Improved virtual orbitals

Flunt W J and Goddard W A III 1969 Excited states of FI2O using improved virtual orbitals Chem. Phys. Lett. 3 414-18... 

The most convenient procedure for attaining the minimum of the second order perturbation expression of the energy, so as to generate the optimized virtual orbitals, depends on the kind of problem being studied. In the case of intermolecular interactions, convergence is quite easy with just a gradient-based procedure. The minimization scheme can be recast in such a way that the coefficients of the improved virtual orbitals can be obtained, at each step, by a resolution of a linear system of NA+NB equations. Specifically. 

We have also added a method of calculating improved virtual orbitals. Our use of this procedure for N electron excited state virtual orbitals (8l) in the framework of the SCF calculation of the N-l electron problem closely resembles those proposed by Huzinaga (82). We have also investigated Huzinagafs recent method for improved virtual orbitals in the extended basis function space (83) This is also a useful procedure where there are convergence problems for the Hartree-Fock calculations for the N-electron occupied space of the excited states. This should also be helpful in optimizing virtual orbitals to use them in perturbation theory expressions. 

Nj molecule for details see Ref. 51 improved virtual orbitals (IVOs) were taken as starting guess. 

Hence the orbital energies of occupied orbitals pertain to interactions appropriate to a total of A electrons, while the orbital energies of virtual orbitals pertain to a system with A+ 1 electrons. This usually makes SCF virtual orbitals not very good for use in subsequent correlation calculations or for use in interpreting electronic excitation processes. To correlate a pair of electrons that occupy a valence orbital requires double excitations into a virtual orbital of similar size the SCF virtual orbitals are too diffuse. For this reason, significant effort has been devoted to developing methods that produce so-called improved virtual orbitals (IVOs) [46] that are of more utility in performing correlated calculations. 

In (18) and (19) the plus and minus signs refer to the singlet and triplet excited states, respectively. The solutions of (19) will be called improved virtual orbitals (IVO) as opposed to the regular virtual orbitals (RVO) solutions of (5). These IVO s, which therefore correspond to variationaly adjusting the orbital in the open-shell HF wave function for the excited state, are considerably different from the RVO s if the basis set is sufficiently flexible. In addition, the orbital (j>i will be different for the singlet and triplet states arising from the same orbital... 

BFGS = Broyden, Fletcher, Goldfarb, and Shannon BLAS = basis linear algebra subprogram DFP = Davidon, Fletcher, and Powell DIIS = direct inversion in interative subspace IVO = improved virtual orbital. 

To improve the perturbative convergence and to eliminate notorious intruder state problems, Freed and coworkers use multiple Fock operators to define the valence orbitals. " In their formalism, all the valence orbitals and orbital energies are obtained from potentials, thereby providing a good first order approximation PHP) to the low lying excited states and thus minimizing the residual corrections to be recovered by the perturbation expansion. The unoccupied valence orbitals are chosen as improved virtual orbitals as described in the next section. Moreover, intruder state problems are further reduced in the IP method by defining the zeroth order Hamiltonian Ho as... 

The generation of improved virtual orbitals (IVOs) and orbital energies presents a non-trivial problem for systems, such as C3H4, C3H4 etc., where the highest occupied molecular orbitals (HOMOs) are doubly degenerate. Unless appropriate Fock operators F are used, the IVOs may break molecular symmetry (see below). 

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