The Effect of Correlation on Spectra of the Lanthanides: Pr3+

The effect of correlation on the spectra of lanthanide ions has been investigated using variational methods based on multiconfiguration Dirac–Hartree–Fock (MCDHF) theory. Results from several computational models are reported for Pr3+. The first assumes an inactive Cd-like 4d10 core with valence electrons in 4 f 25s25p6 subshells. Additional models extend correlation to include core effects. It is shown that, with such models, the difference between computed energy levels and those from observed data increases with the energy of the level, suggesting that correlation among outer electrons should also be based on the correlated core of excited configuration state functions (CSFs). Some M1 transition probabilities are reported for the most accurate model and compared with predictions obtained from semi-empirical methods.


Introduction
The effect of correlation in the atoms and ions of lanthanides and actinides is not well understood.Though O'Malley and Beck [1,2] have studied the effect of valence correlation of lanthanide anion binding energies, very few spectra of lanthanide atoms and ions have been investigated.The periodic table makes atomic structure changes clear, in that new shells appear before an earlier shell is filled.In addition, the number of levels in the f n configurations of lanthanides and actinides increases rapidly with n (n ≤ 7), and resulting spectra consist of numerous closely spaced levels.The simplest case has a 4 f 2 ground configuration with 13 levels.The lanthanides are among the lighter atomic systems where both correlation and relativistic effects require full relativistic treatment.
The spectrum for Pr 3+ is one of the few where levels for several configurations have been classified.Spectra were published in 1965 by both Sugar [3] and Crosswhite et al. [4].Levels of 4 f 2 , 4 f 5d, 4 f 6s, and 4 f 6p configurations for the neutral atom were identified.Theoretical studies in those days, as exemplified by the paper published by Morrison and Rajnak [5], were based on effective operators for the f -shell group used to classify the symmetry of f n states.
In an early publication [6], Pr 3+ was selected as a test case for a multiconfiguration Dirac-Hartree-Fock program capable of including the effects of correlation on wave function expansions in terms of a few thousand configuration state functions.Results based on simplifying assumptions were reported, compared with observation, and important interactions identified.A few years later, Eliav et al. [7] reported results from a relativistic coupled cluster method for both Pr 3+ (4 f 2 ) and U 4+ (5 f 2 ) with improved accuracy.
Seth et al. [8] took a broader approach to the prediction of spectra with errors in the levels less than 500 cm −1 using a multiconfiguration Dirac-Hartree-Fock configuration interaction (MCDHF-CI) method for lanthanide or actinide cases with subshells n f k and additional (n + 1)-subshell electrons.Classes of excitations were considered for the test cases Pr 3+ (4 f 2 )/U 4+ (5 f 2 ), Pm 3+ (4 f 4 )/Np 3+ (5 f 4 ), and Eu 3+ (4 f 6 )/Am 3+ (5 f 6 ) on a trial and error basis.The largest expansion size was about 150,000 CSFs.For Pr 3+ , the average error in the 13 levels was 448 cm −1 .
More recently, Safronova et al. [9] applied higher-order perturbation methods to the study of correlation effects in La, Ce, and lanthanide ions using hybrid methods that combine configuration interaction (CI) with second-order perturbation theory and linearized coupled cluster all-order methods.In Ce 2+ , only the 5 lowest levels were reported with the error increasing with the degree of excitation.
Semi-empirical relativisitic Hartree-Fock methods have been applied to the analysis of some spectra for lanthanides.Wyart et al. [10] used a group theoretical description of energy levels for 4 f n systems in terms of parameters that take into account Coulomb interactions, spin-dependent interactions, and other interactions.Nd 3+ , Pr 3+ , and Nd 4+ were investigated.A least squares fit of parameters to observed energy levels yielded excellent agreement between theory and experiment.For the highest level of Pr 3+ , namely 4 f 2 1 S, an energy level of 48044.66 ± 1.29 cm −1 was predicted, a value that differs appreciably from the 50090.29 cm −1 reported by Crosswhite et al. [4].Sugar [3] did not include this level in his publication, nor is it reported in the Atomic Spectra Database (ASD) [11].These semi-empirical methods were applied by Yoca and Quinet [12] to the study of decay rates in Pr 3+ and by Li et al. [13] to parity forbidden transitions in several lanthanides, including Pr 3+ .
This paper reports preliminary ab initio systematic studies of the many-body effects in the 4 f 2 manifold for Pr 3+ , where observed energy levels have been classified, although the highest 1 S 0 level is uncertain.M1 transition rates are predicted and compared with semi-empirical values.
All calculations were done using the GRASP2K program [14].

Underlying Theory
In the multiconfiguration Dirac-Hartree-Fock (MCDHF) method [15], as implemented in the GRASP2K package [14], the wave function Ψ(γPJ M J ) for a state labeled γPJ M J , where J and M J are the angular quantum numbers and P the parity, is expanded in antisymmetrized and coupled CSFs: The labels {γ j } denote other appropriate information about the CSFs, such as orbital occupancy and coupling scheme.The CSFs are built from products of one-electron orbitals, having the general form where χ ±κ,m (θ, ϕ) are two-component spin-orbit functions.The radial functions {P nκ (r), Q nκ (r)} are represented numerically on a grid.In spectrum calculations, where only energy differences relative to the ground state are important, wave functions for a number of targeted states are determined simultaneously in the extended optimal level (EOL) scheme.Given initial estimates of the radial functions, the energies E and expansion coefficients c = (c 1 , . . ., c M ) t for the targeted states are obtained as solutions to the configuration interaction (CI) problem, where H is the CI matrix of dimension M × M with elements Radial functions are solutions of systems of differential equations that define a stationary state of an energy functional for one or more wavefunction expansions.It is possible to derive the MCDHF equations from the usual variational procedure by varying both the large and small components so that where V(a; r) = V nuc (r) + Y(a; r) + X(a; r) is a potential consisting of nuclear, direct, and exchange contributions arising from both diagonal and off-diagonal Φ α |H DC |Φ β matrix elements [15].In each κ-space, Lagrange-related energy parameters ab = n a n b are introduced to impose orthonormality constraints in the variational process.

Computational Procedures
What distinguishes the f n manifolds computationally is the rapid (almost explosive) increase in the number of CSFs as single-(S) and double-(D) excitations are applied, compared with lighter systems such as the C-, N-, and O-like systems.For the latter, excellent results have been reported both for the spectrum and the transitions rates of interest in astrophysical applications [16,17].Questions then arise about whether such large expansions are needed.The JJGEN program in GRASP2K generates excitations in terms of configurations and, for a given configuration, produces all the CSF basis states associated with the configuration.The 4d 10 4 f 2 manifold has 13 basis states, whereas 4d 8 4 f 4 (produced by the 4d 2 → 4 f 2 excitation) has 3121 basis CSFs over the same range of J, namely J = 0, . . ., 6, which interact with one or more CSFs of 4d 10 4 f 2 .For a small orbital set, these expansions can readily be dealt with on current computers.Difficulties arise with larger orbital sets with multiple "layers" (orbitals with the same "n"), and a range of angular symmetries are used.GRASP calculations are systematic in that the set is increased by successively adding an extra layer of orbitals to an existing set, where each new layer is orthonormal.Associated with this systematic method is the notion of convergence.Thus, at some point, the corrections to an existing result become small.Thus, it is helpful to partition the wave function expansion into a zero-order approximation and its first-order correction whose expansion coefficients are small.In perturbation theory, the interaction between CSFs of the first-order correction for the wave function is ignored.
Two types of expansions may be used-in the past, both have been the same, but for large calculations there are advantages to relaxing this restraint and allowing expansions to be different.
(1) The first is the expansion that determines the radial functions using the RSCF program of the GRASP2K package that determines radial functions.For occupied orbitals, optimized radial functions can be obtained by applying the variational principal to an energy expression or functional.However, for correlation orbitals, the most important interactions appear as contributions to the functions X(a; r) that are in the same region of space as the occupied orbitals.This has been shown in partitioned configuration interaction (PCFI) studies [18].In fact, solutions to the variational equations produce orbitals in a region of space determined by X(a; r) for a given orbital.This can be used effectively in tailoring the orbital to an interaction.(2) The second is an expansion for the RCI program that determines the wavefunction and its associated energy for a given Hamiltonian based on a given orbital basis.In the present work, the Hamiltonian for RCI was the Dirac-Coulomb Hamiltonian (DC) plus the transverse photon interaction (DCB), the vacuum polarization effects as accounted for by the Uehling potential, and electron self-energies as calculated with the screened hydrogenic formula [15,19], referred to as the DCBQ Hamiltonian.The RCI program is relatively simple to parallelize efficiently [20] using message passing.As a result much larger expansions are possible for RCI calculations than RSCF calculations that build the orbital basis.Present calculations were performed using, for larger cases, 48 processors.

A Simple EAL Approximation
The 4 f 2 5s 2 5p 6 configuration of Pr 3+ consists of filled shells and two-electrons in the open 4 f -subshell.When the MCDHF approximation is computed without any Lagrange multipliers, all orbitals will decrease exponentially radially without any extra nodes.GRASP2K provides such a solution when the valence electrons consist only of 4 f 2 and all other subshells are in the inactive core.In the extended-average-level (EAL) approximation, only the diagonal matrix elements define the energy functional, avoiding any cancellation in the definition of the energy functional.Though the difference between extended-optimal-level (EOL) and EAL is small, this study is based on the EAL solution for the Cd-like core.The mean-radii of the orbitals (shown in Table 1) are such that the 4 f -orbitals are like core orbitals in the sense that their mean radii are closer to the other n = 4 orbitals than either 5s or 5p.At the same time, these orbitals define the spectrum.Computationally, it is more convenient to specify the configuration in terms of orbitals in their standard order, as in 4 f 2 5s 2 5p 6 .In this work, the CSF's of the 4 f 2 5s 2 5p 6 configuration define the multireference (MR) set, and 4 f , 5s, 5p electrons are considered to be valence electrons.
The core orbitals 1s, . . ., 4d were fixed in all subsequent calculations.A series of calculations were performed in which the expansions consisted of CSFs that interact with the MR set.The latter were obtained from excitations to orbital sets of increasing size by n, with orbital quantum numbers restricted to l ≤ 4. The set are referred to as ng orbital sets.
Case 1 results from SD excitations from 4 f 2 4s 2 5p 6 to ng orbital sets are given in Table 2.
The n = 4 calculation is an RCI calculation based on the radial functions from the variational EAL calculation.Note that the spectrum has two J = 4 adjacent states that are not in their final order.The valence correlation calculation omitted the very strong interactions within the n = 4 complex, particularly the interactions between 4d 2 and 4 f 2 .By extending the active subshells to include 4d 10 , we are including some of the correlations in the core.At the same time, the expansion size increased significantly by a factor of about four for n = 7.Again, only CSFs that interacted with the MR set were included.For the RSCF calculation, only the interaction with the MR set was used to determine the correlation orbitals.The convergence of these results are shown in Table 3, where the total number of CSFs are also reported for each n.This final calculation extended the active set core to include 4s 2 4p 6 ; however, because of the size of the expansion, some restrictions applied.The n = 4 and 5 expansions were full SD expansions to ng orbital sets.The n = 6 expansion included all SD excitations from 4s 2 4p 6 4d 10 4 f 2 5s 2 5p 6 to the n = 5g orbital set plus SD from 4d 10 4 f 2 5s 2 5p 6 to the n = 6g orbital set, whereas the n = 7 expansion added SD from 4d 10 4 f 2 5s 2 5p 6 to the n = 7g orbital set to the n = 6 expansion and also added CV excitations from 4s 2 4p 6 (core) and 4 f 2 5s 2 5p 6 (valence) shells to the n = 7g orbital set.
The results based on this calculation are reported in Table 4 where they are also compared with data from observation and other theory.Table 4. Case 3: spectrum for expansions from 4s 2 4p 6 4d 10 4 f 2 5s 2 5p 6 to ng orbital sets, n = 4, 5, 6, 7. See text for details.Results are compared with observation and other theory.For n = 7 results, differences from observation are given in parentheses.
4 3 H 0 0 0 0 0.00 0(0) 0( 0) 0(0) 5 3  Table 4 shows theoretically the most accurate results of our study.The differences between our n = 7 results and observations [3,4] and similar information from other theories are also listed.The fine-structure splitting of the lowest 3 H term from the present work is more accurate than the fine-structure reported by Eliav et al. [7], who used a coupled-cluster method.In fact, the fine-structure for the 3 F term from the present work was also well predicted, except for the J = 4 level, which is affected by the interactions with components of the adjacent 1 G 4 level.What is striking is that the difference observed in our present work is related to the energy of the level relative to the ground state.A similar linear dependence in the 3d k levels of tungsten was shown to be related, at least in part, to the omission of core correlation [21] from the calculation.On the other hand, the differences in values reported by Eliav et al. [7] are all negative.A slightly lower energy for the ground term would have improved the accuracy (relative to the ground state) for the entire spectrum.The differences for the last two columns are more random.The present results are not sufficiently accurate for the higher levels to be able to confirm Wyart's prediction of 48044.66 ± 1.29 cm −1 for the highest 1 S level.

M1 Transitions
Table 5 reports the theoretical wavelengths and M1 transition probabilities for all computed levels.The transition with the largest transition rate by far is from the upper 1 S 0 level to 3 P 1 , a transition with the largest error in our computed wavelength.Our results are shown here with the larger semi-empirical transition rates reported by Li et al. [13] as well as those reported by Yoca and Quinet [12].The semi-empirical values are based on observed wavelengths whereas the present rates used computed values.Transition rates for M1 transitions depend largely on wavelengths giving semi-empirical methods an advantage with regard to accuracy.For the larger transition rates, there is greater agreement than expected.Correlation in the core defines the potential (V(a; r) of Equation ( 5)) for the outer electrons.This suggests that all major contributors to the wave function such as 4d 10 4 f 2 5s 2 5p 4 5d 2 should be built on a correlated core but that the correlation between core-correlation CSFs may not be important.This is similar to the CI-RMBPT method [22], where effective operators are used for the calculation of an interaction matrix for the outer correlation.The effective operators include the effect of core correlation.With this model, the expansion for even J = 0 is about 3 million for an n = 5 calculation.Further studies are needed to determine how core correlation can be included efficiently in such cases.