Vector operations

We now consider the connection between the preceding equations and the theory of Aharonov et al. [18] [see Eqs. (51)-(60)]. The tempting similarity between the structures of Eqs. (56) and (90), hides a fundamental difference in the roles of the vector operator A in Eq. (56) and the vector potential a in Eq. (90). The fomrer is defined, in the adiabatic partitioning scheme, as a stiictly off-diagonal operator, with elements (m A n) = (m P n), thereby ensuring that (P — A) is diagonal. By contiast, the Mead-Truhlar vector potential a arises from the influence of nonzero diagonal elements, (n P /i) on the nuclear equation for v), an aspect of the problem not addressed by Arahonov et al. [18]. Suppose, however, that Eq. (56) was contracted between (n and n) v) in order to handle the adiabatic nuclear dynamics within the Aharonov scheme. The result becomes... 

We define the field intensity tensor Fi,c as a function of a so far undetermined vector operator X = Xj, and of the partial derivatives dt... 

Throughout, unless otherwise stated, R and r will be used to represent the nuclear and electronic coordinates, respectively. Boldface is used for vectors and matrices, thus R is the vector of nuclear coordinates with components R. The vector operator V, with components... 

The problem is then reduced to the representation of the time-evolution operator [104,105]. For example, the Lanczos algorithm could be used to generate the eigenvalues of H, which can be used to set up the representation of the exponentiated operator. Again, the methods are based on matrix-vector operations, but now much larger steps are possible. 

Analogous to vector operations the tensorial form of the divergence theorem is written as... 

On a vector computer having vector registers that hold 64 floating-point numbers, this loop would be processed 64 elements at a time. The first 64 elements of Y would be fetched from memory and stored in a vector register. Each iteration of the loop is independent of the previous iteration, so this loop can be fliUy pipelined, with successive iterations started every clock cycle. Once the pipeline is filled, the result, X, will be produced one element per clock cycle and will be stored in another vector register. The results in the vector register will then be stored back into main memory or used as input to a subsequent vector operation. 

A significant amount of machine overhead is involved in setting up a vector operation, with maximum benefit accming if there are a full 64 elements to compute. For short loops, typically eight elements or fewer, vector operations are often slower than doing the computations one at a time in nonvector mode. 

Here, since the measurements were done in an integral reactor, calculation must start with the Conversion vs. Temperature function. For an example see Appendix G. Calculation of kinetic constants starts with listed conversion values as vX and corresponding temperatures as vT in array forms. The Vectorize operator of Mathcad 6 tells the program to use the operators and functions with their scalar meanings, element by element. This way, operations that are usually illegal with vectors can be executed and a new vector formed. The v in these expressions indicates a vector. 

Consider a vector operator V, each component of which is an n-dimensional square matrix. We demand that ftVip shall transform under a rotation like a proper vector. 

Note that we are not transforming the vector operator itself—its form is independent of rotation. Hence, Eq. (7-11) must be satisfied by subjecting the states ijt to a unitary transformation U ... 

This equation must hold for every vector operator. We now show that JxJyJx form a vector operator, J. 

In a manner similar to the above, one verifies that the commutation rules (9-587) and (9-588) guarantee that P as given by Eq. (9-581) is the generator for an infinitesimal spatial translation in the sense that Eq. (9-583) is satisfied for the field operators. Similarly, one verifies that the three components of the vector operator... 

Variance, 269 of a distribution, 120 significance of, 123 of a Poisson distribution, 122 Variational equations of dynamical systems, 344 of singular points, 344 of systems with n variables, 345 Vector norm, 53 Vector operators, 394 Vector relations in particle collisions, 8 Vectors, characteristic, 67 Vertex, degree of, 258 Vertex, isolated, 256 Vidale, M. L., 265 Villars, P.,488 Von Neumann, J., 424 Von Neumann projection operators, 461... 

As a result of the projection theorem [31], the expectation value of the EDM operator d, which is a vector operator, is proportional to the expectation value of J in the angular momentum eigenstate. This fact, in conjunction with Eq. (9), implies that the electric field modifies the precession frequency of the system because of the additional torque experienced by the system due to the interaction between the electric field and the EDM. It can readily be shown that the modified precession frequency is... 

Equation (44) suggests that a vector operator V or nabla (called del") be defined in Cartesian coordinates by... 

The expressions for the various vector operators in spherical coordinates can be derived with the use of the chain rule. Thus, for example,... 

The scalar product of the vector operator V and a vector A yields a scalar quantity, the divergence of A. Thus,... 

To illustrate the use of the vector operators described in the previous section, consider the equations of Maxwell. In a vacuum they provide the basic description of an electromagnetic field in terms of the vector quantifies the electric field and 9C the magnetic field The definition of the field in a dielectric medium requires the introduction of two additional quantities, the electric displacement SH and the magnetic induction. The macroscopic electromagnetic properties of the medium are then determined by Maxwell s equations, viz. 

Section 5.3. There, the operator T> s d/dx was used in the solution of ordinary differential equations. In Chapter 5 the vector operator del , represented by the symbol V, was introduced. It was shown that its algebraic form is dependent on the choice of curvilinear coordinates. 

In this expression, GIX and Ha are both 6x3 rectangular matrices whose elements are known functions of the internal hyperangles pA is a 3 x 1 column vector operator whose elements contain first derivatives with respect to the three q, coordinates and i1X is the 3 x 1 column vector operator whose elements are the components JIX, J,x, and JIX of the system s nuclear motion angular momentum operator J in the IX frame. From these properties, it can be shown that... 

The long but straightforward integration[65] produces the expression for the angular momentum in terms of a new vector operator, s, in the form... 

The solute-solvent system, from the physical point of view, is nothing but a system that can be decomposed in a determined collection of electrons and nuclei. In the many-body representation, in principle, solving the global time-dependent Schrodinger equation with appropriate boundary conditions would yield a complete description for all measurable properties [47], This equation requires a definition of the total Hamiltonian in coordinate representation H(r,X), where r is the position vector operator for all electrons in the sample, and X is the position vector operator of the nuclei. In molecular quantum mechanics, as it is used in this section, H(r,X) is the Coulomb Hamiltonian[46]. The global wave function A(r,X,t) is obtained as a solution of the equation ... 

It is assumed that the field experienced by the charged particle satisfies Eq. (8) and (9). Here V is the vector operator in cylindrical coordinates for an... 

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