Torque and atomic forces for Cartesian tensor atomic multipoles with an application to crystal unit cell optimization

New equations for torque and atomic force are derived for use in flexible molecule force fields with atomic multipoles. The expressions are based on Cartesian tensors with arbitrary multipole rank. The standard method for rotating Cartesian tensor multipoles and calculating torque is to first represent the tensor with n indexes and 3ⁿ redundant components. In this work, new expressions for directly rotating the unique (n + 1)(n + 2)/2 Cartesian tensor multipole components Θ_pqr are given by introducing Cartesian tensor rotation matrix elements X( R ). A polynomial expression and a recursion relation for X( R ) are derived. For comparison, the analogous rotation matrix for spherical tensor multipoles are the Wigner functions D( R ). The expressions for X( R ) are used to derive simple equations for torque and atomic force. The torque and atomic force equations are applied to the geometry optimization of small molecule crystal unit cells. In addition, a discussion of computational efficiency as a function of increasing multipole rank is given for Cartesian tensors. © 2016 Wiley Periodicals, Inc.