Classical Hamiltonian Description

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... 

The quote is from the third volume of Henri Poincare s New Methods of Celestial Mechanics, and is a description of his discovery of homoclinic orbits (see below) in the restricted three-body problem. It is also one of the earliest recorded formal observations that very complicated behavior may be found even in seemingly simple classical Hamiltonian systems. Although Hamiltonian (or conservative) chaos often involves fractal-like phase-space structures, the fractal character is of an altogether different kind from that arising in dissipative systems. An important common thread in the analysis of motion in either kind of dynamical system, however, is that of the stability of orbits. 

It is clear, however, that one may introduce a microscopic description of the solvent into the above formalism. This was done by Straus, Calhoun, and Voth. ° The total classical Hamiltonian is given by... 

F. Halbwachs, F. Piperno, and J. P. Vigier, Relativistic Hamiltonian description of the classical photon behaviour A basis to interpret aspect s experiments, Lett. Nuovo Cimento 33(11) (1982). 

There have been two general directions for classical trajectory studies of IVR (Gomez and Poliak, 1992). One involves the analysis of classical Hamiltonians for molecules and is concerned with the structure of the multidimensional phase space for excited molecules and the mechanism(s) for intramolecular energy transfer (Lichtenberg and Lieberman, 1991). The other is concerned with determining how to use classical mechanics to represent the initially excited zero-order state i and then propagate I (/). There is also an interest in the correspondence between classical and quantal descriptions of IVR (Brumer and Shapiro, 1988). 

In this section, we introduce the Hamiltonian description of the motion which is useful for discussing the underlying geometric structure associated to classical mechanics. We can understand the Hamiltonian formulation as a natural description of classical mechanics from several perspectives. First, it is based directly on the... 

All phenomena of classical nonrelativistic mechanics are solely based on Newton s laws of motion, which are valid in any inertial frame of reference. The natural symmetry operations of classical mechanics are the Galilean transformations, mediating the transition from one inertial coordinate system to another. The fundamental laws of classical mechanics can equally well be formulated applying the elegant Lagrangian and Hamiltonian descriptions based on Hamilton s action principle. Maxwell s equations for electric and magnetic fields are introduced as the basic laws of classical electrodynamics. 

These methods combine a QM representation of solute with a classical continuum description of the solvent [18-23]. The methodology is equivalent to that of classical continuum methods, except that a) the solute charge distribution is allowed to relax by the solvent reaction field, and b) the solute-solvent interaction is computed at the QM level. Most QM continuum methods work within the multipole or apparent surface charge approaches, even though other formalisms are also available [18-23]. The solvent reaction field is introduced into the solute Hamiltonian by means of a perturbation operator (R in equation 22) that couples the solvent reaction field to the solute charge distribution. At this point, it is worth noting that equation 22 is not lineal, since T and R are mutually dependent. This means that a self-consistent process in which both the wavefunction and the reaction field are treated simultaneously is required to solve equation 22. This is the reason why these methods are typically known as self-consistent reaction field (SCRF) methods. 

Miller, Handy, and Adams have recently shown how one can construct a classical Hamiltonian for a general molecular system based on the reaction path and a harmonic approximation to the potential surface about it. The coordinates of this model are the reaction coordinate and the normal mode coordinates for vibrations transverse to the reaction path these are essentially a polyatomic version of the natural collision coordinates introduced by Marcus and by Hofacker for A + BC AB 4- C reactions. One of the important practical aspects of this model is that all of the quantities necessary to define it are obtainable from a relatively modest number of db initio quantum chemistry calculations, essentially independent of the number of atoms in the system. This thus makes possible an ab initio theoretical description of the dynamics of reactions more complicated than atom-diatom reactions. 

Since the theory for the classical description of a two-state system in a laser field has been described previously, only a brief presentation is given here. The classical Hamiltonian for this system has the form... 

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) ... 

A semiclassical description is well established when both the Hamilton operator of the system and the quantity to be calculated have a well-defined classical analog. For example, there exist several semiclassical methods for calculating the vibrational autocorrelation function on a single excited electronic surface, the Fourier transform of which yields the Franck-Condon spectmm [108, 109, 150, 244]. In particular, semiclassical methods based on the initial-value representation of the semiclassical propagator [104-111, 245-248], which circumvent the cumbersome root-search problem in boundary-value-based semiclassical methods, have been successfully applied to a variety of systems (see, for example, Refs. 110, 111, 161, and 249 and references therein). The mapping procedure introduced in Section VI results in a quantum-mechanical Hamiltonian with a well-defined classical limit, and therefore it... 

In this review we shall first establish the theoretical foundations of the semi-classical theory that eventually lead to the formulation of the Breit-Pauli Hamiltonian. The latter is an approximation suited to make the connection to phenomenological model Hamiltonians like the Heisenberg Hamiltonian for the description of electronic spin-spin interactions. The complete derivations have been given in detail in Ref. (21), but turn out to be very involved and are thus scattered over many pages in Ref. (21). For this reason, we aim here at a summary that is as brief and concise as possible so that all relevant connections between different levels of approximation are evident. This allows us to connect present-day quantum chemical methods to phenomenological Hamiltonians and hence to establish and review the current status of these first-principles methods applied to transition-metal clusters. 

Be aware of the fact that we have to consider the non-Markovian version of the quantum master equation to stay at a level of description where the emission rate, Eq. (39), can be deduced. Moreover, to be ready for a translation to a mixed quantum classical description a variant has been presented where the time evolution operators might be defined by an explicitly time-dependent CC Hamiltonian, i.e. exp(—iHcc[t — / M) has been replaced by the more general expression Ucc(t,F). 

Molecular dynamics (MD) and Monte Carlo (MC) are in most cases associated with a discrete description of the solvent and with classical representations of the solute and/or solvent Hamiltonians. However, the same type of sampling engines can be coupled to continuum methods, which implies a loss of detail in the representation of individual solute-solvent interactions, but present two main advantages (i) calculation can be faster since no explicit sampling of solvent is needed, (ii) sampling efficiency of solute movements can be very high because of the neglect of solvent friction. 

Sampling the conformational space of solute(s) by MC or MD algorithms requires many intramolecular and solvation calculations and accordingly simplicity in the solute Hamiltonian and computer efficiency in the continuum method used to compute solvation are key requirements. This implies that, with some exceptions [1], MD/MC algorithms are always coupled to purely classical descriptions of solvation, which in order to gain computer efficiency adopt severe approximations, such as the neglect of explicit electronic polarization contributions to solvation (for a discussion see ref. [1]). In the following we will summarize the major approaches used to couple MD/MC with continuum representations of solvation. 

Gamow waves, even though standard quantum mechanical states remain embedded [6]. Although general Hamiltonians may not subscribe to dilatation analyticity, complex symmetric perturbations, as the specific situation dictates, have an additional appeal. In the forthcoming description, basic quantum mechanical physical law will rule at the same time classical mechanics takes over where and when appropriate. 

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