Non-adiabatic matrix elements

Figure 7 Adiabatic curves (panel a) and non-adiabatic matrix elements (panel b) as a function of p.

An estimate of the coupling between the two surfaces is obtained by calculating the non-adiabatic matrix elements numerically from the Cl wavefunctions [17]. Coupling is mainly induced by the CO rotation and it is an order of magnitude larger than the OH rotation and the radial approach. Numerical values are presented in Table 1. 

In order to get a better description at shorter distances, the non-diagonal matrix elements of the interaction are taken into account inside each of the subspaces. This is done by defining a new basis set, the adiabatic basis set 5), obtained by diagonalization of... 

Some final comments on the relevance of non-adiabatic coupling matrix elements to the nature of the vector potential a are in order. The above analysis of the implications of the Aharonov coupling scheme for the single-surface nuclear dynamics shows that the off-diagonal operator A provides nonzero contiibutions only via the term (n A n). There are therefore no necessary contributions to a from the non-adiabatic coupling. However, as discussed earlier, in Section IV [see Eqs. (34)-(36)] in the context of the x e Jahn-Teller model, the phase choice t / = —4>/2 coupled with the identity... 

The superaiatrix notation emphasizes the structure of the problem. Each diagonal operator drives a wavepaclcet, just as in the adiabatic case of Eq. (10), but here the motion of the wavepackets in different adiabatic states is mixed by the off-diagonal non-adiabatic operators. In practice, a single matrix is built for the operator, and a single vector for the wavepacket. The operator matrix elements in the basis set <() are... 

As shown above in Section UFA, the use of wavepacket dynamics to study non-adiabatic systems is a trivial extension of the methods described for adiabatic systems in Section H E. The equations of motion have the same form, but now there is a wavepacket for each electronic state. The motions of these packets are then coupled by the non-adiabatic terms in the Hamiltonian operator matrix elements. In contrast, the methods in Section II that use trajectories in phase space to represent the time evolution of the nuclear wave function cannot be... 

For diabatic calculations, the equivalent expression uses the diabatic potential matrix elements [218]. When the value of this coupling becomes greater than a pre-defined cutoff, the tiajectory has entered a non-adiabatic region. The propagation is continued from this time, ti, until the trajectoiy moves out of the region at time f2-... 

The elements of the matrix G can be written in terms of F, which is called the non-adiabatic coupling matrix. For a particular coordinate, a, and dropping the subscript for clarity,... 

Figure 11. Results for the C2H molecule as calculated along a circle surroiinding the A -2 A conical intersection. Shown are the geometry, the non-adiabalic coupling matrix elements i(p((p J 2) and the adiabatic-to-diabadc transformation angles y((p J2) as calculated for T] (=CC distance) = 1.35 A and for three values (j 2 is the CH distance) (a) and (i>) = 1.80 A (c) and (tf) = 2.00 A (c) and (/) = 3.35 A. (Note that q = r2.)...

Figure 12, Results for the C2H molecule as calculated along a circle surrounding Che 2 A -3 A conical intersection, The conical intersection is located on the C2v line at a distance of 1,813 A from the CC axis, where ri (=CC distance) 1.2515 A. The center of the circle is located at the point of the conical intersection and defined in terms of a radius < . Shown are the non-adiabatic coupling matrix elements tcp((p ) and the adiabatic-to-diabatic transformation angles y((p i ) as calculated for (ii) and (b) where q = 0.2 A (c) and (d) where q = 0.3 A (e) and (/) where q = 0.4 A. Also shown are the positions of the two close-by (3,4) conical intersections (designated as X34).

The electron capture processes are driven by non-adiabatic couplings between molecular states. All the non-zero radial and rotational eoupling matrix elements have therefore been evaluated from ab initio wavefunctions. 

Fig. 2. a, b, c. Non-adiabatic radial coupling matrix elements for the states of single-electron capture. 

Fig. 5.3, b. Non adiabatic radial coupling matrix elements between states, a) Origin N. b) Origin He. 

Thus, from equation (63), the magnitude of the electronic coupling matrix element may finally be estimated, leading to values of 21 and 24 meV for EDA and perylene, respectively. That these values are quite reasonable derives from the observation that they correspond to moderately non-adiabatic electron transfer at the ground state (with electronic factors of 2 /(1 + P) - 0.5 and 0.6 with EDA and perylene, respectively). 

Multiple spawning, direct molecular dynamics ab initio multiple spawning, 411-414 non-adiabatic coupling, 399-402 Multivalued matrix elements, non-adiabatic coupling ... 

To study the two isolated conical intersections, we have to treat two-state curl equations that are given in Eq. (26). Here, the first 2 x 2 x matrix contains the (vectorial) element, that is, T012 and the second 2 x 2 r matrix contains 1023- As before each of the non-adiabatic coupling terms, r012 and x023 has the following components ... 

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