Actinide chemistry using singlet-paired coupled cluster and its combinations with density functionals

Transcription

1 Atinide hemistry using singlet-paired oupled luster and its ombinations with density funtionals Alejandro J. Garza and Ana G. Sousa Alenar Department of Chemistry, Rie University, Houston, Texas, , USA Gustavo E. Suseria Department of Chemistry and Department of Physis and Astronomy, Rie University, Houston, Texas, , USA (Dated: Deember 3, 015) Singlet-paired oupled luster doubles (CCD0) is a simplifiation of CCD that relinquishes a fration of dynami orrelation in order to be able to desribe stati orrelation. Combinations of CCD0 with density funtionals that reover speifially the dynami orrelation missing in the former have also been developed reently. Here, we assess the auray of CCD0 and CCD0+DFT (and variants of these using Bruekner orbitals) as ompared to well-established quantum hemial methods for desribing ground-state properties of singlet atinide moleules. The f 0 atinyl series (UO +, NpO3+, PuO4+ ), the isoeletroni NUN, and Thorium (ThO, ThO+ ) and Nobelium (NoO, NoO ) oxides are studied. I. INTRODUCTION Atinide hemistry represents a hallenge for experimental approahes due to the high toxiity and radioativity of atinide ompounds. Aurate omputational models are therefore partiularly valuable in this area of hemistry. An example of this was the theoretial predition of NUO + [1] and its subsequent disovery by mass spetrosopy []. However, atinide hemistry is also hallenging for ommon quantum hemial approximations: The presene of multiple degenerate, partially filled f orbitals leads to substantial stati orrelation. Typial tehniques for handling stati orrelation have severe limitations suh as lak of size-onsisteny and size-extensivity and, most restritively, a ombinatorial inrease in omputational ost with system size [3]. Furthermore, many of these approahes may miss important dynami orrelation. Other popular methods of quantum hemistry suh as Kohn Sham density funtional theory (KS-DFT) or single-referene oupled luster (CC) are unreliable when stati orrelation is present [4] (ommon CC methods may even diverge or yield omplex orrelation energies). Reent advanes in omputational atinide hemistry have been reviewed in Refs. [5, 6]. Reently, tehniques that modify the luster operator of CC doubles (CCD) in order to desribe stati orrelation at the ost of negleting some dynami orrelation have been proposed in the literature. These inlude pair CCD [7 1] (pccd), as well as singlet-paired CCD and Bruekner doubles [4, 13] (CCD0 and BD0, respetively). All of these methods have polynomial saling and are size-onsistent and size-extensive (provided that the referene determinant be size-onsistent and size-extensive, whih may demand symmetry breaking [14]). Furthermore, approahes to inorporate the dynami orrelation absent in CCD0 and BD0 via density funtionals (CCD0+DFT) have been developed [13]. A reent study [15] using pccd suggests this new type of CC ansätze to be promising for appliations in atinide hemistry. Here, we assess the auray of CCD0 and CCD0+DFT (and their BD0 variants) as ompared to well-established quantum hemial methods for desribing ground-state properties of singlet atinide moleules. The f 0 atinyl series (UO +, NpO+, PuO+ ), the isoeletroni NUN, and Thorium (ThO, ThO + ) and Nobelium (NoO, NoO ) oxides are studied. II. THEORY AND METHODS A. Singlet-Paired Coupled luster Doubles (CCD0) We give here a minimal desription of CCD0; for further details see Ref. [4]. Like standard CC methods, CCD0 uses an exponential wavefuntion [4] Ψ CCD0 = e T [0] ΦRHF, (1) where Φ RHF is a restrited Hartree Fok determinant and T [0] is the luster operator of singlet-paired double exitations: T [0] = 1 ijab σ ab ij P ab P ij, () where ij and ab are indies for oupied and virtual orbitals, respetively; the amplitudes obey the relation = (ta b )/; and σ ab ij i j + tb a i j P ij = 1 ( j i + i j ). (3) That is, P ab ating on Φ RHF gives a singlet and e T [0] ontains ontributions from all singlet-paired exitations. In standard CCD, the luster operator ontains singletand triplet-paired ontributions, hene apturing more

2 orrelation. However, it is the ombination of the singletand triplet-paired omponents that auses the failure of CCD (and CCSD, CCSDT, et.) in strongly orrelated systems, whih results in unphysial orrelation energies [4] (i.e., too large, divergent, or omplex energies). Thus, CCD0 relinquishes a fration of the orrelation in exhange for safeguard against this breakdown. Singlet-paired Bruekner doubles (BD0) is analogous to CCD0, the only differene being that approximate Bruekner [16 18], rather than RHF, orbitals are used as referene: Ψ BD0 = e T [0] ΦBD. (4) Speifially, the approximate Bruekner orbitals in Φ BD are those whih zero out the amplitudes of single substitutions in a model subspae of single and double substitutions (just like in standard Bruekner doubles). B. Combination with Density Funtionals (CCD0+DFT) The different flavors of CCD0+DFT are disussed in detail in Ref. [13]; we just provide here a brief explanation of these tehniques for the sake of larity. There are two ategories of CCD0+DFT methods: one that adds parallel spin orrelation to CCD0, and another that adds the full ontributions from triplet-paired exitations. The first one is derived by noting that the CCD0 orrelation energy is E CCD0 = ijab σ ab ij v a b i j, (5) where vij ab = ij ab is a two-eletron integral in the Dira notation. Hene, there are no ontributions to the orrelation from pairs of same-spin eletrons. One an thus add (without double ounting) the equal spin orrelation to CCD0 using a density funtional approximation (DFA). For a singlet, the orrelation energy would be E CCD0+pDFT = E CCD0 + E DFA [n, n ], (6) where the p in pdft is for parallel-spin and E DFA αα is the DFA orrelation for the spin-α density. The seond ategory of CCD0+DFT is derived by noting that the full double exitations luster operator T used in the latter an be expressed as T = T [0] + T [1], (7) where T [0] is defined above and T [1] is the triplet-paired omponent of T T [1] = 1 ijab π ab ij Q ab Q ij (8) where Q ij is a vetor Q ij = (Q + ij, Q0 ij, Q ij )t whose omponents are Q + ij = j i, Q ij = j i, (9) Q 0 ij = 1 ( j i i j ) (10) = 1 ( j i + j i ). Thus, CCD0 misses not only parallel spin orrelation, but also the m = 0 hannel of T [1]. For a losed shell, the m = +1, 0, and 1 omponents of of T [1] ontribute equally to the orrelation. We may therefore inorporate the opposite spin orrelation missing in CCD0+pDFT by adding E DFA [n, 0] one more to the (losed-shell) energy E CCD0+tDFT = E CCD0 + 3E DFA [n, n ], (11) where the t in tdft indiates that the full ontributions from the triplet-paired omponent of T are being taken into aount. In the interest of simpliity and intuitiveness, The derivation of CCD0+DFT given here is somewhat heuristi. However, as shown in the Appendix of Ref. [13], it is also possible too derive CCD0+DFT formally using Levy Lieb-type [19, 0] onstrained searh funtionals. The basi idea is that one an divide the eletron-eletron interation in omplementary singletand triplet-pairing omponents. There is then a wavefuntion that solves the Hamiltonian with the singletpairing interation only, and a omplementary density funtional for triplet-pairing interation. This funtional an be written in terms of the parallel spin orrelation of the usual Levy Lieb universal funtional, plus a term orresponding to the m = 0 part of the triplet-pairing ontribution to the energy. If one neglets this last term, a method analogous to CCD0+pDFT is obtained; if it is approximated to be equal to half the ontribution of the parallel spins (whih is reasonable due to the symmetry of the triplet-pairing interation) a method analogous to CCD0+tDFT is obtained. The additional assumptions done in CCD0+DFT are that the wavefuntion that solves for the singlet-pairing interating Hamiltonian is well approximated by CCD0, that the density of the referene determinant is aurate, and that the parallel spin orrelation of an approximate meta-ggas is also aurate. For details, the reader is referred to the Appendix of Ref. [13]; the main purpose of ommenting on this here is to point out that CCD0+DFT an be onsidered as a rigorous way to inorporate dynami orrelation in CCD0 while avoiding double ounting. C. Parallel Spin Funtionals for CCD0+DFT The CCD0+DFT methods desribed above require a spin resolution for DFA orrelation in order to approximate E DFA [n, n ]. For ompleteness, we disuss this

3 3 topi briefly here. The simplest spin resolution for the DFA orrelation is the exhange-like ansatz of Stoll et al. [1] E DFA [n, n ] = E DFA [n, 0]. (1) This equation an be used for the loal density approximation (LDA), generalized gradient approximations (GGAs), and meta-ggas that are rooted on the homogeneous eletron gas. (Not all DFAs have a meaningful spin resolution. The Lee Yang Parr [] funtional, for example, models all orrelation as being opposite spin.) The use of meta-ggas is most desirable beause these funtionals are free of one-eletron self-interation, and thus BD0+DFT with a meta-gga is exat for twoeletron singlets using the spin resolution of Eq. 1. Here, we use two nonempirial meta-ggas in ombination with CCD0 and BD0: the Tao Perdew Staroverov Suseria [3] (TPSS) funtional, and the strongly onstrained and appropriately normed (SCAN) funtional of Sun et al. [4]. Equation 1 is an eduated guess whih is exat only for fully spin-polarized densities and in the high-density limit of the uniform eletron gas [5]. In the ase of SCAN, however, it is possible to ompose an improved guess for the same-spin orrelation [13]. This is beause SCAN onstruts the orrelation energy density ε as [4] ε = ε 1 + f (α) [ ε 0 ε 1 ], (13) where f (α) is a funtion that depends on the kineti energy density (see Supporting Information of Ref. [4]), and ε α=0 and ε α=1 are the single orbital and uniform density limits, respetively, for the orrelation energy density. It is thus logial to write the spin-up orrelation energy density for SCAN as ε = ε 1 + f (α) [ ε 0 ε 1 ]. (14) Furthermore, ε 0 = 0 beause there is no parallel-spin orrelation for two eletrons in the same spatial orbital. Hene, ε = ε 1 f (α)ε 1 (15) so that ε depends only on the uniform density limit of the spin-up orrelation energy density, ε 1. The spin resolution of ε in the uniform eletron gas has been parametrized by Gori-Giorgi and Perdew [5] in terms for frations frations F σσ suh that ε σσ = ε F σσ. Thus, we ompute ε 1 = ε F using the expression for F given in Equation 9 of Ref. [5]. We term the variation of SCAN using this spin resolution as rscan. For onveniene of the reader, Table I summarizes the CCD0+DFT methods desribed and employed in this work. Further details regarding CCD0+DFT, inluding disussion about the spin resolutions, may be onsulted in Ref. [13]. TABLE I: Summary of CC0+DFT methods employed here. The notation and losed-shell energy formulas are given; CC0 an refer to CCD0 or BD0 and the densities are from the RHF or Bruekner referene determinants, respetively; the p in pdft is for parallel spin; the t in tdft is for triplet-pairing omponent; DFT may refer to TPSS or SCAN; and E rscan is the spin-up SCAN orrelation using the spin resolution of Setion II.C. Method CC0+pDFT CC0+tDFT CC0+prSCAN CC0+trSCAN Energy Formula D. Computational Details E CC0 + E DFA [n, 0] E CC0 + 3E DFA [n, 0] E CC0 + E rscan [n, n ] E CC0 + 3E rscan [n, n ] All alulations were arried out using a development version of Gaussian [6] in whih the CCD0 and CCD0+DFT methods have been implemented. As in Ref. [13], CCD0+DFT alulations are done in a nonself-onsistent manner: the DFA orrelation is evaluated in a single-shot, post-ccd0 alulation with the densities from the referene determinant. CCD0+DFT geometry optimizations and harmoni vibrational frequenies were omputed numerially using a onvergene threshold of Hartrees on the CCD0 energy and the largest of the preset grids in Gaussian for integrating the density funtional (Integral=SuperFine keyword). CCD0 geometries and frequenies were evaluated analytially, and we verified that the numerial proedure for determining these properties with CCD0+DFT agreed with analytial results for CCD0. These same speifiations were followed in BD0 and BD0+DFT alulations. Unless otherwise indiated, all alulations employ the Stuttgart relativisti small-ore effetive ore potential [7 9] (RSC ECP) basis for atinide atoms, and the aug--pvdz basis for the light atoms. These basis sets have been shown to be adequate for the type of alulations arried out here [30]. Spin-orbit oupling effets are negleted as they are not important for the losed-shell speies studied in this work [6, 31]. The results reported here are all in gas phase media. III. RESULTS AND DISCUSSION A. Uranyl Cation (UO + ) The uranyl ion UO + is onsidered the most important of the atinyls due to its ubiquity: In pratial appliations, nulear reators usually rely on uranium to fuel nulear hain reations, and UO + is the most ommon form of uranium in aqueous solution. UO + is highly toxi and its study is motivated by the need for

4 4 TABLE II: Bond lengths (R e in Å) and harmoni vibrational frequenies (ω e in m 1 ) for UO + (D h ) alulated by various methods. Results omputed in this work appear in the top part of the table; results ompiled from the literature at the bottom. To the best of our knowledge, no experimental data is available for this speies. This Work CCD CCD0+pTPSS CCD0+pSCAN CCD0+prSCAN CCD0+trSCAN BD BD0+pTPSS BD0+pSCAN BD0+prSCAN BD0+trSCAN CCSD CCSD(T) HF PBE PBEh LC-ωPBE Literature pccd a CCSD(T) b MP a MP CASSCF(10,10) a CASSCF(1,1) a CAS-srPBE(1,1) d CAS-srLDA(1,1) d CASPT(1,1) e CASPT(1,1) f a From Ref. [15] using the -pvdz basis on O. b From Ref. [33]; RSC+3g on U and aug--pvqz on O. From Ref. [34]; RSC+g on U and aug--pvdz on O. d From Ref. [31]; RECP (14s13p10d8f6g)/[6s6p5d4f3g] on U and (4s5p1d)/[s3p1d] on O. e From Ref. [35]; RSC on U and 4s3pd ANO-S on O. f From Ref. [35]; RSC on U and 4s3pd1f ANO-L on O. knowledge regarding soluble atinide omplexes, whih are important for nulear waste disposal and environmental transport [6]. The bare UO + ion has also been observed experimentally via mass-spetrometri tehniques [3]. The uranyl ation has therefore been studied extensively [6, 30, 31, 33 35]. Here, we study this speies in the gas phase due to the availability of data from highlevel alulations to ompare with, as the auray of CCD0 and CCD0+DFT for atinide ompounds has not yet been established. To the best of out knowledge, there is no experimental data for the properties of UO + here alulated. Table II ompiles preditions by various methods for the bond lengths (R e ) and harmoni vibrational frequenies (ω e ) of UO +. Some of these data have been taken from the literature and not all alulations use the same basis set; however, the results should be roughly omparable beause the bases are all of similar, good quality. The highest level methods in this Table are CCSD(T) and CASPT(1,1); CCSD(T) results are onsidered to be reliable for uranyl [6, 33, 34]. CCD0 is in good agreement with CCSD(T) exept for a large overestimation of the bending frequeny ω β. This problem persists in CCD0+pTPSS, but is alleviated by ombinations of CCD0 with SCAN. Compared to CCD0, BD0 provides a muh better estimate of ω β. Whereas BD0+pTPSS tends to give a too short R e and too large frequenies, BD0+SCAN methods are, overall, in exellent agreement with CCSD(T). It is worth noting that the trends for the UO + frequenies are the same as those observed for a set of ten first- and seond-row diatomis in Ref. [13]: CCD0+pTPSS overestimates the frequenies, while CCD0+SCAN ombination improve results. The trend is similar for BD0+DFT methods, whih are more aurate than their CCD0+DFT ounterparts. CCD0, BD0, and their ombinations with DFT (in partiular those with SCAN) fare well against other methods. Results from HF and KS-DFT methods in Table II suggest that the desription of UO + is dependent on the amount of HF exhange in the funtional: more HF exhange leads to shorter bond lengths and higher frequenies. This dependene an make ommon hybrids unreliable for high auray work. The ousin of CCD0, pccd, underestimates R e more than all other methods exept HF. In pccd, a singlet pairing sheme is also employed, but only the diagonal (optimized) terms are retained. A better performane of CCD0 as ompared to pccd ould be expeted beause the former ontains more ontributions in the luster operator: the T operator of pccd is T pccd = ia t a i a a i i, (16) whih is only a part of the T [0] of CCD0. Although pccd normally ompensates for the missing terms via an orbital optimization, this optimization is nontrivial and an have multiple solutions. Nonetheless, pccd has the advantage of having lower saling. The ost of CCD0 is determined by the ost of solving the CCD0 equations with symmetrized amplitudes. Thus, the saling of CCD0 is the same as that of CCD, O(N 6 ). For pccd, the ost of solving the pertinent CC equations is a remarkably low O(N 3 ), although a O(N 5 ) basis transformation transformation is required for the orbital optimization (and this is important for ahieving good results and ensuring size onsisteny). Compared to the traditional, goldstandard CCSD(T), CCD0 and CCD0+DFT are an order of magnitude lower in ost, while providing similar results and being more reliable for stati orrelation [4]. Table II also shows CAS-srDFT results from Ref. [31].

5 5 TABLE III: Bond lengths (R e in Å) and harmoni vibrational frequenies (ω e in m 1 ) for the neptunyl ion NpO 3+ (D h ) alulated by various methods. To the best of our knowledge, no experimental data is available for this speies. This Work CCD CCD0+pTPSS CCD0+pSCAN CCD0+prSCAN CCD0+trSCAN BD BD0+pTPSS BD0+pSCAN BD0+prSCAN BD0+trSCAN Literature CCSD(T) a MP a CASSCF(1,1) a CASSCF(1,16) a a From Ref. [34]; RSC+g on Np and aug--pvdz on O. These methods belong to a lass of tehniques that omplement long-range wavefuntion two-body energy with short-range DFT Hartree exhange orrelation. The idea is to apture the dynami orrelation, whih is shortrange, with DFT and avoid double ounting by range separation; an approah to avoid double ounting that is very different from that used in CCD0+DFT. The CAS-srDFT bond lengths are omparable to those of CCD0+DFT, although the latter has the advantage that it does not neglet effets of stati orrelation in the short-range. CASSCF and CASPT results are also omparable to those from CCD0/BD0 (and their +DFT ombinations), whereas MP gives too large bond lengths. B. Neptunyl and Plutonyl (NpO 3+ and PuO 4+ ) The inreased harge in neptunyl and plutonyl (NpO 3+ and PuO 4+ ) as ompared to the isoeletroni uranyl enhanes degeneraies, bolstering stati orrelation. This makes these ions more hallenging to desribe than UO +. In fat, CCSD(T) results reported in the literature for these systems ome with a warning: The T 1 diagnostis for neptunyl and plutonyl are 0. and 0.35, respetively [34]. Empirially, CCSD(T) preditions are onsidered unreliable when T 1 > 0. [36] (although the norm of T is probably a more reliable and better indiator of stati orrelation [4], we did not find these data in the literature). Table III shows R e and ω e results for NpO 3+. The methods for whih results are shown in this table all predit a linear geometry for neptunyl. The predition of a linear geometry an be onsidered a suess for CCD0+DFT methods beause it has been shown that a orret desription of both exhange and orrelation is neessary for this [31]. Other MR+DFT that avoid double ounting typially do so at the ost of introduing some (inexat) DFT exhange. Popular KS-DFT funtionals like LDA, PBE, and B3LYP predit bent geometries for neptunyl [31]. As a result, MR+DFT ombinations having substantial DFT exhange also tend toward bent geometries [31]. We also note that CCSD(T) gives results omparable to CASSCF, BD0, and BD0+DFT. Considering this and the fat that the T 1 diagnosti of CCSD(T) [34] (T 1 = 0.) is lose to the limit of what is onsidered reliable [36] (0.0), it seems like the CCSD(T) results are salvageable for NpO 3+. MP is not reliable for the f 0 atinyl series isoeletroni to uranyl beyond uranyl itself; the bond lengths in neptunyl appear to be largely overestimated, while the frequenies are underestimated. In the ase of PuO 4+, the CCD0 and BD0 geometry optimizations resulted in linear strutures but with imaginary bending frequenies of 149 and 7 m 1, respetively. Attempts to optimize the struture starting from a bond angle of about 160 (a minimum at a fixed bond length) resulted in the geometry going bak to linear with an extended bond length without ahieving onvergene after many iterations. Straka et al. [34] reported that CAS(1,16) predits the PuO 4+ system to disintegrate and that CCSD(T) results are highly unreliable (T 1 = 0.35). Plutonium has four ommon oxidation states Pu(III), Pu(IV), Pu(V), and Pu(VI); the highest known oxidation state of Pu in aqueous solution is VII and is only stable in strong alkaline medium. However, the formal oxidation state of plutonium in PuO 4+ is VIII. The too-high harge of Pu in PuO 4+ brings the atinide s lowlying f orbitals loser to the atom and makes them less suitable for bonding. Considering all this, it seems very likely that PuO 4+ is not a stable gas phase speies. Nevertheless, Tsushima [37] has theorized a possible synthesis for PuO 5 OH 3 in strong alkaline solution, and Huang et al. [38] have reently reported theoretial evidene for the stability of a PuO F 4 omplex. C. Uranium Dinitride (NUN) The linear NUN moleule has been studied theoretially and and observed experimentally [35, 39 4]. The interest in this ompound stems mostly from its similarity to the important uranyl ion, to whih it is isoeletroni. Results for R e and ω e omputed by various methods here and in previous works are listed on Table IV. General trends are similar to those observed for uranyl: CCD0 and CCD0+DFT methods are in good agreement with available CASPT and CCSD(T) methods, although the CASPT vibrational frequenies are somewhat lower than those predited by oupled lus-

6 6 TABLE IV: Bond lengths (R e in Å) and harmoni vibrational frequenies (ω e in m 1 ) for the NUN moleule (D h ) alulated by various methods. This Work CCD CCD0+pTPSS CCD0+pSCAN CCD0+prSCAN CCD0+trSCAN BD BD0+pTPSS BD0+pSCAN BD0+prSCAN BD0+trSCAN CCSD PBE i PBEh LC-ωPBE Literature HF a CCSD(T) a 1.7 MP a 1.71 CAS-srLDA a CAS-srPBE a CASPT(1,1) b Expt Expt. d Expt. e 1051 b From Ref. [31]; RECP (14s13p10d8f6g)/[6s6p5d4f3g] on U and (4s5p1d)/[s3p1d] on O. b From Ref. [35]; RSC on U and 4s3pd1f ANO-L on O. From Ref. [39]. d From Ref. [40]. d From Ref. [41]. ter methods. Results by KS-DFT methods depend on the amount of exhange, with the bond length being redued as more exhange is inorporated. In addition, PBEh and LC-ωPBE yield a linear geometry but PBE shows an imaginary bending frequeny for the linear struture. Again, HF gives bond lengths that are tooshort, although MP appears to give more reasonable bond lengths for this speies. D. Thorium Oxides ThO and ThO + ThO is the most studied (experimentally and theoretially) of the atinide monoxides [6]. ThO + is also of interest beause Th(IV) is the most ommon oxidation state for thorium. The bonding in ThO + is furthermore peuliar: it forms a triple bond and leaves the lone-pair orbital on the thorium empty (see Fig. 1), as ours in the FIG. 1: The highest three oupied and lowest unoupied (Hartree Fok) moleular orbitals of ThO +. TABLE V: Bond lengths (R e in Å) and harmoni vibrational frequenies (ω e in m 1 ) for ThO and ThO + alulated by various methods. ThO ThO + Method R e ω e R e ω e CCD CCD0+pTPSS CCD0+pSCAN CCD0+prSCAN CCD0+trSCAN BD BD0+pTPSS BD0+pSCAN BD0+prSCAN BD0+trSCAN CCSD PBE PBEh LC-ωPBE CASPT a Expt. b a From Refs. [6, 45]; using all-eletron basis set. b From Refs. [6, 46 48] isovalent TiC [43]. Interest in thorium hemistry arises mainly from potential appliations of the thorium fuel yle the transmutation of the abundant 3 Th into artifiial 33 U, whih is the atual fuel in the nulear hain reation. In fat, thorium has been touted as potential wonder fuel [44] due to ertain advantages over uranium like greater abundane and better resistane to nulear weapons proliferation, albeit the latter advantage has been ontended [44]. Table V ompares the R e and ω e for ThO and ThO +

7 7 TABLE VI: Bond lengths (R e in Å) and harmoni vibrational frequenies (ω e in m 1 ) for NoO and NoO alulated by various methods. For methods that predit geometries that deviate from linearity in NoO, the bond angle is shown in parenthesis. NoO NoO Method R e R e ω as ω s ω β BD BD0+pTPSS BD0+pSCAN BD0+prSCAN BD0+trSCAN CCSD CCSD(T) PBE (15 ) PBEh (19 ) LC-ωPBE (131 ) obtained by various methods. The data from CCD0, BD0, and their ombinations with DFT are all very lose to available CASPT and experimental data from the literature. One more, ombinations using the SCAN funtional tend to be loser to the referene CASPT values, though all the ombinations presented here provide satisfatory results. The trend of DFT methods to shorten the bond length as more Fok exhange is inorporated that was observed in the previous ases is also present here. E. Nobelium Oxides NoO and NoO The hemistry of Nobelium is largely unharaterized, whih offers a possibility for theoretial methods to provide unique insight into it. Table VI shows bond lengths and harmoni frequenies for NoO and NoO omputed by BD0, BD0+DFT, and some standard oupled luster and KS-DFT methods. The ω e for NoO is omitted beause the floppy (long and rather weak) bond of this moleule lead to somewhat large errors in the fitting proedure to obtain the fore onstants; however, they are onsistently estimated to be around 550 m 1. Likewise, CCSD(T) data for NoO is absent due to the diffiulty in onverging these alulations. To the best of our knowledge, there are no aurate referene data for these ompounds in the literature. Ref. [6] reports, based on B3LYP alulations, that the ground states of NoO and NoO are singlet, and we arried out the alulations for singlet states only beause of this and the urrent limitations of applying CCD0 to open shell systems. The most notieable feature of Table VI is that BD0, BD0+DFT, and CCSD predit a linear geometry for NoO, whereas KS-DFT methods yield bent geometries. The bonds in the nobelium ompounds are onsiderably larger than for the f 0 atinide oxides beause the 5f orbitals are ompletely filled in the former. The effet of dynami orrelation is also muh larger: addition of DFT orrelation to BD0 leads to derease in bond lengths of about Å, ompared to hanges the more modest hanges (rarely more than 0.0 Å) observed above. Based on the results for the previous atinide ompounds, we ould expet BD0+pSCAN and BD0+prSCAN to provide aurate estimates for the bond lengths and frequenies for the nobelium oxides. In the ase of NoO, CCSD(T), BD0+pSCAN, and BD0+prSCAN results are highly similar. IV. CONCLUSIONS AND OUTLOOK Singlet-paired oupled luster and its ombinations with DFT an provide aurate estimates for properties of atinide ompounds suh as geometries and vibrational frequenies. Compared to the data from the most aurate estimates available (CASPT, CCSD(T), experiment), typial deviations of CCD0 and CCD0+DFT methods are around 0.01 Å for bond lengths and 0 m 1 for harmoni vibrational frequenies. These deviations are similar to the ones omputed for simple first- and seond-row diatomis in a previous work [13], indiating the wide appliability and generality of the approah. The CCD0 and CCD0+DFT results presented here reinfore preditions by other methods (e.g., CCSD(T) or CASSCF) for speies for whih no experimental data are available (e.g., the important UO + ation), inluding the instability of the PuO 4+ ion. The BD0 and BD0+DFT results for NoO and NoO are probably the most reliable estimates available so far, as previous reports (see Ref. [6]) utilized KS-DFT funtionals whih, aording also to the results here, are not onsistently reliable for atinide ompounds. For most of the moleules studied here, CCSD(T) appears to provide reliable results. CCD0+DFT an provide data of similar quality to CCSD(T) while being a order in magnitude lower in ost and muh more robust for handling stati orrelation. The CCD0 and CCD0+DFT methods are seen to be more aurate than pccd for UO +. Beause pccd has lower ost than CCD0 and misses mostly dynami orrelation, the results here suggest that pccd+dft ombinations analogous to CCD0+DFT (suh as those suggested in Refs. [3, 13, 49]) may be very promising for atinide hemistry. Currently, we are working on some of these pccd+dft ombinations and on extensions of CCD0 for treating open-shells, whih would greatly inrease the appliability of CCD0- based tehniques to atinide hemistry. Aknowledgments This work was supported by the U.S. Department of Energy, Offie of Siene, Offie of Basi Energy Sienes, Heavy Element Chemistry Program under Award Num-

8 8 ber DE-FG0-04ER1553. G.E.S. is a Welh Foundation Chair (C-0036). [1] P. Pyykkö, J.Li, and N. Runeberg, J. Phys. Chem. 98, 4809 (1994). [] C. Heinemann and H. Shwarz, Chem. Eur. J. 1, 7 (1995). [3] A. J. Garza, T. M. Henderson, I. W. Bulik, and G. E. Suseria, Phys. Chem. Chem. Phys. 17, 41 (015). [4] I. W. Bulik, T. M. Henderson, and G. E. Suseria, J. Chem. Theory and Comput. 11, 3171 (015). [5] D. Wang, W. F. van Gunsterenb, and Z. Chai, Chem. So. Rev. 41, 5836 (01). [6] A. Kovás, R. J. M. Konings, J. K. Gibson, I. Infante, and L. Gagliardi, Chem. Rev. 115, 175 (015). [7] P. A. Limaher, P. W. Ayers, P. A. Johnson, S. de Baerdemaker, D. van Nek, and P. Bultink, J. Chem. Theory Comput. 9, 1394 (013). [8] P. A. Limaher, T. D. Kim, P. W. Ayers, P. A. Johnson, S. de Baerdemaker, D. van Nek, and P. Bultink, Mol. Phys. 11, 853 (014). [9] P. Temer, K. Boguslawski, P. A. Johnson, P. A. Limaher, M. Chan, T. Verstraelen, and P. W. Ayers, J. Phys. Chem. A 118, 9058 (014). [10] K. Boguslawski, P. Temer, P. W. Ayers, P. Bultink, S. de Baerdemaker, and D. van Nek, Phys. Rev. B 89, 01106(R) (014). [11] T. Stein, T. M. Henderson, and G. E. Suseria, J. Chem. Phys. 140, (014). [1] T. M. Henderson, I. W. Bulik, T. Stein, and G. E. Suseria, J. Chem. Phys. 141, (014). [13] A. J. Garza, I. W. Bulik, A. G. Sousa Alenar, J. Sun, J. P. Perdew, G. E. Suseria, Mol. Phys. DOI: / [14] C. A. Jimenez-Hoyos, T. M. Henderson, and G. E. Suseria, J. Chem. Theory Comput. 7, 667 (011). [15] P. Temer, K. Boguslawski, and Paul W. Ayers, Phys. Chem. Chem. Phys. 7, 1447 (015). [16] C. E. Dykstra, Chem. Phys. Lett. 45, 466 (1977). [17] N. C. Handy, J. A. Pople, M. Head-Gordon, K. Raghavahari, and G. W. Truks, Chem. Phys. Lett. 164, 185 (1989). [18] R. Kobayashi, N. C. Handy, R. D. Amos, G. W. Truks, M. J. Frish, and J. A. Pople, J. Chem. Phys. 95, 673 (1991). [19] M. Levy., Pro. Natl. Aad. Si. U.S.A. 76, 606 (1979). [0] E. H. Lieb, Int. J. Quantum Chem. 4, 43 (1983). [1] H. Stoll, C. E. Pavlidou, and H. Preuss, Theor. Chim. Ata 49, 143 (1978). [] C. Lee, W. Yang, and R. G. Parr, Phys. Rev. B 37, 785 (1988). [3] J. Tao, J. P. Perdew, V. N. Staroverov, and G. E. Suseria, Phys. Rev. Lett. 91, (003). [4] J. Sun, A. Ruzsinszky, and J. P. Perdew, Phys. Rev. Lett. 115, (015). [5] P. Gori-Giorgi and J. P. Perdew, Phys. Rev. B 69, (R) (004). [6] M. J. Frish, G. W. Truks, H. B. Shlegel et al., Gaussian Development Version, Revision H.1, Gaussian In., Wallingford, CT, 009. [7] A. Bergner, M. Dolg, W. Kuehle, H. Stoll, and H. Preuss, Mol. Phys. 80, 1431 (1993). [8] M. Kaupp, P. v. R. Shleyer, H. Stoll, and H. Preuss, J. Chem. Phys. 94, 1360 (1991). [9] M. Dolg, H. Stoll, H. Preuss, and R.M. Pitzer, J. Phys. Chem. 97, 585 (1993). [30] Y.-K. Han and K. Hirao, J. Chem. Phys. 113, 7345 (000). [31] E. Fromager, F. Réal, P. Wåhlin, U. Wahlgren, and H. J. Aa. Jensen, J. Chem. Phys. 131, (009). [3] H. H. Cornehel, C. Heinemann, J. Marçalo, A. Pires de Matos, and H. Shwarz, Angew. Chem. Int. Ed. Engl. 1996, 35 (891). [33] V. E. Jakson, R. Craium, D. A. Dixon, K. A. Peterson, and W. A. de Jong, J. Phys. Chem. A 11, 4095 (008). [34] M. Straka, K. G. Dyall, and P. Pyykkö, Theor. Chem. A. 106, 393 (001). [35] L. Gagliardi and B. O. Roos, Chem. Phys. Lett. 331, 9 (000). [36] T. J. Lee and P. R. Taylor, Int. J. Quantum Chem., Quant. Chem. Symp., S3 199 (1989). [37] S. Tsushima, J. Phys. Chem. B 11, (008). [38] W. Huang, P. Pyykkö, and J. Li, Inorg. Chem. 54, 885 (015). [39] M. Zhou and L. Andrews, J. Chem. Phys. 111, (1996). [40] R. D. Hunt, J. T. Yustein, and L. Andrews, J. Chem. Phys. 98, 6070 (1993). [41] D. W. Green and G. T. Reedy, J. Chem. Phys. 7, 91 (1976). [4] L. Andrews, X. Wang, Y. Gong, G. P. Kushto, B. Vlaisavljevih, and L. Gagliardi, J. Phys. Chem. A 118, 589 (014). [43] P. Pyykkö, S. Riedel, and M. Patzshke, Chem. Eur. J. 11, 3511 (005). [44] S. F. Ashley, G. T. Parks, W. J. Nuttall, C. Boxall, and R. W. Grimes, Nature 49, 31 (01). [45] A. Kovás and R. J. M. Konings, J. Phys. Chem. A 115, 6646 (011). [46] C. T. Dewberry, K. C. Ethison, and S. A. Cooke, Phys. Chem. Chem. Phys. 9, 4895 (007). [47] G. Edvinsson, and A. Lagerqvist, Phys. Sr. 30, 309 (1984). [48] G. Edvinsson, L.-E. Selvin, and N. Aslund, Ark. Fys. 30, 8 (1965). [49] A. J. Garza, T. M. Henderson, I. W. Bulik, and G. E. Suseria, J. Chem. Phys. 14, (015).

9