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Scientific Reports

Interaction of the hydrogen molecule with the environment: stability of the system and the $${\mathscr{PT}}$$ symmetry breaking

Izabela A. Wrona, M.W. Jarosik, Radosław Szczȩśniak, Kamila A. Szewczyk, M. K. Stala, W. Leoński

Published Jan 14, 2020• 12 citations
Publisher

Abstract

Abstract The stability of the hydrogen molecule interacting with the environment according to the balanced gain and loss energy scheme was studied. We determined the properties of the molecule taking into account all electronic interactions, the parameters of the Hamiltonian being computed by the variational method. The interaction of the hydrogen molecule with the environment was modeled parametrically ( γ ) by means of the non-Hermitian, $${\mathscr{PT}}$$ P T -symmetric Hamiltonian. We showed that the hydrogen molecule is dynamically unstable. Its dissociation time ( T D ) decreases if the γ parameter increases (for γ → 0 we got T D → + ∞). The dynamic instability of the hydrogen molecule is superimposed on the decrease in its static stability as γ increases. Then we can observe the decrease in the dissociation energy value and the existence of the metastable state of the molecule as γ MS reaches 0.659374 Ry. The hydrogen molecule is statically unstable when γ > γ D = 1.024638 Ry. Moreover, we can also observe the $${\mathscr{PT}}$$ P T symmetry breaking effect for the electronic Hamiltonian when $${\gamma }_{{\mathscr{PT}}}$$ γ P T = 0.520873 Ry. This effect does not affect such properties of the hydrogen molecule as: the electronic Hamiltonian parameters, the phonon and the rotational energies, and the values of the electron-phonon coupling constants neither it disturbs the dynamics of the electronic subsystem. However, the number of available quantum states goes down to four.

DOI
10.1038/s41598-019-56849-2

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We study the stability of the hydrogen molecule interacting with the environment according to the balanced gain and loss energy scheme. We determined the properties of the molecule taking into account all electronic interactions, where the parameters of the Hamiltonian have been computed by using the variational method. The interaction of the hydrogen molecule with the environment was modeled parametrically ($γ$) with the help of the non-hermitian operator. We have shown that the hydrogen molecule is dynamically unstable. The dissociation time ($T_{D}$) decreases, if the $γ$ parameter increases (for $γ\rightarrow 0$, we get $T_{D}\rightarrow +\infty$). At the dynamic instability of the hydrogen molecule overlaps its static instability as the coupling constant $γ$ increases. We observed the decrease in the dissociation energy and the existence of the metastable state of the molecule ($γ_{MS}=0.659374$~Ry). The hydrogen molecule is statically unstable for $γ>γ_{D}=1.024638$~Ry. One can also observed the $\mathcal{PT}$ symmetry breaking effect for the electronic Hamiltonian ($γ_{\mathcal {PT}}=0.520873$~Ry). However, it does not affect the properties of the hydrogen molecule, such as: the electronic Hamiltonian parameters, the phonon and rotational energy, and the values of the electron-phonon coupling constants.

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Interaction of the hydrogen molecule with the environment: stability of the system and the PT{\mathscr{PT}}PT symmetry breaking
Abstract The stability of the hydrogen molecule interacting with the environment according to the balanced gain and loss energy scheme was studied. We determined the properties of the molecule taking into account all electronic interactions, the parameters of the Hamiltonian being computed by the variational method. The interaction of the hydrogen molecule with the environment was modeled parametrically ( γ ) by means of the non-Hermitian, PT{\mathscr{PT}}PT P T -symmetric Hamiltonian. We showed that the hydrogen molecule is dynamically unstable. Its dissociation time ( T D ) decreases if the γ parameter increases (for γ → 0 we got T D → + ∞). The dynamic instability of the hydrogen molecule is superimposed on the decrease in its static stability as γ increases. Then we can observe the decrease in the dissociation energy value and the existence of the metastable state of the molecule as γ MS reaches 0.659374 Ry. The hydrogen molecule is statically unstable when γ > γ D = 1.024638 Ry. Moreover, we can also observe the PT{\mathscr{PT}}PT P T symmetry breaking effect for the electronic Hamiltonian when γPT{\gamma }_{{\mathscr{PT}}}γPT​ γ P T = 0.520873 Ry. This effect does not affect such properties of the hydrogen molecule as: the electronic Hamiltonian parameters, the phonon and the rotational energies, and the values of the electron-phonon coupling constants neither it disturbs the dynamics of the electronic subsystem. However, the number of available quantum states goes down to four.
We study the stability of the hydrogen molecule interacting with the environment according to the balanced gain and loss energy scheme. We determined the properties of the molecule taking into account all electronic interactions, where the parameters of the Hamiltonian have been computed by using the variational method. The interaction of the hydrogen molecule with the environment was modeled parametrically (γγγ) with the help of the non-hermitian operator. We have shown that the hydrogen molecule is dynamically unstable. The dissociation time (TDT_{D}TD​) decreases, if the γγγ parameter increases (for γ→0γ\rightarrow 0γ→0, we get TD→+∞T_{D}\rightarrow +\inftyTD​→+∞). At the dynamic instability of the hydrogen molecule overlaps its static instability as the coupling constant γγγ increases. We observed the decrease in the dissociation energy and the existence of the metastable state of the molecule (γMS=0.659374γ_{MS}=0.659374γMS​=0.659374~Ry). The hydrogen molecule is statically unstable for γ>γD=1.024638γ>γ_{D}=1.024638γ>γD​=1.024638~Ry. One can also observed the PT\mathcal{PT}PT symmetry breaking effect for the electronic Hamiltonian (γPT=0.520873γ_{\mathcal {PT}}=0.520873γPT​=0.520873~Ry). However, it does not affect the properties of the hydrogen molecule, such as: the electronic Hamiltonian parameters, the phonon and rotational energy, and the values of the electron-phonon coupling constants.
A theoretical study is conducted on the stability of exotic antiproton-hydrogen molecular anions (p‾H\overline{p}\mathrm{H}p​H), which in the separated atom limit correlate with p‾+H(n=2)\overline{p}+\mathrm{H}(n=2)p​+H(n=2) (nnn being the principal quantum number). The p‾H\overline{p}\mathrm{H}p​H molecule in the electronic ground state is highly unstable due to pair annihilation and autodetachment ($\ensuremath{\rightarrow}\overline{p}p+e$). However, if the p‾H\overline{p}\mathrm{H}p​H molecule is electronically excited, the molecular stability can be drastically improved. Since the excited states of the H\mathrm{H}H atom have accidental degeneracy, the asymptotic form of the Born-Oppenheimer potential curve of p‾+H(n=2)\overline{p}+\mathrm{H}(n=2)p​+H(n=2) behaves as 1/R21/{R}^{2}1/R2 (RRR being the relative distance), which can play a critical role in the bonding and stability of the molecule. It is found that the p‾H\overline{p}\mathrm{H}p​H molecule in the first electronic excited state and in a high rotational state is sufficiently stable and has a lifetime dominated by spontaneous radiative emission ($\ensuremath{\rightarrow}\overline{p}+\mathrm{H}+h\ensuremath{\nu}$). The p‾H\overline{p}\mathrm{H}p​H system is dynamically similar to ${\mathrm{H}}_{2}{\phantom{\rule{0.16em}{0ex}}}^{\ensuremath{-}}$, which had been commonly considered to be unstable due to autodetachment. However, recent measurements of ion-beam sputtering unambiguously verified the existence of long-lived ${\mathrm{H}}_{2}{\phantom{\rule{0.16em}{0ex}}}^{\ensuremath{-}}$ molecules in high rotational states. This suggests that the p‾H\overline{p}\mathrm{H}p​H molecules may actually be created if one employs experimental means such as the chemical sputtering of hydrogen-rich targets with p‾\overline{p}p​ beams.
We report a theoretical study of antihydrogen-hydrogen molecular resonance states consisting of a positron, an antiproton, an electron and a proton. The four particles strongly correlate and show different character from the hydrogen molecule. Because of the non-separability of the positron and electron motions from the antiproton and proton motions, the adiabatic approximation breaks down at the short antiproton-proton distance. Based on a non-adiabatic method, we directly solve the four-body problem and obtain the resonance energies and widths. In order to examine the roles of positron–antiproton and electron–proton correlations as well as positron–electron and antiproton–proton correlations, we introduce two different types of coordinate systems. One is suited for describing an antihydrogen–hydrogen configuration, and the other is a positronium–protonium configuration. The antihydrogen–hydrogen configuration contributes to the existence of the molecular resonance states, and the positronium–protonium configuration makes the resonance states unstable. Mixing of the two configurations results in an antihydrogen-hydrogen molecular resonance state, and the resonance state has an energy of −0.077 9(3) a.u. from the antihydrogen–hydrogen dissociation threshold with its lifetime 16 (2) fs.
We report a theoretical study of antihydrogen-hydrogen molecular resonance states consisting of a positron, an antiproton, an electron and a proton. The four particles strongly correlate and show different character from the hydrogen molecule. Because of the non-separability of the positron and electron motions from the antiproton and proton motions, the adiabatic approximation breaks down at the short antiproton-proton distance. Based on a non-adiabatic method, we directly solve the four-body problem and obtain the resonance energies and widths. In order to examine the roles of positron–antiproton and electron–proton correlations as well as positron–electron and antiproton–proton correlations, we introduce two different types of coordinate systems. One is suited for describing an antihydrogen–hydrogen configuration, and the other is a positronium–protonium configuration. The antihydrogen–hydrogen configuration contributes to the existence of the molecular resonance states, and the positronium–protonium configuration makes the resonance states unstable. Mixing of the two configurations results in an antihydrogen-hydrogen molecular resonance state, and the resonance state has an energy of −0.077 9(3) a.u. from the antihydrogen–hydrogen dissociation threshold with its lifetime 16 (2) fs.
Non-Hermitian systems having parity-time (PT\mathcal{PT}PT) symmetry can undergo a transition, spontaneously breaking the symmetry. Ultracold atomic gases provide an ideal platform to study interaction effects on the transition. We consider a model system of NNN bosons of two components confined in a tight trap. Radio-frequency and laser fields are coupled to the bosons such that the single-particle Non-Hermitian Hamiltonian ${h}_{\mathcal{P}T}=\ensuremath{-}i\mathrm{\ensuremath{\Gamma}}{\ensuremath{\sigma}}_{z}+J{\ensuremath{\sigma}}_{x}$, which has PT\mathcal{PT}PT symmetry, can be simulated in a passive way, namely, in the physical system there is only incurred atom loss but no gain. We show that when interatomic interactions are tuned to maintain the symmetry, the PT\mathcal{PT}PT-symmetry-breaking transition is affected only by the SU(2) variant part of the interatomic interactions. We find that the transition point ${\mathrm{\ensuremath{\Gamma}}}_{\mathrm{tr}}$ decreases as the strength of this interaction part or NNN increases; in the strong strength limit for this interaction part, ${\mathrm{\ensuremath{\Gamma}}}_{\mathrm{tr}}$ scales as the strength to the power of $\ensuremath{-}(N\ensuremath{-}1)$. We also give signatures of the PT\mathcal{PT}PT-symmetric and the symmetry-breaking phases for the interacting bosons in experiment.
The hydrogen spectrum is explored as a sensitive probe of a massive object being in a superposition of two distinct locations. Here I solve the simpler but instructive semiclassical problem of the proton hopping stochastically between two sites but forbidden to be anywhere else, analogous to the superposition over two locations. Although the model is semiclassical, it captures symmetry properties and the structure of the energy spectrum that one would expect in a fully quantum treatment. The hydrogen atom is a well-understood system whose spectroscopy reaches exceptional precision, and its semiclassical description displays symmetry with respect to two fixed locations. I develop a classical model for an electron and a proton, with the proton stochastically hopping between two fixed sites; and describe its classical dynamics mathematically via a one-particle effective Hamiltonian. For this Hamiltonian I identify a set of invariance properties that are characteristic of heteronuclear diatomic molecules and are not as extensive as the SO(4) symmetries present in the traditional models of the hydrogen atom. I connect the classical description of the proton-hopping model with the two-fixed-centre problem with time-dependent charges. I find that there is a lower limit to the hopping distance of the proton below which state-of-the-art measurements cannot resolve the spectral signature predicted; and also an upper limit above which approximation methods are not applicable. I gain meaningful insight into the mechanisms that shape the quantum dynamics of proton tunnelling in hydrogen by analyzing the classical counterpart, leveraging the shared symmetries between quantum and classical analogous systems. My results predict an observable spectral shift in the ground-state energy when the proton is delocalized over two sites separated by a distance large enough to be considered macroscopically distinct. With this research, I support the idea that proton tunnelling inside the hydrogen atom is worth exploring as a probe for a delocalized massive object.