A diabatic process is one in which heat transfer takes place, which is the opposite of an adiabatic process. In

quantum chemistryQuantum chemistry is a branch of chemistry whose primary focus is the application of quantum mechanics in physical models and experiments of chemical systems...

, the

potential energy surfaceA potential energy surface is generally used within the adiabatic or Born–Oppenheimer approximation in quantum mechanics and statistical mechanics to model chemical reactions and interactions in simple chemical and physical systems...

s are obtained within the adiabatic or

Born-Oppenheimer approximationIn quantum chemistry, the computation of the energy and wavefunction of an average-size molecule is a formidable task that is alleviated by the Born–Oppenheimer approximation, named after Max Born and J. Robert Oppenheimer. For instance the benzene molecule consists of 12 nuclei and 42...

. This corresponds to a representation of the molecular wave function where the variables corresponding to the

molecular geometryMolecular geometry or molecular structure is the three-dimensional arrangement of the atoms that constitute a molecule. It determines several properties of a substance including its reactivity, polarity, phase of matter, color, magnetism, and biological activity.- Molecular geometry determination...

and the electronic

degrees of freedomA degree of freedom is an independent physical parameter, often called a dimension, in the formal description of the state of a physical system...

are separated. The

non separable termsIn theoretical chemistry, the vibronic coupling terms, , are proportional to the interaction between electronic and nuclear motions of molecules. The term "vibronic" originates from the concatenation of the terms "vibrational" and "electronic"...

are due to the nuclear kinetic energy terms in the

molecular HamiltonianIn atomic, molecular, and optical physics as well as in quantum chemistry, molecular Hamiltonian is the name given to the Hamiltonian representing the energy of the electrons and nuclei in a molecule...

and are said to couple the

potential energy surfaceA potential energy surface is generally used within the adiabatic or Born–Oppenheimer approximation in quantum mechanics and statistical mechanics to model chemical reactions and interactions in simple chemical and physical systems...

s. In the neighbourhood of an

avoided crossing[Image:Avoided_crossing.png|thumb|right|300px|An avoided energy level crossing in a two level system subjected to an external magnetic field. Note the energies of the diabatic states, \scriptstyle...

or

conical intersectionIn quantum chemistry, a conical intersection of two potential energy surfaces of the same spatial and spin symmetries is the set of molecular geometry points where the two potential energy surfaces are degenerate . Conical intersections are ubiquitous in both trivial and non-trivial chemical...

, these terms cannot be neglected. One therefore usually performs one

unitary transformationIn mathematics, a unitary transformation may be informally defined as a transformation that respects the inner product: the inner product of two vectors before the transformation is equal to their inner product after the transformation....

from the adiabatic representation to the so-called

**diabatic representation** in which the nuclear kinetic energy operator is

diagonalIn linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero. The diagonal entries themselves may or may not be zero...

. In this representation, the coupling is due to the electronic energy and is a scalar quantity which is significantly easier to estimate numerically.

In the diabatic representation, the potential energy surfaces are smoother so that low order

Taylor seriesIn mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point....

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 approximationIn quantum chemistry, the computation of the energy and wavefunction of an average-size molecule is a formidable task that is alleviated by the Born–Oppenheimer approximation, named after Max Born and J. Robert Oppenheimer. For instance the benzene molecule consists of 12 nuclei and 42...

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

, while

indicates dependence on nuclear coordinates. Thus, we assume

with corresponding orthonormal electronic eigenstates

and

.

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},

(

Atomic unitsAtomic units form a system of natural units which is especially convenient for atomic physics calculations. There are two different kinds of atomic units, which one might name Hartree atomic units and Rydberg atomic units, which differ in the choice of the unit of mass and charge. This article...

are used here).

By applying the

Leibniz ruleIn calculus, the general Leibniz rule, named after Gottfried Leibniz, generalizes the product rule. It states that if f and g are n-times differentiable functions, then the nth derivative of the product fg is given by...

for differentiation, the matrix elements of

are (where we suppress coordinates for clarity reasons):

The subscript

indicates that the integration inside the braket is

over electronic coordinates only.

Let us further assume

that all off-diagonal matrix elements

may be neglected except for

*k = 1* and

*p = 2*. Upon making the expansion

the coupled Schrödinger equations for the nuclear part take the form (see the article

Born-Oppenheimer approximationIn quantum chemistry, the computation of the energy and wavefunction of an average-size molecule is a formidable task that is alleviated by the Born–Oppenheimer approximation, named after Max Born and J. Robert Oppenheimer. For instance the benzene molecule consists of 12 nuclei and 42...

)

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** and

where

is the

**diabatic angle**. Transformation of the matrix of nuclear momentum

for

gives for

*diagonal* matrix elements

These elements are zero because

is real

and

is Hermitian and pure-imaginary.

The off-diagonal elements of the momentum operator satisfy,

Assume that a diabatic angle

exists, such that to a good approximation

i.e.,

and

diagonalize the 2 x 2 matrix of the nuclear momentum. By the definition of

Smith

and

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

can be rewritten in the following more familiar form:

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")

is irrotational,

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

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

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

and

are adiabatic PESs obtained from clamped nuclei electronic structure calculations and

is the usual nuclear kinetic energy operator defined above.

Finding approximations for

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

has been found and the coupled equations have been solved, the final vibronic wave function in the diabatic approximation is