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Ab initio SCF MO study of H₆SI₂O₇ at simulated high pressure Ross, Nancy Lee 1981

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AB I N I T I O SCF MO STUDY OF H S I 0 AT SIMULATED HIGH PRESSURE 6  2  7  by  B.Sci., Virginia  Polytechnic  I n s t i t u t e and S t a t e U n i v e r s i t y  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF  SCIENCE  in  THE FACULTY OF GRADUATE STUDIES Department of G e o l o g i c a l  We a c c e p t t h i s  Sciences  t h e s i s as c o n f o r m i n g  to the required  standard  THE UNIVERSITY OF B R I T I S H COLUMBIA August  1981  c Nancy L e e R o s s ,  1981  In p r e s e n t i n g requirements  this thesis  f o r an a d v a n c e d  of  British  it  freely available  agree for  that  Columbia,  f o r reference  the L i b r a r y  shall  and s t u d y .  I  f o r extensive  p u r p o s e s may  f u l f i l m e n t of the  degree a t the U n i v e r s i t y  I agree that  permission  scholarly  in partial  for  that  shall  Geological S c i e n c e s  The U n i v e r s i t y o f B r i t i s h 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 Date  DK-6  (2/79}  August 6.,.19.81  of this  It is thesis  n o t be a l l o w e d w i t h o u t my  permission.  Department o f  thesis  be g r a n t e d by t h e h e a d o f my  copying or p u b l i c a t i o n  f i n a n c i a l gain  further  copying of t h i s  d e p a r t m e n t o r by h i s o r h e r r e p r e s e n t a t i v e s . understood  make  Columbia  written  ABSTRACT  Molecular applied  to  orbital  mineralogical  geometry, e l e c t r o n i c and  bulk  modelled  modulus bonding  increasing  studies  of  calculations. at  these  pressure.  high  successively  electronic  To d a t e ,  atmospheric in  been  e q u i l i b r i u m molecular  pressure  bonding s t u d i e s of m o l e c u l a r  pressure  pressure  have  charge d i s t r i b u t i o n s ,  interest  mineralogy, high  calculations  s t u d i e s have  With  phases groups  spectra  the and  at  ever mantle  simulated  c a n be an i n v a l u a b l e a i d t o u n d e r s t a n d i n g  crystal  chemistry,  bond  energetics  and  high  electronic  spectra. This  investigation  tests  models t o s i m u l a t e p r e s s u r e on  common  metal-oxygen  the c l u s t e r ,  the  feasibility  i n ab i n i t i o  polyhedra.  SCF  MO  Pressure  6  2  i n t h e S i - 0 bond l e n g t h s ,  Changes  force constants are monitored an i n c r e a s e o f 60 k b a r  SiOSi  angle decrease  for  correlation occur  at  a  similar  o f S i - 0 bond one  SiOSi  the b r i d g i n g angles  p r e s s u r e , t h e S i - 0 bond  0.30% a n d 4.5%,  increment length  of and  decrease  observed  pressure.  The  -sec(SiOSi),  force constants  percentage  1:6  the  ratio  up  which in  c-  linear  known  to  In a d d i t i o n ,  t h e S i - 0 s t r e t c h i n g and S i O S i b e n d i n g in  length  respectively,  bar, holds at elevated pressure.  increase  and  with increasing pressure.  c o m p a r e s w e l l w i t h t h e 0.30% a n d 6.6% quartz  atoms  7  oxygen.  and  calculations  H S i 0 , by s y s t e m a t i c a l l y s t e p p i n g h e l i u m the S i - 0 b r i d g i n g v e c t o r s toward  For  various  i s simulated i n  d i r e c t e d ^ along  Si-0  of  t o an  show  a  estimated  p r e s s u r e of  140  kbar.  iv  TABLE OF CONTENTS  Page ABSTRACT L I S T OF TABLES  ii . . ....  v.  L I S T OF FIGURES ACKNOWLEDGEMENTS I.  INTRODUCTION  I I . MOLECULAR ORBITAL METHOD Description MO M e t h o d s III.  CALCULATIONS  I V . MODELS V. RESULTS AND "DISCUSSION  VI.  v i . ,  viii . 1 5 5 11 14 17 22  Model I  22  Model I I  25  CONCLUSIONS  46  REFERENCES  49  V  L I S T OF  TABLES  Table I.  Page Asymmetric stretching force constants ( k ) calculated at 1 bar f o r the c l u s t e r s H S i 0 , H A1 0 - , H Si 0, And H, A l ' S i , 0 , " w i t h a l l S i O S i and A l O S i a n g l e s e q u a l t o 180°.  24  Comparison at 1 bar of c a l c u l a t e d symmetric stretch, v , asymmetric stretch, va , and bending, , frequencies for H S i 0 with those determined f r o m i n f r a r e d and raman s p e c t r a f o r S i 0 " , ( 0 ( S i ( C H ) ) a , and B a T i O S i 0 .  28  M u l l i k e n bond o v e r l a p p o p u l a t i o n s , n ( S i - O b ) and n(Si...Si), and atomic charges on bridging oxygen, Q ( O b ) , and s i l i c o n , Q ( S i ) , f o r H S i 0 at 1 bar,. 60 k b a r and 140 k b a r ; t h e bridging S i - 0 bond and S i O S i a n g l e a r e o p t i m i z e d .  38  a  6  2  ?  II.  ?  2  7  1  7  1 2  5  2  s  6  2  7  6  2  III.  7  3  2  2  7  6  2  7  vi  L I S T OF FIGURES  Figure 1.  2.  3.  Page M o l e c u l a r conformation f o r the dimers studied with model I (note the s t r a i g h t bridging a n g l e ) ; p r e s s u r e i s s i m u l a t e d by d e c r e a s i n g t h e intertetrahedral distance.  19  Molecular conformation f o r H S i 0 studied with model I I . Note t h e bent bridging a n g l e and positioning of h e l i u m atoms used t o s i m u l a t e p r e s s u r e by s y s t e m a t i c a l l y d e c r e a s i n g t h e d ( H e Ojj) d i s t a n c e s .  20  Log of the asymmetric Si-0 stretching force constant, log(ka) , p l o t t e d against the l o gof the i n t e r t e t r a h e d r a l distance, log(d(T...T)), for H S i 0 where log(k*» ) = - 7 . 3 5 1 o g ( S i . . S i ) +4.55 ( r = 0.999) H A1 0 - : l o g ( k a ) = - 7 . 4 2 1 o g ( A l . . . A l ) +4.63 (r =0.998) H, Si 0 : log(k ) = - 7 . 1 2 1 o g ( S i . . . S i ) +4.47 ( r = 0 . 9 9 9 ) and H , A l S i O«: l o g ( k ) =-7.41log(Al...Si)+4.62 (r =0.998).  26  A comparison of asymmetric stretching frequency, v , p l o t t e d a g a i n s t b r i d g i n g bond length, d ( S i - O u ), f o r a group of twelve pyrosilicates ( a ) and H S i 0 ( b ) ; v 's were d e t e r m i n e d from s p e c t r o s c o p i c experiments f o r pyrosilicates whereas i / ' s f o r H S i 0 were calculated.  29  The p o t e n t i a l e n e r g y s u r f a c e s f o r H S i 0 and He H Si 0 a t 1 b a r and 140 k b a r , r e s p e c t i v e l y , plotted as a f u n c t i o n o f t h e b r i d g i n g bond l e n g t h , d(Si-Ofc,), a n d t h e S i O S i a n g l e .  31  A comparison of t h e p o t e n t i a l energy c u r v e s f o r H Si 0 and H e H S i 0 p l o t t e d as a f u n c t i o n of the b r i d g i n g distance, d ( S i - O b ), at 1 bar (upper c u r v e ) a n d 140 k b a r (lower curve), respectively.  33  A comparison of t h e p o t e n t i a l energy c u r v e s f o r H Si 0 and H e H S i 0 p l o t t e d as a f u n c t i o n of t h e S i O S i a n g l e a t 1 b a r ( u p p e r c u r v e ) a n d 140 kbar (lower c u r v e ) , r e s p e c t i v e l y .  34  6  6  2  2  7  7  2  2  6  2  7  2  2  5  a  2  a  2  fl  2  a  4.  a  6  2  Q  7  a  5.  6  2  6.  6  7.  6  6  6  2  2  2  7  7  2  2  7  7  7  2  2  6  6  2  2  7  7  Symmetric, s t r e t c h i n g Si-0 force constant, k , p l o t t e d a g a i n s t the S i O S i angle at 1 bar ( l e f t ) where k = 0 . 0 3 8 ( S i O S i ) + 1 .941 , r = 0.97, and 140 kbar (right) where kg = 0 . 0 4 0 ( S i O S i ) + 3 . 9 6 4 , r =0.93. 5  2  s  2  M u l l i k e n bond overlap p o p u l a t i o n , , n ( S i - 0 ^ ), p l o t t e d a g a i n s t the b r i d g i n g S i - 0 d i s t a n c e at 1 bar (a) and a g a i n s t the symmetric s t r e t c h i n g f o r c e c o n s t a n t a t 1 b a r (b) w i t h r values of 0.997 and 0.989 , respectively; the corresponding r e l a t i o n s h i p s at 140 kbar*" a r e found in ( c ) and (d) w i t h r v a l u e s of 0.999 and 0.971, r e s p e c t i v e l y . 2  2  M u l l i k e n bond o v e r l a p population, n(Si-Ob'), plotted against the b r i d g i n g S i O S i angle at 1 b a r (a) and a g a i n s t t h e p e r c e n t a g e s-character of t h e h y b r i d o r b i t a l s on t h e b r i d g i n g o x y g e n , 1 0 0 / ( 1 ) , a t 1 b a r (b) w i t h t h e c o r r e s p o n d i n g r e l a t i o n s h i p s a t 140 k b a r f o u n d i n ( c ) and ( d ) . The c u r v i l i n e a r t r e n d s of (a) and (c) both become l i n e a r i n (b) and ( d ) . 2  The relationship between the bridging Si-0 d i s t a n c e and - s e c ( S i O S i ) f o r H S i 0 at 1 bar and an e l e v a t e d p r e s s u r e e s t i m a t e d t o be 140 kbar. 6  2  7  A c o m p a r i s o n between t h e a v e r a g e S i - 0 b r i d g i n g distance plotted against -sec(SiOSi) for c o e s i t e ( l e f t ) and H S i 0 (right); at 1 bar and 52 k b a r , t h e r v a l u e s f o r c o e s i t e b a s e d on the experimental data of L e v i e n and P r e w i t t (1981) a r e 0.97 and 0.90, r e s p e c t i v e l y ; t h e r values based on c a l c u l a t i o n s a t 1 b a r and 60 kbar for H Si 0 are 0.97 and 0.98, respectively. 6  2  7  2  2  6  2  7  Illustration of how e s t i m a t e s o f k ^ x roughly equal to 60 kbar pressure were obtained. Modelling changes that occur i n c-quartz at t h i s p r e s s u r e , d ( S i - O b ) was k e p t c o n s t a n t w h i l e d e c r e a s i n g the S i O S i a n g l e from 144° to 134° (path A-C); path B-C shows t h e Ax a s s o c i a t e d w i t h an i n c r e m e n t of 60 k b a r p r e s s u r e .  ACKNOWLEDGEMENTS  S i n c e r e thanks a r e extended guidance, This  support  work  was  Engineering  and  encouragement  supported  Research  t o Dr.  by  Council  E.P.  throughout  the  National  with  NSERC  summer g r a n t s were p r o v i d e d by t h e NAHS. the  computing  centre at the University  also gratefully  this  study.  Science  and  g r a n t 67-7061 a n d  The  cooperation  of  of B r i t i s h Columbia i s  acknowledged.  Appreciation  i s expressed  introducing  me  to  theory  to  Monique  and  Meagher f o r h i s  the  to  exciting Roussy  Dr.  G.V.  Gibbs  w o r l d of m o l e c u l a r for  her  many  for  orbital fruitful  discussions. Finally  I  thank  Gord  d r a u g h t i n g and h i s concern illustrations  used  Hodge  f o r the  i n the text.  f o r h i s deft aesthetic  appeal  hand of  at a l l  I.  Significant five  years  silicate data  INTRODUCTION  a d v a n c e s h a v e been made i n  with  regards  to  the  the  past  twenty  a c c u r a t e d e t e r m i n a t i o n of  s t r u c t u r e s which have, i n t u r n , s u p p l i e d a w e a l t h  for crystal  important  chemical  mineral  investigations  i n v e s t i g a t i o n s of t h i s  group.  have  For  dealt  with  f u n c t i o n of s u b s t i t u e n t c a t i o n recent years, pressure  the  most  radius,  geologically part,  structural  of  these  v a r i a t i o n s as a  temperature,  and  in  ( P a p i k e e t a l . , 1969; Cameron et. a l . ,  1973; L e v i e n and P r e w i t t , 1 9 8 1 ) . Until bonding  r e c e n t l y , i n v e s t i g a t i o n s d e a l i n g w i t h the chemical  in silicate  been b a s e d m a i n l y Ohashi  and  m i n e r a l s h a v e been few i n number and  on t h e e l e c t r o s t a t i c model  Burnham,  1972).  With  s i l i c a t e s have a h i g h c o v a l e n t bonding  (Pauling,  decade toward bonding effort  to understand orbital  extended Huckel  particular,  formalisms  addition  their  chemical  in  silicate  t h e r e h a s been a  concerted  ranging  (Louisnathan  (Meagher  e_t  self-consistent  (Newton a n d G i b b s ,  knowledge t h a t  o r b i t a l methods  the stereochemistry  method  method  sophisticated  In  In  in  1971;  t h e r e h a s been a t r e n d i n t h e p a s t  u t i l i z i n g molecular  studies.  molecular  CNDO/2  1981),  (Whittaker,  the general  character  have  of  using  from t h e s e m i - e m p i r i c a l  and G i b b s ,  a l .,  field  silicates  1979)  (SCF)  ab  1972) to  and the  the more  initio  method  the molecular  orbital  1980). to  the  success  of  method i n s t e r e o c h e m i c a l successively 1980)  and  to  bulk  to the  spectra  1973,  1979;  between m o l e c u l a r silicates possess in  supports  date,  atmospheric  calculations of  the  view  as  a  and  The  derivative important  silicate  studies  minerals.  confining pressures. determinations  currently  molecular  and  Over  in  orbital  of  values  for  groups  those  found  bonding  calculations  and  dK/dP  Recent  these  advances  limited,  at have  (first  employed  valuable  data.  at  The  kbars  the p r e s s u r e  are high  methods  experiments 60  in  structure  diffraction  however, t o a p p r o x i m a t e l y extend  quantities  crystal  are  chemical  especially  in  a t h i g h p r e s s u r e s by x - r a y some  state  high pressure c r y s t a l  experimentally,  are  pressure  range to  200  best. the  past  fifty  r e l a t i o n s h i p s b e t w e e n K and proposed.  to  modelled  ( b u l k modulus)  f o r e s e e a b l e advances w i l l  kbars at  have  Unfortunately  determine  yielded  agreement  the bulk modulus w i t h r e s p e c t t o p r e s s u r e )  research  to  minerals  The  observed  and  been a p p l i e d t o t h e r m o d y n a m i c p r o p e r t i e s of  geophysical  difficult  emission  that i s o l a t e d molecular  parameters i n the e q u a t i o n s  of  (Newton e_t a l . ,  Brown, 1 9 8 0 ) .  studies  quantities K  of  applied  solids.  these  rule,  been  absorption,  silica  D e j o n g and  p r e s s u r e and  minerals.  and  also  l o c a l bonding f o r c e s that are s i m i l a r  To  have  in  has  o r b i t a l c a l c u l a t i o n s and  three dimensional  not,  modulus  interpretation  photoelectronic (Tossell,  studies, i t  Recently  years,  various  empirical  m o l a r v o l u m e s of s o l i d s have  been  i n v e s t i g a t o r s have p r o p o s e d an e m p i r i c a l  relationship polyhedra  between  (Kp)  and  in  t h e Kp  order  mean  (Hazen and  of  these  the  not  Finger,  - cation-anion  1979).  They  relationships  successful  polyhedra  lend  alternative  q u a n t i t i e s Kp and  in  themselves  suggest must know  the  solid.  predicting  s o l i d s a t low c o n f i n i n g  pressures,  for  more  complex  is  will  solids  or  for  whereby  the  utilizing  the  pressures.  approach  d(Kp)/dP  one  one  to  p r e d i c t i o n s of K a t h i g h c o n f i n i n g An  of  c a t i o n - a n i o n d i s t a n c e s at  component  c o m p r e s s i b i l i t i e s of s i m p l e they are  modulus  t o p r e d i c t K of a c o m p l e x s o l i d  values  Although  bulk  the  atmosphere p r e s s u r e that  the  be  proposed computed  relat ionship,  Kp = V 0 E / o r 2  2  ) (dr/dV)  = V(k (dr/dV)  2  (1)  2  s  where V i s t h e volume of t h e p o l y h e d r o n , d i s t a n c e , E i s the t o t a l  energy  k  and  r i s the c a t i o n - a n i o n  i s the s t r e t c h i n g  s  force  constant. This  study  is  the  c o m p r e s s i b i l i t i e s of t h e found  in  the  earth's  future studies i s l a i d  more  models  In  study,  2  7  molecular  monitor  orbital  investigating  metal-oxide  testing  c l u s t e r s of g e o l o g i c a l  6  series  mantle.  molecular  H Si 0  a  c r u s t and  with  we  in common  pressure  this  SCF  by  first  The for  polyhedra  groundwork f o r simulation  calculations.  interest  Among  i s the S i 0 2  7  of the  dimer.  c h a n g e s i n t h e s t e r e o c h e m i s t r y of  as a f u n c t i o n of p r e s s u r e as w e l l as  changes  in  the  stretching  and  with pressure. their  variation  bending  f o r c e c o n s t a n t s of the S i O S i  The c o m p u t a t i o n o f p o l y h e d r a l with  pressure w i l l  linkage  bulk moduli  be c o m p l e t e d  i n work  and now  underway on t h e SiO« a n d AlO„ t e t r a h e d r a and i n f u t u r e work on octahedral silicon.  oxyanion  clusters  Investigations  i n t o the atomic  such  of  magnesium,  aluminum  a s t h e above p r o v i d e  responses t o pressure i n s i l i c a t e  and  insights  structures.  5  II.  MOLECULAR ORBITAL METHOD  Description  The m o l e c u l a r basis  orbital  (MO) method  f o r the calculations  p r o v i d e s an  approximate  forms  i n this  solution  to  the underlying  study.  The MO method  the Schrbdinger  wave  equation,  H*=E*  for  a  many-electron  equivalent central electron  molecule  t o an e i g e n v e c t o r  premise  i n MO  wavefunction,  antisymmetrized  product  c a l l e d moleculer  orbitals,  can  of  or cluster  (*) e i g e n v a l u e  theory *,  (2)  i s that be  of atoms.  (E) problem.  The  t h e c o m p l e x many-  approximated  one-electron  This i s  as  an  wavefunctions,  n  (3)  where n i s t h e t o t a l optimal  number o f e l e c t r o n s  wavefunction,  wavefunction),  will  e n e r g y f o r an a t o m i c  be  *  (also  t h e one  cluster  i n t h e system.  known  as  the  which  minimizes  The  Hartree-Fock the  i n i t s g r o u n d s t a t e , 'E |,  total  f**H*dr  E |= wo  (4)  where * i s t h e m a n y - e l e c t r o n w a v e f u n c t i o n i s the many-electron Hamiltonian Incorporated potential group.  in  energies If  the  atomic  expressed  operator.  hamiltonian . are  of the n u c l e i  are considered  cluster  with  m  approximation fixed,  nuclei  the  the  atomic  i s accepted,  hamiltonian  for  a n d i , j e l e c t r o n s c a n be  2  -V y ^ e V r . J  where X7\ i s t h e L a p l a c i a n o p e r a t o r . kinetic  represents nuclei  k i n e t i c and  i n t h e f o l l o w i n g way,  H =y(-nV2M)V  the  the  and e l e c t r o n s i n  Born-Oppenheimer  whereby t h e n u c l e i an  the  i n (3) a n d H  defined  energy  their  and  the  of  the  potential third  XY(e /r,y) , 2  The f i r s t  electrons,  term  the  (5)  represents  second  term  e n e r g i e s due t o a t t r a c t i o n w i t h t h e  term r e p r e s e n t s  t h e r e p u l s i o n between  electrons. The h a m i l t o n i a n  i s frequently divided  terms, H , and t w o - e l e c t r o n  H =/  The  energy  as t h e c o r e  relating  terms, e /r;', such  H-+>  2  one-electron  that  (6)  /_|e /r^). 2  to the one-electron  hamiltonian) i s  into  operator  (also  known  7  E  where E energy  m V*'* B  (i)H  r e p r e s e n t s t h e sum due  to  an  .-*m of  electron  (i)dT  «'  ( 7 )  the kinetic  occupying o r b i t a l  and  potential  * .  A typical  m  t w o - e l e c t r o n term  representing the repulsive p o t e n t i a l  between e l e c t r o n s  i , ji s  V  •-^•m  ( i )  V.  ( i )  i - V 3>*r» 3>  ( e 2 / r  )  (  (  J  dr  energy  « J " d r  f ( < ( i ) * „ ( i ) ( e V r ^ . ) < ( j ) * ( j ) dr.drjn  where  J  m  n  i s t h e Coulomb  exchange energy.  The  total  repulsive energy  energy of  and K  t h e system  m r !  i sthe  c a n be  expressed as  11  f o r m o l e c u l a r o r b i t a l s m a n d n. After ,  defining  the hamiltonian,  must be f o u n d w h i c h  satisfy  suitable  wavefunctions  the one-electron Schrbdinger  equation,  where t h e o p e r a t o r F i s t h e ' H a r t r e e - F o c k  or  effective  one-  8  electron other  Hamiltonian  words,  eigenvectors energy  e  .  m  and  there of  will  the  be  a  linear  In p r a c t i c e  expanded  in  terms  orbitals,  #,  centered  r  i s the o n e - e l e c t r o n energy.  of  operator  the a  on  series F,  molecular  convenient  of  which  In are  each w i t h a unique  orbitals, basis  *^,  s e t of N  t h e v a r i o u s atoms o f t h e  are atomic  molecule,  N tn  That i s , the m o l e c u l a r combination can  be any The  be  of  orbitals  atomic  general  orbitals  that  give  the  true  reduced the  lowest  respect  energy.  Principle  to  matrix  m  will  This  is in  energy. c  r m  The ,  or  equations  the equal  problem i s  that  yields with  Following this  method,  w h i c h can  written  be  form,  m  C  .  m  ,  i s always g r e a t e r than  e a c h of t h e c o e f f i c i e n t s .  FC =  where  orbitals  i s done by m i n i m i z i n g t h e e n e r g y  t h e c o e f f i c i e n t s must s a t i s f y in  linear  which s t a t e s that  t h e s e t of c o e f f i c i e n t s , This  atomic  energies, t  ground s t a t e e l e c t r o n i c  to f i n d i n g  The  a  single-electron functions.  lowest  v a l u e of t h e c a l c u l a t e d e n e r g y the  as  f o r t h e wavef u n c t i o n s ,  accordance w i t h the V a r i a t i o n  to  expressed  (LCAO).  s e t of s p e c i f i e d  best approximations  those  are  is  a  column  €  m  vector  m a t r i x whose e l e m e n t s a r e d e f i n e d  SC  (12)  m  of MO as  coefficients,  F i s the  9  (13)  where F=H+J-K a n d S i s t h e o v e r l a p  matrix  with  elements,  (14)  The s e c u l a r  equations  ( o r Roothaan  equations),  FC=SCE  are  solved  values  iteratively  u n t i l convergence  In  addition  interested the  total  atomic of  with  to  and  successively  better  c  and  r m  E  (self-consistency) i s achieved.  the  i n the o r b i t a l number o f  (15)  total  molecular  population  electrons  bond c o n t r i b u t i o n s  the t o t a l molecular o r b i t a l  in  energy,  a n a l y s i s which the  system  ( M u l l i k e n , 1955). density  we  are  partitions  into  various  Integration  function  (16)  expanded i n terms of t h e a t o m i c o r b i t a l N  basis,  n  (17) si  yields  the t o t a l  r=i  number o f e l e c r o n s ,  n:  n  (18)  The M u l l i k e n bond o v e r l a p p o p u l a t i o n  for a pair  of atoms, s - t ,  i s d e f i n e d by n (19)  when summed o v e r atomic  orbitals  a l l atomic on  orbitals  center  t.  b e t w e e n two atoms i s p o s i t i v e , they are  they  on  If  the  center  s  overlap  a r e bonded;  and a l l population  if  negative,  antibonded.  The  atomic  obtained  by  orbital  summing  the  population quantity  f o r an atom s, q ( s ) , i s n(s-t)  over  a l l atomic  o r b i t a l s on t :  q(s)  The a t o m i c  c h a r g e o f atom s, Q ( s ) , i s d e f i n e d by  Q(s)  where  q (s) 0  (20)  is  the  = q (s)-q(s)  total  0  number  s t a t e o f t h e f r e e , n e u t r a l atom s.  (21 )  of e l e c t r o n s i n the ground  11  MO M e t h o d s  Molecular  o r b i t a l c a l c u l a t i o n s c a n be c l a s s i f i e d  general c a t e g o r i e s : "approximate and  " ab i n i t i o  molecular  orbital  integrals  involved  c a l c u l a t i o n a r e a p p r o x i m a t e d by known a t o m i c use  of  Hartree-Fock integrals  matrix.  with respect One  "semi-empirical" expressions  and  The  the semi-empirical  to their ability  Neglect  of  ( P o p l e e t a l . , 1965). repulsion  integrals  neglected. to  adopted  expressions  MO  Gibbs,  Overlap  i t s name  values  calculations  energy for  calculations  acid  SiOSi angles  i s the  polymorphs  tend  to drastically  method  electron  overlap" type are  matrix.  disiloxane  1  are used  CNDO/2  (Tossell  and  (Meagher e t a l . ,1979) y i e l d  i n c l o s e agreement w i t h  silica  s e c o n d row e l e m e n t s  on  results.  a l l  In addition, semi-empirical expressions  orbital  these  are evaluated  (CNDO/2)  implies,  the " d i f f e r e n t i a l  1977) a n d p y r o s i l i c i c  minimum  for  methods  c a l c u l a t e the elements of t h e Hartree-Fock  molecular  1  of  q u a n t i t i e s a n d by  t o predict experimental  Differential As  i n the  f o r elements i n the  approximations  of t h e better-known approximate  Complete  methods"  " c a l c u l a t i o n s . I n t h e a p p r o x i m a t e MO m e t h o d s ,  a l a r g e p o r t i o n of t h e e l e c t r o n  the  i n t o two  and  glass.  overestimate  (Marsh and Gordon,  observed  However,  CNDO/2  bond l e n g t h s f o r  1976).  An example of an elecron repulsion integral of the d i f f e r e n t i a l o v e r l a p t y p e i s J*j" *v( 1 ) 4> ( 1 ) (1 f^*^(2) <t> (2 ) d r , d r where # , # , 0 f , a n d * a r e a t o m i c o r b i t a l s . ~>~s  r  5  0  v  z  12  In  recent  initio  SCF  Gaussian  years,  MO  we  calculations  expansions  of  the  electron  full  and  ab  t h e development  computer  initio  electronic  system.  seen  Slater-type  a p p r o x i m a t e MO m e t h o d s , solve  have  kinetic  the  core  integrals for  follows. the  Hartree-Fock  i s made  matrix  Huckel approximation  eigenvalues  2  With (or  eigenvectors  initio  molecular are  2  r 5  until  row e l e m e n t s  al.  1976).  calculated.  C a l c u l a t i o n of the  use  of  orbitals An  a  Gaussian-type expedites  initial  through  orbital solved.  convergence  computations  second  , are  guess  the  of t h e  Huckel or extended of  the  core  energies,e  m  With s u c c e s s i v e l y  )  and better  v a l u e s , t h e s e c u l a r e q u a t i o n s (15) a r e  bond l e n g t h s a n d a n g l e s  ,  f o r a many-  the approximated Hartree-Fock m a t r i x , the  c o e f f i c i e n t s and energy  Ab  S  or through d i a g o n a l i z a t i o n  (c^'s)  iteratively  The  atomic  integrals.  hamiltonian.  to  o n e - e l e c t r o n i n t e g r a l s w h i c h make up  computation of these  solved  equaton  hamiltonian are evaluated next.  wavefunctions  attempt  the  A f t e r d e f i n i n g t h e a t o m i c p o s i t i o n s a n d wave  and p o t e n t i a l  two-electron  using  Unlike  calculations  f u n c t i o n s , a l l atomic o v e r l a p i n t e g r a l s , The  programs  orbitals.  Schrodinger  o f ab  i s achieved.  e n a b l e us t o s o l v e  f o r molecules  w i t h a h i g h degree  Optimized  T-0  for equilibrium  involving of accuracy  distances  first  and  (Collins et  a n d TOT a n g l e s , f o r  W i t h the extended Huckel a p p r o x i m a t i o n , the elements of the Hartree-Fock matrix are approximated with the Valence O r b i t a l I o n i z a t i o n . P o t e n t i a l (VOIP): F =VOIP(u) ; F = K ( V O I P ( u ) + V O I P ( v ) )  example,  compare  polymorphs, 1979,'Newton  well  with  silicates, and  Gibbs,  c a l c u l a t i o n s of q u a d r a t i c polyatomic experimental  molecules trends  r e a s o n s , ab i n i t i o  local  and  geometries  siloxanes  1980).  in  (Meagher  Furthermore,  et ab  silica a l ., initio  f o r c e c o n s t a n t s on a l a r g e number o f  satisfactorily (Newton  et  account  a l .,  c a l c u l a t i o n s were u s e d  for nearly a l l  1970). in this  For study.  these  Ill.  Ab  initio  undertaken with al.  ,  was  adopted is  represented  To  integrals,  the  the  STO  Gibbs  we  are  and  five  functions  are  set i s s u f f i c i e n t 6  orbital  with  (STO)  nine  STO  of  the  two-electron  in  t u r n , expanded  in this  In  E = E  i s expanded  study ( r e f e r r e d to i s r e p r e s e n t e d by a  functions.  when s t u d y i n g 2  Newton  and  t h e bond  length  .  7  themselves r e a d i l y  + (dE/dq)q + 0 . 5 0 E / 9 q ) q  which i s e i t h e r the displacement r-r , 0  or  angle,  to The  i n t e r m s o f q,  2  0  as the  o f f o r c e c o n s t a n t s (Newton e_t a l . , 1 9 7 9 ) .  p o t e n t i a l energy  length,  constituent  functions for  Molecular o r b i t a l c a l c u l a t i o n s lend evaluation  r  (1981) have shown t h a t a STO-3G  and a n g l e r e l a t i o n s h i p s f o r H S i 0  the  <t> ,  basis  s e t ) , e a c h STO  (1980) and G i b b s e t a l . basis  set,  (GTO's) ( H e h r e e t a l . , 1 9 6 9 ) .  s e t c a l c u l a t i o n s used  e_t  STO  ,  c o m b i n a t i o n of t h r e e G a u s s i a n  minimal  the  dealing  computation  m i n i m a l STO-3G b a s i s  linear  of  were  (Binkley  study, a minimal basis  a single Slater-type  for silicon  ease  minimal basis  by  For example,  Gaussian-type o r b i t a l s  a  this  calculations  program  i n which each atomic o r b i t a l  functions  oxygen.  orbital  t h e G a u s s i a n 76 c o m p u t e r  function.  basis  as  molecular  1978). . Throughout  atoms basis  SCF  CALCULATIONS  2  from  9-& , 0  the  2  + • • •  (22)  equilibrium  bond  depending  on  whether  a  stretching i s being  force constant  calculated.  In t h i s  equilibrium  force constant  or'bending force constant,  study,  9 i s the  S i - 0 bond l e n g t h and respective  , k^,  SiOSi  values.  i s twice  r r e f e r s to the angle;  By  r  and  0  definition,  ©  bridging are  0  the  their  quadratic  t h e c o e f f i c i e n t of t h e q u a d r a t i c  k = O E / 3 q ) Nm" z  2  kg,  term:  (23)  1  5  k =(3 E/3q )/r 2  2  Nm"  2  where q qnd  r are defined  found d i r e c t l y curve. and  by  fitting  I n c r e m e n t s of  2°  about the  the p a r a b o l a .  above.  t o the  ranges  of  0.05  A  insignificant.  The  given  is  definition preferred  (force/length)  Three  principal  as  stretching  and  method  outlined  Treating  the  by  Herzberg  cluster  energy  energy length  as  as  an  field  about  higher  (22)  order  it  yields  stretching force  be  determined  XY  2  for  molecule,  model  ,  we  terms  a  constant the  same  for  from by  be  constant.  frequencies  (1945)  the  were f o u n d t o  bending force constants  assuming a v a l e n c e f o r c e potential  because  vibrational  a c i d m o l e c u l e can  8°  of the b e n d i n g f o r c e  the  pyrosilicic  SiOSi  and  b r i d g i n g angle,  t h e e x p a n s i o n of t h e p o t e n t i a l e n e r g y  dimensions  potential  are  e q u i l i b r i u m b r i d g i n g a n g l e were u s e d t o f i t  With  a b o v e (24)  force constants  A a b o u t t h e e q u i l i b r i u m bond  e q u i l i b r i u m bond l e n g t h and in  Thus t h e  a parabola  0.01  (24)  1  b  the  the Si-0  following XY  2  the  molecule.  (0(H Si0 ) ),  and  can  the  3  3  2  express  E'  where and  = 0.5k qr  f r o m t h e e q u i l i b r i u m bond  q© i s t h e d i s p l a c e m e n t The  valence  from  force  the equilibrium  of  qr  a n d q© .  length  bridging  m o d e l . assumes t h a t t h e r e a r e no  c r o s s terms i n t h e p o t e n t i a l energy terms  (25)  2  s  q r i s the displacement  angle.  + 0.5k q©  2  s  i f i t i s expressed  in  W i t h t h e p o t e n t i a l e n e r g y d e f i n e d by  ( 2 4 ) , we c a n d e r i v e t h e f o l l o w i n g e q u a t i o n s  (Herzberg,  1945;  p.169):  4TT IA 2  4rr  (i/  2  = (1 + (2m /m ) s i n ( © / 2 ) ) k / m 2  2  Y  g  2  + i/ )  x  s  0  = (1 + (2m /m ) c o s ( © / 2 ) 2  2  Y  x  0  (26)  y  )  k /m-^ s  + (1 + (2m /m ) s i n ( © / 2 ) ) 2 k / m 2  T  16^"^^  v  where  s  ,v  antisymmetric  a  v  5  , i/g  Y  and  m  K  (26),(27) b  x  s  .  Y  (27)  (28)  s  a r e t h e symmetric S i - 0 s t r e t c h i n g ,  i s t h e mass o f X ( 0 ) , m  3  and v  &  = 2(1 + (2m /m ) ) k k / m ^  ( H O S i ) and a l l other Equations  0  S i - 0 s t r e t c h i n g and SiOSi bending  respectively; 3  ?<  terms and  have  been  r  i s t h e mass o f Y  defined  (28) a r e s o l v e d  frequencies,  previously.  simultaneously f o r  IV.  Basically simulate initial  two d i f f e r e n t m o d e l s  elevated  pressures  m o d e l , w h i c h we w i l l  simulated  by  successively angle.  .simply  in  were u s e d i n an e f f o r t our  refer  locking  calculations.  t o a s model I , p r e s s u r e  the  Si...Si  SiOSi angle.  one  bar  and  asymmetric  elevated pressures  for  the  clusters  H Si 0(, 1 2  and  5  were c a l c u l a t e d monitoring  H , A1S i ,0," . by k e e p i n g  the changes  i n toward the oxygen. other  while  b r i d g i n g oxygen the  effect  of  upon  The  the  2  force  constants  H Si 0 , 6  was  7  H A1 0 ~ 6  force  2  at 2  7  ,  constants  immobile  while  the  Si...Si  the  as  changes We  in  energy  a l s o u s e d model I t o  the study  on t h e S i - 0 f o r c e c o n s t a n t  that substituting force  f o r c e c o n s t a n t s , on  by m a i n t a i n i n g a c o n s t a n t  oscillated.  stretching  force  symmetric  asymmetric  polymerization  w e l l as the e f f e c t  of the  i n e n e r g y as t h e S i atoms were b r o u g h t  monitoring was  SiOSi  stretching  the b r i d g i n g oxygen  h a n d , were c a l c u l a t e d  distance  The  1  2  at  However, model I  i n c o m p a r i n g s y m m e t r i c and a s y m m e t r i c at  was  distance  T h i s m o d e l has r e s t r i c t e d a p p l i c a t i o n s b e c a u s e  constants  to  In the  shorter values while maintaining a straight  need t o m a i n t a i n a s t r a i g h t useful  MODELS  aluminum  constants  at  for silicon one  as has  b a r and a s a  f u n c t i o n of p r e s s u r e . In a l l of the c l u s t e r s s t u d i e d w i t h conformations maintained  were  a t 180°  used .  model  I,  staggered  and t h e S i O S i and A l O S i a n g l e s  In the H S i 0 6  2  7  cluster,  the  0-H  were bond  l e n g t h s were 0.96 A w h i l e t h e SiOH a n d OSiO a n g l e s at  109.47° , r e s p e c t i v e l y  the OSiH and OAlH a n g l e s were  locked  maintained  at  1.49  (Figure 1).  were l o c k e d  In the l a r g e r  clusters,  were 109.47° w h i l e t h e S i - H d i s t a n c e s A.  w i t h i n the SiO  a  Tetrahedral,  T<J ,  symmetry  was  a n d AlO« t e t r a h e d r a t h r o u g h o u t a l l  computations. In pressure inert  t h e s e c o n d m o d e l , w h i c h we w i l l was s i m u l a t e d a b o u t an helium  atoms  systematically oxygen.  along  6  the  2  model  allows  t o a s model I I ,  cluster  7  by  Si-0 bridging  s t e p p i n g t h e two h e l i u m s  This  SiOSi angles  along  H Si 0  refer  toward  f o r pressure  a n d i s more p r e c i s e l y  placing  vector the  and  bridging  s i m u l a t i o n a t bent  a uniaxial  stress  directed  ( F i g u r e 2) was p l a c e d i n a  staggered  the S i - 0 v e c t o r s . The  H Si 0 6  2  conformation bond  7  dimer  w i t h O-H d i s t a n c e s , d ( O - H ) , a n d S i - 0  lengths,  d(Si-Ob  ) , o f 0.96 A a n d 1.65 A,  The  OSiO a n d S i O H a n g l e s  were l i k e w i s e m a i n t a i n e d  and  180° , r e s p e c t i v e l y ,  throughout  At  one  atmosphere,  same w i t h o r w i t h o u t we f o u n d helium silicon,  the  and  nonbridging  respectively. at  109.47°  a l l computations.  e q u i l i b r i u m d i s t a n c e s were t h e  t h e h e l i u m atoms.  t h a t t h e M u l l i k e n bond  nonbridging  At e l e v a t e d p r e s s u r e s ,  overlap  oxygens,  populations  between  n(He-Onb ) , a n d h e l i u m a n d  n ( H e - S i ) , were n e v e r g r e a t e r t h a n  0.004  and  0.007,  stretching  force  respectively. Whereas  model  constants at elevated stretching  force  I  yields  pressure,  constants  asymmetric model  II  yields  symmetric  a t one b a r a n d a t p r e s s u r e .  The  F i g u r e 1. M o l e c u l a r c o n f o r m a t i o n f o r t h e a i m e r s s t u d i e d w i t h model I (note the s t r a i g h t b r i d g i n g a n g l e ) ; p r e s s u r e i s s i m u l a t e d by d e c r e a s i n g the i n t e r t e t r a h e d r a l d i s t a n c e .  Figure 2. Malecular conformation f o r HgSi20y when studied w i t h model I I . Note the bent b r i d g i n g angle and p o s i t i o n i n g of helium atoms used to simulate pressure by s y s t e m a t i c a l l y decreasing the dCHe-Ojj) d i s t a n c e s .  f o l l o w i n g were s t u d i e d equilibrium pressure; constants a  with  stereochemistry  2) c h a n g e s  model of  in  the  with pressure;  and  H Si 0 6  Model  energetics  as w e l l as  3)  the  II i s preferred  s t u d i e d as  pressure  is  the  2  1) 7  stretching  f u n c t i o n of b r i d g i n g bond l e n g t h s  pressures.  II:  bridging  increased.  total and  changes  as and  the  function bending  angles  at  as  elevated  bridging  energetics  of  force  p o t e n t i a l energy  because the bond  a  in  angle  can  be  22  V.  RESULTS AND  DISCUSSION  Model I  As  stated,  model  symmetric,  k  ,  constants.  At  one  are  774  s  Nm"  and  provides  asymmetric  bar,  and  1  I  the  861  a means o f c o m p a r i n g ,  k  calculated k  Nm"  , i s lower than the  1  715  Nm"  1  .  linear bridge and  raman s p e c t r o s c o p i c (Lazarev,  symmetric, v , s  indicate  the  1972).  involves  a  c e n t r a l atom and terminal  3  large a  groups.  terminal  have  T h e d ( S i - O ^ ) =1.65  "The  6  H A1 0 " 6  2  2  2  7  *  small a  7  amplitude  of  of  the  is  ,  630  constant, on  group w i t h  SiOSi  a  of  be  bridge  greater.  The  higher  vibration  should  s  amplitude  low  vibration  since lighter  for  the  since  the  and  the  l a r g e a m p l i t u d e of v i b r a t i o n i n  A in this H S i 0 2  cluster.  7  d ( A l - O ^ ) =1.735 A i n t h i s H A 1 0 " 6  the  s  i s e x p e c t e d t o be  6  3 7  c a l c u l a t e d asymmetric, i / . ,  constant  v  2  Similarly,  data for a S i 0  force  Conversely, a  for H S i 0  a  a m p l i t u d e of v i b r a t i o n f o r t h e  small  c e n t r a l atom has groups  for  stretching frequencies  asymmetric  k  force  w i t h c a l c u l a t i o n s based  The  asymmetric s t r e t c h i n g frequency it  and  symmetric s t r e t c h i n g f o r c e  These r e s u l t s c o n f l i c t  i n f r a r e d and  s  stretching  , respectively.  1  asymmetric s t r e t c h i n g f o r c e constant Nm"  ,  a  the  2  7  2  cluster.  the  23  symmetric  mode ( R o s s , 1 9 7 2 ) .  Model I , however,  predicts  the  o p p o s i t e t o what i s e x p e c t e d . In  a d d i t i o n to comparing  k  and  s  k  , model I was  a  s t u d y t h e e f f e c t s t h a t p o l y m e r i z a t i o n and for  Si  have  upon . t h e  f u n c t i o n of p r e s s u r e . force  constants  H Si 0 6  2  7  and  H Si 0« 1 2  2  and  5  atmospheric  H A1 0 "  ,and  2  7  H, A l S i ,0,,"  increasing polymerization asymmetric  stretching  silicates  show  from  is 6  2  7  constant  Spectroscopic  asymmetric  asymmetric  found  H Si 0  force  increase s i g n i f i c a n t l y .  of  pressure  ,  1  of  Al  substitution  for  stretching the  dimers,  highly-polymerized  2  to  c o n s t a n t s a t one b a r a n d ' a s a  A comparison  at  6  force  used  stretching  to  clusters,  i n Table I. H^SisOj,  of  With ,  the  d ( S i - 0 ] ) does not 0  studies  on  framework  frequencies  for  TOT  l i n k a g e s a r e i n t h e r a n g e 950-1200 cm" ' ( M i l k e y ,  1960;  Moenke,  1962;  these  values  Lyon,  1962;  Moenke, 1 9 6 6 ) .  overlap  the range  (Farmer,  1974)  for H ,  788  Nm"  silicon 2  A l S i "  f o u n d f o r p y r o s i l i c a t e s and c h a i n  thus s u p p o r t i n g our  A decrease from  i n the asymmetric t o 647 Nm"  1  The  i s less  greater  than  that  our  $  results. stretching  cluster  that  f o r the A l - 0  gradual decrease i n k with  than  force  with  stretching for  AlOSi  the  bond s h o w i n g  Si-0  results,  bond  M i l k e y (1960) has n o t e d t h a t  with  shift  linkages,  as t h e aluminum c o n t e n t i n c r e a s e s .  cm"  to  aluminum  to  lower  frequency  and  that there i s a  region  tends  constant  f o r c e c o n s t a n t of  c e n t e r of g r a v i t y of a b s o r p t i o n bands i n the 1  silicates  f o u n d by s u b s t i t u t i n g  , has an a s y m m e t r i c  1  which  keeping  was  1  i n the dimer.  695 Nm"  1  Furthermore,  In the  950-1200 increasing  Table I. Asymmetric s t r e t c h i n g f o r c e c o n s t a n t s ( k ) calculated a t 1 bar f o r the clusters H Si 0 , H A1 0 - , H Si 0„, and H A l S i O „ " with a l l SiOSi and A l O S i a n g l e s e q u a l t o 180°. B  6  2  6  2  1  7  1 2  5  1 2  u  Cluster  kg,(Nm- )  H Si 0  788  6  2  1  7  H A1 0 6  H  2  Si 0  1 2  647  2  7  5  796  4  H, AlSi„O 2  u  1  695  2  7  25  aluminum  content.  Calibration model  I  of p r e s s u r e  was n o t p o s s i b l e .  r e l a t i v e changes increasing  and  pressure  for  clusters  studied  We w e r e , h o w e v e r , a b l e  values by  the  of  the  plotting  force  d(T...T)  pressure  decreases  increases  the  pressure  right  to  asymmetric  stretching  force  constants  consistently  lower  pressure.  In  than those  addition,  for  the  left  as  3).  hence  the  Figure  for  3.  d(Al-Ob )  with  force constants  s i m i l a r l y with decreasing  s e e n by t h e n e a r l y p a r a l l e l  in  d(Si-Ofc )  ( F i g u r e 3a) and t h e h i g h l y - p o l y m e r i z e d increase  (Figure  increases,  from  with  l o g ( d(T...T)  where d ( T . . . T ) i s t h e i n t e r t e t r a h e d r a l d i s t a n c e As  to look at  constants  log(k) verses  with  clusters  The are  increasing  f o r the.dimers (Figure  3b)  intertetrahedral distances  trends.  The u s e of m o d e l I v e r i f i e d  the f e a s i b i l i t y  of  studying  S i - 0 bond e n e r g e t i c s and f o r c e c o n s t a n t s  at simulated  pressures  orbital calculations.  with  ab i n i t i o SCF m o l e c u l a r  The m o d e l was a b a n d o n e d , h o w e v e r , i n f a v o r o f m o d e l allows the  us t o i n c o r p o r a t e  SiOSi  the important  II  which  s t r u c t u r a l v a r i a b l e of  angle.  Model I I  '  With the SiOSi bending force constant stretching  elevated  force constant,  and s y m m e t r i c S i - 0  we c a n s o l v e e q u a t i o n s  (26),  (27)  Figure 3. Log o f the asymmetric S i - 0 s t r e t c h i n g force constant, l o g ( k ) , p l o t t e d against the i n t e r t e t r a h e d r a l distance, l o g ( d(T...T) ), f o r H S i 0 where log(ka)=-7.351og(Si...Si)+4.55 (r2=0.999); H A l 0 - 2 where log(k )=-7.421og(Al...Al)+4.63 (r2=0.998); H i S i 5 0 4 where log(k )=-7.121og(Si...Si) +4.47 (r =0.999); and H A1SI4O4- where log(k )=-7.411og(Al...Si)+4.62 (r =0.998). a  6  6  2  7  2  7  a  2  2  12  0.46  a  1  2  0.48  a  0.50  log ( d (T • • • • T ) )  0.47  0.49  log(d(T •-T))  0.51  and  (28)  for v  simultaneously  a c o m p a r i s o n between the for  H Si 0 6  2  a t one  7  bar  raman s p e c t r o s c o p i c containing  SiOSi  (0(Si(CH ) ), 3  range For  of  example, i / cm"  attributed  solely  The  .  1  considering  we 2  only  The  This  local  involving  same atoms and  Figure  4a,  length  pressure  (Farmer,  (Figure  4b)  2  are the  v  to those  trends  frequency  anion,  siloxane,  2  ,  7  the  display  frequencies. ranges  a  vibrational cm"  even  energetics  forces  Si 0 " 2  agreement  more  further  from  frequency  f o r the  1  6  7  with  encouraging of t h e  SiOSi  SiOSi  linkage in  support in  a  to  the  siloxanes  and  in isolated molecular  clusters  c o o r d i n a t i o n number.  twelve  1974).  v  while  1  i s p l o t t e d against  a  for  where  lends  the average b r i d g i n g  pyrosilicates  A similar different  c a l c u l a t e d e q u i l i b r i u m distances at Both  6  7  Ba TiOSi 0  bonding  similar  S i - 0 bond  2  and  compounds  show a r e a s o n a b l e  comparing  s i l i c a t e s are  In  Si 0 "  calculated  for  w i t h t h e e n e r g e t i c s of t h e  7  the  the  bar  bending  results  c o m p l e x compounds. that  one  presents  from i n f r a r e d  the p r i n c i p a l v i b r a t i o n a l  The  are  in H S i 0  premise  at  The  c a l c u l a t e d values  6  determined  t o S i O S i b e n d i n g i s 169  data.  very  those  pyrosilicate,  for  experimental  linkage  and  Table II  frequencies  v a r i e s f r o m 503-665 cm"  3  1029-1104  anion.  vibrational  linkages.  values  vb•  and  s  experiments  , and  2  ,. v  d  trend  at  atmospheric  i s found f o r  H Si 0 6  d(Si-0^ )  correspond  different  SiOSi  2  7  to  angles.  show a d e c r e a s e i n t h e a s y m m e t r i c S i - 0 s t r e t c h i n g  as  proportional force constant  d(Si-0b to  )  the  (19), k  increases.  Since  s q u a r e r o o t of t h e s  v  a  is  directly  symmetric s t r e t c h i n g  a l s o d e c r e a s e s as d ( S i - O b )  increases.  T a b l e I I . Comparison a t 1 b a r of c a l c u l a t e d symmetric s t r e t c h , v , a n d a s y m m e t r i c s t r e t c h , v-e,, a n d bending, i/b, f r e q u e n c i e s f o r H S i 0 w i t h those determined from i n f r a r e d a n d raman s p e c t r a f o r S i 0 - , (0(Si(CH ) )„, and B a T i O S i 0 . s  6  2  7  6  2  2  3  Exper i m e n t a l Frequencies (cm )  - 1  H Si 0 2  7  - 1  Si 0 2  7  0(Si(CH ) ), 3  2  BaTiOSi 0 2  588  503  547  665  a  1 252  1 029  1 1 04  1039  b  1 33  1 69  5  2  7  Calculated Frequencies (cm )  6  7  G i l l e s p i e and R o b i n s o n ,  1964.  L a z a r e v , 1972. G a b e l i c a - R o b e r t and T a r t e , 1981.  C 7  Figure 4. A comparison of the asymmetric stretching frequency,-v , plotted against the bridging bond length, d(Si-0t>), for a group of twelve p y r o s i l i c a t e s (a) and H S i 0 (b); v ' s were determined from spectroscopic experiments for the p y r o s i l i c a t e s whereas v ' s for H6Si20 were calculated. a  6  a  2  7  a  7  1400  ,1400  H Si 0 6  1300  2  7  1  1200H  1100  (b) 1000 1.60  1.68  d(Si-O) b  1000  1.56  1.64  1.60  d(Si-O) b  30  In  other  words,  incompressible length  the  (that i s , greater  agreement  b e t w e e n our  s t u d i e s at atmospheric proceed  with  pressure  the  bar  not To  (Newton and  represent  units  the  more  bridging  bond  constant  pressures  as  this  H e H S i 0 7 was 2  length  over  A t one  valley  surrounded  method,  a  Ax,  the  equal  fact  of  applied. v  the  symmetric  Ax,  of  studied.  equivalent  i s small. energy  surface  at e l e v a t e d pressure  t o be  140  kbar  The  three  topology  pressure.  sides of  with  distinct  minimum s u r r o u n d e d on significantly  At  the 140  by  f o u r s i d e s by  steeper  than  surface  the  Si-0  explained  long,  steep energy  energy kbar,  a  for  (Figure 5).  by methods  b a r , t h e e n e r g y s u r f a c e shows on  that  t o k-g, Ax were  interval,  potential  SiOSi angle  do  angular  force being  approximations  notably  are  the  interval,  i s estimated  later.  (Figure 5).  different  average  the  widens at  c o n s t r u c t e d a s a f u n c t i o n of t h e b r i d g i n g and  This pressure  for  us  bridging angles.  e m p l o y e d and  the  reasonable  Because  d(He-Ob) v a l u e s  at d i f f e r e n t  was  is  l o n g as the  Using 6  Ax  enough f o r  pressure.  equivalent pressures  stretching  provide  of  experimental  as t h e b r i d g i n g a n g l e  Law  9 V  These  encouraging  p r o p o r t i o n a l to the  of  force  was  pressures  e s t a b l i s h e d where k  which  as  becomes  c a l c u l a t i o n s and  pressures  Hooke's  is directly  Therefore  bond  bond  1980), c o n s t a n t  equivalent  configurations, pressure  Gibbs,  equal  approximate  2  )  5  simulation  e q u i l i b r i u m d(Si-Ob) decreases one  k  Si-0  decreases.  The  to  bridging  narrow barriers changes  s u r f a c e shows a energy  those  at  barriers one  bar.  Figure 5. P o t e n t i a l energy surfaces f o r HgSi20 at 1 bar and 140 kbar p l o t t e d as a f u n c t i o n of the b r i d g i n g d i s t a n c e , d(Si-Ob), and the SiOSi angle. ! 7  /.SiOSi (deg)  32  C o m p a r i n g t h e minimum o f kbar,  we  142°  to  see 132°  and  o  1.585  surface  140  angle  from  a d e c r e a s e of t h e b r i d g i n g S i - 0 bond  from  1.565  A.  to  tripling  of  913 k£  (Figure  Nm"  steepening the at  1  SiOSi  Figure at  140  7).  taking v e r t i c a l  By  1  a n g l e can  be k  s  between and  7  and  that  v  a group of  is  a  and  relation the  that t h i s The the  between v  bridging  infrared  (1972) and glasses  a  at  s  and  pressures  up  cross  V y c o r , and  pressure.  The  Nm"  at  1  an  almost  Nm"  at  1  sections  bar  140  through and  140  kbar.  b r i d g i n g angle  widens  we  and  investigated  (Figure to k .  4)  the  at kg  Therefore  we  are  ( F i g u r e 4)  restating  i n t e r m s of  k  s  predict  pressure. is  supported  s t u d i e s of F e r r a r o and  (1972)  on  t o 58.8  m i x e d O S i O and  Gibbs  that d(Si-O^) i s i n v e r s e l y  with pressure  P y r e x • shows  for  mentioning  Newton and  s  c-quartz  kbar.  i n t e r t e t r a h e d r a l Si-0 s t r e t c h i n g frequency silica,  743  energy  ( F i g u r e 8 ) ; i n a d d i t i o n , we  and  et a l .  t o 20.6  the  d(Si-O^)  spectroscopic  Ferraro  bar  angle.  angle  in k  the  d(Si-Ob) at atmospheric pressure  r e l a t i o n s h i p holds increase  as  Earlier  (1980) have d e m o n s t r a t e d a t one  the  from  s  one  proportional  SiOSi  s i d e s of  s  pyrosilicates  directly  c o r r e l a t e d w i t h the  bar  at  increases  relationship 2  SiOSi  r e l a t i o n s h i p between k  the  studied  pressures.  6  bar  ( F i g u r e 6) and  Nm"  two  H Si 0  in k  kbar  a t one  the  f r o m 8.2  8 shows t h a t  the  of  increase  the p o t e n t i a l energy s u r f a c e s , the  a t one  i s a n a r r o w i n g of t h e  The  i s r e f l e c t e d by  bar  kbar  energy trough  o  A to  one  and.  that there  the  a  Manghnani silicate  They f o u n d t h a t for c-quartz,  positive  SiOSi  and  by  dependence  bending frequency  the  fused with for  c-  33  Figure 6. A comparison of the p o t e n t i a l energy curves f o r HgSi20^ plotted as a function of the bridging distance, d(Si-O^), a t 1 bar (upper curve) and 140 kbar (lower curve), where H e 2 H g S i 0 i s t h e ?  7  high p r e s s u r e phase. -1090.5170  .5171  .5172  .5173  H  k = 743 Nm s  4  d -1097.0968 H  LU .0969  .0970 H  .0971 - 1  k = 9 1 2 Nm s  .0972 1  1.54  1.55  1.56  1.57  1.58  d(Si-O) (A) b  1.59  1.60  1.61  34  F i g u r e 7. A comparison o f t h e p o t e n t i a l energy c u r v e s f o r E^S±20j p l o t t e d as a f u n c t i o n o f t h e S i O S i a n g l e a t 1 b a r (upper c u r v e ; and 140 kbar (lower c u r v e ) , where H e H S i 0 i s the h i g h p r e s s u r e phase.. .. . b d / o  c  o  7  -1091.5172  .0974  1  1 1 130.0  1  1  1  1 1 135.0  1  1  1  1 1 140.0  ASiOSi (deg)  1  1  1  j 145.0  Figure 8. Symmetric Si-0 stretching force constant, k , plotted against the SiOSi angle at 1 bar ( l e f t ) where'k »0.038(SiOSi)+l.941, r2=0.97, and 140 kbar (right) where k =0.040(SiOSi)+3.964, r2=0.93. g  s  g  q u a r t z a l s o shows a p o s i t i v e pressure  dependence  reflects The  t o one  i n d i c a t e that compression  change i n S i O S i a n g l e  glass  i s t h e most p r o m i n e n t e f f e c t o f  i n our c a l c u l a t i o n s , narrowing of  high-pressure  140 k b a r  Between  for a  shown  that  the  a  and  Levien  one  b a r a n d 61.4 k b a r , L e v i e n e t a l . (1980)  decreased  experiments  collapsed  ideally  framework  cooperative t i l t i n g tetrahedra  of  (thereby  comparative  in  d(Si-Ofc )  Jorgensen  vectors.  The r e a s o n  smaller  SiOSi angle  by  (1978)  the  high-pressure  With  an i n c r e a s e i n  reducing  molar  volume)  In helium  for this  by  a  t e t r a h e d r a i n s u c h a way t h a t  remain u n d i s t o r t e d ; the S i O S i a n g l e ,  imposed  while  c o r n e r - l i n k e d t e t r a h e d r a c a n be  of the r i g i d  significantly.  is  performed  0.3%  study, a  i n the bridging angle.  under h y d r o s t a t i c c o n d i t i o n s . the  stress  In t h i s  i n a 0.3% d e c r e a s e  a _ l . (1980)  pressure,  reduced  6.6%.  resulted  4.5% d e c r e a s e e_t  major  on t h e s t r u c t u r e i s t o c l o s e down t h e S i O S i  i n c r e a s e o f 60 k b a r and  Recent  (Jorgensen,  t h a t t h e a v e r a g e S i - 0 bond l e n g t h d e c r e a s e d SiOSi angle  pressure  1.3%.  s t u d i e s of a-quartz  L e v i e n et. a l . , 1980) h a v e a l s o  angle.  7.0%  while d ( S i - 0 ) decreases  crystallographic  e f f e c t of p r e s s u r e  is  of  ( F e r r a r o e t a l . , 1972).  increment  the  primarily  n e t w o r k c h a i n s c a u s i n g t e t r a h e d r a t o move c l o s e r  pressure  found  frequency  The  place  The  the  for this  pressure.  takes  another  1978;  noted  with  t h e change of t h e S i O S i a n g l e , l i n k i n g t h e t e t r a h e d r a .  results  along  dependence  our  calculations,  atoms  placed  intrinsic  with increased pressure  a  however, directed  along the S i - 0  preference  for a  i n the He H Si 07 2  6  2  molecule  i s not  increasing  apparent.  pressure  oxygen  and  silicons silicon  .  Likewise  and  to  i s of  favor  preference  for  (Table  when p r e s s u r e  the  smaller  I I I ) , i t can  bar  and  140  trend  when  with  -sec(SiOSi).  the  valence  variation. I I I ) would  increasing  pressure.  g r o u p shows  an  angles unrelated  be  of H S i 0 2  7  oxygen  of  (Figure 9).  character, investigate  SiOSi  latter  to  expressed  the  it  can  SiOSi  and ; )  )  but  be  angle  of  k  a  at  £  have  curvilinear  is linearly correlated be  related  the b r i d g i n g  to  oxygen  orbitals  on  f o r m s+X.p where X. i s t h e  s-p  shown  that  determines  2  pressure  in (1980)  I f the h y b r i d  1 0 0 / O + X . ) , of e a c h h y b r i d how  shows  and  0 results in  Gibbs  o r b i t a l s on  i n the  increasing  increase  increase  c o r r e l a t i o n can  (Brown e t a l . , 1 9 6 9 ) . are  An  b e t w e e n S i and  n(Si-Oi  the v a l e n c e  coefficient,  furthermore,  that  with  c o r r e l a t e d with. d(Si-Ob)  Newton  against The  change  a concommittant  kbar. bar  plotted  hybridization  mixing  bridging  significant  SiOSi  no  remains constant  d e m o n s t r a t e d a t one  the  the  c h a r g e s on  angle with  exhibits  a s h o r t e r bond l e n g t h and  6  overlap  c h a r g e s on  gross  e l e c t r o n i c overlap population  one  bond  considerations.  pressure s  net  with  seen i n T a b l e I I I .  of n ( S i . . . S i ) ( T a b l e  a wider SiOSi  Although n(Si-Ob)  k  be  i n t e r e s t , however, t h a t the m o l e c u l a r  intrinsic volume  values  adjustments  Mulliken  o x y g e n show no  increasing negative  tend It  no c h a n g e i n t h e  n ( S i - O b ) as w e l l a s t h e  o r b i t a l s of The  electronic  a r e m i n i m a l as can  There i s e s s e n t i a l l y population  The  affects this  X. =-sec ( S i O S i ) ; 2  the p e r c e n t a g e  (McWeeney,  1979).  sTo  r e l a t i o n s h i p , nCSi-O^)  Table I I I . M u l l i k e n bond o v e r l a p p o p u l a t i o n s , n ( S i 0^) a n d n ( S i . . . S i ) , a n d a t o m i c charges on bridging oxygen, Q(0b), and s i l i c o n , Q ( S i ) , f o r H S i 0 at 1 b a r , 6 0 k b a r and 140 k b a r ; b r i d g i n g S i - 0 bonds and SiOSi angle are optimized. 6  (kbar)  2  n(Si-O^)  n (Si ...Si)  Q(O )  Q(Si  +0.50  -0 .058  -0.70  1 .57  60  + 0.50  -0 .060  -0.70  1 .58  1 40  + 0.50  -o .062  -0.71  1 ...59  1x10"  3  b  7  39  Figure 9. Mulliken bond overlap population, n(Si-O^), plotted against the bridging distance, d(Si-Ob), at 1 bar (a) and against the symmetric stretching force constant, k , at 1 bar (b) with r values of 0.997 and 0.989, respectively; the corresponding relationships at 140 kbar are found in (c) and (d) with r values of 0.999 and 0.971, respectively. 2  s  2  —1 1.55  1  d(Si  1 1.57  1  - O ) b  1 1.59  I  1  I  900  1  K  1  1  1  (Nm ) - 1  s  1 1000  40  v a l u e s were p l o t t e d a g a i n s t 140  kbar  ( F i g u r e s 10a  are c u r v i l i n e a r plotted  .  against  On  140  The  other  hand,  percentage  pressures  (Figures  correlation  Newton  increasing  closely  related  and  SiOSi,  Si-0  Gibbs  the  and  bridging  linear  bond  et  al. ,  relationship fails would not with  relation constant  1980;  1 bar;  the  changing  to hold for  pressure.  To  relationship pressure. predictions  well-  r =0.997 at 2  Figure  11  (Gibbs  At  is  bond it  atmospheric  that a s i g n i f i c a n t  linear  —sec.(SiOSi) .  With  decreases.  i n c o e s i t e are p l o t t e d  et a l . , 1977), a  It  and  has  been  well-  increasing pressure. to hold  for  r a t h e r , one lengths  in  be  bar  we  a  structure or  an  - s e c ( S i O S i ) a t an results  bond the at  elevated of  the  elevated  confirming  linear correlation  this  However,  given  undertook a • study  the  that  would expect a  the  suggested  P r e w i t t , 1981)  one  on  When  2  d ( S i - O ^ ) and presents  the  (r =0.96) i s obtained with  pressure;  investigate this, between  above  of t h e h y b r i d o r b i t a l s  lengths  relation  whether  the  d(Si-Ob )  Levien  a l l  t$  and  angles.  to hold with  expect  pressure  bar  correlation  s h o r t bonds i n v o l v i n g w i d e (Levien  I0d),  ( 1 9 8 0 ) have f o u n d t h a t a  s-character  a g a i n s t - s e c ( S i O S i ) a t one developed  was  of t h e b r i d g i n g  -sec(SiOSi).  t h e b r i d g i n g o x y g e n i n c r e a s e s and observed  pressures  obtained.  c o r r e l a t i o n e x i s t s between d ( S i - 0 | o )  length  10b  b a r ; and  n(Si-0 ^ )  s-character  r e l a t i o n s h i p b e t w e e n d ( S i - O b ) and  one  when  2  pressure,  a t one  trends at both  l i n e a r c o r r e l a t i o n s (r =0.996 at  k b a r ) were A  10c).  the  the  o x y g e n a t t h e two developed  and  the b r i d g i n g angle  our  e x i s t s at a  41  F i g u r e 10. M u l l i k e n bond o v e r l a p p o p u l a t i o n , n ( S i - 0 , ) , p l o t t e d a g a i n s t t h e b r i d g i n g S i O S i a n g l e a t 1 bar (a) and a g a i n s t t h e p e r c e n t a g e s c h a r a c t e r o f t h e h y b r i d o r b i t a l s on t h e b r i d g i n g o x y g e n , 100/(1+ ) , a t 1 bar (b) w i t h t h e c o r r e s p o n d i n g r e l a t i o n s h i p s a t 140 k b a r f o u n d i n ( c ) and ( d ) . The c u r v i l i n e a r t r e n d s i n (a) and (c) b o t h become l i n e a r i n (b) and ( d ) . 2  Figure 11. The relationship between the bridging Si-0 distance and -sec(SiOSi) for H,Si„0-, at 1 bar and an elevated pressure estimated to be 140 kbar. 6 27 1.7  # 1 bar  (r -0.97) 2  ® Elev. Pressure  o<  1.5 2.0  1—;  1  1  1  1  1.9 :  1.8  1.7  1.6  1.5  1  1.4  (r  2 s =  0.96)  1 —  1.3  1.1  1.0  -secZ_SiOSi  no  43  given high pressure R e c e n t work coesite this  at  ( r = 0 . 9 6 ) as w e l l as 1 bar  on  the  structure  high pressure  finding.  (r =0.97).  2  and  2  compressibility  ( L e v i e n and P r e w i t t ,  1981).supports  When t h e a v e r a g e S i - 0 b r i d g i n g bond l e n g t h s a r e  p l o t t e d a g a i n s t - s e c ( S i O S i ) a t 51.9 k b a r , a s i g n i f i c a n t correlation for  (r =0.90) i s found.  6  2  a t one b a r and 60  7  experiment  and t h e o r y  Estimates  of  kbar.  pressure. decrease  The  agreement  pressure corresponding  Levien  from  et  143.7° The k  approximated  that  a l . (1980)  a v  to  A-C  by  in  134.2°  A x value keeping  d e c r e a s i n g S i O S i from  occur have  in noted  Figure  144° 13.  for  The  value  data  between  corresponding  of  to  d(Si-0|o ) constant t o 134°. Path  140 k b a r  c-quartz a very in  with slight  the  SiOSi  an i n c r e a s e o f 61.4 61.4  kbar  in H Si 0 6  kbar  2  7  was while  Diagrammatically this B-C  shows  s i g n i f i c a n t Ax a s s o c i a t e d w i t h a c h a n g e i n kbar.  data  t o kg^Ax t e r m s were  i n t h e mean S i - 0 d i s t a n c e and a s h i f t  pressure.  path  the  i s encouraging.  o b t a i n e d by m o d e l l i n g c h a n g e s  angle  F i g u r e 12 c o m p a r e s  2  linear  c o e s i t e a t one b a r and. 51.9 k b a r w i t h t h e c a l c u l a t e d  for H S i 0  of  for k  a y  is  that there i s a  pressure  of  61.4  A x u s e d i n many o f t h e  p r e c e d i n g c a l c u l a t i o n s was e s t i m a t e d by e x t r a p o l a t i o n f r o m t h e 61.4 k b a r  value.  Figure 12. A comparison between the average Si-0 b r i d g i n g distance p l o t t e d against -sec(SiOSi) for c o e s i t e ( l e f t ) and H S i 0 ( r i g h t ) ; at 1 bar and 52 kbar, the r values f o r coesite based on experimental data from Levien and Prewitt (1981) are 0.97 and 0.90, r e s p e c t i v e l y ; the r values based on c a l c u l a t i o n s at 1 bar and 60 kbar f o r HgSi207 are 0.97 and 0.98, r e s p e c t i v e l y . 2  6  2  2  7  Figure 13. I l l u s t r a t i o n of how estimates of k A x roughly equivalent to 60 kbar pressure were obtained. Modelling changes that occur i n ' -quartz at t h i s pressure, d(Si-Ob) was kept constant while decreasing the SiOSi angle from 144° tb 134° (path A-C); path B-C shows the x associated' w i t h an increment of 60 kbar pressure. a V  -secASiOSi  VI.  Molecular o r b i t a l upon  quantum  theory  mechanical  mineralogical electronic  CONCLUSIONS  studies  charge  is a  principles  of  To  formalism  molecular  s p e c t r a and f o r c e  these  studies  have  the ever  increasing  date,  t o one a t m o s p h e r e p r e s s u r e .  interest  i n u l t r a - h i g h p r e s s u r e phases and mantle  can  s t u d i e s of m o l e c u l a r  be  an  invaluable  With  a i d to  understanding  such  studies w i l l  enable  us  to  beyond t h e l i m i t s of c u r r e n t e x p e r i m e n t a l This Si-0  investigation  i s r a t h e r crude directed  Although  stress  axial  pressure  spectra.  simulate  For  to  pressures  technology.  t o t h e study of e q u i l i b r i u m  Furthermore, -sec(SiOSi),  atoms  reasonable  with  are  used  to  apply  a  approximations  of  expected  e x a m p l e , w i t h i n c r e a s i n g p r e s s u r e t h e S i - 0 bond  60 k b a r p r e s s u r e w h i c h  decrease  In  w i t h t h e S i - 0 b r i d g i n g bond l e n g t h , we  l e n g t h and S i O S i a n g l e d e c r e a s e  6.6%  high  t h e method o f a p p l y i n g p r e s s u r e  i n that helium  the r e s u l t s are  trends.  up  pressure  bond l e n g t h s , S i O S i a n g l e s a n d S i - 0 f o r c e c o n s t a n t s  increasing pressure.  feel  i s devoted  been  mineralogy,  groups a t s i m u l a t e d high  c r y s t a l c h e m i s t r y , bond e n e r g e t i c s a n d e l e c t r o n i c addition,  geometry,  electronic  limited  bonding  based  a n d h a s been a p p l i e d t o  equilibrium  distributions,  constant c a l c u l a t i o n s .  bonding  observed the known  0.3% a n d 4.5% ,  c o m p a r e s w e l l w i t h t h e 0.3% a n d  i n c-quartz  (Levien  linear correlation to  occur  respectively,  at  one  e_t  a_l. ,  1980).  o f S i - 0 bond l e n g t h a n d atmosphere,  holds  at  increased high  pressure;  constants  Si-0  estimated  the  relative  pressure decrease  Experimentally constants  Pyrex,  pressures Si-O-Si  up  the  a  t o 58.8  with  of  dependence glasses;  the  frequency  pressure  SiOSi  positive  primarily  force constant Although represents  this the  has  constants,  1  in  polyhedral  bulk  Ferraro  (1972)  general,  have  silica,  glasses  at  a  positive  a corresponding  increase  angles  the mixed shows  the  dependence  a  bending positive  sodium noted  change i n the  silicate for  this  SiOSi  angle  SiOSi  bending  pressure.  focused  on  the H S i 0 6  geologically  we moduli,  are Kp,  7  cluster,  of  studies  2  installment in a series  and  with  bands a t t r i b u t e d t o  Similarly  the  to  force  fused  important  Work i s c u r r e n t l y i n p r o g r e s s  tetrahedra  up  angle.  sparse.  i n d i c a t e s t h a t the  c o m p r e s s i b i l i t i e s of  Hg-AlO/,"  are  f o r a - q u a r t z and  and  1:6  bending  sodium s i l i c a t e  OSiO  pressure  initial  and  Manghnani  indicating  reflects  force  SiOSi  c-quartz,  show,  and  study  the  absorption  i s increasing with  oxygen p o l y h e d r a . and  of  The  vibrations  ( F e r r a r o e t a l . , 1972)  the  and  variety kbar.  bending r a t i o of  and  stretching  s p e c t r a of  with pressure  the  i n the  d(Si-Ob )  s t r e t c h i n g force constant.  frequency  on  i n c o e s i t e at  kbars which i s i n keeping  Ferraro  infrared  and  stretch  dependence  140  SiOSi  at high pressure  and  the  Vycor  of  determined  a l . (1972)  and  increase  in  in silicates  investigated  it  i s a l s o observed  stretching  show a p e r c e n t a g e  an  in  trend  pressures. Symmetric  et  this  on  the  calculating as w e l l a s  the  metalH SiO„ 4  force first  d e r i v a t i v e o f Kp w i t h r e s p e c t t o p r e s s u r e , work  will  be  determinations and  silicon  devoted  to  f o r oxyanion  force clusters  d(Kp)/dP.  constant, of  in octahedral coordination.  Kp and d ( K p ) / d P  magnesium,  c o m p u t e d Kp and b e n d i n g  aluminum  U l t i m a t e l y we hope t o  a p p r o x i m a t e t h e b u l k modulus of a s o l i d phase a t h i g h through  Future  force constants.  pressure  REFERENCES  B i n k l e y , J . S . , R. W h i t e s i d e , P.C. H a r i b a r a n , R. Seeger, W.J. H e h r e , W.A. L a t h a n , M.D. N e w t o n , R. Ditchfield, and J.A. 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