# Born–Huang approximation

The Born–Huang approximation[1] (named after Max Born and Huang Kun) is an approximation closely related to the Born–Oppenheimer approximation. It takes into account diagonal nonadiabatic effects in the electronic Hamiltonian than the Born–Oppenheimer approximation.[2] Despite the addition of correction terms, the electronic states remain uncoupled under Born–Huang approximation, making it an adiabatic approximation.

## Mathematical formula

The Born–Huang approximation asserts that the representation matrix of nuclear kinetic energy operator in the basis of Born–Oppenheimer electronic wavefunctions is diagonal:

$\langle\chi_{k'}(\mathbf{r};\mathbf{R}) | T_\mathrm{n}|\chi_k(\mathbf{r};\mathbf{R})\rangle_{(\mathbf{r})}= \mathcal{T}_\mathrm{k}(\mathbf{R})\delta_{k'k}$

## Consequences

The Born–Huang approximation loosens the Born–Oppenheimer approximation by including some electronic matrix elements, while at the same time maintains its diagonal structure in nuclear equations of motion. As a result, the nuclei still move on isolated surfaces, obtained by addition of a small correction to the potential energy surface.

Under Born–Huang approximation, the Schrödinger equation of the molecular system simplifies to

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

The quantity $\left[E_k(\mathbf{R})+\mathcal{T}_\mathrm{k}(\mathbf{R})\right]$serves as the corrected potential energy surface.

## Born-Huang approximation as a method to evaluate non-adiabatic correction

The evaluation of correction term $\mathcal{T}_\mathrm{k}(\mathbf{R})$ is straightforward and can be obtained with little additional cost when the electronic wavefunction is known. However, the Born–Huang approximation is rarely used as a way to obtain non-adiabatic correction.

The Born–Huang approximation shares the same applicability as the Born–Oppenheimer approximation and requires that the electronic states involved are well separated. However, the diagonal correction from Born–Huang approximation is only large when potential energy surfaces comes close to each other, on which occasion the approximation itself breaks down. In fact, with a small additional cost, the accuracy of energy under Born–Huang approximation is very similar to that obtained under Born–Oppenheimer approximation. These factors prevented Born–Huang approximation from being used as a way to evaluate non-adiabatic corrections to energies.

## Upper-Bound property

The value of Born–Huang approximation is that it provides the upper-bound for the ground state energy.[1] The Born–Oppenheimer approximation, on the other hand, provides the lower-bound for this value.[3]

As a result, the Born–Huang approximation can be used as a very efficient way to validate the Born–Oppenheimer approximation, avoiding the need for explicit treatment of multiple electronic states and vibronic couplings, which is very complicated and computationally demanding. When the energy obtained from Born-Huang and Born–Oppenheimer approximations provide nearly identical values, adiabatic approximations can be safely used for the process. When these two values differ by a significant value, explicit non-adiabatic methods have to be employed to yield correct results.