Fast configuration-interaction calculations for nobelium and ytterbium

TL;DR

This paper introduces an efficient configuration-interaction method to calculate excitation energies of nobelium and ytterbium, aiding spectral predictions and experimental interpretation for these complex atoms.

Contribution

The paper develops and tests a novel, efficient many-electron configuration-interaction approach for heavy atoms with open f-shells, specifically applied to nobelium and ytterbium.

Findings

Accurate excitation energies for nobelium's low states were obtained.

The method was validated on ytterbium, showing reliable results.

Results assist in spectral prediction and experimental analysis for nobelium.

Abstract

We calculate excitation energies for low states of nobelium, including states with open $5f$ subshell. An efficient version of the many-electron configuration-interaction method for treating the atom as a sixteen external electrons system has been developed and used. The method is tested on calculations for ytterbium which has external electron structure similar to nobelium. The results for nobelium are important for prediction of its spectrum and for interpretation of recent measurements. Ytterbium is mostly used to study the features of the method.

Tables

Table 1. Table 1: Comparison of energies (in cm -1 ) and computational times (in minutes) for low states of ytterbium calculated with the use of the CIPT and FCI methods. Note that all states of the same parity and total angular momentum are calculated in one run of the program. Therefore, computational time is shown only for the lowest state. Δ Δ \Delta is the difference between experimental and theoretical energies; Δ t ​ h subscript Δ 𝑡 ℎ \Delta_{th} is the difference between CIPT and FCI energies. All theoretical energies are presented with respect to the FCI ground state. Gain is the ratio of the CIPT computational time to the FCI computational time.

State		Expt.	CIPT			FCI				Gain
		Kramida et al. (2016)	Energy	$\Delta$	time	Energy	$\Delta$	${\Delta }_{th}$	time
		(cm-1)			(m)	(cm-1)			(m)
Even
$4f^{14}6s^2$	1S0	0	73	73	2	0	0	73		$\sim$6
$4f^{14}5d6s$	3D1	24489	27692	-3203	68	27622	-3133	70	3	23
$4f^{14}5d6s$	3D2	24751	27753	-3002	78	27632	-2881	121	3.3	23
$4f^{14}5d6s$	3D3	25271	27873	-2602	55	27812	-2541	61	2.5	22
$4f^{14}5d6s$	1D2	27678	28244	-566		28125	-447	119
Odd
$4f^{14}6s6p$	3P${}_0{}^{}{}_{}{}^o$	17288	17870	-582	215	17820	-532	50	9	24
$4f^{14}6s6p$	3P${}_1{}^{}{}_{}{}^o$	17992	18374	-382	267	18264	-272	110	10	27
$4f^{14}6s6p$	3P${}_2{}^{}{}_{}{}^o$	19710	20076	-366	299	20049	-339	27	13	23
$4f^{13}5d6s^2$	(7/2,3/2)${}_2{}^{}{}_{}{}^o$	23188	24904	-1716		24806	-1618	98
$4f^{13}5d6s^2$	(7/2,3/2)${}_3{}^{}{}_{}{}^o$	27445	27261	184	221	27064	381	197	11	20
$4f^{14}6s6p$	1P${}_1{}^{}{}_{}{}^o$	25068	24433	635		24316	752	117
$4f^{13}5d6s^2$	(7/2,5/2)${}_1{}^{}{}_{}{}^o$	28857	29512	-655		29380	-523	132

Table 2. Table 2: Calculated excitation energies (in cm -1 ) and g 𝑔 g -factors for the lowest states of nobelium. Comparison with earlier calculations and experiment.

State		FCI (this work)		Other Porsev et al. (2018)
		Energy	$g$-factor	Cut CI	CI+all
$5f^{14}7s^2$	1S0	0	0	0
$5f^{14}7s7p$	3P${}_{}{}^o{}_0{}^{}$	20091	0.0000	16360	19567
	3P${}_{}{}^o{}_1{}^{}$	21201	1.4581	18138	21042
	3P${}_{}{}^o{}_2{}^{}$	26177	1.5000	22536	26133
	1P${}_{}{}^o{}_1{}^{}$	29783111Experimental value is 29962 cm-1 Laatiaoui et al. (2016).	1.0423	30237111Experimental value is 29962 cm-1 Laatiaoui et al. (2016).	30203111Experimental value is 29962 cm-1 Laatiaoui et al. (2016).
$5f^{14}7s6d$	3D1	31057	0.5000	31003	28436
	3D2	31132	1.1589	31223	28942
	3D3	31579	1.3333	31608	30183
	1D2	32858	1.0078	37980	33504
$5f^{13}7s^{2}6d$	3P${}_2{}^{}{}_{}{}^o$	42756	1.4414	45720
	3H${}_{}{}^o{}_5{}^{}$	44294	1.0235	49731
	5F${}_{}{}^o{}_3{}^{}$	45452	1.2229	52172
	3H${}_{}{}^o{}_6{}^{}$	45742	1.1667	52415
	5K${}_{}{}^o{}_4{}^{}$	46123	1.1192	53701
	1D${}_{}{}^o{}_2{}^{}$	46718	1.0258	54016
	5K${}_{}{}^o{}_4{}^{}$	47713	1.1143	56597
	3F${}_{}{}^o{}_3{}^{}$	47855	1.0807	56958
	1P${}_{}{}^o{}_1{}^{}$	47952	1.1334	55695

Equations

$$H^{\mathrm{CI}}=\sum _i^{16}{h}_i+\sum _{i<j}^{16}\frac{e^2}{r_{ij}},$$

$$h_i=c\alpha \cdot {\mathbf{p}}_i+(\beta -1)m{c}^2+V^{\mathrm{HF}}(r_i).$$

$$\Psi (r_1,\dots ,r_{16})=\sum _i{c}_i{\Phi }_i(r_1,\dots ,r_{16}).$$

$$(H^{\mathrm{CI}}-EI)X=0,$$

$$⟨i|H^{\mathrm{CI}}|j⟩\to ⟨i|H^{\mathrm{CI}}|j⟩+\sum _k\frac{⟨i|H^{\mathrm{CI}}|k⟩⟨k|H^{\mathrm{CI}}|j⟩}{E-E_k}.$$

$$\sum _k\frac{⟨i|H^{\mathrm{CI}}|k⟩⟨k|H^{\mathrm{CI}}|j⟩}{E-E_k}\approx$$

$$\sum _c\frac{1}{E-E_c}\sum _{k_c}⟨i|H^{\mathrm{CI}}|k_c⟩⟨k_c|H^{\mathrm{CI}}|j⟩.$$

$$\begin{gathered} \sum _{k_c}⟨i|H^{\mathrm{CI}}|k_c⟩⟨k_c|H^{\mathrm{CI}}|j⟩= \\ =\sum _{\alpha {\alpha }^{\prime }}{r}_{ij}^{\alpha {\alpha }^{\prime }}{h}_{\alpha }{h}_{{\alpha }^{\prime }}+\sum _{\alpha \beta }{s}_{ij}^{\alpha \beta }{h}_{\alpha }{q}_{\beta }+\sum _{\beta {\beta }^{\prime }}{t}_{ij}^{\beta {\beta }^{\prime }}{q}_{\beta }{q}_{{\beta }^{\prime }}. \end{gathered}$$

Full text

I Introduction

Configuration interaction (CI) method Grant (1970); Kotochigova and Tupizin (1987) is one of few tools used to calculate electron structure of open-shell many-electron atoms. However, due to huge increase of the computational cost with the number of external electrons, practical application is usually limited to systems with only few (no more than four) external electrons above closed shells. There are no other ab initio methods to deal with more complicated polyvalent systems. On the other hand, the use of different semi-empirical approaches is questionable when experimental data is poor or absent. Superheavy elements ($Z>100$) Schädel (2006); Eliav et al. (2015) and highly charged ions Kozlov et al. (2018) are just two good examples of such systems. Lack of good theoretical approaches is a big obstacle in the use of complicated atomic systems in fundamental research. An important step to address the problem was done in recent work Dzuba et al. (2017). It was demonstrated that neglecting off-diagonal matrix elements in the CI matrix between highly excited states can be used to reduce the CI problem to a matrix eigenvalue problem with relatively small matrix with modified (compared to standard CI approach) matrix elements. Since the corrections to matrix elements were similar to the second-order perturbation theory corrections to the energy, the method was called CI with perturbation theory (CIPT). Similar approaches were later used in a number of works Jönsson et al. (2017); Geddes et al. (2018); Kahl and Berengut (2018), while a somewhat different variant of CI+PT method was developed in Rakhlina et al. (2001); Imanbaeva and Kozlov (2018). This made it possible to perform calculations for such complicated atomic systems as Yb (including states with excitations from $4f$ subshell) Dzuba et al. (2017, 2018), W Dzuba et al. (2017), Ta, Db Lackenby et al. (2018a), Og Lackenby et al. (2018b), etc.

In this work we further develop the method to make it substantially more efficient. We demonstrate that neglecting the difference between energies of the states of the same excited configuration allows one to separate summation over projections of the total angular momentum of single-electron states from summation over other quantum numbers. Since summation over projections is the same for all similar configurations it can be performed only once and then reused for other similar configurations. This reduces computational time for Yb more than twenty times while the effect on the accuracy of the calculations is negligible. We use Yb atom as an example and then apply the method to nobelium. This allows us to predict No spectrum including states with excitations from the $5f$ subshell. It is also important, that we provide the proof of validity of previous calculations used for interpretation of the experimental measurements. The energy, hyperfine structure and isotope shift for the 1P${}_{1}^{\rm o}$ state of several No isotopes were measured Raeder et al. (2018) and used together with atomic calculations to extract nuclear parameters of these isotopes Porsev et al. (2018). Nobelium atom was treated in the calculations as a system with two valence electrons above closed shells. It is known that similar treatment of the 1P${}_{1}^{\rm o}$ state of Yb gives very poor results due to the mixing with a close state containing excitation from the $4f$ subshell. This mixing cannot be properly accounted for in the two-valence-electrons-above-closed-shells calculations. We demonstrate that the situation in No is different and corresponding mixing is small. Therefore, interpretation of the measurements based on the two-valence-electron calculations is correct. New energy levels for low states of No including those with open $5f$ subshell have been calculated.

II Fast configuration interaction method

Fast configuration interaction method (FCI) is a modification of the CIPT (configuration interaction with perturbation theory) method introduced in Ref. Dzuba et al. (2017). We start from its brief description using ytterbium atom as an example. We consider Yb as a system with sixteen electrons above closed shells. The CI Hamiltonian has the form

[TABLE]

where $h_{i}$ is the single-electron part of the Hamiltonian,

[TABLE]

$V^{\rm HF}(r)$ is the self-consistent Hartree-Fock (HF) potential (including nuclear part) obtained in the $V^{N-1}$ approximation, with one $6s$ electron removed from the HF calculations. The many-electron wave function for sixteen external electrons has the form of expansion over single-determinant basis states:

[TABLE]

The basis states $\Phi_{i}(r_{1},\dots,r_{N_{e}})$ are obtained by distributing sixteen electrons over a fixed set of single-electron orbitals. The coefficients of expansion $c_{i}$ and corresponding energies $E$ are found by solving the CI matrix eigenvalue problem

[TABLE]

where $I$ is the unit matrix, the vector $X=\{c_{1},\dots,c_{N_{s}}\}$, and $N_{s}$ is the number of many-electron basis states. It is assumed that a few first terms in the expansion (3) represent a good approximation for the state of interest and the rest of the sum is just a small correction. Then one can neglect the off-diagonal matrix elements between the states from the second part of the expansion and reduce the CI problem to one with the small matrix and the modified matrix elements (see Ref. Dzuba et al. (2017) for details)

[TABLE]

Here $|i\rangle\equiv\Phi_{i}(r_{1},\dots,r_{16})$, $E_{k}=\langle k|H^{\rm CI}|k\rangle$, and $E$ is the energy of the state of interest (the same as $E$ in (4)). Since this energy is not known in advance one needs to perform iterations over it.

Starting from this point we consider further modifications to Eq. (5) which lead to the FCI method. Summation in (5) goes over all states of excited configurations. If we neglect energy difference between states of the same configuration the summation in (5) can be divided in two parts

[TABLE]

First summation is over excited configurations and $E_{c}$ is average energy of each configuration. These energies can be expressed analytically in terms of the radial integrals of the Hamiltonian (1) Grant (1970); Lindgren and Rosén (1975); Tupitsyn et al. (2018), or calculated numerically. Second summation is over many-electron configuration state functions of a given configuration. These functions differ by the values of projections of the total angular momenta of individual electronic states, but have all other quantum numbers fixed. Therefore, second summation can be rewritten as:

[TABLE]

Here $h_{\alpha}$ and $q_{\beta}$ are one- and two-electron radial integrals of the CI Hamiltonian (1), $\alpha$ and $\beta$ are the short notations for corresponding sets of electronic quantum numbers. For example, $\alpha=n_{1}l_{1}j_{1};n_{2}l_{2}j_{2}$. Three terms in (7) correspond to four diagrams on Fig. 1 (there are two diagrams for the last term). Coefficients $r_{ij}^{\alpha\alpha^{\prime}}$, $s_{ij}^{\alpha\beta}$, and $t_{ij}^{\beta\beta^{\prime}}$ do not depend on principal quantum numbers of one-electron states. They can be calculated only once for a whole set of configurations, which differ by the principle quantum numbers. For example, these coefficients are the same for all configurations of the type $4f^{14}nsn^{\prime}s$ ($6<n<n^{\prime}\leq n_{max}$). Since the number of similar configurations can get over a hundred, the effect of reuse of the coefficients is substantial.

The results of calculation for energy levels of Yb are shown in Table 1. All results are obtained by the same computer code which has options to run in either CIPT or FCI mode. Therefore, the difference between CIPT and FCI results ($\Delta_{th}$) is only due to the neglecting the energy difference between states within the same excited configuration. Some difference between the present and previously published CIPT results Dzuba et al. (2017, 2018) is due to the differences in the size of the set of configurations. Present code uses different algorithm to generate excited configurations from the reference configurations. The data in Table 1 shows that switching from the CIPT to FCI approaches brings substantial gain in efficiency (more than twenty times for Yb) while having only negligible effect on the energies. Note also that the use of the FCI instead of CIPT method does not affect the calculation of the matrix elements. The form of the calculated wave function is the same in both methods. Calculation of the matrix elements was considered in Dzuba et al. (2018); Lackenby et al. (2018a).

III Nobelium

Nobelium is the heaviest element ($Z$=102) for which experimental spectroscopic data are available. The frequency of the strong electric dipole transition from the ground state to a state of opposite parity and the first ionization potential have been recently measured Raeder et al. (2018); Chhetri et al. (2018). The measurements Raeder et al. (2018) include hyperfine structure and isotope shifts for three nobelium isotopes 252No, 253No, and 254No. The data were used to extract nuclear parameters, such as nuclear radii, magnetic dipole and electric quadrupole moments. This procedure relies on the atomic calculations. In particular, an advanced combination of the CI with coupled-cluster method was used Raeder et al. (2018); Porsev et al. (2018). Nobelium atom has the electron structure similar to that of ytterbium. Its ground-state is [Ra]$5f^{14}7s^{2}\ {}^{1}$S0. The state for which the frequencies of the transitions were measured was [Ra]$5f^{14}7s7p\ ^{1}$P1. The calculations treated nobelium as a system with two valence electrons above closed shells. However, it is not known in advance whether such calculations produce good results for No. Similar calculations for the $4f^{14}6s6p\ ^{1}$P1 state of Yb give very poor results for hyperfine structure Dzuba et al. (2018) and electric dipole transition amplitude from the ground state Dzuba and Derevianko (2010) due to the strong mixing with the close state of the same parity and $J$ but with an excitation from the $4f$ subshell, $4f^{13}5d6s^{2}$ (7/2,5/2)${}^{\rm o}_{1}$ (last line of Table 1). This mixing cannot be included in the two-valence-electron calculations. However, treating Yb as a sixteen electron system with the CIPT method leads to dramatic improvement of the results Dzuba et al. (2018). The results of the FCI calculations presented in Table 2 show the potentially trouble-making state of the $5f^{13}7s^{2}6d$ configuration in No (last line of Table 2) is significantly higher on the energy scale of No than a similar state in Yb. Energy interval in No is five times larger and mixing is small. The mixing in the 1P1 state of interest is 98% to 2% in No (2% admixture of the state with excitation from $5f$ subshell) and 75% to 25% in Yb. This means that the mixing in No can be neglected and the $7s7p\ ^{1}$P1 state can be treated as a two-valence-electron state.

Note the good agreement of the FCI energies with the only known experimental value and with sophisticated calculations by the CI+all-order method for the two-valence-electron states above closed-shell core of No. There are two major sources of uncertainty in the FCI calculations. One is neglecting core-valence correlations with core states below $5f$. Another is the perturbative treatment of the excited configurations. Both these effects are treated more accurately in the CI+all-order calculations. Therefore, if the mixing with states containing excitations from the $5f$ subshell can be neglected, the CI+all-order calculations are preferable and probably more accurate. From these calculations we know that when No atom is treated as a two-valence-electron system, about 95% of the core-valence correlations come from the $5f$ electrons (this is also true for the $4f$ electrons of Yb). These correlations are included in the FCI calculations. This explains good accuracy of the results.

Acknowledgements.

This work was funded in part by the Australian Research Council and by Russian Foundation for Basic Research under Grant No. 17-02-00216. MGK acknowledges support from the Gordon Godfree Fellowship and thanks the University of New South Wales for hospitality.