Rietveld Fitting with Rigid Bodies

 

Introduction

 

It is common that powder diffraction does not provide enough observations to allow refinement of all atoms independently. This is not only a problem for powder diffraction; it is usually the case in protein refinement as well, as few datasets can be collected to atomic resolution. When the number of parameters is not sufficient to discern individual atomic positions, it is necessary to simplify the crystallographic model to require fewer parameters through use of constraints or to apply restraints, which effectively allow assumptions about the model to be used as if they were experimental observations. Examples of restraints include providing target values for bond distances and angles, or penalizing a refinement if atoms diverge from a plane. A common constraint is to group Uiso values so that a single value is used for all atoms of a particular type or in a common bonding environment.

 

One of the more powerful types of constraints in GSAS-II is to constrain a group of atoms so that their relative positions are fixed. Such a group of atoms is known as a rigid body. The core concept in a rigid body refinement is that (at most) 3 position parameters and 3 orientation parameters are needed to locate a simple rigid body. Sometimes symmetry can reduce the number of refinable parameters. In contrast, 3 position parameters are needed for every atom in a structure on a general position. Thus, a four-atom group requires 12 positional parameters and a 6-atom group requires 18 positional parameters, if modeled with individual atoms, but as rigid-bodies <=6 parameters will be needed.

 

GSAS-II offers two types of rigid bodies. This tutorial will demonstrate use of one type, residue rigid bodies, which allow incorporation of optional torsional parameters, while the second type, vector rigid bodies, offer one or more refinable size parameters, which can allow for refinement of selected bond distances or geometrical factors.

 

In the supplied example GSAS-II project file, a set of diffraction data is supplied for a crystal with two copies of the chiral R-(+)-Carbidopa molecule (shown below) and two water molecules in the asymmetric unit.

Since the phenyl ring defines a plane for the 6 atoms in the ring, as well as the 3 atoms directly bonded to the ring, and since bond distances and angles are well known, the relative positions of these atoms are fixed. The positions for the remaining non-H atom can be determined with only 4 additional parameters, because rotation is possible around the four bonds circled below.

Thus, all 16 non-hydrogen atoms can be positioned with 3 coordinates, 3 orientation parameters and 4 torsions. Compare these 10 parameters to the 48 positional parameters for 16 unconstrained atoms. (Adding 5 more torsions would allow for the hydrogen atoms to be positioned as well, for a total of 30 atoms from 15 parameters.)

 

In this example, the fit in the supplied GSAS-II project file appears quite good with respect to the agreement to the observed data, but closer examination shows that the bond distances and angles of the two molecular groups are quite irregular. Since this distorted structure is not physically reasonable, this model is far from optimal. This well-fitting but poor quality model arises because too many parameters were introduced, which is sometimes called overfitting. In this tutorial, a better model will be developed using rigid bodies for the two molecules, providing a more chemically reasonable representation with fewer parameters -- and a better fit.

 

Step 1. Export Cartesian coordinates for R-(+)-Carbidopa from crystal structure

 

There are two steps to working with rigid bodies in GSAS-II. The first is to obtain an accurate set of coordinates for the molecule, molecular fragment, or ionic moiety, which can be done in a number of ways. One way is to locate an accurate crystal structure from a database or paper. A second method is to use a computational tool to determine an approximate structure regularized for reasonable bond lengths. This can be done with low-accuracy forcefield computations or varying levels of theory. A good tool for force field minimization is the free cross-platform Avogadro program. The R-(+)-Carbidopa can be generated inside Avogadro by drawing the molecule (described here), or since the PubChem entry lists a SMILES description of the atomic bonding [C[C@@](CC1=CC(=C(C=C1)O)O)(C(=O)O)NN], this can also be used using the Build/Insert/SMILES… menu command in that program, but in the next step we will use a more general method to regularize a structure.

 

1A)

For this tutorial, we will export the coordinates from GSAS-II into a form to be read Avogadro. They will be regularized in the next step.

 

Begin by starting GSAS-II. Any .gpx file (GSAS-II project) may be used. If a new project is used, select the Data/“Add new phase” menu item first. Select the Rigid Bodies tab in the data tree.

 

1B)

Use the “Edit Vector Body”/“Extract from file” menu item select the GSAS-II gpx format and read in the distort1.gpx file included in the sample input. This will read in the coordinates for 34 atoms (two carbidopa molecules and two O atoms), from this we want to select only one carbidopa molecule.

 

1C)

Unselect all atoms by pressing the Toggle All button and then select the first sixteen atoms using the “Set Range” button and selecting the first [“0) C”] and the sixteenth atom [“15) O3”]. Note that the selected atoms appear differently from unselected atoms in the plot of the structure.

Then press “Continue”. The rigid body then appears in the plot window. Note how distorted the molecule appears.

 

 

1D)

Export the coordinates by pressing the “export as xyz button on the GSAS-II window.

Make a note of the file name, as it will be needed in the next step. If these coordinates were satisfactory, the rigid body could be created at this point, but since these coordinates are distorted, we will regularize them first. Press Cancel to close this window. It is fine to close GSAS-II, but it can remain open.

 

Step 2: Regularize coordinates in Avogadro

 

For this step, it is assumed that the cross-platform software Avogadro program has been installed. If you do not want to install this program, this step can be skipped. 

 

2A)

Read the coordinates into Avogadro by starting the program and using the Avogadro File/Open command; select the XYZ file type option and read the file produced in step 1D.

 

2B)

Note that the program noted the correct bonding of the carboxylic acid group, where one of the C-O bonds is noted as a double bond, but the phenyl group does not have alternating double bonds. To fix this, we need to click on the bonds, but that is difficult to do unless the atoms are drawn smaller. Make sure that the Display Settings button is selected (shown in blue) and click on the wrench symbol to bring up the “Ball and Sticks Settings” dialog, below.

When the “balls” are smaller, it becomes possible to click on every other bond in the phenyl ring.

 

2C)

Switch to the pencil in the toolbar

 

Click on directly on every other C-C bond in the ring to change the bonds from single to double. (Use care to avoid clicking elsewhere or additional atoms will be created; use a right click to delete them.)

 

2D)

Use the Build/“Add Hydrogens” menu item to add the remaining H atoms to the other end of the molecule.

 

2E)

Optimize the bond distances and angles using Extensions/“Optimize Geometry”. Use it a few times until no further changes are visible. The molecule should appear as below.

 

2F)

Use “Save as” to write the newly generated coordinates out. Be sure to select the file type as “XYZ”. I used name regularized.xyz for the file. Exit Avogadro.

 

Step 3: Create the rigid body in GSAS-II

 

If you skipped step 2, you can download the Avogadro output file, supplied as regularized.xyz

 

3A)

To start, open the distort1.gpx file in GSAS-II. Click on the “Rigid bodies” tree item. Click on the “Edit Residue Body”/“Extract from file” menu item, select the format as XYZ and then select the file written in step 2F (regularized.xyz).

 

Note that if “Import XYZ” were used, the coordinates would be used exactly as in created by Avogadro, but Extract from file allows atoms to be selected, recomputes the origin as the midpoint between atoms and allows rigid body axes and the origin to be defined by user preferences.

 

3B)

Once the file is opened and read; by default all atoms are selected. Note that H atoms are not moved when torsional groups are moved, so we do not want to include any of the non-phenyl H atoms. It is easier to remove all of the H atoms (they can be regenerated later as riders if desired). After selecting only atoms #0-#15 (as below), press Continue.

 

 

3C)

Here all that is needed to select all atoms (done by default) and then use the selected atoms to create a rigid body by pressing the “a Residue Body” button near the bottom of the list of buttons.

 

 

The rigid body is then created.

 

Other things that can be done on the previous window  include defining axes and reordering the atoms by dragging rows in the list. As an example, selecting the atoms in the phenyl ring and pressing “Place in plane” “xy” will orient the z axis as normal to the plane of the ring. Selecting a single atom and pressing “Set origin” will place that atom as the origin, or if two atoms are selected the origin will be the midpoint between the two atoms. None of this is needed here, but such actions can be needed for bodies that will be placed relative to special positions in a unit cell.

 

3D) Find atoms in torsions

Press the Plot button to see the rigid body. Note that dragging with the left mouse button held down is needed to see all the atom labels, as below.

 

 

From this it can be seen that the bonds with possible torsional rotations are C3-C5, C5-C7, C7-N2 and C7-C9.

 

3E) Define the torsions

Press the “Edit Residue Body”/”Define torsion.” This requires selecting an origin atom. Once that is selected, you are offered a choice of atoms that are bonded to the selected atom. Thus for the C3-C5 bond, if C3 is selected first, the options in the second menu are C1, C5 or C8 and C5 is selected. If C5 is selected first, then the choices are C3 and C7.

 

Then repeat this to define a total of 4 torsions, by then adding C5-C7, C7-N2 and C7-C9. The torsions are displayed at the bottom of the window, as shown below:

 

It can be instructive to select each torsional mode in turn and move the slider to see how the mode rotates the riding atoms.

 

Introduction to Steps 4 & 5

 

Now that the rigid body has been defined, the next step is to insert it into the crystal structure model. This means defining where the rigid body is located, oriented and how it is mapped to atoms in the asymmetric unit. The origin of the rigid body Cartesian system is placed at a specified location (in fractional coordinates) and a quaternion defines the rigid body orientation. The same rigid body may be used multiple times in the asymmetric unit. There will need to be a set of atoms, an origin, a quaternion for each insertion of each body. There will also be a set of torsion values angle values associated with each body use.

 

Note that understanding GSAS-II’s 3D visualization can be key to managing rigid bodies. It is valuable to move and rotate the representation, so that distances along all axes can be visualized. For this it is valuable to understand that as the structure is viewed, movement of the mouse while holding down a button (also called dragging) changes the view of the crystal structure, as described here):

 

* Dragging with the right button down: moves the viewpoint, which is kept at the center of the screen (effectively translating the structure)

* Dragging with the left button down H: rotates axes around screen x & y. Rotation is around the viewpoint

* Dragging with the center button down: rotates axes around screen z

* Rotating the scroll wheel: changes “camera position” (zoom in/out)

Note that the screen axes are x: horizontal, y: vertical and z: out of screen. Also, additional plot controls are available by pressing on the “Draw Options” tab.

 

When, as described in the next section, a rigid body is being located into the unit cell, a second set of actions can be accessed with the mouse. This is done by holding the Alt keyboard key down while dragging the mouse. Alt+right mouse dragging changes the rigid body origin, while Alt+middle button and Alt+left button actions rotate the body by changing the quaternion. Visually, the rigid body motions are analogous to what appears when the view of the structure is changed.

 

In this example, we want to place the rigid bodies into the approximate locations where those molecules are already found. There are two ways to do this. One is that if specified atoms in the rigid body are paired to atoms in the crystal, the origin and/or quaternion can be optimized to best collocate the pairs of atoms. This will be shown in Step 4. The other choice is to visually “dock” the rigid body to the location already found, which will be shown in Step 5.

 

Step 4: Insert the first rigid body into the model

 

4A)

To insert a rigid body into a structure, select the desired phase in the data tree (here there is only one choice) and then select the “Edit Body”/“Locate & Insert Rigid Body” menu item. A window is displayed with options to define where the rigid body will be placed

 

and a plot of the structure along with the rigid body location are shown in the plot window

where the asymmetric unit is shown initially with van der Waals spheres and the rigid body is displayed as a ball-and-stick model with green bonds.

 

Change the crystal structure display to either Ball & Sticks or Sticks (your preference) using the buttons on the upper right. This is needed because with van der Waals spheres it is hard to see when the rigid body is close to the location where it will be docked.

 

Note that the table at the lower part of the window shows each atom in the rigid body and the closest atom in the crystal structure and the distance to that atom. In the column labeled as “Assign as atom” to the extreme right of the table, we can specify specific pairing between atoms in the rigid body and atoms in the crystal structure. Scrolling or expanding the window so the entire table is visible will make subsequent steps easier (as seen below). As the rigid body is moved, the distances, and if not paired explicitly, possibly the closest atom will be updated.

 

4B)

We will align the rigid body to the molecule in the cell closest to the upper right corner by matching the atoms to the corresponding atoms.

 

If we click on the 2nd row in the table (with row label “C1”), the row is highlighted

 

Note that atom C1 in the rigid body is highlighted by the ball color changing to green, as well as its matching atom in the crystal structure (C6), as shown below.

 

The crystal structure atom is not the correct atom to match to C1. We can highlight different atoms (of the same type) in the crystal structure by pressing the Tab keyboard key until the correct atom is displayed or by selecting different atoms in the “Crystal Highlight” pull down. This shows that C11 in the crystal structure should be paired to C1 in the rigid body.

 

4C)

Force C11 in the crystal structure should be paired to C1 by clicking in the right-most box in the C2 row of the table and selecting atom C11 from the list. Optional: press “Process Assignments” and the distance between C1 (RB) and C11 (C) will be shown in the table. Likewise, pressing “Set Origin” will translate the rigid body to best match the assigned atoms. Since there is only one, the distance will be zero.

 

4D)

Setting the orientation requires assignment of at least one more atom and preferably a few. Select an easy to identify an atom in the rigid body by clicking on the C4 row in the table. (Alternately, press Alt-Tab to highlight different rigid body atoms.) Using Tab, it can be seen that crystal structure atom C13 should be paired. Select atom C13 in the “Assign as atom” cell for C4.

 

Likewise, rigid body atom C5 should be matched to crystal structure atom C14 by selecting atom C14 in the “Assign as atom” cell for C5.

 

4E)

Then press the “Set Both” button and the phenyl ring positions are approximately matched, as below.

 

4F)

The agreement in atom positions at the other end of the molecule is still not very good, but this can be improved by repositioning the view (dragging with the left mouse button) to look approximately along the plane of the ring. When this is done, it becomes clear that the agreement can be improved significantly by rotating around the first torsion angle by moving the slider. An angle of 22-23 degrees is much brings the atoms much closer, but there are still atoms as much as 1 Å in disagreement, since the crystal structure molecule is rather distorted.

      

 

4G)

Note that the table now shows that the rigid body has crystal structure atoms 16-31 linked consecutively, which is to be expected since the atom order was never changed.  Press Add to define these atoms from the rigid body parameters shown in the window. A warning is displayed, since some of the distances are larger than 0.5 Å, but press Yes since this is about as good as we can do.

The rigid-body constrained molecule is now shown in orange, unless “Show Rigid Bodies” is unchecked on the “Draw Options” tab. Press the c key (with the graphics window active) to see the view below.

 

 

Step 5: Insert the second rigid body into the model

 

As mentioned in the introduction to steps 4 & 5, above, mouse holding a mouse button down (mouse dragging) changes the view of the crystal while holding Alt down while dragging the mouse, moves the rigid body within the cell. We will use this to c

 

5A)

To insert a second copy of the rigid body into the structure, again select the “Edit Body”/“Locate & Insert Rigid Body” menu item. A window is displayed with options to define where the rigid body will be placed

 

5B)

Hold the Alt key down and drag with the middle mouse button to rotate the rigid body by 180 degrees, as shown below.

 

5C)

With the Alt key down and drag with the right mouse button and move the rigid body so that the phenyl rings are aligned.

 

 

5D)

To see how far the rings are separated in the screen z direction, we must rotate the view. (Without pressing Alt) drag the mouse upwards with the left button down to obtain a view similar to the one below.

 

From this view it is clear that a small rotation is needed and a small translation. First with the Alt and the middle mouse button both pressed, drag upwards until the rings molecules are more parallel. Then with the the Alt and the right mouse button both pressed, drag upwards until the molecules are better overlapped.

 

5E)

At the point the rigid body is pretty close to the initial position, with distances in the table ranging from 0.1 to nearly 2 Å, but better agreement can be obtained by making small adjustments with the Alt down and successively pressing each of the three mouse buttons alternated with changing the view to see the molecules from a different perspective. Another choice is to assign several atoms and optimize, as was done in steps 4D and 4E. Choices of atoms to optimize are clear since the atoms are properly paired in the table. Updating the C3-C5 torsion to ~30 degrees will also improve the agreement.

 

5F)

When the agreement is sufficient, press Add, accept the warning and this molecule is also now constrained by the rigid body.

 

Step 6: Optimize the model

 

If the number of cycles is set to zero (in the Controls tree item), the fit will be seen to have gotten significantly worse, with the profile wR jumping from 2.2% with the distorted model to 5.8%. This is not surprising, but a few refinement steps will provide a more realistic model with comparable agreement but far fewer parameters.

 

6A)

Initially, set all atoms to be refined as U only. (Select the Atoms tab, double-click on the refine column label and then select U.  In fact, this change is cosmetic, as the X flag for the atoms in rigid bodies is ignored.

 

 

Then select the RB Models tab. The first rigid body is automatically selected. Select to refine the Origin and the rotation angle/vector with the AV flag.

 

6B)

Then select the second rigid body and repeat the previous actions (setting the refine checkbox and selecting AV for the orientation refinement.)

 

6C)

Set the number of cycles back to 5 and refine. The fit will converge quickly to an wR of 4.0%.

 

6D)

Add refinement of the four torsion angles for each of the two rigid bodies. The wR quickly drops to 2.5%

 

Optional Step 7: Try rigid body TLS motion

 

The motion of rigid bodies has been worked out (Schomaker & Trueblood, Acta Cryst, 1968) known as the TLS description, which consists of three tensors: for translation (movement of all atoms together, described with the T tensor), libration (rotation of all atoms as a unit around the rigid body origin, described with the L tensor) and correlated librational-translational motion (screw-like motion, described with the S tensor). In the full expansion there are 6 T and 6 L and 8 S terms. It is unlikely that all of these terms can be used with powder data. The S terms can be readily omitted if the rigid body origin is at the body center of mass. This is not exactly true here, but the origin is close to the center of mass.

 

The simplest description that can be made with a rigid body is to assume that all motion is translational and isotropic. This is equivalent to constraining all atoms in the rigid body to have the same Uiso value, which has already been implemented in this refinement via constraints. A simpler way to set this up would be to select “Uiso” for the rigid body thermal motion model. This is shown in 7A-7C, below.

 

7A)

Delete the Uiso constraints by clicking on the constraints tree item and deleting each item. Then click on phase and the Atoms tab and remove the U flag for all atoms but the last two (water O atoms).

 

7B)

In the RB Models refine phase tab, set the thermal motion model to Uiso for both rigid bodies and turn on the refine checkbox to optimize the group Uiso value.

 

7C)

Carry out a refinement to confirm that the profile Rw returns to 2.5%.

 

It makes sense to see if further expansion of the rigid body motion/disorder description will produce an even better fit, although it is clear that the crystallographic fit is already quite good considering the limited resolution and high background levels. Refining 24 more parameters (6 T + 6 L x 2 bodies) is clearly impractical. One possible simplification would be to refine only the diagonal T & L elements (which constraints translation and libration directions to crystal axes). That might be appropriate with data more sensitive to group motion, but is still impractical here. One still simpler model is to include translation and libration, but make then isotropic. This can be done by refining only the diagonal T and L terms, but constrain them to be equal for both groups. This means replacing the 2 group Uiso values with 2 terms (a single grouped Tjj and a Ljj term). This is easily done as shown below.

 

7D)

Switch to refine the T & L terms by selecting each body, in the RB Models phase tab, set the thermal motion model to TL for both rigid bodies and then turn on the refine checkbox for the T11, T22, T33, L11, L22 & L33 terms. 

 

7E)

Create constraints on the Tjj and Ljj terms through use of the Constraints tree item. To constrain the Tjj terms, one needs to know their variable name is RBRTjj:b:0 for body #b of type 0.

 

On the Phase tab (loaded by default), use the “Edit Constr.”/“Add Equivalence” menu item and enter RBRT in the search filter. The first term is one of the six that needs to be constrained, so select it as the 1st variable and press OK.

 

On the next window select the remaining 11, 22 and 33 terms

and press OK:

 

 

 

7F)

Create constraints on the Ljj terms through use of the Constraints tree item as was done before. Their variable name is RBRLjj:b:0. Again use the “Edit Constr.”/“Add Equivalence” menu item and enter RBRL in the search filter. The first term is one of the six that needs to be constrained, and on the next screen select the remaining 11, 22 and 33 terms and press OK. Constraints should appear as:

 

 

7G)

Then refine with these parameters and the Rwp remains at 2.5%, but I would argue this is a more reasonable model than the Uiso model as the thermal ellipsoids are plotted, they get larger further away from the origin.

 

7H)

An even more relaxed model would be to allow the Tjj and Ljj terms to differ between the two bodies, with constraints as shown below, but this produces no improvement in the Rwp. Further it has a non-physical negative Ljj value for one of the two bodies, so this is not a better model.