Rayleigh-Schrodinger Perturbation Theory Research Articles

Mechanistic and kinetic study of the oxidation of trifluoroacetonitrile by hydroxyl and oxygen

AbstractTrifluoroacetonitrile (CF3CN) is one of the perfluoronitriles to be used as replacements for electrical insulation, working fluids, and biomedical applications. Potential energy surfaces for the reactions of CF3CN with OH radicals in the presence of molecular O2 have been calculated in details using the second‐order Møller−Plesset perturbation theory (MP2) and coupled‐cluster theory with single and double substitutions (CCSD) for geometrical optimization, and the restricted open‐shell complete basis set quadratic ROCBS‐QB3, multireference second‐order Rayleigh–Schrodinger perturbation theory extrapolated to complete basis set limit (RS2/CBS), and the explicitly‐correlated RS2 theory (RS2‐F12) for energetics. The CF3CN + OH reaction undergoes predominantly via the CO addition/elimination mechanism, leading to CF3C(OH)N radical adduct followed by the formation of CF3 and HOCN. The OH radicals could be recycled through the interception of CF3C(OH)N by O2 to form the orientation‐dependent CF3C(OH)NO2 radicals, leading to CF3C(O)NO + OH via the successive H‐migration and OO bond fission. Under the typical atmospheric conditions, the OH‐recycling mechanism is dominant for the oxidation of CF3CN. The production of CF3 and HOCN can be significant at high temperatures and low pressures. Theoretical rate coefficients are in excellent agreement with the experimental data. The effect of substitution on the reactivity of the perfluoronitriles toward OH was revealed. The present work provides a fundamental understanding on the degradation of CF3CN and serves as a theoretical basis for its secondary chemistry.

Translational and rotational phase ordering of symmetric dimer mesogens: Molecular rigidity effect

A theoretical approach towards molecular rigidity and mesogenic behaviour of symmetric dimer molecules viz., α,ω-bis (4-cyanobiphenyl-4′-yloxycarbonyl) propane (3dimer) and α,ω-bis (4-cyanobiphenyl-4′-yloxycarbonyl) hexane (6dimer) has been analysed with regard to orientational and translational motions. The crystallographic information files of these molecules have been used to report the structures. The formalism of CNDO/2 approach has been executed to estimate the charges and dipole moment components of all atoms. Total configurational interaction energies have been estimated using Rayleigh-Schrodinger perturbation theory. These energy data have been utilized to estimate the configurational probability using the formula given by Maxwell-Boltzmann at phase transition temperature. The translational/ rotational ordering of different configurations has been analysed. The mesogenic behaviour of these compounds has been explained according to their relative order that helps in establishing molecular mesogenic models. As the manipulation of a molecular configuration is significant from the view point of photonics, this manuscript is expected to provide a route in understanding in this feature depending on the parameters reported in this manuscript. The impact of rigidity and aspect ratio upon the liquid crystal transitions has been discussed.

Structure and phase behavior of alkoxy benzoic acids in DMSO for thermodynamic applications: Theoretical investigation

This manuscript emphasizes the phase behavior of p-n-Octyloxy Benzoic acid (OOBA) and p-n-Nonyloxy Benzoic acid (NOBA) which are homologues of alkoxy benzoic acid. Fluctuation of the intermolecular interaction energies have been noticed according to rotational and translational co-ordinates for both the compounds in vacuum and under the influence of polar aprotic solvent Dimethyl sulfoxide (DMSO, ε=47). Structures of the molecules have been framed for both the compounds using published crystallographic data. All the charges of atoms and dipole moments of each atom have been estimated on the basis of complete neglect differential overlap (CNDO/2) method. Long-range intermolecular interactions have been considered by modified Rayleigh–Schrodinger perturbation theory and multicentered-multipole expansion method. Short-range interactions have been described by 6-exponential potential function. Theoretical thermodynamic formulations are chosen for the determination of Helmholtz free energies, entropies and internal energies. Existence of liquid crystal phase and its behavior have been discussed by the combined results achieved from relative free energy (ΔA) and relative entropy (ΔS) data. The possibility of addressing the multi-phase phenomenon is discussed from these relative values. The present data is expected to provide a route to analyze behavior of molecules in various solvents, and property prediction for thermodynamic applications.

Fundamental Analysis of Singular and Resonance Phenomena in Vibrational Polyads of the Difluorosilylene Molecule

The singular structure of lower vibrational states of the difluorosilylene molecule (up to four total excitation quanta) is studied by expansion of energies of each state into series of the high-order Rayleigh–Schrodinger perturbation theory and analysis of their implicit properties of multiple-valuedness using Pade–Hermite fourth-order approximants. The quartic surface of potential energy in dimensionless normal coordinates is calculated quantum-mechanically at the MP2/cc-pVTZ level. It is shown that one of values of multiple-valued approximants reproduces the variational solution with a high accuracy, while other values (beginning with the fourth polyad) in many cases coincide with energies of other states of the same polyad. The Fermi and Darling–Dennison resonances are analyzed based on facts of coincidence of singular complex branch points of the approximants for mutually interacting states inside a unit circle or near it on the complex plane. It is found that a pair of states can have several coinciding branch points of the solutions (in particular, inside the unit circle). A conclusion is made that this approach is an effective method for the determination of the polyad structure of vibrational states. Calculation parameters necessary for the reproducibility of key results are selected. The calculations are carried out by means of a Fortran-based software package using an arithmetic calculation package with a long mantissa of real numbers (200 digits).

Theoretical Investigations on the Oxidation of Heptafluoro-iso-butyronitrile by Atomic Oxygen in Dielectric Breakdown.

Heptafluoro-iso-butyronitrile [(CF3)2CFCN, in abbr., C4] with CO2 gas mixture is a promising dielectric candidate to replace sulfur hexafluoride (SF6) for the sake of environmental concern. Detailed mechanisms for the reactions of C4 with atomic oxygen in both triplet and singlet due to dissociation of CO2 were proposed using high-level ab initio methods including density functional theory, quadratic complete basis set, multireference Rayleigh-Schrodinger perturbation theory, and state-averaged multiconfiguration self-consistent field. The reaction of C4 with O(3P) proceeds via the stepwise C-O and N-O addition/elimination and direct displacement mechanisms. The reaction paths are bifurated into cis and trans conformations. The predominant product channel is the C-O association with the barriers 6.1-6.8 kcal/mol to form the C3F7C(O)N intermediates and followed by the C-C bond cleavage to produce C3F7 and NCO radicals. On the singlet surface, the three-center intramolecular conversion to form the C3F7OCN isomer becomes competitive. The singlet-triplet intersystem crossing is favorable due to the significant spin-orbital coupling. Master equation calculations were carried out to obtain the temperature- and pressure-dependent rate coefficients and yields of the O(3P) + C4 reaction and were compared with the analogous reactions of nitriles. It was found that only the NCO + C3F7 product channel is significant. The overall rate coefficients are pressure-independent in the range 1-10 atm and can be expressed by k∞(T/K)=(6.41 ± 0.22) × 10-13(T/298)1.30±0.01e-(3241.5±24.5)/T cm3 molecule-1 s-1. The present theoretical work provides useful insights into the breakdown diagnostic of the C4/CO2 insulation gas.

Possible Fröhlich surface polaron formation in graphene on a polar substrate

Abstract We theoretically study the formation of Frohlich polaron in monolayer graphene on a substrate made of polar materials. In this calculation, the Rayleigh-Schrodinger perturbation theory is used to obtain analytical expression of polaron energy. The decrease (increase) in the electron (hole) energy due to interaction with surface optical phonons indicates that the formation of a surface polaron is preferred. The charge carrier-phonon interaction in monolayer graphene on a polar substrate does not lead to the formation of a gap between the valence and the conduction bands, as there exist the same energy shifts for both electron and hole at k = 0 . The expression for the renormalized Fermi velocity due to carrier-phonon coupling for small values of k is obtained. Our results show that the influence of the screening effect on polaron energies is significant for k ≤ k F .

3-dimensional eksternal electric field effect (Stark effect) on the ground state energy of Tritium atom

Tritium Atom is a Radioisotope that has a half-life of 12.3 years. This atom produces beta decay with a maximum generated energy of 18.598 KeV and an average energy of 5.7 KeV. Tritium radiation is still relatively safe, so it is widely used in the field of electronics especially in the development of betavoltaics batteries. This research will delve into the effect of 3-dimensional external electric field on Tritium energy on the ground state. The method applied in this research is Rayleigh-Schrodinger Perturbation Theory, which also encompasses second order correction. In addition, by reviewing the movement of electrons fast enough to surround the nucleus, then the relativistic effect will also exert effect to the energy of Tritium atoms. The 3-dimensional electric field effect (in the x, y and z axis) will give different energy shifts with the 1-dimensional perturbation effect. This shift of energy will have an impact on the binding energy of nucleus and electron. The results of mathematical calculations have proven that a significant shift in energy only occurs in a strong external electric field, approximately higher than 1 × 1011 V/m.

Developments and applications of computational infrared spectroscopy in non-bonded systems: Quantal and classical systems

Infrared (IR) spectroscopy, responding to molecular vibrations, is one of the most important tools in understanding the behavior of molecules. For complex systems, because of the relatively dense spectra lines, it could be very difficult to assign IR spectra merely by experience, an instance of which is non-bonded system. For middle-size system, theory and computation, however, can handle part of this problem by construct the potential energy surfaces (PESs) of system following a nuclear motion calculation classic or quantum mechanically. For large-size non-bonded system, although it is hard to calculate the properties of the whole system directly, by decomposing the system into quasi-isolated parts, the spectra can be calculated from the first principle using some approximations, which belong to the toolchain of computational spectroscopy. In this review, we recall some of these methodology based on perturbation theory. (1) In the research of van der Waals complexes, intramolecular vibrations are separated from intermolecular motion via Born-Oppenheimer approximation and first order Rayleigh-Schrodinger perturbation theory. For the ground and first excited state of the chromophores, rigid body intermolecular PESs are build including the relative orientation of the rigid molecules. Based on the intermolecular PESs, rotation and intermolecular radial stretching are treated using Discrete Variable Representation/Finite Basis Representation (DVR/FBR) fully quantum mechanically. The rovibrational spectra are then obtained by the gap between energy levels and the transition dipole moments. (2) Spectroscopy of quantum clusters, where chromophores molecules are caged in helium or hydrogen molecules solvent, are another important topic where perturbation theory and quantum simulation get involved. As previous systems, accurate two-body intermolecular PESs are also needed. The potential of the system is then represented by the many-body expansion, often truncated at two-body term. Quantum effect of the rigid molecules is simulated by finite temperature Path Integral Monte Carlo (PIMC) method. Worm operator is used to take exchange phenomena of the boson into account. Rotation of the chromophore molecule and the boson exchange were found to be very important in the simulation of quantum clusters. By studying IR spectra of quantum cluster, microscopic superfluidity was discovered and confirmed. (3) In bulk system like solution and biological systems, computation of IR spectra remains a challenge. The birth of quantum vibrational perturbation (QVP) theory gives an opportunity of considering the quantum effect and the chemical environment in the same scheme. Regular molecular dynamics simulation (MD) is executed to deal with the solvent fluctuation, and then quantal molecular vibration of chromophore is calculated via DVR quantum solver based on low dimensional cut of intramolecular vibrational coordinate. At each MD step, the instantaneous vibration frequencies are computed by perturbation theory. This mixed quantum mechanical/classic mechanical, or quasi-classical MD simulation will generate the instantaneous frequencies as function of time. Using line shape theory for quasi-classical MD, the spectra line of such system can be obtained. All three situations rely on perturbation theory more or less, which can be very computationally efficient. Because the main part of the IR transition gap can be corrected by experimental data, these methods can focus on transition frequency shift without considering error canceling artifact. In the future, systems containing stronger secondary bond need more efforts in the development of calculation methodology. Kinetic coupling is also an interesting topic to be developed in the coming days. Besides, the new proceeding in electronic structure theory helps a lot in the computational IR spectroscopy.

Mechanistic and Kinetic Investigations on the Thermal Unimolecular Reaction of Heptafluoroisobutyronitrile.

Heptafluoroisobutyronitrile (C4) has been utilized as a dielectric compound to replace sulfur hexafluoride for the sake of environmental concern. Energetic profiles of the potential energy surface for the unimolecular reaction of C4 were calculated using density functional theory (M06-2X), quadratic complete basis set (CBS-QB3), Gaussian-4, multireference Rayleigh-Schrodinger perturbation theory (RS2, RS2C), and state-averaged multiconfiguration self-consistent-field (SA-MCSCF) ab initio methods. Among 38 production channels, the most energetically accessible reaction path is the three-centered rearrangement of cyanide to isocyanide. The C-CF3 bond appears to be the weakest bond in C4(X1A') and the symmetry-breaking C-CF3 bond cleavage undergoes to form the ground-state CF3(X2A1) and CF3CFCN (X2A″) radicals. All the possible isomerization pathways involving F- and CF3-migration together with the concerted elimination and stepwise decomposition routes were revealed for C4 and i-C4. Various isomers and potential toxic byproducts including FCN, CF3CN, C2F5CN, CF2═CFCF3, CF2═CFCN, CF4, C2F6, and alkyne compounds have been identified for the first time. Master equation analysis has been carried out to obtain the temperature- and pressure-dependent thermal rate constants. Theoretical kinetics for the loss of C4 due to pyrolysis is in good agreement with the temperature-ramped flow-tube experiment. The present theoretical work provides new insights on the thermal stability and chemical reactivity of C4. Moreover, i-C4 and C2F6 are proposed to be the potential characteristic gas molecules to monitor the insulation breakdown of C4 in the electric equipment.

A simple and operational test for external connectivity of tensors in many-body methods

Size extensivity of the correlation energy is a crucial property of many-body methods (e.g., coupled cluster, Rayleigh–Schrodinger perturbation theory). It is related to the connectivity of the energy expression which itself relies on the connectivity of the amplitude tensors. Before the introduction of the concept of “generalized extensivity” by Nooijen et al. (Mol Phys 103:2277, 2005) there was no operational test to check for extensivity. The latter probes for connectivity of the energy by testing for separability using an artificial manipulation of the two-electron integrals according to varying partitionings of the orbitals. The present work tests for external connectivity of the wave operator and does not require specially crafted two-electron integrals but access to the values of the individual amplitudes and Coulomb integrals. External connectivity refers to the R-scaling of tensor entries with respect to the distance between the centers of the localized orbital indices. In contrast to “generalized extensivity” the present test does not detect disconnected closed diagrams within a tensor. Since standard coupled cluster tensors are always connected a diagrammatic analysis focuses on tensors related to various configuration interaction methods. In this respect the present work is complementary to that of Lyakh and Bartlett (Mol Phys 112:213, 2014) using algebraic techniques. The theoretical results are confirmed by the proposed operational external connectivity check. The proposed ansatz gives some insight into the property of connectivity especially the meaning of cluster lines and may be easily applied to other, particularly multi-reference coupled cluster, methods to check numerically for proper external connectivity of the amplitudes.

7-Hexyloxy-3-[4’-(3-methylbutyloxy) phenyl]-4H-1-benzopyran-4-one: Study of Smectic behaviour and UV absorption profile

abstractThe intermolecular interaction energies between a pair of 7-Hexyloxy-3-[4’-(3-methylbutyloxy) phenyl]-4H-1-benzopyran-4-one (HMBPB) molecules have been estimated with respect to translational and orientational motions. The complete neglect differential overlap (CNDO/S) method has been employed to calculate the net atomic charge and atomic dipole moment components at each atomic center. The modified Rayleigh-Schrodinger perturbation theory along with multicentered-multipole expansion method has been employed to evaluate the long-range intermolecular interactions, while a ‘6-exp’ potential function has been assumed for short-range interactions. The total interaction energy values obtained through these computations have been used to calculate the probability of each configuration at room temperature (300K), and isotropic-smectic transition temperature (412.6K). Molecular arrangements inside a bulk of materials and smectic behavior of the compound in terms of their relative order have been discussed. Translational and rotational rigidity of the molecule has been analyzed to understand the smectic behaviour. The UV absorption profile has been estimated using CNDO/S, and INDO/S methods.

Probabilistic and thermodynamic approach on phase behavior of nematic liquid crystals: A comparative study

ABSTRACTThe systems chosen for the present investigation, p-n-alkylbenzoic acid (nBAC; n = 6, 7) exhibits nematic-isotropic transition. A comparative study based on quantum probabilistic and statistical thermodynamics has been carried out with respect to translational and orientational motions. The evaluation of net atomic charges and dipole moment at each atomic centre has been carried out through the complete neglect differential overlap (CNDO/2) method. The modified Rayleigh-Schrodinger perturbation theory along with multicentered-multipole expansion method has been employed to evaluate the long-range intermolecular interactions, while the ‘6-exp’ potential function has been assumed for short-range interactions. The total interaction energy values obtained through these computations have been used as input to calculate the probability of occurrence of a particular configuration using the Maxwell-Boltzmann formula. Further, thermodynamic parameters such as Helmholtz free energy, and entropy at room temperature (300 K), and nematic-isotropic transition temperatures have been computed. An attempt has been made to understand the structure-phase behaviour relationship at molecular level.

Correlation of mesogenic properties with intermolecular interaction energy for homologous series of HnCBP

ABSTRACTIntermolecular interaction energy between a pair of molecules of homologous series 4, 4′-disubstituted biphenyl of the general formula HO‒(CH2)n‒O‒C6H4‒C6H4‒CN(n = 3 − 11) (HnCBP) has been evaluated under various interacting conditions viz. stacking, in-plane and terminal interaction. Molecular geometry of the studied molecules was fully optimized without any constraint and checked for imaginary frequencies using hybrid density functional B3LYP combined with 6–31 g** basis set. Electronic structure of the molecules obtained through these calculations has been utilized to calculate electrostatic and polarization energies under Rayleigh-Schrodinger perturbation theory modified with multi-centered multi-pole expansion method. Dispersion and repulsion energies have been evaluated using Kitaigorodskii formula. The identified minimum energy complexes have been further utilized to evaluate interaction energy under super molecular approach by employing M06 and DFT-D methods. A comparative analysis of the results has been reported with a view to examine suitability of different methods to study molecular aggregations in moderately large organic systems.

Classification of vibrational resonances in the energy spectrum of the formaldehyde molecule and Katz’s branch points

The Rayleigh–Schrodinger perturbation theory of high orders and the algebraic Pade–Hermite approximants are used to determine the singular points of a vibrational energy function of the formaldehyde molecule dependent on a complex perturbation parameter as on the argument. It is shown that the Fermi, Darling–Dennison, and other higher-order vibrational resonances are related to Katz’s points—common branch points on the complex plane of the energy of two vibrational states. Analysis of Katz’s points that connect different vibrational states allows one to reveal essential resonance perturbations, to introduce an additional classification for them, and to determine the polyad structure of an energy spectrum.

The Influence of Alkyl Chain Length and Solvents on Configurational Probability of Liquid Crystalline Materials: A Computational Approach

The influence of alkyl chain length and solvents on two liquid crystalline materials, p-n-butylbenzoic acid (4BAC), and p-n-pentylbenzoic acid (5BAC) has been carried out with respect to the translational and orientational motions. The atomic net charge and dipole moment components at each atomic center have been evaluated using the complete neglect differential overlap (CNDO/2) method. The modified Rayleigh–Schrodinger Perturbation theory with the multicentered-multipole expansion method has been employed to evaluate the long-range interactions, and a “6-exp.” potential function has been assumed for the short-range interactions. The minimum energy configurations obtained during the different modes of interactions have been taken as input to calculate the configurational probability using the Maxwell–Boltzmann formula in nonpolar organic solvents, i.e., carbon tetrachloride (CCl4), and chloroform (CHCl3) at room temperature 300 K. It has been observed that the increase of alkyl chain length causes a minimization in the binding energy, and increases the total energy of 5BAC molecule. Further, the molecules produce remarkable property in the solvents.

Theoretical Calculation of the Low-Lying Electronic States of the Molecule BaS

Complete Active Space Self Consistent Field (CASSCF) with Multireference Configuration Interaction (MRCI) and Rayleigh-Schrodinger Perturbation Theory (RSPT2-RS2) methods have been used to investigate the potential energy curves for the 12 low-lying singlet and triplet electronic states in the representation 2s+1Λ(+/-) of the molecule BaS with Davidson corrections. The harmonic frequency we, the internuclear distance Re, the electronic energy with respect to the ground state Te, the rotational constants Be and the permanent dipole moment have been calculated for these electronic states. The eigenvalues Ev, the rotational constants Bv, the centrifugal distortion constant Dv and the abscissas of the turning points Rmin and Rmax have been investigated using the canonical functions approach. Nine new electronic states have been investigated here for the first time. The comparison between the values of the present work and those available in the literature for several electronic states shows a good agreement.

Conformational Behavior and Influence of Organic Solvents on a Strong Polar Group Nematogen at Room Temperature—A Computational Model

The conformational behavior and influence of organic solvents on a nematogen, 4’-n-alkyl-4-cyanobiphenyl, with strong polar group propyl (3CB) that is of commercial and application interest has been studied with respect to the translational and orientational motions. The atomic net charge and dipole moment components at each atomic center have been evaluated using the complete neglect differential overlap (CNDO/2) method. The modified Rayleigh-Schrodinger Perturbation theory with the multicentered-multipole expansion method has been employed to evaluate the long-range interactions, and a “6-exp.” potential function has been assumed for the short-range interactions. The minimum energy configurations obtained during the different modes of interactions have been taken as input to calculate the configurational probability using the Maxwell–Boltzmann formula in nonpolar organic solvents, i.e., carbon tetrachloride (CCl4), and chloroform (CHCl3) at room temperature 300 K. It has been observed that the molecules show the interesting property in the organic solvents. The interaction energies of dimer complexes have been taken into consideration in order to investigate the most energetically stable configuration. An attempt has been made to develop an interesting computational model for nematogen at molecular level.

Calculation of the energy levels of excited vibrational states of the HD16O molecule by summing divergent series of the Rayleigh-Schrödinger perturbation theory. The shift of zero-order levels

The Rayleigh-Schrodinger perturbation theory is applied to calculation of the energy levels of excited vibrational states of the HD16O molecule. The model of coupled anharmonic oscillators is considered, with the anharmonic part of potential energy being taken into account as the perturbation. The calculations are carried out for the vibrational states that correspond to three-to seven-fold vibrational excitations. Since the perturbation series diverge in the case of strong resonance interactions and their approximations by the Pade and Pade-Hermite methods do not yield sufficiently correct results, a calculation technique is applied that allows the zero-order approximation to be modified. The zero-order Hamiltonian is modified by shifting the vibrational frequencies, which decreases the mixing of states. The new Rayleigh-Schrodinger series can be summed using the quadratic Pade-Hermite approximation method.

Boundary perturbations and the Helmholtz equation in three dimensions

We propose an analytic perturbative scheme in the spirit of Lord Rayleigh's work for determining the eigenvalues of the Helmholtz equation in three dimensions inside an arbitrary boundary where the eigenfunction satisfies either the Dirichlet boundary condition or the Neumann boundary condition. Although numerous works are available in the literature for arbitrary boundaries in two dimensions, to the best of our knowledge the formulation in three dimensions is proposed for the first time. In this novel prescription, we have expanded the arbitrary boundary in terms of spherical harmonics about an equivalent sphere and obtained perturbative closed-form solutions at each order for the problem in terms of corrections to the equivalent spherical boundary for both the boundary conditions. This formulation is in parallel with the standard time-independent Rayleigh-Schrodinger perturbation theory. The efficacy of the method is tested by comparing the perturbative values against the numerically calculated eigenvalues for spheroidal, superegg and superquadric shaped boundaries. It is shown that this perturbation works quite well even for wide departure from spherical shape and for higher excited states too. We believe this formulation would find applications in the field of quantum dots and acoustical cavities.

Order–Disorder Phenomenon of Nematogens at Molecular Level—A Computational Approach

The phase behavior of nematogenic p-n-Alkoxy cinamic acids (nOCAC) with alkyl chain carbon atoms (n = 2,4) has been reported with respect to the translational and orientational motions. The atomic net charge and dipole moment components at each atomic center have been evaluated using the complete neglect differential overlap (CNDO/2) method. The modified Rayleigh-Schrodinger Perturbation theory with the multicentered-multipole expansion method has been employed to evaluate the long-range interactions, and a “6-exp.” potential function has been assumed for the short-range interactions. The minimum energy configurations obtained during the different modes of interactions have been taken as input to calculate the configurational probability using the Maxwell–Boltzmann formula in nonpolar organic solvents, i.e., carbon tetrachloride (CCl4), and chloroform (CHCl3) at room temperature 300 K. Further, entropy of each configuration has been computed. It has been observed that the molecules show the remarkable behavior in the solvents. An attempt has been made to develop new and interesting computational model for nematogens in the solvents and order–disorder phenomenon at molecular level.