Density functional theory with approximate kinetic energy functionals applied to hydrogen bonds

Transcription

1 Density functional theory with approximate kinetic energy functionals applied to hydrogen bonds Tomasz Adam Wesolowski Université de Genève, Département de Chimie Physique, 30, quai Ernest-Ansermet, CH-1211 Genève 4, Switzerland Received 22 January 1997; accepted 17 February 1997 Applicability of the approximate kinetic energy functionals to study hydrogen-bonded systems by means of the formalism of Kohn Sham equations with constrained electron density KSCED Cortona, Phys. Rev. B 44, ; Wesołowski and Warshel, J. Phys. Chem. 97, ; Wesołowski and Weber, Chem. Phys. Lett. 248, is analyzed. In the KSCED formalism, the ground-state energy of a molecular complex is obtained using a divide-and-conquer strategy, which is applied to the Kohn Sham-like equations to obtain the electron density of a fragment embedded in a larger system. The approximate kinetic energy functional enters into the KSCED formalism in two ways. First, the effective potential in which the electrons of each fragment move contains a component which is expressed by means of a functional derivative of an approximate kinetic energy functional functional derivative of the non-additive kinetic energy. Second, the KSCED energy functional contains a component non-additive kinetic energy which is expressed using the approximate kinetic energy functional. In this work, the KSCED energies and densities of (H 2 O) 2, (HF) 2, (HCl) 2, and HF NCH are compared to the ones obtained using the standard supermolecule Kohn Sham approach. The following factors determining the agreement between the KSCED and supermolecule Kohn Sham results are analyzed: the analytical form of the gradient-dependent terms in the approximate kinetic energy functional and the number of atom-centered orbitals used to expand electron density of fragments. The best agreement between the supermolecule Kohn Sham and the KSCED results is obtained with the kinetic energy functional derived following the route of Lee, Lee, and Parr Lee et al., Phys. Rev. A 44, from the exchange functional of Perdew and Wang Perdew and Wang, in Electronic Structure of Solids 91, edited by P. E. Ziesche and H. Eschrig Academie Verlag, Berlin, 1991, p. 11. The difference between the KSCED and the supermolecule Kohn Sham energies of studied complexes amounts to less than 0.35 kcal/mol at the equilibrium geometry American Institute of Physics. S I. INTRODUCTION To date, the most successful applications of density functional theory to studies of molecular systems are based on the theory elaborated by Kohn and Sham KS. 1 According to it, the ground-state electron density of a N-electron system in a given external potential V(r) is obtained by minimizing the functional of the total energy, expressed as: E 1 2 i i * r i r 1 dr 1 V r 1 r 1 dr r 1 r 2 dr r 1 r 2 1 dr 2 E xc, where the energy minimum is searched among trial densities taking the form (r) i n i i (r) 2. The electron density minimizing the total energy functional is obtained using a self-consistent procedure involving a set of one-electron equations for i : V r 1 r 2 r 1 r 2 dr 2 V xc i r 1 i i r 1 i 1,N, 1 2 where V xc r E xc. 3 Since the first-principle analytical form of the exchangecorrelation functional E xc is not known, the electron density obtained using the Kohn Sham formalism Equations 1 3 must rely on approximations. Several approximate exchange-correlation functionals developed using different strategies can be found in the literature for a review, see Ref. 2. Equation 1 may be seen as a scheme to partition the functional of the total energy in which the dominant components of E energy of Coulomb repulsion, energy of the interactions with external potential, and kinetic energy of reference system noninteracting electrons are expressed using exact analytical formulas. The remaining part E xc ), the analytic form of which is not known, is only a fraction of the total energy. Owning to this partitioning, the KS formalism has been successfully used to predict properties of chemical systems despite the lack of the knowledge of the exact analytical form of E xc. A number of gradientdependent exchange-correlation functionals developed in the 8516 J. Chem. Phys. 106 (20), 22 May /97/106(20)/8516/11/$ American Institute of Physics

2 Tomasz Adam Wesolowski: Approximate kinetic energy functionals 8517 last decade approximate the exact functional sufficiently well for the KS calculations to enter the domain of modeling the chemical systems for a review, see Ref. 3. To find the electron density that minimizes the total energy functional, alternative routes to the one initiated by Kohn and Sham can be formulated. The most straightforward application of the Hohenberg Kohn theorem 4 would be the direct minimization of the approximate functional of the total energy without using one-electron orbitals to construct trial densities. Compared to the Kohn Sham formalism, additional approximations have to be made because the analytical form of the kinetic energy functional is not known. An approximate kinetic energy functional which yields a sufficiently accurate functional derivative to be applied for the minimization of the energy of chemical systems is yet to be discovered see Ref. 5 for instance. Recently, an intermediate route which uses both the approximate exchange-correlation functional and the approximate kinetic energy functional was introduced for studying solids 6 and molecular interactions. 7 The method, which we have referred to as Kohn Sham equations with constrained electron density KSCED, shall be outlined below Equations 4 7. The kinetic energy of a system formally divided into subsystems is expressed as: T s 1 2 T s 1 T s 2 T s nadd 1, 2, where T s denotes the kinetic energy in a reference system of non-interacting electrons with density. Equation 4 defines the non-additive kinetic energy (T s nadd 1, 2 ). Using the Equation 4 partitioning of the kinetic energy, the Kohn Sham total energy functional can be written as: E E 1 2 T s 1 T s 2 T s nadd 1, r 2 r 1 r 2 r dr dr r r Z A r R A 1 r 2 r dr E xc 1 2, A where the Kohn Sham definition of E xc is retained and the explicit analytical form of the external potential is given V(r) A (Z A / r R A ). The Equation 5 partitioning of the total energy functional makes it possible to find the ground-state electron density of the whole system using a constrained minimization scheme in which the energy is minimized in respect to variations of electron density associated with a selected fragment ( 1 ). Minimizing the total energy functional in respect to the variations of the density 1, formally represented as 1 i N 1 n (1)i (1)i 2, leads to the following Kohn Sham-like one electron equations for (1)i : eff V KSCED 1 r, 2 r 1 i r 1 1 i 1 i r 1 i 1,N 1, 6 where the KSCED effective potential equals to: eff V KSCED A Z A r R A 2 r r r dr 1 r r r dr V xc 1 r 2 r T s nadd 1, Since the electron density 2 (r) enters as a given function into Equations 6 7, its orbital representation is not needed to minimize the total energy with respect to variations of 1 (r). Practical implementations of the KSCED formalism Equations 4 7 involve the following specific issues: 1 The choice for 2 (r): In a general case, neither 2 (r) nor 1 (r) are known and the KSCED electron density can be obtained using the freeze-and-thaw cycle of the KSCED equations in which the electron density of each fragment enters alternatively as 2 (r) or as 1 (r) into Equations 6 7 until the ( 1 (r), 2 (r)) self-consistency is reached. 8 In such a case, the one-electron orbital representations of the 1 and the 2 densities are available N ( m (r) m i n (m)i (m)i (r) 2, m 1,2) and the approximate kinetic energy functional is only needed to express T nadd s 1, 2 which is a small fraction of the total kinetic energy. The electron densities of subsystems may be subjected to approximations such as the one used in our coupled ab initio/molecular dynamics simulations 9,10 where the density of selected subsystems was frozen at an initial shape which corresponded to the electron density of the isolated subsystem Frozen Density Functional Theory, FDFT. 2 The choice of the basis set functions to expand 1 (r) and 2 (r): To represent the supermolecule KS electron density of the whole system obtained using a given set of basis set functions, the same set of functions is needed to expand KSCED electron densities 1 (r) and 2 (r). Hereafter, such a calculation shall be referred to as KSCED s where s indicates that all basis set functions used in the supermolecule KS calculations are used to expand the electron density of fragments. The KSCED calculations in which the electron density of a fragment is expanded using only a subset of the whole set of basis set functions consisting of the functions that are centered on atoms belonging to the considered fragment shall be referred to as KSCED m. Compared to the KSCED s variant, the KSCED m one introduces an additional approximation: In the expansion of the electron density using atom centered basis functions, the 1 i (r)* 2 j (r) terms are neglected. It can be expected, therefore, that the KSCED m is of limited applicability to study complexes involving chemical bonds if standard atomic basis sets are used. Owning to the reduction of the Kohn Sham Fock matrix, the KSCED m calculations repre-

3 8518 Tomasz Adam Wesolowski: Approximate kinetic energy functionals sent a computationally effective implementation of the KSCED formalism realizing the divide-and-conquer strategy. 3 The accuracy of the approximations for T nadd s 1, 2 / 1 and T nadd s 1, 2 : The exact kinetic energy functional applied in the freeze-and-thaw cycle of the KSCED s equations should lead to the KSCED electron density which equals to the supermolecule KS one. Provided the same approximate exchange-correlation functional and the same basis set functions to expand electron density are used, any difference between the electron density obtained from supermolecule Kohn Sham calculations ( KS (r)) and the density obtained by means of the freeze-and-thaw cycle of the KSCED s equations ( KSCED (r) 1 (r) 2 (r)) can be attributed to the accuracy of the used approximate kinetic energy functional. In the case of the freeze-and-thaw cycle the KSCED m equations, the difference may arise due to the insufficient accuracy of the approximate kinetic energy functional and also due to the absence of the 1 i (r)* 2 j (r) terms. It can be expected, therefore, that the KSCED m results are less sensitive than the KSCED s ones on the accuracy of the approximate kinetic energy functional owning to the smaller number of basis functions used to construct trial densities. 11 To study molecular complexes by means of the KSCED equations, the total electron density can be partitioned in such a way that the fragments correspond to the molecules forming the complex. The advantage of such a division is that the overlap between fragments densities decreases with the increasing intermolecular distance, and so is the relative contribution of the non-additive kinetic energy to the total energy. Consequently, the importance of the difference between the approximate kinetic energy functional used in the KSCED equations and the exact one decreases with the intermolecular distance. At small intermolecular separations, although the overlap between the fragments densities is large, the dominant part of T s nadd 1, 2 can be derived from Thomas Fermi theory (T TF ). 12 The comparisons between the supermolecule Kohn Sham and the KSCED electron densities offers therefore a sensitive test of the gradientdependent term in an approximate kinetic energy functional. In our recent paper, it was shown that the KS energies were accurately reproduced by the calculations using the freeze-and-thaw cycle of the KSCED equations and the gradient-dependent kinetic energy functionals for the linear HCN HF complex at intermediate and large intermolecular distances. Considered were approximate kinetic energy functionals of the following general form: T appr /3 5/3 F s d 3 r, where F(s) denotes the enhancement factor, which is a function of scaled density gradient (s /2 k F where k F is the Fermi vector. The analytic form of F(s) determines the gradient-dependency of the approximate kinetic functional. The best agreement between the KS and the KSCED results 8 was obtained using the analytical form of F(s) proposed by Perdew and Wang for the exchange energy: 13 F PW91 s 1 gs sinh 1 gs exp 100s 2 gs 2 1 gs sinh 1 gs 0.004s 4, 9 where g 2 4/3 (3 2 ) 1/3 and,, and are constants. Although the KSCED interaction potential obtained using the T TF functional was attractive, the magnitude of the interaction was significantly underestimated. The gradient-dependent terms in the kinetic energy functional were needed to reduce repulsive contribution of non-additive kinetic energy to the total energy functional, which was overestimated by the Thomas Fermi theory. The present work examines furthermore the applicability in the KSCED calculations of the kinetic energy functionals given by Equation 8. The comparisons between the supermolecule KS and the KSCED results are made for the following hydrogen-bonded systems: (H 2 O) 2, (HF) 2, (HCl) 2, and HF NCH. The complexes differ in the strength of the hydrogen bond, which ranges from about 1 kcal/mol ((HCl) 2 ) to about 7 kcal/mol (HF NCH). II. COMPUTATIONAL DETAILS The freeze-and-thaw cycle was performed using our implementation of the KSCED equations into the demon program developed by Salahub and collaborators. 14 Throughout the text, the KSCED/XXX notation is used in which XXX stands for one of the following approximate kinetic energy functionals used to derive T nadd s : T TF Thomas-Fermi, T LLP Lee Lee Parr 15, T PW86 Perdew Wang, , and T PW91 Perdew Wang, The analytical form of each kinetic energy functional together with its functional derivative has been given elsewhere. 17 The Kohn Sham orbitals were expanded using the Gaussian atomic basis sets with the following contraction patterns: 73111/6111/1* for chlorine, /411/1 for carbon, 19, nitrogen, 20 and oxygen, /311/1 for fluorine; 21 and 41/1 for hydrogen. 19 The associated auxiliary functions were used for fitting the electron density and the exchange-correlation potential 5,4;5,4 for chlorine, 18 5,2;5,2 for carbon, 19 nitrogen, 20 oxygen, 20 and fluorine; 21 and 5,1;5,1 for hydrogen 19. The calculations were made using the fine random grid options FINE and RANDOM in the demon program, and 128 radial grid points. The approximate exchange-correlation functional B88/ P86 was used KS and in the KSCED calculations. Its exchange part B88 was proposed by Becke 22 and the correlation part P86 by Perdew. 23 The B88/P86 functional has been extensively applied to study chemical molecules and their complexes. The properties of hydrogen-bonded systems derived from the Kohn Sham calculations using the B88/ P86 functional are known to agree reasonably well with the experimental results, although in individual cases, other approximate exchange-correlation functionals lead to better

4 Tomasz Adam Wesolowski: Approximate kinetic energy functionals 8519 TABLE I. Equilibrium geometry of considered complexes obtained using the supermolecule KS calculations with the B88/P86 exchange-correlation functional. The experimental results are given in parentheses. See Figure 1 for the notation of atoms. Distances are given in Å and angles in degrees. Complex R DA R DH R AX (ADH) (DAX) (HDAX) H 2 O H 2 O(X H) a 1 10 a a HF HF(X F) b 10 6 c c HCN HF(X C) d 0 d ) 180 d HCl HCl (X Cl) e 0 10 e e a Reference 25. b Reference 26. c Reference 27. d Reference 28. e Reference 29. results. 24 To analyze the differences between the results of the supermolecule KS and the KSCED calculations, the choice of the approximate exchange-correlation functional is of secondary importance, provided the same functional is used in the KS and the KSCED calculations. In the KSCED calculations, the partitioning of the electron density corresponds to the molecules forming the complex. The number of electrons of each fragment was the same as if the fragment was isolated. The KS and the KSCED orbitals were expanded using the same Gaussian atomic basis sets. The KSCED m and KSCED s variants of the electron density expansions were applied. III. RESULTS AND DISCUSSION A. The KSCED results at the KS energy minima For each complex, the comparisons between the KSCED and the KS results energy and electron density are made at the geometry corresponding to the the supermolecule KS energy minimum see Fig. 1 and Table I The KSCED electron density ( KSCED 1 2 ) is obtained by means of the freeze-and-thaw cycle of the KSCED equations using one of the following implementations of the KSCED formalism: KSCED m /TF, KSCED s /TF, KSCED m /LLP, KSCED s /LLP, KSCED m /PW86, KSCED s /PW86, KSCED m /PW91, and KSCED s /PW91. Tables II V collect the supermolecule KS and the KSCED energies Equation 4 of the investigated hydrogenbonded complexes. The basis set superposition error BSSE of the interaction energies derived from the supermolecule Kohn Sham calculations is estimated using the counterpoise method by Boys and Bernardi. 36 It can be seen, from Tables II 5, that all considered gradient-dependent kinetic energy functionals (T LLP, T PW86, and T PW91 ) lead to the KSCED energies which agree significantly better than the KSCED/TF ones with the supermolecule KS energies. In line with our previous results, 17 T TF leads to underestimated energies. The difference between the KSCED s /TF and the supermolecule KS energies amounts to 3.39, 4.12, 5.06, and 2.84 kcal/mol for (H 2 O) 2, (HF) 2, HF NCH, and (HCl) 2, respectively. These numerical values indicate that the non-additive kinetic energy functional derived from Thomas Fermi theory does not provide a sufficiently accurate approximation of the exact one to be applied to study hydrogen-bonded systems. All gradientdepended approximate kinetic energy functionals lead to the significantly better KSCED energies. In particular, the T PW91 functional leads to the best agreement between the KSCED and the KS energies confirming our conclusion based on our previous studies of the linear HCN HF complex. 17 The difference between the KSCED s /PW91 and the KS energies amounts to 0.32, 0.24, 0.17, and TABLE II. The total energies (E) and complexation energies ( E) in atomic units calculated at equilibrium geometry of the H 2 O H 2 O complex. The interaction energies are given in kcal/mol in parentheses. The total energy of the complex at infinite separation amounts to atomic units. The interaction interaction energy derived from the binding enthalpy and vibrational data amounts to kcal/mol Refs. 30 and 31. E CP denotes the energy corrected for the basis set superposition error. Both the KS and the KSCED results are obtained using the B88/P86 exchange-correlation functional. freeze-and-thaw cycle Functional E KSCED s E KSCED s E KSCED m E KSCED m TF LLP PW PW Supermolecule Kohn Sham calculations E E E CP FIG. 1. Structures of the studied molecular complexes

5 8520 Tomasz Adam Wesolowski: Approximate kinetic energy functionals TABLE III. The total energies (E) and complexation energies ( E) in atomic units calculated at equilibrium geometry of the HF HF complex. The interaction energies are given in kcal/mol in parentheses. The total energy of the complex at infinite separation amounts to atomic units. The interaction interaction energy derived from the binding enthalpy and vibrational data amounts to kcal/mol Ref. 26. E CP denotes the energy corrected for the basis set superposition error. Both the KS and the KSCED energies are obtained using the B88/P86 exchange-correlation functional. freeze-and-thaw cycle Functional E KSCED s E KSCED s E KSCED m E KSCED m TF LLP PW PW Supermolecule Kohn Sham calculations E E E CP kcal/mol for (H 2 O) 2, (HF) 2, HF NCH, and (HCl) 2, respectively. For all considered kinetic energy functionals, the KSCED s energies are lower than the KSCED m ones. This reflects the variational principle, underlying the KSCED equations and the fact that the i 1 (r)* j 2 (r) terms are neglected in the KSCED m implementation. For instance, the KSCED m /PW91 energy of the HF NCH complex is TABLE IV. The total energies (E) and complexation energies ( E) in atomic units calculated at equilibrium geometry of the HCN HF complex. The interaction energies are given in kcal/mol in parentheses. The total energy of the complex at infinite separation amounts to atomic units. The interaction interaction energy derived from the binding enthalpy and vibrational data amounts to 7.1 kcal/mol Ref. 32. E CP denotes the energy corrected for the basis set superposition error. Both the KS and the KSCED results are obtained using the B88/P86 exchangecorrelation functional. freeze-and-thaw cycle Functional E KSCED s E KSCED s E KSCED m E KSCED m TF LLP PW PW Supermolecule Kohn Sham calculations E E E CP about 1.3 kcal/mol higher than the KSCED s /PW91 one. For other studied complexes, the magnitude of this effect is smaller. For example, the difference between the KSCED s / PW91 and the KSCED m /PW91 energies amounts to 0.51, 0.80, 0.40 kcal/mol for (H 2 O) 2, (HF) 2, and (HCl) 2, respectively. The differences between the KSCED s and KSCED m energies do not depend on the approximate kinetic energy functional indicating the common origin of the difference. The effect is the smallest for HCl HCl and the largest for HCN HF. This reflects the less significant contribution of the i 1 (r)* j 2 (r) terms into the electron density of HCl HCl in which the molecules forming the complex are separated by a large distance (R ClCl 3.8Å. The KSCED energies depend directly on the approximate kinetic energy functional see Equation 5 but also on the functional derivative of the approximate kinetic energy functional see Equation 7 which determines the electron densities 1 and 2 obtained by means of the freeze-andthaw cycle of the KSCED equations. For a given approximate kinetic energy functional, both the non-additive kinetic energy functional and the functional derivative T s nadd 1, 2 / 1 differ from the exact one. The difference determines the accuracy of the KSCED densities. To analyze the accuracy of T s nadd 1, 2 / 1, the KS and KSCED dipole moments of studied hydrogen-bonded complexes are compared. Table VI collects the magnitudes of dipole moments of the hydrogen-bonded complexes obtained by means of the supermolecule KS calculations and the freeze-andthaw cycle of the KSCED equations. The KSCED calculations are made using either T TF,orT PW91, i.e. the ones giving the worst and the best energies at the equilibrium geometry. In a contrary to the KSCED energies, which were significantly improved upon the introduction of the gradientdependent terms in the kinetic energy functional, the KSCED dipole moments do not depend significantly on the analytic form of the approximate kinetic energy functional see Table VI. The dipole moments depend more on the number of atomic orbitals used for the expansion of the electron density of each fragment. The KSCED s dipole moments agree better than the KSCED m ones with the dipole moments derived from supermolecule KS calculations. The increments of the dipole moment ( AB A B, where (r)rdr ) derived from the supermolecule KS calculations are reproduced in 70% 80% by the freezeand-thaw cycle of the KSCED equations. For all studied complexes and at all considered implementations of the KSCED formalism the freeze-and-thaw cycle of the KSCED equations converge rapidly. In all cases, the convergence of the KSCED energies at the level of 10 6 atomic units was reached in less than seven iterations of the freeze-and-thaw cycle. Table VII illustrates the convergence of the KSCED energy in the KSCED/PW91 calculations. Although the total KSCED energy converges rather rapidly, the first iteration did not provide the accurate estimate of the convergent KSCED energy in any complex investigated. Therefore, the continuation of the freeze-and-

6 Tomasz Adam Wesolowski: Approximate kinetic energy functionals 8521 TABLE V. The total energies (E) and complexation energies ( E) in atomic units calculated at equilibrium geometry of the HCl HCl complex. The interaction energies are given in kcal/mol in parentheses. The total energy of the complex at infinite separation amounts to atomic units. The interaction interaction energy derived from the binding enthalpy and vibrational data range from 1.4 to 2.0 kcal/mol Ref. 29, E CP denotes the energy corrected for the basis set superposition error. Both the KS and the KSCED results are obtained using the B88/P86 exchange-correlation functional. freeze-and-thaw cycle Functional E KSCED s E KSCED s E KSCED m E KSCED m TF LLP PW PW Supermolecule Kohn Sham calculations E E E CP thaw cycle beyond the first iteration is required for the studied hydrogen-bonded complexes. B. KSCED and KS energies at several intermolecular separations TABLE VI. Dipole moments in Debye calculated at equilibrium geometry of each considered hydrogen-bonded complex denotes the sum of dipole moments of isolated molecules oriented as in the complex at the equilibrium geometry. Complex Method x y x H 2 O H 2 O KS KSCED m /TF KSCED s /TF KSCED m /PW KSCED s /PW HF HF KS KSCED m /TF KSCED s /TF KSCED m /PW KSCED s /PW HCN HF KS KSCED m /TF KSCED s /TF KSCED m /PW KSCED s /PW HCl HCl KS KSCED m /TF KSCED s /TF KSCED m /PW KSCED s /PW In our previous study of the linear HCN HF complex, 17 it was shown that the potential energy curve derived from the freeze-and-thaw cycle of the KSCED equations agreed reasonably with the one obtained from the supermolecule Kohn Sham calculations, provided an ap- TABLE VII. The convergence of the KSCED s and the KSCED m energies in the freeze-and-thaw cycle. The energies in atomic units are calculated at the equilibrium geometries of each hydrogen bonded complex for the first five iterations. The T PW91 kinetic energy functional is applied to approximate T s nadd. Cycle No. E KSCED s E KSCED m H 2 O H 2 O HF HF HCN HF HCl HCl

7 8522 Tomasz Adam Wesolowski: Approximate kinetic energy functionals FIG. 2. The energy of interaction between rigid hydrogen-bonded molecules as a function of intermolecular distance. The relative orientation of monomers is similar to that at the equilibrium geometry see the text. Solid lines are used for the supermolecule Kohn Sham results: KS or KS CP where CP stands for the basis set superposition error corrected results. See the text, for the description of density functional theory calculations either KS of KSCED : a H 2 O H 2 O; b HF HF; c HCl HCl. proximate kinetic energy functional used to approximate T nadd s included gradient-dependent terms. Depending on the approximate kinetic energy functional, the KSCED equilibrium intermolecular distances agreed within 0.04 Å with the equilibrium distance derived from the supermolecule KS calculations, whereas the interaction energies at equilibrium separation were within 1.5 kcal/mol. This difference amounts to 20% of the interaction energy. For larger intermolecular distances, the KSCED potential energy curves converged to the potential energy curve derived from the supermolecule KS calculations. The best agreement between the KS and the KSCED equilibrium geometry parameters was obtained using the T PW91 kinetic energy functional to approximate T nadd s 1, 2. The KSCED/PW91 and the KS energies differed less than 0.3 kcal/mol, which amounts to about 3% of the interaction energy. This section presents the results of a similar analysis for the (H 2 O) 2, (HF) 2, (HCl) 2 complexes see Table IX and Fig. 2. Each potential energy curve shows the dependence of the energy of the complex on the intermolecular distance between the rigid molecules forming the complex. The energies are given relative to the energy of monomers at infinite separation which are the same in the KS and in the KSCED formalisms. The orientation of the monomers is similar to the one at the equilibrium geometry whereas their structures are taken from the experiment. Table VIII collects the geometry characteristics of the considered complexes. At each geometry, the electron density is obtained using the supermolecule KS calculations and the freeze-and-thaw cycle of the KSCED equations. Figures 2 a 2 c show the potential energy curves obtained using the supermolecule Kohn Sham calculations and the freeze-and-thaw cycle of the

8 Tomasz Adam Wesolowski: Approximate kinetic energy functionals 8523 TABLE VIII. Geometries of the hydrogen-bonded complexes at each the potential energy curves where obtained. See Figure 1 for notation of atoms. Distances are given in Å and angles in degrees. Complex R DA R DH R AX (ADH) (DAX) (HDAX) H 2 O H 2 O(X H) HF HF (X F) HCl HCl (X Cl) TABLE IX. The minima of the potential energy curves: R DA is the donoracceptor distance at the minimum, E is the interaction energy at the minimum. KSCED equations. The basis set corrected KS energies are calculated following the procedure of Boys and Bernardi. 36 The KSCED energies of the water dimer differ from the KS ones depending on the kinetic energy functional. The T LLP and T PW91 functionals lead to the KSCED energies that are significantly closer to the KS energies than the T TF does. T LLP leads to slightly lower interaction energies and shorter intermolecular distances at the energy minimum. The difference between the KSCED s /PW91 and the KS energies of a water dimer amounts to about 0.5 kcal/mol at the equilibrium geometry and decreases with increasing intermolecular distance. The differences between the KSCED m /PW91 and the basis set superposition error corrected energies are of the same order. The minimum of the KSCED energy is shifted toward smaller intermolecular distances by about 0.05 Å compared to the position of the minimum of the KS energy curve. The T LLP and T PW91 functionals lead also to the KSCED energies of the HF HF complex which are significantly closer to the KS ones than the T TF does. The KSCED s /PW91 and the KS energies of the HF HF complex are in the excellent agreement the largest differences amount to 0.1 kcal/mol. The differences between the KSCED m /PW91 and the supermolecule KS energies corrected for the basis set superposition error are slightly larger up to 0.3 kcal/mol. The T TF functional leads to the repulsive interaction energy curve in the HCl HCl case. The KSCED s /LLP and the KSCED m /PW91 energies are in excellent agreement with the KS ones. The differences between the KSCED s /PW91 and KSCED m /PW91 energies are within 0.5 kcal/mol. The analysis of the potential energy curves shows that the KSCED s energies are systematically lower by about 0.5 kcal/mol than the KSCED m ones at the minima of the potential energy curves. The difference reflects the fact that in the KSCED s electron densities the terms i 1 (r)* j 2 (r) are neglected. Except for the HCl HCl complex, this value is much smaller than the interaction energies and is of the same order of magnitude as the basis set superposition error in standard supermolecule KS calculations. The relatively good KSCED m energies indicate that the assumption that the electron density of each fragment remains within the same fragment upon the complex formation is a good approximation for studied molecular complexes. This assumption introduces a relatively small error into the KSCED energies which are usually very close to the ones obtained from supermolecule KS calculations and corrected for the basis set superposition error. The intermolecular separations corresponding to the minima of the KSCED s /PW91 and the supermolecule KS potential energy curves are in very good agreement for the HF HF and H 2 O H 2 O complexes. The agreement is worse for the HCl HCl complex in which the potential energy curve is very shallow compared to the po- Method R DA Å E kcal/mol (H 2 O) 2 KS KS CP KSCED s /PW KSCED m /PW (HF) 2 KS KS CP KSCED s /PW KSCED m /PW (HCl) 2 KS KS CP KSCED s /PW KSCED m /PW FIG. 3. The non-additive energy T s nadd 1, 2 component of the KSCED energy at several geometries of studied hydrogen-bonded complexes. See Figure 1 and Table VIII for the description of the geometry. The T PW91 kinetic energy functional is used to approximate T s nadd 1, 2.

9 8524 Tomasz Adam Wesolowski: Approximate kinetic energy functionals FIG. 4. The magnitude of the dipole moment of the hydrogen-bonded complexes complex: a H 2 O H 2 O; b HF HF; c HCl HCl. The sum of the dipole moments of isolated monomers oriented as in the complex amounts to: 3.48 Debye, 3.05 Debye, and 1.85 Debye, for H 2 O H 2 O, HF HF, and HCl HCl, respectively. See Figure 1 and Table VIII for the description of the geometry. The T PW91 kinetic energy functional is used to approximate T s nadd 1, 2. tential energy curves of other studied complexes. The positions of the minima at the KSCED m /PW91 potential energy curves are in an excellent agreement with the ones derived from supermolecular KS energies corrected for the basis set superposition error for all studied complexes. The non-additive kinetic energy T s nadd 1, 2 contributes significantly to the KSCED energies of studied complexes. It can be seen from Figure 3 that this contribution to the total energy amounts to 5 9 kcal/mol at geometries around the energy minimum. At smaller intermolecular distances T s nadd 1, 2 increases rapidly resulting in a wall-like repulsive effect whereas it vanishes at larger intermolecular separations. C. KSCED vs KS electron densities at several intermolecular distances The KSCED energies depend on the accuracy of the approximate kinetic functional as well as on the accuracy of its functional derivative determining the KSCED electron densities. In this section, the KS and KSCED electron densities are compared in order to assess the accuracy of the T s nadd 1, 2 / 1 calculated based on approximate kinetic functionals. The KS and KSCED electron densities of each complex are compared at several geometries defined in the previous section. The similarities and differences between the densities are analyzed using two measures: the magnitude of the dipole moment ( (r)rdr ) and the

10 Tomasz Adam Wesolowski: Approximate kinetic energy functionals 8525 FIG. 5. The difference between the exchange energy calculated using the KSCED and the KS electron densities of the studied complexes at several intermolecular distances: a H 2 O H 2 O; b HF HF; c HCl HCl. See Figure 1 and Table VIII for the description of the geometry. The exchange energy is calculated using the B88 Ref. 22 functional. exchange energy (E x (r) x (r)dr). Both and E x are calculated at several geometries of each complex using either the KS or the KSCED electron densities. Figures 4 a 4 c show the magnitude of the electric dipole of each studied complex at several intermolecular distances. For all studied complexes, the KSCED s calculations reproduce at least 70% of the increment of the dipole moment upon the bond formation ( AB A B ) derived from the supermolecule KS calculations. The dipole moments derived from the KS and KSCED densities differ the most at short intermolecular distances whereas they converge one to each other at large intermolecular separations. Among the KSCED s electron densities, a very small effect of the kinetic energy functional parametrization on the dipole moments can be seen. In contrast to the KSCED energies, where the T TF appears to lead to significantly worst results than either T PW91 or T LLP, the KSCED electron densities are at the same accuracy level regardless of the approximate kinetic energy functional. The differences between the KS and the KSCED at short intermolecular distances indicate the inaccuracy of the T nadd s 1, 2 / 1 calculated using all considered approximate kinetic energy functionals. The other measure of the similarities between electron densities, E x, enhances the importance of the electron density gradients. Figures 5 a 5 c show the differences between the E x derived from the KSCED s and KS calculations. In line with the conclusions of the dipole moment analysis, none of the applied approximate kinetic energy functionals leads to accurate T nadd s 1, 2 / 1 at very short intermolecular distances. At intermediate and long distances, T PW91 appears clearly superior as the differences between the E KSCED x and E KS x are the smallest.

11 8526 Tomasz Adam Wesolowski: Approximate kinetic energy functionals IV. CONCLUSIONS In this work the accuracy of approximate kinetic energy functionals has been analyzed within the framework of the KSCED equations, using different hydrogen-bonded complexes as a test. The comparisons between the supermolecule KS and the KSCED results have confirmed our previous observations obtained for a model systems. 17 They can be summarized as follows: All applied approximate gradient-dependent functionals derived from approximate exchange functionals following the route by Lee, Lee, and Parr 15 lead to the KSCED energies in qualitative agreement with the ones derived from the supermolecule KS calculations. The Thomas Fermi functional applied in the KSCED formalism to approximate T s nadd 1, 2 leads to significantly underestimated KSCED energies. Among the gradient-dependent functionals, the kinetic energy functional T PW91, the gradient-dependency of which is such as that of the exchange functional by Perdew and Wang in 1991, 13 leads to the best agreement within 0.5 kcal/ mol between KS and the KSCED energies at the equilibrium geometries. The position of the minima of the KSCED/ PW91 and supermolecule KS potential energy curves agrees reasonably well. The analysis of the differences between the KS and KSCED electron densities shows that the approximate kinetic energy functionals used to derive T s nadd 1, 2 / 1 need further improvement. The differences between the KSCED and KS electron densities decrease with increasing intermolecular distance for all approximate kinetic energy functionals. The T PW91 kinetic energy functional leads to the smallest differences at large and intermediate intermolecular separations. Localizing the electron density of the fragments KSCED m calculations which reduce significantly the computational costs of the KSCED calculations affect the KSCED m energies of studied complexes not significantly. Compared to the KSCED s energies the KSCED m are higher by an amount not increasing 9% of the interaction energy 0.7 kcal/mol in the HCN HF case. The KSCED m energies are in a very good agreement with the supermolecule KS ones corrected for the basis set superposition error. The KSCED m variant can be considered as a method suited for studying weakly interacting molecular complexes. ACKNOWLEDGMENTS The author is grateful to Professor D. R. Salahub for providing the copy of the demon program and to Professor Jacques Weber for helpful discussions. Financial support by the Federal Office for Education and Science, acting as Swiss COST office, is greatly acknowledged. This work is also a part of the Project of the Swiss National Science Foundation. 1 W. Kohn and L. J. Sham, Phys. Rev. A 140, R. Neumann, R. H. Nobes, and N. C. Handy, Mol. Phys. 87, W. Kohn, A. D. Becke, and R. G. Parr, J. Phys. Chem. 100, P. Hohenberg and W. Kohn, Phys. Rev. B 136, L.-W. Wang and M. P. Teter, Phys. Rev. B 45, P. Cortona, Phys. Rev. B 44, T. A. Wesołowski and A. Warshel, J. Phys. Chem. 97, T. A. Wesołowski and J. Weber, Chem. Phys. Lett. 248, T. A. Wesołowski and A. Warshel, J. Phys. Chem. 98, T. A. Wesołowski, R. Muller, and A. Warshel, J. Phys. Chem. 100, T. A. Wesołowski and J. Weber, Int. J. Quant. Chem. 61, D. J. Lacks and R. G. Gordon, J. Chem. Phys. 100, The T PW91 kinetic energy functional is obtained from the PW91 exchange functional Perdew and Wang, in Electronic Structure of Solids 91, edited by P. E. Ziesche and H. Eschrig, Academie Verlag, Berlin, 1991, p.11. using the route proposed by Lee, Lee, and Parr Lee et al., Phys. Rev. A 44, A. St-Amant, Ph.D. Thesis, Université de Montréal H. Lee, C. Lee, and R. G. Parr, Phys. Rev. A 44, The T PW86 kinetic energy functional is obtained from the PW86 exchange functional Perdew and Wang, Phys. Rev. B 33, using the route proposed by Lee, Lee, and Parr Lee et al., Phys. Rev. A 44, T. A. Wesołowski, H. Chermette, and J. Weber, J. Chem. Phys. 105, E. Riuz, D. R. Salahub, and A. Vela, J. Phys. Chem. 100, F. Sim, D. R. Salahub, S. Chin, and M. Dupuis, J. Chem. Phys. 95, F. Sim, A. St-Amant, I. Papai, and D. R. Salahub, J. Am. Chem. Soc. 114, The 6311/311/1 coefficients correspond to the 6-31 G* basis set. The coefficients of the auxiliary basis sets 5,2;5,2 constructed as in N. Godbout, D. R. Salahub, J. Andzelm, and E. Wimmer, Can. J. Chem. 70, taken from the standard input file of the program demon. 22 A. D. Becke, Phys. Rev. A 38, J. P. Perdew, Phys. Rev. B 33, T. A. Wesołowski and J. Weber, in Molecular Orbital Calculations Applied to Biochemical Systems, edited by Anne-Marie Sapse Oxford University Press, in press. 25 J. A. Odutola and T. R. Dyke, J. Chem. 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