Crystal Structures & Lattice Energies

For the purposes of this tutorial, we're going to assume you know what a crystal structure is, and that a molecular crystal consists of (relatively) close-packed complete molecules. Many (most) material properties depend on the crystal structure, but naturally it can be nice to analyse ex post facto why a particular structure might have been energetically favourable.

info

Some would argue that there's no such thing as a lattice energy and we should use the term binding or cohesive energy. They're probably right, but people use the term and that's how language works - so long as we understand what we're talking about (in this case the potential energy difference between an isolated molecule and its crystal environment)

CIF Format: Crystallographic Information Files

Some wise people a long long time ago made the sensible decision to have a standardised format for storing crystallographic data. If only computational chemists could also be so wise... perhaps we need another file format! Anyway, crystal structures are typically stored in CIF (Crystallographic Information File) format, and like it or not (I mostly do) this is the standard™️ for crystallography.

Example: Urea Crystal (CIF)

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Viewer shows the asymmetric unit only

Key information:

• Unit cell dimensions: a=5.582, b=5.582, c=4.686 Å (tetragonal cell)
• Space group: $\text{P}\bar{4}2_1\text{m}$ (International tables number 113) with 8 symmetry operations listed
• Atomic positions: Given in fractional coordinates (0-1 range relative to unit cell)
• Z = 2: Two formula units (urea molecules) per unit cell

tip

Unlike e.g. an xyz file for a molecule, it's a bit harder to reason about what might happen in your head when there are fractional coordinates, symmetry operations involved etc. So, I'd advise not to edit this one if you want chemically meaningful answers - though you can obviously paste in another CIF!

Where to find CIF Files

• Cambridge Structural Database (CSD) - 1M+ experimental organic/organometallic structures
• Crystallography Open Database (COD) - Free, open-access database of experimental structures
• Materials Project - Computed structures for lots of inorganic materials (not so useful for molecular crystals)
• PubChem - Some entries have 3D crystal structures, linking to the CSD!

If it's not in one of the major databases, either you'll have to crystallise it yourself or perform some Crystal Structure Prediction!

Lattice Energy Calculations

The lattice energy is the energy required to separate a crystal into isolated molecules (basically the sublimation enthalpy, ignoring the vibrational bits). It turns out that for neutral molecules a pretty good approximation of this energy is the sum of all pairwise intermolecular interaction energies in the crystal.

Basic ELAT Calculation

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Parameters:

• --model ce-hf - Energy model (CrystalExplorer-HF using HF/3-21g wavefunctions - for quick calcs like this)
• --radius 8 - Include all molecules within 8 Å of the central molecule

Output:

• Energy results and decomposition printed to console
• XYZ files for each unique dimer pair saved in input_dimers/ directory

Expected runtime: From seconds to hours (varies by system size, energy model, and how many cores we throw at it)

Understanding the Output

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2. Pairwise Interactions Table

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Understanding the CE Energy Model

The CrystalExplorer (CE) energy models compute interaction energies using gas-phase SCF (i.e. DFT or HF) calculations on molecular monomers, then put the dimers together and see how they interact. The total interaction energy is computed from several components:

$E_{\text{tot}} = k_{\text{ele}} \times E_{\text{ele}} + k_{\text{pol}} \times E_{\text{pol}} + k_{\text{disp}} \times E_{\text{disp}} + k_{\text{rep}} \times E_{\text{rep}} + k_{\text{ex}} \times E_{\text{ex}}$

Where:

• $E_{\text{ele}}$ (electrostatic) =Coulomb interactions
• $E_{\text{ex}}$ (exchange) = Hartree-Fock exchange interaction (stabilisation from avoiding pauli repulsion)
• $E_{\text{pol}}$ (polarization) = Induction energy from induced dipoles
• $E_{\text{disp}}$ (dispersion) = London/van der Waals attraction
• $E_{\text{rep}}$ (repulsion) = Pauli exclusion repulsion

Important: The k-factors (scaling factors) are model-specific and not equal to 1. This means $E_{\text{total}} \neq E_{\text{coul}} + E_{\text{ex}} + E_{\text{pol}} + E_{\text{disp}} + E_{\text{rep}}$. The scaling factors correct for basis set incompleteness and method deficiencies, and are fitted to high-level benchmark calculations.

Available CE Models

Model	Method	Basis Set	$k_{\text{ele}}$	$k_{\text{ex}}$	$k_{\text{rep}}$	$k_{\text{pol}}$	$k_{\text{disp}}$	Use XDM?
ce-hf	HF	3-21g	1.019	0.811	0.811	0.651	0.901	No
ce-b3lyp	B3LYP	6-31G**	1.057	0.618	0.618	0.740	0.871	No
ce-1p	any (default B3LYP)	def2-SVP	1.0	0.779	0.779	0.779	1.0	Yes

Model selection:

• ce-hf: Fast, good for quick estimates and tutorial examples
• ce-b3lyp: More accurate for organic molecules, uses a larger basis set
• ce-1p: Modern variant using XDM dispersion correction, better for polar systems, transferable across methods

note

The CE-1p model uses the Exchange-Dipole Moment (XDM) dispersion correction instead of what was basically a D2 dispersion + empirical scaling approach. For this model, $k_{\text{ex}} = k_{\text{rep}} = k_{\text{pol}}$ and both electrostatic and dispersion terms are unscaled ($k = 1.0$).

Visualizing and Analyzing Dimers

The calculation above automatically created XYZ files for each unique dimer in the input_dimers/ directory. You can now visualize and explore these structures:

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tip

Visualize dimers in the viewer above to see the N-H···O hydrogen bond geometry. The short H···O contact and linear arrangement confirm this is the dominant interaction stabilizing the urea crystal.

Physical Interpretation

What Makes a Stable Crystal?

The answer, like most chemical problems, is that stability is relative. What we've computed is an estimate of stability/enthalpy for this particular crystal vs. the gas-phase. But this is far from the whole picture, it really depends on the chemical problem. For a crystal grown from solvent, we'd need to know the free energy of solvation, free energy of the crystal, free energy of the gas, the free energy of ... basically all possible accessible states before we could identify the populations of each at equilibrium.

In general though, it might be enough to have some estimate of solvation vs. the crystal, more on that in the section on Crystal Growth & Surface Energies!