Introduction of the explicit long-range nonlocality as an alternative to the gradient expansion approximation for the kinetic-energy functional

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Introduction of the explicit long-range nonlocality as an alternative to the gradient expansion approximation for the kinetic-energy functional

TRAN, Fabien, WESOLOWSKI, Tomasz Adam

Abstract

The approximate nonempirical kinetic-energy functional proposed by Tal and Bader is analyzed for polyatomic systems. The performance of this functional and the functionals derived from the gradient expansion approximation truncated to zeroth, second, and fourth order is investigated for a testing set of 68 neutral and charged molecules. It is shown that the Tal–Bader functional, despite the simplicity of the idea behind its construction, leads to significantly better total kinetic energies than the gradient expansion approximation functionals. The local behavior of the kinetic-energy density derived from the Tal–Bader functional is also discussed.

TRAN, Fabien, WESOLOWSKI, Tomasz Adam. Introduction of the explicit long-range nonlocality as an alternative to the gradient expansion approximation for the kinetic-energy functional. Chemical Physics Letters , 2002, vol. 360, no. 3-4, p. 209-216

DOI : 10.1016/S0009-2614(02)00852-7

Available at:

http://archive-ouverte.unige.ch/unige:3233

Disclaimer: layout of this document may differ from the published version.

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Introduction of the explicit long-range nonlocality as an alternative to the gradient expansion approximation for

the kinetic-energy functional

Fabien Tran, Tomasz A. Wesołowski

^*

Department of Physical Chemistry, University of Geneva, 30 quai Ernest-Ansermet, CH-1211 Geneva 4, Switzerland Received 15 April 2002

Abstract

The approximate nonempirical kinetic-energy functional proposed by Tal and Bader is analyzed for polyatomic systems. The performance of this functional and the functionals derived from the gradient expansion approximation truncated to zeroth, second, and fourth order is investigated for a testing set of 68 neutral and charged molecules. It is shown that the Tal–Bader functional, despite the simplicity of the idea behind its construction, leads to signiﬁcantly better total kinetic energies than the gradient expansion approximation functionals. The local behavior of the kinetic- energy density derived from the Tal–Bader functional is also discussed. Ó 2002 Elsevier Science B.V. All rights reserved.

1. Introduction

Following the Hohenberg–Kohn theorems, all ground-state properties of a given N-electrons system are determined by the electron densityqðrÞ.

In particular, the ground-state energy can be expressed as a functional ofqðrÞ, which is conven- tionally written as [1,2]

E½q ¼T½q þV_ee½q þ Z

vðrÞqðrÞdr; ð1Þ

whereT½qandVee½q, are the kinetic and the electron–electron repulsion energy, respectively, and the last term is the energy due to the external potential vðrÞ. Except for some particular cases, the

exact form of the analytic dependence ofT½qand V_ee½qon qðrÞis unknown. To overcome the problem of the unknown form of T½q, which is a nu- merically signiﬁcant component of the total energy, Kohn and Sham [3] introduced an auxiliary set of orbitalsw_iðrÞrepresenting noninteracting particles.

In the Kohn–Sham formalism, the total electronic energy is represented with the following functional:

E½q ¼T_s½q þ1 2

Z Z qðrÞqðr⁰Þ

jr r⁰j drdr⁰þE_xc½q þ

Z

vðrÞqðrÞdr; ð2Þ

where

T_s½q ¼T_s½fw_ig ¼1 2

X^N

i¼1

Z

jrw_iðrÞj²dr: ð3Þ

Chemical Physics Letters 360 (2002) 209–216

www.elsevier.com/locate/cplett

*Corresponding author. Fax: +41-22-702-6518.

E-mail address:[email protected](T.A.

Wesołowski).

PII: S 0 0 0 9 - 2 6 1 4 ( 0 2 ) 0 0 8 5 2 - 7

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The success of the Kohn–Sham theory as a method to approximate the exact solution of the Schr€oodinger equation, originates from the fact that the only part of the total energy which needs to be approximated, the exchange–correlation energy E_xc½q, is small compared to the total energy.E_xc½q collects all the energetic eﬀects arising from the Pauli exclusion principle and the Coulomb correlation. It is also important to point out thatE_xc½q includes also the component Tc½q ¼T½q Ts½q originating from the eﬀect of electron correlation on the kinetic energy. Tc½q is positive and small compared to the whole T½q.

The introduction of the orbital-dependent kinetic energy T_s½fw_igleads, however, to a formalism in which the number of equations to solve is proportional to the number of electrons. There- fore, computer modeling of large nonperiodic systems remains still a challenging problem. One can envisage an equivalent orbital-free formalism, in which the variational principle leads to the N- independent Euler–Lagrange equation (see [4] for instance). Due to the lack of good approximations for the kinetic-energy functional, a practical im- plementation of such an orbital-free formalism remains a challenge although a signiﬁcant progress has been made in this area leading to computa- tional techniques applicable to large metal clusters [5] and in materials science [6–8].

Approximate orbital-free kinetic-energy functional is also a crucial quantity in the subsystem formulation of density functional theory introduced by Cortona [9] to study solids. This formulation is the basis of the ﬁrst-principle based embedding potential introduced by Wesolowski and Warshel [10] to study embedded molecules in such a way that the molecule under investi- gation and its microscopic environment can be described using approximate theories of diﬀerent level of accuracy (for recent applications, see [11,12]).

In the literature, various approximate functionals have been proposed. They diﬀer in their origin, as diﬀerent theoretical strategies have been applied in various groups to derive a good approximate functional (gradient expansion [13–15], Padee approximation [16], scaling properties [17], conjointness with the exchange functional [18],

linear response theory [6–8]) as well as in the use of empirical parameters in their construction.

In this work, we analyze the performance of the functional introduced by Tal and Bader [19]. The test of this functional on atomic systems reported in the original paper of Tal and Bader showed that this functional performs rather well although no empirical parameters have been used in its construction. We apply the Tal–Bader functional to the case of polyatomic systems and compare its numerical performance with that of other ‘empirical-parameters-free’ functionals derived from the gradient expansion approximation (GEA). The Tal–Bader functional incorporates the long-range nonlocality as the kinetic-energy density at a given pointrdepends not only on the electron density at r(and in its inﬁnitesimal vicinity as in the case of gradient-dependent functionals) but also on the electron density at the position of the nuclei.

2. Kinetic-energy functionals

There are several possible theoretical roads leading to approximate kinetic-energy functionals T_s½qof various level of accuracy. The most relevant to the main issue of this work is the gradient expansion approximation [13–15]. For slowly varying densities, the kinetic energy can be developed in a series involving the densityqðrÞand its derivatives:

T_GEA½q ¼X¹

n¼0

T_2n½q

¼X¹

n¼0

Z

t_2nðqðrÞ;rqðrÞ;. . .Þdr; ð4Þ

t₀¼ 3

10ð3p²Þ²⁼³q⁵⁼³; ð5Þ

t₂¼ 1 72

jrqj²

q ; ð6Þ

t₄¼ð3p²Þ^ð2=3Þ

540 q¹⁼³ r²q q ²

9 8

r²q q

jrqj q 2

þ1 3

jrqj q 4!

: ð7Þ

210 F. Tran, T.A. Wesołowski / Chemical Physics Letters 360 (2002) 209–216

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As shown by Thomas and Fermi (TF) [20,21] the termT₀½qis the exact kinetic-energy functional for the uniform electron gas, but represents a rather poor approximation for atomic and molecular densities, leading to an underestimation of 5–10%

of the total kinetic energy. Adding to T₀½q the second term,T₂½q, leads to the second order gradient expansion which is known to underestimate the kinetic energy by about 0.5–1%, whereas addingT2½q þT4½qresults in an overestimation of the kinetic energy of a similar magnitude. The analytic form ofT6½q is known [15] but is diver- gent for exponentially decaying densities as those of atoms and molecules. The term appearing in the second order of GEA (but with another coeﬃ- cient) was ﬁrst derived by von Weizs€aacker [22]:

TW½q ¼9T2½q ¼1 8

Z jrqðrÞj²

qð Þr dr: ð8Þ

T_W½q yields the exact kinetic energy for one-electron systems and systems with two electrons oc- cupying the same orbital, and the density ð1=8Þjrqj²=qbehaves correctly both at the nuclear cusps and at the long distance from the nuclei [19].

For these reasons, several functionals with TW½q as the leading term were proposed in the literature (see for example [23,24]).

Due to the limited accuracy of the GEA kinetic- energy functionals various approximate functionals have then been developed. Among them, there is the group of the semi-local ones belonging to the generalized gradient approximation (GGA) which can be cast into the form

T_GGA½q ¼ 3

10ð3p²Þ²⁼³ Z

q⁵⁼³ðrÞFðsðrÞÞdr; ð9Þ

wheres¼ jrqj=ð2qkFÞis the scaled density gradient withkF¼ ð3p²qÞ¹⁼³the local Fermi vector and FðsÞthe enhancement factor which determines the gradient-dependency of the functional. Diﬀerent analytical forms for FðsÞ have appeared in the literature, such as the Padee approximation of DePristo and Kress [16] or the two approximate functionals proposed by Ou-Yang and Levy [17]

derived from nonuniform scaling properties considerations. Yet another route to deriveFðsÞ was proposed by Lee et al. [18] who conjectured the

conjointness between the kinetic- and exchange- energy functionals. This conjecture opened new possibilities in constructing approximate gradient- dependent kinetic-energy functionals proﬁting from the progress in the development of approxi- mating the exchange-energy functional [18,25–28].

For the exchange–correlation energy, GGA is a very successful approximation, but the kinetic-energy functionals of the GGA form seem to have several ﬂaws which prevents us to reach desired level of accuracy and especially in formalisms where a good functional derivative of the kinetic energy is required. For recent studies on GGA kinetic-energy functionals, see [29–31].

In this work, we explore an alternative route to approximate the kinetic-energy functional which can potentially overcome the shortcomings of truncated GEA or other semi-local functionals.

We use the functional proposed by Tal and Bader (TB) [19] which takes the form

T_TB½q ¼T_TF½q_s þT_W½q_s þX^N^nuc

a¼1

T_W½q_ra; ð10Þ

withq_raðrÞandq_sðrÞbeing deﬁned by

q_raðrÞ ¼qðRaÞe^2Z^a^{jr R}^a^j ð11Þ and

q_sðrÞ ¼qðrÞ q_rðrÞ ¼qðrÞ X^N^nuc

a¼1

q_raðrÞ; ð12Þ

respectively, where R_a and Z_a are the position and the nuclear charge of the nucleus a in the molecule. q_sðrÞ and q_rðrÞ denote the slowly and rapidly varying components of the total electron density.

It is worthwhile to point out that application of the Tal–Bader functional to polyatomic systems is not straightforward. Due to the inﬁnite range of the exponential function, the q_sðrÞ component deﬁned in Eqs. (11) and (12) cannot be assumed to be always positive (particularly at the position of the nuclei). To avoid using unphysical quantities, we applied an alternative partitioning ofqðrÞinto q_sðrÞandq_rðrÞ:

q_sðrÞ ¼max 0;qðrÞ X^N^nuc

a¼1

q_raðrÞ

!

; ð13Þ

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and consequently

q_rðrÞ ¼qðrÞ q_sðrÞ: ð14Þ This new partitioning makes it possible to apply Eq. (10) not only for atoms but also for polyatomic systems as both componentsq_sðrÞandq_rðrÞ are always positive.

To our knowledge, the only systematic test of the Tal–Bader functional was made by its original authors [19] for atomic systems. Hernaandez and Gaazquez [32] applied the functional in the frame-

work of variational calculations with minimal- basis-set-type electron densities.

3. Results

68 neutral and charged molecules (see Table 1) consisting of second, third, and fourth period atoms in their experimental geometries were chosen to compare the performance of the Tal–Bader functional with that derived from GEA. The ref-

Table 1

The 68 molecules of the testing set

Be2, B2, C2, N2, O2, F2, Na2, Al2, Si2, P2, S2, Cl2, K2, Cr2, Cu2,

BeF, BeS, BO, BF, BCl, CN, CO, CF, CP, CS, NO, NF, NS, NBr, NaF, NaCl, MgO, SiO, SiF, SiS, PN, PO, PS, SO, ClO, ClF, KF, KCl,

BF3, BCl3, CO2, CS2, N2O, NO2, NCO, O3, OCS, F2O, SiF2, SO2, ClNO, ClO2, C^þ₂, C₂, N^þ₂, O^þ₂, O₂, P^þ₂, S^þ₂, Cl^þ₂, NO , PO , CO₂

Fig. 1. Exact and Tal–Bader kinetic-energy density plotted along the internuclear axis of the CO molecule with the C atom atr¼0 and the O atom atr¼2:13 a.u.

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erence electron densities and Kohn–Sham orbitals of each molecule in the testing set were derived using the PW91 [33] exchange–correlation energy functional. The Kohn–Sham orbitals were used to evaluate the exact value of the noninteracting kinetic energy T_s^exact¼T_s½fw_ig (Eq. (3)), and the densities to evaluate the value of the diﬀerent approximate orbital-free expressions T_s½q. For spin-polarized densities, a standard Ansatz was used [34]:

Ts½q_";q_# ¼¹₂Ts½2q_"

þTs½2q_#

: ð15Þ

For each molecule and each considered approximate functional, the percentage of error D¼100ðTs½q T_s^exactÞ=T_s^exact was calculated, and then the mean percentage of error D, the mean percentage of absolute errorDabs, and the standard deviation Dstd were calculated.

Table 2 shows that the TF functional provides, as expected, a very poor approximation leading to

Fig. 2. Exact and Tal–Bader kinetic-energy density plotted along the internuclear axis of the N2molecule with one of the N atom at r¼0 and the other atr¼2:07 a.u.

Table 2

Performance of various approximate kinetic-energy functionals evaluated with the set of 68 molecules of Table 1

Functional D Dabs Dstd

GEA, 0th order, Eqs. (4) and (5) )8.3581 8.3581 1.0992

GEA, 2nd order, Eqs. (4)–(6) )0.7303 0.7303 0.2115

GEA, 4th order, Eqs. (4)–(7) 0.7506 0.7506 0.3067

Tal–Bader, Eqs. (10), (13) and (14) )0.0833 0.3751 0.6389

For the total kinetic energy,Dis the mean error,Dabsis the mean absolute error, andDstdis the standard deviation. All the values are expressed in percentages.

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total kinetic energy underestimated by about 8%.

Taking into account the following terms in the gradient expansion signiﬁcantly reduces the error:

GEA in second order underestimates the kinetic energy by about 0.7%, while GEA in fourth order overestimates it by 0.7%. Employing the Tal–Ba- der functional reduces the value of Dabs to about 0.4%, while D is reduced to the small value of )0.08%, indicating that the Tal–Bader functional is not showing a particular trend to underestimate or overestimate the total kinetic energy. The value of 0.6 forDstd indicates a larger dispersion around Dthan for GEA (second and fourth order).

Turning back to the Tal–Bader functional, its authors noted that the kinetic-energy density ð1=2ÞPN

i¼1jrw_iðrÞj² (this choice for the deﬁnition of the kinetic-energy density implies it to be ev- erywhere positive and ﬁnite) is represented exactly by the von Weizs€aacker term (Eq. (8)) on top of a nucleus and in the asymptotic regions far from the

nuclei. This correct local behavior of the von Weizs€aacker term, motivated Tal and Bader to propose their functional in which the von We- izs€aacker term plays a key role. The local behavior of the exact and Tal–Bader kinetic-energy density (tðrÞ) of CO and N2 molecules is shown in Figs. 1 and 2, respectively. For the CO molecule, tðrÞ is plotted along the internuclear axis with the C atom atr¼0 and the O atom atr¼2:13 a.u. The Tal–

Bader kinetic-energy density reproduces well the exact one in the regions close to the nuclei whereas it slightly overestimates the exact quantity in the mid-bond region. Fig. 2 shows similar trends for the N2molecule and provides also a good example of the correct local behavior of the Tal–Bader functional in the asymptotic region far from nuclei (r! 1).

The good performance of the Tal–Bader functional prompted us to explore its applicability for such cases where considerations underlying the

Fig. 3. The densityqand its rapidlyqrand slowlyqsvarying components in the BH molecule with the B atom atr¼0 and the H atom atr¼2:33 a.u.

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separation of qðrÞ into q_sðrÞ and q_rðrÞ are less applicable. For one- and two-electron atoms, the Tal–Bader construction is not an adequate scheme as the von Weizs€aacker term provides the exact functional. For such cases, the Tal–Bader construction leads obviously to a worse approximation as indicated by the fact that in their original work Tal and Bader showed that their functional performs not as good for the helium atom as it does for heavier atoms. Moreover, the separation into rapidly and slowly varying densities correlates with the fact that the former is determined by the core electrons in the case of heavier atoms. The Tal–Bader functional can be expected, therefore, to be less applicable for hydrogen-containing molecules (not considered in our testing set), which is in fact, what we found. This originates from the qualitative differences in partitioning ofqðrÞ into q_sðrÞandq_rðrÞ. As indicated before, for hydrogen- free molecules, Eqs. (11) and (12) lead to a negative and very small value of q_sðrÞonly at atomic nuclei, resulting in negligible numerical signifi- cance of our modification of the partitioning scheme (Eqs. (13) and (14)) which enforces nonnegativity ofq_sðrÞ. For hydrogen-containing molecules, Eqs. (11) and (12) can lead to negative values of q_sðrÞ in extended regions. Fig. 3 shows the Tal–Bader partitioning scheme ofqðrÞfor the BH molecule, where we can see negative values for q_sðrÞ occurring in the vicinity of the H atom (at r¼2:33 a.u.) in the direction opposite to the B atom (at r¼0). This leads in practice to a very strong dependence of the accuracy of the Tal–

Bader functional on the way the nonnegativity of q_sðrÞis enforced.

4. Conclusions

The numerical results presented in this work show that the Tal–Bader functional applied to polyatomic systems has proven to be an interesting alternative to the gradient expansion approximation. It is based on the separation between the rapidly varying component of the electron density determined by the inner core electrons and the remaining part which is smoother. The overall accuracy of the Tal–Bader functional for the total

kinetic energy is better than the ones obtained with the GEA functionals. Formally, the functional considered in this work can be seen as a simple step towards introduction of long-range nonlocality:

the kinetic-energy density at a given point, obtained using such a functional, is determined not only by the electron density at this point (and possibly in its inﬁnitesimal vicinity) but also by the value of the electron density at several distant points. Missing true nonlocality is an important ﬂaw of local and semi-local approximate functionals (see for example Herring [35] who pointed out the highly nonlocal behavior of the kinetic energy or the recent paper by Wang et al. [30]

exposing ﬂaws of GGA kinetic-energy functionals).

Acknowledgements

This work is part of the Project 21-63645.00 of the Swiss National Science Foundation. T.A.W.

acknowledges the computer equipment grants from Fonds Freedeeric Firmenich et Philippe Chuit andFondation Ernst et Lucie Schmidheiny.

References

[1] P. Hohenberg, W. Kohn, Phys. Rev. 136 (1964) B864.

[2] R.G. Parr, W. Yang, Density-Functional Theory of Atoms and Molecules, Oxford University Press, New York, 1989.

[3] W. Kohn, L.J. Sham, Phys. Rev. 140 (1965) A1133.

[4] G.K.-L. Chan, A.J. Cohen, N.C. Handy, J. Chem. Phys.

114 (2001) 631.

[5] M.D. Glossman, A. Rubio, L.C. Balbaas, J.A. Alonso, Int.

J. Quantum Chem.: Quantum Chem. Symp. 26 (1992) 347.

[6] L.-W. Wang, M.P. Teter, Phys. Rev. B 45 (1992) 13196.

[7] E. Smargiassi, P.A. Madden, Phys. Rev. B 49 (1994) 5220.

[8] Y.A. Wang, N. Govind, E.A. Carter, Phys. Rev. B 58 (1998) 13465;

Phys. Rev. B 60 (1999) 17162(E);

Phys. Rev. B 64 (2001) 129901(E).

[9] P. Cortona, Phys. Rev. B 44 (1991) 8454.

[10] T.A. Wesolowski, A. Warshel, J. Phys. Chem. 97 (1993) 8050.

[11] T.A. Wesolowski, A. Goursot, J. Weber, J. Chem. Phys.

115 (2001) 4791.

[12] J.R. Trail, D.M. Bird, Phys. Rev. B 62 (2000) 16402.

[13] D.A. Kirzhnits, Sov. Phys. JETP 5 (1957) 64.

[14] C.H. Hodges, Can. J. Phys. 51 (1973) 1428.

[15] D.R. Murphy, Phys. Rev. A 24 (1981) 1682.

(9)

[16] A.E. DePristo, J.D. Kress, Phys. Rev. A 35 (1987) 438.

[17] H. Ou-Yang, M. Levy, Int. J. Quantum Chem. 40 (1991) 379.

[18] H. Lee, C. Lee, R.G. Parr, Phys. Rev. A 44 (1991) 768.

[19] Y. Tal, R.F.W. Bader, Int. J. Quantum Chem.: Quantum Chem. Symp. 12 (1978) 153.

[20] L.H. Thomas, Proc. Cambridge Philos. Soc. 23 (1927) 542.

[21] E. Fermi, Z. Phys. 48 (1928) 73.

[22] C.F. von Weizs€aacker, Z. Phys. 96 (1935) 431.

[23] P.K. Acharya, L.J. Bartolotti, S.B. Sears, R.G. Parr, Proc.

Natl. Acad. Sci. USA 77 (1980) 6978.

[24] B.M. Deb, S.K. Ghosh, Int. J. Quantum Chem. 23 (1983) 1.

[25] A.J. Thakkar, Phys. Rev. A 46 (1992) 6920.

[26] D.J. Lacks, R.G. Gordon, J. Chem. Phys. 100 (1994) 4446.

[27] A. Lembarki, H. Chermette, Phys. Rev. A 50 (1994) 5328.

[28] P. Fuentealba, O. Reyes, Chem. Phys. Lett. 232 (1995) 31.

[29] G.K.-L. Chan, N.C. Handy, J. Chem. Phys. 112 (2000) 5639.

[30] B. Wang, M.J. Stott, U. von Barth, Phys. Rev. A 63 (2001) 052501.

[31] S.S. Iyengar, M. Ernzerhof, S.N. Maximoﬀ, G.E. Scuseria, Phys. Rev. A 63 (2001) 052508.

[32] E. Hernaandez, J.L. Gaazquez, Phys. Rev. A 25 (1982) 107.

[33] J.P. Perdew, J.A. Chevary, S.H. Vosko, K.A. Jackson, M.R. Pederson, D.J. Singh, C. Fiolhais, Phys. Rev. B 46 (1992) 6671.

[34] G.L. Oliver, J.P. Perdew, Phys. Rev. A 20 (1979) 397.

[35] C. Herring, Phys. Rev. A 34 (1986) 2614.