Many dimensional

Straub J E and Berne B J 1986 Energy diffusion in many dimensional Markovian systems the consequences of the competition between inter- and intra-molecular vibrational energy transfer J. Chem. Phys. 85 2999 Straub J E, Borkovec M and Berne B J 1987 Numerical simulation of rate constants for a two degree of freedom system in the weak collision limit J. Chem. Phys. 86 4296... 

We are using the term space as defined by one or more coordinates that are not necessarily the a , y, z Cartesian coordinates of space as it is ordinarily defined. We shall refer to 1-space, 2-space, etc. where the number of dimensions of the space is the number of coordinates, possibly an n-space for a many dimensional space. The p and v axes are the coordinates of the density-frequency space, which is a 2-space. 

Most of the molecules we shall be interested in are polyatomic. In polyatomic molecules, each atom is held in place by one or more chemical bonds. Each chemical bond may be modeled as a harmonic oscillator in a space defined by its potential energy as a function of the degree of stretching or compression of the bond along its axis (Fig. 4-3). The potential energy function V = kx j2 from Eq. (4-8), or W = ki/2) ri — riof in temis of internal coordinates, is a parabola open upward in the V vs. r plane, where r replaces x as the extension of the rth chemical bond. The force constant ki and the equilibrium bond distance riQ, unique to each chemical bond, are typical force field parameters. Because there are many bonds, the potential energy-bond axis space is a many-dimensional space. 

There are forces other than bond stretching forces acting within a typical polyatomic molecule. They include bending forces and interatomic repulsions. Each force adds a dimension to the space. Although the concept of a surface in a many-dimensional space is rather abstract, its application is simple. Each dimension has a potential energy equation that can be solved easily and rapidly by computer. The sum of potential energies from all sources within the molecule is the potential energy of the molecule relative to some arbitrary reference point. A... 

We envision a potential energy surface with minima near the equilibrium positions of the atoms comprising the molecule. The MM model is intended to mimic the many-dimensional potential energy surface of real polyatomic molecules. (MM is little used for very small molecules like diatomies.) Once the potential energy surface iias been established for an MM model by specifying the force constants for all forces operative within the molecule, the calculation can proceed. 

Generalizing the Newton-Raphson method of optimization (Chapter 1) to a surface in many dimensions, the function to be optimized is expanded about the many-dimensional position vector of a point xq... 

In planning for railroad-car loading or unloading facilities, many dimensional and weight factors must be dealt with. The common carriers that are to serve the facihty are usually able to provide technical assistance as to clearances and weights to be handled. 

Potential Energy Surface. A many-dimensional function of the energy of a molecule in terms of the geometrical coordinates of the atoms. 

Some coordinate transformations are non-linear, like transforming Cartesian to polar coordinates, where the polar coordinates are given in terms of square root and trigonometric functions of the Cartesian coordinates. This for example allows the Schrodinger equation for the hydrogen atom to be solved. Other transformations are linear, i.e. the new coordinate axes are linear combinations of the old coordinates. Such transfonnations can be used for reducing a matrix representation of an operator to a diagonal form. In the new coordinate system, the many-dimensional operator can be written as a sum of one-dimensional operators. 

Once the "distances" between the animals have been calculated using equation (3.1), we lay the animals out on a piece of paper, so that those that share similar characteristics, as measured by the distance between them, are close together on the map, while those whose characteristics are very different are far apart. A typical result is shown in Figure 3.3. What we have done in this exercise is to squash down the many-dimensional vectors that represent the different features of the animals into two dimensions. 

Computational Study of Many-Dimensional Quantum Energy Flow From Action Diffusion to Localization. 

In order to achieve this, we should in principle calculate the energy of a given aggregate of atoms as a function of their positions in space. The results can be expressed as a many dimensional potential surface, the minima in which correspond to stable molecules, or aggregates of molecules, while the cols separating the minima correspond to the transition states for reactions leading to their interconveision. If such calculations could be carried out with sufficient accuracy, one could not only... 

Chirikov, B. V. (1979), A Universal Instability of Many-Dimensional Oscillator Systems, Phys. Rep. 52, 263. 

The same must be true in dealing with a many-dimensional potential energy diagram (a potential energy surface ). Here all partial derivatives of the energy with respect to each of the independent geometrical coordinates (Ri) are zero. 

Transition states are generalized saddle points on many-dimensional surfaces. 

There are standard methods for finding the positions of minima (or maxima) on many-dimensional surfaces. If there is no foreknowledge of the approximate position of the minimum, which is rare in potential energy problems, then one has to start by a mapping technique or pattern search, the most efficient of which appears to be that known as the Simplex procedure 23, 24). 

The solution of (2.3.69) is a purely mathematical problem well known in the theory of diffusion-controlled processes of classical particles. However, a particular form of writing down (2.3.69) allows us to use a certain mathematical analogy of this equation with quantum mechanics. Say, many-dimensional diffusion equation (2.3.69) is an analog to the Schrodinger equation for a system of N spinless particles B, interacting with the central particle A placed... 

Let us consider a projection of the complex many-dimensional motion (which variables are both concentrations and the correlation functions) onto the phase plane (iVa, iVb). It should be reminded that in its classical formulation the trajectory of the Lotka-Volterra model is a closed curve - Fig. 2.3. In Fig. 8.1 a change of the phase trajectories is presented for d = 3 when varying the diffusion parameter k. (For better understanding logarithms of concentrations are plotted there.)... 

Derivative Methods.—The most well developed of the derivative methods are univariate in nature, that is, they approach the minimum of the multivariate function along a sequence of lines (directions) in the many-dimensional space, and the problem is then to determine an algorithm for the choice of these directions. Usually (but not always) it is required that the current direction be followed until a minimum of the function in that direction is found. One may say that these methods are based on a sequence of onedimensional searches. 

There are some modem methods (such as the memory13 and supermemory14 gradient methods) which are not univariate in nature but which approach the minimum in a sequence of many-dimensional searches. So far, however, such methods have found no use in quantum chemistry and we shall not discuss them further. 

A molecular system consists of electrons and nuclei. Their position vectors are denoted hereafter as rel and qa, respectively. The potential energy function of the whole system is V(rel, qa). For simplicity, we skip the dependence of the interactions on the spins of the particles. The nuclei, due to their larger mass, are usually treated as classical point-like objects. This is the basis for the so called Bom-Oppenheimer approximation to the Schroedinger equation. From the mathematical point of view, the qnuc variables of the Schroedinger equation for the electrons become the parameters. The quantum subsystem is described by the many-dimensional electron wave function rel q ). 

Fig. 1.3. Upper Schematic view (dotted line) of cross-section of many-dimensional highly anharmonic potential energy surfaces for reactants plus solution (R) and (dotted line omitted) for products plus solution (P). TS occurs at the intersection. Lower Plot of free energy G for the above R and P systems vs. the reaction coordinate U.

A cross-section of the many dimensional potential energy surface for reactants in solution (R) and that for products in solution (P), is depicted schematically in Fig. 1.3 (upper). 

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