Toward Chemical Accuracy in ab Initio Thermochemistry and

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Toward Chemical Accuracy in ab Initio Thermochemistry and Spectroscopy of Lanthanide Compounds: Assessing Core-Valence Correlation, SecondOrder Spin-Orbit Coupling, and Higher Order Effects in Lanthanide Diatomics Victor G. Solomonik, and Alexander N Smirnov J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.7b00408 • Publication Date (Web): 19 Sep 2017 Downloaded from http://pubs.acs.org on September 19, 2017

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Journal of Chemical Theory and Computation

Toward Chemical Accuracy in ab Initio Thermochemistry and Spectroscopy of Lanthanide Compounds: Assessing Core-Valence Correlation, Second-Order Spin–Orbit Coupling, and Higher Order Effects in Lanthanide Diatomics Victor G. Solomonik∗ and Alexander N. Smirnov Ivanovo State University of Chemistry and Technology, Ivanovo 153000, Russia E-mail: [email protected] Abstract The higher order (HO) correlation beyond the coupled-cluster single double (triple) CCSD(T) level of theory, second-order spin–orbit coupling (SOC), and core-valence (CV) correlation effects on bond length, re , vibrational frequency, ωe , and dissociation energy, De , are studied for a set of 17 lanthanide containing diatomics, including lanthanum, europium, ytterbium and lutetium oxides and halides. Convergence in the magnitudes of the SOC, CV, and HO corrections with respect to basis set size is examined using a sequence of double, triple, and quadruple-zeta basis sets, with the complete basis set (CBS) limit estimates provided in most cases. The CV effects on De , re , and ωe are calculated to amount up to 1.3 kcal·mol−1 , 0.008 Å, and 5 cm−1 , respectively. A detailed analysis of the origin of the CV effect with a particular accounting for

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various sub-valence shells reveals that, generally, the Ln 4d correlation makes a major contribution, although in some instances the lower-lying 4sp shells contribute largely and even more substantially than the 4d. The second-order SOC effect evaluated via two-component and four-component relativistic techniques proves to be non-negligible, especially for heavier species, e.g., LaI, in which it is as large as 0.8 kcal·mol−1 in De , 0.002 Å in re , and 1.3 cm−1 in ωe . Higher order correlation effects assessed through the CCSDT(Q) level are mostly less than 0.8 kcal·mol−1 , 0.004 Å, and 5 cm−1 , however, in species with prominence of nondynamical correlation, e.g., EuO, YbF, and LuO, the HO correction can amount to 1.2 – 1.6 kcal·mol−1 , 0.005 – 0.008 Å, and 8 – 30 cm−1 in De , re , and ωe , respectively. In general, the [CCSD(T)+CV]/CBS + SOC + HO composite results are in good agreement with the available experimental data, exhibiting a mean absolute deviation of 1.8 kcal·mol−1 in De , 0.0023 Å in re , and 3.5 cm−1 in ωe . A significant experimental outlier, the bond length in YbI, is revealed implying the need for re-examination of the experimental data.

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INTRODUCTION

The importance of electronic structure investigations for the study of lanthanide (Ln) containing species is well known. 1–3 However, quantitatively accurate quantum chemical calculations are still a difficult task for these systems due to the large number of electrons, strong relativistic effects, and often multireference character owing to small energy gap of the 4f , 5d, and 6s atomic orbitals in Ln. Applying highly sophisticated methods capable of incorporating all important contributions to the energy in a single calculation would require too high computational costs and might be unaffordable for the Ln containing polyatomic species. Instead, one might rely upon a composite approach, which is based on additivity arguments, exploits widely varying rates of convergence in the one-particle and n-particle expansions for components of different nature, and strives for obtaining a sufficiently accurate estimate for each component via replacing the single large calculation with a series of much less compu-

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tationally expensive smaller calculations using methods of various sophistication with basis sets of various size. Numerous composite computational chemistry schemes aimed at achieving desirable ("chemical") accuracy in the molecular properties, primarily thermochemical ones, have been developed in recent decades (for a review, see Ref. 4). The term "chemical accuracy" commonly means the capability of a certain computational approach to predict the atomization energy with a deviation from a precise experimental or theoretical benchmark of about 1 kcal·mol−1 . Most of current composite methodologies are able to achieve the prescribed accuracy almost routinely for molecules consisting of relatively light main group elements from the first few rows of the periodic table. 4,5 For transition metal-containing species, however, a less tight threshold of ±3 kcal·mol−1 , termed "transition metal chemical accuracy", has been established. 6 Spectroscopic properties, such as equilibrium structures and vibrational frequencies, were also studied in a composite fashion for both main group and transition metal species, though less commonly. The term "chemical accuracy" was applied to the results of these studies as well, with a threshold chosen (though rather arbitrarily) to be ±0.005 Å and ±15 cm−1 for the deviations in bond lengths and frequencies. 4 Noteworthy, recent computational studies employing the high-level Feller-Peterson-Dixon (FPD) composite approach 4,7,8 have demonstrated the potential for achieving an accuracy in the spectroscopic properties of transition metal-containing molecules well below this threshold. 9,10 For lanthanide compounds, the accuracy that can be achieved by a composite method is still not established well enough. A recent review on the correlation consistent composite approach strategies and applications 5 states that the combination of calculations used in composite approaches allows for energetic predictions to be within 5 kcal·mol−1 for lanthanides. In the FPD composite calculations of the total atomization energies for cerium dihalides and tetrahalides (F, Cl), 11 the error bars on the calculations were estimated to be ±2 kcal·mol−1 , whereas in a subsequent FPD study on a few gadolinium-containing species 12 the error estimate amounted to 3 kcal·mol−1 . This latest estimate was motivated by the ion-

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ization potential results obtained for the Ln atoms, as well as previous CCSD(T) calculations on 3d transition metal molecules 13 and those involving Th and U. 11,14,15 It should be noted, however, that no attempts have yet been made to directly assess the accuracy of the composite procedure for lanthanides by statistically calibrating the composite results against experiment. The present work aims to contribute toward filling this gap. The Ln-containing species are known to exhibit great diversity in bonding pattern and electronic structure. Therefore, while dealing with a variety of these species, one can encounter great difference in the magnitudes and convergence rates of the components entering into a composite scheme. A composite protocol capable of ensuring high accuracy in the molecular properties of Ln compounds should hence allow for flexibility in choosing the level of theory used to describe each component. The flexible, high-level FPD method mentioned above provides a good starting point for an in-depth assessment of the issue. The major components of the FPD procedure are the coupled cluster single, double and perturbative triple CCSD(T) theory or multireference configuration interaction for situations demanding strongly multiconfiguration wave functions, corrections for outer-core valence (CV) and higher order (HO) correlation and spin–orbit coupling (SOC) effects. A role each one of these effects can play in the accurate prediction of spectroscopic and thermochemical properties of lanthanide molecules is not yet well understood. The HO effects in these systems have not been studied so far, as well as basis set convergence in magnitudes of the SOC effects, whereas significance of the Ln core correlation is still subject to controversy. Gomes et al. 16 have noted that correlation of the 4d electrons could be very important due to the compactness of the 4f shell. The results of the recent computational studies on lanthanides mostly indicate significant but not too large sub-valence correlation contributions to the atomization energy: about 0.2 kcal·mol−1 for CeF2 and CeCl2 , 11 0.4 to 0.6 kcal·mol−1 for YbF, 17 Gd2 , and GdF3 , 12 0.9 kcal·mol−1 for GdF, 12 1.3 kcal·mol−1 for CeF4 and CeCl4 . 11 On the other hand, much greater influence of the Ln core correlation on the LnF bond dissociation energy (Ln = Nd, Lu) was reported in Ref. 18: up to 3 kcal·mol−1 on the NdF bond energy and

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even 30 kcal·mol−1 on the LuF bond energy. In this study, we explore the relative importance and basis set convergence rate of various contributions entering into a coupled cluster-based composite scheme for calculating thermochemical and spectroscopic properties of Ln-containing species, taking the example of seventeen diatomics, LnX, in their ground states, namely 1 Σ+ LaX, 2 Σ+ YbX, 1 Σ+ LuX (X = F, Cl, Br, I), 2 Σ+ LaO and LuO, 8 Σ− EuO, 9 Σ− EuF and EuCl. The LnX bond dissociation energies, De , equilibrium bond lengths, re , and harmonic vibrational frequencies, ωe , are studied systematically, with an emphasis on basis set, electron correlation, and relativistic effects. Hopefully, the results will aid in establishing a threshold of chemical accuracy for this kind of molecular systems, and elucidating the routes to control and systematically eliminate errors in theoretical predictions to below this threshold. The paper is arranged as follows. First, starting with a high-level correlation treatment, i.e., employing the CCSD(T) method, the outer-core–valence electron correlation effects are studied in detail. Then, higher-order correlation contributions are evaluated through the CCSDT(Q) level of theory. The convergence in the magnitudes of the CV and HO effects with respect to basis set size is examined and the CBS limit estimates are provided in most cases. Furthermore, relativistic effects are investigated with an emphasis on second-order spin–orbit coupling. The computed molecular properties are compared with the relevant experimental data, thus assessing the accuracy in the theoretical predictions. Finally, summary and conclusions are presented.

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COMPUTATIONAL DETAILS

Core-valence correlation effects were investigated using the MOLPRO program suite 19 at the CCSD(T) level of theory, 20,21 employing the restricted open-shell Hartree–Fock (ROHF) orbitals for the open-shell systems, with a partial treatment of spin adaptation in the CCSD wave function, i.e., R/RCCSD(T). 22,23 Scalar relativistic effects were treated with the third-

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order Douglas–Kroll–Hess 24–27 (DKH or DK) Hamiltonian. The sequences of basis sets, aug-cc-pwCVnZ-DK, of double-, triple-, and quadruple-zeta quality (n = D, T, Q) for the atoms O, F, Cl, 28–32 Br, 33,34 and I, 35 coupled with the respective Sapporo group basis sets, Sapporo-DK3-nZP-2012, for the Ln atoms, 17 were utilized in these calculations. Hereafter, these basis sets are denoted as nZ. The higher order correlation contributions were assessed through coupled-cluster doubles, triples, and quasiperturbative quadruples, CCSDT(Q), 36 with all core AOs kept frozen, using the aug-cc-pVnZ-DK (n = D, T) basis sets for O, F, Cl, Br, 28–31,33 and the Sapporo group sets 37–39 for Ln and I, with the iodine atom basis set fully augmented with diffuse functions. The CCSDT(Q) calculations were performed using the MRCC program of Kállay et al. 40 interfaced to MOLPRO. The open-shell systems were described with unrestricted Hartree–Fock wavefunctions in these calculations, and the HO corrections were defined with respect to the CCSD(T) results also obtained via MRCC. Spectroscopic constants were obtained from a conventional Dunham analysis 41 using fifth- or sixth-order polynomial fits of the total energies calculated for a grid of r(Ln–X) bond lengths at seven equidistant points with a grid step of 0.02 or 0.03 Å. A complete basis set (CBS) limit has been estimated in each case by extrapolation of the total energy via the two-point (ℓ + 1/2)−4 formula, 42

En = ECBS +

A , (n + 1/2)4

(1)

using TZ and QZ energies in the CV calculations, or DZ and TZ energies in the HO calculations. The CBS estimates of the CV effects were also obtained using DZ, TZ, and QZ energies via the three-parameter, mixed Gaussian/exponential expression 43

En = ECBS + Aexp(−(n − 1)) + Bexp(−(n − 1)2 ).

(2)

In the calculations of dissociation energies, the atomic first-order spin–orbit corrections were obtained from the experimental term energies 44 (all in kcal·mol−1 ): -0.22 (O), -0.39 (F), 6


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-0.84 (Cl), -3.51 (Br), -7.25 (I), -1.81 (La), -3.42 (Lu). The second-order SO corrections to the numerical values of molecular properties were obtained using the DIRAC program 45 in two-component (2c) calculations with relativistic effective core potentials (RECP), as well as in four-component (4c) all-electron calculations with a Gaussian nuclear model. An accurate approximation to the full Dirac–Coulomb (DC) Hamiltonian was employed in 4c calculations, with the (SS|SS) integrals neglected and replaced by an interatomic SS correction calculated as a classical repulsion term of tabulated small component atomic charges. 46 The SO correction was defined as the difference between the numerical value of a molecular property obtained from the 4c calculation and its counterpart calculated with the spin-free Hamiltonian of Dyall 47 via the default restricted kinetic balance scheme. The relativistic DZ, TZ, and QZ-quality basis sets of Dyall 16,45,48 were used for the Ln, Br, and I atoms, with diffuse functions included in the basis sets for Br and I. The aug-cc-pVnZ basis sets 28–30 (n = D, T, Q) were used for the O, F, and Cl atoms. Since our preliminary relativistic studies on the LaX diatomics have revealed a deficiency in the description of polarization functions in Dyall’s DZ (24s19p13d2f ) and TZ (30s24p18d5f 2g) basis sets for La, we have modified these basis sets by replacing the original f and higher angular momentum primitives by the sets of primitives 8f 2g and 11f 3g2h taken from the corresponding DZ and TZ basis sets for the Ce atom. 45 In the RECP calculations, the two-component Hartree–Fock orbitals were utilized as the one-electron basis in which spin–orbit coupling is included. 49 The SO correction was defined as the difference between the numerical values of a molecular property calculated with including and omitting the SO parameters of RECPs. For the Ln atoms, the smallcore (28-electron) two-component shape-consistent GRECPs of Mosyagin et al. 50,51 were employed together with the basis sets of Cao and Dolg 52 (14s13p10d8f 6g), augmented with a single h-type function, z = 0.904 (La), 1.757 (Eu), 3.993 (Yb), 4.420 (Lu). For the Br and I atoms, the small-core RECPs 53,54 were used in conjunction with the aug-cc-pVQZ-PP basis sets. 53,54 For the O, F, and Cl atoms, the RECPs of Christiansen, Ross, and Ermler 55

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were utilized, accompanied with the aug-cc-pVQZ basis sets 28–30 with nine s and four p tight functions from Cl, and five s tight functions from O and F removed. In both 4c and 2c calculations, all basis sets were kept uncontracted. The atomic second-order spin–orbit correction to the dissociation energy was defined as the difference between the total spin–orbit correction obtained from a particular relativistic calculation and the spin–orbit correction due to zero-field splitting (ZFS) of the atoms determined at the same level of theory. The correction due to ZFS was calculated by taking the difference between the calculated j-averaged levels and the 3 P2 level for oxygen, the 2 P3/2 levels for the halogens, and the 2 D3/2 levels for La and Lu. In the relativistic calculations, electron correlation was taken into account via CCSD, CCSD(T), and second order Møller–Plesset perturbation theory (MP2) within the frozencore (FC) approximation. For the LaO, LuO, and ytterbium halide molecules, YbHal (Hal = F, Cl, Br, I), the Fock-space CCSD (FSCCSD) 56,57 method was employed as well. The (0,0) sectors required by the FSCCSD method were defined by the closed-shell LnO+ and YbHal+ molecular ions with an active space comprised of two ns spinors in LnO and YbHal, obtained by addition of 1 electron to LnO+ and YbHal+ . To compute the LnX bond dissociation energy at the FSCCSD level, the ground state FSCCSD energies of the Ln, O, and Hal atoms were obtained with the (0,0) sectors defined by the closed-shell La+ , Lu+ (6s2 5d0 ), Yb2+ (4f 14 6s0 ), Hal− and O2− (ns2 np6 ) ions with an active space comprising all ten 5d spinors in La and Lu, two 6s spinors in Yb, or six np spinors in Hal and O, obtained by adding 1 electron to La+ and Lu+ , 2 electrons to Yb2+ , or removing 1 and 2 electrons from Hal− and O2− , respectively. The virtual orbital spaces used in the post-SCF calculations were truncated by deleting all virtual spinors with orbital energies larger than 10 a.u. while using RECPs, 9 a.u. in 4c calculations for closed-shell species, 20 a.u in 4c/DZ calculations for open-shell ones, and 35 a.u. in 4c/TZ for YbF. Convergence tests in the numerical values of the ∆so corrections were performed with all virtuals included in 2c-MP2, 4c-MP2, and 4cCCSD(T)/DZ calculations for closed-shell diatomics, and the virtuals below 50 a.u. included

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in 4c-FSCCSD/DZ for YbF.

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RESULTS AND DISCUSSIONS

3.1

Outer-core–valence correlation effects

In this section, the following issues are addressed: (i) the significance of the CV effects, ∆cv , on De , re , and ωe ; (ii) the CV effect convergence rate with respect to basis set size; (iii) the relative importance of the contributions to ∆cv from correlating various types of atomic outer-core electrons, particularly those of the lanthanide atom. The molecular parameters evaluated at the CCSD(T) level including correlation of the 4s, 4p, 4d (Ln), (n–1)s, (n–1)p, (n–1)d (X) outer-core electrons in addition to the 5s, 5p, 5d, 4f , 6s (Ln), ns, np (X) valence electrons are provided in Table 1. Generally, the resulting values exhibit regular convergence to the CBS limit, mostly with an increase in De and ωe associated with a decrease in re as the basis set cardinal number n increases. The threepoint (DTQ) and two-point (TQ) extrapolation procedures yield very similar CBS limit estimates, with mean absolute deviations in CBS(DTQ) with respect to CBS(TQ) of just 0.1 kcal·mol−1 , 0.0004 Å, and 0.5 cm−1 . The QZ basis set incompleteness error (BSIE) in the LnX bond dissociation energy is estimated to be 1.1 kcal·mol−1 on average, with the largest error (3.0 kcal·mol−1 ) obtained for EuO. The QZ BSIE in ωe is less than 2 cm−1 for all of the LnX diatomics except EuO and LuO, for which it amounts to 5 – 6 cm−1 . The numerical values of the CV effects are listed in Table 2 (the TZ, QZ and CBS(TQ) results) and also, in more detail, in the Supporting Information (Table S1, which includes the DZ and CBS(DTQ) results as well). Although the ∆cv values obtained using DZ basis sets proved inaccurate for some molecules, e.g., YbF, the DTQ and TQ extrapolation procedures yielded virtually identical CBS estimates of ∆cv for all molecules. In general, the ∆cv values converge rapidly with respect to basis set size: even the TZ BSIE appears to be quite small (less than 0.2 kcal·mol−1 , 0.002 Å, and 0.5 cm−1 in ∆cv De , 9



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∆cv re , and ∆cv ωe , respectively), whereas the QZ-quality basis set yields the CV effects closely matching the respective CBS limit estimates. However, a few molecules, namely LaO, EuO, and YbF, exhibit less rapid basis set convergence in ∆cv , hence larger QZ – CBS differences: about 0.1 kcal·mol−1 in ∆cv De (for LaO and EuO), 0.001 Å in ∆cv re (for YbF and LaO), 2.6 cm−1 in ∆cv ωe (for YbF). Correlation contributions from various sub-valence shells, namely 4sp Ln, 4d Ln, (n–1)sp X, and (n–1)d X, have been evaluated at the CCSD(T)/QZ level in a series of particular calculations with the sub-valence shells included in a frozen core sequentially upward, in accord with the orbital energy ordering (see an orbital energy diagram in Figure 1), in few instances, however, with resorting to an orbital rotation foregoing the correlation treatment. The results are detailed in Table 2. One can see substantial contributions to De , re , and ωe from correlating the 3d (Br) and 4d (I) electrons in lanthanide bromides (about 0.7 – 0.9 kcal·mol−1 , –0.008 – –0.009 Å, and 1.0 – 1.5 cm−1 ) and iodides (1.0 – 1.3 kcal·mol−1 , –0.014 – –0.015 Å, and 1.2 – 1.8 cm−1 , respectively). The rationale for this is the fact that the LnBr and LnI molecular orbitals (MO) with a dominant contribution from the (n–1)d orbitals of Br and I are very close in energy to the lowest-lying valence MO arising from the 5s lanthanide atomic orbital, with the 4d iodine orbital in YbI and LuI being higher in energy than the 5s Ln (see Figure 1). Obviously, an inclusion of the (n–1)d electrons of Br and I in a correlation treatment is a prerequisite for accurately predicting the molecular properties of lanthanide bromides and iodides. Considering the CV effects other than those from the (n–1)d (Br, I), it can be seen that the 4d (Ln) correlation makes a major contribution to ∆cv , amounting up to 1 kcal·mol−1 in De , 0.004 Å in re , and 3 cm−1 in ωe . Generally, the lower-lying 4s and 4p Ln shells contribute much less than the 4d and may be excluded from a correlation treatment. Nonetheless, a few exceptions to this rule still exist, e.g., a non-negligible correlation contribution from the 4sp Lu sub-valence shells to the LuHal bond dissociation energy (about –0.2 kcal·mol−1 ), which cancels in part the contribution from the 4d Lu shell. The LaO molecule presents another

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Figure 1: The orbital energies of LaX, LuX, and YbX (X = F, Cl, Br, I).

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interesting exception: it is just the 4sp rather than the 4d La correlation that dominates the CV effect on the LaO bond dissociation energy. Comparing the CV effects in different diatomics under study (Table 2), we can note a decrease in the magnitude of the ∆cv correction on going from lighter (LaX) to heavier (LuX) members of the lanthanide series. While the CV effect on bond length amounts to 0.005 – 0.008 Å in LaX, it is just about 0.002 Å in YbX and LuX. The absolute values of the CV corrections to vibrational frequencies of the EuX, YbX, and LuX molecules are less than 1 cm−1 (except for YbF, exhibiting ∆cv ωe = –1.4 cm−1 ), however, in LaX these can amount to 5 cm−1 . The CV effect on the LaX and EuX bond dissociation energies can be as large as 1.4 kcal·mol−1 , whereas for late lanthanide diatomics, YbX and LuX, the CV correction to De constitutes less than 0.6 kcal·mol−1 . Overall, the CV effect tends to increase the dissociation energy of LaX and LuX, and to decrease that of EuX and YbX. Recently, Lu and Peterson 12 (LP) have developed the correlation-consistent basis set family for the lanthanide atoms, cc-pwCVnZ-DK3 (n = D, T, Q). These new basis sets are apparently more flexible than the Sapporo sets 17 utilized in this study. It is interesting to compare the convergence behavior of the LP and Sapporo basis sets in order to assess the accuracy of the CCSD(T)/CBS results of this work. We have carried out such a comparison for the LaO, LaF, LaCl, EuF, YbF, YbI, and LuF molecules. The results are given in Tables S2 – S5 of the Supporting Information. Although in few instances the convergence of the LP basis sets to the CBS limit is slightly more rapid, the two basis set sequences display very similar convergence behavior, thus providing consistent sets of the CBS limit estimates: the discrepancies between the De , re , and ωe CBS(TQ) values obtained using Sapporo and LP basis sets are 0.4 kcal·mol−1 , 0.0014 Å, and 0.8 cm−1 on average, with the largest deviations obtained for De (LuF) (0.9 kcal·mol−1 ), re (YbI) (0.0032 Å), and ωe (YbF) (1.7 cm−1 ). These numbers can serve as a crude estimate of the errors in the CCSD(T)/CBS values of this study caused by imperfections in the basis sets utilized. The errors associated with ambiguity in choosing the CBS extrapolation formula are expected to be much smaller. For instance, the

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separate two-point (TQ) extrapolations of the HF and correlation energy components using Eq. (1) for the latter and the Karton and Martin formula 58 for the former:

En = ECBS + A(n + 1)exp(−6.57n1/2 ),

(3)

yield the CBS limit estimates for De , re , and ωe which differ from the respective CBS estimates obtained using the total energy extrapolations via Eq.1 by less than 0.2 kcal·mol−1 , 0.0006 Å, and 0.8 cm−1 , respectively. The numerical values of these estimates for selected molecules are provided in Tables S2 – S5 of the Supporting Information. It is advisable to examine if the inclusion of the counterpoise (CP) correction 59 designed to account for the basis set superposition error can improve accuracy in the computed data for the LnX molecules. This issue is of interest particularly because CP corrections have been extensively used in previous computational studies on lanthanide diatomics. 52,60–64 In Figure 2 we show the situation we have found to prevail in the present study, taking the example of the LaF molecule. It can be seen that the CP corrected CBS limit estimates are very close in magnitude to their uncorrected counterparts. Hence, we see no reasons for using the CP corrected energies in basis set extrapolation procedures for the LnX molecules. Then, from Figure 2 it becomes evident that the counterpoise corrected numerical value of a molecular property calculated with a particular basis set deviates more from the respective CBS limit estimate than the uncorrected one, i.e., the inclusion of CP corrections worsens rather than improves accuracy in the computed data. Similar conclusions were drawn in a number of theoretical studies on some other molecules, see Refs. 65–68 and references therein. To conclude this section, we should note that the CV corrections to the LnF bond dissociation energies obtained in this work are close in magnitude to those calculated previously for YbF 17 and GdF, 12 but deviate appreciably from that for the LuF molecule 18 (0.5 kcal·mol−1 in this work compared to 30 kcal·mol−1 reported in Ref. 18). In order to examine whether the Lu inner core electrons, which were included in a correlation treatment 18 together with 13



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Figure 2: Behavior of the CP corrected and uncorrected values of the LaF bond dissociation energy (a), bond length (b), and vibrational frequency (c) as a function of basis set size. The values were obtained at the CCSD(T) level of theory using the aug(F)-cc-pwCVnZ basis sets (n = D, T, Q). The respective CBS(TQ) estimates are shown as well. 14


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the outer-core and valence electrons, could be the reason for the discrepancy noted above, we have estimated the inner core correlation effect on De (LuF) at the CCSD(T) level using the same basis sets as those utilized in Ref. 18: the Sapporo TZ on Lu and cc-pwCVTZ on F. The inner core correlation contribution to De was calculated to be only 0.14 kcal·mol−1 . Replacing the Sapporo set on Lu with cc-pwCVTZ-DK3 also results in a very small inner core effect (0.25 kcal·mol−1 ). These numbers can still be inaccurate due to lack of the inner core correlating functions in both the Sapporo and LP basis sets on Lu. To get rid of this error, we have calculated the inner core correlation correction to De (LuF) using uncontracted cc-pwCVTZ basis sets on Lu and F, with a few tight functions on Lu omitted to avoid linear dependency issues. The resulting correction appeared to be essentially zero.

15


LaO LaF LaCl LaBr LaI 1.9257 1.9006 1.8920 2.1037 2.0891 2.0801 2.5944 2.5839 2.5715

re ————————————————— DZ TZ QZ DTQ TQ 1.8612 1.8372 1.8298 1.8254 1.8254 2.0584 2.0297 2.0245 2.0220 2.0215 2.5336 2.5083 2.5007 2.4966 2.4964 2.6799 2.6630 2.6535 2.6478 2.6481 2.8909 2.8778 2.8704

2.0203 2.4935 2.6493

699.1 498.8 289.5

698.3 497.9 289.3

ωe ——————————————— DZ TZ QZ DTQ TQ 774.7 814.3 817.3 818.0 819.1 565.5 576.7 578.4 579.2 579.5 337.7 341.3 344.4 346.3 346.1 234.7 236.0 238.9 240.9 240.7 184.4 187.9 190.0 665.3 679.7 691.8 494.0 487.3 494.0 280.0 284.0 287.3

847.2 609.2 340.9 228.1 172.7

852.6 613.3 342.0 229.6

852.2 612.6 342.2 229.6 174.5

502.8 504.6 506.4 504.8 285.4 289.0 291.1 290.9 192.2 194.7 196.2 196.1 147.0 149.3 150.5 826.5 838.4 605.0 603.4 328.2 338.6 215.7 225.4 169.7

2.0204 526.5 2.4942 280.5 2.6501 186.3 2.8840 1.7895 1.7895 1.9168 1.9170 2.3767 2.3772 2.5321 2.5326 2.7698

112.2 82.5 68.8

97.7 124.8 93.6

115.0 116.4 117.2 117.2 85.8 87.5 88.5 88.5 72.2 73.8 74.8 74.8 54.8 56.5 57.5

105.0 110.3 113.5 113.3 127.1 129.5 131.0 130.9 96.6 99.6 101.5 101.3

De ——————————————— DZ TZ QZ DTQ TQ 174.5 185.7 189.2 191.2 191.2 152.9 156.3 157.3 157.8 157.8 114.2 117.1 119.4 120.8 120.7 99.4 101.8 104.2 105.8 105.6 82.2 85.0 86.7

151.3 171.1 129.0 112.5

157.7 158.3 158.4 158.6 173.5 171.7 170.5 170.7 132.7 131.8 131.1 131.3 116.2 115.6 115.1 115.3 95.0 94.8 94.7


Table 1: Basis set convergence of the CCSD(T) calculated equilibrium bond lengths, re (in Å), harmonic vibrational frequencies, ωe , (in cm−1 ), and dissociation energies, De (in kcal·mol−1 ), of the LnX diatomics. The corrections due to zero-field splitting of the atoms are included throughout for De .

EuO EuF EuCl 2.0591 2.5248 2.6822

2.0380 2.0270 2.5118 2.5006 2.6697 2.6573 2.9045 2.8914

1.8872 1.8871 2.0747 2.0750 2.5635 2.5644

YbF YbCl YbBr YbI 1.8110 1.9402 2.4035 2.5636

1.7927 1.9207 2.3825 2.5385 2.7764

1.7981 1.9272 2.3915 2.5487 2.7882

LuO LuF LuCl LuBr LuI

16

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Page 16 of 48 Journal of Chemical Theory and Computation

0.2 0.5 0.2

0.7 0.3 0.4

1.1 0.8 0.5

1.2 0.6 0.0 0.0

0.5 0.5 0.5

0.4 -0.2 0.2 -0.5 -0.3

-1.4 0.5 -0.2 0.0

0.0 0.3 0.5

0.01 -0.21 -0.21 -0.20 -0.21

0.00 0.02 0.02 0.02

-0.04 0.00 0.01

-0.11 0.64 0.65 0.64 0.65

-0.39 -0.25 -0.23 -0.21

-1.03 -0.30 -0.29

0.21 0.14 0.06 -0.13 -0.01

0.14 0.00 -0.12 -0.02

0.20 0.15 0.01

0.82 1.17

0.65 0.92

0.18 0.65 0.56 0.42 0.51

-0.33 -0.24 -0.30 -0.18

-0.68 -0.07 -0.21

0.11 0.57 0.50 0.31 0.43

-0.25 -0.23 -0.33 -0.21

-0.87 -0.15 -0.27

0.07 0.53 0.47 0.24 0.38

-0.20 -0.21 -0.35 -0.23

-0.98 -0.20 -0.30

∆cv De ————————————————————— spLn dLn spX dX TZ QZ CBS 0.53 0.08 0.25 0.65 0.86 0.98 0.15 1.07 0.14 1.37 1.35 1.35 0.07 0.67 0.06 0.85 0.80 0.78 0.05 0.56 -0.12 0.75 0.61 0.49 0.43 0.03 0.36 -0.01 1.11 0.48 0.38 0.32

4.3 0.6 0.1 0.0

0.1 -0.1 0.2 -0.4 -0.3


Table 2: The CCSD(T)/TZ, QZ and CBS(TQ) outer-core–valence correlation effects on the LnX bond lengths, ∆cv re (in pm), vibrational frequencies, ∆cv ωe (in cm−1 ), and dissociation energies, ∆cv De (in kcal·mol−1 ), as well as partial contributions to ∆cv (QZ) from correlating the 4sp (Ln), 4d (Ln), (n–1)sp (X), and (n–1)d (X) outer-core electrons.∗

-0.4 -0.3 -0.1

-0.3 -0.1 0.3 -0.2 -0.2

∆cv ωe ——————————————————spLn dLn spX dX TZ QZ CBS 1.4 3.2 0.4 4.8 5.0 5.2 0.5 2.6 0.2 3.4 3.3 3.3 0.3 1.6 0.5 2.0 2.4 2.6 0.2 1.2 -0.2 1.3 1.1 1.2 1.4 0.0 1.0 0.0 1.5 0.9 1.0 1.2

0.01 -0.12 -0.33

∆cv re ————————————————————— spLn dLn spX dX TZ QZ CBS -0.20 -0.21 -0.09 -0.36 -0.50 -0.59 -0.13 -0.37 -0.05 -0.46 -0.55 -0.60 -0.14 -0.39 -0.23 -0.63 -0.76 -0.83 -0.13 -0.40 0.06 -0.82 -0.39 -0.47 -0.52 -0.13 -0.39 -0.05 -1.31 -0.46 -0.57 -0.63 0.02 -0.14 -0.32

LaO LaF LaCl LaBr LaI 0.02 -0.16 -0.31

1.1 1.2

-0.09 -0.06 -0.24

0.3 0.4 -0.1 0.4

0.09 -0.11 -0.12

1.7 0.4 0.3 0.1

0.02 0.03 0.04 -0.8 -0.2 -0.2 -0.5

EuO EuF EuCl 0.05 -0.19 0.15 -0.02

-0.90 -1.47

0.18 -0.13 0.16 0.00

-0.06 -0.23 0.08 -0.04

0.39 -0.05 0.16 0.04

0.16 -0.01 -0.03 -0.07

1.4 1.5

0.08 0.11 0.11 0.11

0.8 0.3 0.6 -0.2 0.2

YbF YbCl YbBr YbI

-0.7 -0.5 -0.5 -0.3 -0.3

-0.89 -1.47

0.0 0.1 0.1 0.1 -0.2

-0.08 -0.05 -0.23 0.07 -0.04

-0.01 0.00 -0.17 0.19 0.07

0.08 0.03 0.08 0.10 0.12

0.01 0.00 -0.14 0.17 0.07

0.01 0.02 0.01 0.00 -0.01

0.02 0.02 -0.11 0.14 0.07

LuO LuF LuCl LuBr LuI

* The 4sp (Ln), 4d (Ln), (n–1)sp (X), and (n–1)d (X) contributions are denoted as sp , d , sp , and d , respectively. The CV effects, ∆ cv Ln Ln X X TZ, QZ and CBS, are defined here as the difference between the numerical value of a molecular property obtained via full outer-core correlation treatment and that with all core orbitals but the (n–1)d of Br and I kept frozen.

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Journal of Chemical Theory and Computation Page 17 of 48


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3.2

Higher order effects

The higher order electron correlation effect, ∆HO, was determined as the difference between the values of molecular properties calculated at the CCSDT(Q) and CCSD(T) levels using the same basis set: ∆HO = CCSDT(Q) – CCSD(T). It combines the effects of iterative triples, ∆T = CCSDT – CCSD(T), and noniterative quadruples, ∆(Q) = CCSDT(Q) – CCSDT. The DZ and TZ basis sets outlined in Section 2 above were employed in these calculations, together with a two-point (DT) CBS extrapolation procedure via Eq. 1. The HO effect on the dissociation energy was calculated using molecular energies obtained at the optimal CCSD(T)/CBS(TQ) geometries listed in Table 1. Since the CCSDT(Q) calculations with the TZ basis set are very demanding computationally, the TZ and CBS(DT) dissociation energies were obtained at the CCSDT(Q) level for only a subset of molecules including LaX, EuO, YbF, and LuO. For the other LnX diatomics, the TZ calculations of dissociation energies were carried out just through the CCSDT level, with the higher order effect on De determined as the sum of the ∆T/CBS(DT) and ∆(Q)/DZ contributions. The HO effect on re and ωe was evaluated using the DZ basis set for all of the species under study, and also the TZ basis set for the LaX, EuO, and YbF molecules, for which the two-point (DT) CBS limit estimates were obtained. The ∆T, ∆(Q), and ∆HO corrections determined in this way are collected in Table 3.

18


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∆T

DZ TZ CBS DZ TZ CBS DZ TZ CBS

LaO 0.10 0.03 0.01 0.27 0.26 0.25 0.37 0.29 0.26 -2.4 -2.8 -3.4 -1.2 -1.5 -1.5 -3.6 -4.3 -4.9 -0.72 -0.98 -1.07 -0.03 -0.13 -0.17 -0.75 -1.11 -1.24

LaF 0.19 0.21 0.22 0.08 0.10 0.09 0.27 0.31 0.31 -2.3 -3.1 -3.1 -0.7 -0.9 -1.1 -3.0 -4.0 -4.2 -0.62 -0.83 -0.91 0.04 0.09 0.11 -0.58 -0.74 -0.80

LaCl 0.34 0.43 0.46 0.07 0.08 0.07 0.41 0.51 0.53 -1.8 -2.2 -2.4 -0.7 -0.9 -0.9 -2.5 -3.1 -3.3 -0.48 -0.65 -0.71 0.03 0.08 0.10 -0.45 -0.57 -0.61

LaBr 0.39 0.50 0.54 0.09 0.11 0.12 0.48 0.61 0.66 -1.6 -2.1 -2.2 -0.7 -0.8 -0.9 -2.3 -2.9 -3.1 -0.29 -0.46 -0.52 0.09 0.18 0.21 -0.20 -0.28 -0.31

LaI 0.47 0.62 0.67 0.14 0.15 0.15 0.61 0.77 0.82 -23.5 -15.2 -9.4 -29.5 -23.5 -20.1 -53.0 -38.7 -29.5 0.41 -0.11 -0.30 1.19 1.12 1.10 1.60 1.01 0.80

EuO -0.13 -0.14 -0.08 0.45 0.40 0.41 0.32 0.26 0.33 0.04

EuF 0.05

0.07

-0.02

EuCl 0.09

-0.2

0.09

-0.4

0.0

0.04 -0.04 -0.07 0.00

-0.3

0.14 0.02 -0.01 0.09

0.04

-0.2

0.23

(-0.07)

-0.7

(0.08)

YbF -1.04 -0.62 -0.44 0.26 0.21 0.19 -0.78 -0.41 -0.25 -11.1 -4.8 -1.8 1.2 0.2 -0.2 -9.9 -4.6 -2.0 1.10 0.77 0.66 0.07 -0.02 -0.05 1.17 0.75 0.61

-1.0

-0.13

0.16

YbCl -0.29

-0.2

-0.4

-0.04

0.15

YbBr -0.19

-0.1

-0.3

0.10

0.11

YbI -0.01

-14.1

-4.7

0.52

0.38

LuO 0.14

-0.7

-1.0

0.13

0.05

LuF 0.08

0.0

-0.7

0.16

0.01

LuCl 0.15

0.0

-0.4

0.16

0.00

LuBr 0.16

-0.01

LuI 0.19

-0.30 -0.42 -0.46 0.16

0.1

-0.35 -0.51 -0.56 0.18

-0.13

-0.4

-0.28 -0.44 -0.50 0.20

-0.17

(-0.30)

-0.7

-0.08

(-0.38)

-1.7

0.27 0.28 0.28 0.06

(-0.30)

-18.8

0.40 0.38 0.38 0.02

0.33

-0.30 -0.65 -0.77 0.85 0.80 0.77 0.55 0.15 0.00

-0.4

0.42 0.36 0.34 0.04

0.42

(0.34)

-0.6

0.46

(0.40)

-0.9

(0.38)

(-0.23)

-0.05

-0.25 -0.37 -0.42 0.19

-0.3

0.1

-0.4

0.18


∆(Q)

∆HO

∆T

∆(Q)

∆HO

∆T

-1.9 -0.8 -0.5 -7.9 -8.0 -7.9 -9.8 -8.8 -8.4 -1.07 -1.54 -1.70 0.28 0.12 0.06 -0.79 -1.42 -1.64

Table 3: Convergence of the higher order corrections to bond lengths re , pm, vibrational frequencies ωe , cm−1 , and dissociation energies De , kcal·mol−1 , with respect to basis set size.

re

ωe

De

∆(Q)

∆HO

DZ TZ CBS DZ TZ CBS DZ TZ CBS DZ TZ CBS DZ TZ CBS DZ TZ CBS

19

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

In general, the numerical values of ∆(Q) converge rapidly with respect to basis set size: across all available comparisons, except EuO and YbF, we find mean (maximum) absolute deviations of the DZ results from the CBS limit estimates of 0.0002 (0.0004) Å in re , 0.2 (0.4) cm−1 in ωe , and 0.11 (0.22) kcal·mol−1 in De , while the respective TZ results deviate from CBS by 0.0001 (0.0001) Å in re , 0.1 (0.2) cm−1 in ωe , and 0.03 (0.06) kcal·mol−1 in De . The triples contribution, ∆T, exhibits slower convergence rate, hence larger deviations of DZ from CBS: 0.0011 (0.0020) Å in re , 0.9 (1.4) cm−1 in ωe , 0.26 (0.71) kcal·mol−1 in De . The respective TZ deviations amount to 0.0004 (0.0006) Å in re , 0.2 (0.6) cm−1 in ωe , 0.07 (0.19) kcal·mol−1 in De . A particular note should be taken of the molecules EuO and YbF. The triples corrections to ωe and re of YbF, and both triples and quadruples corrections to ωe of EuO, exhibit a significant decrease in magnitude on passing from DZ to TZ. This implies that a DZ-quality basis set is not flexible enough to ensure obtaining reliable CBS estimates for these two species via a two-point DZ – TZ extrapolation procedure. Using the TZ – QZ pair of basis sets would be more appropriate, however, higher order calculations with the QZ basis set are currently not feasible due to hardware limitations. Turning now to the CBS estimates of higher order corrections, we note that the LaO and LaF diatomics exhibit the largest HO corrections to De , 1.6 kcal·mol−1 and 1.2 kcal·mol−1 , respectively, while for the other molecules these are calculated to be less than 0.8 kcal·mol−1 . The HO contributions to re are generally less than 0.004 Å, in some instances (LaCl, LaBr, LaI, and LuO), however, amounting to 0.005 – 0.008 Å. Similarly, for most diatomics the absolute value of the HO correction to ωe is calculated to be less than 5 cm−1 , whereas in lanthanide oxides it appears to be as large as 8 cm−1 (LaO), 19 cm−1 (LuO), and even 30 cm−1 (EuO). One might opt to relate a magnitude of the HO correction to a degree of multireference character of a molecule. Indeed, if the wave function is dominated by a single (Hartree–Fock) configuration, the single-reference CCSD and CCSD(T) methods are expected to exhibit a relatively small amount of n-particle error, i.e., the residual correlation effects associated

20


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with higher excitations should be insignificant. On the other hand, the quality of the singlereference CCSD and CCSD(T) methods should degrade if the HF determinant is not a good zeroth order reference. The higher order excitation levels can correct for some amount of multireference character, however, the more appreciable multireference character the species has, the slower the convergence with the level of excitation. Therefore, an extent of the multireference character of a molecule might be judged by the relative magnitude of the higher order effect at a certain level of excitation, e.g., CCSDT(Q). From this perspective one can guess that a few species among those studied here, first of all EuO and LuO, might have a significant amount of multireference character. Bearing in mind the above considerations, it is advisable to compare the calculated HO effects with the single-reference (SR) criteria T1 and %T AE[(T )] commonly used to judge a priori the reliability of a single reference coupled cluster calculation. The T1 diagnostic 69 is computed from the norm of the vector of singles substitution amplitudes, t1 , obtained from a CCSD procedure, scaled by the square root of the number of correlated electrons: ! T1 = ||t1 ||/ Ncorr

(4)

As suggested by Lee and Taylor, 69 T1 magnitudes larger than 0.02 imply that single-reference methods may fail to give reliable predictions. The %T AE[(T )] diagnostic 70,71 presents the percentage of perturbative triples (T) contribution to the total atomization energy,

%T AE[(T )] =

100 × (T AE[CCSD(T)] − T AE[CCSD]) T AE[CCSD(T)]

(5)

Analyzing the computational thermochemistry data, Martin et al. 70 have proposed the following criteria for interpretation of the %T AE[(T )] values: below 2% indicates systems dominated by dynamical correlation; 2% – 5% mild nondynamical correlation; 5% – 10% moderate nondynamical correlation; and in excess of 10% severe nondynamical correlation. Both the T1 and %T AE[(T )] diagnostics were employed in a study on 3d transition metal 21



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Page 22 of 48

species, 72 with the recommended single-reference domination thresholds of T1 < 0.05 and |%T AE(T )| < 10%. The T1 diagnostics of LnX calculated using QZ basis sets at the CCSD(T)/CBS equilibrium bond lengths, are provided in Table 4 together with the CBS limit estimates of %T AE[(T )]. The two sets of the T1 values given there, T1cv and T1f c , were obtained from the respective CCSD calculations with full accounting for outer-core correlation and with all core electrons except (n-1)d (Br,I) neglected in a correlation treatment. It can be seen that for each molecule the T1f c diagnostic is significantly larger in magnitude than T1cv . It should be noted, however, that the ||t1 || norm determined in a CV calculation and then scaled via Eq. 4, assuming the numerical value of Ncorr to be equal to the number of electrons correlated in a respective FC calculation rather than the actual number of correlated electrons, yields T1cv which is close in magnitude to T1f c (cf. the T1cv -rescaled and T1f c sets of numbers in Table 4). This shows that a decrease in the numerical value of T1 upon accounting for core correlation is mainly due to an increase in Ncorr rather than a decrease in the norm of t1 . Nonetheless, the results of the T1 -based diagnostic of the LnX molecules still depend on whether the frozen-core approximation is utilized or not in the T1 evaluation. Looking over the data in Table 4, choosing the T1cv to be a diagnostic, and employing the SR threshold of 0.02, 69 one might conclude that all of the molecules of this study, except EuO, YbF, and LuO, are single-reference dominated. On the contrary, the use of the T1f c diagnostics with the same threshold suggests significant multireference character for most of the molecules studied here. Apparently, the former choice should be preferred in the light of the CCSDT(Q) results discussed above. Although there is not a strong correlation between a large value of T1cv and a magnitude of higher order correction to molecular property, there still exists an appreciable degree of correlation between T1cv and %re [(Q)], the percentage of the (Q) contribution to bond length, as defined by Eq. 6:

%re [(Q)] =

100 × |re [CCSDT(Q)] − re [CCSDT]| re [CCSDT] 22


(6)

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Performing a statistical correlation analysis between the T1cv and %re [(Q)] sets of numbers, we find a correlation coefficient of 0.84 (see also Fig. 3). An analogous analysis involving T1f c yields a correlation coefficient of 0.74, hence favoring the use of the T1cv values rather than T1f c in a SR diagnostic of the Ln-containing species. Considering the %T AE[(T )] values, we can see no obvious correlation between the %T AE[(T )] and T1cv diagnostics and between %T AE[(T )] and a magnitude of the higher order effect on a molecular property. Nonetheless, the EuO and LuO molecules exhibiting the largest %T AE[(T )] values among the species studied here (8.7 and 5.1, respectively), are just two of those three species judged to have appreciable multireference character on the basis of the T1cv diagnostics. These results give us reasons to believe that the %T AE[(T )] diagnostic can still provide useful information in assessing the quality of a particular calculation in the studies of Ln-containing species, as well as the T1cv diagnostic, with the recommended single-reference domination thresholds of %T AE[(T )] < 5 and T1cv < 0.02. Table 4: Single-reference criteria for lanthanide diatomics. T1cv 1 LuBr 0.0132 2 LuI 0.0135 3 LuCl 0.0140 4 LuF 0.0159 5 LaBr 0.0159 6 EuCl 0.0161 7 LaI 0.0164 8 YbBr 0.0166 9 YbI 0.0167 10 LaCl 0.0172 11 YbCl 0.0180 12 EuF 0.0182 13 LaF 0.0185 14 LaO 0.0194 15 16 17 * T cv -rescaled 1

LuO YbF EuO

= T1cv ×

0.0206 0.0212 0.0226

T1cv -rescaled∗ 0.0168 0.0179 0.0188 0.0202 0.0221 0.0233 0.0228 0.0213 0.0213 0.0269 0.0244 0.0247 0.0268 0.0287

T1f c 0.0162 0.0166 0.0182 0.0196 0.0222 0.0229 0.0234 0.0199 0.0201 0.0269 0.0228 0.0243 0.0258 0.0276

%T AE[(T )] 2.3 3.0 2.0 1.9 3.5 0.9 4.7 0.4 0.6 3.0 0.4 1.7 2.6 3.9

0.0265 0.0272 0.0309

0.0262 0.0252 0.0321

5.1 1.0 8.7

! Ncorr,CV /Ncorr,F C , see main text.

23



0.25

17

15

0.20

0.15

% re[(Q)]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 24 of 48

14 0.10

16 8 5 7

0.05

2 3 0.00 0.012

1 0.014

4 6

9

11 13 10

0.016

12 0.018

0.020

0.022

0.024

cv

T1

Figure 3: The percentage of the (Q) contribution to bond length, %re [(Q)], vs. T1cv diagnostic. The numbers shown refer to the numbering of the molecules in Table 4.

24


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3.3 3.3.1

Relativistic effects DK3 vs. DK2 and DK4

In order to examine the effect of the order of the Douglas–Kroll–Hess Hamiltonian chosen in a particular calculation on the results of theoretical predictions, we have studied a few molecules at the CCSD(T)/QZ level using both second- and fourth-order DK Hamiltonians (DK2 and DK4) in addition to the regular third-order DK treatment (DK3). In these calculations, the 4d La, 3d Br, and 4d I core electrons were correlated in addition to valence electrons. The DK2 and DK4 calculated molecular properties are compared with the DK3 counterparts in Table 5. As can be seen, the DK2 – DK3 difference in the numerical values of molecular properties obtained for heavier LnX diatomics is quite noticeable: up to 0.0005 Å in re of LuBr and LuI, 0.16 kcal·mol−1 in De of YbBr and YbI. On the other hand, the DK4 – DK3 difference is practically negligible, thus indicating appropriateness of the thirdorder in the Douglas–Kroll–Hess Hamiltonian for an accurate theoretical treatment of scalar relativistic effects in lanthanide-containing species. Table 5: Differences between the molecular properties calculated at the CCSD(T)/QZ level with the Douglas–Kroll–Hess Hamiltonian of various order (DKn, n = 2, 3, 4)∗ . Units are pm, cm−1 , and kcal·mol−1 . ∆re ∆ωe ∆De ————— ————— ————— 2–3 4–3 2–3 4–3 2–3 4–3 LaBr 0.01 0.00 0.0 0.0 0.01 0.00 LaI 0.00 0.00 0.0 0.0 0.03 0.00 YbBr 0.01 0.00 0.1 0.0 0.15 0.01 YbI 0.00 0.00 0.1 0.0 0.16 0.02 LuBr -0.04 -0.01 0.2 0.0 -0.03 0.00 LuI -0.05 -0.01 0.2 0.1 -0.02 0.00 * The

notations 2 – 3 and 4 – 3 denote the DK2 – DK3 and DK4 – DK3 differences, respectively.

25



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3.3.2

Spin–orbit coupling effect

Spin–orbit effects do not contribute in first order of perturbation theory to the ground electronic state of the molecules we are considering (with Λ = 0). However, the molecules are involved in SO coupling through second order. It is well recognized that second-order SO effects can be appreciable in heavy atom containing species (see, e.g., Refs. 73,74). To understand a role the SO effects can play in achieving high accuracy in ab initio spectroscopy and thermochemistry of lanthanide-containing species, we have calculated the second-order SO corrections to the LnX molecular properties under study using relativistic techniques and basis sets specified in Section 2 above. To assess the accuracy achievable in these calculations, we can compare the calculated zero-field splittings of the O, F, Cl, Br, I, La, and Lu open-shell atoms with experiment. At the 4c-DC-HF/DZ (2c-RECP-HF/QZ) level of theory, the splittings of the 3 P2 , 3 P1 , and 3 P0 levels of oxygen, the 2 P3/2 and 2 P1/2 levels of the halogens, and the 2 D3/2 and 2 D5/2 levels of La and Lu, were calculated to be (all in cm−1 ) 151, 225 (150, 224) for O, 404 (407) for F, 921 (987) for Cl, 3742 (3687) for Br, 7755 (7607) for I, 1042 (1020) for La, and 1428 (1370) for Lu. These numbers can be compared to the experimental values 44 (all in cm−1 ): 158, 227 for O, 404 for F, 882 for Cl, 3685 for Br, 7603 for I, 1053 for La, and 1994 for Lu. As can be seen, in most cases the accuracy of the SO calculations is quite reasonable. Further improvement would require an accounting for electron correlation in the calculations. In particular, going to the FSCCSD level of theory greatly improves accuracy in zero-field splitting for the Lu atom: the ZFS obtained from the 4c-FSCCSD/DZ and 2c-RECP-FSCCSD/QZ calculations amounts to 1987 cm−1 and 1981 cm−1 , respectively. The convergence in the magnitudes of SO corrections with respect to basis set size was examined via the Dirac–Coulomb four-component methodology for a number of LnX molecules, particularly those exhibiting the largest SO effects, for instance, LaI and LuBr, and also YbF, which exemplifies systems with appreciable multireference character. The results collected in Table 6 provide evidence that the second-order SO corrections are nearly basis-set-convergent 26


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already with the DZ basis set, although the YbF case deserves making a special note, see below. Therefore, only the DZ basis sets were employed in the 4c all-electron calculations for the remaining molecules. The 4c/DZ results are compared with those from the RECP-based two-component relativistic calculations in Table 7. As can be seen, the SO corrections determined in 2c calculations agree remarkably well with their 4c counterparts. The effect of electron correlation on the magnitudes of the second-order SO corrections was calculated to be very small, thus indicating that a reasonable estimate of the molecular second-order SO effect can be obtained just at the Dirac–Hartree–Fock level. Looking over the data in Table 7, one can see that the second-order SO effect on the dissociation energy of lanthanide-containing diatomics is small though non-negligible, generally amounting to a few tenths of a kcal·mol−1 . As expected, the largest second-order SO effect on De occurs in heavy-atom containing species: for instance, in LaI it amounts to 0.8 kcal·mol−1 . The SO effects on bond lengths and vibrational frequencies are calculated to be less than 0.003 Å and 1 cm−1 , respectively. A special comment should be made about the YbF molecule. The SO contributions to the YbF bond length and vibrational frequency determined by single-reference coupledcluster methods, CCSD and CCSD(T), appeared to be conspicuously erroneous. The ∆so re (∆so ωe ) values obtained at the 4c-CCSD level using the DZ and TZ relativistic basis sets amount to –0.0113 Å (–23.7 cm−1 ) and –0.0053 Å (–4.8 cm−1 ), respectively, whereas at the CCSD(T) level the respective values are +0.0374 Å (+103.9 cm−1 ) and +0.0104 Å (+33.0 cm−1 ). Similar values were obtained in 2c-CCSD and 2c-CCSD(T) calculations, e.g., – 0.0123 Å (–33.0 cm−1 ) at the CCSD level. All these numbers are not yet convergent with respect to active virtual space size, despite the fact that a fairly large active virtual space was utilized in the calculations. Overall, the relativistic CCSD and CCSD(T) results for YbF provide evidence that still larger basis sets and active virtual spaces, and accounting for higher order excitations are necessary to obtain reasonable ∆so estimates using single27



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reference coupled-cluster methods in difficult cases such as YbF where dynamical correlation can strongly influence the nature of the ground-state wave function. These conclusions are in line with those drawn previously by Gomes, Dyall, and Visscher 16 in their detailed study on the YbF molecule. The FSCCSD method that provides rapid convergence in ∆so with respect to active virtual space size may be suggested as an effective and much less computationally demanding alternative for obtaining accurate estimates of the ∆so values for YbF-like systems, for instance, the other ytterbium halide diatomics (YbCl, YbBr, YbI), where the CCSD(T) calculated SO corrections to re and ωe appeared to be qualitatively incorrect as well. Table 6: Basis set convergence of the second-order SO corrections to the LaI, LuBr, and YbF bond lengths, ∆so re , pm, vibrational frequencies, ∆so ωe , cm−1 , and dissociation energies, ∆so De , kcal·mol−1 , as revealed from the Dirac–Coulomb four-component relativistic calculations.

HF

∆so re ∆so ωe ∆so De ———————– ———————– ———————– LaI LuBr YbF LaI LuBr YbF LaI LuBr YbF DZ 0.24 -0.16 -0.18 -1.3 0.3 0.6 0.77 0.39 0.10 TZ 0.20 -0.18 -0.19 -1.1 0.4 0.6 0.76 0.40 0.10 QZ 0.21 -0.18 -0.19 -1.1 0.4 0.6 0.76 0.40 0.10

MP2

DZ 0.24 TZ 0.22 QZ 0.22

-0.19 -0.20

-0.23 -1.3 -0.24 -1.2 -1.2

CCSD

DZ TZ

0.25 0.25

-1.3 -1.3

CCSD(T)

DZ TZ

0.23 0.22

-1.0 -1.1

FSCCSD

DZ

-0.23

0.5 0.6

1.1 1.0

0.7

28


0.19

LuBr

LuI


Table 7: Second-order SO corrections to the LnX bond lengths, ∆so re , pm, vibrational frequencies, ∆so ωe , cm−1 , and dissociation energies, ∆so De , kcal·mol−1 , obtained from four-component and two-component relativistic calculations.

LuCl

LuO

LuF

YbI

LaI

-0.12 -0.17 -0.11 -0.17 -0.13 -0.14

YbBr

LaBr

-0.16 -0.19 -0.15 -0.18 -0.16 -0.15

YbCl

LaCl

-0.16 -0.19 -0.15 -0.18 -0.16 -0.16

YbF

LaF

-0.12 -0.14

-0.13 -0.13 -0.12 -0.12 -0.13 -0.12 -0.17

0.1 0.2 0.0 0.3 0.2 0.1

-0.21

0.3 0.5 0.3 0.5 0.4 0.4

0.90 0.70

-0.22

0.5 0.7 0.5 0.7 0.6 0.6

0.39 0.23

-0.20

0.6 0.7 0.4 0.7 0.6 0.5

0.23 0.08

-0.1 0.0 -0.1 0.2 0.0

1.1 1.3 1.0 1.1 1.1 1.2 1.0

0.23 0.07

0.2 0.5 0.2 0.0 0.0

0.2

0.63 0.49 0.46

0.4 0.7 0.3 0.7 0.0

0.2

0.60 0.50 0.59

0.6 1.1 0.6 1.2

0.4

0.18 0.10 0.18

EuF

LaO 0.24 0.24 0.19 0.22 0.25 0.24

0.4 1.2

0.7

0.05 -0.02 0.05

-0.11

∆so re

0.02 0.06 0.01 0.03 0.06 0.07 -1.3 -1.3 -1.0 -1.2 -1.4 -1.3

0.09 0.06

0.10 0.03 0.11

-0.09 -0.21 -0.08 -0.18 -0.16

-0.03 0.00 -0.04 -0.01 0.00 0.01 -0.3 -0.5 -0.3 -0.5 -0.7 -0.7

0.77 0.68

-0.19 -0.28 -0.17 -0.27 -0.26

-0.04 -0.03 -0.06 -0.04 -0.04 -0.04 -0.1 -0.3 0.0 -0.3 -0.5 -0.5

0.25 0.20

-0.20 -0.28 -0.18 -0.27 -0.31

0.01 -0.01 -0.01 -0.03 -0.02 -0.03 -0.02 0.2 0.1 0.3 0.1 0.1 0.0 0.11 0.07

-0.18 -0.23 -0.17 -0.21

4c-DC-HF 4c-DC-MP2 2c-RECP-HF 2c-RECP-MP2 2c-RECP-CCSD 2c-RECP-CCSD(T) 2c-RECP-FSCCSD 0.7 0.5 0.8 0.7 0.6 0.7 0.7 0.12 0.08

-0.13 -0.14 -0.15 -0.13 -0.14 -0.15 -0.14

∆so ωe

4c-DC-HF 4c-DC-MP2 2c-RECP-HF 2c-RECP-MP2 2c-RECP-CCSD 2c-RECP-CCSD(T) 2c-RECP-FSCCSD 0.16 0.18 0.13

0.4

∆so De

4c-DC-HF 2c-RECP-HF 2c-RECP-FSCCSD

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3.4

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Final composite results and comparison to experiment

A composite molecular property, Picomp , was evaluated as Picomp = Pi [(CCSD(T ) + CV )/CBS(T Q)] + ∆Pi (HO) + ∆Pi (SO),

(7)

where Pi [(CCSD(T ) + CV )/CBS(T Q)] is the CBS limit estimate of Pi obtained at the CCSD(T) level of theory with full accounting for the outer-core–valence electron correlation (from Table 1); ∆Pi (HO) is the higher order electron correlation effect beyond CCSD(T), and ∆Pi (SO) accounts for second-order spin–orbit coupling. The ∆Pi (SO) and ∆Pi (HO) corrections were taken from Tables 7 and 3, respectively. For ∆Pi (HO), the CBS estimates were employed if available, and the DZ values otherwise. The ∆Pi (SO) corrections utilized in Eq. 7 were those obtained at the 2c-RECP-FSCCSD level of theory for the ytterbium halide diatomics and 2c-RECP-HF for EuF. For the remaining molecules the SO corrections to bond lengths and vibrational frequencies were taken from 2c-RECP-CCSD(T) calculations, whereas the second-order SO contributions to dissociation energies were determined at the 4c-DC-HF level. For the EuO and EuCl molecules, which were not studied in a fully relativistic manner due to computational difficulties, the second-order spin–orbit corrections are expected to be insignificant and hence were neglected. In calculations of the 0 K atomization energies, D0◦ , the required zero point energy corrections were derived from the composite values of vibrational frequencies. The final composite values of D0◦ , re , and ωe , as well as the CCSD(T)/QZ calculated values of the spectroscopic constants ωe xe and αe , are compared with their experimental counterparts in Table 8. The theoretical values of re agree with the respective values obtained from experiment to within a few thousandths of an angstrom, exhibiting a mean absolute deviation (MAD) of 0.0023 Å and a mean signed deviation (MSD) of +0.0015 Å. A notable outlier excluded from the statistics is the YbI bond length reported in Ref. 75, which is shorter by about 0.035 Å than that predicted by high-level theory. This appreciable

30


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discrepancy implies the need for re-examination of the experimental data 75 on re (YbI). The composite harmonic vibrational frequencies ωe are generally in agreement with experiment, with a MAD of 3.5 cm−1 and a MSD of –1.7 cm−1 for all available comparisons. The largest deviations observed for EuO (19 cm−1 ) and LuO (10 cm−1 ) are owing mainly to imperfections in theoretical treatment caused by significant multireference character of these two molecules, as noted in Section 3.2 above. Excluding EuO and LuO from the statistics, we find a MAD and a MSD of just 1.9 cm−1 and +0.1 cm−1 , respectively. The calculated higher order spectroscopic constants ωe xe and αe are also in good agreement with the available experimental data (cf. Table 8). The experimental data on dissociation energies of lanthanide diatomics are far from being complete: the D0◦ values have been experimentally determined for only nine molecules of the seventeen under study. The D0◦ composite values agree with the experimental data to within uncertainties of the latter, exhibiting a MAD of 1.8 kcal·mol−1 and a MSD of –0.7 kcal·mol−1 . In a few instances, the composite values of D0◦ seem to be superior to their experimental counterparts. Particularly, this observation holds for D0◦ (YbI) experimentally derived by Sergeev et al.: 62.3 ± 4.6 kcal·mol−1 (Ref. 76) and 67.6 ± 2.4 kcal·mol−1 (Ref. 77). The latter value is in conspicuous error, whereas the other one hardly overlaps with the composite result of 58.2 kcal·mol−1 . The MADs from experiment in the molecular properties calculated at various levels of theory are depicted in Figure 4. It can be seen that MADs display a gradual decrease as the basis set size increases, except for the ωe case, where a slight increase is observed on going from QZ to CBS. The importance of eliminating the basis set incompleteness error and accounting for higher order correlation and spin–orbit effects is clearly seen in the case of bond lengths, exhibiting a MAD at the CCSD(T)/QZ level almost twice as large as that at the final CCSD(T)/CBS + HO + SO level of theory.

31


D0◦ —————————– Calc. Expt. Ref. 188.6 190.6±2.7 78 155.9 156.6±4.1 81 119.5 105.0 86.9 113.1 114.5±4.0 78 130.4 129±2 88 100.8 97.8±2.2 77 117.2 88.6 87.5±2.2 77 75.1 75.5±2.2 77 58.2 62.3±4.6 76 158.0 160.3±4.0 78 169.7 130.6 115.1 95.2

re ———————– Calc. Expt. Ref. 1.8277 1.8259 79 2.0242 2.0234 82 2.5018 2.4980 82 2.6554 2.6521 82 2.8810 2.8788 82 1.8904 1.89 86 2.0747 2.079b 89 2.5651 2.0159 2.0165 90 2.4907 2.4883 90 2.6476 2.6454 94 2.8832 2.8483 75 1.7932 1.7903 95 1.9171 1.9171 97 2.3772 2.3733 97 2.5327 2.7702

ωe ———————– Calc. Expt. Ref. 811.4 817.0 80 574.6 575.2 83 341.4 341.6 84 236.7 236.2 82 185.6 184.4 85 668.8 687.9a 87 497.6 497.3c 89 289.1 503.5 506.7 91 290.4 290.0 92 195.7 196.5 94 150.5 152.1 93 834.6 844.5 96 611.4 611.8 98 342.1 337 97 229.6 224.9 99 174.3 172.8 99

ωe xe ———————– Calc. Expt. Ref. 2.07 2.13 80 2.10 2.13 83 0.92 0.98 84 0.47 0.53 82 0.32 0.33 85 2.84 2.11 0.84 2.25 2.25 91 0.83 0.89 93 0.41 0.45 94 0.28 0.34 93 3.00 3.1 96 2.60 2.54 98 1.06 1.05 97 0.47 0.51 99 0.25 0.23 99

αe ·104 ———————– Calc. Expt. Ref. 13.90 14.24 79 12.02 12.22 82 3.51 3.64 84 1.28 1.35 82 0.70 0.74 82 15.99 13.65 3.76 15.23 14.99 90 3.85 3.92 90 1.33 1.33 94 0.70 0.69 75 16.62 17.04 95 15.69 15.62 97 4.06 4.13 97 1.40 0.75


Table 8: Final composite versus experimental dissociation energies D0◦ , kcal·mol−1 , and spectroscopic constants re , Å, ωe , ωe xe , and αe , cm−1 , of LnX.

LaO LaF LaCl LaBr LaI EuO EuF EuCl YbF YbCl YbBr YbI LuO LuF LuCl LuBr LuI

a Calculated from ∆G −1 of Ref. 87 using the theoretical value ω x = 2.84 cm−1 . e e 1/2 = 682.2 cm b Calculated from B = 0.2300 cm−1 of Ref. 89 using the theoretical value α = 0.001365 cm−1 . 0 e from ∆G1/2 = 493.1 cm−1 of Ref. 89 using the theoretical value ωe xe = 2.11 cm−1 . c Calculated

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The overall agreement of theoretical predictions with experimental data proves high accuracy of the composite approach for Ln-containing species. There are two main sources of residual errors in the theoretical predictions of this study: first, inaccuracy in the CCSD(T)/CBS estimates discussed in Section 3.1 above, and, second, yet incomplete accounting for higher order correlation effects. Following the discussion of Section 3.3, we might presume the second source to be dominant only for the EuO, LuO, and YbF molecules exhibiting appreciable multireference character. For the other molecules studied here, the predicted bond lengths, vibrational frequencies, and bond dissociation energies are expected to be accurate to within 0.005 Å, 5 cm−1 , and 3 kcal·mol−1 , respectively. These numbers can represent thresholds on chemical accuracy in composite spectroscopy and thermochemistry of lanthanide-containing species.

4

CONCLUSIONS

This paper provides a detailed investigation of a variety of contributions entering into a composite scheme for treating the spectroscopy and thermochemistry of Ln-containing molecules. The composite protocol based on the coupled cluster singles, doubles and perturbative triples method is applied to the test suite of lanthanide diatomics LnX (Ln = La, Eu, Yb, Lu; X = O, F, Cl, Br, I). The outer-core electron correlation, second-order spin–orbit coupling, and higher order correlation effects on bond lengths, vibrational frequencies, and dissociation energies are assessed, with examining basis set convergence in each contribution. The outercore correlation effects are calculated to be quite significant, particularly those from the Ln 4d shell, providing contributions of up to 0.008 Å to re , 5 cm−1 to ωe , and 1.4 kcal·mol−1 to De . The higher order effects are also far from being negligible even for the systems dominated by a single reference configuration, generally of the order of a few thousandths of an angstrom on re , a few wavenumbers on ωe , and a few tenths of a kcal·mol−1 on De , whereas in systems with appreciable nondynamical correlation the HO corrections can be much larger,

33



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Figure 4: Mean absolute deviation of the calculated molecular properties from the experimental data.

34


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e.g., a few dozens of a wavenumber to the vibrational frequency of the EuO molecule. The second-order spin–orbit coupling effects are generally small though non-negligible, particularly for the heavier atom containing species, such as LnBr and LnI, for which the second-order SO correction to re and De can amount to 0.002 Å and 0.6 kcal·mol−1 . The final composite results for re , ωe , and De are expected to be accurate to within 0.005 Å, 5 cm−1 , and 3 kcal·mol−1 , respectively. These numbers can serve as a conservative estimate for the level of accuracy currently attainable in the coupled-cluster-based composite spectroscopy and thermochemistry of single-reference dominated lanthanide-containing species. Some of the composite results of this study represent a significant improvement over the available experimental data.

ASSOCIATED CONTENT Supporting Information The Supporting Information contains the CCSD(T)/DZ, TZ, QZ, CBS(DTQ), and CBS(TQ) core-valence correlation contributions to bond lengths, vibrational frequencies, and atomization energies of LnX diatomics, as well as the individual bond lengths, vibrational frequencies, and atomization energies for a few molecules, as determined in the CCSD(T) calculations both with and without core correlation using Sapporo and LP basis sets. This material is available free of charge via the Internet at http://pubs.acs.org.

AUTHOR INFORMATION Corresponding Author *E–mail: [email protected]

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ACKNOWLEDGMENTS The authors gratefully acknowledge support from the Russian Foundation for Basic Research (Grant Nos. 09-03-01032, 13-03-01051) and the Ministry of Education and Science of the Russian Federation (Project No. 4.3232.2017/4.6).

References (1) Pyykkö, P. Recent developments in the theory of f-element molecules. Inorg. Chim. Acta. 1987, 139, 243–245. (2) Dolg, M.; Stoll, H. In Handbook on the physics and chemistry of rare earths; Gschneidner, K. A., Eyring, L., Eds.; Elsevier, Amsterdam, 1996; Vol. 22; pp 607–729. (3) Dolg, M. In Computational methods in lanthanide and actinide chemistry; Dolg, M., Ed.; Wiley, 2015; pp 425–450. (4) Peterson, K. A.; Feller, D.; Dixon, D. A. Chemical accuracy in ab initio thermochemistry and spectroscopy: current strategies and future challenges. Theor. Chem. Acc. 2012, 131, 1079. (5) Peterson, C.; Penchoff, D. A.; Wilson, A. K. Prediction of thermochemical properties across the periodic table: A review of the correlation consistent composite approach (ccCA) strategies and applications. Ann. Rep. Comp. Chem. 2016, 12, 3–45. (6) DeYonker, N. J.; Peterson, K. A.; Steyl, G.; Wilson, A. K.; Cundari, T. R. Quantitative computational thermochemistry of transition metal species. J. Phys. Chem. A 2007, 111, 11269–11277. (7) Feller, D.; Peterson, K. A.; Dixon, D. A. A survey of factors contributing to accurate theoretical predictions of atomization energies and molecular structures. J. Chem. Phys. 2008, 129, 204105. 36


Page 36 of 48

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(8) Dixon, D. A.; Feller, D.; Peterson, K. A. A practical guide to reliable first principles computational thermochemistry predictions across the periodic table. Ann. Rep. Comp. Chem. 2012, 8, 1–28. (9) Hill, J. G.; Mitrushchenkov, A. O.; Peterson, K. A. Ab initio ro-vibrational spectroscopy of the group 11 cyanides: CuCN, AgCN, and AuCN. J. Chem. Phys. 2013, 138, 134314. (10) Solomonik, V. G.; Smirnov, A. N.; Navarkin, I. S. Composite vibrational spectroscopy of the group 12 difluorides: ZnF2 , CdF2 , and HgF2 . J. Chem. Phys. 2016, 144, 144307. (11) Thanthiriwatte, K. S.; Vasiliu, M.; Battey, S. R.; Lu, Q.; Peterson, K. A.; Andrews, L.; Dixon, D. A. Gas phase properties of MX2 and MX4 (X = F, Cl) for M = Group 4, Group 14, Ce, and Th. J. Phys. Chem. A 2015, 119, 5790–5803. (12) Lu, Q.; Peterson, K. A. Correlation consistent basis sets for lanthanides: The atoms La – Lu. J. Chem. Phys. 2016, 145, 054111. (13) Bross, D. H.; Hill, J. G.; Werner, H.-J.; Peterson, K. A. Explicitly correlated composite thermochemistry of transition metal species. J. Chem. Phys. 2013, 139, 094302. (14) Peterson, K. A. Correlation consistent basis sets for actinides. I. The Th and U atoms. J. Chem. Phys. 2015, 142, 074105. (15) Cox, R. M.; Citir, M.; Armentrout, P. B.; Battey, S. R.; Peterson, K. A. Bond energies of ThO+ and ThC+ : A guided ion beam and quantum chemical investigation of the reactions of thorium cation with O2 and CO. J. Chem. Phys. 2016, 144, 184309. (16) Gomes, A. S. P.; Dyall, K. G.; Visscher, L. Relativistic double-zeta, triple-zeta, and quadruple-zeta basis sets for the lanthanides La – Lu. Theor. Chem. Acc. 2010, 127, 369–381.

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(17) Sekiya, M.; Noro, T.; Koga, T.; Shimazaki, T. Relativistic segmented contraction basis sets with core-valence correlation effects for atoms

57 La

through

71 Lu:

Sapporo-DK-

nZP sets (n = D, T, Q). Theor. Chem. Acc. 2012, 131, 1247. (18) Schoendorff, G.; Wilson, A. K. Low valency in lanthanides: A theoretical study of NdF and LuF. J. Chem. Phys. 2014, 140, 224314. (19) Werner, H. J.; Knowles, P. J.; Lindh, R.; Manby, F. R.; Schütz, M.; Celani, P.; Korona, T.; Mitrushenkov, A.; Rauhut, G.; Adler, T. B.; Amos, R. D.; Bernhardsson, A.; Berning, A.; Cooper, D. L.; Deegan, M. J. O.; Dobbyn, A. J.; Eckert, F.; Goll, E.; Hampel, C.; Hetzer, G.; Hrenar, T.; Knizia, G.; Köppl, C.; Liu, Y.; Lloyd, A. W.; Mata, R. A.; May, A. J.; McNicholas, S. J.; Meyer, W.; Mura, M. E.; Nicklass, A.; Palmieri, P.; Pflüger, K.; Pitzer, R.; Reiher, M.; Schumann, U.; Stoll, H.; Stone, A. J.; Tarroni, R.; Thorsteinsson, T.; Wang, M.; Wolf, A. MOLPRO, a package of ab initio programs. http://www.molpro.net. (20) Raghavachari, K.; Trucks, G. W.; Pople, J. A.; Head-Gordon, M. A fifth-order perturbation comparison of electron correlation theories. Chem. Phys. Lett. 1989, 157, 479–483. (21) Bartlett, R. J.; Watts, J. D.; Kucharski, S. A.; Noga, J. Non-iterative fifth-order triple and quadruple excitation energy corrections in correlated methods. Chem. Phys. Lett. 1990, 165, 513–522. (22) Knowles, P. J.; Hampel, C.; Werner, H.-J. Coupled cluster theory for high spin, open shell reference wave functions. J. Chem. Phys. 1993, 99, 5219–5227. (23) Knowles, P. J.; Hampel, C.; Werner, H.-J. Erratum: Coupled cluster theory for high spin, open shell reference wave functions [J. Chem. Phys. 99, 5219 (1993)]. J. Chem. Phys. 2000, 112, 3106–3107.

38


Page 39 of 48

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(24) Douglas, M.; Kroll, N. M. Quantum electrodynamical corrections to the fine structure of helium. Ann. Phys. 1974, 82, 89–155. (25) Hess, B. A. Relativistic electronic-structure calculations employing a two-component no-pair formalism with external-field projection operators. Phys. Rev. A 1986, 33, 3742–3748. (26) Wolf, A.; Reiher, M.; Hess, B. A. The generalized Douglas–Kroll transformation. J. Chem. Phys. 2002, 117, 9215–9226. (27) Reiher, M.; Wolf, A. Exact decoupling of the Dirac Hamiltonian. II. The generalized Douglas–Kroll–Hess transformation up to arbitrary order. J. Chem. Phys. 2004, 121, 10945–10956. (28) Dunning, T. H. Gaussian basis sets for use in correlated molecular calculations. I. The atoms boron through neon and hydrogen. J. Chem. Phys. 1989, 90, 1007–1023. (29) Kendall, R. A.; Dunning, T. H.; Harrison, R. J. Electron affinities of the first-row atoms revisited. Systematic basis sets and wave functions. J. Chem. Phys. 1992, 96, 6796–6806. (30) Woon, D. E.; Dunning, T. H. Gaussian basis sets for use in correlated molecular calculations. III. The atoms aluminum through argon. J. Chem. Phys. 1993, 98, 1358– 1371. (31) de Jong, W. A.; Harrison, R. J.; Dixon, D. A. Parallel Douglas–Kroll energy and gradients in NWChem: estimating scalar relativistic effects using Douglas–Kroll contracted basis sets. J. Chem. Phys. 2001, 114, 48–53. (32) Peterson, K. A.; Dunning, T. H. Accurate correlation consistent basis sets for molecular core-valence correlation effects: the second-row atoms Al–Ar, and the first-row atoms B–Ne revisited. J. Chem. Phys. 2002, 117, 10548–10560. 39



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(33) Wilson, A. K.; Woon, D. E.; Peterson, K. A.; Dunning, T. H. Gaussian basis sets for use in correlated molecular calculations. IX. The atoms gallium through krypton. J. Chem. Phys. 1999, 110, 7667–7676. (34) DeYonker, N. J.; Peterson, K. A.; Wilson, A. K. Systematically convergent correlationconsistent basis sets for molecular core-valence correlation effects: the third-row atoms gallium through krypton. J. Phys. Chem. A 2007, 111, 11383–11393. (35) Bross, D. H.; Peterson, K. A. Correlation consistent, Douglas–Kroll–Hess relativistic basis sets for the 5p and 6p elements. Theor. Chem. Acc. 2014, 133, 1434. (36) Bomble, Y. J.; Stanton, J. F.; Kállay, M.; Gauss, J. Coupled-cluster methods including noniterative corrections for quadruple excitations. J. Chem. Phys. 2005, 123, 054101. (37) Sekiya, M.; Noro, T.; Osanai, Y.; Koga, T. Contracted polarization functions for the atoms Ca, Ga–Kr, Sr, and In–Xe. Theor. Chem. Acc. 2001, 106, 297–300. (38) Noro, T.; Sekiya, M.; Koga, T.; Saito, S. L. Relativistic contracted Gaussian-type basis functions for atoms K through Xe. Chem. Phys. Lett. 2009, 481, 229–233. (39) Noro, T.; Sekiya, M.; Koga, T. Segmented Gaussian basis set library. [Online]. http://sapporo.center.ims.ac.jp/sapporo. (40) Kállay, M.; Rolik, Z.; Csontos, J.; Ladjánszki, I.; Szegedy, L.; Ladóczki, B.; Samu, G.; Petrov, K.; Farkas, M.; Nagy, P.; Mester, D.; Hégely, B. MRCC, a quantum chemical program suite. See also Rolik, Z.; Szegedy, L.; Ladjánszki, I.; Ladóczki, B.; Kállay, M.; J. Chem. Phys. 2013, 139, 094105, as well as: http://mrcc.hu. (41) Dunham, J. L. The energy levels of a rotating vibrator. Phys. Rev. 1932, 41, 721–731. (42) Martin, J. M. L. Ab initio total atomization energies of small molecules – towards the basis set limit. Chem. Phys. Lett. 1996, 259, 669–678.

40


Page 40 of 48

Page 41 of 48

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(43) Peterson, K. A.; Woon, D.; Dunning, T. H. Benchmark calculations with correlated molecular wave functions. IV. The classical barrier height of the H + H2 → H2 + H reaction. J. Chem. Phys. 1994, 100, 7410–7415. (44) Kramida, A.; Ralchenko Yu.,; Reader, J.; and NIST ASD Team, NIST Atomic Spectra Database (ver. 5.3), [Online]. Available: http://physics.nist.gov/asd [2016, August 16]. National Institute of Standards and Technology, Gaithersburg, MD, 2015. (45) Visscher, L.; H. J. Aa. Jensen,; Bast, R.; Saue, T.; with contributions from V. Bakken, K. G. Dyall, S. Dubillard, U. Ekström, E. Eliav, T. Enevoldsen, E. Faßhauer, T. Fleig, O. Fossgaard, A. S. P. Gomes, T. Helgaker, J. K. Lærdahl, Y. S. Lee, J. Henriksson, M. Iliaš, Ch. R. Jacob, S. Knecht, S. Komorovský, O. Kullie, C. V. Larsen, H. S. Nataraj, P. Norman, G. Olejniczak, J. Olsen, Y. C. Park, J. K. Pedersen, M. Pernpointner, K. Ruud, P. Sałek, B. Schimmelpfennig, J. Sikkema, A. J. Thorvaldsen, J. Thyssen, J. van Stralen, S. Villaume, O. Visser, T.Winther, S. Yamamoto, DIRAC, a relativistic ab initio electronic structure program, Release DIRAC14, 2014. http://www.diracprogram.org. (46) Visscher, L. Approximate molecular relativistic Dirac–Coulomb calculations using a simple Coulombic correction. Theor. Chem. Acc. 1997, 98, 68–70. (47) Dyall, K. G. An exact separation of the spin-free and spin-dependent terms of the Dirac–Coulomb–Breit Hamiltonian. J. Chem. Phys. 1994, 100, 2118–2127. (48) Dyall, K. G. Relativistic quadruple-zeta and revised triple-zeta and double-zeta basis sets for the 4p, 5p, and 6p elements. Theor. Chem. Acc. 2006, 115, 441–447. (49) Park, Y. C.; Lim, I. S.; Lee, Y. S. Two-Component Spin–orbit Effective Core Potential Calculations with an All-electron Relativistic Program DIRAC. Bull. Korean Chem. Soc. 2012, 33, 803–808.

41



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(50) Mosyagin, N. S.; Titov, A. V.; Petrov, A. N.; Zaitsevskii, A. V. Shape-consistent relativistic effective core potentials, [Online]. http://www.qchem.pnpi.spb.ru/recp. (51) Mosyagin, N. S.; Petrov, A. N.; Titov, A. V. The Effect of the Iterative Triple and Quadruple Cluster Amplitudes on the Adiabatic Potential Curve in the Coupled Cluster Calculations of the Ground Electronic State of the Yb Dimer. Int. J. Quant. Chem. 2011, 111, 3793–3798. (52) Cao, X.; Dolg, M. Valence basis sets for relativistic energy-consistent small-core lanthanide pseudopotentials. J. Chem. Phys. 2001, 115, 7348–7355. (53) Peterson, K. A.; Figgen, D.; Goll, E.; Stoll, H.; Dolg, M. Systematically convergent basis sets with relativistic pseudopotentials. II. Small-core pseudopotentials and correlation consistent basis sets for the post-d group 16–18 elements. J. Chem. Phys. 2003, 119, 11113–11123. (54) Peterson, K. A.; Shepler, B. C.; Figgen, D.; Stoll, H. On the Spectroscopic and Thermochemical Properties of ClO, BrO, IO, and Their Anions. J. Phys. Chem. A 2006, 110, 13877–13883. (55) Pacios, L. F.; Christiansen, P. A. Ab initio relativistic effective potentials with spinorbit operators. I. Li through Ar. J. Chem. Phys. 1985, 82, 2664–2671. (56) Kaldor, U. The Fock space coupled cluster method: theory and application. Theor. Chim. Acta 1991, 80, 427–439. (57) Visscher, L.; Eliav, E.; Kaldor, U. Formulation and implementation of the relativistic Fock-space coupled cluster method for molecules. J. Chem. Phys. 2001, 115, 9720– 9726. (58) Karton, A.; Martin, J. M. L. Comment on: Estimating the Hartree–Fock limit from

42


Page 42 of 48

Page 43 of 48

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


finite basis set calculations [Jensen F (2005) Theor Chem Acc 113:267]. Theor. Chem. Acc. 2006, 115, 330–333. (59) Boys, S. F.; Bernardi, F. The calculation of small molecular interactions by the differences of separate total energies. Some procedures with reduced errors. Mol. Phys. 1970, 19, 553–566. (60) Cao, X.; Dolg, M. Segmented contraction scheme for small-core lanthanide pseudopotential basis sets. J. Mol. Struct. (Theochem) 2002, 581, 139–147. (61) Cao, X.; Liu, W.; Dolg, M. Molecular structure of diatomic lanthanide compounds. Sci. China B 2002, 45, 91–96. (62) Heiberg, H.; Gropen, O.; Laerdahl, J. K.; Swang, O.; Wahlgren, U. The performance of density functional theory for LnF (Ln = Nd, Eu, Gd, Yb) and YbH. Theor. Chem. Acc. 2003, 110, 118–125. (63) Cao, X.; Dolg, M. Pseudopotential studies on the electronic structure of lanthanum monohalides LaF, LaCl, LaBr, and LaI. J. Theor. Comp. Chem. 2005, 4, 583–592. (64) Hülsen, M.; Dolg, M.; Link, P.; Ruschewitz, U. Improved valence basis sets for divalent lanthanide 4f-in-core pseudopotentials. Theor. Chem. Acc. 2011, 129, 367–379. (65) Alvarez-Idaboy, J. R.; Galano, A. Counterpoise corrected interaction energies are not systematically better than uncorrected ones: comparison with CCSD(T) CBS extrapolated values. Theor. Chem. Acc. 2010, 126, 75–85. (66) Varandas, A. J. C. Extrapolation to the Complete Basis Set Limit without Counterpoise. The Pair Potential of Helium Revisited. J. Phys. Chem. A 2010, 114, 8505– 8516. (67) Yang, D.-D.; Wang, F. Theoretical investigation for spectroscopic constants of groundstate alkaline-earth dimers with high accuracy. Theor. Chem. Acc. 2012, 131, 1117. 43



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(68) Mentel, L. M.; Baerends, E. J. Can the counterpoise correction for basis set superposition effect be justified? J. Chem. Theory Comput. 2014, 10, 252–267. (69) Lee, T. J.; Taylor, P. R. A diagnostic for determining the quality of single-reference electron correlation methods. Int. J. Quant. Chem. 1989, 36, 199–207. (70) Karton, A.; Rabinovich, E.; Martin, J. M. L.; Ruscic, B. W4 theory for computational thermochemistry: In persuit of confident sub-kJ/mol predictions. J. Chem. Phys. 2006, 125, 144108. (71) Karton, A.; Daon, S.; Martin, J. M. L. W4-11: A high-confidence benchmark dataset for computational thermochemistry derived from first-principles W4 data. Chem. Phys. Lett. 2011, 510, 165–178. (72) Jiang, W.; DeYonker, N. J.; Wilson, A. K. Multireference character for 3d transitionmetal-containing molecules. J. Chem. Theory Comput. 2012, 8, 460–468. (73) Shepler, B. C.; Balabanov, N. B.; Peterson, K. A. Ab initio thermochemistry involving heavy atoms: an investigation of the reactions Hg + IX (X = I, Br, Cl, O). J. Phys. Chem. A 2005, 109, 10363–10372. (74) van Lenthe, E.; Snijders, J. G.; Baerends, E. J. The zero-order regular approximation for relativistic effects: the effect of spin–orbit coupling in closed shell molecules. J. Chem. Phys. 1996, 105, 6505–6516. (75) Noonan, K.; Melville, T. C.; Dickinson, C. S.; Coxon, J. A. High-resolution laser spectroscopy of YbI: the A2 Π ← X2 Σ+ Transition. J. Mol. Spectrosc. 2003, 222, 296– 298. (76) Sergeev, D. N.; Dunaev, A. M.; Butman, M. F.; Ivanov, D. A.; Kudin, L. S.; Krämer, K. W. Energy characteristics of molecules and ions of ytterbium iodides. Int. J. Mass Spectrom. 2014, 374, 1–3. 44


Page 44 of 48

Page 45 of 48

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(77) Sergeev, D. N.; Motalov, V. B.; Butman, M. F.; Kiselev, A. E.; Kudin, L. S.; Krämer, K. W. Atomization Energies of LnX molecules (Ln = Sm, Eu, and Yb; X = Cl, Br, and I). J. Chem. Eng. Data 2014, 59, 4010–4014. (78) Pedley, J. B.; Marshall, E. M. Thermochemical data for gaseous monoxides. J. Phys. Chem. Ref. Data 1983, 12, 967–1031. (79) Törring, T.; Zimmermann, K.; Hoeft, J. The microwave rotational spectrum of X2 Σ LaO. Chem. Phys. Lett. 1988, 151, 520–525. (80) Bernard, A.; Taher, F.; Topouzkhanian, A.; Wannous, G. Laser-excited fluorescence spectra of lanthanum monoxide. Astron. Astrophys. Suppl. Ser. 1999, 139, 163–166. (81) Hildenbrand, D. L.; Lau, K. H. Thermochemical properties of the gaseous scandium, yttrium, and lanthanum fluorides. J. Chem. Phys. 1995, 102, 3769–3775. (82) Rubinoff, D. S.; Evans, C. J.; Gerry, M. C. L. The pure rotational spectra of the lanthanum monohalides, LaF, LaCl, LaBr, LaI. J. Mol. Spectrosc. 2003, 218, 169– 179. (83) Bernard, A.; Effantin, C.; d’Incan, J.; Vergès, J. The (1)1 Π, (2)1 Σ+ → X1 Σ+ transitions of LaF. J. Mol. Spectrosc. 2000, 202, 163–165. (84) Xin, J.; Klynning, L. Fourier transform spectroscopy of LaCl: rotational analyses of the infrared band systems. Phys. Scripta 1994, 49, 209–213. (85) Effantin, C.; d’Incan, J.; Bernard, A.; Shenyavskaya, E. A.; Topouzkhanian, A.; Wannous, G. Electronic States of Lanthanum Monoiodide: Preliminary Data from HighResolution LIF Spectroscopy. J. Mol. Spectrosc. 1998, 192, 394–398. (86) McDonald, S. A. Ph.D. Thesis, M.I.T., USA, 1985; cited in Ref. 100.

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 46 of 48

(87) McDonald, S. A.; Rice, S. F.; Field, R. W.; Linton, C. Laser spectroscopy of low-lying excited states in YbO: Linkage of the Yb2+ f13 s and f14 configurations. J. Chem. Phys. 1990, 93, 7676–7686. (88) Kleinschmidt, P. D.; Lau, K. H.; Hildenbrand, D. L. Thermochemistry of the gaseous fluorides of samarium, europium, and thulium. J. Chem. Phys. 1981, 74, 653–660. (89) Dmitriev, Y. N.;

Kaledin, L. A.;

Kobyliansky, A. I.;

Kulikov, A. N.;

Shenyavskaya, E. A.; Gurvich, L. V. Electronic spectra of diatomic molecules containing f-elements: GdO, EuF and UO. Acta Phys. Hung. 1987, 61, 51–54. (90) Dickinson, C. S.; Coxon, J. A.; Walker, N. R.; Gerry, M. C. L. Fourier transform microwave spectroscopy of the 2 Σ+ ground states of YbX (X = F, Cl, Br): Characterization of hyperfine effects and determination of the molecular geometries. J. Chem. Phys. 2001, 115, 6979–6989. (91) Dunfield, K. L.; Linton, C.; Clarke, T. E.; McBride, J.; Adam, A. G.; Peers, J. R. D. Laser Spectroscopy of the Lanthanide Monofluorides: Analysis of the A2 Π – X2 Σ+ Transition of Ytterbium Monofluoride. J. Mol. Spectrosc. 1995, 174, 433–445. (92) Uttam, K. N.; Gopal, R.; Joshi, M. M. New Band Systems of the YbCl Molecule. J. Mol. Spectrosc. 1997, 185, 8–14. (93) Kramer, J. Spectroscopic characterization of ytterbium monohalide emission in a high pressure electrodeless microwave arc discharge. J. Chem. Phys. 1978, 68, 5370–5377. (94) Dickinson, C. S.; Coxon, J. A. High-resolution laser spectroscopy of YbBr. Part I: the B2 Σ+ ← X2 Σ+ system. J. Mol. Spectrosc. 2004, 224, 18–26. (95) Cooke, S. A.; Krumrey, C.; Gerry, M. C. L. Rotational spectroscopy of lutetium monoxide, LuO. J. Mol. Spectrosc. 2011, 267, 108–111.

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Page 47 of 48

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(96) Bernard, A.; Effantin, C. Further results relating to the electronic spectrum of 175

Lu16 O. Can. J. Phys. 1986, 64, 246–251.

(97) Cooke, S. A.; Krumrey, C.; Gerry, M. C. L. Pure rotational spectra of LuF and LuCl. Phys. Chem. Chem. Phys. 2005, 7, 2570–2578. (98) Effantin, C.; Wannous, G.; d’Incan, J.; Athenour, C. Rotational analysis of selected bands from the electronic spectrum of the LuF molecule. Can. J. Phys. 1976, 54, 279–294. (99) Kramer, J. The detection and characterization of the visible emission spectra of lutetium monochloride, monobromide, and monoiodide. J. Chem. Phys. 1978, 69, 2601–2608. (100) Dolg, M.; Stoll, H.; Preuss, H. Ab initio pseudopotential study of europium monoxide EuO: 8 Σ− ground state and 8 Σ− first excited state. Chem. Phys. 1990, 148, 219–227.

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Toward Chemical Accuracy in ab Initio Thermochemistry and

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