13.5.3 Vibrational degrees of freedom

In the lowest approximation, the molecular vibrations may be described as those of a harmonic oscillator. This can be derived by expanding the energy as a function of the nuclear coordinates in a Taylor series around the equilibrium geometry. For a diatomic molecule, the only relevant coordinate is the internuclear distance R.

E(R) = F(Ro) + d-(R- Ro) + -d__(R - Ro) + _d__(R - Ro) + ... (13.29) dR 2 dR2 6 dRJ

The first term may be taken as zero, since this is just the zero point for the energy. The second term (the gradient) vanishes since the expansion is around the equilibrium geometry. Keeping only the lowest non-zero term results in the harmonic approximation, where k is the force constant.

1 d2E

Including higher order terms leads to anharmonic corrections to the vibration, and such effects are typically of the order of a few percent. The energy levels obtained from the Schrödinger equation for a one-dimensional harmonic oscillator (diatomic system) are given in eq. (13.31).

m1 m2

Here n is a quantum number running from zero to infinity and n is the vibrational frequency given in terms of the force constant k (d2EldR2) and the reduced mass mIn contrast to the translational and rotational energy levels, the spacing between vibrational energy levels is comparable to kT for temperatures around 300 K, and the summation for qvib (eq. (13.10)) cannot be replaced by an integral- Due to the regular spacing, however, the infinite summation can be written in a closed form-

q b = ^ e-en/kT = e-hn/2kT + e-3hv¡2kT + g-5hv¡2kT +____

qvib = e - hvl2kT (1 + e- hvlkT + e-2 hvlkT + ■■■) (13.32)

In the infinite sum, each successive term is smaller than the previous by a constant factor (e-hvlkT, which is <1), and can therefore be expressed in a closed form. Calculating the vibrational partition function for a harmonic oscillator thus requires the second derivative of the energy and the atomic masses.

For a polynuclear molecule, the force constant k is replaced by a 3Natom x 3Natom matrix containing all the second derivatives of the energy with respect to the coordinates. By mass-weighting and transforming to a new coordinate system called the vibrational normal coordinates, this may be brought to a diagonal form (see Section 16.2.2 for details). In the vibrational normal coordinates, the 3N-dimensional Schrödinger equation can be separated into 3N one-dimensional equations, each having the form of a harmonic oscillator. Of these, three describe the overall translation and three (two for a linear molecule) describe the overall rotation, leaving 3N -6(5) vibrations.

If the stationary point is a minimum on the energy surface, the eigenvalues of the force constant matrix are all positive. If, however, the stationary point is a TS, one (and only one) of the eigenvalues is negative. This corresponds to the energy being a maximum in one direction and a minimum in all other directions. The "frequency" for the "vibration" along the eigenvector with a negative force constant will formally be imaginary, as it is the square root of a negative number (eq. (13.31)), and for a TS there are thus only 3N - 7 vibrations.

Within the harmonic approximation, the vibrational degrees of freedom are decoupled in the normal coordinate system. Since the energy of the 3N - 6 vibrations can be written as a sum, the partition function can be written as a product over 3N - 6 vibrational partition functions.

The vibrational frequencies are needed for calculating qvib, and can be obtained from the force constant matrix and atomic masses.

13.5.4 Electronic degrees of freedom

The electronic partition function involves a sum over electronic quantum states. These are the solutions to the electronic Schrodinger equation, i.e. the lowest (ground) state and all possible excited states. In almost all molecules, the energy difference between the ground and excited states is large compared with kT, which means that only the first term (the ground state energy) in the partition function summation (eq. (13.11)) is important.

Defining the zero point for the energy as the electronic energy of the reactant, the electronic partition functions for the reactant and TS is given in eq. (13.35).

The AE* term is the difference in electronic energy between the reactant and TS, and g0 is the electronic degeneracy of the (ground state) wave function. The degeneracy may be either in the spin part (g0 = 1 for a singlet, 2 for a doublet, 3 for a triplet, etc.) or in the spatial part (g0 = 1 for wave functions belonging to an A, B or X representation in the point group, 2 for an E, A or O representation, 3 for a T representation, etc.). The large majority of stable molecules have non-degenerate ground state wave functions, and consequently g0 = 1.

13.5.5 Enthalpy and entropy contributions

Given the partition function, the enthalpy and entropy terms may be calculated by carrying out the required differentiations in eq. (13.18). For one mole of molecules, the results for a non-linear system are (R being the gas constant)

H reactant_0



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