travers_2026_trecx_two_photon#
J. C. Travers, TDSE calculations of the two-photon ionisation and excitation of He and Ar with tRecX (A. Scrinzi, Comp. Phys. Comm. 270, 108146 (2022)), one-electron and haCC models, unpublished (2026)
Source record · Catalogue · Review register
No DOI is recorded.
Source kind: theory
Recorded technique: Numerical solution of the time-dependent Schrodinger equation with tRecX (commit a987d7c), velocity gauge, linear polarisation along z, for optmat’s two-photon TDSE campaign (TC-1048; its brief and two addenda unpublished). Two models. (1) One active electron in Tong and Lin’s potential (tong_2005 Table 1; He 1s, Ar 3p with m = 0 and m = 1): 10 associated Legendre functions in the polar angle (l <= 9), finite elements in r (3 of order 14 on [0, 2], 4 of order 12 on [2, 20], elements of about 4.3 bohr and order 12 to BOX, then 30 polExp functions), BOX 40 (run 1) or 80 (runs 2-6), infinite-range exterior complex scaling from BOX (angle 0.15 for He, 0.3 for Ar), fixed-step RK4 (step 2/Emax, capped at 0.03 au for He and 0.015 au for Ar), cutEnergy 100. (2) Helium in haCC (hybrid anti-symmetrised coupled channels, both electrons antisymmetrised) with the Scrinzi group’s He19_Standard data set, one ion channel (He+ 1s), lMax 10, rMax 40 (haCC fixes cutEnergy 400 and its own complex scaling). Pulses on the vector potential, A(t) = sum_i (E0_i/omega_i) g(t) sin(omega_i t), g = exp(-2 ln2 t^2/T^2) with T the full width at half maximum of the intensity, carrier-envelope phase 0 for each colour, colours sharing the envelope and centre; I0 = (1/2) eps0 c E0^2 per colour; chirp as a quadratic spectral phase on A (phi2 > 0 an up-chirp), I0 then the transform-limited peak. Window [-3T_eff, +3T_eff + 1000 au]. Ionisation read from the iSurff photoelectron spectrum (0 to 40.8 eV, 1200 points; peak integrals over +-0.5 eV (run 1) or +-0.25 eV (runs 2-6) around each two-photon peak), from the bi-orthogonal projection on the field-free bound states (OccDual, one electron), and from the norm-loss (haCC).
Materials in structured entries: Ar, He
Dataset quantities: multiphoton_ionization_cross_section, transition_dipole_moment, two_photon_transition_amplitude
Contents: 6 datasets; 0 formulas.
Curation notes#
UNPUBLISHED CALCULATIONS OF J. C. TRAVERS, TIER 3: validators only (use = check throughout), ranked below published ab initio calculations and measurements; nothing is fitted to them. They answer the campaign’s brief (OEP 6’s Q9 and OEP 8). There is no paper and no library entry: the result tables, the runs’ report, summaries and working notes, the input decks as generated and the scripts that made the inputs and read the outputs are archived in data/external/trecx/two_photon, whose PROVENANCE.md gives the code’s commit, the run-5 patch, the machine (a Slurm cluster), the dates (9-11 October 2026), the sha256 of every file and where the files that are not archived are. The three cross-section files are written from the archived tables by tools/extract/travers_2026_trecx_two_photon.py, which checks every value against the runs’ printed summaries (15 numbers, all equal to their printed digits).
THE MODELS AND THEIR OWN ENERGIES (no NIST value was used in the runs; nothing is shifted here). One-electron He (Tong and Lin): I_p 24.5937 eV (-0.903801 au, the initial state’s energy), 1s2p 21.056 eV, a one-electron 2s at 20.240 eV (not 1s2s 1S: the model has no singlet-triplet distinction). haCC He (He19, one channel): I_p 24.5410 eV (-E of its initial state, -0.901867 au); in run 5’s projection basis 1s2s 20.636 eV and 1s2p 21.209 eV above its ground state. These Occupation energies are not the propagated spectrum: run 5b puts the propagated 1s2s at 20.739 eV (see run 5); the propagated 1s2p, 21.2089 eV (run 5c), equals the Occupation one. One-electron Ar (Tong and Lin, 3p): I_p 15.774 eV (-0.579689 au); 4s 11.267 eV, 4p 12.926 eV, 3d 13.859 eV above 3p. The model has no spin-orbit coupling and no LS terms: its 4s and 4p are one-electron orbitals, not the atom’s 4s[3/2]1, 1P1 or 3p5 4p levels.
THE CROSS SECTIONS (multiphoton_ionization_cross_section, photons = 2). sigma2 = P / int Phi^2 dt, Phi = I(t)/(hbar omega), I(t) = I0 exp(-4 ln2 t^2/T^2), T = 20 fs, so int Phi^2 dt = (I0/hbar omega)^2 T sqrt(pi/(8 ln2)); I0 = 1e12 W/cm2. cm4 s, against the photon energy (eV). The total over final symmetries (1S and 1D for He). The datasets: sigma2_hacc_He (13-20 eV), sigma2_sae_He (13-20 eV), sigma2_sae_Ar (8-11 eV; the atom = 2 P(m = 0) + 4 P(m = 1), the independent-electron combination agreed for the runs). P is proportional to I^2 between 1e11 and 1e13 W/cm2 to 1.4 % (He, both models, from the spectrum peak); for Ar the 1e11 rows give sigma2 within 0.3 % of 1e12 (1.1 % at 11 eV) and the 1e13 rows -3.4 to +1.6 % at 8-10 eV, but 11 eV, 0.27 eV below the model’s 4s, is near-resonant: -7.0 % at 1e13, its m = 0 yield 15 % below I^2 scaling (the runs’ summary). Comparisons with the curated calculations are not made here.
UNCERTAINTY (the curator’s rule, computed by the script’s uncertainties(); level ‘bound’): the larger of (a) the largest change among the family’s convergence runs at one photon energy (one-electron variants each less their own no-field drift) and (b) the row’s disagreement between two readings of the same ionisation. haCC: lMax 15 +0.003 %, rMax 60 -0.25 % (the spectrum peak; -0.23 % on the norm-loss), the spectrum peak and the norm-loss within 0.1 %: 0.25 %. One-electron He: angular functions 15 0.000 %, ECS angle x 2/3 -0.25 %, step x 2/3 and cutEnergy 150 +1.6 % (their no-field drift 1.9e-8 against the base’s 1.43e-7, which is 5 to 11 % of the 1e12 signal), BOX 80 and 120 +0.003 % and +0.02 % on the projection (at most 0.29 % over the five energies); the BOX 80 and BOX 120 spectra within 1.9 % of the projection: 1.6 to 1.9 %. One-electron Ar: angular functions 15, step x 2/3, ECS angle x 2/3 0.000 %, cutEnergy 150 +0.02 %, BOX 120 +0.18 % (9 eV, m = 0, the peak); but the spectrum and the drift-corrected projection disagree by +10.9 % at 8 eV (photoelectrons of 0.23 eV), -1.2 to -3.7 % at 8.5-9.5 eV and +0.5 to +1.1 % at 10-10.5 eV (both m alike), which sets the uncertainty: 0.18 to 10.9 %. Which reading is right is not established; the runs’ summary says the two agree ‘elsewhere <= 2 %’, which holds at 8.5, 10, 10.5 and 11 eV and not at 9 and 9.5 eV (2.4 to 3.7 %).
THE AMPLITUDES (two_photon_transition_amplitude; run 4). One-electron Ar 3p -> 4p with degenerate photons, the lowest-order amplitudes of run4/pt_two_photon.py (Dalgarno-Lewis on a uniform radial grid in the same potential, length gauge): M_0 = +8.914061 au and M_1 = +2.418379 au (dr = 0.002 bohr; dr = 0.004 gives +8.902447 and +2.416898; R_s = +17.06867, R_d = +12.09189 au, M_0 = R_s/3 + 4 R_d/15, M_1 = R_d/5). Rerun by the curator on 2026-10-11 with the archived script, to every printed digit. Each is ONE time ordering, <4p m| z G z |3p m>; the vocabulary’s kernel M_zz sums both orderings, equal for degenerate photons, so m_zz_per_unit = 2 (the run’s P = |M_m|^2 |S|^2, S = int E+(t)^2 exp(i w_fi t) dt, is OEP 8’s P = |M_zz|^2 |S2|^2 / 4). The TDSE (velocity gauge) 4p populations of 30 and 50 fs pulses at 6.463 eV give TDSE/PT = 0.9982 to 0.9996 at 1e10 W/cm2 and 0.9875 to 0.9944 at 1e11 (both m), confirming |M_m| to 0.1 %. The signs are perturbation theory’s, not a projection of the TDSE wave function (a deviation from the brief: tRecX writes psi(T) only in its binary format with an arbitrary eigensolver phase). They are per initial 3p m of one electron, not per LS level of the atom; the recoupling into 1S and 1D strengths is not made here.
HELD IN THE ARCHIVE, NO QUANTITY (runs 2, 3’s two-colour points, 5 and 6; summary_runs2to6.md and the run tables). NOT TAKEN (no quantity): a two-colour continuum kernel is not in the vocabulary (a continuum two-photon amplitude was declined, TC-1040). - Run 2, He two-colour kernel, each colour 1e12 W/cm2, 20 fs, both models (one-electron at BOX 80): the integral of the cross peak at omega1 + omega2 - I_p against a one-colour reference at Omega/2 with the per-colour intensity; the ratio includes the two-colour factor 4. haCC at Omega/2 = 14 eV: 4.005, 4.025, 4.105, 4.451, 5.143 for nu = omega1 - omega2 = 0.5, 1, 2, 4, 6 eV (one electron 4.007 to 5.358); at 17 eV 4.006 to 6.244 (one electron 4.009 to 7.657); at 20 eV 4.106 (nu 0.5), 4.543 (1) and 1.937 (4, omega1 = 22 eV beyond the 1s2p; one electron 1.133). The references reproduce run 1: haCC 12.805 (14 eV) and 10.176 (20 eV) e-52 cm4 s; one electron 7.399 (14 eV spectrum) and 7.328 (14 eV projection, equal to run 1’s at BOX 40), and at 17 eV, which run 1 lacks, haCC 9.774 and one electron 6.268 (spectrum) or 6.312 (projection) e-52 cm4 s. - Run 3, Ar two-colour, 1e12 per colour, 20 fs: the atom’s cross peak at Omega/2 = 8.5, 9.5 (nu 0.5 to 2) and 10.5 eV (nu 0.5), e.g. 6.2158e-4 to 6.6544e-4 at 8.5 eV; its one-colour references at 9.5 and 10.5 eV are the sigma2_sae_Ar rows 9.5 and 10.5, which the runs’ summary does not print (computed here the same way as its five). - Run 5, He 1s2s in haCC at 120.3 nm (10.306 eV per photon, 127 meV below haCC’s propagated two-photon resonance, 20.739 eV; see run 5b): final populations of 1s2s, 1s3s, 1s3d and the ionised fraction for 1, 3 and 10 fs (transform-limited) and 3 fs with +-10 fs2 (9.72 fs, peak field 0.556 E0) at 1e12, 1e13, 1e14 W/cm2. P(1s2s) at 1e12: 3.6453e-7 (1 fs), 2.7243e-6 (3 fs), 1.5568e-5 (10 fs), 8.9781e-7 (+10 fs2), 7.7456e-7 (-10 fs2); at 1e14 and 10 fs 1.2615e-2 with 2.433e-2 ionised. These are single samples at the end of the window; the 16 % chirp difference at 1e12 is a beat (below). haCC’s states 5 (-0.05555 au from He+) and 6 (-0.05474) are 1s3d (populated) and 1s3p (not); state 3 (-0.07776) is never populated and is unidentified. P(1s2s) against time is on the cluster only (run5/hacc/<name>/pop_vs_t.csv). - Run 5b, the campaign’s addendum 3 (11 haCC runs, 11 Oct 2026; run5b/results.csv, run5b/scan_fit.csv). P(1s2s)(t) after the pulse does not decay (<= 0.3 % per 1000 au) but beats at 0.7621 au (20.738 eV) with the ground state, through the OccDual projector’s floor |eps|^2 = 2.862e-9 (a no-field run); so the post-pulse mean, less 2.9e-9, is the population. A 10 fs, 1e12 carrier scan (+-75 meV per photon) fitted to P = K |S2(Omega)|^2 gives Omega = 20.7386 eV (to a few meV) and K = 3.078 au, i.e. |M|^2 = 4K = 12.31 au^2. With these, run 5 is lowest order to <= 1 % at 1e12 for 3 and 10 fs and both chirps (3 % at 1 fs); the +-10 fs2 means differ by 0.07 % at 1e12 (accuracy 1e-9), 0.6 % at 1e13 and 6.6 % at 1e14. Ratios to lowest order at 1e13 and 1e14: 1.013, 0.952 (1 fs); 0.983, 0.748 (3 fs); 0.858, 0.079 (10 fs); 0.989, 0.839 (+10 fs2); 0.983, 0.788 (-10 fs2). Not taken: no quantity holds a population (TC-1051). - Run 5c, the campaign’s addendum 4 (5 haCC runs, 11-12 Oct 2026; run5c/results.csv, run5c/scan_fit.csv): one photon at 21.21 eV + {0, +-50, +-100} meV, 10 fs, 1e9 W/cm2. The P(1s2p) beat asked for cannot exist (the 1P projector cannot see the 1S ground state; floor 1.27e-19), so the beat of <z>(t) after the pulse was fitted instead: Delta = 0.7794133 au = 21.208918 eV in the length and velocity forms (spread 0.9 micro-eV over 10 fits). A fit of P = K |S1(Omega)|^2 to the scan gives 21.20892 eV and K = |z|^2 = 0.169939 au^2; P(1s2p) is first order to 5e-5. The propagated 1s2p is the Occupation set’s to 1e-8 au, so only the 1S excited level is misplaced in the Occupation set, by +0.1 eV, cause unknown. The velocity-to-length ratio of the beat amplitudes is 5.9 % from exact-eigenstate equality: the two dipole forms are not equivalent in this one-channel basis. Taken: dz_1s2p_hacc (TC-1052). - Run 6, one-electron He at 14 eV, broadband: the yield from the spectrum, 1.1180e-7, 2.2234e-7, 5.5483e-7 for 1, 2, 5 fs at 1e12 (1.1162e-5, 2.2200e-5, 5.5405e-5 at 1e13); 1 fs with +5 and -5 fs2 (13.9 fs, peak field 0.269 E0) 7.9470e-9 and 7.9466e-9 (equal to 0.01 %). The projection is biased by the no-field drift there (3.0e-8 over the 1 fs window, longer for the chirped ones): use the spectrum.
NOT TAKEN: the 1e11 and 1e13 rows (the perturbative check above; sigma2 is taken at the one intensity the brief names); the one-electron spectra at BOX 40 (-7.4 to +9.0 % from the projection); run 4’s TDSE populations (they confirm the amplitudes); run 1’s first version (run1_v1_spec_kmax0p02, tRecX’s default spectral range far below the peaks; not archived). Not run: a length-gauge rerun and the optional haCC argon point (mMax 1, 9.5 eV, 6 fs), both available on request (the runs’ report).
Datasets#
travers_2026_trecx_two_photon/sigma2_hacc_He#
{
"id": "sigma2_hacc_He",
"material": "He",
"file": "He/travers_2026_trecx_two_photon__sigma2_hacc_He.csv",
"table": "data/external/trecx/two_photon/run1/results_hacc.csv",
"quantity": "multiphoton_ionization_cross_section",
"unit": "cm4 s",
"abscissa_unit": "eV",
"use": "check",
"uncertainty": {
"relative": 0.002535,
"level": "bound"
},
"parameters": {
"photons": 2,
"polarization": "linear",
"gauge": "velocity",
"intensity_w_cm2": 1000000000000.0,
"pulse_fs": 20,
"pulse_fs_definition": "full width at half maximum of the intensity envelope (the square of the envelope of the vector potential)",
"pulse_envelope": "gaussian",
"envelope_of": "vector potential",
"cep": 0,
"channel": "total",
"method": "TDSE, tRecX a987d7c, haCC (He19_Standard, 1 ion channel, lMax 10, rMax 40), velocity gauge; P the two-photon peak of the iSurff spectrum",
"definition": "sigma2 = P / int Phi^2 dt, Phi = I(t)/(hbar omega), I(t) = I0 exp(-4 ln2 t^2/T^2), T = 20 fs; int Phi^2 dt = (I0/hbar omega)^2 T sqrt(pi/(8 ln2)); I0 = (1/2) eps0 c E0^2",
"energy_axis": "photon energy of the run; the model's own threshold, 24.5410 eV (-E of haCC's initial state), and its 1s2p 21.209 eV; no shift",
"electrons": "both (haCC)",
"model_ionization_energy_ev": 24.541
}
}
Helium in haCC (He19_Standard, one ion channel He+ 1s, lMax 10, rMax 40), run 1: the degenerate generalised two-photon ionisation cross section at 1e12 W/cm2 in a 20 fs Gaussian, P the integral of the iSurff photoelectron spectrum over +-0.5 eV around the two-photon peak (column sigma2_peak2_cm4s, the table’s string; the script recomputes it from P_peak2 to 1e-12). Runs he_<E>eV_I1e12. The norm-loss gives the same to 0.1 %. Photon energies 13, 14, 16, 18, 20 eV, below the model’s 1s2p (21.209 eV); the threshold is the model’s, 24.5410 eV. Total over 1S and 1D. 0.25 %: the record’s rule (rMax 60 the largest change; one photon energy, 16 eV).
travers_2026_trecx_two_photon/sigma2_sae_He#
{
"id": "sigma2_sae_He",
"material": "He",
"file": "He/travers_2026_trecx_two_photon__sigma2_sae_He.csv",
"table": "data/external/trecx/two_photon/run1/results_sae.csv",
"quantity": "multiphoton_ionization_cross_section",
"unit": "cm4 s",
"abscissa_unit": "eV",
"use": "check",
"uncertainty": {
"relative_rows": {
"13.0": 0.01865,
"14.0": 0.01607,
"16.0": 0.01607,
"18.0": 0.01607,
"20.0": 0.01607
},
"level": "bound"
},
"parameters": {
"photons": 2,
"polarization": "linear",
"gauge": "velocity",
"intensity_w_cm2": 1000000000000.0,
"pulse_fs": 20,
"pulse_fs_definition": "full width at half maximum of the intensity envelope (the square of the envelope of the vector potential)",
"pulse_envelope": "gaussian",
"envelope_of": "vector potential",
"cep": 0,
"channel": "total",
"method": "TDSE, tRecX a987d7c, one active electron in Tong and Lin's He potential, velocity gauge, irECS, BOX 40; P the bi-orthogonal projection on the unbound states less the no-field drift",
"definition": "sigma2 = P / int Phi^2 dt, Phi = I(t)/(hbar omega), I(t) = I0 exp(-4 ln2 t^2/T^2), T = 20 fs; int Phi^2 dt = (I0/hbar omega)^2 T sqrt(pi/(8 ln2)); I0 = (1/2) eps0 c E0^2",
"energy_axis": "photon energy of the run; the model's own threshold, 24.5937 eV, and its 1s2p 21.056 eV; no shift",
"electrons": "one active electron",
"potential": "Tong and Lin 2005, Table 1, He: V = -(1 + 1.231 exp(-0.662 r) - 1.325 r exp(-1.236 r) - 0.231 exp(-0.480 r))/r",
"model_ionization_energy_ev": 24.5937
}
}
One-electron helium (Tong and Lin’s potential, 1s), run 1, BOX 40: the degenerate generalised two-photon ionisation cross section at 1e12 W/cm2 in a 20 fs Gaussian, P the bi-orthogonal projection 1 - (bound states, Re E < 0) at the end of the window (P_ion_proj_end) less that of the no-field run he_nofield (1.431520e-7, the discretisation’s drift over 5961 au); the projection is nearly box-independent (BOX 80 and 120 within 0.02 % at 16 eV, 0.29 % at most over the five energies). Runs he_<E>eV_I1e12. The value is computed by the script (6 digits); the runs’ summary prints it to 4 (7.456, 7.328, 6.524, 6.228, 9.346 e-52), all matched. Photon energies 13, 14, 16, 18, 20 eV, below the model’s 1s2p (21.056 eV); the threshold is the model’s, 24.5937 eV. Total over 1S and 1D (one electron: s and d waves). 1.6 to 1.9 %: the record’s rule (step and cutEnergy after their own drift; the BOX 120 spectrum at 13 eV).
travers_2026_trecx_two_photon/sigma2_sae_Ar#
{
"id": "sigma2_sae_Ar",
"material": "Ar",
"file": "Ar/travers_2026_trecx_two_photon__sigma2_sae_Ar.csv",
"table": "data/external/trecx/two_photon/run3/results.csv",
"quantity": "multiphoton_ionization_cross_section",
"unit": "cm4 s",
"abscissa_unit": "eV",
"use": "check",
"uncertainty": {
"relative_rows": {
"8.0": 0.109,
"8.5": 0.01757,
"9.0": 0.02607,
"9.5": 0.0364,
"10.0": 0.005204,
"10.5": 0.0113,
"11.0": 0.001754
},
"level": "bound"
},
"parameters": {
"photons": 2,
"polarization": "linear",
"gauge": "velocity",
"intensity_w_cm2": 1000000000000.0,
"pulse_fs": 20,
"pulse_fs_definition": "full width at half maximum of the intensity envelope (the square of the envelope of the vector potential)",
"pulse_envelope": "gaussian",
"envelope_of": "vector potential",
"cep": 0,
"channel": "total",
"method": "TDSE, tRecX a987d7c, one active electron in Tong and Lin's Ar potential (3p, m = 0 and m = 1), velocity gauge, irECS, BOX 80; P the two-photon peak of the iSurff spectrum; the atom 2 P(m = 0) + 4 P(m = 1)",
"definition": "sigma2 = P_atom / int Phi^2 dt, Phi = I(t)/(hbar omega), I(t) = I0 exp(-4 ln2 t^2/T^2), T = 20 fs; int Phi^2 dt = (I0/hbar omega)^2 T sqrt(pi/(8 ln2)); I0 = (1/2) eps0 c E0^2",
"energy_axis": "photon energy of the run; the model's own threshold, 15.774 eV, and its 4s 11.267 eV; no shift",
"electrons": "one active electron per 3p orbital; the atom as 2 P(m = 0) + 4 P(m = 1)",
"potential": "Tong and Lin 2005, Table 1, Ar: V = -(1 + 16.039 exp(-2.007 r) - 25.543 r exp(-4.525 r) + 0.961 exp(-0.443 r))/r",
"model_ionization_energy_ev": 15.774
}
}
One-electron argon (Tong and Lin’s potential, 3p), run 3, BOX 80: the degenerate generalised two-photon ionisation cross section of the atom at 1e12 W/cm2 in a 20 fs Gaussian, P = 2 P(m = 0) + 4 P(m = 1) (the six 3p electrons as independent; agreed for the runs), each P the integral of its run’s iSurff spectrum over +-0.25 eV around the two-photon peak (P_p11). Runs ar_m0_r3_<E>eV_I1e12 and ar_m1_r3_<E>eV_I1e12. The value is computed by the script (6 digits); the runs’ summary prints 8, 8.5, 9, 10 and 11 eV to 4 (1.921, 1.916, 2.094, 2.503, 3.052 e-50), all matched; 9.5 and 10.5 eV are run 3’s two-colour references, the same pulse and intensity, not printed. The model’s threshold 15.774 eV and 4s 11.267 eV; 11 eV is 0.27 eV below the 4s and near-resonant (P at 1e13 15 % below I^2 scaling for m = 0; 1e11 within 1.1 % of 1e12). Not the atom’s 4s[3/2]1 or 4s’[1/2]1 (no spin-orbit). The uncertainty is the record’s rule: the spectrum against the drift-corrected projection dominates, +10.9 % at 8 eV (0.23 eV photoelectrons).
travers_2026_trecx_two_photon/M_4p_m0#
{
"id": "M_4p_m0",
"material": "Ar",
"table": "data/external/trecx/two_photon/extract_runs.py.txt",
"quantity": "two_photon_transition_amplitude",
"unit": "au",
"abscissa_unit": "eV",
"use": "check",
"uncertainty": {
"relative": 0.0013,
"level": "bound"
},
"parameters": {
"final_level": "4p (one-electron Tong-Lin model; m = 0)",
"gauge": "length",
"polarization": "linear_parallel",
"definition": "M_m = <4p m| z (E_3p + omega - H)^-1 z |3p m> (atomic units), one time ordering, hbar omega = (E_4p - E_3p)/2; M_0 = R_s/3 + 4 R_d/15, M_1 = R_d/5, R_l = int u_4p(r) r f_l(r) dr with (E_3p + omega - H_l) f_l = r u_3p (run4/pt_two_photon.py)",
"phase_convention": "u = rR real, normalised, positive near the origin; spherical harmonics with the Condon-Shortley phase",
"m_zz_per_unit": 2,
"method": "lowest-order perturbation theory (Dalgarno-Lewis) in Tong and Lin's Ar potential, one electron, uniform radial grid dr = 0.002 bohr to 150 bohr, second-order differences; |M_m| confirmed by the velocity-gauge TDSE (tRecX a987d7c) to 0.1 %",
"potential": "Tong and Lin 2005, Table 1, Ar: V = -(1 + 16.039 exp(-2.007 r) - 25.543 r exp(-4.525 r) + 0.961 exp(-0.443 r))/r"
}
}
1 inline points in the source record.
One-electron argon 3p (m = 0) -> 4p (m = 0), degenerate photons at half the grid’s 4p - 3p (6.46351 eV; the model’s 4p 12.926 eV, the TDSE runs at 6.463 eV): the lowest-order length-gauge amplitude of run4/pt_two_photon.py (dr = 0.002 bohr), +8.914061 au as extract_runs.py holds it (M_PT) and as the curator’s rerun prints it (+8.914061e+00); the runs’ summary prints +8.914. ONE time ordering: m_zz_per_unit = 2 (the record’s notes). TDSE/PT of the 4p population 0.9996 (30 fs) and 0.9990 (50 fs) at 1e10 W/cm2. 0.13 %: the change from dr = 0.004 (+8.902447). Sign from perturbation theory, not from the TDSE. Not an LS level of the atom.
travers_2026_trecx_two_photon/M_4p_m1#
{
"id": "M_4p_m1",
"material": "Ar",
"table": "data/external/trecx/two_photon/extract_runs.py.txt",
"quantity": "two_photon_transition_amplitude",
"unit": "au",
"abscissa_unit": "eV",
"use": "check",
"uncertainty": {
"relative": 0.0009,
"level": "bound"
},
"parameters": {
"final_level": "4p (one-electron Tong-Lin model; m = 1)",
"gauge": "length",
"polarization": "linear_parallel",
"definition": "M_m = <4p m| z (E_3p + omega - H)^-1 z |3p m> (atomic units), one time ordering, hbar omega = (E_4p - E_3p)/2; M_0 = R_s/3 + 4 R_d/15, M_1 = R_d/5, R_l = int u_4p(r) r f_l(r) dr with (E_3p + omega - H_l) f_l = r u_3p (run4/pt_two_photon.py)",
"phase_convention": "u = rR real, normalised, positive near the origin; spherical harmonics with the Condon-Shortley phase",
"m_zz_per_unit": 2,
"method": "lowest-order perturbation theory (Dalgarno-Lewis) in Tong and Lin's Ar potential, one electron, uniform radial grid dr = 0.002 bohr to 150 bohr, second-order differences; |M_m| confirmed by the velocity-gauge TDSE (tRecX a987d7c) to 0.1 %",
"potential": "Tong and Lin 2005, Table 1, Ar: V = -(1 + 16.039 exp(-2.007 r) - 25.543 r exp(-4.525 r) + 0.961 exp(-0.443 r))/r"
}
}
1 inline points in the source record.
One-electron argon 3p (m = 1) -> 4p (m = 1), as M_4p_m0: +2.418379 au (extract_runs.py’s M_PT and the curator’s rerun, +2.418379e+00; the summary prints +2.418); M_1/M_0 = +0.2713. ONE time ordering: m_zz_per_unit = 2. TDSE/PT of the 4p population 0.9986 (30 fs) and 0.9982 (50 fs) at 1e10 W/cm2. 0.09 %: the larger of the change from dr = 0.004 (+2.416898, 0.06 %) and the TDSE’s |M_1| (sqrt(0.9982), 0.09 %). Sign from perturbation theory (R_d > 0). Not an LS level of the atom.
travers_2026_trecx_two_photon/dz_1s2p_hacc#
{
"id": "dz_1s2p_hacc",
"material": "He",
"table": "text",
"quantity": "transition_dipole_moment",
"unit": "au",
"abscissa": "label",
"use": "check",
"parameters": {
"states": "1s2 1S0 - 1s2p 1P1",
"component": "z, M = 0, linear polarisation the axis; magnitude",
"method": "two-electron TDSE (tRecX haCC, one ionic channel), first-order fit of a carrier scan"
}
}
1 inline points in the source record.
Run 5c (REPORT.md, addendum 4): haCC He19, one channel, lMax 10, rMax 40. |z| = sqrt(K) from the five-carrier scan fit P = K |S1(Omega)|^2 (K = 0.169939 au^2, gauge-free); the length-form beat amplitude with OccDual populations gives 0.412236. NIST’s f = 0.2762 (f = 2 Delta |z|^2 for M = 0) gives 0.4208: haCC is 2.0 % low (f 0.2649, 4.1 % low). A model’s value: a check.
Formulas#
No formulas are recorded.
Corrections#
No dataset errata are recorded in this source record.
Formula corrections, where recorded, appear with their formulas above. Curation notes may also discuss corrections or unresolved interpretation. Consult the review register for the checks and decisions.