arXiv subjects

J. C. Lopez Vieyra

Publications and source records attributed to J. C. Lopez Vieyra.

At least 19 recordsLinked to original sources

arXiv2023

Ultra-Compact accurate wave functions for He-like iso-electronic sequences and variational calculus. IV. Spin-singlet states $(1s\,ns)$ $n\,{}^1 S$ family of the Helium sequence

As a continuation of Parts I \cite{Part-1:2020}, II \cite{Part-2:2021}, III \cite{Part-3:2022}, where ultra-compact wave functions were constructed for a few low-lying states of He-like and Li-like sequences, the family of spin-singlet $(1s\,ns)$ type excited states $n\,{}^1 S$ of the He-like sequence is studied with an emphasis on the $n=3,4,5$: $3\,{}^1 S, 4\,{}^1 S, 5\,{}^1 S$ states, for nuclear charges $Z \leq 20$. Particular attention is given to finding of critical charges $Z=Z_B$ at which the ultra-compact wave functions lose their square-integrability. For each ${}^1 S$ state an ultra-compact, seven-parametric trial function is constructed, which describes the domain of applicability of the non-relativistic Quantum Mechanics of Coulomb Charges (QMCC) for the total energies (4-5 significant digits (s.d.)) and reproduces 3 decimal digits (d.d.) of the spin-singlet states $n\,{}^1 S$ of He-like ions (in the static approximation with point-like, infinitely heavy nuclei) for $n=1,2,3,\ldots$ and any $Z \leq 20$\,. All energies are well described by second degree polynomials in $Z$ (the Majorana formula). Critical charges $Z=Z_B^{(n)}$, where the ultra-compact trial function for the $n^1 S, n=1,2,3,\ldots$ states loses its square-integrability, are estimated: for all studied states $Z_B^{(n)}$ increases slowly with $n$; it seems they lie in the interval $Z_B(n^1 S) \sim 0.90 - 0.95$, in particular, with $Z_B^{(1)}=Z_B^{(2)}\,=\,0.904$, $Z_B^{(3)}=Z_B^{(4)}\,=\,0.928$, $Z_B^{(5)}\ =\ 0.939$.

physics.atom-ph↗

arXiv2022

Ultra-Compact accurate wave functions for He-like and Li-like iso-electronic sequences and variational calculus. III. Spin-quartet state of the Lithium sequence

As a continuation of Part I, dedicated to the ground state of He-like and Li-like isoelectronic sequences for nuclear charges $Z \leq 20$, and Part II, dedicated to two excited states of He-like sequence, two ultra-compact wave functions in the form of generalized Guevara-Harris-Turbiner functions are constructed for Li-like sequence. They describe accurately the domain of applicability of the Quantum Mechanics of Coulomb Charges (QMCC) for energies (2-3 significant digits (s.d.)) of the spin-quartet state $1^40^+$ of Li-like ions (in static approximation with point-like, infinitely heavy nuclei). Variational parameters are fitted in $Z$ by 2nd degree polynomials. The most accurate ultra-compact function leads to the absolute accuracy $\sim 10^{-3}$\,a.u. for energy, and $\sim 10^{-4}$ for the normalized electron-nuclear cusp parameter for $Z \leq 20$. Critical charge $Z=Z_B$, where the ultra-compact trial function for the $1^40^+$ state looses its square-integrability, is estimated, $Z_B(1^4\,0^+) \sim 1.26 - 1.30$. As a complement to Part I, square integrability for the compact functions constructed for the {\it ground, spin-doublet state} $1^2\,0^+$ of the Li-like sequence is discussed. The critical charge, for which these functions stop to be normalizable, is estimated as $Z_B( 1^2\,0^+) = 1.62 - 1.65$. It implies that at $Z=2$ - the negative helium ion He${}^-$ - both states $1^2\,0^+$ and $1^4\,0^+$ exist as states embedded to continuum.

physics.atom-ph↗

arXiv2021

Ultra-Compact accurate wave functions for He-like and Li-like iso-electronic sequences and variational calculus. II. Spin-singlet (excited) and spin-triplet (lowest) states of the Helium sequence

As a continuation of Part I \cite{Part-1:2020} (Int. Journal of Quantum Chem. 2021; 121: qua.26586), dedicated to the ground state of He-like and Li-like isoelectronic sequences for nuclear charges $Z \leq 20$, a few ultra-compact wave functions in the form of generalized Hylleraas-Kinoshita functions are constructed, which describe the domain of applicability of the Quantum Mechanics of Coulomb Charges (QMCC) for the energies (4-5 significant digits (s.d.)) of two excited states of He-like ions: the spin-singlet (first) excited state $2^1 S$ and the lowest spin-triplet $1^3 S$ state. For both states it provides absolute accuracy for energy $\sim 10^{-3}$\,a.u., exact values for cusp parameters and also for 6 expectation values the relative accuracy $\sim 10^{-2}$. The Bressanini-Reynolds observation about the special form of the nodal surface of the $2^1 S$ state of Helium is confirmed and extended to He-like ions with $Z > 2$. Critical charges $Z=Z_B$, where ultra-compact trial functions lose their square-integrability, are estimated: $Z_B(1^1 S)\approx Z_B(2^1 S)\sim 0.905$ and $Z_B(1^3 S)\sim 0.902$. For both states the Majorana formula - the energy as a second degree polynomial in $Z$ - provides accurately 4-5 significant digits for $Z \leq 20$.

physics.atom-ph↗

arXiv2021

Ultra-Compact accurate wave functions for He-like and Li-like iso-electronic sequences and variational calculus. I. Ground state

Several ultra-compact accurate wave functions in the form of generalized Hylleraas-Kinoshita functions and Guevara-Harris-Turbiner functions, which describe the domain of applicability of the Quantum Mechanics of Coulomb Charges (QMCC), or, equivalently, the Non-Relativistic QED (NRQED), for the ground state energies (4-5 significant digits (s.d.)) of He-like and Li-like iso-electronic sequences in the static approximation with point-like, infinitely heavy nuclei are constructed. It is shown that for both sequences the obtained parameters can be fitted in $Z$ by simple smooth functions: in general, these parameters differ from the ones emerging in variational calculations. For the He-like two-electron sequence the approximate expression for the ground state function, which provides absolute accuracy for the energy $\sim 10^{-3}$\,a.u. and the same relative accuracies $\sim 10^{-2}-10^{-3}$ for both the cusp parameters and the six expectation values, is found. For the Li-like three-electron sequence the most accurate ultra-compact function taken as the variational trial function provides absolute accuracy for energy $\sim 10^{-3}$\,a.u., 2-3 s.d. for the electron-nuclear cusp parameter for $Z \leq 20$ and 3 s.d. for the two expectation values for $Z=3$.

physics.atom-ph↗

arXiv2017

The He${}_2^+$ molecular ion and the He${}^-$ atomic ion in strong magnetic fields

We study the question about existence i.e. stability with respect to dissociation of the spin-quartet, permutation- and reflection-symmetric ${}^4(-3)^+_g$ ($S_z=-3/2, M=-3$) state of the $(ααe e e)$ Coulomb system: the ${\rm He}_2^+$ molecular ion, placed in a magnetic field $0 \le B \le 10000$ a.u. We assume that the $α$-particles are infinitely massive (Born-Oppenheimer approximation of zero order) and adopt the parallel configuration, when the molecular axis and the magnetic field direction coincide, as the optimal configuration. The study of the stability is performed variationally with a physically adequate trial function. To achieve this goal, we explore several Helium-contained compounds in strong magnetic fields, in particular, we study the spin-quartet ground state of ${\rm He}^-$ ion, and the ground (spin-triplet) state of the Helium atom, both for a magnetic field in $100 \leq B\leq 10000$ a.u. The main result is that the ${\rm He}_2^+$ molecular ion in the state ${}^4(-3)^+_g$ is stable towards all possible decay modes for magnetic fields $B \gtrsim 120$ a.u. and with the magnetic field increase the ion becomes more tightly bound and compact with a cigar-type form of electronic cloud. At $B=1000$ a.u., the dissociation energy of ${\rm He}_2^+$ into ${\rm He}^- + α$ is $\sim 701.8$ eV and the dissociation energy for the decay channel to ${\rm He} + α+ e $ is $\sim 729.1$ eV, latter both energies are in the energy window for one of the observed absorption features of the isolated neutron star 1E1207.4-5209.

physics.atom-ph↗

arXiv2016

Three-body quantum Coulomb problem: analytic continuation

The second (unphysical) critical charge in the 3-body quantum Coulomb system of a nucleus of positive charge $Z$ and mass $m_p$, and two electrons, predicted by F~Stillinger has been calculated to be equal to $Z_{B}^{\infty}\ =\ 0.904854$ and $Z_{B}^{m_p}\ =\ 0.905138$ for infinite and finite (proton) mass $m_p$, respectively. It is shown that in both cases, the ground state energy $E(Z)$ (analytically continued beyond the first critical charge $Z_c$, for which the ionization energy vanishes, to $Re Z < Z_c$) has a square-root branch point with exponent 3/2 at $Z=Z_B$ in the complex $Z$-plane. Based on analytic continuation, the second, excited, spin-singlet bound state of negative hydrogen ion H${}^-$ is predicted to be at -0.51554 a.u. (-0.51531 a.u. for the finite proton mass $m_p$). The first critical charge $Z_c$ is found accurately for a finite proton mass $m_p$ in the Lagrange mesh method, $Z^{m_p}_{c}\ =\ 0.911\, 069\, 724\, 655$.

physics.atom-ph↗

arXiv2013

On 1/Z expansion for two-electron systems

The $1/Z$-expansion for the Coulomb system of infinitely massive center of charge Z and two electrons is discussed. Numerical deficiency in Baker et al, {\em Phys. Rev. \bf A41}, 1247 (1990) is indicated which continue to raise doubts in correctness of their calculations of the higher order coefficients in $1/Z$-expansion expressed in Refs.[4-5]. It is shown that a minor modification of a few first coefficients found in Ref.[3] allows to calculate the ground state energies at $Z\ =\ 1, 2,\ldots \ 10$ (as well as at $Z > 10$) with a portion of 15th decimal digit in comparison with highly accurate calculations by C. Schwartz and by Nakashima-Nakatsuji. Ground state energies of two-electron ions $Z=11\ (Na^{9+})$ and $Z=12\ (Mg^{10+})$ are found with 14 decimal digits.

quant-ph↗

arXiv2013

Stable He$^-$ can exist in a strong magnetic field

The existence of bound states of the system $(\al,e,e,e)$ in a magnetic field $B$ is studied using the variational method. It is shown that for $B \gtrsim 0.13\,{\rm a.u.}$ this system gets bound with total energy below the one of the $(\al,e,e)$ system. It manifests the existence of the stable He$^-$ atomic ion. Its ground state is a spin-doublet $^2(-1)^{+}$ at $0.74\, {\rm a.u.} \gtrsim B \gtrsim 0.13\, {\rm a.u.}$ and it becomes a spin-quartet $^4(-3)^{+}$ for larger magnetic fields. For $0.8\, {\rm a.u.} \gtrsim B \gtrsim 0.7\, {\rm a.u.}$ the He$^-$ ion has two (stable) bound states $^2(-1)^{+}$ and $^4(-3)^{+}$.

physics.atom-ph↗

arXiv2013

The ground state of the ${\rm H}_3^+$ molecular ion: physics behind

Five physics mechanisms of interaction leading to the binding of the ${\rm H}_3^+$ molecular ion are identified. They are realized in a form of variational trial functions and their respective total energies are calculated. Each of them provides subsequently the most accurate approximation for the Born-Oppenheimer (BO) ground state energy among (two-three-seven)-parametric trial functions being correspondingly, H$_2$-molecule plus proton (two variational parameters), H$_2^+$-ion plus H-atom (three variational parameters) and generalized Guillemin-Zener (seven variational parameters). These trial functions are chosen following a criterion of physical adequacy. They include the electronic correlation in the exponential form $\sim\exp{(γr_{12})}$, where $γ$ is a variational parameter. Superpositions of two different mechanisms of binding are investigated and a particular one, which is a generalized Guillemin-Zener plus H$_2$-molecule plus proton (ten variational parameters), provides the total energy at the equilibrium of $E=-1.3432$\ a.u. The superposition of three mechanisms: generalized Guillemin-Zener plus (H$_2$ -molecule plus proton) plus (H$_2^+$ -ion plus H) (fourteen parameters) leads to the total energy which deviates from the best known BO energy to $\sim 0.0004$\ a.u., {\it it reproduces two-three significant digits in exact, non-BO total energy}. In general, our variational energy agrees in two-three-four significant digits with the most accurate results available at present as well as major expectation values.

physics.chem-ph↗

arXiv2011

A note about the ground state of the ${\rm H}_3^+$ hydrogen molecular ion

Three simple $7-, (7+3)-, 10-$parametric trial functions for the ${\rm H}_3^+$ molecular ion are presented. Each of them provides subsequently the most accurate approximation for the Born-Oppenheimer ground state energy among several-parametric trial functions. These trial functions are chosen following a criterion of physical adequacy and includes the electronic correlation in the exponential form $\sim\exp{(γr_{12})}$, where $γ$ is a variational parameter. The Born-Oppenheimer energy is found to be $E=-1.340 34, -1.340 73, -1.341 59$\,a.u., respectively, for optimal equilateral triangular configuration of protons with the equilibrium interproton distance $R=1.65$\,a.u. The variational energy agrees in three significant digits (s.d.) with most accurate results available at present as well as for major expectation values.

quant-ph↗

arXiv2009

Charged Hydrogenic, Helium and Helium-Hydrogenic Molecular Chains in a Strong Magnetic Field

A non-relativistic classification of charged molecular hydrogenic, helium and mixed helium-hydrogenic chains with one or two electrons which can exist in a strong magnetic field $B \lesssim 10^{16} $G is given. It is shown that for both $1e-2e$ cases at the strongest studied magnetic fields the longest hydrogenic chain contains at most five protons indicating to the existence of the $\rm{H}_5^{4+}$ and $\rm{H}_5^{3+}$ ions, respectively. In the case of the helium chains the longest chains can exist at the strongest studied magnetic fields with three and four $\al-$particles for $1e-2e$ cases, respectively. For mixed helium-hydrogenic chains the number of heavy centers can reach five for highest magnetic fields studied. In general, for a fixed magnetic field two-electron chains are more bound than one-electron ones.

astro-ph.HE↗

arXiv2008

Sutherland-type Trigonometric Models, Trigonometric Invariants and Multivariate Polynomials

It is conjectured that any trigonometric Olshanetsky-Perelomov Hamiltonian written in Fundamental Trigonometric Invariants (FTI) as coordinates takes an algebraic form and preserves an infinite flag of spaces of polynomials. It is shown that try-and-guess variables which led to the algebraic form of trigonometric Olshanetsky-Perelomov Hamiltonians related to root spaces of the classical $A_N, B_N, C_N, D_N, BC_N$ and exceptional $G_2, F_4$ Lie algebras are FTI. This conjecture is also confirmed for the trigonometric $E_6$ Olshanetsky-Perelomov Hamiltonian whose algebraic form is found with the use of FTI.

math-ph↗

arXiv2005

Hydrogen atom and one-electron molecular systems in a strong magnetic field: are all of them alike?

Easy physics-inspired approximations of the total and binding energies for the ${\rm H}$ atom and for the molecular ions $${\rm H}_2^{(+)} ({\rm ppe}), {\rm H}_3^{(2+)} ({\rm pppe}), ({\rm HeH})^{++} (\al {\rm p e}), {\rm He}_2^{(3+)} (\al \al {\rm e})$$ as well as quadrupole moment for the ${\rm H}$ atom and the equilibrium distances of the molecular ions in strong magnetic fields $> 10^{9}$ G are proposed. The idea of approximation is based on the assumption that the dynamics of the one-electron Coulomb system in a strong magnetic field is governed by the ratio of transverse to longitudinal sizes of the electronic cloud.

physics.atom-ph↗

arXiv2004

A hydrogenic molecular atmosphere of a neutron star

A model of a hydrogenic content of atmosphere of the isolated neutron star 1E1207.4-5209 is proposed. It is based on the assumption that the main component in the atmosphere is the exotic molecular ion $H_3^{2+}$ and that there exists a magnetic field in the range of $(4 \pm 2) \times 10^{14}$ G. Photoionization $H_3^{2+} \rar e + 3p$ and photodissociation $H_3^{2+} \rar H + 2p$ correspond to two absorption features at 0.7 KeV and 1.4 KeV, respectively, discovered by {\it Chandra} observatory (Sanwal et al, 2002). The model predicts one more absorption feature at 80-150 eV corresponding to photodissociation $H_3^{2+} \rar H_2^+ + p$.

astro-ph↗

arXiv2003

The H$_2^+$ ion in a strong magnetic field. Lowest excited states

As a continuation of our previous work ({\it Phys. Rev. A68, 012504 (2003)}) an accurate study of the lowest $1\si_g$ and the low-lying excited $1\si_u$, $2\si_g$, $1π_{u,g}$, $1\de_{g,u}$ electronic states of the molecular ion $H_2^+$ is made. Since the parallel configuration where the molecular axis coincides with the magnetic field direction is optimal, this is the only configuration which is considered. The variational method is applied and the {\it same} trial function is used for different magnetic fields. The magnetic field ranges from $10^9 G$ to $4.414 \times 10^{13} G$ where non-relativistic considerations are justified. Particular attention is paid to the $1\si_u$ state which was studied for an arbitrary inclination. For this state a one-parameter vector potential is used which is then variationally optimized.

astro-ph↗

arXiv2002

$H_2^{+}$ molecular ion in a strong magnetic field: ground state

A detailed quantitative analysis of the system $(ppe)$ placed in magnetic field ranging from $0 - 4.414 \times 10^{13} G$ is presented. The present study is focused on the question of the existence of the molecular ion $H_2^{+}$ in a magnetic field. As a tool, a variational method with an optimization of the form of the vector potential (optimal gauge fixing) is used. It is shown that in the domain of applicability of the non-relativistic approximation the system $(ppe)$ in the Born-Oppenheimer approximation has a well-pronounced minimum in the total energy at a finite interproton distance for $B \lesssim 10^{11} G$, thus manifesting the existence of $H_2^{+}$. For $B \gtrsim 10^{11} G$ and large inclinations (of the molecular axis with respect to the magnetic line) the minimum disappears and hence the molecular ion $H_2^{+}$ does not exist. It is shown that the most stable configuration of $H_2^{+}$ always corresponds to protons situated along the magnetic line. With magnetic field growth the ion $H_2^{+}$ becomes more and more tightly bound and compact, and the electronic distribution evolves from a two-peak to a one-peak pattern. The domain of inclinations where the $H_2^{+}$ ion exists reduces with magnetic field increase and finally becomes $0^o - 25^o$ at $B = 4.414 \times 10^{13} G$. Phase transition type behavior of variational parameters for some interproton distances related to the beginning of the chemical reaction $H_2^{+} \leftrightarrow H + p$ is found.

astro-ph↗

arXiv2002

H_3^{++}$ molecular ion in a strong magnetic field: triangular configuration

The existence of the molecular ion $H_3^{++}$ in a magnetic field in a triangular configuration is revised. Variational method with an optimization of the form of the vector potential (gauge fixing) is used. It is shown that in the range of magnetic fields $10^8 < B < 10^{11} G$ the system $(pppe)$, with protons forming an equilateral triangle perpendicular to magnetic line, has a well-pronounced minimum in the total energy. This configuration is unstable towards decays $H{-atom} + p + p$ and $H_2^+ + p$. Triangular configuration of $H_3^{++}$ complements $H_3^{++}$ in the linear configuration which exists for $B \gtrsim 10^{10} G$.

astro-ph↗

arXiv2000

Bound States from Regge Trajectories in a Scalar Model

The calculation of bound state properties using renormalization group techniques to compute the corresponding Regge trajectories is presented. In particular, we investigate the bound states in different charge sectors of a scalar theory with interaction (phi^dagger phi chi). The resulting bound state spectrum is surprisingly rich. Where possible we compare and contrast with known results of the Bethe-Salpeter equation in the ladder approximation and, in the non-relativistic limit, with the corresponding Schr"odinger equation.

hep-th↗