In order to introduce the diabatic transformation, assume 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, 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
MA, : 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_{\textrm{n}} are (where coordinates are suppressed for clarity): : \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 bracket 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) \begin{pmatrix} E_1(\mathbf{R})+ \mathrm{T_n}(\mathbf{R})_{11}&\mathrm{T_n}(\mathbf{R})_{12}\\ \mathrm{T_n}(\mathbf{R})_{21}&E_2(\mathbf{R})+\mathrm{T_n}(\mathbf{R})_{22}\\ \end{pmatrix} \boldsymbol{\Phi}(\mathbf{R}) = E \,\boldsymbol{\Phi}(\mathbf{R}) \quad \mathrm{with}\quad \boldsymbol{\Phi}(\mathbf{R})\equiv \begin{pmatrix} \Phi_1(\mathbf{R}) \\ \Phi_2(\mathbf{R}) \\ \end{pmatrix} . In order to remove the problematic off-diagonal kinetic energy terms, 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 \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). 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 \begin{pmatrix} T_\mathrm{n}+ \frac{E_{1}(\mathbf{R})+E_{2}(\mathbf{R})}{2} & 0 \\ 0 & T_\mathrm{n} + \frac{E_{1}(\mathbf{R})+E_{2}(\mathbf{R})}{2} \end{pmatrix} \tilde{\boldsymbol{\Phi}}(\mathbf{R}) + \tfrac{E_{2}(\mathbf{R})-E_{1}(\mathbf{R})}{2} \begin{pmatrix} \cos2\gamma & \sin2\gamma \\ \sin2\gamma & -\cos2\gamma \end{pmatrix} \tilde{\boldsymbol{\Phi}}(\mathbf{R}) = E \tilde{\boldsymbol{\Phi}}(\mathbf{R}). 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 : \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}). \tilde{\boldsymbol{\Phi}}(\mathbf{R}) \equiv \begin{pmatrix} \tilde\Phi_1(\mathbf{R}) \\ \tilde\Phi_2(\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} \Phi_1(\mathbf{R}) \\ \Phi_2(\mathbf{R}) \\ \end{pmatrix}. --> ==Adiabatic-to-diabatic transformation==