Orbital occupation numbers, transition

In Tables XVI and XVII we have collected the natural orbital occupation numbers from different CASSCF calculations on a selection of octahedral and tetrahedral molecules of first-row transition metals. In all cases a CASSCF calculation was performed using a basic active space of 10 orbitals. For some of the molecules additional calculations with larger active spaces (up to 14 orbitals) are also presented. Only ground-state results are presented, except for the molecules CrFg , Cr(CN)g , CoFg, and Co(CN)g , for which we have included a selected number of excited states. 

Natural Orbital Occupation Numbers Resulting from Different CASSCF Calculations on Some Representative Octahedral Transition Metal... 

TABLE lb UHF/MCSCF Natural Orbital occupation Numbers Benzene -Dewar Benzene 6 orbital CAS Transition State Geometry ... 

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. 

DFT has come to the fore in molecular calculations as providing a relatively cheap and effective method for including important correlation effects in the initial and final states. ADFT methods have been used, but by far the most popular approach is that based on Slater s half electron transition state theory [73] and its developments. Unlike Hartree-Fock theory, DFT has no Koopmans theorem that relates the orbital energies to an ionisation potential, instead it has been shown that the orbital energy (e,) is related to the gradient of the total energy E(N) of an N-electron system, with respect to the occupation number of the 2th orbital ( , ) [74],... 

The 3(BI)/3bi contributions of the different orbitals are computed through a systematic inspection of the Periodic Table, i.e. by recording the characteristic variation in the bonding indicator when a new outer orbital appears in the atomic electronic configuration. In this way, the s and p contributions are usually assessed. In the transition series , however, the further complication exists that the unsaturated shells across the series may give different contributions to the bonding, i.e. the contribution 9(BI)/3bi depends on the occupancy number Uv,i in the atomic configuration v. 

Instead of the standard Hartree-Fock reference calculation, a grand-canonical Hartree-Fock calculation [35] is used with the occupation number of a single spin-orbital (i.e., the transition spin-orbital) set to 0.5. Upon convergence, appreciable corrections to the relaxation energy are included in the transition spin-orbital s energy [23, 24], Usually a very close agreement with the ASCF method [36] is obtained [26], The second order electron propagator is applied to the ensemble... 

In the first three chapters, instances were noted where the number, symmetry characteristics and occupation numbers of the frontier orbitals of a transition-metal fragment were similar to those of a main-group fragment. Such fragments are said to be isolobal to emphasize similar bonding capabilities. Since its enunciation by Hoffmann and Mingos, the concept has been used effectively for the analysis of both organometallic and cluster problems. Let s explore the idea in a more systematic... 

Figure 3 Plot of correlation energy (Ecor, obtained from a CASSCF calculation see text) versus the M d contribution (in terms of percentage) in the bonding eg orbital in a series of octahedral MLe complexes, with L = HjO, NH3, F, CP, Br, and P, and M a transition metal with a formal (a) or c/ (b) occupation number. Solid lines connect metals with a formal charge (+3) dashed lines connect metals with a formal charge (-I-4). For simplicity, the formal charges on the metals have been omitted from the plots.

This method may be used in conjunction with the various types of Xa calculation. This enables the difference between two state energies to be estimated by calculation of a transition state, which involves occupation numbers half-way between the initial and final states. In the case of ionization energies, this corresponds to removal of half an electron from the appropriate orbital. 

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