV  $  DQG

$  :H ZRXOG OLNH WR FKRRVH WKH WUDQVIRUPDWLRQ PDWULFHV 8 DQG : LQ VXFK D ZD\ WKDW

 $ 

 ZLOO EH GLDJRQDO 7KH DXWKRUV MXVW PHQWLRQHG >@ KDYH VKRZQ WKDW ZH FDQ GR WKLV E\ OHWWLQJ : EH WKH XQLWDU\ PDWUL[ ZKLFK GLDJR QDOL]HV LH WKH VROXWLRQ WR WKH VHFXODU HTXDWLRQ $  ,Q PDWUL[ QRWDWLRQ HTXDWLRQ $ FDQ EH UHZULWWHQ LQ WKH IRUP $  ZKHUH $ LV D GLDJRQDO PDWUL[ ZKRVH GLDJRQDO HOHPHQWV DUH FDQ DOVR ZULWH $ LQ WKH IRUP

$

6LQFH ZH

$ 

LW IROORZV WKDW WKH GLDJRQDO HOHPHQWV DUH SRVLWLYH GHILQLWH ,I WKH DUH DOVR QRQ]HUR ZH FDQ FRQVWUXFW WKH XQLWDU\ PDWUL[

8 6:$ 

$ 

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

$ RU LQ PDWUL[ QRWDWLRQ $ 

 6XEVWLWXWLQJ WKH DGMRLQW RI $ LQWR $  DQG PDNLQJ XVH RI $  ZH KDYH = = $  ZKLFK LV GLDJRQDO EHFDXVH $ LV GLDJRQDO 7KH DXWKRUV RI UHIHUHQFHV >@ DQG >@ ZHUH DEOH WR GHULYH JHQHUDOL]HG 6ODWHU&RQGRQ UXOHV LQ WKH WUDQVIRUPHG EDVLV ,Q RXU QRWDWLRQ WKHVH EHFRPH $ 

$ 

$



Appendix B: Coulomb and exchange integrals for atoms ,Q &KDSWHU  HTXDWLRQV    ZH PHQWLRQHG WKDW LQ RUGHU WR FRQVWUXFW WKH LQWHUHOHFWURQ UHSXOVLRQ PDWUL[ IRU DWRPV ZH PXVW EH DEOH WR HYDOXDWH LQWHJUDOV RI WKH IRUP

%  ZKHUH

% DQG

% 

,Q HTXDWLRQ %  : DQG : DUH SURGXFWV RI VSKHULFDO KDUPRQLFV DQG DV ZH VKDOO VHH EHORZ RQO\ D IHZ RI WKH DQJXODU LQWHJUDOV DO DUH QRQ]HUR ,Q RUGHU WR HYDOXDWH WKH UDGLDO LQWHJUDOV ZH FDQ UHZULWH HTXDWLRQ % LQ WKH IRUP % (DFK RI WKH WZR WHUPV LQ % FDQ EH H[SUHVVHG LQ WHUPV RI DQ LQFRP SOHWH JDPPD IXQFWLRQ

%  DQG

 %  ZKHUH

% 

 ZKLOH

%  7KH ILQDO LQWHJUDWLRQ RYHU GU FDQ EH SHUIRUPHG E\ PHDQV RI IRUPXODV IRU WKH LQWHJUDWLRQ RI LQFRPSOHWH JDPPD IXQFWLRQV ZKLFK FDQ EH IRXQG LQ WKH WDEOHV RI *UDGVKWH\Q DQG 5\VKLN >@ 7KH UHVXOW LV

%  (TXDWLRQ % H[SUHVVHV WKH UDGLDO LQWHJUDO LQ WHUPV RI WKH K\SHUJH RPHWULF IXQFWLRQ GHILQHG E\ HTXDWLRQ   DQG WKXV % JLYHV WKH LQWHJUDO LQ WHUPV RI DQ LQILQLWH VHULHV :H FDQ KRZHYHU WUDQVIRUP WKLV UHVXOW LQWR D SRO\QRPLDO E\ PHDQV RI WKH UHDOWLRQVKLS >@

% /HWWLQJ

%  VR WKDW

%  ZH FDQ WUDQVIRUP WKH ILUVW K\SHUJHRPHWULF IXQFWLRQ RI % 

 %  DQG VLPLODUO\

%  7KXV ZH REWDLQ WKH UDGLDO LQWHJUDO LQ WHUPV RI D SRO\QRPLDO

%  VLQFH LQ DOO FDVHV RI LQWHUHVW O ± M ±  DQG O ± M ±  DUH HLWKHU ]HUR RU HOVH QHJDWLYH LQWHJHUV 7KXV WKH VHULHV GHILQHG E\ HTXDWLRQ  WHUPLQDWHV IRU ERWK WHUPV LQ %  $ SDUWLFXODUO\ VLPSOH FDVH RFFXUV ZKHQ

% 

,Q WKDW FDVH ERWK K\SHUJHRPHWULF IXQFWLRQV DUH HTXDO W R  DQG WKH UDGLDO LQWHJUDO UHGXFHV W R %  VR WKDW

% 

 7KH LQWHJUDOV IRU RWKHU YDOXHV RI M DQG M FDQ EH IRXQG E\ GLIIHUHQWLDWLQJ ERWK VLGHV RI % ZLWK UHVSHFW WR DQG

%  DQG WKLV UHVXOW FDQ EH UHJDUGHG DV DQ DOWHUQDWLYH IRUP RI %  :H PHQWLRQHG DERYH WKDW RQO\ D IHZ RI WKH DQJXODU LQWHJUDOV DO DUH QRQ ]HUR ,Q WKH FDVH RI &RXORPE LQWHJUDOV WKH\ WDNH RQ WKH IRUP %  ZKLOH IRU H[FKDQJH LQWHJUDOV WKH\ KDYH WKH IRUP

6LQFH WKDW

.

=

%  ZH FDQ VHH LPPHGLDWHO\ IURP % DQG % %

)RU RWKHU YDOXHV RI O LW LV KHOSIXO WR XVH WKH UHODWLRQVKLS %  ZKLFK IROORZV IURP HTXDWLRQ   ,Q HTXDWLRQ %  WKH RSHUDWRU 2O LV D SURMHFWLRQ RSHUDWRU FRUUHVSRQGLQJ WR WKH DQJXODU PRPHQWXP TXDQWXP QXPEHU O  )RU WKH SURMHFWLRQ WR EH QRQ]HUR 3O PXVW KDYH WKH VDPH SDULW\ DV : IURP ZKLFK LW IROORZV WKDW =

LI l = RGG

%

DQG WKDW % 

 )LQDOO\ WKH SURMHFWLRQ ZLOO EH ]HUR LI O H[FHHGV WKH RUGHU RI WKH KRPRJH QHRXV SRO\QRPLDO ZKLFK FDQ EH IRUPHG IURP :RU : E\ PXOWLSO\LQJ WKHVH DQJXODU IXQFWLRQV E\ WKH DSSURSULDWH SRZHU RI U 7KHUHIRUH

% DQG

% )RU QRQ]HUR FDVHV WKH KDUPRQLF SURMHFWLRQ IRUPDOLVP RI &KDSWHU  PD\ EH XVHG WR HYDOXDWH WKH DQJXODU LQWHJUDOV DO $OWHUQDWLYHO\ WKHVH LQWHJUDOV PD\ EH HYDOXDWHG E\ PHDQV RI &OHEVFK*RUGDQ FRHIILFLHQWV XVLQJ WKH UHODWLRQVKLSV

% DQG

%  ZKLFK IROORZ IURP HTXDWLRQ  



Solutions to the exercises Exercise 1.1 8VH 7DEOH    W R VKRZ WKDW WKH RQHHOHFWURQ K\GURJHQOLNH 6WXUPLDQ x- REH\V HTXDWLRQV   

Solution

Exercise 1.2 6KRZ WKDW LI N = DQG N REH\ WKH RUWKRJRQDOLW\ UHODWLRQ

WKHQ

xM DQG

xM



Solution

)URP H[HUFLVH  ,

ZKLOH

Exercise 1 . 3

6KRZ WKDW LI N N  N—  WKHQ DQG SRWHQWLDOZHLJKWHG RUWKRQRUPDOLW\ UHODWLRQ

REH\ WKH



6ROXWLRQ

([HUFLVH  8VH HTXDWLRQV  DQG  WR HYDOXDWH WKH GLUHFWVSDFH K\GURJHQOLNH RUELWDOV X s Xs, Xs DQG



6ROXWLRQ

VR WKDW

)RU l



DQG

VR WKDW

([HUFLVH  &DOFXODWH WKH LQWHJUDO LQ HTXDWLRQ  DQG VKRZ WKDW LW \LHOGV WKH UHVXOW VKRZQ  



Solution

Exercise 2.2 6WDUWLQJ ZLWK J FDOFXODWH WKH LQWHJUDOV J2 0 DQG J E\ GLIIHUHQWLDWLQJ ZLWK UHVSHFW WR kµ , DV VKRZQ LQ HTXDWLRQV  DQG  

Solution

Exercise 2.3 6WDUWLQJ ZLWK J XVH WKH UHFXUVLRQ UHODWLRQV RI HTXDWLRQ  WR JHQ HUDWH J, J DQG J   .

Solution



([HUFLVH  8VH WKH LQWHJUDOV LQ 7DEOH  W R HYDOXDWH WKH )RXULHU WUDQVIRUPV RI 6KRZ WKDW WKH WKH GLUHFWVSDFH K\GURJHQOLNH RUELWDOV X  s , Xs DQG WUDQVIRUPV FRUUHVSRQG WR WKH 6ROXWLRQV RI )RFN HTXDWLRQ  

6ROXWLRQ

147

([HUFLVH  Which of the following polynomials in a GGLPHQVLRQDO space are homogeneous? Which are harmonic? r is the hyperradius.) 1. [ ³ + [ ³ ¹ ² 2. [³ + [ ² ² ¹ 3. 2[ ¹³ + [³ ² 4. [³ – [ ³ ¹ ² 5. [ ³ B 3[ U ²/(G + 2) ¹ ¹ 6. [ ² [² [³ – U ²[ ² [ ³ /( G + 4) ¹



6ROXWLRQ KRPRJHQHRXV

KDUPRQLF

([HUFLVH  8VH HTXDWLRQ  WR ILQG H[SUHVVLRQV DQDORJRXV WR  IRU WKH KDU PRQLF GHFRPSRVLWLRQ RI D WKRUGHU SRO\QRPLDO

6ROXWLRQ :KHQ n

 DQG v

  EHFRPHV

:KHQ n  DQG v   EHFRPHV

:KHQ n

 DQG v   EHFRPHV



([HUFLVH  8VH HTXDWLRQ  W R FDOFXODWH WKH QRUPDOL]DWLRQ IDFWRU LQ D GLPHQVLRQDO VSDFH IRU WKH K\SHUVSKHULFDO KDUPRQLF VKRZQ LQ HTXDWLRQ   &RP SDUH WKLV UHVXOW ZLWK 7DEOH 

6ROXWLRQ

)RU d

  EHFRPHV

VR WKDW

([HUFLVH  6KRZ WKDW WKH K\SHUVSKHULFDO KDUPRQLFV ZLWK  LQ 7DEOH  IXOILOO WKH VXP UXOH RI HTXDWLRQ   6KRZ WKDW WKH\ DUH SURSHUO\ QRUPDO L]HG



Solution

From equation (3.49),

([HUFLVH  Use equation (3.72) to show that

Solution



([HUFLVH  8VH HTXDWLRQV   WR GHULYH HTXDWLRQ  

6ROXWLRQ

([HUFLVH  8VH HTXDWLRQ  W R JHQHUDWH WKH DVVRFLDWHG /DJXHUUH SRO\QRPLDOV RI 7DEOH 

6ROXWLRQ

152

Exercise 4.3 From equation (4.38) and the associated Laguerre polynomials in Table 4.2, calculate the parabolic hydrogenlike orbitals shown in Table 4.3. Express these functions as linear combinations of

Solution Let t

kµ r. Then, since

= r + z and

= r–z



([HUFLVH  &DOFXODWH WKH 6KLEX\D:XOIPDQ LQWHJUDOV DQG LQ WHUPV RI WKH XQLYHUVDO IXQFWLRQ E\ PHDQV RI HTXDWLRQ   &RPSDUH \RXU UHVXOWV ZLWK WKRVH VKRZQ LQ 7DEOH  ,V WKH LQWHJUDO UHDO"

6ROXWLRQ



7KH LQWHJUDO LV UHDO

([HUFLVH  &DOFXODWH E\ PHDQV RI HTXDWLRQV    6KRZ WKDW WKH UHVXOW DJUHHV ZLWK 7DEOH  DQG ([HUFLVH 

6ROXWLRQ



([HUFLVH  8VH  WR ZULWH GRZQ DQ DQJXODU LQWHJUDWLRQ WKHRUHP DQDORJRXV WR  IRU WKH FDVH ZKHUH d 

6ROXWLRQ

([HUFLVH  6WDUWLQJ ZLWK HTXDWLRQV  DQG   GHULYH HTXDWLRQV  DQG  

6ROXWLRQ

ZKHUH

^a, µ`

^a, n, l, m` DQG a 



B ZKHUH 5 B ; ;

DQG VLPLODUO\

([HUFLVH  6KRZ WKDW

7DEOH  ZKHUH V

DQG ZKHUH

:KDW LV WKH OLPLW RI WKH LQWHJUDO DV s

6ROXWLRQ /HW W

kµ [, V

kµ 5 DQG

"

 7KHQ

([HUFLVH  &RQVLGHU WZR QXFOHL ZLWK FKDUJHV Z DQG Z :ULWH GRZQ WKH PDWULFHV DQG IRU WKH FDVH ZKHUH WKH EDVLV VHW FRQVLVWV RI D VLQJOH s DWRPLF RUELWDO ORFDOL]HG RQ HDFK FHQWHU :K\ LV WKH VTXDUH RI WKH ILUVW PDWUL[ QRW HTXDO W R WKH VHFRQG"

6ROXWLRQ

158

The inequality K ² (K)² is due to the fact that the basis set is truncated, so that the sum

does not run over all possible values of

([HUFLVH  Calculate

6ROXWLRQ

and

using Table 5.2 and the definition

159

([HUFLVH  Use the results of Exercise 7.1 to calculate the Sturmian overlap integral

6ROXWLRQ

where s = kµ R.

([HUFLVH    Calculate the integral I of Exercise 7.2 using the ellipsoidal coordinates = (ra + = (ra – rb) /R and ø where = [ j . [ j and = . (x j + 5) (x j + 5 ) and where ø has its usual meaning. In ellipsoidal coordinates, the volume element is given by

Compare your answer with the results of Exercise 7.2. Could ellipsoidal coordinates be used to calculate Shibuya-Wulfman integrals?

6ROXWLRQ

160

Ellipsoidal coordinates also offer an alternative method for evaluating Shibuya-Wulfman integrals.

Exercise 8.1 Show that for j = 1/2, l = 0, and M = 1/2, the 4-component solution to the hydrogenlike Dirac equation can be written in the form:

What is the form of the solution corresponding t o j = 1/2, l = 0, and M = –1/2?

Solution From (8.8) we have:

while from (8.9) with l = j – l = 1,

161 Therefore (8.5) yields

Similarly, when

= _ 1/2

Exercise 8.2 Letting bµ = 1, find the values of k , nr , to (8.1) in the n = 1 and n = 2 shells.

N , and

for the solutions

Solution From equations (8.12)-(8.15) and (8.19) we obtain:

Exercise 8.3 The energies calculated in Exercise 8.2 include the electron rest energy mc² and are expressed in units of mc². Subtract the rest energy from the calculated values, and express the results in Hartrees.



6ROXWLRQ ,Q RUGHU WR FRQYHUW IURP XQLWV RI mcð WR +DUWUHHV ZH PXVW PXOWLSO\ E\   ð 7KH UHVXOWLQJ HQHUJLHV

DUH

 %LEOLRJUDSK\  $KOEHUJ 5 DQG /LQGQHU 3 The Fermi correlation for electrons in momentum space, - 3K\V % 9RO    S   $KOHQLXV7RUDQG/LQGQHU3HWHU Semiempirical MO wave functions in momentum space, - 3K\V % 9RO    S    $NKLH]HU $, DQG %HUHVWHWVNLL 9% 4XDQWXP (OHFWURG\ QDPLFV ,QWHUVFLHQFH 1HZ