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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 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 to a and a hexagonal to a hexagonal antiprism eliminate the need for the highest nodality orbitals in the resulting hybrids. A trigonal twist of an Oh 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 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 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 is labeled with the sign of the corresponding lobe and only vertices of opposite sign can be connected by an . 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 , 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 alternative orbitals is sets based of d on 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 orbitals based on is based an 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 symmetrypentagonal [9–12 ] prism One set (Figure of these 1). equivalent These dsets orbitals are isuseful based for on an structures oblate (compressed) with five-fold pentagonal symmetry pentagonalcorresponding prism to (Figure symmetry 1). Thesepoint groupssets are such useful as D 5d for, D 5h structures, 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 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 for d structures orbitals belong with five-fold to 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 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 of set f of orbitals f orbitals is used is used except except for systems for systems with with high high enough enough symmetry symmetry to have to have The three general‐dimensional set of f orbitals irreducible is used representations. except for This systems general with set high consists enough of a unique symmetry 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. graph For andsystems m = ±2, having and± an point f{x(x 2groups − 3y2),−y (3withx2 − threey2)} set−‐dimensional with a withhexagon a orbital orbital graph graph and and mm == ±3.3. ForFor systems systems having having point point groups groups with with three-dimensional three-dimensional hexagonirreducible orbital representations, graph and m =such ±3. ±Foras the systems octahedral having O h point and the groups icosahedral with three Ih point-dimensional groups, the irreducibleirreducible representations, representations such, assuch the octahedral as the octahedralOh and the O icosahedralh and the icosahedIh pointral groups, Ih point the so-called groups, the irreducibleso‐called representations cubic set of f ,orbitals such as is theused. octahedral The cubic O seth and of f theorbitals icosahed consistsral Iofh point the triply groups degenerate, the cubicso set-called of f orbitalscubic set is used.of f orbitals The cubic is used. set of The f orbitals cubic consistsset of f orbitals of the triply consists degenerate of the triply f{x3,y 3degenerate,z3} set so-calledf{x3,y 3cu,z3bic} set set with of flinear orbitals orbital is used. graphs The and cubic a quadruply set of f orbitals degenerate consists f{xyz of , thex(z 2triply − y2), degeneratey(z2 − x2), z (x2 − withf{ linearx3,y3,z orbital3} set with graphs linear and orbital a quadruply graphs degenerateand a quadruply f{xyz, x degenerate(z2 y2), y( zf{2xyzx, 2x),(zz2( x−2 y2),y y2)}(z2 set. − x2), z(x2 − f{x3,y32,z3} set with linear orbital graphs and a quadruply degenerate f{xyz, x(z2 − y2), y(z2 − x2), z(x2 − y2)} set. − − − 2 y )} set. y )} set.

Molecules 2020, 25, x FOR PEER REVIEW 3 of 11 MoleculesMolecules 202020202020,, 2525, ,25, xx , FORFOR 3113 PEERPEER REVIEWREVIEW 33 oo3ff of1111 10 Molecules 2020, 25, x FOR PEER REVIEW 3 of 11 TableTable 2. 2.TheThe polynomials, polynomials, angular angular functions, functions, and and orbital orbital graphs graphs for for both both the the general general and and cubic cubic sets sets of of Table 2.2. The polynomials, angular functions, and orbital graphs for both the general and cubic sets of thethe seven seven f orbitals. f orbitals. thetheTable sevenseven 2. ff The orbitals.orbitals. polynomials, angular functions, and orbital graphs for both the general and cubic sets of |m| the sevenLobes f orbitals. Shape Orbital Graph General Set Cubic Set |m| |m| LobesLobes ShapeShape Orbital Orbital Graph General General SetSet CubicCubic Set Set |m| Lobes Shape Orbital Graph General Set Cubic Set x(x2 − 3y2) 2 2 3 6 Hexagon xx(((xx22 − 33y2))) none 3 36 6 HexagonHexagon y(3x2− − y2) nonenone y(3x22 y22) 2 y(3(3xx(x −− − y 3y)) ) 3 6 Hexagon none y(3x2 − y2) xyz xyz xyz xxyz(z2 − y2) 2 8 Cube xyz x2 (z2 2−xyz y2 ) 2xyzxyz 2 x(zx((zz2y −) y2)) 2 28 8 CubeCube z(x 2 − y2 ) y2 (−z 2− x ) zzz(((xx2 −xyz y2 ))) y(zy((zxz2(xz −2) x−2 y)) 2 ) 2 8 Cube − 2 −2 2 2 2 2 z(xz(x y −2) y ) 2 z(x − y ) zz((−xy2(z − y−2 x)) ) z(x2 − y2) Double x(5z2 − r2) 2 2 1 6 Double xx(5(5(5zz22 − r2)) none 1 16 6 DoubleSquare Square y(5z2 − r2 ) nonenone SquareDouble yy(5(5xz(5z2z −2 r−2 )r) 2) 1 6 y(5(5zz −− rr )) none Square y(5z2 − r2)

x3 3 3 x x3 0 04 4 LinearLinear zz(5(5zz2 − r2)) y3 y3 0 4 Linear z(5z2 − r2) 3 y3x 3 zz(5(5zz − rr )) z y3 2 2 z 3 0 4 Linear z(5z − r ) zz3y Fully using the complete sp3, sp3d5, and sp3d5f7 manifolds in hybridization schemes should lead Fully using the complete sp33, 3sp33d355, and5 sp33d535f77 5manifolds7 in hybridization schemes should3 lead to mostFullyFully nearly using using s phericalthe the complete complete polyhedra sp sp,, spsp, sp(Figured d,, andand, and 2 sp).sp spFord ffd the manifoldsf manifoldsfour-orbital in hybridization in sp hybridization3 system such schemes schemes a should shouldz lead leadis, to most nearly spherical polyhedra3 (Figure3 5 2). For3 the5 7 four-orbital sp33 system3 such a polyhedron is, tooftoto mostmost course, mostFully nearlynearly nearly the using familiarsspherical spherical the complete regular polyhedra polyhedra .sp , (Figuresp (Figured , and 2 ).For2). ). spFor For thed thethef the 16manifolds fourfour- four-orbitalorbital--orbital spin3 dsphybridization5 spf7 systemsystemmanifoldsystem suchsuch such theschemes aa mostpolyhedronpolyhedron a polyhedron should spherical leadis,is, is, of course, the familiar regular tetrahedron. For the 16--orbital sp33d355ff775 manifold37 the most spherical polyhedronofto most course, nearly is the the familiars phericaltetracapped regular polyhedra tetratruncated tetrahedron. (Figure 2tetrahedron, For). For the the 16-orbital four also-orbital of sp Td d sppointf systemmanifold group such symmetry. the a mostpolyhedron sphericalSuch a is, polyhedron is the tetracapped tetratruncated tetrahedron, also of Td3d point5 7 group symmetry. Such a 16polyhedronof-verte course,x the is the familiar tetracapped is the regular largest tetratruncated tetrahedron. Frank-Kasper For tetrahedron, polyhedron the 16-orbital also where ofsp Tdad Frankfpoint manifold-Kasper group the symmetry.polyhedron most spherical Such is a a 16-vertex deltahedron is the largest Frank-Kasper polyhedron where a Frank-Kasper polyhedron is a 16deltahedron16-vertexpolyhedron--vertex deltahedron deltahedron with is the only tetracapped is degree the is the largest 5 largestand tetratruncated Frank degree Frank-Kasper--Kasper 6 vertices tetrahedron, polyhedron polyhedronwith no also wherepair of whereof Ta adjacent dFrank point a Frank-Kasper-- Kaspergroup degree s polyhedronymmetry. 6 vertices polyhedron Such[14] is a. a deltahedron with only degree 5 and degree 6 vertices with no pair of adjacent degree 6 vertices [14]. deltahedronTheis16 -tetracapped averte deltahedronx deltahedron with tetratruncated only with degree is onlythe largest5 tetrahedrondegreeand degree Frank 5 and -6isKasper vertices rarely degree polyhedron found with 6 vertices noin expair whereperimental with of adjacent a no Frank pairmolecular -degreeKasper of adjacent 6 structurespolyhedron vertices degree [14][14] but is.. 6a The tetracapped tetratruncated tetrahedron is rarely found in experimental molecular structures but hasverticesdeltahedron been [ 14 recently]. with The tetracappedonly realized degree 5 tetratruncated and in degree the central 6 tetrahedronvertices Rh with4B12 is no rarelypolyhedron pair foundof adjacent in in experimental degree the rhodaborane6 vertices molecular [14] . has been recently realized in the central Rh44B1212 polyhedron in the rhodaborane Cp*structuresThe3Rh tetracapped3B12H12 butRh(B has tetratruncated4H9 beenRhCp* recently), synthesized tetrahedron realized by inis Ghoshrarely the central found and co Rhin- workersex4Bperimental12 polyhedron [15]. molecularThere in theis nostructures rhodaborane similarly but Cp*33Rh33B1212H1212Rh(B44H99RhCp*)),, synthesized synthesized by by Ghosh Ghosh and and4 co co12 --workers [[15]].. There is no similarly symmetricalCp*has 3Rh been3B12 mostH recently12Rh(B spherical4 H9 realizedRhCp*), 9-vertex synthesized inpolyhedron. the central by The Ghosh most Rh and Bspherical co-workerspolyhedron 9-vertex [15 deltahedron]. in There the is rhodaborane no is similarlythe D3h symmetricalsymmetrical3 3 12 mostmost12 sphericalspherical4 9 99--vertex polyhedron. The most spherical 9--vertex deltahedron is the D33hh tricappedsymmetricalCp* Rh BtrigonalH mostRh(B prism sphericalH RhCp* found 9-vertex) ,experimentally synthesized polyhedron. by in Ghoshthe The MH most and92− (N spherical co =- Tc,workers Re) 9-vertex anions [15]. [ deltahedronThere16,17]. is no is similarly the D3h 22−− tricappedtricapped trigonaltrigonal prismprism foundfound experimentallyexperimentally inin thethe MHMH99 (N2(N == Tc,Tc, Re)Re) anionsanions [[16,17].]. 3h tricappedsymmetrical trigonal most spherical prism found 9-vertex experimentally polyhedron. in theThe MHmost9 −spherical(N = Tc, 9 Re)-vertex anions deltahedron [16,17]. is the D tricapped trigonal prism found experimentally in the MH92− (N = Tc, Re) anions [16,17].

Tetracapped Tetratruncate d T ricapped TTeetrtraaccaappppeedd Tetrahedron T T eTetretrtraatratrhuuenndccraaotetendd T r iTTgricappedricappedonal Prism TTedetr trsayahmheemddrerootrnny TT ed eT trtrseayatrhmhaeemcddarerpootrpnnyed T T rDriig3ghoo snnyaamll PrPrm ieisstrmmy T sym m etry TT Tesstryymamtrmmueentrtrcyayte d TT dd ssyymm mm eetrtryy D Tsricappedym m etry T dd sym m etry DD 33hh ssyymm mm eetrtryy Tetrahedron 4 v Tetrrtiahceedrson 9 T vrigeorntiacl Presism T d sym m etry 1 6 T vd seyrmtimcetrsy 4 verrtiti3ces 9 vD e3hr stiymcem setry sp 9 v sperr3titidc5es 16 ve3rrtiti5c7es 33 3 5 sp d f spsp spsp33d55 33 55 77 4 vertices 9 vertices sp1sp6 vderfftices 3 sp sp3d5 3 5 7 Figure 2. The most spherical deltahedra corresponding to filled 4- orbital sp d spf 3, 9-orbital sp3d5, and Figure 2. The most spherical deltahedra corresponding to filled 4-orbital sp33, 9-orbital sp33d55, and Figure16-orbital 2. spThe3d 5 mostf7 manifolds. spherical deltahedra corresponding to filled 4--orbitalorbital sp sp ,, 9 9--orbitalorbital sp sp d ,, and and 3 5 7 3 3 5 1616Figure--orbitalorbital 2. spsp The33dThe55ff7 7 mostmanifolds. most spherical spherical deltahedra deltahedra corresponding corresponding to tofilled filled 4-orbital 4-orbital sp3,sp 9-orbital, 9-orbital sp3d sp5, andd , and 16-orbital sp3d5f7 manifolds. 16-orbital sp3d5f7 manifolds.

Molecules 2020, 25, x FOR PEER REVIEW 4 of 11 Molecules 2020, 25, 3113 4 of 10

2. Polyhedral Hybridizations 2. Polyhedral Hybridizations 2.1. Polyhedra with Tetragonal Symmetry 2.1. PolyhedraThe simplest with Tetragonal configurat Symmetryion with tetragonal symmetry is the square, found as a coordination polyhedronThe simplest in diamagnetic configuration four with-coordinate tetragonal complexes symmetry of d8 is metals the square, such as found Ni(II), as Pd( a coordinationII), Pt(II), polyhedronRh(I), and inIr(I). diamagnetic Since a square four-coordinate is a two-dimensional complexes structure of d8 metals in an xy such plane, as Ni(II),the polynom Pd(II),ials Pt(II), of the Rh(I), andatomic Ir(I). Sinceorbitals a square for square is a two-dimensional hybridization cannot structure contain in an a xyz varplane,iable. the This polynomials limits the of p the orbital atomic orbitalsinvolvement for square in square hybridization hybridization cannot to containthe p{x} aandz variable. p{y} orbitals This so limits that thea d porbital orbital is involvementrequired for in square hybridization. This d orbital can be either the d{x2 − y2} or d{xy} orbital, depending on square hybridization to the p{x} and p{y} orbitals so that a d orbital is required for square hybridization. whether the x and y axes are chosen to go through the vertices or edge midpoints, respectively, of the This d orbital can be either the d{x2 y2} or d{xy} orbital, depending on whether the x and y axes are square (Table 3). − chosen to go through the vertices or edge midpoints, respectively, of the square (Table3). Table 3. Hybridization schemes for the tetragonal polyhedra. Table 3. Hybridization schemes for the tetragonal polyhedra. Polyhedron Coord. Hybridization (Symmetry)Polyhedron Coord.No. Type s + p Hybridization d f (Symmetry) No. Type s + p d f 2 2 2 Square (D4h) 4 2 sp A1g + Eu B1g 2{x −2 y } — Square (D4h) 4 sp A1g + Eu B1g {x y } — Square Bipy (D4h) 6 3 2sp3d2 A1g + A2u + Eu A1g{2z2} + B−1g2 {x2 −2 y2} — Square Bipy (D4h) 6 sp d A1g + A2u + Eu A1g{z } + B1g {x y } — 3 2 3 2 2 2 2 2 − OctahedronOctahedron (O (hO)h) 6 6 sp dsp d A1g +AT1g1u + T1u EgE{gx{x y− ,yxy,}xy} —— 3 3 2 2− 2 2 Square Prism(D h) 8 sp d f 3 3 A1g + A2u + Eu B {x 2 y }2+ Eg{xz,yz) B { z(x2 y2 )} Square Prism(4D4h) 8 sp d f A1g + A2u + Eu B1g1g {x− − y } + Eg{xz,yz) B2u2u{ z(x −− y )} Cube(O ) 8 3 3 A + T T {xy,xz,yz} A {xyz} h sp d f 3 3 1g 1u 2g 2u Cube(Oh) 8 sp d f A1g + T1u T2g{xy,xz,yz} A2u{3xyz} 3 4 2 2 2 2 A2u(z } + B2u{ Bicapped Cube(D h) 10 sp d f A1g + A2u + Eu A {z } + B {x y } + Eg{xz,yz) 4 1g A1g{z1g2} + B−1g {x2 − y2} + A2uz((zx32} +y B2)}2u{ Bicapped Cube(D4h) 10 sp3d4f2 A1g + A2u + Eu − Square 3 4 2 2 g 2 2 8 sp d A1 + B2 + E1 E2{x y E,xy{}xz+,Eyz3{)xz ,yz) z(x −— y )} Antiprism(D4d) − Square 3 5 2 2 2 3 Bicap Sq Antipr(D d) 10 sp d f 3 4 A + B + E A {z } + E2 {x 2 y ,xy} + E {xz,yz)B {z } 4 8 sp d 1 A21 + B12 + E1 1 E2{x 2 − y−,xy} + E3{3xz,yz) —2 Antiprism(D4d) A1{z2} + E2{x2 − y2,xy} + BicapThe Sq smallest Antipr( three-dimensionalD4d) 10 sp3d polyhedron5f A1 + B with2 + E1 tetragonal symmetry of interest inB this2{z3}context E3{xz,yz) is the regular octahedron with the O containing both three-fold and four-fold axes The smallest three-dimensional polyhedronh with tetragonal symmetry of interest in this context (Figure3). This is the most frequently encountered polyhedron in coordination chemistry and is is the regular octahedron with the Oh point group containing both three-fold and four-fold axes also(Figure the most 3). This spherical is the most 6-vertex frequentlycloso encountereddeltahedron polyhedron in the structures in coordination of boranes chemistry and related and isspecies. also ThetheO mosth point spherical group contains6-vertex two-closo deltahedron and three-dimensional in the structures irreducible of boranes representations. and related species. Under octahedral The Oh symmetry,point group the fivecontains d orbitals two- and split three into- adimensional triply degenerate irreducible T2g{xz representations.,yz,xy} set and a Under doubly octahedral degenerate 2 2 3 Egsymmetry{x y ,xy, }theset. five Octahedral d orbitals split hybridization into a triply supplements degenerate T2g the{xz four-orbital,yz,xy} set and sp a doublyset with degenerate the doubly − 2 2 degenerateEg{x2 − y2, Exyg}{ x set. yOctahedral,xy} set of hybridization d orbitals. supplements the four-orbital sp3 set with the doubly − degenerate Eg{x2 − y2,xy} set of d orbitals.

Bicapped Regular Square Square Bicapped Octahedron Cube (Oh) Antiprism Antiprism (O ) Cube (D4h) (D ) h 4d (D4d)

Figure 3. Tetragonal polyhedra with at least one C4 rotation axis. Figure 3. Tetragonal polyhedra with at least one C4 rotation axis. An octahedral coordination complex can be stretched or compressed along a four-fold axis An octahedral coordination complex can be stretched or compressed along a four-fold axis thereby thereby reducing the symmetry from Oh to D4h by removing the three-fold axes. This corresponds to reducing the symmetry from O to D by removing the three-fold axes. This corresponds to the the Jahn–Teller effect [18] leadingh to 4h a structure that can be designated as a square , Jahn–Teller effect [18] leading to a structure that can be designated as a square bipyramid, analogous to analogous to pentagonal and hexagonal discussed later in this paper. This reduction in pentagonal and hexagonal bipyramids discussed later in this paper. This reduction in symmetry of an symmetry of an octahedral metal complex splits the triply degenerate T1u p orbitals into a doubly octahedral metal complex splits the triply degenerate T1u p orbitals into a doubly degenerate E1u{x,y} 2 2 set and a non-degenerate A {z} orbital and the doubly degenerate Eg{x y ,xy} set of orbitals into 2u − non-degenerate A {z2} and B {x2 y2} orbitals. 1g 1g − Molecules 2020, 25, x FOR PEER REVIEW 5 of 11

Molecules 2020, 25, 3113 5 of 10 degenerate E1u{x,y} set and a non-degenerate A2u{z} orbital and the doubly degenerate Eg{x2 − y2,xy} set of orbitals into non-degenerate A1g{z2} and B1g{x2 − y2} orbitals. TheThe most most symmetrical symmetrical polyhedron polyhedron for for an an 8 8-coordinate-coordinate complex complex is is the the cube, cube, which which is is the the dual dual of of the regular octahedron and, thus, also exhibits the Oh point group (Figure 3). Considering the cube as the regular octahedron and, thus, also exhibits the Oh point group (Figure3). Considering the cube as twotwo squares squares stacked stacked on on top top of of each each other at aa distancedistance leadingleading toto thethe three-fold three-fold symmetry symmetry elements elements of of the Oh point group leads to a simple way of deriving the atomic orbitals for cubic hybridization. the Oh point group leads to a simple way of deriving the atomic orbitals for cubic hybridization. Thus, Thusto form, to a form set of a cubic set of hybrid cubic orbitals, hybrid the orbitals atomic, the orbitals atomic for squareorbitals hybridization for square hybridization are supplemented are supplementedby the four atomic by the orbitals four atomic in which orbitals the polynomials in which the for polynomials square hybridization for square are hybridization multiplied byarez . 2 multipliedIn this way, by a setz. In of pdthis2f way orbitals, a set with of eachpd f atomicorbitals orbital with havingeach atomic one more orbital node having than the one corresponding more node 2 2 thanorbital the incorresponding the sp2d orbital orbital setfor in squarethe sp d hybridization orbital set for is square added hybridization to the sp2d square is added hybridization to the sp d to 2 2 3 3 squareform anhybridization (sp2d)(pd2f) to= formsp3d 3anf set (sp ford)(pd cubicf) hybridization.= sp d f set for cubic This showshybridization. that one This f orbital, shows namely that one the f orbital, namely the A2u{xyz} orbital with a cube orbital graph (Table 2), is required for cube A2u{xyz} orbital with a cube orbital graph (Table2), is required for cube hybridization. This suggests hybridization. This suggests that 8-coordinate ML8 complexes with cubic coordination are likely to that 8-coordinate ML8 complexes with cubic coordination are likely to be restricted to lanthanide be restricted to lanthanide and actinide chemistry, particularly the latter. Reducing the symmetry of and actinide chemistry, particularly the latter. Reducing the symmetry of the cube from Oh to D4h by theelongation cube from or compressionOh to D4h by elongation along a four-fold or compression axis thereby along eliminating a four-fold the axis three-fold thereby symmetry eliminating does the not three-fold symmetry does not eliminate the need for an f orbital in the hybridization of an ML8 eliminate the need for an f orbital in the hybridization of an ML8 complex. complex.The need for an f orbital in the hybridization for an 8-coordinate metal complex can be eliminated The need for an f orbital in the hybridization for an 8-coordinate metal complex can be by converting a cube into a by rotating a square of a cube 45◦ around the C4 axis eliminated by converting a cube into a square antiprism by rotating a square3 4 face of a cube 45° thereby changing the Oh point group into D4d (Figure3 and Table3). The sp d hybridization for the 3 4 aroundsquare the antiprism C4 axis omits thereby the changing d{z2} orbital. the Oh point group into D4d (Figure 3 and Table 3). The sp d hybridization for the square antiprism omits the d{z2} orbital. Capping the square faces of a square antiprism retaining the D4d symmetry leads to the most Capping the square faces of a square antiprism retaining the D4d symmetry2 leads to the 2most spherical closo deltahedron for 10-vertex borane derivatives, such as B10H10 − (Figure3). The d{ z } and 2− 2 sphericalf{z3} orbitals, closo deltahedron both of which for lie 10 along-vertex the borane four-fold derivativesz axis, are, such added as B to10H th10 sp (Figure3d4 hybridization 3). The d{z } of and the 3 3 4 f{squarez } orbitals, antiprism both of to which give the lie sp along3d5f{ zthe3} hybridizationfour-fold z axis of, theare bicappedadded to squareth sp d antiprism. hybridization In a similarof the 3 5 3 squareway the antiprism d{z2} and to f{ zgive3} orbitals the sp cand f{ bez } added hybridization to the sp 3ofd 3thef hybridization bicapped square of the antiprism. cube to give In the a similar sp3d4f 2 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 D4d symmetry. hybridization of the bicapped cube with D4d symmetry. 2.2. Polyhedra with Pentagonal Symmetry 2.2. Polyhedra with Pentagonal Symmetry The simplest structure with pentagonal symmetry is the planar . Using a set of five orbitalsThe withsimplest only structurex and y in with the pentagonal orbital polynomial symmetry leads is tothe sp planar2{x,y}d pentagon.2(x2 y2,xy Using} hybridization a set of five for − orbitalsplanar pentagonwith only coordination x and y in the with orbital minimum polynomial l values. leads Either to thesp2{ prolatex,y}d2(x or2 − oblate y2,xy} setshybridization of equivalent for d planarorbitals pentag (Figureon1 coordination) can provide with alternative minimum d 5 hybridizationl values. Either for the a planarprolate pentagonor oblate structure,sets of equivalent but with dhigher orbitals combined (Figure 1) l valuescan provide for the alternative five-orbital d set5 hybridization and poor overlap for a betweenplanar pentagon the ligand str orbitalsucture, but and withthose higher of the combined central atom. l values for the five-orbital set and poor overlap between the ligand orbitals and thoseThe of smallest the central three-dimensional atom. polyhedron of interest with pentagonal symmetry is the D5h pentagonalThe smallest bipyramid three-dimensional (Figure4). The polyhedron sp 3d3 scheme of interest for with pentagonal pentagonal bipyramidal symmetry coordination is the D5h pentagonalsupplements bipyramid the five-orbital (Figure sp 4).2{x ,y The}d2 (x sp2 3d3y 2scheme,xy} hybridization for pentagonal for the bipyramidal equatorial pentagon coordination with − supplementsthe p{z} and the d{z 2five} orbitals-orbital for sp2 the{x,y linear}d2(x2 − sub-coordination y2,xy} hybridization of the for axialthe equatorial ligands perpendicular pentagon with to the the p{equatorialz} and d{ planarz2} orbitals pentagon for the (Table linear4). sub-coordination of the axial ligands perpendicular to the equatorial planar pentagon (Table 4).

Pentagonal Pentagonal Bicapped Pentagonal Regular Bipyramid Prism (D5h) (D ) (I ) 5h Prism (D5h) (D5h) h

FigureFigure 4. 4. PentagonalPentagonal polyhedra polyhedra with with at at least least one one CC5 5rotatrotationion axis. axis.

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Table 4. Hybridization schemes for the pentagonal polyhedra.

Polyhedron Coord. Hybridization (Symmetry) No. Type s + p d f 2 2 2 2 Pentagon (D5h) 5 sp d A1’ + E1´ E2´{x y } — 3 3 2 − 2 2 Pent Bipy (D5h) 7 sp d A1´ + A2´´ + E1´ A1´{z } + E2´{x y ,xy} — 3 4 2 2 2 − 2 2 Pent Prism (D5h) 10 sp d f A1´ + A2´´ + E1´ E2´{x y } + E1´´{xz,yz}E2´´{xyz,z(x y )} − 3 − Bicap Pent 3 5 3 2 2 2 A2´´{z } + 12 sp d f A1´ + A2´´ + E1´ A1´{z } + E2´{x y ,xy} + E1´´{xz,yz} 2 2 Prism(D5h) − E2´´{xyz,z(x y )} 3 4 2 2 2 − Pent Antiprism(D ) 10 sp d f A + A + Eu E {xz,yz} + E {x y ,xy} E 5d 1g 2u 1g 2g − 2u Bicap Pent 3 5 3 2 2 2 3 12 sp d f A1g + A2u + Eu A1g{z } + E1g{xz,yz} + E2g{x y ,xy} A2u{z } + E2u Antipr(D5d) − 3 5 3 3 3 3 Icosahedron (Ih) 12 sp d f A1g + T1u Hg{all d orbitals} T2u{x ,y ,z }

The 10-vertex of D5h symmetry (Figure4) is encountered experimentally in the 3– + endohedral trianions M@Ge10 (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 stacked on top of each other to preserve the D symmetry supplements the sp2{x,y}d2(x2 y2,xy} of the planar 5h − 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 − − 2 3 in a way to preserve the D5h symmetry adds the d{z } and f{z } 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 D5d symmetry (Figure4). 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 Dnd symmetry of an n-gonal antiprism is preserved by compression or elongation along the major Cn axis to give oblate or prolate polyhedra, respective, as illustrated in Figure1 for the pentagonal antiprisms representing the five equivalent d orbitals. The , 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 D5d symmetry of the bicapped pentagonal antiprism to give the full icosahedral point group Ih. This ascent in symmetry combines some of the irreducible representations of the D5h point group to give irreducible representations of higher degeneracy. As a result, the A1g + E1g + E2g irreducible representations coalesce into the single five-fold degenerate Hg representation of the icosahedral point group. The five d orbitals belong to this Hg representation in . Similarly, the 3 5 3 3 3 3 three f orbitals of the sp d f icosahedral hybridization belong to the single T2u{x ,y ,z } irreducible representation of the Ih point group considering the cubic set of f orbitals (Table2).

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 (Figure5). The sp 3d3f 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 Molecules 2020, 25, x FOR PEER REVIEW 7 of 11 Molecules 2020, 25, 3113 7 of 10

supplementing the sp2{x,y}d2{x2 − y2,xy}f{x(x2 − 3y2)} set of orbitals for the equatorial planar hexagon withwith the cubep{z} and (see d{ above)z2} pair the for feature the two of not additional arising fromaxial any vertices. sp3d4 Thehybridization hexagonal scheme bipyramid but insteadshares withrequiring the cube an f (see orbital above) in a spthe3 dfeature3f hybridization of not arising scheme. from any sp3d4 hybridization scheme but instead requiring an f orbital in a sp3d3f hybridization scheme.

Hexagonal Hexagonal Bicapped Hexagonal Bicapped Bipyramid Prism (D6h) Hexagonal Antiprism Hexagonal (D6h) Prism (D6h) (D6d) Antiprism (D6d)

Figure 5. Hexagonal polyhedra with at least one C6 rotation axis. Figure 5. Hexagonal polyhedra with at least one C6 rotation axis.

TheThe atomic atomic orbitals orbitals required required to to form form a a prism prism combine combine those those for for the the polygonal polygonal face face having having only only xx andand yy inin their their orbital orbital polynomials polynomials with with an equal an equal number number of orbitals of orbitals in which in the which polynomials the polynomials obtained obtainedby multiplying by multiplying the orbital the polynomials orbital polynomials for the planar for face the by planar the z facevariable. by the Thus, z variable. the sp2{ xThus,y}d2,{ xthe2 − spy22,xy{x,}f{y}dx(2x{x22 −3 yy22,xy)} orbitals}f{x(x2 − corresponding3y2)} orbitals corresponding to the hexagonal to facethe hexagonal of the hexagonal face of prism the hexagonal is supplemented prism − isby supplemented the p{z3}d2{xz by,yz the}f2{ xyzp{z,3z}d(x22{xz,yzy2}f)}g{2{xyzxz(,xz(2x2 −3 y22)}g{)} setxz( ofx2 − six 3y orbitals.2)} set of six This orbitals. leads to This the leads interesting to the − − interestingobservation observation that a single that g orbital a single corresponding g orbital corresponding to the B2g irreducible to the B2g irreducible representation representation is required tois requiredprovide ato set provide of 12 hybrida set of orbitals12 hybrid oriented orbital towardss oriented the towards vertices the of vertices a D6h hexagonal of a D6h (Table prism5). (TableNot surprisingly, 5). Not surprisingly, this g orbital this has g a hexagonalorbital has prism a hexagonal orbital graph. prism Capping orbital graph. the hexagonal Capping prism the 2 3 2 3 hexagonalin a way to prism preserve in a way the D to6h preservesymmetry the adds D6h symmetry the d{z } andadds f{ thez } orbitalsd{z } and to f{ thez } orbitals hybrid to leading the hybrid to an leadsp3ding5f3 {toz3 ,anxyz sp,z(3xd25f3{zy32,xyz)} for,z(x the2 − resultingy2)} for the 12-vertex resulting bicapped 12-vertex pentagonal bicapped pentagonal prism. prism. − TableTable 5. HybridizationHybridization schemes schemes for for the the hexagonal hexagonal polyhedra.

PolyhedronPolyhedron Coord. Coord. HybridizationHybridization (Symmetry) No. Type s + p d f g (Symmetry) No. Type s + p d f g 2 2 2 2 2 2 Hexagon (D6h) 6 sp d f A1g + E1u E {x y } B {x(x 3y )} — 2 2 2g 2− 2 1u − 2 2 Hexagon (HexagonalD6h) 6 sp d f A1g + E1u E2g{x − y } B1u{x(x − 3y )} — 8 sp3d3f A + A + E A {z2} + E {x2 y2,xy} B {x(x2 3y2)} — Bipy(D ) 1g 2u 1u 1u 2g 1u Hexagonal Bipy(6h D6h) 8 sp3d3f A1g + A2u + E1u A1u{z2} + E2g{−x2 − y2,xy} B−1u{x(x2 − 3y2)} — Hexagonal 3 4 3 2 2 2 2 12 sp3 d4 f3 g A1g + A2u + E1u E1g{xz,yz} + E2g{x y 2,xy} 2 B1u + E2u{xyz,z(x y )} B2g2 2 HexagonalPrism( Prism(D6h) D6h) 12 sp d f g A1g + A2u + E1u E1g{xz,yz} + E2g−{x − y ,xy} B1u + E2u−{xyz,z(x − y )} B2g Bicap Hex 3 3 55 44 2 2 2 2 2 2 3 3 Bicap Hex Prism(D6h) 1414 spspddff gg A1gA 1g+ +AA2u2u ++ EE1u1u A1u1u{{zz} }+ +E 1gE{xz1g{,yzxz}+,yzE2g}+E{x 2gy{x,xy }− yA,xy2u{z} } +AB2u1u{z+ E} 2u+ B1u + EB2g2u B2g Prism(D6h) − Hex 6d 3 4 4 1 2 1 2 2 2 5 3 4 2 2 Hex Antiprism(D ) 1212 spspd3df4f 4 A A+ +BB++ EE E {{xx2 −y 2y,xy,}xy+ }E +{xz,yz E {}Exz,yz} + E {Exyz ,+z( xE2 {xyzy2)} ,z(x — − y )} — D 1 2 1 2 5 3 4 Antiprism( 6d) 3 5 5 2− 2 2 3 − Bicap HexBicap Antipr( Hex D5d) 14 sp d f A1 + B2 + E1 A1{z } + E2{x − y ,xy} + E5{xz,yz} B2(z ) + E3 + E4 — 14 sp3d5f5 A + B + E A {z2} + E {x2 y2,xy} + E {xz,yz}B (z3) + E + E — Antipr(D ) 1 2 1 1 2 5 2 3 4 Twi5stingd one hexagonal face of a hexagonal prism 30°− around the C6 axis leads to the hexagonal antiprism of D6d symmetry (Figure 5). Going from the D6h symmetry of the hexagonal prism to the D6d symmetryTwisting oneof the hexagonal hexagonal face antiprism of a hexagonal in a 12 prism-vertex 30 hexagonal◦ around thesymCmetry6 axis leadssystem to eliminates the hexagonal the needantiprism for a g of orbitalD6d symmetry in the set of (Figure 12 atomic5). Going orbitals from forming the Dthe6h correspondingsymmetry of the hybrid hexagonal orbitals. prism This tois 3 3 analogousthe D6d symmetry to the conversion of the hexagonal of the cube antiprism requiring in a sp 12-vertexd f hybridization hexagonal to symmetry the square system antiprism eliminates with spthe3d need4 hybridization for a g orbital not requiring in the set an of f 12 orbital atomic by orbitalsrotation forming one square the face corresponding 45° relative hybridto its op orbitals.posite partner.This is analogous Capping tothe the hexagonal conversion faces of theof cubethe hexagonal requiring spantiprism3d3f hybridization in a way to to preserve the square its antiprism six-fold 3 4 symmetrywith sp d leadshybridization to the bicapped not requiring hexagonal an f antiprism, orbital by which rotation is one of significance square face as 45 being◦ relative the mostto its sphericalopposite partner.closo 14-vertex Capping deltahedron the hexagonal in borane faces chemistry. of the hexagonal This capping antiprism process in a adds way tod{z preserve2} and f{z its3} orbitalssix-fold symmetryto the 12-orbital leads to hybridization the bicapped scheme hexagonal for antiprism, the hexagonal which antiprism is of significance leading as to being a sl3d the5f5 hybridizationmost spherical schemecloso 14-vertex for the bicapped deltahedron hexagonal in borane antiprism chemistry. not Thisrequiring capping g orbitals process (Table adds 5). d{z 2} and f{z3} orbitals to the 12-orbital hybridization scheme for the hexagonal antiprism leading to a sl3d5f5 2.4.hybridization Trigonal Twist scheme Processes for the in bicappedPolyhedral hexagonal of Octahedral antiprism Symmetry not requiring g orbitals (Table5).

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Molecules 2020, 25, x FOR PEER REVIEW 8 of 11 2.4. Trigonal Twist Processes in Polyhedral of TheThe octahedron octahedron and and the the cuboctahedron cuboctahedron are are the the simplest simplest two two polyhedra polyhedra with with triangular triangular faces faces and and Oh point group symmetry having both three-fold and four-fold rotation symmetry axes. The Oh point group symmetry having both three-fold and four-fold rotation symmetry axes. The process of convertingprocess of prisms converting of n-fold prisms symmetry of n-fold (n = symmetry4, 5, and 6 including (n = 4, 5, the and cube 6 including as a special the tetragonal cube as prism) a special to tetragonal prism) to the corresponding antiprisms involves a twist of (180/n)° of an n-fold face the corresponding antiprisms involves a twist of (180/n)◦ of an n-fold face around the unique n-fold axis. Analogousaround the processes unique n can-fold be axis.applied Analogous to the three-fold processes axes can in bethe applied regular octahedron to the three and-fold icosahedron. axes in the regular octahedron and icosahedron. Such a twist process converts the Oh regular octahedron into Such a twist process converts the Oh regular octahedron into the D3h trigonal prism and the Oh the D3h trigonal prism and the Oh cuboctahedron into the D3h anticuboctahedron (Figure 6). These cuboctahedron into the D3h anticuboctahedron (Figure6). These processes can be regarded as trigonal twistsprocesses involving can three be symmetry regarded related as diamond-square-diamond trigonal twists involving transformations. three symmetry For octahedral related diamond-square-diamond transformations. For octahedral coordination complexes of the type coordination complexes of the type M(bidentate)3 involving bidentate chelating ligands variations of thisM(bidentate) process are3 involving designated bidentate as Bailar twists chelating [21 ] ligands or Ray-Dutt variations [22] twists. of this process are designated as Bailar twists [21]or Ray-Dutt [22]twists.

3 x dsd

O c ta h e d ron Trigonal Prism Oh D3h

3 x dsd

Cuboctahedron Anticuboctahedron D Oh 3h

Figure 6. Polyhedra with octahedral (Oh) symmetry and their conversions by a triple diamond-square- Figure 6. Polyhedra with octahedral (Oh) symmetry and their conversions by a triple diamond process to a polyhedron with D3h symmetry. diamond-square-diamond process to a polyhedron with D3h symmetry.

Such trigonal twist processes can be studied by first relaxing the symmetry of the original Oh Such trigonal twist processes can be studied by first relaxing the symmetry of the original Oh polyhedron to D3d, which is the subgroup of Oh obtained by removing the C4 axes. Under this polyhedron to D3d, which is the subgroup of Oh obtained by removing the C4 axes. Under this subgroup, the octahedron can be viewed as a trigonal antiprism with the z axis corresponding to subgroup, the octahedron can be viewed as a trigonal antiprism with the z axis corresponding to the the C3 axis rather than the C4 axis in the discussion above of polyhedral with C4 symmetry. Under C3 axis rather than the C4 axis in the discussion above of polyhedral with C4 symmetry. Under D3d D3d symmetry rather than the higher Oh symmetry and with the different location of the z axis, the symmetry rather than the higher Oh symmetry and 2 with2 the different location of the z axis,3 the2 degenerate Eg{xz,yz} pair of d orbitals as well as the Eg(x y ,xy}pair can be used for octahedral sp d degenerate Eg{xz,yz} pair of d orbitals as well as the Eg(x−2 − y2,xy}pair can be used for octahedral sp3d2 hybridization (Table6). When the twist reaches the stage of the D3h trigonal prism only the E´{xz,yz} hybridization (Table 6). When the twist reaches the stage of the D3h trigonal prism only the E´{xz,yz} pair of atomic orbitals is available for the sp3d2 hybridization scheme. Thus, the trigonal twist of an pair of atomic orbitals is available for the sp3d2 hybridization scheme. Thus, the trigonal twist of an x2 2 xy D octahedron to a trigonal prism involves replacement of the Eg{ 2 y ,2 }pair of orbitals in the 3d octahedron to3 2a trigonal prism involves replacement of the Eg{x− − y ,xy}pair of orbitals in the D3d octahedral sp d hybrids with the E´{xz,yz} pair of orbitals in the D3h trigonal prismatic hybrids. octahedral sp3d2 hybrids with the E´{xz,yz} pair of orbitals in the D3h trigonal prismatic hybrids.

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Table 6. Hybridization schemes for polyhedra with Oh symmetry and the D3h polyhedral forme from them by a triple diamond-square-diamond process.

Polyhedron Coord. Hybridization (Symmetry) No. Type s + p d f 3 2 2 2 Octahedron (D3d) 6 sp d A1g + A2u + Eu Eg(x y ,xy}or Eg{xz,yz} — 3 2 − Trigonal Prism (D3h) 6 sp d A1´ + A2´´ + E1´ E´´{xz,yz}— 2 2 2 2 2 3 5 3 2 2 T2u{x(z y ),y(z x ),z(x Cuboctahedron (Oh) 12 sp d f A1g + T1u Eg(x y ,xy} + T {xy,xz,yz} − − − 2g y2)} 2 2 2 − 3 5 3 A1g{z } + Eg(x y ,xy} + 3 Cuboctahedron (D3d) 12 sp d f A1g + A2u + Eu − A2u{z } + Eu Eg{xz,yz} 2 2 2 3 5 3 A1´{z } + E´(x y ,xy} + 2 2 2 2 Anticuboctahedron(D h) 12 sp d f A ´ + A ´´ + E ´ − A ´{x(x 3y } + E´{xz ,yz } 3 1 2 1 E´´{xz,yz} 1 −

3 5 3 The sp d f hybridization for a cuboctahedron using its full Oh symmetry taking the z axis as a C4 axis supplements the full 9-orbital sp3d5 set with the triply degenerate T {x(z2 y2),y(z2 x2),z(x2 2u − − y2)} set of the cubic f orbitals (Table6). Reducing the symmetry of the cuboctahedron to D and − 3d now taking the C3 axis as the z axis splits, the triply degenerate set of f orbitals into a non-degenerate A {z3} orbital and a doubly degenerate Eu set of orbitals, which can be either the Eu{xyz,z(x2 y2)} 1u − 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 D3d cuboctahedron to the D3h 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 Dnh 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 3 3 with four-fold symmetry twisting a square face of a cube with sp d f hybridization by 45◦ around a 3 4 C4 axis to give a square antiprism with sp d 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 3 5 3 the need for a g orbital in the resulting sp d f hybridization. A trigonal twist of an Oh octahedron 3 2 into a D3h trigonal prism can involve a gradual change of the pair of d orbitals in the sp d hybrids. A similar trigonal twist of an Oh cuboctahedron into a D3h cuboctahedron can likewise involve a gradual change in the three f orbitals in the sp3d5f3 hybridization.

Funding: This project did not receive any external funding. Conflicts of Interest: The author declares no conflict of interest. Molecules 2020, 25, 3113 10 of 10

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