Metal orbitals

The main drawback of the chister-m-chister methods is that the embedding operators are derived from a wavefunction that does not reflect the proper periodicity of the crystal a two-dimensionally infinite wavefiinction/density with a proper band structure would be preferable. Indeed, Rosch and co-workers pointed out recently a series of problems with such chister-m-chister embedding approaches. These include the lack of marked improvement of the results over finite clusters of the same size, problems with the orbital space partitioning such that charge conservation is violated, spurious mixing of virtual orbitals into the density matrix [170], the inlierent delocalized nature of metallic orbitals [171], etc. 

The bonding between carbon monoxide and transition-metal atoms is particularly important because transition metals, whether deposited on soHd supports or present as discrete complexes, are required as catalysts for the reaction between carbon monoxide and most organic molecules. A metal—carbon ( -bond forms by overlapping of metal orbitals with orbitals on carbon. Multiple-bond character between the metal and carbon occurs through formation of a metal-to-CO TT-bond by overlap of metal-i -TT orbitals with empty antibonding orbitals of carbon monoxide (Fig. 1). 

Figure 19.18 Schematic representation of the orbital overlaps leading to M-CO bonding (a) a overlap and donation from the lone-pair on C into a vacant (hybrid) metal orbital to form a u M <—C bond, and (b) 7T overlap and the donation from a filled d or dj orbital on M into a vacant antibonding n orbital on CO to form a tt M—> C bond.

Figure 19.23 SchemalicillDslralionofpossiblecom-binplions of orbitals in the rr-allyiic complexes. The bonding dirccrion is taken lo be the z-axis with the M aiom below the Cj plane Approprisle combinations of p, orbitals on the 3 C arc shown in the top half of Ihe figure, and beneath them are the metal orbitals with which Ihey arc mosl likely to form bonding inieraciions.

C. The structure, which involves two bridging carbonyl groups as shown in Fig. 26.8a, can perhaps be most easily rationalized on the basis of a bent Co-Co bond arising from overlap of angled metal orbitals (d sp hybrids). However, in solution this structure is in equilibrium with a second form (Fig. 26.8b) which has no bridging carbonyls and is held together solely by a Co-Co bond. 

The ligands interact with the two orbitals of cr-symmetry modifying the ordering somewhat (Figure 2.37b). As has been pointed out, altering the relative positions of the metal orbitals relative to those of the carboxylates affects the final scheme considerably (Figure 2.38). 

Unsaturated organic molecules, such as ethylene, can be chemisorbed on transition metal surfaces in two ways, namely in -coordination or di-o coordination. As shown in Fig. 2.24, the n type of bonding of ethylene involves donation of electron density from the doubly occupied n orbital (which is o-symmetric with respect to the normal to the surface) to the metal ds-hybrid orbitals. Electron density is also backdonated from the px and dM metal orbitals into the lowest unoccupied molecular orbital (LUMO) of the ethylene molecule, which is the empty asymmetric 71 orbital. The corresponding overall interaction is relatively weak, thus the sp2 hybridization of the carbon atoms involved in the ethylene double bond is retained. 

Variability in metallic valency is also made possible by the resonance of atoms among two or more valence states. In white tin the element has valency approximately 2-5, corresponding to a resonance state between bicovalent tin, with a metallic orbital, and quadricovalent tin, without a metallic orbital, in the ratio 3 to 1 and copper seems similarly in the elementary state to have metallic valency 5-5. 

Finally, the use of simple valence bond theory has led recently to a significant discovery concerning the nature of metals. Many years ago one of us noticed, based on an analysis of the experimental values of the saturation ferromagnetic moment per atom of the metals of the iron group and their alloys, that for a substance to have metallic properties, 0.72 orbital per atom, the metallic orbital, must be available to permit the unsynchronized resonance that confers metallic properties on a substance.34 38 Using lithium as an example, unsynchronized resonance refers to such structures as follows. 

Successive pivoting resonances of a covalent bond allows for electrical conduction to occur, as shown in Figure 1-1. A test of this theory was provided by gray and white tin. Gray tin is not metallic because all its valence orbitals are used for bonding and there is no metallic orbital available. White tin, on the other hand, has the metallic orbital available and therefore has metallic properties. 

The development during the past year of a statistical theory of unsynchronized resonance of covalent bonds in a metal, with atoms restricted by the electroneutrality principle to forming bonds only in number u — 1, u, and v + 1, with u the metallic valence, has led directly to the value 0.70 0.02 for the number of metallic orbitals per atom.39 This theory also has led to the conclusions that stability of a metal or alloy increases with increase in the ligancy and that for a given value of the ligancy, stability is a maxi-... 

The resonating-valence-bond theory of metals discussed in this paper differs from the older theory in making use of all nine stable outer orbitals of the transition metals, for occupancy by unshared electrons and for use in bond formation the number of valency electrons is consequently considered to be much larger for these metals than has been hitherto accepted. The metallic orbital, an extra orbital necessary for unsynchronized resonance of valence bonds, is considered to be the characteristic structural feature of a metal. It has been found possible to develop a system of metallic radii that permits a detailed discussion to be given of the observed interatomic distances of a metal in terms of its electronic structure. Some peculiar metallic structures can be understood by use of the postulate that the most simple fractional bond orders correspond to the most stable modes of resonance of bonds. The existence of Brillouin zones is compatible with the resonating-valence-bond theory, and the new metallic valencies for metals and alloys with filled-zone properties can be correlated with the electron numbers for important Brillouin polyhedra. 

The fundamental structural feature of a metallic system the metallic orbital ... 

However, this uninhibited resonance, involving the conversion of atoms into ions (or the transfer of ionic charges to atoms), requires that the atom receiving a bond have an orbital available for its reception. It is the possession of this extra orbital (the metallic orbital), in addition to the orbitals required on the average for occupancy by unshared... 

In the molecule Li2 the bond involves a hybrid atomic orbital as+bp formed from the 2s orbital and one of the much less stable 2p orbitals. It is shown below that the amount of p character of this bond orbital (equal to b2, with a2 + b2 = 1) is small, being about 8%. On the other hand, if each of the atoms in metallic lithium requires a bond orbital and a metallic orbital and the two are equivalent they will be 2- -p) and 2 t(s —p), with 50 % p character. The analysis of energy quantities supports this conclusion. 

Tin has fourteen electrons outside of its krypton-like core. These may occupy the nine orbitals in the following three most stable ways (atomic electrons are indicated by spin vectors, bonding electrons by dots, the metallic orbitals by open circles) ... 

The metallic form of tin, white tin, consists largely of the bivalent atoms SnB, which have a metallic orbital. In white tin each atom has co-ordination number 6, and the bonds resonate among the alternative positions. It is the energy of this... 

The contribution of Co A is presumably limited to 35 % by the destabilizing effects of absence of a metallic orbital and decreased d character of the bond orbitals, which oppose the stabilizing effect of the quartet atomic state. In the same way nickel involves resonance between the two following structures, in the ratio 30 70 (saturation moment 0-61) ... 

The quinquevalent state with a pure p metallic orbital would have somewhat more d character in the bond orbitals. 

If the assumption is made that the bond orbitals and one metallic orbital (except for the state with maximum valence, which has no metallic orbital) have the same hybrid character, values of the radii for the various pure valence states of the metals of the first ascending branch, from copper to germanium, can be calculated by use of equations (10c) and (10d). These values are given in table 6. There are also given the values interpolated for resonance between the state of maximum valency (with no metallic orbital) and the next state (with valency two less, and with a metallic orbital) in the ratio 25 75, the number of orbitals being included in the calculation as a weight factor. 

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