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Shell structure in free and confined atoms using the density functional theory
Journal of Molecular Structure (Theochem) 501–502 (2000) 183–188 www.elsevier.nl/locate/theochem
Shell structure in free and confined atoms using the density functional theory q J. Garza a,*, R. Vargas a, A. Vela a, K.D. Sen b a
Departamento de Quı´mica, Divisio´n de Ciencias Ba´sicas e Ingenierı´a. Universidad Auto´noma Metropolitana-Iztapalapa. A.P. 55-534, Me´xico Distrito Federal 09340, Mexico b School of Chemistry, University of Hyderabad, Hyderabad 500 046, India
Abstract The average local electrostatic potential function, defined as the electrostatic potential divided by the electron density, is used to study the shell structure in free and confined atoms within Kohn–Sham density functional theory. Several exchangecorrelation functionals have been used to calculate the average potential function. It was observed that the self-interaction correction significantly alters the shell structure along the large radial distances. Many electron atoms confined in a sphere exhibit a gradual loss of the shell structure as the confinement is increased. The loss of structure can be characterized by the sphere radius rc and in the limit rc ! 0; the electron gas behavior is obtained. q 2000 Elsevier Science B.V. All rights reserved. Keywords: Electrostatic potential ; Exchange-correlation functionals; Radial density distribution function
1. Introduction One of the text book examples of a function involving the electron density, r (r), the key parameter of the density functional theory [1], is the radial density distribution function (RDF), D r 4pr 2 r r: Another similar quantity involving the electron density in its definition is the electrostatic potential (ESP), which is defined for a molecular system as:
F r
X a
Z Za r r 0 2 dr 0 ; uRa 2 ru ur 2 r 0 u
1
where Za is the nuclear charge of the a th atom located at Ra . For atoms, the nuclear part in Eq. (1) is given by Z=r: The calculations of ESP have significantly q
Dedicated to Professor R. Ga´spa´r on the occasion of his 80th year. * Corresponding author.
contributed to the understanding of electronic structure of atoms and molecules [2,3]. In the present work we shall focus on the shell structure of atoms. For atoms beyond argon, the RDF does not reveal the complete shell structure, particularly in the outer region. It has been shown that the electrostatic potential divided by the electron density [4] reveals the correct shell structure for all atoms in the form of clear maxima defining the boundaries between the shells. This local quantity is known as the average local electrostatic potential function (ALEPF) and has been studied [5,6] using the Hartree–Fock method [7]. Recently, the effect of the electron correlation on the atomic shell structure has been studied [8–10] using the RDF and ALEPF, respectively. Very recently, a model to simulate the effect of pressure on a many-electron atom has been introduced by confining an atom within a sphere of rigid walls [10]. To our knowledge, a study of the behavior of the
0166-1280/00/$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S0166-128 0(99)00428-5
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J. Garza et al. / Journal of Molecular Structure (Theochem) 501–502 (2000) 183–188
atomic shell structure under these conditions has not been reported so far. In this work, we have analyzed the shell structure in free and confined atoms using the ALEPF derived within the Kohn–Sham (KS) model. In Section 2, the shell structure in free atoms is studied using several exchange-correlation functionals. The shell structure in confined atoms is discussed in Section 3 using a methodology published recently [11]. Finally in Section 4 the conclusions are presented. 2. Shell structure in free atoms using the Kohn– Sham model In terms of KS spin-orbitals, {fsi }; the electron density is written as:
r r
Ns X X
ufsi ru
2
sa;b i1
where s denotes the spin associate to the ith state and the first sum run over all occupied states. Further, in this spin-polarized scheme the total density can be divided as r ra 1 r b : On the contrary, the total energy is expressed as: E
Ns X X
sa;b i1
s
hfi j 2
1 Exc fai ; fbi 1
1 2
72 jfsi i 1
Z
ZZ
drr ry r
dr dr 0
r rr r 0 ur 2 r 0 u (3)
where Exc fai ; fbi denotes the exchange-correlation functional. Since the exact exchange-correlation functional is unknown, several approximations have been proposed in the literature. These approximations can be divided in two groups as the local and semi-local functionals [1], and the hybrid exchange-correlation functional [12,13], respectively. The local approach stems from the electron gas model, and its principal defect is the wrong asymptotic behavior in corresponding potential. On the contrary, in the semi-local functionals such as the generalized gradient approximation, this behavior is corrected to a great measure by the functionals that depend on derivatives of the electron density. In hybrid functionals this defect is corrected by using the Hartree–Fock exchange in a specified manner.
The spin-orbitals that give a minimum in the energy can be shown to satisfy the equations: 2 12 72 1 yseff rfsi r 1si fsi r
4
with
yseff r
Z
dr 0
r r 0 1 ysxc r 1 y r ur 2 r 0 u
5
In the last equation ysxc r represents the functional derivative of the exchange-correlation functional, and y (r) is the external potential. In atoms, the external potential is y r 2Z=r: Several atomic codes are in use to carry out the numerical solutions of Eq. (4) [14–16]. In the present work we have used a modified Herman and Skillman code [14] with several exchange-correlation functionals. As local approximation, we have used the Dirac (D) functional [17] for the exchange energy and the von Barth and Hedin (VH) functional [18] for the correlation energy. Within generalized gradient approximation, the exchange energy has been estimated with Becke (B) functional [19] and the correlation energy with Lee, Yang and Parr (LYP) [20] functional. We have not used the hybrid functionals in the present work. 2.1. The ALEP in free atoms In Fig. 1, we have shown the plots of the ALEPF generated from several exchange-correlation functionals for Li–Rb atoms The exchange-correlation functional used gives significantly different behavior in the ALEPF over the asymptotic region. It is known that in the local approximation, the total potential decays rapidly and this unphysical behavior is also reflected on the electron density and the electrostatic potential, respectively. The semi-local functionals try to correct this deficiency, unfortunately in both schemes of this work the effective potential, yseff r; does not behave like 21=uru when uru ! ∞: We have used the Perdew and Zunger approach [21] to incorporate the self-interaction correction (SIC) in the local approximation, which leads to the correct asymptotic behavior of the total potential. In Fig. 2, a comparison of the several ALEPF derived from the various exchange and correlation functionals including SIC approach, for the Rubidium atom is displayed. In the asymptotic region, for all functionals the curve
J. Garza et al. / Journal of Molecular Structure (Theochem) 501–502 (2000) 183–188
185
Fig. 1. The ALEPF for alkali atoms, Li–Rb, obtained using different exchange-correlation functionals. All values are in a.u.
containing the correlation contribution is found to be below the exchange-only curve. In contrast, including SIC approach does not change the ALEPF behavior in both exchange-only and exchange-correlation functionals. Although there are important differences on the ALEPF in the asymptotic region derived from the different functionals, the topological form is preserved and it remains almost the same for the inner shells. This is corroborated by the data in the Table 1, which shows the total enclosed charge up to the last maximum of the ALEPF, which is computed R according to the integral Qc 4p R0 c dr r r where Rc represents the location of the maximum and Qc the enclosed charge. It is observed from Table 1 that the Rc values obtained with semi-local functionals (B and B-LYP) are quite close to each other. The correlation effects therefore do not significantly change the location of shell boundaries, a conclusion that has been also noted previously [7,8]. Further the results of these func-
tionals are similar to those obtained by the SIC approach. In the local approximation, due to the unphysical long-range behavior in the potential as previously noted, the correlation contributions affect the Rc values. With every functional used in this work the calculated Qc is found to be greater than the Hartree– Fock values, although these values are still close to the ideal core charges and therefore allow the separation of the core and the valence region, respectively.
3. Shell structure in confined atoms The pressure on an atom usually is simulated by confining the system within a sphere of radius rc and imposing an infinite potential on the walls of the sphere [10,11]. With this condition the electron density must be zero on the inner surface of the sphere, which is attained by solving the KS equations subjected to Dirichlet’s boundary conditions [11]. The electron density and the electrostatic potential were
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J. Garza et al. / Journal of Molecular Structure (Theochem) 501–502 (2000) 183–188
Table 1 Charge Qc, contained up to the last maximum Rc of ALEP function Atom
Li Na K Rb a
Rc
Qc
D
B
SIC V-H
D-VH
B-LYP
SIC V-H
HF a
D
B
SIC V-H
D-VH
B-LYP
SIC V-H
HF a
1.82 2.67 3.71 4.18
1.79 2.60 3.71 4.18
1.77 2.60 3.71 4.27
1.79 2.60 3.66 4.10
1.79 2.60 3.71 4.18
1.77 2.60 3.71 4.18
1.63 2.32 3.35 3.76
2.05 10.17 18.23 36.27
2.05 10.15 18.22 36.26
2.05 10.16 18.24 36.30
2.05 10.17 18.25 36.29
2.05 10.16 18.24 36.28
2.05 10.17 18.26 36.30
2.02 10.06 18.10 36.11
Ref. [7].
obtained with several rc values using the numerical approach discussed in detail in Ref. [11]. It is important to mention that in every confinement the total electron number is conserved. In this model the
pressure can be obtained from the virial theorem or from the relationship p 2 2E=2VT [22]. However, as our study of shell structure under pressure is based on finding out whether there are any gradual changes
Fig. 2. The ALEPF for Rb atom obtained using: (a) exchange only functionals; and (b) exchange-correlation functionals. All values are in a.u.
J. Garza et al. / Journal of Molecular Structure (Theochem) 501–502 (2000) 183–188
187
Fig. 3. The RDF and ALEPF for the confined alkali atoms, Li–Rb. The upper curve refers to the higher confinement radius. All values are in a.u.
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J. Garza et al. / Journal of Molecular Structure (Theochem) 501–502 (2000) 183–188
in the topological features of the ALEPF, the exact values of the pressure are not reported. The Dirac functional was used to obtain the electron density and the electrostatic potential, because it has been shown in Section 2 that this functional is enough to give the principal characteristics of these quantities. In Fig. 3, the RDF and the ALEPF are presented for the same alkali metal atoms. Five rc confinements radius have been applied on each atom from 1 to 15 a.u. In Fig. 3 some interesting features can be noted; the upper line belongs to the higher rc. In fact the ALEPF of the free atom represent an upper limit, this is not true to the RDF. However, when the rc is reduced, the external shells are found to collapse and the corresponding maximum points in the ALEPF begin to disappear. This behavior is observed in the radial distribution function too, although in this quantity the minimum points represent the shell boundaries. Further, one can see that in the RDF, the electron gas behavior is reached when rc is small. It means that the electrons are bounded by the sphere box and not by the nuclear Coulombic forces. From this behavior, it may be concluded that inside the sphere an atomic cation and free electrons are enclosed by an infinite potential. 4. Conclusions In this work the shell structure in free and confined atoms has been studied by using several approximate exchange-correlation functionals within the KS density functional theory. To study the shell structure, in free atoms, the ALEPF was used. It was found that in the asymptotic region this function is sensitive to the nature of the exchange-correlation functional used, but all features of the complete shell structures are displayed in each functional. The position of the last maximum in the ALEPF, the core radius, obtained within the KS theory is generally greater than the Hartree–Fock values. However, in both cases core and valence region can be distinguished. For the confined atoms, the shell structure shows several important characteristics, as the sphere volume is reduced, or in others words, when the pressure is increased. For the confinements described by the radius of the enclosing sphere rc it is observed that:
(a) the last maximum in the ALEPF, and the last minimum in the RDF vanished, because only the electrons bounded by the nucleus contribute to the shell structure; (b) the RDF exhibits a behavior like an electron gas, or free particles in a spherical box; (c) the ALEPF in free atoms is an upper limit of those obtained in the confined atoms. We are presently studying these changes in quantitative terms.
Acknowledgements We thank Dr. Marcelo Galva´n and Dr. Andre´s Cedillo for helpful discussions. This research was supported by project Indo-Mexican Scientific Collaboration project. References [1] R.G. Parr, W. Yang, Density-Functional Theory of Atoms and Molecules, Oxford, New York, 1989. [2] P. Politzer, J. Chem. Phys. 72 (1980) 3027. [3] P. Politzer, J. Chem. Phys. 73 (1980) 3264. [4] Electrostatic Potential for Atoms and Molecules, in: J. Murray, K.D. Sen (Eds.), Topics in Computational Chemistry, 3, Elsevier, Amsterdam, 1996. [5] M. Kakkar, K.D. Sen, Chem. Phys. Lett. 226 (1994) 241. [6] R. Krishnaveni, K.D. Sen, J. Chem. Phys. 101 (1994) 7779. [7] K.D. Sen, T.V. Gayatri, R. Krishnaveni, M. Kakkar, H. Toufar, G.O.A. Janssens, B.G. Baekelandt, R.A. Schoonheytdt, W.J. Mortier, Int. J. Quant. Chem. 56 (1995) 399. [8] F. de Proft, P. Geerlings, Chem. Phys. Lett 220 (1994) 405 and references therein. [9] F. de Proft, P. Geerlings, K.D. Sen, Chem. Phys. Lett. 247 (1995) 154. [10] W. Jasko´lski, Phys. Rep. 271 (1996) 1. [11] J. Garza, R. Vargas, A. Vela, Phys. Rev. E 58 (1998) 3949. [12] A.D. Becke, J. Chem. Phys. 98 (1993) 5648. [13] A.D. Becke, J. Chem. Phys. 98 (1993) 1372. [14] F. Herman, S. Skillman, Atomic Structure Calculations, Prentice-Hall, Englewood Cliffs, NJ, 1963. [15] C. Froese-Fischer, The Hartree–Fock Method: a numerical Approach, Wiley, New York, 1977. [16] J.C. Slater, The Calculation of Molecular Orbital, Wiley, New York, 1979. [17] P.A.M. Dirac, Proc. Cambridge Philos. Soc. 26 (1930) 376. [18] U. von Barth, L. Hedin, J. Phys. C 5 (1972) 1629. [19] A.D. Becke, Phys. Rev. A 38 (1988) 3098. [20] C. Lee, W. Yang, R.G. Parr, Phys. Rev. B 37 (1988) 785. [21] J.P. Perdew, A. Zunger, Phys. Rev. B 23 (1981) 5048. [22] E.V. Luden˜a, J. Chem. Phys. 69 (1978) 1770.
Shell structure in free and confined atoms using the density functional theory q J. Garza a,*, R. Vargas a, A. Vela a, K.D. Sen b a
Departamento de Quı´mica, Divisio´n de Ciencias Ba´sicas e Ingenierı´a. Universidad Auto´noma Metropolitana-Iztapalapa. A.P. 55-534, Me´xico Distrito Federal 09340, Mexico b School of Chemistry, University of Hyderabad, Hyderabad 500 046, India
Abstract The average local electrostatic potential function, defined as the electrostatic potential divided by the electron density, is used to study the shell structure in free and confined atoms within Kohn–Sham density functional theory. Several exchangecorrelation functionals have been used to calculate the average potential function. It was observed that the self-interaction correction significantly alters the shell structure along the large radial distances. Many electron atoms confined in a sphere exhibit a gradual loss of the shell structure as the confinement is increased. The loss of structure can be characterized by the sphere radius rc and in the limit rc ! 0; the electron gas behavior is obtained. q 2000 Elsevier Science B.V. All rights reserved. Keywords: Electrostatic potential ; Exchange-correlation functionals; Radial density distribution function
1. Introduction One of the text book examples of a function involving the electron density, r (r), the key parameter of the density functional theory [1], is the radial density distribution function (RDF), D r 4pr 2 r r: Another similar quantity involving the electron density in its definition is the electrostatic potential (ESP), which is defined for a molecular system as:
F r
X a
Z Za r r 0 2 dr 0 ; uRa 2 ru ur 2 r 0 u
1
where Za is the nuclear charge of the a th atom located at Ra . For atoms, the nuclear part in Eq. (1) is given by Z=r: The calculations of ESP have significantly q
Dedicated to Professor R. Ga´spa´r on the occasion of his 80th year. * Corresponding author.
contributed to the understanding of electronic structure of atoms and molecules [2,3]. In the present work we shall focus on the shell structure of atoms. For atoms beyond argon, the RDF does not reveal the complete shell structure, particularly in the outer region. It has been shown that the electrostatic potential divided by the electron density [4] reveals the correct shell structure for all atoms in the form of clear maxima defining the boundaries between the shells. This local quantity is known as the average local electrostatic potential function (ALEPF) and has been studied [5,6] using the Hartree–Fock method [7]. Recently, the effect of the electron correlation on the atomic shell structure has been studied [8–10] using the RDF and ALEPF, respectively. Very recently, a model to simulate the effect of pressure on a many-electron atom has been introduced by confining an atom within a sphere of rigid walls [10]. To our knowledge, a study of the behavior of the
0166-1280/00/$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S0166-128 0(99)00428-5
184
J. Garza et al. / Journal of Molecular Structure (Theochem) 501–502 (2000) 183–188
atomic shell structure under these conditions has not been reported so far. In this work, we have analyzed the shell structure in free and confined atoms using the ALEPF derived within the Kohn–Sham (KS) model. In Section 2, the shell structure in free atoms is studied using several exchange-correlation functionals. The shell structure in confined atoms is discussed in Section 3 using a methodology published recently [11]. Finally in Section 4 the conclusions are presented. 2. Shell structure in free atoms using the Kohn– Sham model In terms of KS spin-orbitals, {fsi }; the electron density is written as:
r r
Ns X X
ufsi ru
2
sa;b i1
where s denotes the spin associate to the ith state and the first sum run over all occupied states. Further, in this spin-polarized scheme the total density can be divided as r ra 1 r b : On the contrary, the total energy is expressed as: E
Ns X X
sa;b i1
s
hfi j 2
1 Exc fai ; fbi 1
1 2
72 jfsi i 1
Z
ZZ
drr ry r
dr dr 0
r rr r 0 ur 2 r 0 u (3)
where Exc fai ; fbi denotes the exchange-correlation functional. Since the exact exchange-correlation functional is unknown, several approximations have been proposed in the literature. These approximations can be divided in two groups as the local and semi-local functionals [1], and the hybrid exchange-correlation functional [12,13], respectively. The local approach stems from the electron gas model, and its principal defect is the wrong asymptotic behavior in corresponding potential. On the contrary, in the semi-local functionals such as the generalized gradient approximation, this behavior is corrected to a great measure by the functionals that depend on derivatives of the electron density. In hybrid functionals this defect is corrected by using the Hartree–Fock exchange in a specified manner.
The spin-orbitals that give a minimum in the energy can be shown to satisfy the equations: 2 12 72 1 yseff rfsi r 1si fsi r
4
with
yseff r
Z
dr 0
r r 0 1 ysxc r 1 y r ur 2 r 0 u
5
In the last equation ysxc r represents the functional derivative of the exchange-correlation functional, and y (r) is the external potential. In atoms, the external potential is y r 2Z=r: Several atomic codes are in use to carry out the numerical solutions of Eq. (4) [14–16]. In the present work we have used a modified Herman and Skillman code [14] with several exchange-correlation functionals. As local approximation, we have used the Dirac (D) functional [17] for the exchange energy and the von Barth and Hedin (VH) functional [18] for the correlation energy. Within generalized gradient approximation, the exchange energy has been estimated with Becke (B) functional [19] and the correlation energy with Lee, Yang and Parr (LYP) [20] functional. We have not used the hybrid functionals in the present work. 2.1. The ALEP in free atoms In Fig. 1, we have shown the plots of the ALEPF generated from several exchange-correlation functionals for Li–Rb atoms The exchange-correlation functional used gives significantly different behavior in the ALEPF over the asymptotic region. It is known that in the local approximation, the total potential decays rapidly and this unphysical behavior is also reflected on the electron density and the electrostatic potential, respectively. The semi-local functionals try to correct this deficiency, unfortunately in both schemes of this work the effective potential, yseff r; does not behave like 21=uru when uru ! ∞: We have used the Perdew and Zunger approach [21] to incorporate the self-interaction correction (SIC) in the local approximation, which leads to the correct asymptotic behavior of the total potential. In Fig. 2, a comparison of the several ALEPF derived from the various exchange and correlation functionals including SIC approach, for the Rubidium atom is displayed. In the asymptotic region, for all functionals the curve
J. Garza et al. / Journal of Molecular Structure (Theochem) 501–502 (2000) 183–188
185
Fig. 1. The ALEPF for alkali atoms, Li–Rb, obtained using different exchange-correlation functionals. All values are in a.u.
containing the correlation contribution is found to be below the exchange-only curve. In contrast, including SIC approach does not change the ALEPF behavior in both exchange-only and exchange-correlation functionals. Although there are important differences on the ALEPF in the asymptotic region derived from the different functionals, the topological form is preserved and it remains almost the same for the inner shells. This is corroborated by the data in the Table 1, which shows the total enclosed charge up to the last maximum of the ALEPF, which is computed R according to the integral Qc 4p R0 c dr r r where Rc represents the location of the maximum and Qc the enclosed charge. It is observed from Table 1 that the Rc values obtained with semi-local functionals (B and B-LYP) are quite close to each other. The correlation effects therefore do not significantly change the location of shell boundaries, a conclusion that has been also noted previously [7,8]. Further the results of these func-
tionals are similar to those obtained by the SIC approach. In the local approximation, due to the unphysical long-range behavior in the potential as previously noted, the correlation contributions affect the Rc values. With every functional used in this work the calculated Qc is found to be greater than the Hartree– Fock values, although these values are still close to the ideal core charges and therefore allow the separation of the core and the valence region, respectively.
3. Shell structure in confined atoms The pressure on an atom usually is simulated by confining the system within a sphere of radius rc and imposing an infinite potential on the walls of the sphere [10,11]. With this condition the electron density must be zero on the inner surface of the sphere, which is attained by solving the KS equations subjected to Dirichlet’s boundary conditions [11]. The electron density and the electrostatic potential were
186
J. Garza et al. / Journal of Molecular Structure (Theochem) 501–502 (2000) 183–188
Table 1 Charge Qc, contained up to the last maximum Rc of ALEP function Atom
Li Na K Rb a
Rc
Qc
D
B
SIC V-H
D-VH
B-LYP
SIC V-H
HF a
D
B
SIC V-H
D-VH
B-LYP
SIC V-H
HF a
1.82 2.67 3.71 4.18
1.79 2.60 3.71 4.18
1.77 2.60 3.71 4.27
1.79 2.60 3.66 4.10
1.79 2.60 3.71 4.18
1.77 2.60 3.71 4.18
1.63 2.32 3.35 3.76
2.05 10.17 18.23 36.27
2.05 10.15 18.22 36.26
2.05 10.16 18.24 36.30
2.05 10.17 18.25 36.29
2.05 10.16 18.24 36.28
2.05 10.17 18.26 36.30
2.02 10.06 18.10 36.11
Ref. [7].
obtained with several rc values using the numerical approach discussed in detail in Ref. [11]. It is important to mention that in every confinement the total electron number is conserved. In this model the
pressure can be obtained from the virial theorem or from the relationship p 2 2E=2VT [22]. However, as our study of shell structure under pressure is based on finding out whether there are any gradual changes
Fig. 2. The ALEPF for Rb atom obtained using: (a) exchange only functionals; and (b) exchange-correlation functionals. All values are in a.u.
J. Garza et al. / Journal of Molecular Structure (Theochem) 501–502 (2000) 183–188
187
Fig. 3. The RDF and ALEPF for the confined alkali atoms, Li–Rb. The upper curve refers to the higher confinement radius. All values are in a.u.
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J. Garza et al. / Journal of Molecular Structure (Theochem) 501–502 (2000) 183–188
in the topological features of the ALEPF, the exact values of the pressure are not reported. The Dirac functional was used to obtain the electron density and the electrostatic potential, because it has been shown in Section 2 that this functional is enough to give the principal characteristics of these quantities. In Fig. 3, the RDF and the ALEPF are presented for the same alkali metal atoms. Five rc confinements radius have been applied on each atom from 1 to 15 a.u. In Fig. 3 some interesting features can be noted; the upper line belongs to the higher rc. In fact the ALEPF of the free atom represent an upper limit, this is not true to the RDF. However, when the rc is reduced, the external shells are found to collapse and the corresponding maximum points in the ALEPF begin to disappear. This behavior is observed in the radial distribution function too, although in this quantity the minimum points represent the shell boundaries. Further, one can see that in the RDF, the electron gas behavior is reached when rc is small. It means that the electrons are bounded by the sphere box and not by the nuclear Coulombic forces. From this behavior, it may be concluded that inside the sphere an atomic cation and free electrons are enclosed by an infinite potential. 4. Conclusions In this work the shell structure in free and confined atoms has been studied by using several approximate exchange-correlation functionals within the KS density functional theory. To study the shell structure, in free atoms, the ALEPF was used. It was found that in the asymptotic region this function is sensitive to the nature of the exchange-correlation functional used, but all features of the complete shell structures are displayed in each functional. The position of the last maximum in the ALEPF, the core radius, obtained within the KS theory is generally greater than the Hartree–Fock values. However, in both cases core and valence region can be distinguished. For the confined atoms, the shell structure shows several important characteristics, as the sphere volume is reduced, or in others words, when the pressure is increased. For the confinements described by the radius of the enclosing sphere rc it is observed that:
(a) the last maximum in the ALEPF, and the last minimum in the RDF vanished, because only the electrons bounded by the nucleus contribute to the shell structure; (b) the RDF exhibits a behavior like an electron gas, or free particles in a spherical box; (c) the ALEPF in free atoms is an upper limit of those obtained in the confined atoms. We are presently studying these changes in quantitative terms.
Acknowledgements We thank Dr. Marcelo Galva´n and Dr. Andre´s Cedillo for helpful discussions. This research was supported by project Indo-Mexican Scientific Collaboration project. References [1] R.G. Parr, W. Yang, Density-Functional Theory of Atoms and Molecules, Oxford, New York, 1989. [2] P. Politzer, J. Chem. Phys. 72 (1980) 3027. [3] P. Politzer, J. Chem. Phys. 73 (1980) 3264. [4] Electrostatic Potential for Atoms and Molecules, in: J. Murray, K.D. Sen (Eds.), Topics in Computational Chemistry, 3, Elsevier, Amsterdam, 1996. [5] M. Kakkar, K.D. Sen, Chem. Phys. Lett. 226 (1994) 241. [6] R. Krishnaveni, K.D. Sen, J. Chem. Phys. 101 (1994) 7779. [7] K.D. Sen, T.V. Gayatri, R. Krishnaveni, M. Kakkar, H. Toufar, G.O.A. Janssens, B.G. Baekelandt, R.A. Schoonheytdt, W.J. Mortier, Int. J. Quant. Chem. 56 (1995) 399. [8] F. de Proft, P. Geerlings, Chem. Phys. Lett 220 (1994) 405 and references therein. [9] F. de Proft, P. Geerlings, K.D. Sen, Chem. Phys. Lett. 247 (1995) 154. [10] W. Jasko´lski, Phys. Rep. 271 (1996) 1. [11] J. Garza, R. Vargas, A. Vela, Phys. Rev. E 58 (1998) 3949. [12] A.D. Becke, J. Chem. Phys. 98 (1993) 5648. [13] A.D. Becke, J. Chem. Phys. 98 (1993) 1372. [14] F. Herman, S. Skillman, Atomic Structure Calculations, Prentice-Hall, Englewood Cliffs, NJ, 1963. [15] C. Froese-Fischer, The Hartree–Fock Method: a numerical Approach, Wiley, New York, 1977. [16] J.C. Slater, The Calculation of Molecular Orbital, Wiley, New York, 1979. [17] P.A.M. Dirac, Proc. Cambridge Philos. Soc. 26 (1930) 376. [18] U. von Barth, L. Hedin, J. Phys. C 5 (1972) 1629. [19] A.D. Becke, Phys. Rev. A 38 (1988) 3098. [20] C. Lee, W. Yang, R.G. Parr, Phys. Rev. B 37 (1988) 785. [21] J.P. Perdew, A. Zunger, Phys. Rev. B 23 (1981) 5048. [22] E.V. Luden˜a, J. Chem. Phys. 69 (1978) 1770.