E4: Kernel Polynomial Method (KPM)

Module: sci_form::experimental::kpmFeature flag: experimental-kpm

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Overview

Computes the density of states (DOS), density matrix, and Mulliken populations via Chebyshev polynomial expansion of the Hamiltonian, avoiding diagonalization entirely. Achieves $O(N)$ scaling for sparse Hamiltonians, enabling electronic structure calculations on systems of thousands of atoms.

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Theory

Chebyshev Expansion

Any function of a Hermitian matrix can be expanded in Chebyshev polynomials:

$$f(H)\approx \sum _{k=0}^M{c}_k{T}_k(\tilde{H})$$

where $\tilde{H}=(H-bI)/a$ is rescaled to $[-1,1]$.

Recursion

The Chebyshev polynomials are computed iteratively:

$$T_0=I,T_1=\tilde{H},T_{k+1}=2\tilde{H}{T}_k-T_{k-1}$$

Each step costs $O(\text{nnz})$ (one sparse matrix-vector product).

Jackson Kernel

Gibbs oscillations are suppressed by the Jackson damping kernel:

$$g_k^{(M)}=\frac{(M-k+1)\cos \frac{\pi k}{M+1}+\sin \frac{\pi k}{M+1}\cot \frac{\pi }{M+1}}{M+1}$$

Stochastic Trace

The trace is estimated stochastically:

$$\text{Tr}[A]\approx \frac{1}{n_v}\sum _{i=1}^{n_v}{r}_i^{T}A{r}_i$$

using $n_v$ random probe vectors $r_i$ with entries $\pm 1$.

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API

use sci_form::experimental::kpm::*;

// Spectral bounds
let (e_min, e_max) = estimate_spectral_bounds(&hamiltonian);

// Rescale matrix to [-1, 1]
let h_scaled = rescale_matrix(&hamiltonian, e_min, e_max);

// Chebyshev expansion (exact trace for small systems)
let expansion = ChebyshevExpansion::from_matrix_exact(&h_scaled, 100);

// Jackson kernel damping
let kernel = jackson_kernel(100);

// DOS at a specific energy
let dos = expansion.dos_at_energy(0.0, &kernel);

// Full KPM DOS computation
let config = KpmConfig { order: 100, n_random: 20, ..Default::default() };
let dos_result = compute_kpm_dos(&hamiltonian, &config);
// dos_result.energies, dos_result.dos, dos_result.homo_lumo_gap

// KPM Mulliken charges
let mulliken = compute_kpm_mulliken(&hamiltonian, &overlap, n_electrons, &config);
// mulliken.charges, mulliken.total_electrons

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Configuration

Parameter	Default	Description
order	100	Number of Chebyshev moments
n_random	20	Random vectors for stochastic trace
temperature	300.0	Electronic temperature (K) for Fermi smearing

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Scaling

System Size	Exact Diag.	KPM
$N=100$	$O({10}^6)$	$O({10}^4)$
$N=1000$	$O({10}^9)$	$O({10}^5)$
$N=10000$	Infeasible	$O({10}^6)$

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Tests

cargo test --features experimental-kpm --test regression -- test_kpm

9 integration tests covering: spectral bounds estimation, Chebyshev recursion stability, Jackson kernel positivity, DOS peak detection, band gap identification, and Mulliken charge conservation.