Multi-electron atoms

1. Multi-electron atoms
Chapter6: Demtröder

2. Building-up Principle of the Electron Shell for Larger Atoms
The population of electrons in atoms with energy levels (n, l,ml ,ms ) occurs in such a way that
1. The Pauli principle is obeyed.
2. The total energy of all electrons is minimum for the atomic ground state.
• Structure of electron shells of all the elements are explained.
• Arrangement of elements in the periodic table.
Without the Pauli principle the electron shells of all atoms would collapse into the 1s shell with the
lowest energy.
In other words: The Pauli principle guarantees the stability of atoms and the great variety of chemical
properties of the different elements.

3. The Model of Electron Shells
The radial distributions of atomic electrons
Rn,l (r ) is the radial part of the wave function for an
electron with principal quantumnumber n and orbital
angular momentum quantum number l.
For each l → (2l+1)degenerate Yl
m angular wave functions.
For each n → n possible values of orbital (angular) quantum number l = 0, 1, 2, . . . , n − 1.
 different states described by the wave functions yn,l,m(r,q, f) that can be
occupied by at most 2n2 electrons with pairs of opposite spins, according to the
Pauli principle.

4. The time-averaged total charge distribution of all 2n2 electrons
with the same principal quantum numbern
is obtained by summation overthe squares of all possible wave
functions with l < n and −l ≤ ml ≤ +l, where C is a normalization
factor.
• The charge distribution has maxima at certain values of the distance r
from the nucleus.
• These values solely depend on the principal quantumnumber n.
• The main part of the electron charge is contained within the spherical
shell between the radii r − Δr/2 and r + Δr/2 .
• Such a spherically symmetric charge distribution is called an electron
shell.

6. Successive Building-up of Electron Shells for Atoms with Increasing Nuclear Charge
• For lithium, with Z = 3 → (1s)2(2s). The quantum numbers of the third electron are n = 2; l = 0, ml = 0; ms = ±1/2 and the
Li ground state is labeled as 22S1/2.
• The Beryllium atom, with Z=4→ (1s)2(2s)2. The 4th electron still occupy the 2s state (n = 2; l = 0; ml = 0; ms = 0 or 1) if
the spin quantum number ms differs fromthat of the third electron. The ground state of the Be atom is therefore 21S0.
• For the fifth electronin the boronatom thestate 2s is alreadyoccupiedand it has to go into the 2p state with n = 2
and l = 1, ms = ±1/2 . Thegroundstate of B is then 22P1/2.

7. Hund's rule states that:
1.Every orbital in a sublevel is singly occupied before any orbital is doubly occupied.
2.All of the electrons in singly occupied orbitals have the same spin (to maximize total spin).
According to the first rule, electrons always enter an
empty orbital before they pair up. Electrons are
negatively charged and, as a result, they repel each
other. Electrons tend to minimize repulsion by
occupying their own orbitals, rather than sharing an
orbital with another electron. Furthermore, quantum-
mechanical calculations have shown that the electrons
in singly occupied orbitals are less effectively
screened or shielded from the nucleus.
For the second rule, unpaired electrons in singly
occupied orbitals have the same spins. Technically
speaking, the first electron in a sublevel could be
either "spin-up" or "spin-down." Once the spin of the
first electron in a sublevel is chosen, however, the
spins of all of the other electrons in that sublevel
depend on that first spin. To avoid confusion,
scientists typically draw the first electron, and any
other unpaired electron,in an orbital as "spin-up."

8. Carbon and Oxygen
According to Hund’s rule → for constructing electron arrangements requires electrons to be placed one at a time in a
set of orbitals within the same sublevel. This minimizes the natural repulsive forces that one electron has for
another.
Hund's rule states that orbitals of equal energy are each occupied by one electron before any orbital is occupied by a
second electron, and that each of the single electrons must have the same spin.

9. For O, F and Ne the three additional electrons in the 2p shell → their spins must be opposite to that of the three
electrons, already occupying this subshell.
The total spin quantum numberdecreases from oxygen (S = 1) to fluorine (S = 1/2) to neon (S = 0).
The quantumnumbers(L, S, and J ) of the atomic ground states are determined by
1. The total orbital angular momentum 𝑳 = σ𝑙𝑖
2. The total spin 𝑺 = σ 𝑠𝑖
3. Their coupling to J = L + S.
The ground state of C → 23P0 because |L | = |l1 + l2| = 1ℏ
The ground state of N → 24S3/2 because σ 𝑙𝑖 = 0
For neon the L-shell with n = 2 is fully occupied.
The total orbital angular momentum→ 𝑳 = σ 𝑙𝑖 = 0
The total orbital angular momentum→ 𝑺 = σ 𝑠𝑖 = 0
The time-averaged electron charge distribution for neon is spherically symmetric.

10. Atomic Volumes and Ionization Energies
Variation of atomic volume 𝑉 =
4
3
𝜋 < 𝑟 > 3 where
< r >= mean atomic radius with the number Z of
electrons. Atomic volumes from the ratio Vmol =
Mmol/S of molar mass Mmol and density S.
Each time a new electron shell starts to be occupied (for the
elements Li, Na, K, Rb and Cs), the atomic volumes jump
upwards.
The new shell with a higher principal quantum number n has a
larger mean radius r than the shells with lower n values.
If the atomic volume
• The atomic volume dependson Z (number of electrons).
• The dependence of volumes on Z shows a periodicity.
where
MM→ the molar mass
ρ → the density
Na→ the Avogadro number
 This will give slight deviation →
As the density depends on
interatomic force too and these
forces can be attractive or repulsive.

11. Ionization energy → shows similar periodicity
The energy necessary to remove the outer electron (which is the
most weakly bound electron) fromits state Y(n,l,ml) to infinity
is
Ionization energy depends on
1. the average distance 𝑟 = 𝑟𝑛 of the electron from the nucleus
2. the effective charge eZeff = e(Z − S), partly shielded by the inner
electrons, with the shielding parameter S.
• Nobel gases → smallest 𝑟 ➔ largest eZeff → highest ionization energy.
• Alkali atoms → largest 𝑟 ➔ smallest eZeff → lowest ionization energy.
• Ionization potential gradually increases until shell is filled and then drops.
• Filled shells are most stable and valence electrons occupy larger, less tightly bound orbits.

12. Experimental evidence for Shell Model

13. Anomalies in filling the subshells:
1. After 3p subshell of the M-shell → two electrons are into the 4s
subshell of the N-shell.
2. 4s shell is filled with 2 electrons → the next 10 electrons will
occupy the 3d shell.
Reason→ QM calculation suggests that the electrons in the 4s shell
have lower energetic order than the electron in the 3d shell.
Radial wavefunctions → for 3d orbital has maximum electron density
at larger distance than 4s orbital.
Similarly, 5s is occupied before 4d; 6s is occupied before 5d and the
reason is same.

15. To memorize the successive filling of orbital

16. Periodic system of the elements Dmitri Mendeleev and J L Meyer
In the periodic table, elements are divided into:
• the s-block (contains reactive metals of Group 1A(1) and 2A (2)),
• the p-block (contains metals and nonmetalsof Group 3A(13) through 8A(18)),
• the d-block (contains transition metals (Group 3B (3) through Group (2B (12)), and
• the f-block (contains lanthanide and actinide series or inner transition metals).

17. • Elements within the same group
o have the same number of electrons in their valence (outermost) shells.
o they have similar electron configurations.
o They exhibit similar chemical properties.
• Elements within the same period
o have different number of electrons in their valence shells (the number is increasing from left to right)
o different valence shell electron configuration.
o Therefore, elements in the same period are chemically different.

18. Alkali atoms
• Alkali atoms: in ground state, contain a set of completely filled subshells with a single valence electron in the next s
subshell.
• They are Hydrogen-like.
• Electrons in p-subshells are not excited in any low-energy processes. s-electron is the single optically active electron and
core of filled subshells can be ignored. This electron is called a “leucht-electron” fromthe German verb leuchten, which
means “to shine”
• In alkali atoms, the l degeneracy is lifted: states with the same principal quantum # n and different orbital quantum# l
have different energies.
• Relative to H-atom terms, alkali terms lie at lower energies.
• For larger values of n (greater orbital radii), the terms are onlyslightly different from hydrogen.
• Electrons with small l are more strongly bound and their terms lie at lower energies.
• Non-coulombic potential breaks degeneracy of levels with the same principal quantumnumber.

19. F(r)

20. rc → the mean radius of the highest closed electron shell,
the potential Φ(r ) of the leucht-electron can be approximated by a
Coulomb potential for all values r > rc. Since the nuclear charge Ze is very
effectively shielded by the (Z − 1) electrons in closed shells, the effective
charge number is Zeff ≈ 1.
For r < rc this is no longer true, because here the outer electron submerges
into the closed shells and the screening of the nuclear charge becomes less
effective.
Here the potential depends on the radial distribution in the electron shell. The
effective radial dependence of the potential Φ(r ) changes from a Coulomb
potential with central charge Ze at small values of r to one with a completely
screened charge (Z − (Z − 1))e = e at very large distances r

21. Example: For Lithium → the outer electron in the 2s state moves in the
potential of the nucleus with charge Q = +3e and the two screening electrons
in the 1s state.
If ri is the distance of the 2s electron from the nucleus and rij to the jth
electron in the 1s state, the potential for the 2s electron is given by
where ψ(rj ): the wave function of the jth electron. The integration is
performed overthe coordinates of the jth electron.
Approximation: The interaction between the two 1s electrons on their wave
function can be neglected and the wave functions ψj can be written as
hydrogenic 1s wave functions. Inserting the wavefunction yields potential
for 2s electron:

22. For r → 0 the potential equals the Coulomb potential for Z = 3,
while for r →∞ the potential becomes a Coulomb potential with Zeff = 1,
➔the two 1s electrons have the screening factor S =2.
➔For the potential energy Epot(r ) = −e · φeff(r ) we therefore obtain:
At small electron-nuclear distances, F(r), has the shape of the unscreened
nuclear Coulomb potential;
at large distances the nuclear charge is screened to one unit of charge.

23. Conclusions:
1. For the H-atom→ levels with the same “n” but different “l” → Degenerate.
→ this n-fold degeneracy is due to the Coulomb potential
→ this no longer holds for other potentials, even if it is spherically symmetric.
2. Therefore,l-degeneracy is lifted in alkali atoms.
3. Energy sequence:
For large principal quantumnumbers n (which means large mean distances
<r> of the electron from the nucleus), where the potential approachesthe
Coulomb potential of the H-atom the energy levels of the alkali atoms can
be described by the modified Rydberg formula
where n is replaced by an effective quantum number
δnl → depends on n and l is called the quantum defect, which expresses the
changes of the energy values Enl against that of the hydrogen atom (δ = 0)
by a dimensionless number.

25. Summary:All the following effects are included in the quantum defects δnl .
The shifts of the alkali energy levels Enl against those of the hydrogen atom are caused by the following effects.
• The deviation of the effective potential from the Coulomb potential, which causes energy shifts ΔEnl that depend on n
and l, because of the n- and l-dependent penetration depths of the outer electron into the core of the other electrons.
• The outer electron interacts with the other electrons in the core and polarizes the electron shell. This leads to a deviation
from the spherical charge distribution even for closed shells. The magnitude of this polarization depends on the angular
momentuml of the outer electron.
• When the outer electron penetrates into the core, it can collide with the other electrons. This may result in an exchange
of the outer with an inner electron, which causes an additional energy shift.
Measured quantum defects δnl for different Rydberg states of the sodium atom

26. The energies shifts ΔE between En of alkali atoms and the energy levels En in the H atom decrease with increasing values of n