Section 2.2.14.6. The density of states (DOS)

The density of states (DOS) is the number of one-electron states (in the HF method or DFT) per unit energy interval and per unit cell volume. It is better to start with the integral quantity , the number of states below a certain energy ,where is the volume of the BZ, the factor 2 accounts for the occupation with spin-up and spin-down electrons (in a non-spin-polarized case), and is the step function, the value of which is 1 if is less than and 0 otherwise. The sum over points has been replaced by an integral over the BZ, since the points are uniformly distributed. Both expressions, sum and integral, are used in different derivations or applications. The Fermi energy is defined by imposing that , the number of (valence) electrons per unit cell.

The total DOS is defined as the energy derivative of as with the normalization where the integral is taken from if all core states are included or from the bottom of the valence bands, often taken to be at zero. This defines the Fermi energy (note that the energy range must be consistent with N). In a bulk material, the origin of the energy scale is arbitrary and thus only relative energies are important. In a realistic case with a surface (i.e. a vacuum) one can take the potential at infinity as the energy zero, but this situation is not discussed here.

The total DOS can be decomposed into a partial (or projected) DOS by using information from the wavefunctions as described above in Section 2.2.14.3. If the charge corresponding to the wavefunction of an energy state is partitioned into contributions from the atoms, a site-projected DOS can be defined as , where the superscript t labels the atom t. These quantities can be further decomposed into -like contributions within each atom to give . As discussed above for the partial charges, a further partitioning of the -like terms according to the site symmetry (point group) can be done (in certain cases) by taking the proper m combinations, e.g. the and manifold of the fivefold degenerate d orbitals in a octahedral ligand field. The latter scheme requires a local coordinate system in which the spherical harmonics are defined (see Section 2.2.13). In this context all considerations as discussed above for the partial charges apply again. Note in particular the difference between site-centred and spatially decomposed wavefunctions, which affects the partition of the DOS into its wavefunction-dependent contributions. For example, in atomic sphere representations as in LAPW we have the decomposition In the case of spin-polarized calculations, one can also define a spin-projected DOS for spin-up and spin-down electrons.