Any Cartesian coordinate system is based on three orthonormal base vectors and an origin point. In the absence of any 500 instructions, ORTEP calculates the base vectors of the reference and working Cartesian systems from the input cell parameters and sets the origin to (0.,0.,0.).

This series of instructions can be used to reorient the reference and working Cartesian systems. Each time the reference system is redefined or rotated, the working system is automatically made coincident with the reference system. The working system can be displaced from the reference system by rotating about the x or y axis of the reference system with a 503 instruction. The working system is always positioned from the reference system and does not depend on any previous working system orientation. After each 500 series instruction, the base vectors of the relevant Cartesian system are printed in the ORTEP output file. These vectors are based on the triclinic coordinate system. The postfactor transformation matrix for converting from triclinic coordinates to Cartesian coordinates is also printed out. The inverse transformation matrix may be formed by placing the three base vectors together in row vector form.

• 501: Explicit Reference Cartesian System Assignment
• 502: Reference Cartesian System Rotation
• 503: Working Cartesian System Rotation (Stereoscopic Rotation)
• 504: Reference Cartesian System Origin Translation
• 505: Reference Cartesian System Origin at Centroid
• 506: Reference Cartesian System Origin at Centroid and Inertial Axis Coordinate System

Instruction 501 allows the user to define the reference Cartesian system explicitly. The origin point in the model (ORGN) is specified with an atom designator code. The three orthonormal base vectors can be described by two non-collinear vectors, and ORTEP provides the two following separate techniques for performing this operation, using vector cross products of the two vectors u and v. Type 1 produces base vectors that are roughly along the general triclinic coordinate axes of the crystal.

	Type 0	Type 1
Base vector 1 (x axis)	u	u
Base vector 2 (y axis)	u v	(u v) u
Base vector 3 (z axis)	u (u v)	u v

The reference system x and y axes will parallel the plotter x and y axes, and the origin point ORGN will lie in the plane of the plotter. The viewer will be looking into the z axis vector of the coordinate system from a distance VIEW in inches directly above the origin point.

The crystal model can be given any desired orientation with a series of rotations of the model about the reference system axes. In general, three rotations (e.g., those of an Eulerian system) are sufficient to achieve any orientation, but for convenience an unlimited number of rotations are permitted in the program. In addition, rotations of 120 about the body diagonal of the reference Cartesian system are permitted (this is achieved by a cyclic permutation of reference base vectors).

Each operation requires two fields on the instruction card. For axial rotations, the first field of each pair will have the number 1, 2, or 3 to indicate rotation about the x, y, or z axis of the reference system, respectively. The second field will have the rotation angle in degrees for a right-handed rotation of the model about the designated axis (i.e., a positive angle signifies a counterclockwise rotation of the structure with the designated axis pointing toward the reader). The body diagonal rotation is designated by either a -1 or a -2 in the first field to indicate a 120 or a 240 right-handed rotation about the body diagonal, respectively, and the second field is blank. A -3 would rotate the structure completely around and thus not change its previous orientation.

If desired, each rotation can be executed with a separate 502 instruction.

Columns	Instruction 502
3	0 (or 1 if continued)
7-9	502
10-18	1, 2, 3, -1, or -2
19-27	(o) (if value in previous field is positive)
28-36	[1, 2, 3, -1, or -2]
37-45	[ (o) (if value in previous field is positive)]
46-54	[...]
55-63	[...]

To define an orientation of the working system that is not coincident with the reference system, a 503 instruction may be used, which allows one rotation about one axis of the reference system. Actually any number of successive rotations can be made, but the effect is not cumulative since the starting point for each rotation is always the reference system. Body diagonal rotations are not permitted.

A 503 rotation normally precedes each member of a stereoscopic pair of plots. The rotation is about axis 2 if the stereo pair is to be viewed with the x axis parallel to the observer's interocular line and about axis 1 if the y axis is to be parallel to that line.

Columns	Instruction 503
7-9	503
10-18	1 or 2
19-27	(o)

Instruction 504 is used to translate the origin of the reference Cartesian system along the x, y, and z axes of the reference system. Stereo by translation of origin can be achieved with instruction 504, which may be used in place of the 503 instruction. However, the 504 instruction should not be used when the ellipsoids have internal structure because the octants selected for shading may not be the same on both stereo views.

This instruction finds the first moment (i.e., centroid or center of gravity) of the atoms in the ATOMS array and makes this point the origin point (ORGN) of the reference and working coordinate systems. The base vectors of the coordinate systems are unchanged from their previous values. A weighting scheme and screening may be applied to the atoms used to calculate the centroid by using Format No. 2 trailer cards. If no trailer cards are used, all atom positions are entered with unit weights.

The calculation described for instruction 505 is performed; then the second moment matrix about the centroid is calculated, and the reference and working coordinate systems are set up along the principal axes of this matrix. This principal axis system of coordinates is along the inertial axis of the configuration of atoms in the ATOMS array. The x axis is along the long axis of the configuration (i.e., the minimum axis of inertia), and the z axis is along the short axis of the configuration (i.e. the maximal axis of inertia). The overlap along the z-view direction is often minimized by this option. Furthermore, the xy plane is the least-squares best plane for the atomic configuration. Format No. 2 trailer cards may be used to supply weights and screening.

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Page last revised: July 11, 1997