Making density functional theory and the quantum theory of ... - Scitation

THE JOURNAL OF CHEMICAL PHYSICS 132, 211101 共2010兲

Communications: Making density functional theory and the quantum theory of atoms in molecules converse: A local approach Vincent Tognetti,1,a兲 Laurent Joubert,2 and Carlo Adamo1 1

Laboratoire d’Electrochimie, Chimie des Interfaces et Modélisation pour l’Energie (UMR 7575), Centre National de la Recherche Scientifique, Chimie ParisTech, 11 rue P. et M. Curie, F-75231 Paris, Cedex 05, France 2 IRCOF, CNRS UMR 6014 & FR 3038, Université de Rouen et INSA de Rouen, 76821Mont St Aignan Cedex, France

共Received 11 February 2010; accepted 16 April 2010; published online 2 June 2010兲 A first 共local兲 bridge between Kohn–Sham density functional theory and the quantum theory of atoms in molecules of Bader is built by means of a second order reduced density gradient expansion of the exchange-correlation energy density at a given bond critical point. This approach leads to the definition of new “mixed” descriptors that are particularly useful for the classification of the chemical interactions for which the traditional atoms in molecules characterization reveals insufficient, as for instance the distinction between hydrogen and agostic bonds. © 2010 American Institute of Physics. 关doi:10.1063/1.3426312兴 I. INTRODUCTION

Although all chemical information is in principle included in the Schrödinger equation and in the derived electron density 共and wave function兲, chemists however still aim to develop interpretative tools that, even if they may contain less information, are very useful to straightforwardly account for 共or even predict兲 a certain number of relevant physicochemical properties. These tools can be either rooted in a mere pragmatic approach or in more fundamental theories. There is nevertheless no reason that these two frameworks should be separate. On the contrary, making them converse often constitutes a fruitful opportunity to validate and extend the experimental observations or a given theory, as epitomized by the conceptual density functional theory 共DFT兲 that enables to physically ground and enrich important empirical chemical concepts and descriptors, such as hardness and electronegativity.1 This appears all the most meaningful that DFT 共Ref. 2兲 has become a choice tool in computational chemistry. As well known, this theory rests on the primary physical observable, the electron density, an ingredient that it shares with the quantum theory of atoms in molecules 共QTAIM兲 developed by Bader and coworkers.3,4 QTAIM notably offers a powerful tool of analysis and rationalization, based on the density gradient field, this latter quantity stemming from theoretical calculations or from experimental analyses. However, despite this common aspect, only few direct links have been evidenced between the two, mainly based on Thomas–Fermi related models5,6 or/and in a Bohmian mechanics perspective.5–7 Similarly, the use of a combined conceptual DFT/QTAIM approach has not been often exploited.8,9 Our purpose here is however slightly different a兲

Author to whom correspondence should be addressed. Present address: Physical Chemistry Institute, University of Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland. Electronic mail: [email protected].

0021-9606/2010/132共21兲/211101/4/$30.00

because we will focus on Kohn–Sham DFT 共KS-DFT兲 共certainly the most popular formulation of DFT兲, and we will give some hints for building a first 共from the best of our knowledge兲 direct 共local兲 bridge between KS-DFT and QTAIM. II. NEW LOCAL QTAIM DESCRIPTORS BASED ON EXACT PROPERTIES OF THE KOHN-SHAM EXCHANGE-CORRELATION FUNCTIONAL

In this communication, we will limit ourselves to a local point of view of QTAIM, focusing on the so-called bond critical points 共BCPs兲 where the density gradient 共but not the density itself兲 vanishes. The local properties at these points are in fact often used to discuss the nature of bonding,10 and we intend now evaluating the behavior of some KS-DFT quantities at these points. To this end, within a sufficiently small sphere near a BCP, the KS exchange and correlation energy densities ex,c can be supposed to be very close to their exact second order expansion for weak inhomogeneous electron densities,11–15 共rជ兲兵1 + ␮xs共rជ兲2 + O共ⵜ4␳共rជ兲兲其, eSx vB共rជ兲 = eUEG x

共1兲

共rជ兲 + ␮c␳共rជ兲t共rជ兲2 + O共ⵜ4␳共rជ兲兲, ecMB共rជ兲 = eUEG c

共2兲

ជ ␳共rជ兲储 / ␳共rជ兲4/3, t共rជ兲 = At储ⵜ ជ ␳共rជ兲储 / ␳共rជ兲7/6, As−1 where s共rជ兲 = As储ⵜ 2 1/3 1/6 = 2共3␲ 兲 , At = 共␲ / 3兲 / 4, ␮x = 10/ 81, ␮c ⬇ 0.067. Such an use of a truncated second order expansion is also a reminiscent of that used for the derivation of the electron localization function 共ELF兲 共the expanded quantity being the spherically average conditional pair probability兲16,17 and of the one used for the study of deformation of materials,18 the expanded quantity being the stress tensor.19,20 Let now ␧uជ be an infinitesimal vector, with 储uជ 储 = 1 defined by the azimuthal ␪ and zenith ␾ angles 共following = deZwillinger’s convention21兲. In the vicinity of a BCP 共H noting the density Hessian and ␭i its eigenvalues兲,

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Tognetti, Joubert, and Adamo

FIG. 1. The considered agostic and hydrogen-bonded systems.

冦

␳共rជc + ␧uជ 兲 ⬇ ␳共rជc兲 +

冧

␧2 uជ · H = 共rជc兲uជ 2 .

ជ ␳共rជc + ␧uជ 兲 ⬇ ␧H ⵜ = 共rជc兲uជ

共3兲

gradient inhomogeneity 共controlled by ␮x,c兲 is multiplied by 〫. Defining 0–2 共rជc兲 Qxc

=

For a local physical property P, we can define the variation

␦ P共␧,uជ 兲rជc = P共rជc + ␧uជ 兲 − P共rជc兲.

共4兲

In the case ␦ P is of ␧ second order, we make the angular and the distance dependences disappear by performing the following spherical average,22 具␦ P典rជ共2兲 = lim c

␧→0

冉 冕冕 1 4␲␧2

2␲

␲

␪=0

␾=0

冊

; 具␦ec典rជ共2兲 c

If we now specify P to be the previously mentioned DFTrelated quantities, one gets 共Cx = −共3 / 4兲共3 / ␲兲1/3, Cc = 1, 关 兴⬘ ⬅ d / d␳兲,

再

UEG ␦ex,c共␧,uជ 兲rជc = ␧2 关ex,c 共␳兲兴rជ⬘c共␭1 sin2 ␾ cos2 ␪

+ ␭2 sin2 ␾ sin2 ␪ + ␭3 cos2 ␾兲/2 2 Cx,c␮x,cAs,t 共␭21 sin2 ␾ cos2 ␪ ␳共rជc兲4/3

␮xq共rជc兲 + 0–2 共rជc兲 = − Qxc

冎

where

〫 ␳共rជc兲 , ⵜ2␳共rជc兲

8␲4/3 ␳共rជc兲5/3 31/3

␲ 2␮ c q共rជc兲 − f共␳共rជc兲兲 3

f共␳兲 =

共9兲

,

共10兲

0.300␳4/3 + 13.454␳7/3 + 100.121␳10/3 . 共0.537 + 7.875␳ − 2.585␳2兲2 共11兲

Equation 共10兲 is exact 共within the limits of starting hypothesis兲 since only properties known to be respected by the exact KS exchange-correlation functional have been used. Obviously, the numerical applications will use approximate 共and so inexact兲 electron densities. Complementary, the following ratio 共that locally compares the exchange and the correlation energy densities up to the second reduced gradient orders兲 can be evaluated, 0–2 共rជc兲 = Pxc

+ ␭22 sin2 ␾ sin2 ␪ + ␭23 cos2 ␾兲 + o共1兲 ,

q共rជc兲 =

one obtains

␦ P共␧,uជ 兲rជc sin ␾d␪d␾ . 共5兲

+

具␦ex典rជ共2兲 c

e0–2 ជ c兲 x 共r e0–2 ជ c兲 c 共r

0 = Pxc 共rជc兲.

共12兲

共6兲 III. APPLICATION TO SELECTED TEST CASES UEG 具␦ex,c典rជ共2兲 = 关ex,c 共␳兲兴rជ⬘cⵜ2␳共rជc兲/6 c

〫 ␳共rជc兲/共3␳共rជc兲 兲, 4/3

共7兲

= 2共rជc兲兲 = 兺 ␭2i 共rជc兲. 〫 ␳共rជc兲 = Tr共H

共8兲

+ where

2 Cx,c␮x,cAs,t

i

The structure of these equations is thus the same for exchange and correlation. The variation of the purely local energy density 共due to the density inhomogeneity兲 is multiplied by the laplacian, whereas the variation stemming from the

In order to have a preliminary analysis of the interest of these new local descriptors, especially for discriminating bond families, we will focus on hydrogen and agostic interactions, whose nature is still controversial but whose importance in biochemistry and in organometallics is fundamental.23 For these bonds, the intervals of variations for ␳共rជc兲 and ⵜ2␳共rជc兲 are thought to be entirely separated, suggesting the use of these descriptors as discriminating indexes.24,25 We have recently revised this last statement by a systematic study on 23 bis共imino兲pyridyl complexes for a

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Making DFT and QTAIM converse

TABLE I. Values of some density-related quantities at various BCPs 共in atomic units兲. Ti共IV兲 Tiu H␤

␳共rជc兲 ⵜ2␳共rជc兲 〫␳共rជc兲 0–2 共rជc兲 Qxc 0–2 Pxc 共rជc兲

ago

0.040 0.141 0.042 ⫺3.297 5.399

V共III兲

Tiu H␦

ago

0.030 0.114 0.029 ⫺2.500 5.134

V u H␤

Cr共II兲 ago

0.047 0.202 0.072 ⫺3.672 5.567

Cru H␦

Ni共II兲 ago

0.030 0.120 0.028 ⫺2.604 5.120

Niu H␦

H2a

H1 ago

0.014 0.046 0.004 ⫺1.946 4.490

1

O...H

O...H

0.053 0.147 0.136 ⫺1.747 5.698

0.042 0.141 0.092 ⫺1.684 5.443

O...H

2

0.050 0.168 0.146 ⫺1.686 5.628

H3

H4

O...H

Cl. . . H

0.028 0.083 0.028 ⫺1.698 5.053

0.013 0.036 0.004 ⫺1.519 4.438

a Contrary to H1 for which the two H bonds are strictly equivalent 共C2h symmetry兲, these bonds are not equivalent in the H2 optimized geometry 共symmetry group: Cs兲.

total of 40 agostic cases,26 leading to a partial reconsideration of this dichotomy. In order to illustrate this point and to motivate the induced need for new local descriptors, let us extract three molecules from this database 共Fig. 1兲. A titanium共IV兲, a vanadium共III兲 and a chromium共II兲 complexes with ␤ 共Ti, V兲 and ␦ 共Ti, Cr兲 agostic bonds, to which we will add a nickel共II兲 complex featuring an agostic H␦ that is nearly anagostic 共or preagostic兲 共⬔C␦H␦Ni⬇ 124°兲 according to Yao’s and al.’s nomenclature.27,28 On the other hand, four hydrogen-bonded systems will be considered, two 共H1 and H2兲 from the formic acid dimer family, and the water and HCl dimers 共Fig. 1兲. As shown in Table I and Fig. 2,29 the 共␳共rជc兲 , ⵜ2␳共rជc兲兲 pair is not sufficient to distinguish the agostic from the H bonds, that is to say one cannot unambiguously draw in this plane continuous zones corresponding to the two bond types. Such a failure is particularly evident for the titanium and the H2 complexes. However, while the laplacian values are confused, the eigenvalues of the density Hessian are quite different at the corresponding BCPs. Indeed, for the O . . . H1 bond, ␭1 = −0.074, ␭2 = −0.070, ␭3 = 0.286, while for the titanium agostic one: ␭1 = −0.047, ␭2 = −0.011, ␭3 = 0.200. Yet, their sums are equal due to numerical compensations between positive and negative values, so that the laplacian cannot discriminate them. As shown in Fig. 3, differentiation becomes possible 0–2 0–2 共rជc兲 , Pxc 共rជc兲兲 representation is considered, when the 共Qxc two distinct and separate groups clearly appearing. This is

0–2 notably due to the fact that the Qxc 共rជc兲 values for the agostic bonds can be almost twice 共in absolute value兲 those for the H bonds. Notably, the almost preagostic complex lies in an intermediate position in this representation, between pure agostic and H bonded systems. However, exchange and correlation always vary following opposite trends, the variation rates of the exchange energy density being higher in absolute value, even if it is less pronounced for H bonds. Let us also recall that the ratio of the total 共integrated兲 exchange energy over the total correlation one is about nine for atoms or molecules. Obviously, this result is a kind of spatial average, and such a ratio is not verified at any point. However, one can show that for the 0–2 C v O bonds in H1 and H2, and the Tiu C␣ bond, Pxc is close to this value. This is not any longer the case for the H and agostic bonds for which the weight of the correlation energy is larger. Interestingly, it can be noticed that we have the same 〫共rជc兲 and ␳共rជc兲 values for the studied bonds in the H3 and 0–2 共rជc兲 also includes a the chromium complexes; but, as Qxc laplacian contribution, our new descriptors succeed in our typology endeavor, proving that the laplacian and the diamond are complementary. In order to show how this works, we must compare the information carried by the corresponding Hessians and, in particular, their eigenvalues. Nevertheless, as the eigenvectors yជ i for a BCP are not generally collinear to the eigenvectors for another BCP, the

FIG. 2. Hydrogen and agostic bonds represented by their ␳共rជc兲 and ⵜ2共rជc兲 values.

0–2 FIG. 3. Hydrogen and agostic bonds classified with respect to their Qxc 共rជc兲 0–2 and Pxc 共rជc兲 values.

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direct comparison of the ␭i has no direct meaning. So, in order not to lose information, the knowledge of the ␭i must be replaced by the one of three 共independent兲 scalar combinations of them that are invariant by rotation. A natural choice would be to use the three symmetric functions that arise from the expansion of the Hessian characteristic polynomial: ␴1 = 兺i␭i, ␴2 = 兺i⬍j␭i␭ j, and ␴3 = 兿i␭i. Nevertheless, one can equivalently privilege the 共␴1 ⬅ ⵜ2 , 〫兲 pair over 共␴1 , ␴2兲, these two similarity invariants being related by ␴2 = 共␴21 − 〫兲 / 2. Such a choice is motivated by the fact that 〫 has a clear physical interpretation since ␦储ⵜ2␳储共␧ , uជ 兲rជc = ␧2共兺i␭2i 共uជ . yជ i兲rជ2c + o共1兲兲, so that: 具␦储ⵜ2␳储典rជ共2兲 = 31 〫 ␳共rជc兲. Thus, 〫 measures c ជ ␳储2 near a given BCP. As 〫 only the variation rate of 储ⵜ involves the eigenvalues squares, there cannot be compensations between negative and positive values; besides small eigenvalues will give a negligible contribution, so that the considered agostic and H bonds may have sufficiently distinguishable 〫 values.30 IV. CONCLUSIONS

In this communication, we have calculated the average variations of the exchange and the correlation energy densities in the vicinity of a BCP, thanks to second order reduced gradient expansions. This approach led to the definition of new descriptors that were then shown to be useful to classify bonds, since they provide crucial supplementary information to that provided by the usual 共␳共rជc兲 , ⵜ2共rជc兲兲 pair that can be not sufficient to univocally characterize chemical bonding. Obviously, extended systematic tests are in progress to confirm this outlined typology based on KS-DFT energetics. These results constitute, from our point of view, incentive arguments to foster a deeper study of the relationships that can exist between the related KS-DFT and QTAIM, and to address the new questions this first local bridge raises. ACKNOWLEDGMENTS

The authors thank the referees for their shrewd comments and suggestions. H. Chermette, J. Comput. Chem. 20, 129 共1999兲. R. G. Parr and W. Yang, Density Functional Theory of Atoms and Molecules 共Oxford University Press, NewYork, 1989兲. 3 R. F. W. Bader, Atoms in Molecules: A Quantum Theory 共Oxford University Press, Oxford, U.K., 1990兲. 4 P. L. A. Popelier, Atoms in Molecules An Introduction 共Pearson Education, Harlow, 2000兲. 5 L. Delle Site, Phys. Lett. A 286, 61 共2001兲. 6 L. Delle Site, Europhys. Lett. 57, 20 共2002兲. 7 H. J. Bohórquez and R. J. Boyd, J. Chem. Phys. 129, 024110 共2008兲. 8 F. A. Bulat, E. Chamarro, P. Fuentealba, and A. Toro-Labbé, J. Phys. 1 2

J. Chem. Phys. 132, 211101 共2010兲

Tognetti, Joubert, and Adamo

Chem. A 108, 342 共2004兲. S. Liu and N. Govin, J. Phys. Chem. A 112, 6690 共2008兲. 10 R. F. W. Bader and H. Essén, J. Chem. Phys. 80, 1943 共1984兲; notice that we will use the common term of “bond” even if its use has been criticized 关see for instance R. F. W. Bader, J. Phys. Chem. A 113, 10391 共2009兲兴. 11 Atomic units will be used unless explicitly otherwise stated. 12 P. S. Svendsen and U. von Barth, Phys. Rev. B 54, 17402 共1996兲. 13 S.-K. Ma and K. A. Brueckner, Phys. Rev. 165, 18 共1968兲. 14 Note that the exchange conventional gauge is used. Besides, it is also assumed that the inhomogeneity of the Laplacian near a BCP will only have a small impact on the energy densities variations. As, from the best of our knowledge, the role of the higher density derivatives 共ⵜnⱖ4␳兲 are unknown at least for correlation, it is anyway not possible to go behind this order; moreover, Eq. 共2兲 is rigorously valid only in the high density limit, see C. D. Hu and D. C. Langreth, Phys. Rev. B 33, 943 共1986兲; T. Thonhauser, V. R. Cooper, S. Li, A. Puzder, P. Hyldgaard, and D. C. Langreth, ibid. 76, 125112 共2007兲; however, we will use it for every density regime, as done for instance by Y. Zhao and D. G. Truhlar, J. Chem. Phys. 128, 184109 共2008兲. 15 UEG ex is the well known Slater’s functional, Slater, The Self-Consistent Field for Molecules and Solids 共McGraw-Hill, New York, 1974兲; ecUEG is not exactly analytically known but the parameterization of J. P. Perdew and Y. Wang, Phys. Rev. B 45, 13244 共1992兲, can be considered a reference; as the BCP density values will typically belong to 关0.007;0.300兴, the 共spin-unpolarized兲 PW92 expression will be reasonably substituted by its second order diagonal Padé approximant calculated at ␳ = 0.06. 16 A. D. Becke and K. E. Edgecombe, J. Chem. Phys. 92, 5397 共1990兲. 17 B. Silvi and A. Savin, Nature 共London兲 371, 683 共1994兲. 18 P. W. Ayers and S. Jenkins, J. Chem. Phys. 130, 154104 共2009兲. 19 R. F. W. Bader, J. Chem. Phys. 73, 2871 共1980兲. 20 A. Holas and N. H. March, Int. J. Quantum Chem. 56, 371 共1995兲. 21 D. Zwillinger, “Spherical Coordinates in Space” §4.9.3 in CRC Standard Mathematical Tables and Formulae 共CRC, Boca Raton, 1995兲. 22 V. Tognetti, L. Joubert, P. Cortona, and C. Adamo, J. Phys. Chem. A 113, 12322 共2009兲. 23 E. Clot and O. Eisenstein, Struct. Bonding 共Berlin兲 113, 1 共2004兲. 24 U. Koch and P. Popelier, J. Phys. Chem. 99, 9747 共1995兲. 25 P. L. A. Popelier and G. Logothetis, J. Organomet. Chem. 555, 101 共1998兲. 26 V. Tognetti, L. Joubert, R. Raucoules, T. De Bruin, and C. Adamo 共unpublished兲. 27 W. Yao, O. Eisenstein, and R. H. Crabtree, Inorg. Chim. Acta 254, 105 共1997兲. 28 J. C. Lewis, J. Wu, R. G. Bergman, and J. A. Ellman, Organometallics 24, 5737 共2005兲. 29 The molecules were fully optimized using Gaussian 03 共Ref. 31兲 at the PBE0 level 共Ref. 32兲, the nonmetallic atoms being described by the standard 6-31+ +G共d , p兲 Pople basis set and the metals by the Wachters+ f one 共Ref. 33兲. The topological analysis has been carried out with the MORPHY98 software 共Refs. 34 and 35兲. 30 Notice that the information carried by ␴3 is lost, but is not relevant for our purpose. The ellipticity 共which only focuses on the plane that is orthogonal to the bond path兲 would constitute an alternative. For the H bonds in Ref. 24, it lies in the 关0.005–1.394兴 range, whereas the one for the agostic bonds in Ref. 25 is 关0.703,1.534兴, so that overlap exists; thus ellipticity cannot be discriminative. 31 M. J. Frisch, G. W. Trucks, H. B. Schlegel et al., GAUSSIAN03, Revision C.02 共Gaussian, Inc., Wallingford, CT, 2004兲. 32 C. Adamo and V. Barone, J. Chem. Phys. 110, 6158 共1999兲. 33 A. J. H. Wachters, J. Chem. Phys. 52, 1033 共1970兲. 34 MORPHY98, a topological analysis program written by P. L. A. Popelier with a contribution from R. G. A. Bone 共UMIST, Engl., EU兲. 35 P. L. A. Popelier, Comput. Phys. Commun. 93, 212 共1996兲. 9