Estimating the three-body correlation function for a liquid using spherical harmonic analysis

Recent studies show how the standard spherical harmonic expanison for the structure factor of a molecular liquid can be used within the Maximum Entropy formalism to extract information about the distribution and relative orientation of neighbouring molecules in a liquid. The expansion is solved by selecting solutions to a many-dimensional problem which are consistent with the supplied diffraction data and which are also the smoothest possible. Because the transform from generalized pair correlation function to structure factor is linear, an ensemble of possible solutions can be built up, and a linear combination of the various particular solutions is itself a possible solution. The distribution of the particular solutions around this average solution gives an idea of the uncertainty associated with the average solution. By treating a pair of atoms in a liquid as a diatomic molecule, it is possible to use the same analysis to map out a three- (or more) body correlation function which is consistent with the data and which is also as smooth as possible. Formally, the only direct access to the three-body correlation function is via the pressure derivative of the structure factor. The method proposed here to extract many-body correlations is in principle much simpler to execute and avoids the need for extra diffraction measurements. The structure factors of liquid argon and liquid water are analysed. Both examples have characteristic features, and in the case of water it is shown that a significant number of near-neighbour molecules are not in the traditional tetrahedrally bonded positions. A wide range of possible solutions is found in both cases and the graphical presentation of the solutions shows that many local configurations of atoms or molecules are present in a liquid. Important qualitative differences between the two liquid structures are highlighted in this way.