Born-Oppenheimer approximation

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In quantum chemistry, the computation of the energy and wave function of an average-size molecule is a formidable task that is alleviated by the Born-Oppenheimer (BO) approximation. For instance the benzene molecule consists of 12 nuclei and 42 electrons. The time-independent Schrödinger equation, which must be solved to obtain the energy and molecular wave function of this molecule, is a partial differential eigenvalue equation in 162 variables—the spatial coordinates of the electrons and the nuclei. The BO approximation makes it possible to compute the wave function in two less formidable, consecutive, steps. This approximation was proposed in the early days of quantum mechanics by Born and Oppenheimer (1927) and is still indispensable in quantum chemistry.

In the first step of the BO approximation the electronic Schrödinger equation is solved, yielding a wave function depending on electrons only. For benzene this wave function depends on 126 electronic coordinates. During this solution the nuclei are fixed in a certain configuration, very often the equilibrium configuration. If the effects of the quantum mechanical nuclear motion are to be studied, for instance because a vibrational spectrum is required, this electronic computation must be repeated for many different nuclear configurations. The set of electronic energies thus computed becomes a function of the nuclear coordinates. In the second step of the BO approximation this function serves as a potential in a Schrödinger equation containing only the nuclei—for benzene an equation in 36 variables.

The success of the BO approximation is due to the high ratio between nuclear and electronic masses. The approximation is an important tool of quantum chemistry, without it only the lightest molecule, H₂, could be handled; all computations of molecular wave functions for larger molecules make use of it. Even in the cases where the BO approximation breaks down, it is used as a point of departure for the computations.

The electronic energies, constituting the nuclear potential, consist of kinetic energies, interelectronic repulsions and electron-nuclear attractions. In a handwaving manner it is sometimes suggested that the nuclear potential is some averaged electron-nuclear attraction. This interpretation of the Born-Oppenheimer approximation is wrong and misleading.

Short description

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

In the first step the nuclear kinetic energy is neglected, that is, the corresponding operator T_n is subtracted from the total molecular Hamiltonian. In the remaining electronic Hamiltonian H_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 r stands for all electronic coordinates. Obviously, the electronic energy eigenvalue E_e depends on the chosen positions R of the nuclei. Varying these positions R in small steps and repeatedly solving the electronic Schrödinger equation, one obtains E_e as a function of R. This is the potential energy surface (PES): E_e(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.

In the second step of the BO approximation the nuclear kinetic energy T_n (containing partial derivatives with respect to the components of R) is reintroduced and the Schrödinger equation for the nuclear motion

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

is solved. This second step of the BO approximation involves separation of vibrational, translational, and rotational motions. This can be achieved by application of the Eckart conditions. The eigenvalue E is the total energy of the molecule, including contributions from electrons, nuclear vibrations, and overall rotation and translation of the molecule.

Derivation of the Born-Oppenheimer approximation

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: $E_{0}(\mathbf {R} )\ll E_{1}(\mathbf {R} )\ll E_{2}(\mathbf {R} ),\cdots$ for all $\mathbf {R}$.

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

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

with

$H_{\mathrm {e} }=-\sum _{i}{{\frac {1}{2}}\nabla _{i}^{2}}-\sum _{i,A}{\frac {Z_{A}}{r_{iA}}}+\sum _{i>j}{\frac {1}{r_{ij}}}+\sum _{A>B}{\frac {Z_{A}Z_{B}}{R_{AB}}}\quad \mathrm {and} \quad T_{\mathrm {n} }=-\sum _{A}{{\frac {1}{2M_{A}}}\nabla _{A}^{2}}.$

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 inertial 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 Z_A and M_A—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:

$T_{\mathrm {n} }=\sum _{A}\sum _{\alpha =x,y,z}{\frac {P_{A\alpha }P_{A\alpha }}{2M_{A}}}\quad \mathrm {with} \quad P_{A\alpha }=-i{\partial \over \partial R_{A\alpha }}.$

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-orbitals (LCAO-MO) approximation, $\chi _{k}\,$ is a 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 the real function $\chi _{k'}(\mathbf {r} ;\mathbf {R} )$ from the left 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 wave function 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 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 for differentiation, the matrix elements of $T_{\textrm {n}}$ 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. 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 $P_{\alpha }^{A}$ is effectively zero. The third term on the right hand side of the expression for the matrix element of T_n (the Born-Oppenheimer diagonal correction) can approximately be written as the matrix of $P_{\alpha }^{A}$ 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.

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 approximation.

Historical note

The Born-Oppenheimer approximation is named after M. Born and R. Oppenheimer who wrote a paper [Annalen der Physik, vol. 84, pp. 457-484 (1927)] entitled: Zur Quantentheorie der Molekeln (On the Quantum Theory of Molecules). This paper describes the separation of electronic motion, nuclear vibrations, and molecular rotation. Somebody who expects to find in this paper the BO approximation—as it is explained above and in most modern textboooks—will be in for a surprise. The reason being that the presentation of the BO approximation is well hidden in Taylor expansions (in terms of internal and external nuclear coordinates) of (i) electronic wave functions, (ii) potential energy surfaces and (iii) nuclear kinetic energy terms. Internal coordinates are the relative positions of the nuclei in the molecular equilibrium and their displacements (vibrations) from equilibrium. External coordinates are the position of the center of mass and the orientation of the molecule. The Taylor expansions complicate the theory and make the derivations very hard to follow. Moreover, knowing that the proper separation of vibrations and rotations was not achieved in this paper, but only 8 years later [by C. Eckart, Physical Review, vol. 46, pp. 383-387 (1935)] (see Eckart conditions), one is not very much motivated to invest much effort into understanding the work by Born and Oppenheimer, however famous it may be. Although the article still collects many citations each year, it is safe to say that it is not read anymore (except perhaps by historians of science).

External links

Resources related to the Born-Oppenheimer approximation:

• The original article (in German)
• Translation by H. Hettema
• Translation by S. M. Blinder