Regularized perturbative series for the ionization potential of atomic ions
We study -electron atoms with nuclear charge . It is well known that, in the cationic () high- region, the atom behaves as a weakly interacting system. The anionic () regime, on the other hand, is characterized by an instability threshold at , below which the atom spontaneously emits an electron. We construct a regularized perturbative series (RPS) for the ionization potential of ions in an isoelectronic sequence that exactly reproduces both, the large and the near limits. The large- expansion coefficients are analytically computed from perturbation theory, whereas the slope of the energy curve at is computed from a kind of zero-range forces theory that uses as input the electron affinity and the covalent radius of the neutral atom with electrons. Relativistic effects, at the level of first-order perturbation theory, are considered. Our RPS formula is to be used in order to check the consistency of the ionization potential values for atomic ions contained in the NIST database.
pacs:32.30.-r, 32.10.Hq, 31.15.-p
Since the foundation of Quantum Mechanics, a huge amount of data on energy levels, linewidths and other properties of atoms have been recorded. Very often, such compilations are still waiting for a qualitative analysis, based on simple models.
In recent articles pra1 (); pra2 (), on the basis of the scaling suggested by Thomas-Fermi theory TF (), we have demonstrated universality in the ionization potentials and the correlation energies of atomic ions.
In the present paper, we construct an analytical expression for the ionization energy of atomic ions, which is to be used in order to detect problematic values NISTrev1 (); NISTrev2 () in the numbers provided by the NIST database.NIST ()
Our expression is a regularized perturbative series (RPS), previously employed in other contexts renormS (). We use perturbation theory in Z-expansion () in order to compute the first two coefficients of the energy series in the large- region. Additionaly, we require our RPS to reproduce the value of the ionization potential at (i.e. the electron affinity) and the slope of the curve at this point. The latter is computed from a kind of zero-range forces theory that uses as input the electron affinity and the covalent radius of the neutral atom with electrons. ourModPhysLett () The RPS continuosly interpolates between the and large- limits for a given isoelectronic sequence.
Ii Atoms near the anionic instability threshold
It is well known that, for large , the attraction of the electrons by the nucleus is stronger than electron-electron repulsion. On the contrary, for the neutral atom both contributions are more or less balanced, and in the anionic domain this balance may even be broken at a given , where the atom spontaneously autoionizes.
First-principle calculations Hogreve () and some extrapolations Kais () indicate that is indeed very close to , excluding the possibility of doubly charged negative ions. A recent result by Gridnev,Gridnev () on the other hand, rigorously states that the wave function is normalized at threshold. If we combine this result with perturbation theory, we get that the binding energy exhibits a linear dependence on near .
In a previous paper,ourModPhysLett () we compute the slope of the curve not at , but at . At this value of the outermost electron weakly interacts with the neutral core and the interaction is short-ranged. It can be shown that conditions are fulfilled for the application of zero-range forces theory. zero-range () The slope of the curve may be computed from:ourModPhysLett ()
where and is the electron affinity of the neutral system with electrons. Atomic units are to be used everywhere in the paper. is related to the size of the core, containing nuclear charge and electrons. For computational purposes, we use the covalent radius of the electron atom as an estimation of .
Iii The large- limit
In the following, we shall construct the large- series for the atomic energy. This is, in fact, a formal limit. In nature, atomic ions become unstable for large , showing a threshold for electron-positron pair production at .pair-production () Performing the scaling in the non-relativistic Hamiltonian, we get:
Notice that the expression inside brackets has a one-particle contribution (kinetic energy plus nuclear attraction) and the two-particle repulsion between electrons. The latter is of order . At large values of , the atom can be described as a system of non-interacting electrons in the central Coulomb field of the nucleus. The energy in this leading approximation is:
Next, we shall include electron repulsions in first order perturbation theory. The energy is written as:
where is the Slater determinant made of hydrogenic functions. Corrections are explicitly given by:
Note that the sums runs over the occupied orbitals and in the Slater determinant, and that and denote, respectively, direct and exchange two-electron Coulomb integrals involving orbitals and . Their explicit expression can be found in Ref. [tesis, ].
Once we constructed a series for the total energy:
one can find also a similar expression for the ionization potential, defined as . We get:
In these equations, is the principal quantum number of the last electronic shell, and – the last occupied orbital.
To end up this section, we shall stress that, in the large- limit:
Eq. (10) comes from analytical estimations, whereas Eq. (11) comes from a fit to the numerical results. These functional forms are consistent with the dependence , suggested by Thomas-Fermi theory.pra1 ()
iii.1 Relativistic corrections
At large , a relativistic approach is required. In the leading approximation, one should solve the Dirac equation for an electron in a central Coulomb field. We choose a simpler approach in which relativistic corrections are computed in first order degenerate perturbation theory,
Both and are eigenstates of the non-relativistic one-electron Hamiltonian. Greek indices label states for which the total angular momentum (orbital plus spin) is a good quantum number. The relativistic perturbation, , includes the kinetic (), spin-orbit (), and the Darwin term () terms.Bransden () The sum in Eq. (13) runs over the space orthogonal to .
where is the last occupied state.
The coefficient must be changed also in accordance with (12). The final expression for reads:
where is the total angular momentum quantum number of the last occupied state and is the fine structure constant.
A summary of matrix elements is given in the Appendix. Details on the derivation of can be found in Ref. [Bransden, ].
Iv Regularizing the perturbative series
Once the region near the anionic threshold and the large- limit are described, one may try to find an interpolation between them. To this end, we use a regularization of the perturbative series, Eq. (7).renormS () The next two formal terms of the series
coming, in principle, from higher order perturbative corrections, are instead used to force that, at , and . That is:
We get a linear system of two equations and two variables ( and ), yielding:
V Detecting problematic points in the NIST database
We would like to show how Eq. (16), with the coefficients and given in (14) and (15), respectively, and and coming from (18), can be used to detect inconsistencies in the NIST data for the ionization potential of atomic ions. We study four isoelectronic systems (Fig. 1-3) in quality of examples. An exhaustive revision will be published elsewhere.NISTrev1 (); NISTrev2 ()
In Figs. 1-3, we plot the non-relativistic and the relativistic RPS, along with the NIST data, for these sequences. The lower panels show the difference NIST - RPS(relativ). The maximum relative errors are near 1% for , 2% for , and rises to around 10% for . In the first case, , our approximate treatment of relativity does not reproduce the correct asymptotics at large . In the rest of the systems, both asymptotics ( and large ) are correct, and the maximum errors are reached at intermediate , as it is common with interpolants renormS ().
We claim that, in spite of the fact that our relativistic RPS does not have spectroscopic precision, abrupt changes in the difference NIST - RPS may be a sign of inconsistency. Indeed, abrupt changes in are related to rearrangements of the electronic spectrum. In the interval between rearrangements or for large enough , the occupancy of orbitals is fixed, and should be smooth. The difference with our smooth RPS interpolant should also be a smooth function of .
Ne-like systems are closed shell, and do not exhibit rearrangements at any . In order to make evident inconsistent points in the NIST data, we construct an average NIST-RPS curve by means of a 5-points running average. In Fig. 1, the point is so far from the average curve, for example, that it should be corrected. We can even give an estimate of the needed correction by measuring the distance to the average curve, which in this case is -0.034 a.u.
In Ni-like ions, the 3d electronic configuration is reached already for . Thus, we expect a smooth dependence from this point on. The point, for example, is deviated from the average curve in 0.141 a.u., and its error bar is only 0.115 a.u. wide. Cu-like ions, on the other hand, show a 3d4s configuration at any . Nd-like ions experience rearrangements at various values, but in the neighbourhood of , a problematic point, the difference should be smooth. Thus, we can undoubtedly distinguish this point.
The main result of the present paper is an analytical expression for the ionization energies of -electron ions () based on first-principles. This formula is exact in the and large regions. In the transition region, the error is only a few percents of the total ionization potential.
Acknowledgements.The authors are grateful to the Caribbean Network for Quantum Mechanics, Particles and Fields (ICTP) for support. G.G. also acknowledge financial support from the European Community’s FP7 through the Marie Curie ITN-INDEX.
Appendix A Explicit matrix elements for relativistic corrections
We can write the relativistically corrected Hamiltonian as:
where the non-perturbed Hamiltonian is given by:
and the perturbation is expressed as a sum of terms:
where is the fine structure constant, and is the orbital angular momentum quantum number.
We consider first-order perturbative corrections due to . The relevant matrix element is , where is an eigenstate of , , , , , and an eigenstate of , , , , . Only states such that enter Eq. (13), thus we restrict ourselves to this case. We have:
where is the total angular momentum quantum number, and , are eigenstates of the Hydrogen () Hamiltonian. and are also scaled energies.
Notice that states can be labelled by (principal quantum number), , and (total angular momentum projection on ), whereas for we need , , (orbital angular momentum projection on ), and (spin angular momentum projection on ). We can expand in terms of by means of the Clebsch-Gordan coefficients:
are non-vanishing only for , and .
The matrix elements entering Eq. (22) are explicitly written as:
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