# Electric Moments of Linear Molecules from Force Microscopy.

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## Abstract

We develop a mathematical formalism to determine possible charge distributions based on atomic force microscope measurements at an array of points. In the real world, the inverse problem is ill-posed; different charge distributions can produce the same experimental results. This is due to error propagation when converting “backwards” from force or potential at observation points to charge distributions. We consider the one-dimensional multipolar expansion to determine if a finite number of terms may be determined from the experimental force measurements. This would avoid the inverse problem for distributions that can be well approximated by a finite number of terms of the multipolar expansion. -- / -- If given values of the electric potential and/or force and a finite number of multipolar terms are expressed as two vectors, they may be related by a square matrix representing a sum over the potential or force due to the charges of each multipole. The values of this square matrix are dependent on the location of the observation points of the AFM microscope. We formulate this square matrix as a product of two matrices, the second of which is a lower triangular one. We first formulate an algorithm to analytically invert the non-triangular matrix. Based on these, we develop two algorithms, one to determine the even terms of the multipolar expansion, and one to determine the odd terms. We then return to the original single square matrix, assign a specific set of observation points, and invert the matrix via Mathematica[i]. We then design simulations to compare the accuracy of the results for different possible accuracies of forces measured by an AFM microscope.