The energy of a molecule.

There are many energies in quantum chemistry. There is the kinetic energy of the electrons, the kinetic energy of the nuclei, the electron-nuclear attraction, the nuclear repulsion, the electron-electron repulsion, the electronic energy, the Coulomb interaction energy, the exchange interaction energy, the total energy of a molecule, and the orbital energy. These are all related as will be shown below.

The structure and stability of a molecule is determined by the total energy E of the molecule. This is just what its name implies: the kinetic energies of all particles forming the molecule and the potential energies of all their interactions. This total energy can be partitioned as follows.

The above partitioning of the total energy E would be the whole story if it weren't for the fact that it is very hard to determine E _el, and that we need to use approximations. The other energy terminology derives from these approximations, and in particular from MO theory or the independent-particle model. In this model the electron-electron repulsion is split in a Coulomb part and an exchange part. The Coulomb part is the Coulomb interaction of the charge cloud that is formed by the electrons with itself. The exchange part is a purely quantum mechanical contribution. It ultimately derives from the Pauli principle, which induces correlation between the positions of electrons with the same spin. The Pauli principle causes the probability for two electrons with the same spin to be at the same position to be zero. Their is no correlation between probabilities to find two electrons with different spin somewhere, and so these electrons have in general a non-zero probability to be at the same place.

The independent-particle model has also orbital energies. These are usually called MO energies when the orbitals are those of a molecule. An orbital energy is also sometimes called the energy of the electron in the corresponding orbital. One can partition the orbital energy also into the kinetic energy of one electron, its attraction with all nuclei, and a Coulomb and an exchange interaction with all other electrons. It is wrong to think that the sum of the energies of all electrons (orbital energies) is the same as the electronic energy. The reason lies in the interaction between the electrons. Suppose that we have just two electrons. The orbital energy for the first electron includes the interaction with the other electron. The orbital energy of the other electron includes the interaction with the first electron. These interactions are the same interaction: it is simply the interaction between the electrons. Thus the interaction between the electron is counted twice in the sum of orbital energies. This is also the case when there are more than two electrons. However, although the sum of orbital energies is not equal to the electronic energy, it is a reasonable approximation for the total energy of the molecule. If S is the sum of the orbital energies, then we have