ELEMENTS OF THE g MATRIX

Recall that the g-density is just the diagonal elements of the g-matrix,... 

Note that there is no simple method for converting the g-density from the spatial representation to the orbital representation and back. The g-density in the spatial representation depends on off-diagonal elements of the g-matrix in the orbital representation and the g-density in the orbital representation depends on off-diagonal elements of the g-matrix in the spatial representation. 

The (Q, R) conditions form a polyhedral hull containing the set of A-representable rg. The (Q, R) conditions either contain, or imply, every necessary condition that can be stated for the diagonal elements of the g-matrix without using more than R distinct orbital indices. In this sense, the (Q, R) conditions are the complete set of R-orbital necessary conditions for the g-density. 

People rarely discuss the A -representabihty of the diagonal of a density matrix. The diagonal elements of a density matrix are A -representable if and only if there exists some A -electron ensemble that reproduces those elements. The preceding statement asserts that if the diagonal elements of the g-matrix, satisfy the (g, K) conditions, then there always exists an A -electron ensemble with these diagonal elements. Thus, if satisfies the (g, K) conditions, then there must be an ensemble-A -representable g-matrix with those... 

Note that the (g, R) conditions apply for any choice of the spin-orbital basis. Because the diagonal elements of the g-matrix in one orbital basis depend on the off-diagonal elements of the g-matrix in another orbital basis. 

Based on the preceding analysis, one can develop an algorithm for generalizing any of the (Q, R) conditions to constrain off-diagonal elements of the g-matrix. In general, an operator with the form of Eq. (29) becomes... 

Wfe now aim to compute the G-matrix 64 elements at a time (this is the optimum for the CRAY, an appropriate vector length for the CDC CYBER 205 would be the size of the G-matrix). We now order the P Supermatrix so that the first 64 elements are Pi-6 wl the second 64 are Pi-6 /2 until all kfc indices are exhausted, when comes P65 128rlt >65-128/2 etc. This ordering permits us to evaluate 64 elements of the G-matrix at a time, where each ki pair index gives rise to the equation... 

The symmetry coordinate iR (y4,) in Eq. 12.21 represents a redundant coordinate (see Eq. 8.1). In such a case, a coordinate transformation reduce.s the order of the matrix by one, since all the G matrix elements related to this coordinate become zero. Conversely, this result provides a general method of finding redundant coordinates. Suppose that the elements of the G matrix are calculated in terms of internal coordinates such as those in Table 1-10. If a suitable combination of internal coordinates is made so that Gy =0 (where j refers to ail the equivalent internal coordinates), such a combination is a redundant coordinate. By using the U matrices in Eqs. 12.20 and 12.21, the problem of solving a tenth-order secular equation for a tetrahedral XY4 molecule is reduced to that of solving two first-order (y4, and E) and one quadratic (F2) equation. 

To take care of the first two points we can drop the use of the macro USE from subroutine scfGR and simply write down the decisions necessary to evaluate the contributions to the distinct elements of the G matrix, using the equalities amongst the labels to decide whether or not to double the relevant contribution. This leads to some very unappealing looking code but it is a good deal more efficient than the original. 

The elements of the G matrix are given in Figure 13. As an example, the G matrix for a nonlinear tri-atomic molecule (Figure 14) is given by... 

Figure 13 Representation of common elements of the G matrix. Key atoms common to both crdinates are double circles in horizontal line number of common atoms as superscript atoms above horizontal belong to first coordinate those below belong...

When this has been done, the result is an expression for the potential energy in terms of a set of n — r independent coordinates. It is then required to construct a G matrix in terms of these same independent coordinates. But examination of the rules for the construction of the elements of the G matrix shows that any element such as Gj,- depends on the nature of the coordinates St and Si and not at all on what other coordinates there may be. Consequently the n — r rows and columns corresponding to the independent coordinates are calculated by the same rules whether or not the redundant coordinates arc utilized. The rule is... 

The elements of the G matrix can be obtained by N-differentiation of the Kohn Sham equations (3) leading to... 

For the G matrix, a lognormal distribution with an error factor of 10 is assumed the confidence in the expert providing the correct value of the HEP. This corresponds to being 90% confident that the expert estimate would deviate within less than a factor of 10 above and below the correct value. The elements of the G matrix (Table 4) are therefore interpreted as follows = probability that the expert assessment of the HEP is x,., when the realization is Xj. For example, element g is the probability that the expert would assess the HEP as Xj (>3.2e-l), when its realization is Xj (le-2). For use in the calculation, the elements of the matrix G are then modified to avoid too low probabilities (i.e. below - le-2) of expert estimates deviating... 

For very large molecules finding the elements of the G matrix becomes hopelessly complicated. Alternatively, one prepares symmetry coordinates and then proceeds directly to the solution of the secular equation and to finding normal coordinates by trial and error. 

In what follows we shall be interested only in contours that surround the above mentioned point of degeneracy. Next it is assumed that for each chosen contour the elements of the g-matrix are arranged in such a way that the vector t(s) is guaranteed to be normalized as follows ... 

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