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I'm using Bilbao Crystallographic Server to find 3D crystallographic point groups of the K points of anatase TiO2.

Since the space group of anatse TiO2 is $I4_1/amd$, on this page, I found the point groups of the K points (in red rectangle).

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It can be seen from the table, there are point groups with dots in their names, for example $m.2m$ and $..2m$, which are not in the complete list of 3D crystallographic point groups.

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What are these point groups? How to map them to the listed 3D crystallographic point groups?

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These are not point groups, rather site symmetries or point symmetries, which do become Hermann–Mauguin point group symbols if the dots are omitted. The dots represent sets of non-contributing equivalent symmetry directions: it's a placeholder or a “dummy” for “unused” directions [1].

Wyckoff position consists of three entities: multiplicity, Wyckoff letter and site symmetry [2, p. 16]:

General and special positions: Below the point group symbol, we find a list of general and special positions (points), the latter lying on a symmetry element, and therefore having fewer ”equivalent points”. Note that the unique point at the center is always omitted. From left to right, we find:

Column 1 The multiplicity, i.e., the number of equivalent points.

Column 2 The Wyckoff letter, starting with $a$ from the bottom up. Symmetry-inequivalent points with the same symmetry (i.e., lying on symmetry elements of the same type) are assigned different letters.

Column 3 The site symmetry, i.e., the symmetry element (always a mirror line for 2D) on which the point lies. The site symmetry of a given point can also be thought as the point group leaving that point invariant. Dots are used to indicate which symmetry element in the point group symbol one refers to. For example, site $b$ of point group $4mm$ has symmetry $..m,$ i.e., lies on the second set of mirror lines, at 45° from the first set.

Simply put, the site symmetry $..m$ is a mirror reflection going through a plane perpendicular to the third symmetry direction $c$. International Tables for Crystallography Vol. A, Section 2.1.3.12. Oriented site-symmetry symbols provides definition for $m.2m$ site symmetry:

Finally, for class $4/mmm$ (full symbol $4/m\;2/m\;2/m),$ the twofold axis of $2mm$ may belong to any of the three kinds of symmetry directions and possible oriented site symmetries are $2mm.,$ $2.mm,$ $m2m.$ and $m.2m.$ In the first two symbols, the twofold axis extends along the single primary direction and the mirror planes occupy either both secondary or both tertiary directions; in the last two cases, one mirror plane belongs to the primary direction and the second to either one secondary or one tertiary direction (the other equivalent direction in each case being occupied by the twofold axis).

References

  1. Donnay, J. D. H.; Turrell, G. Tables of Oriented Site Symmetries in Space Groups. Chemical Physics 1974, 6 (1), 1–18. DOI:10.1016/0301-0104(74)80030-3.

  2. Radaelli, P. G. Symmetry in Crystallography: Understanding the International Tables; IUCr texts on crystallography; Oxford University Press: Oxford; New York, 2011. ISBN 978-0-19-955065-4.

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  • $\begingroup$ Thank you! Since site symmetry can also be thought as point groups, does it mean that if I want to find the subduction of the d atomic orbital to irreps of m.2m, I can refer to the point group mm2? $\endgroup$ Jun 22 at 12:33

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