Lecture 7. Many-Electron Atoms. Pt.5.Good quantum numbers (Terms &

Levels)& the Zeeman effect

References

• 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 Chemistryhttp://vergil.chemistry.gatech.edu/notes/quantrev/quantrev.htmlhttp://vergil.chemistry.gatech.edu/notes/hf-intro/hf-intro.html

Different degrees of electron interaction in many-electron 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 atomswithsignificant 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)

• Since H does not depend on spin, operators depending only on spin commute with H.

• Individual electron quantum numbersl1z 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.

non-relativisticnon-relativistic

Ground state of He (the singlet state) (Review)

1s

Slater determinant

notation

|1s> 1s

S2 (1,2) = (s1 + s2)2 (1,2) = 0, Sz (1,2) = (sz1 + sz2) (1,2) = 0

1s2

Total spin quantum number S = s1 + s2 = ½ - ½ = 0, Ms = 0 (singlet)

Good quantum numbers for many-electron atoms 2.(J, MJ, L, S) & Atomic term symbols 2S+1LJ

• Total angular momentum operator

• 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.

• 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)

0

non-relativistic

not-so-heavy atoms

Different degrees of electron interaction in many-electron 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 atomswithsignificant 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?11.1spin-paired closed shell 1S– We don’t need to worry.

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)

singlettriplet

where

Coulomb integral > 0

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 hasa magnetic moment that will interact with externally applied magnetic fields.

where e (magnetogyric ratio) < 0

Bohr magneton(9.274011024

J/T)

(Zeeman Hamiltonian)

(magnetic moment = area current)

Splitting of spectral line under magnetic field. Zeeman effect

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)

Splitting of spectral line under magnetic field. Zeeman effect (1896)

EPR (Electron Paramagnetic Resonance) & NMR (Nuclear Magnetic Resonance)

Term Report & Presentation (December 18)