Diabatic basis

Sadygov R G and Yarkony D R 1998 On the adiabatic to diabatic states transformation in the presence of a conical intersection a most diabatic basis from the solution to a Poisson s equation. I J. Chem. Rhys. 109 20... 

A perfect diabatic basis would be one for which the first-derivative coupling in Eq. (31) vanishes [10]. From the above mentioned... 

Assuming that the diabatic space can be truncated to the same size as the adiabatic space, Eqs. (64) and (65) clearly define the relationship between the two representations, and methods have been developed to obtain the tians-formation matrices directly. These include the line integral method of Baer [53,54] and the block diagonalization method of Pacher et al. [179]. Failure of the truncation assumption, however, leads to possibly important nonremovable derivative couplings remaining in the diabatic basis [55,182]. 

Setting the diabatic basis equal to the adiabatic basis at the degenerate point, Ro, the expansions can be written in vector notation as... 

The previous treatment relied on the assumption that the transition occurs on a single potential energy surface V(x) characterized by a barrier separating two wells. This potential is actually created from the terms of the initial and final electronic states. The separation of electron and nuclear coordinates in each of these states gives rise to the diabatic basis with nondiagonal Hamiltonian matrix... 

The calculation of the Td matrix, involving second derivatives of the electronie wave-functions, is more expensive and subject to numerical inaccuracy than that of Gd-A simple approximation for the third term in eq.(19) is based on a partial expansion of the identity operator in terms of the diabatic basis ... 

The components of the two vectors ( 1 i 2X when multiplied by the electronic (diabatic) basis set ( cj>i), 14b)), form the corresponding electronic adiabatic basis... 

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... 

B. A. Hess Prof. Jungen, in your talk you emphasized that you don t have to calculate matrix elements of d/dQ or Coriolis coupling. My impression is that this is due to your most appropriate choice of a diabatic basis, which is generally what ab initio quantum chemists do when they want to avoid singularities in the adiabatic basis. On the other hand, the absence of explicit Coriolis coupling matrix elements is due to the transformation to a space-fixed coordinate system. 

The direct numerical solution of the HSCC equations (95) would be impractical because of the sharply peaked nonadiabatic coupling potentials near any of the many avoided crossings. This difficulty is often circumvented by the diabatic-by-sector technique to solve, effectively, the coupled equations (95) [95,96]. By this technique, the region of p up to some large value is divided into many sectors, labeled by i. The adiabatic functions ( 2c / ) at some fixed point p = pt in each sector are employed throughout the sector as the diabatic basis functions 0 ( c) for expansion. The expansion then takes a form (i)(p, 2C) = En p 5/2F (p) (Qc), and the coupled equations in each sector i are... 

The general features of the nonadiabatic coupling and its relation to molecular properties are surveyed. Some consequences of the equation of motion , formally expressing a smoothness of a given molecular property within the diabatic basis, are demonstrated. A particular emphasis is made on the relation between a smoothness of the electronic dipole moment and the generalized Mulliken-Hush formula for the diabatic electronic coupling. 

The other techniques can be divided into a few categories (see also Ref. [14c]). One of them goes back to the idea first coined by Mulliken [41], Hush [42] and Lichten [8] of using molecular properties to determine the diabatic basis and is actually based on the Werner-Meyer-Macfas-Riera formula [32,33] for the adiabatic-to-diabatic mixing angle in terms of electronic dipole moments. This... 

Assuming that a strictly diabatic basis exists and defining the corresponding matrix elements Ay = (( f A(r, R)l Vf)), one arrives at... 

Equation (26) expresses a smoothness of matrix elements of Hermitian operator A in a strictly diabatic basis. It is the key formula for deriving the expression of the ADT angle in terms of a given molecular property [37]. 

Consider the electronic dipole moment matrix M = (mH = ((1P"ilrl1fr/)))H. and project it on a given direction in 1R3. This gives the matrix m = (mkl)kl. Imposing a smoothness condition of this matrix within the two-state diabatic basis, one readily obtains the ADT mixing angle in terms of the electronic dipole moment [32, 33,37,68]... 

In this expression, each individual sum extends over all the Af diabatic basis states. 

Formally, we can write the unitary transformation from the adiabatic to the diabatic basis as... 

Later, in 1990, Kim and Heynes [11] investigated the role of solvent polarization in fast electron transfer processes and pointed out that, when the solvent is instantaneously equilibrated to the quantum charge distribution of the solute, the Hamiltonian itself is a functional of the wave-function, giving a non-linear Schrodinger equation. The resulting solvent contribution to the Hamiltonian matrix on the diabatic basis thus cannot be simply described as in the former EVB method. 

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