drachman_1990#
R. J. Drachman, A. K. Bhatia and A. A. Shabazz, “Two-photon transitions in hydrogen: A test of pseudostate summation”, Phys. Rev. A 42, 6333-6335 (1990)
Source record · Catalogue · Review register
Source kind: theory
Recorded technique: Second-order perturbation theory of the hydrogen 1S-2S two-photon transition with the intermediate p states replaced by N0 pseudostates, the eigenvectors of H0 = -nabla^2 - 2/r (rydberg) in the basis exp(-a r) r^(1+j) P1, j = 0..N0-1 (Eqs. 1-4), and their eigenvalues in the energy denominators; length form D1 (Eq. 5), momentum form D2 (Eq. 6) and the momentum form rewritten by the commutator relation, D3 (Eq. 8). Scale a = 1 (and 0.55); N0 = 5, 10, 15, 20; multiple precision for the larger N0. Non-relativistic, infinite nuclear mass.
Materials in structured entries: H
Dataset quantities: two_photon_transition_amplitude
Contents: 4 datasets; 0 formulas.
Curation notes#
Journal pp. 6333-6335 = PDF pages 1-3 (journal page = PDF page + 6332). Identity checked on the first page (PDF p. 1: ‘PHYSICAL REVIEW A VOLUME 42, NUMBER 11 1 DECEMBER 1990’, Richard J. Drachman and A. K. Bhatia (Goddard), Abdulalim A. Shabazz (Clark Atlanta), received 25 June 1990; footer ‘6333’) on 2026-10-09. Atomic hydrogen is the optmat material H; Table IV is taken under two_photon_transition_amplitude (approved for bound final levels, TC-1040).
THE AMPLITUDE (Eq. 5, p. 6334): D1 = (1/2) sum_N [1/(1 + E_N - omega) + 1/(1/4 + E_N + omega)] R_1S^N R_2S^N, with R the radial integrals of r (Eq. 7), energies in rydberg and omega one photon’s energy, the other 3/4 - omega. It is quattropani_1982’s D_1^n (its Eq. 3a) for n = 2: in atomic units M_zz = (4/3) D and, at omega = 3/8, delta^TP = (16/9) D^2 = 246.7196 au for D = -11.780483 (sigma_0 = 2.246e-35 cm4 in TC-952’s convention); D(omega)/D(3/8) is OEP 8’s two-colour kernel M(omega_a, omega_b)/M(Omega/2, Omega/2) of the 1S-2S level, with omega_a = omega and omega_b = 3/4 - omega (rydberg). Kernel at the printed omega: 1.2505 (0.5250), 3.4929 (0.6750), 4.2178 (0.6875), 5.3189 (0.7000), 7.1750 (0.7125), 10.923 (0.7250), 22.254 (0.7375), 113.27 (0.7475), the last 0.0025 rydberg below the 2p pole at 0.75. Its curvature at omega = 3/8 gives OEP 8’s kappa = 36.83 hartree^-2 and D_eff = 0.879 D_1 (quattropani_1982’s notes).
TAKEN (tools/extract/drachman_1990.py, from the library’s JSON cells, checked against the table image): Table IV (p. 6335, tbl_a9b929131d75), at nine omega (0.3750-0.7475 rydberg), the N0 = 15 lines (a = 1) of D1 (gauge length, use = primary), D2 (gauge velocity, use = primary) and D3 (the velocity form rewritten by the commutator relation, use = check: exact only for true eigenstates, it converges more slowly), and the column ‘Ref. 1’ (use = reproduced, from F. Bassani, J.-J. Forney and A. Quattropani, Phys. Rev. Lett. 39, 1070 (1977), not in the library). m_zz_per_unit = 4/3 (above; quattropani_1982’s notes). The printed digit groups (’-11.780 483’) are joined in value and kept in printed. Not taken: the N0 = 5 and 10 lines (convergence).
CHECKED by the script against the exact amplitude (quattropani_1982’s script, a Dalgarno-Lewis solution in Decimal arithmetic): at all nine omega the N0 = 15 D1 and D2 agree with the exact values to the printed digits (D2 at 0.7475, -1334.3262, against -1334.32614, one in the last digit), and the ‘Ref. 1’ column agrees with them to its printing (e.g. -11.7805, -1334.33). D3 converges more slowly, as the text says (N0 = 15: 2e-6 to 3e-6 relative). Tables II and III (p. 6334-6335, tbl_91240e2dabf7 and tbl_d35f7eb262b9), the cumulative sums at omega = 0.3750 for a = 1 and 0.55, end at -11.78048 for D1 and D2 in both, the exact value; their first term, -17.87845, is the 2p term alone, (1/2)(8/3 + 8/3) R_1S^2p R_2S^2p with R = 1.29027 and -5.19615. The parenthesised ‘Ref. 1’ discrete sums (-13.3824, -4.6924) are reproduced, not checked.
TABLE I (p. 6334, tbl_c412b3ea6782), the pseudostate energies (not physical values; NOT TAKEN), recomputed here from Eqs. (3)-(4) with N0 = 20: the first column (a = 1) to every printed digit (-0.111110988, -0.062144252, ..., 87.071218883), and the second to every printed digit with a = 0.55 (-0.0399937373, ..., 24.376920544), not with the heading’s ‘a = 0.5’ (a misprint; the caption and Table III say 0.55). With a = 0.55 six energies are negative, as printed; the text’s ‘seven’ is what a = 0.5 gives. Eq. (4)’s H and Delta are the matrices of Eq. (1)’s functions with j and k counted from 1 (with j = 0 they hold r^0 exp(-a r), not a p function, and give other energies).
The paper’s omega grid at 0.6750-0.7475 is that of quattropani_1982’s Table I(a) (1s-3s); the two papers share no final level, so they are compared through the exact amplitude, which both match.
Recorded bibliographic identity status: VALIDATED_WITH_WARNINGS. This concerns identification of the source; it does not establish the scientific correctness of its contents.
Datasets#
drachman_1990/D1_N15#
{
"id": "D1_N15",
"material": "H",
"file": "H/drachman_1990__D1_N15.csv",
"table": "tbl_a9b929131d75",
"quantity": "two_photon_transition_amplitude",
"unit": "1",
"abscissa": "wavelength",
"abscissa_unit": "rydberg",
"use": "primary",
"parameters": {
"final_level": "2s",
"gauge": "length",
"polarization": "linear_parallel",
"definition": "D1 = (1/2) sum_N [1/(1 + E_N - omega) + 1/(1/4 + E_N + omega)] R_1S^N R_2S^N (Eq. 5), R_kS^N = int dr r^3 chi_N chi_kS (Eq. 7), energies in rydberg, the second photon 3/4 - omega",
"m_zz_per_unit": 1.3333333333333333,
"method": "pseudostate summation, N0 = 15 eigenvectors of H0 in the basis exp(-a r) r^(1+j) P1, a = 1, eigenvalues in the denominators; non-relativistic, infinite nuclear mass"
}
}
Table IV, column D1, the third line of each omega (N0 = 15, a = 1). Agrees with the exact amplitude to the printed digits (tools/extract/drachman_1990.py). x and row: omega, one photon’s energy as printed (rydberg).
drachman_1990/D2_N15#
{
"id": "D2_N15",
"material": "H",
"file": "H/drachman_1990__D2_N15.csv",
"table": "tbl_a9b929131d75",
"quantity": "two_photon_transition_amplitude",
"unit": "1",
"abscissa": "wavelength",
"abscissa_unit": "rydberg",
"use": "primary",
"parameters": {
"final_level": "2s",
"gauge": "velocity",
"polarization": "linear_parallel",
"definition": "D2 = -2 sum_N [1/(1 + E_N - omega) + 1/(1/4 + E_N + omega)] Q_1S^N Q_2S^N / (omega (3/4 - omega)) (Eq. 6), Q_kS^N = int dr r^2 chi_N chi_kS' (Eq. 7), energies in rydberg, the second photon 3/4 - omega",
"m_zz_per_unit": 1.3333333333333333,
"method": "pseudostate summation, N0 = 15 eigenvectors of H0 in the basis exp(-a r) r^(1+j) P1, a = 1, eigenvalues in the denominators; non-relativistic, infinite nuclear mass"
}
}
Table IV, column D2, the third line of each omega (N0 = 15, a = 1). Agrees with the exact amplitude to the printed digits, within one unit of the last at 0.7475 (-1334.3262 against -1334.32614). x and row: omega, one photon’s energy as printed (rydberg).
drachman_1990/D3_N15#
{
"id": "D3_N15",
"material": "H",
"file": "H/drachman_1990__D3_N15.csv",
"table": "tbl_a9b929131d75",
"quantity": "two_photon_transition_amplitude",
"unit": "1",
"abscissa": "wavelength",
"abscissa_unit": "rydberg",
"use": "check",
"parameters": {
"final_level": "2s",
"gauge": "velocity, rewritten by the commutator relation (Eq. 8)",
"polarization": "linear_parallel",
"definition": "D3 = -(1/2) sum_N [1/(1 + E_N - omega) + 1/(1/4 + E_N + omega)] R_1S^N R_2S^N (1 + E_N)(1/4 + E_N) / (omega (3/4 - omega)) (Eq. 8), Eq. 6 rewritten by nabla = [r, H0]/2, energies in rydberg",
"m_zz_per_unit": 1.3333333333333333,
"method": "pseudostate summation, N0 = 15 eigenvectors of H0 in the basis exp(-a r) r^(1+j) P1, a = 1, eigenvalues in the denominators; non-relativistic, infinite nuclear mass"
}
}
Table IV, column D3, the third line of each omega (N0 = 15, a = 1). The commutator form is exact only for true eigenstates; with pseudostates it converges more slowly (the text): 2e-6 to 3e-6 relative from the exact amplitude at N0 = 15. Kept as a check, not fitted beside D1 and D2. x and row: omega, one photon’s energy as printed (rydberg).
drachman_1990/ref1_bassani_1977#
{
"id": "ref1_bassani_1977",
"material": "H",
"file": "H/drachman_1990__ref1_bassani_1977.csv",
"table": "tbl_a9b929131d75",
"quantity": "two_photon_transition_amplitude",
"unit": "1",
"abscissa": "wavelength",
"abscissa_unit": "rydberg",
"use": "reproduced",
"reproduced_from": "bassani_1977",
"parameters": {
"final_level": "2s",
"gauge": "length (Ref. 1, Table I, D[J0]; TC-1044)",
"polarization": "linear_parallel",
"definition": "the dimensionless 1s-2s amplitude of Ref. 1, the convention of Eq. 5 (Drachman et al. compare it term by term)",
"m_zz_per_unit": 1.3333333333333333,
"method": "exact intermediate states, discrete sum and continuum integral (Bassani, Forney and Quattropani 1977)"
}
}
Table IV, column ‘Ref. 1’: F. Bassani, J.-J. Forney and A. Quattropani, Phys. Rev. Lett. 39, 1070 (1977), held as bassani_1977 (TC-1044): the column is its length-gauge Table I (D[J0]), string for string, including -49.6878 at nu_1 = 0.6875 where its velocity table prints -49.6877. Agrees with the exact amplitude to its printing. x and row: omega, one photon’s energy as printed (rydberg).
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.