Second-order coupling

In a similar fashion, a fraction of the velocity of the molecules with first-order coupling is transmitted to other molecules entangled with the latter. This is called second-order coupling (subscript 2). Still higher orders of effect radiate from the original molecule in the manner suggested by Fig. 2.13. Because of the... 

Finally, we show that the second-order coupling between direct tunneling transitions is subdominant to the already computed quantities. Consider an interaction of the form yy l,-2 )(2,l + H.C. If one repeats simple-mindedly the steps leading to Eq. (75), one obtains the following simple expression for the free energy correction due to interaction between the underlying structural transitions ... 

For example, the double triplet (6 6.16) was obscured due to the poor signal-to-noise and the severe second order coupling at 100 MHz. Simulations (7,8) of the olefinic protons were critical to these assignments (Figure 3). 

Coupling between adjacent single protons, (a) First-order coupling, AX. (b) Second order coupling, AB -> A2. 

The quantum alternative for the description of the vibrational degrees of freedom has been commented by Westlund et al. (85). The comments indicate that, to get a reasonable description of the field-dependent electron spin relaxation caused by the quantum vibrations, one needs to consider the first as well as the second order coupling between the spin and the vibrational modes in the ZFS interaction, and to take into account the lifetime of a vibrational state, Tw, as well as the time constant,T2V, associated with a width of vibrational transitions. A model of nuclear spin relaxation, including the electron spin subsystem coupled to a quantum vibrational bath, has been proposed (7d5). The contributions of the T2V and Tw vibrational relaxation (associated with the linear and the quadratic term in the Taylor expansion of the ZFS tensor, respectively) to the electron spin relaxation was considered. The description of the electron spin dynamics was included in the calculations of the PRE by the SBM approach, as well as in the framework of the general slow-motion theory, with appropriate modifications. The theoretical predictions were compared once again with the experimental PRE values for the Ni(H20)g complex in aqueous solution. This work can be treated as a quantum-mechanical counterpart of the classical approach presented in the paper by Kruk and Kowalewski (161). 

A study of second order coupling in the 13C NMR spectrum of tropine [263] has confirmed the original assignment of the C(6) and... 

C(2) resonances. The identical chemical shifts of the 6er- and 7o-protons together with the strong 13C(6)-1H(6a) coupling and the weak 13C(6)- 11(70) coupling lead to the expectation of the observed second order coupling of the C(6) signals. Since the chemical shifts of the C(2) methylene protons are very different from those of the protons at C(l)... 

Replace the set of second-order coupled equations in the nuclear configuration space by a set of first-order equations appropriate to the time-dependent Schrbdinger equation. 

Fig.l. a The two components (Q, Q ) of the Jahn-Teller active eg vibration of an Cu(II)Lg octahedron. b The E e Jahn-Teller potential energy surfaces with first and second order coupling terms... 

Equation (3) gives an expression for the barrier heights (2p) in terms of the second-order coupling (A2) and the vibrational anharmonicity (K3). However,... 

Figure 5.1 The effect of quadrupole interactions on an / = 3/2 nucleus in a magnetic field. The Zeeman interaction splits the levels by an equal amount, cot, (the Larmor frequency in frequency units). The central +1 /2 to -1 /2 transition is unaffected by first order coupling co j however, second order coupling, co , affects all transitions.

Simulations that have been carried out for a vast number of different rate parameters and mechanistic variants show that only the slow second-order coupling steps between dimers and the subsequent slow elimination of protons can explain the trace-crossing (Fig. 2). In any case, the so-called nucleation loop in cyclic voltammetry is not the result of a nucleation step, but of slow, homogeneous follow-up reactions of soluble oligomers in the diffusion layer. 

Taking into account that the average Hessian of the rotated and unrotated states is the same, the second-order couplings for the unrotated states can be obtained from the average Hessian and the Hessian of one of the rotated states, using for example (30) ... 

For Cr(CO)5 it is interesting to consider the epikernel extrema of C2v symmetry which correspond to saddle-points for pseudo-rotation (Fig. 8). Generally one imagines such features to be associated with second-order coupling terms in the degenerate intersection space of the conical intersection [94,98]. However this is by... 

Fig. 6 The vibronic energy levels as a function of (a) the first-order coupling with (A2 = 0), and (b) the second-order coupling constant for a fixed value of Ai/fm = 1.65...

The Hamiltonian in (36) operates on the ground state vibronic wavefunctions and I g which can be expressed as products of electronic functions and an expansion of the two dimensional vibrational states cpi of appropriate symmetry [41], The linear combinations are found using the E g e and H2 g e vector coupling coefficients (the Hi g e coefficients are of course trivial) following the same procedure as used to construct the vibronic Hamiltonian in (15). It may then be readily shown that for strong linear coupling, p 0 and q 1/2 [10,41]. Note however, for second order coupling q can take values less than 1/2 [42]. 

The effect that is dominant in pure crystals, the Davydov splitting, is missing in mixed crystals. There are no first order effects of the crystal field. Second order coupling terms like (3.3), where the superscripts refer to the lowest transitions in the host and guest molecules, cause intensity transfers. In the b direction the tetracene intensity is increased by a factor of 3 by transfer from the anthracene 250 nm and 380 nm systems in the a direction there is little change. 

Fig. 2. I C NMR spectrum of [1, 2- 3C2] acetate enriched herqueichrysin (as its triacetate) determined at 67.89 MHz. Note the severe overlap of the coupled signals in the aromatic region and the second-order couplings arising from the similarity of the chemical...

Schrodinger s equation leads then to Nx + second-order coupled differential equations with first derivatives. These are solved transforming the set into 2(N + Ne) first-order coupled equations, and generating N2 + N linearly independent solutions by choosing suitable boundary conditions at small rAC. A Bashforth-Moulton fourth-order predictor-corrector algorithm was used in the integration. 

In discussing the alternative theoretical approaches let us limit ourselves to those which have been applied directly to processes in which we are interested in this article, but first of all let us stress once more the importance of the work of Delos and Thorson (1972). They formulated a unified treatment of the two-state atomic potential curve crossing problem, reducing the two second-order coupled equations to a set of three first-order equations. Their formalism is valid in the diabatic as well as the adiabatic representation and also at distances of closest approach near Rc. Moreover the problem of the residual phase x(l) is solved implicitly. They were able to show that a solution of the three first-order classical trajectory equations is not sensitive to all details of the potentials and the coupling term, but to only one function which therefore can be used readily for modelling assumptions. The resulting equations should be solved numerically. Their method has been applied now to the problem of the elastic scattering of He+ + Ne (Bobbio et at., 1973) but unfortunately not yet to any ionization problem. 

Stuckelberg did the most elaborate analysis (15). He applied the approximate complex WKB analysis to the fourth-order differential equation obtained from the original second-order coupled Schrodinger equations. In the complex / -plane he took into account the Stokes phenomenon associated with the asymptotic solutions in an approximate way, and finally derived not only the Landau-Zener transition probability p but also the total inelastic transition probability Pn as... 

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