Theoretical Calculation of the Low-Lying Electronic States of the Molecule PbO

Abstract

The potential energy curves of the lowest 20 electronic states in the representation 2s+1Λ(±) of the molecule PbO have been investigated via ab initio CASSCF and MRCI (single and double excitations with Davidson correction) calculations. The spectroscopic constants such as vibrational harmonic frequency ωe, the internuclear distance at equilibrium Re, the rotational constant Be, and the electronic transition energy Te with respect to the ground state have been calculated along with the permanent dipole moment for the different bound investigated electronic states. By using the canonical functions approach, the eigenvalues Ev, the rotational constant Bv and the abscissas of the turning points Rmin and Rmax have been calculated. The comparison of these values with those available in the literature shows a very good agreement.

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Kaeen, D. , Korek, M. , Abdulal, S. and Awad, R. (2015) Theoretical Calculation of the Low-Lying Electronic States of the Molecule PbO. Journal of Modern Physics, 6, 1171-1183. doi: 10.4236/jmp.2015.68121.

1. Introduction

The lead metal monoxide was the subject of many experimental investigations [1] [2] and several theoretical studies [3] - [7] . It is one of the most technologically important oxide materials which is used in the manufacture of different ceramics products. Because of its electrical and electronic properties, it is used in capacitors and electrophotographic plates, as well as in ferromagnetic and ferroelectric materials. It is also used as an activator in rubber, in oil refining, as an oxidation catalyst in several organic chemical processes, in the production of many lead chemicals, dry colors, soaps, and driers for paint. Moreover it is used in the production of lead salts, particularly those used as stabilizers for plastics, notably polyvinyl chloride materials.

There is a rapid increase and concern in studying the spectra of this molecule which is shown in the different papers written on its low lying electronic states. It has been experimentally proven that the ground state of the molecule PbO is 1S+ [8] - [10] . This result has been theoretically confirmed in a number of studies that used different methods [3] - [14] . The three excited electronic states B3P, A3P, and a3S+ have been investigated by Oldenborg et al. [9] while the electric-dipole moment of (1)3S+ state of PbO molecule has been measured by Hunter et al. [15] . By using the CASSCF/CASPT calculation the six electronic states 3S±, 1S, 1,2D, and 3P of this molecule have been investigated without [12] and with spin orbit coupling [16] .

In the present work 20 low-lying singlet and triplet electronic states of PbO molecule have been investigated by using the ab initio method using two basis sets. The potential energy curves (PECs) together with the transition energy with respect to the minimum energy for the ground state Te, the equilibrium internuclear distance Re, the harmonic frequency ωe, the rotational constant Be, and the permanent dipole moment (DMCs) µ have been obtained for the considered electronic states. Nine and eleven electronic states have been investigated here for the first time using respectively the first and the second basis sets. Taking advantage of the electronic structure of the investigated electronic states of the molecule PbO and by using the canonical functions approach, the eigenvalue Ev, the rotational constant Bv, and the turning points Rmin and Rmax have been calculated for several vibrational levels of the considered singlet and triplet electronic states.

2. Method of Calculations

2.1. Ab Initio Calculation

In the present work we study the low-lying singlet and triplet electronic states of the molecule PbO using state averaged complete active space self consistent field (CASSCF) procedure followed by a multireference configuration interaction (MRDSCI+Q with Davidson correction) treatment for the electron correlation. The entire CASSCF configuration space was used as the reference in the MRDSCI calculations, which were done via the computational chemistry program MOLPRO [17] taking advantage of the graphical user interface GABEDIT [18] . Since we noticed large discrepancies between the calculated values of we in literature, two different ways have been used in the present theoretical study of the PbO molecule. In the first and the second ways the 82 electrons of the lead atom are considered using the effective core potential ECP78MWB basis set for s, p, and d functions. While the oxygen species is treated as a system of 8 electrons by using, for the s, p, and d functions, the 6-311++G** and the DGauss-a2-X fit basis sets respectively for the first and the second ways respectively. For these 2 ways of calculation the potential energy curves and the static dipole moments of the low-lying electronic states of the molecule PbO were generated using the MRSDCI+Q for 350 internuclear distances calculations in the range 1.5 Å ≤ Re ≤ 5.0 Å in the representation 2s+1L(±) where we assumed that, the PbO molecule is mainly ionic around the equilibrium position. These PECs and the DMCs for the different symmetries are given in Figures 1-4.

The electric dipole moment is a fundamental electrostatic property, it is useful in finding the strength of the long-range dipole-dipole forces, and the understanding of the macroscopic properties of imperfect gases, liquids and solids. Our MRCI calculation of the DMCs produces smooth and continuous curves even close to the avoided crossings. At large internuclear distances, the dipole moment of all the investigated electronic states smoothly approaches zero which is theoretically the correct behavior for a molecule that dissociates into natural fragments. It is quite common for the molecular electronic states of the potential energy curves to make crossings or avoided crossings known as conical intersections (Figure 3 and Figure 4).

These points of the potential energy curves of a diatomic molecule are important in photochemistry. In fact, the avoided crossing regions are likely to be a leakage channels along which the molecules flow from the higher down to the lower potential energy curves. Such crossings or avoided crossings can dramatically alter the stability of the molecules. If these crossings are overlooked, then low barrier transitions can be missed and an incorrect chemical picture will arise. In the range of R considered, several avoided crossings have been detected in the potential energy curves of the excited electronic states of the molecule PbO.

The dipole moment function of these states exhibits an abrupt change reflecting the avoided crossing between the two states as also observed at the potential energy curves. The agreement between the positions of the avoided crossings of the PECs of the electronic states (2)1Π and (3)1Π (Figure 1) and (2)3Π and (3)3Π (Figure 2)

(a)(b)

Figure 1. Potential energy curves of the lowest electronic states of the molecule PbO using the first basis set: ECP78MWB and 6-311++G** for Pb and O atoms respectively. (a) Singlet electronic states 1S±, 1D, 1P; (b) Triplet electronic states 3S±, 3D, 3P.

with the positions of the crossing of the corresponding DMCs at the points 2.5 Å (Figure 3) and 2.8 Å (Figure 4) respectively can be considered as confirmation of the accuracy of the present results. In the present calculation of the DMCs we considered the lead atom at the origin. One can notice that, some parts of the DMCs are positive where the lead atom is charged negatively and the other main parts are negative where the charges of the 2

(a)(b)

Figure 2. Potential energy curves of the lowest electronic states of the molecule PbO using the second basis set: ECP78MWB and DGauss-a2-X fit for Pb and O atoms respectively. (a) Singlet electronic states 1S±, 1D, 1P; (b) Triplet electronic states 3S±, 3D, 3P.

(a)(b)

Figure 3. Dipole moment curves of the lowest electronic states of the molecule PbO using the first basis set: ECP78MWB and 6-311++G** for Pb and O atoms respectively. (a) Singlet electronic states 1S±, 1D, 1P; (b) triplet electronic states 3S±, 3D, 3P.

atoms are reversed.

By fitting the calculated energy values around the equilibrium position to a polynomial in terms of the internuclear distance, the spectroscopic constants we, re, Be, and Te have been calculated by using the PECs obtained by the 2 different ways of calculation. These values are given in Table 1 along with the available data given in the literature. The comparison of our results with the calculated values for Re by different techniques in literature for the ground state shows a good agreement with the relative differences 1.3% (Ref. [26] ) ≤ DRe/Re ≤ 5.2% (Ref. [5] ) and 0.61% (Ref. [26] ) ≤ DRe/Re ≤ 4.4% (Ref. [5] ) using the first and second way of calculation respectively. For Be, the comparison of our calculated values with those of Schwenzer et al. [5] , for the ground state, showed the relative differences 9.8% and 8.4% for the first and the second way respectively, while these relative differences showed the good agreement 3.5% and 2.5% by comparing our values with those of Jalbout et al. [26] . For ωe, the comparison of our results with the fourteen values calculated by different techniques in literature for the ground state show the relative differences 8.6% (Ref. [26] ) ≤ Dwe/we ≤ 28.8% (Ref. [4] ) and 4.9% (Ref. [26] )

(a)(b)

Figure 4. Dipole moment curves of the lowest electronic states of the molecule PbO using the second basis set: ECP78MWB and DGauss-a2-X fit for Pb and O atoms respectively. (a) Singlet electronic states 1S±, 1D, 1P; (b) Triplet electronic states 3S±, 3D, 3P.

≤ Dwe/we ≤ 25.9% (Ref. [4] ) for the first and second way respectively. This comparison shows that our values for ωe are in disagreement with those given in Ref. [4] , while there is a good agreement for the theoretical values of Re, Be and we of Jalbout et al. [26] and our calculated values for these constants. The comparison of our calculated values of Te, using the 2 different ways of calculation, for the two states (1)3S+ and (1)3P with those calculated in literature [9] shows a good agreement with the relative difference 1.6% ≤ DTe/Te ≤ 7.9%.

One can notice that the use of 2 ways of calculation for Re and Te have no influence on the calculated values of these constants, but there is an influence on the value of we and large influence on the calculated value of the dipole moment. Moreover our calculated values of dipole moment by using the second way are not in good accuracy with those calculated in literature, while our calculated values by using the first way of calculation are in good agreement with the experimental data of Huber and G. Herzberg [1] . These values are given in Table 1. From this over all agreement between our investigated values of the spectroscopic constants and those found in literature, we can pretend the validity of the used technique of calculation and the accuracy of our calculation for the new investigated electronic states. The future experimental investigation for these new electronic states may confirm our results.

Table 1. Spectroscopic constants for the lowest singlet and triplet electronic states of the molecule PbO.

a1Present work using, for the 82 electrons of the Lead atom, a contracted ECP78MWB basis set for s, p, and d functions. While the oxygen species is treated as a system of 8 electrons by using the 6-311++G** basis set for s, p, and d functions. a2Present work using, for the 82 electrons of the Lead atom, a contracted ECP78MWB basis set for s, p ,and d functions ,while the oxygen species is treated as a system of 8 electrons by using the DGauss-a2-Xfit basis set for s, p, and d functions. bRef. [1] , cRef [12] , dRef. [5] , eRef [4] , fRef. [7] , gRef. [3] , hRef. [19] , kRef. [20] , lRef. [21] , mRef. [6] , nRef. [25] , oRef. [9] , pRef. [26] ,

2.2. The Vibration-Rotation Calculation

By using the canonical functions approach [23] - [25] and the cubic spline interpolation between each two consecutive points of the PECs obtained from the ab initio calculation of the PbO molecule, the eigenvalue Ev, the rotational constant Bv, and the abscissas of the turning point Rmin and Rmax have been calculated for the considered electronic states up to the vibrational levels v = 41. These values for the different electronic states are given in Table 2. The absence of the comparison with other results is because of the calculation of these values here for the first time.

Table 2. Values of the eigenvalues Ev, the rotational constants Bv and the abscissas of the turning points for the different vibrational levels of the singlet and triplet electronic states of the molecule PbO.

3. Conclusion

In the present work, the ab initio investigation for the low-lying singlet and triplet electronic states of the PbO molecule has been performed via CASSCF/MRCI method using 2 different basis sets. The potential energy and the dipole moment curves have been determined along with the spectroscopic constants Te, Re, ωe and the rotational constant Be for these electronic states. The comparison of our results with those obtained theoretically in literature showed an overall good accuracy. By using the canonical functions approach [23] - [25] , the eigenvalue Ev, the rotational constant Bv, and the abscissas of the turning points Rmin and Rmax were calculated up to the vibrational level v = 41. New electronic states have been investigated in the present work for the first time (nine new for first basis and fourteen new for second basis).

NOTES

*Corresponding author.

Conflicts of Interest

The authors declare no conflicts of interest.

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