Effective valence Hamiltonian method

The ACSE has important connections to other approaches to electronic structure including (i) variational methods that calculate the 2-RDM directly [36-39] and (ii) wavefunction methods that employ a two-body unitary transformation including canonical diagonalization [22, 29, 30], the effective valence Hamiltonian method [31, 32], and unitary coupled cluster [33-35]. A 2-RDM that is representable by an ensemble of V-particle states is said to be ensemble V-representable, while a 2-RDM that is representable by a single V-particle state is said to be pure V-representable. The variational method, within the accuracy of the V-representabihty conditions, constrains the 2-RDM to be ensemble N-representable while the ACSE, within the accuracy of 3-RDM reconstruction, constrains the 2-RDM to be pure V-representable. The ACSE and variational methods, therefore, may be viewed as complementary methods that provide approximate solutions to, respectively, the pure and ensemble V-representabihty problems. 

Both the effective valence Hamiltonian method [31, 32] and unitary coupled cluster [33-35] employ a single two-body unitary transformation. In the effective valence Hamiltonian method [31, 32], the unitary transformation, selected by perturbation theory, is applied to the Hamiltonian to produce an effective... 

J. E. Stevens, R. K. Chaudhuri, and K. F. Freed, Global three-dimensional potential energy surfaces of H2S from the ab initio effective valence shell Hamiltonian method. J. Chem. Phys. 105, 8754 (1996). 

In principle, it should also be possible to add a semi-loced potential to the non-relativistic all-electron Hamiltonian to eirrive at a quasi-rela-tivistic all-electron method. One such suggestion has been made by Delley [76], but the resulting method has only been tested for valence properties, which could also have been obtained by valence-only methods. Effective core potential methods have the advantage of a reduced computational effort (compared to all-electron methods) and are a valuable tool as long as one is aware of the limited domain of valence-only methods. Properties for which density variations in the atomic core are important should not be calculated this way. Examples are the electric field gradient at the nucleus or the nuclear magnetic shielding. 

In conclusion, the ab initio effective Hamiltonian method has theoretically justified several basic assumptions of semiempirical models, particularly through recent studies on rr-electron systems [115-118]. It seems to approach the point where it can provide useful theoretical guidance for further improvements in valence-electron semiempirical methods (e.g., with regard to the inclusion of additional interactions in the model or the... 

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. 

Many of the effective potentials (relativistic or non-relativistic) are generated using the Phillips-Kleinman transformation. In this method, the explicit core-valence orthogonality constraints are replaced by a modified valence Hamiltonian. If one replaces the potential generated by core electrons by a potential Fj, then one can write the one-electron valence wave equation as... 

Poly(/7-phenylenevinylene) may be considered a regular copolymer of acetylene and benzene [128]. Electron structures of PA, poly(p-phenylene), and poly(p-phenylenevinylene) were studied by using UV photoelectron spectroscopy and quantum chemical calculations based on the valence effective Hamiltonian method. Excellent agreement between the theory and experiment allows a detailed description of the evolution of the electron structure in this polymer series. 

In recent years, a pseudopotential-like method, the valence effective Hamiltonian method (VEH) (6) has also been used to obtain HF-caliber one electron orbitals for polymers with little computer time and expense. 

The pseudopotential approximation was originally introduced by Hellmann already in 1935 for a semiempirical treatment of the valence electron of potassium [25], However, it took until 1959 for Phillips and Kleinman from the solid state community to provide a rigorous theoretical foundation of PPs for single valence electron systems [26]. Another decade later in 1968 Weeks and Rice extended this method to many valence electron systems [27,28], Although the modern PPs do not have much in common with the PPs developed in 1959 and 1968, respectively, these theories prove that one can get the same answer as from an AE calculation by using a suitable effective valence-only model Hamiltonian and pseudovalence orbitals with a simplified nodal structure [19],... 

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