Rigid rotor harmonic oscillator

The chapter starts with a brief review of thermodynamic principles as they apply to the concept of the chemical equilibrium. That section is followed by a short review of the use of statistical thermodynamics for the numerical calculation of thermodynamic equilibrium constants in terms of the chemical potential (often designated as (i). Lastly, this statistical mechanical development is applied to the calculation of isotope effects on equilibrium constants, and then extended to treat kinetic isotope effects using the transition state model. These applications will concentrate on equilibrium constants in the ideal gas phase with the molecules considered in the rigid rotor, harmonic oscillator approximation. 

Table 4.1 Partition functions evaluated in the rigid rotor harmonic oscillator approximation... 

In Chapter 3, a formula was presented which connects the normal vibrational frequencies of two rigid-rotor-harmonic-oscillator isotopomers with their respective atomic masses m , molecular masses Mi and moments of inertia (the Teller-Redlich product rule). If this identity is substituted into Equation 4.77, one obtains... 

The statement applies not only to chemical equilibrium but also to phase equilibrium. It is obviously true that it also applies to multiple substitutions. Classically isotopes cannot be separated (enriched or depleted) in one molecular species (or phase) from another species (or phase) by chemical equilibrium processes. Statements of this truth appeared clearly in the early chemical literature. The previously derived Equation 4.80 leads to exactly the same conclusion but that equation is limited to the case of an ideal gas in the rigid rotor harmonic oscillator approximation. The present conclusion about isotope effects in classical mechanics is stronger. It only requires the Born-Oppenheimer approximation. 

A2 Corrections to the Rigid Rotor Harmonic Oscillator Approximation in the Calculation of Equilibrium Constants... 

Calculations of isotope effects and isotopic exchange equilibrium constants based on the Born-Oppenheimer (BO) and rigid-rotor-harmonic-oscillator (RRHO) approximations are generally considered adequate for most purposes. Even so, it may be necessary to consider corrections to these approximations when comparing the detailed theory with high precision high accuracy experimental data. 

It should be noted that, in these approximations [usually referred to as the rigid rotor-harmonic oscillator (RRHO) model], all the kinds of motions— electronic vibrational, and rotational—are strictly separated. 

Figure 8. Spectra of CO calculated from the rigid rotor-harmonic oscillator approximation. The top spectrum is CO at 298 K. The bottom is CO at 20 K. This reduction in lines will be very important for simplifying the analysis in larger, more complicated species. 

Principal axes can easily be identified in a molecule which possesses symmetry elements e.g., symmetry axes that coincide with principal ones, and a symmetry plane that is oriented perpendicularly to one of the principal axes. The simplest models discussed here are rigid rotor - harmonic oscillator models, which can be extended on demand to better fit the spectral data. For a more complete coverage, the reader is referred to other text books. As a first introduction to infrared rotation-vibration spectra the author prefers Barrow (1962). The topic is discussed in greater details by publications such as by Allen and Cross (1963), Herzberg (1945, 1950), and Hollas (1982). 

Nucleic acid base pairs 3.3.1 Rigid rotor - harmonic oscillator - ideal gas 845... 

Free energy surface Rigid rotor-harmonic oscillator-ideal gas 850... 

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,... 

Rigid rotor-harmonic oscillator-ideal gas approximation. The AMBER 4.1 free energy values are summarized in Table 7. The entropy term is important and compensates for the interaction energy (enthalpy) term. A similar type of compensation has also been found in the case of DNA base pairs [40]. FI-bonded structure 4 remains the most stable and also HB6 and HBl structures remain as the second and third most stable ones. The following order of stability is however, changed. The H-bonded structure 7 and the T-shaped structure are surprisingly more stable than H-bonded structures 2, 3 and 5. Analyzing veirious... 

---