Cartesian coordinates 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. 

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 ) ... 

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 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 ... 

We can use parity to aid in determining selection rules. Recall (Section 1.8) that the integral vanishes if the integrand is an odd function of the Cartesian coordinates. The operator d [Equation (1.286)] is an odd function. If the wave functions are of definite parity, as is usually true, then if states m and have the same parity, the integrand in mn will be odd. Hence electric-dipole transitions are forbidden between states of the same parity we have the selection rule parity changes. (This is the Laporte rule.)... 

The definition of a reduced dimensionality reaction path starts with the full Cartesian coordinate representation of the classical A-particle molecular Hamiltonian,... 

In a Cartesian coordinate system, by applying the definition of a cross product in this orthogonal system, the unit vectors ex, ey, ez are related as follows ... 

Normal stresses For the exact definition of shear stresses and normal stresses, we use the illustration of the stress components given in Fig. 15.3. The stress vector t on a body in a Cartesian coordinate system can be resolved into three stress vectors h perpendicular to the three coordinate planes In this figure t2 the stress vector on the plane perpendicular to the x2-direction. It has components 21/ 22 and T23 in the X, x2 and x3-direction, respectively. In general, the stress component Tjj is defined as the component of the stress vector h (i.e. the stress vector on a plane perpendicular to the Xj-direction) in the Xj-direction. Hence, the first index points to the normal of the plane the stress vector acts on and the second index to the direction of the stress component. For i = j the stress... 

Here r is a 3n x 1 vector of Cartesian coordinates for the n particles, Lk is an n x n lower triangular matrix of rank n and I3 is the 3x3 identity matrix, k would range from 1 to A where N is the number of basis functions. The Kronecker product with I3 is used to insure rotational invariance of the basis functions. Also, integrals involving the functions k are well defined only if the exponent matrix is positive definite symmetric this is assured by using the Cholesky factorization LkL k. The following simplifications will help keep the notation more compact ... 

Consider a body in the Cartesian coordinates xu x2, and x3. Plane strain is defined as the state the body is in when the displacement vector p.i vanishes and when the orthogonal displacement vector components fi2 and ns are functions of x2 and x3 only (6). Although not rigorously correct, we will substitute strain c for displacement /x in the above definition to simplify the following discussion. 

Figure 2,7-5 Definition of the rotation of a Cartesian coordinate system by an arbitrary angle.

Pq, is expressed in Cartesian coordinates. These polar tensors T), can be derived from experimental intensities by elementary coordinate transformation. If the axes x, y, and z are chosen such that the bonds are oriented along one of the axes, then the derivatives can be used to interpret the changes of the electron clouds during a vibration. Besides, considering the definitions of the axes, it is possible to transfer atomic polar tensors between similar molecules and to estimate their intensities (Person and Newton, 1974 Person and Overend, 1977). 

For some point groups of planar molecules, C2v, >2h> and Dg), it is necessary in order to avoid ambiguity to standardize the definition of the Cartesian coordinate axes (Mulliken, 1955) ... 

The present description will utilize cartesian coordinates = rj,r2,..., r where the Vj locates the / atom taken as distinguishable but with definite, assigned atom-type. We assume that U X") is unchanged when atoms of the same type exchange locations. 

For a formal definition of the Hausdorff distance, first we shall review some relevant concepts. We assume that A and B are subsets of a set X, and for points of W a distance function is already defined. For example, if X is the ordinary, three-dimensional Euclidean space and if the points a and b of A are represented by their three Cartesian coordinates [, 2) 3) and respectively, where a and b can be written as... 

These definitions allow us to express the relations between the generalized coordinates and the 3 N Cartesian coordinates of an arbitrary configuration in the following formal way... 

It can now be shown that the Lagrangian equations are equivalent to the more familiar Newton s second law of motion. If qi = r, the generalized coordinates are simply the Cartesian coordinates. Introducing this definition... 

To obtain the formula for V in cylindrical coordinates we employ the definition of the V-operator in Cartesian coordinates (C.57), eliminate the Cartesian unit vectors by (C.66) and eliminate the Cartesian derivative operators by (C.64). The resulting formula for the V operator in cylindrical coordinates can then be used to calculate all the necessary differential operators in cylindrical coordinates provided that the spatial derivatives of the unit vectors er,eg,ez are used to differentiate the unit vectors on which V operates. 

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