Internal energy from partition function

For AT diatomic molecules that have only a single internal vibration the partition function is easily determined from the approximate formula for the energy levels... 

Computing the Internal Energy from the Partition Function... 

The thermodynamic properties that we have considered so far, such as the internal energy, the pressure and the heat capacity are collectively known as the mechanical properties and can be routinely obtained from a Monte Carlo or molecular dynamics simulation. Other thermodynamic properties are difficult to determine accurately without resorting to special techniques. These are the so-called entropic or thermal properties the free energy, the chemical potential and the entropy itself. The difference between the mechanical emd thermal properties is that the mechanical properties are related to the derivative of the partition function whereas the thermal properties are directly related to the partition function itself. To illustrate the difference between these two classes of properties, let us consider the internal energy, U, and the Fielmholtz free energy, A. These are related to the partition function by ... 

To reiterate a point that we made earlier, these problems of accurately calculating the free energy and entropy do not arise for isolated molecules that have a small number of well-characterised minima which can all be enumerated. The partition function for such systems can be obtained by standard statistical mechanical methods involving a summation over the mini mum energy states, taking care to include contributions from internal vibrational motion. 

C) The error in AE" /AEq is 0.1 kcal/mol. Corrections from vibrations, rotations and translation are clearly necessary. Explicit calculation of the partition functions for anharmonic vibrations and internal rotations may be considered. However, at this point other factors also become important for the activation energy. These include for example ... 

Thus, given sufEcient detailed knowledge of the internal energy levels of the molecules participating in a reaction, we can calculate the relevant partition functions, and then the equilibrium constant from Eq. (67). This approach is applicable in general Determine the partition function, then estimate the chemical potentials of the reacting species, and the equilibrium constant can be determined. A few examples will illustrate this approach. 

From Eqs. (45) and (46) it is apparent that the calculation of the energy and heat capacity of a system depends on the evaluation of the partition function a a function of temperature. In the more general case of molecules with an internal structure, the energy distributions of the various degrees of freedom must bo determined. This problem is outlined briefly in the following section. 

In practice, it proves more convenient to work within a convention where we define tire ground state for each energy component to have an energy of zero. Thus, we view 1/eiec as the internal energy that must be added to I/q, which already includes Eeiec (see Eq. (10.1)), as the result of additional available electronic levels. One obvious simplification deriving from this convention is that the electronic partition function for the case just described is simply eiec = 1, Inspection of Eq. (10.5) then reveals that the electronic component of the entropy will be zero (In of 1 is zero, and the constant 1 obviously has no temperature dependence, so both terms involving eiec are individually zero). 

Note that the zero-point-energy-including difference in internal energies between A and A in the exponential term is easily computable from an electronic structure calculation (for the electronic energy) and a frequency calculation (to determine the ZPVE) for the minimum energy and TS structures corresponding to A and A, respectively. In addition, the availability of frequencies for A permits ready computation of Qa, as described in Chapter 10. Some attention needs to be paid, however, to the nature of the partition function for the activated complex, Q. ... 

The partition function provides the bridge to calculating thermodynamic quantities of interest. Using the molecular partition function and formulas derived in this section, we will be able to calculate the internal energy E, the heat capacity Cp, and the entropy S of a gas from fundamental properties of the molecule, such as its mass, moments of inertia, and vibrational frequencies. Thus, if thermodynamic data are lacking for a species of interest, we usually know, or can estimate, these molecular constants, and we can calculate reasonably accurate thermodynamic quantities. In Section 8.6 we illustrate the practical application of the formulas derived here with a numerical example of the thermodynamic properties for the species CH3. 

For a temperature of 298.15 K, a pressure of 1 bar, and 1 mole of H2S, prepare a table of (1) the entropy (J/mol K), and separately the contributions from translation, rotation, each vibrational mode, and from electronically excited levels (2) specific heat at constant volume Cv (J/mol/K), and the separate contributions from each of the types of motions listed in (1) (3) the thermal internal energy E - Eo, and the separate contributions from each type of motion as before (4) the value of the molecular partition function q, and the separate contributions from each of the types of motions listed above (5) the specific heat at constant pressure (J/mol/K) (6) the thermal contribution to the enthalpy H-Ho (J/mol). 

Given the expression for K(T), one can construct an EOS by modeling the excess free energy density by = HS + u + ID + DI + DD + where is summed over contributions from hard-sphere (HS), ion-ion (II), ion-dipole (ID), dipole-ion (DI), and dipole-dipole interactions (DD), respectively. 4>ex also contains the contribution due to the internal partition function of the ion pair, = — p lnK(T). Pairing theories differ in the terms retained in the expression for ex. 

From differentiation of the partition function and carrying out a reversible change in the internal energy using parameter, X, (H = Ho(l — X) + that goes from 0 to 1, it can be shown that ... 

Given a set of histograms, Hi Ej) from multiple temperature sweeps, Eqs. (14-16) can be solved for Wj seif consistently. We initialize Wj at dj and subsequently iterate these equations sequentially until the total change in W is less than a predetermined limit (set at 10 in our calculations). Once solved, the static thermodynamic properties of the system can be determined from Wj. The partition function, internal energy, specific heat, and entropy can be estimated by... 

We have defined the solvation process as the process of transfer from a fixed position in an ideal gas phase to a fixed position in a liquid phase. We have seen that if we can neglect the effect of the solvent on the internal partition function of the solvaton s, the Gibbs or the Helmholtz energy of solvation is equal to the coupling work of the solvaton to the solvent (the latter may be a mixture of any number of component, including any concentration of the solute s). In actual calculations, or in some theoretical considerations, it is often convenient to carry out the coupling work in steps. The specific steps chosen to carry out the coupling work depend on the way we choose to write the solute-solvent interaction. 

---