Editing Born–Oppenheimer approximation (section)

== Derivation ==
It will be discussed how the BO approximation may be derived and under which conditions it is applicable. At the same time we will show how the BO approximation may be improved by including [[vibronic coupling]]. To that end the second step of the BO approximation is generalized to a set of coupled eigenvalue equations depending on nuclear coordinates only. Off-diagonal elements in these equations are shown to be nuclear kinetic energy terms.

It will be shown that the BO approximation can be trusted whenever the PESs, obtained from the solution of the electronic Schrödinger equation, are well separated:
: <math> E_0(\mathbf{R}) \ll E_1(\mathbf{R}) \ll E_2(\mathbf{R}) \ll \cdots \text{ for all }\mathbf{R}</math>.

We start from the ''exact'' non-relativistic, time-independent molecular Hamiltonian:
: <math>
H = H_\text{e} + T_\text{n}
</math>
with
: <math>
H_\text{e} =
-\sum_{i}{\frac{1}{2}\nabla_i^2} -
\sum_{i,A}{\frac{Z_A}{r_{iA}}} + \sum_{i>j}{\frac{1}{r_{ij}}} + \sum_{B > A}{\frac{Z_A Z_B}{R_{AB}}}
\quad\text{and}\quad T_\text{n} = -\sum_{A}{\frac{1}{2M_A}\nabla_A^2}.
</math>

The position vectors <math>\mathbf{r} \equiv \{\mathbf{r}_i\}</math> of the electrons and the position vectors <math>\mathbf{R} \equiv \{\mathbf{R}_A = (R_{Ax}, R_{Ay}, R_{Az})\}</math> of the nuclei are with respect to a Cartesian [[inertial frame]]. Distances between particles are written as <math>r_{iA} \equiv |\mathbf{r}_i - \mathbf{R}_A|</math> (distance between electron ''i'' and nucleus ''A'') and similar definitions hold for <math>r_{ij}</math> and <math> R_{AB}</math>.

We assume that the molecule is in a homogeneous (no external force) and isotropic (no external torque) space. The only interactions are the two-body Coulomb interactions among the electrons and nuclei. The Hamiltonian is expressed in [[atomic units]], so that we do not see the Planck constant, the dielectric constant of the vacuum, electronic charge, or electronic mass in this formula. The only constants explicitly entering the formula are ''Z<sub>A</sub>'' and ''M<sub>A</sub>'' – the atomic number and mass of nucleus ''A''.

It is useful to introduce the total nuclear momentum and to rewrite the nuclear kinetic energy operator as follows:
: <math> T_\text{n} = \sum_{A} \sum_{\alpha=x,y,z} \frac{P_{A\alpha} P_{A\alpha}}{2M_A}
\quad\text{with}\quad
P_{A\alpha} = -i \frac{\partial}{\partial R_{A\alpha}}. </math>

Suppose we have ''K'' electronic eigenfunctions <math>\chi_k (\mathbf{r}; \mathbf{R})</math> of <math>H_\text{e}</math>; that is, we have solved
: <math>
H_\text{e} \chi_k(\mathbf{r}; \mathbf{R}) = E_k(\mathbf{R}) \chi_k(\mathbf{r}; \mathbf{R}) \quad\text{for}\quad k = 1, \ldots, K.
</math>

The electronic wave functions <math>\chi_k</math> will be taken to be real, which is possible when there are no magnetic or spin interactions. The ''parametric dependence'' of the functions <math>\chi_k</math> on the nuclear coordinates is indicated by the symbol after the semicolon. This indicates that, although <math>\chi_k</math> is a real-valued function of <math>\mathbf{r}</math>, its functional form depends on <math>\mathbf{R}</math>.

For example, in the molecular-orbital-linear-combination-of-atomic-orbitals [[Molecular orbital#Qualitative discussion|(LCAO-MO)]] approximation, <math>\chi_k</math> is a molecular orbital (MO) given as a linear expansion of atomic orbitals (AOs). An AO depends visibly on the coordinates of an electron, but the nuclear coordinates are not explicit in the MO. However, upon change of geometry, i.e., change of <math>\mathbf{R}</math>, the LCAO coefficients obtain different values and we see corresponding changes in the functional form of the MO <math>\chi_k</math>.

We will assume that the parametric dependence is continuous and differentiable, so that it is meaningful to consider
: <math>
P_{A\alpha}\chi_k(\mathbf{r}; \mathbf{R}) = -i \frac{\partial\chi_k(\mathbf{r}; \mathbf{R})}{\partial R_{A\alpha}}
\quad \text{for}\quad
\alpha = x,y,z,
</math>
which in general will not be zero.

The total wave function <math>\Psi(\mathbf{R}, \mathbf{r})</math> is expanded in terms of <math>\chi_k(\mathbf{r}; \mathbf{R})</math>:
: <math>
\Psi(\mathbf{R}, \mathbf{r}) = \sum_{k=1}^K \chi_k(\mathbf{r}; \mathbf{R}) \phi_k(\mathbf{R}),
</math>
with
: <math>
\langle \chi_{k'}(\mathbf{r}; \mathbf{R}) | \chi_k(\mathbf{r}; \mathbf{R}) \rangle_{(\mathbf{r})} = \delta_{k' k},
</math>
and where the subscript <math>(\mathbf{r})</math> indicates that the integration, implied by the [[bra–ket notation]], is over electronic coordinates only. By definition, the matrix with general element
: <math> \big(\mathbb{H}_\text{e}(\mathbf{R})\big)_{k'k} \equiv \langle \chi_{k'}(\mathbf{r}; \mathbf{R})
        | H_\text{e} |
        \chi_k(\mathbf{r}; \mathbf{R}) \rangle_{(\mathbf{r})} = \delta_{k'k} E_k(\mathbf{R})
</math>
is diagonal. After multiplication by the real function <math>\chi_{k'}(\mathbf{r}; \mathbf{R})</math> from the left and integration over the electronic coordinates <math>\mathbf{r}</math> the total Schrödinger equation
: <math>
H \Psi(\mathbf{R}, \mathbf{r}) = E \Psi(\mathbf{R}, \mathbf{r})
</math>
is turned into a set of ''K'' coupled eigenvalue equations depending on nuclear coordinates only
: <math> [\mathbb{H}_\text{n}(\mathbf{R}) + \mathbb{H}_\text{e}(\mathbf{R})] \boldsymbol{\phi}(\mathbf{R}) =
 E \boldsymbol{\phi}(\mathbf{R}).
</math>

The column vector <math>\boldsymbol{\phi}(\mathbf{R})</math> has elements <math>\phi_k(\mathbf{R}),\ k = 1, \ldots, K</math>. The matrix <math>\mathbb{H}_\text{e}(\mathbf{R})</math> is diagonal, and the nuclear Hamilton matrix is non-diagonal; its off-diagonal (''vibronic coupling'') terms <math> \big(\mathbb{H}_\text{n}(\mathbf{R})\big)_{k'k}</math> are further discussed below. The vibronic coupling in this approach is through nuclear kinetic energy terms.

Solution of these coupled equations gives an approximation for energy and wavefunction that goes beyond the Born–Oppenheimer approximation.
Unfortunately, the off-diagonal kinetic energy terms are usually difficult to handle. This is why often a [[Diabatic representation|diabatic]] transformation is applied, which retains part of the nuclear kinetic energy terms on the diagonal, removes the kinetic energy terms from the off-diagonal and creates coupling terms between the adiabatic PESs on the off-diagonal.

If we can neglect the off-diagonal elements the equations will uncouple and simplify drastically. In order to show when this neglect is justified, we suppress the coordinates in the notation and write, by applying the [[Leibniz rule (generalized product rule)|Leibniz rule]] for differentiation, the matrix elements of <math>T_\text{n}</math> as
: <math>
T_\text{n}(\mathbf{R})_{k'k} \equiv
 \big(\mathbb{H}_\text{n}(\mathbf{R})\big)_{k'k}
 = \delta_{k'k} T_\text{n}
 + \sum_{A,\alpha}\frac{1}{M_A} \langle\chi_{k'}|P_{A\alpha}|\chi_k\rangle_{(\mathbf{r})} P_{A\alpha} + \langle\chi_{k'}|T_\text{n}|\chi_k\rangle_{(\mathbf{r})}.
</math>

The diagonal (<math>k' = k</math>) matrix elements <math>\langle\chi_{k}|P_{A\alpha}|\chi_k\rangle_{(\mathbf{r})}</math> of the operator <math>P_{A\alpha}</math> vanish, because we assume time-reversal invariant, so <math>\chi_k</math> can be chosen to be always real. The off-diagonal matrix elements satisfy
: <math>
\langle\chi_{k'}|P_{A\alpha}|\chi_k\rangle_{(\mathbf{r})} =
 \frac{\langle\chi_{k'}| [P_{A\alpha}, H_\text{e}] |\chi_k\rangle_{(\mathbf{r})}}
      {E_{k}(\mathbf{R}) - E_{k'}(\mathbf{R})}.
</math>
The matrix element in the numerator is
: <math>
\langle\chi_{k'}| [P_{A\alpha}, H_\mathrm{e}] |\chi_k\rangle_{(\mathbf{r})} =
iZ_A\sum_i \left\langle\chi_{k'}\left|\frac{(\mathbf{r}_{iA})_\alpha}{r_{iA}^3}\right|\chi_k\right\rangle_{(\mathbf{r})}
\quad\text{with}\quad
\mathbf{r}_{iA} \equiv \mathbf{r}_i - \mathbf{R}_A.
</math>
The matrix element of the one-electron operator appearing on the right side is finite.

When the two surfaces come close, <math>E_{k}(\mathbf{R}) \approx E_{k'}(\mathbf{R})</math>, the nuclear momentum coupling term becomes large and is no longer negligible. This is the case where the BO approximation breaks down, and a coupled set of nuclear motion equations must be considered instead of the one equation appearing in the second step of the BO approximation.

Conversely, if all surfaces are well separated, all off-diagonal terms can be neglected, and hence the whole matrix of <math>P^A_\alpha</math> is effectively zero. The third term on the right side of the expression for the matrix element of ''T''<sub>n</sub> (the ''Born–Oppenheimer diagonal correction'') can approximately be written as the matrix of <math>P^A_\alpha</math> squared and, accordingly, is then negligible also. Only the first (diagonal) kinetic energy term in this equation survives in the case of well separated surfaces, and a diagonal, uncoupled, set of nuclear motion equations results:
: <math>
[T_\text{n} + E_k(\mathbf{R})] \phi_k(\mathbf{R}) = E \phi_k(\mathbf{R})
\quad\text{for}\quad
k = 1, \ldots, K,
</math>
which are the normal second step of the BO equations discussed above.

We reiterate that when two or more potential energy surfaces approach each other, or even cross, the Born–Oppenheimer approximation breaks down, and one must fall back on the coupled equations. Usually one invokes then the [[Diabatic representation|diabatic]] approximation.