Torsion partition function

The partition function of a molecule also contains torsional motions and the construction of such a function requires the knowledge of molecular mass, moments of inertia, and constants describing normal vibration modes. Several of these data may be acquired from infrared and Raman spectra (67SA(A)891 85JST( 126)25), but the procedure has not yet been extensively applied owing to experimental limitations. To characterize the barrier one also needs to know more than one constant, and these are often not available from... 

A very simple view of these reactions would be that the total partition function for the activated state differs from that for the initial state solely by the absence of the contribution 1/(1—e " ) corresponding to the torsional oscillation, all other individual partition functions remaining the same. If this were the case the rate coefficient would be... 

We assume that there is no free internal rotation in the molecule, and the contribution from the torsional oscillation (v. = 182.5 cm" ) is included in the vibrational partition function. Extended Huckel calculations (7) show that the potential... 

When a -0.42 torsional correction to ethane is included we arrive at -30.70 kcal mol 1 which compares with Kistiakowsky s experimental value of Ahyd//298 = -32.58 0.05 kcal mol 1. The discrepancy is well outside the combined error bars of the experiment and the calculation. A calculation by the more recent MM4 force field, however, yields -32.69 kcal mol 1 (Allinger et al., 1996). [For more on the partition function contribution and torsional corrections, see Rogers (2003).]... 

In this expression V x, V and V3 are partition functions for torsional vibration which can be temperature-dependent, 81 S2 and Ss are chemical shifts of the given proton in the three separate rotamers, Ex, Ez and E3 are the energies of the potential maxima from the zero point torsional levels in each separate rotamer. Coupling constants are averaged in a similar way by replacing S in equation (77) by J in order to obtain J averaging. 

Pak and Chang have previously developed a partition function for liquid water applying the modified theory of significant liquid structure proposed by Chang et al. There, it is assumed that Ice-I-like, Ice-III-like, and gas-like molecules exist in liquid water and the molecules like Ice I and Ice III, both of which are oscillating torsionally, are in thermodynamic equilibrium. The equilibrium constant has been taken equal to the ratio of the partition functions of the two species. Various thermodynamic properties and the surface tension of liquid water from the partition function were successfully calculated. 

Also plotted in Fig. 1.2 is the experimental heat capacity of the liquid (at omi-stant pressure) In simple cases, such as polyethylene, the heat capacity of the liquid state could be understood by introducing a heat capacity contribution for the excess volume (hole theory) and by assuming that the torsional skeletal vibration can be treated as a hindered rotator A more general treatment makes use of a separation of the partition function into the vibrational part (approximated for heat capacity by the spectrum of the solid), a conformational part (approximated by the usual conformational statistics) and an external or configurational part. 

In the gas phase, it is usually sufhcient to calculate the partition functions and associated thermal corrections to the enthalpy and entropy using the standard textbook formulae [31] for an ideal gas under the harmonic oscillator-rigid rotor approximation, provided one then makes explicit corrections for low-frequency torsional modes. These modes can be treated instead as one-dimensional hindered internal rotations using the torsional eigenvalue summation procedure described in Ref. [32]. Rate and equilibrium constants can then be obtained from the following standard textbook formulae [31] ... 

As indicated, the heat of formation (HFO) is taken as the sum of three terms. The first is the steric energy (E). The second is the bond enthalpy (BE), which is calculated from normal bond increments. The third is a partition function increment (PFC), which is itself the sum of three terms a conformational population increment (POP), a torsional contribution term (TOR), and a translation / rotation term (T / R). Since the molecular mechanics calculation has considered only one conformation, the POP and TOR terms are defined to be zero for this calculation (but see below). The T/R term is a molecular translational and rotational term that is always taken to be 2.4kcal/mol at room temperature, since even an anfi-butane molecule that is not rotating about the C2-C3 bond will still be undergoing translation and rotation of the entire molecule as a unit. Therefore, HFO is the total of the bond increment contributions (BE = —35.08) plus the steric energy (E = 2.17) plus the partition function increment (PFC = 2.4). The sum, -30.51 kcal/mol, is slightly lower than the literature value of -30.15 kcal/mol. ... 

Having obtained the partition function (or equivalently, the enthalpy and entropy) associated with a low frequency torsional mode, this is used in place of the corresponding harmonic oscillator contribution for that mode. 

Adiabatic Treatment of Torsional Anharmonicity and Mode Coupling in Molecular Partition Functions and Statistical Rate Coefficients Application to Hydrogen Peroxide... 

We first focus on the properties of the bound molecule and consider the effects of non-separability on the molecular partition function and density of states. The potential non-separability is reflected by the variation of the normal mode frequencies as a function of t. Projecting out the torsional motion from the Hessian matrix, we obtain the quadratic form for the potential in terms of the instantaneous normal mode coordinates, ( = 1-5) ... 

For light rotors it may be possible to observe sufficient torsional transitions for the internal rotation partition function to be determined by direction summation (p. 271). If only a few low-lying torsional levels are observed, they may be fitted to a suitable potential from which higher torsional energy levels can be computed. Thermodynamic functions of hydrogen peroxide calculated by use of this procedure give a value of 5 (298.15 K) in excellent agreement with the calorimetric value. ... 

For heavy asymmetric rotors the torsional levels may be sufficiently low for thermodynamic properties to be estimated by using the classical approximation. Gwinn and Pitzer have expressed the classical approximation to the complete internal partition function as... 

Of the 24 possible conformations for cyclohexanethiol, the spectroscopic evidence is that only those with the chair form of the ring are present in high enough concentration to be observed at room temperature. The torsion of the —SH group was treated separately as a 3-fold cosine barrier. There are three conformations with respect to that torsion which are likely to be of high energy and they were excluded. The partition function then becomes... 

The shape of the curve may be explained in terms of the previous considerations as arising from an intrinsic torsional potential and the interaction between the nonbonded methyl groups.There are three minima of the e-nergy, corresponding to T,G, G conformations, the last two being energetically equivalent. For the evaluation of the partition function these are chosen as rotational isomeric states, and their statistical weights are ... 

BE = bond increment GE = group increment HFE = Hartree-Fock energy MH = molar heat content MM2, MMP2, MM3, MM4 = molecular mechanics programs PFC = partition function contribution POP = population contribution SE = structural features increment TOR = torsional contribution. 

The partition function contribution, PFC, in equation (2) includes contributions from statistical thermodynamics that consist of the conformational population contribution (POP), the torsional contribution (TOR), and the contributions from translational rotation and a PV term required to convert the energy into enthalpy (T/R) ... 

---