Diagonalization of the potential energy

We consider a potential energy surface expressed in Cartesian laboratory coordinates, qi (i = 1. n), for a system of n = 3N (where N is the number of nuclei) degrees of freedom. A Taylor expansion of the potential V around the point (q . q ) gives 

We now assume that the expansion of the potential is around a stationary point (stable or unstable, depending on the sign of the second-order derivatives), that is, all the first-order derivatives vanish. The energy is measured relative to the value at equilibrium, and we obtain 

In the potential of Eq. (E.4), we still find that all the coordinates are coupled, i.e., it contains off-diagonal terms of the form with i j. However, since the potential is a quadratic form, we know from mathematics that it is possible to introduce a linear transformation of the coordinates such that the potential takes a diagonal form in the new coordinates. To that end, normal-mode coordinates, Q, are introduced by the following linear transformation of the mass-weighted displacement coordinates  

We have now obtained the desired diagonal form of the potential energy. If all the frequencies, ui2, are positive then the stationary point represents a minimum on 

---

In the presence of reflection symmetry with respect to the diagonal of the potential-energy surface, as in symmetric molecules or in the four-disk scatterer, Burghardt and Gaspard have shown that a further symmetry reduction can be performed in which the symbolic dynamics still contains three symbols A = 0, +, - [10]. The orbit 0 is the symmetric-stretch periodic orbit as before. The orbit + is one of the off-diagonal orbits 1 or 2 while - represents a half-period of the asymmetric-stretch orbit 12. Note that the latter has also been denoted the hyperspherical periodic orbit in the literature. 

This two-dimensional potential energy surface is the lower energy solution obtained from the diagonalization of the potential energy operators in the E 0 e vibronic Hamiltonian acting within a ( 0>, >) electronic basis ... 

The first quartet corresponds to the fundamental level. The second and fourth to antisymmetric torsionally excited levels. The third, fifth and the last to symmetric torsionally excited levels. The antisymmetric torsion modes correspond to a gearing rotation along the principal diagonal of the potential energy surface. They are active in Raman and somewhat in infrared spectra. The symmetric modes correspond to an antigearing rotation along the secondary diagonal and are active in infrared. 

The matrix M contains atomic masses on its diagonal, and the Hessian matrix F contains the second derivatives of the potential energy evaluated at Xq. 

The Newton-Raphson block diagonal method is a second order optimizer. It calculates both the first and second derivatives of potential energy with respect to Cartesian coordinates. These derivatives provide information about both the slope and curvature of the potential energy surface. Unlike a full Newton-Raph son method, the block diagonal algorithm calculates the second derivative matrix for one atom at a time, avoiding the second derivatives with respect to two atoms. 

The usual EVB procedure involves diagonalizing this 3x3 Hamiltonian. However, here we wish to use a very simple model for our reaction and represent the potential surface and wavefunction of the reacting system using only two electronic states. Using a two-state system will preserve most of the important features of the potential energy surface while at the same time provide a simple model that will be more amenable to discussion than the three-state system. For the two-state system we define the following states as the reactant and product wavefunctions ... 

Additional effect of diagonal dynamic disorder. The variations of the electron densities near the centers A and B due to polarization fluctuations and local vibrations lead to changes in the interaction of the electron with the medium and, hence, to changes in the shape of the potential energy surfaces Ut and Uf as compared... 

The curvature of the potential energy surface in the direction Qesk at Res, is measured by the force constant, Kk = Kq + [46], The diagonal matrix ele-... 

A variational treatment yields three block diagonal 2x2 secular determinants, although this result is not initially immediately obvious. The structure of the potential energy matrix elements is discussed in detail in Appendix C. The diagonal matrix elements are readily found to be... 

We obtain the coordinates <2fc in the following way. At point s of the IRC, we diagonalize the Hessian (i.e., the matrix of the second derivatives of the potential energy) and consider all the resulting normal modes are the corresponding frequencies cf. Chapter 7] other than... 

---