Ab Initio Effective Core Potential Calculations on HgI2, PtI2, and PbI2

May 15, 1994 - Ionization potentials (IPS) have also been estimated from Koopmans' theorem. HgI2. This molecule has been studied previously by electro...
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J. Phys. Chem. 1994,98, 61 10-61 13

Ab Initio Effective Core Potential Calculations on HgI2, PtI2, and PbI2 S. T. Howard Department of Chemistry, University of Wales College of CardifJ Cardiff CFI 3TB, U.K. Received: January 21, 1994; In Final Form: April 13, 1994’

Ab initio methods have applied to HgIz, Pt12, and PbI2 to assess the performance of relativistic effective core potentials (RECPs) on ground-state properties of heavy-metal iodides. With electron correlation included via Moller-Plesset (MP) theory up to third order, good agreement with experiment was found for the mercury and lead compounds. PtI2 is also bent in its ground state, with a bond angle near to 96 O . In terms of bond strength, the following order was found (most stable first): Pb-I > Pt-I > Hg-I. Harmonic vibrational frequencies and ionization potentials are also reported. The direction of the first nonzero moment indicates that the metal atom is effectively positively charged in all three molecules. In HgIz, fourth-order MP corrections are shown to be unimportant.

Introduction Little is known about the bonding of iodine to third-row transition metals such as platinum and mercury, and in the literature ab initio studies on such compounds are sparse. This is in spite of the fact that calculations have been feasible for some years, using relativistic effective core potentials (RECPs) such as those developed by Hay and Wadt (hereafter HW)l-3 or Stevens, Basch, Krauss, Jasien, and Cundari.M More recently, Dyall and co-workers7 have developed an all-electron DiracHartree-Fock (DHF) approach. The motivation for this present workis todevelopappropriatebasis sets and test thelikelyaccuracy of RECPs for cluster models of iodine atoms adsorbed onto various metal surfaces, including third-row transition metals and heavier elements. In this first study, we shall focus on three closed-shell diiodides for which some gas-phase spectroscopic and electron diffraction data are available. A single-point restricted Hartree-Fock (RHF) calculation on linear Hg12has been carried out by Zhongxin and Shushan using RECPs,B at an experimental gas-phase value for the geometry of r(Hg-I) = 2.59A. Theaim was tocarry out a population analysis, as well as to predict the photoelectron spectrum. Dyall has made a detailed comparison between the performance of DHF and HW RECP calculations for on PbH2.9 The difference in the predicted equilibrium bond length was 0.017 A at the H F level, but no results including electron correlation were reported. Hein, Theil, and Leelo reported HW RECP calculations on PbH2 with electron correlation at the level of coupled cluster singles and doubles (CCSD). This reduced the bond-lengthdiscrepancywith the DHF result to 0.01 1 A. HgI2 has been the subject of recent dynamical studies using multiphoton spectroscopy.IiJ2 A number of RECP calculations involving platinum have been reported, mainly on the hydrides,I3-15 on PtCO and PtC04I6 (as models of CO adsorbed on platinum), and on platinum phosphyl c~mplexes.’~J* There are no observations of molecular PtIz in the gas phase, although in the solid state a number of adducts have been reported. In most of these, the PtI2 fragment has an I-Pt-I angle near to 90°, and a value of r(Pt-I) close to 2.59 A.19JO Crystals of D-PtI, consist of (approximatelysquare) planar PtI4 units linked through iodine atoms:21 r(Pt-I) varies between 2.58 and 2.61 A, with a mean value of 2.597 A. Like HgI2, PbI2 is a known gas-phase species,22923and RECP calculations at the SCF level have been reported by Ramondo, Rossi, and Benci~enni.~~ Related work includesthat of Sawamura and Ermler, who have carried out RECP calculations on (Pb12)7 clusters, including geometry optimizations, in part to model the bulk solid.25 0

Abstract published in Aduance ACS Abstracts, May 15, 1994.

Computational Details All calculations carried out in this work use the Hay & Wadt (HW) RECPs,I-’ as implemented in GAUSSIAN92.26 Singlet closed-shell molecular ground states have been assumed. Unrestricted Hartree-Fock (UHF)calculations were also carried out on the singlet state, to check for symmetry-broken solutions (none were found). The HW partitioning of the atomic configurationsinto [frozencore] and valence electrons is as follows:

Pt:

[Xe(4f)l0] :(Sd)’(6s)’

10 valence electrons

Hg:

[Xe(4f)’0]:(5d)’0(6s)2

12 valence electrons

Pb:

I:

[Xe(4f) lo( 5d) ‘1 :(S S ) ~6( ~ ) 4~valence electrons [Pd]:(5~)~(5p)’

7 valence electrons

The HW “valence double-r“ (DZV) basis places s, p, and d shells on platinum and mercury, so the p-shell actually provides polarization functions on these atoms, whereas the (s,p]lead basis set is truly valence double-{. Iodine also has an (s,p) DZV basis set without polarization functions. The general procedure followed here is to developan extended basis set by supplementing the DZV HW basis with polarization and diffuse functions and then carry out geometry optimizations with MP2 and MP3 perturbation corrections. No correlated calculations were attempted with the DZV basis sets. Dissociation energies have been computed, corresponding to the reaction MI,

-

M + I(’P)

+ I(*P)

(1)

with the metal atom in the ground state. Projected energies (i.e., PUHF and PUMP3) after removal of the first spin-contaminate were used for the open-shellground-state atoms (Pt(3D);Hg(1S); Pb(3P1, and I(2P)). Zero-point energy corrections have not been included in these De values. The vibrational frequencies are low in these molecules, due to the largeatomicmasses, and appreciable excited state populations would be found at room temperature. However, at 300 K the correction to De due to stored thermal energy would be less than 0.1 eV in all cases, which is much smaller than the expected accuracy in the electronic energy difference with this method. Ionization potentials (IPS) have also been estimated from Koopmans’ theorem. HgI2. This molecule has been studied previously by electron diffraction and IR and electronicspectroscopy(ref 23 summarizes these efforts). These studies have concluded that the molecule is linear, and this is supported by the calculations presented here.

0022-365419412098-61 10%04.50/0 0 1994 American Chemical Society

The Journal of Physical Chemistry, Vol. 98, No. 24, 1994 6111

Ab Inito Calculations on HgI2, PtI2, and PbI2 TABLE 1: Results for HgIz (X12#) property HF/DZV HF/i MP2/i MP3/i MP4/i expt (ref 23) re/A

2.636 -61.0 -70.2 47.8 154.6 225.9 2.3 10.1

2.671

qxx,qyy/DA -61.1 qzz/Dk

-74.2 46.8 149.6 224.3 5.5 10.2

wl/cm-l w2Jcm-2 w3Jcm-l D W IP/eV a

2.607 -60.9 -69.7 49.2 164.1 242.0 3.2

2.621 -61.0 -70.0 48.5 160.2 236.7 2.9

2.620 -61.0 -70.0

2.55-2.61 51 156 237

With the origin at the center-of-mass, the moments given are

( X Z ) , (YZ),( ZZ)

.

TABLE 2: ~eguitsfor PtIz ( 2 1 ~ ~ ) HF/ property DZV HF/ii MP3/ii property re/A 2.582 2.549 2.566 w~/cm-I B,/deg 98.4 99.4 95.8 w3lcm-l p/D 3.6 3.1 3.3 De/eV wl/cm-l 62.1 60.5 51.4 IP/eV

HF/ DZV HF/ii MP3/ii 204.2 208.3 213.3 215.6 2.0 1.2 9.7 9.6

201.2 212.2 4.2

Initial calculationsemployed the DZV basis and assumed a linear geometry. An optimum bond length re = 2.671 A was obtained, which is 0.06 A longer than the greatest experimentalestimate.23 A potential surface scan of the bond angle with r(Hg-I) fixed at the DZV-optimized value verified that the linear geometry is most stable. A Mulliken population analysisZ7at the optimized geometry gives Hg:

(5d) 10~05(6s)1~01(6p)0~63

( 5 s) 1.94 ( 5 p) 5.21

I:

-

which shows substantial s p excitation and charge transfer to iodine from the bonded mercury atom, in agreement with the results of Zhongxin and Shushan.8 (In the Appendix, there is some discussion of the relative merits of population analyses and the different choices of atomiccharges in this study.) The groundstate one-electron configuration is

and the HOMO is a 5dyz(Hg),5py(I) r-bonding orbital. Next, an extended Gaussian basis set (basis set i) was developed. A single-t shell of Cartesian f functions was added to mercury, and a shell of Cartesian d functions to iodine. The exponents for these shells were approximately optimized at the RHF/DZV geometry by carrying out a number of single-point calculations, with the following results: af(Hg) = 0.30(5) and = 0.30( 5 ) . A diffuse single-ts-function (which we shall label as 6s for the purposes of population analysis) was also added to iodine, with anoptimizedexponent of0.045(5). Optimization wascarried out using the Fletcher-Powell method. The result was r(Hg-I) = 2.636 A, much nearer to the experimental values. The oneelectron configuration was unaltered from the DZV result, and population analysis gave Hg:

(5d) io.02(6s) 1.05(6p)0.64(5f)0.05

I:

(5s)

1.92( 5

~ ) ~ 5d)0.02(6s)0.002 .'~(

with net charges on Hg and I of +0.27 and -0.135, respectively. The predicted dissociation energy (see Table 1) more than halves when the basis set is extended from DZV. An MP2 optimization with the improved basis set further shortens the bond length by around 0.03 A, but the MP3 correction cancels out much of this effect. (The column headed "MP3/in in Table 1 refers to the summed effect of perturbation theory up to third order, i.e., MP2 MP3 corrections.) The MP4 correctionhas virtually no further effect oneither thegeometryor chargedistribution,whichsuggests

+

that it may be ignored in the other two molecules. Nuclearcenteredpoint chargeswhich reproduce the calculated quadrupole moment (Table 1) have a direct physical meaning. Using the MP3 qzzvalue, this gives point charges of +0.56 (mercury) and -0.28 (iodine). Harmonic frequencies were computed by numerical differentiation, with a step length of 0.01 au. These are labeled in Tables 1-3 as 01 (bend), 02 (symmetric stretch), and o3 (asymmetric stretch). In the case of linear HgI2, there are of course two degenerate bending normal vibrations). As the study of Ramondo et al. on mixed dihalides of lead and tin showed, reasonable agreement is obtained even at the DZV level of theory (Table 1). At the MP3 level the agreement with experiment is excellent, with a marked improvement in the asymmetric stretch 0 3 (frequencycalculationswere not carried out at the MP4 level). It should be noted that these ab initio frequencies take no account of anharmonicity. This adds a further sourceof uncertainty when comparing them to experimentally-derived frequencies, since the latter are often fitted to spectra simultaneously with extra anharmonic terms. On the whole, it seems that the H F results with the extended basis set (i) will provide an acceptable compromise between accuracy and computational expediency for calculations on bigger molecules. It could be questioned whether 5f-functions are the most appropriate polarization functions for mercury, as opposed to 6d functions. It proves possible to optimize the exponent of an additional shell of single-td-functions on the metal (again, at the DZV optimized geometry), with the result ad(Hg) = 0.035(5). These are very diffuse functions which, when substituted for the 5f polarization functions, give a higher energy and the following optimizedbondlengths: r(Hg-I) = 2.651 A (HFlevel) andr(HgI) = 2.652 A (MP3 level). This demonstrates that the 5f-shell is most appropriate and indeed is essential for accurate structural prediction. The diffuse d-functions should probably be included in calculationson complexes or predictions of the dimer geometry. PtIp At the DZV level the prediction is a strongly bent molecule, with a dipole moment of 3.6 D directed toward Pt in the uv plane, i.e., on this basis Pt is positively charged (although the Mulliken analysis gives -0.10 for platinum and +0.05 for iodine). The electronic configuration is

(al)2(b2)2(al)2(a2)2(a1)2(b1)2(b2)2(a~)2(b1)2(a1)2(b2)2(a2)2 (3) The HOMOof a2 symmetry is a Sd,(Pt),Sp,(I) *-bonding orbital (with the z-axis parallel to the C2 axis). The DZV basis gives the following population analysis: Pt:

I:

(5d)9.11(6s)0.58(6p)0.40

-

(Ss)1.94(sp)5.0'

So there is less s p excitationthan with HgI2, and the implied direction of (slight) charge transfer is from iodine to mercury. In the extended basis set calculations, the iodinebasis set developed for HgI2 was used, and a platinum single-t f-exponent was optimized at the DZV equilibrium geometry, giving ar(Pt) = 0.15(5). We denote this basis set as ii. The energy ordering of molecular orbitals is identical to the DZV results. The Mulliken charges are 4 . 2 2 (platinum) and +0.11 (iodine), still at odds with the sense of the dipole moment. Population analysis now gave

Pt:

(5d)9.08(6s)0.67(6p)0.42( 5f)O.O'

I:

(5s)

1.91( 5p)4.96( 5d)0.03(6s)4'.01

The MP3 correction increases the bond length by 0.017 A (relative to the MP2 result), unlike the 0.015-A decrease found

Howard

6112 The Journal of Physical Chemistry, Vol. 98, No. 24, 1994

TABLE 3: Results for PbIl (%'AI) HF/iii MP31iii property HFJDZV 2.814 2.868 2.821 reIA 0,ldeg

PID wl/cm-l

wz/cm-I

w3lcm-l De/eV IP/eV

102.48 6.0 52.8 15 1.8 154.8 3.1 9.4

101.6 4.7 54.9 157.1 164.3 3.6 9.3

101.7 4.7 52.7 161.1 165.5 5.1

expt [ref] 2.804(4) [28] 99.7(8) [28] 49 [28] 178 [28] 177 [28] 4.3(1) [22]

with HgI2. Nuclear-centered point charges which reproduce the exact dipole at the MP3 level are +0.44 (platinum) and -0.22 (iodine). In the case of platinum, there seems to be less uncertainty that 5f-functionsare the most appropriate polarization functions. The order of electronic states in the metal atom suggests this, since the excited-state configurations are (in order of increasing energy): (5d)*(6~)2;(5d)IO;(5d)9(6p)l and (5d)9(5f)1 (Le., 5f functions are populated before 6d f~nctions).'~ In RECP studies of PtH, 5f-functions have also been used in preference to 6d functions.l4J5 PbI2. In the case of PbI2, both previous calculation^^^ and electron diffraction results28are available for comparison. In agreement with experiment and with the calculationsof Ramondo et al., the DZV prediction is a bent molecule, with a dipole moment of 6.0 D directed toward the metal atom. The Mulliken charges are +0.77 (lead) and-0.385 (iodine). Thedipole-derivedcharges are similar to the population analysis charges in this case: +0.70 (lead) and -0.35 (iodine). The one-electron configuration is

(a1)2(b2)2(a1)2(b2)2(a~)2(bl)2(a2)2(b2)2(al)2 (4) and the HOMO is dominantlya combinationof 6s, 6pz(Pb)hybrid and 5p,(I). The DZV population analysis gives Pb:

(6s) 1.98(6p)1.25

I:

(5s) 1 . 9 7 5p)5.40

-

which indicates considerable metal iodine charge transfer. In the extended basis set calculations, the basis set developed for Hg12was used for iodine. The optimal single-{ d-exponent for lead was found to be ad(Pb) = 0.15( 5) (close to the value found by Thiel et al. of ad(Pb) = 0.19 for lead in PbH2). The Mulliken charges become +0.62 (lead) and -0.31 (iodine). The energy ordering of MOs does not differ from that in the DZV case; the HOMO now has a little 6d,(Pb),5pZ(I) ?r-bonding character. Population analysis yields Pb: I:

( 6 ~ ) ' . ~ ~ (1~25(6d)0.15 6p) (5s)

5 ~ ) ' . ' ~5d)0.02(6s)0.03 (

This shows that extensive use is made of the 6d-functions on lead. This is also reflected in the large equilibrium geometry change of 0.047 A on going from DZV to extended basis sets. Ramondo et al. used a basis set of DZV quality and obtained a bond length some 0.05 A greater than the experimental value as a result. The MP3 prediction of the dissociation energy is 18% larger than a value obtained by tunable laser spectroscopyzz(see Table 3). The experimental error on this result was considered to be only 2%, so this rather large discrepancy must be attributed to the RECP approximation, the finite basis set, the spin projection technique,and the neglect of spin-orbit effects. Illas and Bagus29 have compared all-electronand pseudopotentialmodels for clusters of copper and oxygen and deduced that the RECP approximation alone gives errors of between 0.25-0.5 eV in the Cu-0 bond energy. Here we are dealing with heavier atoms, where the error

TABLE 4 Atomic Properties (eV) with Extended RECP Basis Sets i-iii atom Drowrtv PWHF) P(UMP3) exDt lrefl I IP 9.6 10.1 10.45 [33] I

EA

Pt Hg Pb

IP IP IP

2.0 7.5 8.4 6.6

2.5 8.3 10.0 6.8

3.1 [34] 9.00 [33] 10.43 [33] 7.42 [33]

may be correspondinglygreater (the error is 0.8eV for PbI2). It is possible that the other model approximations, whose reliability is difficult to quantify without carrying out more sophisticated calculations, may in part cancel the errors due to the RECP approximation. Dipole-derived charges at the MP3 level are + O S 5 (lead) and -0.275 (iodine). Vibrational frequencies at this level agree with experiment with an average accuracy of 8%. This represents good agreement, since the experimental values are derived from electron diffraction-the calculated values reported here may be more accurate than the experiment in this case.

Discussion Both mercury and platinum are able to make use of 5f-functions in their diiodides. The inclusion of these polarization functions on the metal atom results in a significantly shorter bond length, which is demonstrably in better agreement with experiment. The most complex bonding occurs in the platinum compound, where the excitation of and charge transfer to the metal atom is sufficientlysmall that population analysis and quadrupole-derived charges give contradictory pictures. Despite considerable differencesin orbital contributions to bonding in the three molecules, atomic charges derived from the first nonzero multipole moment give a fairly consistent picture: qmctalvaries from +0.44to +OS5 electrons,and ( l i d c between only-0.22 and-0.28 electrons.Thus on average half an electron is transferred from the metal atom, with each halogen atom receiving one-quarter. Table 4 lists some atomic IPS and electron affinities (EAs) using the extended basis sets developed in this paper. The properties have been computed from separate UHF calculations on the ions and the neutral atom in their respective ground states, again using spin-projected energies. Clearly, there remain deficiencies in the basis sets, especially for the metal atoms, where the IP is underestimated by 7% on average. The best IP result is obtained for mercury, so this sheds no light on the bond length discrepancy between experiment and theory for HgI2. In iodine, the IP and even the EA (difficult to predict accurately) are quite well represented, given the various approximations of spinaveraged, isotropic RECPs; Gaussian basis sets; and neglect of spin-orbit interactions. Where experimental information is available for comparison, the results are encouraging. However, with HgI2 it should be noted that the more recent (electron diffraction) measurements report a bond length at the shorter end of the 2.55-2.61-A range. In fact, the most recent determination due to Spiridinov et al. yielded a value of 2.57 A. If this is correct, then the best prediction in this work is still 0.05A astray, and further experiments and calculations to resolve this discrepancy should be encouraged. The multipole moment and frequency predictions are quite acceptable at the SCF level. Whether the molecule is linear or bent is predicted correctly at the HF/DZV level, but geometry predictions (and trends in geometricalparameters) are not reliably predicted without a polarized basis set on both the metal and iodine. Bond lengths are consistently too long with the DZV basis, being some 0.05 A greater than the MP3 results with extended basis sets in two cases. The average bond-length discrepancy between SCF and MP3 levels of theory with the extended basis sets is 0.013 A. It is therefore essential to use such extended basis sets in order to obtain reasonable predictions

Ab Inito Calculations on HgI2, PtI2, and PbI2 of equilibrium geometry. Those described here should proveuseful in calculations on larger species such as the tetraiodides and in cluster models of iodine adsorbed on metallic surfaces. Acknowledgment. I express gratitude to Dr. G. Attard for discussions which stimulated this work, to the University of London Computer Centre for a generous allocation of computing time, and to the Science and Engineering Research Council for their support, in the form of an Advanced Research Fellowship. Appendix Atomic Charges and Population Analysis. In symmetric triatomic molecules, it is a simple matter to derive point charges qmetaland qI which reproduce exactly the first nonzero moment (dipolep or quadrupole (Qzz)as computed from the wave function. Thus qmctal= p/(re cos 6,) for bent species, and qmcml= QzZ/2rc2 for linear species. Like the Virial atomic charge scheme of Bader,3O based on interatomic surfaces with normal vectors h placed such that Vp.A = 0

this clearly depends only on the charge distribution p, without additional assumptions of partitioning two-center basis function products. It also has the advantage of reproducing a well-defined physical property. Dipole/quadrupole-based charges were preferred in this work, since to the best knowledge of this author, the Virial scheme has not yet been applied to the valence-only densities associated with pseudopotential methods. The Mulliken analysis has well-known limitations, particularly with regard to overestimation of ionicity. Bagus, Nelin, and Bauschlicher31 have critically analyzed the use of Mulliken population analysis in compounds containing transition metals. Given that this is so, it is pertinent to investigate whether the conclusions on relative charge transfer and excitation might change qualitatively, using an alternative population analysis scheme such as the Lowdin technique. GAUSSIAN92 contains no option for Lowdin population analysis, so some SCF calculationswere repeated using the version of GAMESS by Schmidt et al.32 The calculations employed different RECPs, those of Stevens et al.M In the latter pseudopotentials, the numbers of core and valence electrons for Pt and Hg differ somewhat from the HW set, but for lead and iodinethe SBK RECPs replace thesamenumbers of coreelectrons, and double-c basis sets are provided. Hence this was chosen as the example, since it ought to give results rather similar to the HW RECP calculations described in the main text. The geometry was optimized at the HF/double-flevel, giving re = 2.85 A, Be = 102.0°. The Mulliken (LBwdin) chargesobtained were lead +0.65 (+0.60), iodine433 (-0.30),and the respective population analyses gave Mulliken

Pb: ( 6 ~ ) * . ~ ' ( 6 p ) ' . ~ ~I: (5s)y5p)s.3'

The Journal of Physical Chemistry, Vol. 98, No. 24, 1994 6113 Lowdin

Pb: ( 6 ~ ) ' . ~ ~ ( 6 p ) ' . ~I:~ (5s)

1.y5py.41

Clearly, in this example, qualitatively the same information is obtained from both types of population analysis. References and Notes (1) Hay, P. J.; Wadt, W. R. J. Chem. Phys. 1985.82, 270. (2) Wadt, W. R.; Hay, P. J. J. Chem. Phys. 1985,82, 284. (3) Hay, P. J.; Wadt, W. R. J. Chem. Phys. 1985,82, 299. (4) Stevens,W. J.; Basch, H.; Krauss, M. J . Chem. Phys. 1984,81,6026. (5) Stevens, W. J.; Basch, H.; Krauss, M.; Jasien, P. Can.J. Chem. 1992, 70, 612. (6) Cundari, T. R.; Stevens, W. J. J. Chem. Phys. 1993, 98, 5555. (7) Dyall, K. G.; Faegri, K., Jr.; Taylor, P. R.; Partridge, H. J . Chem. p h p . 1992,95,25a3. (8) Zhongzin, M.; Shushan, D. Acta Phys. Chim. Sinica 1989, 5, 551. (9) Dyall, K. G. J. Chem. Phys. 1992, 96, 1210. (10) Hein, T. A.; Thiel, W.; Lee, T. J. J. Phys. Chem. 1993, 97, 4381. (1 1) Baumert, T.; Pedersen, S.;Zewail, A. H. J . Phys. Chem. 1993, 97. 12441. (12) Pedersen, S.;Baumert, T.; Zewail, A. H. J . Phys. Chem. 1993, 97, 12460. (13) Gropen, 0.; Almlof, J.; Wahlgren,U.J. Chem. Phys. 1992,96,8363. (14) Rohlfing, C. M.; Hay, P. J.; Martin, R. L. J. Chem. Phys. 1986,85, 1447. (15) Wang, S . W. J . Chem. Phys. 1983, 79, 3851. (16) Rohlfing, C. M.; Hay, P. J.; Martin, R. L. J. Chem. Phys. 1985,83, 4641. (17) Noell, J. 0.; Hay, P . J. Inorg. Chem. 1982, 21, 14. (18) Kitaura, K.; Obara, S.;Morokuma, K. Chem. Phys. Lett. 1981,77, 452. (19) Hanan, G. S.;Kickham, J. E.; Lab, S . J. Organometallics 1992,11, 3063. (20) Meganmisibelombe. M.; Endres, H. Acta Crysrallogr. C 1985, 41, 513. (21) Arduengo, A. J.; Stewart, C. A.; Davidson, F. J . Am. Chem. Soc. 1986, 108, 322. (22) Simons, J. W.; Oldenborg, R. C.; Baughcum, S.L. J . Phys. Chem. 1981,91, 3840. (23) Spiridinov, V. P.; Gershikov, A. G.; Butayev, B. S.J . Mol. Struct. 1979, 52, 53. (24) Ramondo, F.; Rassi, V.; Bencivenni, L. J . Mol.Srruct. (THEOCHEM) 1989, 57, 105. (25) Sawamura, M.; Ermler, W. C. J . Phys. Chem. 1990, 94, 7805. (26) Frisch, M. J.; Trucks, G. W.; Head-Gordon, M.; Gill, P. M. W.; Wong, M. W.; Foresman, J. B.; Gonzalez, C.; Martin, R. L.; Fox, D. J.; Defrees, D. J.; Baker, J.; Stewart, J. J. P.; Pople, J. A. Gaussian 92, Reuision B Gaussian, Inc.: Pittsburgh, PA, 1992. (27) Mulliken, R. S.J . Chem. Phys. 1955, 23, 1833. (28) Demidov, A. V.; Gershikov, A. G.; Zasorin, E. 2.;Spiridonov, V. P.; Ivanov, A. A. J. Strucr. Chem. 1983,24, 7. (29) Illas, F.; Bagus, P. S.J . Chem. Phys. 1991, 94, 1236. (30) Bader, R. F. W. Atoms In Molecules; A Quantum Theory; Oxford University Press: Oxford, 1990. (31) Bagus, P. S.;Nelin, C. J.; Bauschlicher, C. W. Phys. Reu. 1983,828, 5423. (32) Schmidt,M.W.;Baldridge,K.K.;Boatz,J.A.;Elbert,S.T.;Gordon, M. S.;Jensen, J. H.; Koseki, S.;Matsunaga, N.; Nguyen, K. A.; Su,S.J.; Windus, T. L.; Dupuis, M.; Montgomery, J. A. J . Compur. Chem. 1993,14, 1347. (33) Gray, H. B. Electrons and ChemicalEonding, W.A. Benjamin Inc.: New York, 1965; p 21. (34) Herzberg, G. Atomic Spectra and Atomic Structure; Dover: New York, 1944; pp 201 & 219.