Cartesian Definition

Equations (56) and (57) give six constrains and define the BF-system uniquely. The internal coordinates qk(k = 1,2, , 21) are introduced so that the functions satisfy these equations at any qk- In the present calculations, 6 Cartesian coordinates (xi9,X29,xi8,Xn,X2i,X3i) from the triangle Og — H9 — Oi and 15 Cartesian coordinates of 5 atoms C2,C4,Ce,H3,Hy are taken. These 21 coordinates are denoted as qk- Their explicit numeration is immaterial. Equations (56) and (57) enable us to express the rest of the Cartesian coordinates (x39,X28,X38,r5) in terms of qk. With this definition, x, ( i, ,..., 21) are just linear functions of qk, which is convenient for constructing the metric tensor. Note also that the symmetry of the potential is easily established in terms of these internal coordinates. This naturally reduces the numerical effort to one-half. Constmction of the Hamiltonian for zero total angular momentum J = 0) is now straightforward. First, let us consider the metric. 

First, we introduce a Cartesian system of coordinates with an origin 0 at the center of mass, Fig. 3.11b. The x and y axes are directed to north and east, respectively, and the z-axis is along the vertical. By definition, the moment of the force F(x, y, z) with components F, Fy, and F with respect to point 0 is... 

It should be noted that in the above presentation of the combination of vectors by addition or subtraction, no reference has been made to their components, although this concept was introduced in the beginning of this chapter. It is, however, particularly useful in the definition of the product of vectors and can be further developed with the use of unit vectors. In the Cartesian system employed in Fig. 1 the unit vectors can be defined as shown in Fig. 4. 

As V is a unitary matrix, Y = VTX is just an equivalent set of Cartesian coordinates, and = UTZ is just an equivalent set of internal coordinates, simply linear combinations of the Zn. The i, , N-6, change independently, in proportion to changes in linear combinations of the Cartesian coordinates. So, locally, we have defined 3N — 6 independent internal coordinates. Every different configuration of the molecule, X, will have a different B matrix, and hence a different definition of local internal coordinates, defined automatically. 

The general definition of a projection has been given on p. 23 in Eq. (2.37). For the purpose of illustration let us write down an example. If s = (Si,Sj,Sk) is a representation of the scattering vector in orthogonal Cartesian coordinates, then the aforementioned ID projection is... 

In the framework of the force field calculations described here we work with potential constants and Cartesian coordinates. The analytical form of the expression for the potential energy may be anything that seems physically reasonable and may involve as many constants as are deemed feasible. The force constants are now derived quantities with the following definition expressed in Cartesian coordinates (x ) ... 

The Hamiltonian function for a system of bound harmonic oscillators is, in the most general form, a sum of two positively definite quadratic forms composed of the particle momentum vectors and the Cartesian projections of particle displacements about equilibrium positions ... 

Vibrations may be decomposed into three orthogonal components Ta (a = x, y, z) in three directions. These displacements have the same symmetry properties as cartesian coordinates. Likewise, any rotation may be decomposed into components Ra. The i.r. spanned by translations and rotations must clearly follow the appropriate symmetry type of the point-group character table. In quantum formalism, a transition will be allowed only if the symmetry product of the initial and final-state wave functions contains the symmetry species of the operator appropriate to the transition process. Definition of the symmetry product will be explained in terms of a simple example. 

Given the definition of the geometry of the transition states in TST as the highest energy point in the minimum energy pathway from reactants to products, the formal definition of MEP is as follows. The MEP is, in one direction, the path of steepest descents from the transition state to reactants while, in the other direction, it is the path of steepest descents from transition state to products. For reasons which will not be discussed here, the formal definition of MEP includes the statement that the pathway is expressed in mass scaled Cartesian coordinates of the position of the atoms (introduced in Chapter 3, e.g. x is replaced by x = ). This simplifies... 

The inversion operator i acts on the electronic coordinates (fr = —r). It is employed to generate gerade and ungerade states. The pre-exponential factor, y is the Cartesian component of the i-th electron position vector (mf. — 1 or 2). Its presence enables obtaining U symmetry of the wave function. The nonlinear parameters, collected in positive definite symmetric 2X2 matrices and 2-element vectors s, were determined variationally. The unperturbed wave function was optimized with respect to the second eigenvalue of the Hamiltonian using Powell s conjugate directions method [26]. The parameters of were... 

To obtain a more compact expession for the Cartesian drift velocity, it is useful to generalize the underlying diffusion equation in the /-dimensional constraint surface to a diffusion equation in the unconstrained 3N dimensional space. To define a mobility tensor throughout the unconstrained space, we adopt Eq. (2.133) as the definition of the constrained Cartesian mobility everywhere. To allow Eqs. (2.133) and (2.134) to be evaluated away from the constraint surface, we must also define n = 0c /0R everywhere, and specify definitions of the... 

Other definitions may be constructed by the following generalization of the relationship between the dynamical reciprocal vectors and the mobility tensor Given any invertible symmetric covariant Cartesian tensor S v with an inverse we may take... 

If this definition of is extended to the foil soft subspace, by letting a,b = 1in Eq. (A.42), then all elements of for which aoib corresponds to a component of the central position may be shown to vanish. Correspondingly, in the Cartesian representation, it is easily confirmed that... 

The function F(l,2) is in fact the space part of the total wave function, since a non-relativistic two-electron wave function can always be represented by a product of the spin and space parts, both having opposite symmetries with respect to the electrons permutations. Thus, one may skip the spin function and use only the space part of the wave function. The only trace that spin leaves is the definite per-mutational symmetry and sign in Eq.(14) refers to singlet as "+" and to triplet as Xi and yi denote cartesian coordinates of the ith electron. A is commonly known angular projection quantum number and A is equal to 0, 1, and 2 for L, II and A symmetry of the electronic state respectively. The linear variational coefficients c, are found by solving the secular equations. The basis functions i(l,2) which possess 2 symmetry are expressed in elliptic coordinates as ... 

In this equation ut should be interpreted as the volumetric flux density (directional flow rate per unit total area). The indexes range from 1 to 3, and repetition of an index indicates summation over that index according to the conventional summation convention for Cartesian tensors. The term superficial velocity is often used, but it is in our opinion that it is misleading because n, is neither equal to the average velocity of the flow front nor to the local velocity in the pores. The permeability Kg is a positive definite tensor quantity and it can be determined both from unidirectional and radial flow experiments [20], Darcy s law has to be supplemented by a continuity equation to form a complete set of equations. In terms of the flux density this becomes ... 

Definition 2.11 Suppose Vi,. ..,Vn are vector spaces over the same scalar field. The Cartesian sum of these vector spaces, denoted Vi V or... 

Exercise 4.3 Show that the set ofly. diagonal special unitary matrices is a group and that it is isomorphic to the group T x T. (See Exercise 4.1 for the definition of the Cartesian product of groups.)... 

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