Understanding the Uniqueness of 2p Elements in Periodic Tables

Abstract The Periodic Table, and the unique chemical behavior of the first element in a column (group), were discovered simultaneously one and a half centuries ago. Half a century ago, this unique chemistry of the light homologs was correlated to the then available atomic orbital (AO) radii. The radially nodeless 1s, 2p, 3d, 4f valence AOs are particularly compact. The similarity of r (2s)≈r(2p) leads to pronounced sp‐hybrid bonding of the light p‐block elements, whereas the heavier p elements with n≥3 exhibit r (ns) ≪ r (np) of approximately −20 to −30 %. Herein, a comprehensive physical explanation is presented in terms of kinetic radial and angular, as well as potential nuclear‐attraction and electron‐screening effects. For hydrogen‐like atoms and all inner shells of the heavy atoms, r (2s) ≫ r (2p) by +20 to +30 %, whereas r (3s)≳r(3p)≳r(3d), since in Coulomb potentials radial motion is more radial orbital expanding than angular motion. However, the screening of nuclear attraction by inner core shells is more efficient for s than for p valence shells. The uniqueness of the 2p AO is explained by this differential shielding. Thereby, the present work paves the way for future physical explanations of the 3d, 4f, and 5g cases.


Introduction
Knowing the trends along as eries of relatedc ompounds is valuable for every chemist. Understanding the underlying physical reasonsi se ven better.T he individual chemical facts can be related to the general physicall aws,s tepwise, by first finding some generalizing empirical rule, and then rationalizing the rule by atomistic and electronic models that can be deduced from aq uantum chemical basis.
The unique chemical behavior of the 2p elements of the second period of the table of elements, in particular of Bt oF , is well known. [1][2][3][4][5][6][7][8][9][10][11][12][13] Yet, this chemical insight still needs better physicalr ationalization, and better integration into the chemical curricula.W eh ere present ac omprehensive analysis of the nsp valence atomic orbitals (AOs) of the p-blocke lements, that is, of the canonical orbitals from Hartree-Fock or Dirac-Fock or Kohn-Sham levels of theory;w hich simulatet he observable spatial and energetic changes in physical ionization and excitation processes.
In 2019, we celebrated the sesquicentenary of the first comprehensive tables of chemical elements, developed by Meyer, Mendeleev,a nd others in the 1860s. [1,14,15] Mendeleev also realized the uniqueness of elements Ha nd Li to F, following earlier notes in Gmelin's handbook of 1843. [16] Although Meyer contributed to the discussion, he didn ot emphasize the aspects of the "uniqueness". [17] It took another century until Jørgensen [2] related the uniqueness of the first elementi nany vertical group of the periodic table to the exceptionally small radial extensions of the 1s, 2p, 3d, and 4f AOs. Orbital functions of all atoms had become known through the then-possible routine atomic structure computations (see also the Supporting Information). The 1s, 2p, 3d and 4f AOs are characterized by having no radial nodes, in contrastt ot he n' AOs with higherp rincipal quantum numbers n >'+ 1, which have nÀ(' + 1) radial nodes (n and ' are the principal and angular quantum numbers). This so-called Radial Node Effect is now linked to aw ell-documented set of empirical chemical phenomena. [2][3][4][5][6][7][8][9] Conceptual Analysis of Observationsand Computations Radial node effect and core screening The radii of the valence AOs, the Radial Node Effect, and the screening of the nuclear attraction potentialb yt he electronic core shellsw eree xplicated by Shchukarev in great chemical detail, and reviewed in the 1970s. [3,4] FollowingJ ørgensen, he was the first to rationalize the comprehensive bulk of empirical chemistry of those elements, where an orbital angular momentum number ' appears for the first time. Shchukarev named the 1s, 2p, 3d, and 4f AOs kaino(ceno)symmetric (Greek: kainó&,k ainos = new) and Pyykkç [5] named them primogenic (Latin:p rimus = first, genitus = born).
Harris and Jones [6] investigated the different geometric and electronic structures and the bondingi ng roup 14 dimers (C 2 to Pb 2 )a nd highlighted the nodelessnesso ft he C2ps hell. In a seminalr eview,K utzelnigg [7] pointedo ut that the distinct hybridization of bondedB ,C ,N ,Oatoms is mainly due to the similar radii of their sa nd pv alenceA Os, occurringd espite the rather different sa nd pA Oe nergies in the second period. The resultingi mpressive difference of structure and bonding of C 2 H 2 and Si 2 H 2 has recently been elucidated by Ruedenberg et al. [18] Only for F, the very different AO energies e(2s) ! e(2p) suppress any significant hybridization. [7] In recent decades, various excellent, chemically oriented reviewsh ave been published by Kaupp, Huheey,a nd others. [8][9][10][11][12][13] Thereby,t he macroscopic chemical observations were realized as empirical trends and qualitatively rationalized at the AO level.
Shchukarev's rationalization was based on two physical mechanisms, [4] of potential and of kinetic type, which in cooperation cause the unique pattern of AO radii and therebyy ield the unique chemistries of each first element of ag roup. The first, potential energy,m echanism is relatedt ot he penetration of the sv alence AOs deep into the atomic core, where the attraction of the effective nuclear Coulomb potential is large. This effect of deep potential energyw ill be quantitatively explored below by an analysiso fc ore screening, using numerical quantum mechanical computations of many-electron atoms within the orbital model (ford etailss ee the Supporting Information, S.2-4). We confirm Shchukarev's educated guess that the pA Os are better shielded from nuclear attraction than the sA Os, meaning an actual deviationf rom Slater's lowest-order approximation of similarscreening of sa nd pAOs. [19] Shchukarev was not fully confident aboutt he second mechanism,r elated to kinetic energy,t hat is, the Radial Node Effect. More radial, instead of more angular,m otion means in wave mechanics that the orbitals have more radial maxima and nodes and fewera ngular maxima (lobes) and nodes. The nodal pattern of aw ave functioni sd etermined by the boundary conditions, the stationary energy,a nd the potentialf unction. This interrelation will be quantitativelye xploredb yu sing mathematical derivations of one-electron atoms with different model potentials. [20,21] We also elucidate the meaning of the so-called non-bondedP auli repulsions by lower-energy occupied (as well as virtual unoccupied) orbitals, represented by pseudopotentials [22] that simulate the orbitalo rthogonality constraint. Ha nd the 2 nd period elements have, respectively, no and ap articularly small 1s 2 Pauli-repelling atomic core shell, as comparedt othe heavierelements.
The second, kinematic mechanism hasb ecome ap opular rationalizationo fthe primogenic effect in chemistry,for example, in refs. [5,8,9],a ssuming that the centrifugal force simply causespAOs to be more expanded than sA Os. However,t he chemicald ifferences and respective physical causes appear ratherc omplicated in the four different sets of 1s, 2p, 3d, and 4f elements. Compare, for example, the 1s case (H 1s 1 vs. Li 2s 1 &F2p 5 ;a nd He 1s 2 vs. Be 2s 2 &N e2p 6 )w ith the 4f case (the 15 lanthanoids La-Gd and Gd-Lu are chemically similart oA ca nd the later actinoids Cm-Lu, whereas the early actinoids Th-Am are more or less akin to de lements Hf-Ir). Clearly,t he chemical diversity is richer than expected on the basis of the Radial Node Effect alone.

The physical problem with the pblock
We will elucidate the physicalm echanismst hat cause the diversityo fns-npr adiip atterns of thec hemically diverse light and heavy p-block elements. The principal quantum number of an AO is [Eq. (1)] where 1 is the quantum number of radial nodes, ' of angular nodes, and the ' + 1' originates from the Heisenberg Uncertainty principle of quantum theory (see also the SupportingI nformation, S.6). Ac ommonc onjecture in chemistry is that the local value of the repulsive centrifugal force for an electron with angular quantum number '>0, [Eq. (2)] (in atomic units, au), moves the outer maximum of an orbital to larger radii. But for hydrogen atoms, sAOs are more extended than pA Os of the same energy ( Figure 1, left). The explicit formula for the hydrogenic hri values [20] is (in au;n ote the minus sign in fronto ft he angular momentum term) [Eq. (3)]: On the other hand, for the second periodp -block elements with somewhat larger and slightly screenedn uclear Coulomb potentials (by the 1s 2 core shell), the valence 2s,2p AOs with somewhat different energies e(2s) < e(2p) have similar radial extensions, r(2s) % r(2p), for various definitions of r (see Ta ble 2), whereas for the heavier elements of the n th period with significantly larger and significantly more screened nuclear Coulombp otentials (by the 1s 2 to (nÀ1)p 6 core shells), the nsa nd npA Os with less different orbital energies e(2s) 9 e(2p) have r(ns) ! r(np) (Figure 1, right). This pattern is summarized in Ta ble 1. Clearly,b oth kinetic and potential energy effects need to be considered in any,even qualitative, explanation.
The 'empirical' finding is: Iso-energetic conversion of radial into angular motion contractshydrogenic orbitals.
The hydrogenic AO energies (in a.u.) [Eq. (5)]: depend only on the principal quantum number n = 1 + ' + 1 [Eq. (1)].T he energy quanta of radial and angularm otionsa re equal (see the Supporting Information, S.6). That does not hold for mosto ther potentials such as fort he harmonic oscillator,t he particle in ab ox, or the linear potential (modeling, e.g., the strong color interaction of quarks with constantf orce at large distances, or the vertical motion of am ass on at rampoline). In general, there are large gaps between the sand porbital levels.F or instance, the energieso ft hree-dimensional harmonic vibrations in potential V(r)~+ r + 2 (comparew ith the Coulombp otential V(r)~Àr À1 ,s ee Eq. (1), Eq. (5)) are [Eq. (6)], [Eq. (7)] In general, one cannot compareq uantized sa nd ps tates at the same or similar energies, as for atoms. For the harmonic oscillator, e(p) is just in the middle betweent he two corresponding e(s) levels.
We now investigate how the shielding of the nuclear attraction of the outer valence sa nd pA Os by the inner core shells emerges. Different relativistic Dirac-Fock and Kohn-Sham density functional calculations yield similar trends. Technical details are given in the Supporting Information (S.4). We increase the nuclear charge of an excited Ha tom and simultaneously add electrons.
First, we add valence electrons to H(ns,np) 1 (H*), obtaining Be ns 2 np 2 (Be**) with an empty core and populated valence shells as in atoms C( n = 2), Si (n = 3), or Ge (n = 4). Since the AO radii vary as~1/Z (compare Eq. (3)), we plot 1/hri versus Z to obtain nearly linear lines ( Figure 3). Up to an n-dependent scale factor,1 / hri had been defined as Z eff by Hartree, [24] that is, Figure 3a ctually shows Z eff versus Z. The steep slopes on the left side indicate strong AO contraction and Z eff increaseu pon increaseo fZ,o wing to the weak screening of the increasing nuclear charge when electrons are added in the same valence shell. This is in accordance with Slater's [19b] medium-small mean shielding constantw ith an average value of s = 0.35 for s,p shells. The screening constant s corrects the nuclear charge Z to ZÀs;t his can be applied to the exponent in the wavefunction [21] or to expressions for orbital energies, or for orbital radii, as suggested by Hartree. He stressed "there is no single 'screening parameter'w hich will represent all the properties. This is perhaps not always sufficiently realized". In quantum defect theory,t wo weakly varying parameters s ' and d ' are neededt od escribe an outer orbital with quantum numbers n,',t he effective screening by ZÀs ' ,a nd the effective phase shift or quantum number by nÀd ' .
Second, we populate the core shells until reaching C, Si, or Ge. Screening by inner shells is more efficient (s!1), whereas screening by outer (Rydberg) shellsw ould be even weaker (s!0).T he radii of the sp valence shells behave approximately as expected for Slater's nuclear screening by the next inner (s s = s p = 0.85) and furtheri n-bound sp shells (s s = s p = 1), corresponding to the flat lines on the right side of Figure 3. However,s omewhat different screening for sthan for pA Os was already noticedb yC lementi et al. [25] This is here reconfirmed by the steeperl ines for nst han for npA Os in Figure 3. This is due to the stronger core penetration of the sA Os. Concerning the AO radii, the differential screening of the valencesversus p AOs by the next inner sp core shell is better described by s s % 0.7 and s p % 0.9 to 1.0, than by Slater's single averaged value of s s,p % 0.85. As ac onsequence, the valence sA Os of heavier p-block atoms are eventually more contracted than the pA Os.

Core vs. valence screenings
The s/p radii ratios Q n for1 -electronic H*, for 4-valencee lectronic Be**, both with empty lower shells,a nd for 1-valence electronic atoms with filled core shells (A = Li, Na, Cu), and for group 14 atoms of the second to fourth period (E = C, Si, and Ge) are displayed in Table 3. The hri and r max ratios show similar trends as sketched in Figure 4. The core-valence inter-shell Figure 3. Inverse radii hn' j r j n'i À1 (in nm À1 )corresponding to Z eff of the atomic n' valenceorbitalsversus nuclear charge number Z (withe lectronic core-holec onfigurations (see Supporting Information, S. [4][5]. The straight lines( full for s, dottedf or p) guidethe eyes from H* n(sp) 1 through Be** ns 2 np 2 to C1s 2 -2s 2 2p 2 or,respectively,Si1s 2 2s 2 2p 6 -3s 2 3p 2 .The smaller slopes for npv ersust he steeper for nsf rom Be onward to Co rS is how that the npv alence orbitals are bettershielded from the (increasing) nuclear charge by the (increasing numbero f) core electrons than the nsorbitals. Note the change of order from H* and Be**(r s > r p )t oCor Si (r s < r p ). Table 2. Ratios Q n = r ns /r np of the variousv alence orbital radii( r max , hri, p hr 2 i)i np eriod n. Resultsf or Hf rom Bethe.

Inferences
When periodic tables of elements were designed1 50 years ago, it was realized that the first element in ac olumn is spe-cial. Ag ood centuryl ater,alarge body of observed facts on the light homologs had been collected, classified, and related to the comparatively small radii of the 1s, 2p, 3d, and 4f valence AOs, resulting in different bonding schemes for the first versus the heavier homologs. Concerning the unique pe lements of the second period, another half century passed until the final step of understanding is now achieved. The physical elucidation reveals why r ns ! r np for the valence shells of most p-block elements (with dominant p-bonding),e xcept r 2s % r 2p in the second period (with dominant sp-hybrid bonding). The hydrogenic relations are in contrast r 2s @ r 2p ,a nd r ns 0 r np for n > 2. Althoughi ndependent-electron orbitals in principle do not exist by themselvesi nm any-electron systems, [29] they have provena sa na pproximate and very useful concept and tool to explain and understand the behavior of chemical systems. In the present context, several important points need to be taken into account: (1) The orbital set of an atom (or molecule) emerges as ac oherents et, describing the quantized motions of electrons in the nuclear Coulomb potential, screened by the other electrons. Canonical orbitals are conveniently chosen as am utuallyo rthogonal set. Owing to the mutual orthogonality and to the atomic potential V(r), the inner node positions of different radial orbitals of given ' occur at similar places, determined by the shape of the potential. The number of radial and angular orbitaln odes and extrema is the number of quanta of the respectiver adial and angular motions. The number of quanta is required either by the Pauli principle and the occupiedc ore shells,o rb ye lectronic excitation into ah ighero rbitala bove unoccupied ones. An orthogonality constraint on lower (occupied or unoccupied) AOs can be simulated by ap seudopotential, which is repulsivei nt he inner region.Apseudopotential is au seful tool for computations and for explanations, it does not represent ap hysical causal effect of mutual orthogonality.
(2) Coulomb potentials~Àr À1 are flattish at larger nuclear distances, with an arrow deep well at the center. This potential shape yields rather large orbital radii of the 2s, 3s, and 4s AOs, that is, significantly larger than the radii of orbitals 2p, 3d, 4f at the same energies. Radial motion (nodes) in an uclear Coulomb potential movest he outer maximum of an orbital to larger radii (the Radial Node Effect) than the centrifugal force, in particulari nt he case of the 2s-2p orbitalp air,w here the 2p AO has no radial node. The radially nodeless 7i (' = 6) AO is smaller than 7s and even than 6s. For the 2s,2pA Os of Ha s well as of heavy atoms with occupied core shells, (r max ) 2s /(r max ) 2p is around 1.3, and hri 2s /hri 2p is around 1.2.
(3) Radial oscillation samples the potentiala tl arge distances as well as in the nuclear vicinity,e speciallyf or the ' = 0s states. For atomic ions Z + q ,t he effective Z eff varies from Z eff = Z at r = 0t oZ eff = 1 + q at large r. The differenceo ft he effective potentiali nside the atomic cores, compared with pure Coulomb potentials, leads to contracted sA Os (the Core Screening Effect). Slater's lowest-order approximation of similarc ore screening for sa nd pv alence AOs is not accurate enough for the chemical problemsa th and. Although ab asically Coulombic potential yields the orders hri 2s @ hri 2p and hri 3s 0 hri 3p 0 hri 3d for the inner core orbitals of all heaviera toms (the Radial Figure 4. Variation of the ratios Q n = r ns /r np of valenceorbitalradii,upon strong differential core-(s,p)valence inter-shell screening( D core %À0.40, Red);upon weak (s,p)valence intra-shell screening (D val %À0.05, Blue); and the double-screening cross term of opposite sign (d(c,v) % + 0.15, Lilac). Table 3. Ratios Q n of ns/npv alence orbital radii of excitedh ydrogen-like states H* (ns,np) 1 ,o fg roup1 or 11 atoms A(ns,np) 1 ,of highly excited empty-coreB e** ns 2 np 2 configurations,a nd of atoms E ns 2 np 2 ,E = C, Si, and Ge.
(4) Some update of chemical explanations appears appropriate. (a) The centrifugal force owing to aq uantum of angular motion (corresponding to an angular node of the AO), is comparable to the expanding effect of aq uantum of radial motion, represented by ar adial node. Which is more effective depends on the shape of the effective potential in thec ore. (b) Electrons in valence sA Os of many-electrona toms are less shielded and more attracted and contracted than their pc ounterparts. (c) Both the energetic and radial patterns of the valence AOs determine the bondingb ehavior of an element.
Other aspects may also be highlighted, [30,31] in particular the diverseP auli repulsions by the divers atomic core shells, [13c, 36] namely the small 1s 2 core of the 2 nd period and the 'standard sized' (n-1)p 6 or (n-1)p 6 d 10 cores of the heavier n th periodsc odetermine the interatomic separations, andt hereby the different valence-orbitalo verlaps. More exotic core interpretations (such as by spurious nodes ando uter tails of inner core orbitals, or by taking formal charges seriously) have been refuted. [32a,b] Conclusion Kinetic and potential energy effects and their interplay need be analyzed together in physicale xplanations of chemistry. There is at endency to explain covalent bonding electrostatically,w hereas the electronic-kinetic aspect is physically dominating. [33] Conversely,t he radii ratio of the s/p valence AOs governing the bonding and chemistry of the p-blocke lements is dominantlyd etermined by the screening of the electrostatic core potential, whereas the kinetic Radial Node Effect has more pedagogic appeal.
The relevance of each term (radial vs. angular motion in a more or less shielded Coulombp otential) can only be judged on the basis of quantitative data, in particular in the more complicated cases of da nd fo rbitals. That is needed for a better future understanding of chemistry over the periodic table. Investigations of differentials creening connected to the Radial Node Effect, asp resented here for the p-block, are still awaiting their turn to trace the physical origin of the chemical peculiarities of the 3d, and 4f,n ot to mention the hypothetical 5g block. [34] The peculiarity of the non-primogenic early 5f elements belongs to this field, too.
In summary,t he 2p elements are known to be qualitativechemically different from their heavierc ongeners. The physical origin is the quantitative interplay of the electronic kinetic and potentiale nergies of the valence orbitals:2 ph as no radial node and little radial kinetic energy,t hus 2p is radially contracted. All so rbitals are weakly shielded from nuclear attraction, thus 2s, 3s, 4s etc. are radially contracted. Therefore r 2s /r 2p % 1, but r ns /r np < 1f or n > 2. The uniqueness of the 2p (as well as the 3d and 4f)b lock elements exhibitingt he quantum primogenice ffect plays as ignificant role in general chemistry.
The effect is also essential for the topicals upport influence in heterogeneous catalysis. [35]