VIBRATZ uses the treatment of the Urey-Bradley potential function for CXn molecules or groups given by Shimanouchi ("Simanouti", J. Chem. Phys. V. 17, p. 245, 1949), in which the potential energy V is:
V = {sum over i} [K'i ri (Dri) + 1/2 Ki (Dri)2]
+ {sum over i and j, j<i} [H'ij rij (Daij) + 1/2 Hij (Daij)^2]
+ {sum over i and j, j<i} [F'ij qij (Dqij) + 1/2 Fij (Dqij)^2]
where ri is the C-Xi bond length, rij = (ri rj )^1/2, aij is the Xi-C-Xj angle and qij is the Xi-Xj distance. Using the law of cosines to derive Dqij to the second order as a function of Drij and Daij, and the redundancy conditions for the angles within the polyhedra (also taken to the second order for tetrahedra and octahedra, but not in other cases), the above expression for V is transformed to one with no linear, only quadratic terms in the bond and angle changes. In the process, Shimanouchi defined the "internal tension", kappa, as
kappa = - (rij)^2 F'ij sin( aij ) - (rij)^2 H'ij
Note that kappa must be constant for a given polyhedron. After many calculations, Shimanouchi later (Pure Appl. Chem. V. 7, p. 131, 1963) set F' empirically at -0.1 F.
The above definition of kappa is more a mathematical necessity for the derivation than a meaningful chemical relationship. The linear angle constant H' was not included in the original Urey-Bradley treatment, and is usually assumed to be zero in perfectly regular polyhedra. In fact, it is possible to assume H' is zero in distorted polyhedra as well, relating the values of F' to the constant kappa, as in the first model described below.
To use Urey-Bradley forces in VIBRATZ it is necessary to set up all the bonds from the central atom to the ligands in the Bonds dialog, as well as X-C-X angles in the Angles dialog, independently of the U-B specifications in the Urey-Bradley Data dialog (all in the Forces menu). Angles should correspond to X-X "bonds" or contacts; e.g. no 180 degree angles in octahedra.
Urey-Bradley models in VIBRATZ. VIBRATZ allows three different ways of handling kappa and F' force constants, to accord with the different ways this has been done by Shimanouchi and others. VIBRATZ can handle polyhedra with up to eight ligands: it uses the second-order angular redundancies for tetrahedra given by Shimanouchi and those for octahedra given by Kim et al. (J. Mol. Spect., V.26, p.46, 1968), but for other polyhedra the redundancies are taken only to first order, and thus kappa is not used for other than tetrahedra and octahedra (except for the first option). Note that second-order redundancy terms vanish anyway for strictly planar configurations.
The first option, F' derived from kappa, is intended primarily for strictly regular polyhedra, in which all ligands are the same. In this case, assuming that H' is zero, kappa and F' are related by the equation
kappa = - r^2 F' sin( alpha )
where alpha is the X-C-X angle. For tetrahedra and octahedra alpha should be 109.47 and 90 degrees respectively. There should thus be only two specifically Urey-Bradley force constants in such cases, F and kappa, and to use U-B force constants from some studies (e.g. early papers by Shimanouchi, Kim et al.) it may be necessary to convert F' to kappa. If the polyhedron is not geometrically regular or the ligands are different, the relation between kappa and F' applies only to individual ligand pairs, and if kappa is constant and H' is always assumed to be zero, F' is variable. This option may thus be a reasonable approximation for distorted polyhedra as well, although this approach does not seem to have been used before.
The second option, Use given F', is the model used originally by Shimanouchi. Note, however, that Shimanouchi and others (e.g. Kim et al.) set H' to zero in the case of perfectly regular polyhedra, so that kappa and F' are directly related with the above equation. If the second option is used in such cases, kappa must be set to agree with F'.
The third option, F' = -0.1F, was used in later work by Shimanouchi and others. This has the advantage over option 2 of greatly reducing the number of adjustable parameters and seems to give satisfactory results in almost all cases. Note that for perfectly regular polyhedra if H' is assumed to be zero it is necessary to adjust the value of kappa with the above equation to agree with F and F' - this is not done automatically in least-squares.
It is possible to use even a lower approximation to Urey-Bradley forces than the first-order approximation used by the VIBRATZ for non-tetrahedral/octahedral coordination. This is done by simply entering the X-X interactions as bonds in the Bonds dialog, and entering nothing in the Urey-Bradley section. This is very roughly equivalent to neglecting F' as well as kappa, although it is not really as simple as this because of redundancies between the various forces. This has been used with some success in silicate crystals (see the quartz and diopside example files), in which even the most elaborate force fields do not produce perfect agreement.
Finally, it should be realized that the Urey-Bradley model is not uniquely distinguishable from other force models, and there is no absolute criterion for correctness of any particular model. Statements to the effect that the pure valence model is inadequate or "fails" in a particular case, whereas the Urey-Bradley model "succeeds", are unjustified in view of the fact that Urey-Bradley forces are normally converted to valence forces and that in the valence model forces can be added until the desired agreement is reached. The advantage of the Urey-Bradley model is that it often uses fewer adjustable force constants. The Urey-Bradley model is sometimes contrasted with the Orbital Valency Force Field of Heath and Linnett (Trans. Faraday Soc. 44, 873, 878, 884;1948), which frames angular or ligand-ligand forces in terms of deviation of central-ligand vectors from the optimum configuration determined by hybrid orbitals on the central atom. However, the original paper by Urey and Bradley (Phys. Review, 38, 1969; 1931) states that "Andrews has made the suggestion that the most promising assumption to make in regard to the restoring forces in molecules is that these forces to a first approximation consist of harmonic forces along the directions of the chemical bonds and perpendicular to them", and in fact their treatment at least in its initial statement is based on this assumption and not directly on the assumption of valence angle forces.