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Journal of Chemical Theory and Computation

Perturbative Coordinate Descent Full Configuration Interaction

Z D Zhang, Mingyu Qiu, Yingzhou Li, Wei Hu

Published May 7, 2026• 0 citations

Abstract

The coordinate descent full configuration interaction (CDFCI) is an accurate and efficient method for calculating ground-state energies of molecular systems. In this work we introduce two perturbative extensions of original CDFCI. We first propose a direct CDFCI + PT to further correct variational energy by perturbation theory. We also propose an iterative perturbative CDFCI (IP-CDFCI) to avoid evaluating the overlarge perturbative space and catch the information encoded in variational stage. In IP-CDFCI, we store three sparse vectors in a Hash table: c, b and d, where c is the variational wave function, b is the compressed approximation of H c and d is the perturbative wave function, respectively. By utilizing the redundant memory in the storage of c, we can calculate perturbation without any extra memory compared to original CDFCI. We also use a strategy to eliminate the expensive repetitive evaluation of diagonal elements of Hamiltonian by creating a vector h which stores the diagonal entries of Hamiltonian. This strategy trades memory usage for computational time and reduces the computing time significantly. Furthermore, we use quadruple-precision floating-point numbers to improve the numerical stability of perturbative energy. We demonstrate the effectiveness of the single-threaded IP-CDFCI and the multithreaded CDFCI + PT, respectively, by performing calculations on several small molecular benchmarks as well as challenging correlated system: the chromium dimer.

DOI: 10.1021/acs.jctc.6c00103

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The coordinate descent full configuration interaction (CDFCI) is an accurate and efficient method for calculating ground-state energies of molecular systems. In this work we introduce two perturbative extensions of original CDFCI. We first propose a direct CDFCI + PT to further correct variational energy by perturbation theory. We also propose an iterative perturbative CDFCI (IP-CDFCI) to avoid evaluating the overlarge perturbative space and catch the information encoded in variational stage. In IP-CDFCI, we store three sparse vectors in a Hash table: c, b and d, where c is the variational wave function, b is the compressed approximation of H c and d is the perturbative wave function, respectively. By utilizing the redundant memory in the storage of c, we can calculate perturbation without any extra memory compared to original CDFCI. We also use a strategy to eliminate the expensive repetitive evaluation of diagonal elements of Hamiltonian by creating a vector h which stores the diagonal entries of Hamiltonian. This strategy trades memory usage for computational time and reduces the computing time significantly. Furthermore, we use quadruple-precision floating-point numbers to improve the numerical stability of perturbative energy. We demonstrate the effectiveness of the single-threaded IP-CDFCI and the multithreaded CDFCI + PT, respectively, by performing calculations on several small molecular benchmarks as well as challenging correlated system: the chromium dimer.

We develop a multithreaded parallel coordinate descent full configuration interaction algorithm (mCDFCI) for the electronic structure ground-state calculation in the configuration interaction framework. The FCI problem is reformulated as an unconstrained minimization problem and tackled by a modified block coordinate descent method with a deterministic compression strategy. mCDFCI is designed to prioritize determinants based on their importance, with block updates enabling efficient parallelization on shared-memory, multicore computing infrastructure. We demonstrate the efficiency of the algorithm by computing an accurate benchmark energy for the chromium dimer in the Ahlrichs SV basis (48e, 42o), which explicitly includes 2.07 × 10 9 variational determinants. We also provide the binding curve of the nitrogen dimer under the cc-pVQZ basis set (14e, 110o). Benchmarks show up to 79.3% parallel efficiency on 128 cores.

We study the effect of compression in Coordinate Descent Full Configuration Interaction (CDFCI) within an unconstrained optimization formulation of the full configuration interaction ground-state problem. Under suitable local assumptions, we prove that the compressed iteration converges linearly to the solution of an associated restricted problem. We also characterize the convergence point of the compressed algorithm. Under an additional exponential decay assumption on the target eigenvector, we show that the resulting eigenvalue error is of order

τ^2

, where

τ

denotes the compression threshold. Numerical results support the analysis.

We propose an efficient deterministic method to calculate the full configuration interaction (FCI) ground state energy. This method leverages the sparseness of the Lanczos basis vectors that span the Krylov subspace associated with the Hamiltonian to rapidly calculate the lowest eigenvalue of the effective Hamiltonian formed in this basis. By iteratively performing the spanning and diagonalization steps, this method is capable of rapidly reaching chemical accuracy for a variety of strongly correlated molecules, including stretched N

_2

and C

_2

, in merely several tens of iterations. To accelerate our algorithm, we implement parallelized sparse matrix-sparse vector multiplication, which features a novel hashing method that establishes a one-to-one correspondence between the set of full configuration Slater determinants and a set of consecutive integers. A near-linear acceleration was observed.

CDFCI is a shared-memory parallel numerical program for computing low-lying eigenpairs of large-scale, non-relativistic fermionic Hamiltonians. The software is designed to handle a broad class of many-body quantum models, including both ab initio electronic structure Hamiltonians and lattice-based Hamiltonians arising in condensed matter physics. CDFCI combines an efficient coordinate-descent-based selected configuration interaction algorithm with dedicated parallelization strategies, achieving high performance on modern multi-core architectures. Benchmark results on representative quantum chemistry and condensed matter test cases demonstrate that CDFCI attains state-of-the-art accuracy with competitive performance compared to established selected configuration interaction (such as CIPSI or SHCI) and DMRG implementations. The software is open-source, extensively documented, and provides a Python interface for seamless integration with PySCF and other many-body simulation workflows.

We present an algorithm related to the full-configuration interaction (FCI) method that makes complete use of the sparse nature of the coefficient vector representing the many-electron wave function in a determinantal basis. Main achievements of the presented sparse FCI (SFCI) algorithm are (i) development of an iteration procedure that avoids the storage of FCI size vectors; (ii) development of an efficient algorithm to evaluate the effect of the Hamiltonian when both the initial and the product vectors are sparse. As a result of point (i) large disk operations can be skipped which otherwise may be a bottleneck of the procedure. At point (ii) we progress by adopting the implementation of the linear transformation by Olsen et al. [J. Chem Phys. 89, 2185 (1988)] for the sparse case, getting the algorithm applicable to larger systems and faster at the same time. The error of a SFCI calculation depends only on the dropout thresholds for the sparse vectors, and can be tuned by controlling the amount of system memory passed to the procedure. The algorithm permits to perform FCI calculations on single node workstations for systems previously accessible only by supercomputers.

Full configuration interaction (FCI) serves as a benchmark tool for evaluating the accuracy of approximate quantum chemical methods. However, its exponential computational cost limits its practical use. The inherently parallel and compute-intensive nature of matrix and vector operations in FCI is well-suited for acceleration on modern GPU architectures. Despite this potential, prior research has only focused on accelerating the matrix and vector multiplication, thus failing to fully exploit the power of GPUs. In this paper, we present a fully GPU-accelerated FCI implementation that uses a CUDA-enabled GPU for the entire computational workflow, achieving end-to-end acceleration. Our implementation incorporates and refines advanced optimization strategies from previous studies, while crucially eliminating redundant computational steps and minimizing CPU-GPU data transfers. To facilitate broad usability and integration, we offer both Python- and C-based implementations. Our C implementation achieves up to a 35.4× speedup over a multithreaded CPU implementation and up to a 15.9× speedup over the current leading GPU-based implementation, respectively. Our Python implementation also delivers substantial acceleration, reaching up to 18.9× and 8.7× speedups over CPU and prior GPU implementations, respectively.