Non-adiabatic coupling matrix element

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... 

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).

Fig. 4. Calculated energies (solid lines with filled symbols) and first-order non-adiabatic coupling matrix elements (NACME) (dotted lines with empty symbols) of the first three excited states in as a function of internuclear distance R.

In an electronic adiabatic representation, however, the electronic Hamiltonian becomes diagonal,i.e. ( a 77e C/3) = da,0Va, where the adiabatic Va potentials for initial (A,B,B ) and final (X) electronic states were described in Ref.[31]. The couplings between different electronic states arises from the matrix elements of the nuclear kinetic operator Tn, giving rise to the so-called non-adiabatic coupling matrix elements (NACME) and are due to the dependence of the electronic functions on the nuclear coordinates. The actual form of these matrix elements depends on the choice of the coordinates. 

Several methods for the determination of -y(R) have been proposed [224,225]. One is the direct computation of the non-adiabatic coupling matrix element ( i l(B 2))(r) by finite difference techniques, which gives the derivative of y (cf. Eq. (10)). Another is by supposing that the diabatic states adiabatic states Xk and Xkt are (almost) pure linear combinations of the two monomer states. This approximation can be made at the orbital level or at the A-electron level (or at both levels simultaneously). Also mixing matrix elements of molecular properties over adiabatic states may be used. 

I.D. Petsalakis, G. Theodorakopoulos, C.A. Nicolaides, R.J. Buenker, Theoretical dipole transition moments for transitions between bound electronic states and non-adiabatic coupling matrix elements between E + states of HeH, J. Phys. B 20 (1987) 5959. 

Generally it is found that only a small number of electronic states interact strongly. For those the so-called non-adiabatic coupling matrix elements... 

If two adiabatic terms Ui and Ug cross, they correspond to the wave functions of different symmetry. Non-adiabatic coupling between these terms occurs in the vicinity of the crossing point where the Massey parameter is zero. In this region, the non-adiabatic coupling matrix element... 

Transitions between terms of different symmetry are usually caused by the motion of nuclei which distort the symmetry of the electronic HamUtonian, e.g. by the molecular axis rotation in the case of two atoms, or by rotation of the plane of a three-atom system. In these cases, the non-adiabatic coupling matrix element Ci,2 is of the order of the rotational quantum. Then, for velocities corresponding to T = 1000 K, the t3q)ical atomic masses 10 and the difference in the term derivatives AF = 2 eV/A we obtain, using Eq. (9.12), P 2 lO -lQ-s. 

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.)...

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 ... 

The theory of electron transfer in chemical and biological systems has been discussed by Marcus and many other workers 74 84). Recently, Larson 8l) has discussed the theory of electron transfer in protein and polymer-metal complex structures on the basis of a model first proposed by Marcus. In biological systems, electrons are mediated between redox centers over large distances (1.5 to 3.0 nm). Under non-adiabatic conditions, as the two energy surfaces have little interaction (Fig. 5), the electron transfer reaction does not occur. If there is weak interaction between the two surfaces, a, and a2, the system tends to split into two continuous energy surfaces, A3 and A2, with a small gap A which corresponds to the electronic coupling matrix element. Under such conditions, electron transfer from reductant to oxidant may occur, with the probability (x) given by Eq. (10),... 

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