Comment on "Simplified LCAO Method for the Periodic Potential Problem"

TL;DR

This paper identifies and corrects two misprints in a widely used tight-binding table from a seminal 1950s paper, aiming to improve the accuracy of related computational methods.

Contribution

It provides specific corrections to a classical tight-binding table, enhancing the reliability of this foundational resource.

Findings

Two misprints in the Slater-Koster table are corrected.

The corrections improve the accuracy of tight-binding calculations.

The paper clarifies the impact of these misprints on related computational work.

Abstract

In this comment we report on two misprints of a classical and still widely used tight-binding table contained in a seminal, 65-years-old paper by Slater and Koster, and suggest the corresponding corrections.

Full text

Perhaps the simplest model of one-electron states in solids is a tight-binding hamiltonian with a few orbitals per atom. As early as 1929, Bloch cast the linear combination of atomic orbitals (LCAO) as an illustration of his theorem in the limit of “strongly bound electrons”; Bloch (1929) 25 years later, Slater and Koster gave a key contribution to the diffusion of this method by (i) suggesting the interpretation of the matrix elements between atomic orbitals as adjustable parameters of a model hamiltonian, (ii) proposing their two-center approximation, and (iii) publishing their tabulation for $spd$ orbitals in cubic crystals.Slater and Koster (2015)

We found two misprints in their Table III, which contains the hamiltonian matrix elements between Bloch sums of atomic orbitals as a function of $\bf k$ for a simple-cubic lattice. They are based on hamiltonian matrix elements between orbitals sitting on nearest, second-nearest, and third-nearest neighboring atoms, calculated according to the two-center approximation; analogous matrix elements between Bloch sums may be immediately deduced from this table for the fcc, bcc, and diamond lattices, too.

Following the Slater-Koster notation (subscripts 1, 2, 3 for hamiltonian matrix elements involving nearest, second-nearest, and third-nearest neighbors):

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in the third row of Table III, within the sum which defines the matrix element ($s/xy$) between two Bloch sums of $s$ and $d_{xy}$ orbitals, the first addend should be $-2\sqrt{3}(sd\sigma)_{2}\sin\xi\sin\eta$ and not $-2\sqrt{3}(sp\sigma)_{2}$ $\sin\xi\sin\eta$, since this $\sigma$ bond derives from $s$ and $d$ orbitals and not from $p$ orbitals;

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in the second-to-last row of Table III, within the sum which defines the term ($3z^{2}-r^{2}/3z^{2}-r^{2}$), the ($dd\sigma$) parameter in one of its addends has no subscript, but it should have a 2; the correct form of the corresponding addend therefore reads $(dd\sigma)_{2}(\cos\xi\cos\eta+\frac{1}{4}\cos\xi\cos\zeta+\frac{1}{4}\cos\eta\cos\zeta)$.