Discrete-variable representation distributions

Bade Z and Light J C 1986 Highly exdted vibrational levels of floppy triatomic molecules—a discrete variable representation—distributed Gaussian-basis approach J. Chem. Phys. 85 4594... 

Bacic Z, Light JC (1986) Highly excited vibrational levels of floppy triatomic molecules A discrete variable representation - Distributed Gaussian approach. J Chem Phys 85 4594... 

Here r(i ) = 0.1(i — lO) 0(i — 10) 2) 2 with 2) being the electronic excited state, and 6 is the Heaviside step function. Time step dt was set to 1, and total propagation time was 1200. The derivative operator and position operator was treated in sine discrete variable representation [92] with 600 grid points equally distributed from R = —3 to 14. We have checked the convergence of the results with respect to grid-density and time step. 

A graphical representation of the discrete distribution f(ji) is shown in Figure 1.3a. Figure 1.3b shows the analogous MWD represented as the (number) distribution ff of the discrete variable (notice the equivalence of the concept of distribution with those of fraction or frequency). 

The purpose of this section is to show the reader how the different degree of reduction of information space (number of new variables) influences the final reproduction. As the procedures for the reduction depend on the type of data, i.e. smooth curves, discret distributions, and 2-dimensional patterns, we shall discuss each of them in appropriate subsections. The common property to all of them is the same representation in which they can be treated by the computer (see equation 4.1). 

While a plotted curve assumes a continuous relationship between the variables by interpolating between individual data points, a histogram involves no such assumptions and is the most appropriate representation if the number of data points is too few to allow a trend line to be drawn. Histograms are also used to represent frequency distributions (p. 265), where the y-axis shows the number of times a particular value of x was obtained (e.g. Fig. 37.3). As in a plotted curve, the x-axis represents a continuous variable which can take any value within a given range, so the scale must be broken down into discrete classes and the scale marks on the x-axis should show either the mid-points (mid-values) of each class (Fig. 37.3), or the boundaries between the classes. 

Typically, electrostatic interactions are represented by some form of Coulomb s law-type function. In this type of calculation, the atoms in the molecule are assigned a charge distribution consisting of discrete point charge approximations centered on each atom. Other variations of these simple functions include exponential-type approximations (12) and variable dielectric approximations (3,11). The point charge approximation is often determined by ab-initio or semi-empirical molecular orbital calculations. The user may select the particular functional representation used in the calculation as well as the parametric form of the function. The variable dielectric function is represented by an approximation to the rigorously derived function (3,11) for computational efficiency. 

In Chapter 5 we described a number of ways to examine the relative frequency distribution of a random variable (for example, age). An important step in preparation for subsequent discussions is to extend the idea of relative frequency to probability distributions. A probability distribution is a mathematical expression or graphical representation that defines the probability with which all possible values of a random variable will occur. There are many probability distribution functions for both discrete random variables and continuous random variables. Discrete random variables are random variables for which the possible values have "gaps." A random variable that represents a count (for example, number of participants with a particular eye color) is considered discrete because the possible values are 0, 1, 2, 3, etc. A continuous random variable does not have gaps in the possible values. Whether the random variable is discrete or continuous, all probability distribution functions have these characteristics ... 

A time discretization step equal to At = 0.01 [s] is used to model the ground acceleration. Thus, the discrete representation of the white noise signal is (o tk) = yinSjAt = 1,. .., 1501, where S = 10 [m /s ] is the spectral density of the white noise and z, k = 1,..., 1501 are independent, identically distributed standard Gaussian variables. 

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