PERSPECTIVES
The PTE is the most fundamental pillar
The periodic table and the physics of chemistry4: molecules, large or small, are
all made of interacting atoms from the PTE,
that drives it forming various types of chemical bonds.
To cite Shaik and colleagues, “The periodic
table gave rise to a central paradigm, which
Peter Schwerdtfeger , Odile R. Smits and Pekka Pyykkö did for chemistry what Newton had done
for physics and Darwin for biology”7.
Abstract | Mendeleev’s introduction of the periodic table of elements is one of the Questions naturally arise from this ordering
most important milestones in the history of chemistry, as it brought order into system: what are the underlying (quantum)
the known chemical and physical behaviour of the elements. The periodic table can principles of the PTE? Where does the
PTE end from an electronic or nuclear
be seen as parallel to the Standard Model in particle physics, in which the elementary
point of view? How far can we go in the
particles known today can be ordered according to their intrinsic properties. The synthesis of new elements and isotopes
underlying fundamental theory to describe the interactions between particles both in the laboratory and in the interstellar
comes from quantum theory or, more specifically, from quantum field theory and its environment? Can we keep using the
inherent symmetries. In the periodic table, the elements are placed into a certain same approach to unambiguously place
period and group based on electronic configurations that originate from the Pauli the elements with nuclear charge Z > 118
into the PTE9 (as, for example, suggested in
and Aufbau principles for the electrons surrounding a positively charged nucleus. 2011 by one of the authors and shown
This order enables us to approximately predict the chemical and physical properties in fig. 1c)?
of elements. Apparent anomalies can arise from relativistic effects, partial-screening In this Perspective, we address
phenomena (of type lanthanide contraction) and the compact size of the first fundamental questions concerning the PTE
shell of every l- value. Further, ambiguities in electron configurations and the and discuss the current status of this field
from a quantum theoretical point of view10,11.
breakdown of assigning a dominant configuration, owing to configuration mixing
We describe the underlying physical
and dense spectra for the heaviest elements in the periodic table. For the short-lived principles that shape the PTE, including
transactinides, the nuclear stability becomes an important factor in chemical the elements up to a certain critical nuclear
studies. Nuclear stability, decay rates, spectra and reaction cross sections are also charge (Z ≈ 172). We focus on anomalies
important for predicting the astrophysical origin of the elements, including the in chemical and physical properties, rather
production of the heavy elements beyond iron in supernova explosions or than on similarities between the elements
within a certain group. Furthermore, we
neutron-star mergers. In this Perspective, we critically analyse the periodic table of discuss the astrophysical origin and nuclear
elements and the current status of theoretical predictions and origins for the stability of the elements, including the
heaviest elements, which combine both quantum chemistry and physics. most recent developments in the field of
nuclear-structure theory.
In 1869, Dmitri Ivanovich Mendeleev wall-hanging PTE (fig. 1a). Mendeleev had
ordered the known elements into what he no knowledge of the internal structure of From fundamental physics to the PTE
termed the ‘periodic table of the elements’ an atom or nucleus; a more detailed picture The PTE is as fundamental to chemists
(PTE) on the basis of their increasing started to emerge only in 1911 with Ernest as the table of elementary particles is to
atomic weight and chemical similarity1. Rutherford’s discovery of the atomic nucleus. physicists (fig. 2). We all know that atoms
Mendeleev’s PTE was proposed 5 years after The development of the PTE over the past interact to form chemical bonds. Note that
Lothar Meyer had organized the 28 known 150 years is nicely illustrated at the Internet the term ‘chemical bond’ is a fuzzy concept,
elements into a table, of which six columns Database of Periodic Tables. In the most because it does not strictly correspond
were labelled with valence number and five recent version of the PTE (fig. 1c), elements to a quantum-mechanical observable.
rows with atomic weight (see Box 1 and, for are ordered according to their atomic However, it is a useful concept derived
a historical account, see refs2–8). number Z (the number of protons inside from quantum-theoretical principles12–14
Mendeleev not only correctly identified the nucleus), thus avoiding irregularities in and can be attributed to the lowering of
several of the then unknown elements, mass numbers due to the different numbers the electronic kinetic energy, concomitant
such as Ge, Sc, Ga and Tc — that were of neutrons inside the nucleus. As of today, with the constructive interference
subsequently discovered in 1876, 1879, 1886 118 elements are experimentally known, between the constituents in the molecular
and 1937, respectively — but also corrected with the most recent additions to the PTE wavefunction15,16. In much the same way,
some erroneous atomic weights, such as for being the main-group elements from Nh fermions (spin 1/2 particles, like the
Be, In, Ce and U. figure 1 shows a collection (Z = 113) to Og (Z = 118), thus successfully electron) interact through (gauge) fields
of PTEs, including an 1885 version of a completing the seventh period of the PTE. described by the exchange of bosons
Nature reviews | Chemistry
, Box 1 | A bit of history: Dmitri ivanovich mendeleev and Lothar meyer
Aufbau principle introduced by Bohr and
Pauli that, together with Hund’s rule, is
the international year of the Periodic table (iYPt) in 2019 commemorated the 1869 papers of Dmitri considered as the second building block of
ivanovich Mendeleev (see the figure, panel a). Five years earlier, Lothar Meyer (see the figure, panel b) the PTE, after the atomic-number ordering.
had introduced in his 1864 book Die Modernen Theorien der Chemie (1st edn., p. 137), a 28-element
Chemical behaviour is the third most
table with six columns labelled by valence numbers and five rows with increasing atomic weight,
important criterion that guides the order
correcting the te/i mass anomaly293. Meyer’s columns correspond to the groups 1–2 and 14–17. He did
not claim new elements and did not have groups 3–13 or explicitly mention periodicity. He also of elements in the PTE and an essential
attributed valencies to certain transition metals from modern groups 4–12. Meyer later commented tool for all chemists. Similarities in the
(translated from German): “recently, Mendeleyeff has shown that such an arrangement can already valence-electron configurations for two
be obtained by simply arranging atomic weights of all elements without random selection into a atoms usually imply similar chemical
single row according to the size of their numerical values, decomposing such a row into sections and properties, although subtle shell-structure
putting them together in the unmodified sequence. the table shown below is essentially identical to effects can lead to anomalies in the
that given by Mendeleyeff”294–296. we refer to an excellent historical discourse into Lothar Meyer’s life chemical and physical behaviour discussed
and work and comments on this issue by Gisela Boeck297. below. The electron configuration of a
a b multi-electron atom or, more precisely, the
configuration list including occupation
numbers for individual one-electron states,
is (together with the atomic number) an
important parameter for placing an element
into the PTE. The Schrödinger equation
gives us the eigenfunctions in the form of
complex many-electron wavefunctions
and the corresponding eigenstates (that
is, the spectrum) of an atom, molecule or
a condensed phase. The solutions of the
Schrödinger equation inform us on physical
properties (such as the dominant electron
configuration), which give us important
insights into the chemical behaviour of the
elements18. Together with thermodynamics
and statistical physics, this differential
equation lies at the very heart of chemistry.
From the solution of the stationary
Credit for figure part a: Granger Historical Picture archive/alamy stock Photo. Credit for figure part b: Schrödinger equation for a hydrogen-like
the History Collection/alamy stock Photo.
atom, we know that (nlml) states with the
same principal quantum number n are
(integer-spin particles). Bosons are the Why do we mention the fundamental energetically degenerate. In the relativistic
carriers of the fundamental forces known in principles of particle physics here? Because case, this degeneracy is partially lifted owing
nature and are accurately described (as far the concept behind the PTE is strongly to spin–orbit coupling, which can become
as we know) by the Standard Model. Here, connected to fundamental physics very large for heavy elements, leading to
the electromagnetic force is mediated by involving not only atomic and molecular the l > 0 levels split into levels of j = l± 1/2.
photons, the weak force responsible for but also particle and nuclear physics, two Quantum electrodynamics (QED) further
the β-decay in nuclei and the existence of the fundamental aspects that are usually not lifts the degeneracy between the s and p1/2
heavier elements in the PTE are mediated part of mainstream chemistry teaching levels by a small amount. This so-called
by charged (W±) and neutral (Z) bosons, and might not be familiar to all chemists. Lamb shift is tiny, 4.372 × 10−6 eV for the
and the strong force responsible for the Starting with the electronic shell structure 2s–2p1/2 splitting in the hydrogen atom, but
existence of protons, neutrons and nuclei is (the nuclear structure is discussed further can approach chemical relevance for heavy
mediated by gluons. The fourth fundamental below), the population of the PTE is elements19,20, such as Au or Og19–21.
interaction in nature, the gravitational force, governed by both the Pauli and the Aufbau Degeneracies are further broken
has yet to be unified with the Standard principles. At a more fundamental level, in the screened Coulomb potential of
Model, which represents one of the major the spin-statistics theorem in physics multi-electron atoms, for example, following
challenges in physics. If the gravitational (formulated by Fierz and Pauli17) demands the Aufbau principle, the 2s levels are filled
force can be quantized, the carriers of that, for fermions — such as the electron — before the 2p levels. Slater was the first
this force would also be bosons, so-called the many-particle wavefunction ψ (ri, t ) has to extend systematically the one-particle
gravitons. All four fundamental forces are to be antisymmetric with respect to the solutions of the Schrödinger equation to a
important for the astrophysical production permutation of two particles, k and j, from multi-electron system22 following earlier
and existence of the elements in the PTE which the Pauli principle in a single-particle work by Zener23. In the so-called mean-field
and, ultimately, for the existence of life in picture (mean-field theories such as model for a multi-electron atom, each
our universe. Finally, the Higgs spin-zero Hartree–Fock or Kohn–Sham) follows. For electron is moving in the field generated
boson provides the mass for the particles chemists, this simply means admitting only by all other electrons and the nucleus
in the Standard Model (except, perhaps, for one electron per single-particle state. This experiencing a reduced nuclear charge,
the neutrinos). mean-field picture then leads to the famous Zeff = Z− σ, which is due to the shielding
, Perspectives
(or screening) by all the other electrons and two one-electron levels is smaller than the A far more accurate determination of
expressed by the screening constant σ. exchange-energy correction, the lowest electron configurations is achieved using
Slater’s rules provide numerical values for energy is obtained for the high-spin mean-field methods such as relativistic
σ in multi-electron systems that enable configuration. Despite its early success, the Hartree–Fock or Kohn–Sham density
the approximate calculation of the total original Slater rules have their limitations. functional theory (DFT). With these
electronic energy. The idea of a screening They did not explain why the 2s level is methods, one can easily obtain low-lying
effect leads to the lifting of degeneracies occupied before the 2p level, as these shared electronic states associated with dominant
and explains why the 4s level is occupied the same screening factor σ, or more subtle electron configurations and effective nuclear
before the 3d levels (for example, in the case differences in electron configurations, charges for a specific electronic shell. For
of K, Zeff = 2.20 for the 4s1 and Zeff = 1.00 as found, for example, in the group-10 example, Hartree defines the screening
for the 3d1 valence configurations). Slater’s elements: Ni (3d84s2), Pd (4d10), Pt (5d96s1) constant as σnl = Z − 〈r 〉H nl /〈r 〉nl for a specific
approach can be seen as the first successful and Ds (6d87s2). However, it is not important nucleus of charge Z and shell (nl), where
quantum-theoretical attempt to place the which screened-Coulomb potential is 〈r 〉H
nl is the unscreened hydrogenic value,
elements correctly into the PTE using chosen for the Aufbau discussion8. For which can be obtained analytically (for the
the Aufbau principle. It places the electrons example, in the past, the Thomas–Fermi relativistic case, we simply extend it to
obeying the Pauli principle into the levels model was used to determine the atomic the shell with quantum numbers (nlj))28.
experiencing the highest effective nuclear number at which l-electrons for a given lmax From relativistic Hartree–Fock calculations
charge first. However, if the gap between first appear24–27. of Li, we obtain Z 2effs = 1.55 and Z 2effp = 1.04,
a b
1 2
H He
3 4 5 6 7 8 9 10
Li Be B C N O F Ne
11 12 13 14 15 16 17 18
Na Mg Al Si P S Cl Ar
19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr
37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54
Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I Xe
55 56 57–71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86
Cs Ba La–Lu Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Po At Rn
87 88 89 90 91 92–106
Fr Ra Ac Th Pa U–(106)
92 93 94 (95) (96) (106)
U Np Pu
57 58 59 60 61 62 63 64 65 66 67 68 69 70 71
La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu
c Group d
Period 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Orbital
1 2
1 H He 1s
3 4 5 6 7 8 9 10
2 Li Be B C N O F Ne 2s2p
11 12 13 14 15 16 17 18
3 Na Mg Al Si P S Cl Ar 3s3p
19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
4 K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr 4s3d4p
37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54
5 Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I Xe 5s4d5p
55 56 57–71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86
6 Cs Ba La-Lu Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Po At Rn 6s5d6p
87 88 89–103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118
7 Fr Ra Ac–Lr Rf Db Sg Bh Hs Mt Ds Rg Cn Nh Fl Mc Lv Ts Og 7s6d7p
121–
8 119 120 156 157 158 159 160 161 162 163 164 139 140 169 170 171 172 8s7d8p
9 165 166 167 168 9s9p
57 58 59 60 61 62 63 64 65 66 67 68 69 70 71
6 La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu 4f
89 90 91 92 93 94 95 96 97 98 99 100 101 102 103
7 Ac Th Pa U Np Pu Am Cm Bk Cf Es Fm Md No Lr 5f
8 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 6f
8 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 5g
Experimentally known elements Experimentally unknown elements
Fig. 1 | Periodic tables. a | Early example of a periodic table of elements (PTE). valence electrons is given by the group number (G) (groups 1–12) or G−10
This PTE was originally printed in 1885 and purchased in 1888 by the (groups 13–18). The length and the location of the rows reflect both the chem-
University of St. Andrews, where it is currently exposed. b | A 1942 PTE by istry of the elements and the shell structure of their atoms. Note the sug-
Glenn T. Seaborg. In this version, the 5f elements were not introduced yet, gested location of a putative 5g series. d | PTE showing the predicted origin
Th is shown in the group below Hf, Pa below Ta and U below W. The actinide of elements in the Solar System. Elements beyond Pu are not included. Part a
series got its proper place in the PTE in 1944. c | PTE in which a single electron is courtesy of the University of St Andrews Library, ms39012, ref.280. Part b is
configuration is assigned to each atom9. The elements in yellow are now adapted from ref.281, World Scientific. Part c is adapted from ref.282, Springer
experimentally known, as opposed to those in grey. The number of Nature Limited. Part d is reprinted with permission from ref.283, AAAS.
Nature Reviews | Chemistry
The PTE is the most fundamental pillar
The periodic table and the physics of chemistry4: molecules, large or small, are
all made of interacting atoms from the PTE,
that drives it forming various types of chemical bonds.
To cite Shaik and colleagues, “The periodic
table gave rise to a central paradigm, which
Peter Schwerdtfeger , Odile R. Smits and Pekka Pyykkö did for chemistry what Newton had done
for physics and Darwin for biology”7.
Abstract | Mendeleev’s introduction of the periodic table of elements is one of the Questions naturally arise from this ordering
most important milestones in the history of chemistry, as it brought order into system: what are the underlying (quantum)
the known chemical and physical behaviour of the elements. The periodic table can principles of the PTE? Where does the
PTE end from an electronic or nuclear
be seen as parallel to the Standard Model in particle physics, in which the elementary
point of view? How far can we go in the
particles known today can be ordered according to their intrinsic properties. The synthesis of new elements and isotopes
underlying fundamental theory to describe the interactions between particles both in the laboratory and in the interstellar
comes from quantum theory or, more specifically, from quantum field theory and its environment? Can we keep using the
inherent symmetries. In the periodic table, the elements are placed into a certain same approach to unambiguously place
period and group based on electronic configurations that originate from the Pauli the elements with nuclear charge Z > 118
into the PTE9 (as, for example, suggested in
and Aufbau principles for the electrons surrounding a positively charged nucleus. 2011 by one of the authors and shown
This order enables us to approximately predict the chemical and physical properties in fig. 1c)?
of elements. Apparent anomalies can arise from relativistic effects, partial-screening In this Perspective, we address
phenomena (of type lanthanide contraction) and the compact size of the first fundamental questions concerning the PTE
shell of every l- value. Further, ambiguities in electron configurations and the and discuss the current status of this field
from a quantum theoretical point of view10,11.
breakdown of assigning a dominant configuration, owing to configuration mixing
We describe the underlying physical
and dense spectra for the heaviest elements in the periodic table. For the short-lived principles that shape the PTE, including
transactinides, the nuclear stability becomes an important factor in chemical the elements up to a certain critical nuclear
studies. Nuclear stability, decay rates, spectra and reaction cross sections are also charge (Z ≈ 172). We focus on anomalies
important for predicting the astrophysical origin of the elements, including the in chemical and physical properties, rather
production of the heavy elements beyond iron in supernova explosions or than on similarities between the elements
within a certain group. Furthermore, we
neutron-star mergers. In this Perspective, we critically analyse the periodic table of discuss the astrophysical origin and nuclear
elements and the current status of theoretical predictions and origins for the stability of the elements, including the
heaviest elements, which combine both quantum chemistry and physics. most recent developments in the field of
nuclear-structure theory.
In 1869, Dmitri Ivanovich Mendeleev wall-hanging PTE (fig. 1a). Mendeleev had
ordered the known elements into what he no knowledge of the internal structure of From fundamental physics to the PTE
termed the ‘periodic table of the elements’ an atom or nucleus; a more detailed picture The PTE is as fundamental to chemists
(PTE) on the basis of their increasing started to emerge only in 1911 with Ernest as the table of elementary particles is to
atomic weight and chemical similarity1. Rutherford’s discovery of the atomic nucleus. physicists (fig. 2). We all know that atoms
Mendeleev’s PTE was proposed 5 years after The development of the PTE over the past interact to form chemical bonds. Note that
Lothar Meyer had organized the 28 known 150 years is nicely illustrated at the Internet the term ‘chemical bond’ is a fuzzy concept,
elements into a table, of which six columns Database of Periodic Tables. In the most because it does not strictly correspond
were labelled with valence number and five recent version of the PTE (fig. 1c), elements to a quantum-mechanical observable.
rows with atomic weight (see Box 1 and, for are ordered according to their atomic However, it is a useful concept derived
a historical account, see refs2–8). number Z (the number of protons inside from quantum-theoretical principles12–14
Mendeleev not only correctly identified the nucleus), thus avoiding irregularities in and can be attributed to the lowering of
several of the then unknown elements, mass numbers due to the different numbers the electronic kinetic energy, concomitant
such as Ge, Sc, Ga and Tc — that were of neutrons inside the nucleus. As of today, with the constructive interference
subsequently discovered in 1876, 1879, 1886 118 elements are experimentally known, between the constituents in the molecular
and 1937, respectively — but also corrected with the most recent additions to the PTE wavefunction15,16. In much the same way,
some erroneous atomic weights, such as for being the main-group elements from Nh fermions (spin 1/2 particles, like the
Be, In, Ce and U. figure 1 shows a collection (Z = 113) to Og (Z = 118), thus successfully electron) interact through (gauge) fields
of PTEs, including an 1885 version of a completing the seventh period of the PTE. described by the exchange of bosons
Nature reviews | Chemistry
, Box 1 | A bit of history: Dmitri ivanovich mendeleev and Lothar meyer
Aufbau principle introduced by Bohr and
Pauli that, together with Hund’s rule, is
the international year of the Periodic table (iYPt) in 2019 commemorated the 1869 papers of Dmitri considered as the second building block of
ivanovich Mendeleev (see the figure, panel a). Five years earlier, Lothar Meyer (see the figure, panel b) the PTE, after the atomic-number ordering.
had introduced in his 1864 book Die Modernen Theorien der Chemie (1st edn., p. 137), a 28-element
Chemical behaviour is the third most
table with six columns labelled by valence numbers and five rows with increasing atomic weight,
important criterion that guides the order
correcting the te/i mass anomaly293. Meyer’s columns correspond to the groups 1–2 and 14–17. He did
not claim new elements and did not have groups 3–13 or explicitly mention periodicity. He also of elements in the PTE and an essential
attributed valencies to certain transition metals from modern groups 4–12. Meyer later commented tool for all chemists. Similarities in the
(translated from German): “recently, Mendeleyeff has shown that such an arrangement can already valence-electron configurations for two
be obtained by simply arranging atomic weights of all elements without random selection into a atoms usually imply similar chemical
single row according to the size of their numerical values, decomposing such a row into sections and properties, although subtle shell-structure
putting them together in the unmodified sequence. the table shown below is essentially identical to effects can lead to anomalies in the
that given by Mendeleyeff”294–296. we refer to an excellent historical discourse into Lothar Meyer’s life chemical and physical behaviour discussed
and work and comments on this issue by Gisela Boeck297. below. The electron configuration of a
a b multi-electron atom or, more precisely, the
configuration list including occupation
numbers for individual one-electron states,
is (together with the atomic number) an
important parameter for placing an element
into the PTE. The Schrödinger equation
gives us the eigenfunctions in the form of
complex many-electron wavefunctions
and the corresponding eigenstates (that
is, the spectrum) of an atom, molecule or
a condensed phase. The solutions of the
Schrödinger equation inform us on physical
properties (such as the dominant electron
configuration), which give us important
insights into the chemical behaviour of the
elements18. Together with thermodynamics
and statistical physics, this differential
equation lies at the very heart of chemistry.
From the solution of the stationary
Credit for figure part a: Granger Historical Picture archive/alamy stock Photo. Credit for figure part b: Schrödinger equation for a hydrogen-like
the History Collection/alamy stock Photo.
atom, we know that (nlml) states with the
same principal quantum number n are
(integer-spin particles). Bosons are the Why do we mention the fundamental energetically degenerate. In the relativistic
carriers of the fundamental forces known in principles of particle physics here? Because case, this degeneracy is partially lifted owing
nature and are accurately described (as far the concept behind the PTE is strongly to spin–orbit coupling, which can become
as we know) by the Standard Model. Here, connected to fundamental physics very large for heavy elements, leading to
the electromagnetic force is mediated by involving not only atomic and molecular the l > 0 levels split into levels of j = l± 1/2.
photons, the weak force responsible for but also particle and nuclear physics, two Quantum electrodynamics (QED) further
the β-decay in nuclei and the existence of the fundamental aspects that are usually not lifts the degeneracy between the s and p1/2
heavier elements in the PTE are mediated part of mainstream chemistry teaching levels by a small amount. This so-called
by charged (W±) and neutral (Z) bosons, and might not be familiar to all chemists. Lamb shift is tiny, 4.372 × 10−6 eV for the
and the strong force responsible for the Starting with the electronic shell structure 2s–2p1/2 splitting in the hydrogen atom, but
existence of protons, neutrons and nuclei is (the nuclear structure is discussed further can approach chemical relevance for heavy
mediated by gluons. The fourth fundamental below), the population of the PTE is elements19,20, such as Au or Og19–21.
interaction in nature, the gravitational force, governed by both the Pauli and the Aufbau Degeneracies are further broken
has yet to be unified with the Standard principles. At a more fundamental level, in the screened Coulomb potential of
Model, which represents one of the major the spin-statistics theorem in physics multi-electron atoms, for example, following
challenges in physics. If the gravitational (formulated by Fierz and Pauli17) demands the Aufbau principle, the 2s levels are filled
force can be quantized, the carriers of that, for fermions — such as the electron — before the 2p levels. Slater was the first
this force would also be bosons, so-called the many-particle wavefunction ψ (ri, t ) has to extend systematically the one-particle
gravitons. All four fundamental forces are to be antisymmetric with respect to the solutions of the Schrödinger equation to a
important for the astrophysical production permutation of two particles, k and j, from multi-electron system22 following earlier
and existence of the elements in the PTE which the Pauli principle in a single-particle work by Zener23. In the so-called mean-field
and, ultimately, for the existence of life in picture (mean-field theories such as model for a multi-electron atom, each
our universe. Finally, the Higgs spin-zero Hartree–Fock or Kohn–Sham) follows. For electron is moving in the field generated
boson provides the mass for the particles chemists, this simply means admitting only by all other electrons and the nucleus
in the Standard Model (except, perhaps, for one electron per single-particle state. This experiencing a reduced nuclear charge,
the neutrinos). mean-field picture then leads to the famous Zeff = Z− σ, which is due to the shielding
, Perspectives
(or screening) by all the other electrons and two one-electron levels is smaller than the A far more accurate determination of
expressed by the screening constant σ. exchange-energy correction, the lowest electron configurations is achieved using
Slater’s rules provide numerical values for energy is obtained for the high-spin mean-field methods such as relativistic
σ in multi-electron systems that enable configuration. Despite its early success, the Hartree–Fock or Kohn–Sham density
the approximate calculation of the total original Slater rules have their limitations. functional theory (DFT). With these
electronic energy. The idea of a screening They did not explain why the 2s level is methods, one can easily obtain low-lying
effect leads to the lifting of degeneracies occupied before the 2p level, as these shared electronic states associated with dominant
and explains why the 4s level is occupied the same screening factor σ, or more subtle electron configurations and effective nuclear
before the 3d levels (for example, in the case differences in electron configurations, charges for a specific electronic shell. For
of K, Zeff = 2.20 for the 4s1 and Zeff = 1.00 as found, for example, in the group-10 example, Hartree defines the screening
for the 3d1 valence configurations). Slater’s elements: Ni (3d84s2), Pd (4d10), Pt (5d96s1) constant as σnl = Z − 〈r 〉H nl /〈r 〉nl for a specific
approach can be seen as the first successful and Ds (6d87s2). However, it is not important nucleus of charge Z and shell (nl), where
quantum-theoretical attempt to place the which screened-Coulomb potential is 〈r 〉H
nl is the unscreened hydrogenic value,
elements correctly into the PTE using chosen for the Aufbau discussion8. For which can be obtained analytically (for the
the Aufbau principle. It places the electrons example, in the past, the Thomas–Fermi relativistic case, we simply extend it to
obeying the Pauli principle into the levels model was used to determine the atomic the shell with quantum numbers (nlj))28.
experiencing the highest effective nuclear number at which l-electrons for a given lmax From relativistic Hartree–Fock calculations
charge first. However, if the gap between first appear24–27. of Li, we obtain Z 2effs = 1.55 and Z 2effp = 1.04,
a b
1 2
H He
3 4 5 6 7 8 9 10
Li Be B C N O F Ne
11 12 13 14 15 16 17 18
Na Mg Al Si P S Cl Ar
19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr
37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54
Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I Xe
55 56 57–71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86
Cs Ba La–Lu Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Po At Rn
87 88 89 90 91 92–106
Fr Ra Ac Th Pa U–(106)
92 93 94 (95) (96) (106)
U Np Pu
57 58 59 60 61 62 63 64 65 66 67 68 69 70 71
La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu
c Group d
Period 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 Orbital
1 2
1 H He 1s
3 4 5 6 7 8 9 10
2 Li Be B C N O F Ne 2s2p
11 12 13 14 15 16 17 18
3 Na Mg Al Si P S Cl Ar 3s3p
19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36
4 K Ca Sc Ti V Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Kr 4s3d4p
37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54
5 Rb Sr Y Zr Nb Mo Tc Ru Rh Pd Ag Cd In Sn Sb Te I Xe 5s4d5p
55 56 57–71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86
6 Cs Ba La-Lu Hf Ta W Re Os Ir Pt Au Hg Tl Pb Bi Po At Rn 6s5d6p
87 88 89–103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118
7 Fr Ra Ac–Lr Rf Db Sg Bh Hs Mt Ds Rg Cn Nh Fl Mc Lv Ts Og 7s6d7p
121–
8 119 120 156 157 158 159 160 161 162 163 164 139 140 169 170 171 172 8s7d8p
9 165 166 167 168 9s9p
57 58 59 60 61 62 63 64 65 66 67 68 69 70 71
6 La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu 4f
89 90 91 92 93 94 95 96 97 98 99 100 101 102 103
7 Ac Th Pa U Np Pu Am Cm Bk Cf Es Fm Md No Lr 5f
8 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 6f
8 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 5g
Experimentally known elements Experimentally unknown elements
Fig. 1 | Periodic tables. a | Early example of a periodic table of elements (PTE). valence electrons is given by the group number (G) (groups 1–12) or G−10
This PTE was originally printed in 1885 and purchased in 1888 by the (groups 13–18). The length and the location of the rows reflect both the chem-
University of St. Andrews, where it is currently exposed. b | A 1942 PTE by istry of the elements and the shell structure of their atoms. Note the sug-
Glenn T. Seaborg. In this version, the 5f elements were not introduced yet, gested location of a putative 5g series. d | PTE showing the predicted origin
Th is shown in the group below Hf, Pa below Ta and U below W. The actinide of elements in the Solar System. Elements beyond Pu are not included. Part a
series got its proper place in the PTE in 1944. c | PTE in which a single electron is courtesy of the University of St Andrews Library, ms39012, ref.280. Part b is
configuration is assigned to each atom9. The elements in yellow are now adapted from ref.281, World Scientific. Part c is adapted from ref.282, Springer
experimentally known, as opposed to those in grey. The number of Nature Limited. Part d is reprinted with permission from ref.283, AAAS.
Nature Reviews | Chemistry
