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arxiv: 2503.14502 · v2 · submitted 2025-03-18 · ⚛️ physics.chem-ph · cond-mat.str-el· quant-ph

Precise Quantum Chemistry calculations with few Slater Determinants

Clemens Giuliani , Jannes Nys , Rocco Martinazzo , Giuseppe Carleo , Riccardo Rossi This is my paper

Pith reviewed 2026-05-22 23:27 UTC · model grok-4.3

classification ⚛️ physics.chem-ph cond-mat.str-elquant-ph
keywords Slater determinantsnon-orthogonal determinantsvariational wavefunctionquantum chemistrymolecular energiescoupled cluster comparison
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0 comments X

The pith

A variational wavefunction of a few hundred optimized non-orthogonal determinants reaches state-of-the-art molecular energies.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that a compact expansion in non-orthogonal Slater determinants, when optimized with a new iterative procedure, can produce variational energies that match or beat coupled-cluster results for small molecules. The approach rests on an exact update rule for each determinant's orbitals that exploits the quadratic form of the energy expectation value. This yields a practical algorithm whose dominant cost is a fourth-power scaling in the number of basis functions. A sympathetic reader would care because current high-accuracy methods either require enormous numbers of determinants or rely on perturbative corrections whose reliability varies with molecular character.

Core claim

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. The method achieves lower variational energies than coupled cluster (CCSD(T)) for several molecules in the double-zeta basis and matches exact full-configuration,

What carries the argument

An iterative orbital optimization that exploits the exact quadratic dependence of the energy on the orbitals of each individual non-orthogonal determinant, paired with a tensor-contraction scheme for the effective Hamiltonian.

If this is right

  • The computational cost remains dominated by a fourth-power scaling with basis-set size rather than factorial growth with the number of determinants.
  • Direct comparison with full-configuration-interaction energies is possible on small systems and confirms the variational upper-bound property.
  • The same wavefunction form yields lower energies than CCSD(T) on several tested molecules without perturbative corrections.
  • The method is variational at every step, so the reported energies are rigorous upper bounds.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same orbital-update rule might be combined with selected configuration interaction to further reduce the number of retained determinants.
  • Because each determinant is optimized independently in the quadratic step, the approach could be parallelized across determinants with little communication.
  • Extension to excited states would require only a change in the target energy functional while retaining the same contraction machinery.

Load-bearing premise

The quadratic dependence on each determinant's orbitals permits an exact iterative update that avoids poor local minima and numerical instabilities for non-orthogonal sets.

What would settle it

A double-zeta calculation on any of the benchmark molecules in which the variational energy lies above the CCSD(T) value would show that the claimed accuracy has not been reached.

Figures

Figures reproduced from arXiv: 2503.14502 by Clemens Giuliani, Giuseppe Carleo, Jannes Nys, Riccardo Rossi, Rocco Martinazzo.

Figure 1
Figure 1. Figure 1: Energy differences with respect to CCSD(T) in the cc-pVDZ basis set. The shaded area indicates results within chemical accuracy (1 kcal/mol) from FCI/DMRG, and the hatched area from CCSD(T). Geometries and FCI reference energies (where available) are taken from Ref. [36], with the exception of the N2 molecule at the distance of 2.118 a.u., where we compare to DMRG results from Ref. [37], and BN with our ow… view at source ↗
Figure 2
Figure 2. Figure 2: Scaling of the number of SDs with bond order. We consider here the four di-atomic molecules C2, BN, BeO and LiF with an identical number of electrons n and basis orbitals m, and BH3 and N2 with similar numbers, all in the cc-pVDZ basis set. The molecules possess different bond orders, ranging from the single bond of LiF to the triple bond of N2. Equilibrium geometries and FCI reference energies from Ref. [… view at source ↗
Figure 4
Figure 4. Figure 4: Dissociation curve of the N2 molecule in the cc￾pVDZ basis. We compare our variational energy results for different numbers of determinants ND with UCCSD(T) (dot￾ted line) and UCISD (dash-dotted line) from our own cal￾culations with PySCF [39]. In the inset we show the error compared to DMRG results from Ref. [37] for the equilibrium bond distance and larger. posed in the wavefunction of Eq. (1), in order … view at source ↗
Figure 5
Figure 5. Figure 5: Triplet ground state and singlet first excited state of the O2 molecule in the cc-pVDZ basis set. The singlet state is found by minimizing the energy of the Hamiltonian with an additive penalty term λ Sˆ2 . The horizontal lines represent CCSD(T) energies for M = 0, S = 0 (dashed) and M = 1, S = 1(dotted). Equilibrium geometry taken from the supple￾mentary material of Ref. [36]. In the inset, we show the av… view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript presents a variational quantum chemistry method using a compact wavefunction of a few hundred optimized non-orthogonal Slater determinants. It introduces an iterative orbital optimization procedure that exploits a claimed quadratic dependence of the variational energy on the orbitals of each individual determinant, together with an O(N^4) tensor-contraction algorithm for the effective Hamiltonian. The central claims are that this ansatz achieves energies comparable to FCI (where available) and lower variational energies than CCSD(T) for several molecules in double-zeta bases.

Significance. If the optimization procedure is shown to be exact and the reported energies are reproducible, the approach would offer a promising route to high-accuracy variational calculations with far fewer determinants than conventional CI expansions, while retaining a direct variational upper-bound character.

major comments (1)
  1. [Abstract] Abstract (optimization method description): the claim that the variational energy exhibits 'quadratic dependence ... on the orbitals of each determinant, enabling an exact iterative optimization' is load-bearing for the entire result. For a general non-orthogonal expansion the energy is E = (c† H c) / (c† S c); fixing all determinants but one, the dependence on that determinant's orbital coefficients enters through det(S) in the denominator and through the cofactors of H and S in the numerator. Both are polynomials of degree equal to the number of electrons, not quadratic. The manuscript must therefore either (i) derive an effective one-particle operator whose stationarity condition is exactly quadratic or (ii) demonstrate that the iteration converges to the true stationary point of the full multi-determinant energy despite the higher-order dependence. Without this derivation or a clear,
minor comments (1)
  1. The abstract asserts benchmark comparisons to FCI and CCSD(T) and lower energies than CCSD(T), yet supplies no numerical values, basis sets, molecule list, or convergence data. A concise table or set of representative numbers should appear in the abstract or the first results paragraph.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the need to clarify the optimization procedure. We address the major comment below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract] Abstract (optimization method description): the claim that the variational energy exhibits 'quadratic dependence ... on the orbitals of each determinant, enabling an exact iterative optimization' is load-bearing for the entire result. For a general non-orthogonal expansion the energy is E = (c† H c) / (c† S c); fixing all determinants but one, the dependence on that determinant's orbital coefficients enters through det(S) in the denominator and through the cofactors of H and S in the numerator. Both are polynomials of degree equal to the number of electrons, not quadratic. The manuscript must therefore either (i) derive an effective one-particle operator whose stationarity condition is exactly quadratic or (ii) demonstrate that the iteration converges to the true stationary point of the full multi-determinant energy despite the higher-order dependence. Without this derivation

    Authors: We agree that the dependence of the total energy on the orbital coefficients of one determinant (with others fixed) is of higher polynomial order, arising from the determinant structure of the overlap and Hamiltonian matrix elements. The abstract phrasing was imprecise and overstated the quadratic character. In the revised manuscript we will (a) remove the 'quadratic dependence' claim from the abstract, (b) add a methods subsection that explicitly constructs an effective one-particle operator from the current multi-determinant coefficients and the fixed determinants, whose stationarity condition matches the derivative of the full variational energy with respect to orbital rotations of the active determinant, and (c) include numerical checks confirming that the iterative procedure reaches a stationary point of the complete energy functional. These additions address both (i) and (ii). revision: yes

Circularity Check

0 steps flagged

No circularity detected; derivation self-contained against external benchmarks

full rationale

The paper introduces a variational ansatz of non-orthogonal determinants and an optimization procedure whose claimed exactness rests on the stated quadratic dependence of the energy on each determinant's orbitals when others are held fixed. Results are obtained by direct minimization and tensor contraction, then compared to independent external references (FCI where available, CCSD(T) otherwise). No load-bearing step reduces a reported energy or accuracy claim to a fitted parameter, a self-citation chain, or a redefinition of the input; the benchmarks remain falsifiable outside the method's internal parameters. The skeptic concern about the precise polynomial degree of the energy functional is a question of correctness of the quadratic claim, not a reduction of the output to the input by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review prevents exhaustive enumeration; the method implicitly relies on standard quantum-chemistry assumptions about basis sets and the variational principle but introduces no new free parameters or invented entities visible at this level.

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