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

A THEORETICAL ANALYSIS OF THE KOHN-SHAM AND HARTREE-FOCK ORBITALS AND THEIR USE IN THE DETERMINATION
OF ELECTRONIC PROPERTIES

Jenny Zevallos and Alejandro Toro-Labbé

Laboratorio de Química Teórica Computacional (QTC), Departamento de Química Física,
Facultad de Química, Pontificia Universidad Católica de Chile, Casilla 306, Correo 22, Santiago, Chile

ABSTRACT

In this work a comparative analysis of the performance of the Hartree-Fock and Kohn-Sham orbitals energies to produce reliable electronic properties is evaluated. Our results suggest that the negative of Hartree-Fock and Kohn-Sham highest occupied orbital define upper and lower limits for the experimental values of the first ionization potential. Chemical potential, hardness and polarizabilities of seventeen representative molecules have been evaluated at the Hartree-Fock and Density Functional Theory levels and a new quasi-analytic model to estimated the energy of the lowest unoccupied molecular orbital emerged from the analysis of these properties.

Key Words: Orbital energies; chemical potencial; hardness; ionization potentials; polarizability; HOMO; LUMO.

RESUMEN

En este trabajo los orbitales Hartree-Fock y Kohn-Sham son evaluados para determinar propiedades electrónicas confiables definidas a partir de las energías de los orbitales frontera. Nuestros resultados sugieren que el negativo de la energía del orbital HOMO Hartree-Fock y Kohn-Sham define los límites superior e inferior para los valores experimentales del primer potencial de ionización. Además, se ha evaluado el potencial químico, dureza y polarizabilidad a nivel Hartree-Fock y Teoría Funcional de la Densidad para diecisiete moléculas representativas de donde emerge un nuevo modelo cuasi-analítico para estimar la energía del primer orbital virtual.

1. INTRODUCTION

In the last years the use of numeric methods based on Density Functional Theory (DFT)^(1-3) to determine different properties of atoms and molecules was increased. In this context it is important to remark the differences among DFT methods and Hartree-Fock method in determining electronic properties that depend on the frontier orbital energies. In this paper a comparative analysis of the eigenvalues and electronic properties at DFT and Hartree-Fock (HF)^(4,5) levels is performed, the main goal of this work is to establish the quality of the different methodologies to determine electronic properties and to check the consistency of theoretical calculations through the determination of the energy of the lowest unoccupied molecular orbital (LUMO).

In the Hartree-Fock method, based on the Molecular Orbital Theory, the physical interpretation of orbital energies is based on the Koopmans theorem^(4,5). On the other hand, in DFT methods the Kohn-Sham (KS)⁽⁶⁾ orbitals are introduced as an auxiliary tool in the calculation of the total energy. In contrast to the HF method, in DFT only the energy of the highest occupied orbital has physical meaning and it corresponds to the first ionization potential, but this is true only for the exact density functional. The theory doesn't give physical meaning of the remaining KS orbitals^(7,8).

2. THEORETICAL BACKGROUND

Hartree-Fock and Kohn-Sham methods has in common that both produce a set of self consistent orbitals with a set of eigenvalues that correspond to the orbital energies ^(1,4-6). The set of eigenfunctions and eigenvalues are obtained through solving N-equations of the type^(4,5,9):

where Ô_i it is the operator that defines the movement of electron i. In the HF method, the operator Ô_iis the Fock operator , that is given by^(4,5)

that includes the kinetic energy (first term), the external potential (V(r_i)) operators and , the Coulomb operator K_j(r_i)=and dr_j, the exchange operator^(4,5). Then each electron moves in a field defined by the Coulomb and exchange potential; the HF method doesn't include the electronic correlation^(4,5,9).

On the other hand, in DFT the fundamental variable used to determine all observables is the electronic density r(r)^(1,6,10), in the Kohn-Sham method the operator Ô is given by the following expression:

where E_xc is the exchange correlation potential; the electronic density r(r) is built up from the occupied orbital f_i:

When analyzing equations 2 and 3 we observe that the KS method can be seen as a generalization of the HF method, in which the exchange potential is replaced by a potential that contains simultaneously the exchange and the electronic correlation⁽⁹⁾.

In HF method, the physical interpretation of orbital energy is based on the Koopmans theorem^(4,11,12), which establishes that the energy of an occupied orbital {e_i}can be approximated to the negative of the ionization potential {I_i}, while the virtual orbital {e_j} energy approaches to the negative of the electronic affinities {A_j}, then:

Ii U-ei (i = occupied orbitals); Aj U-ej (j = virtual orbitgals)

(5)

In the last years there has been a strong increment in the use of methods based on DFT^(1-3) to determine electronic properties that depends upon the frontier orbital energies, in this context it is necessary to assess the reliability of these orbitals. This is an important point that we want to address in this paper.

Since the energy of the highest occupied KS orbital in DFT when the exact density functional is used, corresponds to the first ionization potential^(7,8), I_min=-e_H and since the density functional is not exact, it is not possible to specify the physical meaning for the energy of the KS orbitals. However deviations from the experimental ionization potential are expected to be inversely proportional to the quality of the functional used in the calculations. In this context through the qualitative and quantitative comparison with experimental date and HF orbitals it should be possible to extract information that allows to characterize the KS orbitals to use them safely in the determination of electronic properties.

In this paper we will analyze the HF and KS orbitals with emphasis in the HOMO and LUMO frontier orbitals. Also, we will analyze the chemical potential (m), molecular hardness (h) and polarizability (a) that are electronic properties that characterize the reactivity of a chemical system. These quantities will be used to determine the quality of the frontier orbital and to setup a consistent model to estimate the energy of the first virtual orbital.

Chemical Potential and Hardness. Chemical potential and hardness are the response of the system when the number of electrons is varied at fixed external potential. The definition m of and h given by Parr and Pearson^(1,13,14) and the finite difference approximation lead to the following working expressions for the chemical potential and hardness:

where I and A are the first ionization potential and the electronic affinity respectively. As already mentioned, the Koopmans theorem⁽¹²⁾ allows to write and in terms of the HOMO ( ) and LUMO ( ) frontier orbital energies:

Physically, m characterizes the escaping tendency of electrons from the equilibrium system, it is then related to the electronic charge rearrangement associated to any chemical process; molecular hardness can be understood as a resistance to the charge transfer^(1,13-15). Both, m and m, are global properties of the system and they have been used as descriptors of chemical reactivity. The Principle of Maximum Hardness (PMH) has been established as a reactivity hint, it states that molecular systems in equilibrium should be in a state of maximum hardness^(13,15,16), as corollary it is expected that transition states present a minimum hardness value⁽¹⁷⁾. There are many examples that show that the PMH complements the minimum energy criterion for the stability^(18-20) and it is worth to point out that the PMH connects electronic properties with the energy of the system⁽²⁰⁾.

Polarizability. The polarizability may be used to understand the behavior of the system for changing external potential at constant number of electrons. The polarizability is determined as the average of the tensorial components: aºáa) = 1/3 (a_xx + a_yy + a_zz). Qualitatively it has been found that is inversely proportional to the hardness^(15,21-23), in particular an interesting correlation has been established between the softness (S = 1/(2h) and the cubic square root of a^(24-26). In this context, the polarizability may give the possibility to verify indirectly the quality of the frontier orbitals, in particular it may lead to information about LUMO orbital, this way to characterize the energy of the LUMO is going to be explored in Sections 4.

On the other hand, Chattaraj and collaborators have proposed the Minimum Polarizability Principle (MPP), which establishes that the natural direction of evolution of any system is toward a state of minimum polarizability^{(1,13,24,27,28)}. In general, the conditions of maximum hardness and minimum polarizability complement the minimum energy criterion for the molecular stability. Moreover, the study of these properties may produce alternative or complementary models of reactions and chemical reactivity.

3. Computational Methods

The 17 molecules studied in this paper are presented in Figure 1, full geometry optimization to characterize their properties have been carried out on all of them. All the calculations were carried out using the program Gaussian98⁽³⁰⁾, with a set of double and triple with polarization functions basis sets^(31-41). In DFT calculations, we have first analyzed different exchange and correlation functionals: Becke-Perdew 86 (BP86)^(42,43), Slater (HFS)^(6,10,44), Slater-Vosko-Wilk-Nusair (SVWN)^(6,10,44,45) and Xalpha (Xa) ^(6,10,44); the hybrid functionals: B3LYP made of Becke functional with three parameters with correlation from the Lee-Yang-Parr functional^(42,46-48); the B3PW91 functional that uses correlation from the Perdew-Wang 91 functional^(42,49-51); BHandH made of Becke functional where it is considered half of the HF exchange and the other half of LSDA (Local Spin Density Approximation) using for the correlation the Lee-Yang-Parr functional⁽⁴⁸⁾; and BLYP made of Becke functional with one parameter using for the correlation the Lee-Yang-Parr functional^(52,53).

Figure 1: Sketch of the 17 molecules under study.

4. RESULTS AND DISCUSSION

Functionals Dependency. We begin our study analyzing the exchange and correlation functionals used for the calculation of the KS orbitals energies with the 6-31G* basis set. The functionals used are: 1=B3LYP, 2=B3PW91, 3=BHandH, 4=BLYP, 5=BP86, 6=HFS, 7=SVWN, 8=Xa. To simplify the analysis, we will distinguish occupied and virtual orbitals; among the occupied orbitals are the core (internal) and valency orbitals. In all diagrams, occupied orbitals are shown in black dashes and virtual orbitals in gray dashes, the core orbitals are omitted of the diagrams.

In Figure 2 are shown the energy diagrams of the KS orbitals for all the functional tested in this work and for three representative system (H₂O, CO₂ and t-HNNH) out of the 17 molecules. We first note that the energy of the KS orbitals depend strongly on the functional, it is seen that the gap HOMO-LUMO remains constant, except for the functionals 1, 2 and 3 because these hybrid functionals use HF exchange and a correlation functional DFT⁽⁵⁴⁾. As we will see later on, the HOMO-LUMO gap obtained from an HF calculation is larger than that of a KS calculation with a pure exchange and correlation functional (for more detail see Figure 4), our results show that the energy gap of KS calculated with a hybrid functional lies between these two extreme values. The negative of the ionization potentials of H₂O, CO₂ and t-HNNH are 12.6 eV, 13.8 eV and 9.7 eV, respectively. It is interesting to note that hybrid functionals lead to the best approximation to the experimental ionization potential due to the HF exchange. The conclusion drawn from Figure 2 applies to the remaining 14 molecules not included in the Figure.

Fig. 2: Dependency of the KS orbitals energies with respect to the exchange and correlation functionals for (a) H2O; (b) CO2 and (c) t-HNNH.

Basis Set Dependency. For the analysis of the dependency of the KS orbitals with six basis sets (1=6-31G, 2=6-31G*, 3=6-31G**, 4=6-311G, 5=6-311G*, 6=6-311G**), the functional BP86 that allows an appropriate description of the systems in study and do not contain any of the HF exchange will be used. Figure 3 shows the KS orbital energies for the different basis sets for H₂O, CO₂ and t-HNNH. In general, it is observed that the pattern of occupied orbitals together with the LUMO orbital is conserved for the different basis sets used. This indicated that electronic properties depending on the HOMO-LUMO energy gap are expected to be quite stable upon change of the basis set. The conclusion drawn from Figure 3 applies to the remaining 14 molecules that are not included in the Figure.

Fig. 3: Dependency of the BP86 KS orbital energies with respect to the basis set for (a) H2O; (b) CO2 and (c) t-HNNH.

In the forthcoming analysis the double x with a polarization function 6-31G* basis set and the exchange and correlation functional BP86 will be used.

Comparison HF vs KS. Figure 4 shows the energy diagrams of the HF (HF/6-31G*//HF/6-31G*) and KS (BP86/6-31G*//BP86/6-31G*) orbitals for H₂O, CO₂ and t-HNNH. It is important to highlight the fact that the geometric structures at the HF and DFT levels are very similar and the orbital energies are not affected when using geometries coming from different levels of calculation. In general, the KS occupied orbitals are higher than the HF orbitals, while the virtual KS orbitals are lower than the HF orbitals. Therefore, the HOMO-LUMO energy gap is larger in HF than in KS.

Fig. 4: Correlation diagram of the HF and KS orbitals for (a) H2O; (b) CO2 and (c) t-HNNH.

Figure 5 shows the excellent correlation between the energy of the HF and KS orbitals, this result indicates that it is possible to characterize the energy of the KS orbitals parametrically through the knowledge of the HF orbitals which can be associated to the experimental date of electronic affinities and ionization potentials through the Koopmans theorem.

Fig. 5: Correlation between orbital energies at the HF and BP86 levels for (a) H2O; (b) CO2 and (c) t-HNNH.

Ionization Potentials. With the use of functionals commonly implemented in standard programs of DFT, the energy of the KS HOMO differs significantly of the experimental ionization potential⁽⁵⁶⁾, this difference allows one to characterize the quality of functional used in the actual calculations.

In Table 1, the value of the HOMO energies calculated at HF/6-31G*//HF/6-31G* and BP86/6-31G*//BP86/6-31G* levels are compared to the experimental value of the first ionization potential⁽⁵⁷⁾ for the 17 molecules under study. It is observed that the HF energy overestimates slightly the experimental values while the KS energy underestimates by far these values. The error percentages are also presented in the table to shows the degree of reliability of the calculations. Although Koopmans theorem within DFT calculations fails in giving acceptable approximation for ionization potentials, it is interesting to note that in most cases the experimental value is located within the interval defined by the KS and HF methodologies, thus suggesting that they can be used to identify the range in which the experimental value is expected to be found: |e_X(KS) £ I (exp) £ / e_H(HF)|.

Table 1: HF and KS HOMO energies and vertical ionization potentials for 17 molecules under study.

Molecule	HF	HF	KS	KS	HF	HF	KS	KS	exp
	-eH	% error	-eH	% error	Ivertical	% error	Ivertical	% error	I
	[eV]	[eV]	[eV]	[eV]	[eV]	[eV]	[eV]	[eV]	[eV]
H2CCO	9.753	1.4	5.619	41.6	8.434	12.3	9.632	0.2	9.617
C3H4	9.676	0.1	5.874	39.3	8.533	11.8	9.603	0.7	9.670
NNCH2	11.133	9.4	6.590	35.3	9.824	3.5	10.823	6.3	10.180
H2CO	11.850	8.9	6.023	44.6	9.458	13.1	10.723	1.4	10.880
C2H2	11.007	3.5	6.718	41.1	9.777	14.2	11.200	1.8	11.400
H2O	13.557	7.4	6.320	49.9	10.876	13.8	12.423	1.6	12.621
N2O	13.376	3.8	8.202	36.4	11.217	13.0	12.910	0.2	12.889
F2O	15.095	15.1	7.201	45.1	12.532	4.4	12.524	4.5	13.110
FCN	13.676	2.5	8.413	36.9	12.327	7.8	13.215	0.9	13.340
HCN	13.526	0.6	8.689	36.1	12.375	9.0	13.641	0.3	13.600
CO2	14.693	6.7	8.803	36.1	12.505	9.2	13.612	1.2	13.777
N2	16.874	8.3	10.043	35.6	15.747	1.1	15.418	1.0	15.581
t-HNNH	10.833	12.3	5.403	44.0	9.545	1.1	10.062	4.3	9.650
t-HNNOH	11.590	-	6.067	-	10.122	-	10.425	-	-
c-HONNOH	11.687	-	6.467	-	13.365	-	10.541	-	-
t-HNNPh	9.043	-	5.096	-	8.422	-	8.828	-	-
t-PhNNPh	8.251	2.9	4.992	41.27	7.834	7.7	7.746	8.9	8.500

The above comparison of experimental ionization potentials with HOMO energies within the Koopmans approximation has a qualitative value only since HF and DFT calculation are not directly comparable. For a fair comparison we have calculated the ionization potentials of the 17 molecules using the DSCF procedure in the vertical approximation, the ionization potential is given by I ~ E(N-1) - E(N) and the results are also quoted in Table 1. It is interesting to note that the HF error increase while the KS error decrease dramatically. The HF results of vertical ionization potentials indicate that correlation effects, which are neglected at the HF level, affects the neutral system more strongly than the ionic molecules. In DFT the computed predictions depends on the balanced descriptions of the exchange and correlation contributions in neutral system. In Figure 6 the above discussed results are illustrated through the excellent correlation between calculated and experimental data. Although the correlation with the HF result is good, it is clear that the vertical approximation works better in cases of the DFT scheme.

Fig. 6: Correlation between experimental and vertical ionization potentials calculated at the HF (a) and BP86 (b) levels.

Additionally we calculated the adiabatic ionization potential for the 17 molecules in study at HF and BP86 levels with the 6-311G* basis set, although in few cases we were not able to obtain the geometric optimization of the cations, the results showed that the adiabatic approximation increases slightly the error in the calculated vertical ionization potential, in both HF and DFT calculation.

Chemical potential and Molecular Hardness. With the aim of verifying the quality of the frontier HOMO and LUMO orbitals energies, the chemical potential and the molecular hardness are going to be analyzed in the following paragraphs.

Figure 7 shows a comparison of the HF and DFT chemical potential and hardness. Although acceptable correlations are observed, the m and m obtained from DFT are systematically smaller than those obtained in the HF scheme, thus producing a considerable change in the interpretation of reactivity properties in specific situations.

Fig. 7: Correlation between vs (a) and vs (b), calculated using the frontier orbital energies at HF and BP86 levels.

To determine reliable values of m and h within the Koopmans approximation it is necessary to have reliable values of the LUMO orbital energy, this is a recurrent problem in quantum chemistry that doesn't have direct solution. In the next paragraph we propose an indirect model to estimate the LUMO energy through the knowledge of the polarizability that has been related empirically with the molecules softness⁽²⁶⁾.

Polarizability and Softness. Polarizability is a measure of the change of the electronic density due to the presence of an electric field⁽⁵⁴⁾. In Figure 8, we show the excellent correlation between the polarizability calculated at HF and BP86 levels, this indicates that both calculations can be used for the purpose of characterizing the LUMO orbital energy from polarizability calculations. In Figure 9 we observe that the relation vs S is verified within an acceptable error marge in both calculation, although HF present a better correlation.

Fig. 8: Correlation between HF and BP86 polarizabilities.

Fig. 9: Correlation between vs calculated at HF (a) and BP86 (b) levels.

Now, since , S = 1/e_L - e_H) , then we can write:

where a and b are parameters determined through fitting of a^1/3 vs S. The LUMO energy obtained using equation 10 at the HF level is compared with the calculated LUMO energy of 17 molecules, that with inclusion of the isomeric trans and cis diazenes makes a total of 22 system, the result is shown in Figure 10, where no correlation was observed.

Fig. 10: Comparison of the LUMO energy estimated from equation 10, and from the HF calculation.

In Figure 11 out of the 22 system we consider the diazenes family of molecules, a quite good correlation between a^1/3 and is observed in Figure 11 (a) and in Figure 11 (b) the LUMO energies estimated through equation 10 is compared with the calculated ones, we observe that the results are greatly improved and a good correlation of the LUMO energies is recovered. The LUMO energy obtained from the polarizability compares reasonable well with the value of the virtual orbital energies. Figure 11 (a) shows a correlation coefficient R = 0.959, value that is increased up to 0.987 when omitting molecule t- and c- HNNH, the eight remaining molecules were used in Figure 11 (b). This result shows that that the HF value of the LUMO energy is consistent with the polarizability calculations and the relationship between and is verified within an acceptable range of confidence, this is a quite interesting result since it opens a way to estimate LUMO energies from an independent procedure.

Fig. 11: (a) Correlation between vs calculated at HF level for a family of diazenes and (b) comparison of the LUMO energy estimated from the equation 10 (and ) with the calculated LUMOs at HF level.

5. CONCLUSIONS

We have performed HF and KS calculations on 17 molecules to characterize their orbital energies and electronic properties. Electronic properties depending on the HOMO-LUMO energy gap are found to be quite stable upon change of the basis set. The negative of Hartree-Fock and Kohn-Sham HOMO orbital has been found to define upper and lower limits, respectively, for the experimental values of the first ionization potential.

On the other hand, DFT results indicate that hybrid functionals lead to the best approximation to the experimental ionization indicating that the HF exchange is playing an crucial role whereas correlation effects might be less important.

Finally, based upon the inverse relationship between polarizability and softness a new quasi-analytic model to estimated the energy of the lowest unoccupied molecular orbital has been proposed, this leads to quite good results for the LUMO energies at least for a set of diazene derivative molecules.

ACKNOWLEDGMENTS

The authors wish to dedicate this work to the memory of Profesor Fernando Zuloaga, a fine Professor and a very good friend. This work was financed by FONDECYT, Projects Nº 1020534 and Nº 2010099.

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