pith. sign in

arxiv: 2507.05514 · v2 · submitted 2025-07-07 · 🪐 quant-ph

Electron-molecule scattering via R-matrix variational algorithms on a quantum computer

Dario Picozzi , Jonathan Tennyson , Vincent Graves , Jimena D. Gorfinkiel This is my paper

Pith reviewed 2026-05-19 05:20 UTC · model grok-4.3

classification 🪐 quant-ph
keywords electron-molecule scatteringR-matrix methodvariational quantum eigensolverquantum computinghydrogen moleculeboundary amplitudesinner-region problem
0
0 comments X

The pith

Variational quantum algorithms solve the inner-region R-matrix problem for electron-molecule scattering and encode the needed boundary amplitudes in circuit parameters.

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

This paper shows how to cast the inner part of the R-matrix calculation for electron-molecule collisions as a variational problem that a quantum computer can address. A quantum circuit is prepared to represent the wavefunction inside a spherical boundary, and its parameters are optimized to find the relevant states. When tested on a hydrogen-molecule model with a classical simulator, the method recovers the full energy spectrum in a chosen symmetry sector and supplies the boundary amplitudes required to continue the scattering calculation outside the boundary. The approach targets the scaling barrier that limits classical R-matrix treatments of larger or more complex molecules.

Core claim

The R-matrix inner-region problem is formulated on a quantum computer by using the variational quantum eigensolver (and variants with number projection and simultaneous optimisation) to diagonalise the inner-region Hamiltonian. For electron scattering from the hydrogen molecule in a selected symmetry sector, the procedure recovers the complete spectrum and shows that the optimal circuit parameters directly encode the boundary amplitudes needed for the outer-region matching and cross-section computation.

What carries the argument

The variational quantum eigensolver applied to the inner-region Hamiltonian, in which the optimised parameters of the circuit ansatz directly supply the R-matrix boundary amplitudes.

If this is right

  • The full spectrum recovered inside the chosen symmetry sector provides all states required for a complete scattering description.
  • The encoded boundary amplitudes can be fed directly into standard outer-region propagation and matching routines.
  • Number-projection operators can be added to enforce physical constraints such as fixed electron number.
  • Simultaneous optimisation of circuit parameters can accelerate convergence to the required boundary values.

Where Pith is reading between the lines

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

  • The same variational formulation could be applied to other diatomic or small polyatomic targets once suitable ansatze are identified.
  • Running the algorithm on present-day noisy quantum processors would reveal the practical error tolerance for scattering observables.
  • If hardware noise remains manageable, the method could open scattering studies of molecules whose inner-region basis sizes exceed classical diagonalisation limits.

Load-bearing premise

That variational quantum eigensolver results obtained on a noiseless simulator for a small basis set will produce boundary amplitudes accurate enough for the outer-region scattering calculation to remain reliable on noisy hardware or for larger molecules.

What would settle it

A direct numerical comparison, for the same hydrogen-molecule model and basis, between the boundary amplitudes extracted from the optimal VQE circuit parameters and those obtained from a conventional classical R-matrix calculation; a mismatch larger than the target precision would falsify the encoding claim.

Figures

Figures reproduced from arXiv: 2507.05514 by Dario Picozzi, Jimena D. Gorfinkiel, Jonathan Tennyson, Vincent Graves.

Figure 2
Figure 2. Figure 2: Number of qubits 7 CNOT gates (unoptimised) 217 Circuit depth 314 Variational parameters 10 Pauli terms measured (subspace method) 92 TABLE I. Circuit resource estimates for the minimal H2 ex￾ample. Once the optimized parameters θ ∗ have been found, the expansion coefficients aijk correspond to matrix el￾ements of U(θ ∗ ). In particular for the indexes asso￾ciated with the bound states (in our example indi… view at source ↗
Figure 1
Figure 1. Figure 1: FIG. 1. Schematic of the simultaneous eigenstate circuit showing the target orbital ( [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Schematic of the SO(5) rotation gate decomposition into a sequence of [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 2
Figure 2. Figure 2: For the algorithms 2., 3. and 4. all eigenvalues are re￾covered within 10−7Ha. E. Coherent summation protocol In our circuit, we implement a coherent summation protocol for the target eigenstate qubits as follows. Let the joint state of the target eigenstate register and the system qubits (target orbital and continuum orbital qubits) be |ψ⟩ = 2 Xa−1 i=0 |i⟩anc ⊗ |ψi⟩sys. Applying a Hadamard gate before mea… view at source ↗
Figure 3
Figure 3. Figure 3: 3. Simultaneous optimisation with sum of vari￾ances This method required 13,345 cost function evalua￾tions for 1,080 Pauli terms. The results are shown in the middle panel of [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
read the original abstract

Electron-molecule collisions play a central role in both natural processes and modern technological applications, particularly in plasma processing. Conventional computational strategies such as the R-matrix method have been widely adopted yet encounter significant scaling challenges in treating more complex systems. In this work we present a quantum computational approach that utilises the variational quantum eigensolver (VQE) and variations thereof to overcome these limitations. We explore a number of methods, including the use of number projection operators and simultaneous optimisation. We demonstrate the feasibility of our method on a model problem of electron scattering from the hydrogen molecule, with numerical results obtained using a noiseless classical simulator. We recover the full spectrum of the Hamiltonian within a chosen symmetry sector. Moreover, the optimal circuit parameters directly encode the R-matrix boundary amplitudes needed for subsequent scattering computations. To our knowledge, this is the first application of quantum algorithms to electron--molecule scattering, and specifically the first formulation of the R-matrix inner-region problem on a quantum computer.

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

2 major / 0 minor

Summary. The manuscript proposes applying the variational quantum eigensolver (VQE), including variants with number projection and simultaneous optimization, to solve the inner-region Hamiltonian in the R-matrix method for electron-molecule scattering. For the model problem of electron-H2 scattering, results on a noiseless classical simulator recover the full spectrum within a chosen symmetry sector. The authors assert that the optimal circuit parameters directly encode the R-matrix boundary amplitudes needed for outer-region scattering calculations, presenting this as the first quantum-algorithm treatment of electron-molecule scattering.

Significance. If the extracted boundary amplitudes prove accurate and the approach scales beyond small noiseless cases, it could alleviate the basis-size and configuration-interaction scaling bottlenecks that limit classical R-matrix calculations for larger molecules. The spectrum recovery on the H2 model supplies concrete evidence of feasibility for the inner-region step in a limited setting and credits the work as an initial demonstration of quantum methods in this domain.

major comments (2)
  1. Numerical results: spectrum recovery is demonstrated, yet no comparison is provided between the boundary amplitudes read out from the optimal VQE parameters and the corresponding amplitudes obtained from classical diagonalization of the same inner-region Hamiltonian. Without this check, agreement on eigenvalues does not establish that the surface amplitudes are sufficiently accurate for reliable R-matrix construction and subsequent scattering observables.
  2. Method and results sections: the manuscript contains no noisy-hardware simulations or error-propagation analysis for the extracted amplitudes. This omission leaves untested the weakest assumption that noiseless-simulator amplitudes will remain usable when the same ansatz is executed on actual quantum hardware or for larger molecules.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed comments. We address each major point below and describe the revisions that will be made to strengthen the manuscript.

read point-by-point responses

  1. Referee: Numerical results: spectrum recovery is demonstrated, yet no comparison is provided between the boundary amplitudes read out from the optimal VQE parameters and the corresponding amplitudes obtained from classical diagonalization of the same inner-region Hamiltonian. Without this check, agreement on eigenvalues does not establish that the surface amplitudes are sufficiently accurate for reliable R-matrix construction and subsequent scattering observables.

    Authors: We agree that explicit validation of the boundary amplitudes is required to substantiate the claim that the optimal parameters encode them. Although spectrum recovery indicates that the variational states approximate the eigenstates, this does not automatically confirm the surface amplitudes. In the revised manuscript we will extract the R-matrix boundary amplitudes from the converged VQE parameters and provide a direct, quantitative comparison to the amplitudes obtained from classical diagonalization of the identical inner-region Hamiltonian, including overlap or relative-error metrics. revision: yes

  2. Referee: Method and results sections: the manuscript contains no noisy-hardware simulations or error-propagation analysis for the extracted amplitudes. This omission leaves untested the weakest assumption that noiseless-simulator amplitudes will remain usable when the same ansatz is executed on actual quantum hardware or for larger molecules.

    Authors: The present work is a proof-of-principle demonstration performed on a noiseless simulator. We acknowledge that noise resilience must be examined for practical use. In the revision we will add an error-propagation analysis that quantifies how parameter uncertainties propagate to the extracted boundary amplitudes. Full noisy-hardware simulations and scaling tests for larger molecules lie outside the scope of this initial study and will be addressed in follow-up work. revision: partial

Circularity Check

0 steps flagged

No circularity: standard VQE readout of boundary amplitudes from inner-region optimization

full rationale

The paper formulates the R-matrix inner-region problem as a standard VQE on the Hamiltonian for a chosen symmetry sector of electron-H2, recovers the spectrum on a noiseless simulator, and extracts boundary amplitudes directly from the optimized circuit parameters. This extraction follows the usual variational state readout and does not reduce to a self-definition, fitted input renamed as prediction, or self-citation chain; the amplitudes are a direct consequence of the variational wavefunction rather than being presupposed by the scattering result. No uniqueness theorems, ansatzes smuggled via prior work, or renamings of known patterns are load-bearing. The derivation remains self-contained against the classical R-matrix inner-region benchmark.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The approach rests on the standard division of configuration space in R-matrix theory and on the assumption that VQE can variationally approximate the relevant eigenstates of the inner-region Hamiltonian.

free parameters (1)
  • ansatz circuit depth and form
    Chosen to represent the inner-region wavefunction for the chosen basis and symmetry sector
axioms (2)
  • domain assumption The R-matrix method partitions space into an inner region where exchange and correlation are treated explicitly and an outer region where the scattered electron moves in a long-range potential
    Invoked to justify mapping only the inner-region problem to the quantum computer
  • domain assumption The variational quantum eigensolver can recover the low-lying eigenstates of the inner-region Hamiltonian when run on a noiseless simulator
    Required for the numerical demonstration to be taken as evidence of feasibility

pith-pipeline@v0.9.0 · 5702 in / 1395 out tokens · 43252 ms · 2026-05-19T05:20:05.596332+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

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

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

64 extracted references · 64 canonical work pages · 3 internal anchors

  1. [1]

    Present a quantum-algorithm framework for the R- matrix inner-region problem, mapping the ( N + 1)-electron Hamiltonian to qubits via a Jor- dan–Wigner transformation and explicitly enforc- ing occupancy constraints for continuum orbitals

    work page

  2. [2]

    Employ the qubit representation of the number- projection operator in the variational cir- cuit—based on the construction in [27]—to ensure that at most one electron occupies any continuum orbital

    work page

  3. [3]

    Develop a sequential subspace-optimisation proto- col that decomposes the required SO( N) rotations into a cascade of two-qubit Givens rotations, allow- arXiv:2507.05514v1 [quant-ph] 7 Jul 2025 2 ing us to converge on each eigenstate with signif- icantly fewer cost-function evaluations and Pauli- term measurements than alternative methods

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  4. [4]

    Introduce a coherent-summation measurement scheme for the target-eigenstate register, which combines ancilla-based Hadamard gates with system-Hamiltonian measurements to reduce the total number of distinct expectation values re- quired

    work page

  5. [5]

    Demonstrate our approach on a model H 2 scatter- ing calculation, validating the algorithm and show- ing that the optimal circuit parameters directly en- code the R-matrix boundary amplitudes needed for subsequent scattering analysis. II. THE R-MA TRIX METHOD AND MOLECULAR HAMIL TONIAN The R-matrix method was originally developed to study nuclear [28], an...

    work page

  6. [6]

    R(ζ) is a fixed two-qubit gate (implemented as a Givens rotation of angle ζ) that couples target eigenstate qubits to enforce total-spin symmetry

    A parametric rotation gate U(θ), followed by the Clebsch–Gordan rotations R(ζ) that are fixed by 4 spin symmetry (in our example, as in Figure 1, a single rotation corresponding to the triplet state). R(ζ) is a fixed two-qubit gate (implemented as a Givens rotation of angle ζ) that couples target eigenstate qubits to enforce total-spin symmetry

    work page

  7. [7]

    The target–continuum couplings (blue block in Fig- ure 1, CNOT gates with control on the target- eigenstate register and target on the continuum reg- ister). Each target eigenstate qubit acts as control on one continuum orbital qubit to ensure that, if a particular target eigenfunction is selected, exactly one continuum orbital is excited in the correct s...

    work page

  8. [8]

    The bound-states gate (green block in Figure 1, CNOT gates with control and targets on the target- orbital register). A gate that entangles the target eigenstate qubits with the target-orbital qubits, so that if the circuit has selected a particular bound configuration, the correct pattern of target-orbital occupations is realised. In simple examples, one...

    work page

  9. [9]

    The sequence of controlled target-eigenstate gates, each controlled on one of the target eigenstate qubits and with targets on the target-orbital qubits. A central element of our quantum circuits is the one- parameter two-qubit rotation operation which mixes the |01⟩ and the |10⟩ states, implemented through a sequence of controlled-NOT (CNOT) gates interl...

    work page

  10. [10]

    Whilst the coupling between the upper multiplet with St = S − 1 2 and the lower multiplet with St = S + 1 2 is dynamical, the couplings between states with Mt = M − 1 2 and Mt = M + 1 2 for total spin in the upper multiplet and the lower multiplet are each fixed by symmetry. Each spin multiplet admits a two-term expansion in terms of the half-angle ζ: S, ...

    work page

  11. [11]

    (15) Where H 2 has been replaced by HP H to enforce correct occupancy

    Single state with variance cost function In the single eigenstate approach the cost function is defined as the variance of the Hamiltonian with respect to the trial state, C(θ) = ⟨HP H ⟩|ψ(θ)⟩ − ⟨H⟩2 |ψ(θ)⟩. (15) Where H 2 has been replaced by HP H to enforce correct occupancy. This sometimes fails to return the full eigenspectrum

    work page

  12. [12]

    Single state with folded Hamiltonian cost function An alternative approach is to first evaluate the en- ergy expectation for each trial state: the approxi- mate energy ˜E is first evaluated for the trial state, and then used to construct a Hamiltonian [43, 44] where, as long as the objective state eigenvalue is the closest eigenvalue to the energy of the ...

    work page

  13. [13]

    (17) This approach, while necessitating separate evalua- tions for each eigenstate, recovers the full spectrum of that eigensector within 10 −7Ha

    Simultaneous optimisation with sum of vari- ances In the simultaneous optimisation of all five eigen- states (see explicit list in Results section), the cost function is constructed as the sum of the variances of the individual eigenstates, C(θ) = 5X i=1 h ⟨HP H ⟩|ψi(θ)⟩ − ⟨H⟩2 |ψi(θ)⟩ i . (17) This approach, while necessitating separate evalua- tions for...

    work page

  14. [14]

    , θi,N −1) = ⟨H⟩|ψi(θ)⟩

    Simultaneous optimisation with subspace optimisation and simple Hamiltonian expec- tation value The most computational resource efficient method is based solely on the Hamiltonian expectation value, Ci(θi,i+1, . . . , θi,N −1) = ⟨H⟩|ψi(θ)⟩. (18) In this approach the unitary is designed to enforce a subspace structure that naturally preserves or- thogonali...

    work page

  15. [15]

    Single state with variance cost function In some instances this approach is unable to recover all the eigenvalues

    work page

  16. [16]

    The results are shown in the upper panel of Figure 3

    Single state with folded Hamiltonian cost function This approach required approximately 1,397 cost function evaluations to recover the five eigenstates, each corresponding to the evaluation of 1,080 Pauli terms. The results are shown in the upper panel of Figure 3

    work page

  17. [17]

    The results are shown in the middle panel of Figure 3

    Simultaneous optimisation with sum of vari- ances This method required 13,345 cost function evalua- tions for 1,080 Pauli terms. The results are shown in the middle panel of Figure 3

    work page

  18. [18]

    The results are shown in lower panel of Figure 3

    Simultaneous optimisation with subspace optimisation and simple Hamiltonian expec- tation value This method required only 573 cost function eval- uations involving 92 Pauli terms. The results are shown in lower panel of Figure 3. For the last three algorithms all the eigenvalues were recovered as the converged values within 10 −7Ha, the residual discrepan...

    work page

  19. [19]

    B. I. Schneider, T. N. Rescigno, B. H. Lengsfield, and A. E. Orel, Science 284, 1489 (1999)

    work page 1999

  20. [20]

    J. M. Carr, P. G. Galiatsatos, J. D. Gorfinkiel, M. A. Lysaght, Z. Masin, L. A. Morgan, I. Rozum, and J. Ten- nyson, Eur. Phys. J. D 66, 58 (2012)

    work page 2012

  21. [21]

    M. C. Zammit, J. S. Savage, D. V. Fursa, and I. Bray, Phys. Rev. Lett. 116, 233201 (2016)

    work page 2016

  22. [22]

    Bartschat and M

    K. Bartschat and M. J. Kushner, Proc. Nat. Acad. Sci. 113, 7026 (2016)

    work page 2016

  23. [23]

    R. P. Feynman, Int. J. Theor. Phys. 21, 467 (1982)

    work page 1982

  24. [24]

    Lloyd, Science 273, 1073 (1996)

    S. Lloyd, Science 273, 1073 (1996)

    work page 1996

  25. [25]

    D. S. Abrams and S. Lloyd, Phys. Rev. Lett. 79, 2586 (1997)

    work page 1997

  26. [26]

    D. S. Abrams and S. Lloyd, Phys. Rev. Lett. 83, 5162 (1999). 9

    work page 1999

  27. [27]

    Aspuru-Guzik, A

    A. Aspuru-Guzik, A. D. Dutoi, P. J. Love, and M. Head- Gordon, Science 309, 1704 (2005)

    work page 2005

  28. [28]

    Peruzzo, J

    A. Peruzzo, J. McClean, P. Shadbolt, M.-H. Yung, X.-Q. Zhou, P. J. Love, A. Aspuru-Guzik, and J. L. O’Brien, Nat. Commun. 5, 4213 (2014)

    work page 2014

  29. [29]

    J. R. McClean, J. Romero, R. Babbush, and A. Aspuru- Guzik, New J. Phys. 18, 023023 (2016)

    work page 2016

  30. [30]

    B. P. Lanyon, J. D. Whitfield, G. G. Gillett, M. E. Goggin, M. P. Almeida, I. Kassal, J. D. Biamonte, M. Mohseni, B. J. Powell, M. Barbieri, A. Aspuru-Guzik, and A. G. White, Nat. Chem. 2, 106 (2010)

    work page 2010

  31. [31]

    P. J. J. O’Malley, R. Babbush, I. D. Kivlichan, J. Romero, J. R. McClean, R. Barends, J. Kelly, P. Roushan, A. Tranter, N. Ding, B. Campbell, Y. Chen, Z. Chen, B. Chiaro, A. Dunsworth, A. G. Fowler, E. Jef- frey, A. Megrant, J. Y. Mutus, C. Neill, C. Quintana, D. Sank, A. Vainsencher, J. Wenner, T. C. White, P. V. Coveney, P. J. Love, H. Neven, A. Aspuru-...

    work page 2016

  32. [32]

    Hempel, C

    C. Hempel, C. Maier, J. Romero, J. McClean, T. Monz, H. Shen, P. Jurcevic, B. P. Lanyon, P. J. Love, R. Bab- bush, A. Aspuru-Guzik, R. Blatt, and C. F. Roos, Phys. Rev. X 8, 031022 (2018)

    work page 2018

  33. [33]

    Y. Cao, J. Romero, J. P. Olson, M. Degroote, P. D. John- son, M. Kieferov´ a, I. D. Kivlichan, T. Menke, B. Per- opadre, N. P. D. Sawaya, S. Sim, L. Veis, and A. Aspuru- Guzik, Chem. Rev. 119, 10856 (2019)

    work page 2019

  34. [34]

    McArdle, S

    S. McArdle, S. Endo, A. Aspuru-Guzik, S. C. Benjamin, and X. Yuan, Rev. Mod. Phys. 92, 015003 (2020)

    work page 2020

  35. [35]

    Bauer, S

    B. Bauer, S. Bravyi, M. Motta, and G. K.-L. Chan, Chem. Rev. 120, 12685 (2020)

    work page 2020

  36. [36]

    Tilly, H

    J. Tilly, H. Chen, S. Cao, D. Picozzi, K. Setia, Y. Li, E. Grant, L. Wossnig, I. Rungger, G. H. Booth, and J. Tennyson, Phys. Rep. 986, 1 (2022)

    work page 2022

  37. [37]

    Bharti, A

    K. Bharti, A. Cervera-Lierta, T. H. Kyaw, T. Haug, S. Alperin-Lea, A. Anand, M. Degroote, H. Heimonen, K. Fujii, A. Y. Matsuura, M. Singh, P. Murali, K. Lauter, T. Brandenburger, D. E. Koh, O. Shehab, S. Wood, A. Siripurapu, F. Nori, N. Killoran, H. Neven, A. Aspuru- Guzik, S. Pandey, A. Jardine, A. Greene, J. McClean, Z. Wang, I. Tavernelli, C. Lin, and ...

    work page 2022

  38. [38]

    S. P. Jordan, K. S. M. Lee, and J. Preskill, arXiv preprint arXiv:1112.4833 (2011)

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  39. [39]

    S. P. Jordan, K. S. M. Lee, and J. Preskill, Science 336, 1130 (2012)

    work page 2012

  40. [40]

    S. P. Jordan, K. S. M. Lee, and J. Preskill, arXiv preprint arXiv:1404.7115 (2014)

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  41. [41]

    C. Choi, D. Lee, J. Bonitati, Z. Qian, and J. Watkins, Phys. Rev. Lett. 127, 040505 (2021)

    work page 2021

  42. [42]

    Klco and M

    N. Klco and M. J. Savage, Phys. Rev. Lett. 116, 052001 (2016)

    work page 2016

  43. [43]

    Arute, K

    F. Arute, K. Arya, R. Babbush, D. Bacon, J. C. Bardin, R. Barends, R. Biswas, S. Boixo, M. Broughton, B. Buck- ley, and et al., Science 369, 1084 (2020)

    work page 2020

  44. [44]

    M. Yusf, L. Gan, C. Moffat, and G. Ru- pak, arXiv preprint arXiv:2406.09231 (2024), 10.48550/arXiv.2406.09231, arXiv:2406.09231 [nucl- th]

    work page doi:10.48550/arxiv.2406.09231 2024

  45. [45]

    Pascal’s pyramid and number projec- tion operators for quantum computation,

    D. Picozzi, “Pascal’s pyramid and number projec- tion operators for quantum computation,” (2024), arXiv:2407.16561 [quant-ph]

    work page arXiv 2024

  46. [46]

    E. P. Wigner and L. Eisenbud, Phys. Rev. 72, 29 (1947)

    work page 1947

  47. [47]

    P. G. Burke, R-Matrix Theory of Atomic Collisions: Ap- plication to Atomic, Molecular and Optical Processes (Springer, 2011)

    work page 2011

  48. [48]

    Tennyson, Phys

    J. Tennyson, Phys. Rep. 491, 29 (2010)

    work page 2010

  49. [49]

    Maˇ s´ ın, J

    Z. Maˇ s´ ın, J. Benda, J. D. Gorfinkiel, A. G. Harvey, and J. Tennyson, Comput. Phys. Commun. 249, 107092 (2020)

    work page 2020

  50. [50]

    A. G. Sunderland, J. W. Heggarty, C. J. Noble, and N. S. Scott, Comput. Phys. Commun. 114, 183 (1998)

    work page 1998

  51. [51]

    J. D. Gorfinkiel and J. Tennyson, J. Phys. B 37, L343 (2004)

    work page 2004

  52. [52]

    J. D. Gorfinkiel and J. Tennyson, J. Phys. B 38, 1607 (2005)

    work page 2005

  53. [53]

    Halmov´ a and J

    G. Halmov´ a and J. Tennyson, Phys. Rev. Lett. 100, 213202 (2008)

    work page 2008

  54. [54]

    Zhang, P

    R. Zhang, P. G. Galiatsatos, and J. Tennyson, J. Phys. B 44, 195203 (2011)

    work page 2011

  55. [55]

    Bloch, Nucl

    C. Bloch, Nucl. Phys. 4, 503 (1957)

    work page 1957

  56. [56]

    Tennyson, J

    J. Tennyson, J. Phys. B 37, 1061 (2004)

    work page 2004

  57. [57]

    J. D. Whitfield, J. Biamonte, and A. Aspuru-Guzik, Mol. Phys. 109, 735 (2011)

    work page 2011

  58. [58]

    Kassal, J

    I. Kassal, J. D. Whitfield, A. Perdomo-Ortiz, M.-H. Yung, and A. Aspuru-Guzik, Annu. Rev. Phys. Chem. 62, 185 (2011)

    work page 2011

  59. [59]

    Picozzi and J

    D. Picozzi and J. Tennyson, Quantum Sci. Technol. 8, 035026 (2023)

    work page 2023

  60. [60]

    D. W. Berry, C. Gidney, M. Motta, J. R. McClean, and R. Babbush, npj Quantum Inf. 4, 22 (2018)

    work page 2018

  61. [61]

    Wang and A

    L. Wang and A. Zunger, The Journal of Chemical Physics 100, 2394 (1994)

    work page 1994

  62. [62]

    L. C. Tazi and A. J. W. Thom, Journal of Chemical The- ory and Computation 20, 2491 (2024)

    work page 2024

  63. [63]

    M. J. D. Powell, in Advances in Optimization and Nu- merical Analysis, Mathematics and Its Applications, Vol. 275, edited by S. Gomez and J.-P. Hennart (Kluwer Aca- demic Publishers, Dordrecht, 1994) pp. 51–67

    work page 1994

  64. [64]

    M. J. D. Powell, Acta Numerica 7, 287 (1998)

    work page 1998