Born–Huang approximation

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

${\displaystyle \langle \chi _{k'}(\mathbf {r} ;\mathbf {R} )|T_{\mathrm {n} }|\chi _{k}(\mathbf {r} ;\mathbf {R} )\rangle _{(\mathbf {r} )}={\mathcal {T}}_{\mathrm {k} }(\mathbf {R} )\delta _{k'k}.}$ 

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 the nuclear equations of motion. As a result, the nuclei still move on isolated surfaces, obtained by the addition of a small correction to the Born–Oppenheimer potential energy surface.

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

${\displaystyle \left[T_{\mathrm {n} }+E_{k}(\mathbf {R} )+{\mathcal {T}}_{\mathrm {k} }(\mathbf {R} )\right]\phi _{k}(\mathbf {R} )=E\phi _{k}(\mathbf {R} )\quad {\text{for}}\quad k=1,\ldots ,K.}$ 

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

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]

• Vibronic coupling
• Born–Oppenheimer approximation