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Molecular graphs of Lin , Nan and Cun (n = 6–9) clusters from the density and the molecular electrostatic potential
Computational and Theoretical Chemistry xxx (2014) xxx–xxx
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Molecular graphs of Lin , Nan and Cun (n = 6–9) clusters from the density and the molecular electrostatic potential Gerald Geudtner, Victor Daniel Domínguez-Soria, Patrizia Calaminici ⇑, Andreas M. Köster * Departamento de Quimica, CINVESTAV, Av. Instituto Politecnico Nacional 2508, A.P. 14-740, Mexico D.F. 07000, Mexico
a r t i c l e
i n f o
Article history: Received 15 July 2014 Received in revised form 23 July 2014 Accepted 24 July 2014 Available online xxxx Keywords: Topology analysis Molecular graph MEP Density Clusters
a b s t r a c t A comparative analysis of molecular graphs of the electronic density and the molecular electrostatic potential (MEP) for lithium, sodium and copper clusters with up to nine atoms is presented. Whereas the density molecular graphs are qualitatively different for these clusters the corresponding MEP molecular graphs are surprisingly similar. Only for copper clusters MEP and density molecular graphs are similar. Our analysis indicates that MEP molecular graphs for these metal clusters show connectivities very much in the spirit of Lewis structures. Ó 2014 Elsevier B.V. All rights reserved.
1. Introduction The description of the electronic structure of molecules in terms of an intuitive bond theory is one of the most challenging and controversial topics in modern quantum chemistry. The valence theory of Lewis [1] and its extension to open shell systems by Linnett [2] is still the most common description of chemical bonds due to its simplicity and reliability. Simple extensions like the valence shell electron pair repulsion (VSEPR) model [3] represent practical tools for the ad-hoc prediction of molecular structures. On the other hand, accurate electronic structure calculations have predicted unusual bond situations that have been validated by experimental methods that challenge the original valence theory. This is particularly true for simple metal clusters. Many properties of these systems are better understood in terms of the jellium model [4,5] than in terms of Lewis structures. This raises the question if a structured electron density exist at all in these metal clusters. Common quantum chemical approaches that address this question are bond order concepts based on the analysis of molecular or atomic orbitals [6–28]. Unfortunately, the results of such studies are not always easy to understand, even at a qualitative level. Alternative approaches are based on the topological analysis of molecular scalar fields like the electron ⇑ Corresponding authors. E-mail addresses: [email protected] (P. Calaminici), [email protected] (A.M. Köster).
density [29,30], the Laplacian of the electron density [31,32], the electron localization function [33–35] or the molecular electrostatic potential [36–44]. A clear advantage of the electron density and the molecular electrostatic potential is that they are experimentally amenable by diffraction experiments, e.g. high resolution X-ray crystallography [45,46]. For the question of localized bonds the concept of the molecular graph as introduced by Bader [32] seems most suitable. Recent studies in our laboratories have focused on molecular graphs from the density and the molecular electrostatic potential for small Lin clusters (n = 2–6) [47] and on the comparison of molecular graphs for Lin , Nan and Cun (n = 2–5) clusters [48]. Previously critical points of the MEP of copper clusters with two to ten atoms were studied, too [49]. In particular the comparison of the molecular graphs from the electronic density and the molecular electrostatic potential (MEP) of the small alkali and copper clusters showed characteristic trends that motivated the here presented study of larger clusters. In this article we compare the topology of the electronic density and the MEP of lithium, sodium and copper clusters from the hexamer up to the nonamer. Molecular graphs from both scalar molecular fields are presented for these clusters. The article is organized as follows. In the next section the theoretical background for the calculation of critical points of scalar molecular fields and the corresponding molecular graphs is reviewed. In Section 3 the computational details are given. In Section 4 the obtained molecular graphs are presented and discussed in terms of the topology of the density and the MEP of the studied lithium,
http://dx.doi.org/10.1016/j.comptc.2014.07.021 2210-271X/Ó 2014 Elsevier B.V. All rights reserved.
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sodium and copper clusters. Finally, the conclusions are drawn in the last section. 2. Theoretical background The topological analysis of a scalar molecular field FðRÞ consists of the determination and characterization of critical points (CPs). These points are defined by vanishing gradients of the corresponding field. A CP is characterized by the rank and signature of its Hessian matrix H with the elements:
Hab ¼
@ 2 FðRÞ @Ra @Rb
! ð1Þ R¼RCP
evaluated at the critical point, RCP . In the case of nondegenerated CPs of the electronic density or the MEP the rank is always 3 (the number of nonzero eigenvalues of H) and the signature is the algebraic sum of the eigenvalue signs of H. This sum is different for minima, maxima and saddle points. All nondegenerated CPs of the electronic density and MEP can be characterized as:
ð3; þ3Þ local minimum ð3 positive eigenvaluesÞ ð3; þ1Þ saddle point ð2 positive eigenvalues; 1 negative eigenvalueÞ ð3; 1Þ saddle point ð1 positive eigenvalue; 2 negative eigenvaluesÞ ð3; 3Þ local maximum ð3 negative eigenvaluesÞ: The first number in these parentheses denotes the rank of the CP and the second the signature. For the construction of molecular graphs we are particularly interested in (3, 1) saddle points. In the density ascending gradient paths originating from these CPs are connecting neighboring attractors. The so defined line is referred in the literature as atomic interaction line [50]. This line indicates an accumulation of electron density between the two attractor end points of the atomic interaction line. For this reason the (3, 1) CP and the corresponding gradient path are often called bond critical point and bond path, respectively. The molecular graph is then defined as the union of the closure of these atomic interaction lines. Topologically, the molecular graph is a network of linked attractors. In many molecules with localized bonds this network shows the same connectivity as Lewis structures. Therefore it seems reasonable to use molecular graphs also for the qualitative analysis of the atomic connectivity in metal clusters. A particularity in this respect are the non-nuclear attractors that can appear in the topology of the electronic density [51–60]. As a consequence density molecular graphs of clusters with nonnuclear attractors are very different from common Lewis structures. On the other hand it has been proven that non-nuclear attractors can never occur in the topology of the MEP [36–40]. Therefore, MEP maxima appear always at the position of nuclei. As a consequence the topology of the density and the MEP must differ for systems with non-nuclear attractors in the density. At this point it has to be stressed that the here discussed density and MEP (3, 3) CPs are not true critical points. However, the topological behavior of the neighborhood of these points is indistinguishable from that of true maxima [61] and, therefore, we will name them here (3, 3) CPs or maxima in the sense of their topological behavior. To extend the concept of molecular graphs to the MEP, bond critical points [43,44] and atomic interaction lines [48] must be defined for this scalar molecular field, too. Different from the electronic density, not all (3, 1) CPs of the MEP are connected to topological maxima, i.e. nuclei. To circumvent this problem we assign only those (3, 1) CPs of the MEP as bond critical point that connect topological maxima by a gradient path. By construction the resulting molecular graphs of the MEP connect only nuclei and, therefore, resemble the connectivity of Lewis structures. For most
molecules with localized bonds the above defined MEP molecular graphs are qualitatively identical to their electronic density counterpart. Obviously, this cannot be the case for systems with nonnuclear attractors. 3. Computational details For all calculations the linear combination of Gaussian-type orbitals Kohn–Sham density functional approach as implemented in the program deMon2k [62,63] was employed. The local density approximation (LDA) using the exchange–correlation contributions proposed by Vosko, Wilk and Nusair (VWN) [64] was used in combination with all-electron double zeta valence polarization basis sets [65]. The Coulomb energy was calculated by the variational fitting procedure proposed by Dunlap et al. [66]. For the fitting of the density the automatically generated auxiliary function set GEN-A2 [67] was employed. The exchange–correlation functional was evaluated with this auxiliary density, i.e. the auxiliary density functional theory (ADFT) method was used [68]. The exchange–correlation energy and potential were numerically integrated on an adaptive grid [69]. The grid accuracy was set to 105 a.u. in all calculations. The structures of the clusters were fully optimized using a quasi-Newton method in internal redundant coordinates with analytic energy gradients [70]. The convergence was based on the Cartesian gradient and displacement vectors with a threshold of 104 and 103 a.u., respectively. In order to characterize the optimized structures a vibrational analysis was performed. For this purpose the second derivatives were calculated by numerical differentiation (two-point finite difference) of the analytic energy gradients using a displacement of 0.001 a.u. from the optimized geometry for all Cartesian coordinates and the harmonic frequencies were then obtained by diagonalizing the massweighted Cartesian force constant matrix. The CP search of the density and the MEP was carried out by applying the CP search algorithm as implemented in the deMon2k code. This algorithm is already described in the literature [43,47,48]. It can be used to find critical points of any scalar molecular field because no assumption of the field topology is made. The input for the search algorithm are the scalar fields of the electron density, q, and the MEP, UðRÞ, which are calculated according to:
q¼
X Plm lðRÞmðRÞ l ;m
UðRÞ ¼
X A
X ZA P lm jA Rj l;m
ð2Þ 1 m : l r R
ð3Þ
Here, Plm denotes an element of the converged Kohn–Sham density matrix for the atomic basis functions l and m. Z A denotes the nuclear charge of atom A with the position vector A. The first term on the right side of Eq. (3) contains the contribution of the nuclei to the MEP and the second term the electronic contribution, respectively. 4. Results and discussion In Figs. 1 and 2 the (3, 1) bond critical points (dark dots) along with the corresponding molecular graphs of the MEP and the electronic density, q, for the studied lithium, sodium and copper clusters from the hexamer to the nonamer are illustrated. The cluster structures correspond to optimized ground state structures at the here used theoretical level. The spin multiplicities are singlets for the even and doublets for the odd numbered clusters. For the following discussion it is important to note that the MEP topology arises from the positively charged nuclei and the negative electron distribution, whereas the electronic density topology only
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3
MEP Li
Na
Cu
Fig. 1. MEP molecular graphs of the ground state structures of lithium, sodium and copper clusters from the hexamer to the nonamer.
originates from the electron distribution. Therefore, the topology of the complete MEP is a superposition of the topology of these two separate distributions. Assuming equal molecular structures the basic topology of the contribution of the nuclei to the MEP would be independent of the atom type. In this case differences in the molecular graphs of the MEP can only arise from the distribution of the electrons. Because of the different signs of the nuclear and electronic MEP contributions their balance also depends critically on the distance between the atoms of the molecular structure under study. 4.1. Hexamer Going from Li6 over Na6 to Cu6 the cluster ground state structures change considerably. Whereas Li6 is a square bipyramid, Na6 is a pentagonal pyramid and Cu6 is a planar triangular structure as it is shown in the first row of Figs. 1 and 2. The MEP molecular graph usually connects nearest neighbors in these clusters with a bond critical point in between as depicted in Fig. 1. A notable exception appears in the Li6 cluster. There, no MEP bond path can be found along the four sides of the square. The absence of the corresponding bond critical points in the MEP can be explained by analyzing the critical points and the molecular graph of the electronic density, q, which are shown in Fig. 2. Characteristic for the electronic density topology of Li6 is the appearance of four non-nuclear attractors in the square plane of the Li6 bipyramid (see first structure of first row in Fig. 2). This
is a clear indication that density has been moved away from atomic connection lines into the cluster. In the Na6 cluster the bond paths and bond critical points of the density can be found within the triangular faces of the Na6 pyramid. It is worth to note that the tip atom has no direct connection to the other atoms. For the Cu6 cluster the bond critical points of the density are localized between two neighboring Cu atoms (see third structure of first rows in Fig. 2). As a result the density and MEP molecular graphs of Cu6 are qualitative identical. The bond critical points of the density between the atoms in Cu6 indicate a more localized density distribution as in the corresponding Li and Na clusters. In the MEP this effect is less obvious. However, the bond path curving diminish from Li6 over Na6 to Cu6. In fact this trend holds for all studied clusters. Thus, it seems reasonable to correlate the bond path curving in the MEP molecular graph with the electron delocalization in the here studied clusters. The Li6 cluster possesses three different nearest neighbor distances. The distances between the two pyramid tips is about 2.6 Å. The distance between the pyramid tip and a corner of the square is about 2.8 Å and the length of an edge of the square is about 3.5 Å. The latter distance is significantly larger than the other distances. Also, as mentioned above the electron density in the Li6 cluster is localized more inside the structure which is clearly indicated by the appearance of the non-nuclear attractors in the square plane of the bipyramid. This effect together with the large Li–Li distances results in the disappearance of MEP bond critical points along the edges of the square in the bipyramidal structure.
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ρ Li
Na
Cu
Fig. 2. Density molecular graphs of the ground state structures of lithium, sodium and copper clusters from the hexamer to the nonamer.
4.2. Heptamer The structural picture changes from the clusters with 6 atoms to the clusters with 7 atoms. The ground state structures for Li7, Na7 and Cu7 are all three dimensional and pentagonal bipyramids. The structures are shown in the second row of Figs. 1 and 2. In the MEP all cluster show the same pattern of bond critical points and, thus, the same molecular graph as illustrated in the second row of Fig. 1. The bond paths are connecting two neighboring atoms with a bond critical point in the middle. Even though, the MEP molecular graph topology is the same for all heptamers its appearance is different due to the varying curvatures of the bond paths. The curvature is less and less pronounced going from Li7 to Na7 to Cu7 (see second row of Fig. 1). Quantitative differences appear in the q topology as it can be seen from Fig. 2. The molecular graph of the density for the Li7 cluster lies again completely inside the cluster. But in contrast to the Li6 cluster no non-nuclear attractors appear in the Li7 cluster. This leads to the conclusion that for the Li7 the localization of the electron density is within the cluster and, therefore, the electronic contribution to the MEP is not as dominant as in Li6. As a consequence the MEP molecular graph of Li7 shows the same connectivity as for Cu7. The Na7 cluster possesses a similar density molecular graph structure as the Na6 cluster. As it was previously discussed, in the Na6 cluster the bond critical points of q are located within
the triangular faces of the structure and the bond paths do not directly connect the tip atom with the other atoms. The same is found for the Na7 cluster. Only atoms in the pentagonal plane are connected directly with a bond path. Additionally, there is no bond path between the two tip atoms. In the MEP (see second structure of second row in Fig. 1) also the pyramid tips are connected by a bond path with a critical point in the center of the structure. 4.3. Octamer As shown in the third row of Figs. 1 and 2 the Li8 cluster has a different structure than the Na8 and Cu8 clusters. The structure of the Li8 cluster consists of a trigonal prism with one extra atom in the center of the prism and an other extra atom capping one rectangle of the prism. The structure shown in Figs. 1 and 2 is a view on the trigonal side of the prism with the external capping atom beneath. In Fig. 1 the MEP molecular graph along with the corresponding bond critical points is illustrated for this Li8 cluster. Again, all neighboring atoms are connected except the atoms building the corners of the capped rectangle of the prism. Notice that bond paths connect the central atom with all other atoms. Like in the other Li clusters the distribution of the bond paths and bond critical points of q as shown in Fig. 2 can be interpreted as accumulation of electron density within the cluster cage. Also in this cluster non-nuclear attractors appear which are arranged
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tetrahedrically around the central Li atom. Whereas the distance of the central Li atom to the other atoms is about 2.6 Å the lengths of the edges of the capped rectangle are about 3.3 Å and 3.6 Å. The lengths of the other edges in the structure are about 3.11 Å, 3.24 Å and 3.26 Å. As in the Li6 cluster, along the longest edges no bond critical points can be found in the MEP. A second similarity is the high concentration of bond critical points and non-nuclear attractors in q around the plane which contains these longest edges. The structures of the Na8 cluster and the Cu8 cluster are both snub disphenoids. For Cu8 a congruent picture for the molecular graph and bond critical points of the MEP and electronic density are found, as can be seen by the comparison of the corresponding structures in Figs. 1 and 2, respectively. All bond paths and critical points in the molecular graphs of the MEP and the density are located along the edges of the structure between the atoms. This indicates directed bonds and, thus, a certain covalency in this structure. At first glance the situation is similar for Na8. However, some (3, 1) critical points of the MEP are not existing in the corresponding electronic density. This leads to the assumption that in the Na8 cluster the distribution of the electron density is less structured compared with Cu8 and therefore more homogeneous. 4.4. Nonamers The structures for the Li9, Na9 and Cu9 cluster are shown in the last rows of Figs. 1 and 2. The Li9 cluster consists of a square antiprism with a central atom. Because of a Jahn–Teller effect one square is distorted into a rectangle and the other square is distorted into a rhombus. The bond critical points and the molecular graph of the MEP in Fig. 1 show connections between all neighboring atoms. The bond critical points and the molecular graph of q are accumulated within the cluster cage as illustrated in Fig. 2. A high concentration of these bond critical points and bond paths together with four non-nuclear attractors can be found around a plane which is between and parallel to the two distorted squares. On the other hand, the Na9 and Cu9 cluster show the same structure which consists of a pentagonal bipyramid where two adjacent faces are capped by atoms. These capping creates a second pentagonal bipyramid so that the structures can also be described as two melded pentagonal bipyramids. The molecular graph and bond critical points of the MEP of these clusters show the same behavior. The bond critical points and bond paths are found along the connection lines of neighboring atoms as it is depicted in Fig. 1. In the molecular graph of q for the Cu9 cluster the bond critical points and bond paths are also located along these lines as it is found for all other Cu cluster presented here underlining a certain covalency in these clusters. In the case of the Na9 cluster the molecular graph of q shows a very scanty topology which leads to the conclusion of a rather homogeneous electron density distribution. 5. Conclusions The topological analysis of the larger Li, Na and Cu clusters with up to nine atoms complements a previous study on smaller clusters. As for the smaller clusters non-nuclear attractors were found in some of the Li clusters but not in Na and Cu clusters. We note that this can vary with the theoretical level of description. In any case the density molecular graphs of the Li clusters are always located inside the clusters. This holds even for Li clusters without non-nuclear attractors, e.g. Li7. As a result the density molecular graphs of the Li clusters show connectivities that are not obviously related to Lewis structures. The other extreme are the density molecular graphs of the Cu clusters that show a connectivity in the spirit of Lewis structures. We suggest that this difference
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reflects more directed bonds in the Cu clusters, or vice versa, a more homogeneous electron distribution in the Li clusters. The density molecular graphs of the sodium clusters are always in between these two extremes, sometimes more in the direction of the Li clusters (Na6 and Na9) and sometimes more in the direction of the Cu clusters (Na7 and Na8). The MEP molecular graphs, on the other hand, show always interatomic connectivity in the spirit of Lewis structures. To a good part this originates from their construction. However, they also show clear trends form the Li over the Na to the Cu clusters, mainly by a decreasing convex curvature of the bond paths. By and large the topological equivalence of the MEP and density molecular graph seems to be a good indicator for a more structured electron density with somehow directed bonds. For the here studied systems we find this situation only for the Cu clusters. Acknowledgments Financial support from the CONACYT (Grants 130726 and 179409) is gratefully acknowledged. Some of the calculations were performed at the local HPC resource Xiuhcoatl of CINVESTAV. References [1] G.N. Lewis, Valence and The Structure of Atoms and Molecules, Chemical Catalog, New York, 1923. [2] J.W. Linnett, The Electronic Structures of Molecules, Wiley, New York, 1964. [3] W. L Jolly, Modern Inorganic Chemistry, McGraw-Hill, New York, 1985. [4] W.A. de Heer, The physics of simple metal clusters: experimental aspects and simple models, Rev. Mod. Phys. 65 (1993) 611–676. [5] S.N. Khanna, P. Jena, Atomic clusters: building blocks for a class of solids, Phys. Rev. B 51 (1995) 13705–13716. [6] P.O. Löwdin, On the non-orthogonality problem connected with the use of atomic wave functions in the theory of molecules and crystals, J. Chem. Phys. 18 (1950) 365–375. [7] C.A. Coulson, Valence, Clarendon Press, Oxford, 1952. [8] R.S. Mulliken, Electronic population analysis on LCAO-MO molecular wave functions. 1, J. Chem. Phys. 23 (1955) 1833–1840. [9] L. Pauling, The Nature of the Chemical Bond, Cornell University Press, 1960. [10] K. Jug, A new definition of atomic charges in molecules, Theor. Chim. Acta 31 (1973) 63–73. [11] K. Jug, Charge distributions and multipole moments in molecules, Theor. Chim. Acta 39 (1975) 301–312. [12] R. Heinzmann, R. Ahlrichs, Population analysis based on occupation numbers of modified atomic orbitals (MAOs), Theor. Chim. Acta 42 (1976) 33–45. [13] F.L. Hirshfeld, Bonded-atom fragments for describing molecular charge densities, Theor. Chim. Acta 44 (1977) 129–138. [14] K. Jug, A maximum bond order principle, J. Am. Chem. Soc. 99 (1977) 7800– 7805. [15] C. Ehrhardt, R. Ahlrichs, Population analysis based on occupation numbers II. Relationship between shared electron numbers and bond energies and characterization of hypervalent contributions, Theor. Chim. Acta 68 (1985) 231–245. [16] A.E. Reed, R.B. Weinstock, F. Weinhold, Natural population analysis, J. Chem. Phys. 83 (1985) 735–746. [17] C.M. Smith, G.G. Hall, Optimal population analysis, Int. J. Quant. Chem. 31 (1987) 685–692. [18] A.M. Köster, K. Jug, Multipole moment analysis for hydrides, fluorides, and lithium compounds of first- and second-row elements, Int. J. Quant. Chem. 48 (1993) 295–308. [19] C.A. Coulson, The electronic structure of some polyenes and aromatic molecules VII. Bonds of fractional order by the molecular orbital method, Proc. Roy. Soc. Lond. Ser. A 169 (1939) 413–428. [20] R. Daudel, C. Sandorfy, C. Vioelant, P. Yven, O. Chalvet, Comparaison des methodes statiques et des methodes dynamiques pour la prevision des proprietes chimiques des molecules a partir de leurs diagrammes moleculaires. 18, Bull. Soc. Chim. Fr. 17 (1950) 66–74. [21] K. Wiberg, Application of Pople–Santry–Segal CNDO method to cyclopropylcarbinyl and cyclobutyl cation and to bicyclobutane, Tetrahedron 24 (1968) 1083–1096. [22] D.R. Armstrong, P.G. Perkins, J.J.P. Stewart, Valencies and bond indices for elements from hydrogen to chlorine, J. Chem. Soc. Dalton 21 (1973) 2273– 2277. [23] N.P. Borisova, S.G. Semenov, Molecular orbital definition of chemical bond multiplicity, Trans. Leningrad Univ. 16 (1973) 119–124. [24] M. Giambiagi, M.S. de Giambiagi, D.R. Grempel, C.D. Heynman, Definition of ave bond index with non-orthogonal basis – properties and applications, J. Chim. Phys. 72 (1975) 15–22.
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Molecular graphs of Lin , Nan and Cun (n = 6–9) clusters from the density and the molecular electrostatic potential Gerald Geudtner, Victor Daniel Domínguez-Soria, Patrizia Calaminici ⇑, Andreas M. Köster * Departamento de Quimica, CINVESTAV, Av. Instituto Politecnico Nacional 2508, A.P. 14-740, Mexico D.F. 07000, Mexico
a r t i c l e
i n f o
Article history: Received 15 July 2014 Received in revised form 23 July 2014 Accepted 24 July 2014 Available online xxxx Keywords: Topology analysis Molecular graph MEP Density Clusters
a b s t r a c t A comparative analysis of molecular graphs of the electronic density and the molecular electrostatic potential (MEP) for lithium, sodium and copper clusters with up to nine atoms is presented. Whereas the density molecular graphs are qualitatively different for these clusters the corresponding MEP molecular graphs are surprisingly similar. Only for copper clusters MEP and density molecular graphs are similar. Our analysis indicates that MEP molecular graphs for these metal clusters show connectivities very much in the spirit of Lewis structures. Ó 2014 Elsevier B.V. All rights reserved.
1. Introduction The description of the electronic structure of molecules in terms of an intuitive bond theory is one of the most challenging and controversial topics in modern quantum chemistry. The valence theory of Lewis [1] and its extension to open shell systems by Linnett [2] is still the most common description of chemical bonds due to its simplicity and reliability. Simple extensions like the valence shell electron pair repulsion (VSEPR) model [3] represent practical tools for the ad-hoc prediction of molecular structures. On the other hand, accurate electronic structure calculations have predicted unusual bond situations that have been validated by experimental methods that challenge the original valence theory. This is particularly true for simple metal clusters. Many properties of these systems are better understood in terms of the jellium model [4,5] than in terms of Lewis structures. This raises the question if a structured electron density exist at all in these metal clusters. Common quantum chemical approaches that address this question are bond order concepts based on the analysis of molecular or atomic orbitals [6–28]. Unfortunately, the results of such studies are not always easy to understand, even at a qualitative level. Alternative approaches are based on the topological analysis of molecular scalar fields like the electron ⇑ Corresponding authors. E-mail addresses: [email protected] (P. Calaminici), [email protected] (A.M. Köster).
density [29,30], the Laplacian of the electron density [31,32], the electron localization function [33–35] or the molecular electrostatic potential [36–44]. A clear advantage of the electron density and the molecular electrostatic potential is that they are experimentally amenable by diffraction experiments, e.g. high resolution X-ray crystallography [45,46]. For the question of localized bonds the concept of the molecular graph as introduced by Bader [32] seems most suitable. Recent studies in our laboratories have focused on molecular graphs from the density and the molecular electrostatic potential for small Lin clusters (n = 2–6) [47] and on the comparison of molecular graphs for Lin , Nan and Cun (n = 2–5) clusters [48]. Previously critical points of the MEP of copper clusters with two to ten atoms were studied, too [49]. In particular the comparison of the molecular graphs from the electronic density and the molecular electrostatic potential (MEP) of the small alkali and copper clusters showed characteristic trends that motivated the here presented study of larger clusters. In this article we compare the topology of the electronic density and the MEP of lithium, sodium and copper clusters from the hexamer up to the nonamer. Molecular graphs from both scalar molecular fields are presented for these clusters. The article is organized as follows. In the next section the theoretical background for the calculation of critical points of scalar molecular fields and the corresponding molecular graphs is reviewed. In Section 3 the computational details are given. In Section 4 the obtained molecular graphs are presented and discussed in terms of the topology of the density and the MEP of the studied lithium,
http://dx.doi.org/10.1016/j.comptc.2014.07.021 2210-271X/Ó 2014 Elsevier B.V. All rights reserved.
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sodium and copper clusters. Finally, the conclusions are drawn in the last section. 2. Theoretical background The topological analysis of a scalar molecular field FðRÞ consists of the determination and characterization of critical points (CPs). These points are defined by vanishing gradients of the corresponding field. A CP is characterized by the rank and signature of its Hessian matrix H with the elements:
Hab ¼
@ 2 FðRÞ @Ra @Rb
! ð1Þ R¼RCP
evaluated at the critical point, RCP . In the case of nondegenerated CPs of the electronic density or the MEP the rank is always 3 (the number of nonzero eigenvalues of H) and the signature is the algebraic sum of the eigenvalue signs of H. This sum is different for minima, maxima and saddle points. All nondegenerated CPs of the electronic density and MEP can be characterized as:
ð3; þ3Þ local minimum ð3 positive eigenvaluesÞ ð3; þ1Þ saddle point ð2 positive eigenvalues; 1 negative eigenvalueÞ ð3; 1Þ saddle point ð1 positive eigenvalue; 2 negative eigenvaluesÞ ð3; 3Þ local maximum ð3 negative eigenvaluesÞ: The first number in these parentheses denotes the rank of the CP and the second the signature. For the construction of molecular graphs we are particularly interested in (3, 1) saddle points. In the density ascending gradient paths originating from these CPs are connecting neighboring attractors. The so defined line is referred in the literature as atomic interaction line [50]. This line indicates an accumulation of electron density between the two attractor end points of the atomic interaction line. For this reason the (3, 1) CP and the corresponding gradient path are often called bond critical point and bond path, respectively. The molecular graph is then defined as the union of the closure of these atomic interaction lines. Topologically, the molecular graph is a network of linked attractors. In many molecules with localized bonds this network shows the same connectivity as Lewis structures. Therefore it seems reasonable to use molecular graphs also for the qualitative analysis of the atomic connectivity in metal clusters. A particularity in this respect are the non-nuclear attractors that can appear in the topology of the electronic density [51–60]. As a consequence density molecular graphs of clusters with nonnuclear attractors are very different from common Lewis structures. On the other hand it has been proven that non-nuclear attractors can never occur in the topology of the MEP [36–40]. Therefore, MEP maxima appear always at the position of nuclei. As a consequence the topology of the density and the MEP must differ for systems with non-nuclear attractors in the density. At this point it has to be stressed that the here discussed density and MEP (3, 3) CPs are not true critical points. However, the topological behavior of the neighborhood of these points is indistinguishable from that of true maxima [61] and, therefore, we will name them here (3, 3) CPs or maxima in the sense of their topological behavior. To extend the concept of molecular graphs to the MEP, bond critical points [43,44] and atomic interaction lines [48] must be defined for this scalar molecular field, too. Different from the electronic density, not all (3, 1) CPs of the MEP are connected to topological maxima, i.e. nuclei. To circumvent this problem we assign only those (3, 1) CPs of the MEP as bond critical point that connect topological maxima by a gradient path. By construction the resulting molecular graphs of the MEP connect only nuclei and, therefore, resemble the connectivity of Lewis structures. For most
molecules with localized bonds the above defined MEP molecular graphs are qualitatively identical to their electronic density counterpart. Obviously, this cannot be the case for systems with nonnuclear attractors. 3. Computational details For all calculations the linear combination of Gaussian-type orbitals Kohn–Sham density functional approach as implemented in the program deMon2k [62,63] was employed. The local density approximation (LDA) using the exchange–correlation contributions proposed by Vosko, Wilk and Nusair (VWN) [64] was used in combination with all-electron double zeta valence polarization basis sets [65]. The Coulomb energy was calculated by the variational fitting procedure proposed by Dunlap et al. [66]. For the fitting of the density the automatically generated auxiliary function set GEN-A2 [67] was employed. The exchange–correlation functional was evaluated with this auxiliary density, i.e. the auxiliary density functional theory (ADFT) method was used [68]. The exchange–correlation energy and potential were numerically integrated on an adaptive grid [69]. The grid accuracy was set to 105 a.u. in all calculations. The structures of the clusters were fully optimized using a quasi-Newton method in internal redundant coordinates with analytic energy gradients [70]. The convergence was based on the Cartesian gradient and displacement vectors with a threshold of 104 and 103 a.u., respectively. In order to characterize the optimized structures a vibrational analysis was performed. For this purpose the second derivatives were calculated by numerical differentiation (two-point finite difference) of the analytic energy gradients using a displacement of 0.001 a.u. from the optimized geometry for all Cartesian coordinates and the harmonic frequencies were then obtained by diagonalizing the massweighted Cartesian force constant matrix. The CP search of the density and the MEP was carried out by applying the CP search algorithm as implemented in the deMon2k code. This algorithm is already described in the literature [43,47,48]. It can be used to find critical points of any scalar molecular field because no assumption of the field topology is made. The input for the search algorithm are the scalar fields of the electron density, q, and the MEP, UðRÞ, which are calculated according to:
q¼
X Plm lðRÞmðRÞ l ;m
UðRÞ ¼
X A
X ZA P lm jA Rj l;m
ð2Þ 1 m : l r R
ð3Þ
Here, Plm denotes an element of the converged Kohn–Sham density matrix for the atomic basis functions l and m. Z A denotes the nuclear charge of atom A with the position vector A. The first term on the right side of Eq. (3) contains the contribution of the nuclei to the MEP and the second term the electronic contribution, respectively. 4. Results and discussion In Figs. 1 and 2 the (3, 1) bond critical points (dark dots) along with the corresponding molecular graphs of the MEP and the electronic density, q, for the studied lithium, sodium and copper clusters from the hexamer to the nonamer are illustrated. The cluster structures correspond to optimized ground state structures at the here used theoretical level. The spin multiplicities are singlets for the even and doublets for the odd numbered clusters. For the following discussion it is important to note that the MEP topology arises from the positively charged nuclei and the negative electron distribution, whereas the electronic density topology only
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MEP Li
Na
Cu
Fig. 1. MEP molecular graphs of the ground state structures of lithium, sodium and copper clusters from the hexamer to the nonamer.
originates from the electron distribution. Therefore, the topology of the complete MEP is a superposition of the topology of these two separate distributions. Assuming equal molecular structures the basic topology of the contribution of the nuclei to the MEP would be independent of the atom type. In this case differences in the molecular graphs of the MEP can only arise from the distribution of the electrons. Because of the different signs of the nuclear and electronic MEP contributions their balance also depends critically on the distance between the atoms of the molecular structure under study. 4.1. Hexamer Going from Li6 over Na6 to Cu6 the cluster ground state structures change considerably. Whereas Li6 is a square bipyramid, Na6 is a pentagonal pyramid and Cu6 is a planar triangular structure as it is shown in the first row of Figs. 1 and 2. The MEP molecular graph usually connects nearest neighbors in these clusters with a bond critical point in between as depicted in Fig. 1. A notable exception appears in the Li6 cluster. There, no MEP bond path can be found along the four sides of the square. The absence of the corresponding bond critical points in the MEP can be explained by analyzing the critical points and the molecular graph of the electronic density, q, which are shown in Fig. 2. Characteristic for the electronic density topology of Li6 is the appearance of four non-nuclear attractors in the square plane of the Li6 bipyramid (see first structure of first row in Fig. 2). This
is a clear indication that density has been moved away from atomic connection lines into the cluster. In the Na6 cluster the bond paths and bond critical points of the density can be found within the triangular faces of the Na6 pyramid. It is worth to note that the tip atom has no direct connection to the other atoms. For the Cu6 cluster the bond critical points of the density are localized between two neighboring Cu atoms (see third structure of first rows in Fig. 2). As a result the density and MEP molecular graphs of Cu6 are qualitative identical. The bond critical points of the density between the atoms in Cu6 indicate a more localized density distribution as in the corresponding Li and Na clusters. In the MEP this effect is less obvious. However, the bond path curving diminish from Li6 over Na6 to Cu6. In fact this trend holds for all studied clusters. Thus, it seems reasonable to correlate the bond path curving in the MEP molecular graph with the electron delocalization in the here studied clusters. The Li6 cluster possesses three different nearest neighbor distances. The distances between the two pyramid tips is about 2.6 Å. The distance between the pyramid tip and a corner of the square is about 2.8 Å and the length of an edge of the square is about 3.5 Å. The latter distance is significantly larger than the other distances. Also, as mentioned above the electron density in the Li6 cluster is localized more inside the structure which is clearly indicated by the appearance of the non-nuclear attractors in the square plane of the bipyramid. This effect together with the large Li–Li distances results in the disappearance of MEP bond critical points along the edges of the square in the bipyramidal structure.
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ρ Li
Na
Cu
Fig. 2. Density molecular graphs of the ground state structures of lithium, sodium and copper clusters from the hexamer to the nonamer.
4.2. Heptamer The structural picture changes from the clusters with 6 atoms to the clusters with 7 atoms. The ground state structures for Li7, Na7 and Cu7 are all three dimensional and pentagonal bipyramids. The structures are shown in the second row of Figs. 1 and 2. In the MEP all cluster show the same pattern of bond critical points and, thus, the same molecular graph as illustrated in the second row of Fig. 1. The bond paths are connecting two neighboring atoms with a bond critical point in the middle. Even though, the MEP molecular graph topology is the same for all heptamers its appearance is different due to the varying curvatures of the bond paths. The curvature is less and less pronounced going from Li7 to Na7 to Cu7 (see second row of Fig. 1). Quantitative differences appear in the q topology as it can be seen from Fig. 2. The molecular graph of the density for the Li7 cluster lies again completely inside the cluster. But in contrast to the Li6 cluster no non-nuclear attractors appear in the Li7 cluster. This leads to the conclusion that for the Li7 the localization of the electron density is within the cluster and, therefore, the electronic contribution to the MEP is not as dominant as in Li6. As a consequence the MEP molecular graph of Li7 shows the same connectivity as for Cu7. The Na7 cluster possesses a similar density molecular graph structure as the Na6 cluster. As it was previously discussed, in the Na6 cluster the bond critical points of q are located within
the triangular faces of the structure and the bond paths do not directly connect the tip atom with the other atoms. The same is found for the Na7 cluster. Only atoms in the pentagonal plane are connected directly with a bond path. Additionally, there is no bond path between the two tip atoms. In the MEP (see second structure of second row in Fig. 1) also the pyramid tips are connected by a bond path with a critical point in the center of the structure. 4.3. Octamer As shown in the third row of Figs. 1 and 2 the Li8 cluster has a different structure than the Na8 and Cu8 clusters. The structure of the Li8 cluster consists of a trigonal prism with one extra atom in the center of the prism and an other extra atom capping one rectangle of the prism. The structure shown in Figs. 1 and 2 is a view on the trigonal side of the prism with the external capping atom beneath. In Fig. 1 the MEP molecular graph along with the corresponding bond critical points is illustrated for this Li8 cluster. Again, all neighboring atoms are connected except the atoms building the corners of the capped rectangle of the prism. Notice that bond paths connect the central atom with all other atoms. Like in the other Li clusters the distribution of the bond paths and bond critical points of q as shown in Fig. 2 can be interpreted as accumulation of electron density within the cluster cage. Also in this cluster non-nuclear attractors appear which are arranged
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tetrahedrically around the central Li atom. Whereas the distance of the central Li atom to the other atoms is about 2.6 Å the lengths of the edges of the capped rectangle are about 3.3 Å and 3.6 Å. The lengths of the other edges in the structure are about 3.11 Å, 3.24 Å and 3.26 Å. As in the Li6 cluster, along the longest edges no bond critical points can be found in the MEP. A second similarity is the high concentration of bond critical points and non-nuclear attractors in q around the plane which contains these longest edges. The structures of the Na8 cluster and the Cu8 cluster are both snub disphenoids. For Cu8 a congruent picture for the molecular graph and bond critical points of the MEP and electronic density are found, as can be seen by the comparison of the corresponding structures in Figs. 1 and 2, respectively. All bond paths and critical points in the molecular graphs of the MEP and the density are located along the edges of the structure between the atoms. This indicates directed bonds and, thus, a certain covalency in this structure. At first glance the situation is similar for Na8. However, some (3, 1) critical points of the MEP are not existing in the corresponding electronic density. This leads to the assumption that in the Na8 cluster the distribution of the electron density is less structured compared with Cu8 and therefore more homogeneous. 4.4. Nonamers The structures for the Li9, Na9 and Cu9 cluster are shown in the last rows of Figs. 1 and 2. The Li9 cluster consists of a square antiprism with a central atom. Because of a Jahn–Teller effect one square is distorted into a rectangle and the other square is distorted into a rhombus. The bond critical points and the molecular graph of the MEP in Fig. 1 show connections between all neighboring atoms. The bond critical points and the molecular graph of q are accumulated within the cluster cage as illustrated in Fig. 2. A high concentration of these bond critical points and bond paths together with four non-nuclear attractors can be found around a plane which is between and parallel to the two distorted squares. On the other hand, the Na9 and Cu9 cluster show the same structure which consists of a pentagonal bipyramid where two adjacent faces are capped by atoms. These capping creates a second pentagonal bipyramid so that the structures can also be described as two melded pentagonal bipyramids. The molecular graph and bond critical points of the MEP of these clusters show the same behavior. The bond critical points and bond paths are found along the connection lines of neighboring atoms as it is depicted in Fig. 1. In the molecular graph of q for the Cu9 cluster the bond critical points and bond paths are also located along these lines as it is found for all other Cu cluster presented here underlining a certain covalency in these clusters. In the case of the Na9 cluster the molecular graph of q shows a very scanty topology which leads to the conclusion of a rather homogeneous electron density distribution. 5. Conclusions The topological analysis of the larger Li, Na and Cu clusters with up to nine atoms complements a previous study on smaller clusters. As for the smaller clusters non-nuclear attractors were found in some of the Li clusters but not in Na and Cu clusters. We note that this can vary with the theoretical level of description. In any case the density molecular graphs of the Li clusters are always located inside the clusters. This holds even for Li clusters without non-nuclear attractors, e.g. Li7. As a result the density molecular graphs of the Li clusters show connectivities that are not obviously related to Lewis structures. The other extreme are the density molecular graphs of the Cu clusters that show a connectivity in the spirit of Lewis structures. We suggest that this difference
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reflects more directed bonds in the Cu clusters, or vice versa, a more homogeneous electron distribution in the Li clusters. The density molecular graphs of the sodium clusters are always in between these two extremes, sometimes more in the direction of the Li clusters (Na6 and Na9) and sometimes more in the direction of the Cu clusters (Na7 and Na8). The MEP molecular graphs, on the other hand, show always interatomic connectivity in the spirit of Lewis structures. To a good part this originates from their construction. However, they also show clear trends form the Li over the Na to the Cu clusters, mainly by a decreasing convex curvature of the bond paths. By and large the topological equivalence of the MEP and density molecular graph seems to be a good indicator for a more structured electron density with somehow directed bonds. For the here studied systems we find this situation only for the Cu clusters. Acknowledgments Financial support from the CONACYT (Grants 130726 and 179409) is gratefully acknowledged. Some of the calculations were performed at the local HPC resource Xiuhcoatl of CINVESTAV. References [1] G.N. Lewis, Valence and The Structure of Atoms and Molecules, Chemical Catalog, New York, 1923. [2] J.W. Linnett, The Electronic Structures of Molecules, Wiley, New York, 1964. [3] W. L Jolly, Modern Inorganic Chemistry, McGraw-Hill, New York, 1985. [4] W.A. de Heer, The physics of simple metal clusters: experimental aspects and simple models, Rev. Mod. Phys. 65 (1993) 611–676. [5] S.N. Khanna, P. Jena, Atomic clusters: building blocks for a class of solids, Phys. Rev. B 51 (1995) 13705–13716. [6] P.O. Löwdin, On the non-orthogonality problem connected with the use of atomic wave functions in the theory of molecules and crystals, J. Chem. Phys. 18 (1950) 365–375. [7] C.A. Coulson, Valence, Clarendon Press, Oxford, 1952. [8] R.S. Mulliken, Electronic population analysis on LCAO-MO molecular wave functions. 1, J. Chem. Phys. 23 (1955) 1833–1840. [9] L. Pauling, The Nature of the Chemical Bond, Cornell University Press, 1960. [10] K. Jug, A new definition of atomic charges in molecules, Theor. Chim. Acta 31 (1973) 63–73. [11] K. Jug, Charge distributions and multipole moments in molecules, Theor. Chim. Acta 39 (1975) 301–312. [12] R. Heinzmann, R. Ahlrichs, Population analysis based on occupation numbers of modified atomic orbitals (MAOs), Theor. Chim. Acta 42 (1976) 33–45. [13] F.L. Hirshfeld, Bonded-atom fragments for describing molecular charge densities, Theor. Chim. Acta 44 (1977) 129–138. [14] K. Jug, A maximum bond order principle, J. Am. Chem. Soc. 99 (1977) 7800– 7805. [15] C. Ehrhardt, R. Ahlrichs, Population analysis based on occupation numbers II. Relationship between shared electron numbers and bond energies and characterization of hypervalent contributions, Theor. Chim. Acta 68 (1985) 231–245. [16] A.E. Reed, R.B. Weinstock, F. Weinhold, Natural population analysis, J. Chem. Phys. 83 (1985) 735–746. [17] C.M. Smith, G.G. Hall, Optimal population analysis, Int. J. Quant. Chem. 31 (1987) 685–692. [18] A.M. Köster, K. Jug, Multipole moment analysis for hydrides, fluorides, and lithium compounds of first- and second-row elements, Int. J. Quant. Chem. 48 (1993) 295–308. [19] C.A. Coulson, The electronic structure of some polyenes and aromatic molecules VII. Bonds of fractional order by the molecular orbital method, Proc. Roy. Soc. Lond. Ser. A 169 (1939) 413–428. [20] R. Daudel, C. Sandorfy, C. Vioelant, P. Yven, O. Chalvet, Comparaison des methodes statiques et des methodes dynamiques pour la prevision des proprietes chimiques des molecules a partir de leurs diagrammes moleculaires. 18, Bull. Soc. Chim. Fr. 17 (1950) 66–74. [21] K. Wiberg, Application of Pople–Santry–Segal CNDO method to cyclopropylcarbinyl and cyclobutyl cation and to bicyclobutane, Tetrahedron 24 (1968) 1083–1096. [22] D.R. Armstrong, P.G. Perkins, J.J.P. Stewart, Valencies and bond indices for elements from hydrogen to chlorine, J. Chem. Soc. Dalton 21 (1973) 2273– 2277. [23] N.P. Borisova, S.G. Semenov, Molecular orbital definition of chemical bond multiplicity, Trans. Leningrad Univ. 16 (1973) 119–124. [24] M. Giambiagi, M.S. de Giambiagi, D.R. Grempel, C.D. Heynman, Definition of ave bond index with non-orthogonal basis – properties and applications, J. Chim. Phys. 72 (1975) 15–22.
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