Ball-and-spring model

Figure 1.11(b) illustrates the ball-and-spring model which is adequate for an approximate treatment of the vibration of a diatomic molecule. For small displacements the stretching and compression of the bond, represented by the spring, obeys Hooke s law ... 

The molecular mechanics method, often likened to a ball and spring model of the molecule, represents the total energy of a system of molecules with a set of simple analytical functions representing different interactions between bonded and non-bonded atoms, as shown schematically in Figure 1. 

Molecular mechanics - based on a ball-and-springs model of molecules Ab initio methods - based on approximate solutions of the Schrodinger equation without appeal to fitting to experiment... 

Is it surprising that the geometry and energy (compared to that of other isomers) of a molecule can often be accurately calculated by a ball-and-springs model (molecular mechanics) ... 

Near the equilibrium bond length qe the potential energy/bond length curve for a macroscopic balls-and-spring model or a real molecule is described fairly well by a quadratic equation, that of the simple harmonic oscillator (E = ( /2)K (q — qe)2, where k is the force constant of the spring). However, the potential energy deviates from the quadratic (q ) curve as we move away from qc (Fig. 2.2). The deviations from molecular reality represented by this anharmonicity are not important to our discussion. 

Would you dispute the suggestion that no matter how accurate a set of MM results might be, they cannot provide insight into the factors affecting a chemical problem, because the ball and springs model is unphysical ... 

We have seen three broad techniques for calculating the geometries and energies of molecules molecular mechanics (Chapter 3), ab initio methods (Chapter 5), and semiempirical methods (Chapters 4 and 6). Molecular mechanics is based on a balls-and-springs model of molecules. Ab initio methods are based on the subtler model of the quantum mechanical molecule, which we treat mathematically starting with the Schrodinger equation. Semiempirical methods, from simpler ones like the Hiickel and extended Hiickel theories (Chapter 4) to the more complex SCF semiempirical theories (Chapter 6), are also based on the Schrodinger equation, and in fact their empirical aspect comes from the desire to avoid the mathematical... 

First, the ball and springs model used in molecular mechanics is not completely nonphysical to a fair approximation, molecules really do vibrate and bonds do stretch and bend, as expected from a macroscopic ball and springs model. It is when we want to examine inescapably electronic properties, like, say, UV spectra or the donation of electrons from one species to another to make a bond, that the MM model is completely inadequate. 

The constructions of different approximations will be done in the sections that follow on the basis of the variational principle for molecular electronic energy in the SLG-based approximation. We shall demonstrate that this treatment leads to a mechanistic model which can in a sense be considered a generic or deductive form of MM. It means that although the simple balls-and-springs model can hardly be justified from any general point of view, it does not mean that any other mechanistic model cannot be justified at all. And that is what we shall provide. 

Molecular mechanics - based on a ball-and-springs model of molecules ... 

The easiest way of modelling molecular vibrations is to imagine the atoms in a molecule as balls, and the chemical bonds connecting them as massless springs. Such a ball-and-spring model for a diatomic molecule is illustrated in Fig. 4.1. Let us assume that the masses of the two atoms are and nt2, respectively, and that the restoring force F of the spring is proportional to the displacement X of the atoms from their equilibrium position... 

---