Total-energy differences and eigenvalue sums

We derive a variational expression for the total-energy difference between two crystal structures in terms of the difference in the sum of the one-electron eigenvalues and additional exchange-correlation and Coulomb contributions. These terms can be made to vanish under certain approximations, but for compound formation and for changes in volume, these terms can be of significance. Our derivation clearly shows that it is the same density, not the same potential as is commonly asserted, that is used in calculating the eigenvalue sums of the two structures; this formal distinction is of importance in order to maintain the variational nature of the results. Our results provide a framework in which many commonly used expressions can be justified and generalized.