DOI: 10.5061/dryad.cjsxksnkv
title: Data tables of matrix elements used to calculate the ionization energies of the Rydberg P-states of helium for principal quantum number 2 ≤ n ≤ 35

description: The archive contains data tables used to calculate the nonrelativistic,
 relativistic and quantum electrodynamic energies of the P-states of helium
 up to n = 35. For each n, there is a text file entitled He_nPfinl.txt
 located in He_nPfinl.tar.gz that contains a summary of all matrix elements
 used in the calculations for both singlet and triplet states, together
 with convergence tables and uncertainty estimates. At the end of each text
 file, there is a summary table containing calculated ionization energies
 of the respective atomic states that can be compared directly with
 experimental measurements, in analogy with Table IV and the 11 comparisons
 with experiment contained in the parent article
 https://link.aps.org/doi/10.1103/1dbh-rjjn. The additional tables of
 matrix elements may be useful to others wishing to replicate the present
 results, or search for systematic trends. The matrix elements were
 calculated with respect to two-electron wave functions obtained from
 variational solutions to the quantum mechanical three-body problem in
 fully correlated Hylleraas coordinates, and using extended precision
 arithmetic.
description: ## Theory for the Rydberg states of helium: Results for 2 ≤ n ≤ 35 and
 comparison with experiments for the singlet and triplet P states Dataset
 DOI: [10.5061/dryad.cjsxksnkv](https://doi.org/10.5061/dryad.cjsxksnkv) ##
 Description of the data and file structure The archive contains data
 tables used to calculate the nonrelativistic, relativistic and quantum
 electrodynamic energies of the singlet and triplet P-states of helium up
 to principal quantum number *n* = 35, according to the discussion and
 analysis in the parent article
 [https://link.aps.org/doi/10.1103/1dbh-rjjn](https://link.aps.org/doi/10.1103/1dbh-rjjn). ### Files and variables in He_nPfinl.tar.gz ##### Overview For each *n = 2, 3, 4, ..., 35*, there is a text file in He_nPfinl.tar.gz entitled He_*n*Pfinl.txt that contains a summary of all matrix elements used in the calculations for both singlet and triplet states, together with convergence tables and uncertainty estimates. For example, He_2Pfinl.txt contains the matrix elements and energies for the 1s2p ^1^P and 1s2p ^3^P states of helium, and similarly up to He_35Pfinl.txt for the 1s35p ^1^P and 1s35p ^3^P states of helium. The matrix elements are calculated with variational wave functions in triple Hylleraas basis set. For each n, the spectroscopic notation 1snp ^1^P and 1snp ^3^P for the singlet and triplet P-states is abbreviated to n 1P and n 3P respectively throughout the tables. In summary, the tables are: 1\. spin-independent matrix elements 2\. spin-dependent matrix elements 3\. matrix elements of other common operators 4\. nonrelativistic energies for infinite nuclear mass 5\. nonrelativistic energies for the ^4^He finite nuclear mass 6\. second-order mass polarization corrections 7\. summary of relativistic spin-dependent and total reduced mass corrections (MHz) 8\. anomalous magnetic moment corrections (MHz) 9\. spin-dependent and total α^2^ relativistic corrections (MHz) 10\. fine-structure splittings and the singlet-triplet splitting (MHz) 11\. matrix elements to diagonalize the 2 x 2 singlet-triplet mixing energy matrix 12\. contributions to the quantum electrodynamic energy shift of various orders of α 13\. a final table analogous to Table IV of [https://link.aps.org/doi/10.1103/1dbh-rjjn](https://link.aps.org/doi/10.1103/1dbh-rjjn) giving contributions to the energy and a final total ionization energy in MHz. **This is the main table of interest to most readers.** ##### *Variables* * RY0 is the value of the Rydberg R∞ in units of cm⁻^1^ * RYM is the value of the reduced mass Rydberg R~μ~ = (1 - μ/*M*)R∞ in units of cm⁻^1^ * 1/ALPHA is the inverse fine structure constant 1/α * mu/M is the ratio of the reduced electron mass to the nuclear mass μ/*M = m~e~*/*(M* + *m*~e~), where *m*~e~ is the electron mass and *M* is the nuclear mass for ^4^He * NTERM is the number of terms in the variational basis set for the wave function, with Extp being the extrapolated value at infinity, and +/- being the estimated uncertainty * Units are reduced mass atomic units, with 𝑎~μ~ = (*m*~e~/μ)𝑎~0~ the unit of distance and *E*~μ~ = *e*^2^/(4πε~0~𝑎~μ~) the unit of energy. * For each matrix element ⟨A⟩ in the preceding sections 1 and 2, the tables are divided into two pairs of two blocks each for the singlet-P and triplet-P cases respectively. The first block of each pair gives the value ⟨A∞⟩ for the case of infinite nuclear mass, and the second block of the pair gives the finite mass correction ⟨A~μ~⟩ expressed as the coefficient of μ/*M*, such that the total matrix element has the form ⟨A⟩ = (1-μ/*M*)^k^[ ⟨A∞⟩ + (μ/*M*)⟨A~μ~⟩], with k = 4 for the Breit interaction term ⟨B~1~⟩, and k = 3 for all the others. ###### Table Headings * \ is the matrix element of the mass polarization term ⟨**p**~1~·**p**~2~⟩ in the nonrelativistic Hamiltonian. * \ is the p^4^ term in the Breit interaction defined after Eq. (6) of [https://link.aps.org/doi/10.1103/1dbh-rjjn](https://link.aps.org/doi/10.1103/1dbh-rjjn), divided by α^2^. Note that for small basis sets with Ω &lt; *n* (where Ω is the highest power in the basis set), some of the results become numerically unstable due to large coefficients in the variational wave function and numerical cancellation. These cases do not affect the final accuracy and should b neglected. * \ is the orbit-orbit term in the Breit interaction defined by Eq. (7) of [https://link.aps.org/doi/10.1103/1dbh-rjjn](https://link.aps.org/doi/10.1103/1dbh-rjjn), divided by α^2^ * PI*DLT(R1) denotes the matrix element π⟨δ(**r**~1~)⟩. * PI*DLT(R12) denotes the matrix element π⟨δ(**r**~12~)⟩. * deltaE(POL) denotes the total change in the energy when the mass polarization term (μ/*M*)⟨**p**~1~·**p**~2~⟩ is included in the Hamiltonian, divided by (μ/*M*). * EPOL2 denotes the second (plus higher) order change in the energy due to the mass polarization term, divided by (μ/*M*)^2^ (i.e. the total change minus the first-order change). * \ denotes the recoil correction to the orbit-orbit interaction defined by Eq. (11) of [https://link.aps.org/doi/10.1103/1dbh-rjjn](https://link.aps.org/doi/10.1103/1dbh-rjjn), divided by α^2^μ/*M*. * \ denotes the QED term of order α^3^ defined by Eq. (15) of [https://link.aps.org/doi/10.1103/1dbh-rjjn](https://link.aps.org/doi/10.1103/1dbh-rjjn), and is the corresponding finite-mass correction with r~12~ replaced by r~1~ in Eq. (15). * \ denotes an additional finite mass correction (divided by μ/*M*) to be added to . * \, , and denote the spin-dependent spin-orbit, spin-other-orbit, and spin-spin operators defined by Eqs. (8), (9), and (10) of [https://link.aps.org/doi/10.1103/1dbh-rjjn](https://link.aps.org/doi/10.1103/1dbh-rjjn) respectively, but excluding the anomalous magnetic moment correction 𝑎~e~, and divided by α^2^. * \ denotes the recoil correction to , as defined by Eq. (12) of [https://link.aps.org/doi/10.1103/1dbh-rjjn](https://link.aps.org/doi/10.1103/1dbh-rjjn), divided by α^2^μ/*M*. * Next comes a table summarizing the expectation values of a large number of operators in an obvious notation, with the first number of each pair being the value for infinite nuclear mass, and the second number being the mass polarization correction multiplied by μ/*M*. For example, 1/R1^2 denotes the expectation value ⟨1/r~1~^2^⟩ in reduced mass atomic units 1/𝑎~μ~^2^. BLOG is the Bethe logarithm. * NONRELATIVISTIC ENERGIES shows a convergence table for the nonrelativistic energies for the case of infinite nuclear mass, in atomic units. * ENERGIES WITH DEL1.DEL2 shows a convergence table for the finite nuclear mass case with he mass polarization term (μ/*M*) ⟨**p**~1~·**p**~2~⟩ included explicitly in the Hamiltonian, in educed-mass atomic units. * RELATIVISTIC ENERGIES are the relativistic contributions to the total energies from the Breit interaction terms of order α^2^ a.u. (in MHz) * RELATIVISTIC FINITE MASS are the recoil corrections to the Breit interaction terms due to the mass polarization corrections to the wave functions. * Fine structure splittings, matrix elements to diagonalize the 2 x 2 singlet-triplet mixing energy matrix, and the singlet-triplet mixing angle. * QED Contributions (MHz): gives a breakdown of the various QED contributions to the ionization energy. * Contributions to the 35 P state energies of 4He relative to He+(1s) (MHz ): A final table analogous to Table IV of [https://link.aps.org/doi/10.1103/1dbh-rjjn](https://link.aps.org/doi/10.1103/1dbh-rjjn) giving contributions to the energy and a final total ionization energy in MHz.