Symmetry coordinates, Hamiltonian diagonalization

The two most useful sets are the bond displacements themselves, and the symmetry coordinates. The use of the latter leads naturally to a scheme in which the Hamiltonian for bent molecules is no longer diagonal in the total 0(4) quantum numbers (ti, x2), and thus one loses the simple form of the secular equation (Figure 4.11). The secular equation must be now diagonalized in the full space with dimensions that become rapidly larger. This scheme, developed by Leviatan and Kirson (1988), can be implemented only if the vibron numbers N are relatively small, N < 10. 

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 Equation 7.33 we have written out both the g-value and the zero-field coefficient of the basic S2 interaction term in the form of diagonal 3x3 matrices in which all off-diagonal elements are equal to zero. The diagonal elements were indexed with subscripts x, y, z, corresponding to the Cartesian axes of the molecular axes system. But how do we define a molecular axis system in a (bio)coordination complex that lacks symmetry The answer is that if we would have made a wrong choice, then the matrices would not be diagonal with zeros elsewhere. In other words, if the spin Hamiltonian would have been written out for a different axes system, then, for example, the g-matrix would not have three, but rather six, independent elements ... 

While direct diagonalization of the Hamiltonian matrix works well for situations in which there is a finite number of states, as in Fig. 9.1, it is clearly hopeless to try it in this case. A useful WKB approach was proposed by Edmonds12 and refined by Starace.13 Using the fact that azimuthal symmetry exists, Starace writes the wavefunction of the spinless Rydberg electron in cylindrical coordinates as13... 

A parameterization method of the Hamiltonian for two electronic states which couple via nuclear distortions (vibronic coupling), based on density functional theory (DFT) and Slaters transition state method, is presented and applied to the pseudo-Jahn-Teller coupling problem in molecules with an s2-lone pair. The diagonal and off-diagonal energies of the 2X2 Hamiltonian matrix have been calculated as a function of the symmetry breaking angular distortion modes and r (Td)] of molecules with the coordination number CN = 3... 

To apply these equations, we need the wavefuncdons m> in order to get the dipole moment transition elements and the frequencies spectral series, where only the ground state need be near-exact. This is done by diagonalizing the Hamiltonian matrix formed from a large number of basis functions (which implicitly include the interelectronic coordinate and thus electron correlation). We do this for each symmetry state that is involved. All the ensuing eigenvalues and eigenvectors are then used in the sum-over-states expressions. For helium we require S, P, and D states and for H2 (or D2) E, II, and A states. 

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