Since about 1990, the application of density functional theory (DFT) to chemical problems in general, and transition metal systems in particular, have exploded. The spark was the arrival in 1988 of analytical expressions for the DFT energy derivatives5 which finally allowed DFT computer codes to perform automatic geometry optimizations.
Prior to this, it had already been established that even the simplest forms of DFT, based on the exchange-only Slater or X scheme,6could give good descriptions of the electronic structure of metal complexes and a number of contemporary applications confirmed this.7–11 However, in combination with structure optimization, here at last was a quantum chemical method accurate enough for transition metal (TM) systems and yet still efficient enough to deliver results in a reasonable time. This was in stark contrast to the ‘‘competition’’ which was either based on the single-determinant Hartree–Fock approximation, which had been discredited as a viable theory for TM systems,12 or on more sophisticated electron correlation methods (e.g., second order Mứller–Plesset theory) which are relatively computationally expensive and thus, for the same computer time, treat much smaller systems that DFT.
DFT promised so much more than LFT and over the last decade or so, amazing advances have been made. On the one hand, LFT is restricted to just those energy levels dominated by d(or f) contributions. This limits its application to computing the ‘‘d–d’’ spectra, magnetic susceptibilities, and EPR parameters of essentially classical single-center Werner-type coordination complexes.
DFT, on the other hand, not only treats ‘‘d–d’’ spectra13,14 and EPR parameters15 but also provides information on all the orbitals (not just thedset), can generate charge and spin density distributions, optimize both ground and transition state structures,16,17compute NMR properties of diamagnetic complexes,18 and more. Moreover, DFT is not restricted to Werner complexes.
Seemingly the only advantages left for LFT are:
(i) it is very much quicker to perform LFT calculations and
(ii) the accuracy of LFT transition energies, magnetic susceptibilities, and EPR parameters is usually higher than current DFT methods can achieve.
∆oct ∆oct
CFT MO
Figure 1 Illustration of CFT and MO barycenter energies for octahedral ML6where L is-bonding only.
However, LFT is parametric so it is perhaps not surprising that LFT can deliver good agreement with experiment. The enduring legacy of LFT, apart from its pedagogic value, is that its concepts continue to provide a framework in which to interpret the physical properties of metal complexes in terms of the underlying bonding. It is only natural to explore whether DFT provides an equally viable framework.
2.50.3 LFT vs. DFT
Chapters 2.35 and 2.36 discuss the basis of LFT and suggest that the best way to parameterize the ligand field potential is via a ligand superposition scheme like the AOM. The AOM was originally introduced within the framework of semiempirical MO theory and the Wolfsberg–Helmholtz approximation.19 Later, Gerloch, Woolley, and co-workers developed an alternative theoretical basis for LFT and the AOM which is based on the density functional theorem.20Both approaches reflect the angular momentum properties of the centrald-orbitals and, given identical basis sets, molecular structures, and parameter values, generate identical results. There are some differences in the subsequent interpretation of the values of the parameters but these are not important here.
Recall from Chapter 2.35 that the ligand field model comprises three terms: one for the ligand field potential (HLF); one for d–d interelectronic repulsion (HER); and one for spin-orbit coupling (HSO). The most important feature of the AOM is that it separates each M–L interaction into separateandcontributions which, in principle, provides a direct, albeit qualitative, description of the bonding. Only HLF contains any molecular symmetry information and this term thus determines thed-orbital splittings and hence the AOMeparameter values. In contrast,HERand HSO are spherically symmetric and reflect properties of the complex as a whole. These features have far reaching consequences when comparing thed-orbital energies from a ligand field analysis with the related orbitals from a density functional calculation.
2.50.3.1 Metal-amines and p-Bonding
In the AOM, it is normally assumed, and sometimes verified, that saturated amine ligands have no -bonding capabilities, i.e., e(N)0.21 Hence, for a simple metal hexaamine, [M(NH3)6]nþ, the -type ‘‘t2g’’ orbitals are predicted to be degenerate and nonbonding (Equations (1a) and (1b)).
Moreover, if the complex were subjected to a tetragonal distortion that maintained the effectiveD4h
symmetry, the threedfunctions would remain degenerate and nonbonding (Equations (2a)–(2d)):
Oh: Eðdz2ị ẳ Eðdx2y2ị ẳ 3e ð1aị
Eðdxzị ẳ Eðdyzị ẳEðdxyị ẳ 4e0 ð1bị
D4h: Eðdx2y2ị ẳ 3eðeqị ð2aị
Eðdz2ị ẳeðeq:ị ỵ2eðaxị ð2bị
Eðdxzị ẳEðdyzị ẳ 2eðeqị ỵ2eðaxị 0 ð2cị
Eðdxyị ẳ4eðeqị 0 ð2dị
Comparison of DFT, AOM, and Ligand Field Approaches 645
This result is already counter to the CFT prediction which, for a tetragonal elongation, predicts dxy should rise while dxz and dyz should fall to compensate. This arises from the long-range interaction exerted by point charges and to the fact that point charges implicitly exert bothand effects. In the more sophisticated AOM, the ‘‘t2g’’ degeneracy can only be raised through ML bonding interactions, which are inherently short range and symmetry specific with regard to local and bonding modes.
DFT calculations have been reported for the tetragonally elongated copper(II) diethylenetri- amine complex, [Cu(dien)2]2þ.22Significantly, the ‘‘dxy’’ orbital lies several thousand wavenumbers below thedxz/dyzpair, a result which conflicts with both the AOM, which predicts degeneracy, and electrostatic CFT, which predictsdxyshould be abovedxz/dyzrather than below. Moreover, if one takes the DFT d-orbital energies in an AOM analysis, e would not only be nonzero, but, assuming the magnitude of the axial e parameter is less than the magnitude of the equatorial e, the analysis based on Equations (2a)–(2d) would conclude that e is less than 0, i.e., that amines are acceptors. Evidently, the DFTd-orbitals are not the same as the AOMd-orbitals.
Which model is correct? To answer this question we must also consider: How can we tell? The validity of a theoretical model is based on its ability to reproduce (and predict) experimentally observable quantities. An orbital is not observable per se and in essence is an artifact of the model. For [Cu(dien)2]2þ, both the AOM and DFT reproduce the d–d transition energies and EPR g-values satisfactorily so they are both ‘‘correct.’’ Why, then, are their orbitals different?
2.50.3.2 Spherical versus Non-spherical Potentials
Since orbitals are model dependent, different models will have different orbitals. The basic distinction between DFT d-orbitals and LFT d-orbitals arises from their respective treatments of interelectron repulsions. In LFT,d–drepulsion is treated within a spherical approximation. For d1 and d9 configurations, there is a single free-ion term and hence no need to consider d–d interelectron repulsion at all. In contrast, the Kohn–Sham orbitals in DFT are computed relative to the total molecular potential. For a tetragonald9copper(II) complex, dx2y2is singly occupied while the remainingd-functions are doubly occupied. Hence, to a first approximation, the ‘‘hole’’
in the equatorial plane results in less d–d repulsion in the plane than perpendicular to the plane with the result that the in-plane ‘‘dxy’’ orbital falls relative to the out-of-plane dxz/dyzpair.
Yet, the DFT orbital splitting in the ground state does not translate into different ‘‘d–d’’
excitation energies. LFT predicts two ‘‘d–d’’ bands and when the ‘‘d–d’’ excitation energies are explicitly computed by DFT, via promoting electrons from occupied to the half-filled dx2y2
orbital and subtracting the energies of ground and excited states, only two unique excitation energies result.22Hence, both LFT and DFT provide a reasonable description of the experimen- tally observable data, but the single LFTd-orbital sequence, which applies to all the ligand field states, is different from the DFT ground state sequence.
In LFT, the mere fact of using atom-liked-functions to construct the many-electron multiplet states arising from thed-configuration implies that the potential around the metal is mainly atom like, i.e., essentially spherical. The LFT d-orbitals are defined relative to a notional average potential which affects all possible arrangements of the d-electrons. Thus, the LFT d-orbitals are implicitly designed to describe the ground and all the excited states simultaneously. In contrast, the DFT d-orbitals are defined relative to a particular state. Both DFT and LFT generally give the same ground state but this only requires that the configurations are the same, not that the detailed orbital sequences are identical. Although the DFT Kohn–Sham orbitals look liked-orbitals when plotted in three dimensions,23this does not imply they form the correct basis for a ligand field treatment.
2.50.3.3 Planar[MCl4]2Complexes
Consider, for example, planar tetrachlorometallate complexes. Metal chloride complexes provide a diverse source of experimental data against which to test theoretical models. Planar tetrachloro complexes like [CuCl4]2 and [PdCl4]2 have been well studied24–28 and analyses of the ‘‘dd’’
spectra29 have established a d-orbital sequence of dx2y2>>dxy>dxz/dyz>dz2. Assuming Cl is a
‘‘linear ligator’’ (i.e., a cylindrically symmetric ligand), this sequence is easily accommodated
within the AOM. In particular,dxyis predicted to lie abovedxz/dyzsince the energy of the former is 4e, the latter 2e, ande>0 for a-donor like chloride.
Early applications of DFT based on the exchange-only X models also gave good qualitative agreement with experiment and, for planar [CuCl4]2, the samed-orbital sequence as that derived from ligand field and spectroscopic studies.8 With hindsight, this agreement was fortuitous.
Both the multiple scattering X (MSX) and the discrete variational X (DVX) methods employed spherical approximations when constructing the molecular potential. This tended to smooth the asymmetricd–d repulsion effects and made the DFT procedure more closely emulate LFT with the result that the DFT ground state orbital sequence matched that derived from spectroscopy.
Modern DFT no longer needs to approximate the molecular potential and calculations on [PdCl4]223,30placedxybelowdxzanddyz. Note that both LFT and DFT predict the same ground state configuration, viz,dx2y20
dz2 2dxy
2dxz 2dyz
2, that both models place thedx2y2orbital highest of thed set, and that the Kohn–Sham orbitals look as expected. However, the ground state DFT sequence is clearly not the same as that obtained from LFT with regard to the order ofdxyand dxz/dyz.
The low-spin d8 complex [PdCl4]2is even more susceptible than [Cu(dien)2]2þ to the effects from nonsphericald–d repulsion since thedx2y2orbital is completely empty. The lower in-plane d–d repulsion results in dxy below dxz/dyz in apparent disagreement with the accepted d-orbital sequence. Again, if the DFT d-orbital energies are used directly in an AOM analysis, e would be negative implying a-acceptor role for chloride. Of course, this conclusion is invalid since we cannot use DFT orbitals directly in a purely ligand field context. So, what is the nature of the Pd–Cl interaction? The AOM predicts donation based on an analysis of the d–d spectrum.
The DFT charge distribution is also consistent with donation in that the Cl -orbitals are depleted in the complex relative to the uncoordinated ligand. Moreover, the DFT multiplet state energies agree with the experimental sequence of excited states30 so that, judged by their ability to reproduce experimental data, both DFT and the AOM are equally good.
However, if one wished DFT to make more direct contact with LFT, the former must somehow try to capture the nature of the average spherical potential which defines the ligand field d-functions. Anthon and Scha¨ffer31 are attempting to bridge DFT and LFT in the context of using DFT to compute free-ion interelectron repulsion parameters. Special average-of-configura- tion states are used to define a fixed, spherical atomic potential that is then used to compute multiplet energies of the terms spanned by the configuration. The results are impressive and certainly better than previous estimates based on Hartree–Fock calculations. The goal is to deal with the full Nephelauxetic Effect nonempirically.
2.50.3.4 Excited States
The generality of DFT inevitably leads to compromises. This is particularly evident when computing excited state energies. Within LFT, it is easy to construct proper, single center determinantal descriptions of thed-based multiplet states and explicitly evaluate the configuration interactions between them. The only question is whether the spherical treatment of interelectronic repulsion and spin-orbit coupling is sufficiently accurate. In many instances, this appears to be the case.
In contrast, the construction of general, multicentered determinants is nontrivial14and DFT is still struggling to provide accurate excited state energies consistently. However, DFT has the considerable advantage of being able to compute charge transfer (CT) states which is beyond the scope of LFT. There are notable successes such as [Cr(NH3)6]3þ,32,33 Cr3þ in fluoride lattices,34 and for [Ru(bipy)3]2þ35,36 where the CT state energies can be used to extract d–d excitation energies, but the performance of DFT is still a bit patchy. For example, in the tetrahedral manganese(VI), chromium(V), vanadium(IV), manganese(VII), chromium(VI), and vanadium(V) oxo anions, DFT sometimes places the metal 4sorbital anomalously low in energy which has a knock-on effect on thedexcited states.37
An alternative to the construction of multiplet states is the use of time-dependent DFT (TDDFT, see Chapter 2.40). TDDFT not only offers the potential for computing accurate transition energies but also provides the transition moments so that the intensity of the transition can be assessed. For example, hexacarbonyl complexes of Cr, Mo, and W have been analyzed using relativistic TDDFT.38 In contradiction of the original interpretation of the lowest Comparison of DFT, AOM, and Ligand Field Approaches 647
energy transitions being d–d bands, the TDDFT calculations assign the low energy features to charge transfer. This prompts a critical re-evaluation of the role of ligand field states in CO photodissociation.
2.50.3.5 DFT and LFT Working Together
The previous sections have focused largely on cases where DFT and LFT are apparently at odds and emphasized the differences between them. There are also many instances where the two models work together. After all, it would be good if our understanding of the bonding in a complex was independent of the model we happened to use to interpret the experimental data.
For example, consider the Nephelauxetic Effect. Analysis of the interelectron repulsion para- meters derived from analyzing the d–dspectrum invariably leads to lower values than in the free ion. The interpretation is that, in the complex, thedelectrons are, on average, further apart which is consistent with expandedd-functions in the complex and/or withdelectron delocalization onto the ligands. Analysis of the electron density distribution from X-ray diffraction in trans- [Ni(NH3)4(NO2)2] yields ad-orbital radius larger than that for free Ni2þ.39However, the unpaired electron density derived from polarized neutron diffraction (PND) data yields a d-orbital radius less than for free Ni2þ prompting Figgis to propose an ‘‘anti-Nephelauxetic effect.’’ DFT calculations39 support LFT in that the d-orbitals expand upon complex formation but also provide an explanation of the diffraction data.
For the formallyd8Ni2þcenter, there are five up-spin and three down-spin delectrons. Both sets are expanded relative to the free ion but the extra exchange interaction lowers and contracts the up-spin orbitals relative to down-spin. The X-ray diffraction experiment measures the sum of up-and down-spin densities while the PND experiments measure their difference. As shown qualitatively in Figure 2, the maximum for the sum density occurs at a slightly longer distance than the up-spin density curve, consistent with the experimental measure of a four percent increase in the d-function radius, while the maximum for the difference density occurs at a much shorter distance, consistent with the 8% contraction in thed-orbital radius used to analyze the PND data. Moreover, the small region of negative spin density lying along the NiL vectors is also explained byFigure 2. Thus, there is no anti-Nephelauxetic effect.
There are many other instances of the interplay between experiment, DFT, and LFT. Extensive work on the tetrachlorocuprate anion, [CuCl4]2, has established that the composition of the singly occupieddx2y2orbital is about 70% metal and 30% ligand.40However, DFT calculations suggest the inverse. Szilagyi etal.28 have modified the DFT by including some Hartree–Fock exchange into the Becke/Perdew gradient corrected functional. By tuning the admixture, they make the description of the bonding more ionic which has the added benefit of improving the agreement with computed excited state energies.
SUM
DIFF α-spin
β-spin
Relative electron density Free
ion
Figure 2 Schematic representation of radial density distributions in a high-spind8complex.
DFT calculations on linear dihalides reveal unexpected groundstates. For example, LFT suggests thatdz2should be the highest energyd-orbital for linear CuCl2 while DFT yields a2Eg
state.41 This has prompted a re-evaluation of the ligand field description of this type of molecule and the recognition of the importance of adsconfiguration interaction akin to that for planar ML4systems.42,43