Harmonic approximation, potential energy thermodynamics

Thus, the configurational entropy. S, (7. p) expresses the number of local potential energy minima and hence can be evaluated by potential energy landscape and thermodynamic integration methodology [53]. The vibration entropy was calculated within the framework of a harmonic approximation to each basin [165,166], an approximation that is valid at low temperatures [53]. Three notable results were obtained by Sastry [53] ... 

The harmonic approximation reduces to assuming the PES to be a hyperparaboloid in the vicinity of each of the local minima of the molecular potential energy. Under this assumption the thermodynamical quantities (and some other properties) can be obtained in the close form. Indeed, for the ideal gas of polyatomic molecules the partition function Q is a product of the partition functions corresponding to the translational, rotational, and vibrational motions of the nuclei and to that describing electronic degrees of freedom of an individual molecule ... 

Figure 2. The enthalpy H and its harmonic approximation U, in the vicinity of the minimum, for planar, rigid surfaces separated by a distance d (CE = 0.1 M, 11= 1 x 104 N/m2, T = 3.3 x 10 mol/m2, KD = 0.5 M, b, = 3.08 x 10 22J,b2= 6.28 x 10 14 J/m, b3 = 8.28 x 107 J/m, 64 = 6.13 x 1016 J/m2, br, = -9.00 x 10 23 J, and T = 300 Kj. pi is the distribution of the intersurface distances for interfaces with bending modulus Kc = 2 kT interacting via the potential C/h. This distribution coincides with the Boltzmann distribution of finite pieces of area A. p2 is the Boltzmann distribution of the pieces of area A, but now the enthalpy H and not its harmonic approximation U, is the thermodynamic potential. p3 is calculated using for A the value obtained from the minimization of the Gibbs free energy (eq 11a).

In order to evaluate the thermodynamic functions of the process (5), it is necessary to know the interaction energy, equilibrium geometry and frequencies of the normal vibration modes of the bases and base pairs involved in equilibrium process. Interaction energies and geometries are evaluated using empirical potential or quantum chemically (see next section), and normal vibrational frequencies are determined by a Wilson FG analysis implemented in respective codes. Partition functions, computed from AMBER 4.1, HF/6-31G and MP2/6-31G (0.25) constants (see next section), are evaluated widiin the rigid rotor-harmonic oscillator-ideal gas approximations (RR-HO-IG). We have collected evidence [26] that the use of RR-HO-IG approximations yields reliable thermodynamic characteristics (comparable to experimental data) for ionic and moderately strong H-bonded complexes. We are, therefore,... 

In the previous examples we only considered electronic energy changes and approximated the entropy as all or nothing. In essence, we assumed that gas-phase species have 100% of their standard state entropy and surface species possess no entropy at all. These assumptions can certainly be improved and in order to construct thermodynamically consistent microkinetic models this is not just optional, but absolutely necessary. Entropy and enthalpy corrections for surface species can be calculated using statistical thermodynamics from knowledge of the vibrational frequencies, and the translational and rotational degrees of freedom (DOF). In contrast to gas-phase molecules, adsorbates cannot freely rotate and move across the surface, but the translational and rotational DOF are frustrated within the potential energy well imposed by the surface. In the harmonic limit the frustrated translational and rotational DOF can conveniently be described as vibrational modes, which in turn means that any surface adsorbate will have iN vibrational DOFs that are all treated equally. 

Another technique to obtain the effects of the anharmonic terms on the excitation frequencies and the properties of molecular crystals is the Self-Consistent Phonon (SCP) method [71]. This method is based on the thermodynamic variation principle, Eq. (14), for the exact Hamiltonian given in Eq. (10), with the internal coordinates not explicitly considered. As the approximate Hamiltonian one takes the harmonic Hamiltonian of Eq. (18). The force constants in Eq. (18) are not calculated at the equilibrium positions and orientations of the molecules as in Eq. (19), however. Instead, they are considered as variational parameters, to be optimized by minimization of the Helmholtz free energy according to Eq. (14). The optimized force constants are found to be the thermodynamic (and thus temperature dependent) averages of the second derivatives of the potential over the (harmonic) lattice vibrations ... 

If the inversion barrier is too low for the harmonic oscillator approximation to be valid, or if the potential differs significantly from that described above, thermodynamic contributions should be estimated by direct summation of observed and calculated vibrational energy levels (p. 271). 

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