Point masses

One nice thing about H in mass-scaled coordinates is that it is identical to the Hamiltonian of a mass point movmg in two dimensions. This is convenient for visualizing trajectory motions or wavepackets, so the mass-scaled coordinates are commonly used for plotting data from scattering calculations. 

Almost everything we deal with in this diseussion is based on eoneepts of loeal eause and effeet. Something that happens at a small, but finite, distance away from a given mass point has no direct influence on the material const tu-... 

With this approach, when an element becomes severly distorted, it is eliminated from the computational grid and becomes a free mass point. Clearly, care must be taken to avoid eliminating elements that could potentially influence the problem at some later time. An example of a three-dimensional Lagrangian calculation that uses the eroding element scheme is presented in the next section. 

Name Structure Molar Mass Point Solubility... 

The operator representing the angular momentum of a mass point circulating about the e-axis was given in the specific form... 

Table II contains a rough comparison of execution times for the generation of one data point 6x10 random conformations of chains of 100 mass-points were placed each at 100 equally spaced radial positions of a pore with Aq=0.8. It is obvious that the increase in performance, i.e., a reduction in execution time to 20%, is an excellent return on the investment required to change five lines of a FORTRAN program. We fear, however, that this is a relatively rare situation.

Figure 3. Partition coefficient of freely jointed chains between the bulk solution and a cylindrical pore. The chains have different numbers of mass-points (n) and different bond lengths, and are characterized by the root-mean-square radius of gyration measured in units of the pore radius. See text for details.

Creation and Testing of 6x10 Conformations 100 mass-points, placed at 100 radial positions... 

This quantity is of great importance, since it actually contains all information about electron correlation, as we will see presently. Like the density, the pair density is also a non-negative quantity. It is symmetric in the coordinates and normalized to the total number of non-distinct pairs, i. e., N(N-l).8 Obviously, if electrons were identical, classical particles that do not interact at all, such as for example billiard balls of one color, the probability of finding one electron at a particular point of coordinate-spin space would be completely independent of the position and spin of the second electron. Since in our model we view electrons as idealized mass points with no volume, this would even include the possibility that both electrons are simultaneously found in the same volume element. In this case the pair density would reduce to a simple product of the individual probabilities, i.e.,... 

The theory behind molecular vibrations is a science of its own, involving highly complex mathematical models and abstract theories and literally fills books. In practice, almost none of that is needed for building or using vibration spectroscopic sensors. The simple, classical mechanical analogue of mass points connected by springs is more than adequate. 

Systems involving more mass points are capable of more complex vibrations, since the vibrational modes may involve several to many atoms and all three dimensions are available for vibrational movements. Vibrations where primarily the distances along the bond axis between the involved atoms change during the vibration are called valence vibrations. Vibrations causing a deformation of a bond angle are referred to as deformation vibrations. Deformation movements can also rock , wag or twist a molecular (sub-) structure (Figure 1). 

If there is also an electric field present, the electric force qE must be taken into account as well. The complete force equation for a charged mass point, also known as the Lorentz force, is... 

In the real world the stress tensor never vanishes and so requires a nonvanishing curvature tensor under all circumstances. Alternatively, the concept of mass is strictly undefined in flat Minkowski space-time. Any mass point in Minkowski space disperses spontaneously, which means that it has a space-like rather than a time-like world line. In perfect analogy a mass point can be viewed as a local distortion of space-time. In euclidean space it can be smoothed away without leaving any trace, but not on a curved manifold. Mass generation therefore resembles distortion of a euclidean cover when spread across a non-euclidean surface. A given degree of curvature then corresponds to creation of a constant quantity of matter, or a constant measure of misfit between cover and surface, that cannot be smoothed away. Associated with the misfit (mass) a strain field appears in the curved surface. 

The electronic heat capacity for the free electron model is a linear function of temperature only for T Tp = p / kp. Nevertheless, the Fermi temperature Tp is of the order of 105 K and eq. (8.46) holds for most practical purposes. The population of the electronic states at different temperatures as well as the variation of the electronic heat capacity with temperature for a free electron gas is shown in Figure 8.20. Complete excitation is only expected at very high temperatures, T>Tp. Here the limiting value for a gas of structureless mass points 3/2/ is approached. 

The key idea of the fast torsion angle dynamics algorithm in Dyana is to exploit the fact that a chain molecule such as a protein or nucleic acid can be represented in a natural way as a tree structure consisting of n+1 rigid bodies that are connected by n rotatable bonds (Fig. 2.1) [74, 83]. Each rigid body is made up of one or several mass points (atoms) with invariable relative positions. The tree structure starts from a base, typically... 

Deformation of an elastic solid through which a mass point of the solid with co-ordinates Xi, X2, X-i in the undeformed state moves to a point with co-ordinates xi, X2, X3 in the deformed state and the deformation is defined by... 

Note 1 A strain tensor is a measure of the relative displacement of the mass points of a body. 

The motion of the sliding mass point, which traces the vibrational motion of the excited molecule, is represented by the broken lines. 

Euler s proof of the least action principle for a single particle (mass point in motion) was extended by Lagrange (c. 1760) to the general case of mutually interacting particles, appropriate to celestial mechanics. In Lagrange s derivation [436], action along a system path from initial coordinates P to final coordinates Q is defined by... 

Newton s equations of motion, stated as force equals mass times acceleration , are strictly true only for mass points in Cartesian coordinates. Many problems of classical mechanics, such as the rotation of a solid, cannot easily be described in such terms. Lagrange extended Newtonian mechanics to an essentially complete nonrelativistic theory by introducing generalized coordinates q and generalized forces Q such that the work done in a dynamical process is Qkdqk [436], Since... 

In spherical polar coordinates, for one mass point moving in a central potential V = V(r),... 

The concept of a mass point remains valid, but a time interval dt can no longer be treated as a nondynamical parameter. Einstein s basic postulate [323, 393] is that the interval ds between two space-time events is characterized by the invariant expression... 

The conformational properties of an uncharged molecular chain are well described by a (discrete) semiflexible chain model [33]. The chain is comprised of mass points, each one may represent several monomers, at positions r, (z =0,..., N). The (average) length of a bond is l. The partition function of such a chain is given by... 

One of the problems faced by quantum chemistry is that it is based on a theory borrowed from physics. It therefore is important to note that a given variable or concept may be interpreted very differently in physical and chemical context, respectively. The physicist who is interested in the motion of a molecule in a force-free environment, treats it as a mass point, without any loss of generality. Such a molecule is of no interest to the chemist who studies the interaction of a molecule with its environment. In chemical context the size and shape of the molecule, left undefined in theoretical physics, must be taken into account. The interaction between mass points can simply not account for the observed behaviour of chemical substances. 

The relativistic energy of a mass point is computed by differentiation of the relativistic momentum, to yield the relativistic force... 

The work done by the force in moving the mass point by dZ, defines the energy... 

Advanced theories of physics describe the electron as either a zero-dimensional mass point or an equivalent plane wave, subject to mathematical manipulation, but not to visualization. 

Schrodinger, Einstein, Bohm and others who may have happened to support aspects of the model outlined here, were invariably accused of trying to revive a classical interpretation of non-classical events. The implied sin is that these individuals dared to recognize a causal structure where it is expressly forbidden by the Copenhagen doctrine. The exact opposite is probably closer to the truth What is more non-classical than a particle that consists of vibrations in the aether a particle with the ability to adapt its shape as dictated by the environment a particle that disappears into, or appears from, the wave structure of another particle a particle with continuous non-local mass and charge densities or a particle which is different from the mass points of classical mechanics However, this is an irrelevant argument. Whether a model is classical or quantal is of no consequence -what is important is that it leads to a reasonable interpretation of chemical phenomena. 

It is significant that in both cases Planck s constant appears in the specification of the dynamic variables of angular momentum and energy, associated with wave motion. The curious relationship between mass and energy that involves the velocity of a wave, seems to imply that the motion of mass points also has some wavelike quality. Only because Planck s constant is almost vanishingly small, dynamic variables of macroscopic systems appear to be continuous. However, when dealing with atomic or sub-atomic systems... 

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