Diabatic

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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.

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 subtlety. 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

$\mathbf{r}$ $\chi_1(\mathbf{r};\mathbf{R})\,$

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 differentiation, the matrix elements of T_n are (where we suppress coordinates for clarity reasons):

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

${(\mathbf{r})}$ " src="http://upload.wikimedia.org/math/0/6/5/065022998b886ce3a2b7885cb4a28387.png" /> 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)

$\chi_{1}\,$

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

$\gamma(\mathbf{R})$

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

$\varphi_k$

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

$\gamma(\mathbf{R})$

$\big(P_{A\alpha}\gamma(\mathbf{R})\big)+ \langle\chi_2|\big(P_{A\alpha} \chi_1\big) \rangle_{(\mathbf{r})} = 0$ $\varphi_1$ $\gamma(\mathbf{R})$

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

$F_{A\alpha}(\mathbf{R})$

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

$\gamma(\mathbf{R})$

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

$E_{1}(\mathbf{R})$

$\Psi(\mathbf{r},\mathbf{R}) = \varphi_1(\mathbf{r};\mathbf{R})\tilde\Phi_1(\mathbf{R})+ \varphi_2(\mathbf{r};\mathbf{R})\tilde\Phi_2(\mathbf{R}).$

References

^ F. T. Smith, Physical Review, vol. 179, p. 111 (1969)
^ W. Lichten, Physical Review, vol. 131. p.229 (1963)