Nonadiabatic coupling matrix

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. 

Tq gain insight into requirements for the validity of the adiabatic approximation j/bonsider the case of cp2 = 0. Here the nonadiabatic coupling matrix A is given by... 

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... 

The elements of the nonadiabatic coupling matrix are likewise related ... 

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.

The adiabatic approximation (which was thoroughly investigated in Ref. [57]) i obtained by neglecting the nonadiabatic coupling matrix U dU/dt. This allows u to write Equation 8.28 in the form... 

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... 

Here Tjv denotes the nuclear kinetic energy, V(Q) is the diagonal matrix containing the adiabatic potential energy surfaces Vi(Q) and 1 generally stands for the unit matrix in electronic function space. The vector x(Q) collects the vibrational wavefunctions Xi(Q) of interest, and the elements Aij of the nonadiabatic coupling matrix A can be written as ... 

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