Excitation Energies of UO22+, NUO+, and NUN Based on Equation-of

9 May 2017 - energies with intermediate Hamiltonian Fock space CC method (IHFSCC) are .... SAOP51 potential or the range-separated CAM-B3LYP52...
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Excitation Energies of UO22+, NUO+, and NUN Based on Equation-ofMotion Coupled-Cluster Theory with Spin−Orbit Coupling Shuo Zhang and Fan Wang* Institute of Atomic and Molecular Physics, Key Laboratory of High Energy Density Physics and Technology, Ministry of Education, Sichuan University, Chengdu, People’s Republic of China ABSTRACT: Obtaining reliable excitation energies of the isoelectronic series, UO22+, NUO+, and UN2, is a challenge in quantum chemistry calculations due to importance of electron correlation and spin−orbit coupling (SOC). Vertical spin-free and spin−orbit coupled excitation energies of these molecules are calculated in this work using equation-of-motion coupledcluster approach at the CC singles and doubles level (EOMCCSD). SOC is included in post-SCF calculations and this treatment has been shown to provide SOC effects with high accuracy for heavy elements. Excitation energies for UO22+ with the present approach are in good agreement with previous results using linear response CCSD, multiconfiguration perturbation theory (CASPT2), and multireference configuration interaction (MRCI). As for NUO+ and UN2, excitation energies with CASPT2 from two previous calculations differ significantly and the present results usually lie between these two sets of CASPT2 results. On the other hand, excitation energies with intermediate Hamiltonian Fock space CC method (IHFSCC) are generally too small compared with our results. This work provides new estimates on excitation energies of these molecules and it could be helpful in investigating spectroscopic and luminescence properties of larger uranium complexes.

I. INTRODUCTION The chemistry of uranium-containing systems is important in separation, processing, disposal, and storage of nuclear waste from nuclear energy production,1−3 as well as their applications in catalysis.4,5 Besides radiotoxicity of the involved species, it is also difficult to isolate and characterize individual species experimentally because uranium changes its oxidation state easily. Theoretical modeling thus plays an important role in understanding properties of uranium-containing systems. Calculating excitation energies of uranium-containing molecules are particularly useful in investigating spectroscopic and luminescence properties, probe of uranium compounds, as well as studying photochemical reactions of the involved species. UO22+ is the most common building block of larger uranium complexes, but it is difficult to be formed in gas phase and experimental spectroscopic measurements are carried out in aqueous solution and in crystals. Its environment will inevitably affect its excitation energies and theoretical investigation on related molecules such as [UO2Cl2]6,7 and [UO2Cl42−]8−10 have been reported. On the other hand, calculating excitation energies of the bare UO22+ can still provide useful information. Its isoelectronic homologues, NUO+ and UN2, have been synthesized11 and are also building blocks in larger organometallic systems. Calculating excitation energies of these two species could also be helpful in rationalize spectroscopic and luminescence properties of larger uranium complexes. Besides practical usefulness of computational modeling for the isoelectronic series, study of these molecules is also © 2017 American Chemical Society

theoretically interesting because achieving reliable excitation energies of these species is still challenging. Many electrons or orbitals are involved in bonding and excitations of these molecules, that is, the 5f, 6p, 6d, 7s orbitals of uranium and the 2s, 2p orbitals of oxygen or nitrogen atoms, and electron correlation needs to be taken good care of to obtain reasonable excitation energies. Furthermore, relativistic effects,12−16 which are composed of scalar relativistic (SR) effects and spin−orbit coupling (SOC), need to be considered. SR effects are relatively easy to be tackled, whereas calculating SOC effects is usually much more involved. The ground state of UO22+, NUO+, and UN2 is closed-shell and SOC has only a minor effect on their ground state properties but the excited states are affected by SOC more pronouncedly. Calculating SOC effects perturbatively is certainly nonapplicable for such a heavy element as uranium. A popular method in treating SOC in quantum chemistry calculations is the two-step approach17 where SOC between a bunch of spin-free states are considered. Orbital relaxation is neglected in such calculations and it may still be insufficient in describing SOC effects for excited states of these molecules. On the other hand, including SOC in SCF calculations based on two- or four-component relativistic Hamiltonian18,19 is the most rigorous way in dealing with Received: March 29, 2017 Revised: May 8, 2017 Published: May 9, 2017 3966

DOI: 10.1021/acs.jpca.7b02985 J. Phys. Chem. A 2017, 121, 3966−3975

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The Journal of Physical Chemistry A

IHFSCC results are reported using either two-component ZORA, X2C, or four-component DC Hamiltonian with SOC included in self-consistent field (SCF) calculations.10,22−24,30 According to previous results, excitation energies with different approaches for these molecules do not agree quite well with each other. Spin-free excitation energies of UO22+ with LR-CCSD, MRCI and AQCC are in reasonable agreement with each other, while results of CASPT2 are about 0.3−1.0 eV lower and IHFSCC results are about 0.4 eV lower.20,21 Similar results also hold when SOC is considered. In addition, a recent work23 shows that excitation energies with IHFSCC using a large active space are even lower. As for TDDFT results, excitation energies are generally underestimated pronounced with GGA and hybrid functionals compared with results of ab initio methods.30 On the other hand, TDDFT results using the SAOP51 potential or the range-separated CAM-B3LYP52 functional are improved to some extent. In our previous works,53,54 we implemented a relativistic coupled-cluster approach at the CCSD and CCSD(T)55 levels with SOC included in post-SCF treatment (SOC-CC). This approach was proposed first by Eliav et al.35 and it can be used together with relativistic Hamiltonian where the SOC operator is represented or approximated by a one-electron operator. SOC-CC is rather efficient through the use of real molecular spin−orbitals, spatial symmetry, and time-reversal symmetry.54 Single excitation amplitudes in CC theory can describe orbital relaxation due to SOC effectively and this approach, particularly the SOC-CCSD approach, provides a reliable description on SOC effects of even superheavy elements53,56 compared with the CCSD approach where SOC is considered in the SCF step. EOM-CCSD approaches based on this SOC-CCSD for excitation energies,57 ionization potentials,58,59 electron affinities59,60 as well as double ionization potentials61 of closed-shell systems have also been fulfilled. We propose to report excitation energies of UO22+, NUO+, and UN2 using this EOM-CCSD approach based on SOC-CCSD in this work. Scalar relativistic results will also be presented to facilitate comparison with previous spin-free excitation energies. This paper is organized in the following manner: a brief introduction on the employed approach and computational details, that is, the adopted RECPs and basis set will be provided in Section II. The calculated spin-free and SOC vertical excitation energies will be given in Section III and comparison between our results and previous ones such as those of IHFSCC, CASPT2 will be made in this section. Conclusions will be drawn in Section IV.

SOC but such approaches are usually computationally demanding. Theoretical investigations on excitation energies of UO22+ have been carried out extensively in previous works.8,10,20−24 On the other hand, previous works mainly focus on groundstate properties of NUO+ and UN225−29 and only a few works23,24,30 reported theoretical electronic spectra of these two molecules. Various theoretical methods have been adopted to calculate excitation energies of these molecules. These methods include multireference electron correlation approaches such as complete active space with second order perturbation theory (CASPT2)31 or restrictive active space with second order perturbation theory (RASPT2), multireference configuration interaction method (MRCI) with size-extensivity correction,32 multireference averaged-quadratic coupled-cluster method (MR-AQCC),33 and Fock space coupled-cluster theory (FSCC).34−37 The intermediate Hamiltonian FSCC method (IHFSCC)36−39 is generally adopted in practical FSCC calculations to avoid intruder state problems. The employed single reference approaches are linear response/equation-ofmotion coupled-cluster theory (LR/EOM-CC)40,41 and timedependent density functional theory (TDDFT).42,43 CASPT2 is a popular method in calculating excitation energies and a previous work shows that excitation energies of a set of medium-sized organic molecules with CASPT2 are very close to high level CC3 results.44 However, CASPT2 results depend on the employed active space. The active space becomes too large if all the involved valence orbitals are included in CASPT2 calculations for UO22+, NUO+ and UN2. In previous CASPT2 calculations for these systems20,21,24 the highest 6 bonding orbitals together with their antibonding partners as well as 4 nonbonding f orbitals are usually chosen as the active space with 12 electrons. On the other hand, a minimum active space was adopted in MRCI and MR-AQCC calculations.21 In IHFSCC calculations, the ground state should be described reasonably with CC approach and excitation energies from active occupied orbitals to active virtuals orbitals are calculated in the (1,1) sector of the Fock space. The ground state of these molecules can be described reliably using single reference approaches.30 This ensures applicability of IHFSCC and other single reference approaches such as EOM-CC and TDDFT to excitation energies of these molecules. The most popular EOM-CC approach is that at the CC singles and doubles level (CCSD)45 and error of excitation energies for excited states with a mainly single excitation character is usually about 0.1−0.3 eV using EOM-CCSD.40 TDDFT is nowadays the most popular method for excitation energies due to its compromise between efficiency and accuracy, however, TDDFT results depends critically on the employed exchangecorrelation (XC) potentials as well as on the XC kernel. In previous calculations for these molecules relativistic effects are considered through relativistic effective core potentials (RECPs),46 Douglas−Kroll−Hess Hamiltonian (DKH),47 the ZORA (zeroth order regular approximation) Hamiltonian,48 the eXact 2-component (X2C) Hamiltonian,13,49 or the DiracCoulomb Hamiltonian (DC). These results show that spin-free excitation energies do not change much with respect to the employed scalar relativistic Hamiltonian except that a smallcore RECP instead of large-core RECP should be used.50 Furthermore, difference between results of RECP and allelectron ones becomes larger for higher excited states. As for treatment of SOC, two-step approach is used in previous CASPT224 and LR-CCSD calculations,21 while TDDFT and

II. METHODS AND COMPUTATIONAL DETAILS In EOM-CC approach, excitation energies, ionization potentials, electron affinities, and so forth are calculated based on a CI-like approach from eigenvalues of an effective Hamiltonian e‑THeT in different spaces, where H is the Hamiltonian and T is the cluster operator determined from solving the CC equations. Implementation details of the EOM-CCSD approach for excitation energies based on the SOC-CCSD approach can be found in ref 57. EOM-CC approach is closely related to LRCC approach and these two approaches provide exactly the same excitation energies, while transition properties are different. As a single reference approach, active space is not required in calculating EEs, IPs, and EAs with EOM-CCSD. In fact IPs and EAs with EOM-CCSD have been shown to be exactly the same as those with FSCC,62 although FSCC is a multireference method. On the other hand, excitation energies 3967

DOI: 10.1021/acs.jpca.7b02985 J. Phys. Chem. A 2017, 121, 3966−3975

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The Journal of Physical Chemistry A

description on excitation energies of these molecules. All electrons not treated via ECPs are correlated in the calculations. A bond length of 1.683 Å for U−O in UO22+ is used in some works,20−22 while a bond length of 1.708 Å is adopted in other works.8,21−23,30 According to TDDFT results,22 excitation energies are reduced by about 0.15−0.3 eV when the bond length is increased from 1.683 to 1.708 Å. Excitation energies under these two bond lengths are calculated in the present work to provide a proper comparison with previous works. The U−N bond length is determined to be 1.739 and 1.698 Å in UN2 and NUO+, respectively, and the U−O bond length in NUO+ is calculated to be 1.761 Å with the PBE XC functional and scalar ZORA Hamiltonian in ref 30. On the other hand, these three bond lengths are 1.743, 1.703, and 1.759 Å, respectively, according to CCSD(T) results with RECP.24 Difference in these bond lengths is quite small and excitation energies are affected by less than 0.05 eV according to our tentative calculations. We choose bond lengths determined from CCSD(T) calculations for NUO+ and UN2 in the present work. All calculations are carried out with a locally modified CFOUR71 program package.

with FSCC depend on how the active space is chosen. An exponential ansatz is adopted for the excited state wave function in FSCC and this warrants size-extensivity and sizeintensivity of the obtained excitation energies. However, excitation energies with EOM-CC approach are size-intensive for local excitations and are not size-extensive for charge transfer transitions. In fact sample calculations63,64 showed that this error of EOM-CCSD is less than 0.1 eV. EOM-CCSD usually overestimates excitation energies since the ground state is generally described more accurately than the excited states. In fact, excitation energies are calculated rather reliably using EOM-CCSD for single excitations out from a state with a dominant single reference character. On the other hand, they are overestimated significantly for transitions with a sizable double excitation character. A previous SR calculation using a large-core RECP shows that difference in excitation energies of the lowest five singlet states of UO22+ between EOM-CCSD and EOM-CCSDT is less than 0.2 eV except for the high-lying 1Δu state, where the difference reaches 0.28 eV.50 Another interesting point is that these excitation energies with EOM-CCSD are actually smaller than those with the EOMCCSDT except for the 1Δu state. This may indicate that error for the ground state of UO22+ is even larger than that for the lowest several excited states in EOM-CCSD calculations. It is possible that double excitation character is more pronounced in some higher states such as the 1Δu state and their excitation energies are still overestimated by EOM-CCSD. As for describing SOC effects, SO splitting of the 2Π states of monohydrides up to fifth-row elements can be described with high accuracy using this EOM-CCSD approach for IPs or EAs together with the X2C Hamiltonian according to our previous work.65 In addition, error of excitation energies for some closed-shell sixth row elements such as Ba, Au+, Hg, Tl+, and Rn is calculated to be less than 0.1 eV using this EOM-CCSD approach and the X2C Hamiltonian.66 Unfortunately, all electrons need to be considered in solving the CC equations and a large basis set is required to provide a reliable description on SOC of particularly the inner shell electrons and to achieve highly accurate results in those calculations. The EOM-CC approach based on SOC-CCSD is thus best to be used together with RECPs in practical calculations and satisfactory excitation energies for some closed-shell 6p-block elements are still achieved.57 A recently developed small-core energy-consistent relativistic pseudopotential for uranium67 is adopted in this work. The innermost 60 electrons with n ≤ 4 are replaced by the pseudopotential. The Breit interaction and finite nuclear size effects are considered in determining the parameters in the pseudopotential. It is worth noting that the Gaunt interaction reduces excitation energies of these molecules by about 0.03− 0.06 eV.23 The basis set developed together with the pseudopotential is chosen for uranium. However, it is demonstrated that special care has to be taken in choosing basis set when SOC is present to provide a proper description of valence p and d orbitals.68,69 We use uncontracted basis functions except for g functions, that is, the (14s13p10d8f6g)/ [14s13p10d8f3g] basis set for uranium. As for O and N, the aug-cc-pVQZ basis set70 is employed. Diffuse basis functions are usually necessary particularly for high lying excited states and for excited states of neutral species. A previous work indicated that a minimum of triple-ζ basis set is required to achieve reliable spectroscopic data for [UO22+].20 The employed basis set is flexible enough to provide a reasonable

III. RESULTS AND DISCUSSION Transitions investigated in this work are mainly from the six highest occupied orbitals in these molecules to the virtual fδ and fϕ orbitals of uranium. These highest six occupied orbitals are bonding orbitals between the 2p orbitals of oxygen or nitrogen and the 5f or 6d orbitals of uranium. They are 5σu+, 3πu, 6σg+, and 2πg orbitals in UO22+ and UN2. The ungerade orbitals are bonding orbitals between the 2p orbitals of oxygen or nitrogen and the 5f orbitals of uranium, while the gerade orbitals are bonding orbitals between the 2p orbitals of oxygen or nitrogen and the 6d orbitals of uranium. The ungerade orbitals are higher in energy than the gerade orbitals and the 5σu+ orbital is the HOMO in UO22+ due to antibonding contribution of the 6p orbitals of uranium.8 The situation is different in UN2 and the 5σu+ is lower than the 3πu orbital because contribution of the 6p orbital of uranium to this orbital is less than that in UO22+. In addition, the 6σg+ orbital becomes the HOMO according to our Hartree−Fock results. These six highest occupied orbitals are 4π, 10σ +, 5π, and 11σ+ orbitals in NUO+ and the 11σ+ is the HOMO. The higher 11σ+ and 5π orbitals are bonding orbitals between the 2p orbitals of nitrogen and the 5f, 6d orbitals of uranium, while the lower 10σ+ and 4π orbitals are bonding orbitals between 2p orbitals of oxygen and the 5f, 6d orbitals of uranium. The virutal fδ and fϕ orbitals of uranium are the 1δu and 1ϕu orbitals in UO22+ and UN2 and the 1ϕ and 3δ orbitals in NUO+. These virtual f orbitals are not the LUMO and unoccupied 7σg+ and 2δg orbitals are lower than these virtual fδ and fϕ orbitals in UO22+, while the unoccupied 12σ+, 6π, and 2δ orbitals lie below the virtual fδ and fϕ orbitals in NUO+. The virtual σ and δ orbitals below the fδ and fϕ orbitals are mainly the 7s orbital and 5d orbital of uranium, respectively. Besides the virtual 7s and 5d orbitals of uranium, many more unoccupied orbitals lie below the virtual fδ and fϕ orbitals in UN2. It should be noted that orbital energies of the virtual fδ and fϕ orbitals in UN2 are positive, and this means many unoccupied diffuse orbitals can possibly lie below these two orbitals. A diagram for the related orbitals of these three molecules can be found in Figure 1 of ref 24. A. Vertical Excitation Energies of UO22+. The lowest excited states in UO22+ are transitions out from the highest six occupied orbitals to the virtual fδ and fϕ orbitals. Spin-free 3968

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The Journal of Physical Chemistry A Table 1. Spin-Free Vertical Excitation Energies of UO22+ (All Values in Electronvolts) TDDFT method

EOM-CCSD

r(U−O) in Å state 13Δg 13Φg 11Φg 11Δg 13Πg 13Γg 11Πg 11Γg 13Δu 11Δu 13Φu 11Φu 13Πu 11Πu 11Γu 13Γu

transition 5σu+-1δu 5σu+-1φu 5σu+-1φu 5σu+-1δu 3πu-1δu 3πu-1φu 3πu-1δu 3πu-1φu 6σg+-1δu 6σg+-1δu 6σg+-1φu 6σg+-1φu 2πg-1δu 2πg-1δu 2πg-1φu 2πg-1φu

1.683

1.708

2.986 3.185 3.777 4.165 5.187 5.291 5.326 5.500 5.256 5.272 5.443 5.535 6.129 6.373 6.432 6.455

2.840 2.998 3.571 3.972 4.875 4.944 5.013 5.140 5.017 5.024 5.157 5.241 5.758 5.993 6.005 6.029

IHFSCCSD 1.683

20

2.600 2.858 3.415 3.741 4.839 4.911 4.808 5.137 4.753 4.696 4.967 5.038 5.720 5.719 6.006 6.093

1.708

LR-CCSD 30

1.683

2.480 2.700 3.240 3.570

20

AQCC 1.683

3.022 3.281 3.840 4.204 5.118 5.228 5.224 5.444 5.166 5.152 5.377 5.472 6.012 6.259 6.360 6.339

4.570 4.780 4.760 4.710 4.690 4.740

MRCI+DC

21

1.683

3.185 3.370 3.926 4.419 5.259 5.311 5.349 5.461 5.225 5.262 5.430 5.430 6.170 6.343 6.352 6.416

CASPT2[g1]

21

1.708

3.141 3.197 3.876 4.428 5.213 5.274 5.300 5.449 5.262 5.291 5.485 5.485 6.197 6.424 6.403 6.441

SAOP

8

1.708

2.787 2.937 3.467 3.898 4.610 4.657 4.678 4.857 4.623 4.543 4.713 4.742 5.270 5.355 5.484 5.553

CAM-B3LYP

21

1.70830

3.269 3.052 3.500 4.009 4.709 4.322 4.991 4.506 4.563 4.549 4.193 4.272 5.241 5.679 4.942 4.951

2.560 2.430 3.070 3.690 4.360 4.650 4.910 4.990 4.690 4.810

Table 2. Vertical Excitation Energies for Gerade States with SOC of UO22+ (All Values in Electronvolts) TDDFT22 method

EOM-CCSD

r(U−O) in Å state 1Δg 1Πg 1Φg 2Δg 2Φg 1Γg 3Φg 3Δg 2Πg 2Γg 1Σg− 1Σg+ 4Φg 5Φg 4Δg 3Πg 5Δg 3Γg 5g 4Πg 6Δg 6Φg 7Φg 7Δg

transition 5σu+-1φu/1δu 5σu+-1δu 5σu+-1φu 5σu+-1δu 5σu+-1δu/1φu 5σu+-1φu 5σu+-1φu 5σu+-1δu 3πu-1φu 3πu-1φu 3πu-1δu 3πu-1δu 3πu-1δu 3πu-1φu 3πu-1δu/1φu 3πu-1δu 3πu-1δu/1φu 3πu-1δu/1φu 3πu-1φu 3πu-1δu 3πu-1δu/1φu 3πu-1δu 3πu-1φu 3πu-1φu

principal SOF state 13Φg/13Δg 13Δg 13Φg 13Δg 13Δg /13Φg 13Φg 11Φg 11Δg 23Δg 13Γg/11Γ 13Πg 13Πg 23Φg/21Φg 13Γg 23Φg/23Δg 13Πg 23Φg/23Δg 23Φg/13Γg/11Γg 13Γg 11Πg 13Πg/23Δg 21Φg/23Φg 23Δg 21Δg

SO-LR-CCSD

SO-CASPT2

1.683

1.708

1.68320

SO-IHFSCCSD 1.70823

1.68321

1.7088

1.683

1.708

1.683

2.626 2.670 2.772 2.950 3.312 3.427 3.824 4.139 4.674 4.766 4.795 4.822 4.832 5.075 5.135 5.175 5.195 5.412 5.429 5.446 5.642 5.748 5.820 5.836

2.454 2.525 2.598 2.793 3.158 3.246 3.627 3.947 4.349 4.437 4.494 4.524 4.529 4.720 4.791 4.849 4.864 5.108 5.101 5.144 5.315 5.417

2.310 2.307 2.451 2.614 2.975 3.114 3.488 3.748 4.284 4.405

1.764 1.843 1.919 2.131 2.514 2.592 2.992 3.394 3.853 3.968 4.086 4.122 4.079

2.825 2.848 2.963 3.129 3.448 3.602 4.015 4.303 4.843 4.940 4.981 4.996 4.968 5.054 5.096 5.178 5.126 5.551 5.652 5.598 5.676 5.678 5.773 5.954

2.380 2.493 2.513 2.767 3.154 3.262 3.606 3.882 4.082 4.124

2.660 3.070 2.810 3.290 3.670 3.500 3.830 4.280 4.070 4.180 4.640 4.670 4.420 4.690 4.580 5.030 4.980 5.260 4.900 5.410 5.270 5.240 5.620 5.580

2.490 2.920 2.630 3.140 3.510 3.320 3.650 4.050 3.780 3.870 4.360 4.390 4.100 4.410 4.300 4.730 4.680 4.980 4.600 5.130 4.970 4.920

2.040 2.340 2.210 2.600 3.010 2.920 3.360 3.860 4.050 4.140 4.500 4.560 4.440 4.560 4.600 4.960 4.890 5.160 4.850 5.340 5.320 5.320 5.600 5.580

5.486

vertical excitation energies for these transitions of UO22+ at bond lengths of 1.683 and 1.708 Å with EOM-CCSD are listed in Table 1 together with previous results with LR-CCSD,20 IHFSCCSD,20,30 AQCC,21 MRCI+DC,21 CASPT2,8 and TDDFT using SAOP21 and CAM-B3LYP.30 According to our EOM-CCSD results, excitation energies are reduced by about 0.2−0.3 eV for lower states, while they are reduced by about 0.4 eV for transitions out from the 2πg orbitals when the bond

SAOP

5.280

CAM-B3LYP

length is increased from 1.683 to1.708 Å. This indicates that it is important to compare excitation energies at the same bond length for this molecule. One can see from this table that the present excitation energies for transitions out from the 5σu+ orbital are about 0.05−0.1 eV lower than those of LR-CCSD,20 while they are about 0.1 eV higher for the other transitions. Our EOM-CCSD results are in good agreement with those of LRCCSD and the difference should come from the employed basis 3969

DOI: 10.1021/acs.jpca.7b02985 J. Phys. Chem. A 2017, 121, 3966−3975

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The Journal of Physical Chemistry A Table 3. Vertical Excitation Energies for Ungerade States with SOC of UO22+ (All Values in Electronvolts) TDDFT22 method

EOM-CCSD

r(U−O) in Å state 1Δu 1Πu 2Δu 1Φu 3Δu 2Φu 1Γu 3Φu 1Σu− 1Σu+ 4Φu 2Πu 2Γu 4Δu 5Δu 5Φu 3Πu 3Γu 6Δu 4Πu 6Φu 7Δu 5u 7Φu 4Γu

transition 6σg+-1φu 6σg+-1δu 6σg+-1δu 6σg+-1φu 6σg+-1δu 6σg+-1δu/1φu 6σg+-1φu 6σg+-1φu 2πg-1δu 2πg-1δu 2πg-1δu 2πg-1δu/1φu 2πg-1φu 2πg-1δu 2πg-1φu 2πg-1φu/1δu 2πg-1δu 2πg-1δu 2πg-1σu/1φu 2πg-1δu 2πg-1δu 2πg-1φu 2πg-1φu 2πg-1φu 2πg-1φu

principal SOF state 13Φu 13Δu 13Δu/11Δu 13Φu/11Φu 11Δu/13Δu 13Δu/13Φu 13Φu 11Φu/13Φu 13Πu 13Πu 23Φu 13Πu/23Δu 13Γu/11Γu 23Φu 23Δu/21Δu 13Γu/11Φu 13Πu 23Φu 13Πu 11Πu 21Φu/23Φu 21Δu 13Γu 23Δu 13Γu/11Γu

1.683

1.683

4.994 5.013 5.020 5.025 5.608 5.609 5.750 5.801 5.866 5.887 5.911 5.925 5.945 5.964 6.005 6.047 6.055 6.500 6.546 6.581 6.604 6.685 6.743 6.754 6.807

4.717 4.747 4.774 4.771 5.359 5.361 5.465 5.508 5.494 5.516 5.538 5.525 5.537 5.588 5.598 5.639 5.675 6.118 6.171 6.208 6.219 6.288 6.693 6.352 6.382

SO-LR-CCSD 1.683

21

4.979 5.039 5.047 5.008 5.596 5.590 5.743 5.813 5.900 5.900 5.949 5.916 5.964 6.040 6.054 6.119 6.138 6.499 6.615 6.599 6.658 6.758 6.784 6.871 6.910

SAOP

CAM-B3LYP

SO-IHFSCCSD

1.683

1.683

1.70823

4.010 4.540 4.560 4.040 4.970 4.880 4.840 4.960 5.310 5.340 5.150 4.860 4.860 5.170 5.420 5.370 5.520 5.800 5.710 6.070 5.780 6.030 5.690 6.090 5.940

4.470 4.820 4.850 4.520 5.330 5.310 5.230 5.380 5.500 5.550 5.500 5.220 5.320 5.520 5.580 5.550 5.710 6.170 6.100 6.350 6.150 6.270 6.080 6.320 6.190

4.084

4.125

results are generally smaller than those of EOM-CCSD except for the lowest several states using SAOP. Vertical excitation energies of UO22+ including SOC with EOM-CCSD based on SOC-CCSD are listed in Table 2 for gerade states and in Table 3 for ungerade states. Results of SOC-IHFSCC, 20,23 SO-LR-CCSD, 21 SO-CASPT2, 8 and TDDFT using SAOP and CAM-B3LYP22 are also listed for comparison. SOC will split the triplet states and mix states with the same symmetry of the corresponding double point group. Furthermore, splitting of triplet states due to SOC is larger for states with a higher angular momentum and one can see from these two tables that splitting of 3Φ states is indeed more significant than that of 3Δ and 3Π states. According to results for gerade states in Table 2, the present EOM-CCSD results are about 0.2 eV lower than those of SO-LR-CCSD for states below 5 eV, while they are in quite good agreement with each other for higher states. On the other hand, excitation energies for ungerade states with these two CC approaches are in rather good agreement with each other with a difference of less than 0.05 eV in most cases. Agreement between the present EOMCCSD approach and CASPT2 for gerade states is also quite good and differences are below 0.1 eV for transitions out from the 5σu+ orbital. Compared with results of IHFSCC in ref 20 at bond length of 1.683 Å, the present results are about 0.3−0.4 eV larger, which is similar to that for spin-free results. However, excitation energies from a recent IHFSCC calculation23 at the bond length of 1.708 Å using a larger active space are about 0.7 eV lower for the gerade states and about 0.9 eV lower for the ungerade states than the present results. As for performance of TDDFT with SOC compared with EOM-CCSD, they are similar to that of scalar-relativistic results.

set and ECP. In addition, our results are also in good agreement with those of AQCC21 and MRCI+DC21 and their difference is less than 0.1 eV except for transitions involving the 5σu+ orbital, where some of our results are about 0.1−0.2 eV lower. Excitation energies of EOM-CCSD are always larger than those of IHFSCC.20,30 IHFSCC underestimates excitation energies by about 0.3−0.4 eV for the lowest several states and the underestimation becomes more severe for higher states compared with our results. According to results in this table, excitation energies using CASPT28 for transitions out of the 5σu+ orbital agree rather well with those of EOM-CCSD with a difference of less than 0.1 eV. On the other hand, they are about 0.3−0.5 eV smaller for the other transitions. It is found in previous works21 that excitation energies with CASPT2 are too small compared with results of LR-CCSD, AQCC, and MRCI +DC, and that excitation energies for the lowest several states with CASPT2 are in good agreement with those of IHFSCC.20 In fact, these findings are based on comparison of excitation energies at different bond lengths. Excitation energies of the lowest several states with CASPT2 will agree reasonably with those of LR-CCSD, AQCC and MRCI, while they will be 0.2− 0.3 eV larger than IHFSCC results if excitation energies at the same bond length are compared. As for TDDFT results, one can see that excitation energies depend pronouncedly on the employed XC potential. Excitation energies for transitions out of the 5σu+ orbital with the SAOP potential are about 0.4−0.5 eV larger than those with CAM-B3LYP, while these two XC potentials provide similar results for transitions from the 3πu orbital. On the other hand, CAM-B3LYP results become 0.3− 0.5 eV larger for transitions from the 6σg+ orbital. TDDFT 3970

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The Journal of Physical Chemistry A Table 4. Spin-Free Vertical Excitation Energies of UN2 (All Values in Electronvolts) TDDFT30 state

transition

EOM-CCSD

13Φg 13Δg 11Φg 11Δg 23Δg 13Γg 13Πg 23Φg 21Φg 11Γg 11Πg 21Δg 13Σu+ 11Σu+ 13Δu 11Δu 13Φu 13Πu 11Φu 11Πu 23Δu 21Δu 23Φu 23Πu 21Φu 21Πu

5σu+-1φu 5σu+-1δu 5σu+-1φu 5σu+-1δu 3πu-1φu 3πu-1φu 3πu-1δu 3πu-1δu 3πu-1δu 3πu-1φu 3πu-1δu 3πu-1φu 5σu+-7σg+ 5σu+-7σg+ 5σu+-2δg 5σu+-2δg 6σg+-1φu 3πu-7σg+ 6σg+-1φu 3πu-7σg+ 6σg+-1δu 6σg+-1δu 3πu-2δg 3πu-2δg 3πu-2δg 3πu-2δg

2.015 2.054 2.602 3.135 3.591 3.621 3.716 3.743 3.813 3.832 3.959 4.089 3.090 3.367 3.466 3.696 3.777 3.817 3.919 3.996 4.098 4.134 4.378 4.381 4.428 4.555

TDDFT24

SAOP(DC)

CAM-B3LYP(DC)

SAOP

CASPT230

CASPT224

1.510 1.480 2.060 2.560

1.980 2.460 2.470 3.180

1.650 1.750 2.080 2.830

1.800 1.770 2.300 2.660

3.370

3.090

3.090

3.580

3.350

3.420

2.870 3.140

2.450 2.710

2.830 3.180

3.380

2.930

3.410

3.520

3.040

3.570

1.930 2.430 2.360 2.970 2.910 2.910 3.560 3.560 3.570 3.090 3.830 3.540 2.580 2.830 3.220 3.390 2.760 3.490 2.860 4.340 3.420 3.390 4.160 4.400 4.180 4.740

2.400 2.370 2.910 3.410 3.840 3.830 3.970 3.950 4.000 3.980 4.120 4.300 2.540 2.750 3.020 3.210 4.610 3.550 4.680 3.660 4.700 4.700 4.010 4.050 4.060 4.170

IHFSCC

30

3.040

3.510

2.620 2.850

3.230 3.290

Table 5. Vertical Excitation Energies with SOC of UN2 (All Values in Electronvolts) state

transition

principal SOF state

EOM-CCSD

SO-RASPT224

TDDFT/SAOP24

SO-IHFSCCSD23

1Δg 1Φg 1Πg 2Δg 1Γg 2Φg 3Φg 3Δg 2Πg 2Γg 3Πg 4Φg 1Σu− 1Πu 2Πu 1Σu+ 1Δu 1Φu 2Δu 3Πu 2Φu

5σu+-1φu 5σu+-1φu 5σu+-1δu 5σu+-1δu 5σu+-1φu + 5σu -1δu/5σu+-1φu 5σu+-1φu 5σu+-1δu

1 Φg 13Φg 13Δg 13Δg 13Φg 13Δg/13Φg 11Φg 11Δg 23Δg 13Γg/11Γg 13Σg+ 23Φg/13Γg 13Σu+ 13Σu+ 13Δu 11Σu+ 13Δu 13Δu/13Φu 13Πu/11Δu 13Πu 11Φu/23Φu

1.563 1.710 1.788 2.016 2.266 2.327 2.661 3.095 3.092 3.196 3.285 3.421 2.977 2.981 3.177 3.210 3.260 3.605 3.680 3.750 4.074

1.890 2.030 2.100 2.360 2.750 2.730 3.070 3.370 3.270 3.290 3.980 3.540 2.480 2.480 2.750 2.670 2.820 3.160 3.260 3.500 3.720

1.620 1.760 2.250 2.430 2.340 2.890 2.560 3.210 2.580 2.660

0.956 1.103 1.134 1.398 1.699 1.757 2.076 2.519 2.711 2.844

3πu-1φu 3πu-1φu 6σg+-7σg+ 3πu-1δu/3πu-1φu 5σu+-7σg+ 5σu+-7σg+ 5σu+-2δg 5σu+-7σg+ 5σu+-2δg + 5σu -2δg/6σg+-1φu 3πu-7σg+/5σu+-2δg 3πu-7σg+ + 6σg -1φu/3πu-2δg

3

2.470 2.480 2.850 2.700 2.920 3.280 3.390 3.520 3.770

2.669 2.709 2.749

although the 5σu+ orbital is calculated to be lower than the 3πu orbitals and the 6σg+ orbital is the HOMO of this molecule. On the other hand, transitions from the 5σu+ orbital to the virtual 7sσ orbital and 5dδ orbitals are lower in energy than transitions from the 6σg+ orbital to the virtual fδ and fϕ orbitals, which are in the same energy range as excitations from the 3πu orbitals to the 7sσ orbital.

B. Vertical Excitation Energies of UN2. Spin-free vertical excitation energies of UN2 with EOM-CCSD are listed in Table 4 together with previous results using CASPT2, 24,30 TDDFT,24,30 and IHFSCC.30 Similar to the case of UO22+, the lowest several excited states are transitions from the 5σu+ orbital to the virtual fδ and fϕ orbitals in UN2 followed by excitations from the 3πu orbitals to the virtual fδ and fϕ orbitals, 3971

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The Journal of Physical Chemistry A Table 6. Spin-Free Vertical Excitation Energies of NUO+ (All Values in Electronvolts) TDDFT30 state

transition

EOM-CCSDD

13Φ 13Δ 11Φ 11Δ 23Δ 13Γ 13Π 23Φ 11Γ 21Φ 21Δ 11Π

11σ+-1φ 11σ+-3δ 11σ+-1φ 11σ+-3δ 5π-1φ 5π-1φ 5π-3δ 5π-3δ 5π-1φ 5π-3δ 5π-1φ 5π-3δ

1.902 1.918 2.382 2.715 3.337 3.354 3.399 3.419 3.513 3.514 3.668 3.704

TDDFT24

.

SAOP(DC)

CAM-B3LYP(DC)

SAOP

CASPT230

CASPT224

1.590 1.560 2.060 2.370

1.840 2.280 2.210 2.820

1.370 1.690 1.940 2.560

1.890 1.900 2.320 2.550

3.310 3.340 3.360 3.450 3.380

3.020 3.470 3.470 3.160 3.630

2.850 3.310 3.280 3.120 3.480

3.500

3.830

3.730

1.890 2.280 2.180 2.700 2.890 2.890 3.370 3.370 3.020 3.290 3.410 3.710

2.520 2.400 2.860 3.190 3.470 3.470 3.540 3.540 3.570 3.590 3.730 3.810

IHFSCC

30

3.180 3.200 3.260

3.390

Table 7. Vertical Excitation Energies with SOC of NUO+ (All Values in Electronvolts) state

transition

principal SOF state

EOM-CCSD

SO-RASPT224

SO-IHFSCCSD23

1Δ 1Φ 1Π 2Δ 1Γ 2Φ 3Φ 3Δ 2Π 2Γ 4Φ 1Σ− 1Σ+ 5Φ 4Δ

11σ+-1φ 11σ+-1φ 11σ+-3δ 11σ+-3δ 11σ+-1φ 11σ+-3δ/1φ 11σ+-1φ 11σ-3δ 5π-1φ 5π-1φ 5π-1φ/3δ 5π-3δ 5π-3δ 5π-3δ 5π-1φ

13Φ 13Φ 13Δ 13Δ 13Φ 13Δ/13Φ 11Φ 11Δ 23Δ 13Γ 13Γ/23Φ 13Π 13Π 21Φ/23Φ 23Δ

1.448 1.577 1.651 1.863 2.173 2.208 2.496 2.750 2.854 2.924 3.079 3.090 3.115 3.167 3.172

2.030 2.130 2.190 2.430 2.870 2.840 3.030 3.080 2.920 2.970

1.018 1.147 1.215 1.440 1.778 1.811 2.101 2.361 2.656 2.743 2.932 2.947 2.988 2.987 3.016

CCSD results are about 0.2 eV larger for transitions to the 7sσ orbital than IHFSCC results. As for TDDFT results, excitation energies using the same XC potential already differ by about 0.2 eV for the lowest 1Δg and 1Γg states due to difference in the employed basis set and the relativistic Hamiltonian. Difference between results of TDDFT/SAOP and TDDFT/CAM-B3LYP depends on character of the transitions and their performance for UN2 is the same as that in UO22+. Compared with the present EOM-CCSD results, TDDFT excitation energies are usually too small except for the lowest several states. Excitation energies of UN2 with SOC using EOM-CCSD are listed in Table 5 together with previous results with RASPT2,24 TDDFT,24 and IHFSCC.23 Our results are about 0.3 eV smaller than those of RASPT2 in ref 24 for gerade states which are mainly transitions from the 5σu+ and 3πu orbitals to the virtual f orbtials. On the other hand, they are about 0.5 eV larger for ungerade states which are mainly transitions from the 5σu+ and 3πu orbital to the virtual 7sσ and 5dδ orbitals. Excitation energies with IHFSCC are always lower than our results by about 0.5−0.6 eV. Compared with RASPT2 results, excitation energies using IHFSCC results are about 1 eV smaller for transitions from the 5σu+ orbital to the virtual f orbtials. Results of TDDFT/SAOP are usually about 0.1−0.5 eV larger than the present EOM-CCSD results for transitions from the 5σu+ orbital to the virtual f orbtials, while they are about 0.5 eV smaller for the other transitions. Excitation energies with

One can see from this table that the two sets of CASPT2 results already differ significantly. Excitation energies for transitions from 5σu+ and 3πu orbitals to the virtual f orbitals with CASPT2 from ref 24 are about 0.4−0.8 eV larger than those from ref 30, while agreement between results for transitions from 5σu+ to the 7sσ orbital of these two CASPT2 calculations is much better. Large discrepancy between the two sets of CASPT2 results probably stems from different active space employed in the calculations. Many virtual orbitals lie between the HOMO and the virtual fδ and fϕ orbitals and transitions to the virtual 7sσ and 5dδ orbitals are important in this molecule. This indicates an even larger active space is required in CASPT2 calculations for UN2. Excitation energies with the present EOM-CCSD always lie between these two sets of CASPT2 results for transitions to the virtual fδ and fϕ orbitals. Our results are about 0.3 eV larger than those of CASPT2 in ref 30 and 0.3 eV smaller than those in ref 24 for transitions from 5σu+ orbital to the virtual fδ and fϕ orbitals, while the present excitation energies for transitions from the 3πu orbitals to the virtual fδ and fϕ orbitals agree reasonably with those of CASPT2 in ref 24. Another interesting issue is that our results are always smaller than those of CASPT2 in ref 24 for transitions to the virtual fδ and fϕ orbitals, while they are about 0.3−0.5 eV larger for transitions to the 7sσ and 5dδ orbitals. Compared with IHFSCC results, EOM-CCSD overestimates excitation energies by about 0.3−0.5 eV for transitions to the virtual f orbitals. On the other hand, EOM3972

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The Journal of Physical Chemistry A TDDFT, RASPT2 and IHFSCC are in quite good agreement with each other for ungerade states. C. Vertical Excitation Energies of NUO+. We focus on transitions from the higher σ- and π-orbitals,that is, the bonding orbitals between nitrogen and uranium to the virtual fδ and fϕ orbitals in this molecule. These transitions are also the lowest ones in energy for this molecule. Spin-free vertical excitation energies with EOM-CCSD are listed in Table 6 together with results from CASPT2,24,30 IHFSCC,30 and TDDFT.24 Once again we can see large difference between the two sets of vertical excitation energies with CASPT2. CASPT2 results in ref 24 are about 0.6 eV larger than those reported in ref 30 for transitions from the σ-orbital, whereas they are about 0.3−0.4 eV too large for transitions from the π-orbitals. Similar to the case of UN2, our spin-free EOM-CCSD results always lie between these two sets of CASPT2 results. Agreement between our results and those of CASPT2 in ref 30 is rather good for transitions from the σ-orbital, while our results agree better with those of CASPT2 in ref 24 for transitions from the πorbitals. Compared with IHFSCC results,30 the present results are about 0.4 eV larger for transitions from the σ-orbital and agreement between results of these two approaches are much better for transitions from the π-orbitals. Excitation energies with TDDFT/SAOP are about 0.4 eV larger than those of TDDFT/CAM-B3LYP for the transitions from the σ-orbital and these two TDDFT results agree well with each other for the higher excited states. Our EOM-CCSD results are in good agreement with TDDFT/SAOP results except for the lowest 3 Δ state and the Γ states. Excitation energies including SOC for this molecule has only been investigated using IHFSCC23 and RASPT224 previously and these results together with the present ones are listed in Table 7. One can see from this table that the present excitation energies are about 0.6 eV smaller than those of RASPT2, whereas they are about 0.4 eV larger than those of IHFSCC for the transitions out from the σ-orbital. On the other hand, excitation energies for transitions from the π-orbitals using these three approaches are in reasonable agreement with each other and their difference is within 0.2 eV.

previous CASPT2 results already differ significantly. This is probably because other virtual orbitals such as 7sσ and 5dδ orbitals play a more important role in excitations of these two molecules and an even larger active space is thus required. The present EOM-CCSD results always lie between these two sets of CASPT2 results for transitions to the virtual f orbitals. On the other hand, excitation energies for these molecules with EOM-CCSD are usually much larger than those of IHFSCC. IHFSCC together with Dirac-Coulomb or X2C Hamiltonian has been adopted extensively in calculating excitation energies of heavy element systems such as the actinide-containing molecules. Previous results on light elements show that excitation energies with EOM-CCSD usually agree reasonably well with those of IHFSCC particularly when a large active space is employed.63 In fact, reasonable agreement between excitation energies with EOM-CCSD and IHFSCC was also reported by us for I3− with SOC is included.57 IHFSCC results depend on the active space and one may suspect that sizable difference between results of EOM-CCSD and IHFSCC may be because the active space in IHFSCC calculations is not large enough. However, excitation energies with IHFSCC are usually smaller when a larger active space is employed, as shown in ref 23 and 20 for UO22+. IHFSCC results are already smaller than the present EOM-CCSD results and an even larger active space could not possibly improve agreement between these two approaches. Calculations using a higher level approach such as EOM-CCSDT are necessary to shed light on reliability of the obtained excitation energies for these molecules. Because of lack of experimental data as well as theoretical results of higher level methods, it is difficult to estimate reliability of the obtained results. On the other hand, experimental results exist for some closely related species such as [UO2F42−],72 [UO2F2].73 It would be interesting to evaluate performance of the present approach on excitation energies of these molecules using the present approach.

IV. CONCLUSION Achieving reliable excitation energies for the isoelectronic series: UO22+, NUO+, and UN2 is important in practical application. It is also a challenge in quantum chemistry calculations because many orbitals are involved in bonding and excitations and SOC effects must be considered. Vertical excitation energies of these molecules are calculated using EOM-CCSD with a recently developed small-core relativistic pseudopotential and a large basis set. EOM-CCSD is known to give rise to reasonable excitation energies for single excitations from a reference with a dominant single reference character. SOC is included in post-SCF calculations and this EOM-CCSD approach with SOC has been shown to provide accurate description of SOC effects for heavy elements. Excitation energies of UO22+ are calculated at two different bond lengths and our results show that difference between excitation energies at these two bond lengths could be as large as 0.4 eV. Previous observation on important difference between excitation energies with CASPT2 and those with LR-CCSD, MRCI and AQCC for low lying states mainly comes from this different bond length. Our results for UO22+ are in reasonably agreement with those of CASPT2, LR-CCSD, MRCI, and AQCC. As for NUO+ and UN2, two sets of

ORCID

■

AUTHOR INFORMATION

Corresponding Author

*Phone: 86-15828332921. E-mail: [email protected].

Shuo Zhang: 0000-0001-5182-0277 Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS We thank the National Nature Science Foundation of China (Grants 21473116 and 21273155) for financial support.

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REFERENCES

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