Matta, Boyd. The quantum theory of atoms in molecules

252 9 Atoms in Molecules Theory for Exploring the Nature of the Active Sites on Surfaces

Fig. 9.11 (a) Top view of the 0.002 e A˚ 3 contour of V(r) mapping of the V(r) field values on to colors placed over the IAS defining the basin of the outermost sulfur atoms. The color map tab at the bottom is: dark blue ( 157.530 to 78.765 kJ mol 1), sky blue ( 78.765 to 42.008 kJ mol 1), light blue ( 42.008 to 31.506 kJ mol 1), dark green ( 31.506 to 21.004 kJ mol 1), green ( 21.004 to 10.502 kJ mol 1), light green ( 10.502 to 0.000 kJ mol 1), yellow (0.000 to 26.255 kJ mol 1), yellow cream (26.255 to

52.510 kJ mol 1), orange (52.510 to 65.638 kJ mol 1), brown (65.638 to 78.765 kJ mol 1), purple (78.765 to 91.893 kJ mol 1), light red (91.893 to 105.020 kJ mol 1) and red (105.020 to 144.403 kJ mol 1). White squares indicate the most positive zones on the V(r) contour. (b) Side view of the basin of the outermost sulfur atoms. The 0.002 au contour of V(r) is used to define the border of these open atoms. V(r) mapped on this border enables us to locate the stronger Lewis acid sites.

Acknowledgments 253

in the inter-edge space, forming four-coordinated atoms. Holes (denoted as a white rectangle in Fig. 9.10a) can be observed in the bond path network, between each pair of consecutive four-coordinated S atoms.

The basins for the outermost S atoms of site 3 are shown in Fig. 9.10b. It is apparent that compared with the surface without vacancies (Fig. 9.9b), the basins on the S edge have su ered a drastic change in their shape. These atoms spread to occupy the space liberated in the vacancies and adopt a triangular form like an arrow head. The base of each triangle is defined by the SaS IAS and the opposite angle (head point) penetrates inside the vacancy, occupying the space liberated by the removed S atoms, ending at the middle of the network bond path hole. The region around these head points (denoted by rectangles and circles in Fig. 9.10b) should be the sites with smaller sulfur electronic density on the surface, enabling access to the Mo atoms (the Lewis acid sites). This can be easily checked by means of the electrostatic potential. This let us visualize directly the stronger Lewis acid sites (most positive zones) by surface mapping the V(r) values on to colors on the 0.002 electron A˚ 3 isosurface of r(r). Figure 9.11a shows the superimposition of this color map on the exposed S atom basins of the MoS2 edges model. Starting from the most negative V(r) values (caption of Fig. 9.11a), three kinds of blue and three kinds of green denote the most negative values and two kinds of yellow and one orange denote the positive V(r) values. The orange region corresponds to the most positive site and, as is apparent from Fig. 9.11, it perfectly matches with the hole where two sulfur basin head points almost converge.

Exposed atoms, atoms in an isolated molecule, have substantial open parts that extend to infinity. These atoms are open or unbounded at the exterior of the surface and a practical definition [9] is to cap the atom with an isosurface of the electron density with small r(r) value representing the van der Waals envelope of the system. The 0.002 electron A˚ 3 contour of r(r) shown in Fig. 9.11 just defines the border of the outermost sulfur atoms of the (010) MoS2 surface. Figure 9.11b shows a side view of the atomic shape of these sulfur atoms. It is worthy of note that the stronger Lewis acid sites are located between the sulfur atoms at the bottom of the r(r) 0.002 contour valley. This is the region where the Mo atoms are most accessible to incoming molecules.

In summary, there is a profound relationship between the vacancy creation energy, i.e. the energy necessary to remove exposed sulfur atoms from the MoS2 surface and the nature (number of bond paths and r(r) value at the bond and cage CPs) of the exposed polyhedra defining the surface graph. The structure and shape of the basins of the exposed sulfur atoms also enable us to visualize and locate the Lewis acid sites on this surface.

Acknowledgments

This work was supported by the grant G-2000001512 from the CONICIT (Consejo Nacional de Investigaciones Cientı´ficas y tecnolo´gicas) of Venezuela.

254 9 Atoms in Molecules Theory for Exploring the Nature of the Active Sites on Surfaces

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Part III

Experimental Electron Densities

and Biological Molecules

259

10

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

Vladimir G. Tsirelson

10.1 Introduction

Richard Bader summarized his longstanding studies dealing with development of the quantum theory of atoms in molecules and crystals (QTAMC) in his classic book published in 1990 [1]. He demonstrated that the ground-state electron density rðrÞ, its gradient vector field, ‘rðrÞ, and the Laplacian field, ‘2rðrÞ, enable definition of bond paths, the lines of maximum electron density linking some of the nuclei, which can be identified at the equilibrium geometry, with the chemical bonds, and enable characterization of the type of these bonds in molecules and crystals. Originally, QTAMC was developed using electron density calculated from the wavefunctions. Later, it was demonstrated [2–5] that electron density derived from results from accurate X-ray, g-ray, and synchrotron radiation di raction experiments could also be analyzed in the same manner. Initial application of QTAMC to the experimental electron density of compounds with di erent types of chemical bond [6–11] showed that this function has a similar topology and the same set of critical points as quantum-mechanical r. Thus, the experimental electron density seems to be suitable for the QTAMC analysis of bonding in molecules and crystals, with electron density deduced from the wavefunctions. This approach is now widely used for exploration of experimental features of electron density; a thorough review of the results obtained from this type of work is available elsewhere [12–19].

In addition to electron density and its derivatives, Bader has also described the role of the positively-defined electronic kinetic energy density:

g r	Þ ¼ ð	h2=2m	‘r‘r 0 g r; r0	Þjr ¼	r 0	ð	1	Þ
ð		Þ	ð

and the potential energy density which is negative everywhere, defined by expression:

vðrÞ ¼ ½Zae2=ðr RaÞ&rðrÞ þ e2 ð½Gðr; r1Þ=ðr r1Þ& dr1 þ Vn	ð2Þ
Xa

260 10 Interpretation of Experimental Electron Densities by Combination of the QTAMC and DFT

where gðr; r0Þ and Gðr; r1Þ are one-electron and two-electron density matrices, respectively, Za is the charge the nucleus a, e and m are the electronic charge and mass, respectively, and Vn is the nuclear repulsion. These functions are related by the local form of the virial theorem [20]:

2gðrÞ þ vðrÞ ¼ ðh2=4mÞ‘2rðrÞ

ð3Þ

Finally, the density of the total electronic energy is defined as:

heðrÞ ¼ gðrÞ þ vðrÞ ð4Þ

Exploration of energy distributions in molecules by use of wavefunction-based calculations [21–26] has revealed that analysis of the local electronic energy is a direct approach to characterization of bonding in molecules and crystals. In particular, it facilitates recognition of the type of atomic interactions from the

properties of bond critical points at r			b heðrbÞ	<	0 and gðrbÞ	=r	ðrbÞ	<	1, and
‘	2
		rðrbÞ < 0, are observed for shared-type atomic interactions whereas heðrbÞ q0,

gðrbÞ=rðrbÞ > 1, and ‘2rðrbÞ > 0 are typical for intermediate and closed-shell interactions [16, 18, 23, 27].

Bader has also stressed [26, 28] that the potential energy density represents the field of the virial of the Ehrenfest force [29] acting on an electron at r, the virial field vðrÞ. Irrespective of the type of atomic interaction, each bond path in the electron density at equilibrium geometry is homeomorphically mirrored by a virial path, a line of maximum negative potential energy density linking the same nuclei [24]. The presence of the bond paths and virial path provides, according to Ref. [26], an indicator of bonding atomic interaction. A network of the virial and bond paths defines a molecular graph, which is independent of the nuclear vibrations in a stable system.

It was, of course, a challenge to perform real-space energy analysis of bonding on the basis of the experimental electron density (ED). There is, fortunately, a theory which establishes the interconnection between the electron density and energy densities of di erent kinds. This is the density-functional theory (DFT) [30–40] which exploits r as a main variable and determines all the properties of atoms, molecules, and crystals in the ground electronic state [41]. Thus, DFT is a basis for quantitative characterization of bonding in terms of energy densities and other functions related to electron density. It is, therefore, attractive to combine the formalism of the DFT with experimental electron density to analyze the nature of atomic and molecular interactions in molecules and solids in terms of the local energies. This might, in principle, be done in two di erent ways. The exact functionals connecting r and the energy densities of electrons – the kinetic, potential, exchange and correlation densities – are, in general, unknown [41]. DFT methods therefore use either the Kohn and Sham orbital scheme [42] or approximate functionals with explicit (but nonunique) dependence of these functions on r and its derivatives [31]. The former approach might be achieved by use of an idempotent one-electron density matrix iteratively reconstructed from

10.2 Specificity of the Experimental Electron Density 261

the experimental electron density [12] or by use of Hartree–Fock calculations constrained to obtain the wavefunctions that reproduce experimental X-ray structure factors [43]. The latter approach is closer to the Hohenberg–Kohn formulation of the DFT [41] – it is orbital-free and enables avoidance of the variational determination of wavefunctions. This is the approach used in this chapter – our objective is to demonstrate here that it enables more comprehensive extraction of the information about chemical bonding contained in the experimental electron density.

10.2

Specificity of the Experimental Electron Density

First, we must clarify what is implied by ‘‘experimental’’ electron density. Experimental X-ray structure factors enable reconstruction of the electron density by means of Fourier series. Such density is strongly distorted by series truncation and is hardly suitable for quantitative topological analysis [12]. The electron density is therefore reconstructed from experiment by means of the multipole model in which the electron density of a molecule or crystal is presented as a sum of aspherical pseudoatomic densities, ratomðrÞ, each of which is expanded into a convergent series over the spherical harmonics or over their real combinations, ylmG. There are a few multipole models, di ering in insignificant details [12, 13]. In one of these, the Hansen–Coppens model [44], which we use in this work, the pseudoatomic density has the form:

4				l
Xl		003Rlðk00rÞ	X
ratomðrÞ ¼ rcoreðrÞ þ Pvk03rvalðk0rÞ þ k			m	PlmGylmGðr=rÞ ð5Þ
¼	0			¼
				l

where rcore and rval are the atomic core and valence electron densities, respectively, described by the wavefunctions of the free atoms, k0 and k00 are atomic valence-shell contraction–expansion terms, and Pval and PlmG are the multipole electronic populations. The radial density functions have the exponential form RlðrÞ@r nl expð k00xrÞ, where nl is related to the principal quantum number of the atom. The electronic populations of multipoles and the k terms are determined by least-square fit to the experimental structure factors.

All the multipole models are rather flexible; it is, however, necessary to keep several points in mind. The exponential term, x, in the radial density functions RlðrÞ corresponds to the best model fit to the restricted number of low-angle X- ray structure factors and depends mainly on the electron density of the valence subshells. At the same time, the orbital exponential terms of the Hartree–Fock or Kohn–Sham basis functions are derived from the requirement of a minimum energy of a system; they are, therefore, more sensitive to the distribution of core electrons whose energies are higher. The interatomic density associated with interference of the atomic orbital is, moreover, not accounted for completely in the multipole model consisting of the atomic-like terms. The same is true for the part of the valence electron density localized near the nuclei. Further: