Pith. sign in

REVIEW 3 major objections 4 minor 39 references

First-principles calculation of electron-phonon spectral functions for defects using phonon interpolation

T0 review · 3 major / 4 minor · reviewed 2026-08-01 · deepseek-v4-flash

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

Pith's one-line read A localized transition-force extracted from a 4×4×4 supercell, coupled to phonons interpolated on hypercells of up to 32×32×32, yields converged Huang-Rhys spectral densities and an emission lineshape in agreement with experiment for the ni

desk verdict A clever and mostly sound method for defect phonon sidebands that needs one missing convergence test before I'd fully trust it. read the letter →

arxiv 2607.20788 v1 pith:2ONAUD73 submitted 2026-07-22 cond-mat.mtrl-sci

Zoltan Santha , Gergo Thiering This is my paper · ORCID
classification cond-mat.mtrl-sci
keywords phononinterpolationHuang-Rhysspectraldensitytransition-inducedforcenitrogen-vacancycentreelectron-phononcouplingopticallineshapehypercelldiamonddefects
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 establishes that for charge-conserving optical transitions between localized defect states in a gapped host, the force driving the lattice relaxation is concentrated in a small region around the defect: about seven effective ions and a root-mean-square radius near 2.1 Å in a 4×4×4 supercell. It then shows that this transition-induced force can be reconstructed from a first-principles supercell and used as a localized source that couples the defect to phonons obtained by Fourier-interpolating dynamical matrices on hypercells of up to 32×32×32 (about 17 million atoms). Applied to the nitrogen-vacancy centre in diamond, the procedure recovers a smooth, continuous Huang-Rhys spectral density with a dominant band near 63 meV and a linear low-energy acoustic contribution, and an emission lineshape that agrees with experiment over the full sideband. The approach is presented as a general route to converged defect spectral functions without direct vibrational calculations in prohibitively large cells, with the absorption case showing that final-state vibronic mode mixing remains a limitation of the parallel-mode harmonic treatment.

What carries the argument

The central object is the reconstructed transition-force source F_tr = Φ_f ΔR (mass-weighted f_tr = F_tr/√M), obtained from the relaxed initial- and final-state geometries in a finite supercell. The paper inserts this source into the partial Huang-Rhys factor S_{q,ν} = |Σ_a f^tr_a · e*_{ν,a}(q) e^{-iq·R_a}|² / (2 N_q ℏ ω_ν(q)^3), where e_ν(q) are eigenvectors of the interpolated dynamical matrix. A hypercell is built by repeating the defect supercell M₁×M₂×M₃ times; the supercell force-constant kernel is Fourier-interpolated in reciprocal space and diagonalized on a dense q-point grid. Because the transition force is nonzero in only one reference cell, the factor 1/N_q keeps the total Huang-

What would settle it

Compute the reconstructed transition force and the interpolated Huang-Rhys density with a 5×5×5 or larger first-principles supercell and compare with the 4×4×4 result. If the participation number, RMS localization radius, the position or weight of the 63 meV band, or the acoustic coefficient α change measurably, the claim that the 4×4×4 supercell represents the isolated-defect force source is refuted. A direct check is also to measure the real-space range of the force-constant perturbation itself, which the paper does not isolate.

Watch

Extended reading notes

Core claim

The central discovery is that the transition-induced force—the difference between the Born-Oppenheimer forces of the final and initial electronic states evaluated at the same geometry—is a localized source, and that using this force as the source term of the harmonic relation F_tr = Φ_f ΔR removes the need to resolve the long-range displacement field in a large cell. The paper demonstrates the concept on the NV− centre: the force reconstructed from a modest 4×4×4 density-functional supercell has a participation number of about 6.7 and an RMS localization radius of about 2.1 Å, and coupling it to interpolated hypercell phonons yields converged Huang-Rhys spectral densities with a dominant 63

Load-bearing premise

The load-bearing premise is that the phonons of the periodic array of defect supercells match, in the region where the transition force acts, the vibrational modes of an isolated defect embedded in the perfect host; if the defect-induced force-constant perturbation reaches the supercell boundary, the interpolated phonons and the resulting sidebands inherit the periodic-image error.

Editorial extensions

If this is right

  • A defect transition's full phonon sideband can be computed from an ab initio supercell of a few hundred atoms plus an inexpensive interpolation step, instead of requiring direct vibrational spectra of million-atom cells.
  • Because both the long-wavelength acoustic tail and van Hove features are resolved, finite-temperature lineshapes and the thermal evolution of the zero-phonon-line region become computable within the same first-principles workflow.
  • The linear low-energy scaling of the Huang-Rhys density, S(ℏω)=αℏω with α≈387.7 eV⁻², is a consistency check that the force-source formulation avoids an artificial low-frequency divergence.
  • The total Huang-Rhys factor converges as a power law with hypercell size, enabling systematic extrapolation to the infinite-hypercell limit and yielding a Debye-Waller factor near 2.9% for the NV− transition.
  • Emission is captured well using ground-state phonons, while absorption requires excited-state phonons; a quantitatively complete absorption description needs to go beyond the parallel-mode equal-frequency harmonic approximation.

Reading between the lines

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

  • Because the locality argument rests on a charge-conserving transition in a gapped, weakly polar host, the same strategy should transfer to other colour centres in diamond, silicon carbide, or hexagonal boron nitride whenever the transition-induced density change is localized and the net charge is unchanged.
  • For polar hosts or transitions with a net change of charge, the long-range multipolar tail of the transition force could be added analytically through screened Born effective charges, extending the method beyond its current scope.
  • The emission-absorption asymmetry indicates that applications with strongly vibronically coupled final states will need a full multimode treatment; the interpolation machinery could be paired with such a treatment to test whether the excited-state surface alone explains the residual absorption discrepancy.
  • The interpolated phonon continuum could also be used for other phonon-sampling-limited quantities, such as non-radiative capture rates or temperature-dependent Debye-Waller factors, since those require the same dense vibrational sampling.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper develops a phonon-interpolation method for computing Huang–Rhys spectral densities and optical lineshapes of point defects. The central idea is to reconstruct a localized transition-induced force from DFT supercell calculations via the harmonic relation F̃tr = Φ̃f ΔR̃, then couple that force source to Fourier-interpolated dynamical matrices of a much larger hypercell. The partial Huang–Rhys factor is recast as S_{q,ν} = |Σ_a f^tr_a·e*_{ν,a}(q)e^{-iq·R_a}|²/(2N_qℏω³). The method is benchmarked on the NV⁻ centre in diamond: the transition force is shown to be localized in a 4×4×4 supercell, hypercells up to 32×32×32 are used to resolve the vibrational continuum, and the resulting emission lineshape is compared with experiment, recovering a dominant 63 meV band, linear low-energy acoustic scaling, and van Hove-related high-energy structure.

Significance. If the central equivalence between hypercell phonons and the isolated-defect vibrational continuum is certified, the method would be a practically valuable, systematically improvable route to converged defect spectral functions without prohibitively large direct supercell calculations. The algebraic reformulation in Eq. (22) is clean and appears exact within the stated harmonic, parallel-mode, equal-frequency approximation; the 1/N_q normalization is dimensionally consistent. The paper is also honest about the limitations of the parallel-mode model for absorption. A public implementation is provided, which strengthens reproducibility.

major comments (3)
  1. [§6.2, Fig. 4; Eq. (22)] The central equivalence between the hypercell phonons (a periodic array of defect supercells) and the isolated-defect vibrational continuum is asserted but not tested. The condition that the defect-induced perturbation of the force-constant kernel is converged with respect to defect-image separation is never checked. The locality of the transition force (Fig. 2, Table 1) constrains only the source f^tr, not the kernel Φ. Hypercell-size convergence (Fig. 3) only refines q-point sampling on the same supercell Φ and cannot detect errors in those force constants. The authors should (i) compute the real-space difference ΔΦ = Φ_defect − Φ_pristine in the 4×4×4 cell and show it decays well before the cell boundary, and (ii) compare S(ℏω) obtained from 4×4×4 and at least one larger (e.g. 5×5×5) defect supercell at the same hypercell q-point density. Without such a test, periodic-image contaminat
  2. [§6.2, Fig. 4; Eq. (22)] The linear acoustic scaling S(ℏω)=αℏω is presented as a consistency check, but it is a necessary consequence of Σ_a f^tr_a=0 for a charge-conserving transition and of the ω→0 density of states; it does not certify convergence of the acoustic modes. The coefficient α≈387.7 eV⁻² depends on the first moment of the localized force and on the long-wavelength sound velocities of the hypercell, which are those of a periodic array of defects at finite concentration, not an isolated defect. The authors should test whether α and the low-energy S(ω) are stable when the defect supercell size is increased at fixed hypercell volume, or when the force-constant kernel tail is truncated or compared with the pristine kernel. As it stands, the reported linear scaling is a vanishing-net-force identity, not a convergence test.
  3. [§5; Eq. (17)] The reconstructed transition force is obtained by applying the supercell force-constant matrix to the relaxed displacement field ΔR̃. Because ΔR̃ is the equilibrium displacement of a periodic array of defects, it contains the elastic response of that array. The reconstruction removes the long-range displacement only to the extent that Φ̃f is the exact harmonic kernel of the same array. The paper does not quantify the difference between the reconstructed f^tr and a force directly evaluated from DFT at the initial-state geometry, nor the anharmonic error in using the harmonic relation. A direct comparison in the 4×4×4 cell would test both the reconstruction and the locality assumption on which the method rests.
minor comments (4)
  1. [Fig. 3] The caption and panels are difficult to parse because both the supercell size, hypercell size, and Gaussian smearing width σ vary simultaneously. The apparent convergence may partly reflect the decreasing σ. Please label each panel clearly with all three parameters and, ideally, show fixed-σ comparisons.
  2. [Table 2] The power-law exponent β differs substantially between the 2×2×2 (1.838) and the 3×3×3/4×4×4 cells (≈1.1–1.2). This undercuts the statement that the power-law behaviour is universal. Please discuss the sensitivity of the extrapolated S_tot to the fitting form and to the number of hypercell points included.
  3. [§6.2, Fig. 6] The claim of 'excellent agreement' with experiment is qualitative. Please provide a quantitative discrepancy measure (e.g. integrated absolute difference over the sideband) and state the experimental uncertainty or the sensitivity to the 1 meV broadening choices.
  4. [§5, text near Eq. (23)] The manuscript says the hypercell dynamical matrices are 'diagonalized' for a 32×32×32 hypercell corresponding to ~17 million atoms. In practice one diagonalizes the q-dependent dynamical matrix for each q. Please clarify the wording to avoid implying a direct diagonalization of a 17-million-atom matrix.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Huang-Rhys density is computed from DFT force constants and a DFT transition force; experimental values and empirical fits are output-side only.

full rationale

The central derivation is self-contained and non-circular. The paper computes the transition-induced force F̃tr = Φ̃f ΔR̃ from relaxed DFT geometries in finite supercells (Sec. 5), Fourier-interpolates the supercell force-constant kernel to hypercell dynamical matrices, and evaluates S(ℏω) and lineshapes from the formula S_{q,ν} = |Σ_a f_tr_a · e*_{ν,a}(q)e^{-iq·R_a}|^2/(2N_q ℏω^3). No experimental value enters as an input: the 63 meV band, the linear acoustic slope α≈387.7 eV⁻², and the sideband shape all emerge from the DFT-derived force source and interpolated phonons. The only fitted quantities (1 meV Lorentzian/Gaussian broadenings, the empirical Stot(N)=Stot−αN^−β power-law extrapolation, and the acoustic slope α) are output-side characterizations or display regularizations, not parameters supplied to the derivation. There are no load-bearing self-citations: the cited works on embedding, machine-learned potentials, and standard Huang-Rhys theory are external to the authors [1-8,33,34], and the NV-centre excited-state treatment cites external DFT studies [12]. The paper explicitly acknowledges the main assumption — that the normal modes are formally those of a periodic defect array, becoming indistinguishable from isolated-defect modes only if the force-constant perturbation is converged with respect to defect-image separation (Sec. 5, final paragraph). This is an unverified validation concern about periodic-image contamination, not a circular step: the interpolation does not define its output in terms of the claim, nor does any equation reduce to another by construction. Therefore, the circularity score is 0.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

No invented physical entities: the transition force is a defined physical quantity and the hypercell is a computational construction. The method rests on inherited Huang-Rhys framework axioms (harmonic parallel-mode approximation), DFT/ΔSCF domain assumptions for the JT-active excited state, and the paper-specific hypercell-isolated-defect equivalence, which is only partially validated. All free parameters are output-side smoothing or extrapolation choices; none enter the derivation of S(ω).

free parameters (4)
  • Gaussian smearing width σ for S(ℏω) = 0.1–1.0 meV
    Hand-chosen spectral smearing (Fig. 3 captions); controls the displayed 'continuous' resolution but not integrated spectral weight.
  • Lorentzian + Gaussian broadening of lineshapes = 1 meV each
    Phenomenological broadening for experimental comparison (Figs. 5–7); paper argues they are small relative to the 63 meV scale and do not set the sideband envelope.
  • Power-law extrapolation parameters (Stot, α, β) for Stot(N) = e.g., Stot = 3.557, α = 0.124, β = 1.085 for 4×4×4
    Empirical fit to the hypercell-size series (Table 2), used to quote the extrapolated Stot and the Debye-Waller factor exp(-Stot) ≈ 2.9%.
  • Acoustic slope α = 387.7 eV⁻²
    Fitted to the computed S(ℏω) to demonstrate linear low-energy scaling (Fig. 4); an output characterization and consistency check, not an input to the derivation.
assumptions (6)
  • domain assumption Born-Oppenheimer and Condon approximations separate electronic and vibrational degrees of freedom with a geometry-independent transition dipole.
    Sec. 3 and Appendix A; inherited from standard Franck-Condon theory; required for the lineshape formula W(ℏω) ∝ ω³ A(ℏω).
  • domain assumption Harmonic approximation with parallel modes and equal frequencies in both electronic states (J = δ, Ω_i = Ω_f).
    Sec. 3, displaced-oscillator Hamiltonians; the paper shows this is where the absorption calculation fails (JT-active 3E state), so the approximation is load-bearing and self-flagged.
  • domain assumption Near-sightedness / exponential localization of response kernels and force-constant perturbations in gapped hosts.
    Sec. 5, 'In a gapped host, local electronic response functions... decay exponentially'; invoked to justify the locality of the transition force and truncation at the supercell boundary.
  • domain assumption Constrained-occupation ΔSCF (HSE) gives a valid Born-Oppenheimer surface for the JT-active 3E excited state; PBE gives adequate NV- force constants.
    Sec. 4; the excited-state relaxation is performed without symmetry constraints; no wavefunction-based benchmark is provided.
  • domain assumption The hypercell (periodic array of defect supercells) is equivalent, in the source region, to an isolated defect in a pristine host.
    Sec. 5, final paragraph; central interpolation premise; partially validated by force localization and host-DOS comparison, but the force-constant range itself is not directly verified.
  • domain assumption Real-space force constants of the 4×4×4 supercell contain the full interaction needed for dense q-space interpolation (truncated Fourier sum).
    Sec. 5, D(q) = Σ_L Φ(0,L) e^{iq·(...)}; standard frozen-phonon interpolation assumption; validity depends on the defect-induced perturbation being short-ranged within half the supercell.

how reviews work

0 comments
Cite this review

Pith. "Pith review of First-principles calculation of electron-phonon spectral functions for defects using phonon interpolation." pith.science (2026). https://pith.science/paper/2ONAUD73

@misc{pith2026260720788,
  author       = {Pith},
  title        = {Pith review of: First-principles calculation of electron-phonon spectral functions for defects using phonon interpolation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2ONAUD73}},
  note         = {Machine review of arXiv:2607.20788}
}
abstract

Point defects in wide-band-gap semiconductors exhibit optical spectra strongly shaped by electron-phonon coupling, but direct first-principles calculation of the corresponding phonon sidebands is often limited by the coarse vibrational spectrum of the largest defect supercells accessible by ab-initio electronic structure codes. In this work, we present a phonon-interpolation method for Huang-Rhys spectral densities and optical lineshape functions on hypercells created by extending the defect-containing supercells on arbitrarily dense phonon $q$-point grids. The method is based on reconstructing the transition-induced force associated with the optical excitation and using this localized force source to couple the defect transition to a densely sampled vibrational continuum. In this formulation, the local defect physics is obtained from ab initio supercell calculations, while the long-wavelength acoustic modes and the detailed structure of the host phonon spectrum are recovered by diagonalizing interpolated dynamical matrices in large hypercells. We demonstrate the method on the negatively charged nitrogen-vacancy centre in diamond between its ground $^{3}A_{2}$ and excited $^{3}E$ states. The transition-force is shown to be strongly localized around the defect, with converged localization measures obtained in a $4\times4\times4$ supercell accessible by density functional theory calculations. We interpolate the electron-phonon coupling on hypercells up to $32\times32\times32$ corresponding to approximately 17 million atoms, thereby recovering smooth, continuous Huang-Rhys spectral densities with ultrafine spectral resolution. The dominant coupling band is found near 63~meV; the low-energy acoustic contribution follows the expected linear scaling; and we recover the finer van-Hove-related structures in the optical phonon regime observed in experiments.

Figures

Figures reproduced from arXiv: 2607.20788 by the authors.

Figure 1
Figure 1. Schematic construction of the hypercell. A pristine unit cell is tiled to form a defect supercell, [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗
Figure 2
Figure 2. Localization of the transition-force source for the NV [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. Hypercell convergence of the single-phonon Huang-Rhys density. Single-phonon Huang-Rhys [PITH_FULL_IMAGE:figures/full_fig_p015_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Low-energy scaling of the converged Huang-Rhys spectral density [PITH_FULL_IMAGE:figures/full_fig_p016_4.png]
Figure 5
Figure 5. Figure 5: Temperature dependence of emission and absorption line shapes. Normalized emission and [PITH_FULL_IMAGE:figures/full_fig_p016_5.png]
Figure 6
Figure 6. Figure 6: Comparison of calculated low-temperature spectra with experiment using ground-state phonons. [PITH_FULL_IMAGE:figures/full_fig_p017_6.png]
Figure 7
Figure 7. Figure 7: Improved absorption spectrum using excited-state phonons. Low-temperature absorption line [PITH_FULL_IMAGE:figures/full_fig_p018_7.png]
Figure 8
Figure 8. Figure 8: Hypercell-size scaling of the total Huang-Rhys factor. Total Huang-Rhys factor [PITH_FULL_IMAGE:figures/full_fig_p019_8.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

39 extracted references

  1. [1]

    First-principles theory of the luminescence lineshape for the triplet transition in diamond NV centres.New Journal of Physics, 16(7):073026, July 2014

    Audrius Alkauskas, Bob B Buckley, David D Awschalom, and Chris G Van de Walle. First-principles theory of the luminescence lineshape for the triplet transition in diamond NV centres.New Journal of Physics, 16(7):073026, July 2014

    2014

  2. [2]

    Doherty, Neil B

    Lukas Razinkovas, Marcus W. Doherty, Neil B. Manson, Chris G. Van De Walle, and Audrius Alkauskas. Vibrational and vibronic structure of isolated point defects: The nitrogen-vacancy center in diamond.Physical Review B, 104(4):045303, July 2021

    2021

  3. [3]

    Turiansky, John L

    Mark E. Turiansky, John L. Lyons, and Noam Bernstein. Machine Learning Phonon Spectra for Fast and Accurate Optical Lineshapes of Defects.ACS Nano, 20(9):7454–7463, March 2026

    2026

  4. [4]

    Doherty, Neil B

    Marcus W. Doherty, Neil B. Manson, Paul Delaney, Fedor Jelezko, Jörg Wrachtrup, and Lloyd C.L. Hollenberg. The nitrogen-vacancy colour centre in diamond.Physics Reports, 528(1):1–45, July 2013

    2013

  5. [5]

    Born and R

    M. Born and R. Oppenheimer. Zur Quantentheorie der Molekeln.Annalen der Physik, 389(20):457– 484, January 1927

    1927

  6. [6]

    A Theory of Intensity Distribution in Band Systems.Physical Review, 28(6):1182– 1201, December 1926

    Edward Condon. A Theory of Intensity Distribution in Band Systems.Physical Review, 28(6):1182– 1201, December 1926

    1926

  7. [7]

    The Franck-Condon Principle and Its Application to Crystals.The Journal of Chemical Physics, 20(11):1752–1760, November 1952

    Melvin Lax. The Franck-Condon Principle and Its Application to Crystals.The Journal of Chemical Physics, 20(11):1752–1760, November 1952

    1952

  8. [8]

    Theory of light absorption and non-radiative transitions in F-centres

    Kun Huang and Avril Rhys. Theory of light absorption and non-radiative transitions in F-centres. Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences, 204(1078):406– 423, December 1950

    1950

Show all 39 references
  1. [9]

    Ontheinterpretationofelectronicspectraofpolyatomicmolecules.Acta Physicochim

    F.Duschinsky. Ontheinterpretationofelectronicspectraofpolyatomicmolecules.Acta Physicochim. URSS, 7:551–566, 1937

    1937

  2. [10]

    E. V. Doktorov, I. A. Malkin, and V. I. Man’ko. Dynamical symmetry of vibronic transitions in poly- atomic molecules and the Franck-Condon principle.Journal of Molecular Spectroscopy, 64(2):302– 326, February 1977

    1977

  3. [11]

    Katriel and G

    J. Katriel and G. Adam. On the use of second quantization for the calculation of two-centre harmonic oscillator integrals.Journal of Physics B: Atomic and Molecular Physics, 3(1):13, January 1970

    1970

  4. [12]

    Adam Gali, Maria Fyta, and Efthimios Kaxiras.Ab Initiosupercell calculations on nitrogen-vacancy center in diamond: Electronic structure and hyperfine tensors.Physical Review B, 77(15):155206, April 2008

    2008

  5. [13]

    R. P. Feynman. Forces in Molecules.Physical Review, 56(4):340–343, August 1939

    1939

  6. [14]

    Kresse and J

    G. Kresse and J. Furthmüller. Efficient iterative schemes forab initiototal-energy calculations using a plane-wave basis set.Physical Review B, 54(16):11169–11186, October 1996

    1996

  7. [15]

    Kresse and J

    G. Kresse and J. Hafner.Ab Initiomolecular dynamics for liquid metals.Physical Review B, 47(1):558–561, January 1993

    1993

  8. [16]

    Scuseria, and Matthias Ernzerhof

    Jochen Heyd, Gustavo E. Scuseria, and Matthias Ernzerhof. Hybrid functionals based on a screened Coulomb potential.The Journal of Chemical Physics, 118(18):8207–8215, May 2003

    2003

  9. [17]

    Krukau, Oleg A

    Aliaksandr V. Krukau, Oleg A. Vydrov, Artur F. Izmaylov, and Gustavo E. Scuseria. Influence of the exchange screening parameter on the performance of screened hybrid functionals.The Journal of Chemical Physics, 125(22):224106, December 2006. 21

    2006

  10. [18]

    Perdew, Kieron Burke, and Matthias Ernzerhof

    John P. Perdew, Kieron Burke, and Matthias Ernzerhof. Generalized Gradient Approximation Made Simple.Physical Review Letters, 77(18):3865–3868, October 1996

    1996

  11. [19]

    Heyd-Scuseria-Ernzerhof hybrid functional for calculating the lattice dynamics of semiconductors.Physical Review B, 80(11):115205, September 2009

    Kerstin Hummer, Judith Harl, and Georg Kresse. Heyd-Scuseria-Ernzerhof hybrid functional for calculating the lattice dynamics of semiconductors.Physical Review B, 80(11):115205, September 2009

    2009

  12. [20]

    Kresse and D

    G. Kresse and D. Joubert. From ultrasoft pseudopotentials to the projector augmented-wave method. Physical Review B, 59(3):1758–1775, January 1999

    1999

  13. [21]

    P. E. Blöchl. Projector augmented-wave method.Physical Review B, 50(24):17953–17979, December 1994

    1994

  14. [22]

    Macroscopic polarization in crystalline dielectrics: The geometric phase approach

    Raffaele Resta. Macroscopic polarization in crystalline dielectrics: The geometric phase approach. Reviews of Modern Physics, 66(3):899–915, July 1994

    1994

  15. [23]

    Xavier Gonze and Changyol Lee. Dynamical matrices, Born effective charges, dielectric permittivity tensors, andinteratomicforceconstantsfromdensity-functionalperturbationtheory.Physical Review B, 55(16):10355–10368, April 1997

    1997

  16. [24]

    Electron-phonon interactions from first principles.Reviews of Modern Physics, 89(1):015003, February 2017

    Feliciano Giustino. Electron-phonon interactions from first principles.Reviews of Modern Physics, 89(1):015003, February 2017

    2017

  17. [25]

    Exponential Localization of Wannier Functions in Insulators.Physical Review Letters, 98(4):046402, January 2007

    Christian Brouder, Gianluca Panati, Matteo Calandra, Christophe Mourougane, and Nicola Marzari. Exponential Localization of Wannier Functions in Insulators.Physical Review Letters, 98(4):046402, January 2007

    2007

  18. [26]

    W. Kohn. Density Functional and Density Matrix Method Scaling Linearly with the Number of Atoms.Physical Review Letters, 76(17):3168–3171, April 1996

    1996

  19. [27]

    Prodan and W

    E. Prodan and W. Kohn. Nearsightedness of electronic matter.Proceedings of the National Academy of Sciences, 102(33):11635–11638, August 2005

    2005

  20. [28]

    W. Kohn. Analytic Properties of Bloch Waves and Wannier Functions.Physical Review, 115(4):809– 821, August 1959

    1959

  21. [29]

    Economou.Green ’s Functions in Quantum Physics, volume 7 ofSpringer Series in Solid-State Sciences

    Eleftherios N. Economou.Green ’s Functions in Quantum Physics, volume 7 ofSpringer Series in Solid-State Sciences. Springer Berlin Heidelberg, Berlin, Heidelberg, 2006

    2006

  22. [30]

    Kehayias, M

    P. Kehayias, M. W. Doherty, D. English, R. Fischer, A. Jarmola, K. Jensen, N. Leefer, P. Hemmer, N. B. Manson, and D. Budker. Infrared absorption band and vibronic structure of the nitrogen- vacancy center in diamond.Physical Review B, 88(16):165202, October 2013

    2013

  23. [31]

    NV–N+ pair centre in 1b diamond.New Journal of Physics, 20(11):113037, November 2018

    Neil B Manson, Morgan Hedges, Michael S J Barson, Rose Ahlefeldt, Marcus W Doherty, Hiroshi Abe, Takeshi Ohshima, and Matthew J Sellars. NV–N+ pair centre in 1b diamond.New Journal of Physics, 20(11):113037, November 2018

    2018

  24. [32]

    Davies and M

    G. Davies and M. F. Hamer. Optical studies of the 1.945 eV vibronic band in diamond.Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences, 348(1653):285–298, February 1976

    1976

  25. [33]

    First-principles phonon calculations with phonopy and phono3py.J

    Atsushi Togo. First-principles phonon calculations with phonopy and phono3py.J. Phys. Soc. Jpn., 92(1):012001, 2023

    2023

  26. [34]

    Implementation strategies in phonopy and phono3py.J

    Atsushi Togo, Laurent Chaput, Terumasa Tadano, and Isao Tanaka. Implementation strategies in phonopy and phono3py.J. Phys. Condens. Matter, 35(35):353001, 2023

    2023

  27. [35]

    The quantum theory of the emission and absorption of radiation

    Paul Adrien Maurice Dirac. The quantum theory of the emission and absorption of radiation. Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, 114(767):243–265, March 1927

    1927

  28. [36]

    Fermi’s goldenrule: Itsderivation andbreakdown byan idealmodel.European Journal of Physics, 37(6):065406, October 2016

    JM ZhangandY Liu. Fermi’s goldenrule: Itsderivation andbreakdown byan idealmodel.European Journal of Physics, 37(6):065406, October 2016. 22

    2016

  29. [37]

    Bruno Huttner and Stephen M. Barnett. Quantization of the electromagnetic field in dielectrics. Physical Review A, 46(7):4306–4322, October 1992

    1992

  30. [38]

    G. H. Hardy. Summation of a Series of Polynomials of Laguerre.Journal of the London Mathematical Society, s1-7(2):138–139, 1932

    1932

  31. [39]

    ˆP 2 ν 2 + 1 2 Ω2 ν ˆQ2 ν # in the initial-state and by ˆHf = ∆E+ X ν

    Einar Hille. On Laguerre’s Series.Proceedings of the National Academy of Sciences, 12(4):261–265, April 1926. 23 A Formulation ofW em(ℏω)andW abs(ℏω) For a quantum-mechanical transition between two vibronic states|i, n⟩ → |f, m⟩in then-th andm-th vibrational levels of the init...

    1926

Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.