Ground and Excited Electronic State Analysis of PrF2+ and PmF2+

Nov 14, 2014 - Intruder states were removed via the application of denominator shifts of 0.02 and 0.10 au for the MCQDPT2 and SO-MCQDPT2 calculations,...
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Ground and Excited Electronic State Analysis of PrF2+ and PmF2+ George Schoendorff, Benjamin Chi, Hans Ajieren, and Angela K. Wilson* Department of Chemistry and Center for Advanced Scientific Computing and Modeling (CASCaM), University of North Texas, Denton, Texas 76203-5017, United States ABSTRACT: The ground state and excited state manifolds are computed for PrF2+ and PmF2+ at the CASSCF (n,8) level of theory where the active space spans the Ln 4f orbitals as well as the F 2pz orbital. Dynamical correlation is included using secondorder multireference quasidegenerate perturbation theory (MCQDPT2). The spin− orbit multiplets for each of the excited states are resolved, and spin−orbit coupling constants are computed using the Breit−Pauli spin−orbit operator. Equilibrium geometries for each of the ground and excited states are computed, and the nature of the Ln−F bond is examined. Potential energy curves for the lowest four triplet states and lowest two quintet states are computed for PrF2+, which split into 14 levels upon application of the spin−orbit Hamiltonian. Likewise, the lowest six quintet states are computed for PmF2+ as well as the lowest triplet state and the lowest two septet states. These nine states split into 43 terms upon application of the spin−orbit Hamiltonian.

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INTRODUCTION The lanthanides are unique among the elements due to the nature of their f electrons. Contrary to the chemistry of actinides, it is often assumed that the 4f electrons of the lanthanides are localized on lanthanide and cannot take part in bonding. The resulting ionic character of the lanthanide complexes is well documented.1−5 The difficulty in forming covalent bonds with the 4f electrons arises from the compact nature of the orbitals. The 4f electrons are valence electrons that exist in the subvalence spatial region such that the filled 5s and 5p subvalence orbitals have a greater radial extent. Thus, there is frequently little overlap of the 4f electrons with valence orbitals of adjacent species in a molecular environment, resulting in primarily ionic interatomic interactions. This lack of covalent character leads to great difficulties in the study of lanthanide chemistry. All lanthanides preferentially adopt a +3 oxidation state, resulting in similar chemistry across the series.6 Therefore, it is steric effects resulting from the well-known lanthanide contraction that is the distinguishing feature in lanthanide chemistry. While the compact nature of the 4f orbitals makes the separation of the lanthanides a difficult task, this same feature leads to the novel properties and application of the lanthanides. Essentially, the compact nature of the 4f orbitals allows the lanthanides to maintain spin states that are much higher than possible with any transition metal. Maintaining such a high spin state without affecting the chemistry in terms of covalent bonding has led to the extensive use of lanthanides for magnets and advanced electronics.7 In addition to the high-spin character of the lanthanides, there is the potential to have a low-lying excited-state manifold that may be accessible even at low temperatures. Study of the excited-state manifold is certainly warranted but is complicated by the large number of electrons and relativistic effects in addition to the added complications of accurately © XXXX American Chemical Society

treating the molecular environment. Thus, a small model system is needed to allow for a detailed examination of the excited-state characteristics of Ln3+ in the molecular environment. To this end, LnF2+ may serve as a useful model, and this has already been shown for NdF2+ wherein the low-lying excited states of NdF2+ resemble those of NdF3.8 Such an approach tends to be successful due to the fact that the gasphase LnX3 molecules have energy levels that resemble bare Ln3+ in free ions and in crystals.9 Furthermore, LnF2+ and other LnX2+ species have been the focus of experimental studies as well.10−12 The entire series of LnF2+ (Ln = La−Lu) was studied in solution, and the heats of formation for each were measured.13 It was found that LnF2+ ions are stable in aqueous solution and that the formation of the diatomic dications was driven primarily by an increase in entropy associated with the disruption of hydrogen bonding between the solvating water molecules. More recently, LnF2+ ions were isolated in gas-phase mass spectrometry experiments.14 In the present work, PrF2+ and PmF2+ are studied in an analogous manner to the methods previously used to study NdF2+. The excited-state manifold is computed at a range of internuclear distances, and splittings due to spin−orbit coupling are computed as well as coupling constants between states.

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COMPUTATIONAL DETAILS Potential energy curves were computed for the ground and lowlying excited states of the PrF2+ and PmF2+ diatomic dications in a manner analogous to that used in a prior study of NdF2+.8 Special Issue: 25th Austin Symposium on Molecular Structure and Dynamics Received: August 18, 2014 Revised: November 3, 2014

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employed to determine the orbital energy differences between the Ln 4f and F 2p orbitals in LnF2+. Taken together, the orbital overlap and orbital degeneracy is used to examine the possibility of Ln−F covalent bonding.

All calculations were performed using the GAMESS quantum chemistry software package.15 Model core potentials were used to treat the core electrons and implicitly incorporate scalar relativistic effects at the level of Cowan and Griffin’s quasirelativistic Hartree−Fock method.16 The valence basis sets were of double-ζ quality with a (88111/6121/315/622)/ [5s4p3d3f] contraction scheme for praseodymium and promethium where the 5s, 5p, 4f, 5d, and 6s shells are treated explicitly, and a (31/31/2) contraction scheme was used for fluorine.17−19 These basis sets correspond to the MCP-DZP sets included with GAMESS with the addition of diffuse (s and p) primitive functions also included in the fluorine basis set with exponents of 0.1124467 and 0.0915426, respectively, in keeping with the methodology of the previous study of NdF2+.8 Multiconfigurational SCF (MCSCF) calculations were performed using the full optimized reaction space (FORS)20−22/complete active space SCF (CASSCF)23 wave functions for both molecules. For both PrF2+ and PmF2+, the lanthanide 4f orbitals were included in the active space as well as the fluorine 2pz orbital, resulting in (4,8) and (6,8) active spaces for PrF2+ and PmF2+, respectively. These eight orbitals were chosen for the active space based on the occupations obtained from single-point energy calculations with an (n,16) active space that includes the F 2p orbitals as well as the Ln 4f, 5d, and 6s orbitals. A series of single-point energy calculations at internuclear distances ranging from 1.4 to 2.6 Å show that the F 2px and 2py orbitals are doubly occupied at all internuclear distances (orbital occupations ≥ 1.97 e−), and the Ln 5d and 6s orbitals are unoccupied at all internuclear distances (orbital occupations ≤ 0.03 e−). Thus, the F 2px and 2py orbitals can remain in the core, while the Ln 5d and 6s orbitals can be placed in the virtual space. Spin−orbit coupling and dynamic correlation effects were computed at the spin−orbit second-order multiconfigurational quasidegenerate perturbation theory (SO-MCQDPT2) level of theory using the CASSCF optimized orbitals.24−26 All electrons treated explicitly were correlated at the PT2 level of theory, that is, only the Ln and F cores were frozen because they constitute the MCP core. The Breit−Pauli spin−orbit operator was used for the spin−orbit calculations wherein the full one-electron spin−orbit terms were computed explicitly while the twoelectron core-active terms were included via the MCP but computed explicitly for active−active terms. Accurate explicit computation of spin−orbit coupling is facilitated by the MCP in the fact that all nodes, even radial nodes, are present in the potential rather than needing to rely on a pseudopotential that is fit for spin−orbit coupling. Intruder states were removed via the application of denominator shifts of 0.02 and 0.10 au for the MCQDPT2 and SO-MCQDPT2 calculations, respectively.27,28 Spectroscopic constants for the ground states of PrF2+ and PmF2+ were computed via a Dunham analysis.29 For the Dunham analysis, the ab initio SO-MCQDPT2 energies were used to construct ninth-order polynomial fits to the groundstate potential energy curves. Data points for the polynomial fit were from the region of 1.7−2.2 Å at 0.05 Å increments. Additional points were added in the region of 1.85−1.95 Å to obtain a good fit in the equilibrium region. Because no stable isotope of Pm exists, the Dunham analysis of PmF2+ was performed for 147Pm because it is a relatively long-lived isotope (t1/2 = 2.63 years). Radial distribution functions were computed to determine the extent of orbital overlap between the Ln 4f and F 2p orbitals. The extended Koopmans’ theorem30 was then

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RESULTS Potential energy curves for PrF2+ are shown in Figure 1. Figure 1A shows the four lowest triplet states as well as the two lowest

Figure 1. Potential energy curves for the ground and excited states of PrF2+ calculated at the CASSCF (4,8) level of theory (A) and at the SO-MCQDPT2 level of theory (B). The relative energies are in kcal mol−1, and the zero of energy in both graphs is the minimum energy of the ground-state (X3H) curve at the CASSCF (4,8) level of theory.

quintet states computed at the CASSCF level of theory with a (4,8) active space. The active space spans the Pr 4f orbitals and the F 2pz orbital in order to adequately model the Pr−F bond as well as accounting for the degeneracy of the Pr 4f orbital set. The triplet states are all bound states with the 3H state as the ground state, while the quintet states are repulsive states. All states are doubly degenerate with the exception of the Σ states (A3Σ−, C3Σ−, and b5Σ+). Additionally, all of the bound states are parallel to each other. The bound states result from varying the occupations of the 4f orbitals, and the parallel nature of these states results from the compact local character of the 4f orbitals. Application of the Breit−Pauli spin−orbit operator results in the multiplet-resolved potential energy curves shown B

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in Figure 1B. Spin−orbit coupling splits the five states into 14 terms, yet the bound multiplet-resolved potential energy curves remain parallel to each other. The minimum-energy bond length for the 3H ground state (or 3H4 ground term) is 1.902 Å, and all excited bound states have a minimum within 0.001 Å of the ground-state equilibrium bond length. Transition energies for PrF2+ at the SO-MCQDPT2 level of theory are shown in Table 1. The 3H ground state splits into

no appreciable coupling between the repulsive quintet states and any of the bound triplet states. Thus, there is no apparent route to homolytic bond cleavage. Homolytic dissociation must occur via transition between the 3H ground state and an even higher energy repulsive state. This is quite different from the previously published case of NdF2+, where the 4I ground state moderately couples to the repulsive high-spin 6H excited state.8 Potential energy curves for PmF2+ are shown in Figure 2 at the CASSCF (6,8) level of theory (Figure 2A) and at the SO-

Table 1. PrF2+ Transition Energies (cm−1) at the GroundState Equilibrium Geometry (1.902 Å) Computed at the SOMCQDPT2 Level of Theory Te (cm−1) X3H4 A3H5 B3H6 C3Σ−0 D3Π0 E3Π1 F3Π2 G3Σ−1 a5Φ1 b5Φ2 c5Φ3 d5Φ4 e5Φ5 f5Σ+2

0 1877 3752 4489 9094 9466 9657 24928 42270 42871 43544 44206 44892 46933

three levels with the first excited level 1877 cm−1 above the ground 3H4 term. Despite the splitting of the 3H state, the C3Σ−0 state is still 737 cm−1 above the highest 3H level (3H6) and 4489 cm−1 above the ground state. The next higher state, 3 Π, is split into three levels that range from 9094 to 9657 cm−1 above the ground state. The excited-state manifold has a noticeable gap between the 3Π and the next higher state, which is yet another 3Σ− state (G3Σ−1 ). At the equilibrium geometry, the high-spin quintet states are much higher in energy, with the lowest state (a5Φ1) being 42270 cm−1 above the ground state. Spin−orbit coupling constants for ground and low-lying excited states of PrF2+ are shown in Table 2. The 3H ground Table 2. PrF2+ Spin−Orbit Coupling Constants (cm−1) Computed at the SO-MCQDPT2 Level of Theory 3

XH A3Σ− B3Π C3Σ− a5Φ b5Σ+

X3H

A3Σ−

B3Π

C3Σ−

a5Φ

b5Σ+

2658

81 0

105 1366 532

49 1 193 0

1 7 1 4 1397

8 0 11 0 628 0

Figure 2. Potential energy curves for the ground and excited states of PmF2+ calculated at the CASSCF (6,8) level of theory (A) and at the SO-MCQDPT2 level of theory (B). The relative energies are in kcal mol−1, and the zero of energy in both graphs is the minimum energy of the ground-state (X5H) curve at the CASSCF (6,8) level of theory.

MCQDPT2 level of theory (Figure 2B). The lowest six quintet states were computed as well as the lowest triplet state and the two lowest septet states. The quintet and triplet states are bound states, with the quintet states being the lowest in energy; the 3H state is over 50 kcal mol−1 above the 5H ground state. The ground-state equilibrium bond length is 1.898 Å at the CASSCF (6,8) level, and the bond is shortened by only 0.001 Å when correlation and spin−orbit coupling effects are included. The equilibrium bond length for the excited quintet and triplet states vary by no more than 0.003 Å compared with the ground-state equilibrium bond distance. Thus, as for PrF2+, the

state exhibits the largest zero-field splitting with a magnitude of 2658 cm−1, while the 3Π and 5Φ states split to a much lesser degree, 532 cm−1 and 1397 cm−1, respectively. Significantly, the 3 H ground state couples only weakly with the excited triplet states, with the largest coupling being with the 3Π state. Likewise, the triplet states exhibit only weak coupling with each other, with a notable exception being the coupling between the lowest 3Σ− state and the 3Π state, where the coupling constant is 1366 cm−1. The quintet states couple moderately with each other with a coupling constant of 628 cm−1. However, there is C

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energy 5H state (E5H7). The next higher states, 5Π and 5Γ, are still separated from each other upon splitting due to spin−orbit coupling, with the multiplets of the 5H, 5Σ−, 5Π, and 5Γ occurring in distinct bands. The next higher states, 5Δ and 5Φ, are split such that the multiplets are in the same energy regime from 14780 to 18471 cm−1. The 3H states split such that the multiplets overlap with the energy regime of the 5Φ multiplets at 18385 cm−1 (a3H4). Spin−orbit coupling constants for PmF2+ are shown in Table 4. The X5H ground state does not couple directly with either of the repulsive septet states. In fact, the ground state couples only with the C5Γ state with a magnitude of 3004 cm−1. The C5Γ, in turn, couples moderately to the b7Φ state with a magnitude of 123 cm−1, thus providing a route to achieve homolytic dissociation from the ground state to the lowest repulsive septet state. With the exception of the lowest quintet excited state, A5Σ−, there is strong coupling among the excited quintet states so that there is a high probability of transitions within the quintet excited-state manifold. Transitions from any quintet state to the high-energy, low-spin a3H state do not occur as this triplet state does not couple with any other states that were computed. A Dunham analysis was performed using ninth-order polynomial fits to the ground-state potential energy curves at the SO-MCQDPT2 level of theory for both PrF2+ and PmF2+. Spectroscopic constants for the ground states of both diatomic species are shown in Table 5. The stretching frequencies for

bound state potential energy curves are parallel to the ground state. Again, this is because the low-lying excited states are differentiated from the ground state simply by permutation of the 4f orbital occupations. The septet states are high-energy dissociative states that cross the bound state potential energy curves in the region of 2.1−2.5 Å. The dissociation products resulting from the septet states are the homolytic products, Pm2+ + F•. Spin−orbit coupling splits the original nine states (six quintets, one triplet, and two septets) into 43 terms. From Figure 2A, it can be seen that there exists a high density of states in the low-lying excited-state manifold. The transition energies for each of these terms are shown in Table 3. The 5H Table 3. PmF2+ Transition Energies (cm−1) at the GroundState Equilibrium Geometry (1.897 Å) Computed at the SOMCQDPT2 Level of Theory Te (cm−1) 5

X H3 A5H4 B5H5 C5H6 D5H7 E5Σ−2 F5Π1 G5Π0 H5Π1 I5Π2 J5Π3 K5Γ2 L5Γ3 M5Γ4 N5Γ5 O5Γ6 P5Δ0 Q5Δ1 R5Φ1 S5Δ2 T5Φ2 U5Δ3 V5Δ4 W5Φ3 Y5Φ4 Z5Φ5

Te (cm−1) 3

0 1260 2520 3780 5040 5428 7018 7267 7517 7766 8016 10315 11338 12362 13386 14409 14780 15270 15359 15759 16137 16249 16738 16915 17693 18471

a H4 b3H5 c3H6 d7Φ0 e7Φ1 f7Δ1 g7Φ2 h7Δ0 i7Φ3 j7Δ1 k7Δ2 l7Φ4 m7Δ3 n7Φ3 o7Δ4 p7Φ6 q7Δ5

18385 19817 21249 34246 35004 35430 35763 36040 36521 36652 37263 37280 37874 38038 38486 38796 39097

Table 5. Spectroscopic Constants for the Ground States of PrF2+ (3H4) and PmF2+ (5H3)a PrF2+ PmF2+

ωe

ωeχe

Be

αe

De

D0

723.9 726.0

1.69 2.21

0.2780 0.2794

0.00102 0.00126

57.6 42.2

56.5 41.2

a

The constants were obtained by a Dunham analysis of a ninth-order polynomial fit to the SO-MCQDPT2 ground-state potential energy curves. All values are reported in cm−1 except De and D0, which are reported in kcal mol−1.

PrF2+ and PmF2+ are similar to what was previously computed for NdF2+, that is, ωe = 723.9, 714.0, and 726.0 cm−1 for LnF2+ (Ln = Pr, Nd, Pm). Furthermore, all LnF2+ have zero-point energies of approximately 1 kcal mol−1. The separation between the ground and first excited states is ∼4−5 kcal mol−1; therefore, the ground state is well-defined for all LnF2+. The bound excited-state potential energy curves are parallel to the ground state; therefore, the zero-point energy of the excited states is also ∼1 kcal mol−1. However, the excited states of LnF2+ become so dense that there is no longer such separation

ground state is split into five terms with a range of 5040 cm−1, with the lowest excited state (A5H4) only 1260 cm−1 above the X5H3 ground state. Despite the splitting of the 3H ground state, the next higher state, E5Σ−2 , is 388 cm−1 higher than the highest-

Table 4. PmF2+ Spin−Orbit Coupling Constants (cm−1) Computed at the SO-MCQDPT2 Level of Theory

5

XH A5Σ− B5Π C5Γ D5Δ E5Φ a3H b7Φ c7Δ

X5H

A5Σ−

B5Π

C5 Γ

D5Δ

E5Φ

a3H

b7Φ

c7Δ

4052

11 0

6 4043 799

3004 8 15 3294

7 0 3649 0 1575

0 5 12 2096 246 2509

0 0 0 0 0 0 2208

0 0 0 123 171 22 0 2001

1 0 328 0 20 227 0 1913 1306

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occurs at a high energy (∼50 kcal mol−1 above the ground state). The high-spin bound states lead to heterolytic dissociation (LnF2+ → Ln3+ + F−) rather than undergoing homolytic dissociation (LnF2+ → Ln2+ + F•) for both PrF2+ and PmF2+. Homolytic dissociation occurs via crossing to a dissociative high-spin state, although the dissociative high-spin states that couple to the ground state could not be found due to the high density of states in this region of the excited-state manifold. For PrF2+, the lowest two dissociative high-spin states were computed, but neither state couples appreciably with the ground state or any other of the bound states that were computed, although the dissociative states couple with each other with a spin−orbit coupling constant of 628 cm−1. Thus, homolytic dissociation must occur via transition to a higherenergy dissociative state. Likewise, the 5H ground state of PmF2+ does not couple with either of the lowest two dissociative states. However, a route to dissociation to the lowest-energy repulsive state exists via coupling of the ground state to the C5Γ, which in turn couples moderately to the lowest-energy septet state (b7Φ) with spin−orbit coupling constants of 3004 and 123 cm−1 for coupling of the C5Γ state to the X5H state and the b7Φ state, respectively. The nature of the Ln−F bond was also investigated. The 4f orbital of Pr and Pm exhibits modest overlap of the F 2p orbital, which is enhanced relative to neutral Ln−F due to the short bond length that results from the high positive charge. However, the Pr−F bond remains ionic despite the modest orbital overlap due to a large orbital energy difference. The Pm−F bond, on the other hand, exhibits a 10-fold increase in covalency compared to Pr−F due to a 10-fold decrease of the orbital energy difference between the Ln 4f and F 2p orbitals. While the orbital degeneracy may lead to an appreciable degree of covalency, it is expected that ionic character still dominates the Pm−F bond.

between states in the excited-state manifold as there is with the ground states. The equilibrium bond length for the LnF2+ species exhibits little variation. For example, the ground-state equilibrium bond lengths for LnF2+ at the SO-MCQDPT2 level of theory are 1.902, 1.901,8 and 1.897 Å for PrF2+, NdF2+, and PmF2+, respectively. While the lanthanide contraction is still manifested in these bond lengths, the high positive charge leads to the short bonds with little variation in the bond length. At such short internuclear distances, there is a small yet non-negligible overlap of the Ln 4f and F 2p orbitals. The overlap of the Pm 4f and F 2p orbitals is shown in Figure 3 where the radial

Figure 3. Radial distribution functions for the Pm 4f and F 2p orbitals at an internuclear distance of 1.897 Å. The Pm 4f radial distribution function was computed for Pm2+ with a [Xe] 4f5 configuration corresponding to a 6H state, and the F 2p radial distribution function was computed for neutral F with a [He] 2s2 2p5 configuration corresponding to the 2P ground state.

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distribution functions for these orbitals are plotted with the nuclei fixed at the PmF2+ ground-state equilibrium bond distance of 1.897 Å. While the overlap shown is for PmF2+, a similar overlap of the Ln 4f and F 2p orbitals exists for PrF2+ and NdF2+. Such overlap may allow for covalent mixing of these orbitals provided that the orbital energies match. The difference in the orbital energies of the Pr 4f and F 2p orbitals for PrF2+ is 0.4223 EH (265 kcal mol−1). Such a large energy difference leads to the expected conclusion that the Pr−F bond is ionic. However, the orbital energy difference between the Pm 4f and F 2p orbitals is 0.0411 EH (25.8 kcal mol−1), which is an order of magnitude smaller than the orbital energy difference for PrF2+. Thus, while the Pm−F bond may be largely ionic, it still has 10 times more covalent character than the Pr−F bond. While covalency is primarily driven by orbital overlap, degeneracy-driven covalency is not unprecedented in the f block,31 although any type of covalent bonding is typically unexpected in the 4f series.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel.: 940-565-4296. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This project was funded by the National Science Foundation under Grants CHE-1213874 and CHE-1362479. Computing resources were provided by the Computing and Information Technology Center at the University of North Texas. Support from the United States Department of Energy for the Center for Advanced Scientific Computing and Modeling (CASCaM) is also acknowledged. The authors would also like to acknowledge Christopher South for helpful discussions.

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REFERENCES

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CONCLUSIONS PrF and PmF2+ were studied using multireference methods in an analogous manner to the methods previously employed for the study of NdF2+.8 The ground-state and low-lying excitedstate potential energy curves were computed. The bound state potential energy curves were determined to be parallel in both instances. Likewise, PmF2+ has a bound low-spin state that 2+

E

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