Diagonalized matrix

Note that h is simply the diagonal matrix of zeroth-order eigenvalues In the following, it will be assumed that the zeroth-order eigenfunction a reasonably good approximation to the exact ground-state wavefiinction (meaning that Xfi , and h and v will be written in the compact representations... 

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. 

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... 

In practice, the matrix (iL+R+K) is diagonalized first, with a matrix of eigenvectors, U, as in equation (B2.4.15)), to give a diagonal matrix. A, with the eigenvalues, X, of L down the diagonal. 

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

As with the uncoupled case, one solution involves diagonalizing the Liouville matrix, iL+R+K. If U is the matrix with the eigenvectors as cohmms, and A is the diagonal matrix with the eigenvalues down the diagonal, then (B2.4.32) can be written as (B2.4.33). This is similar to other eigenvalue problems in quantum mechanics, such as the transfonnation to nonnal co-ordinates in vibrational spectroscopy. 

Note that the Liouville matrix, iL+R+K may not be Hennitian, but it can still be diagonalized. Its eigenvalues and eigenvectors are not necessarily real, however, and the inverse of U may not be its complex-conjugate transpose. If complex numbers are allowed in it, equation (B2.4.33) is a general result. Since A is a diagonal matrix it can be expanded in tenns of the individual eigenvalues, X. . The inverse matrix can be applied... 

Let us express the displacement coordinates as linear combinations of a set of new coordinates y >q= Uy then AE = y U HUy. U can be an arbitrary non-singular matrix, and thus can be chosen to diagonalize the synmietric matrix H U HU = A, where the diagonal matrix A contains the (real) eigenvalues of H. In this fomi, the energy change from the stationary point is simply AF. = t Uj A 7- h is clear now that a sufBcient... 

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... 

If we consider G as a unitary transformation matrix that diagonalizes the g matrix and 1 is the diagonal matrix with elements toy, j =, N as... 

Thus D is a diagonal matrix that contains diagonal complex numbers whose nonn is 1. By recalling Eq. (57), we get... 

As the D matrix is a diagonal matrix with a complex number of norm exponent of Eq. (65) has to fulfill the following quantization mle ... 

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... 

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. 

The effective potential matrix for nuclear motion, which is a diagonal matrix for the adiabatic electronic set, is given by... 

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... 

Note that the exact adiabatic functions are used on the right-hand side, which in practical calculations must be evaluated by the full derivative on the left of Eq. (24) rather than the Hellmann-Feynman forces. This forai has the advantage that the R dependence of the coefficients, c, does not have to be considered. Using the relationship Eq. (78) for the off-diagonal matrix elements of the right-hand side then leads directly to... 

The remaining combinations vanish for symmetry reasons [the operator transforms according to B (A") hreducible representation]. The nonvanishing of the off-diagonal matrix element fl+ is responsible for the coupling of the adiabatic electronic states. 

Thus in the lowest order approximation the angle x is eliminated from the off-diagonal matrix elements of [second and third of Eqs. (60)] it solely determines the selection rules for matrix elements of Hg with respect to nuclear basis functions. 

Thus, the neglect of the off-diagonal matrix elements allows the change from mixed states of the nuclear subsystem to pure ones. The motion of the nuclei leads only to the deformation of the electronic distribution and not to transitions between different electronic states. In other words, a stationary distribution of electrons is obtained for each instantaneous position of the nuclei, that is, the elechons follow the motion of the nuclei adiabatically. The distribution of the nuclei is described by the wave function x (R i) in the potential V + Cn , known as the proper adiabatic approximation [41]. The off-diagonal operators C n in the matrix C, which lead to transitions between the states v / and t / are called operators of nonadiabaticity and the potential V = (R) due to the mean field of all the electrons of the system is called the adiabatic potential. 

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, ... 

One of the main outcomes of the analysis so far is that the topological matrix D, presented in Eq. (38), is identical to an adiabatic-to-diabatic transformation matrix calculated at the end point of a closed contour. From Eq. (38), it is noticed that D does not depend on any particular point along the contour but on the contour itself. Since the integration is carried out over the non-adiabatic coupling matrix, x, and since D has to be a diagonal matrix with numbers of norm 1 for any contour in configuration space, these two facts impose severe restrictions on the non-adiabatic coupling terms. 

Since for any closed contour to be a diagonal matrix with (-1-1) and (—1)... 

In Section IV, we introduced the topological matrix D [see Eq. (38)] and showed that for a sub-Hilbert space this matrix is diagonal with (-1-1) and (—1) terms a feature that was defined as quantization of the non-adiabatic coupling matrix. If the present three-state system forms a sub-Hilbert space the resulting D matrix has to be a diagonal matrix as just mentioned. From Eq. (38) it is noticed that the D matrix is calculated along contours, F, that surround conical intersections. Our task in this section is to calculate the D matrix and we do this, again, for circular contours. 

Eigenvalues of the diagonal matrix D will be denoted as Aj. With the transformations... 

---