Multi-Electron Atoms
With more than one electron, several effects need to be considered in addition to
those encountered for one-electron atoms:
! Electron Correlation (! due to electron-electron Coulomb repulsion).
! Electron Exchange (! due to particle indistinguishability).
! Coupling between multiple spins (S) and orbital angular momenta (L).
We will begin by considering these phenomena separately from another; and later
combine them.
Simplest example of a multi-electron atom:
Helium: Two electrons (each or charge -e) and a nucleus (of charge +Ze).
Helium Schrödinger Equation
Consider the nucleus fixed and at the origin.
The time-dependent (electronic Schrödinger Equation):
Kinetic energy
electron 1
Kinetic energy
electron 2
Electronnucleus
Coulomb
interaction
for electron 1
Electron-electron
Coulomb interaction
between electron 1
and 2
Electronnucleus
Coulomb
interaction
for electron 2
Time-dependent
term.
Helium Schrödinger Equation
Separate time
" time-independent Schrödinger equation.
Introduce a simpler notation:
Step 1: Laplacian operators:
Step 2: Replace vectors with particle labels. Combine terms that act on the same subset
of electron. Operator H1 acts on electron 1 only, Potential V12 involves electrons 1 and 2.
Lithium (3 electrons):
Solving the Helium Equations
Lets start with a drastic approximation: Neglect electron-electron Coulomb
interaction V12
Can separate the equation now into (one-electron equations for electron 1 and 2):
Separation of Variables (and use
)
The two equations are really the same, differing only in the labels used. They are
the one-electron atom Schroedinger equation for which we already know the solutions.
Solving the Helium Equations
One-electron atomic equation (Need to be careful with
indices now … here now quantum numbers; not particle
labels)
Calcd.
(no e-e)
Experim.
For the Helium Atom (back to particle labels):
Gives electronic states for helium without electron-electron
interaction (and without exchange and spin-orbit).
Only order of magnitude agreement with experiment due to
drastic approximation. Systematically overestimates the
binding energy.
Particle Indistinguishability
(… what’s the problem with labels ?)
Classical Particle
Trajectory 1
Classical Particle
Trajectory 2
Trajectory of Fuzzy
Quantum Particle
Identical quantum particles are truly
indistinguishable.
(This produces exchange symmetry).
Observable (that is measurable)
properties must be unaffected (i.e.
symmetric) with respect to particle
exchange.
Measurable results should not depend
on the assignment of particle labels.
Remember:
Wave- and eigenfunctions are not
measurable (but their squares are).
Thus, wave- and eigenfunctions are
either symmetric or antisymmetric
w.r.t. particle exchange.
Particle Exchange Symmetry
Introduce the particle exchange operator:
… switches particles i and j.
e.g.
Note: Don’t confuse
the exchange operator
with the parity operator
which also uses symbol
P (… as does the
probability function).
Particle indistinguishability means, we require measurable properties (e.g.
probabilty density) to be symmetric with respect to any particle exchange.
Measurables Must be Symmetric !
e.g. Probability density:
e.g. Expectation value of some measurable M:
Requirement !
Requirement !
It follows that #(1,2) must be either symmetric or antisymmetric:
Symmetric:
In general:
Antisymmetric:
In general:
Consider two independent particles
(e.g. our approximate, non-interacting-electrons solution for Helium)
Two-particle total eigenfunction
Probability Density
Construct symmetric combination:
(symmetric)
Construct antisymmetric combination:
(antisymmetric)
For generalization
use determinant.
Pauli Exclusion Principle
Weak Exclusion Principle (Empirical Finding):
“In a multi electron atom, there can never be more than one electron in the same (n, l, ml, ms)
quantum state.”
Strong Exclusion Principle (strong because it incorporates indistinguishability)
“A system containing several electrons must be described by an antisymmetric total
eigenfunction.”
More general principle: Particles
fall into two kinds that exclusively
described by either antisymmetric
(Fermions)
or
symmetric
(Bosons) total eigenfunctions.
Empirical Finding:
Half-integer spin " Fermion
Integer spin " Boson