Tunneling splitting in double-proton transfer: Direct diagonalization results for porphycene

Abstract

Zero-point and excited level splittings due to double-proton tunneling are calculated for porphycene and the results are compared with experiment. The calculation makes use of a multidimensional imaginary-mode Hamiltonian, diagonalized directly by an effective reduction of its dimensionality. Porphycene has a complex potential energy surface with nine stationary configurations that allow a variety of tunneling paths, many of which include classically accessible regions. A symmetry-based approach is used to show that the zero-point level, although located above the cis minimum, corresponds to concerted tunneling along a direct trans - trans path; a corresponding cis - cis path is predicted at higher energy. This supports the conclusion of a previous paper [Z. Smedarchina, W. Siebrand, and A. Fernández-Ramos, J. Chem. Phys. 127, 174513 (2007)] based on the instanton approach to a model Hamiltonian of correlated double-proton transfer. A multidimensional tunneling Hamiltonian is then generated, based on a double-minimum potential along the coordinate of concerted proton motion, which is newly evaluated at the RI-CC2/cc-pVTZ level of theory. To make it suitable for diagonalization, its dimensionality is reduced by treating fast weakly coupled modes in the adiabatic approximation. This results in a coordinate-dependent mass of tunneling, which is included in a unique Hermitian form into the kinetic energy operator. The reduced Hamiltonian contains three symmetric and one antisymmetric mode coupled to the tunneling mode and is diagonalized by a modified Jacobi-Davidson algorithm implemented in the Jadamilu software for sparse matrices. The results are in satisfactory agreement with the observed splitting of the zero-point level and several vibrational fundamentals after a partial reassignment, imposed by recently derived selection rules. They also agree well with instanton calculations based on the same Hamiltonian.

Entanglement and co-tunneling of two equivalent protons in hydrogen bond pairs

“…This justifies the silencing of the cis tunneling coordinate for this molecule and thus shows that the accurate result for the zero-point splitting obtained in Ref. 40 is not due to a compensation of errors but is a consequence of the proper treatment of the coupling to skeletal modes. In this case, the multidimensional Hamiltonian corresponding to double-proton transfer turns into an imaginary-mode Hamiltonian for the remaining tunneling coordinate X s and several of the skeletal modes coupled to it, but with a position-dependent mass of tunneling.…”

“…A detailed account of this extension of the model will be given elsewhere; 20 in the Appendix we summarize the relations (1,3) for double-proton tunneling in porphycene, in the sudden approximation (SA) for the skeletal modes, adapted from the RI-CC2/cc-pVTZ calculations of Ref. 40: ∆x in Å; G, D in dimensionless units; ∆X s,a in Å amu 1/2 , all other parameters in cm 1 . Middle part: Energies of the stationary configurations, in cm 1 .…”

“…In Ref. 40 we showed that for porphycene the tunneling splitting of the lowest trans level could be calculated with very good accuracy if the cis tunneling coordinate was frozen. The Gaussian form of the probability-flux distribution calculated in Sec.…”

Hindered rotor tunneling splittings: an application of the two-dimensional non-separable method to benzyl alcohol and two of its fluorine derivatives

“…They were also obtained when studying the tunneling splittings due to the double proton transfer in porphycene. 36 The cases of BA and 4FBA are different, because the wells M1 and M2 are equivalent, so there are four wells which are isoenergetic. In principle, the four wells may lead to a quadruplet, but the experiments 12,15 showed that only one doublet has been observed.…”

Der Tunneleffekt von Atomen in der Chemie

“…Die durch den Doppelprotontransfer verursachte Tunnelaufspaltung wurde in verschiedenen Porphycenen untersucht . In diesen Fällen sind der Anfangs‐ und Endzustand äquivalent, und resonantes – daher besonders schnelles – Tunneln setzt ein.…”