9182 J. Chem. Phys. 105 (20), 22 November /96/105(20)/9182/9/$ American Institute of Physics

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1 Accuracy of approximate kinetic energy functionals in the model of Kohn Sham equations with constrained electron density: The FH NCH complex as a test case Tomasz Adam Wesolowski, Henry Chermette, a) and Jacques Weber Department of Physical Chemistry, University of Geneva, 30, quai Ernest-Ansermet, CH-1211 Geneva 4, Switzerland Received 2 May 1996; accepted 21 August 1996 Ground-state properties of a linear hydrogen-bonded FH NCH complex are studied by means of the freeze-and-thaw cycle of Kohn Sham Equations with constrained electron density KSCED T. A. Wesolowski and J. Weber, Chem. Phys. Lett. 248, 71, For several geometries of the complex, the electron density and the total energy are compared to the ones obtained by means of the standard Kohn Sham calculations. The comparisons are made to assess the accuracy of several gradient dependent approximate kinetic energy functionals applied in the KSCED equations. It was found that the closest results to the Kohn Sham ones were obtained with the functional whose analytical form was proposed by Perdew and Wang for exchange energy J. P. Perdew and Y. Wang in Electronic Structure of Solids 91, edited by P. Ziesche and H. Eschrig Academie Verlag, Berlin, 1991, p.11 and parametrized by Lembarki and Chermette for kinetic energy A. Lembarki and H. Chermette, Phys. Rev. A 50, Around the interaction energy minimum as well as for larger intermolecular distances, the freeze-and-thaw cycle of KSCED leads to very similar potential energy surface as the standard supermolecule Kohn Sham calculations American Institute of Physics. S I. INTRODUCTION One of the most challenging problems quantum chemistry is facing is the calculation of the electronic structure of large molecular systems. A straightforward extension of techniques successfully applied to study small molecules does not seem to be a good solution, mainly due to the computer power requirements of such methods. For example, the computation time (t) increases proportionally to the fourth power of the number of Gaussian, Slater, etc. orbitals in the orbital basis set (N)(t N 4 ) in the Hartree Fock approach, t N 5 in the second-order Moller Plesset calculations, and t N 3 in the canonical version of the method of Kohn and Sham KS 1 which is the most widely used application of density functional theory DFT to molecular systems. Many practical implementations of the aforementioned methods reduce to some extent the computer time requirements. This is achieved through the parallelization of the algorithm, special algorithms for diagonalizing large sparse matrices, approximations for the evaluation of the Coulomb integrals, and/or the use of pseudospectral formalism. 2 For handling large molecular systems, DFT offers other solutions that reduce the power dependency of the calculation time compared to the Kohn Sham method. Recently, Yang 3 developed a formalism in which the whole system investigated is divided into fragments. The electron density of each fragment is obtained using a procedure, computer time requirements of which are primarily determined by the a Permanent address: Laboratoire de Chimie-Physique Théorique, Université Claude Bernard LYON 1, et Institut de Recherches sur la Catalyse, UPR 5401 du CNRS, 43, Boulevard du 11 Novembre 1918, F69622 Villeurbanne CEDEX, France. number of electrons in this fragment divide and conquer approach. The electron densities of various fragments are coupled to each other with terms that are derived using the DFT theorems. The idea of calculating the ground-state electronic properties of molecular complexes using fragments can be traced back to the theory developed by Kim and Gordon. 4 This theory was originally applied to complexes made of closed shell atoms and molecules. The fragments correspond to the atoms of molecules, the electron density of which is frozen and equal to the one in gas phase. The electron density of each fragment is kept frozen and the complexation energy is calculated using terms derived from DFT. The approach was subsequently generalized to include electron density deformation upon forming the complex and applied to a variety of molecular complexes. 5 Recently, Wesolowski and Warshel 6 proposed an approach based on DFT which also uses the electron density of fragments Kohn Sham equations with constrained electron density, KSCED. The KSCED formalism is closely related to the Kohn Sham method: its underlying principle is the Euler Lagrange equation applied to the total energy functional and it uses the same definition of the exchangecorrelation functional. However, in addition to the exchangecorrelation functional the analytical form of the approximate kinetic energy functional is needed. The outline of the KSCED formalism and the review of approximate kinetic energy functionals are presented in separate sections. II. THE KSCED FORMALISM The total electron density in the considered system is represented as a sum of two components, 1 and 2, which 9182 J. Chem. Phys. 105 (20), 22 November /96/105(20)/9182/9/$ American Institute of Physics

2 Wesolowski, Chermette, and Weber: Approximate kinetic energy functionals 9183 are determined separately. For a given 2 R, the 1 R is obtained by minimizing the functional of the total energy. As in the original Kohn Sham method, the density 1 is represented using Kohn Sham-type orbitals i ( 1 N 2 1 i 1 1 i * i for closed shell systems. The Euler Lagrange equations lead to the following equations for i V eff R i 1 i i 1, i 1,...,N 1, 1 with the effective potential V eff given by V eff R V nucl R 1 R 2 R R R dr V xc 1 R 2 R T s nadd 1, 2, 2 1 where the exchange-correlation potential V xc is defined as in the original KS method whereas T s nadd 1, 2 is defined as T s nadd 1, 2 T s 1 2 T s 1 T s 2, 3 FIG. 1. Scheme of the freeze-and-thaw cycle of the KSCED equations. where T s [ i ] denotes the kinetic energy of the noninteracting electrons reference system. Atomic units are used in all equations. Equations 1 3 were derived by Cortona 7 to study properties of solids and were introduced by Wesolowski and Warshel to study molecular complexes. 6 Stefanovitch and Truong used them to study the complexes of metal ions with water molecules and interaction between water molecule and the NaCl surface. 8 Both the Coulomb repulsion and the exchangecorrelation terms in Eq. 2 do not depend on the way the total electron density is divided into 1 and 2 components, whereas T nadd s 1, 2 / 1 does. The analytical form of T nadd s 1, 2 is not known yet, therefore T nadd s 1, 2 together with its associated functional derivative are evaluated using an approximate kinetic energy functional (T app s ). The total electronic energy calculated in the KSCED scheme is given as a functional E 1, 2 T s 1 T s 2 V nucl R 1 R 2 R dr R 2 R 1 R 2 R R R dr dr E xc 1 R 2 R T s nadd 1 R, 2 R. Since 1 is represented using one-electron orbitals, it is possible to calculate the accurate value of T s 1 as 4 T s 1 2 i 1 i i. 5 N 1 In Ref. 6, the gas phase electron density of a corresponding fragment was used as 2 in Eqs. 1 3 and it was not modified upon forming a complex. Such an application of Eqs. 1 3 was called FDFT frozen DFT. The approach based on Eqs. 1 3 can be extended to allow that the densities of both fragments are modified upon forming the complex. 9 This can be achieved by iteratively solving Eqs. 1 3 several times in such a way that each fragment electron density enters alternatively as 1 or 2 in Eqs. 1 3 see Fig. 1 until self-consistency the KSCED freeze-andthaw cycle. The KSCED freeze-and-thaw cycle was applied for the calculation of the ground-state electronic energy in a model linear complex made of H 2 and HCN molecules. It was found, that for large and medium intermolecular distances and if the Thomas Fermi T TF kinetic energy functional is used, the KSCED ground-state energy closely approaches the one obtained by means of the standard Kohn Sham equations applied to the whole complex. 9 For practical applications, the freeze-and-thaw cycle of KSCED equations is computationally superior to the standard KS calculations only if the effort for computing T s nadd 1, 2 / 1 is of the same order of magnitude as it is for computing V xc in standard KS method. This imposes severe restrictions on the analytical form of any approximate kinetic energy functional AKF of practical interest. III. APPROXIMATE KINETIC ENERGY FUNCTIONALS One can consider the KSCED method as an intermediate step between the Kohn Sham method, which does not use any AKF, and the direct application of the Euler Lagrange equation which would require accurate functional of the kinetic energy. Despite recent progress in the development of the AKFs, the direct application of the Euler Lagrange equation for obtaining ground-state electron density is not possible yet. In the KSCED approach, only a fraction of the kinetic energy is calculated using the AKF, and only a fraction of the effective potential is calculated using the functional derivative of the AKF. This makes it worthwhile to

3 9184 Wesolowski, Chermette, and Weber: Approximate kinetic energy functionals study the applicability of currently known AKFs in the KSCED approach. Since none of the currently available AKFs are accurate, it is worthwhile to note that the accuracy criteria for the AKF are different in KSCED and are different when one is interested in the total kinetic energy only. As it was demonstrated by Bartolotti and Acharaya 10 the accuracy of T app s and T app s / may differ significantly for a given analytical form of the AKF. Since T nadd s 1, 2 is defined as a linear combination of kinetic energies Eq. 3, a fortuitous error cancellation may take place. Recently, Lacks and Gordon 11 demonstrated that, although the T TF functional underestimates the kinetic energy, it overestimates T nadd s 1, 2 in dimers of rare gas atoms Ne 2 and He 2. For large interatomic distances, they found that the T TF functional is significantly better than,grad dependent functionals known to yield more accurate total kinetic energies. Actually, it was found 12 that the AKFs, which are known to yield similar kinetic energy in model systems, lead to KSCED electron densities that differed significantly from accurate ones. The main directions that where followed in the search for the AKF are reviewed below. The kinetic energy functional for slowly varying electron density can be represented by a gradient expansion 13,14 T t 0 t 2 t 4 dr, 6 where t 0 4/3 3 5/3 10 t / /3 C F 5/3, t 2 1 2, / a The first term t 0 leads to the T TF functional, which yields accurate value of the kinetic energy for the uniform electron density, hardly a valid assumption for molecular systems. The second term t 2 was introduced by von Weizsaecker. 15 The series truncated after the t 2 term is frequently referred to as the second-order expansion. In some approaches, the t 2 term is considered a dominant component in the series. 16 It has correct asymptotic behavior far from nuclei and yields correct kinetic energies for one- and twoelectron systems. As Bartolotti and Acharaya Eq. 18 in Ref. 10 pointed out, it does not fulfill the inhomogeneity condition satisfied by the true functional. It was also found 12 that T nad s 1, 2 / 1 obtained using the second-order expansion terms leads to a very inaccurate KSCED electron densities and consequently energies. The series in Eq. 6 is an asymptotic expansion where the importance of separate terms depends on local values of the scaled density gradient s s r 2 r k F r 1, where k F r 3 2 r 1/3. The t 6 term is divergent for molecules and atoms 17 and the t 4 term leads to divergent functional derivative of T s. 18 The aforementioned properties of the gradient expansion make it impractical in the KSCED or other approaches based on the Euler Lagrange equations. Several alternative strategies were proposed for developing the AKFs. The first one is based on the fact that the asymptotic expansion of the kinetic energy density Eq. 6 requires that the series is truncated depending on the local value of s. Instead of truncating the terms at arbitrarily chosen values of s, several analytical expressions were introduced to smooth the functional Another strategy was applied by Ou-Yang and Levy 22 who proposed that the t i, i 2 terms in Eq. 6 are replaced by another, -dependent term analytical form of which was found taking into account the scaling properties of T s. Two approximate functionals obtained in this way have been proposed. For the spin polarized case,, they take the following form: T s OL1 2 2/3 C F 5/3 1 bs bs dr, 72C F T s OL2 2 2/3 C F 5/3 1 bs bs 72C F 1 2 5/3 bs dr, 8 where the inhomogeneity wave vector is s /2k F, k F is the Fermi vector, equal to k F r 3 2 r 1/3, and b 2 4/ /3. An other route for obtaining the AKF was proposed by Lee, Lee, and Parr, 23 who derived the analytical form of the kinetic energy functional from that of the exchange energy. The analytical form of the kinetic energy density is assumed to be a product of the Thomas Fermi term and a gradient dependent enhancement factor [F(s )] T s app 2 2/3 C F 5/3 F s dr. 9 The analytical form of the enhancement factor F(s) is the same as in the corresponding exchange functional. Starting from different exchange energy functionals, several AKFs have been obtained following this route, for example bs F LLP bs sinh 1 bs, F PW s 2 14s 4 0.2s 6 1/15, 10 11

4 Wesolowski, Chermette, and Weber: Approximate kinetic energy functionals 9185 F PW s sinh 1 bs exp 100s 2 s s sinh 1 bs 0.004s The coefficients in the above formulas may be the same as in the original exchange functionals or may be fitted to the analytical values of the kinetic energies in a sample of atoms or molecules. 26,27 The kinetic energy functional of Thakkar 28 does not enter this category because the analytical form of its factor F was not obtained from a single exchange functional. It combines terms present in the T OL2 and T LLP functionals. Its analytical form is given by bs F T bs sinh 1 bs 0.072bs 1 2 5/3. bs 13 The coefficients of the T T92 functional have been obtained by fitting the numerical values of kinetic energy in a number of molecules. Lacks and Gordon 11 showed that the AKFs obtained using the Lee, Lee, and Parr route yield the most accurate values of T nadd s 1, 2. One can expect that the local properties of the AKF obtained using this strategy are of the same quality as the corresponding exchange energy functional. IV. STUDIED MOLECULAR COMPLEX The total electronic energy calculated by means of the supermolecule KS calculations and by means of the freezeand-thaw cycle of the KSCED equations are compared in a hydrogen bonded HF HCN complex. The comparison is made for several points of the potential energy surface that corresponds to linear geometry and different intermolecular distances around the energy minimum. This system has been selected because of its axial symmetry, which lets us analyze more easily the differences in the electronic and kinetic densities of the complex and its fragments. The experimental data concerning its equilibrium geometry properties are well known 29 and it was extensively studied using ab initio calculations. 30 V. COMPUTATIONAL DETAILS The demon program 31 has been modified to solve Eq. 1 and to perform the freeze-and-thaw cycle of KSCED equations. The KSCED scheme is applied for two subsystems: FH and NCH molecules forming a linear complex FH NCH. The basis set with the contraction pattern 5211/411/1 for carbon and nitrogen and 41/1 for hydrogen was used. This basis set is comparable to the 6-31G** basis set. The following auxiliary basis sets: 32 C 5,2;5,2,O 5,2;5,2, and H 5,1;5,1 were used for fitting the charge density and exchange-correlation potential. The functional derivative of the nonadditive kinetic energy was fitted using auxiliary functions similar C 5,2, O 5,2, and H 5,1 to those used for fitting the exchange-correlation potential. The exchange functional proposed by Becke 33 and the correlation functional proposed by Perdew 34 have been used for E xc and V xc in both KS and KSCED calculations. The iterative procedure Fig. 1 was terminated with the same convergence criterion as that used in the SCF procedure i.e., E n 1 E n 10 6 hartree. Both the numerical grid and the sets of auxiliary functions used in KSCED calculations are the same as for the supermolecule KS calculations 64 radial grid points and fine angular grid everywhere except KSCED/PW86 calculations where 128 radial grid points have been used. The 1 and 2 densities were expanded using the supermolecule basis set. The geometry of the linear complex is defined by a variable intermolecular distance R FN and a rigid conformation of HF as well as HCN molecules R HF nm, R CH nm, R CN nm. The nonadditive kinetic energy functional T s nadd 1, 2 has been evaluated using AKFs defined in Eqs For a given kinetic energy functional taking the form of Eq. 9, its functional derivative is given by T C F 3 5 5/3 F t F t 4tF t 6 2 F t 2tF t, 14 where t 2 / 3/8. For each approximate kinetic energy functional studied, the analytical formula for T nadd s 1, 2 / 1 was derived using Eqs. 2 and 14. In the case of the PW91 functional Eq. 12, the original numerical coefficients were replaced by the ones obtained by Lembarki and Chermette. 26 To indicate this, we will refer to this functional as T LC. For the reference, the KSCED calculations were made also with the T TF functional which does not depend on the density gradient. VI. RESULTS The accuracy of the AKF affects the KSCED density in a different way than it affects the KSCED energies. The KSCED electron density does not depend directly on the AKF used for the evaluation of T s nadd 1, 2 but it depends on its functional derivative. Once the KSCED electron density is evaluated, the corresponding energy is calculated using the AKF. Therefore, the accuracy of the total electronic energy depends on the accuracy of the AKF and on the accuracy of its functional derivative ( T s app / ). The KSCED densities and the KSCED energies are analyzed below. A. The KSCED electron densities In this section, the difference between the KSCED electron density and the KS electron density is analyzed for several intermolecular distances. For large distances, this differ-

5 9186 Wesolowski, Chermette, and Weber: Approximate kinetic energy functionals FIG. 2. The exchange energy error the difference between the KS and KSCED exchange energies in the linear HF NCH complex for several intermolecular distances. In KS and KSCED calculations, the exchange functional of Becke Ref. 33 and correlation energy functional of Perdew Ref. 34 were used. The approximate kinetic energy functionals are described in the text. ence should decrease asymptotically to zero because the electron density of the complex becomes equal to the superposition of the densities of isolated molecules both in the KSCED and in the KS approaches. With the decreasing intermolecular distance, this difference becomes affected by the actual parametrization of the kinetic energy functional in the KSCED approach. The differences between KS and KSCED electron densities are graphically represented using the values of separate energy components V-nuclear attraction, E ee -Coulomb electron electron repulsion, and E x -exchange energy calculated using corresponding electron densities. Figure 2 shows the differences in the exchange energy calculated with the KS and the KSCED densities. The experimental N-F distance equals nm, and the equilibrium distance found in the supermolecule KS calculations is nm, a value which has to be considered as the reference for the KSCED calculations. From Fig. 2 one can see that functionals like T LLP or T T92 systematically lead to densities with overestimated exchange energies whereas functionals like T TF or T PW86 lead to densities with underestimated exchange energies at intermolecular distances exceeding nm. At shorter distances, all considered functionals give electron densities leading to overestimated exchange energies. For large intermolecular distances, the T PW86 functional leads to the results similar to those obtained with the T TF functional. For the T PW86 functional, a spurious numerical noise has been found. This noise has been definitely smoothed by the use of 128 radial grid points. This is probably due to the lack of accuracy in fitting the s 6 term with currently used auxiliary functions and numerical evaluation of its enhancement factor F(s) for large s. 35 The same may explain the related problem of unexpectedly and apparently poor results 36 obtained for other systems with Lacks Gordon exchange functional that contains terms up to s FIG. 3. The nuclear attraction energy error the difference between the KS and KSCED nuclear attraction energies in the linear HF NCH complex for several intermolecular distances see the Fig. 2 caption. Finally, it appears that the T LC functional is clearly superior to the other ones in the whole range of studied intermolecular distances. The superiority of this functional is especially pronounced at intermolecular distances near and larger than the distance corresponding to the minimum of the interaction energy. The variations of the nuclear attraction energy are shown in Fig. 3, whereas the variations of the Coulomb repulsion energy are shown in Fig. 4. Based on these figures, the same conclusions can be drawn as those based on Fig. 2, namely that the KSCED electron densities obtained using the T LC form of the kinetic energy functional are superior to those obtained using other studied forms of the kinetic energy functional. Figures 2 4 also show that, for short intermolecular distances i.e., large 1, 2 -overlap, the T PW86 functional leads to more accurate density than both the T LLP and the T TF FIG. 4. The electron electron repulsion energy error the difference between the KS and KSCED electron electron repulsion energies in the linear HF NCH complex for several intermolecular distances see the Fig. 2 caption.

6 Wesolowski, Chermette, and Weber: Approximate kinetic energy functionals 9187 FIG. 5. Electron density deformation difference between the electron density of the complex FH HCN and the sum of electron densities of isolated FH and HCN molecules calculated along the axis connecting atoms x axis. The x coordinates of atomic nuclei are: X H nm, X C nm, X N 0.0, X H nm, and X F nm. The electron density is calculated using either the supermolecule Kohn Sham calculations or the freeze-and-thaw cycle of the KSCED equations with the T LC kinetic energy functional see text. In KS and KSCED calculations, the exchange functional of Becke Ref. 33 and correlation energy functional of Perdew Ref. 34 were used. functionals. The same trend has been observed in a different molecular complex, 12 where the applicability of different AKFs in the KSCED calculations was examined for conformations characterized by large 1, 2 -overlap. The similarity between the electron density obtained by means of the KSCED/LC where LC stands for the AKF used in the KSCED method to the electron density obtained by means of the supermolecule KS calculations is illustrated in Fig. 5. In this figure the electron density deformation at selected intermolecular distance R NF nm is presented along the axis connecting atoms. The negative value of indicates the reduction of the electron density upon forming the complex, whereas the positive value of it indicates the increase. It can be seen that the KS and KSCED electron density deformations are almost indistinguishable for X 0.05 nm, i.e., close to the HCN molecule. In this region, a well-localized decrease of the electron density at nitrogen atom, a small polarization of the electron density at the C atom, and almost no electron deformation at H atom can be seen. In the intermolecular region and close to the HF molecule, i.e., for X 0.05 nm, the electron density deformation is less localized and the differences between the KS and KSCED densities become more apparent, being still small. One can see a small increase of the electron density in the intermolecular area, the decrease of the electron density at the hydrogen atom, and a small decrease of the electron density at the very center of the fluorine atom accompanied by the increase of electron density more distant from the center. B. The KSCED energies The total energy of the linear FH NCH complex for several intermolecular distances is presented in Fig. 6. The FIG. 6. The total electronic energy in the linear HF NCH complex. The solid line corresponds to the supermolecule Kohn Sham calculations, other lines corresponding to the freeze-and-thaw cycle of KSCED equations. In KS and KSCED calculations, the exchange functional of Becke Ref. 33 and correlation energy functional of Perdew Ref. 34 were used. The approximate kinetic energy functionals are described in the text. energy calculated using the KS method for the whole complex serves as a reference for the KSCED energies obtained using different kinetic energy approximate functionals (T s app ). All considered kinetic energy functional parametrizations lead to the interaction energy curves deviating from the one obtained using the supermolecule KS calculations. The most severely overestimated KSCED energies were obtained with the T TF functional. It appears that the dominant contribution to this inaccuracy is due to the overestimated values of the nonadditive kinetic energy calculated with the T TF functional see Fig. 7. This is in accordance with the findings of Lacks and Gordon who analyzed this component to the interaction energy for Ne and He dimers. 11 All other kinetic energy functional parametrizations lead to FIG. 7. The nonadditive kinetic energy component (T s nadd 1, 2 ) of the total energy in the linear HF NCH complex in hartrees for several intermolecular distances. Each curve corresponds to different approximate kinetic energy functionals and electron densities 1 and 2 at the end of the freeze-and-thaw KSCED cycle. The approximate kinetic energy functionals are described in the text.

7 9188 Wesolowski, Chermette, and Weber: Approximate kinetic energy functionals TABLE I. Equilibrium geometry parameters for the linear HF NCH complex. In KS and KSCED calculations, the exchange functional of Becke Ref. 33 and correlation energy functional of Perdew Ref. 34 were used. The approximate kinetic energy functionals are described in the text. Method R NF nm E int kj/mol KSCED/TF KSCED/LLP KSCED/PW KSCED/LC KSCED/T KS ab initio Ref Exp significantly better energies. The most accurate KSCED energies were obtained with the T LC functional, as expected on the basis of the previous analysis of the electron densities. The equilibrium parameters are listed in Table I. It is noteworthy to note that the T T92 functional, although leading to significantly worse KSCED electron density than the T LC one, leads to almost as accurate KSCED energies as the T LC functional for large intermolecular distances. To stress the importance of the gradient dependent terms in T nadd s 1, 2 the numerical values of the integrand tnad(r) T nadd s 1, 2 tnad(r)dr are reported in Figs. 8 a d. The integrand values were calculated along the NF axis for the geometry corresponding to the intermolecular distance equal to nm. Each figure corresponds to a given kinetic energy functional parametrization and includes: the integrand tnad R, its gradient independent component tnad TF R which is obtained when the F(s) is substituted by a constant equal to 1.0, and its gradient contribution which is equal to the difference tnad R tnad TF R. The maxima of the tnad R correspond to the positions of F, H HF, N,and C atoms. All AKFs considered here lead to very similar values of tnad R everywhere except in the vicinity of the H HF atom. It appears that the gradient dependent components of the tnad R are significant only in the region of large overlap between the two building fragments of the supermolecule. On the basis of Fig. 8, one can conclude that the gradient dependent terms in the T nadd s 1, 2 are less important if either the 1 or the 2 are small. The contributions to the T nadd s 1, 2 of regions where 1 / 2 1 or where 2 / 1 1, although significant, may be accurately approximated using the T TF functional. FIG. 8. The values of the integrand tnad(r)(t s nadd 1, 2 tnad(r)dr) calculated along the FN axis calculated along the FN axis calculated using difference approximate kinetic energy functionals X H HCN nm, X C nm, X N 0.0, X F 0.29 nm, and X H HF nm. The approximate kinetic energy functionals are described in the text.

8 Wesolowski, Chermette, and Weber: Approximate kinetic energy functionals 9189 VII. CONCLUDING REMARKS The comparison of the HF NCH ground-state electron density obtained using the KS approach and the freeze-andthaw cycle of KSCED calculations give direct information on the accuracy of the functional derivative of T s nadd 1, 2 ( T s nadd 1, 2 / 1 ) used in the KSCED equations which is calculated using the AKF. This comparison shows that for T LC, T LLP, and T T92 functionals the difference between the KS and KSCED densities increases with decreasing intermolecular distance. This difference is very small in the case of the T LC functional, which is thus superior to the others and leads to most accurate energies for intermediate around the energy minimum and large distances. The T PW86 and the T TF functionals do not exhibit such a trend. The densities obtained with these functionals are very similar to each other and they differ most significantly from the KS ones for intermediate molecular distances. The total KSCED energy calculated with the T PW86 functional is more accurate then the KSCED energy calculated with the T TF functional for all studied geometries. This is mostly due to the fact that the T TF functional overestimates the T s nadd 1, 2. The T PW86 functional leads to worse total energies then the T LC functional. These results suggest that the T LC is the functional of choice for the KSCED method. The high accuracy of both the density and the energy in the region of intermediate and large intermolecular distances makes it especially suitable in the KSCED formalism for studying molecular interactions. With this functional, the KS and KSCED interaction energies differ less then 0.9 kj/mol, both overestimating the experimental value energy by about 4 kj/mol. The apparent superiority of the T LC functional over T LLP and T T92 is probably due to its better scaling properties. The T LC and T T92 functionals were developed starting from the T LLP one and taking into account the scaling properties in a different way. In the case of T T92 this was done by adding to the T LLP functional a term C*bs /(1 2 5/3 bs ) and fitting the constant (C). For positive values of the constant (C), this term exhibits a correct scaling behavior. Opposite to the T T92 functional, the whole T LC one was developed taking into account scaling properties. It is worthwhile underlining that the T T92 functional contains three parameters fitted in order to reproduce the kinetic energies of several molecules. Indeed, it is a linear combination of two terms, one is the T LLP functional and the other one is the third term in the T OL2 functional. Therefore, it should obey most of the scaling relations satisfied by either the T OL2 or T LLP functionals. However, one can note see Eqs. 10 and 13 that the T T92 functional overweighs the gradient dependent terms of the T LLP functional and therefore contains a negative weight for its T OL2 contribution. Whereas this leads to quite good kinetic energies an integral number, the corresponding potential is much more sensitive to the existence of this minus sign. Indeed one can see from Eq. 13 that the negative term is important only for a small range of s values and that the asymptotic behavior of the T T92 functional for large s is determined by the T LLP component. The behavior of the enhancement factor F at large s i.e., at small densities is an other feature that clearly distinguishes functionals of the T PW91 analytical form from the other ones in this paper. Whereas the enhancement factors F LLP, F PW86, and F T92 are increasing functions of s in the whole range, F PW91 exhibits a different behavior, which makes it more local i.e., gradient dependent terms become negligible at large s see Fig. 1 in Ref. 38 and Eqs Since the difference between F PW91 and F LC lies in the parametrization which comes from the exchange energy of the electron gas, and from kinetic energies of atoms, respectively, F LC behaves similarly to F PW The T PW86 functional, although it leads to a better KSCED electron density at very short distances, leads to much worse densities than the T LC functional for intermediate and large distances. Also, the total energy obtained with this functional is less accurate for all studied geometries. Among studied functionals, this one appears to be the most sensitive to the numerical integration details. The studies presented in this paper were a continuation of similar studies 12 of the performance of different AKFs in a different molecular complex. In Ref. 12 it was shown that the T PW86 functional leads to more accurate KSCED electron density than both the T LLP and the T TF functionals for a very short intermolecular distance large 1, 2 -overlap. The results of the current paper indicate the same trend for large 1, 2 -overlap geometries. The conclusion of this paper, namely that the T LC functional used in the freeze-and-thaw cycle of the KSCED equations leads to very similar electron density and energy to the ones obtained by means of the KS method, was drawn based on one molecular complex. Since both the analytic form of the T LC functional as well as its free parameters are general, one can expect that its superiority stands for other molecular complexes which are characterized by similar 1, 2 -overlaps. The performance of the T LC functional in KSCED calculations of other molecular complexes is currently studied in our laboratory. Finally, we would like to comment on the computational advantages of the KSCED equations over the standard supermolecule KS calculations. It is important to note that the KSCED formalism makes it possible to introduce additional approximations applicable for the particular intermolecular complex under investigation. These approximations may involve expanding 1 and 2 using a limited number of basis functions or improving the initial guess for 2 and terminating the freeze-and-thaw cycle after the first step. Since the aim of this paper was a selection of most accurate approximate kinetic energy functional none of the above simplifications have been made. As the result, the cumulative CPU times of the KSCED calculations were of the same order of magnitude as the standard supermolecule KS calculations. When, however, the aforementioned simplifications are made as in Refs. 40 and 41, the factor reducing the time of the SCF calculations equals approximately 2*N 3 nf /(N nf N f ) 3, where N nf and N f is the number of basis functions used for the expansion of the nonfrozen and frozen density, respectively. The power 3 reflects the scaling of the

9 9190 Wesolowski, Chermette, and Weber: Approximate kinetic energy functionals canonical SCF procedure in the KS equations and the factor 2 reflects the additional computation time needed for calculation of the kinetic energy component of the effective potential in Eq. 2. The value of this factor was numerically confirmed for all gradient dependent kinetic energy functionals considered in this paper. ACKNOWLEDGMENTS The authors are grateful to Professor D. R. Salahub for providing a copy of the demon program. Financial support by the Federal Office for Education and Science, acting as Swiss COST office, is gratefully acknowledged. This work is also a part of the Project No of the Swiss National Science Foundation. 1 W. Kohn and L. J. Sham, Phys. Rev. 140, A R. Friesner, Chem. Phys. Lett. 116, W. Yang, Phys. Rev. Lett. 66, Y. S. Kim and R. G. Gordon, J. Chem. Phys. 56, R. Parr and W. Yang, in Density Functional Theory of Atoms and Molecules Oxford University Press, New York, 1989, pp T. A. Wesolowski and A. Warshel, J. Phys. Chem. 97, P. Cortona, Phys. Rev. B 44, E. V. Stefanovitch and T. N. Truong, J. Chem. Phys. 104, Embeded Density Functional EDFT corresponds to Eqs. 1 4 in this paper. 9 T. A. Wesolowski and J. Weber, Chem. Phys. Lett. 248, L. J. Bartolotti and P. K. Acharaya, J. Chem. Phys. 77, D. J. Lacks and R. G. Gordon, J. Chem. Phys. 100, E k corresponds to T s nadd 1, 2 in this paper. 12 T. A. Wesolowski and J. Weber, Intl. J. Quant. Chem. in press. 13 D. A. Kirzhnits, Sov. Phys. JETP, 5, C. H. Hodges, Can. J. Phys. 51, C. F. von Weizsaecker, Z. Phys. 96, R. Baltin, J. Chem. Phys. 86, D. R. Murphy, Phys. Rev. A 24, D. R. Murphy, Ph.D. thesis, University of North Carolina, E. W. Pearson, Ph.D. thesis, Harvard University, A. E. DePristo and J. D. Kress, Phys. Rev. A 35, G. I. Plindov and S. K. Pogrebnaya, Chem. Phys. Lett. 143, H.Ou-Yang and M. Levy, Intl. J. Quant. Chem. 40, H. Lee, C. Lee, and R. G. Parr, Phys. Rev. A 44, J. P. Perdew and Y. Wang, Phys. Rev. B 33, J. P. Perdew and Y. Wang, in Electronic Structure of Solids 91, edited by P. Ziesche and H. Eschrig Academie Verlag, Berlin, 1991, p A. Lembarki and H. Chermette, Phys. Rev. A 50, P. Fuenteabla and O. Ryes, Chem. Phys. Lett. 234, A. J. Thakkar, Phys. Rev. A 46, A. C. Legon, D. J. Miller, and S. C. Rogers, P. Roy. Soc. London, Ser. A 370, P. Hobza and R. Zahradnik, in Intermolecular Complexes. The Role of van der Waals Systems in Physical Chemistry and the Biodisciplines Elsevier, Amsterdam-Oxford-New York-Tokyo, 1988, p A. St-Amant, Ph.D. thesis, University of Montreal, N. Godbout, D. R. Salahub, J. Andzelm, and E. Wimmer, Can. J. Chem. 70, A. D. Becke, Phys. Rev. A 38, J. P. Perdew, Phys. Rev. B 33, Although for large s the enhancement factor F increases as s 2/5,itmay diverge because the sum of s 2, s 4, and s 6 terms is calculated before taking the 1/15 power. To avoid this catoffs are used. 36 H. Chermette, A. Lembarki, P. Gulbinat, and J. Weber, Intl. J. Quant. Chem. 56, D. J. Lacks and R. G. Gordon, Phys. Rev. A 47, R. Neumann, R. H. Nobes, and N. C. Handy, Mol. Phys. 87, H. Chermette unpublished. 40 T. A. Wesolowski and A. Warshel, J. Phys. Chem. 98, T. A. Wesolowski, R. P. Muller, and A. Warshel, J. Phys. Chem. 100,