Atomic and Molecular Physics: Molecules

Born-Oppenheimer approximation

Treats the nucleus as fixed in position. The mass of the nucleus is much larger than the mass of an electron, and so the nuclei move much more slowly than electrons.

When can the Born-Oppenheimer approximation be applied?

When solving the TISE for a molecule when the motion of the electron(s) is much faster than the motion of the nuclei.

This is generally the case in low energy electronic states when the nuclei are much heavier than the electrons and far from crossings between electronic potential energy curves where the electronic wavefunctions change significantly at very small changes of nuclear positions.

The total wavefunction of a diatomic molecule under the Born-Oppenheimer approximation.

The total wavefunction of a diatomic molecule can be expressed as Ψ(\vec R; \vec r_i) = ν(\vec R)ψ(\vec R; \vec r_i), where \vec R is the internuclear distance, and \vec r_i are the position vectors of individual electrons.

Here ν(\vec R) is the nuclear wavefunction which only depends on the internuclear distance, and ψ (\vec R; \vec r_i) is the electronic wavefunction, which depends on the internuclear distance and the positions of the electrons.

Linear Combination of Atomic Orbitals (LCAO) approximation

Represents the spatial components of molecular orbitals as a linear combination of atomic orbitals \phi_i^{AO} usually centered on atoms: \psi=\sum^N_{i=1}C_i\phi_i^{AO}.

The AOs here are called basis functions.

This approximation assumes that in a molecule electrons are well described by the wave functions of the corresponding atoms.

Most stable nuclear geometry

Occurs at the nuclei separation where potential energy is a minimum.

Given by the minimum energy principle: For a closed system with fixed entropy, the total energy is minimised at equilibrium.

Overlap integral

A measure of the overlap of atomic orbitals centered on two different atoms.

Defined as S(R)=\int\phi^*(\vec r_A)\phi(\vec r_B)\,\mathrm{d}r_A\,\mathrm{d}r_B extending over all of space. When R→∞, the overlap integral tends to 0 and when R →0, it tends to 1.

Lowest energy electronic states of H^+_2

Given by the linear combination of the two atomic orbitals: \psi_+=\frac{1}{\sqrt2}[\phi_{1s}(\vec r_A)+\phi_{1s}(\vec r_B)],\quad \psi_-=\frac{1}{\sqrt2}[\phi_{1s}(\vec r_A)-\phi_{1s}(\vec r_B)] , where \psi_+ is symmetric, and \psi_- anti-symmetric.

Lowest energy electronic states of H₂

Given by the linear combination of the two atomic orbitals: \psi^T=\frac{1}{\sqrt 2}\left[\phi_{1s}(\vec r_{A1})\phi_{1s}(\vec r_{B2})-\phi_{1s}(\vec r_{A2})\phi_{1s}(\vec r_{B1})\right]\,\chi^T, \quad\psi^S=\frac{1}{\sqrt 2}\left[\phi_{1s}(\vec r_{A1})\phi_{1s}(\vec r_{B2})+\phi_{1s}(\vec r_{A2})\phi_{1s}(\vec r_{B1})\right]\,\chi^S, where T and S represent the triplet and singlet states.