Equilibrium Bond Length Research Articles

Solution of Schrödinger Equation for Simple Diatomic Molecules Using One‐Parameter 1s Slater‐Type Orbitals Wave Function

ABSTRACTThe Schrödinger equation for simple homonuclear and heteronuclear diatomic molecules is analytically solved using one‐parameter Slater‐type orbitals (STOs) to approximate the electronic wave function within a molecular orbital (MO)‐like approach. The resulting total energies, equilibrium bond lengths, potential curves, and electron densities are presented in detail. Calculations using a selected orbital exponent accurately reproduce results from standard methods. Furthermore, the optimization of the orbital exponent allows for a more accurate representation of the electronic wave function, leading to the improved results of the total energy and the equilibrium bond length, as well as minimal computational cost. Seen in the heteronuclear diatomic molecule, the use of the one‐parameter STOs allows the transformation of the heteronuclear problem into the homonuclear one, revealing the electron repulsion effect on the orbital exponent parameter.

A new model describing improved bound-state and statistical properties in the framework of deformed Schrödinger equation for the improved Coshine Yukawa potential in 3D-NR(NCPS) symmetries

In this paper, within the three-dimensional non-relativistic non-commutative phase–space (3D-NR(NCPS)) symmetries, we examined the 3D deformed Schrödinger equation (3D-DSE) using the improved Coshine Yukawa potential (ICYP) model composed from Coshine Yukawa potential (CYP) ([Formula: see text]) and other terms produced from the effect of phase–space deformation ([Formula: see text] and [Formula: see text]). For this study, the 3D-DSE in the 3D-NR(NCPS) symmetry is solved and discussed with standard independent time perturbation theory and the generalized Bopp’s shifts method. For the homogeneous (H2 and N2) and heterogeneous (LiH, ScH, and HCL) diatomic molecules, it is obtained that the improved non-relativistic energy equation and eigenfunction for the ICYP in the presence of deformation phase–space are dependent on the discrete atomic quantum numbers ([Formula: see text],m), the dissociation energy [Formula: see text], the equilibrium bond length the screening parameter [Formula: see text], the deformation phase parameters ([Formula: see text]) and the deformation space parameters ([Formula: see text]). Additionally, the thermal properties of the CYP and ICYP with 3D Schrödinger equation (3D-SE) and 3D-DSE are thoroughly examined in the three-dimensional non-relativistic quantum mechanics (3D-NR(QM)) known in the literature symmetry and 3D-NR(NCPS) symmetries, including the partition function, mean energy, free energy, specific heat, and entropy. Furthermore, we discussed particular cases of thermodynamic characteristics for the ICYP model. In our study, we have shown that all the physical values related to energy and thermodynamic properties within the framework of the 3D(NR)NCPS symmetry are equal to the corresponding values within the framework of 3D-NR(QM) symmetry known in the literature, in addition to minor effects resulting from their interaction with the topological properties of the deformed phase–space. This study has multiple applications in various domains, including atomic and molecular physics.

Bond Lengths and Dipole Moments of Diatomic Molecules under Isotropic Pressure with the XP-PCM and GOSTSHYP Models.

While high-pressure chemistry has a well-established history, methods to simulate pressure at the single-molecule level have been somewhat lacking. The current work aims at comparing two static models (XP-PCM and GOSTSHYP) to apply isotropic pressure to single molecules, focusing on the equilibrium bond length and electric dipole moment of diatomic molecules. Numerical challenges arising in the potential energy surface using the XP-PCM method were examined, and a pragmatic approach was followed to mitigate these. The definition of the cavity was scrutinized, and two approaches to retrieve the isotropic character that could potentially be lost when using the standard methodology were suggested. Subsequently, equilibrium bond lengths under pressure were evaluated, showing reasonable agreement between GOSTSHYP and XP-PCM, but some discrepancies persist. A Taylor series analysis introduced elsewhere was then applied to rationalize the observed trends in terms of the bond surface. Finally, the dipole moment was shown to be highly sensitive to the cavity definition, and qualitative agreement necessitates the use of our adapted procedure.

Kekulé Counts, Clar Numbers, and ZZ Polynomials for All Isomers of (5,6)-Fullerenes C52-C70.

We report an extensive tabulation of several important topological invariants for all the isomers of carbon (5,6)-fullerenes Cn with n = 52-70. The topological invariants (including Kekulé count, Clar count, and Clar number) are computed and reported in the form of the corresponding Zhang-Zhang (ZZ) polynomials. The ZZ polynomials appear to be distinct for each isomer cage, providing a unique label that allows for differentiation between various isomers. Several chemical applications of the computed invariants are reported. The results suggest rather weak correlation between the Kekulé count, Clar count, Clar number invariants, and isomer stability, calling into doubt the predictive power of these topological invariants in discriminating the most stable isomer of a given fullerene. The only exception is the Clar count/Kekulé count ratio, which seems to be the most important diagnostic discovered from our analysis. Stronger correlations are detected between Pauling bond orders computed from Kekulé structures (or Clar covers) and the corresponding equilibrium bond lengths determined from the optimized DFTB geometries of all 30,579 isomers of C20-C70.

A new computational framework for spinor-based relativistic exact two-component calculations using contracted basis functions.

A new computational framework for spinor-based relativistic exact two-component (X2C) calculations is developed using contracted basis sets with a spin-orbit contraction scheme. Generally contracted, j-adapted basis sets of p-block elements using primitive functions in the correlation-consistent basis sets are constructed for the X2C Hamiltonian with atomic mean-field spin-orbit integrals (the X2CAMF scheme). The contraction coefficients are taken from atomic X2CAMF Hartree-Fock spinors, thereby following the simple concept of a linear combination of atomic orbitals. Benchmark calculations of spin-orbit splittings, equilibrium bond lengths, and harmonic vibrational frequencies demonstrate the accuracy and efficacy of the j-adapted spin-orbit contraction scheme.

Relativistic Fully Self-Consistent GW for Molecules: Total Energies and Ionization Potentials.

The fully self-consistent GW (scGW) method with an iterative solution of the Dyson equation provides a consistent approach for describing the ground and excited states without any dependence on the mean-field reference. In this work, we present a relativistic version of scGW for molecules containing heavy elements using the exact two-component (X2C) Coulomb approximation. We benchmark SOC-81 data set containing closed shell heavy elements for the first ionization potential using the fully self-consistent GW as well as one-shot GW. The self-consistent GW provides superior results compared to G0W0 with PBE reference and comparable results to G0W0 with PBE0 while also removing the starting point dependence. The photoelectron spectra obtained at the X2C level demonstrate very good agreement with the experimental spectra. We also observe that scGW provides very good estimation of ionization potential for the inner d-shell orbitals. Additionally, using the well-conserved total energy, we investigate the equilibrium bond length and harmonic frequencies of a few halogen dimers using scGW. Overall, our findings demonstrate the applicability of the fully self-consistent GW method for accurate ionization potential, photoelectron spectra, and total energies in finite systems with heavy elements with a reasonable computational scaling.

Theoretical investigation of the silver chalcogenide molecules AgSe and AgTe electronic structure with dipole moment calculation

The spectroscopic constants (Re, Te, ωe, Be) are determined for 46 doublet and quartet low-lying states of the molecules AgTe and AgSe using CASSCF/MRCI (single and double excitations plus Davidson correction) calculations. The equilibrium bond length Re, and the vibrational frequency ωe of the ground state of both molecules have also been investigated using the density functional theory (DFT) incorporating eight different functionals along with different basis sets. The potential energy and permanent dipole moment curves of 23 states of each molecule have been investigated and depicted. This study represents the first theoretical investigation of the molecules AgSe and AgTe.

The CuTe Molecule: Theoretical investigation of the electronic structure and dipole moments of the ground and lowest excited states with rovibrational calculations

The multireference configuration interaction calculation MRSDCI (with Davidson correction) has been performed to investigate the potential energy curves, the transition electronic energies Te, the spectroscopic constants (Re, ωe and Be), the transition and permanent dipole moments, and the vibrational levels corresponding to the ground and 18 excited electronic states of the molecule CuTe in the representation 2S+1Λ(+/-) (neglecting the spin–orbit effects), whereas the PECs and the spectroscopic constants have been calculated for 27 states in the representation Ω(+/-) [including the spin–orbit (SO) effects]. The ground state equilibrium bond length and the vibrational frequency have also been investigated using the density functional theory (DFT), where different density functionals with different basis sets have been tested. This study represents the first theoretical study of the excited states of the CuTe molecule. Comparison of the present results with the available experimental ones shows an overall satisfying agreement.

High Accuracy Ab Initio Potential Energy Curves and Dipole Moment Functions for the X1Σ+ and a3Π Spin States of the CF+ Diatomic Molecule.

Potential energy curves and dipole moment functions constructed using high-accuracy ab initio methods allow for an in-depth examination of the electronic structure of diatomic molecules. Ab initio computations serve as a valuable complement to experimental data, offering insights into the nature of short-lived molecules such as those encountered within the interstellar medium (ISM). While laboratory experiments provide critical groundwork, the ISM's conditions often permit longer lifetimes for lower stability molecules, enabling unique observations. The CF+ diatomic molecule is one such molecule that has been observed spectroscopically in the ISM. Previous experimental and theoretical work have examined different spectroscopic aspects of the CF+ molecule, but the development of newer, more complete potential energy curves and dipole moment functions allows for even greater insight. We constructed both potential energy curves and dipole moment functions for the ground X1Σ+ and first excited a3Π states of CF+ for both the 12C and 13C isotopologues. The potential energy curves were constructed using coupled cluster with single, double, and perturbative triple excitations (CCSD(T)) at the complete basis set limit with corrections from full triple, quadruple, quintuple, and hextuple excitations within a finite-basis coupled cluster wave function as well as corrections from full configuration interaction and relativistic effects. Rovibrational wave functions were calculated using a vibrational Hamiltonian matrix, which moves beyond the harmonic oscillator approximation. The equilibrium bond length, vibrational constant, and rotational constant were reproduced to within 0.00013 Å, 0.28 cm-1, and 0.00045 cm-1, respectively, of experimental values. Experimental transition energies from rovibrational spectra were reproduced with an error of no larger than 0.63 cm-1. The triplet excited state (a3Π) was found to have a longer equilibrium bond length at 1.21069 Å, a vibrational constant of 1611.29 cm-1, and a rotational constant of 1.56376 cm-1. Rovibrational line lists for the 12C and 13C isotopologues for both the X1Σ+ and the excited a3Π states were generated.

Hartree-fock roothaan calculations using optimized huzinaga orbitals on small molecules

We present the ground-state solution of some small molecules using the Hartree–Fock Roothaan method with the optimized Huzinaga basis set. Unlike the previously used least-square methods, the contraction coefficients and exponents of Huzinaga-parameterized primitive Gaussian functions for minimal basis sets are energy-optimized at the atomic level for each molecule. Consequently, as an alternative to and in comparison with standard parameterization, the optimized orbitals significantly improve the total energy and the equilibrium bond length with substantial enhancement shown for heavier nuclei. Despite similar computational cost, the application of our scheme leads to much improved minimal-basis-set Hartree–Fock calculations with less required parameters to match the large basis set calculations. Furthermore, the localization of the electrons near the nuclei which is missing with the standard parameterization is observed with the current scheme.

Formation Mechanisms of Electronically Excited Nitrogen Molecules from N + N2 and N + N + N Collisions Revealed by Full-Dimensional Potential Energy Surfaces.

This work reports six new full-dimensional adiabatic potential energy surfaces (PESs) of the N3 system (four 4A″ states and two 2A″ states) at the MRCI + Q/AVQZ level of theory that correlated to N2(X1Σg+) + N(4S), N2(X1Σg+) + N(2D), N2(A3Σu+) + N(4S), N2(B3Πg) + N(4S), N2(W3Δu) + N(4S), and N(4S) + N(4S) + N(4S) channels. The neural networks with a proper account of the nuclear permutation invariant symmetry of N3 were employed to fit the PESs based on about 4000 ab initio points. The accuracy of the PESs was validated by excellent agreement on the equilibrium bond length, vertical excitation energy, and dissociation energy with experimental values. Two possible mechanisms of the formation of N2(A) were found. One is that the collision occurs between N2(X) and N(4S) in the 14A″ state, followed by a nonadiabatic transition through the conical intersection with the 24A″ PES, resulting in the formation of the N2(A) + N(4S) product. The other takes place in the collision among three N(4S) atoms in the adiabatic 24A″ state, and then, N2(A) + N(4S) is formed. This is the first systematical research of the N3 system focusing on the formation of the excited states of N2 via both adiabatic and nonadiabatic pathways.

Simple, near-universal relationships between bond lengths, strengths, and anharmonicities

Harmonic bond force constants and bond lengths are shown to generally obey the simple relationships, ke=ζ2Re−3 (hydrides) and ke=10ζ1/2Re−4 (all other bond types), where ζ is the reduced nuclear charge and Re is the equilibrium bond length. Equally simple power-law relationships are found for higher-order bond force constants. Although not spectroscopically accurate, these models are nonetheless of significant heuristic value for identifying strongly multireference states of diatomic molecules (including electronically coupled excited states ill-suited for inclusion in laser-cooling schemes), rationalizing the observed trends in vibrational frequencies for diatomics and/or local mode oscillators within molecules or complexes and estimating and/or validating covalent bonding parameters within molecular mechanics force fields. Particular advantages of our approach over other bond length-strength scaling relationships proposed in the literature include its simplicity and generality and its appropriate asymptotic behavior. Notably, the relationships derived in this work can be used to predict harmonic and higher-order force constant bonds between any pair of atoms in the Periodic Table (including transition metals and lanthanides) without requiring row- or column-dependent parameterization, to accuracies commensurate with conventional force field transferability errors. We therefore anticipate that they will expedite force field development for metal-containing complexes and materials, which are structurally well-characterized but challenging to parameterize ab initio.

Defect-Induced Modulation of a 2D ZnO/Graphene Heterostructure: Exploring Structural and Electronic Transformations

This paper presents a theoretical study on the effects of selected defects (oxygen vacancies and substitutional FeZn atoms) on the structural and electronic properties of a 2D ZnO/graphene heterostructure. Spin-polarized Hubbard- and dispersion-corrected density functional theory (DFT) was used to optimize the geometrical configurations of the heterostructure and to analyze the equilibrium distance, interlayer distance, adhesion energy, and bond lengths. Charge density difference (CDD) analysis and band structure calculations were also performed to study the electronic properties of the heterostructure. The results show that the presence of defects affects the interlayer distance and adhesion energy, with structures including oxygen vacancies and FeZn substitutional atoms having the strongest interaction with graphene. It is demonstrated that the oxygen vacancies generate localized defect states in the ZnO bandgap and lead to a shift of both valence and conduction band positions, affecting the Schottky barrier. In contrast, Fe dopants induce strong spin polarization and high spin density localized on Fe atoms and their adjacent oxygen neighbors as well as the spin asymmetry of Schottky barriers in 2D ZnO/graphene. This study presents a comprehensive investigation into the effects of graphene on the electronic and adsorption properties of 2D ZnO/graphene heterostructures. The changes in electronic properties induced by oxygen vacancies and Fe dopants can enhance the sensitivity and catalytic activity of the 2D ZnO/graphene system, making it a promising material for sensing and catalytic applications.

Extremely Long C-C Bonds Predicted beyond 2.0 Å.

A number of conjugated molecules are designed with extremely long single C-C bonds beyond 2.0 Å. Some of the investigated molecules are based on analogues to the recently discovered molecule by Kubo et al. These bonds are analyzed by a variety of indices in addition to their equilibrium bond length including the Wiberg bond index, bond dissociation energy (BDE), and measures of diradicaloid character. All unrestricted DFT calculations indicate no diradical character supported by high-level multireference calculations. Finally, NFOD was computed through fractional orbital density (FOD) calculations and used to compare relative differences of diradicaloid character across twisted molecules without central C-C bonding and those with extremely elongated C-C bonds using a comparison with the C-C bond breaking in ethane. No example of direct C-C bonds beyond 2.4 Å are seen in the computational modeling; however, extremely stretched C-C bonds in the vicinity of 2.2 Å are predicted to be achievable with a BDE of 15-25 kcal mol-1.

Ro-vibrational Spectrum of Linear Dialuminum Monoxide (Al2O) at 10 μm.

Dialuminum monoxide, Al2O, has been investigated in the laboratory at mid-IR wavelengths around 10 μm at high spectral resolution. The molecule was produced by laser ablation of an aluminum target with the addition of gaseous nitrous oxide, N2O. Subsequent adiabatic cooling of the gas in a supersonic beam expansion led to rotationally cold spectra. In total, 848 ro-vibrational transitions have been assigned to the fundamental asymmetric stretching mode ν3 and to five of its hot bands, originating from excited levels of the ν1 symmetric stretching mode and the ν2 bending mode. The measurements encompass 11 vibrational energy states (v1 v2l v3). The ro-vibrational transitions show spin statistical line intensity alternation of 7:5, which is caused by two identical aluminum nuclei of spin I = 5/2 at both ends of the centrosymmetric molecule of structure Al-O-Al. The less effective cooling of vibrational states in the supersonic beam expansion allowed measurement of transitions in excited vibrational states at energies of 1000 cm-1 and higher, while rotational levels within vibrational modes exhibited thermal population, with rotational temperatures around Trot = 115 K. Molecular parameters for 11 vibrational states were derived, including rotation and centrifugal distortion constants and l-type doubling constants for the states (v1 v2l v3) = (0 11 0) and (0 11 1) and an l-type resonance between the states (0 20 0) - (0 22 0) and (0 20 1) - (0 22 1). From the experimental results, rotational correction terms and the equilibrium bond length re were derived. The measurements were supported and guided by high-level quantum-chemical calculations that agree well with the derived experimental results.

Features of molecular structure beneficial for optical pumping

Fast and efficient state preparation of molecules can be accomplished by optical pumping. Molecular structure that most obviously facilitates cycling involves a strong electronic transition, with favorable vibrational branching [diagonal Franck-Condon factors (FCFs)] and without any intervening electronic states. Here, we propose important adjustments to those criteria, based on our experience optically pumping ${\mathrm{SiO}}^{+}$. Specifically, the preference for no intervening electronic states should be revised, and over-reliance on FCFs can miss important features. The intervening electronic state in ${\mathrm{SiO}}^{+}$ is actually found to be beneficial in ground rotational state preparation, by providing a pathway for population to undergo a parity flip. This contribution demonstrates the possibility that decay through intervening states may help state preparation of nondiagonal or polyatomic molecules. We also expand upon the definition of favorable branching. In ${\mathrm{SiO}}^{+}$, we find that the off-diagonal FCFs fail to reflect the vibrational heating versus cooling rates. Since the branching rates are determined by transition dipole moments (TDMs) we introduce a simple model to approximate the TDMs for off-diagonal decays. We find that two terms, set primarily by the slope of the dipole moment function ($d\ensuremath{\mu}/dx$) and offset in equilibrium bond lengths ($\mathrm{\ensuremath{\Delta}}x={r}_{e}^{g}\ensuremath{-}{r}_{e}^{e}$), can add (subtract) to increase (decrease) the magnitude of a given TDM. Applying the model to ${\mathrm{SiO}}^{+}$, we find there is a fortuitous cancellation, where decay leading to vibrational excitation is reduced, causing optical cycling to lead naturally to vibrational cooling.

Multireference Wavefunction-Based Investigation of the Ground and Excited States of LrF and LrO.

Complete active space self-consistent field (CASSCF) and multireference configuration interaction with Davidson correction (MRCI+Q) calculations have been carried out for lawrencium fluoride (LrF) and lawrencium oxide (LrO) molecules, detailing 19 and 20 electronic states for LrF and LrO, respectively. For LrF, two dissociation channels were considered, Lr(2P)+F(2P) and Lr(2D)+F(2P). However, due to the more complex electronic manifold of LrO, three dissociation channels were computed: Lr(2P)+O(3P), Lr(2D)+O(3P), and Lr(2P)+O(1D). In addition, equilibrium bond lengths, harmonic vibrational frequencies ωe, anharmonicity constants ωeχe, ΔG1/2 values, and excitation energies Te for the ground and several excited electronic states were calculated for both molecules, for the first time. Bond dissociation energies (BDEs) were calculated for LrF and LrO using several different levels of theory: unrestricted coupled-cluster with single, double, and perturbative triple excitations (UCCSD(T)), density functional theory (B3LYP, TPSS, M06-L, and PBE), and the correlation-consistent composite approach developed for f-elements (f-ccCA).

The performance of CCSD(T) for the calculation of dipole moments in diatomics.

This work analyzes the accuracy of the coupled cluster with single, double, and perturbative triple excitation [CCSD(T)] method for predicting dipole moments. In particular, we benchmark CCSD(T) predictions for the equilibrium bond length, vibrational frequency, and dipole moment versus accurate experimental data. As a result, we find that CCSD(T) leads to accurate dipole moments. However, in some cases, it disagrees with the experimental values, and the disagreement can not be satisfactorily explained via relativistic or multi-reference effects. Therefore, our results indicate that benchmark studies for energy and geometry properties do not accurately describe other electron density magnitudes.

Theoretical study of low energy electron collisions with the BeO molecule

An R-matrix calculation is reported for low energy electron collision with the beryllium oxide (BeO) molecule at its equilibrium bond length a 0 for incident energies below 15 eV. Using a suitably constructed model to represent the BeO target, detailed scattering calculations are performed to obtain cross sections for many collision processes, namely total cross sections for elastic scattering and electronic excitations to some of its low lying states and cross section for electron impact dissociation. Additionally, we have also calculated differential cross sections for electronic excitations from the X ground state to the first two excited states of the target BeO molecule. BeO is likely to be present as an impurity component in the plasma of fusion devices operating with beryllium as first wall material. We therefore expect the present study to be of substantial importance to the fusion community.

The feasibility of laser cooling: an investigation of ab initio of 88Sr35Cl including the hyperfine structure

ABSTRACT A promising laser cooling candidate molecule is essential for laser cooling experiments. In this study, the possibility of laser cooling an 88Sr35Cl molecule is investigated using an ab initio method. Seven low-lying Λ-S and six Ω electronic states of the 88Sr35Cl molecule are calculated at a multi-reference configuration interaction level of theory. Spectral constants of bound states of fitted values are in good agreement with experimental values, which are better than the previously obtained theoretical data for higher excitation states. The resulting permanent and transition dipole moments near the equilibrium bond length are close to the theoretical values. Potential energy curves and transition dipole moments are used for obtaining highly diagonally distributed Franck-Condon factors for the A2Π → X2Σ+ transition using the LEVEL program. A short radiative lifetime of the A2Π state is determined; a laser cooling scheme is designed in the vibration level that requires a cooling main laser beam and two repumping laser beams. Transition spectral data of hyperfine energy levels with errors relative to the experimental data do not exceed −0.25∼0.19 MHz when using a quantum effective Hamiltonian approach. This study provides a helpful reference for experimental observation and operation of laser cooling the 88Sr35Cl molecule.