Orbitals zero-order

While at first it might seem counterintuitive, this effect is readily explained in terms of the simplest zero-order orbital picture of the electron and electronic excitation transfer processes. Triplet transfer to and from a typical closed-shell organic molecule involves a change in occupancy of both the HOMO and LUMO orbitals of the donor and the acceptor. For an olefinic or aromatic donor or acceptor, this can be represented as ti tc —> (donor) and n n (acceptor). This... 

The Fermi-Sea ( Active Space ) of Zero-Order Orbitals. The SPSA Criterion of oq in Eq. (8)... 

The second notion is concerned with aspects of the issue of the a priori identification of ND and D correlations and the choice of the state-specific set of zero-order orbitals and multiconfigurational wavefunctions in terms of which this identification is assumed and implemented. In this context, 1 use examples from published results and from new computations. 

For molecules, especially in excited states, the choice of the proper, for each problem, extended set of zero-order orbitals and corresponding configurations that would allow, to a good approximation, the recognition, in quantitative terms, of the main features of the wavefunction and the bonds constitutes a challenging problem. For example, such a problem is discussed in Sections 9 and 10, where the exceptional bond of Be2 X E+ is examined in the framework of ND versus D correlation using as reference points the Fermi-seas of the low-lying states of Be. 

For the normal ground states of molecules, the simplest route to choosing a suitable extended set of zero-order orbitals la Hartree-Fock sea is to invoke the sequence of the shell model for the low-lying molecular orbitals in conjunction with the DN-D. Indeed, doing just that, together with the usual trial computations, in 1970 Das and Wahl [85] produced the first reliable results from MCHF calculations on the FES of the... 

THE FERMI-SEA ( ACTIVE SPACE ) OF ZERO-ORDER ORBITALS. THE SPSA CRITERION OF ao 11N EQ. (8)... 

This type of zero-order orbital set was named the Fermi-sea, and depending on symmetry and the states under consideration, the corresponding configurations are to be computed self-consistently. Its heuristic description was given as follows in Refs. [7,45] ... 

The introduction in the early 1970s of the concept and the methodology of the Fermi-sea as the zero-order orbital set for the construction of the state-specific multiconfigurational wavefunction played on the themes... 

An important finding is that the first derivative of the energy contains only the zero-order LCAO coefficients and the zero-order orbital energies. Therefore a second-order property... 

The four zero-order orbitals chosen are a bonding-antibonding pair of a orbitals localized in the new C2-C3 bond lined up along z, and the two combinations of "Pz orbitals on Ci and C4. Of the latter two, through-space interaction is assumed to stabilize tt slightly relative to ttI. 

However, using the McWeeny approach [7], it is sufficient to calculate only the projection P >v/K on the subspace of virtual zero-order orbitals in order to get the second hyperpolarizability tensor. This projection is evaluated via a procedure similar to the one used in solving the first-order equation (21). Taking in (32) the Hermitian product with the unoccupied and using (19), one finds... 

The zero-order orbital approximation employed in Chapter 18 completely neglects the electron-electron repulsions in atoms and therefore gives poor energy values. There are several commonly used methods that go beyond this approximation. The variation method is based on the variation theorem. 

Let us first use the zero-order orbital wave function of Eq. (18.3-2) as a variation function. This is a single function, so no energy minimization can be done, but the procedure illustrates some things about the method. The zero-order function is normalized so that the variation energy is... 

The perturbation method as described in the previous section does not apply if several wave functions correspond to the same zero-order energy (the degenerate case). For example, the zero-order orbital energies of the 2s and 2p hydrogen-like orbitals are all equal, so that all of the states of the (1 X2 ) and (U)(2/ ) helium configurations have the same energy in zero order. A version of the perturbation method has been developed to handle this case. We will describe this method only briefly and present some results for some excited states of the helium atom. There is additional information in Appendix G. 

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