research papers
Can Xray constrained Hartree–Fock wavefunctions retrieve electron correlation?
^{a}CNRS, Laboratoire SRSMC, UMR 7565, Boulevard des Aiguillettes, BP 70239, VandoeuvrelèsNancy, F54506, France, ^{b}Université de Lorraine, Laboratoire SRSMC, UMR 7565, Boulevard des Aiguillettes, BP 70239, VandoeuvrelèsNancy, F54506, France, and ^{c}Department of Chemistry and Biochemistry, University of Bern, Freiestrasse 3, Bern 3012, Switzerland
^{*}Correspondence email: alessandro.genoni@univlorraine.fr, piero.macchi@dcb.unibe.ch
The Xray constrained wavefunction (XCWF) method proposed by Jayatilaka [Jayatilaka & Grimwood (2001), Acta Cryst. A57, 76–86] has attracted much attention because it represents a possible third way of theoretically studying the electronic structure of atoms and molecules, combining features of the more popular wavefunction and DFTbased approaches. In its original formulation, the XCWF technique extracts statistically plausible wavefunctions from experimental Xray diffraction data of molecular crystals. A weight is used to constrain the pure Hartree–Fock solution to the observed Xray structure factors. Despite the wavefunction being a single Slater determinant, it is generally assumed that its flexibility could guarantee the capture, better than any other experimental model, of effects, absent in the Hartree–Fock Hamiltonian but present in the structure factors measured experimentally. However, although the approach has been known for long time, careful testing of this fundamental hypothesis is still missing. Since a formal demonstration is impossible, the validation can only be done heuristically and, to accomplish this task, Xray constrained Hartree–Fock calculations have been performed using amplitudes computed at a very high correlation level (coupled cluster) for selected molecules in isolation, in order to avoid the perturbations due to intermolecular interactions. The results show that a singledeterminant XCWF is able to capture the effects only partially. The largest amount of is extracted when: (i) a large external weight is used (much larger than what has normally been used in XCWF calculations using experimental data); and (ii) the highorder reflections, which carry less information on the are downweighted (or even excluded), otherwise they would bias the fitting towards the unconstrained Hartree–Fock wavefunction.
Keywords: electron correlation; electron density; Xray diffraction; Xray constrained wavefunctions; constrained Hartree–Fock wavefunctions; density functional theory.
1. Introduction
As is well known, Xray scattering is the Fourier image of the dynamic electrondensity distribution. It is now well established that, by fitting suitable multipole models or through i.e. deformations from the spherical distributions around the atoms, are due to the formation of covalent bonds or to the localization of lone pairs. They can be easily observed and they were the first to be visualized in the early experiments of the 1960s (Coppens, 1967). Intra or intermolecular electrostatic interactions induce electrondensity polarizations, which are obviously proportional to the strength of the field. In strong hydrogen bonds, polarization of the lone pairs of acceptor atoms has been visualized since the 1980s (Stevens & Coppens, 1980). More recent studies have demonstrated that even much weaker interactions (like agostic bonds) leave clearly visible fingerprints in the electrondensity distribution (Scherer et al., 2015). The third level of electrondensity deformation features is, in contrast, much subtler, because it is much more difficult to visualize and quantify the deformations produced by Detecting relativistic effects in singlecrystal Xray diffraction experiments is equally difficult, as recently discussed by Bučinský et al. (2016) and by Batke & Eickerling (2016).
estimations, one can reconstruct accurate electrondensity maps from the Xray diffraction of a crystal. Furthermore, the technical developments that have occurred over the past few years constantly encourage the ever deeper exploration of the information contained in Xray diffraction amplitudes. There are three main levels of detail that one may observe: (i) the of the atomic orbitals due to chemical bonding; (ii) the electron polarization caused by inter or intramolecular electrostatic interactions; and (iii) the tiny electrondensity redistribution, which reflects the instantaneous electron–electron repulsions (electron correlation). Details of the first kind,Correct treatment of the ; Helgaker et al., 2000), in particular for obtaining reliable and detailed pictures of the chemical bonding, the electronic transitions and the dynamics of a system. This continuously fosters new theoretical strategies aiming to include as much as possible. In this context, a role of paramount importance is obviously played by the well established postHartree–Fock ab initio techniques (Szabo & Ostlund, 1996; Helgaker et al., 2000), e.g. coupled cluster and manybody perturbation methods. These strategies are based on a multideterminant wavefunction Ansatz and they enable the systematic improvement of the description of manyelectron systems, despite their generally large computational cost. Of course, the preferred methods are those maximizing the marginal utility of these expensive theoretical calculations. A completely different approach is represented by the density functional theory (DFT; Parr & Yang, 1989; Perdew et al., 2009; Jones, 2015) which, relying on the Hohenberg and Kohn theorem (Hohenberg & Kohn, 1964), is in principle exact and has the great computational advantage of using only the threedimensional electron density as a basic physical entity, instead of the multivariable wavefunction. Nevertheless, the exact functional relationship between the groundstate electron density and the groundstate energy of an electronic system remains unknown. Therefore, mainly exploiting the Kohn–Sham approach (Kohn & Sham, 1965), all the currently known DFTbased strategies allow only approximate treatments of the using exchangecorrelation functionals. These are ranked in classes of increasing complexity (Riley et al., 2007), as for example in the `Jacob's ladder' proposed by Perdew & Schmidt (2001). However, unlike the postHartree–Fock methods, moving upwards from one rung of the ladder to the next does not always guarantee a systematic improvement in the calculations.
is crucial for a proper theoretical description of molecules and solids (Szabo & Ostlund, 1996Although not yet commonly adopted by the theoretical chemistry community, a third way of investigating manyelectron systems is the Xray constrained wavefunction (XCWF) approach, originally proposed by Jayatilaka and coworkers (Jayatilaka, 1998, 2012; Jayatilaka & Grimwood, 2001; Grimwood & Jayatilaka, 2001; Bytheway, Grimwood, Figgis et al., 2002; Bytheway, Grimwood & Jayatilaka, 2002; Grimwood et al., 2003). This technique is one of the most successful developments of the pioneering strategies introduced by Clinton and Massa since the 1960s (Clinton et al., 1969; Clinton & Massa, 1972; Clinton et al., 1973; Frishberg & Massa, 1981) and it allows the extraction of plausible model wavefunctions from experimental Xray diffraction data. The XCWF method can be considered as a way of merging wavefunction and DFT methods. In principle, this strategy should intrinsically be able to introduce effects because it uses real observations. This could automatically lead to the definition of `correlation density', which can be regarded as the difference between charge distributions corresponding to Xray constrained and unconstrained Hartree–Fock wavefunctions. This possibility was one of the initial motivations of Jayatilaka's work, but two main drawbacks undermine the ease of this interpretation: (i) an Xray constrained molecular wavefunction is computed, on the one hand, exploiting the Hartree–Fock Hamiltonian for the isolated molecule, which neglects intermolecular fields, and, on the other hand, imposing a constraint on the Xray intensities, which include the crystal field effects; and (ii) the experimental errors that affect the measurements and propagate in the wavefunction coefficients.
Apart from these two caveats, so far no accurate analysis has ever been attempted to demonstrate whether the Xray constrained wavefunctions are indeed able to incorporate the effects of et al., 2010; Genoni, 2013a,b; Dos Santos et al., 2014; Genoni & Meyer, 2016) are actually intrinsically able to capture the effect of on electron density. This is crucial, not only to assess further the capabilities of the Xray constrained wavefunction methods, but also to open up unprecedented perspectives in the search for new density functionals, since, to the best of our knowledge, experimental (or theoretical) Xray structure factors have not been exploited for this purpose.
Therefore, the main goal of this paper is to investigate whether, and to what extent, the Jayatilaka approach and all its later developments (HudákBecause of the abovementioned pitfalls, we identified only one way to answer unequivocally the question in the title of this article. The strategy consists of testing whether, by the reciprocalspace constraint, an intrinsically uncorrelated single Slater determinant wavefunction is able to reproduce the electron distribution of a highly correlated wavefunction for an isolated molecule. This procedure does not make use of experimental data, where the information is actually convoluted with experimental errors and crystal field perturbations and lacks a suitable reference. At the same time, it does not use theoretical structure factors obtained from ab initio periodic calculations, because the effects of and intermolecular interactions would be entangled. All these possibilities are synthetically schematized in Fig. 1.
The paper is organized as follows. First, we will briefly review the theory of the Jayatilaka approach and we will dedicate a section to the computational details of our investigation. We will then show and comment on the obtained results, and finally we will draw general conclusions.
2. Theoretical background
For the sake of completeness, we remind the reader that the XCWF strategy assumes that it is working with an effective molecular crystal constituted by noninteracting units described by electronic wavefunctions that are formally identical and mutually related through the crystal symmetry operations. Moreover, the assumption that each noninteracting unit can be associated with a symmetryunique portion of the crystal N_{m} crystalunit electron distributions ρ_{k}(r), which can be simply obtained from the reference distribution ρ_{0}(r) through the unitcell symmetry operations {R_{k}, r_{k}}
enables the expression of the unitcell electron density as a sum ofEquation (1) is strictly exact if ρ_{0}(r) is an exact partition of the total electron density of the To guarantee this, in Jayatilaka's approach ρ_{0}(r) is the electron density associated with the single Slater determinant that not only variationally minimizes the electronic energy of the reference unit, but also reproduces a set of observed amplitudes , measured experimentally or calculated theoretically. In other words, an external constraint ensures that the global electron density of the fictitious noninteracting crystal is identical to the electron distribution of the corresponding real interacting system (Jayatilaka & Grimwood, 2001). Therefore, the method becomes equivalent to finding the molecular orbitals (MOs) of the Slater determinant that minimize the following functional:
where E_{0} is the energy associated with the Slater determinant of the reference crystal unit, λ_{J} is an external adjustable parameter that is varied during the calculations and which represents the strength of the external constraint, χ^{2} is the measure of statistical agreement between the calculated and experimentally (or theoretically) collected amplitudes, Δ is the desired agreement between those quantities (typically fixed to 1.0 in the case of experimental data) and stresses the functional dependence on the occupied MOs. In particular, χ^{2} is expressed as
with N_{r} the number of collected Xray diffraction data, N_{p} the number of adjustable parameters (in this case only the external multiplier λ_{J}), h the triad of labelling the reflection, σ_{h} the corresponding to each observed amplitude and η a scale factor that, in the case of experimental constraints, is properly determined in order to minimize χ^{2}, whereas in the case of theoretical constraints it is simply set equal to 1.0 (Genoni, 2013b).
Now, exploiting the definition of the
operatorwhere B is the reciprocallattice matrix and and are both hermitian operators, it is easy to show that finding the MOs that minimize the functional of equation (2) is equivalent to solving the following modified Hartree–Fock equation:
where the Fock–Jayatilaka operator is given by
with the usual Fock operator and with the multiplicative constant K_{h} expressed as
3. Computational details
3.1. Computational strategy
To conduct our investigations we considered six small/mediumsized systems, namely the two very simple diatomic molecules N_{2} (nitrogen molecule) and CN^{−} (cyanide anion) and the polyatomic molecules water, urea, benzene and glycine. For each of them we performed both traditional quantum chemistry calculations and Xray constrained wavefunction computations, described below.
3.1.1. Gasphase calculations
The geometries of the different molecules were optimized at the CCSD (coupled cluster with single and double excitations; Čížek, 1966; Purvis & Bartlett, 1982) level with the 6311++G(2d,2p) basis set. Afterwards, using the same set of basis functions and the obtained minimum structure, traditional singlepoint calculations were performed at the RHF (restricted Hartree–Fock), CISD (configuration interaction with single and double excitations; Shavitt, 1977) and DFT levels. In particular, for the DFT calculations we adopted the BLYP (Becke, 1988; Lee et al., 1998), B3LYP (Becke, 1988; Lee et al., 1998; Stephens et al., 1994; Hertwig & Koch, 1997), VSXC (Van Voorhis & Scuseria, 1998) and B1B95 (Becke, 1988; Becke, 1996) functionals, which are GGA (generalized gradient approximation), hybridGGA, metaGGA and hybridmetaGGA functionals, respectively. All these traditional quantum chemistry computations were carried out using the GAUSSIAN09 package (Frisch et al., 2009).
3.1.2. computation
After determination of the CCSD/6311++G(2d,2p) molecular electron densities, for each system we computed theoretical Xray θ/λ = 2.0 Å^{−1}. was sampled with different unitcell sizes, either cubic with a large unitcell edge or adapted to the molecular volume and shape. The unitcell volume played only a negligible role, therefore for the sake of all results presented in this paper refer to a constant type of (a = b = c = 10.0 Å), all tested to be sufficiently large for the molecules under examination.
amplitudes as analytic Fourier transforms up to a resolution sinUsing a gasphase molecular electron density, we avoided any intermolecular interaction in the estimation of the structure factors and, using a large it is easy to observe that, for most of the highangle reflections, the difference between the CCSD and RHF values is very small. This is not surprising, because is expected to modify the contracted (core) electron density only as an indirect effect of the chemical bonding (Gatti et al., 1988; Boyd & Wang, 1989). On the contrary, the largest discrepancies occur at low angles.
we also avoided any artefact due to the superposition of vicinal unitcell densities. The same procedure was used for the calculation of the theoretical amplitudes at the RHF/6311++G(2d,2p) level. The CCSD and RHF reflections were compared afterwards. In Fig. 23.1.3. Xray constrained Hartree–Fock calculations
The ab initio CCSD amplitudes were then exploited to perform Xray constrained restricted Hartree–Fock (XCRHF) calculations (Jayatilaka & Grimwood, 2003) using the same 6311++G(2d,2p) basis set. In particular, all the XCRHF computations were carried out (i) considering a pseudocrystal with P1 and with the same pseudocubic adopted for the computation of the theoretical reflections, and (ii) varying the adjustable parameter λ_{J} from 0 to 10.0 with a step of 0.5. Furthermore, as well as performing calculations with only the complete set of reflections (sinθ/λ ≤ 2.0 Å^{−1}), six further subsets of decreasing maximum resolution were also selected (sinθ/λ ≤ 1.5, 1.2, 0.9, 0.7, 0.5, 0.25 Å^{−1}) in our Xray constrained computations. Since in all the considered cases only purely theoretical amplitudes were used as constraints, the standard uncertainties σ_{h} and the scale factor η [see equation (3)] were set equal to 1.0. Furthermore, treatment of the thermal motion has been completely neglected (i.e. anisotropic displacement parameters set to 0.0). All the XCRHF calculations were carried out using the free software TONTO (Jayatilaka & Grimwood, 2003).
3.2. Comparison of the electron densities
As already mentioned in the Introduction, our main goal is to determine whether and to what extent the Xray constrained wavefunction approach is able to capture the effects of the on electron density. To accomplish this task, we have mainly compared the obtained electron densities by considering two different benchmarks: (i) the RHF charge density, corresponding to a totally uncorrelated electron distribution; and (ii) the CCSD electron density, i.e. the correlated charge distribution from which we calculated the structure factors. In particular, to analyse the differences between the various electron densities, we have used several indicators that are briefly described here.
3.2.1. Topological agreement index
This index, TI, is calculated from the electron densities at the different topological bond critical points (r_{b}) and it is defined as
where ρ_{CCSD}(r_{b}) is the electron density at the CCSD level of theory, ρ_{RHF}(r_{b}) is the unconstrained and uncorrelated Hartree–Fock electrondensity distribution and ρ_{M}(r_{b}) is the charge density to be compared, namely a charge density obtained from one of the considered `correlated methods' (i.e. obtained from an XCWF, a DFT or another postHartree–Fock calculation). Of course, for complete similarity with the CCSD and RHF benchmark values, the topological agreement index is equal to 0 or 100, respectively. A similar index could also be defined for other functions (for example the Laplacian, the electrondensity gradient, the electrostatic potential etc.) evaluated at the bond or at other critical points. Here, however, we report results only for ρ(r_{b}).
3.2.2. Realspace R (RSR) value
The RSR value is calculated on a grid of n_{p} points as (Jones et al., 1991)
In this case, only the CCSD electron density has been used as a reference and complete similarity between the CCSD and the model M charge distribution under examination corresponds to RSR = 0.
3.2.3. Euclidean Carbó distance
The Euclidean distance d_{IJ} between two molecular electron distributions ρ_{I}(r) and ρ_{J}(r) was defined by Carbó and coworkers (Carbó et al., 1980; Carbó & Calabuig, 1992) as
where the overlaplike similarity Z_{IJ} is given by
Of course, complete similarity occurs when d_{IJ} is equal to zero, while smaller similarities correspond to larger values of the index.
3.2.4. Rootmeansquare deviation and mean absolute deviation
For the sake of completeness, the more traditional rootmeansquare deviation (RMSD) and mean absolute deviation (MAD) between two electron densities were also considered. They are defined, respectively, as
and
with n_{p} the number of points on the electrondensity grids.
3.2.5. Attachment and detachment densities
Finally, another indicator of the amount of et al., 1995). The detachment density is originally defined as that part of the groundstate electron distribution that is removed and rearranged into the attachment electron density after an electronic transition. In our study, for all the examined molecules, the unconstrained RHF wavefunctions always represented our `pseudoground states' (starting states), while the correlated wavefunctions (i.e. the CCSD, CISD and XCRHF wavefunctions) were considered as `pseudoexcited states' (final states).
captured by means of the XCWF calculations is provided by the detachment and attachment densities (HeadGordon4. Results and discussion
In agreement with previous studies (Gatti et al., 1988; Boyd & Wang, 1989), the effect of the on ρ(r) is tiny and therefore difficult to capture, especially for methods based on fitting procedures like the Xray constrained wavefunction or the more traditional multipolar expansions. generally reduces the electron density in the bonding regions in favour of the core ones. As a matter of fact, all the correlated methods (i.e. the CCSD, CISD, DFT methods) as well as the Xray constrained Hartree–Fock strategy generally return smaller ρ(r_{b}) values than the uncorrelated RHF technique for all the bonds that we have investigated (see, for instance, the topological agreement index in Tables 1–3 for the N—N, C—N and O—H bond critical points of the nitrogen, cyanide and water molecules, and in Tables S1–S13 of the supporting information for the bond critical points of the other investigated systems). This agrees with the results of previous investigations (Bader & Chandra, 1968; Smith, 1977; Moszyński & Szalewicz, 1987). The trend is also evident when analysing ρ(r) along the covalent bonds of the investigated molecules (see Fig. 3, and Fig. S1 in the supporting information). In fact, the RHF electron density is systematically larger than the CCSD one in the middle of all the bonds.



Obviously, any Xray constrained RHF wavefunction that at least partially includes the effects of is necessary. These indices allowed us to assess the global performances of the Xray constrained wavefunctions and, together with the other more local indicators mentioned above, enabled us to analyse the following features:
should reduce the difference between the CCSD and the uncorrelated RHF electrondensity distributions. The behaviour of partially correlated wavefunctions is quite inhomogeneous and, therefore, the response at one point only (like the bond critical point) or along a line (like the bond path) may not be sufficiently representative. Therefore, for a more comprehensive overview, the whole set of indicators described in Section 3.2(i) The overall similarity between the charge distributions, by means of the realspace R (RSR) value, the Carbó distance and the RMSD and MAD indexes;
(ii) The displacement of electron density due to
visualized by electrondensity differences along representative chemical bonds, as well as by difference density maps and attachment and detachment densities;(iii) The properties of the electron densities at some special points, like the bond critical points (BCPs) of the molecules, that are often used to infer the nature of chemical bonds.
From Figs. 3–8, we can observe some general trends, which are valid for all the investigated systems.
Due to the dual nature of the functional in equation (2), an Xray constrained wavefunction cannot deviate too much from the RHF one if λ_{J} is small. From Fig. 4 (and from Figs. S2–S4 in the supporting information), it is obvious that, given a particular resolution limit, the correlation effects are significantly included only if the external weight λ_{J} is sufficiently large. In fact, a large λ_{J} implies a more precise fitting of the experimental data and a Jayatilaka functional [see equation (2)] dominated by a term that may effectively account for the In traditional XCWF calculations, a λ_{J} ≤ 1 is normally adopted, which implies that the Xray constrained electron densities remain too similar to those associated with the unconstrained and uncorrelated RHF wavefunctions.
As shown in Fig. 5, the amount of effects captured by the XCWFs is also strongly resolution dependent (see also Fig. S5 of the supporting information). For a given weight λ_{J}, the largest recoveries generally occur for calculations only including reflections up to sinθ/λ = 0.5–0.7 Å^{−1}. It is likely that this range may change for molecules containing atoms on the third row of the periodic table or below, because of the different radial distributions of the core and valence densities of those atoms. In order to understand this behaviour, we have to consider that the correlation effects in the valence region of secondrow elements affect the structure factors up to a resolution of ca sinθ/λ = 0.7 Å^{−1}, whereas the correlation effects in the core region affect all reflections up to infinite resolution, well above sinθ/λ = 2.0 Å^{−1}. However, the electron redistribution due to is sufficient to produce an oscillation of the changes, which is clearly visible in Fig. 2. Moreover, the higher resolution reflections are individually less perturbed. A including reflections up to sinθ/λ = 0.7 Å^{−1} comprehensively describes the effects of on the valence density, whereas at lower resolution (e.g. 0.25 Å^{−1}) this fitting is incomplete. On the contrary, at higher resolution, since only part of the core electron density is fitted, the oscillating nature of the perturbation is such that the XCWF charge distributions basically coincide with the RHF one (see Fig. 2), thus losing the correctness of the mediumresolution fit. Only at an extremely high resolution, i.e. well above reasonably affordable limits, may the effects be fully recovered.
For this reason, the differences between electron densities corresponding to XCWF and RHF or CCSD wavefunctions are rather heterogeneously distributed. In the bonding region, the electron density of a mediumresolution XCWF is closer to that associated with the correlated CCSD wavefunction, whereas the agreement is much poorer in the vicinity of the nuclei. In Fig. 3, this is particularly evident for the representative bonds in the cyanide, urea and benzene molecules. As explained above, highresolution XCWFs behave very similarly to the RHF ones. The associated electron densities deviate more substantially from the corresponding CCSD charge distributions, especially in the bonding region. This picture also emerges when analysing the difference densities for the investigated systems. In Fig. 6, we report the representative isosurfaces of the difference density maps computed for the nitrogen molecule using the CCSD electron distribution as a reference. At high resolution, the CCSD/XCWF difference map is practically identical to the CCSD/RHF one, while at medium resolution (sinθ/λ ≤ 0.7 Å^{−1}) the discrepancies in the bonding region disappear and those in the vicinity of the nuclei remain. This figure also confirms what was already observed in Fig. 5, which indicated that the largest recoveries of generally occur when only low to mediumangle reflections are considered in the calculations. As expected, the electron density resulting from the CISD wavefunction significantly approaches the CCSD charge distribution (for example in Fig. 6, consider that the isovalue necessary to visualize the CCSD–CISD difference density maps is much smaller than that used to plot the other isosurfaces). On the other hand, the behaviour of the density functional methods is less systematic. The trends observed in Fig. 6 are common to all the examined systems and, for the sake of proof, in the supporting information we have reported analogous difference density maps for the larger urea molecule (see Fig. S6).
All the observations reported here are quite significant and in part surprising. It is worth stressing that large values of λ_{J} are necessary in order to obtain XCWFs whose associated electron densities fit, at least partially, the effects missing in the Hamiltonian part of the functional given in equation (2) but which are present only in the constraint to the experimental data. Furthermore, it is surprising that a constraint to a set of reflections up to high resolution reduces the ability to capture these effects. As explained above, a full electrondensity fit (an ideal XCWF procedure with λ_{J} → ∞ and sinθ/λ → ∞) should converge to the correct electron density. In fact, the trends of the RSR similarity index as a function of the weight parameter λ_{J} (see Fig. 4) indicate that the highresolution XCWF slowly but constantly deviates from the RHF solution as λ_{J} increases. For relatively small λ_{J} values, a medium/lowresolution XCWF deviates more rapidly from the Hartree–Fock model than does a highresolution XCWF, which shows a much slower convergence (almost perfectly linear up to λ_{J} = 10.0).
The asymptotic behaviour is actually difficult to assess, given that, for very large values of λ_{J}, the convergence of the selfconsistent field cycles can be difficult or even impossible. Anyway, for all the molecules under investigation, we have also performed XCWF test calculations with extremely large λ_{J} values, actually showing that the electron densities computed with all the available diffraction data (sinθ/λ ≤ 2.0 Å^{−1}) almost linearly approach the CCSD density (see Fig. S7 in the supporting information). However, even for λ_{J} = 100.0, which means that the fit accounts for 99% of functional (2) and the RHF part is just 1%, the distance from the CCSD density remains significant, as shown by the RSR indices (see Fig. S7).
All these results seem to indicate that, including only ca 0.7 Å^{−1}, we capture completely the effects of the on the valence component of the electron density. In contrast, the effects on the core electron distribution are spread all over up to highangle reflections. Their complete recovery might not be guaranteed, even using reflections up to a resolution higher than that considered in the present study.
amplitudes up to a medium resolution ofFor a more comprehensive comparison, we also discuss the performances of postHartree–Fock and DFT methods. The topological agreement index reveals that the CISD calculations retrieve a large part of the correlation effects introduced at the CCSD level (see Tables 1–3, and Tables S1–S13 in the supporting information). Of course, this is not surprising, given that many configurations are included in these calculations. At the same time, the computational cost of the CISD method remains very large. Concerning the DFT calculations, the results are quite heterogeneous and difficult to generalize, due to the less systematic nature of the density functionals. In fact, they sometimes provide ρ(r_{b}) values that are even lower than those resulting from the coupled cluster calculations, but in other cases they perform worse than the CISD strategy. In view of this, one might anticipate that an Xray constrained DFT strategy is not necessarily expected to work better than the original XCRHF one. All the trends presented above agree with the analyses of the difference density maps (see Fig. 6, and Fig. S6 in the supporting information).
Finally, in Figs. 7 and 8 we show the detachment and attachment densities (HeadGordon et al., 1995) for the nitrogen and urea molecules (see Figs. S8–S11 in the supporting information for the detachment and attachment densities of the other investigated systems). As discussed in the previous section, in our study the detachment densities are defined as those parts of the unconstrained RHF electron distributions that are removed and rearranged into the correlated attachment electron densities (in our case, those associated with the CCSD, CISD and XCRHF wavefunctions). Here also, the attachment and detachment densities show that the main effect of the is a shift in the charge distribution from the bonding region to the nuclei. This is clearly more pronounced at the CCSD and CISD levels. In fact, the surface isovalues used in Figs. 7 and 8 show that the CCSD and CISD methods provide larger electronic reorganizations of approximately the same order of magnitude. In contrast, for the XCRHF wavefunctions the extent of the rearrangement is definitely smaller, especially when highresolution amplitudes are included (sinθ/λ ≤ 1.5, 2.0 Å^{−1}), in keeping with the results of the similarity indicators discussed above. Not surprisingly, the comparison with the CCSD attachment/detachment densities is better when only low/mediumangle reflections are taken into account (sinθ/λ ≤ 0.5, 0.7 Å^{−1}). Nevertheless, the XCHF approach always provides smaller reorganizations than the CCSD or CISD ones, which is further evidence of its limited ability to recover the entire effect of Therefore, the attachment and detachment densities also confirm the trends already observed in the difference density maps of the investigated systems (see Fig. 6, and Fig. S6 in the supporting information).
5. Conclusions
We have carried out a thorough investigation of Xray constrained wavefunction electron densities, with the aim of ascertaining whether the Xray constrained wavefunction procedure is able to capture the effects of
on electrondensity distributions. The study is based on the use of simulated scattering amplitudes computed for isolated molecules at a very high correlation level, in order to avoid perturbations due to crystal electric fields and/or biases due to experimental errors in the measurements. Under this hypothesis, an XCWF should differ from an uncorrelated and unconstrained RHF wavefunction only by virtue of the effects.Within the framework of the XCWF approach, a perfect density fit requires a very tight constraint (λ_{J} → ∞) and a scan of the entire (sinθ/λ → ∞). For practical reasons, neither condition can be fulfilled. In fact, convergence of the XCWF with large λ_{J} is prohibitive and the resolution of measurable Xray intensities is necessarily finite. Moreover, it is also important to note that simply fitting a wavefunction to a given electron density (which indeed corresponds to an ideal XCWF procedure with λ_{J} → ∞ and sinθ/λ → ∞) would not guarantee finding the desired wavefunction (which was the original goal of the Jayatilaka approach), because, from a theoretical point of view, an infinite number of wavefunctions are actually compatible with a given electron distribution (Gilbert, 1975).
After analysing several molecules and using a realistically obtainable resolution of the Xray data, we can conclude that the effect of λ_{J} is sufficiently large and (ii) only low/mediumangle reflections are used as constraints in the functional to be minimized [see equations (2) and (3)]. In fact, on the one hand a large λ_{J} is necessary because weak effects, like those produced by require a very close fit, while on the other hand, low/mediumresolution data are crucial to recover completely the effect of the on the valence component of the electron density. In contrast, to capture the subtler effects of the on the core electron density, both a large value of λ_{J} and very highresolution reflections might be necessary. A truncation of the resolution, even at the very high value of sinθ/λ = 2.0 Å^{−1}, not only produces an incomplete fit of the core effects but, in the absence of an infinite λ_{J}, it also drastically affects the capability of capturing the effects of on the valence electron density, since the highangle reflections become predominant in the functional to be minimized [see equations (2) and (3)]. As a consequence, a warning emerges from our analysis: the very small λ_{J} values and the highresolution data typically adopted in XCWF calculations with real experimental constraints may not guarantee a significant recovery of the effects. Instead, it appears that a lowresolution fit is more efficient, retrieving at least the effects on the valence density and avoiding the counterproductive result of a partial fit of the correlation effects on the core electron density. Notably, it is plausible that electron distributions associated with frozen core correlated wavefunctions may be more easily fitted by XCWF procedures with either medium or highresolution data.
on electron density can be partially captured by the XCWF method, provided that: (i) the external multiplierWhile the results presented in this paper clearly indicate a trend associated with XCWFN analysis, the `critical' values of λ_{J} and sinθ/λ cannot be generalized, because they depend on the radial distribution of each atom in the molecule. In fact, for complexes of transition metals, one may expect a larger critical resolution because of the more contracted nature of the metal d orbitals that obviously scatter to higher angles.
Further studies are still necessary to clarify better the physical meaning of the adjustable parameter λ_{J} and, consequently, the actual meaning of Xray constrained wavefunctions obtained in cases in which the external (experimental or theoretical) constraint in equation (2) becomes largely predominant compared with the electronic energy of the system.
Furthermore, we envisage the need for other detailed studies where the effects of the crystal field (as distinct from Introduction, the Hartree–Fock part of functional (2) is the Hamiltonian of an isolated unperturbed molecule, whereas the Xray constrained part is generally linked to a periodic electron density. Moreover, the effect of real experimental data and experimental errors should be tested in more detail. In fact, it is very likely that the typical noise of experimental data may significantly affect the possibility of retrieving very small effects such as those due to electron correlation.
effects) can also be quantified and tested. In fact, as explained in theWe believe that the results of this study and the follow up that we are planning could be of great importance, not only to define further the potentialities of the Jayatilaka approach, but also to propose new functionals within the framework of the density functional theory.
Supporting information
Supporting information file. DOI: https://doi.org/10.1107/S2052252516019217/yc5009sup1.pdf
Acknowledgements
We thank Xavier Assfeld for reading the preliminary version of the manuscript and for helpful discussions. PM and LHRS thank the Swiss National Science Foundation (project No. 160157) for financial support.
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