Quantum Mechanical Discussion

It has already been mentioned above that the electron density distribution of a base is the same, independently of whether the base is considered alone or it participates in a polynucleotide. For this reason the discussion here is restricted to the signs of the effective charges. 

Inspection of this table shows the existence of a uniformity, regarding the signs of the effective charges, summarized in Table 41. It can be observed that there is no uniformity regarding the signs of the charges in the last two positions of the third base. If the uniformity criterion is accepted, it must be concluded that those charges are inoperative, at least insofar as the selective role of the codon is concerned. 

In principle, triplets should be given for methionine (A UG) and tryptophan (UGG). One can see, however, that it is sufficient to postulate the modified doublets AU+ and C7G+, which are still different from those found for isoleucine (AC/ ) and cysteine (UG—). 

One has, then, a one-to-one correspondence between modified doublets and aminoacids, except for arginine, leucine, and serine, for which it seems that two different -RNA should be postulated. 

15 The designation triplet and doublet has been used in the literature to denote tri- and dinucleotides, respectively. 

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In the non-relativistic quantum mechanics discussed in this chapter, spin does not appear naturally. Although... 

Another application of the concept of fractional valence bonds has been made in the field of metals and alloys. In the usual quantum mechanical discussion of metals, initiated by W. Pauli (Z. Physik, 41, 81 (1927)) and Sommerfeld (Naturwiss., 15, 825 (1927)), the assumption was made that only a small number of electrons contribute significantly to the binding together of the metal... 

The values of and can be estimated from their internal consistency to be accurate to about 0.1 v.e., the value of 1.71 v.e. for the resonance energy being accurate to about 0.15 v.e. The quantum mechanical discussion of resonance in benzene and naphthalene is given in the preceding paper.1... 

A Quantum Mechanical Discussion of Orientation of Substituents in Aromatic... 

The postulates of quantum mechanics discussed in Section 3.7 are incomplete. In order to explain certain experimental observations, Uhlenbeck and Goudsmit introduced the concept of spin angular momentum for the electron. This concept is not contained in our previous set of postulates an additional postulate is needed. Further, there is no reason why the property of spin should be confined to the electron. As it turns out, other particles possess an intrinsic angular momentum as well. Accordingly, we now add a sixth postulate to the previous list of quantum principles. 

The existence of an attractive force between non-polar molecules was first recognized by van der Waals, who published his classic work in 1873. The origin of these forces was not understood until 1930 when Fritz London (1900-1954) published his quantum-mechanical discussion of the interaction between fluctuating dipoles. He showed how these temporary dipoles arose from the motions of the outer electrons on the two molecules. 

A quantum-mechanical discussion of the structure of these molecules in relation to the dipole moments has been reported by W. N. Lipscomb, J. Chem. Phys. 25, 38 (1956). 

The values for 0ab are obtained from a quantum-mechanical discussion L. Pauling, Z. Krist. 67, 377 (1928). Equation 13-9 and the following equations differ slightly from those originally published in that (rx + tb)w 1 replaces tk + rB)n. 

Nevertheless we must point out that in regard to the problem of orientation Hiickel limited his calculations to the inductive effects. In 1935, Pauling with G. Wheland, and some months later after Bonino s paper, published a paper, in which turned his attention to the problem of orientation in aromatic molecules using the method of molecular orbitals. (See G.W. Wheland and L. Pauling, A quantum mechanical discussion of orientation of substituents in aro-... 

We begin this section with a quite general quantum mechanical discussion of the energy relaxation of an oscillator coupled to a bath. Thus the Hamiltonian is... 

As a preliminary point, we note that the decoupled averaging discussed here in classical views of the potential distribution theorem derives from the denominator of Eq. (3.17), p. 40. This is unchanged in the present quantum mechanical discussion, and thus the sampling of the separated subsystems could be highly quantum mechanical without changing those formalities. 

Quantum-mechanical discussions of the ngauche effect and the "anomeric and exo-anomeric effect have been published ( 10-13 ) which provide a more complete description of these electronic interactions, and the electrostatic repulsions between the vicinal lone-pair electrons. 

Until now, our treatment has been built in exactly the same terms that might have been used in work on normal modes of vibration in the latter part of the nineteenth century. However, it is incumbent upon us to revisit these same ideas within the quantum mechanical setting. The starting point of our analysis is the observation embodied in eqn (5.19), namely, that our harmonic Hamiltonian admits of a decomposition into a series of independent one-dimensional harmonic oscillators. We may build upon this observation by treating each and every such oscillator on the basis of the quantum mechanics discussed in chap. 3. In light of this observation, for example, we may write the total energy of the harmonic solid as... 

An approach very closely related to that of Woodward and Hoffmann is the so-called Hiickel-Mobius approach 35> based on the rule An +2 electron systems prefer Hiickel geometries and An electron systems prefer Mobius geometries 36>. When no symmetry exists and there is no cyclic orbital array the allowedness or forbiddenness of a reaction can be determined by following the form of the MO s during the reaction 37>. A detailed quantum mechanical study of the stereochemistry of thermal and photo cyclo-addition reactions has been reported38), and a quantum mechanical discussion of the applicability of the Woodward-Hoffmann rules can be found in a paper by George and Ross 39>. 

H. Frohlich, A quantum mechanical discussion of the cohesive forces and thermal expansion coefficients of the alkali metals, Proc. Roy. Soc. (London) Ser. A, Math. Phys. Sciences 158 (893) (1937) 97-110. 

The concept of resonance played an important part in the discussion of the behavior of certain systems by the methods of classical mechanics. Very shortly after the discovery of the new quantum mechanics it was noticed by Heisenberg that a quantum-mechanical treatment analogous to the classical treatment of resonating system can be applied to many problems, and that the results of the quantum-mechanical discussion in these cases can be given a simple interpretation as corresponding to a quantum-mechanical resonance phenomenon. It is not required... 

In quantum-mechanical discussions the matrix f corresponding to the dynamical quantity f qit pi) is sometimes defined with the use of the wave functions which include the time (Eq. 51-1), and sometimes with the wave functions with the time factor... 

There have been a number of attempts to carry out more thorough calculations, in which the approximation of step 4 is replaced by more rigorous evaluation of the various interaction integrals. Discussion of these calculations would involve a level of quantum-mechanical discussion inappropriate in this book. The reader is therefore referred to the literature17 for further information. 

For recent treatments of a classical bent see References 47, 15, 14, 58. An interesting quantum mechanical discussion has been given in Reference 93 in the language of the scattering matrix, from which the reaction cross-section may be derived. 

Until now in our quantum mechanical discussions we have described the stationary or time-independent states of a system. Furthermore, our language was such as to imply that we had sufficient information about the system to know that it was in a particular state described by a particular set of quantum numbers. For example, in the case of the harmonic oscillator we spoke as though we knew that the oscillator was in the pth state with wave function i// , and energy = (v A- j)hv or, in the case of the hydrogen atom, that it was in a state described by the set of numbers n, /, m. This approach is very useful in a first discussion of quantum mechanical properties of various kinds of systems. However, we do not have reason to presuppose that a system is in a particular quantum state. 

It will be noted that our Z-matrix for ethane has been defined using the angstrom as the unit of length (1A = 10 m = 100 pm). The mgstrom is a non-SI unit but is a very convenient one to use, as most bond lengths are of the order of 1-2 A. One other very common non-SI unit found in the molecular modelling literature is the kilocalorie (1 kcal =4.1840 kj). Other systems of imits are employed in other types of calculation, such as the atomic imits used in quantum mechanics (discussed in Chapter 2). It is important to be aware of, and familiar with, these non-standard units as they are widely used in the literature and throughout this book. 

Pauling L, Wheland GW (1935) The quantum mechanical discussion of the orientation of substituents in organic molecules. J Am Chem Soc 57 2086-2095... 

The history of coordination chemistry may in a sense be said to have begun with the work of Werner. The early crystal-structure determinations by W. L. and W. H. Bragg showed that in crystals such as sphalerite, ZnS, there is tetrahedral coordination around both zinc and sulfur, and in crystals such as sodium chloride there is octahedral coordination about both the anion and the cation. The modem period may be said to have begun in 1921, with the determination of a cryst containing an octahedral complex by Wyckoff and of crystals containing tetrahedral and square planar complexes (1922) by Dickinson. Later developments include application of quantum mechanics, discussion of hybrid orbitals especially suited to bonding, and detailed interpretation of interatomic distances found by careful X-ray diffraction studies. 

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