Diabatic transformation

In quantum chemistry, the solution of a set of coupled nuclear motion Schrödinger equations  can be simplified by a diabatic transformation. Such a set of coupled equations, describing the (vibrational) motions of nuclei in a molecule, arises when the Born-Oppenheimer approximation  breaks down. The term diabatic was coined in the early 1960s. Around that time shortcomings of the Born-Oppenheimer approximation (also known as the adiabatic approximation) became apparent and improvements of the adiabatic approximation were put forward under the name diabatic approximation. Linguistically the term is unfortunate because there is no connection whatsoever with the Greek word diabasis (going through).

Break-down of Born-Oppenheimer approximation

 * ''See the article Born-Oppenheimer approximation for more details.

The Born-Oppenheimer approximation, designed for the quantum mechanical computation of molecular properties, consists of two steps. In the first step the nuclei of a molecule are fixed in a certain constellation and the nuclear kinetic energies are dropped from the problem, i.e., the nuclei are assumed to be at rest. One or more electronic Schrödinger equations are solved yielding the corresponding (usually the lowest few) electronic energies. Changing sufficiently often the nuclear constellation of the molecule under study and  solving the electronic Schrödinger equations over and over again, gives the electronic energies as functions of the nuclear coordinates. These functions are known as potential energy surfaces.

The second step of the original Born-Oppenheimer approximation consists of the solution of single (uncoupled) Schrödinger equations for the nuclei. In each of these equation the nuclear kinetic energy is reintroduced and one of the single potential energy surfaces (obtained from the first step) serves as potential. This simple approximation breaks down when the potential energy surfaces approach—or maybe even intersect—each other. In that case the nuclear motion equations, which are coupled by nuclear kinetic energy terms, may no longer be taken to be uncoupled, that is, the off-diagonal nuclear kinetic energy terms coupling the equations may no longer be neglected.

In short, when the Born-Oppenheimer approximation breaks down because of the presence of close lying potential energy surfaces, the solution of the nuclear motion problem requires the solution of a coupled set of Schrödinger equations. A diabatic transformation of this set of equations has the purpose of making them easier to solve. It is a linear (usually unitary) transformation applied to the coupled equations that minimizes the off-diagonal nuclear kinetic energy terms. However, at the same time the adiabatic potential energy surfaces that enter the equations (those that approach or intersect each other) are combined linearly to a new set of potentials, the diabatic potentials. In other words, a diabatic transformation is a unitary transformation from the adiabatic representation to the diabatic representation, in which the nuclear kinetic energy operator is a diagonal. In this representation, the coupling between the Schrödinger equations is now due to diabatic potential energy surfaces that are significantly easier to estimate numerically than the nuclear kinetic energy terms that appeared before transformation.

The diabatic potential energy surfaces are smooth, so that low order Taylor series expansions of the surfaces may be applied and capture much of the complexity of the original system. Unfortunately, in general a strictly diabatic transformation does not exist, it is not possible to transform the nuclear kinetic energy rigorously to zero. Hence, diabatic potentials generated from mixing linearly multiple electronic energy surfaces are generally not exact. These surfaces are sometimes called pseudo-diabatic potentials, but generally the term is not used unless it is necessary to highlight this subtlety. Hence, usually pseudo-diabatic potentials are synonymous with diabatic potentials.

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 (do not come close to PES 1 or PES 2); 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 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 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)

$$ \begin{pmatrix} E_1(\mathbf{R})+ \mathrm{T_n}(\mathbf{R})_{11}&\mathrm{T_n}(\mathbf{R})_{12}\\ \mathrm{T_n}(\mathbf{R})_{21}&E_2(\mathbf{R})+\mathrm{T_n}(\mathbf{R})_{22}\\ \end{pmatrix} \boldsymbol{\Phi}(\mathbf{R}) = E \,\boldsymbol{\Phi}(\mathbf{R}) \quad \mathrm{with}\quad \boldsymbol{\Phi}(\mathbf{R})\equiv \begin{pmatrix} \Phi_1(\mathbf{R}) \\ \Phi_2(\mathbf{R}) \\ \end{pmatrix}. $$

In order to remove the problematic off-diagonal kinetic energy terms, 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 $$\varphi_1$$ 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}). $$