Molecular dynamics Hamiltonian systems

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]. 

Here we suggest a different approach that propagates the system using multiple step-sizes, i.e., few steps with step-size At are taken in the slow classical part whereas many smaller steps with step-size 5t are taken in the highly oscillatory quantum subsystem (see, for example, [19, 4] for symplectic multiple-time-stepping methods in the context of classical molecular dynamics). Therefore, we consider a splitting of the Hamiltonian H = Hi +H2 in the following way ... 

Molecular dynamics calculations use equations 25-27. HyperChem integrates equations 26 and 27 to describe the motions of atoms. In the absence of temperature regulation, there are no external sources or depositories of energy. That is, no other energy terms exist in the Hamiltonian, and the total energy of the system is constant. 

To construct Nose-Hoover constant-temperature molecular dynamics, an additional coordinate, s, and its conjugate momentum p, are introduced. The Hamiltonian of the extended system of the N particles plus extended degrees of freedom can be expressed... 

Hpp describes the primary system by a quantum-chemical method. The choice is dictated by the system size and the purpose of the calculation. Two approaches of using a finite computer budget are found If an expensive ab-initio or density functional method is used the number of configurations that can be afforded is limited. Hence, the computationally intensive Hamiltonians are mostly used in geometry optimization (molecular mechanics) problems (see, e. g., [66]). The second approach is to use cheaper and less accurate semi-empirical methods. This is the only choice when many conformations are to be evaluated, i. e., when molecular dynamics or Monte Carlo calculations with meaningful statistical sampling are to be performed. The drawback of semi-empirical methods is that they may be inaccurate to the extent that they produce qualitatively incorrect results, so that their applicability to a given problem has to be established first [67]. 

Evans and Baranyai [51, 52] have explored what they describe as a nonlinear generalization of Prigogine s principle of minimum entropy production. In their theory the rate of (first) entropy production is equated to the rate of phase space compression. Since phase space is incompressible under Hamilton s equations of motion, which all real systems obey, the compression of phase space that occurs in nonequilibrium molecular dynamics (NEMD) simulations is purely an artifact of the non-Hamiltonian equations of motion that arise in implementing the Evans-Hoover thermostat [53, 54]. (See Section VIIIC for a critical discussion of the NEMD method.) While the NEMD method is a valid simulation approach in the linear regime, the phase space compression induced by the thermostat awaits physical interpretation even if it does turn out to be related to the rate of first entropy production, then the hurdle posed by Question (3) remains to be surmounted. 

The statistical perturbation theory arising from the classical work of Zwanzig34 and its detailed implementation in a molecular dynamics program for computation of free energies is described in detail elsewhere.35 36 We give a very brief description of the method for the sake of completeness. The total Hamiltonian of a system may be written as the sum of the Hamiltonian (Ho) of the unperturbed system and the perturbation (Hi) ... 

The molecular motion in MD simulation is deterministic by solving a Hamiltonian system (Allen and Tildesley, 1996). For the precise description of the polymeric systems, Langevin dynamics (Grest, 1996) were employed, where the force acting on the z th bead in the ath molecule can be calculated by the following equation ... 

However, we must ask how reasonable are the approximations employed in its derivation. This can be answered by comparing the above results with those of Monte Carlo (molecular dynamics) methods [240]. With Monte Carlo calculations, no approximations are made other than that of restricting considerations to a finite system (e.g., 1000 chromophores in a polymer lattice of finite dielectric constant). The full Hamiltonian can be used and calculations can be carried be-... 

In order to study a system, one first has to assume a model interaction potential between the particles that are defined as the constituents of the fluid under investigation. Such a modelization is necessary if it is desired not to perform a quantum mechanical description of the system at the level of a first principle Hamiltonian composed of elementary forces. In the latter case, the ab initio molecular dynamics technique, developed by Car and Parrinello [1, 2], was revealed to be a powerful investigation tool that was adopted by many authors the last two decades. 

In a QM/MM calculation [31,32], the system is partitioned into two regions a QM region, typically consisting of a relatively small number of atoms relevant for the specific process being studied, and a MM region with all the remaining atoms. This scheme has been recently implemented in the Amber Molecular Dynamics package with support for various semi-empirical Hamiltonians [33,34], The total Hamiltonian (H) for such a system is written as... 

This chapter highlighted the use of replica exchange molecular dynamics calculations for the investigation of the performance of the PM3, PDDG/PM3 and PM3PC Hamiltonians as applied for the conformational equilibrium of alanine dipeptide in the gas phase and water solution, a convenient model system for the study of biological molecules. 

The exponential in Eq. 2.14 represents the average over the system described by the hamiltonian Hx, and the corresponding series of conformers and configurational isomers is usually created by molecular dynamics or Monte Carlo methods. When the two systems X and Y are very similar, the exponential term vanishes, leading to a very slow convergence of the average in Eq. 2.14. A number of techniques have been described to overcome this problem 43 441. One of the few applications of this method to coordination compounds is the investigation of O2 and CO affinities to iron porphyrins[45]. 

The phase space structure of classical molecular dynamics is extensively used in developing classical reaction rate theory. If the quanmm reaction dynamics can also be viewed from a phase-space perspective, then a quantum reaction rate theory can use a significant amount of the classical language and the quantum-classical correspondence in reaction rate theory can be closely examined. This is indeed possible by use of, for example, the Wigner function approach. For simplicity let us consider a Hamiltonian system with only one DOF. Generalization to many-dimensional systems is straightforward. The Wigner function associated with a density operator /)( / is defined by... 

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