Nonadditive properties of van der Waals interactions in a one-dimensional dipole array

The van der Waals potential energy of a linear dipole array is calculated based on a formula using a determinant of a matrix determining the dipole interactions, which includes all orders of the coupling strength and can be applied to a system with an arbitrary number of dipoles. To consider the nonadditivity of the van der Waals interactions, the van der Waals potential energy is expressed as a series of ratios of the susceptibility of the dipole to a cubic root of the distance between the nearest-neighboring dipoles, up to the fourth order for different numbers of dipoles. We show that the contribution of the terms including nonadditive interactions between four dipoles increases with the total number of dipoles and exceeds that between two dipoles. Furthermore, the error in the series expansion is evaluated by comparing the exact potential energy, including all orders of the coupling strength for a small number of dipoles.