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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 diabatic representation in which the nuclear kinetic energy operator is diagonal. In this representation, the coupling is due to the electronic energy and is a scalar quantity which is much more easy to estimate numerically.

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In 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 pseudo-diabatic potentials, but generally the term is not used unless it is necessary to highlight this subtletly. Hence, pseudo-diabatic potentials are synonymous with diabatic potentials.

Applicability

The 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 beyond the Born-Oppenheimer approximation. This is often the terminology used to refer to the study of nonadiabatic systems.

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 ADT.

Diabatic transformation of two electronic surfaces

In 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 $\mathbf{r}$ , while $\mathbf{R}$ indicates dependence on nuclear coordinates. Thus, we assume $E_1(\mathbf{R}) \approx E_2(\mathbf{R})$ with corresponding orthonormal electronic eigenstates $\chi_1(\mathbf{r};\mathbf{R})\,$ and $\chi_2(\mathbf{r};\mathbf{R})\,$ . 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 A with mass M_A,

$T_\mathrm{n} = \sum_{A} \sum_{\alpha=x,y,z} \frac{P_{A\alpha} P_{A\alpha}}{2M_A} \quad\mathrm{with}\quad P_{A\alpha} = -i \nabla_{A\alpha} \equiv -i \frac{\partial\quad}{\partial R_{A\alpha}}.$

(Atomic units are used here). By applying the Leibniz rule for differentation, the matrix elements of $T n$ are (where we suppress coordinates for clarity reasons):

$\mathrm{T_n}(\mathbf{R})_{k'k} \equiv \langle \chi_{k'} | T_n | \chi_k\rangle_{(\mathbf{r})} = \delta_{k'k} T_{\textrm{n}} + \sum_{A,\alpha}\frac{1}{M_A} \langle\chi_{k'}|\big(P_{A\alpha}\chi_k\big)\rangle_{(\mathbf{r})} P_{A\alpha} + \langle\chi_{k'}|\big(T_\mathrm{n}\chi_k\big)\rangle_{(\mathbf{r})}.$

The subscript ${(\mathbf{r})}$ indicates that the integration inside the braket is over electronic coordinates only. Let us further assume that all off-diagonal matrix elements $\mathrm{T_n}(\mathbf{R})_{kp} = \mathrm{T_n}(\mathbf{R})_{pk}$ may be neglected except for k = 1 and p = 2. Upon making the expansion

$\Psi(\mathbf{r},\mathbf{R}) = \chi_1(\mathbf{r};\mathbf{R})\Phi_1(\mathbf{R})+ \chi_2(\mathbf{r};\mathbf{R})\Phi_2(\mathbf{R}),$

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 diabatic transformation of the adiabatic states $\chi_{1}\,$ and $\chi_{2}\,$

$\begin{pmatrix} \varphi_1(\mathbf{r};\mathbf{R}) \\ \varphi_2(\mathbf{r};\mathbf{R}) \\ \end{pmatrix} = \begin{pmatrix} \cos\gamma(\mathbf{R}) & \sin\gamma(\mathbf{R}) \\ -\sin\gamma(\mathbf{R}) & \cos\gamma(\mathbf{R}) \\ \end{pmatrix} \begin{pmatrix} \chi_1(\mathbf{r};\mathbf{R}) \\ \chi_2(\mathbf{r};\mathbf{R}) \\ \end{pmatrix}$

where $\gamma(\mathbf{R})$ is the diabatic angle. Transformation of the matrix of nuclear momentum $\langle\chi_{k'}|\big(P_{A\alpha}\chi_k\big)\rangle_{(\mathbf{r})}$ for $k', k = 1,2$ gives for diagonal matrix elements

$\langle{\varphi_k} |\big( P_{A\alpha} \varphi_k\big) \rangle_{(\mathbf{r})} = 0 \quad\textrm{for}\quad k=1, \, 2.$

These elements are zero because $\varphi_k$ is real and $P_{A\alpha}\,$ is Hermitian and pure-imaginary. The off-diagonal elements of the momentum operator satisfy,

$\langle{\varphi_2} |\big( P_{A\alpha}\varphi_1\big) \rangle_{(\mathbf{r})} = \big(P_{A\alpha}\gamma(\mathbf{R}) \big) + \langle\chi_2| \big(P_{A\alpha} \chi_1\big)\rangle_{(\mathbf{r})}.$

Assume that a diabatic angle $\gamma(\mathbf{R})$ exists, such that to a good approximation

$\big(P_{A\alpha}\gamma(\mathbf{R})\big)+ \langle\chi_2|\big(P_{A\alpha} \chi_1\big) \rangle_{(\mathbf{r})} = 0$

i.e., $\varphi_1$ and $\varphi_2$ diagonalize the 2 x 2 matrix of the nuclear momentum. By the definition of Smith^[1] $\varphi_1$ and $\varphi_2$ are diabatic states. (Smith was the first to define this concept; earlier the term diabatic was used somewhat loosely by Lichten^[2]).

By a small change of notation these differential equations for $\gamma(\mathbf{R})$ can be rewritten in the following more familiar form:

$F_{A\alpha}(\mathbf{R}) = - \nabla_{A\alpha} V(\mathbf{R}) \qquad\mathrm{with}\;\; V(\mathbf{R}) \equiv \gamma(\mathbf{R})\;\;\mathrm{and}\;\;F_{A\alpha}(\mathbf{R})\equiv \langle\chi_2|\big(iP_{A\alpha} \chi_1\big) \rangle_{(\mathbf{r})} .$

It is well-known that the differential equations have a solution (i.e., the "potential" V exists) if and only if the vector field ("force") $F_{A\alpha}(\mathbf{R})$ is irrotational,

$\nabla_{A\alpha} F_{B\beta}(\mathbf{R}) - \nabla_{B \beta} F_{A\alpha}(\mathbf{R}) = 0.$

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 $\gamma(\mathbf{R})$ leading to pseudo diabatic states.

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

$\langle \varphi_{k'} | T_n | \varphi_k \rangle_{(\mathbf{r})} = \delta_{k'k} T_n.$

On the basis of the diabatic states the nuclear motion problem takes the following generalized Born-Oppenheimer form

It is important to note that the off-diagonal elements depend on the diabatic angle and electronic energies only. The surfaces $E_{1}(\mathbf{R})$ and $E_{2}(\mathbf{R})$ are adiabatic PESs obtained from clamped nuclei electronic structure calculations and $T_\mathrm{n}\,$ is the usual nuclear kinetic energy operator defined above. Finding approximations for $\gamma(\mathbf{R})$ 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 $\gamma(\mathbf{R})$ has been found and the coupled equations have been solved, the final vibronic wave function in the diabatic approximation is