The Schrödinger Equation

It can be argued that the field of quantum chemistry began in 1925, with de Broglie's postulate that electrons possess both wave-like and particle-like characteristics [1]. Later in 1925, Davisson and Germer [2] confirmed de Broglie's conjecture experimentally by producing a diffraction pattern of electrons consistent with the de Broglie relation. Schrödinger then expanded on de Broglie's work, forming the non-relativistic Schrödinger wave equation [3], a mathematical model powerful enough to describe all non-relativistic chemical phenomena. This idea was not fully understood, however, until Heisenberg introduced the Uncertainty Principle (that it is impossible to specify both the linear momentum and position of a particle to arbitrary precision) [4].

The time-dependent Schrödinger equation is
$$\begin{align}\hat{H} \Psi = i \hbar \frac{\partial \Psi}{\partial t}, \end{align}$$
where $\hat{H}$ is the Hamiltonian operator, representing all energy contributions of the system. The wavefunction, $\Psi$, is a function of the nuclear and electron positions, electron spins and time. It is easiest viewed using the Born interpretation [5]:

The probability that a system, described by $\Psi$, will be found in a given state is proportional to $\Psi^{*}\Psi$.

This interpretation generates a constraint that the wavefunction must be square integrable, that is
$$\begin{align}\int{\vert\Psi\vert^{2} \, d\tau} < \infty. \end{align}$$

If we consider a system of M nuclei (each of charge Z_A) and N electrons in the absence of any external field, the Hamiltonian, in atomic units, is given by
$$\begin{align}\hat{H} = \hat{T} + \hat{V}, \end{align}$$
where
$$\begin{align}\hat{T} = - \frac{1}{2}\sum_{A}^{M} \nabla_{A}^{2} - \frac{1}{2}\sum_{i}^{N} \nabla_{i}^{2} \:, \end{align}$$
representing the kinetic energy of the nuclei and electrons respectively, and
$$\begin{align}\hat{V} = - \sum_{A}^{M}\sum_{i}^{N}\frac{Z_{A}}{\vert{\bf R}_{A}-{... ...um_{i}^{N}\sum_{j>i}^{N}\frac{1}{\vert{\bf r}_{i}-{\bf r}_{j}\vert}, \end{align}$$
describing the potentials due to nuclear-electron attraction, nuclear-nuclear repulsion and electron-electron repulsion.

The Hamiltonian above is independent of time, which allows a separation of variables. The wavefunction can then be of the form
$$\begin{align}\Psi(t) = \Psi e^{\frac{-iEt}{\hbar}} \end{align}$$
where $\Psi$ does not depend on time, and the Schrödinger equation reduces to
 $$\begin{align} \hat{H} \Psi = E \Psi, \end{align}$$
also independent of time. This is an important result: if the potential is independent of time and the system is in a state of energy E, all that is required to construct the time-dependent wavefunction from the time-independent wavefunction is multiplication by $e^{-iEt / \hbar}$, which is simply a modulation of its phase.