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Lecture 7. Many-Electron Atoms. Pt.5.
Good quantum numbers (Terms & Levels)
& the Zeeman effect
References
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Ratner Ch.8.6-8.7, 9.7-9.9, Engel Ch.11, Pilar Ch.5.3-5.5,8.1-8.6
Quantum Chemistry, McQuarrie Ch.8.9-8.12
Molecular Quantum Mechanics, Atkins & Friedman (4th ed. 2005), Ch.7
Computational Chemistry, Lewars (2003), Ch. 5
• A Brief Review of Elementary Quantum Chemistry
http://vergil.chemistry.gatech.edu/notes/quantrev/quantrev.html
http://vergil.chemistry.gatech.edu/notes/hf-intro/hf-intro.html
Different degrees of electron interaction in manyelectron atoms and different good quantum numbers
• (n, l, ml, ms) are good quantum numbers for hydrogen-like atoms,
but not for many-electron atoms (due to electron repulsion).
• (L, S, ML, MS) are good quantum numbers for light (Z<50) many-electron atoms
with negligible spin-orbit coupling (or relativistic effect).
Energy depends on L and S, not on ML or MS.
• (L, S, J, MJ) are good quantum numbers for heavy many-electron atoms with
significant spin-orbit coupling (relativistic effect). Energy also depends on J.
• For very heavy atoms, a j-j coupling is needed, where j = l + s for each electron.
Constant of the Motion & Good quantum number
Any operator which commutes with H may have simultaneous eigenfunctions.
The eigenvalues associated with these operators define the “constants of the
motion” that may be used to label the states of the atom.
 a is time-independent.
 a is called a good quantum number.
Good quantum numbers for many-electron atoms 1.
Total orbital/spin angular momentum (L, S, ML, MS)
non-relativistic
• Since H does not depend on spin, operators
depending only on spin commute with H.
non-relativistic
• Individual electron quantum numbers
l1z and l2z are not constants of the motion.
• The sum of angular momentum
operators commutes with H.
• Eigenfunctions can be characterized by quantum numbers (L,ML,S,Ms) as well as E.
Ground state of He (the singlet state) (Review)
notation
 |1s> 
Slater
determinant
1s
Total spin quantum number S = s1 + s2 = ½ - ½ = 0, Ms = 0 (singlet)
S2 (1,2) = (s1 + s2)2 (1,2) = 0, Sz (1,2) = (sz1 + sz2) (1,2) = 0
1s
 1s2
Good quantum numbers for many-electron atoms 2.
(J, MJ, L, S) & Atomic term symbols 2S+1LJ
• Total angular momentum operator
0
• Not possible to label atomic states with 6 quantum numbers (J, MJ, L, ML, S, MS).
• Instead they can be labeled by 4 quantum numbers (L, ML, S, Ms) or (J, MJ, L, S).
• Since both operators commute with H, one choice is not favored over the other.
non-relativistic
• Dominant relativistic contribution to H is one-electron spin-orbit coupling.
• Relativistic contributions are much larger for heavier atoms than for light atoms.
• The operators (L2, Lz, S2, Sz) do not commute with Hso.
• Only J, MJ, and E are good quantum numbers.
• When L and S are “nearly” good quantum numbers,
 Russell-Saunders term symbols to identify states 2S+1LJ (negligible SO coupling)
not-so-heavy atoms
Different degrees of electron interaction in manyelectron atoms and different good quantum numbers
• (n, l ml, ms) are good quantum numbers for hydrogen-like atoms,
but not for many-electron atoms (due to electron repulsion).
• (L, S, ML, MS) are good quantum numbers for light (Z<50) many-electron atoms
with negligible spin-orbit coupling (or relativistic effect).
Energy depends on L and S, not on ML or MS.
• (L, S, J, MJ) are good quantum numbers for heavy many-electron atoms with
significant spin-orbit coupling (relativistic effect). Energy also depends on J.
• For very heavy atoms, a j-j coupling is needed, where j = l + s for each electron.
spin-paired closed shell 1S
(We can ignore them.)
Possible terms 2S+1LJ for various electron
configurations
How many ways to place 2 electrons in 2p orbitals?
What are the values of (ML, MS, L, S) corresponding to these states?
spin-paired closed shell 1S
– We don’t need to worry.
11.1
11.2
Hund’s rule: Relative energies of terms and levels
within a given electron configuration
Energy of the Slater determinant of the He atom:
a “triplet” first excited state (Review)
triplet
singlet
Coulomb integral > 0
where
Exchange integral (>0)
includes in it
wave function
(final solution)!
Aufbau principle for transition metals (Review)
Please don’t be
confused with it!
Hund’s rule predicts the lowest-energy term
within a given electron configuration
According to Hund’s rule, when placing electrons in one-electron orbitals,
the number of unpaired electrons should be maximized. This is why Cr has
the configuration [Ar]4s13d5 rather than [Ar]4s23d4.
Restricted vs. Unrestricted Hartree-Fock (RHF/UHF)
Interaction of electron with magnetic fields.
Splitting of spectral line under magnetic field.
Zeeman effect (1896) – Nobel prize in physics (1902)
Nature 55, 347 (1897) The effect of magnetisation on the nature of light emitted by a substance
Thanks to Zeeman's work it became clear that the oscillating particles (the source of light
emission) were negatively charged. This conclusion was reached well before Thomson's
discovery of the electron. The Zeeman effect thus became an important tool for
elucidating the structure of the atom.
Interaction of electron with magnetic fields
Whenever a charged particle has either spin or orbital angular momentum, it has
a magnetic moment that will interact with externally applied magnetic fields.
(magnetic moment = area  current)
where e (magnetogyric ratio) < 0
(Zeeman Hamiltonian)
Bohr magneton
(9.274011024 J/T)
Splitting of spectral line under magnetic field.
Zeeman effect
Splitting of spectral line under magnetic field.
Zeeman effect (1896)
Energy levels of the 5s orbitals, including fine structure and hyperfine structure. The quantum number
F = J + I, where I is the nuclear spin (for Rb-87, I = 3/2). (MATLAB by plotting the Breit-Rabi equation)
EPR (Electron Paramagnetic Resonance) &
NMR (Nuclear Magnetic Resonance)
Term Report & Presentation (December 18)