Localized molecular orbital studies in momentum space. II. Correspondence between coordinate and momentum space properties of various electron pairs

CHEMICAL

PHYSICS6 (1974) 282-290. Q NORTH-HOLLAND

PUBLISHING

COMPANY

LOCALlZEDMOLECULARORBlTALSTUDlESlNMOMENTUMSPACE.ll. CORRESPONDE~CEBETWEENCOORDlNATEANDMOMENTUMSPACE PROPERTlESOFVARlOUSELECTRONPAlRS M.H. WHANGBO and Vedene H. SMlTH Jr. Department of Chemisny,

Queen’s

University,

Kingston.

Onratio,

Cotul

and WolfgangVON NIESSEN Lthsfuhl I% l%eore,ische

Chemie der Technische

UniversWr

Milnchen.

8~tfUnchen2, Germany Received 9 August 1974

The momentum space propenis of the temlectron systems Ne, HF. H20, NH3 and Cb as well as those of CHSCHJ.CHzNH2.CHjOH and FCHzOHwere investigated by using localized molecular orbitis (LMO) obtained from ab inido selFconsistent-field (SCF) wavefunctions constructed from double zeta quality gaussian basis sets Compton profdes of various LhlO electron pairs (CC, CN. CO, CF; CH. NH, OH, FH bond pairs and C, N, 0, F lone pairs) ate tabulated In order to understand the correspondence between the momentum and Ihe coordinale space properties of those electron pairs, the concept of the sue and the shape of an LMOelectron pair charge disttibutiott has been utilized. The use of the intermediate expectation values of pn is intraduccd for the purpose of interpreting the momentum space properties. The dependence of molecular property partiuonmg on tbfferent localization schemes and on different basis sets is also studied by using the Hz0 protile as an example.

I. Introduction In the previous paper (part I) of this series [ 11, we observed that the momentum.space properties of localkd and canonical core electron pairs are not the same, and also that those of the carbon-carbon single and double bond electron pairs (obtained from the energy localization procedure) are distinguishable in agreement with experiment [2]. The principal objective of the present study is to establish a correlation, though it may be qualitative, betweeh the coordinate and the momentum space properties of molecules. To achieve this goal we have attempted to find a consistent trend by studying a number of closely related molecules and electron pairs. The ten-electron molecular systems HF, H20, NH3 and CH4 were selected in the present study, from

which one obtains CH, NH, OH and FH bond electron pairs as well as N, 0 and F !one pairs (1~). To complete the latter series of e!ectron pairs, the methyl anion CH3 was also considered, although it is a charged species. Another interesting series of bond electron pairs are C-C, C-N, C-O and C-F. The C-C bond electron pair obtained from CH$H, was already considered in part 1. To obtain the C-N and C-O electron pairs, CH$H2 and CH30H were respectively chosen. Finally the C-F bond electron pair was taken from the wavefunction of FCHsOH [3] rather than from that af CH,F, since the former was available to us. 2. calculations

Ab initio SCF MO calculations were carried out for HF, HzO, NH,, CH4W2 and CHsOH by using their

MH. Whangbo et a.& LMO studies in momentum spnce. II

283

experirnentaI geometries* and an extensive basis of gaussiantype functions (CTF) of double zeta quality.

sla = ~$ar&5a,, ,

These GTFs are the same as those employed for CH4

where R, = (@a&a)o. Here Q2, is the trace of the setond moment tensor, that is,

and CH3CH3 in part 1. Namely, the GTFs for C, N, 0 and F were obtained from the work of Huzinaga and Sakai [4], while the GTFs for H from that of Basch et al. [S]t. The localized molecular orbit& (LMO) were then obtained from these canonical molecular orbitals (CMO) through the use of the energy localization procedure developed by Edmiston and Ruedenberg 161. Finally the Dirac-Fourier transformation [7] was ap plied to these coordinate space LMOs to generate

(0

or

their momentum space properties, Further details of the momentum space integration were specified in part 1. Due to the invariance of the one-particle density matrix (within the independent particle model) against any unitary transformation among the occupied molecular orbitah [S], every total molecular property is the same either in the CM0 or in the LMO approach. However, the main advantage of the latter approach is to make possible the quantitative interpretation of molecular properties in terms of core, bond and lone electron pairs.

3. Results and discussion

(2) after an appropriate unitary transformation. This transformation may be viewed as a rotation of the

coordinate axes such that the internuclear axis of a bond electron pair coincides with one of the rotated axes** . Then the diagonalized components (~‘~>, and (z’~)of an LMO electron pair provide us with qualitative information about its charge distribution. These quantities partially account for the directional properties of the electron pair charge distribution that is lost in the detinition of size. In this sense they are

termed as the sfrupeof an electron pair [I 11. 3.1. CJurrgeand momentum disnibution

Coulson [9a] has observed that in H: and H2, the contours of equal momentum density in a plane through the internuclear axis are nearly ellipses with elongation perpendicular to pX, where the x-axis coincides with the internuclear axis. The charge density contours are elongated along the x-axis. The size 52, of an LMO electron pair $a is defined (lo] as its second moment measured at the centroid R, of the charge distribution

FOICHS, the CH bond length of CH.,is used. The final og timized geometry gives 105” as the aHCH. t For the second row atoms, [ 13s7p]primitive GTFs tie contracted to double zeta [4’2t’],whilefor H [4s1primitiveCTFs to double zeta [ 2s]. By using these basis sets, we obtained tbe total energiesof -39.4617, -56.1717, -75.8863, -100.0327, -128.5434, -95.0808 and -115.0111 au for CH;, NHs. HaO, HF. Ne, CHJNHa.C&OH. respectively. l

For the various electron pairs (CC, CN, CO, CF; CH, NH, OH, FH bond pairs and C, N, 0, F lone lairs), our size and shape calculations are summarized in table 1. Here the largest component of the diagonahzed second moments is arbitrarily labelled as (~‘3, and only the average value of and (z’2)is listed because the latter are nearly the sarneti. As one would expect we notice from table 1 that each electron pair charge l

* The chargecentroid of a localized electron pair does not always tie on the given internuclear

axis. However the d&a-

tion from the axis is small [see CA. Naleway and hf.E Schwartz, J. Am. Cbem. Sot 95 (1973) 82351, except for some special cases such as localized electron pairs for muItiple bonds and bigbly strained molecules. tt The axis of the largest component of a bond p& UXTCspends to the internuclear axis. For B lone cle’ctranpair, it correspondsto the direction in the directed lone pair model In special cases such as LMOelectron pairs for multiple and highly strained molecules, the similarity behueen (JJ’~> and k’2) is not sogood as that observed from the ekctron

pairs lisled in table 1.

284 Table 1 Sizes and

shapesof someelectronpairsa)

CH3 Hz0 HF

CiP N IP 0 IP F IP

Q4

C--H

1.120

0.838

NH3

N-H

1.035

NH3

1.146 0.946 0.842 0.736

1.032 0.782 0.634 0.586

1.856 1.455 1.293 1.108

H20

O-H

0.938

cl.741 0.668

HF

F-H

0‘893

0.598

I.631 I.476 1.331 I.230

CH~CHJ Cww CHJOH CSFOH

C-C C-N C-Q C-F

1.269 1.162 1.139 1.019

0.828 0.758 0.689 0.614

1,727 I.581 1.499 1.339

a) ~rou~o~t in au.

tables l-4.

each electron pairisfrom the molecutes listed in the first column of fable 1. Here numerical

distribution is elongated in one direction. Therefore if we consider the shape of an electron pair d/(r),the fo~lo~g inequality is expected to hold: (2) > (y2>6%(21,

f3)

where the superscript’ wasdropped in order to sim-

VakWS

are

Wthin the impulseappro~a~on fl7 j and the independent particle model, the Compton profile .!i(q) of a molecular orbital (MO)i is givenby 00

plify the notation.Then the fo~o~g expre~ions WI (x2,(p21;I;a;, @P&+, k2,(p2Df 2 x Y together with eq. (3) make it plrmsiblethat

(4)

$P;, < (p:) = (p,“>*

W

Since Q.?> = 0 for realwavef~ctio~s, the quantities (p$, ($1 and #> chaI;ict$ze the shape of the momentum distribution of the bond pair,just as cX2& @> and @I do in coor~na~ space. Eq. (4) suggests that a charge distribution elongatedalong the bond (Le., a prolate e~p~id) will cowhand to a momen. turn ~s~bution contracted in the bond direction (i.e., an ablate ellipsoid).This holds irksomecases [sta,13,141,but is not truein general [9b, 151.Thus caution mustbe exercisedin the use of eq. {4) to reachsuch con&sions. In addition,it should not be forgottenthat momentumcontourdiagramsof diatomicmoleculesalso revealsome detailedstructure [15,16j.

where Ii(p) is the radia! momentum probab~~~ disk% bution of the MDi in momentum space, and 4 represents the initial momentum of the electron along the ~tt~~~ vector. The Compton profde and the radial momentum distribution are normaiizedfor each efectron to unity m cx) f f$.PMP= 1 I

0

s J&q)dq = 1 _ -50

0)

The radial moments ~s~butio~s for the bond electron pairs CC,CN, Co and CF (all singlebonds) are shown in fig. 1. As the atomic number of X in the C-X bond electron pair (where X represents C, N, 0 or F atom) increases,the momentum distribution of the C-X bond is shifted more to the large momentum region.This meansthat electrons in the C-X electron pair are more tightly bound to their constituent nuclei with increasing atomic number of X, for X in the

a:CC b:CN

Fig. 1. The mdial rn~rn~n~rn distributions of the CC, CN, CO and CF bond electron p&.

samerow. On the other hand, table I shows that the electron pair size of a C-X bond becomes smaller, as the atomic number of X increases. Therefore we

notice from the C-X bond election pairs &at the more contracted charge distribution in coordinate space co~espon~ to the more diffuse momentum dis-

IX

a:

CH

Fig.2. Theradialmamc~~rndistibutiom of the CH, NH,OHastdFHbandelectranpain

286

a: Clp

P Fig. 3. The radial m~en~rn

~s~b~tio~

t~bution. The same trends as the above are also observed in the bond electron pairs CH, NH, CH and fW and in the lone electron pairs C $, N lp, 0 lp and F Ip (see figs. 2 and 3). 3.3. Qmpron profiies Compton profiles of the various electron pairs considered in the previous section are given in tables 2,3 and 4 and in figs.4,s and 6. It is noted that electron pairswhich have more electronegativeatoms result in lower and broader Compton profiles (see CC, CN, Co, CF; B1, NH, OH, FH; C lp, N Ip, 0 lp, F Ip profdes). By consideringthe radial momentum distributions of these electron pairs, it may then be said that as more electrons are dist~buted in the smallmomentum region, the correspondingprofile becomes sharper. That Compton profdes are mainly due to electron distribution at low momentum can be shown numeticahy by using the intermediate expectation valuesof p” defmed as follows:

of the C, N, 0 and F lone electron

pairs.

where I(p) repre~nts the total radial ~rnentu~~ distribution of a system (within the independent particle model, one can define similarexpectation values for MO).Then the total number of electrons in the system Cp”>is Ipolo, and b” 1, has the mea~g of the number of electrons whose momentum is not less than a. Therefore [p”&‘is the contribution from those electrons to (p”k Particularly interesting cases are Table 2 Comptan

profies

of Mme bond electron

pairs

9

cc

CN

CO

CF

0.0 0.1

0.4844 0.4840

0.4489 0.4486

0.4072 0.4072

O-3636 0.3634

8:: 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.2 1.4 1.6 1.8 2.0

0.4323 0.4776 0.4668 0.4461 0.4155 0.3738 0.3248 0.2726 0.2216 0.1352 0.0768 O&426 0.0243 0.0152

0.4432 0.4412 0.4350 0.4197 0.3957 0.3632 0.3237 0.2803 0.2362 0.1562 0.0962 O.OS?O 0.0339 0.0212

0.4042 0.4064 0.3990 0.3886 0.3715 0.3472 0.3166 0.2817 0.2447 0.1738 0.1160 0.0748 0.0482 0.0320

0.3607 0.3626 O-3565 0.3492 0.3372 63204 0.299 1 0.2740 0.2465 0.1896 0.1379 0.0964 0.0665 0.0466

M.H. whangbo et al, LUO studies in momentum space I1

287

Table 4 Compton profdes of some lone electron pairs

Table 3 Complon profiles of some bond electron pairs 4

CH

NH

OH

FH

4

ClP

NIP

0 IP

F 1~

0.0

0.6005

0.5949

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.2 1.4 1.6

0.5777 0.5478 0.5054 0.4526 0.3935 0.3326 0.2742 0.2210 0.1750 0.1056 0.0620 0.0363

0.5265 0.5228 0.5115 0.4917 0.4630 0.4262 0.3830 0.336 1 0.2886 0.2431 0.2014 0.1331 0.0852

0.4708 0.4678 0.4589 0.4436 0.4218 0.3939 0.3610 0.3249 0.2873 0.2502 0.2150 0.1536 0.1064

0.4189 0.4167 0.4101 0.3988 0.3828 0.3621 0.3375 0.3099 0.2806 0.2501 0.2214 0.1678 0.1236

0.0

0.1

1.8 20

0.02 16 0.0133

0.0537 0.0339 0.02 19

0.0724 0.0492 0.0336

0.0895 0.0643 0.0465

0.6504 0.6469 0.6214 0.5794 0.5231 0.4569 0.3863 0.3168 0.2530 0.1976 0.1517 0.087 1 0.0504 0.0303 0.0191 0.0126

0.4859 0.4835 0.4757 0.4612 0.4390 0.4089 0.3722 0.3312 0.2886 0.2470 0.2083 0.1436 0.0970 0.0654 0.0446 0.0310

0.4295 0.4276 0.4211 0.4109 0.3945 0.3122 0.3449 0.3137 0.2807 0.2476 0.2158 0.1602 0.1169 0.085 1 0.0623 0.0460

0.3640 0.3629 0.3592 0.3521 0.3427 0.3290 0.3119 0.2919 0.2697 0.2465 0.2230 0.1783 0.1394 0.1075 0.0824 0.0631

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.2 1.4 1.6 1.8 2.0

0

lb

D:

NH

0 5-

0

o-

b

Q4-

0

B

20

30 3-

-J

a

0.2-

0

0 I-

0

Fig. 4. The Compton profiles of the CC, CN, CO and CF bond electron pairs.

IO

20

Fig. 5. The Compton profiles of the CH, NH, OH and FH bond electron pairs.

M.H. Whangbo et al, LMi.3 stud& In momentum space u

288 0.

o 0:

\

O!

clp

5: titp c: otp

0:4^D 3

0.

0. 3-

0. 2-

0 .I -

I_ 0.0

IO a

20

Fig. 6. The Compton profiles of the C. N, 0 and F lone electron ptis.

when n = -I and 2. In the former case, we have the relational) = $‘*]q, while in the latter case !Jj& means the contribution to the kinetic energy by the Ip”], electrons. Considerationof Ip-$ and (p2], together with b”], for variousvdues of II revealsthat a Compton profile is very sensitiveto the electron di+ t~bution at Iow moments, but that lhe kinetic energy has quite the opposite behaviour. For example, ap proximateiy two electrons are found in the momentum region 0 Gp 4 1 au in the case of H20. The contribu tion from these electrons amounts to roughly half the total J(O), but only about 1%of the total kinetic energy of HzO. Detailsof this approach will be reported Qsewhere[ 18, !P]. A comparison between our l..MOprofiles and those of Epstein [ZO]is in order. In the latter work Slater type minimal basis sets have been employed. For @d

and Co bond electron pairs, our profiles are lower and broader than Epstein’s,the ~fference being larger than that we found for the hydrocarbon LMOprofiles in part 1.Howeverit is not surprising,because our CN and CO profdes were obtained from CH$WZ and CHJOH,while the correspondingprofiies calculated by Epstein were from the LMOsof HCNand H$O. Thus our profiles represent C-N and C-O single bonds, but Epstein’sprofiles correspond to C=N and C=O multiple bonds. Then the above discrepancy is in agreement with our conclusion in part I that multi. pie bond electron pairs have more expanded charge distributions than do singlebond electron pairs, and thus the Compton profile of a singlebond is lower than that of the correspondingmultiple bond. Significantdiscrepanciesbetween our results and those of Epstein [20] occur for the electron pairs of loneqptircontaining molecules.Our NH and OH profiles have lower peaks than do Epstein’s,while our N Ip and 0 Ip profdes are higher. Whenwe sum those LMOprofdes and construct the total profiles of NH, and HzO,however,Epstein’sand our total proftiesare not as different asare the individualLMOprofiles&e table 5). Thiswould nom&y indicate that a certain molecularproperty, the Compton profile in our case,is differently p~tioned due to different basisset quality, because the same localization procedure (the energy localization) has been employed in Epstein’sand our work. But our NH and N Ip and OH and 0 Ip profies were taken from NH3and H20, respectively,while the co~espon~g valuesof Epstein represent averages over certain molecules.Therefore no meaningful comparisoncan be made for those profiles. Recently von Niessenhas shown [Zi] that the various intrinsic localization methods lead to almost the samepartitioning for poiyatomic moleculeswhen the samebasis set is employed. In order to see the dependence of the molec~ pro~r~ parti~o~g on tiferent localization procedures and on different basis sets, we have calculated the LMOprofiles by using both the energy [6] and Boys [22] localization schemesfor the Hz0 wavefunction of Diercksen1231. This SCF wavefunction is constructed from GTFs which have polarization functions on the H and 0 centers.(The ~l~s~ld~6slP]p~~~veGTFsarecontracted to the [5s4pId/3slP] GTFs.) Table 6 summarizes this result, where the 2nd column contains 1(O) valuestaken from tables 2,3 and 4, and the next two

289

M.H. Whangbo et aL. LMO studies in momentum space. 11

Table 5 Values of I(0)

for the ten-electron

isoelectronic

scnesa)

Molecule

Exp.

Cornille et al. fl

Ne HF

OS22b)

Hz0

0.739c) o.73od) -

2

0.991 e)

EpsteinB)

Present

0.545 0.646 0.721

0.7377

0.5458 0.6391 0.7600

0.923 1.080

0.8646 1.034

0.&192 1.0157 h)

a) Each number represents I/S of the total J(O) vahrc. b) P. Eisenberger. W.H. Hennekcr and P.E. Cade, J. Chem. Phys 56 (1972) 1207. The atomic core contribution is added to the valence protile. See: R.J. Weiss, A. Harvey and W.C. Phillips, Phil. hlag. 17 (1968) 241. C) W.A. Reed and P. Eisenberger (private communication, November 1973). d) R.J. Weiss (privatecommunication, June 1973). e) P. Ejsenbergcr and WK. Marra, Phys. Rev. Leltcrs 27 (197 1) 14 13. D hi. Camille, hf. Roux and B. Tnptine, Acla Cryn. A26 (I 970) 105. These values were ulcuhted from the one-center wavefunc lions of hloccia. See: R. Moccia, J. Chem. Phys. 40 (1964) 2164,2176,2186. g) Ref. [ 181. h)Ref. [I]. Table 6 The dependenceof the molecular property partitioning on locahzation schemes and on basis sets: theJ(0) values of HaOa) Electron

ELhfO

BLMO

par

0 core OH OJP

A

B

B

0.1982 0.9416 0.8590

0.1986 0.9312 0.8698

0.2057 0.9308 0.8666

a) These values are for two electrons. ELMO: BLMO: A: GTF B: GTF

the energy localized molecular orbitals [a]. Boy’s locahxd molecular orbitals [ 221. basis set: [ 13s7p/4sI contracted IO (4S2p12Sj. basis set: I11S7pld/6S1p] contracted to [5S4pld/3Slp].

or the calculated ones from the various quality wavefunctions. Namely, as the number of atoms in the above series increases the profile peak becomes higher. This trend is also related to the molecular charge distribution, which may be qualitatively estimated by molecular sizes [IO]. The square root of these molecular sizes are respectively 2.165,2.593,3.104,3.629 and 4.263 au for Ne, HF, H20, NH3 and CHJ. That is. as a molecule has more atoms, its charge distribution becomes more expanded, which leads IO the concomitant sharpening of its profile. As an example of the application

of our LMO profiles, we list the experimental profile of teflon 1241 as well as our theoretical values (table 7). The agreement between the two is quite excellent.

from the H20 wavefunction of Diercksen. It is to be realized that the property partitioning based upon localization is far less sensitive either to the different localization schemes or to the different basis sets than one might have thought from the previous comparison of our LMO profiles and those of Epstein for the lone-pair containing molecolumns

show those

Acknowledgment

Support of this research by the National Research Council of Canada is gratefully acknowledged. The authors are also thankful to Professors J.W. Graham

cules.

The total J(0) values of the ten-electron molecular systems are listed in table 5. For the purpose of completeness, the J(O) value of the Ne atom has also been included in this table *. One can easily find a consistent relationship between the number of atoms in this series and their correspondingJ(0) values, whether one considers the available experimental J(0) values

l

The GTFs of Huzinagaand Sakai [4] have been used for our computation for neon. We have also carried out the LhIO analysis, and Ihe trends in the sadiaJ momentum distributions and the Compton prof’des for C Ip, N Ip. 0 Ip and F Ip do not change even though the Ne Ip is added to the above series The I(O) values of Ne core and Ne Ip are respectively 0.0775 and 0.3215.

MH. Whangboet al. LMO SNdies

290

Table 7 The Compton proftic of teflon 9

&P.

Present work

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0

7.43 f 0.1 7.43 7.12 6.58 5.60 4.56 3.80 3.01 2.39 1.81 * 0.1 1.48

7.4242 7.3556 7.0908 6.5300 5.6936 4.7312 3.8036 3.0028 2.3596 1.8710 1.5144

and J.B. Moffat and to the staff of the Computing Center of the University of Waterloo for their hospitality and he use of their facilities. The authors wish to thank Drs. R.J. Weiss, WA. Reed and P. Eisen-

berger for communicatingtheir experimental results prior to publication, and Professor I.R. Epstein for his helpful comments and for communicating his unpub lished results.

References (11 V.H. Smith Jr. and hl.H. Whangbo, Chem. Phyr 5 (1974) 234. 121 P. Eisenberger and W.C Mana, Phyr Rev. Letters 27 (1971) 1413. [3] S. Wolfe, A. Rauk, LM. Tel and LG. Csizmadia, J. Chem. Sot. (B) (19711 136. [4] S. Huzinap and Y. Sakai, J. Chem. Phys 50 (1969) 1371.

in momentum spore. II

[S] H. Basch, hlB. Robin and N.A. Kuebler, J. Chem. Phys 47 (1967) 1201. 161 C Edmiston and K. Ruedenberg, J. Chem. Phyc. 43 (1963) 597; Rev. Mod. Phyr 35 (1963) 457. [ 71 R. Bencsch and V.H. Smirb Jr.. in: Wave mechanics The first frfty years, eds WC. Price, S.S. Chissick and T. Ravensdale (Butterworths. London, 1973). LS] T.L Gilbert, in: hlolecular orbhals in chemistry, physits and biology, eds. P.O. Lawdin and B. Pullman (Academic Press,New York, 1964) p. 405. [9] a. CA. Coulaon. Proc Cambridge Phil. Sot 37 (1941) 55; b. C.A. Coulson and WE. Duncanson, Proc Cambridge Phil. Sot 37 (1941167. [ 101 MA. Robb. W.J. Haines and l.C. Csizmadia, 1. Am. Chcm. Sot. 95 (1973) 42. [ 1l] M.H. Wbangbo, S. Wolfe and I.G. Csizmadia, in: The 56th Canadian Chcmicnl Conference, Abstracts of papers No. 164 (1973). 121 A. Messiah, Quantum mechanics (NortJr-Holland, Arnsterdam, 1966)Vol. I, p. 134. 131 I.R. Epstein, Accounls Chem. Res. 6 (1973) 145. 141 I.R. Epstein, in: International Review of Science, phys icai chemistry, Series II, Theoretical Chemistry. 151 P. Kaijser and P. Lindner. TechnicaJ report TN417 from Quantum chemistry group, Uppsala Univ. (19741. 161 W.H. Henneker and P.E. Cade. Cherp. Phyr Letters 2 (1968) 575. 117) P. Eisenberger and P.hl. Platzman, Phys Rev. A2 (1971) 415. [ 181 V.H. Smith Jr. and M.H. Whangbo, to be published. 1191 hl.H. Whangbo. V.H. Smith Jr. and R.E. Brown, to be published. (201 I.R. Epstein, J. Chem. Phyr 53 (1970) 4425. 1211 W. von Niessen, Theoret. Chim. Acta 27 (1972) 9; 29 (1973) 29. (221 S.F. Boys. in: Quantum theory of atoms, molecules and the solid state, ed. P.-Q. Ldwdin (Academic Press, New York, 1966). 1231 G.H. Dierckaen, Theoret. Chim. Acti 21 (1971) 335. 124 I R.J. Weissand J.L Blinger. Phil. Mag. 27 (1973) 989.