How Much Can Density Functional Approximations (DFA) Fail? The

Article pubs.acs.org/JCTC

How Much Can Density Functional Approximations (DFA) Fail? The Extreme Case of the FeO4 Species Wei Huang, Deng-Hui Xing, Jun-Bo Lu, Bo Long, W. H. Eugen Schwarz,* and Jun Li* Department of Chemistry and Key Laboratory of Organic Optoelectronics & Molecular Engineering of the Ministry of Education, Tsinghua University, Beijing 100084, China S Supporting Information *

ABSTRACT: A thorough theoretical study of the relative energies of various molecular Fe·4O isomers with different oxidation states of both Fe and O atoms is presented, comparing simple Hartree− Fock through many Kohn−Sham approximations up to extended coupled cluster and DMRG multiconfiguration benchmark methods. The ground state of Fe·4O is a singlet, hexavalent iron(VI) complex 1C2v-[Fe(VI)O2]2+(O2)2−, with isomers of oxidation states Fe(II), Fe(III), Fe(IV), Fe(V), and Fe(VIII) all lying slightly higher within the range of 1 eV. The disputed existence of oxidation state Fe(VIII) is discussed for isolated FeO4 molecules. Density functional theory (DFT) at various DF approximation (DFA) levels of local and gradient approaches, Hartree−Fock exchange and meta hybrids, range dependent, DFT−D and DFT+U models do not perform better for the relative stabilities of the geometric and electronic Fe·4O isomers than within 1−5 eV. The Fe·4O isomeric species are an excellent testing and validation ground for the development of density functional and wave function methods for strongly correlated multireference states, which do not seem to always follow chemical intuition.

■

INTRODUCTION The common quantum chemical approaches are classified as wave function theory (WFT) or density functional theory (DFT). Systematically improvable WFT comprises single- and multireference methods, which emphasize only dynamic or also static correlation effects. The DFT originated from early quantum statistical efforts of Thomas, Fermi, Dirac and others in the late 1920s and got a new impetus through Hohenberg− Kohn−Sham’s theoretical grounding in the middle 1960s. In the Hohenberg−Kohn DFT, no difference between single- and multireference states is apparent. The Kohn−Sham approach promised all electronic systems to be easily simulated by independent particle approaches, where the Coulomb interaction of semiclassical continuous charge clouds is corrected by a so-called exchange-correlation potential of discrete fermions.1−3 The semiempirical step-by-step improvement of density functionals (DF) has been baptized as climbing the Jacob’s ladder to the heaven of cheap and accurate computational chemistry.4−6 The DF may be approximated by simple integral kernels depending on the local electronic spin density (local spin density approach, LSDA) or also on the density gradient (generalized gradient approach, GGA) or also on occupied orbital gradients (meta-GGA) or also on Hartree− Fock orbital-exchange potentials (hybrid DFs). There are additional approaches that are also dependent on dispersion corrections (DFT+D) or using different approximations for different spatial ranges (range dependent DFs) or adding simple local correlation corrections within the Hückel− © 2016 American Chemical Society

Hubbard framework (DFT+U) etc. ad infinitum. We will here apply many examples of those DFAs, although by far not all proposals in the literature (e.g., Becke’s recent proposal for a new strong-correlation functional) will be explored.7,8 Owing to the often quite reasonable accuracy and low computational costs, DFA has been widely applied in many research fields from molecular chemistry to solid-state physics and to qualitative applications in biology, materials, and engineering. DFA often works quite well for the rather systematic trends in organic and main group chemistry, although difficult nuts such as Be2, F2, O3, or C4H4 are not easily cracked. Owing to their extended d-s and f-d-s valence shells, many “outer” and “inner” transition elements pose harder problems of static-correlation multiconfigurational kind, in particular the compounds of 3d-, of early lanthanide and early actinide metals.9 Because main-group compounds are often experimentally better characterized and often theoretically better understood, they were often chosen as theory testing ground,1,6,10 flaunting accuracies down to 0.1 eV or a few kcal/mol per bond. However, in some specific cases (for example some boron clusters), the errors of the stabilities can sporadically reach 1 eV.11 Very recently, Truhlar’s group tested a series of main-group compounds including also delicate multireference systems with many of the more recent DF models such as M06, M06-2X, O3LYP, TPSS etc., obtaining Received: November 3, 2015 Published: March 3, 2016 1525

DOI: 10.1021/acs.jctc.5b01040 J. Chem. Theory Comput. 2016, 12, 1525−1533

Article

Journal of Chemical Theory and Computation

Figure 1. Stationary Fe·4O isomers: molecular structure, oxidation state of Fe (II−VIII), spin multiplicity (7−1), and spatial symmetry label according to B3LYP geometry optimization.

energies throughout better than 1/4 eV.6 In summary, modern DFA is often believed to achieve nearly chemical accuracy, although in special cases being still unreliable by up to 1 eV per bond or more. In particular, significant effort is still needed toward improving DFA for multiconfigurational states.12,13 We recently studied various Fe·4O structures when investigating maximum valences and oxidation states, applying also DFAs of various types. We were surprised by the more or less unexpected failures of all tested DFs, paralleled by failures of low-level single-reference WFT, the comparison with highquality multireference WFT indicating errors up to 5 eV for the relative binding energies of the various Fe·4O isomers. We hold it worth communicating these failures and suggest the “simple” Fe·4O molecular set of isomers as a crucial test field in future methodology evaluations.14,15 We note that tetrahedral FeO4 is the lighter homologue of rather unproblematic octavalent metal oxides RuO4 and OsO4,16 while isoelectronic with common but also problematic MnO4−.17

differences of 0−3 pm, which would result in energy changes of up to at most a few kcal/mol. Eighteen different DFs were applied comparing the energies among the Fe·4O isomers, namely the local density approximation SVWN,28,29 the three gradient approaches PBE,30 PW91,31−35 and B-LYP,36,37 the meta-GGAs M06L38 and TPSS,39 the three hybrid-GGAs B3LYP,27 PBE0,40 and BH-LYP,36,37 the two hybrid-meta-GGAs M0641 and M062X,41 the double hybrid GGA B2PLYP,42 the two dispersioncorrected DFs PBE-D and PW91-D,43 and two DFT+U approaches44 (LDA+U and PBE+U), where we employed either a common U parameter or a case-adapted one using an ab initio linear response approach.44−46 For the spin-polarized LSDA and PBE approaches, the CP2K program was used, where polarized double-ζ Gaussian basis sets47 (m-DZVP for O and (3s6p12d10f)/[2s2p2d1f] for Fe) were applied for the wave function and the density was expanded in terms of an auxiliary plane-wave basis with a 350 Ry energy cutoff. The core electrons of Fe and O were modeled via relativistic effectivecore pseudopotentials as implemented in CP2K.48 The employed WFT approaches include the single-reference HF,49−51 the perturbative correction MP2,52−55 and the coupled-cluster CCSD, CCSD(T) (using the above-mentioned 3ζ-pol basis sets for O and Fe), CCSDT and CCSDT(Q)56,57 (using 2ζ-pol basis sets cc-pVDZ (9s4p1d)/[3s2p1d] for O, and ECP10MDF-GUESS (8s7p6d)/[6s5p3d] for Fe). Additionally, to recover both the strong near-degeneracy correlation and also some dynamical correlation by second-order perturbation theory or by rather large orbital and CI sets, we also used the multireference CASSCF and CASPT258,59 variational methods and the extended DMRG-CI approach using the 3ζ-pol basis sets. The applied active spaces were CAS(24e,17o) corresponding to the O-2p and Fe-3d valence shells, DMRG(40e,25o) or DMRG(40e,26o) corresponding to the O-2sp and Fe-3spd or 3spd4s shells, and DMRG(40e,50o) including also the virtual shells of types O-3sp and Fe-4spd. The starting MOs of the DMRG-CI calculations were the symmetry-adapted NOs obtained from CASSCF(24e,17o), where the MOs were ordered applying the “genetic algorithm” as implemented in the BLOCK program. While no extensive dynamical electron correlation is included in the DMRG-CI calculations, the remaining defects of the dynamical correlation might more or less cancel out partially between the Fe·4O

■

COMPUTATIONAL DETAILS Most calculations were performed with the GAUSSIAN 0918 and MOLPRO 201219 programs, while the DFT+U calculations were done with CP2K,20 the CCSDT(Q) calculations with MRCC21 combined with MOLPRO, and the DMRG calculations by applying the BLOCK software.22,23 To account for the moderate relativistic effects but keeping the computational expenses under control, scalar-relativistic effective core potentials (ECP) for the Fe electronic cores were applied (“10MDF” of the Stuttgart group).24 Polarized basis sets of Gaussian triple-ζ quality were applied, where the 10MDF set for Fe24,25 includes 2f1g and the aug-cc-pVTZ set for O26 includes 3d2f basis functions. The spin-restricted Hartree− Fock (HF) approach was applied to the open-shell systems to relieve the spin-contaminant issue. Spin-unrestricted energy differences of high-spin isomers differ by only a few kcal/mol (see Table SIII of the Supporting Information). In initial steps, various starting geometric and electronic structures of Fe·4O species were optimized at the hybrid B3LYP self-consistent-field (SCF) approximation27 in order to locate ground and low-lying excited stationary states. All further calculations were then started from those geometric structures. Other DF and MP2 approaches yielded bond lengths 1526


Article

Journal of Chemical Theory and Computation Table 1. Geometric Structures of Fe·4O Isomers at B3LYP Approximation, Bond Lengths R in Å RFe−O OSa II III IV V VI VII VIII

stateb 7

D2d C2v‑planar 1 D2d 3 C2v 1 C2v (3C3v)d 1 Td 7

configurationc d6 d5 d4 d3 d2 d1 d0

π*1 π*1 π*2 π*1 π*2 π*2 π*1 π*2 p1

1

(O)2−

2

(O)1−

RO−O 1

(O2)2− 1.77 1.78

1.60 1.54 1.56 1.57

2

(O2)1− 1.94 1.94

1

(O2)2− 1.44 1.41

1.93 1.75

2

(O2)1− 1.35 1.32 1.28

1.38

1.76

OS = oxidation state of Fe. bSpin multiplicity and structural symmetry. cElectron configuration: d refers to MOs of Fe-3d character, p to O-2p, π* to O2-2pπ*. dRestricting the FeVII species to a triplet biradical with C3v symmetry, otherwise geometrically and electronically unstable. a

are kept by the O atoms yielding four closed-shell O2− ligands, as in the case of MO4 (M = Ru, Os, Ir).9 (ii) Upon deforming to a C3v geometry and “back-donating” one electron from the one Fe−O expanded oxide ion to the central Fe (ligand-to-metal charge transfer, LMCT), we obtain an unstable Fe(VII)-3d1 biradical species •FeVII (O2−)3 (•O1−). The two unpaired electrons, one on the monovalent •O1− ligand and the other on •FeVII, are triplet-coupled, which is analogous, for instance, to uranium−silicon multiradical bonding.62,63 The stationary point on the energy surface has a distortive imaginary vibration frequency of 594i cm−1. The system falls back to a slightly Jahn−Teller distorted triplet of Td-FeVIIO4 type, which can be viewed as an electronically excited LMCT state of the 1Td-FeVIIIO4 species. Furthermore, FeVIIO3O is also unstable with respect to the decomposition reaction 2FeVIIO3O → 2FeVIO3 + O2. As Fe(d1)O3O appears as unstable in different respects, it will not be discussed any further. (iii) Remarkably, two electronic LMCTs are energetically even more favorable. Namely, in the Fe(VI)-3d2 species 1 C2v-••FeVI (O2−)2 (•O1−)2, the two monovalent oxygen atoms can form a stable dimeric ligand, the closed-shell peroxide anion O22−, bound to the bent ferryl unit FeO2. Because the short Fe−O distances cause a “strong oxygen ligand field”, the Fe(d2) species forms a “low spin” singlet. This scenario is reminiscent of the case of the PrO3− ion, which is a •Pr(IV)(O2−)2(•O1−) species with tetravalent praseodymium and open-shell anion • 1− 63 O . (iv) Because FeVI is still comparatively highly oxidized and strongly electron attracting, one can transfer another electron from the O22− ligand to Fe, forming an “intermediate spin” species of Fe(V)−3d3, the 3C2v-•FeV (O2−)2 (•O21−) ferryl superoxide. (v) The next redox step again yields a low-spin complex, the Fe(IV)−3d4 species 1D2d-FeIV (O22−)2 with two peroxide groups. (vi) Common Fe(III)−3d5 forms a peroxide−superoxide complex 7C2v-(O22−) FeIII (•O21−), remarkably being planar. The larger Fe−O distances and lower formal charges on the O atoms no longer support a low-spin complex: FeIII(d5) shows up with a weakly deformed “high spin” 6S ground term, common for oxidic Fe(d5) complexes. (vii) Finally, also an Fe(II)−3d6 complex 7D2d-FeII (•O21−)2 is possible, where the 5D ground term of Fe2+ couples with the two π* holes on the two superoxidic •O21− units, forming also a spin-septet. Energies at the DFT Level. The applied 18 DF models predict similar geometries, but rather different energy orders of

isomers because of the large active space used. No further DMRG calculations were carried out using larger basis and MO sets due to prohibitive demanding on the computational resources. Despite remaining errors of the applied methods, we identify, through extensive state-of-the-art calculations, the ground-state of the Fe·4O species as 1C2v-[FeVIO2]2+(O2)2−, slightly below 1Td-FeVIIIO4 and 7D2d-FeII(O21−)2 and with 7 C2v‑planar-FeIII(O21−)(O22−) as highest in energy. Here the left superscript indicates the spin multiplicity.

■

RESULTS AND DISCUSSION Geometric and Electronic Structures. The Fe·4O manifold of molecular species may be looked upon as formally consisting of one Fe8+-1s2...3p6 closed-shell cation and four closed-shell O4+-1s22s2 cations, embedded in a cloud of 24 valence electrons, distributed over the O-2p and Fe-3d valence shells, with strong radial relaxation and dominant O-3spd and Fe-4spdf polarizations. Figure 1 displays the B3LYP optimized geometric structures of various stationary Fe·4O species in different spin states, exhibiting different oxidation states of Fe from II to VIII, resulting in various anionic states of O2−, O1−, O22−, and O21−. In the geometry optimizations, we had tested as ligands of Fe: single O atoms (Fe−Ox) as well as dioxygen units in η1 (Fe−O2) or η2 coordination (Fe < O2 hereafter). As previous studies of neutral transition metal oxide molecules from the central part of the periodic table had not found lowenergy species with η2-O3 (ozonides) or η2-O4 ligands, we did not attempt the presumably high-energy multioxygen units here.60,61 Except for the planar 7C2v-(O2)FeIII(O2) species, all others are three-dimensional, with two oxygen atoms arranged vertically to the other two in Td, D2d, or C2v symmetry. The corresponding bond length values are listed in Table 1, reflecting the character of the oxygen species: O−O distances around 1.2 Å for dioxygen 3O20 ligands, 1.3 Å for superoxides 2 O21−, 1.4−1.5 Å for peroxides 1O22−, significantly larger O−O separations for single 1O2− or 2O1− ligands. The Fe−O distances decrease with increasing formal charges or oxidation states of Fe and O (Fe−O2−, 1.55−1.6 Å; Fe−O1−, 1.75−1.8 Å; Fe−O0.5−, 1.9−1.95 Å) and varies with the number of (monoor bidentate) ligands and the dn spin state of Fe in the common way. Depending upon the geometric and electronic structures there exist at least seven (meta-)stable isomers, namely with iron in oxidation states II to VIII: (i) The simplest option is Fe(VIII)-3d0 with Fe·4O as tetrahedral FeVIII(O2−)4, where all eight valence electrons of Fe 1527


1528

BH-LYP M06-2X B2PLYP

LDA+Uadp PBE+Uadp

B3LYPc PBE0 M06 LDA+U PBE+U

TPSS M06-Lc

SVWN PBE−D PW91−D PBEc PW91 B-LYP

DF

50 54 54

0 0

20 25 27 0 0

0 0

0 0 0 0 0 0

HF%a

−0− −0− −0− −0− −0− −0−

(0) (0) (0) (0)

−0− −0− −0− −0−

(0) −0− (0) −0− (0) −0−

(0) (0) (0) (0) (0) (0)

(4) −0− (3) −0− (3) −0−

−0− −0− −0− −0− −0− −0− −0−

II/7D2d (4) (4) (4) (4) (4) (4) (4)

16.4 15.2 14.6 14.1 19.4 16 ± 2

(1) (1) (1) (1)

19.6 21.0 20.9 20 ± 1

(1) 13.8 (1) 19.2 (1) 16 ± 3

(3) (1) (2) (1) (1) (1)

(5) 17.9 (5) 17.9 (5) 18 ± 0

16.1 19.1 19.0 19.1 19.0 20.7 19 ± 3

III/7C2v (5) (5) (5) (5) (5) (5) (5)

−22.2 −9.0 −10.0 −9.0 −10.0 −13.1 −12 ± 10

IV/1D2d

22.5 39.3 34.8 28.8 44.8 34 ± 9

(3) (3) (3) (3)

87.8 92.9 99.6 93 ± 6

(3) 51.1 (3) 66.2 (3) 59 ± 8

(4) (4) (5) (4) (4) (5)

(3) −2.1 (4) 3.3 (3) 1 ± 2

(3) (3) (3) (3) (3) (3) (3)

−31.1 −22.1 −22.9 −22.1 −22.9 −26.4 −25 ± 6

V/3C2v

12.6 29.9 24.2 37.8 46.2 30 ± 13

(2) (4) (2) (3)

86.6 93.4 98.2 93 ± 6

(2) 45.6 (2) 54.3 (2) 50 ± 5

(2) (3) (3) (5) (5) (3)

(2) −11.2 (2) −6.3 (2) −9 ± 3

(2) (2) (2) (2) (2) (2) (2)

−64.4 −44.8 −45.6 −44.8 −45.6 −46.1 −49 ± 15

VI/1C2v

1.2 18.6 5.3 18.0 38.4 16 ± 15

(4) (2) (4) (3)

89.2 91.2 104.2 95 ± 8

(4) 61.5 (4) 81.4 (4) 82 ± 10

(1) (2) (1) (2) (3) (1)

(0) −31.5 (0) −30.2 (0) −31 ± 2

(1) (1) (1) (1) (1) (1) (0)

23.8 48.4 29.0 19.4 37.9 32 ± 12

(5) (5) (5) (5)

153.0 155.8 171.5 160 ± 10

(5) 87.6 (5) 105.9 (5) 97 ± 10

(5) (5) (4) (3) (2) (3)

(1) −29.3 (1) −27.6 (1) −28 ± 2

−64.7 −44.9 −45.8 −44.9 −45.8 −47.5 −49 ± 15

VIII/1Td (0) (0) (0) (0) (0) (0) (0)

DF = density functional. HF% = percentage of Hartree−Fock orbital-exchange Coulomb correction added in the DF. The Roman numeral in the headline indicates the oxidation state of Fe. The spinmultiplicity and geometric symmetry are given. In front of the energies, the ordinal of the Fe·4O isomer is given in parentheses: (0) means lowest energy isomer, (1−4) indicate intermediate energy isomers, (5) denotes the highest energy isomer. Near-degenerate isomers get the same number. bU = 5.3 eV. Uadp = U adapted for each species using an ab initio linear response approach:45,46 FeII 5.31 eV, FeIII 5.21 eV, FeIV 6.28 eV, FeV 6.94 eV, FeVI 6.84 eV, FeVIII 7.98 eV. cResults for larger AVQZ basis (ECP10MDF-ANO for Fe with up to 2g, aug-cc-pVQZ for O) show that a polarized VTZ basis is sufficient for DFA (see Table SI of the Supporting Information).

a

hybrid GGA hybrid meta GGA double hybrid GGA hybrid average (DFA-5)

DFT+Uadp average (DFA-4)

DFT+Uadpb

hybrid GGA and +U average (DFA-3)

hybrid meta GGA DFT+Ub

hybrid GGA

meta-GGA average (DFA-2)

meta-GGA

LDA-GGA-D average (DFA-1)

GGA

LDA DFT−D

DF type

Table 2. Relative Energies (in kcal/mol) of Fe·4O Isomers with Respect to [FeII(O2)2], as Obtained with Various Density Functionalsa

Journal of Chemical Theory and Computation Article


Article


Single- and Multireference Wave Function Results. As mentioned before, the 3d elements, and in particular iron, are one of the most delicate cases in the periodic table, owing to rich angular and in−out correlations in the Fe-3d4s valence shell. In addition, there occur static correlations in the oxygen (O2−)4 shell when it becomes oxidized, physically or formally, in competition with Fe8+. To obtain more reliable results, the treatment of both static and dynamic correlation is required by methods such as CASSCF/PT, MR-CCSD(T,Q), or extended CI. Diagnostic parameters of single-reference DFT, CCSD(T) and multireference CAS(24e,17o)-SCF calculations of the six Fe·4O species under discussion are displayed in Table 3, indicating the strong multireference character of all species. Although sometimes KS orbitals yield significantly more compact CI expansions than HF orbitals, in the present case the D1 and T1 parameters on the basis of HF orbitals correlate with the length of the NO expansions. To treat all Fe·4O species on a roughly equal footing, we correlated the full Fe-3d and 4O-2p valence shells at the CAS(24e,17o)SCF level. Natural orbitals (NOs) are displayed in Figure 3 for 1C2v-[FeVIO2](O2), and in Supporting Information, Figures S2−S6, for the other Fe·4O species. There occur several NOs where the occupation numbers appreciably deviate from 2, 1, or 0 by more than 0.1, indicating the multiconfigurational character of the species. Relative energies of the six Fe·4O isomers at various single-reference (SCF-HF, MP2, CCSD, CCSD(T), CCSDT, CCSDT(Q)) and multireference approaches (CASSCF and CASPT2) are displayed in Table 4. Recently, an MR-CI diagnostic has been suggested on the basis of DFT-orbitals.12 We note that the multiconfiguration diagnostics and the DFT orbital gaps indicate intermediate configuration mixing for the Fe(II and III) species and large configuration mixing for the Fe(IV to VI) species. However, the Fe(VIII) species with extremely pronounced static correlation shows only an intermediate orbital gap. Therefore, the current species can also be used for further validating various diagnostic indexes of multiconfigurational characteristics. Both SCF and MP2 results appear completely off, as is the case for a variety of DFAs. Correspondingly, the DFA corrected with Hartree−Fock hybrid funtionals performs badly, too. All single-reference coupled cluster and CAS(24e,17o) approaches do not seem to be reliable enough to determine the energy order of the six Fe·4O isomers unquestionably. Therefore, we finally performed DMRG multiconfiguration calculations on the Fe(II), Fe(VI), and Fe(VIII) isomers (and guessed values for the Fe(III), Fe(IV), and Fe(V) species from the trends in Table 4). We applied an active space of 50 orbitals, derived

the six stable Fe·4O species in Table 2, making it impossible to decide on the ground state of Fe·4O on the basis of DFA. The LDAs and GGAs predict the energetic order of iron oxidation states as III > (II > IV) > V > (VI ≈ VIII)

with common FeIII least stable and rare FeVI and debated FeVIII most stable in isolated tetra-oxo iron complexes. Meta-GGAs interchange II with IV, and VI with VIII, within several kcal/ mol. Adding HF exchange in hybrid DF models, or using DFT +U approaches, a stabilization of the lower Fe oxidation states is obtained, approximately inverting the oxidation-state stability order, though with huge uncertainties of up to several 10 kcal/ mol: VIII > (V ≈ IV) > VI > (III > II)

The largest differences occur between LDA and M06-2X, reaching 155, 150, 61, 32, and 65 kcal/mol for Fe oxidation states II, III, IV, V, and VIII, respectively, in comparison to VI. These surprisingly large errors make DFAs appear having little use for the determination of the energetic order of these Fe·4O isomers. However, for guidance in selecting active spaces in WFT calculations, we show Kohn−Sham orbital level schemes and MO envelope plots in Figure 2 for [FeVIO2](O2), and in Figure S1 of the Supporting Information for [FeVIIIO4].

Figure 2. 1C2v-[FeVIO2](O2). Kohn−Sham orbital energy level scheme (DFT-PBE, energies in eV) and orbital contours (values ± 0.05 au) of MOs of O-2p and Fe-3d character, generating the active correlating space (indicated by the red box). The highest doubly occupied MO is indicated by two red dots.

Table 3. Multi-Configuration Character of Fe·4O Isomersa CASSCFb

CCSD(T) OS(Fe) II III IV V VI VIII

spin mult, geom sym 7

D2d 7 C2v‑planar 1 D2d 3 C2v 1 C2v 1 Td

DFTc

T1

D1

main config coeff

2nd config coeff

sum of weak-occ NOs

HOMO−LUMO gap

0.03 0.04 0.10 0.09 0.06 0.11

0.09 0.16 0.52 0.29 0.20 0.35

0.93 0.92 0.77 0.80 0.78 0.72

0.10 0.17 0.16 0.10 0.18 0.10

0.12 0.14 0.95 0.73 0.93 1.29

4.52 3.28 0.69 0.92 1.23 1.72

a

OS(Fe) = oxidation state of Fe; spin mult = spin multiplicity; geom sym = point group label of geometric structure; T1 and D1 = normalized single substitution amplitudes of CCSD(T) calculations;64−66 bconfig coeff = absolute values of coefficients from CAS(17o,24e)SCF calculations.67 cDFTPBE HOMO/SOMO−LUMO gap in eV. 1529


Article


Figure 3. 1C2v-[FeVIO2](O2). Natural valence orbitals (contour values ± 0.04 au; with dominant AO contributions and occupation numbers, ordered by increasing energy from bottom right to top left) from CAS(17o,24e)SCF calculations.

Table 4. Relative Energies (in kcal/mol) of Fe·4O Isomers with Respect to [FeII(O2)2], Obtained with Various WFT Methods, and Energetic Ordinal Number (Bold, in Parentheses)a method

II/7D2d

III/7C2v

IV/1D2d

V/3C2v

VI/1C2v

VIII/1Td

err

SCF-HF SCF-MP2 SCF-CC-SD SCF-CC-SD(T) SCF-CC-SDTb SCF-CC-SDT(Q)b CAS(24e,17o)SCF CAS(24e,17o)SCF-PT2 DMRG(40e,50o)

(0) −0− (3) −0− (0) −0− (2) −0− (1−2) −0− (3) −0− (0−2) −0− (3) −0− (2−4) −0−

(1) 26.8 (4) 16.9 (1) 16.8 (5) 15.0 (4) 5.9 (5) 15.9 (3−4) 22.3 (5) 14.2 (5) (∼10)

(2) 194.5 (5) 19.0 (4) 53.0 (4) 12.8 (5) 11.5 (4) 7.5 (5) 33.8 (4) 7.8 (2−4) (∼10)

(3) 215.4 (2) −23.2 (3) 48.4 (3) 6.7 (1−2) −1.5 (2) −9.6 (3−4) 22.2 (2) −6.2 (1) (∼−5)

(4) 253.8 (0) −152.0 (2) 38.6 (0) −19.6 (0) −19.7 (0−1) −33.0 (0−2) 1.6 (1) −26.6 (0) −12.4

(5) 374.6 (1) −69.1 (5) 72.4 (1) −9.1b (3) 2.6 (0−1) −35.2 (0−2) 2.5 (0) −32.4 (2−4) −2.0

135 56 26 8 6 15 12 14

a

Headline: Roman numeral = oxidation state of Fe; spin multiplicity and point group label of geometric structure; err = estimated average relative energy error; energies from AVTZ bases. bCalculated with VDZ basis sets, and then the AVTZ-VDZ energy differences from CCSD(T) are added.

al.71 in 2007 were more recently corroborated theoretically by Tran and Hendrickx.72 They proposed that 1C2v-[FeVO2(O2)]− and 1Td-[FeVIIO4]− species can exist in equilibrium at similar energies in the anionic states of [Fe·4O]−, while the neutral species have an energy difference of about 0.54 eV with FeVIII above FeVI. Further, 1.5 eV photons can convert the FeVI species to the FeVIII species, and 2.1 eV photons did reverse it. Following an approach used for assessing the conversion of different isomers of the IrO4+ ion with Ir in IX and VII oxidation states,73 we applied the linear transit approach at the DFT-PBE level (where the two species are nearly degenerate, Figure 5) and the CCSD(T) level (where the FeVIII species is 0.45 eV above the FeVI species) and obtained estimated barriers of ∼1.25 and ∼1.8 eV, respectively. The saddle-point energies at optimized geometries will be somewhat lower. These data are consistent with the experiments.71,72

from the O-2sp3sp and Fe-3spd4spd occupied and virtual shells, while our resources prohibited the inclusion of O-3d and Fe-4f type MOs. To cover some of the dynamic correlation, also the “outer core” shells of O-2s and Fe-3s,3p type were included. On the basis of the energy variations in Table 4 and the literature experiences with iron complexes, we expect our final values to be reliable to a reasonable extent.68−70 Further details are presented in Figures S7−S9 of the Supporting Information. We compare three of the DFA averages from Table 2 and some CASSCF and CC results with our best “benchmark estimates” in Figure 4. The other even worse DFA, the HF and MP2 results are not displayed because they overshoot the ΔE scale. The most reliable energy order of the isomers appears to be

■

III > (II ∼ IV ∼ VIII) > V > VI

CONCLUSIONS Our conclusions refer to three topics, (i) the energetic order and stability of the various molecular species of Fe·4O, (ii) the highest oxidation state of a transition metal atom from the middle of the periodic table, and (iii) the challenge of only

A particularly pestering question is that of the most stable isomer of Fe.4O. Our present theoretical conclusion is that the 1 C2v-[FeVIO2](O2) species is lowest in energy, with the 1Td[FeVIIIO4] species up to about half an eV higher. Deductions from a photoelectron spectroscopic study of FeO4− by Wang et 1530


Article


However, the Fe(VIII) oxidation state seems not to be stable, even for the FeO4 species, under normal chemical conditions. The multiconfiguration character of all Fe·4O molecules means not only that the open-shell Fe(3dn) fragment species are highly correlated but also that the formally closed-shell O2− ligands adopt open-shell character through electron donation into the strongly electron-attracting high-oxidation-states of the FeV−VIII cations. As a general rule, the later transition metal atoms cannot lose all their valence electrons owing to the high energy cost of forming very high oxidation states. For example, fluorides Fe(d0)F8 and Fe(d2)F6 decay to Fe(d4)F4 + n F2 (n = 2, 1) and Pu(f0)F8 decays to Pu(f2)F6 + F2.75 Oxygen ligands OII are more “flexible” than fluorine ligands FI insofar as FI always occurs as F1− while OII can partially transform from divalent O2− to monovalent •O1−+e− through LMCT. In the symmetric Td-FeO4 or D4h-PuO4 molecules,76 it is the “physical oxidation” of the OII ligands (O2− to O1−) that accounts for the local minimum features of the high-symmetry structures. In such cases where the concept of local spin-singlet pair formation breaks down, the Hartree−Fock and Kohn−Sham models based on single-configuration independent particle references yield unreliable results. The HF-MP2 and DF-hybrid approaches with HF exchange are also unsatisfactory in these cases of only apparent closed shell species. The dependence of the energy of the U parameter in DFT+U approaches is unexpectedly large, too. It is thus natural that there is a serious demand to develop DFA techniques that can cover both dynamic and static correlation.77−79 This applies to apparent closed-shell main-group species as well as to compounds of the d- and f-block elements, especially of the later first row transition metals and early actinides,76 which are catalytically and technically particularly important. Our results show that the Fe·4O set can be used as a testing field for the development of both better DF approaches and computationally affordable benchmark approaches.

Figure 4. Relative energies of Fe·4O isomers wrt 7D2d-[FeII(O2)2]: The presumably most reliable DMRG(40e,50o), CAS(24e,17o)/PT2, CCSD(T), and three DFA-1,2,3 averages from Table 2. The errors of HF-SCF and HF-MP2 results and the DFA-4 and DFA-5 averages lie out of the presented ΔE scale and are not displayed.

■

Figure 5. Linear transit (LT) energy lines, illustrating the transitions between 1(3)Td-[FeVIIIO4] (left) and 1(3)C2v-[FeVIO2](O2) (right). DFA-PBE energies (in kcal/mol) with ZPE.

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jctc.5b01040. Software references, orbital energy level scheme and MO envelopes of 1Td-[FeVIIIO4]; natural valence orbitals of 7 D 2d -[Fe II (O 2 ) 2 ], 7 C 2v‑planar -[Fe III (O 2 )(O 2 )], 1 D 2d [FeIV(O2)2], 3C2v-[FeVO2](O2), and 1Td-[FeVIIIO4]; DMRG-CI(40,50) energy extrapolation of 7 D 2d [FeII(O2)2], 1C2v-[FeVIO2](O2), and 1Td-[FeVIIIO4]; relative energies of Fe·4O isomers, DMRG energies from various active spaces, and spin-restricted and -unrestricted energies of Fe·4O isomers (PDF)

apparent closed-shell species for the development of satisfactory density functionals. The low-energy geometric and electronic isomers of isolated Fe·4O molecules with oxidation states of Fe from II to VIII and of O from −II to −1/2 are strongly correlated species, in particular for those with closed-shell fragments. Most contemporary density functional and single-reference wave function approaches cannot reproduce the relative energies of the six (meta-)stable isomers to better than 1 eV or 20 kcal/ mol. Our best results are from extended DMRG-CI calculations that are estimated to be reasonably reliable. The four oxygen ligands are bonded as single atoms or as η2-bonded dimers. We have not considered the possibility of η1-O2 or η 2-O3 ligation with less than four Fe−O interactions because they seem to have energies higher by 1 eV.70 Within a range of 1 eV, the lowest and highest energy isomers seem to be [FeVIO2](η2-O2) and (η2-O2)[FeIII](η2-O2), respectively, with the FeII, FeIV, FeV, and FeVIII species at intermediate energies. We support the conjecture of Hendrickx, Wang, Zhou, and their respective coauthors72,74 that oxidation state VIII is only possible for metastable FeVIIIO4 in vacuum at nonelevated temperatures.

■

AUTHOR INFORMATION

Corresponding Authors

*J.L.: E-mail, [email protected]. phone, +86-10-62795381; fax, +86-10-62797472. *W.H.E.S.: E-mail, [email protected]. Address: Physical and Theoretical Chemistry, University of Siegen, 57068 Germany. Notes

The authors declare no competing financial interest. 1531


Article


■

ACKNOWLEDGMENTS

■

REFERENCES

(21) Kállay, M.; et al., MRCC 2013; Budapest University of Technology and Economics: Budapest, 2013, See http://www.mrcc. hu, and the Supporting Information for full author list. (22) Chan, G. K.-L.; Head-Gordon, M. Highly correlated calculations with a polynomial cost algorithm: A study of the density matrix renormalization group. J. Chem. Phys. 2002, 116, 4462−4476. (23) Olivares-Amaya, R.; Hu, W.; Nakatani, N.; Sharma, S.; Yang, J.; Chan, G. K.-L. The ab-initio density matrix renormalization group in practice J. Chem. Phys. 2015, 142, 034102 10.1063/1.4905329. See http://sanshar.github.io/Block, and the Supporting Information for more references. (24) Dolg, M.; Wedig, U.; Stoll, H.; Preuss, H. Energy-adjusted ab initio pseudopotentials for the first row transition elements. J. Chem. Phys. 1987, 86, 866. (25) Martin, J. M. L.; Sundermann, A. Correlation consistent valence basis sets for use with the Stuttgart-Dresden-Bonn relativistic effective core potentials: The atoms Ga-Kr and In-Xe. J. Chem. Phys. 2001, 114, 3408. (26) Kendall, R. A.; Dunning, T. H., Jr.; Harrison, R. J. Electron affinities of the first-row atoms revisited. Systematic basis sets and wave functions. J. Chem. Phys. 1992, 96, 6796. (27) Becke, A. D. Density-functional thermochemistry. III. The role of exact exchange. J. Chem. Phys. 1993, 98, 5648. (28) Slater, J. C. The Self-Consistent Field for Molecular and Solids, Quantum Theory of Molecular and Solids; McGraw-Hill: New York, 1974. (29) Vosko, S. H.; Wilk, L.; Nusair, M. Accurate spin-dependent electron liquid correlation energies for local spin density calculations: a critical analysis. Can. J. Phys. 1980, 58, 1200. (30) Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized Gradient Approximation Made Simple. Phys. Rev. Lett. 1996, 77, 3865. (31) Perdew, J. P. in Ziesche, P.; Eschrig, H., Eds. Electronic Structure of Solids ’91; Akademie Verlag: Berlin, 1991. (32) Perdew, J. P.; Chevary, J. A.; Vosko, S. H.; Jackson, K. A.; Pederson, M. R.; Singh, D. J.; Fiolhais, C. Atoms, molecules, solids, and surfaces: Applications of the generalized gradient approximation for exchange and correlation. Phys. Rev. B: Condens. Matter Mater. Phys. 1992, 46, 6671. (33) Perdew, J. P.; Chevary, J. A.; Vosko, S. H.; Jackson, K. A.; Pederson, M. R.; Singh, D. J.; Fiolhais, C. Erratum: Atoms, molecules, solids, and surfaces: Applications of the generalized gradient approximation for exchange and correlation. Phys. Rev. B: Condens. Matter Mater. Phys. 1993, 48, 4978. (34) Perdew, J. P.; Burke, K.; Wang, Y. Generalized gradient approximation for the exchange-correlation hole of a many-electron system. Phys. Rev. B: Condens. Matter Mater. Phys. 1996, 54, 16533. (35) Burke, K.; Perdew, J. P.; Wang, Y. In Electronic Density Functional Theory: Recent Progress and New Directions; Dobson, J. F., Vignale, G., Das, M. P., Eds.; Plenum: New York, 1998. (36) Becke, A. D. Density-functional exchange-energy approximation with correct asymptotic behavior. Phys. Rev. A: At., Mol., Opt. Phys. 1988, 38, 3098. (37) Miehlich, B.; Savin, A.; Stoll, H.; Preuss, H. Results obtained with the correlation energy density functionals of Becke and Lee, Yang and Parr. Chem. Phys. Lett. 1989, 157, 200. (38) Zhao, Y.; Truhlar, D. G. A new local density functional for maingroup thermochemistry, transition metal bonding, thermochemical kinetics, and noncovalent interactions. J. Chem. Phys. 2006, 125, 194101. (39) Tao, J.; Perdew, J. P.; Staroverov, V. N.; Scuseria, G. E. Climbing the Density Functional Ladder: Nonempirical Meta−Generalized Gradient Approximation Designed for Molecules and Solids. Phys. Rev. Lett. 2003, 91, 146401. (40) Adamo, C.; Barone, V. Toward reliable density functional methods without adjustable parameters: The PBE0 model. J. Chem. Phys. 1999, 110, 6158. (41) Zhao, Y.; Truhlar, D. G. The M06 suite of density functionals for main group thermochemistry, thermochemical kinetics, noncovalent interactions, excited states, and transition elements: two new

We are grateful to Prof. Garnet Chan and Dr. Qiming Sun at Princeton University for helpful discussions on the DMRG approach. This work was financially supported by the National Natural Science Foundation of China (grant nos. 91426302, 21590792, and 21433005) and was performed using supercomputers at Tsinghua National Laboratory for Information Science and Technology, and the Supercomputing Center, Computer Network Information Center of the Chinese Academy of Sciences.

(1) Gombás, P. Pseudopotentiale; Springer: Wien, Austria, 1967. (2) Labanowski, J. K.; Andzelrn, J. Density Functional Methods in Chemistry; Springer, New York, 1991. (3) Ziegler, T. Approximate density functional theory as a practical tool in molecular energetics and dynamics. Chem. Rev. 1991, 91, 651− 667. (4) Perdew, J. P.; Schmidt, K. Jacob’s ladder of density functional approximations for the exchange-correlation energy. AIP Conf. Proc. 2000, 577, 1. (5) Becke, A. D. Perspective: Fifty years of density-functional theory in chemical physics. J. Chem. Phys. 2014, 140, 18A301. (6) Yu, H.; Truhlar, D. G. Components of the Bond Energy in Polar Diatomic Molecules, Radicals, and Ions Formed by Group-1 and Group-2 Metal Atoms. J. Chem. Theory Comput. 2015, 11, 2968−2983. (7) Becke, A. D. Density functionals for static, dynamical, and strong correlation. J. Chem. Phys. 2013, 138, 074109. (8) Becke, A. D. Calibration of a strong-correlation density functional on transition-metal atoms. J. Chem. Phys. 2013, 138, 161101. (9) Huang, W.; Xu, W.−H.; Schwarz, W. H. E.; Li, J. On the Highest Oxidation States of Metal Elements in MO4 Molecules. Inorg. Chem. 2016, submitted for publication. (10) Johnson, B. G.; Gill, P. M. W.; Pople, J. A. The performance of a family of density functional methods. J. Chem. Phys. 1993, 98, 5612. (11) Pan, L.-L.; Li, J.; Wang, L.-S. Low-lying isomers of the B9− boron cluster: The planar molecular wheel versus three-dimensional structures. J. Chem. Phys. 2008, 129, 024302. (12) Grimme, S.; Hansen, A. A Practicable Real-Space Measure and Visualization of Static Electron-Correlation Effects. Angew. Chem., Int. Ed. 2015, 54, 12308. (13) Cohen, A. J.; Mori-Sánchez, P.; Yang, W. Challenges for Density Functional Theory. Chem. Rev. 2012, 112, 289−320. (14) Altun, A.; Breidung, J.; Neese, F.; Thiel, W. Correlated Ab Initio and Density Functional Studies on H2 Activation by FeO+. J. Chem. Theory Comput. 2014, 10, 3807−3820. (15) Swart, M. Spin states of (bio)inorganic systems: Successes and pitfalls. Int. J. Quantum Chem. 2013, 113, 2−7. (16) Wells, A. F. Structural Inorganic Chemistry, 5th ed.; Oxford University Press: Oxford, UK, 1984. (17) (a) Buijse, M. A.; Baerends, E. J. Analysis of Nondynamical Correlation in the metal-ligand bond. Pauli Repulsion and Orbital Localization in MnO4−. J. Chem. Phys. 1990, 93, 4129−4141. (b) Su, J.; Xu, W.-H.; Xu, C.-F.; Schwarz, W. H. E.; Li, J. Theoretical Studies on the Photoelectron and Absorption Spectra of MnO4− and TcO4−. Inorg. Chem. 2013, 52, 9867−9874. (18) Frisch, M. et al. GAUSSIAN 2009; Gaussian, Wallingford CT, 2010. See http://www.gaussian.com, and the Supporting Information for full author list. (19) Werner, H.-J. et al. MOLPRO; University College Cardiff Consultants: Cardiff, UK, 2012. See http://www.molpro.net, and the Supporting Information for full author list. (20) CP2K 2010; Developers Group under the terms of the GNU General Public License, 2010. See http://www.cp2k.org, and the Supporting Information for full author list. 1532


Article


anions. Sci. China: Chem., 2016, 59, 1−10, doi: 10.1007/s11426-0155481-z. (64) Lee, T. J.; Taylor, P. R. A diagnostic for determining the quality of single-reference electron correlation methods. Int. J. Quantum Chem. 1989, 36, 199. (65) Janssen, C. L.; Nielsen, I. M. B. New diagnostics for coupledcluster and Møller-Plesset perturbation theory. Chem. Phys. Lett. 1998, 290, 423. (66) Lee, T. Comparison of the T1 and D1 diagnostics for electronic structure theory: a new definition for the open-shell D1 diagnostic. Chem. Phys. Lett. 2003, 372, 362. (67) Werner, H. J.; Follmeg, B.; Alexander, M. H. Adiabatic and diabatic potential energy surfaces for collisions of CN (X 2Σ+, A 2Π) with He. J. Chem. Phys. 1988, 89, 3139. (68) Kurashige, Y.; Chan, G. K.; Yanai, T. Entangled quantum electronic wavefunctions of the Mn4CaO5 cluster in photosystem II. Nat. Chem. 2013, 5, 660−666. (69) Sharma, S.; Sivalingam, K.; Neese, F.; Chan, G. K. Low-energy spectrum of iron-sulfur clusters directly from many-particle quantum mechanics. Nat. Chem. 2014, 6, 927−933. (70) Saitow, M.; Kurashige, Y.; Yanai, T. Fully Internally Contracted Multireference Configuration Interaction Theory Using Density Matrix Renormalization Group: A Reduced-Scaling Implementation Derived by Computer-Aided Tensor Factorization. J. Chem. Theory Comput. 2015, 11, 5120−5131. (71) Wu, H.; Desai, S. R.; Wang, L.-S. Observation and Photoelectron Spectroscopic Study of Novel Mono- and Diiron Oxide Molecules: FeOy− (y = 1−4) and Fe2Oy− (y = 1−5). J. Am. Chem. Soc. 1996, 118, 5296−5301. (72) Tran, V. T.; Hendrickx, M. F. A. Description of the geometric and electronic structures responsible for the photoelectron spectrum of FeO4−. J. Chem. Phys. 2011, 135, 094505. (73) Wang, G.; Zhou, M.; Goettel, J. T.; Schrobilgen, G. J.; Su, J.; Li, J.; Schlöder, T.; Riedel, S. Identification of an Iridium-containing compound with a formal oxidation state of IX. Nature 2014, 514, 475− 477. (74) Wang, C.; Jian, J.; Wang, G.; Li, Z.; Zhou, M. Infrared Photodissociation Spectroscopy of Oxygen-Rich Fe(O2)n+ (n = 3−5) Cation Complexes. J. Phys. Chem. A 2014, 118 (25), 4519−4526. (75) Huang, W.; Pyykkö, P.; Li, J. Is Octavalent Pu(VIII) Possible? Mapping the Plutonium Oxyfluoride Series PuOnF8−2n (n = 0−4). Inorg. Chem. 2015, 54, 8825−8831. (76) Huang, W.; Xu, W.-H.; Su, J.; Schwarz, W. H. E.; Li, J. Oxidation States, Geometries, and Electronic Structures of Plutonium Tetroxide PuO4 Isomers: Is Octavalent Pu Viable? Inorg. Chem. 2013, 52, 14237−14245. (77) Savin, A. Towards a systematic way to correct density functional approximations. J. Chem. Phys. 2014, 140, 18A509. (78) Mardirossian, N.; Head-Gordon, M. ωB97X-V: A 10-parameter, range-separated hybrid, generalized gradient approximation density functional with nonlocal correlation, designed by a survival-of-thefittest strategy. Phys. Chem. Chem. Phys. 2014, 16, 9904−9924. (79) Mardirossian, N.; Head-Gordon, M. Mapping the genome of meta-generalized gradient approximation density functionals: The search for B97M-V. J. Chem. Phys. 2015, 142, 074111.

functionals and systematic testing of four M06-class functionals and 12 other functionals. Theor. Chem. Acc. 2008, 120, 215. (42) Grimme, S. Semiempirical hybrid density functional with perturbative second-order correlation. J. Chem. Phys. 2006, 124, 034108. (43) Grimme, S.; Antony, J.; Ehrlich, S.; Krieg, H. A consistent and accurate ab initio parametrization of density functional dispersion correction (DFT-D) for the 94 elements H-Pu. J. Chem. Phys. 2010, 132, 154104. (44) Liechtenstein, A. I.; Anisimov, V. I.; Zaanen, J. Densityfunctional theory and strong interactions: Orbital ordering in MottHubbard insulators. Phys. Rev. B: Condens. Matter Mater. Phys. 1995, 52, R5467(R). (45) Cococcioni, M.; de Gironcoli, S. Linear response approach to the calculation of the effective interaction parameters in the LDA+U method. Phys. Rev. B: Condens. Matter Mater. Phys. 2005, 71, 035105. (46) Kulik, H. J.; Cococcioni, M.; Scherlis, D. A.; Marzari, N. Density Functional Theory in Transition-Metal Chemistry: A Self-Consistent Hubbard U Approach. Phys. Rev. Lett. 2006, 97, 103001. (47) VandeVondele, J.; Hutter, J. Gaussian basis sets for accurate calculations on molecular systems in gas and condensed phases. J. Chem. Phys. 2007, 127, 114105. (48) Hartwigsen, C.; Goedecker, S.; Hutter, J. Relativistic separable dual-space Gaussian pseudopotentials from H to Rn. Phys. Rev. B: Condens. Matter Mater. Phys. 1998, 58, 3641. (49) Roothaan, C. C. J. New Developments in Molecular Orbital Theory. Rev. Mod. Phys. 1951, 23, 69. (50) Pople, J. A.; Nesbet, R. K. Self-Consistent Orbitals for Radicals. J. Chem. Phys. 1954, 22, 571. (51) McWeeny, R.; Dierksen, G. Self-Consistent Perturbation Theory. II. Extension to Open Shells. J. Chem. Phys. 1968, 49, 4852. (52) Suhai, S.; Ladik, J. Perturbation theoretical calculation of the correlation energy in an infinite metallic hydrogen chain. J. Phys. C: Solid State Phys. 1982, 15, 4327. (53) Sun, J.-Q.; Bartlett, R. J. Second-order many-body perturbationtheory calculations in extended systems. J. Chem. Phys. 1996, 104, 8553. (54) Hirata, S.; Iwata, S. Analytical energy gradients in second-order Møller-Plesset perturbation theory for extended systems. J. Chem. Phys. 1998, 109, 4147. (55) Hirata, S.; Grabowski, I.; Tobita, M.; Bartlett, R. J. Highly accurate treatment of electron correlation in polymers: coupled-cluster and many-body perturbation theories. Chem. Phys. Lett. 2001, 345, 475. (56) Pople, J. A.; Krishnan, R.; Schlegel, H. B.; Binkley, J. S. Electron correlation theories and their application to the study of simple reaction potential surface. Int. J. Quantum Chem. 1978, 14, 545. (57) Pople, J. A.; Head-Gordon, M.; Raghavachari, K. Quadratic configuration interaction. A general technique for determining electron correlation energies. J. Chem. Phys. 1987, 87, 5968. (58) Frisch, M. J.; Ragazos, I. N.; Robb, M. A.; Schlegel, H. B. An evaluation of three direct MC-SCF procedures. Chem. Phys. Lett. 1992, 189, 524−528. (59) Yamamoto, N.; Vreven, T.; Robb, M. A.; Frisch, M. J.; Schlegel, H. B. A direct derivative MC-SCF procedure. Chem. Phys. Lett. 1996, 250, 373−378. (60) Xu, B-F.; Yang, C-L.; Wang, M-S.; Ma, X-G. Structural, electronic, and magnetic properties of FenOm+ (n+m = 5) clusters. J. Mol. Struct.: THEOCHEM 2010, 957, 26−30. (61) Pradhan, K.; Gutsev, G. L.; Weatherford, C. A.; Jena, P. A systematic study of neutral and charged 3d-metal trioxides and tetraoxides. J. Chem. Phys. 2011, 134, 144305. (62) Hu, H.-S.; Wei, F.; Wang, X.-F.; Andrews, L.; Li, J. Actinide− Silicon Multiradical Bonding: Infrared Spectra and Electronic Structures of the Si (μ-X) AnF3 (An = Th, U; X = H, F) Molecules. J. Am. Chem. Soc. 2014, 136, 1427−1437. (63) Su, J.; Hu, S.-X.; Huang, W.; Zhou, M.-F.; Li, J. On the oxidation states of metal elements in MO3− (M = V, Nb, Ta, Db, Pr, Gd, Pa) 1533


How Much Can Density Functional Approximations (DFA) Fail? The

Recommend Documents