Quantum mechanics Hamiltonian approach

A more elegant model of quantum mechanical computers was introduced by Feynman ([feyii85], [feynSb]). Although the formalism is similar to Benioff s - both approaches seek to define an appropriate quantum mechanical Hamiltonian H whose time evolution effectively represents the execution of a desired computation... 

The inter/intramolecular potentials that have been described may be viewed as classical in nature. An alternative is a hybrid quantum-mechanical/classical approach, in which the solute molecule is treated quantum-mechanically, but interactions involving the solvent are handled classically. Such methods are often labeled QM/MM, the MM reflecting the fact that classical force fields are utilized in molecular mechanics. An effective Hamiltonian Hefl is written for the entire solute/solvent system ... 

The mapping procedure introduced in Sec. 6 results in a quantum-mechanical Hamiltonian with a well-defined classical limit, and therefore extends the applicability of the established semiclassical approaches to nonadiabatic dynamics. The thus obtained semiclassical version of the mapping approach, as well as the equivalent formulation that is obtained by requantizing the classical electron analog model of Meyer and Miller, have been applied to a variety of systems with nonadiabatic dynamics in the recent years. It appears that this approach is so far the only fully semiclassical method that allows a numerical treatment of truly multidimensional nonadiabatic dynamics at conical intersections. 

The treatment developed here is based on the density matrix of quantum mechanics and extends previous work using wavefunctions.(42 5) The density matrix approach treats all energetically accessible electronic states in the same fashion, and naturally leads to average effective potentials which have been shown to give accurate results for electronically diabatic collisions. 19) The approach is taken here for systems where the dynamics can be described by a Hamiltonian operator, as it is possible for isolated molecules or in models where environmental effects can be represented by terms in an effective Hamiltonian. 

In the full quantum mechanical approach [8], one uses Eq. (22) and considers both the slow and fast mode obeying quantum mechanics. Then, one obtains within the adiabatic approximation the starting equations involving effective Hamiltonians characterizing the slow mode that are at the basis of all principal quantum approaches of the spectral density of weak H bonds [7,24,25,32,33,58, 61,87,91]. It has been shown recently [57] that, for weak H bonds and within direct damping, the theoretical lineshape avoiding the adiabatic approximation, obtained directly from Hamiltonian (22), is the same as that obtained from the RR spectral density (involving adiabatic approximation). 

In the quantum mechanical continuum model, the solute is embedded in a cavity while the solvent, treated as a continuous medium having the same dielectric constant as the bulk liquid, is incorporated in the solute Hamiltonian as a perturbation. In this reaction field approach, which has its origin in Onsager s work, the bulk medium is polarized by the solute molecules and subsequently back-polarizes the solute, etc. The continuum approach has been criticized for its neglect of the molecular structure of the solvent. Also, the higher-order moments of the charge distribution, which in general are not included in the calculations, may have important effects on the results. Another important limitation of the early implementations of this method was the lack of a realistic representation of the cavity form and size in relation to the shape of the solute. 

In the classical approach, it is relatively simple to calculate the solvation energies. However, in the quantum mechanical formulations, the electronic structure of the solute molecule depends on the reaction held and the reaction held in turn depends on the structure of the solute. It is a typical nonlinear problem and has to be solved iteratively. Several approaches have been proposed for solving this problem [8-11]. All of them are based on the modihcation of the Hamiltonian in the following equation ... 

The notion of chaos is interwoven with the discussion of time evolution, which we do not pursue in this volume. It is worthwhile, however, to note that it is, by now, well understood that a quantum-mechanical system with a finite Hamiltonian matrix cannot satisfy many of the purely mathematical characterizations of chaos. Equally, however, over long periods of time such systems can manifest many of the qualitative features that one associates with classically chaotic systems. It is not our intention to follow this most interesting theme. Instead we seek a more modest aim, namely, to forge a link between the elementary notions of classical nonlinear dynamics and the algebraic approach. This turns out to be possible using the action-angle variables of classical mechanics. In this section we consider only the nonlinear dynamics aspects. We complete the bridge in Chapter 7. 

The quantum mechanical approach to the problem is based on the Hamiltonian of Equation 3.19b with velocities Qi replaced by momenta P conjugate to the coordinate Qi. P is classically defined... 

An approximate quantum mechanical expressions- that allows one to calculate the electrostatic surface potential around atoms, radicals, ions, and molecules by assuming that the ground-state electron density uniquely specifies the Hamiltonian of the system and thereby all the properties of the ground state. This approach greatly facilitates computational schemes for exact calculation of the ground-state energy and electron density of orbitals. 

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