Continuous symmetry measures for electronic wavefunctions

A continuous symmetry measure (CSM) which gives the magnitude of the deviation of an electronic wavefunction from a given symmetry element S is derived from the quantum mechanical expectation value of a symmetry operator Ŝ. The functional is normalized in the interval [0,100], i.e. CSM(Ŝ)=0 means that Ŝ commutes with the Hamiltonian. Intermediate values indicate the percentage deviation of the wavefunction from the corresponding symmetry. First applications of the method using ab initio Hartree–Fock SCF, density functional and semi-empirical AM1 wavefunctions are presented to quantify the chirality (degree of deviation from improper rotational symmetry) of organic molecules.