Multidimensional Hamiltonian for tunneling with position-dependent mass

Fernández‐Ramos, Antonio; Smedarchina, Zorka; Siebrand, Willem

doi:10.1103/physreve.90.033306

Cited by 8 publications

(13 citation statements)

References 34 publications

(53 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…16,17 To solve this Hamiltonian, which consists of a double-minimum potential along the concerted tunneling path coupled to a number of skeletal vibrations, we use a new method of direct diagonalization. 18 The generation of the PES at RI-CC2/cc-pVTZ level of theory is discussed in Sec. III and the diagonalization results are reported in Sec.…”

Section: Introductionmentioning

confidence: 99%

“…IV. We recently showed 18 that such a Hamiltonian can be diagonalized if the number of coupled vibrations is kept within certain limits. This we achieve in two ways: first by restricting, as before, coupling between tunneling and skeletal modes to terms linear in the skeletal modes, which eliminates all modes that are neither symmetric nor antisymmetric to the dividing plane of the transfer reaction; and second, by treating modes that are fast on the time scale of tunneling in the adiabatic approximation.…”

Section: Introductionmentioning

confidence: 99%

“…A unique Hermitian form for this operator was constructed and tested in Ref. 18. The diagonalization procedure is carried out by a modified Jacobi-Davidson algorithm implemented in the Jadamilu software for sparse matrices.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

Smedarchina

Siebrand

Fernández‐Ramos

2014

The Journal of Chemical Physics

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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

Smedarchina

Siebrand

Fernández‐Ramos

2014

The Journal of Chemical Physics

Self Cite

View full text Add to dashboard Cite

show abstract

“…The method we will adopt is based on direct diagonalization of an imaginary-mode Hamiltonian of a type we introduced earlier for 1D tunneling. [7][8][9] A fundamental assumption is that this Hamiltonian can be written as the sum of two equivalent single-proton Hamiltonians plus a coupling. In these single-proton Hamiltonians, the proton moves in a symmetric double-minimum potential.…”

Section: And References Therein)mentioning

confidence: 99%

“…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. [7][8][9]40 Hence these calculations remain very demanding computationally.…”

Section: The One-instanton Approximationmentioning

confidence: 99%

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

Smedarchina

Siebrand

Fernández‐Ramos

2017

The Journal of Chemical Physics

Self Cite

View full text Add to dashboard Cite

A theoretical study is reported of a system of two identical symmetric hydrogen bonds, weakly coupled such that the two mobile protons can move either separately (stepwise) or together (concerted). It is modeled by two equivalent quartic potentials interacting through dipolar and quadrupolar coupling terms. The tunneling Hamiltonian has two imaginary modes (reaction coordinates) and a potential with a single maximum that may turn into a saddle-point of second order and two sets of (inequivalent) minima. Diagonalization is achieved via a modified Jacobi-Davidson algorithm. From this Hamiltonian the mechanism of proton transfer is derived. To find out whether the two protons move stepwise or concerted, a new tool is introduced, based on the distribution of the probability flux in the dividing plane of the transfer mode. While stepwise transfer dominates for very weak coupling, it is found that concerted transfer (co-tunneling) always occurs, even when the coupling vanishes since the symmetry of the Hamiltonian imposes permanent entanglement on the motions of the two protons. We quantify this entanglement and show that, for a wide range of parameters of interest, the lowest pair of states of the Hamiltonian represents a perfect example of highly entangled quantum states in continuous variables. The method is applied to the molecule porphycene for which the observed tunneling splitting is calculated in satisfactory agreement with experiment, and the mechanism of double-proton tunneling is found to be predominantly concerted. We show that, under normal conditions, when they are in the ground state, the two porphycene protons are highly entangled, which may have interesting applications. The treatment also identifies the conditions under which such a system can be handled by conventional one-instanton techniques.

show abstract

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

Alves

Simón-Carballido

Ornellas

et al. 2016

Phys. Chem. Chem. Phys.

Self Cite

View full text Add to dashboard Cite

In this work we present a novel application of the two-dimensional non-separable (2D-NS) method to the calculation of torsional tunneling splittings in systems with two hindered internal rotors. This method could be considered an extension of one-dimensional methods for the case of compounds with two tops. The 2D-NS method includes coupling between torsions in the kinetic and potential energy. Specifically, it has been applied to benzyl alcohol (BA) and two of its fluorine derivatives: 3-fluorobenzyl alcohol (3FBA) and 4-fluorobenzyl alcohol (4FBA). These molecules present two torsions, i.e., about the -CH2OH (ϕ1) and -OH (ϕ2) groups. The electronic structure calculations to build the two-dimensional torsional potential energy surface were performed at the DF-LMP2-F12//DF-LMP2/cc-pVQZ level of theory. For BA and 4FBA the calculated ground-state vibrational level splittings are 429 and 453 MHz, respectively, in good agreement with the experimental values of 337.10 and 492.82 MHz, respectively. In these two cases there are four equivalent wells and the tunneling splitting is the result of transitions between the two closer minima along ϕ1. The analysis of the wavefunctions, as well as the previous experimental work on the system, supports this conclusion. For 3FBA the observed ground-state splitting is 0.82 MHz, whereas in this case the calculated value amounts only to 0.02 MHz. The 2D-NS method, through the analysis of the wavefunctions, shows that this tiny tunneling splitting occurs between the two most stable minima of the potential energy surface. Additionally, we predict that the first vibrationally excited tunneling splitting will also be small and exclusively due to the interconversion between the second lowest minima.

show abstract

Multidimensional Hamiltonian for tunneling with position-dependent mass

Cited by 8 publications

References 34 publications

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

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

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

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

Contact Info

Product

Resources

About