Discretized Hamiltonian

The continuum model with the Hamiltonian equal to the sum of Eq. (3.10) and Eq. (3.12), describing the interaction of electrons close to the Fermi surface with the optical phonons, is called the Takayama-Lin-Liu-Maki (TLM) model [5, 6], The Hamiltonian of the continuum model retains the important symmetries of the discrete Hamiltonian Eq. (3.2). In particular, the spectrum of the single-particle states of the TLM model is a symmetric function of energy. 

Here, for notational convenience, we have assumed that Vnm = We would like to emphasize that the mapping to the continuous Hamiltonian (88) does not involve any approximation, but merely represents the discrete Hamiltonian (1) in an extended Hilbert space. The quantum dynamics generated by both Hamilton operators is thus equivalent. The Hamiltonian (88) describes a general vibronically coupled molecular system, whereby both electronic and nuclear DoF are represented by continuous variables. Contrary to Eq. (1), the quantum-mechanical system described by Eq. (88) therefore has a well-defined classical analog. 

The Chebyshev filtering in Step 3 costs 0(s N/p) flops. The discretized Hamiltonian is sparse and each matrix-vector product on one processor costs 0 N/p) flops. Step 3 requires m s matrix-vector products, at a total cost of 0(s m N/ p) where the degree w of the polynomial is small (typically between 8 and 20). 

Compute the upper bound b p of the spectrum of the current discretized Hamiltonian H (call Algorithm 6.5 in Sect. 6.4.2). 

Here, we discuss the theoretical aspect of our discrete Hamiltonian to show nonlinear K-G equation and then deal with quantirm mechanical approach for phonon boimd state or QB state to explain the possible dependence of criticality on the Landau coefficient through quantum route. 

Our discrete Hamiltonian gives a general treatment of the mode dynamics in the array, particularly for modes, which are strongly localized over a small number of domains in the array. For such modes, Eq. (1) can be split as ... 

In Chapter 11, Bandyopadhyay et al. have reported on the non-linear Klein-Gordon equation that is based on their discrete Hamiltonian in a typical array of ferroelectric domains. The effect of second quantization, in a particular environment, toward the nonlinearity has been described. This is considered useful for a future study in this new field of investigation of quantum breathers in ferroelectrics. 

The electronic Hamiltonian and the comesponding eigenfunctions and eigenvalues are independent of the orientation of the nuclear body-fixed frame with respect to the space-fixed one, and hence depend only on m. The index i in Eq. (9) can span both discrete and continuous values. The q ) form... 

For bound state systems, eigenfunctions of the nuclear Hamiltonian can be found by diagonalization of the Hamiltonian matiix in Eq. (11). These functions are the possible nuclear states of the system, that is, the vibrational states. If these states are used as a basis set, the wave function after excitation is a superposition of these vibrational states, with expansion coefficients given by the Frank-Condon overlaps. In this picture, the dynamics in Figure 4 can be described by the time evolution of these expansion coefficients, a simple phase factor. The periodic motion in coordinate space is thus related to a discrete spectrum in energy space. 

The classical microscopic description of molecular processes leads to a mathematical model in terms of Hamiltonian differential equations. In principle, the discretization of such systems permits a simulation of the dynamics. However, as will be worked out below in Section 2, both forward and backward numerical analysis restrict such simulations to only short time spans and to comparatively small discretization steps. Fortunately, most questions of chemical relevance just require the computation of averages of physical observables, of stable conformations or of conformational changes. The computation of averages is usually performed on a statistical physics basis. In the subsequent Section 3 we advocate a new computational approach on the basis of the mathematical theory of dynamical systems we directly solve a... 

Backward Analysis In this type of analysis, the discrete solution is regarded as an exact solution of a perturbed problem. In particular, backward analysis of symplectic discretizations of Hamiltonian systems (such as the popular Verlet scheme) has recently achieved a considerable amount of attention (see [17, 8, 3]). Such discretizations give rise to the following feature the discrete solution of a Hamiltonian system is exponentially close to the exact solution of a perturbed Hamiltonian system, in which, for consistency order p and stepsize r, the perturbed Hamiltonian has the form [11, 3]... 

This means that the discrete solution nearly conserves the Hamiltonian H and, thus, conserves H up to 0 t ). If H is analytic, then the truncation index N in (2) is arbitrary. In general, however, the above formal series diverges as jV —> 00. The term exponentially close may be specified by the following theorem. 

In this paper, we discuss semi-implicit/implicit integration methods for highly oscillatory Hamiltonian systems. Such systems arise, for example, in molecular dynamics [1] and in the finite dimensional truncation of Hamiltonian partial differential equations. Classical discretization methods, such as the Verlet method [19], require step-sizes k smaller than the period e of the fast oscillations. Then these methods find pointwise accurate approximate solutions. But the time-step restriction implies an enormous computational burden. Furthermore, in many cases the high-frequency responses are of little or no interest. Consequently, various researchers have considered the use of scini-implicit/implicit methods, e.g. [6, 11, 9, 16, 18, 12, 13, 8, 17, 3]. 

The form of the Hamiltonian impedes efficient symplectic discretization. While symplectic discretization of the general constrained Hamiltonian system is possible using, e.g., the methods of Jay [19], these methods will require the solution of a nontrivial nonlinear system of equations at each step which can be quite costly. An alternative approach is described in [10] ( impetus-striction ) which essentially converts the Lagrange multiplier for the constraint to a differential equation before solving the entire system with implicit midpoint this method also appears to be quite costly on a per-step basis. 

Thus random interfaces on lattices can be investigated rather efficiently. On the other hand, much analytical work has concentrated on systems described by Hamiltonians of precisely type (21), and off-lattice simulations of models which mimic (21) as closely as possible are clearly of interest. In order to perform such simulations, one first needs a method to generate the surfaces 5, and second a way to discretize the Hamiltonian (21) in a suitable way. 

The discreteness in this new theory refers to the discrete nature of all measurements. Each measurement fixes a particle s position Xn and time Both and tn are allowed to take on any value in the spectrum of continuous eigenvalues of the operators (xn)op and (tn)op. Notice also that in this discrete theory there is no Hamiltonian and no Lagrangian only Action. 

For quantum chemistry the expansion of e in a Gaussian basis is, of course, much more important than that of 1/r. The formalism is a little more lengthy than for 1/r, but the essential steps of the derivation are the same. For an even-tempered basis one has a cut-off error exp(—n/i) and a discretization error exp(-7//i), such that results of the type (2.15) and (2.16) result. Of course, e is not well represented for r very small and r very large. This is even more so for 1/r, but this wrong behaviour has practically no effect on the rate of convergence of a matrix representation of the Hamiltonian. This is very different for basis set of type (1.1). Details will be published elsewhere. 

These ideas can be applied to electrochemical reactions, treating the electrode as one of the reacting partners. There is, however, an important difference electrodes are electronic conductors and do not posses discrete electronic levels but electronic bands. In particular, metal electrodes, to which we restrict our subsequent treatment, have a wide band of states near the Fermi level. Thus, a model Hamiltonian for electron transfer must contains terms for an electronic level on the reactant, a band of states on the metal, and interaction terms. It can be conveniently written in second quantized form, as was first proposed by one of the authors [Schmickler, 1986] ... 

To calculate Mossbauer spectra, which consist of a finite number of discrete lines, the nuclear Hamiltonian, and thus also Hsu, has to be set up and solved independently for the nuclear ground and excited states. The electric monopole interaction, that is, the isomer shift, can be omitted here since it is additive and independent of Mj. It can subsequently be added as an increment 5 to the transition energies of each of the obtained Mossbauer lines. 

This procedure follows, in effect, the derivation of Jarzynski s identity in discrete time [2,18], as outlined in Sect. 5.5. Finally, for Hamiltonian dynamics, one can use (5.23) and calculate the work directly from the difference in total energy between trajectory start and end points. 

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