Completely diabatic basis

The pure reactant and product states are given by the coefficient vectors (10 0) and (010), respectively. A simple view of a chemical process will be elicited by a -path leading over time from one state to the other. The amplitudes in the diabatic basis functions are obtained by diagonalizing the matrix of the complete Hamiltonian operator H = + V(q, A). 

Figure 20. The (So —> S2) absorption spectrum of pyrazine for reduced three- and four-dimensional models (left and middle panels) and for a complete 24-vibrational model (right panel). For the three- and four-dimensional models, the exact quantum mechanical results (full line) are obtained using the Fourier method [43,45]. For the 24-dimensional model (nearly converged), quantum mechanical results are obtained using version 8 of the MCTDH program [210]. For all three models, the calculations are done in the diabatic representation. In the multiple spawning calculations (dashed lines) the spawning threshold 0,o) is set to 0.05, the initial size of the basis set for the three-, four-, and 24-dimensional models is 20, 40, and 60, and the total number of basis functions is limited to 900 (i.e., regardless of the magnitude of the effective nonadiabatic coupling, we do not spawn new basis functions once the total number of basis functions reaches 900).

Let f Ld(r q ) refer to that alternate basis set. Assuming that it is complete in r and orthonormal in a manner similar to Eq. (10), we can use it to expand the total orbital wave function of Eq. (11) in the diabatic version of Bom-Huang expansion as... 

In coordinate representation, there exists alternative base representations, adiabatic and diabatic. Both representations are equivalent if the basis are complete. For a thorough discussion on adiabatic-diabatic electronic state transformations the reader is referred to the work by Baer [49, 50], see also the work by Chapuisat et al. [51] In this... 

If one had a complete basis set then, for the stationary geometry, the corresponding electronic waveftmctions obtained by actual calculation would be models for the diabatic base functions. The latter statement has to be understood in the sense that the set of nodal planes so obtained must be kept fixed. This was the procedure used to extract diabatic base states for cis and trans ethylene. 

It is assumed that target states p are indexed for each value of q such that a smooth diabatic energy function Ep(q) is defined. This requires careful analysis of avoided crossings. The functions should be a complete set of vibrational functions for the target potential Vp = Ep, including functions that represent the vibrational continuum. All vibrational basis functions are truncated at q = qd, without restricting their boundary values. The radial functions fra should be complete for r < a. 

The derivation up to this point involves no approximations if the vibronic basis set is complete in the closed hypervolume, including its surface. If a dissociation channel exists, it can be approximated by projection onto a single diabatic state < d(q xN+1), assumed to be well defined as a discrete state on the cap surface q = qd.. The projection integrals on this surface are... 

Apart from the selection rules for the electronic coupling matrix element, spin-forbidden and spin-allowed nonradiative transitions are treated completely analogously. Nonradiative transitions caused by spin-orbit interaction are mostly calculated in the basis of pure spin Born-Oppenheimer states. With respect to spin-orbit coupling, this implies a diabatic behavior, meaning that curve crossings may occur in this approach. The nuclear Schrodinger equation is first solved separately for each electronic state, and the rovibronic states are spin-orbit coupled then in a second step. 

While the definition of an adiabatic state is straightforward, as an eigenfunction of the Hamiltonian within the complete set of VB structures, the concept of diabatic state is less clear-cut and accepts different definitions. Strictly speaking, a basis of diabatic states (fi, ...) should be such that eq 21 is satisfied for any variation <5Q of the geometrical coordinates. 

It is clear from (A.8) and (A.9) that the gradient difference and derivative coupling in the adiabatic representation can be related to Hamiltonian derivatives in a quasidiabatic representation. In the two-level approximation used in Section 2, the crude adiabatic states are trivial diabatic states. In practice (see (A.9)), the fully frozen states at Qo are not convenient because the CSF basis set l Q) is not complete and the states may not be expanded in a CSF basis set evaluated at another value of Q (this would require an infinite number of states). However, generalized crude adiabatic states are introduced for multiconfiguration methods by freezing the expansion coefficients but letting the CSFs relax as in the adiabatic states ... 

An alternative approach to the selection of A(q) is to consider an electronically adiabatic expansion truncated at a small number X of terms and require A(q) to be an (X X >T)-dimensional matrix. In this case, neither the adiabatic nor the diabatic electronic basis set is complete, but we assume that the adiabatic expansion in X terms is sufficiently accurate for our purposes. We now wish to select this A(q) so as to minimize the effect of the term in (75) containing W(1)ad(R). Ideally, we would like to force this matrix to vanish identically. Unfortunately, this is not always possible, as we shall now show. 

In this expression, W 1)ad(R) (k = i, j) is the X X N matrix whose row n and column n element is the k element of the W(n1, 1ad(R) vector, i.e., [W ad(R)], and the brackets in its right-hand side denote the commutator of the two matrices within. When n and n are allowed to span the complete infinite set of adiabatic electronic quantum numbers, condition (102) is satisfied [24,26], (99) has a solution, and the resulting A(q) leads to the q-independent diabatic electronic basis set mentioned in connection with (83). For the small values of X case being considered here, (102) is in general not satisfied and (98) does not have a solution. On the other hand, the equation obtained by replacing in (99) W(1)ad(R) by its longitudinal part VRd><1)ad(q) [see remark after (98)], namely... 

For class-1 states, a simple harmonic representation of U leads to a complete set of eigenfunctions ( ) this harmonic oscillator basis set is used to diagonalize equation (6). In this case, it is sufficient to construct U( 4>k) using a standard approach involving mass fluctuation (or nuclear ) coordinates and the corresponding electronic state dependent Hessian. The higher terms in the Taylor expansion define anharmonic contributions to the transition moments. These diabatic states are confining and only one stationary point in -space would be found for each... 

In the context of traditional electronic calculations, the GED approach would correspond to a multi-configuration Cl wave function written in a complete basis of direct products of one-electron states, as discussed in Section 3.1 [13]. The basis set must be in principle invariant with respect to the choice of electronic Hamiltonian. In practice, however, care must be exercised when using different electronic diabatic states that have been derived from distinct stationary geometries of the PCB. 

The diabatic and the adiabatic electronic states are simply two choices from the basis set in non-adiabatic calculations. If the sets were complete, the results would be identical. The first choice underlines the importance of the chemical bond pat-... 

The most straightforward numerical technique for the solution of Eq. (1) is based on the expansion of the state vector (t)) in a complete set of time-independent basis functions. Such a complete basis can be constructed as the direct product of diabatic electronic basis states l n) and suitable orthonormal states xiyj) for each nuclear degree of freedom (see Chapter 7)... 

Thus the perfect diabatic representation does not in fact generally exist unless the basis set is complete [198]. Practical theories of diabatic representation, including those for multi-dimensional systems, have been studied extensively by Smith [376] and Baer [27, 28]. Another scheme was proposed using a different perspective [23, 292]. In this approximate but practical treatment, a diabatization is pursued by requiring the basis states to retain their individual characters smoothly, rather than minimizing the magnitude of derivative coupling. 

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