Diagonal elements matrix

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... 

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... 

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 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, ... 

The diagonal matrix element F is broken up into three parts... 

The diagonal matrix elements of H in the CSF basis multiplied by the appropriate Cl... 

Xv reside on neighboring atoms that are ehemieally bonded. If X i and Xv reside on atoms that are not bonded neighbors, then the off-diagonal matrix element is set equal to zero. 

The simplest scheme that accounts for the destruction of phase coherence is the so-called stochastic interruption model [Nikitin and Korst 1965 Simonius 1978 Silbey and Harris 1989]. Suppose the process of free tunneling is interrupted by a sequence of collisions separated by time periods vo = to do After each collision the system forgets its initial phase, i.e., the off-diagonal matrix elements of the density matrix p go to zero, resulting in the density matrix p ... 

The potential surfaces Eg, Hn, and H22 of the HF molecule are described in Fig. 1.6. These potential surfaces provide an instructive example for further considerations of our semiempirical strategy (Ref. 5). That is, we would like to exploit the fact that Hn and H22 represent the energies of electronic configurations that have clear physical meanings (which can be easily described by empirical functions), to obtain an analytical expression for the off-diagonal matrix element H12. To accomplish this task we represent Hn, H22, and Eg by the analytical functions... 

The approach presented above is referred to as the empirical valence bond (EVB) method (Ref. 6). This approach exploits the simple physical picture of the VB model which allows for a convenient representation of the diagonal matrix elements by classical force fields and convenient incorporation of realistic solvent models in the solute Hamiltonian. A key point about the EVB method is its unique calibration using well-defined experimental information. That is, after evaluating the free-energy surface with the initial parameter a , we can use conveniently the fact that the free energy of the proton transfer reaction is given by... 

These matrix elements are in a form that can be evaluated using standard quantum chemical methods. This evaluation is tedious and the earlier assumptions that we made will lead to significant errors in the matrix elements. On the other hand, we can conveniently use experimental information to approximate the diagonal matrix elements. 

In order to construct free-energy surfaces for this system we start by defining the diagonal matrix elements, or the force fields , for each resonance structure ... 

The various approximated HtJ are given in Table 6.3 together with the parameters for the diagonal matrix elements. 

A detailed study of the various possibilities in the choice of the partition to be used in performing the perturbation falls outside the scope of the present contribution (see reference [34]) here we will limit the discussion to the widely used Moller-Plesset partition [7] in which the diagonal matrix elements are defined by ... 

The ansatz for the diagonal matrix element of the QMP appearing in eq. (2.5) corresponds to assmne the validity of eq. (2.6) also for potentials v(F) other than the... 

The summation runs over all M K,L. The diagonal matrix elements are ... 

In general, the equations for the density operator should be solved to describe the kinetics of the process. However, if the nondiagonal matrix elements of the density operator (with respect to electron states) do not play an essential role (or if they may be expressed through the diagonal matrix elements), the problem is reduced to the solution of the master equations for the diagonal matrix elements. Equations of two types may be considered. One of them is the equation for the reduced density matrix which is obtained after the calculation of the trace over the states of the nuclear subsystem. We will consider the other type of equation, which describes the change with time of the densities of the probability to find the system in a given electron state as a function of the coordinates of heavy particles Pt(R, q, Q, s,...) and Pf(R, q, ( , s,... ).74,77 80... 

Bond strengths are essentially controlled by valence ionization potentials. In the well established extended Hiickel theory (EHT) products of atomic orbital overlap integrals and valence ionization potentials are used to construct the non-diagonal matrix elements which then appear in the energy eigenvalues. The data in Table 1 fit our second basic rule perfectly. 

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