Abstract
Slater determinants have underpinned quantum chemistry for nearly a century, yet their full potential has remained challenging to exploit. In this work, we show that a variational wavefunction composed of a few hundred optimized non-orthogonal determinants can achieve energy accuracies comparable to the state of the art. This is obtained by introducing an optimization method that leverages the quadratic dependence of the variational energy on the orbitals of each determinant, enabling an exact iterative optimization, and uses an efficient tensor-contraction algorithm to evaluate the effective Hamiltonian with a computational cost that scales as the fourth power of the number of basis functions. We benchmark the accuracy of the proposed method with exact full-configuration interaction results where available, and we achieve lower variational energies than coupled cluster (CCSD(T)) for several molecules in the double-zeta basis.
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The data generated in this study are publicly available on Zenodo at https://doi.org/10.5281/zenodo.18390677.
Code availability
The code used to generate the results reported in this study is publicly available on Zenodo at https://doi.org/10.5281/zenodo.18390552.
References
Heitler, W. & London, F. Wechselwirkung neutraler Atome und homöopolare Bindung nach der Quantenmechanik. Z. Phys. https://doi.org/10.1007/bf01397394 (1927).
Coulson, C. & Fischer, I. XXXIV. Notes on the molecular orbital treatment of the hydrogen molecule. Lond. Edinb. Dublin Philos. Mag. J. Sci. https://doi.org/10.1080/14786444908521726 (1949).
Goddard, W. A. Improved quantum theory of many-electron systems. I. Construction of Eigenfunctions of S2 which satisfy Pauli principle. Phys. Rev. https://doi.org/10.1103/PhysRev.157.73 (1967).
Gerratt, J. General theory of spin-coupled wave functions for atoms and molecules. Adv. Atomic Mol. Phys. https://doi.org/10.1016/S0065-2199(08)60360-7 (1971).
Cooper, D. L., Gerratt, J. & Raimondi, M. Applications of spin-coupled valence bond theory. Chem. Rev. https://doi.org/10.1021/cr00005a014 (1991).
Lawley, K. P., Cooper, D. L., Gerratt, J. & Raimondi, M. Modern valence bond theory. AB Initio Methods Quantum Chem. II https://doi.org/10.1002/9780470142943.CH6 (2007).
Wu, W. Su, P., Shaik, S. & Hiberty, P. C. Classical valence bond approach by modern methods. Chem. Rev. https://doi.org/10.1021/cr100228r (2011).
McWeeny, R. Methods of Molecular Quantum Mechanics (Academic Press, 1996).
Pople, J. A., Head-Gordon, M. & Raghavachari, K. Quadratic configuration interaction. A general technique for determining electron correlation energies. J. Chem. Phys. https://doi.org/10.1063/1.453520 (1987).
Coester, F. & Kümmel, H. Short-range correlations in nuclear wave functions. Nucl. Phys. https://doi.org/10.1016/0029-5582(60)90140-1 (1960).
Thom, A. J. W. & Head-Gordon, M. Hartree-Fock solutions as a quasidiabatic basis for nonorthogonal configuration interaction. J. Chem. Phys. https://doi.org/10.1063/1.3236841 (2009).
Siegbahn, P., Heiberg, A., Roos, B. & Levy, B. A comparison of the Super-CI and the Newton-Raphson scheme in the complete active space SCF method. Phys. Scr. https://doi.org/10.1088/0031-8949/21/3-4/014 (1980).
Olsen, J. Novel methods for configuration interaction and orbital optimization for wave functions containing non-orthogonal orbitals with applications to the chromium dimer and trimer. J. Chem. Phys. https://doi.org/10.1063/1.4929724 (2015).
Bremond, B. Hartree-Fock methods for degenerate cases. Nuclear Phys. https://doi.org/10.1016/0029-5582(64)90579-6 (1964).
Koch, H. & Dalgaard, E. Linear superposition of optimized non-orthogonal Slater determinants for singlet states. Chem. Phys. Lett. https://doi.org/10.1016/0009-2614(93)87129-q (1993).
Schmid, K., Kyotoku, M., Grümmer, F. & Faessler, A. Small amplitude vibrations in symmetry-projected quasi-particle mean fields. Ann. Phys. https://doi.org/10.1016/0003-4916(89)90264-9 (1989).
Jiménez-Hoyos, C. A., Rodríguez-Guzmán, R. & Scuseria, G. E. Multi-component symmetry-projected approach for molecular ground state correlations. J. Chem. Phys. https://doi.org/10.1063/1.4832476 (2013).
Fukutome, H. Theory of resonating quantum fluctuations in a fermion system: Resonating Hartree-Fock approximation. Prog. Theor. Phys. https://doi.org/10.1143/ptp.80.417 (1988).
Ikawa, A., Yamamoto, S. & Fukutome, H. Orbital optimization in the resonating Hartree-Fock approximation and its application to the one dimensional Hubbard model. J. Phys. Soc. Jpn https://doi.org/10.1143/jpsj.62.1653 (1993).
Tomita, N., Ten-no, S. & Tanimura, Y. Ab initio molecular orbital calculations by the resonating Hartree-Fock approach: superposition of non-orthogonal Slater determinants. Chem. Phys. Lett. https://doi.org/10.1016/s0009-2614(96)01266-3 (1996).
Mahler, A. D. & L. Thompson, M. Orbital optimization in nonorthogonal multiconfigurational self-consistent field applied to the study of conical intersections and avoided crossings. J. Chem. Phys. https://doi.org/10.1063/5.0053615 (2021).
Thouless, D. Stability conditions and nuclear rotations in the Hartree-Fock theory. Nuclear Phys. https://doi.org/10.1016/0029-5582(60)90048-1 (1960).
Sun, C., Gao, F. & Scuseria, G. E. Selected nonorthogonal configuration interaction with compressed single and double excitations. J. Chem. Theory Comput. https://doi.org/10.1021/acs.jctc.4c00240 (2024).
Miller, E. R. & Parker, S. M. Numerically stable resonating Hartree-Fock. J. Chem. Phys. https://doi.org/10.1063/5.0246790 (2025).
Van Lenthe, J. & Balint-Kurti, G. The valence-bond SCF (VB SCF) method. Chem. Phys. Lett. https://doi.org/10.1016/0009-2614(80)80623-3 (1980).
Rashid, Z. & van Lenthe, J. H. A quadratically convergent VBSCF method. J. Chem. Phys. https://doi.org/10.1063/1.4788765 (2013).
Meyer, H.-D., Manthe, U. & Cederbaum, L. The multi-configurational time-dependent Hartree approach. Chem. Phys. Lett. https://doi.org/10.1016/0009-2614(90)87014-i (1990).
Lode, A. U., Lévêque, C., Madsen, L. B., Streltsov, A. I. & Alon, O. E. Colloquium: multiconfigurational time-dependent Hartree approaches for indistinguishable particles. Rev. Mod. Phys. https://doi.org/10.1103/RevModPhys.92.011001 (2020).
Al-Saidi, W., Zhang, S. & Krakauer, H. Auxiliary-field quantum Monte Carlo calculations of molecular systems with a Gaussian basis. J. Chem. Phys. https://doi.org/10.1063/1.2200885 (2006).
Lee, J., Pham, H. Q. & Reichman, D. R. Twenty years of auxiliary-field quantum monte carlo in quantum chemistry: an overview and assessment on main group chemistry and bond-breaking. J. Chem. Theory Comput. https://doi.org/10.1021/acs.jctc.2c00802 (2022).
Scemama, A., Applencourt, T., Giner, E. & Caffarel, M. Quantum Monte Carlo with very large multideterminant wavefunctions. J. Comput. Chem. https://doi.org/10.1002/jcc.24382 (2016).
Ceperley, D., Chester, G. V. & Kalos, M. H. Monte Carlo simulation of a many-fermion study. Phys. Rev. B https://doi.org/10.1103/PhysRevB.16.3081 (1977).
Pfau, D., Spencer, J. S., Matthews, A. G. & Foulkes, W. M. C. Ab initio solution of the many-electron schrödinger equation with deep neural networks. Phys. Rev. Res. https://doi.org/10.1103/PhysRevResearch.2.033429 (2020).
Hermann, J., Schätzle, Z. & Noé, F. Deep-neural-network solution of the electronic schrödinger equation. Nat. Chem. https://doi.org/10.1038/s41557-020-0544-y (2020).
Liu, A.-J. & Clark, B. K. Neural network backflow for ab initio quantum chemistry. Phys. Rev. B https://doi.org/10.1103/PhysRevB.110.115137 (2024).
Bortone, M., Rath, Y. & Booth, G. H. Simple fermionic backflow states via a systematically improvable tensor decomposition. Commun. Phys. https://doi.org/10.1038/s42005-025-02083-4 (2025).
Gao, H., Imamura, S., Kasagi, A. & Yoshida, E. Distributed implementation of full configuration interaction for one trillion determinants. J. Chem. Theory Comput. https://doi.org/10.1021/acs.jctc.3c01190 (2024).
Chan, G. K., Kállay, M. & Gauss, J. State-of-the-art density matrix renormalization group and coupled cluster theory studies of the nitrogen binding curve. J. Chem. Phys. https://doi.org/10.1063/1.1783212 (2004).
Scuseria, G. E., Jiménez-Hoyos, C. A., Henderson, T. M., Samanta, K. & Ellis, J. K. Projected quasiparticle theory for molecular electronic structure. J. Chem. Phys. https://doi.org/10.1063/1.3643338 (2011).
Hohenstein, E. G., Parrish, R. M. & Martínez, T. J. Tensor hypercontraction density fitting. I. Quartic scaling second- and third-order møller-plesset perturbation theory. J. Chem. Phys. https://doi.org/10.1063/1.4732310 (2012).
Peng, B. & Kowalski, K. Highly efficient and scalable compound decomposition of two-electron integral tensor and its application in coupled cluster calculations. J. Chem. Theory Comput. https://doi.org/10.1021/acs.jctc.7b00605 (2017).
Burton, H. G. A. & Thom, A. J. W. Reaching full correlation through nonorthogonal configuration interaction: a second-order perturbative approach. J. Chem. Theory Comput. https://doi.org/10.1021/acs.jctc.0c00468 (2020).
Löwdin, P.-O. Quantum theory of many-particle systems. II. Study of the ordinary Hartree-Fock approximation. Phys. Rev. https://doi.org/10.1103/physrev.97.1490 (1955).
Malmqvist, P. Å. Calculation of transition density matrices by nonunitary orbital transformations. Int. J. Quantum Chem. https://doi.org/10.1002/qua.560300404 (1986).
Broer, R. & Nieuwpoort, W. C. Broken orbital symmetry and the description of valence hole states in the tetrahedral [CrO4]2 anion. Theor. Chim. Acta https://doi.org/10.1007/bf00527744 (1988).
Verbeek, J. & Van Lenthe, J. H. The generalized Slater-Condon rules. Int. J. Quantum Chem. https://doi.org/10.1002/qua.560400204 (1991).
Mazziotti, D. A. Comparison of contracted schrödinger and coupled-cluster theories. Phys. Rev. A https://doi.org/10.1103/physreva.60.4396 (1999).
Dijkstra, F. & van Lenthe, J. H. Gradients in valence bond theory. J. Chem. Phys. https://doi.org/10.1063/1.482021 (2000).
Tomita, N. Many-body wave functions approximated by the superposition of spin-projected nonorthogonal Slater determinants in the resonating Hartree-Fock method. Phys. Rev. B https://doi.org/10.1103/physrevb.69.045110 (2004).
Brouder, C. Matrix elements of many-body operators and density correlations. Phys. Rev. A https://doi.org/10.1103/physreva.72.032720 (2005).
Utsuno, Y., Shimizu, N., Otsuka, T. & Abe, T. Efficient computation of Hamiltonian matrix elements between non-orthogonal Slater determinants. Comput. Phys. Commun. https://doi.org/10.1016/j.cpc.2012.09.002 (2013).
Rodriguez-Laguna, J., Robledo, L. M. & Dukelsky, J. Efficient computation of matrix elements of generic Slater determinants. Phys. Rev. A https://doi.org/10.1103/PhysRevA.101.012105 (2020).
Burton, H. G. A. Generalized nonorthogonal matrix elements: unifying Wick’s theorem and the Slater-Condon rules. J. Chem. Phys. https://doi.org/10.1063/5.0045442 (2021).
Burton, H. G. A. Generalized nonorthogonal matrix elements. II: extension to arbitrary excitations. J. Chem. Phys. https://doi.org/10.1063/5.0122094 (2022).
Bertsch, G. F. & Robledo, L. M. Symmetry restoration in Hartree-Fock-Bogoliubov-based theories. Phys. Rev. Lett. https://doi.org/10.1103/physrevlett.108.042505 (2012).
Chen, G. P. & Scuseria, G. E. Robust formulation of Wick’s theorem for computing matrix elements between Hartree-Fock-Bogoliubov wavefunctions. J. Chem. Phys. https://doi.org/10.1063/5.0156124 (2023).
King, H. F., Stanton, R. E., Kim, H., Wyatt, R. E. & Parr, R. G. Corresponding orbitals and the nonorthogonality problem in molecular quantum mechanics. J. Chem. Phys. https://doi.org/10.1063/1.1712221 (1967).
Sunet, Q. et al. Recent developments in the PySCF program package. J. Chem. Phys. https://doi.org/10.1063/5.0006074 (2020).
Zhai, H. et al. Block2: a comprehensive open source framework to develop and apply state-of-the-art DMRG algorithms in electronic structure and beyond. J. Chem. Phys. https://doi.org/10.1063/5.0180424 (2023).
Smith, D. G. A. et al. PSI4 1.4: open-source software for high-throughput quantum chemistry. J. Chem. Phys. https://doi.org/10.1063/5.0006002 (2020).
Acknowledgements
We acknowledge insightful discussions with F. Vicentini and L. Viteritti. We thank H. Koch and G. Scuseria for valuable remarks. This work was supported by the Swiss National Science Foundation under Grant No. 200021_200336 (C.G.). This research was also supported by SEFRI through Grant No. MB22.00051 (NEQS - Neural Quantum Simulation) (R.R.).
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R.R. conceived the overall idea, C.G. derived the final theoretical formulation, implemented the algorithm, performed the simulations, and curated the data for the article. C.G., J.N., R.N., G.C., and R.R. contributed to the choice of studied problems, discussions, and to the writing and review of the paper.
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Giuliani, C., Nys, J., Martinazzo, R. et al. Precise Quantum Chemistry calculations with few Slater Determinants. Nat Commun (2026). https://doi.org/10.1038/s41467-026-70255-z
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DOI: https://doi.org/10.1038/s41467-026-70255-z
