Molecular dynamics Hamiltonian

Two areas in classical mechanics have contributed greatly to developments in our understanding of molecular Hamiltonian dynamics. The first is nonlinear... 

We have alluded to the comrection between the molecular PES and the spectroscopic Hamiltonian. These are two very different representations of the molecular Hamiltonian, yet both are supposed to describe the same molecular dynamics. Furthemrore, the PES often is obtained via ab initio quairtum mechanical calculations while the spectroscopic Hamiltonian is most often obtained by an empirical fit to an experimental spectrum. Is there a direct link between these two seemingly very different ways of apprehending the molecular Hamiltonian and dynamics And if so, how consistent are these two distinct ways of viewing the molecule ... 

The first classical trajectory study of iinimoleciilar decomposition and intramolecular motion for realistic anhannonic molecular Hamiltonians was perfonned by Bunker [12,13], Both intrinsic RRKM and non-RRKM dynamics was observed in these studies. Since this pioneering work, there have been numerous additional studies [9,k7,30,M,M, ai d from which two distinct types of intramolecular motion, chaotic and quasiperiodic [14], have been identified. Both are depicted in figure A3,12,7. Chaotic vibrational motion is not regular as predicted by tire nonnal-mode model and, instead, there is energy transfer between the modes. If all the modes of the molecule participate in the chaotic motion and energy flow is sufficiently rapid, an initial microcanonical ensemble is maintained as the molecule dissociates and RRKM behaviour is observed [9], For non-random excitation initial apparent non-RRKM behaviour is observed, but at longer times a microcanonical ensemble of states is fonned and the probability of decomposition becomes that of RRKM theory. 

Chapuisat, X., Natus, A., and Brunet, J.-P. (1991), Exact Quantum Molecular Hamiltonians. Part I. Application to the Dynamics of Three Particles, Mol. Phys. 72,1. 

A. H. Zewail If we solve for the molecular Hamiltonian, we will be theorists I do, of course, understand the point by Prof. Quack and the answer comes from the nature of the system and the experimental approach. For example, in elementary systems studied by femtosecond transition-state spectroscopy one can actually clock the motion and deduce the potentials. In complex systems we utilize a variety of template-state detection to examine the dynamics, and, like every other approach, you/we use a variety of input to reach the final answer. Solving the structure of a protein by X-ray diffraction may appear impossible, but by using a number of variant diffractions, such as the heavy atom, one obtains the final answer. 

All of the analyses described above are used in a predictive mode. That is, given the molecular Hamiltonian, the sources of the external fields, the constraints, and the disturbances, the focus has been on designing an optimal control field for a particular quantum dynamical transformation. Given the imperfections in our knowledge and the unavoidable external disturbances, it is desirable to devise a control scheme that has feedback that can be used to correct the evolution of the system in real time. A schematic outline of the feedback scheme starts with a proposed control field, applies that field to the molecular system that is to be controlled, measures the success of the application, and then uses the difference between the achieved and desired final state to design a change that improves the control field. Two issues must be addressed. First, does a feedback mechanism of the type suggested exist Second, which features of the overall control process are most efficiently subject to feedback control ... 

The Hamiltonians and the energy functionals for molecules interacting with a structured environment method are obtained by dividing a large system into two subsystems. One of these subsystems is the molecular system of interest and that part of the system is described by quantum mechanics. The other subsystem is not of principal interest and it is therefore treated by a much coarser method. Approaches along these lines have been presented within quantum chemistry [13,14,45,46-77] and molecular reaction dynamics [62,78-81],... 

The ASEP/MD method, acronym for Averaged Solvent Electrostatic Potential from Molecular Dynamics, is a theoretical method addressed at the study of solvent effects that is half-way between continuum and quantum mechanics/molecular mechanics (QM/MM) methods. As in continuum or Langevin dipole methods, the solvent perturbation is introduced into the molecular Hamiltonian through a continuous distribution function, i.e. the method uses the mean field approximation (MFA). However, this distribution function is obtained from simulations, i.e., as in QM/MM methods, ASEP/MD combines quantum mechanics (QM) in the description of the solute with molecular dynamics (MD) calculations in the description of the solvent. 

In many cases the continuum may have structures that are narrower than the bandwidth of the pulse. Such structures may be due to either the natural spectrum of the molecular Hamiltonian [327, 328] or to the interaction with the strong external field [195, 197-199, 329]. Under such circumstances we expect the SVCA approximation to break down, yielding nonmonotonic decay dynamics. 

Once this discussion of the space-inversion operator in the context of optically active isomers is accepted, it follows that a molecular interpretation of the optical activity equation will not be a trivial matter. This is because a molecule is conventionally defined as a dynamical system composed of a particular, finite number of electrons and nuclei it can therefore be associated with a Hamiltonian operator containing a finite number (3 M) of degrees of freedom (variables) (Sect. 2), and for such operators one has a theorem that says the Hamiltonian acts on a single, coherent Hilbert space > = 3 (9t3X)51). In more physical terms this means that all the possible excitations of the molecule can be described in . In principle therefore any superposition of states in the molecular Hilbert space is physically realizable in particular it would be legitimate to write the eigenfunctions of the usual molecular Hamiltonian, Eq. (2.14)1 3 in the form of Eq. (4.14) with suitable coefficients (C , = 0. Moreover any unitary transformation of the eigen-... 

It is easily seen that for such trial functions the minimization of the Hamiltonian K, Eq. (5.1), may be replaced by the minimization of a specified nonlinear functional 6(0) of the molecular states 0 alone. In the following we refer to either formulation as seems convenient. This argument also enables one to connect these field theoretical models with the earlier suggestion of mine that molecular structure states can be associated with those solutions of the Schrodinger equation for the full molecular Hamiltonian ft that satisfy certain subsidiary conditions3,35), if the latter are associated with the nonlinearity in the functional 6(0). As we shall see, it may happen that 6(0) has two degenerate minima and it is in this sense that the dynamics gives rise to a double-well structure. 

Here <5FS signifies the fluctuating solvent force on the coordinate qs, while < qs (t) is the Heisenberg time-dependent operator, with dynamics governed by the full internal anharmonic molecular Hamiltonian, associated with the fluctuation <5qs = qs — (i qs f). Finally, the prefactor yi( is (2)... 

A typical problem of interest at Los Alamos is the solution of the infrared multiple photon excitation dynamics of sulfur hexafluoride. This very problem has been quite popular in the literature in the past few years. (7) The solution of this problem is modeled by a molecular Hamiltonian which explicitly treats the asymmetric stretch ladder of the molecule coupled implicitly to the other molecular degrees of freedom. (See Fig. 12.) We consider the the first seven vibrational states of the mode of SF (6v ) the octahedral symmetry of the SF molecule makes these vibrational levels degenerate, and coupling between vibrational and rotational motion splits these degeneracies slightly. Furthermore, there is a rotational manifold of states associated with each vibrational level. Even to describe the zeroth-order level states of this molecule is itself a fairly complicated problem. Now if we were to include collisions in our model of multiple photon excitation of SF, e wou d have to solve a matrix Bloch equation with a minimum of 84 x 84 elements. Clearly such a problem is beyond our current abilities, so in fact we neglect collisional effects in order to stay with a Schrodinger picture of the excitation dynamics. 

Normally the TDSE cannot be solved analytically and must be obtained numerically. In the numerical approach we need a method to render the wave function. In time-dependent quantum molecular reaction dynamics, the wave function is often represented using a discrete variable representation (DVR) [88-91] or Fourier Grid Hamiltonian (FGH) [92,93] method. A Fast Fourier Transform (FFT) can be used to evaluate the action of the kinetic energy operator on the wave function. Assuming the Hamiltonian is time independent, the solution of the TDSE may be written... 

In order to investigate the energies and molecular properties of molecules interacting with aerosol particles, it is crucial to establish the Hamiltonians and the energy functionals for the two structural environment methods. The basic principle for both structural environment methods is the same and it is one that has been utilized successfully within quantum chemistry [2-33] and molecular reaction dynamics [19,68-71,96], we divide a large system into two subsystems. The focus is... 

Initiated by the work of Bunker [323,324], extensive trajectory simulations have been performed to determine whether molecular Hamiltonians exhibit intrinsic RRKM or non-RRKM behavior. Both types have been observed and in Fig. 43 we depict two examples, i.e., classical lifetime distributions for NO2 [271] and O3. While Pd t) for NO2 is well described by a single-exponential function — in contrast to the experimental and quantum mechanical decay curves in Fig. 31 —, the distribution for ozone shows clear deviations from an exponential decay. The classical dynamics of NO2 is chaotic, whereas for O3 the phase space is not completely mixed. This is in accord with the observation that the quantum mechanical wave... 

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. 

X. Chapuisat, A. Nauts, and J.-P. Brunet, Exact quantum molecular Hamiltonians. Part 1. Application to the dynamics of three particles. Mol. Phys. 72, 1-31 (1991). 

The use of non-Hamiltonian dynamical systems has a long history in mechanics [8] and they have recently been used to study a wide variety of problems in molecular dynamics (MD). In equilibrium molecular dynamics we can exploit non-Hamiltonian systems in order to generate statistical ensembles other than the standard microcanonical ensemble NVE) that is generated by traditional Hamiltonian dynamics. These ensembles, such as the canonical (NVT) and isothermal-isobaric (NPT) ensembles, are much better than the microcanonical ensemble for representing the actual conditions under which experiments are carried out. 

It is clear, from the discussion thus far, that typical molecular Hamiltonians display features characteristic of chaotic classical motion. The logical order followed previously, that is, the introduction of well-defined concepts of chaotic behavior in classical ideal systems, followed by an examination of realistic molecular models, does not follow through to quantum mechanics. The primary difficulty is that quantum mechanics always predicts, for bound-state dynamics, quasiperiodic motion. Several aspects of quantum chaos are discussed in Section IV. We note at this point, however, that this quantum-... 

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