Stretching, bond

To calculate the bonded interaction of two atoms, a Morse function is often used. It has the form described by Eq, (19). 

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)). 

In this case, only two parameters (k and Iq) per atom pair are needed, and the computation of a quadratic function is less expensive. Therefore, this type of expression is used especially by biomolecular force fields (AMBER, CHARMM, GROMOS) dealing with large molecules like proteins, lipids, or DNA. 

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  

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. 

Here is the force constant, lo is the natural bond length, and / is the actual bond length. 

In the MM2 program, bond stretching is a little more complicated than the harmonic approximation as indicated in Eq. [3]. A cubic term has been added to better reproduce the Morse curve in the region where bonds are being pulled apart. 

Equation [3] looks complicated at first sight, but is just Eq, [2] to which a cubic term has been added. The cubic constant has the value - 2.00 times the quadratic constant. The factor of 143.88 converts the units to kcal/mol. Judicious. selection of the force constant parameter for this cubic expression allowed for accurate treatments of bond length deformations in a wide variety of molecules. 

In MM3, this problem of trying to compute as efficiently as possible has been corrected by adding a quartic term. In this way, the possibility of having the potential energy curve invert is eliminated. Moreover, the new curve is a better approximation to a Morse potential over a longer distance. Accordingly, MM3 has one additional term to describe bond stretching as shown in Eq. [4]. 

The factor 7/12 is used because it is obtained when a Morse potential is expanded in a power series. 

Dg is the depth of the potential energy minimum and a = wy /n/2De), where /n is the reduced mass and w is the frequency of the bond vibration, w is related to the stretching constant of the bond. If, by w = is the reference value of the bond. The Morse potential is not 

Fig 4 4 Variation in bond energy with interatomic separation. 

The forces between bonded atoms are very strong and considerable energy is required to cause a bond to deviate significantly from its equilibrium value. This is reflected in the magnitude of the force constants for bond stretching some typical values from the MM2 force field are shown in Table 4.1, where it can be seen that those bonds one would 

300 kcal mol A would cause the energy of the system to rise by 12kcal/mol. 

Fig 4 6 A cubic bond-stretdiing potential passes throng a maximum but gives a better approximation to the Morse curve close to tfie equilibrium structure than the quadratic form 

Before we go on to consider functional forms for all of the components of a molecule s total steric energy, let us consider the limitations of Eq. (2.2) for bond stretching. Like any truncated Taylor expansion, it works best in regions near its reference point, in this case req. Thus, if we are interested primarily in molecular structures where no bond is terribly distorted from its optimal value, we may expect Eq. (2.2) to have reasonable utility. However, as the bond is stretched to longer and longer r, Eq. (2.2) predicts the energy to become infinitely positive, which is certainly not chemically realistic. The practical solution to such inaccuracy is to include additional terms in the Taylor expansion. Inclusion of the cubic term provides a potential energy function of the form 

the simple, practical solution is to include the next term in the Taylor expansion, namely the quartic term, leading to an expression of the form 

The alert reader may wonder, at this point, why there has been no discussion of the Morse function 

Even in these instances, however, there is some utility to considering the Morse function. If we approximate the exponential in Eq. (2.5) as its infinite series expansion truncated at the cubic term, we have 

Typically, the simplest parameters to determine from experiment are ab and Dab- With these two parameters available, aAB can be determined from Eq. (2.8), and thus the cubic and quartic force constants can also be determined from Eqs. (2.4) and (2.7). Direct measurement of cubic and quartic force constants requires more spectral data than are available for many kinds of bonds, so this derivation facilitates parameterization. We will discuss parameterization in more detail later in the chapter, but turn now to consideration of other components of the total molecular energy. 

Squaring the quantity in braces and keeping only terms through quartic gives 

We begin by defining the individual terms of the equation for Etot, as well as presenting some discussion of the nature of the various parameters. The total energy, i produced by a molecular mechanics calculation is also referred to as the steric energy. It is not to be confused with strain energy, a very different quantity, as we will elaborate below. 

The individual terms in Eq. 2.40 can each be viewed as a potential function, and they have the same mathematical forms as those for stretches, bends, and torsions that we discussed earlier in this chapter. It is important to remember, however, that the parameters used in the equations that describe the real degrees of freedom of molecules do not necessarily have any relation to the parameters used in the equations of the molecular mechanics method. Moreover, whereas the potential surfaces that describe the vibrational degrees of freedom in molecules derive from the forces that hold the atoms together, the potential functions in molecular mechanics are derived simply to get the right answer. 

The standard equation for bond stretching is Eq. 2.41, where r is the length of the bond being evaluated, kr is analogous to a force constant, and r is the natural bond length. 

However, at greater values of r—when a bond is stretched—the approximation is quite poor. For this reason, many force fields add a cubic term to the stretching potential function (Eq. 2.42). 

This expansion introduces another parameter (k/), but it does improve the force field. For the highest possible precision in calculations of organic molecules, such additional terms are usually included. However, in a force field for proteins or nucleic acids (see below), structures that rarely deviate substantially from standard bonding parameters, cubic terms are often unnecessary. 

Having obtained structural models of high accuracy, we can now turn to our main purpose, the enumeration of topological constraints from the atomic scale trajectories obtained by MD simulations (Fig. 11.8). 

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One can note some interesting features from these trajectories. For example, the Mulliken population on the participating atoms in Figure 1 show that the departing deuterium canies a full electron. Also, the deuterium transferred to the NHj undergoes an initial substantial bond stretch with the up spin and down spin populations separating so that the system temporarily looks like a biradical before it settles into a normal closed-shell behavior. 

Figure 1, Coordinates used for describing the dynamics of a) H -I- H2 (6) NOCl, (c) butatriene, (a), (b) Are Jacobi coordinates, where and are the dissociative and vibrational coordinates, respectively, (c) Shows the two most important normal mode coordinates, Qs and Q a, which are the torsional and central C—C bond stretch, respectively.

Fig. 1. The time evolution (top) and average cumulative difference (bottom) associated with the central dihedral angle of butane r (defined by the four carbon atoms), for trajectories differing initially in 10 , 10 , and 10 Angstoms of the Cartesian coordinates from a reference trajectory. The leap-frog/Verlet scheme at the timestep At = 1 fs is used in all cases, with an all-atom model comprised of bond-stretch, bond-angle, dihedral-angle, van der Waals, and electrostatic components, a.s specified by the AMBER force field within the INSIGHT/Discover program.

For example, the SHAKE algorithm [17] freezes out particular motions, such as bond stretching, using holonomic constraints. One of the differences between SHAKE and the present approach is that in SHAKE we have to know in advance the identity of the fast modes. No such restriction is imposed in the present investigation. Another related algorithm is the Backward Euler approach [18], in which a Langevin equation is solved and the slow modes are constantly cooled down. However, the Backward Euler scheme employs an initial value solver of the differential equation and therefore the increase in step size is limited. 

In an atomic level simulation, the bond stretch vibrations are usually the fastest motions in the molecular dynamics of biomolecules, so the evolution of the stretch vibration is taken as the reference propagator with the smallest time step. The nonbonded interactions, including van der Waals and electrostatic forces, are the slowest varying interactions, and a much larger time-step may be used. The bending, torsion and hydrogen-bonding forces are treated as intermediate time-scale interactions. 

In this article we describe an extension of SISM to a system of molecules for which it can be assumed that both bond stretching and angle bending describe satisfactorily all vibrational motions of the molecule. The SISM presented here allows the use of an integration time step up to an order of magnitude larger than possible with other methods of the same order and complexity. 

SISM Treatment of Bond Stretching and Angle Bending Terms... 

For the model Hamiltonian used in this study it was assumed that the bond stretching and angle i)ending satisfactorily describe all vibrational motions... 

SISM Treatment of only Bond Stretching Term... 

SISM for an Isolated Linear Molecule An efficient symplectic algorithm of second order for an isolated molecule was studied in details in ref. [6]. Assuming that bond stretching satisfactorily describes all vibrational motions for linear molecule, the partitioned parts of the Hamiltonian are... 

As for bond stretching, the simplest description of the energy necessary for a bond angle to deviate firom the reference value is a harmonic potential following Hooke s law, as shown in Eq. (22). 

Intensive use of cross-terms is important in force fields designed to predict vibrational spectra, whereas for the calculation of molecular structure only a limited set of cross-terms was found to be necessary. For the above-mentioned example, the coupling of bond-stretching (f and / and angle-bending (B) within a water molecule (see Figure 7-1.3, top left) can be calculated according to Eq. (30). 

Figure 7-13. Cross-terms combining internal vibrational modes such as bond stretch, angle bend, and bond torsion within a molecule.

Directly bonded (a 1-2 bond stretch relationship) Geniinal to each other (a 1-3 angle bending relationship)... 

In addition to these basic term s. force fieldsoften h ave cross term s that combine the above interactions. For example there may be a term which causes ati angle bend to interact with a bond stretch term (opening a bond angle may tend to lengthen the bonds in volved). 

In stead, the electrostatic con tribn tion conies from definin g a set of bond dipole moments associated woth polar bonds. These bond moments are defined in the m m psir.LxL(dbf) file along with the bond stretching parameters and are given in units of Debyes. The cen ter of th e dipole Is defined to be th e m Idpoint of the bond an d two dipoles p. and pj. separated by Rjj. as shown beltnv ... 

Bond Stretch and Angle Bending Cross Term... 

Th c fun ction al form for bon d stretch in g in HlOa, as in CHARMM, is quadratic only and is identical to that shown in equation (1 1) on page I 75. Th e bond stretch in g force con stan ts are in units of... 

The default parameters for bond stretching are an ec iiilibriiim bond length an d a stretch in g force eon starit. fb e fun etion al form isjiist that of the. M.M+ force field including a correction for cubic stretches. The default force constant depends only on the bond... 

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