Hamiltonian potential dependence

Since the potential depends only on the interatomic distances Rab and Rbc (or R and r) and not on S, the center-of-mass of the triatomic molecule moves with constant velocity along the x-axis. If we transform to a new system, whose origin moves with the center-of-mass, the first term in (2.37) vanishes and the Hamiltonian reduces to ... 

The three-center integrals [mn k]Erei in the definition of the DKH Hamiltonian, variationaUy obtained from the energy expression, are closely related to the three-center integrals [mnjjfelvrei defining the transformed Hartree potential of the fitted density. Indeed, analysis of the Hartree potential dependent terms in the Hamiltonians h oLeia > Eqs. (23), and > Eq. (24), establishes the equiva-... 

The relativistic elimination of small components (RESC) is another two-component scheme that was applied to the DKS problem [69,102]. After transforming the DKS Hamiltonian in the same fashion as in the regular approximations [based on the generator X(e), Eq. (44)], the potential dependence of the relativistic terms is decomposed using the identity... 

The determination of minima and saddle points on the potential-energy surface of a molecule plays an important role (Schaefer and Miller, 1977, Chapter 4) in describing the electronic structure and chemical reactivity of molecules. In this section, we show how such stationary points on a molecule s potential energy surface may be found by using an approach similar to that employed in Section 5.B. We first consider how the electronic Hamiltonian changes when the nuclear positions are changed from an initial set of positions, to R, i.e., R - R + u. The electron-nuclear interaction is the only term in the Hamiltonian that depends explicitly on the nuclear position. Performing a Taylor expansion of this potential about the point R, we obtain... 

Here p, is the reduced mass, and are the Hamiltonians defined in Equation 11.1 for each atom, and Vint is the effective interaction potential depending on the relative position of the atoms, r. For many applications, such as the description of broad scattering resonances and their associated Feshbach molecules, it is sufficient to include in Vint only the rotationally symmetric singlet and triplet Born-Oppenheimer potentials, Vs=o and V5=i, respectively. Their labels 5 = 0 and 5=1 refer to the possible values of the angular-momentum quantum number associated with the total spin of the two atomic valence electrons, S = si -E S2. In this approximation, the interaction part of Equation 11.4 can be represented by [8,29]... 

To conclude, we have seen that for a given wave function and Hamiltonian, the Ehrenfest theorem can be instrumentalized to derive explicit expressions for the density and current-density distributions by rewriting it in such a way that the continuity equation results. We will rely on this option in the relativistic framework in chapters 5, 8, and 12 to define these distributions for relativistic Hamiltonian operators and various approximations of N-particle wave functions. From the derivation, it is obvious that the definition of the current density is determined by the commutator of the Hamiltonian operator with the position operator of a particle. All terms of the Hamiltonian which depend on the momentum operator of the same particle will produce contributions to the current density. In section 5.4.3 we will encounter a case in which the momentum operator is associated with an external vector potential so that an additional term will show up in the commutator. Then, the definition of the current density has to be extended and the additional term can be attributed to an (external-field) induced current density. 

If V is the total Coulombic potential between all the nuclei and electrons in the system, then, in the absence of any spin-dependent terms, the electronic Hamiltonian is given by... 

In Table I, 3D stands for three dimensional. The symbol symbol in connection with the bending potentials means that the bending potentials are considered in the lowest order approximation as already realized by Renner [7], the splitting of the adiabatic potentials has a p dependence at small distortions of linearity. With exact fomi of the spin-orbit part of the Hamiltonian we mean the microscopic (i.e., nonphenomenological) many-elecbon counterpart of, for example, The Breit-Pauli two-electron operator [22] (see also [23]). 

The quantum degrees of freedom are described by a wave function /) = (x, t). It obeys Schrodinger s equation with a parameterized coupling potential V which depends on the location q = q[t) of the classical particles. This location q t) is the solution of a classical Hamiltonian equation of motion in which the time-dependent potential arises from the expectation value of V with regard to tp. For simplicity of notation, we herein restrict the discussion to the case of only two interacting particles. Nevertheless, all the following considerations can be extended to arbitrary many particles or degrees of freedom. 

Let us introduce the coherent potential Vk(E) which is thought to be dependent on energy E and exciton momentum k. The coherent potential is translational invariant in the site representation. The Hamiltonian (1) is transformed with the coherent potential taken into account as... 

For the kind of potentials that arise in atomic and molecular structure, the Hamiltonian H is a Hermitian operator that is bounded from below (i.e., it has a lowest eigenvalue). Because it is Hermitian, it possesses a complete set of orthonormal eigenfunctions ( /j Any function spin variables on which H operates and obeys the same boundary conditions that the ( /j obey can be expanded in this complete set... 

Hamiltonian contains (fe2/2me r ) 32/3y2 whereas the potential energy part is independent of Y, the energies of the moleeular orbitals depend on the square of the m quantum number. Thus, pairs of orbitals with m= 1 are energetieally degenerate pairs with m= 2 are degenerate, and so on. The absolute value of m, whieh is what the energy depends on, is ealled the X quantum number. Moleeular orbitals with = 0 are ealled a orbitals those with = 1 are 7i orbitals and those with = 2 are 5 orbitals. 

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