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.

Read more about this topic:  Tight Binding

Famous quotes containing the words table, matrix and/or elements:

    “the moderate Aristotelian city
    Of darning and the Eight-Fifteen, where Euclid’s geometry
    And Newton’s mechanics would account for our experience,
    And the kitchen table exists because I scrub it.”
    —W.H. (Wystan Hugh)

    ““The matrix is God?”
    “In a manner of speaking, although it would be more accurate ... to say that the matrix has a God, since this being’s omniscience and omnipotence are assumed to be limited to the matrix.”
    “If it has limits, it isn’t omnipotent.”
    “Exactly.... Cyberspace exists, insofar as it can be said to exist, by virtue of human agency.””
    —William Gibson (b. 1948)

    “The Laws of Nature are just, but terrible. There is no weak mercy in them. Cause and consequence are inseparable and inevitable. The elements have no forbearance. The fire burns, the water drowns, the air consumes, the earth buries. And perhaps it would be well for our race if the punishment of crimes against the Laws of Man were as inevitable as the punishment of crimes against the Laws of Nature—were Man as unerring in his judgments as Nature.”
    —Henry Wadsworth Longfellow (1807–1882)