Operators valence projection

Every operator must now be bracketed by valence projection operators, including the metric operator, so we must insert the valence projection operator in every place where we would normally compute an overlap integral. This converts every one-electron operator into an -electron operator. 

This is not as bad as it sounds. If we ensure that the spinors are orthonormal on the valence projection operator,... 

At this point, we have a valence wave function given in terms of pseudospinors that have little or no core contribution, and a set of valence operators that only operate on the valence space. We do however have explicit use of the valence projection operators, which are composed of infinite sums. What we would like to do is to write the Hamiltonian in terms of the normal Hamiltonian and a correction, which is termed a pseudopotential. This we can do by replacing with 1 - and extracting out the unprojected Hamiltonian,... 

If 1 , is a valence spinor, we may insert the valence projection operator as defined by (20.8) into (20.17) to obtain a projected equation that is satisfied for any valence spinor. 

First, we note that the determination of the exact many-particle operator U is equivalent to solving for the full interacting wavefunction ik. Consequently, some approximation must be made. The ansatz of Eq. (2) recalls perturbation theory, since (as contrasted with the most general variational approach) the target state is parameterized in terms of a reference iko- A perturbative construction of U is used in the effective valence shell Hamiltonian theory of Freed and the generalized Van Vleck theory of Kirtman. However, a more general way forward, which is not restricted to low order, is to determine U (and the associated amplitudes in A) directly. In our CT theory, we adopt the projection technique as used in coupled-cluster theory [17]. By projecting onto excited determinants, we obtain a set of nonlinear amplitude equations, namely,... 

All electron calculations were carried out with the DFT program suite Turbomole (152,153). The clusters were treated as open-shell systems in the unrestricted Kohn-Sham framework. For the calculations we used the Becke-Perdew exchange-correlation functional dubbed BP86 (154,155) and the hybrid B3LYP functional (156,157). For BP86 we invoked the resolution-of-the-iden-tity (RI) approximation as implemented in Turbomole. For all atoms included in our models we employed Ahlrichs valence triple-C TZVP basis set with polarization functions on all atoms (158). If not noted otherwise, initial guess orbitals were obtained by extended Hiickel theory. Local spin analyses were performed with our local Turbomole version, where either Lowdin (131) or Mulliken (132) pseudo-projection operators were employed. Broken-symmetry determinants were obtained with our restrained optimization tool (136). Pictures of molecular structures were created with Pymol (159). 

The generalization of the pseudopotential method to molecules was done by Boni-facic and Huzinaga[3] and by Goddard, Melius and Kahn[4] some ten years after Phillips and Kleinman s original proposal. In the molecular pseudopotential or Effective Core Potential (ECP) method all core-valence interactions are approximated with l dependent projection operators, and a totally symmetric screening type potential. The new operators, which are parametrized such that the ECP operator should reproduce atomic all electron results, are added to the Hamiltonian and the one electron ECP equations axe obtained variationally in the same way as the usual Hartree Fock equations. Since the total energy is calculated with respect to this approximative Hamiltonian the separability problem becomes obsolete. 

Bonifacic and Huzinaga[3] use explicit core orbital projection operators, while orbital angular momentum projection operators are used by Goddard, Kahn and Melius[4], by Barthelat and Durand[5] and others. Explicit core orbital projection operators can, in the full basis set, be viewed as shift operators which ensure that the first root in the Fock matrix really corresponds to a valence orbital. However, in applications the basis set is always modified and the role of the core orbital projection operators thus partly changes. 

Explicit orthogonality constraints can be removed by transformingthe hamiltonian so that it only acts on a specific subspace (e.g. the valence space) of the all-electron space. This can be done formally by the use of the projection operator method (see Huzinaga and Cantu18 or Kahn, Baybutt, and Truhlar20). If the core function is written as in equation (6) a projection operator may be defined for each valence electron p ... 

The action of the projection operator (-e +A) 0 ><0 is to raise the eigenvalue of the core orbital % to the value A. A new lower bound for the eigenvalue for the pseudo-orbital xl can be shown to be the lower of A and ej. In practice the core eigenvalues are usually shifted so as to be degenerate with the lowest valence eigenvalues of the same symmetry. The coefficients a in equation (37) can now assume values which allow the pseudo-orbital Xt to be nodeless and thus capable of representation by a smaller basis set expansion. 

The derivation of the pseudopotentials discussed above has been developed from equation (12), in which the effect of core projection operators on the valence-valence electron repulsion has been neglected. The error introduced by this approximation can be largely removed by adding to the core repulsion operators the repulsion from the difference in valence electron density in the reference atoms between the all-electron and valence-electron calculations 33... 

Matrix elements for the valence functions were taken with the effective core potential the coulomb and exchange terms were handled exactly, numerically, without any parameterization and a Phillips-Kleinman projection operator term was also used. Spin-orbit coupling effects amongst the valence orbitals were treated semi-empirically using the operator... 

In theory, an infinite number of calculations for highly excited states is required to complete the expansion of the EP given by Eq. (24), since there are only a few occupied valence orbitals in neutral atoms. This difficulty also exists in the nonrelativistic case and is resolved by using the closure property of the projection operator with the assumption that radial parts of EPs are the same for all orbitals having higher angular momentum quantum numbers than are present in the core. The same approximation is applicable in the present... 

Furthermore, since the same operators appear in all the n-valence sectors of the Fock-space, the projections onto the wave-functions can be performed at the very end, and it is possible to treat systems with varying n (i.e., different degrees of ionization) on the same footing, and in an explicitly size-extensive manner with respect to the valence electrons. As it has turned out, only the Fock—space approach has the potentiality to furnish explicitly connected size-extensive theories for a general incomplete model space/91-96/, which shows its flexibility and generality. 

The ECP s are constructed based on the frozen orbital ECP technique (12). In this technique some of the core orbitals are expanded in the valence basis set and frozen in atomic shapes. This reduces the demand on the accuracy of the ECP potentials and the projection operators. One-electron ECP s constructed by this technique for nickel and copper have been shown to give results of quantitative accuracy for surface problems, particularly for hydrogen chemisorption which is treated here (13,14). In the previous studies the one-electron ECP s included a frozen 3s orbital. In the present case, states with a large occupancy of 4p appeared for the s type configurations in a cluster surrounding. 

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