Atomic Configurations and Term Symbols
Δ
charge correlation
energy
(e-e
repulsion!)
p
y
=
A
p
x
=
B
1.6
Term Symbols
A
brief general review
of
atomic
configurations:
There are
four
different interactions
in
any atom that determine
the
relative energies
of
atomic configurations (in order
of
importance):
a)
Electrostatic attraction between electrons
and the
nucleus
Electrostatic energy
1/
n
2
, where
n
=
principal quantum
number
b)
Electron-electron repulsion
Leads
to Hund's
rule
of
maximum
multiplicity
Because
of
Coulomb repulsion
, pairing
of
electrons
in one
orbital costs
energy:
Charge Correlation
Energy
Exchange
energy:
Electrons
of
like
spin
in
energy degenerate orbitals
can
“exchange”
leading
to
an
overall
lower
energy
(quantum
mechanical
resonance
phenomenon.)
Another way
of
looking at this
is to
consider
a
diagram
of the
p
x
and p
y
orbitals
of
an
atom.
Although
these orbitals are orthogonal (
zero overlap integral
), there
is
nevertheless
a
region
of
actual orbital overlap
in
space
(shown in
red
).
Consider
one
electron
in
each orbital.
If
they have
opposite
spin, their spatial wave
function can
put
them
in
exactly
the
same space at
the
same
time (
high
electrostatic
repulsion
).
If
they have
the
same spin, there
must be a
constraint
on
their spatial
wave function
so
as
to not put
them
in
exactly
the
same space at
the
same
time
(therefore
lower electrostatic repulsion
if spin is the
same
)
Δ
E
=
exchange
energy
E
p
x
p
y
p
z
p
x
p
x
p
y
p
z
p
z
p
y
In electronic spectroscopy, an atomic term symbol specifies
a certain electronic state of an atom (usually a multi-
electron one), by briefing the quantum numbers for the
angular momenta of that atom. The form of an atomic term
symbol implies
Russell-Saunders
coupling. Transitions
between two different atomic states may be represented
using their term symbols, to which certain rules apply.
In the Russell-Saunders coupling scheme, term symbols are
in the form of
2
S
+1
L
J
, where
S
represents the total spin
angular momentum,
L
specifies the total orbital angular
momentum, and
J
refers to the total angular momentum. In
a term symbol,
L
is always an upper-case from the sequence
"s, p, d, f, g, h, i, k...", wherein the first four letters stand
for sharp, principal, diffuse and fundamental, and the rest
follow in an alphabetical pattern. Note that the letter j is
omitted.
c)
Spin-orbit coupling
i)
Russel-Saunders
for light
atoms
:
J = L + S, L + S –1, …, |L
–S|
L =
l
1
+
l
2
, l
1
+
l
2
–1, …,
|l
1
-l
2
|
S =
s
1
+
s
2
, s
1
+
s
2
–1, …,
|s
1
–
s
2
|
ii)
jj-coupling
for
heavy atoms (heavier
than
bromine) j
1
=
l
1
+
s
1
, l
1
+
s
1
– 1, …,
|l
1
–
s
1
|
.
j
n
=
l
n
+
s
n
, l
n
+
s
1
– 1, …,
|l
n
–
s
n
|
J =
j
1
+
j
2
, j
1
+
j
2
–1, …,
|j
1
–
j
2
|
d)
Spin-spin interactions
Leads
to the
Pauli-Principle
:
No two
Fermions
in
any spatially/energetically
confined system can have
the
same
four
quantum numbers
(
n, l,
m
l
,
m
s
).
SEE SUPP. INFO ON
HUND’S
RULE
In
many-electron-atoms
each electronic configuration can
be
described
by a
unique
term
symbol.
E.g., for
scandium
3d
1
4s
2
:
3d
2
4s
1
:
2
D
3/2
,
2
D
5/2
(only
one
unpaired electron
easy
to
solve)
2
S,
2
D,
2
G,
2,4
P,
2,4
F (more difficult
to figure
out)
The
relative energies
of the
terms depend
on
spin their multiplicity
(2
S
+1)
and their
orbital angular momentum
(
L
).
Based
on Hund's
rule,
the
lowest energy terms
are the
ones with
the
largest spin
multiplicity
and among
these,
the
term with
the
highest angular momentum
is the
lowest.
In
polyelectronic atoms
the
motions and spins
of the
individual electrons are
correlated
due to
their
electrostatic
and
spin-spin
interactions.
Consequences:
a)
Only electronic configurations that
do not
violate
the
Pauli Principle
are
allowed.
b)
The
different arrangements
of
x
electrons occupying
y
orbitals are
not
all equal
in
energy
due to the
different electron-electron repulsion energies
in
these
microstates
.
This leads
to
electronic
fine
structure
of the
atom.
E.g., the
d
2
configuration:
Two
electrons can occupy any
of the five
d-orbitals depending
on
their quantum
numbers
m
l
and
m
s
.
One
possible microstate would be:
m
l
m
s
…
Obviously there are many
more
possible…
Deriving Spectroscopic Terms
Clearly, there
is a
finite number
of
(allowed) electronic configurations
for 2
electrons
in 5
degenerate orbitals (and
for
any
x
electrons
in
y
orbitals).
Each unique configuration, called
a
microstate
,
is
defined
by a
unique combination
of
quantum
numbers.
Individual microstates may have
different energies
because each represents
a
different
spatial
distribution
of
electrons within
the
atom resulting
in
different inter-electronic
repulsions.
Microstates
of the
same energy (degenerate) are grouped together into
terms
.
A
general
term
symbol
that uniquely describes
a
specific electronic configuration
looks like
this:
(2
S
+1)
L
J
where
2
S
+ 1 is the
spin multiplicity (and
S
is the
total spin angular
momentum.)
L
is the
total orbital angular
momentum
J
is the
total angular momentum (spin
+
orbital)
S
= 0
“Singlet”
S
= ½
“Doublet”
S
= 1
“Triplet”
etc.
NOTE:
We
use
lower case letters
to
define single electron quantum numbers, and
upper case letters
to
define multiple electron
terms.
Microstates can
be
visualized
through the
vector model
of the
atom.
Each electron has an orbital angular momentum
l
and
a
spin angular momentum
s
.
The
single electron orbital
angular
momentum
l
(and hence
the
total orbital
angular momentum
L
)
can only have certain orientations
quantization.
The
total
orbital
angular
momentum
L
of a group of
electrons
in
an atom
is
given
by a
vector
sum
of the
individual orbital angular momenta
l
.
Two
simple examples are p
2
and
d
2
:
(Source: “Molecular Symmetry and
Group
Theory”, R.L. Carter, Wiley,
1998)
The
total angular
momentum
J
is
related
to the
energy, i.e. different combinations
of
l
and
s
will
result
in
different energies
or
terms.
→
SPIN-ORBIT
COUPLING!
There are
two
ways
of
defining
J
:
1)
Russel-Saunders coupling:
-
Couple all individual orbital angular momenta
l
to
give
a
resultant total orbital
angular momentum
L
. (
L
=
l
)
-
Couple all individual spin angular momenta
s
to
give
a
resultant total spin angular
momentum
S
. (
S
=
s
)
-
Finally couple
L
and
S
to
give
the
total angular momentum
J
for the
entire
atom.
-
Russel-Saunders coupling
works
well
for the light
elements
up to
bromine.
2) j-j
coupling
-
Couple individual orbital
l
and spin
s
angular momenta
first to the
complete electron
angular momentum
j
. (
j
=
l
+
s
)
-
Couple all
j
to
give
the
total angular momentum
J
. (
J
=
j
)
-
j-j
coupling
is
much
more
complicated
to
treat,
but
should
be
used
for
elements heavier
than bromine.
Applying
the
Russel-Saunders
Scheme:
Need
to know the
values
of
L
and
S
:
L
=
total orbital angular momentum quantum number associated
with
collection
of
microstates
with
L
.
S
=
total spin angular momentum quantum number associated
with
collection
of
microstates
with
S
.
L
and
S
define
the
spectroscopic
term
.
L
=
maximum
M
L
M
L
= 0, ±1, ±2, …,
L
S
=
maximum
M
S
M
S
=
S
,
S
-1,
S
-2, …,
-
S
and
M
L
=
m
l
M
S
=
m
s
where
m
l
and
m
s
are values
for
individual electrons
in a
given
microstate
#
microstates
n
!
e
!
h
!
In
order
to find the
terms
L
and
S
we
have
to sum up
m
l
and
m
s
of
all possible
microstates.
There are
2
L
+1
possible orientations
of
L
and
2
S
+1
possible orientations
of
S
.
Therefore,
the
total number
of
microstates
in one
term
given
L
and
S
will
be
(2
L
+ 1)
(2
S
+
1)
This
must be so
as
the
possible values
of
M
L
and
M
S
are:
M
L
= 0,
1,
2, …,
L
M
S
=
S
,
S
–1,
S
-2, …,
-
S
and
L
=
max.
M
L
and
S
=
max.
M
S
By
convention
the
atomic term symbols
are assigned as follows (function
of
L
and
S
):
Example
of a
Many Electron
Atom
:
Carbon
in its
ground-state
[He]2s
2
2p
2
First an important
realization:
Closed (sub)shells make zero contribution
to
angular momentum.
If
all orbitals are
filled
with 2
electrons,
S
= 0
and
L
=
0
→
i.e., all ang. momenta CANCEL
OUT!
for
[He]2s
2
2p
2
that leaves
the
p
2
configuration
to be
considered.
Each
p
electron can
have:
n
= 2,
l
=
1
m
l
= 1, 0,
-1
m
s
= ½, -
½
(2p
orbitals)
(3
possible values: p
x
, p
y
,
p
z
)
Determining
the
number
of
microstates:
n
=
total
#
sites available (i.e.,
2 x # of
orbitals)
e
=
number
of
electrons
h
=
number
of
holes (i.e.,
n –
e
)
For
p
2
, all possible combination
of 2
electrons gives
6!/2!4! = 15
combinations, which
are best collected and visualized
in a
table:
m
l
m
s
The
total angular momentum quantum numbers
L
and
S
are
the
largest possible
values
of
M
L
and
M
S
We
now
have
to
consider that
just
as
for
l
and
m
l
,
M
L
and
M
S
can
have
M
L
=
L
,
L
-1,
L
-2, …,
-
L
i.e. M
L
= 0,
1,
2, …,
L
; for
any given
L
there are
2
L
+1
microstates
and
M
S
=
S
,
S
-1,
S
-2, …
-
S
i.e
.
M
S
= 0,
1,
2, …,
S
; for
any given
S
there are
2
S
+1
microstates
Now
start
with the
maximum
M
L
:
We see
from the
table that
the
maximum
M
L
here
is 2,
and that
it
only occurs
in
combination
with
M
S
=
0.
Therefore,
we must
have
a
term
with
L
= 2
and
S
= 0:
1
D
This term accounts
for (2
L
+
1)(2
S
+ 1) = 5
(
M
L
,
M
S
) microstates
(2,0), (1,0), (0,0), (-1,0),
(-2,0)
and leaves ten microstates
to be
accounted
for.
Cross
off the
five microstates
that we have
accounted
for
and
we
are left
with:
The
maximum value
of
M
L
is now 1,
and
it
occurs
with a
maximum
M
S
=
1:
Therefore,
we must
have
a
term
with
L
= 1
and
S
= 1:
3
P
This term accounts
for
nine
microstates:
(1,1), (1,0), (1,-1), (0,1), (0, 0), (0,-1),
(-1,1),
(-1,0), (-1,-1)
and leaves only
one
microstate
(0,0) to
account
for:
Therefore,
we must
have
a
term
with
L
= 0
and
S
= 0:
1
S
Each term (
1
D,
3
P,
and
1
S) defines
a state
(group
of
microstates
of the same
energy.)
Total Angular
Momentum
in
Many Electron
Atoms: Finding
J
and
M
J
Applying
the
Russel-Saunders Scheme:
to our C
example…
As the
carbon atom
is a light
atom
we
can
now use the
Russell-Saunders
coupling scheme
to
account
for
spin-orbit coupling and
its
effects.
Considering only
the 9
microstates
of the
3
P term
we find the
values
of
J
.
Again
we
will use a
table
to
visualize
the
possible
combinations:
The
largest
M
J
= 2,
i.e.
the
largest
J
= 2
;
there are
2
J
+1 = five
states:
M
J
= 2, 1, 0, -1,
-2
If we
cross these
M
J
values
off the
above table,
we
are left
with:
The
largest remaining
M
J
=1,
therefore
J
=1
:
there are three
M
J
states:
1, 0,
-1
Finally, all that
is
left
is
M
J
= 0,
therefore
J
=0
:
there
is
only
one
M
J
state:
0
The
term
3
P consists
of
three terms
with
different total angular
momenta:
3
P
2
3
P
1
3
P
0
Each
of the
terms
is
degenerate
by
(2
L
+1)(2
S
+1)
=
(2
J
+1) giving
the
total
15
microstates
we
started
with:
(2
J
+1)
1
5
5
3
1
15
1
S
o
1
D
2
3
P
2
3
P
1
3
P
0
Total
The
graphs above give
a
graphical summary
of the
whole
process.
Source: Purcell
&
Kotz, Inorganic Chemistry,
1977;
P.W. Atkins, Physical Chemistry,
3
rd
Edition,
1987
**HOMEWORK: Derive
ALL the
term symbols
for the
Ti
2+
ion in the gas
phase.
The
correct answer
is
given
in the
table
below.
Q.
Which term represents
the
ground
state?
A.
This can
be
determined
using
Hund’s
Rules
:
1)
The
term
with the
highest multiplicity
(=
microstates
with
highest number
of
unpaired
electrons)
is
lowest
in
energy.
2)
For a
term
of
given multiplicity,
the
greater
the
value
of
L
, the
lower
the
energy.
Classical explanation:
The
higher
L
, the
better correlated
the
orbital motion
of the
electrons,
the
less repulsion,
the
lower
the
energy.
For the
d
2
configuration term energies
we find the
theoretical
order:
3
F
<
3
P
<
1
G
<
1
D
<
1
S
Experimentally
found is:
3
F
<
1
D
<
3
P
<
1
G
<
1
S
Hund’s
rules are
not
always reliable regarding
the
largest
L
term,
but
are always reliable
in
establishing
the
ground
state.
How to
quickly
find the ground
state
:
-
Find the
microstate
with the
highest
multiplicity.
-
Find the
highest possible
M
L
for
that
microstate
Ground
state
term.
Finding
the ground
state
in
free
ions –
e.g. d
4
and
d
7
:
m
s
=
+1/2
+1/2
+1/2
+1/2
m
s
=
+1/2
-1/2
+1/2
-1/2
+1/2
NOTE:
Rule #3.
For
less than half-filled sub-shells,
the
term
with the
lowest value
of
J
is
lowest
in
energy.
m
l
=
+2
+1
0
-
1
-
2
M
S
=
1/2
+
1/2
+
1/2
+1/2
=
2
M
L
= +2 +
1
+
0
+
-1
=2
so...
L
=
2;
S
=
2
5
D
m
l
=
+2
+1
0
-
1
-
2
M
S
=
1/2-1/2
+
1/2-1/2
+
1/2
+1/2
+1/2
=
3/2
M
L
= +2 +
2
+
1
+
1
+
0
+
-1+ -2
=3
so...
L
=
3;
S
=
3/2
4
F
+1
/2
+1/2
Applications
of
term
symbols:
a)
Term symbols allow
a
quick energetic ordering
of
atomic microstates
(Hund’s
Rules):
b)
Spectroscopic selection rules tell
us
which transitions are expected
to
have zero
intensity based
on the
harmonic oscillator
approach.
Selection rules
for
electronic transitions can
be
expressed
using
term
symbols:
Δ
S
=
0
(electrons are Fermions
with
half-integral spin; photons are
bosons with
integral spin
light
cannot affect
spin)
Δ
L
= 0,
1 with
Δ
l
=
1
(the orbital angular momentum
of
an electron
must
change,
but
this
does
not
necessarily affect
the
overall momentum)
Δ
J
=
1, 0
(but
J
= 0
cannot combine
with
J
=0)
The energy of atomic configurations is determined by electrostatic attraction between electrons and the nucleus, electron-electron repulsion, spin-orbit coupling, and spin-spin interactions. Term symbols in electronic spectroscopy specify atomic states using quantum numbers. Hund's rule and the Pauli principle play essential roles in describing atomic configurations. The relative energies of terms depend on spin multiplicity and angular momentum. Various coupling schemes like Russell-Saunders and jj-coupling are used for different atoms.
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Presentation Transcript
1.6 Term Symbols A brief general review of atomic configurations: There are four different interactions in any atom that determine the relative energies of atomic configurations (in order of importance): a) Electrostatic attraction between electrons and the nucleus Electrostatic energy 1/n2, where n = principal quantum number b) Electron-electron repulsion Leads to Hund's rule of maximum multiplicity Because of Coulomb repulsion, pairing of electrons in one orbital costs energy: Charge CorrelationEnergy px py pz E = charge correlationenergy (e-e repulsion!) py = A px py pz E = exchangeenergy px = B px py pz Exchange energy: exchange leading to an overall lower energy (quantum mechanical resonance phenomenon.) Electrons of like spin in energy degenerate orbitals can Another way of looking at this is to consider a diagram of the pxand pyorbitals of an atom. Although these orbitals are orthogonal (zero overlap integral), there is nevertheless a region of actual orbital overlap in space (shown in red). Consider one electron in each orbital. If they have opposite spin, their spatial wave function can put them in exactly the same space at the same time (high electrostatic repulsion). If they have the same spin, there must be a constraint on their spatial wave function so as to not put them in exactly the same space at the same time (therefore lower electrostatic repulsion if spin is the same)
In electronic spectroscopy, an atomic term symbol specifies a certain electronic state of an atom (usually a multi- electron one), by briefing the quantum numbers for the angular momenta of that atom. The form of an atomic term symbol implies Russell-Saunders coupling. Transitions between two different atomic states may be represented using their term symbols, to which certain rules apply. In the Russell-Saunders coupling scheme, term symbols are in the form of 2S+1LJ, where S represents the total spin angular momentum, L specifies the total orbital angular momentum, and J refers to the total angular momentum. In a term symbol, L is always an upper-case from the sequence "s, p, d, f, g, h, i, k...", wherein the first four letters stand for sharp, principal, diffuse and fundamental, and the rest follow in an alphabetical pattern. Note that the letter j is omitted.
c) Spin-orbit coupling i) Russel-Saunders for light atoms: J = L + S, L + S 1, , |L S| L = l1 + l2, l1 + l2 1, , |l1-l2| S = s1 + s2, s1 + s2 1, , |s1 s2| ii) jj-coupling for heavy atoms (heavier than bromine) j1 = l1 + s1, l1 + s1 1, , |l1 s1| . jn = ln + sn, ln + s1 1, , |ln sn| J = j1 + j2, j1 + j2 1, , |j1 j2| d) Spin-spin interactions Leads to the Pauli-Principle: No two Fermions in any spatially/energetically confined system can have the same four quantum numbers (n, l, ml, ms). SEE SUPP. INFO ON HUND S RULE In many-electron-atoms each electronic configuration can be described by a unique term symbol. E.g., for scandium 3d14s2: 2D3/2, 2D5/2 (only one unpaired electron easy to solve) 3d24s1: 2S, 2D, 2G, 2,4P, 2,4F (more difficult to figure out) The relative energies of the terms depend on spin their multiplicity (2S+1) and their orbital angular momentum (L). Based on Hund's rule, the lowest energy terms are the ones with the largest spin multiplicity and among these, the term with the highest angular momentum is the lowest.
In polyelectronic atoms the motions and spins of the individual electrons are correlated due to their electrostatic and spin-spin interactions. Consequences: a) Only electronic configurations that do not violate the Pauli Principle are allowed. b) The different arrangements of x electrons occupying y orbitals are not all equal in energy due to the different electron-electron repulsion energies in these microstates. This leads to electronic fine structure of the atom. E.g., the d2configuration: Two electrons can occupy any of the five d-orbitals depending on their quantum numbers ml andms. One possible microstate would be: ml 2 1 0 -1 -2 ms +1/2 +1/2 Obviously there are many more possible Deriving Spectroscopic Terms Clearly, there is a finite number of (allowed) electronic configurations for 2 electrons in 5 degenerate orbitals (and for any x electrons in y orbitals). Each unique configuration, called a microstate, is defined by a unique combination of quantum numbers. Individual microstates may have different energies because each represents a different spatial distribution of electrons within the atom resulting in different inter-electronic repulsions. Microstates of the same energy (degenerate) are grouped together into terms.
A general term symbol that uniquely describes a specific electronic configuration looks like this: (2S+1)LJ where 2S + 1 is the spin multiplicity (and S is the total spin angular momentum.) L is the total orbital angular momentum J is the total angular momentum (spin + orbital) Singlet S = Doublet Triplet S = 0 S = 1 etc. NOTE: We use lower case letters to define single electron quantum numbers, and upper case letters to define multiple electron terms. Microstates can be visualized through the vector model of the atom. Each electron has an orbital angular momentum l and a spin angular momentum s. The single electron orbital angular momentum l (and hence the total orbital angular momentum L) can only have certain orientations quantization. The total orbital angular momentum L of a group of electrons in an atom is given by a vector sum of the individual orbital angular momenta l. Two simple examples are p2 andd2: (Source: Molecular Symmetry and Group Theory , R.L. Carter, Wiley, 1998)
The total angular momentum J is related to the energy, i.e. different combinations of l and s will result in different energies or terms. SPIN-ORBIT COUPLING! There are two ways of defining J: 1) Russel-Saunders coupling: - Couple all individual orbital angular momenta l to give a resultant total orbital angular momentum L. (L = l) - Couple all individual spin angular momenta s to give a resultant total spin angular momentum S. (S = s) - Finally couple L and S to give the total angular momentum J for the entire atom. - Russel-Saunders coupling works well for the light elements up to bromine. 2) j-j coupling -Couple individual orbital l and spin s angular momenta first to the complete electron angular momentum j. (j = l + s) - Couple all j to give the total angular momentum J. (J = j) -j-j coupling is much more complicated to treat, but should be used for elements heavier than bromine. Applying the Russel-Saunders Scheme: Need to know the values of L and S: L = total orbital angular momentum quantum number associated with collection of microstates with L. S = total spin angular momentum quantum number associated with collection of microstates with S. L and S define the spectroscopic term. L = maximumML ML = 0, 1, 2, ,L S = maximum MS MS = S, S-1, S-2, ,-S and ML = ml MS = ms where ml and ms are values for individual electrons in a givenmicrostate
In order to find the terms L and S we have to sum up ml and ms of all possible microstates. There are 2L+1 possible orientations of L and 2S+1 possible orientations of S. Therefore, the total number of microstates in one term given L and S will be (2L + 1) (2S + 1) This must be so as the possible values of ML and MSare: ML = 0, 1, 2, , L MS = S, S 1, S-2, ,-S and L = max. ML and S = max. MS By convention the atomic term symbols are assigned as follows (function of L and S): L = 0 S 0 1 1 P 2 2 D 1 3 3 F 4 G 2 5 5 H 5/2 6 6 I 6 7 S = 2S+1 3/2 4 Example of a Many Electron Atom: Carbon in its ground-state [He]2s22p2 First an important realization: Closed (sub)shells make zero contribution to angular momentum. If all orbitals are filled with 2 electrons, S = 0 and L = 0 i.e., all ang. momenta CANCELOUT! for [He]2s22p2 that leaves the p2 configuration to be considered. Each p electron can have: n = 2, l =1 ml = 1, 0,-1 ms = , - (2p orbitals) (3 possible values: px, py, pz) # microstates =n! Determining the number of microstates: e!h! n = total # sites available (i.e., 2 x # of orbitals) e = number of electrons h = number of holes (i.e., n e) For p2, all possible combination of 2 electrons gives 6!/2!4! = 15 combinations, which are best collected and visualized in a table:
ml ms - - - 1 x x x x x x x x x x x 0 x x x x x x x x x x -1 x x x x x -2 0 x 0 0 x 0 -1 x -1 0 x -1 -1 ML = ml MS = ms 2 0 1 1 1 0 0 1 1 0 1 -1 0 0 0 0 -1 1 -1 0 The total angular momentum quantum numbers L and S are the largest possible values of ML andMS We now have to consider that just as for l and ml, ML and MS canhave ML = L, L-1, L-2, ,-L i.e. ML = 0, 1, 2, , L; for any given L there are 2L+1 microstates and MS = S, S-1, S-2, -S i.e. MS = 0, 1, 2, , S; for any given S there are 2S+1microstates Now start with the maximum ML: We see from the table that the maximum ML here is 2, and that it only occurs in combination with MS =0. 1D Therefore, we must have a term with L = 2 and S = 0: This term accounts for (2L + 1)(2S + 1) = 5 (ML,MS) microstates (2,0), (1,0), (0,0), (-1,0), (-2,0) and leaves ten microstates to be accounted for. Cross off the five microstates that we have accounted for and we are left with: 1 1 0 1 1 0 1 -1 0 0 0 -1 0 0 -1 1 -1 0 -1 -1 ML MS Therefore, we must have a term with L = 1 and S = 1: The maximum value of ML is now 1, and it occurs with a maximum MS =1: 3P This term accounts for nine microstates: (-1,0), (-1,-1) and leaves only one microstate (0,0) to account for: (1,1), (1,0), (1,-1), (0,1), (0, 0), (0,-1),(-1,1), 1S Therefore, we must have a term with L = 0 and S = 0: Each term (1D, 3P, and 1S) defines a state (group of microstates of the same energy.)
Total Angular Momentum in Many Electron Atoms: Finding J and MJ Applying the Russel-Saunders Scheme: to our C example As the carbon atom is a light atom we can now use the Russell-Saunders coupling scheme to account for spin-orbit coupling and its effects. Considering only the 9 microstates of the 3P term we find the values of J. Again we will use a table to visualize the possible combinations: ML MS MJ 1 1 1 0 0 0 -1 -1 -1 1 0 -1 1 0 -1 1 0 -1 2 1 0 1 0 -1 0 -1 -2 The largest MJ = 2, i.e. the largest J = 2; there are 2J+1 = five states: MJ = 2, 1, 0, -1,-2 If we cross these MJ values off the above table, we are leftwith: The largest remaining MJ =1, therefore J=1: there are three MJ states: 1, 0,-1 Finally, all that is left is MJ = 0, therefore J=0: there is only one MJ state:0 The term 3P consists of three terms with different total angular momenta: 3P2 3P1 3P0 Each of the terms is degenerate by (2L+1)(2S+1) = (2J+1) giving the total 15 microstates we started with: (2J+1) 1 5 5 3 1 1So 1D2 3P2 3P1 3P0 Total 15
The graphs above give a graphical summary of the whole process. Source: Purcell & Kotz, Inorganic Chemistry, 1977; P.W. Atkins, Physical Chemistry, 3rd Edition,1987 **HOMEWORK: Derive ALL the term symbols for the Ti2+ ion in the gasphase. The correct answer is given in the table below. 1G 3F 1D 3P 1S 9 21 5 9 1 45 Total # of microstates in d2 Q. Which term represents the ground state? A. This can be determined using Hund s Rules: 1) The term with the highest multiplicity (= microstates with highest number of unpaired electrons) is lowest in energy. 2) For a term of given multiplicity, the greater the value of L, the lower the energy.
Classical explanation: The higher L, the better correlated the orbital motion of the electrons, the less repulsion, the lower the energy. For the d2 configuration term energies we find the theoreticalorder: 3F < 3P < 1G < 1D < 1S Experimentally found is: 3F < 1D < 3P < 1G < 1S Hund s rules are not always reliable regarding the largest L term, but are always reliable in establishing the ground state. How to quickly find the ground state: - Find the microstate with the highest multiplicity. - Find the highest possible ML for thatmicrostate Ground state term. Finding the ground state in free ions e.g. d4 andd7: +1/2 +1/2 ms = +1/2 -1/2 +1/2 -1/2 +1/2 ms= +1/2 +1/2 +1/2 +1/2 ml= +2 +1 0 -1 -2 ml= +2 +1 0 -1 -2 MS = 1/2 + 1/2 + 1/2 +1/2 = 2 ML = +2 + 1 + 0 + -1 =2 so... L = 2; S =2 MS = 1/2-1/2 + 1/2-1/2 + 1/2 +1/2 +1/2 = 3/2 ML = +2 + 2 + 1 + 1 + 0 + -1+ -2 =3 so... L = 3; S =3/2 5D 4F NOTE: Rule #3. For less than half-filled sub-shells, the term with the lowest value of J is lowest in energy.
Applications of term symbols: a) Term symbols allow a quick energetic ordering of atomic microstates (Hund s Rules): b)Spectroscopic selection rules tell us which transitions are expected to have zero intensity based on the harmonic oscillator approach. Selection rules for electronic transitions can be expressed using term symbols: S =0 (electrons are Fermions with half-integral spin; photons are bosons with integral spin light cannot affect spin) L = 0, 1 with l = 1 (the orbital angular momentum of an electron must change, but this does not necessarily affect the overall momentum) J = 1, 0 (but J = 0 cannot combine with J =0)
