ENSE~t\.'\ZA REVISTA MEXICANA DE FíSICA.H ,615hX~575 DICIEMBRE 2001 Thc discrctc variable rcprcscntation mcthod tilr hound statc cigenvalucs and cigcnt'unctions Frederico V. Prudente. Antonio Rig;'lJlelli. and I Antónin J.e. Varandas De'p"rtllll1('lIfO de' Q//{1JIlCll. Ullil'('1"sidadc de CoÍ/IIhm P-3049 Coimhm Cmlex. Portllgal ('.l1Ia il: I varwula.\'@qtvsl.qui.ucIJI Hedhido el 14 lIe marzo de 2<X)I:accplaJo el 25 dc ~cp(icmhre Je 20() I Tlle ollc-Jllllell~llJnal radial SchrliJingercquatioll is ~olvcJ llllllh.'ric.dly hy mcans olthc Jl~nClC vanahJc rerrC~ClHa[lOnmClho(J. This is ba~cd l)jl <1 ~ml rqm:'~l'lHati{)n (lflhe qualllum mechanical wavcfum:tion í1mlmay he introduccd (111 lile h;l~i .... (11;1rigorou:-.varwlional principIe. The Illetlllld h;l'" !leen applied lo {he ca1culation 01"thc l:11ergyspcctrulll 01"tllree rerrcscllIalive 1Tludcls:l\uantlltll panidc movln}! 1Il a quadratic PI1[l"lllía1. hlllllld "'[;lle energy Ic\'el\ ()f a dialomic lllolel"ulc. ami ~pcclnllll uf hydrogcn al\lll1. I":"l"ll"IJII/' (}llalHlllll l1leCh¡lllic~:\'ariationnl principie; houlld ~tale:-.:discrcle \',:¡riahle repre",clllatioll Illt:lhod L:llTlI:tl.I()1lr:ldi:t1l1J1iuilllell~I{)I1:I1 de Schrüuingcr e~ n:~uelta llllllléncalllcllle por medIO dellllctodo dt"1:1rl'ple~Cnlación oc variahlc Ji~ereta. Este 1lll'1()(lo~e h;l... a en la rcprescntación de la fUllclón de onda sohrc ulla rejilla y pucdc ~ef introducido rigurmamcllIc usando el principio \mi¡lCH)nal. Iknws clnplead\l dicho método CI1el dlculo del espectro ellcrgétil"O dt: Irc~ l\lodclllS rcprescntati\'m. panícula cu¡íntica en UI1 plllcll,.lal cll;ldr;illco. C~tilJO:.cilla/aJos de una molécula diatómÍl"a y CSPCl'trodel ¡¡{omo Jc hidrógello. /)(,.".I"II'/Ol"{'\' \kdnica cu;ínlil"¡¡; principio variacional; estados cnlatado ... : mélodo Je representación de vanahlc di~ncta 1,,,es: o2.70.-c: lJ3.có.lic; 31.1).pf 1. lntrodllt'lion Tlle radial ScllI'ódinger equation is a second order differential IYPl: l'qllillioll Ihat arises in thc modeling of mDIlY physical phCJ101l1Cllaal ¡¡tomic sc¡¡le. Hüwever. this eqllation can be "nlved analylically ollly for so me simple potential Illodels which arl' normally ellcoulltered in textbooks of qU<.\lltum mechanio 11-:l}. Thus. Ihc use of numerical ll1ethods to solve lhlo' Schrildinger cqllalion is fundamental for sluclying realistic quallllll1l prohlL'IllS. These can be eilher scattcring nI" bOllnd sial..: ami resonanc..: processes. In p;'lrticlllar. Ih..: "llldy' lit" olll'-dilll..:nsional bOllnd-state systems which will be discll~ ..• l.'d !ll're lIsing the time-independellt fnnnalism present ... Ollll: inll'l"c,,1 frolll a pedagogical point of vie\\'. There are tlm,'c lIIaill reasons for Ihis interest: i) Tilc I\\,'o-particle proccsses are governed by llIlC-di- 11l1'Ilsional Schrodingcr eqllalions. ¡jI Tilc me[hod providcs a rclati\'e simple introdllction to Ihe s()llIlion of Illany-dimensional problcllls. iii) 11 ¡llIows iI pedagogical dcscriplion of Ilumerical me- II10d.... used in quanlul1lmecilanics. Indeed. :-'lIi.."h¡\Il illteresl is corrobor;'lIl'd by the anicles which haV\.' ;Ippcarcd trom timc to lime in cducational jom- nals 11-1:11. NUllllo'riGI! Illclhods to solve one-dimensional eigenvalllc problcllls call be classiflcd ilHO three types: nUll1crical inle- ~ratiolls (or shouting Illclhods) [13.9,1-1), llIatrix or global arrroachl's [11,"¡.15}. and Rayleigh-Rilz varialional me. thods 17,((¡-IR]. An approximale Raylcigll-Rilz variational IIlcthod lhal ha:,> dClllollslraled high cffkicllCY ror solving une- amllllany-dilllcnsional Scluüdinger equations is the dis- cn~te variable reprcsenlalion (DVR) melhod [ID. 20). This method was originally introdllccd by Harris ('1 "l. [21] slrictly rOl"cvalll<lling Illllt.:nlial 1ll;'ltri.\ ckmcnts in thlo: fiO's. It was Ihen by Did.inson ami CIo:Hain [22) shown lo be rclated to gaussian quadralure methods throllgh a simple lInitary trans- fonnalion. Ilowl:\'er. il was only in the SO's tlwt Light and l'o-\\'orkers [2:~-2;jl generalized the DVR Illethod and applied il to a nUlllber of intcresting quantulll mechanical problems. 1\n advanlage of the DVR IIlclhod is tllat its computer imple- lllenlalioll is simple. This allows undergraduatc students of qllantum mechanical aJ1(l/m compulalional phYSlCS courses lo use such a IIlclllOd as ¡¡ \\'orking 1001 to Ireal l1umerically quanlulll sySll'llls. This ability was shown to be very impor- lant when applicd to srccitic rcscarch tielcls sllch as atomic, molecular. ;'l1ld lIIore recently snlid stale physics 12GI. Yet, no ¡¡nicle has appearcd lo dalc in an cducational .intlmal ;'lbout Ih~ DVR Illethod and its applications. [n the present work. \Ve ailll lo cover tllis gap by inlroducing the DVR method fmm a pcdagogical point of vie\l•...Since lhe emphasis will be 011 Ihe COIllPUlatiollal lechniqlle and ils physical jllstificalion r¡¡lhlo:rthan on Ihe energy eigcnv;:lluc Ihemselves, we apply Ihe mclhod lo Ihree \Vcll known protolypes in physics. First. we "olve [he radial Schrodinger cqu;ltion for a quantum particle in all harmonic POlclllial. Thlo:ll. wc use Ihe sall\c methodol- ogy to calculale the bound states of the Ro! molecule. Finally. wc calculale cncrgy spcclrum of the hydrogcn atom. The pa- pel' is organized as follo\Vs. In Sec. 2 we describe lhe general [heory orthc I)VR IJlclhod. v•.hile Sec, J discllsses the equa\ly spaccd discrcte variable reprcscntatioll. Thc applic~ltiollS are in Seco -l. and Ihe cOllclllding rCll1arks in Seco S.
Transcript
ENSE~t\.'\ZA REVISTA MEXICANA DE FíSICA.H ,615hX~575 DICIEMBRE 2001
L:llTlI:tl.I()1lr:ldi:t1l1J1iuilllell~I{)I1:I1de Schrüuingcr e~ n:~uelta llllllléncalllcllle por medIO dellllctodo dt" 1:1rl'ple~Cnlación oc variahlc Ji~ereta.Este 1lll'1()(lo~e h;l...a en la rcprescntación de la fUllclón de onda sohrc ulla rejilla y pucdc ~ef introducido rigurmamcllIc usando el principio\mi¡lCH)nal. Iknws clnplead\l dicho método CI1el dlculo del espectro ellcrgétil"O dt: Irc~ l\lodclllS rcprescntati\'m. panícula cu¡íntica en UI1plllcll,.lal cll;ldr;illco. C~tilJO:.cilla/aJos de una molécula diatómÍl"a y CSPCl'trodel ¡¡{omoJc hidrógello.
/)(,.".I"II'/Ol"{'\' \kdnica cu;ínlil"¡¡; principio variacional; estados cnlatado ...: mélodo Je representación de vanahlc di~ncta
1,,,es: o2.70.-c: lJ3.có.lic; 31.1).pf
1. lntrodllt'lion
Tlle radial ScllI'ódinger equation is a second order differentialIYPl: l'qllillioll Ihat arises in thc modeling of mDIlY physicalphCJ101l1Cllaal ¡¡tomic sc¡¡le. Hüwever. this eqllation can be"nlved analylically ollly for so me simple potential Illodelswhich arl' normally ellcoulltered in textbooks of qU<.\lltummechanio 11-:l}. Thus. Ihc use of numerical ll1ethods tosolve lhlo' Schrildinger cqllalion is fundamental for sluclyingrealistic quallllll1l prohlL'IllS. These can be eilher scattcringnI" bOllnd sial..: ami resonanc..: processes. In p;'lrticlllar. Ih..:"llldy' lit" olll'-dilll..:nsional bOllnd-state systems which will bediscll~ ..•l.'d !ll're lIsing the time-independellt fnnnalism present...Ollll: inll'l"c,,1 frolll a pedagogical point of vie\\'. There aretlm,'c lIIaill reasons for Ihis interest:
i) Tilc I\\,'o-particle proccsses are governed by llIlC-di-11l1'Ilsional Schrodingcr eqllalions.
¡jI Tilc me[hod providcs a rclati\'e simple introdllction toIhe s()llIlion of Illany-dimensional problcllls.
iii) 11 ¡llIows iI pedagogical dcscriplion of Ilumerical me-II10d....used in quanlul1lmecilanics.
Indeed. :-'lIi.."h¡\Il illteresl is corrobor;'lIl'd by the anicleswhich haV\.' ;Ippcarcd trom timc to lime in cducational jom-nals 11-1:11.
NUllllo'riGI! Illclhods to solve one-dimensional eigenvalllcproblcllls call be classiflcd ilHO three types: nUll1crical inle-~ratiolls (or shouting Illclhods) [13.9,1-1), llIatrix or globalarrroachl's [11,"¡.15}. and Rayleigh-Rilz varialional me.thods 17, ((¡-IR]. An approximale Raylcigll-Rilz variationalIIlcthod lhal ha:,> dClllollslraled high cffkicllCY ror solving
une- amllllany-dilllcnsional Scluüdinger equations is the dis-cn~te variable reprcsenlalion (DVR) melhod [ID. 20). Thismethod was originally introdllccd by Harris ('1"l.[21] slrictlyrOl"cvalll<lling Illllt.:nlial 1ll;'ltri.\ ckmcnts in thlo: fiO's. It wasIhen by Did.inson ami CIo:Hain [22) shown lo be rclated togaussian quadralure methods throllgh a simple lInitary trans-fonnalion. Ilowl:\'er. il was only in the SO's tlwt Light andl'o-\\'orkers [2:~-2;jlgeneralized the DVR Illethod and appliedil to a nUlllber of intcresting quantulll mechanical problems.1\n advanlage of the DVR IIlclhod is tllat its computer imple-lllenlalioll is simple. This allows undergraduatc students ofqllantum mechanical aJ1(l/m compulalional phYSlCS courseslo use such a IIlclllOd as ¡¡ \\'orking 1001 to Ireal l1umericallyquanlulll sySll'llls. This ability was shown to be very impor-lant when applicd to srccitic rcscarch tielcls sllch as atomic,molecular. ;'l1ldlIIore recently snlid stale physics 12GI. Yet, no¡¡nicle has appearcd lo dalc in an cducational .intlmal ;'lboutIh~ DVR Illethod and its applications. [n the present work.\Ve ailll lo cover tllis gap by inlroducing the DVR methodfmm a pcdagogical point of vie\l •...Since lhe emphasis will be011Ihe COIllPUlatiollal lechniqlle and ils physical jllstificalionr¡¡lhlo:rthan on Ihe energy eigcnv;:lluc Ihemselves, we apply Ihemclhod lo Ihree \Vcll known protolypes in physics. First. we"olve [he radial Schrodinger cqu;ltion for a quantum particlein all harmonic POlclllial. Thlo:ll. wc use Ihe sall\c methodol-ogy to calculale the bound states of the Ro! molecule. Finally.wc calculale cncrgy spcclrum of the hydrogcn atom. The pa-pel' is organized as follo\Vs. In Sec. 2 we describe lhe general[heory orthc I)VR IJlclhod. v•.hile Sec, J discllsses the equa\lyspaccd discrcte variable reprcscntatioll. Thc applic~ltiollS arein Seco -l. and Ihe cOllclllding rCll1arks in Seco S.
FREDERICO y. PRUDENTE, ANTONIO RIGANELLL AND ANTONIO J.c. VARAN DAS 569
2.1. I{~lyl{'igh.n.itz\'nriational principie
2, J)VR mclhod
Tlle IlOIl-rdativistií,.' time-independclll radial Schrodingcrt:quation assumes in atomic units lhe form
\\'hcrc 11 is lhe reduccLl m<.lSS ofthe system, \/(1') is lile ínter-;H.:lion potential. 1 is Ihe angular momentuTn quantuJn nUIll-ht:r. J.:: is the energy cigenv~lue. and t/J(1') is lhe corrcspond-ing wavc fUllction. Within lhe spirit ofthe Rayleigh-Ritz \";:¡ri-alional principie. lhe problcm is transformed ioto that ol' find-ing (he stationary solutions of lhe functional .J[l/,j givcn hy
values and eigenwctors of the generalized eigenvalue pro-hlelll in Eq. (4). Once the hamiltonian matrix has been built.the next step \\'ill he to obtain its cigenvalues.
The integrals in Eqs,(6), (7) and (8) can seldorn be sol vedi.malytically. For eX<lmple. the POlclHial energy marrix ele-llll'nts [Eq. O)J can he obtained analytico:dly only fOI" a fewpotcntial models ami b¡ISis sets. Thll~. a numerical slralc-gy will oc nCL:e~sary 10 calculatc slIch integrals. For prac-tical reasons lhe scmi.intinitc interval 01' the radial coordi.nate ¡1. E (O, (0)] has then to be rcstricted lo a finite ran-ge [,. E lo. iJ)]. where we assume Ihal ¡j,(a) = ¡j1(iJ) = U,One efficient \Vay to evaluate one-dimcnsional integrals is bylIsing the gaussian quadrature techniquc. which will be dis-cllssed in ncxt subseclion.
\\ht:l"l" \tll(,.):: '"(1"1 T l(l + l)/'2Jl/:'! is rheeffecrive po.ll'llllallor Ihe I-th rotalional state.
To oblain nUJ11crically the eigenenergies and eigenfullc-liullS, Ihe wavc function is fjrst expanded using a finite basis,el j J¡(r)} which hclongs to a L2-space:
For ¡¡ gencral {llle-dimcnsional integral. rhe gaussian quadra-turc formula ilSSllllles lhe forl11 [27.28]
(9)
wllcre {r:i ~ •.1I.C the cxpansion coefficicnts. Then, .1[1,/)] is re-quired to be stationary under var¡ation of such coefficicnts.Thll~. lhe variarional procedure converts the problcm into that01 :-.olving a generalized eigcnvalue problem. which in matrixlIot •.llioll assumes Ihe form
sucll that the following ol1hogonality and completeness rcla-,ions are satislicd:
where lr" 1, ,,",1 lo',,} (o = 1. .. , ,1\) are the pivots andweights which define the gaussian quadraturc. respeclively.E;¡ch set of pivols •.ll1d wcights is associated to a particularset of 1\" basis fUllctions !/¡(r), Ci = l •... , I<) which areorthonormaJ in fhe ((l. fJ) interva!.
'o' '1'(1' )'1.(" )=''/'(,. )wr/2wrn'J(")L- <l.' ". I n L-. I n o o.) ••0=1 <>=1
i,j=1,'" ,1\, (11)
o./I=I, .. ',K. (12)
are kinetic energy matrix (T) elements.
(7)
It is important to point out that there is a unique set of pointsand weights which satisfy these relations for each particularset 01' orthogonal polynomials {!Ji}. Using the Dirae nota-tion [29)
.,re averlap rnatrix (5) elements. In otiler words, the Ray-1cigh.H.ilz variational procedure implies obtaining the eigen-
¡Ife potcntial cncrgy malrix (V) elements. and
lS}', = 1= dI' 1;(1')1,(1')(J
(8)
(rlh) = h(r),
(hlr) = h'(,,),
(h,lh2) = / dr h;(r)h2(,'),
( 13)
( 14)
( 15)
Re.', Mex, fú. 47 (6) (2(X)I) 56X-)75
57U 1111;I>lSCRETE VARIAnLE REPRESENTATlO:'>l METlIOD FOR nOUND ST,\TE EIGENVALUES AND EI(jENFUNC"TIO;\lS
\"'here 11(1') i~.lI1 arbitrary fUl1ction, we can rewrite Eq. (11);lIld Eq. (12) illlhc 1'01"111:
2,3, Discn'te variahle reprcsentatioll llu'1hod
where (in arc Ihe col'!"ticients \\,:hich depcnds on the particularquadraturechoice Isee Eq, (29)]. and
\VL' disCllSS llL'xl lhL' gencral DVR lllL'lhod starling from lhegaussiall quadraturl' IcchnilJuc prcsl'lllL'd in thc previous scc-lion. Considcr ;1 particular gaussian quadratllre with pi-\ols {",,} and WL'ighls ~W"}: (1 = 1,. . h-. The DVRIllclhod consi",ls 01 huilding a ba",is set which satisncs thenmdilions
Usin!; sucll a basis lo l'-'pand thL' waH." ftlllClion according loEq. l3). UIlC obtain ...•
'" "h'Olll Eq. (16) <lnd (17) we can definc the unit operator inIhL' coordinale repreSL'IlI<llion ami the projeclioll opcratnr forlllL' l ,'1, } spal".'e. re ...•peclin~ly gi\'cll by
1,'
" 1,. )'" (,. I = 1.L" ,,(\,,== J
;llld
f~,(j = l.' ,J<.
h "_ " ,'Ir )"(1')=" ",,1,,(")= L -'-"-1,,(1').L (/
,1 1 ,,-1"
(24)
('26)
Equillions ( 1 I ) .llld ( 12) [or Eqs. (1 S) ami ( 19)] sho\l,. Ihen>In re that Ihe inlinile ;lne! conlinllous feprcsentalion 01' Ihe..•paCl" has hl'L'1l ;lppI"Il.\imated by a linite and discretc n:pn:-:--L'lllallOIl.
11is I"."as)"Iu IWIL'Ihat Eqs. (11) amI (12) ddine :111orlho-~{)I1;dIrallsforll1alion. ¡.C'..
whcre thl' rel;¡lioll bctwCL'1l lhe cucflicicllts {("(}} <lnd Ihl' \'a-lues of lhc W;¡\'l' fUllctioll on Ihe pivols quadrature {I;"(",,)}is easi1y nhlained dlJing ,. = r in Fq. (26) and employ-ing E'l. (2"'). NIlIL' l!lat Ihe \\;l\L' fUIlClioll al the quadralUrcpoi nI is dirL'clly e\alllated frOI11the cocfticients of the cxpan-sion ohtained hy soh.ing lhe generali/úl eigen\'allle problem.Eq. 1~1.
"¿ 1.'I,)(fl,1 = 1i== J
(1 ~)
( 20 I
To ohlaill\VI.." expand{!!,} (i = 1.
lhe hasis function sel {.{,.} ((\ = l .... , /\')il III Icrnb of lile known funclions
./\') wllich ",atisf)" Eqs.lIO)-( 12)
..•tldl thal
lJ J U = UU 1 = ,/,.1,
f.,(') = (,II,,) '" ¿("1.'1,)('1, 11,,).i=l
(271
\l,hel"l:"1/\ is 1111".' unity matrix of J\' x J\' dimension.ln particll-Idr. 1)id:insllll ilnd CL'rtain [22] dCllIollslrated thal this ortho-~onal transforlllalioll diagonalizes tlle pOlential cnergy matrix(in ~L'neral. allY 1l1alri\ representalion 01"the coordinall' 0PL'-l'atorl \\ilh tlK' di;¡~on;ll ~1~ll1ents hL'ing ~i\'l:11 by lhe value nIIhL' pOlellljal enl"Tgy at ~¡¡ch qlladr.lIl1rc poinl. Thus. lhL' Sl't 01gatl:--.••iall qlladr:lIl1rl" poinls associaled wilh Ihe {y,} functionsl\1;lYhe' I;lkl'n as Ihe L'igenvalues of ¡¡ lIlatrix (X) witll J\' X /\.djlllensioll Ihc elt:lllcnls of which are dclined by
1X f" = .r d,. yi(r),.y,(,,) = (y,lél.'!). 1221
\\ hen: /. is tlll' Il():--ition operalOr. 1100\'L'\'Cr. the gaus:--ianljuadr;l1url"." plllllb ;111"." lr;lditionally IhL' rools of thc .ti" . 1\ 1" I
Illlhonol'm;d IUIll"lion in Ihe interv;¡1 (1/. IJ). In holh casL'",. 7.ll"
\\ hCl"e Eq. (2"') ha", also been used. The COllstants{(/,,} (o = 1. .... /\') can now hl." GllclIlated slraightfor-wardly from Eq. (25):
~ "j . .r.~(,.)f,("1 ,¡,.=¿ <I'.,/,;(",)f,'(',)=II',,";,"J6,,". " I
(.'9)
UC'\'. Me.\". "'ú. .t7 (h) (20tJl) 5()X-575
FREDERICO V. PRUDENTE. ANTONIO RIGANELLI. AND ANTÓNIO J C. VARANDAS 571
SlIbstituling E'1. (28) in E'1. (27) with {ao} given by E'1. (29),Ihe DVR basis funetion sel {Jn} is given by
Thus, th~ DVR Illcthod COllsists 01'expanding lhe wavcfunc-tion lIsing lhe basis set in Eq. (24) and cvaluating lile appro-prialC integrals by gaussian quadraturc.
The intcgrals which are necess;:lry to evaluate in arder lo"pply Ihe DVR proeedure are shown in E'1s. (6)-(8). The po"tClllial cnergy matrix clements are givcn by
1,
j,,(r) = I:.;w:: !J;(ro)!J.("),i=J
o = 1. .... h". (30)
3, ElIually spaccd DVR
\Ve illustrate 110W 110\.1,'to COIlSlrllct the DVR hasis I"unc-liollS (;lllJ Ihe respeclive malrices) for a particular Case: lheequ;¡lly spaced qlladralure. which is Ihe simplest choice of amesh {1!J.:t1.:!O]. Tbis procedllrc will then be employed inth(: Ilext section 10 calculate Ihe cncrgy speelrum 01"fhe threereprcscntalivc systellls.
The set 01' Ihe K orthonormal funclions on. the ((l, b) in-ten'.tl (rhe DVR primitivc funetians) that generate an equallyspaced C)lIadralllre is simply Ihe set of paJ1icle-in-a-box fune-lions [1)
In Ibis casc, Ihe quadrature points are determined by Ihe Ira-dilional 1'01"111.¡.c.. lhey are the zeros nI' the 9/\+1(1') eigen-fllnction [:34]
where S 1\' + l. Now. lIsing Eq. (23), the qu;¡dratllreweiglll ....arc giVL~1lby
\Vilb Ihis choie\.:' 01"Ihe J)VR primitive fllllctions, Ihe DvRbasis set in Eq. (0) b\.:'comes straightforwardly
(34)
OS)
(1 = 1,'" ,K,
b-a11'" == IIJ = ----¡;-.
(31)
I ¡." d'{TI"" = -" d,. J~(")-,.,J'j(")
4./' . 11 ( /
v.'hcrc we llave cmployed Eqs. (24) and (29). Similarly. lhe)...¡neliccnergy matrix elements are built as
\\,:hcrewc have utilizcd lhe detinilioll oflhe DVR basis fUllc.¡ion in Eq. (30). Note rhat if one chooscs as DVR primitivel"unclions {.fl¡} any knO\\'n sel of orthononnal polynomi;¡ls.
lhe integral .J::' d1" .'J;(r) (Pyj(r)/dr'l can be oblaincd analy-lically (see Scc. 3). The overlar rnatrix elements defincd inEq. (8) are calclllatcd Jirectly by making use of the propertygi\'cn in Eq. (25). \Vilh the results of Eqs. (31) and (32), \vehi~hlight Ihe fact that the poten ti al cnergy matrix is diagonal;tIId lhe kinelic encrgy malrix can be evaluated analyticallywhell we utilizc Ihe DvR methoJ witll ;:¡ particular choice of1)Vf{ primilive fllnctiolls!Ji (see Seco 3). Note that the quadra-IIJn..' poinls and wcights of the DvR basis functions {in}dcpcnd direclly uron Ihe choice of the set of OvR primi-lin.' functions {f/¡}. In the literature. Ihere is a grcat nlll11-her of DVR primitivc functions which have been utilizcd logL'llcrate Ihe DvR basis functions (sec. for examplc, Ref. 30rOl" ;¡ discussiOll1. including nurncric¡¡Jly oplimized eigcn-fUllclions [31,3:2j. In lhe nex[ section. we preselll Ihe DvRUI(:t1JOdrOl" rhe case of an equally spaced gaussian C)uadra-ItlIe.
Tlle Slllll over £ in Ec¡. (37) can be evaluated analytically. Pol-lowing the steps poinled out in Ref. 33, lhe kinetic energy
Figure 1a shO\\is JI (,.) rol' the case 01' I{ = 9. and Fig. Ihshows lhe complete .1",,(,,) set ((1 = l ..... 9). Note lhalrhe Kronccker dt'lta property of Eq. (24) bceomes clea •. fmmFig. J.
SubstillHing Ihe J)VR primitivo funelions {g,} IE'1. (.13)J.in Eq. (2), we oblajn lhc kinetie energy matrix elemcnts:
572 TIIE DISCRETE VARIABLE REPRESENTATION METHOD FOR "aUND STATE EIGENVALllES AND EIGENFUNCTIONS
lllalrix elelllLllls beco me
lIsed the rsg subroutinc of the ElSPACK numerieal li-brary r;~;)j,As \Ve will sec in the Ilcxt scetion the biggestca1culatioll \Ve pcrfol"med implies iI matrix of dimcnsion!)OO x !)OO. Even in Ihis case the ca1cuialions may be earricdoul on a desktop computer within a fc\\' minutes.
TABI.L 1. Exacl allu calculaled "'¡lh [)VR cigcnvalucs 01' lhe har-1ll00lil' o~l'i lIalor 1m dlllcrclII 111llnhcr 01 ha"],, ~~L Thc colulllnmarkct.J wi¡h (a) are lhe rc:-.ult..•from Eg. (41 J.
As poinled out in Ihe illtroduction we llave considered threeexamples to illustrare Ihe prcsent mcthodologyll.
for off-diagonal clclllcnts (o "# {J), and ..1.1. The harmonit.' nscillator
ror the diagonal ones, \Ve remind that the potential energymalrix i:-.di •.lgonal \vith the eJcmcnts being givcll by Eq. (31).¡\ step by stcp algorithm to implcment the describcd proce-dure is givcn in appendix.. The CPU time rcquired to run theprogr<.llll depends mainly by the size of the hamiltonian ma-trix to be diagonalized since the lime spent lo asselllblc thelllatrix ami other operations is negligible cOlllpared to that.Among lhe variolls available tcehniques to this end, we have
wherc :.v is the angular frequency (w = Jk7I"t). Table 1gathers the ealculatcd valllcs Ilumerically t1sing DVR and theanalytic results, adopting different numbt.:1' uf basis fUTlct;ons;the calculations wcre done laking k = Jl = 1, and the range
rol' Illilny Illodel ca!clil;¡lions in physics, the harmonic oscil-lator potential has provcd to be very use fui [12]. For this easethe potenlial energy f[lnetion in Eq, (1) is \f = 1/2kx"l where:r = ,. - "','q and rl'q ;:-, the diswllce 01' the Illinimum 01' thepUlential curve. The ilnalytic solutiolls are
(41 )/1. = 0, 1,2,'
(40)= 1 1 ,,' [2N' + 1 1 121'(1'-")' 2 3 ,("O)
SlIl N
Rt"'. Mex. FiJ. 47 (6) (2IX}I) 568-575
fREDERICO y. PRUDENTE. ANTONIO RIGANELLI. AND ANTÓN 10 J C. VARANDAS
the following analytic 1'01'111[1]
573
.J.3. The h)'drogen atom
This Icads lO a total 01'17 vibrational bound siates for ground-stale H:!, NUlllcrical c<llculations of all such states wereperforllled elllploying Ihe DVR Illelhod using 500 equallyspaced DVR basis fUl1ctions for different integration ranges.The results are in Table lB and show c1early thal lhe discre-pancy bctwecn lhe calculated and exact results which oc-curs for lhe last eigcnvalues tends to vanish by increasingthe integration interval. This lllay be rationalized from the1'aetthal lhe integration region has been restricted to a finiterange (a. b). and a small interval truneates the excited eigen-functioll~ al regions whcn: they did nol yet vanish.
The hydrogcn atom problcm is lhe only one túHy resolvedanalytically in qualllulll I11cehanics [2]. It is <lIso presentedas the simplest electronic structure theory calculation.ln thiscase \ '(r) = -1/r (Coulomb potential in atomic units). Thean.llytic cigenvalue~ are given by
for .1' was -lOan :S :t :S 1() ito' Table 1shows how lhe calcu-I¡¡tiolls. ror a given rilngc uf :17. are sensitive lo Ihe Ilumher ofuscd oasis fUl1ctions. In particular. for J( = 20, the DVR rc-sults silo\\' discrepancics slarting al 11= 3. On lhe other !land.rOl" 1\" = .to. ol1ly lhe highest values show significant discre-pancics. Finally, ror 1\' = GO. lhe calculated DVR eigenvaJuesare in perfect agreemcnl \vith lhe exact ones.
\Ve no\\' I:.Icklc lhe problcm of calculating of bound statesrOl" a diatolllic Illolecuk described by a realistic potential. Wechoosc as test case lhe H.! l1lolecule for which we employ lheMorse potential [3GJ dclined by
\ '(1') = D{ 1 - "xI' [ - w(r - 1',.,,)]}", (42)
\\here J) i:-. lhe dissoci.llioll cnergy of lhe Illolecule. w is<1parall1elCr and, ""q lhe equilibriu111 interatomic distance:Tahle 11galhers lhe l1ulll\:rical values of lhe involved paramc-ters. For this case. lhe vibralional (l ::::::O) eigenvalucs assumc
I
( l)fifW" ( 1)!W2
E =D "'+- -- 11+- -...tn 2 D ji 2 2Jl'
fOf /1 = (l. 1. .. '. lindel' lhe condition
J2¡ID 1,,< --- --.w 2
11 = 1,2,3, ...
(43)
(44)
(45)
TAHU: 111.The vihr¡llionill ~la{CSal' lhe H2 molcculc. Comparison oclwcen the exact eigcnvalucs ¡¡nJ lhe DVR calculalions for differenlr¡¡dial illlcgralioll rangcs. Thc numbcrs in parcnthcscs are lhe illlcrvals (a,h). Encrgies are in rm-I ano Jislances in éltl. Columns wilh (a)are lhc unhound cigcllslatcs rO( (his inlegralioll rangc anJ (.'olumn markcd with (l.) arc lhc n:sults !"mmE4. (43).
Table IV shows the calculated values for lhe first six k'vels.The optillllllll numher of basis functions 90tJ. rangc 01' ,. inthis case: lJ ~ ,. ::; ,jU flllo Thcse llulIlbers are Illuch largerIhan Ihe ones used in Ihe harmonic oscillator and H:.!exalll-plt: ...•.This is tille \O lhe very long range tail of this potentialwhich requirc!'>a I••rger space rangc and Ilumber of basis fUllc-¡iuns. A \Va)' lo cope \Vith this difllcu1ty would be lo cmployil Illore suilahlc: DVR basis set al' fUJ1ctions but lhis problemwill 110( be addrcssed here: sce Ref. 20 for a discussion.
5. C()l1dLldil1~rcmarks
\Ve han." prcscntcd a pedagogically simple and accuratc al.~(lrithll1 lO calculatc the bound-state energies alld corres-p()nding wan:functions of the one-dimcnsional Schrodinger('quation. nalllcly the DVR mcthod. \Vithout greal diftl-clllly. the Illclh(ld can be extended to salve lTlulti-dimensionalprubklll:--. Indccd. it is currcntly (lile 01' the most powerfulIllt:thndolo~ics in molecular and chcmical physics to COI1l-
pUle rhe rnvihralional energy levels 01' slTlall polyntolllic111p!t:.'cule:--(Rc!'s. J2, .,7-J'.>. :\lld rcferences therein). f\:-- a¡.;omplllt:T e.\perilllcnt. it also :1IIo••••'s Ihe srudent lO solvc IheSchrodinger eqllation hy means of a 11111llerical techniqucwhich is strai~htforward to implement. Moreover. it allo\Vs11Illdamelllal l'f'llcepts sllch as Ihe variational principie. hasistUllction e.\pansion:--. ami gaussian <juadralure integration tohL' inlroduccd in lllldcrgradllate qu:mtulll mechanical or COI1l.
putational physics courses.
lhe rde\':llH H >RTRAt\ l..'oJes wilh eX:llllple applic:atiulIs ;lre;l\'ail<lhlc trolll tll~ l..'orrcsponJIIlg. :lulhor upon re4UCSI.
L. P<ll1lin~ ¡1I1dE.B. \Vilson. IntmdllClúm lo Quanll1l1l A/rdulI/.1("\. l.\k(,raw-lIill. Ncv,' York. II)J5).
l. U. Btll\';i:--<llId J.i\1. lIuwctt..1. 0/('11I. I~'''I/c.ft(1(19X3l "207.
1, e l\.lIh¡lCh. I ("/¡¡'III. 1:'£1111". 611 (19X3) 212.
Thc i1uthors thank Dr. J.J. Soares Neto and Dr. L.S. Costa(Ulliversidadc de Ilrasílial ro.- hclpful discussions. This workhas Ihe suppon oí" Funda\üo para a Ciéncia e Tecnologia. Por-tugal. under programme PRAXIS XXI. lt has also henefiredfrom an EC grantunder conlract # CHRX-CT '.> •.•. 0..•36. r-VPacknowledges Ihe I1n<lncial •.lid through partial granr fromf'unda<;üo Coordcna<;ao dt' Apt'rfei<;oamento de Pessoal deNível Superior (CAPES. Bra/.il).
Input: K (numoer uf basis funclion). a,b (interval for r).mu (rcduced Illass)OUlput: Eigt'l1~I;¡le and codlicicllls uf Eq. (26)STEP l. Deline lile pOlential funclion f'OT(R)STEP 2. Create lile arrays fOl T(K.K). V(K.K). H(K.K).E(K). eOEF( K.K)STEP 3. Initialize the arraysSTEP ..•. Build the pivot vector: For alpha= l.K EvaluateX(alphal using Eq.(3"')STEP 5. Build the hamiltonian rnatrix H=T +Y:
For alpila= I.KFor ilela= 1. KEvaluate '[(alpila.beta) using Eqs. (39H40)Evaluate V(alpha.betalll~in!! Eq. n II BlIild the h.amihonianH(alpila.ileta I=TI alpila.bela 1+ VI alpila.bela ISTEP 6. CALL rsg to diagonalizc HIK.K)
STEf' 7. For alpila= l.KOUlPUI E(alpila)ror beta:::::: I.KOulput COEFlalpila.beta)
STOP
j. 1). Seark~ ami E.1. Nat!y-Fetsoouki. Am . .1. I'!I".\'. 5(. (t t)SS)
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