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 `__ · :doc:`Catalogue ` · :doc:`Review register ` DOI: `10\.1103/PhysRevA\.42\.6333 `__ **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 ~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "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 ~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "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 ~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "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 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ .. code-block:: text { "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 :doc:`review register ` for the checks and decisions.