Systematics of Atomic Orbital Hybridization of Coordination Polyhedra: Role of f Orbitals

Abstract

The combination of atomic orbitals to form hybrid orbitals of special symmetries can be related to the individual orbital polynomials. Using this approach, 8-orbital cubic hybridization can be shown to be sp³d³f requiring an f orbital, and 12-orbital hexagonal prismatic hybridization can be shown to be sp³d⁵f²g requiring a g orbital. The twists to convert a cube to a square antiprism and a hexagonal prism to a hexagonal antiprism eliminate the need for the highest nodality orbitals in the resulting hybrids. A trigonal twist of an O_h octahedron into a D_3h trigonal prism can involve a gradual change of the pair of d orbitals in the corresponding sp³d² hybrids. A similar trigonal twist of an O_h cuboctahedron into a D_3h anticuboctahedron can likewise involve a gradual change in the three f orbitals in the corresponding sp³d⁵f³ hybrids.

1. Introduction

In a series of papers in the 1990s, the author focused on the most favorable coordination polyhedra for sp³dⁿ hybrids, such as those found in transition metal complexes. Such studies included an investigation of distortions from ideal symmetries in relatively symmetrical systems with molecular orbital degeneracies [1] In the ensuing quarter century, interest in actinide chemistry has generated an increasing interest in the involvement of f orbitals in coordination chemistry [2,3,4,5,6,7]. This has prompted me to revisit such issues, adding the new feature of f orbital involvement as might be expected to occur in structures of actinide complexes and relating hybridization schemes to the polynomials of the participating orbitals.

The single s and three p atomic orbitals have simple shapes. Thus, an s orbital, with the quantum number l = 0, is simply a sphere equivalent in all directions (isotropic), whereas the three p orbitals, with quantum number l = 1, are oriented along the three axes with a node in the perpendicular plane. For higher nodality atomic orbitals with l > 1 nodes the situation is more complicated since there is more than one way of choosing an orthogonal 2 l + 1 set of orbitals. In this connection, a convenient way of depicting the shapes of atomic orbitals with two or more nodes is by the use of an orbital graph [8]. Such an orbital graph has vertices corresponding to its lobes of the atomic orbital and the edges to nodes between adjacent lobes of opposite sign. Orbital graphs are necessarily bipartite graphs in which each vertex is labeled with the sign of the corresponding lobe and only vertices of opposite sign can be connected by an edge. For clarity in the orbital graphs depicted in this paper, only the positive vertices are labeled with plus (+) signs. The unlabeled vertices of the orbital graphs are considered negative.

Table 1. The polynomials, angular functions, and orbital graphs for the five d orbitals.

Polynomial	Angular Function	Appearance and Orbital Graph	Shape
xy x2 − y2 xz yz	sin2 sin sin2 cos2 sin cos cos sin cos sin		square
2z2 − r2 (abbreviated as z2)	(3cos2 − 1)		linear

shows the orbital graphs for the commonly used set of five d orbitals as well as the corresponding orbital polynomials. The orbital graphs for four of this set of five d orbitals are squares, whereas that for the fifth d orbital is a linear configuration with three vertices and two edges. The exponent of the variable z corresponds to l-m, i.e., 2-m for this set of d orbitals. This set of d orbitals is not the only possible set of d orbitals. Two alternative sets of d orbitals have all five d orbitals of equivalent shape corresponding to rectangular orbital graphs oriented to exhibit five-fold symmetry [9,10,11,12] One set of these equivalent d orbitals is based on an oblate (compressed) pentagonal antiprism, whereas the other set of equivalent d orbitals is based on a prolate (elongated) pentagonal prism (Figure 1). These sets are useful for structures with five-fold symmetry corresponding to symmetry point groups such as D_5d, D_5h, and I_h in the structures studied here. For the icosahedral group I_h all five d orbitals belong to the five-dimensional H_g irreducible representation.

Two different sets of f orbitals are used depending on the symmetry of the system (

Table 2. The polynomials, angular functions, and orbital graphs for both the general and cubic sets of the seven f orbitals.

|m|	Lobes	Shape	Orbital Graph	General Set	Cubic Set
3	6	Hexagon		x(x2 − 3y2) y(3x2 − y2)	none
2	8	Cube		xyz z(x2 − y2)	xyz x(z2 − y2) y(z2 − x2) z(x2 − y2)
1	6	Double Square		x(5z2 − r2) y(5z2 − r2)	none
0	4	Linear		z(5z2 − r2)	x3 y3 z3

) [13]. The general set of f orbitals is used except for systems with high enough symmetry to have three-dimensional irreducible representations. This general set consists of a unique f{z³} orbital with a linear orbital graph and m = 0, an f{xz²,yz²} pair with a double square orbital graph and m = ± 1, an f{xyz, z(x² − y²)} pair with a cube orbital graph and m = ±2, and an f{x(x² − 3y²),y(3x² − y²)} set with a hexagon orbital graph and m = ±3. For systems having point groups with three-dimensional irreducible representations, such as the octahedral O_h and the icosahedral I_h point groups, the so-called cubic set of f orbitals is used. The cubic set of f orbitals consists of the triply degenerate f{x³,y³,z³} set with linear orbital graphs and a quadruply degenerate f{xyz, x(z² − y²), y(z² − x²), z(x² − y²)} set.

Fully using the complete sp³, sp³d⁵, and sp³d⁵f⁷ manifolds in hybridization schemes should lead to most nearly spherical polyhedra (Figure 2). For the four-orbital sp³ system such a polyhedron is, of course, the familiar regular tetrahedron. For the 16-orbital sp³d⁵f⁷ manifold the most spherical polyhedron is the tetracapped tetratruncated tetrahedron, also of T_d point group symmetry. Such a 16-vertex deltahedron is the largest Frank-Kasper polyhedron where a Frank-Kasper polyhedron is a deltahedron with only degree 5 and degree 6 vertices with no pair of adjacent degree 6 vertices [14]. The tetracapped tetratruncated tetrahedron is rarely found in experimental molecular structures but has been recently realized in the central Rh₄B₁₂ polyhedron in the rhodaborane Cp*₃Rh₃B₁₂H₁₂Rh(B₄H₉RhCp*), synthesized by Ghosh and co-workers [15]. There is no similarly symmetrical most spherical 9-vertex polyhedron. The most spherical 9-vertex deltahedron is the D_3h tricapped trigonal prism found experimentally in the MH₉²⁻ (N = Tc, Re) anions [16,17].

2. Polyhedral Hybridizations

2.1. Polyhedra with Tetragonal Symmetry

The simplest configuration with tetragonal symmetry is the square, found as a coordination polyhedron in diamagnetic four-coordinate complexes of d⁸ metals such as Ni(II), Pd(II), Pt(II), Rh(I), and Ir(I). Since a square is a two-dimensional structure in an xy plane, the polynomials of the atomic orbitals for square hybridization cannot contain a z variable. This limits the p orbital involvement in square hybridization to the p{x} and p{y} orbitals so that a d orbital is required for square hybridization. This d orbital can be either the d{x² − y²} or d{xy} orbital, depending on whether the x and y axes are chosen to go through the vertices or edge midpoints, respectively, of the square (

Table 3. Hybridization schemes for the tetragonal polyhedra.

Polyhedron	Coord.	Hybridization
(Symmetry)	No.	Type	s + p	d	f
Square (D4h)	4	sp2	A1g + Eu	B1g {x2 − y2}	—
Square Bipy (D4h)	6	sp3d2	A1g + A2u + Eu	A1g{z2} + B1g {x2 − y2}	—
Octahedron (Oh)	6	sp3d2	A1g + T1u	Eg{x2 − y2,xy}	—
Square Prism(D4h)	8	sp3d3f	A1g + A2u + Eu	B1g {x2 − y2} + Eg{xz,yz)	B2u{ z(x2 − y2)}
Cube(Oh)	8	sp3d3f	A1g + T1u	T2g{xy,xz,yz}	A2u{xyz}
Bicapped Cube(D4h)	10	sp3d4f2	A1g + A2u + Eu	A1g{z2} + B1g {x2 − y2} + Eg{xz,yz)	A2u(z3} + B2u{ z(x2 − y2)}
Square Antiprism(D4d)	8	sp3d4	A1 + B2 + E1	E2{x2 − y2,xy} + E3{xz,yz)	—
Bicap Sq Antipr(D4d)	10	sp3d5f	A1 + B2 + E1	A1{z2} + E2{x2 − y2,xy} + E3{xz,yz)	B2{z3}

).

The smallest three-dimensional polyhedron with tetragonal symmetry of interest in this context is the regular octahedron with the O_h point group containing both three-fold and four-fold axes (Figure 3). This is the most frequently encountered polyhedron in coordination chemistry and is also the most spherical 6-vertex closo deltahedron in the structures of boranes and related species. The O_h point group contains two- and three-dimensional irreducible representations. Under octahedral symmetry, the five d orbitals split into a triply degenerate T_2g{xz,yz,xy} set and a doubly degenerate E_g{x² − y²,xy} set. Octahedral hybridization supplements the four-orbital sp³ set with the doubly degenerate E_g{x² − y²,xy} set of d orbitals.

An octahedral coordination complex can be stretched or compressed along a four-fold axis thereby reducing the symmetry from O_h to D_4h by removing the three-fold axes. This corresponds to the Jahn–Teller effect [18] leading to a structure that can be designated as a square bipyramid, analogous to pentagonal and hexagonal bipyramids discussed later in this paper. This reduction in symmetry of an octahedral metal complex splits the triply degenerate T_1u p orbitals into a doubly degenerate E_1u{x,y} set and a non-degenerate A_2u{z} orbital and the doubly degenerate E_g{x² − y²,xy} set of orbitals into non-degenerate A_1g{z²} and B_1g{x² − y²} orbitals.

The most symmetrical polyhedron for an 8-coordinate complex is the cube, which is the dual of the regular octahedron and, thus, also exhibits the O_h point group (Figure 3). Considering the cube as two squares stacked on top of each other at a distance leading to the three-fold symmetry elements of the O_h point group leads to a simple way of deriving the atomic orbitals for cubic hybridization. Thus, to form a set of cubic hybrid orbitals, the atomic orbitals for square hybridization are supplemented by the four atomic orbitals in which the polynomials for square hybridization are multiplied by z. In this way, a set of pd²f orbitals with each atomic orbital having one more node than the corresponding orbital in the sp²d orbital set for square hybridization is added to the sp²d square hybridization to form an (sp²d)(pd²f) = sp³d³f set for cubic hybridization. This shows that one f orbital, namely the A_2u{xyz} orbital with a cube orbital graph (Table 2), is required for cube hybridization. This suggests that 8-coordinate ML₈ complexes with cubic coordination are likely to be restricted to lanthanide and actinide chemistry, particularly the latter. Reducing the symmetry of the cube from O_h to D_4h by elongation or compression along a four-fold axis thereby eliminating the three-fold symmetry does not eliminate the need for an f orbital in the hybridization of an ML₈ complex.

The need for an f orbital in the hybridization for an 8-coordinate metal complex can be eliminated by converting a cube into a square antiprism by rotating a square face of a cube 45° around the C₄ axis thereby changing the O_h point group into D_4d (Figure 3 and Table 3). The sp³d⁴ hybridization for the square antiprism omits the d{z²} orbital.

Capping the square faces of a square antiprism retaining the D_4d symmetry leads to the most spherical closo deltahedron for 10-vertex borane derivatives, such as B₁₀H₁₀²⁻ (Figure 3). The d{z²} and f{z³} orbitals, both of which lie along the four-fold z axis, are added to th sp³d⁴ hybridization of the square antiprism to give the sp³d⁵f{z³} hybridization of the bicapped square antiprism. In a similar way the d{z²} and f{z³} orbitals can be added to the sp³d³f hybridization of the cube to give the sp³d⁴f² hybridization of the bicapped cube with D_4d symmetry.

2.2. Polyhedra with Pentagonal Symmetry

The simplest structure with pentagonal symmetry is the planar pentagon. Using a set of five orbitals with only x and y in the orbital polynomial leads to sp²{x,y}d²(x² − y²,xy} hybridization for planar pentagon coordination with minimum l values. Either the prolate or oblate sets of equivalent d orbitals (Figure 1) can provide alternative d⁵ hybridization for a planar pentagon structure, but with higher combined l values for the five-orbital set and poor overlap between the ligand orbitals and those of the central atom.

The smallest three-dimensional polyhedron of interest with pentagonal symmetry is the D_5h pentagonal bipyramid (Figure 4). The sp³d³ scheme for pentagonal bipyramidal coordination supplements the five-orbital sp²{x,y}d²(x² − y²,xy} hybridization for the equatorial pentagon with the p{z} and d{z²} orbitals for the linear sub-coordination of the axial ligands perpendicular to the equatorial planar pentagon (

Table 4. Hybridization schemes for the pentagonal polyhedra.

Polyhedron	Coord.	Hybridization
(Symmetry)	No.	Type	s + p	d	f
Pentagon (D5h)	5	sp2d2	A1’ + E1´	E2´{x2 − y2}	—
Pent Bipy (D5h)	7	sp3d3	A1´ + A2˝ + E1´	A1´{z2} + E2´{x2 − y2,xy}	—
Pent Prism (D5h)	10	sp3d4f2	A1´ + A2˝ + E1´	E2´{x2 − y2} + E1˝{xz,yz}	E2˝{xyz,z(x2 − y2)}
Bicap Pent Prism(D5h)	12	sp3d5f3	A1´ + A2˝ + E1´	A1´{z2} + E2´{x2 − y2,xy} + E1˝{xz,yz}	A2˝{z3} + E2˝{xyz,z(x2 − y2)}
Pent Antiprism(D5d)	10	sp3d4f2	A1g + A2u + Eu	E1g{xz,yz} + E2g{x2 − y2,xy}	E2u
Bicap Pent Antipr(D5d)	12	sp3d5f3	A1g + A2u + Eu	A1g{z2} + E1g{xz,yz} + E2g{x2 − y2,xy}	A2u{z3} + E2u
Icosahedron (Ih)	12	sp3d5f3	A1g + T1u	Hg{all d orbitals}	T2u{x3,y3,z3}

).

The 10-vertex pentagonal prism of D_5h symmetry (Figure 4) is encountered experimentally in the endohedral trianions M@Ge₁₀^3– (M = Fe [19], Co [20]), isolated as K(2,2,2-crypt)⁺ salts and structurally characterized by X-ray crystallography. Considering the pentagonal prism as two pentagons stacked on top of each other to preserve the D_5h symmetry supplements the sp²{x,y}d²(x² − y²,xy} of the planar pentagon with an additional set of five orbitals with the orbital polynomials multiplied by z, i. e. the p{z}d³(xz,yz}f²{xyz,z(x² − y²)} set. Thus, the set of 10 orbitals for the pentagonal prism are sp³d⁴{x² − y²,xy,xz,yz} f{xyz,z(x² − y²)} without involvement of the d{z²} orbital. Capping the pentagonal prism in a way to preserve the D_5h symmetry adds the d{z²} and f{z³} orbitals to the hybrid leading to an sp³d⁵f³{z³,xyz,z(x² − y²)} set for the resulting 12-vertex bicapped pentagonal prism.

Twisting one pentagonal face of a pentagonal prism 36° around the C₅ axis leads to the pentagonal antiprism of D_5d symmetry (Figure 4). Unlike the analogous conversion of the cube to the square antiprism, this process does not change the sp³d⁴f² hybridization. Capping the two pentagonal faces of the pentagonal antiprism while preserving five-fold symmetry leads to the bicapped pentagonal antiprism. This adds the d(z²) and f(z³) orbitals to give an sp³d⁵f³ 12-orbital hybridization scheme. This process is exactly analogous to the conversion of the square antiprism to the bicapped square antiprism discussed above.

The special D_nd symmetry of an n-gonal antiprism is preserved by compression or elongation along the major C_n axis to give oblate or prolate polyhedra, respective, as illustrated in Figure 1 for the pentagonal antiprisms representing the five equivalent d orbitals. The regular icosahedron, so important in several areas of chemistry, including polyhedral borane chemistry, is a special case of the bicapped pentagonal antiprism with a specific degree of compression/elongation to add three-fold symmetry to the D_5d symmetry of the bicapped pentagonal antiprism to give the full icosahedral point group I_h. This ascent in symmetry combines some of the irreducible representations of the D_5h point group to give irreducible representations of higher degeneracy. As a result, the A_1g + E_1g + E_2g irreducible representations coalesce into the single five-fold degenerate H_g representation of the icosahedral point group. The five d orbitals belong to this H_g representation in icosahedral symmetry. Similarly, the three f orbitals of the sp³d⁵f³ icosahedral hybridization belong to the single T_2u{x³,y³,z³} irreducible representation of the I_h point group considering the cubic set of f orbitals (Table 2).

2.3. Polyhedra with Hexagonal Symmetry

The simplest structure with hexagonal symmetry is the planar hexagon itself. The restriction of the atomic orbitals to those containing no z term in their polynomials requires the use of a single f atomic orbital to supplement the planar five-orbital sp²{x,y}d²(x² − y²,xy} set with either f orbital with a planar hexagon orbital graph, namely the f{x(x² − 3y²)} or the f{y(3x² − y²)} orbital depending on the {x,y,z} coordinate system chosen.

The smallest three-dimensional figure with hexagonal symmetry is the 8-vertex hexagonal bipyramid (Figure 5). The sp³d³f set of hybrid orbitals for the hexagonal bipyramid is obtained by supplementing the sp²{x,y}d²{x² − y²,xy}f{x(x² − 3y²)} set of orbitals for the equatorial planar hexagon with the p{z} and d{z²} pair for the two additional axial vertices. The hexagonal bipyramid shares with the cube (see above) the feature of not arising from any sp³d⁴ hybridization scheme but instead requiring an f orbital in a sp³d³f hybridization scheme.

The atomic orbitals required to form a prism combine those for the polygonal face having only x and y in their orbital polynomials with an equal number of orbitals in which the polynomials obtained by multiplying the orbital polynomials for the planar face by the z variable. Thus, the sp²{x,y}d²{x² − y²,xy}f{x(x² − 3y²)} orbitals corresponding to the hexagonal face of the hexagonal prism is supplemented by the p{z³}d²{xz,yz}f²{xyz,z(x² − y²)}g{xz(x² − 3y²)} set of six orbitals. This leads to the interesting observation that a single g orbital corresponding to the B_2g irreducible representation is required to provide a set of 12 hybrid orbitals oriented towards the vertices of a D_6h hexagonal prism (

Table 5. Hybridization schemes for the hexagonal polyhedra.

Polyhedron	Coord.	Hybridization
(Symmetry)	No.	Type	s + p	d	f	g
Hexagon (D6h)	6	sp2d2f	A1g + E1u	E2g{x2 − y2}	B1u{x(x2 − 3y2)}	—
Hexagonal Bipy(D6h)	8	sp3d3f	A1g + A2u + E1u	A1u{z2} + E2g{x2 − y2,xy}	B1u{x(x2 − 3y2)}	—
Hexagonal Prism(D6h)	12	sp3d4f3g	A1g + A2u + E1u	E1g{xz,yz} + E2g{x2 − y2,xy}	B1u + E2u{xyz,z(x2 − y2)}	B2g
Bicap Hex Prism(D6h)	14	sp3d5f4g	A1g + A2u + E1u	A1u{z2} + E1g{xz,yz}+E2g{x2 − y2,xy}	A2u{z3} + B1u + E2u	B2g
Hex Antiprism(D6d)	12	sp3d4f4	A1 + B2 + E1	E2{x2 − y2,xy} + E5{xz,yz}	E3 + E4{xyz,z(x2 − y2)}	—
Bicap Hex Antipr(D5d)	14	sp3d5f5	A1 + B2 + E1	A1{z2} + E2{x2 − y2,xy} + E5{xz,yz}	B2(z3) + E3 + E4	—

). Not surprisingly, this g orbital has a hexagonal prism orbital graph. Capping the hexagonal prism in a way to preserve the D_6h symmetry adds the d{z²} and f{z³} orbitals to the hybrid leading to an sp³d⁵f³{z³,xyz,z(x² − y²)} for the resulting 12-vertex bicapped pentagonal prism.

Twisting one hexagonal face of a hexagonal prism 30° around the C₆ axis leads to the hexagonal antiprism of D_6d symmetry (Figure 5). Going from the D_6h symmetry of the hexagonal prism to the D_6d symmetry of the hexagonal antiprism in a 12-vertex hexagonal symmetry system eliminates the need for a g orbital in the set of 12 atomic orbitals forming the corresponding hybrid orbitals. This is analogous to the conversion of the cube requiring sp³d³f hybridization to the square antiprism with sp³d⁴ hybridization not requiring an f orbital by rotation one square face 45° relative to its opposite partner. Capping the hexagonal faces of the hexagonal antiprism in a way to preserve its six-fold symmetry leads to the bicapped hexagonal antiprism, which is of significance as being the most spherical closo 14-vertex deltahedron in borane chemistry. This capping process adds d{z²} and f{z³} orbitals to the 12-orbital hybridization scheme for the hexagonal antiprism leading to a sl³d⁵f⁵ hybridization scheme for the bicapped hexagonal antiprism not requiring g orbitals (Table 5).

2.4. Trigonal Twist Processes in Polyhedral of Octahedral Symmetry

The octahedron and the cuboctahedron are the simplest two polyhedra with triangular faces and O_h point group symmetry having both three-fold and four-fold rotation symmetry axes. The process of converting prisms of n-fold symmetry (n = 4, 5, and 6 including the cube as a special tetragonal prism) to the corresponding antiprisms involves a twist of (180/n)° of an n-fold face around the unique n-fold axis. Analogous processes can be applied to the three-fold axes in the regular octahedron and icosahedron. Such a twist process converts the O_h regular octahedron into the D_3h trigonal prism and the O_h cuboctahedron into the D_3h anticuboctahedron (Figure 6). These processes can be regarded as trigonal twists involving three symmetry related diamond-square-diamond transformations. For octahedral coordination complexes of the type M(bidentate)₃ involving bidentate chelating ligands variations of this process are designated as Bailar twists [21] or Ray-Dutt [22] twists.

Such trigonal twist processes can be studied by first relaxing the symmetry of the original O_h polyhedron to D_3d, which is the subgroup of O_h obtained by removing the C₄ axes. Under this subgroup, the octahedron can be viewed as a trigonal antiprism with the z axis corresponding to the C₃ axis rather than the C₄ axis in the discussion above of polyhedral with C₄ symmetry. Under D_3d symmetry rather than the higher O_h symmetry and with the different location of the z axis, the degenerate E_g{xz,yz} pair of d orbitals as well as the E_g(x² − y²,xy}pair can be used for octahedral sp³d² hybridization (

Table 6. Hybridization schemes for polyhedra with O_h symmetry and the D_3h polyhedral forme from them by a triple diamond-square-diamond process.

Polyhedron	Coord.	Hybridization
(Symmetry)	No.	Type	s + p	d	f
Octahedron (D3d)	6	sp3d2	A1g + A2u + Eu	Eg(x2 − y2,xy}or Eg{xz,yz}	—
Trigonal Prism (D3h)	6	sp3d2	A1´ + A2˝ + E1´	E˝{xz,yz}	—
Cuboctahedron (Oh)	12	sp3d5f3	A1g + T1u	Eg(x2 − y2,xy} + T2g{xy,xz,yz}	T2u{x(z2 − y2),y(z2 − x2),z(x2 − y2)}
Cuboctahedron (D3d)	12	sp3d5f3	A1g + A2u + Eu	A1g{z2} + Eg(x2 − y2,xy} + Eg{xz,yz}	A2u{z3} + Eu
Anticuboctahedron(D3h)	12	sp3d5f3	A1´ + A2˝ + E1´	A1´{z2} + E´(x2 − y2,xy} + E˝{xz,yz}	A1´{x(x2 − 3y2} + E´{xz2,yz2}

). When the twist reaches the stage of the D_3h trigonal prism only the E´{xz,yz} pair of atomic orbitals is available for the sp³d² hybridization scheme. Thus, the trigonal twist of an octahedron to a trigonal prism involves replacement of the E_g{x² − y²,xy}pair of orbitals in the D_3d octahedral sp³d² hybrids with the E´{xz,yz} pair of orbitals in the D_3h trigonal prismatic hybrids.

The sp³d⁵f³ hybridization for a cuboctahedron using its full O_h symmetry taking the z axis as a C₄ axis supplements the full 9-orbital sp³d⁵ set with the triply degenerate T_2u{x(z² − y²),y(z² − x²),z(x² − y²)} set of the cubic f orbitals (Table 6). Reducing the symmetry of the cuboctahedron to D_3d and now taking the C₃ axis as the z axis splits, the triply degenerate set of f orbitals into a non-degenerate A_1u{z³} orbital and a doubly degenerate Eu set of orbitals, which can be either the Eu{xyz,z(x² − y²)} or the Eu{x(3x² − y²),y(3x² − y²)} set. Thus, for the cuboctahedron, like the octahedron under D_3d symmetry, there are two alternative pairs of the required highest l value orbitals in the minimum l value hybrid that can be used in the hybrid. Converting the D_3d cuboctahedron to the D_3h anticuboctahedron eliminates the Eu{x(3x² − y²),y(3x² − y²)} pair of f orbitals from the sp³d⁵f³ hybrid requiring use of the E´{xz²,yz²} pair.

3. Conclusions

The combination of atomic orbitals to form hybrid orbitals of special symmetries can be related to the individual orbital polynomials. Thus, planar hybridizations such as the square, pentagon, and hexagon can only use atomic orbitals with only two variables in their polynomials, conventionally designated as x and y. Since there are only two orbitals for each non-zero l value and one orbital (the s orbital) for l = 0, square planar hybridization requires one d orbital, pentagonal planar hybridization requires two d orbitals, and hexagonal planar hybridization requires an f orbital as well as two d orbitals. Prismatic configurations with two parallel n-gonal faces and D_nh symmetry combine the set of atomic orbitals for the n-gonal face with only the two x and y variables with a set of equal size corresponding to the orbitals having polynomials in which the polynomials of the planar n-gon orbitals are multiplied by a third z variable. As a result, cubic hybridization can be shown to require an f orbital and hexagonal prismatic hybridization can be shown to require a g orbital. For 8-coordinate systems with four-fold symmetry twisting a square face of a cube with sp³d³f hybridization by 45° around a C₄ axis to give a square antiprism with sp³d⁴ hybridization eliminates the need for an f orbital in the resulting sp³d⁴ hybridization. Similarly, for 12-coordinate systems with six-fold symmetry, twisting a hexagonal face of a hexagonal prism with sp³d⁵f²g hybridization to a hexagonal antiprism eliminates the need for a g orbital in the resulting sp³d⁵f³ hybridization. A trigonal twist of an O_h octahedron into a D_3h trigonal prism can involve a gradual change of the pair of d orbitals in the sp³d² hybrids. A similar trigonal twist of an O_h cuboctahedron into a D_3h cuboctahedron can likewise involve a gradual change in the three f orbitals in the sp³d⁵f³ hybridization.