Harmonic free energy

Figure Schematic diagram to show the reorganization energy for isotopic reactions for harmonic free energy profiles. Ci i, and Gf, represent the initial (reactant) and the final (product) system free energy respectively.

Figure 2. Dissociation pressure of ethane hydrate at 273.15 K. Solid, dashed, and dash-dot lines show the dissociation pressures for the nonspherical guest evaluated by anharmonic and harmonic free energy, for spherical guest molecule evaluated by only harmonic free energy, and for spherical guest according to the original vdWP theory, respectively. The horizontal lines show the chemical potential difference between ice and empty hydrate, dashed harmonic-t-anharmonic, dotted harmonic.

Fig. 2. Two opposite limits for the behaviour of the Fourier transform f(q) of the harmonic free energy given by Equ. (23). The Fourier componet ip (q) of ip (r) is kept independent of q in the figure.

Calculation of entropies of both types is straightforward and efficient because simulations are not required. This allows one to study the conformational stability of many localized microstates of the same molecule by comparison of their harmonic free energies. However, the method is limited only to models of macromolecules in vacuum. 

The temperatures are provided in the first row. The last two rows indicate the harmonic free energy (kcal/mol) and the potential energy value (kcal/mol), respectively. 

Relative free energies were also calculated for clusters of low-energy conformers. This analysis is useful because it is difficult to capture the tme accessibiUty of individual structures based on a pointwise approximation of entropic effects. That is, the harmonic free energy approximation does not provide a continuous free energy landscape. By clustering structures into larger... 

Mesoscale simulations model a material as a collection of units, called beads. Each bead might represent a substructure, molecule, monomer, micelle, micro-crystalline domain, solid particle, or an arbitrary region of a fluid. Multiple beads might be connected, typically by a harmonic potential, in order to model a polymer. A simulation is then conducted in which there is an interaction potential between beads and sometimes dynamical equations of motion. This is very hard to do with extremely large molecular dynamics calculations because they would have to be very accurate to correctly reflect the small free energy differences between microstates. There are algorithms for determining an appropriate bead size from molecular dynamics and Monte Carlo simulations. 

In the Hamiltonian Eq. (3.39) the first term is the harmonic lattice energy given by Eq. (3.12). It depends only on A iU, i.e., the part of the order parameter that describes the lattice distortions. On the other hand, the electron Hamiltonian Hcl depends on A(.v), which includes the changes of the hopping amplitudes due to both the lattice distortion and the disorder. The free electron part of Hel is given by Eq. (3.10), to which we also add a term Hc 1-1-1 that describes the Coulomb interne-... 

Exercise 3.4. Evaluate the free-energy difference between two one-dimensional harmonic oscillators with potentials U1 = (X- l)2 and U2 = 0.3(X —... 

The harmonic approximation can also be used to provide an estimate of the vibrational free energy, using (Refs. 1 and 6). 

The Gibbs free energy (computed in the harmonic approximation) were converted from the 1 atm standard state into the standard state of molar concentration (ideal mixture at 1 molL-1 and 1 atm). 

Instead of the quantity given by Eq. (15), the quantity given by Eq. (10) was treated as the activation energy of the process in the earlier papers on the quantum mechanical theory of electron transfer reactions. This difference between the results of the quantum mechanical theory of radiationless transitions and those obtained by the methods of nonequilibrium thermodynamics has also been noted in Ref. 9. The results of the quantum mechanical theory were obtained in the harmonic oscillator model, and Eqs. (9) and (10) are valid only if the vibrations of the oscillators are classical and their frequencies are unchanged in the course of the electron transition (i.e., (o k = w[). It might seem that, in this case, the energy of the transition and the free energy of the transition are equal to each other. However, we have to remember that for the solvent, the oscillators are the effective ones and the parameters of the system Hamiltonian related to the dielectric properties of the medium depend on the temperature. Therefore, the problem of the relationship between the results obtained by the two methods mentioned above deserves to be discussed. 

If the size of the complex is rather small and the intramolecular vibrations along the coordinate Q may be described in the harmonic approximation, the free energy surfaces of the initial and final states may be written in the form... 

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