Compressive Sensing Lattice Dynamics (CSLD)

• CSLD is an information-theoretically motivated method that parametrizes the lattice dynamical Hamiltonian using sparse regression and symmetry constraints.
• It employs convex optimization to recover both harmonic and anharmonic interatomic force constants, enabling precise phonon dispersions and thermal transport predictions.
• The technique has been validated on diverse materials, demonstrating efficient performance with significantly fewer DFT calculations compared to traditional approaches.

Compressive Sensing Lattice Dynamics (CSLD) is an information-theoretically motivated, non-perturbative methodology for parametrizing the lattice dynamical Hamiltonian of crystals from first-principles data. CSLD reframes the extraction of interatomic force constants (IFCs)—harmonic and anharmonic—as a sparse linear-inverse problem, using a convex optimization over a rigorously symmetrized tensor basis. By exploiting the sparsity of physically relevant force constants, CSLD recovers both harmonic (second-order) and anharmonic (higher-order) parameters from a small set of density functional theory (DFT) calculated atomic forces, enabling accurate and efficient calculations of phonon dispersions, lifetimes, and thermal transport, even in complex or strongly anharmonic materials.

1. Theoretical Foundation and Formalism

CSLD is rooted in a systematic expansion of the Born–Oppenheimer potential energy surface of a crystal in terms of atomic displacements $u_{a,i}$ from equilibrium: $$U = U_0 + \frac{1}{2} \sum_{ab,ij} \Phi_{ij}(ab)\,u_{a,i}\,u_{b,j} + \frac{1}{3!} \sum_{abc,ijk} \Phi_{ijk}(abc)\,u_{a,i}\,u_{b,j}\,u_{c,k} + \dots$$ where $\Phi^{(2)}$, $\Phi^{(3)}$, etc., are the harmonic and anharmonic force constant tensors (FCTs), respectively. The lattice Hamiltonian becomes

$$H = \frac{1}{2} \sum_{a,i} m_a \dot{u}_{a,i}^2 + U(u)$$

with quadratic terms governing phonon spectra and higher-order terms controlling phonon–phonon interactions.

A compact "cluster" notation is adopted: any $n$-tuple of atomic sites and Cartesian indices ($\alpha$ and $I$) defines a term

$$U = \sum_{\alpha} \frac{1}{\alpha!}\, \Phi_I(\alpha)\, u_I^\alpha$$

where space-group, permutation, and translational symmetries reduce the number of independent parameters from the naïve dimensionality (post-symmetrization, $\sim$hundreds to thousands per structure).

2. Mathematical Structure and Optimization Problem

The core of CSLD is the formulation of the force–displacement relation as a sparse regression: $\mathbf{F} = \mathbf{A}'\,\Phi_S$ where $\mathbf{F}$ stacks all Cartesian force components from DFT for different configurations, $\Phi_S$ collects all distinct, symmetrized FCTs, and $\mathbf{A}'$ encodes monomials of displacements (including symmetry projection operators).

Symmetry and invariance constraints ($\mathbf{B}\,\Phi_S=0$) are incorporated by reducing $\Phi_S$ to a set of independent parameters $\phi$, yielding a linear system $\mathbf{F} = \mathbf{A}\,\phi$.

To exploit sparsity, CSLD solves

$$\min_{\phi} \|\phi\|_1 + \frac{\mu}{2}\| \mathbf{F} - \mathbf{A}\,\phi \|_2^2$$

where $\|\phi\|_1$ promotes sparsity and $\mu$ tunes fidelity-to-sparsity tradeoff. The dual (basis-pursuit–denoising) form,

$$\min \|\phi\|_1 \quad \text{subject to} \quad \| \mathbf{F} - \mathbf{A}\,\phi\|_2 \leq \varepsilon,$$

is equivalent for appropriate $\lambda$, $\mu$ pairing. Optimization is performed by split-Bregman, FISTA, or related convex solvers; optimal $\mu$ is determined by cross-validation (e.g., leave-out tests).

Scaling of FCTs is applied to ensure uniform physical dimensions and appropriate weighting of higher-order terms. For an $n$th-order FCT, scaling by $u_0^{n-1}$ (with $u_0$ representing typical thermal displacement amplitude) harmonizes all elements in units of force.

3. Sampling and Construction of Training Data

High-quality recovery of sparse FCTs demands that the design matrix $\mathbf{A}$ be maximally incoherent. Instead of sampling configurations from low-energy minima alone (which induces high column correlations in $\mathbf{A}$), CSLD prescribes:

• Running short ab initio molecular dynamics (AIMD) at a target $T$
• Extracting $\sim$10–20 snapshots separated by several picoseconds
• Adding small random displacements (0.1–0.2 Å) to every atom for each snapshot

This produces a training set where each supercell delivers %%%%26$n$27%%%% force equations ($N_a=$ atomic count in supercell), enabling fitting with only tens of moderately sized configurations.

4. Computational Workflow

The end-to-end CSLD workflow involves:

1. Supercell construction with designated cluster order/cutoff.
2. Generation of $M$ quasi-random training configurations.
3. DFT force computation for each configuration.
4. Assembly of the global design (sensing) matrix $\mathbf{A}$ and target vector $\mathbf{F}$.
5. Imposition and enforcement of symmetry and invariance constraints (null-space basis reduction).
6. Solution of the $\ell_1$ optimization problem for $\phi$.
7. Reconstruction of the full FCT set and application of symmetry operations to populate all clusters.
8. Validation via test-set force prediction, calculation of phonon dispersions, lifetimes ($\tau_\lambda$) via perturbative methods + Boltzmann transport, or direct molecular dynamics using the fitted expansion.

The entire procedure differs fundamentally from traditional finite-difference or DFPT methods by enabling direct, unified, and sparse recovery of high-order IFCs across potentially large, low-symmetry, multicomponent, or strongly anharmonic systems.

5. Benchmark Studies and Applications

CSLD's capabilities have been established across a range of materials:

• Elemental and Binary Systems: For Si, NaCl, and Al, extraction of 3rd-order FCTs from 2 configurations of $3\times3\times3$ supercells (random 0.03 Å displacements) achieves $\lesssim$5 meV/Å RMSE and rapid decay of $\Phi^{(3)}$ beyond first-neighbor distances.
• Complex and Strongly Anharmonic Materials: In Cu$_{12}$Sb$_4$S$_{13}$ (tetrahedrite, 29 atoms/cell, $>55,000$ FCTs), symmetry reduces unknowns to $3,188$; a three-step fit (radial pair-field basis for strong covalent bonds and displacements of varying magnitudes) enables stable MD at 300 K, outperforming direct Taylor expansion approaches.
• Clathrate Phases: For type-I Si$_{46}$ clathrates (including guest atoms Na$_8$, Ba$_8$), 20 $2\times2\times2$ configurations sufficed (total $2,760$ force equations) versus $\sim$600 individual DFPT displacements by direct methods. Key observations include robust recovery of bond softening, guest–phonon avoided crossings in dispersions, and $\kappa_L$ suppression consistent with experiment.

Table 1 summarizes case paper results.

System	IFC Rank/Range	Force RMSE / Accuracy
Si, Al, NaCl	up to 3rd-neigh.	$\lesssim$5 meV/Å; spectra $<$0.1 THz error
Tetrahedrite	up to 6th order	4% RMS error; anharmonic spectra
Si$_{46}$ clathrates	2nd/3rd order, 2nd neighbor	$\kappa_L$ w/in 10–20% of experiment

In strongly anharmonic compounds or large-unit-cell systems, CSLD reproduces ab initio forces with high fidelity using an order-of-magnitude fewer DFT calculations than direct displacement or linear-response approaches.

6. Extensions, Computational Considerations, and Limitations

CSLD supports:

• Arbitrary expansion order and cluster cutoffs (by expanding the $\phi$ basis)
• Alloy, defect, and interface modeling by increasing problem dimensionality
• Export of FCTs to phonon/thermal transport codes (e.g., SHENGBTE format)
• Real-space long-range interaction corrections for polar insulators, including analytic dipole–dipole splitting and adjustments to ensure sum-rule and Hermiticity (Zhou et al., 2018)

Computational cost is dominated by DFT force calculations; fitting up to $M\sim10^4$ IFCs is tractable using modern convex solvers and memory resources.

Principal limitations include:

• Necessity of refitting (choosing new $\lambda,\mu$) when training set or system parameters change
• Truncation by expansion order (neglect of 4th and higher order may hinder accuracy at high $T$)
• Sensitivity to quality of DFT forces (e.g., k-point sampling, PAW errors)
• Scaling challenges for $N_{\text{atoms}}>200$ or when seeking very long-range interactions

Nonetheless, the method's automatic sparsity enforcement, unified tensor treatment, robust cross-validation selection of regularization, and systematic workflow yield compact, physically transparent models amenable to both perturbative (Boltzmann transport equation) and non-perturbative (MD/Green–Kubo/HNEMD) analysis of lattice dynamics.

7. Context, Impact, and Outlook

CSLD is a general lattice-dynamical modeling technique that brings information-theory rigor to the parametrization of high-order force-constant potentials. Its ability to systematically and automatically select only those FCTs needed to reproduce microscopic forces obviates the need for manual truncation or prior "guesswork," accommodates chemical complexity, and is well-suited to thermoelectric, strongly anharmonic, and large-unit-cell materials. Contemporary research leverages CSLD for thermal conductivity prediction, phonon scattering and lifetimes, phase transitions, and effective-potential construction for molecular or path-integral dynamics. Advanced implementations exploit its flexibility for studies in temperature-dependent lattice potentials, defect and alloy effects, and machine-learning-enabled force field development.

An open-source implementation is maintained at https://github.com/LLNL/csld. Major methodological developments, benchmarking, and extensions to diverse materials classes are described in (Zhou et al., 2014, Zhou et al., 2018, Zhou et al., 2018), and (Xia et al., 2017).