Report No xxxx - Instytut Fizyki JÄ drowej PAN

LOW TEMPERATURES DYNAMICS IN SYSTEMS CONSISTING
OF HYDROGEN BOND AND METHYL GROUPS
L. Latanowicz, W. Medycki*, J. Boguszyńska*, E. C. Reynhardt**
Department of Biophysics, Institute of Biotechnology and Environmental Sciences, Zielona Góra University,
Monte Cassino 21 B, 65-561 Zielona Gora, Poland; *Institute of Molecular Physics, Polish Academy of
Sciences, Smoluchowskiego 17, 60-179 Poznań, Poland; **Department of Physics, University of South Africa,
P O Box 392, Pretoria, 0003, SA
The paper deals with the two different approaches to the spectral densities functions of
proton transfer and methyl group rotation.
The transfer of a hydrogen atom in a hydrogen bond is due to classical jumps over the
potential barrier and incoherent tunnelling. The concerted jumps of two hydrogen bonded
protons (or deuterons) along a hydrogen bond of benzoic acid can be visualised as an
exchange motion between potential energy minima associated with tautomers A and B with
different sets of rate constants, the first set obeying Arrhenius' law (rate constants of the
jumps over the barrier) and the second set given by Skinner and Trommsdorff [1] or Nagaoka
et al [2] (rate constants of the incoherent tunnelling). The preexponential factor of the rate of
tunneling jumps (k tu 0 ) is usually in range 10 8 Hz for hydrogen (10 6 Hz for deuterium) while
this of the rate of jumps over the barrier (k ov 0 ) is in range 10 12 Hz (10 10 Hz). If the interaction
Hamiltonian is modulated by a series of independent stochastic processes, the total correlation
function of such a complex motion should be calculated [3]. The equations for spectral
densities functions of complex motion, which consists of jumps over the barrier and
incoherent tunnelling were derived by Reynhardt and Latanowicz [4]. That the spectral
density of incoherent tunnelling is eliminated above a certain temperature, leaving only the
spectral density of the classical jump motion contributing to spin-lattice relaxation at higher
temperatures has been revealed in the paper [5]. A comparison (Fig. 1) of measured (open
circles [2]) and calculated T 1 (theory [4] - #1, jumps over the barrier - #2) for benzoic acid
confirms this discovery. We have found that this event has justification in the quantum
mechanics. Quantum mechanics involves probability of transfer of the particle through the
barrier, which is serious different then probability of jumps over the barrier. This probability
can be defined as D = I / I 0 , where I is intensity of de Broglie's wave (which describe the
particle) outside of the potential wall and I 0 is this intensity inside the potential wall. The
formula, which describes a coefficient D depends on the shape of the potential barrier. For the
2L
rectangular barrier D = D exp[ − 2m( E E ) ] , where E AB and L are height and width
0 AB
−
of the barrier, E and m are energy and mass of the particle. The value (E AB - E) under square
root is negative for E > E AB and the probability D is lost. The thermal energy of the Avogadro
number of particles equals E = C p T [kJ/mol] where C p is molar heat capacity and T is
temperature in the Kelvin scale. When C p T = E AB this should be the last value of coefficient D
in the temperature dependence.
In opposition to presented equations, the expression for spectral density functions of single
motion with total rate constant, approximated by two exponents k AB = k ov tu
0 exp(-H/RT) + k 0
exp(-∆/RT) (H and ∆ are the high and low temperature slopes of T 1 ) has been used in number
of papers. This last one theoretical approach assumes the smooth transition between the over
the barrier rate constant and incoherent tunneling rate constant. Here the question appears
how this is possible that the Arrhenius rate constant k ov = k 0 exp(-H/RT), goes to zero value
above temperature zero Kelvin and classical motion ceases? Moreover, when "single motion
approach" to the total spectral density functions is applied then T 1 looses the frequency
dependence at minimum value for low resonance frequencies of spectrometer. The
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