The Born-Oppenheimer Approximation

C. David Sherrill
Department of Chemistry
University of California, Berkeley
September 1996

We may write the nonrelativistic Hamiltonian for a molecule as a sum of five terms:

$${\hat H} = - \frac{\hbar^2}{2m} \sum_{i} \nabla^2_i - \s... ... \frac{Z_A Z_B e^2}{R_{AB}} + \sum_{i>j} \frac{e^2}{r_{ij}},$$ (1)

where i, j refer to electrons and A, B refer to nuclei. In atomic units, this is just

$${\hat H} = - \frac{1}{2} \sum_{i} \nabla^2_i - \sum_{A} \... ..._{A>B} \frac{Z_A Z_B}{R_{AB}} + \sum_{i>j} \frac{1}{r_{ij}}.$$ (2)

The Schrödinger equation may be written more compactly as

$${\hat H} = {\hat T}_N(\mathbf{R}) + {\hat T}_e(\mathbf{r}) ... ...R}) + {\hat V}_{NN}(\mathbf{R}) + {\hat V}_{ee}(\mathbf{r}),$$ (3)

where R is the set of nuclear coordinates and r is the set of electronic coordinates. If spin-orbit effects are important, they can be added through a spin-orbit operator ${\hat H}_{so}$.

Unfortunately, the ${\hat V}_{eN}(\mathbf{r, R})$ term prevents us from separating ${\hat H}$ into nuclear and electronic parts, which would allow us to write the molecular wavefunction as a product of nuclear and electronic terms, $\Psi(\mathbf{r,R}) = \Psi(\mathbf{r}) \chi(\mathbf{R})$. We thus introduce the Born-Oppenheimer approximation, by which we conclude that this nuclear and electronic separation is approximately correct. The term ${\hat V}_{eN}(\mathbf{r, R})$ is large and cannot be neglected; however, we can make the R dependence parametric, so that the total wavefunction is given as $\Psi(\mathbf{r;R}) \chi(\mathbf{R})$. The Born-Oppenheimer approximation rests on the fact that the nuclei are much more massive than the electrons, which allows us to say that the nuclei are nearly fixed with respect to electron motion. We can fix R, the nuclear configuration, at some value R_a, and solve for the electronic wavefunction $\Psi(\mathbf{r; R_a})$, which depends only parametrically on R. If we do this for a range of R, we obtain the potential energy curve along which the nuclei move.

We now show the mathematical details. Initially, ${\hat T}_N(\mathbf{R})$ can be neglected since ${\hat T}_N$ is smaller than ${\hat T}_e$ by a factor of $M_A / \mu_e$, where $\mu_e$ is the reduced mass of an electron. Thus for a fixed nuclear configuration, we have

$${\hat H}_{el} = {\hat T}_e(\mathbf{r}) + {\hat V}_{eN}(\m... ...}) + {\hat V}_{NN}(\mathbf{R}) + {\hat V}_{ee}(\mathbf{r})$$ (4)

such that

$${\hat H}_{el} \Psi(\mathbf{r; R}) = E_{el} \Psi(\mathbf{r; R})$$ (5)

This is the ``clamped-nuclei'' Schrödinger equation. Quite frequently ${\hat V}_{NN}(\mathbf{R})$ is neglected in the above equation, which is justified since in this case R is just a parameter so that ${\hat V}_{NN}(\mathbf{R})$ is just a constant and shifts the eigenvalues only by some constant amount. Leaving ${\hat V}_{NN}(\mathbf{R})$ out of the electronic Schrödinger equation leads to a similar equation,

$${\hat H}_e = {\hat T}_e(\mathbf{r}) + {\hat V}_{eN}(\mathbf{r;R}) + {\hat V}_{ee}(\mathbf{r})$$ (6)

$${\hat H}_e \Psi(\mathbf{r;R}) = E_e \Psi_e(\mathbf{r;R})$$ (7)

For the purposes of these notes, we will assume that ${\hat V}_{NN}(\mathbf{R})$ is included in the electronic Hamiltonian. Additionally, if spin-orbit effects are important, then these can be included at each nuclear configuration according to

$${\hat H}_0 = {\hat H}_{el} + {\hat H}_{so}$$ (8)

$${\hat H}_0 \Psi(\mathbf{r;R}) = E_0 \Psi(\mathbf{r;R})$$ (9)

Consider again the original Hamiltonian (1). An exact solution can be obtained by using an (infinite) expansion of the form

$$\Psi(\mathbf{r, R}) = \sum_k \Psi_k(\mathbf{r;R}) \chi_k(\mathbf{R}),$$ (10)

although, to the extent that the Born-Oppenheimer approximation is valid, very accurate solutions can be obtained using only one or a few terms. Alternatively, the total wavefunction can be expanded in terms of the electronic wavefunctions and a set of pre-selected nuclear wavefunctions; this requires the introduction of expansion coefficients:

$$\Psi^i(\mathbf{r, R}) = \sum_{kl} c^i_{kl} \Psi_k(\mathbf{r;R}) \chi_{kl}(\mathbf{R})$$ (11)

where the superscript i has been added as a reminder that there are multiple solutions to the Schrödinger equation.

Expressions for the nuclear wavefunctions $\chi_k(\mathbf{R})$ can be obtained by inserting the expansion (10) into the total Schrödinger equation yields

$${\hat H} \sum_{k'} \Psi_{k'}(\mathbf{r;R}) \chi_{k'}(\mathbf{... ... E \sum_{k''} \Psi_{k''}(\mathbf{r;R}) \chi_{k''}(\mathbf{R})$$ (12)

or

$$\int d\mathbf{r} \Psi_k^*(\mathbf{r;R}) {\hat H} \sum_{k'} ... ...(\mathbf{r;R}) \chi_{k'}(\mathbf{R}) = E \chi_{k}(\mathbf{R})$$ (13)

if the electronic functions are orthonormal. Simplifying further,

$$E \chi_{k}(\mathbf{R}) = \sum_{k'} \int d\mathbf{r} \Psi_{k... ...H}_{so} \right] \Psi_{k'}(\mathbf{r;R}) \chi_{k'}(\mathbf{R})$$ (14)

$$\sum_{k'} \left\{ \langle \Psi_{k}(\mathbf{r;R}) \vert {\hat H}_{... ...mathbf{r;R}) \vert {\hat H}_{so} \vert \Psi_{k'}(\mathbf{r;R}) \rangle \right\}$$ =

$$\sum_{k'} \int d\mathbf{r} \Psi_{k}^*(\mathbf{r;R}) {\hat T}_N \Psi_{k'}(\mathbf{r;R}) \chi_{k'}(\mathbf{R})$$ +

The last term can be expanded using the chain rule to yield

$$\sum_A \frac{-1}{2M_A} \left[ \nabla^2_A \chi_{k}(\mathbf... ... \vert \nabla^2_A \vert \Psi_k(\mathbf{r;R}) \rangle \right]$$ (15)

$$\sum_A \frac{1}{2M_A} \sum_{k' \neq k} \left[ 2 \langle \Psi_k(\m... ...k(\mathbf{r;R}) \vert \nabla^2_A \vert \Psi_{k'}(\mathbf{r;R}) \rangle \right].$$ -

At this point, a more compact notation is very helpful. Following Tully [1], we introduce the following quantities:

U_kk'(R) = U^(el)_kk'(R) + U^(so)_kk'(R) (16)

$$U^{(el)}_{kk'}(\mathbf{R}) = \langle \Psi_k(\mathbf{r;R}) \vert {\hat H}_{el} \vert \Psi_{k'}(\mathbf{r;R}) \rangle$$ (17)

$$U^{(so)}_{kk'}(\mathbf{R}) = \langle \Psi_k(\mathbf{r;R}) \vert {\hat H}_{so} \vert \Psi_{k'}(\mathbf{r;R}) \rangle$$ (18)

$${\hat T}_{kk'}'(\mathbf{R}) = \sum_A \frac{-1}{M_A} \mathbf{d}_{kk'}^{(A)}(\mathbf{R}) \cdot \nabla_A$$ (19)

$$T''_{kk'}(\mathbf{R}) = \sum_A \frac{-1}{2M_A} D_{kk'}^{(A)}(\mathbf{R})$$ (20)

$${\mathbf{d}}_{kk'}^{(A)}(\mathbf{R}) = \langle \Psi_k({\mathbf{r;R}}) \vert \nabla_A \vert \Psi_{k'}({\mathbf{r;R}}) \rangle$$ (21)

$$D_{kk'}^{(A)}(\mathbf{R}) = \langle \Psi_k({\mathbf{r;R}}) \vert \nabla^2_A \vert \Psi_{k'}({\mathbf{r;R}}) \rangle$$ (22)

Note that equation (18) of reference [1] should not contain a factor of 1/2 as it does. Now we can rewrite equations (14) and (15) as

$$\left[ {\hat T}_N + \sum_A \left( \frac{-1}{2M_A} \right) \... ...k}^{(A)} \right\} + U_{kk} - E \right] \chi_k({\mathbf{R}})$$ (23)

$$- \sum_{k \neq k'} \left[ U_{kk'} + \sum_{A} \left( \frac{-1}{2M_... ...}}_{kk'}^{(A)} \nabla_A + D_{kk'}^{(A)} \right\} \right] \chi_k'({\mathbf{R}}),$$ =

or

$$\left[ {\hat T}_N + {\hat T}'_{kk} + T''_{kk} + U_{kk} - E ... ...+ {\hat T}'_{kk'} + T''_{kk'} \right] \chi_{k'}({\mathbf{R}})$$ (24)

This is equation (14) of Tully's article [1]. Tully simplifies this equation by one more step, removing the term ${\hat T}'_{kk}$. By taking the derivative of the overlap of $\langle \Psi_k({\mathbf{r;R}}) \vert \Psi_k({\mathbf{r;R}}) \rangle$ it is easy to show that this term must be zero when the electronic wavefunction can be made real. If we use electronic wavefunctions which diagonalize the electronic Hamiltonian, then the electronic basis is called adiabatic, and the coupling terms U_kk'vanish.¹ This is the general procedure. However, the equation above is formally exact even if other electronic functions are used. In some contexts it is preferable to minimize other coupling terms, such as ${\hat T}'_{kk'}$; this results in a diabatic electronic basis. Note that the first-derivative nonadiabatic coupling matrix elements ${\hat T}'_{kk'}$ are usually considered more important than the second-derivative ones, T''_kk'.

In most cases, the couplings on the right-hand side of the preceeding equation are small. If they can be safely neglected, and assuming that the wavefunction is real, we obtain the following equation for the motion of the nuclei on a given Born-Oppenheimer potential energy surface:

$$\left[{\hat T}_N + T''_{kk} + U_{kk} \right] \chi_k({\mathbf{R}}) = E \chi_k({\mathbf{R}})$$ (25)

This equation clearly shows that, when the off-diagonal couplings can be ignored, the nuclei move in a potential field set up by the electrons. The potential energy at each point is given primarily by U_kk (the expectation value of the electronic energy), with a small correction factor T''_kk. Following Steinfeld [2], we can estimate the magnitude of the term T''_kk as follows: a typical contribution has the form $1/(2M_A) \nabla^2_A \Psi({\mathbf{r;R}})$, but $\nabla_A \Psi({\mathbf{r;R}})$ is of the same order as $\nabla_i \Psi({\mathbf{r;R}})$ since the derivatives operate over approximately the same dimensions. The latter is $\Psi({\mathbf{r;R}})p_e$, with p_e the momentum of an electron. Therefore $1/(2M_A) \nabla^2_A \Psi({\mathbf{r;R}}) \approx p_e^2/(2M_A) = (m/M_A)E_e$. Since $m/M_A \sim 1/10000$, this term is expected to be small, and it is usually dropped. For methylene, this term, called the Born-Oppenheimer diagonal correction (BODC), has an effect of about 40 cm^-1 to the singlet-triplet energy gap [3].