S. Scandolo, ICTP

Lesson number 8(Chemical bonds/Covalent bond)

In this lesson the covalent bond is studied in details, particularly it is focused for simplicity in the solution of the problem of the covalent bond of two hydrogen atoms where the electronic interaction is neglected. Without the electron-electron interaction the solution of the problem have to be a combination of the solutions for the single atom. Then an approximation to the solution of the problem when the two atoms are far apart, that is the presence of the second atom is considered a perturbation to the state of the first one, the solution is then assumed to be a linear combination of the atomics wave functions. The Bra-ket notation was recall as a tool used to solve the problem.

As the atoms are far it is assumed that the overlap of the atomic wave functions is approximately zero. Then it is obtained an eigenvalue problem for the coefficients of the suggested solution, a standard 2 by 2 quantum mechanics problem. This problem was solved and the “on-site” energy and the so called “hopping” term were introduced. The limitcase of two isolated atoms was considered proving that the general solution is consistent with the approximations made at the beginning of the problem. If atoms are isolated then we have two electrons at the atomic ground state Eo and this state is degenerate. As atoms are putting closer and closer this degeneracy start to split and a new ground state with energy lower than Eo appears and the two electrons can be arranged in this state with opposite spin. The energy gain is proportional to the overlap of the wave functions of the single atoms (how much the wave function feels the presence of the other atom). A second consequence is that the density of electrons at the midpoint between the atoms is doubled.