Classical terms

The expansion of the generalized momentum operator only involves the magnetic interactions in the electronic part of the wave function. Since the corresponding nuclear part has been separated out by the Bom-Oppenheimer approximations, we need to add a few terms corresponding to the (classical) interaction of the nuclear magnetic moments with an external magnetic field and between nuclei. 

The term involving two nuclei is analogous to the electron-electron spin-dipole term in eq. (10.77). The corresponding Fermi contact term disappears since nuclei cannot occupy the same position at energies relevant for chemistry. Note that the direct spin-spin coupling is independent of the electronic wave function it only depends on the molecular geometry. 

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The first three terms in Eq. (10-26), the election kinetic energy, the nucleus-election Coulombic attraction, and the repulsion term between charge distributions at points Ti and V2, are classical terms. All of the quantum effects are included in the exchange-correlation potential... 

Another phenomenon that was inexplicable in classical terms was the photoelectric effect discovered by Hertz in f 887. When ultraviolet light falls on an alkali metal surface, electrons are ejected from the surface only when the frequency of the radiation reaches the threshold... 

The NMR phenomenon can be quantitatively described in classical terms. This was first done by Bloch.The approach is helpful in developing an understanding of nuclear relaxation processes. 

Species 3.6 is therefore indeed a complex which, in classical terms, may be called a nitrosyl ion solvated with one molecule of water. The complex is calculated to be more stable than its fragments by 75 kJ mol-1. It is likely, as the authors say, that this calculated value is a little smaller than the true value. 

The total energy of the electrostatic interaction between a nucleus with charge Ze and charges around this nucleus may be expressed in classical terms as [1], [9] in Chap. 1 ... 

Remarkably, only one nuclear constant, Q, is needed in (4.17) to describe the quadrupole moment of the nucleus, whereas the full quadrupole tensor Q has five independent invariants. The simplification is possible because the nucleus has a definite angular momentum (7) which, in classical terms, imposes cylindrical symmetry of the charge distribution. Choosing x, = z as symmetry axis, the off-diagonal elements Qij are zero and the energy change caused by nuclear... 

Let us explore first the nature of the integrand E cl for the limiting cases. At X = 0 we are dealing with an interaction free system, and the only component which is not included in the classical term is due to the antisymmetry of the fermion wave function. Thus, E j° is composed of exchange only, there is no correlation whatsoever.19 Hence, the X = 0 limit of the integral in equation (6-25) simply corresponds to the exchange contribution of a Slater determinant, as for example, expressed through equation (5-18). Remember, that E ° can... 

In classic terms, the elastic modulus of a material is the stress divided by the strain (i.e., the slope) of the linear portion of its force versus elongation curve at low strain. In this region, the material is assumed to behave in a Hookean fashion, i.e., stress and strain are linearly proportional, as illustrated in Fig. 8.5 a). Most polymers do not behave in this manner. 

The Raman effect arises when a photon is incident on a molecule and interacts with the electric dipole of the molecule. In classical terms, the interaction can be viewed as a perturbation of the molecule s electric field. In quantum mechanics the scattering is described as an excitation to a virtual state lower in energy than a real electronic transition with nearly coincident de-excitation and a change in vibrational energy. The scattering event occurs in 10 14 seconds or less. The virtual state description of scattering is shown in Figure 1. 

The probability of a particular reaction (or scattering) taking place is measured by the relevant cross-section, which can be thought of in classical terms as the cross-sectional area of a sphere that has the same probability of being hit. The corresponding nuclear dimensions would be of order... 

Spin-orbit coupling problems are of a genuine quantum nature since a priori spin is a quantity that only occurs in quantum mechanics. However, already Thomas (Thomas, 1927) had introduced a classical model for spin precession. Later, Rubinow and Keller (Rubinow and Keller, 1963) derived the Thomas precession from a WKB-like approach to the Dirac equation. They found that although the spin motion only occurs in the first semiclassical correction to the relativistic classical electron motion, it can be expressed in merely classical terms. 

The density of He I at the boiling point at 1 atm is 125 kg m 3 and the viscosity is 3 x 10 6 Pa s. As we would anticipate, cooling increases the viscosity until He II is formed. Cooling this form reduces the viscosity so that close to 0 K a liquid with zero viscosity is produced. The vibrational motion of the helium atoms is about the same or a little larger than the mean interatomic spacing and the flow properties cannot be considered in classical terms. Only a quantum mechanical description is satisfactory. We can consider this condition to give the limit of De-+ 0 because we have difficulty in defining a relaxation when we have the positional uncertainty for the structural components. 

There is a fundamental difference between a chirality center and a pseudoasymmetric center and that is that reflection and permutation of ligands have the same effect for the chirality center, but not for the pseudoasymmetric center. This is because, on reflection of the latter, the chirality senses of both the center, seen when the bonds are numbered, and each of the enantiomorphic ligands are reversed. Another way of stating the difference is that Re/Si and rejsi descriptors specify absolute and relative configuration, respectively. Pseudoasymmetric is a most unfortunate term, and in order to avoid it, the classical terms chirality center and pseudoasymmetric center would perhaps best be replaced by more neutral terms, such as stereogenic centers of type 1 and type 2, respectively, in order to emphasize the aspect of stereogenicity. 

Finally, double-bond isomers must be discussed [type (5) in Table 1], Modern views of the term geometric (see Section 1.1.2.1.) deprives us of the classical term geometric isomers and excludes the term geometric enantiomers [type (5b) in Table 1]. This, unfortunately, leaves a void for a suitable term. The term double-bond isomers is used here for want of a better one. It is not perfect because it does not include structures with a partial double bond, e.g., amides. The term /Z-isomers has been used, however, this implies diastereomerism and does not apply to type (5 b) which involves enantiomerism. 

Obviously, permutation and reflection do not, in general, give identical results. Unfortunately, the important classical terms asymmetric atom (center) and pseudoasymmetric atom (center)... 

It should be noted that some of the Izumi - Tai terms are also known by other designations. For example, the term enantiomer-differentiating reaction is equivalent to the classical term kinetic resolution, and as both are sufficiently clear they will be used here. 

To obtain the RFR term on this level, the same method is used for complexvalued A. This gives an extra classical term, or expectation value, which can be written as... 

Crowell discovered a variety of effects numerically, including modified Rabi flopping, which has an inverse frequency dependence similar to that observed in the solid state in reciprocal noise [73]. The latter is also explained by Crowell [17] using a non-Abelian model. A variety of other effects of RFR on the quantum electrodynamical level was also reported numerically [17]. The overall result is that the occurrence, classically, of the B V> field means that there is a quantum electrodynamical Hamiltonian generated by the classical term proportional to 3 2. This induces transitional behavior because it contributes to the dynamics of probability amplitudes [17]. The Hamiltonian is a quartic potential where the value of determines the value of the potential. The latter has two minima one where B = 0 and the other for a finite value of the B i) field, corresponding to states that are invariants of the Lagrangian but not of the vacuum. 

Thus, die electrostatic interaction term of Eq. (13.2) has been separated into an operator acting on die QM electrons (the first tenn on the r.h.s. of Eq. (13.3)) and the classical term for the interacdon of the MM atoms widi the solute nuclei. The Lennard-Jones term is the same in Eqs. (13.2) and (13.3) (although the parameters may certainly be different from one model to another). 

As the (XH - Y) frequency is not fixed but varies somewhat with the amplitude of the vibration in classical terms, so each of these sub-bands has a finite width,... 

In classical terms, if we use the mass-weighted Cartesian displacement coordinates, the kinetic energy of the moving nuclei isf... 

The electronic structure of the carbonyl group is perturbed by the rest of the molecule. In classical terms, a carbonyl compound may be represented as a resonance hybrid of the canonical structure I to V, viz. 

In classical terms ( mixing can be thought of in a simple way. When an Ar atom, for example, enters the Rydberg electron s orbit the electron can scatter elastically from the Ar. The reorientation of the orbit corresponds to a new (. If we pursue this notion a little further we can understand qualitatively the dependence... 

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