Model potentials Hamiltonian

Nevertheless, very-long-lived quasi-stationary-state solutions of Schrodinger s equation can be found for each of the chemical structures shown in (5.6a)-(5.6d). These are virtually stationary on the time scale of chemical experiments, and are therefore in better correspondence with laboratory samples than are the true stationary eigenstates of H.21 Each quasi-stationary solution corresponds (to an excellent approximation) to a distinct minimum on the Born-Oppenheimer potential-energy surface. In turn, each quasi-stationary solution can be used to construct an alternative model unperturbed Hamiltonian //(0) and perturbative interaction L("U),... 

In this section, we introduce the model Hamiltonian pertaining to the molecular systems under consideration. As is well known, a curve-crossing problem can be formulated in the adiabatic as well as in a diabatic electronic representation. Depending on the system under consideration and on the specific method used, both representations have been employed in mixed quantum-classical approaches. While the diabatic representation is advantageous to model potential-energy surfaces in the vicinity of an intersection and has been used in mean-field type approaches, other mixed quantum-classical approaches such as the surfacehopping method usually employ the adiabatic representation. 

The Hamiltonian underlying the calculations of Fig. 1 is based on a diabatic model potential [7-9] of system-bath type, H = HS + HB, where the system and bath parts are coupled via the conical intersection. Here, Hs refers to a four-mode core composed of three tuning type modes Q. Q, Qz and a coupling mode Q4,... 

The model potential displayed in Figure 8.2 had originally been used by Kulander and Light (1980) to study, within the time-independent R-matrix formalism, the photodissociation of linear symmetric molecules like C02. It will become apparent below that in this and similar cases the time-dependent approach, which we shall pursue in this chapter, has some advantages over the time-independent picture. The motion of the ABA molecule can be treated either in terms of the hyperspherical coordinates defined in (7.33) or directly in terms of the bond distances Ri and i 2 The Hamiltonian for the linear molecule expressed in bond distances... 

Seijo [120] has performed relativistic ab initio model potential calculations including spin-orbit interaction using the Wood-Boring Hamiltonian. Calculations ere performed for several atoms up to Rn, and several dimer... 

Let us now turn to the problem switching on a model potential V(r) to the Hamiltonian used above. Denoting the canonical density matrix calculated there by = C(V =0), the simplest approximation is to follow the ideas of the Thomas-Fermi (TF) method. Then, with slowly varying V(r) for which the assumptions of this approximation are valid, one can return to the definition at Eq. (2.2), and simply move all eigenvalues a,- by the same (almost constant— ) amount F(r), the wavefunctions ( i(r) being unaffected to the same order of approximation. Hence one can write for the diagonal form of the canonical density matrix... 

Figure 3. Born Oppenheimer surfaces generated by the model electronic Hamiltonian in Equation (5) as the hydrogen is displaced from the origin in the -direction. The inset at the right schematically shows the model which electron is harmonically bound to a point at the origin of coordinates while the electron and proton interact via a Coulomb potential. The wave function is expanded as a linear combination of three basis functions, hydrogen Is, 2s and 2pz eigenstates.

The effect of the core electrons is taken into account through the use of a modified model potential (M0DP0T) Hamiltonian. 

Figure 11. Comparison of the ratio of the reactive correlation functions for two Hamiltonians, Ci(t)/C2(t), as a function of time t. Hamiltonian Hi is the model potential with y, = 0.1, y = 0.2. Hamiltonian H2 is the model potential with = 0.1, y = 0.3. The solid line is the computation with the kinetic cycle method the dotted line is the computation done in the conventional manner (running many trajectories with different initial conditions and counting how many are in region B at time t. 

The AMI (Austin Model 1) Hamiltonian available in Version 3.10 of the MOP AC (Molecular Orbital PACkage) program was used to carry out QM calculations, for the purpose of estimating the partial atomic charges to be used in the electrostatic portion of the potential energy for the calculations on the interactions between pairs of model molecules. [34] The AMI calculations also yielded quantum mechanical estimates of the heats of formation of the model molecules. These heats of formation generally followed similar trends to the MM2 steric energies. 

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. 

Modern relativistic effective core potentials provide a useful tool for accurate quantum chemical investigations of heavy atom systems. If sufficiently small cores are used to minimize frozen-core and other errors, they are able to compete in accuracy with the more rigorous all-electron approaches and still are, at the same time, economically more attractive. Successful developments in the field of valence-only Hamiltonians turned relativistic effects into a smaller problem than electron correlation in practical calculations. Both the model potential and the pseudopotential variant have advantages and disadvantages, and the answer to the question which approach to follow may be a matter of personal taste. Highly accurate correlated all-electron calculations are becoming... 

All structures were optimized at the CASSCF(8,8) level with the cc-pVDZ basis set. For multireference calcidations involving bromine and iodine, the Cowan-Griffin ab initio model potential with a relativistic effective core potential was used.222 CASPT2 calculations were performed on all optimized CASSCF(8,8)/cc-pVDZ geometries, using the CASSCF wave functions as the reference wave functions. SOCs were computed by using the Pauli-Breit Hamiltonian. 

Finally, the normal coordinates need not be linear combinations of the internal extension coordinates. In the results reported in this chapter, we use Simons-Parr-Finlan (SPF) (68) or Morse coordinates (2) for describing the stretching degrees of freedom. The normal coordinates are then defined as the appropriate linear combination of these coordinates. When we expand the coordinate dependent terms of the Hamiltonian in a Taylor series to a given order, the normal coordinates based on the SPF or Morse coordinates lead to a more accurate representation of both the model potential and the G matrix elements than do expansions based on the usual internal extension coordinates. In the case of the SPF coordinates, these expansions are exact at fourth order. Likewise, an appropriate choice of bending coordinates can also provide a more rapid convergence of these terms (49). 

The main idea behind the ab initio model potential method is taking a well defined Hamiltonian, identifying cumbersome operators in it, and, if possible. 

Table 1 shows the spin-orbit splittings of the neutral mercury atom from ZORA calculations. Different potentials have been used to construct the ZORA kinetic energy operator. In the column labelled ZORA/NUC, only the nuclear potential has been used while the column ZORA/FULL reports results from fully self-consistent ZORA calculations, in which the ZORA Hamiltonian is reconstructed in each SCF iteration using the full effective potential including the Hartree and exchange-correlation term. Note that these results are identical to a model potential calculation since for a neutral atom, the model potential is just the full self-consistent effective potential. The entries in the column labelled ZORA/COUL have been calculated using only the Coulombic parts of the model potential, i. e. the nuclear and Hartree potentials. 

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