Nuclear magnetic shielding constants of Dirac one-electron atoms
in some low-lying discrete energy eigenstates
We present tabulated data for the nuclear magnetic shielding constants () of the Dirac one-electron atoms with a pointlike, motionless and spinless nucleus of charge . Utilizing the exact general analytical formula for derived by us [P. Stefańska, Phys. Rev. A. 94 (2016) 012508/1-15], valid for an arbitrary discrete energy eigenstate, we have computed the numerical values of the magnetic shielding factors for the ground state and for the first and the second set of excited states, i.e.: 2s, 2p, 2p, 3s, 3p, 3p, 3d, and 3d, of the relativistic hydrogenic ions with the nuclear charge numbers from the range . The comparisons of our results with the numerical values reported by other authors for some atomic states are also presented.
Keywords: Hydrogenlike atom, Shielding constant, Magnetic field, Screening factor, Nuclear magnetic resonance
Published as: At. Data Nucl. Data Tables 120 (2018) 352–372
*[1ex] doi: 10.1016/j.adt.2017.05.005
When an atom or a molecule is placed in an external magnetic field, then electrons circling around their nuclei will begin to interact with the perturbing field. In this way, an additional magnetic field, oriented opposite to the external field, is generated in the system. Consequently, the strength of the magnetic field “effectively sensed” by the nucleus decreases. This phenomenon is called nuclear magnetic shielding. The changes that occur in the location of the nucleus can be fully characterized by the magnetic shielding constant (or the nuclear screening factor). This physical quantity – of course, depending on the electron density – is directly related to the chemical shift, by which the position of the signal in the NMR (Nuclear Magnetic Resonance) spectrum is determined. This makes the magnetic shielding constant () one of the most important atomic parameters currently used in chemistry and medicine.
There are many experimental results for the magnetic screening constants for many-electron atoms and molecules. But relativistic theoretical investigations of that quantity for such complex systems are quite complicated – very often they come down to performing numerical calculations, which almost always require access to specialized software. The numerical values for for some closed-shell atoms and ions were reported, among others, in Refs. [1, 2, 3]. However, for the simplest systems, like one-electron atoms, the purely analytical relativistic calculations for the magnetic shielding constant (and also for another properties of the atom) are possible, and this was suggested already in 1969 by Hegstrom in Ref. . One of such a technique which allows one to derive analytically the formula for the shielding factor () is the one based on the perturbation theory combined with the Sturmian expansion of the first-order generalized Dirac–Coulomb Green function, proposed in Ref. . Using the method presented in that article, some time ago we have found the closed-form expression for for the ground state of relativistic hydrogenlike atom , in agreement with corresponding formulas derived earlier by other authors in completely different ways. (For other applications of the technique proposed by Szmytkowski in Ref.  to some electromagnetic properties of the ground state of Dirac one-electron atoms, see Refs. [7, 8, 9, 10, 11, 12, 13, 14, 15].).
Quite recently, we have shown that the usefulness of the aforementioned method goes beyond the study of the atomic ground state. In Refs. [17, 18, 19, 16, 20] we have considered an arbitrary discrete energy eigenstate of relativistic hydrogenlike atom, in order to obtain analytical expressions (and further – the numerical values) for some properties illustrating the influence of the external electromagnetic perturbations on the atom. In particular, in Ref.  one can find a detailed derivation of the formula for the magnetic shielding constant of Dirac one-electron atom being in an arbitrary energy eigenstate. The quantum state of the electron is generally characterized by the set of quantum numbers , in which denotes the radial quantum number, the Dirac quantum number is an integer different form zero, whereas is the magnetic quantum number. The final closed-form expression for the shielding factor for such a state we have arrived at in Ref.  reads as follows:
In Ref.  we have proved analytically that the above result is valid for an arbitrary atomic state. However, in the aforementioned article, there are no tables with the numerical values of the relativistic shielding factors of hydrogenic ions. This kind of data may be useful – among others – to those who deal with the spectroscopic methods, for example, the NMR technique. This fact, as well as the lack of sufficiently large data sets in the literature, prompted us to carry out numerical calculations for the magnetic shielding constant for one-electron atoms ions with the nuclear charge numbers from the whole range . Our results will be discussed briefly in the next section.
Discussion of results
Numerical results presented in this work has been computed with the help of the exact analytical formula given in Eqs. (1)–(4). In Table 1 we have included the values of the magnetic shielding factors for the ground state (1s) of Dirac one-electron atoms. Tables 2–5 contain the results for states belonging to the first set of excited state, i.e.: 2s, 2p, and 2p. The second excited atomic states (i.e.: 3s, 3p, 3p, 3d, and 3d) are presented in Tables 6–14. For the excited states, all possible values of the magnetic quantum number were considered. The Reader should observe that for the states with , there is a restriction for the nuclear charge number, i.e.: (see Tables 1–3, 6, and 7); a detailed explanation of this limitation can be found just follow Eq. (3.17) in Ref. .
The value of the inverse of the fine-structure constant we have used during creating Tables 1–14 was , and was taken from the newest CODATA 2014 report on Recommended Values of the Fundamental Physical Constants . However, in order to make the most accurate comparison of our exact results with the numerical results reported earlier by other authors for some atomic states [22, 23], we have performed two additional tables with the values of obtained using (from CODATA 1986) .
States with zero radial quantum number have been considered in Table 15, where we have confronted our results with the values published by Moore in Ref. . Actually, to be able to compare these numbers, we had to add the corresponding numerical components provided by her for a given atomic state. The resulting sums of the three ingredients for each state (the lower entries in Table 15) appear to be in a pretty good agreement with our exact results (the upper entries therein).
In contrast to the values of the screening constants presented in Tables 1–15 (where they were given in units of ), the numbers included in Table 16 should be multiplied only by the commonly used factor of . We have rescaled in this way a part of the data from Tables 1 and 3–5 (i.e. for states 1s, 2p, and 2p), because we wanted to compare them with the corresponding numerical values provided by Pyper and Zhang . The agreement between their results and ours is also very good, and this has been shown in Table 16. The value of the magnetic shielding factor for the ground state of relativistic hydrogen atom () can be found – among others – in the paper by Feiock and Johnson . Their result, which is (not shown in Table 16), also agrees with our result for this case.
I am indebted to Professor A. Rutkowski for his suggestion to publish present results. I also thank Professor R. Szmytkowski for technical assistance in carrying out the calculations.
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