Diabatic
Encyclopedia
A diabatic process is one in which heat transfer takes place, which is the opposite of an adiabatic process. In quantum chemistry
, the potential energy surface
s 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 surface
s. 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 significantly easier 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.
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.
, 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 MA,
(Atomic units
are used here).
By applying the Leibniz rule
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 approximation
)
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
Quantum chemistry
Quantum 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 surface
Potential energy surface
A 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 approximation
Born-Oppenheimer approximation
In 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 geometry
Molecular geometry
Molecular 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 freedom
Degrees of freedom (physics and chemistry)
A 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 terms
Vibronic coupling
In 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 Hamiltonian
Molecular Hamiltonian
In 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 surface
Potential energy surface
A 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
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 intersection
Conical intersection
In 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 transformation
Unitary transformation
In 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 diagonal
Diagonal matrix
In 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 series
Taylor series
In 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 approximationBorn-Oppenheimer approximation
In 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 MA,
(Atomic units
Atomic units
Atomic 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 rule
Leibniz rule (generalized product rule)
In 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 isover electronic coordinates only.
Let us further assume
that all off-diagonal matrix elements
may be neglected except for k = 1 andp = 2. Upon making the expansion
the coupled Schrödinger equations for the nuclear part take the form (see the article Born-Oppenheimer approximation
Born-Oppenheimer approximation
In 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 elementsThese elements are zero because
is realand
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 approximationi.e.,
and diagonalize the 2 x 2 matrix of the nuclear momentum. By the definition ofSmith
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