## Outline of Thermodynamics and Auxiliary Disciplines

This chapter contains several items of textbook knowledge. The reason for this is that some potential users of theoretical tools are, to a smaller or greater extent, familiar with the individual steps of the overall treatment but essential features of the overall procedure escape them (Fig. 2).

QUANTUM MECHANICS

(energy and structure)

STATISTICAL MECHANICS

COMPUTER EXPERIMENTS

MC MD

THERMODYNAMICS

Equilibrium constants of the processes under study

(vibrational modes)

THEORY OF

SMALL VIBRATIONS

MC MD

Figure 2. Theories and procedures used in computational treatment of equilibria (MC and MD are Monte Carlo-type computer experiments and molecular dynamics, respectively).

Some fundamental relationships of reversible thermodynamics are summarized in Appendix 1. Boltzmann introduced statistical mechanics a century ago. It is the partition function (a magnitude related to the Boltzmann distribution function) which assumes a central position in statistical mechanics and thermodynamics: its important feature is that it can be easily evaluated in terms of universal constants and molecular constants and characteristics (see Appendix 2). These magnitudes can either be obtained from the analysis of experimental molecular spectra (rotational, vibrational, and electronic) or can be generated by means of molecular quantum mechanics (solution of the Schrodinger equation) and by solving the vibrational problem [10] (Wilson's FG analysis). A fascinating feature of statistical thermodynamics is that it connects the structure of a substance with its thermodynamic behavior. Evaluation of the partition functions of molecules is a simple and straightforward task for processes taking place in an ideal gas phase. It is true that the fundamental equations of statistical mechanics have been available for many years also for condensed media, but their evaluation is hopelessly complicated. In spite of great differences between ideal gas phase and real condensed phase, the former represents mostly a valuable starting point.

The introduction of computer experiments about 30 years ago, represented a great step forward. That which is called the Monte Carlo (MC) procedure [11] is well suited for describing the structures of liquids and solutions and also equilibria in solutions. However, another procedure, based on classical mechanics, is capable of doing the same job but it also describes the dynamic features of processes in solutions [12], which is a very valuable feature. Two types of procedure are elaborated, which permit us to calculate a change in the Helmholtz energy (definition: A = U - TS; cf. Abbreviations) of the system under study when passing from state i to state j (AA, ,j). The first procedure is named the thermodynamic perturbation theory [13] and the second, the thermodynamic integration method [14] (Appendix 3). Calculations of A A are carried out for an NVT ensemble (i.e., constant number of particles, N, constant volume, V and temperature, T). Passing from the NVT ensemble to the NPT ensemble (P is constant pressure) is connected with passing from the Helmholtz to the Gibbs energy (definition: G = H — TS; cf. Abbreviations). A specific illustration of application of the relative free energy calculation method [15] to the evaluation of the difference in the solvation free energies of methanol and ethane [16] is given in section 3.5.

Molecular quantum mechanics represents a tool [3] that makes the molecular constants and total energy readily accessible (Appendix 4). All the necessary molecular constants can be obtained from the total wave function. In a majority of instances, we are interested in the lowest energy value (i.e., energy value of the ground state) resulting from solution of the determinant equation.

Although solving the Schrodinger equation now represents a standard and routine procedure, a good-quality level for bigger polyatomic molecules is still a rather demanding task. Therefore, it is expedient even now to have several levels of complexity at our disposal [3, 17]. Firstly, there are nonempirical and semiempirical quantum chemistry methods. Pragmatically speaking, the variation method is mostly a versatile tool; its combination with the Hartree-Fock-Roothaan (HFR) procedure is the most widespread technique. It is fair to admit that it is only rarely a reliable tool at a nonempirical level. The beyond-HFR procedure is always necessary when a covalent or van der Waals bond is formed or split. When investigating polyatomic species, complete geometry optimization is a standard procedure. When stationary points are located on the potential energy surface of the system under study, a decision must be made as to whether they represent minima (stable isomers) or saddle points (Eyring's activated complexes). This is an easy task because we need to carry out vibrational analysis anyway (Appendix 5) and this analysis permits us to make the decision.

The remaining two procedures are based on classical mechanics. This is amazing because our treatment deals with particles of the microcosmos, which is, of course, a realm for quantum and not for classical mechanis. It turns out, however, that this requirement is strictly valid only for electrons. Much heavier nuclei can frequently be properly described in terms of classical mechanics. This has significant practical consequences: vibrational spectra (Appendix 5) and the dynamics of molecules can be successfully described in terms of Newton's mechanics (or in terms of its version originating from the 19th century, Lagrange and Hamilton mechanics) (Appendix 6).

In order to be able to analyze the vibrational motion of a polyatomic molecule, it is necessary to perform the Wilson's matrix analysis. The formulation of characteristic equations (Appendix 5) requires the evaluation of the elements of the potential (b¡¡) and kinetic energy (ay) matrices. From a formal point of view the problems solved in Appendices 4 and 5 are identical. The characteristic values represent the energies of the individual normal modes of vibration (X) and the associated characteristic vectors offer information on the nature of the individual vibration modes (the so-called harmonic approximation). The vibrational energies permit as to evaluate the zero-point energies of the molecules under study, to obtain their vibrational partition functions, and to decide on the nature of the localized stationary points on the potential energy surface of the molecule under study. (As mentioned before, the partition function is related to the Boltzmann distribution function. The total partition function for a system can be approximated as a product of its partial partition functions, which are related to rotational, vibrational, electronic, and translational energy).

As already mentioned, classical mechanics represents a valuable tool for investigating molecular dynamics (Appendix 6). Clearly, when using this tool, one does not obtain information on quantum effects like quantum mechanical tunneling. More specific information will be presented in section 3.5.

0 0