HUND'S RULES
Hund’s Rules relate the energy ordering of the various atomic terms
1
1
arising from a single configuration
configuration, e.g.
e g 2p2. 3P0,1,2
0 1 2 < D2 < S0
They have a simple physical interpretation.
1. The term with the highest multiplicity (2S+1) lies lowest in energy.
2. For
2
F terms
t
with
ith the
th same multiplicity,
lti li it th
the one with
ith th
the llargestt L li
lies llowestt iin
energy.
Terms with the same L and S have very nearly the same energy for light atoms.
They do differ in the total J, and each J state has a very slightly different
energy, referred
f
d tto as th
the fi
fine structure
t
t
splitting
litti iin th
the atom.
t
3
3
3
e.g. P2, P1, and P0
Hunds third rule applies to the ordering of these J states.
3 ) If th
3a)
the sub
b shell
h ll iis lless th
than h
halflf filled,
fill d th
the llowestt J llevell iis th
the mostt
stable.
3b) If the sub shell is more than half- filled, the highest J level is the most
stable.
F h
For
heavy atoms,
t s this L-S
L S coupling
li scheme
s h
is nott a very good
d approximation,
i ti
and
d
we employ j-j coupling of angular momenta.
As a final note, interaction with the nuclear spins produces further very slight
splitti of
splitting
f the
th iindividual
di id l J levels,
l
ls an effect
ff t known
k
ass the
th h
hyperfine
p fi splitti
splitting. H
Hund
d'ss
last rule tells that the order of the fine structure levels is inverted for the more
than half-filled shells.
Electron Correlation and Quantum Chemistry Programs
To recapitulate earlier results, we have first seen that we can
employ
l a self
lf consistent f
field
ld (SCF)
( CF) approach
h for
f an n-electron
l
atom
system to obtain the best possible wave function of the product form,
N
ψ ( r1 , r2 , θ1 , θ2 , φ1 , φ2 …) = ∏ R , (ri) Y , (θ i, φ i) .
m
n
i =1
i i
i
i
This SCF wavefunction has each electron moving in the electrostatic
potential given by the nuclear attraction and the electrostatic charge
cloud arising from the ϕ
ϕ*ϕ
ϕ of the other n-1 electrons.
electrons Only an average
electron correlation is accounted for in this limit, but the concept of
orbitals for each electron is still present. We can do a variational
calculation
calculat
on w
with
th a bas
basiss set to obtain
obta n the best value for each ϕI w
within
th n
the confines of that basis set. If the basis set were complete, then we
would obtain the Hartree Fock SCF limit.
To go beyond
y
this limit, we add more terms to account for
instantaneous electron correlation. These could be terms of the form
This method of adding terms that look like other orbitals is referred to
as configuration interaction (CI), and it can give an exact nonrelativistic
l ti i ti solution
l ti tto th
the wave equation.
ti
H
However th
the conceptt of
f
individual orbitals is lost, and we now have only a very complex function.
This is by far the hardest part of a given calculation.
He atom total energy and ionization energy calculations
M th d
Method
Total
E
Energy/au*
/ *
Ionization
E
Energy/au*
/ *
Perturbation calculations
Complete
p
neglect
g
of
f the interelectronic repulsion term
First-order perturbation theory
Second-order perturbation theory
Thirteenth order perturbation theory
Thirteenth-order
Variational calculations
(ls)2 with ξ = 1.6875
(ns)2 with ξ = 1.61162
n = 0.995
Hartree-Fock
Hy11eras, 10 parameters
Kinoshita 39 parameters
Kinoshita,
Pekeris, 1078 parameters
*au = atomic unit of energy
-4.00
-2.75
2.91
- 2.903
2 903 7243 3
-2.8477
-2.8542
-2.8617
-2.90363
-2 9037225
-2.9037225
-2.9037243 75
2.00
0.75
0.91
0 904
0.904
0.848
0.854
0.862
0.904
0 904
0.904
0.904
1 au = 2*IP(H) = 2 Hartree ≅ 27.21 eV
Multielectron Atoms
We have seen how to employ the variational method to improve the
wave
v functi
function
n for
f r the
th helium
h lium atom.
t m We
W have
h v seen
s n that
th t with
ith hard
h rd work
rk we
can do almost arbitrarily well for the properties of the helium atom. The
first step was to obtain the best independent particle (separable) product
form of the wave function.
function The starting point was our independent
electron (neglect r12) model, and the best possible result of this form was
called the Hartree-Fock limit. The results were much better, but still not
of chemically relevant accuracy, so we had to devise methods to include
the correlated motion of the electrons. The method described in class was
called configuration interaction, and we demonstrated that the results
could be very
y good.
g
In the process,
p
we of course took into account the
couplings between electrons, and so the original naive concept of separate
good quantum numbers n, , m and ms for each electron has gone down the
drain.
W are left
We
l f with
i h the
h fact
f
that
h only
l totall energy, totall angular
l
momentum are rigorously conserved. We now make a brief foray into
writing out the Schrödinger equation and its independent electron solution
for an n-electron
n electron system.
system
Multielectron Atoms
Consider an atom with nuclear charge +Ze, nuclear mass M and N
electrons. The Hamiltonian is given by
2
⎡
⎤
Ze
e
2
2
ˆ
H=−
+ ∑∑
∇M + ∑ ⎢ −
∇i −
⎥
2M
r i ⎦ i=1
ii=11 ⎣ 2 m e
i 1 j> i r ij
2
N
2
2
N
N
What do we do now?? First, we separate out the center of mass
motion,
ti
taking
t ki care of
f 3 of
f the
th 3(N+1)
3(N 1) degrees
d
s of
f freedom.
f
d
N
Next,
t ass
before, we assume the nuclear mass to be much greater than all of the
electrons. This places the center of mass on the nucleus, dropping the
first term and it makes the electron- nucleus attractions central
forces.
The problem, as with helium, is now separable if we neglect the
electron-electron
electron
electron repulsion terms. We have N independent +Ze
Ze ((-e)
e)
one electron problems.
As before the solutions are given by
2
En ,n ,...n = − Z e
1
2
2
2a o
N
n1 = 1,2,…
n2 = 1,2,…
n3 …nN
N
1
∑n
i=1
i
2
N
where ψ = ∏ R , (r i )Y , (θ i , φi )
mi
ni i
i
i =1
Next we must include spin, and make the resulting
function antisymmetric. As earlier, we do this using Slater
Determinants.
Determinants
Now on to molecules!
MOLECULAR STRUCTURE
The Born
Born-Oppenheimer
Oppenheimer Approximation
The general molecular Schrödinger equation, apart from
electron spin effects, is
(T
n
)
+Te +Vnn +Vee +Ven ψ = Eψ
where the operators in the Hamiltonian are the kinetic energy
operators of the nuclei and the electrons,
electrons and then the potential
energy operators between the nuclei, between the electrons, and
between the nuclei and electrons.
Coordinates are defined below for a three nucleus, three electron molecule.
Definition of distances in a molecule
In general, the explicit forms of these operators are as follows
Tn =
Te =
nuclei
∑
α
−
2
2 Mα
electrons
∑
i =1
−
2
∇α
2
2 me
nuclei
2
∇i
Z Z e
Vnn = ∑ ∑ α β
R αβ
α β>α
Vee =
Ven =
2
e2
∑
∑
i =1
j >i r
ijj
electrons
nuclei
∑
α
electrons
∑i
−
Zα e
2
ri α
There is a repulsive interaction among the nuclear charges and a
repulsive charge-charge interaction among the electrons. However, the
interaction potential between electrons and nuclei is attractive, since the
particles have opposite charges. This particular interaction couples the
motions of the electrons and the motions of the nuclei.
Born-Oppenheimer
Born
Oppenheimer Approximation
The wavefunctions that satisfy this Hamiltonian must be functions of both
the electron position coordinates and the nuclear position coordinates, and
this differential equation is not separable. In principle, true solutions
could be found, but this is a formidable differential equation to work with.
An
n alternative
a t rnat
iss an approximate
appro mat ((but
ut generally
g n ra y a very
ry
good approximation) separation of the differential equation
based upon the very large difference between the mass of an
electron and the masses of the nuclei.
nuclei The difference
suggests that the nuclei will be sluggish in their motions
relative to the electron motions. Over a brief period of time,
the
h electrons
l
will
ll "see"
"
" the
h nuclei
l as if
f fixed
f d in space; the
h
nuclear motions will be relatively slight. The nuclei, on the
g of
f a blur,,
other hand,, will "see" the electrons as something
given their fast motions.