American Mineralogist, Volume 66, pages 1237-1249, l98I

A molecular orbital study of rings in silicates and siloxanes

Bnyex C. CnerouMAKos

Department of Geological SciencesVirginia Polytechnic Institute and State University

Blacksburg, Virginia 24061

R. J. Hnr

CSIRO Division of Mineral ChemistryPort Melbourne, Victoriq, Australia

eNo G. V. Grsss

Department of Geological SciencesVirginia Polytechnic Institute and State University

Blacksburg, Virginia 24061

AbstractAb initio STO-3G.molecular orbital theory has been used to calculate energy-optimized

geometries for cyclotrisiloxane and cyclotetrasiloxane. The resulting SiOSi angles are inclose correspondence with experimental SiOSi angle frequency distributions for tetrahedralrings in silicates and siloxanes. Experimental correlations between SiO bond length andbridging SiOSi angle for rings in silicates and siloxanes are reproduced by the theory. Thestable configurations of 4-membered tetrahedral rings are predicted to have the exactreverse sequence to that indicated by previous electrostatic potential energy calculations.The observed configurations of the tetrahedral sheets in gillespite and apophyllite conformwith the calculated stable configurations of 4-membered rings with Cnn point symmetry.Ascribing the configuration of the silicate sheets to the constitution of the interlayer ionsdoes not appear to be necessary. CNDO/2 calculations for Dn6 and Cnu hydroxycyclosilox-ane molecules were used to compare the relative stability of different size silicate rings. The3-membered ring is indicated to be unstable relative to larger rings due to strained SiOSiangles -130", consistent with its breakdown during a trimethylsilylation reaction. Itspresence in silicate glasses and melts must also be infrequent. Larger rings assumenonplanar configurations to achieve minimum-energy SiOSi angles averaging 147".

IntroductionIn an attempt to account for the relative frequen-

cies of n-membered tetrahedral rings in silicateminerals, Zoltai and Buerger (1960) undertook acalculation of the electrostatic energies of modelring structures (D,,6 point symmetry) as a functionof the number of tetrahedra in the ring. Each ringwas constructed from a loop of alternating Sia* and02- ions and from Fr- ions replacing the nonbridg-ing oxide ions to achieve tetrahedral coordination ofSia* and electrostatic neutrality of the final unit.Using Coulomb's law, the potential energy perSiOF2 unit was calculated by summing the electro-

0003-004x/8 l / | | 12-1237 $02.00

static interactions between the ions in the ring,assuming regular tetrahedra. By constraining thesymmetry of each ring to be Dn6, the SiOSi angle iscompletely determined by the number of tetrahe-dra, n, in the ring (Fig. 1). Upon calculating theelectrostatic energy as a function of n, they foundthat a 5-membered ring possessed the lowest ener-gy, This led to the assertion that a silicate tetrahe-dral structure composed of S-membered ringsshould be more stable than one composed, forexample, of 4- or 6-membered rings. Nevertheless,an examination of the SiOSi angles calculated forthe n-membered rings (Fig. 1) shows that the 5-

1237

1238 CHAKOUMAKOS ET AL.: R/NGS IN SILICATES AND SILOXANES

ato6{

TETRAHEDRA PER RING

Fig l. The SiOSi angle variation for different size Dn6symmetry rings (solid points using the left-hand scale) and thecorresponding electrostatic energies per SiOF2 unit (open pointsusing the right-hand scale) (after Zoltai and Buerger, 1960). Notethat the energy units were not reported. The minimum energyring occurs for five tetrahedra with a SiOSi angle of 178.5'. For n< 5, SiOSi : 250.53' - 360Yn and for n > 5, SiOSi : 10D.47' +360Yn.

membered ring possesses a SiOSi angle nearest180o, where adjacent highly charged Sia* cationscomprising the angle are at maximum separation. Infact, each of the remaining rings is indicated to bedestabilized relative to the 5-membered ring be-cause of the closer contacts made in these ringsbetween the Sia* cations. Other silicate rings, de-parting from D,6 symmetry, were also examined byZoltai and Buerger. When the rings were distoftedwithin the constraints of C,,, point symmetry whereall tetrahedral apices point up (or down), a 6-membered ring was found to possess the lowestenergy. For even-membered rings having apicespointing alternatively up and down, again a six-membered ring was found to be most stable. For 4n-membered rings having alternating pairs of tetrahe-dra up and down, the most stable ring was found tohave eight tetrahedra. Similarly for each of thesedistortions, the minimum energy ring was found tobe the one in which the SiOSi angle is the widestand the separation between the highly charged Sia+cations is maximal. For the rings having more thanone of these distortions, the D,6 configurationwhere the Sia* are at maximum separation was

always indicated to be the most stable whereas theone with C., configuration where the cations are incloser contact was indicated to be the least stable.

Inasmuch as the SiO bond possesses significantcovalent character (Newton and Gibbs, 1980), weundertook a series of molecular orbital (MO) calcu-lations for a variety of siloxane and silicate rings toexplore their energetics and to determine whetherthe optimized geometries conform with those ob-tained by Zoltaiand Buerger (1960). In addition, theoptimized SiO bond lengths and SiOSi angles arecompared with experimental values for cyclosilox-anes and silicates.

Molecular ring models and methods of calculation

Suitable molecular models to represent silicatetetrahedral rings should be neutral and smallenough to make the calculations feasible, yet stillretain the stereochemistry which is to be represent-ed. Ring molecules with the compositions (SiO3);2"n = 3 are not suitable because of their unrealisticlarge negative charge. The compositons [(OH)zSiO]"are among the best conceivable models, howevertheir large size greatly restricts the level of the MOmethod. Less suitable, but more tractable, are thecompositions (FzSiO)" and (HzSiO)".

The molecules cyclotrisiloxane (HzSiOh andcyclotetrasiloxane (H2SiO)4 were chosen to repre-sent 3- and 4-membered tetrahedral rings, respec-tively (Fig. 2).To our knowledge, only the tetramerhas been isolated as a volatile gas-phase species(Campbell-Ferguson, 1965), the structure of whichhas been determined by electron difraction meth-ods (Glidewell. et al., 1970). The MO calculationsfor these two molecules were made by ab initiomethods (Richards and Horsley, 1970) using thecomputer program cAUssIAN 70 (Pople et al., qcyn11,236, 1973), employing a minimal STO-3G basisset (Hehre et al., 1969,1970).

In choosing the most appropriate set of basisfunctions to do MO calculations for siloxanes andsilicic acid molecules, considerable discussion hasbeen devoted to the extent of involvement of Si d-orbitals in the SiO bond (Cruickshank, 1961; Baur,1971; Colins et al., 1972,1976; Sauer and Zurawski,1979; Meier and Ha, 1980; Newton and Gibbs ,1979,1980). The consensus is that a STO-3G* basis whichincludes the d-type basis functions, reproducesaccurate bond angles, energy barriers and dipolemoments whereas the minimal STO-3G basis yieldsmoderately accurate bond lengths (Newton andGibbs, 1980). However, using available computer

I

. Ft u ,

oE,

(JlrlJ

Z lt)

LOTO = 109.47D16 POINT SYMMETRY

CHAKOUMAKOS ET AL.: R/NGS IN SILICATES AND SILOXANES 1239

S i H = l . 5 lt H S i H = l H S i O = 1 O 9 . 4 7 o

Fig. 2. Cyclotetrasi loxane molecule, (HzSiO)+, D+r,configuration with d(SiH) fixed at l.5A and angles HSiH = HSiO= 109.47". The larger spheres represent oxygen, the smallerspheres represent hydrogen and the intermediate-sized spheresrepresent silicon. No particular physical significance is ascribedto these sphere sizes.

programs, the additional five d-orbitals on each Siin molecules of this size precludes ab initio calcula-tions for (H2SiO)" rings larger than the trimer. Tomake the calculations tractable. the SiH distancewas fixed At l.5A and the angles OSiH : HSiH setequal to 1W.47".

Since the cAUSsTAN 70 computer program ascurrently dimensioned cannot accommodate thenumber of orbitals required for similar moleculeswith five or more tetrahedra, approximate molecu-lar orbital calculations.were made for larger rings.-For these calculations, molecules of the composi--tion [(OH)zSiO]", where n : 3 through 10, wereused. The calculations were made with the cNDo/2method (Pople and Beveridge, 1970) using the com-puter program cNINDo (Pople and Beveridge, 1970,p. 163-205). A minimum-valence sp basis was usedfor O and Si and an s basis was used for H. For theensuing calcu^lations the geometries were fixed:d(SiO) : l .62A,d(OH): 1.0A, OSiH: 180", OSiO: 109.47" (Fig. 3).

To obtain comparable energies for different-sizedrings, the total energy calculated for each moleculewas divided by n, the number of tetrahedra in thering. Direct comparison of ab initio and cNoo/2energies cannot be readily made inasmuch as onlythe valence electrons are used in the latter. Totalenergies are reported in atomic units (a.u.) oftencalled the Hartree, which is defined as twice the

energy of the hydrogen atom, 2625.46 kJ/mole,27.21 eY, or 627.50 kcaUmole.

Ab initio calculations for (HzSiO)r and (HzSiO)rcyclosiloxanes

The Si-O bond lengths, d(SiO), and the angles,SiOSi and OSiO, were optimized for the 3-mem-bered ring in (HzSiO)r while maintaining D36 pointsymmetry. Subsequently, the tetrahedra were keptrigid but alowed to tilt about their bridging oxy-gen edges while preserving C3, symmetry. Thistilting, for any size ring, can be described by 7, theangle between the HH vector within each tetrahe-dron and the n-fold symmetry axis of the ring(Peterson et a1.,1979). For rings with D6 slmme-try, y: 0'(Figs. 2,3), and when lateral faces oftetrahedra in the ring are coplanar, 7: t35.3o. Thevariation of the energy per tetrahedron with 7 ispresented in Figure 4. Since d(SiO) correlates in-versely with SiOSi angle (Brown et al., 1969), thetetrahedra should not be expected to remain rigidduring the tilting and the distortion of the SiOSiangle. Re-optimizing the bond lengths and anglesindicates this to be the case with the SiO bondslengthening as the SiOSi angles narrow. Table Isummarizes the features of the final optimizedconfiguration, which occurs when 7 : -r 17". Thecalculated barrier to tilting is very small, 4 kJ/mole;

S i O = 1 . 6 2 AOH = I.OOA

tS |OH = lSOof OS |O = lO9 .47o

Fig. 3. Hydroxycyclopentasiloxane molecule, [(OH)zSiO]s,D56 configuration with fixed geometry: d(SiO) : 1.624, d(OH) =1.0A, SiOH : 1E0" and OSiO = 109.49". The nearly straightSiOSi angles are conspicuous in the S-membered ring, purportedby Zoltu and Buerger to be most stable. Sphere size assignmentsare the same as in Figure 2.

1240

t _lrlo-

CHAKOUMAKOS ET AL,: R/NGS IN SILICATES AND SILOXANES

9 - r e o e e r szolrl

< -360 6625E.Ft!F

D+r I

3;;1,"oza l.

TILT ANGLE 7

Fig. 4. Potential energy curves (Hartree units) calculated as afunction of tilt angle for the Cs" distortion of planar (HzSiO)r andDza, Czr and Ca, distortions of planar (H2SiO)4. Here, rigidtetrahedra are assumed, whereas for the dala in Table I thiscondition is not imposed. For symmetry reasons, the curves aresymmetrical atrout 7 : 0o.

consequently, the 3-membered ring is indicated tobe very flexible to this distortion over a range of fvalues from -20o to *20'.

The bond lengths and angles of the 4-memberedring in cyclotetrasiloxane (H2SiO)4 were similarlyoptimized, but several configurations are possible,depending upon the signs of 7 for each tetrahedron.When fangles of the tetrahedra in the 4-memberedring have the signs + + + + the point symmetry ofthe ring is C,u, when they have the signs + - + - thering has D26 point symmetry, and when they havethe signs ++-- the ring has C26 symmetry.

Table I summarizes the features of the optimizedgeometries. The order of increasing stability is D+n,Dza, Czn, Ca" (Fig. 5), the exact opposite to thesequence indicated by the electrostatic calculationsof Zoltai and Buerger (1960). These later threeconfigurations comprise three local minima on theenergy hypersurface of (H2SiO)a, but to assure thatthe lowest energy configuration has been obtainedwould require complete relaxation of the symmetryconstraints. In fact, the electron diffraction profileof the gas-phase tetramer has been suggested to beconsistent with Sa point symmetry (Glidewell et al.,1970). The proposed structure (Fig. 6) is similar tothe D26 configuration, but the bridging oxygensalternate up and down in the ring. Even so, thecalculated bond lengths and angles for the D26tetramer compare well with those observed 0 forthe gas phase molecule: d(Si-O) : 1.61(1.628A),SiOSi : 142(148.6!1.2'), OSiO = lll(112.0-F0.9'),adjacent d(O . . .O) : 2.66(2.704) and adjacentd(Si . . . Si) : 3.05(3.13A). Note, however, thattheD26 tetramer is not the lowest energy configurationcalculated.

Interactions governing the ring geometries of(H2SiO)3 and (HzSiO)r

The frequency distribution of observed SiOSiangles in silicates spans a range of values from 120'to 180" with an average of - 145' (Tossell and Gibbs,1978) so that the inherent tendency of the SiOSiangle to adopt a bent bond configuration is clearly

- a - l

3;X 1,".. 'o,oaot

(,ff -reozl!

Table l. Summary of silicate ring calculations using a STO-3G basis

. . a

telrehedra per r lng

(H25lo) 2

(H2Sro) 3

1

(n2s1o) 4 1 {on; rsro1 u"

6

polnt €ynmetry Dzt D3r, c3r D4t D2a czr,

-1081.99133 -1442.64762 -1442.65511 -1442,65789

-360.65377 -360.66190 -360.66377 -360.66447t o t a l e n e r g y ( a . u . ) -720.99238 -1081,98970

energy per te l rahedron (a ,u . ) -360.49619 -360 '66323

a ( s r o ) ( l ) l . 7 t r . 6 4 0

d ( s l . . . s t ) 1 l ; a a l a c e n t

d ( o . . . 0 ) 1 . { . ; r d l a c e n t

rs los l (c )

r,osro ( ')t t l t a n g l e y ( ' ) 0 0

b a r r l e r t o p l a n a r l t y ( k J / n o l e ) b 0 O

SIOSI bending force constantf a ( N / n )

r . 6 4 4

2.9L6

2 . 6 8 3

L 2 4 . 9

109 .4

r74 . 2 6

8 , 8 3

1.6003 . r 41

2 .655

r57 ,9

112. l -

o

0

c ,

-L442.66t45

-360.66536

r . .619

2.990

2 . 6 6 2

L34.7

1 r 0 . 5

3 2

36 .31

20.48

-3Str.gZtt Sd

-59.15359o

L.62e

3 . 0 4 5

2.645e

140

1 0 9 . 5

3 7

50.80

16,99

2.630

2 , L 6 2

101. 2

2 . 9 7 2

2.687

1 3 0 . O

1 ,613

3.054

2.665

1 . 6 1 3

3.L23

2.660

r 4 2 . 3 1 5 0 , 9 , 1 3 2 . 3

7 8 . 8 1 1 0 , 0 11 r .4

49 34

1 9 . 6 6 2 6 , 9 6

6 . 9 1 1 0 . 3 2

a for the fir|t Lhree rcLecul,ee d(siH) = l.so E, osin = HSi.n = 109.4?"

o nnengy di.fferences tith respect to the Dnh configmtion

" CUAO/Z calculati,on (Peterson et al., 7979)d

uoln*n eleetrons onlye

1ired ua\re

CHAKOUMAKOS ET AL.: R/NGS IN SILICATES AND SILOXANES

sio lstosi losio

f .6 f A 142" l l lo

r.6rA j! i: nr"

1 .62 A l 35c l l l "

Fig. 5. Final optimized geometries f6r (HzSiO)+ with the enthalpy changes for the distortions from planar symmetry, with thecalculated bond lengths and angles.

F.

an intrinsic and governing feature of the SiOSidisiloxy unit. For the cyclic trimer, (HzSiO)r, thecurve for total energy versus fangle is very flatover the range -20o to *20", but at -r17o doesexhibit a minimum lower in energy by 4 kJ/molethan when 7 : 0o. Moreover, as 7 increases from 0oto 17", the SiOSi angle contracts from 130o to 125",contrary to the tendency ofthe SiOSi angle to adopta wider angle near 145'. A net bonding interactioninvolving the terminal hydrogens across the ringcould stabilize the C3, configuration relative to D35,but this is apparently not the case. The geminalMulliken overlap population per tetrahedron, de-fined to be the sum of the overlap populationsbetween the two hydrogens and the silicon within atetrahedron and those atoms outside the tetrahe-dron (Bartell et aI.,1970), increases non-linearly asthe tilt angle increases. As a result the optimumconfiguration, when l: !I7", exhibits more repul-sive interaction than the D36 configuration. SinceSTO-3G calculations for disiloxane (HrSi)zO (Sauerand Zurawski, 1979; Newton and Gibbs, 1979;Meier and Ha, 1980) generate an optimum SiOSiangle of 126', the optimum SiOSi for (HzSiO)r ofI25" may not be surprising. If this correspondenceis true, then there exists a transferability of thedisiloxy configuration in disiloxane to that in cy-clotrisiloxane, (HzSiO)r. However, the structure ofdisiloxane examined by electron diffraction (Almen-ningen et a1.,1963) and infrared and Raman spec-troscopy (Durig et al.,1977; Durig and Kalasinsky,

1978) is consistent with the SiOSi angle bent to144.0=0.8' and 149=2.0o, respectively, and an SiOdistance of 1.634=0.0024.

Meier and Ha (1979) have investigated basis setdependence by comparing calculations for disilox-ane using four different basis sets: STO-3G, STO-3G*, 4-3lG and 6-6-31**. The STO-3G and 4-3lGcalculations most closely reproduce the SiO dis-tance, the 6-6-31** calculation of Newton andGibbs (1979) and Ernst et al. (1981) yields the bestSiOSi angle, and overall the 6-6-3lx* calculationbest matches the experimental structure. In gener-al, STO-3G calculations underestimate SiOSi angles

Fig. 6. Structure of cyclotetrasiloxane, (H2SiO)4, with Sa pointsymmetry, suggested by Glidewell et al. (1970) to be consistentwith the electron diftaction profile of the gas-phase molecule.

t24l

(H2SiO)4

s4 SiO =1 .634S iH =1 .484

/os io = I l2 'ZSiOSi =149"LH SiO = 109 'LHS iH = l1 l '

1242 CHAKOUMAKOS ET AL.: RINGS IN SILICATES AND SILOXANES

in siloxanes; therefore, they overestimate the stabil-ity of small cyclic rings (Sauer and Zurawski, 1979)like cyclotrisiloxane (H2SiO)3, in which the SiOSiangles are necessarily restricted to narrow values.Often small rings are strained (unstable) due to theconstriction of minimum-energy bond angles (e.9.,Newton, 1977), which would be the case for cy-clotrisiloxane if the STO-3G basis did yield a mini-mum-energy SiOSi angle of 145'. The results indi-cate the D36 3-membered ring to be more stable thanthe Da1,4-membered ring, and the C3u 3-memberedring to be as stable as the D26 4-membered ring(Table 1). For the 4-membered rings, decreasingtotal energy corresponds to decreasing averageSiOSi: Da6, 157.9'; D2a, 142.3"; Czn, 14l.6o; Cnu,134.7".In fact, the depths of the potential energywells for all four distortions of the Dnh configura-tions correlate linearly (f : 0.97) with the changein the SiOSi angle from the D,6 configuration. Forthe 3-membered ring the potential energy curveover the range 7 : -20o to *20" is very flat becausethe SiOSi angle is not deformed appreciably. Incontrast, the sharper wells in the potential curvesfor the 4-membered rings reflect the SiOSi angle'stendency to approach 125', but falling short as thenonbonded repulsions quickly dominate.

Calculations using a STO-3G* basis (d-orbitalfunctions included for Si) were made for the 3-membered ring as a function of 7 to see if the totalenergy curve would exhibit a minimum at 7: 6o. 6planar equilibrium geometry does result, with widerbond angles and shorter bonds. Therefore, we con-clude that the similarity in energy of the 3-mem-bered ring to those of the 4-membered rings must bean artifact of the STO-3G basis.

The energy per tetrahedron of the D4h 4-mem-bered ring is 435 kJ/mol lower than that of the 2-membered ring in the dimer (H2SiO)2 (Table l). TheSiOSi angle in the dimer,10l.2o, is most assuredlystrained, explaining the absence of silicate mineralsof 2-membered silicate rings consisting of two tetra-hedra sharing a common edge.

These results are consistent only within the STO-3G level of calculation; they must be accepted withcaution until better basis sets and models vieldsimilar results.

CNDO/2 calculations for [(OH)2SiO],n-membered silicate rings

Fixed geometries as described previously werecalculated for n-membered rings of compositions[(OH)2SiO]", r : 3 through 10. Note that these

molecules difer from the previous ones in that theterminal hydrogens are replaced by OH ligands.Figure 7 presents the energy per tetrahedron as afunction of ring size for rings with D,r6 point symme-try. Note the sharp drop in energy from three tofour tetrahedral per ring (the 2-ring is some 105 kcaUmole less stable than the 3-ring), at variance withthe ab initio results. Figure 7 contrasts markedlywith the electrostatic energies (Fig. 1) calculated byZoltai and Buerger inasmuch as no minimum occursover this ring size interval. Since OSiO angles werefixed at 109.47", the SiOSi angles are constrained,and, as the ring size increases, the SiOSi passes amaximum angle of 180' at five tetrahedra andbeyond narrows to the limit of 109.47' (i.e., re-entrant SiOSi angles). A minimum should result atsome size greater than the interval shown in Figure7, due to the eventual close approach of the Siatoms. When the ring contains an infinite number oftetrahedra it is equivalent to a chain (c/. Meagher,1980, for a CNDO/2 study of the instability of asilicate chain of this configuration). The slight cur-vature of the points beyond six tetrahedra per ringprojects to a minimum energy ring having 13 tetra-hedra. For a D,6 ring containing 13 tetrahedra withequal length SiO distances and OSiO angles fixed at109.47", the SiOSi angle is constrained to value of137'. Considering the tendency for the SiOSi angleto be bent near 145o, this result agrees with the

3 5 7 9 l lTelrohedro Per Ring (n)

Fig. 7. The energy per tetrahedron (Hartree units) for D"1symmetry [(OH)2SiO]. composition rings plotted against ringsize, constituting the revision ofFigure I obtained by Zoltai andBuerger.

on; [ton).sto]nCNDO/2 colculotion3

CHAKOUMAKOS ET AL.: R/NGS IN SILICATES AND SILOXANES

observed better than the electrostatic calculationsby Zoltai and Buerger (1960). However, for thelarger rings, ideal geometries with planar bridgingoxygen arrangements would not be expected, there-fore comparisons of this kind may not be particular-ly meaningful.

Since the ab initio calculations indicate the C,,configurations for 3- and 4-membered tetrahedralrings to be most stable, the energies for Cn, configu-rations of [(OH)zSiO]., D : 3 through 8, have beencalculated for different tilt angles (Fig. 8). The formof the energy curve for the 3-membered ring differsnoticeably from that obtained inthe'ab initio calcu-lations. All of the C3, configurations have energieswell above those for the 4-membered ring, consis-tent with the 3-membered ring being more strainedthan the 4-membered ring. The minimum energyconfiguration occurs for a tilt angle of 0o, and thecurve is flat over a smaller interval, y - - 10" to +10". The CNDO/2 result may be more reasonablethan the ab initio calculation in this case, probablydue to the presence of OH rather than H ligands.Note that when the six terminal H atoms of H6Si2Oare replaced by OH, the SiOSi angle opens to 140'in a STO-3G calculation (Newton and Gibbs, 1979;Meagher et al., 1980; Gibbs er al., l98l).In fact, anoptimization of the cyclic trimer, [(OH)zSiO]r, withSTO-3G methods does yield a Tvalue of 0', inagreement with the CNDO/2 results.

For the 4-membered ring, the form of energycurve is very similar to the one obtained with the abinitio method. Compare the minimum energy re-sults with those for the ab initio calculation ( ): tiltangle : 27 .6(32"), barrier to planarity : 7 .9(36.3 kllmole), SiOSi : 140(135), and d(Si. . . Si) :

3.04(2.eeL).A minimum energy Cnu ring was not found in the

interval 3 to 8 tetrahedra per ring as shown byFigure 9. The potential energy curve decreasesrapidly to 6 tetrahedra, then flattens, but no mini-mum occurs at 6 as determined by Zoltai andBuerger. Optimizing the tilt angle Tfor the C,' ringsenables the SiOSi angle to adopt an equilibriumvalue. The number of tetrahedra in the ring, theoptimum tilt angle and the optimum SiOSi anglewere obtained as follows: 3, 0o, 130.5"; 4, 27.6",139.8 ' ; 5 , 35 .5o , 139.0" ; 6 ,33 .9" , 142.5" ;7 ,34 .6o ,140.8"; 8, 33.7', 139.5". Zoltai and Buerger hadconsidered only those ring models with 7 fixed at35.3', which requires all tetrahedral faces to becoplanar. Peterson et al. (1979) have demonstratedby methods identical with ours for the same mole-cules that the Co, 6-membered ring is not onlymore stable than the D66 ring, but that it is addition-ally stabilized by a trigonal distortion. To meaning-fully compare the stability of different size ringsrequires completely optimized geometries for eachring determined without symmetry constraints, butas yet these requirements have not been met usingab initio STO-NG basis set calculations because ofthe excessive computer time and the large numberof calculations required to accomplish this goal. So-called approximate methods (Halgren et al., 1978;

? - 5 e 1 4 5

gzp -ss ' "2oUJI

f, -ss.r+sFlrlF

E - 5 9 . 1 5 1

ulo

> - - 59 . r53(,Elrlz

-=gzoTEolrj-TEFtdF

CElrJ(L

(9EUJzlrJ

[roH t.sio]no n = 3o n = 4a n = 5e n = 6L-"o

- t - .

- 5s. ls46lo 20 30 40 50

T I L T A N G L E 7Fig. E. The energy per tetrahedron (Hartree units) calculated

as a function of tilt angle 7 for C," symmetry [(OHz)SiO]" rings.

l rJ -59.155i

TETRAHEDRA PER RING (N)

Fig. 9. The energy per tetrahedron (Hartree units) for theminimum energy C""[(OH)zSiO]" rings plotted against ring size.

-ur,root\o<\\.,/

1244

Dewar and Ford, 1979) are promising alternativesfor future research.

A comparison of silicate and siloxane rings

Bond length and angle trends in ring structures

An inverse correlation betwen the average SiOdistance and SiOSi angle has been corroborated byexperimental data (Cannillo et a1.,1968; Brown eral.,1969; Hill and Gibbs, 1979), semi-empirical MOcalculations and ab initioMO calculations (Gibbs elal., 1972; Louisnathan and Gibbs, I972a,b; Cohen etal . , 1977; Meagher et a1.,1979; Gibbs et a1.,1974,1977, l98l; Newton and Gibbs, 1980). Additionaldocumentation is made here by comparing thevariation of average SiO bond length, <d(SiO)>,versus the percent fractional s-character of thebridging SiO bonds Vof": 100(1 + )r2), where )r2 :-sec SiOSi, for tetrahedral rings in siloxanes andsilicates with that calculated ab initio for the follow-ing cyclosiloxane molecules: (H2SiO)3, D3h, C:, and(H2SiO)4h, Ctu, Dza (Fie. 10). Note that the C26configuration for (H2SiO)a was omitted because themolecule's symmetry does not necessarily con-strain the d(SiO) distances to be equivalent, yet thiscondition was imposed. Figure 10a is similar toFigure 9b of Newton and Gibbs (1980), differing in

CHAKOUMAKOS ET AL.: R/NGS IN SILICATES AND SILOXANES

that it possesses more data which are restricted tosilicate rings (Chakoumakos, 1981). The signifi-cance of this linear correlation is interpreted else-where (Newton and Gibbs, 1980; Newton, 1981)using classical hybridization arguments.

SiOS, angle frequency distribution for ring struc-tures

The stereochemistry of the organosiloxanes isremarkably similar to that of the silicates. Noll(1968, p. 246-331) describes this correspondence,noting similar anion configurations and bondlength-angle variations. Shklover and Struchkov(1980) review the structural chemistry of the organ-ocyclosiloxanes. As expected, the SiOSi angle fre-quency distribution for Si-containing tetrahedralrings in siloxanes and silicates are quite similar (Fig.11). The paucity of data for the hexamers is due tothe lack of structure refinements. In contrast, thelack of data for the pentamers of siloxanes andsilicates reflects their rare occuffence. In preparingthe plots for the silicate data, silicate rings incyclosilicates and as portions of more condensedanions were used. The true relative abundance oftetrahedral ring sizes for both siloxanes and silicatesis not completely represented by the histograms,since structural data are lacking for the larger rings.

^ 1 6 4

Ia

! t t . 6 l

O O

50 30

PERCENTAGE S-CHARACTER = IOO/(1+ 12)

Fig. 10. Scatter diagrams of the average SiO bond length, <d(SiO)>, to a bridging oxygen for rings in (a) cyclosiloxane moleculesand molecular crystals, and (b) silicates, plotted against the percent fractional s-character,Vof": 100/(l + )r21, ofthe SiOSi bond. Thesolid lines correspond to the regression equations for the observed data, (a) <d(SiO)> = -3.5 x l0-3 Vof, + 1.78, f : 0.34; and (b)<d(SiO)> = -5.6 x lO-3Vof,+ 1.87,f = 0.64. Thedashedlinecorrespondstotheregressionequation <d(SiO)>: -4.0 x l0-3Vof, + 1.79, f : 0.%, calculated for the following molecules (indicated by solid circles): (H2SiO)3, Drr,, Cr" and (HzSiO)+, D+r,, C+,,D2a, whose geometries were optimized ab initio. Note that )\2 = -sec SiOSi, so that for f" = 0.5, SiOSi : lEO', and for 0.3, SiOSi =I 15".

SILICATES

CHAKOUMAKOS ET AL.: R/NGS IN SILICATES AND SILOXANES

SILOXANES6 - MEMBERED

RING

4 - MEMBERED

3 - MEMBERED RING

Fig. 11. The SiOSi angle frequency distributions for Sicontaining tetrahedral rings observed for silicates and siloxanes(Chakoumakos, lgEl). The bold arrows indicate the mean values and the other arrows with Schoenflies notation mark the SiOSiangles calculated for the ab initio optimized geometries for cyclotrisiloxane and cyclotetrasiloxane.

1245

IDsh

Actually the 6-membered rings are more abundantfor the silicates and the 4-membered rings for thesiloxanes. In general, the SiOSi angles of the 3-membered rings are narrower (-130) and have amore restricted range of values than those for the 4-membered rings. The SiOSi angle distributions forthe larger rings are quite broad yet center aroundt47".

Occurrence of ring structures

3-membered rings. Occurrence of 3-memberedrings in the silicates is restricted almost entirely tocyclosilicates like benitoite, catapleiite, margaro-sanite, walstromite, wadeite and high- pressureCaSiO3. Eudialyte is unusual because its structurehas discrete 3- and 9-membered rings. Rarely dothey occur as portions of more condensed silicateanions except, for instance, as part of the sheetanion of zussmanite. In all cases, configurations

close to D36 and C3y &r€ observed. The 3-memberedsilicate ring in walstromite (Glasser and Glasser,1968) is reported to have a configuration close toC3" with y: =35o, which requires an average SiOSiangle of 125'. Our MO calculations indicate thisconfiguration to be relatively unstable. Additionalminerals, with suspected 3-membered silicate rings,are pabstite (Gross et al.,1965) and bazirite (Younget al., 1978). Three-membered silicate rings arestabilized by replacing one or two of the Si atomswith Be or B. which have Rarrower minimum-energy TOSi angles to alleviate the ring strain.

Cyclotrisiloxane has yet to be isolated, but thering structure of hexamethylcyclotrisiloxane hasbeen determined to be consistent with D35 pointsymmetry, with a flat energy curve for tilting overthe range y : =8o (Oberhammer et a1.,1973). Allcyclotrisiloxane rings are nearly planar. Note thatin Figure 11 the SiOSi angles obtained in ab initio

SILICATES

r301Dsn

1246 CHAKOUMAKOS ET AL.: R/NGS IN SILICATES AND SILOXANES

optimized (H2SiO)3 conform with the observed dis-tributions.

4-membered rings. Many configurations are ob-served in addition to the D+n, Dzn, C26 and Cauconfigurations examined previously. Hence, it isnot possible to directly correlate portions of thefrequency distribution with particular configura-tions. The SiOSi angles, obtained in ab initio opti-mized (H2SiO)4, effectively span the range of ob-served SiOSi angles, and for the siloxanes theobserved configurations actually exhibit a prepon,derance of configurations close to D26.

S-membered rungs. These are very uncommon insilicates, but they do occur in frameworks suchas keatite (Shropshire et al., 1959) and some syn-thetic high-silica zeolites (e.g., silicalite, Flanigen e/al., 1978) and in the sheet anion of okenite (Mame-dov and Belov, 1958) and nekoite (Alberti and Galli,1980). Five-membered rings with one or more of theSi atoms replaced by other cations (Li, Be, B, Al,Fe3+) are more frequent.

In siloxanes 5-memberd tetrahedral rings are alsouncommon. They occur in decamethylcyclopentasi-loxane (Oberhammer et aI.,1973) and as part of theSir2OlS- cage of dodeca(phenylsilasesquioxane)(Clegg et al., 1980). Discrete double S-memberedrings occur in decamethylsilasesquioxane (Baidinaet al., 1980) and they have been isolated fromcrystal l ine mixtures in the systems N(n-C+Hq)+,OH-SiOz-H2O and N(i-C5Hr)4OH-SiO2-H2O (Hoebbel et al.,1975).

Silicate rings in glasses and melts. Rings oftetrahedra in the silicate anions of glasses andmelts conform with popular models for those vitre-ous states; however, quantitative structural studiesare as yet not possible. Discrete 4-membered sili-cate rings in some lead orthosilicate glasses havebeen detected by combined mass spectroscopy andchromatography applied to trimethylsilyl deriva-tives (Gotz et al.,1972,1976). Improvements of thismethod (Masson, 1977; Sharma and Hoering,1977)now allow semi-quantitative determination of theanionic constitution in silicate structures; however,the silylation techniques are applicable only tosilicate structures with small anions with easilyreplaceable metallic ions. A common problem en-countered in the silylation process is that sidereactions give rise to new anions not present in theoriginal structure (Lenz, 1964, 1966 ; Mas son, 1 977).In particular, the trimethylated cyclic trimer is verydfficult to recover in high yields inasmuch as itundergoes cleavage during trimethylsilylation,

transforming to the linear trimer (Masson, 1977).This experimental observation is not surprisingbecause the cyclotrisiloxane ring is unstable due tostrained SiOSi angles. In contrast, the linear tetra-mer, SiaOlS-, undergoes cyclisation to the cyclictetramer, SroO?t, during trimethylsilylation ofAg16SiaO13 (Calhoun et al., L980).

Raman spectra of trimethylsilyl derivative mix-tures are not easily deciphered since the majorfrequency shifts are insensitive to the size of thepolymer (Sharma and Hoering, 1977). In contrast,Raman spectroscopy applied directly to glasses andquenched melts may provide a means of identifyingspecific anion structures (Mysen et al.,1980l'Yirgoet al., 1980). Moreover, unique sets of anionicspecies in equilibrium in silicate melts may beidentified for specific ranges of nonbridging oxygento silicon ratios. Our MO calculations suggest thatthe 3-membered tetrahedral ring should be rare, ifnot totally absent.

Inherent stability of silicate anions

The silicate anions in gillespite (Pabst, 1943) andapophyllite (Chao, l97l) are SisO9g sheets that canbe viewed as tesselations of Ca, 4-membered sili-cate rings (Liebau, 1968) (Fig. l2).ln gillespite theapices of the Ca" rings point at each other whereasin apophyllite the tetrahedral bases of the Ca, ringsface each other. Of the 4-membered ring distortionsexamined by MO calculations, the Ca, configura-tion is indicated to be the most stable, so the sheetsin gillespite and apophyllite are simply two possiblestable condensations of the Cau ring. Attributing theconfiguration of the apophyllite sheet to the consti-tution of the interlayer ions (Liebau, 1968; Chao,1971) is not necessary. The distortions in the layerstructures mica, kaolinite and serpentine and thoseof single and multiple chain structures are alsoindicated to be inherently stable configurations (Pe-terson et al., 1979, Meagher, 1980).

Conclusions

Over-estimation of the stability of cyclotrisilox-ane results from using a STO-3G basis set. Thehexahydroxycyclosiloxane molecule is a bettermodel for the 3-membered ring as evinced by theSTO-3G and CNDOI2 calctlations. The 3-mem-bered ring is less stable than larger rings, since theSiOSi angles are strained, and their optimum con-figurations are necessarily restricted to be nearlyplanar. Their instability is consistent with theirrarity in silicate structures and their breakdown

APOPHYTTITE GII. I ,ESPITEsi8o2o8- sHEETs

Fig. 12. Silicate sheet anions, SitOlo, in gillespite andapophyllite can be viewed as two different tesselations of the Ca,ring (Liebau, l96E).

during a trimethylsilylation reaction, but they maybe stabilized by high pressure. Their presence insilicate glasses and melts must also be infrequent.

Larger rings are indicated to assume nonplanarconfigurations in order to achieve minimum energySiOSi angles averaging 147'. For 4-membered tetra-hedral rings, three local minima on the energyhypersurface of (H2SiO)+ have been determined tobe associated with the configurations Dzo, Czn, Cqu,in order of increasing stability. For C,, rings, aminimum energy ring does not occur at 4 or 6tetrahedra, in disagreement with the frequency dis-tribution of ring sizes in siloxanes and silicates. Therelative stability of different size silicate rings canonly be meaningfully compared using completelyoptimized geometries for each ring. As larger calcu-lations become more feasible, they should be car-ried out for [(OH)zSiO]" ring molecules using alarger basis set than STO-3G or STO-3G.*

Frequency distributions of SiOSi angles and<d(SiO)> versus their percentage s-character intetrahedral rings containing only Si are very similarfor siloxanes and silicates, further substantiatingthe correspondence between these two chemicalgroups.

A premise of the conceptual framework used todescribe the stability and structure of silicate anionsand siloxanes must be that observed structures areinherently stable, and not solely consequent of theinteraction with their local crystal field (Noll, 1968,p.246-33r).

Acknowledgments

The National Science Foundation is thanked for providingsupport for this study with grant EAR-7723114 to G.V. Gibbs

1247

and P.H. Ribbe to study bonding in materials. Donald K.Swanson and the Virginia Tech Computer Center User Servicesare sincerely thanked for their assistance with program manipu-lations. Suggestions by Professors J.W. Viers, P.H. Ribbe andF.D. Bloss improved the readability of the text. Critical com-ments by Dr. M.S.T. Bukowinski helped to clarify its intent. Weare grateful to Ramonda Haycocks for typing the manuscript andSharon Chiang and Martin Eiss for skillfully illustrating thefigures.

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