Molecular harmonic oscillator approximation

The coordinates of interest to us in the following discussion are Qx and Qy, which describe the distortion of the molecular triangle from Dy, symmetry. In the harmonic-oscillator approximation, the factor in the vibrational wave... 

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. 

Molecular structure enters into the rotational entropy component, and vibrational frequencies into the vibrational entropy component. The translational entropy component cancels in a (mass) balanced reaction, and the electronic component is most commonly zero. Note that the vibrational contribution to the entropy goes to oo as v goes to 0. This is a consequence of the linear harmonic oscillator approximation used to derive equation 7, and is inappropriate. Vibrational entropy contributions from frequencies below 300 cm should be treated with caution. 

Within the harmonic oscillator approximation, the energy of the lowest vibrational level can be determined from Eq. (9.47) as ha>/2 where h is Planck s constant (6.6261 x 10- J s) and a> is the vibrational frequency. The sum of all of these energies over all molecular vibrations defines the zero-point vibrational energy (ZPVE). We may then define the internal energy at 0 K for a molecule as... 

The fundamental frequencies 9t (t = 1, 2,... 3tf—6) are related to and since Xt are the roots of det B—XE) — 0, r, are related to the matrix B and to the molecular force constants Bif. Hence the vibrational energy levels for a non-linear polyatomic molecule in the harmonic oscillator approximation are given by... 

For the vibrational term qivib, a classical high-T continuum approximation is seldom valid, and evaluation of the discrete sum over states is therefore required over the quantum vibrational distribution. (As pointed out in Sidebar 5.13, accurate treatment of molecular vibrations is crucial for accurate assessment of entropic contributions to AGrxn.) A simple quantum mechanical model of molecular vibrations is provided by the harmonic oscillator approximation for each of the 3N — 6 normal modes of vibration of a nonlinear polyatomic molecule of N atoms (cf. Sidebar 3.8). In this case, the quantum partition function can be evaluated analytically as... 

Geometric distortion is perhaps the least controversial of the simple dimensions because each such dimension is really a molecular vibration carried to an extreme such a distortion should be expected to be readily calculated, and should follow a parabolic equation to the extent that the harmonic oscillator approximation is valid. 

Molecular vibrations are actually rather complex. Generally, all the atoms in a molecule contribute to a vibration. Fortunately, some molecular vibrations can be treated by considering the motion of a few atoms relative to one another, ignoring the rest of the atoms in the molecule. To a useful approximation (the harmonic oscillator approximation) the vibration frequency of a bond is related to the masses of the vibrating atoms and the force constant, f, of the vibrating bond by the following equation ... 

It follows from the preceding discussion that the equilibrium constant for complex formation evaluated using the rigid rotor-harmonic oscillator approximation, with molecular constants derived from ab initio SCF calculations with a medium basis set (of DZ quality), is not very accurate. Comparison of the AG° values calculated using extended and medium basis sets indicates that the major uncertainty in AG is derived from AH . TASP is not as dependent on the basis set used. Furthermore, it is evident that the entropy term plays an extremely important rote in complex formation neglecting it may result not only in quantitative, but even in qualitative failure. 

The thermodynamic functions were estimated from those in the present table for HgS(g) (6 ) by adding those for DgS(g) and subtracting those for HgS(g), where both the added and subtracted functions were generated using the rigid-rotor harmonic oscillator approximation. In this calculation the molecular constants for DgS were taken from reference (2). 

IR-frequencies were determined by Christe et al. [72CHR/SCH] and, together with molecular constant data, they computed thermodynamic properties of SeFsC g) using the rigid-rotor, harmonic-oscillator approximation. The following temperature dependence... 

Al, Mo, and W are obtained from the stretching frequencies of 2360 cm 1,1670 cm 1, 1847 cm 1, and 1896 cm 1, respectively, for the corresponding hydrogen compounds by utilizing the isolated harmonic oscillators approximation k values for the other atoms are taken from E. B. Wilson, Jr., J. C. Decius, and P. C. Cross, Molecular Vibrations, p. 175 (McGraw-Hill, New York, 1955). The q s are taken from Table 2 with a conversion factor of 1/657. ... 

The thermodynamic energy of a collection of N harmonic oscillators (approximate representations of molecular vibrations) is given by... 

Using the rigid-rotor harmonic-oscillator approximation on the basis of molecular constants and the enthalpies of formation, the thermodynamic functions C°p, S°, — G° —H°o)/T, H° — H°o, and the properties of formation Af<7°, and log K°(to 1500 K in the ideal gas state at a pressure of 1 bar, were calculated at 298.15 K and are given in Table 9 <1992MI121, 1995MI1351>. Unfortunately, no experimental or theoretical data are available for comparison. From the equation log i = 30.25 - 3.38 x /p t, derived from known reactivities (log k) and ionization potential (fpot) of cyclohexane, cyclohexanone, 1,4-cyclohexadiene, cyclohexene, 1,4-dioxane, and piperidine, the ionization potential of 2,4,6-trimethyl-l,3,5-trioxane was calculated to be 8.95 eV <1987DOK1411>. 

In this section, we shall propose to the intramolecular vibrational relaxation. We shall first describe the problem associated with the harmonic approximation of molecular vibration. In the harmonic oscillator approximation, we have... 

In the harmonic-oscillator approximation, the quantum-mechanical energy levels of a polyatomic molecule turn out to be vib 2, (v, + )hv,, where the v s are the frequencies of the normal modes of vibration of the molecule and v, is the vibrational quantum number of the ith normal mode. Each v, takes on the values 0,1,2,... independently of the values of the other vibrational quantum numbers. A linear molecule with n atoms has 3n — 5 normal modes a nonlinear molecule has 3n — 6 normal modes. (See Levine, Molecular Spectroscopy, Chapter 6 for details.)... 

Decide whether entropic effects are likely to be important (for example if charged species are released to the solvent) and, if so, decide on whether a quantum chemical approach (calculating the partition function within a harmonic-oscillator approximation) may be used or whether a molecular dynamics-based approach (e.g., free-energy perturbation theory) should be used to properly sample phase space. 

The rigid-rotor harmonic oscillator approximation provides an adequate description of molecular properties. 

The selected molecular constants and electronic state excitation energies were used to calculate the thermodynamic functions in the rigid rotator-harmonic oscillator approximation over the temperature range 298.15-3000 K at standard pressure. These functions are gathered in Table A2 in the form of the coefficients of the polynomial (see Appendix). 

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