- Email: [email protected]
Icosahedral hybrid orbitals
14 April 2000
Chemical Physics Letters 320 Ž2000. 523–526 www.elsevier.nlrlocatercplett
Icosahedral hybrid orbitals Emilio Martınez ´ Torres ) Department of Physical Chemistry, UniÕersity of Castilla-La Mancha, 13071 Ciudad Real, Spain Received 20 December 1999; in final form 10 March 2000
Abstract A set of twelve equivalent icosahedral hybrid orbitals pointing from the centre to the corners of a regular icosahedron has been obtained. Such hybrids can be used to explain the geometry of twelve-coordinate complexes of a rare-earth atom. Using group theoretical considerations, it is shown that these hybrids can be constructed by linear combination of one s, three p, five d and three f-orbitals. Bearing in mind that the twelve hybrids have identical shape but are oriented differently in space, their mathematical expressions have been obtained by applying geometrical transformations to the sp 3d 5 f 3 hybrid pointing along the positive z-axis. In order to obtain elegant mathematical expressions, the x, y and z axes have been chosen to be coincident with three orthogonal binary axes of the icosahedron. q 2000 Elsevier Science B.V. All rights reserved.
1. Introduction Hybridization is an important concept in the conventional valence bond method w1x. Thus, the qualitative discussion of the process of mixing ‘pure’ atomic orbitals to obtain sp, sp 2 and sp 3 hybrid orbitals provides an intuitive justification for the geometry of some simple molecules, as Cl 2 Be, BF3 , CH 4 , . . . , whose shapes are different from those expected using s and p-orbitals separately. Hybrids involving the participation of d-orbitals, such as dsp 3 , d 2 sp 3, . . . , are also used to explain the shape of both molecules and transition-metal complexes w2–4x. Icosahedral symmetry is becoming of increasing interest to chemists. However, although twelve-coordinate complexes with icosahedral structure have been described in the literature for a long time, no hybrids
associated with this geometry have been described yet. The most common twelve-coordinate complexes are those containing six bidentate ligands: wMŽbidentate. 6 x. The regular icosahedron is expected to be the most stable structure for such complexes because it gives minor repulsion energy coefficients w5x. In fact, structures which approximate to a regular icosahedron have been observed for a number of hexanitrato complexes of the rare earths, such as wLaŽNO 3 . 6 x 3y, wThŽNO 3 . 6 x 2y, wNdŽNO 3 . 6 x 3y, etc. w6x, as well as for the praseodymiumŽIII. naphthyridine complex, wPrŽC 8 H 6 N2 . 6 x 3y w7x. Fig. 1 shows the chemically relevant structural isomers for such complexes.
2. Attainment of icosahedral hybrids
)
Fax: q34-926-295315; e-mail: [email protected]
The equivalent hybrids pointing to the twelve corners of a regular icosahedron form the basis for a
0009-2614r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 0 0 . 0 0 3 0 2 - X
524
E. Martınez ´ Torresr Chemical Physics Letters 320 (2000) 523–526 Table 1 Attainment of the icosahedral hybrid orbitals from c z
c 1 sC ywcz c4 s sy z c 1 c 7 s ic1 c 10 s sx y c 1
c 2 sC31 c 1 c5 s sx z c 3 c8 s ic2 c 11 s sx y c 3
c 3 sC31 c 2 c6 s sx z c 2 c9 s ic3 c 12 s sy z c 2
Fig. 1. Twelve-coordinate complexes with icosahedral structure.
reducible representation, G , of the icosahedral point group, I, which reduces into w8x
G s A [ T1 [ T2 [ H
Ž 1.
On the other hand, the reductions of the representations based on the s, p, d, and f-orbitals under the symmetry operations of the I group are w8x
Gs s A
in the I group, together with the high dimension of some of its irreducible representations, the use of such a method in the attainment of icosahedral hybrids requires laborious and careful work. Hence we will employ a simpler method that makes use of the fact that the icosahedral hybrids are equivalent; i.e. they have identical shape but are oriented differently in space. Therefore, the twelve icosahedral hybrids can be obtained starting from a sp 3 d 5 f 3 hybrid, cz , pointing along the positive z-axis:
Gp s T1 cz s
Gd s H Gf s T 2 [ G
Ž 2.
From Eqs. Ž1. and Ž2. we conclude that one s, three p, five d and three f-orbitals can be combined in twelve different ways to produce twelve sp 3 d 5 f 3 equivalent icosahedral hybrids. Note that the three f-orbitals suitable for the construction of such hybrids must be a basis for the T2 representation. The usual procedure to obtain explicit mathematical expressions for hybrids as linear combinations of atomic orbitals makes use of the projection operator technique. Such methods are described in standard textbooks on group theory w9x. However, because of the large number of symmetry operations contained
1
' ' '12 Ž s q 3 pz q 5 d z
2
q '3 f z 3 .
Ž 3.
In order to obtain elegant mathematical expressions, we have chosen three orthogonal binary axes of the icosahedron as coordinate axes Žsee Fig. 2.. Table 1 shows how the twelve icosahedral hybrids can be obtained from cz , where C yw is an anti-clockwise rotation by an angle w s siny1 w1r 'F q 2 x Žwhere F s Ž1 q '5 .r2 is the ‘golden number’. about the y-axis, C31 is an anti-clockwise rotation by 1208 about the axis passing through the center of the coordinate system and the point Ž1, 1, 1., i is the inversion, and sx y , sx z , sy z are reflections in the Table 2 Expressions for the atomic orbitals as functions of Cartesian coordinates neglecting invariant common factors w10x ss1 py s y d z 2 s Ž1r2. =Ž2 z 2 y x 2 y y 2 . d x y s'3 xy d y z s'3 yz f x z 2 s Ž'6 r4. =Ž4 z 2 y x 2 y y 2 . x f x y z s'15 xyz f yŽ3 x 2yy 2 . s Ž'10 r4. =Ž3 x 2 y y 2 . y
Fig. 2. Numbering and coordinate system for the icosahedron.
px s x pz s z d x 2yy 2 s Ž'3 r2.Ž x 2 y y 2 . d x z s'3 xz f z 3 s Ž1r2.Ž2 z 2 y3 x 2 y3 y 2 . z f y z 2 s Ž'6 r4.Ž4 z 2 y x 2 y y 2 . y f x Ž x 2yy 2 . s Ž'15 r2.Ž x 2 y y 2 . z f x Ž x 2y3 y 2 . s Ž'10 r4.Ž x 2 y3 y 2 . x
E. Martınez ´ Torresr Chemical Physics Letters 320 (2000) 523–526
525
Table 3 Icosahedral hybrids as linear combinations of atomic orbitalsa
c 1 s Ž1r2'3 . s q a p x q b p z q g d z 2 q r d x 2yy 2 q Ž1r2. d x z y b f x 3q3F y 1 x y 2 y3F x z 2 q a f z 3q3F y 1 z x 2 y3F z y 2 c 2 s Ž1r2'3 . s q b p x q a p y y d d z 2 q Ž1r4. d x 2yy 2 q Ž1r2. d x y q a f x 3q3F y 1 x y 2 y3F x z 2 y b f y 3 q3F y 1 y z 2y3F y x 2 c 3 s Ž1r2'3 . s q b p y q a p z y ´ d z 2 y s d x 2yy 2 q Ž1r2. d y z q a f y 3q3F y 1 y z 2 y3F y x 2 y b f z 3q3F y 1 z x 2 y3F z y 2 c4 s Ž1r2'3 . s y a p x q b p z q g d z 2 q r d x 2yy 2 y Ž1r2. d x z q b f x 3q3F y 1 x y 2 y3F x z 2 q a f z 3q3F y 1 z x 2y3F z y 2 c 5 s Ž1r2'3 . s y b p y q a pz y ´ d z 2 y s d x 2yy 2 y Ž1r2. d y z y a f y 3q3F y 1 y z 2y3F y x 2 y b f z 3q3F y 1 z x 2y3F z y 2 c6 s Ž1r2'3 . s q b p x y a p y y d d z 2 q Ž1r4. d x 2yy 2 y Ž1r2. d x y q a f x 3q3F y 1 x y 2y3F x z 2 q b f y 3q3F y 1 y z 2 y3F y x 2 c 7 s Ž1r2'3 . s y a p x y b p z q g d z 2 q r d x 2yy 2 q Ž1r2. d x z q b f x 3q3F y 1 x y 2y3F x z 2 y a f z 3q3F y 1 z x 2 y3F z y 2 c 8 s Ž1r2'3 . s y b p x y a p y y d d z 2 q Ž1r4. d x 2yy 2 q Ž1r2. d x y y a f x 3q3F y 1 x y 2 y3F x z 2 q b f y 3 q3F y 1 y z 2y3F y x 2 c 9 s Ž1r2'3 . s y b p y y a p z y ´ d z 2 y s d x 2yy 2 q Ž1r2. d y z y a f y 3q3F y 1 y z 2 y3F y x 2 q b f z 3q3F y 1 z x 2 y3F z y 2 c 10 s Ž1r2'3 . s q a p x y b p z q g d z 2 q r d x 2yy 2 y Ž1r2. d x z y b f x 3q3F y 1 x y 2 y3F x z 2 y a f z 3q3F y 1 z x 2y3F z y 2 c 11 s Ž1r2'3 . s q b p y y a p z y ´ d z 2 y s d x 2yy 2 y Ž1r2. d y z q a f y 3q3F y 1 y z 2y3F y z 2 q b f z 3q3F y 1 z x 2y3F z y 2 c 12 s Ž1r2'3 . s y b p x q a p y y d d z 2 q Ž1r4. d x 2yy 2 y Ž1r2. d x y y a f x 3q3F y 1 x y 2 y3F x z 2 y b f y 3 q3F y 1 y z 2y3F y x 2 a
as
The hybrid given by the function c i points to the ith corner of the icosahedron Žsee Fig. 2.. 1 2'F q 2
F , bs
2'F q 2
, gs
Fq1 4'3
, ds
'5 4'3
, ´s
2yF 4'3
coordinate planes. Making use of the expressions for the atomic orbitals as functions of Cartesian coordinates given in Table 2 w10x, it is easy to obtain the
, rs
Fy1 4
F , ss
4
.
expressions for the icosahedral hybrids shown in Table 3, where the following icosahedral f-orbitals have been used: f x 3y3F y 1 x y 2y3F x z 2 s y Ž F'6 r4 . f x z 2 q Ž '2 r4F 2 . f xŽ x 2y3 y 2 . f y 3y3F y 1 y z 2y3F y x 2 s y Ž '6 r4F . f y z 2 y Ž F 2'2 r4 . f yŽ3 x 2yy 2 . f z 3y3F y 1 z x 2y3F z y 2 s Ž 1r2 . f z 3 q Ž '3 r2 . f zŽ x 2yy 2 . ,
Fig. 3. Icosahedral hybrids viewed down a 5-fold axis.
Ž 4.
which are linear combinations of those obtained by Boyle et al. w11x. Figs. 3 and 4 show two views of the icosahedral hybrids.
3. Conclusions
Fig. 4. Icosahedral hybrids viewed down a 3-fold axis.
A set of twelve hybrid orbitals pointing from the centre to the corners of a regular icosahedron have been obtained. By group theoretical considerations it has been shown that these icosahedral hybrids can be constructed by linear combinations of one s, three p, five d and three f-orbitals. Since such hybrids have identical shape but are oriented differently in space, their mathematical expressions have been obtained by applying geometrical transformations to the
526
E. Martınez ´ Torresr Chemical Physics Letters 320 (2000) 523–526
sp 3d 5 f 3 hybrid pointing along the positive z-axis. The three coordinate axes have been chosen to be coincident with three orthogonal binary axes of the icosahedron.
References w1x L. Pauling, The Nature of the Chemical Bond and the Structure of Molecules and Crystals: An Introduction to Modern Structural Chemistry, 3rd edn., Cornell University Press, Ithaca, NJ, 1979. w2x R. McWeeny, Coulson’s Valence, 3rd edn., Oxford University Press, Oxford, 1979.
w3x R.L. De Rock, H.B. Gray, Chemical Structure and Bonding, 2nd edn., University Science Books, Mill Valley, CA, 1989. w4x F.A. Cotton, G. Wilkinson, Advanced Inorganic Chemistry, 5th edn., Wiley, New York, 1988. w5x D.L. Kepert, in: G. Wilkinson, R.D. Gillard, J.A. McCleverty ŽEds.., Comprehensive Coordination Chemistry, Vol. I, Pergamon, Oxford, 1987. w6x D.L. Kepert, Inorganic Stereochemistry, Springer, Berlin, 1982. w7x A. Clearfield, R. Gopal, R.W. Olsen, Inorg. Chem. 16 Ž1977. 911. w8x E. Martınez-Torres, J.J. Lopez-Gonzalez, M. Fernandez, ´ ´ ´ ´ Comp. Math. Applic. 26 Ž1993. 67. w9x F.A. Cotton, Chemical Applications of Group Theory, 2nd edn., Wiley-Interscience, New York, 1971. w10x S.E. Harnung, C.E. Schaffer, Struct. Bond. 12 Ž1972. 201. ¨ w11x L.L. Boyle, Z. Ozgo, Int. J. Quantum Chem. 7 Ž1973. 383.
Chemical Physics Letters 320 Ž2000. 523–526 www.elsevier.nlrlocatercplett
Icosahedral hybrid orbitals Emilio Martınez ´ Torres ) Department of Physical Chemistry, UniÕersity of Castilla-La Mancha, 13071 Ciudad Real, Spain Received 20 December 1999; in final form 10 March 2000
Abstract A set of twelve equivalent icosahedral hybrid orbitals pointing from the centre to the corners of a regular icosahedron has been obtained. Such hybrids can be used to explain the geometry of twelve-coordinate complexes of a rare-earth atom. Using group theoretical considerations, it is shown that these hybrids can be constructed by linear combination of one s, three p, five d and three f-orbitals. Bearing in mind that the twelve hybrids have identical shape but are oriented differently in space, their mathematical expressions have been obtained by applying geometrical transformations to the sp 3d 5 f 3 hybrid pointing along the positive z-axis. In order to obtain elegant mathematical expressions, the x, y and z axes have been chosen to be coincident with three orthogonal binary axes of the icosahedron. q 2000 Elsevier Science B.V. All rights reserved.
1. Introduction Hybridization is an important concept in the conventional valence bond method w1x. Thus, the qualitative discussion of the process of mixing ‘pure’ atomic orbitals to obtain sp, sp 2 and sp 3 hybrid orbitals provides an intuitive justification for the geometry of some simple molecules, as Cl 2 Be, BF3 , CH 4 , . . . , whose shapes are different from those expected using s and p-orbitals separately. Hybrids involving the participation of d-orbitals, such as dsp 3 , d 2 sp 3, . . . , are also used to explain the shape of both molecules and transition-metal complexes w2–4x. Icosahedral symmetry is becoming of increasing interest to chemists. However, although twelve-coordinate complexes with icosahedral structure have been described in the literature for a long time, no hybrids
associated with this geometry have been described yet. The most common twelve-coordinate complexes are those containing six bidentate ligands: wMŽbidentate. 6 x. The regular icosahedron is expected to be the most stable structure for such complexes because it gives minor repulsion energy coefficients w5x. In fact, structures which approximate to a regular icosahedron have been observed for a number of hexanitrato complexes of the rare earths, such as wLaŽNO 3 . 6 x 3y, wThŽNO 3 . 6 x 2y, wNdŽNO 3 . 6 x 3y, etc. w6x, as well as for the praseodymiumŽIII. naphthyridine complex, wPrŽC 8 H 6 N2 . 6 x 3y w7x. Fig. 1 shows the chemically relevant structural isomers for such complexes.
2. Attainment of icosahedral hybrids
)
Fax: q34-926-295315; e-mail: [email protected]
The equivalent hybrids pointing to the twelve corners of a regular icosahedron form the basis for a
0009-2614r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 Ž 0 0 . 0 0 3 0 2 - X
524
E. Martınez ´ Torresr Chemical Physics Letters 320 (2000) 523–526 Table 1 Attainment of the icosahedral hybrid orbitals from c z
c 1 sC ywcz c4 s sy z c 1 c 7 s ic1 c 10 s sx y c 1
c 2 sC31 c 1 c5 s sx z c 3 c8 s ic2 c 11 s sx y c 3
c 3 sC31 c 2 c6 s sx z c 2 c9 s ic3 c 12 s sy z c 2
Fig. 1. Twelve-coordinate complexes with icosahedral structure.
reducible representation, G , of the icosahedral point group, I, which reduces into w8x
G s A [ T1 [ T2 [ H
Ž 1.
On the other hand, the reductions of the representations based on the s, p, d, and f-orbitals under the symmetry operations of the I group are w8x
Gs s A
in the I group, together with the high dimension of some of its irreducible representations, the use of such a method in the attainment of icosahedral hybrids requires laborious and careful work. Hence we will employ a simpler method that makes use of the fact that the icosahedral hybrids are equivalent; i.e. they have identical shape but are oriented differently in space. Therefore, the twelve icosahedral hybrids can be obtained starting from a sp 3 d 5 f 3 hybrid, cz , pointing along the positive z-axis:
Gp s T1 cz s
Gd s H Gf s T 2 [ G
Ž 2.
From Eqs. Ž1. and Ž2. we conclude that one s, three p, five d and three f-orbitals can be combined in twelve different ways to produce twelve sp 3 d 5 f 3 equivalent icosahedral hybrids. Note that the three f-orbitals suitable for the construction of such hybrids must be a basis for the T2 representation. The usual procedure to obtain explicit mathematical expressions for hybrids as linear combinations of atomic orbitals makes use of the projection operator technique. Such methods are described in standard textbooks on group theory w9x. However, because of the large number of symmetry operations contained
1
' ' '12 Ž s q 3 pz q 5 d z
2
q '3 f z 3 .
Ž 3.
In order to obtain elegant mathematical expressions, we have chosen three orthogonal binary axes of the icosahedron as coordinate axes Žsee Fig. 2.. Table 1 shows how the twelve icosahedral hybrids can be obtained from cz , where C yw is an anti-clockwise rotation by an angle w s siny1 w1r 'F q 2 x Žwhere F s Ž1 q '5 .r2 is the ‘golden number’. about the y-axis, C31 is an anti-clockwise rotation by 1208 about the axis passing through the center of the coordinate system and the point Ž1, 1, 1., i is the inversion, and sx y , sx z , sy z are reflections in the Table 2 Expressions for the atomic orbitals as functions of Cartesian coordinates neglecting invariant common factors w10x ss1 py s y d z 2 s Ž1r2. =Ž2 z 2 y x 2 y y 2 . d x y s'3 xy d y z s'3 yz f x z 2 s Ž'6 r4. =Ž4 z 2 y x 2 y y 2 . x f x y z s'15 xyz f yŽ3 x 2yy 2 . s Ž'10 r4. =Ž3 x 2 y y 2 . y
Fig. 2. Numbering and coordinate system for the icosahedron.
px s x pz s z d x 2yy 2 s Ž'3 r2.Ž x 2 y y 2 . d x z s'3 xz f z 3 s Ž1r2.Ž2 z 2 y3 x 2 y3 y 2 . z f y z 2 s Ž'6 r4.Ž4 z 2 y x 2 y y 2 . y f x Ž x 2yy 2 . s Ž'15 r2.Ž x 2 y y 2 . z f x Ž x 2y3 y 2 . s Ž'10 r4.Ž x 2 y3 y 2 . x
E. Martınez ´ Torresr Chemical Physics Letters 320 (2000) 523–526
525
Table 3 Icosahedral hybrids as linear combinations of atomic orbitalsa
c 1 s Ž1r2'3 . s q a p x q b p z q g d z 2 q r d x 2yy 2 q Ž1r2. d x z y b f x 3q3F y 1 x y 2 y3F x z 2 q a f z 3q3F y 1 z x 2 y3F z y 2 c 2 s Ž1r2'3 . s q b p x q a p y y d d z 2 q Ž1r4. d x 2yy 2 q Ž1r2. d x y q a f x 3q3F y 1 x y 2 y3F x z 2 y b f y 3 q3F y 1 y z 2y3F y x 2 c 3 s Ž1r2'3 . s q b p y q a p z y ´ d z 2 y s d x 2yy 2 q Ž1r2. d y z q a f y 3q3F y 1 y z 2 y3F y x 2 y b f z 3q3F y 1 z x 2 y3F z y 2 c4 s Ž1r2'3 . s y a p x q b p z q g d z 2 q r d x 2yy 2 y Ž1r2. d x z q b f x 3q3F y 1 x y 2 y3F x z 2 q a f z 3q3F y 1 z x 2y3F z y 2 c 5 s Ž1r2'3 . s y b p y q a pz y ´ d z 2 y s d x 2yy 2 y Ž1r2. d y z y a f y 3q3F y 1 y z 2y3F y x 2 y b f z 3q3F y 1 z x 2y3F z y 2 c6 s Ž1r2'3 . s q b p x y a p y y d d z 2 q Ž1r4. d x 2yy 2 y Ž1r2. d x y q a f x 3q3F y 1 x y 2y3F x z 2 q b f y 3q3F y 1 y z 2 y3F y x 2 c 7 s Ž1r2'3 . s y a p x y b p z q g d z 2 q r d x 2yy 2 q Ž1r2. d x z q b f x 3q3F y 1 x y 2y3F x z 2 y a f z 3q3F y 1 z x 2 y3F z y 2 c 8 s Ž1r2'3 . s y b p x y a p y y d d z 2 q Ž1r4. d x 2yy 2 q Ž1r2. d x y y a f x 3q3F y 1 x y 2 y3F x z 2 q b f y 3 q3F y 1 y z 2y3F y x 2 c 9 s Ž1r2'3 . s y b p y y a p z y ´ d z 2 y s d x 2yy 2 q Ž1r2. d y z y a f y 3q3F y 1 y z 2 y3F y x 2 q b f z 3q3F y 1 z x 2 y3F z y 2 c 10 s Ž1r2'3 . s q a p x y b p z q g d z 2 q r d x 2yy 2 y Ž1r2. d x z y b f x 3q3F y 1 x y 2 y3F x z 2 y a f z 3q3F y 1 z x 2y3F z y 2 c 11 s Ž1r2'3 . s q b p y y a p z y ´ d z 2 y s d x 2yy 2 y Ž1r2. d y z q a f y 3q3F y 1 y z 2y3F y z 2 q b f z 3q3F y 1 z x 2y3F z y 2 c 12 s Ž1r2'3 . s y b p x q a p y y d d z 2 q Ž1r4. d x 2yy 2 y Ž1r2. d x y y a f x 3q3F y 1 x y 2 y3F x z 2 y b f y 3 q3F y 1 y z 2y3F y x 2 a
as
The hybrid given by the function c i points to the ith corner of the icosahedron Žsee Fig. 2.. 1 2'F q 2
F , bs
2'F q 2
, gs
Fq1 4'3
, ds
'5 4'3
, ´s
2yF 4'3
coordinate planes. Making use of the expressions for the atomic orbitals as functions of Cartesian coordinates given in Table 2 w10x, it is easy to obtain the
, rs
Fy1 4
F , ss
4
.
expressions for the icosahedral hybrids shown in Table 3, where the following icosahedral f-orbitals have been used: f x 3y3F y 1 x y 2y3F x z 2 s y Ž F'6 r4 . f x z 2 q Ž '2 r4F 2 . f xŽ x 2y3 y 2 . f y 3y3F y 1 y z 2y3F y x 2 s y Ž '6 r4F . f y z 2 y Ž F 2'2 r4 . f yŽ3 x 2yy 2 . f z 3y3F y 1 z x 2y3F z y 2 s Ž 1r2 . f z 3 q Ž '3 r2 . f zŽ x 2yy 2 . ,
Fig. 3. Icosahedral hybrids viewed down a 5-fold axis.
Ž 4.
which are linear combinations of those obtained by Boyle et al. w11x. Figs. 3 and 4 show two views of the icosahedral hybrids.
3. Conclusions
Fig. 4. Icosahedral hybrids viewed down a 3-fold axis.
A set of twelve hybrid orbitals pointing from the centre to the corners of a regular icosahedron have been obtained. By group theoretical considerations it has been shown that these icosahedral hybrids can be constructed by linear combinations of one s, three p, five d and three f-orbitals. Since such hybrids have identical shape but are oriented differently in space, their mathematical expressions have been obtained by applying geometrical transformations to the
526
E. Martınez ´ Torresr Chemical Physics Letters 320 (2000) 523–526
sp 3d 5 f 3 hybrid pointing along the positive z-axis. The three coordinate axes have been chosen to be coincident with three orthogonal binary axes of the icosahedron.
References w1x L. Pauling, The Nature of the Chemical Bond and the Structure of Molecules and Crystals: An Introduction to Modern Structural Chemistry, 3rd edn., Cornell University Press, Ithaca, NJ, 1979. w2x R. McWeeny, Coulson’s Valence, 3rd edn., Oxford University Press, Oxford, 1979.
w3x R.L. De Rock, H.B. Gray, Chemical Structure and Bonding, 2nd edn., University Science Books, Mill Valley, CA, 1989. w4x F.A. Cotton, G. Wilkinson, Advanced Inorganic Chemistry, 5th edn., Wiley, New York, 1988. w5x D.L. Kepert, in: G. Wilkinson, R.D. Gillard, J.A. McCleverty ŽEds.., Comprehensive Coordination Chemistry, Vol. I, Pergamon, Oxford, 1987. w6x D.L. Kepert, Inorganic Stereochemistry, Springer, Berlin, 1982. w7x A. Clearfield, R. Gopal, R.W. Olsen, Inorg. Chem. 16 Ž1977. 911. w8x E. Martınez-Torres, J.J. Lopez-Gonzalez, M. Fernandez, ´ ´ ´ ´ Comp. Math. Applic. 26 Ž1993. 67. w9x F.A. Cotton, Chemical Applications of Group Theory, 2nd edn., Wiley-Interscience, New York, 1971. w10x S.E. Harnung, C.E. Schaffer, Struct. Bond. 12 Ž1972. 201. ¨ w11x L.L. Boyle, Z. Ozgo, Int. J. Quantum Chem. 7 Ž1973. 383.