Molecular shape and electronic states — VSEPR, hybridization, term symbols

Predict a molecule’s shape from its valence-electron count alone, and use the same logic to pin a $^{2S+1}L_J$ label onto an atom’s ground state.

Opening

Chapter 7 worked with Slater determinants and molecular orbitals (MOs) for many-electron molecules. This chapter comes down to a more tangible picture — why does a molecule take the shape it does? The reason H $_2$ O bends, NH $_3$ is umbrella-shaped, and CH $_4$ is a perfect tetrahedron, once unpacked, is all a consequence of one sentence: electron pairs try to get as far apart as they can. After this chapter you can predict a molecule’s shape immediately from the number of bonded atoms and lone pairs around the central atom, see how that prediction meshes with the mathematical picture of hybrid orbitals, and explain in one line how the same reasoning, at the atomic level, organizes into the term symbol $^{2S+1}L_J$ .

Main 1 — VSEPR: electron pairs choose the positions

The VSEPR (valence-shell electron-pair repulsion) model rests on a single premise. The electron pairs around a central atom — whether bonding pairs or lone pairs — repel one another, so they adopt the arrangement that maximizes their separation. The shape follows from that one line.

The quantity we count is the steric number (SN).

\text{SN} = (\text{number of bonded atoms}) + (\text{number of lone pairs})

The standard geometry for each steric number is as follows.

SN = 2 → linear, bond angle 180°. Examples: BeCl $_2$ , CO $_2$ .
SN = 3 → trigonal planar, 120°. Example: BF $_3$ .
SN = 4 → tetrahedral, 109.5°. Example: CH $_4$ . With one lone pair it collapses to trigonal pyramidal, giving NH $_3$ (~107°). With two lone pairs it narrows further to bent, giving H $_2$ O (~104.5°).
SN = 5 → trigonal bipyramidal, 90° axial / 120° equatorial. Example: PCl $_5$ .
SN = 6 → octahedral, 90°. Example: SF $_6$ .

There is a reason H $_2$ O’s bond angle is 104.5°, tighter than the ideal tetrahedral 109.5°. A lone pair is not tied down to a proton, so it is more spatially diffuse and takes up more room than a bonding pair. The lone pairs therefore push the bonding pairs inward from the outside, squeezing the angle. NH $_3$ ’s 107° is a weaker version of the same effect. VSEPR gets this far quantitatively without invoking quantum mechanics at all.

Main 2 — Hybridization: how to re-shape atomic orbitals

If VSEPR tells you what shape you get, hybridization tells you how to recombine the central atom’s atomic orbitals (AOs) to match that shape.

Carbon’s ground-state configuration is $1s^2 2s^2 2p^2$ , but when it forms four equivalent bonds as in CH $_4$ , we mix one $2s$ and three $2p$ AOs into four equivalent $sp^3$ hybrid orbitals. These four point to the corners of a tetrahedron. For a trigonal-planar case — BF $_3$ , or the carbons of ethylene — we mix $2s$ + $2p_x$ + $2p_y$ into three $sp^2$ hybrids plus one leftover $2p_z$ ; for a linear case — the carbons of acetylene — we mix only $2s$ + $2p_x$ into two $sp$ hybrids plus two leftover $2p$ . The rule is one line.

\text{steric number} = \text{number of hybrid orbitals}

The leftover unhybridized $p$ orbitals meet the matching $p$ on a neighboring atom and form $\pi$ bonds (chapter 6). A double bond is $sp^2$ + $\pi$ ; a triple bond is $sp$ + $\pi \times 2$ .

One caution. Hybridization is not a measured fact but a post-hoc description. We take the experimental result that a molecule is tetrahedral and then choose an orbital basis to fit it. Even so, this description lets us organize bond angles, the presence of $\pi$ bonds, and planarity all at once.

Main 3 — Atomic term symbols $^{2S+1}L_J$

Organizing the same reasoning at the atomic level gives the term symbol $^{2S+1}L_J$ . Call the total orbital angular momentum of an atom $L$ , the total spin $S$ (capital, summed over all electrons), and the total angular momentum $J = L + S$ . The notation reads as follows.

The superscript $2S+1$ — the multiplicity.
The central letter — the $L$ value. $L = 0, 1, 2, 3 \to$ S, P, D, F.
The subscript — $J$ .

This label distinguishes the slightly different states that share one electron configuration. Which one is the ground state is decided by Hund’s rules.

Maximize $S$ .
For fixed $S$ , maximize $L$ .
For a less-than-half-filled shell, minimize $J$ (for more than half full, the opposite).

Let us do nitrogen ( $1s^2 2s^2 2p^3$ ) directly. Three electrons go into the three $2p$ slots $m_\ell = -1, 0, +1$ ; Hund’s first rule aligns all three to the same spin, so $S = 3/2$ . The sum of the magnetic quantum numbers is $-1 + 0 + 1 = 0$ , so $L = 0$ → the letter is S. $J$ runs from $|L - S|$ to $|L + S|$ , and for a less-than-half-filled shell we take the smallest value: $J = 3/2$ . The result is

^4 S_{3/2}

That is nitrogen’s ground-state label. Multiplicity 4 means there are $2S+1 = 4$ microstates of the same energy.

Molecules carry symbols of the same kind. The ground state of H $_2$ is $^1\Sigma_g^+$ ; the ground state of O $_2$ is $^3\Sigma_g^-$ (which is why the oxygen molecule is paramagnetic). Here $\Sigma, \Pi, \Delta$ are the $L$ -analog for angular momentum along the molecular axis, $g/u$ labels inversion symmetry, and $\pm$ labels mirror symmetry. The deeper classification reappears in chapter 9 on spectroscopy.

In Python

# A VSEPR classifier: from bond count and lone-pair count alone,
# return the steric number and the shape. We also print the
# experimental bond angle alongside.

SHAPES = {
    (2, 0): ("linear",               180.0),
    (3, 0): ("trigonal planar",      120.0),
    (4, 0): ("tetrahedral",          109.5),
    (3, 1): ("trigonal pyramidal",   107.0),
    (2, 2): ("bent",                 104.5),
    (5, 0): ("trigonal bipyramidal",  90.0),
    (6, 0): ("octahedral",            90.0),
}

def vsepr(bonding, lone):
    sn = bonding + lone
    shape, angle = SHAPES[(bonding, lone)]
    return sn, shape, angle

cases = [
    ("H2O",   2, 2),
    ("NH3",   3, 1),
    ("CH4",   4, 0),
    ("BeCl2", 2, 0),
    ("BF3",   3, 0),
    ("PCl5",  5, 0),
    ("SF6",   6, 0),
]

print(f"{'formula':<6}  {'SN':>2}  {'shape':<22}  angle")
print("-" * 46)
for formula, b, l in cases:
    sn, shape, angle = vsepr(b, l)
    print(f"{formula:<6}  {sn:>2}  {shape:<22}  {angle:5.1f}°")

If each line of the output matches the shape and angle predicted in the text, you have taken hold of the geometry of seven molecules with the one-line rule that electron pairs spread apart.

To the next chapter

In Chapter 9: How molecules interact with light we see which state a molecule’s electronic state — its shape, hybridization, term symbol — can transition to when it meets light, and how the selection rules for that are fixed. The photon picture of chapter 4 and the state picture of this chapter merge there.