Atomic transferability within the exchange-correlation density
Atomic Transferability within the Exchange-Correlation Density XAVIER FRADERA,1 MIQUEL DURAN,1 JORDI MESTRES2 1 2
Institute of Computational Chemistry, University of Girona, 17071 Girona, Catalonia, Spain Molecular Design & Informatics, N. V. Organon, 5340 BH OSS, The Netherlands
Received 19 April 2000; accepted 29 May 2000
ABSTRACT: Starting from either the exchange or the exchange-correlation density together with Bader’s definition of an atom in a molecule, an atomic hole density function can be defined. Contour maps of atomic hole density functions are able to show how the electron density of each atom in a molecule is partially delocalized into the rest of atoms in the molecule. The degree of delocalization of the atomic density ultimately depends on the nature of the atom studied and its environment. Atomic hole density functions are also used to define an atomic similarity measure, which allows for the quantitative assessment of the degree of atomic transferability in different molecular environments. In this article, contour maps for the N atom in the (N2 , CN− , NO+ ) series and the O atom in the (CO, H2 CO, and HCOOH) series are presented at the Hartree–Fock and CISD levels of theory. Moreover, the transferability of N and O within the two series is studied c 2000 John Wiley & Sons, Inc. by means of atomic similarity measures. J Comput Chem 21: 1361–1374, 2000 Keywords: molecular similarity; atomic transferability; Fermi hole; exchange-correlation hole; electron-pair density
Introduction
S
ince the introduction of the Lewis model in 1916,1 the idea that electrons in atoms and molecules are arranged in pairs, and that the atoms in a molecule can share pairs of electrons between them to form covalent bonds, have been fundamental concepts in chemistry. From a physical point of Correspondence to: J. Mestres; e-mail: [email protected]. akzonobel.nl Contract/grant sponsor: Spanish DGICYT; contract/grant number: PB98-0457-C02-01
view, the pairing of electrons is a consequence of the antisymmetry of the electronic wave function with respect to the interchange of the space-spin coordinates of any pair of electrons. Same-spin electrons are strongly correlated between them, and the probability that two electrons with the same spin are near in space is very low, thus allowing for the formation of localized pairs of electrons of different spin. Correlation between same-spin electrons is usually referred to as Fermi or exchange correlation, whereas correlation between electrons of different spin is called Coulomb correlation. Fermi correlation is basic for an accurate description of the electronic structure of atoms and mole-
Journal of Computational Chemistry, Vol. 21, No. 15, 1361–1374 (2000) c 2000 John Wiley & Sons, Inc.
FRADERA, DURAN, AND MESTRES cules. At the Hartree–Fock (HF) level, this is the only kind of electron correlation that is taken into account, while ab initio post-Hartree–Fock methods, such as Møller–Pleset (MP) or Configuration Interaction (CI), introduce also Coulomb correlation between electrons of different spin. In any case, the pairing and correlation of the electrons in an atom or molecule is determined by the exchange density, at the HF level, or exchange-correlation density, for more general wave functions, which accounts for the differences between the electron-pair density and the simple product of one-electron densities.2 The exchange-correlation density contains a complete description of the way in which the electrons in a system are mutually correlated. However, because of its dependence on the coordinates of two electrons, the exchange-correlation density is a sixdimensional function, which makes it difficult to visualize directly. Consequently, different approaches have been proposed to reduce the dimensionality of the original exchange-correlation density. A widely used approach is to define a reference electron, fixed at a given position in real space. The resulting threedimensional function is called a Fermi hole, or, more generally, exchange-correlation hole. Integration of the Fermi hole over all space yields −1, corresponding to the removal of one electron. A Fermi hole for a given reference position shows the instantaneous effect of an electron located at that point on the rest of electrons of the molecule. The analysis of Fermi holes in molecules has received considerable attention, and has been used to illustrate the localization of electrons in regions of real space.3, 4 An alternative approach for extracting quantitative information from the exchange-correlation density is the definition of localization and delocalization indices. Within Bader’s theory of Atoms in Molecules,5 atomic basins are defined as regions in real space delimited by zero-flux surfaces in the one-electron density.5 Thus, one can obtain a localization index for an atom by integrating the two electron coordinates of the exchange or exchangecorrelation density over the same atomic basin. Similarly, a delocalization index for a pair of atoms can be obtained by integrating each of the two electron coordinates over a different atomic basin.6, 7 Another possibility is to integrate only one of the electron coordinates over an atomic basin.8 This gives an atomic hole density that is a function of the second electron coordinate, with the property that it is not dependent on the particular position of a reference electron but on the choice of a reference atomic basin. Studies so far have focused on calculations of atomic hole densities at the semiempirical
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or Hartree–Fock levels of theory. The purpose of this article is not only to show that post-Hartree–Fock atomic hole densities can be computed efficiently and routinely, but also to analyze the important changes they exhibit when increasing the level of theory (e.g., CI with Singles and Doubles, CISD). Such an insight will be gained through the analysis of atomic holes for the N and O atoms in different molecular environments. Transferability is a fundamental concept in the theory of Atoms in Molecules,5 that is, atoms or functional groups with similar properties should have similar electron densities.9 Therefore, the definition of a similarity measure between atoms or functional groups provides a means for assessing quantitatively the degree of transferability of those atoms or functional groups in molecules. Similarity measures have been already succesfully applied to assess the degree of transferability of atoms in molecules from the point of view of the one-electron density.11 In this article, the degree of transferability of atoms in molecules will be analyzed from the point of view of the exchange-correlation density. In particular, similarities between the atomic holes of the N atom in the CN− , N2 , and NO+ series of molecules and the carbonyl O atom in the CO, H2 CO, and HCOOH series of molecules will be studied at the HF and CISD levels of theory. The effect of Fermi and Coulomb correlation on the atomic holes, and thus, on the degree of transferability of atoms along those molecular series, will be also discussed.
Methodology EXCHANGE-CORRELATION AND HOLE-DENSITY FUNCTIONS At any level of theory, a function describing how the electrons are mutually correlated in atoms or molecules can be defined from the one-electron and electron-pair densities as, f (r1 , r2 ) = 20(r1 , r2 ) − ρ(r1 )ρ(r2 ); Z f (r1 , r2 ) dr1 dr2 = −N,
(1)
where ρ(r) and 0(r1 , r2 ) are the first- and secondorder density functions, respectively, and N is the number of electrons in the system. As stated above, there is no correlation between electrons with antiparallel spin within the HF approximation. At this level of theory, f (r1 , r2 ) is the Fermi or exchange density. By fixing one of the two electron coordinates, r1 , and dividing by the value of ρ(r) at that point, the
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ATOMIC TRANSFERABILITY Fermi or exchange hole can be defined as,2 h(r1 ; r2 ) =
20(r1 , r2 ) f (r1 , r2 ) = − ρ(r2 ); ρ(r1 ) ρ(r1 ) Z h(r1 ; r2 ) dr2 = −1. (2)
When necessary, we will refer to this function as the point Fermi hole, to distinguish it from the atomic Fermi hole, to be described later. h(r1 ; r2 ) shows how the presence of the reference electron at r1 lowers the probability that a second electron with the same spin be in a region nearby. Within the restricted HF approximation, it holds that h(r1 ; r1 ) = −ρ(r1 )/2.13 Usually, Fermi holes are defined for either α or β electrons only, and then hαα (r1 ; r1 ) = ρ α (r1 ) an similarly for β electrons, showing that the electron exclusion at the reference point is complete for same-spin electrons. The Fermi hole and related functions have been employed to visualize how the exchange correlation between same spin electrons determines the distribution of electrons in real space.3, 4 Plots of the Fermi hole for fixed positions of the reference electron show in which regions the exchange density is highly localized, or where it is delocalized over a large region of space. It is normally found that the exchange density is highly localized in regions occupied by core electrons, and delocalized in bonding regions. Although molecular Fermi holes have been mainly investigated within the HF approximation, the definitions in eqs. (1) and (2) are completely general, and can be applied to any methodology for which the one-electron and electron-pair densities can be obtained. In the general case, when both Fermi and Coulomb correlation are taken into account, the exchange density becomes the exchangecorrelation density, and the Fermi hole becomes the exchange-correlation hole. Because f (r1 , r2 ) is a two-electron function, it can be partitioned into monoatomic and biatomic contributions, X X fA (r1 , r2 ) + fA,B (r1 , r2 ), (3) f (r1 , r2 ) = A
AB
where A and B are atomic basins as defined in the theory of Atoms in Molecules; that is, regions in real space delimited by zero-flux surfaces in the one-electron density.5 The monoatomic terms reflect how much the electrons are localized into atoms, and the biatomic terms account for the delocalization or sharing of electrons between pairs of atoms. By integrating the fA (r1 , r2 ) and fA,B (r1 , r2 ) terms over the two-electron coordinates, one obtains atomic localization and delocalization indices, respectively.7
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A second possibility for the study of electronic localization in atoms and molecules, recently proposed by R. Ponec,8 is to integrate the exchange or exchange-correlation density over one-electron coordinate within a reference atomic basin, Z f (r1 , r2 ) dr2 ; gA (r) ≡ gA (r1 ) = A Z gA (r) dr = −NA . (4) By analogy with the point Fermi hole function described above, Ponec calls gA (r) a charge-weighted Fermi hole. It is a three-dimensional function, which integrates to −NA , the atomic population of atom A, and shows how the electron density of a given atom excludes NA electrons, in the same way that a Fermi or exchange-correlation hole excludes one electron. Thus, the gA (r) function can be considered as an atomic Fermi or exchange correlation hole, or atomic hole, for short, for the atom A. Also, starting from the one-electron density matrix, instead of the pair-density function, Fulton et al. have defined basin–basin and basin–point sharing indices.10 The basin–point sharing index is, in fact, a function depending on the space-spin coordinates of an electron. At the HF level, the basin–basin sharing index is equivalent to the delocalization index defined in ref. 7, and the basin–point sharing index is equivalent to the atomic-hole density function, gA (r). However, for post-HF methods this equivalence is not preserved. An interesting property of gA (r) is that, in a LCAO framework, it can be expressed in closed analytical form as X ∗ PA (5) gA (r) = µν ϕµ (r)ϕν (r), µν
where ϕµ (r) are basis functions, and PA µν are obtained as: X (2Pµνλσ − Pµν Pλσ )SA (6) PA µν = λσ . λσ
Pµν and Pµνλσ are the elements of the first and second-order density matrices of the molecule, respectively, and SA µν stands for the overlap of a pair of basis functions, µ and ν, within the basin of the atom A. Within the restricted HF approximation, eq. (6) can be simplified to yield: 1X (Pµσ Pνλ )SA (7) PA µν = − λσ . 2 λσ Following Bader’s definition of an atom in a molecule, the one-electron density of any molecule can be partitioned into a set of disjoint atomic
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FRADERA, DURAN, AND MESTRES fragments. Similarly, by using the definition of the charge-weighted or atomic hole in eq. (4), the exchange-correlation density can also be partitioned into a set of atomic holes. Taking into account that the integration of the exchange-correlation density over the electron coordinate r2 over all space is equivalent to the one-electron density at the point r1 ,14 Z (8) f (r1 , r2 ) dr2 = −ρ(r1 ), one obtains that: ρ(r) =
X A
ρA (r) = −
X
gA (r).
(9)
A
Although the definitions of both ρA (r) and gA (r) fragments involve zero-flux surfaces in the one-electron density, there is an important difference between them. Each ρA (r) density fragment has nonzero values only within its own atomic basin, i.e., is limited in space. Each atomic basin is separated from its neighbors by closed boundaries, coincident with the zero-flux surfaces in the one-electron density. On the contrary, each gA (r) density is extended over all space, and every pair of atomic holes in a molecule has some degree of overlap. From a computational point of view, gA (r) has the advantage that it has an analytical LCAO expression analogous to that of the one-electron density, eq. (5). However, there is no such possibility for ρA (r): the fact that it has closed boundaries does not allow to express it as a linear combination of basis functions extended over all space. The Fermi or exchange-correlation hole density shows how the presence of an electron at the reference position, r1 , excludes a second electron, by diminishing the probability of finding another electron in a region nearby. Accordingly, an atomic hole shows how the one-electron density associated to a reference atom excludes the same amount of one-electron density. In both cases, the spatial localization of the corresponding hole density function will be related to the degree of localization of the reference electron or the one-electron density of the reference atom, respectively. The fact that atomic holes are not dependent on the position of a reference electron, makes them particularly suitable for studying the localization and delocalization of electrons from an atomic point of view. For any atom, a picture of the corresponding atomic hole will show how much of the electron density is localized in that atom, and how much is delocalized towards other atoms, the same information that can be obtained in a quantitative way by means of localization and delocalization indices.7
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SIMILARITY MEASURES BETWEEN ATOMIC HOLES For an atom in a molecule, the associated electron charge density determines all of its properties, including the atomic charge, atomic energy, etc. Atoms or functional groups having similar charge density distributions must have similar properties. Thus, the degree of transferability of any two atoms can be assessed in a quantitative way by calculating a similarity measure between the corresponding charge distributions.11 Taking into account that similar atoms must make similar contributions to the molecular exchange-correlation density, the comparison of atomic hole densities appears also as a possibility for the quantitative comparison of atoms in molecules. Moreover, because atomic holes and molecular one-electron densities have the same analytical expressions (5), similarities between atomic hole densities can be calculated using exactly the same methodology and computer codes used for the calculation of similarities between molecular densities, including the possibility of computing approximate similarity measures at a low computational cost.15 An overlap-like similarity measure between two molecules, X and Y, is given by:12 Z ZXY = ρX (r)ρY (r) dr, (10) where ρX (r) and ρY (r) are the one-electron densities of the molecules X and Y to be compared. Recently, a similarity measure between atoms in molecules has been defined as11 Z (1) ZAB = ρX (r)ρY (r) dr, (11) AB
being A and B two atoms belonging to molecules X and Y, respectively. The definition of a similarity measure between two atomic holes is straightforward: Z (2) (12) ZAB = gA (r)gB (r) dr, gA (r) and gB (r) being the atomic holes of the two atoms, A and B, to be compared. Note that, in eq. (11), the integration is carried out only within the subset of space defined by the union of the atomic basins of atoms A and B, AB . On the contrary, the similarity measures defined in eqs. (10) and (12) are calculated by carrying out integrations throughout all space. The superscripts (1) and (2) , in eqs. (11) and (12), respectively, are used to stress that the (1) (2) ZAB and ZAB measures are ultimately derived from one and two-electron densities, respectively.
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ATOMIC TRANSFERABILITY The values of the integrals in eqs. (10), (11), and (12) depend on the relative spatial orientation of the densities being compared. Thus, for molecular similarity measures, the mutual alignment of molecules X and Y must be optimized to maximize the ZXY values;16 the same applies for similarity (1) measures between atomic densities (ZAB ) or atomic (2) holes (ZAB ). By computing the overlap of a density (1) with itself, a self-similarity measure, ZXX , ZAA or (2) ZAA is also obtained. Once a similarity measure and two self-similarities computed for a pair of densities, it is possible to define an euclidean distance index between these densities, which is equal to zero for identical densities and has no upper limit.12 For instance, using molecular similarity measures, one obtains p (13) DXY = ZXX + ZYY − 2ZXY . Analogous expressions can be obtained for (1) (2) atomic euclidean distances, DAB and DAB , by using the similarity measures defined in eqs. (11) and (12), respectively. COMPUTATIONAL DETAILS Molecular geometries for all the studied molecules have been completely optimized at the HF and CISD levels of theory by using the Gaussian21 and GAMESS packages,22 respectively. The 6-311+G(2d) basis set has been used for the N2 , CN− , and NO+ molecules, whereas the 6-311G∗∗ basis set for CO, H2 CO, and HCOOH.23 To obtain density difference maps, CISD calculations have been also performed on the HF optimized geometry for each molecule (CISD//HF). The Atomic Overlap Matrices for all the atoms have been obtained by means of the program PROAIM,24 and the density matrix elements have been computed for all the atomic holes considered, following eqs. (6) and (7). Density maps have been calculated on the plane defined by the molecular axis and an axis perpendicular to it, for the cylindrical molecules, or the plane defined by the nuclei, for the H2 CO and HCOOH molecules. For each atom, similarities between the atomic holes obtained at the HF, CISD//HF, and CISD levels of theory have been calculated. Also, similarities between different atomic holes have been calculated at each level of theory. For the calculation of similarities between atomic holes, the alignment has been performed by matching the nuclei of the two atoms to be compared, while keeping the molecular axis of the two molecules coincident. Within these restrictions, there are
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two possible alignments for each pair of atoms. We decided to keep always the orientation having more chemical sense, that is, placing the nonmatching nuclei as close as possible, although, for some pairs, (2) this does not lead to the maximal ZAB value. How(2) ever, in this cases the differences in ZAB obtained for the two orientations are always very small, and the choice of one orientation or the other has no practical consequences on the subsequent discussion of the results.
Results and Discussion First of all, some representative point Fermi holes for the N2 molecule at the HF level are discussed and compared to the atomic hole obtained for one of the N atoms. Differences between HF and CISD//HF calculations are also presented for point and atomic Fermi holes. Then, the transferability of the atomic holes corresponding to the N atom in the N2 , CN− , and NO+ series, and to the carbonyl O in the CO, H2 CO, and HCOOH series are discussed. For both series, similarities between HF, CISD//HF, and CISD atomic holes are also discussed, as well as molecular similarities based on one-electron densities. HOLE DENSITIES FOR THE N2 MOLECULE The N2 molecule will be taken as an example to make a qualitative comparison between point and atomic Fermi or exchange-correlation holes. Figure 1a, b, and c shows three point Fermi holes for N2 , calculated at the HF level, for the reference electron being fixed at three different locations. In Figure 1a, where the reference electron is located at the bond critical point (bcp) between the two N atoms, the Fermi hole density is symmetrically spread within the bond region between the two atoms. On the other hand, in Figure 1b, the reference electron is located at the position of one of the N nuclei. In this case, the Fermi hole density is highly localized in a small region around the nucleus. Finally, in Figure 1c, the reference point is located 0.83 a.u. below the position of a N nucleus on the molecular axis, which is the position of maximal accumulation of lone-pair electron density, according to the Laplacian of the one-electron density. The hole density becomes now localized mainly in this lone-pair region. Figure 1d, e, f shows the differences between the Fermi holes calculated at the HF and CISD//HF levels of theory, for the reference electron being at
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FIGURE 1. Fermi holes at the HF level and differences between HF Fermi holes and CISD//HF exchange correlation holes for N2 , for different positions of the reference electron. The position of the reference electron in each map is marked with a star. HF Fermi holes are depicted with contours in steps of 0.05 a.u., and difference maps with steps of 0.001 · 2n a.u., n = 1, 2, . . . . (a) N2 (HF); reference electron located at the bond critical point; (b) N2 (HF); reference electron located at one N nucleus; (c) N2 (HF); reference electron located in the lone pair region of a N atom; (d) N2 (HF–CISD//HF); reference electron located at the bond critical point; (e) N2 (HF–CISD//HF); reference electron located at one N nucleus; (f) N2 (HF–CISD//HF); reference electron located in the lone pair region of a N atom.
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ATOMIC TRANSFERABILITY the bcp, the N nucleus and the lone-pair region, respectively. At the CISD//HF level, the minimum of the Laplacian of the one-electron density in the lone-pair region is 0.88 a.u. below the N atom; however, the CISD//HF hole used in the difference map in Figure 1f has been calculated with the reference electron located 0.83 a.u. below the N atom, as in the HF calculation. The (HF-CISD//HF) difference maps show the effect of adding Coulomb correlation on the hole densities: positive and negative values are found in regions where CISD accumulates more or less hole density, respectively, with respect to HF (for a proper interpretation of difference maps, it must be taken into account that hole density functions exhibit negative values throughout all space). For all the maps, CISD adds hole density around the position of the reference electron, in agreement with the fact that the probability of having two electrons of any spin close in space is decreased. Because the Coulomb hole integrates to zero over all space, the regions of accumulation of hole density are always compensated by regions of hole density depletion. The sizes and shapes of the zones of hole density accumulation and depletion are different for the three reference positions considered, and depend on the degree of localization of the Fermi hole at each reference point. For the reference electron at the N nucleus (Fig. 1e), the difference map also reveals a shell structure for the N atom. The atomic hole for a N atom in N2 at the HF level (Fig. 2a) shows that the atomic hole density is localized mainly within the basin of the reference atom; however, there is also a significant accumulation of hole density in the basin of the neighbor atom. By comparing the N atomic hole to the reference-fixed Fermi holes in Figure 1, and taking into account that different normalization conventions are used for the two kinds of functions, one realizes that the atomic hole is approximately an average of the three reference-fixed Fermi holes. However, the atomic hole is able to display in a single plot relevant information about the localization of the electron density for an atom in a molecule without the need to select representative locations for a reference electron. TRANSFERABILITY OF N WITHIN THE N2 , CN− , AND NO+ SERIES Figure 2 gathers pictures of the atomic holes for the N atoms in the N2 , CN− , and NO+ molecules, calculated at the HF level. For a clearer discussion of the atomic hole densities, these atoms will be referred to as N(N2 ), N(CN− ), and N(NO+ ). For all the
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atoms, there is a maximal accumulation of atomic hole density within the basin of the reference atom, although the three atomic hole densities also spread into the basin of the neighboring atom. However, there are significant differences in the shapes and sizes of the three atomic holes. For instance, the electron populations assigned to the N atomic holes in the N2 , NO+ , and CN− molecules, are −7.00, −8.47, and −5.23, respectively. Note that, because of the way in which the atomic hole has been defined, the population of an atomic hole corresponds exactly to the negative of the Bader’s electron population of the atom. Difference maps between HF and CISD//HF holes (Fig. 2d, e, and f) show the same trends for all the N atoms: the effect of adding Coulomb correlation is to increase the amount of hole density localized within the basin of the reference atom (positive values in the difference maps) and decrease the hole density delocalized into the basin of the neighboring atom (negative values in the difference maps). Similarity, matrices for the N(N2 ), N(CN− ), and N(NO+ ) atomic holes, calculated at the HF, CISD//HF and CISD levels, are reported in Table I. The relative ordering of the three atomic holes at the HF and CISD levels of theory is depicted in Scheme 1, where the N(N2 ) atomic holes calculated at the two levels of theory have been superimposed arbitrarily. CISD//HF results, not presented in Scheme 1, are equivalent to the CISD ones. As shown visually in Scheme 1, when using the HF method, the atomic holes are ordered as N(CN− ) < N(NO+ ) < N(N2 ), while, at the CISD//HF and CISD levels, they are ordered as N(CN− ) < N(N2 ) < N(NO+ ). The similarity values obtained at the CISD//HF and CISD levels are very close (see Table I), showing that allowing the molecule to relax from the HF to the CISD-optimized geometries has a minor effect on the atomic hole densities. The reasons for the differences between the HF and the correlated similarity matrices have to be found mainly in the bad description of the N–N interaction at the HF level; which largely overestimates the electron delocalization between the two N atoms. Thus, at this level of theory, the N(N2 ) hole appears to be quite different to that of the N(CN− ) and N(NO+ ). Such an overestimation is corrected at the CISD level, so one finds the expected result of the N(CN− ) and N(NO+ ) holes being the most different. Finally, it is also worth saying that the euclidean distances obtained at the CISD//HF and CISD levels are systematically lower than the HFs. This is a consequence of the reduction of charge transfer and electron delocalization at the
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FIGURE 2. Atomic holes for the N(N2 ), N(CN− ), and N(NO+ ) atoms at the HF level, in contours of 0.05 a.u., and differences between HF and CISD//HF holes, in contours of 0.001 · 2n a.u., n = 1, 2, . . . . (a) N(N2 ) HF; (b) N(CN− ) HF; (c) N(NO+ ) HF; (d) N(N2 ) (HF–CISD//HF); (e) N(CN− ) (HF–CISD//HF); (f) N(NO+ ) (HF–CISD//HF).
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ATOMIC TRANSFERABILITY TABLE I. Similarity Matrices for the N Atomic Holes in the (N2 , CN− , NO+ ) Series, at the HF, CISD//HF, and CISD Levels of Theory. N(N2 )
N(CN− )
N(NO+ )
I.a HF N(N2 ) N(CN− ) N(NO+ )
50.1187 50.6695 50.3025
0.9009 52.0318 51.3616
0.6752 0.5007 50.9421
I.b CISD//HF N(N2 ) N(CN− ) N(NO+ )
51.9439 52.1037 51.6959
0.3372 52.3772 51.8868
0.3766 0.4396 51.5897
I.c CISD N(N2 ) N(CN− ) N(NO+ )
51.9363 52.0852 51.7320
0.3366 52.3474 51.9212
0.3712 0.4128 51.6654
(2)
Similarity measures (ZAB ), in roman type, are in the lower tri(2)
angular matrix; euclidean distances (DAB ), in italics, are in the (2)
upper triangular matrix; self-similarities (ZAA ), in bold type, are in the diagonal.
CISD level, with respect to HF, which makes all the atoms more transferable between them. Solà et al.17 recently proposed the use of molecular self-similarities as a means of analyzing electron density distributions. They found that molecular self-similarities depend not only on the number of electrons of the atom or molecule studied, but also on how the electron density of the system studied is locally concentrated. For instance, for several isoelectronic series, a good correlation was found between molecular volumes and self-similarity measures. The same kind of analysis can be carried out here for the atomic hole densities. It is to be (2) expected that ZAA self-similarities will also reflect the local concentration of atomic hole density functions. Interestingly, at the HF level, atomic hole selfsimilarities are ordered as N(CN− ) > N(NO+ ) > N(N2 ), while, at the CISD//HF and CISD levels,
they are ordered as N(CN− ) > N(N2 ) > N(NO+ ), changing the relative position of N(N2 ) and N(NO+ ) (see Table I). The low self-similarity value obtained for N(N2 ) at the HF level is due again to the overestimation of the electron delocalization between the two atoms in N2 . Difference maps between the HF and CISD//HF atomic holes for the N atoms in the three molecules considered show the same trends: CISD accumulates more hole density in the basin of the reference atom, especially at the position of the nucleus, and removes it from the second atom. The differences between molecular densities or between atomic holes calculated at different levels of theory can also be assessed by means of similarity measures (see Table II). Distance indices between HF and CISD//HF atomic holes are 0.73, 0.11, and 0.19 for N(N2 ), N(CN− ), and N(NO+ ). The same results, within two digits, are obtained for the comparison of HF and CISD atomic holes, showing that the effect of allowing nuclear relaxation on the atomic holes is very low (see Table II). Table II also reports similarity measures between molecular densities calculated at different levels of theory. The HF and CISD//HF molecular one-electron densities are very similar for all the molecules, with DXY values between 0.03 and 0.04. Molecular similarities between one-electron densities are known to be extremely sensitive to slight nuclear rearrangements.16 This is reflected in the DXY values obtained when comparing HF and CISD densities, which are 2.15, 1.11, and 2.52, for N2 , CN− , and NO+ , respectively. According to the DXY values between HF and CISD//HF molecular densities, which are between TABLE II. Similarities and Euclidean Distances between HF and CISD//HF and between HF and CISD, for the Atomic (2) (2) Hole Densities of the N Atoms (ZAB , DAB ) and for the Molecular One-Electron Densities (ZXY , DXY ), in the (N2 , CN− , NO+ ) Series. (HF–CISD//HF) Atoms N(N2 ) N(CN− ) C(NO+ ) Molecules
SCHEME 1.
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N2 CN− NO+
(2) ZAB
50.7643 52.1981 51.2479
(2) DAB
0.7308 0.1129 0.1899
(HF–CISD) (2) ZAB
50.7634 52.1833 51.2852
(2)
DAB 0.7269 0.1126 0.1929
ZXY
DXY
ZXY
DXY
105.3975 84.2583 134.4295
0.0350 0.0300 0.0418
103.0597 83.6194 131.1939
2.1451 1.1138 2.5238
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FRADERA, DURAN, AND MESTRES 0.03 and 0.04 for the three molecules, consideration of Coulomb correlation leads to analogous electronic redistributions in N2 , CN− , and NO+ (see Table II). Furthermore, from the point of view of the N atomic holes, N(N2 ) suffers the greatest electronic redistribution upon inclusion of Coulomb correlation. Thus, when comparing the HF and CISD//HF (2) hole densities for N(N2 ), a DAB value of 0.73 is obtained, compared to 0.11 and 0.19 for N(CN− ) and N(NO+ ), respectively. These results show that the effect of Coulomb correlation can be different for molecular charge densities and for atomic holes. The fact that N2 is a diatomic homonuclear molecule means that the one-electron atomic selfsimilar(1) ity ZAA is exactly equivalent to half the molecular selfsimilarity, ZXX . This will allow to compare the (1) (2) values of the ZAA and ZAA atomic self-similarities for the N atom. The same applies to the comparisons between N(N2 ) atoms calculated at the HF and CISD//HF levels of theory. Altogether, these comparisons will show that different results can be obtained when using atomic similarity measures based on the one-electron density or the exchange(1) (2) correlation density. Table III collects ZAA and ZAA values for N(N2 ) at the HF, CISD//HF, and CISD (2) levels of theory. At the three levels of theory, ZAA is (1) smaller than ZAA . Moreover, for the HF–CISD//HF (2) (1) similarity, ZAB is also smaller than ZAB . These results are consistent with the fact that atomic hole densities are partially spread onto the basins of neighbor atoms, in contrast to the one-electron atomic densities, which cannot extend beyond interatomic boundaries.
TABLE III. Atomic Self-Similarities for the N Atom in the N2 Molecule. Self-Similarities
(1)
(2)
ZAA
ZAA
HF CISD//HF CISD
52.6616 52.6936 52.6574
50.1187 51.9439 51.9363
Similarity
ZAB
ZAB
52.6987
50.7643
HF–CISD//HF
(1)
(2)
The self-similarity values corresponding to the atomic hole (2) (1) (ZAA ) and to the atomic one-electron density (ZAA ) are presented, at the HF, CISD//HF, and CISD levels of theory. The (1) (2) similarity values (ZAB , ZAB ) between the N atoms at the HF and CISD//HF level are reported also.
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TRANSFERABILITY OF THE CARBONYL O WITHIN THE CO, H2 CO, AND HCOOH SERIES Figure 3 depicts the atomic holes corresponding to the carbonyl O atoms in the CO, H2 CO, and HCOOH series, at the HF level of theory, as well as difference maps between the atomic holes calculated using the HF and CISD approximations. From now on, the atoms in this series will be denoted as O(CO), O(H2 CO), and O(HCOOH). As expected, all maps show a maximum accumulation of hole density within the basin of the O atom, with some delocalization towards the C atom. For O(H2 CO) and O(HCOOH), the delocalization towards H atoms or OH groups is very low. In fact, only the hydroxyl O in HCOOH appears to have a significant accumulation of hole density, which is not apparent for the H atoms in any molecule (Fig. 3b and c). Difference maps between atomic holes calculated using the HF and CISD approximations also show the same trends for the three atoms in the series. Thus, upon consideration of Coulomb correlation, there is a buildup of hole density in a small region around the O nucleus (positive values in the difference maps) and a depletion of hole density outside these regions (negative values in the difference maps). However, for O(CO), CISD adds hole density also in a small region corresponding approximately to the valence shell of the C atom. Atomic hole similarity matrices at the HF, CISD//HF, and CISD levels of theory are presented in Table IV. According to the euclidean distance values, O(H2 CO) and O(HCOOH) are highly trans(2) ferable between them (DAB = 0.056), at the HF level of theory. In contrast, O(CO) appears to be (2) less transferable, as indicated by its DAB values (2) when compared to O(H2 CO) (DAB = 0.237) and (2) O(HCOOH) (DAB = 0.206). These values are in agreement with the different hybridizations of the C atom in CO, in one hand, and H2 CO and HCOOH, in the other hand. Note, also that the O carbonyl atomic holes in this series are more transferable than the N atomic holes in the previous series, calculated at the same level of theory. The higher transferability within this second series reflects the fact that the O(CO), O(H2 CO), and O(HCOOH) atoms are bonded to the same atom (C), and share a common molecular environment, except for the differences due to second-neighbor atoms. In contrast, each of the N(N2 ), N(CN− ), and N(NO+ ) atoms is bonded to a different atom, leading to a higher variability of the molecular environment of the N atom in this series.
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ATOMIC TRANSFERABILITY
FIGURE 3. Atomic holes for the O(CO), O(H2 CO), and O(HCOOH) carbonyl oxygens at the HF level, in contours of 0.1 a.u., and differences between HF and CISD//HF holes, in contours of 0.001 · 2n a.u., n = 1, 2 . . . . (a) CO(CO) HF; (b) CO(H2 CO) HF; (c) CO(HCOOH) HF; (d) CO(CO) (HF–CISD//HF); (e) CO(H2 CO) (HF–CISD//HF); (f) CO(HCOOH) (HF–CISD//HF).
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FRADERA, DURAN, AND MESTRES TABLE IV.
TABLE V.
Similarity Matrices for the O Holes in the CO, H2 CO, and HCOOH Series.
Similarities and Euclidean Distances between HF and CISD//HF and CISD, for the Atomic Hole Densities of (2) (2) the Carbonyl O (ZAB , DAB ) and for the Molecular One-Electron Densities (ZXY , DXY ), in the CO, H2 CO, and HCOOH Series.
O(CO)
O(H2 CO)
O(HCOOH)
IV.a HF O(CO) O(H2 CO) O(HCOOH)
80.8384 80.7313 80.7469
0.2370 80.6804 80.6876
0.2063 0.0557 80.6979
IV.b CISD//HF O(CO) O(H2 CO) O(HCOOH)
81.2216 81.1943 81.1881
0.2052 81.2090 81.1961
0.1763 0.0486 81.1855
IV.c CISD O(CO) O(H2 CO) O(HCOOH)
81.5093 81.3301 81.2090
0.2002 81.1910 81.1855
0.1676 0.0503 81.1721
Plain values, in the lower triangle, refer to similarity measures (2) (ZAB ); values in italics, in the upper triangle refer to euclidean (2)
distances (DAB ), values in bold, in the diagonal, refer to selfsimilarities
(2) (ZAA ).
The transferability of the carbonyl O atomic holes is preserved at the HF, CISD//HF, and CISD levels of theory (see Table IV). Indeed, O atomic holes are found to be slightly more transferable at the CISD//HF and CISD levels of theory. As for the self-similarity values, they increase at the CISD//HF and CISD levels of theory, with respect to the HF results. Thus, Coulomb correlation increases the concentration of atomic hole density into the reference atoms. Both trends agree with those found above for the N atomic holes in the first series of molecules. Finally, we consider the quantitative comparison between O atomic holes calculated using different approximations. Table V gathers the similarity measures and indices between O atomic holes calculated at different levels of theory. As shown by the difference maps in Figure 3, the inclusion of electron correlation has the same effect on the three atomic (2) holes. Accordingly, the DAB values obtained after comparison of HF and CISD//HF results are very close for the three atomic holes: 0.102, 0.097, and 0.087, for O(CO), O(H2 CO), and O(HCOOH), respectively. Geometry relaxation at the CISD level only affects meaningfully the O(CO) atomic hole, (2) for which the DAB value between the HF and CISD hole densities is 0.136, while the corresponding (2) DAB values for O(H2 CO) and O(HCOOH) are 0.096
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(HF–CISD//HF) Atoms O(CO) O(H2 CO) O(HCOOH) Molecules CO H2 CO HCOOH
(2) ZAB
81.0248 80.9401 80.9380
(2) DAB
0.1019 0.0965 0.0868
(HF–CISD) (2) ZAB
81.1646 80.9311 80.9313
(2)
DAB 0.1357 0.0957 0.0857
ZXY
DXY
ZXY
DXY
113.4055 113.1643 194.4723
0.0337 0.0307 0.0340
113.5865 112.3833 191.3265
1.2598 1.2232 2.4826
and 0.087, very close to the results obtained for the comparison of the HF and CISD//HF atomic holes. Table V also collects the similarities between the CO, H2 CO, and HCOOH molecular densities calculated at different theoretical levels. Again, molecular similarity measures between one-electron densities appear to be very dependent on nuclear rearrangements, rather than on electronic redistributions. Thus, DXY values for the comparison between HF and CISD//HF results are very low for all the molecules (ca. 0.03), while the values corresponding to the comparison of HF and CISD results are much higher (1.26, 1.22, and 2.48 for CO, H2 CO, and HCOOH, respectively).
Conclusions The definition of an atomic hole density function allows for the separation of the exchangecorrelation density into atomic and biatomic contributions, following the theory of Atoms in Molecules. The shape of an atomic hole for an atom or functional group is related to the way in which the electron density of that atom or functional group is localized. In particular, electron delocalization between an atom and its neighbors is made evident on its atomic hole. In general, for a given atom or functional group in different molecular environments, the shape of the atomic hole is partially preserved, depending on the degree of transferability of the atom within the different molecules.
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ATOMIC TRANSFERABILITY Similarity measures appear to be a convenient tool for the quantitive comparison of atomic holes. For instance, the degree of transferability of an atom in different molecules can be easily quantified. Also, for a given atom, atomic holes calculated at different levels of theory can be compared to assess the effect of introducing electron correlation. An additional advantage is that similarities between atomic holes (2) (ZAB ) are not heavily dependent on slight geometrical rearrangements, in contrast to the similarity measures between molecular one-electron densities (ZXY ). Thus, they allow one to focus on genuine two-electron effects, rather than on geometrical rearrangements. As a practical application, the transferability of N in the N2 , CN− , and NO+ series, and of the CO functional group in the CO, H2 CO, and HCOOH series have been studied, using HF and CISD wave functions. It has been found that atomic holes calculated at the HF, CISD//HF, and CISD levels of theory are quite similar, although CISD correlation tends to concentrate more electron hole density around the nucleus, with respect to HF. Similarity measures between atomic holes show that the N and O atoms are highly transferable, within the two series studied. Atomic transferability is not decreased by the addition of Coulomb correlation. On the contrary, atomic holes appear to be more transferable at the CISD//HF and CISD levels than at the HF level of theory. Moreover, Coulomb correlation has been shown to be necessary for a correct ordering of the N atomic holes in the first series. The transferability of the exchange-correlation density must be parallel to that of the electrondensity itself. Thus, it is expected that qualitatively equivalent results could be obtained from the calculation of similarities between atoms in molecules defined as the overlap of the one-electron densities corresponding to two atomic basins. More work on this subject, including a comparison of atomic similarity measures obtained from the one-electron and the exchange-correlation density, is under way in our laboratory.
Acknowledgments X.F. benefits from a doctoral fellowship from the University of Girona.
References 1. Lewis, G. N. J Am Chem Soc 1916, 38, 762. 2. McWeeny, R. Rev Mod Phys 1960, 32, 335.
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3. Cooper, I. L.; Pounder, C. N. M. Theor Chim Acta 1973, 47, 51; Doggett, G. Mol Phys 1977, 34, 1739; Luken, W. L.; Beratan, D. N. Theor Chim Acta 1982, 61, 265; Luken, W. L. Croat Chem Acta 1984, 57, 1283. 4. Bader, R. F. W.; Stephens, M. E. J Am Chem Soc 1975, 97, 7391; Bader, R. F. W.; Streitwieser, A.; Neuhaus, A.; Laidig, K. E.; Speers, P. J Am Chem Soc 1996, 118, 4959; Bader, R. F. W.; Johnson, S.; Tang, T.-H. J Phys Chem 1996, 100, 15398; Gillespie, R. J.; Bayles, D.; Platts, J.; Heard, G. L.; Bader, R. F. W. J Phys Chem A 1996, 102, 3407; Bader, R. F. W.; Heard, G. L. J Chem Phys 1999, 111, 8789. 5. Bader, R. F. W. Atoms in Molecules: A Quantum Theory; Claredon: Oxford, 1990. 6. Ponec, R.; Uhlik, F. J Mol Struc (Theochem) 1997, 391, 159. 7. Fradera, X.; Austen, M. A.; Bader, R. J Phys Chem A 1999, 103, 304. 8. Ponec, R. J Math Chem 1997, 21, 323; Ponec, R. J Math Chem 1998, 23, 85; Ponec, R.; Duben, A. J. J Comput Chem 1999, 20, 760. 9. Chang, C.; Bader, R. F. W. J Phys Chem 1992, 96, 1654; Bader, R. F. W.; Popelier, P. L. A.; Keith, T. A. Angew Chem Int Ed Eng 1994, 33, 620; Popelier, P. L. A.; Bader, R. F. W. J Phys Chem 1994, 98, 4473. 10. Fulton, R. L.; Mixon, S. T. J Phys Chem 1995, 99, 9768; Fulton, R. L.; Perhacs, P. J Phys Chem 1998, 102, 8988; Fulton, R. L.; Perhacs, P. J Phys Chem 1998, 102, 9001. 11. Cioslowski, J.; Nanayakkara, A. J Am Chem Soc 1993, 115, 11213; Cioslowski, J.; Stefanov, B.; Constans, P. J Comput Chem 1996, 17, 1352; Stefanov, B. E.; Cioslowski, J. Can J Chem 1996, 74, 1263. 12. Carbó, R.; Leyda, L.; Arnau, M. Int J Quantum Chem 1980, 17, 1185. 13. For post-HF wavefunctions, it is usually found that h(r1 ; r1 ) < −ρ(r1 )/2, because the probability of coalescence of two electrons of different spin, 0 αβ (r1 , r1 ) + 0 βα (r1 , r1 ), is lower than at the HF level, where this probability is given by a product of first order density functions. 14. Within the HF approximation, it is easy to check that R f (r1 , r2 ) dr2 = ρ(r1 ). Since the integration of the Coulomb hole over all space is equal to zero, this identity holds also for post-HF wavefunctions. 15. Mestres, J.; Solà, M.; Duran, M.; Carbó, R. J Comput Chem 1994, 15, 1113; Constans, P.; Carbó, R. J Chem Inf Comput Sci 1995, 35, 1046. 16. Constans, P.; Amat, L.; Carbó, R. J Comput Chem 1997, 18, 826. 17. Solà, M.; Mestres, J.; Oliva, J. M.; Duran, M.; Carbó, R. Int J Quantum Chem 1996, 58, 361. 18. Solà, M.; Mestres, J.; Carbó, R. Duran, M. J Chem Phys 1996, 104, 636. 19. Boyd, R. J.; Wang, L.-C. J Comput Chem 1989, 10, 367. 20. Wiberg, K. B.; Hadad, C. M.; LePage, T. J.; Breneman, C. M.; Frisch, M. J. J Phys Chem 1992, 96, 671. 21. Frisch, M. J.; Trucks, G. W.; Schlegel, H. B.; Gill, P. M. W.; Johnson, B. G.; Robb, M. A.; Cheeseman, J. R.; Keith, T.; Petersson, G. A.; Montgomery, J. A.; Raghavachari, K.; AlLaham, M. A.; Zakrzewski, V. G.; Ortiz, J. V.; Foresman, J. B.; Peng, C. Y.; Ayala, P. Y.; Chen, W.; Wong, M. W.; Andrés, J. L.; Replogle, E. S.; Gomperts, R.; Martin, R. L.; Fox, D. J.; Binkley, J. S.; Defrees, D. J.; Baker, J.; Stewart, J. P.; Head–Gordon,
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FRADERA, DURAN, AND MESTRES M.; Gonzalez, C.; Pople, J. A. Gaussian 94; Gaussian, Inc.: Pittsburgh, PA, 1995. 22. Schmidt, M. W.; Baldridge, K. K.; Boate, J. A.; Elbert, S. T.; Gordon, M. S.; Jensen, J. H.; Koseki, S.; Matsunaga, N.; Nguyen, K. A.; Su, S. S.; Windus, T. L.; Dupuis, M.; Montgomery, J. A. J Comput Chem 1993, 14, 1347.
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23. Krishan, R.; Binkley, J. S.; Seeger, R.; Pople, J. A. J Chem Phys 1980, 72, 650; Clark, T.; Chandreshakar, J.; Spitznagel, G. W.; von Schleyer, P. J Comput Chem 1983, 4, 294; Frisch, M. J.; Pople, J. A.; Binkley, J. S. J Chem Phys 1984, 80, 3265. 24. Biegler–König, F. W.; Bader, R. F. W.; Tang, T.-H. J Comput Chem 1982, 3, 317.
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Institute of Computational Chemistry, University of Girona, 17071 Girona, Catalonia, Spain Molecular Design & Informatics, N. V. Organon, 5340 BH OSS, The Netherlands
Received 19 April 2000; accepted 29 May 2000
ABSTRACT: Starting from either the exchange or the exchange-correlation density together with Bader’s definition of an atom in a molecule, an atomic hole density function can be defined. Contour maps of atomic hole density functions are able to show how the electron density of each atom in a molecule is partially delocalized into the rest of atoms in the molecule. The degree of delocalization of the atomic density ultimately depends on the nature of the atom studied and its environment. Atomic hole density functions are also used to define an atomic similarity measure, which allows for the quantitative assessment of the degree of atomic transferability in different molecular environments. In this article, contour maps for the N atom in the (N2 , CN− , NO+ ) series and the O atom in the (CO, H2 CO, and HCOOH) series are presented at the Hartree–Fock and CISD levels of theory. Moreover, the transferability of N and O within the two series is studied c 2000 John Wiley & Sons, Inc. by means of atomic similarity measures. J Comput Chem 21: 1361–1374, 2000 Keywords: molecular similarity; atomic transferability; Fermi hole; exchange-correlation hole; electron-pair density
Introduction
S
ince the introduction of the Lewis model in 1916,1 the idea that electrons in atoms and molecules are arranged in pairs, and that the atoms in a molecule can share pairs of electrons between them to form covalent bonds, have been fundamental concepts in chemistry. From a physical point of Correspondence to: J. Mestres; e-mail: [email protected]. akzonobel.nl Contract/grant sponsor: Spanish DGICYT; contract/grant number: PB98-0457-C02-01
view, the pairing of electrons is a consequence of the antisymmetry of the electronic wave function with respect to the interchange of the space-spin coordinates of any pair of electrons. Same-spin electrons are strongly correlated between them, and the probability that two electrons with the same spin are near in space is very low, thus allowing for the formation of localized pairs of electrons of different spin. Correlation between same-spin electrons is usually referred to as Fermi or exchange correlation, whereas correlation between electrons of different spin is called Coulomb correlation. Fermi correlation is basic for an accurate description of the electronic structure of atoms and mole-
Journal of Computational Chemistry, Vol. 21, No. 15, 1361–1374 (2000) c 2000 John Wiley & Sons, Inc.
FRADERA, DURAN, AND MESTRES cules. At the Hartree–Fock (HF) level, this is the only kind of electron correlation that is taken into account, while ab initio post-Hartree–Fock methods, such as Møller–Pleset (MP) or Configuration Interaction (CI), introduce also Coulomb correlation between electrons of different spin. In any case, the pairing and correlation of the electrons in an atom or molecule is determined by the exchange density, at the HF level, or exchange-correlation density, for more general wave functions, which accounts for the differences between the electron-pair density and the simple product of one-electron densities.2 The exchange-correlation density contains a complete description of the way in which the electrons in a system are mutually correlated. However, because of its dependence on the coordinates of two electrons, the exchange-correlation density is a sixdimensional function, which makes it difficult to visualize directly. Consequently, different approaches have been proposed to reduce the dimensionality of the original exchange-correlation density. A widely used approach is to define a reference electron, fixed at a given position in real space. The resulting threedimensional function is called a Fermi hole, or, more generally, exchange-correlation hole. Integration of the Fermi hole over all space yields −1, corresponding to the removal of one electron. A Fermi hole for a given reference position shows the instantaneous effect of an electron located at that point on the rest of electrons of the molecule. The analysis of Fermi holes in molecules has received considerable attention, and has been used to illustrate the localization of electrons in regions of real space.3, 4 An alternative approach for extracting quantitative information from the exchange-correlation density is the definition of localization and delocalization indices. Within Bader’s theory of Atoms in Molecules,5 atomic basins are defined as regions in real space delimited by zero-flux surfaces in the one-electron density.5 Thus, one can obtain a localization index for an atom by integrating the two electron coordinates of the exchange or exchangecorrelation density over the same atomic basin. Similarly, a delocalization index for a pair of atoms can be obtained by integrating each of the two electron coordinates over a different atomic basin.6, 7 Another possibility is to integrate only one of the electron coordinates over an atomic basin.8 This gives an atomic hole density that is a function of the second electron coordinate, with the property that it is not dependent on the particular position of a reference electron but on the choice of a reference atomic basin. Studies so far have focused on calculations of atomic hole densities at the semiempirical
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or Hartree–Fock levels of theory. The purpose of this article is not only to show that post-Hartree–Fock atomic hole densities can be computed efficiently and routinely, but also to analyze the important changes they exhibit when increasing the level of theory (e.g., CI with Singles and Doubles, CISD). Such an insight will be gained through the analysis of atomic holes for the N and O atoms in different molecular environments. Transferability is a fundamental concept in the theory of Atoms in Molecules,5 that is, atoms or functional groups with similar properties should have similar electron densities.9 Therefore, the definition of a similarity measure between atoms or functional groups provides a means for assessing quantitatively the degree of transferability of those atoms or functional groups in molecules. Similarity measures have been already succesfully applied to assess the degree of transferability of atoms in molecules from the point of view of the one-electron density.11 In this article, the degree of transferability of atoms in molecules will be analyzed from the point of view of the exchange-correlation density. In particular, similarities between the atomic holes of the N atom in the CN− , N2 , and NO+ series of molecules and the carbonyl O atom in the CO, H2 CO, and HCOOH series of molecules will be studied at the HF and CISD levels of theory. The effect of Fermi and Coulomb correlation on the atomic holes, and thus, on the degree of transferability of atoms along those molecular series, will be also discussed.
Methodology EXCHANGE-CORRELATION AND HOLE-DENSITY FUNCTIONS At any level of theory, a function describing how the electrons are mutually correlated in atoms or molecules can be defined from the one-electron and electron-pair densities as, f (r1 , r2 ) = 20(r1 , r2 ) − ρ(r1 )ρ(r2 ); Z f (r1 , r2 ) dr1 dr2 = −N,
(1)
where ρ(r) and 0(r1 , r2 ) are the first- and secondorder density functions, respectively, and N is the number of electrons in the system. As stated above, there is no correlation between electrons with antiparallel spin within the HF approximation. At this level of theory, f (r1 , r2 ) is the Fermi or exchange density. By fixing one of the two electron coordinates, r1 , and dividing by the value of ρ(r) at that point, the
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ATOMIC TRANSFERABILITY Fermi or exchange hole can be defined as,2 h(r1 ; r2 ) =
20(r1 , r2 ) f (r1 , r2 ) = − ρ(r2 ); ρ(r1 ) ρ(r1 ) Z h(r1 ; r2 ) dr2 = −1. (2)
When necessary, we will refer to this function as the point Fermi hole, to distinguish it from the atomic Fermi hole, to be described later. h(r1 ; r2 ) shows how the presence of the reference electron at r1 lowers the probability that a second electron with the same spin be in a region nearby. Within the restricted HF approximation, it holds that h(r1 ; r1 ) = −ρ(r1 )/2.13 Usually, Fermi holes are defined for either α or β electrons only, and then hαα (r1 ; r1 ) = ρ α (r1 ) an similarly for β electrons, showing that the electron exclusion at the reference point is complete for same-spin electrons. The Fermi hole and related functions have been employed to visualize how the exchange correlation between same spin electrons determines the distribution of electrons in real space.3, 4 Plots of the Fermi hole for fixed positions of the reference electron show in which regions the exchange density is highly localized, or where it is delocalized over a large region of space. It is normally found that the exchange density is highly localized in regions occupied by core electrons, and delocalized in bonding regions. Although molecular Fermi holes have been mainly investigated within the HF approximation, the definitions in eqs. (1) and (2) are completely general, and can be applied to any methodology for which the one-electron and electron-pair densities can be obtained. In the general case, when both Fermi and Coulomb correlation are taken into account, the exchange density becomes the exchangecorrelation density, and the Fermi hole becomes the exchange-correlation hole. Because f (r1 , r2 ) is a two-electron function, it can be partitioned into monoatomic and biatomic contributions, X X fA (r1 , r2 ) + fA,B (r1 , r2 ), (3) f (r1 , r2 ) = A
AB
where A and B are atomic basins as defined in the theory of Atoms in Molecules; that is, regions in real space delimited by zero-flux surfaces in the one-electron density.5 The monoatomic terms reflect how much the electrons are localized into atoms, and the biatomic terms account for the delocalization or sharing of electrons between pairs of atoms. By integrating the fA (r1 , r2 ) and fA,B (r1 , r2 ) terms over the two-electron coordinates, one obtains atomic localization and delocalization indices, respectively.7
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A second possibility for the study of electronic localization in atoms and molecules, recently proposed by R. Ponec,8 is to integrate the exchange or exchange-correlation density over one-electron coordinate within a reference atomic basin, Z f (r1 , r2 ) dr2 ; gA (r) ≡ gA (r1 ) = A Z gA (r) dr = −NA . (4) By analogy with the point Fermi hole function described above, Ponec calls gA (r) a charge-weighted Fermi hole. It is a three-dimensional function, which integrates to −NA , the atomic population of atom A, and shows how the electron density of a given atom excludes NA electrons, in the same way that a Fermi or exchange-correlation hole excludes one electron. Thus, the gA (r) function can be considered as an atomic Fermi or exchange correlation hole, or atomic hole, for short, for the atom A. Also, starting from the one-electron density matrix, instead of the pair-density function, Fulton et al. have defined basin–basin and basin–point sharing indices.10 The basin–point sharing index is, in fact, a function depending on the space-spin coordinates of an electron. At the HF level, the basin–basin sharing index is equivalent to the delocalization index defined in ref. 7, and the basin–point sharing index is equivalent to the atomic-hole density function, gA (r). However, for post-HF methods this equivalence is not preserved. An interesting property of gA (r) is that, in a LCAO framework, it can be expressed in closed analytical form as X ∗ PA (5) gA (r) = µν ϕµ (r)ϕν (r), µν
where ϕµ (r) are basis functions, and PA µν are obtained as: X (2Pµνλσ − Pµν Pλσ )SA (6) PA µν = λσ . λσ
Pµν and Pµνλσ are the elements of the first and second-order density matrices of the molecule, respectively, and SA µν stands for the overlap of a pair of basis functions, µ and ν, within the basin of the atom A. Within the restricted HF approximation, eq. (6) can be simplified to yield: 1X (Pµσ Pνλ )SA (7) PA µν = − λσ . 2 λσ Following Bader’s definition of an atom in a molecule, the one-electron density of any molecule can be partitioned into a set of disjoint atomic
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FRADERA, DURAN, AND MESTRES fragments. Similarly, by using the definition of the charge-weighted or atomic hole in eq. (4), the exchange-correlation density can also be partitioned into a set of atomic holes. Taking into account that the integration of the exchange-correlation density over the electron coordinate r2 over all space is equivalent to the one-electron density at the point r1 ,14 Z (8) f (r1 , r2 ) dr2 = −ρ(r1 ), one obtains that: ρ(r) =
X A
ρA (r) = −
X
gA (r).
(9)
A
Although the definitions of both ρA (r) and gA (r) fragments involve zero-flux surfaces in the one-electron density, there is an important difference between them. Each ρA (r) density fragment has nonzero values only within its own atomic basin, i.e., is limited in space. Each atomic basin is separated from its neighbors by closed boundaries, coincident with the zero-flux surfaces in the one-electron density. On the contrary, each gA (r) density is extended over all space, and every pair of atomic holes in a molecule has some degree of overlap. From a computational point of view, gA (r) has the advantage that it has an analytical LCAO expression analogous to that of the one-electron density, eq. (5). However, there is no such possibility for ρA (r): the fact that it has closed boundaries does not allow to express it as a linear combination of basis functions extended over all space. The Fermi or exchange-correlation hole density shows how the presence of an electron at the reference position, r1 , excludes a second electron, by diminishing the probability of finding another electron in a region nearby. Accordingly, an atomic hole shows how the one-electron density associated to a reference atom excludes the same amount of one-electron density. In both cases, the spatial localization of the corresponding hole density function will be related to the degree of localization of the reference electron or the one-electron density of the reference atom, respectively. The fact that atomic holes are not dependent on the position of a reference electron, makes them particularly suitable for studying the localization and delocalization of electrons from an atomic point of view. For any atom, a picture of the corresponding atomic hole will show how much of the electron density is localized in that atom, and how much is delocalized towards other atoms, the same information that can be obtained in a quantitative way by means of localization and delocalization indices.7
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SIMILARITY MEASURES BETWEEN ATOMIC HOLES For an atom in a molecule, the associated electron charge density determines all of its properties, including the atomic charge, atomic energy, etc. Atoms or functional groups having similar charge density distributions must have similar properties. Thus, the degree of transferability of any two atoms can be assessed in a quantitative way by calculating a similarity measure between the corresponding charge distributions.11 Taking into account that similar atoms must make similar contributions to the molecular exchange-correlation density, the comparison of atomic hole densities appears also as a possibility for the quantitative comparison of atoms in molecules. Moreover, because atomic holes and molecular one-electron densities have the same analytical expressions (5), similarities between atomic hole densities can be calculated using exactly the same methodology and computer codes used for the calculation of similarities between molecular densities, including the possibility of computing approximate similarity measures at a low computational cost.15 An overlap-like similarity measure between two molecules, X and Y, is given by:12 Z ZXY = ρX (r)ρY (r) dr, (10) where ρX (r) and ρY (r) are the one-electron densities of the molecules X and Y to be compared. Recently, a similarity measure between atoms in molecules has been defined as11 Z (1) ZAB = ρX (r)ρY (r) dr, (11) AB
being A and B two atoms belonging to molecules X and Y, respectively. The definition of a similarity measure between two atomic holes is straightforward: Z (2) (12) ZAB = gA (r)gB (r) dr, gA (r) and gB (r) being the atomic holes of the two atoms, A and B, to be compared. Note that, in eq. (11), the integration is carried out only within the subset of space defined by the union of the atomic basins of atoms A and B, AB . On the contrary, the similarity measures defined in eqs. (10) and (12) are calculated by carrying out integrations throughout all space. The superscripts (1) and (2) , in eqs. (11) and (12), respectively, are used to stress that the (1) (2) ZAB and ZAB measures are ultimately derived from one and two-electron densities, respectively.
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ATOMIC TRANSFERABILITY The values of the integrals in eqs. (10), (11), and (12) depend on the relative spatial orientation of the densities being compared. Thus, for molecular similarity measures, the mutual alignment of molecules X and Y must be optimized to maximize the ZXY values;16 the same applies for similarity (1) measures between atomic densities (ZAB ) or atomic (2) holes (ZAB ). By computing the overlap of a density (1) with itself, a self-similarity measure, ZXX , ZAA or (2) ZAA is also obtained. Once a similarity measure and two self-similarities computed for a pair of densities, it is possible to define an euclidean distance index between these densities, which is equal to zero for identical densities and has no upper limit.12 For instance, using molecular similarity measures, one obtains p (13) DXY = ZXX + ZYY − 2ZXY . Analogous expressions can be obtained for (1) (2) atomic euclidean distances, DAB and DAB , by using the similarity measures defined in eqs. (11) and (12), respectively. COMPUTATIONAL DETAILS Molecular geometries for all the studied molecules have been completely optimized at the HF and CISD levels of theory by using the Gaussian21 and GAMESS packages,22 respectively. The 6-311+G(2d) basis set has been used for the N2 , CN− , and NO+ molecules, whereas the 6-311G∗∗ basis set for CO, H2 CO, and HCOOH.23 To obtain density difference maps, CISD calculations have been also performed on the HF optimized geometry for each molecule (CISD//HF). The Atomic Overlap Matrices for all the atoms have been obtained by means of the program PROAIM,24 and the density matrix elements have been computed for all the atomic holes considered, following eqs. (6) and (7). Density maps have been calculated on the plane defined by the molecular axis and an axis perpendicular to it, for the cylindrical molecules, or the plane defined by the nuclei, for the H2 CO and HCOOH molecules. For each atom, similarities between the atomic holes obtained at the HF, CISD//HF, and CISD levels of theory have been calculated. Also, similarities between different atomic holes have been calculated at each level of theory. For the calculation of similarities between atomic holes, the alignment has been performed by matching the nuclei of the two atoms to be compared, while keeping the molecular axis of the two molecules coincident. Within these restrictions, there are
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two possible alignments for each pair of atoms. We decided to keep always the orientation having more chemical sense, that is, placing the nonmatching nuclei as close as possible, although, for some pairs, (2) this does not lead to the maximal ZAB value. How(2) ever, in this cases the differences in ZAB obtained for the two orientations are always very small, and the choice of one orientation or the other has no practical consequences on the subsequent discussion of the results.
Results and Discussion First of all, some representative point Fermi holes for the N2 molecule at the HF level are discussed and compared to the atomic hole obtained for one of the N atoms. Differences between HF and CISD//HF calculations are also presented for point and atomic Fermi holes. Then, the transferability of the atomic holes corresponding to the N atom in the N2 , CN− , and NO+ series, and to the carbonyl O in the CO, H2 CO, and HCOOH series are discussed. For both series, similarities between HF, CISD//HF, and CISD atomic holes are also discussed, as well as molecular similarities based on one-electron densities. HOLE DENSITIES FOR THE N2 MOLECULE The N2 molecule will be taken as an example to make a qualitative comparison between point and atomic Fermi or exchange-correlation holes. Figure 1a, b, and c shows three point Fermi holes for N2 , calculated at the HF level, for the reference electron being fixed at three different locations. In Figure 1a, where the reference electron is located at the bond critical point (bcp) between the two N atoms, the Fermi hole density is symmetrically spread within the bond region between the two atoms. On the other hand, in Figure 1b, the reference electron is located at the position of one of the N nuclei. In this case, the Fermi hole density is highly localized in a small region around the nucleus. Finally, in Figure 1c, the reference point is located 0.83 a.u. below the position of a N nucleus on the molecular axis, which is the position of maximal accumulation of lone-pair electron density, according to the Laplacian of the one-electron density. The hole density becomes now localized mainly in this lone-pair region. Figure 1d, e, f shows the differences between the Fermi holes calculated at the HF and CISD//HF levels of theory, for the reference electron being at
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FIGURE 1. Fermi holes at the HF level and differences between HF Fermi holes and CISD//HF exchange correlation holes for N2 , for different positions of the reference electron. The position of the reference electron in each map is marked with a star. HF Fermi holes are depicted with contours in steps of 0.05 a.u., and difference maps with steps of 0.001 · 2n a.u., n = 1, 2, . . . . (a) N2 (HF); reference electron located at the bond critical point; (b) N2 (HF); reference electron located at one N nucleus; (c) N2 (HF); reference electron located in the lone pair region of a N atom; (d) N2 (HF–CISD//HF); reference electron located at the bond critical point; (e) N2 (HF–CISD//HF); reference electron located at one N nucleus; (f) N2 (HF–CISD//HF); reference electron located in the lone pair region of a N atom.
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ATOMIC TRANSFERABILITY the bcp, the N nucleus and the lone-pair region, respectively. At the CISD//HF level, the minimum of the Laplacian of the one-electron density in the lone-pair region is 0.88 a.u. below the N atom; however, the CISD//HF hole used in the difference map in Figure 1f has been calculated with the reference electron located 0.83 a.u. below the N atom, as in the HF calculation. The (HF-CISD//HF) difference maps show the effect of adding Coulomb correlation on the hole densities: positive and negative values are found in regions where CISD accumulates more or less hole density, respectively, with respect to HF (for a proper interpretation of difference maps, it must be taken into account that hole density functions exhibit negative values throughout all space). For all the maps, CISD adds hole density around the position of the reference electron, in agreement with the fact that the probability of having two electrons of any spin close in space is decreased. Because the Coulomb hole integrates to zero over all space, the regions of accumulation of hole density are always compensated by regions of hole density depletion. The sizes and shapes of the zones of hole density accumulation and depletion are different for the three reference positions considered, and depend on the degree of localization of the Fermi hole at each reference point. For the reference electron at the N nucleus (Fig. 1e), the difference map also reveals a shell structure for the N atom. The atomic hole for a N atom in N2 at the HF level (Fig. 2a) shows that the atomic hole density is localized mainly within the basin of the reference atom; however, there is also a significant accumulation of hole density in the basin of the neighbor atom. By comparing the N atomic hole to the reference-fixed Fermi holes in Figure 1, and taking into account that different normalization conventions are used for the two kinds of functions, one realizes that the atomic hole is approximately an average of the three reference-fixed Fermi holes. However, the atomic hole is able to display in a single plot relevant information about the localization of the electron density for an atom in a molecule without the need to select representative locations for a reference electron. TRANSFERABILITY OF N WITHIN THE N2 , CN− , AND NO+ SERIES Figure 2 gathers pictures of the atomic holes for the N atoms in the N2 , CN− , and NO+ molecules, calculated at the HF level. For a clearer discussion of the atomic hole densities, these atoms will be referred to as N(N2 ), N(CN− ), and N(NO+ ). For all the
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atoms, there is a maximal accumulation of atomic hole density within the basin of the reference atom, although the three atomic hole densities also spread into the basin of the neighboring atom. However, there are significant differences in the shapes and sizes of the three atomic holes. For instance, the electron populations assigned to the N atomic holes in the N2 , NO+ , and CN− molecules, are −7.00, −8.47, and −5.23, respectively. Note that, because of the way in which the atomic hole has been defined, the population of an atomic hole corresponds exactly to the negative of the Bader’s electron population of the atom. Difference maps between HF and CISD//HF holes (Fig. 2d, e, and f) show the same trends for all the N atoms: the effect of adding Coulomb correlation is to increase the amount of hole density localized within the basin of the reference atom (positive values in the difference maps) and decrease the hole density delocalized into the basin of the neighboring atom (negative values in the difference maps). Similarity, matrices for the N(N2 ), N(CN− ), and N(NO+ ) atomic holes, calculated at the HF, CISD//HF and CISD levels, are reported in Table I. The relative ordering of the three atomic holes at the HF and CISD levels of theory is depicted in Scheme 1, where the N(N2 ) atomic holes calculated at the two levels of theory have been superimposed arbitrarily. CISD//HF results, not presented in Scheme 1, are equivalent to the CISD ones. As shown visually in Scheme 1, when using the HF method, the atomic holes are ordered as N(CN− ) < N(NO+ ) < N(N2 ), while, at the CISD//HF and CISD levels, they are ordered as N(CN− ) < N(N2 ) < N(NO+ ). The similarity values obtained at the CISD//HF and CISD levels are very close (see Table I), showing that allowing the molecule to relax from the HF to the CISD-optimized geometries has a minor effect on the atomic hole densities. The reasons for the differences between the HF and the correlated similarity matrices have to be found mainly in the bad description of the N–N interaction at the HF level; which largely overestimates the electron delocalization between the two N atoms. Thus, at this level of theory, the N(N2 ) hole appears to be quite different to that of the N(CN− ) and N(NO+ ). Such an overestimation is corrected at the CISD level, so one finds the expected result of the N(CN− ) and N(NO+ ) holes being the most different. Finally, it is also worth saying that the euclidean distances obtained at the CISD//HF and CISD levels are systematically lower than the HFs. This is a consequence of the reduction of charge transfer and electron delocalization at the
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FIGURE 2. Atomic holes for the N(N2 ), N(CN− ), and N(NO+ ) atoms at the HF level, in contours of 0.05 a.u., and differences between HF and CISD//HF holes, in contours of 0.001 · 2n a.u., n = 1, 2, . . . . (a) N(N2 ) HF; (b) N(CN− ) HF; (c) N(NO+ ) HF; (d) N(N2 ) (HF–CISD//HF); (e) N(CN− ) (HF–CISD//HF); (f) N(NO+ ) (HF–CISD//HF).
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ATOMIC TRANSFERABILITY TABLE I. Similarity Matrices for the N Atomic Holes in the (N2 , CN− , NO+ ) Series, at the HF, CISD//HF, and CISD Levels of Theory. N(N2 )
N(CN− )
N(NO+ )
I.a HF N(N2 ) N(CN− ) N(NO+ )
50.1187 50.6695 50.3025
0.9009 52.0318 51.3616
0.6752 0.5007 50.9421
I.b CISD//HF N(N2 ) N(CN− ) N(NO+ )
51.9439 52.1037 51.6959
0.3372 52.3772 51.8868
0.3766 0.4396 51.5897
I.c CISD N(N2 ) N(CN− ) N(NO+ )
51.9363 52.0852 51.7320
0.3366 52.3474 51.9212
0.3712 0.4128 51.6654
(2)
Similarity measures (ZAB ), in roman type, are in the lower tri(2)
angular matrix; euclidean distances (DAB ), in italics, are in the (2)
upper triangular matrix; self-similarities (ZAA ), in bold type, are in the diagonal.
CISD level, with respect to HF, which makes all the atoms more transferable between them. Solà et al.17 recently proposed the use of molecular self-similarities as a means of analyzing electron density distributions. They found that molecular self-similarities depend not only on the number of electrons of the atom or molecule studied, but also on how the electron density of the system studied is locally concentrated. For instance, for several isoelectronic series, a good correlation was found between molecular volumes and self-similarity measures. The same kind of analysis can be carried out here for the atomic hole densities. It is to be (2) expected that ZAA self-similarities will also reflect the local concentration of atomic hole density functions. Interestingly, at the HF level, atomic hole selfsimilarities are ordered as N(CN− ) > N(NO+ ) > N(N2 ), while, at the CISD//HF and CISD levels,
they are ordered as N(CN− ) > N(N2 ) > N(NO+ ), changing the relative position of N(N2 ) and N(NO+ ) (see Table I). The low self-similarity value obtained for N(N2 ) at the HF level is due again to the overestimation of the electron delocalization between the two atoms in N2 . Difference maps between the HF and CISD//HF atomic holes for the N atoms in the three molecules considered show the same trends: CISD accumulates more hole density in the basin of the reference atom, especially at the position of the nucleus, and removes it from the second atom. The differences between molecular densities or between atomic holes calculated at different levels of theory can also be assessed by means of similarity measures (see Table II). Distance indices between HF and CISD//HF atomic holes are 0.73, 0.11, and 0.19 for N(N2 ), N(CN− ), and N(NO+ ). The same results, within two digits, are obtained for the comparison of HF and CISD atomic holes, showing that the effect of allowing nuclear relaxation on the atomic holes is very low (see Table II). Table II also reports similarity measures between molecular densities calculated at different levels of theory. The HF and CISD//HF molecular one-electron densities are very similar for all the molecules, with DXY values between 0.03 and 0.04. Molecular similarities between one-electron densities are known to be extremely sensitive to slight nuclear rearrangements.16 This is reflected in the DXY values obtained when comparing HF and CISD densities, which are 2.15, 1.11, and 2.52, for N2 , CN− , and NO+ , respectively. According to the DXY values between HF and CISD//HF molecular densities, which are between TABLE II. Similarities and Euclidean Distances between HF and CISD//HF and between HF and CISD, for the Atomic (2) (2) Hole Densities of the N Atoms (ZAB , DAB ) and for the Molecular One-Electron Densities (ZXY , DXY ), in the (N2 , CN− , NO+ ) Series. (HF–CISD//HF) Atoms N(N2 ) N(CN− ) C(NO+ ) Molecules
SCHEME 1.
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N2 CN− NO+
(2) ZAB
50.7643 52.1981 51.2479
(2) DAB
0.7308 0.1129 0.1899
(HF–CISD) (2) ZAB
50.7634 52.1833 51.2852
(2)
DAB 0.7269 0.1126 0.1929
ZXY
DXY
ZXY
DXY
105.3975 84.2583 134.4295
0.0350 0.0300 0.0418
103.0597 83.6194 131.1939
2.1451 1.1138 2.5238
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FRADERA, DURAN, AND MESTRES 0.03 and 0.04 for the three molecules, consideration of Coulomb correlation leads to analogous electronic redistributions in N2 , CN− , and NO+ (see Table II). Furthermore, from the point of view of the N atomic holes, N(N2 ) suffers the greatest electronic redistribution upon inclusion of Coulomb correlation. Thus, when comparing the HF and CISD//HF (2) hole densities for N(N2 ), a DAB value of 0.73 is obtained, compared to 0.11 and 0.19 for N(CN− ) and N(NO+ ), respectively. These results show that the effect of Coulomb correlation can be different for molecular charge densities and for atomic holes. The fact that N2 is a diatomic homonuclear molecule means that the one-electron atomic selfsimilar(1) ity ZAA is exactly equivalent to half the molecular selfsimilarity, ZXX . This will allow to compare the (1) (2) values of the ZAA and ZAA atomic self-similarities for the N atom. The same applies to the comparisons between N(N2 ) atoms calculated at the HF and CISD//HF levels of theory. Altogether, these comparisons will show that different results can be obtained when using atomic similarity measures based on the one-electron density or the exchange(1) (2) correlation density. Table III collects ZAA and ZAA values for N(N2 ) at the HF, CISD//HF, and CISD (2) levels of theory. At the three levels of theory, ZAA is (1) smaller than ZAA . Moreover, for the HF–CISD//HF (2) (1) similarity, ZAB is also smaller than ZAB . These results are consistent with the fact that atomic hole densities are partially spread onto the basins of neighbor atoms, in contrast to the one-electron atomic densities, which cannot extend beyond interatomic boundaries.
TABLE III. Atomic Self-Similarities for the N Atom in the N2 Molecule. Self-Similarities
(1)
(2)
ZAA
ZAA
HF CISD//HF CISD
52.6616 52.6936 52.6574
50.1187 51.9439 51.9363
Similarity
ZAB
ZAB
52.6987
50.7643
HF–CISD//HF
(1)
(2)
The self-similarity values corresponding to the atomic hole (2) (1) (ZAA ) and to the atomic one-electron density (ZAA ) are presented, at the HF, CISD//HF, and CISD levels of theory. The (1) (2) similarity values (ZAB , ZAB ) between the N atoms at the HF and CISD//HF level are reported also.
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TRANSFERABILITY OF THE CARBONYL O WITHIN THE CO, H2 CO, AND HCOOH SERIES Figure 3 depicts the atomic holes corresponding to the carbonyl O atoms in the CO, H2 CO, and HCOOH series, at the HF level of theory, as well as difference maps between the atomic holes calculated using the HF and CISD approximations. From now on, the atoms in this series will be denoted as O(CO), O(H2 CO), and O(HCOOH). As expected, all maps show a maximum accumulation of hole density within the basin of the O atom, with some delocalization towards the C atom. For O(H2 CO) and O(HCOOH), the delocalization towards H atoms or OH groups is very low. In fact, only the hydroxyl O in HCOOH appears to have a significant accumulation of hole density, which is not apparent for the H atoms in any molecule (Fig. 3b and c). Difference maps between atomic holes calculated using the HF and CISD approximations also show the same trends for the three atoms in the series. Thus, upon consideration of Coulomb correlation, there is a buildup of hole density in a small region around the O nucleus (positive values in the difference maps) and a depletion of hole density outside these regions (negative values in the difference maps). However, for O(CO), CISD adds hole density also in a small region corresponding approximately to the valence shell of the C atom. Atomic hole similarity matrices at the HF, CISD//HF, and CISD levels of theory are presented in Table IV. According to the euclidean distance values, O(H2 CO) and O(HCOOH) are highly trans(2) ferable between them (DAB = 0.056), at the HF level of theory. In contrast, O(CO) appears to be (2) less transferable, as indicated by its DAB values (2) when compared to O(H2 CO) (DAB = 0.237) and (2) O(HCOOH) (DAB = 0.206). These values are in agreement with the different hybridizations of the C atom in CO, in one hand, and H2 CO and HCOOH, in the other hand. Note, also that the O carbonyl atomic holes in this series are more transferable than the N atomic holes in the previous series, calculated at the same level of theory. The higher transferability within this second series reflects the fact that the O(CO), O(H2 CO), and O(HCOOH) atoms are bonded to the same atom (C), and share a common molecular environment, except for the differences due to second-neighbor atoms. In contrast, each of the N(N2 ), N(CN− ), and N(NO+ ) atoms is bonded to a different atom, leading to a higher variability of the molecular environment of the N atom in this series.
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ATOMIC TRANSFERABILITY
FIGURE 3. Atomic holes for the O(CO), O(H2 CO), and O(HCOOH) carbonyl oxygens at the HF level, in contours of 0.1 a.u., and differences between HF and CISD//HF holes, in contours of 0.001 · 2n a.u., n = 1, 2 . . . . (a) CO(CO) HF; (b) CO(H2 CO) HF; (c) CO(HCOOH) HF; (d) CO(CO) (HF–CISD//HF); (e) CO(H2 CO) (HF–CISD//HF); (f) CO(HCOOH) (HF–CISD//HF).
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FRADERA, DURAN, AND MESTRES TABLE IV.
TABLE V.
Similarity Matrices for the O Holes in the CO, H2 CO, and HCOOH Series.
Similarities and Euclidean Distances between HF and CISD//HF and CISD, for the Atomic Hole Densities of (2) (2) the Carbonyl O (ZAB , DAB ) and for the Molecular One-Electron Densities (ZXY , DXY ), in the CO, H2 CO, and HCOOH Series.
O(CO)
O(H2 CO)
O(HCOOH)
IV.a HF O(CO) O(H2 CO) O(HCOOH)
80.8384 80.7313 80.7469
0.2370 80.6804 80.6876
0.2063 0.0557 80.6979
IV.b CISD//HF O(CO) O(H2 CO) O(HCOOH)
81.2216 81.1943 81.1881
0.2052 81.2090 81.1961
0.1763 0.0486 81.1855
IV.c CISD O(CO) O(H2 CO) O(HCOOH)
81.5093 81.3301 81.2090
0.2002 81.1910 81.1855
0.1676 0.0503 81.1721
Plain values, in the lower triangle, refer to similarity measures (2) (ZAB ); values in italics, in the upper triangle refer to euclidean (2)
distances (DAB ), values in bold, in the diagonal, refer to selfsimilarities
(2) (ZAA ).
The transferability of the carbonyl O atomic holes is preserved at the HF, CISD//HF, and CISD levels of theory (see Table IV). Indeed, O atomic holes are found to be slightly more transferable at the CISD//HF and CISD levels of theory. As for the self-similarity values, they increase at the CISD//HF and CISD levels of theory, with respect to the HF results. Thus, Coulomb correlation increases the concentration of atomic hole density into the reference atoms. Both trends agree with those found above for the N atomic holes in the first series of molecules. Finally, we consider the quantitative comparison between O atomic holes calculated using different approximations. Table V gathers the similarity measures and indices between O atomic holes calculated at different levels of theory. As shown by the difference maps in Figure 3, the inclusion of electron correlation has the same effect on the three atomic (2) holes. Accordingly, the DAB values obtained after comparison of HF and CISD//HF results are very close for the three atomic holes: 0.102, 0.097, and 0.087, for O(CO), O(H2 CO), and O(HCOOH), respectively. Geometry relaxation at the CISD level only affects meaningfully the O(CO) atomic hole, (2) for which the DAB value between the HF and CISD hole densities is 0.136, while the corresponding (2) DAB values for O(H2 CO) and O(HCOOH) are 0.096
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(HF–CISD//HF) Atoms O(CO) O(H2 CO) O(HCOOH) Molecules CO H2 CO HCOOH
(2) ZAB
81.0248 80.9401 80.9380
(2) DAB
0.1019 0.0965 0.0868
(HF–CISD) (2) ZAB
81.1646 80.9311 80.9313
(2)
DAB 0.1357 0.0957 0.0857
ZXY
DXY
ZXY
DXY
113.4055 113.1643 194.4723
0.0337 0.0307 0.0340
113.5865 112.3833 191.3265
1.2598 1.2232 2.4826
and 0.087, very close to the results obtained for the comparison of the HF and CISD//HF atomic holes. Table V also collects the similarities between the CO, H2 CO, and HCOOH molecular densities calculated at different theoretical levels. Again, molecular similarity measures between one-electron densities appear to be very dependent on nuclear rearrangements, rather than on electronic redistributions. Thus, DXY values for the comparison between HF and CISD//HF results are very low for all the molecules (ca. 0.03), while the values corresponding to the comparison of HF and CISD results are much higher (1.26, 1.22, and 2.48 for CO, H2 CO, and HCOOH, respectively).
Conclusions The definition of an atomic hole density function allows for the separation of the exchangecorrelation density into atomic and biatomic contributions, following the theory of Atoms in Molecules. The shape of an atomic hole for an atom or functional group is related to the way in which the electron density of that atom or functional group is localized. In particular, electron delocalization between an atom and its neighbors is made evident on its atomic hole. In general, for a given atom or functional group in different molecular environments, the shape of the atomic hole is partially preserved, depending on the degree of transferability of the atom within the different molecules.
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ATOMIC TRANSFERABILITY Similarity measures appear to be a convenient tool for the quantitive comparison of atomic holes. For instance, the degree of transferability of an atom in different molecules can be easily quantified. Also, for a given atom, atomic holes calculated at different levels of theory can be compared to assess the effect of introducing electron correlation. An additional advantage is that similarities between atomic holes (2) (ZAB ) are not heavily dependent on slight geometrical rearrangements, in contrast to the similarity measures between molecular one-electron densities (ZXY ). Thus, they allow one to focus on genuine two-electron effects, rather than on geometrical rearrangements. As a practical application, the transferability of N in the N2 , CN− , and NO+ series, and of the CO functional group in the CO, H2 CO, and HCOOH series have been studied, using HF and CISD wave functions. It has been found that atomic holes calculated at the HF, CISD//HF, and CISD levels of theory are quite similar, although CISD correlation tends to concentrate more electron hole density around the nucleus, with respect to HF. Similarity measures between atomic holes show that the N and O atoms are highly transferable, within the two series studied. Atomic transferability is not decreased by the addition of Coulomb correlation. On the contrary, atomic holes appear to be more transferable at the CISD//HF and CISD levels than at the HF level of theory. Moreover, Coulomb correlation has been shown to be necessary for a correct ordering of the N atomic holes in the first series. The transferability of the exchange-correlation density must be parallel to that of the electrondensity itself. Thus, it is expected that qualitatively equivalent results could be obtained from the calculation of similarities between atoms in molecules defined as the overlap of the one-electron densities corresponding to two atomic basins. More work on this subject, including a comparison of atomic similarity measures obtained from the one-electron and the exchange-correlation density, is under way in our laboratory.
Acknowledgments X.F. benefits from a doctoral fellowship from the University of Girona.
References 1. Lewis, G. N. J Am Chem Soc 1916, 38, 762. 2. McWeeny, R. Rev Mod Phys 1960, 32, 335.
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