6. Separating Electronic and Nuclear Motions The Born-Oppenheimer Approximation All Computational Chemistry rests on a fundamental assumption called the Born-Oppenheimer Approximation in which the motions of the electrons and atomic nuclei can be treated separately. The motion of electrons is much faster than that of the corresponding nuclei such that the nuclei can be considered as fixed (clamped nuclei). The Born-Oppenheimer Approximation assumes that the motions of electrons and nuclei can be decoupled and treated separately giving rise to two separate equations. where H nucl and H elec correspond to the nuclear and electronic components of the Hamiltonian, respectively. The electronic wavefunction, depends parametrically on the nuclear configuration (R), as well as the electronic coordinates (r), and is the solution of the electronic Schroedinger equation: H elec is the electronic Hamiltonian and corresponds to the motion of electrons (r) in the field of fixed nuclear positions (R). Note that E(R) is obtained by solving the electronic Schroedinger equation, and there exists an implicit dependence of the electronic energy (E) on the nuclear geometry (R).

7. Potential Energy Surfaces The sum of the two terms, V nn (R) + E(R), where V nn (R) represents the nuclear-nuclear repulsion energy and E(R) the electronic energy, is often referred to as the potential energy surface or the potential energy function and is obtained when the electronic Schroedinger equation is computed at many different nuclear configurations. For example, if one were to compute the electronic energy of butane as a function of the C-C-C-C torsion angle using the Schroedinger equation above, a torsional energy curve (V nn (R) + E(R)) for the C-C-C-C dihedral angle would be obtained.