Diabatic transformation: Difference between revisions

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imported>Paul Wormer
(New page: 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...)
 
imported>Paul Wormer
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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.
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 ==
== Mathematical formulation ==
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 <math>\mathbf{r}</math>, while  <math>\mathbf{R}</math> indicates  dependence on  nuclear coordinates.  Thus, we assume <math>E_1(\mathbf{R}) \approx
In order to introduce mathematically the diabatic transformation we assume now, for the sake of argument, that only two adiabatic potential energy surfaces (PES), ''E''<sub>1</sub> and ''E''<sub>2</sub>, approach each other and that all other surfaces are well separated (do not come close to ''E''<sub>1</sub> or ''E''<sub>2</sub>); the argument can be generalized to more surfaces.  
E_2(\mathbf{R})</math> with corresponding orthonormal electronic eigenstates
 
<math>\chi_1(\mathbf{r};\mathbf{R})\,</math> and <math>\chi_2(\mathbf{r};\mathbf{R})\,</math>.  
Let the collection of electronic coordinates be indicated by '''r''', while  '''R''' indicates  dependence on  nuclear coordinates.  Thus, we assume {{nowrap|''E''<sub>1</sub>('''R''') &asymp; ''E''<sub>2</sub>('''R''')}} with corresponding orthonormal electronic eigenstates &chi;<sub>1</sub>('''r''';'''R''') and &chi;<sub>2</sub>('''r''';'''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.
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''<sub>A</sub>,
The nuclear kinetic energy is a sum over nuclei ''A'' with mass ''M''<sub>A</sub>,
:<math> T_\mathrm{n} = \sum_{A} \sum_{\alpha=x,y,z} \frac{P_{A\alpha} P_{A\alpha}}{2M_A}  
:<math>  
T_\mathrm{n} = \sum_{A} \sum_{\alpha=x,y,z} \frac{P_{A\alpha} P_{A\alpha}}{2M_A}  
\quad\mathrm{with}\quad  
\quad\mathrm{with}\quad  
P_{A\alpha} = -i \nabla_{A\alpha} \equiv -i \frac{\partial\quad}{\partial R_{A\alpha}}. </math>
P_{A\alpha} = -i \nabla_{A\alpha} = -i \frac{\partial\quad}{\partial R_{A\alpha}}.  
([[Atomic units]] are used here).
</math>
By applying the [[Leibniz rule (generalized product rule)|Leibniz rule]] for differentiation, the matrix elements of  <math>T_{\textrm{n}}</math> are (where we suppress coordinates for clarity reasons):
([[Atomic units]] are used here and &nabla;<sub>''A''&alpha;</sub> is the a component of the [[gradient]] operator, short-hand for a differential.)
By applying the [[Leibniz rule (generalized product rule)|Leibniz rule]] for differentiation, the matrix elements of  ''T''<sub>n</sub> are (where coordinates are suppressed for clarity reasons):
:<math>  
:<math>  
\mathrm{T_n}(\mathbf{R})_{k'k}
\mathrm{T_n}(\mathbf{R})_{k'k}
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         + \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})}.
         + \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})}.
</math>
</math>
The subscript <math>{(\mathbf{r})}</math> indicates that the integration inside the braket is  
The subscript '''r''' indicates that the integration inside the braket is over electronic coordinates only. The round brackets indicate the range of differentiation.
over electronic coordinates only.
 
Let us further assume
Assume that the off-diagonal matrix elements
that all off-diagonal matrix elements
<math>\mathrm{T_n}(\mathbf{R})_{12} = \mathrm{T_n}(\mathbf{R})_{21}
<math>\mathrm{T_n}(\mathbf{R})_{kp} = \mathrm{T_n}(\mathbf{R})_{pk}
</math> may not be neglected (in agreement with the assumption that only two surfaces approach each other,  off-diagonal matrix elements with ''k'', ''k''&prime; > 2 are negligible, so that only a set of two coupled equations has to be considered). Upon making the expansion
</math> may be neglected except for ''k = 1'' and
''p = 2''. Upon making the expansion
:<math>
:<math>
\Psi(\mathbf{r},\mathbf{R}) = \chi_1(\mathbf{r};\mathbf{R})\Phi_1(\mathbf{R})+
\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}),
\chi_2(\mathbf{r};\mathbf{R})\Phi_2(\mathbf{R}),
</math>
</math>
the coupled Schrödinger equations for the nuclear part take the form (see the article [[Born-Oppenheimer approximation]])
the two coupled nuclear Schrödinger equations take the form (see the article [[Born-Oppenheimer approximation]])


<math>
:<math>
\begin{pmatrix}
\begin{pmatrix}
E_1(\mathbf{R})+ \mathrm{T_n}(\mathbf{R})_{11}&\mathrm{T_n}(\mathbf{R})_{12}\\
E_1(\mathbf{R})+ \mathrm{T_n}(\mathbf{R})_{11}&\mathrm{T_n}(\mathbf{R})_{12}\\
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\Phi_1(\mathbf{R}) \\
\Phi_1(\mathbf{R}) \\
\Phi_2(\mathbf{R}) \\
\Phi_2(\mathbf{R}) \\
\end{pmatrix} .
\end{pmatrix} , \qquad\qquad(1)
</math>
</math>
where ''E'' is the ''total'' (electronic plus nuclear motion) energy of the molecule.


In order to remove the problematic off-diagonal kinetic energy terms, we
In order to remove the problematic off-diagonal kinetic energy terms,  
define two new orthonormal states by a ''diabatic transformation'' of the ''adiabatic states'' <math>\chi_{1}\,</math> and <math>\chi_{2}\,</math>
two new orthonormal states are defined by a ''diabatic transformation'' of the ''adiabatic states'' &chi;<sub>1</sub>('''r''';'''R''') and &chi;<sub>2</sub>('''r''';'''R''')
:<math>  
:<math>  
\begin{pmatrix}
\begin{pmatrix}
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\varphi_2(\mathbf{r};\mathbf{R}) \\
\varphi_2(\mathbf{r};\mathbf{R}) \\
\end{pmatrix}
\end{pmatrix}
=
\equiv
\begin{pmatrix}
\begin{pmatrix}
  \cos\gamma(\mathbf{R}) & \sin\gamma(\mathbf{R}) \\
  \cos\gamma(\mathbf{R}) & \sin\gamma(\mathbf{R}) \\
Line 75: Line 75:
\end{pmatrix}
\end{pmatrix}
</math>
</math>
where  <math>\gamma(\mathbf{R})</math> is the ''diabatic angle''. Transformation of the matrix of nuclear momentum <math>\langle\chi_{k'}|\big(P_{A\alpha}\chi_k\big)\rangle_{(\mathbf{r})}</math> for <math>k', k =1,2</math> gives for ''diagonal'' matrix elements
where  &gamma;('''R''') is the ''diabatic angle''. Transformation of the matrix of nuclear momentum <math>\langle\chi_{k'}|\big(P_{A\alpha}\chi_k\big)\rangle_{(\mathbf{r})}</math> for ''k''&prime;, ''k'' =1,2 gives for diagonal matrix elements:
:<math> \langle{\varphi_k} |\big( P_{A\alpha} \varphi_k\big) \rangle_{(\mathbf{r})} = 0 \quad\textrm{for}\quad k=1, \, 2.
:<math>  
\langle{\varphi_k} |\big( P_{A\alpha} \varphi_k\big) \rangle_{(\mathbf{r})} = 0 \quad\textrm{for}\quad k=1, \, 2.
</math>
</math>
These  elements are zero because <math>\varphi_k</math> is real
These  elements are zero because <math>\varphi_k</math> is real
and <math>P_{A\alpha}\,</math> is Hermitian and pure-imaginary.
and <math>P_{A\alpha}\,</math> is Hermitian and pure-imaginary.
The off-diagonal elements of the momentum operator satisfy,
The off-diagonal elements of the momentum operator satisfy,
:<math>  \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})}.
:<math>   
\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})}.
</math>
</math>


Assume that a diabatic angle <math>\gamma(\mathbf{R})</math> exists, such that to a good approximation
Assume that a diabatic angle &gamma;('''R''') exists, such that to a good approximation the right-hand side of the last equation vanishes,
:<math> \big(P_{A\alpha}\gamma(\mathbf{R})\big)+ \langle\chi_2|\big(P_{A\alpha} \chi_1\big) \rangle_{(\mathbf{r})} = 0 </math>
:<math>
\big(P_{A\alpha}\gamma(\mathbf{R})\big)+ \langle\chi_2|\big(P_{A\alpha} \chi_1\big) \rangle_{(\mathbf{r})} = 0  
</math>


i.e., <math>\varphi_1</math> and  <math>\varphi_2</math> diagonalize the 2 x 2 matrix of the nuclear momentum. By  the definition of
i.e., <math>\varphi_1</math> and  <math>\varphi_2</math> diagonalize the 2 x 2 matrix of the nuclear momentum. By  the definition of Felix Smith <math>\varphi_1</math> and  <math>\varphi_2</math> are ''diabatic states''.<ref>F. T. Smith, ''Diabatic and Adiabatic Representations for Atomic Collision Problems'', Physical Review, vol. '''179''', p. 111&ndash;123 (1969) [http://dx.doi.org/10.1103/PhysRev.179.111 DOI]]</ref> (Smith was the first to define this concept; earlier the term ''diabatic'' was used somewhat loosely by Lichten.<ref>W. Lichten, ''Resonant Charge Exchange in Atomic Collisions'', Physical Review, vol. '''131''',  p. 229&ndash;238 (1963) [http://dx.doi.org/10.1103/PhysRev.131.229 DOI]</ref>)
Smith<ref>F. T. Smith, Physical Review, vol. 179, p. 111 (1969)</ref> <math>\varphi_1</math> and  <math>\varphi_2</math> are '''diabatic states'''. (Smith was the first to define this concept; earlier the term ''diabatic'' was used somewhat loosely by Lichten<ref>W. Lichten, Physical Review, vol. 131. p.229 (1963)</ref>).


By a small change of notation these differential equations for <math>\gamma(\mathbf{R})</math> can be rewritten in the following more familiar form:
By a small change of notation these differential equations for &gamma;('''R''') can be rewritten in the following more familiar form reminiscent of Newton's equations,
:<math>
:<math>
F_{A\alpha}(\mathbf{R}) = - \nabla_{A\alpha} V(\mathbf{R})  
\nabla_{A\alpha} V(\mathbf{R}) + F_{A\alpha}(\mathbf{R}) = 0 
\qquad\mathrm{with}\;\; V(\mathbf{R}) \equiv \gamma(\mathbf{R})\;\;\mathrm{and}\;\;F_{A\alpha}(\mathbf{R})\equiv
\quad\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})} .
\langle\chi_2|\big(iP_{A\alpha} \chi_1\big) \rangle_{(\mathbf{r})} .
</math>
</math>
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")
It is well-known that the differential equations have a solution (i.e., the "potential" &nbsp; ''V'' exists) if and only if the vector field ("force") &nbsp; ''F''<sub>''A''&alpha;</sub>('''R''')
<math>F_{A\alpha}(\mathbf{R})</math> is [[irrotational]],  
is [[irrotational]],  
:<math>
:<math>
\nabla_{A\alpha} F_{B\beta}(\mathbf{R}) - \nabla_{B \beta} F_{A\alpha}(\mathbf{R}) = 0.
\nabla_{A\alpha} F_{B\beta}(\mathbf{R}) - \nabla_{B \beta} F_{A\alpha}(\mathbf{R}) = 0.
Line 104: Line 107:
transformation rarely ever exists. It is common to use approximate functions <math>\gamma(\mathbf{R})</math> leading to ''pseudo diabatic states''.
transformation rarely ever exists. It is common to use approximate functions <math>\gamma(\mathbf{R})</math> 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
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,  
the assumption of "strict" (not pseudo) diabaticity,  
it can be  shown that
it can be  shown that
:<math>
:<math>
Line 111: Line 114:
</math>
</math>


On the basis of the diabatic states
On the basis of the diabatic states the nuclear motion problem [Eq. (1)] &nbsp;  takes the following form
the nuclear motion problem takes the following ''generalized Born-Oppenheimer'' form
:<math>
<math>
\left[
\begin{pmatrix}
\begin{pmatrix}
T_\mathrm{n}+
T_\mathrm{n}+
Line 120: Line 123:
     \frac{E_{1}(\mathbf{R})+E_{2}(\mathbf{R})}{2}
     \frac{E_{1}(\mathbf{R})+E_{2}(\mathbf{R})}{2}
\end{pmatrix}
\end{pmatrix}
\tilde{\boldsymbol{\Phi}}(\mathbf{R})
+
+
\tfrac{E_{2}(\mathbf{R})-E_{1}(\mathbf{R})}{2}
\frac{E_{2}(\mathbf{R})-E_{1}(\mathbf{R})}{2}
\begin{pmatrix}
\begin{pmatrix}
\cos2\gamma
\cos2\gamma(\mathbf{R})
  & \sin2\gamma \\
  & \sin2\gamma(\mathbf{R}) \\
\sin2\gamma &
\sin2\gamma(\mathbf{R}) &
-\cos2\gamma
-\cos2\gamma(\mathbf{R})
\end{pmatrix}
\end{pmatrix} \right]
\tilde{\boldsymbol{\Phi}}(\mathbf{R})
\tilde{\boldsymbol{\Phi}}(\mathbf{R})
= E \tilde{\boldsymbol{\Phi}}(\mathbf{R}).
= E \tilde{\boldsymbol{\Phi}}(\mathbf{R}).
</math>
</math>


It is important to note that the off-diagonal elements depend on the diabatic angle and electronic energies only. The surfaces  <math>E_{1}(\mathbf{R})</math> and <math>E_{2}(\mathbf{R})</math> are adiabatic PESs obtained from clamped nuclei electronic structure calculations and <math>T_\mathrm{n}\,</math> is the usual nuclear kinetic energy operator defined above.
It is important to note that the off-diagonal elements (that appear only in the second term on the left-hand side) depend on the diabatic angle and adiabatic electronic energies only. The adiabatic surfaces  ''E''<sub>1</sub>('''R''') and ''E''<sub>2</sub>('''R''') are PESs obtained from clamped nuclei electronic structure calculations and ''T''<sub>n</sub> is the usual nuclear kinetic energy operator defined above.
Finding approximations for <math>\gamma(\mathbf{R})</math> 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  <math>\gamma(\mathbf{R})</math> has been found and the coupled equations have been solved, the final vibronic wave function in the diabatic approximation is
 
Finding approximations for &gamma;('''R''') is the remaining problem before a solution of the coupled nuclear Schrödinger equations can be attempted. Much of the current research in quantum chemistry is devoted to this determination. Once  &gamma;('''R''') has been found and the coupled equations have been solved, the final vibronic (vibration—i.e., nuclear motion—plus electronic) wave function in the diabatic approximation is
:<math>
:<math>
\Psi(\mathbf{r},\mathbf{R}) = \varphi_1(\mathbf{r};\mathbf{R})\tilde\Phi_1(\mathbf{R})+
\Psi(\mathbf{r},\mathbf{R}) = \varphi_1(\mathbf{r};\mathbf{R})\tilde\Phi_1(\mathbf{R})+

Revision as of 08:12, 6 May 2010

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.

Mathematical formulation

In order to introduce mathematically the diabatic transformation we assume now, for the sake of argument, that only two adiabatic potential energy surfaces (PES), E1 and E2, approach each other and that all other surfaces are well separated (do not come close to E1 or E2); the argument can be generalized to more surfaces.

Let the collection of electronic coordinates be indicated by r, while R indicates dependence on nuclear coordinates. Thus, we assume E1(R) ≈ E2(R) with corresponding orthonormal electronic eigenstates χ1(r;R) and χ2(r;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,

(Atomic units are used here and ∇Aα is the a component of the gradient operator, short-hand for a differential.) By applying the Leibniz rule for differentiation, the matrix elements of Tn are (where coordinates are suppressed for clarity reasons):

The subscript r indicates that the integration inside the braket is over electronic coordinates only. The round brackets indicate the range of differentiation.

Assume that the off-diagonal matrix elements may not be neglected (in agreement with the assumption that only two surfaces approach each other, off-diagonal matrix elements with k, k′ > 2 are negligible, so that only a set of two coupled equations has to be considered). Upon making the expansion

the two coupled nuclear Schrödinger equations take the form (see the article Born-Oppenheimer approximation)

where E is the total (electronic plus nuclear motion) energy of the molecule.

In order to remove the problematic off-diagonal kinetic energy terms, two new orthonormal states are defined by a diabatic transformation of the adiabatic states χ1(r;R) and χ2(r;R)

where γ(R) is the diabatic angle. Transformation of the matrix of nuclear momentum for k′, k =1,2 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 γ(R) exists, such that to a good approximation the right-hand side of the last equation vanishes,

i.e., and diagonalize the 2 x 2 matrix of the nuclear momentum. By the definition of Felix Smith and are diabatic states.[1] (Smith was the first to define this concept; earlier the term diabatic was used somewhat loosely by Lichten.[2])

By a small change of notation these differential equations for γ(R) can be rewritten in the following more familiar form reminiscent of Newton's equations,

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")   FAα(R) 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" (not pseudo) diabaticity, it can be shown that

On the basis of the diabatic states the nuclear motion problem [Eq. (1)]   takes the following form

It is important to note that the off-diagonal elements (that appear only in the second term on the left-hand side) depend on the diabatic angle and adiabatic electronic energies only. The adiabatic surfaces E1(R) and E2(R) are PESs obtained from clamped nuclei electronic structure calculations and Tn is the usual nuclear kinetic energy operator defined above.

Finding approximations for γ(R) is the remaining problem before a solution of the coupled nuclear Schrödinger equations can be attempted. Much of the current research in quantum chemistry is devoted to this determination. Once γ(R) has been found and the coupled equations have been solved, the final vibronic (vibration—i.e., nuclear motion—plus electronic) wave function in the diabatic approximation is

References

  1. F. T. Smith, Diabatic and Adiabatic Representations for Atomic Collision Problems, Physical Review, vol. 179, p. 111–123 (1969) DOI]
  2. W. Lichten, Resonant Charge Exchange in Atomic Collisions, Physical Review, vol. 131, p. 229–238 (1963) DOI