Topology of the Electron Density in Open-Shell Systems - The Journal

Dec 4, 2009 - Abstract Image. This work describes the decomposition of the electron density field of open-shell molecular systems into physically mean...
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J. Phys. Chem. A 2010, 114, 1200–1206

Topology of the Electron Density in Open-Shell Systems Rosana M. Lobayan,† Diego R. Alcoba,‡,# Roberto C. Bochicchio,*,‡,# Alicia Torre,§ and Luis Lain§ Facultad de Ingenierı´a, UniVersidad de la Cuenca del Plata, LaValle 50, 3400, Corrientes, Argentina, Departamento de Física, Facultad de Ciencias Exactas y Naturales, UniVersidad de Buenos Aires, Ciudad UniVersitaria 1428, Buenos Aires, Argentina, and Departamento de Quı´mica Fı´sica, Facultad de Ciencia y Tecnologı´a, UniVersidad del Paı´s Vasco, Apdo. 644 E-48080 Bilbao, Spain ReceiVed: October 16, 2009; ReVised Manuscript ReceiVed: NoVember 16, 2009

This work describes the decomposition of the electron density field of open-shell molecular systems into physically meaningful contributions. The new features that the open-shell nature of the wave function introduces into these fields are topologically studied and discussed, showing the charge concentration and depletion regions within the system. The localized character (field concentration only close to the nuclear positions of the system) or the nonlocalized character (concentration in other regions of the system) is used to study the reliability of the Lewis model of bonding to describe chemical bonding phenomenon in open-shell systems. Numerical examples are reported for molecular systems at a correlated level of approximation, in the single-double configuration interaction wave function approach. 1. Introduction The interpretation of chemical data is directly related to the extraction of the information contained in the wave function of molecular systems, which is generally represented by means of the electron density. The theory of Atoms in Molecules (AIM), introduced a long time ago,1,2 describes the topology of the electron density in molecular systems, providing important information which is essentially contained in two devices, the electron density, F(r), and its associated Laplacian field, ∇2F(r).1,2 The density is characterized by the localization and classification of its critical points, cp, (points of the distribution where the gradient of the density vanishes, ∇F(r) ) 0), that is, maxima, minima, or saddle points. The classification of the cp’s (nuclear, bond, ring, and cage type) and the value of the Laplacian of the charge density at these points turn out to be essential to describe the electronic structure of the molecules.1-3 The cp’s of the charge distribution are classified according to the characteristics of the Hessian matrix of the function F(r), that is, the sign and number of its non-null eigenvalues.1,2 The cp’s that exhibit three non-null negative eigenvalues are called nuclear critical points, ncp’s (3, -3), describing the electron density concentration around the nuclear position. These points together with the denominated bond critical points, bcp’s (3, -1) (two negative and one positive eigenvalues) are used to define the topological bonding region as the region around the line joining two nuclei and possessing a bcp. This approach will be denominated hereafter the local formulation of the description of the electron distribution. The other main derived quantity is the Laplacian function, that is, the sum of the curvatures in the electron density along any orthogonal coordinate axes at the point r. Its sign indicates whether the electron density is locally depleted (positive) or locally concentrated * To whom correspondence should be addressed. E-mail: [email protected]. † Universidad de la Cuenca del Plata. ‡ Universidad de Buenos Aires. § Universidad del Paı´s Vasco. # Fellow of the Consejo Nacional de Investigaciones Cientı´ficas y Te´cnicas (CONICET) Repu´blica Argentina.

(negative), and thus, it constitutes valuable information to describe and classify the behavior of the density around a local point.1,2,4 The pth-order reduced density matrix (p-RDM) corresponding to an N-electron system is derived as an average of the N-electron density matrix.5,6 The procedure to obtain a lowerorder RDM from a higher-order one is called contraction mapping.7 Thus, the electron density, expressed as the diagonal part of the first-order reduced density matrix (1-RDM), can be derived from application of the contraction mapping to the second-order reduced density matrix (2-RDM). This algorithm allows one to perform a natural decomposition of the electron density F(r) into two parts, F(p)(r) and F(u)(r), which correspond to the effectively paired and effectively unpaired electron densities, respectively.8,9 Hence, this procedure provides the appropriate method for studying the topology of each component in an independent way and consequently for drawing out all of the features of the electron density. In previous works, we have described the behavior of both contributions of the electron density and the shift of their (3, -3) cp’s and (3, -1) cp’s in comparison with the ncp’s and bcp’s of the total density.8,9 The Laplacian fields for both effectively paired and unpaired densities have been studied in order to describe its behavior in normal systems9 with conventional bonds (covalent and ionic ones) and in systems with nonclassical patterns of bonding such as boron hydrides of closed-shell structure.10 These studies have shown that the Lewis conjecture of the chemical bond as a fermion pairing phenomenon11 can be quantum mechanically related to the local concentration/depletion of the effectively paired and unpaired density fields at the bonding regions for closed-shell systems8,9 exhibiting slight deviations for systems with nonclassical patterns of bonding.10 However, this statement is not clear for the open-shell case, and thus, to study the validity of this conjecture is the main goal of this work. Hence, in the present paper, we perform the decomposition of the electron density for open-shell systems within the RDM framework.12-14 In these systems, the density incorporates new terms, that is, an explicit spin-dependent part and an irreducible cumulant

10.1021/jp909935j  2010 American Chemical Society Published on Web 12/04/2009

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contribution, related to the spin exchange and the of 2-RDM cumulant matrix, respectively,15-21 which deserve to be characterized in detail. These fields reveal finer details of the density structure, providing a physical picture in the understanding of chemical bonding from a many-body physics point of view. The organization of this article is as follows. The second section is devoted to the theoretical framework of the partitioning of the electron density, the relationships between the density components, their Laplacian fields, and the associated topological quantities. The third section describes the computational details of the calculations performed over a set of selected molecules and the discussion of the theoretical and numerical results. Finally, the last section is devoted to the concluding remarks. 2. Theory: Decomposition of the Electron Density The matrix elements of the second-order reduced density matrix (2-RDM), 2D, corresponding to an N-electron system in an i, j, k, l, ... orthogonal spin orbital basis set is

Dklij )

2

11 i 1 j 1 1 D D - 1Dli1Djk + ∆klij 2 k l 2 2

(1)

where 1Dki and ∆klij are the matrix elements of the first-order reduced density matrix (1-RDM) and the cumulant of the 2-RDM, respectively.5,16-25 The terms on the right-hand side (rhs) of eq 1 stand for the Coulomb, exchange, and cumulant terms, respectively. Spin variable integration of eq 1 leads to the spin-free 2-RDM, which reads12,13

Dklij )

2

11 i 1 j 1 1 1 Dk Dl - 1Dli1Djk - 1D(s)li1D(s)jk + Γklij 2 4 4 2

(2) where 2Dijkl, 1Dik ) 1DRik + 1Dβik, 1D(s)ik ) 1DRik - 1Dβik, and Γijkl stand for the matrix elements of the spin-free 2-RDM, 1-RDM, spin density matrix, and 2-RDM cumulant matrix, respectively, in the i, j, k, l, ... orthogonal orbital basis set. 1DR and 1Dβ are the spin-up and spin-down density matrices, respectively. The exchange term (second term on the rhs of eq 1) splits into a particle-particle exchange term and an explicit spin density exchange contribution (third term on the rhs of eq 2), thus segregating the many-body effects and the irreducible spin contributions of the cumulant in the last term of eq 2. The spin invariance properties of the density matrices are expressed by the relations25,26

Dklij(S, Sz) ) 2Dklij(S, S)

2

1

Dji(S, Sz) ) 1Dji(S, S)

(3) 1

2

for all Sz, that is, D and D are independent of the spin projection and are only a function of the total spin S (Sz ) S, highest projection), while it is not true either for 1(s)D or for Γ, k that is, the terms 2Dijkl, (1/2)1Dik1Djl, and (1/4)1Di1 l Dj are independent of the Sz component of the multiplet state as well as ij (s)j the sum (1/4)1(s)Di1 l D k + (1/2)Γkl, whereas the terms 1 (s)i1 (s)j ij (1/4) D l D k and (1/2)Γkl are, individually, Sz-dependent. It may be noted that all of the results in this article are symmetric under the inversion of Sz projections, that is, the results are symmetric for the same module of Sz despite the sign of the projection; hence, we will only make use of the positive projections. In agreement with the invariant properties for openshell systems, we will use all of the terms in the density matrices evaluated at the maximum projection Sz ) S.12 The 1-RDM may be written from the contraction mapping of the 2-RDM as27,28

1

Dik )

2 N-1

∑ 2Dkjij

(4)

j

leading to12

Dik )

1

1 2

∑ 1Dji1Djk + 21 ∑ 1D(s)ji1D(s)jk - ∑ Γkjij j

j

(5)

j

Thus, regarding the closed-shell case,8 the first term of the rhs of eq 5 stands for the effectively paired density; the second and third terms represent the spin density exchange and a cumulant density, respectively. These last two terms stand for the effectively unpaired density matrix in the open-shell case regarding the associated u matrix structure whose matrix elements are defined by uik )[1(D(s))2]ik - 2∑j Γijkj.23,29 In the coordinate representation, the 1-RDM is expressed as 1 D(r|r′), where r and r′ are the spatial coordinates. The density stands for the diagonal terms of the 1-RDM as F(r) ) 1D(r|r), and hence, its trace (integration over the whole physical space) is equal to the number of electrons in the system. The effectively unpaired density component of the total electron density F(r) is F(u)(r) ) (1/2)u(r|r), were u(r|r) is the diagonal element of the above-mentioned effectively unpaired density matrix. Following the same procedure as that employed to decompose the closed-shell density8 and introducing the structure of the effectively unpaired density,29 the electron density may be written as

F(r) ) F(p)(r) + Fs(u)(r) + F(u) c (r)

(6)

or, equivalently, in a more convenient way to preserve the structure of the closed-shell case and hence to directly compare with Lewis model of pairing8

F(r) ) Fopen(r) + F(u) c (r)

(7)

with Fopen(r) defined as the sum of the first two terms on the rhs in eq 6, where F(p)(r) stands for the effectively paired density expressed by

F(p)(r) )

1 2

∫ dr′1D(r|r′)1D(r′|r)

(8)

Fs(u)(r) is the spin density contribution defined by

Fs(u)(r) )

1 2

∫ dr′1D(s)(r|r′)1D(s)(r′|r)

(9)

and F(u) c (r), the cumulant density, represents the irreducible spin and many-body effects expressed by

F(u) c (r) ) -

∫ dr′ Γ(rr′|rr′)

(10)

It may be noted that Fopen(r) defined by eq 7 coincides with F(p)(r) in the closed-shell case because the spin density vanishes and Fs(u)(r) is null in this case, that is, eqs 6 and 7 become eq 12 in ref 8. The new term (eq 9) which arises from the open-shell structure of the wave function with respect to the closed-shell case depends on the spin density matrix (which is of net unpaired nature) through a spin density exchange, and therefore it is of nondefined nature and needs to be characterized within the local topology formalism.30,31 In ref 12, such spin exchange density has been considered as distributed into one-center and two-center contributions within the scenario of the integrated formalism. That approach led to rigorously obtain a bond order definition for open-shell systems, clarifying the role of the electron spin density in the bonding phenomena in the open-shell case.12 Hence, we will attempt in the present article to properly describe

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(p) (u) (u) (u) Figure 1. L(r) contour maps of O2 molecule in 3Σg state for (a) F(r); (b) F (r); (c) F (r); (d) Fopen(r); (e) Fs (r); and (f) Fc (r). Positive and negative values are denoted by solid and dashed lines, respectively.

(u) (u) (p) (u) Figure 2. L(r) contour maps of linear HBBH molecule in 3Σg state for (a) F(r); (b) F (r); (c) F (r); (d) Fopen(r); (e) Fs (r); and (f) Fc (r). Positive and negative values are denoted by solid and dashed lines, respectively.

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Figure 3. L(r) contour maps of planar CH3 molecule in the 2A2′′ state in the plane of the molecule for (a) F(r); (b) F(p)(r); (c) F(u)(r); (d) Fopen(r); (e) Fs(u)(r); and (f) F(u) c (r). Positive and negative values are denoted by solid and dashed lines, respectively.

the behavior of the spin exchange density in the framework of the local formalism. 3. Computational Details, Results, and Discussion The systems O2, HBBH, and CH3 have been chosen as simple examples with nonsinglet ground states to study the features of electron density topology in the open-shell case as well as the reliability of these features to describe bonding phenomena. The wave functions used in this work to describe the molecular systems were calculated at the level of configuration interaction with single and double excitations (CISD) generated from the restricted open-shell Hartree-Fock (ROHF) states using the PSI3 package32 with the basis set 6-31G. The geometries for all systems were taken from experimental data.33 The densities, their critical points, and their Laplacian fields were determined by appropriately modified AIMPAC modules.34 The reported systems have been chosen to study fundamental and excited states supporting an open-shell structure. For practical reasons, we will use the function L(r) ) -∇2F(r) in the discussion of the results as an indicator of concentration (positive value) or depletion (negative value) of the densities at the point r.8,9 Because of the complex structure of the F(u)(r) topology, we will only deal with critical points associated with its valence shells (vs) in the corresponding systems, and no reference will be made to those of the inner shells of this density; in fact, only the former ones are involved into bonding phenomena. The terminology vs(3,-1)cp, vs(3,+1)cp, and vs(3,+3)cp will refer to (3, -1), (3, +1), and (3, +3) critical points of the F(u)(r) valence shell, in analogy with the bcp (bond critical points), rcp (ring critical points), and ccp (cage critical points) of the

total density, respectively. Nevertheless, it is important to note that such points are not sensu strictu bcp’s, rcp’s or ccp’s because only the cp’s of the total density are able to define a bond in the AIM topological formalism.1,2 Figure 1 shows the L(r) field for the densities of the O2 system in its 3Σg- triplet state. Figure 1a represents this function for the total density field F(r); it shows a shell structure for the oxygen atoms exhibiting the highest density concentration (oblique line), which indicates the localization of the electrons close to the nucleus, as expected. The concentration of the density in the bonding region (vertical line) and the existence of one bcp between the nuclei indicate a typical structure associated with the bond phenomena.8,9 The structure of the effectively paired density F(p)(r) in Figure 1b is similar to that of F(r) but exhibits higher concentration at the bonding region than the total density due to the delocalization of the atomic valence shells (oblique line) and the polarization of the inner shells out of the topological bonding region (horizontal line). The F(u)(r) field shows a marked depletion (Figure 1c) in the internuclear region. This depletion is more effective than the F(p)(r) concentration, and because F(r) differs from F(p)(r) only by the F(u)(r) contribution, it explains the strong lowering (strangulation) of the bonding concentration of F(r) (vertical line in Figure 1a) with respect to F(p)(r) and permits interpretation of this difference as a transference of electrons from bonding regions to atomic regions. The Fopen(r) field (Figure 1d), which collects the effectively paired and spin exchange density contributions, is similar to that of F(r) and shows zones of higher concentration (oblique line) than F(p)(r) due to the spin exchange density (Figure 1e, oblique line). The F(u) c (r) field (cumulant density) is

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Figure 4. L(r) contour maps of CH3 molecule in the 2A′′2 state in a plane perpendicular to the molecular plane containing the CH bond for (a) F(r); (b) F(p)(r); (c) F(u)(r); (d) Fopen(r); (e) Fs(u)(r); and (f) F(u) c (r). Positive and negative values are denoted by solid and dashed lines, respectively.

completely concentrated at the nuclei (oblique line in Figure 1f) and vanishes at the internuclear region, as shown in Figure 1f, reinforcing the conjecture of its atomic localization.8,9,13,23,35 This description is in agreement with the important exchange density contribution to the bond order from the nonlocal formulation point of view (cf. ref 12). Moreover, it may be observed that the spin exchange contribution (Figure 1e) is concentrated at the nuclear positions exhibiting a π-like symmetry when projected on an arbitrary plane (oblique line); a three-dimensional analysis would show a local concentration with the toroidal form, with the internuclear line as its axis, that is, the F(u)(r) field concentrates in the space with symmetry similar to that of a molecular orbital of 2p2 hybridization. The cumulant density shows a typical shell structure as found previously for closed-shell compounds8-10 and exhibits spherical symmetry (oblique line at Figure 1f). Figure 2 collects the L(r) contour maps of the densities for the HBBH system in its 3Σg- triplet state. Figure 2a shows this map for the total density F(r), which clearly shows a shell structure for the boron atoms, that is, core and valence shell concentration and depletion. The existence of bcp’s for BB and BH sequences indicates that such atoms are bonded. This is graphically shown by the valence shell charge concentration over these regions, exhibiting a marked encirclement of the boron atoms because of the high density concentration in the bonding region (oblique lines). These are indicators of classical bonds.9 Let us analyze the contributions to the density coming from its component fields. The F(p)(r) field shows similar behavior (Figure 2b), although its valence shell density concentration shows boron nuclei less encircled than in the case of

F(r) (vertical line in Figure 2b) and Fopen(r) (oblique lines in Figure 2d). The local concentration of F(u)(r) also at the bonding regions (vertical line at Figure 2c), obtained by the overlap of lobular boron concentrations, leads to the marked encirclement and strangulation of the bonding concentration of F(r) as well as Fopen(r) around the boron nuclei (Figure 2d, oblique lines). This is due to the markedly different behavior of Fs(u)(r) (Figure 2e) from that of the O2 system (cf. Figure 1e), which exhibits concentration over all nuclear positions (oblique line indicating boron concentration over the nuclei), and it models the F(u)(r) density, which possesses a projected π-type symmetry on an arbitrary plane (vertical lines, Figure 2e) joining the boron atoms, which represents a toroidal structure in the physical space. The contributions of the paired and spin exchange density parts of the particle density, delocalized over the internuclear region, exhibit a high concentration that reflects the local view of the nonlocal formalisms to bonding between atoms.12 The cumulant density (Figure 2f) shows the typical shell structure found for closed-shell compounds with nuclear concentrations (oblique line), spherical valence shell concentrations, and depletion at the bond midpoint (vertical line);8,9 its slight delocalization over the bonding internuclear region confirms its localized character13,35 and a typical behavior of systems containing B atoms.10 Figures 3 and 4 are dedicated to show the L(r) contour maps for the CH3 doublet state 2A2′′ in the molecular plane and in a perpendicular plane to the molecule containing a CH bond, respectively. The C atom shows shell structure for F(r), F(p)(r), Fopen(r) (Figures 3a,b,d and 4a,b,d, respectively), which may be noted from the spherical core concentration and depletion

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TABLE 1: Density and L(r) of the Total Density G Field Components G(p), Gopen, Gs(u), and G(u) c at the Nuclear Critical Points of Total Density at the CISD/6-31G Level of Calculation (all quantities in atomic units) system

state

nucleus

field

density

L(r)

δa

Db

O2

3 Σg

O

3 Σg

B

2.919 × 102 2.919 × 102 7.694 × 10-3 1.200 × 10-1 6.684 × 101 6.684 × 101 2.236 × 10-3 5.199 × 10-2 3.973 × 10-1 3.978 × 10-1 5.301 × 10-4 1.219 × 10-2 1.182 × 102 1.182 × 102 5.810 × 10-3 8.222 × 10-2 4.155 × 10-1 4.158 × 10-1 5.050 × 10-4 1.579 × 10-2

1.931 × 106 1.931 × 106 2.697 × 101 7.652 × 102 1.695 × 105 1.695 × 105 5.768 × 100 1.475 × 102 1.810 × 101 1.812 × 101 2.524 × 10-2 7.241 × 10-1 4.367 × 105 7.862 × 10-1 2.023 × 101 2.984 × 102 1.871 × 101 1.872 × 101 2.582 × 10-2 9.765 × 10-1

9.324 × 10-6 9.316 × 10-6 2.847 × 10-4 1.053 × 10-3 2.572 × 10-6 2.569 × 10-6 7.464 × 10-5 3.018 × 10-4 2.354 × 10-2 2.349 × 10-2 3.810 × 10-3 5.939 × 10-4 0.000 0.000 0.000 0.000 3.212 × 10-2 3.207 × 10-2 1.517 × 10-2 8.960 × 10-3

8.897 × 10-6

HBBH (linear)

F(p) Fopen Fs(u) F(u) c F(p) Fopen Fs(u) F(u) c

H

CH3

2

A2′′

C

F(p) Fopen Fs(u) F(u) c

H

a

2.829 × 10-6

2.254 × 10-2

0.000

2.987 × 10-2

Distance between the nuclei and the corresponding vs(3,-3)cp of each density. b Distance between the nuclei and the corresponding ncp.

TABLE 2: Density and L(r) of the Total Density G Field Components G(p), Gopen, Gs(u), and G(u) c at the Bond Critical Points of Total Density at the CISD/6-31G Level of Calculation (all quantities in atomic units) system

state

bond

field

density

L(r)

O2

3 Σg

OO

HBBH (linear)

3 Σg

BB

F(p) Fopen Fs(u) F(u) c F(p) Fopen Fs(u) F(u) c

4.523 × 10-1 4.523 × 10-1 1.308 × 10-5 7.506 × 10-3 1.868 × 10-1 1.869 × 10-1 1.300 × 10-4 3.300 × 10-3 1.753 × 10-1 1.754 × 10-1 1.300 × 10-4 3.260 × 10-3 2.560 × 10-1 2.561 × 10-1 1.100 × 10-4 3.730 × 10-3

2.005 × 10-1 2.055 × 10-1 -4.935 × 10-3 -1.250 × 10-1 5.650 × 10-1 3.870 × 10-1 -6.835 × 10-2 -1.810 × 10-3 4.324 × 10-1 3.870 × 10-1 -5.129 × 10-2 5.956 × 10-3 8.081 × 10-1 7.862 × 10-1 -4.394 × 10-2 -4.359 × 10-2

BH

CH3

a

2

A2′′

CH

F(p) Fopen Fs(u) F(u) c

δa 0.000 0.000 0.000 0.000 0.000 0.000 6.214 × 10-3 6.060 × 10-3 2.660 × 10-1 4.126 × 10-3 4.064 × 10-3

εb 0.000 0.000 c 0.000 0.000 0.000 c 0.000 0.000 0.000 c 0.000 5.825 × 10-2 2.306 × 10-2 c c

Distance between the bcp and the vs(3,-1)cp of the corresponding field. b Ellipticity. c No vs(3,-1)cp exist for this density.

regions in the molecular plane; the valence shell of charge concentration spreads over the CH bonds. The view of these densities from the perpendicular plane filters the description of the CH bonds and the individual atomic behavior from the whole molecule, as may be noted from their contour maps (Figures 4). Figures 3a and 4a show the total particle density; the observation of the concentration of the component F(p)(r) (oblique lines in Figures 3b and 4b) and the depletion of the unpaired one F(u)(r) (Figures 3c and 4c) in the bonding region indicates a typical covalent distribution.8 Figure 3b,d shows a similar structure for F(p)(r) and Fopen(r), respectively, in the molecular plane. However, there is a difference between them, as shown in Figure 4b,d, in a plane perpendicular to that one that is, as Fopen(r) contains the exchange spin density F(u) s (r) and it is localized on the C atom, it becomes totally encircled (cf. vertical lines in Figure 4b,d). The cumulant density F(u) c (r) is concentrated over this atom (cf. Figures 3f and 4f) and also contributes to the encirclement. The density at the CH bonding region in Figures 3a,b,d and 4a,b,d (oblique lines) is similar for F(p)(r) and Fopen(r) because the spin exchange and cumulant contributions to the density

show depletion there and are localized close to the nuclei, in both planes (Figures 3e and 4e) and (Figures 3f and 4f). These last mentioned figures also permit one to reveal the shell structure of the atoms and exhibit a 2p-type symmetry on the unpaired components on the C atom and a spherical one for the H with a high concentration of these fields, denoting their atomic localization on this compound. Table 1 shows the numerical values describing the local behavior of the field densities of the systems at the ncp’s of each nucleus. As expected, all fields are concentrated at the nuclear positions and exhibit a very small deviation of their corresponding ncp’s and vs(3,-1)cp’s from the nuclei positions. It may be noted that these deviations are of the same order of magnitude as those in the closed-shell case.8 Table 2 shows the results for the densities at the total density bcp. The paired density F(p)(r) possesses one vs(3,-1)cp and shows concentration at the bcp’s for all two-atom sequences in the systems, reveling typical bonding situations. The same observation is also verified for Fopen(r); nevertheless, the spin exchange density shows depletion at the bcp’s for all two-atom sequences. The densities exhibit no shift in their associated

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vs(3,-1)cp values, in case they exist, with respect to the bcp positions in O2 and in the BB sequence in HBBH because of their symmetry; however, a marked one appears for the cumulant density at the BH bond. This last find is not surprising because of the electron-deficient nature of the boron compounds10 agreeing with a cumulant density concentration at the B-H bcp (Table 2). It is shown graphically in Figure 2f, where this density spreads over the BH bonding region. The behavior of the densities around the bcp’s indicates that the regions in which they are concentrated are mainly located close to the nuclei, similarly to what has been reported for the closed-shell systems.8 Therefore, the quantum description of the bonding phenomena fits well with the pairing Lewis conjecture of chemical bonds and the role of “odd” electrons for openshell molecular systems.11 The ellipticity of the electron distribution at the bcp’s reported in the last column of Table 2 shows that it vanishes in the O2 and HBBH systems because of their symmetry and even is very small for CH3, thus denoting that the density contributions are not deformed. 4. Concluding Remarks The electron density for systems described by open-shell wave functions has been decomposed into contributions of pairing and unpairing nature. This procedure has been performed in order to search for the degree of fitness of the Lewis hypothesis of chemical bonding as an electron pairing phenomena with the electron cloud localization in atomic and bonding regions. This goal has required extension of the relationships between the traditional chemical concepts and the quantum mechanical ones in closed-shell systems8-10 to the open-shell case. The main theoretical result is that the pairing density F(p)(r) always shows concentration at the bonding region, while the spin exchange density Fs(u)(r) in some cases concentrates and in other depletes, but both densities may be considered for interpreting the chemical bond (cf. Figures 1e-4e and Table 2). Hence, it indicates the fundamental importance of the spin exchange densities for describing the bonding phenomena and leads one to conclude that both F(p)(r) and Fs(u)(r) fields are responsible for the linkage of the atoms. These considerations permit incorporation of the spin electron cloud to understand open-shell systems. The cumulant density F(u) c (r) possesses shell structure and is localized around the atomic nuclei,3,9 thus confirming that this density does not contribute to bonding. This work may be considered as the local formulation of the bond order theory, which, similar to the integrated formulation,12 provides the description of the bonding concept in open-shell systems. Acknowledgment. This work has been partly supported by the Projects X-24 (Universidad de Buenos Aires), the Spanish Ministry of Science and Innovation (Grant No. CTQ2009-07459/ BQU), and the Universidad del Paı´s Vasco (Grant No. GIU06/ 03). We thank the Universidad del Paı´s Vasco for the allocation of computational resources. R.M.L. acknowledges aid from the

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