arXiv subjects
Publications and source records attributed to Krzysztof Pachucki.
A general approach to evaluation of two-centre two-electron exponential integrals with arbitrary parameters is presented. The results for the Born-Oppenheimer potential for various excited states of molecular hydrogen with Koł os-Wolniewicz functions are obtained with precision exceeding previous values by about 3 orders of magnitude.
The extremely precise extraction of the proton radius by Pohl et al. from the measured energy difference between the 2P and 2S states of muonic hydrogen disagrees significantly with that extracted from electronic hydrogen or elastic electron-proton scattering. This is the proton radius puzzle. The origins of the puzzle and the reasons for believing it to be very significant are explained. Various possible solutions of the puzzle are identified, and future work needed to resolve the puzzle is discussed.
The fundamental ground tone vibration of H2, HD, and D2 is determined to an accuracy of 2 x 10^-4 cm^-1 from Doppler-free laser spectroscopy in the collisionless environment of a molecular beam. This rotationless vibrational splitting is derived from the combination difference between electronic excitation from the X^1Sg+, v=0 and v=1 levels to a common EF^1g+, v=0 level. Agreement within $1σ$ between the experimental result and a full ab initio calculation provides a stringent test of quantum electrodynamics in a chemically-bound system.
High-precision results are reported for the energy levels of $2{^1S}$ and $2{^1P}$ states of the beryllium atom. Calculations are performed using fully correlated Gaussian basis sets and taking into account the relativistic, quantum electrodynamics (QED), and finite nuclear mass effects. Theoretical predictions for the ionization potential of the beryllium ground state $75\,192.699(7) \icm$ and the $2{^1P} \rightarrow 2{^1S}$ transition energy $42\,565.441(11) \icm$ are compared to the known but less accurate experimental values. The accuracy of the four-electron computations approaches that achieved for the three-electron atoms, which enables determination of the nuclear charge radii and precision tests of QED.
Various properties of the general two-center two-electron integral over the explicitly correlated exponential function are analyzed for the potential use in high precision calculations for diatomic molecules. A compact one dimensional integral representation is found, which is suited for the numerical evaluation. Together with recurrence relations, it makes possible the calculation of the two-center two-electron integral with arbitrary powers of electron distances. Alternative approach via the Taylor series in the internuclear distance is also investigated. Although numerically slower, it can be used in cases when recurrences lose stability. Separate analysis is devoted to molecular integrals with integer powers of interelectronic distances $r_{12}$ and the vanishing corresponding nonlinear parameter. Several methods of their evaluation are proposed.
We demonstrate high accuracy calculations for the HeH$^+$ molecule using newly developed analytic formulas for two-center two-electron integrals with exponential functions. The Born-Oppenheimer potential for the ground electronic $^1Σ^+$ state is obtained in the range of 0.1 -- 60 au with precision of about $10^{-12}$ au. As an example at the equilibrium distance $r=1.463\,283$ au the Born-Oppenheimer potential amounts to $-2.978\,708\,310\,771(1)$. Obtained results lay the ground for theoretical predictions in HeH$^+$ with spectroscopic precision.
The muonic hydrogen experiment measuring the $2P-2S$ transition energy [R. Pohl {\em et al.}, Nature {\bf 466}, 213 (2010)] is significantly discrepant with theoretical predictions based on quantum electrodynamics. A possible approach to resolve this conundrum is to compare experimental values with theoretical predictions in another system, muonic deuterium $μ$D. The only correction which might be questioned in $μ$D is that due to the deuteron polarizability. We investigate this effect in detail and observe cancellation with the elastic contribution. The total value obtained for the deuteron structure correction in the $2P-2S$ transition is 1.680(16) meV.
We present recent progress in the calculation of the helium fine-structure splitting of the 2^3P_J states, based on the quantum electrodynamic theory. Apart from the complete evaluation of mα^7 and m^2/Mα^6 corrections, we have performed extensive tests by comparison with all experimental results for light helium-like ions and with the known large nuclear charge asymptotics of individual corrections. Our theoretical predictions are still limited by the unknown mα^8 term, which is conservatively estimated to be 1.7 kHz. However, comparison with the latest experimental result for the 2^3P_0 - 2^3P_2 transition [M. Smiciklas and T. Shiner, Phys. Rev. Lett. 105, 123001 (2010)] suggests that the higher-order contribution is in fact much smaller than the theoretical estimate. This means that the spectroscopic determination of αcan be significantly improved if another measurement of the 2^3P_0 - 2^3P_2 transition in helium-like Li^+ or Be^{2+} ion is performed.
Calculational results are presented for the fine-structure splitting of the 2^3P state of helium and helium-like ions with the nuclear charge Z up to 10. Theoretical predictions are in agreement with the latest experimental results for the helium fine-structure intervals as well as with the most of the experimental data available for light helium-like ions. Comparing the theoretical value of the 2^3P_0-2^3P_1 interval in helium with the experimental result [T. Zelevinsky et al. Phys. Rev. Lett. 95, 203001 (2005)], we determine the value of the fine-structure constant αwith an accuracy of 31 parts per billion.
The Born-Oppenheimer potential for the $^1Σ_g^+$ state of H$_2$ is obtained in the range of 0.1 -- 20 au, using analytic formulas and recursion relations for two-center two-electron integrals with exponential functions. For small distances James-Coolidge basis is used, while for large distances the Heitler-London functions with arbitrary polynomial in electron variables. In the whole range of internuclear distance about $10^{-15}$ precision is achieved, as an example at the equilibrium distance $r=1.4011$ au the Born-Oppenheimer potential amounts to $-1.174\,475\,931\,400\,216\,7(3)$. Results for the exchange energy verify the formula of Herring and Flicker [Phys. Rev. {\bf 134}, A362 (1964)] for the large internuclear distance asymptotics. The presented analytic approach to Slater integrals opens a window for the high precision calculations in an arbitrary diatomic molecule.
We demonstrate the applicability of four-body exponentially correlated functions for the accurate calculations of relativistic effects in lithium-like atoms and present results for matrix elements of various operators which involve negative powers of interparticle distances.
Rigorous quantum electrodynamical calculation is presented for energy levels of the 1^1S, 2^1S, 2^3S, 2^1P_1, and 2^3P_{0,1,2} states of helium-like ions with the nuclear charge Z=3...12. The calculational approach accounts for all relativistic, quantum electrodynamical, and recoil effects up to orders mα^6 and m^2/Mα^5, thus advancing the previously reported theory of light helium-like ions by one order in α.
We report a calculation of the fine structure splitting in light helium-like atoms, which accounts for all quantum electrodynamical effects up to order α^5 Ry. For the helium atom, we resolve the previously reported disagreement between theory and experiment and determine the fine structure constant with an accuracy of 31 ppb. The calculational results are extensively checked by comparison with the experimental data for different nuclear charges and by evaluation of the hydrogenic limit of individual corrections.
In order to interpret precise measurements of molecular properties the finite nuclear mass corrections to the Born-Oppenheimer approximation have to be accounted for. It is shown that they can be obtained systematically in the perturbative approach. The formulae for the leading corrections to the relativistic contribution to energy, the transition electric dipole moment, the electric polarizability, and the magnetic shielding constant are presented.
We compare the explicitly correlated Hylleraas and exponential basis sets in the evaluations of ground state of Li and Be$^+$. Calculations with Hylleraas functions are numerically stable and can be performed with the large number of basis functions. Our results for ground state energies $-7.478 060 323 910 10(32)$, $-14.324 763 176 790 43(22)$ of Li and Be$^+$ correspondingly, are the most accurate to date. When small basis set is considered, explicitly correlated exponential functions are much more effective. With only 128 functions we obtained about $10^{-9}$ relative accuracy, but the severe numerical instabilities make this basis costly in the evaluation.
We present an efficient approach to evaluate two-center two-electron integrals with exponential functions and with an arbitrary polynomial in electron-nucleus and electron-electron distances. We show that the master integral with the single negative power of all distances can be obtained from the second order differential equation in $r$, the distance between nuclei. For particular values of nonlinear parameters corresponding to the James-Coolidge basis, we find a fully analytic expression. For integrals with arbitrary powers of all distances, we construct recursion relations which starts from the master integral. The presented approach opens a window for the high precision calculations of relativistic effects in diatomic molecules.
The magnetic shielding $σ$ of $^3$He is studied. The complete relativistic corrections of order $O(α^2)$, leading QED corrections of order $O(α^3 \lnα)$, and finite nuclear mass effects of order $O(m/m_{\rm N})$ are calculated with high numerical precision. The resulting theoretical predictions for $σ= 59.967~43(10)\cdot 10^{-6}$ are the most accurate to date among all elements and support the use of $^3$He as a NMR standard.
Accurate calculations of the fine and hyperfine splitting of the 2P state in Li and Be$^+$ isotopes using the explicitly correlated Hylleraas basis set are presented. Theoretical predictions including the mixing of $P_{1/2}$ and $P_{3/2}$ states, relativistic and quantum electrodynamics effects on hyperfine interactions, are compared with experimental values. It is concluded that precise spectroscopic determination of the nuclear magnetic moments requires elimination of nuclear structure effects by combining measurements for two different states.