Molecules, properties energy levels

Equation (5.20) is the basis for calculation of absolute entropies. In the case of an ideal gas, for example, it gives the probability ft for the equilibrium distribution of molecules among the various quantum states determined by the translational, rotational, and vibrational energy levels of the molecules. When energy levels are assigned in accord with quantum mechanics, this procedure leads to a value for the energy as well as for the entropy. From these two quantities all other thermodynamic properties can be evaluated from definitions (of H. G,... 

Here (p, cp2Q and (p2 are the waveflmctions of tire non-excited and excited molecules if tliere is no interaction between tliem. In tire case we consider tire molecules do interact and as a result the dimer exlribits properties different from Arose of tire monomers it comprises. In particular, tire energy level of tire excited state is different from tire monomer—it is split into two states ... 

In Chapter 2, a brief discussion of statistical mechanics was presented. Statistical mechanics provides, in theory, a means for determining physical properties that are associated with not one molecule at one geometry, but rather, a macroscopic sample of the bulk liquid, solid, and so on. This is the net result of the properties of many molecules in many conformations, energy states, and the like. In practice, the difficult part of this process is not the statistical mechanics, but obtaining all the information about possible energy levels, conformations, and so on. Molecular dynamics (MD) and Monte Carlo (MC) simulations are two methods for obtaining this information... 

The most extensive calculations of the electronic structure of fullerenes so far have been done for Ceo- Representative results for the energy levels of the free Ceo molecule are shown in Fig. 5(a) [60]. Because of the molecular nature of solid C o, the electronic structure for the solid phase is expected to be closely related to that of the free molecule [61]. An LDA calculation for the crystalline phase is shown in Fig. 5(b) for the energy bands derived from the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) for Cgo, and the band gap between the LUMO and HOMO-derived energy bands is shown on the figure. The LDA calculations are one-electron treatments which tend to underestimate the actual bandgap. Nevertheless, such calculations are widely used in the fullerene literature to provide physical insights about many of the physical properties. 

In Chapter 10, we will make quantitative calculations of U- U0 and the other thermodynamic properties for a gas, based on the molecular parameters of the molecules such as mass, bond angles, bond lengths, fundamental vibrational frequencies, and electronic energy levels and degeneracies. 

Tables 10.1, 10.2, and 10.3e summarize moments of inertia (rotational constants), fundamental vibrational frequencies (vibrational constants), and differences in energy between electronic energy levels for a number of common molecules or atoms/The values given in these tables can be used to calculate the rotational, vibrational, and electronic energy levels. They will be useful as we calculate the thermodynamic properties of the ideal gas.

Table A4.1 summarizes the equations needed to calculate the contributions to the thermodynamic functions of an ideal gas arising from the various degrees of freedom, including translation, rotation, and vibration (see Section 10.7). For most monatomic gases, only the translational contribution is used. For molecules, the contributions from rotations and vibrations must be included. If unpaired electrons are present in either the atomic or molecular species, so that degenerate electronic energy levels occur, electronic contributions may also be significant see Example 10.2. In molecules where internal rotation is present, such as those containing a methyl group, the internal rotation contribution replaces a vibrational contribution. The internal rotation contributions to the thermodynamic properties are summarized in Table A4.6.

This results in a property which we call fluid viscosity since the moment of each molecule interacts with all of its nearest neighbors. Yet, the same vibrational and rotational states are stUl present but in a different form. That is, they are mutated forms of the same energy levels that we found in the gaseous state. This is illustrated in the following diagram ... 

Fluorescence is a process that occurs after excitation of a molecule with light. It involves transitions of the outermost electrons between different electronic states of the molecule, resulting in emission of a photon of lower energy than the previously absorbed photon. This is represented in the Jablonski diagram (see Fig. 6.1). As every molecule has different energy levels, the fluorescent properties vary from one fluorophore to the other. The main characteristics of a fluorescent dye are absorption and emission wavelengths, extinction... 

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