Diagonal product

We will first evaluate the energy of a single Slater detenninant. It is convenient to write it as a sum of permutations over the diagonal of the determinant. We will denoted the diagonal product by IT, and use the symbol to represent the determinant wave function. 

The determinant is found by cross-multiplying the diagonal elements in a matrix and subtracting one diagonal product from the other, such that... 

Consider, for example, the permuted orbital product of the second line of the table, —baab, and let us integrate the monoelectronic Hamiltonian h( 1). This Hamiltonian applies to the first orbital of the orbital products a for the diagonal product, and b for the permuted product. This yields the term hab. This term is multiplied by a product of overlaps between the remaining orbitals ... 

The energy term in the fourth row is determined in the same manner, but now all the orbitals of the permuted product are different from those of the diagonal product. As a consequence, all energy terms are of the hab type, and all overlaps are different from unity. 

Each half-determinant is a determinant made of a diagonal product of spin-orbitals followed by a signed sum of all the permutations of this product, with the restriction that a half-determinant must involve spin-orbitals that are all of the same spin. It will be convenient for the problem at hand to consider a full Slater determinant as the union of its two constituent half-determinants ... 

Consequently, Eqs. (43) and (59) are identical, for applications in a 3D parameter space, except that the vector product in the former is expressed as a commutator in the latter. Both are computed as diagonal elements of combinations of strictly off-diagonal operators and both give gauge independent results. Equally, however, both are subject to the limitations with respect to the choice of surface for the final integration that are discussed in the sentence following Eq. (43). 

The eigenvalues of this mabix have the form of Eq. (68), but this time the matrix elements are given by Eqs. (84) and (85). The symmetry arguments used to determine which nuclear modes couple the states, Eq. (81), now play a cracial role in the model. Thus the linear expansion coefficients are only nonzero if the products of symmebies of the electronic states at Qq and the relevant nuclear mode contain the totally symmebic inep. As a result, on-diagonal matrix elements are only nonzero for totally symmebic nuclear coordinates and, if the elecbonic states have different symmeby, the off-diagonal elements will only... 

The ti eatment of the Jahn-Teller effect for more complicated cases is similar. The general conclusion is that the appearance of a linear term in the off-diagonal matrix elements H+- and H-+ leads always to an instability at the most symmetric configuration due to the fact that integrals of the type do not vanish there when the product < / > / has the same species as a nontotally symmetiic vibration (see Appendix E). If T is the species of the degenerate electronic wave functions, the species of will be that of T, ... 

To do this, multiply the binomials at the top left and bottom right (the principal diagonal) and then, from this product, subtract the product of the remaining two elements, the off-diagonal elements (42 — 9x). The difference is set equal to zero ... 

I = identity matrix, which has ones on diagonal, zeros elsewhere M = mih matrix, which transforms mill-feed-size distrihiition into mill-product-size distrihiition... 

While the F-N curve is a cumulative illustration, the risk profile shows the expected frequency of accidents of a particular category or level of consequence. The diagonal line is a line of constant risk defined such that the product of expected frequency and consequence is a constant at each point along the line. " As the consequences of accidents go up, the expected frequency should go down in order for the risk to remain constant. As the example illustrates, if a portion of the histogram sticks its head up above the line (i.e., a particular type of accident contributes more than its fair share of the risk), then that risk is inconsistent with the risk presented by other accident types. (Note There is no requirement that you use a line of constant risk other more appropriate risk criteria for your application can be easily defined and displayed on the graph.)... 

Let us now turn to the surfaces themselves to learn the kinds of kinetic information they contain. First observe that the potential energy surface of Fig. 5-2 is drawn to be symmetrical about the 45° diagonal. This is the type of surface to be expected for a symmetrical reaction like H -I- H2 = H2 -h H, in which the reactants and products are identical. The corresponding reaction coordinate diagram in Fig. 5-3, therefore, shows the reactants and products having the same stability (energy) and the transition state appearing at precisely the midpoint of the reaction coordinate. 

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