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Register a new userA regularized inverted perturbation approach method: potential energy curve of the $4^1Sigma^+_u$ state in $Na_2$
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We describe a modification of the inverted perturbation approach method allowing to construct physically sensible potential energy curves for electronic states of diatomic molecules even when some parts of the potential are not adequately characterized by the experimental data. The method is based on a simple regularization procedure, imposing an additional constraint on the constructed potential curve. In the present work it is applied to the double minimum 4$^1Sigma^{+}_{mathrm u}$ state of Na$_2$, observed experimentally by polarization labeling spectroscopy technique.
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The X$^{1}Sigma ^{+}$ state of NaRb was studied by Fourier transform spectroscopy. An accurate potential energy curve was derived from more than 8800 transitions in isotopomers $^{23}$Na$^{85}$Rb and $^{23}$Na$^{87}$Rb. This potential reproduces the experimental observations within their uncertainties of 0.003 rcm to 0.007 rcm. The outer classical turning point of the last observed energy level ($v=76$, $J=27$) lies at $approx 12.4$ AA, leading to a energy of 4.5 rcm below the ground state asymptote.
The 4503 rovibronic term values belonging to the mutually perturbed $A^1Sigma^+_u$ and $b^3Pi_u$ states of Cs$_2$ were extracted from laser induced fluorescence (LIF) $Asim brightarrow X^1Sigma^+_g$ Fourier transform spectra with the 0.01 cm$^{-1}$ uncertainty. The experimental term values of the $A^1Sigma^+_usim b^3Pi_u$ complex covering the rotational levels $Jin [4,395]$ in the excitation energy range $[9655,13630]$ cm$^{-1}$ were involved into coupled-channel (CC) deperturbation analysis. The deperturbation model takes explicitly into account spin-orbit coupling of the $A^1Sigma^+_u(A0^+_u)$ and $b^3Pi^+_{0_u}(b0^+_u)$ states as well as spin-rotational interaction between the $Omega=0$, $1$ and $2$ components of the $b^3Pi^+_{Omega_u}$ state. The emph{ab initio} relativistic calculations on the low-lying electronic states of Cs$_2$ were accomplished in the framework of Fock space relativistic coupled cluster (FSRCC) approach to provide the interatomic potentials of the interacting $A0^+_u$ and $b0^+_u$ states as well as the relevant $Asim b$ spin-orbit coupling function. To validate the present CC deperturbation analysis solely obtained by energy-based data, the $Asim b to X(v^{primeprime}_X)$ LIF intensity distributions were measured and compared with their theoretical counterparts obtained by means of the non-adiabatic vibrational wave functions of the $Asim b$ complex and the FSRCC $Asim b to X$ transition dipole moments calculated by the finite-field method.
We observed the $A^1Sigma^+sim b^3Pito a^3Sigma^+/X^1Sigma^+$ laser-induced fluorescence (LIF) spectra of the RbCs molecule excited from the ground $X^1Sigma^+$ state by the Ti:Sapphire laser. The LIF radiation from the common perturbed levels of the singlet-triplet $Asim b$ complex was recorded by the Fourier-transform (FT) spectrometer with the instrumental resolution of 0.03~cm$^{-1}$. The relative intensity distribution in the rotationally resolved $Asim bto a^3Sigma^+(v_a)/X^1Sigma^+(v_X)$ progressions was measured, and their branching ratio was found to be about of 1$div$5$ times$10$^{-4}$ in the bound region of the $a^3Sigma^+$ and $X^1Sigma^+$ states. The experiment was complemented with the scalar and full relativistic calculations of the $A/b - X/a$ transition dipole moments (TDMs) as functions of internuclear distance. The relative systematic error in the resulting emph{ab initio} TDM functions evaluated for the strong $A - X$ transition was estimated as few percent in the energy region, where the experimental LIF intensities are relevant. The relative spectral sensitivity of the FT registration system, operated with the InGaAs diode detector and CaF beam-splitter, was calibrated in the range $[6~500,12~000]$~cm$^{-1}$ by a comparison of experimental intensities in the long $Asim bto X(v_X)$ LIF progressions of the K$_2$ and KCs molecules with their theoretical counterparts evaluated using the emph{ab initio} $A - X$ TDMs. Both experimental and theoretical transition probabilities can be employed to improve the stimulated Raman adiabatic passage process, $ato Asim b to X$, which is exploited for a laser assembling of ultracold RbCs molecules.
The intercombination $a^3Pi - X^1Sigma^+$ Cameron system of carbon monoxide has been computationally studied in the framework of multi-reference Fock space coupled cluster method with the use of generalized relativistic pseudopotential model for the effective introducing the relativity in all-electron correlation treatment. The extremely weak $a^3Pi_{Omega=0^+,1} - X^1Sigma^+$ transition probabilities and radiative lifetimes of the metastable $a^3Pi$ state were calculated and compared with their previous theoretical and experimental counterparts. The impact of a presumable variation of the fine structure constant $alpha=e^2/hbar c$ on transition strength of the Cameron system has been numerically evaluated as well.
Balancing the privacy-utility tradeoff is a crucial requirement of many practical machine learning systems that deal with sensitive customer data. A popular approach for privacy-preserving text analysis is noise injection, in which text data is first mapped into a continuous embedding space, perturbed by sampling a spherical noise from an appropriate distribution, and then projected back to the discrete vocabulary space. While this allows the perturbation to admit the required metric differential privacy, often the utility of downstream tasks modeled on this perturbed data is low because the spherical noise does not account for the variability in the density around different words in the embedding space. In particular, words in a sparse region are likely unchanged even when the noise scale is large. %Using the global sensitivity of the mechanism can potentially add too much noise to the words in the dense regions of the embedding space, causing a high utility loss, whereas using local sensitivity can leak information through the scale of the noise added. In this paper, we propose a text perturbation mechanism based on a carefully designed regularized variant of the Mahalanobis metric to overcome this problem. For any given noise scale, this metric adds an elliptical noise to account for the covariance structure in the embedding space. This heterogeneity in the noise scale along different directions helps ensure that the words in the sparse region have sufficient likelihood of replacement without sacrificing the overall utility. We provide a text-perturbation algorithm based on this metric and formally prove its privacy guarantees. Additionally, we empirically show that our mechanism improves the privacy statistics to achieve the same level of utility as compared to the state-of-the-art Laplace mechanism.
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