Li Rui, Liang Gui-Ying, Lin Xiao-He, Zhu Yu-Hao, Zhao Shu-Tao, Wu Yong. Explicitly correlated configuration interaction investigation on low-lying states of SiO+ and SiO. Chinese Physics B, 2019, 28(4): 043102
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Explicitly correlated configuration interaction investigation on low-lying states of SiO+ and SiO
Li Rui1, 2, †, Liang Gui-Ying2, Lin Xiao-He2, Zhu Yu-Hao2, Zhao Shu-Tao3, Wu Yong2, 4, ‡
Department of Physics, College of Science, Qiqihar University, Qiqihar 161006, China
Institute of Applied Physics and Computational Mathematics, Beijing 100088, China
School of Physics and Electronic Science, Fuyang Normal University, Fuyang 236037, China
HEDPS, Center for Applied Physics and Technology, Peking University, Beijing 100084, China
SiO+ and SiO, which play vital roles in astrophysics and astrochemistry, have long attracted considerable attention. However, accurate information about excited states of SiO+ is still limited. In this work, the structures of 14Λ–S states and 30 Ω states of SiO+ are computed with explicitly correlated configuration interaction method. On the basis of the calculated potential energy curves of those Λ–S states and Ω states, the spectroscopic constants of bound states are evaluated, which are in good agreement with the latest experimental results. The predissociation mechanism of state is illuminated with the aid of spin–orbit coupling matrix elements. On the basis of the calculated potential energy curves and transition dipole moments, the radiative lifetime for each of low-lying vibrational states and is estimated. The laser cooling scheme of SiO+ is proposed by employing transition. Finally, the vertical ionization energy values from SiO ( to ionic states: SiO+, , , and are calculated, which agree well with experimental measurements.
SiO+ and SiO play important roles in astrophysics and astrochemistry, which have been photographed in hot circumstellar regions,[1,2] interstellar clouds,[3–8] meteor emission.[9] The SiO+ cation was also readily generated by shock-heated dense molecular clouds.[10,11] In order to illuminate the physical and chemical phenomenons of SiO+ in celestial body, it is necessary to study the electronic structures and spectroscopic properties of the ion. In recent years, SiO+ has also aroused growing interest in scientific research due to the application in the laser cooling of molecular ions.[12] The accurate control of internal and external degree of laser cooling molecular ion has potential applications in metrology, quantum information, and quantum computation.[13–16] Stollenwerk et al.[12] obtained a trapped sample of SiO+ by employing supersonic expansion. The transition dipole moment and radiative lifetime of transition were investigated in detail. Apart from state, the other low-lying states and have been studied by Rosner et al.[17–21] through utilizing the approach of fast-ion beam, and the interaction and rotation vibration spectra of the two low-lying states were analyzed in detail. The spectroscopic constant for each of , , and was evaluated in their investigations, and fine-structure constant of was obtained from the high-resolution spectrum.
Along with experimental investigations, a series of theoretical studies have been carried out to study the electronic structures and spectroscopic properties of the low-lying states of SiO+. In 1978, Colbourn et al.[22] used multiple-scattering transition-state method to determine the vertical ionization potential for each of , , and of SiO+ relative to the ground state of SiO. Later, Werner et al. used the multiconfiguration self-consistent field (MCSCF) method to calculate the potential energy curves (PECs) and dipole moment (DM) curves of of SiO+,[23] which predicted the ion to be a good IR emitter. Cai and François[24] studied the electronic structures of the low-lying states associated with the two lowest dissociation limits, and gave the spin-allowed transition dipole moments and radiative lifetimes for the three low-lying vibrational levels of . Nguyen and Odom[25] utilized the calculated PEC and transition probability for each of , , to simulate laser cooling scheme of SiO+. Chattopadhyaya et al.[26] adopted the relativistic configuration interaction method to investigate the electronic spectrum of SiO+ within 7 eV. The spectroscopic constants of six low-lying Ω states (, , , , , and ) were obtained in their calculations. However, the previous theoretical investigations only considered the spin–orbit coupling (SOC) of the four low-lying states (, , , and ), and the spectroscopic properties of low-lying states affected by SOC effect have been still unclear.
In this work, we use the high-level ab initio method to calculate the low-lying electronic states of SiO+. The PECs of 14 Λ–S states and 30 Ω states are determined by the theoretical calculation. In the SOC calculation, the coupling effects of the 14 Λ–S states are simultaneously taken into account. Based on the computed PECs, the spectroscopic constant for each of bound Λ–S and Ω state is evaluated by the numerical method. Based on the calculated PECs and transition dipole moments, the radiative lifetimes of several low-lying vibrational states of and electronic states are computed. On the basis of transition, the feasibility of laser cooling of SiO+ is discussed. Finally, the vertical ionization energy values from SiO () to several ionic states of SiO+ are determined.
2. Methods and computational details
In the present work, the accurate electronic structure of SiO+ is computed with MOLPRO 2012 suite of quantum chemical program.[27] Owing to the limit of MOLPRO program, the Abel point group is applied, and the low-lying electronic states of SiO+ are expressed as follows: , , , and . For Si and O, the Gaussian-type all-electron basis sets cc-pVQZ-F12 are selected.[28] To obtain the accurate single point energy of electronic state, the electronic structure calculation is performed by the three steps below. First, Hartree–Fock (HF) method is used to compute the singlet-configuration wavefunction of the ground state. Second, utilizing the HF wavefunction as the initial guess, a multi-reference wavefunction is generated by the state-averaged complete active space self-consistent field (SA-CASSCF) computation.[29,30] Finally, utilizing the CASSCF wavefunction as reference, the singlet point energy values of Λ–S states are computed by the explicitly correlated multireference configuration interaction method (MRCI-F12).[31] The construction of active space in the CASSCF computation is crucial for the precision of electronic structure. We test different active spaces, and finally select 8 molecular orbitals (MOs) as active space. The 8 MOs, which correspond to the 3s, 4s, px, 3py, 3pz atomic orbitals of Si+ and 2px, 2py, 2pz atomic orbitals of O, are referred to as (9e,8o). In the following MRCI-F12 computation, the 1s22s22p6 electrons of Si+ and 1s2 electrons of O are set to be core orbitals, while 2s22p1 electrons of Si+ and 2s22p4 electrons of O are correlated with single and double excitations. In order to overcome the size-inconsistency problem caused by the MRCI-F12 method, the Davidson correction is taken into consideration.
To improve the precision of our calculation, the SOC effect of electronic states is considered as a perturbation via a two-step procedure.[32,33] The values of SO matrices are extracted from the SOC calculation.[34] The off-diagonal SO matrix elements are generated by the MRCI-F12 wave function, while the diagonal SO matrix elements are obtained from the MRCI-F12+Q computation. The PECs of Λ–S states and Ω states are plotted by connecting the energy points of low-lying states with the aid of the avoided crossing principle.
Based on the computed PECs, the spectroscopic constants, vibrational energy levels, vibrational wavefunctions and Franck–Condon factors (FCFs) of SiO+ are given by the numerical solution of nuclear Schrödinger equation through employing LEVEL procedure.[35] The transition dipole moments are determined at the MRCI-F12 level in theory. On the basis of the computed transition dipole moments, energy gaps of vibrational levels, and FCFs, the radiative lifetime of and state are estimated.
3. Results and discussion
3.1. Electronic structures and spectroscopic properties of Λ–S states
The 14Λ–S states associated with the two lowest dissociation limits and are computed at MRCI-F12 level by employing the cc-pVQZ-F12 basis sets. The adiabatic PECs of the 14Λ–S are presented in Fig. 1, which are constructed by 29 singlet point energy values in an internuclear distance range of 1.3 Å–3.5 Å.
Fig. 1. Potential energy curves of (a) doublet Λ–S states of SiO+ and (b) quartet Λ–S states of SiO+.
From the PECs of Fig. 1, it can be seen that most of Λ–S states are typical bound states except for , , , and . The shoulder in PEC of may be attributed to the avoided crossing phenomenon with state. The and form an avoided crossing point around R=1.6 Å, leading to the irregular shapes of PECs of the 4Π sates. On the basis of these calculated PECs, the spectroscopic constants of bound states are determined by numerical method and given in Table 1. The dominant electronic configurations of bound states at equilibrium distance are also listed in Table 1.
Table 1.
Table 1.
Table 1.
Spectroscopic constants of bound Λ–S states of SiO+.
Spectroscopic constants of bound Λ–S states of SiO+.
.
For the SiO+ molecular ion, the dominated electronic state of the ground state and the first excited state states are (73%) and (92%), respectively, which means that the transition corresponds to the one electron promotion . Compared with previous results, our MRCI-F12 calculated results Re (1.519 Å) and (1159 cm−1) of are close to the experimental results (1.516 Å and 1162 cm−1[17]). For the spectroscopic constant of state, the Te is calculated to be 2397 cm−1, which is in good accordance with experimental value of 2242 cm−1.[17] The calculated Re is 1.641 Å, fitting very well with 1.645 Å derived from the photoelectron spectrum of SiO.[22] The calculated value of 929 cm−1 is in good agreement with the experimental result 944 cm−1 determined from the analysis of the two band systems and ,[17] and is somewhat smaller than the predicted value 1030 cm−1 obtained from the photoelectron spectrum.[22]
Since the transition plays an important role in laser-cooling investigation of SiO+,[12] the transition is well studied theoretically and experimentally. The dominated electronic states of are (63%) and (25%), corresponding to and singlet electron transition relative to the state. The computed values of Be and Re of are 0.7065 cm−1 and 1.531 Å, which only differ from the experimental value by 0.0038 cm−1 and 0.004 Å, respectively.[36] The Te values of is computed to be 25828 cm−1, which is closer to the recently experimental value of 26030 cm−1, than to previous theoretical values of 25484[26] and 25722 cm−1.[24] For the other bound states, our present spectroscopic constants are consistent with the theoretical values calculated recently at the CI and coupled cluster (CCSD(T)) level of theory,[24,26] which can motivate the future experiment to verify the correctness of these predictions.
3.2. Transition properties and scheme for laser cooling of SiO+
The spin-allowed transition dipole moments and transitions of SiO+ are calculated with the MRCI-F12 method. In the Franck–Condon region, the averaged transition dipole moments of and have large values, which are about 0.65 a.u. and 0.21 a.u. (the unit a.u. is short for atomic unit), respectively. Our theoretical values of the transition dipole moments of these two transitions are close to the recent experimental values of 0.64 a.u. and 0.08 a.u.[12] On the basis of the calculated PECs of Λ–S states, FCFs and transition dipole moments, the lifetime of low-lying vibrational state of bound state is determined by the following formula:
where Einstein coefficient is determined by
In formulas (1) and (2), the unit of is in unit of s−1, the unit of energy gap is in unit of cm−1, the unit of transition dipole moment (TDM) is a.u. Vibrational branching ratio is determined from .
Oscillator strength is evaluated from the FCFs and TDM by
where and TDM are in atomic unit.
According to formulas (1) and (2), we calculate the lifetime of and state. For vibrational state of , the lifetime is computed to be 68.2 ns, which is in very good agreement with experimental measurement of (66±2) ns,[12] demonstrating the reliability of our calculation. The lifetime of vibrational state of is calculated to be 8.3 ms, which agrees with theoretical result (4.1 ms)[26] in magnitude. The FCFs, Einstein coefficient , vibrational branching ratio , oscillator strength , transition energy , and transition wavelength λ for the transition are calculated and listed in Table 2. As shown in Table 2, the FCFs of transition are highly diagonal. The diagonal FCFs q00 and q11 are 0.972 and 0922, respectively, which are at least one order of magnitude larger than that of non-diagonal FCFs. Moreover, the diagonal Einstein coefficient and vibrational branching ratio demonstrate that the transition is dominant by those ( vibrational transitions. The A00, R00, and f00 are computed to be , 0.976, and , respectively, illuminating that the scattering probability to hot bands can be neglected.
Table 2.
Table 2.
Table 2.
Spectroscopic quantities for transitions of SiO+.
.
0
1.10
0.17
0.976
0.030
b
0.97
0.026
0.002
25807
26917
28014
29098
30168
31225
λ/nm
387.5
371.5
357.0
343.7
331.5
320.3
b
26030
1
9.97
0.023
0.925
0.056
0.003
b
0.030
0.027
0.082
0.002
24655
25765
26862
27946
29016
30073
b
24870
λ/nm
405.6
388.1
372.3
357.8
344.6
332.5
a Note: Values of and FCFs smaller than 0.001 are omitted in the table.
The laser cooling candidate of diatomic molecule requires a transition with two particular properties:[39,40] highly diagonal FCFs and a short radiative lifetime suitable for rapid laser cooling. These calculated results indicate that the transition of SiO+ satisfies the two particular properties. According to our calculated spectroscopic quantities of transition of SiO+, we propose a laser cooling scheme for the molecular ions by using two lasers at wavelengths around 400 nm. The wavelength of main pump laser is chosen to be 387.5 nm, which is near-resonance wavelength of the ()– () transition. To suppress the unwanted vibrational branching loss, an additional laser (λ =405.6 nm) is used to repump the accumulated population at the vibrational levels of the .
In Table 3, we summarize the dominant parameters of laser cooling of SiO+, together with the corresponding parameters of other monohalides containing Group-IIA and Group-III A atoms which have already been suggested as molecular laser-cooling candidates in the previously theoretical investigations.[39,40] For comparison, the corresponding results for diatomic molecules that have been experimentally laser-cooled are also given in Table 3.[13,41,42] We can see that the q00 and lifetime of transition of SiO+ are of the same order of experimentally laser-cooled molecules SrF, YO and CaF, illuminating the possibility of the proposed laser-cooling scheme of SiO+.
Table 3.
Table 3.
Table 3.
Comparison of q00, R00, , and τ between SiO+ and other molecules suitable for laser cooling.
Comparison of q00, R00, , and τ between SiO+ and other molecules suitable for laser cooling.
.
3.3. Influence of spin–orbit coupling on spectroscopic properties
As shown in Fig. 1, the state crosses ten electronic states (, , , , , , , , , and ) in a bond length range of R=1.75 Å–2.35 Å. Since the value of the SOC for each of –, –, –, –, and – is close to zero, for clarity, none of the PECs of , , , , and are not shown in the amplified view of the crossing regions of Fig. 2(a). The SO matrix elements between the and , , , , are computed, which are used to qualitatively analyze the perturbation and predissociation of caused by the SOC effect. The computed SO matrix elements each as a function of internuclear distance in the vicinity of the curve crossing regions are given in Fig. 2(b). The SO matrix element value of –, –, –, and – are calculated to be 13, 30, 20, and 13 cm−1 at their crossing points, respectively, which can cause the the vibrational levels of to be perturbed.
Fig. 2. (a) Magnified crossing regions together with vibrational levels of and (b) corresponding SO matrix element values including .
Previous investigations show that the SOC effects are very important for spectroscopic properties of low-lying states, even when only light atoms are included.[43–46] By considering the SOC effects, different Λ–S states with common Ω components will be recombined and change the shape of PECs. Figure 3 only shows the PECs of the low-lying 15 Ω states. For clarity, the Ω =1/2, and 3/2, 5/2, 7/2 are shown in Figs. 3(a) and 3(b), respectively. The calculated spin–orbit energy splitting for (O atom), (O atom) and (Si+ ion) are 154, 231, and 290 cm−1, respectively, which agree well with the existing experimental results of 158, 227, and 287 cm−1.[47]
Fig. 3. Potential energy curves of the low lying Ω states of SiO+: Ω =1/2 (a) and Ω =3/2, 5/2, and 7/2 (b).
Based on the PECs of the Ω states, the spectroscopic constants of SiO+ molecular ion are numerically determined and presented in Table 4. For the (1)1/2 state, the calculated values of , Be, and Re agree with the experimental values,[24,48] but the spectroscopic constant (12.8 cm−1) is 5.9 cm−1 bigger than the experimental result 6.9 cm−1.[48] The SOC effect causes splits into and . Both and the ground state have Ω =1/2 component, so the two states form an avoided crossing point around R=1.65 Å. However, the does not have Ω =3/2 component, so the (1)3/2 state ( state) mainly originates from . For the state, the Te, , and Re are computed to be 2422 cm−1, 915 cm−1, and 1.630 Å, which differ from the experimental results by less than 180 cm−1, 29 cm−1, and 0.006 Å, respectively.[17] The state crosses the state nearby R=1.75 Å. Considering the SOC effects, the and states both have Ω =1/2 component, so the two states form an avoided crossing point around R= 1.75 Å. The Te, ωe, and Re of are computed to be 25454 cm−1, 1140 cm−1, and 1.535 Å, which are consistent with experimental results of 26016 cm−1, 1137 cm−1, and 1.545 Å, respectively.[17,22]
Table 4.
Table 4.
Table 4.
Spectroscopic constants of bound Ω states of SiO+.
Spectroscopic constants of bound Ω states of SiO+.
.
3.4. Ionization energy of SiO
To determine the ionization process of SiO, the PECs of the ground state for the SiO molecule are calculated with the MRCI-F12 method through utilizing the same active space and basis set as those of SiO+ calculation. The PECs of low-lying states of SiO and SiO+ are plotted in Fig. 4, and these PECs are given by energy with respect to the minimum energy point of the of SiO molecule. On the basis of the calculated PEC of the of SiO, the and Re are computed to be 1.514 Å and 1228 cm−1, which are in good agreement with experimental values of 1.5097 Å and 1241.55 cm−1.[36] Utilizing the vibrational wave function of low-lying states of SiO+ and SiO, the FCFs between the ν =0 state of the and vibrational states of the are calculated. The FCFs’ variations with ionization energy are presented in Fig. 5. As shown in Fig. 5, the vertical ionization energy of is 11.5 eV, which is in reasonable agreement with experimental value of 11.61 eV.[22] As for the excited states and of SiO+, the vertical ionization energy values are calculated to be 12.1 eV and 14.7 eV, respectively, which are about 0.1 eV smaller than previous measurements of 12.19 eV and 14.80 eV.[22] As shown in Fig. 4, the equilibrium distance of the and states of SiO+ are very similar to that of the state of SiO, so the FCFs of and transitions are highly diagonal, and the q00 of the two transitions in Fig. 5 are both larger than 0.9. The equilibrium distance of the is 0.14 Å larger than that of the , so the vertical ionization process of corresponds to the SiO+ (, , transition. The vertical ionization energy of is computed to be 16.1 eV, corresponding to SiO+ (, ) (, ) transition. Since the equilibrium distance of the and states of SiO+ are very close to that of the state of SiO, the vertical ionization process of (SiO) and ( (SiO) are favorable, as compared with that of ( (SiO) and ( (SiO).
Fig. 5. Franck–Condon factors of transition between state of SiO and , , , states of SiO+ relative to ionization energy.
4. Conclusions
The electronic structures of low-lying states for SiO+ are studied by the MRCI-F12 method with cc-pVQZ-F12 basis set. It can be seen from the calculated PECs that the state crosses , , , , states in a internuclear distance range of 1.88 Å–2.25 Å. The calculated SO matrix between state and these states are in a range of 13 cm−1–30 cm−1 around the crossing points. With the help of the calculated SO matrices including state, the predissociation mechanism of state is explained in detail. On the basis of the calculated vibrational level and transition dipole moments, the radiative lifetimes of low-lying vibrational levels of are computed, which are consistent with the most recent experimental results. Our calculations indicate that the FCFs and vibrational branching ratios of transition are highly diagonal, and the radiative lifetime of is short. We propose a laser cooling scheme of SiO+ that employs the transition. The vertical ionization energy values from SiO () to SiO+ (, and ) agree with the previous experimental results. The present theoretical investigations will provide an in-depth understanding of the structure and spectroscopic properties of the SiO/SiO+.
Acknowledgment
We thank the High Performance Computing Center of Qiqihar University for supercomputing time.
Le RoyR J2002LEVEL 7.5: a Computer Program for Solving the Radial Schröinger Equation for Bound and Quasibound LevelsUniversity of Waterloo, Chemical Physics Research Report CP-655
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