Answer

For f electrons, m_l = 3, 2, 1, 0, –1, –2, –3 so L = 3 + 3 = 6

S = ½ + ½ = 1

Use the notation (m_l^±, m_l^±)

The total number of microstates = [2(2(3)+1)]!/2![12]! = 91

Then the matrix of microstates becomes:

ML\MS	1	0	–1
6		(3+, 3–)
5	(3+, 2+)	(3+, 2–) (3–, 2+)	(3–, 2–)
4	(3+, 1+)	(3+, 1–) (3–, 1+) (2+, 2–)	(3–, 1–)
3	(3+, 0+) (2+, 1+)	(3+, 0–) (3–, 0+) (2+, 1–) (2–, 1+)	(3–, 0–) (2–, 1–)
2	(3+, –1+) (2+, 0+)	(3+, –1–) (3–, –1+) (2+, 0–) (2–, –1+) (1+, 1–)	(3–, –1–) (2–, 0–)
1	(3+, –2+) (2+, –1+) (1+, 0+)	(3+, –2–) (3–, –2+) (2+, –1–) (2–, –1+) (1+, 0–) (1–, 0+)	(3–, –2–) (2–, –1–) (1–, 0–)
0	(3+, –3+) (2+, –2+) (1+, –1+)	(3+, –3–) (3–, –3+) (2+, –2–) (2–, –2+) (1+, –1–) (1–, –1+) (0+, 0–)	(3–, –3–) (2–, –2–) (1–, –1–)
–1	(–3+, 2+) (–2+, 1+) (–1+, 0+)	(–3+, 2–) (–3–, 2+) (–2+, 1–) (–2–, 1+) (–1+, 0–) (–1–, 0+)	(–3–, 2–) (–2–, 1–) (–1–, 0–)
–2	(–3+, 1+) (–2+, 0+)	(–3+, 1–) (–3–, 1+) (–2+, 0–) (–2–, –1+) (–1+, –1–)	(3–, –1–) (2–, 0–)
–3	(–3+, 0+) (–2+, –1+)	(–3+, 0–) (–3–, 0+) (–2+, –1–) (–2–, –1+)	(–3–, 0–) (–2–, –1–)
–4	(–3+, –1+)	(–3+, –1–) (–3–, –1+) (–2+, –2–)	(–3–, –1–)
–5	(–3+, –2+)	(–3+, –2–) (–3–, –2+)	(–3–, –2–)
–6		(–3+, –3–)

Find the different terms:

Black microstates

M_L = 5, M_S = 1 giving ³H (accounting for 3×11 = 33 microstates), J = 6, 5, 4

Red microstates

M_L = 3, M_S = 1 giving ³F (accounting for 3×7 = 21 microstates), J = 4, 3, 2

Blue microstates

M_L = 1, M_S = 1 giving ³P (accounting for 3×3 = 9 microstates), J = 2, 1, 0

Green microstates

M_L = 6, M_S = 0 giving ¹I (accounting for 1×13 = 13 microstates), J = 6

Magenta microstates

M_L = 4, M_S = 0 giving ¹G (accounting for 1×9 = 9 microstates), J = 4

Brown microstates

M_L = 2, M_S = 0 giving ¹D (accounting for 1×5 = 5 microstates), J = 2

Violet microstates

M_L = 0, M_S = 0 giving ¹P (accounting for 1×1 = 1 microstate), J = 0

Predicted energy order: ³H₄ < ³H₅ < ³H₆ < ³F₂ < ³F₃ < ³F₄ < ³P₀ < ³P₁ < ³P₂ < ¹I₆ < ¹G₄ < ¹D₂ < ¹S₀