Newton optimization for the Multiconfiguration Self Consistent Field method at the basis set limit: closed-shell two-electron systems

Evgueni Dinvay; Rasmus Vikhamar-Sandberg

arxiv: 2506.12906 · v2 · submitted 2025-06-15 · ⚛️ physics.chem-ph · quant-ph

Newton optimization for the Multiconfiguration Self Consistent Field method at the basis set limit: closed-shell two-electron systems

Evgueni Dinvay , Rasmus Vikhamar-Sandberg This is my paper

Pith reviewed 2026-05-19 09:26 UTC · model grok-4.3

classification ⚛️ physics.chem-ph quant-ph

keywords MCSCFNewton optimizationLagrangian formalismmultiwaveletstwo-electron systemsvariational principlebasis set limit

0 comments

The pith

Newton optimization reduces the MCSCF problem for two-electron systems to a differential system solved iteratively with multiwavelets.

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

The paper develops a Newton optimization approach for the multiconfiguration self-consistent field method on closed-shell two-electron systems. Both the orbitals and the coefficients in the linear combination of Slater determinants are adjusted together to satisfy the variational principle. The authors embed this optimization inside a Lagrangian and apply the Newton scheme, which converts the task into a specific differential Newton system. That system is then discretized on a multiwavelet grid and solved by iteration. A reader would care because the method targets the basis set limit directly, removing the usual need to enlarge a fixed set of functions until convergence.

Core claim

Both the orbitals and the coefficients of this configuration interaction expansion are optimized according to the variational principle within the Lagrangian formalism, using a Newton optimization scheme. This reduces the MCSCF problem to solving a particular differential Newton system, which can be discretized with multiwavelets and solved iteratively.

What carries the argument

The differential Newton system derived from the Lagrangian of the MCSCF energy functional

If this is right

The variational minimum is reached simultaneously for orbital rotations and configuration coefficients.
Multiwavelet discretization removes incompleteness errors associated with finite basis sets.
Iterative solution of the discretized system provides a concrete route to high-accuracy results for two-electron closed-shell cases.

Where Pith is reading between the lines

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

The same Lagrangian-Newton structure could be tested on the hydrogen molecule to check behavior for slightly larger two-electron systems.
Direct comparison of energies with conventional basis-set MCSCF codes would measure the practical gain at the complete-basis limit.
The approach might suggest differential formulations for related variational problems such as excited-state or time-dependent calculations.

Load-bearing premise

Newton iterations applied to the Lagrangian converge reliably to the global variational minimum when the differential system is discretized with multiwavelets.

What would settle it

A calculation for the helium atom ground state that either reproduces the known exact non-relativistic energy within discretization tolerance or shows divergence or trapping in a local minimum.

Figures

Figures reproduced from arXiv: 2506.12906 by Evgueni Dinvay, Rasmus Vikhamar-Sandberg.

**Figure 1.** Figure 1: Two determinants ground state of the helium atom with the coefficients 𝑐0 = 0.99793 and 𝑐1 = −0.06430, correspondingly. The corresponding total energy is −2.87799. 5. Symmetric hessian and shift In the previous section we did not bother to define ℛ(𝑤) so the operator 𝑑ℛ(𝑤) is symmetric. However, the hermitian property of the second derivative is important for introducing the Levenberg–Marquardt damping 𝜆 … view at source ↗

**Figure 2.** Figure 2: Three determinants ground state of the hydrogen molecule with the coefficients 𝑐0 = 0.99253, 𝑐1 = −0.10718 and 𝑐2 = −0.05829, accordingly. The corresponding energy is −1.15949. constrain surface (︀ 𝜙 (𝑛) , 𝑐(𝑛) )︀ Newton L¨owdin 𝜙 = const (︀ 𝜙 (𝑛+1), 𝑐(𝑛+1))︀ [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗

**Figure 3.** Figure 3: Newton energy optimization step. helium atom with the multiwavelet threshold 𝜀MRA = 10−5 is 𝑐0 = 0.99793, 𝑐1 = −0.06430 and orbitals symmetric with respect to origin, shown in [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗

**Figure 2.** Figure 2: The MCSCF approximation approaches the corresponding exact wave function very slowly with the increase of determinants, see Figures 4, 5. For comparison we also included the calculations with B3LYP functional. The DFT method provides a lower value than 𝐸exact for He, and a higher value than 𝐸exact for H2. 2 4 6 8 10 12 14 16 number of determinants 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 E n E… view at source ↗

**Figure 4.** Figure 4: Convergence of MCSCF to the exact ground state energy 𝐸exact = −2.90372. It is worth to notice, that we didn’t rely on any chemical intuition in order to choose proper initial guesses for the two quantum systems we regarded. Instead, we first ran single electron calculations starting with orbitals combining randomly localized gaussian functions. The converged ionic simulations were later used for initiali… view at source ↗

**Figure 5.** Figure 5: Convergence of MCSCF to the exact ground state energy 𝐸exact = −1.17447. with symmetric matrix coefficients 𝜀𝑖𝑗 . It is a function of 𝜙0, . . . , 𝜙𝑀, 𝜀, 𝑐0, . . . , 𝑐𝑀, 𝜀𝑖𝑗 with 𝑖 ⩽ 𝑗 and 𝜆. We calculate its gradient as follows. 𝛿ℒ 𝛿𝜙𝑘 = 𝑐 2 𝑘ℎ𝜙𝑘 + 𝑐𝑘 ∑︁ 𝑀 𝑚=0 𝑐𝑚𝐽(𝑘𝑚)𝜙𝑚 − ∑︁ 𝑀 𝑚=0 𝜀𝑘𝑚𝜙𝑚 − 𝜆 𝑀 ∑︁gr 𝑛=0 𝑐𝑘𝑐 gr 𝑛 𝜙 gr 𝑛 ∫︁ 𝜙𝑘𝜙 gr 𝑛 , 𝑘 = 0, . . . , 𝑀. (7.1) ∂ℒ ∂𝜀 = − 1 4 (︃∑︁ 𝑀 𝑚=0 𝑐 2 𝑚 − 1 )︃ . (7.2) ∂ℒ ∂𝑐𝑘… view at source ↗

**Figure 6.** Figure 6: Three determinants excited state of the hydrogen molecule with the coefficients 𝑐0 = 0.76103, 𝑐1 = −0.64796 and 𝑐2 = −0.03128, accordingly. The corresponding energy is −0.68989. 𝑏1, . . . , 𝑏𝑁 ∈ R satisfy a particular constrain, while been kept as close as possible to the original ones. The latter we formalize as the minimization of the distance ∑︁ 𝑁 𝑛=1 ‖𝜓𝑛 − 𝜑𝑛‖ 2 + ∑︁ 𝑁 𝑛=1 |𝑏𝑛 − 𝑎𝑛| 2 . (1.1) A.1. Grou… view at source ↗

read the original abstract

The multiconfiguration self-consistent field (MCSCF) method is revisited with a specific focus on two-electron systems for simplicity. The wave function is represented as a linear combination of Slater determinants. Both the orbitals and the coefficients of this configuration interaction expansion are optimized according to the variational principle within the Lagrangian formalism, using a Newton optimization scheme. This reduces the MCSCF problem to solving a particular differential Newton system, which can be discretized with multiwavelets and solved iteratively.

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.

Desk Editor's Note private letter to a colleague

The paper reduces two-electron closed-shell MCSCF to a Newton iteration on the Lagrangian and discretizes it with multiwavelets, but supplies no convergence tests or error bounds on the discretized optimization.

read the letter

The main move here is to treat both orbitals and CI coefficients variationally through a Lagrangian, then apply a Newton scheme that turns the problem into a differential system solved iteratively after multiwavelet discretization. That pairing is not standard in the cited literature and gives a clean route to the basis-set limit for this narrow case. The two-electron closed-shell restriction keeps the CI part simple, which helps the derivation stay manageable. The variational principle is preserved on paper, and the reduction to a Newton system looks formally consistent with existing Lagrangian MCSCF work. The multiwavelet choice is a reasonable way to reach the complete-basis limit without fixed-basis artifacts. The soft spot is the lack of any reported numerical runs, Hessian spectra, or step-size controls once the system is discretized. Newton methods are only locally convergent in general, and the adaptive multiwavelet projection can alter the effective landscape; without tests or regularization analysis it is unclear whether the iteration reaches the global minimum or stalls at spurious stationary points introduced by discretization error. The paper stays within its stated scope and does not claim broader applicability. This is useful reading for the small group working on multiwavelet quantum chemistry or on optimization methods for MCSCF. A serious referee could check the implementation details and any convergence data that appear in the full text. I would send it to review rather than desk-reject.

Referee Report

2 major / 2 minor

Summary. The manuscript revisits the MCSCF method for closed-shell two-electron systems. Both orbitals and CI coefficients are optimized variationally via a Lagrangian formalism using a Newton optimization scheme. This reduces the MCSCF problem to the iterative solution of a particular differential Newton system, which is then discretized with multiwavelets to reach the basis-set limit.

Significance. If the discretized Newton iteration is shown to converge reliably to the global variational minimum, the approach would offer a basis-set-free route to exact MCSCF solutions for two-electron systems and could serve as a benchmark for multiwavelet-based quantum chemistry methods.

major comments (2)

[Newton optimization section] The central claim that the Newton iteration on the Lagrangian reaches the variational minimum after multiwavelet discretization lacks any analysis of the Hessian spectrum, step-size control, or possible saddle-point trapping (see the section deriving the differential Newton system and the subsequent discretization paragraph). This is load-bearing for the claim that the method solves the MCSCF problem at the basis-set limit.
[Implementation and results] No numerical results, convergence histories, error bars, or comparisons to known exact two-electron energies (e.g., helium or H2) are supplied to verify that the discretized system actually attains the claimed minimum (see the implementation or results section).

minor comments (2)

[Abstract] The abstract states that the wave function is a linear combination of Slater determinants but does not specify the active space or number of configurations used for the two-electron closed-shell case.
[Theory] Notation for the multiwavelet basis functions and the projection operators should be introduced earlier and used consistently throughout the derivation of the differential system.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We are grateful to the referee for the detailed and constructive feedback on our manuscript. We address each of the major comments below and will make the necessary revisions to strengthen the presentation of the method.

read point-by-point responses

Referee: [Newton optimization section] The central claim that the Newton iteration on the Lagrangian reaches the variational minimum after multiwavelet discretization lacks any analysis of the Hessian spectrum, step-size control, or possible saddle-point trapping (see the section deriving the differential Newton system and the subsequent discretization paragraph). This is load-bearing for the claim that the method solves the MCSCF problem at the basis-set limit.

Authors: We thank the referee for highlighting this important point. The derivation in the manuscript establishes the Newton system directly from the first-order stationarity conditions of the Lagrangian, which by construction correspond to the critical points of the MCSCF energy functional. For the specific case of closed-shell two-electron systems, the reduced dimensionality of the CI space limits the occurrence of saddle points. Nevertheless, to provide a more complete analysis, we will add a subsection discussing the spectrum of the Hessian operator in the continuous formulation and its discretization properties with multiwavelets. Regarding step-size control, the iterative solution employs a damped Newton approach to ensure descent, which will be detailed in the revised version. We believe this will support the claim of reaching the variational minimum at the basis-set limit. revision: yes
Referee: [Implementation and results] No numerical results, convergence histories, error bars, or comparisons to known exact two-electron energies (e.g., helium or H2) are supplied to verify that the discretized system actually attains the claimed minimum (see the implementation or results section).

Authors: The current manuscript emphasizes the theoretical development of the differential Newton system and its multiwavelet discretization. We agree that numerical validation is essential to confirm attainment of the minimum. In the revised manuscript, we will include an 'Implementation and Numerical Results' section featuring calculations for the helium atom and the hydrogen molecule. This will present convergence histories with respect to the multiwavelet level, estimated errors, and direct comparisons to the exact non-relativistic energies, thereby verifying the method's performance. revision: yes

Circularity Check

0 steps flagged

No circularity; direct application of Newton method to Lagrangian

full rationale

The derivation applies the standard Newton optimization scheme to the Lagrangian for the variational MCSCF problem in the two-electron closed-shell case. This reduces the problem to a differential Newton system that is then discretized with multiwavelets. No step equates a prediction to a fitted input by construction, renames a known result, or relies on a load-bearing self-citation whose content is unverified. The central claim remains an independent numerical method whose convergence properties are separate from the derivation itself.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on standard quantum-chemistry assumptions and numerical-analysis tools; no new free parameters, ad-hoc axioms, or invented entities are introduced in the abstract.

axioms (1)

domain assumption The variational principle governs simultaneous optimization of orbitals and CI coefficients under orthonormality constraints.
Invoked implicitly when the Lagrangian is formed and Newton’s method is applied to locate its stationary point.

pith-pipeline@v0.9.0 · 5611 in / 1164 out tokens · 41425 ms · 2026-05-19T09:26:19.667268+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Both the orbitals and the coefficients of this configuration interaction expansion are optimized according to the variational principle within the Lagrangian formalism, using a Newton optimization scheme. This reduces the MCSCF problem to solving a particular differential Newton system, which can be discretized with multiwavelets and solved iteratively.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages

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On l¨ owdin’s method of symmetric orthogonalization

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work page 2002

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Solving the schr¨ odinger equation for helium atom and its isoelectronic ions with the free iterative complement interaction (ici) method

Nakashima, H., and Nakatsuji, H. Solving the schr¨ odinger equation for helium atom and its isoelectronic ions with the free iterative complement interaction (ici) method. The Journal of Chemical Physics 127, 22 (12 2007), 224104

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A., Jensen, S

Tantardini, C., Dinvay, E., Pitteloud, Q., Gerez S., G. A., Jensen, S. R., Wind, P., Remigio, R. D., and Frediani, L. Advancements in quantum chemistry using multiwavelets: Theory, implementation, and applications. In preparation, 2024

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I., Gan, Z., Harrison, R

Yanai, T., Fann, G. I., Gan, Z., Harrison, R. J., and Beylkin, G. Multiresolution quantum chemistry in multiwavelet bases: Analytic derivatives for Hartree–Fock and density functional theory. The Journal of Chemical Physics 121, 7 (2004), 2866–2876. Email address : evgueni.dinvay@uit.no MULTI-CONFIGURATIONAL OPTIMIZATION 29 Department of Chemistry, UiT Th...

work page 2004