Transcript Chapter 10 - Lecture 3

Atomic Structure and Atomic
Spectra
Chapter 10
Structures of many-electron atoms
• Because of electron correlation, no simple analytical
expression 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 of
other electrons
Structures of many-electron atoms
Pauli exclusion principle – no more than two electrons
may occupy an atomic orbital, and if so, must be of opposite
spin
Structures of many-electron atoms
• In many-electron atoms, subshells are not
degenerate. Why?
• Shielding and penetration
Fig 10.19 Shielding and effective nuclear charge, Zeff
• Shielding from core electrons
reduces 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 atoms
• In many-electron atoms, subshells are not
degenerate. 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)
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Be
Li
B5
C
3
64electrons
electrons
22s
222s
22p
12 1
BBe
Li1s1s
1s
2s
H
He12electron
electrons
He
H 1s
1s12
Structure of many-electron atoms
• In many-electron atoms, subshells are not
degenerate. Why?
• Shielding and penetration
• The building-up principle (Aufbau)
• Mnemonic:
• Hund’s rule of maximum multiplicity
• Results from spin correlation
The most stable arrangement of electrons in
subshells is the one with the greatest number of
parallel spins (Hund’s rule).
Ne97
C
N
O
F
6
810
electrons
electrons
electrons
22s
222p
22p
5
246
3
Ne
C
N
O
F 1s
1s222s
Fig 10.21 Electron-electron repulsions in Sc atom
Reduced repulsions
with configuration
[Ar] 3d1 4s2
If configuration was
[Ar] 3d2 4s1
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-
I1 first ionization energy
I2 + X+(g)
X2+(g) + e-
I2 second ionization energy
I3 + X2+(g)
X3+(g) + e-
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 atoms
• Energy levels not solely given by energies
of 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
Three ways to obtain nonzero spin
Multiplicity = (2S + 1)
= (2·1 + 1)
=3
Triplet state
Spins are partially
parallel
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
Total angular momentum (j) = orbital (l) + spin (s)
e.g., for
for
l=0→j=½
l = 1 → j = 3/2, ½
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 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
El,s,j = 1/2hcA{ j(j+1) – l(l+1) – s(s+1) }
where A is the spin-orbit coupling constant