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:

    “But hospitality must be for service, and not for show, or it pulls down the host. The brave soul rates itself too high to value itself by the splendor of its table and draperies. It gives what it hath, and all it hath, but its own majesty can lend a better grace to bannocks and fair water than belong to city feasts.”
    —Ralph Waldo Emerson (1803–1882)

    “As all historians know, the past is a great darkness, and filled with echoes. Voices may reach us from it; but what they say to us is imbued with the obscurity of the matrix out of which they come; and try as we may, we cannot always decipher them precisely in the clearer light of our day.”
    —Margaret Atwood (b. 1939)

    “In spite of all their kind some elements of worth
    With difficulty persist here and there on earth.”
    —Hugh MacDiarmid (1892–1978)