Valence Cross-Terms

Some of the modern force fields also include cross-terms to account for bond or angle distortions caused by nearby atoms. These terms are required to accurately reproduce experimental vibrational frequencies of molecules. Cross-terms may include stretch-stretch, stretch-bend-stretch, bend-bend, torsion-stretch, torsion-bend-bend, bend-torsion-bend, stretch-torsion-stretch terms. 

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The consistent force field (CFF) was developed to yield consistent accuracy of results for conformations, vibrational spectra, strain energy, and vibrational enthalpy of proteins. There are several variations on this, such as the Ure-Bradley version (UBCFF), a valence version (CVFF), and Lynghy CFF. The quantum mechanically parameterized force field (QMFF) was parameterized from ah initio results. CFF93 is a rescaling of QMFF to reproduce experimental results. These force fields use five to six valence terms, one of which is an electrostatic term, and four to six cross terms. 

Empirical force field (EFF) is a force field designed just for modeling hydrocarbons. It uses three valence terms, no electrostatic term and five cross terms. 

MMl, MM2, MM3, and MM4 are general-purpose organic force fields. There have been many variants of the original methods, particularly MM2. MMl is seldom used since the newer versions show measurable improvements. The MM3 method is probably one of the most accurate ways of modeling hydrocarbons. At the time of this book s publication, the MM4 method was still too new to allow any broad generalization about the results. However, the initial published results are encouraging. These are some of the most widely used force fields due to the accuracy of representation of organic molecules. MMX and MM+ are variations on MM2. These force fields use five to six valence terms, one of which is an electrostatic term and one to nine cross terms. 

Optimized potentials for liquid simulation (OPES) was designed for modeling bulk liquids. It has also seen significant use in modeling the molecular dynamics of biomolecules. OPLS uses five valence terms, one of which is an electrostatic term, but no cross terms. 

We have seen that for our calculations essentially two types of force fields have to be considered VFF- and UBFF-expressions. The main difference with repect to spectroscopic force fields consists in the superposition of nonbonded interactions. The force fields used so far for our purposes are almost exclusively simple valence force fields without cross terms, and a veriety of UB-force fields. Only recently could experiences be gathered with a valence force field that includes a number of important cross terms (79). Vibrational spectroscopic force fields of both types have been derived and tested with an overwhelming amount of experimental data. The comprehensive investigations of alkanes by Schachtschneider and Snyder (26) may be mentioned out of numerous examples. The insights gained from this voluminous spectroscopic work are important also when searching for suitable potentials for our force-field calculations. 

These two terms play an important role among the cross terms in valence... 

A stretch-bend cross-term is now widely used in valence force Helds but is not needed in the Urey-Bradley force Helds. 

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. 

EAS (Engler, Andose, Schleyer) [184] is quite an old force field designed to model alkanes exclusively. The harmonic potential is used for the bond stretching and cubic anharmonic for the valence angle bending. No out of plane, electrostatic or cross terms are included. The nonbonded interactions are represented by the Buchingham potential. 

EFF (Empirical force field) [186] has been designed just for modeling hydrocarbons. It uses the quartic anharmonic potential for the bond stretching, and the cubic anharmonic for the valence angle bending. No out of plane or electrostatic terms are involved, although the cross terms, except torsion-torsion and bend-torsion ones, are included. 

ESFF has been designed to be universal [187]. The Morse potential is employed for bond stretching, the potential quadratic in the cosine of the valence angles for their bending and the harmonic potential for the out of plane force field. The 6-9 with the charge based electrostatic potential is used for nonbonding interactions. No cross terms are involved. 

MM2 [189] uses cubic anharmonic potential to represent the bond stretching, up to sixth power expansion for the valence angle bending, and harmonic field for the out-of-plane deformations. The stretch-bending cross term is included. 

The essential difference between the S.V F.F. and the complete haimonic oscillator treatment does not of course lie in the choice of aience coordinates it lies in the fact that the former represents the potential energy as the sum of a set ol squared terms only that is, no interactions in the form of cross-products are allowed. It is possible to obtain a complete description of the force field to the harmonic oscillator approximation using valence coordinates if all the cross terms of the form /crA,B G c inserted. (It should... 

Linnett o gives a discussion of the use of valence force fieid with the addition ol selected cross terms. One method of reducing the number of constants to Tdc determined from the frequencies is to carry over from molecule to molecule certain force constants for squared terms and even for cross terms. Linnett mentions in this connection the work of Crawford and Brinkley who studied acetylene, ethane, methylacetylene, dimethylacetylene, hydrogen cyanide, methyl cyanide and the methyl halides in this way, and were able, for all the molecules, to account for 84 frequencies with 31 constants. Linnetttreated some of these compounds using a different force field. He was able to account satisfactorily for 25 frequencies using 11 force constants. From our point of view the trouble with these results is that Linnett obtained a value for the C - C force constant in these acetylene derivatives which was different from that obtained by Crawford and Brinkley. For C - C in methyl cyanide for example, Linnett obtained... 

Figure 7 Schematic representation of the more important cross terms identified for valence force fields.

This part of the correction arises from the use of parity-mixed valence orbitals in the expression for the RPA amplitude. There is also a much smaller cross term caused by the parity-mixing of the RPA amplitudes themselves, -I- This cross term contributes... 

Neglecting all of the higher-order (cubic and quartic) cross terms in eq. (32) yields the harmonic-general-plus-anharmonic valence force... 

In general, the total potential energy of a system in a simulation can be expressed as a sum of valence (or bond) energy, cross terms which account for such factors as bond or angle distortions caused by nearby atoms, and non-bond interactions (Accelrys, 2003) ... 

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