The dⁿ Number in Transition Metal Chemistry: Its Utility and Limitations

Abstract

The dⁿ number or configuration is a very useful parameter in transition metal chemistry because it conveys information about the expected properties of a d-block metal complex, including its shape, magnetism, thermodynamic stability, kinetic lability, and spectroscopic properties. The dⁿ number can be determined from either the oxidation state (OS) or valence number (VN) of the metal centre, and since, in most cases, these are numerically equal, the derived dⁿ value is the same. However, examples are discussed where the OS and VN are not equal, which results in unavoidable ambiguities in dⁿ values. Following a discussion of these examples, a revised definition of dⁿ is proposed based on the occupation of the frontier molecular orbitals of the complex.

1. Introduction

The importance of the d-block or transition elements in, for example, synthesis, catalysis, biological inorganic chemistry, and in numerous solid-state materials would be hard to overstate, and understanding their chemistry is, therefore, both a central pillar of any undergraduate chemistry course and essential to many research programmes (Appendix A Note 1). In seeking to understand and explain the chemistry of these elements, an appreciation of the characteristics of electrons in d orbitals is obviously indispensable, and most textbooks and undergraduate courses begin with an account of Crystal Field Theory (CFT) before progressing to a more detailed treatment using Ligand Field Theory (LFT), the latter being essentially a molecular orbital (MO) description of transition metal complexes. Several current and many older texts offer a detailed explication of both CFT and LFT, which will be no more than briefly adumbrated here, but a key feature of any analysis is the nature and extent of d orbital splitting in a crystal/ligand field and the number of electrons formally assigned to the d orbitals (Appendix A Note 2) [1,2,3,4,5]. This latter feature is generally enumerated as n in the term dⁿ, and it is widely accepted that assigning a dⁿ number is particularly useful for rationalising many of the properties of transition metal complexes. There are, nevertheless, some intrinsic ambiguities in dⁿ assignments in some instances, not least because of certain differences in how n is determined according to the oxidation state (OS) and the Covalent Bond Classification (CBC) methods. The primary objective of this article will be to discuss these ambiguities and offer some suggestions on how to resolve them with a principal focus on how this topic is taught at the undergraduate level whilst recognising that many of the basic principles are relevant to several current areas of research.

2. A Definition of the dⁿ Number and Why It Is a Useful Parameter

A general definition of the dⁿ number can be ascertained from the following representative literature quotes:

• ‘The dⁿ configuration is the number of electrons that are housed in metal d orbitals that are either primarily non-bonding or have metal–ligand antibonding character.’ (Appendix A Note 3) [6,7].
• ‘The number of d-electrons generally describes the number of electrons not involved in the primary metal–ligand bonding interactions.’ [8].
• ‘…in the dⁿ configuration using the definition that this quantity refers to electrons in d-based molecular orbitals that are not considered to be bonding.’ [9].
• ‘…the dⁿ configuration corresponds to the number of electrons in d-based molecular orbitals that are formally localised on the metal after bond formation, i.e., those that are not components of metal–ligand bonding orbitals.’ [9].

The key distillate from these four definitions is that the electrons included in the dⁿ number should not be involved in significant metal–ligand bonding interactions or, to put it another way, should reside only in orbitals that are non-bonding or metal–ligand antibonding. However, the adjective ‘primary’ and the adverb ‘primarily’ employed in the first two quotes are inherently subjective, with the consequence that the assignment of a dⁿ number remains open to interpretation, at least in some circumstances. Whilst there is a general consensus on the usefulness of dⁿ for the great majority of complexes, divergences arise in determining the value of n in two broad areas, namely: (a) when the oxidation state (OS) and valence number (VN) of the metal in the complex are not the same and (b) when a dⁿ number has been determined computationally that differs from the value assigned using either OS or VN methods (Appendix A Note 4) [10,11,12,13] (Appendix A Note 5) [14,15,16,17]. In the sections that follow, many of the general areas of divergence will be discussed before concluding with some final observations and a proposed, modified definition of the dⁿ number.

Before proceeding, however, it is important to reiterate that a dⁿ configuration is a very useful heuristic, especially in an educational setting, because it allows the following features (the list is not exhaustive) to be readily understood, interpreted, rationalised, and, in some cases, predicted, at least to a first approximation:

(i)

The magnetic properties of complexes based on the number of unpaired electrons present, which includes a consideration of whether the complex is low spin or high spin according to the relative magnitude of the crystal field splitting energy (Δ) vs. the electron pairing energy (P) (Appendix A Note 6).

(ii)

The expectation that d⁰ complexes should not be susceptible to oxidation at the metal centre.

(iii)

The observation that many four-coordinate d⁸ metal complexes are square planar, whereas four-coordinate d¹⁰ and d⁰ complexes are almost invariably tetrahedral (Appendix A Note 7) [18].

(iv)

The observation that octahedral coordination is particularly favourable for metals with a d⁶ or a d³ configuration.

(v)

Whether or not complexes are kinetically labile or inert with respect to ligand substitution reactions according to the magnitude of the crystal/ligand field stabilisation energy (CFSE/LFSE) and the derived crystal/ligand field activation energy.

(vi)

The rationalisation of trends in lattice energies and hydration energies according to the magnitude of the CFSE/LFSE.

(vii)

The mechanisms of homogeneous catalysis by late transition metal complexes, which frequently involve interchange between d⁶, d⁸ and d¹⁰ species.

(viii)

Jahn-Teller distortions from idealised geometries for particular dⁿ configurations; for example, d⁹ octahedral complexes.

(ix)

The interpretation of the electronic spectra of complexes using qualitative Orgel or more quantitative Tanabe–Sugano diagrams.

(x)

The isolobal principle and the fragment orbital analysis from which it is derived [19].

Detailed explanations of features (i)–(ix) can be found in the texts cited in refs. [1,2,3,4,5] (and in other standard inorganic chemistry texts) whilst ref. [19] provides a comprehensive description of the isolobal principle (x).

3. Crystal Field Theory and Ligand Field Theory

Despite its vintage and well-known limitations, the starting point for understanding the properties of d orbitals and their associated electrons in transition metal chemistry is usually Crystal Field Theory (CFT). In outline, the basis of CFT considers a metal ion to be surrounded by a collection of ligands arranged in a particular geometry, providing an electrostatic field that gives rise to a characteristic splitting of the valence d orbitals resulting from coulombic repulsions between the metal d electrons and the ligand lone pairs. In the simplest and the most commonly first taught example, an octahedral coordination geometry results in the splitting of the five d orbitals into two groups, one containing three orbitals at lower energy and the other containing two orbitals at higher energy. These orbitals are then allocated the symmetry labels t_2g and e_g taken from the character table for the O_h point group in this instance. This arrangement of two groups of orbitals in an octahedral crystal field is illustrated in Figure 1a, which also highlights the term Δ_oct that denotes the energy separation between them, usually termed the crystal field splitting energy or parameter. The value of n in the dⁿ number, which indicates the occupancy of these orbitals, is then normally assigned according to Equation (1) where m is the Group number of the metal in the periodic table and OS is the oxidation state; for example, n for the complex [Fe(OH₂)₆]³⁺ is 8 − 3 = 5; i.e., Fe(III) is therefore assigned a d⁵ configuration (Appendix A Note 5) [14,15,16,17] (Appendix A Note 8).

n = m − OS

(1)

The concept of the Crystal Field Stabilisation Energy (CFSE, sometimes referred to as Ligand Field Stabilisation Energy or LFSE) is then derived from the energies of the t_2g and e_g orbitals relative to a hypothetical spherical field (Figure 1a) and this is widely used to explain and rationalise the features outlined in points (i) and (iv)–(vi) listed in Section 2.

Useful though the CFT model is, its treatment of the interaction between the metal ion and the ligands as purely electrostatic ignores any covalency in the metal–ligand bonding. This shortcoming is generally addressed using an LFT/MO approach, which results, for an octahedral complex, in the qualitative molecular orbital energy level diagram illustrated in Figure 1b. The MO diagram shown in Figure 1b and its derivation will not be discussed in detail except to note that the part of it highlighted in the red box corresponds to the d orbital splitting arrangement obtained from CFT, as a comparison with Figure 1a reveals. The crucial difference, however, is that the e_g orbitals of the CFT approach are labelled e_g* in the LFT/MO diagram since they are now metal–ligand σ antibonding in character and are, therefore, no longer the pure d orbitals of the CFT model. This point is made explicitly in the following quote taken from ref. [1]: ‘The main difference between the MO and the crystal field results is that the e_g* orbitals, as they are obtained in the MO treatment, are not pure metal d orbitals.’ [original emphasis]. Nevertheless, these e_g* orbitals are closer in energy to the metal d orbitals than they are to the ligand-based orbitals since the elements through which the ligands are bonded to the metal are usually more electronegative than the metals to which they are bonded, such that the ligand orbitals lie at lower energy than the metal orbitals. Although this is something of an oversimplification (as discussed in Section 11), the usual approach adopted for electron accounting purposes is for the a_1g, t_1u and e_g orbitals (see Figure 1b) to accommodate the ligand-based electrons, leaving the metal-based electrons to occupy the t_2g and e_g* orbitals. Accordingly, using the example above, an Fe(III) complex would still be designated as d⁵.

The diagram shown in Figure 1b represents an octahedron in which only metal–ligand σ bonding has been considered such that, whilst the e_g* orbitals now incorporate some ligand character (and are σ antibonding as noted above), the t_2g orbitals are non-bonding and therefore remain as pure metal d orbitals. As soon as any metal–ligand π bonding is considered, however, these t_2g orbitals become involved, in a manner shown qualitatively for π donors and π acceptors in Figure 1c and d respectively. Accordingly, they can no longer be considered as pure metal d orbitals but, for the same reasons as outlined above for the e_g* orbitals, they remain close in energy to the metal d orbitals from which they were derived (Appendix A Note 9) [5]. It therefore still makes sense to assign a dⁿ number based on the combined occupancy of these metal-based t_2g and e_g orbitals whilst recognising that neither are any longer pure metal d orbitals, nor can they be described as non-bonding orbitals. Although the precise arrangement and labelling of the frontier orbitals will change depending on the coordination geometry of the metal and the point group of the complex, the same OS-based principles for determining the dⁿ number apply for a range of coordination numbers and geometries.

The brief outline of CFT and LFT given above reveals how the dⁿ number is usually determined using the OS method, but the value of this term in relation to the points listed in Section 2 is that (for an octahedron) the t_2g and e_g orbitals (shown in the red boxes of Figure 1) are also the frontier orbitals and therefore play a central rôle in the chemistry of the complex. For complexes with other geometries, the frontier orbitals will have different symmetry labels, but in each case, they will have the same symmetries as the d-orbitals from which they are derived.

Before leaving this section on the CFT and LFT/MO approaches to understanding transition metal chemistry, we note that it has been argued that, for highly covalent compounds, a sd/sd² hybridisation model derived from a Natural Bond Order (NBO) Theory analysis is intuitively more useful from a teaching perspective than LFT derived MO diagrams [20] (Appendix A Note 10) [21,22]. This hybridisation approach considers only ns and (n − 1)d orbitals (on the basis that the np orbitals are deemed too high in energy to be available for bonding) and makes use of a three-centre, four-electron (3c,4e) model involving sd or sd² hybrid orbitals at the metal centre to describe much of the metal–ligand bonding [23]. However, since this sd/sd² hybridisation approach does not differ from the CFT/LFT analysis in terms of how to determine the dⁿ number, it will not be considered further in this article.

4. Covalent Bond Classification

Having outlined a simple CFT/LFT approach to determining dⁿ which explicitly employs the OS of the metal as shown in Equation (1), there is an alternative methodology for assigning the dⁿ number, which, in certain circumstances, results in a different value of n. This is the Covalent Bond Classification (CBC) method, which focusses on what is termed the Valence Number (VN) instead of the OS, and it is therefore important to establish where and why the CBC and OS approaches differ [24,25,26].

The CBC methodology categorises compounds in terms of a central element, M, bound to L, X and Z ligands, which formally donate two, one and zero electrons, respectively, to the M centre and leads to a description of the type ML_lX_xZ_z, where the subscripts l, x and z refer to the number of each of the L, X and Z ligands. Full details of the history and application of the CBC method are set out in the papers of Green and Parkin listed in refs. [24,25,26] and will not be reproduced here except with regard to what is defined as the Equivalent Neutral Class (ENC), which is used to render any charged complexes (and neutral complexes containing both L and Z ligands) in terms of an equivalent neutral format according to the transformations set out in Scheme 1.

The ENC feature within the CBC method is important since the resulting neutral ML_lX_xZ_z designation leads directly to the assignment of the VN according to Equation (2), from which the value of n in the dⁿ number is then determined using Equation (3) where m is the metal’s Group number in the periodic table.

VN = x + 2z

(2)

n = m − VN

(3)

For the great majority of mononuclear complexes, VN and OS are numerically equal, and the dⁿ number assigned to the complex is, therefore, the same whichever method is used, but as shown in Figure 2, the derived dⁿ numbers do differ in some circumstances [24,26]. Thus, for the example compounds 1–5, the dⁿ numbers are the same, whereas for compounds 6–10, they are not. Specifically, complexes where OS ≠ VN occur when: (i) σ Z (and sometimes π Z) ligands are present; (ii) the ligands are formally cationic; (iii) the metal is in a negative oxidation state; (iv) the complexes are binuclear and contain metal–metal bonds. It is these differences and how they may be reconciled, which are the principal focus of the discussions that follow, but first, the ambiguities in the dⁿ assignment for carbene and alkene ligand complexes are considered as these illustrate that a one-to-one mapping of ligands via either OS or VN is not always chemically defensible, which can therefore be a source of divergent dⁿ assignments (Appendix A Note 11).

5. Carbene and Alkene Complexes

Metal carbene complexes provide a useful illustration of potential ambiguities in the assignment of a dⁿ number. Such complexes have traditionally been categorised as either Fischer or Schrock carbenes (Appendix A Note 11) [23,28], which are distinguished by the nature of the metal-carbon bonding, as shown generically in Figure 3. Thus, in classical, heteroatom-stabilised Fischer carbene complexes such as 11 (Figure 4), the carbene carbon is electrophilic, which is rationalised on the basis that carbene ligands are good σ donors (and moderate π acceptors) derived from singlet carbenes (Figure 3a); accordingly, they are treated as neutral, L-type ligands and have no consequence when determining the OS or VN of the metal. In contrast, in Schrock carbenes such as 12 (Figure 4), the carbene carbon is nucleophilic, the ligand being formally derived from a triplet carbene fragment (Figure 3b); they are therefore treated as dianionic ligands for the purposes of determining the metal OS and as X₂ ligands within the CBC framework.

It is important to recognise, however, that Fischer and Schrock carbenes are effectively two extremes at the ends of a continuum in terms of the properties of a general CR₂ ligand, and many instances are known of carbene ligands that fall between these extremes. Thus, carbene complexes such as 13 have been described as ambiphilic [30], and Fischer-type carbenes are known that do not have heteroatom substituents [31,32]. Moreover, within this smorgasbord of examples must be included not only complexes of Arduengo’s N-heterocyclic carbenes (NHCs) [33] and Bertrand’s cyclic alkyl amino carbenes (CAACs) [34], but also, the Grubbs’ ruthenium complexes (typically incorporating the carbene ligand, CHPh). Grubbs’ carbenes are often considered to be intermediate between the Fischer and Schrock extremes, although, as argued in ref. [35], they are rather closer to the latter type than to the former.

It is clear that the substituents on the carbene ligand greatly affect the nature of the metal-carbene complex, but it is important to recognise that the metal centre to which the carbene is bonded also has a profound effect. This is obvious when comparing the known carbene complexes of Zr with those of Pd, but it is also clear that the ancillary ligands can decisively control the properties of the carbene ligand as illustrated by the electrophilic vs. nucleophilic nature of the CF₂ ligand in the complexes [RuCl₂(CF₂)(CO)(PPh₃)₂] and [Ru(CF₂)(CO)₂(PPh₃)₂] respectively [36,37] (Appendix A Note 12) [38,39].

How, then, should metal carbenes that are situated between the Fischer-Schrock extremes be classified (i.e., L vs. X₂) and what dⁿ number should be assigned? It is possible that the classification of all the ‘in-between’ carbene complexes could be arbitrated by computational studies and a single value of dⁿ assigned on a case-by-case basis. More generally, however, this dichotomy exposes a fundamental limitation in the simple dⁿ model (as well as within the OS and CBC models) by offering a binary choice between two values that represent the extremes of a chemical continuum, a point made, albeit not in relation to dⁿ numbers, by the authors of ref. [35], viz., ‘Of course, loss of detail is the inevitable result when undertaking a discrete classification where nature offers a continuum of variation.’

Similar arguments can be made for compounds with carbyne/alkylidyne (CR) ligands and also for metal alkene complexes according to the nature of the alkene substituents and the metal centre to which the alkene is bound [8,23]. For example (see Figure 5), C₂H₄ may be considered as an L ligand in 14, whereas C₂(CN)₄ can be treated as a dianionic or X₂ ligand (i.e., part of a metallacyclopropane) in 15 as evidenced, in each case, by the C–C bond lengths indicated. In complex 16, however, the C₂H₄ ligand is described as ‘intermediate along the continuum between the Ti(II)-ethylene and Ti(IV)-metallacyclopropane limiting structures.’ [40] (Appendix A Note 13) [41].

Thus, in the case of metal carbene and metal alkene complexes (and in other classes of compounds, including metal carbynes, metal alkynes and some of those discussed below) it is apparent that a limitation of the dⁿ formalism results from it being a digital assignment applied to what are analogue chemical bonding systems (Appendix A Note 14) [42]. Coherent MO bonding schemes for metal carbenes and metal alkenes that apply across the Fischer/Schrock and metal-alkene/metallacyclopropane ranges all have two components comprising σ donation and π back-donation (according to the classic Dewar-Chatt-Duncanson (DCD) model described in all standard texts), the relative contributions of each being controlled by the nature of the metal fragment and the ligand substituents [1,2,3,4,5,19,23].

We note that similar arguments for carbene and alkene ligands to those outlined above have been presented by Parkin in ref. [9] in terms of an LZ’ designation where Z’ denotes a varying degree of ligand π acceptor character. Thus, a minimal Z component is consistent with a simple L ligand, whereas a substantial Z component leads to a ligand designation of LZ, which transforms [Scheme 1 (iii)] to X₂.

6. σ Acceptor (σ Z) Ligands: Boranes

As noted in Section 4, a circumstance where the OS and CBC methods diverge in the assignment of the dⁿ number arises for complexes that contain Z ligands. For the situation where one σ Z ligand is coordinated to the metal (π Z ligands are discussed in the following section), the OS-derived dⁿ number determined using Equation (1) becomes dⁿ⁻² according to Equation (3) where VN is employed, as exemplified in complex 7 (Figure 2), which contains the σ Z BH₃ ligand. This difference arises because, in the OS approach, a Z ligand is usually considered as neutral and, therefore, has no effect on the OS, whereas Equation (2) reveals that the VN increases by two for each Z ligand present (since the metal uses two further electrons to bond to the Z ligand) with the consequences for the value of n in dⁿ then evident from Equation (3). It should be noted, however, that if the Z ligand is treated as a dianionic group, the OS- and CBC-derived dⁿ numbers would then be in agreement.

Examples of σ Z ligands are commonly encountered in a range of ambiphilic ligands, and the discrepancy in dⁿ values that arises from using an OS or VN determination can be appreciated with reference to the examples shown in Figure 6 (Appendix A Note 15) [43]. Thus, the gold complex 17 is observed to be square planar with a short Au–B bond distance of 2.309(8) Å and significant pyramidalisation around the B atom, all of which are consistent with a d⁸ configuration at a Au(III) centre. However, Mössbauer spectroscopy and MO calculations support a Au(I) d¹⁰ configuration for complex 17 [44]. In addition, the anionic platinum complex 18 has recently been reported, which is isoelectronic with 17 and also square planar but with a substantially shorter Pt–B bond distance of 2.219(2) Å. Whilst this is consistent with a Pt(II) d⁸ configuration, the X-ray photoelectron spectrum of 18 and MO calculations support a Pt(0) d¹⁰ assignment [45]. Furthermore, the gold complexes 19a and 19b each contain a P,B-chelate but with different B substituents that result in the Lewis acidity of the borane function in 19b being considerably greater than in 19a, as reflected in the Au–B bond distance being much shorter in 19b [2.663(8) Å] than in 19a [2.995(8) Å] albeit both rather long and therefore weak [46].

The large variation in M–B bond distances for the complexes 17, 18, 19a and 19b (which span almost 0.8 Å), depending on the nature of M and the B-substituents R, indicates that the strength of the M→BR₃ interaction varies considerably, as shown in Figure 7, with attendant consequences for the determination of the dⁿ configuration of M; this is reminiscent of the situation with Fischer/Schrock carbene ligands discussed in Section 5. We note that Hill has recognised these complexities and, in order to address the problem, has advocated that the formulae of complexes containing a M → BR₃ bond be written with an associated (M → B)ⁿ descriptor to indicate the presence of the BR₃ ligand [47]. Following Hill’s suggestion (which has been criticised by Parkin [6]), more generally, a formula for a compound containing one Z ligand would be presented as [ML_lX_xZ] (M → Z)ⁿ where n is the n in dⁿ determined using Equation (1), assuming the Z ligand to be neutral. As Hill states, this descriptor is analogous to one proposed by Enemark and Feltham (and widely used) for metal nitrosyl complexes (see below) [48,49] (Appendix A Note 16).

As a further example, a recent paper has offered a detailed computational and spectroscopic analysis of a terminal iron(III) nitride complex P₃^BFe(N) where P₃^B is the boratrane ligand, tris(o-di-isopropylphosphinophenyl)borane [50]. This complex is described as a Fe(III) d⁵ complex and, to quote more specifically, ‘The unusual d⁵-configuration is stabilized by significant delocalisation (≈50%) of the unpaired electron onto the axial boron and nitrogen ligands, with a majority of the spin residing on boron.’ We highlight this OS-derived d⁵ configuration, with the subsequent description of the single unpaired electron being delocalised onto both B and N atoms, since it contrasts with a CBC approach which would seek to assign a specific X and/or Z designation to each of these ligating atoms (B and N) and derive a dⁿ number on that basis. The merits of each approach will be discussed further in Section 12.

Finally in this section, we note that the issues associated with ambiphilic ligand complexes have received detailed attention from Karen, McArdle and Takats, who discuss several more examples, together with related SO₂ complexes, where they compare and contrast the OS and CBC approaches to σ Z ligands [14].

7. π Acceptor (π Z) Ligands: CO

Having examined the situation where the metal is bonded to the Z ligand through a σ bond (σ Z ligands), it is important to address how the OS and CBC approaches deal with π acceptor ligands (π Z ligands) and the extent to which the π Z component should be considered in the determination of the dⁿ number. This matter is usefully exemplified by considering the homologous series of isoelectronic, metal hexacarbonyl complexes [M(CO)₆]^Q where Q varies from −2 to +3 [9,26].

For the situation where M is a 5d element, taking the two complexes at the ends of this series, [Hf(CO)₆]²⁻ and [Ir(CO)₆]³⁺, the extent of π back-donation from the metal d orbitals (the t_2g set in a regular octahedron) into the CO π* orbitals decreases very substantially from the Hf complex to the Ir complex, as shown by computational studies, and evidenced by an increase of almost 500 cm⁻¹ in the ν_CO stretching frequency on progressing from [Hf(CO)₆]²⁻ to [Ir(CO)₆]³⁺. Figure 8 shows the trend in the ν_CO stretching frequency together with the σ bonding and π back-bonding contributions for each complex in the series [9,51,52,53,54] (Appendix A Note 17) [55].

From an OS perspective, the oxidation state of the metal M for the series [M(CO)₆]^Q is numerically equal to Q, and each complex would be assigned a d⁶ configuration according to Equation (1). However, it is argued in ref. [9] that in circumstances where the π back-donation is deemed significant, a (CO)₆ ligand combination should be classified not simply as L₆ but rather, with an incorporated Z component (on the basis of symmetry considerations), as L₆Z₃. This L₆Z₃ designation then transforms under the ENC rules [Scheme 1 (iii)] to L₃X₆. Specifically, for [Hf(CO)₆]²⁻, this [ML₆]²⁻ species would be considered as [ML₆Z₃]²⁻ and since Z⁻ is equivalent to X [Scheme 1 (ii)], this becomes ML₆X₂Z. Then, because LZ is equivalent to X₂ [Scheme 1 (iii)], the final MLXZ designation of [Hf(CO)₆]²⁻ is ML₅X₄ from which, according to Equation (3), a d⁰ assignment follows (Appendix A Note 18). If one ignores (reasonably) the π back-bonding in [Ir(CO)₆]³⁺, a CBC-derived d⁶ configuration is obtained ([ML₆]³⁺ transforms to ML₃X₃, leading to d⁶ where M = Ir), which therefore begs the question, at what point is the π Z component deemed sufficiently large for it to be taken into account in assigning the dⁿ number?

A descriptive answer to this question is offered in the following text taken from ref. [9].

Thus, the t_2g orbitals transform from being classified as largely non-bonding in [Ir(CO)₆]³⁺ to bonding in [Hf(CO)₆]²⁻. This progression is necessarily accompanied by a decrease in the dⁿ configuration using the definition that this quantity refers to electrons in d-based molecular orbitals that are not considered to be bonding.

It is further suggested that the π Z component should be explicitly included for metal carbonyl anions, which therefore implies that the transition from d⁶ to d⁰ in the [M(CO)₆]^Q series should occur between [W(CO)₆] and the anionic [Ta(CO)₆]⁻ [9]. However, as the plot shown in Figure 8 demonstrates, the switch in the relative importance of the σ L and π Z components of the M–CO bonding actually occurs between cationic [Re(CO)₆]⁺ and [W(CO)₆]. Thus, for [W(CO)₆], the reasonable assumption of significant π back-bonding (see Figure 8) would result in an ENC of ML₃X₆ (see above), which has a d⁰ configuration, in contrast to the more usually encountered classification of ML₆, which is d⁶. For the cationic species [Re(CO)₆]⁺, [Os(CO)₆]²⁺ and [Ir(CO)₆]³⁺, the data in Figure 8 are in accord with a d⁶ assignment. We note that this matter has previously received detailed consideration in refs. [9,26], to which we will return in Section 12.

The trend towards increasing π back-donation from [Ir(CO)₆]³⁺ to [Hf(CO)₆]²⁻ shown in Figure 8 is readily understood in terms of the increasing electron richness of the metal centre, which is illustrated in the qualitative MO diagram shown in Figure 9 (Appendix A Note 19) [56]. Thus, for [Ir(CO)₆]³⁺, the lowering in energy of the occupied metal t_2g level due to π back-bonding is negligible as a consequence of a 3+ charge, which results in a large energy separation between the metal t_2g orbitals and the π* orbitals (of t_2g symmetry) on the CO ligands. [Ir(CO)₆]³⁺ is therefore often classed as a ‘non-classical’ metal carbonyl, and the frontier t_2g orbitals are essentially non-bonding. By contrast, for [Hf(CO)₆]²⁻, the lowering in energy of the occupied metal t_2g level due to π back-bonding is substantial because of the negative charge on the complex leading to the energy of the metal t_2g orbitals and CO π* orbitals being significantly closer. Moreover, as shown in Figure 9, because Hf is early and Ir is late in the d-block, the filled HOMO (t_2g) energy level will be energetically more accessible for [Hf(CO)₆]²⁻ than for [Ir(CO)₆]³⁺, consistent with the experimental observation that metal carbonyl anions are much more readily oxidised than non-classical metal carbonyl cations [51,52] (Appendix A Note 20) [57].

8. Cationic Ligands: Linear NO and Tropylium

It is well recognised that there are difficulties in finding a consistent description of the bonding in metal nitrosyl complexes using simple models, and this has led to much discussion over several decades. Moreover, although this debate has generally been a topic primarily of academic interest, the potential applications of metal nitrosyls, including in medicine, have recently attracted considerable attention [58,59].

Only a brief mention will be made here of the categorisation of complexes containing nitrosyl (NO) ligands because this topic is considered in more detail in refs. [9,60]. The OS method treats linear NO ligands as NO⁺ and bent NO ligands as NO⁻ with the consequent impact on the dⁿ number. This classification has been criticised as simplistic based on structural and computational evidence, which shows that there are exceptions to such assignments and several examples where the M–N–O angle lies between the expected extremes of 180° (for NO⁺) and ~120° (for NO⁻). As a result, Enemark and Feltham proposed the {MNO}ⁿ notation (cited earlier) where the value of n is the dⁿ number of the complex when the NO ligands are treated as NO⁺ [48,49] (Appendix A Note 16). Nevertheless, the NO⁺/NO⁻ classification remains a topic of continued interest, as demonstrated in the recent publications by Klüfers et al. concerning whether or not a range of {CoNO}⁸ complexes should be classified as Co(III)–NO⁻ or Co(I)–NO⁺ [61,62] (Appendix A Note 21) [63,64,65,66,67,68].

Whatever the issues around an OS-derived classification of the NO ligand, the CBC method explicitly requires that the π Z component be considered in the ML_lX_xZ_z designation for linear NO complexes when assigning the ENC and VN (and hence the dⁿ number) since the N–O π* orbitals are both the LUMO and the HOMO, which therefore have the same energy (see Figure 10) [26].

As set out in some detail in ref. [9] (Appendix A Note 22) [26,65], the precise CBC description of coordinated NO ligands depends on their number and the symmetry of the complex in which they reside. Thus, mononitrosyls are designated as X₃, dinitrosyls [as a (NO)₂ group in an octahedral complex] are designated L₂X₂ for trans or LX₄ for cis, and tetranitrosyls [as a (NO)₄ group in a tetrahedral complex] are together designated L₆, which leads to the (sometimes substantial) differences in the dⁿ assignments based on CBC and OS methods illustrated in Figure 11 for complexes 20–23.

In the case of the cycloheptatrienyl ligand, for the purposes of assigning a dⁿ number using the OS formalism, it is treated as either [C₇H₇]³⁻ or [C₇H₇]⁺ since both are aromatic systems with 10π and 6π electrons respectively (Appendix A Note 25). In contrast, in the CBC method, the π Z component associated with the C₇H₇ ligand leads to a singular assignment as an L₂X₃ ligand in all cases, and the variance between the two approaches is clear from a comparison of complexes 24 and 25 (Figure 12). From an OS perspective, the η⁷-C₇H₇ ligand is judged to be [C₇H₇]³⁻ in 24 and [C₇H₇]⁺ in 25. This is not an arbitrary judgement since the highly electropositive Ti centre in 24 makes [C₇H₇]³⁻ a chemically reasonable choice and is in line with MO calculations [70], whereas the alternative [C₇H₇]⁺ assignment for 25 is consistent with both its chemistry and its XPS spectrum and is supported by DFT calculations [71]. Both OS and CBC methods, therefore, agree on the d⁰ configuration assigned to 24 (Figure 12), but a marked difference is seen for 25, which would be assigned a d² configuration using the CBC method, whereas the OS-derived configuration is d⁶. More generally, the exclusive L₂X₃ classification for the η⁷-C₇H₇ ligand within the CBC framework will give rise to a dⁿ number consistent with the OS approach where it is appropriate to consider the ligand as [C₇H₇]³⁻, but the OS method will result in a dⁿ⁺⁴ assignment where it is deemed more appropriate to consider the ligand as [C₇H₇]⁺.

The issue of alternative η⁷-C₇H₇ assignments was also discussed in ref. [60] in relation to [Mo(η⁷-C₇H₇)(CO)₃]⁺ (9 in Figure 2), where it was argued that since this species displays tropylium-like chemistry for the C₇H₇ ligand, a [C₇H₇]⁺ assignment is preferred leading to a d⁶ configuration for 9 rather than the d² configuration derived using the CBC method.

Cationic ligands are not common, but many examples are nevertheless known (in addition to [NO]⁺ and [C₇H₇]⁺) for which the assignment of a dⁿ number using the OS method is generally straightforward [72,73,74,75,76]. However, it is important to recognise that the CBC method treats cationic ligands rather differently since, in general, an L⁺ designation transforms to X according to Scheme 1 (i), which then has consequences for the assignment of a dⁿ number (Appendix A Note 26).

9. Negative Oxidation States

How the OS and CBC methods deal with negative oxidation states is a matter we have considered before and will not reprise in detail here, but the contrast in the two approaches is illustrated by the difference between the derived dⁿ assignments for [Fe(CO)₄]²⁻ (26) and [Fe(CO)₆]²⁺ (27) shown in Figure 13 [60]. In terms of the OS description, 26 is a tetrahedral, d¹⁰-Fe(–II) complex, whereas 27 is an octahedral, d⁶-Fe(II) species. The OS description clearly signals that these two iron carbonyl species would be expected to have very different chemistries, which they certainly do: the dicationic 27 has been characterised as a ‘superelectrophile’ whilst the dianionic 26 is a powerful nucleophile that is readily oxidised [77,78]. According to the CBC method, 26 [using L⁻ → LX, Scheme 1 (ii)] and 27 [using L⁺ → X, Scheme 1 (i)] are both members of the same ENC of ML₄X₂ from which a d⁶ configuration follows [9] (Appendix A Note 27).

10. Compounds Containing Metal–Metal Bonds

Figure 14 highlights a selection of binuclear metal complexes (28–30) that feature unsupported metal–metal bonds along with the OS and CBC derived dⁿ configurations. The binuclear complex [Mn₂(CO)₁₀] (28) has previously been discussed [60], but in brief, the OS method views 28 as a Mn(0) compound comprising two d⁷-Mn(CO)₅ fragments that are linked by a σ² Mn–Mn single bond. In contrast, the CBC method treats both Mn centres as ML₅X since each Mn(CO)₅ unit is counted as an X ligand as far as the other Mn centre is concerned, resulting in a VN of 1 and a derived d⁶ configuration for each Mn according to Equation (3) [24,25,26].

Similarly, using the OS method, the ditungsten complex W₂(O^tBu)₆ (29) features two W(III) centres linked by a σ²π⁴ W≡W triple bond formed by combination of two d³-W(O^tBu)₃ fragments whilst the classic dirhenium(III) complex [Re₂Cl₈]²⁻ (30a) features a σ²π⁴δ² Re≣Re quadruple bond arising from the combination of two d⁴-[ReCl₄]⁻ units [2,79,80,81]. In contrast, it is assumed, based on the treatment of [Mn₂(CO)₁₀] outlined above, that the CBC method would classify the metal centres in 29 as MX₆ and in 30a as MLX₇ (consistent with a standard assignment of their valence) resulting in a d⁰ configuration in each case based on the VN numbers of 6 and 7 respectively. The discrepancy between the two methods is quite general, such that within the CBC framework, compounds containing metal–metal bonds will have dⁿ numbers for each metal centre that are reduced in direct proportion to the metal–metal bond order compared to the dⁿ numbers derived from an OS analysis.

Much of the chemical literature, especially for those compounds with metal–metal multiple bonds, adopts the OS approach of counting the M–M bonding electrons as part of the dⁿ count in large part because they are predominantly localised on the metal centres rather than on the ligands [79]. Our preference is the OS approach since we contend that it gives a greater insight into the chemistry of dimetallic complexes. For example, the observation that the species [Cl₄Mo≣MoCl₄]⁴⁻ (30b) is readily oxidised chemically or electrochemically can be understood in terms of its OS description as a d⁴-d⁴ Mo(II) complex [82,83] but less readily in terms of the alternative CBC description as a d⁰-d⁰ complex. Similarly, the dimolybdenum complex 31 undergoes the reaction shown in Figure 15 to give 32, which has been characterised as an oxidative addition of the peroxide RO–OR to the d³-d³ Mo(III) complex 31 to give compound 32 containing two d² Mo(IV) centres linked by a Mo=Mo double bond, albeit with two bridging alkoxides [84]. Such a rationalisation does not naturally emerge from a d⁰-d⁰ configuration that would be assigned to both 31 and 32 according to the CBC method, although one could characterise the reaction shown in Figure 15 as a Mo≡Mo/RO–OR bond metathesis reaction [85]. It should nevertheless be stated in any OS-based discussion concerning dⁿ numbers for M–M bonded compounds that some of the dⁿ electrons are involved in M–M bonding, the number of which is equal to the M–M bond order (at least to a first approximation), a caveat that is unnecessary when using the CBC approach (Appendix A Note 29) [86].

There are numerous examples of binuclear compounds with formal metal–metal bonds (single or multiple), which contain bridging (μ) ligands that are not considered here (apart from 32 above) since, in many cases, debates continue regarding the actual magnitude of the bond order [26]. Likewise, cluster compounds that incorporate metal–metal bonds are best described using cluster bonding models such as Polyhedral Skeletal Electron Pair Theory (PSEPT), sometimes referred to as the Wade-Mingos rules, since the OS and dⁿ number of individual metal centres within the cluster are less meaningful, especially as the number of the metal atoms in the cluster increases (Appendix A Note 30) [87].

11. Inverted Ligand Fields

The dⁿ configuration of the anionic complex [Cu(CF₃)₄]⁻ (33) has been the source of much, and at times strenuous, debate for over a decade [88]. The application of OS and CBC methods is apparently straightforward for this species, and, as shown in Figure 16, both methods agree on a d⁸ assignment, which is consistent with the observed square planar geometry at the Cu centre. However, based on computational analysis of the frontier orbitals in 33, it has been suggested that the resonance structures denoted as 33* in Figure 16 are a more accurate representation of the bonding with an attendant d¹⁰ Cu(I) description.

The conventional ordering of the orbitals in a square planar complex has the HOMO and LUMO localised primarily on the metal, which is a consequence of the energy of the orbitals on the ligands being lower than the energy of the metal valence orbitals; Figure 17a illustrates this for the LUMO in a normal D_4h ligand field. It is argued, however, that the CF₃ ligands in a complex such as 33 provide an inverted ligand field such that the HOMO and LUMO are localised primarily on the ligand due to the energy of the ligand orbitals being higher than the energy of the metal valence orbitals; Figure 17b illustrates this for the LUMO in an inverted D_4h field. It is notable that for the inverted ligand field, the electrons in the b_1g orbital are included in the dⁿ count, although this runs counter to the definitions of the dⁿ number given in Section 2 since this b_1g orbital is a σ bonding orbital.

The debate over whether 33 should be treated as d¹⁰ Cu(I) or d⁸ Cu(III) has even been referred to as the ‘Oxidation State Wars’, and a recent analysis provided by Klein et al. has led to the interesting suggestion that [Cu(CF₃)₄]⁻ could be described as having a quasi-d¹⁰ configuration (based on their computational analysis) but a formal-d⁸ configuration [90]. There seems to be little debate in a recent publication regarding a d⁸-Ag(III) assignment for the complex [Ag(CF₃)₄]⁻ [91].

The analogous square planar nickel(II) complex 34 is isoelectronic with 33 and would be assigned a d⁸ configuration by applying simple OS or CBC rules. However, computational and spectroscopic analysis of 34 has led to the suggestion [92] that it is better physically described as a d⁹ metal complex. In our view, this is a problematic assignment because when employing a conventional use of the dⁿ term, a d⁹ assignment for mononuclear complex 34 would carry the implication that it is paramagnetic, which it is not, although we note that the authors rationalize the observed diamagnetism as being due to anti-ferromagnetic coupling between an electron on the metal and an electron delocalized over all four CF₃ ligands.

Another example related to compound 33 is the fluorogold complex 35 (Figure 18), which is a well-characterised square planar, d⁸ Au(III) (or trivalent MLX₃) species. More recently, the oxo-gold (36) and nitrido-gold (37) complexes have also been described. These complexes are ostensibly examples of d⁷ Au(IV)/MLX₄ and d⁶ Au(V)/MLX₅ species, respectively, but a computational analysis suggests that both 36 and 37 are best treated as d⁸ Au(III) species, with one and two unpaired electrons localised on the O and N ligands respectively. This is illustrated in Figure 18 using dots to indicate the salient electrons on the F/O/N atoms, and it is argued that 36 and 37 are further examples of inverted ligand field complexes where the frontier orbitals are ligand-based [93].

In general, inverted ligand field complexes can be viewed as a subset of complexes that feature non-innocent ligands where neither the formal oxidation state nor the ML_lX_xZ_z descriptor (and associated dⁿ number) reflect the electronic structure evident from a detailed analysis of experimental (e.g., spectroscopic data) or computational studies.

12. Concluding Discussion and a Modified Definition of dⁿ

The dⁿ configuration of a transition metal complex is a unifying concept that pervades much of d-block chemistry, from organometallic/coordination chemistry and catalysis to bioinorganic chemistry and solid-state materials. Indeed, the great strength of the dⁿ heuristic lies in its simplicity and its amenability to useful generalisations that help to rationalise many of the properties of d-block metal complexes, such as those listed in Section 2. Nevertheless, there are some fundamental ambiguities associated with how a dⁿ number is determined, which arise for the following reasons: (a) dⁿ values can be intrinsically ambiguous where they denote one or other extreme in a metal–ligand bonding scheme that is actually a continuum (see Section 5 for examples); (b) in circumstances where spectroscopic and/or computational results indicate a dⁿ value that differs from dⁿ values determined according to OS- or CBC-derived rules (see Section 11); and (c) where differences arise between OS and CBC methodologies (despite the fact that the rules of each method are internally self-consistent).

There is no general resolution for any ambiguities in the assignment of a dⁿ number associated with reasons (a) and (b) since they reflect the inherent limitations of the simple dⁿ model. With regard to reason (c), for the most part, OS- and CBC-derived dⁿ numbers determined using either Equation (1) or Equation (3) are the same. However, as set out in Section 4, and the following sections, and addressed in refs. [6,24,25,26], the OS and CBC methods do differ in their dⁿ assignment for the following categories:

(i)

Complexes containing σ Z and, in some instances, π Z ligands (Section 6 and Section 7).

(ii)

Complexes containing cationic ligands (Section 8).

(iii)

Complexes in which the metal is in a negative oxidation state (Section 9).

(iv)

Complexes with metal–metal bonds (Section 10).

That such differences should occur is not unexpected given that simple or approximate models, by their very nature, will not be applicable in all situations, and the fact that two such models might diverge in certain instances is unsurprising. Having said this, we draw attention to the text below taken from ref. [6].

However, whereas oxidation numbers depend on how one chooses to deconstruct the molecule, the dⁿ configuration is a function of the molecule and must be independent of how one decides to deconstruct it. As such, a system of evaluating molecules that results in different dⁿ configurations for the same molecule according to the preference of the author is unsatisfactory.

[original emphasis]

Considering the first sentence, it is true that there are some circumstances where ambiguities arise in determining an oxidation state. More fundamentally, however, the statement, ‘…the dⁿ configuration is a function of the molecule and must be independent of how one decides to deconstruct it.’, is one we would question since, as we have argued above and will discuss further below, any definition of dⁿ is necessarily somewhat subjective and, therefore, comes with inevitable uncertainties that are inherent in any approximate model. With regard to the second sentence in the above-quoted text, we are inclined to agree and, accordingly, propose the following definition for dⁿ:

The dⁿ number of a transition metal complex is assigned based on the number of electrons that occupy the frontier orbitals, which have the same symmetry as metal d-orbitals.

This broader definition removes any reference to dⁿ electrons having to be in orbitals that are primarily non-bonding (or antibonding) and recognises (a) the fact that dⁿ is itself something of a misnomer since the electrons of interest occupy molecular orbitals that, whilst they have the same symmetry as d orbitals, are not d atomic orbitals and (b) that dⁿ is effectively a mnemonic for the number of electrons in the frontier orbitals of the complex, i.e., those electrons that are involved in the chemistry of the complex. In addition, we argue that this definition of dⁿ has the following advantages for mononuclear complexes:

(1)

It removes the issues around the subjective interpretation of the terms ‘primary’ or ‘primarily’ with regard to whether or not electrons reside in bonding orbitals (see Section 2).

(2)

It preserves the useful link between the value of dⁿ and the structural and chemical properties listed in Section 2.

(3)

It takes no account of any Z ligands and, therefore, in the case of π Z ligands, avoids any abrupt changes in dⁿ configurations as a consequence of whether π back-bonding is judged to be significant or not (see Section 7, Section 8 and Section 9).

Points (1) and (2) are self-explanatory and require no further comment. The essence of point (3) is that, according to the proposed definition of dⁿ, we start by determining a dⁿ number based on the metal OS and then subsequently add whatever caveats are necessary in relation to the compound of interest. For example, compounds with σ Z ligands are assigned an OS-derived dⁿ number, following which, it is made clear in any discussion that two of these electrons are associated, whether strongly or weakly, with the M→Z bond (see Section 6) (Appendix A Note 31). Likewise, for compounds with π Z ligands, such as in the homologous series [M(CO)₆]^Q, a d⁶ assignment is preferred, followed by an indication of the extent of any π back-donation from the metal to the CO π* orbitals, in line with the discussion presented in Section 7. Moreover, a d⁶ assignment preserves the link with the expected redox chemistry of metal carbonylate anions that are easily oxidised [54,94,95] (Appendix A Note 32) [96]. A similar caveat was discussed in Section 10 regarding an OS-derived dⁿ number for metal–metal bonded compounds and those electrons associated with the M–M bonding (Appendix A Note 33).

Much of the discussion above highlights a salient and important difference between the OS and CBC approaches, which is that the former is (obviously) based on the oxidation state, whereas the latter employs valence. Thus, if one accepts Sidgwick’s original definition for the valence of an atom as the number of electrons that the atom uses in bonding, then in complexes such as [Hf(CO)₆]²⁻ and [Re₂Cl₈]²⁻, it is inevitable that the value of n in the term dⁿ will decrease to zero since all of the metal-based electrons are involved in bonding (π back-bonding or metal–metal bonding) to some degree [10,11,12,13,97]. Furthermore, in using the OS method, an OS is first determined and the dⁿ number is assigned directly from the application of Equation (1). Any judgement about the validity of this dⁿ number then follows according to the nature of the compound, based on specific experimental and/or spectroscopic data and the insights afforded by computational studies, as illustrated with the examples given in Section 6, Section 7, Section 8, Section 9 and Section 10 above. In contrast, the CBC method requires rather less judgement and fewer caveats, but in the case of π Z ligands, the judgement required is made before a dⁿ number is assigned.

In conclusion, we emphasise that simple models will always have limitations, and any expectation that they will be applicable in all circumstances is chimerical [60]. It is nevertheless important to recognise the great value of the OS and CBC methodologies for the systematisation and classification of, particularly, transition metal compounds, in both teaching and research contexts, despite the limitations they each may have. In this regard, as we have done before [60], we will quote Frenking, Schwerdtfeger, and co-workers who offer the apposite apothegm: ‘Chemical bonding models are not right or wrong, they are more or less useful’ [98].

OS assignments continue to attract discussion (and sometimes controversy) in the chemical literature, but whatever one’s preference in terms of OS vs. CBC, we close with a final recommendation that is in line with a broadly similar statement made by Parkin in ref. [9]. Namely, in any discussion where the dⁿ number is being used to gain some insight into the chemistry of a particular compound, it is important, for reasons that must now be obvious, to specify whether n has been determined using the OS or CBC method and also to what extent any assignment derives from spectroscopic and/or computational insights (Appendix A Note 34) [99,100].