We provide analytical solutions of the Continuous Symmetry Measure (CSM) equation for several symmetry point-groups, and for the associated Continuous Chirality Measure (CCM), which are quantitative estimates of the degree of a symmetry-point group or chirality in a structure, respectively. We do it by solving analytically the problem of finding the minimal distance between the original structure and the result obtained by operating on it all of the operations of a specific G symmetry point group. Specifically, we provide solutions for the symmetry measures of all of the improper rotations point group symmetries, S(n), including the mirror (S(1), C(S)), inversion (S(2), C(i)) as well as the higher S(n)s (n > 2 is even) point group symmetries, for the rotational C(2) point group symmetry, for the higher rotational C(n) symmetries (n > 2), and finally for the C(nh) symmetry point group. The chirality measure is the minimal of all S(n) measures.

Chiral zeolites are porous materials of prime importance because of their enantioselective potential applications in catalysis, in separation-science and in chromatography. We report a surprising observation made: as many as many 20 chiral silicate zeolites, none of which were recognized as such in their original publication, exist. In other words, the ''holy grail'' of having a library of such silicates was right under our nose. To be of practical use, such zeolites must reveal molecular level enantioselectivity; we show, for the first time, that chiral silicate zeolites are capable of enantioselective recognition of enantiomers, as revealed from comparative adsorption calorimetry experiments with either D-or L-histidine. The question of how to assign handedness to a chiral zeolite is addressed.

A common perception of many chemists is that non-biological chiral crystals comprise a small fraction of all crystals, as is the case of chiral non-biological molecules (~10%). We show that the proportion of non-biological chiral crystals is as high as 23%; and only ~6% of these are labelled as chiral.

We introduce a new mathematical tool for quantifying the symmetry contents of molecular structures: the Symmetry Operation Measures. In this approach, we measure the minimal distance between a given structure and the structure which is obtained after applying a selected symmetry operation on it. If the given operation is a true symmetry operation for the structure, this distance is zero; otherwise it gives an indication of how different the transformed structure is from the original one. Specifically, we provide analytical solutions for measures of all the improper rotations, S n p, including mirror symmetry and inversion, as well as for all pure rotations, C n p. These measures provide information complementary to the Continuous Symmetry Measures (CSM) that evaluate the distance between a given structure and the nearest structure which belongs to a selected symmetry point-group.