Chemical Bonding

Orbitals from two or more atoms overlap and the electrons rearrange to minimize energy. Everything else — shape, polarity, boiling point, drug binding, the shape of water — is bookkeeping on that single idea.

1. Why bonding matters

Atoms by themselves do almost nothing interesting. It's when they bond that chemistry begins. A bond is the energetically favorable sharing or transfer of electrons between two nuclei — it exists because the combined system (two nuclei plus the shared electrons) sits at a lower total energy than the two isolated atoms.

That's the whole story, physically. Everything else in this page is a practical consequence: which electrons get shared, what shape they prefer, how unequal the sharing is, how strong the resulting bond is, and how the molecules then interact with each other.

Why should you care?

• Materials. Diamond and graphite are both pure carbon. One is the hardest natural material, the other is a soft lubricant you write with. The difference is 100% bonding geometry.
• Drugs and proteins. Drug-target binding is almost never a covalent event — it's a cloud of intermolecular forces (hydrogen bonds, van der Waals contacts, hydrophobic collapse) that turn a geometry match into a selective grip. Every drug designer works at this layer.
• Water. Water is liquid at room temperature because of hydrogen bonds, which come directly from O-H polarity. Without them, H$_2$O would boil near $-80$ $^\circ$C and life would never have started.
• Batteries and photovoltaics. Band gaps are the outputs of molecular-orbital theory scaled up to a crystal. Picking a cathode is picking a bonding pattern that holds a certain voltage.

2. Vocabulary cheat sheet

3. The three kinds of bond

Historically, chemists carved bonding into three pictures. They are caricatures — real bonds almost always blend them — but each one is clean enough to reason with.

Ionic

One atom is much more electronegative than the other. The less electronegative atom hands its valence electron(s) over and becomes a cation. The more electronegative atom keeps them and becomes an anion. The two resulting ions then hold together by plain Coulomb attraction. Sodium chloride is the textbook case: Na (Pauling 0.93) gives its $3s^1$ electron to Cl (3.16), becoming Na$^+$ and Cl$^-$, and the two ions stack into a rock-salt crystal.

The energy of an ion pair at distance $r$ is well approximated by the Coulomb law:

In a crystal, you don't just have one ion pair — you have a whole lattice. The total Coulomb energy per ion depends on the geometry through a dimensionless factor called the Madelung constant, typically $1.6$–$1.8$ for common ionic solids. The electromagnetism page covers the derivation.

Covalent

Two atoms of similar electronegativity share one or more pairs of electrons. The shared pair sits in a bonding molecular orbital, a new orbital formed by constructive interference between the two atomic orbitals. The competing antibonding combination sits at higher energy and stays empty in stable molecules.

For two hydrogen atoms forming H$_2$, the simplest MO picture writes the two-atom orbital as a linear combination of the individual $1s$ orbitals:

For H$_2$, both electrons pair up in $\psi_+$ (spins opposite, Pauli-allowed) and the molecule is stable at $r_e = 74$ pm with a bond energy of $436$ kJ/mol. That's the smallest covalent-bond calculation you can do, and it already captures the key ideas: bonding orbitals lower the electrons' energy, antibonding raise it, and the bond order is $(n_{\text{bonding}} - n_{\text{antibonding}})/2$.

Metallic

In a metal, many atoms contribute their valence electrons to a common pool — a sea of electrons that belongs to the whole crystal rather than to any individual bond. The positive ion cores sit at regular lattice sites, and the delocalized electrons fill a continuous set of bands up to the Fermi level.

That picture explains metallic properties in one sentence each:

• Conductivity. The delocalized electrons respond to an electric field — no bond to break, just a drift.
• Malleability. The sea doesn't care if you slide one row of ions over another; the bonds don't have fixed partners.
• Opacity and reflectivity. The continuous band of electronic states absorbs photons across the visible spectrum and re-emits them, giving the shiny luster.
• Thermal conductivity. The same mobile electrons carry heat as well as charge, which is why metals are both electrical and thermal conductors.

4. Lewis structures and formal charge

A Lewis structure is a back-of-envelope diagram of where the valence electrons live: which pairs are bonding, which are lone pairs, and which atoms are connected. It's 100 years old, coarse, and still the fastest way to predict the geometry and reactivity of a molecule you haven't seen before.

The recipe, which works for most small molecules:

1. Count valence electrons. Add up group numbers for each atom, subtract for positive charge, add for negative charge.
2. Pick a skeleton. Usually the least electronegative atom (not H) goes in the center.
3. Put single bonds between the central atom and each terminal atom.
4. Distribute the remaining electrons as lone pairs, starting with the most electronegative terminals, until each has an octet (H gets 2).
5. If the central atom is short, form double or triple bonds by moving a terminal lone pair into the bond.
6. Check formal charges. If needed, reshape to minimize them.

The formal charge on an atom in a Lewis structure is

Worked example: CO$_2$

Carbon dioxide has $4 + 2(6) = 16$ valence electrons. Carbon goes in the middle. With two double bonds (O=C=O), the structure uses all 16 electrons: four shared in each double bond (8 total), plus two lone pairs on each oxygen (8 total).

Check formal charges:

• Carbon: $V = 4$, $L = 0$, $B = 8$. FC $= 4 - 0 - 4 = 0$.
• Each oxygen: $V = 6$, $L = 4$, $B = 4$. FC $= 6 - 4 - 2 = 0$.

All zeros — this is the right Lewis structure. Alternatives like O-C$\equiv$O (single and triple) give nonzero formal charges and are worse.

Resonance

Sometimes no single Lewis structure fits. Ozone (O$_3$), for example, has two equivalent structures with the double bond on either side. The real molecule is neither — it is a superposition (resonance hybrid) in which both O-O bonds have identical length and strength, intermediate between a single and double bond. You draw both structures with a double-headed arrow between them and understand that the truth is the average.

Resonance is not "the molecule flickers between these forms." It means the wavefunction is not well-described by any one of them and the best single-reference description is the linear combination. Same idea as the bonding MO for H$_2$, scaled up to multi-site delocalization.

5. VSEPR geometry

Once you have a Lewis structure, the shape of the molecule follows from one principle: electron pairs around a central atom repel each other and arrange themselves as far apart as possible. That's the valence shell electron pair repulsion (VSEPR) model. It's not a physical law, but it's a startlingly accurate pattern-match for most main-group compounds.

Count the "electron domains" on the central atom — each bond (single, double, or triple counts as one domain) and each lone pair — then look up the geometry:

Two refinements that come up constantly:

• Lone pairs take more space than bonding pairs. They're held by only one nucleus, so they fan out more. That's why H$_2$O (two lone pairs) has an H-O-H angle of $104.5°$ instead of the tetrahedral $109.5°$, and NH$_3$ (one lone pair) has $107°$.
• Double and triple bonds also take more space than single bonds, so geometries distort slightly around them. The effect is smaller than lone-pair effects but measurable in high-resolution crystal structures.

VSEPR predicts shape but not bond angles to better than a degree or two. For quantitative work you need an actual electronic-structure calculation; see quantum chemistry.

6. Interactive: molecular shape explorer

Pick a molecule and the figure below draws its VSEPR geometry with the central atom at the origin. Bonds are drawn as lines; lone pairs as small pink clouds. The numbers show the approximate angle between neighboring bonds. Hover the control to see what VSEPR predicts vs. what experiment measures.

7. Hybridization and sigma/pi bonds

Pure atomic orbitals almost never explain molecular geometry directly. Carbon's ground state is $1s^2\,2s^2\,2p^2$ — two unpaired $p$ electrons — but methane has four equivalent C-H bonds at the tetrahedral angle. That's not what two $p$ electrons should give you.

The fix is hybridization: mix the $2s$ and the three $2p$ orbitals on the carbon to make four equivalent $sp^3$ hybrid orbitals, each one a blend pointing at a tetrahedral corner. Then each hybrid overlaps with an H $1s$ to form a C-H bond. You get four equivalent bonds at 109.5° — exactly what methane has.

The three common hybridizations for main-group atoms:

• $sp$ — one $s$ + one $p$ gives two linear orbitals 180° apart. Leaves two unhybridized $p$ orbitals perpendicular to the bond axis. Used in triple bonds (C$\equiv$C in acetylene) and CO$_2$.
• $sp^2$ — one $s$ + two $p$ gives three trigonal planar orbitals 120° apart. Leaves one unhybridized $p$ orbital perpendicular to the plane. Used in double bonds (C=C in ethene), benzene, and carbonyl groups.
• $sp^3$ — one $s$ + three $p$ gives four tetrahedral orbitals 109.5° apart. No leftover $p$ orbitals. Used in saturated carbons (methane, ethane, diamond) and in ammonia and water (with lone pairs in some of the hybrids).

When a hybrid orbital on atom $A$ points at and overlaps a hybrid on atom $B$, the bond is cylindrically symmetric around the line joining the nuclei. That's a $\sigma$ bond. It's the strong, "head-on" kind of overlap and you find one in every chemical bond.

When an unhybridized $p$ orbital on atom $A$ overlaps sideways with a $p$ orbital on atom $B$, the result has a nodal plane containing the bond axis. That's a $\pi$ bond. It's weaker than $\sigma$ because the overlap is smaller, and it locks the molecule against twisting around the $\sigma$ axis.

Hybridization vs. domain count

Quick map from VSEPR domains to hybridization:

• 2 domains → $sp$ (linear)
• 3 domains → $sp^2$ (trigonal planar, including bent with one lone pair)
• 4 domains → $sp^3$ (tetrahedral, including pyramidal/bent)
• 5 or 6 domains → "hypervalent" territory, traditionally $sp^3d$ or $sp^3d^2$, but the $d$-orbital picture is out of favor; it's better to think of these as multicenter bonds. (See quantum chemistry.)

8. Bond polarity and dipole moments

If the two atoms in a bond have different electronegativities, the shared electrons spend more time near the more electronegative one. That atom picks up a partial negative charge ($\delta^-$) and the other picks up a partial positive charge ($\delta^+$). The bond is polar covalent, and the separation of charge over the bond length creates a dipole moment.

For a polyatomic molecule, the total dipole moment is the vector sum of the individual bond dipoles. This is where geometry matters:

• CO$_2$ has two strongly polar C=O bonds but they point in opposite directions along the same line. They cancel, and CO$_2$ has $\mu = 0$ — it's a nonpolar molecule. That's why it's a poor greenhouse absorber at the symmetric stretch (IR-inactive) and only the bending and asymmetric-stretch modes catch infrared light.
• H$_2$O also has two polar O-H bonds, but the molecule is bent at 104.5°. The bond dipoles don't cancel; they add up to a substantial molecular dipole $\mu \approx 1.85$ D. That's why water is such a good solvent for ionic compounds.
• CH$_4$ has four weakly polar C-H bonds pointed at tetrahedral corners. They cancel exactly by symmetry, so methane is nonpolar.
• NH$_3$ has three polar N-H bonds in a pyramidal arrangement plus a lone pair on nitrogen; the dipole is $\mu \approx 1.47$ D, pointing away from the nitrogen along the pyramid axis.

Worked example: the polarity of HCl

HCl has a bond length of $127.5$ pm and an experimental dipole moment of $1.08$ D. What fraction of an elementary charge is transferred?

1. $1.08$ D $= 1.08 \times 3.336\times 10^{-30}$ C·m $= 3.60\times 10^{-30}$ C·m.
2. $q = \mu / r = 3.60\times 10^{-30} / 1.275\times 10^{-10} = 2.83\times 10^{-20}$ C.
3. In units of $e$: $q/e = 2.83\times 10^{-20} / 1.602\times 10^{-19} \approx 0.177$.

So about 18% of a full charge is transferred — the bond is 18% ionic, 82% covalent in character. That single-digit percentage is the full story behind "polar covalent" and maps cleanly onto the Pauling $\Delta\chi = 3.16 - 2.20 = 0.96$ rule of thumb.

9. Intermolecular forces

So far everything has been about what holds a single molecule together. Equally important: what holds a collection of molecules together. These are called intermolecular forces (IMFs), and they set the phase behavior of matter — whether something is a gas, liquid, or solid at a given temperature.

IMFs are a whole order of magnitude weaker than real bonds (think $1$–$40$ kJ/mol, compared to $200$–$1000$ for a covalent bond), but they're what makes water boil at 100°C instead of $-80$, what makes oil refuse to mix with water, and what makes a drug lodge in its target protein. In order of increasing strength:

London dispersion (van der Waals)

Every atom, even a noble gas, has an electron cloud that instantaneously fluctuates. Those instantaneous fluctuations induce correlated fluctuations in a nearby atom, and the two transient dipoles attract each other. The energy of this dispersion attraction falls off as $1/r^6$:

Dipole-dipole

Two polar molecules orient their $\delta^+$ and $\delta^-$ ends toward each other. This dipole-dipole attraction is stronger than dispersion for molecules of similar size and falls off as $1/r^3$ between fixed dipoles. For freely rotating molecules, thermal averaging gives an effective $1/r^6$ dependence — still dispersion-like in range, but with a prefactor proportional to $\mu_1^2\mu_2^2$.

Hydrogen bonding

A special, extra-strong case of dipole-dipole. When hydrogen is bonded to a very electronegative atom (F, O, or N), the H bears a large $\delta^+$ and is small enough to get very close to the lone pairs of a neighboring F, O, or N. The result is a hydrogen bond: about $5$–$40$ kJ/mol, with a strong directionality preference (the H$\cdots$A angle likes to be near 180°).

Hydrogen bonds are why:

• Water is liquid at room temperature. Each H$_2$O molecule can donate two and accept two H-bonds, building a three-dimensional network. The extra energy needed to break it out of the network is why water boils so high.
• Ice floats. The open hexagonal H-bonded lattice is less dense than liquid water, where molecules can tumble closer together.
• DNA has base pairing. Adenine-thymine is two H-bonds, guanine-cytosine is three. Those specific geometries are why the genetic code copies cleanly.
• Proteins fold. $\alpha$-helices and $\beta$-sheets are stabilized by the exact same H-bond between a backbone carbonyl oxygen and an amide N-H.

Ion-dipole

When an ion sits next to a polar molecule, the ion attracts the appropriately oriented end of the dipole. Ion-dipole interactions are what dissolve salt in water: each Na$^+$ gets surrounded by water molecules pointed $\delta^-$ (oxygen) end toward it; each Cl$^-$ gets surrounded by water molecules pointed $\delta^+$ (hydrogen) end toward it. That "hydration shell" is strong enough to overcome the lattice energy of NaCl, and the entropy gain of dispersing the ions seals the deal.

10. Bonding in code

Three small calculations. The first computes formal charges from a Lewis-style inventory. The second computes a total dipole moment by adding bond-dipole vectors. The third compares boiling points of a family of compounds to their dispersion-force proxy (molecular mass or polarizability).

import numpy as np

# ---------- 1. Formal charge from (V, L, B) ----------
def formal_charge(valence, lone, bonding):
    # FC = V - L - B/2
    return valence - lone - bonding // 2

# CO2: central C with two double bonds, no lone pairs
#   V=4, L=0, B=8 (four pairs, each double bond = 4 shared e-)
print(f"FC(C in CO2)  = {formal_charge(4, 0, 8)}")   # 0
# Each O: V=6, L=4 (two lone pairs), B=4
print(f"FC(O in CO2)  = {formal_charge(6, 4, 4)}")   # 0

# NH4+: central N with four single bonds, no lone pairs
#   V=5, L=0, B=8
print(f"FC(N in NH4+) = {formal_charge(5, 0, 8)}")   # +1

# ---------- 2. Total molecular dipole ----------
def total_dipole(bond_vectors_A, mu_per_bond_D):
    # bond_vectors_A: (n, 3) unit vectors from central atom toward each bonded atom
    # mu_per_bond_D : scalar dipole magnitude of each bond in debyes
    v = np.asarray(bond_vectors_A, dtype=float)
    v = v / np.linalg.norm(v, axis=1, keepdims=True)
    total = (mu_per_bond_D * v).sum(axis=0)
    return np.linalg.norm(total)

# CO2: two C=O bonds pointing exactly opposite; should cancel.
co2 = [[1,0,0], [-1,0,0]]
print(f"CO2 dipole  = {total_dipole(co2, 2.3):.2f} D")   # 0.00

# H2O: two O-H bonds at 104.5 degrees
a = np.deg2rad(104.5 / 2)
oh = [[np.sin(a),  np.cos(a), 0],
      [-np.sin(a), np.cos(a), 0]]
print(f"H2O dipole  = {total_dipole(oh, 1.5):.2f} D")   # ~1.83

# ---------- 3. Boiling point vs. dispersion proxy ----------
# Noble gases: higher molar mass -> more polarizability -> higher bp
gases   = ["He", "Ne", "Ar", "Kr", "Xe"]
mmass   = [4.0, 20.2, 39.9, 83.8, 131.3]      # g/mol
bp_K    = [4.2, 27.1, 87.3, 119.8, 165.0]     # boiling point, K
# A crude log fit: bp ~ a * log(M) + b
coeffs = np.polyfit(np.log(mmass), bp_K, 1)
print(f"fit slope = {coeffs[0]:.1f} K per log(M)")

import math

# Formal charge without NumPy.
def formal_charge(V, L, B):
    return V - L - B // 2

# Ozone center O in O=O-O Lewis form: V=6, L=2, B=6 -> FC = +1
print(f"center O in O3 = {formal_charge(6, 2, 6):+d}")
# Terminal single-bonded O: V=6, L=6, B=2 -> FC = -1
print(f"term   O in O3 = {formal_charge(6, 6, 2):+d}")

# Bond angle from dot product of unit vectors
def angle_deg(u, v):
    d = sum(a * b for a, b in zip(u, v))
    return math.degrees(math.acos(max(-1.0, min(1.0, d))))

# Tetrahedral H-C-H angle from methane coordinates:
# pick two of the four corners of a cube at (1,1,1),(1,-1,-1),(-1,1,-1),(-1,-1,1)
def norm(v):
    m = math.sqrt(sum(x * x for x in v))
    return [x / m for x in v]

a = norm([ 1,  1,  1])
b = norm([ 1, -1, -1])
print(f"tetrahedral angle = {angle_deg(a, b):.2f} deg")  # 109.47

# Convert a dipole moment in debye to partial charge in units of e
def partial_charge(mu_D, bond_length_pm):
    mu_SI = mu_D * 3.336e-30               # C*m
    r_m   = bond_length_pm * 1e-12
    q_C   = mu_SI / r_m
    return q_C / 1.602e-19              # in units of e

print(f"HCl partial q = {partial_charge(1.08, 127.5):.3f} e")  # ~0.177
print(f"HF  partial q = {partial_charge(1.83, 91.7):.3f} e")   # ~0.415

Three observations:

• Formal charges are a dirt-cheap check on a Lewis structure. Any atom with FC > $|1|$ deserves a second look.
• The dipole-vector sum is exactly the way you'd compute a net force, which is not an accident — a dipole in an external field is a force problem.
• Boiling point correlates with polarizability, which in noble gases is a clean function of electron count. For more complex molecules, shape matters too, which is why $n$-pentane boils higher than neopentane even though they have the same formula.

11. Cheat sheet

See also