Internal partition function

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... 

From the grand canonical partition function internal energy and pressure are defined by using (d0/d7 )y we find... 

Since translational and internal energy (of rotation and vibration) are independent, the partition function for the gas can be written... 

Thus the kinetic and statistical mechanical derivations may be brought into identity by means of a specific series of assumptions, including the assumption that the internal partition functions are the same for the two states (see Ref. 12). As discussed in Section XVI-4A, this last is almost certainly not the case because as a minimum effect some loss of rotational degrees of freedom should occur on adsorption. 

Once the partition function is evaluated, the contributions of the internal motion to thennodynamics can be evaluated. depends only on T, and has no effect on the pressure. Its effect on the heat capacity can be... 

There is an inunediate coimection to the collision theory of bimolecular reactions. Introducing internal partition functions excluding the (separable) degrees of freedom for overall translation. 

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 ... 

LS now consider the problem of calculating the Helmholtz free energy of a molecular 1. Our aim is to express the free energy in the same functional form as the internal that is as an integral which incorporates the probability of a given state. First, we itute for the partition function in Equation (6.21) ... 

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. 

Once the partition function is known, thermodynamic functions such as the internal energy U and Helmholtz free energy A may be calculated according to... 

The above treatment has made some assumptions, such as harmonic frequencies and sufficiently small energy spacing between the rotational levels. If a more elaborate treatment is required, the summation for the partition functions must be carried out explicitly. Many molecules also have internal rotations with quite small barriers, hi the above they are assumed to be described by simple harmonic vibrations, which may be a poor approximation. Calculating the energy levels for a hindered rotor is somewhat complicated, and is rarely done. If the barrier is very low, the motion may be treated as a free rotor, in which case it contributes a constant factor of RT to the enthalpy and R/2 to the entropy. 

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 ... 

With equations (10.138) and (10.139) the partition function for free rotation can be written. However, when the internal rotation can be described by a... 

Table A4.6 gives the internal rotation contributions to the heat capacity, enthalpy and Gibbs free energy as a function of the rotational barrier V. It is convenient to tabulate the contributions in terms of VjRTagainst 1/rf, where f is the partition function for free rotation [see equation (10.141)]. For details of the calculation, see Section 10.7c.

The model [39] was developed using three assumptions the conformers are in thermodynamic equilibrium, the peak intensities of the T-shaped and linear features are proportional to the populations of the T-shaped and linear ground-state conformers, and the internal energy of the complexes is adequately represented by the monomer rotational temperature. By using these assumptions, the temperature dependence of the ratio of the intensities of the features were equated to the ratio of the quantum mechanical partition functions for the T-shaped and linear conformers (Eq. (7) of Ref. [39]). The ratio of the He l Cl T-shaped linear intensity ratios were observed to decay single exponentially. Fits of the decays yielded an approximate ground-state binding... 

Partition functions are very important in estimating equilibrium constants and rate constants in elementary reaction steps. Therefore, we shall take a closer look at the partition functions of atoms and molecules. Motion, or translation, is the only degree of freedom that atoms have. Molecules also possess internal degrees of freedom, namely vibration and rotation. 

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. 

In the following we consider a surface with adsorbed atoms or molecules that react. We will leave out the details of the internal coordinates of these adsorbed species, but note that their partition functions can be found using the schemes presented above. Let us assume that species A reacts with B to form an adsorbed product AB via an activated complex AB ... 

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. 

Here an additional distinction is to be made between thermodynamic averages of a conformational observable such as the internal energy, which converges well if potential minima are correctly sampled, and statistical properties such as free energies, which depend on the entire partition function. 

The chemical potential pa on the left is the full chemical potential including ideal and excess parts. In this chapter we will scale the chemical potentials by (3 and often refer to this unitless quantity as the chemical potential. 3/ia yields the absolute activity. The first term on the right is the ideal-gas chemical potential, where pa is the number density, Aa is the de Broglie wavelength, and q 1 is the internal (neglecting translations) partition function for a single molecule without interactions with any other molecules. 

Show that, for the bimolecular reaction A + B - P, where A and B are hard spheres, kTsr is given by the same result as jfcSCT, equation 6.4-17. A and B contain no internal modes, and the transition state is the configuration in which A and B are touching (at distance dAR between centers). The partition functions for the reactants contain only translational modes (one factor in Qr for each reactant), while the transition state has one translation mode and two rotational modes. The moment of inertia (/ in Table 6.2) of the transition state (the two spheres touching) is where p, is reduced mass (equation 6.4-6). 

Using harmonic oscillator partition functions to describe both internal and external modes, the logarithmic Q ratios introduced above, ln(Qg/Qc/QgQc/) = ln(Qc/QcO + ln(Qg7Qg), become... 

The Kieffer model correctly predicts the systematic change of the reduced partition functions of various minerals with structure, as indicated by Taylor and Epstein (1962). For anhydrous sihcates, the decrease in the sequence framework-chain-orthosilicate reflects the decreasing frequency of antisymmetric Si-O stretching modes. The internal frequencies of the carbonate ion give a high reduced partition function at all T. The value for rutile is low because of the low frequencies of the Ti-0 modes (Kieffer, 1982). 

The temperature ranges in which these simple behaviours are approximated depend on the vibrational frequencies of the molecules involved in the reaction. For the calculation of a partition function ratio for a pair of isotopic molecules, the vibrational frequencies of each molecule must be known. When solid materials are considered, the evaluation of partition function ratios becomes even more complicated, because it is necessary to consider not only the independent internal vibrations of each molecule, but also the lattice vibrations. 

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). 

Evaluation of die rotational components of the internal energy and entropy using the partition function of Eq. (10.19) gives... 

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. ... 

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