Abstract
Automatic detection of point groups as well as symmetrisation of molecular geometry and wavefunctions are useful tools in computational quantum chemistry. Algorithms for developing these tools as well as an implementation are presented. The symmetry detection algorithm is a clustering algorithm for symmetry invariant properties, combined with logical deduction of possible symmetry elements using the geometry of sets of symmetrically equivalent atoms. An algorithm for determining the symmetry adapted linear combinations (SALCs) of atomic orbitals is also presented. The SALCs are constructed with the use of projection operators for the irreducible representations, as well as subgroups for determining splitting fields for a canonical basis. The character tables for the point groups are auto generated, and the algorithm is described. Symmetrisation of molecules use a projection into the totally symmetric space, whereas for wavefunctions projection as well and partner function determination and averaging is used. The software has been released as a stand-alone, open source library under the MIT license and integrated into both computational and molecular modelling software.Graphical abstract.
Highlights
In the early days of quantum chemistry, the use of molecular symmetry was motivated by necessity to reduce the size of the problem at hand
In this article we present a new library for detecting the molecular symmetry and using it for the symmetrisation of wavefunctions
The conjugacy class for each operation needs to be computed. This can be achieved by testing each matrix product in the cartesian basis or directly when generating the point group elements. The former will always generate different classes for operations whose characters are complex, in e.g. the C3 point group, whereas the latter allows for the classes to be the same, if linear combinations of the representations are used, as is the case when dealing with real spherical harmonic basis functions
Summary
In the early days of quantum chemistry, the use of molecular symmetry was motivated by necessity to reduce the size of the problem at hand. Some calculations were only possible due to the ability to split the Hamiltonian into smaller portions, determined by the irreducible representations (irreps) of the symmetry group. Equivalence set information can be used to restrict the possible symmetry elements to a point where no search is required.
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