Philip Hofmann - Solid State Physics

(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 and . With this, Eq. (2.6) becomes

(2.7)

This equation system has a nontrivial solution when the determinant of the coefficient matrix vanishes. This gives two possible solutions

(2.8)

With these solutions, we can calculate the relation between the coefficients and from Eq. (2.7), and we find that for the “ ” solution and for the “ ” solution (see Problem 3 and note that ).

The energy levels and wave functions from this calculation are illustrated in Figure 2.2a. For a large distance between the nuclei, the overlap integral and the matrix element vanish and the only solution is . This makes sense as the second proton is far away from , so is approximately the ground state energy of atomic hydrogen. A corresponding solution does, of course, exist for the electron located on the other proton. For a short distance , on the other hand, this energy level splits up into the bonding and antibonding molecular states . The single electron will occupy the lower level and this will result in the energy gain of covalent bonding. The wave function corresponding to this is the one for which , which increases the electron density between the nuclei. The wave function corresponding to the antibonding upper energy level is constructed by choosing , and it shows the characteristics of an antibonding state with a node between the two nuclei.