Matta, Boyd. The quantum theory of atoms in molecules

202 7 Solid State Applications of QTAIM and the Source Function

and NðWÞ may thus be decomposed in an inner contribution Ni(W) and in a outer contribution No(W):

NðWÞ ¼ MðW; WÞ þ	X0	ð9Þ
	MðW; W0Þ ¼ NiðWÞ þ NoðWÞ
	W 0W

MðW; WÞ represents the contribution to the electron population of W from its own basin, whereas MðW; W0Þ is the contribution to this same population from basin W0. The matrix M is not usually symmetric and defines a full population analysis based only on the observable ‘2r. Preliminary results, and the noteworthy numerical di culties encountered in implementation of this unambiguous population analysis, have recently been presented [85] and will soon be published [86]. Interesting correlations of MðW; WÞ and MðW; W0Þ with, respectively, the localization and the delocalization indices, have been observed in several diatomic molecules and along the ethane, ethylene, acetylene series. Such population analysis may potentially be applied, with no approximations, to the experimental electron densities.

Extension of the SF approach and of population analysis based on the SF to the a and b-components of the spin-polarized densities is planned for the near future.

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48F. J. Himpsel, P. M. Marcus, R. Tromp, I. P. Batra, M. R. Cook, F. Jona, H. Liu, Structure analysis of Si(111)2x1 with low-energy electron di raction, Phys. Rev. 1984, B30, 2257–2259.

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207

8

Topology and Properties of the Electron Density in Solids

Vı´ctor Luan˜a, Miguel A. Blanco, Aurora Costales, Paula Mori-

Sa´nchez, and Angel Martı´n Penda´s

8.1 Introduction

Most chapters in this book explore the foundation and consequences of the quantum theory of atoms in molecules [1–17] (QTAIM) when applied to gas-phase molecules. Although most of the theory remains unaltered for solids, there are some significant di erences that should be taken into account.

First, solids are an exciting challenge to physical bonding theories, because of their wide diversity of macroscopic and microscopic behavior. Insulators and electrical conductors; ionic, covalent, and molecular crystals; a rich variety of magnetic conductors; impurities and defects that can modify local and even bulk properties; significant di erences between bulk and surface electronic properties; and a yet to be understood e ect of grain shape and size on electronic and mechanical solid properties. The field o ers a large collection of problems awaiting and deserving a careful look.

Topologically the main di erence between solids and gas phase molecules is that atomic basins always have a finite size inside solids. As a consequence, the electron density of solids always has a rich collection of ring and cage critical points, in addition to the usual bond and nuclear points that dominate the topological description of molecules. Finite basins also imply well defined atomic radii in every geometrical direction. In this way the topology of the electron density provides a firm foundation for the important concept in the solid state theory of the atomic (ionic) radius, which played a prominent role in the early theories of phase stability. We will observe how the radius concept emerges again in topological analysis of the electron density as we rationalize the di erent topologies observed in families of compounds.

The large variability and mixture of types of bonding that may be found together in a given crystal makes the solid state a sort of paradise for complex topological behavior. This includes all kinds of nonstandard connectivities, for

208 8 Topology and Properties of the Electron Density in Solids

example bonds to rings, or bond to bonds, that are stabilized in high-symmetry situations [18], nonnuclear attractors which are rather uncommon in isolated molecules, and topological polytypism, i.e. systems with the same atomic arrangement with electron densities with di erent topologies.

Solids also change volume and shape, and undergo electronic and structural phase transitions under the influence of external thermodynamic conditions, most notably pressure and temperature. Several metastable phases can, in fact, coexist for the same thermodynamic regime. The graphite, diamond, and buckminsterfullerene allotropes of carbon come to mind as simple and well known examples of this polymorphism. The availability of external variables that may be controlled at will introduces a wide scenario that lies outside the capabilities of experimental molecular physics. Pressure, the state variable that is most easily simulated, may be used to push atoms well past their equilibrium positions in the gas phase, so many profound questions that are still the subject of debate may find a natural answer here. One example is the role of unusual bond paths, for example those occurring among the anions in ionic crystals. These are intrinsic bond paths, surviving the large geometrical changes that are introduced on compression. Another is the e ect of the virial of the nuclear forces in determining atomic energies, which is obscured in geometrically constrained isolated molecules.

The mechanics of quantum atoms in solids also provides a rigorous microscopic formulation of thermodynamic properties [19, 20]. The exhaustive partition of volume into atomic contributions, for example, enables us to immediately partition some important physical properties of crystals, for example the macroscopic compressibility. This generates fruitful chemical images of atoms being di erentially compressed under application of an external pressure, in accordance with their intrinsic, atomic compressibility.

Last, but not least, we cannot forget that the electron density in crystals is an experimental observable. The applications of the quantum theory of atoms in molecules have enjoyed an important boost in the last decade, coupled with the availability of increasingly accurate experimental densities [21]. Many, if not most, experimentalists have now embraced the QTAIM to rationalize their data, leaving behind all the problems associated with use of density di erence maps. The day that experiments provide equivalent or even more accurate densities than careful computation is approaching very quickly.

Gatti and coworkers embraced QTAIM methods when discussing the electron density of l-alanine [22, 23], although Zou’s work in Bader’s laboratory was, probably, the first attempt to derive specifically crystalline concepts, for example the Wigner–Seitz cell, from the electron density topology [24, 25]. Whereas approximately 15 years have passed since these seminal topological works in crystals, the field is still in its infancy, and whole terrains remain unexplored. For example, the very-high-pressure regime, in which large bonding changes are expected to occur, has not been appropriately simulated, and the always di cult simulation of temperature has simply been ignored. These di cult problems, together with many others, guarantee plenty of free room for future work.

8.2 The Electron Density Topology and the Atomic Basin Shape 209

It is not our purpose in this chapter to provide a thorough review of the QTAIM work performed for solid materials. There are already excellent accounts that cover that work (for example, the review by Koritzansky and Coppens [21] on experimental densities, Gatti’s excellent paper on computational results [26], and Chapters 10–12 of this book). We will focus, in contrast, on some of the basic topological information obtained when studying solids that is not present at the isolated gas phase level. For the rest of this chapter we will assume the solid to be in a crystalline state, so space-group symmetry will be an essential aspect of the determination and analysis of electron density. Our discussion will also be limited to ideal bulk properties like most of the studies published so far. (See Chapters 7 and 9 of this book for authoritative discussions of surfaces).

8.2

The Electron Density Topology and the Atomic Basin Shape

Solid state theory assumes an ideal crystal model is formed as a translationally invariant repetition of a parallelepipedic unit cell. Such an ideal crystal has no borders, but the þy and y limits along each crystallographic direction are connected to form a ring. Whereas a finite molecule inhabits S3, the ordinary three-dimensional space, the ideal crystal belongs in R3, the three-dimensional torus. As a consequence, the Poincare´–Hopf formula, that connects the number of critical points (CP) of any molecular scalar field, is substituted here by the Morse relationships [27]:

n; c b1; b; r b3; n b þ r c ¼ 0;

ð1Þ

where n, b, r, and c denote the number of ð3; 3Þ –nuclear–, ð3; 1Þ –bond–, ð3; þ1Þ –ring–, and ð3; þ3Þ –cage–, CPs, respectively, that can be found in each crystal unit cell. Apart from this traditional (rank, signature) notation, the four types of nondegenerate CP can be distinguished in terms of the dimensionality of their attraction and repulsion basins, from 0D (points) to 3D (volumes), as summarized in Table 8.1.

Space group symmetry plays an essential part in the determination and analysis of the crystalline electron density. The gradient vector field must fulfill the point group symmetry within the unit cell; it must, therefore, be aligned with the symmetry axes, be contained within the symmetry planes, and be zero wherever an inversion point occurs. As a consequence, the local point group symmetry of some positions within the unit cell ensures the gradient vector field will be zero there. For example, the coincidence of three noncollinear symmetry axes, of an axis and a perpendicular plane, or the existence of a single inversion center will force that particular place to be a CP. Table 8.2 summarizes the crystallographic symmetries that guarantee the presence of a CP.

The points with special symmetry within the unit cell have been catalogued for the 230 space groups [28, 29] under the denomination of Wycko positions.

2108 Topology and Properties of the Electron Density in Solids

Table 8.1 Critical points (CP) of the electron density classified by rank, signature, and the dimensions of its attraction and repulsion basins (AB and RB, respectively). The attraction (repulsion) basin is defined as the geometrical place of all uphill gradient vector field lines ending in (starting from) the CP. Only the nondegenerate CPs, i.e. those with rank 3, should occur in an ordinary molecule and crystal, the appearance of a degenerate point would indicate structural instability.

(r, s)	AB	RB	QTAIM name	Description
ð3; 3Þ	3D	0D	Nucleus (n)	Local maximum
ð3; 1Þ	2D	1D	Bond (b)	First-order saddle
ð3; þ1Þ	1D	2D	Ring (r)	Second-order saddle
ð3; þ3Þ	0D	3D	Cage (c)	Local minimum
ð2; 2Þ	2D	0D		Degenerated 2D maximum
ð2; 0Þ	1D	1D		2D saddle
ð2; þ2Þ	0D	2D		Degenerated 2D minimum
ð1; 1Þ	1D	0D		Degenerated 1D maximum
ð1; þ1Þ	0D	1D		Degenerated 1D minimum
ð0; 0Þ	0D	0D		Inflection point

Close inspection of Table 8.2 reveals that the sites ensured to be a CP are nothing but those Wycko positions having their three crystallographic coordinates fixed by symmetry. It is also interesting to notice that all other special positions, though not guaranteeing the occurrence of a CP, strongly limit their possible location. One-parameter and two-parameter special positions may have a null gradient point at selected values of those parameters, though the actual point may coincide with a higher symmetry position. In that way, the search for some hard- to-find CPs can be bracketed.

Table 8.3 shows the electron-density topology of face-centered cubic (FCC) Al. The unit cell of this metallic phase contains four Al atoms at Wycko ’s 4a posi-

Table 8.2 Symmetry of fixed-point positions ensuring the presence of a critical point.

Triclinic

Cið1Þ

Monoclinic

C2hð2=mÞ

D2hðmmmÞ

Orthorhombic

D2ð222Þ

Tetragonal

C4hð4=mÞ

D4

ð422Þ

D2dð

42mÞ

D4hð4=mmmÞ

Trigonal

C3ið

3

Þ

D3

ð32Þ

D3dð

3

mÞ

D3hð62mÞ

D6hð6=mmmÞ

Hexagonal

C3hð6Þ

C6hð6=mÞ

D6ð622Þ

Tð23Þ

Thðm3Þ

Oð432Þ

Tdð

Ohðm

mÞ

Cubic

43mÞ

3

8.2 The Electron Density Topology and the Atomic Basin Shape 211

Table 8.3 Electron density topology of Al for the experimental FCC crystal structure at ambient conditions (space group Fm 3 m, a ¼ 4:0495 A˚ ). The results correspond to a wien [30] fpLAPW calculation using the PBE96 [31] generalized gradient approach (Section 8.7).

Wycko			Pos.						CP
4a	m3m		ð0; 0; 0Þ						Al	1.5181103	þ2.236810 2
4b	m3m		ð1=2; 1=2; 1=2Þ						c1	1.656710 2
8c			ð1=4; 1=4; 1=4Þ						c2	2.797510 2	þ8.466810 3
	43m
24d	m:mm		ð	0; 1=4;			Þ		b	2.995410 2
			ð			1=4					1.212710 3
32f	:3m			x; x; x	Þ	x	¼	0:28574	r	2.808310 2	þ
											7.211810 3

tions, 24 AlaAl bond CPs, 32 ring CPs, and two di erent types of cage point, one at the 4b octahedral site and a second at the 8c tetrahedral position. Four out of these five types of CP occur at positions completely fixed by the local symmetry. Only the ring CPs happen along the ðx; x; xÞ line, with a single free parameter.

As happens in molecules, the crystal volume is divided, by means of zero flux surfaces, into quantum regions in which all quantum observables are well defined and all quantum laws can be locally applied [1, 7, 14–16, 32]. No gradient field line can cross those surfaces except the single 1D repulsion (attraction) line of the bond (ring) CP whose 2D attraction (repulsion) basin is the origin of the surface.

The smallest quantum region in a crystal is a primary bundle [33], formed by all the gradient vector field lines of the electron density going uphill from a given cage CP to a given nuclear CP. The bundle is topologically equivalent to a polyhedron, with vertices, edges and faces in numbers that fulfills Euler’s invariant formula:

faces þ vertices ¼ edges þ 2:

ð2Þ

Apart from the nuclear and the cage points that originate the bundle, the vertices are bond and ring CPs. The faces correspond to the surface of the 2D attraction basins of the bond, and to the 2D repulsion basins of the ring CPs. The intersection between these surfaces gives rise to the edges of the bundles. The most basic topological structure of the crystal is that of its distinct primary bundles, and of their interconnections.

Primary bundles are, however, not reported in common topological analyses of the electron density. They are normally collected to introduce coarser partitions of space in which bigger zero-flux bounded regions are taken as the primary objects of study.

Union of all the primary bundles sharing the same maximum gives rise to an atomic basin. The interior of this object is nothing but the attraction basin of its