Born-Oppenheimer approximation

Rationale

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 indispensable in quantum chemistry and ubiquitous in large parts of computational physics.

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.

Short description

The Born-Oppenheimer approximation consists of two steps:

In the first step the nuclear kinetic energy is neglected,^[1] that is, the corresponding operator T_n is subtracted from the total molecular Hamiltonian. In the remaining electronic Hamiltonian H_e the nuclear positions still appear, as fixed parameters. The electron-nucleus attractions 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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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.^[2] 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.

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^[3]

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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.

Remark: Very often the BO approximation is presented in a simple, but wrong, way. The Hamiltonian is often erroneously assumed to be separable in a nuclear and electronic part, which implies that the total wave function can be written^[4] as a (tensor) product of a function depending on electrons only times a function depending on the nuclei:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Psi_\mathrm{total} = \psi_\mathrm{e} \times \psi_\mathrm{n} } .

This is incorrect, because, as was pointed out above, the Coulombic electron-nucleus attraction terms are not removed from the electronic Hamiltonian, so that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_\mathrm{total} \ne H_\mathrm{e} + H_\mathrm{n} \quad\hbox{and}\quad \Psi_\mathrm{total} \ne \psi_\mathrm{e} \times \psi_\mathrm{n}. }

The essential feature of the wave function factorization introduced below is that the electronic factor ψ(r; R) depends on the electron coordinates r and also on the nuclear ooordinates R.

Footnotes

↑ This step is often justified by stating that "the heavy nuclei move more slowly than the light electrons." Classically this statement makes sense only if one assumes in addition that the momentum p of electrons and nuclei is of the same order of magnitude. In that case m_nuc >> m_elec implies p²/(2m_nuc) << p²/(2m_elec). 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 p²/(2m) , 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).
↑ 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 discontinuity (jump) in the PES if electronic state-switching would occur.
↑ This equation is time-independent and stationary wave functions for the nuclei are obtained, nevertheless it is traditional to use the word "motion" in this context, although classically motion implies time-dependence.
↑ Wikipedia article retrieved 13 April 2009

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: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle E_0(\mathbf{R}) \ll E_1(\mathbf{R}) \ll E_2(\mathbf{R}), \cdots } for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \mathbf{R}} .

We start from the total (electrons and nuclei) exact non-relativistic, time-independent molecular Hamiltonian:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H= H_\mathrm{e} + T_\mathrm{n} \, }

with

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \mathbf{r}\equiv \{\mathbf{r}_i\}} of the electrons and the position vectors Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle r_{iA} \equiv |\mathbf{r}_i - \mathbf{R}_A|} (distance between electron i and nucleus A) and similar definitions hold for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle r_{ij}\;} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle 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 electric constant, 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 mass of nucleus A.

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \chi_k (\mathbf{r}; \mathbf{R})} of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle H_\mathrm{e}\,} , that is, we have solved

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \chi_k\,} on the nuclear coordinates is indicated by the symbol after the semicolon. This indicates that, although Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \chi_k\,} is a real-valued function of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \mathbf{r}} , its functional form depends on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \mathbf{R}} . For example, in the molecular-orbital-linear-combination-of-atomic-orbitals (LCAO-MO) approximation, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \mathbf{R}} , the LCAO coefficients obtain different values and we see corresponding changes in the functional form of the MO Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \chi_k\,} . We will assume that the parametric dependence is continuous and differentiable, so that it is meaningful to consider

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \Psi(\mathbf{R},\mathbf{r})} is expanded in terms of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \chi_k (\mathbf{r}; \mathbf{R})} :

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Psi(\mathbf{R}, \mathbf{r}) = \sum_{k=1}^K \chi_k(\mathbf{r};\mathbf{R}) \phi_k(\mathbf{R}) , }

with

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle\,\chi_{k'}(\mathbf{r};\mathbf{R})\,|\, \chi_k(\mathbf{r};\mathbf{R})\rangle_{(\mathbf{r})} = \delta_{k' k} }

and where the subscript Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle (\mathbf{r})} indicates that the integration, implied by the bra-ket notation, is over electronic coordinates only. By definition, the matrix with general element

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \chi_{k'}(\mathbf{r};\mathbf{R})} from the left and integration over the electronic coordinates Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \mathbf{r}\,} the total Schrödinger equation

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \boldsymbol{\phi}(\mathbf{R})} has elements Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \phi_k(\mathbf{R}),\; k=1,\ldots,K} . The matrix Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \mathbb{H}_\mathrm{e}(\mathbf{R})} is diagonal and the nuclear Hamilton matrix is non-diagonal with the following off-diagonal (vibronic coupling) terms,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle T_{\mathrm{n}}} as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle k'=k \,} ) matrix elements Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \langle\chi_{k}|\big(P_{A\alpha}\chi_k\big)\rangle_{(\mathbf{r})}} of the operator Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle P_{A\alpha}\,} vanish, because this operator is Hermitian and purely imaginary. The off-diagonal matrix elements satisfy

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle {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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle P^{A}_\alpha} 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 Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle 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.

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. A reader who expects to find in this paper the BO approximation—as it is explained above and in most modern textbooks—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 eight years later [by C. Eckart, Physical Review, vol. 46, pp. 383-387 (1935)] (see Eckart conditions), working chemists and molecular physicists are 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 maybe by historians of science.

External links

Resources related to the Born-Oppenheimer approximation:

[1] This step is often justified by stating that "the heavy nuclei move more slowly than the light electrons." Classically this statement makes sense only if one assumes in addition that the momentum p of electrons and nuclei is of the same order of magnitude. In that case m_nuc >> m_elec implies p²/(2m_nuc) << p²/(2m_elec). 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 p²/(2m) , 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 discontinuity (jump) in the PES if electronic state-switching would occur.

[3] This equation is time-independent and stationary wave functions for the nuclei are obtained, nevertheless it is traditional to use the word "motion" in this context, although classically motion implies time-dependence.

[4] Wikipedia article retrieved 13 April 2009

[1]

[2]

[3]

[4]

Born-Oppenheimer approximation

Contents

Rationale

Short description

Footnotes

Derivation of the Born-Oppenheimer approximation

Historical note

External links

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Born-Oppenheimer approximation

Rationale

Short description

Footnotes

Derivation of the Born-Oppenheimer approximation

Historical note

External links

Navigation menu

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