Tight Binding - Table of Interatomic Matrix Elements

Table of Interatomic Matrix Elements

In 1954 J.C. Slater and F.G. Koster published, mainly for the calculation of transition metal d-bands, a table of interatomic matrix elements

which, with a little patience and effort, can also be derived from the cubic harmonic orbitals straightforwardly. The table expresses the matrix elements as functions of LCAO two-centre bond integrals between two cubic harmonic orbitals, i and j, on adjacent atoms. The bond integrals are for example the, and for sigma, pi and delta bonds.

The interatomic vector is expressed as

where d is the distance between the atoms and l, m and n are the direction cosines to the neighboring atom.

E_{x,x^2-y^2} = \frac{\sqrt{3}}{2} l (l^2 - m^2) V_{pd\sigma} + l (1 - l^2 + m^2) V_{pd\pi}
E_{y,x^2-y^2} = \frac{\sqrt{3}}{2} m(l^2 - m^2) V_{pd\sigma} - m (1 + l^2 - m ^2) V_{pd\pi}
E_{z,3z^2-r^2} = n V_{pd\sigma} + \sqrt{3} n (l^2 + m^2) V_{pd\pi}
E_{xy,xy} = 3 l^2 m^2 V_{dd\sigma} + (l^2 + m^2 - 4 l^2 m^2) V_{dd\pi} + (n^2 + l^2 m^2) V_{dd\delta}
E_{xy,yz} = 3 l m^2 nV_{dd\sigma} + l n (1 - 4 m^2) V_{dd\pi} + l n (m^2 - 1) V_{dd\delta}
E_{xy,zx} = 3 l^2 m n V_{dd\sigma} + m n (1 - 4 l^2) V_{dd\pi} + m n (l^2 - 1) V_{dd\delta}
E_{xy,x^2-y^2} = \frac{3}{2} l m (l^2 - m^2) V_{dd\sigma} + 2 l m (m^2 - l^2) V_{dd\pi} + l m (l^2 - m^2) / 2 V_{dd\delta}
E_{yz,x^2-y^2} = \frac{3}{2} m n (l^2 - m^2) V_{dd\sigma} - m n V_{dd\pi} + m n V_{dd\delta}
E_{zx,x^2-y^2} = \frac{3}{2} n l (l^2 - m^2) V_{dd\sigma} + n l V_{dd\pi} - n l V_{dd\delta}
E_{xy,3z^2-r^2} = \sqrt{3} \left[ l m (n^2 - (l^2 + m^2) / 2) V_{dd\sigma} - 2 l m n^2 V_{dd\pi} + l m (1 + n^2) / 2 V_{dd\delta} \right]
E_{yz,3z^2-r^2} = \sqrt{3} \left[ m n (n^2 - (l^2 + m^2) / 2) V_{dd\sigma} + m n (l^2 + m^2 - n^2) V_{dd\pi} - m n (l^2 + m^2) / 2 V_{dd\delta} \right]
E_{zx,3z^2-r^2} = \sqrt{3} \left[ l n (n^2 - (l^2 + m^2) / 2) V_{dd\sigma} + l n (l^2 + m^2 - n^2) V_{dd\pi} - l n (l^2 + m^2) / 2 V_{dd\delta} \right]
E_{x^2-y^2,x^2-y^2} = \frac{3}{4} (l^2 - m^2)^2 V_{dd\sigma} + V_{dd\pi} + V_{dd\delta}
E_{x^2-y^2,3z^2-r^2} = \sqrt{3} \left[ (l^2 - m^2) V_{dd\sigma} / 2 + n^2 (m^2 - l^2) V_{dd\pi} + (1 + n^2)(l^2 - m^2) / 4 V_{dd\delta}\right]
E_{3z^2-r^2,3z^2-r^2} = ^2 V_{dd\sigma} + 3 n^2 (l^2 + m^2) V_{dd\pi} + \frac{3}{4} (l^2 + m^2)^2 V_{dd\delta}

Not all interatomic matrix elements are listed explicitly. Matrix elements that are not listed in this table can be constructed by permutation of indices and cosine directions of other matrix elements in the table.

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