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

Program Jones webpage
Lattice Sums

Program JONES
(Version April 14, 2022)

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

by Peter Schwerdtfeger, Shaun Cooper 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)

Lattice Sums

The program uses fast convergent series for lattice sums Ls with real exponents s>3 for inverse power (Lennard-Jones type) series (~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).
5) By defining input lattice vectors or the symmetric components of th 3D Gram matrix the Terras decomposition is used.
6) A general cuboidal lattice with the quadratic form Q=A(i+j)^2+(i+k)^2+(j+k)^2 and prefactor (A+1)^s with A>0.
7) hcp structure with variable c/a ratio.
8) The Madelung constant in three and more generally in N dimensions.
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). For citations please use:
- A. Burrows and P. Schwerdtfeger, Program Jones - A Fortran program for calculating lattices sums for cubic lattices and the hexagonal close-packed structure, Massey University, 2019. (pdf file with more details available upon request).
- A. Burrows, S. Cooper, E. Pahl, and P. Schwerdtfeger, Analytical methods for fast converging lattice sums for cubic and hexagonal close-packed structurea, J. Math. Phys. 61, 123503-1-35 (2020).
- A. Burrows, S. Cooper, and P. Schwerdtfeger, The cuboidal lattices and their lattice sums, arXiv:2105.08922[math-ph] (2021).
For an introduction to extended Lennard-Jones potentials and the use of corresponding lattice sums for solid-state properties see:
- A. Burrows, S. Cooper, and P. Schwerdtfeger, Instability of the Body-Centered Cubic Lattice within the Sticky Hard Sphere and Lennard-Jones Model obtained from Exact Lattice Summations, Phys. Rev. E 104, 035306-1-10 (2021).
- A. Burrows, S. Cooper, and P. Schwerdtfeger, The Madelung constant in N dimensions, paper available upon request (2022).
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).
A more detailed paper for applications see P. Schwerdtfeger, A. Burrows, O. R. Smits, "The Lennard Jones Potential Revisited: Analytical Expressions for Vibrational Effects in Cubic and Hexagonal Close-Packed Lattice", J. Phys. Chem. A 125, 3037-3057 (2021).
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 and 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. For the Madelung constant the paper by Grandall is recommended: New representations for the madelung constant, Experimental Mathematics 8, 367 (1999). Note that we have recently succeeded to find a fast convergent series for the N-dimensional Madelung constant.

Note: This is work in progress. We are currently working on lattice sums in N dimensions and its analytical continuations.
Maintained by Peter Schwerdtfeger | Last updated: June 2019 | Copyright 2014 | Massey University