Multidimensional integrals

This delta function can be used in the expression for R-p to constrain the multidimensional integral over vibration-rotetion coordinates (denoted Q) to those specific values which obey the energy conservation condition... 

Brychkov YA, Glaeske HJ, Prudnikov AP, Tuan VK (1992) Multidimensional integral transformations. Gordon Breach, Philadelphia... 

If there were no intramolecular interactions (such as bonding or excluded volume), then V(R) = 0, and the next guess for the density profile can be obtained directly from Eq. (75). The presence of V(R) necessitates either a multidimensional integration or (more conveniently) a single-chain simulation. 

The obstacle to simultaneous quantum chemistry and quantum nuclear dynamics is apparent in Eqs. (2.16a)-(2.16c). At each time step, the propagation of the complex coefficients, Eq. (2.11), requires the calculation of diagonal and off-diagonal matrix elements of the Hamiltonian. These matrix elements are to be calculated for each pair of nuclear basis functions. In the case of ab initio quantum dynamics, the potential energy surfaces are known only locally, and therefore the calculation of these matrix elements (even for a single pair of basis functions) poses a numerical difficulty, and severe approximations have to be made. These approximations are discussed in detail in Section II.D. In the case of analytic PESs it is sometimes possible to evaluate these multidimensional integrals analytically. In either case (analytic or ab initio) the matrix elements of the nuclear kinetic energy... 

The example above of numerically integrating a one-dimensional function can be summarized in three main points that also apply to multidimensional integrals ... 

The present state in the theory of time-dependent processes in liquids is the following. We know which correlation functions determine the results of certain physical measurements. We also know certain general properties of these correlation functions. However, because of the mathematical complexities of the V-body problem, the direct calculation of the fulltime dependence of these functions is, in general, an extremely difficult affair. This is analogous to the theory of equilibrium properties of liquids. That is, in equilibrium statistical mechanics the equilibrium properties of a system can be found if certain multidimensional integrals involving the system s partition function are evaluated. However, the exact evaluation of these integrals is usually extremely difficult especially for liquids. 

The FC factor depends on the quantum numbers describing the initial state and the state of the photofragments and its evaluation enables one to obtain a state-to-state description of polyatomic photodissociation. The FC factor is, in general, a multidimensional integral ... 

In the systematic development of distributed systems it is necessary to use the basic system models using the interface, distribution and state transition approaches. Each of these fundamental parameters is very helpful and plays an important role in the systems development process. For large systems, the development is carried out through several levels of abstraction. And the same time for creation of such kind of development processes of the modular systems it is obvious to estimate and select the refinement steps, which give the possibility to build the effective multilevel and multidimensional integrated distribution. 

Makri and Miller [1987b], Doll and Freeman [1988], Doll et al. [1988] (see also the review by Makri [1991b]) exploited the stationary phase approximation (i.e., the semiclassical limit) as an initial approximation to the path integral. For example, for the multidimensional integral of the form J dx exp[iS(x)], one may obtain the following approximation [Makri, 1991b] ... 

The multidimensional integrals in the definition of the potential of mean force can be evaluated directly using the Monte Carlo method (see Appendix I). 

The Monte Carlo method is a very powerful numerical technique used to evaluate multidimensional integrals in statistical mechanics and other branches of physics and chemistry. It is also used when initial conditions are chosen in classical reaction dynamics calculations, as we have discussed in Chapter 4. It will therefore be appropriate here to give a brief introduction to the method and to the ideas behind the method. 

This becomes even more clear when we consider the multidimensional integrals encountered in statistical mechanics. Then only a very small fraction of the points in phase space are accessible, that is, correspond to states with a non-zero Boltzmann factor. So, even if L is very large, only pL of the points will be in such a region corresponding to a sampling of the function with only pL points, and thus a variance which behaves as l/(pL) rather than 1/L. Here p is the fraction of points in phase space accessible to the system. The variance in a random sampling may indeed be very large when it is observed that for a liquid of 100 atoms it has been estimated that the Boltzmann factor will be non-zero for 1 out of about 10260 points in phase space, that is, p = 10 260. 

Unfortunately, the simple importance sampling as described above cannot be used to sample multidimensional integrals over configuration space as in Eq. (1.1). The reason is simply that we do not know how to construct the transformation in Eq. (1.7) that will enable us to generate points in configuration space with a probability density as given by the Boltzmann factor. In fact, in order to do so, we must be able to compute analytically the partition function of the system. If we could do that, there would hardly be any need for computer simulations ... 

More recently, Goldman has introduced two additional modifications to MCI. First, modified radial functions are introduced which are functions of r> and r[54]. In these variables, all multidimensional integrals reduce to simple, one-electron integrals a simplification even over Cl, where coupled two-electron integrals always appear. Secondly, modified angular functions are introduced which implicitly contain an infinite number of coupled harmon-ics[55]. Several examples of such angular functions are given, one of which has an obvious connection to the ECG basis ... 

Recent years have seen the extensive application of computer simulation techniques to the study of condensed phases of matter. The two techniques of major importance are the Monte Carlo method and the method of molecular dynamics. Monte Carlo methods are ways of evaluating the partition function of a many-particle system through sampling the multidimensional integral that defines it, and can be used only for the study of equilibrium quantities such as thermodynamic properties and average local structure. Molecular dynamics methods solve Newton s classical equations of motion for a system of particles placed in a box with periodic boundary conditions, and can be used to study both equilibrium and nonequilibrium properties such as time correlation functions. 

The multidimensional integral (72) is reduced to a simple summation over the centers t of the cavity tesserae (their number is NT) of the normal component of the simple auxiliary function ... 

Precision and Calculation Time. To calculate one point on the instrumental profile we need to calculate a multidimensional integral. For the case where absorption can be neglected this integral reduces to a four-dimensional integral. To estimate how the number of points on the calculation grid affects the precision and calculation time, the calculations of the total instrumental profile were performed for four cases. They are given in Table 6.3. 

H. Conroy, Molecular Schrodinger equation. VIII. A new method for the evaluation of multidimensional integration. J. Chem. Phys., 1967, 47, 5307-5318. 

Eventually, the current considerations will serve to express the partition function Q as a (multidimensional) integral over configuration amd momentum space. It is therefore necessary to investigate the effect of Pk on integrals of the general type... 

Because MC is a numerical technique to calculate multidimensional integrals like the one over configuration space in Eq. (5.1), we begin by discretizing configuration space and rewrite the integral as... 

The important point is that the multidimensional integrals over... 

FIG. 7 IT vs. L curves for finite disk-shaped substrates. Both tip and substrate reactions are diffusion-controlled. Filled symbols calculated using multidimensional integral equations open symbols are from ADI simulation. The h (as/a) values are as indicated. Dashed line is simulation for H = oo from Ref. 1. The lines through the symbols are drawn as a guide. (Reprinted with permission from Ref. 3. Copyright 1992 American Chemical Society.)... 

SECM theory has been developed for lour mechanisms with homogeneous chemical reactions coupled with electron transfer, i.e., a first-order irreversible reaction (ErQ mechanism) (5), a second-order irreversible dimerization (ErC2i mechanism) (36), ECE and DISP1 reactions (38). [The solution obtained for a EqCr mechanism in terms of multidimensional integral equations (2) has not been utilized in any calculations.] While for ErC, and ErC2i mechanisms analytical approximations are available (39), only numerical solutions have been reported for more complicated ECE and DISP1 reactions (38). 

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