Coupling matrix element

Here the distortion (diagonal) and back coupling matrix elements in the two-level equations (section B2.2.8.4) are ignored so that = exp(ik.-R) remains an imdistorted plane wave. The asymptotic solution for ij-when compared with the asymptotic boundary condition then provides the Bom elastic ( =f) or inelastic scattering amplitudes... 

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... 

These coupling matrix elements are scalars due to the presence of the scalar Laplacian V. in Eq. (25). These elements are, in general, complex but if we require the to be real they become real. The matrix unlike its... 

Thus, as the adiabatic PES become degenerate the adiabatic coupling matrix elements become singular. 

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 first-order MPPT wavefunction can be evaluated in terms of Slater determinants that are excited relative to the SCF reference function k. Realizing again that the perturbation coupling matrix elements I>k H i> are non-zero only for doubly excited CSF s, and denoting such doubly excited i by a,b m,n the first-order... 

The evaluation of the radial coupling matrix elements between molecular states of the same symmetry 3... 

The rotational coupling matrix elements between Z-IT and fl-A states have been evaluated analytically by use of the L. and L. operators. 

The main features of the radial coupling matrix elements are presented in Fig. 2. In correspondence with the avoided crossings between the potential energy curves of singleelectron capture, sharp peaked functions appear at respectively 6.35, 7.50 and 8.30 a.u.. They are approximately 1.23, 2.53 and 12.21 a.u. high and respectively 0.75, 0.50 and less than 0.10 a.u. wide at half height. 

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

Fig. 3. Rotational coupling matrix element between the ll (N (3p) + He (ls)) state and the states of single-electron capture.

In relation with these avoided crossings, the radial coupling matrix elements present sharp peaks at respectively 5.4, 6.6, 7.55 and 9.5 a.u. (Fig. 5). We may notice that these radial couplings are almost insensitive to the choice of the origin of electronic coordinates. The most sensitive one is the g23 function at short internuclear distance range, but we may expect weak translational effects for such electron capture processes dominated by collisions at large distance of closest approach. 

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

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... 

For the intersystem crossing T Si, the electronic coupling matrix element (T il so S ) often vanishes. In this case, we have to take into account the higher-order terms in Eq. (3.69), that is,... 

The immediate question is where (in phase space) to place the newly spawned basis functions. The optimal choice will maximize the absolute value of the coupling matrix element between the existing basis function (i.e., the... 

The probability P in equations (61) and (62) may be related to the electronic coupling matrix element through equation (63) by application of the Landau-Zener model ... 

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). 

The adiabatic coupling matrix elements, Fy, can be evaluated using an off-diagonal form of the Hellmann-Feynman theorem... 

Note that, if the donor and acceptor s and p orbitals refer to the same atomic center, the coupling matrix elements and /pp- are identically zero, and hybridization cannot lower the energy. Hence, atomic hybridization is intrinsically a bonding effect. 

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