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arXiv:2010.02027 · The Journal of Chemical Physics

Approaching the full configuration interaction ground state from an arbitrary wavefunction with gradient descent and quasi-Newton algorithms

Carlos A. Jiménez-Hoyos

Published Dec 18, 2020• 3 citations
arXivHTMLPDF

Abstract

We consider gradient descent and quasi-Newton algorithms to optimize the full configuration interaction (FCI) ground state wavefunction starting from an arbitrary reference state |0⟩. We show that the energies obtained along the optimization path can be evaluated in terms of expectation values of |0⟩, thus avoiding explicit storage of intermediate wavefunctions. This allows us to find the energies after the first few steps of the FCI algorithm for systems much larger than what standard deterministic FCI codes can handle at present. We show an application of the algorithm with reference wavefunctions constructed as linear combinations of non-orthogonal determinants.

DOI
10.1063/5.0031958

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We consider gradient descent and quasi-Newton algorithms to optimize the full configuration interaction (FCI) ground state wavefunction starting from an arbitrary reference state |0⟩. We show that the energies obtained along the optimization path can be evaluated in terms of expectation values of |0⟩, thus avoiding explicit storage of intermediate wavefunctions. This allows us to find the energies after the first few steps of the FCI algorithm for systems much larger than what standard deterministic FCI codes can handle at present. We show an application of the algorithm with reference wavefunctions constructed as linear combinations of non-orthogonal determinants.
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.
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 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 N2_22​ and C2_22​, 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.
A method is implemented wherein numerical approximations to the ground and first few excited states of a quantum mechanical N -body 1D harmonic oscillator are found through variational methods, representing the states as a linear combination of normalized pseudo-states which are themselves linear combinations of non-orthogonal Slater determinants. These states are then used as a low energy basis for configuration interaction. An expression is derived for an analytical matrix derivative of the energy functional, in order to improve the speed of the variation. The speed and accuracy using the analytical derivative is compared to that of the numerical derivative, and a number of different gradient descent methods are tried and compared.
The convergence of full configuration interaction quantum Monte Carlo (FCIQMC) is accelerated using a quasi-Newton propagation (QN) which can also be applied to coupled cluster Monte Carlo (CCMC). The computational scaling of this optimized propagation is, keeping the additional computational cost to a bare minimum. Its effects are investigated deterministically and stochastically on a model system, the uniform electron gas, with Hilbert space size up to 10 40, and shown to accelerate convergence of the instantaneous projected energy by over an order of magnitude in the FCIQMC test case. Its capabilities are then demonstrated with FCIQMC on an archetypical quantum chemistry problem, the chromium dimer, in an all-electron basis set with Hilbert space size of about 10 22 yielding highly accurate FCI energies.