## Appendices

Appendix 1. Classical Reversible Thermodynamics First Law

Second Law n

Second Law n

Third Law lim 5 = 0 (pure, crystalline substance)

Classical thermodynamics: A, phenomenological science completely independent of concepts of the structure of the matter. U, internal energy; H, enthalpy; P, pressure; V, volume; Cv and Cp, heat capacity at constant volume and pressure, respectively; S entropy; qrev, heat accompanying a reversible process, A, Helmholtz energy; G, Gibbs energy; AG0, standard change of Gibbs energy; R, gas constant; T, absolute temperature; K, thermodynamic equilibrium constant of the process.

Appendix 2. Statistical Mechanics

1. Rigid rotor - harmonic oscillator approximation

The number of molecules n, having energy £,:

Q = Y,K>e £''kr ■ ■ ■ partition function i

(N is Avogadro's number, k is the Boltzmann constant, T is the absolute temperature, gi is the degeneracy of the 8; level; for other symbols, see Appendix 1)

2. Evaluation of Partition Function: Ideal gas

The definition equation of G (see Appendix I) is used directly for the evaluation of the electronic partition function, Qt.

G, is partition function associated with the i-th type of motion, i.e., the translational, rotational, vibrational, and electronic motion. The Ej's are the corresponding energies. IN is a component of the total moment of inertia, h is Planck's constant, v, is the frequency of the i-th vibrational mode.

Appendix 3. Statistical Mechanics: Liquids and Solutions

Change in the Helmholtz energy AA,^, when the system passes from state i to state ;:

1. Thermodynamic perturbation theory (PT)

2. Thermodynamic integration method (TI)

The coupling parameter A (0, 1) stands for a true or hypothetical reaction coordinate.

Appendix 4. Molecular Quantum Mechanics (Time-independent nonrelativistic Schrodinger Equation).

where n

Application of the variation principle:

IcJHllv - E,SIIV) = 0-> c,„, i.e., Aj det H,n - E,S„f |=0 E,

H is the «-electron Hamiltonian (operator making total energy calculations possible), V is the «-electron wavefunction, E is the total energy of the system. The wave function xj) has the form of a Slater determinant or of a linear combination of Slater determinants constructed, from molecular orbitals (</>,), which are expressed in the LCAO form (x,, is an atomic orbital, AO). Ht,v are the matrix elements of the Hamiltonian, S/lr are the matrix elements of the overlap matrix. The linear equations for c(> (i.e., for the wavefunctions) have nontrivial solutions only for those values of Et which satisfy the secular equation det | H^ - E,S,n, | =0.

Appendix 5. Vibrational problem (theory of small vibrations). Wilson's matrix analysis:

X Aj(bij - Xflij) = 0 vectors A, describe the motion associated != 1 with the individual vibrational modes

det | b,j - Xflij | = 0 vibrational energy Aj bij and a,t are the elements of the potential and kinetic energy matrices, Aj's are the energies of the vibrational modes, which are described by vectors. Aj. The diagonal elements of the potential energy matrix are the force constants, the nondiagonal elements are the interaction constants.

Appendix 6. Classical mechanics. Newton's 2nd principle:

m is the mass of a mass point moving along the jc-coordinate, F is the force, x = dx/dt is the velocity, x = d2x/df = dx/di is the acceleration.

Lipophilicity in Drug Action and Toxicology edited by Vladimir PliSka,Bernard Testa & Han van de Waterbeemd Copyright© VCH Verlagsgesellschaft mbH,1996

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