VIBRATZ uses traditional valence forces, but the approach is different in certain respects from that often used in the past. First, rather than frame the calculations in terms of a specific number of internal coordinates, which are the bonds and angles in which valence forces are specified, VIBRATZ converts all the bond and angle forces to forces in the x, y and z Cartesian coordinates of each atom, and interactions thereof. The solution to the secular equation is done in terms of these Cartesian coordinates rather than internal coordinates. This eliminates any direct restrictions on the number of bonds and angles and interactions thereof, whereas when using internal coordinates the number of internal coordinates must be 3n-3 for crystals or 3n-6 for molecules where n is the number of atoms in the unit cell or molecule; or if the number of internal coordinates is greater, redundancy equations must be supplied. Results are exactly the same as if internal coordinates are used directly.
Second, the bonds and angles can be identified by providing general specifications in terms of the nature of the atoms involved and the values of bond distances and angles. VIBRATZ will then automatically locate all bonds and angles in the structure - this ensures consistency with symmetry. Alternately, for molecules (not crystals), the bonds and angles can be specified by entering the individual atoms involved.
In principle, these two considerations together allow for a general overall specification of forces which is not based on symmetric or geometric aspects of any particular structure. In fact VIBRATZ has options to read and write files containing such specifications, for purposes of tranfer from one structure to another.
The addition of Badger's rule variation to valence-bond and Urey-Bradley repulsion should cut down or eliminate in many cases the necessity for differentiating these forces on the basis of bond length.
Hierarchy and Ambiguity of Forces.
The strongest forces in any structure are the valence bonds. Bonds between nearest neighbors are usually easily identified, although in ionic structures there may be no sharp cutoff between bonded and non-bonded pairs.
In covalent structures, 3-atom valence angles may be next in importance. These are also easy to identify; the problem with valence angle forces is that it may be difficult or impossible to distinguish them from non-bonded repulsion forces.
Interactions involving bonds and angles may often be identified. For example lengthening one bond in a coordination polyhedron normally increases the bond order of the other bonds, increasing the force required to stretch those other bonds. For this reason, symmetric stretch vibrations typically occur at higher frequencies than antisymmetric stretches.
In many structures the attraction of valence bonding forces non-bonding atoms into contact or repulsive relation with one another. The general model use in VIBRATZ for dealing with such repulsion is the Urey-Bradley model for CXn coordination polyhedra. In this model, the linear X-X repulsive forces are converted to quadratic C-X bond and X-C-X angle forces, and bond-bond and bond-angle interactions. Thus it is often impossible to tell whether forces of the latter type which may be used in a non-Urey-Bradley model are true valence forces, or attributable wholly or in part to X-X repulsion. Note also that in some situations it is not possible to say whether X-X repulsion is due to electrostatic repulsion of charged ions, or to "short-range" interaction involving the electronic orbitals.
Nature of Bending forces
Consider the bending forces in a simple A-B-C chain of bonded atoms. In a simple isolated (gaseous) triatomic bent molecule such as H2O it is necessary to consider bending only in the plane of the atoms in equilibrium configuration, because bending out of the plane can always be resolved into bending in the plane and rotation of the molecule as a whole. Likewise in a straight triatomic molecule like CO2 it is theoretically necessary to consider bending in only a single plane, but in setting up the equations it is necessary to choose arbitrarily the orientation of that plane, and this choice must be consistent with the symmetry analysis if only one member of a degenerate group is calculated.
Now consider a situation in which the end atoms are fixed in a large molecule or crystal. Overall rotation is no longer possible, and it becomes necessary to account explicitly for possible bending in two directions, which can conveniently be specified as in-plane and out-of-plane in the case of a non-linear configuration, and which are arbitrary (but presumably mutually perpendicular) in the case of a linear configuration. Actually, we could still resolve such motion into bending in a single plane, combined with rotation of that plane about the axis of the end atoms, but it would still be necessary to introduce additional forces or internal coordinates to account for this rotation. Accounting for bond-bending in more than one dimension can be done in a number of ways with traditional valence forces, including tau and psi four-atom angles, and also bonding forces between non-adjacent atoms - there is no one correct way to do this.
VIBRATZ provides a means of accounting for 3-atom bending forces in both directions, which can eliminate the necessity for using more complex forces such as tau and psi 4-atom angles in some situations. Each such configuration can be assigned two bending coordinates, with s-vectors (displacement directions) perpendicular to each other. See the Angle Coordinate Data dialog for details of these coordinates.
In considering the X-C-X angles of CXn coordination polyhedra, the strictly geometric insufficiency of in-plane bending forces is removed for n = 4 (tetrahedra) and for larger polyhedra there is even a redundancy of bending forces. There is no need to remove this redundancy in VIBRATZ, nor should there be any need to provide extra coordinates, provided it is understood that values of angle forces are strictly applicable only for a particular coordination number, which implies a particular redundancy. Of course we are not likely to try to transfer angle force constants between say, tetrahedra and octahedra because the bond strengths and equilibrium angle values are normally different anyway.
The example of the CO3 radical (files CO3A.VBR, with psi angles, and CO3B.VBR, with external 3-atom angles) shows how the out-of-plane 3-atom bending forces may be used in place of psi 4-atom angles. Although no rigorous analysis has been attempted, it would seem that psi angles can always be replaced by out-of-plane 3-atom bending. The converse will usually, but perhaps not always, be the case.
In geometrically ambiguous situations such as discussed above the valence forces or coordinates used, for example out-of-plane 3-atom angles versus 4-atom torsion, may be a matter of preference, but whatever combination is used, it should be used in a consistent way or the forces will not be transferable or comparable between structures. It may in some cases be helpful to start from the lowest possible symmetry of the compound in question (which might be attained by chemical or isotopic substitution) to see the nature of the force constants required.