Density of states analyses based on Mulliken population analysis

Total density of states

The total density of states TDOS at energy E is written as

TDOS: N(E) = ∑_i δ(E-ε_i)   (3.3.8)

so the integral of N(E) over an energy interval E₁ to E₂ gives the number of one-electron states in that interval. In practice the delta functions are approximated by Lorentzians,

TDOS: N(E) = ∑_i L(E-ε_i) = ∑_i {σ/π ⋅ 1/[(E-ε_i)²+σ²]}   (3.3.9)

A plot of N(E) versus E reveals energetic regions where many levels are located. The width parameter s determines of course the appearance of the plot. A typical value is 0.25 eV (used as default in dos).

Partial (gross population and projected) density of states

In order to find out if a given function χ_μ contributes strongly to one-electron levels at certain energies, one may weigh a one-electron level with the percentage χ_μ character. We usually determine the χ_μ character by the gross populations, obtaining the GPDOS form of the partial density of states,

GPDOS: N_μ(E) = ∑_i GP_i,μ L(E-ε_i)   (3.3.10)

If the weight factor is determined by projection of φ_i against χ_μ, we obtain the projected density of states PDOS,

PDOS: N_μ(E) = ∑_i |⟨ χ_μ| φ_i ⟩|² L(E-ε_i)   (3.3.11)

One should not use the PDOS for d-type or f-type primitive basis functions ('BAS'). A d-type function consists of 6 Cartesian functions, while there can of course be only 5 true d-type functions among them: one (linear combination) of them is in fact an s-type function (x2+y2+z2). Similarly, there are 10 f-type Cartesian functions, 3 of which are in fact p-functions. The PDOS is calculated for the 6 d-type and 10 f-type Cartesian functions, which leads to undesired results. An PDOS for SFOs does not suffer from this problem.

Overlap population density of states (OPDOS)

If the delta function representing orbital φ_i is weighed with the overlap population between χ_μ and χ_ν in φ_i, the overlap population density of states OPDOS is obtained,

OPDOS: N_μν(E) = ∑_i OP_i,μν L(E-ε_i)   (3.3.12)

If an orbital φ_i at energy ε_i is strongly bonding between χ_μ and χ_ν the overlap population is strongly positive and OPDOS(e) will be large and positive around E=ε_i. Similarly, OPDOS(E) will be negative around energy ε_i when there is antibonding between χ_μ and χ_ν in φ_i.

The OPDOS(E) has been used under the name coop (crystal orbital overlap population) in Extended-Hückel solid state calculations by Hoffmann and coworkers [2].
[2] R. Hoffmann, A chemist's view of bonding in extended structures (VCH Publishers, New York, 1988).