Classical descriptive methods

A classical description of M can for example be a standard force field with (partial) atomic charges, while a quantum description involves calculation of the electronic wave function. The latter may be either a semi-empirical model, such as AMI or PM3, or any of the ab initio methods, i.e. HF, MCSCF, CISD, MP2 etc. Although the electrostatic potential can be derived directly from the electronic wave function, it is usually fitted to a set of atomic charges or multipoles, as discussed in Section 9.2, which then are used in the actual solvent model. 

Section 2 mainly focuses on the current efforts to improve the accuracy of quantum calculations using simplified empirical model forms. McNamara and Hillier, in Chapter 5, summary their work on improving the description of the interactions in biological systems via their optimized semiempirical molecular models. Piquemal and co-workers present recent advances in the classical molecular methods, aiming at better reproduction of high-level quantum descriptions of the electtostatic interactions in Chapter 6. In Chatper 7, Cui and Elstner describe a different semiempir-... 

Percolation theory describes [32] the random growth of molecular clusters on a d-dimensional lattice. It was suggested to possibly give a better description of gelation than the classical statistical methods (which in fact are equivalent to percolation on a Bethe lattice or Caley tree, Fig. 7a) since the mean-field assumptions (unlimited mobility and accessibility of all groups) are avoided [16,33]. In contrast, immobility of all clusters is implied, which is unrealistic because of the translational diffusion of small clusters. An important fundamental feature of percolation is the existence of a critical value pc of p (bond formation probability in random bond percolation) beyond which the probability of finding a percolating cluster, i.e. a cluster which spans the whole sample, is non-zero. 

In spite of the frequency-shifted excitation, the quantized PIP inevitably excites multiple sidebands located at n/At ( = 1, 2,...) from the centre band. An attempt was made16 to calculate the excitation profile of multiple bands created by a PIP of a constant RF field strength, using an approximate method based on the Fourier analysis. The accuracy of the method relies partly on the linear response of the spin system, which is, unfortunately, not true in most cases except for a small angle excitation. In addition, the spins inside a magnet consitute a quantum system, which is sensitive not only to the strengths but also to the phases of the RF fields. Any classical description is doomed to failure if the quantum nature of the spin system emerges. 

In this chapter, we are concerned with various theoretical formulations that allow us to treat nonadiabatic quantum dynamics in a classical description. To introduce the main concepts, we first give a brief overview of the existing methods and then discuss their application to ultrafast molecular photoprocesses. 

The goal of this chapter is twofold. First we wish to critically compare—from both a conceptional and a practical point of view—various classical and mixed quantum-classical strategies to describe non-Born-Oppenheimer dynamics. To this end. Section II introduces five multidimensional model problems, each representing a specific challenge for a classical description. Allowing for exact quantum-mechanical reference calculations, aU models have been used as benchmark problems to study approximate descriptions. In what follows, Section III describes in some detail the mean-field trajectory method and also discusses its connection to time-dependent self-consistent-field schemes. The surface-hopping method is considered in Section IV, which discusses various motivations of the ansatz as well as several variants of the implementation. Section V gives a brief account on the quantum-classical Liouville description and considers the possibility of an exact stochastic realization of its equation of motion. 

The previous two sections of this review deal with classical simulation methods. A description of the activation of adsorbates by acidic sites, together with any bond breaking or bond formation that may take place, is the realm of quantum mechanical (QM) simulations. These types of calculations are particularly well-suited to zeolite-adsorbate systems when the cluster approximation is used. The active acidic site in the zeolite is modeled by a molecular cluster, formed by cutting out a small portion of... 

Classical Path. Another approach to scattering calculations uses a quantum-mechanical description of the internal states, but classical mechanics for the translational motion. This "classical path" method has been popular in line-shape calculations (37,38). It is almost always feasible to carry out such calculations in the perturbation approximation for the internal states (37). Only recently have practical methods been developed to perform non-perturbative calculations in this approach (39). 

Recent work by Pritchard has concentrated on a state-to-state description of unimolecular reactions229 and an examination by classical trajectory methods of the effects of overall molecular rotation on the unimolecular rate. The latter calculations have revealed a most interesting aspect of computing in chaotic systems, namely, that the same algorithm gives different results on different machines for a trajectory with identical initial conditions, or even on the same machine with different releases of the same compiler. However, the ensemble average behavior, with an ensemble comprising 100 or more trajectories, is acceptably the same each time.230... 

Abstract. A rigorous derivation of quantum-classical equations of motion is still lacking. The framework proposed so far to describe in a consistent way the dynamics of a mixed quantum-classical system using systematic approximations have failed. A recent attempt to solve the inconsistencies of quantum-classical approximated methods by introducing a group-theoretical approach is discussed in detail. The new formulation which should restore the consistency of the proposed quantum-classical dynamics and statistical mechanics will be shown to produce, instead, a purely classical description. In spite of that, the discussed approach remains interesting since it could produce non-trivial formulations. 

The question then arises if a convenient mixed quantum-classical description exists, which allows to treat as quantum objects only the (small number of) degrees of freedom whose dynamics cannot be described by classical equations of motion. Apart in the limit of adiabatic dynamics, the question is open and a coherent derivation of a consistent mixed quantum-classical dynamics is still lacking. All the methods proposed so far to derive a quantum-classical dynamics, such as the linearized path integral approach [2,6,7], the coupled Bohmian phase space variables dynamics [3,4,9] or the quantum-classical Li-ouville representation [11,17—19], are based on approximations and typically fail to satisfy some properties that are expected to hold for a consistent mechanics [5,19]. 

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. 

Now we are ready to start the derivation of the intermediate scheme bridging quantum and classical descriptions of molecular PES. The basic idea underlying the whole derivation is that the experimental fact that the numerous MM models of molecular PES and the VSEPR model of stereochemistry are that successful, as reported in the literature, must have a theoretical explanation [21], The only way to obtain such an explanation is to perform a derivation departing from a certain form of the trial wave function of electrons in a molecule. QM methods employing the trial wave function of the self consistent field (or equivalently Hartree-Fock-Roothaan) approximation can hardly be used to base such a derivation upon, as these methods result in an inherently delocalized and therefore nontransferable description of the molecular electronic structure in terms of canonical MOs. Subsequent a posteriori localization... 

MFTA often gives models that are comparable in quality of description and prediction to models based on the widely used classical QSAR methods and 3D approaches. 

We show how the quantum-classical evolution equations of motion can be obtained as an approximation to the full quantum evolution and point out some of the difficulties that arise because of the lack of a Lie algebraic structure. The computation of transport properties is discussed from two different perspectives. Transport coefficient formulas may be derived by starting from an approximate quantum-classical description of the system. Alternatively, the exact quantum transport coefficients may be taken as the starting point of the computation with quantum-classical approximations made only to the dynamics while retaining the full quantum equilibrium structure. The utility of quantum-classical Liouville methods is illustrated by considering the computation of the rate constants of quantum chemical reactions in the condensed phase. 

AIMD data were used to improve the classical description of C mim Cl] by applying the force matching approach [72], A self-consistent optimization method for the generation of classical potentials of general functional form was presented and applied. A force field that better reproduces the observed first-principles forces was obtained [72],... 

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