This is a brief set of notes summarizing some of the original Slater-Koster tight-binding paper.[] The notation is somewhat different from the original paper. In particular, I'll denote the tight-binding wavefunctions by ψα(r), where α is an index running from 1 through 9 for an spd basis. The form of the wavefunctions is, as usual, a spherically symmetric function times an angular term. The form of each function is given in the Table:

Table I: Assumed functional forms for the tight-binding atomic-like basis functions. The radial functions are normalized so that each wave function is normalized to unity. The index lα is the total angular momentum of the αth state, and r = |r|. Note that these wave functions are orthogonal on the same site, i.e., <ψα (r) | ψβ(r) > δαβ.

α Index lα |ψα (r) >
1 s 0 1/√(4π)   φs (r)
2 p 1 √[3/(4π)] [x/r] φp (r)
3 p 1 √[3/(4π)] [y/r] φp (r)
4 p 1 √[3/(4π)] [z/r] φp (r)
5 d 2 √[15/(4π)] [xy/r2] φd (r)
6 d 2 √[15/(4π)] [yz/r2] φd (r)
7 d 2 √[15/(4π)] [zx/r2] φd (r)
8 d 2 √[15/(16π)] [(x2-y2)/r2] φd(r)
9 d 2 √[5/(16π)] [(2z2-x2-y2)/r2] φd (r)

The two-center Slater-Koster tight-binding integrals between an orbital at the origin and an orbital at the site R will be denoted

< ψα (r - R) |H| ψβ (r) >   , (1)

with

< ψα (r - R) |H| ψβ (r) > = (-1)lα + lβ < ψβ (r - R) |H| ψα (r) >   , (2)

Note that this depends only on α, β, R = |R|, and the orientation of R. Following Slater and Koster, we write this last quantity in terms of direction cosines l,m,n, where

R = R ( l ^
x
 
 + m ^
y
 
 + n ^
z
 
 )   . (3)

The 45 independent two-center integrals of the form (1) can be written in terms of 10 Slater-Koster parameters Pll'μ (R), as shown in Table II. Note that this same form works for the overlap matrix

< ψα (r - R) | ψβ (r) >   , (4)

with, of course, different forms for the P functions.

Table II: Slater-Koster parameters as derived in their paper but in a slightly different notation.

α β < ψα (r - R) |H| ψβ (r) >
1 1 Pssσ (R)
1 2 l Pspσ (R)
1 3 m Pspσ (R)
1 4 n Pspσ (R)
2 2 Pppπ (R) + l2 (Pppσ (R) - Pppπ (R))
2 3 l m (Pppσ (R) - Pppπ (R))
2 4 l n (Pppσ (R) - Pppπ (R))
3 3 Pppπ (R) + m2 (Pppσ (R) - Pppπ (R))
3 4 m n (Pppσ (R) - Pppπ (R))
4 4 Pppπ (R) + n2 (Pppσ (R) - Pppπ (R))
1 5 √3 l m Psdσ (R)
1 6 √3 m n Psdσ (R)
1 7 √3 n l Psdσ (R)
1 8 [(√3)/ 2] (l2-m2) Psdσ (R)
1 9 [n2 - ½ (l2+m2)] Psdσ (R)
2 5 m[Ppdπ (R) + l2 (√3 Ppdσ(R) - 2Ppdπ(R))]
2 6 lmn(√3 Ppdσ(R) - 2 Ppdπ(R))
2 7 n[Ppdπ (R) + l2 (√3 Ppdσ(R) - 2Ppdπ(R))]
2 8 l[Ppdπ (R) + ½ (l2-m2) (√3 Ppdσ(R)- 2 Ppdπ(R))]
2 9 [1/( √3)] l ( ½ (3 n2-1) (√3 Ppdσ(R) - 2 Ppdπ(R)) - Ppdπ (R))
3 5 l[Ppdπ (R) + m2 (√3 Ppdσ(R) - 2Ppdπ(R))]
3 6 n[Ppdπ (R) + m2 (√3 Ppdσ(R) - 2Ppdπ(R))]
3 7 lmn(√3 Ppdσ(R) - 2 Ppdπ(R))
3 8 m[-Ppdπ (R) + ½ (l2-m2) (√3 Ppdσ(R) - 2 Ppdπ(R))]
3 9 [1/( √3)] m ( ½ (3 n2-1) (√3 Ppdσ(R) - 2 Ppdπ(R)) - Ppdπ (R))
4 5 lmn(√3 Ppdσ(R) - 2 Ppdπ(R))
4 6 m[Ppdπ (R) + n2 (√3 Ppdσ(R) - 2Ppdπ(R))]
4 7 l[Ppdπ (R) + n2 (√3 Ppdσ(R) - 2Ppdπ(R))]
4 8 n(l2-m2) (√3 Ppdσ(R) - 2 Ppdπ(R)) / 2
4 9 [1/( 2 √3)] n [(3n2-1) (√3 Ppdσ(R)- 2 Ppdπ(R)) + 4 Ppdπ (R)]
5 5 l2 m2 (3 Pddσ (R) - 4 Pddπ (R) +Pddδ (R)) + (l2+m2) (Pddπ (R) - Pddδ (R)) +Pddδ (R)
5 6 l n [m2 (3 Pddσ (R) - 4 Pddπ (R) +Pddδ (R)) + (Pddπ (R) - Pddδ (R))]
5 7 m n [l2 (3 Pddσ (R) - 4 Pddπ (R) +Pddδ (R)) + (Pddπ (R) - Pddδ (R))]
5 8 ½ l m (l2-m2) (3 Pddσ (R) - 4 Pddπ (R)+ Pddδ (R))
5 9 [1/( 2√3)] l m [(3n2 - 1) (3 Pddσ (R) - 4 Pddπ (R) + Pddδ (R)) - 4 (Pddπ (R) - Pddδ(R))]
6 6 m2 n2 (3 Pddσ (R) - 4 Pddπ (R) +Pddδ (R)) + (m2+n2) (Pddπ (R) - Pddδ (R)) +Pddδ (R)
6 7 l m [n2 (3 Pddσ (R) - 4 Pddπ (R) +Pddδ (R)) + (Pddπ (R) - Pddδ (R))]
6 8 m n [ ½ (l2-m2) (3 Pddσ (R) - 4 Pddπ(R) + Pddδ (R)) - (Pddπ (R) - Pddδ (R))]
6 9 [1/( 2√3)] m n [ (3n2-1) (3 Pddσ (R) - 4 Pddπ (R) + Pddδ (R)) + 2 (Pddπ (R) - Pddδ(R)) ]
7 7 l2 n2 (3 Pddσ (R) - 4 Pddπ (R) +Pddδ (R)) + (l2+n2) (Pddπ (R) - Pddδ (R)) +Pddδ (R)
7 8 l n [ ½ (l2-m2) (3 Pddσ (R) - 4 Pddπ(R) + Pddδ (R)) + (Pddπ (R) - Pddδ (R))]
7 9 [1/( 2√3)] l n [ (3n2-1) (3 Pddσ (R) - 4 Pddπ (R) + Pddδ (R)) + 2 (Pddπ (R) - Pddδ(R)) ]
8 8 ¼ (l2-m2)2 (3 Pddσ (R) - 4 Pddπ (R) +Pddδ (R)) + (l2+m2) (Pddπ (R) - Pddδ (R)) +Pddδ (R)
8 9 [1/( 4√3)] (l2-m2) ((3n2-1) (3 Pddσ (R) - 4 Pddπ (R) + Pddδ (R)) - 4 (Pddπ (R) - Pddδ(R)))
9 9 [1/ 12] (3n2-1)2 (3 Pddσ (R) - 4 Pddπ (R) +Pddδ (R)) + (1/3) (3n2+1) (Pddπ (R) - Pddδ(R)) + Pddδ (R)

References

[1] J. C. Slater and G. F. Koster, ``Simplified LCAO Method for the Periodic Potential Problem,'' Phys. Rev. 94, 1498-1524 (1954).

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