Electron configurations and the periodic table

Hydrogen has one electron, and lesson 6 described its orbitals exactly. Every other atom has several electrons, which repel one another as well as being attracted to the nucleus. Schrödinger's equation cannot be solved exactly for such atoms, but good approximations show that their electrons occupy orbitals much like hydrogen's, with one important difference: orbitals in the same shell no longer all have the same energy. This lesson shows how electrons fill orbitals, writes electron configurations for atoms and ions, and explains why the periodic table has the shape it does.

Orbital energies in many-electron atoms

In hydrogen, the 2s and 2p orbitals have the same energy, as do 3s, 3p and 3d. In a many-electron atom they do not. The reason is shielding. Each electron is attracted by the nucleus but repelled by the other electrons, especially those closer to the nucleus. The inner electrons partly cancel, or shield, the nuclear charge felt by an outer electron. The net charge it experiences is the effective nuclear charge, Z_eff, which is less than the actual nuclear charge Z.

Orbitals differ in how well their electrons penetrate the inner electron cloud. An s electron has some probability of being very close to the nucleus, inside the inner shells, so it experiences a larger Z_eff than a p electron of the same shell. A p electron, in turn, penetrates better than a d electron. As a result, within a shell the subshell energies increase in the order s < p < d < f. The effect is large enough that the 4s orbital lies slightly below 3d for potassium and calcium, which is why the fourth period starts with the 4s orbital before the 3d orbitals fill.

Three rules for filling orbitals

The ground-state electron configuration of an atom, the arrangement of its electrons with the lowest total energy, follows three rules.

The Pauli exclusion principle, stated by Wolfgang Pauli in 1925, says no two electrons in an atom can have the same set of four quantum numbers. Since an orbital is defined by n, l and mₗ, two electrons in the same orbital must have opposite spins (mₛ = +½ and −½). Each orbital therefore holds at most two electrons, and a subshell with 2l + 1 orbitals holds 2(2l + 1) electrons: 2 for s, 6 for p, 10 for d and 14 for f.

The aufbau principle (German for “building up”) says electrons occupy the lowest-energy orbitals available. Filling follows the approximate order 1s, 2s, 2p, 3s, 3p, 4s, 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 5f, 6d, 7p. This order can be remembered with the diagonal diagram below, or, better, read directly from the periodic table.

Hund's rule says that when electrons occupy orbitals of equal energy, such as the three 2p orbitals, they spread out singly with parallel spins before any orbital receives a second electron. Electrons in separate orbitals repel one another less, and parallel spins lower the energy further through a quantum effect called exchange.

Writing electron configurations

An electron configuration lists occupied subshells with the number of electrons as a superscript. An orbital diagram shows each orbital as a box with arrows for electrons, making spins visible. To save space, the configuration of the preceding noble gas can be written in brackets: sodium, 1s² 2s² 2p⁶ 3s¹, becomes [Ne] 3s¹. The electrons in the noble-gas core are core electrons; the electrons beyond it in the outermost shell are valence electrons, the ones involved in chemical bonding. (For transition metals, the d electrons just beyond the core are often counted as valence electrons too.)

The program below generates ground-state configurations by filling subshells in aufbau order. It includes the two most common exceptions among the first 36 elements, discussed next.

ORDER = ["1s", "2s", "2p", "3s", "3p", "4s", "3d", "4p", "5s", "4d", "5p",
         "6s", "4f", "5d", "6p", "7s", "5f", "6d", "7p"]
CAP = {"s": 2, "p": 6, "d": 10, "f": 14}
SYMBOLS = ("H He Li Be B C N O F Ne Na Mg Al Si P S Cl Ar K Ca Sc Ti V Cr Mn Fe "
           "Co Ni Cu Zn Ga Ge As Se Br Kr").split()
EXCEPTIONS = {24: {"4s": 1, "3d": 5}, 29: {"4s": 1, "3d": 10}}   # Cr and Cu

def configuration(Z):
    left, filled = Z, {}
    for sub in ORDER:
        if left == 0:
            break
        n = min(CAP[sub[-1]], left)
        filled[sub] = n
        left -= n
    filled.update(EXCEPTIONS.get(Z, {}))
    return " ".join(f"{s}{n}" for s, n in filled.items() if n)

for Z in (1, 6, 8, 11, 17, 20, 24, 26, 29, 35):
    print(f"{SYMBOLS[Z - 1]:2s} (Z = {Z:2d}): {configuration(Z)}")

H  (Z =  1): 1s1
C  (Z =  6): 1s2 2s2 2p2
O  (Z =  8): 1s2 2s2 2p4
Na (Z = 11): 1s2 2s2 2p6 3s1
Cl (Z = 17): 1s2 2s2 2p6 3s2 3p5
Ca (Z = 20): 1s2 2s2 2p6 3s2 3p6 4s2
Cr (Z = 24): 1s2 2s2 2p6 3s2 3p6 4s1 3d5
Fe (Z = 26): 1s2 2s2 2p6 3s2 3p6 4s2 3d6
Cu (Z = 29): 1s2 2s2 2p6 3s2 3p6 4s1 3d10
Br (Z = 35): 1s2 2s2 2p6 3s2 3p6 4s2 3d10 4p5

Exceptions: chromium and copper

A few configurations break the simple filling order. Chromium is [Ar] 4s¹ 3d⁵ rather than [Ar] 4s² 3d⁴, and copper is [Ar] 4s¹ 3d¹⁰ rather than [Ar] 4s² 3d⁹. The 4s and 3d orbitals are so close in energy that small effects decide the arrangement: a half-filled or completely filled d subshell has particularly favourable exchange energy and reduced electron repulsion. The same pattern appears below them in the table (molybdenum, silver, gold), and there are further exceptions among the heavier transition metals and the lanthanides and actinides. In this course, you should know chromium and copper; the others are best looked up.

Configurations of ions

For main-group elements, ions usually have the configuration of the nearest noble gas. Sodium loses its 3s electron to form Na⁺, [Ne]; chlorine gains one electron to form Cl⁻, [Ar]. Species with the same configuration, such as O²⁻, F⁻, Ne, Na⁺ and Mg²⁺, are isoelectronic.

Transition metals are different in one important way: when they form cations, they lose their outermost s electrons before their d electrons, even though the s orbital filled first. Iron, [Ar] 4s² 3d⁶, forms Fe²⁺ as [Ar] 3d⁶ and Fe³⁺ as [Ar] 3d⁵. Once the metal is ionized, the 3d orbitals drop below 4s in energy, so the 4s electrons are the easiest to remove.

Magnetism

Unpaired electrons give a substance paramagnetism: it is drawn into a magnetic field. Substances whose electrons are all paired are diamagnetic and are weakly repelled by a field. Measuring magnetic behaviour counts unpaired electrons and so tests predicted configurations. Fe³⁺, with five unpaired d electrons, is strongly paramagnetic; Zn²⁺ ([Ar] 3d¹⁰) is diamagnetic. Ferromagnetism, the much stronger permanent magnetism of iron, cobalt and nickel, arises when unpaired spins on neighbouring atoms line up across whole domains of a solid. Lesson 10 uses paramagnetism to test a bonding theory: liquid oxygen sticks to the poles of a strong magnet, which simple bonding pictures cannot explain.

Effective nuclear charge in numbers

Slater's rules, devised by John Slater in 1930, give a quick estimate of the shielding constant. For an electron in an s or p orbital: electrons in higher shells contribute nothing; other electrons in the same ns, np group contribute 0.35 each (0.30 for 1s); electrons in the n − 1 shell contribute 0.85 each; and electrons in shells n − 2 and below contribute 1.00 each. The results are approximate but capture the trends well.

# Slater's-rules estimate of Z_eff for the outermost s/p electron of main-group atoms
def zeff_valence(Z, config):
    """config: list of (n, electrons in ns+np) shells from inner to outer, e.g. [(1,2),(2,8),(3,1)]"""
    n_out, e_out = config[-1]
    S = (e_out - 1) * (0.30 if n_out == 1 else 0.35)
    if len(config) >= 2:
        S += config[-2][1] * 0.85
    S += sum(e for n, e in config[:-2]) * 1.00
    return Z - S

atoms = {
    "Li": (3,  [(1, 2), (2, 1)]),
    "C":  (6,  [(1, 2), (2, 4)]),
    "F":  (9,  [(1, 2), (2, 7)]),
    "Na": (11, [(1, 2), (2, 8), (3, 1)]),
    "Cl": (17, [(1, 2), (2, 8), (3, 7)]),
    "K":  (19, [(1, 2), (2, 8), (3, 8), (4, 1)]),
}
for sym, (Z, cfg) in atoms.items():
    print(f"{sym:2s}: Z = {Z:2d}, Z_eff on valence electron = {zeff_valence(Z, cfg):.2f}")

Li: Z =  3, Z_eff on valence electron = 1.30
C : Z =  6, Z_eff on valence electron = 3.25
F : Z =  9, Z_eff on valence electron = 5.20
Na: Z = 11, Z_eff on valence electron = 2.20
Cl: Z = 17, Z_eff on valence electron = 6.10
K : Z = 19, Z_eff on valence electron = 2.20

Two patterns stand out. Across a period, from lithium to fluorine, Z_eff rises steadily, because each added proton is only partly shielded by the added electron in the same shell. Down a group, from lithium to sodium to potassium, Z_eff on the valence electron changes little, but the electron is in a larger shell, farther from the nucleus. These two facts explain most of the periodic trends in lesson 8.

The shape of the periodic table explained

The periodic table is a map of electron configurations. Its blocks correspond to the subshell being filled. The s block (groups 1 and 2, plus helium) fills an s orbital; the p block (groups 13 to 18) fills p orbitals; the d block (the transition metals, groups 3 to 12) fills d orbitals; and the f block (lanthanides and actinides) fills f orbitals. The widths of the blocks, 2, 6, 10 and 14 columns, are exactly the capacities of the subshells. The period number gives the principal quantum number of the outermost s and p electrons, while the d subshell being filled in period n is (n − 1)d and the f subshell is (n − 2)f.

Elements in the same group have the same number and type of valence electrons, which is why they behave alike. All the alkali metals end in ns¹, so all lose one electron readily and react vigorously with water. All the halogens end in ns² np⁵, one electron short of a noble-gas configuration, so all gain an electron readily. The noble gases have filled ns² np⁶ shells (1s² for helium) and are very unreactive. Mendeleev built his table from chemical behaviour without knowing anything about electrons; quantum mechanics explained, half a century later, why his arrangement works.