Chapter 10

Atomic Structure and Atomic Atomic Structure and Atomic SpectraSpectra

Structures of many-electron atoms

• Because of electron correlation, no simple analyticalexpression for orbitals is possible

• Therefore ψ(r1, r2, ….) can be expressed as ψ(r1)ψ(r2)…

• Called the orbital approximation

• Individual hydrogenic orbitals modified by presence ofother electrons

Structures of many-electron atoms

Pauli exclusion principle – no more than two electronsmay occupy an atomic orbital, and if so, must be of oppositespin

Structures of many-electron atomsStructures of many-electron atoms

• In many-electron atoms, subshells are notdegenerate. Why?

• Shielding and penetration

Fig 10.19 Shielding and effective nuclear charge, Zeff

• Shielding from core electronsreduces Z to Zeff

Zeff = Z – σ

where σ ≡ shielding constant

Fig 10.20 Penetration of 3s and 3p electrons

• Shielding constant different for s and p electrons

• s-electron has greater

penetration and is bound more

tightly bound

• Result: s < p < d < f

Structure of many-electron atomsStructure of many-electron atoms

• In many-electron atoms, subshells are notdegenerate. Why?

• Shielding and penetration

• The building-up principle (Aufbau)

• Mnemonic:

Order of orbitals (filling) in a many-electron atom

1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s < 4d < 5p < 6s

“Fill up” electrons in lowest energy orbitals (Aufbau principle)

H 1 electron

H 1s1

He 2 electrons

He 1s2

Li 3 electrons

Li 1s22s1

Be 4 electrons

Be 1s22s2

B 5 electrons

B 1s22s22p1

C 6 electrons

? ?

Structure of many-electron atomsStructure of many-electron atoms

• In many-electron atoms, subshells are notdegenerate. Why?

• Shielding and penetration

• The building-up principle (Aufbau)

• Mnemonic:

• Hund’s rule of maximum multiplicity

• Results from spin correlation

C 6 electrons

C 1s22s22p2

N 7 electrons

N 1s22s22p3

O 8 electrons

O 1s22s22p4

F 9 electrons

F 1s22s22p5

Ne 10 electrons

Ne 1s22s22p6

The most stable arrangement of electrons in subshells is the one with the greatest number of parallel spins (Hund’s rule).

Fig 10.21 Electron-electron repulsions in Sc atom

If configuration was

[Ar] 3d2 4s1

Reduced repulsions

with configuration

[Ar] 3d1 4s2

Ionization energy (I) - minimum energy (kJ/mol) required to remove an electron from a gaseous atom in its ground state

I1 + X(g) X+

(g) + e-

I2 + X+(g) X2+

(g) + e-

I3 + X2+(g) X3+

(g) + e-

I1 first ionization energy

I2 second ionization energy

I3 third ionization energy

I1 < I2 < I3

Mg → Mg+ + e− I1 = 738 kJ/mol

Mg+ → Mg2+ + e− I2 = 1451 kJ/mol

Mg2+ → Mg3+ + e− I3 = 7733 kJ/mol

For Mg2+

1s22s22p6

Fig 10.22 First ionization energies

N [He] 2s2 2p3 I1 = 1400 kJ/mol

O [He] 2s2 2p4 I1 = 1314 kJ/mol

Spectra of complex atomsSpectra of complex atoms

• Energy levels not solely given by energiesof orbitals

• Electrons interact and make contributions to E

Fig 10.18 Vector model for paired-spin electrons

Multiplicity = (2S + 1)

= (2·0 + 1)

= 1

Singlet state

Spins are perfectly

antiparallel

Fig 10.24 Vector model for parallel-spin electrons

Multiplicity = (2S + 1)

= (2·1 + 1)

= 3

Triplet state

Spins are partiallyparallel

Three ways to obtain nonzero spin

Fig 10.25 Grotrian diagram for helium

Singlet – triplet transitions

are forbidden

Fig 10.26 Orbital and spin angular momenta

Spin-orbit

coupling

Magnetogyric ratio

Fig 10.27(a) Parallel magnetic momenta

Total angular momentum (j) = orbital (l) + spin (s)

e.g., for l = 0 → j = ½

Fig 10.27(b) Opposed magnetic momenta

e.g., for l = 0 → j = ½

for l = 1 → j = 3/2, ½

Total angular momentum (j) = orbital (l) + spin (s)

Fig 10.27 Parallel and opposed magnetic momenta

Result: For l > 0, spin-orbit

coupling splits a configuration

into levels

Fig 13.30 Spin-orbit coupling of a d-electron (l = 1)

j = l + 1/2

j = l - 1/2

Energy levels due to spin-orbit couplingEnergy levels due to spin-orbit coupling

• Strength of spin-orbit coupling depends on

relative orientations of spin and orbital

angular momenta (= total angular momentum)

• Total angular momentum described in terms of

quantum numbers: j and mj

• Energy of level with QNs: s, l, and j

where A is the spin-orbit coupling constant

El,s,j = 1/2hcA{ j(j+1) – l(l+1) – s(s+1) }