Nonadiabatic coupling matrix elements

The C matrix, the columns ofwhich, Cj(, are the eigenvectors of H, is normally not too different from the matrix defined above. However, the QDPT treatment, applied either to an adiabatic or to a diabatic zeroth-order basis, is necessary in order to prevent serious artefacts, especially in the case of avoided crossings [27]. The preliminary diabatisation makes it easier to interpolate the matrix elements of the hamiltonian and of other operators as functions of the nuclear coordinates and to calculate the nonadiabatic coupling matrix elements ... 

In this section we give the relations between the nonadiabatic coupling matrix elements in the quasi-diabatic and adiabatic representations. We do not obtain simple... 

When the quantum-classical Liouville equation is expressed in the adiabatic basis, the most difficult terms to simulate come from the off-diagonal force matrix elements, which give rise to the nonadiabatic coupling matrix elements. As described above, contributions coming from this term were computed using the momentum-jump approximation in the context of a surface-hopping scheme. 

Lengsfield III, B.H., Saxe, P., and Yarkony, D.R. (1984). On the evaluation of nonadiabatic coupling matrix elements using SA-MCSCF/CI wavefunctions and analytic gradient methods. I, J. Chem. Phys. 81, 4549-4553. 

The nonadiabatically coupled system shown in Fig. 6.10 consists of two nonadiabatic transition processes one is cl <-> feO and the other is cO bl. Other processes, cO -o- bO and cl -o- bl, were omitted. This is because nonadiabatic coupling matrix elements between the two vibronic states with equal vibrational quantum numbers approximately give zero for the displaced harmonic potential model with dimensionless potential displacement A fcl < 1. 

Here, Vd,w is the nonadiabatic coupling matrix element between vibronic state cl with energy and vibronic state bQ with energy In the displaced and undistorted potential model, En - Ebo = Ed - Eco is satisfied. The solution is given by solving the equation... 

Figure 2. For the Henon-Heiles potential with X=80 (equation (2) upper broken curve, ridge profile (6=0, 2tt/3 4tt/3) lower broken curve, valley bottom profile (6=Tr/3, ir, 5tt/3)), adiabatic potential curves (p) (equation 12) and corresponding nonadiabatic coupling matrix elements Pj j (p) (equation (13)) as a function of radial coordinate p for and A2 symmetry. Positions of levels indicated by continuous segments for those identified as quasiperiodic [3l] and by dotted segments for those not identified as quasiperiodic.

Thus the first derivative nonadiabatic coupling matrix element consists of two terms... 

We now consider the CSF contribution to the first derivative nonadiabatic coupling matrix element. A CSF, 06, is an antisymmetric, symmetry-adapted product of molecular orbitals, defined in Eq. (5b) so that its derivative has the form... 

In this subsection the determination of the second derivative nonadiabatic coupling matrix elements is considered. Real-valued wave functions... 

The nonadiabatic coupling matrix element is a consequence of the Born-Oppenheimer... 

Following Stephens, we can write the vibronic wavefunction as a combination of a pure Born-Oppenheimer zero-order wavefunction, and the first-order perturbed wavefunction due to the nonadiabatic coupling matrix elements as... 

One can see that the coupling in equation (5), when substituted from equation (6), gives rise to the denominator in equation (2). From equation (6) it is clear that the coupling depends on the velocity and the nonadiabatic coupling matrix element. Since... 

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