Effective valence shell Hamiltonian

M. G. Sheppard and K. F. Freed, Effective valence shell Hamiltonian calculations using third-order quasi-degenerate many-body perturbation theory. J. Chem. Phys. 75, 4507 (1981). 

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,... 

Effective valence-shell Hamiltonian [19, 25], Assumes degenerate valence space. 

P. Strodel and P. Tavan. A revised MRCI-algorithm coupled to an effective valence-shell Hamiltonian. 11. Application to the valence excitations of butadiene, J. Chem. Phys., 117 4677 683 (2002). 

Freed KF, Sheppard MG (1982) Ab initio treatments of quasidegenerate many-body perturbation theory within the effective valence shell Hamiltonian formalism. J Phys Chem 86 2130-2133... 

Since the basic formalism of the effective valence shell Hamiltonian (H ) method is presented elsewhere, we only provide a brief overview of the approach. As in conventional many-body perturbation theory, the iiT method begins with the decomposition of the exact Hamiltonian H into the zeroth order Hamiltonian Hq and the perturbation V,... 

Semi-empirical Hamiltonians and operators are taken to be state independent [56] and have the same Hermiticity as their true counterparts. Consequently, the valence shell effective Hamiltonians and operators they mimic must also have these two properties. Table I shows that the effective Hani iltonian and operator definitions H and A, as well as H and either A or a fulfill these criteria. Thus, these definition pairs may be used to derive the valence shell effective Hamiltonians and operators mimicked by the semi-empirical methods. Table III indicates that the commutation relation (4.12) is preserved by all three definition pairs. Hence, the validity of the relations derived from the semi-empirical version of (4.12) depends on the extent to which the semi-empirical Hamiltonians and operators actually mimic, respectively, exact valence shell effective Hamiltonians and operators. In particular, the latter Hamiltonians and operators contain higher-body terms which are neglected, or ignored, in semi-empirical theories. These nonclassical higher body interactions have been shown to be nonnegligible for the valence shell Hamiltonians of many atoms and molecules [27, 145-149] and for the dipole moment operators of some small molecules [56-58]. There is no a... 

For a closed-shell system the effective valence hamiltonian (we follow the work of Weeks, Hazi, and Rice13) can be written in the general form... 

Both of these methods essentially use ab initio wavefunctions to deduce the composition of the MOs. An alternative approach is to use semi-empirical functions, obtained from the MINDO or CNDO methods, to calculate the constants directly from the hamiltonian. A major problem in such work is that in many cases only valence shell electrons are included and consequently spin-other orbit effects cannot be calculated directly. Hinkley, Walker, and Richards122 have shown from ab initio calculations that this shielding for a given atom often bears a constant ratio to theZ/r3 terms. Such a ratio could be... 

Since the model potential approach yields valence orbitals which have the same nodal structure as the all-electron orbitals, it is possible to combine the approach with an explicit treatment of relativistic effects in the valence shell, e.g., in the framework of the DKH no-pair Hamiltonian [118,119]. Corresponding ab initio model potential parameters are available on the internet under http //www.thch.uni-bonn.de/tc/TCB.download.html. 

To demonstrate the effect of different orders in DKH calculations. Table 16.4 presents results obtained for SnO and CsH (different ansatze for the electronic wave function were employed). As can be seen from the table, all spectroscopic parameters converge fast with increasing DKH order. The accuracy is mostly determined by the quality of the wave-function approximation. Note that DKHn denotes the scalar-relativistic variant, which has also been called the spin-averaged (i.e., the spin-free) DKH approach. Also, the two-electron terms have not been transformed. Since the nuclear charge numbers of Sn and Cs are not very high, these elements are not ultrarelativistic cases. However, also for Au2 it was found that the spectroscopic parameters are already converged with the second-order DKH2 Hamiltonian [1127], which is typical for such valence-shell dominated properties. 

In actual applications of the method, Snijders and Baerends used a frozen core and incorporated core relativistic effects by using atomic all-electron relativistic orbitals. This is because the Pauli Hamiltonian is not bounded from below. The orthogonality requirement against the core prevents the orbitals from collapsing. With this approach they were able to reproduce quite accurately the valence shell orbital energies from fully relativistic all-electron Dirac-Slater calculations for atoms. 

---