Classical treatment

The classical scattering amplitude of an electron was derived in section 1.2.1 as 

The amplitude A can be separated into real and imaginary parts by multiplication of both the numerator and the denominator by coj — co2 — ikto. For unit value of E0, in units of —e/mc2, the result is 

The anomalous contribution to the real part of the scattering amplitude can be separated by subtraction of the classical Thompson scattering, —e2/mc2, from the first term of Eq. (1.29) to give 

Including resonance effects, the atomic scattering factor for a many-electron atom is written as 

For values of co cos, that is, for negligible effect of the resonance, the real contribution / + / = g(s), and f - 0. Thus, the real contribution becomes 

The harmonic oscillator is an idealized one-dimensional physical system in which a single particle of mass m is attracted to the origin by a force F proportional to the displacement of the particle Ifom the origin 

The proportionality constant k is known as the force constant. The minus sign in equation (4.1) indicates that the force is in the opposite direction to the direction of the displacement. The typical experimental representation of the oscillator consists of a spring with one end stationary and with a mass m 

In classical mechanics the particle obeys Newton s second law of motion 

According to equation (4.3), the particle oscillates sinusoidally about the origin with frequency v and maximum displacement A. 

The potential energy Fof a particle is related to the force F acting on it by the expression 

If we look at the nucleus as a positive charge distribution of density n(r) surrounded by a negative charge distribution Qe ) the Coulomb interaction energy in atomic units is written as 

The common expansion of the reciprocal distance in terms of spherical harmonics and collection of the corresponding orders (monopole, dipole, quadrupole,. .., interactions) leads to 

The superscript S denotes the spherical expansion and the range of the nuclear coordinate now only comprises the spatial region of the nucleus. This point needs some clarification concerning the assumption that AV = 0. The Cartesian expansion of the term is written as 

Obviously, the spherical and Cartesian expressions for cannot be identical since the Cartesian expansion parameters a and /3 are not confined to specific spatial regions or have to fulfill an inequality such as r r. A more explicit analysis which can be found in [28] states the relation between these two expansions as 

The expression for is then formulated as a scalar product (one-index contraction) of two second-rank spherical tensors, the nuclear 

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The classical treatment of the Ising model makes no distinction between systems of different dimensionality, so, if it fails so badly for d= 2, one might have expected that it would also fail for [Pg.644]

No system is exactly unifomi even a crystal lattice will have fluctuations in density, and even the Ising model must pemiit fluctuations in the configuration of spins around a given spin. Moreover, even the classical treatment allows for fluctuations the statistical mechanics of the grand canonical ensemble yields an exact relation between the isothemial compressibility K j,and the number of molecules Ain volume V ... 

Markovic N and Billing G D 1997 Semi-classical treatment of chemical reactions extension to 3D wave packets Chem. Phys. 224 53... 

In what is called BO MD, the nuclear wavepacket is simulated by a swarm of trajectories. We emphasize here that this does not necessarily mean that the nuclei are being treated classically. The difference is in the chosen initial conditions. A fully classical treatment takes the initial positions and momenta from a classical ensemble. The use of quantum mechanical distributions instead leads to a seraiclassical simulation. The important topic of choosing initial conditions is the subject of Section II.C. 

For the case of intramolecular energy transfer from excited vibrational states, a mixed quantum-classical treatment was given by Gerber et al. already in 1982 [101]. These authors used a time-dependent self-consistent field (TDSCF) approximation. In the classical limit of TDSCF averages over wave functions are replaced by averages over bundles of trajectories, each obtained by SCF methods. 

Prevailing interest rates probably tend to reflect an estimate of future inflation and contain a component that can be attributed loosely to inflationary expectations. However, the classical treatment is to assume that an inflation-free interest rate, r and average inflation rate, r, over the project lifetime can be identified. A discount factor (1 + r) can be modified (25) so that... 

The plasticity equations presented so far are still more general than the equations usually considered in the classical theory of plasticity. Linearity and symmetry assumptions, inherent in most classical treatments, are yet to be made. Particularly simple assumptions are made here to serve as an example. 

The molecular mechanics calculations discussed so far have been concerned with predictions of the possible equilibrium geometries of molecules in vacuo and at OK. Because of the classical treatment, there is no zero-point energy (which is a pure quantum-mechanical effect), and so the molecules are completely at rest at 0 K. There are therefore two problems that I have carefully avoided. First of all, I have not treated dynamical processes. Neither have I mentioned the effect of temperature, and for that matter, how do molecules know the temperature Secondly, very few scientists are interested in isolated molecules in the gas phase. Chemical reactions usually take place in solution and so we should ask how to tackle the solvent. We will pick up these problems in future chapters. 

The concept of a potential energy surface has appeared in several chapters. Just to remind you, we make use of the Born-Oppenheimer approximation to separate the total (electron plus nuclear) wavefunction into a nuclear wavefunction and an electronic wavefunction. To calculate the electronic wavefunction, we regard the nuclei as being clamped in position. To calculate the nuclear wavefunction, we have to solve the relevant nuclear Schrddinger equation. The nuclei vibrate in the potential generated by the electrons. Don t confuse the nuclear Schrddinger equation (a quantum-mechanical treatment) with molecular mechanics (a classical treatment). 

We assume familiarity with the classical treatment of grand ensembles,24 and know that the probability has a logarithm equal to... 

An interesting improvement from the classical treatment of the bond under stress was proposed by Crist et al, [101], Considering the chain as a set of N-coupled Morse oscillators, these authors determined the elongation and time to failure as a function of the axial stress. The results, reported in Fig. 20, show a decreasing correlation between the total elastic strain before failure and the level of applied force with the chain length. To break a chain within some reasonable time interval (for example <10-3s) requires, however, the same level of stress (a0.7 fb) as found from the simpler de Boer s model. 

Therefore, the simplest classical treatment in which the propagator exp(it (T+V) ) is approximated in the product form exp(it (T) ) exp(it (V)/fc) and die nuclear kinetic energy T is conserved during the transition produces a nonsensical approximation to the non BO rate. This should not be surprising because (a) In the photon absorption case, the photon induces a transition in the electronic degrees of freedom which subsequently cause changes in the vibration-rotation energy, while (b) in the non BO case, the electronic and vibration-... 

Improving on the semi-classical treatment of the vibration-rotation motion only slightly allows Rt to be recast in a form... 

In their classic treatment of the compact-layer capacitances, MacDonald and Barlow12 affirmed that the thickness of the space charge or penetration region in the metallic electrode is so small for a good conductor that its effect may be neglected. Their theory... 

In the classical treatment of the paracrystal, Hosemann [5] refers to the quantity oc/dc as g-factor . 

The classical treatment of diffuse SAXS (analysis and elimination) is restricted to isotropic scattering. Separation of its components is frequently impossible or resting on additional assumptions. Anyway, curves have to be manipulated one-by-one in a cumbersome procedure. Discussion of diffuse background can sometimes be avoided if investigations are resorting to time-resolved measurements and subsequent discussion of observed variations of SAXS pattern features. A background elimination procedure that does not require user intervention is based on spatial frequency filtering (cf. p. 140). 

The deposition of sub-micron aerosols in a hollow cast of human bronchi has recently been measured under realistic conditions (Cohen et al., in press). Typical data are shown in Figure 4. These are inconsistent with convective enhancement of deposition but support the classical treatment of deposition by diffusion (Chamberlain and Dyson, 1956). 

The model presented here is simplified in several ways (harmonic approximation, purely classical treatment of inner-sphere reorganization), and it says little about the pre-exponential factor A. But it does... 

In order to better understand the NMR techniques described in this paper, let us first briefly review some fundamental concepts in NMR. (For more details, see Reference 8.) Throughout the discussion, we will use a classical treatment. 

In Eqs. (25) and (26), the summations are over the incremental steps in going from X to Y in the gas phase or in solution. The Hj are the intermediate Hamiltonians (or force fields in a classical treatment). Thus, Hi=0,gas = Hx,gas, Hi=Njgas = HY,gas, etc. It is of course desirable that the molecules X and Y be structurally similar, so that the perturbation of X that produces Y be small. Another option is to let Y be composed of noninteracting dummy) atoms,75 so that its free energy of solvation is zero. Then Eq. (24) gives the absolute free energy of solvation of X ... 

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