Autocorrelation function operators

This expression is derived as the Fourier transform of a time-dependent one-particle autocorrelation function (26) (i.e. propagator), and cast in matrix form G(co) over a suitable molecular orbital (e.g. HF) basis, by means of the related set of one-electron creation (ai" ") and annihilation (aj) operators. In this equation, the sums over m and p run over all the states of the (N-1)- and (N+l)-electron system, l P > and I P " respectively. Eq and e[ represent the energy of the... 

The linear response theory [50,51] provides us with an adequate framework in order to study the dynamics of the hydrogen bond because it allows us to account for relaxational mechanisms. If one assumes that the time-dependent electrical field is weak, such that its interaction with the stretching vibration X-H Y may be treated perturbatively to first order, linearly with respect to the electrical field, then the IR spectral density may be obtained by the Fourier transform of the autocorrelation function G(t) of the dipole moment operator of the X-H bond ... 

Now, let us look at the autocorrelation function of the dipole moment operator within the adiabatic approach. In the representation I it is... 

Here, t/(f) is the reduced time evolution operator of the driven damped quantum harmonic oscillator. Recall that representation II was used in preceding treatments, taking into account the indirect damping of the hydrogen bond. After rearrangements, the autocorrelation function (45) takes the form [8]... 

Langevin equation. RR [58] have obtained for the dipole moment operator the following autocorrelation function which may be written after a correction (corresponding to a zero-point-energy they neglected) [8] ... 

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... 

The Tr operation denotes a classical integration over all coordinates. Apart from the mean potential, the particle also feels a random force , = which is due to all the bath degrees of freedom. This random force has zero mean, and one can compute its autocorrelation function. The mapping of the tme dynamics onto the GLE is then completed by assuming that the random force (t) is Gaussian and its autocorrelation function is ( ,(t) ,(t )) = Y(t t ) where p =... 

Since the velocity relaxation time, m/J, is typically 0.1 ps, t is rather shorter than that estimated from the decay of the velocity autocorrelation function. As an operational convenience, rrel — mjl can be deduced from the decay time re of the velocity autocorrelation functions. However, this procedure still does not entirely adequately describe the details of Brownian motion of particles over short times. The velocity relaxes in a purely exponential manner characteristic of a Markovian process. Further comments on the reduction of the Fokker—Planck equation to the diffusion equation have been made by Harris [526] and Tituiaer [527]. 

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... 

If F is an operator working on the system S alone this autocorrelation function describes the fluctuations in S under influence of the bath. We compute it to second order in the coupling constant a. 

However, if the initial state is a thermal state, such as the canonical den-sity matrix p - (1/Z)exp(- 3//), the autocorrelation is no longer given by a single quantum amplitude but becomes a sum of quantum amplitudes in which quantum phases are randomized. In the classical limit ft - 0, the leading expression becomes the purely classical autocorrelation function with the dynamics being ruled by the classical Liouvillian operator ci = Hci> ... 

In reality, these functions are more complex and the operator has to use the trial and error mode. Practical criteria which improve the likelihood of correct selection of the parameters of the ARIMA model are the autocorrelation and the partial autocorrelation function of the errors of the resulting ARIMA fit. If they do not have significant spikes the model is satisfactory. 

Qr is the partition function of the reactants, and v = 2 and 1 for a bimolecular and a unimolecular reaction, respectively. Note that in the flux autocorrelation function, the position of the Boltzmann operator differs from the form given in Eq. (5.115). 

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. 

Let us then derive an expression for the matrix element of the flux operator in the coordinate representation, an expression we need in order to develop the time autocorrelation function of the flux operator in the coordinate representation. We use the axiom for the matrix element of the momentum operator in the coordinate representation, and obtain... 

Here, G(t) is the quantum autocorrelation function (ACF) of the dipole moment operator responsible for the dipolar absorption transition, whereas oo is the angular frequency and t is the time. Equation (1) has been used, for example, by Bratos [45] and Robertson and Yarwood [46] in their semiclassical studies of H-bonded species within the linear response theory. 

T.l.D.l. Complex Autocorrelation Function (ACF) G(t) of the Dipole Moment Operator of the Fligh Frequency Mode... 

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