the nuclei fixed at these positions and denote the distance in between them by . The Hamiltonian for the hydrogen molecule can then be written as
(2.3)
where and are the coordinates of the electrons belonging to nucleus A and B, respectively. The first two terms describe the kinetic energies of the two electrons. The operators and act only on the coordinates and , respectively. The electrostatic term contains the repulsion between the two nuclei and the repulsion between the two electrons as well as the attraction between each electron and each nucleus.
Calculating the energy eigenvalues and wave functions for this Hamiltonian is a formidable problem, mostly because of the interactions between the two electrons (the second term in the curly brackets). We shall return to this problem later. For now, we exploit the fact that the essence of covalent bonding can already be understood by considering just one electron, i.e. by simplifying Eq. (2.3) so that it describes the molecular ion:
(2.4)
This Hamiltonian is that of an electron moving in the Coulomb potential of two protons separated by a distance . The electrostatic repulsion of the nuclei (the term) does not depend on the position of the electron and just leads to an energy offset that could be treated separately. We choose to leave it in the Hamiltonian, though, as we want to inspect the dependence of the resulting energy levels on later.
We can calculate an approximate ground‐state solution to the Schrödinger equation by writing as a linear combination of the atomic 1s wave functions of the two atoms, and . This approach is commonly known as linear combination of atomic orbitals (LCAO). Our ansatz is thus
(2.5)
where and are constants. Multiplying this equation from the left with either or and integrating gives two algebraic equations
(2.6)
where we have introduced the so‐called overlap integral , as well as the abbreviations and correspondingly for and . As the two nuclei at and are completely equivalent, we can simplify this by noticing that
