Hooke potential

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)). 

Compared with the Morse potential, Hooke s law performs reasonably well in the equilibrium area near If, where the shape of the Morse function is more or less quadratic (see Figure 7-9 in the minimum-energy region). To improve the performance of the harmonic potential for non-equilibrium bond lengths also, higher-order terms can be added to the potential according to Eq. (21). 

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). 

Comparison of the simple harmonic potential (Hooke s law) with the Morse curve. 

Hooke s law functional form is a reasonable approximation to the shape of the potential gy curve at the bottom of the potential well, at distances that correspond to bonding in md-state molecules. It is less accurate away from equilibrium (Figure 4.5). To model the se curve more accurately, cubic and higher terms can be included and the bond- ching potential can be written as follows ... 

As a simple example of a normal mode calculation consider the linear triatomic system ir Figure 5.16. We shall just consider motion along the long axis of the molecule. The displace ments of the atoms from their equilibrium positions along this axis are denoted by It i assumed that the displacements are small compared with the equilibrium values Iq and th( system obeys Hooke s law with bond force constants k. The potential energy is given by ... 

Just as the electrical behaviour of a real diatomic molecule is not accurately harmonic, neither is its mechanical behaviour. The potential function, vibrational energy levels and wave functions shown in Figure f.i3 were derived by assuming that vibrational motion obeys Hooke s law, as expressed by Equation (1.63), but this assumption is reasonable only... 

Hazard An inherent chemical or physical characteristic that has the potential for cansing damage to people, property, or the environment. In this hook, it is typically the comhination of a hazardons material, an operating environment, and certain nnplanned events that conld resnlt in an accident. 

A simple eigenvalue problem can be demonstrated by the example of two coupled oscillators. The system is illustrated in fug. 2. It should be compared with the classical harmonic oscillator that was treated in Section 5.2.2. Here also, the system will be assumed to be harmonic, namely, that both springs obey Hooke s law. The potential energy can then be written in the form... 

The classical harmonic oscillator in one dimension was illustrated in Seetfon 5.2.2 by the simple pendulum. Hooke s law was employed in the fSfin / = —kx where / is the force acting on the mass and k is the force constant The force can also be expressed as the negative gradient of a scalar potential function, V(jc) = for the problem in one dimension [Eq. (4-88)]. Similarly, the three-dimensional harmonic oscillator in Cartesian coordinates can be represented by the potential function... 

The family Caprifoliaceae comprises approx 400 species, of which Lonicera japonica Thunb., Lonicera affinis Hook and Arn, Lonicera confusa DC, Sambucus javanica Reinw. ex. Bl, Sambucus sieboldiana (Miq.) Graebn, and Weigela floribunda (Sieb. and Zucc.) K. Koch, are used to treat inflammatory conditions in Asia and the Pacific. There is an expanding body of evidence to suggest that biflavonoids from this family might hold some potential as phospholipase A2 inhibitors. One such compound is ochnaflavone from Lonicera japonica Thunb. 

Theoretical models include those based on classical (Newtonian) mechanical methods—force field methods known as molecular mechanical methods. These include MM2, MM3, Amber, Sybyl, UFF, and others described in the following paragraphs. These methods are based on Hook s law describing the parabolic potential for the stretching of a chemical bond, van der Waal s interactions, electrostatics, and other forces described more fully below. The combination assembled into the force field is parameterized based on fitting to experimental data. One can treat 1500-2500 atom systems by molecular mechanical methods. Only this method is treated in detail in this text. Other theoretical models are based on quantum mechanical methods. These include ... 

The half-wave potentials of K+, Tl+ and Ca2+ in water are slightly more negative and thosefor Zn2+, Cd2+, Mn2+, Ni2+ and Co2+ considerably more negative than is expected according to the donicity rule. It has been shown in the previous sections that water is a rather unique solvent. The effect in question may be interpreted by the so-called Katzin-effect according to which water forms a royal core of coordinated water molecules which are hooked together by hydrogen bonds 70,71>122,1231. 

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...

Fig. 5.1 A schematic projection of the 3n dimensional (per molecule) potential energy surface for intermolecular interaction. Lennard-Jones potential energy is plotted against molecule-molecule separation in one plane, the shifts in the position of the minimum and the curvature of an internal molecular vibration in the other. The heavy upper curve, a, represents the gas-gas pair interaction, the lower heavy curve, p, measures condensation. The lighter parabolic curves show the internal vibration in the dilute gas, the gas dimer, and the condensed phase. For the CH symmetric stretch of methane (3143.7 cm-1) at 300 K, RT corresponds to 8% of the oscillator zpe, and 210% of the LJ well depth for the gas-gas dimer (Van Hook, W. A., Rebelo, L. P. N. and Wolfsberg, M. /. Phys. Chem. A 105, 9284 (2001))...

Kluwe WM, Hook JB. 1981. Potentiation of acute chloroform nephrotoxicity by the glutathione depletor diethyl maleate and protection by the microsomal enzyme inhibitor piperonyl butoxide. Toxicol Appl Pharmacol 59 457-466. 

The first edition of Automatic Chemical Analysis was published in 1974 and was written in coUahoration with the late Jim Foreman. It was well received and became a set course book in many universities. Its aim was to provide a basic grounding in the subject and to cover the wider issues of economic assessments, educational requirements and specification of systems. TTiere are now other books on laboratory automation in print [1,2], but none of these will help potential users of automation to evaluate the economic viability of a project indeed, they may even put them off. The other hooks are scholarly rather than practical. 

If position X of the particle corresponds to its displacement from the origin, the restoring force acting on the particle by virtue of the harmonic potential is given by Hooke s law ... 

Differences in the potential energy t/ of a Hooke s Law spring can be calculated as the product of the restoring force (—/) and the distance dx over which the force acts. Thus, by substituting -kx in place of/, we get... 

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