Ground-state wave function effective Hamiltonians

A discussion is given of electron correlations in d- and f-electron systems. In the former case we concentrate on transition metals for which the correlated ground-state wave function can be calculated when a model Hamiltonian is used, i.e. a five-band Hubbard Hamiltonian. Various correlation effects are discussed. In f-electron systems a singlet ground-state forms due to the strong correlations. It is pointed out how quasiparticle excitations can be computed for Ce systems. 

Contributions originating in the Darwin, mass-velocity and spin-orbit corrections to the ground state wave function are obtained in agreement with previous works where the Darwin and mass-velocity scalar effects were included within the unperturbed molecular Hamiltonian. ... 

The function Eq. (2) is very general and does not correspond to the assumed wave function of the hybrid QM/MM method. First of all the numbers of electrons in the subsystems must be fixed to apply computational schemes to separate parts on the legal ground. Second, we assume that the M-subsystem is treated with use of the MM, i.e, the PES of the M-subsystem is evaluated without explicit invocation of its wave function. The parameters of the M-subsystem must be transferable, Le, applicable to combination with any i -subsystem and even in the absence of it. For this purpose we should use the wave function of the ground state of the effective Hamiltonian for the M-subsystem since it is in a certain sense close the wave function calculated without any i -subsystem [73]. Thus the required wave function is represented by the antisymmetrized product of... 

Once an effective Hamiltonian for the system has been defined, the wave function T and energy can be evaluated by minimizing the Vsb in Eq. (8.6) with respect to the molecular orbital coefhcients of the ground state wave function using a self-consistent (SCF) procedure to solve the effective Schrodinger equation. 

Other quantum simulations involve simulations with effective Hamiltonians [261-263] or the simulation of ground state wave properties by Green s function Monte Carlo or diffusion Monte Carlo for reviews and further references on these methods see Refs. 162, 264-268. 

The theory of solvent effects on the electronic structure of a given solnte leads to a representation of the subsystem embedded in a larger one with the help of effective Hamiltonians, wave functions, and eigenvalues. Since the whole electronic system is quantum mechanical in nature, and in principle nonseparable, the theory for the ground electronic state permits defining under which conditions the solnte and solvent separability is an acceptable hypothesis. It is possible to distinguish passive from... 

In this framework, the requirement needed in order to incorporate the solvent effects into the reactant (and product) wavefunctions is automatically fulfilled by using the effective Hamiltonian defined in Equation (3.155) and by adopting an iterative procedure until the wave-function and the solvent reaction field induced by the Cl density matrix of the state of interest reach self-consistency. One must note that this procedure is valid for ground and excited states fully equilibrated with the solvent, while the inclusion of nonequilibrium effects needs some further refinements, as indicated, for example, in ref. [35],... 

Averaging the interaction operators in eq. (1.246) - they are both two-electronic ones - over the ground states of each subsystem does not touch the fermi-operators of the other subsystem. The averaging of the two-electron operators PWCP and PwrrP yields the one-electron corrections to the bare subsystem Hamiltonians. The wave functions and d, )7 are calculated in the presence of each other. The effective operator iTff describes the electronic structure of the R-system in the presence of the medium, whereas HIf describes the medium in the presence of the R-system. 

Wave function of electrons in quantum R-system Ap satisfies the Schrodinger equation with the effective Hamiltonian iTff eq. (1.246), which is obtained by averaging the interaction operators in eq. (1.232) over the ground state of the M-system, i.e. over Ap, and acts on the quantum numbers (variables) of electrons in the R-system. 

In the frame of the hybrid methods it must be computed by a QM method. The Schrodinger equation with the effective Hamiltonian Hjf has multiple solutions, which describe excited states of the R-system provided the M-system is frozen in its ground state. Electronic energy of the system in the state expressed by the wave function eq. (1.231), has the form [29,30] ... 

Estimates of the electronic energy of the complex system employed in the expressions eqs. (1.254), (1.256) for its PES can be further improved. For this let us notice that the solutions of the self consistent system eq. (1.246) are used as multipliers in the basis functions eq. (1.216) of the subspace Im/. It turns out that the effective Hamiltonian Heff eq. (1.232) has nonvanishing matrix elements between the ground state of eq. (1.246) and the basis product states of the subspace ImP, differing from it by two multipliers simultaneously by the wave function for the R-system and by that for the M-system ( A p. p f 0). Indeed ... 

---