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Polarized basis sets for accurate predictions of molecular electric properties. Electric moments of the LiH molecule
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A method for the determination of large polarized GTO/CGlO basis ssts for the cahlati~n of moIectdar prop&tis.is vay ac+ta tt SCF and CAS SCJTcaXculationsof the dipole and quadrupole moments of the LiH mokcuk Without any ojhnization of orbital urponcnts the GTO/CGTO HF SCF results are nearly the same as those rccemly nbtaincd by ~!ICnumerical _jx+graLionof HF equations The CAS Sq rcsulq. aft& roviimtional avcrz@+ &e in an :
procntd andFaustnted by
ex~t~caich~a~Icexpcrimedraldafa_
-
~. thoseparainetes. and those regions of- the electron density which make the largest contribution to the energy. lowering and the- resuhing ~wavefunction. Most computational methods which are curmay not be accurate enough for the caIcu+ion oLrently employed to study the~electronicstructure average values of operators which do not appear of many-ekctron atoms and molecuks * involve, directIy in the hat&o&n [8]. This is usuaUy. at least as an intermediate step, the concept of remedied by augmenting the given basis set with s&k-particle states [2]_ Their determination is additionai diffuse and polarization functions [83, usuaIIy achieved by what is known as the basis set. who&z orbi~talexpOnents are selected according to expansion method [3]. In principle the basis set the present computational experience+This proce: expansion techniques can provide solutions of dure has been to some- extent ~standardized by arbitrarily high accuracy_ However, for practical Werner and Meyer [9] who devised severaI rules reasons the number of basis set functions has to be for the determination of basis sets for. accurate made fiite, and-most fi=equentIy~ reIativeIyim&_ caI&Iations -of. &+eIation energies, dipole. mo- .~ Even with rather short expansions the accuraz+c. of merits, and dipole poIarizabi@ies_ An alternative .singI&particlestates can be improved by the optimethod has rec+tIy been proposkd [lo] for -the mization-of non-linear parameters with respect-to determination of gaussian basis sets. for. highlythe. total energy of the given syst&- This tech-, accurate cakulations of molecuIarquadrupolemo- _ nique has resuhed in a.variety of energy-optimized ments Ho&v&r, in both cases some opmtion basis sets of either Sk&r-type (SIO) [4] or gausof orbital exponents is i.uvoIved[9,10]. T-‘-. _. -... .’ siari-type (GTOj 15:81’ orbitak ‘which are, nowa-~ Any one-eIectron_operator whose expectation days used in atomic and molectdar calculations. value. is to be determined from the giveri~wave-. . : The ~energy-orientedoptimizaiion: of. basis set -- function can be reIat& though sometimes art.& functions-stresses obviousIy the:importance of. ,_ dally, to.some~externaiperturbation On the-other . . .. .. _ -. z hand, for severalphysicahy important~one-&ectron perturbations-one. can devise ‘an ez@Iicit,de@et$ : ; : F&r rcccnt rwi~ of &ffaa&kmp&aIi?naI mehock yt _&[ll_ <- r-1 ;_ _ .I- -,. .. _: :: iience of’ basis set functions on the ekt&aIpek.~ ~. . _-~. _ -. 0301-0104/85/$0330 0 Eh+er Science Pubhshers B-G:. 1z ‘_ -‘. ym T (North-HoBand Physi~_Pub&hiug Division) ---__:r :. ;_ _ .~_ AL -[‘ ‘~___. ;_ ‘. _:_‘. ~. -: 1,Introduction
turbation strength [11,12]_ Then, the form of basis set functions, which should bc included in the basis set in order to ensure that both the unperturbed energy and the property of interest are of similar quahty. can be determined throu$r the +en order of perturbation theory_ This method will be used in the present paper to obtain basis sets for ItighIy accurate cakulations of mokcular ekctric proper&s. The performance of a basis set determined from the power series expansion of what has been termed the electric-fiskI variant (EFV) GTOs [11,121 wiII be illustrated by cakulations of the dipole and quadrupole moment of LiH_
the corresponding properties, can be directly obtained without any additional optimization. The proposal concerning the form of the explicit dependence of ba&s set. functions on the extemai ekctric field strength .F, [l&12] follows from the consideration of the harmonic osciilator in the external ekctric field_~IL has been proposed [ll] that the initial se’ of GTOs, {x,[r.A,(O), a,]}, with orbital exponents aP and origins at Ap(0)_ is transformed in the presence of the external eketric field into the EFV GTO set {xJ(
of poIaiized basis sets for cakuIation of moIecuIar ektric properties
2
D&vauion
the
Suppose &at the basis set functions depend in some way_ through their parameters_ on the external perturbation strength Then, the HelhnannFeynman theorem is no ionger satisfied [13-lS]_ Even the Ers~-order properties must then be determined as the first-order derivatives of the total perturbation-dependent energy_ The use of the expeetation-value definition does SOL aammt for the fuli fkxibihty of the =avefunction and only in the Iimit of a compIete basis set the two definitions of the first-order property wiII give the same result_ The additional terms which have to be accounted for when computing the fiit-order properties as the energy derivatives involve, in the present case, the first-order derivatives of basis set functions with respect to the perturbation strength [13,14]_ Hence, if those derivatives were present in the initiai basis set, there would be no contibution of the correspondin terms through the wavefunction derivatiwz_ The above argument can be easiIy extended to higherorder properties Thus, in order to caIcuIate some property of the given system at the same IeveI of accuracy as its totaI energy* the initiai basis set must be extcmtded by the approI.niaLe derivatives though the nth order in the perturbation strength_ Once the basis set dependence on the perturbation strenght is known, its counterpwhich is appropriate for the calculation of
0)
where X is a common scale factor [11.12]. Later on this concept has been extended for perturbations
due to the electric field gradients [16.17) as well as for the ST0 basis sets [lZlS]_ Both the EFV GTOs and EFV STOs have been found particularly usefuI in caIcuIations of electric poIarizabi& ties with relatively smaI1 basis sets [11,12,1S-19]_ In the present paper we shall take the advantage of the known dependence of basis set functions on the electric Iield strength rather than LO use the JSV functions exphcitly in cakulations of mokcuIar energies and properties. For the initial GTO basis set comprising s, p, d * ___ primitive GTOs its extension appropriate for the calculation of dipole moments amount to adding the corresponding fmt-order derivative functions_ It foIIows from the definition of EFV GTOs that the initial s-type functions wiII contribute p-type GTOs, the p-type GTOs &II lead to s-
and d-t_vpe functions, and so on, ali having the same orbital exponents_ Hence, no optimization of orbital exponents is needed if the basis set is derived from the consideration of the con-espending EFW GTO basis set Obviously, the symmetry of orbit& LO be included in the basis set follows directly from the consideration of the symmetry of the perturbation operator and the present scheme does not add too much in this respect_ The major advantage of our approach is that we can also predict the changes in the relative conttibution of different primitive GTOs in their Iinear
combination_ ‘The latter ,could be a contracted GTO or an orbital determined inSCF calculation_$_ ., _:: Let us -consider some linear. combination of initial -basis set functions .in the absence of the external electric field: :
h(r;O) ==Cc,(O)x,(r;O), P
where x,(r.O) denotes a basis set function for F = O_ The F_ counierpart of the function (2) determined in the presence of the external electric field will be
u(r;F) =Cc,(F) x,(M).
(3)
P
and its fmt-order contribution can be written as u”‘(r;O) = 1 ]c~‘(O, x,(r;O) P
+c,(0)
$‘(r;O)]
_ (4)
The fiit term on the rhs of eq_ (4) involves the functions of the initial basis set and for this reason is rather irrelavant. The first-order derivatives of EFV GTOs, xa’(r;O), can be expressedin terms of normalized GTOs with the GTO “quatum numbers” shifted up or down by one, i.e., xz*(~;O) and X,1-: r,O), respectively xa’(F;o) = a;rx;r(‘;o)
-+a ,'&'(r;O)_
(5)
The down-shifted term is again of rather minute importance since either it aheady appears in the initial basis set or contributes another GTO to the energy optimized subset_ If one is interested in accui%te calculations, then the energy optimized initial GTO subset must correspond to a rather dense spectrum of orbital exponents. Hence, adding one more GTO is _of no importance and it is only the polariza~on part of (5) which must be taken into account when constructing a linear combination of polarized GTGs which corresponds to the ix&al function (2) Therefore, for the function (2) present in. the initial basis set the extended set which is to be used for the calculation of the first electric moment should contain the following term ~~~(O)~~'(r;0)=~~~(0)~~'X~'(r;O). p .-_.. P
I- (6)
By using eq. (1) and the GTO-no&a&ttion fac;J ’ torsonecaneasilyshd~that -_ -:.- :.-: ; _-_-I ~~(&&j),qy,
r_ :
_I
:-._
-(7);
and as a consequence the imp&tar& of &OS with low orbital exponents is &rea&i- at the. expense ‘of .GTOs with high orbital expone&. This form of the combined polarization function agrees with qualitative ideas concerning the sensitivity-of different GTOs land different regions of. the electron density distribution towards the polarization by the external electr&field_ Similar considerations are also valid for EFV basis sets appropriate for the electricfield gradient perturbation [16-U] and can be used to obtain’ basis sets suitable for highIy accuratecalculationsof molecular quadrupole moments In the next section this technique is used to devise a huge polarized basis set for the calculation of the dipole and quadrupole moment of LiH. 3. CaIculations of electric moments of the LiH molecuIe The LiH molecule is one of the most frequently used test systems for checking different approzcimations Very accurate calculations with a basis set of elliptical functions have recently been reported by Bishop and Cheung [20,21]. More recently theirresultshave been super-cededby Handy et al. [22], who employed a very huge GTO basis set comprising s, p, d f, and g GTOs on Li and s, p.d,andfGTOsonH_Inboththesecasesthe main objective was to calculate the total energy of LiH as accurately 5s possibleThe aim of the present paper is to obtain the most accuratevalues of the dipole and quadrupole moment by usingmoderately large and rather standard initial blasts sets_ Moreover, owing to recent adva.nc& in the numerical integration of molecular Hart&e-rock (HE) equations [23-26!, the HF limits for the total energy, dipole moment and_ quadrupole moment have been established_They provide a direct check on the accuracy of our results obtained at the HE SCF Ievel of approximation_With accurate HE SCE orbit& determined by using a polarized basis set the correlation effects
46
l&o_ Ron% Af_ sldkj
/ Eluuk-
ar&taken into account with the aid of the CAS SCF method [27,28]_ The convergence of correiation corrections to the dipoIe and quadrupole moment is investigated by considering different choices of the (active) orbital subspace in which a full four-ekctron CI wavefunction is constructed_ For the most accurate CAS SCF wavefunction, the caIcuIations are repeated for several geometries in the vicinity of the experimentaI equilibrium configuration and roviirational averages of both the dipole and quadrupoIe moments are obtained and compared with other caIcuIations and avaiIabIe experimental data It should be mentioned that for the present choice of the basis set-functions both the HF SCF and the CAS SCF method satisfy the HeIImann-Feynmann theorem_ Hence, the fit-order properties can be legitimately cakulated [15.29] as the average values of the appropriate operators. 3J_ The pohrized GTO basis set for LiH The ElV bases have initiahy been proposed in order to make feasibIe the calculation of moIecuIar electric properties with rather small basis sets [l&12]_ Since the present study is designed for highIy accurate caIcuIations of electric moments of the LiH moIecuIe, it is obvious that sufIicientIy Ianle basis sets have to be used from the very beggnning Pn order to make our scheme for the determination of pohuized basis sets as systematic as possiiIe. the initial GTO basis sets are assumed to foIIow from atomic caIcuIations_ For both Li and H we have started with the energyoptimized basis sets of van Duijneveldt [7J; the 13s GTO set for Li and the 10s GTO set for H To take into accouut the diffuseness of the eIe&ron density distribution on the hy&=en side of the Lili molecule, the H atom GTO basis set has been further augmented with two soft GTOs, whose orbital exponents follow from the geometric progression assumption appkd to the originaI van DuineveIdt*s basis set Additionally some contraction of GTOs with high orbitaI exponents has been performed with the contraction coefficients taken from the corresponding atomic SCF orbita& The finat. GTO/CGTO basis sets for Li and H which have been employed for the determination of the corre-
mnmam o/Ike Lix
i#lok-dc
_
spending polarized sets can be written as [13_/8] and [12/S_]. respectively_ Tke p subsets of polarized basis sets have been obtained by the method described in the previous secticn with the contraction coefficients foilowing from the change in the LCAO SCF coefficients for atomic orbitak Since the pohuization of very inner reggons of the ekctron density distriiution has a negggible effect on the caIcuIated properties, some of the resulting p GTOs with the -highest vaIues of orbital exponents have been deleted from the p subset_ If de above sp basis sets are employed in mokcular caIcuIations, the p orbitals make a considerable contribution to the bonding Hence, their further polarization should be considered. This kads to polarized d subsets for both atoms. Again, some basis set functions which either correspond to relatively high orbital exponents or folIow from the polarization of those initial GTOs which appear in the restthing molecular orbitaIs with very smaII LCAO coefficients, have been deleted from the d subset_ By~a similar consideration aIso a subset of two f GTOs contracted to a singIe C-0 has been produced for Li. The final poIarked basis sets, the [13_8.62./8.5.3.1.] set for Li and the [12.8_5./85.3.] GTO/CGTO set for H, are presented in table l_ It is important to point out that by the method of the derivation of polarized basis sets, the orbital exponents for poIarized functions cover a wide range cf vahtes_ Hence, our pohuized basis sets should be simuhaneously suitable for the caIcuIation of correIation energies This is aIso one of the reasons that higher than p pohuized components have been taken into account They are simpIy needed for the poIa.tization of orbit& which appear in other than the ground state configurations when the CI wavefunction is buik Let us also recall that neither the -orbital expoxiexk nor ,the contraction-coeffxcient optimization are invokd in constructing the moIecuIar basis set lof table 1_ 3-L HF SCF reszdts _ The HF SCF resuhs for the to-d energy, di&e moment and quadrupole moment of -the LiH mokcuIe at the experimental equiIiirium distance
i416.8112 321.45994 91.124163 w-999891 ~11.017631 4.372801 1.831256 O-802261 0.362648 0.113995 0_051237 0.022468 psubxt 1
11.017631 4.372801 1.831~256
Oman075 0_@062 OBOO584
1776.7756 254.01771 54-698039
o_om372 OBO2094
0_012605 O.&Q356
IS.018344
0_008s63
0.114780 0239381
4-915078 1.794924
0_030540 0.090342
1 1 1 1 1 1
0_7107l6 0301802 0_138W6 0.062157 O-027%7 0_012583
I 1 1' 1 1 1
0_0346
4-915078 1.794924 O-710716
0.0138
0.1145 0.2fGl4
0.2S29
0.802261 0362648
0.1117 0.1470
0_3M802
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3
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1 1 I
0_138iM6 0.062157 O-017957
1 1 I
4372801 I.831256 0.802261
0.0547 0.1954 03783
1.794924 0.710716
0.0503 0.3000
2
0.113995 O-051237
0.1816 1.1978
0304802 0.138046
0.1156 O-2460
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05292 0.8151
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of 3.015 au, caIcul&d with the &arized GTO/ CGTO basis set of table J, arc shown in table 2. They are compared. with recent results of -seminiimerjcd [23_24] and fully-nume&aV two-dimensional [25,26j HF SCF calculations The latter data can be amsidg-ed as the near-e%act~HF SCF values for the properties listed in &bIe 2: Also some sekcted au&ate re$ul& of +&%a HF .!XF
calculations- with. ST0 -and -GTO basis sets are incIude& _. A comparison of our pr-nt k&s with those of the ntieri&l HF SCF caki&tic& allows us to draw some preJ‘ ‘y ConclusionS about the.quality of the- @~Miz.ed~ GTO)CGTO -b&s se?. cjerived in .this p&erfrom-standard energy_Optimize$ has+ sets for Li and EL ~bviotily, it i$not-‘: _
s_o_Raos. it-r_ skftq- / Elarric
3s Tabk
momoln
of Kite LiH
nsdde
2
moment of the LiH mokwk
RaulCSoCaCCtZUC HFSCFcAahiomofLhcW~_dipokmoment~dquadrupok ~R-3_015au.Mlvalue~inatomicunit~ inremrrdcy E-
chisa-ork accuraccreferencesresulls oumaicaI SCF SCF
EWGTOKF GTOSCF
lW PI [3i] =’ 1191dB [32] *
GCF
PSCF
-7_9!n27s
23625
-7987352 - 7_987352* -7_9s7313 -7%7072 -7_987063
23618 23618105 ‘x61 2361 23620
at the
(y&a) -33719
- 3_37uo6 L’
-
momem -mr taken at the nuckar antrc of mass for ‘Li’H_ All dcfinitio~ according to *Theori@Forthequadstpok =d=Q&== !33l. b’ C&t& fm &c v&c of the dccuonic contriicn tt the molecular midpoint and the dipole momalt whu2 of ref_ [30). 0 ~a4-E c&uhtions pith cnergs-dptin&cd basis set of six s_ four pa. two do. and one fa orbitak for Li and *&~czs and ollt PO orbimIs for H 131). s) CahWiom uiih &C [11_63_/5_63_] EW GfO/CGTO basis set for Li sod the [115_/75.] ElT GTO/cGTO basis set for H [191_ * HF SCF d&W wish rhc [14_i_3_/9_73_] GTO/CGiO basis sex for Li and the [10_6_/3_6_] GTO/CGTO basis set for H [32k
surprisingthat our SCF energy is very cIose to the best HF SCF result obtained within the finite basis set approximation_ Also in comparison with the numerical HF SCF energy value we miss less than lo-’ au_ In addition both the dipole moment and the quadrupole moment calculated in this paper differ from the numen_cal HF SCF data by = 0.03 and 0.0695, respectively_ The HF SCF results of Lazeretti et aL 1321obtained with a huge but rather ad hoc selected GTO/CGf’O basis set appear to be slightly better for the dipole moment than those calculated in the present paper_ Howexr_ the aaxna cy of their dipole moment is not accompanied by a similar accuracy of the total SCF energy_ In the case of polarized basis sets the method of their derivation suggests that the ekctric moments should be ahnost as accurate as the HF SCF energies, though r&able measure of their relative accum ties is rather difficuh to establish_ It is of interest to note that the results of HF SCF calculations with reIatively sti EFV GTO/ CGTO basis sets [19] provide the dipoie moment values of quite high accuracy_ This gives a further support to the reliability of our method for the basis set det emrination for calculations of mokcular electric properties_ Quite unfortunately the HF SCF data are not repotted for the largest GTO/CGTO basis set for the LiH molecule (113_11_83_1_/11_11_83_1_]on Li
and [1453_2_/10_5_32_3 on H) employed recently by Handy et al. [22]_ However, from the comparison of their orbital exponents with those of tabIe 1 we can conclude that in both cases the HF SCF energies should be nearly the same The accuracy of HF SCF dipole and quadrupole moments which would follow from the basis set of Handy et aL 1221is more difficult to predicr, though it should also be nearly the same as that achieved in our calculations_ 3_3_ CAS SCF resuffs The complete active space (CAS) SCF approach_ whose detailed description can be found elsewhere [27,28], is a multiconfiguration SCF scheme based on a full CI wavefunction built within some limited number of (active) orbitals. The orbitak_ whose occupation numbers have fiied non-zero value are classified into what is called the inactive orbital subspace_ The CAS SCF wavefunction can be uniquely specified by the number of electrons in both subspaces and the list of inactive and active orbitak The symbols used to denote the CAS SCF ~avefunctions for the LiH molecule are based on the classification of molecuiar orbitals according to the C, symmetry group and their detaikd description can be found in our earlier papers [19_34]_:
B_O_ it-
Li-
Scdkj
/
EIe&c
tn&em o; the -Liwmokcde
:.
-
’
_.4g.--:
:
The use of a fuII CI wavefunction in the CAS SCF app_roach is undoubtedly one- of its major adVantages: Above aU, no a priori selection of what are usuaIIy izalkd the important ~contigura- ‘tions is invoIved_ Moreover. the CAS SCF wavefunction for_ perturbation-independent basis sets satisfies the Helhnkn-Feynnran theorem and facilitates the caktdation of fmt-order properties_ However. similarIy to aII other computational methods of quantum chemistry, it also suffers from some difficulties_ The major problem is the Iength of the fuII CI expansion which considerably knits the size of the active orbital subspace Hence, a careful consideration of the choice of active orbi& followed, whenever possible, by the appropriate convergence study_ is necessary. Such a study has been carried out in the present paper prior to the selection of the most reliable active space of orbitak and the corresponding data are reported in tabIe 3. In the first CXS SCF calculation the valenceshell approximation has been made and the active space has been Iimited to three G and one c orbit& as iu our previous calculations of electric properties of the IiH molecule [19]_ In aII other
caIc$ations the valence-shell approximation h&s ‘: been_-released sinceit % not expectedto-:provide high enou& a&racy because of-: rehit.iveIy imiportant (Is, 2s) correlation effects-The second &AS -SCF wavefunction based on the active orbital sub-space corn&-king six (J and two = orbita& has also been investigated previously [19] and found to be highly satisf&tory in tams of both the total mokcular energy and electric properties of LiH. Since the occupancies of the most weakly occupied. naturai orbitals are still of the order. of. 10m3, a further extension of both the G and k orbitat subspaces has been made, resulting in the (OOOO/ 8330) CAS SCF wavefunction which comprises 993 configuration state functions (in GV symme-. try)_ As can be seen from the data of table 3, the occupancy of the most weakly occupi&l natural orbitals becomes reduced to = 10mJ_ -. -.-A further increase of the GS active subspace is not expected to bring signitkmt changes in either the dipole or quadrupole moments_ The occupancy of additional natural orbitak wilI be less than IO-” and the changes of occupancies of natural orbitak which are already used in the (0000/8330) CAS SCF wavefunction are expected to be even
Table 3 Total energies, dipoIe momems quadmpole moments_and occupation numbers of natural orbirds in difkres for the LiH mokcuIe at R = 3.015 au_All valuesin atomic units
CAS SCF ca!cuhtions
Inacri\_c/ti%-esubspacc
-crgy dipole moment quadrupole moment a) occupation numbers cc, lo 20 3o 40 50 6a 70 8a lZ&.lq 2n,-, 3%3-, lq~.lq=___~
(1000/3110)
(ONO/6220)
wo0~1)
-8.021109 23058 -3.1161
- 8.057253
- 8.059499 22921 - 3.0699
-so57774 23001 - 3.0938
- 3.0967
SyNnerry) ZOO000 1.93979 0.03088 o.OOs43 -_
0.01045
-
I-99370 Z-93976 0.03099 0.00856 0_00244 0_00117
0.01047 0.00123
__-
a Origin for Ihe quadmpole moment operator at the nude;lr~ccntrcof.-
I-99320 I.93973 0.03113 0_00846 0.00256 0_00122 0_00020 O-GO006 0.01033 0_00129 o_OcKy
I_99361 1.93983 0.03091 0.00853 0_00244 0_00121 0.01046 0.00123
.-
0-m
for 7Li’H;-
--
1
50
B-O_ Roar. AI _ so&j
/ Ei?xuic momentroftJlefiHmoIde
smaIIer_ The corresponding orbitaI contributions to *be dipore and quadrupoIe moments turn out to be Iess than 10-s au. Moreover, multiconfiguration SCF optimization~of very weakly occupied natural orbit& represents a ratherdifficuIt numerical probkm because of numerical instabiIities_ Hence, BXZdid not attempt further extensions of the ez active orbital subspace_ In principIe one may expect that including S orbitaIs in the active subspace could be of some importance_ For this reason we have investigated the (OOOO/T221) CAS SCF wavefunction obtained by exteuding the active subspace of the (WOO/ 6220) function with one S orbital_ It foIIo\kisfrom the data of table 3 that the corresponding effect on the dipole moment, quadrupole moment and occupation numbers is ahnost negIigiiIe_ However, in terms of the t0ta.Ienergy incIuding a 6 orbital in the active orbitaI subspace resuhs in a noticeable improvement_ Since the total moIecuIar energy is not the principal objective of our caIcuIations. deleting the S orbital from the active subspace appears to be quite acceptabIe_ Thus, the rest of OUrCii scussia wili be concerned with CAS SCF rest&s obtained witb the (0000/8330) wavefunction_ The correspoilding summary of our data at the e__perimental equilibrium distance of 3-015 au and their comparison with recent most accurate vahxs of other authors_ are presented in table 4_ It is obvious -that one cannot expect a very high accuracy of the totaI CAS SCF energy cakulated within a reIativeIy smaII active orbital subspace AIso a part of the difference between our result and the estimate of the exact non-relativistic Born-Oppeeeimer energy of LiH comes from the fiite basis set appnximation and reIativeIy poor vahte of the core correlation energy_ However, the missing part of the core correIaticn effects is rather unimportant for the accurate prediction of the dipoIe and quadrupoIe moments_ It can be seen from our (1000/3110) CA!S SCF results that the major part of the coixIation correction to these two properties comes from the vaknce-sheII correIation effects_ It has aheady been mentioned that the CAS SCF method for perturbation-independent basis sets satisfies the HeIImann -Feynman theorem, and thus. the expectation values of one-ekctron op-
Table 4
conlparhn of_difraulr aL_ruraKCfhcorcIica!results ror the Liz-I mokcuk at the inumuckar distance R - 3.015 au All xalucs in atomic &its
ibis work b’ rcr_[35] rek_ [20_2l] C’ ref_ [zz] csact
-
S-059499 8.063866=’ 8_065& &06904 5.07049@
Dipole momcnr
Quaanrpole xnomcnt*
x97-1 2_2w2~ 22917 725 x93h’
- 3.0699 - 3.0655 0 -
=’ The origin for the quadmpok moment opemor at the nuckar ccntrc of mass for ‘Li’H_ b’ Rcsuh cakulaccd with xhe
43-53
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A method for the determination of large polarized GTO/CGlO basis ssts for the cahlati~n of moIectdar prop&tis.is vay ac+ta tt SCF and CAS SCJTcaXculationsof the dipole and quadrupole moments of the LiH mokcuk Without any ojhnization of orbital urponcnts the GTO/CGTO HF SCF results are nearly the same as those rccemly nbtaincd by ~!ICnumerical _jx+graLionof HF equations The CAS Sq rcsulq. aft& roviimtional avcrz@+ &e in an :
procntd andFaustnted by
ex~t~caich~a~Icexpcrimedraldafa_
-
~. thoseparainetes. and those regions of- the electron density which make the largest contribution to the energy. lowering and the- resuhing ~wavefunction. Most computational methods which are curmay not be accurate enough for the caIcu+ion oLrently employed to study the~electronicstructure average values of operators which do not appear of many-ekctron atoms and molecuks * involve, directIy in the hat&o&n [8]. This is usuaUy. at least as an intermediate step, the concept of remedied by augmenting the given basis set with s&k-particle states [2]_ Their determination is additionai diffuse and polarization functions [83, usuaIIy achieved by what is known as the basis set. who&z orbi~talexpOnents are selected according to expansion method [3]. In principle the basis set the present computational experience+This proce: expansion techniques can provide solutions of dure has been to some- extent ~standardized by arbitrarily high accuracy_ However, for practical Werner and Meyer [9] who devised severaI rules reasons the number of basis set functions has to be for the determination of basis sets for. accurate made fiite, and-most fi=equentIy~ reIativeIyim&_ caI&Iations -of. &+eIation energies, dipole. mo- .~ Even with rather short expansions the accuraz+c. of merits, and dipole poIarizabi@ies_ An alternative .singI&particlestates can be improved by the optimethod has rec+tIy been proposkd [lo] for -the mization-of non-linear parameters with respect-to determination of gaussian basis sets. for. highlythe. total energy of the given syst&- This tech-, accurate cakulations of molecuIarquadrupolemo- _ nique has resuhed in a.variety of energy-optimized ments Ho&v&r, in both cases some opmtion basis sets of either Sk&r-type (SIO) [4] or gausof orbital exponents is i.uvoIved[9,10]. T-‘-. _. -... .’ siari-type (GTOj 15:81’ orbitak ‘which are, nowa-~ Any one-eIectron_operator whose expectation days used in atomic and molectdar calculations. value. is to be determined from the giveri~wave-. . : The ~energy-orientedoptimizaiion: of. basis set -- function can be reIat& though sometimes art.& functions-stresses obviousIy the:importance of. ,_ dally, to.some~externaiperturbation On the-other . . .. .. _ -. z hand, for severalphysicahy important~one-&ectron perturbations-one. can devise ‘an ez@Iicit,de@et$ : ; : F&r rcccnt rwi~ of &ffaa&kmp&aIi?naI mehock yt _&[ll_ <- r-1 ;_ _ .I- -,. .. _: :: iience of’ basis set functions on the ekt&aIpek.~ ~. . _-~. _ -. 0301-0104/85/$0330 0 Eh+er Science Pubhshers B-G:. 1z ‘_ -‘. ym T (North-HoBand Physi~_Pub&hiug Division) ---__:r :. ;_ _ .~_ AL -[‘ ‘~___. ;_ ‘. _:_‘. ~. -: 1,Introduction
turbation strength [11,12]_ Then, the form of basis set functions, which should bc included in the basis set in order to ensure that both the unperturbed energy and the property of interest are of similar quahty. can be determined throu$r the +en order of perturbation theory_ This method will be used in the present paper to obtain basis sets for ItighIy accurate cakulations of mokcular ekctric proper&s. The performance of a basis set determined from the power series expansion of what has been termed the electric-fiskI variant (EFV) GTOs [11,121 wiII be illustrated by cakulations of the dipole and quadrupole moment of LiH_
the corresponding properties, can be directly obtained without any additional optimization. The proposal concerning the form of the explicit dependence of ba&s set. functions on the extemai ekctric field strength .F, [l&12] follows from the consideration of the harmonic osciilator in the external ekctric field_~IL has been proposed [ll] that the initial se’ of GTOs, {x,[r.A,(O), a,]}, with orbital exponents aP and origins at Ap(0)_ is transformed in the presence of the external eketric field into the EFV GTO set {xJ(
of poIaiized basis sets for cakuIation of moIecuIar ektric properties
2
D&vauion
the
Suppose &at the basis set functions depend in some way_ through their parameters_ on the external perturbation strength Then, the HelhnannFeynman theorem is no ionger satisfied [13-lS]_ Even the Ers~-order properties must then be determined as the first-order derivatives of the total perturbation-dependent energy_ The use of the expeetation-value definition does SOL aammt for the fuli fkxibihty of the =avefunction and only in the Iimit of a compIete basis set the two definitions of the first-order property wiII give the same result_ The additional terms which have to be accounted for when computing the fiit-order properties as the energy derivatives involve, in the present case, the first-order derivatives of basis set functions with respect to the perturbation strength [13,14]_ Hence, if those derivatives were present in the initiai basis set, there would be no contibution of the correspondin terms through the wavefunction derivatiwz_ The above argument can be easiIy extended to higherorder properties Thus, in order to caIcuIate some property of the given system at the same IeveI of accuracy as its totaI energy* the initiai basis set must be extcmtded by the approI.niaLe derivatives though the nth order in the perturbation strength_ Once the basis set dependence on the perturbation strenght is known, its counterpwhich is appropriate for the calculation of
0)
where X is a common scale factor [11.12]. Later on this concept has been extended for perturbations
due to the electric field gradients [16.17) as well as for the ST0 basis sets [lZlS]_ Both the EFV GTOs and EFV STOs have been found particularly usefuI in caIcuIations of electric poIarizabi& ties with relatively smaI1 basis sets [11,12,1S-19]_ In the present paper we shall take the advantage of the known dependence of basis set functions on the electric Iield strength rather than LO use the JSV functions exphcitly in cakulations of mokcuIar energies and properties. For the initial GTO basis set comprising s, p, d * ___ primitive GTOs its extension appropriate for the calculation of dipole moments amount to adding the corresponding fmt-order derivative functions_ It foIIows from the definition of EFV GTOs that the initial s-type functions wiII contribute p-type GTOs, the p-type GTOs &II lead to s-
and d-t_vpe functions, and so on, ali having the same orbital exponents_ Hence, no optimization of orbital exponents is needed if the basis set is derived from the consideration of the con-espending EFW GTO basis set Obviously, the symmetry of orbit& LO be included in the basis set follows directly from the consideration of the symmetry of the perturbation operator and the present scheme does not add too much in this respect_ The major advantage of our approach is that we can also predict the changes in the relative conttibution of different primitive GTOs in their Iinear
combination_ ‘The latter ,could be a contracted GTO or an orbital determined inSCF calculation_$_ ., _:: Let us -consider some linear. combination of initial -basis set functions .in the absence of the external electric field: :
h(r;O) ==Cc,(O)x,(r;O), P
where x,(r.O) denotes a basis set function for F = O_ The F_ counierpart of the function (2) determined in the presence of the external electric field will be
u(r;F) =Cc,(F) x,(M).
(3)
P
and its fmt-order contribution can be written as u”‘(r;O) = 1 ]c~‘(O, x,(r;O) P
+c,(0)
$‘(r;O)]
_ (4)
The fiit term on the rhs of eq_ (4) involves the functions of the initial basis set and for this reason is rather irrelavant. The first-order derivatives of EFV GTOs, xa’(r;O), can be expressedin terms of normalized GTOs with the GTO “quatum numbers” shifted up or down by one, i.e., xz*(~;O) and X,1-: r,O), respectively xa’(F;o) = a;rx;r(‘;o)
-+a ,'&'(r;O)_
(5)
The down-shifted term is again of rather minute importance since either it aheady appears in the initial basis set or contributes another GTO to the energy optimized subset_ If one is interested in accui%te calculations, then the energy optimized initial GTO subset must correspond to a rather dense spectrum of orbital exponents. Hence, adding one more GTO is _of no importance and it is only the polariza~on part of (5) which must be taken into account when constructing a linear combination of polarized GTGs which corresponds to the ix&al function (2) Therefore, for the function (2) present in. the initial basis set the extended set which is to be used for the calculation of the first electric moment should contain the following term ~~~(O)~~'(r;0)=~~~(0)~~'X~'(r;O). p .-_.. P
I- (6)
By using eq. (1) and the GTO-no&a&ttion fac;J ’ torsonecaneasilyshd~that -_ -:.- :.-: ; _-_-I ~~(&&j),qy,
r_ :
_I
:-._
-(7);
and as a consequence the imp&tar& of &OS with low orbital exponents is &rea&i- at the. expense ‘of .GTOs with high orbital expone&. This form of the combined polarization function agrees with qualitative ideas concerning the sensitivity-of different GTOs land different regions of. the electron density distribution towards the polarization by the external electr&field_ Similar considerations are also valid for EFV basis sets appropriate for the electricfield gradient perturbation [16-U] and can be used to obtain’ basis sets suitable for highIy accuratecalculationsof molecular quadrupole moments In the next section this technique is used to devise a huge polarized basis set for the calculation of the dipole and quadrupole moment of LiH. 3. CaIculations of electric moments of the LiH molecuIe The LiH molecule is one of the most frequently used test systems for checking different approzcimations Very accurate calculations with a basis set of elliptical functions have recently been reported by Bishop and Cheung [20,21]. More recently theirresultshave been super-cededby Handy et al. [22], who employed a very huge GTO basis set comprising s, p, d f, and g GTOs on Li and s, p.d,andfGTOsonH_Inboththesecasesthe main objective was to calculate the total energy of LiH as accurately 5s possibleThe aim of the present paper is to obtain the most accuratevalues of the dipole and quadrupole moment by usingmoderately large and rather standard initial blasts sets_ Moreover, owing to recent adva.nc& in the numerical integration of molecular Hart&e-rock (HE) equations [23-26!, the HF limits for the total energy, dipole moment and_ quadrupole moment have been established_They provide a direct check on the accuracy of our results obtained at the HE SCF Ievel of approximation_With accurate HE SCE orbit& determined by using a polarized basis set the correlation effects
46
l&o_ Ron% Af_ sldkj
/ Eluuk-
ar&taken into account with the aid of the CAS SCF method [27,28]_ The convergence of correiation corrections to the dipoIe and quadrupole moment is investigated by considering different choices of the (active) orbital subspace in which a full four-ekctron CI wavefunction is constructed_ For the most accurate CAS SCF wavefunction, the caIcuIations are repeated for several geometries in the vicinity of the experimentaI equilibrium configuration and roviirational averages of both the dipole and quadrupoIe moments are obtained and compared with other caIcuIations and avaiIabIe experimental data It should be mentioned that for the present choice of the basis set-functions both the HF SCF and the CAS SCF method satisfy the HeIImann-Feynmann theorem_ Hence, the fit-order properties can be legitimately cakulated [15.29] as the average values of the appropriate operators. 3J_ The pohrized GTO basis set for LiH The ElV bases have initiahy been proposed in order to make feasibIe the calculation of moIecuIar electric properties with rather small basis sets [l&12]_ Since the present study is designed for highIy accurate caIcuIations of electric moments of the LiH moIecuIe, it is obvious that sufIicientIy Ianle basis sets have to be used from the very beggnning Pn order to make our scheme for the determination of pohuized basis sets as systematic as possiiIe. the initial GTO basis sets are assumed to foIIow from atomic caIcuIations_ For both Li and H we have started with the energyoptimized basis sets of van Duijneveldt [7J; the 13s GTO set for Li and the 10s GTO set for H To take into accouut the diffuseness of the eIe&ron density distribution on the hy&=en side of the Lili molecule, the H atom GTO basis set has been further augmented with two soft GTOs, whose orbital exponents follow from the geometric progression assumption appkd to the originaI van DuineveIdt*s basis set Additionally some contraction of GTOs with high orbitaI exponents has been performed with the contraction coefficients taken from the corresponding atomic SCF orbita& The finat. GTO/CGTO basis sets for Li and H which have been employed for the determination of the corre-
mnmam o/Ike Lix
i#lok-dc
_
spending polarized sets can be written as [13_/8] and [12/S_]. respectively_ Tke p subsets of polarized basis sets have been obtained by the method described in the previous secticn with the contraction coefficients foilowing from the change in the LCAO SCF coefficients for atomic orbitak Since the pohuization of very inner reggons of the ekctron density distriiution has a negggible effect on the caIcuIated properties, some of the resulting p GTOs with the -highest vaIues of orbital exponents have been deleted from the p subset_ If de above sp basis sets are employed in mokcular caIcuIations, the p orbitals make a considerable contribution to the bonding Hence, their further polarization should be considered. This kads to polarized d subsets for both atoms. Again, some basis set functions which either correspond to relatively high orbital exponents or folIow from the polarization of those initial GTOs which appear in the restthing molecular orbitaIs with very smaII LCAO coefficients, have been deleted from the d subset_ By~a similar consideration aIso a subset of two f GTOs contracted to a singIe C-0 has been produced for Li. The final poIarked basis sets, the [13_8.62./8.5.3.1.] set for Li and the [12.8_5./85.3.] GTO/CGTO set for H, are presented in table l_ It is important to point out that by the method of the derivation of polarized basis sets, the orbital exponents for poIarized functions cover a wide range cf vahtes_ Hence, our pohuized basis sets should be simuhaneously suitable for the caIcuIation of correIation energies This is aIso one of the reasons that higher than p pohuized components have been taken into account They are simpIy needed for the poIa.tization of orbit& which appear in other than the ground state configurations when the CI wavefunction is buik Let us also recall that neither the -orbital expoxiexk nor ,the contraction-coeffxcient optimization are invokd in constructing the moIecuIar basis set lof table 1_ 3-L HF SCF reszdts _ The HF SCF resuhs for the to-d energy, di&e moment and quadrupole moment of -the LiH mokcuIe at the experimental equiIiirium distance
i416.8112 321.45994 91.124163 w-999891 ~11.017631 4.372801 1.831256 O-802261 0.362648 0.113995 0_051237 0.022468 psubxt 1
11.017631 4.372801 1.831~256
Oman075 0_@062 OBOO584
1776.7756 254.01771 54-698039
o_om372 OBO2094
0_012605 O.&Q356
IS.018344
0_008s63
0.114780 0239381
4-915078 1.794924
0_030540 0.090342
1 1 1 1 1 1
0_7107l6 0301802 0_138W6 0.062157 O-027%7 0_012583
I 1 1' 1 1 1
0_0346
4-915078 1.794924 O-710716
0.0138
0.1145 0.2fGl4
0.2S29
0.802261 0362648
0.1117 0.1470
0_3M802
1
3
0.113995 0_051237 0.022468
1 1 I
0_138iM6 0.062157 O-017957
1 1 I
4372801 I.831256 0.802261
0.0547 0.1954 03783
1.794924 0.710716
0.0503 0.3000
2
0.113995 O-051237
0.1816 1.1978
0304802 0.138046
0.1156 O-2460
3
0_07E468
1
0_051237 0_022468
05292 0.8151
dsubscr 1
fsubset 1
-.
0.068-t
2
4 5
... {I
~0.052157
1
.
of 3.015 au, caIcul&d with the &arized GTO/ CGTO basis set of table J, arc shown in table 2. They are compared. with recent results of -seminiimerjcd [23_24] and fully-nume&aV two-dimensional [25,26j HF SCF calculations The latter data can be amsidg-ed as the near-e%act~HF SCF values for the properties listed in &bIe 2: Also some sekcted au&ate re$ul& of +&%a HF .!XF
calculations- with. ST0 -and -GTO basis sets are incIude& _. A comparison of our pr-nt k&s with those of the ntieri&l HF SCF caki&tic& allows us to draw some preJ‘ ‘y ConclusionS about the.quality of the- @~Miz.ed~ GTO)CGTO -b&s se?. cjerived in .this p&erfrom-standard energy_Optimize$ has+ sets for Li and EL ~bviotily, it i$not-‘: _
s_o_Raos. it-r_ skftq- / Elarric
3s Tabk
momoln
of Kite LiH
nsdde
2
moment of the LiH mokwk
RaulCSoCaCCtZUC HFSCFcAahiomofLhcW~_dipokmoment~dquadrupok ~R-3_015au.Mlvalue~inatomicunit~ inremrrdcy E-
chisa-ork accuraccreferencesresulls oumaicaI SCF SCF
EWGTOKF GTOSCF
lW PI [3i] =’ 1191dB [32] *
GCF
PSCF
-7_9!n27s
23625
-7987352 - 7_987352* -7_9s7313 -7%7072 -7_987063
23618 23618105 ‘x61 2361 23620
at the
(y&a) -33719
- 3_37uo6 L’
-
momem -mr taken at the nuckar antrc of mass for ‘Li’H_ All dcfinitio~ according to *Theori@Forthequadstpok =d=Q&== !33l. b’ C&t& fm &c v&c of the dccuonic contriicn tt the molecular midpoint and the dipole momalt whu2 of ref_ [30). 0 ~a4-E c&uhtions pith cnergs-dptin&cd basis set of six s_ four pa. two do. and one fa orbitak for Li and *&~czs and ollt PO orbimIs for H 131). s) CahWiom uiih &C [11_63_/5_63_] EW GfO/CGTO basis set for Li sod the [115_/75.] ElT GTO/cGTO basis set for H [191_ * HF SCF d&W wish rhc [14_i_3_/9_73_] GTO/CGiO basis sex for Li and the [10_6_/3_6_] GTO/CGTO basis set for H [32k
surprisingthat our SCF energy is very cIose to the best HF SCF result obtained within the finite basis set approximation_ Also in comparison with the numerical HF SCF energy value we miss less than lo-’ au_ In addition both the dipole moment and the quadrupole moment calculated in this paper differ from the numen_cal HF SCF data by = 0.03 and 0.0695, respectively_ The HF SCF results of Lazeretti et aL 1321obtained with a huge but rather ad hoc selected GTO/CGf’O basis set appear to be slightly better for the dipole moment than those calculated in the present paper_ Howexr_ the aaxna cy of their dipole moment is not accompanied by a similar accuracy of the total SCF energy_ In the case of polarized basis sets the method of their derivation suggests that the ekctric moments should be ahnost as accurate as the HF SCF energies, though r&able measure of their relative accum ties is rather difficuh to establish_ It is of interest to note that the results of HF SCF calculations with reIatively sti EFV GTO/ CGTO basis sets [19] provide the dipoie moment values of quite high accuracy_ This gives a further support to the reliability of our method for the basis set det emrination for calculations of mokcular electric properties_ Quite unfortunately the HF SCF data are not repotted for the largest GTO/CGTO basis set for the LiH molecule (113_11_83_1_/11_11_83_1_]on Li
and [1453_2_/10_5_32_3 on H) employed recently by Handy et al. [22]_ However, from the comparison of their orbital exponents with those of tabIe 1 we can conclude that in both cases the HF SCF energies should be nearly the same The accuracy of HF SCF dipole and quadrupole moments which would follow from the basis set of Handy et aL 1221is more difficult to predicr, though it should also be nearly the same as that achieved in our calculations_ 3_3_ CAS SCF resuffs The complete active space (CAS) SCF approach_ whose detailed description can be found elsewhere [27,28], is a multiconfiguration SCF scheme based on a full CI wavefunction built within some limited number of (active) orbitals. The orbitak_ whose occupation numbers have fiied non-zero value are classified into what is called the inactive orbital subspace_ The CAS SCF wavefunction can be uniquely specified by the number of electrons in both subspaces and the list of inactive and active orbitak The symbols used to denote the CAS SCF ~avefunctions for the LiH molecule are based on the classification of molecuiar orbitals according to the C, symmetry group and their detaikd description can be found in our earlier papers [19_34]_:
B_O_ it-
Li-
Scdkj
/
EIe&c
tn&em o; the -Liwmokcde
:.
-
’
_.4g.--:
:
The use of a fuII CI wavefunction in the CAS SCF app_roach is undoubtedly one- of its major adVantages: Above aU, no a priori selection of what are usuaIIy izalkd the important ~contigura- ‘tions is invoIved_ Moreover. the CAS SCF wavefunction for_ perturbation-independent basis sets satisfies the Helhnkn-Feynnran theorem and facilitates the caktdation of fmt-order properties_ However. similarIy to aII other computational methods of quantum chemistry, it also suffers from some difficulties_ The major problem is the Iength of the fuII CI expansion which considerably knits the size of the active orbital subspace Hence, a careful consideration of the choice of active orbi& followed, whenever possible, by the appropriate convergence study_ is necessary. Such a study has been carried out in the present paper prior to the selection of the most reliable active space of orbitak and the corresponding data are reported in tabIe 3. In the first CXS SCF calculation the valenceshell approximation has been made and the active space has been Iimited to three G and one c orbit& as iu our previous calculations of electric properties of the IiH molecule [19]_ In aII other
caIc$ations the valence-shell approximation h&s ‘: been_-released sinceit % not expectedto-:provide high enou& a&racy because of-: rehit.iveIy imiportant (Is, 2s) correlation effects-The second &AS -SCF wavefunction based on the active orbital sub-space corn&-king six (J and two = orbita& has also been investigated previously [19] and found to be highly satisf&tory in tams of both the total mokcular energy and electric properties of LiH. Since the occupancies of the most weakly occupied. naturai orbitals are still of the order. of. 10m3, a further extension of both the G and k orbitat subspaces has been made, resulting in the (OOOO/ 8330) CAS SCF wavefunction which comprises 993 configuration state functions (in GV symme-. try)_ As can be seen from the data of table 3, the occupancy of the most weakly occupi&l natural orbitals becomes reduced to = 10mJ_ -. -.-A further increase of the GS active subspace is not expected to bring signitkmt changes in either the dipole or quadrupole moments_ The occupancy of additional natural orbitak wilI be less than IO-” and the changes of occupancies of natural orbitak which are already used in the (0000/8330) CAS SCF wavefunction are expected to be even
Table 3 Total energies, dipoIe momems quadmpole moments_and occupation numbers of natural orbirds in difkres for the LiH mokcuIe at R = 3.015 au_All valuesin atomic units
CAS SCF ca!cuhtions
Inacri\_c/ti%-esubspacc
-crgy dipole moment quadrupole moment a) occupation numbers cc, lo 20 3o 40 50 6a 70 8a lZ&.lq 2n,-, 3%3-, lq~.lq=___~
(1000/3110)
(ONO/6220)
wo0~1)
-8.021109 23058 -3.1161
- 8.057253
- 8.059499 22921 - 3.0699
-so57774 23001 - 3.0938
- 3.0967
SyNnerry) ZOO000 1.93979 0.03088 o.OOs43 -_
0.01045
-
I-99370 Z-93976 0.03099 0.00856 0_00244 0_00117
0.01047 0.00123
__-
a Origin for Ihe quadmpole moment operator at the nude;lr~ccntrcof.-
I-99320 I.93973 0.03113 0_00846 0.00256 0_00122 0_00020 O-GO006 0.01033 0_00129 o_OcKy
I_99361 1.93983 0.03091 0.00853 0_00244 0_00121 0.01046 0.00123
.-
0-m
for 7Li’H;-
--
1
50
B-O_ Roar. AI _ so&j
/ Ei?xuic momentroftJlefiHmoIde
smaIIer_ The corresponding orbitaI contributions to *be dipore and quadrupoIe moments turn out to be Iess than 10-s au. Moreover, multiconfiguration SCF optimization~of very weakly occupied natural orbit& represents a ratherdifficuIt numerical probkm because of numerical instabiIities_ Hence, BXZdid not attempt further extensions of the ez active orbital subspace_ In principIe one may expect that including S orbitaIs in the active subspace could be of some importance_ For this reason we have investigated the (OOOO/T221) CAS SCF wavefunction obtained by exteuding the active subspace of the (WOO/ 6220) function with one S orbital_ It foIIo\kisfrom the data of table 3 that the corresponding effect on the dipole moment, quadrupole moment and occupation numbers is ahnost negIigiiIe_ However, in terms of the t0ta.Ienergy incIuding a 6 orbital in the active orbitaI subspace resuhs in a noticeable improvement_ Since the total moIecuIar energy is not the principal objective of our caIcuIations. deleting the S orbital from the active subspace appears to be quite acceptabIe_ Thus, the rest of OUrCii scussia wili be concerned with CAS SCF rest&s obtained witb the (0000/8330) wavefunction_ The correspoilding summary of our data at the e__perimental equilibrium distance of 3-015 au and their comparison with recent most accurate vahxs of other authors_ are presented in table 4_ It is obvious -that one cannot expect a very high accuracy of the totaI CAS SCF energy cakulated within a reIativeIy smaII active orbital subspace AIso a part of the difference between our result and the estimate of the exact non-relativistic Born-Oppeeeimer energy of LiH comes from the fiite basis set appnximation and reIativeIy poor vahte of the core correlation energy_ However, the missing part of the core correIaticn effects is rather unimportant for the accurate prediction of the dipoIe and quadrupoIe moments_ It can be seen from our (1000/3110) CA!S SCF results that the major part of the coixIation correction to these two properties comes from the vaknce-sheII correIation effects_ It has aheady been mentioned that the CAS SCF method for perturbation-independent basis sets satisfies the HeIImann -Feynman theorem, and thus. the expectation values of one-ekctron op-
Table 4
conlparhn of_difraulr aL_ruraKCfhcorcIica!results ror the Liz-I mokcuk at the inumuckar distance R - 3.015 au All xalucs in atomic &its
ibis work b’ rcr_[35] rek_ [20_2l] C’ ref_ [zz] csact
-
S-059499 8.063866=’ 8_065& &06904 5.07049@
Dipole momcnr
Quaanrpole xnomcnt*
x97-1 2_2w2~ 22917 725 x93h’
- 3.0699 - 3.0655 0 -
=’ The origin for the quadmpok moment opemor at the nuckar ccntrc of mass for ‘Li’H_ b’ Rcsuh cakulaccd with xhe