Ground State Of Hydrogen Atom Research Articles

Strong error bounds for Trotter and strang-splittings and their implications for quantum chemistry

Efficient error estimates for the Trotter product formula are central in quantum computing, mathematical physics, and numerical simulations. However, the Trotter error's dependency on the input state and its application to unbounded operators remains unclear. Here, we present a general theory for error estimation, including higher-order product formulas, with explicit input state dependency. Our approach overcomes two limitations of the existing operator-norm estimates in the literature. First, previous bounds are too pessimistic as they quantify the worst-case scenario. Second, previous bounds become trivial for unbounded operators and cannot be applied to a wide class of Trotter scenarios, including atomic and molecular Hamiltonians. Our method enables analytical treatment of Trotter errors in chemistry simulations, illustrated through a case study on the hydrogen atom. Our findings reveal the following: (i) for states with fat-tailed energy distribution, such as low-angular-momentum states of the hydrogen atom, the Trotter error scales worse than expected (sublinearly) in the number of Trotter steps; (ii) certain states do not admit an advantage in the scaling from higher-order Trotterization and, thus, the higher-order Trotter hierarchy breaks down for these states, including the hydrogen atom's ground state; (iii) the scaling of higher-order Trotter bounds might depend on the order of the Hamiltonians in the Trotter product for states with fat-tailed energy distribution. Physically, the enlarged Trotter error is caused by the atom's ionization due to the Trotter dynamics. Mathematically, we find that certain domain conditions are not satisfied by some states so higher moments of the potential and kinetic energies diverge. Our analytical error analysis agrees with numerical simulations, indicating that we can estimate the state-dependent Trotter error scaling genuinely. Published by the American Physical Society 2024

State-selective electron capture in Ar16++H(1s) collisions for charge-exchange recombination spectroscopy

Abstract State-selective electron capture in collisions of Ar16+ ions with ground-state hydrogen atoms has been modeled using the two-center wave-packet convergent close-coupling approach. The partially stripped He-like projectile ion is represented using a model potential. Experimental measurements are not available for this collision system and to date, only the classical trajectory Monte Carlo (CTMC) method has been applied to calculate cross sections. The calculated total electron-capture cross section (TECS) is in good agreement with the previous CTMC results at the low energies but slightly larger at higher energies. This is likely because, in this work, we account for capture into highly excited states, which contribute significantly to the TECS at the intermediate energies. The n-resolved electron-capture cross sections have also been presented for capture into states with n = 6 − 19 , where n is the final-state principal quantum number. The most important of these are the cross sections for capture into the n = 14 − 17 states, which are used in charge-exchange recombination spectroscopy techniques. For these cross sections, a significant difference is observed between the present and previously published data. The cross sections differ by an order of magnitude in the 10−60 keV u−1 energy range. The agreement between the calculations is observed at the energies above 70 keV u−1. The n ℓ -resolved electron-capture cross sections have also been presented at 15, 60, 100 and 200 keV u−1 projectile energies, where ℓ is the final-state angular momentum quantum number.

A comparative study of transition oscillator strengths and static polarizabilities of the hydrogen atom confined in Gaussian potential

Abstract The multipole (dipole, quadrupole, and octopole) photon-absorption transition oscillator strengths for the ground state of hydrogen atom confined in Gaussian potential are investigated for a great variety of potential depths and confining radii. It is interestingly found that at fixed potential depth the gradual increase of confining radius shows first destructive and then constructive effect on the multipole oscillator strengths. Such an effect can be understood from the overlap between the initial and final states. Multipole polarizabilities of the system are obtained through the sum-over-states formalism where the contributions from both the bound and continuum spectra of the system are included. Although the separate bound and continuum contributions can not be determined accurately, due to the long-range nature of the Coulomb potential introduced by the nucleus, their summations can be obtained to reasonably good accuracy, leading to fast convergence of numerical calculations of multipole polarizabilities. The present results are compared with previous calculations available in the literature. Although good agreement is observed for the dipole polarizability, significant differences exist in the quadrupole polarizability and orders-of-magnitude differences appear in the octopole polarizability. The possible reason for such large differences is analyzed by comparing the sum rule of corresponding oscillator strengths.

Rayleigh and Raman scattering cross-sections and phase matrices of the ground-state hydrogen atom, and their astrophysical implications

ABSTRACT We present explicit expressions for Rayleigh and Raman scattering cross-sections and phase matrices of the ground 1s state hydrogen atom based on the Kramers–Heisenberg–Waller dispersion formula. The Rayleigh scattering leaves the hydrogen atom in the ground-state while the Raman scattering leaves the hydrogen atom in either ns (n ≥ 2; s-branch) or nd (n ≥ 3; d-branch) excited state, and the Raman scattering converts incident ultraviolet (UV) photons around the Lyman resonance lines into optical-infrared (IR) photons. We show that this Raman wavelength conversion of incident flat UV continuum in dense hydrogen gas with a column density of NH > 1021 cm−2 can produce broad emission features centred at Balmer, Paschen, and higher level lines, which would mimic Doppler-broadened hydrogen lines with the velocity width of ≳1000 km s−1 that could be misinterpreted as signatures of active galactic nuclei, supernovae, or fast stellar winds. We show that the phase matrix of the Rayleigh and Raman s-branch scatterings is identical to that of the Thomson scattering while the Raman d-branch scattering is more isotropic, thus the Paschen and higher level Raman features are depolarized compared to the Balmer features due to the flux contribution from the Raman d-branch. We argue that observations of the line widths, line flux ratios, and linear polarization of multiple optical/IR hydrogen lines are crucial to discriminate between the Raman-scattered broad emission features and Doppler-broadened emission lines.

Flavor Symmetry of Hydrogen Atoms Potentially Affecting the Proton Radius Deduced from the Electron-Hydrogen Scattering

Precise knowledge of such fundamental quantity as the proton charge radius rp is extremely important both for the quantum chromodynamics (for quark-gluon structure) and for atomic physics (for atomic hydrogen spectroscopy). Yet the ambiguity in measuring rp persists for over a dozen of years by now—from the time when in 2010 the muonic hydrogen spectroscopy experiment yielded rp ≈ 0.84 fm in contrast to the form factor experiment by the Mainz group that produced rp ≈ 0.88 fm. Important was that this difference corresponded to about seven standard deviations and therefore was inexplicable. In the intervening dozen of years, more experiments of various kinds were performed in this regard. Nevertheless, the controversy remains, which is why several different types of new experiments are being prepared for measuring rp. In one of our previous papers, we pointed out the factor that was never taken into account by the corresponding research community: the flavor symmetry of electronic hydrogen atoms, whose existence was confirmed by four kinds of atomic or molecular experiments and also evidenced by two kinds of astrophysical observations. Specifically, in that paper there was discussed the possible presence of the second flavor of muonic hydrogen atoms (in the corresponding experimental gas) and its effect on the shift of the ground state of muonic hydrogen atoms due to the proton finite size. In the present paper we analyze the effect of the flavor symmetry of electronic hydrogen atoms on the corresponding elastic scattering cross-section and on the proton charge radius rp deduced from the cross-section. As an example, we use our analytical results for reconciling two distinct values of rp obtained in different elastic scattering experiments: 0.88 fm and 0.84 fm (which is by about 4.5% smaller than 0.88 fm). We show that if the ratio of the second flavor of hydrogen atoms to the usual hydrogen atoms in the experimental gas would be about 0.3, then the extraction of rp from the corresponding cross-section would yield by about 4.5% smaller value of rp compared to its true value. We also derive the corresponding general formulas that can be used for interpreting the future electronic and muonic experiments.

Splitting of the atomic hydrogen 21-cm line induced by gravitational waves from a supermassive black hole binary

We investigate the characteristic of the hyperfine splitting of a ground-state hydrogen atom subjected to a field of a gravitational wave. In our consideration, hydrogen atoms are in the source’s local wave zone. The interaction between the atoms and the gravitational wave induces the shifts of some energy levels in the hyperfine structure, computed by the first-order perturbation theory. The splitting of energy levels leads to more than one hyperfine transition. Our analysis suggests that a gravitational wave emitted by supermassive black hole binaries could be strong enough to induce a significant effect on the hyperfine splitting. This radio-wave spectrum provides the gravitational-wave signature and thus offers a way of identifying the merger of supermassive black hole binaries.

The effects of Heisenberg constraint on the classical cross sections in proton hydrogen collision

The interaction between a proton and a ground state hydrogen atom is studied using a standard three-body classical trajectory Monte Carlo (CTMC) and a quasi-classical trajectory Monte Carlo (QCTMC) model where the quantum feature of the collision system is mimicked using the model potential in the Hamiltonian as was proposed by Kirschbaum and Wilets (1980 Phys. Rev. A 21 834). The influence of the choice of the model potential parameters (α, ξ) on the initial radial and momentum distribution of the electron are analyzed and optimized. We found that although these distributions may not be as close to the quantum results as the distribution of standard CTMC results, we can find the combination of the (α, ξ) where the calculated cross sections are closer to the experimental data and closer to the results obtained quantum mechanically. We show that the choice of 3 < α < 5 is reasonable. To validate our observation, we present cross sections for ionization, excitation, charge exchange (CX), and state selective CX to the projectile bound state. Calculations are carried out in the projectile energy range between 10 and 1000 keV amu−1.

Photoelectron momentum distribution of hydrogen atoms in a superintense ultrashort high-frequency pulse

We use a numerically solved time-dependent Schrödinger equation for calculating the photoelectron momentum distribution of ground-state hydrogen atoms in the presence of superintense ultrashort high-frequency pulses. It is demonstrated that the dynamic interference effect within a superintense XUV laser beam has the ability to significantly alter the photoelectron momentum distribution. In our work, a clearly visible dynamic interference pattern is observed when hydrogen atoms are exposed to a superintense circularly polarized laser pulse with a photon energy of ℏω = 53.605 eV, which has previously been found for linearly polarized pulses or the weakly bounded model H− system for circularly polarized pulses. Angular-distorted interference arises for linear superintense XUV pulses of similar intensity. The significant differences in photoelectron momentum distributions that have been seen by linearly and circularly polarized XUV pulses are caused by the Coulomb rescattering phenomenon.

Nl-Selective Classical Charge-Exchange Cross Sections in Be4+ and Ground State Hydrogen Atom Collisions

Charge-exchange cross sections in Be4+ + H(1s) collisions are calculated using the three-body classical trajectory Monte Carlo method (CTMC) and the quasi-classical trajectory Monte Carlo method of Kirschbaum and Wilets (QCTMC) for impact energies between 10 keV/amu and 300 keV/amu. We present charge-exchange cross sections in the projectile n = 2 and nl = 2s, 2p states. Our results are compared with the previous quantum-mechanical approaches. We found that the QCTMC model is a powerful classical model to describe the state-selective charge-exchange cross sections at lower impact energies and the QCTMC results are in good agreement with previous observations.

Experiments on the Electron Impact Excitation of the 2s and 2p States of Hydrogen Atoms Confirm the Presence of Their Second Flavor as the Candidate for Dark Matter

For the excitation of the n = 2 states of hydrogen atoms due to electron impact, we compared the experimental and theoretical ratios of the cross-sections σ2s/σ2p. We found this theoretical ratio to be systematically higher than the corresponding experimental ratio by about 20%—far beyond the experimental error margins. We suggest that this discrepancy can be explained by the presence of the Second Flavor of Hydrogen Atoms (SFHA) in the experimental hydrogen gas. The explanation is based on the fact that, in the experiments, the cross-section σ2s was determined by using the quenching technique—by applying an electric field that mixed the 2s and 2p states, followed by the emission of the Lyman-alpha line from the 2p state. However, the SFHA only had the s-states, so the quenching technique would not count the excitation of the SFHA in the 2s state and, thus, lead to the underestimation of the cross-section σ2s. We estimates the share of the SFHA in the experimental hydrogen gas required for eliminating the above discrepancy and found this share to be about the same as the share of the usual hydrogen atoms. Thus, our results constitute the third proof from atomic experiments that the SFHA does exist, the first proof being related to the experimental distribution of the linear momentum in the ground state of hydrogen atoms, and the second proof being related to the experimental cross-section of charge exchange between hydrogen atoms and low-energy protons.

State-selective electron capture in collisions of fully stripped neon ions with ground-state hydrogen

Electron capture and ionisation in bare neon ion collisions with ground-state atomic hydrogen are modelled over the energy range from 1 to 2000 keV/u using the two-center semiclassical wave-packet convergent close-coupling method. The calculated total electron-capture cross section agrees very well with the molecular and atomic orbital close-coupling calculations at low and intermediate energies. Our results slightly overestimate the experimental results by Meyer et al [1985 Phys. Rev. A 32 3310], but underestimate the measurements by Panov et al [1983 Phys. Scr. T3 124] available only below 10 keV/u. At higher energies, where there are no measurements, the results also agree very well with the classical trajectory Monte-Carlo results. Partial n and nl-resolved electron-capture cross sections, important for fusion plasma diagnostics, have also been calculated for final states up to n = 10, where n and l are the final state principal and angular momentum quantum numbers, respectively. The results are generally in good agreement with the atomic calculations. However, due to the finer energy grid used, we are able to detect pronounced oscillations in the state-selective cross sections for n ⩾ 8 at energies below 10 keV/u. Our results for the total ionisation cross section are overall in good agreement with the latest classical trajectory Monte-Carlo results.

State-selective electron capture cross sections in collision between fully striped ions and ground state hydrogen atom—Classical treatment of the collision

We present a state-selective electron capture cross sections database from the ground state hydrogen atom. A standard three-body classical trajectory Monte Carlo (CTMC) and quasi-classical trajectory Monte Carlo (QCTMC) models were employed for impact energies between 10 and 200 keV/amu. The projectile ions are H+, He2+, Li3+, Be4+, B5+, C6+, N7+, and O8+. The cross sections are tabulated for each value of the final quantum numbers n, l, m, summed over m for the given n and l and over l and m for the given n. The maximum quantum number nmax is set to 4 for H+, He2+ and Li3+, as well as to 5 for Be4+, B5+, C6+, N7+ and O8+, respectively.

Distinctive Features of Charge Exchange Involving the Second Flavor of Hydrogen Atoms—The Candidates for Dark Matter

The second flavor of hydrogen atoms (SFHA) refers to the kind of hydrogen atoms that have only the states of the zero orbital angular momentum (the S-states), both in the discrete and continuous spectra. They were first discovered theoretically in one of my earlier papers, where a proof of their existence was also provided by analyzing atomic experiments concerning the high-energy tail of the linear momentum distribution in the ground state of hydrogen atoms. From a theoretical point of view, the discovery was based on the standard Dirac equation for hydrogen atoms without changing the existing physical laws. Recently, the existence of the SFHA was seemingly also confirmed by two types of astrophysical observations: the allowance for the SFHA explained the puzzling results concerning both the anomalous absorption of the redshifted 21 cm spectral line from the early Universe, and the observations by the Dark Energy Survey (DES) team where it was found that the distribution of dark matter in the Universe is noticeably smoother than predictions employing Einstein’s relativity. In the present review, we exhibit results from two recent papers where attention was brought to a visible difference in the cross-sections of the resonant charge exchange for collisions of the SFHA with incoming protons, compared to collisions of the usual hydrogen atoms with incoming protons. It was shown that, after taking into account the SFHA, there is a better agreement with the corresponding experimental cross-section. Coupled with the previous evidence of the existence of the SFHA, deduced from the analysis of the other kind of atomic experiments, and evidenced by two different kinds of astrophysical observations, this strengthens the standing of the SFHA as the most probable candidate for all or a part of dark matter.

Signatures of stabilization in the angle-resolved photoemission by an ultrashort intense XUV laser pulse

We explore features of stabilization in the angle-resolved spectra of photoelectrons when ground-state atomic hydrogen is exposed to an intense ($\ensuremath{\sim}{10}^{18} \mathrm{W}/{\mathrm{cm}}^{2}$) linearly polarized ultrashort XUV pulse. In this regime, atomic stabilization can be traced to the temporal destructive interference between wave packets released by the intense field at different instants of time. By a comparison between the numerical solution of the time-dependent Schr\"odinger equation and a semianalytical model, we can identify the observed oscillations in the momentum distribution and the suppression of photoemission into the laser polarization direction as a direct result of stabilization. We also explore the effect of nondipole corrections on the angular distribution.

Excitations of the nS States of Atomic Hydrogen by Electron Impact, Excitation Rate Coefficients, and Phase Shifts: Comparison with Positron Impact Excitation

The excitation cross-sections of the nS states of atomic hydrogen, n = 2 to 6, by electron impact on the ground state of atomic hydrogen were calculated using the variational polarized-orbital method at various incident electron energies in the range 10 to 122 eV. Converged excitation cross-sections were obtained using sixteen partial waves (L = 0 to 15). Excitation cross-sections to 2S state, calculated earlier, were calculated at higher energies than before. Results obtained using the hybrid theory (variational polarized orbital method) are compared to those obtained using other approaches such as the Born–Oppenheimer, close-coupling, R-matrix, and complex-exterior scaling methods using only the spherical symmetric wave functions. Phase shifts and elastic cross-sections are given at various energies and angular momenta. Excitation rate coefficients were calculated at various electron temperatures, which are required for plasma diagnostics in solar and astrophysics to infer plasma parameters. Excitation cross-sections are compared with those obtained by positron impact excitation.

Evidence of the dominant production mechanism of ammonia in a hydrogen plasma with parts per million of nitrogen

Absolute ground state atomic hydrogen densities were measured, by the utilization of two-photon absorption laser induced fluorescence, in a low-pressure electron cyclotron resonance plasma as a function of nitrogen admixtures—0 to 5000 ppm. At nitrogen admixtures of 1500 ppm and higher, the spectral distribution of the fluorescence changes from a single Gaussian to a double Gaussian distribution; this is due to a separate, nascent contribution arising from the photolysis of an ammonia molecule. At nitrogen admixtures of 5000 ppm, the nascent contribution becomes the dominant contribution at all investigated pressures. Thermal loading experiments were conducted by heating the chamber walls to different temperatures; this showed a decrease in the nascent contributions with increasing temperature. This is explained by considering how the temperature influences recombination coefficients, and from which, it can be stated that the Langmuir–Hinshelwood recombination mechanism is dominant over the Eley–Rideal mechanism.

Analysis of Experimental Cross-Sections of Charge Exchange between Hydrogen Atoms and Protons Yields More Evidence of the Existence of the Second Flavor of Hydrogen Atoms

Measurements of cross-sections of charge exchange between hydrogen atoms and low energy protons (down to the energy ~10 eV) revealed a noticeable discrepancy with previous theories. The experimental cross-sections were systematically slightly higher—beyond the error margins—than the theoretical predictions. In the present paper, we study whether this discrepancy can be eliminated or at least reduced by using the Second Flavor of Hydrogen Atoms (SFHA) in calculations. We show that for the SFHA, the corresponding cross-section is noticeably larger than for the usual hydrogen atoms. We demonstrate that the allowance for the SFHA does bring the theoretical cross-sections in a noticeably better agreement with the corresponding experiments within the experimental error margins. This seems to constitute yet another evidence from atomic experiments that the SFHA is present within the mixture of hydrogen atoms. In combination with the first corresponding piece of evidence from the analysis of atomic experiments (concerning the distribution of the linear momentum in the ground state of hydrogen atoms), as well as with the astrophysical evidence from two different kinds of observations (the anomalous absorption of the redshifted 21 cm radio line from the early universe and the smoother distribution of dark matter than that predicted by the standard cosmology), the results of the present paper reinforce the status of the SFHA as the candidate for dark matter, or at least for a part of it.

State selective classical electron capture cross sections in Be4+ \u2009+\u2009H(1s) collisions with mimicking quantum effect

We present state-selective electron capture cross sections in collision between Be4+ and ground state hydrogen atom. The n- and nl-selective electron capture cross sections are calculated by a three-body classical trajectory Monte Carlo method (CTMC) and by a classical simulation schema mimicking quantum features of the collision system. The quantum behavior is taken into account with the correction term in the Hamiltonian as was proposed by Kirschbaum and Wilets (Phys Rev A 21:834, 1980). Calculations are carried out in the projectile energy range of 1–1000 keV/amu. We found that our model for Be4+ + H(1s) system remarkably improves the obtained state-selective electron capture cross sections, especially at lower projectile energies. Our results are very close and are in good agreement with the previously obtained quantum–mechanical results. Moreover, our model with simplicity can time efficiently carry out simulations where maybe the quantum mechanical ones become complicated, therefore, our model should be an alternative way to calculate accurate cross sections and maybe can replace the quantum–mechanical methods.

Chirp-dependent ionization of hydrogen atoms in the presence of super-intense laser pulses* *Project supported by the National Natural Science Foundation of China (Grant Nos. 11774131 and 91850114).

We perform a theoretical study on dynamic interference in single photon ionization of ground state hydrogen atoms in the presence of a super-intense ultra-fast chirped laser pulse of different chirp types (equal-power and equal-FWHM laser pulses) by numerically solving the time-dependent Schrödinger equation in one dimension. We investigate the influences of peak intensity and chirp parameters on the instantaneous ionization rate and photoelectron yield, respectively. We also compare the photoelectron energy spectra for the ionization by the laser pulses with different chirp types. We find that the difference between the instantaneous ionization rates for the ionization of hydrogen atom driven by two different chirped laser pulses is originated from the difference in variation of vector potentials with time.

Combined crossed molecular beams and computational study on the N(2D) + HCCCN(X1Σ+) reaction and implications for extra-terrestrial environments

The reaction of the nitrogen atom (N) in its first electronically excited state (2D) with cyanoacetylene (HC3N) has been investigated under single-collision conditions by using the crossed molecular beam method with mass spectrometric detection at a collision energy of 31 kJ mol−1. With the support of electronic structure calculations, we found that this reaction proceeds via the barrierless addition of the N(2D) atom to the carbon–carbon triple bond of HC3N, followed by the formation of a cyclic intermediate adduct HC(N)CCN, which dissociates to C(N)CCN + H products or isomerises to a more stable intermediate HNCCCN by H-migration and ring-opening processes. The long-lived HNCCCN complex produces the linear 3Σg – ground state dicyanocarbene (NCCCN) radical plus atomic hydrogen through a barrierless unimolecular dissociation accompanied by a negligible competitive channel forming the NCCCN radical (1A1) with a bent C 2v structure plus H. The main product of this neutral-neutral reaction is the 3NCCCN radical that could be a potential precursor to form other nitriles (C2N2, C3N) or more complex organic species in planetary atmospheres, such as that of Titan and Pluto, in cometary comas, and in UV irradiated interstellar environments.