SME
Physical Chemistry II: Quantum & Spectroscopy · Problem type
Microstates of an open subshell
Use this for “How many microstates does the d³ configuration of V²⁺ have?”
Diagram
Row 6: 1, 6, 15, 20, 15, 6, 1; they add to 2⁶ = 64.
Entry 2 of row 6: C(6, 2) = 15.
The two above add to it: C(5, 1) + C(5, 2) = 5 + 10 = 15.
2 places in order: 6 × 5 = 30 orders.
Each
group
of
2
comes
in
2!
=
2
orders:
30
2
=
15
groups.
This is an example. Type your own values to start a new problem.
N
=
2(2
×
ℓ
+
1)
=
N
M
=
C(
N
,
n
)
=
M
Formulas
N
=
2
×
(2
×
ℓ
+
1)
6 = 2 × (2 × 1 + 1)
M
=
C(
N
,
n
)
15 = C(6, 2)
Assumptions
•
Only the open subshell counts: filled shells have one arrangement each and add nothing.
•
The subshell has 2ℓ + 1 orbitals, each with spin up or spin down, so 2(2ℓ + 1) spin-orbitals.
•
Pauli: at most one electron in each spin-orbital, and electrons cannot be told apart, so the count is C(2(2ℓ + 1), n).
Step-by-step
WE KNOW
ℓ = 1, n = 2
FIND
spin-orbitals (N), microstates (M)
Step 1 · Find spin-orbitals (N)
Use N = 2 × (2 × ℓ + 1)
Each
of
the
2
ℓ
+
1
orbitals
holds
two
spin-orbitals,
spin
up
and
spin
down.
N
=
2
×
(2
×
1
+
1)
N
=
2
×
3
N
=
6
Step 2 · Find microstates (M)
Use M = C(N, n)
Choose
which
n
of
the
N
spin-orbitals
hold
the
electrons:
N
!
n
! (
N
−
n
)!
.
M
=
C(6,
2)
M
=
15
→
2
electrons
leave
4
empty
spin-orbitals,
and
4
electrons
would
give
the
same
count:
holes
count
like
electrons.
Check
Put the numbers back in. Both sides should match.
6 = 2 × (2 × 1 + 1) ✓
15 = C(6, 2) ✓
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Variational and perturbation methods
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