What is Electronic Structure Theory?

Electronic Structure Theory describes the motions of electrons in atoms or molecules. Generally this is done in the context of the Born-Oppenheimer Approximation, which says that electrons are so much lighter (and therefore faster) than nuclei that they will find their optimal distribution for any given nuclear configuration. The electronic energy at each nuclear configuration is the potential energy that the nuclei feel, so solving the electronic problem for a range of nuclear configurations gives the potential energy surface.

Because the electrons are so small, one needs to use quantum mechanics to solve for their motion. Quantum mechanics tells us that the electrons will not be localized at particular points in space, but they are best thought of as ``matter waves'' which can interfere. The probability of finding a single electron at a given point in space is given by $\Psi^*(x)\Psi(x)$ for its wavefunction $\Psi$ at the point $x$. The wavefunction can be determined by solving the time-independent Schrödinger equation ${\hat H} \Psi = E \Psi$. If the problem is time-dependent, then the time-dependent Schrödinger equation $\imath \hbar \frac{\partial \Psi}{\partial t} = {\hat H} \Psi$ must be used instead; otherwise, the solutions to the time-independent problem are also solutions to the time-dependent problem when they are multiplied by the energy dependent phase factor $e^{-iEt/\hbar}$. Since we have fixed the nuclei under the Born-Oppenheimer approximation, we solve for the nonrelativistic electronic Schrödinger equation:

$${\hat H} = - \frac{\hbar^2}{2m} \sum_{i} \nabla^2_i - \sum... ...n_0 R_{AB}} + \sum_{i>j} \frac{e^2}{4 \pi \epsilon_0 r_{ij}},$$ (1)

where $i, j$ refer to electrons and $A, B$ refer to nuclei. In atomic units, this simplifies to:

$${\hat H} = - \frac{1}{2} \sum_{i} \nabla^2_i - \sum_{A} \f... ...m_{A>B} \frac{Z_A Z_B}{R_{AB}} + \sum_{i>j} \frac{1}{r_{ij}}.$$ (2)

This Hamiltonian is appropriate as long as relativistic effects are not important for the system in question. Roughly speaking, relativistic effects are not generally considered important for atoms with atomic number below about 25 (Mn). For heavier atoms, the inner electrons are held more tightly to the nucleus and have velocities which increase as the atomic number increases; as these velocities approach the speed of light, relativistic effects become more important. There are various approaches for accounting for relativistic effects, but the most popular is to use relativistic effective core potentials (RECPs), often along with the standard nonrelativistic Hamiltonian above.