A Simple Model Hamiltonian

Analysis of Green s functions can be useful in seeking to establish model hamil-tonians with the property of giving approximately correct propagators, when put in the equations of motion. In this section, we explore a particularly simple model in order to familiarize the reader with various molecular orbital concepts using the terminology of Green s function theory. We employ the Hartree-Fock approximation and seek the molecular Fock operator matrix elements 

As a guide to this calculation, we consider the limit of separated atoms, for which we may write 

This result is similar to the widely used approximations suggested using different arguments by Mulliken and by Wolfsberg and Helmholz. The equation of motion for the Green s function [Eq. (4.15)] 

It is often the case that all the elements Wr are negative, so we can write 

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The Hessian, IJ. can also be used in other ways in optimizing the structure. First, Eq. (33) gives a simple estimate for the optimized structure, once the Hessian has been calculated. Fernandez-Serra el al.24 suggested to introduce a Hessian obtained from a simple model Hamiltonian as an efficient way of obtaining reasonable estimates for how to change the structure towards a local total-energy minimum. They showed that this approach could reduce considerably the number of steps before convergence was reached. 

Since the pioneering work of Anderson and Mott it is known that the wave functions of disordered systems are localized. The degree of this so-called Anderson localization increases with increasing disorder. Anderson s original argument was based on a simple model Hamiltonian. In a number of cases different authors > carried out numerical studies on disordered systems still using different simplified model Hamiltonians. 

In a seminal paper [4], Su, Schrieffer, and Heeger discussed the energetics of solitons using a simple model of free electrons interacting with the lattice described by the Hamiltonian ... 

The usual EVB procedure involves diagonalizing this 3x3 Hamiltonian. However, here we wish to use a very simple model for our reaction and represent the potential surface and wavefunction of the reacting system using only two electronic states. Using a two-state system will preserve most of the important features of the potential energy surface while at the same time provide a simple model that will be more amenable to discussion than the three-state system. For the two-state system we define the following states as the reactant and product wavefunctions ... 

As a simple model, we confine our attention just to a single mode Ha(t) of the Hamiltonian (23). Note that neither any instantaneous eigenstate of Ha(t) is an exact quantum state nor e-/3ii W is a density operator. To calculate the thermal expectation value of an operator A, one needs either the Heisenberg operator Ah or the density operator pa(t) = UapaUa Now we use the time-dependent creation and annihilation operators (24), invariant operators, to construct the Fock space. 

To account for photochemical processes, we adopt a simple model that was proposed by Seidner and Domcke for the description of cis-trans isomerization processes [164]. In addition to the normal-mode expansion above, they introduced a Hamiltonian exhibiting torsional motion. The diabatic matrix elements of the Hamiltonian are given as... 

We consider the same reaction model used in previous studies as a simple model for a proton transfer reaction. [31,57,79] The subsystem consists of a two-level quantum system bilinearly coupled to a quartic oscillator and the bath consists of v — 1 = 300 harmonic oscillators bilinearly coupled to the non-linear oscillator but not directly to the two-level quantum system. In the subsystem representation, the partially Wigner transformed Hamiltonian for this system is,... 

In the present simulation we apply a simple model consisting of two vibrational modes in the relevant system. The first one will contribute to the Stokes shift as well as to the Herzberg-Teller correction, while the second one only to the Stokes shift. Next, we will assume a mutual coupling of the modes in the bilinear form Q Qi for the excited state. This type of coupling is usually omitted in literature. The Hamiltonian 77s is written as... 

Here we employ the quantum kicked rotor as a simple model of quanmm chaos systems. The Hamiltonian of a kicked rotor is written as... 

In order to demonstate the power of the formalism described above we applied the scheme to a simple model system studied recently by Mavri and Berendsen with the DME method [17]. The model consists of a quantum oscillator with reduced mass p immersed in a bath of 79 argon atoms, which are treated classically. The Hamiltonian for this system takes the form... 

In Section VII we conclude our results and discuss several issues arising from our proposals. We revisit our original motivation—that is, to find a simple model, in the sense of dynamical systems, that captures several common aspects of slow dynamics in liquid water, or more generally supercooled liquids or glasses. Our attempt is to make clear the relation and compatibility between the potential energy landscape picture and phase space theories in the Hamiltonian dynamics. Importance of heterogeneity of the system is discussed in several respects. Unclarified and unsolved points that still remain but should be considered as crucial issues in slow dynamics in molecular systems are listed. 

As a simple model for Hamiltonian dynamical system with several degrees of freedom, we have chosen Froeschle map [9-13], given by... 

As an application of this formalism, we consider a two-level quantum system coupled to a classical bath as a simple model for a transfer reaction in a condensed phase environment. The Hamiltonian operator of this system, expressed in the diabatic basis L), P), has the matrix form [43]... 

The internal spin interaction Hamiltonian Hmt can be decomposed into spatial Tm[ ua(l) ] and spin Sm degrees of freedom Hin (t) = 2mTm[ ua(t) ]Sm. The spatial contribution, hereafter an NMR interaction rank-2 tensor T, is a stochastic function of time Tm[ ua(t) because it depends on generalized coordinates < ( ) of the system (atomic and molecular positions, electronic or ionic charge density, etc.) that are themselves stochastic variables. To clarify the role of these coordinates in the NMR features, a simple model is developed below.19,20 At least one physical quantity should distinguish the parent and the descendant phase after a phase transition. For simplicity, we suppose that the components of the interaction tensor only depend on one scalar variable u(t) whose averaged value is modified from m to m + ( at a phase transition. To take into account the time fluctuations, this variable is written as the sum of three terms, i.e. u(t) — m I I 8us(t). The last term is a stationary stochastic process such that — 0, where <.) denotes a... 

Here Ef is the amplitude, t the duration, and co the frequency of the ith pulse. This scheme has been applied in Ref [46] to a generic two-dimensional HT model which incorporated a H-atom reaction coordinate as well as a low-frequency H-bond mode. In a subsequent work [47] the approach has been specified to a simple model of HT in thioacetylacetone. The Hamiltonian was tailored to the form of Eq. (4.1) based on the information available for the stationary points, that is, the energetics as well as the normal modes of vibration. From these data an effective two-dimensional potential was constructed including the H-atom coordinate as well as a coupled harmonic oscillator, which describes the 0-S H-bond motion. Although perhaps oversimplified, this model allowed the study of some principle aspects of laser-driven H-bond motion in an asymmetric low-barrier system. 

The starting point for the study is a simple model of the coupled process. This model is found from a generalization of the Hamiltonian in Eq. (9.2) to include the modulation of hydrogen transfer potential as a result of electron transfer. 

A simple model of a nano-dimensional structure in the form of a neutral spherical SNc of radius a and permittivity si, embedded in a medium with permittivity ej, has been discussed elsewhere. An electron e and hole h with elfective masses and m i were assmned to travel within this SNc (we use r and rh to denote the distances of the electron and the hole, respectively, from the center of the SNc). We assume that the two permittivites are such that E2 ei, and that the electron and hole bands are parabolic in shape. In this model, and subject to these approximations and the effective mass approximation, the exciton Hamiltonian takes the... 

Here we consider a simple model of six noninferacfing resonances of zero-order energies E° = f - 1, (f = 1, 2,. .., 6) and of identical partial width Tq. The effective Hamiltonian is written as... 

A simple ID Hamiltonian model of a decaying particle, including a full description of the dynamics from the preparation region, consists of a flat well surrounded by equal square barriers in units h = 2m = l, we write V(x) = -Vo when -a d. The initial state is chosen symmetric T o(T) = l/Va... 

Ogata et al. attacked the same nucleic acid conformation problem, but replaced the buildup scheme of Lucasius with a local filter that is equivalent to the use of a rotamer library. In both cases, these methods must deal with the fact that this is an underconstrained problem because several of the dihedrals have no NOEs associated with them. Schuster earlier treated a simple model of RNA to predict three-dimensional (3D) conformations, using a variant on a spin-glass Hamiltonian as his fitness function. The simple model used allowed for the analysis of the complexity of the fitness landscape, couched in terms of the genotype-to-phenotype mapping. 

Badiali and coworkers started by applying this method to a combination of the primitive model, in which the ions are considered as point charges and the solution is treated as a dielectric continuum, and a simple model for the metal in which the charge distribution is taken as uniform in the direction parallel to the surface. In this model the Hamiltonian depends only on... 

To keep the story simple, let us limit ourselves to the single electron mentioned above. First, let us define the two diabatic states (the basis set) of the system only the 3s orbital of Na (when the electron resides on Na we have atoms) denoted by 35) and the 3p orbital of Cl (when the electron is on Cl we have ions) 3p). Now. what about the Hamiltonian 7Y Well, a reasonable model Hamiltonian may be taken as ... 

Considering a simple model for a bound pair of atoms with positions qi and Q2, moving in a position dependent potential field [Pg.96]

The results of a simple model calculation will be presented here which demonstrate quite dramatically the dependence of the convergence properties of the perturbation series on the basis set employed. Consider the model problem of a hydrogenic atom with nuclear charge Z perturbed by the potential — Z /r, i.e. the problem with Hamiltonian... 

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