Using Wannier functions to improve solid band gap predictions in density functional theory

Enforcing a straight-line condition of the total energy upon removal/addition of fractional electrons on eigen states has been successfully applied to atoms and molecules for calculating ionization potentials and electron affinities, but fails for solids due to the extended nature of the eigen orbitals. Here we have extended the straight-line condition to the removal/addition of fractional electrons on Wannier functions constructed within the occupied/unoccupied subspaces. It removes the self-interaction energies of those Wannier functions, and yields accurate band gaps for solids compared to experiments. It does not have any adjustable parameters and the computational cost is at the DFT level. This method can also work for molecules, providing eigen energies in good agreement with experimental ionization potentials and electron affinities. Our approach can be viewed as an alternative approach of the standard LDA+U procedure.

: The energy correction (λ) for the Si VBM state in the 4 × 4 × 4, 5 × 5 × 5, and 6 × 6 × 6 supercells. The energy difference between the 4 × 4 × 4 and 6 × 6 × 6 supercells is 45 meV. If we fit the data to A + B/L (A and B are fitting parameters and L is the supercell length), the energy correction is 0.52 eV, as L tends to infinity. Thus the error of λ and the resulting eigen energy in the 4 × 4 × 4 supercell is ∼ 0.1 eV. We find the same for the conduction band.

Supplementary Note
In the Hartree-Fock (HF) method, the many body wave function is expressed as a Slater deter- Here the {ϕ i } are orthonormal single-particle orbitals, but not necessarily eigen orbitals because a unitary transform of the orbitals does not change the slater determinant. The HF total energy can then be expressed as: Here It is impossible to write down a N ± s l electron many body wave function for fractional s l except in a grand canonical ensemble as will be discussed later. However, intuitively, to describe a orbital ϕ l which is partially occupied by 1 − s l electrons (in below we will only discuss removing s l electrons from the ϕ l orbital for simplicity), it makes sense to replace ϕ l by √ 1 − s l ϕ l in Eq. (S2), and ρ = ∑ i̸ =l |ϕ i | 2 + (1 − s l )|ϕ l | 2 . We find that the self interaction energy in the Coulomb term cancels out with that in the exchange term. As a result, we have Here E l = ⟨ϕ l |H HF |ϕ l ⟩. H HF is the HF single-particle Hamiltonian, and E l is the expectation value of H HF by the ϕ l orbital. Note, ϕ l does not need to be a HF eigen orbital. In the derivations above, just like in the orignial Koopmans' theorem, we assume {ϕ i } do not change with s l .
term (which only exists in the self Coulomb interaction and exchange interaction terms under the frozen orbital approximation) has been cancelled out. If we take s l = 0 and ϕ l to be the VBM state, Eq. (S3) is the original Koopmans' theorem. The Eq. (S3) can be applied to arbitrary orbital ϕ l in the occupied subspace, and it shows that without the self-interaction term, E HF (N − s l ) should be a linear function of s l . The same is true for the unoccupied subspace.
The plausibility of applying the straight-line energy condition to Wannier functions can also be viewed from the a grand canonical ensemble point of view for the total energy E(N ± s l ), similar to that in Ref. [S1]. In Ref. [S1], the authors constructed a grand canonical ensemble, which is a statistical mixture of N -electron pure states Ψ N and (N + 1)-electron pure states Ψ N +1 . For the state with N + s l electrons [s l ∈ (0, 1)], the probability of Ψ N is 1 − s l and that of Ψ N +1 is s l , respectively. According to the variational principle, the lowest energy of the ensemble is are the ground state energies of the N -electron and (N + 1)-electron pure states respectively [S1]. Thus the total energy is a linear function of s l . To extend it to the Wannier functions, we can construct a similar grand canonical ensemble with Ψ N and constrained Ψ c N +1 (we first discuss the conduction band). To ensure a Wanner function ϕ l is fully contained in the constrained (N + 1)-electron wave function Ψ c N +1 (here ϕ l is within the conduction band subspace and thus orthogonal to Ψ N , i.e. ∫ which indicates the probability of electrons on ϕ l is one (i.e., the Wannier function is fully occupied).
One can use Ψ c N +1 that satisfies Eq. (S4) to variationally minimize the total energy. We may call the resulting energy E c N +1 the constrained ground-state energy. According to the variational principle, the statistic total energy of the grand canonical ensemble with N + s l electrons is (1 − s l )E N + s l E c N +1 . It shows that after a fractional-electron addition onto a Wannier function (constructed within the conduction band subspace), the total energy follows a linear function of s l . Similarly, to remove s l electrons from a Wannier function ϕ l (constructed within the valence band subspace), we require a constraint on the (N − 1)-electron wave function Ψ c N −1 : ∫ dr 1 ϕ * l (r 1 )Ψ c N −1 (r 1 , r 2 , · · · , r N −1 ) = 0 Where < w l |ϕ l >= 0 and < w l |φ j >= 0 (i.e, w l is the residual). Then we can construct: It follows One can then easily re-diagonalize the Hamiltonian H w using {φ j , ϕ l }, and the eigen states will be the valence orbitals of H w . In other words, {φ j , ϕ l } will span the valence band subspace of H w . For small s l , it is easy to show the H w should have a gap, since when s l = 0, H w equals to H LDA [N ] that has a gap. From this argument, we realize that when we discuss the valence band subspace (during the process of removing electrons from a Wannier function), we need to be clear what the single particle Hamiltonian is. We like to emphasize that, at the current stage, we are treating the Wannier function non-self-consistently during the removing of electrons. We feel that even if a self-consistent treatment is used, the change for the Wannier function could be small. For example, if we take δ electrons from the canonical orbital (the usual ∆SCF approach), during the self-consistent calculation for the N − δ electron system, the most relaxation effect comes from the N − 1 valence state, instead of the valence band maximum (VBM) state. Thus, even if we fix the VBM state (like we fix the Wannier function here), the error in E(N − δ) is usually rather small.
Nevertheless, it might be interesting in the future to test the effects of the self-consistency during the removing of electrons. But even if ϕ l (sl) depends on s l , in Eq.
(2) the ϕ l must be ϕ l (s l = 0) when calculating band gap correction. Thus, all these might only change the procedure to calculate λ l .
If the major effect of the screening comes from the other N − 1 electrons (as we discussed above), the self-consistent effects for ϕ l should be quite small.

Supplementary Table
Supplementary