The Born-Oppenheimer Hamiltonian

Abstract

The Hamiltonian for two interacting particles is, as known

$$\hat H = - \frac{{{\hbar ^2}}}{{2{m_1}}}{\Delta _1} - \frac{{{\hbar ^2}}}{{2{m_2}}}{\Delta _2} + V({\vec r_1},{\text{ }}{\vec r_2}),$$

(1.1)

where

$${\Delta _1} \equiv \nabla _1^2 = \frac{{{\partial ^2}}}{{\partial x_1^2}} + \frac{{{\partial ^2}}}{{\partial y_1^2}} + \frac{{{\partial ^2}}}{{\partial z_1^2}}$$

(1.2)

and Δ₂ is analogously defined. In a closed system the potential energy V depends only on the relative position of the particles, so we may write

$$\hat H = - \frac{{{\hbar ^2}}}{{2{m_1}}}{\Delta _1} - \frac{{{\hbar ^2}}}{{2{m_2}}}{\Delta _2} + V({\vec r_1} - {\vec r_2}).$$

(1.3)