Molecular systems quantization

For reactions between atoms, the computation needs to model only the translational energy of impact. For molecular reactions, there are internal energies to be included in the calculation. These internal energies are vibrational and rotational motions, which have quantized energy levels. Even with these corrections included, rate constant calculations tend to lose accuracy as the complexity of the molecular system and reaction mechanism increases. 

The aim of this section is to familiarize the reader with the second quantization and the many-body diagrammatic techniques which are now widely used in up-to-date quantum chemistry. These techniques are very efficient since they permit the formulation of the problem by means of diagrams from which the explicit formula can be obtained. Another advantage is that the problem of spin can be handled very simply. This approach also permits us to have a microscopic view of the problem (as will be seen in the study of ionization potentials, excitation energies, interaction of two molecular systems etc.). 

Similar to quantum mechanics, which can be formulated in terms of different quantities in addition to the traditional wave function formulation, in quantum chemistry a number of alternative tools are developed for this purpose, which may be useful in the context of the present book. We have already described different approximate models of representing the electronic structure using (many-electronic) wave functions. The coordinate and second quantization representations were employed to get this. However, the entire amount of information contained in the many-electron wave function taken in whatever representation is enormously large. In fact it is mostly excessive for the purpose of describing the properties of any molecular system due to the specific structure of the operators to be averaged to obtain physically relevant information and for the symmetry properties of the wave functions the expectation values have to be calculated over. Thus some reduced descriptions are possible, which will be presented here for reference. 

We indicated in the Introduction that the reasons for writing this article are two-fold (i) to present a survey of the various conical intersections which govern potential transitions between electronic states and (ii) to establish the 3-state quantization of the NACM for molecular systems. 

Having seen how the operators of second quantization can be used to express wavefunctions and quantum-mechanical operators, let us now move on to the problem of choosing wavefunctions that yield optimum descriptions, in an energy optimization sense, of the stationary states of atomic and molecular systems. 

Quantum mechanics olecular modeling method that examines the electronic structure and energy of molecular systems based on various schemes for solving the Schrodinger equation based on the quantized nature of electronic configurations in atomic and molecular orbitals. 

One noticeable exception is molecular spectroscopy, more specifically infrared spectroscopy, where, almost by definition, quantization of the vibrational states cannot be neglected since it concerns the measurement of the transitions between the quantized vibrational states [15]. Due to its importance in chemistry, for instance for the detection of functional groups in organic chemistry, infrared spectroscopy is one of the very few domains where the students in both physics and chemistry experience the application of a full-quantum mechanical treatment for both the electrons and the nuclei in a molecular system. However, there is growing evidence that a significant number of various chemical reactions are impacted by strong quantum-mechanical effects involving nuclei [16]. 

There are a few other analytically solvable systems, but most are variations on the themes presented here and in the last chapter. For now, we will halt our treatment of model systems and move on to a system that is more obviously relevant chemically. But before we do, it is important to reemphasize a few conclusions about the systems we have treated so far. (1) In all of our model systems, the total energy (kinetic -I- potential) is quantized. This is a result of the postulates of quantum mechanics. (2) In some of the systems, other observables are also quantized and have analytic expressions for their quantized values (like momentum). Whether other observables have analytic expressions for their quantized values depends on the system. Average values, rather than quantized values, may be all that can be determined. (3) All of these model systems have approximate analogs in reality, so that the conclusions obtained from the analysis of these systems can be applied approximately to known chemical systems (much in the same way ideal gas laws are applied to the behavior of real gases). (4) Classical mechanics was unable to rationalize these observations of atomic and molecular systems. It is this last point that makes quantum mechanics worth understanding in order to understand chemistry. 

The partition function is the key quantity in the calculation of the Boltzmann equilibrium distribution of all molecular energies, and paves the way to a calculation of the total internal energy of any molecular system at equilibrium. Consider a system made of N molecules, each of which has a set of quantized energy levels Sj, known by the solution of some quantum chemical secular equation, as shown in Chapter 3. Energies are distributed over the accessible energy levels. With a modicum of mathematical derivation the following basic equations can be obtained ... 

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