Efficient full Newton–Raphson technique for the solution of molecular integral equations – example of the SPC/E water-like system

Abstract

It is shown how a full Newton–Raphson technique speeds up in impressive proportions the iterative resolution of molecular integral equations and makes it possible to reach quadratically complete convergence down to machine precision in a very few cycles. The technique generalises what has been originally proposed by Zerah and extensively used since then with great success for various fluids and mixtures of spherical objects. At each main iteration, the linearised cycle obtained by differentiating the Ornstein–Zernike and the integral equations is itself solved iteratively in terms of Δg^mnl_μν(r) projections. Its solution is reached very rapidly thanks to the powerful biconjugate gradient method and to the absence of any Euler angle manipulation. The virial equation is written in a shape formally different from the standard one, which allows a much higher numerical precision for the pressure without extra numerical work. The complete scheme is illustrated on the popular SPC/E water model.

Keywords:

• integral equation
• Ornstein–Zernike
• molecular HNC
• Newton–Raphson convergence

Acknowledgements

L. Belloni has been initiated to the world of IEs by Pierre Turq a long time ago and is pleased to contribute to this special issue in his honour.

Additional information

Funding

Support from ANR [grant number ANR-06-BLAN-0276] is gratefully acknowledged.