The importance of the total energy

Figure 1 illustrates the modern scientific view of a typical material system such as a molecule or solid. The nuclei are small, point-like objects, and we denote their locations in $d=3$ dimensional space as $\vec X_1$, $\vec X_2$, .... The electrons, which have much smaller mass, move much more quickly and spread out into ``electron clouds'' which we learn below to describe in terms of orbitals.

Figure 1: Modern view of a molecular or solid system: nuclei (large dots), electron clouds (grey shaded regions).

The large difference in the mass of the nuclei and electrons allows us to consider the nuclei as nearly stationary compared to the electrons so that we may consider total energy of the system $E(\vec X_1, \vec X_2, \ldots)$ for each particular arrangement of the nuclei, a quantity which gives us a tremendous knowledge of the system. For instance, the minimum value of $E$ over all possible nuclear positions $\vec X_1, \vec X_2, \ldots$ gives the binding energy of the system. The arrangement of the nuclei for which the energy is minimized gives the equilibrium structure, including all of the bond lengths, bond angles and, for solids, the lattice parameters. The derivatives of $E$ with respect to the coordinates of the nuclei give the forces $\vec F_i= - \nabla_{\vec X_i} E$ on the nuclei, which determine how the system evolves in time. From this, we can compute the vibrational frequencies and even the melting point of the system. These are just a few of the things which we can compute ab initio, using only a few fundamental constants of nature as experimental input!