Diagonal elements

The diagonal elements of this matrix approximate the variances of the corresponding parameters. The square roots of these variances are estimates of the standard errors in the parameters and, in effect, are a measure of the uncertainties of those parameters. 

The off-diagonal elements of the variance-covariance matrix represent the covariances between different parameters. From the covariances and variances, correlation coefficients between parameters can be calculated. When the parameters are completely independent, the correlation coefficient is zero. As the parameters become more correlated, the correlation coefficient approaches a value of +1 or -1. 

It is important to realize that while the uppennost diagonal elements of these matrices are numbers, the other diagonal element is a matrix of dimension N. Specifically, these are the matrix representations of Hq and Fin the basis q which consists of all the original set, apart from i.e. 

The off-diagonal elements in this representation of h and v are the zero vector of lengtii (for h) and matrix elements which couple the zeroth-order ground-state eigenfunction members of the set q (for v) ... 

It is well known that the trace of a square matrix (i.e., the sum of its diagonal elements) is unchanged by a similarity transfonnation. If we define the traces... 

Note that the diagonal elements of the matrix, ap and hp, correspond to the populations in the energy levels, a and b, and contain no time dependence, while the off-diagonal elements, called the coherences, contain all the time dependence. 

The populations, /Q, appear on the diagonal as expected, but note that there are no off-diagonal elements—no coherences this is reasonable since we expect the equilibrium state to be time-independent, and we have associated the coherences with time. 

It follows that there are two kinds of processes required for an arbitrary initial state to relax to an equilibrium state the diagonal elements must redistribute to a Boltzmaim distribution and the off-diagonal elements must decay to zero. The first of these processes is called population decay in two-level systems this time scale is called Ty The second of these processes is called dephasmg, or coherence decay in two-level systems there is a single time scale for this process called T. There is a well-known relationship in two level systems, valid for weak system-bath coupling, that... 

The exponential of a diagonal matrix is again a diagonal matrix with exponentials of the diagonal elements, equation (B2.4.17)). 

A better set of parameters can be chosen as the above-diagonal elements of an antisynnnetric matrix K, used... 

The EF algoritlnn [ ] is based on the work of Cerjan and Miller [ ] and, in particular, Simons and coworkers [70,1Y. It is closely related to the augmented Hessian (rational fiinction) approach[25]. We have seen in section B3.5.2.5 that this is equivalent to addmg a constant level shift (damping factor) to the diagonal elements of the approximate Hessian H. An appropriate level shift effectively makes the Hessian positive definite, suitable for minimization. 

The symmetry argument actually goes beyond the above deterniination of the symmetries of Jahn-Teller active modes, the coefficients of the matrix element expansions in different coordinates are also symmetry determined. Consider, for simplicity, an electronic state of symmetiy in an even-electron molecule with a single threefold axis of symmetry, and choose a representation in which two complex electronic components, e ) = 1/v ( ca) i cb)), and two degenerate complex nuclear coordinate combinations Q = re " each have character T under the C3 operation, where x — The bras e have character x. Since the Hamiltonian operator is totally symmetric, the diagonal matrix elements e H e ) are totally symmetric, while the characters of the off-diagonal elements ezf H e ) are x. Since x = 1, it follows that an expansion of the complex Hamiltonian matrix to quadratic terms in Q. takes the form... 

Comparison with Eq. (43) is illuminating. By the method of constmction, the matrix elements of A aie identical with the off-diagonal elements of P thus, with the help of Eqs. (41) and (42)... 

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). 

We now consider the connection between the preceding equations and the theory of Aharonov et al. [18] [see Eqs. (51)-(60)]. The tempting similarity between the structures of Eqs. (56) and (90), hides a fundamental difference in the roles of the vector operator A in Eq. (56) and the vector potential a in Eq. (90). The fomrer is defined, in the adiabatic partitioning scheme, as a stiictly off-diagonal operator, with elements (m A n) = (m P n), thereby ensuring that (P — A) is diagonal. By contiast, the Mead-Truhlar vector potential a arises from the influence of nonzero diagonal elements, (n P /i) on the nuclear equation for v), an aspect of the problem not addressed by Arahonov et al. [18]. Suppose, however, that Eq. (56) was contracted between (n and n) v) in order to handle the adiabatic nuclear dynamics within the Aharonov scheme. The result becomes... 

The sum over all m is justified by the fact that the diagonal elements (t P ji ) vanish in a real representation. It is also evident from the factorization of n) and... 

Requiring l/f (r qx) to be real, the matrix W (Rx) becomes real and skew-symmetiic (just like its adiabatic counterpart) with diagonal elements equal to zero. Similarly, W (Rx) is an n X u diabatic second-derivative coupling matrix with elements defined by... 

By using this equation and the fact that for real electronic wave functions the diagonal elements of W (q ) vanish, it can be shown that... 

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 familiar BO approximation is obtained by ignoring the operators A completely. This results in the picture of the nuclei moving over the PES provided by the electrons, which are moving so as to instantaneously follow the nuclear motion. Another common level of approximation is to exclude the off-diagonal elements of this operator matrix. This is known as the Bom-Huang, or simply the adiabatic, approximation (see [250] for further details of the possible approximations and nomenclature associated with the nuclear Schrodinger equation). 

The basis consisting of the adiabatic electronic functions (we shall call it bent basis ) has a seiious drawback It leads to appearance of the off-diagonal elements that tend to infinity when the molecule reaches linear geometry (i.e., p 0). Thus it is convenient to introduce new electronic basis functions by the transformation... 

The diagonal elements of the matrix [Eqs. (31) and (32)], actually being an effective operator that acts onto the basis functions Ro,i, are diagonal in the quantum number I as well. The factors exp( 2iAct)) [Eqs. (27)] determine the selection rule for the off-diagonal elements of this matrix in the vibrational basis—they couple the basis functions with different I values with one another (i.e., with I — l A). 

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