Graph-based block-diagonalization of full configuration interaction Hamiltonian: H$_2$ chains study

TL;DR

This paper introduces a graph-based block-diagonalization method for the full configuration interaction Hamiltonian, enabling efficient and accurate calculation of low-energy eigenstates in molecular systems like hydrogen chains.

Contribution

The paper presents a novel graph-based approach to decompose the Hamiltonian into blocks, improving computational efficiency for exact eigenvalue calculations in molecular quantum chemistry.

Findings

Accurate eigenvalues for H$_2$ chains up to H$_{12}$

Excellent agreement with exact solutions

Efficient decomposition of Hamiltonian into blocks

Abstract

We developed a graph-based block-diagonalization (GBBD) method for the full configuration interaction Hamiltonian of molecular systems to efficiently calculate the exact eigenvalues of low-energy states. In this approach, the non-zero matrix elements of the Hamiltonian are represented as edges on a graph, which naturally decomposes into disconnected clusters. Each cluster corresponds to an independent block in the block-diagonalized form of the Hamiltonian. The eigenvalues in the low-energy sector were obtained by solving the eigenvalue problem for each block matrix and by solving a modified Hamiltonian subject to orthonormality constraints with respect to previously computed lower-energy eigenstates. We applied the GBBD method to linear hydrogen H chains ranging from H$_2$ to H$_{12}$. The results showed excellent agreement with exact ones, confirming both the accuracy and efficiency of the proposed method. Finally, we discussed several physical properties with respect to the number of H$_2$ molecules.