Pith. sign in

REVIEW 3 major objections 5 minor 133 references

Quantum Dynamics of Dissipative Polarizable Media

T0 review · 3 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper shows that the classical equation of motion for dissipative polarizable media can be quantized exactly as a finite quadratic Hamiltonian, reproducing the classical dynamics as Ehrenfest means and yielding a master equation for…

desk verdict A serious non-Hermitian quantization of classical polarizable media, but the derived master equation lacks a complete-positivity proof and the applications re-derive known spectra. read the letter →

arxiv 2501.12070 v2 pith:HBFUUC4B submitted 2025-01-21 quant-ph cond-mat.mtrl-scicond-mat.other

Frank Ernesto Quintela Rodriguez , Piero Lafiosca , Tommaso Giovannini , Chiara Cappelli This is my paper · ORCID
classification quant-phcond-mat.mtrl-scicond-mat.other
keywords quantumpolarizablemediumdampedharmonicoscillatorpseudo-bosonsopensystemsmasterequationplasmonicspolarizabilityGaussianstates
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper sets out to show that the dissipative polarizable media used throughout computational chemistry and plasmonics, normally treated as classical equations of motion with friction, are the mean-field limit of a genuine quantum system with the same finite number of degrees of freedom. It constructs a quadratic Hamiltonian in an extended phase space, quantizes it under the condition that expectation values follow the classical damped equation, and calls the result the Quantum Polarizable Medium (QPM). From there it derives a spectral decomposition of the medium's polarizability, a self-consistent semiclassical equation for the re-emitted electric field, and a master equation for a quantum emitter embedded in the medium, with correlation functions valid for arbitrary Gaussian states. If these derivations are right, standard polarizable models inherit a fully quantum description of decoherence and energy exchange without invoking an infinite bath or external noise.

What carries the argument

The load-bearing object is the matrix square root of the extended dynamical matrix, $\sqrt{\mathcal{K}} = i\begin{pmatrix}0&-I\\K&2\Gamma\end{pmatrix}$, whose eigenvalues coincide, up to a possible null mode, with the poles of the polarizability kernel $(\omega^2 + 2i\omega\Gamma - K)^{-1}$. This matrix converts the second-order damped equation into a first-order system and supplies the spectrum used to decompose absorption and dispersion. On top of it the paper builds a quadratic Hamiltonian $H = \frac{1}{2}\pi^T A\pi + \frac{1}{2}x^T A^{-1}\mathcal{K}x + x^T A^{-1}F(t)$, with $A$ the symmetric matrix realizing $\mathcal{K} = A\mathcal{K}^T A^{-1}$; quantization of $x,\pi$ under the canonical commutation relation makes the classical equation the Ehrenfest limit. Pseudo-boson operators $b,\tilde b$ diagonalize $H_0$ and generate bi-coherent states, while the symplectic evolution $\Lambda_t = e^{JBt}$ in phase space carries the Gaussian-state calculations that lead to the self-consistent field equation and the correlation functions.

What would settle it

Compute the rate matrix $\gamma_{\alpha\beta}(\omega)$ from Eq. (98) for the simplest QPM, a single damped harmonic oscillator with $\Gamma>0$ in a thermal state, and check whether $\gamma(\omega)$ is positive semidefinite for all $\omega$; a negative eigenvalue anywhere would show that the derived dissipator is not of the standard completely positive form, so the master equation could not describe a physical open system.

Watch

Extended reading notes

Core claim

The central claim is that the damped oscillator equation $\ddot u + 2\Gamma\dot u + Ku + f=0$, which underlies fluctuating-charge, fluctuating-dipole, discrete-interaction, and continuum polarizable models, can be embedded in an extended phase space whose quadratic Hamiltonian $H = \frac{1}{2}\pi^T A\pi + \frac{1}{2}x^T A^{-1}\mathcal{K}x + x^T A^{-1}F(t)$ is finite and time-independent when no external field acts. Quantizing this Hamiltonian under the canonical commutation relations defines the Quantum Polarizable Medium; the classical damped equation is recovered as the Ehrenfest mean-value dynamics, and the time-independent part $H_0$ is a constant of motion that is identically zero on-shell, so the isolated medium's energy comes only from the interaction term. The spectrum of the extended matrix $\sqrt{\mathcal{K}} = i\begin{pmatrix}0&-I\\K&2\Gamma\end{pmatrix}$ is shown to coincide with the poles of the polarizability kernel $(\omega^2 + 2i\omega\Gamma - K)^{-1}$, and $H_0$ is diagonalized by pseudo-boson operators whose bi-coherent states evolve by a simple phase-space rule. On this basis the paper obtains a self-consistent semiclassical equation for the emitted electric field and a Markovian master equation for an embedded quantum system, with correlation functions computed over arbitrary Gaussian states.

Load-bearing premise

The argument rests on the assumption that the standard open-quantum-system master-equation derivation remains valid when the environment Hamiltonian is non-Hermitian; if the correlation rates $\gamma_{\alpha\beta}(\omega)$ from Eq. (98) can be negative, the master equation is not completely positive and would not describe a physical system.

Editorial extensions

If this is right

  • Any classical polarizable medium of the form $\ddot u + 2\Gamma\dot u + Ku + f=0$ acquires a finite-dimensional quantum counterpart, so dissipation no longer requires an infinite bath or an added noise term.
  • The polarizability of a plasmonic structure decomposes into absorptive and dispersive eigenmode contributions of $\sqrt{\mathcal{K}}$; the intercept filter reconstructs the main spectral features with very few modes, up to 4 for the regular graphene disk.
  • The first-order solution of the self-consistent field equation expresses electric-field enhancement as a sum of Gaussian modes in $k$-space weighted by the resonances of $(\omega E + iJB)^{-1}$, connecting the medium's quantum spectrum to near-field enhancement.
  • A quantum emitter coupled to the QPM obeys the derived master equation, with rates determined by the $K$ and $\Gamma$ matrices and correlation functions for thermal or arbitrary Gaussian states.
  • Because $H_0$ vanishes on-shell, the average energy of the isolated medium is exactly $E(t)=\mathrm{tr}\{H_1(t)\rho\}$; only the interaction with external fields contributes to the energetics.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The paper does not check whether the rate matrix $\gamma_{\alpha\beta}(\omega)$ is positive semidefinite; a single-oscillator calculation would settle whether the derived master equation is physically admissible, a question the authors leave open.
  • A direct extension would use the same finite-dimensional non-Hermitian Hamiltonian as a reservoir in numerically exact methods such as quantum trajectories or hierarchical equations of motion, rather than stopping at the Markovian master equation; this would test the Markovian approximation when the environment itself has non-Hermitian spectral features.
  • The intercept filter suggests a practical model-order reduction: keeping only the eigenpairs with the largest intercept coefficients should reproduce time-domain polarization dynamics with far fewer variables than the full $\sqrt{\mathcal{K}}$ matrix, a numerical strategy the paper illustrates but does not propose as a general algorithm.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The manuscript develops a Hamiltonian and quantum-mechanical formulation of dissipative classical polarizable media governed by Eq. (1). It introduces auxiliary variables v=u˙, a matrix √K whose nonzero spectrum matches the poles of (ω^2+2iωΓ−K)^{-1}, and a quadratic Hamiltonian (Eq. 11) that reproduces Eq. (1) as Ehrenfest mean dynamics after canonical quantization. The resulting Quantum Polarizable Medium (QPM) is analyzed via pseudo-boson operators and bi-coherent states (Sec. II.C), applied to reproduce the polarizability spectra of Ag147 and graphene-disk nanostructures using eigenpair filtering (Sec. III.B), and used to derive a self-consistent semiclassical field equation (Eq. 84) and a Born-Markov master equation for a quantum system embedded in the QPM (Eqs. 95-98), with correlation functions for Gaussian states (Eqs. 104-110). The paper also derives algebraic decompositions of the polarizability in terms of the eigenvalues and eigenvectors of √K.

Significance. The paper is ambitious and addresses a genuine need: putting classical polarizable embedding models on a quantum footing for open-system simulations. The spectral decomposition of the polarizability in terms of √K is original and potentially useful, and the phase-space treatment of quadratic non-Hermitian Hamiltonians is carefully developed in the appendices. The authors are explicit about several limitations, including the unresolved gauge dependence of the interaction term and the loss of Hermitian correlation-function symmetries. However, the central open-system result, the master equation, rests on an unproven extension of the standard GKSL derivation to a non-Hermitian finite bath, and the thermal-state covariance formula is used without proof of validity for the indefinite B that appears here. These gaps are load-bearing rather than presentational. If the positivity of the rates and the validity of the thermal state can be established, the framework would be a substantial contribution to the theory of dissipative quantum polarizable media.

major comments (3)
  1. [Section V.B, Eqs. (95)-(99)] The derivation of the master equation assumes that the standard Born-Markov-Lindblad construction (Ref. [114]) extends to a non-Hermitian environment Hamiltonian H0. The paper defines the rates γαβ(ω)=Ξαβ(ω)+Ξ*βα(ω) in Eq. (98) but does not prove that the matrix γ(ω) is positive semidefinite. For a Hermitian bath, this follows from stationarity and x†(t)=x(t), which makes γ the full Fourier transform of a positive-definite kernel. The paper explicitly states before Sec. V.A that "the correlation functions will lack the symmetries present in the Hermitian case," but it does not address the positivity consequence. Without positivity, the dissipator (96) is not in Lindblad form and can generate non-positive reduced states. Additionally, the QPM has a finite number of modes, so the correlation functions in Eq. (104) are finite sums of exponentials and do not decay; the Markovian limit invoked by the Breuer-Petruccione construction is not justified for such a finite environment, and the "limit η→0+" in Eq. (106) is a distributional identity rather than a physical Markov approximation. This is load-bearing because the master equation is one of the three headline results and the only concrete open-system application.
  2. [Section IV.A, Eq. (107)] The thermal Gaussian state covariance M0 = -ℏ/2 cot(ℏβ J B /2) J^T is used as an input for the correlation functions in Eqs. (108)-(110). In the standard phase-space derivation, this formula requires the quadratic Hamiltonian to be Hermitian and the matrix JB to have appropriate spectral properties so that M0 is a legitimate covariance matrix (positive definite). Here B is only symmetric and JB is generally non-Hermitian, and the paper provides no proof that the coth formula yields a valid Gaussian state with a positive Wigner function (Eq. 61). If M0 fails to be positive definite, the "arbitrary Gaussian states" claim and the correlation functions derived from it are not physically meaningful. A proof of validity under explicit conditions on K and Γ, or a restriction to parameter regimes where the formula is known to hold, is needed.
  3. [Section II.B, Eq. (11)] The Hamiltonian quantization is defined only up to the similarity matrix A, and the paper states that "the situation is, however, less clear for the interaction term H1(t), an issue that will remain open in this work." This unresolved gauge ambiguity propagates into the later results that use H1(t): the self-consistent electric field equation (84) and the open-system interaction Hamiltonian in Sec. V.A (Eq. 88) both depend on the interaction term. Furthermore, the claim that H0 is identically zero on-shell relies on the complex relation π(t)=iA^{-1}√K x(t); for a non-Hermitian H0, the physical meaning of "zero on-shell" as an energy is not established. The authors acknowledge the H1 issue, but it should either be resolved or explicitly stated as a condition restricting the validity of the subsequent derivations.
minor comments (5)
  1. [Section II.A, Eq. (10)] The text describes a "spectral equivalence" between √K and Eq. (1), but Eq. (10) states an inclusion Sp{−√K} ⊆ {ω : det(ω^2+2iωΓ−K)=0} ∪ {0}. Since the applications rely on the eigenvalues of √K for filtering, please clarify whether equality is intended and how zero eigenvalues are handled.
  2. [Section III.B, Fig. 2 caption] The red vertical lines in Fig. 2 are described as the real part of the eigenvalues of √K, but the caption does not state the normalization or the binning/density of the eigenvalue distribution; please clarify.
  3. [Section V.B, Eq. (96)] The dissipator is written with a summation over α,β=1,...,N, but the operators O_α(ω) are defined for α up to 2n=N in Eq. (97); please ensure the index ranges are consistent.
  4. [General] Many key derivations are relegated to the Supplemental Material (e.g., Refs. [83], [98], [105], [111]-[113]); as a standalone journal article, at least the main steps of these derivations should be summarized in the main text or the supplement made openly available.
  5. [Section III.A, Eqs. (36)-(37)] The notation for the tensor product in Eq. (36)-(37) alternates between f(ω)⊗R_s and f(ω)R_s^T; please unify the notation and explicitly define the dimensions of the matrices involved.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the QPM is explicitly constructed to reproduce Eq. (1), and the master equation and field equation are genuine extensions whose non-Hermitian positivity gap is a correctness question, not a circularity.

full rationale

The paper's central construction is transparently definitional rather than circular: 'a quadratic Hamiltonian is proposed and quantized under the criterion that its mean dynamics coincide with Eq. (1), thereby defining the Quantum Polarizable Medium (QPM).' Because this is an explicit construction criterion, the later statement that the QPM reproduces the classical CPM dynamics is a consistency check, not a hidden prediction obtained from the input. The spectral equivalence between sqrt(K) and the classical response kernel A(omega) is proven in Appendix C, not assumed from a cited uniqueness theorem. The applications in Section III recompute the already-known Ag147 and GD10 spectra from the classical response quantities and from parameters taken from earlier works; no parameters are fitted in this paper, so the spectra are validation of the reformulation rather than a fitted-input-called-prediction. The self-consistent field equation (84) is self-consistent by design and follows from Maxwell's equations and exact phase-space expectation values, so the target field is not used as an input to derive itself. The master equation in Section V.B is derived by applying the standard Breuer-Petruccione construction to the QPM bath; the failure to prove complete positivity of gamma(omega) for the non-Hermitian environment is a real validity gap and a correctness risk, but it is not a circularity because the rates are computed from the QPM correlation functions and are not chosen to force the dissipator. The only self-citation that could be load-bearing, Ref. [84], supplies the elementary identity [q,H] = -i hbar J grad H, which is also verified directly in Appendix D, so the argument does not reduce to an unverified self-citation. No step was found in which a claimed prediction is equivalent by construction to an input, a fitted parameter, or a self-citation chain.

Assumptions & free parameters 4 free parameters · 5 assumptions · 2 invented entities

The framework pulls the CPM equations and model parameters from prior literature, adds an auxiliary-variable construction, and quantizes via CCR. The main new assumptions are the validity of non-Hermitian quantization and the extension of standard open-system formulas to a non-Hermitian environment.

free parameters (4)
  • Random displacement amplitude for perturbed nanostructures = 0.5 Å
    Chosen by hand to generate 'random' structures; no rationale or statistics over realizations.
  • Eigenvalue Filter frequency ranges = e.g., 2-5 eV; ranges centered at plasmon peaks
    Thresholds chosen to isolate eigenpairs; convergence is non-uniform for Ag147.
  • Intercept Filter thresholds = e.g., 0.01 for GD10; other thresholds not all listed
    Thresholds chosen to select eigenpairs; affects reconstruction quality.
  • ωFQFμ and ωFQ model parameters (from Refs [12,47,26]) = Not re-fit here
    Inputs from prior literature; the framework treats them as given matrices K and Γ.
assumptions (5)
  • domain assumption The CPM equation of motion, Eq (1), is an accurate model of dissipative polarizable media.
    Starting point of the whole framework; taken from classical polarizable models.
  • standard math For every matrix K, there exists a symmetric invertible A such that K = A K^T A^{-1} (Taussky-Zassenhaus theorem).
    Used to define the Hamiltonian in Eq (11) and the B matrix; construction via Jordan form in Appendix D.
  • domain assumption Canonical quantization with CCR, Eq (16), and Heisenberg equations reproduce Eq (1) as Ehrenfest dynamics.
    This is the quantization criterion; implicitly assumes that non-Hermitian Hamiltonians can be quantized and generate physical dynamics with U^{-1} evolution.
  • ad hoc to paper The thermal Gaussian state covariance formula M0 = -hbar/2 cot(hbar beta J B /2) J^T is valid for the possibly indefinite matrix B.
    Used in Eq (107); no proof of existence or positivity for non-Hermitian or indefinite B.
  • ad hoc to paper The standard Born-Markov master equation derivation (Ref [114]) extends to a non-Hermitian environment with modified correlation functions.
    Section V.B applies the Lindblad-like form while explicitly dropping Hermiticity of x(t); complete positivity is not demonstrated.
invented entities (2)
  • Quantum Polarizable Medium (QPM)
    purpose: A quantized version of the classical polarizable medium, used as a finite environment for open quantum dynamics.
    Defined by construction in Section II.B; no falsifiable observable independent of the framework is proposed.
  • Auxiliary variables v = dot u and the pseudo-boson operators b, b-tilde
    purpose: Lift the second-order CPM equation to a first-order Hamiltonian system and diagonalize H0 via pseudo-bosons.
    Mathematical auxiliaries, not physical degrees of freedom; their dynamical information is fully determined by Eq (1).

how reviews work

0 comments
Cite this review

Pith. "Pith review of Quantum Dynamics of Dissipative Polarizable Media." pith.science (2026). https://pith.science/paper/HBFUUC4B

@misc{pith2026250112070,
  author       = {Pith},
  title        = {Pith review of: Quantum Dynamics of Dissipative Polarizable Media},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HBFUUC4B}},
  note         = {Machine review of arXiv:2501.12070}
}
read the original abstract

Classical polarizable approaches have become the gold standard for simulating complex systems and processes in the condensed phase. These methods describe intrinsically dissipative polarizable media, requiring a formal definition within the framework of open quantum systems. We present a Hamiltonian formulation for the quantum dynamics of polarizable sources based on a generalized theory of the damped harmonic oscillator, using pseudo-boson theory to characterize their coherent state dynamics. We then apply our theory to the study of the optical response of two plasmonic systems. Furthermore, by exploiting the phase space formulation of quantum mechanics and the integrability of quadratic Hamiltonians, we derive a self-consistent relation for the emitted electric field of the polarizable medium under the semiclassical approximation, based on exact formulas for medium polarization. Finally, we derive the master equation describing the open dynamics of a quantum system interacting with the quantum polarizable medium, along with analytical expressions for correlation functions calculated over arbitrary Gaussian states.

Figures

Figures reproduced from arXiv: 2501.12070 by the authors.

Figure 1
Figure 1. FIG. 1: Schematic representation of the [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Simulated optical response of two plasmonic systems: a Silver cluster composed of 147 atoms in an icosahedral [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Reconstruction of the optical response of the [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Reconstruction of the optical response of the [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

133 extracted references · 76 canonical work pages

  1. [114]

    See Supplemental Material (Section VIII) at http:// for the derivation of the differential equation for the external field interacting with the new variables vα(t)

    work page

  2. [1]

    they depend on all the eigenvectors P of K, which is indicative of a collective phenomenon, but are independent of the eigenvalues

  3. [2]

    regular” structure with the highest symmetry and a “random

    they are in a one-to-one correspondence to the eigenvalues, allowing us to use them to “weight” the contributions of λk to the polarization. 7 If the external force acting on the polarization sources is a kick pulse, i.e. f (ω) = f is a frequency-independent constant, the coefficients SA ⃗ sk(ω) and SD ⃗ sk(ω) become a linear function of the frequency ω, ...

  4. [3]

    Intercept Filter (IF), in which the pair ( λk, vk) is selected if | Re Ik| exceeds a given threshold, with Ik given by Eq. (44). The EF is in principle able to select all the relevant contributions of the spectrum to the polarization function in a given frequency range, given the decay properties of Dk(ω) and Ak(ω) (see Eqs. (40b) and (40c)). On the other...

  5. [4]

    L. E. Ratcliff, S. Mohr, G. Huhs, T. Deutsch, M. Masella, and L. Genovese, Challenges in large scale quantum mechanical calculations, Wiley Interdiscip. Rev. Comput. Mol. Sci. 7, e1290 (2017). 16

    2017

  6. [5]

    Catlow, P

    R. Catlow, P. Burke, J. Goodfellow, D. Tildesley, M. Wilson, and K. Morokuma, New challenges in quan- tum chemistry: quests for accurate calculations for large molecular systems, Philos. Trans. Royal Soc. A 360, 1149 (2002)

    2002

  7. [6]

    Sisto, C

    A. Sisto, C. Stross, M. W. van der Kamp, M. O’Connor, S. McIntosh-Smith, G. T. Johnson, E. G. Hohenstein, F. R. Manby, D. R. Glowacki, and T. J. Martinez, Atom- istic non-adiabatic dynamics of the lh2 complex with a gpu-accelerated ab initio exciton model, Phys. Chem. Chem. Phys. 19, 14924 (2017)

    2017

  8. [7]

    Karplus, M

    M. Karplus, M. Levitt, and A. Warshel, https://www.nobelprize.org/prizes/chemistry/ 2013/press-release/ (2013), [Online, accessed by date 12.09.2024]

    2013

Show all 133 references
  1. [8]

    Warshel and M

    A. Warshel and M. Levitt, Theoretical studies of en- zymic reactions: dielectric, electrostatic and steric sta- bilization of the carbonium ion in the reaction of lysozyme, J. Mol. Biol. 103, 227 (1976)

    1976

  2. [9]

    M. J. Field, P. A. Bash, and M. Karplus, A combined quantum mechanical and molecular mechanical poten- tial for molecular dynamics simulations, J. Comput. Chem. 11, 700 (1990)

    1990

  3. [11]

    M. F. Horstemeyer, Multiscale modeling: A review, in Practical Aspects of Computational Chemistry..., edited by J. Leszczynski and M. K. Shukla (Springer Netherlands, Dordrecht, 2010) pp. 87–135

    2010

  4. [12]

    Weinan, Principles of multiscale modeling (Cam- bridge University Press, 2011)

    E. Weinan, Principles of multiscale modeling (Cam- bridge University Press, 2011)

    2011

  5. [13]

    Reuter, A

    N. Reuter, A. Dejaegere, B. Maigret, and M. Karplus, Frontier bonds in qm/mm methods: A comparison of different approaches, J. Phys. Chem. A 104, 1720 (2000)

    2000

  6. [14]

    H. M. Senn and W. Thiel, Qm/mm methods for biomolecular systems, Angew. Chem. Int. Ed. 48, 1198 (2009)

    2009

  7. [15]

    Giovannini, L

    T. Giovannini, L. Bonatti, P. Lafiosca, L. Nicoli, M. Castagnola, P. G. Illobre, S. Corni, and C. Cappelli, Do we really need quantum mechanics to describe plas- monic properties of metal nanostructures?, ACS Pho- tonics 9, 3025 (2022)

    2022

  8. [16]

    Bondanza, M

    M. Bondanza, M. Nottoli, L. Cupellini, F. Lipparini, and B. Mennucci, Polarizable embedding qm/mm: the future gold standard for complex (bio) systems?, Phys. Chem. Chem. Phys. 22, 14433 (2020)

    2020

  9. [17]

    Tomasi, B

    J. Tomasi, B. Mennucci, and R. Cammi, Quantum me- chanical continuum solvation models, Chem. Rev. 105, 2999 (2005)

    2005

  10. [18]

    Mennucci and S

    B. Mennucci and S. Corni, Multiscale modelling of pho- toinduced processes in composite systems, Nat. Rev. Chem. 3, 315 (2019)

    2019

  11. [19]

    Lin and D

    H. Lin and D. G. Truhlar, Qm/mm: what have we learned, where are we, and where do we go from here?, Theor. Chem. Acc. 117, 185 (2007)

    2007

  12. [20]

    C. M. Baker, Polarizable force fields for molecular dy- namics simulations of biomolecules, Wiley Interdiscip. Rev. Comput. Mol. Sci. 5, 241 (2015)

    2015

  13. [21]

    T. P. Senftle, S. Hong, M. M. Islam, S. B. Kylasa, Y. Zheng, Y. K. Shin, C. Junkermeier, R. Engel- Herbert, M. J. Janik, H. M. Aktulga, et al., The reaxff reactive force-field: development, applications and fu- ture directions, Npj Comput. Mater. 2, 1 (2016)

    2016

  14. [22]

    Coccia, J

    E. Coccia, J. Fregoni, C. Guido, M. Marsili, S. Pipolo, and S. Corni, Hybrid theoretical models for molecular nanoplasmonics, J. Chem. Phys. 153 (2020)

    2020

  15. [23]

    B. T. Draine and P. J. Flatau, Discrete-dipole approxi- mation for scattering calculations, J. Opt. Soc. Am. A 11, 1491 (1994)

    1994

  16. [24]

    F. G. De Abajo and A. Howie, Retarded field calcula- tion of electron energy loss in inhomogeneous dielectrics, Phys. Rev. B 65, 115418 (2002)

    2002

  17. [25]

    L. L. Jensen and L. Jensen, Atomistic electrodynamics model for optical properties of silver nanoclusters, J. Phys. Chem. C 113, 15182 (2009)

    2009

  18. [26]

    V. I. Zakomirnyi, Z. Rinkevicius, G. V. Baryshnikov, L. K. Sørensen, and H. ˚Agren, Extended discrete inter- action model: plasmonic excitations of silver nanopar- ticles, J. Phys. Chem. C 123, 28867 (2019)

    2019

  19. [27]

    Giovannini, M

    T. Giovannini, M. Rosa, S. Corni, and C. Cappelli, A classical picture of subnanometer junctions: an atom- istic drude approach to nanoplasmonics, Nanoscale 11, 6004 (2019)

    2019

  20. [28]

    F. J. Garcia de Abajo, Graphene plasmonics: challenges and opportunities, ACS Photonics 1, 135 (2014)

    2014

  21. [29]

    Giovannini, L

    T. Giovannini, L. Bonatti, M. Polini, and C. Cappelli, Graphene plasmonics: Fully atomistic approach for re- alistic structures, J. Phys. Chem. Lett. 11, 7595 (2020)

    2020

  22. [30]

    D. Loco, L. Lagard` ere, O. Adjoua, and J.-P. Piquemal, Atomistic polarizable embeddings: energy, dynamics, spectroscopy, and reactivity, Acc. Chem. Res. 54, 2812 (2021)

    2021

  23. [31]

    Giovannini and C

    T. Giovannini and C. Cappelli, Continuum vs. atomistic approaches to computational spectroscopy of solvated systems, Chem. Commun. 59, 5644 (2023)

    2023

  24. [32]

    J. L. Payton, S. M. Morton, J. E. Moore, and L. Jensen, A hybrid atomistic electrodynamics–quantum mechan- ical approach for simulating surface-enhanced raman scattering, Acc. Chem. Res. 47, 88 (2014)

    2014

  25. [33]

    Lafiosca, L

    P. Lafiosca, L. Nicoli, L. Bonatti, T. Giovannini, S. Corni, and C. Cappelli, Qm/classical modeling of surface enhanced raman scattering based on atomistic electromagnetic models, J. Chem. Theory Comput. 19, 3616 (2023), pMID: 37278989

    2023

  26. [34]

    Fregoni, T

    J. Fregoni, T. S. Haugland, S. Pipolo, T. Giovannini, H. Koch, and S. Corni, Strong coupling between local- ized surface plasmons and molecules by coupled cluster theory, Nano Lett. 21, 6664 (2021)

    2021

  27. [35]

    C. A. Guido, M. Rosa, R. Cammi, and S. Corni, An open quantum system theory for polarizable continuum models, J. Chem. Phys. 152 (2020)

    2020

  28. [36]

    See Supplemental Material (Section I) at http:// for a definition of the acronyms and further details on the diverse models

  29. [37]

    S. W. Rick, S. J. Stuart, and B. J. Berne, Dynamical fluctuating charge force fields: Application to liquid wa- ter, J. Chem. Phys. 101, 6141 (1994)

    1994

  30. [38]

    S. W. Rick, S. J. Stuart, J. S. Bader, and B. Berne, Fluctuating charge force fields for aqueous solutions, J. Mol. Liq. 65-66, 31 (1995)

    1995

  31. [39]

    Thole, Molecular polarizabilities calculated with a modified dipole interaction, Chem

    B. Thole, Molecular polarizabilities calculated with a modified dipole interaction, Chem. Phys. 59, 341 (1981)

    1981

  32. [40]

    Ohno, Some remarks on the pariser-parr-pople method, Theor

    K. Ohno, Some remarks on the pariser-parr-pople method, Theor. Chim. Acta 2, 219 (1964)

    1964

  33. [41]

    Mayer, Formulation in terms of normalized propaga- 17 tors of a charge-dipole model enabling the calculation of the polarization properties of fullerenes and carbon nanotubes, Phys

    A. Mayer, Formulation in terms of normalized propaga- 17 tors of a charge-dipole model enabling the calculation of the polarization properties of fullerenes and carbon nanotubes, Phys. Rev. B 75, 045407 (2007)

    2007

  34. [42]

    Giovannini, A

    T. Giovannini, A. Puglisi, M. Ambrosetti, and C. Cap- pelli, Polarizable qm/mm approach with fluctuating charges and fluctuating dipoles: The qm/fqf µ model, J. Chem. Theory Comput. 15, 2233 (2019)

    2019

  35. [43]

    M. A. Thompson, Qm/mmpol: A consistent model for solute/solvent polarization. application to the aqueous solvation and spectroscopy of formaldehyde, acetalde- hyde, and acetone, J. Phys. Chem. 100, 14492 (1996)

    1996

  36. [44]

    J. M. Olsen, K. Aidas, and J. Kongsted, Excited states in solution through polarizable embedding, J. Chem. Theory Comput. 6, 3721 (2010)

    2010

  37. [45]

    Jensen, P

    L. Jensen, P. T. Van Duijnen, and J. G. Snijders, A dis- crete solvent reaction field model within density func- tional theory, J. Chem. Phys. 118, 514 (2003)

    2003

  38. [46]

    J. W. Ponder, C. Wu, P. Ren, V. S. Pande, J. D. Chodera, M. J. Schnieders, I. Haque, D. L. Mobley, D. S. Lambrecht, R. A. J. DiStasio, M. Head-Gordon, G. N. I. Clark, M. E. Johnson, and T. Head-Gordon, Current status of the amoeba polarizable force field, J. Phys. Chem. B 114,...

    2010

  39. [47]

    Pipolo, S

    S. Pipolo, S. Corni, and R. Cammi, The cavity electro- magnetic field within the polarizable continuum model of solvation: An application to the real-time time depen- dent density functional theory, Comput. Theor. Chem. 1040, 112 (2014)

    2014

  40. [48]

    Klamt and G

    A. Klamt and G. Sch¨ u¨ urmann, Cosmo: a new approach to dielectric screening in solvents with explicit expres- sions for the screening energy and its gradient, J. Chem. Soc., Perkin Trans. 2 , 799 (1993)

    1993

  41. [49]

    Cances, B

    E. Cances, B. Mennucci, and J. Tomasi, A new in- tegral equation formalism for the polarizable contin- uum model: Theoretical background and applications to isotropic and anisotropic dielectrics, J. Chem. Phys. 107, 3032 (1997)

    1997

  42. [50]

    Lafiosca, L

    P. Lafiosca, L. Nicoli, S. Pipolo, S. Corni, T. Giovan- nini, and C. Cappelli, Real-time formulation of atom- istic electromagnetic models for plasmonics, The Jour- nal of Physical Chemistry C 128, 17513 (2024)

    2024

  43. [51]

    Lafiosca, T

    P. Lafiosca, T. Giovannini, M. Benzi, and C. Cappelli, Going beyond the limits of classical atomistic modeling of plasmonic nanostructures, J. Phys. Chem. C 125, 23848 (2021)

    2021

  44. [52]

    Pipolo and S

    S. Pipolo and S. Corni, Real-time description of the electronic dynamics for a molecule close to a plasmonic nanoparticle, J. Phys. Chem. C 120, 28774 (2016)

    2016

  45. [53]

    Dall’Osto, G

    G. Dall’Osto, G. Gil, S. Pipolo, and S. Corni, Real- time dynamics of plasmonic resonances in nanoparticles described by a boundary element method with generic dielectric function, J. Chem. Phys. 153, 184114 (2020)

    2020

  46. [54]

    P. K. Ghosh, Taming hamiltonian systems with bal- anced loss and gain via lorentz interaction: general re- sults and a case study with landau hamiltonian, J. Phys. A 52, 415202 (2019)

    2019

  47. [55]

    P. K. Ghosh, Classical hamiltonian systems with bal- anced loss and gain, J. Phys. Conf. Ser. 2038, 012012 (2021)

    2021

  48. [56]

    Nottoli and F

    M. Nottoli and F. Lipparini, General formulation of po- larizable embedding models and of their coupling, J. Chem. Phys. 153, 224108 (2020)

    2020

  49. [57]

    Taussky and H

    O. Taussky and H. Zassenhaus, On the similarity trans- formation between a matrix and its transpose., Pac. J. Math. 9, 893 (1959)

    1959

  50. [58]

    R. A. Horn and C. R. Johnson, Matrix analysis (Cam- bridge university press, 2012)

    2012

  51. [59]

    Cappelli, Integrated QM/polarizable MM/continuum approaches to model chiroptical properties of strongly interacting solute–solvent systems, Int

    C. Cappelli, Integrated QM/polarizable MM/continuum approaches to model chiroptical properties of strongly interacting solute–solvent systems, Int. J. Quantum Chem. 116, 1532 (2016)

    2016

  52. [60]

    See Supplemental Material (Section I A 1) at http:// for a derivation of the FQ model

  53. [61]

    Caldirola, Quantum theory of nonconservative sys- tems, Nuovo Cimento B 77, 241 (1983)

    P. Caldirola, Quantum theory of nonconservative sys- tems, Nuovo Cimento B 77, 241 (1983)

    1983

  54. [62]

    Kanai, On the Quantization of the Dissipative Sys- tems, Prog

    E. Kanai, On the Quantization of the Dissipative Sys- tems, Prog. Theor. Phys. 3, 440 (1948)

    1948

  55. [63]

    C. I. Um, K. H. Yeon, and W. H. Kahng, The quantum damped driven harmonic oscillator, J. Phys. A 20, 611 (1987)

    1987

  56. [64]

    See Supplemental Material (Section II) at http:// for a derivation of the expanding coordinates Hamiltonian

  57. [65]

    Goldstein, C

    H. Goldstein, C. Poole, and J. Safko, Classical mechanics (American Association of Physics Teachers, 2002)

    2002

  58. [66]

    Bateman, On dissipative systems and related varia- tional principles, Phys

    H. Bateman, On dissipative systems and related varia- tional principles, Phys. Rev. 38, 815 (1931)

    1931

  59. [67]

    R. S. Langley, A generalization of the La- grange–Hamilton formalism with application to non-conservative systems and the quantum to classical transition, J. Math. Phys. 62, 033503 (2021)

    2021

  60. [68]

    A. O. Caldeira and A. J. Leggett, Influence of dissi- pation on quantum tunneling in macroscopic systems, Phys. Rev. Lett. 46, 211 (1981)

    1981

  61. [69]

    Weiss, Quantum dissipative systems (World Scien- tific, 2012)

    U. Weiss, Quantum dissipative systems (World Scien- tific, 2012)

    2012

  62. [70]

    L. H. Yu and C.-P. Sun, Evolution of the wave function in a dissipative system, Phys. Rev. A 49, 592 (1994)

    1994

  63. [71]

    Sun and L.-H

    C.-P. Sun and L.-H. Yu, Exact dynamics of a quantum dissipative system in a constant external field, Phys. Rev. A 51, 1845 (1995)

    1995

  64. [72]

    Schuch, Quantum theory from a nonlinear perspective (Springer, 2018)

    D. Schuch, Quantum theory from a nonlinear perspective (Springer, 2018)

    2018

  65. [73]

    Schuch, J

    D. Schuch, J. Guerrero, F. F. L´ opez-Ruiz, and V. Al- daya, Interrelations between different canonical descrip- tions of dissipative systems, Phys. Scr. 90, 045209 (2015)

    2015

  66. [74]

    Schuch and M

    D. Schuch and M. Blasone, Connections between ’t hooft’s beables and canonical descriptions of dissipative systems, J. Phys. Conf. Ser. 880, 012050 (2017)

    2017

  67. [75]

    K. A. Foss, Co-ordinates which uncouple the equations of motion of damped linear dynamic systems, Journal of Applied Mechanics 25, 361 (2021)

    2021

  68. [76]

    R. W. Traill-Nash, Modal methods in the dynamics of systems with non-classical damping, Earthquake Engi- neering & Structural Dynamics 9, 153 (1981)

    1981

  69. [77]

    J. W. Sanders, A dual-oscillator approach to complex- stiffness damping based on fourth-order dynamics, Non- linear Dynamics 109, 285 (2022)

    2022

  70. [78]

    Ashida, Z

    Y. Ashida, Z. Gong, and M. U. and, Non-hermitian physics, Advances in Physics 69, 249 (2020)

    2020

  71. [79]

    Krovi, Quantum algorithms to simulate quadratic classical hamiltonians and optimal control (2024), arXiv:2404.07303 [quant-ph]

    H. Krovi, Quantum algorithms to simulate quadratic classical hamiltonians and optimal control (2024), arXiv:2404.07303 [quant-ph]

    2024 arXiv

  72. [80]

    Kaczorek, A relation between the concepts of trans- position, similarity, and symmetry, as well as matrix equations, Math

    T. Kaczorek, A relation between the concepts of trans- position, similarity, and symmetry, as well as matrix equations, Math. Appl. 33, 21 (2005)

    2005

  73. [81]

    G´ oger, A

    S. G´ oger, A. Khabibrakhmanov, O. Vaccarelli, D. V. 18 Fedorov, and A. Tkatchenko, Optimized quantum drude oscillators for atomic and molecular response properties, J. Phys. Chem. Lett. 14, 6217 (2023)

    2023

  74. [82]

    Ditte, M

    M. Ditte, M. Barborini, L. Medrano Sandonas, and A. Tkatchenko, Molecules in environments: Toward sys- tematic quantum embedding of electrons and drude os- cillators, Phys. Rev. Lett. 131, 228001 (2023)

    2023

  75. [83]

    Sadhukhan and F

    M. Sadhukhan and F. R. Manby, Quantum mechanics of drude oscillators with full coulomb interaction, Phys. Rev. B 94, 115106 (2016)

    2016

  76. [84]

    A. P. Jones, J. Crain, V. P. Sokhan, T. W. Whitfield, and G. J. Martyna, Quantum drude oscillator model of atoms and molecules: Many-body polarization and dispersion interactions for atomistic simulation, Phys. Rev. B 87, 144103 (2013)

    2013

  77. [85]

    M. S. Tame, K. R. McEnery, S ¸. K.¨Ozdemir, J. Lee, S. A. Maier, and M. S. Kim, Quantum plasmonics, Nature Physics 9, 329 (2013)

    2013

  78. [86]

    See Supplemental Material (Section IV B) at http:// for a derivation of the Ehrenfest dynamics

  79. [87]

    F. E. Q. Rodriguez, Quantum dynamics in the self-consistent quadratic approximation (2024), arXiv:2403.11327 [quant-ph]

    2024 arXiv

  80. [88]

    G. H. Golub and C. F. Van Loan, Matrix computations (JHU press, 2013)

    2013

  81. [89]

    J. R. Silvester, Determinants of block matrices, The Mathematical Gazette 84, 460–467 (2000)

    2000

  82. [90]

    Bagarello, Pseudo-Bosons and Their Coherent States, Vol

    F. Bagarello, Pseudo-Bosons and Their Coherent States, Vol. 6 (Springer, 2022)

    2022

  83. [91]

    D. A. Trifonov, Pseudo-boson coherent and fock states (2009), arXiv:0902.3744 [quant-ph]

    2009 arXiv

  84. [92]

    Bagarello, More mathematics for pseudo-bosons, Journal of Mathematical Physics 54, 063512 (2013)

    F. Bagarello, More mathematics for pseudo-bosons, Journal of Mathematical Physics 54, 063512 (2013)

    2013

  85. [93]

    Bagarello, Pseudobosons, Riesz bases, and coher- ent states, Journal of Mathematical Physics 51, 023531 (2010)

    F. Bagarello, Pseudobosons, Riesz bases, and coher- ent states, Journal of Mathematical Physics 51, 023531 (2010)

    2010

  86. [94]

    S. T. Ali, F. Bagarello, and J. P. Gazeau, Modified Landau levels, damped harmonic oscillator, and two- dimensional pseudo-bosons, Journal of Mathematical Physics 51, 123502 (2010)

    2010

  87. [95]

    B. M. Villegas-Mart ´ ınez, H. M. Moya-Cessa, and F. Soto-Eguibar, Exact solution for the time-dependent non-hermitian generalized swanson oscillator, Indian Journal of Physics 97, 3957 (2023)

    2023

  88. [96]

    Elaihar, W

    L. Elaihar, W. Koussa, Y. Bouguerra, and M. Maa- mache, Time-dependent non-hermitian systems: pseudo-squeezed coherent states, Journal of Physics A: Mathematical and Theoretical 54, 175301 (2021)

    2021

  89. [97]

    Bagarello and J

    F. Bagarello and J. Feinberg, Bicoherent-state path in- tegral quantization of a non-hermitian hamiltonian, An- nals of Physics 422, 168313 (2020)

    2020

  90. [98]

    N. Mana, O. Zaidi, and M. Maamache, Time-dependent pseudo-bosonic coherent states, Journal of Mathemati- cal Physics 61, 102103 (2020)

    2020

  91. [99]

    Bagarello, F

    F. Bagarello, F. Gargano, and S. Spagnolo, Bi-squeezed states arising from pseudo-bosons, Journal of Physics A: Mathematical and Theoretical 51, 455204 (2018)

    2018

  92. [100]

    [12], Eqs

    See the definition of the complex polarizability in the Supplemental Material of Ref. [12], Eqs. (S10) and (S11)

  93. [101]

    See Supplemental Material (Section V) at http:// for the derivation of Eq. (43)

  94. [102]

    Mukamel, Principles of nonlinear optical spectroscopy, 6 (Oxford University Press, USA, 1995)

    S. Mukamel, Principles of nonlinear optical spectroscopy, 6 (Oxford University Press, USA, 1995)

    1995

  95. [103]

    Akhundova, V

    E. Akhundova, V. Dodonov, and V. Man’ko, Wigner functions of quadratic systems, Physica A 115, 215 (1982)

    1982

  96. [104]

    See Supplemental Material (Section VI) at http:// for a derivation of the von Neumann equation for non- Hermitian Hamiltonians

  97. [105]

    V. V. Dodonov and V. I. Man’ko, Theory of nonclassical states of light (CRC Press, 2003)

    2003

  98. [106]

    Gadella, Moyal formulation of quantum mechanics, Fortschritte der Phys

    M. Gadella, Moyal formulation of quantum mechanics, Fortschritte der Phys. 43, 229 (1995)

    1995

  99. [107]

    M. A. De Gosson, The Wigner Transform (World Sci- entific Publishing Company, 2017)

    2017

  100. [108]

    See Supplemental Material (Section III A) at http:// for the spectral analysis of JB

  101. [109]

    Lalisse, G

    A. Lalisse, G. Tessier, J. Plain, and G. Baffou, Quanti- fying the efficiency of plasmonic materials for near-field enhancement and photothermal conversion, The Jour- nal of Physical Chemistry C 119, 25518 (2015)

    2015

  102. [110]

    Baffou and R

    G. Baffou and R. Quidant, Nanoplasmonics for chem- istry, Chem. Soc. Rev. 43, 3898 (2014)

    2014

  103. [111]

    R´ acz, Z

    P. R´ acz, Z. P´ apa, I. M´ arton, J. Budai, P. Wr´ obel, T. Ste- faniuk, C. Prietl, J. R. Krenn, and P. Dombi, Measure- ment of nanoplasmonic field enhancement with ultrafast photoemission, Nano Letters 17, 1181 (2017)

    2017

  104. [112]

    Z.-K. Zhou, J. Liu, Y. Bao, L. Wu, C. E. Png, X.-H. Wang, and C.-W. Qiu, Quantum plasmonics get ap- plied, Progress in Quantum Electronics 65, 1 (2019)

    2019

  105. [113]

    J. D. Cox and F. J. Garc ´ ıa de Abajo, Nonlinear graphene nanoplasmonics, Accounts of Chemical Re- search 52, 2536 (2019)

    2019

  106. [115]

    See Supplemental Material (Section VIII A) at http:// for the definition and algebraic properties of the symbol ⊛

  107. [116]

    See Supplemental Material (Section IX A) at http:// for the calculation of the polarization in the k, ωspace

  108. [117]

    Breuer and F

    H.-P. Breuer and F. Petruccione, The theory of open quantum systems (Oxford Uni- versity Press, USA, 2002). CONTENTS I. Introduction 1 II. Generalized theory of the damped harmonic oscillator 3 A. Classical Dynamics 3 B. Hamiltonian Formulation 4 C. Pseudo-Boson Decomposition ...

    2002

  109. [118]

    Fluctuating Charges/Fluctuating Dipoles force fields 20 b

    Non-dissipative CPM 20 a. Fluctuating Charges/Fluctuating Dipoles force fields 20 b. Induced dipole force fields 20 c. Continuum models 21

  110. [119]

    Atomistic CPM 21 b

    Inclusion of decoherence 21 a. Atomistic CPM 21 b. Continuum CPM 21 B. Hamiltonian in the expanding coordinates 22 C. Classical Dynamics 23

  111. [120]

    Hamiltonian Formulation in the Extended Variables 25

    Spectral Analysis 24 D. Hamiltonian Formulation in the Extended Variables 25

  112. [121]

    The Similarity Transformation 25

  113. [122]

    Decomposition of dispersive and absorptive amplitudes 27 F

    Ehrenfest Dynamics 26 E. Decomposition of dispersive and absorptive amplitudes 27 F. von Neumann equation for non-Hermitian Hamiltonians 27 G. The Delta Function 28 H. Field interacting with the Auxiliary Variables 29

  114. [123]

    Quantities in the k, ωspace 30

    Formal integration 30 I. Quantities in the k, ωspace 30

  115. [124]

    Spatial Distribution and the Interaction Term 32

  116. [125]

    List of Fourier Transforms 32 20 Appendix A: Equations of Motion for the Classical Polarizable Medium (CPM)

  117. [126]

    polarization catastrophe

    Non-dissipative CPM a. Fluctuating Charges/Fluctuating Dipoles force fields Let us consider a set of n atoms, located at (fixed) positions Ri with i = 1, . . . , n. To simulate the polarization of this system to the electromagnetic field, the Fluctuating Charges (FQ) force fie...

  118. [127]

    Atomistic CPM The FQ force field has been recently extended to describe the optical response of plasmonic substrates under the action of an external oscillating electric field

    Inclusion of decoherence a. Atomistic CPM The FQ force field has been recently extended to describe the optical response of plasmonic substrates under the action of an external oscillating electric field. The resulting model is called the Frequency-Dependent Fluctuating Charge...

  119. [128]

    We aim to show that Sp − √ K ⊆ ω : det ω2 + 2iωΓ − K = 0 ∪ {0}, (C14) which up to the constant solution ( ω = 0) correspond to those of Eq

    Spectral Analysis Let Sp{M } denote the set of eigenvalues of the matrix M . We aim to show that Sp − √ K ⊆ ω : det ω2 + 2iωΓ − K = 0 ∪ {0}, (C14) which up to the constant solution ( ω = 0) correspond to those of Eq. (C1). Proof. This can be seen by noting that if A is inverti...

  120. [129]

    By the Jordan normal form theorem, ∀ K ∈ Cn×n there exists a non- singular matrix P1 ∈ Cn×n such that K = P1JKP −1 1 (D8) JK = diag J1 J2

    The Similarity T ransformation To find the matrix A symmetric and invertible that achieves the similarity transformation ofK into KT, namely [54] K = AKTA−1, (D7) we can follow the strategy outlined in [77]. By the Jordan normal form theorem, ∀ K ∈ Cn×n there exists a non- sin...

  121. [130]

    ,2N }, (D16) where J = 0 I −I 0 is the standard symplectic matrix in R2N ×2N , with I the identity in RN ×N and J T = J −1 = −J, we reinterpret Eq

    Ehrenfest Dynamics By imposing the canonical quantization relation [qξ, qη] = −iℏJξη , ξ, η ∈ {1, . . . ,2N }, (D16) where J = 0 I −I 0 is the standard symplectic matrix in R2N ×2N , with I the identity in RN ×N and J T = J −1 = −J, we reinterpret Eq. (C11) (and consequently E...

  122. [131]

    (I24)) in Eq

    F ormal integration Taking the Fourier Transform (Eq. (I24)) in Eq. (H9) we have gi(t) = nX k=1 F { ˜Lik(ω)˜hk(ω)} = nX k=1 F { ˜Lik(ω)˜hk(ω)} = nX k=1 1 2π Z R dω exp{−iωt} ˜Lik(ω)˜hk(ω) = nX k=1 1 2π Z R dω exp{−iωt} ˜Lik(ω) h Z R dτ exp{iωτ }hk(τ ) i = nX k=1 Z R dτ hk(τ ) ...

  123. [132]

    Polarization Defining Λ± t ≡ exp{±JBt} we have F {Λ± t }(ω) = Z R dt exp{i(ωE ∓ iJB)t} = 2πδ(ωE ∓ iJB). (I1) Taking the Fourier transform of ⟨q⟩t = Λ−1 t (⟨q⟩0 − ∆t), (I2) we have ⟨q⟩ (ω) = F {Λ−1 t ⟨q⟩0 − Λ−1 t ∆t}(ω) (I3a) = 2πδ(ωE + iJB) ⟨q⟩0 − 1 2π (Λ−1 t )(ω) ⊛ ∆(ω) (I3b)...

  124. [133]

    − 1 (2π)3 3X j=1 Rjβ Z R3 dk ˜Ej(k, ω) Z R3 dr exp ikTr G(r; Rβ, Σβ) # = 1 (2π) Z R dω exp{−iωt}

    Spatial Distribution and the Interaction T erm The Fourier transform of the Gaussian function G(r; Rα, Σα) = | det{2πΣα}|− 1 2 exp − 1 2 (r − Rα)TΣ−1 α (r − Rα) (I17) to the k space, Eq. (I27), reads ˜G(k; Rα, Σα) = Z R3 dr exp −ikTr G(r; Rα, Σα) = | det{2πΣα}|− 1 2 Z R3 dr ex...

  125. [134]

    (I25) 33 In the k space, V (r) = 1 (2π)3 Z R3 dk exp ikTr ˜V (k) (I26) ˜V (k) = Z R3 dr exp −ikTr V (r)

    List of F ourier T ransforms The Fourier transform in the ω space reads [99] F (t) = 1 2π Z R dω exp{−iωt} ˜F (ω) (I24) ˜F (ω) = Z R dt exp{iωt}F (t). (I25) 33 In the k space, V (r) = 1 (2π)3 Z R3 dk exp ikTr ˜V (k) (I26) ˜V (k) = Z R3 dr exp −ikTr V (r). (I27) In the k, ωspac...

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.