Intermolecular Interaction Energies in Transition Metal Coordination Compounds

The PIXEL method has been parameterised and validated for transition metals, extending its applicability from ~40% to ~85% of all published crystal structures. Abstract Parameters required to perform PIXEL energy calculations, a semi-empirical method for evaluating intermolecular interactions, have been defined for the transition metals. Using these parameters, lattice energies of thirty-two 1st row, five 2nd row and six 3rd row transition metal complexes have been calculated and compared to experimental values giving correlations of calculated sublimation enthalpies comparable to those obtained for organic crystal structures. Applications of the method are illustrated by analysis of the intermolecular interactions in chromium hexacarbonyl, stacking interactions in bis (acetylacetonato)-oxo-vanadium(IV) and dihydrogen bonding. The results extend the applicability of the PIXEL method from organic materials ( ca. 40% of the Cambridge Structural Database (CSD)) to a much wider range of organic and organometallic systems ( ca. 85% of the CSD).


Introduction
Methods for interpretation of molecular crystal structures have advanced considerably in the past decade. While analysis of intermolecular interactions using graphical tools such as Mercury 1 can be achieved in a matter of seconds using fast algorithms based on geometry, it is increasingly common to evaluate intermolecular interaction energies using ab initio methods, [2][3][4][5][6][7] PIXEL calculations, 8 symmetry-adapted perturbation theory 9 or force fields. 10 The results can be used to visualise contributing energy terms using Hirshfeld surfaces 11 or energy vectors or frameworks [12][13][14] , This progress has been applied to areas such as polymorphism 15 and energy landscapes, 16 cocrystals and solvates, 17 crystal engineering, 18 molecular recognition 19 and extreme conditions research. 20 The techniques listed above have been applied extensively in work on organic materials. 21 There is, nevertheless, substantial interest in intermolecular interactions in metal-containing systems. 22 Crystal engineering and supramolecular chemistry frequently make use of strategies involving metals. For example, Orpen and co-workers have shown that H-bond acceptors based on metal halides and oxalates can be used to form more reliable and reproducible supramolecular building blocks than those based on purely organic ligands. 23,24 Use of diplatinum thiocarboxylate complexes in bottom-up assembly of conductive one-dimensional nanostructures has been suggested on the basis of the strong (~50 kJ mol −1 ) intermolecular Pt···Pt interactions that occur in these systems. 25 Density functional theory (DFT) has been used to study the different intermolecular energies of alternative Ru(II) hydrogenation catalysis pathways, [26][27][28] while more recently Li et al. performed calculations to investigate the performance of cobalt and copper analogues of a pre-existing nickel catalyst for olefin purification. 29 A great deal of computational effort has been invested in the study of the binding affinities and selectivities of metalloproteinligand interactions in these systems, [30][31][32] and the desire to find more efficient methods of drug design means that computational analysis of metal-based pharmaceuticals is an ever expanding field, 33 exemplified by the analysis of the interactions of zinc ions with several anti-inflammatory drugs. 34 Metal-organic frameworks, large porous structures consisting of metal ions linked by organic ligands, are increasingly being studied as potential gas storage and separation materials, and a variety of computational methods are used to study the adsorption of small molecules in these systems. 35-37 3 The development of the PIXEL method, 8,21,38,39 a semi-empirical technique for evaluating intermolecular interactions based on integrations over calculated electron densities of molecules, has allowed energetic analysis in organic crystal structures to be carried-out quickly with an accuracy comparable to high level quantum mechanical methods. 40,41 PIXEL calculations yield the total lattice energy partitioned into individual molecule-molecule energies, which are themselves partitioned into four terms: Coulombic, polarisation, dispersion and repulsion. The separation of contributions allows for the character of individual interactions and overall crystal packing to be inferred from the dominant terms, providing chemical insight. The PIXEL method requires definition of certain atomic parameters, but these are mostly physically measurable quantities such as ionisation potentials, and one of the most appealing features of the method is the transferability of parameters across many different chemical systems. The PIXEL method is thus a potentially valuable addition to established techniques applied to metal-containing systems, such as ab initio methods and calculations based on force-fields.
The aims of this paper are (i) to define a validated PIXEL parameter set for use with the dblock metals, and (ii) to illustrate possible applications of the method in interpreting intermolecular interactions in metal-containing structures. This expands the potential applicability of the PIXEL method from organic materials (~40% of the CSD) to the majority of organic and organometallic structures (~85% of the CSD).

Definition of Metal Parameters
The four energy terms evaluated during a PIXEL calculation depend on a small number of fundamental atomic parameters. Values of these parameters for atoms common in organic chemistry are embedded in the PIXEL code. In the present work we have defined new values for transition metals and validated them against experimental sublimation enthalpies (ΔH sub ) using the convention that lattice energies are approximately equal to -ΔH sub . This procedure assumes that there are no intramolecular structural changes on passing from the solid state to the gas phase. The names used to refer to the parameters in the following sections are those used in the PIXEL program documentation and Gavezzotti's publications, where full details of their definition, use and significance can be found. 21,38,39 Definition of some parameters is straightforward. ZTOT and ZVAL are the total number of electrons and the number of valence electrons in the neutral atom; POTIO is the first ionisation energy (in atomic units) and WEIGHT is the atomic weight. 42 For other parameters a choice among several possibilities needs to be made. Unless otherwise specified, values used for non-metallic elements in the compounds were the program defaults.
Dispersion energies are calculated in a London-type expression in which the ionisation energy of a pixel is used to approximate its 'oscillator strength'. DIFA (also given the symbol β in Gavezzotti's papers) is a "variable ionisation" parameter which controls the diminution of the ionisation energy of a pixel as the distance from the nucleus increases. 21  The covalent radii of atoms, RINTER, are used in PIXEL to check for short internuclear distances, and not used to calculate energies. Values were taken from Emsley's compilation. 42 RAVDW, the van der Waals radius, is used to assign pixels of electron density to atomic basins. The sets of values reported by Nag et al. 43 and Batsanov 44 were tested.
The atomic polarisability, POLZE or α in Å 3 , appears in the calculation of both the polarisation and dispersion terms. For non-metallic species the PIXEL method makes use of the Slater-Kirkwood approximation to estimate α according to Equation 1, where a 0 is the Bohr radius and R vdW is the van der Waals radius. The Nag and Batsanov radii were tested. The Clausius-Mossotti relation (Equation 2) is another simple method for estimating atomic polarisabilities: where V m is the atomic volume, ε is the dielectric constant of the species and ε 0 is the permittivity of free space. Atomic volumes were obtained from the crystal structures of the elemental metals at room temperature and pressure. For pure metals, the dielectric constant ε → ∞, so that the first term of this equation tends to unity, giving α = 3V m /4π. Variation of α(Ti) between 3.5 and 5.0 Å 3 yielded values of the TiCl 4 lattice energy between −45.9 and −57.1 kJmol −1 .
i Test calculations were performed using parameter set 5 (see below) The electronegativity, ELNEG, is used in the calculation of the repulsion energy. Both Pauling and Allred-Rochow values 42 were investigated.

PIXEL Calculations and the Treatment of Ligand Parameters
PIXEL calculations can be carried-out in one of two ways using the programs PIXELc or PIXELd of Gavezzotti's CLP package of modelling and structure analysis software. 45 PIXELd is intended for the calculation of interaction energies of discrete dimers. PIXELc is used for calculations on a complete crystal structure, yielding a lattice energy which is further broken-down into the contributions from individual molecule-molecule pairs out to a specified cluster cut-off distance.
PIXELc calculations can be carried-out straightforwardly on crystal structures with one or two molecules in the asymmetric unit. More complex cases can be handled with some userintervention. For example, where a molecule lies on a special position the space group symmetry should be lowered and the structure specified using whole molecules. Calculations on disordered structures can be carried-out using permutations of PIXELc calculations considering different molecular pairs. 46 We thank a referee to this paper for pointing-out that some structures may in fact be more readily amenable for processing using a series of PIXELd dimer calculations, while PIXELc can also be used to calculate dimer energies for a user-specified list of symmetry operations if the cluster cut-off distance is set to zero. Although both of these procedures would fail to reflect the manybody nature of the polarisation contribution to the lattice energy, they could be applied to the interpretation of crystal packing. We also note that a correction is applied to PIXELc lattice energies of structures in polar space groups. 47  densities which lead to systematic shortening of distances involving hydrogen atoms when determined by X-ray diffraction. The electron density was obtained in a single-point calculation with a B3LYP functional and a 6-31G** basis-set (Gaussian09) 50 for main-group elements and firstrow transition metals. Second-row transition metal species were treated with the LanL2DZ basis-set, and third-row metals used the LanL2DZ basis-set with pseudopotentials to model the core orbitals of the metal atom. The "cube" format electron density files were then used in PIXEL calculations. Unless otherwise specified, the pixel size for all calculations was 0.16 x 0.16 x 0.16 Å (corresponding to 'condensation level ' 4), and the cluster cut-off distance was kept at the default value generated by PIXEL for each structure. 21 While atomic parameters in PIXEL calculations are intended to be transferable between different compounds, values of atomic polarisabilities may be varied depending on chemical bonding. For instance, three different atomic polarisabilities are used for carbon, depending on whether it is aliphatic, aromatic or 'bridging aromatic' as in naphthalene. While such differences might ideally be taken into account for other species (e.g. carbonyl and ether oxygen atoms), the dominance of carbon in organic compounds means that it is much more important to account for variation in its different chemical environments than it is for less abundant atomic species. While in practice only α(C) is usually varied, modification of the atomic polarisabilities of other species has been applied previously by Gavezzotti, 51 for example for chloride ions in ionic organic crystals.
Although carbon is a common constituent of many ligands, it may be necessary to consider alternative values of polarisabilities of non-carbon atoms in cases such as homoleptic carbonyl complexes where the molecular surface is composed of exposed oxygen atoms. PIXEL analysis of molecular carbon monoxide using the default parameters in the program yields a lattice energy of −7.9 kJ mol −1 . The experimental sublimation enthalpy is 7.9(2) kJ mol −1 (average value from three determinations). However, when carbon monoxide acts as a ligand, PIXEL results were found to be around 20 kJ mol −1 lower than the literature value when the default value of α(O) (0.75Å 3 ) was used (e.g. Cr(CO) 6 , literature sublimation enthalpy 69.6 kJ mol −1 , calculated lattice energy −47.8 kJ mol −1 ). Carbonyl oxygen therefore, like carbon, seems to require its own value of α(O) depending on whether the CO is ligating or not. By testing different values of atomic polarisability of O, a value of α(O) = 1.0 Ǻ 3 was chosen for this species when carbon monoxide is acting as a ligand, yielding a lattice energy of −70.5 kJ mol −1 for Cr(CO) 6 . Support for these adjustments was obtained by calculation (AIMALL) 52 of atomic polarisabilities in CO and Cr(CO) 6  were arithmetically averaged with no weighting after elimination of any egregious outliers. The full validation data set, which contains 43 different compounds, is given in Table 2, and chemical structures are given in the supplementary data. Also listed in Table 2 are the CSD refcodes, along with average experimental sublimation enthalpies calculated from data in ref. 55 and PIXEL calculated lattice energies. All complexes investigated had centrosymmetric crystal structures, and no polarisation corrections 47 were necessary.

Calculations of Individual Intermolecular Interactions
Dimers displaying a variety of intermolecular interaction types were selected to compare the PIXEL results with those calculated using higher level computational methods. A range of interactions was investigated involving chromium hexacarbonyl, vanadyl stacking and metal hydrides participating in dihydrogen bonding. [60][61][62] For each system a combination of Mercury 3.5 and Materials Studio V5 63 was used to obtain a structural model which was then optimised using Gaussian09. In cases where calculations had been previously reported, we used the same level of theory and basis-set as in the literature study. The optimised structures were then used for PIXELc calculations as described above. Further computational details are given in the relevant sections below, but Hirshfeld surface analysis was carried-out with CrystalExplorer 3.1, 64 where the required wavefunctions were calculated with the program TONTO 3.2 rev. 4048, 65 while analysis of PIXEL results was accomplished using processPIXEL. 12

Parameter Set Selection and Reproduction of Experimental Sublimation Enthalpy Data
Five different parameter sets were constructed using different combinations of methods for estimation of the van der Waals radii, polarisability and electronegativity, as defined in Section  Table 2 to the calculated lattice energies. The straight-line fitting statistics are listed in Table S2; the fits used unit weights and were constrained to intercept at the origin.
The data in Table S2 show that there is little difference between the performance of the different parameter sets, demonstrating the robustness of the PIXEL method to different choices of reasonable parameters. As the volume (and therefore polarisability) of an elemental metal and the Allred-Rochow electronegativities are more unambiguously defined in terms of readily accessible experimental data than the quantities used to define parameters in other sets, set 5, the parameters of which are shown in Table 3, was used in further analyses. Furthermore, the gradient for 2 nd and 3 rd row transition metals for set 5 is nearer unity than the other sets. Overall, while the data presented above indicate that the PIXEL method can be applied with some confidence to first-row metal complexes, more data are needed to establish this for compounds containing heavier metals.
It is important to note that the assumption made in calculating a lattice energy by the PIXEL method is that no change in molecular structure occurs on passing from the crystalline state to the gas phase. PIXEL calculations of the lattice energies of amino acids, for example, are in poor agreement with experimental values because amino acids exist as zwitterions in the solid state but as neutral molecules in the gas phase. The energy associated with the transfer of a proton from the ammonium to the carboxylate group does not form part of the PIXEL analysis, and would need to be calculated separately in any calculation aiming to reproduce the experimental sublimation energy. This consideration may account for some of the differences between observed and calculated data presented in Table 2.
The experimental sublimation energies of the bis-cyclopentadienyl complexes of Cr, Fe, Ni, Ru and Os all lie between 70 and 80 kJ mol -1 , and are reproduced to within 10 kJ mol -1 in the PIXEL calculations. By contrast the relative difference between the experimental and PIXEL values for Cp 2 V is the poorest in Table 2

Chromium Hexacarbonyl
Chromium hexacarbonyl (CSD refcode FOHCOU02) 67 Table 2). The three principal intermolecular interactions are to the molecules labelled A, B and C in Figure 3; Table 4 shows the breakdown of the component energy terms and the Cr…Cr distance for each interaction.
The data in Table 4  onto its Hirshfeld surface. 11,68 Favourable overlap occurs when the negatively-charged (red) regions of one molecule are in contact with the positively-charged (blue) regions of a neighbouring molecule, and the white lines separating these regions are contiguous. This arrangement is seen for interactions A and C, but in the interaction with molecule B there is some overlap between negative regions. This is the source of the more negative (stabilising) Coulombic terms for interactions A and C, which is not apparent from geometric analysis alone.

Bis(acetylacetonato)-oxo-vanadium(IV)
In the crystal structure of bis(acetylacetonato)-oxo-vanadium(IV) (VO(acac) 2 , CSD refcode ACACVO12), 69  A metal-bound hydride may be sufficiently negatively charged such that it can act as a hydrogen bond acceptor, forming a dihydrogen bond. 74 Theoretical analysis of such interactions has shown that they are attractive and are predominantly electrostatic in character. 75 Interaction energies for a series of dihydrogen-bonded dimers comprising a Ru-hydride complex and carboxylic acids or alcohols with different levels of fluorination have been evaluated using either DFT (B3LYP) or Hartree-Fock (HF) calculations with the LanL2DZ basis-set. 60-62 PIXEL calculations were carried-out using electron densities calculated with the same geometries, functionals and basis-sets as reported for the DFT calculations, and the results are compared in Figure 7 and Table   5.
The level of agreement between the PIXEL and DFT results is similar to that found for the sublimation enthalpies: the gradient of Figure 7 is 0.91 and the correlation coefficient 0.96; differences are of the order of 5 kJ mol −1 . The data in Table 5 show that the interactions are predominantly electrostatic in character, in accordance to observations made by Liu and Hoffmann, and that the Coulombic term increases with increasing fluorination of the donor as  strong acid-base hydrogen bond such as benozoic acid -imidazole, 21 and explains why stacking interactions are a recurring feature of vanadyl crystal structures. Application to guest-binding in metal-organic frameworks will be described in a future paper.

Conclusions
The gradient and correlation coefficient of the fit between sublimation enthalpies and PIXEL lattice energies for second and third-row complexes are 1.02(3) and 0.92, respectively, but these figures are based on a much less extensive set of data than those for the first-row systems.
The parameters for 2 nd and 3 rd row transition metals should be used with caution, but they are consistent with DFT results for dihydrogen bonds involving ruthenium hydride complexes and intermolecular embraces.
The PIXEL method assumes that an interaction is truly non-covalent, and it cannot be applied to interactions involving, for example, partial bond formation or other electron-sharing regimes for which full quantum mechanical treatments are necessary. Distinction between noncovalent and more complex interactions is often quite intuitive in organic chemistry, but this may not be the case in transition metal chemistry, and so caution is also necessary in this context. It would not be appropriate, for example, to apply PIXEL calculations to modelling the transition between long and short Jahn-Teller distortions or to aurophilic interactions, which both demand high-level quantum mechanical treatments even though the interatomic distances which characterise them extend to 3 Å and beyond. This said, comparison of PIXEL and quantum mechanical treatments for such systems might be used to detect the presence of complex behaviour.
The accurate and efficient quantification of crystal-packing and intermolecular energies is having a transformative effect on solid-state organic chemistry. It is no longer necessary to base the interpretation of a structure or a crystal engineering strategy on the assumption that short interactions are strong interactions. 77 Systematic quantification of molecule-molecule energies further reveals important interactions that are easily missed on the basis of analysis of distances alone, a particular issue with electrostatic and van der Waals interactions which lack characteristic geometric signatures. 78 While the PIXEL calculations do not replace quantum mechanical methods, they have proved to be an extremely valuable tool for interpretation of the thermodynamic stability of crystalline organic phases. The parameterisation described here should enable these advantages to be extended to systems containing metals, an extension which dramatically increases the domain of applicability for the PIXEL method.  Table   3.