arXiv subjects

Horacio Olivares-Pilón

Publications and source records attributed to Horacio Olivares-Pilón.

8 recordsLinked to original sources

arXiv2024

Towards the "puzzle" of Chromium dimer Cr$_2$: predicting the Born-Oppenheimer rovibrational spectrum

The experimentally-observed non-trivial electronic structure of the Cr$_2$ dimer has made the calculation of its potential energy curve a theoretical challenge in the last decades. By matching the perturbation theory at small internuclear distances $R$ and the multipole expansion at large distances $R$ (supposedly both of asymptotic nature), and by adding a few Rydberg-Klein-Rees (RKR) turning points, extracted from experimental data by Casey-Leopold (1993), the analytic form of the potential energy curve for the ground state $X^1Σ^+$ of the Cr$_2$ dimer is found for the first time for the whole range of internuclear distances $R$. This has the form of a two-point Padé approximant and provides an accuracy of 3-4 decimal digits in 29 experimental vibrational energies. The resulting ground state $X^1Σ^+$ potential curve supports 19694 rovibrational states with a maximal vibrational number $ν_\text{max}=104$ at zero angular momentum and with a maximal angular momentum $L_\text{max}=312$ with energies $> 10^{-4}$ { hartree}, and additionally 218 weakly-bound states (close to the dissociation limit) with energies $< 10^{-4}$ { hartree}.

physics.chem-ph↗

arXiv2023

HCl, DCl and TCl diatomic molecules in their ground state: predicting Born-Oppenheimer rovibrational spectra

The analytic Born-Oppenheimer (B-O) potential curve for the ground state $X^1Σ^+$ of the molecule (H,D,T)Cl is constructed for the whole range of internuclear distances $R \in [0,\infty)$ with an accuracy of 3-5 figures in comparison with the RKR-style potential curve derived from available experimental data on vibrational energies. With an accuracy of 3-4 significant figures in the energies, it is predicted for HCl (DCl, TCl) the 836 (1625, 2366) B-O rovibrational bound states with maximal vibrational number $ν_{max} = 20\, (29, 35)$ and maximal angular momentum $L_{max} = 64\, (90, 109)$ including 24 (46, 63) weakly-bound states (close to the dissociation limit) with energies $\lesssim 10^{-4}$ Hartree. Insufficiency of existing experimental data is indicated and a prediction of the bulk of missing rovibrational states is made for all HCl, DCl, TCl molecules.

physics.chem-ph↗

arXiv2022

Towards the analytic theory of Potential Energy Curves for diatomic molecules. Studying He${}_2^+$ and LiH diatomics as illustration

Following the first principles the elements of the analytic theory of potential curves for diatomic molecules (diatomics) are presented. It is based on matching the perturbation theory at small internuclear distances $R$ and multipole expansion at large distances, modified by instanton-induced trans-series for homonuclear case, with addition of the phenomenologically described equilibrium configuration, if exists. It leads to a new class of (generalized) meromorphic potentials (modified by exponential terms) with difference in degrees of polynomials in numerator and denominator equal to 4 (6) for positively charged (neutral) diatomics. As examples the He$_2^+$ and LiH diatomics in Born-Oppenheimer (adiabatic) approximation are considered. For ${}^{4}$He$_2^+$ (${}^{3}$He$_2^+$) diatomics it is found the approximate analytic expression for the potential energy curves (analytic PEC) $V(R)$ for the ground state $X^2 Σ_u^+$ and the first excited state $A^2 Σ_g^+$. It provides 3-4 s.d. correctly for distances $R \in [1, 10]$\,a.u. with some irregularities for $A^2 Σ_g^+$ PEC at small distances (much smaller than equilibrium distances) probably related to level crossings which may occur there. The analytic PEC for the ground state $X^2 Σ_u^+$ supports 829 (626) rovibrational states with 3-4 s.d. of accuracy in energy, which is only by 1 state less (more) than 830 (625) reported in the literature. In turn, the analytic PEC for the excited state $A^2 Σ_g^+$ supports all 9 reported weakly-bound rovibrational states. Entire rovibrational spectra is found in a single calculation using the code based on the Lagrange mesh method.

physics.atom-ph↗

arXiv2021

HF, DF, TF: Approximating potential curves, calculating rovibrational states

An analytical representation for the potential energy curve for the ground state $X^1Σ^+$ of the hydrogen fluoride molecule (HF) is presented in the frame of the Born-Oppenheimer approximation. The analytical expression for the potential energy curve is based in a two point Padé approximant which correctly reproduces the asymptotic behaviors at small $R\rightarrow 0$ and large $R\rightarrow\infty$ internuclear distances, obtaining not less than 4-5 s.d. when compared with experimental results. The rovibrational spectra of the diatomic molecule HF is calculated by solving the Schrödinger equation for nuclear motion. The ground state $X^1Σ^+$ contains 21 vibrational states ($ν,0$) and 722 rovibrational states ($ν,L$). A slight modification in the differential equation for nuclear motion allows us to obtain the rovibrational spectrum of the ground state of the deuterium fluoride (DF) and tritium fluoride (TF) molecules. Full spectra is presented for this two isotopologues species of HF.

physics.chem-ph↗

arXiv2021

Helium-like ions in $d$-dimensions: analyticity and generalized ground state Majorana solutions

Non-relativistic Helium-like ions $(-e,-e,Ze)$ with static nucleus in a $d-$dimensional space $\mathbb{R}^d$ ($d>1$) are considered. Assuming $r^{-1}$ Coulomb interactions, a 2-parametric correlated Hylleraas-type trial function is used to calculate the ground state energy of the system in the domain $Z \leq 10$. For odd $d=3,5$, the variational energy is given by a rational algebraic function of the variational parameters whilst for even $d=2,4$ it is shown for the first time that it corresponds to a more complicated non-algebraic expression. This twofold analyticity will hold for any $d$. It allows us to construct reasonably accurate approximate solutions for the ground state energy $E_0(Z,d)$ in the form of compact analytical expressions. We call them generalized Majorana solutions. They reproduce the first leading terms in the celebrated $\frac{1}{Z}$ expansion, and serve as generating functions for certain correlation-dependent properties. The (first) critical charge $Z_{\rm c}$ vs $d$ and the Shannon entropy $S_{r}^{(d)}$ vs $Z$ are also calculated within the present variational approach. In the light of these results, for the physically important case $d=3$ a more general 3-parametric correlated Hylleraas-type trial is used to compute the finite mass effects in the Majorana solution for a three-body Coulomb system with arbitrary charges and masses. It admits a straightforward generalization to any $d$ as well. Concrete results for the systems $e^-\,e^-\,e^+$, $H_2^+$ and $H^-$ are indicated explicitly. Our variational analytical results are in excellent agreement with the exact numerical values reported in the literature.

quant-ph↗

arXiv2017

{H$_2^+$, HeH and H$_2$}: approximating potential curves, calculating rovibrational states

Analytic consideration of the Bohr-Oppenheimer (BO) approximation for diatomic molecules is proposed: accurate analytic interpolation for potential curve consistent with its rovibrational spectra is found. It is shown that in the Bohr-Oppenheimer approximation for four lowest electronic states $1sσ_g$ and $2pσ_u$, $2p π_u$ and $3d π_g$ of H$_2^+$, the ground state X$^2Σ^+$ of HeH and the two lowest states $^1Σ^+_g$ and $^3Σ^+_u$ of H$_2$, the potential curves can be analytically interpolated in full range of internuclear distances $R$ with not less than {4-5-6} figures. Approximation based on matching the Taylor-type expansion at small $R$ and a combination of the multipole expansion with one-instanton type contribution at large distances $R$ is given by two-point Padé approximant. The position of minimum, when exists, is predicted within 1$\%$ or better. For the molecular ion H$_2^+$ in the Lagrange mesh method, the spectra of vibrational, rotational and rovibrational states $(ν,L)$ associated with $1sσ_g$ and $2pσ_u$, $2p π_u$ and $3d π_g$ potential curves is calculated. In general, $1sσ_g$ electronic curve contains 420 rovibrational states, which increases up to 423 when we are beyond BO approximation. For the state $2pσ_u$ the total number of rovibrational states (all with $ν=0$) is equal to 3, within or beyond Bohr-Oppenheimer approximation. As for the state $2pπ_u$ within the Bohr-Oppenheimer approximation the total number of the rovibrational bound states is equal to 284. The state $3dπ_g$ is repulsive, no rovibrational state is found. The ground state potential curve of the heteronuclear molecule HeH does not support rovibrational states. Accurate analytical expression for the potential curves of the hydrogen molecule H$_2$ for the states $^1Σ^+_g$ and $^3Σ^+_u$ is presented.

physics.atom-ph↗

arXiv2016

The H$_2^+$ molecular ion: low-lying states

Matching for a wavefunction the WKB expansion at large distances and Taylor expansion at small distances leads to a compact, few-parametric uniform approximation found in {\it J. Phys. B44, 101002 (2011)}. The ten low-lying eigenstates of H$_2^+$ of the quantum numbers $(n,m,\La,\pm)$\, with $n=m=0$ at $\La=0,1,2$, with $n=1$, $m=0$ and $n=0$, $m=1$ at $\La=0$ of both parities are explored for all interproton distances $R$. For all these states this approximation provides the relative accuracy $\lesssim 10^{-5}$ (not less than 5 s.d.) locally, for any real coordinate $x$ in eigenfunctions, when for total energy $E(R)$ it gives 10-11 s.d. for $R \in [0,50]$~a.u. Corrections to the approximation are evaluated in the specially-designed, convergent perturbation theory. Separation constants are found with not less than 8 s.d. The oscillator strength for the electric dipole transitions $E1$ is calculated with not less than 6~s.d. A dramatic dip in the $E1$ oscillator strength $f_{1s\si_g-3p\si_u}$ at $R \sim R_{eq}$ is observed. The magnetic dipole and electric quadrupole transitions are calculated for the first time with not less than 6~s.d. in oscillator strength. For two lowest states $(0,0,0,\pm)$ (or, equivalently, $1s\si_g$ and $2p\si_u$ states) the potential curves are checked and confirmed in the Lagrange mesh method within 12~s.d. Based on them the Energy Gap between $1s\si_g$ and $2p\si_u$ potential curves is approximated with modified Pade $R e^{-R} [Pade(8/7)](R)$ with not less than 4-5 figures at $R \in [0, 40]$\,a.u. Sum of potential curves $E_{1s\si_g} + E_{2p\si_u}$ is approximated by Pade $1/R [Pade(5/8)](R)$ in $R \in [0, 40]$\,a.u. with not less than {3-4} figures.

quant-ph↗

arXiv2011

The H$_2^+$ molecular ion: a solution

Combining the WKB expansion at large distances and Perturbation Theory at small distances it is constructed a compact uniform approximation for eigenfunctions. For lowest states $1s\si_{g}$ and $2p\si_{u}$ this approximation provides the relative accuracy $\lesssim 10^{-5}$ (5 s.d.) for any real $x$ in eigenfunctions and for total energy $E(R)$ it gives 10-11 s.d. for internuclear distances $R \in [0,50]$. Corrections to proposed approximations are evaluated. Separation constants and the oscillator strength for the transition $1s\si_{g} \rar 2p\si_{u}$ are calculated and compared with existing data.

quant-ph↗