J. Phys. Chem. 1994,98, 3688-3693
3688
Kinetic Energies and Exchange Model Potentials Calculated with a Simple Analytical Representation of Atomic Densities L. Fernandez Pacios Dep. Quimica y Bioquimica, E TSI Montes, Universidad Politecnica de Madrid, 28040 Madrid, Spain Received: November 24, 1993; In Final Form: January 24, 1994"
A simple analytical representation of H a r t r e e F o c k atomic densities recently devised defines p ( r ) as a sum of exponentially decaying functions. This basis set-independent representation is here applied to the calculation of kinetic energies, making use of various well-known kinetic energy functionals. Despite its extreme simplicity, these functions provide highly reliable representations of p ( r ) , allowing the calculation of kinetic energies to within 1% or less. This representation of the density is also analyzed in exchange-only functional calculations. A semiempirical model exchange potential developed before (Pacios, L. F. J . Plays. Chem. 1992, 96, 7294) is here modified and analyzed. This potential presents three valuable features: reproduces the exact exchange energies, exhibits the correct asymptotic trend when r - 03, and displays a spatial behavior close to the numerical exchange optimized potential model by Talman et al.
Introduction Density functional (DF) theory provides a methodology to study atomic and molecular systems in terms of the electron density p ( r ) instead of the many-particle wave function P. Recent conceptual advances along with parallel software developments have led to greatly increased use of D F procedures in atomic and molecular calculations.1 The Hohenberg-Kohn existence theorem2 provides the basic framework on which D F formalisms are developed. However, since the exact E[p] is not known, the use of approximations is inherent to such treatments. This poses the need to evaluate the reliability and effectiveness of DF calculations, comparing their results with reference values, ab initio or experimental. Aside from the variety of exchange correlation and kinetic energy functionals proposed, the practical question remains as to whether the electron density is represented in terms of basis functions or numerically, which is of interest in quantum chemical a p p 1 i ~ a t i o n s . l ~ ~ ~ ~ A crucial issue of D F applications is precisely the quality of the description of p ( r ) . To avoid all the undesired sources of deviations typically associated with basis sets, a number of D F calculations have resorted to numerical procedures. Thus, one excludes from the comparative studies the need to explore also the influence of distinct bases on different properties of atomic and molecular systems. This approach, however, presents the obvious inconveniences associated with numerical methods. From this perspective, I have recently devised5v6a very simple analytical representation of Hartree-Fock (HF) atomic densities,' intended for DF-type application in quantum chemical studies of large molecular systems. These analytically modeled densities, hereafter denoted AMD, are defined by a sum of few (from 1 for I jto 4 for Na-Kr atoms) exponential terms, and their parameters are fixed by reproducing exactly the HF-limit atomic potential energies (nuclear attraction, electron repulsion and exchange) for every atom. AMD functions, recently published for main group atoms H-Kr,5 constitute a refinement and extension of a previous procedure6 developed to represent analytically the numerical atomic densities obtained with the optimized potential model (OPM) .8 Two issues of importance for further D F applications of the AMD are addressed in this work. The first is treated in the next section and refers to the reliability of these functions for obtaining accurate kinetic energies. Since the conditions imposed to determine the parameters (a brief account follows below) refer 0
Abstract published in Aduonce ACS Abstracts, March 1, 1994.
0022-365419412098-3688$04.50/0
exclusively to potential energies, the performance of AMD when implemented in kinetic energy functionals has not be analyzed. The other issue concerns exchange: a semiempirical exchange potential for use with the AMD was proposed before for the OPM atomic calculations.6 By construction, this analytical model exchange potential fulfills the proper asymptotic behavior a t large radial distances, has a finite value at r = 0, and, in its latest ~ e r s i o nreproduces ,~ "exact" HF exchange energies. In the third section this potential is modified so that only one parameter remains to define it. An interesting correlation between this parameter and the atomic number is presented. Results obtained with AMD functions and the widely used exchange functional by Becke are also analyzed. The paper ends with some relevant conclusions.
Kinetic Energy Functionals The motivation to search simple analytical representations of the atomic electron density p ( r ) was setting a first step toward basis set-free applications of D F methods, especially suited for calculations on large biomolecular systems. The function selected to represent p(r) in a physically meaningful form is a simple sum of exponential terms
This choice is grounded on the well-known monotonic decaying behavior exhibited by the spherically averaged charge density p ( r ) for atoms in their ground state, having only one maximum at the nucleus. Note that the exact solution for hydrogen is the particular case M = 1 with A1 = l/?r and B1 = 2. Details on the size M procedure to fix the parameters and an overall analysis of performance of the model are presented in refs 5 and 6. In brief, the set (A,Bi;i = 1,M) for every atom is obtained under conditions imposed to reproduce HF-limit values7 for nuclear attraction potential energy, VN,,interelectron Coulomb potential energy, VC,and density at the nucleus and radial expectation values ( r p ) , with p = 1 or 2 depending on the size M . These conditions are supplemented with the normalization of the AMD and some criteria to represent the shape of radial distribution functions, 4ar*p(r), a t large distances from the nucleus. The electrostatic potential related with the density (1) via Poisson's equation is also obtained in analytical form; it happens to closely match the electrostatic potential computed with the original H F 0 1994 American Chemical Society
Simple Analytical Representation of Atomic Densities
The Journal of Physical Chemistry, Vol. 98, No. 14, 1994 3689
densities. Final sets of parameters {A&) for H through Kr main group elements were listed in ref 5. If the correlation contribution is initially discarded, the total atomic potential energy is V = + VC+ V,, where V, is the HF exchange energy (also reproduced in our model; see below). AMD functions are then obtained by employing only potential energies without any consideration of kinetic terms. (It is worth pointing out that the correlation energy computed with some widely used functionals is also satisfactorily determined with AMD function^.^) If further molecular applications have to be explored, it is necessary to study the reliability of such approximated p(r) in context not settled in their construction. This analysis is next presented for the determination of kinetic energies using two D F schemes: the gradient expansion on the one side and other recent functionals depending on the local inhomogeneity of the density on the other. The simplest and best known kinetic energy D F is provided by the local Thomas Fermi (TF) theory: in which the kinetic energy of a nonuniform electron gas is approximated in terms of the density p by
T(TF) = (3/ i O ) ( 3 ~ ~ ) ~ / ~di.J p ’ / ~ (As in the rest of the paper, atomic units are used and spinaveraged atomic densities assumed.) Although successful in computing some global properties, the TF model is locally inaccurate since it contains no term to prevent discontinuities in the electron density. The next step in improving the model was taken by von Weizsackerlo by adding a term which yields the kinetic energy in the case of a singly occupied level. This correction led to the so-called inhomogeneity kinetic energy. Subsequently, other authorsllJ2 derived quantum corrections to the TF approximation in a series of ascending powers of gradients of the electron density, so that the total kinetic energy T[p] may be written as T[Pl = To[PI + TJPI
+ T4[P1 +
e..
(3)
where the zeroth-order leading term TO is given by (2). In particular, Kirzhnits” showed that the first-order term is exactly 1/9 of the von Weizsiicker formula, and the first inhomogeneity correction is then (4)
Jones and Young12 confirmed this result, suggesting that the one-ninth correction has to be associated with weakly varying electron gas, whereas the original von Weizsacker term Gust formula 4 with the numerical coefficient 1/8 instead of 1/72) is relevant in strongly inhomogeneous regions of the electron gas. Hodges13 proposed a higher-order quantum extension of the TF approximation following the method of Kirzhnits to derive an equation whose iteration gives corrections to the density matrix in terms of gradients of the potential. The correction to power (4) in the kinetic energy gradient expansion is then obtained in the form
L(B)4] 3 P
di. (5)
The next term, T6[p],was obtained by Murphy,14 who showed that it diverges in atomic systems. Therefore, for atomic charge distributions it makes no sense to carry out the gradient expansion any further than fourth order.I5 Because of the divergence problems in the gradient expansion, other functionals have been recently reported. To assess the
physical validity of our proposed AMD, let us now consider three of the most promising functionals which depend on the local inhomogeneity measure
Depristo and Kress (DK)16 have proposed a PadQtype functional
where
The parameters were fixed by combining theoretical arguments and empirical fitting of HF kinetic energies for noble gases He, Ne, Ar, and Kr, and their values are al = 0.95, a2 = 14.281 11, a3 = -19.579 62, a4 = 9b3, bl = -0.05, b2 = 9.998 02, and b3 = 2.960 85.
Ou-Yang and levy (OL)17corrected the second-order gradient expansion substituting the T4 term by a functional deduced from nonuniform scaling requirements. They proposed two different corrections,
with c = 0.001 87 and D = 0.0245 chosen to fit the HF kinetic energy for some selected atoms. Very recently Lee, Lee, and Parr (LLP)I*extended the widely used Becke exchange functional19 and suggested a kinetic energy functional correction to the Thomas Fermi term in conjunction with the exchange functional. Their result gave place to the proposal
with the parameters a = 0.004 418 8 and y = 0.0253 obtained after imposing minimization of errors respect to HF kinetic energies for He-Ne atoms. In a study on kinetic energies of a large number of polyatomic molecules determined with these and other functionals, Thakkar20 has found this LLP formula to yield the best results. We have computed two sets of kinetic energies using our AMD funotions for p(r) and assuming spherical symmetry for the main group atoms He-Kr. One set corresponds to the gradient expansion truncated as T(G) = TO+ T2 + T4, with these terms given by eqs 2, 4, and 5, respectively. The other set is formed by the kinetic energies T(DK), T(OLl), T(OL2), and T(LLP) as defined by eqs 7, 8, and 9, respectively. All the calculations presented in this paper have been performed with the MATHEMATICA software system.21 Table 1 lists the values of the three terms in the truncated gradient expansion (4) as well as their sum T(G). For all the atoms included in this table, the total kinetic energies agree with HF-limit values to an average 0.65%. This average error reduces to 0.34% if the first three atoms are excluded: the lighter atoms are obviously the worst cases for a statistical theory, and He, Li, and Be are the only atoms showing deviations above 1%, with a maximum of 3.6% for helium. From carbon to krypton no difference greater than 0.6% is found in spite of the rather simple representation provided by our AMD. These results are in
Pacios
3690 The Journal of Physical Chemistry, Vol. 98, No. 14, 1994 TABLE 1: Kinetic Energy Components in the Gradient Expansion UD to Fourth Order, Computed with the A M P
4O
r
I
He Li Be B C N 0 F Ne Na Mg A1 Si P
S C1 Ar
K Ca Ga Ge As Se Br Kr
~
2.5597 6.6958 13.023 21.844 33.506 48.457 66.959 89.670 117.18 147.58 182.52 221.68 264.82 312.78 365.68 423.39 486.26 554.44 626.55 1804.1 1951.7 2107.7 2265.0 2427.3 2593.5
0.3 1798 0.08707 0.81301 0.17749 0.31518 1SO02 0.50348 2.4151 0.72167 3.5236 0.97519 4.8373 1.2546 6.3425 1.5631 8.0385 1.8991 9.9244 2.2016 11.840 2.5962 14.152 3.0411 16.700 3.4765 19.231 3.9564 22.084 4.4684 25.101 5.0305 28.375 5.6013 3 1.860 6.1346 37.213 6.7913 41.365 15.948 102.29 16.946 109.59 18.041 117.73 19.180 125.12 20.460 132.78 21.993 140.13
2.9647 7.6863 14.838 24.763 37.751 54.269 74.556 99.272 129.00 161.62 199.27 241.42 287.53 338.82 395.25 456.79 523.72 591.79 674.71 1922.3 2078.2 2243.4 2409.3 2580.5 2755.6
2.8617 7.4328 14.573 24.529 37.688 54.401 74.809 99.409 128.55 161.86 199.61 241.87 288.85 340.71 397.51 459.46 526.81 599.16 676.76 1923.3 2075.4 2234.2 2399.9 2572.4 2752.0
+ +
0 T(G) = TO 7'2 T4. Hartree-Fock kinetic energies T(HF) taken from ref 7. All energies in atomic units.
TABLE 2: Kinetic Energies Computed with the AMD and Various Density Functionals' T(OL1) T(OL2) T(LLP) T(HF) 2.9069 2.8865 2.8617 2.8773 2.9133 He 7.6012 7.5851 7.5433 7.4328 Li 7.3713 14.573 14.672 14.624 14.701 14.309 Be 24.450 24.529 24.508 24.551 23.976 B 37.410 37.361 37.688 36.764 37.469 C 54.401 53.844 53.817 53.916 53.124 N 74.060 74.073 74.809 74.141 0 73.310 98.723 98.791 99.409 98.807 F 97.974 128.55 28.43 128.57 127.69 128.51 Ne 161.15 61.09 161.35 161.86 Na 160.43 98.73 199.10 199.61 198.78 197.94 MI3 241.87 40.88 241.34 AI 239.97 240.91 287.65 288.85 287.04 Si 286.14 287.03 339.14 340.71 338.34 338.39 P 337.47 397.51 394.90 395.79 S 393.88 394.80 456.53 457.56 459.46 C1 455.38 456.37 524.79 526.81 523.36 523.59 Ar 522.31 599.16 597.89 599.20 K 596.33 597.68 676.76 674.96 676.42 Ca 673.20 674.66 1923.3 1926.6 1929.5 Ga 1919.8 1924.4 2075.4 2083.1 2086.1 Ge 2075.6 2080.6 2234.2 2248.9 2251.9 As 2240.7 2246.1 2399.9 2415.3 2418.4 Se 2406.4 2412.3 2590.1 2572.4 2583.6 2587.1 Br 2577.3 2762.5 2765.4 2752.0 Kr 2151.6 2758.7 Hartree-Fock kinetic energies T(HF) are taken from ref 7. All energies are in atomic units. atom
T,+T,+T,
o
3.0 ~
T(DK)
complete agreement with the calculations by Murphy and Yang,Is who reported similar kinetic energies for HF-limit densities, finding for the same sequence of elements in Table 1 an average relative error of 0.87% and a maximum of 3.5% for He. Table 2 displays the kinetic energies computed with the DK, OL, and LLP functionals. The comparison with the H F values shows again a very reasonable quality as the following results point out. An average error of 0.94% and a maximum deviation of 2.4% (in C) are found in DK results. OL functionals show average deviations of 0.61% and 0.632, with maximum errors of 2.3% and 2.05% (both for Li) for OL1 and OL2 functionals, respectively. The LLP functional provides the best method to
T(DK)
- 3 . O i '
4
'
I
6
"
8
'
' 10
'
'
12
I
0
~(0~1)
0
T(LLP)
'
14
,
z
' , 16
I
18
'
' 20
'
31
33
35
I
'
Figure 1. Relative errors in the kinetic energies computed with various density functionals and the AMD p(r) with respect to the Hartree-Fock kinetic energies taken from ref 7.
compute kinetic energies with AMD functions, even better than the gradient expansion: average error 0.49% and maximum 1.5% (Li) are found. In a recent study by ThakkarZ0for a set of 77 molecules ranging from diatomics to molecules containing 12 atoms and exhibiting different bonding situations, the LLP functional was found to give the most accurate kinetic energies among many other available functionals. The variation with Z of the relative errors in the four types of kinetic energy functionals analyzed is plotted in Figure 1. The sign of the deviation is that of the difference T(functiona1) T(HF), with Hartree-Fock data taken from the Clementi-Roetti tabulation.' As expected from the statistical theory underlying these kinetic energies, the lighter elements present the greater deviations, while errors become closer to zero as Z increases. Starting from 2 = 10, the four curves exhibit a remarkable closeness with all their points placed within a band of f l % error. This study allows one to state some remarks on the validity of the AMD p(r) as defined by eq 1. This simple representation of p(r) not only is physically meaningful but also provides a firm starting point for further searching of analytical molecular densities in terms of functions not depending on basis sets. Since AMD were constructed under the condition of reproducing potential energies (without any fitting to the H F electron densities), the calculation of kinetic energies with some of the best kinetic energy functionals available is a valuable test of reliability. It must be also emphasized that functionalsdepending on the inhomogeneity (6) have been employed with their original parameters. Undoubtedly, a reparametrization suited to our densities should give still better results, but it is remarkable that even with only four terms in the expansion (1) for representing p(r) in atoms with Z = 3 1-36, more than 99.6% of the HF-limit kinetic energy values can be easily obtained.
Model Exchange Potential Generalized gradient approximations for the exchange energy functional as introduced for example by Becke19have substantially reduced the known errors associated with the local density approximation (LDA). The nonlocal energy functional of Becke is probably the most widely used exchange functional in DF applications. We begin this section by testing our analytical approximate densities in the Becke functional. Let us recall its formIg for a spin saturated density p
where
Simple Analytical Representation of Atomic Densities
The Journal of Physical Chemistry, Vol. 98, No. 14, 1994 3691 0.40
TABLE 3 Exchange Energies Computed with the LDA Approach and Becke Functional, Using AMD Functions (LDA-AMD and B-AMD, Respectively), Compared with Reference LDA Values from Ref 23 and Hartree-Fock
ElU?rsies' atom H He Li Be B C N 0
F Ne Na Mg AI
Si P
S
c1 Ar
K Ca Ga Ge A8
Se Br Kr 0
LDA-AMD
LDA-Ref
B-AMD
HF
-0.2680 -0.8840 -1.5426 -2.2981 -3.2740 -4.4962 -5.9740 -7.3633 -9.0409 -1 1.009
-0.2680 -0.8841 -1.5379 -2.3124 -3.2898 -4.4785 -5.8932 -7.3734 -9.0829 -1 1.034
-0.3098 -1.0254 -1.7797 -2.6418 -3.7271 -5.0625 -6.6577 -8.1821 -9.9965 -1 2.104
-0.3125 -1.0257 -1.78 12 -2.6670 -3.7437 -5.0447 -6.5967 -8.1748 -10.003 -12.108
-12.751 -14.565 -16.501 -18.582 -20.821 -22.982 -25.302 -27.785
-12.786 -14.61 2 -16.544 -18.603
-13.976 -15.931 -18.008 -20.221 -22.597 -24.905 -27.373 -30.002
-14.018 -15.994 -18.070 -20.28 1 -22.642 -25.003 -27.509 -30.188
-29.927 -32.3 13 -68.979 -72.771 -76.674 -80.447 -84.326 -88.384
-30.204 -32.594
-32.295 -34.831 -73.254 -77.219 -81.301 -85.242 -89.286 -93.503
-32.675 -35.216 -73.522 -77.483 -81.515 -85.536 -89.635 -94.073
-25.371 -27.864
-88.629
All energies are in atomic units.
0.20
-
015
-
0.10
-d
0
a
Oo5
'
LDA Model Potential
1
'
4'
b'
Z 10.0
'
'lO'l'L'1'4'1'6'18'2'0'~~'24.2'6'2'8'30'32'34'36'
,
,
'"1 //
Xa Model Potential
1.0
'
1 4 '
'
1
6 ' 6 '1b'1~'1'4'16'1~'2b'~2'24'26'28'3b'~2'~4'36'
Z
Figure 2. Parameters @ in the semiempirical model exchange potential (12) for H-Kr main group atoms: (a) LDA-type model potential; (b) Xu-type model potential with a = I/J.
and the gradient term y is (6) multiplied by 2J13. Original parameters in (10) are 8 = y = 0.0042. More recently, Lee and ZhouZZhave reparametrized the Becke functional by including more atomic data and proposed new parameters C, = 0.9345, B = 0.003 54, and y = 0.002 52. For the atomic results considered here, however, no noticeable differences between both sets of parameters have been found. Table 3 shows exchange energies computed with the local density approximation formula in eq 11 (LDA-AMD) and the Becke functional with the original parameters (B-AMD), using our AMD functions. These results are compared with HartreeFockexchangeenergies,(HF),and reference LDAvaluesZ3(LDARef) where available. A good agreement is again found despite the extreme simplicity of the atomic densities used. The average relative error in LDA energies respect to the reference values is 0.40%, essentially the same that the average error associated with Becke functional energies with respect to H F data: 0.44%. AMD functions thus seem sufficiently reliable for use with conventional exchange functionals. However, let us now consider a distinct exchange functional definition. One of the original additional motivations to develop AMD functions was precisely to set a simple analytical representation of the OPM numerical exchange potential.* (As it has been repeatedly emphasized," OPM reference values are the most appropriate for comparison in DF exchange-only studies.) To this end, a semiempirical exchange potential was defined in ref 6 to be used with atomic radial densities as
This potential behaves like -I/r when r value at the nucleus given by
h 025 -
+d
-+
and has a finite
The exchange energy is then obtained as
As r tends to infinity, p(r) approaches exp(-BMr) and V,(r) approaches - l / r , and so E, correctly behaves like -exp(-BMr)/ 2r. BM is the numerically lower exponent in our analytical expansion (l), and as it was demonstrated,s the value of BM is in good agreement with the proper d(84asymptotic density limit. If the coefficient in (12) is made equal to C, in (1 l), the first term gives the usual LDA exchange energy, while the second represents an ad hoc correction to yield the proper asymptotic behavior. Admittedly, this term was introduced without any theoretical justification: it was just added to the p ( r ) l l J term to force its long-range behavior to match the correct numerical OPM values. The approach followed before for fixing C, and ,9576 was to let both as adjustable parameters and set their value under the condition to reproduce the H F exchange energy and the OPM exchange potential at r = 0. This way, using the analytical representation (1) for p(r), every atom has two parameters in the semiempirical exchange potential. Two alternative procedures to set these parameters are here presented. First, assume C, is exactly the LDA coefficient and then compute LDA energies with the AMD p(r) and determine the only parameter ,9 with the condition to reproduce the H F exchange energy with eqs 12 and 14. This option, which may be termed the LDA model exchange potential, has thus a unique parameter B characteristic for every atom. In Figure 2a, this B is plotted versus the atomic number: no regular trend is exhibited except that, starting from carbon, it increases with 2. Due to the small values encountered for this exponential parameter 8, the LDA model potential approaches very slowly to - l / r at large radial distances. On the other hand, when the values of the
3692 The Journal of Physical Chemistry, Vol. 98, No. 14, 1994
Pacios
TABLE 4 Parameters @ in the Xa Model Exchange
Potential (12) and Values of at the Nucleus for the Model, Exchange Potentials' Ux-(0), and OPM, UxopM(0), H
He Li Be B
C N 0 F
Ne Na Mg
A1 Si P
S
c1
Ar K
Ca Ga Ge As Se Br Kr a
0.4941 0.8084 1.165 1.422 1.651 1.857 2.075 2.273 2.468 2.684 2.964 3.214 3.470 3.722 3.977 4.224 4.471 4.732 5.053 5.291 7.982 8.269 8.547 8.826 9.083 9.406
-0.8 12 -1.940 -2.9 15 -3.848 -4.710 -5.527 -6.342 -7.247 -8.115 -8.982 -9.909 -10.82 -1 1.73 -12.63 -13.52 -14.44 -15.37 -16.30 -17.28 -18.18 -28.16 -29.10 -30.03 -30.99 -3 1.92 -32.91
-1 .ooo -1,687 -2.303 -3.131 -3.976 -4.779 -5.579 -6.381 -7.184 -7.990 -8.574 -9.379 -10.22 -1 1.05 -11.88 -12.72 -13.55 -14.39 -15.06 -15.88 -24.91 -25.75 -26.58 -27.42 -28.26 -29.10
All values are in atomic units.
exchange potential at the nucleus, eq 13, are compared with the corresponding OPM data, very large discrepancies (typically a factor of 2) were found for all the atoms analyzed. In an attempt to reduce these differences, I removed the condition to set C, to its LDA value and explored other choices, although the requirement of keeping this coefficient constant for all the atoms was maintained. A second approach, which may be denoted Xa-type model exchange potential, was then developed. The Xa exchange functional introduced by Slater has formally the same form that the LDA functional except that the coefficient is
As it is well-known, when a = 2/3 this functional coincides with the LDA. If the model potential (12) is adapted to a Xa-type definition by simply taking a coefficient C, like (15 ) , an interesting new feature for j3 comes out when a is set to '13. Fixing again this exponential parameter under the condition of reproducing HF exchange energies, and plotting the new set of j3 values versus Z, Figure 2b, a straight line with equation /3 = 0.28442
+ 0.249962
(16)
is obtained. A strikingcorrelation coefficient of 0.999 67 is found, and the slope is surprisingly close to 1/4. Parameters j3 for this Xa model potential are listed in Table 4 as well as both exchange potential values at r = 0, the model expression (13), column Uxmdel(0), and OPM reference data, U,OPM(O).S From a physical point of view, the precise value of the exchange potential at the nucleus is not expected to play an important role, particularly if one considers the main dominant effect, namely, the nuclear Coulomb attraction. However, since the OPM exchange potential is taken as reference, a reasonable agreement between the values of U, at r = 0 must be regarded as a desirable feature of our semiempirical potential. (Remember that only reproduction of HF exchange energies is imposed in the fit.) Table 4 shows that the relative errors typically around 100% for the LDA model potential are reduced to an average error of
?"?
00 -2.0
'? - 4 0 v
3-
-6.0
-8
0
-1001' 0 001
'
'
"""'
0 01
'
'
"""'
01
Lo3 r
'
'
"""'
'
' " ' .10A
Figure 3. Exchange potentials for the neon atom: solid line, XCYmodel exchangepotential ( 12); dashed line, OPM numericalexchangepotential. The curve -1 / r is plotted for reference.
15% for the Xa version, which must be taken as an additional proof of the better overall performance of this latter semiempirical exchange potential. The spatial behavior of the Xa potential is plotted with the OPM numerical potential for neon in Figure 3. The -1 / r curve is also included in this plot to illustrate long-range trends. The good behavior of the semiempirical model potential is evident, especially in the region where the exchange potential reduces to the asymptotic limit -l/r. Of course, the model potential lacks the reproduction of the inner radial structure exhibited by the OPM in the form of "kinks", but this is expectable if one considers the naivetC of the definition (12). This potential should be regarded as a fit in the form of an easy-to-use representation which encodes the essential physical information of the true atomic exchange potential, while yielding correct HF exchange energies. It is intended as a practical tool for quantum chemical applications concerning exchange effects in systems where simplifications are computationally mandatory.
Conclusions A simple analytical representation of HF-limit atomic densities in terms of a small expansion of exponential terms has been developed.s.6 The interest in that work lies in the further construction of simple representations of molecular electron densities not depending on choices of basis sets. This should be of great utility in density functional applications in large molecules like those present in biomolecular systems. Although these analytically modeled densities (AMD) were developed without any condition regarding kinetic energies, we have shown in this work that the use of AMD with the most precise functionals available in the literature allowsvery reliable estimations of atomic kinetic energies at the HF-limit level. In terms of these AMD, a semiempirical exchange model potential is also proposed (eq 12). This potential consists of one Xa-type term and a correction which guarantees the proper asymptotic limit, while presenting values at the nucleus close to those given by the OPMS method. With only a parameter B characteristic of every atom, this model potential reproduces the HF exchange energy and exhibits an overall spatial behavior in good agreement with the OPM exchange potential. An interesting additional feature of the model potential is that the exponential parameters B display a very good correlation with the atomic number. An extension of this exchange potential to (large) molecular environments thus seems plausible.
Acknowledgment. Financial support by the Spanish Direcci6n General de Investigacih Cientifica y Tknica, DGICYT, under Grant PB92-033 1 is gratefully acknowledged.
Simple Analytical Representation of Atomic Densities
References and Notes (1) (a) Parr, R. G.; Yang, W. &nsi?y-FunctioM/ Theory of Atoms and Molecules; Oxford: New York, 1989. (b) March, N.H. Electron Denrity Theory of Atoms and Molecules; Academic Press: New York, 1992. (2) Hohenberg, P.; Kohn, W. Phys. Rev. 1964, 136,B864. (3) Zicgler, T. Chem. Rev. 1991, 91, 651. Johnson, B. G.; Gill, P. M. W.; Pople, J. J. Chem. Phys. 1992,97,7846.Politzer, P.; Seminario, J. M.; Concha, M. C.; Murray, J. S. Theor. Chim. Acta 1993,85, 127. (4) Becke, A. D. J. Chem. Phys. 1992,97,9173;1993.98, 1372. (5) Pacios, L. F. J . Compur. Chem. 1993,14,410. (6) Pacios, L. F. J . Phys. Chem. 1991, 95, 10653; 1992,96,7294. (7) Clementi, E.; Roetti, C. At. Data Nucl. Data Tables 1974,14,177. (8) Talman, J. D.; Shadwick, W. F. Phys. Rev. A 1976,14,36. Talman, J. D. Comput. Phys. Commun. 1989,54, 25. (9) Thomas, L.H. Proc. Cambridge Philos, Soc. 1926,23,542. Fermi, E. Rend. Licei, 1927,6,602. (10) von WeizsBcker, C. F. Z.Phys. 1935, 96,431. (1 1 ) ( a ) Kirzhnits, D. A. Sw.Phys. JETP 1957,5,64. (b) Stoddart, J. C.; Beattie, A. M.; March, N. H. In?. J. Quantum Chem. Symp. 1970, IS, 35.
The Journal of Physical Chemistry, Vol. 98, No. 14, 1994 3693 (12) Jones, W.; Young, W. H. J. Phys. C 1971.4, 1322. (13) Hodges, C. H. Can. J. Phys. 1973.51, 1428. (14) Murphy, D. R. Phys. Rev. A 1981, 24, 1682. (15) Murphy, D. R.; Yang, W. J. Chem. Phys. 1980,72,429. (16) Depristo, A. E.; Krees, J. D. Phys. Rev. A 1987,35,438. (17) Ou-Yang, H.; Lcvy, M. Inr. J . Quantum Chem. 1991, 40, 379. (18) Lee, H.;Lee,C.; Parr, R. G. Phys. Rev. A 1991,44,768. (19) Becke,A. D.J. Chem. Phys. 1986,84,4524;Phys. Rev. A 1988,38, 3098. (20) Thakkar, A. J. Phys. Rev. A 1992,46,6920. (21) Wolfram, S.J. Mathematica. A Systemfor Doing Mathematics by Computers, 2nd ed.;Addison-Wesley: Redwood City, CA, 1991. (22) Lee, C.;Zhou, 2.Phys. Rev. A 1991,44,1536. (23) March, N.H. Reference lb, Appendix 4.1. (24) Sahni, V.;Gruenebaum, J.; Perdew, J. P. Phys. Rev. B 1982, 26, 4371. Langrcth,D.C.;Mehl,M. J. Phys.Rev.B1983,28,1809. Antoniewicz, P. R.; Kleinman, L.Phys. Rev. B 1985,31,6779. Levy,M.; Perdew, J. P. Phys. Rev. A 1985,32,2010. Engcl, E.; Chevary, J. A.; Macdonald, L.D.; Vosko, S.H. Z.Phys. D 1992,23, 7.
