Gaussian Basis Sets for the Calculation of Some States of the Lanthanides

Gaussian Basis Sets for the Calculation of Some States of the Lanthanides

P.R. Librelon and F.E. Jorge^*

Departamento de Física

Universidade Federal do Espirito Santo

29060-900 Vitória, ES, Brasil

Received on 19 October, 2000 2

Highly accurate adapted Gaussian basis sets are used to study the ground and some excited states for the neutral atoms and also some corresponding 6s and 4f ionized states from Cs through Lu. Our total energies are compared with those calculated with a numerical Hartree-Fock method. The mean error of our energy results is equal to 0.74 mhartree. Our calculations reproduce the experimental trend to increase or to decrease the 6s and 4f ionization potentials with increasing atomic number, although they are respectively smaller and larger than the experimental values.

I Introduction

In this last decade lanthanide chemistry and physics have experienced tremendous growth, for example in the field of catalysts [1] and high temperature superconductors [2]. Thus, it would be highly desirable to elucidate the electronic structure of lanthanide atoms at least in the Hartree-Fock (HF) approximation. For these atoms, numerical HF (NHF) calculations [3-5] were performed mainly on the ground states.

In this work, the adapted Gaussian basis sets [AGBSs - one different set of Gaussian-type function (GTF) exponents for each atomic species for the atoms from Cs (Z=55) through Lu (Z=71) [6] are initially augmented until saturation is achieved for each symmetry of each atom and then, using the generator coordinate HF (GCHF) [7] method, they are reoptimized for each atomic species. Next the energies for the atoms Cs-Lu and their positive ions are calculated and compared with those obtained with a NHF [5] method. The ionization potentials (IPs) are also computed and compared with the corresponding experimental values [8, 9].

II The method

An approach to select the basis sets arises from the GCHF method [7]. In the GCHF method the one-electron functions are integral transforms, i.e.,

where f_i are the generator functions (GTFs in our case), f_i are the unknown weight functions, and a is the generator coordinate. The application of the variational principle to calculate the energy expectation value built with such one-electron functions leads to the Grifftn-Hill-Wheeler-HF (GHWHF) equations [7]. The GHWHF equations are integrated using a procedure known as integral discretization (ID) [10]. The ID technique is implemented through a relabelling of the generator coordinate space, i.e.,

where A is a numerically determined scaling factor. In the new generator coordinate space W, an equally spaced N-point mesh {W_i} is selected, and the integration range is characterized by a starting point W_min, an increment DW, and N (number of discretization points). The highest value (W_max) for the generator coordinate is given by

The choice of the discretization points determines the exponents of the GTFs.

In the last four years, the GCHF [7] method was successfully tested in the generation of basis sets for atomic and molecular systems [11-16].

III III Results and discussion

By employing the GCHF method we have generated AGBSs for the atomic species presented in Table 1. Throughout the calculations we have used the scaling factor A (see Eq. (2)) equal to 6.0, and for all atomic species we have sought the best discretization parameter (W_min and DW) values for each s, p, d and f symmetries. All calculations were carried out using a modified version of the ATOMSCF program [17], and for each atomic species the optimization process is repeated until the total energy value is stabilized within ten significant figures. The resulting wave functions are available by request through the e-mail address [e-mail].

Table I shows the ground and some excited state HF total energies (in hartrees) for the neutral atoms and some cations from Cs (Z=55) through Lu (Z=71) computed with our AGBSs and with a NHF [5] method. Our basis set sizes are presented in the seventh column. We recall that the AGBSs are generated from the basis sets of Ref. [6]. First, we augmented these basis sets until saturate each symmetry of each atom, and second, using the GCHF [7] method, we reoptimized each AGBS of each atomic species studied here. From Table I, we can see that our total energies, for all atomic specie of interest, are in good agreement with the corresponding NHF [5] values and that our energy errors do not exceed 1.72 mhartree. Here it is important to say that the vector coupling coefficients used in the calculations of the open-shell configurations have been taken from the tabulation by Malli and Olive [18]. These tables show the vector coupling coefficients for the electron configurations s, pⁿ, spⁿ, dⁿ, sdⁿ, p^mdⁿ, sp^mdⁿ and fⁿ. The HF total energies of the ground states of the atoms Ce and Gd and of some states of the cations Pr⁺, Nd⁺, Pm⁺, Sm⁺, Eu⁺, Tb⁺, Dy⁺, Ho⁺, Er⁺ and Tm⁺ are not calculated here, because the electron configurations of these atomic species have 5d and 4f and 6s and 4f open shells, respectively. The electron configuration of Lu⁺ (³H) has 5d and 4f open shells, and thus the wave function for this cation is not generated here.

Table II contains the IPs (in eV) computed by using the Koopmans theorem (e is our orbital energy), the total energy difference DE = E(X⁺) - E(X) [X is the atomic symbol and E(X⁺) and E(X) are our total energies respectively for the cation and the neutral atom presented in Table I], and the experimental values (E_expt.) [8, 9].

From Table II we can see that the differences between our IP's calculated through -e (see the fourth column) and through DE (see the fifth column) are small for 6s orbital, indicating that the Koopmans theorem works for the 6s ionization. Besides this, for the 6s orbitals, our IPs calculated with these two approaches are very similar to those computed with a NHF method (see the sixth column). For all lanthanide atoms presented in Table II, and from our results for -e, we can see that the 6s orbitals are more diffuse than the corresponding 4f orbitals, that is, the 6s IPs are smaller than the 4f lPs. For these atoms, it is known that the mean values of r for the 6s orbitals are larger than those for the 4f orbitals, that is, the 6s electrons are far from the nucleus than the 4f electrons. From La through Eu, both the calculated -e ( ~ 4.4-4.6 eV) and the experimental (~ 5.4-5.8 eV) 6s IPs are almost constant. After Tb, the -e and experimental [8, 9] IPs gradually increase. The experimental IPs are always larger than the -e values. To correct this discrepancy, it is necessary to include in the calculations electron correlation effects and relativistic corrections, but this is outside the scope of this work. Here, it is important to say that Jorge et al. have developed the generator coordinate Dirac-Fock (GCDF) [19, 20] method for closed-shell atoms and a segmented contraction methodology for relativistic Gaussian basis sets [21, 22]. From Table II, only Yb (Z=70) has closed-shell, thus, for the other atoms presented in this Table, we cannot use the GCDF method to calculate the relativistic IPs.

Besides this, Table II shows that for the lanthanides, the 4f IPs calculated by us through -e and through DE give very different results. The ionization of the electrons in the outermost 6s shell causes small reorganization on the whole electron distribution, whereas the inner 4f electron ionization causes larger reorganization effects because of the appearance of a hole in the inner shell. Thus, for these atoms, it is not appropriated to use the Koopmans theorem to calculate the 4f electron ionization. For all lanthanide atoms, the 4f IPs calculated by us (DE) are in good agreement with the corresponding values obtained with a NHF [5] method, and although the calculated 4f IPs are 1-3 eV greater than the corresponding experimental values [8, 9], NHF and our DE calculations describe the experimental trend well.

IV Conclusions

In this work we have generated AGBSs for the 45 atomic species presented in Table I with the GCHF [7] method. The largest difference between the total energies calculated by us and by a NHF [5] method is equal to 1.72 mhartree for Lu. Although our 6s and 4f IPs (-e and DE) are respectively smaller and larger than the corresponding experimental values [8, 9], our calculations reproduce the experimental trends on the 6s and 4f electron ionizations well. For the 4f IPs, our DE results are better than those computed with the Koopmans theorem, whereas for 6s IPs the two approaches give similar results.

Acknowledgements

We acknowledge the financial support of CNPq and Dr. L.T. Peixoto for valuable discussions.

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Publication Dates

• Publication in this collection
  05 Mar 2002
• Date of issue
  June 2001

History

• Received
  19 Oct 2000