molecules Article Systematics of Atomic Orbital Hybridization of Coordination Polyhedra: Role of f Orbitals R. Bruce King Department of Chemistry, University of Georgia, Athens, GA 30602, USA; [email protected] Academic Editor: Vito Lippolis Received: 4 June 2020; Accepted: 29 June 2020; Published: 8 July 2020 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 Oh octahedron into a D3h trigonal prism can involve a gradual change of the pair of d orbitals in the corresponding sp3d2 hybrids. A similar trigonal twist of an Oh cuboctahedron into a D3h anticuboctahedron can likewise involve a gradual change in the three f orbitals in the corresponding sp3d5f3 hybrids. Keywords: coordination polyhedra; hybridization; atomic orbitals; f-block elements 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 [2–7]. 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. Table1 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 Molecules 2020, 25, 3113; doi:10.3390/molecules25143113 www.mdpi.com/journal/molecules Molecules 2020, 25, x FOR PEER REVIEW 2 of 11 Molecules 2020, 25, x FOR PEER REVIEW 2 of 11 Molecules 2020, 25, x FOR PEER REVIEW 2 of 11 Moleculesd orbitals2020, 25 ,is 3113 not the only possible set of d orbitals. Two alternative sets of d orbitals have 2all of 10five d d orbitals is not the only possible set of d orbitals. Two alternative sets of d orbitals have all five d d orbitalsorbitals is ofnot equivalent the only possibleshape corresponding set of d orbitals. to rectangular Two alternative orbital s etsgraphs of d orientedorbitals haveto exhibit all five five d‐ fold orbitals of equivalent shape corresponding to rectangular orbital graphs oriented to exhibit five-fold orbitalssymmetry of equivalent [9–12] shape One correspondingset of these equivalent to rectangular d orbitals orbital graphsis based oriented on an to oblateexhibit (compressed)five-fold orbitalssymmetry is not the[9–12] only One possible set of set these of d orbitals.equivalent Two d alternativeorbitals is setsbased of don orbitals an oblate have (compressed) all five d symmetrypentagonal [9–12] antiprism, One set whereas of these the equivalent other set of d equivalent orbitals is d orbitalsbased on is basedan oblate on a prolate(compressed) (elongated) orbitalspentagonal of equivalent antiprism shape, whereas corresponding the other to set rectangular of equivalent orbital d orbitals graphs is oriented based on to a exhibit prolate five-fold (elongated) pentagonalpentagonal antiprism prism, whereas (Figure the 1). other These set ofsets equivalent are useful d orbitals for structures is based on with a prolate five‐ (elongated)fold symmetry symmetrypentagonal [9–12 ]prism One set (Figure of these 1). equivalent These dsets orbitals are isuseful based for on anstructures oblate (compressed) with five-fold pentagonal symmetry pentagonalcorresponding prism to(Figure symmetry 1). Thesepoint groupssets are such useful as D 5dfor, D 5hstructures, and Ih in thewith structures five-fold studied symmetry here. For antiprism,corresponding whereas to the symmetry other set ofpoint equivalent groups dsuch orbitals as D is5d, basedD5h, and on aIh prolatein the structures (elongated) studied pentagonal here. For correspondingthe icosahedral to symmetry group pointIh all groups five dsuch orbitals as D5 d,belong D5h, and to Ih ithen the five structures‐dimensional studied H heg re.irreducible For prismthe (Figure icosahedral1). These group sets areIh all useful five ford structuresorbitals belong with five-foldto the symmetryfive-dimensional corresponding Hg irreducible to the representation.icosahedral group Ih all five d orbitals belong to the five-dimensional Hg irreducible representation. representation.symmetry point groups such as D5d, D5h, and Ih in the structures studied here. For the icosahedral group Ih all fiveTable d orbitals 1. The belong polynomials, to the angular five-dimensional functions, and Hg orbitalirreducible graphs representation. for the five d orbitals. Table 1. The polynomials, angular functions, and orbital graphs for the five d orbitals. Table 1. The polynomials, angular functions, and orbital graphs for the five d orbitals. Table 1. The polynomials, angular functions, andAppearance orbital graphs and for the Orbital five d orbitals. Polynomial Angular Function Appearance and Orbital Shape Polynomial Angular FunctionAppearance andGraph Orbital Shape Polynomial Angular Function Graph Shape Polynomial Angular Function AppearanceGraph and Orbital Graph Shape xy sin2 sin xy 2sin2 sin xy xy 2 2 sinsin2 sinsin 2 x2 − y2 sin2 cos2 x y22 x 2 − y sin2sincos22 cos2 square − x − y xz sin2 cos2 squaresquare xz xz sin cossin coscos cos square xz yz sin cos cos yz yz sinsin cos cossin cos sincos sin yz sin cos sin 2 2 sin cos sin 2z2 − r2 2 2z2 − r 2z2 r22z(abbreviated − r − (abbreviated (3cos(3cos2 1)2 − 1) linearlinear (abbreviated(abbreviated as z2) 2 (3cos− 2 − 1) linear as z2) (3cos2 − 1) linear 2as z ) as z ) Figure 1. The oblate and prolate pentagonal antiprisms on which the two sets of five equivalent d FigureFigure 1. The 1. The oblate oblate and and prolate prolate pentagonal pentagonal antiprisms antiprisms on which on which the twothe two sets sets of five of equivalentfive equivalent d d Figureorbitals 1. The are oblate based and indicating prolate thepentagonal amounts antiprisms of compression on which and elongation,the two sets respectively. of five equivalent d orbitals are based indicating the amounts of compression and elongation, respectively. orbitals are based indicating the amountsamounts of compression and elongation, respectively. Two different sets of f orbitals are used depending on the symmetry of the system (Table 2) [13]. TwoTwo different different sets ofsets f orbitalsof f orbitals are usedare used depending depending on the on symmetrythe symmetry of the of the system system (Table (Table2)[ 13 2)]. [13]. TheTwo general different set sets of of f f orbitals areis usedused dependingexcept for onsystems the symmetry with high of theenough system symmetry (Table 2) [13to ].have TheThe general general set ofset fof orbitals f orbitals is used is used except except for systemsfor systems with with high high enough enough symmetry symmetry to haveto have The threegeneral‐dimensional set of f orbitals irreducible is used representations. except for Thissystems general with set high consists enough of a uniquesymmetry f{z3 }to orbital have with three-dimensionalthree-dimensional irreducible irreducible representations. representations. This This general general set set consists consists ofof aa unique f{ f{zz33}} orbital orbital with threea- dimensionallinear orbital ir graphreducible and representations.m = 0, an f{xz2,yz This2} pair general with a set double consists square of a orbitalunique graph f{z3} orbital and m with = ± 1, an witha a linear linear orbital orbital graph graph and m = 0, an f{xz22,yz2}} pair pair with a doubledouble squaresquare orbitalorbital graph graph and andm m= = ±1, 1, an a linearf{xyz orbital, z(x2 − graphy2)} pair and with m = 0,a cubean f{ xzorbital2,yz2} pairgraph with and a mdouble = ±2, andsquare an orbitalf{x(x2 − graph3y2),y (3andx2 − my 2=)} ± set 1,± anwith a an f{f{xyzxyz, ,z z(x(x22 − yy22)} pair with a cube orbital graph and and m = ±2,2, and and an an f{ f{xx(x(x2 2− 3y32y),2y),(3yx(32 x−2 y2)}y 2set)} setwith a f{xyzhexagon, z(x2 − y 2−)}orbital pair withgraph a cube and orbitalm = ±3.