A General Geometric Representation
of Sphere-Sphere Interactions
Ho-Kei Chan, Eric B. Lindgren, Anthony J. Stace
and Elena Bichoutskaia
Abstract A general geometric representation of sphere-sphere interactions is derived
using the bispherical coordinate system. It presents a dimensionless, scaled surface-tosurface separation parameter s
à , which is valid for all possible combinations of sphere
size and separation distance. The proposed geometric description is not limited to
sphere-sphere interactions, but also describes interactions that involve a point particle
or a plane. The surface-to-surface separation parameter approaches the limit of s
Ã
¼ 1
if the radii of both spheres are much smaller than the actual surface-to-surface
separation distance s, i.e. in the limit of two point particles. On the other hand, the
geometric limit of s
Ã
¼ 0 corresponds to two planes, namely when the radii of both
spheres are much larger than s.
Keywords Sphere-sphere interactions Á Bispherical coordinates Á Geometric
description Á Surface-to-surface separation Á Inverse points
1 Introduction
Much research has been carried out on pairwise interactions, notably in the area of
classical electrostatics [1–7], in which each interacting body is geometrically
described as a point particle, a sphere or a plane. As shown in Fig. 1, the geometries
include the interaction between a pair of point particles (as described by Coulomb’s
law in the case of two point charges), the interaction between two spheres, as well
H.-K. Chan (&) Á E.B. Lindgren Á A.J. Stace Á E. Bichoutskaia (&)
Department of Physical and Theoretical Chemistry, School of Chemistry,
University of Nottingham, University Park, Nottingham NG7 2RD, UK
e-mail: epkeiyeah@yahoo.com.hk
E. Bichoutskaia
e-mail: elena.bichoutskaia@nottingham.ac.uk
© Springer International Publishing Switzerland 2015
M.A.C. Nascimento et al. (eds.), Frontiers in Quantum Methods and Applications
in Chemistry and Physics, Progress in Theoretical Chemistry and Physics 29,
DOI 10.1007/978-3-319-14397-2_2
29
of Sphere-Sphere Interactions
Ho-Kei Chan, Eric B. Lindgren, Anthony J. Stace
and Elena Bichoutskaia
Abstract A general geometric representation of sphere-sphere interactions is derived
using the bispherical coordinate system. It presents a dimensionless, scaled surface-tosurface separation parameter s
à , which is valid for all possible combinations of sphere
size and separation distance. The proposed geometric description is not limited to
sphere-sphere interactions, but also describes interactions that involve a point particle
or a plane. The surface-to-surface separation parameter approaches the limit of s
Ã
¼ 1
if the radii of both spheres are much smaller than the actual surface-to-surface
separation distance s, i.e. in the limit of two point particles. On the other hand, the
geometric limit of s
Ã
¼ 0 corresponds to two planes, namely when the radii of both
spheres are much larger than s.
Keywords Sphere-sphere interactions Á Bispherical coordinates Á Geometric
description Á Surface-to-surface separation Á Inverse points
1 Introduction
Much research has been carried out on pairwise interactions, notably in the area of
classical electrostatics [1–7], in which each interacting body is geometrically
described as a point particle, a sphere or a plane. As shown in Fig. 1, the geometries
include the interaction between a pair of point particles (as described by Coulomb’s
law in the case of two point charges), the interaction between two spheres, as well
H.-K. Chan (&) Á E.B. Lindgren Á A.J. Stace Á E. Bichoutskaia (&)
Department of Physical and Theoretical Chemistry, School of Chemistry,
University of Nottingham, University Park, Nottingham NG7 2RD, UK
e-mail: epkeiyeah@yahoo.com.hk
E. Bichoutskaia
e-mail: elena.bichoutskaia@nottingham.ac.uk
© Springer International Publishing Switzerland 2015
M.A.C. Nascimento et al. (eds.), Frontiers in Quantum Methods and Applications
in Chemistry and Physics, Progress in Theoretical Chemistry and Physics 29,
DOI 10.1007/978-3-319-14397-2_2
29