Force diabatization

For the a- and / -th adiabatic states having the energy Ea and E/g, respectively, the element of the force matrix is given as 

With this matrix and its eigenforces, denoted as /x if = 1, , M, we consider a diabatization. (By force diabatization, we do not mean the ordinary diabatization, in which the derivative coupling matrix elements Xpj are all zero.) To examine the global feature of them, we calculate the spatial distribution of the eigenforces fpc and (A A ), with A  

R 3 Bohrs, which is pretty far from the avoided crossing region. The asymmetric spatial distribution of the nonadiabatic coupling element Xap having the peak aroimd R 3 Bohrs accounts for this deviation. However, the truly remarkable featme of (A A ) is that the crossing takes 

The above observation can be confirmed more clearly in the similar results arising from the three-state model, which are depicted in panels (c) and (d) in Fig. 6.3. Corresponding to the three avoided crossing points at f = 2.3, 4.8 and 7.3 Bohrs for the curves of the eigenforces in panel (a), Xk A/f) imdergo crossing at three places (panel (b)). Again, 

Chemical Theory Beyond the Born- Oppenheimer Paradigm 

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The transformation T we adopt is induced by the wave function normalization condition which, in terms of the weights, reads w + W3 = 1. From (3.5), it is apparent that if T sends the vvm set into a new set wm with ivi = vvi + iv3 = 1 as one of its elements, then both the first row and the first column of the transformed polarization component of the solvent force constant matrix K, "/ = T. Kp°r. T (T = T) are zero, since the derivatives of wi are zero. Given the normalization condition and the orthogonality requirement — with the latter conserving the original gauge of the solvent coordinates framework — one can calculate T for any number of diabatic states [42], The transformation for the two state case is... 

The two diabatic energy profiles are expressed in terms of harmonic forms having a common force constant as ... 

In chemical reactions there is an electronic reordering in which some bonds are broken to form new ones. A full description of a chemical process thus requires the understanding of the electronic change involved since it will determine the main forces appearing along the process. Using the electronic states of reactants and products as a diabatic basis set representation, the reactions take place when... 

Here, the diabatic PES for the motion of the skeleton modes are Ea(Q) = Ea — faaQ + 1/2QKqqQ and the coupling between the diabatic states is given by V Q) = — fa0Q + 1/2QKq/3Q. The fo0 and Ko0 are the diabatic state matrix elements of the forces and the Hessian for the skeleton modes, respectively. Exemplary PES are shown in Fig. 1. Note that Eq. (1) is exact within the reaction surface approximation, i.e., no assumption concerning an adiabatic separation of H-atom and skeleton motions has been made. 

In the Landau-Zener model, dynamics is described by a single trajectory which due to the constant force undergoes an accelerated motion in the crossing region. The probability of a transition from diabatic state 1 to state 2 is denoted by P 2l which is also the probability of remaining in the lower adiabatic state, and the transition probability from the lower to the upper adiabatic state is then Pnonadia. = 1 — -P12, which is given by [16,17]... 

If simultaneously W k = 0 and <

= 0 in a region of M, then the nuclei move in the field of force the potential of which is given by the energy Wm of a single state of the electronic subsystem and there is no difference between the description of the electronic subsystem in the adiabatic or diabatic basis sets, i.e., Wam = W, . The corresponding behavior of the polyatomic system is referred to as electronically adiabatic ... 

Thermodynamic cost analysis relates the thermodynamic limits of separation systems to finite rate processes, and considers the environmental impact through the depletion of natural resources within the exergy loss concept. Still, economic analysis and thermodynamic analysis approaches may not be parallel. For example, it is estimated that a diabatic column has a lower exergy loss (39%) than that of adiabatic distillation however, this may not lead to a gain in the economic sense, yet it is certainly a gain in the thermodynamic sense. The minimization of entropy production is not always an economic criterion sometimes, existing separation equipment may be modified for an even distribution of forces or an even distribution of entropy production. Thermodynamic analysis requires careful interpretation and application. 

The equipartition principle is mainly used to investigate binary distillation columns, and should be extended to multicomponent and nonideal mixtures. One should also account for the coupling between driving forces since heat and mass transfer coupling may be considerable and should not be neglected especially in diabatic columns. 

The general patterns of the energy contributions to (FC) are more easily discussed than the entropy contributions, and the energy contributions will be emphasized here. Equations (15-18) and (29) predict a complex dependence of ket on the reaction driving force. In the diabatic limit this corresponds to variations in the intersections of the reactants and products PE surfaces with the differences in their PE minima this is illustrated in Figure 2 for the diabatic (a) and weak-coupling (b) limits. 

Some important free-energy relationships are presented in terms of the diabatic energy profiles G, and Gf in Figure 3. The vertical and horizontal shifts of the G/ profile relative to that for C, correspond, respectively, to the driving force of the ET process (—AG,y ) and the reorganization energy (z) of nuclear modes (shifts of equilibrium coordinate values). 

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