On physical analysis of entropy measures for sodium oxide via statistical models

This study delves into a comprehensive physical analysis of entropy measures applied to sodium oxide ${\text{Na}}_2\text{O}$. A key idea in information theory and thermodynamics, entropy is essential to comprehending the stability of a system. The study clarifies the intricate relationships and physical properties of ${\text{Na}}_2\text{O}$ by combining theoretical analysis with statistical methods, providing important knowledge for material design and industrial operations. In a chemical graph, atoms are shown by vertices while their bonding are illustrated by edges. Sodium oxide is a major contributor to manufacture glass, it also has potential applications in ${\text{CO}}_2$ sequestration, transparent materials, biomedical devices and nano grating glass. A topological index is a relationship between the molecular graph and its topology. We have computed various graph entropies based on different topological indices of chemical graph of sodium oxide. Further we have integrated these graph entropies with distinct thermodynamical measures of sodium oxide by developing mathematical models between both quantities. We have developed these mathematical frameworks in MATLAB. All the models are selected relying on the least mean squared error or sum of squared error.     

Comparative analysis of modified reverse degree topological indices for certain carbon nanosheets using entropy measures and multi criteria decision-making analysis

The diverse applications of various carbon nanostructures have sparked significant research interest. The potential significance of carbon nanosheets lies in their exceptional mechanical, electrical, and thermal properties, enabling advancements in various technological applications such as energy storage, electronics, and engineering, playing important roles in material science and nanotechnology. In the field of mathematical chemistry, topological indices play a crucial role by providing numerical insights into molecular structures. These insights facilitate predictive correlations with chemical properties and reactivity, ultimately finding utility in areas like drug design and material sciences. Expanding on this, the study delves into the application of entropy-based methodologies that originates from the arrangement of chemical structures. By assessing the complexity and other features of these structures, graph entropies are translated into information-theoretic metrics. This article examines modified reverse degree-based topological indices with its universal applicability to all degree-based indices. The standout feature is the adaptable parameter “ $k$” effectively shaping the molecular graphs degree sequence to best fit each dataset with their unique physicochemical properties, distinguishing it from conventional fixed degree methodologies. Additionally, we investigate graph entropies and examine the impact of varying the parameter “ $k$” on entropy measures across a range of nanosheet structures. The research focuses on characterizing carbon nanosheets, employing effective (MCDM) multiple criteria decision-making methods like VIKOR, TOPSIS, and SAW. Through these techniques, a comprehensive comparative analysis of the nanosheets is conducted with the aim of establishing optimized rankings for each type based on their unique attributes and characteristics.     

Topological defects on the energy spectra of diatomic molecules embedded with shifted Deng–Fan oscillator potential under AB flux field

A study on the effect of point like global monopole topological defects on the energy eigenvalues of the diatomic molecules $H_2$, $LiH$, $CO$, $HCl$ embedded with Shifted Deng–Fan Oscillator Potential under the influence of Aharonov–Bohm flux field has been made here. Asymptotic Iteration Method (AIM) is used to find out the bound state solutions for arbitrary $l$ states by solving the non-relativistic Schrödinger equation. A Pekeris-type approximation has been used to approximate the centrifugal barrier term. It is observed that, energy levels of the diatomic molecules is significantly affected by the global effects of the point like global monopole, flux field and effective potential field.     

Cn,CnH and their anions: Quest for linearity with n≤8 even versus odd,and beyond

Using the concept of quasi-molecule (“tile”) and the database of quasi-molecules embedded on a parent molecule, it is discussed whether the latter can attain linear form or otherwise. Besides anew accurate optimization of all tiles (quasi-triatomics) at various levels of ab initio theory and basis-sets, the nature of the predicted stationary points for the title parent molecules is probed through a priori calculations here too reported. Also discussed is the common rule that even- $n$ ${\text{C}}_n{\text{H}}^{-}$ anions are linear while odd-numbered ones tend to have nonlinear isomers. The reported quasi-molecule approach is general, and allow the prediction of linearity or otherwise of the parent systems prior to calculations on them. When based on an extension of the bisection method (Varandas, Int. J. Quantum Chem. 2023 , 123, e27036.), it is easy to use even for large parent molecules, as illustrated for neutral and anionic carbon clusters with $n\ge 9$.     

The confined helium atom: An information–theoretic approach

In this article, we study the helium atom confined in a spherical impenetrable cavity by using informational measures. We use the Ritz variational method to obtain the energies and wave functions of the confined helium atom as a function of the cavity radius $r_0$. As trial wave functions we use one uncorrelated function and five explicitly correlated basis sets in Hylleraas coordinates with different degrees of electronic correlation. We computed the Shannon entropy, Fisher information, Kullback–Leibler entropy, Tsallis entropy, disequilibrium and Fisher–Shannon complexity, as a function of $r_0$. We found that these entropic measures are sensitive to electronic correlation and can be used to measure it. As expected these entropic measures are less sensitive to electron correlation in the strong confinement regime ( a.u.).     

Physical versus non-physical effective nuclear masses for non-adiabatic corrections to molecular spectra

Two types of developments for very accurate non-adiabatic corrections to rovibrational molecular energy levels, one of a formal nature and the other of a heuristic nature, lead to fundamentally different approaches for effective nuclear masses. The former yields effective masses that have non-physical interpretation at some ranges of nuclear distances. The later uses physical masses obtained from electronic structure calculations. This paper contains a brief review of the subject and proposes procedures to improve and generalize the heuristic approach. Comparisons are made of the results obtained by the two approaches for the H ${}_2$ molecule, since no further calculations were found with the proper accuracy, but some issues involving the HeH ${}^{+}$ ion and the water molecule are discussed. The conclusion is that the heuristic approach has many advantages over the formal one, namely, equivalent accuracy and physically grounded qualitative interpretation. But, moreover, it seems to be presently the only method that allows non-adiabatic calculations for well isolated states of larger molecules.     

Topological indices and QSPR analysis of some chemical structures applied for the treatment of heart patients

In this study, various beta-blocker drugs used for heart disease were analyzed, and their degree-based topological indices derived from the M-polynomial were calculated. Linear and quadratic regression analysis was used to obtain quantitative structure-property relationship models between the topological indices and eight the physicochemical properties of the drugs to determine their effectiveness. The results show that the harmonic index was the best predictor for boiling point, flashpoint, and enthalpy of vaporization, while the redefined third Zagreb index was effective for polarizability, molar refractivity, and molar volume. The inverse sum indeg index was found to be effective for molar refractivity, and the second modified Zagreb index was surface tension in linear regression models. In addition, the redefined third Zagreb index was the best predictor for polarizability and molar refractivity, while the second modified Zagreb index was effective for molar volume. The SDD index was found to be effective for surface tension in quadratic regression models.     

Spherically confined hydrogenic atoms under classical non-ideal plasmas: Scaling law for the critical cage size

An elegant calculation is carried out to investigate the effects of the non-ideality of classical plasma on the energy levels of the hydrogenic atoms held in a spherical cage. Organized effect of the non-ideal classical plasma is described by an analytical pseudopotential which contains the Debye length $D$ and non-ideality parameter $\gamma$ as parameters. Convergent results for the bound states are obtained variationally by utilizing a large trail function containing cosine term which automatically takes care of the requisite boundary conditions. For the plasma-free case, our results are in excellent agreement with the most accurate results available in the literature. An inclusive study is made to explore the changes emerging in the energy levels due to the variation of the plasma parameters and cage size. Special emphasis is made on the determination of critical cage size precisely. The present study specifically reveals that the increasing plasma non-ideality leads to the elongation of the critical cage size. Moreover, it is empirically found that the critical cage size for a given hydrogenic atom can be obtained from a scaling law.     

Nonstationarity and related measures for time‐dependent hartree–fock and multiconfigurational models

Based on an earlier article (Eberly and Singh, Phys. Rev. D 1973 , 7, 359) and related works on short‐time evolution, this article proposes a many‐electron formulation for the nonstationarity degree which can be assigned to quantum system at each time point. The key measure introduced, , is a nonstationarity index that can be thought of as an inverse nominal lifetime at each instance of time. The index is directly computed from the time derivative of one‐electron density matrix and is a size‐consistent quantity. In this article, the approach is developed for the time‐dependent Hartree–Fock (TDHF), single‐excitation (TDCIS), and time‐dependent full configuration interaction (TDFCI) models. As a rule, nonstationarity effects are more pronounced in correlated electron systems, and a joint analysis of and the multiconfigurational character of wave functions apparently provide a deeper insight into dynamical molecular processes. The performed calculations on small molecules in laser fields show a preference for the TDCIS model when comparing TDCIS and TDHF with the “exact” TDFCI model. © 2013 Wiley Periodicals, Inc.     

Theoretical study of metal‐free organic dyes based on different configurations for efficient dye‐sensitized solar cells

Considering different solar dyes configuration, four novel metal‐free organic dyes based on phenoxazine as electron donor, thiophene and cyanovinylene linkers as the ‐conjugation bridge and cyanoacrylic acid as electron acceptor were designed to optimize open circuit voltage and short circuit current parameters and theoretically inspected. Density functional theory and time‐dependent density functional theory calculations were used to study frontier molecular orbital energy states of the dyes and their optical absorption spectra. The results indicated that D2‐4 dyes can be suitable candidates as sensitizers for application in dye sensitized solar cells and among these three dyes, D3 showed a broader and more bathochromically shifted absorption band compared to the others. The dye also showed the highest molar extinction coefficient. This work suggests optimizing the configuration of metal‐free organic dyes based on simple D‐ ‐A configuration containing alkyl chain as substitution, starburst conformation, and symmetric double D‐ ‐A chains would produce good photovoltaic properties.     

Effect of thermal fluctuations on the electronic excitation energies of linear polyenes: A combined molecular dynamics and TD-DFT study

In the present study, thermally averaged electronic excitation energies (the so-called dynamic approach) are quantitatively assessed for a model molecular system composed of a series of increasingly longer chains of simple linear polyenes. The molecular dynamics trajectories are calculated using the semi-empirical “Geometry, Frequency, Noncovalent, eXtended Tight Binding” (GFN2-xTB) method, while TD-DFT vertical excitation energies of selected snapshots are obtained with the spin-component-scaled double hybrid density functional SCS-wPBEPP86. Boltzmann weighted average excitation energies were then calculated in order to take into account the contribution of thermal motions. Excitation energies via the dynamic approach are compared with that of the static approach in which thermal fluctuations are not considered. The differences between the dynamic and the static approaches were found to be around the range $-0.01$ to $+0.01$ eV for polyenes up to 44 carbon atoms. This range of errors is relatively small in comparison with typical errors produced by using different density functionals and/or basis sets. Therefore, the electronic excited states of small to medium lengths of linear polyenes (less than 50 carbon atoms) can be safely modeled via the static approach. However, extrapolation of the results to longer polyenes indicates that the difference between both approaches is estimated to be 0.05 eV for polyenes containing 100 carbon atoms, which suggests that considering thermal motions in the calculations of excitation energies is recommended for such long conjugated molecular systems.     

Explicitly correlated variational estimates of the energy levels of negative hydrogen ion under spatial confinement

Comprehensive investigations on the structural modifications of negative hydrogen ion within an impenetrable spherical domain has been performed in the framework of Ritz variational method. Electron correlation plays a major role in the formation of H^– ion. The Hylleraas‐type basis set expansion of wave function considered here incorporates the effect of electron correlation in an explicit manner. Energy values of and 1sn states of H^– ion within confined domain have been calculated. Although the singly excited states do not exist for a “free” H^– ion, well converged energy values of such states have been found within a wide range of confinement radius. The thermodynamic pressure felt by the ion inside the sphere is also estimated. The general trend shows successive destabilization of the excited energy levels with increase of pressure. The contribution of angular correlation in the energy values have been estimated. Evolution of and energy levels of H^– ion as quasi‐bound states are being reported.     

Geometry of the canonical Van Vleck transformation

Bloch's transformation from the zeroth‐order space for a perturbation problem to the corresponding space of exact eigenvectors, was found as a geometrically defined alternative to the algebraically constructed Van Vleck transformation. Klein's theorem of uniqueness transferred some of this geometrical interpretation to its canonical form . Quite recently Kvaal has taken a large step further by writing as a product of commuting planar rotations, obtained by describing and in terms of certain principal vectors and canonical angles. Kvaal's approach is now developed further, using a new commutation relation which simplifies algebraic manipulations substantially. It allows for a simple definition of an operator for the angle between and which has Kvaal's vectors and angles as eigenvectors and eigenvalues. Klein's theorem is refined in various ways. The impact of the approach on a number of previous results is considered. © 2015 Wiley Periodicals, Inc.     

Ritz variational calculation for the singly excited states of compressed two‐electron atoms

A detailed analysis on the effect of spherical impenetrable confinement on the structural properties of two‐electron ions in states has been performed. The energy values of 1sns [ ] ( ) states of helium‐like ions ( ) are estimated within the framework of Ritz variational method using explicitly correlated Hylleraas‐type basis sets. The correlated wave functions used here are consistent with the finite boundary conditions due to spherical confinement. A comparative study between the singlet and triplet states originating from a particular electronic configuration shows incidental degeneracy and the subsequent level‐crossing phenomena. The thermodynamic pressure felt by the ion inside the sphere pushes the energy levels toward continuum. Critical pressures for the transition to strong confinement regime (where the singly excited two‐electron energy levels cross the corresponding one‐electron threshold) as well as for the complete destabilization are also estimated.     

Presentation of a new index for estimation of aromaticity in halo‐ and cyanobenzenes: The role of potential energy in aromaticity

A linear correlation has been obtained between average values of Hamiltonian kinetic energy ( ) and potential energy ( ) calculated at the bond critical points using atoms in molecules method. This relation was used to introduce a new index ( ) for estimation of aromaticity in halo‐ and cyanobenzenes. Potential energy has different terms such as attraction between nuclei and electrons, also repulsion of electrons which affect the inertia and mobility of electrons, respectively. Therefore, contribution of potential energy in this relation must be controlled. Contribution of potential energy in aromaticity has been managed using a fitting parameter. This parameter was obtained by fitting the aromaticity stabilization energy data with values of aromaticity calculated by index for halo‐ and cyanobenzenes. The contribution of potential energy in index is complete when molecule is nonaromatic and is negligible when molecule is antiaromatic. Indeed, molecule is aromatic when contribution of potential energy in index lies between above limits. © 2013 Wiley Periodicals, Inc.