Motion, internal

As we shall see, in molecules as well as atoms, the interplay between the quantum description of the internal motions and the corresponding classical analogue is a constant theme. However, when referring to the internal motions of molecules, we will be speaking, loosely, of the motion of the atoms in the molecule, rather than of the fiindamental constituents, the electrons and nuclei. This is an extremely fundamental point to which we now turn. 

The result is that, to a very good approxunation, as treated elsewhere in this Encyclopedia, the nuclei move in a mechanical potential created by the much more rapid motion of the electrons. The electron cloud itself is described by the quantum mechanical theory of electronic structure. Since the electronic and nuclear motion are approximately separable, the electron cloud can be described mathematically by the quantum mechanical theory of electronic structure, in a framework where the nuclei are fixed. The resulting Bom-Oppenlieimer potential energy surface (PES) created by the electrons is the mechanical potential in which the nuclei move. Wlien we speak of the internal motion of molecules, we therefore mean essentially the motion of the nuclei, which contain most of the mass, on the molecular potential energy surface, with the electron cloud rapidly adjusting to the relatively slow nuclear motion. 

We will now treat the internal motion on the PES in cases of progressively increasing molecular complexity. We start with the simplest case of all, the diatomic molecule, where the notions of the Bom-Oppenlieimer PES and internal motion are particularly simple. 

It must be pointed out that another type of internal motion is the overall rotation of the molecule. The vibration and rotation of the molecule are shown schematically in figure Al.2.2. 

Flowever, if only the Darling-Deimison conplmg is important for the conpled stretches, what is its importance telling ns about the internal molecnlar motion It turns out that the right kind of analysis of the spectroscopic fitting Flamiltonian reveals a vast amonnt about the dynamics of the molecnle it allows ns to decipher the story encoded in the spectrum of what the molecule is really doing in its internal motion. We will approach this spectral cryptology from two complementary directions ... 

Once the partition function is evaluated, the contributions of the internal motion to thennodynamics can be evaluated. depends only on T, and has no effect on the pressure. Its effect on the heat capacity can be... 

Rotational diffusion coefficient, Dg, internal motion rate parameter, angle between the internal rotation axis and the internuclear axis... 

When relaxation of the internal motion during the collision is fast compared with the slow collision speed v, or when the relaxation time is short compared with the collision time, the kinetic energy operator... 

When relaxation of the internal motion is slow compared with the fast relative speed v., then T is expanded in temrs of the known unperturbed (diabatic) ortironomral eigenstates j(V> ... 

To enable an atomic interpretation of the AFM experiments, we have developed a molecular dynamics technique to simulate these experiments [49], Prom such force simulations rupture models at atomic resolution were derived and checked by comparisons of the computed rupture forces with the experimental ones. In order to facilitate such checks, the simulations have been set up to resemble the AFM experiment in as many details as possible (Fig. 4, bottom) the protein-ligand complex was simulated in atomic detail starting from the crystal structure, water solvent was included within the simulation system to account for solvation effects, the protein was held in place by keeping its center of mass fixed (so that internal motions were not hindered), the cantilever was simulated by use of a harmonic spring potential and, finally, the simulated cantilever was connected to the particular atom of the ligand, to which in the AFM experiment the linker molecule was connected. 

The explicit definition of water molecules seems to be the best way to represent the bulk properties of the solvent correctly. If only a thin layer of explicitly defined solvent molecules is used (due to hmited computational resources), difficulties may rise to reproduce the bulk behavior of water, especially near the border with the vacuum. Even with the definition of a full solvent environment the results depend on the model used for this purpose. In the relative simple case of TIP3P and SPC, which are widely and successfully used, the atoms of the water molecule have fixed charges and fixed relative orientation. Even without internal motions and the charge polarization ability, TIP3P reproduces the bulk properties of water quite well. For a further discussion of other available solvent models, readers are referred to Chapter VII, Section 1.3.2 of the Handbook. Unfortunately, the more sophisticated the water models are (to reproduce the physical properties and thermodynamics of this outstanding solvent correctly), the more impractical they are for being used within molecular dynamics simulations. 

It is possible (see, for example, J. Nichols, H. E. Taylor, P. Schmidt, and J. Simons, J. Chem. Phys. 92, 340 (1990) and references therein) to remove from H the zero eigenvalues that correspond to rotation and translation and to thereby produce a Hessian matrix whose eigenvalues correspond only to internal motions of the system. After doing so, the number of negative eigenvalues of H can be used to characterize the nature of the... 

The time step should be at least an order of magnitude lower than the shortest period of internal motion. If hydrogens are present in a system near room temperature a value of 0.5 femtoseconds (0.0005 ps) is usually appropriate. 

Transitions. Samples containing 50 mol % tetrafluoroethylene with ca 92% alternation were quenched in ice water or cooled slowly from the melt to minimise or maximize crystallinity, respectively (19). Internal motions were studied by dynamic mechanical and dielectric measurements, and by nuclear magnetic resonance. The dynamic mechanical behavior showed that the CC relaxation occurs at 110°C in the quenched sample in the slowly cooled sample it is shifted to 135°C. The P relaxation appears near —25°C. The y relaxation at — 120°C in the quenched sample is reduced in peak height in the slowly cooled sample and shifted to a slightly higher temperature. The CC and y relaxations reflect motions in the amorphous regions, whereas the P relaxation occurs in the crystalline regions. The y relaxation at — 120°C in dynamic mechanical measurements at 1 H2 appears at —35°C in dielectric measurements at 10 H2. The temperature of the CC relaxation varies from 145°C at 100 H2 to 170°C at 10 H2. In the mechanical measurement, it is 110°C. There is no evidence for relaxation in the dielectric data. 

In the equation referred to above, it is assumed that there is 100 percent transmission of the shear rate in the shear stress. However, with the slurry viscosity determined essentially by the properties of the slurry, at high concentrations of slurries there is a shppage factor. Internal motion of particles in the fluids over and around each other can reduce the effective transmission of viscosity efficiencies from 100 percent to as low as 30 percent. 

Direct experiment-simulation quasielastic neutron scattering comparisons have been perfonned for a variety of small molecule and polymeric systems, as described in detail in Refs. 4 and 18-21. The combination of simulation and neutron scattering in the analysis of internal motions in globular proteins was reviewed in 1991 [3] and 1997 [4]. 

A dynamic transition in the internal motions of proteins is seen with increasing temperamre [22]. The basic elements of this transition are reproduced by MD simulation [23]. As the temperature is increased, a transition from harmonic to anharmonic motion is seen, evidenced by a rapid increase in the atomic mean-square displacements. Comparison of simulation with quasielastic neutron scattering experiment has led to an interpretation of the dynamics involved in terms of rigid-body motions of the side chain atoms, in a way analogous to that shown above for the X-ray diffuse scattering [24]. 

Having demonstrated that our simulation reproduces the neutron data reasonably well, we may critically evaluate the models used to interpret the data. For the models to be analytically tractable, it is generally assumed that the center-of-mass and internal motions are decoupled so that the total intermediate scattering function can be written as a product of the expression for the center-of-mass motion and that for the internal motions. We have confirmed the validity of the decoupling assumption over a wide range of Q (data not shown). In the next two sections we take a closer look at our simulation to see to what extent the dynamics is consistent with models used to describe the dynamics. We discuss the motion of the center of mass in the next section and the internal dynamics of the hydrocarbon chains in Section IV.F. 

Two physically reasonable but quite different models have been used to describe the internal motions of lipid molecules observed by neutron scattering. In the first the protons are assumed to undergo diffusion in a sphere [63]. The radius of the sphere is allowed to be different for different protons. Although the results do not seem to be sensitive to the details of the variation in the sphere radii, it is necessary to have a range of sphere volumes, with the largest volume for methylene groups near the ends of the hydrocarbon chains in the middle of the bilayer and the smallest for the methylenes at the tops of the chains, closest to the bilayer surface. This is consistent with the behavior of the carbon-deuterium order parameters,. S cd, measured by deuterium NMR ... 

Another way to describe deviations from the simple BPP spectral density is the so-called model-free approach of Lipari and Szabo [10]. This takes account of the reduction of the spectral density usually observed in NMR relaxation experiments. Although the model-free approach was first applied mainly to the interpretation of relaxation data of macromolecules, it is now also used for fast internal dynamics of small and middle-sized molecules. For very fast internal motions the spectral density is given by ... 

Worse was to come. Boltzmann in 1872 made the same weird statistical equality hold for every mode in a dynamical system. It must, for example, apply to any internal motions that molecules might have. Assuming, as most physicists did by then, that the sharp lines seen in the spectra of chemical elements originate in just such internal motions, any calculation now of Cp/C would yield a figure even lower than 1.333. Worse yet, as Maxwell shatteringly remarked to one student, equipartition must apply to solids and liquids as well as gases Boltzmann has proved too much. ... 

Some molecules undergo an internal motion in which one part of the molecule rotates about a bond connecting it with the rest of the molecule. Some examples are... 

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