NITROGEN FIXATION VIA A TERMINAL IRON(IV) NITRIDE
5.2 Results and Discussion
5.2.4.2 Multireference Character from Broken-symmetry DFT and CASSCF
could be explained, for example, in terms of a S= 1/2 [NNMe2]•− ligand either coupled ferromagnetically to an S =1/2 or antiferromagnetically to anS = 3/2 Fe center. In turn, this suggests that the Stot = 0 ground state of (P3B)Fe(NNMe2) may possesses open-shell singlet character.
134 the singlet, triplet, and quintet states (Table 5.2). It should be noted that the energies of the BS solutions are not corrected for spin contamination, and these calculations do not include thermal or environmental corrections. In light of this, the small absolute adiabatic singlet-triplet gaps predicted by the 0% and 10% HF calculations appear to provide the best accordance with experiment, although in the 10% case the ground state is incorrectly predicted to be3Γ0,0.
Table 5.2: Relative Electronic Energies (cm−1) for (P3B)Fe(NNMe2) from VT NMR, DFT, and NEVPT2
State VT NMR 0% HF 10% HF 25% HF NEVPT2 Singlet geometry
1Γ0,0RK S – 0 0 693 –
1Γ0,0 0 – – 0a 0
3Γ0,0 – 5526b 3137b −577b 6324c
5Γ0,0 – 20050 16063 10117 23816c
Triplet geometry
3Γ0,0 1266±7 1330b −1415b −6003b 6074d
5Γ0,0 – 14386 9931 3067 18819d
aApproximatingE(1Γ) ≈ E(1ΓBS).
bApproximatingE(3Γ) ≈ E(3ΓBS).
cAn NEVPT2 calculation was performed using a SA- CASSCF(10,10) reference including three roots corresponding to the lowest energy singlet, triplet, and quintet states.
dAn NEVPT2 calculation was performed using a SA- CASSCF(10,10) reference including two roots corresponding to the lowest energy triplet and quintet states.
To understand the nature of the spin-coupling in this system, the magnetic orbitals from the 10% HF 3Γ0BS,0 solution in the triplet geometry are shown in Figure 5.8, A. In both the singlet and triplet geometries, two pairs of magnetic orbitals can be identified that clearly correspond to strong antiferromagnetic coupling between the Fe 3dz2 and B 2pz orbitals, and the Fe 3dyz andπ*NN orbitals. In the singlet geometry, the SOMOs are largely Fe-centered and consist of the 3dx2−y2 and (3dxz − πN) orbitals. Upon relaxation to the triplet geometry, the antiferromagnetic couplings weaken, as judged by the overlap
between the corresponding α and β spin orbitals (hα|βi), concomitant with a bending of the Fe–N–N angle from 174◦ to 164◦. This angular distortion is quite similar to that observed crystallographically in the reduction of (P3B)Fe(NNMe2) to [(P3B)Fe(NNMe2)]−, and is accompanied by decreased overlap between the 3dxz andπN orbitals, as judged by a Löwdin population analysis. Indeed, upon geometric relaxation, the Fe character of the 3dxz-derived SOMO increases from 69% to 78%, which can be understood in terms of a partial re-hybridization of the Nα πN lone-pair to avoid unfavorable π* interactions with the Fe ion.
β α β
α
α
α
〈α|β〉= 0.92
〈α|β〉= 0.97
〈α|β〉= 0.88
〈α|β〉= 0.95
A B BS: 10% HF
Fe: 2.34 Nβ py: −0.11 Nα px: 0.068
Nα py: −0.15
B pz: −0.12
CASSCF(10,10)
Fe: 2.20 Nβ py: −0.031 Nα px: 0.026
Nα py: −0.10
B pz: −0.085 x
z y
Figure 5.8: (A) Magnetic orbitals from the3Γ0,0BS state of (P3B)Fe(NNMe2) computed with 10% HF in the triplet geometry (isovalue=0.075 a.u.), along with qualitative MO diagrams.
α spins are shown in red, while β spins are shown in blue. (B) Spin density isosurfaces of the3Γ0,0 state of (P3B)Fe(NNMe2) in the triplet geometry (isovalue =0.005 a.u.), from a 10% HF BS DFT calculation (top), and from a ground state specific CASSCF(10,10) wavefunction (bottom). α density is shown in red, while β density is shown in green.
Selected Löwdin spin populations are shown.
These BS solutions can be understood in terms of the modified MO diagram of Figure 5.8, A, where the reduced overlap of the 3dyz/π*NN and 3dz2/B 2pz interactions produces an electronic structure most concisely described as a high-spin, S = 2, Fe(II) center antiferromagnetically coupled to both a S = 1/2 borane radical anion ([R3B]•−)
136 and a S = 1/2 [NNMe2]•− ligand in the 3Γ0,0 state. This electronic structure rationalizes the VT NMR data presented above, as it results in negative π-symmetry spin density at both N atoms due to population of the π*NN orbital. In addition, Nα is predicted to bear orthogonal, positive π-symmetry spin density due to delocalization via the 3dxz-based SOMO, producing an unusual, rhombic spin distribution about this atom (Figure 5.8, B).
According to this analysis, the 1Γ0,0state should correspond to the pairing of the two largely Fe-centered spins. However, the DFT calculations are ambiguous with respect to the open-shell singlet character of this state. To address this, we performed calculations based on the CASSCF ansatz using a CAS(10,10) reference with an active space composed of the five 3d, B 2pz, πN, and π*NNorbitals, with an additional two second-shell 3d’ orbitals to provide greater flexibility for the occupied 3dxy and 3dx2−y2 orbitals.73 From a ground- state-specific calculation, the1Γ0,0state does indeed exhibit multireference character, with the closed-shell configuration,
|(3dxy)2(3dx2−y2)2(3dxz +πN)2(3dz2 +2pz)2(3dyz+π*NN)2i
composing only 79.8% of the zeroth order wavefunction. The next three most important configurations (comprising 8.7% of the wavefunction) involve single and double excitations from the bonding (3dxz+πN), (3dz2+ 2pz), and (3dyz+π*NN) orbitals into their antibonding counterparts, which indicates antiferromagnetic character. The antiferromagnetic nature of these interactions is hinted at from localization of the active space orbitals, which results in strong spatial separation of the Fe 3d and πNiv/π*NN orbitals;45 the B 2pz orbital becomes largely B-centered, although the degree of localization is less, consistent with greater relative covalency (Figure 5.9). Unfortunately, in the localized basis, the CI expansion of the wavefunction becomes very diffuse, but an examination of the occupation numbers of the active space orbitals suggests a dominant configuration,
|(3d)6(2pz)1(πN)2(π*NN)1i
ivTheπNorbital is admixed with aσ-type phosphorous group orbital.
While an analysis of such localized orbitals has been used to argue for the presence of metal-ligand antiferromagnetic coupling,45,46this can be misleading because configurations corresponding to a normal covalent bond and an antiferromagnetic exchange coupling interaction cannot be distinguished.44
1.93 1.89 1.83
0.09 0.11 0.17 Natural Orbitals
0.51 1.17 1.00
1.52 0.93 0.99 Localized Orbitals
3dxz ± πN
3dyz ± π*NN
3dz2 ± B 2pz
Figure 5.9: Active space orbitals (isovalue = 0.075) corresponding to the Fe–NNMe2πand Fe–Bσ interactions from a ground state specific CASSCF(10,10) calculation of the1Γ0,0 state. Both the natural and localized orbital bases are presented. Occupations numbers are given below each orbital.
An unbiased, quantitative measure of antiferromagnetic character of the 1Γ0,0 state can be obtained from the natural orbital occupation numbers (NOONs) of the bond- ing/antibonding NOs (n±) describing the Fe–NNMe2 π and Fe–B σ bonding. A mea- sure of the diradical character (Y) of these interactions can be defined from their effective bond order (beff = (n+ − n−)/2), as a covalent bond has beff = 1 and a pure diradical
138 has beff = 0.43,44 Using the NOONs from the entire active space, we can also ascertain the open-shell singlet character from the number of “effectively unpaired” electrons (using the Davidson–Yamaguchi definition, ND = Í
ini(2−ni), or the Head-Gordon definition NU =Í
i1− |1−ni|, wherei runs over the active space orbitals).74–76Using these indices, we have tabulated the diradical character of the Fe–NNMe2π and Fe–Bσbonding and the number of effectively unpaired electrons for the1Γ0,0state in Table 5.3. Although there are only about 1 to 2 effectively unpaired electrons, which would seemingly correspond to a singlet diradical, the combined (unnormalized) polyradical character of 28% has significant contributions from both theπ and theσbonding.
Table 5.3: NOON-based Chemical Bonding Indices for (P3B)Fe(NNMe2) from CASSCF Calculations
1Γ0,0 3Γ0,0b
NOs n+ n− Ya ND NU n+ n− Ya ND NU
3dz2 ±B 2pz 1.89 0.11 0.11 0.42 0.22 1.79 0.21 0.21 0.75 0.42 3dyz± π*NN 1.83 0.17 0.17 0.62 0.34 1.72 0.28 0.28 0.96 0.56
CAS – – – 1.76 0.93 – – – 3.94 3.08
aY =1− n+−n−
2 . Note that for a spatially-matched bond/antibond pair,Y = n−.
bComputed in the triplet geometry.
Repeating these calculations for the 3Γ0,0 state in the triplet geometry shows that, consistent with the BS DFT calculations, the antiferromagnetic character of the zeroth order wavefunction increases significantly, which is reflected in the increased polyradical character of the wavefunction (49%, Table 5.3), although the number of effectively unpaired electrons remains approximately 1 to 2 in excess of those expected for a triplet. Also consistent with the BS calculations, the overlap between the Fe 3dxz andπN orbitals decreases in the triplet state such that their antibonding combination is essentially a pure 3d orbital (87%
Fe). The weakened antiferromagnetic couplings increase the multireference character of the wavefunction, with the configuration,
|(πN)2(3dxy)2(3dz2+2pz)2(3dyz+π*NN)2(3dx2−y2)1(3dxz)1i
comprising only 69.6% of the ground state wavefunction, with the next two most important configurations,
|(πN)2(3dxy)2(3dz2+2pz)2(3dyz+π*NN)0(3dx2−y2)1(3dxz)1(3dyz−π*NN)2i
|(πN)2(3dxy)2(3dz2+2pz)2(3dyz+π*NN)1(3dx2−y2)1(3dxz)1(3dyz−π*NN)1i
having a weight of 13%. These configurations are responsible, in part, for the antiferromag- netic character of theπbonding, and produce a spin density distribution that is, qualitatively, in agreement with the BS DFT results. Comparing the 10% HF-calculated spin density with that predicted by the CASSCF(10,10) wavefunction (Figure 5.8, B), it can be seen that these methods predict the same spin topology, but differ quantitatively. This can be attributed to the spin-contamination of the BS DFT solution (hSˆ2i= 2.34 for the 10% HF calculation).77
To determine whether these multireference calculations provide an accurate basis for the static correlation effects in the bonding in (P3B)Fe(NNMe2), we have performed second-order N-electron valence perturbation theory (NEVPT2) calculations on top of state- averaged (SA) CASSCF(10,10) reference wavefunctions to predict the energetic ordering of low-lying excited states. As can be seen in Table 5.2, these NEVPT2 calculations correctly predict the ground state multiplicity, although the adiabatic singlet-triplet gap is overestimated. However, as shown in Figure 5.6, D, the NEVPT2-calculated electronic spectrum is in nearly quantitative agreement with experiment. The overestimated singlet- triplet gap can thus be attributed, in part, to inaccuracy in the DFT-predicted geometry of the triplet state, rather than deficiencies in the zeroth order CASSCF reference or the perturbative treatment of dynamic correlation.
On the basis of these calculations, we can assign the optical transitions of (P3B)Fe- (NNMe2) as being due principally to transitions from the filled 3dxy and 3dx2−y2 orbitals into the (3dxz − πN) and (3dyz − π*NN) orbitals, as postulated. One-electron excitations from these orbitals into the (3dxz −πN) orbital compose 60 to 70% of the wavefunctions of
140 the first two excited singlet states, 1Γ0,1and1Γ0,2. The next two states,1Γ0,3and1Γ0,4, are of more mixed character, containing contributions from one-electron excitations from the 3dxy, 3dx2−y2, and (3dz2 + 2pz) orbitals into both the (3dxz−πN) and (3dyz−π*NN) orbitals, although the latter is the dominant acceptor orbital.
These calculations validate the schematic MO description of the 3Γ0,0 excited state proposed in Figure 5.8, A, and reveal the open-shell character of the singlet ground state,
1Γ0,0. While the electronic structure of the ground state is clearly multiconfigurational, the dominant antiferromagnetic terms involve the 3dyz/π*NN and 3dz2/B 2pz interactions, leading to a succinct description of the 1Γ0,0 state as an intermediate-spin, S = 1 Fe(II) center coupled antiferromagnetically toS =1/2 [NNMe2]•−andS =1/2 [R3B]•−ligands.
5.2.5 Electronic Structures of [(P3B)Fe(NNMe2)]+/− from Pulsed EPR Studies