Are atoms spherical?
RRIKEN-QHP-480, RIKEN-iTHEMS-Report-20, KUNS-2835
Are atoms spherical?
Tomoya Naito ( 内藤 智 也 ),
1, 2, ∗ Shimpei Endo ( 遠 藤 晋 平 ),
3, 4, † Kouichi Hagino ( 萩 野 浩 一 ), ‡ and Yusuke Tanimura ( 谷 村 雄 介 )
3, 6, § Department of Physics, Graduate School of Science,The University of Tokyo, Tokyo 113-0033, Japan RIKEN Nishina Center, Wako 351-0198, Japan Department of Physics, Tohoku University, Sendai 980-8578, Japan Frontier Research Institute for Interdisciplinary Science, Tohoku University, Sendai 980-8578, Japan Department of Physics, Kyoto University, Kyoto 606-8502, Japan Graduate Program on Physics for the Universe, Tohoku University, Sendai 980-8578, Japan (Dated: September 15, 2020)Atomic nuclei can be spontaneously deformed into non-spherical shapes as many-nucleon systems.We discuss to what extent a similar deformation takes place in many-electron systems. To this end,we employ several many-body methods, such as the unrestricted Hartree-Fock method, post-Hartree-Fock methods, and the density functional theory, to compute the electron distribution in atoms. Weshow that the electron distribution of open-shell atoms is deformed due solely to the single-particlevalence orbitals, while the core part remains spherical. This is in contrast to atomic nuclei, whichcan be deformed collectively. We qualitatively discuss the origin for this apparent difference betweenatoms and nuclei by estimating the energy change due to deformation.
I. INTRODUCTION
Atoms and atomic nuclei share common features asquantum many-body systems of interacting fermions [1].In atoms, the inter-particle interaction is the repulsiveCoulomb interaction between electrons, and there existsthe spherical external Coulomb potential due to the nu-cleus. On the other hand, in atomic nuclei, the inter-particle interaction is the attractive nuclear force be-tween two or more nucleons, with no external potential.Despite the differences in the fundamental interactions,many similar properties have been observed in both sys-tems, such as the shell structure and the magic numbers[2, 3].One of the most important properties of atomic nu-clei is the collective nuclear deformation, which is ev-idenced by the characteristic rotational spectra [4, 5].Here, atomic nuclei are deformed as a whole, resulting ina strong enhancement in the transition probability fromthe first excited state to the ground state. This is inmarked contrast to trivial deformation seen in nuclei witha valence nucleon outside a doubly-magic nucleus [6–8].Since doubly-magic nuclei are, in general, spherical, suchdeformation is entirely due to the valence nucleon, thus ofsingle-particle nature, unless the valence nucleon inducesa strong polarization effect of the core nucleus [9]. Itis known that the deformation is much more significantfor the collective deformation. Atomic nuclei may alsobe deformed due to the α -cluster formation [10–19], eventhough it is out of the scope of this paper. See Refs. [4, 5]for more details. ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] A natural question then arises: Can atoms be deformedcollectively as in atomic nuclei? Notice that calculationsof the atomic structure have been usually carried out byassuming spherical symmetry [20–22]. Likewise, spheri-cal symmetry has been usually assumed to construct thepseudopotential for valence electrons in calculations ofmolecules and solids [23–26]. It is thus widely believedthat atoms are rather spherical, in stark contrast to de-formed atomic nuclei. What is the physical origin of thisdifference? To what extent can electron distributions inatoms be deformed collectively?To answer these questions, in this paper we calculateelectron distributions of atoms and their deformation pa-rameters using various numerical methods without as-suming spherical symmetry. Namely, we use and com-pare the unrestricted Hartree-Fock methods, the densityfunctional theory, and several post-Hartree-Fock meth-ods. We shall show that atoms are not deformed collec-tively, and the electron distribution may, at most, be de-formed only by a few valence electrons. We also presenta qualitative model to explain the result. We shall ar-gue that the collective deformation may take place onlyif the inter-particle interaction has an attractive featureas in nuclear systems. We, thus, show that the differencebetween atoms and atomic nuclei on their deformabilityoriginates from the different nature of the inter-particleinteractions.The paper is organized as follows. In Sec. II, we firstdetail the deformation of many-body systems, and intro-duce a parameter β to characterize the deformation. Wethen carry out numerical calculations for isolated atomsand discuss their deformations in Sec. III. In Sec. IV, wequalitatively compare between atoms and nuclei usinga model to estimate a change in the energy, and discussthe origin for collective deformation. We then summarizethe paper in Sec. V. In Appendix A, we discuss the defor-mation of neutron drops, that is, many-neutron systems a r X i v : . [ phy s i c s . a t o m - ph ] S e p trapped in a harmonic oscillator potential, and exem-plify that nuclear systems can be deformed collectively.Appendix B summarizes the numerical data for atomicdeformation calculated in the main part of the paper. II. COLLECTIVE DEFORMATION OFMANY-FERMION SYSTEMSA. Deformation parameters
Before we present the results of our calculations, letus briefly summarize the basic concepts of collective de-formation. Atomic nuclei in the rare earth region, suchas
Sm and
Er, as well as those in the actinide re-gion, such as
U, often exhibit characteristic spectra, inwhich the energy of the state with the total angular mo-mentum I ( I = 0, 2, 4, . . . ) is proportional to I ( I + 1).These spectra are interpreted as that those nuclei aredeformed in the ground state and show the rotationalspectra given by E rot = I ( I + 1) (cid:126) I , (1)where I is the moment of inertia for the rotational mo-tion. An important fact is that a large number of nucle-ons in those nuclei are involved in the deformation, andthus such deformation is referred to as collective deforma-tion. Indeed, the electromagnetic transition probabilityfrom e.g., the I = 2 state to the I = 0 state (the groundstate) is significantly enhanced due to the collectivity ofdeformation [5].The collective deformation is characterized by a non- zero value of the quadrupole moment Q ij defined by Q ij = (cid:90) (cid:0) r i r j − δ ij r (cid:1) ρ ( r ) d r ( i , j = x , y , z ) , (2)where ρ ( r ) is the density distribution. Note that thereexist several normalizations for the quadrupole momenttensor Q . Here, we take the normalization often usedin nuclear physics [5], which is sometimes referred to as“traceless quadrupole moment” in the context of quan-tum chemistry.The symmetric matrix Q ij can be diagonalized to de-fine the cartesian axes, x , y , and z in the intrinsic frame.With the eigenvalues of Q ij , the deformation parameters β k can be defined as β k = (cid:114) π Q k N tot (cid:104) r (cid:105) ( k = x , y , z ) , (3)where Q k is an eigenvalue of Q ij and (cid:10) r (cid:11) is the mean-square radius given by (cid:10) r (cid:11) = (cid:82) r ρ ( r ) d r (cid:82) ρ ( r ) d r = 1 N tot (cid:90) r ρ ( r ) d r , (4)with N tot being the total number of particle in the sys-tem. For atoms, N tot is the total number of electrons(the atomic number), Z , whereas for atomic nuclei it isthe total number of nucleons (the mass number), A .The deformation parameter β j is closely related to theangle-dependent radius of the system, R ( θ, φ ). The ra-dius can be expanded in multipoles with spherical har-monics Y lm as [5] R ( θ, φ ) = R (cid:34) ∞ (cid:88) l =2 l (cid:88) m = − l a lm Y ∗ lm ( θ, φ ) (cid:35) . (5)The monopole part ( l = 0) in the expansion is absorbedin the definition of R , while the dipole part ( l = 1) neednot be considered since, to the first order, it merely shiftsthe center of mass of the whole system without changingits shape. If one considers only the quadrupole deforma-tion with l = 2, after transforming to the intrinsic frame,Eq. (5) is reduced to R ( θ, φ ) = R (cid:20) β cos γY ( θ, φ ) + 1 √ β sin γY ( θ, φ ) + 1 √ β sin γY − ( θ, φ ) (cid:21) = R (cid:34) (cid:114) π β (cid:110) cos γ (cid:0) θ − (cid:1) + √ γ sin θ cos 2 φ (cid:111)(cid:35) , (6)with a (cid:48) = β cos γ , a (cid:48) = a (cid:48) − = √ β sin γ , and a (cid:48) = a (cid:48) − = 0, where the primes denote the quantities in theintrinsic frame. Here, β > γ represent the degrees of elongation and triaxiality [5]. Due to the D symmetry, γ can be restricted to γ ∈ [0 , π/ γ = π/ β > γ = 0 with β <
0. Hence, if
Spherical(β=0)Oblate(β<0) Prolate(β>0) z FIG. 1. A schematic picture of the collective deformation. one only considers the axially symmetric case with bothpositive and negative β , Eq. (6) can be simplified furtheras R ( θ, φ ) = R (cid:34) (cid:114) π β (cid:0) θ − (cid:1)(cid:35) , (7)with axially symmetric shape about the z axis. In thiscase, the radii in the x , y , and z directions are given by R x = R y = R ( π/ , φ ) = R (cid:32) − (cid:114) π β (cid:33) , (8) R z = R (0 , φ ) = R (cid:32) (cid:114) π β (cid:33) , (9)respectively. Thus, the density distribution is sphericalfor β = 0, while for β > β < β , the z axis makes the longest axisas shown schematically in Fig. 2.The quadrupole deformation parameter β here cor-responds to β z defined by Eq. (3) (to the leading or-der, at least for a sharp-cut density distribution withthe radius R ( θ, φ ) [5]). Note that the condition of Q x = Q y = − Q z /
2, i.e., β x = β y = − β z / − β/ z axis) and the system would show spherical distribution,even when it is deformed in the intrinsic frame.Microscopically, the collective deformation discussed inthis section is intimately related to the mean-field theory.In this picture, particles move independently in a meanfield potential, which is formed self-consistently by theinteraction among the particles. The collective deforma-tion occurs as a consequence of spontaneous breaking ofrotational symmetry. That is, a system may be intrinsi-cally deformed and breaks the rotational symmetry evenif the total Hamiltonian has the symmetry. This is ac-tually a good advantage of the self-consistent mean-fieldtheory used in Sec. III, since a large part of the many-body correlations can be taken into account while keepingthe simple picture of independent particles [5, 27–29]. z x R R z R x FIG. 2. A schematic picture of a collective deformation inthe xz plane. The axially symmetric shape along the z axis isassumed. The dashed and the solid curves correspond to thedeformation parameter of β = 0 and 0 .
25, respectively.
B. Single-particle contribution to deformation
In connection to the deformation in atoms discussed inthe next section, it is instructive to compute the defor-mation parameters discussed in the previous subsectionfor a case with a spherical core plus a valence particle. Inthis case, the quadrupole moment of the system entirelycomes from the valence particle, since Eq. (2) vanishesfor the spherical density distribution. If one assumes thatthe wave function for the valence particle is given by ψ nlms ( r ) = u nl ( r ) r Y lm ( θ, φ ) χ s , (10)as a single-particle orbital in a spherical (effective) po-tential, where χ s is the spinor of the valence parti-cle, the quadrupole moments of the whole system read Q ij = q (sp) ij with q (sp) ij = (cid:90) (cid:0) r i r j − δ ij r (cid:1) | ψ nlms ( r ) | d r = (cid:90) (3 s i s j − δ ij ) r | ψ nlms ( r ) | d r . (11)Here, we have introduced the notation r i = r s i with s i = sin θ cos φ ( i = x ) , sin θ sin φ ( i = y ) , cos θ ( i = z ) . (12)Substituting Eq. (10), one finds q (sp) ij = (cid:90) ∞ r | u nl ( r ) | dr × (cid:90) (3 s i s j − δ ij ) | Y lm ( θ, φ ) | d Ω , (13)where d Ω is the angular part of d r . The deformationparameter for the valence particle, that is, the single-particle deformation parameter, then reads, β (sp) i = (cid:114) π q (sp) ii (cid:104) r (cid:105) nl = (cid:114) π (cid:90) (cid:0) s i − (cid:1) | Y lm ( θ, φ ) | d Ω . (14)Notice that the radial integral in Eq. (13) is cancelledwith the mean-square radius (cid:10) r (cid:11) nl = (cid:90) ∞ r | u nl ( r ) | dr. (15)The deformation parameter of the whole system is thengiven by β i = (cid:114) π q (sp) i N tot (cid:104) r (cid:105) , (16)where (cid:10) r (cid:11) = (cid:2) ( N tot − (cid:10) r (cid:11) core + (cid:10) r (cid:11) nl (cid:3) /N tot is themean-square radius of the whole system with (cid:10) r (cid:11) core being that of the core. If one assumes the radius of thewhole system is similar to the radius of the valence or-bital, (cid:10) r (cid:11) (cid:39) (cid:10) r (cid:11) nl , which is reasonable in the case ofatomic nuclei, the deformation parameter is given by β i (cid:39) β (sp) i N tot . (17)In contrast, in the case of atoms, valence electrons havea larger spatial distribution and contribute more signif-icantly than inner core electrons to the radius (cid:10) r (cid:11) . Ifone assumes (cid:10) r (cid:11) (cid:39) (cid:10) r (cid:11) nl /N tot , i.e., only one electronmainly contributes to (cid:10) r (cid:11) , one obtains β i (cid:39) β (sp) i . (18)Notice that when the number of valence electrons is N val ,the deformation parameter is, instead, approximatelygiven by β i (cid:39) (cid:80) val β (sp) i N val . (19)Let us evaluate explicitly the single-particle deforma-tion parameter for the pure p and d orbitals. To this end,we define the indices of the real spherical harmonics as follows [30]: Y xp ( θ, φ ) = (cid:114) π sin θ cos φ, (20a) Y yp ( θ, φ ) = (cid:114) π sin θ sin φ, (20b) Y zp ( θ, φ ) = (cid:114) π cos θ, (20c)and Y xyd ( θ, φ ) = (cid:114) π sin θ sin 2 φ, (21a) Y yzd ( θ, φ ) = (cid:114) π sin 2 θ sin φ, (21b) Y zxd ( θ, φ ) = (cid:114) π sin 2 θ cos φ, (21c) Y x − y d ( θ, φ ) = (cid:114) π sin θ cos 2 φ, (21d) Y z d ( θ, φ ) = (cid:114) π (cid:0) θ − (cid:1) . (21e)Using these notations, the deformation parameter β (sp) i for Y jp is found to be β (sp) i = (cid:40) (cid:112) π = 0 . i = j ) , − (cid:112) π = − . . (22)For the d -orbitals, the deformation parameter for Y jkd (( j, k ) = ( x, y ), ( y, z ), ( z, x )) is given by β (sp) i = (cid:40) − (cid:112) π = − . i (cid:54) = j and i (cid:54) = k ) , (cid:112) π = 0 . , (23a)while the deformation parameters for Y x − y d and Y z d read β (sp) i = (cid:40) − (cid:112) π = − . i = z ) , (cid:112) π = 0 . , (23b)and β (sp) i = (cid:40) (cid:112) π = 0 . i = z ) , − (cid:112) π = − . , (23c)respectively. Note that due to higher-order deformations,the d orbitals are not axially symmetric except the d z orbital, even though triaxiality vanishes ( γ = 0). III. NUMERICAL CALCULATION
In this section, the possibility of the deformation ofisolated atoms is studied numerically with the computa-tional code
Gaussian [31]. In the
Gaussian code, theground-state wave function and energy of atomic andmolecular systems are numerically calculated by usingthe Gaussian-type basis expansion. In the calculation,all-electron calculation can be performed for wide rangeof atoms ( Z ≤ m e = (cid:126) = e = a B = 4 πε = 1 is used, where m e is the mass ofelectrons, ε is the permittivity of vacuum, and a B =5 . . . . × − m is the Bohr radius [32, 33]. Theunity of the electric quadrupole moment in the Hartreeatomic unit corresponds to 4 . . . . × − C m =1 . . . . Debye ˚A. Here, we use quadrupole moment of density itself, instead of charge density, and thus Q k inthis paper corresponds to − e ( <
0) times of the electricquadrupole moment.
A. Calculation setup and deformation parameters
An isolated neutral atom with the atomic number Z in the non-relativistic scheme is described by the Hamil-tonian H = − Z (cid:88) j =1 ∆ j + Z (cid:88) j =1 V ext ( r j ) + (cid:88) ≤ j B. Systematic behaviors of atomic deformationsand their basis dependence Let us now numerically investigate deformation inatoms. We first show in Fig. 3 the deformation parameter β for atoms from Li ( Z = 3) to Kr ( Z = 36) calculatedwith the unrestricted Hartree-Fock method with the 6-31+G basis. See Table IV in Appendix B for the actualvalues. In Fig. 3, we use different symbols to classify thenature of the atoms: The filled circles show noble-gas(i.e., closed-shell) atoms. The filled squares show half-closed-shell atoms; atoms with three valence electronsoccupying the outer-most p orbitals, or with five valenceelectrons occupying the outer-most d orbitals. The tri-angles, the inverse triangles, and the diamonds show theothers; atoms with s , p , and d orbitals for the outer-mostopen shell, respectively [66]. Note that the octupole mo-ments for all the calculated atoms are found to be zero. Spherical atoms —According to Fig. 3, all the noble-gas atoms are spherical ( β = 0), as is expected. Suchtrivial results are guaranteed by the fact that all the or-bitals for any given principal and azimuth quantum num-bers, n and l , are either fully occupied or unoccupied, andthus their distributions are, in total, spherical. In otherwords, (cid:80) lm = − l | Y lm ( θ, φ ) | = (2 l + 1) / π is a constant,independent of the angles θ and φ .For the other atoms, the electronic configuration oftheir cores is the same as that of the noble-gas atoms.Hence, as long as the core density is not deformed due tonon-trivial many-body effects, i.e., the polarization dueto the interaction between the core and valence electrons,the deformation of the atom is expected to come onlyfrom the valence electrons. Possibility of such non-trivialdeformation will be studied in details in Sec. III D, but wecan easily conclude before going to such details that the s -block and half-closed-shell atoms should be spherical( β = 0), as shown in Fig. 3. The valence electrons ofthe s -block atoms by definition only occupy an s orbital,and the many-body effects between the valence electronsand the spherically symmetric core necessarily results ina spherical electron distribution. This is the case also forthe half-closed-shell atoms: Atom with three electronsoccupying valence p orbitals, or those with five electronsoccupying valence d orbitals, are spherical in both cases,since α -spin electrons occupy all the (2 l + 1) states with m = − l , − l + 1, . . . , l due to the Hund rule [67–71]. Open-shell atoms —For the other open-shell atoms( p -block and d -block atoms), the deformation parame-ters β are generally non-zero. The deformation param-eters for the atoms with the same group show similartendencies, i.e., | β | (cid:39) . . p -block atoms and | β | (cid:46) . 01 for the d -block atoms. This shows that the na-ture of the few valence open-shell electrons plays a majorrole in determining β of the whole atom. In the context ofnuclear physics, the deformation parameter with ± . | β | (cid:38) . p -block atomscan be understood as the valence p orbitals mainly con-tributing to the deformation. Indeed, | β | (cid:39) . . p -block atoms in Fig. 3 are similar values in order ofmagnitude as that of a single p orbital given by Eqs. (18)and (22). On the other hand, the deformations of the d -block atoms are much smaller than those of the p -blockatoms (see and compare with Eq. (23)). This implies thatone cannot regard the deformation of d -block atoms asoriginating only from the single-orbital deformation. Themechanism of the deformation and the physical origin ofthis difference will be explained in Sec. III D: We will con- clude there that the p -block atoms can be regarded as analmost inert core plus valence p -electrons, while many-body effects between the core and the valence electronsare relevant in the d -block atoms. In the d -block atoms,the core electrons tend to collectively cancel the defor-mation of the valence d orbitals.Exceptionally β = 0 open-shell atoms are V ( Z = 23),Co ( Z = 27), Cu ( Z = 29), and Zn ( Z = 30). In the caseof Cu and Zn atoms, ten of the valence electrons com-pletely occupy the valence d orbital and the remainingone or two valence electrons occupy the s orbital. Thus,they are trivially spherical, in the same manner as the s -block atoms. In the case of V and Co atoms, on theother hand, three or eight valence electrons occupy the d orbitals. At first sight, such open-shell d -block atoms donot seem to be trivially β = 0. However, we note thatthey can be β = 0 if d x − y and d z orbitals are occu-pied (unoccupied) at the same time [72]. For example, d occupation may lead to β = 0 if the unoccupied orbitalsare d x − y and d z orbitals. This is the case for Co atom.For V atom, d occupation leads to β = 0 because d x − y and d z orbitals are occupied at the same time. Basis dependence —In order to guarantee that ourresults presented in this section and in Fig. 3 do not de-pend on a specific choice the basis in the calculation, wealso perform calculations with different basis sets: In ad-dition to the 6-31+G basis used in Fig. 3 and Table IV inAppendix B, we also use the dAug-CC-pV5Z and STO-3G bases. The results are shown in Table V and TableVI in Appendix B, respectively. We find that as long asthe configurations of the valence electrons are the same,the deformation parameters β are essentially the same,hence basis independent. We note, however, that thereare a few results with the STO-3G basis which are dif-ferent from those with the dAug-CC-pV5Z and 6-31+Gbases, even when the configurations are the same. Thismay be because the STO-3G basis set spans a smallerspace than the other basis sets, and thus the tail regionof the electron density distribution ρ ( r ) is not accuratelydescribed compared with the others. C. Many-body method dependence Our results presented in the previous section essen-tially remain the same, even if we employ other many-body methods than the unrestricted Hartree-Fock (UHF)method. As benchmark examples, Al ( Z = 13), Cu( Z = 29), and Ga ( Z = 31) are selected. In Fig. 4, weshow the deformation parameters β for those atoms cal-culated with a wide variety of different many-body meth-ods with the 6-31+G basis. The data are shown in Ta-ble VII in Appendix B. We find that the deformation pa-rameters β calculated with the post-Hartree-Fock meth-ods (CCD, CCSD, CID, CISD, MP2, MP3, and MP4)are almost the same as those with the UHF. The excel-lent agreement of the UHF and the post-Hartree-Fockmethods in Fig. 4 implies that the correlations beyond − . − . − . . . . . β Z Noble gasHalf-closed shell s -block p -block d -block FIG. 3. Deformation parameter β as a function of theatomic number Z calculated with the unrestricted Hartree-Fock method with the 6-31+G basis. The filled circles andthe filled squares show noble-gas and half-closed-shell atoms,respectively. The triangles, the inverse-triangles, and the di-amonds show the others: atoms with the outer-most open-shell s orbitals ( s block), p orbitals ( p block), and d orbitals( d block), respectively. The vertical dashed lines denote theclosed-shell atoms. Hartree-Fock may be irrelevant to the quadrupole defor-mation.We have also performed the DFT calculation (indi-cated by LDA in Fig. 4) and found that the resultalso agrees excellently with the UHF and post-Hartree-Fock methods. Since the DFT method is not directlylinked to the UHF calculation, there is no a priori rea-son why β should be the same. In principle, owing tothe Hohenberg-Kohn theorem, the DFT calculation pro-vides the exact, i.e., the full CI, density, if the exactexchange-correlation functional were known. In prac-tice, the DFT results should depend on the choice ofthe employed exchange-correlation functional, since onlyapproximated functional have been known. The excel-lent agreement between the DFT (LDA) and the othermethods in Fig. 4, therefore, guarantees that finite de-formation parameters presented in the previous sectionand following sections are not at all artifacts of a specificmany-body approximation method.We note that all the above results and discussions inthis section hold true for the other bases: We have per-formed the same calculation also with dAug-CC-pV5Zbasis as shown in Table VIII in Appendix B, and ob-tained essentially the same results as those with the 6-31+G basis in Fig. 4 and Table VII. . . . . β AlCuGa FIG. 4. Dependence of the deformation parameter β onmany-body methods with the 6-31+G basis. Data for Al ( Z =13), Cu ( Z = 29), and Ga ( Z = 31) are shown with circles,squares, and triangles, respectively. For more details, see thetext. D. Analyses with single-particle orbitals To understand further the physical mechanism of theatomic deformations, let us define the deformation pa-rameter of each single-particle orbital β (sp)max by β (sp) i forthe direction of the symmetry axis of the orbital. Here,the symmetry axis of the orbital may be different fromthat of the whole atoms, i.e., the i axis for p i orbital, the k axis for d ij orbital ( i (cid:54) = j (cid:54) = k ), and the z axis for theother d orbitals. Angular-momentum mixing can occurdue to the deformation, so that single-particle orbitalsin these methods can, in general, show β different fromthose of the pure spherical harmonics. Specifically, aswe will see, an s orbital can show a non-zero β due tothe mixture with d orbitals. Note that β (sp)max is the de-formation parameter for each single-particle orbital (seeEq. (14)), which is divided by the radius of each orbital.Therefore, β i of the whole atom is not a simple sum of β (sp)max . Thus, a care must be taken when using them, butthey are still useful in understanding the mechanics ofthe deformation microscopically. Deformation of single-particle orbitals —Table Ishows β (sp)max for selected valence orbitals of some open-shell atoms. All the data are calculated by the unre-stricted Hartree-Fock calculation with the 6-31+G basis.First, we find that the deformation parameters of thevalence p and d orbitals are almost the same as thoseof the pure p and d orbitals given by Eqs. (22)–(23c).This suggests that even when the atom is deformed, thesingle-particle orbitals of the valence p and d electronsare described almost as a product of the radial functionand the pure spherical harmonics. In other words, thevalence p and d electrons feel an almost perfectly spheri-cal effective (Hartree-Fock or Kohn-Sham) potential. Wecan, therefore, conclude that the density of the electronsis almost perfectly spherical in the central region dueto the strong Coulomb force of the nucleus, resulting inthe spherical effective potential for the valence electrons.The electron density is deformed only in the surface andthe tail regions of the atoms described by the valenceelectrons. Deformations in the surface and degeneracylifting —Such deformations lead to a non-spherical effec-tive potential in the surface and the tail regions. Thereare several evidences for this. Firstly, the outer-most s orbital can have non-zero β and become non-spherical.This is shown in Table I as a non-zero β of the 4 s orbitals,which tend to cancel the deformation of the valence 3 d orbitals. This is due to the screening effect. That is,the outer-most 4 s electrons try to cancel the deformedelectron density of the open-shell valence 3 d electrons.We will discuss this s -orbital deformation in more detailslater in this section taking Sc atom as an example.Secondly, the orbitals with the same n and l becomenon-degenerate in some cases. This can be seen in Ta-ble II, where the quadrupole moment of q (sp)max , the squareradius (cid:10) r (cid:11) , and the eigenenergies of the single-particleorbitals ε j , for Sc atoms are shown. As one can see,the eigenenergies of the p z states become different fromthose of p x and p y states. Since the deformation is ax-ially symmetric and thus there is only one specific di-rection, i.e., the orbitals along x and y axes ( p x and p y orbitals or d yz and d zx orbitals) remain degenerate. Thisis similar to the Stark effect [73]. We note, however, thatthe energy difference for the core orbitals is the order of0 . 05 Hartree (cid:39) p z orbital. If the system is completelyspherical, the 3 p z orbital is degenerate with the 3 p x and3 p y orbitals, while due to the deformation the 3 p x and3 p y orbitals are non-degenerate to the 3 p z orbital. Theeigenenergy of the 3 p z orbital is − . p x and 3 p y orbitals are 0 . p z orbital instead of the equal filling of the 3 p x ,3 p y , and 3 p z orbitals, leading to the deformation. It isnoted that this degeneracy lifting of p states results fromthe second-order perturbation: It is caused by the defor-mation of the core and the 3 s orbital, which is inducedby the existence of the 3 p z electron.These results suggest that the dominant part of the ef-fective potential is created by the spherical central region.The non-spherical contribution to the effective potentialof the valence deformed electrons is sub-dominant andcan be treated as small correction. Screening effect —Let us scrutinize deformations ofeach orbital with Sc ( Z = 21) atom in Table II (resultsfor other atoms are also shown in Table XI-IX in Ap-pendix B). Here, Sc atom has filled 1 s –3 p orbitals, in ad-dition to one 3 d xy electron and two 4 s electrons. Thequadrupole moments q (sp) z of the filled 1 s –3 p orbitals TABLE I. Deformation parameters for the selected single-particle orbitals.Atom Orbital β j max Al α p z . α p z . α p x . α d yz − . α s − . β s . α p x . α p z . α d zx − . α s − . β s − . α p z . α d x − y − . α d xy − . α d yz − . α d z . α s − . α p x . α p z . α d x − y − . α d yz − . α d z . α s − . α p z . completely cancel each other, resulting in undeformedspherical core. The square radii of the 1 s –3 p orbitals aremuch smaller than those of 3 d xy and 4 s orbitals, so thatwe can essentially regard Sc atom as a system of one 3 d xy electron and two 4 s electrons orbiting around an almostperfectly inert small spherical core. The electron in the3 d xy in the α -spin state is oblate q (sp) z = − . < q (sp) z = − . d xy electron. This can beexplained by the screening effect: We can see in TableII that the electron in the 4 s orbital with β -spin statedeforms so significantly that the quadrupole moment ofthe β s electron cancels that of the α d xy orbital. Tobe more precise, (cid:12)(cid:12)(cid:12) q (sp) z (cid:12)(cid:12)(cid:12) of β s is almost the same butslightly smaller than (cid:12)(cid:12)(cid:12) q (sp) z (cid:12)(cid:12)(cid:12) of α d xy , which suggests thatthe quadrupole moment of the α d xy electron is almostbut not perfectly cancelled, resulting in a small but finitedeformation of Sc atom. On the other hand, the 4 s elec-tron in the other spin state (i.e., α s ) is deformed muchless than β s electron, and does not contribute so muchto the screening. This is because the Coulomb repulsion,which is the key for the screening effect (see Sec. IV),is relevant between different spin states, while the Pauliprinciple makes the effects of the Coulomb repulsion be-tween the same spins weaker.Similar situations also occur in the other atoms pre-sented in Table I. For Ti atom (see also Table XI in Ap-pendix B), the large deformation is due to the α d yz andthe α d zx orbitals, and in total the deformation is pro-late ( β > β s orbital is deformed to cancel thedeformation of the α d orbitals. The α s orbital is alsodeformed but because the effects of the Coulomb repul-sion is weakened by the Pauli principle, the deformationis much smaller. For Ni atom (Table XII in Appendix B),the large prolate deformation created by the β d z orbitalis slightly cancelled by the α s electron [74].In the case of Ga atom (see also Table XIII in Ap-pendix B), the large prolate deformation is due to the α p z orbital q (sp) z = 11 . > 0. With the same ar-gument, we might expect that the β s orbital shouldbe largely deformed and cancel that of the α p z orbital.However, the β s orbital has rather small quadrupolemoment q (sp) z = 0 . α p z orbital, resulting in Q z = 11 . (cid:10) r (cid:11) of the 4 s orbital is much smaller than that of the α p z orbital, and thus the wave function of the 4 s or-bital does not have enough overlap region with the α p z orbital to cancel its deformation.We can also see the importance of the radii by com-paring Sc, Ti, and Ni data in Tables II, XI, XII. ForSc atom, the quadrupole moment of the valence d elec-tron is almost perfectly cancelled. On the other hand, itis marginally cancelled for Ti atom, while it is slightlycancelled for Ni atom. This behavior can be easily un-derstood by noting that radius of the valence d orbitalgets smaller as the atomic number increases: (cid:10) r (cid:11) = 3 . . 9, and 1 . for Sc, Ti, and Ni atoms, respectively.The overlap between the valence d orbital and the 4 s orbital gets smaller, which explains why the screening ef-fects gets weaker as the atomic number increases in theseatoms.Moreover, the deformation of the d -block atoms be-comes further smaller due to the radius of the valencesingle-particle orbitals, compared to that of the p -blockatoms. The order of (cid:80) m (cid:10) r (cid:11) n = 3 (cid:10) r (cid:11) n for the va-lence p orbital is the same as that of Z (cid:10) r (cid:11) for the wholeatom and thus | β | of the p -block open-shell atoms arelarge, whereas the order of (cid:80) m (cid:10) r (cid:11) n = 5 (cid:10) r (cid:11) n for thevalence d orbital is smaller than that of Z (cid:10) r (cid:11) , since the s orbital with the principal quantum number n + 1 is, ingeneral, also occupied. Systematic behavior of β —With all these physicalarguments, we can now fully understand the results inFig. 3: We have found there that the p -block and d -block atoms can be deformed, but the deformation ismuch smaller for the d -block atoms compared with the p -block atoms showing single-particle-like deformation.This result can be explained by noting that the screeningeffects of the outer-most s orbital are rather significantfor the d -block atoms, as discussed above for Sc, Ti, andNi atoms. On the other hand, the outer-most valence p orbital of the p -block atom has much smaller overlapwith the outer-most s orbital, so that the deformation TABLE II. Mean-square radius (cid:10) r (cid:11) nl , single-particlequadrupole moment q (sp) z , and deformation parameter β (sp) z for single-particle orbitals of Sc atom calculated by the unre-stricted Hartree-Fock method with 6-31+G basis. The single-particle energy of each orbital ε j and corresponding naturalorbital (NO) are also shown.Spin NO ε j (cid:10) r (cid:11) nl q (sp) z β (sp) z α s − . . . . α s − . . . . α p z − . . . . α p x − . . − . − . α p y − . . − . − . α s − . . − . − . α p x − . . − . − . α p y − . . − . − . α p z − . . . . α d xy − . . − . − . α s − . . − . − . β s − . . . . β s − . . . . β p z − . . . . β p x − . . − . − . β p y − . . − . − . β s − . . . . β p z − . . . . β p x − . . − . − . β p y − . . − . − . β s − . . . . . − . caused by the valence p orbital remains unscreened.Since deformation of the p -block atoms originates fromsingle-particle-like deformation, atoms whose configura-tions of valence electrons are p and p always show pro-late ( β > 0) deformation and the last electron occupies p z orbital. In contrast, atoms whose configurations of va-lence electrons are p and p always show oblate ( β < p x and p y orbital. Since deformation of the d -block atoms inducesthe screening as discussed above, the deformation is smalland the tendency is rather complicated. E. Short conclusion Before going to the qualitative discussion which willaugment our results in this section by a qualitativemodel, let us summarize the results of the numerical cal-culations and the discussions in this section.Firstly, as shown in Fig. 3 the closed-shell and half-closed-shell atoms are exactly spherical. The density dis-tribution of the core electrons are spherical since it is thesame as the noble-gas electronic configuration. Thus,the deformation comes from open-shell valence electrons.Since the valence orbital of the s -block atoms are alsoexactly spherical, only the p - or d -block atoms can bedeformed, in principle. We find numerically that the de-0formation of the p -block atom is as large in order of mag-nitude as that of a single p -orbital deformation, while d -block atoms are much less deformed. The extremely tinydeformation of the d -block atoms are explained by thescreening effect, where the outer-most s electrons are de-formed to cancel the deformations of the inner d valenceelectrons and the radii of the valence electrons. The p -block atom is almost intact by the screening effect sincethe radius of the p -orbital is larger than that of s elec-trons.Therefore, the deformations of the electron density inatoms are at most originating from the single-particleorbitals of a few valence electrons, and there is no col-lective deformation. Rather, many-body effects in atomsdisfavor deformations. With such tiny deformations, theeffective potential is slightly deformed only in the surfaceand the tail (i.e., the valence electron) regions, but thedominant part remains spherical. This justifies the use ofspherically symmetric effective potentials or density func-tionals in the conventional atomic structure calculations[22, 25, 75].This is in stark contrast with nuclear systems. It is wellknown that nuclei can be deformed significantly via col-lective many-body effects, and they often have | β | (cid:38) . IV. QUALITATIVE DISCUSSION In the previous section and in Appendix A, we haveshown that electrons in atoms are much less likely todeform than nucleons in nuclei. We argue in this sec-tion that this difference physically originates from thenature of the inter-particle interactions: the repulsiveCoulomb interaction between the electrons and an at-tractive nuclear force between the nucleons. To illustratethis point, we closely follow Refs. [77, 78] and considerdeforming a spherical wave function | Ψ (cid:105) of the Hamil-tonian H = T + V with the following canonical transfor-mation: | Ψ β (cid:105) = e β [ H,Q ] | Ψ (cid:105) (27) Q = m N (cid:88) i =1 (cid:20) z i − (cid:0) x i + y i (cid:1)(cid:21) (28)where m and x i , y i , z i are the mass and coordinates ofthe particles. This transformation induces quadrupoledeformation of the many-body wave function in the formofΨ β ( x , y , z , x , y , z , . . . )= Ψ (cid:0) e β x , e β y , e − β z , e β x , e β y , e − β z , . . . (cid:1) , (29)that is, it makes the wave function shrunken in the x - and y -axes, and elongated in the z -axis directions when β is positive. When the absolute value of the deformationparameter β is small, the one-body density ρ β of thesystem | Ψ β (cid:105) is, indeed, found to be proportional to β : ρ β ( x, y, z ) = ρ (cid:0) e β x, e β y, e − β z (cid:1) = ρ ( x, y, z ) − β (cid:114) π rY ( θ, φ ) dρ ( x, y, z ) dr + O (cid:0) β (cid:1) . (30)Let us consider how the energy of the system changeswhen we induce an infinitesimal quadrupole deforma-tion | β | (cid:28) ρ ( r ) = ρ ( r ). The kinetic part T = − (cid:80) Ni =1 (cid:126) ∇ i m of the Hamiltonian changes as (cid:104) Ψ β | T | Ψ β (cid:105) = (cid:18) e − β + 23 e β (cid:19) (cid:104) Ψ | T | Ψ (cid:105) . (31)We can therefore see that the kinetic energy increaseswith the deformation (cid:104) Ψ β | T | Ψ β (cid:105) − (cid:104) Ψ | T | Ψ (cid:105) (cid:39) β (cid:104) Ψ | T | Ψ (cid:105) > . (32)The interaction part, on the other hand, can favor de-formation depending on the nature of the interaction.To see this point, let us consider the direct part of thetwo-body interaction energy characterized by one-bodydensity ρ β ( r ):12 (cid:90) V ( | r − r (cid:48) | ) ρ β ( r ) ρ β ( r (cid:48) ) d r d r (cid:48) = 2 π (cid:90) (cid:90) V ( r, r (cid:48) ) ρ ( r ) ρ ( r (cid:48) ) r dr r (cid:48) dr (cid:48) + 16 πβ (cid:90) (cid:90) V ( r, r (cid:48) ) dρ ( r ) dr dρ ( r (cid:48) ) dr (cid:48) r dr r (cid:48) dr (cid:48) + O (cid:0) β (cid:1) , (33)where we have used the partial wave expansion of theinteraction V ( | r − r (cid:48) | ) = (cid:80) l, m V l ( r, r (cid:48) ) Y l,m ( r ) Y ∗ l,m ( r (cid:48) )and Eq. (30). For both electrons in atoms and nucleonsin nuclei, the one-body density is generally a decreasingfunction of r , except for a surface region (this will bediscussed later in this section), so that it is reasonable toassume dρ /dr ≤ V < 0. For electrons in atoms, theCoulomb interaction between the electrons reads V ( r, r (cid:48) ) = 4 π r < r > , (34)where r < and r > are smaller and greater ones of r and r (cid:48) ,respectively. Since V is positive, the electrons in atomsare unlikely to deform. Notice that the potential betweenthe electrons and the nucleus in an atom is spherical,and does not contribute to the β -dependence of the total1energy, as long as the expansion in Eq. (30) is valid ( | β | (cid:28) V < 0. We note that the interac-tion between the nucleons are not yet rigorously known,but it basically comprises of the long-range Yukawa at-tractive force originating from one-pion exchange and theshort-range repulsive force [79–83]. These combined to-gether have a net attractive effect between the nucleons,which results in a bound state of a proton and a neutron(i.e., deuteron) [84, 85], and in a large negative scatteringlength between the neutrons [81, 82]. We also note thatan effective in-medium nucleon-nucleon interaction usedin nuclear many-body calculations in general has a netattractive effect [86]. For the purpose of our qualitativediscussion in this section, a simple delta-function attrac-tive force V ( r ) = − gδ (3) ( r ) with a parameter g > V ( r, r (cid:48) ) = − gδ ( r − r (cid:48) ) /rr (cid:48) ≤ 0, leading to a decreaseof the interaction energy with the deformation. If thisdecrease is larger than the increase of the kinetic energyin Eq. (32), the system is likely to deform.From these qualitative discussions, we can see that dif-ference in the nature of the interaction is the key in un-derstanding why atoms do not tend to deform collec-tively while nuclei do. As the above discussions for thequadrupole deformation also apply to the higher ordermultipole deformations at any order, we can make thefollowing general statement: The multipole componentof the inter-particle interaction must be attractive forthe system to undergo significant collective deformations.The Coulomb force between the electrons have repulsivemultipole components for all orders, and therefore theyare unlikely to deform in any shape, as numerically foundin the previous section. On the other hand, the nuclearforce has attractive multipole components, and a nucleusis likely to deform in a shape to maximize the interactionenergy gain.The above model discussion is closely related to thescreening effects of electrons numerically found and dis-cussed in Sec. III. The screening effect occurs due tothe repulsive Coulomb interaction, which aims to de-crease any charge imbalance and attains charge neutral-ity. Therefore, the above argument based on the multi-pole component of the interaction is consistent with thescreening effect. We have also found in Sec. III that thedeformations of atoms occur only in the valence electronregion for p - and d -block atoms. We can alternativelyexplain this by using the above argument: We have as-sumed in the above model argument that dρ /dr ≤ ψ nlm ( r, θ, φ ), which showoscillations with r when n > 1. Thus, dρ ( r ) dr dρ ( r (cid:48) ) dr in Eq. (33) can be negative, so that the deformations maybe energetically allowed in the valence region. This qual-itatively explains why atoms can be deformed only in thevalence electron region.We note that we have neglected in the above discussionthe exchange-correlation part in evaluating the interac-tion energy [Eq. (33)]. In the Hartree-Fock theory andthe density functional theory, the exchange or exchange-correlation part is generally found to be sub-dominantthan the direct part contributions for electrons in atoms[25, 41, 75, 87]. We have, therefore, only consideredthe direct part in our basic qualitative discussions above,while one needs to include the exchange-correlation partfor more elaborated descriptions. In particular, for elec-trons in the surface region of atoms (i.e., the valence or-bital), the exchange-correlation part becomes more im-portant compared with the central region, because thedensity is rather small and thus the electrons are stronglycorrelated [41, 75, 88]. Our model discussion on the sur-face region of atoms, therefore, should be at best quali-tative one. In the nuclear DFT, on the other hand, thedirect and exchange parts are usually not considered sep-arately. Rather, a short-range attractive effective interac-tion V ( | r − r (cid:48) | ) in Eq. (33) is often used whose shape andparameters are chosen to reproduce well nuclear prop-erties measured in experiments [86, 89, 90]. Thus, theabove argument based on Eq. (33), ostensibly consider-ing only the direct part of interaction energy, naturallycontains many-body correlation effects, including thosedue to the three-body and tensor interaction, and there-fore should be valid even though atomic nuclei are ratherstrongly correlated.While we have shown that atoms in their ground statesare unlikely to be deformed significantly, we may expectlarger deformations in more exotic atomic systems. Ascan be seen in our qualitative discussion above, atomsmay be deformed if the condition dρ /dr ≤ V. CONCLUSION We have studied numerically whether the electron dis-tribution of atoms can be deformed. We have calculatedthe deformation parameters for various atoms with awide variety of different many-body methods, and havefound that the noble-gas, half-shell-closed, and s -blockatoms are spherical, while the p - and d -block atoms are2deformed. The deformations of the p - and d -block atomsare neither collective nor significantly deformed. We haveshown that the core part remains spherical even for theseatoms, and their deformations originate from a few va-lence electrons. Therefore, the deformations of the atomsare at most of a single-particle nature of a few valanceelectrons: Atoms do not deform collectively, in contrastto nuclei. We have also shown that the many-body ef-fects of electrons in atoms tend to make the deformationssmaller due to the screening effect.Owing to the deformation, the self-consistent effectivemean-field potential for p - and d -block atoms is slightlydeformed in the surface and the tail region. Therefore,the core electrons are still eigenstates of angular mo-mentum and are described by single spherical harmon-ics, whereas the valence electrons are not. We havefound numerically that the degeneracy of the valenceelectrons are indeed lifted due to this effect, but the ef-fect is 0 . 05 Hartree or much less. Hence, calculations ofthe atomic structure with spherical symmetry are stilljustified.We have compared the atomic deformation to the nu-clear deformation by using a qualitative model. We haveargued that the difference between them originates fromthe properties of the inter-particle interactions. On onehand, in the case of atoms, the interaction is purely repul-sive, and the collective deformation does not occur. Onthe other hand, in the case of atomic nuclei, the interac-tion is of attractive in net, and the collective deformationcan occur.While the collective deformations of nuclei have clearlybeen observed by the gamma-ray spectroscopy of theirrotational spectra of Eq. (1), the atomic deformationsstudied in this work are neither significant nor collective,and therefore should not show similar rotational spectra.One rather needs to directly probe the density distri-bution of the electrons to experimentally test the defor-mations found by our numerical calculations. It wouldbe rather challenging but we expect that they may be di-rectly observed with recent atomic and molecular physicstechnology, such as photo-ionization microscopy [94, 95],or tomographic imaging [96]. ACKNOWLEDGMENTS We thank Kenichi Yoshida for discussions at theearly stage of this work. The RIKEN iTHEMS pro-gram, the JSPS Grant-in-Aid for JSPS Fellows underGrant No. JP19J20543, and JSPS KAKENHI GrantNo. JP19K03861 are acknowledged. The numerical cal-culations were performed on cluster computers at theRIKEN iTHEMS program. Appendix A: Deformation of neutron drop in aharmonic trap In this Appendix, we discuss an illustrative case ofdeformed systems: that is, a neutron drop trapped ina spherical harmonic potential. Even though neutrondrops are fictitious, they provide useful systems to bench-mark many-body theories [97–103], as the density can becontrolled by changing the strength of the trapping po-tential. They have also been utilized in connection withneutron-rich nuclei and neutron stars [104].Although the interaction between neutrons is attrac-tive, in contrast to the interaction between electrons, typ-ical nuclear interactions are not strong enough so thatsystems consisting solely of neutrons are not bound bythemselves. To study properties of many-neutron sys-tems, one thus needs to introduce an external confiningpotential, analogously to the electron-nucleus externalpotential in atoms. In this Appendix, to localize theneutrons in neutron drops, we use an isotropic harmonicoscillator potential, V ext ( r ) = 12 m n Ω r , (A1)where m n is the neutron mass. To take into account theinteraction between neutrons, we employ the relativisticmean-field theory, which is based on a meson-exchangemodel [76, 105]. We adopt the PK1 parameter set [106]for the nucleon-meson couplings. In our numerical cal-culations, the single-particle orbitals are represented on3-dimensional lattice in the real space [107, 108].Table III shows the root-mean-square radii (cid:10) r (cid:11) / and the quadrupole deformation parameters β z for the n , n , and n systems for different strengths of theexternal harmonic potential given by Eq (A1). Here, N n denotes the system with N neutrons. Only β z isshown in the table, since all the systems studied hereare found to be axially symmetric with respect to the z axis, that is, β x = β y = − β z / 2. The n system isalways spherical because of the shell closure at N = 8,which is one of the well-known magic numbers of nuclei(2, 8, 20, . . . ). The neutron configuration in this case is (cid:0) s / (cid:1) (cid:0) p / (cid:1) (cid:0) p / (cid:1) . As is expected, the radius be-comes smaller as the external potential becomes strong.In contrast to n , the n system is well deformed pro-lately for any value of (cid:126) Ω examined here. The deforma-tion as well as the radius decreases with the strength ofthe external potential, (cid:126) Ω, since the role of the spher-ical confining potential increases as a function of (cid:126) Ω.The n system is softer against deformation than n .It is oblately deformed when (cid:126) Ω is not too large. For (cid:126) Ω = 10 MeV, it becomes spherical with the configura-tion of (cid:0) s / (cid:1) (cid:0) p / (cid:1) (cid:0) p / (cid:1) (cid:0) d / (cid:1) .Figures 5 and 6 show the density distributions of the n and n systems, respectively. The density distri-bution of n (Fig. 5) becomes more compact as (cid:126) Ω in-creases, while keeping similar shapes. On the other hand,3 TABLE III. The root-mean-squared radii (cid:10) r (cid:11) / and thequadrupole deformation parameters β z of the neutron drops n , n , and n for different values of the curvature (cid:126) Ω of theconfining potential. All the solutions shown here are found tobe axially symmetric.System (cid:126) Ω (MeV) (cid:10) r (cid:11) / (fm) β z n . 45 0 . 008 2 . 95 0 . . 75 0 . n . 67 0 . 348 3 . 14 0 . . 91 0 . n . − . 318 3 . − . . 08 0 . the shape of n drastically changes with (cid:126) Ω from oblateto spherical shape.Evidently, neutrons trapped in an isotropic harmonic-oscillator potential can be collectively deformed for open-shell systems, such as n and n , in contrast to elec-trons in atoms, as is argued in Sec. IV. As expected, thespherically symmetric external field tends to suppress thedeformation parameter as well as the radius of the sys-tem. Appendix B: Numerical data All the numerical data related to the main part of thepaper are shown in Tables IV–XIII. -4 -2 0 2 4y (fm)-4-2 0 2 4 z (f m ) ρ ( , y , z ) (f m - ) (a) h Ω = 10 MeV--4-2 0 2 4 z (f m ) ρ ( , y , z ) (f m - ) (a) h Ω = 8 MeV--4-2 0 2 4 z (f m ) ρ ( , y , z ) (f m - ) (a) h Ω = 5 MeV- FIG. 5. The neutron density distribution of n on the yz plane for (a) (cid:126) Ω = 5 MeV, (b) (cid:126) Ω = 8 MeV, and (c) (cid:126) Ω =10 MeV. The vertical axis corresponds to the symmetry axis. TABLE IV. The root-mean radii, Z (cid:10) r (cid:11) , the quadrupole moments, Q , and the deformation parameters, β , calculated withthe unrestricted Hartree-Fock method with the 6-31+G basis. Configurations of the valence electrons calculated by the naturalorbital analysis are also shown. Atom Z Mult. Energy Z (cid:10) r (cid:11) Q x Q y Q z β x β y β z Config.Li 3 2 − . . . . . . . . s . Be 4 1 − . . . . . . . . s . B 5 2 − . . − . − . . − . − . . s . p . C 6 3 − . . . . − . . . − . s . p . N 7 4 − . . . . . . . . s . p . O 8 3 − . . − . − . . − . − . . s . p . p . F 9 2 − . . . . − . . . − . s . p . Ne 10 1 − . . . . . . . . s . p . Na 11 2 − . . . . . . . . s . Mg 12 1 − . . . . . . . . s . Al 13 2 − . . − . − . . − . − . . s . p . Si 14 3 − . . . . − . . . − . s . p . P 15 4 − . . . . . . . . s . p . S 16 3 − . . − . − . . − . − . . s . p . Cl 17 2 − . . . . − . . . − . s . p . Ar 18 1 − . . . . . . . . s . p . K 19 2 − . . . . . . . . s . Ca 20 1 − . . . . . . . . s . Sc 21 2 − . . . . − . . . − . s . d . Ti 22 3 − . . − . − . . − . − . . s . d . V 23 4 − . . . . . . . . s . d . s . Cr 24 7 − . . . . . . . . s . d . Mn 25 6 − . . . . − . . . − . s . d . d . Fe 26 5 − . . − . − . . − . − . . s . d . d . Co 27 4 − . . . . . . . . s . d . Ni 28 3 − . . − . − . . − . − . . s . d . Cu 29 2 − . . . . . . . . s . d . Zn 30 1 − . . . . . . . . s . d . Ga 31 2 − . . − . − . . − . − . . s . p . Ge 32 3 − . . . . − . . . − . s . p . As 33 4 − . . . . . . . . s . p . Se 34 3 − . . − . − . . − . − . . s . p . Br 35 2 − . . . . − . . . − . s . p . Kr 36 1 − . . . . . . . . s . p . TABLE V. Same as Table IV, but with the dAug-CC-pV5Z basis. Atom Z Mult. Energy Z (cid:10) r (cid:11) Q x Q y Q z β x β y β z Config.Li 3 2 − . . . . . . . . s . Be 4 1 − . . . . . . . . s . B 5 2 − . . − . − . . − . − . . s . p . d . C 6 3 − . . . . − . . . − . s . p . N 7 4 − . . . . . . . . s . p . O 8 3 − . . − . − . . − . − . . s . p . p . F 9 2 − . . . . − . . . − . s . p . Ne 10 1 − . . . . . . . . s . p . Na 11 2 − . . . . . . . . s . Al 13 2 − . . − . − . . − . − . . s . p . d . Si 14 3 − . . . . − . . . − . s . p . d . P 15 4 − . . . . . . . . s . p . S 16 3 − . . − . − . . − . − . . s . p . d . Cl 17 2 − . . . . − . . . − . s . p . Ar 18 1 − . . . . . . . . s . p . Sc 21 2 − . . . . − . . . − . s . d . Ti 22 3 − . . − . − . . − . − . . s . d . V 23 4 − . . . . . . . . s . d . s . Cr 24 7 − . . . . . . . . s . d . Mn 25 6 − . . . . − . . . − . s . d . d . Fe 26 5 − . . − . − . . − . − . . s . d . d . Co 27 4 − . . . . . . . . s . d . Ni 28 3 − . . . . − . . . − . s . d . Cu 29 2 − . . − . − . . . − . . s . d . Zn 30 1 − . . . . . . . . s . d . Ga 31 2 − . . − . − . . − . − . . s . p . d . Ge 32 3 − . . . . − . . . − . s . p . d . As 33 4 − . . . . . . . . s . p . Se 34 3 − . . − . − . . − . − . . s . p . Br 35 2 − . . . . − . . . − . s . p . Kr 36 1 − . . . . . . . . s . p . TABLE VI. Same as Table IV, but with the STO-3G basis. Atom Z Mult. Energy Z (cid:10) r (cid:11) Q x Q y Q z β x β y β z Config.Li 3 2 − . . . . . . . . s . Be 4 1 − . . . . . . . . s . B 5 2 − . . − . − . . − . − . . s . p . C 6 3 − . . . . − . . . − . s . p . N 7 4 − . . . . . . . . s . p . O 8 3 − . . − . − . . − . − . . s . p . F 9 2 − . . . . − . . . − . s . p . Ne 10 1 − . . . . . . . . s . p . Na 11 2 − . . . . . . . . s . Mg 12 1 − . . . . . . . . s . Al 13 2 − . . − . − . . − . − . . s . p . Si 14 3 − . . . . − . . . − . s . p . P 15 4 − . . . . . . . . s . p . S 16 3 − . . − . − . . − . − . . s . p . Cl 17 2 − . . . . − . . . − . s . p . Ar 18 1 − . . . . . . . . s . p . K 19 2 − . . . . . . . . s . Ca 20 1 − . . . . . . . . s . Sc 21 2 − . . . . − . . . − . s . d . Ti 22 3 − . . − . − . . − . − . . s . d . V 23 4 − . . . . . . . . s . p . Cr 24 7 − . . − . − . . − . − . . s . d . p . Mn 25 6 − . . . . . . . . s . d . Fe 26 5 − . . − . − . . − . − . . s . d . p . Co 27 4 − . . . . − . . . − . s . d . p . Ni 28 3 − . . . . − . . . − . s . d . p . Cu 29 2 − . . − . − . . − . − . . s . d . p . Zn 30 1 − . . − . − . . − . − . . s . d . p . Ga 31 2 − . . − . − . . − . − . . s . p . Ge 32 3 − . . . . − . . . − . s . p . As 33 4 − . . . . . . . . s . p . Se 34 3 − . . − . − . . − . − . . s . p . Br 35 2 − . . . . − . . . − . s . p . Kr 36 1 − . . . . . . . . s . p . Rb 37 2 − . . . . . . . . s . Sr 38 1 − . . . . . . . . s . Y 39 2 − . . . . − . . . − . s . d . Zr 40 3 − . . − . − . . − . − . . s . d . Nb 41 6 − . . . . − . . . − . s . d . Mo 42 7 − . . . . . . . . s . d . Tc 43 6 − . . . . . . . . s . d . Ru 44 5 − . . . . − . . . − . s . d . Rh 45 4 − . . − . − . . − . − . . s . d . Pd 46 1 − . . . . − . . . − . s . d . Ag 47 2 − . . − . − . . − . − . . s . d . Cd 48 1 − . . . . . . . . s . d . In 49 2 − . . − . − . . − . − . . s . p . Sn 50 3 − . . . . − . . . − . s . p . Sb 51 4 − . . . . . . . . s . p . Te 52 3 − . . − . − . . − . − . . s . p . I 53 2 − . . . . − . . . − . s . p . Xe 54 1 − . . . . . . . . s . p . TABLE VII. The root-mean radii, Z (cid:10) r (cid:11) , the quadrupole moments, Q , and the deformation parameters, β , for Al, Cu, andGa calculated with the several many-body calculation methods with the 6-31+G basis. Configurations of the valence electronscalculated by the natural orbital analysis are also shown. Atom Method Energy Z (cid:10) r (cid:11) Q x Q y Q z β x β y β z Config.Al UHF − . . − . − . . − . − . . s . p . s . p . Al CCD − . . − . − . . − . − . . s . p . s . p . Al CCSD − . . − . − . . − . − . . s . p . s . p . Al CID − . . − . − . . − . − . . s . p . s . p . Al CISD − . . − . − . . − . − . . s . p . s . p . Al MP2 − . . − . − . . − . − . . s . p . s . p . Al MP3 − . . − . − . . − . − . . s . p . s . p . Al MP4 − . . − . − . . − . − . . s . p . s . p . Al LDA − . . − . − . . − . − . . s . p . s . p . Cu UHF − . . . . . . . . d . s . p . d . Cu CCD − . . . . . . . . d . s . p . d . Cu CCSD − . . . . . . . . d . s . p . d . Cu CID − . . . . . . . . d . s . p . d . Cu CISD − . . . . . . . . d . s . p . d . Cu MP2 − . . . . . . . . d . s . p . d . Cu MP3 − . . . . . . . . d . s . p . d . Cu MP4 − . . . . . . . . d . s . p . d . Cu LDA − . . . . . . . . d . s . p . d . Ga UHF − . . − . − . . − . − . . s . p . s . p . Ga CCD − . . − . − . . − . − . . s . p . s . p . Ga CCSD − . . − . − . . − . − . . s . p . s . p . Ga CID − . . − . − . . − . − . . s . p . s . p . Ga CISD − . . − . − . . − . − . . s . p . s . p . Ga MP2 − . . − . − . . − . − . . s . p . s . p . Ga MP3 − . . − . − . . − . − . . s . p . s . p . Ga MP4 − . . − . − . . − . − . . s . p . s . p . Ga LDA − . . − . − . . − . − . . s . p . s . p . TABLE VIII. Same as Table VII, but with the dAug-CC-pV5Z basis. Atom Method Energy Z (cid:10) r (cid:11) Q x Q y Q z β x β y β z Config.Al UHF − . . − . − . . − . − . . s . p . d . p . Al CCD − . . − . − . . − . − . . s . p . d . p . Al CCSD − . . − . − . . − . − . . s . p . d . p . Al CID − . . − . − . . − . − . . s . p . d . p . Al CISD − . . − . − . . − . − . . s . p . d . p . Al MP2 − . . − . − . . − . − . . s . p . d . p . Al MP3 − . . − . − . . − . − . . s . p . d . p . Al MP4 − . . − . − . . − . − . . s . p . d . p . Al LDA − . . − . − . . − . − . . s . p . d . p . Cu UHF − . . − . − . . − . − . . d . s . p . d . f . s . p . Cu CCD − . . . − . . . − . . d . s . p . d . f . s . p . Cu CCSD − . . . − . . . − . . d . s . p . d . f . s . p . Cu CID − . . . − . . . − . . d . s . p . d . f . s . p . Cu CISD − . . . − . . . − . . d . s . p . d . f . s . p . Cu MP2 − . . − . − . . − . − . . d . s . p . d . f . s . p . Cu MP3 − . . − . − . . − . − . . d . s . p . d . f . s . p . Cu MP4 − . . . − . . . − . . d . s . p . d . f . s . p . Cu LDA − . . . . . . . . d . s . p . d . f . s . p . Ga UHF − . . − . − . . − . − . . s . p . d . f . s . p . d . Ga CCD − . . − . − . . − . − . . s . p . d . f . s . p . d . Ga CCSD − . . − . − . . − . − . . s . p . d . f . s . p . d . Ga CID − . . − . − . . − . − . . s . p . d . f . s . p . d . Ga CISD − . . − . − . . − . − . . s . p . d . f . s . p . d . Ga MP2 − . . − . − . . − . − . . s . p . d . f . s . p . d . Ga MP3 − . . − . − . . − . − . . s . p . d . f . s . p . d . Ga MP4 − . . − . − . . − . − . . s . p . d . f . s . p . d . Ga LDA − . . − . − . . − . − . . s . p . d . f . s . p . d . -4 -2 0 2 4y (fm)-4-2 0 2 4 z (f m ) ρ ( , y , z ) (f m - ) (a) h Ω = 10 MeV--4-2 0 2 4 z (f m ) ρ ( , y , z ) (f m - ) (a) h Ω = 8 MeV--4-2 0 2 4 z (f m ) ρ ( , y , z ) (f m - ) (a) h Ω = 5 MeV- FIG. 6. Same as Fig. 5, but for n . TABLE IX. The single-particle quadrupole moments q (sp) i and the single-particle root-mean-square radii (cid:10) r (cid:11) nl for theoccupied orbitals in the Al atom calculated by the unre-stricted Hartree-Fock method with the 6-31+G basis. Thesingle-particle energy of each orbital, ε j , and the correspond-ing natural orbital (NO) are also shown. Spin NO ε j (cid:10) r (cid:11) nl q (sp) x q (sp) y q (sp) z α s − . . . . . α s − . . . . . α p z − . . − . − . . α p x − . . . − . − . α p y − . . − . . − . α s − . . . . . α p z − . . − . − . . β s − . . . . . β s − . . . . . β p x − . . . − . − . β p y − . . − . . − . β p z − . . − . − . . β s − . . . . . . − . − . . TABLE X. Same as Table IX, but for the Sc atom. Spin NO ε j (cid:10) r (cid:11) nl q (sp) x q (sp) y q (sp) z α s − . . . . . α s − . . . . . α p z − . . − . − . . α p x − . . . − . − . α p y − . . − . . − . α s − . . . . − . α p x − . . . − . − . α p y − . . − . . − . α p z − . . − . − . . α d xy − . . . . − . α s − . . . . − . β s − . . . . . β s − . . − . − . . β p z − . . − . − . . β p x − . . . − . − . β p y − . . − . . − . β s − . . − . − . . β p z − . . − . − . . β p x − . . . − . − . β p y − . . − . . − . β s − . . − . − . . . . . − . TABLE XI. Same as Table IX, but for the Ti atom. Spin NO ε j (cid:10) r (cid:11) nl q (sp) x q (sp) y q (sp) z α s − . . . . . α s − . . . . . α p x − . . . − . − . α p y − . . − . . − . α p z − . . − . − . . α s − . . − . − . . α p z − . . − . − . . α p x − . . . − . − . α p y − . . − . . − . α d yz − . . − . . . α d zx − . . . − . . α s − . . . . − . β s − . . . . . β s − . . . . − . β p x − . . . − . − . β p y − . . − . . − . β p z − . . − . − . . β s − . . . . − . β p x − . . . − . − . β p y − . . − . . − . β p z − . . − . − . . β s − . . . . − . . − . − . . TABLE XII. Same as Table IX, but for the Ni atom. Spin NO ε j (cid:10) r (cid:11) nl q (sp) x q (sp) y q (sp) z α s − . . . . . α s − . . . . − . α p x − . . . − . − . α p y − . . − . . − . α p z − . . − . − . . α s − . . . . − . α p x − . . . − . − . α p y − . . − . . − . α p z − . . − . − . . α d x − y − . . . . − . α d xy − . . . . − . α d yz − . . − . . . α d zx − . . . − . . α d z − . . − . − . . α s − . . . . − . β s − . . . . . β s − . . . . . β p x − . . . − . − . β p y − . . − . . − . β p z − . . − . − . . β s − . . − . − . . β p z − . . − . − . . β p x − . . . − . − . β p y − . . − . . − . β d xy − . . . . − . β d yz − . . − . . . β d zx − . . . − . . β d z − . . − . − . . . − . − . . TABLE XIII. Same as Table IX, but for the Ga atom. Spin NO ε j (cid:10) r (cid:11) nl q (sp) x q (sp) y q (sp) z α s − . . . . . α s − . . . . . α p z − . . − . − . . α p x − . . . − . − . α p y − . . − . . − . α s − . . . . − . α p x − . . . − . − . α p y − . . − . . − . α p z − . . − . − . . α d x − y − . . . . − . α d xy − . . . . − . α d yz − . . − . . . α d zx − . . . − . . α d z − . . − . − . . α s − . . . . − . α p z − . . − . − . . β s − . . . . . β s − . . . . . β p x − . . . − . − . β p y − . . − . . − . β p z − . . − . − . . β s − . . . . − . β p x − . . . − . − . β p y − . . − . . − . β p z − . . − . − . . β d x − y − . . . . − . β d xy − . . . . − . β d yz − . . − . . . β d zx − . . . − . . β d z − . . − . − . . β s − . . − . − . . . − . − . . [1] J. A. Maruhn, P.-G. Reinhard, and E. Suraud, Sim-ple Models of Many-Fermion Systems (Springer-Verlag,Berlin Heidelberg, 2010).[2] K. Hagino and Y. Maeno, A nuclear periodic table,Found. Chem. , 267 (2020).[3] Y. Maeno, K. Hagino, and T. Ishiguro, Three relatedtopics on the periodic tables of elements, in press.[4] A. Bohr and B. R. Mottelson, Nuclear Deformations ,Nuclear Structure, Vol. 2 (W. A. Benjamin, 1975).[5] P. Ring and P. Schuck, The Nuclear Many-Body Prob-lem (Springer-Verlag, Berlin Heidelberg, 1980).[6] T. Minamisono, S. Fukuda, T. Ohtsubo, A. Kitagawa,Y. Nakayama, Y. Someda, S. Takeda, M. Fukuda,K. Matsuta, and Y. Nojiri, Quadrupole moment of thedoubly-closed-shell-plus-one-nucleon nucleus Sc andits core deformation, Nucl. Phys. A , 239 (1993).[7] T. Minamisono, K. Matsuta, K. Minamisono, S. Kudo,M. Ogura, S. Fukuda, K. Sato, M. Mihara, andM. Fukuda, Precise Quadrupole Moment of the DoublyClosed Shell Plus One Proton Nucleus Sc, HyperfineInteract. , 225 (2001).[8] T. Ohtsubo, N. J. Stone, J. R. Stone, I. S. Towner, C. R.Bingham, C. Gaulard, U. K¨oster, S. Muto, J. Nikolov,K. Nishimura, G. S. Simpson, G. Soti, M. Veskovic,W. B. Walters, and F. Wauters, Magnetic Dipole Mo-ment of the Doubly-Closed-Shell Plus One Proton Nu-cleus Sc, Phys Rev Lett , 032504 (2012).[9] H. Sagawa and B. Alex Brown, E2 core polarization for sd -shell single-particle states calculated with a skyrme-type interaction, Nucl. Phys. A , 84 (1984).[10] H. Morinaga, Interpretation of Some of the ExcitedStates of 4 n Self-Conjugate Nuclei, Phys. Rev. , 254(1956).[11] H. Morinaga, On the spin of a broad state around10 MeV in C, Phys. Lett. , 78 (1966).[12] K. Ikeda, N. Takigawa, and H. Horiuchi, The SystematicStructure-Change into the Molecule-like Structures inthe Self-Conjugate 4 n Nuclei, Prog. Theor. Phys. Suppl. E68 , 464 (1968).[13] Y. Kanada-En’yo, H. Horiuchi, and A. Dot´e, Structureof excited states of Be studied with antisymmetrizedmolecular dynamics, Phys. Rev. C , 064304 (1999).[14] N. Itagaki, S. Okabe, K. Ikeda, and I. Tanihata,Molecular-orbital structure in neutron-rich C isotopes,Phys. Rev. C , 014301 (2001).[15] W. von Oertzen, M. Freer, and Y. Kanada-En’yo, Nu-clear clusters and nuclear molecules, Phys. Rep. , 43(2006).[16] M. Itoh, H. Akimune, M. Fujiwara, U. Garg,N. Hashimoto, T. Kawabata, K. Kawase, S. Kishi,T. Murakami, K. Nakanishi, Y. Nakatsugawa, B. K.Nayak, S. Okumura, H. Sakaguchi, H. Takeda,S. Terashima, M. Uchida, Y. Yasuda, M. Yosoi, andJ. Zenihiro, Candidate for the 2 + excited Hoyle state at E x ∼ 10 MeV in C, Phys. Rev. C , 054308 (2011).[17] M. Freer and H. Fynbo, The Hoyle state in C, Prog.Part. Nucl. Phys. , 1 (2014).[18] T. Baba and M. Kimura, Structure and decay pattern ofthe linear-chain state in C, Phys. Rev. C , 044303(2016). [19] T. Baba and M. Kimura, Three-body decay of linear-chain states in C, Phys. Rev. C , 064318 (2017).[20] R. C. Johnson and R. R. Rettew, Shapes of atoms, J.Chem. Educ. , 145 (1965).[21] I. Cohen, Letters, J. Chem. Educ. , 397 (1965).[22] H. Friedrich, Theoretical Atomic Physics (Springer-Verlag, Berlin Heidelberg, 2006).[23] B. J. Austin, V. Heine, and L. J. Sham, General Theoryof Pseudopotentials, Phys. Rev. , 276 (1962).[24] S. G. Louie, S. Froyen, and M. L. Cohen, Nonlinearionic pseudopotentials in spin-density-functional calcu-lations, Phys. Rev. B , 1738 (1982).[25] R. M. Martin, Electronic Structure (Cambridge Univer-sity Press, Cambridge, 2004).[26] P. Schwerdtfeger, The Pseudopotential Approximationin Electronic Structure Theory, ChemPhysChem ,3143 (2011).[27] S. G. Nilsson, Binding states of individual nucleons instrongly deformed nuclei, Dan. Mat. Fys. Medd. , 1(1955).[28] B. R. Mottelson and S. G. Nilsson, Classification of theNucleonic States in Deformed Nuclei, Phys. Rev. ,1615 (1955).[29] I. Ragnarsson and S. G. Nilsson, Shapes and Shells inNuclear Structure (Cambridge University Press, Cam-bridge, 2005).[30] M. A. Blanco, M. Fl´orez, and M. Bermejo, Evaluationof the rotation matrices in the basis of real sphericalharmonics, J. Mol. Struct. , 19 (1997).[31] M. J. Frisch, G. W. Trucks, H. B. Schlegel, G. E.Scuseria, M. A. Robb, J. R. Cheeseman, G. Scal-mani, V. Barone, G. A. Petersson, H. Nakatsuji, X. Li,M. Caricato, A. V. Marenich, J. Bloino, B. G. Janesko,R. Gomperts, B. Mennucci, H. P. Hratchian, J. V. Or-tiz, A. F. Izmaylov, J. L. Sonnenberg, D. Williams-Young, F. Ding, F. Lipparini, F. Egidi, J. Go-ings, B. Peng, A. Petrone, T. Henderson, D. Ranas-inghe, V. G. Zakrzewski, J. Gao, N. Rega, G. Zheng,W. Liang, M. Hada, M. Ehara, K. Toyota, R. Fukuda,J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Ki-tao, H. Nakai, T. Vreven, K. Throssell, J. A. Mont-gomery, Jr., J. E. Peralta, F. Ogliaro, M. J. Bearpark,J. J. Heyd, E. N. Brothers, K. N. Kudin, V. N.Staroverov, T. A. Keith, R. Kobayashi, J. Normand,K. Raghavachari, A. P. Rendell, J. C. Burant, S. S.Iyengar, J. Tomasi, M. Cossi, J. M. Millam, M. Klene,C. Adamo, R. Cammi, J. W. Ochterski, R. L. Martin,K. Morokuma, O. Farkas, J. B. Foresman, and D. J.Fox, Gaussian 16 Revision A.03 (2016), Gaussian Inc.Wallingford CT.[32] P. J. Mohr, D. B. Newell, and B. N. Taylor, CODATArecommended values of the fundamental physical con-stants: 2014, Rev. Mod. Phys. , 035009 (2016).[33] E. Tiesinga, P. J. Mohr, D. B. Newell, and B. N. Tay-lor, The 2018 CODATA Recommended Values of theFundamental Physical Constants (2019).[34] C. C. J. Roothaan, New Developments in Molecular Or-bital Theory, Rev. Mod. Phys. , 69 (1951).[35] J. A. Pople and R. K. Nesbet, Self-Consistent Orbitalsfor Radicals, J. Chem. Phys. , 571 (1954). [36] G. Berthier, Extension de la m´ethode du champmol´eculaire self-consistent `a l’´etude des ´etats `a couchesincompl`etes, C. R. Acad. Sci. , 91 (1954).[37] R. McWeeny and G. Diercksen, Self-Consistent Pertur-bation Theory. II. Extension to Open Shells, J. Chem.Phys. , 4852 (1968).[38] F. Jensen, Introduction to Computational Chemistry ,3rd ed. (John Wiley & Sons, Chichester, 2017).[39] P. Hohenberg and W. Kohn, Inhomogeneous ElectronGas, Phys. Rev. , B864 (1964).[40] W. Kohn and L. J. Sham, Self-Consistent EquationsIncluding Exchange and Correlation Effects, Phys. Rev. , A1133 (1965).[41] W. Kohn, Nobel Lecture: Electronic structure ofmatter—wave functions and density functionals, Rev.Mod. Phys. , 1253 (1999).[42] J. P. Perdew and A. Zunger, Self-interaction correctionto density-functional approximations for many-electronsystems, Phys. Rev. B , 5048 (1981).[43] J. A. Pople, J. S. Binkley, and R. Seeger, Theoreticalmodels incorporating electron correlation, Int. J. Quan-tum Chem. , 1 (1976).[44] J. A. Pople, R. Seeger, and R. Krishnan, Variationalconfiguration interaction methods and comparison withperturbation theory, Int. J. Quantum Chem. , 149(1977).[45] R. Krishnan, H. B. Schlegel, and J. A. Pople, Derivativestudies in configuration-interaction theory, J. Chem.Phys. , 4654 (1980).[46] J. A. Pople, Nobel Lecture: Quantum chemical models,Rev. Mod. Phys. , 1267 (1999).[47] F. Coester, Bound states of a many-particle system,Nucl. Phys. , 421 (1958).[48] J. ˇC´ıˇzek, On the Correlation Problem in Atomic andMolecular Systems. Calculation of Wavefunction Com-ponents in Ursell-Type Expansion Using Quantum-Field Theoretical Methods, J. Chem. Phys. , 4256(1966).[49] J. ˇC´ıˇzek and J. Paldus, Correlation problems in atomicand molecular systems III. Rederivation of the coupled-pair many-electron theory using the traditional quan-tum chemical methodst, Int. J. Quantum Chem. , 359(1971).[50] C. Møller and M. S. Plesset, Note on an ApproximationTreatment for Many-Electron Systems, Phys. Rev. ,618 (1934).[51] R. Krishnan and J. A. Pople, Approximate fourth-orderperturbation theory of the electron correlation energy,Int. J. Quantum Chem. , 91 (1978).[52] M. Head-Gordon, J. A. Pople, and M. J. Frisch, MP2energy evaluation by direct methods, Chem. Phys. Lett. , 503 (1988).[53] G. W. Trucks, E. Salter, C. Sosa, and R. J. Bartlett,Theory and implementation of the MBPT density ma-trix. An application to one-electron properties, Chem.Phys. Lett. , 359 (1988).[54] K. Raghavachari, J. A. Pople, E. S. Replogle, andM. Head-Gordon, Fifth order Moeller-Plesset perturba-tion theory: comparison of existing correlation methodsand implementation of new methods correct to fifth or-der, J. Phys. Chem. , 5579 (1990).[55] W. J. Hehre, R. Ditchfield, and J. A. Pople, Self-Consistent Molecular Orbital Methods. XII. Further Ex-tensions of Gaussian—Type Basis Sets for Use in Molec- ular Orbital Studies of Organic Molecules, J. Chem.Phys. , 2257 (1972).[56] T. Clark, J. Chandrasekhar, G. W. Spitznagel, andP. V. R. Schleyer, Efficient diffuse function-augmentedbasis sets for anion calculations. III. The 3-21+G basisset for first-row elements, Li–F, J. Comput. Chem. ,294 (1983).[57] D. E. Woon and T. H. Dunning, Gaussian basis setsfor use in correlated molecular calculations. III. Theatoms aluminum through argon, J. Chem. Phys. ,1358 (1993).[58] K. A. Peterson, D. E. Woon, and T. H. Dunning, Bench-mark calculations with correlated molecular wave func-tions. IV. The classical barrier height of the H + H → H + H reaction, J. Chem. Phys. , 7410 (1994).[59] W. J. Hehre, R. F. Stewart, and J. A. Pople, Self-Consistent Molecular-Orbital Methods. I. Use of Gaus-sian Expansions of Slater-Type Atomic Orbitals, J.Chem. Phys. , 2657 (1969).[60] A. Kramida, Yu. Ralchenko, J. Reader, and and NISTASD Team, NIST Atomic Spectra Database (ver. 5.7.1),[Online]. Available: https://physics.nist.gov/asd [2020, July 30]. National Institute of Standards andTechnology, Gaithersburg, MD. (2019).[61] For example, in the case of B, 2 p x , 2 p y , and 2 p z or-bitals are occupied with the probability of 1 / 3. Noticethat the wave function in the filling approximation isdifferent from the coherent superposition of those or-bitals, | val (cid:105) = ( | p x (cid:105) + | p y (cid:105) + | p z (cid:105) ) / √ 3, even thoughboth of them lead to a spherical density distribution.[62] J. P. Foster and F. Weinhold, Natural hybrid orbitals,J. Am. Chem. Soc. , 7211 (1980).[63] J. Carpenter and F. Weinhold, Analysis of the geometryof the hydroxymethyl radical by the “different hybridsfor different spins” natural bond orbital procedure, J.Mol. Struct. , 41 (1988).[64] A. E. Reed, L. A. Curtiss, and F. Weinhold, Intermolec-ular interactions from a natural bond orbital, donor-acceptor viewpoint, Chem. Rev. , 899 (1988).[65] E. D. Glendening, A. E. Reed, J. E. Carpenter, andF. Weinhold, NBO Version 3.1, Gaussian Inc. Walling-ford CT.[66] F. A. Cotton, G. Wilkinson, and P. L. Gauss, BasicInorganic Chemistry (John Wiley & Sons, Chichester,1995).[67] F. Hund, Zur Deutung verwickelter Spektren, insbeson-dere der Elemente Scandium bis Nickel, Z. Phys. ,345 (1925).[68] F. Hund, Zur Deutung verwickelter Spektren. II., Z.Phys. , 296 (1925).[69] J. C. Slater, The Theory of Complex Spectra, Phys.Rev. , 1293 (1929).[70] K. Hongo, R. Maezono, Y. Kawazoe, H. Yasuhara,M. D. Towler, and R. J. Needs, Interpretation of Hund’smultiplicity rule for the carbon atom, J. Chem. Phys. , 7144 (2004).[71] T. Oyamada, K. Hongo, Y. Kawazoe, and H. Yasuhara,Unified interpretation of Hund’s first and second rulesfor 2 p and 3 p atoms, J. Chem. Phys. , 164113 (2010).[72] Higher-order deformations, such as hexadecapole defor-mation, exist even though they are considerably small.[73] L. Schiff, Quantum Mechanics , 3rd ed. (McGraw-Hill,New York, 1968). [74] Since the eigenenergies of d x − y and d z are identi-cal if the system is completely spherical, the state withthe last outer-most electron occupying d x − y is almostdegenerate with the state with d z . The deformation pa-rameters β of these two states may have opposite sign,since β (sp) of d x − y and d z orbitals have the same ab-solute value but with the opposite sign (see Eqs. (23b)and (23c)). Therefore, there exist two states close in en-ergy to each other which have opposite sign of β withalmost the same absolute value.[75] R. G. Parr, Density functional theory of atoms andmolecules, in Horizons of quantum chemistry (Springer,Dordrecht, 1980) p. 5.[76] B. D. Serot and J. D. Walecka, The Relativistic NuclearMany-Body Problem, in Advances in Nuclear Physics ,Vol. 16, edited by J. Negele and E. Vogt (Plenum Press,1986).[77] O. Bohigas, A. Lane, and J. Martorell, Sum rules fornuclear collective excitations, Phys. Rep. , 267 (1979).[78] G. F. Bertsch and H. Feldmeier, Variational approachto anharmonic collective motion, Phys. Rev. C , 839(1997).[79] H. Yukawa, On the Interaction of Elementary Particles.I, Proc. Phys. Math. Soc. Jpn. Third , 48 (1935).[80] V. G. J. Stoks, R. A. M. Klomp, C. P. F. Terheggen,and J. J. de Swart, Construction of high-quality NNpotential models, Phys. Rev. C , 2950 (1994).[81] R. B. Wiringa, V. G. J. Stoks, and R. Schiav-illa, Accurate nucleon-nucleon potential with charge-independence breaking, Phys. Rev. C , 38 (1995).[82] R. Machleidt, High-precision, charge-dependent Bonnnucleon-nucleon potential, Phys. Rev. C , 024001(2001).[83] N. Ishii, S. Aoki, and T. Hatsuda, Nuclear Force fromLattice QCD, Phys. Rev. Lett. , 022001 (2007).[84] K. S. Krane, Introductory Nuclear Physics (John Wiley& Sons, Chichester, 1988).[85] B. Povh, K. Rith, C. Scholz, and F. Zetsche, Particlesand Nuclei (Springer-Verlag, Berlin Heidelberg, 2008).[86] M. Bender, P.-H. Heenen, and P.-G. Reinhard, Self-consistent mean-field models for nuclear structure, Rev.Mod. Phys. , 121 (2003).[87] E. Engel and R. M. Dreizler, Density FunctionalTheory—An Advanced Course , Theoretical and Math-ematical Physics (Springer-Verlag, Berlin, Heidelberg,2011).[88] E. Wigner, On the Interaction of Electrons in Metals,Phys. Rev. , 1002 (1934).[89] D. Vautherin and D. M. Brink, Hartree-Fock Calcu-lations with Skyrme’s Interaction. I. Spherical Nuclei,Phys. Rev. C , 626 (1972).[90] D. Vautherin, Hartree-Fock Calculations with Skyrme’sInteraction. II. Axially Deformed Nuclei, Phys. Rev. C , 296 (1973).[91] T. F. Gallagher, Rydberg atoms , Vol. 3 (Cambridge Uni-versity Press, Cambridge, 2005).[92] M. Saffman, T. G. Walker, and K. Mølmer, Quantuminformation with Rydberg atoms, Rev. Mod. Phys. , 2313 (2010).[93] P. J. J. Luukko and J.-M. Rost, Polyatomic TrilobiteRydberg Molecules in a Dense Random Gas, Phys. Rev.Lett. , 203001 (2017).[94] A. S. Stodolna, A. Rouz´ee, F. L´epine, S. Cohen, F. Ro-bicheaux, A. Gijsbertsen, J. H. Jungmann, C. Bordas,and M. J. J. Vrakking, Hydrogen Atoms under Magni-fication: Direct Observation of the Nodal Structure ofStark States, Phys. Rev. Lett. , 213001 (2013).[95] S. Cohen, M. M. Harb, A. Ollagnier, F. Robicheaux,M. J. J. Vrakking, T. Barillot, F. L´epine, and C. Bordas,Photoionization microscopy of the lithium atom: Wave-function imaging of quasibound and continuum Starkstates, Phys. Rev. A , 013414 (2016).[96] J. Itatani, J. Levesque, D. Zeidler, H. Niikura, H. P´epin,J. C. Kieffer, P. B. Corkum, and D. M. Villeneuve, To-mographic imaging of molecular orbitals, Nature ,867 (2004).[97] B. S. Pudliner, A. Smerzi, J. Carlson, V. R. Pandhari-pande, S. C. Pieper, and D. G. Ravenhall, NeutronDrops and Skyrme Energy-Density Functionals, Phys.Rev. Lett. , 2416 (1996).[98] A. Smerzi, D. G. Ravenhall, and V. R. Pandharipande,Neutron drops and neutron pairing energy, Phys. Rev.C , 2549 (1997).[99] S. Gandolfi, J. Carlson, and S. C. Pieper, Cold Neu-trons Trapped in External Fields, Phys. Rev. Lett. ,012501 (2011).[100] S. K. Bogner, R. J. Furnstahl, H. Hergert, M. Korte-lainen, P. Maris, M. Stoitsov, and J. P. Vary, Testingthe density matrix expansion against ab initio calcula-tions of trapped neutron drops, Phys. Rev. C , 044306(2011).[101] P. Maris, J. P. Vary, S. Gandolfi, J. Carlson, and S. C.Pieper, Properties of trapped neutrons interacting withrealistic nuclear Hamiltonians, Phys. Rev. C , 054318(2013).[102] S. Shen, H. Liang, J. Meng, P. Ring, and S. Zhang,Relativistic Brueckner-Hartree-Fock theory for neutrondrops, Phys. Rev. C , 054312 (2018).[103] S. Shen, G. Col`o, and X. Roca-Maza, Skyrme functionalwith tensor terms from ab initio calculations of neutron-proton drops, Phys. Rev. C , 034322 (2019).[104] P. W. Zhao and S. Gandolfi, Radii of neutron dropsprobed via the neutron skin thickness of nuclei, Phys.Rev. C , 041302 (2016).[105] P. Ring, Relativistic mean field theory in finite nuclei,Prog. Part. Nucl. Phys. , 193 (1996).[106] W. Long, J. Meng, N. Van Giai, and S.-G. Zhou, New ef-fective interactions in relativistic mean field theory withnonlinear terms and density-dependent meson-nucleoncoupling, Phys. Rev. C , 034319 (2004).[107] Y. Tanimura, K. Hagino, and H. Z. Liang, 3D mesh cal-culations for covariant density functional theory, Prog.Theor. Exp. Phys. , 073D01 (2015).[108] Y. Tanimura, Clusterization and deformation of multi-Λhypernuclei within a relativistic mean-field model, Phys.Rev. C99