Diagonal force fields

Within the harmonic approximation the choice of a system of internal coordinates is irrelevant provided they are independent and that a complete potential function is considered ). For example, the vibrations of HjO can be analysed in terms of valence coordinates (r, >2, or interatomic coordinates (r, r, 3) and any difference in the accuracy to which observed energy levels are fitted (considering all the isotopic species H2O, HDO and D2O) will be due to the neglect of anharmonic terms. If one makes the approximation of a diagonal force field so that one is comparing the two potentials... 

COSMIC [181] method uses only harmonic potential with respect to displacements in the diagonal force fields. Neither improper torsion nor cross terms are included. The nonbonding interactions are the sum of van der Waals interactions represented by the Morse potential and the charge-based Coulomb energy. 

CVFF [182] is the valence version of CFF. It uses only harmonic expansion with respect to displacements in the diagonal force fields and reduces cross terms selection to some extent. 

In the original force fields, AlUnger and ofhers used Hooke s-law harmonic potentials for a diagonal force field (Eq. 4). If /o, 6t, and are the strain-free bond distances, angles, and torsion angles and fen and are the relevant force constants, then the potential function for the molecule is given by... 

A few force fields have an accuracy worse than that of semiempirical methods. CVFF was developed from an initially diagonal force field by adding a large number of cross terms, with insufficient reparameterization. It is obvious that this resulted in a force field with low predictivity, and its use cannot be recommended for any application. UFF was intended to cover the entire periodic table and is still the only published force field that can accomplish this task. However, the accuracy for organic molecules was sacrificed in the process the MAE for UFF is ca. 3 kcal/mol. 

The different force fields developed over the years differ considerably with respect to types and number of cross-terms. The TRIPOS, Chem-X, CHARMm, DREIDING, UFF, and COSMIC force fields do not include any cross-terms at all, and accordingly they are all purely diagonal force fields. MM2, the MM2-based force fields, and MMFF93 include a single cross-term (a stretch-bend function), whereas MM3 includes stretch-bend, bend-bend, and torsion-stretch terms. CVFF and CFF91 additionally include stretch-stretch, torsion-bend-bend, and other cross-terms. 

Figure 8 clearly shows that MMFF93, MM2, MM3, and the force fields based on them [ MM2 (Chem3D Plus), MMX (PCMODEL), MM2 , and MM3 ] have the best overall results. The harmonic and diagonal force fields including the TRIPOS force fields in Alchemy and SYBYL and the DREIDING... 

In electronic structure calculations, the zero point for the energy function has all particles (electrons and nuclei) infinitely removed from each other, and this common reference state allows energies for systems with different numbers of particles to be directly compared. If the same reference is used in force field methods, the energy function becomes an absolute measure of molecular stability. The difference relative to the normal reference state for force field functions is the sum of all bond dissociation energies, at least for a simple diagonal force field. If correction terms are added to the... 

Vibrational spectroscopy was a forerunner of molecular mechanics in the sense that spectroscopists used many of the basic methods used in molecular mechanics to calculate spectra, long before it was possible to calculate stiuctures. It was early found that a purely diagonal force field was less accurate than desired, and that cross terms could improve the quality of the spectra calculated. One of these cross terms was a stretch-bend term. It has the form shown in Eq. (4.12) ... 

For a very large number of variables, the question of storing the approximate Hessian or inverse Hessian F becomes important. Wavefunction optimization problems can have a very large number of variables, a million or more. Geometry optimization at the force field level can also have thousands of degrees of freedom. In these cases, the initial inverse Hessian is always taken to be diagonal or sparse, and it is best to store the... 

The first five terms in the general energy expression, eq. (2.1), are common to all force fields. The last term, crossi covers coupling between these fundamental, or diagonal,... 

The approach presented above is referred to as the empirical valence bond (EVB) method (Ref. 6). This approach exploits the simple physical picture of the VB model which allows for a convenient representation of the diagonal matrix elements by classical force fields and convenient incorporation of realistic solvent models in the solute Hamiltonian. A key point about the EVB method is its unique calibration using well-defined experimental information. That is, after evaluating the free-energy surface with the initial parameter a , we can use conveniently the fact that the free energy of the proton transfer reaction is given by... 

In order to construct free-energy surfaces for this system we start by defining the diagonal matrix elements, or the force fields , for each resonance structure ... 

In the construction of the matrix F of Eq. (63), the symmetrical equivalence of the two O-H bonds was taken into account. Nevertheless, it contains four independent force constants. As the water molecule has but three fundamental vibrational frequencies, at least one interaction constant must be neglected or some other constraint introduced. If all of the off-diagonal elements of F are neglected, the two principal constants, f, and / constitute the valence force field for this molecule. However, to reproduce the three observed vibrational frequencies this force field must be modified to include the interaction constant... 

From the above example it is apparent that there may be (and usually are) more unknown force constants than observed frequencies. If no additional sources of data are available, it is necessary to make some assumptions to simplify the force field. Often all or some of the off-diagonal elements in the F matrix [Eq. (63)] are neglected, leading to the so-called valence force field (VFF) or modified valence force field (MVFF), respectively. 

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