Bond harmonic function

In a Urey-Bradley force field, angle bending is achieved using 1,3 non-bonded interaction rather than an explicit angle-bending potential. The stretch-bond term in such a forci field would be modelled by a harmonic function of the distance between the 1,3 atoms ... 

HyperChem uses harmonic functions to calculate potentials for bonds and bond angles (equation 9). 

This term is associated with deformation of a bond from its standard equilibrium length. For small displacements from equilibrium, a harmonic function is often used ... 

As mentioned in Section 2.2.3, the out-of-plane energy may also be described by an improper torsional angle. For the example shown in Figure 2.6, a torsional angle ABCD may be defined, even though there is no bond between C and D. The out-of-plane oop may then be described by an angle for example as a harmonic function... 

The parameters Pim , Pcore, and k can be refined within a least square procedure, together with positional and thermal parameters of a normal refinement to obtain a crystal structure. In the Hansen and Coppens model, the valence shell is allowed to contract or expand and to assume an aspherical form [last term in (11)], as it is conceivable when the atomic density is deformed by the chemical bonding. This is possible by refining the k and k radial scaling parameters and population coefficients Pim of the multipolar expansion. Spherical harmonics functions yim are used to describe the deformation part. Several software packages [68-71] are available for multipolar refinement of the electron density and some of them [68, 70, 72] also compute properties from the refined multipolar coefficients. 

A harmonic potential is a good approximation of the bond stretching function near the energy minimum (Fig. 2.7). Therefore, most programs use this approximation (see Eq. 2.6) however the limits of the simplification have to be kept in mind, in those cases where the anharmonicity becomes important. Apart from the possibility of including cubic terms to model anharmonicity fsee the second term in Eq. 2.14), which is done in the programs MM2 and MM3[1,2,2 241, the selective inclusion of 1,3-nonbonded interactions can also be used to add anharmonicity to the total potential energy function. 

ABCD may be defined, even though there is no bond between C andTt. The out-of-plane Eoop may then be described by an angle for example as a harmonic function... 

Orbitals (GTO). Slater type orbitals have the functional form e, if) = NYi, d, e- -- (5.1) is a normalization constant and T are the usual spherical harmonic functions. The exponential dependence on the distance between the nucleus and the electron mirrors the exact orbitals for the hydrogen atom. However, STOs do not have any radial nodes. centre of a bond. 5.2 Classification of Basis Sets Having decided on the type of function (STO/GTO) and the location (nuclei), the most important factor is the number of functions to be used. The smallest number of functions... 

Spherical harmonic functions are important in many problems in Chemistry and Physics. Spherical harmonic functions are central in discussions of rotation, motion in a central potential, multipole expansion, cluster bonding, spherical wave expansions and many more topics. The calculation of symmetrized powers of representations give a way of obtaining the... 

A general equation can be derived that describes the variation in direction of the valence electron density about the nucleus. The distortion from sphericity caused by valence electrons and lone-pair electrons is approximated by this equation, which includes a population parameter, a radial size function, and a spherical harmonic function, equivalent to various lobes (multipoles). In the analysis the core electron density of each atom is assigned a fixed quantity. For example, carbon has 2 core electrons and 4 valence electrons. Hydrogen has no core electrons but 1 valence electron. Experimental X-ray diffraction data are used to deri e the parameters that correspond to this function. The model is now more complicated, but gives a better representation of the true electron density (or so we would like to think). This method is useful for showing lone pair directionalities, and bent bonds in strained molecules. Since a larger number of diffraction data are included, the geometry of the molecular structure is probably better determined. 

Figure 28-7 In force field calculations, different levels of approximations are used to reproduce the stretching and compression of chemical bonds The plot shows a Morse potential energy function siipenmposed with various power series approximations (quadratic, cubic, and quartic functions) Note that the bottoms of the curves, representing the bond length for most chemical bonds of interest to medicinal chemists, almost overlap exactly. This nearly perfect fit in the bonding region is the reason simple harmonic functions can be used to calculate tx>nd lengths for unstrained molecular structures in the force lield method.

For many-electron systems the C02 has been taken as a prototype(80). Using a basis set containing higher harmonics on the nuclear centres, together with bond centred functions originally developed for JV2 and later applied to CO as a universal basis set, a molecular basis set was developed for CO2 which may be designated... 

The best description of a bond stretch is a Morse function but, as this is computationally expensive, a simpler harmonic function is usually used in molecular modelling. Many force fields use extra terms in the equation to improve the accuracy of the function which, at its simplest, has the form ... 

Angles are treated in a similar way to bonds by using a harmonic function based on ... 

To study the bonding in transition metal cluster compounds, a new type of Spherical Harmonic, with tensor properties, is required. This is because the metal d orbitals have (in addition to 1 o and 2 jt components) 2 8 components (d and dx2 y2) which are doubly noded in the plane perpendicular to the radial vector (see Fig. 16c) and which, therefore, behave as tensors. Two Tensor Surface Harmonic functions may be obtained from each Scalar Spherical Harmonic as follows146 ... 

Although in this case the form of the bond potential is well known, this simple example clearly demonstrates the way in which the technique can be used to extract information about the functional form from ab initio calculations. If we did not know how the bond stretch should be adequately represented, this analysis would clearly confirm that a Morse function is more accurate than a harmonic function. 

The twin internal orbitals are uninodal (i.e., n-type)and lead to surface bonding described by the vector surface harmonics. Two vector surface harmonic functions can be generated from each Yu, as follows ... 

The calculated harmonic vibrational frequencies for N2O are shown in Table 23. The derived experimental values are also quoted ". As expected, the MP2 N—N (2159 cm ) and N—O (1251 cm ) stretches are underestimated, with the triple-bond harmonic mode having the larger calculated-experiment gap. The observed fundamental vibrational frequencies(2224 and 1285 cm respectively) for these two modes are naturally smaller than the derived harmonic values, and are therefore closer to the calculated stretch frequencies. The bending mode is very well calculated, despite the absence of f-type functions from the basis set, as has been shown necessary for acetylene. 

Note that a combination of various types of potentials can be activated. For example, the coordination geometry for octahedral transition metal compounds can be modeled by 1,3-non-bonded interactions in combination with a multiple harmonic function. This is the approach used in the Momec force field for a number of metal... 

Build the bispidine complex using Momec and refine the corresponding cobalt(III) and cobalt(II) complexes (note that the force fields are not optimized for tetrahedral chromophores, but our aim here is just to compute relative metal—ligand distances). Note that all these structural optimizations need to be performed with 1,3-non-bonded interactions alone for the angular geometry around the metal ions (i.e., deactivate the multiple harmonic functions for both metal atom types in the force field) compare your results with those in Table 17.16. 

The TRIPOS force fields in the SYBYL and Alchemy programs and the Chem-X, CHARMm, and COSMIC force fields all employ a simple harmonic potential (Eq. [1]) for bond stretching/compression. The CVFF, DREIDING, and UFF force fields support a Morse potential as well as a harmonic potential. The harmonic function is the default in DREIDING and UFF. 

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