^{1}

^{2}

^{1}

^{*}

Via CASSCF/MRCI and RSPT2 calculations (single and double excitation with Davidson correction) the potential en- ergy curves of 20 electronic states in the representation ^{2S+1}Λ^{（±）}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.

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 [

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 [

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 X^{1}Σ^{+} vary as

,

[^{3}Σ^{+}, (1)^{3}Σ^{+}, (2)^{3}Σ. 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 [^{–}^{1}. In this work, we investigate the potential energy curves (PECs), the electric dipole moment and spectroscopic constants for the 20 ^{2S+1}Λ^{±} 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.

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 [_{2}-Xfit basis set including s, p and d functions [

The PECs for the 20 electronic states in the representation ^{2S+1}Λ^{(±)} 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_{e}x_{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 _{e}ω_{e}, and B_{e} for the ground state X^{1}Σ^{+} with those given in literature, either theoretical or experimental, shows an excellent agreement with the relative differences

,

,

^{a1}For present work with MRCI calculation; ^{a2}For present work with RSPT2 calculation; ^{b}Ref.[^{c1(experimental)}Ref. [^{c2(B3-LYP1)}Ref.[^{c3(B3-LYP2)}Ref.[^{c4(BP86)}Ref.[^{c5(MP2_1)}Ref.[^{c6(MP2_2)}Ref.[^{d}Ref.[^{e1(MRCI+Q)}Ref.[^{e2(Fit)}Ref.[^{f1(SCF)}Ref.[^{f2(CI-SD)}Ref.[^{f3CEPA-1)}Ref.[^{g(SCF+CI)}Ref.[^{h1(SCF)}Ref.[^{h2(MCSCF30)}Ref.[^{h3(MCSCF-CI)}Ref.[^{i(exp)}Ref.[^{j(exp)}Ref.[^{k(exp)}Ref.[^{m}Ref.[^{n}Ref.[^{p}Ref.[^{r1}Ref.[^{r2}Ref.[^{ s1}Ref.[^{s2}Ref.[^{t}Ref.[^{u}Ref.[^{v}Ref.[^{w}Ref.[^{x}Ref.[^{y}Ref.[^{z}Ref.[^{q}Ref.[^{ab}Ref.[^{ac}Ref.[^{ad}Ref.[^{ }N.B experimental value c1 (Ref.[ 31]) is in solid methane.

.

The agreement becomes less by comparing our calculated value of ω_{e}x_{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_{e}x_{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)^{3}Σ^{+}, (1)^{3}Σ, (1)^{3}Σ^{–}, (1)^{1}Σ^{–}, (1)^{1}Σ, (2)^{1}Σ 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)^{3}Σ^{+}, (2)^{1}Σgiven 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.[^{1}Σ. Concerning the assignment of our calculated value of the ^{1}Σ state, it is in good agreement with the calculated values of G^{1}Σ state [32,38] and acceptable agreement with the experimental value given in Ref.[_{e} with the newly published theoretical work by using the MRCI approach [^{3}Σ^{+} and (1)^{1}Σ^{–} 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 G^{1} 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_{e}ω_{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)^{3}Σ^{+} (Ref.[^{3}Σ

(Ref.[^{3}Σ^{–} (Ref.

[

for the state (2)^{1}Σ^{+} (Ref.[

for the state (1)^{3}Σ (Ref.[

for the state (1)^{1}Σ (Ref.[^{3}Σ^{+} and (1)^{1}Σ^{–} 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 ^{1}Σ, 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 [

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.

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

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

with e_{0} = E_{v}, e_{1} = B_{v}, e_{2} = –D_{v}···, f_{0} 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]

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 f_{0 }and find

Once e_{0} 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 X^{1}Σ^{+}, (1)^{1}∆, (1)^{3}Σ, (1)^{3}Π, (2)^{3}Π and (2)^{1}Π (as illustration) are given in _{v}, B_{v}, r_{min} and r_{max} with the experimental data of Ref.[^{1}Σ^{+} 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. [

_{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].

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_{e}x_{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.

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[^{*}Corresponding author.

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