Ⅲ-3. Radiochemistry & Nuclear Chemistry

RIKEN Accel. Prog. Rep. 53

Relativistic density functional theory with the finite-light-speed
correction for Coulomb interaction†
T. Naito,∗1,∗2 R. Akashi,∗1 H. Z. Liang,∗1,∗2 and S. Tsuneyuki∗1,∗3
The periodic table of elements is determined by theelectronic conﬁguration and represents the periodicity of fundamental atomic properties. However, thepositions of super-heavy elements (SHEs) in the periodic table are tentative because their chemical properties have not been established. Owing to the shorthalf-lives of SHEs, experimental measurements of theirchemical properties are diﬃcult. Thus, ﬁrst-principles
numerical simulations are complementary tools to such
experimental measurements.
The ﬁrst-principles calculation for SHEs requires
proper treatment of the relativistic eﬀect. The relativistic eﬀect can modify the Coulomb interaction
between the charged particles, which is signiﬁcant inSHEs. The non-relativistic form is of the order O (Zα),
whereasdominant contribution is of the or( the second)2der O (Zα) , which is called the Breit correction.1)
Here, α ≃ 1/137 is the ﬁne-structure constant. TheBreit correction is usually called the relativistic eﬀect
or ﬁnite-light-speed eﬀect. The lowest order of relativistic eﬀects
incorporatedusing the Dirac equation
()
( 2)
2
is also O (Zα)∼ O 1/c . Once the relativisticeﬀects incorporated by the Dirac equation are considered, the Breit correction between two electrons should
also be considered to keep consistency. Hereafter, we
refer to both the Breit correction and eﬀects incorporated using the Dirac equation as “relativistic eﬀect”
for simplicity.
So far, the electronic structure() calculation with therelativistic eﬀects in O 1/c2 has been performed
based on wave-function theory. In contrast, density
functional theory (DFT)2,3) with the consistent(inclu-)
sion of the relativistic eﬀects up to the order O 1/c2
is unprecedented. Although the accuracy of DFT islower than that of wave-function theory, DFT withthe relativistic eﬀects is highly desired, because DFT
is applicable to larger systems such as molecules andcoordination complex ions, the experimental synthesesof which are ongoing, and even solids.
In this paper, we formulate the Hartree-exchangecorrelation functional) for electron systems with the(correction of O 1/c2 . As a benchmark calculation,
the all-electron calculation of atoms is performed.
We calculated the energy of the lawrencium atomwith two electronic conﬁgurations, where one valence
electron occupies the 6d orbital ([Rn] 5f 14 6d1 7s2 ) or
7p orbital ([Rn] 5f 14 7s2 7p1 ). For comparison, that of†

∗1
∗2
∗3

Condensed from the article in arXiv: 1911.10670
Department of Physics, The University of Tokyo
RIKEN Nishina Center
Institute for Solid State Physics, The University of Tokyo- 161 -Table 1. Total energies for lutetium and lawrencium
atoms. All values are in the Hartree atomic unit.

LuLr

Outermost
5d1
6p1
6d1
7p1

Non-rel−13848.19912−13848.12376−33551.48205−33551.38274

SRel-CB−14527.46807−14527.42700
Unbound−37331.31054

the lutetium atom is also calculated, where one valence
electron occupies the 5d orbital ([Xe] 4f 14 5d1 6s2 ) or
6p orbital ([Xe] 4f 14 6s2 6p1 ). The energies calculatedwith these conﬁgurations in the non-relativistic (Nonrel) and the newly-developed relativistic (SRel-CB)
schemes are listed in Table 1. The smaller values forthe respective approximations in each atom are written
in bold. In the lutetium atom, the outermost valence
electron prefers to occupy the d orbital rather than thep orbital with all schemes. In contrast, in the lawrencium atom, the electron prefers to occupy the p orbital
rather than the d orbital with the SRel-CB scheme,
whereas it still prefers the d orbital in the Non-rel
scheme, as in the lutetium case. The occupation of thep orbital could be the origin of the anomalous behaviorof lawrencium.4) This result is consistent with previous calculations using wave-function theory.5,6) Notethat we obtained the consistent results with far lesscomputational cost than those previous calculations.
It should also be noted that the p-block elements aredeﬁned as those in which the outer-most electrons occupy the p orbitals.7) Lawrencium, thus, belongs to
the p block, rather than d according to the deﬁnition.
It seems, however, more appropriate to regard thatthe concept of “block” is ambiguous for the heavy andsuper-heavy elements, because the electronic conﬁguration of lawrencium is almost the same as that oflutetium, and the only diﬀerence is in the outermost
electron. Therefore, reconsideration of the classiﬁcation appropriate for SHEs is mandatory.
References
1) G. Breit, Phys. Rev. 34, 553 (1929). ibid 36, 383 (1930).
2) P. Hohenberg, W. Kohn, Phys. Rev. 136, B864 (1964).
3) W. Kohn, L. J. Sham, Phys. Rev. 140, A1133 (1965).
4) T. K. Sato et al ., Nature 520, 209 (2015).
5) E. Eliav, U. Kaldor, Y. Ishikawa, Phys. Rev. A 52, 291(1995).
6) X. Cao, M. Dolg, H. Stoll, J. Chem. Phys. 118, 487(2003).
7) F. A. Cotton, G. Wilkinson, P. L. Gauss, Basic Inorganic Chemistry (John Wiley & Sons, 1995).