Low valency in lanthanides: A theoretical study of NdF and LuF George Schoendorff and Angela K. Wilson Citation: The Journal of Chemical Physics 140, 224314 (2014); doi: 10.1063/1.4882135 View online: http://dx.doi.org/10.1063/1.4882135 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/140/22?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Matrix isolation ESR and theoretical studies of metal phosphides J. Chem. Phys. 133, 164311 (2010); 10.1063/1.3491501 Anharmonic vibrational frequencies and vibrationally averaged structures and nuclear magnetic resonance parameters of F H F − J. Chem. Phys. 128, 214305 (2008); 10.1063/1.2933284 Dissociation of ground and n σ * states of C F 3 Cl using multireference configuration interaction with singles and doubles and with multireference average quadratic coupled cluster extensivity corrections J. Chem. Phys. 127, 164320 (2007); 10.1063/1.2800020 Characterization of singlet ground and low-lying electronic excited states of phosphaethyne and isophosphaethyne J. Chem. Phys. 125, 104306 (2006); 10.1063/1.2222356 Potential curves and spectroscopic properties for the ground state of ClO and for the ground and various excited states of ClO − J. Chem. Phys. 117, 9703 (2002); 10.1063/1.1516803
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THE JOURNAL OF CHEMICAL PHYSICS 140, 224314 (2014)
Low valency in lanthanides: A theoretical study of NdF and LuF George Schoendorff and Angela K. Wilsona) Department of Chemistry and Center for Advanced Scientific Computing and Modeling (CASCaM), University of North Texas, Denton, Texas 76203-5017, USA
(Received 9 April 2014; accepted 27 May 2014; published online 11 June 2014) The ground and low-lying excited state potential energy curves of neodymium monofluoride were calculated using multireference (CASSCF) and single reference (EOM-CR-CCSD(T)) methods. Optimized bond lengths were obtained and accurate bond dissociation energies were computed. The EOM-CR-CCSD(T) method was used to determine the bond dissociation energy of lutetium monofluoride, and it is shown that core correlation is required to produce bond dissociation energies in agreement with experiment. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4882135] INTRODUCTION
Low valency lanthanide monofluorides were studied with both multireference and single reference methods. Neodymium monofluoride was chosen as the model compound since it was expected to be a challenging species to model accurately due to the multireference character arising from the near degeneracy of the partially filled 4f, 5d, and 6s orbitals. NdF is also the lanthanide monofluoride (LnF) species with the most reliable experimental bond dissociation energy available1, 2 making it a natural choice for gauging the accuracy of computational methods. Methods that successfully model NdF were then applied to LuF. Experimental transition energies and splittings for the ground and excited states of Nd+ can provide guidance for calculations of Nd+ in molecular systems.3 Neodymium in the +1 oxidation state has a ground state electron configuration of [Xe] 4f4 6s1 corresponding to a 6 I state. Spin-orbit coupling splits this into six terms with a zero-field splitting of 5086 cm−1 . The first excited state is a 4 I state that also has an electron configuration of [Xe] 4f4 6s1 but with the 6s electron antiparallel to the spin of the 4f electrons. The 4 I state splits into four terms with a zero-field splitting of 4336 cm−1 with the lowest quartet state (4 I9/2 ) only 1650 cm−1 above the ground state. Thus, both the sextet and quartet states must be considered for NdF, especially if ionic character dominates (i.e., Nd+ F− ). In addition to the proximity of the 4 I state to the 6 I ground state of Nd, the first excited sextet state, 6 L, appears just 4438 cm−1 above the ground state. Spin-orbit coupling also splits this into six states with a zero-field splitting of 6079 cm−1 . The electron configuration of this excited sextet state is [Xe] 4f4 5d1 . Thus, any active space employed must include both the 5d and 6s orbitals along with the partially filled 4f orbitals. NdF is expected to be among the most challenging LnF species to accurately model due to the multireference character arising from the near degeneracy of the partially filled 4f, 5d, and 6s orbitals. Thus, neodymium monofluoride was a) Author to whom correspondence should be addressed. Electronic mail:
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chosen as the model compound, and methods that successfully model NdF were then applied to LuF. In contrast to NdF, LuF should be a much simpler system. Lu+ has a ground state electron configuration of [Xe] 4f14 6s2 that corresponds to a 1 S ground state. The first excited state is a triplet state 11 796 cm−1 higher in energy. It is likely then that the 5d orbitals will have a negligible role in the electronic structure of LuF. Furthermore, the large separation between the ground state and the lowest triplet state of Lu+ suggests that the ground state of LuF should be a singlet. Prior theoretical studies of both NdF and LuF have been performed with a variety of methods. An early study modeled LnF as Ln2+ F− with an electron promoted to σ *-type orbital localized primarily on the metal.4 It was determined that the orbital of σ * character results from a polarized 6s orbital, thus supporting the contention that the 6s orbital may play a role in bonding in LnF. Ligand field theory has been applied to the monofluorides to determine bond dissociation energies and ionization potentials with remarkable success,5, 6 which indicates that treating LnF as predominantly Ln+ F− is a reasonable approximation. Density functional theory (DFT) also has been applied to the monohalides both to determine suitable functionals7 and to examine the ground electronic states of the strongly multireference species (LnF where Ln = Pr – Yb).8 Dirac-Fock-Roothaan calculations have been performed on all LnF species to determine the impact of relativistic effects on the ground states.9 Fully relativistic calculations indicate a sextet ground state for NdF and a singlet ground state for LuF. However, the Nd+ in NdF is predicted to have a [Xe] 4f3 5d1 6s1 ground state whereas ligand field theory5 and CISD calculations10 using pseudopotentials predict Nd+ in NdF to have a [Xe] 4f4 6s1 ground state in agreement with an ionic model. In the present study, the ground and excited states of NdF are determined using multireference methods as well as highlevel single references methods. The ground electronic state and the bond dissociation energy are determined and compared with experiment. The computational methods that produce the most accurate bond dissociation energies for NdF are then applied to LuF to determine bond lengths and accurate bond dissociation energies.
140, 224314-1
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COMPUTATIONAL DETAILS
DISCUSSION
All multireference calculations were performed using full optimized reaction space (FORS)11–13 /complete active space self-consistent field (CASSCF)14 methods implemented within the GAMESS quantum chemistry software package.15 The Sapporo-DKH3-TZP-2012 basis set16 was used for the lanthanide (Nd) while the Sapporo-TZP-2012 basis set17 was used for fluorine. The Sapporo-2012 sets are all electron basis sets developed in a correlation consistent manner. The 2012 sets also include functions for core and valence correlation. Scalar relativistic effects were included via use of the infinite order two-component method (IOTC) of Barysz et al.18–21 Multireference calculations were performed with a 12 orbital active space that spanned the F 2pz , Nd 4f, 6s, and 5d orbitals. Two 5d orbitals with B1 and B2 symmetry in the C2v point group were omitted due to negligible occupation at all internuclear distances. Potential energy curves (PECs) for low lying states of each spin multiplicity were computed via single point energies at internuclear distances ranging from 1.3 Å to 8.0 Å. Gradient-driven geometry optimizations were performed to obtain accurate equilibrium bond lengths, and minimum energy crossing points were also computed. Spin-orbit coupling constants were computed at the CAS-CI level of theory with the full Breit-Pauli one and two-electron spin-orbit operator22, 23 in the C∞v point group. Zero-point energy corrections to the total energy were obtained from CASSCF Hessian calculations at the equilibrium bond distance for each of the computed states. Coupled cluster calculations were performed using the NWChem software suite.24, 25 The Sapporo-DKH3-TZP-2012 basis sets were used for the lanthanides (Nd and Lu)17 while the Sapporo-TZP-2012 basis set was used for fluorine. Relativistic effects were incorporated a priori using the full third order Douglas-Kroll method26–28 (DK3full as implemented in NWChem29–31 ) which includes scalar relativistic effects and energy lowering due to spin-orbit interactions via the inclusion of the (pxVp) cross product terms in the DK Hamiltonian, although the result remains J-averaged due to the neglect of the odd terms. C2v symmetry was enforced during calculation of the reference wave function as well as during the coupled cluster calculations. The EOMCR-CCSD(T) variant of coupled cluster was employed.32, 33 The equation of motion (EOM) implementation was required to compute excited states, and the completely renormalized (CR) method was chosen due to its performance for systems with strong static correlation resulting from stretched bonds and near degeneracies.34, 35 Potential energy curves were computed for the lowest state within each irreducible representation and for each spin multiplicity wherein initially only the valence electrons were correlated and then both the valence electrons and the Ln 5s and 5p subvalence electrons were correlated. Gradient-driven optimizations were performed with the valence electrons correlated, with the valence and the subvalence electrons correlated, and again with all electrons correlated. Dissociation energies were computed at an internuclear separation of 8.0 Å in all cases.
Multireference calculations were performed on the neutral NdF diatomic molecule. The initial choice for an active space was a 16 orbital active space that included the 2p orbital set centered on fluorine as well as the 4f, 5d, and 6s orbital sets centered on neodymium. An initial scan of the potential energy surface was performed with single point energies computed at internuclear distances of 1.9 Å, 2.3 Å, 2.7 Å, 3.1 Å, and 4.0 Å for both the sextet and quartet states. C2v symmetry was enforced, so single point energy calculations were performed for each irreducible representation within the point group. The lowest eight roots were computed subject to spin and symmetry constraints to ensure that the lowest energy state was found for each spin and symmetry combination. Natural orbital occupation numbers for the lowest state of each spin and symmetry were then tabulated at each of the aforementioned internuclear distances to determine if a smaller active space could be used. In all cases the fluorine 2px and 2py orbitals had an occupation number of 2.0 e− . Thus, the F 2px and 2py orbitals could be placed in the core while the 2pz orbital must remain active. The neodymium 4f orbitals had occupation numbers that ranged from 0.0 to 2.0 e− , but each 4f orbital had a non-zero occupation for at least one of the states examined. Likewise, the neodymium 6s orbital had occupation numbers ranging from 1.0 to 2.0 e− . Therefore, both the complete Nd 4f and 6s orbital sets must remain active. The 5d orbitals exhibited non-negligible occupation for only three of the five orbitals. The 5dxz and 5dyz orbitals (B1 and B2 symmetry, respectively, in C2v ) had occupations of less than 10−5 e− for the lowest two states of each irreducible representation and multiplicity, so these orbitals were excluded from the active space in subsequent calculations resulting in a truncated active space of (7,12) where the 12 active orbitals were the F 2pz , Nd 4f, Nd 6s, Nd 5dz2 , Nd 5dxy , and Nd 5dx2 -y2 . The CASSCF (7,12) potential energy curves are shown in Figure 1. The lowest two quartet and two sextet states for each irreducible representation in the C2v point group are shown. Most of these states are separated by less than 50 kcal mol−1 near the equilibrium bond lengths, but the first excited sextet states of A1 and A2 symmetry are much higher in energy, lying over 200 kcal mol−1 above the other states depicted. Some of the states dissociate to the homolytic products, Nd + F• , as would be expected in the gas phase. Homolytic dissociation occurs at an internuclear distance near 5 Å. Other states dissociate to the heterolytic products, Nd+ + F− , a process that is not stable in the gas phase, and thus the energy continues to increase even at an internuclear separation of 8.0 Å. If only the lowest state of each symmetry and multiplicity is considered, then every quartet state and one of the sextet states (6 A2 ) undergo heterolytic rather than homolytic dissociation. Figure 2 shows the lowest states for each multiplicity and symmetry near the equilibrium region (1.8–2.3 Å) and Table I shows the equilibrium bond distances, stretching frequencies, and energies relative to the lowest state for each of the states shown in Figure 2. The states in Figure 2 and Table I are in the C∞v representation, thus the 4 A1 and 6 A1 states become 4 + and 6 − , respectively, the 4 A2 and 6 A2
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FIG. 1. Potential energy curves computed at the CASSCF (7,12) level of theory with an active space that spans the F 2pz , Nd 4f, 6s, and 5d (excluding the 5d orbitals with B1 and B2 symmetry). The lowest two states for each irreducible representation for both the quartet and sextet states of NdF are shown. In all cases, states with B1 and B2 symmetry are degenerate. The lowest 4 B1 and 4 B2 states comprise a 4 state and the lowest 6 B1 and 6 B2 states comprise a 6 state in the C∞v point group. Likewise, the A1 and A2 states correspond to + and - states, respectively, in the C∞v point group.
states become 4 − and 6 − , respectively. The degenerate 4 B1 and 4 B2 states comprise the 4 state in C∞v symmetry, and the degenerate 6 B1 and 6 B2 states comprise the 6 state in C∞v symmetry. Contrary to the NdF2+ case where Nd is in the common +3 oxidation state,36 the PECs for the neutral species with Nd formally in the +1 oxidation state are not parallel. In general, the quartet state minima are generally
at slightly shorter internuclear distances than the sextet state minima. The + states tend to have the longest bond lengths of 2.132 Å and 2.133 Å for the quartet and sextet states, respectively. The 6 − and 4 − PECs are parallel with an equilibrium bond length of 2.058 Å. The 4 and 6 PECs have short bond lengths relative to the + states with the 4 state having the shortest bond length of 2.027 Å and the 6 state
FIG. 2. Potential energy curves near the equilibrium region (1.8–2.3 Å). The lowest quartet and sextet states for NdF are shown. The 4 state is predicted to be the ground state at the CASSCF (7,12) level of theory.
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TABLE I. Equilibrium bond lengths (Å), vibrational frequencies (cm−1 ), and relative energies (kcal mol−1 ) for each of the lowest quartet and sextet states of NdF. Re (Å) 4+ 4− 4 6+ 6− 6
2.132 2.058 2.027 2.133 2.058 2.067
ωe (cm−1 ) 471 524 565 472 530 523
TABLE II. Gross atomic orbital populations (GAOPs) for the lowest quartet and sextet states at the equilibrium bond distances.
Erel (kcal mol−1 )
GAOP
4+
F 2pz 1.99 ; Nd 4f3.17 F 2pz 0.97 ; Nd 4f4.98 F 2pz 1.99 ; Nd 4f3.12 F 2pz 1.99 ; Nd 4f3.99 F 2pz 1.72 ; Nd 4f3.08 F 2pz 0.98 ; Nd 4f5.02
4−
12.0 12.7 0.0 2.4 3.5 3.5
having a slightly longer bond length of 2.67 Å. All of the PECs have relatively flat potential energy wells, which is reflected in the small magnitude of the stretching frequencies. The + states have the flattest PECs with stretching frequencies of 471 cm−1 and 472 cm−1 for the 4 + and 6 + states, respectively. All other states have slightly more curved potential energy wells with stretching frequencies ranging from 523 cm−1 to 565 cm−1 . Since the stretching frequency is the only non-zero mode, the zero point energy is easily computed as one half of the frequency. Thus, the zero point energy for NdF ranges from 235 cm−1 to 283 cm−1 (0.67– 0.81 kcal mol−1 ). All the states shown in Figure 2 are relatively close in energy with a separation of no more than 12.7 kcal mol−1 . The ground state at the CASSCF (7,12) level is predicted to be the 4 state. The lowest sextet state, 6 + , is 2.4 kcal mol−1 above the quartet ground state, and the remaining sextet states are all 3.5 kcal mol−1 above the ground state. However, the 4 + and 4 − states are significantly higher in energy than the ground state at 12.0 kcal mol−1 and 12.7 kcal mol−1 , respectively. Gross atomic orbital populations (GAOPs) for the lowest quartet and sextet states of NdF are listed in Table II. In most cases, there is evidence of electron transfer from Nd to F so that the formal charges are Nd+ and F− . This occurs for the 4 + 4 , , 6 + , and 6 − states. In the case of the 4 + and 4 states, the donated electron is taken primarily from the 4f shell with 0.83 e− coming from the 4f and 0.17 e− from the 6s for the 4 + state and 0.88 e− from the 4f and 0.12 e− from the 6s for the 4 state. The 6 + state also exhibits electron donation to fluorine. In this case the electron is donated from the 6s orbital. As this is the most diffuse orbital, it is expected that the 6s orbital should be the most important Nd orbital for covalent bond formation. The 6 − orbital also exhibits some degree of electron donation to fluorine albeit only 0.72 e− . The remaining electron density from the Nd 6s orbital is transferred to the 5d. Furthermore, an additional electron is promoted from the 4f to the 5d. Of the six low-lying states presented in Table II, only the 6 − state has any appreciable electron density in the 5d shell. The remaining two states, 4 − and 6 , do not exhibit any charge transfer to fluorine so that NdF is formally neutral Nd and F• . The differences in the electronic structure arise from the electronic configuration of Nd. In both cases there is electron transfer from the 6s orbital to the 4f shell so that 5 electrons occupy the 4f shell. Spin-orbit coupling constants at the CAS CI level are shown in Table III. The coupling constants are related to the
State
4 6+ 6− 6
6s1.83 6s0.97 6s1.81 6s1.00 6s0.83 6s0.93
5d0.01 5d0.08 5d0.08 5d0.02 5d1.37 5d0.03
probability of transition between states. In the C∞v representation, transitions between the quartet and sextet states are forbidden, hence the spin-orbit coupling constants are zero. There is only week coupling among the quartet states. The 6 + state couples weakly to the excited 6 − and 6 states with coupling constants of 1 and 12 cm−1 , respectively. The only appreciable coupling occurs between the 6 − and 6 states with a coupling constant of 559 cm−1 . Zero-field splitting for the 4 states is only 18 cm−1 while the zero-field splitting for the 6 state is 582 cm−1 . As the largest splitting is less that 2 kcal mol−1 , multiplets were not resolved for the potential energy curves. While neodymium monofluoride is clearly shown to be a molecule that requires a multireference treatment, it also would be advantageous to ascertain whether there is a single reference method that can be applied with suitable accuracy. To this end, potential energy curves for NdF were also computed using EOM-CR-CCSD(T), a highly correlated single reference method.32, 33 As the coupled cluster expansion converges to the Full CI limit faster than the CI expansion, inclusion of up to perturbative triples in conjunction with a completely renormalized coupled cluster method may sufficiently capture enough of the static correlation to provide reliable energetic predictions.34, 35 Potential energy curves were computed for the lowest quartet and sextet states of each irreducible representation in the C2v point group. Potential energy curves in the equilibrium bonding region (1.7–2.4 Å) are shown in Figure 3. Initially, the EOM-CR-CCSD(T) calculations were performed using a frozen core where only the valence electron were correlated (i.e., Nd 4f, 5d, and 6s; F 2s and 2p). The PECs resulting from the frozen core calculations are shown in Figure 3(a). Upon visual inspection of the PECs, it is clear that there are deficiencies that arise from correlating only the valence electrons. Most notable is the curvature of the TABLE III. Spin-orbit coupling constants (cm−1 ) between the lowest quartet and sextet states of NdF computed at the CAS-CI level of theory.
4+ 4+ 4− 4 6+ 6− 6
0
Spin-orbit coupling constants (cm−1 ) 4− 4 6+ 1 0
9 8 18
0 0 0 0
6−
6
0 0 0 1 0
0 0 0 12 559 582
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FIG. 3. Potential energy curves in the equilibrium region (1.7–2.4 Å) for the lowest quartet and sextet states of NdF for each irreducible representation in the C2v point group obtained with the EOM-CR-CCSD(T) method correlating the valence only (a) or the valence and the subvalence (b). Correlation of the 5s and 5p subvalence orbitals (b) lowers the sextet states relative to the quartet states and smoothes the sextet potential energy curves near 1.9 Å as is evident in going from Figure 3(a) to Figure 3(b).
sextet state curves at internuclear distances of 1.9 Å and less. This behavior can be explained by examination of the radial functions for the Nd 4f, 5s, 5p, and the F 2p orbitals (Figure 4). The Nd 4f and F 2p orbitals have non-negligible overlap at an internuclear distance of 1.9 Å, so it is essential that these electrons are correlated as is the case in the frozen core EOM-CR-CCSD(T) calculations. However, the Nd 5s and 5p orbitals have a much greater penetration into the F 2p region than is the case with the Nd 4f orbitals, yet the Nd 5s and 5p electrons are not correlated in the frozen core calculations. Potential energy curves produced by EOM-CR-CCSD(T) calculations with the same reference
wave function are shown in Figure 3(b), with both the Nd 5s and 5p subvalence electrons and the valence electrons correlated. With the subvalence electrons now correlated, the sextet PECs are smooth curves throughout, even at short internuclear distances. Correlation of the Nd 5s and 5p subvalence electrons also has a sizeable impact on the ordering and relative positions of the low-lying electronic states of NdF. When only the valence electrons are correlated, the quartet states are predicted to be the lowest energy states, which is in agreement with the CASSCF (7,12) results. However, EOM-CR-CCSD(T) with a frozen core gives the 4 A2 (4 − ) state as the lowest state
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FIG. 4. Radial distribution functions for the Nd 4f, 5s, and 5p orbitals as well as the F 2p orbitals at an internuclear distance of 1.9 Å. The radial distribution functions were computed for neutral Nd with a [Xe] 4f4 6s2 configuration (i.e. the 5 I ground state), and the F 2p radial distribution function was computed for neutral F with a [He] 2s2 2p5 configuration (i.e., the 2 P ground state).
followed by the degenerate 4 B1 and 4 B2 states (4 ) and then by the 4 A1 (4 + ) state, whereas CASSCF (7,12) predicts the 4 A2 (4 − ) state to be 12.7 kcal mol−1 above the 4 B1 and 4 B2 states (4 ). The ordering of the quartet states is in agreement with CASSCF (7,12) when the Nd 5s and 5p subvalence electrons are correlated. The ordering of the sextet states also is impacted by the correlation of the Nd 5s and 5p subvalence electrons. The 6 A1 (6 + ) state is always the lowest energy sextet state, but when the subvalence electrons are correlated, all the sextet states are stabilized relative to the quartet states. This stabilization is so significant that the lowest energy state becomes the 6 A1 (6 + ) state. It is also noteworthy that EOM-CR-CCSD(T) predicts the 6 A2 , 6 B1 , and 6 B2 states to be degenerate, whereas CASSCF (7,12) breaks this degeneracy by shifting the 6 A2 minimum to a shorter bond length. The EOM-CR-CCSD(T) 6 A2 , 6 B1 , and 6 B2 states correspond to the CASSCF 6 − and 6 states, respectively. The relative positions of the equilibrium bond lengths can be seen by visual inspection of Figures 2 and 3 but are also listed in Table IV. In all cases the sextet states favor longer bond lengths than the quartet states, but this difference in bond length is much more pronounced when EOM-CR-CCSD(T) is used rather than CASSCF. Within each multiplicity, the potential energy curves become nearly parallel with the sex-
tet states, each having bond lengths ∼0.01 Å longer than the quartet states when EOM-CR-CCSD(T) is used. The number of electrons correlated has a negligible impact on the bond lengths obtained with coupled-cluster with a variation of at most 0.008 Å in the case of the 4 A2 state, but otherwise with a variation of 0.003 Å or less. Crossing points between states are shown in Table V. EOM-CR-CCSD(T) curves have crossings between the same states regardless of the number of electrons correlated, which is a result of the small variation in the positions of the equilibrium geometries. However, the positions of the crossing points vary, sometimes greatly, due to the change in the relative order of the states when more electrons are correlated, which is most easily seen for crossing points involving the 4 − state. The large variation in the computed crossing point is a result of the artificial stabilization of the 4 − when the Nd 5s and 5p orbitals are not correlated. Likewise, there is a shift in the crossing points of sextet/quartet crossings due to the increased stabilization of all the sextet states relative to the quartets when the subvalence electrons are correlated. There is less agreement between the CASSCF (7,12) and the EOM-CR-CCSD(T) results crossing points. While some of the crossing points are predicted with both methods. CASSCF predicts additional crossing points and a lack of the 6 − /4 −
TABLE IV. Equilibrium bond lengths (Å) of NdF calculated with CASSCF (7,12) and EOM-CR-CCSD(T) correlating the valence electrons only, correlating the valence plus subvalence electrons, and correlating all electrons.
CASSCF (7,12) EOM-CR-CCSD(T) Valence only EOM-CR-CCSD(T) Valence + subvalence EOM-CR-CCSD(T) all correlated
4 + (4 A ) 1
4 − (4 A ) 2
4 (4 B /4 B ) 1 2
6 + (6 A ) 1
6 − (6 A ) 2
6 (6 B /6 B ) 1 2
2.022 1.981 1.982 1.978
2.058 1.980 1.988 1.980
2.027 1.986 1.988 1.985
2.133 2.087 2.090 2.088
2.058 2.087 2.091 2.091
2.065 2.090 2.093 2.093
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TABLE V. NdF crossing points (Å) calculated with CASSCF (7,12) and EOM-CR-CCSD(T) correlating the valence electrons only and correlating the valence plus subvalence electrons. The 4 + , 4 − , 4 , 6 + , 6 − , and 6 CASSCF states correspond to the 4 A , 4 A , 4 B /4 B , 6 A , 6 A , and 1 2 1 2 1 2 6 B /6 B states, respectively, in the EOM-CR-CCSD(T) calculations with C 1 2 2v symmetry.
6 + /6 − 6 + /6 6 − /6 6 + /4 + 6 + /4 − 6 + /4 6 − /4 + 6 − /4 − 6 − /4 6 /4 + 6 /4 − 6 /4 4 + /4
CASSCF (7,12)
EOM-CR-CCSD(T) valence only
EOM-CR-CCSD(T) valence + subvalence
2.057 2.046 2.065 2.085 1.780 2.111 2.191 ... 2.243 2.214 ... 2.120 2.208
... ... ... 2.118 2.160 2.146 2.184 2.229 2.218 2.184 2.229 2.218 ...
... ... ... 1.975 1.854 2.017 2.017 1.893 2.060 2.017 1.893 2.060 ...
and 4 + /4 crossing points due to shifts in the bond lengths of the states involved. Bond dissociation energies for NdF are shown in Table VI. Only the dissociation energies for the sextet states are shown since the quartet states all dissociate to the ionic products rather than the neutral products. Moreover, there is always one sextet state that leads to the ionic products. Thus, dissociation was computed at an internuclear distance of 8.0 Å. The experimental bond dissociation energy is 130.3 ± 3.0 kcal mol−1 .1 It appears that CASSCF (7,12) achieves this with the 6 state, except that it is the 6 + state that is the ground state. The 6 + state is roughly 9 kcal mol−1 lower than the experimental bond dissociation energy, and this increases to nearly 10 kcal mol−1 from experiment if zeropoint energy is included. EOM-CR-CCSD(T) bond dissociation energies computed with a frozen core can be just as deceptive. The 6 state (6 B1 and 6 B2 ) is almost within experimental uncertainty, except once again these are excited states. The bond dissociation energy for the 6 A1 (6 + ) ground state is ∼20 kcal mol−1 higher than the experimental dissociation energy. The situation greatly improves when the Nd 5s and 5p subvalence electrons are correlated as well as the va-
lence electrons. When the Nd 5s and 5p subvalence electrons are correlated, the 6 A1 (6 + ) ground state bond dissociation energy is within experimental uncertainty, and even the excited 6 state (6 B1 and 6 B2 ) has bond dissociation energies are within experimental uncertainty. But as zero-point energy is not included in the computed dissociation energies, these all fall short of the experimental value by a further 0.6–0.8 kcal mol−1 . However, highly accurate bond dissociation energies are achieved best when all electrons are correlated using the EOM-CR-CCSD(T) method. Correlation of all electrons increases the separation between the 6 A1 (6 + ) ground state and the excited states and results in a bond dissociation energy of 132.6 kcal mol−1 for the 6 A1 ground state, a value that is well within experimental uncertainty and that is in even better agreement with experiment if zero-point energy is included as well. Given the exceptional performance of the EOM-CRCCSD(T) method with NdF, this method was also applied to the LuF molecule. In principle, LuF is presumed to be a simpler molecule to study given the fact that it is a closed shell singlet. The equilibrium bond length was computed with EOM-CR-CCSD(T) with the valence electrons correlated as well as with the subvalence 5s and 5p electrons correlated. The 1 A1 state (corresponding to the 1 + ground state) has a computed bond length of 1.917 Å. This is fortuitously in exact agreement with the experimental bond length.37 The bond lengths for the lowest states of the other irreducible representations ranged from 1.950–1.967 Å. However, these states were neglected for the remainder of the study as they are at least 89.1 kcal mol−1 above the ground state (or 62.6 kcal mol−1 above the ground state if all electrons are correlated). The large separation of the ground state from the excited states is in rough agreement with experiment, though the first excited state was not computed as it is shown experimentally that both states have the same symmetry, i.e., 1 + state with 1 A1 symmetry.37 Moreover, the elongated excited state bond lengths are in agreement with experiment to within 0.01 Å.38, 39 The experimental gas phase bond dissociation energy of LuF is 135 ± 10 kcal mol−1 .2 However, the bond dissociation energy computed with EOM-CR-CCSD(T) is 171.3 kcal mol−1 , which is not near the experimental result even considering the large experimental uncertainty. However, if all electrons are correlated, then the computed bond dissociation energy is 139.6 kcal mol−1 , which is well within the experimental uncertainty. Thus, while the EOM-CR-CCSD(T)
TABLE VI. Bond dissociation energies (kcal mol−1 ) of NdF computed at the CASSCF (7,12) level as well as with EOM-CR-CCSD(T) with only the valence electrons correlated, the valence and subvalence electrons correlated, and all electrons correlated. Bond dissociation energies are computed at an internuclear distance on 8.0 Å, and the bond dissociation energies do not include zero-point energy (
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THE JOURNAL OF CHEMICAL PHYSICS 140, 224314 (2014)
Low valency in lanthanides: A theoretical study of NdF and LuF George Schoendorff and Angela K. Wilsona) Department of Chemistry and Center for Advanced Scientific Computing and Modeling (CASCaM), University of North Texas, Denton, Texas 76203-5017, USA
(Received 9 April 2014; accepted 27 May 2014; published online 11 June 2014) The ground and low-lying excited state potential energy curves of neodymium monofluoride were calculated using multireference (CASSCF) and single reference (EOM-CR-CCSD(T)) methods. Optimized bond lengths were obtained and accurate bond dissociation energies were computed. The EOM-CR-CCSD(T) method was used to determine the bond dissociation energy of lutetium monofluoride, and it is shown that core correlation is required to produce bond dissociation energies in agreement with experiment. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4882135] INTRODUCTION
Low valency lanthanide monofluorides were studied with both multireference and single reference methods. Neodymium monofluoride was chosen as the model compound since it was expected to be a challenging species to model accurately due to the multireference character arising from the near degeneracy of the partially filled 4f, 5d, and 6s orbitals. NdF is also the lanthanide monofluoride (LnF) species with the most reliable experimental bond dissociation energy available1, 2 making it a natural choice for gauging the accuracy of computational methods. Methods that successfully model NdF were then applied to LuF. Experimental transition energies and splittings for the ground and excited states of Nd+ can provide guidance for calculations of Nd+ in molecular systems.3 Neodymium in the +1 oxidation state has a ground state electron configuration of [Xe] 4f4 6s1 corresponding to a 6 I state. Spin-orbit coupling splits this into six terms with a zero-field splitting of 5086 cm−1 . The first excited state is a 4 I state that also has an electron configuration of [Xe] 4f4 6s1 but with the 6s electron antiparallel to the spin of the 4f electrons. The 4 I state splits into four terms with a zero-field splitting of 4336 cm−1 with the lowest quartet state (4 I9/2 ) only 1650 cm−1 above the ground state. Thus, both the sextet and quartet states must be considered for NdF, especially if ionic character dominates (i.e., Nd+ F− ). In addition to the proximity of the 4 I state to the 6 I ground state of Nd, the first excited sextet state, 6 L, appears just 4438 cm−1 above the ground state. Spin-orbit coupling also splits this into six states with a zero-field splitting of 6079 cm−1 . The electron configuration of this excited sextet state is [Xe] 4f4 5d1 . Thus, any active space employed must include both the 5d and 6s orbitals along with the partially filled 4f orbitals. NdF is expected to be among the most challenging LnF species to accurately model due to the multireference character arising from the near degeneracy of the partially filled 4f, 5d, and 6s orbitals. Thus, neodymium monofluoride was a) Author to whom correspondence should be addressed. Electronic mail:
[email protected]
0021-9606/2014/140(22)/224314/8/$30.00
chosen as the model compound, and methods that successfully model NdF were then applied to LuF. In contrast to NdF, LuF should be a much simpler system. Lu+ has a ground state electron configuration of [Xe] 4f14 6s2 that corresponds to a 1 S ground state. The first excited state is a triplet state 11 796 cm−1 higher in energy. It is likely then that the 5d orbitals will have a negligible role in the electronic structure of LuF. Furthermore, the large separation between the ground state and the lowest triplet state of Lu+ suggests that the ground state of LuF should be a singlet. Prior theoretical studies of both NdF and LuF have been performed with a variety of methods. An early study modeled LnF as Ln2+ F− with an electron promoted to σ *-type orbital localized primarily on the metal.4 It was determined that the orbital of σ * character results from a polarized 6s orbital, thus supporting the contention that the 6s orbital may play a role in bonding in LnF. Ligand field theory has been applied to the monofluorides to determine bond dissociation energies and ionization potentials with remarkable success,5, 6 which indicates that treating LnF as predominantly Ln+ F− is a reasonable approximation. Density functional theory (DFT) also has been applied to the monohalides both to determine suitable functionals7 and to examine the ground electronic states of the strongly multireference species (LnF where Ln = Pr – Yb).8 Dirac-Fock-Roothaan calculations have been performed on all LnF species to determine the impact of relativistic effects on the ground states.9 Fully relativistic calculations indicate a sextet ground state for NdF and a singlet ground state for LuF. However, the Nd+ in NdF is predicted to have a [Xe] 4f3 5d1 6s1 ground state whereas ligand field theory5 and CISD calculations10 using pseudopotentials predict Nd+ in NdF to have a [Xe] 4f4 6s1 ground state in agreement with an ionic model. In the present study, the ground and excited states of NdF are determined using multireference methods as well as highlevel single references methods. The ground electronic state and the bond dissociation energy are determined and compared with experiment. The computational methods that produce the most accurate bond dissociation energies for NdF are then applied to LuF to determine bond lengths and accurate bond dissociation energies.
140, 224314-1
© 2014 AIP Publishing LLC
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COMPUTATIONAL DETAILS
DISCUSSION
All multireference calculations were performed using full optimized reaction space (FORS)11–13 /complete active space self-consistent field (CASSCF)14 methods implemented within the GAMESS quantum chemistry software package.15 The Sapporo-DKH3-TZP-2012 basis set16 was used for the lanthanide (Nd) while the Sapporo-TZP-2012 basis set17 was used for fluorine. The Sapporo-2012 sets are all electron basis sets developed in a correlation consistent manner. The 2012 sets also include functions for core and valence correlation. Scalar relativistic effects were included via use of the infinite order two-component method (IOTC) of Barysz et al.18–21 Multireference calculations were performed with a 12 orbital active space that spanned the F 2pz , Nd 4f, 6s, and 5d orbitals. Two 5d orbitals with B1 and B2 symmetry in the C2v point group were omitted due to negligible occupation at all internuclear distances. Potential energy curves (PECs) for low lying states of each spin multiplicity were computed via single point energies at internuclear distances ranging from 1.3 Å to 8.0 Å. Gradient-driven geometry optimizations were performed to obtain accurate equilibrium bond lengths, and minimum energy crossing points were also computed. Spin-orbit coupling constants were computed at the CAS-CI level of theory with the full Breit-Pauli one and two-electron spin-orbit operator22, 23 in the C∞v point group. Zero-point energy corrections to the total energy were obtained from CASSCF Hessian calculations at the equilibrium bond distance for each of the computed states. Coupled cluster calculations were performed using the NWChem software suite.24, 25 The Sapporo-DKH3-TZP-2012 basis sets were used for the lanthanides (Nd and Lu)17 while the Sapporo-TZP-2012 basis set was used for fluorine. Relativistic effects were incorporated a priori using the full third order Douglas-Kroll method26–28 (DK3full as implemented in NWChem29–31 ) which includes scalar relativistic effects and energy lowering due to spin-orbit interactions via the inclusion of the (pxVp) cross product terms in the DK Hamiltonian, although the result remains J-averaged due to the neglect of the odd terms. C2v symmetry was enforced during calculation of the reference wave function as well as during the coupled cluster calculations. The EOMCR-CCSD(T) variant of coupled cluster was employed.32, 33 The equation of motion (EOM) implementation was required to compute excited states, and the completely renormalized (CR) method was chosen due to its performance for systems with strong static correlation resulting from stretched bonds and near degeneracies.34, 35 Potential energy curves were computed for the lowest state within each irreducible representation and for each spin multiplicity wherein initially only the valence electrons were correlated and then both the valence electrons and the Ln 5s and 5p subvalence electrons were correlated. Gradient-driven optimizations were performed with the valence electrons correlated, with the valence and the subvalence electrons correlated, and again with all electrons correlated. Dissociation energies were computed at an internuclear separation of 8.0 Å in all cases.
Multireference calculations were performed on the neutral NdF diatomic molecule. The initial choice for an active space was a 16 orbital active space that included the 2p orbital set centered on fluorine as well as the 4f, 5d, and 6s orbital sets centered on neodymium. An initial scan of the potential energy surface was performed with single point energies computed at internuclear distances of 1.9 Å, 2.3 Å, 2.7 Å, 3.1 Å, and 4.0 Å for both the sextet and quartet states. C2v symmetry was enforced, so single point energy calculations were performed for each irreducible representation within the point group. The lowest eight roots were computed subject to spin and symmetry constraints to ensure that the lowest energy state was found for each spin and symmetry combination. Natural orbital occupation numbers for the lowest state of each spin and symmetry were then tabulated at each of the aforementioned internuclear distances to determine if a smaller active space could be used. In all cases the fluorine 2px and 2py orbitals had an occupation number of 2.0 e− . Thus, the F 2px and 2py orbitals could be placed in the core while the 2pz orbital must remain active. The neodymium 4f orbitals had occupation numbers that ranged from 0.0 to 2.0 e− , but each 4f orbital had a non-zero occupation for at least one of the states examined. Likewise, the neodymium 6s orbital had occupation numbers ranging from 1.0 to 2.0 e− . Therefore, both the complete Nd 4f and 6s orbital sets must remain active. The 5d orbitals exhibited non-negligible occupation for only three of the five orbitals. The 5dxz and 5dyz orbitals (B1 and B2 symmetry, respectively, in C2v ) had occupations of less than 10−5 e− for the lowest two states of each irreducible representation and multiplicity, so these orbitals were excluded from the active space in subsequent calculations resulting in a truncated active space of (7,12) where the 12 active orbitals were the F 2pz , Nd 4f, Nd 6s, Nd 5dz2 , Nd 5dxy , and Nd 5dx2 -y2 . The CASSCF (7,12) potential energy curves are shown in Figure 1. The lowest two quartet and two sextet states for each irreducible representation in the C2v point group are shown. Most of these states are separated by less than 50 kcal mol−1 near the equilibrium bond lengths, but the first excited sextet states of A1 and A2 symmetry are much higher in energy, lying over 200 kcal mol−1 above the other states depicted. Some of the states dissociate to the homolytic products, Nd + F• , as would be expected in the gas phase. Homolytic dissociation occurs at an internuclear distance near 5 Å. Other states dissociate to the heterolytic products, Nd+ + F− , a process that is not stable in the gas phase, and thus the energy continues to increase even at an internuclear separation of 8.0 Å. If only the lowest state of each symmetry and multiplicity is considered, then every quartet state and one of the sextet states (6 A2 ) undergo heterolytic rather than homolytic dissociation. Figure 2 shows the lowest states for each multiplicity and symmetry near the equilibrium region (1.8–2.3 Å) and Table I shows the equilibrium bond distances, stretching frequencies, and energies relative to the lowest state for each of the states shown in Figure 2. The states in Figure 2 and Table I are in the C∞v representation, thus the 4 A1 and 6 A1 states become 4 + and 6 − , respectively, the 4 A2 and 6 A2
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J. Chem. Phys. 140, 224314 (2014)
FIG. 1. Potential energy curves computed at the CASSCF (7,12) level of theory with an active space that spans the F 2pz , Nd 4f, 6s, and 5d (excluding the 5d orbitals with B1 and B2 symmetry). The lowest two states for each irreducible representation for both the quartet and sextet states of NdF are shown. In all cases, states with B1 and B2 symmetry are degenerate. The lowest 4 B1 and 4 B2 states comprise a 4 state and the lowest 6 B1 and 6 B2 states comprise a 6 state in the C∞v point group. Likewise, the A1 and A2 states correspond to + and - states, respectively, in the C∞v point group.
states become 4 − and 6 − , respectively. The degenerate 4 B1 and 4 B2 states comprise the 4 state in C∞v symmetry, and the degenerate 6 B1 and 6 B2 states comprise the 6 state in C∞v symmetry. Contrary to the NdF2+ case where Nd is in the common +3 oxidation state,36 the PECs for the neutral species with Nd formally in the +1 oxidation state are not parallel. In general, the quartet state minima are generally
at slightly shorter internuclear distances than the sextet state minima. The + states tend to have the longest bond lengths of 2.132 Å and 2.133 Å for the quartet and sextet states, respectively. The 6 − and 4 − PECs are parallel with an equilibrium bond length of 2.058 Å. The 4 and 6 PECs have short bond lengths relative to the + states with the 4 state having the shortest bond length of 2.027 Å and the 6 state
FIG. 2. Potential energy curves near the equilibrium region (1.8–2.3 Å). The lowest quartet and sextet states for NdF are shown. The 4 state is predicted to be the ground state at the CASSCF (7,12) level of theory.
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TABLE I. Equilibrium bond lengths (Å), vibrational frequencies (cm−1 ), and relative energies (kcal mol−1 ) for each of the lowest quartet and sextet states of NdF. Re (Å) 4+ 4− 4 6+ 6− 6
2.132 2.058 2.027 2.133 2.058 2.067
ωe (cm−1 ) 471 524 565 472 530 523
TABLE II. Gross atomic orbital populations (GAOPs) for the lowest quartet and sextet states at the equilibrium bond distances.
Erel (kcal mol−1 )
GAOP
4+
F 2pz 1.99 ; Nd 4f3.17 F 2pz 0.97 ; Nd 4f4.98 F 2pz 1.99 ; Nd 4f3.12 F 2pz 1.99 ; Nd 4f3.99 F 2pz 1.72 ; Nd 4f3.08 F 2pz 0.98 ; Nd 4f5.02
4−
12.0 12.7 0.0 2.4 3.5 3.5
having a slightly longer bond length of 2.67 Å. All of the PECs have relatively flat potential energy wells, which is reflected in the small magnitude of the stretching frequencies. The + states have the flattest PECs with stretching frequencies of 471 cm−1 and 472 cm−1 for the 4 + and 6 + states, respectively. All other states have slightly more curved potential energy wells with stretching frequencies ranging from 523 cm−1 to 565 cm−1 . Since the stretching frequency is the only non-zero mode, the zero point energy is easily computed as one half of the frequency. Thus, the zero point energy for NdF ranges from 235 cm−1 to 283 cm−1 (0.67– 0.81 kcal mol−1 ). All the states shown in Figure 2 are relatively close in energy with a separation of no more than 12.7 kcal mol−1 . The ground state at the CASSCF (7,12) level is predicted to be the 4 state. The lowest sextet state, 6 + , is 2.4 kcal mol−1 above the quartet ground state, and the remaining sextet states are all 3.5 kcal mol−1 above the ground state. However, the 4 + and 4 − states are significantly higher in energy than the ground state at 12.0 kcal mol−1 and 12.7 kcal mol−1 , respectively. Gross atomic orbital populations (GAOPs) for the lowest quartet and sextet states of NdF are listed in Table II. In most cases, there is evidence of electron transfer from Nd to F so that the formal charges are Nd+ and F− . This occurs for the 4 + 4 , , 6 + , and 6 − states. In the case of the 4 + and 4 states, the donated electron is taken primarily from the 4f shell with 0.83 e− coming from the 4f and 0.17 e− from the 6s for the 4 + state and 0.88 e− from the 4f and 0.12 e− from the 6s for the 4 state. The 6 + state also exhibits electron donation to fluorine. In this case the electron is donated from the 6s orbital. As this is the most diffuse orbital, it is expected that the 6s orbital should be the most important Nd orbital for covalent bond formation. The 6 − orbital also exhibits some degree of electron donation to fluorine albeit only 0.72 e− . The remaining electron density from the Nd 6s orbital is transferred to the 5d. Furthermore, an additional electron is promoted from the 4f to the 5d. Of the six low-lying states presented in Table II, only the 6 − state has any appreciable electron density in the 5d shell. The remaining two states, 4 − and 6 , do not exhibit any charge transfer to fluorine so that NdF is formally neutral Nd and F• . The differences in the electronic structure arise from the electronic configuration of Nd. In both cases there is electron transfer from the 6s orbital to the 4f shell so that 5 electrons occupy the 4f shell. Spin-orbit coupling constants at the CAS CI level are shown in Table III. The coupling constants are related to the
State
4 6+ 6− 6
6s1.83 6s0.97 6s1.81 6s1.00 6s0.83 6s0.93
5d0.01 5d0.08 5d0.08 5d0.02 5d1.37 5d0.03
probability of transition between states. In the C∞v representation, transitions between the quartet and sextet states are forbidden, hence the spin-orbit coupling constants are zero. There is only week coupling among the quartet states. The 6 + state couples weakly to the excited 6 − and 6 states with coupling constants of 1 and 12 cm−1 , respectively. The only appreciable coupling occurs between the 6 − and 6 states with a coupling constant of 559 cm−1 . Zero-field splitting for the 4 states is only 18 cm−1 while the zero-field splitting for the 6 state is 582 cm−1 . As the largest splitting is less that 2 kcal mol−1 , multiplets were not resolved for the potential energy curves. While neodymium monofluoride is clearly shown to be a molecule that requires a multireference treatment, it also would be advantageous to ascertain whether there is a single reference method that can be applied with suitable accuracy. To this end, potential energy curves for NdF were also computed using EOM-CR-CCSD(T), a highly correlated single reference method.32, 33 As the coupled cluster expansion converges to the Full CI limit faster than the CI expansion, inclusion of up to perturbative triples in conjunction with a completely renormalized coupled cluster method may sufficiently capture enough of the static correlation to provide reliable energetic predictions.34, 35 Potential energy curves were computed for the lowest quartet and sextet states of each irreducible representation in the C2v point group. Potential energy curves in the equilibrium bonding region (1.7–2.4 Å) are shown in Figure 3. Initially, the EOM-CR-CCSD(T) calculations were performed using a frozen core where only the valence electron were correlated (i.e., Nd 4f, 5d, and 6s; F 2s and 2p). The PECs resulting from the frozen core calculations are shown in Figure 3(a). Upon visual inspection of the PECs, it is clear that there are deficiencies that arise from correlating only the valence electrons. Most notable is the curvature of the TABLE III. Spin-orbit coupling constants (cm−1 ) between the lowest quartet and sextet states of NdF computed at the CAS-CI level of theory.
4+ 4+ 4− 4 6+ 6− 6
0
Spin-orbit coupling constants (cm−1 ) 4− 4 6+ 1 0
9 8 18
0 0 0 0
6−
6
0 0 0 1 0
0 0 0 12 559 582
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FIG. 3. Potential energy curves in the equilibrium region (1.7–2.4 Å) for the lowest quartet and sextet states of NdF for each irreducible representation in the C2v point group obtained with the EOM-CR-CCSD(T) method correlating the valence only (a) or the valence and the subvalence (b). Correlation of the 5s and 5p subvalence orbitals (b) lowers the sextet states relative to the quartet states and smoothes the sextet potential energy curves near 1.9 Å as is evident in going from Figure 3(a) to Figure 3(b).
sextet state curves at internuclear distances of 1.9 Å and less. This behavior can be explained by examination of the radial functions for the Nd 4f, 5s, 5p, and the F 2p orbitals (Figure 4). The Nd 4f and F 2p orbitals have non-negligible overlap at an internuclear distance of 1.9 Å, so it is essential that these electrons are correlated as is the case in the frozen core EOM-CR-CCSD(T) calculations. However, the Nd 5s and 5p orbitals have a much greater penetration into the F 2p region than is the case with the Nd 4f orbitals, yet the Nd 5s and 5p electrons are not correlated in the frozen core calculations. Potential energy curves produced by EOM-CR-CCSD(T) calculations with the same reference
wave function are shown in Figure 3(b), with both the Nd 5s and 5p subvalence electrons and the valence electrons correlated. With the subvalence electrons now correlated, the sextet PECs are smooth curves throughout, even at short internuclear distances. Correlation of the Nd 5s and 5p subvalence electrons also has a sizeable impact on the ordering and relative positions of the low-lying electronic states of NdF. When only the valence electrons are correlated, the quartet states are predicted to be the lowest energy states, which is in agreement with the CASSCF (7,12) results. However, EOM-CR-CCSD(T) with a frozen core gives the 4 A2 (4 − ) state as the lowest state
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J. Chem. Phys. 140, 224314 (2014)
FIG. 4. Radial distribution functions for the Nd 4f, 5s, and 5p orbitals as well as the F 2p orbitals at an internuclear distance of 1.9 Å. The radial distribution functions were computed for neutral Nd with a [Xe] 4f4 6s2 configuration (i.e. the 5 I ground state), and the F 2p radial distribution function was computed for neutral F with a [He] 2s2 2p5 configuration (i.e., the 2 P ground state).
followed by the degenerate 4 B1 and 4 B2 states (4 ) and then by the 4 A1 (4 + ) state, whereas CASSCF (7,12) predicts the 4 A2 (4 − ) state to be 12.7 kcal mol−1 above the 4 B1 and 4 B2 states (4 ). The ordering of the quartet states is in agreement with CASSCF (7,12) when the Nd 5s and 5p subvalence electrons are correlated. The ordering of the sextet states also is impacted by the correlation of the Nd 5s and 5p subvalence electrons. The 6 A1 (6 + ) state is always the lowest energy sextet state, but when the subvalence electrons are correlated, all the sextet states are stabilized relative to the quartet states. This stabilization is so significant that the lowest energy state becomes the 6 A1 (6 + ) state. It is also noteworthy that EOM-CR-CCSD(T) predicts the 6 A2 , 6 B1 , and 6 B2 states to be degenerate, whereas CASSCF (7,12) breaks this degeneracy by shifting the 6 A2 minimum to a shorter bond length. The EOM-CR-CCSD(T) 6 A2 , 6 B1 , and 6 B2 states correspond to the CASSCF 6 − and 6 states, respectively. The relative positions of the equilibrium bond lengths can be seen by visual inspection of Figures 2 and 3 but are also listed in Table IV. In all cases the sextet states favor longer bond lengths than the quartet states, but this difference in bond length is much more pronounced when EOM-CR-CCSD(T) is used rather than CASSCF. Within each multiplicity, the potential energy curves become nearly parallel with the sex-
tet states, each having bond lengths ∼0.01 Å longer than the quartet states when EOM-CR-CCSD(T) is used. The number of electrons correlated has a negligible impact on the bond lengths obtained with coupled-cluster with a variation of at most 0.008 Å in the case of the 4 A2 state, but otherwise with a variation of 0.003 Å or less. Crossing points between states are shown in Table V. EOM-CR-CCSD(T) curves have crossings between the same states regardless of the number of electrons correlated, which is a result of the small variation in the positions of the equilibrium geometries. However, the positions of the crossing points vary, sometimes greatly, due to the change in the relative order of the states when more electrons are correlated, which is most easily seen for crossing points involving the 4 − state. The large variation in the computed crossing point is a result of the artificial stabilization of the 4 − when the Nd 5s and 5p orbitals are not correlated. Likewise, there is a shift in the crossing points of sextet/quartet crossings due to the increased stabilization of all the sextet states relative to the quartets when the subvalence electrons are correlated. There is less agreement between the CASSCF (7,12) and the EOM-CR-CCSD(T) results crossing points. While some of the crossing points are predicted with both methods. CASSCF predicts additional crossing points and a lack of the 6 − /4 −
TABLE IV. Equilibrium bond lengths (Å) of NdF calculated with CASSCF (7,12) and EOM-CR-CCSD(T) correlating the valence electrons only, correlating the valence plus subvalence electrons, and correlating all electrons.
CASSCF (7,12) EOM-CR-CCSD(T) Valence only EOM-CR-CCSD(T) Valence + subvalence EOM-CR-CCSD(T) all correlated
4 + (4 A ) 1
4 − (4 A ) 2
4 (4 B /4 B ) 1 2
6 + (6 A ) 1
6 − (6 A ) 2
6 (6 B /6 B ) 1 2
2.022 1.981 1.982 1.978
2.058 1.980 1.988 1.980
2.027 1.986 1.988 1.985
2.133 2.087 2.090 2.088
2.058 2.087 2.091 2.091
2.065 2.090 2.093 2.093
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G. Schoendorff and A. K. Wilson
J. Chem. Phys. 140, 224314 (2014)
TABLE V. NdF crossing points (Å) calculated with CASSCF (7,12) and EOM-CR-CCSD(T) correlating the valence electrons only and correlating the valence plus subvalence electrons. The 4 + , 4 − , 4 , 6 + , 6 − , and 6 CASSCF states correspond to the 4 A , 4 A , 4 B /4 B , 6 A , 6 A , and 1 2 1 2 1 2 6 B /6 B states, respectively, in the EOM-CR-CCSD(T) calculations with C 1 2 2v symmetry.
6 + /6 − 6 + /6 6 − /6 6 + /4 + 6 + /4 − 6 + /4 6 − /4 + 6 − /4 − 6 − /4 6 /4 + 6 /4 − 6 /4 4 + /4
CASSCF (7,12)
EOM-CR-CCSD(T) valence only
EOM-CR-CCSD(T) valence + subvalence
2.057 2.046 2.065 2.085 1.780 2.111 2.191 ... 2.243 2.214 ... 2.120 2.208
... ... ... 2.118 2.160 2.146 2.184 2.229 2.218 2.184 2.229 2.218 ...
... ... ... 1.975 1.854 2.017 2.017 1.893 2.060 2.017 1.893 2.060 ...
and 4 + /4 crossing points due to shifts in the bond lengths of the states involved. Bond dissociation energies for NdF are shown in Table VI. Only the dissociation energies for the sextet states are shown since the quartet states all dissociate to the ionic products rather than the neutral products. Moreover, there is always one sextet state that leads to the ionic products. Thus, dissociation was computed at an internuclear distance of 8.0 Å. The experimental bond dissociation energy is 130.3 ± 3.0 kcal mol−1 .1 It appears that CASSCF (7,12) achieves this with the 6 state, except that it is the 6 + state that is the ground state. The 6 + state is roughly 9 kcal mol−1 lower than the experimental bond dissociation energy, and this increases to nearly 10 kcal mol−1 from experiment if zeropoint energy is included. EOM-CR-CCSD(T) bond dissociation energies computed with a frozen core can be just as deceptive. The 6 state (6 B1 and 6 B2 ) is almost within experimental uncertainty, except once again these are excited states. The bond dissociation energy for the 6 A1 (6 + ) ground state is ∼20 kcal mol−1 higher than the experimental dissociation energy. The situation greatly improves when the Nd 5s and 5p subvalence electrons are correlated as well as the va-
lence electrons. When the Nd 5s and 5p subvalence electrons are correlated, the 6 A1 (6 + ) ground state bond dissociation energy is within experimental uncertainty, and even the excited 6 state (6 B1 and 6 B2 ) has bond dissociation energies are within experimental uncertainty. But as zero-point energy is not included in the computed dissociation energies, these all fall short of the experimental value by a further 0.6–0.8 kcal mol−1 . However, highly accurate bond dissociation energies are achieved best when all electrons are correlated using the EOM-CR-CCSD(T) method. Correlation of all electrons increases the separation between the 6 A1 (6 + ) ground state and the excited states and results in a bond dissociation energy of 132.6 kcal mol−1 for the 6 A1 ground state, a value that is well within experimental uncertainty and that is in even better agreement with experiment if zero-point energy is included as well. Given the exceptional performance of the EOM-CRCCSD(T) method with NdF, this method was also applied to the LuF molecule. In principle, LuF is presumed to be a simpler molecule to study given the fact that it is a closed shell singlet. The equilibrium bond length was computed with EOM-CR-CCSD(T) with the valence electrons correlated as well as with the subvalence 5s and 5p electrons correlated. The 1 A1 state (corresponding to the 1 + ground state) has a computed bond length of 1.917 Å. This is fortuitously in exact agreement with the experimental bond length.37 The bond lengths for the lowest states of the other irreducible representations ranged from 1.950–1.967 Å. However, these states were neglected for the remainder of the study as they are at least 89.1 kcal mol−1 above the ground state (or 62.6 kcal mol−1 above the ground state if all electrons are correlated). The large separation of the ground state from the excited states is in rough agreement with experiment, though the first excited state was not computed as it is shown experimentally that both states have the same symmetry, i.e., 1 + state with 1 A1 symmetry.37 Moreover, the elongated excited state bond lengths are in agreement with experiment to within 0.01 Å.38, 39 The experimental gas phase bond dissociation energy of LuF is 135 ± 10 kcal mol−1 .2 However, the bond dissociation energy computed with EOM-CR-CCSD(T) is 171.3 kcal mol−1 , which is not near the experimental result even considering the large experimental uncertainty. However, if all electrons are correlated, then the computed bond dissociation energy is 139.6 kcal mol−1 , which is well within the experimental uncertainty. Thus, while the EOM-CR-CCSD(T)
TABLE VI. Bond dissociation energies (kcal mol−1 ) of NdF computed at the CASSCF (7,12) level as well as with EOM-CR-CCSD(T) with only the valence electrons correlated, the valence and subvalence electrons correlated, and all electrons correlated. Bond dissociation energies are computed at an internuclear distance on 8.0 Å, and the bond dissociation energies do not include zero-point energy (
