Density matrices path integrals

Nevertheless, the density matrix—path integral description allows the general formulation for the many-electronic density through the so-called canonical density algorithm it prescribes that the system is firstly solved for the single electron evolution under the concerned potential for which the time-space density matrix is analytically formulated, in an evolution manner, as the propagator (x, x, tj then, the partition function is com-... 

The book s simple and illustrative presentation of eoneepts and analyses inelude both basic physical-chemical quantum principles and observabihty at each level of matter s oiganization as well as advanced (more abstract, thus most necessary) formalisms of density matrix, path integrals, Hartree-Fock (as self-consistent quantum methods), original Heisenberg uncertainty by a sub-quantum extension (with more quantum fluctuation insight for the free evolution modeling), eventually leading to a novel undulatory/corpuscular characterization of the Si-based nano/ mesosystems. 

Figure 9.30. Kinetics of the photoinduced back ET reaction for (NH3)5Fe"(CN)(Ru) "(CN)5 in water at BOOK with F = 2500cm , — AG = 3900cm and , = 3800cm . Ohmic spectral density with exponential cutoff at 220 cm is assumed for the solvent. Exact enumeration method (dashed) and the transfer matrix path integral approach (solid) are compared with the Golden Rule prediction (dash-dotted). (Reproduced from [132] with permission. Copyright (1998) by the American Institute of Physics.)...

In the light of the path-integral representation, the density matrix p Q-,Q-,p) may be semi-classically represented as oc exp[ —Si(Q )], where Si(Q ) is the Eucledian action on the -periodic trajectory that starts and ends at the point Q and visits the potential minimum Q = 0 for r = 0. The one-dimensional tunneling rate, in turn, is proportional to exp[ —S2(Q-)], where S2 is the action in the barrier for the closed straight trajectory which goes along the line with constant Q. The integral in (4.32) may be evaluated by the method of steepest descents, which leads to an optimum value of Q- = Q. This amounts to minimization of the total action Si -i- S2 over the positions of the bend point Q. ... 

The influence functional theory, as it was formulated by Feyman and Vernon, relies on the additional assumption concerning factorization of the total (system and bath) density matrix in the past. Without this assumption the theory requires a triple path integral, with one thermal integration over the imaginary time axis [Grabert et al. 1988]. 

In coordinate space, the diagonal elements of the canonical density matrix in the Fourier path integral representation are given by [20]... 

Fig. 6. All paths leading from the initial to the final points in time t contribute an interfering amplitude to the path sum describing the resultant probability amplitude for the quantum propagation. In this double slit free particle case, two paths of constant speed are local functional stationary points of the action, and these two dominant paths provide the basis for a (semiclassical) classification of subsets of paths which contribute to the path integral. In the statistical thermodynamic path expression, the path sum is equal to the off-diagonal electronic thermal density matrix...

The most simple procedure can be carried out in the case of path integral for the equilibrium density matrix, i.e., with it = j6. Specifically, the Metropolis algorithm [Metropolis et al., 1953] applies to n-dimension-al integrals of the general form... 

So far, one can be much more successful in calculating a rate constant when one knows in advance that it exists, than in answering the question of whether it exists. A considerable breakthrough in this area was the solution of the spin-boson problem, which, however, has only limited relevance to any problem in chemistry because it neglects the effects of intrawell dynamics (vibrational relaxation) and does not describe thermally activated transitions. A number of attempts have been made to go beyond the two-level system approximation, but the basic question of how vibrational relaxation affects the transition from coherent oscillations to exponential decay awaits a quantitative solution. Such a solution might be obtained by numerical computation of real-time path integrals for the density matrix using the influence functional technique. 

By increasing pressure and/or decreasing temperature, ionic quantum effects can become relevant. Those effects are important for hydrogen at high pressure [7, 48]. Static properties of quantum systems at finite temperature can be obtained with the Path Integral Monte Carlo method (PIMC) [19]. We need to consider the ionic thermal density matrix rather than the classical Boltzmann distribution ... 

Warshel and Chu [42] and Hwang et al. [60] were the first to calculate the contribution of tunneling and other nuclear quantum effects to PT in solution and enzyme catalysis, respectively. Since then, and in particular in the past few years, there has been a significant increase in simulations of quantum mechanical-nuclear effects in enzyme and in solution reactions [16]. The approaches used range from the quantized classical path (QCP) (for example. Refs. [4, 58, 95]), the centroid path integral approach [54, 55], and variational transition state theory [96], to the molecular dynamics with quantum transition (MDQT) surface hopping method [31] and density matrix evolution [97-99]. Most studies of enzymatic reactions did not yet examine the reference water reaction, and thus could only evaluate the quantum mechanical contribution to the enzyme rate constant, rather than the corresponding catalytic effect. However, studies that explored the actual catalytic contributions (for example. Refs. [4, 58, 95]) concluded that the quantum mechanical contributions are similar for the reaction in the enzyme and in solution, and thus, do not contribute to catalysis. 

The DPI representation of the path integral that was developed in the preceding section is not unique. Another path-integral representation is often used that has come to be known as the Fourier representation [33,34,36-42,44,85]. Like the DPI representation, the Fourier representation transforms the path integral into an infinite-dimensional Riemann integral. In this formalism, we consider the paths to be periodic signals that can be represented as a Fourier series. Consider the density matrix p(x, x j8). Since the partition function is the trace of the density matrix, we have... 

We follow Doll et al. [42] and begin by writing a Fourier path-integral (FPI) formula for the ratio of the density matrix of the full system to that of a free-particle system (zero potential energy). The Jacobian and other prefactors cancel in the ratio, and we obtain... 

Recently, there has been some progress in generalizing the path-integral method to treat fermion systems, which is called restricted PIMC (RPIMC) [24]. One can also apply the fixed-node method to the density matrix. The fermion density matrix is given by... 

In order to obtain an expression suitable for numerical calculations, we express the density matrix using the discretized path integral as [1]... 

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