High level multireference calculations were performed for LuF for a total of 132 states, including four dissociation channels Lu(2D) + F(2P), Lu(2P) + F(2P), and two Lu(4F) + F(2P). The 6s, 5d, and 6p orbitals of lutetium, along with the valence 2p and 3p orbitals of fluorine, were included in the active space, allowing for the accurate description of static and dynamic correlation. The Lu(4F) + F(2P) channel has intersystem spin crossings with the Lu(2P) + F(2P) and Lu(2D) + F(2P) channels, which are discussed herein. To obtain spectroscopic constants, bond lengths, and excited states, multi-reference configuration interaction (MRCI) was used at a quadruple-ζ basis set level, correlating also the 4f electrons and corresponding orbitals. Core spin–orbit (C-MRCI) calculations were performed, revealing that 13Π0− is the first excited state closely followed by 13Π0+. In addition, the dissociation energy of LuF was determined at different levels of theory, with a range of basis sets. A balance between core correlation and a relativistic treatment of electrons is fundamental to obtain an accurate description of the dissociation energy. The best prediction was obtained with a combination of coupled-cluster single, double, and perturbative triple excitations /Douglas–Kroll–Hess third order Hamiltonian methods at a complete basis set level with a zero-point energy correction, which yields a dissociation value of 170.4 kcal mol−1. Dissociation energies using density functional theory were calculated using a range of functionals and basis sets; M06-L and B3LYP provided the closest predictions to the best ab initio calculations.

The accurate description of ground and excited state properties of lanthanides provides a route toward understanding their fundamental chemical reactivity. The high density of states and partially filled 4f and 5d orbitals are hurdles that need to be properly addressed in order to achieve such predictions. The use of multireference methods in lanthanide electronic structure calculations is of paramount importance and allows for an accurate description of static and dynamic correlation. Additionally, an appropriate choice of methods to account for correlation and spin–orbit effects is necessary for both the ground and excited states.

Lutetium, the last element in the lanthanide series, is also generally regarded as the first element of the sixth period transition metals due to its full 4f and partially filled 5d orbitals. Recently, interest in lutetium has grown, with one of its main applications in the radiopharmaceutical industry, more specifically, with the use of 177Lu as a radionuclide.1 Small molecules, such as peptides and steroids, have been radiolabeled with 177Lu in the treatment of a number of diseases. For example, 177Lu-labeled DOTA-Tyr3-octreotate, which is a somatostatin analog peptide, is currently being used to treat neuroendocrine tumors.1 Lutetium also has been linked to astrophysics. It has been discovered in the composition of the metal-poor stars CS 31062-050 and CS 22892-052 and in the enriched star BD+17 3248.2–5 The Lu+ spectra has been investigated by Hartog and co-workers revealing the presence of an excited state at 28 503.16 cm−1, which corresponds to a 6s6p, 3P1 configuration.6 Lanthanide species, in general, are also being used and considered in a broad range of applications, such as in electrodes and optical telecommunications (i.e., NaLuF4). With such a wide range of applications, it is important to better understand lutetium at a fundamental level and the methodologies needed to describe its complex electron manifold.

In considering the ground and excited state properties of LuF, the available experimental data are from the 1960s, 1970s, and 1980s. In 1968, Zmbov extrapolated the dissociation energy of LuF from other lanthanide monofluorides by means of mass spectroscopy and obtained 136 ± 12 kcal mol−1.7 The authors estimated the dissociation energy of lutetium fluoride using both the heats of sublimation and the enthalpies of other lanthanide fluorides. Their estimation came from fluorine-exchange reactions of Sm, Eu, Gd, and Dy, and Er. Kaledin et al. predicted the dissociation energy of LuF to be 124 kcal mol−1. The authors used ligand field theory and extrapolated the dissociation energy, utilizing a fitting model and experimentally determined ionization potentials for other lanthanide fluorides.8 Since the 1970s, several experimental studies have targeted the vibrational and rotational spectrum of lutetium fluoride.9–12 D’Incan et al. and Effantin et al. reported dissociation energies for LuF (105 kcal mol−1) and assigned the lowest lying electronic excited states for LuF. The symmetry and spin were labeled either 1Σ or 1Π for all the excited states.9,10,12 These results were later compiled by Huber and Herberg in an extensive review of molecular spectra.11 In the 1980s, Rajamanickam and Narasimhamurthy and Reddy et al. obtained experimental dissociation energies of  96.0 ± 2.4 and 79 kcal mol−1, respectively.13,14 These authors used the experimental spectroscopic constants of the ground state (ωe, ωeχe, etc, obtained from the work of Effantin et al.12), calculated the vibrational potential energy curve (PEC), fitted it with different empirical formulas, and calculated the dissociation energy.

Theoretical studies are useful in describing the spectroscopic properties of lanthanides. There are a number of recent studies on lanthanide monohalides (LnX, X = F, Cl, Br, I).15–22 In the 1990s, a number of theoretical studies focused on the spectroscopic properties of lanthanides and actinides. Wang et al. and Küchle et al. studied diatomics, lanthanide, and lanthanide and actinide contractions and were the first to use density functional theory (DFT) along with coupled cluster (CC) methods to calculate ground state properties and bond lengths for some of these molecules.23,24 Cooke et al. investigated the rotation spectra of LuF and used DFT to compare with their ground state experimental values. Their theoretical prediction of the dissociation energy of 96.6 kcal mol−1 was based on a statistical average of orbital potentials.25 Density functional theory with scalar-relativistic ZORA and Douglas–Kroll–Hess approaches have been used by Hong et al. to calculate the dissociation energy of LuF. The authors obtained values in the range of 167–176 kcal mol−1.26 In 2016, Grimmel et al. determined for the Ln54 set, a set of 54 enthalpies of formation and bond dissociation energies of small lanthanides, using 22 different DFT functionals and employing the Douglas–Kroll–Hess Hamiltonian in combination with a triple-ζ level basis set [Sapporo-Douglas–Kroll–Hess third order Hamiltonian (DKH3)-TZP-2012 for Ln and cc-pVTZ-DK or cc-pV(T+d)Z-DK for the ligands], resulting on average, overall energy errors for the set on the order of 1 eV, even with the most popular and well-utilized functionals for the lanthanides.27 Aebersold et al. reexamined the energies of the Ln54 set using the same functionals employed by Grimmel and co-workers, considering the several impacts including the introduction of effective core potential (ECP) and DKH3 approaches.27,28 In terms of ab initio studies, the equation of motion completely renormalized coupled-cluster single, double and perturbative triple excitations [CCSD(T)] [EOM-CR-CCSD(T)] was used in a study of NdF and LuF.29 The authors reported that the use of a full valence shell rather than the traditional frozen core approximation can result in a dramatic change in the dissociation energy of LuF (a change of ∼35 kcal mol−1). Ab initio composite methods have also been employed in the prediction of ground state properties of lanthanides. Solomonik and Smirnov calculated the bond dissociation of LuF as 169.7 kcal mol−1 and Qing computed the same as 172.4 kcal mol−1,30,31 which are near to our recent prediction of 170.2 kcal mol−1 in a large scale study of lanthanides.32 In considering the prior experimental and theoretical studies, as overviewed, there are substantial differences in the predictions. It is important to note that the dissociation energies reported from experiments are not direct measurements but are instead based on empirical models.7–11 

In terms of excited states, a complete understanding of the potential energy surface of LuF and its bonding patterns allows for the probing of possible chemical reactivity routes using excited state dissociation channels. Toward this goal, in 2009, Hamade et al.33 used CASSCF (complete active space self-consistent field) and MRCI (multi-reference configuration interaction), for the first low-lying excited states of LuF, using a pseudopotential for lutetium of 60 electrons. The authors determined 26 electronic states, including the spectroscopic constants and bond lengths for each state; however, these calculations did not account for spin–orbit effects. The authors assigned the first and second excited states as 3Π and 3Δ instead of the 1Σ and 1Π states, respectively, previously assigned in the literature.9–11 In 2019, Assaf et al. used multireference methods (CASSCF and MRCI+Q) to calculate spectroscopic constants and bond lengths for ground and excited states.34 The authors considered a 28 electron pseudopotential (ECP28MWB), which allowed for a more accurate treatment of electron correlation. In addition, sub-valence electrons (4f) were also correlated, though not included in the active space. The latter step enables the prediction of bond lengths within 0.1 Å of experiment. The active space utilized in this study did not include the bonding orbitals of fluorine, which are important in the construction of the full potential energy curves. However, spin–orbit effects were considered, and spectroscopic constants were calculated for the low-lying excited states using the Breit–Pauli Hamiltonian.

Although there have been a number of studies on lutetium fluoride, detailed insight about its dissociation channels and binding patterns have not yet been provided. For this work, 132 states were investigated using multireference methods and double-, triple-, and quadruple-ζ level basis sets. The results herein provide important insight into the higher energy channels that play a role in the excited state surface of LuF. MRCI calculations were performed to recover dynamic correlation of the system beyond what CASSCF can obtain. Valence, sub-valence, and inner core levels of correlation were probed, detailing their effects on the energetics of the ground and excited states. The second part of this work (See Sec. III C) focuses on the dissociation energy (D0) using a range of DFT functionals and also ab initio methods, including coupled-cluster and CASSCF. Complete basis set (CBS) extrapolation was also considered for the ab initio methods.

Multireference calculations were performed using MOLPRO 2018.35 As MOLPRO does not use full linear molecule symmetries, the C2v point group symmetry was utilized and the molecular orbitals were optimized using CASSCF. For this step, the active space used was composed of eight electrons and fifteen orbitals (8, 15). The 15 orbitals correspond to 6 a1 [5dz2, 5dx2y2, 6s, 6pz (Lu), 2pz, 3pz (F)], 4 b1 [5dxz, 6px (Lu), 2px, 3px (F)], 4 b2 [5dyz, 6px (Lu), 2px, 3px (F)], and 1 a2 [5dxy (Lu)], which correspond to the 6s and 5d of lutetium and to 2p and 3p of fluorine. The inclusion of the additional 3p orbitals of fluorine was deemed necessary to obtain smooth potential energy curves (PECs).

MRCI and MRCI+Q were employed to calculate spectroscopic constants.36–39 Harmonic vibrational frequencies, anharmonicities, and ΔG1/2 values were calculated solving the rovibrational Schrödinger equation numerically using the Dunham approach.40 Due to the computational cost, the 2p and 3p orbitals of fluorine were not included in the active space and thus were not optimized at the CASSCF level, within the MRCI calculations. The active space for MRCI consists of the following orbitals: 4 a1 [5dz2, 5dx2y2, 6s, 6pz (Lu)], 2 b1 [5dxz, 6px (Lu)], 2 b2 [5dyz, 6px (Lu), and 1 a2 (5dxy (Lu)]. However, the 2p orbitals of fluorine were included in the MRCI calculations as “core” (per MOLPRO 2018) by allowing the electrons to be promoted to the active and virtual spaces through single and double excitations. Considering the CI vectors, for the equilibrium bond region, there are not significant contributions that correspond to the promotion of electrons from the 2p orbitals of fluorine. In addition, for the MRCI calculations, sub-valence correlation effects were also described by including the 4f14 orbitals of Lu by also allowing single and double excitations to the active and virtual spaces. Since a pseudopotential was considered for the metal (see next paragraph), the remaining 52 electrons (9 from fluorine and 43 of lutetium) were also correlated for MRCI calculations. The Davidson correction or MRCI+Q as implemented within MOLPRO was used to account for size extensivity issues.36–39 To account for spin–orbit coupling, the Breit–Pauli Hamiltonian was diagonalized in the basis of the MRCI wavefunction. For this step, two levels of correlation were considered for inclusion in the core: 4f14 (Lu) and 2p5(F) orbitals, and 4d10 5s2 5p6 4f14 (Lu) and 2s2 2p5 of (F) orbitals. The latter describe the effects of inner-shell correlation.

For CASSCF calculations, a segmented contracted basis set along with a pseudopotential (ECP28MWB) developed by Cao and Dolg was employed (triple-ζ level).41,42 For fluorine, the aug-cc-pVTZ basis set was utilized.43 For MRCI and spin–orbit calculations, the def2-QZVPP basis set was employed for lutetium with a pseudopotential (ECP28MWB), while fluorine was described with aug-cc-pVQZ.42–45 

For the second part of this work (See Sec. III C), the geometry optimization step was carried out with CCSD(T) in combination with a contracted basis set by Cao and Dolg, which was used for lutetium, and the aug-cc-pVTZ basis set for fluorine.41–43 The frequency was also obtained at the same level to ensure a minimum at the potential energy surface. The geometry was then used to evaluate dissociation energies at different levels of theory, and the energy was corrected for the zero-point vibrational energy (ZPE). CCSD(T) and the completely renormalized [CR-CCSD(T)] approach with DKH3 in combination with Sapporo double-, triple-, and quadruple-ζ basis set for lutetium and fluorine have been utilized.46 The effect of a four-component Hamiltonian on the dissociation energy was also probed with CCSD(T) using a Dirac–Coulomb (DC) Hamiltonian. In addition, the Perdew-Burke-Ernzerhof (PBE),47 the Becke, 3-parameter, Lee -Yang -Parr (B3LYP),48,49 the Minnesota 2006 local functional (M06-L)50 and the Tao, Perdew, Staroverov, Scuseria (TPSS)51 functionals were utilized to predict dissociation energies, employing a DKH3 Hamiltonian. These functionals were chosen as they are either widely utilized or were among the better functionals for the prediction of enthalpy of formation and dissociation energies for lanthanide complexes.27,28

Moreover, these functionals will provide some level of comparison between the generalized gradient approximation (GGA): PBE; meta-GGA: TPSS, M06-L; and hybrid-GGA: B3LYP on the prediction of the dissociation energy. The double-, triple-, and quadruple-ζ level Sapporo basis sets for lutetium and fluorine were used (noted Sap-nz) and the Dyall augmented double-, triple-, and quadruple-ζ (noted Dyall-nz) for the Dirac–Coulomb Hamiltonian where n = D, T, Q.52 

The dissociation energy was calculated using the methods described above and at each level of basis set as well. Extrapolations of the total energies to the complete basis set limit were performed using a mixed exponential/Gaussian three point scheme developed by Peterson,53 

En=ECBS+Ben1+Cen12,
(1)

where B and C are constants determined in the scheme, n is the basis set level (n = D, T, Q), En represents the energy for each basis set level, and ECBS represents the energy at the CBS limit. Unfortunately, it was not possible to obtain values at a quadruple-ζ basis set for CCSD(T) and MP2 with the Dirac–Coulomb Hamiltonian due to the very high computational cost. Thus, the complete basis set limit using the following two-point extrapolation (Dyall.dz and Dyall.tz) scheme by Martin54 was used:

E=ECBS+Bn+0.54.
(2)

This scheme has been shown to provide reliable extrapolated energies for molecules containing lighter elements when compared to experiment.55,56 The final dissociation energy is calculated by adding the zero-point vibrational energy to the final energy. The 95% confidence limit has been investigated, and results (Table S1) are given in the supplementary material. In addition to evaluating the 95% confidence intervals, the error from basis set superposition (BSSE) was calculated utilizing Boys and Bernardi’s counterpoise correction approach (Sec. III C).57 

Due to the large number of electrons, it is important to consider different frozen-core spaces, i.e., the number of electrons explicitly correlated. Thus, two frozen-core spaces have been considered: (FC)-val and FC-subval. FC-val corresponds to a space where only the valence electrons (6s and 5d of Lu and 2s and 2p of F) are treated and the rest is frozen. The FC-subval describes the space where the valence and sub-valence electrons are explicitly treated (5s, 5p of Lu). All calculations using the DKH3 Hamiltonian were performed with NwChem 6.1,58 while the Dirac–Coulomb calculations were done using DIRAC18.59 

The PECs calculated at the CASSCF level are displayed in Figs. 1 and 2. The former portrays the Lu (2D; 5d16s2) + F(2P) and Lu (2P; 6s25p1) + F(2P) channels, and the latter shows the two upper binding Lu (4F; 5d16s25p1) + F(2P) channels. In Fig. 3, MRCI+Q curves are provided with selected states spanning the equilibrium bond region. The zero of the energy scale in Figs. 13 is taken as the energy of the lowest energy asymptote Lu (2D) + F(2P). In Table I, detailed spectroscopic information of the ground and 22 excited states is shown, which includes spectroscopic constants, harmonic vibrational frequencies, ωeχe, ΔG1/2, and Te (excitation energies).

FIG. 1.

CASSCF PECs of LuF with respect to the Lu–F distance.

FIG. 1.

CASSCF PECs of LuF with respect to the Lu–F distance.

Close modal
FIG. 2.

Example of intersystem crossing from upper dissociation channels of LuF at the CASSCF level.

FIG. 2.

Example of intersystem crossing from upper dissociation channels of LuF at the CASSCF level.

Close modal
FIG. 3.

MRCI+Q PECs of LuF with respect to the Lu–F distance.

FIG. 3.

MRCI+Q PECs of LuF with respect to the Lu–F distance.

Close modal
TABLE I.

Computational method, bond length Req (Å), harmonic vibrational frequencies ωe (cm−1), anharmonicity ωeχe (cm−1), ΔG1/2 (cm−1) values, and excitation energy Te (cm−1) for the lowest electronic excited states of 175Lu19F. MRCI, MRCI+Q and CCSD(T) calculations were performed using an ECP28MWB/Def2-QZPP for Lu and aug-cc-pVQZ for F and CASSCF with ECP28MWB/ANO-TZ for Lu and aug-cc-pVTZ for F.

StatesMethodologyReqωeωeχeΔG1/2Te
X1Σ+ Exp.11  1.9171 611.79 2.54 ⋯ 
Exp.12  1.9165 611.79 2.54  
CASSCF ⋯ ⋯ ⋯ ⋯ 
MRCI 1.916 613.9 2.67 608.6 
MRCI+Q 1.914 611.7 2.82 606.1 
CCSD(T) 1.917 610.4 2.51 605.4 
CCSD(T) 1.917 610.8 ⋯ ⋯ 
MRCI+Q34  1.913 618.9 2.5 ⋯ 
MRCI+Q33  1.922 606.6 3.3 ⋯ 
13Δ Exp.11  1.9319 587.95 2.58 ⋯ 16 165 
Exp.12  1.9313 587.95 2.58  16 153 
CASSCF ⋯ ⋯ ⋯ ⋯ 18 000 
MRCI 1.947 573.4 2.54 568.3 14 917 
MRCI+Q 1.945 570.5 2.45 565.6 14 676 
MRCI+Q34  1.947 576.0 2.7 ⋯ 14 927 
MRCI+Q33  1.952 596.2 3.0 ⋯ 17 904 
13Π Exp.11  1.9361 576.08 2.5 ⋯ 16 800 
Exp.12  1.933 581.3 2.6  16 785 
CASSCF ⋯ ⋯ ⋯ ⋯ 17 155 
MRCI 1.928 570.8 3.88 563.0 15 630 
MRCI+Q 1.930 570.0 3.75 562.5 15 805 
CCSD(T) def2 1.943 574.8 2.50 569.6 18 528 
MRCI+Q34  1.933 579.2 2.7 ⋯ 15 959 
MRCI+Q33  1.923 567.1 2.6 ⋯ 16 165 
13Σ+ Exp.9,10  605.5 2.5 ⋯ 18 894 
CASSCF ⋯ ⋯ ⋯ ⋯ 19 900 
MRCI 1.957 600.4 2.47 595.4 17 947 
MRCI+Q 1.958 590.5 2.53 585.4 18 181 
MRCI+Q34  1.961 559.6 2.5 ⋯ 18 856 
MRCI+Q33  1.953 567.1 2.6 ⋯ 19 131 
11Δ Exp.11  1.948 569.7 2.5  20 048 
Exp.12  ⋯ ⋯ 2.6  20 027 
CASSCF ⋯ ⋯ ⋯ ⋯ 21 612 
MRCI 1.954 567.0 2.14 562.7 19 392 
MRCI+Q 1.953 564.4 2.11 560.2 19 060 
MRCI+Q34  1.955 567.7 2.8 ⋯ 19 471 
MRCI+Q33  1.956 555.0 2.5 ⋯ 21 634 
11Π Exp.11  1.9584 543.42 2.28  24 474 
Exp.12  1.9584 543.42 2.28  24 440 
CASSCF ⋯ ⋯ ⋯ ⋯ 27 049 
MRCI 1.966 554.0 2.42 549.1 23 371 
MRCI+Q 1.969 546.7 2.53 541.7 23 065 
MRCI+Q34  1.972 525.3 2.2 ⋯ 23 708 
MRCI+Q33  1.945 544.7 2.6 ⋯ 25 538 
21Σ+ Exp.11  1.9520 555.59 2.6  25 832 
Exp.12  1.9514 560.8 2.6  25 806 
CASSCF ⋯ ⋯ ⋯ ⋯ 29 240 
MRCI 1.959 548.5 4.41 539.7 25 628 
MRCI+Q 1.957 543.2 3.82 535.5 25 292 
MRCI+Q34  1.959 553.0 2.5  25 932 
MRCI+Q33  1.947 563.8 2.8  26 524 
23Π CASSCF ⋯ ⋯ ⋯ ⋯ 34 583 
MRCI 1.983 570.9 2.90 565.1 29 091 
MRCI+Q 1.978 559.9 1.67 556.5 28 870 
MRCI+Q34  1.981 577.7 2.3  29 354 
MRCI+Q33  1.995 525.7 3.4  30 681 
21Π Exp.11  1.951 599.1 2.6 ⋯ 33 226 
CASSCF ⋯ ⋯ ⋯ ⋯ 38 511 
MRCI 1.948 593.9 3.09 587.7 32 809 
MRCI+Q 1.944 606.4 3.12 600.1 32 517 
MRCI+Q34  1.951 614.7 2.9  32 968 
MRCI+Q33  1.961 579.3 2.5  33 378 
13Φ CASSCF ⋯ ⋯ ⋯ ⋯ 38 846 
MRCI 1.944 565.2 −0.33 565.9 33 566 
MRCI+Q 1.944 571.2 0.28 570.7 33 499 
MRCI+Q34  1.944 570.5 2.7  34 248 
MRCI+Q33  1.942 570.8 3.2  36 401 
33Π CASSCF ⋯ ⋯ ⋯ ⋯ 42 188 
MRCI 1.960 554.9 2.54 549.8 36 422 
MRCI+Q 1.959 543.7 2.63 538.4 36 123 
MRCI+Q34  1.956 545.0 2.8 ⋯ 36 896 
MRCI+Q33  1.957 552.4 3.2 ⋯ 39 048 
23Δ CASSCF ⋯ ⋯ ⋯ ⋯ 42 211 
MRCI 1.974 573.3 5.50 562.3 36 674 
MRCI+Q 1.974 592.2 7.18 577.8 36 323 
MRCI+Q34  1.976 540.8 3.0 ⋯ 37 162 
MRCI+Q33  1.969 541.8 2.3 ⋯ 39 569 
13Σ CASSCF ⋯ ⋯ ⋯ ⋯ 41 126 
MRCI 1.974 534.3 0.34 533.6 36 683 
MRCI+Q 1.974 522.2 0.24 521.8 36 338 
MRCI+Q34  1.973 544.0 2.6 ⋯ 37 338 
MRCI+Q33  1.949 551.3 3.6 ⋯ 39 216 
21Δ CASSCF ⋯ ⋯ ⋯ ⋯ 46 419 
MRCI 1.955 567.0 1.85 563.3 40 151 
MRCI+Q 1.956 557.6 1.47 554.6 39 524 
MRCI+Q34  1.956 558.5 2.6  40 954 
MRCI+Q33  1.946 566.6 3.3  45 661 
31Σ+ CASSCF ⋯ ⋯ ⋯ ⋯ ⋯ 
MRCI ⋯ ⋯ ⋯ ⋯ ⋯ 
MRCI+Q ⋯ ⋯ ⋯ ⋯ ⋯ 
MRCI+Q34  1.942 550.1 3.0 ⋯ 42 847 
MRCI+Q33  1.917 588.9 2.8 ⋯ 42 763 
11Σ CASSCF     46 100 
MRCI 1.953 565.3 2.40 560.5 43 049 
MRCI+Q 1.959 557.9 2.33 553.2 41 310 
MRCI+Q34  ⋯ ⋯ ⋯ ⋯ ⋯ 
MRCI+Q33  ⋯ ⋯ ⋯ ⋯ ⋯ 
11Φ CASSCF ⋯ ⋯ ⋯ ⋯ 51 158 
MRCI 1.952 566.9 2.41 562.1 43 048 
MRCI+Q 1.951 562.5 2.41 557.7 41 767 
MRCI+Q34  1.942 564.2 2.7 ⋯ 43 231 
MRCI+Q33  1.950 567.7 2.4 ⋯ 45 152 
23Σ CASSCF ⋯ ⋯ ⋯ ⋯ ⋯ 
MRCI 1.983 522.9 2.12 518.6 42 275 
MRCI+Q 1.983 510.5 2.16 506.2 41 714 
MRCI+Q34       
MRCI+Q33       
31Π CASSCF ⋯ ⋯ ⋯ ⋯ 52 481 
MRCI 1.963 600.1 2.79 594.5 44 083 
MRCI+Q 1.955 555.6 2.04 551.5 42 790 
MRCI+Q34  1.941 550.4 2.9 ⋯ 44 678 
MRCI+Q33  1.944 574.2 2.8 ⋯ 45 319 
43Π CASSCF ⋯ ⋯ ⋯ ⋯ 49 935 
MRCI 1.968 553.4 4.25 544.9 44 648 
MRCI+Q 1.972 545.9 4.06 537.8 44 453 
MRCI+Q34  1.972 553.5 3.0 ⋯ 44 849 
MRCI+Q33  1.957 553.4 3.2 ⋯ 45 454 
31Δ CASSCF ⋯ ⋯ ⋯ ⋯ 53 789 
MRCI 1.996 517.3 −3.92 525.1 45 578 
MRCI+Q 1.982 525.52 2.79 519.9 44 774 
MRCI+Q34  1.975 546.6 3.3 ⋯ 43 806 
MRCI+Q33  1.965 540.3 2.1 ⋯ 47 006 
21Σ CASSCF ⋯ ⋯ ⋯ ⋯ ⋯ 
MRCI 1.980 529.5 2.38 524.6 45 660 
MRCI+Q 1995 500.6 −6.47 513.59 45 461 
MRCI+Q34  ⋯ ⋯ ⋯ ⋯ ⋯ 
MRCI+Q33  ⋯ ⋯ ⋯ ⋯ ⋯ 
23Σ+ CASSCF ⋯ ⋯ ⋯ ⋯ ⋯ 
MRCI ⋯ ⋯ ⋯ ⋯ ⋯ 
MRCI+Q ⋯ ⋯ ⋯ ⋯ ⋯ 
MRCI+Q34  1.900 556.4 2.7 ⋯ 47 316 
MRCI+Q33  1.871 664.6 2.1 ⋯ 43 031 
StatesMethodologyReqωeωeχeΔG1/2Te
X1Σ+ Exp.11  1.9171 611.79 2.54 ⋯ 
Exp.12  1.9165 611.79 2.54  
CASSCF ⋯ ⋯ ⋯ ⋯ 
MRCI 1.916 613.9 2.67 608.6 
MRCI+Q 1.914 611.7 2.82 606.1 
CCSD(T) 1.917 610.4 2.51 605.4 
CCSD(T) 1.917 610.8 ⋯ ⋯ 
MRCI+Q34  1.913 618.9 2.5 ⋯ 
MRCI+Q33  1.922 606.6 3.3 ⋯ 
13Δ Exp.11  1.9319 587.95 2.58 ⋯ 16 165 
Exp.12  1.9313 587.95 2.58  16 153 
CASSCF ⋯ ⋯ ⋯ ⋯ 18 000 
MRCI 1.947 573.4 2.54 568.3 14 917 
MRCI+Q 1.945 570.5 2.45 565.6 14 676 
MRCI+Q34  1.947 576.0 2.7 ⋯ 14 927 
MRCI+Q33  1.952 596.2 3.0 ⋯ 17 904 
13Π Exp.11  1.9361 576.08 2.5 ⋯ 16 800 
Exp.12  1.933 581.3 2.6  16 785 
CASSCF ⋯ ⋯ ⋯ ⋯ 17 155 
MRCI 1.928 570.8 3.88 563.0 15 630 
MRCI+Q 1.930 570.0 3.75 562.5 15 805 
CCSD(T) def2 1.943 574.8 2.50 569.6 18 528 
MRCI+Q34  1.933 579.2 2.7 ⋯ 15 959 
MRCI+Q33  1.923 567.1 2.6 ⋯ 16 165 
13Σ+ Exp.9,10  605.5 2.5 ⋯ 18 894 
CASSCF ⋯ ⋯ ⋯ ⋯ 19 900 
MRCI 1.957 600.4 2.47 595.4 17 947 
MRCI+Q 1.958 590.5 2.53 585.4 18 181 
MRCI+Q34  1.961 559.6 2.5 ⋯ 18 856 
MRCI+Q33  1.953 567.1 2.6 ⋯ 19 131 
11Δ Exp.11  1.948 569.7 2.5  20 048 
Exp.12  ⋯ ⋯ 2.6  20 027 
CASSCF ⋯ ⋯ ⋯ ⋯ 21 612 
MRCI 1.954 567.0 2.14 562.7 19 392 
MRCI+Q 1.953 564.4 2.11 560.2 19 060 
MRCI+Q34  1.955 567.7 2.8 ⋯ 19 471 
MRCI+Q33  1.956 555.0 2.5 ⋯ 21 634 
11Π Exp.11  1.9584 543.42 2.28  24 474 
Exp.12  1.9584 543.42 2.28  24 440 
CASSCF ⋯ ⋯ ⋯ ⋯ 27 049 
MRCI 1.966 554.0 2.42 549.1 23 371 
MRCI+Q 1.969 546.7 2.53 541.7 23 065 
MRCI+Q34  1.972 525.3 2.2 ⋯ 23 708 
MRCI+Q33  1.945 544.7 2.6 ⋯ 25 538 
21Σ+ Exp.11  1.9520 555.59 2.6  25 832 
Exp.12  1.9514 560.8 2.6  25 806 
CASSCF ⋯ ⋯ ⋯ ⋯ 29 240 
MRCI 1.959 548.5 4.41 539.7 25 628 
MRCI+Q 1.957 543.2 3.82 535.5 25 292 
MRCI+Q34  1.959 553.0 2.5  25 932 
MRCI+Q33  1.947 563.8 2.8  26 524 
23Π CASSCF ⋯ ⋯ ⋯ ⋯ 34 583 
MRCI 1.983 570.9 2.90 565.1 29 091 
MRCI+Q 1.978 559.9 1.67 556.5 28 870 
MRCI+Q34  1.981 577.7 2.3  29 354 
MRCI+Q33  1.995 525.7 3.4  30 681 
21Π Exp.11  1.951 599.1 2.6 ⋯ 33 226 
CASSCF ⋯ ⋯ ⋯ ⋯ 38 511 
MRCI 1.948 593.9 3.09 587.7 32 809 
MRCI+Q 1.944 606.4 3.12 600.1 32 517 
MRCI+Q34  1.951 614.7 2.9  32 968 
MRCI+Q33  1.961 579.3 2.5  33 378 
13Φ CASSCF ⋯ ⋯ ⋯ ⋯ 38 846 
MRCI 1.944 565.2 −0.33 565.9 33 566 
MRCI+Q 1.944 571.2 0.28 570.7 33 499 
MRCI+Q34  1.944 570.5 2.7  34 248 
MRCI+Q33  1.942 570.8 3.2  36 401 
33Π CASSCF ⋯ ⋯ ⋯ ⋯ 42 188 
MRCI 1.960 554.9 2.54 549.8 36 422 
MRCI+Q 1.959 543.7 2.63 538.4 36 123 
MRCI+Q34  1.956 545.0 2.8 ⋯ 36 896 
MRCI+Q33  1.957 552.4 3.2 ⋯ 39 048 
23Δ CASSCF ⋯ ⋯ ⋯ ⋯ 42 211 
MRCI 1.974 573.3 5.50 562.3 36 674 
MRCI+Q 1.974 592.2 7.18 577.8 36 323 
MRCI+Q34  1.976 540.8 3.0 ⋯ 37 162 
MRCI+Q33  1.969 541.8 2.3 ⋯ 39 569 
13Σ CASSCF ⋯ ⋯ ⋯ ⋯ 41 126 
MRCI 1.974 534.3 0.34 533.6 36 683 
MRCI+Q 1.974 522.2 0.24 521.8 36 338 
MRCI+Q34  1.973 544.0 2.6 ⋯ 37 338 
MRCI+Q33  1.949 551.3 3.6 ⋯ 39 216 
21Δ CASSCF ⋯ ⋯ ⋯ ⋯ 46 419 
MRCI 1.955 567.0 1.85 563.3 40 151 
MRCI+Q 1.956 557.6 1.47 554.6 39 524 
MRCI+Q34  1.956 558.5 2.6  40 954 
MRCI+Q33  1.946 566.6 3.3  45 661 
31Σ+ CASSCF ⋯ ⋯ ⋯ ⋯ ⋯ 
MRCI ⋯ ⋯ ⋯ ⋯ ⋯ 
MRCI+Q ⋯ ⋯ ⋯ ⋯ ⋯ 
MRCI+Q34  1.942 550.1 3.0 ⋯ 42 847 
MRCI+Q33  1.917 588.9 2.8 ⋯ 42 763 
11Σ CASSCF     46 100 
MRCI 1.953 565.3 2.40 560.5 43 049 
MRCI+Q 1.959 557.9 2.33 553.2 41 310 
MRCI+Q34  ⋯ ⋯ ⋯ ⋯ ⋯ 
MRCI+Q33  ⋯ ⋯ ⋯ ⋯ ⋯ 
11Φ CASSCF ⋯ ⋯ ⋯ ⋯ 51 158 
MRCI 1.952 566.9 2.41 562.1 43 048 
MRCI+Q 1.951 562.5 2.41 557.7 41 767 
MRCI+Q34  1.942 564.2 2.7 ⋯ 43 231 
MRCI+Q33  1.950 567.7 2.4 ⋯ 45 152 
23Σ CASSCF ⋯ ⋯ ⋯ ⋯ ⋯ 
MRCI 1.983 522.9 2.12 518.6 42 275 
MRCI+Q 1.983 510.5 2.16 506.2 41 714 
MRCI+Q34       
MRCI+Q33       
31Π CASSCF ⋯ ⋯ ⋯ ⋯ 52 481 
MRCI 1.963 600.1 2.79 594.5 44 083 
MRCI+Q 1.955 555.6 2.04 551.5 42 790 
MRCI+Q34  1.941 550.4 2.9 ⋯ 44 678 
MRCI+Q33  1.944 574.2 2.8 ⋯ 45 319 
43Π CASSCF ⋯ ⋯ ⋯ ⋯ 49 935 
MRCI 1.968 553.4 4.25 544.9 44 648 
MRCI+Q 1.972 545.9 4.06 537.8 44 453 
MRCI+Q34  1.972 553.5 3.0 ⋯ 44 849 
MRCI+Q33  1.957 553.4 3.2 ⋯ 45 454 
31Δ CASSCF ⋯ ⋯ ⋯ ⋯ 53 789 
MRCI 1.996 517.3 −3.92 525.1 45 578 
MRCI+Q 1.982 525.52 2.79 519.9 44 774 
MRCI+Q34  1.975 546.6 3.3 ⋯ 43 806 
MRCI+Q33  1.965 540.3 2.1 ⋯ 47 006 
21Σ CASSCF ⋯ ⋯ ⋯ ⋯ ⋯ 
MRCI 1.980 529.5 2.38 524.6 45 660 
MRCI+Q 1995 500.6 −6.47 513.59 45 461 
MRCI+Q34  ⋯ ⋯ ⋯ ⋯ ⋯ 
MRCI+Q33  ⋯ ⋯ ⋯ ⋯ ⋯ 
23Σ+ CASSCF ⋯ ⋯ ⋯ ⋯ ⋯ 
MRCI ⋯ ⋯ ⋯ ⋯ ⋯ 
MRCI+Q ⋯ ⋯ ⋯ ⋯ ⋯ 
MRCI+Q34  1.900 556.4 2.7 ⋯ 47 316 
MRCI+Q33  1.871 664.6 2.1 ⋯ 43 031 

In Table II, the CI vectors of the studied ground and excited states are shown. For the calculations, CASSCF, MRCI, and MRCI+Q were used, and for states that were deemed to be single reference in nature, CCSD(T) was employed. For the first part of this work (Fig. 1), state averaged CASSCF was used for the 132 states, which aids in describing intersystem crossings that come from upper channels and merge with the Lu (2D) + F(2P) channel. This is the first time such a level of detail is considered for LuF, providing insight into how the dissociation channels are formed and describing some of the higher energy, upper channel intersystem crossings. In addition, no evidence of the presence of the ionic channel (Lu+ + F) was found in the MCSCF calculations. In addition, from the 132 states studied, none of them converged to Lu+ + F at infinity, demonstrated by its CI vectors. The orbital pictures included in the active space at an equilibrium bond length (1.92 Å) and at 6 Å are shown in Figs. 4 and 5, respectively.

TABLE II.

CI vectors at equilibrium bond length for LuF were obtained through CASSCF using the ECP28MWB/ANO-TZ for Lu and aug-cc-pVTZ for F.

StatesCoeff1πz1δx2y21δz22πz3πz1πx1δxz2πx3πx1πy1δyz2πy3πy1δxy
X1Σ+ 0.91 
13Δ 0.94 α α 
13Π 0.91 α α 
13Σ+ 0.96 α α 
11Δ 0.59 β α 
−0.59 α β 
11Π −0.35 β α 
0.35 α β 
−0.50 β α 
0.50 α β 
21Σ+ −0.65 β α 
0.65 α β 
23Π 0.86 α α 
2 1Π −0.50 α β 
0.50 β α 
13Φ 0.66 α α 
0.66 α α 
33Π −0.54 α α 
0.54 α α 
23Δ 0.94 α α 
13Σ 0.64 α α 
−0.65 α α 
21Δ 0.59 β α 
−0.59 α β 
31Σ+ 0.56 
0.57 
−0.30 
−0.30 
11Σ −0.69 α β 
0.69 β α 
11Φ 0.43 α β 
−0.43 β α 
0.43 α β 
−0.43 β α 
23Σ 0.71 α α 
−0.34 α α 
−0.34 α α 
0.45 α α 
31Π −0.30 β α 
0.30 α β 
43Π 0.80 α α 
0.30 α α 
−0.30 α 
31Δ 0.49 α β 
−0.49 β α 
21Σ 0.48 α β 
−0.48 β α 
−0.48 α β 
0.48 β α 
23Σ+ 0.94 α α 
StatesCoeff1πz1δx2y21δz22πz3πz1πx1δxz2πx3πx1πy1δyz2πy3πy1δxy
X1Σ+ 0.91 
13Δ 0.94 α α 
13Π 0.91 α α 
13Σ+ 0.96 α α 
11Δ 0.59 β α 
−0.59 α β 
11Π −0.35 β α 
0.35 α β 
−0.50 β α 
0.50 α β 
21Σ+ −0.65 β α 
0.65 α β 
23Π 0.86 α α 
2 1Π −0.50 α β 
0.50 β α 
13Φ 0.66 α α 
0.66 α α 
33Π −0.54 α α 
0.54 α α 
23Δ 0.94 α α 
13Σ 0.64 α α 
−0.65 α α 
21Δ 0.59 β α 
−0.59 α β 
31Σ+ 0.56 
0.57 
−0.30 
−0.30 
11Σ −0.69 α β 
0.69 β α 
11Φ 0.43 α β 
−0.43 β α 
0.43 α β 
−0.43 β α 
23Σ 0.71 α α 
−0.34 α α 
−0.34 α α 
0.45 α α 
31Π −0.30 β α 
0.30 α β 
43Π 0.80 α α 
0.30 α α 
−0.30 α 
31Δ 0.49 α β 
−0.49 β α 
21Σ 0.48 α β 
−0.48 β α 
−0.48 α β 
0.48 β α 
23Σ+ 0.94 α α 
FIG. 4.

Molecular orbitals for LuF at 1.92 Å.

FIG. 4.

Molecular orbitals for LuF at 1.92 Å.

Close modal
FIG. 5.

Molecular orbitals for LuF at 6.0 Å.

FIG. 5.

Molecular orbitals for LuF at 6.0 Å.

Close modal

At 6 Å, the orbitals resemble atomic ones, with no mixing between fluorine and lutetium, providing insight into dissociation. The radial distribution using CR-CCSD(T) is plotted in Fig. 6.

FIG. 6.

Radial distribution functions at the CR-CCSD(T) level.

FIG. 6.

Radial distribution functions at the CR-CCSD(T) level.

Close modal

The large orbital overlap near the equilibrium bond length (1.92 Å) shows that the 3p orbitals need to be included at the CASSCF level to describe the full dissociation channels from infinity to equilibrium smoothly. An active space with 15 orbitals in the calculation of full potential energy curves for LuF was deemed necessary to obtain smooth curves. Accounting for the irreducible representation for each spin generates hundreds of thousands of configuration state functions (CSFs), increasing both the complexity of the calculations and the computational time.

According to the Witmer–Wigner angular momentum coupling rules, the four channels generate the following manifolds of states:

  • first—Lu (2D) + F(2P): 1,3+(2), Π (3), Δ (2), Φ, Σ];

  • second—Lu (2P) + F(2P): 1,3+(2), Π (2), Δ, Σ];

  • third and fourth—Lu (4F) + F(2P): 3,5+(2), Π (3), Δ (3), Φ (2), Γ, Σ].

The calculations show that the ground state is a well separated 1Σ+, a closed shell singlet, which is in agreement with experiment.9–11 In the ground state, the unpaired 5d1 (Lu, at infinity) electron couples with the unpaired electron on the 2pz orbital of fluorine (see Table II). The spectroscopic constants calculated with CCSD(T) and MRCI/MRCI+Q are all within 1 cm−1 of experiment. The next two states were assigned as either 3Π or 3Δ in previous literature. Hamade et al. predicted the 3Π to be the first excited state, and Assaf et al. predicted the 3Δ as the first excited state.33,34 According to our calculations, for CASSCF, 13Π is followed by 13Δ and their separation is 845 cm−1. However, for MRCI and MRCI+Q, the 3Δ is the first excited state followed by the 3Π. The separation of states for MRCI and MRCI+Q is 713 cm−1 and 1129 cm−1, respectively (see Table II). Both states are a product of electron promotion from the lutetium 6s (at infinite separation) to its 5d orbitals (see Figs. 4 and 5). In order to generate the 13Δ state, an electron populates the 5dx2y2 (Lu), while for 3Π, it occupies the 5dxz (Lu). These two states are very close in energy and both were assigned a different spin and symmetry in previous experimental data. In the present work, the two experimental values from the literature were assigned to 13Δ and 13Π.9–12 Previous theoretical data from Hamed et al. and Assaf et al. do not compare the first experimental excited state energy with their first calculated excited state.11,12,33,34 Assaf et al. assigns their second excited state to A and B from the literature, 1Σ+ and 1Π, respectively.11,12,34 The 13Π is in good agreement with experiment for bond lengths and spectroscopic constants, but the 3Δ is ∼1000 cm−1 below the experimental value. However, when both 13Δ and 13Π are corrected for spin–orbit effects (see Sec. III B), the range of Ω-state energies spans over 3000 cm−1 (Table III). The next excited state is 13Σ+, which corresponds to a promotion of an electron from the 6s of lutetium to the 5dz2. In fact, electronic excitations from 6s → 5d orbitals occur until ∼33 000 cm−1, as per Table II. States 11Δ and 11Π are the corresponding open-shell singlets of 13Δ and 13 Π, respectively, and are 4384 and 7260 cm−1 above the aforementioned, according to MRCI+Q.

11Δ and 11Π are also 1000 cm−1 below their assigned experimental states, but their bond length is within 0.01 Å from experiment. The next three states, 21Σ+, 23Π, and 21Π, also correspond to the promotion of an electron from the 6s (Lu) into the 5d orbitals (Lu). 13Φ is 33 566 and 33 499 cm−1 above the ground state according to MRCI and MRCI+Q, respectively, and it is the first excited state that has two electrons promoted from the 6s (Lu) into 5d and 6p (Lu) orbitals. There is a ∼3000 cm−1 gap in which there are no populated states, but in the 36 000 cm−1 region, there are three excited states within 200 cm−1 of one another according to MRCI+Q (33Π, 23Δ, and 13Σ). From 36 000 to 50 000 cm−1, there is a large agglomeration of states, which show mixing from the first two dissociation channels. In this 14 000 cm−1 or 30 kcal mol−1 region, nine states overlap each other. The first state in this region is 21Δ, followed by 31Σ+. The latter belongs to the next binding channel, Lu (2P; 6s25p1) + F(2P) (see Fig. 2). This channel is not displayed in Fig. 1 due to the very large mix of states from upper channels, so only the binding region (2.7–1.4 Å) is plotted. The other states displayed in Fig. 1, which belong to the Lu (2P; 6s25p1) + F(2P) channel, are 23Σ, 43Π, 31Δ, and 21Σ. The last states that belong to the first binding channel are 11Σ, 11Φ, and 23Σ+. The first 1Σ state undergoes intersystem crossings, as shown in Fig. 2. There is a range of singlet and triplet states that couple together after 45 000 cm−1 (∼175 kcal mol−1) from three different dissociation channels, which originate multiple avoided and intersystem crossings.

Spin–orbit calculations were performed on the ground state and the first eight excited states of LuF, which cover a region of ∼100 kcal mol−1 or ∼36 000 cm−1. The first nine 2S+1Λ states split into the Ω-states as follows: X1Σ+ → X1Σ+0+; 13Δ → 3Δ1, 3Δ2,3Δ3; 13Π → 3Π0−,3Π0+, 3Π1,3Π2; 13Σ+3Σ+0+, 3Σ+1; 11Δ → 1Δ2; 11Π → 1Π1; 21Σ+ → 21Σ+0+, 23Π → 3Π0−, 3Π0+, 3Π1, 3Π2, 21Π → 1Π1; and 13Φ → 3Φ2, 3Φ3, 3Φ4. For singlet states, Λ = 0 is expected to be minimal. The C-MRCI spin–orbit PECs are depicted in Fig. 7 (spin–orbit states with the same Ω value have the same color), and MRCI spin–orbit are depicted in supplementary material (Fig. S1. and Table S2). The bond lengths and spectroscopy constants are included in Table III, and the decomposition of the spin–orbit states is included in Table IV.

FIG. 7.

Core-spin–orbit MRCI (C-MRCI) PECs of LuF with respect to the Lu–F distance.

FIG. 7.

Core-spin–orbit MRCI (C-MRCI) PECs of LuF with respect to the Lu–F distance.

Close modal
TABLE III.

Methodology, bond length Req (Å), harmonic vibrational frequencies ωe (cm−1), anharmonicity ωeχe (cm−1), ΔG1/2 (cm−1) values, and excitation energy Te (cm−1) for the lowest electronic excited states of 175Lu19F at the spin–orbit level. The states are ordered according to C-MRCI energetics. MRCI and C-MRCI calculations were performed using an ECP28MWB/Def2-QZPP for Lu and aug-cc-pVQZ for F.

StatesMethodologyReqωeωeχeΔG1/2Te
X1Σ+0+ MRCI 1.917 614.9 2.24 610.4 
C-MRCI 1.913 618.2 2.52 613.2 
MRCI34  1.914 619.4 2.53 ⋯ 
13Π0− MRCI 1.933 584.7 2.72 579.3 13 831 
C-MRCI 1.924 595.6 2.59 590.4 14 377 
MRCI34  1.938 573.3 2.58 ⋯ 14 629 
13Π0+ MRCI 1.928 590 2.62 584.9 14 270 
C-MRCI 1.919 601.3 2.57 596.2 14 788 
MRCI34  1.935 577.3 2.69 ⋯ 15 003 
13Δ1 MRCI 1.952 567.2 2.81 561.6 13 866 
C-MRCI 1.939 565.0 1.62 561.8 14 943 
MRCI34  1.949 572.4 2.57 ⋯ 13 513 
13Π1 MRCI 1.935 585.9 1.76 582.4 15 142 
C-MRCI 1.932 601.3 3.29 594.7 15 844 
MRCI34  1.938 571.3 2.59 ⋯ 15 600 
13Δ2 MRCI 1.953 569.3 2.30 564.7 14 781 
C-MRCI 1.943 571.3 2.88 565.6 15 890 
MRCI34  1.949 572.7 2.82 ⋯ 14 435 
13Π2 MRCI 1.929 589.8 2.48 584.8 16 774 
C-MRCI 1.922 601.0 2.62 595.7 17 313 
MRCI34  1.931 580.5 2.66 ⋯ 16 884 
13Δ3 MRCI 1.947 576.4 2.43 571.5 16 748 
C-MRCI 1.941 579.7 2.20 575.2 17 641 
MRCI34  1.946 576.4 2.53 ⋯ 16 170 
13Σ+1 MRCI 1.952 574.6 2.51 569.5 18 955 
C-MRCI 1.945 581.6 2.52 576.5 19 520 
MRCI34  1.956 567.7 2.80 ⋯ 19 101 
13Σ+0− MRCI 1.952 575.9 2.74 570.4 19 238 
C-MRCI 1.945 581.6 2.48 576.7 19 782 
MRCI34  1.955 567.9 2.77 ⋯ 19 352 
11Δ2 MRCI 1.953 570.6 2.07 566.5 19 902 
C-MRCI 1.946 576.8 2.38 572.0 21 180 
MRCI34  1.954 570.2 2.54 ⋯ 19 702 
11Π1 MRCI 1.967 534.1 2.00 530.1 24 403 
C-MRCI 1.954 544.1 2.03 540.1 25 493 
MRCI34  1.969 528.65 2.31 ⋯ 23 839 
21Σ+0− MRCI 1.955 562.7 2.71 557.3 26 211 
C-MRCI 1.946 572.5 2.82 566.9 27 042 
MRCI34  1.959 553.1 2.60 ⋯ 26 037 
23Π0− MRCI 1.993 532.1 ⋯ 556.6 28 782 
C-MRCI 1.991 523.8 ⋯ 566.08 29 920 
MRCI34  1.984 573.6 2.61 ⋯ 28 744 
23Π0+ MRCI 1.991 536.8 ⋯ 557.5 28 818 
C-MRCI 1.989 523.4 ⋯ 568.3 29 959 
MRCI34  1.984 574.5 2.58 ⋯ 28 730 
23Π1 MRCI 1.988 570.2 ⋯ 556.8 29 369 
MRCI 1.986 567.8 ⋯ 556.0 30 461 
C-MRCI34  1.982 577.5 2.64 ⋯ 29 291 
23Π2 MRCI 1.981 656.3 ⋯ 586.9 30 345 
C-MRCI 1.979 611.7 ⋯ 564.2 31 373 
MRCI34  1.978 583.2 2.66 ⋯ 30 095 
13Φ2 MRCI 1.948 512.8 ⋯ 536.9 31 774 
C-MRCI 1.939 531.6 ⋯ 553.4 33 444 
MRCI34  1.951 560.6 2.41 ⋯ 31 877 
21Π1 MRCI 1.949 577.15 ⋯ 579.3 32 891 
C-MRCI 1.946 584.1 ⋯ 590.5 33 812 
MRCI34  1.957 594.72 2.35 ⋯ 32 921 
13Φ3 MRCI 1.946 597.7 ⋯ 586.9 34 013 
C-MRCI 1.937 608.9 ⋯ 595.5 35 587 
MRCI34  1.946 565.9 2.41 ⋯ 33 965 
13Φ4 MRCI 1.942 571.8 ⋯ 568.7 36 287 
C-MRCI 1.934 583.0 ⋯ 578.7 37 762 
MRCI34  1.949 494.93 2.53 ⋯ 36 218 
StatesMethodologyReqωeωeχeΔG1/2Te
X1Σ+0+ MRCI 1.917 614.9 2.24 610.4 
C-MRCI 1.913 618.2 2.52 613.2 
MRCI34  1.914 619.4 2.53 ⋯ 
13Π0− MRCI 1.933 584.7 2.72 579.3 13 831 
C-MRCI 1.924 595.6 2.59 590.4 14 377 
MRCI34  1.938 573.3 2.58 ⋯ 14 629 
13Π0+ MRCI 1.928 590 2.62 584.9 14 270 
C-MRCI 1.919 601.3 2.57 596.2 14 788 
MRCI34  1.935 577.3 2.69 ⋯ 15 003 
13Δ1 MRCI 1.952 567.2 2.81 561.6 13 866 
C-MRCI 1.939 565.0 1.62 561.8 14 943 
MRCI34  1.949 572.4 2.57 ⋯ 13 513 
13Π1 MRCI 1.935 585.9 1.76 582.4 15 142 
C-MRCI 1.932 601.3 3.29 594.7 15 844 
MRCI34  1.938 571.3 2.59 ⋯ 15 600 
13Δ2 MRCI 1.953 569.3 2.30 564.7 14 781 
C-MRCI 1.943 571.3 2.88 565.6 15 890 
MRCI34  1.949 572.7 2.82 ⋯ 14 435 
13Π2 MRCI 1.929 589.8 2.48 584.8 16 774 
C-MRCI 1.922 601.0 2.62 595.7 17 313 
MRCI34  1.931 580.5 2.66 ⋯ 16 884 
13Δ3 MRCI 1.947 576.4 2.43 571.5 16 748 
C-MRCI 1.941 579.7 2.20 575.2 17 641 
MRCI34  1.946 576.4 2.53 ⋯ 16 170 
13Σ+1 MRCI 1.952 574.6 2.51 569.5 18 955 
C-MRCI 1.945 581.6 2.52 576.5 19 520 
MRCI34  1.956 567.7 2.80 ⋯ 19 101 
13Σ+0− MRCI 1.952 575.9 2.74 570.4 19 238 
C-MRCI 1.945 581.6 2.48 576.7 19 782 
MRCI34  1.955 567.9 2.77 ⋯ 19 352 
11Δ2 MRCI 1.953 570.6 2.07 566.5 19 902 
C-MRCI 1.946 576.8 2.38 572.0 21 180 
MRCI34  1.954 570.2 2.54 ⋯ 19 702 
11Π1 MRCI 1.967 534.1 2.00 530.1 24 403 
C-MRCI 1.954 544.1 2.03 540.1 25 493 
MRCI34  1.969 528.65 2.31 ⋯ 23 839 
21Σ+0− MRCI 1.955 562.7 2.71 557.3 26 211 
C-MRCI 1.946 572.5 2.82 566.9 27 042 
MRCI34  1.959 553.1 2.60 ⋯ 26 037 
23Π0− MRCI 1.993 532.1 ⋯ 556.6 28 782 
C-MRCI 1.991 523.8 ⋯ 566.08 29 920 
MRCI34  1.984 573.6 2.61 ⋯ 28 744 
23Π0+ MRCI 1.991 536.8 ⋯ 557.5 28 818 
C-MRCI 1.989 523.4 ⋯ 568.3 29 959 
MRCI34  1.984 574.5 2.58 ⋯ 28 730 
23Π1 MRCI 1.988 570.2 ⋯ 556.8 29 369 
MRCI 1.986 567.8 ⋯ 556.0 30 461 
C-MRCI34  1.982 577.5 2.64 ⋯ 29 291 
23Π2 MRCI 1.981 656.3 ⋯ 586.9 30 345 
C-MRCI 1.979 611.7 ⋯ 564.2 31 373 
MRCI34  1.978 583.2 2.66 ⋯ 30 095 
13Φ2 MRCI 1.948 512.8 ⋯ 536.9 31 774 
C-MRCI 1.939 531.6 ⋯ 553.4 33 444 
MRCI34  1.951 560.6 2.41 ⋯ 31 877 
21Π1 MRCI 1.949 577.15 ⋯ 579.3 32 891 
C-MRCI 1.946 584.1 ⋯ 590.5 33 812 
MRCI34  1.957 594.72 2.35 ⋯ 32 921 
13Φ3 MRCI 1.946 597.7 ⋯ 586.9 34 013 
C-MRCI 1.937 608.9 ⋯ 595.5 35 587 
MRCI34  1.946 565.9 2.41 ⋯ 33 965 
13Φ4 MRCI 1.942 571.8 ⋯ 568.7 36 287 
C-MRCI 1.934 583.0 ⋯ 578.7 37 762 
MRCI34  1.949 494.93 2.53 ⋯ 36 218 
TABLE IV.

Spin–orbit composition at the C-MRCI level (1.92 Å) for the lowest excited states of 175Lu19F.

StateComposition
X1Σ+0+ 99.87% X1Σ+, 0.06% 13Π, 0.08% 23Π 
13Π0− 89.24% 13Π, 10.76% 13Σ+ 
13Π0+ 98.44% 13Π, 1.49% 21Σ+,0.06% X1Σ+ 
13Δ1 42.49% 13Δ, 53.19% 13Π, 3.81% 13Σ+, 0.26% 11Π, 0.16% 21Π,0.08% 23Π 
13Π1 45.12% 13Π, 46.13% 13Δ, 7.32% 13Σ+, 1.34% 11Π, 0.07% 23Π, 0.02% 21Π 
13Δ2 69.16% 13Δ, 25.20% 13Π, 5.30% 11Δ, 0.24% 3Φ, 0.08% 23Π 
13Π2 74.0% 13Π, 25.83% 13Δ, 0.12% 11Δ, 0.02% 23Π, 0.02% 13Φ 
13Δ3 99.95% 13Δ, 0.05% 13Φ 
13Σ+1 86.83% 13Σ, 11.78% 13Π, 1.14% 11Π, 0.24% 13Δ, 0.01% 23Π 
13Σ+0− 89.22% 13Σ+, 10.76% 13Π, 0.02% 23Π 
11Δ2 93.78% 11Δ, 4.84% 13Δ, 0.78% 13Π, 0.36% 23Π, 0.24 13Φ% 
11Π1 97.21% 11Π, 2.03% 13Σ+, 0.57% 13Π, 0.14% 13Δ, 0.06% 23Π 
21Σ+0+ 98.32% 21Σ+, 1.50% 13Π, 0.18% 23Π 
23Π0− 99.98% 23Π, 0.02% 13Π, 0.01% 13Σ+ 
23Π0+ 99.74% 23Π, 0.19% 21Σ+, 0.07% X1Σ+ 
23Π1 94.6%, 5.26% 11Π, 0.08%, 0.01% 
23Π2 99.28% 23Π, 0.46% 11Δ, 0.14% 13Φ, 0.13% 13Δ 
13Φ2 99.52% 13Φ, 0.34% 11Δ, 0.10% 13Π, 0.05% 13Δ 
21Π1 94.56% 21Π, 5.19% 23Π, 0.22% 13Δ, 0.03% 13Π 
13Φ3 99.95% 13Φ, 0.06% 13Δ 
13Φ4 100% 13Φ 
StateComposition
X1Σ+0+ 99.87% X1Σ+, 0.06% 13Π, 0.08% 23Π 
13Π0− 89.24% 13Π, 10.76% 13Σ+ 
13Π0+ 98.44% 13Π, 1.49% 21Σ+,0.06% X1Σ+ 
13Δ1 42.49% 13Δ, 53.19% 13Π, 3.81% 13Σ+, 0.26% 11Π, 0.16% 21Π,0.08% 23Π 
13Π1 45.12% 13Π, 46.13% 13Δ, 7.32% 13Σ+, 1.34% 11Π, 0.07% 23Π, 0.02% 21Π 
13Δ2 69.16% 13Δ, 25.20% 13Π, 5.30% 11Δ, 0.24% 3Φ, 0.08% 23Π 
13Π2 74.0% 13Π, 25.83% 13Δ, 0.12% 11Δ, 0.02% 23Π, 0.02% 13Φ 
13Δ3 99.95% 13Δ, 0.05% 13Φ 
13Σ+1 86.83% 13Σ, 11.78% 13Π, 1.14% 11Π, 0.24% 13Δ, 0.01% 23Π 
13Σ+0− 89.22% 13Σ+, 10.76% 13Π, 0.02% 23Π 
11Δ2 93.78% 11Δ, 4.84% 13Δ, 0.78% 13Π, 0.36% 23Π, 0.24 13Φ% 
11Π1 97.21% 11Π, 2.03% 13Σ+, 0.57% 13Π, 0.14% 13Δ, 0.06% 23Π 
21Σ+0+ 98.32% 21Σ+, 1.50% 13Π, 0.18% 23Π 
23Π0− 99.98% 23Π, 0.02% 13Π, 0.01% 13Σ+ 
23Π0+ 99.74% 23Π, 0.19% 21Σ+, 0.07% X1Σ+ 
23Π1 94.6%, 5.26% 11Π, 0.08%, 0.01% 
23Π2 99.28% 23Π, 0.46% 11Δ, 0.14% 13Φ, 0.13% 13Δ 
13Φ2 99.52% 13Φ, 0.34% 11Δ, 0.10% 13Π, 0.05% 13Δ 
21Π1 94.56% 21Π, 5.19% 23Π, 0.22% 13Δ, 0.03% 13Π 
13Φ3 99.95% 13Φ, 0.06% 13Δ 
13Φ4 100% 13Φ 

The ground state of LuF (X1Σ+), 11Δ, 11Π, and 21Σ+ remain almost unaffected due to zero first order spin–orbit effects. Without spin–orbit effects, the 13Δ is the first excited state followed by 13Π, which is ∼1200 cm 1 higher in energy according to MRCI+Q. However, with spin–orbit correction, the ordering of Ω-states is more complex to assess due to the closeness of the energetics gaps. The 3Δ and 3Π states, spin–orbit corrected at the MRCI and C-MRCI level, follow the same ascending order: 3Π0−, 3Π0+, 3Π1, 3Π2 and 3Δ1, 3Δ2, 3Δ3. According to C-MRCI, the 13Π0− is the first excited state followed by 13Π0+, which is ∼400 cm−1 above in energy. However, for MRCI, the 13Δ1, is the second excited followed by 13Π0+. The effect of the core orbitals is also felt on the bond lengths of 13Π0−, 13Π0+, and 13Δ1, which drop by ∼0.01 Å when using C-MRCI. For C-MRCI, the third excited state is 3Δ1, followed by 3Π1, 3Δ2, 3Π2, and 3Δ3. The Ω-states of 13Δ and 13Π span over a range of more than 3000 cm−1, which shows a large spin–orbit contribution and the importance of including inner core correlation.

When comparing this work with Assaf et al., their state ordering is different, and the 13Δ1 is their first excited state followed by 13Δ2 and then 13Π0−. These differences can be attributed to the use of a more state specific approach in the CASSCF and MRCI calculations, a higher level basis set in the present study. The inner orbitals of lutetium and fluorine were not considered in their calculations, but only the lutetium sub-valence 4f14 was included along with the 2p5 of fluorine. The use of inner core orbitals results in significant differences in bond lengths and spectroscopic constants.

In terms of composition (Table IV), Ω-states = 1, 2 for 13Δ and 13Π are heavily mixed, but 13Δ3 can only mix with 13Φ3. The next excited is 3Σ+, which splits into 3Σ+0+ and 3Σ+1. The bond length dropped ∼0.07 Å when using C-MRCI, and the Te is ∼500 cm−1 for both Ω states above MRCI. The next three states have minimal spin–orbit effects, but the inclusion of the core orbitals for C-MRCI changed their bond lengths by almost 0.1 Å, and the Te is ∼1000 cm−1 above MRCI. The last three states considered in Fig. 3 are 23Π, 21Π, and 13Φ. The 23Π follows the same ordering for its Ω states as the 13Π. When comparing MRCI and C-MRCI, the bond length for this state only varies 0.02 Å on average. C-MRCI still is ∼1000 cm−1 above MRCI. 21Π1 is in between the 13Φ Ω states. 13Φ2 is a heavily mixed state as reported in Table IV. 13Φ3 can only mix with 13Δ3, but 13Φ4 is a pure state. For the 13Φ splitting, C-MRCI also drops the bond length by almost ∼0.1 Å for the three Ω states. The Te for C-MRCI is also on average 1000 cm−1 above MRCI.

Dissociation energies calculated in this work as well as those reported previously from both theoretical and experimental studies are included in Table V.

TABLE V.

Dissociation energy of LuF in kcal mol−1 with different levels of theory and a range of basis sets.

Relativistic
MethodFrozen-coretreatmentD0 (dz)D0 (tz)D0 (qz)D0 CBS
CR-CCSD(T) FC-val DKH3 177.93 180.35 180.38 180.3 
CR-CCSD(T) FC-subval DKH3 163.01 167.20 167.66 167.9 
CCSD(T) FC-val DKH3 178.99 182.16 182.40 182.3 
CCSD(T) FC-subval DKH3 164.16 169.25 169.96 170.4 
CCSD(T) FC-val ECP28-Def2 ⋯ 172.35 169.47 167.9 
CCSD(T) FC-subval DC 158.33 165.63 ⋯  
MP2  DC 158.78 167.19 ⋯  
HF  DC 134.15 137.49 137.74 137.8 
CASSCF(8,15)  ECP28-ANO ⋯ 159.74 ⋯ ⋯ 
PBE  DKH3 174.40 172.29 176.04 ⋯ 
TPSS  DKH3 170.64 167.87 171.73 ⋯ 
M06-L  DKH3 169.02 169.77 171.93 ⋯ 
B3LYP  DKH3 167.72 166.09 169.97 ⋯ 
Other theoretical values       
Composite30       169.7 
Composite31       173.32 
PP-CCSD(T)24  Valence ECP60    173 
PP-MRACPF24  Valence ECP60    175 
DFT: SOAP25      96.6 ⋯ 
PBE26   ZORA    174 
DFT27,a  DKH3    195.3–161.6 
EOM-CR-CCSD(T)29  Valence DKH3    171.3 
EOM-CR-CCSD(T)29  Full DKH3    139.6 
Experimental value       
Mass spectroscopy7  136 ± 12 
Ligand field theory8  124 
Fitting PES14  79 
Fitting PES9  105 
Fitting PES13  96.0 ± 2.4 
Relativistic
MethodFrozen-coretreatmentD0 (dz)D0 (tz)D0 (qz)D0 CBS
CR-CCSD(T) FC-val DKH3 177.93 180.35 180.38 180.3 
CR-CCSD(T) FC-subval DKH3 163.01 167.20 167.66 167.9 
CCSD(T) FC-val DKH3 178.99 182.16 182.40 182.3 
CCSD(T) FC-subval DKH3 164.16 169.25 169.96 170.4 
CCSD(T) FC-val ECP28-Def2 ⋯ 172.35 169.47 167.9 
CCSD(T) FC-subval DC 158.33 165.63 ⋯  
MP2  DC 158.78 167.19 ⋯  
HF  DC 134.15 137.49 137.74 137.8 
CASSCF(8,15)  ECP28-ANO ⋯ 159.74 ⋯ ⋯ 
PBE  DKH3 174.40 172.29 176.04 ⋯ 
TPSS  DKH3 170.64 167.87 171.73 ⋯ 
M06-L  DKH3 169.02 169.77 171.93 ⋯ 
B3LYP  DKH3 167.72 166.09 169.97 ⋯ 
Other theoretical values       
Composite30       169.7 
Composite31       173.32 
PP-CCSD(T)24  Valence ECP60    173 
PP-MRACPF24  Valence ECP60    175 
DFT: SOAP25      96.6 ⋯ 
PBE26   ZORA    174 
DFT27,a  DKH3    195.3–161.6 
EOM-CR-CCSD(T)29  Valence DKH3    171.3 
EOM-CR-CCSD(T)29  Full DKH3    139.6 
Experimental value       
Mass spectroscopy7  136 ± 12 
Ligand field theory8  124 
Fitting PES14  79 
Fitting PES9  105 
Fitting PES13  96.0 ± 2.4 
a

DFT functionals used are SVWN, BP86, BLYP, PW91, PBE, B97-D, SSB-D, M06-L, TPSS, PBE0, B3LYP, BHLYP, B3P86, MPW1K, B97-1, X3LYP, M06, M06-2X, TPSSh, M11, CAM-B3LYP, and B2PLYP.

For the correlation, two approaches to the valence space were considered: FC-val, which includes only valence electrons (6s2, 5d1 of Lu and 2s2, 2p5 of F), and FC-subval, which includes sub-valence orbitals (5s2, 5p6 of Lu). In addition, the effects of using a full relativistic Hamiltonian and ECPs (28 electrons) were probed. For ab initio calculations, CCSD(T), CR-CCSD(T), and MP2 were utilized. For DFT, a variety of functionals were considered: PBE, TPSS, M06-L, and B3LYP.

The dissociation energy difference between the Sapporo-DZ and Sapporo-TZ for CR-CCSD(T) is 2 kcal mol−1 with FC-val, while between Sapporo-TZ and Sapporo-QZ basis sets, the energy difference drops to 0.03 kcal mol−1, which implies that the energy is almost converged at the triple-ζ level. The same trend is observed for CCSD(T), where the energy at the triple-ζ level is almost converged. When the sub-valence electrons from Lu are added (FC-subval results), the dissociation energy with CR-CCSD(T)/Sapporo-DZ dropped by 14 kcal mol−1 and by ∼13 kcal mol−1 at the CBS limit. At the CCSD(T) level of theory, the difference between FC-val and FC-subval dissociation energies is ∼14 and ∼12 kcal mol−1 with the Sapporo-DZ and at the CBS limit, respectively. Such a large difference arising from the choice of valence indicates that the electron correlation arising from the sub-valence electrons is important in the overall energy.

The basis set superposition error has been investigated by using the counterpoise method suggested by Boys and Bernardi for CCSD(T) and CR-CCSD(T) at the CBS limit for FC-val and FC-subval.57 For both FC-val calculations, considering CCSD(T) and CR-CCSD(T), the BSSE extrapolated to the CBS limit using a mixed exponential/Gaussian by Peterson is 0.87 kcal mol−1.53 For CCSD(T) and CR-CCSD(T) using sub-valence electrons, 0.81 and 0.59 kcal mol−1 were obtained, respectively, for BSSE corrections at CBS. As an example, for CCSD(T)/FC-subval at a double-, triple-, and quadruple-ζ basis set levels, the BSSE is 6.82, 3.52, and 1.21 kcal mol−1, respectively, which at CBS yields 0.81 kcal mol−1.

In addition, the dissociation energy of LuF was evaluated using the ECP28MWB pseudopotential and Def2-TZVPP, Def2-QZVPP (Lu) and aug-cc-pVTZ, aug-cc-PVQZ (F) basis sets. The value obtained at the quadruple-ζ level is very close to DKH3 predictions mentioned earlier, while the triple-ζ result is slightly higher than the DKH3 dissociation reported. The pseudopotential used for lutetium accounts for relativistic effects arising from the inner-core electrons. To evaluate the spin–orbit contribution to the ground state, the Dirac–Coulomb (DC) four component Hamiltonian was utilized. CCSD(T), MP2, and HF were probed for this step. The utility of the double- and triple-ζ CBS extrapolation by Martin54 has been considered for CCSD(T)/FC-subval/DKH3. This double-, triple-ζ CBS extrapolation scheme results in a dissociation energy of 171.1 kcal mol−1, while considering a two-point scheme extrapolation with triple-ζ and quadruple-ζ basis sets, 170.4 kcal mol−1 is obtained. Considering the unextrapolated triple-ζ basis set, the value obtained is 169.3 kcal mol−1. For CCSD(T)/FC-subval/DKH3, the double-, triple-ζ CBS extrapolated energy is closer to the triple-, quadruple-ζ extrapolated energy than the unextrapolated triple-ζ energy.

This shows that the spin–orbit contribution is small to the ground state, which is expected for a 1Σ+. In terms of calculations at the Hartree–Fock level, the necessary electron correlation is not present, so its dissociation energy prediction is very far from the best estimate. Finally, CASSCF was also used to calculate the dissociation energy by using the state-averaged wavefunction utilized to construct Fig. 1. The prediction is 159.74 kcal mol−1 at a triple-ζ level, which is ∼9 kcal mol−1 from the CCSD(T)/DKHH3/FC-subval dissociation energy.

CR-CCSD(T) and CCSD(T) results obtained in this study are in good agreement with other theoretical dissociation energies from the literature. When comparing the current results with Solomonik and Smirnov, a difference of 2 kcal mol−1 is obtained when using a sub-valence space correlation.30 Solomonik’s dissociation energy was obtained with a composite scheme based on CCSD(T)/CBS with core–valence correlation energy, spin–orbit, and scalar relativistic effects. The CCSD(T)/CBS results herein are in very good agreement with previous work from Lu.31 A composite scheme utilizing the Feller–Peterson–Dixon scheme renders a value of 173.32 kcal mol−1, which is only ∼3 kcal mol−1 from our best CCSD(T)/CBS results and 5 kcal mol−1 from CR-CCSD(T). Küchle et al.24 used the multireference averaged coupled-pair functional (MRACPF), and their dissociation energy is 4 and 7 kcal mol−1 higher than the results obtained in the CCSD(T)/CBS and CR-CCSD(T)/CBS predictions herein, respectively. However, both CCSD(T)/CBS and CR-CCSD(T)/CBS dissociation energies are quite distant from reported experimental values. In Table IV, the smallest difference in dissociation energy between experiment and our predictions was obtained by mass spectroscopy (Zmbov and Margrave,7 136 kcal mol−1). The other experimental values presented in Table IV have large energetic differences from our calculated values, with a maximum ΔE of ∼90 kcal mol−1. This shows the large discrepancy between experiment and theory.

Additionally, the potential utility of several DFT functionals in the determination of the dissociation energy of LuF has been considered. The PBE, TPSS, M06-L, and B3LYP functionals have been used, along with a DKH3 Hamiltonian and the Sap-nz basis set. The PBE dissociation energy obtained in this study is in agreement with the one predicted by Hong et al.26 using PBE and the ZORA Hamiltonian. The dissociation energies obtained with the functionals are in a range between 166 and 176 kcal mol−1. B3LYP at the triple-ζ level results in the lowest dissociation energy (166.09 kcal mol−1), while PBE with the quadruple-ζ basis set leads to the largest dissociation (176.04 kcal mol−1). These results largely compare with the DFT dissociation energies reported by Grimmel et al.27 However, in their study, a larger range of functionals were used, with SVWN leading to the largest dissociation energy at 195.3 kcal mol−1 and BHLYP resulting in the lowest energy at 161.6 kcal mol−1. Moreover, from the prior effort, B97-1 predicted a dissociation energy that is the closest to our CCSD(T)/CBS with DKH3/FC-subval dissociation energy. Finally, when comparing PBE, TPSS, M06-L, and B3LYP dissociation energies from our work and Grimmel et al., PBE has the largest dissociation energy among the four functionals and B3LYP the lowest. (To note, the differences between the Grimmel study and the present one are the use of a larger basis set (quadruple-ζ) in this study as well as a different type of basis set for the ligand.)

The bond lengths, spectroscopic constants, energetics, and potential energy curves are reported, which include four dissociation channels and detailed information concerning intersystem and avoided crossings. In addition, spin–orbit effects are calculated at a level of correlation that can aid experimentalists in further pursuits of the description of the ground and excited states and their spectroscopic data. The use of sub-valence orbitals at spin–orbit demonstrated that they are necessary to recover the necessary correlation to obtain results that are in agreement with experiment, especially for the low-lying excited states. The first excited state of LuF at spin–orbit C-MRCI is 13Π0−, followed by 13Π0+ and 13Δ1, which shows the importance of considering sub-valence and inner core orbitals to calculate spectroscopic constants and bond lengths.

In the second part of this work, the sub-valence orbitals are of paramount importance for predicting dissociation energies and can shift the dissociation energy by up to ∼13 kcal mol−1. CR-CCSD(T) and CCSD(T) at the CBS limit estimate the dissociation energy as 167.9 and 170.4 kcal mol−1, respectively. Utilizing a four-component Hamiltonian (Dirac–Coulomb) resulted in a dissociation energy ∼2 kcal mol−1 lower than the DKH3 calculations. The DFT calculations are overall in good agreement with our best estimate (from ∼1 to ∼6 kcal mol−1 to 170.40 kcal mol−1). Due to the large discrepancies between the results in this study as well as other theoretical data and the experiment, the experimental dissociation energy might need to be revisited. Finally, while in this case DFT gave similar dissociation than ab initio methods, a study of an open-shell molecule with a multi-reference character at the ground state might need more robust methods, such as ab initio methods.

Overall, lanthanide species are difficult to investigate from both theoretical and experimental perspectives. The high density of states, which can be very close in energy (herein, 132 states, most of which are bound and in a ∼55 000 cm−1 range, just below the dissociation energy), the effect of spin–orbit on the ground and excited states, as well as the influence of the sub-valence electrons are effects that should be included in a detailed analysis. Ab initio methods, as utilized herein, are vital to the description of the complex electronic manifold. Already for diatomics, such an analysis is significantly demanding and requires judicious selection of the active space, the electron correlation space, and the method.

MRCI spin–orbit potential energy curves along with the corresponding energies and distances are provided in the supplementary material. As well, an analysis of an error estimate in the CBS extrapolation scheme is provided, considering the 95% confidence limit.

This material is based upon work supported by the National Science Foundation under Grant No. CHE-1900086. This work utilized computational facilities at the Center for Advanced Scientific Computing and Modeling (CASCaM) at the University of North Texas, which, in part, were supported by the NSF (Grant No. CHE-1531468), as well as the Institute for Cyber-Enabled Research (ICER) at Michigan State University. We also gratefully acknowledge support from The Extreme Science and Engineering Discovery Environment (XSEDE) supercomputer, which is supported by the National Science Foundation (Grant No. ACI-1548562). The XSEDE staff request the acknowledgment of the following publication: DOI: https://doi.org/10.1109/MCSE.2014.80.

The authors declare no competing financial interests.

The data that support the findings of this study are available within the article and its supplementary material and from the corresponding author upon reasonable request.

1.
S.
Banerjee
,
M. R. A.
Pillai
, and
F. F.
(Russ) Knapp
,
Chem. Rev.
115
,
2934
(
2015
).
2.
S. G.
Hacker
,
Astrophys. J.
83
,
140
(
1936
).
3.
J. A.
Johnson
and
M.
Bolte
,
Astrophys. J.
605
,
462
(
2004
).
4.
C.
Sneden
,
J. J.
Cowan
,
J. E.
Lawler
,
I. I.
Ivans
,
S.
Burles
,
T. C.
Beers
,
F.
Primas
,
V.
Hill
,
J. W.
Truran
,
G. M.
Fuller
,
B.
Pfeiffer
, and
K. L.
Kratz
,
Astrophys. J.
591
,
936
(
2003
).
5.
I. U.
Roederer
,
C.
Sneden
,
J. E.
Lawler
, and
J. J.
Cowan
,
Astrophys. J. Lett.
714
,
L123
(
2010
).
6.
E. A.
Den Hartog
,
J. J.
Curry
,
M. E.
Wickliffe
, and
J. E.
Lawler
,
Sol. Phys.
178
,
239
(
1998
).
7.
K. F.
Zmbov
and
J. L.
Margrave
,
Mass Spectrometry in Inorganic Chemistry
(
ACS, Washington, DC
,
1968
), Vol. 72, p.
267
.
8.
L. A.
Kaledin
,
M. C.
Heaven
, and
R. W.
Field
,
J. Mol. Spectrosc.
193
,
285
(
1999
).
9.
J.
D’Incan
,
C.
Effantin
, and
R.
Bacis
,
J. Phys. B: At. Mol. Phys.
5
,
L189
(
1972
).
10.
C.
Effantin
,
G.
Wannous
, and
J.
D’Incan
,
Can. J. Phys.
55
,
64
(
1977
).
11.
K. P.
Huber
and
G.
Herzberg
,
Molecular Spectra and Molecular Structure
(
Springer, Boston, MA
,
1979
).
12.
C.
Effantin
,
G.
Wannous
,
J.
D’Incan
, and
C.
Athenour
,
Can. J. Phys.
54
,
279
(
1976
).
13.
N.
Rajamanickam
and
B.
Narasimhamurthy
,
Acta Phys. Hung.
56
,
67
(
1984
).
14.
R. R.
Reddy
,
A. R.
Reddy
, and
T. V. R.
Rao
,
Indian J. Pure Appl. Phys.
23
,
424
(
1985
).
15.
A.
Simon
,
2.3 Lanthanide Compounds with Low Valence
(
Walter de Gruyter GmbH & Co. KG
,
2020
).
16.
Y.
Hamade
and
A.
El Sobbahi
,
J. Mol. Model.
24
,
100
(
2018
).
17.
W.
Chmaisani
,
N.
El-Kork
,
S.
Elmoussaoui
, and
M.
Korek
,
ACS Omega
4
,
14987
(
2019
).
18.
W.
Chmaisani
and
M.
Korek
,
J. Quant. Spectrosc. Radiat. Transfer
217
,
63
(
2018
).
19.
S.
Yamamoto
and
H.
Tatewaki
,
J. Chem. Phys.
142
,
094312
(
2015
).
20.
M. P.
Plokker
and
E.
van der Kolk
,
J. Lumin.
216
,
116694
(
2019
).
21.
J.
Assaf
,
F.
Taher
, and
S.
Magnier
,
J. Quant. Spectrosc. Radiat. Transfer
189
,
421
(
2017
).
22.
C.
South
,
G.
Schoendorff
, and
A. K.
Wilson
,
Int. J. Quantum Chem.
116
,
791
(
2016
).
23.
S. G.
Wang
and
W. H. E.
Schwarz
,
J. Phys. Chem.
99
,
11687
(
1995
).
24.
W.
Küchle
,
M.
Dolg
, and
H.
Stoll
,
J. Phys. Chem. A
101
,
7128
(
1997
).
25.
S. A.
Cooke
,
C.
Krumrey
, and
M. C. L.
Gerry
,
Phys. Chem. Chem. Phys.
7
,
2570
(
2005
).
26.
G.
Hong
,
M.
Dolg
, and
L.
Li
,
Chem. Phys. Lett.
334
,
396
(
2001
).
27.
S.
Grimmel
,
G.
Schoendorff
, and
A. K.
Wilson
,
J. Chem. Theory Comput.
12
,
1259
(
2016
).
28.
L. E.
Aebersold
,
S. H.
Yuwono
,
G.
Schoendorff
, and
A. K.
Wilson
,
J. Chem. Theory Comput.
13
,
2831
(
2017
).
29.
G.
Schoendorff
and
A. K.
Wilson
,
J. Chem. Phys.
140
,
224314
(
2014
).
30.
V. G.
Solomonik
and
A. N.
Smirnov
,
J. Chem. Theory Comput.
13
,
5240
(
2017
).
31.
Q.
Lu
, “
Development and applications of relativistic correlation consistent basis sets for lanthanide elements and accurate ab initio thermochemistry
,”
Ph.D. thesis
(
Washington State University
,
2017
).
32.
B. K.
Welch
and
A. K.
Wilson
, “
Extending ccCA to the lanthanides: f-ccCA
,” (unpublished).
33.
Y.
Hamade
,
F.
Taher
,
M.
Choueib
, and
Y.
Monteil
,
Can. J. Phys.
87
,
1163
(
2009
).
34.
J.
Assaf
,
S.
Zeitoun
,
A.
Safa
, and
E. C. M.
Nascimento
,
J. Mol. Struct.
1178
,
458
(
2019
).
35.
H.-J.
Werner
,
P. J.
Knowles
,
F. R.
Manby
,
J. A.
Black
,
K.
Doll
,
A.
Heßelmann
,
D.
Kats
,
A.
Köhn
,
T.
Korona
,
D. A.
Kreplin
,
Q.
Ma
,
T. F.
Miller
,
A.
Mitrushchenkov
,
K. A.
Peterson
,
I.
Polyak
,
G.
Rauhut
, and
M.
Sibaev
,
J. Chem. Phys.
152
,
144107
(
2020
).
36.
P. J.
Knowles
and
H.-J.
Werner
,
Chem. Phys. Lett.
145
,
514
(
1988
).
37.
H. J.
Werner
and
P. J.
Knowles
,
J. Chem. Phys.
89
,
5803
(
1988
).
38.
H. J.
Werner
and
P. J.
Knowles
,
J. Chem. Phys.
82
,
5053
(
1985
).
39.
H.-J.
Werner
,
Adv. Chem. Phys.
69
,
1
(
1987
).
40.
J. L.
Dunham
,
Phys. Rev.
41
,
721
(
1932
).
41.
X.
Cao
and
M.
Dolg
,
J. Mol. Struct.: THEOCHEM
581
,
139
(
2002
).
42.
X.
Cao
and
M.
Dolg
,
J. Chem. Phys.
115
,
7348
(
2001
).
43.
R. A.
Kendall
,
T. H.
Dunning
, and
R. J.
Harrison
,
J. Chem. Phys.
96
,
6796
(
1992
).
44.
R.
Gulde
,
P.
Pollak
, and
F.
Weigend
,
J. Chem. Theory Comput.
8
,
4062
(
2012
).
45.
F.
Weigend
,
Phys. Chem. Chem. Phys.
8
,
1057
(
2006
).
46.
M.
Sekiya
,
T.
Noro
,
T.
Koga
, and
T.
Shimazaki
,
Theor. Chem. Acc.
131
,
1247
(
2012
).
47.
J. P.
Perdew
,
K.
Burke
, and
M.
Ernzerhof
,
Phys. Rev. Lett.
77
,
3865
(
1996
).
48.
J. P.
Perdew
,
Phys. Rev. B
33
,
8822
(
1986
).
49.
A. D.
Becke
,
J. Chem. Phys.
98
,
1372
(
1993
).
50.
Y.
Zhao
and
D. G.
Truhlar
,
J. Chem. Phys.
125
,
194101
(
2006
).
51.
J.
Tao
,
J. P.
Perdew
,
V. N.
Staroverov
, and
G. E.
Scuseria
,
Phys. Rev. Lett.
91
,
146401
(
2003
).
52.
A. S. P.
Gomes
,
K. G.
Dyall
, and
L.
Visscher
,
Theor. Chem. Acc.
127
,
369
(
2010
).
53.
K. A.
Peterson
,
D. E.
Woon
, and
T. H.
Dunning
,
J. Chem. Phys.
100
,
7410
(
1994
).
54.
J. M. L.
Martin
,
Chem. Phys. Lett.
259
,
669
(
1996
).
55.
D. H.
Bross
and
K. A.
Peterson
,
J. Chem. Phys.
141
,
244308
(
2014
).
56.
C.
Peterson
,
D. A.
Penchoff
, and
A. K.
Wilson
,
Annu. Rep. Comput. Chem.
12
,
3
(
2016
).
57.
S. F.
Boys
and
F.
Bernardi
,
Mol. Phys.
19
,
553
(
1970
).
58.
M.
Valiev
,
E. J.
Bylaska
,
N.
Govind
,
K.
Kowalski
,
T. P.
Straatsma
,
H. J. J.
Van Dam
,
D.
Wang
,
J.
Nieplocha
,
E.
Apra
,
T. L.
Windus
, and
W. A.
de Jong
,
Comput. Phys. Commun.
181
,
1477
(
2010
).
59.
DIRAC18, a relativistic ab initio electronic structure program,
T.
Saue
,
L.
Visscher
,
H. J. A.
Jensen
,
R.
Bast
,
V.
Bakken
,
K. G.
Dyall
,
S.
Dubillard
,
U.
Ekström
,
E.
Eliav
,
T.
Enevoldsen
,
E.
Faßhauer
,
T.
Fleig
,
O.
Fossgaard
,
A. S. P.
Gomes
,
E. D.
Hedegård
,
T.
Helgaker
,
J.
Henriksson
,
M.
Iliaš
,
C. R.
Jacob
,
S.
Knecht
,
S.
Komorovský
,
O.
Kullie
,
J. K.
Lærdahl
,
C. V.
Larsen
,
Y. S.
Lee
,
H. S.
Nataraj
,
M. K.
Nayak
,
P.
Norman
,
G.
Olejniczak
,
J.
Olsen
,
J. M. H.
Olsen
,
Y. C.
Park
,
J. K.
Pedersen
,
M.
Pernpointner
,
R.
di Remigio
,
K.
Ruud
,
P.
Sałek
,
B.
Schimmelpfennig
,
A.
Shee
,
J.
Sikkema
,
A. J.
Thorvaldsen
,
J.
Thyssen
,
J.
vanStralen
,
S.
Villaume
,
O.
Visser
,
T.
Winther
, and
S.
Yamamoto
, available at , see also http://www.diracprogram.org,
2018
.

Supplementary Material