Equilibrium Internuclear Separation Research Articles

New bound state solutions of diatomic molecules and mass spectrum of heavy quarkonium with improved modified Kratzer potential plus screened Coulomb potential model in the framework of three-dimensional nonrelativistic noncommutative phase-space symmetries

Approximate analytical solutions of the modified Schrodinger equation have been obtained with a combination of two proposed potentials, called the improved modified Kratzer potential plus screened Coulomb (IMKSC) potential, using the approximation scheme to the centrifugal term for any l-states. We applied the generalized Bopp’s shift method and standard perturbation theory. IMKSC is a superposition of modified Kratzer potential plus screened Coulomb (MKSC) potential, and new terms appear as a result of the effects of the noncommutativity properties of space and phase potential on the MKSC potential. The new bound state energy eigenvalues for some diatomic molecules were obtained in terms of the discrete atomic quantum numbers (j,n,s,m), the infinitesimal NC phase-space parameters, the dissociation energy, the equilibrium internuclear separation, the depth of the potential, and the range of the potential. Furthermore, we have shown that the NC-Hamiltonian operator is the sum of the Hamiltonian operator of the MKSC potential, the perturbed spin-orbit, and the modified Zeeman operator. The new mass spectra of heavy quarkonium systems are calculated using the new energy spectrum expression. It appears as a combination of corresponding values in the literature plus an infinitesimal part that describes the deformation phase-space.

Investigating the Heaviest Halogen: Lessons Learned from Modeling the Electronic Structure of Astatine’s Small Molecules

We present a systematic study of electron-correlation and relativistic effects in diatomic molecular species of the heaviest halogen astatine (At) within relativistic single- and multireference coupled-cluster approaches and relativistic density functional theory. We establish revised reference ab initio data for the ground states of At2, HAt, AtAu, and AtO+ using a highly accurate relativistic effective core potential model and in-house basis sets developed for accurate modeling of molecules with large spin-orbit effects. Spin-dependent relativistic effects on chemical bonding in the ground state are comparable to the binding energy or even exceed it in At2. Electron-correlation effects near the equilibrium internuclear separation are mostly dynamical and can be adequately captured using single-reference CCSD(T). However, bond elongation in At2 and, especially, AtO+ results in rapid manifestation of its multireference character. While useful for evaluating the spin-orbit effects on the ground-state bonding and properties, the two-component density functional theory lacks predictive power, especially in combination with popular empirically adjusted exchange-correlation functionals. This drawback supports the necessity to develop new functionals for reliable quantum-chemical models of heavy-element compounds with strong relativistic effects.

Diabatic Valence-Hole States in the C2 Molecule: “Putting Humpty Dumpty Together Again”

Despite the long history of spectroscopic studies of the C$_2$ molecule, fundamental questions about its chemical bonding are still being hotly debated. The complex electronic structure of C$_2$ is a consequence of its dense manifold of near-degenerate, low-lying electronic states. A global multi-state diabatic model is proposed here to disentangle the numerous configuration interactions within four symmetry manifolds of C$_2$ ($^{1}\Pi_g$, $^{3}\Pi_g$, $^{1}\Sigma_u^+$, and $^{3}\Sigma_u^+$). The key concept of our model is the existence of two "valence-hole" configurations, $2\sigma_g^22\sigma_u^11\pi_{u}^33\sigma_g^2$ for $^{1,3}\Pi_g$ states and $2\sigma_g^22\sigma_u^11\pi_{u}^43\sigma_g^1$ for $^{1,3}\Sigma_u^+$ states that derive from $3\sigma_g\leftarrow2\sigma_u$ electron promotion. The lowest-energy state from each of the four C$_2$ symmetry species is dominated by this type of valence-hole configuration at its equilibrium internuclear separation. As a result of their large binding energy (nominal bond order of 3) and correlation with the 2s$^2$2p$^2$+2s2p$^3$ separated-atom configurations, the presence of these valence-hole configurations has a profound impact on the $global$ electronic structure and unimolecular dynamics of C$_2$.

Rotational characterization of the [formula omitted](5s6s 3S1) Rydberg state of CdNe van der Waals complex via selective J-excitation in OODR process

Spectroscopic rotational characterization of two υ′=0 and υ′=1 vibrational components of the E3Σ1+(5s6s3S1) Rydberg state of CdNe van der Waals (vdW) complex has been performed using specially designed approach. In optical–optical double resonance (OODR) process, a selective J-excitation has been realized using the E3Σ1+←A3Π0+(5s5p3P1) transition, with υ″=0 level of the A3Π0+ state as an intermediate in the excitation from υ=0 of the X1Σ0+(5s21S0) state. As a result, Bυ=0′ and Bυ=1′ rotational constants have been determined as well as the Re′ equilibrium internuclear separation in the E3Σ1+ - state potential well.

WKB approximation for analyzing quantum tunneling effect through negative Kratzer potential

Kratzer potential is a potential for bound states in molecular bonds, while negative Kratzer potential is a potential for scattering states which has a barrier form, and which clearly has properties opposite to the bound state potential. This type of potential is the main subject of this paper. When a non-relativistic particle which has lower energy moving towards the potential, then there is a probability for the particle to break through the potential. In this paper, we use the WKB (Wenztel-Kramers-Brillouin) method to analyze the physical conditions that must be met by the system in order for the tunneling processes to occur. We present these conditions in the form of the relationship among several quantum variables such as the mass of the particle, the energy of the particle, the maximum height of the barrier potential, equilibrium internuclear separation, and the enactment domain of the potential field effect on the particle when the particle breaks through that potential.

Line list for the ground state of CaF

The molecular potential energy function and electronic dipole moment function for the ground state of CaF were studied with MRCI, ACPF, and RCCSD(T) ab initio calculations. The RCCSD(T) potential function reproduces the experimental vibrational intervals to within ∼2 cm−1. The RCCSD(T) dipole moment at the equilibrium internuclear separation agrees well with the experimental value. Over a wide range of internuclear separations, far beyond the range associated with the observed spectra, the ab initio dipole moment functions are similar and highly linear. An extended Morse oscillator (EMO) potential function was also obtained by fitting the observed lines of the laboratory vibration-rotation and pure rotation spectra of the 40CaF X2Σ+ ground state. The fitted potential reproduces the observed transitions (v ≤ 8, N ≤ 121, Δv = 0, 1) within their experimental uncertainties. With this EMO potential and the RCCSD(T) dipole moment function, line lists for 40CaF, 42CaF, 43CaF, 44CaF, 46CaF, and 48CaF were computed for v ≤ 10, N ≤ 121, Δv = 0–10. The calculated emission spectra are in good agreement with an observed laboratory spectrum of CaF at a sample temperature of 1873 K.

Charge Migration in Eyring, Walter and Kimball's 1944 Model of the Electronically Excited Hydrogen-Molecule Ion.

In an elementary variational treatment of the electronic structure of H2+, Eyring, Walter, and Kimball (EWK) serendipitously discovered charge migration (CM) in 1944. Using an analytic expression for the electronic probability density (EPD), they found that if the electron is initially localized on one of the protons (by taking the initial state to be a superposition of the ground and first excited electronic energy eigenstates), then it oscillates adiabatically between fixed protons with a period T inversely proportional to the energy gap between the eigenstates. At the equilibrium internuclear separation, T = 550.9 as. As shown here, the EWK model also yields an analytic formula for the electronic flux density (EFD). While the EPD indicates where the electron is at any instant, the EFD reveals the pathways the electron follows during its migration. Thus, the EFD complements the EPD, providing valuable new insight into the mechanism of CM. The formula for the EFD is a simple product of a time factor and a spatial factor. This factoring exposes a plethora of spatial-temporal symmetry relations which imply novel and surprising properties. An especially significant finding is that, in contrast to multielectron systems, where electron correlation may play a role in CM, in the EWK model of H2+, CM is due strictly to quantum interference between the ground and first excited electronic states.

Exploration of the origin of anomalous dependence for near-threshold harmonics in on the ellipticity of driving laser fields

The anomalous dependence of near-threshold harmonics in the molecular ion on the ellipticity of the driving near-infrared laser field is studied theoretically based on accurate solution of the time-dependent Schrödinger equation in prolate spheroidal coordinates with the help of the generalized pseudospectral method. For these harmonics, the maximum radiation energy corresponds to a non-zero ellipticity of the driving field. Our analysis reveals that the origin of the phenomenon lies in the near-resonant excitation of π-symmetry molecular orbitals. The excited states responsible for the anomalous ellipticity dependence of different near-threshold harmonics are identified. The effect is confirmed at the equilibrium internuclear separation R = 2 a.u. as well as for stretched molecules at R = 3 a.u.

Above- and below-threshold high-order-harmonic generation ofH2+in intense elliptically polarized laser fields

We present an ab initio three-dimensional precision calculation and analysis of high-order-harmonic generation (HHG) of the hydrogen molecular ion subject to intense elliptically polarized laser pulses by means of the time-dependent generalized pseudospectral method in two-center prolate spheroidal coordinates. The calculations are performed for the ground and first excited electronic states of ${\mathrm{H}}_{2}{}^{+}$ at the equilibrium internuclear separation $R=2$ a.u. as well as for the stretched molecule at $R=7$ a.u. The spectral and temporal structures of the HHG signal are explored by means of the wavelet time-frequency analysis. Several aspects of ellipticity-dependent dynamical behaviors are uncovered. We found that the production of above-threshold harmonics for nonzero ellipticity is generally reduced, as compared with linearly polarized fields. However, below-threshold harmonics still appear quite strong except when the polarization plane is perpendicular to the molecular axis. Weak even harmonics are detected in the HHG spectra of stretched molecules. This effect can be explained by the broken inversion symmetry due to dynamic localization of the electron density near one of the nuclei. Multiphoton resonance and two-center interference effects are analyzed for the exploration of the quantum origin of the predicted HHG spectral and dynamical behavior.

Improved direct potential fit analyses for the ground electronic states of the hydrogen halides: HF/DF/TF, HCl/DCl/TCl, HBr/DBr/TBr and HI/DI/TI

The potential energy and Born–Oppenheimer breakdown functions for the X1Σ+ ground electronic states of the hydrogen halides HF, HCl, HBr, and HI are reported in full analytic form. All available pure rotational and vibrational–rotational spectroscopic data for the various isotopologues of the four HX molecules, as well as the B1Σ+→X1Σ+ emission band system data for HF/DF and HCl/DCl, were employed in direct potential fit determinations of the various radial functions. Significant improvements over previous work have been made to the mathematical models for these functions, particularly with respect to the behavior of the potential energy in the long-range region where dispersion forces between the component atoms dominate. The MLR3 model for the potential energy is employed, allowing for constraint of the three leading dispersion coefficients C6, C8, and C10, for the X states of HF and HCl. Quantum-mechanical rotational and centrifugal distortion constants are calculated for all isotopologues considered in the non-linear least-squares fitting procedures. Computer code is provided in order that precise calculation of all functions for all isotopologues is possible for a user-specified radial grid. Precise estimates are obtained for X state equilibrium internuclear separations for the principal isotopologues of all four halides, namely re(HF)=0.91683897±0.00000004Å, re(H35Cl)=1.27454677±0.00000006Å, re(H79Br)=1.4144292±0.0000001Å, and re(HI)=1.6290588±0.0000004Å.

Transport properties for systems with deep potential wells: H + O2.

Transport properties for collisions of the oxygen molecule with hydrogen atoms are computed by means of quantum scattering calculations. Because two potential energy surfaces (PESs) arise from the interaction of H(2S) with O2(X3Σg+), namely 2A″ and 4A″, collision integrals were computed using both PESs and then averaged with weighting by their respective spin multiplicities. A PES for the 4A″ state was computed for the interaction of O2, frozen at its equilibrium internuclear separation, with a hydrogen atom, using a coupled-cluster method that includes all single and double excitations as well as perturbative contributions of connected triple excitation. A PES of similar quality was taken from Kłos et al. [J. Chem. Phys.2008, 129, 064306] for the 2A″ state. Because the 2A″ state correlates with the deep HO2(X̃2A″) well, statistical capture boundary conditions [Rackham et al., J. Chem. Phys.2003, 119, 12895] were applied to compute the S matrix, and then the transport properties, for this state.

A quantum-chemical study of intermediates of the 1O2 photogeneration sensitized by buckminsterfullerene and accompanying photochemical reactions

The intermediates of hypothetical photochemical reactions that accompany the quenching of the 3C60* triplet state by triplet oxygen are studied by the (U)PBE0 quantum-chemical method. The diradical C60-O-O formed from 3O2 and photoexcited buckminsterfullerene 3C60* is characterized by a negative binding energy −1.11 eV (with respect to C60 and 3O2), the singlet-triplet splitting ΔEST of 0.07 eV, and the dipole moment of 3.2 D at the equilibrium internuclear separations 1.522 A (CO) and 1.294 A (OO). Its decay produces 1O2. The formation of a dioxetane circle lowers the energy by 0.8 eV. The ground-state energy of diketone C58(C=O)2 is 2.0 eV lower than the energy of C60-O-O. The metastable centrosymmetric diradical C60-C60, formed upon ineffective light absorption by clusters (C60)N, has a single interpolyhedral C-C bond (1.657 A). Its triplet state T1 lies 0.16 eV higher than the S1 singlet. The S1 → S0 relaxation leads to the formation of a stable C60-C60 dimer with a shorter (1.584 A) bis-single exothermic (+0.24 eV) bond of polyhedra. The photoexcited C60-C60 dimer is able to form isomeric metastable diradicals C60-C60-O-O.

Exploring the Nature of the H2 Bond. 2. Using Ab Initio Molecular Orbital Calculations To Obtain the Molecular Constants

A project for students in an upper-level course in quantum or computational chemistry is described in which they are introduced to the concepts and applications of a high quality, ab initio treatment of the ground-state potential energy curve (PEC) for H2 and D2. Using a commercial computational chemistry application and a scientific spreadsheet, students learn how to obtain the energy of H2 (D2) employing configuration interaction with single and double excitations (CISD) and an extrapolation technique to estimate the energy in the complete basis set (CBS) limit. They construct the PEC from the CISD/CBS energies obtained at several internuclear separations. Students obtain the equilibrium dissociation energy and internuclear separation, as well as the rovibrational molecular constants, from the regression parameters of a sixth-order polynomial fit to the PEC. Results are in nearly exact agreement with experimental data.

Probing two-center interference in H2+using chirped pulses

The influence of intermediate resonant dissociative channels in the few-photon ionization dynamics of H${}_{2}$${}^{+}$ is demonstrated in a pump-probe scenario. Two-photon ionization of H${}_{2}$${}^{+}$ by two sequentially applied pump and probe vuv femtosecond ${10}^{11}$ W/cm${}^{2}$ laser pulses is reported. The kinetic energy distribution of the ejected protons is calculated by solving the time-dependent Schr\"odinger equation within the Born-Oppenheimer approximation, including the electronic three-dimensional and vibrational one-dimensional motion. Population is effectively transferred from the $1s{\ensuremath{\sigma}}_{g}$ to the $3p{\ensuremath{\sigma}}_{u}$ potential surface, via resonant one-photon absorption, by applying a chirped pump pulse. The molecule is ultimately ionized by the probe pulse. It is found that a double-peaked structure appears in the resulting kinetic energy release spectra of the nuclear fragments. A corresponding modulated structure also appears in the dissociative channels, merely demonstrating that the double-peaked structure in the spectra originates from the molecular $1s{\ensuremath{\sigma}}_{g}$-$3p{\ensuremath{\sigma}}_{u}$ electronic dynamics, and an inherent two-center interference in the underlying electric-dipole coupling. It turns out that the dipole coupling between the $1s{\ensuremath{\sigma}}_{g}$ and $3p{\ensuremath{\sigma}}_{u}$ electronic states vanishes at an internuclear distance of $2.25$ a.u., i.e., close to the equilibrium internuclear separation. The resulting node in the $R$-dependent dipole coupling imposes a minimum in the corresponding kinetic energy release spectra of the dissociating nuclei. By creating a negative or positive chirp in the applied pump pulse, it is found that the modulations can be made more or less salient.

He 2++ molecular ion in a strong time-dependent magnetic field: A current-density functional study

The He 2++ molecular ion exposed to a strong ultrashort time-dependent (TD) magnetic field of the order of 10(9) G is investigated through a quantum fluid dynamics (QFD) and current-density functional theory (CDFT) based approach using vector exchange-correlation (XC) potential and energy density functional that depend not only on the electronic charge-density but also on the current density. The TD-QFD-CDFT computations are performed in a parallel internuclear-axis and magnetic field-axis configuration at the field-free equilibrium internuclear separation R = 1.3 au with the field-strength varying between 0 and 10(11) G. The TD behavior of the exchange- and correlation energy of the He 2++ is analyzed and compared with that obtained using a [B-TD-QFD-density functional theory (DFT)] approach based on the conventional TD-DFT under similar computational constraints but using only scalar XC potential and energy density functional dependent on the electronic charge-density alone. The CDFT based approach yields TD exchange- and correlation energy and TD electronic charge-density significantly different from that obtained using the conventional TD-DFT based approach, particularly, at typical magnetic field strengths and during a typical time period of the TD field. This peculiar behavior of the CDFT-based approach is traced to the TD current-density dependent vector XC potential, which can induce nonadiabatic effects causing retardation of the oscillating electronic charge density. Such dissipative electron dynamics of the He 2++ molecular ion is elucidated by treating electronic charge density as an electron-"fluid" in the terminology of QFD.

Spectroscopic investigations on BH+ (X2 Σ+ ) ion using MRCI method and correlation-consistent sextuple basis set augmented with diffuse functions

The highly accurate valence internally contracted multireference configuration interaction (MRCI) approach in combination with a series of correlation-consistent basis sets, cc-pVnZ and aug-cc-pVnZ (n = 2, 3, 4, 5, 6), in the valence range is employed to investigate the equilibrium internuclear separations, harmonic frequencies and potential energy curves (PECs) of the BH+(X2Σ+) ion. The PECs are all fitted to the Murrell–Sorbie function, which are used to reproduce the spectroscopic parameters, such as De, ωeχe, αe, Be, and D0. By comparison with the available experiments, The PEC, which is determined at the sextuple basis set augmented with diffuse functions, aug-cc-pV6Z, is found to most suitably determine the spectroscopic properties of the ground-state BH+ ion. The D0, De, Re, ωe, ωeχe, αe, and Be at this basis set are of 1.9823 eV, 2.1364 eV, 0.12041 nm, 2520.81 cm−1, 61.80 cm−1, 0.4824 cm−1, and 12.5932 cm−1, respectively, which are in excellent agreement with the available measurements. With the interaction potential determined at the MRCI/aug-cc-pV6Z level of theory, by numerically solving the radial Schrodinger equation of nuclear motion using the Numerov method, the total of 16 vibrational states is predicted when the rotational quantum number J is equal to zero (J = 0). The complete vibrational levels, classical turning points, inertial rotation and centrifugal distortion constants are determined when J = 0 for the first time, which agree well with the available experiments. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011

Formula omitted] embedded in a Debye plasma: Electronic and vibrational properties

The effect of plasma screening on the electronic and vibrational properties of the H 2 + molecular ion was analyzed within the Born–Oppenheimer approximation. When a molecule is embedded in a plasma, the plasma screens the electrostatic interactions. This screening is accounted in the Schrödinger equation by replacing the Coulomb potentials with Yukawa potentials that incorporate the Debye length as a screening parameter. Variational expansions in confocal elliptical coordinates were used to calculate energies of the 1 s σ g and 2 p σ u states over a range of Debye lengths and bond distances. When the Debye length is comparable to the equilibrium bond distance, the dissociation energy is reduced while the equilibrium internuclear separation is increased. Expectation values, static dipole polarizabilities and spectroscopic constants were calculated for the 1 s σ g state.

Accurate potential energy function and spectroscopic study of the X2Σ+, A2Π and B2Σ+ states of the CP radical**Project supported by the National Natural Science Foundation of China (Grant No. 10874064) and the Program for Science & Technology Innovation Talents in Universities of Henan Province in China (Grant No. 2008HASTIT008).

This paper calculates the equilibrium internuclear separations, the harmonic frequencies and the potential energy curves of the X2Σ+, A2Π and B2Σ+ states of the CP radical by the highly accurate valence internally contracted multireference configuration interaction method with correlation-consistent basis sets (aug-cc-pV6Z for C atom and aug-cc-pVQZ for P atom). The potential energy curves are all fitted with the analytic potential energy function by the least-square fitting. Employing the analytic potential energy function, we determine the spectroscopic constants (Be, αe and ωeχe) of these states. For the X2Σ+ state, the obtained values of De, Be, αe, ωeχe, Re and ωe are 5.4831 eV, 0.792119 cm−1, 0.005521 cm−1, 6.89653 cm−1, 0.15683 nm, 12535.11 cm−1, respectively. For the A2Π state, the present values of De, Be, αe, ωeχe, Re and ωe are 4.586 eV, 0.703333 cm−1, 0.005458 cm−1, 6.03398 cm−1, 0.16613 nm, 1057.89 cm−1, respectively. For the B2Σ+ state, the present values of De, Be, αe, ωeχe, Re and ωe are 3.506 eV, 0.677561 cm−1, 0.00603298 cm−1, 5.68809 cm−1, 0.1696 nm, 822.554 cm−1, respectively. For these states, the vibrational states with the rotational quantum number J equals zero (J = 0) are studied by solving the radial nuclear Schrödinger equation using the Numerov method. For each vibrational state, the vibrational level, the classical turning points, the rotational inertial and the centrifugal distortion constants are calculated. Comparison is made with recent theoretical and experimental results.

Spectroscopic investigations on AlH(X1Σ+) radical using multireference configuration interaction theory in combination with correlation‐consistent quintuple basis set augmented with diffuse functions

AbstractThe equilibrium internuclear separations, harmonic frequencies, and potential energy curves (PECs) of the AlH(X1Σ+) radical have been investigated using the highly accurate valence internally contracted multireference configuration interaction (MRCI) approach in combination with a series of correlation‐consistent basis sets in the valence range. The PECs are all fitted to the Murrell‐Sorbie function, which are used to reproduce the spectroscopic parameters (De, D0, ωeχe, αe, and Be). By comparison with the available experimental results, the PEC obtained at the basis set, aug‐cc‐pV5Z, is selected to investigate the vibrational manifolds. The present D0, De, Re, ωe, ωeχe, αe, and Be are of 3.0731 eV, 3.1775 eV, 0.16510 nm, 1683.37 cm−1, 29.3786 cm−1, 0.18876 cm−1, and 6.3663 cm−1, respectively, which almost perfectly conform to the available measurements. With the potential determined at the MRCI/aug‐cc‐pV5Z level of theory, the total of 26 vibrational states is predicted when the rotational quantum number J is set to equal zero (J = 0) by numerically solving the radial Schrödinger equation of nuclear motion with the Numerov method. The complete vibrational levels, classical turning points, inertial rotation, and centrifugal distortion constants are determined when J = 0 for the first time, which are in excellent agreement with the available experimental data. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2011

Spectroscopic investigations on HBr+(X2Π) ion using CCSD(T) theory in combination with cc-pV5Z basis set

The equilibrium internuclear separations, harmonic frequencies, and potential energy curves (PECs) of the HBr+(X2Π) ion have been investigated using the coupled-cluster singles-doubles-approximate-triples [CCSD(T)] theory in combination with the series of correlation-consistent basis sets in the valence range. The PECs are all fitted to the Murrell-Sorbie function, which are used to reproduce the spectroscopic parameters such as De, ωeχe, αe, Be, and D0. The present D0, De, Re, ωe, ωeχe, αe, and Be obtained at the cc-pV5Z basis set are of 3.9730 eV, 4.1243 eV, 0.14427 nm, 2462.970 cm−1, 47.6979 cm−1, 0.2361 cm−1, and 8.13591 cm−1, respectively, which almost perfectly conform to the measurements. With the potential obtained at the UCCSD(T)/cc-pV5Z level of theory, a total of 22 vibrational states is predicted when the rotational quantum number J is set to equal zero (J = 0) by numerically solving the radial Schrodinger equation of nuclear motion. The complete vibrational levels and their corresponding classical turning points, inertial rotation, and centrifugal distortion constants are determined when J = 0 for the first time, which are in excellent agreement with the available experiments. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2011