Off-diagonal

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. 

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) ... 

In solving the secular equation it is important to know which of the off-diagonal matrix elements " I wanish since this will enable us to simplify the equation. 

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... 

Figure Al.6.18. Liouville space lattice representation in one-to-one correspondence with the diagrams in figure A1.6.17. Interactions of the density matrix with the field from the left (right) is signified by a vertical (liorizontal) step. The advantage to the Liouville lattice representation is that populations are clearly identified as diagonal lattice points, while coherences are off-diagonal points. This allows innnediate identification of the processes subject to population decay processes (adapted from [37]).

Since the vibrational eigenstates of the ground electronic state constitute an orthonomial basis set, tire off-diagonal matrix elements in equation (B 1.3.14) will vanish unless the ground state electronic polarizability depends on nuclear coordinates. (This is the Raman analogue of the requirement in infrared spectroscopy that, to observe a transition, the electronic dipole moment in the ground electronic state must properly vary with nuclear displacements from... 

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... 

In other words, A is a strictly off-diagonal operator that can be evaluated as the difference between P and its diagonal parts... 

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... 

Some final comments on the relevance of non-adiabatic coupling matrix elements to the nature of the vector potential a are in order. The above analysis of the implications of the Aharonov coupling scheme for the single-surface nuclear dynamics shows that the off-diagonal operator A provides nonzero contiibutions only via the term (n A n). There are therefore no necessary contributions to a from the non-adiabatic coupling. However, as discussed earlier, in Section IV [see Eqs. (34)-(36)] in the context of the x e Jahn-Teller model, the phase choice t / = —4>/2 coupled with the identity... 

Symmetry considerations forbid any nonzero off-diagonal matrix elements in Eq. (68) when f(x) is even in x, but they can be nonzero if f x) is odd, for example,/(x) = x. (Note that x itself hansforms as B2 [284].) Figure 3 shows the outcome for the phase by the continuous phase tracing method for cycling... 

Here, the integrand is the off-diagonal gradient mahix element between adiabatic electronic states,... 

The method shown affords easy generalization to higher order coupling in the important case where a single mode is engaged, that is, G i = g i = (l/i /2) e . Then the two off-diagonal terms derived above are, after physics-based constant coefficients have been affixed, in the upper right comer... 

The first and second terras contain phase factors identical to those previously met in Eq. (82). The last term has the new phase factor [Though the power of q in the second term is different from that in Eq. (82), this term enters with a physics-based coefficient that is independent of k in Eq. (82), and can be taken for the present illustration as zero. The full expression is shown in Eq. (86) and the implications of higher powers of q are discussed thereafter.] Then a new off-diagonal matrix element enlarged with the third temi only, multiplied by a (new) coefficient X, is... 

To see that this phase has no relation to the number of ci s encircled (if this statement is not already obvious), we note that this last result is true no matter what the values of the coefficients k, X, and so on are provided only that the latter is nonzero. In contrast, the number of ci s depends on their values for example, for some values of the parameters the vanishing of the off-diagonal matrix elements occurs for complex values of q, and these do not represent physical ci s. The model used in [270] represents a special case, in which it was possible to derive a relation between the number of ci s and the Berry phase acquired upon circling about them. We are concerned with more general situations. For these it is not warranted, for example, to count up the total number of ci s by circling with a large radius. 

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