Electronic structure with dipole moment calculations of the high-lying electronic states of BeH, MgH and SrH molecules

By using the complete active space self consistent field (CASSCF) with multi-reference configuration interaction MRCI + Q method including single and double excitations with Davidson correction, the 26, 27 and 25 low-lying doublet and quartet electronic states in the representation 2s+1Λ(+/−) (without spin orbit interaction) of the molecules BeH, MgH and SrH have been investigated. The potential energy curves, the internuclear distance Re, the harmonic frequency ωe, the permanent dipole moment μ, the rotational constant Be and the electronic transition energy with respect to the ground state Te are calculated. Using the canonical approach the eigenvalue Ev, the rotational constant Bv and the abscissas of the turning points Rmin and Rmax have been calculated for the investigated electronic states. The comparison between the values of the present work and those available in the literature for several electronic states shows very good agreement.


Introduction
The mono-hydrides of the alkaline-earth metals are expected to be present in sunspots, stars, nebulae, and the interstellar medium. These species, have received considerable attention from both the experimentalists and the theoreticians, because the molecules SrH, MgH and CaH are relatively easy to synthesize in the gas-phase, have interesting ground states, due to astrophysical significance [1][2][3][4][5] and the apparent simplicity of their electronic structure. Since the excited electronic states of the considered three molecules showed complex spectra their electronic states have been the subject of much interest from both experimental spectroscopists and quantum chemists. Since the collision process between cold atoms allowing high precision measurements with hydrogen isotopes, precise determination of molecular potentials and atomic lifetimes the alkaline-earth-metal hydrides have been the subject of extensive research in the past years. These molecules are ideal candidates for the production of the polar molecules and direct laser cooling [6][7][8][9][10][11][12][13][14][15][16] as well as for the ultracold fragmentation. Moreover, these Hydrides containing alkaline-earth metals offer a wide variety of interesting applications. For example, they are prototypes for hydrogen storage materials. They have been predicted to be stable at pressures that can be achieved in a diamond anvil cell and they are predicted to display high-temperature superconductivity. With small amounts of hydrogen (0.1%-1%) the Hydrides containing alkaline-earth metals are investigated as viable magnetic, thermoelectric or semiconducting materials.
The BeH molecule is less popular for experimentalists because of the toxicity of the Be-containing molecules, while it is extensively studied theoretically by using ab initio method. The early work on the electronic emission spectra of BeH molecule has been done by Watson and Fredrickson [17]. Recently, there is more demanding of this molecule due to the existence of near-degeneracy effects and low-lying states [18] and it is being studied in the surfaces of fusion reactors ITER (International Thermonuclear Experimental Reactor) through the Joint European Torus (JET) of the European Community Fusion Program.
Since the visible emission of MgH was observed in the Sun and then in many stars and because of the relative abundances of the magnesium isotopes in stellar atmospheres, this molecule attracted the attention of the
In the present work, we employed the ab-initio method to investigate the potential energy curves (PECs) and the static dipole moment curves (DMCs) of the low-lying 26, 27 and 25 doublet and quartet electronic states of the BeH, MgH and SrH molecules respectively. The spectroscopic constants as the equilibrium internuclear distance R e , the harmonic frequency ω e , the vibrational constant B e , the transition energy with respect to the ground state minimum T e , and the permanent dipole moment μ are calculated for the bound states of these molecules. For the study of the rovibrational problem there are many important theories and techniques in literature with computer programs as LEVEL [33][34][35][36] or the Duo program [37]. In order to obtain high order precision there is a need for large order centrifugal distortion constant as D v , H v , K Q v K [38][39][40][41][42][43][44]. By using the canonical function approach one can obtain these constants with the high values of vibrational levels even near dissociation by one single and simple routine. This method provides strong evidence for our assumption that the higher-order of D v , H v , L v K. and the higher vibrationel v and rotational level J for any electronic state and any type of potential energy curves (either experimental, empirical or theoretical) are as accurate as the low-order values. In this technique we use the compact form e n =〈Φ 0 RΦ n−1 〉 (R=1/r) of a CDC (e 1 =B v , e 2 =D v , e 3 =H v K..) of any order n where Φ n is the solution of the differential equation Φ″ n +f(r)Φ n =s(r) where f(r) and s(r) are given, as well as their initial values at an arbitrary origin. This is done by using a simple method for the computation of the functions Φ n , as successive solutions of the 'rotational Schrödinger equations' by taking Φ n orthogonal to Φ 0 (the vibrational wavefunction), and by deriving exact values of Φ n (r o ) and Φ n ′(r o ) which are the initial values of Φ n at an arbitrary point r o [38,[45][46][47]].

Ab initio calculation
In the present work an ab initio calculation of the lowest-lying electronic states of BeH, MgH and SrH has been performed via CASSCF and MRCI (single and double excitation with Davidson correction) calculations. Multireference CI calculations (MRCI) were performed to determine the correlation effects. The potential energy calculations for the states L  ( ) of the molecules have been carried by using CAS-SCF method. The calculations have been performed via the computational chemistry program MOLPRO [48] taking the advantage of the graphical user interface GABEDIT [49]. This software is intended for high level accuracy correlated ab initio calculations. MOLPRO has been run on a PC-computer with UNIX-type operating systems. For the three studied molecules BeH, MgH and SrH, the one electron hydrogen atom is treated using for s, p, d, and f functions the correlation-consistent polarized Quadruple-Zeta basis set, augmented with sets of diffuse functions aug-cc-pVQZ [50].  In the BeH molecule, the beryllium atom is treated in all electron schemes, the 4 electrons are considered using for s, p, d, and f functions of the triple zeta basis set VTZ [51]. The quality of chosen basis sets for the H, Be, Mg, and Sr isolated atoms is checked by comparing our calculation of the lowest energy values of the asymptotic energy at each dissociative asymptote with those obtained experimentally by NIST Atomic Spectra Database [52] (table 1). This comparison shows a relative difference ranges between 0.7%ΔE/E11.7%. For some of the highest electronic states the dissociation limits are not obtained because of the undulation of potential energy curves of these states. For these undulations the short-range electronic interactions are significant at certain energy points. They are related to the singularities in the electronic Hamiltonian operator and giving rise to the Coulomb cusp in the electronic wave function and appearance of cusps in the exact wave function. Also, these undulations are explained by the breakdown of the Born-Oppenheimer (B.O) approximation since the interactions between electronic states are significant at certain energy points, and the responsible term for the socalled 'non-adiabatic effects' can be very important and it cannot be neglected. However, the overall good relative error given in table 1 can ensure the accuracy of our calculated data.
An heteronuclear diatomic molecule belong to C ∞v group. Since MOLPRO can handle only Abelian pointgroups the linear molecules are treated in C 2v instead of C ∞v . The states Σ + , Π z , Δ 0 , and Δx2−y2 belong to the irreducible representation number 1, the states Π x , Δ xy belong to the irreducible representation 2, the states Π y , Δ yz belong to the irreducible representation 3, and Δ zx belong to the irreducible representation 4. Among the 5 electrons explicitly considered for BeH (4 electrons for beryllium and 1 electron for hydrogen) two inner electrons were frozen in subsequent calculations so that 3 valence electrons were explicitly treated. The active space contains 7s ( ) s p s p s d H s Be: 2 , 2 , 3 , 3 , 4 , 3 ; : 1 , Be: 3 2 orbital which corresponds to 15 active molecular orbitals distributed into irreducible representation a1, b1, b2, and a2, in the following way: 8a1, 3b1, 3b2, 1a2, noted [1,3,8].
In the molecule SrH, the strontium species is considered using for s, p, d and f functions the Effective Core Potential ECP28MWB basis set, where 28 electrons are considered as inner electrons and the remaining 10 electrons are considered as valence electrons. Among the 11 electrons, 8 electrons were frozen in subsequent calculations, so that 3 electrons were explicitly treated. The active space contains 2σ (Sr: 4d 0 ; H: 1s), 1π(Sr: 4d ±1 ),  and 1δ (Sr: 4d ±2 ) orbitals in the C 2v symmetry; this corresponds to 6 active molecular orbitals distributed into irreducible representation a 1 , b 1 , b 2 , a 2 in the following way: 3a 1 , 1b 1 , 1b 2 , 1a 2 , noted [1, 3].

Potential energy curves
For the BeH molecule, the calculations have been performed for 442 internuclear distances in the range 0.67 ÅR7.70 Å for 26 electronic states in the representation 2s+1 Λ (±) . The number of electronic states obtained in the present work is 14 doublet electronic states (8 in symmetry 1, 5 in symmetry 2 and1 in symmetry 4), and 12 quartet electronic states (6 in symmetry 1, 3 in symmetry 2 and 3 in symmetry 4). The potential energy with the dipole moment curves of these electronic states are given in figures 1-4 while the complete tables with the figures of these potential energy curves are given in the supplementary materials. In the considered range of R, some crossings and avoided crossings occur between some potential energy curves of at different values of internuclear distances for the doublet and quartet electronic states. The positions of these crossings and avoided crossings are given in table 2.
The calculations for MgH molecule have been performed for 488 internuclear distances in the range 0.80 ÅR8.9 Å in the representation 2s+1 Λ (±) . The number of electronic states obtained in the present work is 17 doublet states (9 in symmetry one, 6 in symmetry two, and 2 in symmetry four) and 13 quartet electronic states (7 in symmetry one, 4 in symmetry two, and 2 in symmetry four). The potential energy with the dipole moment curves of these electronic states are given in figures 5-8 while the complete tables with the figures of these potential energy curves are given in the supplementary materials. In the considered range R, crossings and avoided crossings occur between some of the investigated potential energy curves. The positions of these crossings and avoided crossings are given in table 2. The calculations of the SrH molecule have been performed for 649 internuclear distances in the range 1.2 ǺR7.3 Ǻ in the representation 2s+1 Λ (±) . The number of electronic states obtained in the present work is 10 doublet electronic states (6 in symmetry one and 4 in symmetry two), and 11 quartet states (7 in symmetry one and 4 in symmetry two). The potential energy with the dipole moment curves of these electronic states are

BeH molecule
The dipole moment operator is among the most reliably predicted physical properties. The expectation value of this operator is sensitive to the nature of the least energetic and most chemically relevant valence electrons. The HF dipole moment is usually large, as the HF wave function over estimates the ionic contribution. To obtain the best accuracy of this operator, multireference configuration interaction (MRCI) wave function were constructed using multi configuration Self-consistent field (MCSCF) active space. All the calculation were performed with  the MOLPRO [48] program. The variation of the static dipole moment curves in term of the internuclear R are plotted with potential energy curves in the same figures in order to show the agreement between the positions of the avoided crossings of the potential energy curves and the crossing of the dipole moment curves. This agreement, which is represented by vertical lines, is a criteria of the validity and the accuracy of the present work for the titled molecules.
The static dipole moments curves with the potential energy curves for the 3 considered molecules BeH, MgH, and SrH are given as function of the internuclear distance R in figures 1-12 while the complete tables with the figures of these static dipole moments curves are given in the supplementary materilas. The dipole moment curves of the electronic states (4) 2 Σ + , (5) 2 Π, (4) 4 Σ + of BeH, (4) 2 Σ + , (4) 4 Σ + of MgH, and (6) 2 Σ + of SrH dissociate at infinity as ionic character, while these dipole moment curves of the other electronic states tends to zero at infinity.

BeH molecule
The calculations of the spectroscopic constants such as the vibrational harmonic constant ω e , the internuclear distance at equilibrium r e , the rotation constant B e , the centrifugal distortion constant D e , and the electronic transition energy with respect to the ground state T e have been done by fitting the energy values around the equilibrium position to a polynomial in terms of the internuclear distance, the degrees of these polynomials are determined from the evaluation of the statistical error for the coefficients. These values are displayed in table 3 together with the available data in literature either theoretical or experimental. Due to the avoided crossing at their minima, we have not carried out the calculations for the spectroscopic constants for the states 3 2 ∑ + , 2 2 ∏, 3 2 ∏, 2 4 ∑ + , 1 4 Δ, and 2 4 Δ.

Comparison with the experimental values
As shown in table 3 the comparison of the present values of T e for the electronic states A 2 ∏, C 2 ∑ + , and 4 2 ∏ with the experimental data given by Colin and DeGreef, and Colin et al [54,64,60], showed a very good agreement with relative differences δT e /T e =1.31%, 0.52%, 10.36%, respectively. Similar very good agreement is obtained by comparing our value of T e for the state A 2 ∏ with that given by O'Neil and Schaefer [60, 65, 66] with relative     difference δT e /T e =1.15%. For the 4 2 ∑ + state, the comparison between our calculated value of T e with that given by Lefebvre-Brion and Colin, and Pitarch-Ruiz et al [56,67], showed a less agreement with a relative error of δT e /T e =13.12%. The comparison between our calculated values for R e, ω e , and B e for the electronic states states X 2 ∑ + , A 2 ∏, and C 2 ∑ + with those obtained experimentally given by Colin and DeGreef [64], showed a very good agreement with relative differences δR e /R e =0.33%, δω e /ω e =0.82%, and δB e /B e =0.6% for X 2 ∑ + , δR e /R e =0.34%, δω e /ω e =0.8%, and δB e /B e =0.74% for A 2 ∏, andδR e /R e =0.043%, δω e /ω e =5.25%, and δB e /B e =0.4%. . By comparing our calculated values for R e, ω e , and B e for the electronic states states X 2 ∑ + , A 2 ∏, and C 2 ∑ + with those obtained theoretically by Petsalakis et al [68], one can find an excellent agreement with relative differences δR e /R e =0.33%, δω e /ω e =0.25%, and δB e /B e =0.1% for X 2 ∑ + , δR e /R e =0.03%, δω e /ω e =0.3%, and δB e /B e =0.1% for A 2 ∏, and δR e /R e =0.28%, δω e /ω e =4.44%, and δB e /B e =0.14% for C 2 ∑ + . Similar excellent agreement is obtained by comaparing our values of these constants with those of Focsa et al [61] with the relative differences 0.3% and 0.37% for X 2 ∑ + and A 2 ∏ states respectively. The comparison between the present values for R e and those given by Cade and Huo, and Herzberg [57,69] showed an excellent agreement with the relative differences 0.3% and 0.37% for X 2 ∑ + and A 2 ∏ states respectively.

MgH molecule
In the present work, the calculated spectroscopic constants of the molecule MgH are given in table 4 along with those found in the literature either experimentally or theoretically. The comparison of our calculated values of the vibrational harmonic frequency constant ω e for the ground state X 2 ∑ + with those given in literature shows a very good agreement with the relative difference Δω e /ω e equal 2.22% [70], 0.15% [71], 0.54% [72,73], 0.58% [74]. A less accuracy is obtained for ω e by comparing our values with those given in [73] by using MC and HF methods where Δω e /ω e are respectively 6.97% and 7.43%. For the values of R e , there is a very good agreement between our result and those given in [70][71][72][73][74][75] where 0.07%ΔR e /R e 1.79%. As for the values of B e , there is also a very good agreement between our result and the ones given in [72,73,75] [73]. The comparison between our results and those found in literature for the excited state (2) 2 ∑ + shows also the very good agreement with the relative differences 0.55%ΔT e /T e 1.06% (except for the result given in [75] where the relative error increases to 16  compared to [70]. The results of B e are also found for the first time for the considered state. The values obtained for the internuclear distance at equilibrium R e and the electronic transition energy with respect to the ground states T e for first minima of both states (2) 2 ∏ and (5) 2 ∑ + shows a good agreement compared to [70], but the accuracy becomes less for the values of the vibrational harmonic constant ω e compared to the same reference. The accuracy deteriorates by comparing our results with those given by Mestdagh et al [70] for the states (3) 2 ∏ and (1) 2 Δ. It was not possible to obtain such constants for some states that made avoided crossing with a neighbor one, these states are given in table 2.

SrH molecule
The comparison of our calculated values in the present work with those given in literature (table 5) for the constants ω e , and B e for the ground state X 2 Σ + shows very good agreemenst with the relative differences Δω e /ω e equal 0.0% [76], 3.2% [32], 3.3% [27] and ΔB e /B e =0.6% [45], 5.8% [32], 4.7% [27]. The comparison of our results for the state (1) 2 ∏ with those given theoretically by Leininger and Jeung [64] shows a very good agreement for T e , ω e and B e with the relative difference ΔT e /T e =5%, Δω e /ω e =3.3%, and ΔB e /B e =0.2%. While the comparison of these constants with those obtained experimentally [27] shows also very good agreement with the relative differences ΔT e /T e =2.8%, Δω e /ω e =3.1% ΔB e /B e =3.7%. The comparison between our results and those obtained theoretically in literature for the excited state (2) 2 Σ + shows also the very good agreement with the relative difference: ΔT e /T e =1.4%, Δω e /ω e =7.4% and ΔB e /B e =4.3% with the results of Leininger and Jeung [76]. Similar results are obtained by comparing our data with those obtained experimentally [27] with the relative difference ΔT e /T e =2.9%, Δω e /ω e =3.4% and ΔB e /B e =2.9%. The spectroscopic constants of the state (1) 2 Δ are compared only with the theoretical data given by Leininger and Jeung [76] since there is no experimental data is available yet. It shows a less agreement for T e with a relative difference ΔT e /T e =13.7% which is may be due to an upward displacement in the potential energy. However, the relative error for ω e shows somehow a good agreement with a relative difference Δω e /ω e =8.4% and an excellent agreement for B e with a relative difference ΔB e /B e =0.6%. The three published values for R e for the state X 2 Σ + shows a very good agreement with our calculated value with relative differences 0.0% [22,76] <ΔR e /R e <3% [27,32]. For (1) 2 Π, (2) 2 Σ + , and (1) 2 Δ the calculated values of R e in the present work are in good with agreement with the values in the literature in which the relative difference are 0.5% [76] <ΔR e /R e <2.4% [27], 1.8% [17,22] <ΔR e /R e <2% [19,63] and ΔR e /R e =2% [27] respectively. The comparison for the calculated values for the other states is not possible since they are given here for the first time.
Because of the overall good agreement between our calculated data for the three molecules and those given in literature either theoretical or experimental we may confirm the accuracy of these data for the new investigated electronic states. These results may stimulate new experimental works for the high-lying electronic states for these three molecules.

Transition dipole moment
The transition dipole moment (TDM) between two electronic states is a very useful data as: designing some cooling experiments, to model the electronic spectroscopy of the molecule in all situations and it is then about where the molecules can be found. For the three considered molecules we present in figures 13-15 the TDM between the ground and 2 Π and 2 Σ + excited states. From these figures one can notice, some curves tend to zero due to spin forbidden transitions between two atomic orbitals at the asymptotic limits, while the other curves tend to constant values because of the allowed transitions between the corresponding atomic orbitals. For the readers, it should be noted that there is no guarantee that the relative signs were extracted from MOLPRO correctly and the risk of getting it wrong should be understood. For example, changing the sign of 〈X|μ|(2) 2 Σ + 〉 (BeH) might be attributed to the phase change of either |X〉 or |(2) 2 Σ + 〉, which should be then propagated to other transition dipole moments consistently.

The vibration-rotation calculation
For a given electronic state the vibration-rotation motion of a diatomic molecule is governed by the radial Schrödinger equation within the Born-Oppenheimer approximation Table 6. Values of the eigenvalues E v , the abscissas of the turning points R min , R max , the rotational constants B v , and the centrifugal distortion constants D v for the different vibrational levels of the investigated electronic states of the BeH molecule.          Table 7. Values of the eigenvalues E υ , the abscissas of the turning points R min and R max , the rotational constants B υ , and the centrifugal distortion constants D υ for the different vibrational levels of the investigated electronic states of the molecule MgH.      Table 8. Values of the eigenvalues E υ , the abscissas of the turning points R min and R max , the rotational constants B υ , and the centrifugal distortion constants D υ for the different vibrational levels of the investigated electronic states of the molecule SrH.