Energy of a molecule

It is of interest in the present context (and is useful later) to outline the statistical mechanical basis for calculating the energy and entropy that are associated with rotation [66]. According to the Boltzmann principle, the time average energy of a molecule is given by... 

A force field does not consist only of a mathematical eiqjression that describes the energy of a molecule with respect to the atomic coordinates. The second integral part is the parameter set itself. Two different force fields may share the same functional form, but use a completely different parameterization. On the other hand, different functional forms may lead to almost the same results, depending on the parameters. This comparison shows that force fields are empirical there is no "correct form. Because some functional forms give better results than others, most of the implementations within the various available software packages (academic and commercial) are very similar. 

The following sections give an overview of the functional form of the PFF and a short explanation of the various contributions to the total force field energy of a molecule or molecular system. 

The absolute energy of a molecule in molecular nicchari ics has no iulriusic physical mean mg values are useful on ly for com-... 

The difference between the energy of a molecule at 0 K and its enthalpy at 298 depends on the thermal contr ibution due to vibration at the two temperatures. If the molecule in question is rigid, with few vibrational degrees of freedom, this contribution will be small, as it is for propene and cyclopropane. For larger molecules with a good deal of vibrational freedom, the difference will be conespondingly larger. 

The premise behind DFT is that the energy of a molecule can be determined from the electron density instead of a wave function. This theory originated with a theorem by Hoenburg and Kohn that stated this was possible. The original theorem applied only to finding the ground-state electronic energy of a molecule. A practical application of this theory was developed by Kohn and Sham who formulated a method similar in structure to the Hartree-Fock method. 

Both molecular and quantum mechanics methods rely on the Born-Oppenheimer approximation. In quantum mechanics, the Schrodinger equation (1) gives the wave functions and energies of a molecule. 

Infrared Spectra for Molecules and Polyatomic Ions The energy of infrared radiation is sufficient to produce a change in the vibrational energy of a molecule or polyatomic ion (see Table 10.1). As shown in Figure 10.14, vibrational energy levels are quantized that is, a molecule may have only certain, discrete vibrational energies. The energy for allowed vibrational modes, Ey, is... 

Vibrational energy, which is associated with the alternate extension and compression of die chemical bonds. For small displacements from the low-temperature equilibrium distance, the vibrational properties are those of simple harmonic motion, but at higher levels of vibrational energy, an anharmonic effect appears which plays an important role in the way in which atoms separate from tire molecule. The vibrational energy of a molecule is described in tire quantum theory by the equation... 

Another approach to calculating molecular geometry and energy is based on density functional theory (DFT). DFT focuses on the electron cloud corresponding to a molecule. The energy of a molecule is uniquely specified by the electron density functional. The calculation involves the construction of an expression for the electron density. The energy of the system is then expressed as... 

A set of equations defining how the potential energy of a molecule varies with the locations of its component atoms. 

There are many ways to express the energy of a molecule. Most common to organic chemists is as a heat of formation, AHf. This is the heat of a hypothetical chemical reaction that creates a molecule from so-called standard states of each of its constituent elements. For example, AHf for methane would be the energy required to create CH4 from graphite and H2, the standard states of carbon and hydrogen, respectively. 

Potential Energy Surface. A many-dimensional function of the energy of a molecule in terms of the geometrical coordinates of the atoms. 

In order to calculate q (Q) all possible quantum states are needed. It is usually assumed that the energy of a molecule can be approximated as a sum of terms involving translational, rotational, vibrational and electronical states. Except for a few cases this is a good approximation. For linear, floppy (soft bending potential), molecules the separation of the rotational and vibrational modes may be problematic. If two energy surfaces come close together (avoided crossing), the separability of the electronic and vibrational modes may be a poor approximation (breakdown of the Bom-Oppenheimer approximation. Section 3.1). 

Rotational Energy Levels The rotational energy of a molecule depends upon the molecular geometry. For a linear molecule that behaves as a rigid rotator,3... 

Since the free energy of a molecule in the liquid phase is not markedly different from that of the same species volatilized, the variation in the intrinsic reactivity associated with the controlling step in a solid—liquid process is not expected to be very different from that of the solid—gas reaction. Interpretation of kinetic data for solid—liquid reactions must, however, always consider the possibility that mass transfer in the homogeneous phase of reactants to or products from, the reaction interface is rate-limiting [108,109], Kinetic aspects of solid—liquid reactions have been discussed by Taplin [110]. 

The following assumptions are made (i) the activated complexes are in equilibrium with the reactants, (ii) the energy of a molecule is not altered when an activated complex is substituted for a nearest neighbour, and (iii) the products do not affect the course of reaction, except to define a boundary in surface processes. The various cases can be recognized from the magnitude of the pre-exponential term and calculated values [515] are summarized in Table 7. Low values of A indicate a tight surface complex whereas higher values are associated with a looser or mobile complex. 

A plot of the Maxwell distribution for the same gas at several different temperatures shows that the average speed increases as the temperature is raised (Fig 4.27). We knew that already (Section 4.9) but the curves also show that the spread of speeds widens as the temperature increases. At low temperatures, most molecules of a gas have speeds close to the average speed. At high temperatures, a high proportion have speeds widely different from their average speed. Because the kinetic energy of a molecule in a gas is proportional to the square of its speed, the distribution of molecular kinetic energies follows the same trends. 

FIGURE 4.29 Variation in the potential energy of a molecule as it approaches another molecule. The potential energy rises sharply once the two molecules come into direct contact. 

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