Rotation-invariant functions

The above function is a one-center correlated Gaussian with exponential coefficients forming the symmetric matrix A]. <1) are rotationally invariant functions as required by the symmetry of the problem—that is, invariant with respect to any orthogonal transformation. To show the invariance, let U be any 3x3 orthogonal matrix (any proper or improper rotation in 3-space) that is applied to rotate the r vector in the 3-D space. Prove the invariance ... 

The elements of Tr wq restrictions of rotation-invariant functions to the ball of radius R. NIq will apply the Stone-Weierslrass Theorem (Theorem 3.2) and Proposition 3.7 to show that Tr 0 y spans Br), which will imply thatJ 0 y spans... 

T complex scalar product space of rotation-invariant functions in... 

And what if the basis functions are centred on different atoms The CNDO solution to the problem is to take all possible integrals such as those above to be equal, and to assume that they depend only on the atoms A and B on which the basis functions are centred. This satisfies the rotational invariance requirement. In CNDO theory, we write the two-electron integrals as pab and they are taken to have the same value irrespective of the basis functions on atom A and/or atom B. They are usually calculated exactly, but assuming that the orbital in question is a Is orbital (for hydrogen) or a 2s orbital (for a first row atom). 

The Kronecker product with the identity ensures rotational invariance (sphericalness) elliptical Gaussians could be obtained by using a full n x n A matrix. In the former formulation of the basis function, it is difficult to ensure the square integrability of the functions, but this becomes easy in the latter formulation. In this format, all that is required is that the matrix, A, be positive definite. This may be achieved by constructing the matrix from a Cholesky decomposition A), = Later in this work we will use the notation... 

In general, for rotational invariant systems, the gap function can be expanded into polynomial harmonics... 

To see why this is the case, we first consider the portion of the response that arises from llsm. According to Equation (10), we can express (nsm(t) nsm(0)> in terms of derivatives of llsm with respect to the molecular coordinates. Since in the absence of intermolecular interactions the polarizability tensor of an individual molecule is translationally invariant, FIsm is sensitive only to orientational motions. Since the trace is a linear function of the elements of n, the trace of the derivative of a tensor is equal to the derivative of the trace of a tensor. Note, however, that the trace of a tensor is rotationally invariant. Thus, the trace of any derivative of with respect to an orientational coordinate must be zero. As a result, nsm cannot contribute to isotropic scattering, either on its own or in combination with flDID. On the other hand, although the anisotropy is also rotationally invariant, it is not a linear function of the elements of 11. The anisotropy of the derivative of a tensor therefore need not be zero, and nsm can contribute to anisotropic scattering. 

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 ... 

Notice that the density of a complete p, d, f, atomic subshell, or an incomplete sub shell in the central field approximation is rotationally invariant [58]. Thus only s-type charge density fitting functions are needed in any atomic central-field calculation. However if the central-field approximation is not invoked then very-high angular momenta are required to fit the density. From a practical point of view it might be better to set off center s-type fitting functions. 

To handle these variables, we expand each correlation function in an angle-dependent basis set of rotational invariants [258]. Taking advantage of the fact that, in an globally isotropic system, the direction of the wavevector k does not matter, we choose k to be parallel to the -axis of the space-fixed coordinate system. The resulting "fc-frame expansion is then defined by [258]... 

A property of the autocorrelation function is that it does not change when the origin of the x variable is shifted. In effect, autocorrelation descriptors are considered TRI descriptors, meaning that they have translational and rotational invariance. 

Another very useful way of testing our results has been to use two independent routes to numerically evaluate Vq. The first route is reviewed in Appendix C. It consists of the direct evaluation of the vector elements in the coupled basis which largely involves numerical integration of the function This direct approach is convenient for the case of rotational invariance which is characteristic of the physical systems we have studied. A second route has previously been recommended by Moro and Freed [50] and Schneider and Freed [42]. They consider the following expression... 

The local sixfold bond orientational order parameter is defined in Eq. (3.3). g FpFj) is divided out of Eq. (3.15) in order to remove translational correlations from the bond orientational correlation function. In the homogeneous and isotropic liquid phase gl (r,F2) reduces to a function of Fj, only, which we will denote by g r), and a corresponding translation- and rotation-invariant quantity can be defined for the solid phase by performing suitable averages. 

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