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One of the guiding principles in modern chemical dynamics and spectroscopy is that the motion of the nuclei in a molecule is slow compared to that of its electrons. This is justified by the large disparity between the mass of an electron and the typical mass of a nucleus and leads to the Born-Oppenheimer approximation and the idea that the structure and dynamics of a chemical species are largely determined by nuclear motion on potential energy surfaces. The
potential energy surface A potential energy surface (PES) describes the energy of a system, especially a collection of atoms, in terms of certain parameters, normally the positions of the atoms. The surface might define the energy as a function of one or more coordinates; ...
s are obtained within the adiabatic or
Born–Oppenheimer approximation In quantum chemistry and molecular physics, the Born–Oppenheimer (BO) approximation is the best-known mathematical approximation in molecular dynamics. Specifically, it is the assumption that the wave functions of atomic nuclei and elect ...
. This corresponds to a representation of the molecular
wave function A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements mad ...
where the variables corresponding to the
molecular geometry Molecular geometry is the three-dimensional arrangement of the atoms that constitute a molecule. It includes the general shape of the molecule as well as bond lengths, bond angles, torsional angles and any other geometrical parameters that determ ...
and the electronic
degrees of freedom Degrees of freedom (often abbreviated df or DOF) refers to the number of independent variables or parameters of a thermodynamic system. In various scientific fields, the word "freedom" is used to describe the limits to which physical movement or ...
are separated. The non separable terms are due to the nuclear kinetic energy terms in the
molecular Hamiltonian In atomic, molecular, and optical physics and quantum chemistry, the molecular Hamiltonian is the Hamiltonian operator representing the energy of the electrons and nuclei in a molecule. This operator and the associated Schrödinger equation p ...
and are said to couple the
potential energy surface A potential energy surface (PES) describes the energy of a system, especially a collection of atoms, in terms of certain parameters, normally the positions of the atoms. The surface might define the energy as a function of one or more coordinates; ...
s. In the neighbourhood of an
avoided crossing In quantum physics and quantum chemistry, an avoided crossing (sometimes called intended crossing, ''non-crossing'' or anticrossing) is the phenomenon where two eigenvalues of an Hermitian matrix representing a quantum observable and depending on ...
or
conical intersection In quantum chemistry, a conical intersection of two or more potential energy surfaces is the set of molecular geometry points where the potential energy surfaces are degenerate (intersect) and the non-adiabatic couplings between these states are ...
, these terms cannot be neglected. One therefore usually performs one
unitary transformation In mathematics, a unitary transformation is a transformation that preserves the inner product: the inner product of two vectors before the transformation is equal to their inner product after the transformation. Formal definition More precisely, ...
from the adiabatic representation to the so-called diabatic representation in which the nuclear kinetic energy operator is
diagonal In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Greek δ ...
. In this representation, the coupling is due to the electronic energy and is a scalar quantity that is significantly easier to estimate numerically. In the diabatic representation, the potential energy surfaces are smoother, so that low order
Taylor series In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor serie ...
expansions of the surface capture much of the complexity of the original system. However 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 In quantum chemistry and molecular physics, the Born–Oppenheimer (BO) approximation is the best-known mathematical approximation in molecular dynamics. Specifically, it is the assumption that the wave functions of atomic nuclei and elect ...
breaks down, 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 expanding 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 element In geometry, a diagonal is a line segment joining two vertices of a polygon or polyhedron, when those vertices are not on the same edge. Informally, any sloping line is called diagonal. The word ''diagonal'' derives from the ancient Greek δΠ...
s 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, while \mathbf indicates dependence on nuclear coordinates. Thus, we assume E_1(\mathbf) \approx E_2(\mathbf) with corresponding orthonormal electronic eigenstates \chi_1(\mathbf;\mathbf)\, and \chi_2(\mathbf;\mathbf)\,. 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 = \sum_ \sum_ \frac \quad\mathrm\quad P_ = -i \nabla_ \equiv -i \frac. (
Atomic units The Hartree atomic units are a system of natural units of measurement which is especially convenient for atomic physics and computational chemistry calculations. They are named after the physicist Douglas Hartree. By definition, the following f ...
are used here). By applying the Leibniz rule for differentiation, the matrix elements of T_ are (where we suppress coordinates for clarity reasons): : \mathrm(\mathbf)_ \equiv \langle \chi_ , T_n , \chi_k\rangle_ = \delta_ T_ + \sum_\frac \langle\chi_, \big(P_\chi_k\big)\rangle_ P_ + \langle\chi_, \big(T_\mathrm\chi_k\big)\rangle_. The subscript indicates that the integration inside the bracket is over electronic coordinates only. Let us further assume that all off-diagonal matrix elements \mathrm(\mathbf)_ = \mathrm(\mathbf)_ may be neglected except for ''k = 1'' and ''p = 2''. Upon making the expansion : \Psi(\mathbf,\mathbf) = \chi_1(\mathbf;\mathbf)\Phi_1(\mathbf)+ \chi_2(\mathbf;\mathbf)\Phi_2(\mathbf), the coupled Schrödinger equations for the nuclear part take the form (see the article
Born–Oppenheimer approximation In quantum chemistry and molecular physics, the Born–Oppenheimer (BO) approximation is the best-known mathematical approximation in molecular dynamics. Specifically, it is the assumption that the wave functions of atomic nuclei and elect ...
) \begin E_1(\mathbf)+ \mathrm(\mathbf)_&\mathrm(\mathbf)_\\ \mathrm(\mathbf)_&E_2(\mathbf)+\mathrm(\mathbf)_\\ \end \boldsymbol(\mathbf) = E \,\boldsymbol(\mathbf) \quad \mathrm\quad \boldsymbol(\mathbf)\equiv \begin \Phi_1(\mathbf) \\ \Phi_2(\mathbf) \\ \end . 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_\, and \chi_\, : \begin \varphi_1(\mathbf;\mathbf) \\ \varphi_2(\mathbf;\mathbf) \\ \end = \begin \cos\gamma(\mathbf) & \sin\gamma(\mathbf) \\ -\sin\gamma(\mathbf) & \cos\gamma(\mathbf) \\ \end \begin \chi_1(\mathbf;\mathbf) \\ \chi_2(\mathbf;\mathbf) \\ \end where \gamma(\mathbf) is the diabatic angle. Transformation of the matrix of nuclear momentum \langle\chi_, \big(P_\chi_k\big)\rangle_ for k', k =1,2 gives for ''diagonal'' matrix elements : \langle , \big( P_ \varphi_k\big) \rangle_ = 0 \quad\textrm\quad k=1, \, 2. These elements are zero because \varphi_k is real and P_\, is Hermitian and pure-imaginary. The off-diagonal elements of the momentum operator satisfy, : \langle , \big( P_\varphi_1\big) \rangle_ = \big(P_\gamma(\mathbf) \big) + \langle\chi_2, \big(P_ \chi_1\big)\rangle_. Assume that a diabatic angle \gamma(\mathbf) exists, such that to a good approximation : \big(P_\gamma(\mathbf)\big)+ \langle\chi_2, \big(P_ \chi_1\big) \rangle_ = 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) can be rewritten in the following more familiar form: : F_(\mathbf) = - \nabla_ V(\mathbf) \qquad\mathrm\;\; V(\mathbf) \equiv \gamma(\mathbf)\;\;\mathrm\;\;F_(\mathbf)\equiv \langle\chi_2, \big(iP_ \chi_1\big) \rangle_ . 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_(\mathbf) is
irrotational In vector calculus, a conservative vector field is a vector field that is the gradient of some function. A conservative vector field has the property that its line integral is path independent; the choice of any path between two points does not c ...
, : \nabla_ F_(\mathbf) - \nabla_ F_(\mathbf) = 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) 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_ , T_n , \varphi_k \rangle_ = \delta_ T_n. On the basis of the diabatic states the nuclear motion problem takes the following ''generalized Born–Oppenheimer'' form \begin T_\mathrm+ \frac & 0 \\ 0 & T_\mathrm + \frac \end \tilde(\mathbf) + \tfrac \begin \cos2\gamma & \sin2\gamma \\ \sin2\gamma & -\cos2\gamma \end \tilde(\mathbf) = E \tilde(\mathbf). It is important to note that the off-diagonal elements depend on the diabatic angle and electronic energies only. The surfaces E_(\mathbf) and E_(\mathbf) are adiabatic PESs obtained from clamped nuclei electronic structure calculations and T_\mathrm\, is the usual nuclear kinetic energy operator defined above. Finding approximations for \gamma(\mathbf) 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) has been found and the coupled equations have been solved, the final vibronic wave function in the diabatic approximation is : \Psi(\mathbf,\mathbf) = \varphi_1(\mathbf;\mathbf)\tilde\Phi_1(\mathbf)+ \varphi_2(\mathbf;\mathbf)\tilde\Phi_2(\mathbf).


Adiabatic-to-diabatic transformation

Here, in contrast to previous treatments, the non-Abelian case is considered. Felix Smith in his article considers the adiabatic-to-diabatic transformation (ADT) for a multi-state system but a single coordinate, \mathrm. In Diabatic, the ADT is defined for a system of two coordinates \mathrm and \mathrm, but it is restricted to two states. Such a system is defined as Abelian and the ADT matrix is expressed in terms of an angle, \gamma (see Comment below), known also as the ADT angle. In the present treatment a system is assumed that is made up of ''M'' (> 2) states defined for an ''N''-dimensional configuration space, where ''N'' = 2 or ''N'' > 2. Such a system is defined as non-Abelian. To discuss the non-Abelian case the equation for the just mentioned ADT angle, \gamma (see Diabatic), is replaced by an equation for the MxM, ADT matrix, \mathbf: : \nabla \mathbf where \mathbf is the force-matrix operator, introduced in Diabatic, also known as the Non-Adiabatic Coupling Transformation (NACT) matrix: : \ \mathbf_ = \langle \chi_j\mid \nabla \chi_k\rangle;\qquad j,k=1,2,\ldots,M Here \nabla is the ''N''-dimensional (nuclear) grad-operator: : \nabla = \left\ and , \chi_k( \mathbf ) \rangle;\ k= 1,M,are the electronic adiabatic eigenfunctions which depend explicitly on the electronic coordinates \mathbf and parametrically on the nuclear coordinates \mathbf . To derive the matrix \mathbf one has to solve the above given first order differential equation along a specified contour \Gamma . This solution is then applied to form the diabatic potential matrix \mathbf : : \mathbf = \mathbf^\mathbf where \mathbf_j ; ''j'' = 1, ''M'' are the Born–Oppenheimer adiabatic potentials. In order for \mathbf to be single-valued in configuration space, \mathbf has to be analytic and in order for \mathbf to be analytic (excluding the pathological points), the components of the vector matrix, \mathbf , have to satisfy the following equation: : G_= \frac - \frac - \left \mathbf_ , \mathbf_ \right= 0. where \mathbf is a
tensor field In mathematics and physics, a tensor field assigns a tensor to each point of a mathematical space (typically a Euclidean space or manifold). Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis ...
. This equation is known as the non-Abelian form of the
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Equation. A solution of the ADT matrix \mathbf along the contour \Gamma can be shown to be of the form: : \mathbf\left( \mathbf, \Gamma \right) = \hat \exp : \left( - \int_\mathbf^\mathbf\mathbf \left( \mathbf\mid \Gamma \right) \cdot d\mathbf\right) (see also
Geometric phase In classical and quantum mechanics, geometric phase is a phase difference acquired over the course of a cycle, when a system is subjected to cyclic adiabatic processes, which results from the geometrical properties of the parameter space of the Ha ...
). Here \hat is an ordering operator, the dot stands for a
scalar product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebra ...
and \mathbf and \mathbf are two points on \Gamma. A different type of solutions is based on quasi-Euler angles according to which any \mathbf-matrix can be expressed as a product of Euler matrices. For instance in case of a tri-state system this matrix can be presented as a product of three such matrices, \mathbf_(\gamma_) (''i'' < ''j'' = 2, 3) where e.g. \mathbf_ (\gamma_) is of the form: : \mathbf_ = \begin \cos \gamma_ & 0 & \sin\gamma_\\0 & 1 & 0\\ -\sin\gamma_ & 0 & \cos\gamma_ \end The product \mathbf = \mathbf_ \mathbf_ \mathbf_ which can be written in any order, is substituted in Eq. (1) to yield three first order differential equations for the three _-angles where two of these equations are coupled and the third stands on its own. Thus, assuming: \mathbf = \mathbf_ \mathbf_ \mathbf_ the two coupled equations for _ and _ are: : \nabla \gamma_ = -F_ - \tan_ (- F_ \cos\gamma_ + F_\sin \gamma_) : \nabla \gamma_ = - (F_ \cos\gamma_ + F_ \sin\gamma_) whereas the third equation (for \gamma_) becomes an ordinary (line) integral: : \nabla \gamma_ =(\cos\gamma_)^(- F_\cos\gamma_ + F_\sin\gamma_) expressed solely in terms of \gamma_ and \gamma_. Similarly, in case of a four-state system \mathbf is presented as a product of six 4 x 4 Euler matrices (for the six quasi-Euler angles) and the relevant six differential equations form one set of three coupled equations, whereas the other three become, as before, ordinary line integrals.


A comment concerning the two-state (Abelian) case

Since the treatment of the two-state case as presented in Diabatic raised numerous doubts we consider it here as a special case of the Non-Abelian case just discussed. For this purpose we assume the 2 × 2 ADT matrix \mathrm to be of the form: : \mathrm = \begin\cos\gamma & - \sin\gamma\\\sin\gamma & \cos\gamma \end Substituting this matrix in the above given first order differential equation (for \mathrm) we get, following a few algebraic rearrangements, that the angle \gamma fulfills the corresponding first order differential equation as well as the subsequent line integral: : \nabla \mathbf \cdot \Longrightarrow \cdot\gamma\left( \mathbf\mid \Gamma \right) = -\int_\mathbf^\mathbf\mathbf_ \left( \mathbf\mid \Gamma \right) \cdot d\mathbf where \mathrm_{12} is the relevant NACT matrix element, the dot stands for a scalar product and \Gamma is a chosen contour in configuration space (usually a planar one) along which the integration is performed. The line integral yields meaningful results if and only if the corresponding (previously derived)
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-equation is zero for every point in the region of interest (ignoring the pathological points).


References

Quantum chemistry