The Quantum-Mechanical Treatment

We now need to investigate the quantum-mechanical treatment of vibrational motion. Consider then a diatomic molecule with reduced mass /c- His time-independent Schrodinger equation is 

We normally take the constant of integration i7o to be zero. Solution of the time-independent Schrodinger equation can be done exactly. We don t need to concern ourselves with the details, I will just give you the results. 

First of all, the vibrational energy is quantized, and we write the single quantum number v. This quantum number can take values 0, 1, 2. 

The normalized vibrational wavefunctions are given by the general expression 

The Hermite polynomials are well known in science and engineering. 

---

The quantum mechanical treatment of a hamionic oscillator is well known. Real vibrations are not hamionic, but the lowest few vibrational levels are often very well approximated as being hamionic, so that is a good place to start. The following description is similar to that found in many textbooks, such as McQuarrie (1983) [2]. The one-dimensional Schrodinger equation is... 

At longer distances weak attractions are expected due to induced dipoles and so-called dispersion forces. These are not included in the quantum-mechanical treatment outlined above but could be covered in principle by the inclusion of large numbers of excited... 

Both quantum mechanical and classical theories of Raman scattering have been developed. The quantum mechanical treatment of Kramers and Heisenberg 5) preceded the classical theory of Cabannes and Rochard 6). 

The third common level is often invoked in simplified interpretations of the quantum mechanical theory. In this simplified interpretation, the Raman spectrum is seen as a photon absorption-photon emission process. A molecule in a lower level k absorbs a photon of incident radiation and undergoes a transition to the third common level r. The molecules in r return instantaneously to a lower level n emitting light of frequency differing from the laser frequency by —>< . This is the frequency for the Stokes process. The frequency for the anti-Stokes process would be + < . As the population of an upper level n is less than level k the intensity of the Stokes lines would be expected to be greater than the intensity of the anti-Stokes lines. This approach is inconsistent with the quantum mechanical treatment in which the third common level is introduced as a mathematical expedient and is not involved directly in the scattering process (9). 

But why linearly and why with a slope of-1, or something thereabout, the reader may righteously ask. In anticipation of the quantum mechanical treatment in Chapter 5 we can briefly discuss here a simple electrostatic model which fully accounts for the observed behaviour. After all, as the detailed quantum mechanical treatment has shown, direct electrostatic... 

The quantum-mechanical treatment previously applied to benzene, naphthalene, and the hydrocarbon free radicals is used in the calculation of extra resonance energy of conjugation in systems of double bonds, the dihydro-naphthalenes and dihydroanthracenes, phenylethylene, stilbene, isostilbene, triphenylethylene, tetraphenylethyl-... 

There are two principal methods available for the quantum mechanical treatment of molecular structure, the valence bond method and the molecular orbital method. In this paper we shall make use of the latter, since it is simpler in form and is more easily adapted to quantitative calculations.3 We accordingly consider each electron... 

The quantum mechanics treatment of diamagnetism has not been published. It seems probable, however, that Larmor s theorem will be retained essentially, in view of the marked similarity between the results of the quantum mechanics and those of the classical theory in related problems, such as the polarisation due to permanent electric dipoles and the paramagnetic susceptibility. f Thus we are led to use equation (30), introducing for rK2 the quantum mechanics value... 

It is thus evident that the experimental results considered in sect. 4 above are fully consistent with the interpretation based on absolute reaction rate theory. Alternatively, consistency is equally well established with the quantum mechanical treatment of Buhks et al. [117] which will be considered in Sect. 6. This treatment considers the spin-state conversion in terms of a radiationless non-adiabatic multiphonon process. Both approaches imply that the predominant geometric changes associated with the spin-state conversion involve a radial compression of the metal-ligand bonds (for the HS -> LS transformation). 

This list of postulates is not complete in that two quantum concepts are not covered, spin and identical particles. In Section 1.7 we mentioned in passing that an electron has an intrinsic angular momentum called spin. Other particles also possess spin. The quantum-mechanical treatment of spin is postponed until Chapter 7. Moreover, the state function for a system of two or more identical and therefore indistinguishable particles requires special consideration and is discussed in Chapter 8. 

The harmonic oscillator is an important system in the study of physical phenomena in both classical and quantum mechanics. Classically, the harmonic oscillator describes the mechanical behavior of a spring and, by analogy, other phenomena such as the oscillations of charge flow in an electric circuit, the vibrations of sound-wave and light-wave generators, and oscillatory chemical reactions. The quantum-mechanical treatment of the harmonic oscillator may be applied to the vibrations of molecular bonds and has many other applications in quantum physics and held theory. 

We now apply the results of the quantum-mechanical treatment of generalized angular momentum to the case of orbital angular momentum. The orbital angular momentum operator L, defined in Section 5.1, is identified with the operator J of Section 5.2. Likewise, the operators I , L, Ly, and are identified with J, Jx, Jy, and Jz, respectively. The parameter j of Section 5.2 is denoted by I when applied to orbital angular momentum. The simultaneous eigenfunctions of P and are denoted by Im), so that we have... 

Thus, the quantum-mechanical treatment of generalized angular momentum presented in Section 5.2 may be applied to spin angular momentum. The spin operator S is identified with the operator J and its components Sx, Sy, Sz with Jx, Jy, Jz- Equations (5.26) when applied to spin angular momentum are... 

N. F. Mott and H. S. W. Massey (1965) The Theory of Atomic Collisions, 3rd edition (Oxford University Press, Oxford). The standard reference for the quantum-mechanical treatment of collisions between atoms. 

The general theory of the quantum mechanical treatment of magnetic properties is far beyond the scope of this book. For details of the fundamental theory as well as on many technical aspects regarding the calculation of NMR parameters in the context of various quantum chemical techniques we refer the interested reader to the clear and competent discussion in the recent review by Helgaker, Jaszunski, and Ruud, 1999. These authors focus mainly on the Hartree-Fock and related correlated methods but briefly touch also on density functional theory. A more introductory exposition of the general aspects can be found in standard text books such as McWeeny, 1992, or Atkins and Friedman, 1997. As mentioned above we will in the following provide just a very general overview of this... 

One of die most important second-order, homogeneous differential equations is that of Hennite It arises in the quantum mechanical treatment of the harmonic oscillator. Schrfidinger s equation for the harmonic oscillator leads to the differential equation... 

B. T. Thole and P. T. van Duijnen, On the quantum mechanical treatment of solvent effects, Theor. Chim. Acta 55 307 (1980). 

In order to motivate the quantum mechanical treatment of a system with the energy functional Genp, we first consider the functional... 

A rigorous mathematical formalism of chemical bonding is possible only through the quantum mechanical treatment of molecules. However, obtaining analytical solutions for the Schrodinger wave equation is not possible even for the simplest systems with more than one electron and as a result attempts have been made to obtain approximate solutions a series of approximations have been introduced. As a first step, the Bom-Oppenheimer approximation has been invoked, which allows us to treat the electronic and nuclear motions separately. In solving the electronic part, mainly two formalisms, VB and molecular orbital (MO), have been in use and they are described below. Both are wave function-based methods. The wave function T is the fundamental descriptor in quantum mechanics but it is not physically measurable. The squared value of the wave function T 2dT represents probability of finding an electron in the volume element dr. 

What was the distinction between quantum chemistry and chemical physics After the Journal of Chemical Physics was established, it was easy to say that chemical physics was anything found in the new journal. This included molecular spectroscopy and molecular structures, the quantum mechanical treatment of electronic structure of molecules and crystals and the problem of chemical binding, the kinetics of chemical reactions from the standpoint of basic physical principles, the thermodynamic properties of substances and calculation by statistical mechanical methods, the structure of crystals, and surface phenomena. 

In all results of the MFT method discussed so far, all vibrational modes have been treated classically. As has been mentioned above, an improved approximation of the overall dynamics can be obtained if some of the vibrational modes are included in the quantum-mechanical treatment. Figure 9... 

Studies of gas-phase S"n2 reactions at sp carbon have been made by Fourier transform ion cyclotron resonance mass spectrometry (FTlCRMS) and complemented by both semiempirical and ab initio MO calculations. The particular processes of interest involved intramolecular reactions in which neutral nucleophiles displace neutral leaving groups within cationic substrates, e.g. A-(2-piperidinoethyl)-2,4,6-triphenylpyridinium cation (59), in which the piperidino moiety is the nucleophile and 2,4,6-triphenylpyridine (60) is the leaving group. No evidence has been obtained for any intermolecular gas-phase 5) 2 reaction involving a pyridine moiety as a leaving group. The quantum mechanical treatments account for the intramolecular preference. 

Figure 2b depicts a strong acceptor bond for a Na atom. It is formed from the weak bond depicted in Fig. 2a, for example, as a result of the capture and localization of a free electron, that is, as a result of the transformation of a Na+ ion of the lattice serving as an adsorption center, into a neutral Na atom. We obtain a bond of the same type as in the molecules H2 or Na2. This is a typically homopolar two-electron bond formed by a valence electron of the adsorbed Na atom and an electron of the crystal lattice borrowed from the free electron population. The quantum-mechanical treatment of the problem 2, 8) shows that these two electrons are bound by exchange forces which in the given case are the forces keeping the adsorbed Na atom at the surface and at the same time holding the free electron of the lattice near the adsorbed atom. 

The quantum-mechanical treatment of the interaction of a foreign molecule with a crystal lattice, as carried out for the simplest models and generalized to more complex systems, leads to a number of results that may be regarded as the fundamental propositions of the theory. The following are the principal ones ... 

The quantum mechanical treatment of harmonic oscillators is described in essentially all books on quantum mechanics. Several good examples are ... 

The classical model predicts thermal motion to vanish at very low temperatures, in contradiction to the zero-point vibrations which follow from the quantum-mechanical treatment of oscillators. For temperatures at which hv % kBT, the spacing of the discrete energy levels cannot be neglected, so the classical model is no longer valid. 

---