Morse force constant

For each pair of interacting atoms (/r is their reduced mass), three parameters are needed D, (depth of the potential energy minimum, k (force constant of the par-tictilar bond), and l(, (reference bond length). The Morse ftinction will correctly allow the bond to dissociate, but has the disadvantage that it is computationally very expensive. Moreover, force fields arc normally not parameterized to handle bond dissociation. To circumvent these disadvantages, the Morse function is replaced by a simple harmonic potential, which describes bond stretching by Hooke s law (Eq. (20)). 

A larger value for the stretch force constant Kj. leads to a greater tendency for the bond to remain at its equilibrium distance rg Higher powers of r - rg, giving cubic, quartic, or higher terms are also common. A Morse function might also be employed. 

Some additional comments regarding Equation 4 are in order. The factor 143.88 converts the units to kcal/mol. There are two additional constants. The first is ks, which is the stretching force constant parameter in units of md A 1. The second constant is cs, which is the cubic term with a unitless value of 2.55. When a Morse potential is expanded in a power series, the factor 7/12 is obtained. 

Fig. 3.1 Born-Oppenheimer vibrational potentials for a diatomic molecule corresponding to the CH fragment. The experimentally realistic anharmonic potential (solid line) is accurately described by the Morse function Vmorse = De[l — exp(a(r — r0)]2 with De = 397kJ/mol, a = 2A and ro = 1.086 A (A = Angstrom = 10 10m). Near the origin the BO potential is adequately approximated by the harmonic oscillator (Hooke s Law) function (dashed line), Vharm osc = f(r — ro)2/2. The harmonic oscillator force constant f = 2a2De...

The attractive Morse potential has three variable parameters which can be fitted to the dissociation energy, the harmonic force constant and the equilibrium distance. The hi er force constants are then in error, but zsHulburt mdHirschfelder 123) showed these can be corrected by multiplying the repulsive part of the potential by a polynomial as follows... 

The harmonic oscillator model does not take into account the real nature of chemical bonds, which are not perfect springs. The force constant k decreases if the atoms are pulled apart and increases significantly if they are pushed close together. The vibrational levels, instead of being represented by a parabolic function as in equation (10.3), are contained in an envelope. This envelope can be described by the Morse equation (Fig. 10.5) ... 

The frequency-time correlation function is dependent on the frequency and the force constants of the vibrational mode whose dephasing is being considered. They are determined by fitting the vibrational bond energies to a Morse potential of the following form ... 

Derive an expression for the (quadratic) force constant of a Morse potential... 

It is common practice to present the cubic and quartic force constants f3 and /4 in a slightly different way, in terms of so-called Morse parameters.0 6 The Morse function is an empirical diatomic potential function of the form... 

Kuchitsu and co-workers5 7 were the first to introduce what is perhaps the simplest and most generally useful model, in which they assume all anharmonic force constants in curvilinear co-ordinates to be zero with the exception of cubic and quartic bond-stretching constants. These may be estimated from the corresponding diatomics, or from a Morse function, or they may be adjusted to give the best fit to selected spectroscopic constants to which they make a major contribution. This is often called the valence-force model. It is clear from the results on general anharmonic force fields quoted above that this model is close to the truth, and in fact summarizes 80 % of all that we have learnt so far about anharmonic force fields. 

This treats the bond as a mechanical spring whose force constant is strong for small and weak for large interatomic distances. The disadvantage of using a Morse function in empirical force field calculations is that an exponential in addition to the square function and three parameters are involved, increasing the time requirement for the minimization process and the complexity of the force field parameterization. 

To establish the relationship between a and the force constants of eq. (11), the Morse function is expanded as an exponential series, using the same zero point in energy ... 

With the quadratic force constant ke in units of mdyneA-1 and De in eV, the Morse constant in atomic units (a x) is calculated as... 

If this Morse function is used to represent any single bond, not necessarily of a diatomic molecule, the constant a calculated from the harmonic force constant may not be entirely appropriate, and especially not over the entire range of r. Before deriving multiple-bond properties from the single-bond curve it is therefore useful to optimize the Morse constant empirically to improve the match between calculated and observed single-bond values of De and re. 

Calculate the harmonic force constant 4 and the Morse parameter j8 for the two states. Using the known r J value (0.2666 nm), plot the Morse curve for the ground electronic state of I2. Compare this with the harmonic potential calcnlated from k. ... 

Fig. 2.27. (a) The experimental potential curve for H. (full curve) the dotted curve represents a Morse fit to the data, and the dashed curve shows a harmonic-oscillator function with the force constant taken at r = (0.7414 A). The first... 

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