Change of Coordinate System

In many cases, it is possible to simplify a problem by choosing a particular coordinate system. It is therefore important to be able to describe how vectors and matrices change when switching from one coordinate system to another. 

For the 2 X 2 case, the new coordinates x and y are related to the original x and y coordinates by means of a 2 x 2 matrix containing cosines and sines of the rotational angle a. 

The rotation matrix is a unitary (orthogonal) matrix U, since the determinant is equal to 1 (cos or-t sin a= 1). The significance of a unitary matrix is that it describes a rotation of the coordinate system without changing the length of the coordinate axes. A unitary matrix with a determinant of -1 describes a rotation of the coordinate system, followed by inverting the directions of the coordinate axis, i.e. an improper rotation in the language of point group symmetry. 

The connection between the primed and unprimed coordinate systems is given by the unitary matrix in eq. (16.22), and can be written as in eq. (16.23). 

The inverse operation U corresponds to rotation with the angle -a, and back-transforms the primed coordinates to the unprimed ones. 

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Suppose that 4> x,y,z) is a scalar point function, that is, a scalar function that is uniquely defined in a given region. Under a change of coordinate system to, say, x y z, it will take on another form, although its value at any point remains the same. Applying the chain rule (Section 2.12),... 

An important purpose of tensor analysis is to describe any physical or geometrical quantity in a form that remains invariant under a change of coordinate system. The simplest type of invariant is a scalar. The square of the line element ds of a space is an example of a scalar, or a tensor of rank zero. 

It follows that the quantities A = d(j)/dx3 (e.g. gradients) transform under a change of coordinate system from xl to xl like... 

SCF and MSCF electric field dependence of the magnetizability and nuclear magnetic shielding have been studied by Rizzo et al (17-19) within GIAO basis sets. The use of London orbitals guarantees invariance of theoretical estimates in a change of coordinate system, which is a basic requirement in the computation of magnetic response properties. 

Transformation Laws for the Current Density in a Change of Coordinate System. In the coordinate transformation... 

In a change of coordinate system 21, the contributions to the shielding polarizabilities transform... 

CHANGE OF COORDINATE SYSTEM CHANGE OF COORDINATE SYSTEM 311... 

The second problem is to hnd a description of geometrical objects that remains invariant under a change of coordinate system. The mathematical solution is to dehne such an object as a tensor a concept which hnds wide application in the analysis of anisotropic media. 

A more special case with c = o and Q constant is physically trivial because it expresses the change of coordinate system only. Therefore, a change of coordinates (in Rem. 4) is not the same as the much more general change of frame (where time and its transformation (3.26) and shifts in origins are moreover considered). 

The simulation space and boundary conditions for a microband is exactly the same as that for a microdisc, as depicted in Figure 9.3, except for the change of coordinate system. 

The total current density vector field induced in a molecule by an external magnetic field, Eq. (41), is a function of position J = J (r), whose origin can arbitrarily be chosen in the case of exact and optimal variational wave-functions [7]. In a change of coordinate system, Eq. (61), diamagnetic and paramagnetic contributions change according to Eqs. (63) and (67), but the total function should remain the same [14, 15], that is,... 

The total quantum mechanical current density is an invariant quantity, mapped onto itself in a gauge transformation, in the ideal case of electronic wavefunctions satisfying hypervirial theorems, for example, optimal variational eigenfunctions [7]. In particular, it remains the same in a change of coordinate system, as recalled above. Then comparison between Eqs. (41) and (93) necessarily impUes that the new, formally paramagnetic, term should be equivalent to the diamagnetic contribution in the conventional formulation, that is. 

In the special instance of a molecule perturbed by a stationary homogenous magnetic field, the transverse vector potential is usually chosen according to the Coulomb prescription, and the gauge transformations most extensively studied so far are those associated merely with a change of coordinate system. 

In particular, a change of coordinate system described by an origin shift d, i.e.,... 

Accordingly, invariance of second-, third-, and fourth-rank tensors in a change of coordinate system is fulfilled if the state functions are exact eigenfunctions to a model Hamiltonian, and satisfy the hypervirial theorems for position and second-moment operators. 

Therefore, the conditions (101) for charge and current conservation in the presence of electric and magnetic fields are identical to the sum rules for gauge invariance in a change of coordinate system, i.e., equation (56), and equation (64) to first order in E. 

The relationships describing the transformation of diamagnetic and paramagnetic contributions to the current density within a change of coordinate system (44) are... 

In order to investigate the origin dependence of CTOCD-DZ properties, we observe thaL in the change of coordinate system... 

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