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## Diabatic*See also adiabatic process, a concept in thermodynamics*
In quantum chemistry, the potential energy surfaces are obtained within the adiabatic or Born-Oppenheimer approximation. This corresponds to a representation of the molecular wave function where the variables corresponding to the molecular geometry and the electronic degrees of freedom are separated. The non separable terms are due to the nuclear kinetic energy terms in the molecular Hamiltonian and are said to couple the potential energy surfaces. In the neighbourhood of an avoided crossing or conical intersection, these terms cannot be neglected. One therefore usually performs one unitary transformation from the adiabatic representation to the so-called ## Additional recommended knowledgeIn the diabatic representation, the potential energy surfaces are smoother so that low order Taylor series expansions of the surface capture much of the complexity of the original system. Unfortunately, strictly diabatic states do not exist in the general case. Hence, diabatic potentials generated from transforming multiple electronic energy surfaces together are generally not exact. These can be called ## ApplicabilityThe motivation to calculate diabatic potentials often occurs when the Born-Oppenheimer approximation does not hold, or is not justified for the molecular system under study. For these systems, it is necessary to go A well-known approach involves recasting the molecular Schrödinger equation into a set of coupled eigenvalue equations. This is achieved by expansion of the exact wave function in terms of products of electronic and nuclear wave functions (adiabatic states) followed by integration over the electronic coordinates. The coupled operator equations thus obtained depend on nuclear coordinates only. Off-diagonal elements in these equations are nuclear kinetic energy terms. A diabatic transformation of the adiabatic states replaces these off-diagonal kinetic energy terms by potential energy terms. Sometimes, this is called the "adiabatic to diabatic transformation", abbreviated
## Diabatic transformation of two electronic surfacesIn order to introduce the diabatic transformation we assume now, for the sake of argument, that only two Potential Energy Surfaces (PES), 1 and 2, approach each other and that all other surfaces are well separated; the argument can be generalized to more surfaces. Let the collection of electronic coordinates be indicated by , while indicates dependence on nuclear coordinates. Thus, we assume with corresponding orthonormal electronic eigenstates and . In the absence of magnetic interactions these electronic states, which depend parametrically on the nuclear coordinates, may be taken to be real-valued functions. The nuclear kinetic energy is a sum over nuclei (Atomic units are used here).
By applying the Leibniz rule for differentation, the matrix elements of The subscript indicates that the integration inside the braket is
over electronic coordinates only.
Let us further assume
that all off-diagonal matrix elements
may be neglected except for the coupled Schrödinger equations for the nuclear part take the form (see the article Born-Oppenheimer approximation) In order to remove the problematic off-diagonal kinetic energy terms, we
define two new orthonormal states by a where is the These elements are zero because is real and is Hermitian and pure-imaginary. The off-diagonal elements of the momentum operator satisfy, Assume that a diabatic angle exists, such that to a good approximation i.e., and diagonalize the 2 x 2 matrix of the nuclear momentum. By the definition of
Smith By a small change of notation these differential equations for can be rewritten in the following more familiar form: It is well-known that the differential equations have a solution (i.e., the "potential" It can be shown that these conditions are rarely ever satisfied, so that a strictly diabatic
transformation rarely ever exists. It is common to use approximate functions leading to Under the assumption that the momentum operators are represented exactly by 2 x 2 matrices, which is consistent with neglect of off-diagonal elements other than the (1,2) element and the assumption of "strict" diabaticity, it can be shown that On the basis of the diabatic states
the nuclear motion problem takes the following It is important to note that the off-diagonal elements depend on the diabatic angle and electronic energies only. The surfaces and are adiabatic PESs obtained from clamped nuclei electronic structure calculations and is the usual nuclear kinetic energy operator defined above. Finding approximations for is the remaining problem before a solution of the Schrödinger equations can be attempted. Much of the current research in quantum chemistry is devoted to this determination. Once has been found and the coupled equations have been solved, the final vibronic wave function in the diabatic approximation is
## References |

This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Diabatic". A list of authors is available in Wikipedia. |