How accurate are electronic structure methods for actinoid chemistry

REGULAR ARTICLE

How accurate are electronic structure methods for actinoidchemistry?

Boris B. Averkiev • Manjeera Mantina •

Rosendo Valero • Ivan Infante • Attila Kovacs •

Donald G. Truhlar • Laura Gagliardi

Received: 2 December 2010 / Accepted: 16 February 2011 / Published online: 8 March 2011

� Springer-Verlag 2011

Abstract The CASPT2, CCSD, and CCSD(T) levels of

wave function theory and seven density functionals were

tested against experiment for predicting the ionization

potentials and bond dissociation energies of actinoid

monoxides and dioxides with their cations. The goal is to

guide future work by enabling the choice of an appropriate

method when performing calculations on actinoid-con-

taining systems. We found that four density functionals,

namely MPW3LYP, B3LYP, M05, and M06, and three

levels of wave function theory, namely CASPT2, CCSD,

and CCSD(T), give similar mean unsigned errors for acti-

noid–oxygen bond energies and for ionization potentials of

actinoid oxides and their cations.

Keywords Actinoids � Electronic structure � Density

functional theory � CASSCF � CASPT2

1 Introduction

Actinoid chemistry is important for reactor waste man-

agement [1–3], and actinoids serve as catalysts for a variety

of reactions [4–7], but understanding actinoid chemistry

poses formidable challenges to both experimental and

computational chemists. From a computational point of

view, in order to be able to treat actinoid chemistry, we

need electronic structure methods capable of properly

describing both electron correlation and relativistic effects

in systems that often have large multireference character.

Many electronic structure studies have appeared in the

literature, involving both wave function theory (WFT) and

density functional theory (DFT). The complete active space

(CAS) self-consistent field (SCF) method [8] followed by

multireference second-order perturbation theory (CASPT2)

[9] and DFT with the original B3LYP [10–13] density

functional (which is based on the VWN3 approximation for

the local spin density component) were tested against

experiment for the prediction of bond distances and

vibrational frequencies in uranium triatomic molecules and

cations [14, 15]. Successively, the CASPT2 [9], relativistic

coupled cluster calculations with single and double exci-

tations (CCSD) [17, 18], and relativistic coupled cluster

calculations with single and double excitations and

Dedicated to Professor Pekka Pyykko on the occasion of his 70th

birthday and published as part of the Pyykko Festschrift Issue.

Electronic supplementary material The online version of thisarticle (doi:10.1007/s00214-011-0913-0) contains supplementarymaterial, which is available to authorized users.

B. B. Averkiev � M. Mantina � R. Valero �D. G. Truhlar � L. Gagliardi (&)

Department of Chemistry and Supercomputing Institute,

University of Minnesota, 207 Pleasant St. SE, Minneapolis,

MN 55455-0431, USA

e-mail: [email protected]

D. G. Truhlar

e-mail: [email protected]

I. Infante

Kimika Fakultatea, Euskal Herriko Unibertsitatea

and Donostia International Physics Center (DIPC),

P.K. 1072, 20080 Donostia, Euskadi, Spain

A. Kovacs

European Commission, Joint Research Centre,

Institute for Transuranium Elements,

P.O. Box 2340, 76125 Karlsruhe, Germany

A. Kovacs

Research Group for Materials Structure and Modeling

of the Hungarian Academy of Sciences,

Budapest University of Technology and Economics,

Szt. Gellert ter 4, 1111 Budapest, Hungary

123

Theor Chem Acc (2011) 129:657–666

DOI 10.1007/s00214-011-0913-0

http://dx.doi.org/10.1007/s00214-011-0913-0

quasiperturbative treatment of connected triple excitations

(CCSD(T)) [19, 20] levels of WFT were tested against

experiment for ionization potentials of actinoid monoxides

and dioxides [16]. In the current paper, we present tests for

a larger set of experimental bond dissociation energies in

which we include actinoid dioxides and their cations and

dications, and, in addition, we test newer density func-

tionals. In particular, the MPW3LYP [21], MOHLYP [22],

M05 [23], M06-L [24], and M06 [25] density functionals

are tested against both some of the previous [14, 16] data

and all of the new test sets. We also checked the perfor-

mance of an additional older functional, namely the PW91

[26] functional, for these systems, because it was shown

[27] that it performs well for vibrational frequencies of the

triatomic molecules NUN, NUO?, and CUO. In related

work, Austin et al. [28] tested the B3LYP, M05, M06-L,

and M06 density functionals and two others against

experiment and compared them with CCSD(T) and CAS-

PT2 results for actinoid reaction energies; they found that

‘‘the M06 functional definitely stands out as being more

successful than the other five functionals tested’’ and ‘‘is

competitive with high-level ab initio methods’’ for these

reaction energies. The present tests should be particularly

useful in that all density functionals and WFT methods

mentioned so far are compared to the same sets of data,

allowing a consistent comparison.

2 Methods

In the following, we will describe the details of the WFT-

based calculations followed by the details of the DFT

calculations.

2.1 WFT calculations of bond dissociation energies

Calculations were performed using CASSCF and CASPT2.

Scalar relativistic effects were included using the Douglas-

Kroll-Hess Hamiltonian to second order [29, 30]. For these

calculations, we used a relativistic ANO-RCC basis set of

triple zeta quality [31, 32], as implemented in the MOL-

CAS 7.4 package [33].

For actinoid monoxides AnO (where An denotes an

actinoid, in particular, Th, Pa, U, Np, Pu, Am, or Cm), a

full valence active space includes 17 orbitals, in particular

13 from the actinoid and four from the oxygen. In the

CASSCF calculations, we used an active space of 16

orbitals obtained by omitting the 2s orbitals of oxygen. For

monoxides, the largest number of electrons in these active

space orbitals is 14 for CmO. In the subsequent CASPT2

calculations, the orbitals up to the 5d of the actinoid and the

1s of oxygen were kept frozen, while the remaining

valence and semi-valence orbitals, now including the

6s and 6p of the actinoid and the 2s of oxygen, were cor-

related. The full valence active space of actinoid dioxides

AnO2 contains 21 orbitals, but calculations with a complete

active space this large are not feasible, so the 2s orbitals of

oxygen, the two lowest 6dpg bonding orbitals, the corre-

sponding antibonding orbitals, and one rg* antibonding

orbital are treated as inactive. Thus, the active space is

composed of 14 orbitals, occupied primarily by the 5f, 6d,

and 7s orbitals of the actinoid atom and the 2p orbitals of

the oxygen atoms. By applying the same truncation along

the whole series, the largest active space in all our calcu-

lations contains 14 electrons in 14 orbitals for CmO2.

Spin–orbit coupling was included using the complete-

active-space state-interaction method, CASSI [34, 35],

which employs an effective one-electron spin–orbit (SO)

Hamiltonian, based on a mean field approximation to the

two-electron contribution [36]. Note that a CASSI calcu-

lation, like a CASSCF calculation, does not include

dynamical correlation. To include this, the spin-free

CASPT2 energies are used as diagonal matrix elements in

the CASSI treatment; this is called spin–orbit-corrected

CASPT2 (SO-CASPT2).

To assist in judging the validity of the CASPT2 results,

we performed exact 2-component (X2C) [37, 38] relativ-

istic calculations at the CCSD and CCSD(T) levels using

the DIRAC08 program [39]. Note that X2C is also called

infinite-order two-component (IO2C). In these coupled

cluster calculations, we correlated the electrons in the 6s,

6p, 5f, 6d, and 7s orbitals of the actinoid atom and the four

valence electrons of the 2p orbitals of the oxygen. We then

restricted the set of virtual orbitals to those most important

for valence and sub-valence correlation by deleting virtual

orbitals with an orbital energy above 10 Eh.

Note that Dyall’s transformation [38] allows one to

separate a four-component relativistic treatment into spin-

free and spin–orbit contributions. The X2C transformation

[37] to a two-component treatment that is used here is

based only on the spin-free part, i.e., we use the Dirac-

Coulomb operator without the Breit spin–orbit term.

Therefore, we add the spin–orbit contribution to the close

coupling calculations as the difference between the

SO-CASPT2 result and the spin-free CASPT2 result.

For the CCSD and CCSD(T) calculations, we used an

uncontracted double-zeta 26s23p17d13f2g basis set [40]

for actinoid atoms and the cc-pVTZ basis set [41] for

oxygen atoms.

The geometries for all WFT calculations are SO-CAS-

PT2 geometries. (Please note that the statement in [16] that

some geometries are spin-free CASPT2 is a typo.) The

zero-point vibrational energy (ZPE) change for the WFT

calculations was taken to be the same as in the M06 cal-

culations (calculated from scaled M06 frequencies as

described in Sect. 2.3).

658 Theor Chem Acc (2011) 129:657–666

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https://www.researchgate.net/publication/245933276_DIRAC04_a_relativistic_ab_initio_electronic_structure_program_release_dirac040?el=1_x_8&enrichId=rgreq-92111d85a947f492fe81d07d093d1ed5-XXX&enrichSource=Y292ZXJQYWdlOzIyNTA4OTExNztBUzo5NzY3NjExMjY5NTMwMUAxNDAwMjk5MjA4NjQx

2.2 WFT calculations of ionization potentials

The WFT calculations of ionization potentials (IPs) are

updated in two respects from those published [16] previ-

ously. Previously, PuO2, PuO2?, and PuO2

2? were com-

puted by RASPT2 [42, 43]; in the present article, all PT2

results are CAS, not RAS. Previously, the change in ZPE

was limited to the symmetric stretching mode; here, a full

ZPE correction is made, based on DFT, as mentioned at the

end of Sect 2.1.

2.3 DFT calculations

For ionization potentials and bond dissociation energies of

actinoid monoxides and dioxides, DFT calculations were

performed with the relativistic MWB60 small-core effec-

tive core potential [44] and the Stuttgart/Dresden-seg-

mented ECP60MWB_SEG basis set [45] for actinoids and

the 6-311?G(2df) basis set [46] for oxygen atoms. All

calculations were carried out with Gaussian09 [47].

Geometries for these calculations were optimized by

DFT for each functional tested. (For additional compar-

isons, we also carried out single-point energy calcula-

tions at SO-CASPT2 geometries, but these results are

given only in supporting information). For all DFT cal-

culations, we used an integration grid that has 400 radial

shells and 974 angular points per shell. All DFT calcu-

lations were performed without enforcing symmetry on

the orbitals. All calculations are spin unrestricted,

although for some of the singlets, the optimized wave

functions retain doubly occupied orbitals as in the

restricted theory even when we started with a guess that

destroys spin symmetry.

Harmonic ZPEs were scaled using the following scale

factors: M05, 0.977; M06, 0.981; M06-L, 0.978; B3LYP,

0.983; MOHLYP, 1.022; MPW3LYP, 0.983, and PW91,

1.007 [48, 49].

We added the spin–orbit contribution to all DFT cal-

culations as the difference between the SO-CASPT2 result

and the spin-free CASPT2 result (although readers who are

interested in the results without such additions may find

them in the supporting information).

3 Results

All numerical results in the article proper include spin–

orbit and ZPE contributions, and all DFT results in the

article proper are at DFT geometries. For the specialist,

the supporting information gives results without spin–

orbit, without ZPE, and without both. The supporting

information also gives DFT results at spin-free CASPT2

geometries.

Tables 1, 2, 3, and 4 report the first and second ioni-

zation potentials (IPs) of AnO and AnO2 obtained with

WFT and DFT, and Tables 5 and 6 give bond dissociation

energies for breaking An–O bonds. Because the values in

Tables 1, 2, 3, 4, 5, and 6 include ZPEs, they are enthalpy

changes at 0 K.

In tables, MUE(7) means the mean unsigned error for

the seven previous rows. MUE(x) in the last row of a table

means the mean unsigned for all x rows of data in the table.

By error, we mean the deviation from experiment. In this

regard, the reader should be aware that most of the

experimental data have an error bar large enough that it

must be taken into account in drawing conclusions about

the comparison of theory to experiment; we cannot state

that a method is better than another when their difference is

within the experimental error bar. We will therefore con-

centrate our attention mainly on analyzing which theoret-

ical models predict results that agree with experiment

within the error bars.

4 Discussion

In discussing numerical results, we will discuss only those

including spin–orbit and ZPE contributions, and we will

discuss DFT results only for calculations at DFT

Table 1 First and Second Ionization Potentials (in eV) of AnO by

WFT

CASPT2 CCSD CCSD(T) Expa

ThO 6.56 6.39 6.53 6.6035 ± 0.0008

PaO 6.28 6.18 6.31 5.9 ± 0.2

UO 6.05 5.92 5.99 6.0313 ± 0.0006

NpO 5.97 5.97 6.04 6.1 ± 0.2

PuO 6.17 5.95 6.00 6.1 ± 0.2

AmO 6.22 6.02 6.04 6.2 ± 0.2

CmO 6.68 6.42 6.46 6.4 ± 0.2

MUE(7) 0.13 0.15 0.13 0.14b

ThO? 11.94 11.92 11.99 11.8 ± 0.7

PaO? 12.10 12.36 12.46 11.8 ± 0.7

UO? 13.08 12.41 12.39 12.4 ± 0.6

NpO? 13.44 13.28 13.23 14.0 ± 0.6

PuO? 14.36 14.11 14.00 14.0 ± 0.6

AmO? 15.04 15.83 15.71 14.0 ± 0.6

CmO? 15.88 15.08 15.08 15.8 ± 0.4

MUE(7) 0.45 0.58 0.58 0.63b

MUE(14) 0.29 0.37 0.36 0.39b

a Most of the experimental values were obtained through indirect

measurements [57]; Values for ThO and UO were accurately mea-

sured by Heaven and co-workers using REMPI method [50, 51]b The MUE(x) for the experiment is referred to the average error bar

Theor Chem Acc (2011) 129:657–666 659

123

geometries. Before discussing the tables, though, we will

briefly discuss the characteristics of the ground states and

the Kohn–Sham determinants.

4.1 Ground states and Kohn–Sham determinants

The ground states and dominant electron configurations

in the CASSCF calculations for the molecules in Tables 1,

2, 3, 4, 5, and 6 are summarized in [16] and that infor-

mation will not be repeated here. For the CCSD and

CCSD(T) calculations, we have converged the Hartree–

Fock single-configuration reference wave functions to

the dominant determinant identified by the CASSCF

calculations.

For the DFT calculations on the molecules in Tables 1,

2, 3, 4, 5, and 6, in every case, we checked all possible

values of the spin projection quantum number MS to find

the lowest-energy spin state and orbital composition. We

compared our findings for the lowest-energy spin state in

DFT to the lowest spin state in spin-free CASPT2, as given

in [16]. We found that it agreed in every case except for

M05, M06, M06-L, and MOHLYP calculations of neutral

UO, for which it turned out that a triplet state is lower than

the quintet state that is predicted to be the ground state by

WFT calculations.

Spin contamination is not severe (see Table S35 of

Supporting Information for a detailed table). For example,

consider the dioxides. For neutral AnO2, the expectation

value of S2 is within 1% of MS (MS ? 1) in all cases, where

MS, as usual, denotes the projection quantum number of

electron spin. The AnO2? monocations satisfy this condi-

tion within 1% for MS B 3/2 and within 2% for other MS.

Table 2 First and second ionization potentials (eV) of AnO from DFT Calculations

M05 M06 M06-L B3LYP MOHLYP MPW3LYP PW91 Expa

ThO 6.61 6.84 6.45 6.52 6.17 6.57 6.47 6.6035 ± 0.0008

PaO 6.31 6.67 5.31 6.30 5.78 6.32 8.40 5.9 ± 0.2

UO 5.71 6.09 5.55 6.18 5.84 6.21 6.22 6.0313 ± 0.0006

NpO 5.63 5.79 5.42 6.26 5.85 6.29 6.28 6.1 ± 0.2

PuO 5.75 5.88 5.54 6.38 5.97 6.42 6.39 6.1 ± 0.2

AmO 6.04 6.08 5.77 6.50 6.21 6.54 6.59 6.2 ± 0.2

CmO 6.14 6.27 5.87 6.67 6.19 6.71 6.63 6.4 ± 0.2

MUE(7) 0.28 0.26 0.49 0.23 0.19 0.25 0.56 0.14b

ThO? 11.63 11.84 11.36 12.21 11.82 12.24 12.18 11.8 ± 0.7

PaO? 12.36 12.42 12.86 12.63 14.81 12.67 12.69 11.8 ± 0.7

UO? 13.01 12.95 12.75 13.08 12.63 13.11 13.01 12.4 ± 0.6

NpO? 13.54 13.60 13.29 13.75 13.14 13.78 13.56 14.0 ± 0.6

PuO? 14.19 14.38 14.05 14.42 13.80 14.44 14.22 14.0 ± 0.6

AmO? 15.29 15.37 14.93 15.33 14.69 15.36 15.12 14.0 ± 0.6

CmO? 15.28 15.40 15.15 15.44 15.11 15.47 15.49 15.8 ± 0.4

MUE(7) 0.54 0.54 0.60 0.61 0.81 0.63 0.57 0.63b

MUE(14) 0.41 0.40 0.54 0.42 0.50 0.44 0.56 0.39b

a Most of the experimental values were obtained through indirect measurements [57]; Values for ThO and UO were accurately measured by

Heaven and co-workers using REMPI method [50, 51]b The MUE(x) for the experiment is referred to the average error bar

Table 3 First and second ionization potentials (in eV) of AnO2 from

WFT calculations

CASPT2 CCSD CCSD(T) Expa

ThO2 8.49 8.58 8.58 8.9 ± 0.4

PaO2 5.71 5.93 6.04 5.9 ± 0.2

UO2 6.24 6.05 6.07 6.128 ± 0.003

NpO2 6.30 5.91 6.01 6.33 ± 0.18

PuO2 6.23 5.62 5.73 7.03 ± 0.12

AmO2 6.80 6.66 6.55 7.23 ± 0.15

CmO2 8.31 7.77 7.60 8.5 ± 1.0

MUE(7) 0.31 0.51 0.53 0.29b

ThO2? 15.09 15.86 15.39 16.6 ± 1.0

PaO2? 16.98 16.93 16.87 16.6 ± 0.4

UO2? 14.38 14.13 14.01 14.6 ± 0.4

NpO2? 15.59 15.44 15.26 15.1 ± 0.4

PuO2? 15.38 15.84 15.62 15.1 ± 0.4

AmO2? 16.29 16.45 16.33 15.7 ± 0.6

CmO2? 16.17 16.10 15.95 17.9 ± 1.0

MUE(7) 0.74 0.74 0.76 0.60b

MUE(14) 0.53 0.62 0.65 0.45b

a Most of the experimental values were obtained through indirectmeasurements [57]; Value for UO2 was accurately determined by Hanet al. [52] using REMPI methodb The MUE(x) for the experiment is referred to the average error bar

660 Theor Chem Acc (2011) 129:657–666

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For the AnO22? dications, the condition is satisfied within

1% for MS B �, within 2% for MS = 1, within 4% for

MS = 3/2, and within 6% for MS = 2. The only exception

is singlet ThO22? molecule, for which S2 has high values

(0.52–0.70) except for the MOHLYP functional

(S2 = 0.09). For the B3LYP functional, the optimization of

the ThO22? as closed-shell singlet (i.e. with constrained

S2 = 0.0) results in Th–O distance 1.851 A, while the

optimization as an open-shell singlet with S2 = 0.7 results

in Th–O distance 1.909 A. The latter structure is 7.6 kcal/

mol lower than the former one. For analogous M06 cal-

culations of ThO22? the distance of closed-shell singlet is

1.800 A, for open-shell singlet is 1.867 A. The latter

structure is 2.2 kcal/mol lower than the former one.

Comparing the orbital character of the DFT determi-

nants to the orbital composition of the CASSCF wave

function is harder because in most cases where the con-

tribution of the Hartree–Fock determinant in the CASSCF

wave function is less than 80% (see Tables 3 and 4 of

[16]), the Kohn–Sham orbitals break symmetry. In several

cases, even when the system can be described to better than

90% by a single configuration, two singly occupied orbitals

(for example, the d and / orbitals in the cases of NpO2?

and the isoelectronic PuO22?) have very similar orbital

energies, and the Kohn–Sham orbitals are mixtures of these

orbitals. When the orbitals do not break symmetry, we can

compare the DFT orbital character to the dominant orbital

composition case of the CASSCF wave function. As an

example, consider the case of M06 calculations for AnO2

and their cations. In most cases, for M06 method, the DFT

orbitals correspond to the CASPT2 orbitals of AnO2 and

their cations. The exceptions are UO2, NpO2, and PuO2

molecules. For NpO2, M06 predicts a 4Uu ground state

when compared with 4Hg for CASPT2, and for PuO2, M06

predicts a 5Rg ground state when compared with 5Uu for

CASPT2. For the UO2 molecule, the M05, M06, and M06-

L functionals give the ground state as 3Hg, while other DFT

methods and WFT methods give the 3Uu state as the ground

state. The difference is due to the HOMO orbital, which is

the 7s orbital of U for the 3Uu state and 5fd orbital of U for

the 3Hg state. In the latter case, singly occupied d and /orbitals are mixed, and their linear combinations constitute

two broken-symmetry orbitals with the same energy.

4.2 Ionization potentials

When comparing WFT results with DFT results, we have to

contextualize the values with respect to available experi-

mental data. The most accurate experiments are for ThO

[50], UO [51], and UO2 [52], where the error on the IP is less

than or equal to 0.003 eV. CASPT2 and CCSD(T) perform

rather well for these cases, with an MUE(3) of 0.06 eV

(where MUE(x) denotes mean unsigned error averaged over

x molecules), while CCSD performs worse—the MUE(3) is

0.13 eV. CASPT2 overcomes this deficiency by a multi-

configurational treatment of static correlation [8, 9], and

Table 4 First and second ionization potentials (eV) of AnO2 from DFT calculations

M05 M06 M06-L B3LYP MOHLYP MPW3LYP PW91 Expa

ThO2 8.31 8.52 8.00 8.56 8.02 8.62 8.39 8.9 ± 0.4

PaO2 5.82 6.13 6.59 6.34 7.15 6.35 6.27 5.9 ± 0.2

UO2 6.08 6.07 5.81 6.25 5.89 6.29 6.27 6.128 ± 0.003

NpO2 6.31 6.21 5.77 6.36 6.08 6.36 6.40 6.33 ± 0.18

PuO2 6.46 6.60 6.17 6.61 6.29 6.61 6.58 7.03 ± 0.12

AmO2 6.98 6.96 6.58 7.17 6.22 7.17 6.84 7.23 ± 0.15

CmO2 8.21 8.22 7.74 8.27 7.46 8.29 7.82 8.5 ± 1.0

MUE(7) 0.27 0.25 0.67 0.24 0.77 0.23 0.38 0.29b

ThO2? 16.34 16.43 16.07 16.30 15.94 16.32 16.38 16.6 ± 1.0

PaO2? 16.70 17.03 16.44 16.93 17.47 17.00 16.70 16.6 ± 0.4

UO2? 15.35 15.14 14.86 15.08 14.49 15.09 14.96 14.6 ± 0.4

NpO2? 16.30 16.27 16.02 16.21 15.45 16.23 15.94 15.1 ± 0.4

PuO2? 16.45 16.40 15.94 16.32 15.72 16.34 16.13 15.1 ± 0.4

AmO2? 16.50 16.59 16.18 16.44 15.90 16.45 16.25 15.7 ± 0.6

CmO2? 16.41 16.48 16.00 16.52 15.60 16.53 16.04 17.9 ± 1.0

MUE(7) 0.85 0.84 0.73 0.80 0.73 0.81 0.71 0.60b

MUE(14) 0.56 0.55 0.70 0.52 0.75 0.52 0.54 0.45b

a Most of the experimental values were obtained through indirect measurements [57]; Value for UO2 was accurately determined by Han et al.

[52] using REMPI methodb The MUE(x) for the experiment is referred to the average error bar

Theor Chem Acc (2011) 129:657–666 661

123

density functional theory includes some static correlation in

the exchange potential [53–56]. On the same molecules, the

M05, M06, MPW3LYP, and B3LYP functionals give

MUE(3) = 0.12 eV, PW91 gives 0.16 eV, MOHLYP gives

0.29 eV, and M06-L gives 0.32 eV.

For the other set of AnO molecules, there is no direct

measurement of the IP, and the experimental error bar of

the first ionization potential is rather large, about 0.2 eV

[57]. Inspection of Table 1 shows that the mean unsigned

error (MUE) for CASPT2, CCSD, and CCSD(T) for the

first IP of AnO is, respectively, 0.13, 0.15, and 0.13 eV.

Thus, among the wave function methods, all three perform

within the experimental uncertainty. The DFT results for

IPs are reported in Table 2. For first IPs, the MUE is 0.28,

0.26, 0.49, 0.23, 0.19, 0.25, and 0.56 eV for M05, M06,

M06-L, B3LYP, MOHLYP, MPW3LYP, and PW91,

respectively.

For the second IP, the experiments show even larger

error bars, ranging from 0.4 eV for CmO? to 0.6–0.7 eV

for the remaining AnO monocationic species [57]. Almost

all WFT and DFT methods agree with experiment within

these limits (the only exception being MOHLYP), in par-

ticular the WFT methods give MUE values of 0.45

(CASPT2), 0.58 (CCSD), and 0.58 eV (CCSD(T)). The

MUEs are 0.54, 0.54, and 0.60 eV for M05, M06, M06-L,

respectively, and 0.61, 0.81, 0.63, and 0.57 eV for B3LYP,

MOHLYP, MPW3LYP, and PW91, respectively.

Now we consider the AnO2 systems. The IPs of these

molecules have been measured with three different meth-

ods, each of them bearing a different error bar. The only

very accurate direct measurement (using the REMPI

method) has been carried out for UO2 [52]. The second

type of experiment employed to measure the first ionization

potential of some of the actinoid species is a well-estab-

lished electron-transfer bracketing technique, which is an

indirect approach that carries an error in the range of

0.1–0.2 eV. In particular, this method has been employed

by Marcalo and Gibson [57] to obtain the IPs of NpO2,

PuO2, and AmO2, with error bars of 0.18, 0.12, and

0.15 eV, respectively. Table 4 shows that WFT methods

tend to give results lower than experiment by more than the

experimental error bar, more prominently for the CCSD

approaches. This has been discussed previously [16], in

particular for the PuO2 molecule, where the first IP was

computed to be considerably lower than the experiment.

The first IP of PuO2 is computed by DFT to range from a

minimum of 6.17 eV for M06-L to a maximum of 6.61 eV

for MPW3LYP and B3LYP. We notice that these values

still fall far from the experimental error bar, and quantum

chemical methods therefore suggest that the actual value of

the first IP of PuO2 is closer to 6–6.5 eV than to 7 eV.

CASPT2 and five of the density functionals give very

similar values for the IP of NpO2, all in a small window of

energies, between 6.21 and 6.36 eV. The 6.21–6.36 eV

window agrees well with the experiment. The CCSD,

CCSD(T), MOHLYP, and M06-L calculations give a lower

IP, outside the experimental error bar.

For AmO2, we encounter a similar situation to PuO2,

with all computed first ionization potentials below the

experimental error bar; however, they are closer to exper-

iment than in the case of PuO2. WFT shows the largest

differences, while B3LYP and MPW3LYP give IPs inside

the experimental error bar. Further experimental work to

confirm that the actual ionization potentials are lower than

the one measured will be difficult due to the hazardous

nature of the materials, and the bracketing technique will

probably remain the most reliable experimental method for

some time.

The third method used to measure the IP of actinoid

species is an indirect method based on the reactions of

actinoid molecules with species of known IPs [57]. In this

case, the error bars are quite large, 0.2 eV for PaO2,

Table 5 Bond Energies (in kcal/mol) for AnO2 AnO2? and AnO2

2?

from WFT Calculations

CASPT2 CCSD CCSD(T) Exp.a

ThO2 147 139 144 164 ± 3

PaO2 190 172 181 187 ± 11

UO2 146 170 183 180 ± 3

NpO2 169 143 158 152 ± 10

PuO2 134 128 144 144 ± 5

AmO2 119 96 110 122 ± 16

CmO2 108 81 95 97 ± 17

MUE(7) 14 17 7 9b

ThO2? 103 88 96 111 ± 9

PaO2? 203 177 185 187 ± 7

UO2? 181 171 182 178 ± 3

NpO2? 162 144 156 146 ± 5

PuO2? 131 130 143 122 ± 9

AmO2? 117 84 98 98 ± 13

CmO2? 73 51 88 49 ± 14

MUE(7) 13 9 13 9b

ThO22? 30 -4 16 0 ± 41

PaO22? 92 73 85 76 ± 26

UO22? 143 127 140 127 ± 7

NpO22? 113 94 109 121 ± 2

PuO22? 104 92 107 97 ± 23

AmO22? 67 67 83 61 ± 31

CmO22? 41 28 47 0 ± 36

MUE(7) 18 10 19 24

MUE(21) 15 12 13 14

a The experimental values from Marcalo and Gibson work [57]

obtained using Born cycle utilizing the measured ionization potentialsb The MUE(x) for the experiment is referred to the average error bar

662 Theor Chem Acc (2011) 129:657–666

123

increasing to 0.4 for ThO2 and to 1.0 eV for CmO2. For

these cases, WFT calculations yield values falling within

the experimental limits. For PaO2, M05 and M06 methods

give values falling within the experimental limits, while

other DFT functionals overestimate the experimental value

of 5.9 eV, especially MOHLYP (7.15 eV). For ThO2, M06,

B3LYP, and MPW3LYP give values falling within the

experimental limits, while other functionals underestimate

the experimental value 8.9 eV.

For the second IPs, the experimental error bars are rather

large, ranging from 0.4 to 1.0 eV [57]. As in the case of the

monoxides, the calculations show significant discrepancies

from the middle experimental value. On average, WFT and

DFT have roughly the same MUE(7); however, in specific

cases, they may differ from each other by a significant

amount. For example, looking at the second IP of UO2, the

CASPT2 calculation gives 14.38 eV, CCSD(T) gives

14.01 eV, and M05, which has performed well in cases

discussed above, yields a value 1 eV higher, 15.35 eV.

Such discrepancies occur quite often in Tables 3 and 4.

Unfortunately, due to the large error bars, the experimental

values do not help much in deciding which method is most

reliable.

The overall performance of methods for the first and

second ionization potentials of AnO and AnO2 is summa-

rized by MUE(AIP28) in Table 7 (The other rows of this

table are discussed below).

4.3 Bond energies

We now consider the performance of the various meth-

ods for bond dissociation energies (BDEs), bearing in

mind that their experimental values are much more

uncertain than those for ionization potentials. This is due

to the fact that the BDEs are computed by a Born cycle

using the measured ionization potentials [57], hence

inheriting their errors along with the errors for other

steps in the cycle. Furthermore, we must consider the

following issue regarding the CASPT2 calculations.

When calculating the reaction AnO2 ? AnO ? O, it is

difficult to keep a balanced active space between the

AnO2 and its dissociation products because a full valence

Table 6 Bond Energies (kcal/mol) for AnO2, AnO2?, and AnO2

2? from DFT Calculations

M05 M06 M06-L B3LYP MOHLYP MPW3LYP PW91 Exp.a

ThO2 159 155 166 157 163 158 176 164 ± 3

PaO2 165 169 199 179 186 181 146 187 ± 11

UO2 172 170 179 171 182 172 193 180 ± 3

NpO2 163 167 172 154 168 155 178 152 ± 10

PuO2 148 153 159 141 155 142 165 144 ± 5

AmO2 109 115 122 108 112 109 128 122 ± 16

CmO2 104 106 116 101 114 102 125 97 ± 17

MUE(7) 10 11 10 7 8 6 21 9b

ThO2? 121 116 130 111 121 112 132 111 ± 9

PaO2? 176 181 169 178 154 179 195 187 ± 7

UO2? 163 171 173 169 180 170 191 178 ± 3

NpO2? 146 156 163 150 161 151 173 146 ± 5

PuO2? 126 131 139 130 141 131 154 122 ± 9

AmO2? 88 95 104 93 112 94 123 98 ± 13

CmO2? 58 62 74 65 86 67 98 49 ± 14

MUE(7) 8 8 15 7 18 7 25 9b

ThO22? 12 11 22 17 25 18 35 0 ± 41

PaO22? 72 76 88 79 94 81 103 76 ± 26

UO22? 102 115 119 118 132 119 141 127 ± 7

NpO22? 84 94 100 93 107 94 117 121 ± 2

PuO22? 81 87 98 88 99 90 112 97 ± 23

AmO22? 61 67 74 67 83 69 97 61 ± 31

CmO22? 28 34 51 37 71 39 82 0 ± 36

MUE(7) 18 14 18 16 23 16 30 24b

MUE(21) 12 11 14 10 16 10 25 14b

a The experimental values from Marcalo and Gibson work [57] obtained using Born cycle utilizing the measured ionization potentialsb The MUE(x) for the experiment is referred to the average error bar

Theor Chem Acc (2011) 129:657–666 663

123

active space is impractical for AnO2, and we need to

truncate the space (see Sect. 2). This lack of balance at

the CASSCF level could be reflected in the description

of the correlation energy, and the error at the CASPT2

level can be larger than what is expected on the basis

of calculations where a full valence active space is

employed. Inspection of Table 5 confirms this by

showing that the MUE for CASPT2 is slightly larger

than that for CCSD(T), especially for the dissociation of

the neutral AnO2 (14 vs. 7 kcal/mol).

DFT methods perform well, especially B3LYP and

MPW3LYP. The M06 functional shows slightly larger

differences from the experiment, but still performs well.

The overall performance of methods for dissociation

energies of AnO2 is shown by MUE(ABDE21) in Table 7.

The bad performance of PW91 is related to systematic

overestimation of the dissociation energies of AnO2 by this

method, with the exception of PaO2, for which PW91

underestimates the bond dissociation energy by 41 kcal/

mol.

4.4 Bond distances

The Ac–O bond distance was optimized for all oxides with

the various density functionals. In Fig. 1, we report the

deviations of DFT bond distances from the CASPT2 bond

distances (A). The absolute values are reported in Sup-

porting Information. The chart shows a nearly systematic

deviation of bond distances predicted by the various

exchange–correlation functionals from those predicted by

CASPT2.

4.5 Overall assessment

The overall performance for both ionization potentials and

bond dissociation energies is given by MUE(AC49) in

Table 7. Four of the density functionals (those containing a

nonzero fraction of Hartree–Fock exchange) and all three

WFT methods show very similar average performance.

Among the three functionals without Hartree–Fock

exchange, the M06-L and MOHLYP functionals perform

better than PW91.

5 Conclusions and outlook

The systems discussed in this paper pose non-trivial chal-

lenges because of their complex electronic structure and

relativistic effects. We have compared the performance of

different methods with the hope of making future users

more aware of their strengths and limitations.

The CASPT2 wave function method used here has been

used successfully in studying many actinoid-containing

systems [58–63]. Recent developments of the Cholesky

decomposition (CD) approach (they have not been

employed in the present study), in combination with the

Table 7 Performance of Tested Methods for 28 Actinoid Ionization Potentials MUE(AIP28) for 21 Actinoid Bond Dissociation Energies

MUE(ABDE21) and for the Actinoid Composite Energy Database of all 49 Ionization Potential and Bond Dissociation Energy Data

MUE(ACE49)

CASPT2 CCSD CCSD(T) M05 M06 M06-L B3LYP MOHLYP MPW3LYP PW91

In kcal/mol

MUE(ABDE21) 15 12 13 12 11 14 10 16 10 25

In eV

MUE(AIP28) 0.41 0.50 0.50 0.49 0.47 0.62 0.47 0.63 0.48 0.55

MUE(ABDE21) 0.65 0.53 0.56 0.52 0.47 0.63 0.43 0.71 0.43 1.10

MUE(AC49) 0.51 0.51 0.52 0.50 0.47 0.63 0.45 0.66 0.46 0.79

Fig. 1 Deviations of DFT bond

distances from the CASPT2

bond distances (A)

664 Theor Chem Acc (2011) 129:657–666

123

CASPT2 method [64], have extended the applicability of

the CASPT2 method to larger molecular systems. A CD-

CASPT2 calculation with 1000 basis functions is nowadays

affordable [65]. In addition, CD-CASPT2 calculations can

be performed at even lower computational costs and on

large systems by invoking the frozen natural orbital (FNO)

approximation [66, 67] to effectively reduce the size of the

virtual orbital space with no loss of accuracy. However,

increasing the molecular size is not sufficient. There are

many quantum-chemical problems whose accurate solution

is out of reach for CASPT2 because the adequate number

of MOs cannot be included in the active space [42]. Wave

function methods such as CCSD and CCSD(T) are even

more expensive for large systems. Density functional the-

ory on the other hand remains quite affordable for large

systems, and it has had many successes for transition metal

chemistry [56, 68]. Here, we find that DFT, at least with

well-chosen functionals such as MPW3LYP, B3LYP, M06,

and M05, is, on average, as accurate as or more accurate

than CASPT2, CCSD, and CCSD(T) methods for bond

energies to actinoids. These four functionals also provide

comparable accuracy for ionization potentials of actinoid

compounds.

Acknowledgments This work was supported by the Director, Office

of Basic Energy Sciences, U.S. Department of Energy under Contract

No. USDOE/DE-SC002183, by the Air force Office of Scientific

Research under grant no. FA9550-08-1-0183, by the National Science

Foundation under grant No. CHE09-56776, by 7th Framework Pro-

gramme of the European Commission (Collaboration Project No.

211690), by the Hungarian Scientific Research Foundation (OTKA

No. 75972), and by the University of Minnesota Supercomputing

Institute. We would like to thank John K. Gibson, Berkeley National

Laboratory, for useful discussion.

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S1

Supporting Information for

How accurate are electronic structure methods for actinoid chemistry?

Boris Averkiev, Manjeera Mantina, Rosendo Valero, Ivan Infante, Attila Kovacs, Donald G. Truhlar, and Laura Gagliardi

Date of preparation of supporting information: Feb. 11, 2011

Number of pages, including this page: 27

Contents

Tables S1–S2. Calculated bond distances

Tables S3–S4. Calculated zero point vibrational energies

Table S5–S34. Ionization potentials and bond energies

In Tables S5–S34 the MUE for the theoretical columns is the mean unsigned deviation

from the experimental values, whereas the MUE entry for the experimental column is

the average error bar. All numerical results in the article proper include spin–orbit

coupling and zero point vibrational energy (ZPE) contributions, and all DFT results in

the article proper are at DFT geometries. For the specialist, this supporting information

gives results without spin-orbit coupling, without ZPE, and without both. The

supporting information also gives DFT results at spin-free CASPT2 geometries.

Table S35. Calculated expectation values of the square of the total electron spin

Table S36. Bond distances for AnO2 calculated with the aug-cc-pVTZ basis set for oxygen

S2

Table S1. Bond Distances (Ǻ) for AnO

CASPT2 M05 M06 M06-L B3LYP MOHLYP MPW3LYP PW91 ThO 1.863 1.805 1.812 1.818 1.833 1.845 1.832 1.835 PaO 1.818 1.779 1.786 1.792 1.804 1.823 1.803 1.818 UO 1.838 1.788a 1.773a 1.812a 1.842 1.825a 1.842 1.838 NpO 1.839 1.813 1.811 1.822 1.836 1.838 1.836 1.831 PuO 1.820 1.804 1.804 1.822 1.829 1.841 1.829 1.828 AmO 1.801 1.847 1.820 1.841 1.838 1.861 1.837 1.845 CmO 1.836 1.832 1.827 1.839 1.843 1.851 1.843 1.840 ThO+ 1.827 1.778 1.776 1.786 1.800 1.812 1.799 1.803 PaO+ 1.804 1.750 1.754 1.767 1.777 1.792 1.790 1.792 UO+ 1.796 1.780 1.784 1.792 1.800 1.802 1.800 1.796 NpO+ 1.798 1.770 1.770 1.784 1.793 1.799 1.792 1.791 PuO+ 1.789 1.763 1.767 1.778 1.792 1.797 1.792 1.788 AmO+ 1.782 1.763 1.765 1.778 1.782 1.797 1.782 1.787 CmO+ 1.792 1.785 1.783 1.794 1.798 1.812 1.798 1.802 ThO2+ 1.790 1.740 1.744 1.753 1.764 1.776 1.764 1.768 PaO2+ 1.733 1.723 1.727 1.737 1.747 1.762 1.747 1.755 UO2+ 1.720 1.700 1.703 1.716 1.725 1.743 1.725 1.736 NpO2+ 1.723 1.690 1.694 1.714 1.717 1.739 1.716 1.730 PuO2+ 1.731 1.693 1.698 1.728 1.723 1.755 1.722 1.743 AmO2+ 1.808 1.734 1.742 1.796 1.778 1.796 1.777 1.778 CmO2+ 1.791 1.868 1.854 1.827 1.842 1.789 1.841 1.780 a distances for lowest (triplet) states. Optimized distances for pentet states are 1.821, 1.822,

1.830, and 1.845 Ǻ for M05, M06, M06-L, and MOHLYP, respectively.

S3

Table S2. Bond Distances (Ǻ) for AnO2 and Angle (degrees) for ThO2

CASPT2 M05 M06 M06-L B3LYP MOHLYP MPW3LYP PW91 ThO2 1.923 1.868 1.872 1.883 1.898 1.909 1.897 1.898 ThO2

a 111.9 121.0 120.8 118.3 119.0 119.0 119.0 118.5 PaO2 1.816 1.779 1.786 1.806 1.806 1.833 1.805 1.813 UO2 1.827 1.813 1.816 1.830 1.789 1.818 1.789 1.808 NpO2 1.761 1.791 1.795 1.809 1.821 1.815 1.821 1.803 PuO2 1.744 1.782 1.786 1.805 1.814 1.813 1.813 1.802 AmO2 1.807 1.793 1.800 1.828 1.827 1.856 1.827 1.827b CmO2 1.832 1.809 1.815 1.839 1.839 1.852 1.838 1.840 ThO2

+ 1.832 1.814 1.827 1.847 1.868 1.879 1.868 1.870 PaO2

+ 1.767 1.740 1.745 1.759 1.767 1.812 1.767 1.777 UO2

+ 1.745 1.725 1.730 1.745 1.753 1.773 1.753 1.764 NpO2

+ 1.723 1.704 1.708 1.724 1.732 1.754 1.731 1.745 PuO2

+ 1.704 1.686 1.692 1.709 1.714 1.737 1.714 1.728 AmO2

+ 1.721 1.684 1.690 1.714 1.716 1.743 1.716 1.733 CmO2

+ 1.746 1.709 1.716 1.748 1.748 1.770 1.747 1.759 ThO2

2+ 1.903 1.781 1.867 1.848 1.909 1.864 1.909 1.856

PaO22+ 1.726 1.728 1.738 1.756 1.772 1.792 1.772 1.785

UO22+ 1.710 1.667 1.672 1.688 1.693 1.717 1.693 1.710

NpO22+ 1.700 1.659 1.665 1.684 1.688 1.715 1.688 1.708

PuO22+ 1.675 1.646 1.650 1.670 1.674 1.704 1.673 1.696

AmO22+ 1.679 1.636 1.642 1.664 1.666 1.696 1.666 1.688

CmO22+ 1.674 1.647 1.653 1.687 1.688 1.718 1.688 1.707

a angle values for ThO2 molecule, which has a bent geometry With C2v symmetry bMolecule has a bent geometry with C2v symmetry and a bond angle of 175.1 deg.

S4

Table S3. Zero Point Energies (kcal/mol) for AnO

M05 M06 M06-L B3LYP MOHLYP MPW3LYP PW91 ThO 1.36 1.35 1.31 1.30 1.25 1.30 1.27 PaO 1.38 1.37 1.34 1.34 1.28 1.34 1.30 UO 1.25a 1.30a 1.22a 1.20 1.21a 1.20 1.20 NpO 1.21 1.23 1.18 1.19 1.16 1.19 1.18 PuO 1.21 1.22 1.14 1.16 1.10 1.16 1.14 AmO 1.04 1.12 1.06 1.11 1.04 1.11 1.08 CmO 1.18 1.20 1.15 1.17 1.12 1.17 1.15 ThO+ 1.39 1.45 1.39 1.38 1.33 1.38 1.35 PaO+ 1.47 1.47 1.42 1.37 1.36 1.34 1.31 UO+ 1.34 1.34 1.31 1.31 1.25 1.31 1.26 NpO+ 1.33 1.33 1.29 1.29 1.25 1.29 1.27 PuO+ 1.33 1.31 1.26 1.28 1.21 1.28 1.24 AmO+ 1.29 1.29 1.23 1.26 1.19 1.26 1.22 CmO+ 1.29 1.31 1.27 1.28 1.21 1.27 1.23 ThO2+ 1.55 1.54 1.50 1.48 1.43 1.48 1.45 PaO2+ 1.57 1.55 1.50 1.49 1.43 1.49 1.45 UO2+ 1.56 1.55 1.48 1.48 1.39 1.48 1.42 NpO2+ 1.52 1.50 1.41 1.44 1.33 1.44 1.36 PuO2+ 1.41 1.39 1.21 1.32 1.17 1.32 1.22 AmO2+ 1.01 0.98 0.76 0.89 0.93 0.90 1.00 CmO2+ 0.90 0.90 0.87 0.92 1.03 0.92 1.04 aZPEs for lowest (triplet) states. ZPEs for pentet states are 1.23, 1.24, 1.20, and 1.18

kcal/mol for M05, M06, M06-L, and MOHLYP, respectively

S5

Table S4. Zero Point Energies (kcal/mol) for AnO2

M05 M06 M06-L B3LYP MOHLYP MPW3LYP PW91 ThO2 2.58 2.56 2.50 2.49 2.41 2.49 2.45 PaO2 2.97 2.96 2.74 2.86 2.62 2.86 2.78 UO2 2.68 2.69 2.56 2.99 2.69 3.00 2.76 NpO2 2.89 2.88 2.75 2.69 2.47 2.69 2.51 PuO2 2.99 2.98 2.79 2.78 2.71 2.78 2.77 AmO2 2.81 2.76 2.50 2.53 2.31 2.54 2.24 CmO2 2.70 2.42 2.43 2.43 2.37 2.44 2.39 ThO2

+ 2.55 2.36 2.24 2.06 2.19 2.05 2.23 PaO2

+ 3.24 3.23 3.08 3.09 2.77 3.08 2.97 UO2

+ 3.42 3.39 3.21 3.22 3.04 3.22 3.08 NpO2

+ 3.64 3.61 3.44 3.40 3.19 3.40 3.24 PuO2

+ 3.77 3.73 3.54 3.53 3.31 3.52 3.37 AmO2

+ 3.77 3.72 3.44 3.47 3.22 3.47 3.30 CmO2

+ 3.40 3.31 2.97 3.00 2.90 3.00 2.97 ThO2

2+ 3.06 2.09 2.24 2.20 2.53 2.20 2.41 PaO2

2+ 3.12 2.88 2.72 2.45 2.57 2.45 2.59 UO2

2+ 3.83 3.78 3.54 3.60 3.28 3.60 3.32 NpO2

2+ 3.87 3.81 3.53 3.60 3.25 3.60 3.29 PuO2

2+ 4.03 3.99 3.72 3.75 3.38 3.75 3.45 AmO2

2+ 4.12 4.06 3.80 3.82 3.52 3.82 3.59 CmO2

2+ 3.79 3.75 3.35 3.36 3.20 3.36 3.30

S6

Table S5. First and Second Ionization Potentials (eV) of AnO from WFT Calculations at the CASPT2 Geometry Without ZPE

CASPT2 CCSD CCSD(T) CASPT2 CCSD CCSD(T) Exp. Without Spin–Orbit Coupling With Spin–Orbit Coupling

ThO 6.58 6.41 6.55 6.56 6.39 6.53 6.6035 ± 0.0008 PaO 6.48 6.38 6.51 6.28 6.18 6.31 5.9 ± 0.2 UO 6.04 5.91 5.98 6.05 5.92 5.99 6.0313 ± 0.0006 NpO 5.98 5.98 6.05 5.97 5.97 6.04 6.1 ± 0.2 PuO 6.16 5.94 5.99 6.17 5.95 6.00 6.1 ± 0.2 AmO 6.24 6.04 6.06 6.21 6.01 6.03 6.2 ± 0.2 CmO 6.68 6.42 6.46 6.68 6.42 6.46 6.4 ± 0.2 MUE(7) 0.16 0.18 0.15 0.13 0.16 0.13 0.14 ThO+ 11.94 11.92 11.99 11.94 11.92 11.99 11.8 ± 0.7 PaO+ 11.91 12.17 12.27 12.10 12.36 12.46 11.8 ± 0.7 UO+ 12.99 12.32 12.30 13.07 12.40 12.38 12.4 ± 0.6 NpO+ 13.50 13.34 13.29 13.43 13.27 13.22 14.0 ± 0.6 PuO+ 14.56 14.31 14.20 14.36 14.11 14.00 14.0 ± 0.6 AmO+ 15.12 15.91 15.79 15.05 15.84 15.72 14.0 ± 0.6 CmO+ 15.90 15.20 15.20 15.90 15.10 15.10 15.8 ± 0.4 MUE(7) 0.45 0.58 0.58 0.46 0.58 0.58 0.63 MUE(14) 0.30 0.38 0.37 0.29 0.37 0.36 0.39

Table S6. First and Second Ionization Potentials (eV) of AnO from WFT Calculations at the CASPT2 Geometry With ZPE


ThO 6.58 6.41 6.55 6.56 6.39 6.53 6.6035 ± 0.0008 PaO 6.48 6.38 6.51 6.28 6.18 6.31 5.9 ± 0.2 UO 6.04 5.91 5.98 6.05 5.92 5.99 6.0313 ± 0.0006 NpO 5.98 5.98 6.05 5.97 5.97 6.04 6.1 ± 0.2 PuO 6.16 5.94 5.99 6.17 5.95 6.00 6.1 ± 0.2 AmO 6.25 6.05 6.07 6.22 6.02 6.04 6.2 ± 0.2 CmO 6.68 6.42 6.46 6.68 6.42 6.46 6.4 ± 0.2 MUE(7) 0.16 0.18 0.15 0.13 0.15 0.13 0.14 ThO+ 11.94 11.92 11.99 11.94 11.92 11.99 11.8 ± 0.7 PaO+ 11.91 12.17 12.27 12.10 12.36 12.46 11.8 ± 0.7 UO+ 13.00 12.33 12.31 13.08 12.41 12.39 12.4 ± 0.6 NpO+ 13.51 13.35 13.30 13.44 13.28 13.23 14.0 ± 0.6 PuO+ 14.56 14.31 14.20 14.36 14.11 14.00 14.0 ± 0.6 AmO+ 15.11 15.90 15.78 15.04 15.83 15.71 14.0 ± 0.6 CmO+ 15.88 15.18 15.18 15.88 15.08 15.08 15.8 ± 0.4 MUE(7) 0.44 0.58 0.58 0.45 0.58 0.58 0.63 MUE(14) 0.30 0.38 0.37 0.29 0.37 0.36 0.39

S7

Table S7. First and Second Ionization Potentials (eV) of AnO from DFT Calculations at the CASPT2 Geometry Without ZPE and Without Spin–Orbit Coupling

M05 M06 M06-L B3LYP MOHLYP MPW3LYP PW91 Exp. ThO 6.61 6.87 6.46 6.54 6.19 6.58 6.48 6.6035 ± 0.0008 PaO 6.55 6.91 5.53 6.51 5.98 6.52 8.61 5.9 ± 0.2 UO 5.66 6.00 5.53 6.16 5.82 6.20 6.21 6.0313 ± 0.0006 NpO 5.64 5.79 5.42 6.26 5.86 6.30 6.28 6.1 ± 0.2 PuO 5.75 5.88 5.53 6.36 5.95 6.40 6.37 6.1 ± 0.2 AmO 6.03 6.10 5.76 6.50 6.18 6.54 6.58 6.2 ± 0.2 CmO 6.14 6.27 5.86 6.66 6.19 6.70 6.63 6.4 ± 0.2 MUE(7) 0.33 0.29 0.46 0.26 0.19 0.28 0.58 0.14 ThO+ 11.65 11.83 11.36 12.21 11.81 12.24 12.17 11.8 ± 0.7 PaO+ 11.90 12.16 12.64 12.42 14.64 12.48 12.51 11.8 ± 0.7 UO+ 12.86 12.87 12.67 12.99 12.56 13.02 12.93 12.4 ± 0.6 NpO+ 13.69 13.67 13.35 13.82 13.21 13.84 13.63 14.0 ± 0.6 PuO+ 14.61 14.59 14.25 14.62 14.01 14.64 14.43 14.0 ± 0.6 AmO+ 15.47 15.48 15.00 15.43 14.77 15.46 15.21 14.0 ± 0.6 CmO+ 15.34 15.44 15.18 15.48 15.11 15.51 15.50 15.8 ± 0.4 MUE(7) 0.51 0.52 0.58 0.60 0.75 0.61 0.56 0.63 MUE(14) 0.42 0.41 0.52 0.43 0.47 0.44 0.57 0.39

Table S8. First and Second Ionization Potentials (eV) of AnO from DFT Calculations at the DFT Geometry Without ZPE and Without Spin–Orbit Coupling


S8

Table S9. First and Second Ionization Potentials (eV) of AnO from DFT Calculations at the CASPT2 Geometry With ZPE and Without Spin–Orbit Coupling


Table S10. First and Second Ionization Potentials (eV) of AnO from DFT Calculations at the DFT Geometry With ZPE and Without Spin–Orbit Coupling


S9

Table S11. First and Second Ionization Potentials (eV) of AnO from DFT Calculations at the CASPT2 Geometry Without ZPE and With Spin–Orbit Coupling


Table S12. First and Second Ionization Potentials (eV) of AnO from DFT Calculations at the DFT Geometry Without ZPE and With Spin–Orbit Coupling


S10

Table S13. First and Second Ionization Potentials (eV) of AnO from DFT Calculations at the CASPT2 Geometry With ZPE and With Spin–Orbit Coupling


Table S14. First and Second Ionization Potentials (eV) of AnO from DFT Calculations at the DFT Geometry With ZPE and With Spin–Orbit Coupling


S11

Table S15. First and Second Ionization Potentials (eV) of AnO2 from WFT Calculations at the CASPT2 Geometry Without ZPE


ThO2 8.50 8.59 8.59 8.50 8.59 8.59 8.9 ± 0.4 PaO2 5.70 5.92 6.03 5.70 5.92 6.03 5.9 ± 0.2 UO2 6.20 6.01 6.03 6.21 6.02 6.04 6.128 ± 0.003 NpO2 6.33 5.94 6.04 6.27 5.88 5.98 6.33 ± 0.18 PuO2 6.43 5.82 5.93 6.20 5.59 5.70 7.03 ± 0.12 AmO2 6.71 6.57 6.46 6.76 6.62 6.51 7.23 ± 0.15 CmO2 8.39 7.85 7.68 8.27 7.73 7.56 8.5 ± 1.0 MUE(7) 0.27 0.48 0.50 0.32 0.53 0.55 0.29 ThO2

+ 15.10 15.87 15.40 15.10 15.87 15.40 16.6 ± 1.0 PaO2

+ 16.99 16.94 16.88 16.99 16.94 16.88 16.6 ± 0.4 UO2

+ 14.51 14.26 14.14 14.36 14.11 13.99 14.6 ± 0.4 NpO2

+ 15.37 15.22 15.04 15.58 15.43 15.25 15.1 ± 0.4 PuO2

+ 15.48 15.94 15.72 15.37 15.83 15.61 15.1 ± 0.4 AmO2

+ 16.75 16.91 16.79 16.28 16.44 16.32 15.7 ± 0.6 CmO2

+ 15.97 15.90 15.75 16.15 16.08 15.93 17.9 ± 1.0 MUE(7) 0.80 0.80 0.84 0.74 0.74 0.76 0.60 MUE(14) 0.54 0.64 0.67 0.53 0.63 0.66 0.45

Table S16. First and Second Ionization Potentials (eV) of AnO2 from WFT Calculations at the CASPT2 Geometry With ZPE


ThO2 8.49 8.58 8.58 8.49 8.58 8.58 8.9 ± 0.4 PaO2 5.71 5.93 6.04 5.71 5.93 6.04 5.9 ± 0.2 UO2 6.23 6.04 6.06 6.24 6.05 6.07 6.128 ± 0.003 NpO2 6.36 5.97 6.07 6.30 5.91 6.01 6.33 ± 0.18 PuO2 6.46 5.85 5.96 6.23 5.62 5.73 7.03 ± 0.12 AmO2 6.75 6.61 6.50 6.80 6.66 6.55 7.23 ± 0.15 CmO2 8.43 7.89 7.72 8.31 7.77 7.60 8.5 ± 1.0 MUE(7) 0.26 0.46 0.48 0.31 0.51 0.53 0.29 ThO2

+ 15.09 15.86 15.39 15.09 15.86 15.39 16.6 ± 1.0 PaO2

+ 16.98 16.93 16.87 16.98 16.93 16.87 16.6 ± 0.4 UO2

+ 14.53 14.28 14.16 14.38 14.13 14.01 14.6 ± 0.4 NpO2

+ 15.38 15.23 15.05 15.59 15.44 15.26 15.1 ± 0.4 PuO2

+ 15.49 15.95 15.73 15.38 15.84 15.62 15.1 ± 0.4 AmO2

+ 16.76 16.92 16.80 16.29 16.45 16.33 15.7 ± 0.6 CmO2

+ 15.99 15.92 15.77 16.17 16.10 15.95 17.9 ± 1.0 MUE(7) 0.80 0.80 0.83 0.74 0.74 0.76 0.60 MUE(14) 0.53 0.63 0.66 0.53 0.62 0.65 0.45

S12

Table S17. First and Second Ionization Potentials (eV) of AnO2 from DFT Calculations at the CASPT2 Geometry Without ZPE and Without Spin–Orbit Coupling

M05 M06 M06-L B3LYP MOHLYP MPW3LYP PW91 Exp. ThO2 8.17 8.40 7.94 8.57 8.07 8.61 8.40 8.9 ± 0.4 PaO2 5.77 6.10 6.57 6.33 7.19 6.35 6.28 5.9 ± 0.2 UO2 6.06 6.03 5.78 6.18 5.91 6.22 6.27 6.128 ± 0.003 NpO2 6.32 6.20 5.71 6.28 6.08 6.32 6.42 6.33 ± 0.18 PuO2 6.62 6.73 6.22 6.62 6.41 6.66 6.73 7.03 ± 0.12 AmO2 6.95 6.92 6.48 7.07 6.43 7.11 6.77 7.23 ± 0.15 CmO2 8.34 8.32 7.84 8.37 7.59 8.41 7.95 8.5 ± 1.0 MUE(7) 0.26 0.24 0.69 0.22 6.86 7.15 7.02 0.29 ThO2

+ 16.49 16.46 16.11 16.25 15.90 16.28 16.38 16.6 ± 1.0 PaO2

+ 16.68 17.02 16.49 17.07 17.63 17.10 16.86 16.6 ± 0.4 UO2

+ 15.58 15.35 15.02 15.23 14.60 15.25 15.09 14.6 ± 0.4 NpO2

+ 16.17 16.11 15.82 16.00 15.21 16.03 15.71 15.1 ± 0.4 PuO2

+ 16.59 16.53 16.04 16.42 15.83 16.44 16.23 15.1 ± 0.4 AmO2

+ 17.00 17.07 16.65 16.91 16.37 16.93 16.72 15.7 ± 0.6 CmO2

+ 16.19 16.27 15.81 16.33 15.47 16.36 15.90 17.9 ± 1.0 MUE(7) 0.96 0.97 0.82 0.92 15.91 16.39 16.18 0.60 MUE(14) 0.61 0.61 0.75 0.57 11.38 11.77 11.60 0.45

Table S18. First and Second Ionization Potentials (eV) of AnO2 from DFT Calculations at the DFT Geometry Without ZPE and Without Spin–Orbit Coupling


+ 16.31 16.44 16.07 16.29 15.94 16.32 16.38 16.6 ± 1.0 PaO2

+ 16.70 17.04 16.45 16.96 17.47 17.00 16.70 16.6 ± 0.4 UO2

+ 15.48 15.27 14.99 15.21 14.64 15.24 15.11 14.6 ± 0.4 NpO2

+ 16.08 16.05 15.81 15.99 15.24 16.02 15.73 15.1 ± 0.4 PuO2

+ 16.55 16.50 16.04 16.42 15.84 16.45 16.24 15.1 ± 0.4 AmO2

+ 16.96 17.04 16.64 16.90 16.37 16.92 16.73 15.7 ± 0.6 CmO2

+ 16.21 16.29 15.81 16.32 15.42 16.35 15.86 17.9 ± 1.0 MUE(7) 0.95 0.94 0.82 0.90 0.80 0.91 0.81 0.60 MUE(14) 0.60 0.58 0.73 0.55 0.76 0.55 0.57 0.45

S13

Table S19. First and Second Ionization Potentials (eV) of AnO2 from DFT Calculations at the CASPT2 Geometry With ZPE and Without Spin–Orbit Coupling


+ 16.52 16.45 16.11 16.25 15.90 16.28 16.38 16.6 ± 1.0 PaO2

+ 16.67 17.01 16.48 17.04 17.63 17.10 16.86 16.6 ± 0.4 UO2

+ 15.60 15.37 15.04 15.24 14.60 15.25 15.09 14.6 ± 0.4 NpO2

+ 16.17 16.12 15.83 16.01 15.21 16.03 15.71 15.1 ± 0.4 PuO2

+ 16.60 16.54 16.05 16.43 15.83 16.44 16.23 15.1 ± 0.4 AmO2

+ 17.01 17.09 16.66 16.92 16.37 16.93 16.72 15.7 ± 0.6 CmO2

+ 16.20 16.29 15.83 16.35 15.47 16.36 15.90 17.9 ± 1.0 MUE(7) 0.96 0.97 0.82 0.92 0.81 0.93 0.82 0.60 MUE(14) 0.60 0.60 0.75 0.56 0.76 0.57 0.58 0.45

Table S20. First and Second Ionization Potentials (eV) of AnO2 from DFT Calculations at the DFT Geometry With ZPE and Without Spin–Orbit Coupling


+ 16.34 16.43 16.07 16.30 15.94 16.32 16.38 16.6 ± 1.0 PaO2

+ 16.70 17.03 16.44 16.93 17.47 17.00 16.70 16.6 ± 0.4 UO2

+ 15.50 15.29 15.01 15.23 14.64 15.24 15.11 14.6 ± 0.4 NpO2

+ 16.09 16.06 15.81 16.00 15.24 16.02 15.73 15.1 ± 0.4 PuO2

+ 16.56 16.51 16.05 16.43 15.83 16.45 16.24 15.1 ± 0.4 AmO2

+ 16.97 17.06 16.65 16.91 16.37 16.92 16.72 15.7 ± 0.6 CmO2

+ 16.23 16.30 15.82 16.34 15.42 16.35 15.86 17.9 ± 1.0 MUE(7) 0.95 0.94 0.83 0.90 0.80 0.91 0.81 0.60 MUE(14) 0.59 0.57 0.73 0.55 0.76 0.55 0.57 0.45

S14

Table S21. First and Second Ionization Potentials (eV) of AnO2 from DFT Calculations at the CASPT2 Geometry Without ZPE and With Spin–Orbit Coupling


+ 16.49 16.46 16.11 16.25 15.90 16.28 16.38 16.6 ± 1.0 PaO2

+ 16.68 17.02 16.49 17.07 17.63 17.10 16.86 16.6 ± 0.4 UO2

+ 15.43 15.20 14.87 15.08 14.45 15.10 14.94 14.6 ± 0.4 NpO2

+ 16.38 16.32 16.03 16.21 15.42 16.24 15.92 15.1 ± 0.4 PuO2

+ 16.48 16.42 15.93 16.31 15.72 16.33 16.12 15.1 ± 0.4 AmO2

+ 16.53 16.60 16.18 16.44 15.90 16.46 16.25 15.7 ± 0.6 CmO2

+ 16.37 16.45 15.99 16.51 15.65 16.54 16.08 17.9 ± 1.0 MUE(7) 0.86 0.87 0.72 0.82 0.75 0.83 0.72 0.60 MUE(14) 0.58 0.58 0.73 0.55 0.75 0.54 0.55 0.45

Table S22. First and Second Ionization Potentials (eV) of AnO2 from DFT Calculations at the DFT Geometry Without ZPE and With Spin–Orbit Coupling


+ 16.31 16.44 16.07 16.29 15.94 16.32 16.38 16.6 ± 1.0 PaO2

+ 16.70 17.04 16.45 16.96 17.47 17.00 16.70 16.6 ± 0.4 UO2

+ 15.33 15.12 14.84 15.06 14.49 15.09 14.96 14.6 ± 0.4 NpO2

+ 16.29 16.26 16.02 16.20 15.45 16.23 15.94 15.1 ± 0.4 PuO2

+ 16.44 16.39 15.93 16.31 15.73 16.34 16.13 15.1 ± 0.4 AmO2

+ 16.49 16.57 16.17 16.43 15.90 16.45 16.26 15.7 ± 0.6 CmO2

+ 16.39 16.47 15.99 16.50 15.60 16.53 16.04 17.9 ± 1.0 MUE(7) 0.85 0.84 0.72 0.80 0.73 0.81 0.71 0.60 MUE(14) 0.57 0.56 0.71 0.52 0.75 0.52 0.54 0.45

S15

Table S23. First and Second Ionization Potentials (eV) of AnO2 from DFT Calculations at the CASPT2 Geometry With ZPE and With Spin–Orbit Coupling


+ 16.52 16.45 16.11 16.25 15.90 16.28 16.38 16.6 ± 1.0 PaO2

+ 16.67 17.01 16.48 17.04 17.63 17.10 16.86 16.6 ± 0.4 UO2

+ 15.45 15.22 14.89 15.09 14.45 15.10 14.94 14.6 ± 0.4 NpO2

+ 16.38 16.33 16.04 16.22 15.42 16.24 15.92 15.1 ± 0.4 PuO2

+ 16.49 16.43 15.94 16.32 15.72 16.33 16.12 15.1 ± 0.4 AmO2

+ 16.54 16.62 16.19 16.45 15.90 16.46 16.25 15.7 ± 0.6 CmO2

+ 16.38 16.47 16.01 16.53 15.65 16.54 16.08 17.9 ± 1.0 MUE(7) 0.86 0.87 0.72 0.82 0.75 0.83 0.72 0.60 MUE(14) 0.57 0.57 0.72 0.54 0.75 0.54 0.55 0.45

Table S24. First and Second Ionization Potentials (eV) of AnO2 from DFT Calculations at the DFT Geometry With ZPE and With Spin–Orbit Coupling


+ 16.34 16.43 16.07 16.30 15.94 16.32 16.38 16.6 ± 1.0 PaO2

+ 16.70 17.03 16.44 16.93 17.47 17.00 16.70 16.6 ± 0.4 UO2

+ 15.35 15.14 14.86 15.08 14.49 15.09 14.96 14.6 ± 0.4 NpO2

+ 16.30 16.27 16.02 16.21 15.45 16.23 15.94 15.1 ± 0.4 PuO2

+ 16.45 16.40 15.94 16.32 15.72 16.34 16.13 15.1 ± 0.4 AmO2

+ 16.50 16.59 16.18 16.44 15.90 16.45 16.25 15.7 ± 0.6 CmO2

+ 16.41 16.48 16.00 16.52 15.60 16.53 16.04 17.9 ± 1.0 MUE(7) 0.85 0.84 0.73 0.80 0.73 0.81 0.71 0.60 MUE(14) 0.56 0.55 0.70 0.52 0.75 0.52 0.54 0.45

S16

Table S25. Bond Energies (kcal/mol) for AnO2, AnO2+, and AnO22+ from WFT Calculations at the CASPT2 Geometry Without ZPE


ThO2 148 140 145 148 140 145 164±3 PaO2 200 182 191 192 174 183 187±11 UO2 147 171 184 147 171 184 180±3 NpO2 172 146 161 171 145 160 152±10 PuO2 136 130 146 136 130 146 144±5 AmO2 126 103 117 121 98 112 122±16 CmO2 108 81 95 109 82 96 97±17 MUE(7) 15 13 6 13 15 7 9 ThO2

+ 104 89 97 104 89 97 111±9 PaO2

+ 218 192 200 205 179 187 187±7 UO2

+ 183 173 184 183 173 184 178±3 NpO2

+ 165 147 159 164 146 158 146±5 PuO2

+ 133 132 145 133 132 145 122±9 AmO2

+ 125 91 106 119 86 100 98±13 CmO2

+ 70 48 85 75 53 90 49±14 MUE(7) 17 7 16 15 9 14 9 ThO2

2+ 31 -3 17 31 -3 17 0±41 PaO2

2+ 101 82 94 93 74 86 76±26 UO2

2+ 145 129 142 145 129 142 127±7 NpO2

2+ 123 104 119 115 96 111 121±2 PuO2

2+ 107 95 110 107 95 110 97±23 AmO2

2+ 67 67 83 70 70 86 61±31 CmO2

2+ 46 33 52 44 31 50 0±36 MUE(7) 20 10 20 19 11 20 24

MUE(21) 17 10 14 16 11 14 14

S17

Table S26. Bond Energies (kcal/mol) for AnO2, AnO2+, and AnO22+ from WFT Calculations at the CASPT2 Geometry With ZPE.


ThO2 147 139 144 147 139 144 164±3 PaO2 198 180 189 190 172 181 187±11 UO2 146 170 183 146 170 183 180±3 NpO2 170 144 159 169 143 158 152±10 PuO2 134 128 144 134 128 144 144±5 AmO2 124 101 115 119 96 110 122±16 CmO2 107 80 94 108 81 95 97±17 MUE(7) 15 15 6 14 17 7 9 ThO2

+ 103 88 96 103 88 96 111±9 PaO2

+ 216 190 198 203 177 185 187±7 UO2

+ 181 171 182 181 171 182 178±3 NpO2

+ 163 145 157 162 144 156 146±5 PuO2

+ 131 130 143 131 130 143 122±9 AmO2

+ 123 89 104 117 84 98 98±13 CmO2

+ 68 46 83 73 51 88 49±14 MUE(7) 16 8 14 13 9 13 9 ThO2

2+ 30 -4 16 30 -4 16 0±41 PaO2

2+ 100 81 93 92 73 85 76±26 UO2

2+ 143 127 140 143 127 140 127±7 NpO2

2+ 121 102 117 113 94 109 121±2 PuO2

2+ 104 92 107 104 92 107 97±23 AmO2

2+ 64 64 80 67 67 83 61±31 CmO2

2+ 43 30 49 41 28 47 0±36 MUE(7) 18 9 18 18 10 19 24

MUE(21) 16 11 13 15 12 13 14

S18

Table S27. Bond Energies (kcal/mol) for AnO2, AnO2+, and AnO22+ from DFT Calculations at the CASPT2 Geometry Without ZPE and Without Spin–Orbit Coupling

M05 M06 M06-L B3LYP MOHLYP MPW3LYP PW91 Exp. ThO2 159 154 166 158 163 159 177 164±3 PaO2 174 178 209 189 195 190 156 187±11 UO2 174 173 181 171 183 173 194 180±3 NpO2 165 169 173 153 168 155 179 152±10 PuO2 149 153 158 138 153 139 164 144±5 AmO2 116 122 128 114 125 116 136 122±16 CmO2 104 106 116 101 115 103 125 97±17 MUE(7) 8 9 12 5 8 5 21 9 ThO2

+ 123 119 132 111 120 112 132 111±9 PaO2

+ 192 197 185 193 168 194 209 187±7 UO2

+ 165 173 175 171 181 172 193 178±3 NpO2

+ 149 160 166 153 163 154 175 146±5 PuO2

+ 129 134 142 132 142 133 156 122±9 AmO2

+ 95 103 112 101 119 102 131 98±13 CmO2

+ 53 58 71 62 82 63 95 49±14 MUE(7) 7 9 14 7 18 7 29 9 ThO2

2+ 11 12 22 18 26 19 35 0±41 PaO2

2+ 82 85 96 86 99 87 109 76±26 UO2

2+ 102 115 120 119 134 121 143 127±7 NpO2

2+ 92 103 109 103 116 104 127 121±2 PuO2

2+ 83 89 100 91 101 92 114 97±23 AmO2

2+ 60 66 74 67 82 68 96 61±31 CmO2

2+ 34 39 56 42 74 44 85 0±36 MUE(7) 17 15 19 16 23 16 33 24

MUE(21) 11 11 15 9 16 9 27 14

S19

Table S28. Bond Energies (kcal/mol) for AnO2, AnO2+, and AnO22+ from DFT Calculations at the DFT Geometry Without ZPE and Without Spin–Orbit Coupling


+ 122 117 131 112 122 112 133 111±9 PaO2

+ 190 196 184 193 169 194 209 187±7 UO2

+ 165 173 175 171 182 172 193 178±3 NpO2

+ 149 159 166 153 164 154 176 146±5 PuO2

+ 129 133 142 132 144 133 156 122±9 AmO2

+ 97 104 112 101 120 102 131 98±13 CmO2

+ 55 59 71 62 83 63 95 49±14 MUE(7) 6 9 14 7 18 7 29 9 ThO2

2+ 14 11 23 17 26 18 36 0±41 PaO2

2+ 82 85 97 88 103 90 112 76±26 UO2

2+ 105 117 121 120 134 121 143 127±7 NpO2

2+ 94 105 110 103 117 104 127 121±2 PuO2

2+ 84 89 100 90 101 92 115 97±23 AmO2

2+ 61 67 74 67 83 68 96 61±31 CmO2

2+ 33 39 55 42 76 43 86 0±36 MUE(7) 17 14 19 16 24 16 33 24

MUE(21) 10 11 15 9 17 9 28 14

S20

Table S29. Bond Energies (kcal/mol) for AnO2, AnO2+, and AnO22+ from DFT Calculations at the CASPT2 Geometry With ZPE and Without Spin–Orbit Coupling


+ 122 118 131 110 119 111 131 111±9 PaO2

+ 190 195 183 192 166 192 208 187±7 UO2

+ 163 171 173 169 180 170 191 178±3 NpO2

+ 147 157 164 151 161 152 173 146±5 PuO2

+ 126 131 139 130 140 131 154 122±9 AmO2

+ 93 100 110 99 117 100 129 98±13 CmO2

+ 51 56 69 60 81 62 93 49±14 MUE(7) 6 8 14 6 16 6 27 9 ThO2

2+ 10 11 22 17 24 18 34 0±41 PaO2

2+ 80 83 95 85 98 86 108 76±26 UO2

2+ 100 113 118 117 132 119 141 127±7 NpO2

2+ 90 101 107 100 115 102 125 121±2 PuO2

2+ 81 86 98 88 98 90 112 97±23 AmO2

2+ 57 63 71 64 80 66 94 61±31 CmO2

2+ 31 36 54 40 72 41 83 0±36 MUE(7) 18 15 18 15 21 16 31 24

MUE(21) 11 10 14 9 15 9 26 14

S21

Table S30. Bond Energies (kcal/mol) for AnO2, AnO2+, and AnO22+ from DFT Calculations at the DFT Geometry With ZPE and Without Spin–Orbit Coupling


+ 121 116 130 111 121 112 132 111±9 PaO2

+ 189 194 182 191 167 192 208 187±7 UO2

+ 163 171 173 169 180 170 191 178±3 NpO2

+ 147 157 164 151 162 152 174 146±5 PuO2

+ 126 131 139 130 141 131 154 122±9 AmO2

+ 94 101 110 99 118 100 129 98±13 CmO2

+ 53 57 69 60 81 62 93 49±14 MUE(7) 6 7 14 5 17 6 27 9 ThO2

2+ 12 11 22 17 25 18 35 0±41 PaO2

2+ 80 84 96 87 102 89 111 76±26 UO2

2+ 102 115 119 118 132 119 141 127±7 NpO2

2+ 92 102 108 101 115 102 125 121±2 PuO2

2+ 81 87 98 88 99 90 112 97±23 AmO2

2+ 58 64 71 64 80 66 94 61±31 CmO2

2+ 30 36 53 39 73 41 84 0±36 MUE(7) 17 14 18 16 22 16 32 24

MUE(21) 10 10 15 9 16 9 26 14

S22

Table S31. Bond Energies (kcal/mol) for AnO2, AnO2+, and AnO22+ from DFT Calculations at the CASPT2 Geometry Without ZPE and With Spin–Orbit Coupling


+ 123 119 132 111 120 112 132 111±9 PaO2

+ 179 184 172 180 155 181 196 187±7 UO2

+ 165 173 175 171 181 172 193 178±3 NpO2

+ 148 159 165 152 162 153 174 146±5 PuO2

+ 129 134 142 132 142 133 156 122±9 AmO2

+ 89 97 106 95 113 96 125 98±13 CmO2

+ 58 63 76 67 87 68 100 49±14 MUE(7) 9 8 16 7 19 7 26 9 ThO2

2+ 11 12 22 18 26 19 35 0±41 PaO2

2+ 74 77 88 78 91 79 101 76±26 UO2

2+ 102 115 120 119 134 121 143 127±7 NpO2

2+ 84 95 101 95 108 96 119 121±2 PuO2

2+ 83 89 100 91 101 92 114 97±23 AmO2

2+ 63 69 77 70 85 71 99 61±31 CmO2

2+ 32 37 54 40 72 42 83 0±36 MUE(7) 18 15 19 16 23 16 31 24

MUE(21) 12 11 15 10 16 10 26 14

S23

Table S32. Bond Energies (kcal/mol) for AnO2, AnO2+, and AnO22+ from DFT Calculations at the DFT Geometry Without ZPE and With Spin–Orbit Coupling


+ 122 117 131 112 122 112 133 111±9 PaO2

+ 177 183 171 180 156 181 196 187±7 UO2

+ 165 173 175 171 182 172 193 178±3 NpO2

+ 148 158 165 152 163 153 175 146±5 PuO2

+ 129 133 142 132 144 133 156 122±9 AmO2

+ 91 98 106 95 114 96 125 98±13 CmO2

+ 60 64 76 67 88 68 100 49±14 MUE(7) 9 8 16 7 20 8 27 9 ThO2

2+ 14 11 23 17 26 18 36 0±41 PaO2

2+ 74 77 89 80 95 82 104 76±26 UO2

2+ 105 117 121 120 134 121 143 127±7 NpO2

2+ 86 97 102 95 109 96 119 121±2 PuO2

2+ 84 89 100 90 101 92 115 97±23 AmO2

2+ 64 70 77 70 86 71 99 61±31 CmO2

2+ 31 37 53 40 74 41 84 0±36 MUE(7) 17 14 19 16 24 16 32 24 MUE(21) 12 11 15 10 18 10 27 14

S24

Table S33. Bond Energies (kcal/mol) for AnO2, AnO2+, and AnO22+ from DFT Calculations at the CASPT2 Geometry With ZPE and With Spin–Orbit Coupling


+ 122 118 131 110 119 111 131 111±9 PaO2

+ 177 182 170 179 153 179 195 187±7 UO2

+ 163 171 173 169 180 170 191 178±3 NpO2

+ 146 156 163 150 160 151 172 146±5 PuO2

+ 126 131 139 130 140 131 154 122±9 AmO2

+ 87 94 104 93 111 94 123 98±13 CmO2

+ 56 61 74 65 86 67 98 49±14 MUE(7) 8 8 15 7 18 7 25 9 ThO2

2+ 10 11 22 17 24 18 34 0±41 PaO2

2+ 72 75 87 77 90 78 100 76±26 UO2

2+ 100 113 118 117 132 119 141 127±7 NpO2

2+ 82 93 99 92 107 94 117 121±2 PuO2

2+ 81 86 98 88 98 90 112 97±23 AmO2

2+ 60 66 74 67 83 69 97 61±31 CmO2

2+ 29 34 52 38 70 39 81 0±36 MUE(7) 18 15 18 16 22 16 30 24 MUE(21) 12 11 14 10 15 10 25 14

S25

Table S34. Bond Energies (kcal/mol) for AnO2, AnO2+, and AnO22+ from DFT Calculations at the DFT Geometry With ZPE and With Spin–Orbit Coupling


+ 121 116 130 111 121 112 132 111±9 PaO2

+ 176 181 169 178 154 179 195 187±7 UO2

+ 163 171 173 169 180 170 191 178±3 NpO2

+ 146 156 163 150 161 151 173 146±5 PuO2

+ 126 131 139 130 141 131 154 122±9 AmO2

+ 88 95 104 93 112 94 123 98±13 CmO2

+ 58 62 74 65 86 67 98 49±14 MUE(7) 8 8 15 7 18 7 25 9 ThO2

2+ 12 11 22 17 25 18 35 0±41 PaO2

2+ 72 76 88 79 94 81 103 76±26 UO2

2+ 102 115 119 118 132 119 141 127±7 NpO2

2+ 84 94 100 93 107 94 117 121±2 PuO2

2+ 81 87 98 88 99 90 112 97±23 AmO2

2+ 61 67 74 67 83 69 97 61±31 CmO2

2+ 28 34 51 37 71 39 82 0±36 MUE(7) 18 14 18 16 23 16 30 24 MUE(21) 12 11 14 10 16 10 25 14

S26

Table S35. S2 Expectation Values for AnO2 and AnO for Single-Point DFT Calculations at the CASPT2 Geometry

2S+1 M05 M06 M06-L B3LYP MOHLYP MPW3LYP ThO2 1 0.00 0.00 0.00 0.00 0.00 0.00 PaO2 2 0.76 0.75 0.76 0.75 0.76 0.75 UO2 3 2.01 2.01 2.01 2.01 2.01 2.01 NpO2 4 3.77 3.77 3.77 3.77 3.77 3.77 PuO2 5 6.04 6.03 6.05 6.04 6.04 6.04 AmO2 6 8.83 8.83 8.85 8.83 8.83 8.83 CmO2 7 12.09 12.10 12.09 12.09 12.05 12.09 ThO2

+ 2 0.76 0.75 0.75 0.75 0.75 0.75 PaO2

+ 1 0.00 0.00 0.00 0.00 0.17 0.00 UO2

+ 2 0.76 0.75 0.76 0.75 0.75 0.75 NpO2

+ 3 2.01 2.01 2.01 2.01 2.01 2.01 PuO2

+ 4 3.78 3.78 3.79 3.78 3.78 3.78 AmO2

+ 5 6.10 6.09 6.13 6.10 6.09 6.10 CmO2

+ 6 8.95 8.94 8.95 8.96 8.87 8.96 ThO2

2+ 1 0.68 0.70 0.52 0.70 0.09 0.70 PaO2

2+ 2 0.76 0.75 0.75 0.75 0.75 0.75 UO2

2+ 1 0.06 0.00 0.00 0.00 0.00 0.00 NpO2

2+ 2 0.76 0.76 0.76 0.76 0.76 0.76 PuO2

2+ 3 2.04 2.03 2.03 2.03 2.02 2.03 AmO2

2+ 4 3.87 3.86 3.86 3.86 3.82 3.86 CmO2

2+ 5 6.34 6.28 6.27 6.30 6.16 6.30 ThO 1 0.00 0.00 0.02 0.00 0.42 0.00 PaO 2 0.78 0.76 0.76 0.76 0.96 0.76 UO 5 6.01 6.01 6.01 6.01 6.01 6.01 NpO 6 8.77 8.77 8.77 8.76 8.77 8.76 PuO 7 12.04 12.04 12.04 12.03 12.03 12.02 AmO 8 15.80 15.80 15.80 15.80 15.79 15.80 CmO 9 20.02 20.02 20.02 20.02 20.01 20.02 ThO+ 2 0.75 0.75 0.75 0.75 0.75 0.75 PaO+ 3 2.03 2.01 2.01 2.01 2.01 2.00 UO+ 4 3.76 3.76 3.76 3.76 3.76 3.76 NpO+ 5 6.02 6.02 6.02 6.01 6.02 6.01 PuO+ 6 8.80 8.78 8.81 8.78 8.80 8.78 AmO+ 7 12.10 12.09 12.11 12.08 12.08 12.08 CmO+ 8 15.77 15.76 15.77 15.76 15.76 15.76 ThO2+ 1 0.00 0.00 0.00 0.00 0.00 0.00 PaO2+ 2 0.76 0.76 0.76 0.75 0.75 0.75 UO2+ 3 2.04 2.02 2.03 2.02 2.02 2.02 NpO2+ 4 3.85 3.81 3.81 3.81 3.82 3.81 PuO2+ 5 6.20 6.18 6.21 6.13 6.15 6.13 AmO2+ 6 9.38 9.33 9.30 9.30 9.14 9.30 CmO2+ 7 12.71 12.68 12.56 12.63 12.33 12.63

S27

Table S36. Bond Distances (Ǻ) for AnO2 Obtained by Using a Different Basis Set for Oxygen , in Particular, the Correlation-Consistent aug-cc-pVTZ Basis

B3LYP B3P86 MPW1-PW91 PBE0 PBE TPSS HCTH BLYP BP86

ThO2a 1.898 1.884 1.882 1.881 1.899 1.900 1.879 1.917 1.902

PaO2 1.806 1.793 1.789 1.787 1.828 1.813 1.802 1.847 1.833 UO2 1.789 1.814 1.812 1.810 - b - b - b - b - b NpO2 1.821 1.804 1.801 1.800 - b - b - b - b - b PuO2 1.814 1.797 1.794 1.792 - b - b - b - b - b AmO2 1.826 1.809 1.807 1.805 - b 1.835 - b - b 1.835 CmO2 1.839 1.820 1.819 1.817 1.839 1.844 1.826 1.863 1.844 ThO2

+ 1.868 1.851 1.847 1.846 1.870 1.872 1.845 1.868 1.874 PaO2

+ 1.767 1.756 1.752 1.751 1.777 1.776 1.757 1.793 1.780 UO2

+ 1.753 1.741 1.737 1.736 1.764 1.763 1.743 1.781 1.768 NpO2

+ 1.732 1.720 1.715 1.714 1.745 1.744 1.724 1.763 1.749 PuO2

+ 1.714 1.700 1.695 1.694 1.728 1.726 1.708 1.747 1.731 AmO2

+ 1.716 1.701 1.695 1.694 1.733 1.732 1.716 1.754 1.736 CmO2

+ 1.748 1.729 1.726 1.724 1.759 1.762 1.746 1.783 1.762 ThO2

2+ 1.850 1.832 1.827 1.825 1.855 1.856 1.829 1.880 1.861 PaO2

2+ 1.772 1.759 1.753 1.752 1.784 1.784 - b 1.804 1.788 UO2

2+ 1.693 1.683 1.678 1.677 1.710 1.707 1.690 1.725 1.713 NpO2

2+ 1.688 1.677 1.671 1.670 1.707 1.705 1.687 1.724 1.711 PuO2

2+ 1.673 1.661 1.655 1.654 1.694 1.691 1.675 1.713 1.698 AmO2

2+ 1.666 1.652 1.646 1.644 1.687 1.684 1.670 1.707 1.691 CmO2

2+ 1.688 1.668 1.664 1.663 1.706 1.706 1.696 1.732 1.710 aMolecule has bent geometry (C2v symmetry) With angle 111.9 degrees. b SCF did not converge. The test calculations in Table S36 show that changing the basis set for oxygen makes only negligible differences (for example, only in the fourth digit after the decimal point for B3LYP calculations) in the bond distances. These results have been obtained with the Gaussian 03 program.

How accurate are electronic structure methods for actinoid chemistry

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