Tight Binding - Table of Interatomic Matrix Elements | Table 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:

“A child who is not rigorously instructed in the matter of table manners is a child whose future is being dealt with cavalierly. A person who makes an admiral’s hat out of linen napkins is not going to be in wild social demand.”
—Fran Lebowitz (20th century)

“In all cultures, the family imprints its members with selfhood. Human experience of identity has two elements; a sense of belonging and a sense of being separate. The laboratory in which these ingredients are mixed and dispensed is the family, the matrix of identity.”
—Salvador Minuchin (20th century)

“Nature confounds her summer distinctions at this season. The heavens seem to be nearer the earth. The elements are less reserved and distinct. Water turns to ice, rain to snow. The day is but a Scandinavian night. The winter is an arctic summer.”
—Henry David Thoreau (1817–1862)