2.3 Ring-Polymer instanton method
2.3.3 Expression for the tunneling splitting
treated separately. The result is that each of the zero-frequency modes contributes a factor of √
βNℏSkink to the path integral, where Skink is the action integral of a single kink.16, 17, 20, 111
Chapter 3
Methyl group rotation and tunneling in condensed phase
In this chapter, tunneling and librational dynamics of the methyl groups within a model of rotational potential will be discussed. A general approach is introduced for the calculation of the methyl rotor quantum states in actual crystal structures.
The specific focus is on γ-picoline and toluene, which are chemically very similar, however, by contrast γ-picoline obtains a larger tunnel splitting than toluene. The rotational potential energy surfaces of the methyl groups are computed using a set of first principle calculations combined with the nudged elastic method. The tunnel splitting is calculated by an explicit diagonalization of the one-dimensional time-independent Hamiltonian matrix with the obtained potential energy profile.
3.1 Methyl group dynamics within a parameter model po-tential
A methyl rotor in a molecular crystal performs hindered rotation around the carbon-molecule bond axis. The resultant rotational potential has a typical three-fold sym-metry because of the indistinguishability of the protons. At low temperatures, the motion of the methyl group is reduced to a libration around the identical minima of the rotational potential. As the wave-like characteristics of the methyl group gives rise to quantum tunneling through connected potential minima, the three-fold degenerate ground state of methyl groups, E0, splits into double degenerate levels Ea/b and a non-degenerate ground state A, where the symmetry labels refer to the irreducible representation of the methyl rotor with C3(M) molecular symmetry.112 Applying Pauli principle to this group imposes that the spatialA state must be as-sociated with the nuclear triplet spin state 3/2. Accordingly, the spatialEa and Eb states must be associated with the nuclear singlet spin state 1/2, respectively. The correlation between spin and rotational quantum states has important consequences for tunneling spectroscopy because a change in rotational state requires a change in spin state.
The tunneling dynamics of hydrogen atoms in methyl rotors can be modeled in terms of a methyl group rotating around its C-C bond axis in a rigid environment. A single rotational coordinatesϕ describes the orientation of protons for single methyl rotation.24 The one-dimensional Hamiltonian is defined as,
H =−ℏ2 2I
∂2
∂ϕ2 +V(ϕ) (3.1)
where I, ϕ, and V(ϕ) stand for the moment of inertia of the methyl group, the rotation angle, and rotational potential, respectively. The motion of the methyl group can be approximately described in terms of a potential V(ϕ) composed of an intramolecular term and intermolecular van der Waals and Coulomb terms. An overall contribution of these terms can often result in a simplification where the threefold symmetry of the methyl group constrains the symmetry of the total inter-molecular potential to be at least threefold.113 Therefore, the rotational potential can be written as a Fourier series in which only those terms whose order is a multiple of three occur due to the three fold symmetry of the methyl group,
V(ϕ) =
∑∞ i=1
V3i
2 [1−cos(3iϕ+α)] (3.2) This series is usually truncated after the second term without loss of generality, yielding,
V(ϕ) = V3
2 [1−cos(3ϕ+α)] + V6
2 [1−cos(6ϕ+α)] (3.3) where α is the phase angle. In Appendix A, the ground state tunnel splitting and librational energies of methyl group in such potential of general shape are given.
Here, the discussion is restricted to the contribution of the three fold potential with α = 0. Accordingly, the following Hamiltonian is considered,
H =−ℏ2 2I
∂2
∂ϕ2 + V3
2 (1−cos3ϕ) (3.4)
which is solved numerically using matrix representation of Numerov method.114 The dependence of eigenvalues of the methyl group on the barrier height over a broad range of V3 is shown in Fig. 3.1. In the following, three limits of high, intermediate, and low strength of V3 for the methyl motion are discussed.
A. High barrier
In this limit, small deviations from one of the three equilibrium positions results in
nearly harmonic motion with the usual quantized energy levels, En= (nLIB+1
2)ℏω0 (3.5)
where ω0 is the oscillation frequency. An estimation of the oscillation frequency of the motion at the lowest energy state can be obtained in the harmonic approximation via Taylor series expansion aboutϕ = 0 for the cosine potential term,
V(ϕ) = V3
2 (1−cos3ϕ)≈ V3
2 (1−1 + (3ϕ)2
2! +. . .) = 1 2
9V3
2 ϕ2 (3.6) A comparison between the above potential and harmonic potential gives,
ω0 = 3
√
V3
2I (3.7)
Thus, the librational transitions, the energy difference between the librational levels, are approximately constant, given by
∆En=En−En−1 =ℏω0 = 3ℏ
√
V3
2I (3.8)
B. Intermediate barrier
In the intermediate regime, the rotational dynamics can be simplified to the energy-level diagram shown in Fig. 3.2, in which the lowest two librational energy-levels, nLIB, and their tunnel split sub levels are shown. The transition from the ground state to the first excited librational state, denoted by ℏω01 can differ from the ground state tunnel splittingℏωtunnel0 by several orders of magnitude, which is an important practical point in the measurements of rotational tunneling. This means that at low temperature where the thermal population of the excited librational states is negligible, the system can be approximated by a two-state model.113
C. Low barrier
In this limit, the energy levels are approximately the free rotation levels,En=Bn2, whereB is the rotational constant. The librational transitions are given by
∆En =En−En−1 = (2n−1)B for n= 1,2,3, . . . (3.9)
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
0 10 20 30 40 50
E (meV)
V3 (meV)
|3 , A >
|3 , Ea > = |3 , Eb >
|2 , Ea > = |2 , Eb >
|2 , A >
|1 , A >
|1 , Ea > |1 , Eb >
|0 , Ea > = |0 , Eb >
|0 , A >
Figure 3.1: Eigenvalues for a methyl group as a function of the barrier height, V3.
A Ea E
b
Ea E
b
A
ħω
01nLIB= 0 nLIB= 1
ħω
0tunnel
Figure 3.2: Schematic illustration of the first two lowest librational levels and tunnel split. The symmetries of the eigenstates are denoted by labels A,Ea and Eb.