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Investigation of the Electronic Structure and Optical Spectra of Uranium (IV), (V), and (VI) Complexes Using Multiconfigurational Methods
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A: Structure, Spectroscopy, and Reactivity of Molecules and Clusters

Investigation of the Electronic Structure and Optical Spectra of Uranium (IV), (V), and (VI) Complexes Using Multiconfigurational Methods
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Cite this: J. Phys. Chem. A 2022, 126, 36, 6059–6066
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https://doi.org/10.1021/acs.jpca.2c03314
Published September 6, 2022

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Abstract

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Interpreting electronic spectra of uranium-containing compounds is an important component of fundamental chemistry as well as in the assessment of waste streams in the nuclear fuel cycle. Here we employ multiconfigurational calculations with CASSCF or DMRGSCF methods on exemplar uranium molecules [UVIO2Cl4]2–, [UV(TRENTIPS)(N)]−, and [UIVCl5(THF)]−, featuring an array of geometries and oxidation states, to determine their effectiveness in predicting electronic spectra, compared to literature calculations and experimental data. For [UVIO2Cl4]2–, DMRGSCF alone shows poor agreement with experiment, which can be improved by adding corrections for dynamic correlation with MC-PDFT to give results of similar quality to TD-DFT. However, for [UV(TRENTIPS)(N)]− the addition of dynamical correlation via MC-PDFT or CASPT2 made no improvements over CASSCF, suggesting that perhaps other factors such as solvation effects could be more important in this case. Finally, for [UIVCl5(THF)]−, dynamical correlation included via MS-CASPT2 on top of CASSCF calculations is crucial to obtaining a quantitatively correct spectrum. Here, MC-PDFT fails to even qualitatively describe the spectrum, highlighting the shortcomings of single-state methods in cases of near-degeneracy.

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1. Introduction

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The decommissioning of legacy nuclear reactor sites in the U.K., such as Sellafield, is an expensive and time-consuming task. (1) Due to poor record keeping, the composition of the waste stored at such facilities is mostly unknown. Hence, the determination of the chemical species (speciation) in legacy wastes is of crucial importance, and an effective proposed technique is luminescence spectroscopy. (2) Luminescence spectroscopy is ideal in this case, as the nature of the emission spectra of actinide compounds carries an imprint of both electronic and vibrational states of the molecules within the waste, providing information on elements, oxidation states, and chemical structure. Through comparison to a library of luminescence spectra, the experimental spectrum of a mixture could thus be deconvoluted. While a good idea in theory, the preparation of a range of model compounds in different conditions is complicated due to radiolysis and changes in oxidation state due to disproportionation (3) or other redox processes. Thus, one route would be to generate an accurate library of luminescence spectra using computational methods.
To accurately calculate a theoretical luminescence spectrum, one must accurately determine both the electronic structure and the vibrational spectrum of the complex (for both ground and excited states) as well as account for an accurate representation of the molecular environment (e.g., solution, surface immobilization, or solid). When it comes to actinide complexes, the confluence of a large number of electrons, strong relativistic effects, orbital degeneracy, and accessible redox states makes even the first step (an accurate calculation of electronic structure) difficult. There are several approaches to the calculation of electronic structure and excitation energies, but by far the most utilized method is time-dependent density functional theory (TD-DFT), because it circumvents the calculation of excited state wave functions and instead uses the change in electronic density in response to an external potential to derive excited-state energies, (4) so it is therefore relatively fast. Indeed, it can also be reasonably accurate depending on the functional chosen, such as the long-range corrected functional CAM-B3LYP. (5) However, a poor choice of functional can also lead to poor results, so calculations are usually tailored to specific molecules and therefore one cannot use such methods to generate a reliable spectral reference library. DFT is also inherently a single-configuration approach and therefore cannot account for orbital degeneracy effects which are common in both ground and excited states of actinide complexes.
An alternative strategy is to use explicit wave function-based methods, which also allow a more rigorous inclusion of spin–orbit coupling which plays a crucial role in the electronic spectra of actinide complexes. As we wish to calculate both ground- and excited-state properties as well as account for orbital degeneracy, this necessitates a multireference method. A very common choice is the complete active space self-consistent field (CASSCF) method, (6) which treats a subset of the molecular orbital space with full configuration interaction (FCI) and the remaining orbitals with Hartree–Fock theory. (7) Such methods have found extensive use for probing actinide chemistry; examples include Vallet et al. investigating the mechanism of water exchange of actinyl aquo ions, (8) Gagliardi et al. and Heit et al. calculating electronic structure and spectra, (9−11) Mounce et al. interpreting nuclear magnetic resonance data, (12,13) and Autschbach et al. predicting magnetic properties. (14) Theoretical methods are also able to probe the limits of our understanding of bonding at the bottom of the periodic table, exemplified by the definition of a quintuple bond in U2 by Roos et al. (15) and recently countered to in fact be a quadruple bond when relativistic effects are considered more rigorously. (16)
Using CASSCF allows the inclusion of the “important” orbitals in optical excitation, without the computational burden of FCI in the entire orbital space. The advantage of this method over TD-DFT is the inclusion of static correlation as well as relativistic effects. However, CASSCF suffers from a lack of dynamical correlation (DFT attempts to include some dynamic correlation effects with the exchange-correlation functional) and thus often needs to be corrected using perturbative methods such as complete active space second-order perturbation theory (CASPT2) (17) or the more recent multiconfiguration pair-density functional theory (MC-PDFT). (18) The cost of the calculations also increases very quickly when the active space is expanded, though this can be alleviated by using the restricted active space self-consistent field (RASSCF) methods to limit the number of excitations or, alternatively, an approximate CI solver such as a density matrix renormalization group (DMRG) (19−21) can be used for much larger active spaces. Unlike CASSCF, DMRG uses matrix product states allowing the number of configurations to be reduced by reducing the size of the matrices via a single-value decomposition. However, all of these multiconfigurational methods are far from the “black-box” approach of DFT methods, and results can vary significantly given details of the active space and dynamic correlation corrections.
Thus, we set out to systematically compare TD-DFT, CASSCF, DMRG, CASPT2, and MC-PDFT methods for calculating the excitation spectra of some uranium complexes. We find that TD-DFT and DMRG-MC-PDFT methods seem to be appropriate for uranyl U(VI) compounds, while we find that minimal CASSCF with or without CASPT2 corrections seems most appropriate for U(V) and U(VI) compounds. However, we caution that the representation of the environment of the uranium compound, in this case, the solvent, must be crucial to obtaining experimental accuracy and hence building libraries of spectra.

2. Methods

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DFT optimizations were performed in the gas phase using Gaussian 09 (22) with the PBE0 functional (23) and the D3 version of Grimme’s dispersion. (24) The Stuttgart RSC 1997 (25−28) basis set and ECP were used for uranium, and the cc-pVTZ basis set (29) was used for all other atoms. Vibrational frequency calculations were performed with the HPModes option. CASSCF calculations were performed using OpenMolcas 19.11 (30,31) with the geometries obtained from DFT optimization. ANO-RCC-VTZP, VDZP, and VDZ basis sets (32,33) were used for the uranium, first coordination sphere, and other atoms, respectively. Cholesky decomposition of the two-electron integrals at a threshold of 10–8 was used for all calculations. The spin–orbit coupling and dipole transition strengths were calculated using the RASSI module. DMRGSCF calculations were performed using the QCMaquis DMRG program suite (19−21) with a maximum bond dimension of 512. Dynamical correlation was added using either singlet-state CASPT2 (SS-CASPT2), (17) multistate CASPT2 (MS-CASPT2), (34) extended multistate CASPT2 (XMS-CASPT2), (35) or MC-PDFT.

3. Results and Discussion

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3.1. [UO2Cl4]2– – U(VI)

Uranium is commonly found in the +6 oxidation state as uranyl compounds in environmental settings and as such is an important class of molecules to consider. Uranium(VI) compounds are 5f0; therefore, optical transitions arise from ligand–metal and/or metal–ligand charge-transfer excitations. The [UO2Cl4]2– anion (Figure 1) has been investigated both experimentally and computationally, so it is a good benchmark system for our purposes. In this compound, excitation is supposed to occur from the ground singlet state (S0) to the first excited triplet state (T1) from which emission occurs back to S0 after internal conversion. Pierloot, Van Besien, and colleagues found good agreement between their SOC-CASPT2 excitation energies and experimental data, thus they were able to show that the excitation occurs from a σu orbital (S0) to a nonbonding 5fδ orbital (T1) and that two-component TD-DFT approaches were also quite accurate. (36,37) Tecmer et al. (38) and most recently Oher et al. (39) have also shown that TD-DFT with the CAM-B3LYP functional is quite accurate compared to SOC-CASPT2, SOC-CI, and experimental data, agreeing on the nature of the excitation. Here we sought to use larger active spaces to better treat electron correlation in the optical excitation and emission processes.

Figure 1

Figure 1. Molecular structure of [UO2Cl4]2–. Red = oxygen, green = chlorine, and blue = uranium.

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We have optimized geometries for the singlet ground state and first excited triplet state using the same methodology and functional as Oher et al., using their optimized structures as initial guesses. CASSCF calculations were then performed for the ground-state (S0) and excited-state (T1) geometries. An active space of 2 electrons in 9 orbitals, herein CAS(2,9), was chosen initially, which included the 7 5f orbitals as well as the σ bonding and antibonding “yl” orbitals, where 45 singlet and 36 triplet roots were considered, followed by mixing with SOC. These initial results showed that the lowest-lying spin–orbit excitation is S0 yl-σ to T1 5fδ (Figure 2) at 16 800 cm–1. However, after these calculations we found that the active space had significantly changed from our original selection. For the singlets, many unoccupied 5f orbitals had rotated out (including 5fδ), in favor of unoccupied ligand orbitals, while some 5f orbitals remained in the triplet states (specifically, the occupied 5fδ). Because the active spaces differed considerably between spin multiplicities, the amount and nature of electron correlation is different, which is not ideal for the evaluation of transition energies. Hence, in an attempt to maintain a consistent active space, we attempted to expand the active space using RASSCF methods. We added the closest-energy secondary orbitals (uranium 6d orbitals) to RAS3, allowing a maximum of 2 electrons in this space, while the oxygen 2p orbitals were added to the RAS2 space for a total of 6 RAS2 orbitals with 12 electrons (RAS(12,0,2:0,6,11)). However, this was still not able to stabilize a consistent active space between singlet and triplet multiplicities, and larger active spaces quickly hit the computational limits of RASSCF.

Figure 2

Figure 2. (Left) Highest occupied orbital in the ground state S0 (σu) and (right) the newly singly occupied molecular orbital in the first excited state T1 (5fδ) for [UO2Cl4]2–.

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To expand further, we attempted DMRG(16,17)SCF calculations and started with the largest feasible active space we obtained from RASSCF, consisting of the oxygen 2s and 2p orbitals as well as the uranium 5f (0, ±1, ±2) and 6d (±1, ±2) orbitals for both multiplicities at both geometries considering two roots per multiplicity to account for low-lying excitations. However, similar to the RASSCF calculations, the active space still differed between the singlet and triplet multiplicities. It was not until DMRG(16,40)SCF calculations that we could stabilize the 5fδ orbitals in the active space in both spin multiplicities: these calculations were incredibly time-consuming and so were not pursued further. Comparing the vertical S0–T1 absorption energy calculated with DMRG(16,17)SCF to that found in the literature (39) (Table 1), we find our result to be ca. 5000 cm–1 above the experimental and TD-DFT values. However, the nature of our transition is very similar.
Table 1. Vertical Absorption Energies for [UO2Cl4]2– Calculated with DMRG(16,17)SCF and Compared to Experimental and TD-DFT Data (39)
 transitionexperimental (39)TD-DFT (39)SOC-CASPT2 (37)DMRG(16,17)SCF
absorption/cm–1σu (S0) → 5fδ (T1)20 09620 73720 28025 207
Given these relatively poor results, we reasoned that our strategy of enlarging the active space was detrimental to the accuracy of the calculated excitation energy and, given the good accuracy of TD-DFT, that perhaps it could be more important to consider dynamical correlation. The latter is normally added on top of CASSCF calculations using CASPT2; however, this is not an option for DMRGSCF in OpenMolcas, so we opted to try MC-PDFT, (40) which should be much faster than CASPT2. However, like other DFT methods, it is also dependent on the choice of the on-top functional, so some benchmarking is required. Therefore, MC-PDFT calculations were performed (Table 2) for each density functional available, four translated functionals (tLSDA, tPBE, tBLYP, and trevPBE) and their “fully-translated” variants (41) (ftLSDA, ftPBE, ftBLYP, and ftrevPBE) on top of the DMRG(16,17)SCF calculations (Table 2). These calculations are reported spin-free as the addition of spin–orbit coupling does not change the energy of the vertical excitation.
Table 2. Vertical Absorption and Emission Energies for [UO2Cl4]2– Calculated with DMRG(16,17)SCF-MC-PDFT, Compared to Experimental and TD-DFT Data (39)
 transitionexperimental (39)TD-DFT (39)SOC-CASPT2 (37)DMRG(16,17)SCFtPBEtLSDA
absorption/cm–1σu (S0) → 5fδ (T1)20 09620 73720 28025 20721 96422 492
emission/cm–15fδ (T1) → σu (S0)-19 924-23 68621 24821 854
  tBLYPtrevPBEftBLYPftrevPBEftPBEftLSDA
absorption/cm–1σu (S0) → 5fδ (T1)21 84821 87423 18323 04623 18723 839
emission/cm–15fδ (T1) → σu (S0)21 07521 15822 20222 12622 26823 073
The addition of MC-PDFT greatly improves the agreement with the experimental data, with the translated functionals performing slightly better than their fully translated variants. This suggests that dynamical correlation is more important when calculating transition energies than obtaining a commensurate active space between spin multiplicities. While the use of MC-PDFT requires the choice of the functional such as for other DFT methods, we found that all translated functionals give very similar results so that the choice is not extremely important. It is clear, however, that our results are no more accurate than TD-DFT, and hence the benefits of multiconfigurational methods are not necessarily required or realized in U(VI) compounds.

3.2. [U(TRENTIPS)(N)]− – U(V)

Uranium(V) has a single 5f electron in the ground state, which may require multiconfigurational methods in cases of high symmetry where some 5f orbitals are degenerate, and SOC must be considered to be a non-negligible perturbation to the electronic structure. Here we have chosen to look at [U(TRENTIPS)(N)]− (Figure 3), studied previously by King et al., (42) for which experimental and CASSCF excitation energies are available. In this case, we are interested in the low-energy f–f absorptions between the degenerate spin–orbit doublet ground state to the excited spin–orbit doublet states. The ground doublet is either mJ = ±3/2 or ±5/2, as derived from the doubly degenerate orbital pairs of ml = ±2 or ±3, respectively, which are rather close in energy, necessitating a multiconfigurational approach with SOC. Therefore, TD-DFT is not an appropriate method for obtaining excitation energies here. The choice of the starting active space is simple, considering the seven 5f orbitals (CAS(1,7)SCF) in the original work.

Figure 3

Figure 3. 2D (left, reproduced from ref (42) under a CC BY 4.0 license) and 3D (right) molecular structure of [U(TrenTIPS)(N)]−. Purple = nitrogen, yellow = silicon, blue = uranium, and gray = carbon.

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As the experimental data are solution-phase, we have optimized the structure using the crystal structure as a starting point and then performed a CAS(1,7)SCF calculation for seven doublets (Table 3); we note that DFT optimization is often an acceptable approach for structural optimization, which is not very sensitive to orbital degeneracy and SOC effects. (43) Unsurprisingly, the results are very similar to the original CAS(1,7)SCF values, where the differences (ca. 100 cm–1) are due to changes in the optimized geometry (cf. the crystal structure). As we found MC-PDFT corrections to be valuable for the U(VI) example above, we added them on top of the CAS(1,7)SCF results using the tPBE functional. However, in this case the agreement with the experimental data worsened, considerably overestimating the energies of the doublets (Table 3). As the active space is very small, CASPT2 calculations are affordable, and thus we used single-state (SS), multistate (MS), and extended multistate (XMS) variants to provide further estimates of the dynamical correlation (Table 3). In every case, dynamical correlation worsened the agreement with the experimental data over the CASSCF results (Figure 4).
Table 3. Spin-Orbit Doublet Energies of [U(TrenTIPS)(N)]− from CAS(1,7)SCF-CASPT2 Calculations for Singlet-State, Multistate, and Extended Multistate Methods Compared to the Experimental Data (42)
CAS(1,7)SCFMC-PDFT (tPBE)SS-CASPT2MS-CASPT2XMS-CASPT2experimental
7598587429661025 
472460835355561659234700
672568076692675167796000
743984018155778879706900
950211 44310 81310 34210 7928900
16 69022 48820 60119 70620 56818 000

Figure 4

Figure 4. Absorption spectra for [U(TrenTIPS)(N)]− calculated with CAS(1,7)SCF (top), CAS(1,7)SCF-MS-CASPT2 (middle), and CAS(1,7)SCF-MC-PDFT (bottom) compared to the experiment. (42) Theoretical line widths are scaled to match the experimental peak widths.

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In this case, the addition of dynamical correlation did not improve the agreement with experimental data. To try to improve further, we expanded the active space by adding the nitride 2p orbitals (RAS(7,1,1;3,7,3), seven roots) restricted to only single excitations. The agreement with the experimental data worsened (Table 4), which could indicate a poor choice of active space. Further inclusion of the 2px/y orbitals from the equatorial nitrogen donor atoms and the 2pz orbital from the axial nitrogen donor in a RAS(21,1,1;10,7,10) calculation leads to even worse agreement, except for the highest energy state. This is likely because the highest energy state is the mJ = ±1/2 doublet arising from the ml = 0 function which is formally the σ antibonding U-nitride orbital. Further expansions exceed computational limitations for RASSCF, and using seven roots is beyond the scope of the DMRG implementation in OpenMolcas, which is advisible only for the lowest few roots.
Table 4. Results Obtained in cm–1 from the RAS(7,1,1;3,7,3) and RAS(21,1,1;10,7,10) Calculations on [U(TrenTIPS)(N)]− Compared to the CAS(1,7) Calculation and Experimental Data (42)
experimentalCAS (1,7)RAS (7,1,1;3,7,3)RAS (21,1,1;10,7,10)
 000
 759800863
4700472449875082
6000672566596966
6900743974517704
8900950296689916
18 00016 69017 09617 319
Thus, we can conclude that the discrepancies between experimental and calculated excitation energies arise here from either extensive correlation effects that cannot be captured by the perturbative methods attempted or, more likely, from solvent effects such as screening, polarization, and geometric changes. Two methodologies to overcome these solvent effects would be to either perform molecular dynamics calculations or include explicit solvent molecules in a structural optimization.

3.3. [UCl5(THF)]− – U(IV)

Further increasing in complexity, uranium(IV) has a 5f2 ground configuration, meaning that there are two electrons in the 5f orbitals, leading to a ground-state triplet (3H4). The extra electron considerably complicates the electronic spectrum but also moves the electronic structure into a regime more akin to the trivalent lanthanide ions. Again, we consider only the f–f transitions; however, this time the excited states have both singlet and triplet multiplicities. Previous work by Hashem et al. (44) studied the emission and absorption of uranium(IV) compounds experimentally and computationally using CASSCF and CASPT2. In their work, they investigated the f–d and LMCT transitions and concluded that the emission spectra of simple U(IV) compounds could be used as a diagnostic tool to deconvolute experimental luminescence spectra in the presence of [UO2]2+. For one complex in particular, [UCl5(THF)]− (Figure 5), they provide a fully assigned experimental absorption spectrum for the f–f transitions, thus providing a useful benchmark.

Figure 5

Figure 5. Molecular structure of [UCl5(THF)]−. Red = oxygen, green = chlorine, and blue = uranium.

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Starting with a minimal active space CAS(2,7)SCF and considering 28 singlet and 21 triplet roots, we have followed up with MS-CASPT2 and MC-PDFT(tPBE) calculations. The spectra were calculated on the basis of CASSCF/CASPT2 isotropic Einstein coefficients after spin–orbit coupling and were assigned purely ab initio by transforming the spin–orbit states into a basis of well-defined spin, orbital, and total angular momentum by successive block diagonalization of the appropriate operators (45) (Figure 6). Both the CAS(2,7)SCF and CAS(2,7)SCF-MS-CASPT2 calculations generally show good agreement with the experimental data (Figure 6), but the CAS(2,7)SCF-MC-PDFT results differ significantly (Figure S1). This is likely because MC-PDFT is a single-state method, which does not model how the dynamic correlation perturbation induces the interaction between nearly degenerate states in the f2 configuration. (46) XMS-CASPT2 and extended dynamically weighted CASPT2 (47) (XDW-CASPT2) calculations were also performed (Figures S2 and S3), although they did not agree with the experimental data as well as MS-CASPT2 calculations.

Figure 6

Figure 6. Absorption spectra of [UCl5(THF)]− calculated by (left) CAS(2,7)SCF and (right) CAS(2,7)SCF-MS-CASPT2, compared to the experimental data (black). All spectra are normalized to the intensity of the 3F3/3F4 transition, and the calculated spectra are plotted with a FWHM line width of 9 nm.

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The 3H5 peak in the experimental spectrum is very broad (1300–1500 nm) due to crystal field splitting, and the appearance of multiple peaks in both CASSCF (∼1200–1600 nm) and MS-CASPT2 (∼1200–1450 nm) calculations is simply due to an arbitrary line width being chosen to produce the theoretical spectrum, though the calculated crystal field splitting is undoubtedly imperfect. In the CASSCF spectrum, some of these peaks have a moderate contribution from the 3F2 state which is incorrect in comparison to the experimental data, which shows the 3F2 states between 1800 and 2000 nm; MS-CASPT2 calculations appear to correct this. The main 3F3/3F4 peak in the experiment occurs at 1100 nm, whereas in CASSCF this peak appears at ∼900 and 1000 nm for MS-CASPT2. While the theoretical spectra have two peaks corresponding to these states, the experimental spectrum only has one, although there is a small unlabeled peak at ca. 900 nm which could have some 3F3/3F4 contribution. The states at lower wavelengths are generally blue-shifted with respect to experiment: the 1I6 state is shifted 100 nm lower for CASSCF and 30 nm lower for MS-CASPT2 than its experimental position. These data clearly show that corrections for dynamical correlation are important when calculating the electronic transitions in U(IV) complexes.
While the peak positions are generally in good agreement for the MS-CASPT2 calculations, the relative intensities are not as accurate. For some features such as the 1D2 and 3H6 transitions, the relative intensities are in good agreement, but for others such as 1I6/3P1 and one of the 3H5 peaks, the intensities are much greater than the experimental data.

4. Conclusions

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Herein we have calculated the absorption spectra for uranium compounds in the +4, +5, and +6 oxidation states using different levels of theory. We have found that DMRGSCF is a good alternative to CASSCF for larger active spaces, but when used alone it is not able to accurately calculate transition energies for U(VI), implying that dynamical correlation is crucially important. It seems that DMRGSCF-MC-PDFT approaches are just as accurate as TD-DFT in this case; however, the single-state MC-PDFT method is not appropriate in cases of orbital degeneracy, such as the U(VI) example herein, where CASPT2 approaches are more robust. This conclusion could be altered by adopting state-interaction pair-density functional theory methods (46) in future work. In the case of U(V) [U(TrenTIPS)(N)]−, neither enlarging the active space nor adding dynamical correlation (by any means) was able to improve agreement with the experimental data over minimal CAS(1,7)SCF, suggesting that solvent effects must be crucial in this case. It is likely that this is generally true of U(V) compounds, where crystal field effects are on the same level of importance as spin–orbit coupling and electron correlation. While accurate environmental effects appear to be less crucial for the structure of U(IV) absorption spectra, we caution that the details of the fine structure contain compound-specific information, and hence we suggest that a crucial aspect of the calculation of uranium electronic spectra for U(V) and U(IV) must be the inclusion of an environmental model. We acknowledge that this is, computationally, a very costly predicament. Overall, it is clear that calculating electronic transitions accurately for actinide complexes using multiconfigurational methods is not a trivial task, and work must continue to find a generally reliable and consistent approach across different oxidation states and compounds. Furthermore, this work highlights the difficulty in accurately reproducing electronic spectroscopy in solution using the current state-of-the-art methods.

Supporting Information

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The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.jpca.2c03314.

  • Absorption spectra of [UCl5(THF)]− (calculated and compared to experimental data) (PDF)

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Author Information

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Acknowledgments

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We thank The University of Manchester and the Royal Society (URF191320 to NFC) for supporting this work and the Computational Shared Facility at The University of Manchester for computational resources. We also thank Dr. Robert Baker (Trinity College Dublin) for supplying experimental data for the spectrum in Figure 6.

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    Physical Review Letters (1996), 76 (8), 1212-15CODEN: PRLTAO; ISSN:0031-9007. (American Physical Society)
    A new d.-functional approach to calc. the excitation spectrum of many-electron systems is proposed. It is shown that the full linear d. response of the interacting system, which has poles at the exact excitation energies, can rigorously be expressed in terms of the response function of the noninteracting (Kohn-Sham) system and a frequency-dependent exchange-correlation kernel. Using this expression, the poles of the full response function are obtained by systematic improvement upon the poles of the Kohn-Sham response function. Numerical results are presented for Be, Mg, Ca, Zn, Sr, and Cd atoms.
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  5. 5
    Yanai, T.; Tew, D. P.; Handy, N. C. A New Hybrid Exchange–Correlation Functional Using the Coulomb-Attenuating Method (CAM-B3LYP). Chem. Phys. Lett. 2004, 393 (1–3), 51– 57,  DOI: 10.1016/j.cplett.2004.06.011
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    A new hybrid exchange-correlation functional using the Coulomb-attenuating method (CAM-B3LYP)
    Yanai, Takeshi; Tew, David P.; Handy, Nicholas C.
    Chemical Physics Letters (2004), 393 (1-3), 51-57CODEN: CHPLBC; ISSN:0009-2614. (Elsevier Science B.V.)
    A new hybrid exchange-correlation functional named CAM-B3LYP is proposed. It combines the hybrid qualities of B3LYP and the long-range correction presented by Tawada et al. [J. Chem. Phys., in press]. We demonstrate that CAM-B3LYP yields atomization energies of similar quality to those from B3LYP, while also performing well for charge transfer excitations in a dipeptide model, which B3LYP underestimates enormously. The CAM-B3LYP functional comprises of 0.19 Hartree-Fock (HF) plus 0.81 Becke 1988 (B88) exchange interaction at short-range, and 0.65 HF plus 0.35 B88 at long-range. The intermediate region is smoothly described through the std. error function with parameter 0.33.
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  6. 6
    Roos, B. O.; Taylor, P. R.; Sigbahn, P. E. M. A Complete Active Space SCF Method (CASSCF) Using a Density Matrix Formulated Super-CI Approach. Chem. Phys. 1980, 48 (2), 157– 173,  DOI: 10.1016/0301-0104(80)80045-0
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    6
    A complete active space SCF method (CASSCF) using a density matrix formulated super-CI approach
    Roos, Bjoern O.; Taylor, Peter R.; Siegbahn, E. M.
    Chemical Physics (1980), 48 (2), 157-73CODEN: CMPHC2; ISSN:0301-0104.
    A d. matrix formulation of the super-CI MCSCF method is presented. The MC expansion is assumed to be complete in an active subset of the orbital space, and the corresponding CI secular problem is solved by a direct scheme using the unitary group approach. With a d. matrix formulation the orbital optimization step becomes independent of the size of the CI expansion. It is possible to formulate the super-CI in terms of d. matrices defined only in the small active subspace; the doubly occupied orbitals (the inactive subspace) do not enter. Further, in the unitary group formalism it is straightforward and simple to obtain the necessary d. matrices from the symbolic formula list. It then becomes possible to treat very long MC expansions, the largest so far comprising 726 configurations. The method is demonstrated in a calcn. of the potential curves for the 3 lowest states (1.sum.g+, 3.sum.u+ and 3πg) of the N2 mol., using a medium-sized gaussian basis set. 7 Active orbitals were used yielding the following results: Dc:8.76(9.90), 2.43(3.68) and 3.39 (4.90) eV; rc:1.108 (1.098), 1.309(1.287) and 1.230 (1.213) Å; ωe: 2333 (2359), 1385 (1461) and 1680 (1733) cm-1, for the 3 states (exptl. values within parentheses). The results of these calcns. indicate that it is important to consider not only the dissocn. limit but also the united atom limit in partitioning the occupied orbital space into an active and an inactive part.
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  7. 7
    Hartree, D. R. The Wave Mechanics of an Atom with a Non-Coulomb Central Field. Part I. Theory and Methods. Math. Proc. Cambridge Philos. Soc. 1928, 24 (1), 89– 110,  DOI: 10.1017/S0305004100011919
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  8. 8
    Vallet, V.; Privalov, T.; Wahlgren, U.; Grenthe, I. The Mechanism of Water Exchange in AmO2(H2O) 52+ and in the Isoelectronic UO2(H 2O)5+ and NpO2(H2O) 52+ Complexes as Studied by Quantum Chemical Methods. J. Am. Chem. Soc. 2004, 126 (25), 7766– 7767,  DOI: 10.1021/ja0483544
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    8
    The Mechanism of Water Exchange in AmO2(H2O)52+ and in the Isoelectronic UO2(H2O)5+ and NpO2(H2O)52+ Complexes as Studied by Quantum Chemical Methods
    Vallet, Valerie; Privalov, Timofei; Wahlgren, Ulf; Grenthe, Ingmar
    Journal of the American Chemical Society (2004), 126 (25), 7766-7767CODEN: JACSAT; ISSN:0002-7863. (American Chemical Society)
    The water exchange mechanism for AmO2(H2O)52+ and the isoelectronic UO2(H2O)5+ and NpO2(H2O)52+ ions has been investigated using quantum chem. methods and compared to previous findings in the uranyl(VI) system (Vallet, V. et al. J. Am. Chem. Soc. 2001, 123, 11999). There are substantial and predictable changes in the geometry and preferred exchange pathways between uranyl(V) and the different actinyl(VI) complexes. The smaller charge on the uranyl(V) center makes the dissociative pathway more favorable than the associative/interchange pathways in the actinyl(VI) systems.
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  9. 9
    Gagliardi, L.; Roos, B. O.; Malmqvist, P. Å.; Dyke, J. M. On the Electronic Structure of the UO2Molecule. J. Phys. Chem. A 2001, 105 (46), 10602– 10606,  DOI: 10.1021/jp012888z
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    9
    On the Electronic Structure of the UO2 Molecule
    Gagliardi, Laura; Roos, Bjoern O.; Malmqvist, Perke; Dyke, John M.
    Journal of Physical Chemistry A (2001), 105 (46), 10602-10606CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)
    The structure and vibrational frequencies of the UO2 mol. have been detd. using multiconfigurational wave functions (CASSCF/CASPT2), together with a newly developed method to treat spin-orbit coupling. The mol. has been found to have a (5fφ)(7s), 3Φu, Ω = 2 ground state with a U-O bond distance of 1.77 Å. The computed antisym. stretching σu frequency is 923 cm-1 with a 16/18 isotope ratio of 1.0525 which compares with the exptl. values of 915 cm-1 and 1.0526, resp. Calcns. of the first adiabatic ionization energy gave the value 6.17 eV, which is 0.7 eV larger than the currently accepted exptl. result. Reasons for this difference are suggested.
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  10. 10
    Gagliardi, L.; Heaven, M. C.; Krog, J. W.; Roos, B. O. The Electronic Spectrum of the UO2 Molecule. J. Am. Chem. Soc. 2005, 127 (1), 86– 91,  DOI: 10.1021/ja044940l
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    10
    The Electronic Spectrum of the UO2 Molecule
    Gagliardi, Laura; Heaven, Michael C.; Krogh, Jesper Wisborg; Roos, Bjoern O.
    Journal of the American Chemical Society (2005), 127 (1), 86-91CODEN: JACSAT; ISSN:0002-7863. (American Chemical Society)
    The electronic spectrum of the UO2 mol. was detd. using multiconfigurational wave functions together with the inclusion spin-orbit coupling. The mol. has a (5fφ)(7s), 3Φ2u, ground state. The lowest state of gerade symmetry, 3H4g, corresponding to the electronic configuration (5f)2 was found 3330 cm-1 above the ground state. The computed energy levels and oscillator strengths were used for the assignment of the exptl. spectrum in the energy range 17,000-19,000 and 27,000-32,000 cm-1.
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  11. 11
    Heit, Y. N.; Gendron, F.; Autschbach, J. Calculation of Dipole-Forbidden 5f Absorption Spectra of Uranium(V) Hexa-Halide Complexes. J. Phys. Chem. Lett. 2018, 9 (4), 887– 894,  DOI: 10.1021/acs.jpclett.7b03441
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    11
    Calculation of Dipole-Forbidden 5f Absorption Spectra of Uranium(V) Hexa-Halide Complexes
    Heit, Yonaton N.; Gendron, Frederic; Autschbach, Jochen
    Journal of Physical Chemistry Letters (2018), 9 (4), 887-894CODEN: JPCLCD; ISSN:1948-7185. (American Chemical Society)
    Restricted-active-space wave function calcns. including spin-orbit coupling, in combination with Kohn-Sham d. functional calcns. of vibrational modes, were used to det. the vibronic and electronic absorption intensities of the near-IR elec. dipole-forbidden 5f-5f transitions of representative U(V) hexa-halide complex ions. The agreement with exptl. assigned vibronic and electronic transitions measured for powder or soln. samples of salts of the complex ions is reasonable overall and excellent for the exptl. best-resolved E5/2u → E'5/2u bands. The intensity of the vibronic transitions may be borrowed from ligand-to-metal charge-transfer excitations as well as 5f-to-6d metal-centered transitions. Magnetically allowed electronic transitions contribute to the 2 lower-frequency bands of the ligand-field spectrum.
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  12. 12
    Mounce, A. M.; Yasuoka, H.; Koutroulakis, G.; Lee, J. A.; Cho, H.; Gendron, F.; Zurek, E.; Scott, B. L.; Trujillo, J. A.; Slemmons, A. K. Nuclear Magnetic Resonance Measurements and Electronic Structure of Pu(IV) in [(Me)4N]2PuCl6. Inorg. Chem. 2016, 55 (17), 8371– 8380,  DOI: 10.1021/acs.inorgchem.6b00735
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    12
    Nuclear Magnetic Resonance Measurements and Electronic Structure of Pu(IV) in [(Me)4N]2PuCl6
    Mounce, Andrew M.; Yasuoka, Hiroshi; Koutroulakis, Georgios; Lee, Jeongseop A.; Cho, Herman; Gendron, Frederic; Zurek, Eva; Scott, Brian L.; Trujillo, Julie A.; Slemmons, Alice K.; Cross, Justin N.; Thompson, Joe D.; Kozimor, Stosh A.; Bauer, Eric D.; Autschbach, Jochen; Clark, David L.
    Inorganic Chemistry (2016), 55 (17), 8371-8380CODEN: INOCAJ; ISSN:0020-1669. (American Chemical Society)
    The synthesis, electronic structure, and characterization via single crystal x-ray diffraction, NMR spectroscopy, and magnetic susceptibility of (Me4N)2PuCl6 are reported. NMR measurements were performed to both search for the direct 239Pu resonance and to obtain local magnetic and electronic information at the Cl site through 35Cl and 37Cl spectra. No signature of 239Pu NMR was obsd. The temp. dependence of the Cl spectra were simulated by diagonalizing the Zeeman and quadrupolar Hamiltonians for 35Cl, 37Cl and 14N isotopes. Electronic structure calcns. predict a magnetic Γ5 triplet ground state of Pu(IV) in the cryst. elec. field of the undistorted PuCl6 octahedron. A tetragonal distortion would result in a very small splitting (∼20 cm-1) of the triplet ground state into a nonmagnetic singlet and a doublet state. The Cl shifts have an inflection point at T ∼ 15 K, differing from the bulk susceptibility, indicating a nonmagnetic crystal field ground state. The Cl spin-lattice relaxation time is const. down to T = 15 K, below which it rapidly increases, also supporting the nonmagnetic crystal field ground state.
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  13. 13
    Gendron, F.; Autschbach, J. Ligand NMR Chemical Shift Calculations for Paramagnetic Metal Complexes: 5f1 vs 5f2 Actinides. J. Chem. Theory Comput. 2016, 12 (11), 5309– 5321,  DOI: 10.1021/acs.jctc.6b00462
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    13
    Ligand NMR Chemical Shift Calculations for Paramagnetic Metal Complexes: 5f1 vs 5f2 Actinides
    Gendron, Frederic; Autschbach, Jochen
    Journal of Chemical Theory and Computation (2016), 12 (11), 5309-5321CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
    Ligand paramagnetic NMR (pNMR) chem. shifts of the 5f1 complexes UO2(CO3)35- and NpO2(CO3)34-, and of the 5f2 complexes PuO2(CO3)34- and (C5H5)3UCH3 are investigated by wave function theory calcns., using a recently developed sum-over-states approach within complete active space and restricted active space paradigm including spin-orbit (SO) coupling. The exptl. 13C pNMR shifts of the actinyl tris-carbonate complexes are well reproduced by the calcns. The results are rationalized by visualizing natural spin orbitals (NSOs) and spin-magnetizations generated from the SO wave functions, in comparison with scalar relativistic spin densities. The anal. reveals a complex balance between spin-polarization, spin and orbital magnetization delocalization, and spin-compensation effects due to SO coupling. This balance creates the magnetization due to the electron paramagnetism around the nucleus of interest, and therefore the pNMR effects. The calcd. proton pNMR shifts of the (C5H5)3UCH3 complex are also in good agreement with exptl. data. Because of the nonmagnetic ground state of (C5H5)3UCH3, the 1H pNMR shifts arise mainly from the magnetic coupling contributions between the ground state and low-energy excited states belonging to the 5f manifold, along with the thermal population of degenerate excited states at ambient temps.
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  14. 14
    Autillo, M.; Guerin, L.; Bolvin, H.; Moisy, P.; Berthon, C. Magnetic Susceptibility of Actinide(III) Cations: An Experimental and Theoretical Study. Phys. Chem. Chem. Phys. 2016, 18 (9), 6515– 6525,  DOI: 10.1039/C5CP07456H
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    14
    Magnetic susceptibility of actinide(III) cations: an experimental and theoretical study
    Autillo, Matthieu; Guerin, Laetitia; Bolvin, Helene; Moisy, Philippe; Berthon, Claude
    Physical Chemistry Chemical Physics (2016), 18 (9), 6515-6525CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)
    In a previous paper, the influence of radioactive decay (α and β-) on magnetic susceptibility measurements by the Evans method has been demonstrated by the study of two americium isotopes. To characterize more accurately this phenomenon and particularly its influence on the Curie law, a new study has been performed on two uranium isotopes (238U and 233U) and on tritiated water (3H2O). The results on the influence of α emissions have established a relationship between changes in the temp. dependence and the radioactivity in soln. Regarding the β- emissions, less influence was obsd. while no temp. dependence linked to this kind of radioactive emission could be identified. Once magnetic susceptibility measurements of actinide(III) cations were cor. from radioactivity effects, methods of quantum chem. have been used on free ions and aquo complexes to calc. the electronic structure explaining the magnetic properties of Pu(III), Am(III) and Cm(III). The ligand field effect on the magnetic behavior (the Curie const. and temp.-independent susceptibilities) was analyzed by considering different solvation environments.
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  15. 15
    Gagliardi, L.; Roos, B. O. Quantum Chemical Calculations Show That the Uranium Molecule U2 Has a Quintuple Bond. Nature 2005, 433 (7028), 848– 851,  DOI: 10.1038/nature03249
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    15
    Quantum chemical calculations show that the uranium molecule U2 has a quintuple bond
    Gagliardi, Laura; Roos, Bjoern O.
    Nature (London, United Kingdom) (2005), 433 (7028), 848-851CODEN: NATUAS; ISSN:0028-0836. (Nature Publishing Group)
    Covalent bonding is commonly described by Lewis's theory, with an electron pair shared between two atoms constituting one full bond. Beginning with the valence bond description for the hydrogen mol., quantum chemists have further explored the fundamental nature of the chem. bond for atoms throughout the periodic table, confirming that most mols. are indeed held together by one electron pair for each bond. But more complex binding may occur when large nos. of AOs can participate in bond formation. Such behavior is common with transition metals. When involving heavy actinide elements, metal-metal bonds might prove particularly complicated. To date, evidence for actinide-actinide bonds is restricted to the matrix-isolation of uranium hydrides, including H2U-UH2, and the gas-phase detection and preliminary theor. study of the uranium mol., U2. Here we report quantum chem. calcns. on U2, showing that, although the strength of the U2 bond is comparable to that of other multiple bonds between transition metals, the bonding pattern is unique. We find that the mol. contains three electron-pair bonds and four one-electron bonds (i.e., 10 bonding electrons, corresponding to a quintuple bond), and two ferromagnetically coupled electrons localized on one U atom each-so all known covalent bonding types are contributing.
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  16. 16
    Knecht, S.; Jensen, H. J. A.; Saue, T. Relativistic Quantum Chemical Calculations Show That the Uranium Molecule U2 Has a Quadruple Bond. Nat. Chem. 2019, 11 (1), 40– 44,  DOI: 10.1038/s41557-018-0158-9
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    16
    Relativistic quantum chemical calculations show that the uranium molecule U2 has a quadruple bond
    Knecht, Stefan; Jensen, Hans Joergen Aa.; Saue, Trond
    Nature Chemistry (2019), 11 (1), 40-44CODEN: NCAHBB; ISSN:1755-4330. (Nature Research)
    Understanding the bonding, reactivity and electronic structure of actinides is lagging behind that of the rest of the periodic table. This can be partly explained by the challenges that one faces in exptl. studies of such radioactive compds. and also by the need to properly account for relativistic effects in theor. studies. A further challenge is the very complicated electronic structures encountered in actinide chem., as vividly illustrated by the naked diuranium mol. U2. Here we report a computational study of this emblematic mol. using state-of-the-art relativistic quantum chem. methods. Notably, the variational inclusion of spin-orbit interactions leads not only to a different electronic ground state, but also to a lower bond multiplicity compared with those in previous studies.
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  17. 17
    Andersson, K.; Malmqvist, P. Å.; Roos, B. O.; Sadlej, A. J.; Wolinski, K. Second-Order Perturbation Theory with a CASSCF Reference Function. J. Phys. Chem. 1990, 94 (14), 5483– 5488,  DOI: 10.1021/j100377a012
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    Second-order perturbation theory with a CASSCF reference function
    Andersson, Kerstin; Malmqvist, Per Aake; Roos, Bjoern O.; Sadlej, Andrzej J.; Wolinski, Krzysztof
    Journal of Physical Chemistry (1990), 94 (14), 5483-8CODEN: JPCHAX; ISSN:0022-3654.
    Second-order perturbation theory based on a CASSCF ref. state is derived and implemented. The first-order wave function includes the full space of interacting states. Expressions for the contributions to the second-order energy are obtained in terms of up to four-particle d. matrixes for the CASSCF ref. state. The zeroth-order Hamiltonian reduces to the Moeller-Plesset Hamiltonian for a closed-shell ref. state. The limit of the implementation is given by the no. of active orbitals, which dets. the size of the d. matrixes. It is presently around 13 orbitals. The method is illustrated in a series of calcns. on H2, H2O, CH2, and F-, and the results are compared with corresponding full CI results.
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  18. 18
    Gagliardi, L.; Truhlar, D. G.; Li Manni, G.; Carlson, R. K.; Hoyer, C. E.; Bao, J. L. Multiconfiguration Pair-Density Functional Theory: A New Way To Treat Strongly Correlated Systems. Acc. Chem. Res. 2017, 50 (1), 66– 73,  DOI: 10.1021/acs.accounts.6b00471
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    18
    Multiconfiguration Pair-Density Functional Theory: A New Way To Treat Strongly Correlated Systems
    Gagliardi, Laura; Truhlar, Donald G.; Li Manni, Giovanni; Carlson, Rebecca K.; Hoyer, Chad E.; Bao, Junwei Lucas
    Accounts of Chemical Research (2017), 50 (1), 66-73CODEN: ACHRE4; ISSN:0001-4842. (American Chemical Society)
    A review. The electronic energy of a system provides the Born-Oppenheimer potential energy for internuclear motion and thus dets. mol. structure and spectra, bond energies, conformational energies, reaction barrier heights, and vibrational frequencies. The development of more efficient and more accurate ways to calc. the electronic energy of systems with inherently multiconfigurational electronic structure is essential for many applications, including transition metal and actinide chem., systems with partially broken bonds, many transition states, and most electronically excited states. Inherently multiconfigurational systems are called strongly correlated systems or multireference systems, where the latter name refers to the need for using more than one ("multiple") configuration state function to provide a good zero-order ref. wave function. The present account describes (MC-PDFT), which was developed as a way to combine the advantages of wave function theory (WFT) and d. functional theory (DFT) to provide a better treatment of strongly correlated systems. First we review background material: the widely used Kohn-Sham DFT (which uses only a single Slater determinant as ref. wave function), multiconfiguration WFT methods that treat inherently multiconfigurational systems based on an active space, and previous attempts to combine multiconfiguration WFT with DFT. Then we review the formulation of MC-PDFT. MC-PDFT is a generalization of Kohn-Sham DFT in that the electron kinetic energy and classical electrostatic energy are calcd. from a ref. wave function, with the rest of the energy obtained from a d. functional. However, there are two main differences: (i) The ref. wave function is multiconfigurational rather than being a single Slater determinant. (ii) The d. functional is a function of the total d. and the on-top pair d. rather than being a function of the spin-up and spin-down densities. In work carried out so far, the multiconfigurational wave function is a multiconfiguration self-consistent-field wave function. The new formulation has the advantage that the ref. wave function has the correct spatial and spin symmetry and can describe bond dissocn. (of both single and multiple bonds) and electronic excitations in a formally and phys. correct way. We then review the formulation of d. functionals in terms of the on-top pair d. Finally we review successful applications of the theory to bond energies and bond dissocn. potential energy curves of main-group and transition metal bonds, to barrier heights (including pericyclic reactions), to proton affinities, to the hydrogen bond energy of water dimer, to ground- and excited-state charge transfer, to valence and Rydberg excitations of mols., and to singlet-triplet splittings of radicals. We find that MC-PDFT can give accurate results not only with complete-active-space multiconfiguration wave functions, but also with generalized-active-space multiconfiguration wave functions, which are practical for larger nos. of active electrons and active orbitals than are complete-active-space wave functions. The sepd.-pair approxn., which is a special case of generalized active space self-consistent-field theory, is esp. promising. MC-PDFT, because it requires much less computer time and storage than previous WFT methods, has the potential to open larger and more complex strongly correlated systems to accurate simulation.
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  19. 19
    Keller, S.; Dolfi, M.; Troyer, M.; Reiher, M. An Efficient Matrix Product Operator Representation of the Quantum Chemical Hamiltonian. J. Chem. Phys. 2015, 143 (24), 244118,  DOI: 10.1063/1.4939000
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    19
    An efficient matrix product operator representation of the quantum chemical Hamiltonian
    Keller, Sebastian; Dolfi, Michele; Troyer, Matthias; Reiher, Markus
    Journal of Chemical Physics (2015), 143 (24), 244118/1-244118/10CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    We describe how to efficiently construct the quantum chem. Hamiltonian operator in matrix product form. We present its implementation as a d. matrix renormalization group (DMRG) algorithm for quantum chem. applications. Existing implementations of DMRG for quantum chem. are based on the traditional formulation of the method, which was developed from the point of view of Hilbert space decimation and attained higher performance compared to straightforward implementations of matrix product based DMRG. The latter variationally optimizes a class of ansatz states known as matrix product states, where operators are correspondingly represented as matrix product operators (MPOs). The MPO construction scheme presented here eliminates the previous performance disadvantages while retaining the addnl. flexibility provided by a matrix product approach, for example, the specification of expectation values becomes an input parameter. In this way, MPOs for different symmetries - Abelian and non-Abelian - and different relativistic and non-relativistic models may be solved by an otherwise unmodified program. (c) 2015 American Institute of Physics.
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  20. 20
    Keller, S.; Reiher, M. Spin-Adapted Matrix Product States and Operators. J. Chem. Phys. 2016, 144 (13), 134101,  DOI: 10.1063/1.4944921
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    20
    Spin-adapted matrix product states and operators
    Keller, Sebastian; Reiher, Markus
    Journal of Chemical Physics (2016), 144 (13), 134101/1-134101/9CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    Matrix product states (MPSs) and matrix product operators (MPOs) allow an alternative formulation of the d. matrix renormalization group algorithm introduced by White. Here, we describe how non-Abelian spin symmetry can be exploited in MPSs and MPOs by virtue of the Wigner-Eckart theorem at the example of the spin-adapted quantum chem. Hamiltonian operator. (c) 2016 American Institute of Physics.
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    Knecht, S.; Hedegård, E. D.; Keller, S.; Kovyrshin, A.; Ma, Y.; Muolo, A.; Stein, C. J.; Reiher, M. New Approaches for Ab Initio Calculations of Molecules with Strong Electron Correlation. Chimia (Aarau). 2016, 70 (4), 244– 251,  DOI: 10.2533/chimia.2016.244
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    New Approaches for ab initio Calculations of Molecules with Strong Electron Correlation
    Knecht Stefan; Hedegard Erik Donovan; Keller Sebastian; Kovyrshin Arseny; Ma Yingjin; Muolo Andrea; Stein Christopher J; Reiher Markus
    Chimia (2016), 70 (4), 244-51 ISSN:0009-4293.
    Reliable quantum chemical methods for the description of molecules with dense-lying frontier orbitals are needed in the context of many chemical compounds and reactions. Here, we review developments that led to our new computational toolbox which implements the quantum chemical density matrix renormalization group in a second-generation algorithm. We present an overview of the different components of this toolbox.
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    Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Scalmani, G.; Barone, V.; Mennucci, B.; Petersson, G. A., Gaussian 09, Revision D.01; Gaussian, Inc.: Wallingford, CT, 2013.
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    Adamo, C.; Barone, V. Toward Reliable Density Functional Methods without Adjustable Parameters: The PBE0Model. J. Chem. Phys. 1999, 110 (13), 6158– 6170,  DOI: 10.1063/1.478522
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    Toward reliable density functional methods without adjustable parameters: the PBE0 model
    Adamo, Carlo; Barone, Vincenzo
    Journal of Chemical Physics (1999), 110 (13), 6158-6170CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    We present an anal. of the performances of a parameter free d. functional model (PBE0) obtained combining the so called PBE generalized gradient functional with a predefined amt. of exact exchange. The results obtained for structural, thermodn., kinetic and spectroscopic (magnetic, IR and electronic) properties are satisfactory and not far from those delivered by the most reliable functionals including heavy parameterization. The way in which the functional is derived and the lack of empirical parameters fitted to specific properties make the PBE0 model a widely applicable method for both quantum chem. and condensed matter physics.
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    Grimme, S.; Antony, J.; Ehrlich, S.; Krieg, H. A Consistent and Accurate Ab Initio Parametrization of Density Functional Dispersion Correction (DFT-D) for the 94 Elements H-Pu. J. Chem. Phys. 2010, 132 (15), 154104,  DOI: 10.1063/1.3382344
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    A consistent and accurate ab initio parametrization of density functional dispersion correction (DFT-D) for the 94 elements H-Pu
    Grimme, Stefan; Antony, Jens; Ehrlich, Stephan; Krieg, Helge
    Journal of Chemical Physics (2010), 132 (15), 154104/1-154104/19CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    The method of dispersion correction as an add-on to std. Kohn-Sham d. functional theory (DFT-D) has been refined regarding higher accuracy, broader range of applicability, and less empiricism. The main new ingredients are atom-pairwise specific dispersion coeffs. and cutoff radii that are both computed from first principles. The coeffs. for new eighth-order dispersion terms are computed using established recursion relations. System (geometry) dependent information is used for the first time in a DFT-D type approach by employing the new concept of fractional coordination nos. (CN). They are used to interpolate between dispersion coeffs. of atoms in different chem. environments. The method only requires adjustment of two global parameters for each d. functional, is asymptotically exact for a gas of weakly interacting neutral atoms, and easily allows the computation of at. forces. Three-body nonadditivity terms are considered. The method has been assessed on std. benchmark sets for inter- and intramol. noncovalent interactions with a particular emphasis on a consistent description of light and heavy element systems. The mean abs. deviations for the S22 benchmark set of noncovalent interactions for 11 std. d. functionals decrease by 15%-40% compared to the previous (already accurate) DFT-D version. Spectacular improvements are found for a tripeptide-folding model and all tested metallic systems. The rectification of the long-range behavior and the use of more accurate C6 coeffs. also lead to a much better description of large (infinite) systems as shown for graphene sheets and the adsorption of benzene on an Ag(111) surface. For graphene it is found that the inclusion of three-body terms substantially (by about 10%) weakens the interlayer binding. We propose the revised DFT-D method as a general tool for the computation of the dispersion energy in mols. and solids of any kind with DFT and related (low-cost) electronic structure methods for large systems. (c) 2010 American Institute of Physics.
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    Pritchard, B. P.; Altarawy, D.; Didier, B.; Gibson, T. D.; Windus, T. L. New Basis Set Exchange: An Open, Up-to-Date Resource for the Molecular Sciences Community. J. Chem. Inf. Model. 2019, 59 (11), 4814– 4820,  DOI: 10.1021/acs.jcim.9b00725
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    New Basis Set Exchange: An Open, Up-to-Date Resource for the Molecular Sciences Community
    Pritchard, Benjamin P.; Altarawy, Doaa; Didier, Brett; Gibson, Tara D.; Windus, Theresa L.
    Journal of Chemical Information and Modeling (2019), 59 (11), 4814-4820CODEN: JCISD8; ISSN:1549-9596. (American Chemical Society)
    A review. The Basis Set Exchange (BSE) has been a prominent fixture in the quantum chem. community. First publicly available in 2007, it is recognized by both users and basis set creators as the de facto source for information related to basis sets. This popular resource has been rewritten, utilizing modern software design and best practices. The basis set data has been sepd. into a stand-alone library with an accessible API, and the Web site has been updated to use the current generation of web development libraries. The general layout and workflow of the Web site is preserved, while helpful features requested by the user community have been added. Overall, this design should increase adaptability and lend itself well into the future as a dependable resource for the computational chem. community. This article will discuss the decision to rewrite the BSE, the new architecture and design, and the new features that have been added.
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    Cao, X.; Dolg, M.; Stoll, H. Valence Basis Sets for Relativistic Energy-Consistent Small-Core Actinide Pseudopotentials. J. Chem. Phys. 2003, 118 (2), 487– 496,  DOI: 10.1063/1.1521431
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    Valence basis sets for relativistic energy-consistent small-core actinide pseudopotentials
    Cao, Xiaoyan; Dolg, Michael; Stoll, Hermann
    Journal of Chemical Physics (2003), 118 (2), 487-496CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    Gaussian (14s13p10d8f6g)/[6s6p5d4f3g] at. natural orbital valence basis sets were generated for relativistic energy-consistent small-core actinide pseudopotentials of the Stuttgart-Bonn variety. Effective valence spin-orbit operators supplementing the scalar-relativistic pseudopotentials were derived from multiconfiguration Dirac-Hartree-Fock ref. data. Pseudopotentials, basis sets and spin-orbit operators were used to det. the first and second ionization potentials of all actinide elements at the multiconfiguration SCF and multireference averaged coupled-pair functional level. Comparison was made to results obtained from large-scale calcns. using uncontracted basis sets up to i-type functions and extrapolation to the basis set limit as well as to exptl. data. Mol. calibration studies using the coupled-cluster singles, doubles, and perturbative triples approach are reported for the ground states of AcH, AcO, AcF, and ThO.
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    Cao, X.; Dolg, M. Segmented Contraction Scheme for Small-Core Actinide Pseudopotential Basis Sets. J. Mol. Struct. THEOCHEM 2004, 673 (1–3), 203– 209,  DOI: 10.1016/j.theochem.2003.12.015
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    Segmented contraction scheme for small-core actinide pseudopotential basis sets
    Cao, Xiaoyan; Dolg, Michael
    Journal of Molecular Structure: THEOCHEM (2004), 673 (1-3), 203-209CODEN: THEODJ; ISSN:0166-1280. (Elsevier Science B.V.)
    Gaussian (14s13p10d8f6g)/[10s9p5d4f3g] valence basis sets using a segmented contraction scheme have been derived for relativistic energy-consistent small-core actinide pseudopotentials of the Stuttgart-Koln variety. The present basis sets are only slightly larger than previously published (14s13p10d8f6g)/[6s6p5d4f3g] at. natural orbital basis sets, which use a generalized contraction scheme, and achieve a similar accuracy in at. and mol. calcns. For calibration purposes multi-configuration SCF and subsequent multi-ref. averaged coupled-pair functional calcns. are presented for the first to fourth ionization potentials of all actinide elements. Th Pa U Np Pu Am Cm Bk Cf Es Fm Md No Lr. In addn., results of mol. calibration studies using the coupled-cluster singles, doubles and perturbative triples approach as well as gradient-cor. d. functional theory are reported for the monohydrides, monoxides and monofluorides of actinium and lawrencium.
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    Küchle, W.; Dolg, M.; Stoll, H.; Preuss, H. Energy-adjusted Pseudopotentials for the Actinides. Parameter Sets and Test Calculations for Thorium and Thorium Monoxide. J. Chem. Phys. 1994, 100 (10), 7535– 7542,  DOI: 10.1063/1.466847
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    Energy-adjusted pseudopotentials for the actinides. Parameter sets and test calculations for thorium and thorium monoxide
    Kuechle, W.; Dolg, M.; Stoll, H.; Preuss, H.
    Journal of Chemical Physics (1994), 100 (10), 7535-42CODEN: JCPSA6; ISSN:0021-9606.
    The authors present nonrelativistic and quasirelativistic energy-adjusted pseudopotentials, the latter augmented by spin-orbit operators, as well as optimized (12s11p10d8f)/[8s7p6d4f]-Gaussian-type orbitals (GTO) valence basis sets for the actinide elements actinium through lawrencium. At. excitation and ionization energies obtained by the use of these pseudopotentials and basis sets in SCF calcns. differ by less than 0.2 eV from corresponding finite-difference all-electron results. Large-scale multiconfiguration SCF (MCSCF), multireference CI (MRCI), and multireference averaged coupled-pair functional (MRACPF) calcns. for thorium and thorium monoxide yield results in satisfactory agreement with available exptl. data. Preliminary results from spin-orbit CI calcns. for the low-lying electronic states of thorium monoxide are also reported.
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    Kendall, R. A.; Dunning, T. H.; Harrison, R. J. Electron Affinities of the First-row Atoms Revisited. Systematic Basis Sets and Wave Functions. J. Chem. Phys. 1992, 96 (9), 6796– 6806,  DOI: 10.1063/1.462569
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    Electron affinities of the first-row atoms revisited. Systematic basis sets and wave functions
    Kendall, Rick A.; Dunning, Thom H., Jr.; Harrison, Robert J.
    Journal of Chemical Physics (1992), 96 (9), 6796-806CODEN: JCPSA6; ISSN:0021-9606.
    The authors describe a reliable procedure for calcg. the electron affinity of an atom and present results for H, B, C, O, and F (H is included for completeness). This procedure involves the use of the recently proposed correlation-consistent basis sets augmented with functions to describe the more diffuse character of the at. anion coupled with a straightforward, uniform expansion of the ref. space for multireference singles and doubles configuration-interaction (MRSD-CI) calcns. A comparison is given with previous results and with corresponding full CI calcns. The most accurate EAs obtained from the MRSD-CI calcns. are (with exptl. values in parentheses): H 0.740 eV (0.754), B 0.258 (0.277), C 1.245 (1.263), O 1.384 (1.461), and F 3.337 (3.401). The EAs obtained from the MR-SDCI calcns. differ by less than 0.03 eV from those predicted by the full CI calcns.
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    Fdez. Galvan, I.; Vacher, M.; Alavi, A.; Angeli, C.; Aquilante, F.; Autschbach, J.; Bao, J. J.; Bokarev, S. I.; Bogdanov, N. A.; Carlson, R. K.; Chibotaru, L. F.; Creutzberg, J.; Dattani, N.; Delcey, M. G.; Dong, S. S.; Dreuw, A.; Freitag, L.; Frutos, L. M.; Gagliardi, L.; Gendron, F.; Giussani, A.; Gonzalez, L.; Grell, G.; Guo, M.; Hoyer, C. E.; Johansson, M.; Keller, S.; Knecht, S.; Kovacevic, G.; Kallman, E.; Li Manni, G.; Lundberg, M.; Ma, Y.; Mai, S.; Malhado, J. P.; Malmqvist, P. A.; Marquetand, P.; Mewes, S. A.; Norell, J.; Olivucci, M.; Oppel, M.; Phung, Q. M.; Pierloot, K.; Plasser, F.; Reiher, M.; Sand, A. M.; Schapiro, I.; Sharma, P.; Stein, C. J.; Sørensen, L. K.; Truhlar, D. G.; Ugandi, M.; Ungur, L.; Valentini, A.; Vancoillie, S.; Veryazov, V.; Weser, O.; Wesołowski, T. A.; Widmark, P.-O.; Wouters, S.; Zech, A.; Zobel, J. P.; Lindh, R. OpenMolcas: From Source Code to Insight. J. Chem. Theory Comput. 2019, 15 (11), 5925– 5964,  DOI: 10.1021/acs.jctc.9b00532
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    OpenMolcas: From Source Code to Insight
    Fdez. Galvan, Ignacio; Vacher, Morgane; Alavi, Ali; Angeli, Celestino; Aquilante, Francesco; Autschbach, Jochen; Bao, Jie J.; Bokarev, Sergey I.; Bogdanov, Nikolay A.; Carlson, Rebecca K.; Chibotaru, Liviu F.; Creutzberg, Joel; Dattani, Nike; Delcey, Mickael G.; Dong, Sijia S.; Dreuw, Andreas; Freitag, Leon; Frutos, Luis Manuel; Gagliardi, Laura; Gendron, Frederic; Giussani, Angelo; Gonzalez, Leticia; Grell, Gilbert; Guo, Meiyuan; Hoyer, Chad E.; Johansson, Marcus; Keller, Sebastian; Knecht, Stefan; Kovacevic, Goran; Kaellman, Erik; Li Manni, Giovanni; Lundberg, Marcus; Ma, Yingjin; Mai, Sebastian; Malhado, Joao Pedro; Malmqvist, Per Aake; Marquetand, Philipp; Mewes, Stefanie A.; Norell, Jesper; Olivucci, Massimo; Oppel, Markus; Phung, Quan Manh; Pierloot, Kristine; Plasser, Felix; Reiher, Markus; Sand, Andrew M.; Schapiro, Igor; Sharma, Prachi; Stein, Christopher J.; Soerensen, Lasse Kragh; Truhlar, Donald G.; Ugandi, Mihkel; Ungur, Liviu; Valentini, Alessio; Vancoillie, Steven; Veryazov, Valera; Weser, Oskar; Wesolowski, Tomasz A.; Widmark, Per-Olof; Wouters, Sebastian; Zech, Alexander; Zobel, J. Patrick; Lindh, Roland
    Journal of Chemical Theory and Computation (2019), 15 (11), 5925-5964CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
    In this Article we describe the OpenMolcas environment and invite the computational chem. community to collaborate. The open-source project already includes a large no. of new developments realized during the transition from the com. MOLCAS product to the open-source platform. The paper initially describes the tech. details of the new software development platform. This is followed by brief presentations of many new methods, implementations, and features of the OpenMolcas program suite. These developments include novel wave function methods such as stochastic complete active space SCF, d. matrix renormalization group (DMRG) methods, and hybrid multiconfigurational wave function and d. functional theory models. Some of these implementations include an array of addnl. options and functionalities. The paper proceeds and describes developments related to explorations of potential energy surfaces. Here we present methods for the optimization of conical intersections, the simulation of adiabatic and nonadiabatic mol. dynamics, and interfaces to tools for semiclassical and quantum mech. nuclear dynamics. Furthermore, the Article describes features unique to simulations of spectroscopic and magnetic phenomena such as the exact semiclassical description of the interaction between light and matter, various X-ray processes, magnetic CD, and properties. Finally, the paper describes a no. of built-in and add-on features to support the OpenMolcas platform with postcalcn. anal. and visualization, a multiscale simulation option using frozen-d. embedding theory, and new electronic and muonic basis sets.
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    Aquilante, F.; Autschbach, J.; Baiardi, A.; Battaglia, S.; Borin, V. A.; Chibotaru, L. F.; Conti, I.; De Vico, L.; Delcey, M.; Fdez; Galván, I. Modern Quantum Chemistry with [Open]Molcas. J. Chem. Phys. 2020, 152 (21), 214117,  DOI: 10.1063/5.0004835
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    Modern quantum chemistry with [Open]Molcas
    Aquilante, Francesco; Autschbach, Jochen; Baiardi, Alberto; Battaglia, Stefano; Borin, Veniamin A.; Chibotaru, Liviu F.; Conti, Irene; De Vico, Luca; Delcey, Mickael; Fdez. Galvan, Ignacio; Ferre, Nicolas; Freitag, Leon; Garavelli, Marco; Gong, Xuejun; Knecht, Stefan; Larsson, Ernst D.; Lindh, Roland; Lundberg, Marcus; Malmqvist, Per Ake; Nenov, Artur; Norell, Jesper; Odelius, Michael; Olivucci, Massimo; Pedersen, Thomas B.; Pedraza-Gonzalez, Laura; Phung, Quan M.; Pierloot, Kristine; Reiher, Markus; Schapiro, Igor; Segarra-Marti, Javier; Segatta, Francesco; Seijo, Luis; Sen, Saumik; Sergentu, Dumitru-Claudiu; Stein, Christopher J.; Ungur, Liviu; Vacher, Morgane; Valentini, Alessio; Veryazov, Valera
    Journal of Chemical Physics (2020), 152 (21), 214117CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    MOLCAS/OpenMolcas is an ab initio electronic structure program providing a large set of computational methods from Hartree-Fock and d. functional theory to various implementations of multiconfigurational theory. This article provides a comprehensive overview of the main features of the code, specifically reviewing the use of the code in previously reported chem. applications as well as more recent applications including the calcn. of magnetic properties from optimized d. matrix renormalization group wave functions. (c) 2020 American Institute of Physics.
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    Roos, B. O.; Lindh, R.; Malmqvist, P. Å.; Veryazov, V.; Widmark, P. O. Main Group Atoms and Dimers Studied with a New Relativistic ANO Basis Set. J. Phys. Chem. A 2004, 108 (15), 2851– 2858,  DOI: 10.1021/jp031064+
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    Main Group Atoms and Dimers Studied with a New Relativistic ANO Basis Set
    Roos, Bjoern O.; Lindh, Roland; Malmqvist, Per-Aake; Veryazov, Valera; Widmark, Per-Olof
    Journal of Physical Chemistry A (2004), 108 (15), 2851-2858CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)
    New basis sets of the at. natural orbital (ANO) type have been developed for the main group and rare gas atoms. The ANO's have been obtained from the av. d. matrix of the ground and lowest excited states of the atom, the pos. and neg. ions, and the dimer at its equil. geometry. Scalar relativistic effects are included through the use of a Douglas-Kroll Hamiltonian. Multiconfigurational wave functions have been used with dynamic correlation included using second-order perturbation theory (CASSCF/CASPT2). The basis sets are applied in calcns. of ionization energies, electron affinities, and excitation energies for all atoms and the ground-state potentials for the dimers. These calcns. include spin-orbit coupling using the RASSCF State Interaction (RASSI-SO) method. The spin-orbit splitting for the lowest at. term is reproduced with an accuracy of better than 0.05 eV, except for row 5, where it is 0.15 eV. Ionization energies and electron affinities have an accuracy better than 0.2 eV, and at. polarizabilities for the spherical atoms are computed with errors smaller than 2.5%. Computed bond energies for the dimers are accurate to better than 0.15 eV in most cases (the dimers for row 5 excluded).
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    Roos, B. O.; Lindh, R.; Malmqvist, P. Å.; Veryazov, V.; Widmark, P. O. New Relativistic ANO Basis Sets for Actinide Atoms. Chem. Phys. Lett. 2005, 409 (4–6), 295– 299,  DOI: 10.1016/j.cplett.2005.05.011
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    New relativistic ANO basis sets for actinide atoms
    Roos, Bjoern O.; Lindh, Roland; Malmqvist, Per-Aake; Veryazov, Valera; Widmark, Per-Olof
    Chemical Physics Letters (2005), 409 (4-6), 295-299CODEN: CHPLBC; ISSN:0009-2614. (Elsevier B.V.)
    New basis sets of the at. natural orbital (ANO) type have been developed for the actinide atoms Ac-Cm. The ANOs have been obtained from the av. d. matrix of the ground and lowest excited states of the atom, the pos. ions, and the atom in a elec. field. Scalar relativistic effects are included through the use of a Douglas-Kroll-Hess Hamiltonian. Multiconfigurational wave functions have been used with dynamic correlation included using second order perturbation theory (CASSCF/CASPT2). The basis sets are applied in calcns. of ionization energies and some excitation energies. Computed ionization energies have an accuracy better than 0.2 eV in most cases. The lowest multiplet levels have been computed. These calcns. include spin-orbit coupling using a variation-perturbation approach. The at. polarizability of the spherically sym. americium atom has been computed to be 116 au3.
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    Finley, J.; Malmqvist, P. Å.; Roos, B. O.; Serrano-Andrés, L. The Multi-State CASPT2Method. Chem. Phys. Lett. 1998, 288 (2–4), 299– 306,  DOI: 10.1016/S0009-2614(98)00252-8
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    The multi-state CASPT2 method
    Finley, James; Malmqvist, Per-Ake; Roos, Bjorn O.; Serrano-Andres, Luis
    Chemical Physics Letters (1998), 288 (2,3,4), 299-306CODEN: CHPLBC; ISSN:0009-2614. (Elsevier Science B.V.)
    An extension of the multiconfigurational second-order perturbation approach CASPT2 is suggested, where several electronic states are coupled at second order via an effective-Hamiltonian approach. The method has been implemented into the MOLCAS-4 program system, where it will replace the single-state CASPT2 program. The accuracy of the method is illustrated through calcns. of the ionic-neutral avoided crossing in the potential curves for LiF and of the valence-Rydberg mixing in the V-state of the ethylene mol.
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    Granovsky, A. A. Extended Multi-Configuration Quasi-Degenerate Perturbation Theory: The New Approach to Multi-State Multi-Reference Perturbation Theory. J. Chem. Phys. 2011, 134 (21), 214113,  DOI: 10.1063/1.3596699
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    Extended multi-configuration quasi-degenerate perturbation theory: The new approach to multi-state multi-reference perturbation theory
    Granovsky, Alexander A.
    Journal of Chemical Physics (2011), 134 (21), 214113/1-214113/14CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    The distinctive desirable features, both math. and phys. meaningful, for all partially contracted multi-state multi-ref. perturbation theories (MS-MR-PT) are explicitly formulated. The original approach to MS-MR-PT theory, called extended multi-configuration quasi-degenerate perturbation theory (XMCQDPT), having most, if not all, of the desirable properties is introduced. The new method is applied at the second order of perturbation theory (XMCQDPT2) to the 11A' - 21A' conical intersection in allene mol., the avoided crossing in LiF mol., and the 11A1 to 21A1 electronic transition in cis-1,3-butadiene. The new theory has several advantages compared to those of well-established approaches, such as second order multi-configuration quasi-degenerate perturbation theory and multi-state-second order complete active space perturbation theory. The anal. of the prevalent approaches to the MS-MR-PT theory performed within the framework of the XMCQDPT theory unveils the origin of their common inherent problems. We describe the efficient implementation strategy that makes XMCQDPT2 an esp. useful general-purpose tool in the high-level modeling of small to large mol. systems. (c) 2011 American Institute of Physics.
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    Pierloot, K.; Van Besien, E. Electronic Structure and Spectrum of UO22+ and UO2Cl42–. J. Chem. Phys. 2005, 123 (20), 204309,  DOI: 10.1063/1.2121608
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    Electronic structure and spectrum of UO22+ and UO2Cl42-
    Pierloot, Kristine; van Besien, Els
    Journal of Chemical Physics (2005), 123 (20), 204309/1-204309/10CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    A theor. study is presented of the electronic spectra of the UO22+ and UO2Cl42- ions, based on multiconfigurational perturbation theory (CASSCF/CASPT2), combined with a recently developed method to treat spin-orbit coupling [P.-A. Malmqvist et al., Chem. Phys. Lett. 357, 230 (2002); B. O. Roos and P.-A. Malmqvist, Phys. Chem. Chem. Phys. 6, 2919 (2004)]. The results are compared to the exptl. spectroscopic data obtained for uranyl ions in Cs2UO2Cl4 crystals from Denning [Struct. Bonding (Berlin) 79, 215 (1992)] and to previous theor. calcns. performed using a combined configuration-interaction spin-orbit treatment [Z. Zhang and R. M. Pitzer, J. Phys. Chem. A 103, 6880 (1999); S. Matsika and R. M. Pitzer, J. Phys. Chem. A. 105, 637 (2001)]. As opposed to the latter results, the calcns. performed in this work point to a significant effect of the weakly bound equatorial chlorine ligands on the excitation energies.
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    Pierloot, K.; Van Besien, E.; Van Lenthe, E.; Baerends, E. J. Electronic Spectrum of UO22+ and [UO2Cl4]2– Calculated with Time-Dependent Density Functional Theory. J. Chem. Phys. 2007, 126 (19), 194311,  DOI: 10.1063/1.2735297
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    Electronic spectrum of UO2+2 and [UO2Cl4]2- calculated with time-dependent density functional theory
    Pierloot, Kristine; van Besien, Els; van Lenthe, Erik; Baerends, Evert Jan
    Journal of Chemical Physics (2007), 126 (19), 194311/1-194311/8CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
    The electronic spectra of UO2+2 and [UO2Cl4]2- are calcd. with a recently proposed relativistic time-dependent d. functional theory method based on the two-component zeroth-order regular approxn. for the inclusion of spin-orbit coupling and a noncollinear exchange-correlation functional. All excitations out of the bonding σu+ orbital into the nonbonding δu or φu orbitals for UO2+2 and the corresponding excitations for [UO2Cl4]2- are considered. Scalar relativistic vertical excitation energies are compared to values from previous calcns. with the CASPT2 method. Two-component adiabatic excitation energies, U-O equil. distances, and sym. stretching frequencies are compared to CASPT2 and combined configuration-interaction and spin-orbit coupling results, as well as to exptl. data. The compn. of the excited states in terms of the spin-orbit free states is analyzed. The results point to a significant effect of the Cl ligands on the electronic spectrum, thereby confirming the CASPT2 results: The excitation energies are shifted and a different luminescent state is found.
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    Tecmer, P.; Bast, R.; Ruud, K.; Visscher, L. Charge-Transfer Excitations in Uranyl Tetrachloride ([UO 2Cl 4] 2-): How Reliable Are Electronic Spectra from Relativistic Time-Dependent Density Functional Theory?. J. Phys. Chem. A 2012, 116 (27), 7397– 7404,  DOI: 10.1021/jp3011266
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    38
    Charge-Transfer Excitations in Uranyl Tetrachloride ([UO2Cl4]2-): How Reliable are Electronic Spectra from Relativistic Time-Dependent Density Functional Theory?
    Tecmer, Pawel; Bast, Radovan; Ruud, Kenneth; Visscher, Lucas
    Journal of Physical Chemistry A (2012), 116 (27), 7397-7404CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)
    Four-component relativistic time-dependent d. functional theory (TD-DFT) is used to study charge-transfer (CT) excitation energies of the uranyl mol. as well as the uranyl tetrachloride complex. Adiabatic excitation energies and vibrational frequencies of the excited states are calcd. for the lower energy range of the spectrum. The results for TD-DFT with the CAM-B3LYP exchange-correlation functional for the [UO2Cl4]2- system are in good agreement with the exptl. obsd. spectrum of this species and agree also rather well with other theor. data. Use of the global hybrid B3LYP gives qual. correct results, while use of the BLYP functional yields results that are qual. wrong due to the too low CT states calcd. with this functional. The applicability of the overlap diagnostic of Peach et al. to identify such CT excitations is investigated for a wide range of vertical transitions using results obtained with three different approx. exchange-correlation functionals: BLYP, B3LYP, and CAM-B3LYP.
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    Oher, H.; Réal, F.; Vercouter, T.; Vallet, V. Investigation of the Luminescence of [UO 2 × 4 ] 2– (X = Cl, Br) Complexes in the Organic Phase Using Time-Resolved Laser-Induced Fluorescence Spectroscopy and Quantum Chemical Simulations. Inorg. Chem. 2020, 59 (9), 5896– 5906,  DOI: 10.1021/acs.inorgchem.9b03614
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    Investigation of the Luminescence of [UO2X4]2- (X = Cl, Br) Complexes in the Organic Phase Using Time-Resolved Laser-Induced Fluorescence Spectroscopy and Quantum Chemical Simulations
    Oher, Hanna; Real, Florent; Vercouter, Thomas; Vallet, Valerie
    Inorganic Chemistry (2020), 59 (9), 5896-5906CODEN: INOCAJ; ISSN:0020-1669. (American Chemical Society)
    The luminescence properties of the [UO2Cl4]2- complex in an org. phase, esp. the influence of large org. countercations, have been studied by time-resolved laser-induced fluorescence spectroscopy (TRLFS) and ab initio modeling. The exptl. spectrum was assigned by vibronic Franck-Condon calcns. on quantum chem. methods on the basis of a combination of relativistic d. functional approaches. The shape of the luminescence spectrum of the uranyl tetrachloride complex is detd. by sym. vibrations and geometrical change upon emission. The possible change in the luminescence properties depending on the first and second uranyl coordination spheres was predicted theor. for the [UO2Br4]2- and [R4N]2[UO2Cl4] ([R4N] = [Bu4N], [A336]) systems. The computations reveal that, for U(VI), the second coordination sphere has little influence on the spectrum shape, making speciation of uranyl complexes with identical first-coordination-sphere ligands tedious to discriminate. The computed structural changes agreed well with exptl. trends; theor. spectra and peak attributions are in good accordance with TRLFS and magnetic CD (MCD) data, resp. Luminescence spectra of [UO2Cl4]2- complexes were successfully computed by vibrationally resolved ab initio calcns. on various chem. models. By comparison with spectra recorded by time-resolved laser-induced fluorescence spectroscopy (TRLFS), we rationalize to which extent the first and second uranyl coordination spheres, as well as the solvent, influence the spectral shapes and vibrational progressions. The present theor. investigations support that TRLFS spectroscopy may not be sensitive enough to discriminate long-range interactions above the first coordination sphere of uranyl.
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    Li Manni, G.; Carlson, R. K.; Luo, S.; Ma, D.; Olsen, J.; Truhlar, D. G.; Gagliardi, L. Multiconfiguration Pair-Density Functional Theory. J. Chem. Theory Comput. 2014, 10 (9), 3669– 3680,  DOI: 10.1021/ct500483t
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    Multiconfiguration Pair-Density Functional Theory
    Li Manni, Giovanni; Carlson, Rebecca K.; Luo, Sijie; Ma, Dongxia; Olsen, Jeppe; Truhlar, Donald G.; Gagliardi, Laura
    Journal of Chemical Theory and Computation (2014), 10 (9), 3669-3680CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
    We present a new theor. framework, called Multiconfiguration Pair-D. Functional Theory (MC-PDFT), which combines multiconfigurational wave functions with a generalization of d. functional theory (DFT). A multiconfigurational self-consistent-field (MCSCF) wave function with correct spin and space symmetry is used to compute the total electronic d., its gradient, the on-top pair d., and the kinetic and Coulomb contributions to the total electronic energy. We then use a functional of the total d., its gradient, and the on-top pair d. to calc. the remaining part of the energy, which we call the on-top-d.-functional energy in contrast to the exchange-correlation energy of Kohn-Sham DFT. Because the on-top pair d. is an element of the two-particle d. matrix, this goes beyond the Hohenberg-Kohn theorem that refers only to the one-particle d. To illustrate the theory, we obtain first approxns. to the required new type of d. functionals by translating conventional d. functionals of the spin densities using a simple prescription, and we perform post-SCF d. functional calcns. using the total d., d. gradient, and on-top pair d. from the MSCSF calcns. Double counting of dynamic correlation or exchange does not occur because the MCSCF energy is not used. The theory is illustrated by applications to the bond energies and potential energy curves of H2, N2, F2, CaO, Cr2, and NiCl and the electronic excitation energies of Be, C, N, N+, O, O+, Sc+, Mn, Co, Mo, Ru, N2, HCHO, C4H6, c-C5H6, and pyrazine. The method presented has a computational cost and scaling similar to MCSCF, but a quant. accuracy, even with the present first approxns. to the new types of d. functionals, that is comparable to much more expensive multireference perturbation theory methods.
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    Carlson, R. K.; Truhlar, D. G.; Gagliardi, L. Multiconfiguration Pair-Density Functional Theory: A Fully Translated Gradient Approximation and Its Performance for Transition Metal Dimers and the Spectroscopy of Re 2 Cl 8 2–. J. Chem. Theory Comput. 2015, 11 (9), 4077– 4085,  DOI: 10.1021/acs.jctc.5b00609
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    Multiconfiguration Pair-Density Functional Theory: A Fully Translated Gradient Approximation and Its Performance for Transition Metal Dimers and the Spectroscopy of Re2Cl82-
    Carlson, Rebecca K.; Truhlar, Donald G.; Gagliardi, Laura
    Journal of Chemical Theory and Computation (2015), 11 (9), 4077-4085CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
    We extend the on-top d. functional of multiconfiguration pair-d. functional theory (MC-PDFT) to include the gradient of the on-top d. as well as the gradient of the d. We find that the theory is reasonably stable to this extension; furthermore, it provides improved accuracy for mols. contg. transition metals. We illustrate the extended on-top d. functionals by applying them to Cr2, Cu2, Ag2, Os2, and Re2Cl82- as well as to our previous database of 56 data for bond dissocn. energies, barrier heights, reaction energies, proton affinities, and the water dimer. The performance of MC-PDFT is comparable to or better than that of CASPT2.
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    King, D. M.; Cleaves, P. A.; Wooles, A. J.; Gardner, B. M.; Chilton, N. F.; Tuna, F.; Lewis, W.; McInnes, E. J. L.; Liddle, S. T. Molecular and Electronic Structure of Terminal and Alkali Metal-Capped Uranium(V) Nitride Complexes. Nat. Commun. 2016, 7 (1), 13773,  DOI: 10.1038/ncomms13773
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    Molecular and electronic structure of terminal and alkali metal-capped uranium(V) nitride complexes
    King, David M.; Cleaves, Peter A.; Wooles, Ashley J.; Gardner, Benedict M.; Chilton, Nicholas F.; Tuna, Floriana; Lewis, William; McInnes, Eric J. L.; Liddle, Stephen T.
    Nature Communications (2016), 7 (), 13773CODEN: NCAOBW; ISSN:2041-1723. (Nature Publishing Group)
    A review. Detg. the electronic structure of actinide complexes is intrinsically challenging because inter-electronic repulsion, crystal field, and spin-orbit coupling effects can be of similar magnitude. Moreover, such efforts have been hampered by the lack of structurally analogous families of complexes to study. Here we report an improved method to U≃N triple bonds, and assemble a family of uranium(V) nitrides. Along with an isoelectronic oxo, we quantify the electronic structure of this 5f1 family by magnetometry, optical and ESR (EPR) spectroscopies and modeling. Thus, we define the relative importance of the spin-orbit and crystal field interactions, and explain the exptl. obsd. different ground states. We find optical absorption linewidths give a potential tool to identify spin-orbit coupled states, and show measurement of UV···UV super-exchange coupling in dimers by EPR. We show that obsd. slow magnetic relaxation occurs via two-phonon processes, with no obvious correlation to the crystal field.
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    Reta, D.; Ortu, F.; Randall, S.; Mills, D. P.; Chilton, N. F.; Winpenny, R. E. P.; Natrajan, L.; Edwards, B.; Kaltsoyannis, N. The Performance of Density Functional Theory for the Description of Ground and Excited State Properties of Inorganic and Organometallic Uranium Compounds. J. Organomet. Chem. 2018, 857, 58– 74,  DOI: 10.1016/j.jorganchem.2017.09.021
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    The performance of density functional theory for the description of ground and excited state properties of inorganic and organometallic uranium compounds
    Reta, Daniel; Ortu, Fabrizio; Randall, Simon; Mills, David P.; Chilton, Nicholas F.; Winpenny, Richard E. P.; Natrajan, Louise; Edwards, Bryan; Kaltsoyannis, Nikolas
    Journal of Organometallic Chemistry (2018), 857 (), 58-74CODEN: JORCAI; ISSN:0022-328X. (Elsevier B.V.)
    Mol. uranium complexes are the most widely studied in actinide chem., and make a significant and growing contribution to inorg. and organometallic chem. However, reliable computational procedures to accurately describe the properties of such systems are not yet available. In this contribution, 18 exptl. characterized mol. uranium compds., in oxidn. states ranging from III to VI and with a variety of ligand environments, are studied computationally using d. functional theory. The computed geometries and vibrational frequencies are compared with X-ray crystallog., and infra-red and Raman spectroscopic data to establish which computational approach yields the closest agreement with expt. NMR parameters and UV-vis spectra are studied for three and five closed-shell U(VI) compds. resp. Overall, the most robust methodol. for obtaining accurate geometries is the PBE functional with Grimme's D3 dispersion corrections. For IR spectra, different approaches yield almost identical results, which makes the PBE functional with Grimme's D3 dispersion corrections the best choice. However, for Raman spectra the dependence on functional is more pronounced and no clear recommendation can be made. Similarly, for 1H and 13C NMR chem. shifts, no unequivocal recommendation emerges as to the best choice of d. functional, although for spin-spin couplings, the LC-ωPBE functional with solvent corrections is the best approach. No form of time-dependent d. functional theory can be recommended for the simulation of the electronic absorption spectra of uranyl (VI) compds.; the orbitals involved in the transitions are not calcd. correctly, and the energies are also typically unreliable. Two main approaches are adopted for the description of relativistic effects on the uranium centers: either a relativistic pseudopotential and assocd. valence basis set, or an all-electron basis set with the ZORA Hamiltonian. The former provides equal, if not better, agreement with expt. vs all-electron basis set calcns., for all properties investigated.
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    Hashem, E.; Swinburne, A. N.; Schulzke, C.; Evans, R. C.; Platts, J. A.; Kerridge, A.; Natrajan, L. S.; Baker, R. J. Emission Spectroscopy of Uranium(Iv) Compounds: A Combined Synthetic, Spectroscopic and Computational Study. RSC Adv. 2013, 3 (13), 4350,  DOI: 10.1039/c3ra22712j
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    Emission spectroscopy of uranium(IV) compounds. A combined synthetic, spectroscopic and computational study
    Hashem, Emtithal; Swinburne, Adam N.; Schulzke, Carola; Evans, Rachel C.; Platts, James A.; Kerridge, Andrew; Natrajan, Louise S.; Baker, Robert J.
    RSC Advances (2013), 3 (13), 4350-4361CODEN: RSCACL; ISSN:2046-2069. (Royal Society of Chemistry)
    Emission spectroscopy was used for the 1st time in a spectroscopic study of a family of U(IV) halide complexes in non-aq. media. The room temp. electronic absorption spectra of the simple coordination compds. [Li(THF)4][UX5(THF)] (X = Cl, Br, I), [Et4N]2[UCl6] and UCl4 in THF were recorded and all transitions assigned with the aid of a comprehensive computational study using CASSCF and CASPT2 techniques. Excitation into a band of f-d and LMCT character followed by energy transfer into the 5f-orbital manifold accounts for the UV-visible radiative transitions obsd. in the emission spectra, which were fully assigned as arising from transitions from the 5f16d1 electronic configuration to envelopes of states arising from the ground state 5f2 configuration. The bonding in [Li(THF)4][UCl5(THF)] was further elucidated utilizing NBO and AIM calcns. which describe the nature of the U-Cl bond as predominantly ionic with some dative covalent character and substantial overlap between the Cl 3p orbitals and 5f and 6d orbitals on uranium. These studies indicate that the emission spectral fingerprint of simple U(IV) compds. of Oh, C4v, and C2v symmetry are similar and characteristic and may be used as a diagnostic tool to assign U(IV) species in soln. and by inference, in the environment, in the presence of [UO2]2+.
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    Walisinghe, A. J.; Chilton, N. F. Assessment of Minimal Active Space CASSCF-SO Methods for Calculation of Atomic Slater–Condon and Spin–Orbit Coupling Parameters in d- and f-Block Ions. Dalt. Trans. 2021, 50 (40), 14130– 14138,  DOI: 10.1039/D1DT02346B
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    Bao, J. J.; Zhou, C.; Varga, Z.; Kanchanakungwankul, S.; Gagliardi, L.; Truhlar, D. G. Multi-State Pair-Density Functional Theory. Faraday Discuss. 2020, 224 (0), 348– 372,  DOI: 10.1039/D0FD00037J
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    Multi-state pair-density functional theory
    Bao, Jie J.; Zhou, Chen; Varga, Zoltan; Kanchanakungwankul, Siriluk; Gagliardi, Laura; Truhlar, Donald G.
    Faraday Discussions (2020), 224 (New Horizons in Density Functional Theory), 348-372CODEN: FDISE6; ISSN:1359-6640. (Royal Society of Chemistry)
    Multi-configuration pair-d. functional theory (MC-PDFT) has previously been applied successfully to carry out ground-state and excited-state calcns. Here we propose two new methods, called extended-multi-state-PDFT (XMS-PDFT) and variational-multi-state-PDFT (VMS-PDFT), that generate the intermediate states in a balanced way with a single set of orbitals. The former uses the intermediate states proposed by Granovsky for extended multi-configuration quasi-degenerate perturbation theory (XMC-QDPT); the latter obtains the intermediate states by maximizing the sum of the MC-PDFT energies for the intermediate states. We also propose a Fourier series expansion to make the variational optimizations of the VMS-PDFT method convenient, and we implement this method (FMS-PDFT) both for conventional configuration-interaction solvers and for d.-matrix-renormalization-group solvers. The new methods are tested for eight systems, exhibiting avoided crossings among two to six states. The FMS-PDFT method is successful for all cases for which it has been tested (all cases in this paper except O3 for which it was not tested), and XMS-PDFT is successful for all eight cases except the mixed-valence case. Since both XMS-PDFT and VMS-PDFT are less expensive than XMS-CASPT2, they will allow well-correlated calcns. on much larger systems for which perturbation theory is unaffordable.
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    Battaglia, S.; Lindh, R. Extended Dynamically Weighted CASPT2: The Best of Two Worlds. J. Chem. Theory Comput. 2020, 16 (3), 1555– 1567,  DOI: 10.1021/acs.jctc.9b01129
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    Extended Dynamically Weighted CASPT2: The Best of Two Worlds
    Battaglia, Stefano; Lindh, Roland
    Journal of Chemical Theory and Computation (2020), 16 (3), 1555-1567CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
    We introduce a new variant of the complete active space second-order perturbation theory (CASPT2) method that performs similarly to multistate CASPT2 (MS-CASPT2) in regions of the potential energy surface where the electronic states are energetically well sepd. and is akin to extended MS-CASPT2 (XMS-CASPT2) in case the underlying zeroth-order refs. are near-degenerate. Our approach follows a recipe analogous to that of XMS-CASPT2 to ensure approx. invariance under unitary transformations of the model states and a dynamic weighting scheme to smoothly interpolate the Fock operator between state-specific and state-av. regimes. The resulting extended dynamically weighted CASPT2 (XDW-CASPT2) methodol. possesses the most desirable features of both MS-CASPT2 and XMS-CASPT2, i.e., the ability to provide accurate transition energies and correctly describe avoided crossings and conical intersections. The reliability of XDW-CASPT2 is assessed on a no. of mol. systems. First, we consider the dissocn. of lithium fluoride, highlighting the distinctive characteristics of the new approach. Second, the invariance of the theory is investigated by studying the conical intersection of the distorted allene mol. Finally, the relative accuracy in the calcn. of vertical excitation energies is benchmarked on a set of 26 org. compds. We found that XDW-CASPT2, albeit being only approx. invariant, produces smooth potential energy surfaces around conical intersections and avoided crossings, performing equally well to the strictly invariant XMS-CASPT2 method. The accuracy of vertical transition energies is almost identical to MS-CASPT2, with a mean abs. deviation of 0.01-0.02 eV, in contrast to 0.12 eV for XMS-CASPT2.
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  1. Leyla R. Valerio, Sabyasachi Roy Chowdhury, Rob Lewis, Kathryn E. Knowles, Bess Vlaisavljevich, Ellen M. Matson. Photoluminescence of a Uranium(IV) Alkoxide Complex. JACS Au 2025, 5 (1) , 332-342. https://doi.org/10.1021/jacsau.4c01022
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  • Abstract

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    Figure 1

    Figure 1. Molecular structure of [UO2Cl4]2–. Red = oxygen, green = chlorine, and blue = uranium.

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    Figure 2

    Figure 2. (Left) Highest occupied orbital in the ground state S0 (σu) and (right) the newly singly occupied molecular orbital in the first excited state T1 (5fδ) for [UO2Cl4]2–.

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    Figure 3

    Figure 3. 2D (left, reproduced from ref (42) under a CC BY 4.0 license) and 3D (right) molecular structure of [U(TrenTIPS)(N)]−. Purple = nitrogen, yellow = silicon, blue = uranium, and gray = carbon.

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    Figure 4

    Figure 4. Absorption spectra for [U(TrenTIPS)(N)]− calculated with CAS(1,7)SCF (top), CAS(1,7)SCF-MS-CASPT2 (middle), and CAS(1,7)SCF-MC-PDFT (bottom) compared to the experiment. (42) Theoretical line widths are scaled to match the experimental peak widths.

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    Figure 5

    Figure 5. Molecular structure of [UCl5(THF)]−. Red = oxygen, green = chlorine, and blue = uranium.

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    Figure 6

    Figure 6. Absorption spectra of [UCl5(THF)]− calculated by (left) CAS(2,7)SCF and (right) CAS(2,7)SCF-MS-CASPT2, compared to the experimental data (black). All spectra are normalized to the intensity of the 3F3/3F4 transition, and the calculated spectra are plotted with a FWHM line width of 9 nm.

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  • References

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      Drobot, Bjoern; Steudtner, Robin; Raff, Johannes; Geipel, Gerhard; Brendler, Vinzenz; Tsushima, Satoru
      Chemical Science (2015), 6 (2), 964-972CODEN: CSHCCN; ISSN:2041-6520. (Royal Society of Chemistry)
      This study of aq. metal speciation is an advanced combination of theor. and exptl. methods. Continuous wave (CW) and time-resolved laser-induced fluorescence spectroscopy (TRLFS) data of uranyl(VI) hydrolysis were analyzed using parallel factor anal. (PARAFAC). Distribution patterns of five major species were thereby derived under a fixed uranyl concn. (10-5 M) over a wide pH range from 2 to 11. UV (180 nm to 370 nm) excitation spectra were extd. for individual species. Time-dependent d. functional theory (TD-DFT) calcns. revealed ligand excitation (water, hydroxo, oxo) in this region and ligand-to-metal charge transfer (LMCT) responsible for luminescence. Thus excitation in the UV region is extreme ligand sensitive and specific. Combining findings from PARAFAC and DFT the [UO2(H2O)5]2+ cation (aquo complex 1 : 0) and four hydroxo complexes (1 : 1, 3 : 5, 3 : 7 and 1 : 3) were identified. The methodol. concept used here is applicable to luminescent metals in general and thus enables acquisition of refined structural and thermodynamical data of lanthanide and actinide complexation.
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      A new d.-functional approach to calc. the excitation spectrum of many-electron systems is proposed. It is shown that the full linear d. response of the interacting system, which has poles at the exact excitation energies, can rigorously be expressed in terms of the response function of the noninteracting (Kohn-Sham) system and a frequency-dependent exchange-correlation kernel. Using this expression, the poles of the full response function are obtained by systematic improvement upon the poles of the Kohn-Sham response function. Numerical results are presented for Be, Mg, Ca, Zn, Sr, and Cd atoms.
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      Yanai, T.; Tew, D. P.; Handy, N. C. A New Hybrid Exchange–Correlation Functional Using the Coulomb-Attenuating Method (CAM-B3LYP). Chem. Phys. Lett. 2004, 393 (1–3), 51– 57,  DOI: 10.1016/j.cplett.2004.06.011
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      A new hybrid exchange-correlation functional using the Coulomb-attenuating method (CAM-B3LYP)
      Yanai, Takeshi; Tew, David P.; Handy, Nicholas C.
      Chemical Physics Letters (2004), 393 (1-3), 51-57CODEN: CHPLBC; ISSN:0009-2614. (Elsevier Science B.V.)
      A new hybrid exchange-correlation functional named CAM-B3LYP is proposed. It combines the hybrid qualities of B3LYP and the long-range correction presented by Tawada et al. [J. Chem. Phys., in press]. We demonstrate that CAM-B3LYP yields atomization energies of similar quality to those from B3LYP, while also performing well for charge transfer excitations in a dipeptide model, which B3LYP underestimates enormously. The CAM-B3LYP functional comprises of 0.19 Hartree-Fock (HF) plus 0.81 Becke 1988 (B88) exchange interaction at short-range, and 0.65 HF plus 0.35 B88 at long-range. The intermediate region is smoothly described through the std. error function with parameter 0.33.
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      Roos, B. O.; Taylor, P. R.; Sigbahn, P. E. M. A Complete Active Space SCF Method (CASSCF) Using a Density Matrix Formulated Super-CI Approach. Chem. Phys. 1980, 48 (2), 157– 173,  DOI: 10.1016/0301-0104(80)80045-0
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      A complete active space SCF method (CASSCF) using a density matrix formulated super-CI approach
      Roos, Bjoern O.; Taylor, Peter R.; Siegbahn, E. M.
      Chemical Physics (1980), 48 (2), 157-73CODEN: CMPHC2; ISSN:0301-0104.
      A d. matrix formulation of the super-CI MCSCF method is presented. The MC expansion is assumed to be complete in an active subset of the orbital space, and the corresponding CI secular problem is solved by a direct scheme using the unitary group approach. With a d. matrix formulation the orbital optimization step becomes independent of the size of the CI expansion. It is possible to formulate the super-CI in terms of d. matrices defined only in the small active subspace; the doubly occupied orbitals (the inactive subspace) do not enter. Further, in the unitary group formalism it is straightforward and simple to obtain the necessary d. matrices from the symbolic formula list. It then becomes possible to treat very long MC expansions, the largest so far comprising 726 configurations. The method is demonstrated in a calcn. of the potential curves for the 3 lowest states (1.sum.g+, 3.sum.u+ and 3πg) of the N2 mol., using a medium-sized gaussian basis set. 7 Active orbitals were used yielding the following results: Dc:8.76(9.90), 2.43(3.68) and 3.39 (4.90) eV; rc:1.108 (1.098), 1.309(1.287) and 1.230 (1.213) Å; ωe: 2333 (2359), 1385 (1461) and 1680 (1733) cm-1, for the 3 states (exptl. values within parentheses). The results of these calcns. indicate that it is important to consider not only the dissocn. limit but also the united atom limit in partitioning the occupied orbital space into an active and an inactive part.
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      Hartree, D. R. The Wave Mechanics of an Atom with a Non-Coulomb Central Field. Part I. Theory and Methods. Math. Proc. Cambridge Philos. Soc. 1928, 24 (1), 89– 110,  DOI: 10.1017/S0305004100011919
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    8. 8
      Vallet, V.; Privalov, T.; Wahlgren, U.; Grenthe, I. The Mechanism of Water Exchange in AmO2(H2O) 52+ and in the Isoelectronic UO2(H 2O)5+ and NpO2(H2O) 52+ Complexes as Studied by Quantum Chemical Methods. J. Am. Chem. Soc. 2004, 126 (25), 7766– 7767,  DOI: 10.1021/ja0483544
      8
      The Mechanism of Water Exchange in AmO2(H2O)52+ and in the Isoelectronic UO2(H2O)5+ and NpO2(H2O)52+ Complexes as Studied by Quantum Chemical Methods
      Vallet, Valerie; Privalov, Timofei; Wahlgren, Ulf; Grenthe, Ingmar
      Journal of the American Chemical Society (2004), 126 (25), 7766-7767CODEN: JACSAT; ISSN:0002-7863. (American Chemical Society)
      The water exchange mechanism for AmO2(H2O)52+ and the isoelectronic UO2(H2O)5+ and NpO2(H2O)52+ ions has been investigated using quantum chem. methods and compared to previous findings in the uranyl(VI) system (Vallet, V. et al. J. Am. Chem. Soc. 2001, 123, 11999). There are substantial and predictable changes in the geometry and preferred exchange pathways between uranyl(V) and the different actinyl(VI) complexes. The smaller charge on the uranyl(V) center makes the dissociative pathway more favorable than the associative/interchange pathways in the actinyl(VI) systems.
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    9. 9
      Gagliardi, L.; Roos, B. O.; Malmqvist, P. Å.; Dyke, J. M. On the Electronic Structure of the UO2Molecule. J. Phys. Chem. A 2001, 105 (46), 10602– 10606,  DOI: 10.1021/jp012888z
      9
      On the Electronic Structure of the UO2 Molecule
      Gagliardi, Laura; Roos, Bjoern O.; Malmqvist, Perke; Dyke, John M.
      Journal of Physical Chemistry A (2001), 105 (46), 10602-10606CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)
      The structure and vibrational frequencies of the UO2 mol. have been detd. using multiconfigurational wave functions (CASSCF/CASPT2), together with a newly developed method to treat spin-orbit coupling. The mol. has been found to have a (5fφ)(7s), 3Φu, Ω = 2 ground state with a U-O bond distance of 1.77 Å. The computed antisym. stretching σu frequency is 923 cm-1 with a 16/18 isotope ratio of 1.0525 which compares with the exptl. values of 915 cm-1 and 1.0526, resp. Calcns. of the first adiabatic ionization energy gave the value 6.17 eV, which is 0.7 eV larger than the currently accepted exptl. result. Reasons for this difference are suggested.
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    10. 10
      Gagliardi, L.; Heaven, M. C.; Krog, J. W.; Roos, B. O. The Electronic Spectrum of the UO2 Molecule. J. Am. Chem. Soc. 2005, 127 (1), 86– 91,  DOI: 10.1021/ja044940l
      10
      The Electronic Spectrum of the UO2 Molecule
      Gagliardi, Laura; Heaven, Michael C.; Krogh, Jesper Wisborg; Roos, Bjoern O.
      Journal of the American Chemical Society (2005), 127 (1), 86-91CODEN: JACSAT; ISSN:0002-7863. (American Chemical Society)
      The electronic spectrum of the UO2 mol. was detd. using multiconfigurational wave functions together with the inclusion spin-orbit coupling. The mol. has a (5fφ)(7s), 3Φ2u, ground state. The lowest state of gerade symmetry, 3H4g, corresponding to the electronic configuration (5f)2 was found 3330 cm-1 above the ground state. The computed energy levels and oscillator strengths were used for the assignment of the exptl. spectrum in the energy range 17,000-19,000 and 27,000-32,000 cm-1.
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    11. 11
      Heit, Y. N.; Gendron, F.; Autschbach, J. Calculation of Dipole-Forbidden 5f Absorption Spectra of Uranium(V) Hexa-Halide Complexes. J. Phys. Chem. Lett. 2018, 9 (4), 887– 894,  DOI: 10.1021/acs.jpclett.7b03441
      11
      Calculation of Dipole-Forbidden 5f Absorption Spectra of Uranium(V) Hexa-Halide Complexes
      Heit, Yonaton N.; Gendron, Frederic; Autschbach, Jochen
      Journal of Physical Chemistry Letters (2018), 9 (4), 887-894CODEN: JPCLCD; ISSN:1948-7185. (American Chemical Society)
      Restricted-active-space wave function calcns. including spin-orbit coupling, in combination with Kohn-Sham d. functional calcns. of vibrational modes, were used to det. the vibronic and electronic absorption intensities of the near-IR elec. dipole-forbidden 5f-5f transitions of representative U(V) hexa-halide complex ions. The agreement with exptl. assigned vibronic and electronic transitions measured for powder or soln. samples of salts of the complex ions is reasonable overall and excellent for the exptl. best-resolved E5/2u → E'5/2u bands. The intensity of the vibronic transitions may be borrowed from ligand-to-metal charge-transfer excitations as well as 5f-to-6d metal-centered transitions. Magnetically allowed electronic transitions contribute to the 2 lower-frequency bands of the ligand-field spectrum.
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    12. 12
      Mounce, A. M.; Yasuoka, H.; Koutroulakis, G.; Lee, J. A.; Cho, H.; Gendron, F.; Zurek, E.; Scott, B. L.; Trujillo, J. A.; Slemmons, A. K. Nuclear Magnetic Resonance Measurements and Electronic Structure of Pu(IV) in [(Me)4N]2PuCl6. Inorg. Chem. 2016, 55 (17), 8371– 8380,  DOI: 10.1021/acs.inorgchem.6b00735
      12
      Nuclear Magnetic Resonance Measurements and Electronic Structure of Pu(IV) in [(Me)4N]2PuCl6
      Mounce, Andrew M.; Yasuoka, Hiroshi; Koutroulakis, Georgios; Lee, Jeongseop A.; Cho, Herman; Gendron, Frederic; Zurek, Eva; Scott, Brian L.; Trujillo, Julie A.; Slemmons, Alice K.; Cross, Justin N.; Thompson, Joe D.; Kozimor, Stosh A.; Bauer, Eric D.; Autschbach, Jochen; Clark, David L.
      Inorganic Chemistry (2016), 55 (17), 8371-8380CODEN: INOCAJ; ISSN:0020-1669. (American Chemical Society)
      The synthesis, electronic structure, and characterization via single crystal x-ray diffraction, NMR spectroscopy, and magnetic susceptibility of (Me4N)2PuCl6 are reported. NMR measurements were performed to both search for the direct 239Pu resonance and to obtain local magnetic and electronic information at the Cl site through 35Cl and 37Cl spectra. No signature of 239Pu NMR was obsd. The temp. dependence of the Cl spectra were simulated by diagonalizing the Zeeman and quadrupolar Hamiltonians for 35Cl, 37Cl and 14N isotopes. Electronic structure calcns. predict a magnetic Γ5 triplet ground state of Pu(IV) in the cryst. elec. field of the undistorted PuCl6 octahedron. A tetragonal distortion would result in a very small splitting (∼20 cm-1) of the triplet ground state into a nonmagnetic singlet and a doublet state. The Cl shifts have an inflection point at T ∼ 15 K, differing from the bulk susceptibility, indicating a nonmagnetic crystal field ground state. The Cl spin-lattice relaxation time is const. down to T = 15 K, below which it rapidly increases, also supporting the nonmagnetic crystal field ground state.
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    13. 13
      Gendron, F.; Autschbach, J. Ligand NMR Chemical Shift Calculations for Paramagnetic Metal Complexes: 5f1 vs 5f2 Actinides. J. Chem. Theory Comput. 2016, 12 (11), 5309– 5321,  DOI: 10.1021/acs.jctc.6b00462
      13
      Ligand NMR Chemical Shift Calculations for Paramagnetic Metal Complexes: 5f1 vs 5f2 Actinides
      Gendron, Frederic; Autschbach, Jochen
      Journal of Chemical Theory and Computation (2016), 12 (11), 5309-5321CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
      Ligand paramagnetic NMR (pNMR) chem. shifts of the 5f1 complexes UO2(CO3)35- and NpO2(CO3)34-, and of the 5f2 complexes PuO2(CO3)34- and (C5H5)3UCH3 are investigated by wave function theory calcns., using a recently developed sum-over-states approach within complete active space and restricted active space paradigm including spin-orbit (SO) coupling. The exptl. 13C pNMR shifts of the actinyl tris-carbonate complexes are well reproduced by the calcns. The results are rationalized by visualizing natural spin orbitals (NSOs) and spin-magnetizations generated from the SO wave functions, in comparison with scalar relativistic spin densities. The anal. reveals a complex balance between spin-polarization, spin and orbital magnetization delocalization, and spin-compensation effects due to SO coupling. This balance creates the magnetization due to the electron paramagnetism around the nucleus of interest, and therefore the pNMR effects. The calcd. proton pNMR shifts of the (C5H5)3UCH3 complex are also in good agreement with exptl. data. Because of the nonmagnetic ground state of (C5H5)3UCH3, the 1H pNMR shifts arise mainly from the magnetic coupling contributions between the ground state and low-energy excited states belonging to the 5f manifold, along with the thermal population of degenerate excited states at ambient temps.
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    14. 14
      Autillo, M.; Guerin, L.; Bolvin, H.; Moisy, P.; Berthon, C. Magnetic Susceptibility of Actinide(III) Cations: An Experimental and Theoretical Study. Phys. Chem. Chem. Phys. 2016, 18 (9), 6515– 6525,  DOI: 10.1039/C5CP07456H
      14
      Magnetic susceptibility of actinide(III) cations: an experimental and theoretical study
      Autillo, Matthieu; Guerin, Laetitia; Bolvin, Helene; Moisy, Philippe; Berthon, Claude
      Physical Chemistry Chemical Physics (2016), 18 (9), 6515-6525CODEN: PPCPFQ; ISSN:1463-9076. (Royal Society of Chemistry)
      In a previous paper, the influence of radioactive decay (α and β-) on magnetic susceptibility measurements by the Evans method has been demonstrated by the study of two americium isotopes. To characterize more accurately this phenomenon and particularly its influence on the Curie law, a new study has been performed on two uranium isotopes (238U and 233U) and on tritiated water (3H2O). The results on the influence of α emissions have established a relationship between changes in the temp. dependence and the radioactivity in soln. Regarding the β- emissions, less influence was obsd. while no temp. dependence linked to this kind of radioactive emission could be identified. Once magnetic susceptibility measurements of actinide(III) cations were cor. from radioactivity effects, methods of quantum chem. have been used on free ions and aquo complexes to calc. the electronic structure explaining the magnetic properties of Pu(III), Am(III) and Cm(III). The ligand field effect on the magnetic behavior (the Curie const. and temp.-independent susceptibilities) was analyzed by considering different solvation environments.
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    15. 15
      Gagliardi, L.; Roos, B. O. Quantum Chemical Calculations Show That the Uranium Molecule U2 Has a Quintuple Bond. Nature 2005, 433 (7028), 848– 851,  DOI: 10.1038/nature03249
      15
      Quantum chemical calculations show that the uranium molecule U2 has a quintuple bond
      Gagliardi, Laura; Roos, Bjoern O.
      Nature (London, United Kingdom) (2005), 433 (7028), 848-851CODEN: NATUAS; ISSN:0028-0836. (Nature Publishing Group)
      Covalent bonding is commonly described by Lewis's theory, with an electron pair shared between two atoms constituting one full bond. Beginning with the valence bond description for the hydrogen mol., quantum chemists have further explored the fundamental nature of the chem. bond for atoms throughout the periodic table, confirming that most mols. are indeed held together by one electron pair for each bond. But more complex binding may occur when large nos. of AOs can participate in bond formation. Such behavior is common with transition metals. When involving heavy actinide elements, metal-metal bonds might prove particularly complicated. To date, evidence for actinide-actinide bonds is restricted to the matrix-isolation of uranium hydrides, including H2U-UH2, and the gas-phase detection and preliminary theor. study of the uranium mol., U2. Here we report quantum chem. calcns. on U2, showing that, although the strength of the U2 bond is comparable to that of other multiple bonds between transition metals, the bonding pattern is unique. We find that the mol. contains three electron-pair bonds and four one-electron bonds (i.e., 10 bonding electrons, corresponding to a quintuple bond), and two ferromagnetically coupled electrons localized on one U atom each-so all known covalent bonding types are contributing.
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    16. 16
      Knecht, S.; Jensen, H. J. A.; Saue, T. Relativistic Quantum Chemical Calculations Show That the Uranium Molecule U2 Has a Quadruple Bond. Nat. Chem. 2019, 11 (1), 40– 44,  DOI: 10.1038/s41557-018-0158-9
      16
      Relativistic quantum chemical calculations show that the uranium molecule U2 has a quadruple bond
      Knecht, Stefan; Jensen, Hans Joergen Aa.; Saue, Trond
      Nature Chemistry (2019), 11 (1), 40-44CODEN: NCAHBB; ISSN:1755-4330. (Nature Research)
      Understanding the bonding, reactivity and electronic structure of actinides is lagging behind that of the rest of the periodic table. This can be partly explained by the challenges that one faces in exptl. studies of such radioactive compds. and also by the need to properly account for relativistic effects in theor. studies. A further challenge is the very complicated electronic structures encountered in actinide chem., as vividly illustrated by the naked diuranium mol. U2. Here we report a computational study of this emblematic mol. using state-of-the-art relativistic quantum chem. methods. Notably, the variational inclusion of spin-orbit interactions leads not only to a different electronic ground state, but also to a lower bond multiplicity compared with those in previous studies.
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    17. 17
      Andersson, K.; Malmqvist, P. Å.; Roos, B. O.; Sadlej, A. J.; Wolinski, K. Second-Order Perturbation Theory with a CASSCF Reference Function. J. Phys. Chem. 1990, 94 (14), 5483– 5488,  DOI: 10.1021/j100377a012
      17
      Second-order perturbation theory with a CASSCF reference function
      Andersson, Kerstin; Malmqvist, Per Aake; Roos, Bjoern O.; Sadlej, Andrzej J.; Wolinski, Krzysztof
      Journal of Physical Chemistry (1990), 94 (14), 5483-8CODEN: JPCHAX; ISSN:0022-3654.
      Second-order perturbation theory based on a CASSCF ref. state is derived and implemented. The first-order wave function includes the full space of interacting states. Expressions for the contributions to the second-order energy are obtained in terms of up to four-particle d. matrixes for the CASSCF ref. state. The zeroth-order Hamiltonian reduces to the Moeller-Plesset Hamiltonian for a closed-shell ref. state. The limit of the implementation is given by the no. of active orbitals, which dets. the size of the d. matrixes. It is presently around 13 orbitals. The method is illustrated in a series of calcns. on H2, H2O, CH2, and F-, and the results are compared with corresponding full CI results.
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    18. 18
      Gagliardi, L.; Truhlar, D. G.; Li Manni, G.; Carlson, R. K.; Hoyer, C. E.; Bao, J. L. Multiconfiguration Pair-Density Functional Theory: A New Way To Treat Strongly Correlated Systems. Acc. Chem. Res. 2017, 50 (1), 66– 73,  DOI: 10.1021/acs.accounts.6b00471
      18
      Multiconfiguration Pair-Density Functional Theory: A New Way To Treat Strongly Correlated Systems
      Gagliardi, Laura; Truhlar, Donald G.; Li Manni, Giovanni; Carlson, Rebecca K.; Hoyer, Chad E.; Bao, Junwei Lucas
      Accounts of Chemical Research (2017), 50 (1), 66-73CODEN: ACHRE4; ISSN:0001-4842. (American Chemical Society)
      A review. The electronic energy of a system provides the Born-Oppenheimer potential energy for internuclear motion and thus dets. mol. structure and spectra, bond energies, conformational energies, reaction barrier heights, and vibrational frequencies. The development of more efficient and more accurate ways to calc. the electronic energy of systems with inherently multiconfigurational electronic structure is essential for many applications, including transition metal and actinide chem., systems with partially broken bonds, many transition states, and most electronically excited states. Inherently multiconfigurational systems are called strongly correlated systems or multireference systems, where the latter name refers to the need for using more than one ("multiple") configuration state function to provide a good zero-order ref. wave function. The present account describes (MC-PDFT), which was developed as a way to combine the advantages of wave function theory (WFT) and d. functional theory (DFT) to provide a better treatment of strongly correlated systems. First we review background material: the widely used Kohn-Sham DFT (which uses only a single Slater determinant as ref. wave function), multiconfiguration WFT methods that treat inherently multiconfigurational systems based on an active space, and previous attempts to combine multiconfiguration WFT with DFT. Then we review the formulation of MC-PDFT. MC-PDFT is a generalization of Kohn-Sham DFT in that the electron kinetic energy and classical electrostatic energy are calcd. from a ref. wave function, with the rest of the energy obtained from a d. functional. However, there are two main differences: (i) The ref. wave function is multiconfigurational rather than being a single Slater determinant. (ii) The d. functional is a function of the total d. and the on-top pair d. rather than being a function of the spin-up and spin-down densities. In work carried out so far, the multiconfigurational wave function is a multiconfiguration self-consistent-field wave function. The new formulation has the advantage that the ref. wave function has the correct spatial and spin symmetry and can describe bond dissocn. (of both single and multiple bonds) and electronic excitations in a formally and phys. correct way. We then review the formulation of d. functionals in terms of the on-top pair d. Finally we review successful applications of the theory to bond energies and bond dissocn. potential energy curves of main-group and transition metal bonds, to barrier heights (including pericyclic reactions), to proton affinities, to the hydrogen bond energy of water dimer, to ground- and excited-state charge transfer, to valence and Rydberg excitations of mols., and to singlet-triplet splittings of radicals. We find that MC-PDFT can give accurate results not only with complete-active-space multiconfiguration wave functions, but also with generalized-active-space multiconfiguration wave functions, which are practical for larger nos. of active electrons and active orbitals than are complete-active-space wave functions. The sepd.-pair approxn., which is a special case of generalized active space self-consistent-field theory, is esp. promising. MC-PDFT, because it requires much less computer time and storage than previous WFT methods, has the potential to open larger and more complex strongly correlated systems to accurate simulation.
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    19. 19
      Keller, S.; Dolfi, M.; Troyer, M.; Reiher, M. An Efficient Matrix Product Operator Representation of the Quantum Chemical Hamiltonian. J. Chem. Phys. 2015, 143 (24), 244118,  DOI: 10.1063/1.4939000
      19
      An efficient matrix product operator representation of the quantum chemical Hamiltonian
      Keller, Sebastian; Dolfi, Michele; Troyer, Matthias; Reiher, Markus
      Journal of Chemical Physics (2015), 143 (24), 244118/1-244118/10CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      We describe how to efficiently construct the quantum chem. Hamiltonian operator in matrix product form. We present its implementation as a d. matrix renormalization group (DMRG) algorithm for quantum chem. applications. Existing implementations of DMRG for quantum chem. are based on the traditional formulation of the method, which was developed from the point of view of Hilbert space decimation and attained higher performance compared to straightforward implementations of matrix product based DMRG. The latter variationally optimizes a class of ansatz states known as matrix product states, where operators are correspondingly represented as matrix product operators (MPOs). The MPO construction scheme presented here eliminates the previous performance disadvantages while retaining the addnl. flexibility provided by a matrix product approach, for example, the specification of expectation values becomes an input parameter. In this way, MPOs for different symmetries - Abelian and non-Abelian - and different relativistic and non-relativistic models may be solved by an otherwise unmodified program. (c) 2015 American Institute of Physics.
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    20. 20
      Keller, S.; Reiher, M. Spin-Adapted Matrix Product States and Operators. J. Chem. Phys. 2016, 144 (13), 134101,  DOI: 10.1063/1.4944921
      20
      Spin-adapted matrix product states and operators
      Keller, Sebastian; Reiher, Markus
      Journal of Chemical Physics (2016), 144 (13), 134101/1-134101/9CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      Matrix product states (MPSs) and matrix product operators (MPOs) allow an alternative formulation of the d. matrix renormalization group algorithm introduced by White. Here, we describe how non-Abelian spin symmetry can be exploited in MPSs and MPOs by virtue of the Wigner-Eckart theorem at the example of the spin-adapted quantum chem. Hamiltonian operator. (c) 2016 American Institute of Physics.
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    21. 21
      Knecht, S.; Hedegård, E. D.; Keller, S.; Kovyrshin, A.; Ma, Y.; Muolo, A.; Stein, C. J.; Reiher, M. New Approaches for Ab Initio Calculations of Molecules with Strong Electron Correlation. Chimia (Aarau). 2016, 70 (4), 244– 251,  DOI: 10.2533/chimia.2016.244
      21
      New Approaches for ab initio Calculations of Molecules with Strong Electron Correlation
      Knecht Stefan; Hedegard Erik Donovan; Keller Sebastian; Kovyrshin Arseny; Ma Yingjin; Muolo Andrea; Stein Christopher J; Reiher Markus
      Chimia (2016), 70 (4), 244-51 ISSN:0009-4293.
      Reliable quantum chemical methods for the description of molecules with dense-lying frontier orbitals are needed in the context of many chemical compounds and reactions. Here, we review developments that led to our new computational toolbox which implements the quantum chemical density matrix renormalization group in a second-generation algorithm. We present an overview of the different components of this toolbox.
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    22. 22
      Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Scuseria, G. E.; Robb, M. A.; Cheeseman, J. R.; Scalmani, G.; Barone, V.; Mennucci, B.; Petersson, G. A., Gaussian 09, Revision D.01; Gaussian, Inc.: Wallingford, CT, 2013.
      There is no corresponding record for this reference.
    23. 23
      Adamo, C.; Barone, V. Toward Reliable Density Functional Methods without Adjustable Parameters: The PBE0Model. J. Chem. Phys. 1999, 110 (13), 6158– 6170,  DOI: 10.1063/1.478522
      23
      Toward reliable density functional methods without adjustable parameters: the PBE0 model
      Adamo, Carlo; Barone, Vincenzo
      Journal of Chemical Physics (1999), 110 (13), 6158-6170CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      We present an anal. of the performances of a parameter free d. functional model (PBE0) obtained combining the so called PBE generalized gradient functional with a predefined amt. of exact exchange. The results obtained for structural, thermodn., kinetic and spectroscopic (magnetic, IR and electronic) properties are satisfactory and not far from those delivered by the most reliable functionals including heavy parameterization. The way in which the functional is derived and the lack of empirical parameters fitted to specific properties make the PBE0 model a widely applicable method for both quantum chem. and condensed matter physics.
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    24. 24
      Grimme, S.; Antony, J.; Ehrlich, S.; Krieg, H. A Consistent and Accurate Ab Initio Parametrization of Density Functional Dispersion Correction (DFT-D) for the 94 Elements H-Pu. J. Chem. Phys. 2010, 132 (15), 154104,  DOI: 10.1063/1.3382344
      24
      A consistent and accurate ab initio parametrization of density functional dispersion correction (DFT-D) for the 94 elements H-Pu
      Grimme, Stefan; Antony, Jens; Ehrlich, Stephan; Krieg, Helge
      Journal of Chemical Physics (2010), 132 (15), 154104/1-154104/19CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      The method of dispersion correction as an add-on to std. Kohn-Sham d. functional theory (DFT-D) has been refined regarding higher accuracy, broader range of applicability, and less empiricism. The main new ingredients are atom-pairwise specific dispersion coeffs. and cutoff radii that are both computed from first principles. The coeffs. for new eighth-order dispersion terms are computed using established recursion relations. System (geometry) dependent information is used for the first time in a DFT-D type approach by employing the new concept of fractional coordination nos. (CN). They are used to interpolate between dispersion coeffs. of atoms in different chem. environments. The method only requires adjustment of two global parameters for each d. functional, is asymptotically exact for a gas of weakly interacting neutral atoms, and easily allows the computation of at. forces. Three-body nonadditivity terms are considered. The method has been assessed on std. benchmark sets for inter- and intramol. noncovalent interactions with a particular emphasis on a consistent description of light and heavy element systems. The mean abs. deviations for the S22 benchmark set of noncovalent interactions for 11 std. d. functionals decrease by 15%-40% compared to the previous (already accurate) DFT-D version. Spectacular improvements are found for a tripeptide-folding model and all tested metallic systems. The rectification of the long-range behavior and the use of more accurate C6 coeffs. also lead to a much better description of large (infinite) systems as shown for graphene sheets and the adsorption of benzene on an Ag(111) surface. For graphene it is found that the inclusion of three-body terms substantially (by about 10%) weakens the interlayer binding. We propose the revised DFT-D method as a general tool for the computation of the dispersion energy in mols. and solids of any kind with DFT and related (low-cost) electronic structure methods for large systems. (c) 2010 American Institute of Physics.
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    25. 25
      Pritchard, B. P.; Altarawy, D.; Didier, B.; Gibson, T. D.; Windus, T. L. New Basis Set Exchange: An Open, Up-to-Date Resource for the Molecular Sciences Community. J. Chem. Inf. Model. 2019, 59 (11), 4814– 4820,  DOI: 10.1021/acs.jcim.9b00725
      25
      New Basis Set Exchange: An Open, Up-to-Date Resource for the Molecular Sciences Community
      Pritchard, Benjamin P.; Altarawy, Doaa; Didier, Brett; Gibson, Tara D.; Windus, Theresa L.
      Journal of Chemical Information and Modeling (2019), 59 (11), 4814-4820CODEN: JCISD8; ISSN:1549-9596. (American Chemical Society)
      A review. The Basis Set Exchange (BSE) has been a prominent fixture in the quantum chem. community. First publicly available in 2007, it is recognized by both users and basis set creators as the de facto source for information related to basis sets. This popular resource has been rewritten, utilizing modern software design and best practices. The basis set data has been sepd. into a stand-alone library with an accessible API, and the Web site has been updated to use the current generation of web development libraries. The general layout and workflow of the Web site is preserved, while helpful features requested by the user community have been added. Overall, this design should increase adaptability and lend itself well into the future as a dependable resource for the computational chem. community. This article will discuss the decision to rewrite the BSE, the new architecture and design, and the new features that have been added.
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    26. 26
      Cao, X.; Dolg, M.; Stoll, H. Valence Basis Sets for Relativistic Energy-Consistent Small-Core Actinide Pseudopotentials. J. Chem. Phys. 2003, 118 (2), 487– 496,  DOI: 10.1063/1.1521431
      26
      Valence basis sets for relativistic energy-consistent small-core actinide pseudopotentials
      Cao, Xiaoyan; Dolg, Michael; Stoll, Hermann
      Journal of Chemical Physics (2003), 118 (2), 487-496CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      Gaussian (14s13p10d8f6g)/[6s6p5d4f3g] at. natural orbital valence basis sets were generated for relativistic energy-consistent small-core actinide pseudopotentials of the Stuttgart-Bonn variety. Effective valence spin-orbit operators supplementing the scalar-relativistic pseudopotentials were derived from multiconfiguration Dirac-Hartree-Fock ref. data. Pseudopotentials, basis sets and spin-orbit operators were used to det. the first and second ionization potentials of all actinide elements at the multiconfiguration SCF and multireference averaged coupled-pair functional level. Comparison was made to results obtained from large-scale calcns. using uncontracted basis sets up to i-type functions and extrapolation to the basis set limit as well as to exptl. data. Mol. calibration studies using the coupled-cluster singles, doubles, and perturbative triples approach are reported for the ground states of AcH, AcO, AcF, and ThO.
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    27. 27
      Cao, X.; Dolg, M. Segmented Contraction Scheme for Small-Core Actinide Pseudopotential Basis Sets. J. Mol. Struct. THEOCHEM 2004, 673 (1–3), 203– 209,  DOI: 10.1016/j.theochem.2003.12.015
      27
      Segmented contraction scheme for small-core actinide pseudopotential basis sets
      Cao, Xiaoyan; Dolg, Michael
      Journal of Molecular Structure: THEOCHEM (2004), 673 (1-3), 203-209CODEN: THEODJ; ISSN:0166-1280. (Elsevier Science B.V.)
      Gaussian (14s13p10d8f6g)/[10s9p5d4f3g] valence basis sets using a segmented contraction scheme have been derived for relativistic energy-consistent small-core actinide pseudopotentials of the Stuttgart-Koln variety. The present basis sets are only slightly larger than previously published (14s13p10d8f6g)/[6s6p5d4f3g] at. natural orbital basis sets, which use a generalized contraction scheme, and achieve a similar accuracy in at. and mol. calcns. For calibration purposes multi-configuration SCF and subsequent multi-ref. averaged coupled-pair functional calcns. are presented for the first to fourth ionization potentials of all actinide elements. Th Pa U Np Pu Am Cm Bk Cf Es Fm Md No Lr. In addn., results of mol. calibration studies using the coupled-cluster singles, doubles and perturbative triples approach as well as gradient-cor. d. functional theory are reported for the monohydrides, monoxides and monofluorides of actinium and lawrencium.
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    28. 28
      Küchle, W.; Dolg, M.; Stoll, H.; Preuss, H. Energy-adjusted Pseudopotentials for the Actinides. Parameter Sets and Test Calculations for Thorium and Thorium Monoxide. J. Chem. Phys. 1994, 100 (10), 7535– 7542,  DOI: 10.1063/1.466847
      28
      Energy-adjusted pseudopotentials for the actinides. Parameter sets and test calculations for thorium and thorium monoxide
      Kuechle, W.; Dolg, M.; Stoll, H.; Preuss, H.
      Journal of Chemical Physics (1994), 100 (10), 7535-42CODEN: JCPSA6; ISSN:0021-9606.
      The authors present nonrelativistic and quasirelativistic energy-adjusted pseudopotentials, the latter augmented by spin-orbit operators, as well as optimized (12s11p10d8f)/[8s7p6d4f]-Gaussian-type orbitals (GTO) valence basis sets for the actinide elements actinium through lawrencium. At. excitation and ionization energies obtained by the use of these pseudopotentials and basis sets in SCF calcns. differ by less than 0.2 eV from corresponding finite-difference all-electron results. Large-scale multiconfiguration SCF (MCSCF), multireference CI (MRCI), and multireference averaged coupled-pair functional (MRACPF) calcns. for thorium and thorium monoxide yield results in satisfactory agreement with available exptl. data. Preliminary results from spin-orbit CI calcns. for the low-lying electronic states of thorium monoxide are also reported.
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    29. 29
      Kendall, R. A.; Dunning, T. H.; Harrison, R. J. Electron Affinities of the First-row Atoms Revisited. Systematic Basis Sets and Wave Functions. J. Chem. Phys. 1992, 96 (9), 6796– 6806,  DOI: 10.1063/1.462569
      29
      Electron affinities of the first-row atoms revisited. Systematic basis sets and wave functions
      Kendall, Rick A.; Dunning, Thom H., Jr.; Harrison, Robert J.
      Journal of Chemical Physics (1992), 96 (9), 6796-806CODEN: JCPSA6; ISSN:0021-9606.
      The authors describe a reliable procedure for calcg. the electron affinity of an atom and present results for H, B, C, O, and F (H is included for completeness). This procedure involves the use of the recently proposed correlation-consistent basis sets augmented with functions to describe the more diffuse character of the at. anion coupled with a straightforward, uniform expansion of the ref. space for multireference singles and doubles configuration-interaction (MRSD-CI) calcns. A comparison is given with previous results and with corresponding full CI calcns. The most accurate EAs obtained from the MRSD-CI calcns. are (with exptl. values in parentheses): H 0.740 eV (0.754), B 0.258 (0.277), C 1.245 (1.263), O 1.384 (1.461), and F 3.337 (3.401). The EAs obtained from the MR-SDCI calcns. differ by less than 0.03 eV from those predicted by the full CI calcns.
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    30. 30
      Fdez. Galvan, I.; Vacher, M.; Alavi, A.; Angeli, C.; Aquilante, F.; Autschbach, J.; Bao, J. J.; Bokarev, S. I.; Bogdanov, N. A.; Carlson, R. K.; Chibotaru, L. F.; Creutzberg, J.; Dattani, N.; Delcey, M. G.; Dong, S. S.; Dreuw, A.; Freitag, L.; Frutos, L. M.; Gagliardi, L.; Gendron, F.; Giussani, A.; Gonzalez, L.; Grell, G.; Guo, M.; Hoyer, C. E.; Johansson, M.; Keller, S.; Knecht, S.; Kovacevic, G.; Kallman, E.; Li Manni, G.; Lundberg, M.; Ma, Y.; Mai, S.; Malhado, J. P.; Malmqvist, P. A.; Marquetand, P.; Mewes, S. A.; Norell, J.; Olivucci, M.; Oppel, M.; Phung, Q. M.; Pierloot, K.; Plasser, F.; Reiher, M.; Sand, A. M.; Schapiro, I.; Sharma, P.; Stein, C. J.; Sørensen, L. K.; Truhlar, D. G.; Ugandi, M.; Ungur, L.; Valentini, A.; Vancoillie, S.; Veryazov, V.; Weser, O.; Wesołowski, T. A.; Widmark, P.-O.; Wouters, S.; Zech, A.; Zobel, J. P.; Lindh, R. OpenMolcas: From Source Code to Insight. J. Chem. Theory Comput. 2019, 15 (11), 5925– 5964,  DOI: 10.1021/acs.jctc.9b00532
      30
      OpenMolcas: From Source Code to Insight
      Fdez. Galvan, Ignacio; Vacher, Morgane; Alavi, Ali; Angeli, Celestino; Aquilante, Francesco; Autschbach, Jochen; Bao, Jie J.; Bokarev, Sergey I.; Bogdanov, Nikolay A.; Carlson, Rebecca K.; Chibotaru, Liviu F.; Creutzberg, Joel; Dattani, Nike; Delcey, Mickael G.; Dong, Sijia S.; Dreuw, Andreas; Freitag, Leon; Frutos, Luis Manuel; Gagliardi, Laura; Gendron, Frederic; Giussani, Angelo; Gonzalez, Leticia; Grell, Gilbert; Guo, Meiyuan; Hoyer, Chad E.; Johansson, Marcus; Keller, Sebastian; Knecht, Stefan; Kovacevic, Goran; Kaellman, Erik; Li Manni, Giovanni; Lundberg, Marcus; Ma, Yingjin; Mai, Sebastian; Malhado, Joao Pedro; Malmqvist, Per Aake; Marquetand, Philipp; Mewes, Stefanie A.; Norell, Jesper; Olivucci, Massimo; Oppel, Markus; Phung, Quan Manh; Pierloot, Kristine; Plasser, Felix; Reiher, Markus; Sand, Andrew M.; Schapiro, Igor; Sharma, Prachi; Stein, Christopher J.; Soerensen, Lasse Kragh; Truhlar, Donald G.; Ugandi, Mihkel; Ungur, Liviu; Valentini, Alessio; Vancoillie, Steven; Veryazov, Valera; Weser, Oskar; Wesolowski, Tomasz A.; Widmark, Per-Olof; Wouters, Sebastian; Zech, Alexander; Zobel, J. Patrick; Lindh, Roland
      Journal of Chemical Theory and Computation (2019), 15 (11), 5925-5964CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
      In this Article we describe the OpenMolcas environment and invite the computational chem. community to collaborate. The open-source project already includes a large no. of new developments realized during the transition from the com. MOLCAS product to the open-source platform. The paper initially describes the tech. details of the new software development platform. This is followed by brief presentations of many new methods, implementations, and features of the OpenMolcas program suite. These developments include novel wave function methods such as stochastic complete active space SCF, d. matrix renormalization group (DMRG) methods, and hybrid multiconfigurational wave function and d. functional theory models. Some of these implementations include an array of addnl. options and functionalities. The paper proceeds and describes developments related to explorations of potential energy surfaces. Here we present methods for the optimization of conical intersections, the simulation of adiabatic and nonadiabatic mol. dynamics, and interfaces to tools for semiclassical and quantum mech. nuclear dynamics. Furthermore, the Article describes features unique to simulations of spectroscopic and magnetic phenomena such as the exact semiclassical description of the interaction between light and matter, various X-ray processes, magnetic CD, and properties. Finally, the paper describes a no. of built-in and add-on features to support the OpenMolcas platform with postcalcn. anal. and visualization, a multiscale simulation option using frozen-d. embedding theory, and new electronic and muonic basis sets.
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    31. 31
      Aquilante, F.; Autschbach, J.; Baiardi, A.; Battaglia, S.; Borin, V. A.; Chibotaru, L. F.; Conti, I.; De Vico, L.; Delcey, M.; Fdez; Galván, I. Modern Quantum Chemistry with [Open]Molcas. J. Chem. Phys. 2020, 152 (21), 214117,  DOI: 10.1063/5.0004835
      31
      Modern quantum chemistry with [Open]Molcas
      Aquilante, Francesco; Autschbach, Jochen; Baiardi, Alberto; Battaglia, Stefano; Borin, Veniamin A.; Chibotaru, Liviu F.; Conti, Irene; De Vico, Luca; Delcey, Mickael; Fdez. Galvan, Ignacio; Ferre, Nicolas; Freitag, Leon; Garavelli, Marco; Gong, Xuejun; Knecht, Stefan; Larsson, Ernst D.; Lindh, Roland; Lundberg, Marcus; Malmqvist, Per Ake; Nenov, Artur; Norell, Jesper; Odelius, Michael; Olivucci, Massimo; Pedersen, Thomas B.; Pedraza-Gonzalez, Laura; Phung, Quan M.; Pierloot, Kristine; Reiher, Markus; Schapiro, Igor; Segarra-Marti, Javier; Segatta, Francesco; Seijo, Luis; Sen, Saumik; Sergentu, Dumitru-Claudiu; Stein, Christopher J.; Ungur, Liviu; Vacher, Morgane; Valentini, Alessio; Veryazov, Valera
      Journal of Chemical Physics (2020), 152 (21), 214117CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      MOLCAS/OpenMolcas is an ab initio electronic structure program providing a large set of computational methods from Hartree-Fock and d. functional theory to various implementations of multiconfigurational theory. This article provides a comprehensive overview of the main features of the code, specifically reviewing the use of the code in previously reported chem. applications as well as more recent applications including the calcn. of magnetic properties from optimized d. matrix renormalization group wave functions. (c) 2020 American Institute of Physics.
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    32. 32
      Roos, B. O.; Lindh, R.; Malmqvist, P. Å.; Veryazov, V.; Widmark, P. O. Main Group Atoms and Dimers Studied with a New Relativistic ANO Basis Set. J. Phys. Chem. A 2004, 108 (15), 2851– 2858,  DOI: 10.1021/jp031064+
      32
      Main Group Atoms and Dimers Studied with a New Relativistic ANO Basis Set
      Roos, Bjoern O.; Lindh, Roland; Malmqvist, Per-Aake; Veryazov, Valera; Widmark, Per-Olof
      Journal of Physical Chemistry A (2004), 108 (15), 2851-2858CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)
      New basis sets of the at. natural orbital (ANO) type have been developed for the main group and rare gas atoms. The ANO's have been obtained from the av. d. matrix of the ground and lowest excited states of the atom, the pos. and neg. ions, and the dimer at its equil. geometry. Scalar relativistic effects are included through the use of a Douglas-Kroll Hamiltonian. Multiconfigurational wave functions have been used with dynamic correlation included using second-order perturbation theory (CASSCF/CASPT2). The basis sets are applied in calcns. of ionization energies, electron affinities, and excitation energies for all atoms and the ground-state potentials for the dimers. These calcns. include spin-orbit coupling using the RASSCF State Interaction (RASSI-SO) method. The spin-orbit splitting for the lowest at. term is reproduced with an accuracy of better than 0.05 eV, except for row 5, where it is 0.15 eV. Ionization energies and electron affinities have an accuracy better than 0.2 eV, and at. polarizabilities for the spherical atoms are computed with errors smaller than 2.5%. Computed bond energies for the dimers are accurate to better than 0.15 eV in most cases (the dimers for row 5 excluded).
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    33. 33
      Roos, B. O.; Lindh, R.; Malmqvist, P. Å.; Veryazov, V.; Widmark, P. O. New Relativistic ANO Basis Sets for Actinide Atoms. Chem. Phys. Lett. 2005, 409 (4–6), 295– 299,  DOI: 10.1016/j.cplett.2005.05.011
      33
      New relativistic ANO basis sets for actinide atoms
      Roos, Bjoern O.; Lindh, Roland; Malmqvist, Per-Aake; Veryazov, Valera; Widmark, Per-Olof
      Chemical Physics Letters (2005), 409 (4-6), 295-299CODEN: CHPLBC; ISSN:0009-2614. (Elsevier B.V.)
      New basis sets of the at. natural orbital (ANO) type have been developed for the actinide atoms Ac-Cm. The ANOs have been obtained from the av. d. matrix of the ground and lowest excited states of the atom, the pos. ions, and the atom in a elec. field. Scalar relativistic effects are included through the use of a Douglas-Kroll-Hess Hamiltonian. Multiconfigurational wave functions have been used with dynamic correlation included using second order perturbation theory (CASSCF/CASPT2). The basis sets are applied in calcns. of ionization energies and some excitation energies. Computed ionization energies have an accuracy better than 0.2 eV in most cases. The lowest multiplet levels have been computed. These calcns. include spin-orbit coupling using a variation-perturbation approach. The at. polarizability of the spherically sym. americium atom has been computed to be 116 au3.
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    34. 34
      Finley, J.; Malmqvist, P. Å.; Roos, B. O.; Serrano-Andrés, L. The Multi-State CASPT2Method. Chem. Phys. Lett. 1998, 288 (2–4), 299– 306,  DOI: 10.1016/S0009-2614(98)00252-8
      34
      The multi-state CASPT2 method
      Finley, James; Malmqvist, Per-Ake; Roos, Bjorn O.; Serrano-Andres, Luis
      Chemical Physics Letters (1998), 288 (2,3,4), 299-306CODEN: CHPLBC; ISSN:0009-2614. (Elsevier Science B.V.)
      An extension of the multiconfigurational second-order perturbation approach CASPT2 is suggested, where several electronic states are coupled at second order via an effective-Hamiltonian approach. The method has been implemented into the MOLCAS-4 program system, where it will replace the single-state CASPT2 program. The accuracy of the method is illustrated through calcns. of the ionic-neutral avoided crossing in the potential curves for LiF and of the valence-Rydberg mixing in the V-state of the ethylene mol.
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    35. 35
      Granovsky, A. A. Extended Multi-Configuration Quasi-Degenerate Perturbation Theory: The New Approach to Multi-State Multi-Reference Perturbation Theory. J. Chem. Phys. 2011, 134 (21), 214113,  DOI: 10.1063/1.3596699
      35
      Extended multi-configuration quasi-degenerate perturbation theory: The new approach to multi-state multi-reference perturbation theory
      Granovsky, Alexander A.
      Journal of Chemical Physics (2011), 134 (21), 214113/1-214113/14CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      The distinctive desirable features, both math. and phys. meaningful, for all partially contracted multi-state multi-ref. perturbation theories (MS-MR-PT) are explicitly formulated. The original approach to MS-MR-PT theory, called extended multi-configuration quasi-degenerate perturbation theory (XMCQDPT), having most, if not all, of the desirable properties is introduced. The new method is applied at the second order of perturbation theory (XMCQDPT2) to the 11A' - 21A' conical intersection in allene mol., the avoided crossing in LiF mol., and the 11A1 to 21A1 electronic transition in cis-1,3-butadiene. The new theory has several advantages compared to those of well-established approaches, such as second order multi-configuration quasi-degenerate perturbation theory and multi-state-second order complete active space perturbation theory. The anal. of the prevalent approaches to the MS-MR-PT theory performed within the framework of the XMCQDPT theory unveils the origin of their common inherent problems. We describe the efficient implementation strategy that makes XMCQDPT2 an esp. useful general-purpose tool in the high-level modeling of small to large mol. systems. (c) 2011 American Institute of Physics.
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      https://chemport.cas.org/services/resolver?origin=ACS&resolution=options&coi=1%3ACAS%3A528%3ADC%252BC3MXntV2rur4%253D&md5=7409ccdd89a7ee06e60252fe22ea7586
    36. 36
      Pierloot, K.; Van Besien, E. Electronic Structure and Spectrum of UO22+ and UO2Cl42–. J. Chem. Phys. 2005, 123 (20), 204309,  DOI: 10.1063/1.2121608
      36
      Electronic structure and spectrum of UO22+ and UO2Cl42-
      Pierloot, Kristine; van Besien, Els
      Journal of Chemical Physics (2005), 123 (20), 204309/1-204309/10CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      A theor. study is presented of the electronic spectra of the UO22+ and UO2Cl42- ions, based on multiconfigurational perturbation theory (CASSCF/CASPT2), combined with a recently developed method to treat spin-orbit coupling [P.-A. Malmqvist et al., Chem. Phys. Lett. 357, 230 (2002); B. O. Roos and P.-A. Malmqvist, Phys. Chem. Chem. Phys. 6, 2919 (2004)]. The results are compared to the exptl. spectroscopic data obtained for uranyl ions in Cs2UO2Cl4 crystals from Denning [Struct. Bonding (Berlin) 79, 215 (1992)] and to previous theor. calcns. performed using a combined configuration-interaction spin-orbit treatment [Z. Zhang and R. M. Pitzer, J. Phys. Chem. A 103, 6880 (1999); S. Matsika and R. M. Pitzer, J. Phys. Chem. A. 105, 637 (2001)]. As opposed to the latter results, the calcns. performed in this work point to a significant effect of the weakly bound equatorial chlorine ligands on the excitation energies.
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    37. 37
      Pierloot, K.; Van Besien, E.; Van Lenthe, E.; Baerends, E. J. Electronic Spectrum of UO22+ and [UO2Cl4]2– Calculated with Time-Dependent Density Functional Theory. J. Chem. Phys. 2007, 126 (19), 194311,  DOI: 10.1063/1.2735297
      37
      Electronic spectrum of UO2+2 and [UO2Cl4]2- calculated with time-dependent density functional theory
      Pierloot, Kristine; van Besien, Els; van Lenthe, Erik; Baerends, Evert Jan
      Journal of Chemical Physics (2007), 126 (19), 194311/1-194311/8CODEN: JCPSA6; ISSN:0021-9606. (American Institute of Physics)
      The electronic spectra of UO2+2 and [UO2Cl4]2- are calcd. with a recently proposed relativistic time-dependent d. functional theory method based on the two-component zeroth-order regular approxn. for the inclusion of spin-orbit coupling and a noncollinear exchange-correlation functional. All excitations out of the bonding σu+ orbital into the nonbonding δu or φu orbitals for UO2+2 and the corresponding excitations for [UO2Cl4]2- are considered. Scalar relativistic vertical excitation energies are compared to values from previous calcns. with the CASPT2 method. Two-component adiabatic excitation energies, U-O equil. distances, and sym. stretching frequencies are compared to CASPT2 and combined configuration-interaction and spin-orbit coupling results, as well as to exptl. data. The compn. of the excited states in terms of the spin-orbit free states is analyzed. The results point to a significant effect of the Cl ligands on the electronic spectrum, thereby confirming the CASPT2 results: The excitation energies are shifted and a different luminescent state is found.
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    38. 38
      Tecmer, P.; Bast, R.; Ruud, K.; Visscher, L. Charge-Transfer Excitations in Uranyl Tetrachloride ([UO 2Cl 4] 2-): How Reliable Are Electronic Spectra from Relativistic Time-Dependent Density Functional Theory?. J. Phys. Chem. A 2012, 116 (27), 7397– 7404,  DOI: 10.1021/jp3011266
      38
      Charge-Transfer Excitations in Uranyl Tetrachloride ([UO2Cl4]2-): How Reliable are Electronic Spectra from Relativistic Time-Dependent Density Functional Theory?
      Tecmer, Pawel; Bast, Radovan; Ruud, Kenneth; Visscher, Lucas
      Journal of Physical Chemistry A (2012), 116 (27), 7397-7404CODEN: JPCAFH; ISSN:1089-5639. (American Chemical Society)
      Four-component relativistic time-dependent d. functional theory (TD-DFT) is used to study charge-transfer (CT) excitation energies of the uranyl mol. as well as the uranyl tetrachloride complex. Adiabatic excitation energies and vibrational frequencies of the excited states are calcd. for the lower energy range of the spectrum. The results for TD-DFT with the CAM-B3LYP exchange-correlation functional for the [UO2Cl4]2- system are in good agreement with the exptl. obsd. spectrum of this species and agree also rather well with other theor. data. Use of the global hybrid B3LYP gives qual. correct results, while use of the BLYP functional yields results that are qual. wrong due to the too low CT states calcd. with this functional. The applicability of the overlap diagnostic of Peach et al. to identify such CT excitations is investigated for a wide range of vertical transitions using results obtained with three different approx. exchange-correlation functionals: BLYP, B3LYP, and CAM-B3LYP.
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    39. 39
      Oher, H.; Réal, F.; Vercouter, T.; Vallet, V. Investigation of the Luminescence of [UO 2 × 4 ] 2– (X = Cl, Br) Complexes in the Organic Phase Using Time-Resolved Laser-Induced Fluorescence Spectroscopy and Quantum Chemical Simulations. Inorg. Chem. 2020, 59 (9), 5896– 5906,  DOI: 10.1021/acs.inorgchem.9b03614
      39
      Investigation of the Luminescence of [UO2X4]2- (X = Cl, Br) Complexes in the Organic Phase Using Time-Resolved Laser-Induced Fluorescence Spectroscopy and Quantum Chemical Simulations
      Oher, Hanna; Real, Florent; Vercouter, Thomas; Vallet, Valerie
      Inorganic Chemistry (2020), 59 (9), 5896-5906CODEN: INOCAJ; ISSN:0020-1669. (American Chemical Society)
      The luminescence properties of the [UO2Cl4]2- complex in an org. phase, esp. the influence of large org. countercations, have been studied by time-resolved laser-induced fluorescence spectroscopy (TRLFS) and ab initio modeling. The exptl. spectrum was assigned by vibronic Franck-Condon calcns. on quantum chem. methods on the basis of a combination of relativistic d. functional approaches. The shape of the luminescence spectrum of the uranyl tetrachloride complex is detd. by sym. vibrations and geometrical change upon emission. The possible change in the luminescence properties depending on the first and second uranyl coordination spheres was predicted theor. for the [UO2Br4]2- and [R4N]2[UO2Cl4] ([R4N] = [Bu4N], [A336]) systems. The computations reveal that, for U(VI), the second coordination sphere has little influence on the spectrum shape, making speciation of uranyl complexes with identical first-coordination-sphere ligands tedious to discriminate. The computed structural changes agreed well with exptl. trends; theor. spectra and peak attributions are in good accordance with TRLFS and magnetic CD (MCD) data, resp. Luminescence spectra of [UO2Cl4]2- complexes were successfully computed by vibrationally resolved ab initio calcns. on various chem. models. By comparison with spectra recorded by time-resolved laser-induced fluorescence spectroscopy (TRLFS), we rationalize to which extent the first and second uranyl coordination spheres, as well as the solvent, influence the spectral shapes and vibrational progressions. The present theor. investigations support that TRLFS spectroscopy may not be sensitive enough to discriminate long-range interactions above the first coordination sphere of uranyl.
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    40. 40
      Li Manni, G.; Carlson, R. K.; Luo, S.; Ma, D.; Olsen, J.; Truhlar, D. G.; Gagliardi, L. Multiconfiguration Pair-Density Functional Theory. J. Chem. Theory Comput. 2014, 10 (9), 3669– 3680,  DOI: 10.1021/ct500483t
      40
      Multiconfiguration Pair-Density Functional Theory
      Li Manni, Giovanni; Carlson, Rebecca K.; Luo, Sijie; Ma, Dongxia; Olsen, Jeppe; Truhlar, Donald G.; Gagliardi, Laura
      Journal of Chemical Theory and Computation (2014), 10 (9), 3669-3680CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
      We present a new theor. framework, called Multiconfiguration Pair-D. Functional Theory (MC-PDFT), which combines multiconfigurational wave functions with a generalization of d. functional theory (DFT). A multiconfigurational self-consistent-field (MCSCF) wave function with correct spin and space symmetry is used to compute the total electronic d., its gradient, the on-top pair d., and the kinetic and Coulomb contributions to the total electronic energy. We then use a functional of the total d., its gradient, and the on-top pair d. to calc. the remaining part of the energy, which we call the on-top-d.-functional energy in contrast to the exchange-correlation energy of Kohn-Sham DFT. Because the on-top pair d. is an element of the two-particle d. matrix, this goes beyond the Hohenberg-Kohn theorem that refers only to the one-particle d. To illustrate the theory, we obtain first approxns. to the required new type of d. functionals by translating conventional d. functionals of the spin densities using a simple prescription, and we perform post-SCF d. functional calcns. using the total d., d. gradient, and on-top pair d. from the MSCSF calcns. Double counting of dynamic correlation or exchange does not occur because the MCSCF energy is not used. The theory is illustrated by applications to the bond energies and potential energy curves of H2, N2, F2, CaO, Cr2, and NiCl and the electronic excitation energies of Be, C, N, N+, O, O+, Sc+, Mn, Co, Mo, Ru, N2, HCHO, C4H6, c-C5H6, and pyrazine. The method presented has a computational cost and scaling similar to MCSCF, but a quant. accuracy, even with the present first approxns. to the new types of d. functionals, that is comparable to much more expensive multireference perturbation theory methods.
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    41. 41
      Carlson, R. K.; Truhlar, D. G.; Gagliardi, L. Multiconfiguration Pair-Density Functional Theory: A Fully Translated Gradient Approximation and Its Performance for Transition Metal Dimers and the Spectroscopy of Re 2 Cl 8 2–. J. Chem. Theory Comput. 2015, 11 (9), 4077– 4085,  DOI: 10.1021/acs.jctc.5b00609
      41
      Multiconfiguration Pair-Density Functional Theory: A Fully Translated Gradient Approximation and Its Performance for Transition Metal Dimers and the Spectroscopy of Re2Cl82-
      Carlson, Rebecca K.; Truhlar, Donald G.; Gagliardi, Laura
      Journal of Chemical Theory and Computation (2015), 11 (9), 4077-4085CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
      We extend the on-top d. functional of multiconfiguration pair-d. functional theory (MC-PDFT) to include the gradient of the on-top d. as well as the gradient of the d. We find that the theory is reasonably stable to this extension; furthermore, it provides improved accuracy for mols. contg. transition metals. We illustrate the extended on-top d. functionals by applying them to Cr2, Cu2, Ag2, Os2, and Re2Cl82- as well as to our previous database of 56 data for bond dissocn. energies, barrier heights, reaction energies, proton affinities, and the water dimer. The performance of MC-PDFT is comparable to or better than that of CASPT2.
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    42. 42
      King, D. M.; Cleaves, P. A.; Wooles, A. J.; Gardner, B. M.; Chilton, N. F.; Tuna, F.; Lewis, W.; McInnes, E. J. L.; Liddle, S. T. Molecular and Electronic Structure of Terminal and Alkali Metal-Capped Uranium(V) Nitride Complexes. Nat. Commun. 2016, 7 (1), 13773,  DOI: 10.1038/ncomms13773
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      Molecular and electronic structure of terminal and alkali metal-capped uranium(V) nitride complexes
      King, David M.; Cleaves, Peter A.; Wooles, Ashley J.; Gardner, Benedict M.; Chilton, Nicholas F.; Tuna, Floriana; Lewis, William; McInnes, Eric J. L.; Liddle, Stephen T.
      Nature Communications (2016), 7 (), 13773CODEN: NCAOBW; ISSN:2041-1723. (Nature Publishing Group)
      A review. Detg. the electronic structure of actinide complexes is intrinsically challenging because inter-electronic repulsion, crystal field, and spin-orbit coupling effects can be of similar magnitude. Moreover, such efforts have been hampered by the lack of structurally analogous families of complexes to study. Here we report an improved method to U≃N triple bonds, and assemble a family of uranium(V) nitrides. Along with an isoelectronic oxo, we quantify the electronic structure of this 5f1 family by magnetometry, optical and ESR (EPR) spectroscopies and modeling. Thus, we define the relative importance of the spin-orbit and crystal field interactions, and explain the exptl. obsd. different ground states. We find optical absorption linewidths give a potential tool to identify spin-orbit coupled states, and show measurement of UV···UV super-exchange coupling in dimers by EPR. We show that obsd. slow magnetic relaxation occurs via two-phonon processes, with no obvious correlation to the crystal field.
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    43. 43
      Reta, D.; Ortu, F.; Randall, S.; Mills, D. P.; Chilton, N. F.; Winpenny, R. E. P.; Natrajan, L.; Edwards, B.; Kaltsoyannis, N. The Performance of Density Functional Theory for the Description of Ground and Excited State Properties of Inorganic and Organometallic Uranium Compounds. J. Organomet. Chem. 2018, 857, 58– 74,  DOI: 10.1016/j.jorganchem.2017.09.021
      43
      The performance of density functional theory for the description of ground and excited state properties of inorganic and organometallic uranium compounds
      Reta, Daniel; Ortu, Fabrizio; Randall, Simon; Mills, David P.; Chilton, Nicholas F.; Winpenny, Richard E. P.; Natrajan, Louise; Edwards, Bryan; Kaltsoyannis, Nikolas
      Journal of Organometallic Chemistry (2018), 857 (), 58-74CODEN: JORCAI; ISSN:0022-328X. (Elsevier B.V.)
      Mol. uranium complexes are the most widely studied in actinide chem., and make a significant and growing contribution to inorg. and organometallic chem. However, reliable computational procedures to accurately describe the properties of such systems are not yet available. In this contribution, 18 exptl. characterized mol. uranium compds., in oxidn. states ranging from III to VI and with a variety of ligand environments, are studied computationally using d. functional theory. The computed geometries and vibrational frequencies are compared with X-ray crystallog., and infra-red and Raman spectroscopic data to establish which computational approach yields the closest agreement with expt. NMR parameters and UV-vis spectra are studied for three and five closed-shell U(VI) compds. resp. Overall, the most robust methodol. for obtaining accurate geometries is the PBE functional with Grimme's D3 dispersion corrections. For IR spectra, different approaches yield almost identical results, which makes the PBE functional with Grimme's D3 dispersion corrections the best choice. However, for Raman spectra the dependence on functional is more pronounced and no clear recommendation can be made. Similarly, for 1H and 13C NMR chem. shifts, no unequivocal recommendation emerges as to the best choice of d. functional, although for spin-spin couplings, the LC-ωPBE functional with solvent corrections is the best approach. No form of time-dependent d. functional theory can be recommended for the simulation of the electronic absorption spectra of uranyl (VI) compds.; the orbitals involved in the transitions are not calcd. correctly, and the energies are also typically unreliable. Two main approaches are adopted for the description of relativistic effects on the uranium centers: either a relativistic pseudopotential and assocd. valence basis set, or an all-electron basis set with the ZORA Hamiltonian. The former provides equal, if not better, agreement with expt. vs all-electron basis set calcns., for all properties investigated.
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    44. 44
      Hashem, E.; Swinburne, A. N.; Schulzke, C.; Evans, R. C.; Platts, J. A.; Kerridge, A.; Natrajan, L. S.; Baker, R. J. Emission Spectroscopy of Uranium(Iv) Compounds: A Combined Synthetic, Spectroscopic and Computational Study. RSC Adv. 2013, 3 (13), 4350,  DOI: 10.1039/c3ra22712j
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      Emission spectroscopy of uranium(IV) compounds. A combined synthetic, spectroscopic and computational study
      Hashem, Emtithal; Swinburne, Adam N.; Schulzke, Carola; Evans, Rachel C.; Platts, James A.; Kerridge, Andrew; Natrajan, Louise S.; Baker, Robert J.
      RSC Advances (2013), 3 (13), 4350-4361CODEN: RSCACL; ISSN:2046-2069. (Royal Society of Chemistry)
      Emission spectroscopy was used for the 1st time in a spectroscopic study of a family of U(IV) halide complexes in non-aq. media. The room temp. electronic absorption spectra of the simple coordination compds. [Li(THF)4][UX5(THF)] (X = Cl, Br, I), [Et4N]2[UCl6] and UCl4 in THF were recorded and all transitions assigned with the aid of a comprehensive computational study using CASSCF and CASPT2 techniques. Excitation into a band of f-d and LMCT character followed by energy transfer into the 5f-orbital manifold accounts for the UV-visible radiative transitions obsd. in the emission spectra, which were fully assigned as arising from transitions from the 5f16d1 electronic configuration to envelopes of states arising from the ground state 5f2 configuration. The bonding in [Li(THF)4][UCl5(THF)] was further elucidated utilizing NBO and AIM calcns. which describe the nature of the U-Cl bond as predominantly ionic with some dative covalent character and substantial overlap between the Cl 3p orbitals and 5f and 6d orbitals on uranium. These studies indicate that the emission spectral fingerprint of simple U(IV) compds. of Oh, C4v, and C2v symmetry are similar and characteristic and may be used as a diagnostic tool to assign U(IV) species in soln. and by inference, in the environment, in the presence of [UO2]2+.
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    45. 45
      Walisinghe, A. J.; Chilton, N. F. Assessment of Minimal Active Space CASSCF-SO Methods for Calculation of Atomic Slater–Condon and Spin–Orbit Coupling Parameters in d- and f-Block Ions. Dalt. Trans. 2021, 50 (40), 14130– 14138,  DOI: 10.1039/D1DT02346B
      There is no corresponding record for this reference.
    46. 46
      Bao, J. J.; Zhou, C.; Varga, Z.; Kanchanakungwankul, S.; Gagliardi, L.; Truhlar, D. G. Multi-State Pair-Density Functional Theory. Faraday Discuss. 2020, 224 (0), 348– 372,  DOI: 10.1039/D0FD00037J
      46
      Multi-state pair-density functional theory
      Bao, Jie J.; Zhou, Chen; Varga, Zoltan; Kanchanakungwankul, Siriluk; Gagliardi, Laura; Truhlar, Donald G.
      Faraday Discussions (2020), 224 (New Horizons in Density Functional Theory), 348-372CODEN: FDISE6; ISSN:1359-6640. (Royal Society of Chemistry)
      Multi-configuration pair-d. functional theory (MC-PDFT) has previously been applied successfully to carry out ground-state and excited-state calcns. Here we propose two new methods, called extended-multi-state-PDFT (XMS-PDFT) and variational-multi-state-PDFT (VMS-PDFT), that generate the intermediate states in a balanced way with a single set of orbitals. The former uses the intermediate states proposed by Granovsky for extended multi-configuration quasi-degenerate perturbation theory (XMC-QDPT); the latter obtains the intermediate states by maximizing the sum of the MC-PDFT energies for the intermediate states. We also propose a Fourier series expansion to make the variational optimizations of the VMS-PDFT method convenient, and we implement this method (FMS-PDFT) both for conventional configuration-interaction solvers and for d.-matrix-renormalization-group solvers. The new methods are tested for eight systems, exhibiting avoided crossings among two to six states. The FMS-PDFT method is successful for all cases for which it has been tested (all cases in this paper except O3 for which it was not tested), and XMS-PDFT is successful for all eight cases except the mixed-valence case. Since both XMS-PDFT and VMS-PDFT are less expensive than XMS-CASPT2, they will allow well-correlated calcns. on much larger systems for which perturbation theory is unaffordable.
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    47. 47
      Battaglia, S.; Lindh, R. Extended Dynamically Weighted CASPT2: The Best of Two Worlds. J. Chem. Theory Comput. 2020, 16 (3), 1555– 1567,  DOI: 10.1021/acs.jctc.9b01129
      47
      Extended Dynamically Weighted CASPT2: The Best of Two Worlds
      Battaglia, Stefano; Lindh, Roland
      Journal of Chemical Theory and Computation (2020), 16 (3), 1555-1567CODEN: JCTCCE; ISSN:1549-9618. (American Chemical Society)
      We introduce a new variant of the complete active space second-order perturbation theory (CASPT2) method that performs similarly to multistate CASPT2 (MS-CASPT2) in regions of the potential energy surface where the electronic states are energetically well sepd. and is akin to extended MS-CASPT2 (XMS-CASPT2) in case the underlying zeroth-order refs. are near-degenerate. Our approach follows a recipe analogous to that of XMS-CASPT2 to ensure approx. invariance under unitary transformations of the model states and a dynamic weighting scheme to smoothly interpolate the Fock operator between state-specific and state-av. regimes. The resulting extended dynamically weighted CASPT2 (XDW-CASPT2) methodol. possesses the most desirable features of both MS-CASPT2 and XMS-CASPT2, i.e., the ability to provide accurate transition energies and correctly describe avoided crossings and conical intersections. The reliability of XDW-CASPT2 is assessed on a no. of mol. systems. First, we consider the dissocn. of lithium fluoride, highlighting the distinctive characteristics of the new approach. Second, the invariance of the theory is investigated by studying the conical intersection of the distorted allene mol. Finally, the relative accuracy in the calcn. of vertical excitation energies is benchmarked on a set of 26 org. compds. We found that XDW-CASPT2, albeit being only approx. invariant, produces smooth potential energy surfaces around conical intersections and avoided crossings, performing equally well to the strictly invariant XMS-CASPT2 method. The accuracy of vertical transition energies is almost identical to MS-CASPT2, with a mean abs. deviation of 0.01-0.02 eV, in contrast to 0.12 eV for XMS-CASPT2.
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    Supporting Information

    The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.jpca.2c03314.

    • Absorption spectra of [UCl5(THF)]− (calculated and compared to experimental data) (PDF)


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