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Approximation to the classical Coulomb energy for atoms
TRAN, Fabien
TRAN, Fabien. Approximation to the classical Coulomb energy for atoms. Physics Letters. A , 2003, vol. 310, no. 2-3, p. 177-181
DOI : 10.1016/S0375-9601(03)00344-X
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Approximation to the classical Coulomb energy for atoms
Fabien Tran
Department of Physical Chemistry, University of Geneva, 30 quai Ernest-Ansermet, CH-1211 Geneva 4, Switzerland Received 21 January 2003; accepted 10 February 2003
Communicated by V.M. Agranovich
Abstract
An approximate one-electron functional for the classical Coulomb energyJ[ρ]is presented. The analytical form of the terms appearing in the functional is justified by the scaling relations of the exact form of the classical Coulomb energy, and, the coefficients in front of each term are determined by a least-squares fit of the exact values for rare-gas atoms. It is shown that the approximation, tested on a set of neutral atoms (2Z54), can predicts energies with accuracy and leads to a potential vJ(r)=δJ[ρ]/δρ(r)which is in qualitative agreement with the exact one.
2003 Elsevier Science B.V. All rights reserved.
PACS: 31.15.Ew
Keywords: Density functional theory; Classical Coulomb energy; Scaling relations
The evaluation of the classical Coulomb energy of an electronic system of densityρ(r), represented by the two-electron functional
(1) J[ρ] =1
2
ρ(r)ρ(r)
|r−r| drdr and its potential by
(2) vJ(r)=δJ[ρ]
δρ(r)=
ρ(r)
|r−r|dr,
is computationally demanding for its evaluation (see, for example, Manby and Knowles [1]), and therefore, development of accurate one-electron functionals for J[ρ]is of interest. In comparison with the exchange- correlation and kinetic energies in density functional theory [2,3], a very small number of one-electron
E-mail address: fabien.tran@chiphy.unige.ch (F. Tran).
functionals have so far been proposed as approxima- tions toJ[ρ](or to the total electron–electron repul- sion energyVee[ρ]). Several of these approximations have the local form [4–6]
(3) A(N )
ρ4/3(r) dr
or the gradient corrected form [7,8]
(4) A(N )
ρ4/3(r) dr+B(N )
|∇ρ(r)|2
ρ4/3(r) dr,
whereA(N )andB(N )are functions of the number of electronsN. Other forms have been proposed [9–12].
The two following scaling relations are satisfied by the exact functional (Eq. (1)): (a) homogeneity with respect to density scaling
(5) J[λρ] =λ2J[ρ],
0375-9601/03/$ – see front matter 2003 Elsevier Science B.V. All rights reserved.
doi:10.1016/S0375-9601(03)00344-X
178 F. Tran / Physics Letters A 310 (2003) 177–181
or equivalently,
(6) J[ρ] =1
2
ρ(r)δJ[ρ] δρ(r)dr,
and (b) homogeneity with respect to coordinate scal- ing
(7) J[ρλ] =λJ[ρ],
or equivalently [13],
(8) J[ρ] = −
ρ(r)r· ∇δJ[ρ] δρ(r)dr, whereρλ(r)=λ3ρ(λr).
In this Letter a solution to Eqs. (6) and (8) is proposed:
(9) J[ρ] = ρβ(r)rγdr
α
,
wherer= |r|is the distance from the nucleus andα, β, andγ are to be determined. It is straightforward to demonstrate that if we want Eq. (9) to satisfy Eqs. (6) and (8),β=2/αandγ=5/α−3 for a given value forα. This leads us to consider a solution forJ[ρ]of the form
(10) J[ρ] =
M
i=1
ci ρ2/pi(r)r5/pi−3dr pi
,
where the constantsci andpiare to be determined.
The term in Eq. (10) corresponding top=5/3 has the same analytical form as the upper bound derived by Thulstrup and Linderberg [14]
J[ρ]4 3
2 π
1/3
ρ6/5(r) dr 5/3
≈1.1470 (11)
ρ6/5(r) dr 5/3
,
and has already been proposed as an approximation forJ[ρ]by Liu and Parr [11] (withc=1.0829) and by Gadre and Bendale [9] (withc=1). Restrictions on the choice of thepi in Eq. (10) can be made if we consider the behavior of the exact potential given by Eq. (2). The approximate potential is given by vJ(r)=δJ[ρ]
δρ(r)
= M
i=1
ci ρ2/pi(r)(r)5/pi−3dr pi−1
×2ρ2/pi−1(r)r5/pi−3. (12)
At the position of the nucleus (r =0), the exact potential vJ(r) has a finite positive value and this implies to have the term with p=5/3 present, and for the other terms (if M2), to havep∈ ]0,5/3] in order to avoid divergence. With these restrictions for the pi the condition limr→∞vJ(r)=0 is also satisfied.
The performance of the approximate functional J[ρ] (Eq. (10)) was studied using different values forM, whereas the values of thepi were chosen to be pi =5/3−(i−1) /3 (several other choices for the pi were studied, but not leading to qualitative improvements). The coefficients ci, collected in Ta- ble 1, were obtained by least-squares fits of percent- age differences between exact and approximate val- ues of J[ρ]for five rare-gas atoms (He, Ne, Ar, Kr, and Xe) using Hartree–Fock densities [15,16]. The ap- proximate functionals were then tested on a set of 53 atomic (2Z54) densities [15,16] (which were spherically symmetrized for this work). The results presented in Tables 2 and 3 show that the approxi-
Table 1
Coefficientsci of the approximate functionalJ[ρ](Eq. (10) with different values forM) obtained with least-squares fits of percentage difference between exact and approximate values ofJ[ρ]for He, Ne, Ar, Kr, and Xe rare-gas atoms
M pi ci
1 5/3 1.0914
2 5/3 0.8016
4/3 1.0588
3 5/3 1.8912
4/3 −6.6322
1 12.8731
4 5/3 1.9476
4/3 −7.4330
1 16.3716
2/3 −4.6066
5 5/3 1.6500
4/3 −4.2842
1 4.9002
2/3 12.1110
1/3 −7.9462
Table 2
Exact (Eq. (1)) and approximate (Eq. (10) with different values forM) values of the classical Coulomb energyJ[ρ]using Hartree–Fock densities. The values are expressed in atomic units
Atom Exact M=1 M=2 M=3 M=4 M=5
He 2.052 1.963 2.044 2.052 2.052 2.052
Li 4.062 3.979 4.045 4.095 4.088 4.095
Be 7.156 7.190 7.221 7.194 7.191 7.198
B 11.583 11.765 11.752 11.595 11.611 11.620
C 17.804 18.132 18.087 17.773 17.811 17.831
N 26.147 26.590 26.545 26.081 26.134 26.153
O 36.632 37.143 37.122 36.547 36.603 36.612
F 49.852 50.359 50.420 49.778 49.822 49.822
Ne 66.148 66.556 66.778 66.128 66.145 66.148
Na 80.035 80.105 80.209 80.242 80.219 80.162
Mg 95.811 95.837 95.691 96.173 96.079 96.057
Al 112.82 113.01 112.57 113.36 113.20 113.22
Si 131.99 132.55 131.73 132.58 132.37 132.42
P 153.43 154.49 153.28 153.96 153.72 153.77
S 176.89 178.53 176.90 177.32 177.09 177.12
Cl 202.90 205.25 203.19 203.19 203.00 202.99
Ar 231.61 234.78 232.30 231.76 231.65 231.61
K 257.31 260.27 257.27 257.50 257.43 257.22
Ca 284.90 288.21 284.48 285.37 285.29 285.05
Sc 318.45 322.43 318.07 318.75 318.73 318.51
Ti 355.47 360.12 355.16 355.51 355.59 355.39
V 395.72 401.01 395.49 395.48 395.66 395.52
Cr 444.14 450.35 444.50 443.23 443.58 443.57
Mn 486.50 492.91 486.39 485.69 486.09 486.10
Fe 536.54 543.42 536.40 535.46 535.94 536.03
Co 590.45 597.71 590.27 589.13 589.69 589.85
Ni 648.20 655.76 647.95 646.67 647.30 647.50
Cu 716.71 724.89 717.20 714.71 715.45 715.72
Zn 775.89 783.75 775.45 774.16 774.87 775.09
Ga 835.18 842.78 833.77 834.03 834.66 834.82
Ge 897.25 904.91 894.96 896.57 897.08 897.28
As 962.10 970.07 959.02 961.67 962.03 962.34
Se 1029.1 1037.5 1025.3 1028.8 1029.0 1029.4
Br 1099.2 1108.3 1094.7 1098.8 1098.9 1099.4
Kr 1172.3 1182.4 1167.5 1171.8 1171.8 1172.3
Rb 1239.3 1248.3 1232.4 1238.7 1238.7 1238.9
Sr 1308.5 1317.6 1300.3 1309.0 1308.9 1308.8
Y 1384.3 1394.3 1375.3 1385.2 1385.0 1384.9
Zr 1464.3 1475.4 1454.7 1465.3 1465.0 1464.8
Nb 1553.0 1566.3 1543.7 1553.3 1552.9 1552.9
Mo 1641.1 1656.2 1631.9 1641.0 1640.7 1640.7
Tc 1725.9 1741.6 1715.6 1726.0 1725.7 1725.4
Ru 1826.4 1845.2 1817.4 1825.5 1825.4 1825.1
Rh 1924.8 1945.7 1916.1 1923.3 1923.4 1923.0
Pd 2034.5 2059.8 2028.4 2032.4 2032.7 2032.5
Ag 2133.3 2158.8 2125.7 2130.6 2131.2 2130.5
Cd 2234.0 2260.2 2225.1 2231.3 2232.0 2231.1
In 2334.6 2361.3 2324.2 2332.2 2333.0 2331.8
Sn 2438.2 2466.1 2426.6 2436.4 2437.3 2436.0
Sb 2544.8 2574.2 2532.1 2543.5 2544.4 2543.1
Te 2653.4 2684.6 2639.9 2652.6 2653.5 2652.3
I 2765.1 2798.5 2750.9 2764.8 2765.6 2764.6
Xe 2879.9 2915.9 2865.4 2880.0 2880.8 2879.9
180 F. Tran / Physics Letters A 310 (2003) 177–181
Table 3
The mean error∆¯=1/5353
i=1∆i, the mean absolute error∆¯abs=1/5353
i=1|∆i|, and the maximum absolute error∆max=maxi|∆i|,
∆i=100(Japp−Jexact)/Jexact, calculated for a set of neutral atoms (2Z54).Jexactis Eq. (1) andJappis Eq. (10) with different values forMor another approximate functional taken from the literature
Functional ∆¯ ∆¯abs ∆max
Eq. (10) withM=1 0.8555 1.0961 4.3254
Eq. (10) withM=2 −0.0645 0.4314 1.5883
Eq. (10) withM=3 0.0145 0.1498 0.8124
Eq. (10) withM=4 0.0289 0.0996 0.6323
Eq. (10) withM=5 0.0317 0.1059 0.8166
GBH1a 0.1039 1.9575 16.6981
GBH2b 0.0358 1.2624 6.1803
BPc 3.2749 3.3659 8.8000
GBd −7.5908 7.5908 12.3378
LPe 0.0699 0.4889 5.0706
NLPf −2.2145 2.4882 22.4335
a Eq. (iii) in Table 1 of Gadre et al. [5]; b Eq. (iv) in Table 1 of Gadre et al. [5]; cEq. (16) of Bartolotti and Parr [8]; d Eq. (2) of Gadre and Bendale [9]; e Eq. (20) of Liu and Parr [11]; f Eq. (13) of Nagy et al. [12].
mate functionalJ[ρ]given by Eq. (10) performs very well. Among the considered choices for the value ofM (from 1 to 5), the one withM=4 gives the best results with a value of less than 0.1% for the mean absolute er- ror∆¯absand a maximum absolute error of about 0.6%.
The results obtained with M=3 and M=5 show similar trends and, as expected, having only one term in Eq. (10) leads to the worst results.
Several other approximate functionals taken from the literature were also tested, and the results are presented in Table 3. The functional of Liu and Parr (LP) [11], which has the same form as our functional with M =1, but a different value for c (Liu and Parr obtained a fitted value of 1.0829 for the first 36 atoms (1Z36) using Hartree–Fock densities), has smaller values for the mean error ∆¯ and the mean absolute error∆¯absthan our functional (M=1), but a larger maximum absolute error ∆max of 5%
(about 4% with our coefficientc=1.0914). Choosing c=1 forM=1, as proposed by Gadre and Bendale (GB) [9], leads to large errors. The functionals of Gadre et al. [5] (GBH1 and GBH2 have one and two parameters fitted using Hartree–Fock densities, 2Z 54, respectively), which have the form of Eq. (3), lead to values larger than 1% for ∆¯abs and large maximum absolute error. The other approximate functionals (BP [8] and NLP [12]) produce large values for∆,¯ ∆¯abs, and∆max.
In Figs. 1, 2, and 3 the approximate potentialvJ(r) given by Eq. (12) withM=1,M=2, andM=4 is
Fig. 1. Comparison of the exact (Eq. (2)) and approximate (Eq. (12) withM=1,M=2, andM=4) Coulomb potentialsvJ(r)of the He atom.
compared with the exact one (Eq. (2)) for He, Ar, and Xe atoms, respectively (a logarithmic scale as been adopted for the distance r from the nucleus). From these examples we can observe that the approximate potential, despite slight oscillations around the exact one, reproduces fairly well the exact one, and this, particularly in the regions beyond r =0.1 a.u. In the very near vicinity of the nucleus the approximate potential can present deviations from the exact one, and we can see that for M =4 the deviations are important.
Fig. 2. Comparison of the exact (Eq. (2)) and approximate (Eq. (12) withM=1,M=2, andM=4) Coulomb potentialsvJ(r)of the Ar atom.
Fig. 3. Comparison of the exact (Eq. (2)) and approximate (Eq. (12) withM=1,M=2, andM=4) Coulomb potentialsvJ(r)of the Xe atom.
In summary, it is shown that the one-electron func- tional given by Eq. (10), proposed to approximate the classical Coulomb energy, can achieve good accuracy and has a functional derivative which shows a qual- itatively good behavior. With a number of terms M
of at least three a mean absolute error of the order of 0.1% is obtained. A disadvantage of the functional given by Eq. (10) is that it depends explicitly on the distancerfrom the nucleus (see for example Nagy et al. [12] and Liu and Parr [17] for kinetic-, exchange-, and correlation-energy functionals depending explic- itly onr). This means that a generalization of Eq. (10) has to be found in order to make it applicable for poly- atomic systems, but this is far from being a trivial problem.
Acknowledgements
The author thanks Tomasz A. Wesołowski for reading the manuscript and helpful comments. This work is part of the Project 21-63645.00 of the Swiss National Science Foundation.
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