Chemical Bonding

A chemical bond is a durable pattern of quantum-mechanical stabilization and structure that makes it useful to regard atoms or fragments as one aggregate. The word compresses several kinds of evidence: a bound or metastable state, an equilibrium geometry, a dissociation energy, characteristic forces and spectra, and reproducible changes in electronic density or response.

No universal bond potential is added to the molecular Hamiltonian. In the standard nonrelativistic model, electrons and nuclei have kinetic energy and charged particles interact through Coulomb terms. Bonding emerges after the many-particle state adjusts subject to symmetry, antisymmetry, boundary conditions, and the chosen environment.

This immediately gives two guardrails:

• the existence and strength of binding are total-energy or free-energy questions;
• orbital pictures, Lewis structures, atomic charges, bond orders, bond paths, and energy components are analyses of a state, not interchangeable observables.

The distinction does not make chemical language arbitrary. A good bonding description is constrained by converged calculations, spectroscopy, thermochemistry, scattering, structure, response, and limiting cases. It does mean that every numerical bond claim needs an operational definition.

Required background. Molecular Hamiltonian and potential-energy surfaces define the energy comparison. Molecular orbitals, valence-bond theory, and the worked H₂ bond supply the representations compared here.

What Is a Bond in Quantum Mechanics?

Start from a dissociation channel

For fixed nuclear geometry $\mathbf R$, let an electronic state satisfy

$$\hat H_e(\mathbf R) |\Psi_k(\mathbf R)\rangle = E_k(\mathbf R) |\Psi_k(\mathbf R)\rangle.$$

If the electronic convention excludes internuclear repulsion, the corresponding Born–Oppenheimer surface is

$$U_k(\mathbf R) = E_k(\mathbf R) +V_{NN}(\mathbf R).$$

Choose a physically allowed fragment channel $A+B$ with the same conserved total charge and all required spin and symmetry labels. Its asymptotic threshold is

$$E_{\mathrm{th}}^{A+B} = \lim_{R\to\infty}U_k(R),$$

provided that this adiabatic surface correlates with the stated fragments. For a diatomic minimum at $R_e$, the electronic well depth is

$$D_e = E_{\mathrm{th}}^{A+B} -U_k(R_e).$$

A positive $D_e$ says that the bottom of the chosen electronic well lies below that threshold. It does not yet prove that nuclear motion has a bound level. The nuclear Hamiltonian on the surface,

$$\hat H_{\mathrm{nuc}} = \hat T_{\mathrm{nuc}} +U_k(\mathbf R),$$

must support a normalizable internal eigenstate. An energy below the lowest allowed dissociation threshold guarantees stability against dissociation. Levels above a coupled threshold generally decay; exceptional bound states embedded in a continuum require an additional decoupling mechanism. If its lowest rovibrational energy is $E_{v=0}$, then

$$D_0 = E_{\mathrm{th}}^{A+B} -E_{v=0}, \qquad D_0<D_e$$

for an ordinary bound well with positive zero-point energy.

A local minimum above a lower threshold can instead describe a metastable resonance. Its physical characterization requires a lifetime or width, not merely a geometry. At finite temperature, in solution, or under pressure, the relevant stability criterion may be a Helmholtz or Gibbs free-energy difference rather than an isolated-molecule energy.

Forces and equilibrium

For an exact differentiable electronic eigenstate, the Hellmann–Feynman relation gives

$$\frac{\partial E_k}{\partial R_A} = \left\langle\Psi_k\left| \frac{\partial\hat H_e}{\partial R_A} \right|\Psi_k\right\rangle.$$

The force on nucleus $A$ is

$$\mathbf F_A = -\nabla_{\mathbf R_A}U_k.$$

At an unconstrained equilibrium geometry the internal forces vanish, and the internal-coordinate Hessian is positive definite for a nondegenerate minimum. A zero eigenvalue requires higher-order analysis; a strict minimum can be quartically flat. Vanishing net force does not mean that attractive and repulsive contributions have disappeared. It means that their total derivative balances at that geometry.

A geometry-dependent finite basis contributes Pulay terms even at a converged variational stationary point. Additional parameter-response terms arise when the wavefunction is not fully stationary. A visually plausible density is not a substitute for a converged force calculation.

Evidence and interpretation

Three questions are often conflated:

Question	Evidence and qualification
Does an aggregate exist?	Use a bound level, resonance, or thermodynamically stable phase; state the channel, environment, and lifetime.
How hard is it to separate?	Specify $D_e$, $D_0$, a bond-dissociation enthalpy, or a free energy; these can use different products.
Where and what kind is the bond?	Combine densities, spectra, response, orbitals, structures, and indices; no single analysis is universal.

IUPAC definitions deliberately include a pragmatic stability criterion: an aggregate must be sufficiently stable to be treated as an independent species in the chemical context. Quantum mechanics sharpens that idea through thresholds, lifetimes, and free energies, but it does not create one context-free numerical boundary between “bonded” and “not bonded.”

There is no universal bond operator

An observable in quantum mechanics is represented by an operator or measurement protocol. There is no state-independent Hermitian operator $\hat B_{AB}$ whose eigenvalues count chemical bonds between every possible pair of atoms in every molecule, crystal, excited state, solvent, and reaction.

Bond graphs are therefore a physically disciplined coarse-graining. They are successful because many electronic and nuclear states display robust locality, energy gaps, recurring valences, and transferable response patterns. The graph can remain stable under small changes in method even though its detailed orbital or density partition is not unique.

Covalent Bonding

H₂⁺ shows how a one-electron state can bind two nuclei through coherent mixing and orbital relaxation. H₂ adds electron repulsion and spin symmetry: the spatially symmetric singlet permits an energy-lowering state, while its restricted single-determinant approximation fails at dissociation.

These examples do not make exchange a new force or identify a density interference term with a bond energy. Kinetic, Coulomb, and nuclear-repulsion contributions must be compared together. Localized VB and delocalized MO expansions represent the same state when they span the same complete many-electron space; their truncations can have very different errors.

Covalency also includes three-center bonds, electron-deficient clusters, and aromatic delocalization. Such systems need more than a single two-center electron pair.

Polar covalency

Homonuclear and purely ionic limits are endpoints, not exhaustive categories. Unequal orbital energies and environment-dependent polarization usually mix neutral and charge-transfer structures. A polar covalent bond may have:

• appreciable density sharing and orbital mixing;
• a nonzero dipole moment;
• unequal atomic populations in a declared partition;
• both covalent and ionic amplitudes in a VB expansion.

The percentage labels depend on the basis and population scheme. Observable dipole moments and dissociation products constrain the interpretation more directly.

Ionic Bonding

Charge-transfer energetics

Consider a donor $A$ and acceptor $B$. At large separation, the energy cost of the ionic channel $A^+B^-$ relative to neutral fragments is approximately

$$\Delta_\infty = I_A-\mathcal A_B,$$

where $I_A$ is the ionization energy of $A$ and $\mathcal A_B$ is the electron affinity of $B$. At finite separation, define the point-charge contribution

$$\Delta_C(R) = -\frac{e^2}{4\pi\epsilon_0R}.$$

Let

$$\Delta_{IN}(R) \equiv E_I(R)-E_N(R).$$

Then

$$\Delta_{IN}(R) \simeq{} \Delta_\infty +\Delta_C(R) +\Delta E_{\mathrm{pol}}(R) +\Delta E_{\mathrm{overlap}}(R) +\cdots .$$

The attractive Coulomb term can make the ionic channel competitive, while polarization, finite charge distributions, orbital overlap, short-range antisymmetry, and correlation modify the point-charge picture.

Neutral–ionic mixing

In an orthonormal diabatic basis $\{|N\rangle,|I\rangle\}$, write

$$\mathbf H_d(R) = \begin{pmatrix} E_N(R)&V(R)\\ V(R)&E_I(R) \end{pmatrix}.$$

Define

$$\bar E=\frac{E_N+E_I}{2}, \qquad \Delta=E_I-E_N.$$

The adiabatic energies are

$$E_\pm = \bar E \pm \frac12\sqrt{\Delta^2+4V^2}.$$

For real $V$ and a normalized lower eigenstate, the ionic weight in this specified orthonormal basis is

$$w_I = \frac12 \left( 1-\frac{\Delta}{\sqrt{\Delta^2+4V^2}} \right).$$

When $\Delta\gg|V|$, the state is mostly neutral; when $-\Delta\gg|V|$, it is mostly ionic; at a diabatic crossing it is an equal mixture and the adiabatic gap is $2|V|$. The numerical value of $w_I$ is not unique if the diabatic structures are changed or are nonorthogonal, but the avoided crossing, adiabatic energies, dipole, and response are physical.

Ions in solids and environments

An ionic crystal’s energy includes lattice electrostatics, overlap, polarization, electronic correlation, and lattice motion, as well as the cost of creating ions from neutral atoms. Gas-phase pair energies alone cannot decide its stability.

Formal oxidation states are exact bookkeeping rules within a chosen chemical convention. Partial atomic charges are partition-dependent. Neither should be confused with measuring a point charge located on an atom.

Solvation can reverse gas-phase preferences because ionic and neutral channels receive different polarization free energies. A statement such as “this bond is ionic” is incomplete without the state, geometry, phase, and diagnostic.

Metallic Bonding Preview

Metallic cohesion is an extended many-electron problem. If one orbital per atom is coupled along a periodic one-dimensional chain, the nearest-neighbor tight-binding model gives

$$\varepsilon(k) = \varepsilon_0-2t\cos(ka).$$

As the number of atoms grows, discrete bonding and antibonding levels become a band. A partially filled band has low-energy particle–hole excitations near its Fermi surface and can carry current when scattering and interactions are treated appropriately.

The Tight-Binding Model derives this reduction. Several cautions matter for bonding:

• a delocalized Bloch orbital is not a classical electron trajectory through the lattice;
• a completely filled band can be strongly cohesive and electrically insulating;
• conductivity is not a definition of bond strength;
• localized Wannier functions can represent the same occupied band subspace;
• the total cohesive energy also contains ion–ion terms, electron interactions, screening, correlation, and double-counting corrections appropriate to the method.

The phrase “electron sea” is a useful limiting image for simple $s$-band metals, not a universal microscopic description. Transition metals, narrow bands, magnetism, Mott physics, and directional orbital hybridization require more structure.

Hydrogen Bonding and Weak Interactions

Hydrogen bonds require evidence

A hydrogen bond is an attractive interaction in which hydrogen bound to an atom or group $X$ interacts with an atom or group $Y$, with experimental or theoretical evidence that a specific association has formed. Modern usage is broader than the mnemonic “H attached to N, O, or F,” although those donors and acceptors are common.

Angular preferences, vibrational shifts and intensities, NMR signals, isotope effects, and a favorable interaction or association energy can support the assignment. These measurements constrain different aspects of the interaction and need not all be present.

No universal distance cutoff proves a hydrogen bond. A short contact can be imposed by sterics or a crystal lattice, while a real weak interaction can be longer than a tabulated sum of radii.

An interaction is a balance, not one mechanism

For two fragments, symmetry-adapted perturbation theory organizes the interaction energy schematically as

$$E_{\mathrm{int}} ={}E_{\mathrm{elst}}+E_{\mathrm{exch}} +E_{\mathrm{ind}}+E_{\mathrm{disp}} +E_{\mathrm{exch\text{-}ind}} +E_{\mathrm{exch\text{-}disp}} +\cdots .$$

The terms have clear definitions only within the stated SAPT order, monomer method, basis, and partition. Other energy-decomposition analyses divide the same total differently.

Mechanism	Origin and qualification
electrostatics	Permanent charge distributions and multipoles interact; penetration matters when densities overlap.
induction	One fragment polarizes another; the assigned term depends on the fragment response space.
dispersion	Correlated quantum fluctuations attract; ordinary single-determinant Hartree–Fock misses the long-range term.
exchange overlap	Antisymmetrization changes overlapping fragment states; this is not a new pair potential and usually decays exponentially.

For nondegenerate isotropic ground-state fragments at zero temperature, in the nonretarded long-range limit,

$$E_{\mathrm{disp}}(R) \sim -\frac{C_6}{R^6}.$$

In Hartree atomic units, the Casimir–Polder representation is

$$C_6 = \frac3\pi \int_0^\infty \alpha_A(i\omega) \alpha_B(i\omega) \,d\omega,$$

where $\alpha_A$ and $\alpha_B$ are dynamic dipole polarizabilities evaluated at imaginary frequency. At asymptotically larger separations, electromagnetic retardation changes the power law; ordinary molecular separations are usually in the nonretarded regime.

Hydrogen bonds, halogen bonds, $\pi$ stacking, ion–molecule complexes, and dispersion-bound dimers all combine several of these mechanisms. Naming the interaction identifies a recurring geometry and chemical pattern; it does not isolate one term of the Hamiltonian.

Cooperativity and many-body effects

Pairwise additivity is an approximation. A third molecule can polarize both members of a pair, alter charge-transfer amplitudes, change exchange overlap, and contribute three-body dispersion. Hydrogen-bond networks can therefore be cooperative or anticooperative depending on geometry and electronic response.

In condensed phases, entropy and solvent reorganization can be as important as the isolated interaction energy. A deep gas-phase minimum need not imply a favorable solution-phase association free energy.

Exchange, Correlation, and Electrostatics

One density is not enough for every energy

Let the spin-summed pair density $P(\mathbf r,\mathbf r')$ be normalized by

$$\int d^3r\,d^3r'\, P(\mathbf r,\mathbf r') = N(N-1).$$

The exact electron–electron interaction energy is

$$E_{ee} = \frac{e^2}{8\pi\epsilon_0} \int \frac{ P(\mathbf r,\mathbf r') }{ |\mathbf r-\mathbf r'| } \,d^3r\,d^3r'.$$

The one-electron density $\rho(\mathbf r)$ determines many observables, but the pair density records conditional electron avoidance. A single determinant includes the exchange hole required by antisymmetry. Correlated states modify opposite-spin and same-spin pair structure beyond that determinant.

Exchange and Correlation develops the direct, exchange, pair-hole, and density-functional distinctions. Reduced Density Matrices provides the general subsystem language.

What “Pauli repulsion” means

When closed-shell fragments overlap, the product of their states must be antisymmetrized. Orbitals need not themselves be orthogonal, but linear dependence is excluded from a nonzero determinant; orthogonalization is a convenient representation of the same occupied space. The resulting energy increase is often called Pauli, exchange, or steric repulsion. These phrases summarize changes in kinetic and Coulomb expectation values caused by the restricted state space.

There is no additional repulsive term in the exact Coulomb Hamiltonian labeled “Pauli.” An exponential short-range potential can be a useful effective fit, but it is not a new fundamental force.

Correlation can strengthen or weaken binding

For a fixed Hamiltonian and complete one-electron basis,

$$E_{\mathrm{corr}} = E_{\mathrm{exact}}-E_{\mathrm{HF}} \leq0.$$

The correlation contribution to a binding energy is instead a difference of differences. Define

$$E_{\mathrm{corr}}(A+B) = E_{\mathrm{corr}}(A) +E_{\mathrm{corr}}(B).$$

Then

$$\Delta D_e^{\mathrm{corr}} = E_{\mathrm{corr}}(A+B) -E_{\mathrm{corr}}(AB).$$

It can be positive or negative. Correlation is essential for long-range dispersion and for symmetry-preserving dissociation of stretched H₂, but it does not follow that correlation strengthens every bond. Fragment and molecular correlation energies compete.

Hartree–Fock Approximation derives the determinant mean field and explains precisely what it includes and omits.

Total-energy partitions

For the nonrelativistic Coulomb Hamiltonian, one can write the change on assembly as

$$\Delta E = \Delta T +\Delta V_{eN} +\Delta V_{ee} +\Delta V_{NN}.$$

These expectation values are defined once the Hamiltonian, state, geometry, and fragment comparison are fixed. Their changes can be individually large and cancel strongly. Further labels such as orbital interaction, charge transfer, polarization, steric energy, covalency, and preparation energy depend on a chosen path and partition.

Energy decomposition is valuable when it answers a declared comparative question and remains stable under reasonable methodological changes. Its components should not be reported as directly measurable fractions of a bond.

Bond Order and Electron Density

Strength, stiffness, and multiplicity differ

Several quantities are informally called bond strength:

$$D_e = E_{\mathrm{fragments}} -E_{\mathrm{minimum}},$$ $$k_e = \left. \frac{d^2U}{dR^2} \right|_{R=R_e},$$

and a thermochemical bond-dissociation enthalpy for a specified homolytic or heterolytic reaction. A deep well, a large harmonic force constant, a short equilibrium distance, and a large electronic bond index often correlate within a chemical family, but none determines all the others.

Zero-point energy distinguishes $D_0$ from $D_e$. Fragment relaxation distinguishes a vertical separation from an adiabatic dissociation. Spin–orbit levels, charge states, temperature, and phase must be included when they matter.

Orbital bond order

For a chosen bonding–antibonding orbital pair, the familiar index is

$$b_{\mathrm{MO}} = \frac{ N_{\mathrm{bonding}} -N_{\mathrm{antibonding}} }{2}.$$

This is effective bookkeeping in small orbital diagrams. It is not a universal observable. The restricted $g^2$ description of H₂ has $b_{\mathrm{MO}}=1$ at every separation even though it dissociates incorrectly.

For an orthonormal atomic-orbital partition, one Wiberg-type closed-shell index has the form

$$W_{AB} = \sum_{\mu\in A} \sum_{\nu\in B} |P_{\mu\nu}|^2,$$

where $P$ is a spin-summed one-particle density matrix in the stated normalization. Mayer-type indices adapt the construction to a nonorthogonal basis through the overlap matrix $S$. Numerical factors and open-shell generalizations vary by convention.

Such indices can be excellent comparative descriptors along a reaction or within a method-consistent series. They depend on the basis, orbital partition, density matrix, and population prescription.

Density differences

A common visualization is the frozen-fragment density difference

$$\Delta\rho(\mathbf r) = \rho_{AB}(\mathbf r) -\rho_A^{\mathrm{frag}}(\mathbf r) -\rho_B^{\mathrm{frag}}(\mathbf r).$$

If the fragment densities contain the same total number of electrons as the aggregate,

$$\int\Delta\rho(\mathbf r)\,d^3r=0.$$

Positive and negative regions show redistribution, not creation and destruction of charge. The picture depends on fragment geometries, electronic states, alignment, and whether the fragment densities are frozen or relaxed.

Density accumulation between nuclei can support a covalent interpretation, but it is neither necessary nor sufficient for every bond. Polar, multicenter, charge-shift, closed-shell, and transition-metal interactions can have different signatures.

Density topology and localization

In the quantum theory of atoms in molecules, atomic basins are bounded by zero-flux surfaces of $\nabla\rho$, and a bond path passes through a rank-three, signature-minus-one critical point when that topology exists. This is a precise statement about a chosen electron density.

A bond critical point is not automatically:

• a measure of bond energy;
• proof of a conventional two-center bond;
• a unique chemical graph for every purpose;
• interchangeable with an orbital bond order.

Electron-localization functions, natural bond orbitals, localized MOs, delocalization indices, source functions, and pair-density analyses answer other questions. Agreement among independent diagnostics is stronger evidence than a single colorful isosurface.

Match the diagnostic to the question

Diagnostic	Use and limitation
dissociation or association energy	Tests stability against a specified channel, not local multiplicity in a polyatomic molecule.
force constant and spectrum	Probes local curvature and nuclear dynamics, not the asymptotic dissociation energy.
density and density difference	Shows charge distribution and redistribution, not a unique energy decomposition.
orbital or density-matrix index	Tracks multiplicity within a convention, not a basis-independent observable bond count.
interaction-energy decomposition	Compares mechanisms in one scheme, not unique measurable percentages.

Finite-Basis Binding Errors

In a finite atom-centered basis, a dimer can use functions on both partners more flexibly than an isolated fragment can use its own functions. This basis-set superposition error can exaggerate binding. A counterpoise calculation evaluates fragment energies with both centers’ basis functions present, including unoccupied ghost centers. Compare that estimate with systematic basis convergence; neither removes correlation or geometry errors.

Exercises

1. Electronic well depth versus nuclear binding

A diatomic Born–Oppenheimer surface has its minimum $0.180\,\mathrm{eV}$ below the lowest dissociation threshold. A harmonic calculation estimates the vibrational zero-point energy as $0.125\,\mathrm{eV}$ above the minimum. Find $D_e$ and the harmonic estimate of $D_0$. Would a zero-point estimate of $0.195\,\mathrm{eV}$ support a bound ground vibrational level?

Solution

The electronic well depth is

$$D_e=0.180\,\mathrm{eV}.$$

The dissociation energy from the lowest vibrational level is

$$\begin{aligned} D_0 &=D_e-E_{\mathrm{ZPE}}\\ &=0.180-0.125\\ &=0.055\,\mathrm{eV}. \end{aligned}$$

Within the harmonic estimate, the lowest level lies below threshold and is bound. If $E_{\mathrm{ZPE}}=0.195\,\mathrm{eV}$, the estimate places that level $0.015\,\mathrm{eV}$ above threshold, so it does not support a bound level. The full anharmonic nuclear Schrödinger equation is needed for a definitive conclusion near threshold.

2. Neutral–ionic avoided crossing

Take the neutral diabatic energy as zero, the ionic diabatic energy as $\Delta=4.0\,\mathrm{eV}$, and the coupling as $V=1.0\,\mathrm{eV}$. Find the lower adiabatic energy relative to the neutral state and its ionic weight. What happens when $\Delta=0$?

Solution

With $E_N=0$ and $E_I=\Delta$,

$$E_- = \frac{\Delta}{2} -\frac12\sqrt{\Delta^2+4V^2}.$$

Therefore

$$\begin{aligned} E_- &=2-\frac12\sqrt{20}\\ &\simeq-0.236\,\mathrm{eV}. \end{aligned}$$

The ionic weight is

$$\begin{aligned} w_I &=\frac12 \left(1-\frac4{\sqrt{20}}\right)\\ &\simeq0.0528. \end{aligned}$$

Thus a state that is mostly neutral is stabilized by ionic mixing. At $\Delta=0$, both adiabatic states are equal-magnitude neutral–ionic mixtures and their gap is $2|V|=2.0\,\mathrm{eV}$.

3. Derive the single-frequency dispersion coefficient

Model each imaginary-frequency polarizability by

$$\alpha_A(i\omega)=\frac{\alpha_A(0)\omega_A^2}{\omega_A^2+\omega^2},$$

and similarly for $B$, with positive characteristic frequencies. Evaluate $C_6$ and its equal-frequency limit.

Solution

Partial fractions and $\int_0^\infty d\omega/(\omega^2+a^2)=\pi/(2a)$ give

$$\int_0^\infty \frac{d\omega}{(\omega^2+\omega_A^2)(\omega^2+\omega_B^2)} =\frac{\pi}{2\omega_A\omega_B(\omega_A+\omega_B)}.$$

Consequently,

$$C_6=\frac32\alpha_A(0)\alpha_B(0) \frac{\omega_A\omega_B}{\omega_A+\omega_B}.$$

For $\omega_A=\omega_B=\omega_0$ this becomes $3\alpha_A(0)\alpha_B(0)\omega_0/4$, including for unlike static polarizabilities. The finite equal-frequency limit follows continuously, despite the apparent singular denominators in the partial-fraction decomposition.

4. Does correlation strengthen the bond?

Suppose the correlation energy of a molecule is $-0.40\,\mathrm{eV}$ and the sum of the correlation energies of its separated fragments is $-0.30\,\mathrm{eV}$. What is the correlation contribution to $D_e$? Repeat if the fragment sum is $-0.50\,\mathrm{eV}$.

Solution

Define the separated-fragment correlation energy by

$$E_{\mathrm{corr}}(A+B) = E_{\mathrm{corr}}(A) +E_{\mathrm{corr}}(B).$$

The correlation contribution is

$$\Delta D_e^{\mathrm{corr}} = E_{\mathrm{corr}}(A+B) -E_{\mathrm{corr}}(AB).$$

For the first case,

$$\begin{aligned} \Delta D_e^{\mathrm{corr}} &=(-0.30)-(-0.40)\\ &=+0.10\,\mathrm{eV}. \end{aligned}$$

Correlation strengthens the binding by $0.10\,\mathrm{eV}$. For the second case,

$$\begin{aligned} \Delta D_e^{\mathrm{corr}} &=(-0.50)-(-0.40)\\ &=-0.10\,\mathrm{eV}, \end{aligned}$$

so correlation weakens the binding by the same amount even though every individual correlation energy is negative.

5. Bandwidth and cohesion

For $\varepsilon(k)=\varepsilon_0-2t\cos(ka)$, find the bandwidth. Show that the average of $\varepsilon(k)$ over a completely filled Brillouin zone is $\varepsilon_0$. What does this imply about using bandwidth or conductivity as a direct measure of metallic cohesion?

Solution

The band extrema occur at $\cos(ka)=\pm1$, so

$$W = \varepsilon_{\max}-\varepsilon_{\min} = 4|t|.$$

Over a full Brillouin zone,

$$\frac{a}{2\pi} \int_{-\pi/a}^{\pi/a} \cos(ka)\,dk =0.$$

Hence the average one-electron band energy is $\varepsilon_0$. A partially filled band can conduct, whereas a completely filled isolated band cannot carry a dc current in the ideal band picture. Neither fact alone determines cohesion: the total energy also includes ionic terms, electron interactions, screening, orbital relaxation, and the reference atomic energies.

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