Lecture 3. Many-Electron Atom. Pt.1
Electron Correlation, Indistinguishableness,
Antisymmetry & Slater Determinant
References
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Ratner Ch. 7.1-7.2, 8.1-8.5, Engel Ch. 10.1-10.3, Pilar Ch.7
Modern Quantum Chemistry, Ostlund & Szabo, Ch. 2.1-2.2
Molecular Quantum Mechanics, Atkins & Friedman (4th ed. 2005), Ch.7
Quantum Chemistry, McQuarrie, Ch. 7-8
Computational Chemistry, Lewars (2003), Ch.4
• A Brief Review of Elementary Quantum Chemistry
http://vergil.chemistry.gatech.edu/notes/quantrev/quantrev.html
• Slater, J.; Verma, H.C. (1929) Phys. Rev. 34, 1293-1295.
Helium Atom First (1 nucleus + 2 electrons)
• Electron-electron repulsion
• Indistinguishability
newly introduced
1. Electron-electron repulsion (correlation)
~H atom electron
~H atom electron
at r1
: Correlated, coupled
at r2
The r12 term removes the spherical symmetry in He.
We cannot solve this Schrödinger equation analytically.
(Two electrons are not separable nor independent any more.)
 A series of approximations will be introduced.
Approximation #1.
To first approximation, electrons are treated independently.
Many-electron (many-body) wave function
1-electron wave function ~ H atom orbital
An N-electron wave function is approximated by
a product of N one-electron wave functions (orbitals)
(a so-called Hartree product).
Orbital Approximation or Hartree Approximation
or Single-particle approach or One-body approach
This does not mean that electrons do not sense each other.
(We’ll see later.)
Electron spin & Spin angular momentum
The Stern-Gerlach experiment shows two beams.
Multiplicity = 2 (doublet)
Electron has an “intrinsic spin” angular momentum, which
has nothing to do with the orbital angular momentum in an atom.
l = 0, 1, 2, …; multiplicity (= the number of allowed ml values) = 1, 3, 5, …
Spin operator, Eigenfunctions, and Eigenvalues
Spin (angular momentum) operator, s2 and sz:
- just like orbital angular momentum operator, L2 and Lz
Two eigenstates only {, } or {,} – an orthonormal set:
- eigenfunction  with eigenvalue (s, ms) = (½, ½)
- eigenfunction  with eigenvalue (s, ms) = (½, -½)
- We don’t know (don’t care) the form of the eigenfunction  and .
 ()
½
-½
3/2
 ()
How is spin integrated into wave function?
space
spin
new degree of freedom
(4th quantum number) with
only two values (1/2, -1/2)
- Wolfgang Pauli (1924)
 |1s> 
1s
e.g.
 |1s> 
1s
Just a product of spatial orbital  spin orbital, because
the non-relativistic Hamiltonian operator does not include spin.
(Space & spin variables are separated, and [H,s2] = [H,sz] = [s2,sz] = 0)
Orthogonal to each other (integration now over r, , , and )
History of Quantum Mechanics
• 1885 – Johann Balmer – Line spectrum of hydrogen
• 1886 – Heinrich Hertz – Photoelectric effect experiment
• 1897 – J. J. Thomson – Discovery of electrons from cathode rays experiment
• 1900 – Max Planck – Quantum theory of blackbody radiation
• 1905 – Albert Einstein– Quantum theory of photoelectric effect
• 1910 – Ernest Rutherford – Scattering experiment with -particles
• 1913 – Niels Bohr – Quantum theory of hydrogen spectra
• 1923 – Arthur Compton – Scattering experiment of photons off electrons
• 1924 – Wolfgang Pauli – Exclusion principle – Ch. 10
particle
wave & spin
• 1924 – Louis de Broglie – Matter waves
• 1925 – Davisson and Germer – Diffraction experiment on wave properties of electrons
• 1926 – Erwin Schrodinger – Wave equation – Ch. 2
• 1927 – Werner Heisenberg – Uncertainty principle – Ch. 6
• 1927 – Max Born – Interpretation of wave function – Ch. 3
A Little History of Spin in Quantum Mechanics
• 1922 – Otto Stern & Walter Gerlach – The existence of spin angular momentum is inferred
from their experiment, in which particles (Ag atoms) are observed to possess angular
momentum that cannot be accounted for by orbital angular momentum alone.
• 1924 – Wolfgang Pauli – proposed a new quantum degree of freedom (or quantum
number) with two possible values and formulated the Pauli exclusion principle.
• 1925 – Ralph Kronig, George Uhlenbeck & Samuel Goudsmit – identified Pauli's new degree
of freedom as electron spin and suggested a physical interpretation of particles spinning
around their own axis.
• 1926 – Enrico Fermi & Paul Dirac – formulated (independently) the Fermi-Dirac statistics,
which describes distribution of many identical particles obeying the Pauli exclusion principle
(fermions with half-integer spins – contrary to bosons satisfying the Bose-Einstein statistics)
• 1926 – Erwin Schrödinger – formulated his non-relativistic Schrödinger equation, but it
incorrectly predicted the magnetic moment of H to be zero in its ground state.
• 1927 – T.E. Phipps & J.B. Taylor – reproduced the effect using H atoms in the ground state,
thereby eliminating any doubts that may have been caused by the use of Ag atoms.
• 1927 – Wolfgang Pauli – worked out on mathematical formulation of spin (22 matrices).
• 1928 – Paul Dirac – showed that spin comes naturally from his relativistic Dirac equation.
2. Electrons (in a He atom) are indistinguishable.
 Probability doesn’t change.
Two possibilities in wave function
e.g.
asymmetric
not good!
ok
1s
ok
Antisymmetry of electrons (or other fermions)
Electrons (s = ½) are fermion (s = half-integer).  antisymmetric wavefunction
Quantum postulate 6 (Pauli Principle; 1924-1925):
[H, P12] = 0
Wave functions describing a many-electron system should
- change sign (be antisymmetric) under the exchange of any two electrons.
- be an eigenfunction of the exchange operator P12 with the eigenvalue of -1.
exchange operator P12

ok
not ok!
Ground state of He (the singlet state)
notation
 |1s> 
Slater
determinant
(1929)
1s
1s
 1s2
Slate determinants provide a convenient way to antisymmetrize
many-electron wave functions built with the Hartree approximation.
S2 (1,2) = (s1 + s2)2 (1,2) = 0, Sz (1,2) = (sz1 + sz2) (1,2) = 0
Excited state of He
Slater determinant and Pauli exclusion principle
• A determinant changes sign when two rows (or columns) are exchanged.
“antisymmetric”
 Exchanging two electrons leads to a change in sign of the wave function.
• A determinant with two identical rows (or columns) is equal to zero.
=0
=0
 No two electrons can occupy the same state. “Pauli’s exclusion principle”
4 quantum numbers
(space and spin)
 We cannot put more than two electrons in one space orbital (nlml).
N-electron wave function: Slater determinant
• N-electron wave function is approximated by
a product of N one-electron wave functions (hartree product).
but not antisymmetric!
• It should be antisymmetrized.
Ground state of Lithium
Total angular momentum of many-electron atom
Add li (or si) vectors to form L (or S) vector.
LS coupling (contrary to jj coupling)
for non-relativistic, no-spin-orbit-coupling cases
Ground state of He (the singlet state)
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
Excited state of He (singlet and triplet states)
Excited state of He (singlet and triplet states)
+
S = s1 + s2 = ½ - ½ = 0, Ms = 0 (singlet)
(S,Ms)
4 =
(0,0)
2 =
(1,1)
3 =
(1,-1)
-4 =
S = s1 + s2 = ½ + ½ = 1,
Ms = 1, 0, -1 (triplet)
(1,0)