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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/S0717-97072003000400018 

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

CORRECTING THE ATOMIC HIGHEST OCCUPIED ORBITAL ENERGY WITHIN AN HYBRID DENSITY FUNCTIONAL MODEL.

P. FUENTEALBA AND O. REYES

Departamento de Física
Facultad de Ciencias, Universidad de Chile, Santiago de Chile, Chile.
E-mail: pfuentea@uchile.cl

Received: July 9, 2003 - Accepted: September 16, 2003)

ABSTRACT

A modification in the parameter fitting of an hybrid functional yields a new functional which is able to improve the atomic highest orbital energy. According to the Janak’s theorem it should be equal to the ionization potential. Results for the atoms hydrogen to argon confirms the improvement. The dipole polarizabilities are also improved. We discuss and show that such a modification can not be applied to molecular systems. Further, it is shown that long range interactions can be also improved. As an example, the classical transition barrier of the H + H2 H2 + H reaction has been calculated.

INTRODUCTION

Density functional models are increasingly used in the study of the electronic structure of atoms, molecules, clusters and solids. The accuracy of the calculated properties depends to a great extend on the used exchange-correlation functional, and this is the main difference among the many density functional models currently applied in computational quantum chemistry. The simplest and first implemented functional is the Local Density Approximation (LDA) [1], in which the exchange-correlation functional of the uniform electron gas of density r is used at every point , with the density r() of the studied system. Several corrections for the nonuniform behaviour of atomic and molecular densities have been proposed. The initial gradient expansion, unfortunately, diverges. Hence, corrections to the gradient expansion have been put forward leading to the GGA (Generalized Gradient Approximations) models [2]. They are generally expressed in terms of the dimensionless reduced gradient x=|Ñr()|/r()4/3.. The results obtained by the GGA models are comparable in accuracy to the ones obtained by the second order Moller-Ploesset (MP2) perturbation theory approach. One of the qualitative failure of this functionals is the so called self-interaction-energy. Its importance in chemistry was clearly recognized by Zuloaga et al [3].The next improvement to the exchange-correlation functional model cames by the inclusion of some amount of Hartree-Fock like exchange calculated with the Kohn-Sham orbitals [4]. The methods are based on the adiabatic connection formula and they are known as hybrid functionals. More recently, a new type of functionals including the kinetic energy density have been proposed [5]. In this paper we will focus on the hybrid density functional models.

Several hybrid schemes have been proposed [6]. The most widely used has three empirical parameters. They are generally written as:

(1)

where the subindices x and c stand for the exchange and the correlation contribution respectively. The upper indices stand for the type of model functional. The three empirical parameters are a1, a2 and a3.. Among those functional models the most popular in computational quantum chemistry is the B3LYP [7] which is claimed to yield an accuracy of a couples of kcal/mol.

Perdew and collaborators [8] have next shown that it is possible to simplify it to only one parameter which can be fixed a priori using a density functional perturbation theory to fourth order. The new family of functionals can be written as:

(2)

The parameter a has a theoretical value of 1/4. Recent studies have shown that among them the B1LYP variant gives results comparable to the ones obtained with the B3LYP model[9].

In spite of the great success of the mentioned functionals in the calculation of a variety of electronic structure properties of atoms, molecules, clusters and solids, there is, however, a very simple test they can not pass. According to the Janak’s theorem [10] the HOMO energy must be exactly equal to the ionization potential. No one of the mentioned functionals yield reasonable HOMO energies when compared with experimental ionization potentials. For instance, the B3LYP model gives an average error of 3.82 eV. in the ionization potential of the atoms H to Ar (see Table 1). The failure is not only in the ionization potentials, it is also reflected in many response functions such as polarizabilities and Rydberg excitation energies. The reason of the failures are well understand but there is no formal way to correct it. The error is due to the incorrect asymptotic behaviour of the model exchange potentials. It has been demonstrated that it is impossible to construct an exchange functional of the GGA form that, at the same time, reproduces the correct asymptotic behaviour of the functional and its respective potential [11]. To solve, in part, the problem, some phenomenological exchange potentials and empirical functionals have been proposed [12, 13, 14, 15, 16].

In this paper, we explore the possibility of improving the asymptotic behaviour of the exchange potential in an empirical way, mainly by modifying the empirical parameter a of Eq. (2) in order to match the experimental ionization potential of Ne atom with the calculated energy of the HOMO. This possibility is based on the known relationship between the ionization potential and the exponentially decaying behaviour of the electron density at large distances.


Table 1.Calculated and experimental ionization potential (IP) of atoms (in eV).


Atom

IP(B1LYP)a)

IP(B1LYPm)b)

eH(B1LYP)

eH(B1LYPm)

eH(B3LYP)

IP(exp)c)


H

13.55

13.58

8.909

12.15

8.754

13.60

He

24.75

24.67

18.27

23.49

17.91

24.58

Li

5.520

5.508

3.635

4.895

3.647

5.39

Be

8.801

8.759

6.333

8.117

6.300

9.32

B

8.557

8.497

5.210

7.753

5.107

8.296

C

11.38

11.35

7.445

10.76

7.286

11.26

N

14.50

14.49

9.946

14.05

9.732

14.54

O

13.85

13.46

9.286

13.30

9.059

13.61

F

17.48

17.12

12.41

17.28

12.12

17.42

Ne

21.51

21.60

15.87

21.60

15.50

21.56

Na

5.277

5.214

3.457

4.609

3.475

5.138

Mg

7.594

7.522

5.285

6.790

5.282

7.644

Al

5.842

5.861

3.554

5.257

3.539

5.984

Si

7.970

8.022

5.277

7.435

5.230

8.149

P

10.27

10.37

7.161

9.740

7.082

11.00

S

10.30

10.24

7.100

9.720

7.012

10.36

Cl

12.89

12.88

9.328

12.43

9.201

13.01

Ar

15.70

15.78

11.76

15.36

11.59

15.76


a) energy difference between neutral and ionic atom calculated with B1LYP functional.
b) energy difference between neutral and ionic atom calculated with B1LYPm functional modified.
c) experimental ionization potential from Ref. [20].

RESULTS AND DISCUSSION

All calculations have been done using the G98W program [17]. The B1LYP model has been constructed using the IOp(5/45=07500250) values. The default B1LYP keyword was also used to check the validity of the results. Then the parameter a was varied until the HOMO energy of the Ne atom matched to within 1% the experimental ionization potential. The tzv basis set developed in Ref. [18] was used. In this way a value of a=0.73 was obtained. The calculations with the modified parameter are called B1LYPm in the following Tables and Figures. In Table 1 and Figs. 1 and 2 the calculated ionization potentials of the atoms from hydrogen to argon are shown. They were calculated as the HOMO energy

eH, and also as the energy difference between the neutral atom and the ion. The improvement in eH obtained with the B1LYPm is impressive as can be seen in the Figures. The standard deviation drop down from 3.668 eV. in the standard B1LYP model to 0.662 eV. in the modified version. It is surprising that adjusting just only one atom the whole curve moves in the right direction. However, it is to note that the absolute errors are still considerable, and the calculations by means of energy differences are still better, as can be seen in Table 1. It is also interesting to note that the energy differences calculated with the modified version are very similar to the original ones. This means we have improved the orbital energy of the HOMO without affecting the total energy of the atomic systems.

Now, we look for other atomic property which is not directly related to the orbital energy but to the asymptotic behaviour of the exchange potential. In Table 2 the calculated atomic dipole polarizabilities are compared with the recommended values. The improvement is also visible starting with the dipole polarizability of the hydrogen atom, which falls down from 5.063 a.u. to 4.644 a.u. in much better agreement with the exact value of 4.5 a.u. . Excepting the alkaline earth atoms the modified version change the values in the right direction. The alkaline earth atoms present a near degenerancy effect which is very difficult to mimic using any density functional model. It is however to note that for some atoms the correction is overemphasized.

Having in mind, that the empirically corrected exchange potential proposed by Leuween and Baerends [12] also improves the atomic response properties but fails in the description of bonding, we have calculated some molecular properties using the modified B1LYP functional. In Table 3, some calculated properties for the N2, O2 and F2 molecules are compared with the experimental values. All properties, excepting the dipole polarizabilities, were calculated using the 6-311G** basis set. For the calculation of the dipole polarizabilities the Sadlej’s basis set [19] were used. Excepting for the mean value of the polarizability which is slightly improved, all other properties are clearly worse. Specially, the dissociation energies. As it has been discussed elsewhere [13, 15], it seems that the asymptotic behaviour of the exchange potential is important for atomic properties but the molecular electronic structure is mainly dictated by the bonding properties, and therefore, for the behaviour of the exchange and correlation potentials in the bond region..


Fig 1. Calculated ionization potential (IP) as the HOMO energy for the atoms H to Ne, in comparison to the experimental values.


Fig 2. Calculated ionization potential (IP) as the HOMO energy for the atoms Na to Ar, in comparison to the experimental values.


Table 2. Calculated and recommended mean values of the atomic dipole polarizability.


a(B1LYP)

a(B1LYPm)

Exp.a)


H

5.063

4.644

4.521

He

1.410

1.316

1.383

Li

139.0

148.4

164.0

Be

42.56

42.57

37.79

B

22.23

20.45

20.45

C

12.47

11.15

11.88

N

7.737

6.912

7.423

O

5.468

4.783

5.412

F

3.844

3.346

3.779

Ne

2.024

1.882

2.672

Na

146.1

158.1

159.3

Mg

72.64

74.86

71.53

Al

61.48

57.66

56.28

Si

40.34

37.01

36.31

P

26.81

24.78

24.50

S

20.43

18.85

19.57

Cl

15.22

14.09

14.71

Ar

10.00

9.536

11.07


a)Results from Ref. [21]


Table 3.Calculated and experimental dipole polarizability <a>, bond length Re, dissociation energy De, vibrational frequency ne and vertical ionization potential IPv of N2, O2 and F2.


B1LYP

B1LYPM

Exp


N2

<a>

(au)

11.78

11.23

11.74a

Re

(au)

1.094

1.071

1.098b

De

(eV)

9.497

8.074

9.91b

ne

(cm-1)

2467

2706

2360b

IPv

(eV)

15.69

16.51

15.5b

O2

<a>

(au)

10.65

9.95

10.59a

Re

(au)

1.203

1.160

1.208b

De

(eV)

3.266

3.475

5.21b

ne

(cm-1)

1650

1958

1580b

IPv

(eV)

12.93

13.44

13.1b

F2

<a>

(au)

8.592

7.682

8.38a

Re

(au)

1.403

1.335

1.435b

De

(eV)

1.272

-0.1824

1.66b

ne

(cm-1)

998.1

1194

892b

IPv

(eV)

15.84

16.64

-


a) from Ref. [22]
b) from Ref. [23]

Another situation where the asymptotic behaviour can be important is for long range interactions like hydrogen bonds. As a known case where the currently used functionals fail we studied the classical barrier heights of the H2 + H H + H2 reaction. The results are shown in Table 4. Again, the modified version of the B1LYP functional shows an impressive improvement.


Table 4. Calculated and experimental classical barrier heights (Kcal/mol) of H2+H H+H2 reaction.

 

Method

Classsical barrier


B3LYP

4.25

B1LYP

4.79

B1LYPm

8.00

Exp

9.7a

 

a) Results from Ref. [24]

Concluding, in this work has been shown that a small modification of an hybrid functional leads to a model satisfying the Janak’s theorem in atomic systems. Further, the calculation of any atomic property depending on the asymptotic behaviour of the exchange potential is improved. So are also the long range interactions. It is, however, important to note, that the modified version of the B1LYP functional presented here, is not able to reproduce molecular properties.

ACKNOWLEDGMENTS

This work is dedicated in the memory of Professor Fernando Zuloaga. He was one of the pioneer in recognizing the importance of density functional theory in chemistry. Part of this work has been supported by FONDECYT (Fondo de Desarrollo Cientifico y Tecnologico), Grant 1010649.

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