Matta, Boyd. The quantum theory of atoms in molecules

26210 Interpretation of Experimental Electron Densities by Combination of the QTAMC and DFT

the cusp condition is not necessary preserved during model fitting;

the multipoles on di erent centers are not orthogonal (the Hirshfeld multipole model [45] is an exception); and

the orthogonality of the core and valence s-type density functions of the same pseudoatom is not fulfilled in the Hansen–Coppens multipole model.

An attempt to correct the last shortcoming [46] failed.

Generalizing previous experience [11–19] we can conclude that X-ray di raction experiments yield a quasi-static model of electron density extrapolated to infinite resolution, which is, typically, as precise around the bond critical point as @0.05 e A˚ 3. The experimental error becomes larger closer to nuclei and increases with the atomic number in the vicinity of nuclei; therefore ‘‘internal’’ atomic regions (RA0.3 A˚ ) are normally excluded from consideration. Despite this, it is well-documented that experimentally derived electron density distributions have similar topology and the same set of the critical points as the corresponding quantum-mechanically derived densities [12–19].

Recent studies have shown the situation is not as encouraging for the Laplacian of the electron density [47–49]. For the closed-shell and intermediate atomic interactions [1], the Laplacian is restored from experiment in reasonable agreement with direct wavefunction calculations. In this instance contraction of the density toward the bond path is small, ‘2rðrbÞ > 0 and the atomic-like presentation of the electron density by the multipole model is quite reasonable. For shared atomic interaction ð‘2rðrbÞ < 0Þ, the electron density curvatures perpendicular to the bond path, l1 and l2, are determined with good accuracy; the multipole model has, however, failed to correctly describe the electron-density curvature along the bond path, l3. As a result, the overall error in the experimental Laplacian ‘2rðrbÞ ¼ l1 þ l2 þ l3 in this example can reach 50% [48].

Thus, the experimental electron density reconstructed with current multipole models can be regarded as an approximate homeomorphic image of the ‘‘true’’ r derived from the first principles, which is reasonably accurate for the closed-shell and intermediate atomic interactions and has significant quantitative uncertainty along the bond line for shared atomic interactions.

10.3

Approximate Electronic Energy Densities

10.3.1

Kinetic and Potential Energy Densities

The key problem in DFT is to express the kinetic, potential, and total electronic energy in terms of rðrÞ [31, 32, 38]. The problem is the same for the energy densities. One way of solving this problem for kinetic energy density uses the fact

10.3 Approximate Electronic Energy Densities 263

that the one-electron density matrix gðr; r0Þ, which defines the kinetic energy by use of Eq. (1), is related to the one-particle Green function by the inverse Laplace transform [38]. The gradient expansion of the Green function around the classical Thomas–Fermi approximation [50] leads to the following approximate expression for kinetic energy density:

Application of this formula, which is valid for smooth (but not necessarily small) variation of the electron density, to real systems implies the local homogeneity approximation [41] – the energy density at each point r is supposed to be the same as that of a homogeneous electron gas with the same electron density, which is equal to rðrÞ everywhere. Note the role of Laplacian term in Eq. (6) [32, 51–60] – although it does not a ect the average total and atomic energies its presence provides a description of the electronic shells and improves the local kinetic energy behavior of the valence electrons.

Consideration of the asymptotic properties of the kinetic energy density, Eq. (1), derived from the one-electron density matrix shows [20] that the long-range behavior of the approximate gDFT ðrÞ, Eq. (6), is physically acceptable [60]. In contrast, the function gðrÞ, Eq. (1), becomes ð1=2ÞZ2riðRiÞ as r ! Ri, where riðRiÞ is the value of the electron density at the positions of the nuclei, Ri, while approximate gDFT ðrÞ, Eq. (6), becomes minus infinity as r ! Ri, because of the Laplacian term. The radius of the negative hole around the nuclei is maximum for the hydrogen atom (0.15 A˚ ), is less than 0.02 A˚ for atoms with Z b11 and reaches 0.005 A˚ for Z ¼ 36 (Kr). This observation fits the 1/Z-dependency of this radius [55]. From the consideration given above it follows that the physically meaningless negative gDFT regions are completely within the region of uncertainty of this function, because of experimental and model errors in ED; they should therefore be excluded from the discussion during interpretation of the approximate gDFT ðrÞ. In other points of the position space, gDFT ðrÞ, Eq. (6), is quite close to the quantum mechanical gðrÞ, Eq. (1) [52, 61]. Thus, the use of the experimental electron density and its derivatives to determine the kinetic energy density has a physical basis.

Approximation Eq. (6) opens the way to determination of the potential energy density from X-ray experiments. It has been postulated [62, 63] that the model electron density derived by the fit to experimental structure factors does obey the local virial theorem, Eq. (3), the same as the quantum-mechanical rðrÞ does. Then, using gDFT ðrÞ, Eq. (6), and ‘2rðrÞ, it is possible to obtain the potential energy density by use of Eq. (3) and calculate electronic energy density heðrÞ, Eq. (4), from the experimental ED. This approximation is nonevident; subsequent studies [61, 64, 65] have shown, however, that gDFT ðrÞ, Eq. (6), calculated from the experimental electron density leads to physically reasonable (negative everywhere) potential energy density.

264 10 Interpretation of Experimental Electron Densities by Combination of the QTAMC and DFT

After other work [58, 62, 63, 66–68], the approach summarized above became a popular tool for determination of the energy characteristics at the bond critical points of crystalline systems [14–19, 67–71]. It is, however, necessary to mention that the above-mentioned inability of the current multipole models to correctly describe the curvature of the experimental electron density along the bond path for shared atomic interaction influences all existing QTAMC bonding descriptors which contain the Laplacian term. Fortunately, electron density for the closedshell and intermediate atomic interactions is reconstructed correctly and corresponding kinetic energy density approximated by use of Eq. (6) behaves properly in the internuclear space [72]. Equations (6), (2), and (4) are therefore completely applicable to systems with closed shells and intermediate atomic interactions.

To illustrate the actual situation with the applicability of the gradient expansion, Eq. (6), to the determination of the energy densities, let us consider these functions for urea, CO(NH2)2. The electron density in a single urea molecule removed from a crystal was reconstructed from data from two di raction experiments – an X-ray four-circle di ractometer experiment at 148 K [73] and an X-ray synchrotron experiment at 123 K [74]. Multipole data derived from X-ray and synchrotron results were taken from Refs [73] and [75]. The approximate kinetic energy density, gEXP=DFT ðrÞ, and the potential energy density, vEXP=DFT ðrÞ, were calculated from the multipole data by use of Eq. (3) and Eq. (6). We then calculated the wavefunction for the urea molecule by the Hartree–Fock method in the 6-311G** basis set using the PC version [76] of the GAMESS software [77]. The optimized molecule geometry was taken from Ref. [61]. First, the gradient expansion Eq. (6) was used to calculate the kinetic energy density gHF=DFT ðrÞ from the wavefunction and the local potential energy vHF=DFT ðrÞ was calculated by means of the local virial theorem, Eq. (3), which is valid in the Hartree–Fock theory. Second, the same functions were also calculated directly from the Hartree–Fock wavefunctions using the AIMPAC software suite [78]. The later functions will subsequently be referred to as gHFðrÞ and vHFðrÞ, respectively.

By comparing electron densities derived for urea by di erent methods (Fig. 10.1), we see that both experimental EDs are in very reasonable agreement, excluding the vicinity of the hydrogen nuclei (we ignore small distortions of the density present on the periphery of molecules because of the e ect of the crystalline environment). More complete and accurate synchrotron-measured electron density is also in quantitative agreement with the Hartree–Fock electron density practically everywhere in the position space. Functions gEXP=DFT ðrÞ derived from the data from both experiments (Fig. 10.2) are in remarkable mutual quantitative agreement (again excluding the vicinity of the hydrogen nuclei); they di er, however, from results from direct Hartree–Fork calculation, gHFðrÞ (note that the geometry of the urea molecule in a crystal depends on the temperature of the experiments and is di erent from that in the free state). A similar di erence is observed between gHFðrÞ and gHF=DFT ðrÞ. The approximate function gHF=DFT ðrÞ and both experimental kinetic energy densities reconstructed in the same manner via the multipole model are in close agreement.

10.3 Approximate Electronic Energy Densities 265

Fig. 10.1 Electron densities of urea derived by di erent methods:

(a)reconstructed from the synchrotron di raction experiment [74],

(b)reconstructed from the X-ray di raction experiment [73], (c) calcu-

lated theoretically by use of the Hartree–Fock /6-311G** method. Line intervals are ð2; 4; 8Þ 10n e A˚ 3 ( 2 an a2).

Both functions vEXP=DFT ðrÞ derived from the experimental data (Fig. 10.3) are also in close agreement; they di er from vHFðrÞ only in secondary details.

Let us now compare the experimental and Hartree–Fock energy densities for ionic LiF crystal. Multipole-model data were taken from Ref. [79]. Hartree–Fock (6-311G*) calculations were performed for the cubic-like cluster Li14Fþ13 of optimized geometry [80] surrounding the central fluorine ion to simulate a crystal. Experimental and theoretical electron densities of LiF are in evident agreement (Figs 10.4a and 10.4b). Comparison of the approximate DFT-based local energy densities gEXP=DFT ðrÞ and vEXP=DFT ðrÞ with results from direct Hartree–Fock calcu-

10.3 Approximate Electronic Energy Densities 267

Fig. 10.3 Distribution of the potential energy in the urea molecule. Left and right parts of each figure represent the local potential energies obtained from di erent sources: (a) left, synchrotron data; right, Hartree–Fock result; (b) left, X-ray di raction data; right, Hartree–Fock result. Line intervals are ð2; 4; 8Þ 10n atomic units ( 2 an a2).

vature along the bond path for shared atomic interactions is, however, a major source of distortion of the energy distribution around the bond critical points. At the same time it is essential that the local energy functions gEXP=DFT and vEXP=DFT have the same topology as the corresponding Hartree–Fock functions. We expect that use of more flexible radial functions in the multipole model, as it discussed elsewhere [47, 48], will remove this defect.

This shortcoming of the model density is less vital for shared atomic interactions (because of partial compensation of contributions) if, in accordance with Ref. [21], one calculates the di erence functions:

and

(the su x ‘‘pro’’ denotes functions calculated for a procrystal – a hypothetical system consisting of spherical noninteracting atoms placed in the same positions as real atoms). These di erence functions reveal, at the semiquantitative level, the changes in corresponding energy densities caused by formation of a crystal from the atoms. Figs 10.5 and 10.6 depicting the functions dgðrÞ and dvðrÞ for crystalline urea and LiF, respectively, explicitly demonstrate the di erence between the covalent and ionic bonding mechanisms as reflected in the energy distribution in

268 10 Interpretation of Experimental Electron Densities by Combination of the QTAMC and DFT

10.3 Approximate Electronic Energy Densities 269

Fig. 10.5 Crystalline urea – the di erence functions dgðrÞ ¼ gðrÞ gproðrÞ

(a) and dvðrÞ ¼ vðrÞ vproðrÞ (b) characterizing changes in the kinetic and potential energy densities, correspondingly, caused by formation of a crystal from the atoms. Line intervals are 0.05 a.u. Solid lines correspond to excessive (positive) kinetic energy density and (negative) potential energy density.

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H

Fig. 10.4 Distributions of electron density and kinetic and potential energy in the (100) plane of crystalline LiF: (a) experimental electron density, (b) Hartree–Fock (cluster) electron density, (c) experimental kinetic energy density, (d) Hartree–Fock kinetic energy density, (e) experimental potential energy density, (f ) Hartree–Fock potential energy density. Line intervals are Gð2; 4; 8Þ 10n a.u. ( 2 an a2).

270 10 Interpretation of Experimental Electron Densities by Combination of the QTAMC and DFT

Fig. 10.6 Crystalline LiF: the di erence functions dgðrÞ ¼ gðrÞ gproðrÞ

(a) and dvðrÞ ¼ vðrÞ vproðrÞ (b). Line intervals are 0.05 a.u. Solid lines correspond to excessive (positive) kinetic energy density and (negative) potential energy density.

position space. In urea, the kinetic energy density increases strongly in the internal electronic shells of nonhydrogen atoms and only slightly (and irregularly) in the intramolecular bonds and electron lone-pair regions. In contrast, the potential energy density regularly increases in the intramolecular bonds and in the electron lone pairs (and within the cores of nonhydrogen atoms). Distributions of both functions in the hydrogen bond areas are close to the superimposition of the free atoms. Thus, distributions dgðrÞ and dvðrÞ show the stabilizing enhancement in the potential energy along the bond lines and in the electron lone pairs of the oxygen atom and details of destabilizing contributions of the local kinetic energy during formation of crystalline urea from atoms.

The distribution of functions dgðrÞ and dvðrÞ in the ionic LiF crystal (Fig. 10.6) is dramatically di erent. Formation of this crystal from the neutral atoms is accompanied by concentration of the kinetic energy in the Li atom basins and more pronounced enhancement in the (negative) local potential energy within the F atomic basins. This reveals the stabilizing role of the anions during LiF crystal formation. Note that the areas of the energy concentration/depletion in LiF are close to spherical.

Consideration of the functions dgðrÞ and dvðrÞ for some organic compounds and cubic perovskite SrTiO3 [82, 83] led to similar conclusions. It was also noted [82] that partial covalence of the TiaO bond in SrTiO3 manifests itself in the noticeable dipole-type dvðrÞ distribution and small excessive dipole-type dgðrÞ distribution around the O atoms directed to the Ti atoms; it explicitly exhibits the polar bonding contribution in the TiaO closed-shell interaction. Thus, the energy distribution features can help in explicit topological electron density characterization of the polar shared atomic interactions (or partially ionic bonds) and provide a new insight into bonding mechanisms from data of the X-ray di raction experiment.

10.3 Approximate Electronic Energy Densities 271

The energy distributions depicted in Figs. 10.1–10.6 show the importance of consideration of all the position space in studies of bonding in molecular systems. No significant details are seen in the position–space energy distributions in ionic LiF crystal, therefore information about atomic interactions, which is concentrated in the bond critical points, is su cient for bond description. In contrasts, many energy-density features reflecting the bonding mechanism in urea (for example, the electron lone pairs) cannot be taken in account if only the bond critical points are considered.

It is well known that the rapid variation of the electron density in the vicinity of the nuclei and its slow variation in the valence electron shells makes it di cult to find a functional for kinetic energy density which provides a good description everywhere in the position space [32, 60]. We have tested a few other approximations for describing the kinetic energy density via the electron density and its derivatives; our observations can be briefly summarized as follows. Inclusion of the 4th-order correction to the gradient expansion Eq. (6) [84] did not result in discernible improvement. The Lee–Lee–Parr formula [85] (plus the Laplacian term) yielded gDFT ðrÞ in quantitative agreement with the gradient expansion. Zero-order presentation of the Green function in the mean-path approximation using the Feynman path–integral method [54] yields an expression di ering from Eq. (6) by the numerical coe cients in front of the gradient corrections; this resulted in physically meaningless areas with vDFT ðrÞ > 0 in the low-electron-density regions, for example the periphery of free molecules or the centers of the faces of a cubic unit cell of LiF. Kinetic energy density derived using the virial theorem relationships of density-functional theory [57] seemed to depend on the origin of the coordinate system; this makes its application to molecules and crystals di cult. And, finally, the Weizsacker approximation [86], which corresponds to rapidly oscillating density (as happens in atomic cores), has failed to describe the middle-bond areas.

It would be interesting to test the hybrid orbital-free energy functionals [87] to search for the better energy-density description.

10.3.2

Exchange and Correlation Energy Densities

In DFT, the exchange-correlation energy, Exc, describes a contribution of nonclassical electron–electron interaction to total energy [88]. Exc is usually decomposed into the exchange, Ex, and the correlation, Ec, parts [31]; correspondingly, the exchange, ex, and correlation, ec, energy densities are often discussed. These functions are not uniquely defined (they can, for example, be altered by addition of any functional of r that integrates to zero over the density, or by coordinate transformation [88]); they also have a di erent physical content in di erent versions of DFT [88]. The densities ex and ec nevertheless play an important role in DFT; they also extend naturally a set of the functions, which are considered in QTAMC.

Many approximations have been developed which enable expression of ex and