Phillips-Kleinman operator

Non-uniqueness of the Pseudopotential.—The Phillips-Kleinman pseudopotential contains operators 2( v- k) k)<.p which shifts the eigenvalues of the core func-... 

In the Reporters pseudopotential calculations31 a different approach is taken by replacing the Phillips-Kleinman term with an operator,... 

However, it is known that the open-shell hamiltonian itself is non-unique.34 In particular this means that the eigenvalues are dependent on the particular form of the hamiltonian for which the orbitals are canonical. Any attempt to use the extended form of the Phillips-Kleinman projection operator, which for open-shell cases can be written... 

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... 

Extensive introductions to the effective core potential method may be found in Ref. [8-19]. The theoretical foundation of ECP is the so-called Phillips-Kleinman transformation proposed in 1959 [20] and later generalized by Weeks and Rice [21]. In this method, for each valence orbital (pv there is a pseudo-valence orbital Xv that contains components from the core orbitals and the strong orthogonality constraint is realized by applying the projection operator on both the valence hamiltonian and pseudo-valence wave function (pseudo-valence orbitals). In the generalized Phillips-Kleinman formalism [21], the effect of the projection operator can be absorbed in the valence Pock operator and the core-valence interaction (Coulomb and exchange) plus the effect of the projection operator forms the core potential in ECP method. 

The ECP method dates back to 1960, when Phillips and Kleinman suggested an approximation scheme for discarding core orbitals in band calculations [1]. They replaced the full Fock-operator with the following operator ... 

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. 

There are essentially two types of ECP s in general use, one which follows Phillips and Kleinmans original suggestion and uses explicit core orbitals in the projection operators, and one which uses projection operators on the orbital angular momentum with... 

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