L2. Statistical Thermodynamics
of Large-Amplitude Torsions: Beyond the Conventional Separability
Assumption
Massachusetts Institute of Technology
77 Massachusetts Avenue
Cambridge, MA 02139-4307 USA
E-mail: oamazyar@mit.edu
E-mail: whgreen@mit.edu
The partition function Z(T) is the principal quantity used to connect quantum chemical calculations (and spectroscopic data) with the crucial technological parameters: thermochemical quantities (entropies, heat capacities) and (via transition state theory) reaction rates. The thermochemical quantities and rates are used in a variety of computations and simulations which strongly influence governmental regulations, business decisions, and the design, hazard assessment, control, and improvement of industrial processes. As quantum chemical calculations are becoming both accurate and computationally affordable, they are becoming the method of choice for estimating reaction rates, and are becoming competitive with other methods for estimating thermochemical data. It is therefore important to carefully consider how the approximations made in conventional partition function calculations affect these rate and thermo estimates.
The conventional statistical-mechanical approach to computing the partition functions of polyatomic molecules and transition states involves assuming that the many vibrational modes of the system are separable, i.e. one typically assumes
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where Zi is the partition function of the ith vibrational mode. Each vibrational mode is usually assumed to be either a harmonic oscillator, a free internal rotor, or a hindered internal rotor. The implicit assumption of Eq.1 is that there exists a separable Hamiltonian
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which is an accurate enough 0th order approximation to the true Hamiltonian that the Z computed using Eq.1 is a close approximation to the true partition function of the system. This approach is widely used for computing reaction rates, and is carefully presented from both a classical and a quantum point of view in H.S. Johnston’s classic text “Gas-Phase Reaction Rate Theory”.
If all modes can be described as small-amplitude motions, and particularly if the vibrational frequencies are all greater than kT/h, a normal-mode harmonic oscillator treatment is very reasonable, and one would expect the Z computed using Eq.1 to accurate reproduce the true Z. However, in many cases there are modes such as internal rotors which are known to have large amplitudes even at very low energies, and then it is much less clear that Eq.1 should be accurate.
Here we present a formalism for computing the effect of the coupling of all the small-amplitude vibrational modes to a large-amplitude torsion j, allowing assessment of the accuracy of the conventional separability assumption used in computing partition functions, thermochemical properties, and reaction rates.
Our approach is based on the assumption that the molecule only makes small-amplitude displacements, j, away from a minimum energy path, j(j), i.e. the geometry appropriate for each value of the large-amplitude torsional coordinate jj. These small displacement coordinates {tj} are defined in Eq.3:
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where Rj is each of the 3 N – 7 internal coordinates other than j used to describe the system. Further, we define normal coordinates for each value of j:
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so the 0th-order potential becomes diagonal:
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where Eel(j) is the electronic energy of the system.
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In the (j,
{Qj(j)}) coordinate
system, the J = 0 kinetic energy operator associated with motion of the nuclei is
where Si are 3 N cartesian coordinates referenced to a space-fixed frame.--> Note that the coefficients in front of the derivatives in Eq. 6 are functions of j and all the Q’s. For the 0th-order Hamiltonian, we neglect the Q-dependence of the coefficients, evaluating all the derivatives at Qj = 0, i.e. Rj = Reqj(j).
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By assuming that the small-amplitude coordinates can be considered adiabatic
relative to the large-amplitude torsion, we show that the approximate eigenvalues of the
Hamiltonian determined by Eqs.5 and 6 can be
obtained by repetitive diagonalization of the Hamiltonians
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where nj are
frequencies of 3 N – 7 normal modes, {nj} are different sets of vibrational
quantum numbers, and
The total number of diagonalizations increases dramatically with the increase of the number of the internal degrees of freedom, thus making the solution a computationally intensive endeavor.
Alternatively, we show that the approximate treatment of energy levels using the first-order nondegenerate perturbation theory allows fast computation of the partition function for the internal degrees of freedom:
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in which Ei is the ith
energy level of the hindered rotor with variable moment of inertia, I(f) (see Eq.8),
and the potential energy
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and yi is the corresponding unperturbed normalized wavefunction of this hindered rotor.
Several example cases of computing partition functions, thermochemical properties, and reaction rates are presented.
References
(1.) Gas Phase Reaction Rate Theory. New York, The Ronald Press Company.
(2.) Bramley, M. J.; Green Jr, W. H.; Handy, N. C. Molecular Physics 1991, 73, 1183-1208.