Quantization in riemannian space

Elyutin, P. V.; Krivchenkov, V. D.

doi:10.1007/bf01042434

Cited by 2 publications

(3 citation statements)

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“…However, it has been shown that, in the two-parameter family of Hermitian operators, there is a unique form which is compatible with the additional conditions of Galilean invariance and probabilistic interpretation of the wave function. [33][34][35] It is given by the symmetric form…”

Section: Diagonalization Of the Hamiltonian Of Concerted Tunnelinmentioning

confidence: 99%

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

Smedarchina

Siebrand

Fernández‐Ramos

2014

The Journal of Chemical Physics

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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.

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Section: Diagonalization Of the Hamiltonian Of Concerted Tunnelinmentioning

confidence: 99%

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

Smedarchina

Siebrand

Fernández‐Ramos

2014

The Journal of Chemical Physics

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“…It has been asserted by von Roos [12] that the general Hermitian form of the kinetic energy operator T [m(x)] is not unique and that consequently, the effective mass approach is not suitable. However, it has been shown that there is a unique form of this operator which is compatible with the additional conditions of Galilean invariance and probabilistic interpretation of the wave function [13][14][15], and this is the form adopted in the present study.…”

Section: Introductionmentioning

confidence: 77%

“…The additional conditions that make a unique choice possible are obtained from the general requirements of the probabilistic interpretation of the wave function and Galilean invariance [13][14][15]. Therefore in the applications of Sec.…”

Section: Tunneling Of a Coordinate-dependent Mass And Hermiticitymentioning

confidence: 99%

Multidimensional Hamiltonian for tunneling with position-dependent mass

2014

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A multidimensional Hamiltonian for tunneling is formulated, based on the mode with imaginary frequency of the transition state as a reaction coordinate. To prepare it for diagonalization, it is transformed into a lower-dimension Hamiltonian by incorporating modes that move faster than the tunneling into a coordinate-dependent kinetic energy operator, for which a Hermitian form is chosen and tested for stability of the eigenvalues. After transformation to a three-dimensional form, which includes two normal modes strongly coupled to the tunneling mode, this Hamiltonian is diagonalized in terms of a basis set of harmonic oscillator functions centered at the transition state. This involves a sparse matrix which is easily partially diagonalized to yield tunneling splittings for the zero-point level and the two fundamental levels of the coupled modes. The method is tested on the well-known benchmark molecule malonaldehyde and a deuterium isotopomer, for which these splittings have been measured. Satisfactory agreement with experiment results is obtained.

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Quantization in riemannian space

Cited by 2 publications

References 0 publications

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

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

Multidimensional Hamiltonian for tunneling with position-dependent mass

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