Fixed bond

A diatomie moleeule with fixed bond length R rotating in the absenee of any external potential is deseribed by the following Sehrodinger equation ... 

As given in Chapter 3, the Schrodinger equation for the angular motion of a rigid (i.e., having fixed bond length R) diatomic molecule is... 

For non-linear molecules, when treated as rigid (i.e., having fixed bond lengths, usually taken to be the equilibrium values or some vibrationally averaged values), the rotational Hamiltonian can be written in terms of rotation about three axes. If these axes (X,Y,Z) are located at the center of mass of the molecule but fixed in space such that they do not move with the molecule, then the rotational Hamiltonian can be expressed as ... 

The semi-empirical methods of HyperChem are quantum mechanical methods that can describe the breaking and formation of chemical bonds, as well as provide information about the distribution of electrons in the system. HyperChem s molecular mechanics techniques, on the other hand, do not explicitly treat the electrons, but instead describe the energetics only as interactions among the nuclei. Since these approximations result in substantial computational savings, the molecular mechanics methods can be applied to much larger systems than the quantum mechanical methods. There are many molecular properties, however, which are not accurately described by these methods. For instance, molecular bonds are neither formed nor broken during HyperChem s molecular mechanics computations the set of fixed bonds is provided as input to the computation. 

This kind of perfect flexibility means that C3 may lie anywhere on the surface of the sphere. According to the model, it is not even excluded from Cj. This model of a perfectly flexible chain is not a realistic representation of an actual polymer molecule. The latter is subject to fixed bond angles and experiences some degree of hindrance to rotation around bonds. We shall consider the effect of these constraints, as well as the effect of solvent-polymer interactions, after we explore the properties of the perfectly flexible chain. Even in this revised model, we shall not correct for the volume excluded by the polymer chain itself. 

Figure 1.7 Placement of successive polymer segments connected at fixed bond angles, (a) Carbon 3 can lie anywhere on the rim of the cone, (b) This effect is illustrated for a longer portion of chain. [Panel (a) reprinted from Ref. 4, p. 118.]...

Figure 4 Time step dependence of the average total energy for two models of a partially hydrated dodecamer DNA duplex. Thinner traces show results for virtually harmonic conditions when temperature was lowered to 1 K. The DNA molecule has fixed bond lengths, rigid bases, and fixed valence angles except for the mtra- and extracyclic bond angles m sugars, (a) No modifications of inertia (b) inertia modified as explained m the text. (From Ref. 54.)...

Sukigara and Kikuchi interpreted these experimental data by carrying out Pariser-Parr (PP) calculations on the benzenediazonium ion. On the basis of Romming s X-ray structure of the benzenediazonium ion (1963), with fixed bond integrals for the CC and CN bonds (/ Cc and / CN), but an adjustable bond integral / NN, they were able to ascribe the 259 nm and 295 nm bands to the lAx — xAi and transi-... 

Fig. 3.11. 3D-trajectory of a point in the moving residue during an entire revolution of the driving angle 0. The three trajectories correspond to slightly different values for the fixed bond angles... 

Fig. 3.12. a Displacement of a point in the moving residue compared with the fixed bond length, b First set of solutions for the torsion angles and c Second set of solutions for the torsion angles... 

We choose the potentials used in [146] with a fixed bond length and harmonic potentials for the cosine of the bond angles Of. 

In the application of the above treatment to the present model, account must be taken of the condition of fixed bond angle which disallows the overlap between two successive bonds in the chain. This feature can be accommodated in the scheme by replacing the quantities and ++ in the transition... 

Amines with three different substituents are potentially chiral because of the pseudotetrahedral arrangement of the three groups and the lone-pair electrons. Under normal conditions, however, these enantiomers are not separable because of the rapid inversion at the nitrogen center. As soon as the lone-pair electrons are fixed by the formation of quaternary ammonium salts, tertiary amide N-oxide, or any other fixed bonding, the inversion is prohibited, and consequently the enantiomers of chiral nitrogen compounds can be separated. 

Figure 2.8 shows part of a polyethylene glycol chain and illustrates the type of problem encountered with real chains. First, there are fixed bond angles (in this case the tetrahedral angle), and second, there is the physical volume occupied by each atom so that the walk cannot for example cross itself. Both of these constraints mean that the random... 

Usually, MD methods are applied to polymer systems in order to obtain short-time properties corresponding to problems where the influence of solvent molecules has to be explicitly included. Then the models are usually atomic representations of both chain and solvent molecules. Realistic potentials for non-bonded interactions between non-bonded atoms should be incorporated. Appropriate methods can be employed to maintain constraints corresponding to fixed bond lengths, bond angles and restricted torsional barriers in the molecules [117]. For atomic models, the simulation time steps are typically of the order of femtoseconds (10 s). However, some simulations have been performed with idealized polymer representations [118], such as Bead and Spring or Bead and Rod models whose units interact through parametric attractive-repulsive potentials. 

Additivity schemes with fixed bond energy (or enthalpy) parameters plus a host of corrective factors reflecting nonbonded steric interactions have a long history in the prediction of thermochemical properties, such as the classical enthalpy of formation of organic molecules. AUen-type methods, for example, nicely Ulustrate the usefulness of empirical bond additivity approaches [1,2]. 

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