It is fairly common that a rigid body will have internal symmetry, such as the 6/m symmetry of the C atoms in a phenyl group; it is somewhat less common for the crystal symmetry to enforce some or all of the symmetry of the rigid body. When this does occur, the rigid body must be placed at a special location (a point, axis or plane) where that crystallographic symmetry is present. As will be discussed here, placement of the rigid body and its refinement will require special attention, but that will require that the rigid body coordinates be defined to be compatible with that symmetry and that the rigid body then be incorporated into the unit cell axes in a manner that preserves the symmetry. The rigid body definition must align the Cartesian axes and/or origin to the internal symmetry in the body:
The following sections of this chapter will discuss details for how the the rigid body is inserted into the unit cell. However, before we get to that, let’s consider how symmetry affects this process.
When a rigid body is placed at a general position in fractional coordinates, the x,y & z coordinates for the rigid body positions are all refined, but symmetry may require a constraint such as x=0 or x=y. Note that when individual atoms occur on a high-symmetry site, GSAS-II will generate the constraints that force the atom to remain on that symmetry location, but for rigid bodies, these constraints must be generated manually. These are created using the Constraints data tree entry where you can place a “Hold” (see §24.2.4) on the parameters that should not change. This requires knowing the names for the rigid body location parameters, which are determined from two numbers that are assigned to each rigid body in a project and in a structure. Each rigid body that is created in a project is assigned a number. These numbers start at 0; the number is shown on the data window for vector and residue rigid body, as seen in the example in Fig. 26.11. In addition, each rigid body can be used multiple times within a phase. These are also numbered and are known as the insertion number. The parameter names for residue bodies locations are RBRPx:n:m, RBRPy:n:m, and RBRPz:n:m for the x, y & z fractional coordinates of the rigid body origin, respectively, where n is the rigid body number and m is the insertion number. For vector bodies parameter names are RBVPx:n:m, RBVPy:n:m, and RBVPz:n:m, for the x, y & z fractional coordinates of the rigid body origin, respectively.
The possible ways that symmetry can interact with rigid bodies follows:
Center of symmetry The Cartesian origin will need to be the \(\overline {1}\) position, so the rigid body origin will need to be placed at the site of symmetry for the rigid body. This will always be either on an atom in the group or at the midpoint between two (or, occasionally more) atoms. There is a choice here with respect to which atoms will be included. One can include only one from each symmetry-related pair of atoms in the body, or both. If symmetry-related atoms are included, GSAS-II will recognize them and will set the occupancy to zero for one atom of each pair of duplicates. Note that in this case (\(\overline {1}\)), there are no requirements for the orientation of axes. In crystal coordinates: when the rigid body is inserted into the structure, the origin location will need to selected as a \(\overline {1}\) site in the structure. Since that symmetry location is a fixed point in fractional coordinates, no coordinates can be refined. The body orientation can be refined in “AV” mode (no orientational constraints).
Mirror plane The rigid body must be defined so that the normal to the mirror plane will be along a fixed Cartesian axis for the body; use of x, y, or z for that axis is recommended. The Cartesian origin must be defined so that it lies in the mirror plane. I.e., if z will be the direction normal to the mirror plane, then the rigid body’s internal mirror plane must be placed at z=0. In crystal coordinates: One coordinate in the rigid body origin will need to be fixed on the mirror plane. So that the origin may be refined, you will need to place a constraint hold (see §24.2.4) on parameter RBRPx:n:m, RBRPy:n:m, or RBRPz:n:m. If located on in a diagonal mirror planes, you will need to place an equivalence between x, y and/or z. The rigid body orientation vector will need to be fixed along the mirror plane normal, so this can’t be refined. Only the rigid body azimuth angle can be refined (mode “A”). Thus there are three free parameters: two for position and one for orientation.
Rotation axis The Cartesian origin must be defined on the rigid body’s symmetry axis, which will be on an atom or at a midpoint between two or more atoms. Align the rigid body’s symmetry axis along a fixed Cartesian axis; use of x, y, or z for that axis is recommended. In crystal coordinates: the coordinates for the rigid body origin will need to be constrained to stay on the axis by placing holds (see §24.2.4) or constraints on two origin parameters (RBRPx:n:m, RBRPy:n:m, and/or RBRPz:n:m.) The rigid body orientation vector will need to be fixed along the axis in crystal coordinates. For orientation only the rigid body azimuth angle can be refined (mode “A”). Thus there are two free parameters for the body: one for position and one for orientation.
Improper rotation axes define both an axis and a perpendicular plane. It will be uncommon to find a rigid body on an improper rotation axis, but if this occurs, the Cartesian origin must be fixed on the point where the axis meets the plane and a Cartesian axis must be chosen along the symmetry axis for the rigid body. In crystal coordinates, the origin is fixed to be at the point where the axis and plane intersect in fractional coordinates. The rigid body Cartesian axis must be aligned with the crystallographic rotation axis. The only degree of freedom for refinement will be the rigid body azimuth angle (mode “A”).
Glide planes and screw axes are unlikely to occur with rigid bodies, as the molecular fragment for the body would have to have internal symmetry that includes translational symmetry; the rigid body would need to be part of an infinite-length molecule. I would be very interested to see an example of this.
A linear rigid body has two orientational degrees of freedom rather than three, since a rotation of the object along the linear axis leaves it unchanged. The linear rigid body will usually be defined so that the linear axis falls on one of the Cartesian axes. (Note exception below for a body in a symmetry plane). The specific requirements depend on the symmetry for the location where the inear rigid body will be placed.
General position: when a linear rigid body is placed in a location with no symmetry, there are no restrictions on the origin location or the orientation. The origin can be refined freely and the orientation can be refined in “V” mode (with two degrees of freedom) for a total of five refinable parameters.
Linear Rigid Body in a Symmetry Plane This provides a minor challenge, in that there can be only one orientational degree of freedom, since rotation along the rigid body linear axis leaves the body invariant and the body must remain in the plane. The solution to this is that the body orientation must be defined so that the azimuthal axis is normal to the plane so that if refined in “A” mode, the body rotates in the plane. The origin can be placed anywhere along the body, but is best in the middle. The origin for the rigid body will have two degrees of freedom (use appropriate constraints).
Linear Rigid Body Perpendicular to a Symmetry Plane In this case, there are no orientational degrees of freedom. The origin must be placed in the middle of the body, and will have two degrees of freedom so that the body can move in the plane (use appropriate constraints).