arXiv subjects

A. E. Sitnitsky

Publications and source records attributed to A. E. Sitnitsky.

At least 19 recordsLinked to original sources

arXiv2026

Weiner's theory for exactly solvable Schrödinger equation with symmetric double well potential

The Weiner's theory (WT) is developed on the basis of the exactly solvable Schrödinger equation with trigonometric double-well potential (TDWP). The symmetric case of TDWP is considered. This modified version of WT (mWT) enables one to eliminate some severe approximations of the original Weiner's approach and to obtain more accurate results. An analytic formula is derived which provides the calculation of the proton transfer rate with the help of elements implemented in {\sl {Mathematica}}. We exemplify the application of mWT by calculating the proton transfer rate constant in the hydrogen bond of the proton-bound ammonia dimer cation ${\rm{N_2H_7^{+}}}$ (${\rm{H_3N\cdot\cdot\cdot H^{+} \cdot\cdot\cdot NH_3}}$). The parameters of the model for this object are extracted from available literature data on IR spectroscopy and quantum chemical calculations. The approach yields the transition from the Arrhenius-like exponential temperature dependence characteristic of thermal activation to that of quantum tunneling. Besides it is well suited for describing the phenomenon of vibrationally enhanced tunnelling.

physics.chem-ph↗

arXiv2025

Schrödinger equation with Pauli-Fierz Hamiltonian and double well potential as model of vibrationally enhanced tunneling for proton transfer in hydrogen bond

A solution of the two-dimensional Schrödinger equation with Pauli-Fierz Hamiltonian and trigonometric double-well potential is obtained within the framework of the first-order of adiabatic approximation. The case of vibrational strong coupling is considered which is pertinent for polariton chemistry and (presumably) for enzymatic hydrogen transfer. We exemplify the application of the solution by calculating the proton transfer rate constant in the hydrogen bond of the Zundel ion ${\rm{H_5O_2^{+}}}$ (oxonium hydrate) within the framework of the Weiner's theory. An analytic formula is derived which provides the calculation of the proton transfer rate with the help of elements implemented in {\sl {Mathematica}}. The parameters of the model for the Zundel ion are extracted from the literature data on IR spectroscopy and quantum chemical calculations. The approach yields a vivid manifestation of the phenomenon of vibrationally enhanced tunneling, i.e., a sharp bell-shaped peak of the rate enhancement by the external vibration at its symmetric coupling to the proton coordinate. The results obtained testify that the effect of resonant activation in our model is robust and stable to variations in the types of the quadratically coupled mode (vibrational strong coupling or symmetric one).

physics.chem-ph↗

arXiv2023

Model for vibrationally enhanced tunneling of proton transfer in hydrogen bond

Theoretical analysis of the effect of an external vibration on proton transfer (PT) in a hydrogen bond (HB) is carried out. It is based on the two-dimensional Schrödinger equation with trigonometric double-well potential. Its solution obtained within the framework of the standard adiabatic approximation is available. An analytic formula is derived that provides the calculation of PT rate with the help of elements implemented in {\sl {Mathematica}}. We exemplify the general theory by calculating PT rate constant for the intermolecular HB in the Zundel ion ${\rm{H_5O_2^{+}}}$ (oxonium hydrate). This object enables one to explore a wide range of the HB lengths. Below some critical value of the frequency of the external vibration the calculated PT rate yields extremely rich resonant behavior (multiple manifestations of bell-shaped peaks). It takes place at symmetric coupling of the external vibration to the proton coordinate. This phenomenon is absent for anti-symmetric and squeezed mode couplings.

physics.chem-ph↗

arXiv2021

Exactly solvable double-well potential in Schrödinger equation for inversion mode of phosphine molecule

The reduced mass of the effective quantum particle for the inversion mode $ν_2$ of phosphine molecule ${\rm{PH_3}}$ is known to be a position dependent one. In the present article the inversion spectrum of ${\rm{PH_3}}$ is considered with the help of the Schrödinger equation (SE) with position dependent mass and corresponding modified double-well potential. The SE is shown to be exactly solvable. The results are used for the analysis of the pertinent experimental data available in literature. We are based on the reliable value $ν_2=E_2-E_0=992.1\ {\rm cm^{-1}}$ ($2ν_2=E_4-E_0=1972.5\ {\rm cm^{-1}}$; $3ν_2=E_6-E_0=2940.8\ {\rm cm^{-1}}$; $4ν_2=E_8-E_0=3895.9\ {\rm cm^{-1}}$) obtained by \v Spirko et al. Also we use the value for the barrier height $E_b=12300 \ {\rm cm^{-1}}$ that seems to be commonly accepted at present and the hypothetical value for the energy splitting of the 11-th doublet of the $10ν_2$ band $s_{10}=E_{21}-E_{20}\approx 7.2\ {\rm cm^{-1}}$ suggested in the literature. Definite predictions are derived for the energy splitting of the 4-th doublet of the $3ν_2$ band $s_3=E_{7}-E_{6}$ that is a test one for the observation in the experiment of Okuda et al. SE with position dependent mass provides self-consistently the required values of $\{ν_2; E_b;s_{10}\}$ yielding $s_3= 6.21\cdot 10^{-12}\ {\rm cm^{-1}}$.

physics.chem-ph↗

arXiv2020

Calculation of IR absorption intensities for hydrogen bond from exactly solvable Schrödinger equation

A theoretical description of IR spectroscopy data for a hydrogen bond (HB) is constructed on the base of trigonometric double-well potential for which an exact analytic solution of the one-dimensional Schrödinger equation (SE) is available. The wave functions (full orthogonal basis) are expressed via the spheroidal function while its spectrum of eigenvalues yields the corresponding energy levels (both special functions are implemented in {\sl {Mathematica}}). Then an approximate solution of two-dimensional SE taking into account the excitation state of heavy atoms stretching mode in HB is obtained. It is constructed by decomposing over the above mentioned basis within the framework of standard adiabatic separating the proton motion from that of the heavy atoms. We exemplify the general theory by calculating the IR relative absorption intensities for HB in the Zundel ion ${\rm{H_5O_2^{+}}}$ (oxonium hydrate).

physics.chem-ph↗

arXiv2019

Analytic treatment of IR-spectroscopy data for double well potential

A theoretical scheme for the analysis of experimental data on IR spectroscopy for a quantum particle in a double well potential (DWP) is suggested. The analysis is based on the trigonometric DWP for which the exact analytic solution of the Schrödinger equation is available. The corresponding energy levels along with their wave functions are expressed via special functions implemented in {\sl {Mathematica}} (spheroidal function and its spectrum of eigenvalues). As a result trigonometric DWP makes the calculation of the energy levels an extremely easy procedure. It contains three parameters allowing one to model the most important characteristics of DWP (barrier height and the distance between the minima of the potential) along with the required asymmetry. Our approach provides an accurate calculation of the energy spectrum for hydrogen bonds in chromous acid (CrOOH) and potassium dihydrogen phosphate (${\rm{KH_2PO_4}}$) along with their polarizability in agreement with available experimental data.

physics.chem-ph↗

arXiv2018

Analytic calculation of ground state splitting in symmetric double well potential

The exact solution of the one-dimensional Schrödinger equation with symmetric trigonometric double-well potential (DWP) is obtained via angular oblate spheroidal function. The results of stringent analytic calculation for the ground state splitting of ring-puckering vibration in the 1,3-dioxole (as an example of the case when the ground state tunneling doublet is well below the potential barrier top) and 2,3-dihydrofuran (as an example of the case when the ground state tunneling doublet is close to the potential barrier top) are compared with several variants of approximate semiclassical (WKB) ones. This enables us to verify the accuracy of various WKB formulas suggested in the literature: 1. ordinary WKB, i.e., the formula from the Landau and Lifshitz textbook; 2. Garg's formula; 3. instanton approach. We show that for the former case all three variants of WKB provide good accuracy while for the latter one they are very inaccurate. The results obtained provide a new theoretical tool for describing relevant experimental data on IR spectroscopy of ring-puckering vibrations.

physics.chem-ph↗

arXiv2018

Exact solution of Schrödinger equation with symmetric double-well potential versus WKB: accuracy for ground state splitting

The one-dimensional Schrödinger equation with symmetric trigonometric double-well potential (DWP) is exactly solved via angular oblate spheroidal function. The results of stringent analytic calculation for the ground state splitting of hydrogen bond in malonaldehyde are compared with several variants of approximate semiclassical (WKB) ones. This enables us to compare the accuracy of various WKB formulas suggested in the literature: 1. ordinary WKB, i.e., the formula from the Landau and Lifshitz textbook; 2. Garg's formula; 3. instanton approach. The results obtained provide a new theoretical tool for the precise quantitative description of experimental data on IR spectroscopy of malonaldehyde.

physics.chem-ph↗

arXiv2017

Analytic description of inversion vibrational mode for ammonia molecule

The one-dimensional Schrödinger equation with symmetric trigonometric double-well potential is exactly solved via angular prolate spheroidal function. Although it is inferior compared with multidimensional counterparts and its limitations are obvious nevertheless its solution is shown to be analytic rather than commonly used numerical or approximate semiclassical (WKB) one. This comprises the novelty and the merit of the present work. Our exact analytic description of the ground state splitting can well be a referee point for comparison of the accuracy of numerous WKB formulas suggested in the literature. The approach reasonably well suits for the inversion mode in the ammonia molecule $NH_3$ and thus yields a new theoretical tool for its description. The results obtained provide good quantitative description of relevant experimental data on microwave and IR spectroscopy of $NH_3$.

physics.chem-ph↗

arXiv2017

Exactly solvable Schrödinger equation with double-well potential for hydrogen bond

We construct a double-well potential for which the Schrödinger equation can be exactly solved via reducing to the confluent Heun's one. Thus the wave function is expressed via the confluent Heun's function. The latter is tabulated in {\sl {Maple}} so that the obtained solution is easily treated. The potential is infinite at the boundaries of the final interval that makes it to be highly suitable for modeling hydrogen bonds (both ordinary and low-barrier ones). We exemplify theoretical results by detailed treating the hydrogen bond in $KHCO_3$ and show their good agreement with literature experimental data.

physics.chem-ph↗

arXiv2016

Probability distribution function for reorientations in Maier-Saupe potential

Exact analytic solution for the probability distribution function of the non-inertial rotational diffusion equation, i.e., of the Smoluchowski one, in a symmetric Maier-Saupe uniaxial potential of mean torque is obtained via the confluent Heun's function. Both the ordinary Maier-Saupe potential and the double-well one with variable barrier width are considered. Thus, the present article substantially extends the scope of the potentials amenable to the treatment by reducing Smoluchowski equation to the confluent Heun's one. The solution is uniformly valid for any barrier height. We use it for the calculation of the mean first passage time. Also the higher eigenvalues for the relaxation decay modes in the case of ordinary Maier-Saupe potential are calculated. The results obtained are in full agreement with those of the approach developed by Coffey, Kalmykov, Déjardin and their coauthors in the whole range of barrier heights.

cond-mat.stat-mech↗

arXiv2014

Exact solution of Smoluchowski's equation for reorientational motion in Maier-Saupe potential

The analytic treatment of the non-inertial rotational diffusion equation, i.e., of the Smoluchowski's one (SE), in a symmetric genuinely double-well Maier-Saupe uniaxial potential of mean torque is considered. Such potential may find applications to reorientations of the fragments of structure in polymers and proteins. We obtain the exact solution of SE via the confluent Heun's function. The solution is uniformly valid for any barrier height. We apply the obtained solution to the calculation of the mean first passage time and the longitudinal correlation time and obtain their precise dependence on the barrier height. In the intermediate to high barrier (low temperature) region the results of our approach are in full agreement with those of the approach developed by Coffey, Kalmykov, Déjardin and their coauthors. In the low barrier (high temperature) region our results noticeably distinguish from the predictions of the literature formula and give appreciably greater values for the transition rates from the potential well. The reason is that the above mentioned formula is obtained in the stationary limit. We conclude that for very small barrier heights the transient dynamics plays a crucial role and has to be taken into account explicitly. When this requirement is satisfied (as, e.g, at the calculation of the longitudinal correlation time) we obtain absolute identity of our results with the literature formula in the whole range of barrier heights. The drawbacks of our approach are its applicability only to the symmetric potential and its inability to yield an analytical expression for the smallest non-vanishing eigenvalue.

cond-mat.stat-mech↗

arXiv2012

Relationship between preexponent and distribution over activation barrier energies for enzymatic reactions

A relationship between the preexponent of the rate constant and the distribution over activation barrier energies for enzymatic/protein reactions is revealed. We consider an enzyme solution as an ensemble of individual molecules with different values of the activation barrier energy described by the distribution. From the solvent viscosity effect on the preexponent we derive the integral equation for the distribution and find its approximate solution. Our approach enables us to attain a twofold purpose. On the one hand it yields a simple interpretation of the solvent viscosity dependence for enzymatic/protein reactions that requires neither a modification of the Kramers' theory nor that of the Stokes law. On the other hand our approach enables us to deduce the form of the distribution over activation barrier energies. The obtained function has a familiar bell-shaped form and is in qualitative agreement with the results of single enzyme kinetics measurements. General formalism is exemplified by the analysis of literature experimental data.

q-bio.BM↗

arXiv2011

Model for crankshaft motion of protein backbone in nonspecific binding site of serine proteases

The consequences of recent experimental finding that hydrogen bonds of the anti-parallel $β$-sheet in nonspecific binding site of serine proteases become significantly shorter and stronger synchronously with the catalytic act are examined. We investigate the effect of the transformation of an ordinary hydrogen bond into a low-barrier one on the crankshaft motion a peptide group in the anti-parallel $β$-sheet. For this purpose we make use of a realistic model of the peptide chain with stringent microscopically derived coupling interaction potential and effective on-site potential. The coupling interaction characterizing the peptide chain rigidity is found to be surprisingly weak and repulsive in character. The effective on-site potential is found to be a hard one, i.e., goes more steep than a harmonic one. At transformation of the ordinary hydrogen bond into the low-barrier one the frequency of crankshaft motion of the corresponding peptide group in the anti-parallel $β$-sheet is roughly doubled.

q-bio.BM↗

arXiv2011

Analytic treatment of nuclear spin-lattice relaxation for diffusion in a cone model

We consider nuclear spin-lattice relaxation rate resulted from a diffusion equation for rotational wobbling in a cone. We show that the widespread point of view that there are no analytical expressions for correlation functions for wobbling in a cone model is invalid and prove that nuclear spin-lattice relaxation in this model is exactly tractable and amenable to full analytical description. The mechanism of relaxation is assumed to be due to dipole-dipole interaction of nuclear spins and is treated within the framework of the standard Bloemberger, Purcell, Pound - Solomon scheme. We consider the general case of arbitrary orientation of the cone axis relative the magnetic field. The BPP-Solomon scheme is shown to remain valid for systems with the distribution of the cone axes depending only on the tilt relative the magnetic field but otherwise being isotropic. We consider the case of random isotropic orientation of cone axes relative the magnetic field taking place in powders. Also we consider the cases of their predominant orientation along or opposite the magnetic field and that of their predominant orientation transverse to the magnetic field which may be relevant for, e.g., liquid crystals. Besides we treat in details the model case of the cone axis directed along the magnetic field. The latter provides direct comparison of the limiting case of our formulas with the textbook formulas for free isotropic rotational diffusion. The dependence of the spin-lattice relaxation rate on the cone half-width yields results similar to those predicted by the model-free approach.

cond-mat.soft↗

arXiv2011

Nuclear spin-lattice relaxation from fractional wobbling in a cone

We consider nuclear spin-lattice relaxation rate resulted from a fractional diffusion equation for anomalous rotational wobbling in a cone. The mechanism of relaxation is assumed to be due to dipole-dipole interaction of nuclear spins and is treated within the framework of the standard Bloemberger, Purcell, Pound - Solomon scheme. We consider the general case of arbitrary orientation of the cone axis relative the magnetic field. The BPP-Solomon scheme is shown to remain valid for systems with the distribution of the cone axes depending only on the tilt relative the magnetic field but otherwise being isotropic. We consider the case of random isotropic orientation of cone axes relative the magnetic field taking place in powders. Also we consider the case of their predominant orientation along or opposite the magnetic field and that of their predominant orientation transverse to the magnetic field which may be relevant for, e.g., liquid crystals. Besides we treat in details the model case of the cone axis directed along the magnetic field. The latter provides direct comparison of the limiting case of our formulas with the textbook formulas for ordinary isotropic rotational diffusion. We show that the present model enables one to obtain naturally the well known power law for Larmor frequency dependence of the spin-lattice relaxation rate. The latter is observed in some complex systems. From this law the dependence of the fractional diffusion coefficient on the fractional index is obtained to have a rather simple functional form. The dependence of the spin-lattice relaxation rate on the cone half-width for the case of ordinary rotational diffusion yields results similar to those predicted by the model-free approach.

cond-mat.stat-mech↗

arXiv2009

Model for solvent viscosity effect on enzymatic reactions

Why reaction rate constants for enzymatic reactions are typically inversely proportional to fractional power exponents of solvent viscosity remains to be already a thirty years old puzzle. Available interpretations of the phenomenon invoke to either a modification of 1. the conventional Kramers' theory or that of 2. the Stokes law. We show that there is an alternative interpretation of the phenomenon at which neither of these modifications is in fact indispensable. We reconcile 1. and 2. with the experimentally observable dependence. We assume that an enzyme solution in solvent with or without cosolvent molecules is an ensemble of samples with different values of the viscosity for the movement of the system along the reaction coordinate. We assume that this viscosity consists of the contribution with the weight $q$ from cosolvent molecules and that with the weight $1-q$ from protein matrix and solvent molecules. We introduce heterogeneity in our system with the help of a distribution over the weight $q$. We verify the obtained solution of the integral equation for the unknown function of the distribution by direct substitution. All parameters of the model are related to experimentally observable values. General formalism is exemplified by the analysis of literature experimental data for oxygen escape from hemerythin.

q-bio.BM↗

arXiv2008

Anomalous diffusion coefficient in disordered media from NMR relaxation

Application of fractional calculus to the description of anomalous diffusion and relaxation processes in complex media provided one of the most impressive impulses to the development of statistical physics during the last decade. In particular the so-called fractional diffusion equation enabled one to capture the main features of anomalous diffusion. However the price for this achievement is rather high - the fractional diffusion coefficient becomes an involved function of a characteristic of the media (e.g., that of the radius of pores in the case of the porous one). Revealing this dependence from the first principles is one of the main problems in this field of science. Another one still remains that of extracting this dependence from the experiment. The latter problem is tackled in the present paper. Our aim is to provide detailed and pedagogical deriving the relationship of the fractional diffusion coefficient with experimentally observable value from nuclear magnetic resonance (NMR) spin-lattice relaxation data. The result obtained promotes the NMR relaxation method to become a powerful tool in solving the problem of experimental measuring the fractional diffusion coefficient. Also the merits and limitations of NMR relaxation method and pulsed-field gradient (PFG) NMR for the research of anomalous diffusion are compared and discussed.

cond-mat.soft↗