Simplified Hamiltonian

Let US consider the simplified Hamiltonian in which the nuclear kinetic energy term is neglected. This also implies that the nuclei are fixed at a certain configuration, and the Hamiltonian describes only the electronic degrees of freedom. This electronic Hamiltonian is... 

As a naive or zero-order approximation, we can simply ignore the V12 term and allow the simplified Hamiltonian to operate on the Is orbital of the H atom. The result is... 

This is a simplified Hamiltonian that ignores the direct interaction of any nuclear spins with the applied field, B. Because of the larger coupling, Ah to most transition metal nuclei, however, it is often necessary to use second-order perturbation theory to accurately determine the isotropic parameters g and A. Consider, for example, the ESR spectrum of vanadium(iv) in acidic aqueous solution (Figure 3.1), where the species is [V0(H20)5]2+. 

Spin-Orbit Coupling For the derivation of selection rules, it is sufficient to employ a simplified Hamiltonian. To this end, we rewrite each term in the microscopic spin-orbit Hamiltonians in form of a scalar product between an appropriately chosen spatial angular momentum 2 and a spin angular momentum S... 

Using second-order perturbation theory and evaluating the matrix elements of the vibrational and rotational operators occuring in our simplified Hamiltonian... 

The solution to the fiill Hamiltonian is rendered difficult by the electron-electron repulsion term that depends on ri 2. The full solution can be approximated by initially ignoring this term, solving the remaining simplified Hamiltonian, and then reintroducing the term as a perturbation. 

In Eq. (IV.56h) we have chosen the space fixed F-axis to point in the direction of the virtual electric field Ets- The eigenvalue problem which corresponds to the above simplified Hamiltonian may be solved by a perturbation treatment. For this purpose the corresponding Hamiltonian matrix is calculated in a basis of functions built from products between electronic functions cz,..., aNe,bxfg,CNe) and rotational functions iptoi ,0,x)- As electronic functions we will use the eigenfunctions of Eq. (IV.56a), i.e.,... 

Assuming only axial asymmetry (Et = 0), the simplified Hamiltonian matrix becomes... 

As a first example we consider a nondegenerate single-channel system with a simplified Hamiltonian in which continuum-continuum interactions have been neglected ... 

In the simple Hiickel theory of planar hydrocarbons, each tt MO contains contributions from one Ipir AO on each carbon atom. In the extended Hiickel treatment of nonplanar hydrocarbons, each valence MO contains contributions from four AOs on each carbon atom (one 2s and three 2p s) and one Is AO on each hydrogen atom. The AOs used are usually Slater-type orbitals with fixed orbital exponents determined from Slater s rules (Problem 15.79). For the simplified Hamiltonian (16.76), the problem separates into several one-electron problems ... 

The coefficients are combined with the radial term to construct a simplified Hamiltonian describing the d-orbital energies. Note that represents a radial correction due to LFSE only, and that the primary contribution to the radial dependence of the ligand-metal interaction energy comes from the standard... 

A The simplified Hamiltonian operator can be divided into two parts, one relating to electron 1, and the other to electron 2 ... 

The present review is devoted to two main approaches that may lead in a controlled way from the exact Hamiltonian to simplified Hamiltonians, which are more easy to handle and on which deductive derivations may be easier to draw. One approach proceeds through projections of some exact wavefunc-tions into a relevant reduced subspace and leads to the ejfective Hamiltonian methodology. The techniques will be described in Section II.A, and their applications in Section III. The other procedure may be considered as a simulation of the considered exact Hamiltonian by a simpler Hamiltonian, the efficiency of the simulation being measured through a reduced distance, taken on a small subspace. This procedure, described in Section II.B, leads to the definition of pseudo-Hamiltonians or pseudo-operators. The corresponding applications are reviewed in Section IV. 

Since in EPR only transitions between the lowest levels arc studied, it seems to be a good approximation to treat them as isolated levels. Starting from this assumption. Abragam and Pryce have shown [56, 60] that a simplified Hamiltonian... 

Semiempirical calculations generally use a simplified Hamiltonian operator and incorporate experimental data or a set of parameters that can be adjusted to fit experimental data. 

The classical fluid systems are characterized with simplified Hamiltonian in which the semiempirical pairwise-additive interaction between two particles are included. Those semiempirical interactions substantially arise from the Pauh exclusion of two electrons at the same quantum state and from the electrostatic interactions among electrons and nuclei. As a result, both repulsion and attraction appear in the two-body interaction. Specifically, when all involved particles are spherical ones like atoms, ions, or coarsegrained beads, the systems are called simple fluids. Obviously, the pair interactions in simple fluid systems are simply distance-dependent. Toward the investigation of simple fluid systems, atomic DFT is developed. A notable merit of atomic DFT is that the contributions to the free energy functional from different interaction parts can be treated separately. To demonstrate, below we present the DFT investigations for the simple systems of HS fluids, LJ fluids, and charged systems. 

In the case of polymer systems, most often only two first terms need to be considered, and the spectra are adequately described by the simplified Hamiltonian ... 

For some model hamiltonians it is easy to obtain an explicit expression which tells us something about the importance of the friction felt by a small system, due to inelastic processes in the large system to which it is dynamically coupled. Consider, for example, the following simplified hamiltonian [90]... 

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