Accurate tight-binding Hamiltonian matrices from ab initio calculations: Minimal basis sets

Projection of Bloch states obtained from quantum-mechanical calculations onto atomic orbitals is the fastest scheme to construct ab initio tight-binding Hamiltonian matrices. However, the presence of spurious states and unphysical hybridizations of the tight-binding eigenstates has hindered the applicability of this construction. Here we demonstrate that those spurious effects are due to the inclusion of Bloch states with low projectability. The mechanism for the formation of those effects is derived analytically. We present an improved scheme for the removal of the spurious states which results in an efficient scheme for the construction of highly accurate ab initio tight-binding Hamiltonians.

Figure 1

Projector matrix $|P_B|$ of benzene on a minimal AO basis set for the 23 molecular orbitals of lowest energy. The diagonal elements are the projectability numbers $p_n$. The presence of nonzero off-diagonal elements $\langle B_m|B_n\rangle \left(m\ne n\right)$ reflects the nonorthonormality of the vectors $|B_n\rangle$.

Figure 2

(a) Evolution of the tight-binding eigenvalues with increasing number of low-projectability Bloch states. $N$ is the number of states used in the construction of the TB matrix, where the first 17 states are those of high projectability. In all cases, the null states (not seen) are shifted by $\kappa =8$ eV. (b) Zoom-in around the low-projectability TB eigenvalues. The red dots mark the TB eigenenergy corresponding to the 18th Bloch state. (c) Energy variation of each TB state $n$ against increasing $N$. The reference energy ${\overline{E}}_n^{*}$ corresponds to the Hamiltonian without low-projectability states $\left(N=17\right)$.

Figure 3

Behavior of the null and low-projectability TB eigenstates under the shifting operation using (a) the exact $Q_{\mathcal{N}}=I_M-A{\left(A^{†}A\right)}^{-1}{A}^{†}$ or (b) the approximated $Q_{\mathcal{N}}\approx I_M-A{A}^{†}$ shifting matrix.

Figure 4

(a) Minimum projectability $p_n$ per band $n$ over all $\mathbf{k}$ points for ${\mathrm{CoSb}}_3$ on the chosen minimal basis set. A sharp decline of the projectability $\left(p_n<0.2\right)$ is seen for states $n\ge 58$. (b) Projector matrix $|P_B|$ at ${\mathbf{k}}_0=(0.4,-0.4,0.5)$, in reciprocal coordinates. Only the matrix elements from $44\le n\le 58$ are shown.

Figure 5

Tight-binding eigenenergies for ${\mathrm{CoSb}}_3$. (a) Accurate TB bands are obtained when adopting a high-projectability filtering criterion $\left(p_n>0.87\right)$ in the TB Hamiltonian. The band $n=55$ is shown in red. (b) The inclusion of a low-projectability band $\left(p_{58}=0.1474\right)$ introduces the eigenenergies seen in blue, which induces unphysical hybridizations with the band $n=55$. The arrows in the inset, whose size is $|\mathrm{\Delta }{\overline{E}}_{58}|$, illustrate the level repulsion at ${\mathbf{k}}_0=(0.4,-0.4,0.5)$, in reciprocal coordinates, due to hybridization. (c) The TB Hamiltonian is built with the approximated shifting matrix $Q_{\mathcal{N}}$ and $\kappa =1.0$ eV to show the distinct dependencies on $\kappa$ of the eigenstates. The reference DFT bands are shown with gray lines. Brillouin-zone integration follows the AFLOW standard as discussed in Ref. [31].