Adiabatic Hessian

Figure 8.3 ECD spectra of the S2 So electronic transition of ax-R3MCP computed with FCIAH (adiabatic Hessian) and FCHTIAH approaches, which span an energy range of 0-4000 cm with respect to the 0-0 transition.

In such a case the last choice is to take the direction of the eigenvector of the only one nonzero eigenvalue of the rank one Hessian matrix of the difference between the two adiabatic potential energies [51]. In the vicinity of conical intersection, the topology of the potential energy surface can be described by the diadiabatic Hamiltonian in the form... 

Here, the diabatic PES for the motion of the skeleton modes are Ea(Q) = Ea — faaQ + 1/2QKqqQ and the coupling between the diabatic states is given by V Q) = — fa0Q + 1/2QKq/3Q. The fo0 and Ko0 are the diabatic state matrix elements of the forces and the Hessian for the skeleton modes, respectively. Exemplary PES are shown in Fig. 1. Note that Eq. (1) is exact within the reaction surface approximation, i.e., no assumption concerning an adiabatic separation of H-atom and skeleton motions has been made. 

The leading quantum correction to the static JT energetics is given by the zero-point energy gain due to the softening of the vibrational frequency at the JT-distorted minima [5]. To obtain this information, by finite differences we compute the Hessian matrix of the second-order derivatives of the lowest adiabatic potential sheet, at one of the static JT minima Q,mn... 

Newton s second law, 2, 383 NMR shielding, 239 Non-adiabatic coupling elements, 55 Non-bonded energy, in force held methods, 18 Non-bonded list, in force held methods, 43 Non-local DFT methods, 184 Norm-extended Hessian ophmizahon method, 320... 

So far, we have only separated out the HX vibrational motion. Generally, such a solution of the stationary Schiodinger equation is not computationally feasible for clusters with more atoms. Therefore, other approximations have to be employed. At the same time, all phenoniena important for the cluster structure have to be properly included. We have performed an adiabatic separation of the HX libra-tional motion from the motion of the heavy particles, i.e., from the cage modes. Moreover, the cage modes have been calculated within the harmonic approach, i.e., by a diagonalization of the Hessian matrix. Formally, the wavefunction is expressed as... 

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