CTCP
Centre for Theoretical Chemistry and Physics
at Massey University (Albany Campus), New Zealand





Program Jones webpage
Lattice Sums

Program JONES
(Version June 19, 2019)


A Fortran Program for Lattice Sums
for sc, bcc, fcc and hcp 3D lattices

by Peter Schwerdtfeger and Antony Burrows
Centre for Theoretical Chemistry and Physics
The Quantum Chemistry and Physics Group
The New Zealand Institute for Advanced Study
Massey University Auckland
0732 Auckland, New Zealand (Middle Earth)


Equation
Lattice Sums

The program uses fast convergent series for lattice sums Ls with real exponents s>3 for Lennard-Jones type of potentials (~r -s) associated with the simple cubic, face-centred cubic, body-centred cubic and hexagonal closed-packed lattices. Four different methods are used:
1) Direct summation (good for large exponents);
2) Dirichlet series (number theoretical treatment, good for large exponents);
3) Terras' decomposition of the Epstein zeta function for pure quadratic forms (good for small exponents);
4) Van der Hoff - Benson expansion (good for small exponents).

This is an open-source code and free for distribution (except for commercial purposes as it contains a modified version for the Bessel function published in Numerical Recipes). Please cite the following paper if you use this program for publishing data:
- Antony Burrows, Shaun Cooper, Elke Pahl and Peter Schwerdtfeger, Fast Converging Lattice Sums for the Cubic and Hexagonal Closed Packed Lattices, to be published (pdf file available upon request).
For an introduction to extended Lennard-Jones potentials and the use of corresponding lattice sums for solid-state properties see:
- P. Schwerdtfeger, N. Gaston, R. P. Krawczyk, R. Tonner, and G. E. Moyano, Extension of the Lennard-Jones potential: Theoretical investigations into rare-gas clusters and crystal lattices of He, Ne, Ar and Kr using many-body interaction expansions, Phys. Rev. B 73, 064112 (2006).
For a general introduction into lattice sums see:
- J. M. Borwein, M. Glasser, R. McPhedran, J. Wan, and I. Zucker, Lattice sums then and now, 150 (Cambridge University Press, 2013).
To download the fortran program click here:

For the number theoretical treatment in terms of the Dirichlet series a very small database containing the number of representations for an integer r3(n) with n≤100 is included in the program. Sloane's bigger data base for the four lattices with n≤10,000 can also be used and can be downloaded here (recreated by using Program Jones):
Download for the Sloane Database
A very large database (533.4 MB) for n≤20,000,000 well exceeding Sloane's data can be used as well and can downloaded here:
Download for Extended Sloane Database (compressed 178.1 MB)

History: A detailed historical survey on lattice sums can be found in Borwein's book. For the sc, bcc, and fcc lattices, the first attempt to calculate the corresponding lattice sums to sufficient accuracy was by Lennard-Jones and Ingham, who used expansions in terms of Bessel functions. Details can also be found in the work by M. Born R. D. Misra [On the stability of crystal lattices. iv, Math. Proc. Cambridge Phil. Soc. 36, 466-478 (1940)]. In 1953 B. M. E. Van der Hoff and G. C. Benson introduced an expansion in terms of Bessel functions for lattice sums [A method for the evaluation of some lattice sums occurring in calculations of physical properties of crystals, Can. J. Phys. 31, 1087-1094 (1953)]. In 1973, A. A. Terras introduced an expansion of quadratic forms in terms of Bessel functions to successively reduce the dimension N in the Epstein zeta function by block-diagonalizing the matrix S of the quadratic form [Bessel series expansions of the Epstein zeta function and the functional equation, Trans. Am. Math. Soc. 183, 477{486 (1973)] shown at the top of the webpage. Further improvement came from the work by I. J. Zucker in 1975 who evaluated lattice sums and Madelung constants for cubic and tetragonal lattices to 10 significant digits by using a Mellin transformation in conjunction with Jacobian θ-functions (Madelung constants and lattice sums for invariant cubic lattice complexes and certain tetragonal structures, J. Phys. A: Math. Gen. 8, 1734 (1975); and many other papers]. The lattice sum for the hexagonal closed packed structure was given early by G. Kane and M. Goeppert-Mayer in 1940 [Lattice summations for hexagonal close-packed crystals, J. Chem. Phys. 8, 642 (1940)], but has never been evaluated to high precision as fast convergent expansions for this lattice were unknown so far.

Note: We are currently working on lattice sums in N dimensions using Terras' decomposition of the Epstein zeta function and analytical continuations.
Maintained by Peter Schwerdtfeger | Last updated: June 2019 | Copyright 2014 | Massey University