Deriving Term Symbols for dⁿ Configurations

Introduction

The list of terms for d² is not a collection of labels to memorise without explanation. It can be derived by arranging two electrons among ten d spin-orbitals and sorting the allowed arrangements by their total orbital and spin projections. This procedure makes the Pauli principle visible, explains why both triplet and singlet terms occur, and gives a completeness check. It is the bridge from electron-box diagrams to the many-electron states used in electronic spectroscopy.

Core explanation

A d electron has m l values −2, −1, 0, +1 or +2, each available with m s=+1/2 or −1/2. For a pair, choose any two different spin-orbitals: there are C(10,2)=45 microstates. For each pair calculate M L=m l(1)+m l(2) and M S=m s(1)+m s(2). Place a tally in the cell at those coordinates. The table is symmetric under simultaneous reversal of projections, but its entries are not all equal. For instance, M S=+1 requires two spin-up electrons in different spatial orbitals. M L=4 would require both electrons in m l=+2, impossible when their spins are the same. Thus the highest M L in the M S=+1 row is 3, produced by m l=+2 and +1.

The highest remaining M S and M L identify a term with S=M S and L=M L. Starting at M S=1, M L=3 gives ^3F. One ^3F term accounts for one microstate at every combination M S=−1,0,+1 and M L=−3,…,+3: 3×7=21 entries. Subtract that rectangular pattern from the table. The largest remaining triplet projection is M L=1, giving ^3P, with 3×3=9 microstates. The remaining singlet patterns successively yield ^1G, ^1D and ^1S. Their dimensions are 9, 5 and 1. The sum 21+9+9+5+1=45 matches C(10,2), so no state is lost or counted twice.

Why do we subtract a rectangle rather than merely remove its highest cell? A quantum state of definite L has every orbital projection from −L to +L; a state of definite S has every spin projection from −S to +S. One term therefore occupies all combinations of those projections. Even where two terms share an M L,M S cell, the corresponding microstates are distinct linear combinations of electron assignments. The tally records dimension, not a unique pictorial electron pair for each term.

The procedure generalises to dⁿ by selecting n of ten spin-orbitals, constructing projection totals, extracting terms and checking against C(10,n). Its arithmetic grows rapidly, so published term tables and computational algorithms are practical for larger n. The logic is unchanged. Complementary electron and hole configurations, such as d² and d⁸, have related sets of LS terms because two holes in a filled d shell can be counted similarly to two electrons. Their spin–orbit level ordering need not be identical.

Step-by-step reasoning

List the ten d spin-orbitals and make all unordered electron pairs without repetition. Calculate M L and M S for every pair. Locate the maximum surviving M S and, within it, maximum M L; these give S and L of one term. Remove one tally throughout the full (2S+1) by (2L+1) projection rectangle. Repeat until every cell is empty, then add term dimensions and compare with C(10,2).

Visual explanation

Imagine a grid whose columns run M L=−4 through +4 and rows run M S=−1,0,+1. Write the number of pairs in each cell. Shade a 3-row by 7-column rectangle centred on M L=0 for ^3F. After reducing each shaded count by one, shade a 3-by-3 rectangle for ^3P. The leftover one-row strips have widths 9, 5 and 1, displaying ^1G, ^1D and ^1S.

Real-world analogy

A seating chart counts how many valid pairs of people can occupy seats with particular combined properties. A named group may occupy several cells of that chart, and different groups may contribute to the same cell. Subtracting a complete term is like removing one systematically related set of seating arrangements, not crossing out just its most conspicuous arrangement.

Real-world example

V³⁺ has a d² configuration. Its octahedral aqua ion begins with free-ion d² terms, including ^3F and ^3P, before the ligand field splits them into octahedral states. Recognising several triplet parent terms is essential when assigning more than one spin-allowed absorption band; a simple t₂g-to-e g arrow cannot describe the full spectrum.

Why?

Why are two electrons in m l=+2 allowed for M S=0 but forbidden for M S=+1? Opposite spins occupy different spin-orbitals even when the spatial orbital is the same. Two spin-up electrons in that same spatial orbital would have identical four quantum numbers and violate Pauli exclusion.

Common misconception

“Forty-five microstates mean forty-five spectral bands.” A microstate is a basis-state count, not an observed transition. Electron repulsion groups the microstates into five LS terms; a ligand field, selection rules, degeneracy, and broadening then govern the bands that can actually appear.

Worked example

After extracting ^3F and ^3P from the d² table, 45−21−9=15 microstates remain. A surviving M S=0, M L=4 cell identifies ^1G: S=0, L=4, dimension 1×9=9. Six remain. The largest remaining orbital projection is 2, so ^1D takes five. The final one is ^1S. The completed decomposition is ^3F+^3P+^1G+^1D+^1S, with dimension 21+9+9+5+1=45.

Quick check

1. How many pairs of spin-orbitals must a complete d² table contain? Answer: C(10,2)=45. Selecting ordered first and second electrons would incorrectly double-count each pair.

Exam focus

Show the projection logic and the dimension sum. A term list without the 45-state check may hide a duplicated or omitted term. Do not infer a term merely from two orbital boxes without enforcing Pauli and including all projections.

Advanced insight

Antisymmetrised Slater determinants provide the microstate basis, while LS-coupled functions are symmetry-adapted linear combinations of those determinants. The rectangular subtraction is a character-counting procedure for decomposing the many-electron representation into irreducible angular-momentum sectors; it does not imply that individual determinants themselves possess definite L and S.

Summary

The d² configuration has 45 Pauli-allowed microstates. Sorting by M L and M S and subtracting complete projection sets gives ^3F, ^3P, ^1G, ^1D and ^1S. Their dimensions reproduce 45 and establish the parent terms later split by a ligand field.

Practice questions

1. Why is an M S=+1 pair with M L=4 impossible for d²? Answer: Both electrons would need m l=+2 and spin up, placing them in the same spin-orbital, contrary to Pauli exclusion. 2. What terms and dimensions remain after removing the two triplets from d²? Answer: Singlets ^1G, ^1D and ^1S with dimensions 9, 5 and 1; together they account for the remaining 15 microstates. 3. Does a cell with count two necessarily represent two different terms? Answer: It represents two linearly independent microstates of the same M L and M S. They may belong to different terms, but extraction of complete projection patterns is needed to assign them. 4. Why does ^3P remove nine table entries? Answer: Triplet S=1 supplies three M S values, and P means L=1 with three M L values; their product is nine.