Ab initio

2 downloads 30 Views 2MB Size Report
bond length and angle variations calculated for pyrosilicic acid and those ...... postulate. In addition, difference electron density maps calculated for planes.
Phys. Chem. Minerals 6, 221 246 (1980)

PHYSICS CHEMISTRY ]IRIHERALS © by Springer-Verlag 1980

Ab Initio Calculated Geometries and Charge Distributions

for H4SiO4 and H6Si207 Compared with Experimental Values for Silicates and Siloxanes M.D. Newton 1 and G.V. Gibbs 2 1 Departmentof Chemistry,BrookhavenNational Laboratories, Upton, New York 11973, U.S.A. z Departmentof GeologicalSciences,VirginiaPolytechnicInstitute and State University, Blacksburg, Virginia24061, U.S.A. Abstract. Ab initio STO-3G molecular orbital theory has been used to calcu-

late energy-optimized S i - O bond lengths and angles for molecular orthosilicic and pyrosilicic acids. The resulting bond length for orthosilicic acid and the nonbridging bonds for pyrosilicic acid compare well with S i - O H bonds observed for a number of hydrated silicate minerals. Minimum energy S i - O bond lengths to the bridging oxygen of the pyrosilicic molecule show a close correspondence with bridging bond length data observed for the silica polymorphs and for gas phase and molecular crystal siloxanes when plotted against the SiOSi angle. In addition, the calculations show that the mean S i - O bond length of a silicate tetrahedron increases slightly as the SiOSi angle narrows. The close correspondence between the S i - O bond length and angle variations calculated for pyrosilicic acid and those observed for the silica polymorphs and siloxanes substantiates the suggestion that local bonding forces in solids are not very different from those in molecules and clusters consisting of the same atoms with the same coordination numbers. An extended basis calculation for H4SiO 4 implies that there are about 0.6 electrons in the 3 d-orbitals on Si. An analysis of bond overlap populations obtained from STO-3G* calculations for H6Si207 indicates that S i - O bond length and SiOSi angle correlations may be ascribed to changes in the hybridization state of the bridging oxygen and ( d - p ) 7z-bonding involving all five of the 3d AO's of Si and the lone-pair AO's of the oxygen. Theoretical density difference maps calculated for H6SizO 7 show a build-up of charge density between Si and O, with the peak-height charge densities of the nonbridging bonds exceeding those of the bridging bonds by about 0.05e fi-3. In addition, atomic charges (+1.3 and -0.65) calculated for Si and O in a SiO 2 moiety of the low quartz structure conform reasonably well with the electroneutrality postulate and with experimental charges obtained from monopole and radial refinements of diffraction data recorded for low quartz and coesite. 0342-1791/80/0006/0221/$05.20

222

M.D. Newton and G.V. Gibbs

Introduction

In recent years it has been found that S i - O bond length and SiOSi and OSiO valence angle variations in silicates and siloxanes can be rationalized moderately well in terms of semi-empirical molecular orbital theory. For example, Gibbs et al. (1972, 1974, 1977a and b)and Louisnathan and Gibbs (1972a-c) completed extended Hfickel Theory (EHT) calculations for fragments isolated from a large number of silicates and found that observed S i - O bond lengths, d(Si - O), correlate inversely with Mulliken bond overlap populations, n(Si-O), obtained from calculations based on observed angles and constant S i - O distances. In spite of the latter constraint, the variations in the overlap populations indicate the directions that the bond distances distort when allowed to relax. The calculations also show that n(Si-O) is strongly dependent on the OSiO angle and LOSiO3 (the average of the three OSiO angles common to each bond) with shorter bonds predicted to involve wider angles. More recently, CNDO/2 molecular orbital calculations completed for the gas phase molecule disiloxane (Tossell and Gibbs 1977) and various silicic acid molecules (DeJong and Brown 1975; Meagher et al. 1979; Lasaga 1979) have yielded minimum energy SiOSi angles in close agreement with observed values and the average value recorded for the silica polymorphs and glass. However, because CNDO/2 calculations tend to drastically overestimate bond lengths for second row elements (Marsh and Gordon 1976), no attempts were made in these studies to reproduce bond lengths. With the recent development of ab initio SCF-MO calculations and computer programs using Gaussian expansions of Slater-type orbitals, it is now possible to reproduce the bond lengths and angles for molecules involving first and second row elements with a moderately high degree of accuracy. Collins et al. (1976), for example, have shown that the bond lengths for P-, S- and C1containing molecules can be reproduced with a mean absolute error of 0.043 •, provided that their STO-3G minimal sp-basis sets were augmented with d-type functions (see also Schmiedekamp et al. 1979; Wallmeier and Kutzelnigg 1979). On the other hand, bond lengths calculated for molecules containing the more electropositive elements Na through Si proved to be in better agreement with experiment when a minimal STO-3G basis set was used. Moreover, Collins et al. found that bond angles could be reproduced in moderately good agreement with experiment, but, unlike the generation of bond lengths, they were found to be relatively insensitive to the inclusion of the d-functions in the basis sets. However, a minimal basis set optimization of the geometry for the Si207 6 anion resulted in an equilibrium SiOSi angle of 180°, nearly 30° wider than that observed ( ~ 150°) on .the average for silica glass (Zupan and Buh 1978). Because of the large negative charge on the anion, the failure of the calculation to generate an angle close to that in silica glass may not be simply ascribed to an inadequate basis set. The Coulombic repulsion between the - 3 charges on opposite ends of the anion must have dominated in the calculation, requiring the energy to be minimized when the angle is straight. Recent STO-3G calculations for disiloxane (Sauer and Zurawski 1979) have yielded a S i - O bond length of 1.656 A and a SiOSi angle of 124°, compared with the experimental

Ab Initio Calculated Geometries and Charge Distributions

223

values of 1.634 A and 150° (Durig et al. 1977), respectively. Although no attempt was made in the Sauer-Zurawski study to include d-type orbitals in their basis sets, Newton and Gibbs (1979) have since found that 6-6-31G* calculations for the molecule with d-type orbitals on both Si and O generate a minimum energy angle of 154°, a barrier to linearity of 0.2 kcal compared with 0.3 kcal measured by Durig et al. (1977), and a dipole moment of 0.32 D compared with 0.24 D measured by Varma et al. (1964). Nevertheless, the minimum energy S i - O bond length (1.58 A) generated for the molecule was found to be in poorer agreement with the observed value when d-type orbitals were added to the extended basis set. These results suggest that an extended basis set with d-type basis functions may be requisite to reproduce accurate bond angles, energy barriers and dipole moments but that a minimal STO-3G basis set may suffice in the reproduction of moderately accurate bond lengths in siloxanes and other Si-containing molecules. As the S i - O bond is the strongest and the most important bond in many silicate minerals and siloxanes, we have optimized the geometries of orthosilicic acid, H4SiO~, and pyrosilicic acid, H6Si207, using ab initio theory, and have explored the utility of the resulting wave functions in reproducing bond length and angle trends observed for the silica polymorphs, silicates and siloxane gas phase molecules and molecular crystals. Atomic charges and charge density distributions calculated from the resulting wave functions for the two silicic acid molecules will be compared with those measured for the silica polymorph coesite. In addition, the extent to which the 3d-type functions on Si may be involved in re-back bonding with ligand oxygen atoms will be examined (Newton and Gibbs 1979).

Molecular Orbital Methods In the molecular orbital (MO) method which forms the basis of this paper, the total molecular wavefunction is taken essentially as a product of one-electron molecular orbitals. Strictly speaking the MO product or configuration must be "antisymmetrized" so as to conform to the indistinguishability of electrons, the most common manifestation of which is the familiar Pauli exclusion principle. The individual MO's, ~bi, are generally expanded as a linear combination of atomic orbital (AO) basis functions, Zk,

q~ = ~ c~z~

(1)

k

and the MO coefficients, Cki, are determined by solution of the Hartree-Fock matrix equations. Aside from the total molecular energy, which is the expectation value of the Schr6dinger Hamiltonian, many molecular properties of interest can be expressed in terms of the charge density p(r),

p(r)=~,~(r) J

(2)

224

M.D. Newton and G.V. Gibbs

where the sum is over all occupied MO's. While the integral of the charge density yields the total number of electrons, it is often useful to partition the total charge into various atomic or bond populations which arise naturally from the expansion of the MO's in terms of the atomic orbitals [Eq. (1)]. For the case of closed shell species, which we are concerned with in the present study, one is led finally to the following expression for the atomic orbital population, qk, and overlap population, nkl, associated with the atomic orbitals Zk and Zl (Mulliken 1955):

qk = 2 ~ Skt cki c u

(3)

l,i

and

(4)

n~l -- 2Skz ~, % % i

Sk~ ( = S ZkZldr) in Eqs. (3) and (4) quantitativelymeasures the extent of overlap of atomic orbitals Zk and Z~. It is also of interest to define corresponding quantities summed over atomic orbitals on particular atoms. Thus, the atomic population on atom X is defined as Q ( X ) = ~ , q k where k includes all atomic orbitals on k

atom X and the Mulliken overlap population for a pair of atoms X - Y is defined as n ( X - Y ) = ~ nkl where k and 1 refer to the atomic orbitals on X and Y,, respectively, k, In the present ab initio calculations, all necessary molecular integrals over the AO basis functions are evaluated accurately, using standard computer programs (Binkley et al. 1978), and it only remains to discuss the selection of the AO basis functions. While the true Hartree-Fock limit requires a very elaborate basis set, we shall limit our studies to bases which are considerably simpler, but which have been shown to be very useful in studying molecular properties. The simplest basis set is the minimal set designated STO-3G (Hehre et al. 1969, 1970), in which each atomic orbital of an atom is represented by a so-called Slater-type orbital (STO), expanded as a linear combination of three Gaussian-type orbitals (GTO's). We shall also make use of more flexible basis sets designated 4-31G (for hydrogen and oxygen) (Ditchfield et al. 1971) or 6-6-31G (Binkley 1979) (for Si; the 6-6-31G basis is an elaboration of 4-4-31G basis cf., Hehre and Lathan 1972), in which the inner shell electrons are represented by minimal sets (expanded in terms of four or six GTO's), while a double set of AO basis functions is employed for the valence electrons, i.e., a fixed combination of three GTO's and an additional single GTO. Both the STO-3G and the 6-6-31G sp basis sets for Si will be supplemented in some cases with a set of 3d AO's, leading to the designation STO-3G* (Collins et al. 1976) and 6-6-31G*. In the case of Si, or other second row atoms, 3d AO's are of potentially great importance in facilitating 7c-back bonding from ligand atoms such as oxygen (Pauling 1952; Cruickshank 1961; Collins et al. 1972, 1976; Schmiedekamp et al. 1979). For a recent general discussion of applications of ab initio molecular orbital theory, the reader is referred to the work edited by Schaefer (1977).

Ab Initio Calculated Geometries and Charge Distributions

225

Fig. 1. Orthosilicic acid molecule, H4SiO4, D 2 d conformation with SiOH angles of 109.5 °. The larger spheres represent oxygen, the smallpheres represent hydrogen and the central intermediate-sized sphere represents Si. No particular physical significance is ascribed to these sphere sizes

-583.35

\

!

!

-'~ STO-3G(D2d)

-583.36 .

Fig. 2. Potential energy curves (Hartree units) calculated as a function of the S i - O bond length for orthosilicic acid (Fig. I). The minimum energy S i - O bond lengths are indicated by arrows. Optimization of each O - H bond and each SiOH angle of the molecule yielded d ( O - H ) = 0 . 9 9 A and LSiOH = 108 °, values essentially identical with those assumed in the calculation of the minimum energy S i - O bond length

6 v

I-583.62

ST0-3G*

\.&/ (D2d)

g

/"

(.o _

-585.6

°~.

/"

/e

~.....°J° I

1,60

I

1.65

d(Si-O)l

S i - O H Bond Length and OSiO Angle Calculations for H4Si04 The structure of orthosilicic acid was taken in this study to consist of a central silicon atom bonded to four oxygen atoms arranged at the corners of a tetrahedron. A hydrogen atom was placed at 0.96 ~ from each oxygen atom and each SiOH angle was bent at 109.5 °, reducing the symmetry of the molecule to D2d as shown in Fig. 1 (cf. Collins et al. 1972). Ab initio STO-3G calculations completed for the molecule as a function of the S i - O H bond length generated the potential energy curve drawn in Fig. 2 (Table 1). The minimum energy bond length (1.65 A) obtained by fitting a parabola to the curve is in close agreement with the mean S i - O H bond length (1.67 A) measured for S i - O H bonds in a number of sodium silicate hydrates, N a 2 0 . S i O 2 . x H 2 0 , x = 5 , 6,

226

M.D. Newton and G.V. Gibbs

Table 1. The computed total energy as a function of the S i - O bond length for orthosilicic acid, H4SiO4, and m i n i m u m energy bond lengths, d ( S i - O ) e

STO-3G d ( S i - O)

1.62

I

-

Er(a.u.) a

1.535 - 583.31727 1.580 - 583.35018 1.585 -583.35022 1.600 -583.35824 1.620 -583.36397 1.640 - 583.36676 1.650 -583.36727 1.660 - 583.36717 1.685 -583.36170 d(Si - O)e = 1.65 A

1 atomic u n i t = 6 2 7 . 5 kcal 1.63

STO-3G ~

I

I

I

.,:.,

..:._.-.:..

d(Si

O)

Er(a.u.)"

1.580 1.600 1.620 1.630 1.640 1.650

-583.62808 - 583.63045 -583.62971 -583.62828 -583.62617 - 583.62344

1.660

- 583.62009

d ( S i - O ) e = 1.60 A

I

-

"-:! ":"".; "

A

1.61

0

o•lm~



I

~

~lluo







omo ooo o ~ • •



•0%





r

• •o



"a 1.60

i

1.59

1.58

I 107

TRIPLE

I 108

ANGLE

I 109

I I10

AVERAGE

I

III LOSi03

112

Fig. 3. Scatter diagram of the d ( S i - O) vs the triple angle average of the three OSiO angles c o m m o n to the bond for the silica polymorphs coesite (Gibbs et al. 1977), low cristobalite (Dollase 1965) and low tridymite (Konnert and Appleman 1978). The dashed line locates the regression line, d ( S i - O ) = 2 . 1 7 - 0 . 0 0 5 1 LOSiO3, fit to the experimental data for the silica polymorphs and the solid one is the line calculated from ab initio theory for H6Si20 7 (see Table 2)

7, 9 (Glasser and Jamieson 1976), Na3HSiO4.5H20 (Smolin et al. 1973), pectolite, HNaCazSi30 9 (Tak6uchi et al. 1976) and rosenhahnite, CaaSi3Os(OH)2 (Wan et al. 1977). On the other hand, the bond length obtained in a STO-3G* basis set calculation for the molecule (Fig. 2) is somewhat shorter (1.60 4) and in poorer agreement. As noted above, Collins et al. (1976) observed a similar deterioration in the accuracy of calculated bond lengths when d-type functions were included in the STO-3G basis sets for several other silicon containing molecules. EHT calculations for orthosilicic acid (see Fig. 1 c of Louisnathan and Gibbs, 1972b) have shown that bond overlap populations calculated for each bond are strongly dependent upon LOSiO3. Inasmuch as larger n ( S i - O ) values were

Ab Initio Calculated Geometries and Charge Distributions

227

Table 2. Total energy of H4SiO4 (see Fig. 1c of Louisnathan and Gibbs 1972) calculated as a function of OSiO~,for three different sets of apical, Si- Oa, and basal, Si Ob, bonds (Louisnathan and Gibbs 1972b). The minimum energy triple angle average calculated for each apical bond is denoted by L(OSi03)e d(Si- O)a = 1.50 A. d(Si-- Oh) = 1.66 .A.

d(Si-Q) =d(Si--Ob) = 1.62/~

d(Si-O.) =1.68 A d(Si-Q)= 1.60 A

LOSiO3

LOSiO3

LOSiO3

Er(a.u.) a

109° -583.61191 112 -583.61200 115 -583.60704 L(OSiO3)e=110.6° a

Er(a.u.)

106° -583.62642 109 -583.62965 112 - 583.62792 L(OSiO3)e= 109.4°

Er(a.u.)

103° - 583.61666 106 -583.62388 109 -583.62623 L(OSiO3)e= 108.9°

a.u. =atomic units

calculated for bonds involved in the wider angles, it was asserted, as observed in Fig. 3 for the silica polymorphs, that there should be a tendency for short bonds to involve wide angles. To discover whether the bond length-LOSiO3 correlations can be reproduced by ab initio methods, calculations were completed for the molecules studied by Louisnathan and Gibbs (1972b). Despite the small variation calculated for d ( S i - O) and LOSiO3 (Fig. 3), the calculations corroborate the E H T result that short bonds should involve wide angles (Table 2). Nevertheless, the observed S i - O bond length variations in the silica polymorphs appear to show a significantly greater dependence on LOSiO3 variations than predicted by theory based on calculations for an isolated orthosilicic acid molecule.

S i - O Bond Length and SiOSi Angle Calculations for H6Si207 Bond Overlap Populations and Isovalent Hybridization Mulliken bond overlap populations calculated with extended Htickel theory for the bridging S i - O bonds in Si20)-6 as a function of the SiOSi angle (for the range 130 ° to 180 °) have been shown to vary linearly when plotted against -secLSiOSi (cf. Fig. 4, Gibbs et al. 1972). To learn how bond overlap populations generated with STO-3G theory vary with -secLSiOSi, calculations were completed for a pyrosilicic acid molecule (Fig. 4) with all its S i - O bonds fixed at 1.62 A, all its O - H bonds fixed at 0.96 A and all its SiOH angles bent at 109.5 °. The resulting n ( S i - O ) values for the bridging bonds (Table 3) are plotted against the SiOSi angle in Fig. 5a where a nonlinear curve obtains. On the other hand, when n ( S i - O ) is plotted against -secLSiOSi (Fig. 5b) a well-developed linear correlation (r2=0.98) results as predicted by extended Htickel theory. Nevertheless, when the n ( S i - O ) value calculated at 110 ° is included in the regression analysis, the r 2 value is reduced to 0.88, indicating that the n ( S i - O ) vs -secLSiOSi correlation departs somewhat from linearity when a larger range of angle data is used in the analysis. The correlation between n ( S i - O ) and SiOSi angle can be related to the hybridization of the valence orbitals on the bridging oxygen of H6Si20 7"(Gaskell

228

M.D. Newton and G.V. Gibbs

f Fig. 4. Pyrosilicic acid molecule, H6Si207, C2v point symmetry and bent SiOH angle=109.5 °. The large spheres represent oxygen, the intermediate-sized spheres represent silicon and the small ones represent hydrogen. No particular physical significance is ascribed to these sphere sizes

Table3. STO-3G Mulliken bond overlap population, n(Si-O), calculated for the bridging bond of H6Si20~ as a function of the SiOSi angle. 2 is the coefficient of mixing of a hybrid orbital 2s+22p on the oxygen and 100/(1 +22) is the percentage s-character of the oxygen determined by the SiOSi angle n ( S i - O) a

LSiOSi

22= - secLSiOSi

100/(1 +~2)

0.579 0.572 0.562 0.550

180° 160 150 140

1.000 1.064 1.155 1.305

50.0 48.4 46.4 43.4

0.543 0.536 0.502

135 130 110

1.414 1.556 2.924

41.4 39.1 25.5

a The contribution associated with the oxygen l s and silicon ls, 2s, and 2p basis functions is small and essentially constant in magnitude (-0.04)

1966; Bokii and Struchkov 1968; Brown et al. 1969). If each of the two bond hybrid orbitals on the oxygen is expressed in the form 2s+22p where 2 is a mixing coefficient, then it can be shown (Coulson 1961) that the percentage s-character [100/(1 +)2)] of each hybrid is determined by the SiOSi angle where )42= -secLSiOSi. Since the radius for the radial charge density of the 2s orbital of oxygen is smaller than that of the 2p orbitals, it follows that the effective radius of the oxygen atom should decrease and that the S i - O(br) bond should shorten as the SiOSi angle widens. Moreover, since the 2s orbital lies below the 2/) orbital in energy, the electronegativity and accordingly the effective charge on the bridging oxygen should increase as the SiOSi angle widens. It is important to note in the above analysis that the oxygen hybrids are always directed along the silicon-oxygen bond vectors.

Ab Initio Calculated Geometries and Charge Distributions

0.58

0.58

A

0 0.56 I

b3 ¢-

0.54

/

/

/

/

I

co

(a) I

140

160

180

/ 0.54

./"

(c)

/ I

I

I

40

45

5O

I00/(1 + Xz)

ZSiOSi ,

/ Fz = 0 . 9 9 5 / "

0 0.56

I

0.58

229

,

/)

0 0.56 I

b3 e--

0.54

(b) r

i

J

1.4

1.2

1.0

X2 = -

Fig. 5a-e. Mulliken bond overlap populations, n(Si-O), calculated for the bridging bond in H68izO v plotted against (a) kSiOSi, (b) 2 2 = - secLSiOSi and (c) percentage s-character = 100/(1 + 2 z) of the bridging oxygen atom; regression equation is n ( S i O)=3.1 x 10 -3 (% s-character)+0.419.

secZSiOSi

Since the OSiO angles were fixed in tile calculation.at 109.5°, we assumed that the hybridization of Si remained fixed at sp 3. Hence, the n(Si-O) values obtained for H6Si207 should depend only on the hybridization of valence orbitals on oxygen. Figure 6 shows that the orbital overlap populations (Table 4) of the O 2s (and also 2p) with the Si 3s and 3p orbitals are highly correlated with the percentage s-character of the oxygen. On the other hand, the overlap population of the O 2p with the Si 3s and 3p orbitals shows a concomitant decrease with angle and s-character as expe,cted. Since the S i - O(br) bond overlap population is obtained by summing each of these orbital populations, the linear correlation (r 2= 0.99) that results between n(Si- O) and percentage s-character of the bridging oxygen is expected (Fig. 5c). We note that the inclusion of the 110° n(Si-O) value (Table 3) into the regression analysis has an insignificant effect (r2=0.98), indicating that the n(Si-O) vs percentage s-character correlation remains linear (unlike the n(Si--O) vs ) 2 correlation) over the entire angular range from 110° to 180°. Pantelides and Harrison (1976) have assumed in their bond orbital descriptions of silica that the 2s AO's of the oxygen atom behave essentially as core electrons. On the other hand, our calculations (Table 4) indicate that the overlap populations involving the O 2s and the valence AO's on Si are about 0.1, indicating that the electrons of the 2s AO's are bonding electrons. Furthermore, even though the O atom has appreciable

230

M.D. Newton and G.V. Gibbs

1','3 ¢J') 0 , 0 2 I

0.00

/

/

r 2 = 7//" /"

I

I

I

140

160

180

[

I

40

I

I

I

45

50

I

I

r 2 = 0.998

0.14 A

/

roro U'~(D ,

I

/

0.12

OJoJ

o.o. "

4"

""

0.10

/ 0.0~

I

I

I

140

160

180

40

I

L

45

50 992

0.34

.\

,

Od ,~, 0.32 140

I 160

I 180

Z SiOSi

40

I 45

I 50

100/(1+ X2)

Fig. 6. Orbital overlap populations, n[O(2s, 2 p ) - S i ( 3 s , 3p)], of the valence 2s- and 2p-atomic orbital of the bridging oxygen and the 3s- and 3p-atomic orbitals of Si calculated for H6SizO v and plotted against LSiOSi and 100/(1 +22). The overlap populations of the AO's representing the core electrons and the valence shell electrons were neglected in our analysis as they were calculated to be small and independent of SiOSi angle

Table 4. The orbital overlap populations of the oxygen 2s and 2p atomic orbitals with the 3s and 3p atomic orbitals on silicon calculated for the bridging oxygen of H6SizO7 at SiOSi angles of I40 °, 160 °, and 180 °

Atomic orbital overlap populations

n[O(2 s) -- Si(3 s)] n[O(2 s) - Si(3p)] n[O(2p) - Si(3 s)]

n[O(Zp)- Si(3p)]

/.SiOSi 140

160

180

0.009 0.087 0.145 0.348

0.027 0.118 0.132 0.335

0.033 0.130 0.127 0.329

Ab Initio Calculated Geometries and Charge Distributions

Table 5. The total energy, E r , calculated as a function of the S i O S i angle for pyrosilicic acid, H 6 S i 2 O v , and minimum energy SiOSi angle, LSiOSie

d(Si-()br)=

231

1.59 A , d ( S i - - O n b r ) = 1.63 A

110

- 1091.74363

150

i40

-1091.76642

160

- 1091.76636 - I091.76551

i45

-1091.76656

180

- 1091.76434

LSiOSie=145 ° d(Si - Obr) = d(Si -- Onbr)

=

1.62 A

110

- 1091.74618

150

- 1091.76121

130

- 1091.76221

160

- 1091.75932

I35

-- 1 0 9 1 . 7 6 2 7 4

180

- 1091.75726

140

- 1091.76261 /-SiOSie = 136 °

d(Si - O b r ) = 1.65, d(Si - O,b~) = 1.61 A 110

- 1091.74448

I35

- 1091.75592

125

- 1091.75562

160

- 1091.74938

130

- 1091.75619

180

- 1091.74644

L S i O S i ~ = 131 °

negative charge, its 2s orbital contains 1.75e as compared with 2.0e for a true lone pair. However, in contrast with our results, Yip and Fowler (1974) have concluded from LCLO-MO calculations for such large clusters as Si2Ov, SisO 4 and S i s O 7 that the sp hybridization of the bridging oxygen is negligibly small.

Potential Energy Curves for the SiOSi Angle

To clarify the nature of the correlation between S i - O bond length and SiOSi angle as predicted by ab initio theory, three STO-3G calculations each were made for pyrosilicic acid as a function of the SiOSi angle. In the calculations, the bridging bonds were set successively at 1.59, 1.62 and 1.65 • whereas the nonbridging bonds were set at 1.63, 1.62 and 1.61 ~ , respecctively, to maintain the mean S i - O bond length of the molecule at 1.62 A. The potential energy curves generated in the calculations (Table 5) are drawn as a function of the SiOSi angle in Fig. 7. The overall shape of each of these curves differs at wide angles from the one generated in an ab initio calculation for the Si20~ 6 ion (Zupan and Buh 1978). Like the curves in Fig. 7, the Si2076 curve rises steeply at narrow angles, yet, unlike our curves, it shows a well-defined minimum at 180 °. Actually, the shapes of the curves in Fig. 7 are qualitatively similar to those calculated for various s]licic acid molecules and disiloxane using C N D O / 2 theory (Tossell and Gibbs 1978; Meagher et al. 1979). They show a relatively flat segment at wide angles from 160 ° to 180 ° and a very steep segment at angles narrower than the minimum energy angle, ~ 1 4 0 °.

232

M.D. Newton and G.V. Gibbs - 1091.74

/

I

I

I

I

,o J ÷ f ÷ ~o ~

t37 - 1091.76 I,-'-

o-~.i~5 ~ - - x

ILl

X_ t _ X ..----"X o

+ d(Si-Obr)

- 1091.78

-I00

= 1.65A

o

d(Si-Obr)=

1.62~,

x

d(Si-Obr)=

1.59,~

I

I

I

I

120

140

160

180

Fig. 7. Potential energy curves (atomic units) calculated as a function of the SiOSi angle in the pyrosilicic acid molecule (cf. Fig. 3) for bridging S i - O bond lengths of 1.65, 1.62 and 1.59 A. The minimum energy LSiOSi angles are indicated by the numbers next to the a r r o w s

ZSiOSi

The steep segment of the curves at narrow angles is consistent with the fact that SiOSi angles less than 120 ° are unknown in silicates and silica glass. In addition, the flatness of the curves and the relatively low energy barrier to linearity (4-6 kT) conform with the large range of SiOSi angles (120 ° to 180 °) exhibited by silicates, siloxanes and silica glass. Interestingly, the application of high pressure to the low quartz (Jorgensen 1978; Levien et al. 1980) and coesite structures (Levien and Prewitt, in preparation) produces significant narrowing of the SiOSi angles;yet the narrowest angle attained, 133 °, is nearly 15° wider than the steep and rapidly rising portion of the curves in Fig. 7 at angles less than about 120% It is possible upon compression of the angle below 120 ° that the non-bonded repulsions between adjacent Si atoms (O'Keefe and Hyde 1976, 1978; O'Keefe et al. submitted) will destabilize the structure in favor of one with six-coordinated Si. If true, this may explain in part why four-coordinate silica polymorphs and glasses possess SiOSi angles of 120 ° or wider. Lazarev et al. (1967) have observed for a number of siloxanes that the SiOSi bending force constant decreases and the S i - O stretching force constant increases as the SiOSi angle widens. The curvatures of the curves in Fig. 7 indicate as observed for the siloxanes that the SiOSi bending force constant for siloxane groups in silicates should decrease in a regular way as the bridging S i - O distance decreases. Moreover, the recent calculation of the bulk modulus of low quartz by O'Keefe et al. (submitted) using the SiOSi angle bending force constant calculated for H6Si207 indicates that the interatomic forces in crystals like low quartz are very similar to those in molecules like pyrosilicic acid. As predicted by semi-empirical theory, the minimum energy values for each curve in Fig. 7 show a progressive shift to narrower angles as the length of the bridging bond increases from 1.59 to 1.65 ~. Moreover, when minimum energy percent s-character values are plotted as a function of the bridging bond lengths, a linear correlation obtains (r2=0.99). Because bond length is

Ab Initio C a l c u l a t e d G e o m e t r i e s a n d C h a r g e D i s t r i b u t i o n s

233

Table 6. S T O - 3 G c o m p u t e d t o t a l energy as a function of Si O b o n d l e n g t h (A) for pyrosilicic acid, H6Si207, for SiOSi angles of 180 °, 160 °, a n d 140 o with S i O H = I 0 9 . 5 ° a n d d ( O - H ) = 0 . 9 5 A Er(a.u.)

Si - O(br)

Si - O(nbr)

Er(a. u.)

Si - O(br)

Si - O(nbr)

1.67 1.67 1.62

1.6l 1.66 1.72

- 1091.75333 1091.76333 - 1091.76852

1.62 1.62 1.57

1.61 1.66 1.66

1.67 1.67 1.62

1.6l 1.66 1.7l

-1091.75537 - 1091.76545 - 1091.76914

1.62 1.62 1.57

1.61 1.66 1.66

1.67 1.62 1.62

1.61 1.71 1.61

--1091.76162 -1091.76845 --1091.76880

1.67 1.57 1.62

1.66 1.66 1.66

LSiOSi = 180 ° - 1091.74051 - 1091.75059 - 1091.75224 L S i O S i = 160 ° - 1091.74408 - 1091.75424 - 1091.75435 L S i O S i = 140 ° - 1091.7513l - 1091.75777 1091.75863

Table 7. S T O - 3 G m i n i m u m energy S i - O b o n d lengths for H6Si207 c a l c u l a t e d as a function of the SiOSi angle

L SiOSi

d(Si - Obr)

d ( S i - O,br)

(d(Si-- O))

140 160 180

1.60 1.57 1.56

1.66 1.66 1.66

1.645 1.637 1.635

indicated to shorten in a regular way w:ith angle, the bond stretching force constant for siloxane groups in silicates, silicate glasses and siloxanes should also increase in a regular way with angle (Revesz 1971).

S i - 0 Bond Length and Angle Variations We assumed in the above calculations that the mean S i - O bond length of a silicate tetrahedron is essentially constant and independent of the SiOSi angle between condensed tetrahedra. However, semi-empirical MO calculations completed for clusters isolated from coesite (Gibbs et al. 1977b) and other silica polymorphs (Meagher et al. 1977, 1979) and low cordierite (Cohen et al. 1977) suggest that the mean S i - O bond length, ( d ( S i - O ) ) , of a silicate tetrahedron depends upon the SiOSi angle, with larger tetrahedra predicted to involve narrower angles. As a further test of this trend, STO-3G calculations were completed as functions of the bridging and nonbridging S i - O bond lengths for three H6Si207 molecules whose SiOSi angles were clamped at 140°, 160°, and 180° (Table 6), respectively. The minimum energy S i - O bond lengths obtained in the calculations and given in Table 7 show that the Si-O(br) bond shortens progressively as the SiOSi angle is widened with the Si-O(nbr) bond length remaining unchanged.

234

M.D. Newton and G.V. Gibbs "~ I

I

J

I 4

J

1.62

o

1.60

I

i "{3 1.58

1.56

l (a)

I

i

140

160

/SiOSi

+-I i

180

I

(b)

t 40

I 45

I00/(

x~X* I 50

1 +X a )

Fig. 8a and b. The observed S i - O bond lengths in coesite (upper curves) and those calculated for the bridging bonds in H6SizO 7 plotted against LSiOSi (a) and 100/(1+)~ z) (b) were )2= -secLSiOSi, The brackets' about each data point for coesite denote the estimated standard deviation of the observed bond length (Gibbs et al. 1977a); regression equation for H 6 S i 2 0 v data plotted in (5): d ( S i - O) = - 6.046 × 10- 3 (% s-character) + 1.862

Accordingly, the calculations show that the mean S i - O bond length of the silicate tetrahedra decreases from 1.645 A when the SiOSi angle is 140° to 1.635 • when the angle is 180°. A linear regression analysis of " / / .-_'2 "--~-- "~--~-~i;-7" .~:~:;I," '.. \\ \,£:"i(::~'::!?':~ ~ ......... ~ "\ ":::%?'.'.:i!i)':/ / .."

_-

oo o o 5 ~

, '...

237

.... #

,

i .

:_.. ~-

~ "

-.,

Fig. 10a and b. Theoretical charge density difference m a p (a) calculated for the plane of the bridging SiOSi bonds of H6Si~O v compared with an experimental difference F o u r i e r m a p (b) obtained for a typical SiOSi b o n d in coesite. The atoms that lie in the plane of each m a p are connected by solid lines• Bonds to atoms situated above (or below) the plane of t h e m a p are drawn as dashed lines. The n u m b e r s in parentheses next to each a t o m located out of the plane give the distance of the a t o m f r o m the plane• The n u m b e r s on the contours in (a) refer to electrons per cubic A; the c o n t o u r interval in (b) is 0.04 e/:~ 3

on Si and O(br) show a slight but systematic decrease as the bridging bond is lengthened and the SiOSi angle is narrowed. These results agree with the constructs of hybridization but are at variance with the charge variations obtained in extended Hfickel calculations for silicate clusters (Gibbs et al. 1977b) which suggested that the charge on O(br) increases as the SiOSi angle narrows. In addition, it is noteworthy that a population analysis of the minimum basis STO-3G calculation completed for H4SiO4 (D2e) gave atomic charges (Si, + 1.36; O(nbr), - 0.52; H, + 0.18) in close agreement with those obtained for H6Si207.

Theoretical

Deformation

Map

Figure 10a shows a theoretical (STO-3G) density difference map for H6Si207 obtained by subtracting from the molecular charge density the superposed densities of the constituent atoms, centered at their respective positions in the

238

M.D. Newton and G.V. Gibbs

molecule. It is well known that a basis set as limited as STO-3G is not reliable for quantitative electron density studies. Nevertheless, we can use properly scaled STO-3G wave functions to illustrate a few simple, qualitative features. The atomic densities for the calculation were obtained from spherically-averaged ground state STO-3G wavefunctions. It is important to note that the molecular and atomic densities involved in the theoretical difference map are based on Slater orbital exponents optimized, respectively, for the molecular and free atom situations (Hehre et al. 1969, 1970). The major rescaling effect occurs for the valence AO's of Si, with the standard molecular value (~(3 s)= ~(3p)= 1.75 ao 1) being appreciably greater than the energy optimized free atom value (~(3 s)= ~(3p)= 1.52 a o 1). Difference maps obtained for H6Si20 7 from free atom densities which are based on molecular STO exponents showed virtually no density build-up in the bond region. On the other hand, difference maps (Fig. 10 a) calculated with the free atom densities based on optimized exponents show a small but significant build-up of charge density (peak heights of 0.1 e A - 3) along both the bridging and nonbridging bonds. In addition, the map shows a negative peak at the position of the bridging oxygen adjacent to a relatively large positive lone-pair peak situated at about 0.5 ~ from the oxygen on the back side of the SiOSi angle, and negative peaks at the two Si positions. For purposes of comparison, a difference Fourier map computed for a plane defined by a typical SiOSi linkage in coesite is shown in Figure 10b (Gibbs et al. 1978). The build-up of electron density along the S i - O bond conforms reasonably well with that obtained for the theoretical map. However, there is no peak on the backside of the SiOSi angle that may be ascribed to the lone-pair electrons of oxygen.2 The larger peak heights calculated for the nonbridging bonds of pyrosilicic acid (Fig. 10a) conform with the observation that the nonbridging bonds in forsterite (Lager et al. 1979) and orthoenstatite (Ghose 1979) show a larger concentration of electron density in the nonbridging bonds than observed for the bridging bonds in coesite. The 3 d-Orbitals of Si

Since Pauling's original proposal, there has been a prolonged debate about the involvement of the 3 d orbitals in the wavefunctions of the S i - O bond (cf. Craig et al. 1954; Stone and Seyferth 1955); Cruickshank 1961; Brown et al. 1969; Baur 1971, 1977, 1978; Gibbs et al. 1972; Louisnathan and Gibbs 1972b). More recently, Collins et al. (1972) concluded from ab initio calculations completed for the silicate ion and orthosilicic acid that there is considerable d orbital involvement, with a rather large number of electrons in the 3d-orbitals on Si. Moreover, they found that the addition of these orbitals to their minimal basis set calculations produced calculated L2,3 X-ray fluorescence spectra in close agreement with those observed for silica. Further support for the importance of 3 d orbitals was indicated by a recent extended basis SCF ab initio calculation for molecular S i O 2 which yielded a substantial population of 3d electrons (0.6) (Pacansky and Hermann 1978). Contrary to these results, a popuz One of the reviewers (DWJC) suggested that the lack of an oxygen lone-pair peak in the experimental map may be due to the refinement process moving the coordinate of the oxygen atom towards the lone pair

Ab Initio Calculated Geometries and Charge Distributions

239

Table 9. Atomic charges, Q, and Si3d orbital populations for H4SiO, (D2d), d(Si-O)=1.635 Basis set

Q(Si)

Q(O)

Q(H)

Population of the Si3d type orbitals

-0.52 -0.40

0.18 0.20

0.80

0.46 0,44

0.64

Minimal sp STO-3G STO-3G*

1.36 0.81

Split valence sp 6-6-31 6-6-31G*

1.84 1.10

-0.92 -0.72

lation analysis obtained in an extended basis ab initio calculation for the linear Si20 molecule yielded only 0.2 electrons in the 3 d-type orbitals (Gilbert et al. 1973). Tossell et al. (1973) also failed to find significant electron density in the Si3d-type orbitals in an X ~-scattered wave calculation performed on the silicate ion surrounded by a Watson sphere of +4 charge assumed to approximate the stabilizing electrostatic field of a silicate crystal structure. To explore the extent to which the 3d orbitals on Si may be involved in the wave functions of orthosilicic acid, extended basis 6-6-31G and 6-6-31G* as well as STO-3G and STO-3G* ab initio calculations were completed for H4SiO4 (D2d) with d(Si-O) clamped at 1.635 ~. As shown in Table 9, both sets of calculations yield appreciable 3 d-orbital populations of similar magnitudes in close agreement with that obtained by Pacansky and Hermann (1978) in their extended basis calculation for molecular SiO2. Since we obtained essentially the same populations from the extended and the minimal sp basis calculations, we believe that the 3d-orbital populations obtained in this study and by Pacansky and Hermann (1978) are qualitatively significant and not an artifact of a poor basis set.

The ( d - p ) ~-Bonding Model In 1961 Cruickshank asserted from symmetry and group overlap integral arguments that for the purposes of understanding S i - O bond length variations, one need only consider the ~z-bonding MO's formed with the e-type (3dz 2, 3dx2y~ ) orbitals of Si and a pair of 2p orbitals of O. Utilizing a recipe prescribed by simple valence bond theory, he concluded that the ~-bond order of a bridging bond should be maximized (i.e., S i - O bond length minimized) when the SiOSi angle is straight and the pair of 2p-orbitals on O, oriented perpendicular to the SiOSi bond, join fully in the ~-systems. Notwith standing the assertion that the t2 orbitals of ~z-symmetry (3 d~y, 3 dyz, 3 dzx) on Si are only weakly bonding, Collins et al. (1972) have since discovered from ab initio calculations that all five 3 d-type AO's of Si are ofcomparable importance in the wavefunction of the molecule. In spite of this important result, workers continue to ascribe Si - O bond length variations in silicates and siloxanes to changes in the ~z-bond order of the e-type MO's induced by variations in the SiOSi angle. To clarify the role played by the 3 d-type functions in representing a S i - O

240

M.D. Newton and G.V. Gibbs

Table 10. Mulliken bond overlap populations for the Si-O bridging bond and atomic charges for H6Si207 calculated as a function of the SiOSi angle with a STO-3G* basis Si orbitals

Total sp d

da &r

Overlap populations

LSiOSi 140

160

180

n[Si(3s, 3p, 3c0-O(2s, 2p)] n[Si(3s, 3p)- O(2s, 2p)] n[Si(3d)-O(2s, 2p)]

0.832 0.556 0.276

0.854 0.572 0.282

0.862 0.577 0.285

n [Si(3d) - O(2p, 2s)]G n [Si(3d) - O(2p)]~

0.113 0.163

0.114 0.169

0.114 0.171

d~(e)

n [Si(3de) - O(2p)]~

dTr(t2)

n [Si(3dr2)- O(2p)]~

0.108 0.063

Atomic charges Q(Si) Q(Obr) Q(Onbr) Q(H)

+0.834 - 0.423 -0.393 +0.185

+0.844 - 0.439 -0.393 +0.185

+0.847 - 0.445 -0.393 +0.185

bond as a function of its angle, STO-3G* calculations were completed for H6Si20 7 with the SiOSi angle fixed successively at 140 °, 160 °, and 180 °. The Mulliken bond overlap populations obtained for the bridging bond of the molecule are given in Table 10 as a function of the SiOSi angle. As observed in the STO-3G calculations discussed earlier (see Table 4), the overlap population, n[Si(3s, 3 p ) - O ( 2 s , 2p)], of the 3s and 3p AO's of Si with the 2s and 2p AO's of oxygen is linearly (r 2= 0.99) correlated with the percentage s-character of the bridging oxygen. Of the total overlap population associated with the Si3s and 3p orbitals, only about 15% arises from re-overlap. However, most of the 7c-overlap population comes from the 3 d-orbitals and is linearly correlated (r2=0.99) with 100/(1+22). On the other hand, the 3d a-overlap population is independent of the SiOSi angle. Inasmuch as n[Si(3s, 3 p ) - O ( 2 s , 2p)] and n[Si(3 d) - O(2p)]~ both vary linearily with 100/(1 + .~2) and n[Si(3 d) - 0(2 s, 2p)]o is independent of the angle, the total overlap population, n[Si(3 s, 3p, 3 d ) - O(2s, 2p)], of the 3s, 3p and 3d AO's of Si with the valence AO's on O must also vary linearily with the percentage s-character of the bridging bond. This result indicates that the linear correlation that obtains between observed S i - O bond length data and 100/(1 + ) 2 ) (Fig. 9) is expected to hold despite the participation of the 3d AO's in the wave functions of the silicate anion. Finally, the values calculated for the n-overlap populations of the e- and t2-type AO's of Si with the lone-pair orbitals of the oxygen indicate that all five of the 3d orbitals of Si play an important role in ( d - p ) 7c-bonding and S i - O bond length variations in silicates and siloxanes, as indicated in Table 10 for the 180 ° case. Atomic Charges

The atomic charges calculated for H6Si20 7 with a STO-3G* basis are similar to those obtained in our STO-3G* calculation for H4SiO4. Moreover, as

Ab Initio Calculated Geometries and Charge Distributions

241

Fig. 11. A stereo-pair drawing of a SiO2 moiety of low quartz where the immediate environment of the moiety is simulated by self-consistent point charges. The SiO2 moiety is depicted as two large spheres (oxygen) and bonded to a common smaller sphere (Si). No particular physical significance is ascribed to these sphere sizes. The small two-coordinated solid spheres represent the sites in the simulated low quartz structure where the self-consistent charges of oxygen are located, the four-coordinated solid spheres correspond to lhe sites where the self-consistent charges of Si are located, and the peripheral one-coordinated spheres represent the sites of the point-charge H atoms whose charges are defined to make the H12SisO16 cluster electrically neutral

observed in our STO-3G calculations, the charge on the bridging oxygen is indicated to increase progressively as the SiOSi angle widens (Table 10). However, the increase in Q(O) is small (-0.423 to -0.445) compared with the large increase (-1.00 to -1.25) calculated for a comparable angle change by Pantelides and Harrison (1976). In assessing the Mulliken atomic charges in Table 9 and elsewhere in this paper, one should not place too much quantitative significance on the absolute magnitude, especially for the extended 6-6-31G basis sets, where the Mulliken definition of charge clearly seems to yield an exaggerated picture of the electronegativity of oxygen. Nevertheless, these simple measures of atomic population are of considerable qualitative utility, especially in comparing a given atom (with a given basis set) in different molecular environments. The principal qualitative implication of Table 9 is that the silicate oxygen bond has an appreciable 3d population, a result which seems to obtain irrespective of the particular level used to represent the valence s and p orbitals (e.g., STO or 6-6-31G). The data of Table 9, of course, refer to free molecules, and one must recognize the fact that the Coulombic interactions in the solid state could stabilize a more ionic type of S i - O bonding. As a simple test of environmental effects, we have completed calculations for a SiO2 moiety of low quartz, with the immediate environment of the SiO2 simuhtted by point charges placed on the positions occupied by the nearest and next nearest neighbor atoms as in the crystal (Fig. 11). Overall charge neutrality was maintained by "protonating" the peripheral oxygen atoms in the cluster model. The protons were simulated by appropriate point charges, with d ( O - H ) taken as 0.96 A and LSiOH set equal to 80°. The point charges for all al~oms of a given type (Si, O or H) were constrained to have a common value, and the final values were determined by requiring that the point charges in the simulated environment be the same

242

M.D. Newton and G.V. Gibbs

as the Mulliken atomic charges obtained from the SiOa wavefunctions calculated in the presence of this environment. This self consistency was achieved through a series of iterative STO-3G calculations, which converged to the following atomic charges : Si, + 1.30 ; O, - 0.65. Since the corresponding Si and O charges for free SiO2 (with the geometry appropriate to a fragment of low quartz) are significantly smaller (+0.74 and - 0 . 3 7 , respectively), it is clear that the crystalline environment makes ionic bonding character relatively more favorable, even though the bonding should still be thought of as primarily covalent in character. In spite of the different formalisms used to estimate atomic charges in a Mulliken population analysis or a refinement of precise X-ray diffraction data (Stewart 1976; Yang et al. in preparation), the charges calculated for the Si and O atoms of the SiO 2 moiety (in the simulated low quartz environment) are in satisfactory agreement with experimental values (Q (Si)~ + 1.0; Q (O)~ 0.5) obtained in X-ray refinements of the crystal structures of the SiO2 polymorphs low quartz (Stewart, personal communication) and coesite (Gibbs et al. 1978). Nevertheless, it should be emphasized that the calculated charges are based on a limited model for the environment and thus we are by no means claiming to have converged to the results that would be obtained from the full electrostatic field of a quartz crystal,

Conclusions

Equilibrium geometries generated for orthosilicic and pyrosilicic acids conform with bond length and angle trends observed for the silica polymorphs and silonal groups in silicate hydrates. The broad asymmetric SiOSi angle distribution observed for silica and the variation of the SiOSi bending force constant with S i - O distance are in harmony with potential energy vs SiOSi angle curves generated for H6Si207. In addition, a theoretical deformation map shows a concentration of electron density in the S i - O bond in accordance with experimental maps. Atomic charges calculated for the Si and O atoms of the molecule are small, in agreement with Pauling's electroneutrality principle and experimental charges obtained in X-ray diffraction experiments. The close correspondence that obtains between observables generated for the molecules and those exhibited by the silica polymorphs and silicate hydrates indicates that local bonding forces in solids may not be very different from those in molecules and small clusters consisting of the same atoms with identical ligancies (see also Gibbs et al. 1972, 1974, 1977; Tossell and Gibbs 1976, 1977, 1978; Hill et al. 1979; Meagher et al. 1979; McLarnan et al. 1979; Peterson et al. 1979; Burdett 1980; Meagher 1980; O'Keeffe et al. submitted).

Acknowledgments. We are grateful to the National Science Foundation for supporting this study with grant EAR7723114, to Ramonda Haycocks for typing the manuscript and to Sharon Chiang for drafting the figures. This research was carried out in part under contract with the U.S. Department of Energy and supported by its Division of Basic Energy Sciences, and part was done while GVG was a Visiting Distinguished Professorin the Chemistryand GeologyDepartments at Arizona State University at Tempe during the 1979 Winter term. Professors D.W.J. Cruickshank of the

Ab Initio Calculated Geometries and Charge Distributions

243

Theoretical Chemistry Department at U.M.I.S.T., M. O'Keeffe of the Chemistry Department of the Arizona State University and Dr. A.G. Rcvesz of COMSAT Laboratories read the manuscript and made a number of helpful suggestions for its improvement.

References Appelo, C.A.J.: Layer deformation and crystal energy of micas and related minerals. I. Structural models for 1 M and 2M 1 polytypes. Am. Mineral. 63, 782-792 (1978) Baur, W.H.: The prediction of bond length variations in silicon-oxygen bonds. Am. Mineral. 56, 1573-1599 (1971) Baur, W.H.: Silicon-oxygen bond lengths, bridging angles, SiOSi and synthetic low tridymite. Acta Crystallogr. Sect. B: 33, 2615~619 (1977) Baur, W.H. : Variation of mean S i - O bond lengths in silicon-oxygen tetrahedra. Acta Crystallogr. Sect. B: 34, 1751-1756 (1978) Binkley, J.S. : The 6-6-31G basis is an elaboration of the 4-4-31G basis (private communication) (1979) Binkley, J.S., Whiteside, R., Haribaran, P.C., Seeger, R., Hehre, W.J., Lathan, W.A., Newton, M.D., Ditchfield, R., Pople, J.A. : Gaussian 76 an ab initio molecular orbital program. Quantum Program Chemistry Exchange, Bloomington, IN (1978) Bokii, N.G., Struchkov, Y.T.: Structural chemistry of organic compounds of the nontransition elements of group IV (Si, Ge, Sn, Pb). Zh. Strukt. Khim. 4, 722 (1968) Brown, G.E., Gibbs, G.V., Ribbe, P.H.: The nature and variation in length of the S i - O bond and A 1 - O bonds in framework silicates. Am. Mineral. 54, 1044-1061 (1969) Burdett, J.K.: How some simple solids hold together: The use of the fragment formalism in crystal chemistry. J. Am. Chem. Soc. in press (1980) Clark, J.R., Appleman, D.E., Papike, J.J.: Crystal-chemical characterization of clinopyroxenes based on eight new structure refinements. M.S.A Spec. Publ. No. 2, 31-50 (1969) Cohen, J.P., Ross, F.K., Gibbs, G.V.: An X-ray and neutron diffraction study of hydrous low cordierite. Am. Mineral. 62, 67-78 (1977) Collins, G.A.D., Cruickshank, D.W.J., Breeze, A. : Ab initio calculations on the silicate ion, orthosilicic acid and their L2, 3 X-ray spectra. J. Chem. Soc. Faraday Trans. 2 68, 1189 1195 (1972) Collins, J.B., Schleyer, P. von R., Binktey, J.S., Pople, J.A.: Self-consistent molecular orbital methods. XVII. Geometries and binding energies of second-row molecules. A comparison of three basis sets. J. Chem. Phys. 64, 5142-5151 (1976) Coulson, C.A. : Valence. London: Oxford University Press 1961 Craig, D.P., Maccoll, A., Nyholm, R.S., Orgel, L.E., Sutton, L.E.: Chemical bonds involving d-orbitals. J. Chem. Soc. 1954, 332-357 (1954) Cruickshank, D.W.J.: The role of 3d-orbitals in z-bonds between (a) silicon, phosphorus, sulfur, or chlorine and (b) oxygen or nitrogen. J. Chem. Soc. 1077, 5486-5504 (1961) Dewar, M.J.S., Schmeising, H.N. : Resonance and conjugation-II. Factors determining bond lengths and heats of formation. Tetrahedron 11, 96-120 (1960) DeJong, B.H.W.S., Brown, G.E. : Molecular orbital study of (Si, A1)zO~'- groups in silicates (abstr.). Eos 56, 1076- 1077 (1975) Ditchfield, R., Hehre, W.J., Pople, J.A. : Self-consistent molecular-orbital methods. IX. An extended Gaussian-type basis for molecular-orbital studies of organic molecules. J. Chem. Phys. 54, 724-728 (1971) Dollase, W.A.: Reinvestigation of the structure of low cristobalite. Z. Kristallogr. 121, 369 377 (1965) Durig, J.R., Flanagan, M.J., Kalasinsky, V.F. : The determination of the potential function governing the low frequency bending mode of disiloxane. J. Chem. Phys. 66, 2775-2785 (1977) Gait, R.I., Ferguson, R.B.: Electrostatic charge distributions in the structure of low albite, NaA1Si3Os. Acta Crystallogr. Sect. B:26, 68-77 (1970) Gaskell, P.H. : Thermal properties of silica Part 2. Thermal expansion coefficient of vitreous silica. Trans. Faraday Soc. 62, 1505-1510 (1966) Ghose, S.: Charge density distribution in enstatite, Mg2Si206 - preliminary results (abstr.). Eos 60, 415 (1979)

244

M.D. Newton and G.V. Gibbs

Gibbs, G.V., Hamil, M.M., Louisnathan, S.J., Bartell, L.S., Yow, H. : Correlations between S i - O bond length, SiOSi angle and bond overlap populations calculated using extended H/ickel molecular orbital theory. Am. Mineral. 57, 1578 1613 (1972) Gibbs, G.V., Hill, R.J., Ross, F.K., Coppens, P. : Net charge distributions and radial dependences of the valence electrons on the Si and O atoms in coesite (abstr.). Abstr. Progr, 3, 407 (1978) Gibbs, G.V,, Louisnathan, S.J., Ribbe, P.H., Phillips, M.W. : Semiempirical molecular orbital calculations for the atoms of the tetrahedral framework in anorthite, low albite, maximum microcline and reedmergnerite. In: The Feldspars, Mackenzie, W.S., Zussman, J. (eds.). Manchester: Manchester Univ. Press 1974, pp. 49-67 Gibbs, G.V., Prewitt, C.T., Baldwin, K.J.: A study of the structural chemistry of coesite. Z. Kristallogr. 145, 108 123 (1977a) Gibbs, G.V., Meagher, E.P., Smith, J.V., Pluth, J.J.: Molecular orbital calculations for atoms in the tetrahedral frameworks of zeolites. In: Molecular Sieves-II, Katzer, J.R. (ed.). New York: Am. Chem. Soc. 1977b, pp. 19~9 Gilbert, T.L., Stevens, W.J., Schrenk, H., Yoshimine, M., Bagus, P.S.: Chemical bonding effects in the oxygen Kc~ X-ray emission bands of silica. Phys. Rev. B 8, 5977-5998 (1973) Glasser, L.S Dent, Jamieson, P.B.: Sodium silicate hydrates, V. The crystal structure of Na20-SiO2.8H20. Acta Crystallogr. Sect. B: 32, 705 710 (1976) Harrison, W.A.: Is silicon dioxide covalent or ionic? In: Physics of SiO2, Pantelides, S.T. (ed.). New York: Pergamon Press 1978, pp. 105-110 Hehre, W.J., Ditchfield, R., Stewart, R.F., Pople, J.A. : Self-consistent molecular orbital methods. IV. Use of Gaussian expansions of Slater-type orbitals. Extension to second row molecules. J. Chem. Phys. 52, 2769-2773 (1970) Hehre, W.J., Lathan, W.A. : Self-consistent molecular orbital methods. XIV. An extended Gaussiantype basis for molecular orbital studies of organic molecules. Inclusion of second row elements. J. Chem. Phys. 56, 5255-5257 (1972) Hehre, W.J., Stewart, R.F., Pople, J.A. : Self-consistent molecular-orbital methods. I. Use of Gaussian expansions of Slater-type atomic orbitals. J. Chem. Phys. 51, 2657-2664 (1969) Hill, R.J., Gibbs, G.V. : Variation in d ( T - O ) , d(T... T) and /.TOT in silica and silicate minerals, phosphates and aluminates. Acta Crystallogr. Sect. B: 35, 25-30 (1979) Hill, R.J., Gibbs, G.V., Peterson, R.C. : A molecular orbital study of the stereochemistry of pentacoordinated aluminum. Aust. J. Chem. 32, 231~41 (1979) Iishi, K.: The analysis of the phonon spectrum of ~ quartz based on a polarizable ion model. Z. Kristallogr. 144, 289-303 (1976) lishi, K. : Lattice dynamics of forsterite. Am. Mineral. 63, 1198-1208 (1978) Jorgensen, J.D.: Compression mechanisms in c~-quartz structures SiO2 and GeO2. J. Appl. Phys. 49, 5473 5478 (1978) Konnert, J.H., Appleman, D.E.: The crystal structure of low tridymite. Acta Crystallogr. Sect. B: 34, 391-403 (1978) Lager, G.A., Hill, R.J., Ross, F.K., Gibbs, G.V.: A difference-Fourier study of electron density in forsterite, Mg2SiO ¢ (abstr.). Eos 60, 415 (1979) Lasaga, A.C.: Optimization of CNDO for molecular calculations on silicates (abstr.). Eos 60, 414-415 (1979) Lazarev, A.N., Poiker, K.R., Tenisheva, R.F.: The force constants of S i - O bonds and their dependence on the nature of substituents on silicon and oxygen. Akad. Nauk SSSR Dokl. Nov. Ser. 175, 1322-1324 (1967) Levien, L., Prewitt, C.T.: High-pressure crystal structure and compressibility of coesite. In press (1980) Levien, L., Prewitt, C.T., Weidner, D.J.: Single-crystal X-ray study of quartz at pressure. In press (1980) Louisnathan, S.J., Gibbs, G.V.: Bond length variation in TO~- tetrahedral oxyanions of the third row elements: T=A1, Si, P, S and C1. Mater. Res. Bull. 7, 1281-1292 (1972a) Louisnathan, S.J., Gibbs, G.V. : The effect of tetrahedral angles on S i - O bond overlap populations for isolated tetrahedra. Am. Mineral. 57, 1614 1642 (1972b) Louisnathan, S.J., Gibbs, G.V. : Variation of S i - O distances in olivines, soda melilite and sodium metasilicate as predicted by semiempirical molecular orbital calculations. Am. Mineral. 57, 1643-1663 (1972c)

Ab Initio Calculated Geometries and Charge Distributions

245

Marsh, F.J., Gordon, M.S. : Second row molecular orbital calculations III. Semi-empirical calculations of geometries. J. Mol. Struct. 31, 345 357 (1976) McLarnan, T.J., Hill, R.J., Gibbs, G.V. : A CNDO/2 molecular orbital study of shared tetrahedral edge conformations in olivine-type compounds. Aust. J. Chem. 32, 949-959 (1979) Meagher, E.P.: A study of the stereochemistry and energies of single two-repeat silicate chains. Am. Mineral. 65, in press (1980) Meagher, E.P., Tossell, J.A., Gibbs, G.V. : A CNDO/2 molecular orbital study of the silica polymorphs (abstr.). Abstr. Progr. 9, 1092 (1977) Meagher, E.P., Tossell, J.A., Gibbs, G.V. : A CNDO/2 molecular orbital study of the silica polymorphs quartz, cristobalite, and coesite. Phys. Chem. Minerals 4, 11-2I (1979) Mulliken, R.S.: Electronic population analysis on L C A O - M O molecular wavefunctions. I. J. Chem. Phys. 23, 1833-1840 (1955) Newton, M.D., Gibbs, G.V.: A calculation of bond length and angles, force constants, vertical ionization potentials and charge density distributions for the silicate ion in H4SiO4, H6SizO7 and H6Si20 (abstr.). Eos 60, 415 (1979) O'Keeffe, M., Hyde, B.G.: Cristobalites and topologically-related structures. Acta Crystallogr. Sect. B: 32, 2923~936 (1976) O'Keeffe0 M., Hyde, B.G.: On S i - O - S i configurations in silicates. Acta Crystallogr. Sect. B: 34, 27-32 (1978) O'Keeffe, M., Newton, M.D., Gibbs, G.V.: Ab initio calculation of interatomic force constants in H6SizO 7 and the bulk modulus of cz-quartz and ¢~-cristobalite. Phys. Chem. Minerals (1980) Pacansky, J., Hermann, K.: Ab initio SCF calculations on molecular silicon dioxide. J. Chem. Phys. 69, 963-967 (1978) Pantelides, S.T., Harrison, W.A. : Electronic structure, spectra and properties of 4:2-coordination materials. I. Crystalline and amorphous SiO2 and GeO> Phys. Rev. Sect. B: 13, 2667-2691 (1976) Papike, J.J., Ross, M., Clark, J.R.: Crystal-chemical characterization of clinoamphiboles based on five new structure refinements. Mineral. Soc. Am. Spec. Pap. 2, 117-136 (1969) Pauling, L.: The principles determining the structure of complex ionic crystals. J. Am. Chem. Soc. 51, 1010-1026 (1929) Pauling, L.: The Nature of the Chemical Bond, 1st edn. Ithaca, N.Y.: Cornell Univ. Press 1939 Pauling, L. : The modern theory of valency J. Chem. Soc. 1948, 1461-1467 (1948) Pauling, L. : Interatomic distances and bond character in the oxygen acids and related substances. J. Phys. Chem. 56, 361 365 (1952) Pauling, L.: The Nature of the Chemical Bond, 3rd edn. Ithaca, N.Y. : Cornell Univ. Press 1960 Peterson, R.C., Hill, R.J., Gibbs, G.V.: A molecular orbital study of distortions in the layer structures brucite, gibbsite and serpentine. Can. Mineral. 17, 703-711 (1979) Revesz, A.G.: r~-bonding and delocalization effects in SiO 2 polymorphs. Phys. Rev. Lett. 27, 1578-1581 (1971) Rosenberg, M., Martino, F., Reed, W.A., Eisenberger, P. : Compton-profile studies of amorphous and single-crystal SiO2. Phys. Rev. Sect. B: 18, 844-850 (1978) Sauer, J., Zurawski, B.: Molecular and electronic structure of disiloxane. An ab initio MO study. Chem. Phys. Lett. 65, 587-591 (1979) Schaefer, N.F. (editor): Modern Theoretical Chemistry, Vol. 4, Applications of Electronic Structure Theory. New York: Plenum Press 1977 Schmiedekamp, A., Cruickshank, D.W.J., Skaarup, S., Pulay, P., Hargittai, I., Boggs, J.E. : Investigation of the basis of the valence shell electron pair repulsion model by ab initio calculation of geometry variations in a series of tetrahedral and related molecules. J. Am. Chem. Soc. 101, 2002-2010 (1979) Smolin, Yu.I., Shepelev, Yn.F., Butikova, K.: Crystal structure of sodium hydrosilicate Na3HSIO4.5H20. Sov. Phys. Crystallogr. 18, 173 176 (1973) Stewart, R.F.: Electron population analysis with rigid pseudoatoms. Acta Crystallogr. Sect. A: 32, 565 574 (1976) Stone, F.G.A., Seyferth, D.: The chemistry of silicon involving probable use of d-type orbitals. J. Inorg. Nucl. Chem. 1, 112-118 (1955) Tak6uchi, Y., Kudoh, Y.: Hydrogen bonding and cation ordering in Magnet Cove pectolite. Z. Kristallogr. 146, 281-292 (1977)

246

M.D. Newton and G.V. Gibbs

Tak6uchi, Y., Kudoh, Y., Yamanaka, T.: Crystal chemistry of the serandite-pectolite series and related minerals. Am. Mineral. 61, 229-237 (1976) Tossell, J.A., Gibbs, G.V. : A molecular orbital study of shared-edge distortions in linked polyhedra. Am. Mineral. 61, 287-294 (1976) Tossell, J.A., Gibbs, G.V.: Molecular orbital studies of geometries and spectra of minerals and inorganic compounds. Phys. Chem. Minerals 2, 21 57 (1977) Tossell, J.A., Gibbs, G.V.: The use of molecular-orbital calculations on model systems for the prediction of bridging-bond-angle variations in siloxanes, silicates, silicon nitrides and silicon sulfides. Acts Crystallogr. Sect. A: 34, 463472 (1978) Tossell, J.A., Vaughan, D.J., Johnson, K.H. : X-ray photoelectron, X-ray emission and UV spectra of SiO2 calculated by the SCF X~ scattered wave method. Chem. Phys. Lett. 20, 329-334 (1973) Urusov, V.S. : Chemical bonding in silica and silicates. Geokhimiya 4, 399-412 (1967) Varma, R., MacDiarmid, A.G., Miller, J.G.: The dipole moments and structures of disiloxane and methoxysilane. Inorg. Chem. 3, 1756-1757 (1964) Verhoogen, J.: Physical properties and bond type in M g - A 1 oxides and silicates. Am. Mineral. 43, 552-579 (1958) Wallmeier, H., Kutzelnigg, W. : Nature of the semipolar XO bond. Comparative Ab initio study of H3NO, H2NOH, H3PO, H2POH, H2P(O)E, H2SO, HSOH, HC10, ArO, and related molecules. J. Am. Chem. Soc. 101, 2804-2814 (1979) Wan, C., Ghose, S., Gibbs, G.V. : Rosenhahnite, Ca3Si3Os(OH)2: crystal structure and the stereochemical configuration of the hydroxylated trisilicate group, [Si3Os(OH)2 ]. Am. Mineral. 62, 503-512 (1977) Wenk, H.-R., Raymond, K.N.: Four new structure refinements of olivine. Z. Kristallogr. 137, 86-105 (1973) Yip, K.L., Fowler, W.B.: Electronic structure of SiO 2. II. Calculations and results. Phys. Rev. Sect. B: 10, 1400-1408 (1974) Zupan, J., Buh, M.: Ab initio calculation on the (Si207) 6- bitetrahedra. J. Non-Cryst. Solids 27, 127-133 (1978)

Received December 14, 1979