Matta, Boyd. The quantum theory of atoms in molecules

222 8 Topology and Properties of the Electron Density in Solids

pressible which, together with the weaker electrostatics, results in less stress over the halide anions. Thus, partitioning of thermodynamic data into atomic components aids interpretation of their values and provides a powerful tool for qualitative predictions based on atomistic intuition.

8.7

Obtaining the Electron Density of Crystals

Most solid state electronic structure methods make use of pseudopotential techniques to avoid the explicit calculation of the atomic-like core regions. The valence electron density obtained from those calculations can, in principle, be completed by addition of the missing atomic core densities [54–56]. It must be noticed, however, that there are many di erent strategies for obtaining pseudopotentials, usually involving quite di erent partitions of the atomic density into core-like and pseudovalence contributions. An inappropriate match between the core and valence portions can result in spurious topological features. Whereas the field is very promising, some e ort is still needed, in our opinion, to compare the results from the main types of pseudopotential currently in use. There is, therefore, a limited number of methods that provide the all electron crystalline electron density in and analytical form suitable for high-quality QTAIM studies.

Under the promolecular approximation the crystalline electron density is obtained as a sum of atomic contributions:

rð~rÞ ¼ Xj	rjð~r ~RjÞ	ð6Þ
	~	~
where the sum runs over all atoms, and rjðr		RjÞ is the electron density of the

~

atom situated at Rj. This simple formula not only enables rapid approximate estimation but lies at the core of the experimental determination of the electron density from X-ray and neutron di raction experiments. As a further simplification, the atomic densities can be assumed to retain the spherical symmetry of the free atoms. Although the promolecular model lacks the internuclear electron density accumulation characteristic of true covalent bonding, it has been observed to frequently retain the topological features of the density in nonmolecular crystals. Even such fine details as the tiny nonnuclear maxima of some alkaline metals [18] can be retained. In contrast, the promolecular model has been shown to miss intermolecular bond CPs, i.e. give a di erent topology, in some molecular crystal studies [57].

The ab initio perturbed ion (aiPI) [58, 59] method has been used quite successfully for calculation of the electron density of many ionic crystals [33–35, 37, 44, 51, 60]. The method solves the Hartree–Fock equations of a solid in a localized Fock space by partitioning the crystal wavefunction into local, weakly overlapping, group functions, each containing a fixed number of electrons and a single nucleus. The aiPI crystal density can be described as a promolecular-like formula

8.7 Obtaining the Electron Density of Crystals 223

where the atomic contributions have been self consistently adapted to the crystal environment.

The crystal software of Dovesi et al. [61, 62] exploits the same MO-LCAO (crystal orbitals by linear combination of atomic Gaussian functions should be the motto here) approach that dominate quantum chemical studies in the molecular realm. Hartree–Fock (HF) and density functional theory (DFT) calculations have been used to examine the electron density topology of many ionic, covalent, molecular, and metallic crystals [26, 63–65]. It must be remarked, however, that use of the rich and flexible bases that are common within the molecular regime usually gives rise to linear dependencies and other technical problems that render the crystal calculation unsuitable. The molecular basis sets must then be truncated and Gaussian exponents less than 0.03–0.05 bohr 2, approximately, become forbidden. To compensate for this problem, the lowest retained Gaussian exponents are usually massaged or optimized within the crystal. When those di culties are taken into account, the method remains one of the two best techniques available to obtain the crystalline electron density. The next version of the software, release of which is planned for 2006, promises a definitive solution to the basis set problem and could produce a significant breakthrough in the field.

The other candidate for the best technique crown is the full potential linearized augmented plane wave (fpLAPW) formalism, implemented in the wien [30, 66] software by Blaha et al. The method uses di erent treatment for nonoverlapping spherical regions close to the nuclei, mu n tins, than for the internuclear regions. Atomic orbitals inside each mu n tin are described as the product of spherical harmonics and fully relativistic radial functions. Plane waves and a scalar relativistic treatment is applied to the internuclear region. The wave functions of both pieces are forced to match on value and slope at the mu n boundaries, but perfect agreement is not possible, because it would require inclusion of infinite spherical harmonics within each atomic sphere. Some care is then required to ensure that any remaining discontinuity in the density or its slope is not seen by the topological algorithms. Saturating the internuclear region with plane waves is, in contrast, both easy and inexpensive.

The fpLAPW, HF-LCAO, and aiPI topological properties of MgO and ZnO, two representative examples of the crystals that can be examined with the three techniques, are compared in Table 8.6. As usually occurs, the three techniques arrive at the same topology, which validates the very rapid and very stable aiPI procedure for analyzing trends and exploring the e ect of geometry changes. They also agree on the properties of the main bond critical points. The aiPI calculation, however, underestimates the electron density at the ring and cage CPs and, in general, in the low-density regions of the crystal. The e ect is more significant, because the average coordination index becomes smaller. The basin integrated charges provide clear evidence of this – the charges in the oxide basin agree for MgO (LCAO: 1.795, and aiPI: 1.852e) but di er substantially for ZnO (LCAO: 1.496, and aiPI: 1.800e). The di erence between the fpLAPW and HF-LCAO results is mainly because of the correlation e ects taken into account in the first calculation: DFT-LCAO results (not shown in the table) close the gap

2248 Topology and Properties of the Electron Density in Solids

Table 8.6 Comparison of the crystal topologies obtained by use of fpLAPW (PBE96 [31] GGA), HF-LCAO, and aiPI calculations on MgO (rock-salt structure, a ¼ 4:210 A˚ ) and ZnO (wurtzite structure,

a ¼ 3:250 and c ¼ 5:207 A˚ ). The (n,b,r,c) column provides the number of CPs of each type found in the unit cell. The rA/rB column gives the distance from the nuclei to a bond CP along the bond path. Atomic units are used throughout.

Crystal	Method	(n, b, r, c)	CP	r(~rc)	‘2r	rA/rB
MgO	fpLAPW	ð8; 48; 48; 8Þ	bðMg; OÞ	0.03911	0.21775	1.716/2.262
			bðO; OÞ	0.01770	0.04553	2.813/2.813
			r	0.01713	0.05651
		ð8; 48; 48; 8Þ	c	0.00998	0.02402
MgO	HF-LCAO		bðMg; OÞ	0.03599	0.25421	1.694/2.284
			bðO; OÞ	0.01611	0.04900	2.813/2.813
			r	0.01507	0.06309
		ð8; 48; 48; 8Þ	c	0.00723	0.03096
MgO	aiPI		bðMg; OÞ	0.03266	0.28543	1.704/2.274
			bðO; OÞ	0.01183	0.05363	2.813/2.813
			r	0.01075	0.06737
		ð4; 10; 8; 2Þ	c	0.00444	0.02460
ZnO	HF-LCAO		bðZn; OÞ	0.07870	0.47364	1.789/1.941
			bðZn; OÞ	0.07554	0.44617	1.802/1.962
			bðZn; OÞ	0.00650	0.02995	2.903/3.173
			r	0.00649	0.02990
			r	0.00437	0.01926
		ð4; 10; 8; 2Þ	c	0.00177	0.00850
ZnO	aiPI		bðZn; OÞ	0.07961	0.51117	1.797/1.932
			bðZn; OÞ	0.07568	0.48603	1.811/1.953
			bðZn; OÞ	0.00363	0.02172	2.809/3.268
			r	0.00363	0.02171
			r	0.00261	0.01354
			c	0.00082	0.00466

between the LCAO and LAPW techniques. The correlation tends to increase the electron density population of the bond CPs at the expense of the population of the low-density regions.

The theoretical electron densities can be corrected by using the crystal form

ð~Þ

factors, Fobs h , derived from experimental di raction measurements [67, 68]. In principle, the electron density of the crystal can be obtained directly from the form factors:

		Xh		~
r	~r	F	~h e i2ph~r		7	Þ
	obsð Þ ¼		obsð Þ		ð

~

8.7 Obtaining the Electron Density of Crystals 225

where the sum runs over the infinite lattice planes. In practice, however, the number of form factors determined is severely limited by the experimental conditions. Equation (7) comprises a series of alternating terms and truncation of the sum to any finite number of terms, no matter how large, produces a density that does not satisfy the required analytical conditions. In particular, robsð~rÞ can have unphysical negative values and is full of spurious hills and valleys that render it not useful the gradient vector field.

The electron density obtained as a Fourier di erence synthesis has much better analytical properties [71]:

X

ð Þ ¼ ð Þ þ ð~Þ r ~r rm ~r DF h e

~		8
i2ph ~r	ð		Þ

~

h

where rmð~rÞ is a smooth model density and

ð ð ð

~ ~ ~

DFðhÞ ¼ FobsðhÞ rmð~rÞei2ph ~r d~r ð9Þ

is the di erence between the values measured and calculated using rm for plane

~

h. A particular form of promolecular model constitutes the basis for the Hansen– Coppens multipole refinement model [72], used on most current studies of the experimentally determined electron density [21], but nothing prevents use of crystal’s LCAO or wien’s fpLAPW as the source of the rm. The theoretical electron density supplies the information missing on the incomplete set of experimental form factors and the experimental data correct any e ect not accounted for in the theoretical calculation, for example imperfect treatment of electronic correlation or an incomplete basis set. This combination of theoretical and experimental data can be of great help when investigating problems that lie at the resolution limit of either approach.

This is illustrated for the Si crystal in Fig. 8.5, in which the electron density along a nearest-neighbors SiaSi line is plotted. Addition of purely atomic densities (i.e. the promolecular model) produces a bond CP at the ð0; 0; 0Þ position with a very low density, 0.05e bohr 3. HF and BPW91-DFT LCAO calculations introduce nonnuclear attractors between nearest-neighbor silicon atoms that have been the subject of much speculation [73]. The HF result provides a high density, 0.098e bohr 3 at the SiaSi midpoint, which is slightly reduced to 0.090e bohr 3 at the correlated level. When these three model densities are corrected with the consolidated set of Cumming and Hart [69] and the Sake and Kato [70] set of form factors, all nonnuclear maxima disappear, and a well developed bond point appears. The corrected densities are, in fact, very similar, di ering by less than 0.0015e bohr 3 in the bonding region.

The meaning of those results should be assessed with some care. The SiaSi distance in the RTP (room temperature and pressure) diamond phase lies at the upper limit of a window of existence of NNM [38]. This is one of the reasons silicon

226 8 Topology and Properties of the Electron Density in Solids

Fig. 8.5 Electron density of the Si crystal along the xxx crystallographic direction. This is the line connecting two nearest neighbors situated on x ¼ 1/8 and x ¼ 1/8, respectively. Lines (a) through (c) are the raw promolecular, HF-LCAO, and DFT-LCAO electron densities, respectively. Lines (d) through (f ) correspond to the same model densities, now corrected with the experimental form factors [69, 70].

is an interesting target. The use of a small or unbalanced basis set, di erences in the approximated treatment of the correlation, and other small computational details can modify the theoretical equilibrium distance enough to make the NNM appear or disappear. The experimental form factors are corrected, on their own, to avoid some well known e ects, for example the directionally dependent X-ray absorption or the di erent time expended on measurement of the di erent

ð~Þ

Fobs h . The experimental form factors also unavoidably contain some thermal motion noise, even if they have been deconvoluted from thermal agitation. Neither quantum mechanical calculations nor experimental di raction data are thus free from trouble when examining problematic systems lying at the edge of topological regions. The interesting thing is that both approaches can be combined in several ways, giving the researcher new tools to examine the stability of di cult topological features.

Acknowledgements

We thank the Spanish Ministerio de Ciencia y Tecnologı´a for financial support under project BQU2003-06553.

			References	227
	References
1	R. F. W. Bader, Atoms in Molecules.	21	T. S. Koritsanszky, P. Coppens,
	A Quantum Theory, Oxford		Chem. Rev. 101 (2001) 1583–1627.
	University Press, Oxford, 1990.	22	R. Destro, R. Bianchi, C. Gatti, F.
2	R. F. W. Bader, Int. J. Quantum		Merati, Chem. Phys. Lett. 186 (1991)
	Chem. 56 (1995) 409–419.		47–52.
3	R. F. W. Bader, Phys. Rev. B 49 (1994)	23	C. Gatti, R. Bianchi, R. Destro, F.
	13348–13356.		Merati, J. Mol. Struct. (Theochem) 87
4	R. F. W. Bader, Int. J. Quantum		(1992) 409–433.
	Chem. 49 (1994) 299–308.	24	P. F. Zou, R. F. W. Bader, Acta Cryst.
5	R. F. W. Bader, P. L. A. Popelier, Int.		A 50 (1994) 714–725.
	J. Quantum Chem. 45 (1993)	25	V. G. Tsirelson, P. F. Zou, T. H. Tang,
	189–207.		R. F. W. Bader, Acta Cryst. A 51
6	P. F. Zou, R. F. W. Bader, Int. J.		(1995) 143–153.
	Quantum Chem. 43 (1992) 677–	26	C. Gatti, Z. Kristall, 220 (2005)
	699.		399–457.
7	R. F. W. Bader, Chem. Rev. 91 (1991)	27	M. Morse, S. S. Cairns, Critical point
	893–928.		Theory in Global Analysis and
8	R. F. W. Bader, J. Chem. Phys. 91		Di erential Geometry, Academic
	(1989) 6989–7001.		Press, New York, 1969.
9	R. F. W. Bader, Prog. Appl. Chem. 60	28	T. Hahn (Ed.), International Tables
	(1988) 145–155.		for X-Ray Crystallography. A. Space-
10	R. F. W. Bader, P. J. Macdougall,		group symmetry, Vol. A of
	J. Am. Chem. Soc. 107 (1985) 6788–		International Tables for X-Ray
	6795.		Crystallography, D. Reidel, Berlin,
11	R. F. W. Bader, Acc. Chem. Res. 18		1983.
	(1985) 9–15.	29	M. I. Aroyo, J. M. Pe´rez-Mato, C.
12	R. F. W. Bader, H. Essen, J. Chem.		Capillas, E. Kroumova, S. Ivantchev,
	Phys. 80 (1984) 1943–1960.		G. Madariaga, A. Kirov, H.
13	R. F. W. Bader, T. T. Nguyen-Dang, Y.		Wondratschek. Zeitschrift fu¨r
	Tal, Rep. Prog. Phys. 44 (1981) 893–		Kristallographie 221 (2006) 15–27,
	948.		http://www.cryst.ehu.es/.
14	R. F. W. Bader, T. T. Nguyen-Dang,	30	P. Blaha, K. Schwarz, J. Luitz,
	Adv. Quantum Chem. 14 (1981)		WIEN97, a Full Potential Linearized
	63–124.		Augmented Plane Wave package for
15	R. F. W. Bader, S. Srebrenik, T. T.		calculating crystal properties, (K.
	Nguyen-Dang, J. Chem. Phys. 68		Schwartz, Techn. Univ. Wien,
	(1978) 3680–3691.		Vienna). Updated version of P. Blaha,
16	S. Srebrenik, R. F. W. Bader, T. T.		K. Schwarz, P. Sorantin, and S. B.
	Nguyen-Dang, J. Chem. Phys. 68		Trickey, Comput. Phys. Commun. 59,
	(1978) 3667–3679.		399, 1990 (1999).
17	R. F. W. Bader, Acc. Chem. Res. 8	31	J. P. Perdew, S. Burke, M. Ernzerhof,
	(1975) 34–40.		Phys. Rev. Lett. 77 (1996) 3865.
18	V. Luan˜a, P. Mori-Sa´nchez, A.	32	A. Martı´n Penda´s, M. A. Blanco, E.
	Costales, M. A. Blanco, A. Martı´n		Francisco, Chem. Phys. Letters 417
	Penda´s, J. Chem. Phys. 119 (2003)		(2005) 16–21.
	6341–6350.	33	A. M. Penda´s, A. Costales, V.
19	A. M. Penda´s, A. Costales, M. A.		Luan˜a, Phys. Rev. B 55 (1997) 4275–
	Blanco, J. M. Recio, V. Luan˜a, Phys.		4284.
	Rev. B. 62 (21) (2000) 13970–13978.	34	M. A. Blanco, A. Costales, A. M.
20	A. M. Penda´s, J. Chem. Phys. 117		Penda´s, V. Luan˜a, Phys. Rev. B. 62
	(2002) 965–979.		(2000) 12028–12039.

228 8 Topology and Properties of the Electron Density in Solids

35V. Luan˜a, A. Costales, A. Martı´n Penda´s, Phys. Rev. B 55 (1997) 4285–4297.

36A. Costales, Topologı´a de la densidad electro´nica en cristales. una teorı´a cua´ntica del enlace cristalino., Ph.D. thesis, Universidad de Oviedo (1998). URL http://web.uniovi.es/qcg/

37A. Martı´n Penda´s, A. Costales, V. Luan˜a, J. Phys. Chem. B 102 (1998) 6937–6948.

38A. Martı´n Penda´s, M. A. Blanco, A. Costales, P. Mori-Sa´nchez, V. Luan˜a, Phys. Rev. Lett. 83 (1999) 1930–1933.

39G. K. H. Madsen, P. Blaha, K. Schwarz, J. Chem. Phys. 117 (2002) 8030–8035.

40J. Friis, G. K. H. Madsen, F. K. Larsen, B. Jiang, K. Marthinsen, R. Holmestad, J. Chem. Phys. 119 (2003) 11359–11366.

41P. Mori-Sa´nchez, A. Martı´n Penda´s,

V.Luan˜a, J. Am. Chem. Soc. 124 (2002) 14721–14723.

42P. Mori-Sa´nchez, A. Martı´n Penda´s,

V.Luan˜a, Phys. Rev. B 63 (2001) 125103–1–4.

43R. F. W. Bader, P. J. Macdougall,

C.D. H. Lau, J. Am. Chem. Soc. 106 (1984) 1594–1605.

44E. Francisco, J. M. Recio, M. A. Blanco,

A.Martı´n Penda´s, A. Costales, J. Phys. Chem. A 102 (1998) 1595–1601.

45E. Espinosa, E. Molins, C. Lecomte, Chem. Phys. Lett. 285 (1998) 170–173.

46E. Espinosa, M. Souhassou, H. Lachekar, C. Lecomte, Acta Cryst. B 55 (1999) 563–572.

47E. Espinosa, I. Alkorta, J. Elguero, E. Molins, J. Chem. Phys. 117 (2002) 5529–5542.

48O. Ga´lvez, P. C. Go´mez, L. F. Pacios,

J.Chem. Phys. 115 (2001) 11166– 11184.

49R. F. W. Bader, Atoms in Molecules. A Quantum Theory, Oxford University Press, Oxford, 1990.

50A. Costales, M. A. Blanco, A. M. Penda´s, A. K. Kandalam, R. Pandey, J. Am. Chem. Soc. 124 (2002) 4116–23.

51V. Luan˜a, A. Costales, P. MoriSa´nchez, A. Martı´n Penda´s, J. Phys. Chem. B 107 (2003) 4912–4921.

52A. Costales, M. A. Blanco, P. MoriSa´nchez, V. Luan˜a, A. Martı´n Penda´s,

J.Phys. Chem. A 108 (2004) 2794– 2801.

53P. Mori-Sa´nchez, Densidad electro´nica y enlace quı´mico. De la mole´cula al cristal, Ph.D. thesis, Universidad de Oviedo (2002). URL http://web.uniovi.es/qcg/

54J. Cioslowski, P. Piskorz, Chem. Phys. Lett. 255 (1996) 315–319.

55S. F. Vyboishchikov, A. Sierraalta, G. Frenking, J. Comput. Chem. 18 (1997) 416–429.

56A. Sierraalta, A. Herize, J. Mol. Struct. (Theochem) 529 (2000) 173–182.

57C. Gatti, E. May, R. Destro, F. Cargnoni, J. Phys. Chem. A 106 (2002) 2707–2720.

58V. Luan˜a, L. Pueyo, Phys. Rev. B 41 (1990) 3800–3814.

59M. A. Blanco, V. Luan˜a, A. M. Penda´s, Comput. Phys. Commun. 103 (1997) 287–302.

60V. Luan˜a, A. Costales, A. M. Penda´s,

L.Pueyo, J. Phys.: Condens. Matter 11 (1999) 6329–6336.

61R. Dovesi, V. R. Saunders, C. Roetti,

M.Causa`, N. M. Harrison, R. Orlando, C. M. Zicovich-Wilson, CRYSTAL98 User’s Manual (1998).

62V. R. Saunders, R. Dovesi, C. Roetti,

R.Orlando, C. M. Zicovich-Wilson,

N.M. Harrison, K. Doll, B. Civalleri,

I.J. Bush, CRYSTAL03 User’s Manual, http://www.crystal.unito.it/ (2003).

63G. K. H. Madsen, C. Gatti, B. B. Iversen, L. Damjanovic, G. D. Stucky,

V.I. Srdanov, Phys. Rev. B 59 (1999) 12359–12369.

64C. Gatti, V. R. Saunders, C. Roetti,

J.Chem. Phys. 101 (1994) 10686– 10696.

65B. Silvi, C. Gatti, J. Phys. Chem. A 104 (2000) 947–953.

66P. Blaha, K. Schwarz, G. Madsen, D. Kvasnicka, J. Luitz, WIEN2k, an Augmented Plane Wave plus Local Orbitals program for calculating crystal properties, Vienna University of Technology, Institute of Physical

			References	229
	and Theoretical Chemistry (2001).	69	S. Cumming, M. Hart, Aust. J. Phys.
	URL http://www.wien2k.at/		41 (1988) 423.
67	G. H. Stout, L. H. Jensen, X-ray	70	T. Saka, N. Kato, Acta Crystallogr. A
	structure determination: A Practical		42 (1986) 469.
	Guide, 2nd Edition, John Wiley &	71	R. Hosemann, S. N. Bagchi, Nature
	Sons, New York, 1989.		171 (1953) 785–787.
68	C. Giacovazzo, H. L. Monaco, D.	72	N. K. Hansen, P. Coppens, Acta
	Viterbo, F. Scordari, G. Gilli, G.		Cryst. A 34 (1978) 909.
	Zanotti, M. Catti, Fundamentals of	73	J. M. Zuo, P. Blaha, K. Schwarz,
	Crystallography, Oxford UP, Oxford,		J. Phys. Condens. Matter. 9 (1997)
	UK, 1992.		7541–7561.

231

9

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

Yosslen Aray, Jesus Rodrı´guez, and David Vega

9.1 Introduction

The atoms in molecules theory provides a rigorous definition of chemical bonds for all types of molecules and solids [1–10] and has proven useful in analysis of the physical properties of insulators, pure metals, and alloys [4–8]. In particular, it has been observed that the strength of the bonding between a given pair of atoms in a molecule correlates with the values of the electron density at the bond critical point, rb [1]. A simple relationship between the binding energy and rb in periodic solids has also been reported [4, 6, 8]. The AIM theory also leads to unique partition of three-dimensional space into a collection of chemically identifiable regions called atomic basins (the atoms in a molecule or in a crystal). These are the most transferable pieces one can define in exhaustive partitioning of the real space [1]. In this chapter we describe the implementation of an algorithm that uses the r(r) topological information to determine the main elements of the AIM theory for periodic systems, and discuss an application to nanocatalysis.

9.2

Implementing the Determination of the Topological Properties of r(r) from a Three-dimensional Grid

The CPs are usually calculated using the Newton–Raphson (NR) technique [11]. The NR algorithm starts from a truncated Taylor expansion at a point r ¼ r0 þ h, about r0 of a multidimensional scalar function ð‘rðrÞÞ:

‘rðrÞ ¼ ‘rðr0Þ þ H0h þ higher order terms

ð1Þ

where H is the Hessian (the Jacobian of ‘rðrÞÞ at point r0. The best step, h, to move from the initial r0 to the CP is h ¼ H 1‘rðr0Þ. This correction is then used to obtain a vector rnew ¼ rold þ th (t is a small value) and the process is iter-