Bonding indicators from electron pair density functionals - CiteSeerX

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www.rsc.org/faraday_d | Faraday Discussions

Bonding indicators from electron pair density functionals Miroslav Kohout Received 26th April 2006, Accepted 8th June 2006 First published as an Advance Article on the web 14th September 2006 DOI: 10.1039/b605951c

The bonding analysis of a chemical system is usually based on some descriptors. Distinct approaches are used to generate the bonding descriptors, whereby the usefulness of a particular approach is emphasized by the desire to yield a description consistent with the examined effects. Thus, whereas the bond path from the electron density gradient field yields the connectivity of the atomic fragments, the orbital picture can easily rationalize the rotational rigidity of a double bond. On the other hand none of the former is able to describe the volume demand of the bonds, which can be accessed by descriptors originating from approaches using space partitioning. As a conceivable way to describe the bonding situation a class of functionals based on the electron pair density integrals in both, direct and momentum space, is proposed. The localizability indicators defined by those functionals are examined on several molecules.

1. Introduction The knowledge of the wavefunction of a system enables to recover all desired expectation values of the corresponding operators. Besides the expectation values there is also a demand to understand how the form of the wavefunction itself influences the expectation values. Of course, such analysis of the wavefunction is hindered by the large number of coordinates (moreover, only an approximate form of the wavefunction is available). To overcome the large dimensionality problem different ways of reduction are used. One possibility is the use of mean field approaches. In that case a particle is described by a mathematical function reducing the influence of all other particles into some mean field. Such orbital based analyses are widely used and certainly have advantages in the easy accessible pictorial description in 3 dimensions as well as the often relatively straightforward connection to the wavefunction and the energy. Another possibility to lower the dimensionality is to define the 2-matrix by the reduction of higher order density matrix through averaging over the coordinates of the corresponding number of particles. Although the reduction removes the direct connection to the wavefunction, the 2-matrix can recover all expectation values of two particle operators. The diagonal elements of the continuous representation of the 2-matrix is the electron pair density. Further reduction yields the 1-matrix with the electron density as the diagonal part. The analysis of these quantities is performed in a coordinate space. In the case of electron density the analysis is carried out utilizing the density vector field, density Laplacian and charge determination over basins,1 or using different potentials computed from the electron density. Max-Planck-Institut fu¨r Chemische Physik fester Stoffe, No¨thnitzer Str. 40, 01187 Dresden, Germany This journal is


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Additionally, instead of potentials, the integrals of electron density or electron pair density (populations) over regions given by restricted partitioning can be used to reveal information about the density matrices.2 The analysis of the wavefunction and the corresponding density matrices also serves the purpose to understand what is termed ‘the bonding situation’. From the view of a chemist this means statements about the reasons for the found molecular geometry, bond polarity, bond strength, rigidity, atomic electronic balances, as well as questions concerning the transferability of molecular fragments, etc. Many different indicators are used for this purpose, because the evaluation of the bonding situation is a wide field with partially unclear definitions. The obtained values are not simply connected to an expectation value. The usefulness of such indicators is often determined by the practice, regardless whether the physical background of the indicator is well understood. For instance, the topological analysis of a scalar field is a rigorous mathematical prescription yielding specific indicators. If such indicators have some practical advantages, they will be used. Nevertheless, the question about the physics behind the indicator should be always asked (even if there is no clear answer at the time). In the following the restricted populations, which is a class of functionals based on the electron pair density integrals, will be described and examined in both direct and momentum space.

2. Background 2.1.

Restricted populations

Density matrices are central objects of quantum mechanics.3,4 They are connected with the N electron wavefunction of the system and describe a probability distribution. The integration of the N-th order density matrix over the coordinates of chosen number m r N of electrons inside given region O and the remaining (N  m) electrons outside O yields event probabilities Pm.5,6 Those event probabilities were utilized elsewhere in the bonding analysis.7,8 The integration of the N-th order density matrix over the coordinates of (N  m) electrons over the whole space and the summation over all spins yields the m-th order spinless reduced density matrix. For spin-free one-electron resp. two-electron operators the reduced 2-matrix r2(r 0 1r 0 2,r1r2) is sufficient to compute the expectation values of such operators.9 The action of such an operator Fˆ on the 2-matrix results in a function f(r 0 1r 0 2,r1r2). The evaluation of the expectation value F = hFˆi involves the equating of the primed and unprimed coordinates, yielding f(r1,r2), and the integration of this function over the whole space. If the space is divided into mutually exclusive space filling regions Oi, then the integration can be performed over each region separately. Of course, for one-electron operators, where the information is comprised in a function f(r1), the sum of the values 1Fi for each region Oi will yield the expectation value. For two-electron operators the expectation value will be given as the sum of the terms 2Fi = Fii, with both electrons inside Oi, and Fij for the situation that the coordinate r1 is integrated over Oi and r2 over Oj, respectively. The basis of the restricted populations approach is the space division into mutually exclusive space filling regions. Imposing an additional constraint, namely that the regions are compact, ensures the locality of the examined effects. Furthermore, demanding that the integral of some chosen property over each region Oi yields always the same value o results in o-restricted partitioning.2 The whole space is at once partitioned into regions enclosing the same amount o of the ‘probe’ quantity. The choice of the value o and the initial region fixes the complete partitioning. Then, the analysis of the discrete distribution of the chosen ‘sampling’ quantity over the regions of the partitioning yields the ‘response’ of sort of ‘samequality’ regions. As a consequence, the analyzed discrete distribution of values have certain spatial density. 44 | Faraday Discuss., 2007, 135, 43–54

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2.2.

Electron localizability indicator

The electron localizability indicator (ELI) is determined by o-restricted partitioning based on the condition that each region of the partitioning encloses the same fraction of an electron pair, i.e., the probability to find an electron pair is the same for all regions.10,11 The ELI value is proportional to the charge found in the respective regions. From this procedure, several indicators can be deduced, depending on the choice of electron pair which determines the partitioning, i.e., same-spin,10 opposite-spin,11 or electron pairs formed by singlet-coupled resp. triplet-coupled electrons.2 Similar decomposition applies to the charge, i.e., the average number of electrons, sampled over the regions. Additionally, the populations can be computed not only in direct coordinate space, but also in the momentum space as well. For instance, partitioning the direct space in such a way that each region encloses the same fraction of a same-spin electron pair Dss and then determining the s-spin charge Qsi in each region Oi (centered around the position ai) yields the electron localizability indicator with the values UDs(ai). In the following the subscript D is omitted, i.e., the symbol Us(ai) is used instead, because the restriction of the fixed fraction of an electron pair is used throughout the present paper. Loosely speaking, this indicator shows how many s-spin electrons (average number) are needed to form a same-spin pair. This nicely reflects the idea of correlation of electronic motion.12,13 If the motion of same-spin electrons is highly correlated, i.e., the electrons try to avoid each other, then a high average number of electrons is needed to form a pair. Another pictorial view of ELI values is given by the event probabilities mentioned above. It is possible to show that the Us(ai) values are proportional to the event probability Ps1 , that reflects the extent to which the charge in Oi, that form a same-spin pair, represents a single electron.2,14 This view motivates the choice of the term localizability for the indicator Us(ai). Of course, to perform the space partitioning according to the chosen restriction and then compute the corresponding integrals over a large number of regions is not tractable and even not necessary. If the regions are small enough, then the integrals over pair density can be approximated by a Taylor expansion around a reference point ai. This leads to the Fermi hole curvature gs(ai) at the position ai, which for 1-determinantal Ansatz reads: gs ðai Þ ¼

s X

gij ðai Þgij ðai Þ;

ð1Þ

ioj

with gij(ai) = fi(ai) rfj(ai)  fj(ai) rfi(ai),

(2) 15

where the sum runs over the occupied s-spin orbitals f. With the Fermi hole curvature the volume Vi of the region Oi enclosing Dss same-spin electron pairs is approximately given by:2
 Ds s 3=8 Vi  12 : ð3Þ gs ðai Þ At the same time the s-spin charge Qsi in Oi is approximately given by rs(ai) Vi. Then, utilizing eqn (3) the s-spin charge in Oi can be now written as: " #3=8 8=3 rs ðai Þ s s s 3=8 Qi  ½D  12 : ð4Þ gs ðai Þ The above charge distribution is dependent on the fixed pair population Dss which is a constant for all the regions of the o-restricted partitioning. ELI is the This journal is


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Dss independent part of the above expression for Qsi :
 12 3=8 s U ðai Þ  rs ðai Þ : gs ðai Þ

ð5Þ

Thus, ELI is proportional to a discrete distribution of charges (with positive values, not squeezed into an [0–1] interval, i.e. unlike the value range of the electron localization function (ELF) of Becke and Edgecombe16). At Hartree–Fock (HF) level: gs ¼ rs

s X k

1 jrfk j2  ðrrs Þ2 : 4

ð6Þ

Thus, at HF level the ELI formula for the same-spin electron pairs resembles the kernel of the ELF formula of Becke and Edgecombe, but without any reference to the homogeneous electron gas. Moreover, the above ELI formula is valid for correlated wavefunctions as well.11 The scheme was also applied to o-restricted partitioning based on a fixed number of triplet coupled electron pairs2 (the triplet coupled electron pairs originate from the decomposition of the 2-matrix into a symmetric and antisymmetric part17,18). In this case the resulting localizability indicator U(t)(ai) is proportional to the total charge that is needed to form a pair of electrons coupling to a triplet. Interestingly, at HF level U (t)(ai) resembles the kernel of the widely used ‘closed-shell’ ELF formula,19,20 where instead of the originally suggested use of spin-dependent quantities the total density is fed into the ELF. In spin polarized case the ELI for triplet coupled electrons, U (t)(ai), allows to analyze just a single diagram instead of two diagrams (separate for each spin part). 2.3.

ELI in momentum space

ELI is defined as a functional of electron pair density. Although the electron pair density can be formulated in both direct and momentum space, it is only the direct space formulation of ELI (resp. ELF) that experienced widespread use in the bonding analysis. The present paper focuses on the examination of ELI in the momentum space. Kulkarni21 presented the electron momentum localization function (EMLF) by exchanging the orbitals in ELF formula with the momentals22 (i.e., orbitals in momentum space). Unlike this, the scheme of restricted populations allows consistent derivation of ELI in momentum space.23 The derivation yielding ELI as a measure of the charge that is needed to form a pair again involves the Fermi hole, taking into account the complex character of the momentals. Actually this means using the complex momentals as the input to eqn (1), yielding the Fermi hole curvature gs(pi) at the momentum (pi). Then, with the s-spin momentum density ps(pi), the ELI value Us(pi) for region Oi centered around pi in momentum space is given by:23
 12 3=8 U s ðpi Þ  ps ðpi Þ : ð7Þ gs ðpi Þ Because each orbital has complex dependence on the atomic coordinates,24 additional terms arise compared to the simple exchange of orbitals and momentals. Those terms are not present for atomic calculations. 2.4.

Comments on electron localizability indicators

Recently proposed spin-pair composition function25 is based on the ratio between the number of same-spin electron pairs to the opposite-spin electron pairs in an arbitrary chosen volume around a reference point. Thus, it is defined at any point. It 46 | Faraday Discuss., 2007, 135, 43–54

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differs from the restricted populations approach, where the volume partitioning is given at once for the whole system and the volumes have to obey specified restrictions. Furthermore, unlike ELI, the spin-pair composition function does not measure the charge in the arbitrary volumes. Of course, for closed-shell systems at HF level the spin-pair composition function is formally identical with ELI for samespin electron pairs. This is due to the fact that in this case the number of oppositespin electrons per fixed charge is a constant, and the spin-pair composition function actually measures only the same-spin electron pairs. For correlated wavefunctions the ELI formula11 differs from the one for the spin-pair composition function (the corresponding formula can be simply deduced2 from the respective formulae for same-spin pairs, ELI,11 and opposite-spin pairs, ELIA,26 which was derived using the scheme of the restricted populations). It would be interesting to examine the differences in bonding patterns at highly correlated level. In most cases time independent functions are analyzed in direct space. Considering the time dependency, an ELF formula was derived by Burnus et al.,27 revealing an additional term, namely the current density, in the ELF formula. This demands a comment: For formulae based on electron pair density, like ELI or ELF of Becke and Edgecombe, the current density term naturally arises simply by feeding the complex time dependent orbitals into the expression g for the Fermi hole curvature, cf. eqn (1) as well as previous Section 2.3 for ELI in momentum space. For ELF formulae based on the electron density, like ELF of Savin19 or ELF of Tsirelson,28 the current density term is not present if the time dependent orbitals are used. This should not be understood as a failure or troublesome defect. On the contrary, the inclusion of the current density into those formulae is equivalent to the disapproval of kinetic energy density as the basis of ELF and, consequently, force ELF to originate from pair density.

3. Results and discussion 3.1.

ELI as a tool

Tools based on the idea of measuring the extent to which an electron is spatially localized are widely used in the analysis of chemical bonding. ELI can be connected to the correlation of electronic motion or, in a certain sense, with a measure to what extent an electron occupies alone a region that encloses specified electron pair population. Although one can argue that chemical bonding certainly has something to do with the pairing of opposite-spin electrons resp. the avoidance of same-spin electrons, ELI (and all ELF variants as well) offers only hints about the bonding situation. Up to now, there is no sound physical background that would guide the researcher through the analysis. Such developed Ansatz is found in the quantum theory of atoms in molecules (QTAIM) of Bader. The topological analysis used nowadays for ELF is just an application of prescriptions to a scalar field. It could be applied, for instance, to the density Laplacian or the one-electron potential as well. Yet, we cannot predict the ELF topology nor give clear reasoning for the topological analysis. Why do we choose ELF basins (besides the analogy to Baders approach)? What electron counts in the ELF basin should be expected for exact wavefunctions? In more complex compounds the number of ELF maxima is sometimes a hardly explainable fact, where it is not even clear whether the situation is due to a failure of the underlying quantum chemical code. In the unfavorable cases, it does not help much to use interpretations based on orbitals, i.e., to reduce possible high level calculation back to the mean field picture. Then it is more faithful to abide with the orbitals. Nevertheless, the results shown by ELI (or ELF), being so close to what is expected by chemists, are very appealing. The topology of ELI seems to reveal most of the chemical descriptors, i.e., the bonds, lone pairs, or atomic shells (surely, such relationship is based on our expectation and experience). Thus, there is nothing This journal is


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wrong with the usage of ELI as a bonding descriptor, as long as the limitations are taken into consideration. Even more, the search for new relations could possibly reveal the real physical background. In contrast to ELF, ELI partially removes the arbitrariness in the definitions. The o-restricted partitioning is connected with a unique density of regions resulting from the partitioning. ELI is proportional to discrete distribution of charges, from which it follows that the sum of all ELI values is proportional to the total number of (s-spin) electrons. In the following the electron localizability in both direct and momentum space will be analyzed for the Ar atom as well as the molecules N2, C2H4, and C6H6. Of course, for those simple examples the ELF analysis (i.e., direct space) is known. The ELI analysis in the direct space is shown for two reasons: (1) to make the reader familiar with the value range, because the Lorentzian transformation is not used here for ELI; (2) to enable the guidance in the momentum space, as the molecular structure, reflected by the electron density, has another signature as in the direct space. 3.2.

Ar atom

In direct space the ELI for Ar atom shows 3 atomic shells, cf. Fig. 1a for U(t)(r) computed29 with the basis set of Clementi and Roetti.30 ELI reaches the maximal value U(t)(r) = 7.55 at the position of the nucleus. The ELI values of the following two minima and maxima are roughly around 1 and 2, respectively. ELI approaches zero at large distances from the Ar nucleus. In momentum space, cf. Fig. 1b for U(t)(p) computed with the same basis set, ELI exhibits again 3 atomic shells23 (the core K shell has ELI maximum U(t)(p) = 4.34 at the momentum p = 47.83  h bohr1). For an atom it is very simple to establish the correspondence between the respective shells in direct and momentum space, knowing that the momentum increases with decreasing distance to the nucleus. However, there is another approach,31 which becomes favorable to find this correlation also for molecules. In both direct and momentum space, ELI is proportional to the charge in a compact region O, cf. eqn (5) and (7). Thus, ELI can be mathematically exactly decomposed into charge contributions, precisely in the same way as the charge in O, irrespectively of the formalism used for the partial charges (i.e., charges from orbital densities, both canonical or localized, s and p contributions, or a-spin and b-spin contributions to the total charge). In direct space:
 X 12 3=8 U ðrÞ  rk ðrÞ : ð8Þ gðrÞ k The Fermi hole curvature g(r) is of course computed from the corresponding 2matrix part, i.e., from the same-spin part in the case of Us(r) resp. from the antisymmetric part in case of U(t)(r). The total HF density of the Ar atom, which

Fig. 1 The ELI for the Ar atom in (a) direct and (b) momentum space representation. Solid line: U(t)(r) from the total electron density; dotted line: K shell contribution to U(t)(r); dashed line: L shell contribution to U(t)(r); dash-dotted line: M shell contribution to U(t)(r). 48 | Faraday Discuss., 2007, 135, 43–54

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is used to compute U contributions:

(t)

(r), can be decomposed at any position in space into 3 shell

r(r) = rls(r) + r2s2p(r) + r3s3p(r).

(9)

With this, ELI can be decomposed into 3 shell contributions, as shown in Fig. 1a. It can be seen that around each shell maximum the ELI is mainly due to the charge originating from the corresponding shell, whereas, for instance, the ELI minimum between the second and third shell is based on roughly equal contributions of the neighboring shell charges, cf. the arrow in Fig. 1a. Equivalently to the direct space representation, ELI in momentum space can again be decomposed into charge contributions:
 X 12 3=8 U ðpÞ  pk ðpÞ : ð10Þ gðpÞ k The Fourier–Dirac transformation of the one-determinantal HF wavefunction of Ar atom yields again a determinant composed of the corresponding momentals, i.e., direct space orbitals transformed into the momentum space.24 This easily allows us to trace the ELI contributions in the respective coordinate representations. Fig. 1b shows each of the shell contributions to ELI in momentum space. The correlation between the shells in both space representations is obvious (although trivial in this case). The diagram also emphasizes the extent of the penetration of the K shell orbitals into the M shell ELI region. 3.3.

N2 molecule

In case of molecules, the classification of ELI domains in momentum space is not trivial by far. To outline the increasing complexity let us turn to the diatomic molecule N2. The molecule was computed at the bond distance of 107 pm with Gaussian 0332 using the aug-cc-pVTZ basis set (the molecule was oriented along the z axis). The ELI diagram in direct space, cf. Fig. 2a, shows the 1.8-localization domains, i.e., regions of space enclosed by isosurfaces with U(t)(r) = 1.8. For each atom a separate core and lone pair region, respectively, is visible. Between the core domains there is 1 localization domain around the ELI maximum at the bond midpoint that can be connected with the triple bond (the absence of the s–p separation is usually found in direct space ELI). The charges originating from orbitals attributed to the lone pairs (2 orbitals), the s bond (1 orbital), and the p bond (2 orbitals), respectively, were used to decompose ELI into the respective contributions. Fig. 2b presents the 1.8-localization domains (red) for the lone pair contributions, together with 1.1-localization domains for the s (blue) and p (green) contributions. One could maybe wonder that the ELI contributions show separate localization domains for the s and p bonds. However, this is similar to maxima in orbital density which are not expected to emerge in the total density. The ELI value U(t)(r) = 1.8 was used only for the lone pair contributions, because for the s and p bonds the ELI contributions do not reach this value. This is especially true for the p bond, where the ELI contribution attains maximally the value U(t)(r) = 1.12 (inside the green colored ring). Fig. 2c shows at low momentum, i.e., in the valence region, three separate 2.7localization domains. One domain encloses the ELI ring maximum in the pxpy plane around the pz axis. The other two domains correspond to ELI maxima located on the pz axis (because of the inversion symmetry in the momentum space those two localization domains on the pz axis represent only a single feature). In momentum space the K shells of both nitrogen atoms occupy the same high momentum regions. There are ELI maxima U(t)(p) = 1.06 at pz = 4.48  h bohr1, not shown in Fig. 2c, that can be attributed to the nitrogen cores. Thus, similarly to the direct space representation, there are two topological features in momentum valence space (as This journal is


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Fig. 2 The ELI for the N2 molecule. (a) ELI in direct space: 1.8-localization domains for total ELI. (b) ELI contributions in direct space: 1.8-localization domains for the lone pair contributions (red); 1.1-localization domains for the s bond (blue) and p bond (green) contributions. (c) ELI in momentum space: 2.7-localization domains for total ELI. (d) ELI contributions in momentum space: 0.7-localization domains for the s bond (blue) contributions; 2.1-localization domains for the p bond (green) and lone pair (red) contributions.

already mentioned the two domains along the pz axis reflect the inversion symmetry, i.e., for a component with moments in a particular direction there must be a component with moments in the opposite direction). To decide which domains represent specific bonding features, as given by ELI in direct space, the orbital contribution to ELI were used. Fig. 2d shows ELI contributions based on charges originating from momentals corresponding to the orbitals chosen for the ELI decomposition in direct space (the color code is the same as in Fig. 2b). For localization domains corresponding to the lone pair and p bond contributions the isovalue U(t)(p) = 2.1 was chosen, whereas the s bond contribution is shown by 0.7localization domain (which is slightly below the maximal value for this contribution). From the diagram it is clear that the lone pairs are represented by the total ELI localization domains along the pz axis. The ring shaped domain can be attributed to the triple bond, where again the s and p contributions are not separated in the total property. In contrast to the direct space, the p contribution attains a higher value than the s contribution and controls the position of the ELI maximum for the triple bond. 3.4.

C2H4 molecule

The complexity of the bonding situation is increased in the C2H4 molecule by additional C–H bonds. The calculation was performed with the aug-cc-pVTZ basis set at the C–C bond distance of 131.8 pm and C–H bond distance of 107.7 pm. The ELI 1.8-localization domains in Fig. 3a can be attributed to the carbon cores, the four C–H bond, and the ‘banana’-like double bond between the carbon atoms (there is no ELI maximum at the bond midpoint). To prepare the analysis of ELI in momentum space, localized orbitals were used to determine the ELI contribution of the C–H resp. C–C bonds (Boys localization with Gaussian 03). The localization procedure yields the two ‘banana’ bonds instead of the s and p combination. The ELI contributions of the localized C–H bonds are shown in Fig. 3b by the 1.8localization domains (red). The 1.8-localization domain (green) is based on the 50 | Faraday Discuss., 2007, 135, 43–54

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Fig. 3 The ELI for the C2H4 molecule. (a) ELI in direct space: 1.8-localization domains for total ELI. (b) ELI contributions in direct space: 1.8-localization domains for the C–H bond (red) and C–C bond (green) contributions. (c) ELI in momentum space: 2.54-localization domains for total ELI. (d) ELI contributions in momentum space: 2.0-localization domains for the C–H bond (red) contributions; 0.85-localization domains for the C–C bond (green) contributions.

charge of the localized C–C bonds (the separation into two irreducible localization domains could be achieved at U(t)(r) E 2.16, but the resulting domains would be very small for the graphical representation). The localization domains for the ELI contributions are almost identical with the respective domains of total ELI. Such resemblance is not always obtained by the Boys localization procedure. The momentum space ELI shows in the valence region (low momentum) six maxima in the pypz plane and two maxima along the pz axis, cf. Fig. 3c (because of the inversion symmetry, the eight maxima correspond to four different features). Similar to the N2 molecule, there are ELI maxima at higher moments, not shown in the diagram, corresponding to the carbon cores. The ELI contributions computed for the momentals from the Fourier–Dirac transformed localized orbitals are presented in Fig. 3d, using the same color code as in the direct space. The 2.0localization domains (red) shows that the four diagonal total ELI maxima in the pypz plane (Fig. 3c), can be attributed to the C–H bonds. This could be rationalized by the idea that there are two possible directions for the movement of the C–H bond electrons along the bond direction (i.e., the same direction for the trans-C–H bond). The 0.85-localization domains (green) for the ELI contribution originate from the ‘banana’-like orbitals. The diagram show large domains (green) along the px axis and small ones (green) along the pz axis. Again, this could be rationalized as the p component (movement perpendicular to the molecular plane, i.e., in the x direction) and the s component (movement along the C–C bond axis, i.e., in z direction) of the double bond. Interestingly, the small domains along the pz axis are revealed neither by the ELI contribution of the canonical s orbital nor the canonical p orbital (not even by the sum of both). This is in contrast to the direct space, where the ELI bond feature (i.e., ELI maximum below and above the molecular plane) is revealed alone by the ELI contribution of the canonical p orbital. The direct space ELI contribution This journal is


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of the canonical s orbital is maximal at the bond midpoint, however, being too low to emerge in total ELI. It should be once more emphasized that the discussion of the orbital charge contributions to ELI serves only as a guide to reveal the possible correspondence between the direct and momentum space ELI representations. 3.5.

C6H6 molecule

The benzene molecule was computed with the aug-cc-pVTZ basis set using the C–C bond distance of 139 pm, resp. 107 pm for the C–H bonds. The complexity of the bonding situation as described by ELI in direct space, cf. Fig. 4a presenting the 1.8localization domains, seemingly does not increase to much when going from C2H4 to the benzene molecule. The pattern of core and C–H bond localization domains is just repeated over the structure (instead of the double ‘banana’ bond in ethene, ELI shows a maximum at the C–C bond midpoint). Using orbitals prepared by the Boys localization procedure the ELI contribution in direct space were computed. The 1.8localization domains in Fig. 4b show the ELI contributions for the localized C–H bonds (red), the 3 C–C single bonds (blue), and the three C–C double bonds (green). The situation is significantly different for ELI in momentum space. The outstanding result is that the carbon cores are no more resolved by separate ELI maxima (there is only a change in slope at higher momentum). The core electrons of the six cores share similar high momentum regions, which apparently hinders the electron localization. In the valence region at low momentum, ELI exhibits six localization domains in the pxpy plane, cf. Fig. 4c with the 2.2-localization domains (the molecule is oriented in the xy plane). Above and beneath the pxpy plane there are ELI rings, each of which collapses into six small irreducible domains at

Fig. 4 The ELI for the C6H6 molecule. (a) ELI in direct space: 1.8-localization domains for total ELI. (b) ELI contributions in direct space: 1.8-localization domains for the Boys localized C–H bond (red), double C–C bond (green), and single C–C bond (blue) contributions. (c) ELI in momentum space: 2.2-localization domains for total ELI. (d) ELI contributions in momentum space: 1.3-localization domains for the Boys localized C–H bond (red) and double C–C bond (green) contributions; 0.64-localization domains for the single C–C bond (blue) contributions. 52 | Faraday Discuss., 2007, 135, 43–54

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U (t)(p) E 2.34, pointing between the localization domains located in the pxpy plane. Fig. 4d presents the ELI contributions in momentum space based on the charge originating from Boys localized orbitals. It can, at least partially, reveal the similarity with the direct and momentum space for the total ELI. The 1.3-localization domain (green), describing the ELI contribution from the C–C double bonds, confirms that this region can be attributed to the p-like character (electrons circulating perpendicular to the C–C bond). The 0.64-localization domains (blue) stemming from the C–C single bonds are located near the total ELI maxima in the pxpy plane. However, the C–C single bonds contribute by merely about 30% to the value of those total ELI maxima (roughly the same as the C–C double bond contribution). The position of the 1.3-localization domain (red) for the ELI contribution of the C–H bonds proves that there are no separate total ELI maximum for this bond type. Closer analysis shows that the C–H contribution to the total ELI maximum in the pxpy plane is about 40% (cf. the other contribution given above). In the direct space there is usually no separation between the s and p bonds. It seems that separate ELI descriptors in momentum space also remain hidden for other groups of bonding characteristics.

4. Conclusions The Electron Localizability Indicator (ELI) in direct space reflects certain features of the bonding situation in a molecule. The ELI bonding signatures of the system are evaluated mainly in accordance with ‘chemical’ experience. The ELI is based on a restricted population approach which allows consistent derivation, not only for different ELI types (for instance same-spin, opposite-spin, or triplet coupled), but also transformation into the momentum space representation. It was shown that in momentum space the bonding situation can be analyzed by ELI, at least to a certain extent. However, in contrast to the ‘extensive’ character of ELI in direct space, where the number of ELI signatures increases with molecular size, in momentum space all the signatures would share the common, relatively small, valence region of low momentum. With increasing molecular size the contributions of electron density resp. electron pair density, corresponding to all equivalent direct space ELI signatures, are transformed into similar momentum regions, which hinders the localization. On the other hand, this could enable a sort of molecular fingerprint, showing the similarity between the systems.

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