Scholarly article on topic 'Theoretical study of actinide monocarbides (ThC, UC, PuC, and AmC)'

Theoretical study of actinide monocarbides (ThC, UC, PuC, and AmC) Academic research paper on "Chemical sciences"

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Academic research paper on topic "Theoretical study of actinide monocarbides (ThC, UC, PuC, and AmC)"

Theoretical study of actinide monocarbides (ThC, UC, PuC, and AmC)

Peter Pogany, Attila Kovacs, Lucas Visscher, and Rudy J. M. Konings

Citation: J. Chem. Phys. 145, 244310 (2016); doi: 10.1063/1.4972812 View online: http://dx.doi.org/10.1063Z1.4972812 View Table of Contents: http://aip.scitation.org/toc/jcp/145/24 Published by the American Institute of Physics

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Theoretical study of actinide monocarbides (ThC, UC, PuC, and AmC)

Peter Pogany,1,2 Attila Kovacs,1,3,a) Lucas Visscher,4 and Rudy J. M. Konings1

1 European Commission, Joint Research Centre, P.O. Box 2340, 76125 Karlsruhe, Germany 2Spectroscopic Research Department, Gedeon Richter Plc., Gyömro'i ut 19-21, 1103 Budapest, Hungary 3 Department of Inorganic and Analytical Chemistry, Budapest University of Technology and Economics, Szt. Gellert ter 4, 1111 Budapest, Hungary

4Department of Theoretical Chemistry, Vrije Universiteit Amsterdam, De Boelelaan 1083, NL-1081 HV Amsterdam, The Netherlands

(Received 4 October 2016; accepted 8 December 2016; published online 28 December 2016)

A study of four representative actinide monocarbides, ThC, UC, PuC, and AmC, has been performed with relativistic quantum chemical calculations. The two applied methods were multireference complete active space second-order perturbation theory (CASPT2) including the Douglas-Kroll-Hess Hamiltonian with all-electron basis sets and density functional theory with the B3LYP exchange-correlation functional in conjunction with relativistic pseudopotentials. Beside the ground electronic states, the excited states up to 17 000 cm1 have been determined. The molecular properties explored included the ground-state geometries, bonding properties, and the electronic absorption spectra. According to the occupation of the bonding orbitals, the calculated electronic states were classified into three groups, each leading to a characteristic bond distance range for the equilibrium geometry. The ground states of ThC, UC, and PuC have two doubly occupied n orbitals resulting in short bond distances between 1.8 and 2.0 A, whereas the ground state of AmC has significant occupation of the antibonding orbitals, causing a bond distance of 2.15 A. © 2016 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). [http://dx.doi.org/10.1063/L4972812]

I. INTRODUCTION

Recent increasing interest in actinide (An) carbides is due to their potential application as nuclear fuels in future fast reactors1-3 and propulsion reactors for space.4 In fact, three of the six current Generation IV nuclear reactor designs under development are fast reactors.5 The main advantage of the carbide fuels compared with the widely used UO2 and MOX (mixed U-Pu oxide) fuels is their higher heavy metal density, higher thermal conductivity,6 and thus higher breeding gain and smaller plutonium inventory.7 Currently only one reactor, the Indian fast breeder test reactor is operated with a (mixed U-Pu) carbide fuel. For safe application, additional extensive research elucidating all of the relevant properties of these potential fuels is required.

While for the solid carbides a considerable amount of data has been collected,8 the available reliable information is much less for the gaseous species. Although the rim of the fuel pellet has a moderate temperature, well below the melting point of carbides, in the middle zone of the fuel the temperature is much higher (can reach 2500 K), at which already some sublimation takes place. This refers particularly to the gas-cooled fast reactors, where the gas coolants (He and CO2) have a very poor heat capacity9 and locally overheated areas can easily occur. Thus, the above solid-gas phase transitions belong to the operational features of the reactors; moreover, they become especially important in close-to-accidental situations

a)Electronic mail: attila.kovacs@

ec.europa.eu

for which we have to be prepared properly. Therefore their thermodynamics should be explored in detail, which includes also the determination of reliable molecular parameters of the gas-phase species.

Until recently very few experimental studies dealt with the actinide carbides in the gaseous phase. They include a few experiments performed with Knudsen-cell mass spectrome-try10-14 and single filament surface ionisation mass spectrometry15,16 published mostly in the 1960s and 1970s for ThCx and UCx (x = 1-6). These articles focused on the partial pressures, equilibrium constants between different oligocarbides, and atomisation energies and determined some thermodynamic functions according to second and third law methods. Some of these thermodynamic parameters were calculated assuming linear or planar oligocarbide structures and estimated bond distances. Recent quantum chemical studies showed the irrelevance of some assumed oligocarbide structures and the unreliability of their estimated molecular parameters (e.g., Th-C bond distances of around 2.4 A17,18 compared to the estimated 1.9 A11,14 for ThC4).

One of the most abundant species in the vapour above actinide carbides is the monocarbide. Yet, only four theoretical studies dealt with AnC molecules (UC,19,20 NpC,21 and PuC22) until now, providing very limited information on their molecular properties. In contrast, the di-,21,23-26 tri-,27,28 and tetracarbides18,20,21,29 of Th, U, Np, Pu, and Am have been quite extensively explored by advanced quantum chemical calculations. The lack of reliable information on the monocarbides is the motivation to our present study in which we apply relativistic multireference and density functional

0021-9606/2016/145(24)/244310/9

145, 244310-1

© Author(s) 2016 I

theory (DFT) methods to elucidate the electronic structure and other molecular characteristics of the most important mono-carbides (ThC, UC, PuC, and AmC). This includes the determination of ground and low-lying excited electronic states, bond distances, and electronic spectra.

II. COMPUTATIONAL DETAILS

A. Complete active space self-consistent field (CASSCF)/complete active space second-order perturbation theory (CASPT2)

Our all-electron relativistic multireferencecalculations were performed using the code MOLCAS 7.4, patch 097.30-32 The complete active space (CAS) SCF method33 was used to generate the molecular orbitals (MOs) and reference functions for subsequent multiconfigurational second-order perturbation theory calculations for the dynamic correlation energy (this spin-orbit-free (SF) level is denoted in the following as CASPT2).34,35 Active spaces of 8/14,10/16,12/16, and 13/16 electrons/orbitals were used for ThC, UC, PuC, and AmC, respectively. All-electron basis sets of atomic natural orbital type developed for relativistic calculations (ANO-RCC) with the Douglas-Kroll-Hess Hamiltonian36,37 were used for all the atoms. For Th a primitive set of 27s24p18d14f6g3h was contracted to 9s8p6d4f2g1h, and for the rest of the actinides (U, Pu, and Am) a primitive set of 26s23p17d13f5g3h was contracted to 9s8p6d5f2g1h;38 whereas for carbon a primitive set of 14s9p4d3f2g was contracted to 4s3p2d1f39 achieving VTZP quality.

The Douglas-Kroll-Hess Hamiltonian accounts for the scalar relativistic effects in CASSCF calculations. In subsequent second-order perturbation treatments (CASPT2), the orbitals up to 5d of the actinides and 1s of carbon were kept frozen, while the remaining valence and semi-valence orbitals (including 6s and 6p of the actinides and 2s of carbon) were correlated. Effective bond orders (EBOs) were determined from bonding minus antibonding occupancies divided by two.40 The spin-orbit (SO) coupling was taken into account by the complete active space state interaction (CASSI) method41,42 employing an effective one-electron spin-orbit Hamiltonian based on a mean field approximation to the two-electron con-tribution.43 In order to include the dynamical correlation in the CASSI treatments, the spin-orbit-free (SF) CASPT2 energies were used as diagonal matrix elements (SO-corrected CASPT2, denoted in the following as SO-CASPT2).

In MOLCAS, non-abelian symmetries are treated in lower-symmetry groups, and our previous experiences showed that the calculated electronic states can be symmetry-broken. Therefore, for the actinide monocarbides (with symmetry CTCv) the C1 point group was used in the calculations and, with a script,44 the symmetry-broken orbitals were checked and corrected. The axis of the carbide molecules is used to define the z axis, so the atomic orbital (AO) indices 0, i±, 2±, and 3± refer to the angular part of the orbitals with a, n, 6, and 0 symmetry, respectively.

B. Density functional theory

The density functional theory studies (B3LYP45,46) were performed by means of the Gaussian0947 code. For

these calculations the relativistic pseudopotentials of the Stuttgart-Cologne group (ECP60MWB48) were applied with segmented-contracted valence basis sets49,50 for actinides and the cc-pVTZ basis51 set for carbon. The primitive 14s13p10d8f6g valence basis sets were contracted to 10s9p5d4f3g. The Stuttgart-Cologne pseudopotentials include the scalar relativistic effects. All the calculations were performed within the unrestricted scheme. The stability of the unrestricted SCF solution was checked and if instability was found, it was corrected with the stable = opt keyword, followed by geometry optimisation. This keyword searches for lower-energy electronic states (trying to find the ground electronic state) by altering some high-energy occupied and unoccupied orbitals. The Natural Bond Orbital (NBO) analysis52 and the calculation of Wiberg bond indices (BIs) were performed using the NBO 5.G code. The Wiberg bond index is a wave function-based approach using the components of the density matrix for estimation of the bond order (number of covalent bonds

formed) in molecules.53,54

III. RESULTS AND DISCUSSION

The molecular geometries of the four monocarbides (ThC, UC, PuC, and AmC) were optimized at the CASPT2 and B3LYP levels. To investigate the effect of the initial geometry on the outcome, we started the calculations with bond distances of 1.9 and 2.4 A, covering the broad variation of the actinide-carbon bond distances reported in the literature.19,20,22,55,56 Different spin multiplicities (m) were probed for these compounds: in the case of ThC m = 1, 3, and 5, in the case of UC m = 1,3,5,7, and 9, in the case of PuC m = 3,5,7, and 9, and in the case of AmC m = 4, 6, 8,10, and 12. The relative energies, the spectroscopic terms characterising the electronic states, the optimised bond distances, and the vibrational frequencies (at the B3LYP level) of the calculated equilibrium geometries at different spin multiplicities are compiled in Table I. (More detailed tables can be found in the supplementary material as Tables S1 and S3.)

The most stable SF states are: triplet for ThC, quintet for UC, septet for PuC, and sextet for AmC. We note that the 8 and 10 spin multiplicities of AmC are quite close in energy (within 10 kJ mol"1) to the ground electronic state (m = 6) at the CASPT2 level and a quite large discrepancy was found between CASPT2 and B3LYP results for PuC indicating a more complex electronic structure of these two carbides than those of ThC and UC.

A. Spin-orbit-free CASPT2 calculations: The ground-state structure and geometry

The optimised bond distances of different electronic states of actinide monocarbides vary in a large range (from 1.8 A to 2.4 A). There is a correlation between the electronic structure and the bond distances and on this basis we can distinguish between three groups of states:

1. Group 1: Formal bonding configuration a2n4 or a1 n4 with bond distances between 1.8 A and 2.0 A.

2. Group 2: Formal bonding configuration a1 n3 with bond distances between 2.0 A and 2.1 A.

TABLE I. Calculated properties of AnC by CASPT2 (plain) and B3LYP

(italics).

State aEa r b

Thr1 1z+ /12+ 8.2/6.3 1.953/1.932 833

ThC 3 z+ /3 2+ 0.0/0.0 1.967/1.929 874

1H/1 X 39.9/39.5 1.889/1.790 779

3H/3 X 22.9/12.6 1.797/1.814 832

UC 5H/5 X 0.0/0.0 1.870/1.859 928

7I/7 X 66.4/48.9 2.075/2.075 707

9K/9 X 149.2/174.3 2.285/2.354 481

3 n/3 X 53.4/12.6 1.839/1.906 508

Pur1 5 a/5 X 33.6/10.4 1.945/1.876 709

puc 7 o/7 X 0.0/0.2 1.913/1.894 748

9 n/9 X 34.3/0.0 2.036/2.055 664

4 z+ /4 2+ 31.2/48.6 2.151/2.217 341

6 z+ /6 2+ 0.0/0.0 2.150/2.307 347

AmC 8 /8 X 3.1/45.3 2.210/2.305 363

10 /10 2+ 8.7/18.2 2.261/2.402 412

12 z-/12 2 - 16.5/19.1 2.351/2.400 430

aRelative energy in kJ mol_1 for the optimised geometries. bBond distance in Â.

cHarmonic frequencies (B3LYP) in cm1.

3. Group 3: Formal bonding configuration <anb n*c (0 < a < 2;0 < b - c < 1) with bond distances between 2.1 Â and 2.4 Â.

Note that in all the molecules the active space contains

an almost completely occupied semi-core orbital (<sc2) built up mostly from actinide 6p^ and carbon 2s orbitals which is

not discussed here (see the supplementary material). The most

important bonding and antibonding orbitals are depicted in Figure 1 in which one can see the increasing importance of the

5f orbitals in the n bonds going from Th to Am.

Both the effective bond orders (EBOs, from CASPT2 cal-

culations, Table S3 in the supplementary material) and the Wiberg bond indices (BIs, from B3LYP calculations, Table S1 in the supplementary material) decrease with increasing bond distances. For the <1 n4 configurations (group 1), the average EBO is 2.35 and the average BI is 2.56, both suggesting a covalent bond order between 2 and 3 in these cases. For the

<1 n3 configurations (group 2), bond orders slightly below 2 were calculated (EBO = 1.75 and BI = 1.54) while for group

3 the bond orders are below 1 (EBO = 0.72 and BI = 0.58).

The SF ground electronic states of the four AnC molecules can be characterised as follows (for the molecular constants see Table I):

The triplet3 ground state of ThC belongs to group 1. It has two singly occupied a orbitals, one with 6d, another with 7s character, and two doubly occupied n orbitals. The bonding orbitals have strong 6d and weak 5f character. The state consists of one main configuration with a weight of 90%.

UC has a quintet 5H SF ground electronic state also belonging to group 1. Besides the two n bonds, which have considerable 6d and weaker 5f character, the state has a singly occupied a orbital and a singly occupied non-bonding 7 s and four 5f orbitals (2 x 5f^ and 2 x 5f^) with occupations of 0.5e.

The SF ground state of PuC is a septet 7 O, belonging to group 1. The bonding configuration for PuC is ca. a0 5 n3 5 and the EBO value is 1.90. Contrarily to ThC and UC, it has one semi-occupied a and one a * with occupation around 0.5e and two n orbitals occupied each with ca. 1.7e but the n orbitals now have predominant 5f character. From the non-bonding orbitals the 7 s orbital is singly occupied, whereas all the 5f orbitals have either an occupation number close to 0.75 (2 x and 2 x 5f0) or to 0.5 (2 x 5fn).

For AmC the occupation changes as the 5f energy falls below that of 6d. The n bonds weaken and antibonding orbitals get occupied. Our computations at the SO-CASPT2 level predicted the sextet 6state as ground state, belonging to the bonding group 3. We note that the first excited state 8is very close in energy, hence an interchange at higher theoretical levels cannot be excluded. These low-energy states of AmC are strongly multiconfigurational. The ground state, e.g., has a main configuration with a weight of only 15%. It has partial occupations of all the bonding orbitals: 1.4e in the a and both n MOs. At the same time it has 0.6e in the antibonding a * MO and in both the n* MOs. The 5f^ and 7s MOs together form a singly occupied non-bonding orbital. Both 5f^ and 5f^ are singly occupied and non-bonding, while 5fn takes part in the bonding and antibonding n orbitals. The different nature of the ground state is also reflected in the markedly longer bond length of AmC compared to that of PuC.

B. SO calculations and electronic spectra

The most characteristic low-energy SF and SO states are summarised in Tables II-V for ThC, UC, PuC, and AmC,

i t t T T

<xd (ThC) 2pz,6d0,5f0

c-si (ThC) 7s,2s,2pz

C7f (PuC) 5f0,2pz,6d0

cjsf (AmC) 2pz,7s,5f0

<f (AmC) 2pz,7s,5f0

* * * &

TTdx, 7rdy (ThC) 6di±,2pi±

TTfdx, TTfdy (PuC) 2pi±,5fi±,6di±

7i"fx, TTfy (AmC) 5fi±,2pi±

wjy (AmC) 2pi±,5fi±

FIG. 1. Characteristic MOs from CASPT2 calculations of AnC at 0.02 e a.u.-3 isodensity value. The main atomic valence orbitals forming the given MOs are given below each one. Orbitals (d, (s1, and ndx have been selected from the ones of ThC, (f and nfdx from the ones of PuC while (sf, (f, nfx, and nf from the ones of AmC.

TABLE II. The lowest-energy vertical SF and SO states of ThC.a

aE S D Main composition

SF states

TABLE III. The lowest-energy vertical SF and SO states of UC.a More detailed data can be found in Tables S4 and S5 of the supplementary material.

0.0 0 3£+(1) 1 88% x,an? n,2 s1 d dx dy

8.5 706 1Z+(1) 1 81% xßx?n,2 s1 d dx dy

62.7 5 237 3n(1) 2 56% x2n,2 n,a 7sa d dx dy

68.0 5 688 1Z+(2) 1 73% x2n.2 n.2 d dx dy

88.9 7 434 1 n 2 88% x2n.2 nß 7sa d dx dy 50% xan.2 7s2 d dx dy

97.8 8 177 3n(2) 2

111.9 9 356 1z+(3) 1 63% (X2.n.2 s1 dx dy

119.0 9 946 1a 2 89% x,an.2 6dß d dx dy 2±

165.5 13 831 3£+(2) 1 90% (X2(X2. n? n? d s1 dx dy

SO states

0.0 0 1 98% 3 £+(1)

0.3 22 3£+(1) 2 98% 3 £+(1)

8.5 714 1 £+ £0 1 98% 1 £+(1)

57.9 4 838 X0(1) 1 52% 3 n(1) + 47% 1z+(2)

60.1 5 024 3n2(1) 2 97% 3 n(1)

64.3 5 377 3n1(1) 2 96% 3 n(1)

70.2 5 870 X(1) 1 98% 3 n(1)

79.1 6 614 X0(2) 1 49% 1 £+(2) + 44% 3 n(2)

91.2 7 627 1 n 2 87% 1 n + 8% 3n(2)

120.1 10 038 X0(3) 1 74% 1 £+(3) + 26% 3 n(2)

aSF and SO states computed at the bond distance of 1.967 A. AE: Relative energy: first column in kJ moP1, second column in cm1; S: Term symbol of the electronic state; D: Degeneracy due to spatial symmetry. For the r and n orbitals see Figure 1. a and & represent the different spin of the electrons.

respectively. Note that the parameters of the SF and SO states are given at the SF and SO ground state geometry, respectively. Accordingly, the electronic absorption spectra represent vertical transitions at the SO ground state geometry. More detailed tables with the calculated bond distances are available in the supplementary material (Tables S4 and S5).

ThC has a triplet32+ SO ground state (see Table II). The 32 SF ground state splits due to SO coupling into Q = 0 and Q = 1 components. The zero-field splitting (which is due to second order SO coupling effects) between these two states is very small (0.3 kJ mol 1). The lowest-energy singlet state is

lying 8.5 kJ mol"1 (SO-CASPT2) above the ground state. Both the3 2+ and states consist primarily of one determinant (>80%) in which both the n bonding orbitals are fully occupied and the ^s1 and orbitals are singly occupied. In the case of the12+ state these latter two are filled with opposite spin electrons. States in which the n orbitals are depopulated are found at significantly higher energy, with generally only small mixing of the SF states. The SO splitting of the lowest-energy 3n state is ca. 6 kJ mol"1 (between the Q = 0 and Q = 1 states). In this triplet state the 7s orbital is singly occupied. For the doubly degenerate 1n state the orbital is doubly occupied. The proximity of the 12+ and 3n states leads to some mixing making it difficult to characterise the states in terms of an exclusive LS-coupled state (i.e., higher-energy X0 states).

For UC the 5H SF ground state contains two major configurations (in these configurations the doubly degenerate 5f^ and 5f^ are singly occupied, cf. Table III). Three quintet and one triplet excited states (5r,52+, 5A, and 3H) are found below

aE S D Main composition

SF states

0.0 0 5H 2 47% X® n2. 7sa d dx dy 5fa 5f2+ 5fa 5f3+

5.8 482 5 r 2 44% xa n. n. 7sa d dx dy 5fa 1+ 5fa 3+

16.3 1 360 5£+(1) 1 57% xa n2. n2. 7sa d dx dy 5fa 2- 5fa 2+

18.3 1 532 5a 2 36% xa n2. n2. 7sa d dx dy 5fa 5f1- 5fa 5f3+

34.0 2 846 3H 2 45% x2.n2. n2. 5f? d dx dy 2- 5fa 3-

40.7 3 405 1H 2 35% xa n2 n2 7sß d dx dy 5fß 2+ 5fa 3+

48.7 4 043 5n 2 43% xan2. n2. 7sa d dx dy 5fa 2+ 5f3a-

50.1 4 185 3 £+ 1 59% x2.n2. n2. 5f? d dx dy 2- 5fa 2+

64.4 5 383 1 £+ 1 40% xa n2. n2. 7sß d dx dy 5fa 2- 5fß 2+

SO states

0.0 0 5X3(1) 2 88% 5H + 12% 5r

22.6 1 888 5 r 2 95% 5r

25.3 2 114 5X4(1) 2 82% 5H + 17% 5r

40.4 3 379 3H4 2 93% 3H

48.3 4 038 5X0(1) 1 56% 5a + 22% 5n + 21% 5£(1)

51.3 4 292 5X3(2) 2 88% 5r + 12% 5H

53.5 4 468 5XQ(2) 1 90% 5a + 8% 5n

55.4 4 630 5X1(1) 2 38% 5a + 35% 5£(1) + 26% 5n

69.3 5 800 5X1(2) 2 50% 5a + 38% 5£(1) + 12% 5n

85.8 7 170 5X3(3) 2 84% 5a + 16% 5n

103.7 8 666 1H5 2 81% 1H

104.1 8 705 5 a4 2 85% 5a

105.2 8 795 5X1(3) 2 86% 5n + 13% 5£(1)

113.0 9 445 3£1 2 100% 3£

113.0 9 445 3 £q 1 100% 3£

127.3 10 642 1 £q 1 100% 1 £

129.5 10 826 5X3(4) 2 84% 5 n + 16% 5 a

aSF and SO states computed at the bond distance of 1.870 A. AE: Relative energy: first column in kJ mol 1 , second column in cm 1 ; S: Term symbol of the electronic state; D: Degeneracy due to spatial symmetry. For the r and n orbitals see Figure 1. a and & represent the different spin of the electrons.

40 kJ mol"1 giving rise to ahigher density of states as compared to the simpler ThC case discussed above. The state 5X3(1) has been identified by SO-CASPT2 calculations as the electronic ground state. The energy difference between the lowest spinorbit states 5X3(1) and 5r2 at the 1.870 A bond distance of the ground state is 22.6 kJ mol"1. The equilibrium bond distances for all the calculated quintet, triplet, and singlet SF states are very similar (cf. Table I). The shorter bond distance of both the SF and SO triplet states (ca. 1.80 A, compared to ca. 1.87 A for quintet and ca. 1.89 A for singlet states) is caused by the higher occupation of the orbitals. For the quintet and singlet states investigated, both n orbitals are fully occupied and the <rd and 7s orbitals are singly occupied, as in the ground state of ThC. The two additional electrons occupy 5f orbitals. Our results are in good agreement with those of Wang et al.19 who reported an Q = 3 state as the lowest in energy using SO-CASPT2 with a smaller active space (10/12) than used in our work. Both results agree upon placing UC in group 1 with a covalent bond order between 2 and 3 and a relatively short bond length.

Moving now to PuC we find a high density of states as there are 6 electrons to occupy a range of (primarily) 5f orbitals

TABLE IV. Selected low-energy vertical SF and SO states of PuC.a More detailed data can be found in Tables S4 and S5 of the supplementary material.

aE S D Main composition

SF states

7.6 635

15.7 1316

28.8 2 408 34.7 2 897

35.5 2 971 44.7 3 733

56.1 4 691

56.2 4 700

57.0 4 766 58.2 4 867

60.6 5 067 68.4 5 717

71.1 5 947

SO states

o.o o 7X1(1) 2 46% 7r + 4o% 7$ + 1o% 7a

5.3 442 7X2(1) 2 42% 7r + 17% 7$ + 1o% 7a

9.6 8o2 7Xo(1) 2 63% 7$ + 3o% 7a + 3% 7n

16.9 1 41o 7X3(1) 2 4o% 7$ + 38% 7r + 12% 7a

25.2 2 1o7 7X1(2) 2 36% 7a + 24% 7r + 24% 7n

28.4 2 37o 7X1(3) 2 76% 7a + 2o% 7 n + 1%9$

45.9 3 839 5X1(1) 2 54% 5$ + 25% 5a + 14% 7r

48.2 4 o3o 9Xo(1) 1 3 CO % 9n(1) + 28% 9a + 2o% 9e

53.1 4 443 7X5(1) 2 46% 7$ + 34% 7 r + 1o% 7H

87.4 7 3o7 7X5(2) 2 35% 7a + 26% 7r + 19% 7H

125.4 1o 486 7X7 2 84% 7r + 16% 7H

146.1 12 214 X5(3) 2 75% 5$ + 6% 7 h + 6%7 a + 3%7r

169.4 14 163 X8(1) 2 71% 9r + 28% 7H

197.4 16 5o1 X8(2) 2 72% 7H + 28% 9 r

aSF states computed at the bond distance of 1.913 Ä and SO states computed at the bond distance of 1.898 Ä. AE: Relative energy: first column in kJ mop1, second column in cm1; S: Term symbol of the electronic state, D: Degeneracy due to spatial symmetry. For the t and n orbitals see Figure 1. 22 and ß represent the different spin of the electrons.

7$ 7 a

7 r 7 n

9e+ 9,

23% 12% 32% 15% 38% 46% 32% 30% 22% 41% 15% 24% 22% 48%

tf ^ f 7s2 f 5fJ+ (5f3_,5f3+)Q _2 7ca cf<a

7sa 5^ 5fif_ 5fa+ (5f^,5f3+)a

T f n fdx " fdy

t f f f 7sa 5fa+ f 5fa+ (5f^,5f3+)a t ? f f 7sa (5f^,5f1+)a 5fa+ 5fa_ 5fa+

T2sf f f (5fo,7s)a (5f^,5f2+)a 5f3a_ 5f3a+ T2f f f (5fo,7s)a f 5f2+ (5f3-,5f3+)a T2sf n fdx2 n^yf-5fa+ f 5fa+ T f f f 7sa 5f2- 5f2+ 5fa+ (5f^,5f3+)a

T? ndx n2dy (7s,6do)a 5f2_ 5f2+ 5f2- 5f2+ (5f^,5f3+)a tf T*a ndx n2dy (7s,6do)« 5f2- 5ff+ 5f+ 5f2+ t? ndy (7s,6do)a (5f^,5f1+)a 5^ 5fa+ 5^ 5^+ ts t*2 ndy (7s,6do)a 5^ 5^ 5fa+ (5f^,5f3+)a Ts ^ ndy (7s,6do)a 5f2f 5f2+ 5f2f 5f2+ (5f3-,5f3+)a t J T*fa ndx n2dy (7s,6do)f 5f2f 5f2+ 5f2f 5f2+

that are close in energy. The SO coupling furthermore couples the low-lying quintet, septet, and nonet states found in the SF calculations. Altogether 14 SF states were investigated which resulted in 105 SO states (cf. Tables S4 and S5 in the supplementary material). The two lowest-lying SO states are within 6 kJ mol"1 (cf. Table IV) with a 7Xi(1) ground state formed mostly from the 7r and 7® states and with some contribution from 7 A. The equilibrium bond distance of the 7X1(1) is 1.898 A whereas the equilibrium bond distance of the SF 7® ground state is 1.913 A. The slight bond distance shortening during SO coupling is caused by the presence of the 7r state in the SO ground state which has a larger occupation of the n bonding orbitals giving a final bonding scheme of a061 n3 62 for the 7X1(1) state. Both the septet ground and the lowest-energy quintet states have similar bond distances with doubly occupied n bonding orbitals. The first non-septet SO states 5X1(1) and 9X0(1) have vertical excitation energies of 45.9 and 48.2 kJ mol"1, respectively. The bonding schemes of these compounds differ from the septet states. In the case of 5X1(1), the a orbitals are occupied with a single

electron (more precisely the difference between the occupation of a and a* orbitals, cf. Figure S1 of the supplementary material), whereas each n orbital is similarly occupied with ca. 1.7e. In the case of 9X0(1), the occupation number of each n orbital is decreased by 0.3-0.4e (compared to the SO ground state) causing an increased SO bond distance of 2.018 A. Performing the SO calculations at the equilibrium bond distances, the adiabatic energy of the 5X1(1) state is 40.4 kJ mol"1 and the adiabatic energy of the 9X0(1) state is 20.9 kJ mol"1.

There is a previous DFT study on PuC22 reporting a 5S- ground state for PuC with a bond distance of 2.462 A. We tried to identify this state among our CASPT2 results. According to our CASPT2 calculations, the first52- SF state appears 44.1 kJ mol"1 higher in energy and with a quite short optimised bond distance (1.850 A) compared to the septet ground state. Inspecting the PuC states having long bond distances, the longest one predicted by our CASPT2 calculations is 2.369 A for an excited undectet state (112-, group 3) which lies over 100 kJ mol"1 higher in energy than the ground state. This 112- state has low occupation of the

TABLE V: The lowest-energy vertical SF and SO states of AmC.a More detailed data can be found in Tables S4 and S5 of the supplementary material.

aE S Db Main composition

SF states

0.0 0 6£+(1)

4.7 394 8£+(1)

14.4 1202 10£+(1)

31.2 2608 4£+

40.3 3366 8 £+(2)

40.9 3419 6£+(2)

46.5 3884 10£+(2)

47.2 3943 12£-

15% x2sf n2fx nf 7sa (5f^,5f2+)a (5f2+,5f2_)a 5^ 5f3a+ 9% x2sf n « n2fy f (5fo,7s)a f - 5fa+ 5fa- 5fa+

19% x f xf n fx 4 f 5fa 5fa- 5fa+ f- 5fa+

9% x2f nf nf 7sß (5f^,5f2+)a (5f2+,5f^)a 5f^- 5f3a+

6% ^ n fx f n fy f (5fQ,7s)2 5f2a- 5f2a+ 5f3a- 5fa+

4% xsf nf n2fy 7s2 (5f2_,5f2+)a (5f2+,5f^)a 5^ 5f3+ 41% x2f na n'3 ni n*a 5f? 5f? 5f? 5f? 5f?

sf fx fx fy fy 0 2- 2+ 3- 3+

98% xsf xf nfy nfx (5fo,7s)a 5f<3- 5fa+ 5Ç- 5^+ 5f3a- 5f3a+

SO states

0.0 0 6£+(1) 6 100% 6£+(1)

4.7 394 8£+(1) 8 100% 8£+(1)

14.4 1202 10£+(1) 10 100% 1q£+(1)

31.2 2611 4£+ 4 100% 4£+

40.3 3366 8 £+(2) 8 100% 8 £+(2)

40.9 3419 6£+(2) 6 100% 6£+(2)

46.5 3884 10£+(2) 10 100% 1q£+(2)

47.2 3943 12£- 12 100% 12£-

aStates computed at the bond distance of 2.150 A. AE: Relative energy: first column in kJ mop1, second column in cm1; S: Term symbol of the electronic state, D: Degeneracy due to spatial symmetry. For the r and n orbitals see Figure 1. a and & represent the different spin of the electrons.

bThe Q values are not listed for the SO states because 2 states are not split by spin-orbit coupling.

bonding and high occupation of the antibonding orbitals. Therefore we suspect that Ref. 22 results may be an artifact due to the large 78-electron core relativistic effective core potential (RECP) used for PuC compared to the 60-electron core size that we used. It has been shown that the large-core RECPs in conjunction with DFT can provide unrealistic structures.57-60 We note that because of the high number of low-lying electronic states found for PuC, only a selection of characteristic states is listed in Table IV. More detailed information on the SF and SO states at the two bond distances considered (1.913 and 1.898 A, respectively) can be found in the supplementary material (Tables S4 and S5).

For AmC the picture becomes somewhat simpler (cf. Table V). With the 6d being too high in energy to play a role in the electronic configurations, the seven non-bonding electrons are distributed among pure 5f orbitals. The f7 configuration is highly stable61 causing the first non-2 state (10A) to be 94.3 kJ mol"1 higher in energy than the ground state. In the 62+ ground state, the n bonding orbitals are fully occupied, while the higher spin multiplicity states, with all the 5f orbitals filled, require the excitation of electrons to the antibonding <** and n* orbitals. At the level of our calculations, the SO states of AmC listed in Table V have no splitting. Each state is doubly degenerate.

The electronic absorption spectra of the four monocar-bides have been calculated at the SO-CASPT2 level. The spectra predicted for high-temperature gas-phase measurements (3000 K) are shown in Figure 2 up to 12 000 cm1 while the detailed description of the most important vertical transitions is given in Table VI. We note that several transitions

in the spectra start from low-lying excited states, the populations of which depend on the temperature. Therefore different temperatures would result in different relative spectral intensities.

For the three most intense ThC absorption lines, the donor state is 2, whereas the acceptor states are either n or mixed from 2 and n states. The most intense transitions start from the12+ excited state. All the excitations from the ground state have intensities below 40%. The changes in orbital occupations are in accordance with the 2 ^ n transitions; the populations of both 5fr and 6dr orbitals increase, while the populations of n-type MOs decrease.

In the spectrum of UC, the three most intense lines represent Q value changes by one digit between the donor and acceptor states. The most intense transitions occur from excited states 5A4,5X0(2), and 5X3(1), the two former ones being sufficiently populated only at a high temperature. These intense lines correspond to partial population changes between the 5fn and 5f^ MOs.

Due to the high density of excited states, the spectrum of PuC is the richest of the studied monocarbides. All the intense transitions occur from excited states which are sufficiently populated only at a high temperature. The most intense lines correspond to partial population changes between the non-bonding 7s and bonding MOs with < and n character.

The three most important transitions of AmC occur from the first three electronic states, the one from the ground state having the smallest intensity of the three at the modelling temperature (3000 K). Due to the previously mentioned strong multiconfigurational nature, only minor population changes in

FIG. 2. Calculated electronic absorption spectra of AnC molecules for 3000 K.

the MOs can be recognised during the transitions. As can be seen in Table VI, they affect mainly the bonding and antibond-ing MOs.

The transition probabilities, described by the Einstein coefficients, are given in Table VI. The considerably different Einstein coefficients of the four actinide carbides can

TABLE VI. Significant transitions in the absorption electronic spectra of AnC.a

D A aE Einst. Int. Transition

1 1 1 3 ^a) 1 n X0(2) X0(3) 3 n1(1) 6913 5901 9325 5377 29 795 34 ggg 21 372 15 05g 1.00 1.00 0.97 0.39 7s0.936d0rg05f«r256d».745f«-27 ^ 7s1-026d°r995f0cr326dn.555fn.23 6d0.806d0.74 ^ 6d^fg«6dn:64 7s0.936d0rg06d«746d0-01 ^ 7Sl:l06d(¿856d«:676dO:06 7s0.926d2^g26dn.715f|n.27

5 a4 5X0(2) 5X3(1) 5X3(4) 5X1(3) 5 r 2121 4327 1ggg 3 g3g 3 590 4 037 1.00 0.95 0.94 5f4695f<000 ^ 5f0:395f0:33 5f°/55f°.°4 ^ 5fПr355fOr39 5f0:375f<).35 ^ 5f0:685f«.00

7X7 7XI(2) 7X5(2) Xg(1) Xg(2) 5X1(1) X5(3) Xg(2) 6015 1732 4906 2339 1 055 3 3gg 1 17g 2 2g3 1.00 0.92 0.91 0.g4 5f0:735f0.84 ^ 5f0:9l5ffl:59 7s0.696d0.426dft545f0.505fa6g ^ 7s0:g76d(¿336d0п4g5f<¿g15fП56 7s0.696dOr416dП575fа505f«:745f«:76 ^ 7s0:g66d^346dп495f^805fп5g5f0:g2 5f0r385f<П865f0.75 ^ 5f0r445fn915f0.59

g (1) 10 (1) 6 (1) g (2) 10 (2) 6 (2) 2973 26g1 3419 232 227 77 1.00 0.75 0.35 xLf26 n ;.-22 n f77 ^ x Lf16 n'-16 n *fag4 sf foy foy sf foy foy xLf12 xT1 n^1 n*agg sf sf foy foy ^ x1,33 xt66 n1..00 nf99 sf sf foy foy x 1v41 x^ ^ xVg xf76 sf sf sf sf

aD: donor state; A: acceptor state; ÀE: Energy in cm 1; Einst.: Einstein coefficients [x1016 m3 J 1 s 2]; Int.: Relative intensity referring to 3000 K. The orbital populations are Mulliken populations from CASPT2 calculations. For the Q values see Table V.

TABLE VII. Comparison of selected molecular data of Th, U, Np, Pu, and Am monocarbides.

ThC UC NpCa PuC AmC

State 3 Z0+(1) 5X3(1) H3.5(1) 7Xi(1) 6S+.54.5 2.5(1)

r(An-C)b 1.967 1.870 1.774 1.898 2.150

De (^An + C)c 562 402 332 365 221

EBOd 2.38 2.37 2.57 2.07 1.14

nAne 0.99 0.86 0.80 0.92 0.80

BIf 2.57 2.62 2.52 2.36 0.77

aData taken from Ref. 21. bSO-CASPT2 bond distances are given in Â. cSO-CASPT2 dissociation energies are given in kJ mop1. dEBO: Effective bond order from CASPT2 calculations. eNatural atomic charges in e from NBO calculations (DFT). fBI: Wiberg bond index (DFT).

be explained by the selection rules. In the non-relativistic approximation transitions with angular momentum changes of only AA = 0, ± 1 at no spin change (AS = 0) are allowed. In rel-ativistic theory the only selection rule for allowed electronic transitions is AQ = 0, ± 1.

The main differences in the Einstein coefficients in Table VI can partly be related to the behaviour of spin: the transitions of UC and ThC are quite strong. For UC the spin symmetry is kept for the intense transitions; in the case of ThC the most intense transition obeys both non-relativistic rules, whereas in the other two the angular momentum change is below one. No changes of spin can be observed in the listed most intense transitions of AmC (cf. Table VI), while these transitions have a very low probability.

C. Bonding properties and dissociation

Table VII summarises some molecular parameters for the studied actinide monocarbides considering also the previously studied NpC.21 The calculated NBO charges of actinides in the ground-state structures were 0.99, 0.86, 0.80, 0.92, and 0.80e for ThC, UC, NpC, PuC, and AmC, respectively. The latter quite small atomic charges of the actinide atoms point to the importance of MO interactions in actinide monocarbides. The interactions in AmC (belonging to group 3, vide supra) differ considerably from those of the other monocarbides (belonging to group 1). The much weaker bond in AmC is evident from both its considerably smaller vibrational frequency (cf. Table I) and the small bond order values (nearly half of those of the other AnC molecules).

The stability of the five monocarbides can be compared on the basis of the computed (SO-CASPT2) dissociation energies (Table VII), showing that ThC is the most stable molecule of the group. Although UC, NpC, and PuC have shorter bond distances, their dissociation energy is significantly smaller. In the case of AmC the dissociation energy decreases further. The trend is similar to the one found for the monoxides62,63 and is considerably influenced by the stabilisation of the actinide atom (as a dissociation product).

IV. CONCLUSIONS

In this paper we have reported the first systematic study of the most important actinide monocarbides (ThC, UC, PuC,

and AmC). We have elucidated their ground electronic states and low-lying excited states by relativistic multireference calculations. The multiconfigurational character of the carbides increases from ThC to AmC. Particularly in PuC and AmC (due to the larger number of 5f electrons), the states are very close energetically, contributing to the complexity of the electronic structure.

The computed Einstein coefficients facilitate the evaluation of the absorption electronic spectra. At the experimentally relevant temperature (3000 K), due to their increased populations, the most intense transitions occur with excitations from the lowest excited states. For many AmC transitions, the wave functions involved do not have large terms obeying the non-relativistic selection rules, so the transitions have low intensities.

The molecular geometries are the most relevant of the computed molecular properties. According to the occupation of bonding orbitals, the calculated electronic states were classified into three groups, each determining a characteristic bond distance range of the equilibrium geometry. The ground states of ThC, UC, and PuC have two doubly occupied n orbitals causing a bond distance between 1.8 and 2.0 Â, whereas the ground state of AmC has significant occupation of the antibonding orbitals, causing a bond distance of 2.15 Â.

SUPPLEMENTARY MATERIAL

See supplementary material for a figure showing characteristic molecular orbitals from CASPT2 calculations and six tables with: detailed B3LYP results and NBO characteristics; composition of AnC Kohn-Sham orbitals for chosen electronic states; detailed data of the lowest-energy CASPT2 spin-free states; extended list of low-energy spin-free and spin-orbit states as well as of the vertical transitions in the absorption electronic spectra.

ACKNOWLEDGMENTS

The 7th Framework Programme of the European Commission (Grant No. 323300, TALISMAN) and the Hungarian Scientific Research Foundation (OTKA No. 75972) are acknowledged for financial support.

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