- Email: [email protected]
Charge densities and bond orders for zero differential overlap wavefunctions: Comparison with ab-initio calculations and diffraction results
Volume,JX,,nu&G
.
,:
1
‘.
:
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.I :
,'
:. ...
.;.
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‘.
.
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’ . ‘.15 September 1971
_: .’ ‘. ,_ : ‘..
.’
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1. ,..
C!KEMICALPHYSI~CSLETTERS .,_
.
~~GEDENs~~SANDEO~D;DRDERSFOR:
:
~EROD.~E~TIGLOvlERLApWA~F~r?7QNS:
'.
..
'._J.W. McIVER Jr,, P. COPPk and D. N&AK ,Depvtmen~ ofChemistry,State Univerdy of New York it Buffdo, Buffalo; Ne,w York 14214,USA :>
Received 9 July 1971
:
~I.
’Z&differential overlap calculations are compared with minimal basis set ab-initio calculations in order to assess their validity in the i.r,~terpretation of experimental electron densip maps. Various indices of bond character and thermally averaged electron density maps are used as criteria of comparison. It is wnciuded that charge distributions obtained from the ZDO methods will be valuable in the interpretation of experimental results,
The,development of X-ray cry&liographic refmement techniques is rapidly approaching the stage where ‘the distribution of electrons in the bonding regions of moiecules can be measured experimentally. In partic,. ular, i technique developed by Stewart [ 13 and Coppens [2] is gaining recognition as a tool for the refinemknt of X-ray diffraction data. This method, ‘,which involves the fitting
of structure
facto=
(1) I’where the Pi+ are “population coefficients”’ and the x8+) are atomic orbitals. These orbit& are u&ally taken as a minimal &t of Hartree-Fock or Slater-type atomic orbitals wh&h are least-squares fitted to a Iinear combination of&Man functions. in pririciple the identification of and refative strengths (ii terms of number of electrons) of the bonds in the molecule can be obtained by the pdpulation analysis [3] procedure, which partitions the number of electrons, via the normalization condition into contributions from net atomic populztions& and overlap populationspAh, ‘N=
c A
atoms atoms pAN+
F.
gA
PA*,
;
:
.A A p~=~~pii, i /
(3)
and
Fourier-transformed atomic orbital products, gives the resulting one-electron density function as
atoms
where
‘- m
wh:re Sij is an element of the atomic orbital’overlap matrix. Total, or gross, atomic populations pA are obtained as,
The form of eq. (1) for the exberime&l charge density distribution and the population analysis technique used to obtain information about the bonds naturally invites comparison withcharge densities &lculated by quantum mechan.icaI methods. The first order density mat& then plays tie role of P’. For rninind basis set molecular orbital methods which, include overlap, such as ab-initio and,extended Hiickel methdds, the compa&on presents no great difficulty __ since the one electron density function is obtained in thC same form as eq.‘( 1) with the normalization con$tiqn of eq. (2). However, the, relatively large size of , .,
:’
_. ‘.
.,-
._
“’
:
,‘,.
‘_
-Volume 11, number 1
&MICAL
PHYSICS LETl-ERS
15 September 1971
many molecules studied experimentally by X-ray diffraction precludes, at this time, the use of ab-initio methods; and the extended Hickel methods are not
expected to give a proper account of charge pokization effects due to the omission of eIectron repulsion
terms. The zero differential overlap @DO) methods, such as CNDO/2 [4] and INDO [.5] , on the other hand can be easily applied to large molecules, and would be expected to give a reasonable accounting of the charge density distribution, primarily because of their success in correlating a number of molecular properties ‘mch as dipole moments [6] with experimental results. The ZDO methods, however, satisfy the normahsation condition, atoms
N=TrP=
c
atoms A qA=
c A
A
cPii, i
(6)
where Pii is an element of the fust order density matrix calculated by the ZDO method and qA represents the charge density on atom A. Since overlap populations cannot be extracted from eq. (6), the results of ZDO calculations cannot be directly compared with the experimentaI X-ray results and the identification of bonds is not straightforward. We have thus undertaken a limited numerical study of some alternative. simple methods of corrkting
atoms
atoms-’
c
qT+
A ,.’
where
.,
q&3=$+‘;It may be noted that the total atomic charge obtained by this procedure is just qA of eq. (6). (ii) The MuIIiken overlap popuIation pAB obtained from the assumption [4] that the ZDO wavefunctionis equivalent to an ab-initio wavefunction obtained with a Lijwdin-orthogonalized basis set [IO]. L’nder this assumption the density matrix P’corresponding to the original basis set is related to the ZDO P by the Liiwdin transformation P’= S-r”PS-L’2. If the matrix P’rather than P is used, then the charge density distribution is in the form of eq. (I) and the analysis of eqs. (2)-(S) CXI be used to obtain PAB. (iii) The bond order bAB used by Ehrenson and Seltzer [ 1 l] . These authors have reported extensive study of bond strengths in hydrocarbons using the CNDG/2 method and they con&& that the quantity
this difficulty.
INDO molecular orbital and ab-initio calcuIations were performed on the hydrocarbons ethane, ethylene, acetylene and butadiene, using the experimental geometry [7]. The ab-initio calculations were taken to represent a minimal basis set of Slater-type orbit& by the STO-3G approximation of Hehre et al. [8]. The two sets of caIcuIations were compared by evaluating three previously defmed indices of bonding which are applicable to ZDO wavefunctions: (i) The bond index qAB used by Wiberg [9]. This quantity can be regarded as an “overlap” population which is obtained by a population analysis of the ZDO idempotency condition P = f PP. Thus,
N=TrP=$TrPdP=
and
.‘.
c B>A
--. qAB,
(7) .
bAB =
5i 5j Pii sii
is a valid and useful index of bond strength. The results of our similar study using the LNDO method differ only negligibly from those of the CND0/2 calculations and they wiIl therefore not be reported here. Although similar in form to the overlap population of eq. (4), GAB is different in two respects. In the first place the ZDO density matrix P rather than P’is used is obtained from a partitioning of and S%OIldy bAS the molecular energy and is thus not formally related to the charge density. In this latter respect bAB ‘resembles the- bond order often used in pi-electron theory. The results for the .hydrocarbons are given in table 1. As seen from table 1, aU of the indices for the carbon-carbon btinds for the MD0 c’aIcu1arion show a substantial correlation with one another and with the overlap populations of the ab-initio calculations. In 83
Volume Il,nurnber
1
:,
CHBMICALPHYS&S LETTERS
Caldied
&anti&
be
4c!c
PCC
4;
ethane
2.90
butadiene c-c* butadiene
2.18
0.72 0.80
0.53 ., 9.53
0.66 0.89
3.95 ‘4.00
2.54.
I.10
0.97
1.28
,3.96
1.14 1.44 ..
i.oi . ‘: 1.49"
1.33 1.71
4.00 4.06
,. 2.52 2.28 .,
’ k&bne
-.
"
‘: 1 iS September 1971 :’
.. =p&l+$j, : ., 1. ., &I a number of C-C tend feogths. $&$s.
RCC (au)
C&* ‘ctli&oe~
‘.
: :. 1. i as deEned ir@xt
:
;
,, i’:
PE..
.PCy:
4.02 .4.05.
0=70 ‘0.84,.
4.37 x 4.27
,3.98’
1.20
4.14
4.04 pm ..
1.22’ 1.7k
4228 4.18,
;
PE
:
a):The ab-mitio charge de&ties were equitiitated with the Ml30 results by ekninating the density matrix elements corres$mding to 1s orbit& which we Pot use$ in INDO. _’ ;
, ., 2.3
2-4
2.5
2.6
2.7
2.8
2.9
3.0
R(AU.1
F$g. 1. INDO bond order GAB (a), bond POPu~tiOn PAB ix)' andabinitio bond population (0) versus kxnd length for car: .. bon-carbon bonds in wrne bydroca+ns . . ._ : _&,,..‘.
‘,.,
.’
..‘.
Fig. 2. Theoretical density mapJ’in the plane of the cyanuric acid ino@cuIe(a) a-i& to STQ-3G ab-iitio calculations, ($4 according to Liiwdin-ksfon~~ed LNDO caIcuIatioqr _-Both’maps are thermal& averaged with thi tempe.$ure paritieters of,a 10S3K neutron diffraction ka&sis. The other .’ ‘hiIf of the cyanuri$ acid m&cull is symmetiy related to the .._ p&t*ow& __ ._’ : ,._.
particular the overlap populations obtained from the Lijwdin transformed INDO density matrix agree fairly welI with those of the ab-initio method; a notabie exception being ethane for which the overlap populations of the two methods differ by 0.16 electrons. The Wiberg bond index.q& gives the poorest correlation with the bond lengths, the same value being obtained for the C-C single bond in both ethane and butadiene for example. The bond population for the INDO calculation correlates better, but in this series. of molecules not quite as good as the bond order bAB used by Ehrenson and Selbzer (fig. 1). The result is reminiscent of the bond-length-bond-order relationships obtained in pi-electron theory [ 121. Itis worth noting that the most linear correlation with the bond
lengths is obtained’for the abinitio bond population pee. Such correlations are of considerable interest for the interpretation of experimentat electron density maps. However, the most detailed analysis of the experimental iesults as compared with those obtained from D?DO and abinitio calculations.is not a comparison ’ of Mull&en populations, but rather a comparison of the charge density distributions themselves. To determine whether the MD0 matrix P’ as defied above is adequate for this purpose we have obtained p(r) in the plane of the molecule of cyanuric acid from eq. (1). The INDO and ab-initio maps shown in fig. 2 are difference density maps in which neutral spherical atoms have been subtracted. They have been evaluated by calculating the X-ray structure factors for the actua! unit cell of cyanuric acid, corresponding to the theore-
tical values of P’and with a thermal smearing function as determined experimentally by neutron diffraction at liquid-nitrogen temperature [ 131. The maps are the Fourier transforms of these thermally-smeared theoretical
structure
lS,Septemker 1971
CHEMICAL PHYSICS LBTIERS
Volume 11; number 1
factors
and they
are the only
which cun be realistically compared with the experimental densities As is seen by a comparison of the two figures, the contour maps are very similar in overall topology, pa.rticuIarly in the atomic regions. The STO-3G calculation appears to give slightly larger density in the bonding region. Itis worthwhile mentioning that the experimental density differs from both maps primarily in the amount of charge being somewhat smaller in the lone pair regions, and larger
firnctions
in the bonds, as wiIl be discussed in.i$bsequent pub-* lication. We conclude from this study that charge density distributions calcutated by the iND0 method, with the assumption that the results are for a Lowdin-orthogonad abinitio caIcuIation, give a reasonable representation of the charge distribution, which can be compared with experimentally obtained results when more sophisticated calculations are not feasible. We thank Professor Robert F. Stewart for the STO-3G results for cyanuric acid and Mr. D. Pautler for the computation of *Lhecyanuric acid difference maps. We also *hank the donors of the Petroleum Research Fund aiiministered by the American Chemical Society for support given to I.M.‘and P.C. Computer time was generously donated by the Computing Center of the State University of New York at Buffalo.
References 111R.F.Stewart, J. Chcm. Phyr 50 (1969) 2485. 121P.CoppenS T.V.Wfloughby and L_N.Csonka, Acta Cryst., to be published; P.Coppeas, DSautk znd JGriffii, I. Am. Chem. Sot. 93 (1971) 1051. 23 (19.55) 1833. I31 R.S.MuUiken,J.Chem.Phys. I41 J.A.Pople and G.i%.Segal, I. Chem. Phys. 43 (1965J S136. ISI J.A.Pople, D.B.Bweridge and P.A.Dobosh, 3. Chem. Phyr 47 (1967) 2026. I61 J.A.Pop!e and M.Cordon, I. Am. Chem. S-c. 89 (1967) 42.53.
171 L.S.BarteU and H.K.Hig@nbotham,
I. Chem. Phys. 42 (1’365) 851; W.Haugen and M.Traettebeq, Acta Chem. Stand. 20 (1966) 1726; L.S.BarteU, E.A.Roth, CD.HoUoweU. K.Kuchistsu and J.E.Young Jr., I. Chem. Phys. 42 (L96Sj 2683; J.HI.Womon and BPStoicheff. Cm_ J_ Phys. 35 (1957)
373. I81 WJ.Hehre, RE.Stcwart and J.ASople, J. Chem. Phys. Sl(L969)
2657.
I91 K.B.WZ+zg, Tetrahedron 24 (1968) 1083. r101 P.O.L~wdin, J. G-v&. Phys. 18 (1950) 365. I111 S.Ehtenson and S.SeItzer. Thipret. Chim. Acta 20 (1971) 17. I121 CA.Coulsan. J. Phys Chem. 56 (1952) 363. I131 P.Coppens and A.Vos, Acta Cryst. 927 (197 1) 146; D.S.Jones. D.PautIer aud P.Copgens, to be published.
.
85
.
,:
1
‘.
:
_’ :: 1’
.I :
,'
:. ...
.;.
-. ,_
‘.
.
‘.
:
’ . ‘.15 September 1971
_: .’ ‘. ,_ : ‘..
.’
.: : ..
1. ,..
C!KEMICALPHYSI~CSLETTERS .,_
.
~~GEDENs~~SANDEO~D;DRDERSFOR:
:
~EROD.~E~TIGLOvlERLApWA~F~r?7QNS:
'.
..
'._J.W. McIVER Jr,, P. COPPk and D. N&AK ,Depvtmen~ ofChemistry,State Univerdy of New York it Buffdo, Buffalo; Ne,w York 14214,USA :>
Received 9 July 1971
:
~I.
’Z&differential overlap calculations are compared with minimal basis set ab-initio calculations in order to assess their validity in the i.r,~terpretation of experimental electron densip maps. Various indices of bond character and thermally averaged electron density maps are used as criteria of comparison. It is wnciuded that charge distributions obtained from the ZDO methods will be valuable in the interpretation of experimental results,
The,development of X-ray cry&liographic refmement techniques is rapidly approaching the stage where ‘the distribution of electrons in the bonding regions of moiecules can be measured experimentally. In partic,. ular, i technique developed by Stewart [ 13 and Coppens [2] is gaining recognition as a tool for the refinemknt of X-ray diffraction data. This method, ‘,which involves the fitting
of structure
facto=
(1) I’where the Pi+ are “population coefficients”’ and the x8+) are atomic orbitals. These orbit& are u&ally taken as a minimal &t of Hartree-Fock or Slater-type atomic orbitals wh&h are least-squares fitted to a Iinear combination of&Man functions. in pririciple the identification of and refative strengths (ii terms of number of electrons) of the bonds in the molecule can be obtained by the pdpulation analysis [3] procedure, which partitions the number of electrons, via the normalization condition into contributions from net atomic populztions& and overlap populationspAh, ‘N=
c A
atoms atoms pAN+
F.
gA
PA*,
;
:
.A A p~=~~pii, i /
(3)
and
Fourier-transformed atomic orbital products, gives the resulting one-electron density function as
atoms
where
‘- m
wh:re Sij is an element of the atomic orbital’overlap matrix. Total, or gross, atomic populations pA are obtained as,
The form of eq. (1) for the exberime&l charge density distribution and the population analysis technique used to obtain information about the bonds naturally invites comparison withcharge densities &lculated by quantum mechan.icaI methods. The first order density mat& then plays tie role of P’. For rninind basis set molecular orbital methods which, include overlap, such as ab-initio and,extended Hiickel methdds, the compa&on presents no great difficulty __ since the one electron density function is obtained in thC same form as eq.‘( 1) with the normalization con$tiqn of eq. (2). However, the, relatively large size of , .,
:’
_. ‘.
.,-
._
“’
:
,‘,.
‘_
-Volume 11, number 1
&MICAL
PHYSICS LETl-ERS
15 September 1971
many molecules studied experimentally by X-ray diffraction precludes, at this time, the use of ab-initio methods; and the extended Hickel methods are not
expected to give a proper account of charge pokization effects due to the omission of eIectron repulsion
terms. The zero differential overlap @DO) methods, such as CNDO/2 [4] and INDO [.5] , on the other hand can be easily applied to large molecules, and would be expected to give a reasonable accounting of the charge density distribution, primarily because of their success in correlating a number of molecular properties ‘mch as dipole moments [6] with experimental results. The ZDO methods, however, satisfy the normahsation condition, atoms
N=TrP=
c
atoms A qA=
c A
A
cPii, i
(6)
where Pii is an element of the fust order density matrix calculated by the ZDO method and qA represents the charge density on atom A. Since overlap populations cannot be extracted from eq. (6), the results of ZDO calculations cannot be directly compared with the experimentaI X-ray results and the identification of bonds is not straightforward. We have thus undertaken a limited numerical study of some alternative. simple methods of corrkting
atoms
atoms-’
c
qT+
A ,.’
where
.,
q&3=$+‘;It may be noted that the total atomic charge obtained by this procedure is just qA of eq. (6). (ii) The MuIIiken overlap popuIation pAB obtained from the assumption [4] that the ZDO wavefunctionis equivalent to an ab-initio wavefunction obtained with a Lijwdin-orthogonalized basis set [IO]. L’nder this assumption the density matrix P’corresponding to the original basis set is related to the ZDO P by the Liiwdin transformation P’= S-r”PS-L’2. If the matrix P’rather than P is used, then the charge density distribution is in the form of eq. (I) and the analysis of eqs. (2)-(S) CXI be used to obtain PAB. (iii) The bond order bAB used by Ehrenson and Seltzer [ 1 l] . These authors have reported extensive study of bond strengths in hydrocarbons using the CNDG/2 method and they con&& that the quantity
this difficulty.
INDO molecular orbital and ab-initio calcuIations were performed on the hydrocarbons ethane, ethylene, acetylene and butadiene, using the experimental geometry [7]. The ab-initio calculations were taken to represent a minimal basis set of Slater-type orbit& by the STO-3G approximation of Hehre et al. [8]. The two sets of caIcuIations were compared by evaluating three previously defmed indices of bonding which are applicable to ZDO wavefunctions: (i) The bond index qAB used by Wiberg [9]. This quantity can be regarded as an “overlap” population which is obtained by a population analysis of the ZDO idempotency condition P = f PP. Thus,
N=TrP=$TrPdP=
and
.‘.
c B>A
--. qAB,
(7) .
bAB =
5i 5j Pii sii
is a valid and useful index of bond strength. The results of our similar study using the LNDO method differ only negligibly from those of the CND0/2 calculations and they wiIl therefore not be reported here. Although similar in form to the overlap population of eq. (4), GAB is different in two respects. In the first place the ZDO density matrix P rather than P’is used is obtained from a partitioning of and S%OIldy bAS the molecular energy and is thus not formally related to the charge density. In this latter respect bAB ‘resembles the- bond order often used in pi-electron theory. The results for the .hydrocarbons are given in table 1. As seen from table 1, aU of the indices for the carbon-carbon btinds for the MD0 c’aIcu1arion show a substantial correlation with one another and with the overlap populations of the ab-initio calculations. In 83
Volume Il,nurnber
1
:,
CHBMICALPHYS&S LETTERS
Caldied
&anti&
be
4c!c
PCC
4;
ethane
2.90
butadiene c-c* butadiene
2.18
0.72 0.80
0.53 ., 9.53
0.66 0.89
3.95 ‘4.00
2.54.
I.10
0.97
1.28
,3.96
1.14 1.44 ..
i.oi . ‘: 1.49"
1.33 1.71
4.00 4.06
,. 2.52 2.28 .,
’ k&bne
-.
"
‘: 1 iS September 1971 :’
.. =p&l+$j, : ., 1. ., &I a number of C-C tend feogths. $&$s.
RCC (au)
C&* ‘ctli&oe~
‘.
: :. 1. i as deEned ir@xt
:
;
,, i’:
PE..
.PCy:
4.02 .4.05.
0=70 ‘0.84,.
4.37 x 4.27
,3.98’
1.20
4.14
4.04 pm ..
1.22’ 1.7k
4228 4.18,
;
PE
:
a):The ab-mitio charge de&ties were equitiitated with the Ml30 results by ekninating the density matrix elements corres$mding to 1s orbit& which we Pot use$ in INDO. _’ ;
, ., 2.3
2-4
2.5
2.6
2.7
2.8
2.9
3.0
R(AU.1
F$g. 1. INDO bond order GAB (a), bond POPu~tiOn PAB ix)' andabinitio bond population (0) versus kxnd length for car: .. bon-carbon bonds in wrne bydroca+ns . . ._ : _&,,..‘.
‘,.,
.’
..‘.
Fig. 2. Theoretical density mapJ’in the plane of the cyanuric acid ino@cuIe(a) a-i& to STQ-3G ab-iitio calculations, ($4 according to Liiwdin-ksfon~~ed LNDO caIcuIatioqr _-Both’maps are thermal& averaged with thi tempe.$ure paritieters of,a 10S3K neutron diffraction ka&sis. The other .’ ‘hiIf of the cyanuri$ acid m&cull is symmetiy related to the .._ p&t*ow& __ ._’ : ,._.
particular the overlap populations obtained from the Lijwdin transformed INDO density matrix agree fairly welI with those of the ab-initio method; a notabie exception being ethane for which the overlap populations of the two methods differ by 0.16 electrons. The Wiberg bond index.q& gives the poorest correlation with the bond lengths, the same value being obtained for the C-C single bond in both ethane and butadiene for example. The bond population for the INDO calculation correlates better, but in this series. of molecules not quite as good as the bond order bAB used by Ehrenson and Selbzer (fig. 1). The result is reminiscent of the bond-length-bond-order relationships obtained in pi-electron theory [ 121. Itis worth noting that the most linear correlation with the bond
lengths is obtained’for the abinitio bond population pee. Such correlations are of considerable interest for the interpretation of experimentat electron density maps. However, the most detailed analysis of the experimental iesults as compared with those obtained from D?DO and abinitio calculations.is not a comparison ’ of Mull&en populations, but rather a comparison of the charge density distributions themselves. To determine whether the MD0 matrix P’ as defied above is adequate for this purpose we have obtained p(r) in the plane of the molecule of cyanuric acid from eq. (1). The INDO and ab-initio maps shown in fig. 2 are difference density maps in which neutral spherical atoms have been subtracted. They have been evaluated by calculating the X-ray structure factors for the actua! unit cell of cyanuric acid, corresponding to the theore-
tical values of P’and with a thermal smearing function as determined experimentally by neutron diffraction at liquid-nitrogen temperature [ 131. The maps are the Fourier transforms of these thermally-smeared theoretical
structure
lS,Septemker 1971
CHEMICAL PHYSICS LBTIERS
Volume 11; number 1
factors
and they
are the only
which cun be realistically compared with the experimental densities As is seen by a comparison of the two figures, the contour maps are very similar in overall topology, pa.rticuIarly in the atomic regions. The STO-3G calculation appears to give slightly larger density in the bonding region. Itis worthwhile mentioning that the experimental density differs from both maps primarily in the amount of charge being somewhat smaller in the lone pair regions, and larger
firnctions
in the bonds, as wiIl be discussed in.i$bsequent pub-* lication. We conclude from this study that charge density distributions calcutated by the iND0 method, with the assumption that the results are for a Lowdin-orthogonad abinitio caIcuIation, give a reasonable representation of the charge distribution, which can be compared with experimentally obtained results when more sophisticated calculations are not feasible. We thank Professor Robert F. Stewart for the STO-3G results for cyanuric acid and Mr. D. Pautler for the computation of *Lhecyanuric acid difference maps. We also *hank the donors of the Petroleum Research Fund aiiministered by the American Chemical Society for support given to I.M.‘and P.C. Computer time was generously donated by the Computing Center of the State University of New York at Buffalo.
References 111R.F.Stewart, J. Chcm. Phyr 50 (1969) 2485. 121P.CoppenS T.V.Wfloughby and L_N.Csonka, Acta Cryst., to be published; P.Coppeas, DSautk znd JGriffii, I. Am. Chem. Sot. 93 (1971) 1051. 23 (19.55) 1833. I31 R.S.MuUiken,J.Chem.Phys. I41 J.A.Pople and G.i%.Segal, I. Chem. Phys. 43 (1965J S136. ISI J.A.Pople, D.B.Bweridge and P.A.Dobosh, 3. Chem. Phyr 47 (1967) 2026. I61 J.A.Pop!e and M.Cordon, I. Am. Chem. S-c. 89 (1967) 42.53.
171 L.S.BarteU and H.K.Hig@nbotham,
I. Chem. Phys. 42 (1’365) 851; W.Haugen and M.Traettebeq, Acta Chem. Stand. 20 (1966) 1726; L.S.BarteU, E.A.Roth, CD.HoUoweU. K.Kuchistsu and J.E.Young Jr., I. Chem. Phys. 42 (L96Sj 2683; J.HI.Womon and BPStoicheff. Cm_ J_ Phys. 35 (1957)
373. I81 WJ.Hehre, RE.Stcwart and J.ASople, J. Chem. Phys. Sl(L969)
2657.
I91 K.B.WZ+zg, Tetrahedron 24 (1968) 1083. r101 P.O.L~wdin, J. G-v&. Phys. 18 (1950) 365. I111 S.Ehtenson and S.SeItzer. Thipret. Chim. Acta 20 (1971) 17. I121 CA.Coulsan. J. Phys Chem. 56 (1952) 363. I131 P.Coppens and A.Vos, Acta Cryst. 927 (197 1) 146; D.S.Jones. D.PautIer aud P.Copgens, to be published.
.
85