ABSTRACT

Via CASSCF/MRCI and RSPT2 calculations (single and double excitation with Davidson correction) the potential energy curves of 20 electronic states in the representation of the molecule SiO have been calculated. By fitting these potential energy curves to a polynomial around the equilibrium internuclear distance r_e, the harmonic frequency ω_e, the rotational constant B_e, and the electronic energy with respect to the ground state T_e have been calculated. For the considered electronic states the permanent dipole moment μ have been plotted versus the internuclear distance r. Based on the canonical functions approach, the eigenvalues E_v, the rotational constant B_v and the abscissas of the turning points r_min and r_max have been calculated. The comparison of these values to the experimental and theoretical results available in the literature is presented. In the present work 8 higher electronic states have been studied theoretically for the first time.

1. Introduction

The silicon monoxide SiO molecule is of considerable astrophysical interest, it is detected in the interstellar medium and in a variety of astrophysical objects which are mostly associated with warm, dense, and shocked gas [1]. Because of the interaction between high velocities jets emerging from a young star and the surrounding molecular environment a large fraction of the silicon monoxide relative to hydrogen molecule are found in the high velocity gas components of molecular outflows [2]. This abundance of SiO comes from the sputtering of dust grains in shocked regions and the subsequent release of Si-bearing material into the gas phase [3-7].

In recent years silica nanoparticles has attracted considerable attention due to their potential applications in many fields including ceramics, chromatography, catalysis and chemical mechanical polishing [8], nanodevices and mesoscopic research [9]. Altman et al. [9] probed the behavior of light absorption of silica nanoparticles at high temperatures in the Urbach region, and compare it with that in bulk materials. They assumed that, the SiO vapor emission does not contribute significantly to the flame radiation in visible. This can be easily justified by considering that the lower state of a SiO molecule involved in transitions in visible light is not a ground state of the SiO molecule, but a highly excited one [10].

By studying the published data in literature on the molecule SiO, one can notice the large discrepancy between these values either theoretical or experimental. The values of the electronic transition energy T_e with respect to the ground state X¹Σ⁺ vary as

,

[12]and [12] respectively for the electronic states (1)³Σ⁺, (1)³Σ⁺, (2)³Σ. Similar data can be found for different spectroscopic constants of different electronic states. Stimulated by these discrepancies, the important connection between energy relations of solids and molecules [13], and based on our previous theoretical calculation [14-23], we performed an ab initio study of the low-lying electronic states of the molecule SiO below 132,500 cm^–1. In this work, we investigate the potential energy curves (PECs), the electric dipole moment and spectroscopic constants for the 20 ^2S+1Λ^± lowlying electronic states of this molecule obtained by MRCI and RSPT2 calculations. Taking advantage of the electronic structure of the investigated electronic states of the SiO molecule and by using the canonical functions approach [24-26], the eigenvalues E_v, the rotational constant B_v and the abscissas of the turning points r_min and r_max have been calculated up to the vibrational level v = 52.

2. Computational Approach

2.1. Ab Initio Calculation

The PECs of the lowest-lying electronic states of SiO molecule have been investigated via CASSCF method. MRCI and RSPT2 calculations (single and double excitations with Davidson corrections) were performed. Silicon atom is treated in all electron schemes where the 14 electrons of the silicon atom are considered using the cc-PVTZ basis set including s, p, d and f functions [27]. The oxygen atom is treated in all electron schemes where the 8 electrons of the oxygen atom are considered using the DGauss-a₂-Xfit basis set including s, p and d functions [28]. Among the 22 electrons explicitly considered for the SiO molecule (14 electrons for Si and 8 for O), 18 inner electrons were frozen in subsequent calculations so that 4 valence electrons were explicitly treated. This calculation has been performed via the computational chemistry program MOLPRO [29] taking advantage of the graphical user interface GABEDIT [30].

The PECs for the 20 electronic states in the representation ^2S+1Λ^(±) obtained from MRCI calculation have been obtained for 222 internuclear distances in the range 1.06 Å ≤ r ≤ 4.00 Å. These potential energy curves for the singlet, triplet and quintet electronic states in the different symmetries are given, respectively in Figures 1-3.

The spectroscopic constants such as the vibration harmonic constants ω_e and ω_ex_e, the internuclear distance at equilibrium r_e, the rotational constant B_e, and the electronic transition energy with respect to the ground state T_e have been calculated by fitting the energy values around the equilibrium position to a polynomial in terms of the internuclear distance. These values are given in Table 1 together with the available values in the literature either theoretical or experimental. The comparison of our MRCI calculated values of r_eω_e, and B_e for the ground state X¹Σ⁺ with those given in literature, either theoretical or experimental, shows an excellent agreement with the relative differences

,

,

Table 1. Spectroscopic constants for the electronic states of the molecule SiO.

^a1For present work with MRCI calculation; ^a2For present work with RSPT2 calculation; ^bRef.[10]; ^{c1(experimental)}Ref. [31]; ^c2(B3-LYP1)Ref.[31]; ^c3(B3-LYP2)Ref.[31]; ^c4(BP86)Ref.[31]; ^c5(MP2_1)Ref.[31]; ^c6(MP2_2)Ref.[31]; ^dRef.[32]; ^e1(MRCI+Q)Ref.[6]; ^e2(Fit)Ref.[6]; ^f1(SCF)Ref.[33]; ^f2(CI-SD)Ref.[33]; ^f3CEPA-1)Ref.[33]; ^g(SCF+CI)Ref.[34]; ^h1(SCF)Ref.[35]; ^h2(MCSCF30)Ref.[35]; ^h3(MCSCF-CI)Ref.[35]SCF; ^i(exp)Ref.[36]; ^j(exp)Ref.[37]; ^k(exp)Ref.[38]; ^mRef.[39]; ⁿRef.[40]; ^pRef.[44]; ^r1Ref.[11]; ^r2Ref.[11]; ^s1Ref.[45]; ^s2Ref.[45]; ^tRef.[46]; ^uRef.[47]; ^vRef.[48]; ^wRef.[49]; ^xRef.[50]; ^yRef.[51]; ^zRef.[52]; ^qRef.[53]; ^abRef.[54]; ^acRef.[55]; ^adRef.[56]; N.B experimental value c1 (Ref.[ 31]) is in solid methane.

.

The agreement becomes less by comparing our calculated value of ω_ex_e with the experimental values of literature [10,32,36-38,44,55] where the relative difference. One can notice that, the theoretical value of ω_ex_e published in literature varies between 5.0 cm^–1 and 8.0 cm^–1 for the ground state [6,33-35]. The comparison of our calculated values by using the RSPT2 and MRCI techniques with those available in literature for the spectroscopic constants r_e, ω_e and B_e shows the average values

, ,

, ,

,.

From theseresults one can find that, the RSPT2 technique may gives better value for ω_e while MRCI technique gives better values for r_e and B_e for the ground state of the molecule SiO.

By comparing our calculated values of T_e for the states (1)³Σ⁺, (1)³Σ, (1)³Σ^–, (1)¹Σ^–, (1)¹Σ, (2)¹Σ with those obtained experimentally in literature one can find an overall acceptable agreement with relative difference

in Refs.[10,36] and larger relative difference for the states (1)³Σ⁺, (2)¹Σgiven in Refs.[3339,43] with relative difference.

One can notice that, the comparison of our calculated value of T_e, for the considered electronic states, with those calculated in literature shows an excellent agreement by using one technique of calculation with

(Ref.[11]) and disagreement by using another technique with (Ref.[11]) for the same state (1)¹Σ. Concerning the assignment of our calculated value of the ¹Σ state, it is in good agreement with the calculated values of G¹Σ state [32,38] and acceptable agreement with the experimental value given in Ref.[39]. The comparison of our values of T_e with the newly published theoretical work by using the MRCI approach [56] shows an acceptable agreement for the 2 excited electronic states (1)³Σ⁺ and (1)¹Σ^– with relative differences 11.9% and 8.2% respectively.

The comparison of our calculated values of r_e, ω_e, and B_e, for the excited states, with those given in literature experimentally [10,48,51,54] shows that, our values of r_e and B_e are in very good agreement for all the investigated states with

and

except the value of B_e for the state G¹ where. Our values of ω_e are also in very good agreement with the experimental values for the electronic states with and becomes larger for the other investigated electronic states with

.

Similarly, by comparing our calculated values of r_eω_e, and B_e with those calculated in literature, one can notice that an excellent agreement by using one technique of calculation with relative differences for the state (1)³Σ⁺ (Ref.[40]), for the state (1)³Σ

(Ref.[49]), and for the state (1)³Σ^– (Ref.

[50]) and disagreement by using another technique with

for the state (2)¹Σ⁺ (Ref.[11]),

for the state (1)³Σ (Ref.[11]), and

for the state (1)¹Σ (Ref.[11]). These discrepancies in the theoretical results can be referred to the basis used in this calculation, the number of valence electron, the software used, the bad assignment of a state ….etc. While the comparison with the recent results of Ref.[56] for the 2 states (1)³Σ⁺ and (1)¹Σ^– by using an approach similar to that we used in the present work shows an excellent agreement for the values of r_e and B_e and good agreement for ω_e. For the state ¹Σ, there is an excellent agreement with the value of ω_e and good agreement with the values of r_e and B_e. The SiO molecule possesses sizable dipole moments of 3.0982 D [41]. Such magnitudes of the dipole moment should be sufficient for sustaining dipole-bound states (DBSs). Extensive experimental and computational studies [42,43] of an extra-electron attachment to a number of polar molecules have shown the critical value of the dipole moment required to support a DBS to be 2.5 D. The electric dipole moment is also of great utility in the construction of molecular orbital based models of bonding and helping in the search for an understanding of the macroscopic properties of imperfect gases, liquids and solids. The expectation value of this operator is sensitive to the valence electrons and the general predictive quality of the computational methodology.

For the investigated electronic states, we calculated in the present work the permanent dipole μ(r) for 1.2 Å ≤ r ≤ 4 Å (Figures 4 and 5). Each time an adiabatic state loses its ionic character, it becomes again neutral and the corresponding dipole moment tends towards zero.

2.2. The Vibration-Rotation Calculation

Within the Born-Oppenheimer approximation, the vibration rotation motion of a diatomic molecule in a given electronic state is governed by the radial Schrödinger equation

(1)

where r is the internuclear distance, v and J are respectively the vibrational and rotational quantum numbers, , and are respectively the eigenvalue and the eigenfunction of this equation. In the perturbation theory these functions can be expanded as

(2)

(3)

with e₀ = E_v, e₁ = B_v, e₂ = –D_v···, f₀ is the pure vibration

wave function and f_n its rotational corrections. By replacing Equations (2) and (3) into Equation (1) and since this equation is satisfied for any value of l, one can write [24-26]

(4)

(5-1)

(5-2)

(5-n)

where, the first equation is the pure vibrational Schrödinger equation and the remaining equations are called the rotational Schrödinger equations. One may project Equations (7) onto f₀ and find

(6-1)

(6-2)

(6-n)

Once e₀ is calculated from Equation (4), can be obtained by using alternatively Equations (5) and (6). By using the canonical functions approach [24-26] and the cubic spline interpolation between each two consecutive points of the PECs obtained from the ab initio calculation of the SiO molecule, the eigenvalue E_v, the rotational constant B_v, the distortion constant D_v, and the abscissas of the turning point r_min and r_max have been calculated up to the vibration level v = 52. These values for the state X¹Σ⁺, (1)¹∆, (1)³Σ, (1)³Π, (2)³Π and (2)¹Π (as illustration) are given in Table 2. The comparison of our calculated values of E_v, B_v, r_min and r_max with the experimental data of Ref.[55] for the ground state X¹Σ⁺ shows an excellent agreement for the 35 considered vibrational levels. Similar results are obtained by comparing our calculated values of B_v with the calculated Shi et al. [56] for the considered states.

Table 2. Values of E_v, B_v, D_v and r_min and r_max of the SiO molecule.

^*First entry for the present work; ^**Second entry Refs. [51,54].

3. Conclusion

In the present work, the ab initio investigation for the 20 low-lying singlet and triplet electronic states of the SiO molecule has been performed via CAS-SCF/MRCI method. The potential energy and the dipole moment curves have been determined along with the spectroscopic constants T_e, r_e, ω_e, ω_ex_e and the rotational constant B_e for the lowest-lying electronic states. The comparison of our results, for the ground and excited states, with those obtained experimentally in literature shows an overall very good agreement, while the agreement with the theoretical data depends on the technique of calculation. By using the canonical functions approach [24-26], the eigenvalue E_v, the rotational constant B_v, and the abscissas of the turning points r_min and r_max have been calculated up to the vibrational level v = 52 with an excellent agreement by comparing with the available results in literature. Eight electronic states have been investigated in the present work for the first time. These newly obtained results maybe confirmed by the investigation of new experimental works on this molecule.

REFERENCES

[57]    NOTES

[58]     

[59]    ^*Corresponding author.

[60]