pith. sign in

arxiv: 2607.02058 · v1 · pith:M3RMOUL4new · submitted 2026-07-02 · ⚛️ physics.chem-ph · quant-ph

On the Symplectic Propagation of the Spin-MInt Algorithm for Non-Adiabatic Quantum Dynamics

James R. Rampton , Lauren E. Cook , Timothy J. H. Hele This is my paper

Pith reviewed 2026-07-03 03:58 UTC · model grok-4.3

classification ⚛️ physics.chem-ph quant-ph
keywords spin-mappingsymplecticitynon-adiabatic dynamicsLie-Poisson algebracoadjoint orbitSpin-MInt algorithmmonodromy matrixquantum dynamics
0
0 comments X

The pith

The Spin-MInt algorithm preserves symplecticity on the coadjoint orbit for any number of electronic states.

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

The paper proves that the Spin-MInt algorithm for propagating spin-mapping variables remains symplectic when applied to systems with a general number K of electronic states. It does this by explicitly checking that the monodromy matrix satisfies MJM transpose equals J, using the Lie-Poisson structure of the su(K) algebra. A reader would care because symplectic methods respect the geometric structure of the underlying classical-like equations, which often yields better long-term stability and conservation in non-adiabatic simulations. The result extends an earlier proof that held only for two states.

Core claim

The symplecticity of the Spin-MInt algorithm on the associated coadjoint orbit is shown via an explicit verification of the symplecticity condition MJM^T = J exploiting the Lie-Poisson structure of the system. To our knowledge, this is the first time the monodromy matrix for the Spin-MInt algorithm has been explicitly stated using canonical coordinates on the coherent state manifold for a general number of states.

What carries the argument

The Lie-Poisson structure of the su(K) algebra on its coadjoint orbit, which the Spin-MInt update rules are shown to respect so that the symplectic condition can be verified directly.

If this is right

  • The algorithm can be applied to multi-state non-adiabatic problems while preserving the symplectic form.
  • The explicit monodromy matrix can be used to develop further classical-like spin-mapping integrators.
  • Similar symplectic propagators can be constructed for other coupled or uncoupled Lie-Poisson systems.
  • Long-time numerical simulations of spin-mapping dynamics gain improved geometric fidelity.

Where Pith is reading between the lines

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

  • The same Lie-Poisson verification strategy could be tried on other mapping-variable algorithms.
  • Adoption of this integrator might reduce artificial energy drift in extended quantum-dynamics trajectories.
  • Canonical-coordinate expressions for the monodromy matrix open the door to variational or shadow-Hamiltonian analyses.

Load-bearing premise

The Spin-MInt update rules exactly respect the Lie-Poisson structure and coadjoint orbit geometry of the su(K) algebra for the chosen mapping.

What would settle it

A direct computation of the monodromy matrix M for K greater than 2 that yields MJM^T not equal to J would show the claimed symplecticity does not hold.

read the original abstract

Mapping methods are often used for the numerical simulation of nonadiabatic systems by propagating classical mapping variable trajectories. A recently popularised mapping method is spin-mapping, whose mapping variables arise from quantum mechanical operators with symmetries described by a Lie-Poisson algebra. Simulating the classical-like dynamics of spin-mapping systems accurately is generally challenging, with many methods unable to preserve the underlying geometric structure of the symplectic form. The Spin-MInt algorithm is a recently proposed algorithm propagating spin-mapping variables, with a direct proof of symplecticity existing only for 2 electronic states. Here, we directly prove the symplecticity of the Spin-MInt algorithm for a general $K$ electronic states. A review of the symplectic nature of coadjoint orbits of the $\mathfrak{su}(K)$ Lie-Poisson algebra provides the framework needed to understand symplecticity of the Spin-MInt algorithm in this general case. The symplecticity of the method on the associated coadjoint orbit is then shown for what we believe to be the first time via an explicit verification of the symplecticity condition $\mathbf{MJ}\mathbf{M}^\textrm{T}=\mathbf{J}$ exploiting the Lie-Poisson structure of the system. To our knowledge, this is the first time the monodromy matrix for the Spin-MInt algorithm has been explicitly stated using canonical coordinates on the coherent state manifold for a general number of states. We hope that this will assist the development of classical-like spin-mapping methods which might utilise elements of the monodromy matrix, and inform future work on similar symplectic algorithms for coupled and uncoupled Lie-Poisson systems.

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 / 2 minor

Summary. The paper claims to extend the known symplecticity of the Spin-MInt integrator from K=2 to general K electronic states. It reviews the symplectic geometry of coadjoint orbits of the su(K) Lie-Poisson algebra, maps the spin-mapping variables to canonical coordinates on the coherent-state manifold, explicitly constructs the monodromy matrix M of the integrator, and verifies the symplecticity condition MJM^T = J by direct algebraic manipulation that exploits the underlying Lie-Poisson structure.

Significance. If the verification is complete and the discrete flow is shown to remain on the orbit, the result supplies the first general-K proof of symplecticity together with an explicit expression for the monodromy matrix in canonical coordinates. This is a concrete technical contribution that can support the construction of higher-order or structure-preserving variants for non-adiabatic dynamics and the use of the monodromy matrix in linear-response or stability analyses.

major comments (2)
  1. [Section describing the Spin-MInt update rules and the subsequent proof of MJM^T = J] The central algebraic verification of MJM^T = J presupposes that the Spin-MInt update rules map the coadjoint orbit to itself exactly. No separate demonstration is supplied that the discrete flow remains on the orbit for arbitrary K (or that any O(Δt) drift vanishes identically). Without this step the cancellation that produces the symplecticity identity is not guaranteed to hold for the actual numerical map.
  2. [Section introducing the mapping to canonical coordinates] The canonical coordinate chart on the coherent-state manifold is stated to be valid for general K, yet the paper does not address whether the chart remains regular for all points on the orbit when K > 2 or whether the Poisson tensor is faithfully represented everywhere the integrator is applied.
minor comments (2)
  1. [Abstract] The abstract states that the monodromy matrix is given 'for what we believe to be the first time'; a brief comparison with prior explicit expressions for K=2 would clarify the novelty.
  2. [Review of coadjoint-orbit geometry] Notation for the Lie-Poisson bracket and the matrix J should be introduced once with an explicit definition before being used in the verification.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and insightful comments on our manuscript. We address each major comment below and will revise the manuscript accordingly where appropriate.

read point-by-point responses

  1. Referee: [Section describing the Spin-MInt update rules and the subsequent proof of MJM^T = J] The central algebraic verification of MJM^T = J presupposes that the Spin-MInt update rules map the coadjoint orbit to itself exactly. No separate demonstration is supplied that the discrete flow remains on the orbit for arbitrary K (or that any O(Δt) drift vanishes identically). Without this step the cancellation that produces the symplecticity identity is not guaranteed to hold for the actual numerical map.

    Authors: We agree with the referee that an explicit demonstration of orbit preservation by the discrete Spin-MInt map is necessary to rigorously justify the symplecticity proof for general K. While the update rules are designed based on the Lie-Poisson structure, which typically preserves the orbit, we did not provide a separate verification in the original manuscript. In the revised version, we will include a proof that the Spin-MInt updates preserve the Casimir functions of the su(K) algebra, thereby ensuring the numerical flow remains exactly on the coadjoint orbit. This addition will strengthen the foundation of our algebraic verification. revision: yes

  2. Referee: [Section introducing the mapping to canonical coordinates] The canonical coordinate chart on the coherent-state manifold is stated to be valid for general K, yet the paper does not address whether the chart remains regular for all points on the orbit when K > 2 or whether the Poisson tensor is faithfully represented everywhere the integrator is applied.

    Authors: The referee correctly notes that we did not explicitly discuss the regularity of the canonical coordinate chart for K > 2. The chart is constructed from the standard parametrization of the coherent states on the su(K) orbit, and it is regular almost everywhere on the orbit (with singularities at a set of measure zero). We will add a paragraph clarifying the domain of validity of these coordinates and confirming that the Poisson tensor is faithfully represented in the regions relevant to the integrator's application, ensuring the symplecticity condition holds where the map is defined. revision: yes

Circularity Check

0 steps flagged

No significant circularity; explicit algebraic verification stands independently.

full rationale

The paper's core claim is a direct, explicit verification of the symplecticity condition MJM^T = J for general K by exploiting the Lie-Poisson structure of su(K) coadjoint orbits after mapping to canonical coordinates. This is presented as an algebraic proof extending a prior result limited to K=2, with no reduction of the result to fitted parameters, self-definitional mappings, or load-bearing self-citations whose validity is presupposed without external support. The derivation chain consists of reviewing the symplectic geometry of the algebra and then performing the matrix verification; neither step collapses to its inputs by construction. The result is therefore self-contained against external benchmarks such as the Lie-Poisson framework itself.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper is a pure mathematical proof with no fitted parameters or new postulated entities; it rests on standard properties of the su(K) Lie-Poisson algebra and the definition of the Spin-MInt update rules.

axioms (1)
  • domain assumption Spin-mapping variables obey the Lie-Poisson algebra of su(K) and evolve on its coadjoint orbit
    This is the foundational geometric structure invoked to frame the symplecticity proof.

pith-pipeline@v0.9.1-grok · 5845 in / 1259 out tokens · 27877 ms · 2026-07-03T03:58:22.757330+00:00 · methodology

discussion (0)

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

Reference graph

Works this paper leans on

121 extracted references · 89 canonical work pages · 4 internal anchors

  1. [1]

    Zhu, X.-Y., Podzorov, V.: Charge Carriers in Hybrid Organic–Inorganic Lead Halide Perovskites Might Be Protected as Large Polarons. J. Phys. Chem. Lett. 6(23), 4758–4761 (2015) https://doi.org/10.1021/acs.jpclett.5b02462

    work page doi:10.1021/acs.jpclett.5b02462 2015

  2. [2]

    Cheng, Y.-C., Fleming, G.R.: Dynamics of Light Harvesting in Photosynthe- sis. Annu. Rev. Phys. Chem.60(1), 241–262 (2009) https://doi.org/10.1146/ annurev.physchem.040808.090259

    work page arXiv 2009

  3. [3]

    Marcus, R.A., Sutin, N.: Electron transfers in chemistry and biology. Biochim. Biophys. Acta, Rev. Bioenerg.811(3), 265–322 (1985) https://doi.org/10.1016/ 0304-4173(85)90014-X

    1985

  4. [4]

    Runeson, J.E., Fay, T.P., Manolopoulos, D.E.: Exciton dynamics from the mapping approach to surface hopping: comparison with F¨ orster and Redfield theories. Phys. Chem. Chem. Phys.26(6), 4929–4938 (2024) https://doi.org/10. 1039/D3CP05926J

    2024

  5. [5]

    M¨ uh, F., Renger, T.: Refined structure-based simulation of plant light-harvesting complex II: Linear optical spectra of trimers and aggregates. Biochim. Biophys. Acta, Bioenerg.1817(8), 1446–1460 (2012) https://doi.org/10.1016/j.bbabio. 2012.02.016

    work page doi:10.1016/j.bbabio 2012

  6. [6]

    Domcke, W., Yarkony, D.R.: Role of Conical Intersections in Molecular Spec- troscopy and Photoinduced Chemical Dynamics. Annu. Rev. Phys. Chem.63(1), 325–352 (2012) https://doi.org/10.1146/annurev-physchem-032210-103522

    work page doi:10.1146/annurev-physchem-032210-103522 2012

  7. [7]

    Hammes-Schiffer, S., Stuchebrukhov, A.A.: Theory of Coupled Electron and Proton Transfer Reactions. Chem. Rev.110(12), 6939–6960 (2010) https://doi. org/10.1021/cr1001436

    work page doi:10.1021/cr1001436 2010

  8. [8]

    Beck, M.: The multiconfiguration time-dependent Hartree (MCTDH) method: a highly efficient algorithm for propagating wavepackets. Phys. Rep.324(1), 1–105 (2000) https://doi.org/10.1016/S0370-1573(99)00047-2

    work page doi:10.1016/s0370-1573(99)00047-2 2000

  9. [9]

    Shin, S., Metiu, H.: Multiple Time Scale Quantum Wavepacket Propagation: Electron-Nuclear Dynamics. J. Phys. Chem.100(19), 7867–7872 (1996) https: //doi.org/10.1021/jp952498a 35

    work page doi:10.1021/jp952498a 1996

  10. [10]

    Wang, H., Thoss, M.: Multilayer formulation of the multiconfiguration time- dependent Hartree theory. J. Chem. Phys.119(3), 1289–1299 (2003) https:// doi.org/10.1063/1.1580111

    work page doi:10.1063/1.1580111 2003

  11. [11]

    Mukherjee, S., Lassmann, Y., Mattos, R.S., Demoulin, B., Curchod, B.F.E., Barbatti, M.: Assessing Nonadiabatic Dynamics Methods in Long Timescales. J. Chem. Theory Comput.21(1), 29–37 (2025) https://doi.org/10.1021/acs.jctc. 4c01349

    work page doi:10.1021/acs.jctc 2025

  12. [13]

    Ananth, N.: Mapping Variable Ring Polymer Molecular Dynamics: A Path- Integral Based Method for Nonadiabatic Processes. J. Chem. Phys.139(12), 124102 (2013) https://doi.org/10.1063/1.4821590

    work page doi:10.1063/1.4821590 2013

  13. [14]

    Richardson, J.O., Thoss, M.: Communication: Nonadiabatic ring-polymer molec- ular dynamics. J. Chem. Phys.139(3), 031102 (2013) https://doi.org/10.1063/ 1.4816124

    2013

  14. [15]

    An Electronically Non-Adiabatic Generalization of Ring Polymer Molecular Dynamics

    Hele, T.J.H.: An Electronically Non-adiabatic Generalization of Ring Poly- mer Molecular Dynamics. MChem thesis, Exeter College, University of Oxford, Oxford (2011). https://doi.org/10.48550/arXiv.1308.3950

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.1308.3950 2011

  15. [16]

    Matsubara dynam- ics

    Hele, T.J.H., Willatt, M.J., Muolo, A., Althorpe, S.C.: Boltzmann-conserving classical dynamics in quantum time-correlation functions: “Matsubara dynam- ics”. J. Chem. Phys.142(13), 134103 (2015) https://doi.org/10.1063/1.4916311

    work page doi:10.1063/1.4916311 2015

  16. [17]

    Chowdhury, S.N., Huo, P.: Non-adiabatic matsubara dynamics and non- adiabatic ring-polymer molecular dynamics. J. Chem. Phys.154(12), 124124 (2021) https://doi.org/10.1063/5.0042136

    work page doi:10.1063/5.0042136 2021

  17. [18]

    Van Vleck, J.H.: The Correspondence Principle in the Statistical Interpretation of Quantum Mechanics. Proc. Natl. Acad. Sci. U.S.A.14(2), 178–188 (1928) https://doi.org/10.1073/pnas.14.2.178

    work page doi:10.1073/pnas.14.2.178 1928

  18. [19]

    Herman, M.F., Kluk, E.: A semiclasical justification for the use of non-spreading wavepackets in dynamics calculations. Chem. Phys.91(1), 27–34 (1984) https: //doi.org/10.1016/0301-0104(84)80039-7

    work page doi:10.1016/0301-0104(84)80039-7 1984

  19. [20]

    Miller, W.H.: The Semiclassical Initial Value Representation: A Potentially Practical Way for Adding Quantum Effects to Classical Molecular Dynamics Simulations. J. Phys. Chem. A105(13), 2942–2955 (2001) https://doi.org/10. 1021/jp003712k 36

    2001

  20. [21]

    Church, M.S., Hele, T.J.H., Ezra, G.S., Ananth, N.: Nonadiabatic semiclassical dynamics in the mixed quantum-classical initial value representation. J. Chem. Phys.148(10) (2018) https://doi.org/10.1063/1.5005557

    work page doi:10.1063/1.5005557 2018

  21. [22]

    Kay, K.G.: The Herman–Kluk approximation: Derivation and semiclassical corrections. Chem. Phys.322(1-2), 3–12 (2006) https://doi.org/10.1016/j. chemphys.2005.06.019

    work page doi:10.1016/j 2006

  22. [23]

    Sun, X., Miller, W.H.: Forward–backward initial value representation for semi- classical time correlation functions. J. Chem. Phys.110(14), 6635–6644 (1999) https://doi.org/10.1063/1.478571

    work page doi:10.1063/1.478571 1999

  23. [24]

    Atsango, A.O., Montoya-Castillo, A., Markland, T.E.: An accurate and efficient Ehrenfest dynamics approach for calculating linear and nonlinear electronic spec- tra. J. Chem. Phys.158(7), 074107 (2023) https://doi.org/10.1063/5.0138671

    work page doi:10.1063/5.0138671 2023

  24. [25]

    Faraday Discuss.153, 105 (2011) https: //doi.org/10.1039/c1fd00034a

    Shalashilin, D.V.: Multiconfigurational Ehrenfest approach to quantum coherent dynamics in large molecular systems. Faraday Discuss.153, 105 (2011) https: //doi.org/10.1039/c1fd00034a

    work page doi:10.1039/c1fd00034a 2011

  25. [26]

    Tully, J.C.: Perspective: Nonadiabatic dynamics theory. J. Chem. Phys.137(22), 22–301 (2012) https://doi.org/10.1063/1.4757762

    work page doi:10.1063/1.4757762 2012

  26. [27]

    Zimmermann, T., Van´ ıˇ cek, J.: Efficient on-the-flyab initiosemiclassical method for computing time-resolved nonadiabatic electronic spectra with surface hop- ping or Ehrenfest dynamics. J. Chem. Phys.141(13), 134102 (2014) https: //doi.org/10.1063/1.4896735

    work page doi:10.1063/1.4896735 2014

  27. [28]

    Lieberherr, A.Z., Kelly, J., Runeson, J.E., Markland, T.E., Manolopoulos, D.E.: Two-dimensional electronic spectra from trajectory-based dynamics: Pure-state Ehrenfest, spin-mapping, and mean classical path approaches. J. Chem. Phys. 163(21), 214111 (2025) https://doi.org/10.1063/5.0299204

    work page doi:10.1063/5.0299204 2025

  28. [29]

    Kapral, R.: Surface hopping from the perspective of quantum–classical Liouville dynamics. Chem. Phys.481, 77–83 (2016) https://doi.org/10.1016/j.chemphys. 2016.05.016

    work page doi:10.1016/j.chemphys 2016

  29. [30]

    Mannouch, J.R., Richardson, J.O.: A mapping approach to surface hopping. J. Chem. Phys.158(10), 104111 (2023) https://doi.org/10.1063/5.0139734

    work page doi:10.1063/5.0139734 2023

  30. [31]

    Shakib, F.A., Huo, P.: Ring Polymer Surface Hopping: Incorporating Nuclear Quantum Effects into Nonadiabatic Molecular Dynamics Simulations. J. Phys. Chem. Lett.8(13), 3073–3080 (2017) https://doi.org/10.1021/acs.jpclett. 7b01343

    work page doi:10.1021/acs.jpclett 2017

  31. [32]

    Tully, J.C., Preston, R.K.: Trajectory Surface Hopping Approach to Nonadia- batic Molecular Collisions: The Reaction of H+ with D2. J. Chem. Phys.55(2), 37 562–572 (1971) https://doi.org/10.1063/1.1675788

    work page doi:10.1063/1.1675788 1971

  32. [33]

    Meyer, H.-D., Miller, W.H.: A classical analog for electronic degrees of freedom in nonadiabatic collision processes. J. Chem. Phys.70(7), 3214–3223 (1979) https://doi.org/10.1063/1.437910

    work page doi:10.1063/1.437910 1979

  33. [34]

    131, 1st edn., pp

    Stock, G., Thoss, M.: Classical Description of Nonadiabatic Quantum Dynamics vol. 131, 1st edn., pp. 243–375. Wiley, Chichester (2005). https://doi.org/10. 1002/0471739464.ch5

    2005

  34. [35]

    Bossion, D., Chowdhury, S.N., Huo, P.: Non-Adiabatic Ring Polymer Molecular Dynamics with Spin Mapping Variables. J. Chem. Phys.154(18) (2021) https: //doi.org/10.1063/5.0051456

    work page doi:10.1063/5.0051456 2021

  35. [36]

    Stock, G., Thoss, M.: Semiclassical Description of Nonadiabatic Quantum Dynamics. Phys. Rev. Lett.78(4), 578–581 (1997) https://doi.org/10.1103/ PhysRevLett.78.578

    1997

  36. [39]

    Cook, L.E., Rampton, J.R., Hele, T.J.H.: The Spin-MInt algorithm: An accurate and symplectic propagator for the spin-mapping representation of nonadiabatic dynamics. J. Chem. Phys.164(14), 144112 (2026) https://doi.org/10.1063/5. 0314688

    work page doi:10.1063/5 2026

  37. [40]

    Cook, L.E., Runeson, J.E., Richardson, J.O., Hele, T.J.H.: Which Algorithm Best Propagates the Meyer–Miller–Stock–Thoss Mapping Hamiltonian for Non- Adiabatic Dynamics? J. Chem. Theory Comput.19(18), 6109–6125 (2023) https: //doi.org/10.1021/acs.jctc.3c00709

    work page doi:10.1021/acs.jctc.3c00709 2023

  38. [41]

    Thoss, M., Stock, G.: Mapping approach to the semiclassical description of nonadiabatic quantum dynamics. Phys. Rev. A59(1), 64–79 (1999) https: //doi.org/10.1103/PhysRevA.59.64

    work page doi:10.1103/physreva.59.64 1999

  39. [42]

    Dover books on physics

    Feynman, R.P., Hibbs, A.R., Styer, D.F.: Quantum Mechanics and Path Inte- grals, Emended edn. Dover books on physics. Dover publications, Mineola (2010)

    2010

  40. [43]

    Miller, W.H.: Classical S Matrix: Numerical Application to Inelastic Collisions. J. Chem. Phys.53(9), 3578–3587 (1970) https://doi.org/10.1063/1.1674535 38

    work page doi:10.1063/1.1674535 1970

  41. [44]

    Gutzwiller, M.C.: Periodic Orbits and Classical Quantization Conditions. J. Math. Phys.12(3), 343–358 (1971) https://doi.org/10.1063/1.1665596

    work page doi:10.1063/1.1665596 1971

  42. [45]

    Faraday Discuss.195, 269–289 (2016) https://doi.org/10.1039/C6FD00106H

    Hele, T.J.H., Ananth, N.: Deriving the exact nonadiabatic quantum propagator in the mapping variable representation. Faraday Discuss.195, 269–289 (2016) https://doi.org/10.1039/C6FD00106H

    work page doi:10.1039/c6fd00106h 2016

  43. [46]

    Faraday Discuss.195, 311–344 (2016) https://doi.org/10.1039/C6FD90078J

    Althorpe, S.C., Ananth, N., Angulo, G., Astumian, R.D., Beniwal, V., Blum- berger, J., Bolhuis, P.G., Ensing, B., Glowacki, D.R., Habershon, S., Hammes- Schiffer, S., Hele, T.J.H., Makri, N., Manolopoulos, D.E., McKemmish, L.K., Miller Iii, T.F., Miller, W.H., Mulholland, A.J., Nekipelova, T., Pollak, E., Richardson, J.O., Richter, M., Roy Chowdhury, P., ...

    work page doi:10.1039/c6fd90078j 2016

  44. [47]

    Ananth, N., Venkataraman, C., Miller, W.H.: Semiclassical description of elec- tronically nonadiabatic dynamics via the initial value representation. J. Chem. Phys.127(8), 084114 (2007) https://doi.org/10.1063/1.2759932

    work page doi:10.1063/1.2759932 2007

  45. [48]

    Miller, W.H.: Electronically Nonadiabatic Dynamics via Semiclassical Initial Value Methods. J. Phys. Chem. A113(8), 1405–1415 (2009) https://doi.org/ 10.1021/jp809907p

    work page doi:10.1021/jp809907p 2009

  46. [49]

    Sun, X., Wang, H., Miller, W.H.: Semiclassical theory of electronically nona- diabatic dynamics: Results of a linearized approximation to the initial value representation. J. Chem. Phys.109(17), 7064–7074 (1998) https://doi.org/10. 1063/1.477389

    1998

  47. [50]

    Sun, X., Wang, H., Miller, W.H.: On the semiclassical description of quantum coherence in thermal rate constants. J. Chem. Phys.109(11), 4190–4200 (1998) https://doi.org/10.1063/1.477025

    work page doi:10.1063/1.477025 1998

  48. [52]

    Moscato, D., Gandolfi, M., Ceotto, M.: A time averaged semiclassical approach to the computation of nonadiabatic vibronic absorption spectra. J. Chem. Phys. 162(23), 234108 (2025) https://doi.org/10.1063/5.0276220

    work page doi:10.1063/5.0276220 2025

  49. [53]

    Craig, I.R., Manolopoulos, D.E.: Quantum statistics and classical mechanics: Real time correlation functions from ring polymer molecular dynamics. J. Chem. Phys.121(8), 3368–3373 (2004) https://doi.org/10.1063/1.1777575

    work page doi:10.1063/1.1777575 2004

  50. [54]

    Cook, L.E., Hele, T.J.H.: On the electronic path integral normal modes of the Meyer–Miller–Stock–Thoss representation of nonadiabatic dynamics. J. Chem. 39 Phys.163(17), 174113 (2025) https://doi.org/10.1063/5.0297571

    work page doi:10.1063/5.0297571 2025

  51. [55]

    Richardson, J.O., Meyer, P., Pleinert, M.-O., Thoss, M.: An analysis of nonadi- abatic ring-polymer molecular dynamics and its application to vibronic spectra. Chem. Phys.482, 124–134 (2017) https://doi.org/10.1016/j.chemphys.2016.09. 036

    work page doi:10.1016/j.chemphys.2016.09 2017

  52. [56]

    Richardson, J.O., Lawrence, J.E., Mannouch, J.R.: Nonadiabatic dynam- ics with the mapping approach to surface hopping (MASH). Annu. Rev. Phys. Chem.76(1), 663–687 (2025) https://doi.org/10.1146/ annurev-physchem-082423-120631

    2025

  53. [58]

    Runeson, J.E., Richardson, J.O.: Quantum Entanglement from Classical Tra- jectories. Phys. Rev. Lett.127(25), 250403 (2021) https://doi.org/10.1103/ PhysRevLett.127.250403

    2021

  54. [59]

    Bossion, D., Ying, W., Chowdhury, S.N., Huo, P.: Non-adiabatic mapping dynamics in the phase space of theSU(N) lie group. J. Chem. Phys.157(8) (2022) https://doi.org/10.1063/5.0094893

    work page doi:10.1063/5.0094893 2022

  55. [60]

    Mannouch, J.R., Richardson, J.O.: A partially linearized spin-mapping approach for nonadiabatic dynamics. II. Analysis and comparison with related approaches. J. Chem. Phys.153(19), 194110 (2020) https://doi.org/10.1063/5.0031173

    work page doi:10.1063/5.0031173 2020

  56. [61]

    Bengtsson, I., ˙Zyczkowski, K.: Geometry of Quantum States: An Introduction to Quantum Entanglement, 2nd edn., pp. 103–137. Cambridge University Press, Cambridge (2017). https://doi.org/10.1017/CBO9780511535048

    work page doi:10.1017/cbo9780511535048 2017

  57. [62]

    Oh, P., Kim, M.-H.: Action-angle variables for complex projective space and semiclassical exactness. Mod. Phys. Lett. A09(36), 3339–3346 (1994) https: //doi.org/10.1142/S0217732394003166

    work page doi:10.1142/s0217732394003166 1994

  58. [63]

    Cambridge University Press, Cambridge (2005)

    Leimkuhler, B., Reich, S.: Simulating Hamiltonian Dynamics, 1st edn. Cambridge University Press, Cambridge (2005). https://doi.org/10.1017/ CBO9780511614118

    2005

  59. [64]

    Kirillov, A.A.: Unitary representations of nilpotent lie groups. Russ. Math. Surv. 17(4), 53–104 (1962) https://doi.org/10.1070/RM1962v017n04ABEH004118

    work page doi:10.1070/rm1962v017n04abeh004118 1962

  60. [65]

    In: Taam, C.T

    Kostant, B.: Quantization and unitary representations. In: Taam, C.T. (ed.) Lectures in Modern Analysis and Applications III vol. 170, pp. 87–208. Springer, Berlin, Heidelberg (1970). https://doi.org/10.1007/BFb0079068 40

    work page doi:10.1007/bfb0079068 1970

  61. [66]

    Souriau, J.-M.: Structure of Dynamical Systems vol. 149, pp. 111–117. Springer, Boston (1997)

    1997

  62. [67]

    BIT Numerical Mathematics35(4), 572–587 (1995) https://doi.org/10.1007/BF01739828

    Munthe-Kaas, H.: Lie-Butcher theory for Runge-Kutta methods. BIT Numerical Mathematics35(4), 572–587 (1995) https://doi.org/10.1007/BF01739828

    work page doi:10.1007/bf01739828 1995

  63. [68]

    In: Cucker, F., Shub, M

    Zanna, A., Munthe-Kaas, H.: Iterated Commutators, Lie’s Reduction Method and Ordinary Differential Equations on Matrix Lie Groups. In: Cucker, F., Shub, M. (eds.) Foundations of Computational Mathematics, pp. 434–441. Springer, Berlin, Heidelberg (1997). https://doi.org/10.1007/978-3-642-60539-0 36

    work page doi:10.1007/978-3-642-60539-0 1997

  64. [69]

    BIT Numerical Math- ematics38(1), 92–111 (1998) https://doi.org/10.1007/BF02510919

    Munthe-Kaas, H.: Runge-Kutta methods on Lie groups. BIT Numerical Math- ematics38(1), 92–111 (1998) https://doi.org/10.1007/BF02510919

    work page doi:10.1007/bf02510919 1998

  65. [70]

    Munthe-Kaas, H.: High order Runge-Kutta methods on manifolds. Appl. Numer. Math.29(1), 115–127 (1999) https://doi.org/10.1016/S0168-9274(98)00030-0

    work page doi:10.1016/s0168-9274(98)00030-0 1999

  66. [71]

    Acta Numer.9, 215–365 (2000) https://doi.org/10.1017/S0962492900002154

    Iserles, A., Munthe-Kaas, H.Z., Nørsett, S.P., Zanna, A.: Lie-group methods. Acta Numer.9, 215–365 (2000) https://doi.org/10.1017/S0962492900002154

    work page doi:10.1017/s0962492900002154 2000

  67. [72]

    Lasagni, F.M.: Canonical Runge-Kutta methods. Z. Angew. Math. Phys.39(6), 952–953 (1988) https://doi.org/10.1007/BF00945133

    work page doi:10.1007/bf00945133 1988

  68. [73]

    BIT Numer

    Sanz-Serna, J.M.: Runge-kutta schemes for Hamiltonian systems. BIT Numer. Math.28(4), 877–883 (1988) https://doi.org/10.1007/BF01954907

    work page doi:10.1007/bf01954907 1988

  69. [74]

    Suris, Y.B.: The canonicity of mappings generated by runge-kutta type methods when integrating the systems. Comput. Math. Math. Phys.29(1), 138–144 https: //doi.org/10.1016/0041-5553(89)90058-X

    work page doi:10.1016/0041-5553(89)90058-x

  70. [75]

    Sakurai, J.J., Napolitano, J.: Modern Quantum Mechanics, 3rd edn., pp. 149–

  71. [76]

    https://doi.org/10.1017/ 9781108587280

    Cambridge University Press, Cambridge (2020). https://doi.org/10.1017/ 9781108587280

    2020

  72. [77]

    Bertlmann, R.A., Krammer, P.: Bloch vectors for qudits. J. Phys. A: Math. Theor.41(23), 235303 (2008) https://doi.org/10.1088/1751-8113/41/23/ 235303

    work page doi:10.1088/1751-8113/41/23/ 2008

  73. [78]

    Fano, U., Rau, A.R.P.: Symmetries in Quantum Physics, 1st edn., pp. 127–156. Academic Press, San Diego (1996)

    1996

  74. [79]

    In: Wightman, A.S

    Wigner, E.P.: Einige Folgerungen aus der Schr¨ odingerschen Theorie f¨ ur die Termstrukturen. In: Wightman, A.S. (ed.) The Collected Works of Eugene Paul Wigner, pp. 53–81. Springer, Berlin, Heidelberg (1993). https://doi.org/10.1007/ 978-3-662-02781-3 5

    1993

  75. [80]

    M¨ uller, U., Stock, G.: Flow of zero-point energy and exploration of phase space in 41 classical simulations of quantum relaxation dynamics. II. application to nonadi- abatic processes. J. Chem. Phys.111(1), 77–88 (1999) https://doi.org/10.1063/ 1.479255

    1999

  76. [81]

    Cotton, S.J., Miller, W.H.: Trajectory-adjusted electronic zero point energy in classical meyer-miller vibronic dynamics: Symmetrical quasiclassical application to photodissociation. J. Chem. Phys.150(19), 194110 (2019) https://doi.org/ 10.1063/1.5094458

    work page doi:10.1063/1.5094458 2019

  77. [82]

    Saller, M.A.C., Kelly, A., Richardson, J.O.: On the identity of the identity oper- ator in nonadiabatic linearized semiclassical dynamics. J. Chem. Phys.150(7), 071101 https://doi.org/10.1063/1.5082596

    work page doi:10.1063/1.5082596

  78. [83]

    Graduate Texts in Mathe- matics, vol

    Fulton, W., Harris, J.: Representation Theory. Graduate Texts in Mathe- matics, vol. 129. Springer, New York, NY (2004). https://doi.org/10.1007/ 978-1-4612-0979-9

    2004

  79. [84]

    , TITLE =

    Hall, B.C.: Lie Groups, Lie Algebras, and Representations: An Elementary Intro- duction. Graduate Texts in Mathematics, vol. 222, pp. 109–140. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-13467-3

    work page doi:10.1007/978-3-319-13467-3 2015

  80. [85]

    Graduate Texts in Mathematics, vol

    Humphreys, J.E.: Introduction to Lie Algebras and Representation Theory. Graduate Texts in Mathematics, vol. 9, pp. 107–144. Springer, New York, NY (1972). https://doi.org/10.1007/978-1-4612-6398-2

    work page doi:10.1007/978-1-4612-6398-2 1972

Showing first 80 references.