Quantum mechanics autocorrelation

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... 

It is noted that in the results of Suzuki s propagator given in Fig. 7 of Ref. 100, an incorrect normalization factor was used, which resulted in a strong deviation of the overall magnitude of the autocorrelation function from the quantum-mechanical result. 

Abbreviations MD, molecular dynamics TST, transition state theory EM, energy minimization MSD, mean square displacement PFG-NMR, pulsed field gradient nuclear magnetic resonance VAF, velocity autocorrelation function RDF, radial distribution function MEP, minimum energy path MC, Monte Carlo GC-MC, grand canonical Monte Carlo CB-MC, configurational-bias Monte Carlo MM, molecular mechanics QM, quantum mechanics FLF, Hartree-Fock DFT, density functional theory BSSE, basis set superposition error DME, dimethyl ether MTG, methanol to gasoline. 

The rate coefficient kernel of eqn. (369) and the initial condition term of eqn. (370) are those given by Northrup and Hynes [ 103]. These expressions are cast into the form of correlation functions (cf. the velocity autocorrelation function) and have a close similarity to the matrix elements in quantum mechanic applications. While they are quite easy to derive and... 

First, one has to decide what the quantum-mechanical translation is of the autocorrelation function, Eq. (4). Let Q be the operator corresponding to the physical quantity q, and let... 

The computation of spectra from the dipole autocorrelation function, Eq. 2.66, does not impose such stringent conditions on the integrand as our derivation based on Fourier transform suggests. Equation 2.66 is, therefore, a favored starting point for the computation of spectral moments and profiles the relationship is also valid in quantum mechanics as we will see below. 

Density expansion. The method of cluster expansions has been used to obtain the time-dependent correlation functions for a mixture of atomic gases. The particle dynamics was treated quantum mechanically. Expressions up to third order in density were given explicitly [331]. We have discussed similar work in the previous Section and simply state that one may talk about binary, ternary, etc., dipole autocorrelation functions. 

Le Quere and Leforestier (1990, 1991) calculated the autocorrelation function directly using the same PES and found fair agreement in regard of the recurrence times while the amplitudes were in remarkable disagreement. This may be due to either deficiencies of the calculated PES or the neglect of nonzero total angular momentum states in the theory. If the recurrences in the autocorrelation function are rescaled, the quantum mechanically calculated spectrum agrees well with experiment. 

The quantum mechanical wavepacket closely follows the main classical route. It slides down the steep slope, traverses the well region, and travels toward infinity. A small portion of the wavepacket, however, stays behind and gives rise to a small-amplitude recurrence after about 40-50 fs. Fourier transformation of the autocorrelation function yields a broad background, which represents the direct part of the dissociation, and the superimposed undulations, which are ultimately caused by the temporarily trapped trajectories (Weide, Kiihl, and Schinke 1989). A purely classical description describes the background very well (see Figure 5.4), but naturally fails to reproduce the undulations, which have an inherently quantum mechanical origin. 

In the following we present the axioms or basic postulates of quantum mechanics and accompany them by their classical counterparts in the Hamiltonian formalism. We begin the presentation with a brief summary of some of the mathematical background essential for the developments in the following. It is by no means a comprehensive presentation, and the reader is supposed to have some basic knowledge about quantum mechanics that may be obtained from any of the many introductory textbooks in quantum mechanics. The focus here is on results of particular relevance to the subjects of this book. We consider, for example, a derivation of a formal expression for the flux density operator in quantum mechanics and its coordinate representation. A systematic way of generating any representation of any combination of operators is set up, and is of immediate usage for the time autocorrelation function of the flux operator used to determine the rate constants of a chemical process. 

Three of the experiments are completely new, and all make use of optical measurements. One involves a temperature study of the birefringence in a liquid crystal to determine the evolution of nematic order as one approaches the transition to an isotropic phase. The second uses dynamic laser light scattering from an aqueous dispersion of polystyrene spheres to determine the autocorrelation function that characterizes the size of these particles. The third is a study of the absorption and fluorescence spectra of CdSe nanocrystals (quantum dots) and involves modeling of these in terms of simple quantum mechanical concepts. 

The quantum flux-flux autocorrelation formalism, developed by Miller, Schwartz, and Tromp [78] and by Yamamoto [79], represents an exact quantum mechanical expression for a chemical reaction rate constant. According to the flux-flux autocorrelation formalism, the thermally averaged rate constant k T) is given by... 

Though Eq. (27) is exact in a perturbative sense, it is demanding to calculate the quantum mechanical force autocorrelation function 5(f) even for small molecular systems. Hence, many computational schemes have been developed to approximate the quantum mechanical force autocorrelation function. 

The quantities that hold the complete information, the states vectors T and 0(f), can be derived from fhe sysfem operator S2. This, in turn, means that the entire sought informatioiyis also present in or, equivalently, in U(f). The total evolution operator U(f) itself is the major physical content of C(f). Hence, fhe stated quantum-mechanical postulate on completeness implies that everything one could possibly learn about any considered system is also contained in the autocorrelation function C(f). Despite the fact that the same full information is available from T, 0(f), S2,U(f), and C(f), fhe autocorrelation functions are more manageable in practice, since they are observables. As a scalar, the quantity C(f) has a functional form fhaf is defined by its... 

The treatment up to this point has been fully quantum mechanical Vjj is an operator in the bath degrees of freedom. For many calculations on liquids, however, one wants to treat these degrees of freedom (rotations and translations) classically the question then arises of what is the best way to replace a quantum correlation function with a classical one. A classical autocorrelation function is an even function of the time, a property shared by the anticommutator in (2.11) but not by the one sided correlation function of (2.10). It thus appears that the best place to make a classical approximation is in (2.11) in addition, doing so gives... 

---