Comparison of DAFH and FALDI-like approaches

Two complementary methodologies for extracting useful insights into electronic structure and bonding from contemporary wavefunctions are compared. The first of these, known as the analysis of domain-averaged Fermi holes (DAFH), mostly provides visually appealing descriptions of the role and the extent of electron sharing in chemical bonding. The second one, known as the fragment, atom, localized, delocalized and interatomic (FALDI) charge density decomposition scheme, uses the partitioning of certain localization and delocalization indices to focus on highly visual contributions associated with individual domains and with pairs of domains, respectively. Four variants of a FALDI-like approach are investigated here in some detail, mostly to establish which of them are the most reliable and the most informative. In addition to ‘full’ calculations that use the correlated pair density, the consequences for the DAFH and FALDI-like procedures of using instead a popular one-electron approximation are explored. Additionally, the geometry dependence of the degree of acceptability of the errors that this introduces for delocalization indices is assessed for different formal bond multiplicities. The familiar molecular test systems employed for these various linked investigations are the breaking of the bonds in H2 and in N2, as well as the nature of the bonding in B2H6, as a simple example of multicenter bonding. One of the key outcomes of this study is a clear understanding of how DAFH analysis and a particular variant of FALDI-like analysis could be most profitably deployed to extract complementary insights into more complex and/or controversial bonding situations.


Introduction
Advances in understanding electronic structure and bonding continue to rely not only upon the development and application of computational strategies for carrying out accurate calculations, but also upon the development and application of reliable techniques for extracting useful insights from contemporary calculations. Some 6 decades ago, Charles Coulson suggested that a key role of quantum chemistry is to understand concepts and to show what are the essential features of chemical behavior [1] and the motto of Richard Hamming's textbook on numerical methods is that the purpose of computing is insight, not numbers [2]. Eugene Wigner has often been quoted as saying: 'It is nice to know that the computer understands the problem. But I would like to understand it too.' Somewhat more recently, Frank Neese and co-workers have reminded all of us that nowadays we need both the insights and the numbers [3].
Unsurprisingly an impressive array of techniques has been developed for extracting useful chemical insights from contemporary calculations, even though some of the most useful concepts in electronic structure and bonding do tend to be difficult to pin down with unambiguous precise definitions. Examples of such broad families of approaches include various energy partitioning schemes, those which rely on partitioning densities and/or density matrices, conceptual density functional theory and procedures for extracting descriptions of the valence bond type, just to mention but a few. Often, it turns out that combinations of techniques give the most useful and reliable insights, with one such example being EDA-NOCV [4,5] which combines a particular form of energy decomposition analysis with highly visual natural orbitals for chemical valence. In the present work, we focus on two methodologies that are linked to the partitioning of density matrices, with one of the aims being to find a combination in which they provide complementary useful information.
The two methodologies considered here have both proved to be particularly useful for visualizing the actual bonding situation in molecular systems. One of them is domain-averaged Fermi hole (DAFH) analysis [6][7][8][9][10][11][12][13][14] which is especially useful for the manner in which it depicts the role and the extent of electron sharing in bonding situations. The other is linked to the original fragment, atom, localized, delocalized and interatomic (FALDI) charge density decomposition scheme [15][16][17][18][19]. The FALDI scheme provides, amongst other things, information about the partitioning of shared electrons into the contributions of individual bonded atoms.
In essence, DAFH analysis involves integration over a chosen domain of one of the electron coordinates in the exchange-correlation density, xc 1 , 2 , so as to generate a one-electron quantity that can loosely be labeled a 'hole.' Although the eigenvectors and associated eigenvalues of a matrix representation of that 'hole' could in principle be interpreted directly as domain natural orbitals, they are more usually transformed using an isopycnic (meaning 'same density') localization procedure [20]. Examination of the forms of the resulting (nonorthogonal) one-electron DAFH functions and their corresponding occupation numbers then provides highly visual interpretations of chemical bonding. In general, it proves to be particularly useful to perform DAFH analysis for 'holes' that are averaged over (combinations of) individual quantum theory of atoms in molecules (QTAIM) [21] domains.
Except for the special case of closed-shell restricted Hartree-Fock wavefunctions, the actual construction of xc 1 , 2 does of course require use of the pair density, which is not always readily available. Fortunately, it has been shown that the significant simplifications that arise from the use instead of a convenient one-electron approximation do not lead to significant changes in the forms of the dominant DAFH functions [22], but there are modest changes to the associated occupation numbers.
Integration of the 'hole' over the same or a different domain generates quantities known as localization and delocalization indices [23], respectively. Bader argued that the information provided by these delocalization indices is independent of any association with chemical bonding between the atoms involved and, as such, the delocalization index should not be identified as a bond order [23]. Furthermore, it was argued that the delocalization index does not determine the number of Lewis-bonded pairs [23], except in the special case of equally shared electron pairs. Nonetheless, such delocalization indices have been widely used as bond orders or as surrogates for bond orders in a wide variety of applications. In particular, it has been strongly suggested relatively recently in a study of nearly 200 inorganic and organic molecules that such delocalization measures in real space could be used to revitalize the whole concept of bond order [24].
Whether or not it should actually be considered a bond order, the delocalization index is clearly an example of an electron sharing index [25], quantifying the extent of the sharing of the electron distribution between two QTAIM domains. Accordingly, we have chosen to use instead the term shared-electron distribution index (SEDI) [26], thereby also avoiding any possible confusion with alternative delocalization measures or electron sharing indices. Indeed, purely for convenience, we will refer to the corresponding localization indices as diagonal SEDI values, whereas the off-diagonal ones are of course delocalization indices. It proves straightforward to resolve such values of SEDI into contributions that involve individual DAFH functions [27] (see also [12,13]).
The original FALDI scheme [15][16][17][18][19] quantifies pseudosecond-order contributions arising from electrons within QTAIM domains. One of its outcomes is in essence a means of visualizing SEDI values (albeit calculated within the usual one-electron approximation) in terms of sets of one-electron functions and their associated eigenvalues. We note that it has been found that whereas the significant simplifications provided by the one-electron approximation introduce errors in off-diagonal SEDI values that are less than 5% for many bonds, the corresponding errors for high bond multiplicities, such as the formal triple bonds in N 2 and HCN, can exceed 10% [24]. Provided that it is readily available, it would of course be straightforward to avoid the one-electron approximation that is implicit in the original formulations of FALDI [15][16][17][18][19] by using instead the actual pair density; we could also choose to localize the resulting one-electron functions by means of an isopycnic transformation [20], exactly as in DAFH analysis. The variants of FALDI considered here have been labeled FALDI-like, to distinguish them from the original, but it is our expectation that any lessons learnt here can be carried back to the original formulation.
The main purpose of the present work is to try to provide answers to a series of interrelated questions associated with DAFH analysis, the FALDI-like approach and SEDI values: • For the different variants of the FALDI-like approach, as explained later, which are the most useful? • How similar/different are the functions generated by the DAFH and FALDI-like approaches for a particular molecule at a given level of theory and how similar/different are the corresponding eigenvalues? • Is there a combination of the DAFH and FALDI-like approaches in which they provide complementary useful information? • Can the FALDI-like approach give a more compact (or more 'efficient') expansion of SEDI values than does DAFH analysis? • Given the extent to which use of the usual one-electron approximation has been shown to affect SEDI values [24], would its use in the DAFH and FALDI-like approaches affect also the relative importance of the different functions in the expansions of SEDI values? • Do the answers to any of the above questions depend on the formal bond multiplicity and do any of the answers change for a given system when nuclear separations are increased/ decreased?
Along the way, we also examine for representative systems the extent to which the accuracy of the one-electron approximation for calculating SEDI values varies with nuclear separation.

Theoretical and computational details
A convenient starting point for introducing DAFH analysis is to define the 'hole' g Ω for a particular domain Ω as follows: (1) where ρ (1) and ρ (2) are spinless one-and two-electron densities and the I are (real) orthonormal natural orbitals with occupation numbers I . It is straightforward to construct this matrix representation Ω of the 'hole' g Ω by combining elements of the (spinless) one-and two-electron density matrices, expressed in this natural orbital basis, with socalled domain-condensed overlap integrals: A convenient one-electron approximation that can be traced back to work by Müller [28], as well as to certain bond orders introduced by Fulton [29], has been rediscovered many times and continues to be used for various purposes, especially when the two-electron density is not readily available. This one-electron approximation, which is exact in the special case of closed-shell restricted Hartree-Fock wavefunctions, effectively reduces the expression for G Ω (I, J) to √ I J S Ω (I, J). Whether we do the full calculation using the pair density or we use instead the much simpler form based on the one-electron approximation mentioned above, the eigenvectors and eigenvalues of Ω are transformed using an implementation of Cioslowski's isopycnic localization procedure [20], resulting for the chosen domain Ω in a set of (real) DAFH functions iΩ with occupation numbers n iΩ . In effect, the so-called hole g Ω for domain Ω has been re-expressed in the following simple form where the d iΩI is the expansion coefficient of the DAFH functions iΩ in the basis of the natural orbitals I : It is the inspection of visual depictions of the DAFH functions iΩ for various domains in the molecule, alongside a consideration of the corresponding occupation numbers n iΩ , which provides insights into the chemical bonding, including direct links to familiar chemical concepts such as bonds, lone pairs and so on. The DAFH analysis detects any electron pairs that remain intact within a given domain and provides information about the broken or dangling valences that are created by the (formal) bond splitting that would be required to isolate that domain from the rest of the molecule. It straightforwardly elucidates the manner and the extent to which the electrons associated with a given domain (often an atom) are involved in interactions with those for other domains in the molecule.
As was mentioned in the Introduction, further integration of g Ω over the same or a different domain generates localization and delocalization indices, respectively. Specifically, we may write: and then define 'diagonal' (i.e., one-domain) elements SEDI(Ω, Ω) = k ΩΩ and the corresponding off-diagonal (i.e., two-domain) quantities SEDI Ω, Ω � = k ΩΩ � + k Ω � Ω . It follows from Eqs. 3 and 5 that the values of k ΩΩ and k ΩΩ � (and thus SEDI) can straightforwardly be expressed [27] (see also [12,13]) as the simple summation of terms P i that involve individual DAFH functions iΩ . These P i are very straightforward to compute given the following expression: We turn now to the FALDI-like approach, which involves the generation of the eigenfunctions and corresponding eigenvalues of Ω Ω � rather than those of Ω . Whereas the matrices Ω and Ω � are symmetric, their product Ω Ω � need not be (unless Ω and Ω � commute) and so Ω Ω � can exhibit different sets of left-and right-hand eigenvectors, albeit with the same eigenvalues. Instead of solving directly the right-hand FALDI-like eigenvalue equation, namely or the corresponding left-hand FALDI-like eigenvalue problem we can in principle recast both of these equations in the more convenient symmetric form: 1∕2 Ω � from the symmetric matrix Ω � , and provided that Ω � is also positive definite, we can also easily generate −1∕2 Ω � . A simple pragmatic alternative to the use of Eqs. 7-9 is to find instead the eigenvectors and corresponding eigenvalues of the following symmetric problem: 1∕2 We note that such a strategy does in fact mimic the approach adopted in the original FALDI studies [15][16][17][18][19] that were based on a one-electron approximation to Ω . Regardless of which of Eqs. 7-10 we use, the eigenvalues must sum to k ΩΩ � (see Eq. 5) so that we can easily identify the principal contributions to a given SEDI value.
In common with what is done in DAFH analysis, we have chosen here to carry out a subsequent isopycnic transformation [20] of the corresponding eigenvectors and then to analyze the transformed functions with the largest eigenvalues. The full sets of eigenvalues will of course still sum to k ΩΩ � in each case, and the sum of all diagonal (i.e., onedomain) and off-diagonal (i.e., two-domain) k values for a given system must still match the trace of the one-electron density matrix, i.e., the number of active electrons. It is important to note a fundamental difference between the DAFH and FALDI-like approaches when it comes to the individual contributions to these different k values. On the one hand, FALDI and FALDI-like approaches involve solving separate one-domain and two-domain eigenvalue problems, thereby generating two different sets of functions, with the former related only to diagonal k values and the latter only to off-diagonal k values. On the other hand, DAFH functions, which are obtained from one-domain eigenvalue problems, contribute (whether significantly or not) both to diagonal and off-diagonal k values.
It is useful to notice that whereas full DAFH analysis requires construction of the matrix representation Ω of the 'hole' g Ω , as was described earlier, we can regenerate that matrix with minimal effort when we then go on to the FALDI-like analysis. This is because we can identify from Eq. 3 that which involves little more than the calculation of dot products. (Note that it is important in the summations in Eqs. 3 and 11 to include all of the functions, labeled by i, whether or not they were actually included in the isopycnic transformation, which deals only with nonnegative eigenvalues [20].) In order to provide answers to various questions posed in the Introduction, we first examine the breaking of the single bond in H 2 so as to compare DAFH analysis with variants of FALDI-like approaches. We also examine the geometry dependence of the reliability of the usual oneelectron approximation to the full treatment. We then do much the same for the breaking of the formal triple bond in N 2 , not least to discover whether various observations for H 2 also apply for higher bond multiplicity. Finally, we use B 2 H 6 near its equilibrium geometry as an example that features bonds that are not only heteronuclear but that also (11) involves a bonding pattern which transcends the conventional two-center paradigm. It is also useful to note that whereas the isopycnic transformation actually does relatively little to the DAFH eigenfunctions and eigenvalues in the cases of H 2 and N 2 , it does somewhat more in the case of B 2 H 6 ; these observations also turn out to be true for the FALDI-like approaches we considered. It is important to stress that our emphasis in the present work is on the testing of (combinations of) methodologies rather than on looking for new insights into the electronic structure and bonding in the chosen intentionally familiar molecular systems.
It is also important to notice that whereas for the DAFH and diagonal FALDI-like approaches, we solve a separate eigenvalue problem for each domain Ω in turn, the offdiagonal FALDI-like case does of course involve two such domains, Ω and Ω′. Although k ΩΩ′ and k Ω′Ω must coincide, the FALDI-like partitioning of them leads to two different sets of functions. Depending on their forms, we can thus expect to be able to identify (by visual inspection) the FALDI-like functions which are mostly associated with onedomain (Ω) or with the other one (Ω′). For homonuclear systems, such as H 2 and N 2 , the functions in the two sets are trivially related to one another by symmetry. On the other hand, for (say) a B-H bond in a system such as B 2 H 6 , the two sets of functions will be different from one another so that, depending on the forms of the functions, we can say that they provide a view of the bonding from either the B or the H perspective.
As is well known, a variety of definitions of bond orders have proved to be especially useful in attempts to understand electronic structure and bonding, with variants of the Wiberg-Mayer (W-M) index being particularly widely used. For the special case of correlated singlet systems, Mayer introduced a so-called improved definition [30] of such a two-center W-M index which, when re-expressed in QTAIM-generalized form [31], can be written [32]: It proves useful for the bond breaking in H 2 and N 2 to compare the geometry dependence of W−M Ω, Ω � and SEDI Ω, Ω � .
We used here full configuration interaction (FCI) descriptions of H 2 based on a standard cc-pVTZ basis. In practice, these FCI/cc-pVTZ wavefunctions for H 2 were generated for a range of nuclear separations (R) by means of '2 electrons in 28 orbitals' complete active space self-consistent field (CASSCF) calculations. We also used standard cc-pVTZ basis sets, always in spherical form, for the other systems we studied, with all of the various calculations carried out in D 2h symmetry. In the case of N 2 , we employed full-valence CASSCF ('10 electrons in 8 orbitals') descriptions, again for a range of nuclear separations, and we also carried out fullvalence CASSCF ('12 electrons in 14 orbitals') calculations for B 2 H 6 near its equilibrium geometry. (The coordinates used for the symmetry-unique atoms are available in Table S1 in the electronic supplementary material.) All of the wavefunctions required for the present work were calculated using MOLPRO [33,34] and the subsequent DAFH, FALDI-like, SEDI and bond order analysis including isopycnic transformations used our own codes, with the QTAIM analysis [21] carried out with AIMAll [35]. Using the same isovalue throughout, pictorial depictions of DAFH and FALDI-like functions were produced using Virtual Reality Markup Language (VRML) files generated with Molden [36].

Results and discussion
We start with an examination of the bond-breaking process in H 2 . Then, taking account of what we find for H 2 , we carry out similar analysis for the breaking of the formal triple bond in N 2 before moving on to consider the bonding situation in B 2 H 6 .

H 2
The geometry dependence of (off-diagonal) SEDI values and of the results of DAFH analysis for H 2 have been described in detail several times. In particular, wavefunctions at exactly the same level of theory as used in the present work were deployed by Cooper and Ponec [14] to show how combinations of DAFH and bond order analysis can be used to provide insights into the electron reorganization that accompanies the making and breaking of chemical bonds. We report in Table 1 the values of SEDI H, H ′ as a function of R, calculated using the full methodology as well as with the usual one-electron approximation. (Given that SEDI(H, H) + 1∕2 SEDI H, H � = 1 in this case, the diagonal SEDI elements provide no additional information.) We observe that the sign of the errors introduced for this system by the one-electron approximation is consistent throughout and that the magnitudes remain rather small, increasing to no more than 1.1% for the largest of the R values that we have considered. Also reported in Table 1 are values of the QTAIM-generalized 'improved' W-M index, as defined in Eq. 12. We observe that the reduction in the value of SEDI H, H ′ with increasing R is similar to the corresponding behavior of the W-M index, although the two curves are not parallel. Even so, it is clear that SEDI H, H ′ behaves very much like a bond order.
Dominant DAFH functions associated with the domain of one of the H atoms in H 2 are shown in Fig. 1 for three representative values of R. As in previous work, we use here the term 'dominant functions' to signify all of those with eigenvalues (or occupation numbers) of at least 0.1. Also displayed in Fig. 1 for each function is the corresponding occupation number as well as the proportion of k HH � = 1∕2 SEDI H, H � that is contributed by the value of P i (see Eq. 6). Quoting these proportions (expressed as percentages) rather than the actual numerical values of P i proves to be somewhat more convenient for direct comparisons with the corresponding results when using instead the one-electron approximation. In each case, the occupation number is fairly close to unity and the proportion of k HH ′ is very high. As can be seen from Figure S1 in the electronic supplementary material, the analogous DAFH functions and numerical values generated using the one-electron approximation are quite similar to those presented here for the full calculation.
Near equilibrium geometry, the dominant DAFH function for each H atom domain has an occupation close to unity (0.992). These two functions have a high overlap (0.984), with each of them resembling a slightly asymmetric version of a 1σ g molecular orbital. Clearly, such a DAFH description corresponds rather closely to an almost doubly occupied 1σ g orbital. This is of course consistent with the occupation number of the corresponding natural orbital, which is also close to two (1.964). Increasing the nuclear separation leads to further asymmetry in the dominant DAFH functions and to a reduction in the overlap between them, corresponding to a decrease in the extent of sharing, but it has relatively little effect on their occupation numbers (which gradually move slightly closer to unity).
Of course, from a purist point of view, there is something slightly unsettling about some of these results not least because, in the absence of same-spin interactions, there can be no Fermi hole [37]. Nonetheless, the DAFH analysis does lead to functions that can easily be interpreted in terms of the R-dependence of the bonding and to values of SEDI H, H ′ which, in general terms, display the sort of behavior anticipated for a bond index.
We now turn to the results of the FALDI-like approaches, for which we have considered the four variants introduced above: A. Direct use of z ΩΩ � and ΩΩ � from the solution of Eq. 9. B. Use of v ΩΩ � and s ΩΩ � from the artificially symmetrized problem in Eq. 10. C. Use of the left-hand eigenvectors x ΩΩ � = −1∕2 Ω � z ΩΩ � following solution of Eq. 9. D. Use of the right-hand eigenvectors y ΩΩ � = 1∕2 Ω � z ΩΩ � following solution of Eq. 9.
For the same nuclear separations as are considered in Fig. 1, the resulting dominant FALDI-like functions that can be associated (by visual inspection) with the domain of one of the H atoms are shown in Fig. 2 together with their eigenvalues and the relative contributions to the relevant k values. For each value of R, the first row corresponds to partitioning of the (diagonal, i.e., one-domain) k HH value and the second one to the corresponding partitioning of the (off-diagonal, i.e., two-domain) k HH ′ value. The columns are labeled A-D according to the variant of the approach that was used. As can be seen from Figure S2 in the electronic supplementary material, the analogous results when using the one-electron approximation are rather similar to those presented here for the full calculation. An immediate observation from Fig. 2 is that scheme B, namely use of the artificially symmetrized problem (Eq. 10), can produce eigenvalues that exceed the relevant k values. They are of course balanced by contributions from functions with negative eigenvalues, but the fact that the dominant functions can contribute more than 100% seems to be an unwelcome distraction. As such, we do not consider any further the results for H 2 from scheme B. For somewhat different reasons, we also do not dwell on the results for H 2 from scheme D, namely use of the right-hand eigenvectors y ΩΩ � = 1∕2 Ω � z ΩΩ � following solution of Eq. 9. This is because the forms of these functions turn out to be rather similar to those from the DAFH analysis and so it seems unlikely that the results of scheme D will provide much additional information. This leaves us with Schemes A and C, for which the two sets of results turn out to be rather similar. From a purely computational point of view, Scheme A (direct use of the functions from the solution of Eq. 9) is slightly preferable to Scheme C because it avoids the requirement to generate S −1∕2 Ω � (as would be required for the construction of Focusing on the results of Scheme A, we observe in each case that the dominant FALDI-like function for a given value of R accounts for a high proportion (> 97%) of the relevant k value. Except for the largest of the three nuclear separations considered in Fig. 2, the dominant functions from the partitioning of diagonal (i.e., one-domain) and offdiagonal (i.e., two-domain) k values are remarkably similar to one another. Looking first at the dominant function for the diagonal case (top row for each value of R), we notice that it becomes increasingly s-like at larger R and that it is somewhat more localized for all three R values than was the case for the corresponding DAFH functions. Turning now to the dominant FALDI-like functions for the off-diagonal case, we notice that these are also somewhat more localized than are the corresponding DAFH functions, but that there is a deviation from pure s-like character for the largest value of R shown in Fig. 2-exactly the same behavior is observed for the Scheme C functions.
Having examined the breaking of the single bond in H 2 , we can now move onto a comparison of our observations with those that we can make for the formal triple bond in N 2 .

N 2
It is clear from the SEDI N, N ′ values for N 2 which are reported in Table 2 that the magnitudes of the errors introduced for this system by the one-electron approximation are somewhat larger than was the case for H 2 , as might have been anticipated from the larger number of electrons. Furthermore, the errors show much more dependence on R for breaking the formal triple bond in N 2 than they do for the single bond in H 2 . An error of 18% near the equilibrium geometry of N 2 increases to 24% at intermediate R before changing sign at larger nuclear separations. Also reported in Table 2 are values of the 'improved' W-M index and we observe that the reduction in the value of SEDI N, N ′ in the region from 1.5 to 2 Å is somewhat more gradual than is the corresponding behavior of the W-M index. (Given for these calculations with 10 active electrons that SEDI(N, N) + 1∕2 SEDI N, N � = 5 , the diagonal SEDI elements provide no additional information.) Dominant DAFH functions associated with the domain of one of the N atoms are shown in Fig. 3 for three representative values of R with the smallest of the three values (1.1 Å) being close to R e . Also displayed for each of these functions is the corresponding occupation number as well as the proportion of k NN � = 1∕2 SEDI N, N � that is contributed by the value of P i (see Eq. 6). As can be seen from Figure  S3 in the electronic supplementary material, the analogous DAFH functions and numerical values generated using the one-electron approximation are similar to those presented here for the full calculation.
It is clear from Fig. 3 that we observe a consistent pattern for the dominant DAFH functions for N 2 at the different values of R. The first function, with an occupancy approaching two, resembles a nonbonding 'lone pair' σ function near equilibrium geometry and evolves into an s orbital at larger R. It accounts for ~ 50% of k NN � = 1∕2 SEDI N, N � near equilibrium, dropping toward 40% as the two atoms are moved apart. Next, we observe a degenerate pair of functions of π symmetry, each with an occupation number that is close to unity. Each of these functions accounts for 18% of 1∕2 SEDI N, N � near the equilibrium geometry, but this proportion increases slightly as R is increased. Finally, we observe another σ function, again with an occupancy that is close to unity. It evolves from a σ bonding function near equilibrium geometry into a 2p z function at larger R. Although it contributes less than 16% of 1∕2 SEDI N, N � near the equilibrium geometry, this proportion increases with R (albeit to a slightly larger fraction of a somewhat smaller total number).
For the same nuclear separations as are considered in Fig. 3, the resulting dominant Scheme A FALDI-like functions that can be associated (by visual inspection) with the domain of one of the N atoms are shown in Fig. 4 together with their eigenvalues as well as the relative contributions that these make to the relevant k values. The corresponding results when using instead Schemes B-D (see  Looking at the dominant Scheme A FALDI-like functions for the two-domain off-diagonal (NN′) case for R = 1.1 Å (near R e ), we observe in Fig. 4 a σ function that accounts for nearly 40% of k NN ′ and then a pair of degenerate π functions, each of which accounts for nearly 30% of k NN ′ . Moving on to R = 1.5 Å, we find that the π functions become somewhat more localized on the relevant N atom domain and now contribute a slightly reduced proportion of k NN ′ , with an increase to 45% for the σ function. It is clear for both of these values of R that the dominant Scheme A FALDIlike functions provide a more efficient representation of k NN ′ than was the case for the corresponding DAFH functions. This is because just three FALDI-like functions (one σ and two π) account for most of k NN ′ , whereas we require instead four DAFH functions (two σ and two π). Moving to larger R (2.5 Å), the value of k NN � = 1∕2 SEDI N, N � drops below 0.1 (see Table 2) and so it is not surprising that none of the Scheme A FALDI-like functions reached our chosen threshold (eigenvalue > 0.1) to be displayed.
Moving on to the one-domain diagonal (NN) case, we observe from Fig. 4 that there are striking similarities to the outcome of the DAFH analysis (see Fig. 3). Indeed, the dominant Scheme A FALDI-like functions account for almost the same proportion of k NN = SEDI(N, N) as did the DAFH functions for k NN � = 1∕2 SEDI N, N � . Focusing instead on the differences, we observe in particular that the degenerate  . 3 Dominant DAFH functions associated with the domain of one of the N atoms in N 2 at three representative nuclear separations. Also shown for each function is the corresponding occupation number as well as the proportion of k NN � = 1∕2 SEDI N, N � which can be assigned to a term P i (see Eq. 6) that involves this function pair of π functions is much more localized to a single N atom domain that are the corresponding DAFH functions. As our final example, we move on to B 2 H 6 near its equilibrium geometry. Not only does this system feature bonds between different types of atom, unlike the cases of H 2 and N 2 , but it also presents a bonding pattern that goes beyond the usual two-electron two-center paradigm.

B 2 H 6
Various off-diagonal SEDI Ω, Ω � values for B 2 H 6 are reported in Table 3. The corresponding diagonal elements are also listed (last three rows of Table 3) for a terminal hydrogen atom (H t ), a bridging hydrogen atom (H b ) and for a boron atom. We have used primes to distinguish between atoms of the same type, such that SEDI H t , H t ′ corresponds to the value for two different terminal H atoms that are attached to the same boron center. For the directly bonded atoms, the error from use of the one-electron approximation is somewhat smaller than was the case for N 2 , varying only from −1.4 to + 2.1%. On the other hand, it is certainly true that there are large deviations for some of the other cases, but it is important to remember that the corresponding SEDI Ω, Ω � values are actually very small, with the consequence being that large percentage changes correspond to rather small absolute changes. This is especially true for the case of SEDI B, B ′ .
The dominant DAFH functions for various QTAIM domains in B 2 H 6 are depicted in the first column of Fig. 5, together with occupation numbers and the proportion of a relevant k ΩΩ � = 1∕2 SEDI Ω, Ω � value. The function in the top row (occupation 1.526), which is for the domain of one of the terminal hydrogen atoms, represents the main contribution from that atom to one of the B-H t bonds and it accounts for nearly 94% of 1∕2 SEDI B, H t . The function in the second row (occupation 0.346), which is instead for the domain of the corresponding boron atom, is the complementary contribution to this B-H t bond, and it accounts for over 95% of 1∕2 SEDI B, H t . (Notice also that 1.526 + 0.346 is almost 1.9, consistent with conventional notions of a twocenter two-electron B-H t bond.) The function in the third row (occupation 0.192) is for the domain of the same boron atom and represents the main contribution from this center to the bonding that involves one of the bridging hydrogen atoms, but it only accounts for ~ 85% of 1∕2 SEDI B, H b . Finally, the function in the bottom row (occupation 1.419) is for the domain of that bridging hydrogen and it does of course represent the main contribution from that atom to the anticipated three-center two-electron (3c-2e) B-H b -B′ bonding. (Notice also that 2 × 0.192 + 1.419 slightly exceeds 1.8, consistent with notions of 3c-2e bonding.) This function only accounts for ~ 79% of each 1∕2 SEDI B, H b value, being somewhat more suited to the diagonal term SEDI H b , H b , for which it accounts for 97.7%.
The dominant Scheme A FALDI-like functions are depicted in the second and third columns of Fig. 5. The corresponding results using Schemes B-D (see Figures S11-S13 in the electronic supplementary material) again show that the dominant Scheme B functions tend to overestimate k values, that the outcome of Scheme D is strongly reminiscent of the results of the DAFH analysis, and that it is the results from Schemes A and C which are rather similar to one another. Additionally, the corresponding functions and numerical values for Schemes A-D when using instead the usual one-electron approximation are again similar to those from the full calculations (see Figures S14-S17 in the electronic supplementary material).
Looking first at the dominant Scheme A FALDI-like functions for the off-diagonal (i.e., two-domain) SEDI values (middle column of Fig. 5), the top two functions correspond to one of the B-H t linkages, but seen from the H t and B perspectives, respectively. Each function accounts for a high proportion of 1∕2 SEDI B, H t . In a similar fashion, the final two functions in the middle column correspond to one of the B-H b linkages, as seen from the B and H b perspectives, but both functions account for a slightly lower proportion of 1∕2 SEDI B, H b than was the corresponding case for B-H t . Nonetheless, in general terms, we observe that the Scheme A FALDI-like functions do provide a slightly more efficient expansion of SEDI values that do those from the corresponding DAFH analysis. The third column of Fig. 5 shows depictions of the dominant Scheme A FALDI-like functions for the diagonal (i.e. one-domain) cases, specifically H t and H b , each of which accounts for a high proportion of the relevant SEDI(Ω, Ω) element. The largest eigenvalue for the B domain was below our usual cutoff (0.1), and so this function has not been shown. (It is for much the same reason that we have not depicted here the FALDI-like functions for the two-domain BB′ and HH′ cases.) We notice that the 'diagonal' FALDI-like functions shown in the third column of Fig. 5 are rather reminiscent of the DAFH functions associated with the same atomic domains.

Conclusions
We have presented here a systematic comparison of the performance of two complementary methodologies, namely DAFH analysis and FALDI-like analysis, both of which aim to provide useful insights into electronic structure and bonding by means of partitioning the densities and/or the density matrices provided by contemporary calculations. Whereas DAFH analysis provides information about the role and the extent of electron sharing in bonding situations, the complementary FALDI-like picture is in some senses somewhat richer. In particular, FALDI-like analysis provides detailed insights into the partitioning of localization and delocalization indices (collectively called SEDI) into contributions associated with individual domains and with pairs of domains, respectively.
We are now in a strong position to provide some answers to the series of interrelated questions that were posed in the Introduction. We wondered just how similar/different are the functions generated by the DAFH and FALDIlike approaches for a particular molecule at a given level of theory, and how similar/different are the corresponding eigenvalues. A minor complication is that Ω Ω � need not be symmetric, so that we are faced with more than one FALDI-like scheme. A simple pragmatic approach (which we called Scheme B) of using instead 1∕2 Ω Ω � + Ω � Ω turns out not be wholly satisfactory because the combined contributions from the dominant functions often exceed 100%. The left-hand eigenfunctions of Ω Ω � (Scheme D) do turn out to be rather similar to those generated by DAFH analysis, but the eigenvalues are somewhat different, in one case being a direct measure of the contribution to the relevant k value and in the other being an occupation number. On the other hand, the right-hand eigenfunctions of Ω Ω � (Scheme C) are somewhat different so that they could, in principle, provide additional, complementary information that is not already available from the DAFH analysis. Our computationally preferred approach (Scheme A), which is based on the convenient symmetric form 1∕2 Ω � Ω 1∕2 Ω � , produces results which are rather similar to those from Scheme C. The consequences of our findings as to which of the different variants of the FALDI-like approach are the most useful can of course now be carried back to implementations of the original FALDI scheme [15][16][17][18][19]. In the remainder of this discussion, we consider only FALDI-like Scheme A.
One aim of the FALDI-like approach is to provide a compact description corresponding to expansions of the SEDI values. As such, an obvious question is whether such expansions are more compact (or more 'efficient') than are those provided by DAFH analysis via the values of P i Ω, Ω � calculated using Eq. 6. As might have been anticipated, we found for H atom domains in H 2 at each value of R that there is a single dominant DAFH function and that there is a single dominant FALDI-like function. Each of these accounts for most of the total k HH � = 1∕2 SEDI H, H � value, but the proportions are slightly higher with the FALDI-like approach, especially for larger nuclear separations. In this sense, the preferred FALDI-like approach (Scheme A) could be said to provide a slightly more efficient expansion of two-domain SEDI values than does DAFH analysis.
The errors in SEDI Ω, Ω � values from use of the usual one-electron approximation turned out to be rather small (up to 1.1%) for breaking the formal single bond in H 2 , and they all have the same sign. On the other hand, the situation turned out to be somewhat different for breaking the formal triple bond in N 2 . Not only did an error of 18% near the equilibrium geometry increase to 24% at intermediate R, but it then changed sign at larger nuclear separations. The corresponding errors at a fixed geometry of B 2 H 6 were somewhat smaller for the directly bonded atoms, varying only from −1.4 to + 2.1%. Nonetheless, it turns out that the forms of the dominant DAFH and FALDI-like functions from a given Scheme show rather little sensitivity to use of the one-electron approximation. Furthermore, the differences in the relative importance of the different functions in the expansions of the nontrivial SEDI values (when expressed as percentages) were always small, regardless of the formal bond multiplicity or of variations in the nuclear separations.
Having now established which of the four FALDI-like variants that we have considered provides information that complements DAFH and bond order analysis, we are now in a strong position to deploy such a combination of techniques for studies of systems with unknown and/or controversial bonding patterns. Based on the various observation in the present work, our preference will be for the full treatments, whenever they are possible, but we do now have clear indications of the extents to which we can rely instead on the somewhat more convenient one-electron approximation.

Funding
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