A =A +VA, A =d A, A =A (r r'). Within the Coulomb gauge, the transverse vector potential associated to spatially uniform, time-independent

Transcription

1 PHYSICAL REVIEW A VOLUME 45, NUMBER 9 1 MAY 199 Calculatio of molcular magtic proprtis withi th Ladau gaug M. B.Frraro ad T. E. Hrr Dpartamto d F&sica, Facultad d Cicias Exactas y Naturals, Uivrsidad d Buos Airs, 148 Ciudad Uivrsitaria, I'abllo I, Buos Airs, Argtia P. Lartti, M. Malagoli, ad R. Zaasi Dipartimto di Chimica dll Uivrsita dgli Studi di Moda, Via Campi 183, 41100Moda, Italy (Rcivd 8 Octobr 1991) A thortical procdur for calculatig magtic suscptibility ad uclar magtic shildigs i molculs i th prsc of a spatially uiform tim-idpdt magtic fild, withi th Ladau gaug for th vctor pottial, is dscribd. Sum ruls for charg ad currt cosrvatio ad gaug ivariac hav b drivd. A computatioal schm basd o th radom-phas approximatio has b usd to calculat magtic suscptibility ad shildig tsors i th watr molcul, adoptig fairly flxibl Gaussia basis sts of larg siz. Th rsults show that vry accurat molcular wav fuctios ar dd to obtai paramagtic cotributios to th magtic suscptibility of th sam quality as thos obtaiabl withi th Coulomb gaug for th vctor pottial. Th thortical uclar magtic shildig tsors i th Ladau gaug ar charactrizd by th sam quality as thos i th Coulomb gaug. PACS umbr{s): s, 35.0.My, 75.0.Ck I. INTRODUCTION Thortical dtrmiatios of magtic proprtis of molculs i th prsc of a tim-idpdt ad spatially uiform magtic fild ar usually carrid out withi th Coulomb prscriptio [1] for th gaug of vctor pottial A. Accordigly, th molcular Bor- Opphimr Hamiltoia is writt i trms of a divrgclss A (r)=, 'B Xr. Th magtic fild B=V X A is ivariat udr a gaug trasformatio [1,], Ac A+VA whr A(r) is a arbitrary fuctio of th positio coordiat. Udr th chag of gaug, th wav fuctio P ad th Hamiltoia h of a particl with charg q udrgo corrspodig gaug trasformatios []: g~g'= g xp lq A Pic h ~h'=xp A h xp A Pic Ac lavig ivariat th Schrodigr quatio h g=sp of th particl. Hc, molcular magtic proprtis valuatd via quatum-mchaical approachs must b gaug idpdt, which is a fudamtal physical rquirmt. Whvr approximat mthods ar adoptd, th dgr to which th costraits for gaug ivariac ar satisfid provids a yardstick of accuracy for th computatioal schm ad a fudamtal tst of quality for th molcular wav fuctio. I particular, if th algbraic approximatio is usd, th sstial faturs of a basis st, i.., Lq optimum siz, flxibility, dgr of compltss, ad ability to dscrib th magtic prturbatio, ca b asily judgd by aalyzig appropriat sum ruls [3,4]. I practic, th gaug trasformatios that hav b studid so far amout mrly to a chag of origi of th coordiat systm r'~r" =r'+ d, whr d is a arbitrary vctor, i.., to a limitd class of fuctios A, such that A =A +VA, A =d A, A =A (r r'). Much mor gral choics for th gaug fuctio A ca, of cours, b mad. A particularly itrstig gaug trasformatio of vctor pottial has b dvisd by Ladau [1,,5], ladig to a quit simpl form for th Hamiltoia. Th prst papr sts out to ivstigat th suitability of th Ladau gaug as a altrativ choic with rspct to th customary Coulomb's gaug for calculatig molcular magtic proprtis. A sstial aim is also that of obtaiig a st of sum ruls for gaug ivariac of calculatd magtic proprtis which may srv as idpdt a priori critria for th accuracy of molcular wav fuctios, i.., wh compariso with xprimtal data is difficult. To this d, a sris of basis sts of Gaussia fuctios hav b cosidrd for xtdd studis o th watr molcul. Prturbd coupld Hartr-Fock (CHF) ad radom-phas-approximatio (RPA) mthods implmtd i fficit computr programs [6 8] hav b mployd to calculat magtic suscptibility ad uclar magtic shildig tsors. II. MAGNETIC SUSCEPTIBILITY WITHIN THE LANDAU GAUGE Withi th Coulomb gaug, th trasvrs vctor pottial associatd to spatially uiform, tim-idpdt Oc 199 Th Amrica Physical Socity

2 45 CALCULATION OF MOLECULAR MAGNETIC PROPERTIES magtic fild 8 is, usig tsor otatio [1], C Aa = EaPyBP1y (sum ovr rpatd Grk idics is implid throughout this papr). Th Ladau gaug [1,] is obtaid itroducig th gaug fuctio A=, '(B yz+8 zx+b,xy) ad th gaug trasformatio (lavig B~= & V&A ~ ivariat) A". = A. '+V.x=a~r,, (3) whr a, P, ad y ar a v cyclic prmutatio of x, y, ad z. Accordig to Eq. (3), also, th Ladau vctor pottial is divrgclss, i.., V.A". =0. (4) Th Hamiltoia withi th Coulomb gaug of a lctro, with mass m charg, positio r, ad liar momtum p, ' 8 h = p + A udr th gaug trasformatio whr h 1 m~ ' m a a (3) bcoms hp+hg (6) h&'= A p = (8 zp +B,xp~+B yp, ), m, c m, c (By+Bz+Bx ) (9) Lt us ow cosidr a molcul with lctros ad N ucli. W dot by r;, p;, ad /; =r, Xp,, th positio ad liar ad agular momtum of th ith lctro ad by RI ad ZI, th positio ad charg of th Ith u- clus. Th uprturbd Bor-Opphimr Hamiltoia of th lctros is p. Ho= g m N ' g Z, /r, R, / +' g fr, r, [ jai ~ N N + g Q ZI+J IRr I=1 J=1 JAI (10) with igfuctios ~j). I th prsc of th magtic fild 8, th first- ad scod-ordr Hamiltoias withi th Ladau gaug bcom H = g(a p);, mc I HM z B gy; +B gz; +8, g x,. mc so that th total Hamiltoia is Hg =Hp+H~ +, 'H~ (13) I th rfrc stat ~a ) th diamagtic to th magtic suscptibility tsor [9] is (a [ 'H~~)a )=- a' xx g8 C mc cotributio gyz a (14) Th othr tsor compots ar obtaid by cyclic prmutatio of th idics x, y, ad z. It is worthy of otic that th off-diagoal compots of th diamagtic cotributio ar vaishig for ay arbitrary coordiat systm withi th Ladau gaug. Th paramagtic cotributios to th magtic suscptibility tsor [9] ar obtaid from prturbatio thory. Thy ar ' R a g(yp, ); j j g(yp, ), a m C R j(aa) ~ja i =1 8 y~y= g R a g(yp),. j j g(zp ); a C '5 j(xa) ~ja i =1 (15) with cyclic prmutatio of th idics for th othr tsor compots. I ths formulas %co., =E. E, is th trasitio rgy btw th ~j) xcitd stat ad th ~a ) rfrc stat. Th paramagtic cotributio is a symmtric tsor, as i th Coulomb gaug. Whras th diamagtic trm (14) is always diagoal, th paramagtic trm (15) is diagoal oly withi its pricipal axis systm. As a cosquc of th fact that th diamagtic cotributio (14) is diagoal, th off-diagoal compots of th suscptibility tsor ar paramagtic, ad, quit rmarkably, thy ar origi idpdt withi th Ladau

3 674 FERRARO, HERR, LAZZERETTI, MALAGOLI, AND ZANASI 45 gaug. From Eq. (15) o ca raliz that paramagtic suscptibilitis i th Ladau gaug ar simplr to calculat tha i th Coulomb's gaug wh a Gaussia basis st is usd. I fact, opratig with th caoical momtum p= iav o a giv Gaussia, i.., diffrtiatig, givs ris to a liar combiatio of two Gaussias, which is vry asy to hadl, vry typ of itgral apparig i Eq. (15) big vtually rducibl to a sum of ovrlap itgrals. whr III. NUCLEAR MAGNETIC SHIELDING IN THE LANDAU GAUGE Lt us suppos that uclus I carris a itrisic magtic dipol pi. Th vctor pottial actig o lctro i du to such a uclar dipol is p I 1 A i a &apy PIPEIy ~ Rr (16) fr, R, f' (17) is th lctric fild of lctro i o uclus I. I th prsc of th uclar magtic dipol two xtra trms tr Hamiltoia (13): ad m, c (A 'A ),, whr th oprator for th magtic fild of lctros o uclus Ii th absc of a xtral magtic fild is 1 ~Ia MIa& MIa ~apr g Elp piy m, c (0) Withi th Ladau gaug for th vctor pottial, th diamagtic cotributio to th uclar magtic shildig [10] of uclus I bcoms a api-bb m a a gyei odi~= (a fh "' fa) Bp BB mc a gyes (1) Othr ovaishig compots ar obtaid by cyclic prmutatio of th idics x, y, ad z. It is rmarkabl that, for ay coordiat systm, d d d O ~xy yz zx () i.., th diamagtic cotributio to th magtic shildig tsor of uclus Ii th Ladau gaug is rprstd by a scod-rak asymmtric tsor with six ovaishig compots i th absc of symmtry. Th paramagtic cotributio to th magtic shildig [10] of uclus I is pix xx pil ~xy mc A j (~a) ja mc A j (Wa) R (a fmi fj ) R ja X j g(yp);a (a I~I" Ij& X j g(zp); a (3) This cotributio is a asymmtric tsor with i idpdt compots i th absc of molcular symmtry. Similarly to th cas of magtic suscptibility, th off-diagoal compots O. x, O. yz ad o, ar fully paramagtic ad origi idpdt. ~df ap dic ~ap IV. CONNECTIONS BETWEEN MAGNETIC PROPERTIES IN LANDAU AND COULOMB GAUGES Withi th Coulomb gaug th diamagtic cotributios to th magtic suscptibility ad to th uclar magtic shildig tsors ar [3,4] 4mC mc a g(r 5p rrp}; a (4} a g (r; EI 5 p r, aeip) a. (5) Thrfor, o ca asily fid, from Eqs. (14) ad (1), +d C' ] +dx dic i ( d+ d ) +xx +xx +yy dic 1 d yx yx ~ ~ ~ aa dic~ d +aa (6) I ordr to obtai th rlatioships btw paramagtic cotributios withi th diffrt gaugs, w cosidr th idtity ' g (xp };=il, +, g (xp +yp ),, L, = g I,, (7)

4 45 CALCULATION OF MOLECULAR MAGNETIC PROPERTIES ad th commutator Ho, g (xy),. ia g (xp +yp ),. 7?l For th off-diagoal matrix lmts, o gts a g (xp ), j =, '(a)-l, j~& ai,, a g(xy);j (g) (9) By cyclic prmutatio of x, y ad z, th formulas for all tsor compots ar obtaid. From th dfiitios of th paramagtic cotributios to th magtic proprtis withi th Coulomb gaug [3,4], pic 4' C '5 J(~ ) jg mc ad usig formulas R(&a~L j~&(j~lp~a&), X y R(&a~MI" ~j &&j~lg~a &), ) (~a) ~J~ (9), o gts X' =X' +-'(X' 3X'") xy =+xy + px pc ~ dc C Xxy ++xy =+xy px~ dx ~ C pc~ dc ~ap ~+ap =&ap +ap=~ap++ap opic opip+ i ( d dij' ) ~xx Oxx Oyy ~ IX pix pic + dic IC' +xy = ~xy ~xy ~xy = ~xy pix pic dic yx yx yx ~pix+ dij ~IJ ~IC ~pic + ~dic +ap +ap = +ap +ap = +ap +ap (30) (31) (3) (33} It may b usful to rcall that i ay calculatio basd o th algbraic approximatio, Eqs. (6) ar idtically satisfid for ay basis st. O th othr had, Eqs. (3) ad (33), coctig th paramagtic cotributios withi diffrt gaugs for th vctor pottial, ar obyd if ad oly if th off-diagoal rlatios (9) ar satisfid, i.., if th hyprvirial thorm for th scod momt oprator holds [11]. For xampl, Eqs. (3) ad (33) ar valid for th xact ~a & ad ~j & Hartr-Fock igstats [11]. If a fiit basis st is mployd accordig to th slf-cosistt-fild schm, th dgr to w»ch Eqs. (3) ad (33) ar fulfilld givs a masur of accuracy of th calculatio, i.., iformatio o th quality of th basis st ad its dgr of compltss with rspct to th oprators ivolvd. I particular, total magtic proprtis ar ivariat udr th gaug trasformatio (3), [s th last li of Eqs. (3) ad (33)) oly if th basis st is complt. V. ORIGIN DEPENDENCE OF THE MAGNETIC PROPERTIES IN THE LANDAU GAUGE Th thortical tota1 magtic proprtis must to b idpdt of th origi of th coordiat systm [9,10], i.., th variatio of th paramagtic cotributios should xactly cacl th corrspodig variatio of th diamagtic cotributios. A arbitrary shift d of origi r'~r" =r'+d (34) ca b thought of as a gaug trasformatio of th Ladau vctor pottial whr (35} X =B d, x +B,d y +B d»z. (36) Accordigly, o fids for th magtic suscptibility, =y, (r')+ a g (y; y') a y», ( r" ) = y» ( r' ) y»» (r" ) =y» (r') [ ( T P,),d» (P P,), d ],??l C Al C [ ( T,P ),d, +(T,P,),d (P P ), d d, ], ad for th magtic shildig of uclus I, whr o (r")=o" (r') (a ie" ia &d C odi~(r")=o, ~di(r') +', &a~zi" ~a &d,, mc P =gp,, Al C m c Ei =QEJ (37) (38)

5 676 FERRARO, HERR, LAZZERETTI, MALAGOLI, AND ZANASI 45 ad (T P ) I 1 j (+g) ja 1 j (Xa) j+ R(&.IP. IJ &&~IP,I.&), r XR a g [(x x')p ], j (39) x&j IP, Ia& (40) 1 (Ml~ Pp) i= X R(&MIMI" ji&&j IPpI~ &) j (Aa) ja Th coditios X p(r" }=X p(r'), o (r" ) =o (r') (41) for ivariac of thortical magtic proprtis udr a coordiat trasformatio imply that (P,Pp), =m, 5 p, (4) (P T ),=m, a g (y, y') a (P T),=(T P,) i=0, m (P,Mlp) i= pr&~ I&lr Ia &. (43) (44) Ths quatios ar dirctly obtaid from (14), (15), (1), (3), (37), ad (38). Thy ar also dirctly obtaid from th dfiitios (39) (41) usig simpl off-diagoal hyprvirial rlatios [4]. Equatios (4) ad (44) ar th sam as thos obtaid withi th Coulomb gaug [3,4]; Eq. (43) is w. Th hyprvirial coditios udr which (4) (44) ar satisfid ar similar to thos aalyzd bfor i th cas of Eqs. (3) ad (33). I particular, thy ar obyd by xact Hartr-Fock wav fuctios [11]. Th costraits for i variac ar vry gral quatum-mchaical rlatioships for istac, Eq. (4) is th Thomas-Rich-Kuh sum rul [1] withi th dipol vlocity formalism. Thy also rstat commutatio formulas, coditios for currt cosrvatio, hyprvirial thorms, tc. [3,4, 11]. VI. OUTLINES OF CALCULATION A sris of Gaussia basis sts of icrasig quality has b xamid i th prst study. Th mai faturs of ths basis sts ar availabl i Tabl I. Basis st I is obtaid from th 6-31G basis [13],polarizd accordig to a rcip prviously discussd [6,7]. Basis st II, tak from Rf. [14], is spcially dsigd to calculat ar- Hartr-Fock dipol polarizabilitis i th lgth gaug. It has b xamid hr i viw of th fact that, accordig to Eqs. (15), a good rprstatio of th dipol lgth oprator is also dd to gt accurat Ladau paramagtic suscptibilitis. Basis st III is costructd from th (lls7p/5s) substrat from va Duijvldt's tabls [15], by addig two sts of 3d fuctios with xpots ad o oxyg ad o st of p fuctios with xpot 1.16 to hydrog. Basis st IV adopts th (13sgp/8s) substrat from th sam tabls [15];th xpots for th 3d fuctios o oxyg ar.51691, , ad ; th xpots for th p fuctios o hydrog ar ad I basis st V th sam (13s8p/8s) substrat has b augmtd by two di6'us s fuctios o oxyg, with xpots ad 30666, to improv th dscriptio of th tail rgios of th molcular domai. Ths fuctios ar xpctd to yild sigificat cotributios to th trasitio matrix of th dipol lgth oprator. Th 3d xpots for oxyg ar 4.0, , , ad 0.1; th p xpots for hydrog ar 1.5, 0.4, ad 0.1. Basis st VI is th sam as V, with o st of 4f o oxyg, with xpot 0.41 ad o st of 3d fuctios o hydrog, with xpot Th ovrall quality of th Gaussia basis sts ca b judgd from th slf-cosistt-fild (SCF) rgis rportd i Tabl I. Supplmtary iformatio ca b obtaid from Rf. [8], whr wav fuctio V has b usd to calculat a umbr of proprtis. Th scodordr magtic tsors ad sum rul (43) hav b calculatd via th sysmo suit of computr programs [6 8], implmtig a RPA sctio dsigd for th Ladau gaug. From th rsults displayd i Tabl II, o ca s that th dipol-vlocity Thomas-Rich-Kuh sum rul (4) is satisfid almost xactly by basis sts V ad VI, quit good TABLE I. Spcificatio of basis sts ad SCF rgy. GTO stads for Gaussia-typ orbitals; CGTO for cotractd Gaussia-typ orbitals. Basis st I II III IV V VI Cotractio GTO (13s13p4d /4s4p) (10s6p4d /6s4p) (11s7pd /5s 1p) ( 13s8p 3d /Ss p) (15s8p4d /10s 3p) 115s8p4d 1f/10s3p ld) schm CGTO [4s4p d /s p] [Ss3p d /3sp] [6s5pd /3s 1p] [8s6p 3d /6sp] Ucotr actd Ucotractd Numbr of Numbr of GTO CGTO's SCF rgy (a.u.)

6 45 CALCULATION OF MOLECULAR MAGNETIC PROPERTIES TABLE II. Sum ruls [coordiats i bohr: d(h, ) =(0, , ), d(o) =( )] for charg ad currt cosrvatio ad gaug ivariac of magtic suscptibility (a.u.). Basis st I II III IV V VI (Tx Py) ( Ty, P ) (P,P ) (P,P ) (P P,) '(P, P ) valus havig alrady b obtaid via basis st IV. O th othr had, costrait (43) is much mor difficult to fulfill allowig for th basis sts rtaid hr. I th cas of basis sts I ad III, th sig of th thortical prdictios is wrog. Ev th rsults yildd by vry larg basis sts ar far from big fully satisfactory. I fact, whras som of th coditios (T~,P&),=0 ar satisfid by symmtry, th coditios (T~,P ) i=(z) ad (T,P ),=0 ar fulfilld with good accuracy oly via basis st VI. This mas that sum rul (43) is a quit svr prob of accuracy for molcular wav fuctios. I particular, as ca b ralizd from dfiitio (40}, th basis st ought to b suitabl to rprst both dipol lgth ad vlocity oprators at th sam tim, which may b difbcult to obtai via ordiary Gaussia basis sts. I fact, it is worth oticig that Sadlj basis sts [14], spcially dvlopd to calculat ar-hartr-fock lctric polarizabilitis, i.., to rprst th positio oprator fairly accuratly, giv th right sig, v if thir xtsio is smallr tha that of basis sts IV-VI. From ths fidigs o ifrs that ad hoc basis sts ar to b costructd by carfully tailorig th xpots of th polarizatio fuctios i ordr for sum rul (43) to b fulfilld to a rasoabl xtt. It sms to b madatory to iclud 4f fuctios o havy atoms ad 3d fuctios o hydrog for a basis st to guarat accurat simultaous rprstatio of lgth ad vlocity oprators. Th sum ruls for charg cosrvatio ad origi idpdc of th uclar magtic shildigs ar rportd i Tabl III. Th thortical xpctatio valu of th lctric filds at oxyg, (Eo), ad hydrog, (EH), is littl affctd by basis-st quality, as ca b achivd by ispctio of Tabl III, but, as xpctd, tsors (MH, P), ad (Mo, p), ar much mor dpdt o th umbr of polarizatio fuctios ad thir xpots. W ca s that sum rul (44} is poorly obyd by basis sts of lowr quality ad oly basis sts V ad VI provid accurat rsults for oxyg. For hydrog, basis st IV alrady sms to b fairly adquat to guarat a good dgr of origi idpdc. This is du to th fact that th Gaussia sts w hav tak ito accout ar, i gral, mor suitabl to dscrib th lctroic viromt of hydrog tha of oxyg. Th diamagtic cotributios to th magtic suscptibility withi th Ladau prscriptio for th gaug of th vctor pottial [s th first li of Eqs. (6)], ar xpctd to b roughly two tims largr tha th corrspodig quatitis i th Coulomb gaug. I fact, th thortical valus for th lattr, calculatd via basis st V [8], y = , gys = , y = , show that, at last i th cas of watr, th diamagtic cotributio is ot highly aisotropic; thrfor, for th diamagtic cotributios withi th Ladau gaug, o xpcts y +=g ",tc., which holds xactly for th trac of th tsor [s th third li of Eqs. (6}]. From th coditio for gaug ivariac of total magtic suscptibility [s th last li of Eqs. (3)], paramagtic cotributios withi th Ladau gaug much largr tha th corrspodig os withi th Coulomb gaug ar also achivd [s also th first of Eqs. (3)]. Accordigly, total magtic suscptibilitis withi th Ladau gaug ar th diffrc btw two larg umbrs. Now, th diamagtic cotributios ar mrly xpctatio valus ovr th uprturbd wav fuctio ad ar usually rathr accurat v for rlativly small basis sts. Th paramagtic cotributios, howvr, ar much mor dpdt o th quality of th basis sts. As a cosquc, for a giv basis st, paramagtic cotributios ad total suscptibilitis i th Ladau gaug ar xpctd to b grally gaug. wors tha thos i th Coulomb TABLE III. Sum ruls [coordiats i bohr: d(h) :d(h, )=(0, , )] for charg ad currt cosrvatio ad gaug ivariac of uclar magtic shildigs (a.u.). Basis st (E~ ) (M i', P» L, (Mii,Pz), (EH, ) (M, Py ) (MH, P ) (Eo",) (MQ Py ) (M(),P ) I II III IV V VI

7 678 FERRARO, HERR, LAZZERETTI, MALAGOLI, AND ZANASI Basis st I II III IV V YE"- LZ' Expt. " Expt. ' TABLE IV. Magtic suscptibility Xxx P d P Zi, '8 34K.69' ' ' ' of H&O i ppm (a.u.). X P / X 'Th covrsio factor from ppm (a.u. ) pr molcul to usual ppm (cgs) pr mol is X 10 Paramagtic suscptibilitis i th Coulomb gaug from th sam basis st: g~ =6.318 ppm g~ = ppm (a.u.). 'Thortical suscptibility i th Coulomb gaug from Rf. [8]. Exprimtal valus from Rf. [16]. 'Exprimtal valus from Rf. [17]. d Xav &44.46K TABLE V. Magtic shildig at proto H& i ppm from basis sts I, II, III, IV, ad V. Compot " (c.m. ) o~ (c.m. ) " (H, ) o~ (H)) o (c.m. ) o (Hi) XX yz zy hr yz zy Ao. yz zy ho XX yy yz Zy Ao. XX yy yz Zy ho Basis st I Basis st II Basis st III Basis st IV Basis st V Xav P Xav & (a.u.), g~ =7.588 ppm (a.u. ),

8 45 CALCULATION OF MOLECULAR MAGNETIC PROPERTIES TABLE VI. Magtic shildig at proto H& i ppm from basis st VI ad compariso with prvious thortical ad xprimtal rsults. Expt. ' Calc. Compot yz zy 60. Lo. cr (c.m. ) u~ (c.m. ) o "(Hi) ~(H) ) (c.m. ) g (H)) 'Exprimtal valus from Rfs. [18]ad [19]. bgto calculatios from Rf. [8]. Th rsults rportd i Tabl IV largly cofirm ths xpctatios. Th diamagtic cotributios i th Ladau gaug ar fairly idpdt of basis-st quality ad ar roughly two tims largr tha i th Coulomb gaug. Ladau paramagtic suscptibilitis ar also much largr, roughly o ordr of magitud, ad xtrmly dpdt o basis-st quality: It ca b obsrvd that, v for th largst basis sts V ad VI, th rsults hav ot fully covrgd. As a mattr of fact, total Ladau magtic suscptibilitis ar usually lss accurat tha TABLE VII. Magtic shildig at oxyg 0 i ppm from basis sts I-V. Coordiats i bohr: d(o) =(0,0, ). Compot cr" (c.m. ) cr J' (c.m. ) 0 (0) u (c.m. ) o(o) hr Basis st I ha Basis st II XX yy hr Basis st III ho Basis st IV ho Basis st V

9 680 FERRARO, HERR, LAZZERETTI, MALAGOLI, AND ZANASI 45 TABLE VIII. Magtic shildig at oxyg 0 i ppm from basis st VI ad compariso with prvious thortical ad xprimtal rsults. Compot o" (c.m. ) o (c.m. ) Expt. ' Calc. b yy ZZ Ao. ho. 'Exprimtal valus from Rf. [18]. GTO calculatios from Rf. [8] cr~ (c.m. ) o. (0) corrspodig quatitis i th Coulomb gaug [8], which hav virtually attaid th Hartr-Fock limit ad ar clos to th xprimtal valus rportd for y ad x ~ O th othr had, as ca b xpctd by ispctio of Eqs. (6) ad (33), th accuracy of thortical uclar magtic shildigs furishd by a giv basis st should b comparabl withi Coulomb ad Ladau gaugs, at last i th cas of diagoal compots. I fact, from th fourth li of Eq. (6), w obsrv that th diamagtic cotributios should hav comparabl magitud i particular, th trac of th tsor must b th sam withi th diffrt gaugs [s th last li of Eqs. (6)]. Thrfor, from th gaug ivariac for total valus [s th last li of Eqs. (33)], o could prdict that similar quality could b obtaid for th paramagtic cotributios i th Coulomb ad Ladau gaugs. Th rsults from th prst calculatio do i fact support this hypothsis. Th paramagtic cotributios to th uclar magtic shildigs of hydrog ad oxyg (s Tabls V VIII) ar cotiuously improvig from basis st I to VI. Th thortical prdictios from basis sts V ad VI ar vry clos to thos obtaid i our bst prvious calculatios of th sam quatitis i th Coulomb gaug [8], usig agular momtum ad torqu formalisms. Accordigly, w bliv that th prst stimats ar of ar-hartr-fock quality. Compariso with th availabl xprimtal avrag uclar shildigs (s Tabls VI ad VIII) also cofirms th vry good quality of th calculatd valus. VII. CONCLUSIONS Ths fidigs ar a pic of vidc showig that accurat paramagtic suscptibilitis i th Ladau gaug ar obtaiabl oly by mas of vry larg Gaussia basis sts, icludig 4f polarizatio fuctios o havy atoms ad 3d fuctios o hydrog, carfully costructd for this quatity. Togthr with sum rul (43), Ladau paramagtic suscptibilitis provid a xtrmly svr tst o th quality of molcular wav fuctios ad thir ability to rprst magtic prturbatios. I ordiary calculatios aimd at prdictig accurat magtic suscptibilitis, it is advisabl to adopt th Coulomb gaug for th vctor pottial. Th opposit may b tru i th cas of uclar magtic shildig withi th Ladau gaug, owig to th advatags i valuatig itgrals ovr a Gaussia basis discussd i Sc. II. As a mattr of fact, uclar magtic shildig calculatd via Gaussia basis sts is charactrizd by comparabl accuracy withi th Coulomb ad Ladau gaugs. ACKNOWLEDGMENTS Grous allocatio of computr rsourcs providd by SEYCAD, Armada Argtia, ad th CICAIA of th Uivrsity of Moda ad fiacial support for th prst rsarch from th Argtiia CONICET, th Italia CNR (Progtto Fialiato Sistmi Iformatici Calcolo Paralllo), ad th Italia Miistry of Public Educatio ar gratfully ackowldgd. [1] L. Ladau ad E. Lifshitz, Thoric d Champs, 3rd d. (Mir, Moscow, 1970). [] L. Ladau ad E. Lifshitz, Mcaiqu Quatiqu (Mir, Moscow, 1966). [3] P. Lartti ad R. Zaasi, Phys. Rv. A 3, 607 (1985). [4] P. Lartti, Adv. Chm. Phys. 75, 507 (1987). [5] F. Bloch, i W. Hisbrg ud di Physik usrr Zit (Viwg, Brauschwig, 1961). [6] P. Lartti ad R. Zaasi, J. Chm. Phys. 84, 3916 (1986). [7] P. Lartti ad R. Zaasi, J. Chm. Phys. 85, 593 (1986). [8] P. Lartti ad R. Zaasi, Phys. Rv. A 33, 377 (1986). [9] J. H. Va Vlck, Th Thory of Elctric ad Magtic Sus cptibilitis (Oxford Uivrsity Prss, Nw York, 193). [10]J. A. Popl, W. G. Schidr, ad H. T. Brsti, High Rsolutio Nuclar Magtic Rsoac Spctroscopy (Mcoraw-Hill, Nw York, 1959). [11]S. T. Epsti, Th Variatio Mthod i Quatum Chmis try (Acadmic, Nw York, 1974). [1] H. A. Bth ad E. W. Salptr, Quatum Mchaics of O- ad Two-Elctro Atoms (Sprigr-Vrlag, Brli, 1957). [13]W. J. Hhr, L. Radom, P. vo R. Schlyr, ad J. A. Po-

10 CALCULATION OF MOLECULAR MAGNETIC PROPERTIES pi, Ab Iitio Molcular Orbital Thory (Wily, Nw York, 1986). [14] A. J. Sadlj, Collct. Czch. Chm. Commu. 53, 1995 (1988). [15]F. B. va Duijvldt, IBM Rs. Rp. RJ 945 (1971). [16] H. Taft ad B. P. Dayly, J. Chm. Phys. 51, 100 (1969). [17]W. H. Flygar, Chm. Rv. 74, 653 (1974). [18] C. J. Jamso, i Nuclar Magtic Rsoac, Vol. of Spcialist Priodical Rports (Royal Chmical Socity, Lodo, 1983). [19]P. Lartti ad R. Zaasi, J. Chm. Phys. 68, 153 (1978).