Born-Oppenheimer approximation

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The Born-Oppenheimer approximation (BO) is ubiquitous in quantum chemical calculations of molecular wavefunctions. It consists of two steps.

In the first step the nuclear kinetic energy is neglected,[1] that is, the corresponding operator T_\mathrm{n}\, is subtracted from the total molecular Hamiltonian. In the remaining electronic Hamiltonian H_\mathrm{e}\, the nuclear positions enter as parameters. The electron-nucleus interactions are not removed and the electrons still "feel" the Coulomb potential of the nuclei clamped at certain positions in space. (This first step of the BO approximation is therefore often referred to as the clamped nuclei approximation.)

The electronic Schrödinger equation

H_\mathrm{e} \; \chi(\mathbf{r}) = E_\mathrm{e} \; \chi(\mathbf{r})

is solved (out of necessity approximately) with a fixed nuclear geometry as input. The quantity \mathbf{r} stands for all electronic coordinates. Obviously, the electronic energy eigenvalue E_\mathrm{e}\, depends on the chosen positions \mathbf{R} of the nuclei. Varying these positions \mathbf{R} in small steps and repeatedly solving the electronic Schrödinger equation, one obtains E_\mathrm{e}\, as a function of \mathbf{R}. This is the potential energy surface (PES): E_\mathrm{e}(\mathbf{R}). Because this procedure of recomputing the electronic wave functions as a function of an infinitesimally changing nuclear geometry is reminiscent of the conditions for the adiabatic theorem, this manner of obtaining a PES is often referred to as the adiabatic approximation and the PES itself is called an adiabatic surface.[2]

In the second step of the BO approximation the nuclear kinetic energy T_\mathrm{n}\, (containing partial derivatives with respect to the components of \mathbf{R}) is reintroduced and the Schrödinger equation for the nuclear motion[3]

\left[ T_\mathrm{n} + E_\mathrm{e}(\mathbf{R})\right] \phi(\mathbf{R}) = E \phi(\mathbf{R})

is solved. This second step of the BO approximation is far from trivial and involves separation of vibrational and rotational motion. The real number E\, is the total energy of the molecule, including contributions from electrons, nuclear vibrations, and rotations.

[edit] Beyond the Born-Oppenheimer approximation

It will be discussed how the BO approximation may be improved by including vibronic coupling. At the same time this discussion will clarify under which conditions the BO approximation is applicable. We will recast the molecular Schrödinger equation into a set of coupled eigenvalue equations. 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: E_0(\mathbf{R}) \ll E_1(\mathbf{R}) \ll E_2(\mathbf{R}), \cdots for all \mathbf{R}.

We start from the exact non-relativistic, time-independent molecular Hamiltonian:

H= H_\mathrm{e} + T_\mathrm{n} \,

with

H_\mathrm{e}= -\sum_{i}{\frac{1}{2}\nabla_i^2}- \sum_{i,A}{\frac{Z_A}{r_{iA}}} + \frac{1}{2}\sum_{i>j}{\frac{1}{r_{ij}}}+ \frac{1}{2}\sum_{A > B}{\frac{Z_A Z_B}{R_{AB}}} \quad\mathrm{and}\quad T_\mathrm{n}=-\sum_{A}{\frac{1}{2M_A}\nabla_A^2}.

The position vectors \mathbf{r}\equiv \{\mathbf{r}_i\} of the electrons and the position vectors \mathbf{R}\equiv \{\mathbf{R}_A = (R_{Ax},\,R_{Ay},\,R_{Az})\} of the nuclei are with respect to a Cartesian molecule-fixed frame. Distances between particles are written as r_{iA} \equiv |\mathbf{r}_i - \mathbf{R}_A| (distance between electron i and nucleus A) and similar definitions hold for r_{ij}\; and R_{AB}\,. We assume that the molecule is in a homogeneous (no external force) and isotropic (no external torque) space. The only interactions are the Coulomb interactions between the electrons and nuclei. The Hamiltonian is expressed in atomic units, so that we do not see Planck's constant, the dielectric constant of the vacuum, electronic charge, or electronic mass in this formula. The only constants explicitly entering the formula are ZA and MA—the atomic number and charge of nucleus A.

It is useful to introduce the total nuclear momentum and to rewrite the nuclear kinetic energy operator as follows:

T_\mathrm{n} = \sum_{A} \sum_{\alpha=x,y,z} \frac{P_{A\alpha} P_{A\alpha}}{2M_A} \quad\mathrm{with}\quad P_{A\alpha} = -i \partial /\partial R_{A\alpha}.

Suppose we have K electronic eigenfunctions \chi_k (\mathbf{r}; \mathbf{R}) of H_\mathrm{e}\,, that is, we have solved

H_\mathrm{e}\;\chi_k (\mathbf{r};\mathbf{R}) = E_k(\mathbf{R})\;\chi_k (\mathbf{r};\mathbf{R}) \quad\mathrm{for}\quad k=1,\ldots, K.

The electronic wave functions \chi_k\, will be taken to be real, which is possible when there are no magnetic or spin interactions. The parametric dependence of the functions \chi_k\, on the nuclear coordinates is indicated by the symbol after the semicolon. This indicates that, although \chi_k\, is a real-valued function of \mathbf{r}, its functional form depends on \mathbf{R}. For example, in the molecular-orbital-linear-combination-of-atomic-orbital (LCAO-MO) approximation, \chi_k\, is an 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 \mathbf{R}, the LCAO coefficients obtain different values and we see corresponding changes in the functional form of the MO \chi_k\,. We will assume that the parametric dependence is continuous and differentiable, so that it is meaningful to consider

P_{A\alpha}\chi_k (\mathbf{r};\mathbf{R}) = - i \frac{\partial\chi_k (\mathbf{r};\mathbf{R})}{\partial R_{A\alpha}} \quad \mathrm{for}\quad \alpha=x,y,z,

which in general will not be zero.

The total wave function \Psi(\mathbf{R},\mathbf{r}) is expanded in terms of \chi_k (\mathbf{r}; \mathbf{R}):

\Psi(\mathbf{R}, \mathbf{r}) = \sum_{k=1}^K \chi_k(\mathbf{r};\mathbf{R}) \phi_k(\mathbf{R}) ,

with

\langle\,\chi_{k'}(\mathbf{r};\mathbf{R})\,|\, \chi_k(\mathbf{r};\mathbf{R})\rangle_{(\mathbf{r})} = \delta_{k' k}

and where the subscript (\mathbf{r}) indicates that the integration, implied by the bra-ket notation, is over electronic coordinates only. By definition, the matrix with general element

\big(\mathbb{H}_\mathrm{e}(\mathbf{R})\big)_{k'k} \equiv \langle \chi_{k'}(\mathbf{r};\mathbf{R}) | H_\mathrm{e} | \chi_k(\mathbf{r};\mathbf{R})\rangle_{(\mathbf{r})} = \delta_{k'k} E_k(\mathbf{R})

is diagonal. After multiplication by \chi_{k'}(\mathbf{r};\mathbf{R}) and integration over the electronic coordinates \mathbf{r} the total Schrödinger equation

H\;\Psi(\mathbf{R},\mathbf{r}) = E \; \Psi(\mathbf{R},\mathbf{r})

is turned into a set of K coupled eigenvalue equations depending on nuclear coordinates only

\left[ \mathbb{H}_\mathrm{n}(\mathbf{R}) + \mathbb{H}_\mathrm{e}(\mathbf{R}) \right] \; \boldsymbol{\phi}(\mathbf{R}) = E\; \boldsymbol{\phi}(\mathbf{R}).

The column vector \boldsymbol{\phi}(\mathbf{R}) has elements \phi_k(\mathbf{R}),\; k=1,\ldots,K. The matrix \mathbb{H}_\mathrm{e}(\mathbf{R}) is diagonal and the nuclear Hamilton matrix is non-diagonal with the following off-diagonal (vibronic coupling) terms,

\big(\mathbb{H}_\mathrm{n}(\mathbf{R})\big)_{k'k} = \langle\chi_{k'}(\mathbf{r};\mathbf{R}) | T_\mathrm{n}|\chi_k(\mathbf{r};\mathbf{R})\rangle_{(\mathbf{r})}.

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 transformation is applied, which moves the nuclear kinetic energy terms to the diagonal and creates coupling terms between the adiabatic PESs on the off-diagonal.

If we could neglect the off-diagonal elements the equations would uncouple and simplify drastically. In order to show when this neglect is justified, we suppres the coordinates in the notation and write, by applying the Leibniz rule for differentation, the matrix elements of Tn as

\mathrm{H_n}(\mathbf{R})_{k'k}\equiv \big(\mathbb{H}_\mathrm{n}(\mathbf{R})\big)_{k'k} = \delta_{k'k} T_{\textrm{n}} + \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})}

The diagonal (k' = k) matrix elements \langle\chi_{k}|\big(P_{A\alpha}\chi_k\big)\rangle_{(\mathbf{r})} of the operator P_{A\alpha}\, vanish, because this operator is Hermitian and purely imaginary. The off-diagonal matrix elements satisfy

\langle\chi_{k'}|\big(P_{A\alpha}\chi_k\big)\rangle_{(\mathbf{r})} = \frac{\langle\chi_{k'} |\big[P_{A\alpha}, H_\mathrm{e}\big] | \chi_k\rangle_{(\mathbf{r})}} {E_{k}(\mathbf{R})- E_{k'}(\mathbf{R})}.

The matrix element in the numerator is

\langle\chi_{k'} |\big[P_{A\alpha}, H_\mathrm{e}\big] | \chi_k\rangle_{(\mathbf{r})} = iZ_A\sum_i \;\langle\chi_{k'}|\frac{(\mathbf{r}_{iA})_\alpha}{r_{iA}^3}|\chi_k\rangle_{(\mathbf{r})} \;\;\mathrm{with}\;\; \mathbf{r}_{iA} \equiv \mathbf{r}_i - \mathbf{R}_A .

The matrix element of the one-electron operator appearing on the right hand side is finite. Let us assume that there is no electronic degeneracy and that E_k \ne E_{k'}\,. When the two surfaces come close, {E_{k}(\mathbf{R})\approx E_{k'}(\mathbf{R})}, 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 P^{A}_\alpha is effectively zero. The third term on the right hand side of the expression for the matrix element of Tn can be written as the matrix of P^{A}_\alpha 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,

\left[ T_\mathrm{n} +E_k(\mathbf{R})\right] \; \phi_k(\mathbf{R}) = E \phi_k(\mathbf{R}) \quad\mathrm{for}\quad k=1,\ldots, K,

which are the normal second-step of the BO equations discussed above.

[edit] Footnotes

  1. ^ This step is often justified by stating that "the heavy nuclei move more slowly than the light elecrons." Classically this statement makes only sense if one additionally assumes that the momentum p = mv of electrons and nuclei is of the same order of magnitude. Quantum mechanically it is not unreasonable to assume that the momenta of the electrons and nuclei in a molecule are comparable in magnitude (recall that the corresponding operators do not contain mass and think of the molecule as a box containing the electrons and nuclei and see particle in a box). Since the kinetic energy is proportional to the square of the absolute value of the momentum and inversely proportional to the mass of a particle, it follows that, indeed, the kinetic energy of the nuclei in a molecule is usually much smaller than the kinetic energy of the electrons, the mass ratio being on the order of 104).
  2. ^ It is assumed, in accordance with the adiabatic theorem, that the same electronic state (for instance the electronic ground state) is obtained upon small changes of the nuclear geometry. The method would give a non-continuous PES if electronic state-switching would occur.
  3. ^ This equation is time-independent and stationary wavefunctions for the nuclei are obtained, nevertheless it is traditional to use the word "motion" in this context, although classically motion implies time-dependence.


[edit] See also

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