A Density Functional Investigation of the Geometrical and Electronic

Why Are SiX5 and GeX5 (X = F, Cl) Stable but Not CF5 and CCl5? Marzena Marchaj , Sylwia Freza , and Piotr Skurski. The Journal of Physical Chemistry A...
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1570

J . Phys. Chem. 1994,98, 1570-1575

A Density Functional Investigation of the Geometrical and Electronic Structure of the SiCl, and SiCl, Series, n = 1-5 G . L. Gutsevt Chemistry Department, University of Arizona, Tucson, Arizona 85721 Received: May 13, 1993"

The electronic and geometrical structures of the ground and some low-lying excited states of the SiC1, and SiC1,series, n = 1-5, are calculated within a local spin density approximation (LSDA) augmented with the nonlocal gradient corrections to the exchange functional. Energies of fragmentation through different decay channels are evaluated for both series, and a n estimation of the electronic affinity for all the neutral silicon chlorides is made for the first time. All the anions considered are found to have excited states which are stable toward dissociation, except for SiClZ-. According to the results of the calculations, the stability of the Sic15 radical is near zero threshold, whereas its anion is rather stable and possesses a dissociation energy of about 30 kcal/mol.

Introduction For neutral silicon halides with coordination numbers from 1 to 4 there are plenty of experimental data on their ground-state geometry thermochemistry.' Silicon fluorides were the subject of intensive quantum chemical calculations; see ref 2 and references therein. Silicon chlorides appear to have received much less attention due to the necessity of using larger basis set expansions. The only study on the atomization energies for all the series SiCl,, n = 1-4, has been carried out at the HF/6-3 lG* level.3 Very little is known about thestructureof the negatively charged silicon chlorides. Only the SiC13- anion was considered within an UMP2/6-31 +G* approach.4 For the adiabatic electron affinity (E&) of the neutral silicon chlorides, only rough experimental estimates seem to exist in the literature.5 The Sic&-anion was observed in the gas phase quite long ago;6 however, data on its stability as well as of its neutral precursor are lacking in the literature. This paper is aimed a t a systematic investigation on the geometrical and electronic structure of both the series SiCl, and SiC1,-, n = 1-5, within the same approach based on the density functional theory. Our approach has been applied early to the study of the carbon chlorides CC1, and CC1,- series,7 as well as heavier carbon halides,* and was shown to be rather reliable for reproduction of experimental data on geometry and thermochemistry of the carbon halides. Calculational Details The original package of the LCAO-HFS programs9 is used. Geometry optimizations are carried out at the Hartree-FockSlater (HFS) level of theory with the employment of a gradient searchlo for stationary states on the potential energy surface (PES). Our approach is based on the extensive use of effective numerical cellular integration schemes" for evaluating the matrix elements of the Hamiltonian. A standard triple-{ STO basis set12 is augmented by two sets of polarization 3d functions with the same exponents of 2.5 and 1.0 for all the atoms. The local spin density approximation (LSDA)13 represents the exchange and correlation functionals as a function of the electronic densities p a ( r ) , u = cy or p. The well-known approximation of Vosko, Wilk, and Nusair (WVN)14 is used in the present work. The explicit expression for the WVN exchange-correlation f Permanent address: InstituteofChemicalPhysicsoftheRussian Academy of Sciences, Chernogolovka, Moscow Region 142432, Russia. Abstract published in Advance ACS Absrracrs, January 15, 1994.

0022-3654/94/2098- 1510%04.50/0

potential may be found elsewhere.15 The correlation part is modified according to prescriptions of Stoll et a1.16 This approach will be referred as to LSDA. The total energies of configurations optimized at the HFS level have been recalculated at both the LSDA level and with the inclusion of the nonlocal gradient correction to the exchangeI7

E,NL= -b

x,2

S[pa(r)l4/' o=a,@

1

+ 6bXa sinh-'

dr

(1)

Xu

where b = 0.0042, and X u reads

This approximation will be referred to as LSDA/NL. It should be noted that the self-consistent inclusion of this correction in the course of geometry optimizations results usually'* in minor changes in comparison with the HFS approach. The vertical and adiabatic EA of the neutral are given by

and the first (vertical) IP of the anion reads

FIP = Etot(M,Re-)- Etot(M-,R;) (5) In eqs 3-5 M stands for the neutral, M- for its anion, Re is the equilibrium ground-state geometry of the neutral, and Re- is the equilibrium ground-state geometry of the anion. Let us remember that the first adiabatic IP of an anion is equal to the adiabatic EA of the neutral precursor by definition. For calculations on EA,,, and FIP it is convenient to use the Slater transition-state concept19which is validzoin any local density approximation and allows for substitution of eqs 4 an3 5 with the energy of the highest occupied MO with the fractional occupation number. If one makes inclusion of nonlocal gradient corrections, then it is necessary for the separate calculations on a neutral and its anion to be carried out for the same geometrical configuration. It leads usually to small (-0.2 eV) deviations7J from the LSDA values. The atomic EA values calculated within C,, constraintz1 on the atomic wave functions are as follows: C1, 3.17 (3.24) eV; Si, 1.11 (1.17) eV a t the LSDA (LSDA/NL) level of theory. They have to be compared with experimentalvaluesZZof 3.63 and 1.385 eV, respectively. As is seen, the maximal discrepancy does not exceed 0.4 eV on the highest, LSDA/NL, level. Approximately 0 1994 American Chemical Society

The Journal of Physical Chemistry, Vol. 98, No. 6, 1994 1571

The SiCl, and SiCln- Series TABLE 1: Separation Energies (T,) of the Ground and First Excited States of Sic1 and SiCISic1 property

method

Te EA,efi/ FIP

LSDA LSDA/NL LSDA

211 0.0 0.0 0.57b

SiCl42-

3Z-

lA

IZ-

3.34 3.43 3.86c

0.0 0.0 0.90d

0.49 0.35

1.13 1.24

The vertical EA of the radical and the first IP of the radical-anion. All values in electronvolts. Thevertical EA, AE(211-3Z-). Thevertical EA, AE(4Z--3Z-). The first vertical IP, AE(3Z--211).

the same deviation is to be anticipated for the molecular EA as well, bearing in mind that the atomic EA calculations are a crucial test23for any quantum chemical method. Because we shall consider some excited states, let us remember that the LSDA will give a correct d e ~ c r i p t i o nof~ ~the lowest excited state of each symmetry.

c,,

SiCI,

sic$

C,

’A,

3 ~ ,

SiCl;

C,,

SiCl;

B4

c,,

,A,

Figure 1. Optimized configurationsof the ground and first excited states of Sic1 and Sic12 and their anions.

and pure states: Electronic and Geometrical Structure of the Neutral and Singly Negatively Charged Silicon Fluorides To evaluate the EAad, it is necessary to obtain the equilibrium ground-state geometryof both neutrals and anions. The structure of low-lying excited states of both series is also of much interest because of allowing, for example, the knowledge on the anionic states which are stable toward dissociationand detachment of an extra electron. Let us consider both series in the order of increasing the coordination number. Sic1 and SiCI-. The ground-state 211 of the Sic1 radical possesses the electronic configuration 3 a227r1. Our calculated bond length (see Figure 1) is in good accord with the experimental of 2.061 A. The first excited state 42- of the radical corresponds to promotion of a 3a electron to the 2a MO. The energy gap between two states is rather large (being equal to 3.43 eV on the LSDA/NL level of theory); see Table 1. Attachment of an extra electron to the 27r HOMO of Sic1 leads to formation of the anion with the electronic configuration 3a22a2. The latter generates three pure spectroscopicstates: 32-, ]A, and ]E-. Two singlet states may be separated according to a “sum rule”.26 Although it is a rather straightforward application of this rule, let us see how it works in this case. Denoting by X and Y the componentsof a 27r orbital, one may write the pure wave functions for the multiplet arising from the 2a2 configuration as

E(W,Y@I) = 1 / 2 [ E ( 3 v + E(’A)l ~(W,Y,I)

Carrying out three separate self-consistent calculations with these determinantal functions, one will be able then to separate between two singlet states. The results of such a procedure applied to SiCl- are presented in Table 1. As is seen, the excited state lA is separated from the ground triplet state by a relatively small gap, and the IZ-state is the highest in energy. However, separation energies are about 3 times lower than in the neutral radical where the ground and first excited states possess different electronic configurations. The EAvert of Sic1 is close to the FIP of the anion (see Table l), which indicates the proximity of both species PESs near their equilibriums. Sic12 and SiC12-. Geometrical parameters for the AI ground state of the Sic12 molecule have been obtained by microwave ~pectroscopy~~ and electron diffraction28methods. Our calculated Si-Cl bond length (see Figure 1) is in good agreement with that of both experimental investigations (where close agreement on the ground-state bond length has been obtained), and our bond angle is closer to that from electron diffraction experiments.28a Another high-resolution spectroscopy has confirmed the microwave value of the bond angle, 101So. Thus, both our value and that deduced from the ED experiments28seem to be overstated by -2’. Geometrical parameters of the first excited state 3B1of Sic12 are available from the HF30 and post-HF3I calculations. They are in fairly good accord with ours presented in Figure 1. It is worth noting that promotion of an electron to the a-type MO 2bl (transition X1A1 a3B1)leads to a slight shortening of the Si-Cl bond length and to a considerable change in the bond angle. The singlet-triplet splitting in Sic12 has been calculated within an extended Hiickel type (EHT) method quite long and was experimentally determined recently.33 Our values obtained at both LSDA and LSDA/NL levels of theory are in good agreement with both the experimental data and the results of the ab initio calculations (see Table 2). Geometrical parameters of the SiC12- anion are obtained for its ground (2B1) and first excited (2A1) states, see Figure 2. Attachment of an extra electron to the 2b*1 MO of the molecule results in elongating the Si-Cl bond lengths without changing the bond angle, because of the a-type of the MO, while attachment to the a-system leads to a less elongated bond length but to a considerable enlargement of the bond angle in analogy with the -+

Taking into account that matrix elements of the Hamiltonian between functions of different symmetry are zero, one may write down the following relations between energiesof the determinants

= E(32-)

1572

The Journal of Physical Chemistry, Vol. 98, No. 6, 1994

TABLE 2: Separation Energies ( Te) of the Ground and First Excited States of the Sic12 Molecule and the SiCl2- Anions Sic12 Te

LSDA LSDA/NL LSDA

EAvefi/FIP

Gutsev TABLE 3 Separation Energies (T') of the Pyramidal and Flat Configurations of the Sic13 Radical and the SiC13- Anion. Sic13

SiC12-

'A1

3B1

2B1

'A1

0.0 0.0 O.3gc

2.45 2.4gb -0.51d

0.0 0.0 0.96e

3.36 3.50 -0.77f

The vertical EA of the molecule and the first IP of the radical-anion. All values in electronvolts. Experimental values: 2.35 eV (see ref 33a), 2.29 f 0.0006 eV (see ref 33b). Theoretical estimations: 2.84 eV (EHT, see ref 32), 2.36 eV (CISDQ, see ref 31), 2.38 eV (UMP2/6-31G*, see ref 31), 1.60 eV (HF/6-31G*, see ref 30). The vertical EA, AE('B1lA1). The vertical EA AE('AI-~B~).e The first vertical IP, AE(3B1'B1). /The first vertical IP, AE(1A1-2AI).

Te

LSDA LSDA/NL UHF/6-31+ G* UMP2/6-31+ G* UHF/STO-3G* LSDA

EAvefi

Te

0.0 0.0 0.0 0.0 3.07g

LSDA LSDA/NL UHF/6-3 1+ G* UMP2/6-3 1 G* LSDA

+

c3,

2 ~ ,

SiCl;

0.85 0.78 1.71 1.47 1.31e

0.86 0.78 1.99 1.55 1.71 1.29

SiCl3-

FIP

Sic!,

0.0 0.0 0.0 0.0 0.0 1.41d

1.03 0.88 1.41 1.41 3.04h

1.27 0.97 2.05 1.87 2.82'

=The vertical EA of the neutral and the first IP of the anion. All valuesinelectronvolts. Seeref4. Seeref 35. TheverticalEA,AE(2A1lA1); 2.5 eV (MS-X,, see ref 35). e Thevertical EA, AE(2Al-1Al)fThe vertical EA, AE(2Ar1-1Ar1).g The first IP, AE(2A1-1A1). The first IP, AE(2A1-1A1). The first IP, hE(2A'l-1Ar1).

C3v 'A,

Q N

sic!,

c,

2 ~ ,

SiCI,

T,

A1

sicli

c,,

*A~

SiCli

c,"

*A~

Figure 3. Optimized configurations of the Sic14 molecule and the two states of its anion.

Both pyramidal and flat configurations are considered for the SiC13- anion. Geometrical parameters of the (C3", 'AI) ground state (see Figure 2) are close to those of Hopkinson et al.,4although agreement is not so good for the flat configurations. As follows from Table 3, the Cb configuration is separated from the D3h one by the gap of -0.1 eV, and it appears the edge inversion to be realized in the anion. The EAvertof the neutral is by 1.6 eV higher than the FIP of the anion that may be related to the drastic change in geometry of the radical occurring after attachment of an extra electron. SIC14 and SiCId-. The Sic14 molecule possesses tetrahedral shape and its bond length was determined by various experimental methods36to be enclosed within 2.014-2.017 A. The bond length obtained in the H F calc~lations3~ is overestimated, and our HFS value is closer to the experimental data (see Figure 3). Geometry optimizations carried out for C2, and C3" configurations led to the ground-state configuration. The structure of the SiCl4- anion was studied by methods of ESR in tetramethylsilane matrices38and mass spectrometry in the gas phase.39 Electron beam collisions40with the target gas of Sic14 or impact of the gas by alkali-metal atoms41 did not allow the SiC14- anion to be detected. According to data of the ESR investigation^,^^ this anion possesses Cb symmetry, as does42 its isoelectronic radical PC4. Our optimized configuratioqs of SiC14-are presented in Figure 3, and energies of their separation from the ground state of the neutral molecule are given in Table 4. Both configurations of the anion are close in energy to each other at the LSDA level and are nearly degenerate at the LSDA/ NL level, that suggests the anion nuclear frame is nonrigid. It is worth noting that the C3v state of Sic&- has been optimized recentlV3at the H F level, and the geometrical parameters, R(SiClax)= 2.1 15 A,R(Si-Cl,) = 2.225 A (the bond angle was not indicated in the paper) are in good accord with ours. The tetrahedral configuration of the anion is placed higher

-

Figure 2. Optimized configurations of Sic13 and SiCl3-.

-

singlet-triplet transition in the neutral. The splitting energy AE(2A1- 2B1)is by 1 eV higher than that of Sic12 (see Table 2). Sic13and SiC13-. Experimental electron spin resonance (ESR) investigation^^^ on the ground state of the Sic13 radical allowed the conclusion on its C3" shape with the bond angle of 110'. In the ab initio calculation^^^^^ the bond lengths of 2.086 and 2.047 A as well as bond angles of 108' and 109.7" (see refs 4 and 35, respectively) for the ground state of the radical were determined. Our parameters are in nice agreement with those obtained by Hopkinson et al.4 In both theoretical paper^^.^^ the inversion barrier height was investigated, and in the more recent paper of Hopkinson et al.4 a new type of inversion through an intermediate C2, flat state was proposed. According to the results of our calculations, the D3h and C2vconfiguration are almost degenerate in energy, although the C2, configuration is slightly closer to the ground state, in agreement with the results of Hopkinson et al.4 However, our separation energies are substantially lower. As seen from Table 3, taking into account of the electronic correlation4 leads to a decrease of the gap, prompting a further decrease if the electronic correlation will be included at the highest level of theory.

-

The Journal of Physical Chemistry, Vol. 98, No. 6, 1994 1573

The SiC1, and SiCln- Series TABLE 4: Separation Energies (T') of the Sic14 Ground State and Two Configurations of the SiCl4- Anion. SiCl4Te EA,/FIP

LSDA LSDA/NL LSDA

-0.67 -1.50 1.93c

0.0 0.0 -1.226

TABLE 6 Adiabatic and Vertical EA (in electronvolts) of the SiCI,, Series, n = 1-5 EA

method

Sic1

Sic12

Sic13

Sic14

Sic15

EAad

LSDA LSDA/NL Other

0.84 1.17

2.72 3.27 ~1.1" 2.6W

0.67 1.51 -0.3d

4.72 4.50 4.2e

EA/

LSDA

0.57

0.79 1.25 0.64" 0.90" >2S6 0.38

-0.58 -1.49 1S d

a The vertical EA of the molecule and the first IP of its anion. All values in electronvolts. The vertical EA, AE(2A1-1A1) for a tetrahedral *anionconfiguration. The vertical first IP, AE(2A1-1A1). The vertical first IP, AE(2A1-1A1).

3.07

3.30

Experimental; see ref 45b. Experimental; see ref 45c. The theoretical estimate from the UMP2/6-3 1+G* calculations;see ref 4. The experimental estimate of the vertical EA; see ref 46. e From DVM-X, calculations; see ref 47. f The vertical EA of the ground states.

Both the EA,,, of the radical and the FIP of the anion are close to each other, Le., the adiabatic corrections to the EAad of Sic15 are small. Electron Affinity of Silicon Chlorides SiCI,,, n = 1-5

SiCI,

caV 2

~

2

SiCl;

c,

'A,

Figure 4. Optimized configurations of Sic15 and SiCls-.

TABLE 5 Separation Energies (2'') of Different Configurations of the Sic15 Radical and Its Anion.

Te

LSDA LSDA/NL EAVb LSDA F I F LSDA

0.0 0.0 3.73

0.15 0.10 4.09

0.33 -0.36 3.30

0.0 0.0

0.08 0.09

4.21

3.99

The vertical EA of the radical and the first vertical IP of the anion. All values in electronvolts. The vertical EA of the radical, the extra electron is attached to the singly occupied HOMO. The first vertical IP of the anion (from the 2e"2 MO in the D3h configuration and from the la2 MO in the Cb one.

than the ground state of the neutral. Because attachment of an extra electron results in the change of the nuclear frame symmetry, the adiabatic corrections to the EAad of SiC14, Le., the differences between the EAad and EA,,,, or the EAad and FIP of the anion, are large. Sic15 and SICIS-. The SiCls- anion was observed6in a SiC14Et4NC1 mixture, although stable salts containing Sic&-have not been synthesized. No data appear to be available about the structure of its neutral precursor Sic&. We have optimized the geometry of the radical under several symmetry constraints: C4u, D3h, and C3, (see Figure 4). The ground state of the radical possesses the square-pyramidal shape at the LSDA level, but at the LSDA/NL level the most stable is a C3, configuration of the adduct SiC14:Cl type (see Table 5 ) . In the latter case, the ligand hole is localizedat a distant chlorine, while in the Cb configuration it is delocalized over the four basal chlorines; for separation energies see Table 5 . Optimization started with a C3u configuration of the anion converged to a D3h configuration (see Figure 4), which is the ground state of the anion. The Berry pseudorotation barrier44 in the anion is rather small, being equal to -2 kcal/mol.

Experimental estimations on the EAad are known for the Sic12 m o l e ~ u l e and ~ ~ .the ~ ~ Sic13 radical.40 For the Sic14 molecule there is an estimate on the EAvert.46 Theoretical estimations have been carried out for SiC134 and As seen from Table 6, previous theoretical v a l ~ e s ~are v ~in~ agreement with the LSDA/NL ones, and only one experimental estimate40is close to ours (that for SiC12). The EAad behavior is nonmonotonous when going along the series. It increases from Sic1to SiC13, drops at SiC14, and increases at S3iC15.The EAad of the two first members, Sic1 and SiC12, is close to the EA of silicon, whereas the EAad of Sic13 is close to the EA of chlorine. The extra electron in Sic13has the possibility of using the last valency of the central atom while in Sic14 all valencies are saturated. This results in a sharp decrease in the EAad value Of sic&. A considerable increase in the EA when going from Sic14 to Sic15 is explainedby the hypervalent character of the latter species, in which a hole accepting an extra electron has to be delocalized over the C15frame possessing a high collectiveelectronegativity.48 The high EAad of Sic15 may be related to the following cycle (data on dissociationenergies are from Tables 7 and 8, the LSDA/ N L results):

-

De = 0.06 eV

SiCl, EA = 4.50 e V t

SiC1,-

De = 1.27 eV

SiC1,+ Cl ?EA = 3.19 eV (calc)

SiC1,

+ Cl-

The excess of the fragmentation energy of the anion over that of the neutral warrants the corresponding predominance of the radical EAad over the chlorine atomic EA. The SiF5- anion is by -2 eV more stable2 than its chlorine congener, and it leads to an approximately the same difference in the EAadof the correspondingneutrals. The decreasein stability when going from fluorides to chlorides may be related at a qualitative level to a higher interligand repulsion in chlorides due to their larger covalent radii. Comparison of separation energies (see Tables 1-5) with the EAadvalueSallows the conclusionthat all the odd-n silicon chloride anions may possess the excited states which are stable towards the loss of an extra electron. The SiC12- anion appears to be the only exception in the series. Stability of the Silicon Chlorides and Their Anions Energies of fragmentation through various decay channels are calculated as the differences in the total energies of the ground-

1574 The Journal of Physical Chemistry, Vol. 98, No. 6, 1994

TABLE 7: Fragmentation Energies (in electronvolts) of the Neutral Silicon Chlorides SiCI, n = 1-5

--

channel

SiCl Sic12

+

LSDA LSDAJNL

Si CI Sic1 CI -Si + C12 -Si 2CI Sic13 Sic12 CI SiCl CI2 Si 3C1 Sic14 SiClp C1 Sic12 C12 -Si+4CI Sic15 Sic14 + CI Sic13 + C12 -Si+5CI

-----

+ +

+

+

+

+ +

4.79 4.80 6.60 9.59 3.01 4.82 12.60 4.84 4.86 17.44 0.24 2.09 17.68

3.91 3.95 5.83 7.86 2.06 3.98 9.92 3.70 3.73 13.62 0.06 1.73 13.68

other 3.95;'~~ 3.79;c 4.80d 4.94'-b 7.82c 2.86;' 3.08;b2.69c 10.29c 4.81;' 4.55;b 4.6 f 0.17f 14.63c

TABLE 8 Fragmentation Energies (in electronvolts) of SiCI;, n = 1-5 Sick

----------

SiC12-

SiCl3-

SiCl4-

SiCls-

-

+ + +

Si CISi- CI SiCl C1SiCI- CI Si (212Si- + C12 Sic12 C1SiC12- CI SiCl Cl2S i c k Cl2 Sic13 C1SiC13- CI Sic12 Cl2SiC12- Cl2 Sic14 CISiC14- CI Sic13 + Cl2SiC13- Clz

+

+

+ + + + + + + + + +

+

LSDA

LSDAJNL

2.54 4.52 2.50 4.75 5.16 6.28 2.64 4.94 5.31 6.70 2.42 2.79 3.30 4.73 1.87 4.29 4.58 4.09

1.79 3.91 1.92 4.03 4.04 5.91 2.07 4.08 4.21 6.08 1.92 1.94 2.21 3.99 1.27 3.05 3.19 2.96

other

2.12'

Theoretical; see ref 4.

state configurations of initial compounds and its decay products, Le., without taking into account the zero-point energies ( Z P E ) which are usually small. Thus, for Sic13 the ZPE equals 3.4 kcal/moL4 Comparison of the Si-Cl bond-cleavage energies calculated at the LSDA and LSDA/NL levels with the experimental data39349 shows the LSDA to present closer agreement with the experiment for n = 2-4, while the LSDA/NL leads to underestimating of 1 eV (see Table 7). For the first member of the series the trend is opposite and the LSDA presents overestimated value as it was found earlier.50 Atomization energies calculated on the LSDA/ NL level are in good agreement with those obtained by Melius et al.3 The Sic15 radical is weakly stable at both levels of theory. Fragmentation energies for the anion series are given in Table 8. T h e only known estimate4 for t h e detachment of a C1- is in nice accord with our L S D A / N L value. As seen from Table 8, the energy of the C1- detachment is practically independent of coordination number within the range 1 I n I 4 and slightly decreases at SiCls-. The same behavior was observed for the silicon fluorides as we11.2 On the whole, the anions are less stable than their neutral precursors except for SiC13-, where the energy of t h e C1- detachment is close to t h a t of the Si-Cl bond rupture in SiCl3. T h e SiCls- anion is quite stable at both levels of theory a n d even at the LSDA/NL level, which seems t o m a k e an underestimation of fragmentation energies, its stability is not lower than 30 kcal/mol. Comparing the separation energies between t h e ground a n d excited states from Tables 1-5 and the energies of fragmentation

-

through the uppetmost channels of the anions, one may conclude that all the anions, except for SiClz-, may possess excited states stable against dissociation. Acknowledgment. This investigation was supported by the National Science and Engineering Research Council of Canada (NSERC), whose grant for international collaboration is greatly acknowledged. I am appreciative to the University of Calgary for a copious supply of computer time and access to their Cyber205 facilities. I wish to express my gratitude to Professor Tom Ziegler for hospitality and assistance. I would like also to thank the referees for useful remarks and suggestions. References and Notes

Experimental; see ref 49a. Experimental; see ref 39. e Theoretical; seeref 3. Theoretical; see ref 50. e Theoretical; seeref 4. fExperimenta1; see ref 49b.

channel

Gutsev

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