Bonding and Interatomic Potentials

This article covers: why atoms form bonds, types of solids, interatomic potentials (Lennard-Jones, Morse, Coulomb), harmonic approximation and force constants, thermal expansion from anharmonicity, and cohesive energy of crystals.

Why Do Atoms Form Bonds?

Chemical bonding is the fundamental mechanism that holds atoms together to form molecules and solids. Bohr’s model describes electrons in fixed orbits but does not account for bonding. To understand bonding, we need quantum mechanics.

Quantum Mechanics Description

Atoms are described by quantum mechanics. They have three-dimensional “wave functions” whose density can tell us the probability of where to find electrons.

Time-independent Schrodinger Equation (TISE):

For a hydrogen atom with Coulomb potential:

Solving the Schrodinger equation for a system of many atoms determines the allowed atomic configurations in a solid. The equilibrium structure is set up by the balance of attractive and repulsive forces.

Classifying Types of Solids

Different solutions to the Schrodinger equation give rise to distinct types of bonding.

Molecular Solids

Weak van der Waals forces hold molecules together, but the molecules themselves have strong covalent structure. Atoms remain essentially unchanged from the gas phase. Low melting/boiling points.

Covalent Solids

Electrons shared between atoms in new orbitals. Very strong directional bonds. Very high melting points.

Ionic Solids

Electron transfer creates oppositely charged ions held by electrostatic attraction. Crystalline, brittle, poor conductors.

Metallic Solids

Delocalised electrons form a “sea” around positive ion cores. Malleable, ductile, good conductors.

Molecular Solids in Detail

The atoms or molecules involved in molecular bonding remain essentially unaltered from their state in the gas phase. They are bound together by weak attractive interactions known as van der Waals forces.

Van der Waals Forces

Van der Waals interactions bind molecular solids. Types include: London dispersion forces, dipole-dipole interactions, and hydrogen bonding. They all relate to electrostatic forces by electric dipole interactions.

London Dispersion Force

The origin of the attractive force between neutral particles is charge fluctuation. A neutral atom has fluctuating electron distribution on a timescale of s, behaving as a tiny electric dipole. This dipole can induce an oppositely directed dipole on a neighbour, creating a net attractive force.

Deriving the Dependence

Dipole electric fields decay one more power of than point charges. In the far-field approximation:

Consider two atoms: Atom 1 has dipole moment , generating an electric field at distance :

Atom 2 feels this field and acquires an induced dipole , where is the molecular polarisability. Thus is also proportional to .

The potential energy of a dipole in an electric field is:

Interaction energy:

Repulsive Interaction

At very short distances, there is a strong repulsion. Two common forms:

The exponential form is physically justified by wave function decay; the form is mathematically easier and commonly used.

Interatomic Potential 1: Lennard-Jones

The Lennard-Jones potential energy function is defined as:

Alternate (more common) form:

where is the distance where (approximately atomic size), and is the depth of the potential well (potential magnitude when ).

Pair Potentials

The LJ potential is an example of a pair potential (pairwise interaction). The total potential energy is a sum over all atom pairs:

The force between each pair of atoms is independent of the presence of other atoms. This is approximate but very useful.

Covalent Solids in Detail

Covalently-bonded solids are very different from molecular solids. Electrons no longer occupy isolated atomic orbitals – instead they occupy new orbitals distributed around at least one other atom, creating much stronger bonds. Examples: silicon, germanium, diamond.

Quantum Mechanics and Covalent Bonding

Two atomic orbitals can add (bonding, ) or subtract (anti-bonding, ). When they add, there is more electron density between the two nuclei, creating an attractive interaction – the origin of covalent bonding.

Interatomic Potential 2: Morse Potential

The Morse potential is a good approximation for diatomic covalent bonds and provides a good fit to experimental data. Its exponential form is similar to how atomic wave functions decay with distance.

Definition of Morse Potential:

Phase Diagrams for Solids

Many materials can exist in more than one solid phase. This information can be shown on a phase diagram.

Polar Covalent Bonds and Electronegativity

The two idealised extremes of chemical bonding:

(1) Covalent bonding: electrons shared equally between two atoms.

(2) Ionic bonding: electrons transferred completely from one atom to another.

Most compounds have polar covalent bonds – electrons shared unevenly. Electronegativity measures the ability of an atom to attract electrons. A bond is nonpolar if atoms have equal electronegativities; otherwise it is polarised towards the more electronegative atom. The trend: electronegativity increases going up a group and right across a period.

Ionic Solids in Detail

Ionic bonding is the extreme form of polar covalent bonding. One or more electrons are transferred completely, creating ions of opposite charge held together by electrostatic forces.

Properties: Typically crystalline, high melting and boiling points, brittle, poor heat and electricity conductivity.

Interatomic Potential 3: Coulomb Potential

For ionic solids, we replace the attractive term (van der Waals) with the Coulomb potential.

The Coulomb Potential:

The Coulomb force is far longer range than van der Waals, falling off only as instead of .

Note: attractive for opposite charges (), repulsive for like charges ().

Metallic Solids in Detail

In metallic bonding, electrons are delocalised across the entire material, forming a “sea of electrons” that surround positively charged metallic ions.

Properties: High melting/boiling points, malleable and ductile, good electrical and thermal conductivity.

Fermi Energy and Free Electron Model

Coulomb potentials of ions combine to approximate a square-well potential. Valence electrons occupy quantum particle-in-a-box states, delocalised throughout the crystal.

Fermi Energy:

where is the number density of atoms.

Total energy of electrons in volume :

As volume decreases, energy increases – this “degeneracy pressure” contributes to the bulk modulus of metals (and prevents white dwarf stars from collapsing).

Small Oscillations and Force Constants

Atoms oscillate about their equilibrium positions at Hz. We approximate the potential near equilibrium by a harmonic potential (Taylor expansion to 2nd order):

Harmonic Approximation:

The force constant is given by the second derivative at equilibrium:

Force Constant for L-J Potential

We find:

Differentiating again:

At (where ):

Vibrational Frequency of H2

Given and :

Reduced mass:

Experimental value: Hz. The LJ potential overestimates because H2 bonding is dominated by covalent (not molecular) interactions.

Thermal Expansion

Most solids expand when heated.

Coefficient of Volume Expansion:

Coefficient of Linear Expansion:

For isotropic materials: .

Why Does Thermal Expansion Occur?

The harmonic approximation does NOT explain thermal expansion – in a symmetric potential, the average position stays at . We must include the anharmonicity (asymmetry) of the potential.

Cohesive Energy of a Crystal

The cohesive energy is the energy required to completely separate all constituent atoms/molecules from each other to infinite distance, starting from their equilibrium positions.

Cohesive Energy Per Atom:

The factor accounts for double-counting of pair interactions.