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vol.48 número4CHEMICAL BASIS FOR THE ANTIMICROBIAL ACTIVITY OF ACETANILIDESA THEORETICAL ANALYSIS OF THE KOHN-SHAM AND HARTREE-FOCK ORBITALS AND THEIR USE IN THE DETERMINATION OF ELECTRONIC PROPERTIES índice de autoresíndice de materiabúsqueda de artículos
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Journal of the Chilean Chemical Society

versión On-line ISSN 0717-9707

J. Chil. Chem. Soc. v.48 n.4 Concepción dic. 2003

http://dx.doi.org/10.4067/S0717-97072003000400006 

J. Chil. Chem. Soc., 48, N 4 (2003) ISSN 0717-9324

ELECTRONIC PROPERTIES OF ATOMS AND COVALENT RADIUS DETERMINED BY MEANS OF AN EXCHANGE POTENTIAL NEW MODEL CONTAINING SELF INTERACTION AND GRADIENT CORRECTIONS

MAURICIO BARRERA* AND FERNANDO ZULOAGA

Facultad de Quimica,Pontificia Universidad Catolica de Chile,Santiago,Chile e-mail : m_barrera@mi.terra.cl
(Received: June 10, 2003 - Accepted: July 28, 2003)

ABSTRACT

Within the Density Functional Theory a new model of exchange potential containing corrective terms for self interaction and long range behavior is proposed . This potential model contains three constants: a , b and g. The first two are determined through experimental data while the third is calculated by means of a polynomial function of six degrees depending on the total number of electrons.

When this potential is introduced in the effective potential and Kohn-Sham equations are solved by iterations, calculated total energies and eigen values reproduce with high accuracy the experimental values. Total electronic densities obtained with this scheme are employed to find numerically the point where the derivatives of the kinetic energy respect to the total electronic density equal either the exchange correlation potential or half the effective Kohn-Sham potential.

Solutions of this equality come as a singular point in the radial mesh and are correlated with atomic and covalent radius.

I. INTRODUCTION

Density functional Theory (DFT) has become a standard approach to the structure of many electronic systems such as atoms, molecules and solids1. Since last decade, an increasingly interest on DFT has been observed by chemists2 partly due to the possibility of studding large molecules without increasing excessively computational cost. On the other hand, it has provided a theoretical support for some popular qualitative chemical concepts like chemical potential, electronegativity and hardness.

The DFT theory has its origin in the Hohemberg-Kohn theorem3 who in 1964 stated that the total energy for the ground state of an electronic system subject to an external potential is a unique function of its electronic density . The theorem does not specify the nature of the functionl and much of the work done by researchers has been devoted to find appropriate approximations.

One step further was achieved by Kohn and Sham (KS) in 1965 ,who developed a method4 to obtain numerical data. The method introduced a set of independent particles moving in an effective potential (Kohn-Sham potential) ,such that the particle density distribution is the same as the electron density of the interacting electronic system of interest. The DFT then relates the energy of this independent system to the desired ground state energy of the interacting system.

The Kohn-Sham potential contains a summation of terms that takes into account both the nucleus-electron attraction and the electron-electron repulsion. This last term is decomposed into a coulombic term plus another term derived from the Exchange-Correlation energy of yet not known exact function. The local density approximation (LDA) is obtained by substituting the exchange-correlation energy of the inhomogenous system with a functionality originated in the uniform electron gas theory. When LDA is used with polarized densities the improved local spin density (LSD) is achieved. The KS method with LSD has been the basis of many studies of the properties of solids, surfaces, cluster , atoms and molecules5.

In spite of its success the LSD shows failures that cannot be overcome easily. For example, calculated total energies are too high, unbound states are predicted for negative ions and orbitals' eigenvalues are not connected with removal energies due to the presence of self interaction.

In this paper a series of corrective terms for the exchange potential is introduced in the KS potential. As a result the overall values of LSD calculations for the eigenvalues and total energies simultaneously are improved. Within this theoretical framework, the introduction of the new exchange potential in the equation for thechemical potential gives rise to some equalities that are proved numerically. The resulting turnover points are compared with atomic and covalent radii.

II. THEORETICAL BACKGROUND

The fundamental variable of DFT is the charge density r (r) , that can be expressed as a sum over the ground state occupied orbitals , r (r)= FALTA IMAGEN ni|fi(r)|2. Accordingly, the chemical potential m for atoms is given by6


is the Kohn-Sham potential in the presence of the classical electrostatic potential Vel (r) , the exchange potential Vxs (r) and the correlation potential Vcs (r),


In equation 1, TS is the non-interacting kinetic energy, EX is taken as the s -polarized exchange energy and EC the correlation energy. All these terms conform the total energy EDFT that amounts to


Here J [r ] is the Coulombic repulsion energy and v(r) is the external potential . The local spin exchange energy is ,


where ex (r) is the exchange energy density.

However, since Vks density dependent, the Kohn-Sham eigenvalue equation must be solved iterating up to selfconsistency once a suitable form for VXs (r) and VCs ( (r) is established. Up to now a drastic decision has to be made when choosing these approximate exchange and correlation terms: to focus either on the total energy calculations or on appropriate eigenvalues, hopefully in agreement with experimental ionization potentials. Unfortunately it has not been possible to have either total or orbital energies within the experimental limits . Generally, authors7,8,9 have preferred to look for exchange functional that provide better total energies values rather than good eigenvalues estimates.

Another property considered out of the many electron systems is the classical turnover points of eq 1 terms at different values of the radial distance from nucleus. To begin with, the right hand side of eq 1 combined with eq 2 has four terms which individually vary with position r in the non homogeneous electron cloud of either the atom or molecule. Notwithstanding, the constant value for µ holds when the total energy is a minimum. March10 relates the chemical potential to the ionization potential in atoms by going to infinite distance from the nucleus to keep only the kinetic energy term ( as a functionl of r(r) ), plus a correlation energy correction per electron. However, another possibility is to determine the radial distance rµ at which


since at that point the chemical potential equals the electrostatic potential,


The two terms of the right hand side of eq 10 have opposite effects together they create a real physical property at the point rm which is exactly equal to the electrostatic interaction energy between the static charge distribution of the atomic system and a positive unit charge located at rm . Politzer et al.11 used this approach for neutral atoms along with Thomas Fermi methods to state that when r(r) = 0.0087 a.u. the r-position for any atom is fixed so that the electrostatic potential evaluated at this point, equals m . A similar approach was proposed by Deb12 to calculate a characteristic atomic radius on the basis of an improved Thomas-Fermi-Dirac method but always looking for the above quoted value for the charge density. It has been stated also that rm can serve as a basis to determine approximate Wigner-Seitz radius13( rs )and to find absolute hardness at the covalent radius from the electrostatic potential for atoms14 . Finally, electrostatic potential values evaluated at the molecular electronegativity isosurface in molecules reacting regiospecifically, have also been investigated from the hard and soft acid base theory (HSAB) point of view to find molecular hardness response values because of chemical attack at specific sites15,16.

In this paper, we investigate solutions of equation (10) within the KS framework. First we identify the left hand side term with the single particle kinetic operator coming from the KS’s single particle equation and for the right hand side we employ an exchange correlation model potential with correct asymptotic behavior.

A second interesting possibility is to employ equation (1) and to look for an rm value that satisfy :


The later equation implies that the chemical potential is now located in the KS effective potential


It will be shown through this work that the rm obtained numerically are a unique radial position and that the equalities given in eq 10 and 12 necessarily imply that all potential functions cross each other at a single point in r-space. Next section gives details for the exchange and correlation functions used.

III. NUMERICAL IMPLEMENTATION

Unless specified, atomic units are used elsewhere. Calculations were done over Desclaux29 atomic program modified to accept all the features required in this work.

The evaluation of the total exchange energy in the local spin density approximation (LSD) has been considered


where the exchange energy density ex(r) can be identified as the total electron density to one third power. It is well known that good results for total energies are obtained if the asymptotic behavior ex(r) ® - 1/2 when r® ¥. is obeyed. One of the best functional that comply with this requirement is Becke's 88 functional17 for the total energy.

On the other hand, the requirement for good eigenvalues18 is that Vx,s (r) ® - 1/r when r® ¥. However, the use of eq 4 yields

for the exchange potential ; so that

this semiempirical functional should not provide good eigenvalues at all. In fact, from the mentioned statement it can be concluded that ex (r) and V (r) are not compatible to obtain good eigenvalues and exact total energies simultaneously. One way to overcome the problem is to incorporate Self-Interaction Correction terms (SIC) because it is well known that there are no cancellation effects in the KS potential when solving the eigenvalue value problem19. To deal with this problem we used Rae20 and Cortona’s21 approaches that consist in introducing the classical expression for the homogenous electron gas multiplied by a g function depending on the Ns spin orbitals .This new term compensates the self interaction originated in the integration of the coulombic energy when the condition i¹ j is not obeyed, as required by equation 8.

On the other hand a potential containing exchange plus correlation was built. The correlation was taken from the PW91 expression18 .Also, the exchange model potential used includes a term to correct the self interaction term mentioned above, and a generalized gradient correction (GGC) type expression improving the long range behavior in the sense of Van-Leeuween18.


Here x =and a and b are two numerical constants optimized in regard to experimental values for total energies and ionization potentials for atoms. Their values are a = 0.600 and b =0.0206 . Furthermore, g has been written as a polynomial including six constants23.


When all of this formulation are inserted in eq 2 to solve the spin-polarized Kohn-Sham eigenvalue equation, the


density rs (r) can be built, and the total energy calculated by means of equation 6. Because the BZ model exchange potential is not a functional derivative, the exchange energy can be calculated with the Levy-Perdew equation24


that hold aproximately.

According to the described analysis, several equations can be written now to find the different classical turning points. Each of the several possibilities available are going to be analyzed hereafter.

1. Equation 17 can be written in the form


where tsis (r) is the single particle kinetic energy operator

which is also


Veff(r) is the effective Kohn-Sham potential, that contains explicitly all terms discussed above, and eis is the spin polarized eigenvalue. When the equality eis = emax holds, it is found that emax = m , so that it can immediately state that for the equality


exist for a given value r, then at that value of r the chemical potential is given by


The value r = rm1 fixes the first radial point to determine the chemical potential. Incidentally, the use of eq 20 in terms of is

not arbitrary beause it can be deduced21 from the DFT relationship between the total kinetic energy functional Ts[r] and the total potential


so that by taking the explicit functional derivative of Ts[r] against r(r) it allows us to use eq 20.

Thus, a plot of FALTA IMAGEN as given by eq 20 and emax versus r should define a crossing point r = rm1 at which the three functions meet with each other.

2. The possibility to determine the radial distance rm at which -tsis (r) = vxs (r) + vcs (r) states that at rm2 the chemical potential equals the electrostatic potential,


and this defines a second crossing point for rµ. It is clear that rm2 at which

requires also that

cross the eigenvalue eis at exactly the same rm2 value so that -tsis (r) , vxs (r) + vcs (r) , Vel (r) and eis do not meet each other at a common point on the r-surface, rather they define two crossing points at the same rm value.

3. Equation 1 can be rewritten in the explicit manner :


in which all the different terms have been defined previously. Since VFALTA IMAGEN(r) and Vcs (r) are corrective terms to improve the KS equation because of correlation effects and asymptotic behavior for the potential, a "pseudo" Kohn-Sham equation will define:


so that when the following relation holds :


In the specific case that there exists a rm value at which the above equality holds true.

then the pseudo chemical potential is determined also by the electrostatic potential value at that given r-position,


as stated before. Obviously, rµ3 from eq (26) is not equal to rµ2 from eq (24) because of the shift established in eq (26).

4. For the frontier orbital ( the homo level for the molecular case) it is also possible to write from eq (25)



IV. RESULTS AND DISCUSSION

Figure 1shows plots for two different atoms of the gama-sic parametrized exchange potential with non local correction. It is possible to observe the difference between a local potential (LSD) that tends to zero at large distance, and a non local one (LB) with asymptotic behavior that tends to -1. The model potential developed in this work (thereafter called BZ) follows the general form of a LB type potential. The main differences between both potentials do not appear in the outermost region, but near the nuclei, between 1 and 2 were they show different shapes. This difference is caused by the predominance of the term a b x2r2/ 3 over 3bx(sinh-1x). Note that this effect occurs in a region of special interest for the chemical reactivity ,because bonding occur within these limits and it has been pointed out that the LB model potential fails in reproducing accurately bond length in molecules

In table 1 the calculated Total Energies for two series of atoms are compared to experimental data. Experimental Total Energies are calculated as the sum of all the Ionization Potentials for each atom . The results obtained with the BZ potential are compared with data obtained from two well known exchange potentials in order to observe relative differences. The differences are measured as percent error ,and a statistical absolute mean for 19 atoms is calculated showing 1.34% for LSD, 0.52% for LB and 0.08% for BZ.


Fig. 1.- r* Exchange potential for Beryllium and Calcium
---- Exchange potential with LSD
o Exchange potential with Van Leuween-Baerends
correction
* This work




Table 1: Total Energies ( in Hartree) calculated by different methods
compared with experimental data for two series of atoms.


ATOM

Experiment1

LSD2

LB3

BZ4


He -2.904 -2.755 -2.860 -2.906

Li

-7.478

-7.254

-7.447

-7.483

Be

-14.669

-14.319

-14.678

-14.664

B

-24.659

-24.197

-24.763

-24.630

C

-37.858

-37.285

-38.070

-37.831

N

-54.615

-53.925

-54.931

-54.642

O

-75.112

-74.251

-75.654

-75.088

F

-99.811

-98.779

-100.553

-99.769

Ne

-129.057

-127.843

-129.969

-129.067

Na

-162.439

-161.046

-163.496

-162.530

Mg

-200.336

-198.702

-201.598

-200.590

Al

-242.741

-240.861

-244.091

-243.039

Si

-289.904

-287.739

-291.275

-290.159

P

-341.999

-339.500

-343.315

-342.11

S

-399.101

-396.184

-400.448

-399.230

Cl

-461.404

-458.063

-462.719

-461.458

Ar

-529.139

-525.292

-530.295

-528.957

K

-599.566

-597.542

-602.993

-601.503

Ca

-680.227

-675.047

-681.058

-679.403


% Error
 
1.34
0.52
0.08

1Ref. 26 2 LSD with PW91 3 Baerend-Leuween with PW91 4This work

Figure 2 shows the general trend of the error as function of the atomic number. The most interesting features of this graph is that the error remains constant along the ordinate for BZ model exchange potential and predicted Total Energies for light atom (He, Li, Be) are in excellent accordance with experimental data.


Fig 2. % Error between Calculated and Experimental Energies for 19 atoms :
* this work
o LB exchange potential
* LSD

In table 2, the highest eigenvalues are compared with the first Ionization Potential for 27 atoms. The error is calculated as the difference between two values and the absolute mean of this difference, for the series of 27 atoms are 0.157 for LSD, 0.015 for LB and 0.014 for BZ. Again the statistical analysis confirms the good performance obtained with the BZ exchange potential when predicting an experimental property as the Ionization Potential.


Table 2: Comparison of First Ionization Potentials of series of 27 atoms with highest eigenvalues (in au)


ATOM

- IP 1

LSD2

LB3

BZ4


He

-0.903

-0.565

-0.846

-0.860

Li

-0.198

-0.110

-0.187

-0.190

Be

-0.343

-0.202

-0.316

-0.320

B

-0.305

-0.149

-0.294

-0.288

C

-0.414

-0.225

-0.399

-0.391

N

-0.534

-0.306

-0.508

-0.507

O

-0.501

-0.263

-0.506

-0.480

F

-0.640

-0.377

-0.640

-0.614

Ne

-0.792

-0.493

-0.777

-0.765

Na

-0.189

-0.108

-0.200

-0.199

Mg

-0.281

-0.172

-0.285

-0.284

Al

-0.220

-0.109

-0.214

-0.218

Si

-0.300

-0.168

-0.288

-0.296

P

-0.385

-0.228

-0.365

-0.376

S

-0.381

-0.220

-0.401

-0.396

Cl

-0.476

-0.299

-0.485

-0.488

Ar

-0.579

-0.378

-0.574

-0.582

K

-0.160

-0.092

-0.177

-0.177

Ca

-0.224

-0.138

-0.236

-0.237

Ga

-0.220

-0.108

-0.228

-0.218

Ge

-0.290

-0.163

-0.291

-0.277

As

-0.347

-0.217

-0.355

-0.338

Se

-0.358

-0.210

-0.392

-0.377

Br

-0.434

-0.277

-0.456

-0.438

Kr

-0.514

-0.342

-0.524

-0.502

Rb -0.154 -0.089 -0.173 -0.162
Sr -0.209 -0.129 -0.224 -0.212

Absolute Error   0.157 0.015 0.014

1Ref. 26 2LSD with PW91 3Baerend-Leuween with PW91 4This work

Figure 3 depicts the overall trend of the error for three series of atoms for the BZ and LB exchange potentials. From this graph it is possible to observe the same general behavior of the error for both model potential. Differences appear for Oxygen and Fluorine were the error is greater for the BZ scheme than for LB. Another interesting feature is that for the 3rd series the deviation for BZ values keep constant while for LB it ls greater. Data for LSD values are not included because they are out of scale.

Electron Affinities were also calculated and found as the negative of the highest occupied eigenvalue for the negative single charged ion. Results are displayed in Table 3 and compared with experimental values. Eigenvalues were obtained through a multistep procedure consisting of adding small amounts of charge to the neutral atom, iterate to reach self-consistency and followed by constant increments of charge until the single ion electronic occupancy is reached. Attempts to calculate ionization in one step was only possible for the Halide series.


Fig 3. Difference between Ionization Potential Highest Eigenvalue
for 27 atoms,
* this work
o LB exchange potential


Table 3 : Experimental Electron Affinities and negative of highest orbital energy
of the single negative ion ( in eV )


ATOM

Exp 1 1

BZ 2


Li

0.62

0.46

B

0.28

0.30

C

1.27

1.25

N

0.07

0.06

O

1.47

1.50

F

3.42

3.51

Na

0.55

0.52

Al

0.46

0.49

Si

1.40

1.39

P

0.73

1.28

S

2.08

2.53

Cl

3.65

3.93

K

0.50

0.57

Br

3.36

3.32

I 3.06 3.05

Absolute Error

 

0.12


1 Ref- 26

According to these results, the different potential vs. the radial distance were plotted and the intersection from the different curves located as predicted from equations (19) to (30).In figure 4 a representative picture of this equations for Carbon is shown. The expected turning points are depicted with number in circles. The interpretation of this map starts with the identification of the curve for the negative of the one electron kinetic energy operator (-tS) ,that moves from high to low energies as the radius is increased. Beginning at the left down corner of figure 5 and moving upwards the first turning point appears, namely rµ2 , which results from the intersection of -tS with Vxc ( the exchange correlation potential) .At the same radii but at lower energies emax intercepts Vel , the electrostatic potential as stated in paragraph 2 .Moving a step forward along the curve of -tS ,the intersection with Vxs, the local SIC exchange only potential, defines rµ3 . Again, moving up over the energy ordinate and according to equation 25 , the electrostatic potential intercepts the m' function. The next turning point found is rµ1 and it occurs as -tS intercepts half the effective Kohn-Sham potential and the highest eigenvalue ( equation 19 and 20 ).Finally the last crossing point, rµ4, appears when m' and 1/2Vss converge with -tS .


Fig 4. Graphical representation of the 4 tuning points founds for Carbon

Table 4 summarizes results for the four radii obtained by solving the equations .In general they appear following the order : rµ2 < rµ3 » rµ1 < rµ4 and no crossing occur between them except for rµ3 and rµ1 that present a little influence of the outermost shell occupancy; for closed shell rµ3 > rµ1 while for half filled, rµ1 is greater. An attempt to attribute a chemical meaning to these values is achieved by comparing them with well-known radius scales. From table 5 it can be seen that rµ-values are near the atomic radii, Rat24 ,a theoretical parameter determined as the maximum radial probability density of the outermost electron orbital. Linear correlation between this parameter calculated and radius is higher for r µ3 and rµ1.

Next we compared our results with an experimental parameter ,the Sanderson’s non polar covalent radius obtained though experimental bond length of homonuclear molecules. Again rµ3 and rµ1 show higher correlation than rµ2 and rµ4 but considering the four values it seems that numerically the most correlated is rµ1 . Finally comparison is done with the radius of PPM11 .Those values were calculated by equalizing the Mullikan’s electronegativity with an electrostatic potential obtained with Hartree-Fock densities. Surprisingly correlation coefficient lies above 0.95 for all four radii while rµ3 and rµ1 continue being the most correlated, but from a numerical viewpoint it is rµ3 the most correlated.


Table 4 : Comparison between calculated Rµ with different radius scales ( in Angstrom)


ATOM

Rca

Rppmb bBb

Rat c

Rµ1

Rµ2

Rµ3

Rµ4


Li

1.334

1.357

1.586

1.257

1.076

1.163

1.423

Be

0.886

-

1.040

0.996

0.922

1.044

1.163

B

0.822

1.091

0.776

1.1636

0.922

1.110

1.468

C

0.772

0.912

0.620

0.966

0.790

0.908

1.128

N

0.734

0.814

0.521

0.814

0.687

0.777

0.922

O

0.702

0.765

0.450

0.777

0.666

0.802

0.936

F

0.681

0.671

0.395

0.676

0.597

0.607

0.666

Na

1.539

1.463

1.713

1.358

1.060

1.297

1.538

Mg

1.373

-

1.279

1.181

1.060

1.219

1.358

Al

1.258

1.487

1.312

1.586

1.219

1.468

2.001

Si

1.169

1.296

1.068

1.358

1.110

1.257

1.586

P

1.107

1.185

0.919

1.181

1.012

1.110

1.337

S

1.049

1.120

0.810

1.093

0.966

1.128

1.277

Cl 0.994
0.999
0.725 0.981 0.894 0.996
1.110

K

1.962

1.802

2.162

1.725

1.303

1.669

1.968

Ca

1.740

-

1.690

1.538

1.337

1.586

1.768

Ga

1.256

-

1.254

1.542

1.277

1.459

1.962

Ge

1.223

-

1.090

1.401

1.110

1.317

1.636

As

1.194

1.258

1.001

1.277

0.981

1.219

1.445

Se

1.167

1.209

0.918

1.163

1.044

1.238

1.379

Br

1.142

1.116

0.851

1.093

0.996

1.128

1.238

Rb

2.160

1.924

2.287

1.900

1.436

1.900

2.185

Sr

1.910

-

1.836

1.741

1.487

1.803

1.967

Sn

1.420

1.492

1.240

1.562

1.297

1.514

1.823

Sb

1.389

1.433

1.193

1.445

1.257

1.401

1.611

Te

1.360

1.381

1.111

1.337

1.219

1.445

1.562

I

1.333

1.299

1.044

1.277

1.181

1.337

1.423

Cs

2.350

-

2.518

2.152

1.661

2.152

2.495

Ba

1.980

-

2.060

1.968

1.697

2.103

2.246


Corr ( Rc )

0.934

0.927

0.949

0.889


Corr ( Rppm)

0.985

0.956

0.977

0.958


Corr ( Rat )       0.919 0.882 0.913
0.886

a Ref 27 b Ref. 14 c Ref. 28

       

V. CONCLUSION

The BZ model exchange potential reproduces Ionization Potential, Electron Affinities and Total Experimental Energies with less deviation from experimental data than previous models. This enhanced performance is due to the inclusion in the exchange potential of corrective terms for the self interaction and asymptotic behavior.

The KS single particle equation combined with the chemical potential equation predicts some equalities that have to be reach by the different potentials used therein .When these equalities are solved numerically, four single points in the radial mesh are obtained. These points, called rm values, can be correlated with covalent radii of neutral atoms. From the above analysis two points are of particular interest : rµ1 obtained when the chemical potential is equal to half the KS potential and rµ3 occurs when the derivative of the kinetic energy meet the local component of the exchange potential.

ACKNOWLEDGMENT

This work is dedicated to the memory of Dr. Fernando Zuloaga; the seeds of knowledge he sowed among us will never be forgotten. I would like acknowledge to Dr. Barbara Loeb for encouraging me to finish this paper and for her suggestions and also Dr. Alejandro Toro and Dr. Patricio Fuentealba for helpful criticism.

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23.- This constants were calculated by a procedure that involve two steps.In the first step exact values of g are determined by matching the calculated total energy with experimental data of a series of atom including He,Be,Ne,Mg,Ar and Ca.The gama values are introduced in the program has adjustable parameters.The second step consist in introducing previously determined gama values in a Kaleidagraph program and fit them against N-1/3 using a six degree polynomial curve fitting option.Coeficients obtained are C0= 21.95, C1= -247.979, C2= 1115.451 , C3= -2560.222, C4=3180.237, C5= - 2036.248, C6= 527.001

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