Cite this Article
Wang Qiao-Xia, Wang Yu-Min, Ma Ri, Yan Bing. Accurate all-electron calculation on the vibrational and rotational spectra of ground states for O2 and its ions. Chinese Physics B, 2019, 28(7): 073101  
Permissions
Accurate all-electron calculation on the vibrational and rotational spectra of ground states for O2 and its ions
Wang Qiao-Xia, Wang Yu-Min, Ma Ri †, Yan Bing ‡
Key Laboratory of Applied Atomic and Molecular Spectroscopy, Jilin Province, Institution of Atomic and Molecular Physics, Jilin University, Changchun 130012, China

 

† Corresponding author. E-mail: rma@jlu.edu.cn yanbing@jlu.edu.cn

Project supported by the National Key Research and Development Program of China (Grant No. 2017YFA0403300), the National Natural Science Foundation of China (Grant Nos. 91750104, 11574114, and 11874177), and the Natural Science Foundation of Jilin Province, China (Grant No. 20160101332JC).

Abstract

The potential energy curves, spectroscopic constants, and low-lying vibration–rotation levels of ground-state O2 and its cation and anion were calculated with the explicitly correlated multireference configuration interaction method. The zeroth-order reference wavefunction was treated with the complete active space multiconfigurational self-consistent field method, in which the active space was carefully selected, and an additional molecular orbital was added into the full valence active space. The electron correlation of the 1s core in the oxygen atom was considered in the computations. The Davidson correction on molecular energy was considered to account for higher electron excitation. The relativistic effects, including the scalar relativistic effect and spin–orbit coupling, were considered in the computation of potential energy curves. These physical effects on the spectroscopic constants were examined. The low-lying levels of vibration–rotation spectra of O2 and its ions were determined based on the computed potential energy curves. Comparisons with available experiments were made and excellent agreement was obtained for the vibrational and rotational parameters. The spectroscopic constants and vibration–rotation spectrum of , which is sparse in experiments, were provided. Our study will shed some light on further theoretical and experimental studies on these simple but important molecular systems.

PACS: 31.15.A-;31.15.V-;31.15.vn;31.50.Bc
Keyword:O2 molecule/molecular ion;explicitly correlated multireference configuration interaction;spectroscopic constant;vibration-rotation spectrum
1. Introduction

Oxygen is one of the most important gas-phase molecules in the earth’s atmosphere, and its positive and negative molecular ions are also very important to the physics and chemistry of upper atmospheric, life science, or life evolution processes.[1,2] The neutral and ionic compositions of O2 exist in different altitudes, which manifestly plays a diverse role in the maintenance of life on earth. The potential energy curves (PECs) of molecules can be constructed by measuring the molecular spectra so as to further understand the electronic structures of molecules. Theoretical calculations of the potential energy curve, spectroscopic constants, and vibration–rotation levels of a molecular electronic state are also important to test the theory of electronic structure and to aid experimental measurements of the molecular spectrum.[3–9]

Over the past decades, the potential curve, spectroscopic constants, and vibration–rotation spectrum of the triplet ground state of O2 molecule have attracted extensive experimental and theoretical interest. For example, Cheung et al.[7] obtained the accurate spectroscopic constants and energy levels of the state through Schumann–Runge absorption spectroscopy. Bytautas et al.[8,9] performed high-level ab initio calculations on the potential energy curves of O2 with consideration of various physical effects, and obtained an accurate vibrational and rotational spectrum of the state. For the ground state of , the vibrational constants 0.7 cm−1 and 0.02 cm−1 were obtained by a high resolution threshold photoelectron spectrum experiment[10] using synchrotron radiation. More accurate information of the ground state of was given by the – Fourier transform emission spectroscopy,[11] the results of which were , , Be=1.689824±0.00091 cm−1, and Re=1.116877±0.000030 Å. According to these spectroscopic constants, the vibrational and rotational levels of the state were also given in the experiments.

The adiabatic electron affinity of O2 is about 0.45 eV.[12] It is indicated that the only stable electronic state of the anion is the ground electronic state. The characterization and understanding of the anion are far less than those of the neutral O2 and cation , especially the accurate spectroscopic constants and vibrational spectrum information. Huber and Herzberg et al.[13] compiled and evaluated the experimental data of spectroscopic constants of the ground state of in early studies with low-resolution spectroscopic measurements. Later, Travers[14] and Ervin[12] obtained some spectroscopic parameters of the ground state by using negative ion photoelectron spectroscopy and extreme ultraviolet photoelectron spectroscopy, respectively. Theoretical studies on the electronic states of were also performed using multireference configuration interaction methods.[15]

In this paper, we focus on the ground electronic states of O2 and its ions ( and . The ground state wave function was optimized by using the appropriate active space multi-configuration self-consistent field method. The system energy was calculated through the explicitly correlated multireference configuration interaction method including the electron dynamic correlation. The potential energy curves of the ground state of the molecular systems were calculated by considering various physical correction and accurate spectroscopic constants and vibrational and rotational levels were obtained.

2. Methods and computational details

In this work, the explicitly correlated multireference configuration interaction (MRCI-F12) method[16] implemented in Molpro package[17] was carried out to study the electronic structure computation. Firstly, the Hartree–Fock (HF) self-consistent field method was used to construct the starting single configurational molecular orbital wavefunctions. The wave function of a single electronic ground state was further optimized using complete active space multiconfigurational self-consistent field (CASMCSCF), and finally the total energy including the electron dynamic correlation energy of the molecular system was estimated with the MRCI-F12 method. In the calculation of CASMCSCF, two active spaces were used to test computational results, one is full valence active space including all 2s2p electrons and the corresponding valence molecular orbital, which is denoted as CAS-I; the other consists of valence and an extra molecular orbital, denoted as CAS-II. The CAS-I is comprised of the 2s , 2p , 2s , 2p , 2p , and 2p molecular orbitals, totally, 2ag+1b2u+1b3u+2au orbitals in D2h symmetry are included. And in the CAS-II, an additional molecular orbital corresponding to 3p is added, that is, 2ag+2b2u+2b3u+2au orbitals in D2h symmetry are included. In the calculation with MRCI-F12, the correlations of all electrons in the molecular system were considered. The correlation effect of 1s orbital was also taken into account in this work. Davidson correction (+Q) was considered in computation to overcome the size-consistency problem. The correlation consistent basis sets cc-pCVXZ-F12 (X=T(3),Q(4))[18] including core–valence (CV) and core–core correlation effect were used in the calculations. The scalar relativistic (SR) correction effect was evaluated with MRCI method and uncontracted aug-cc-pVQZ basis set by using second order Douglas–Kroll–Hess (DKH) Hamiltonian.[19–21] The spin–orbit coupling (SOC) was calculated at the CASMCSCF/aug-cc-pVQZ level with state-interacting technique.[22]

In total 145 single energy points were calculated in the range of internuclear R=0.8–200 Å, and the potential energy curve V(R) of the system was obtained. The eigenvalues were then obtained by solving the one-dimensional radial Schrödinger equation listed as follows: where (μ represents the reduced mass of the system, v represents the vibrational quantum number, and J is the rotational quantum number. The rotational sublevels of a given vibrational level can be expressed as where G(v) is the pure vibrational term, and Bv and Dv are the first two terms of the centrifugal distortion constants. The Level program[23] was used to solve the radial Eq. (1).

3. Result and discussion
3.1. Spectroscopic constants

In Table 1, taking as an example, we show various physical effects on some spectroscopic constants. The results are compared with the available experimental values. Our previous study[24] indicates that the CV electron correlation on the spectroscopic constants is important, thus the CV effect is directly included in the computation by using the corresponding CV basis sets and singly, doubly electron excitation of 1s core in MRCI-F12 calculations. First of all, we discuss the influence of zero-order reference wave functions with different active spaces on the computational results of MRCI-F12. The results of CAS-II calculations are marked with MRCI-F12( ) in Table 1. The contribution of additional orbital in active space is estimated and denoted as in Table 1. When the Davidson’s correction is not included, the additional orbital of the active space improves the equilibrium internuclear distance Re and harmonic vibrational constant by 0.0014 Å and (∼9 cm−1, respectively. Little change in the anharmonic vibrational constant is observed, and the rotational constant Be is slightly improved. It can be seen that the configuration generated by the additional molecular orbital through electronic excitation is necessary for accurate determination of spectroscopic constants. When considering the Davidson correction, the computed Re and are improved by 0.0004 Å and (∼3 cm−1, respectively, which indicates that both Davidson correction and the additional orbital in active space improve the accuracy of computed spectroscopy. The account of Davidson correction involves the contribution of excitation of three and four electrons to the energy, and the orbital in active space increases the electron configurations of zero-order reference wavefunction and further improves the correlation energy calculation. The fact that the contribution of additional orbitals improves the potential energy and electronic transition properties has also been indicated in previous computational studies.[25,26] The correction of relativistic effect is also considered. The corrections of scalar relativistic effects on Re and are 0.0001 Å and (∼3 cm−1, respectively. The SOC only slightly modifies the potential energy curve since it does not have much effect on and . After comprehensive consideration of all of the above effects, our best estimated value of Re coincides with the experimental value to five significant digits, with an error less than 0.0001 Å ( ), 1 cm−1 (0.5‰), 0.2 cm−1, and 0.0004 cm−1 for Re, , , and Be, respectively.

Table 1.

The comparison of spectroscopic constants of under different approaches. The digits in parentheses are the experimental errors.

.

The spectroscopic constants of neutral and anion molecular systems are given in Table 2 and compared with the available experimental values. For the spectroscopic constants of O2, by comparing with accurate experimental values,[7] it is found that the accuracy of the current calculated results for O2 is very close to that of , with the error being only 0.0001 Å and (∼0.1 cm−1 for Re and , respectively. For the negative ion , the present best estimated result is in good agreement with the earlier compiled experimental value,[13] while it differs from the other two experimental values by 0.0021 Å[12] and 0.0031 Å[14] for Re. The experimental measurement error of of is relatively large (20 cm−1 and 50 cm−1), which is mainly limited by the number of the observed vibrational energy levels. The error bar of early complied value is not determined; our present result differs from the complied value by (∼26 cm−1, and is close to the lower limit of early experimental value with negative ion photoelectron spectrum, but differs from the ultraviolet photoelectron spectrum experimental value by (∼8 cm−1. Our calculated spectroscopic constants of are in reasonable agreement with recent experiment,[14] and the values are within the experimental error range. Due to the limitation of the number for observed vibration–rotation levels (only 29 levels observed in the latest experiment), the measurement error of the experiment is relatively large; on the other hand, accurate ab initio calculation on the vibration–rotation energy levels is much more important.

Table 2.

The spectroscopic constants (bond distance Re, harmonic and anharmonic frequency ( ), and rotational constant Be) of ground states of O2 and molecular systems.

.
3.2. Vbrational and rotational levels

On the basis of PECs deduced from the MRCI-F12( Q/CVQZ+SR+SOC computations, the low-lying vibrational terms G(v), rotational constant Bv, and Dv of the ground states for neutral O2 and ions are obtained and listed in Tables 3, 4, and 5, respectively, with available experimental values for comparison. Table 3 shows the low-lying v = 0–11 vibrational energy levels of these molecular systems. The zero-point vibrational energy (ZPVE) of O2 molecule calculated in this work is 787.8 cm−1, which is in good agreement with the experimental value[9] of 787.2 cm−1. The ZPVE values of the cation and anion are calculated to be 947.8 cm−1 and 556.4 cm−1, respectively. In the present calculation, the mean absolute deviation (MAD) of the first 12 vibrational levels v = 0–11 is 6.46 cm−1 (less than 1%), which is smaller than that of the most recent ab initio calculations (12.84 cm−1), indicating the improved accuracy of the present computational scheme for the low-lying vibrational levels of the ground state.

Table 3.

The vibrational levels (v = 0–11, J = 0) for neutral and ionic molecular systems (in the unit of cm−1).

.
Table 4.

The rotational constants Bv values for vibrational levels v = 0–11 (in the unit of cm−1).

.
Table 5.

The rotational constants Dv values for vibrational levels v = 0–11 (in the unit of 10−6 cm−1).

.

The MAD of v = 0–11 vibrational levels for is also small, which is only 9.77 cm−1 (0.88%). We note that the absolute deviation increases with the vibrational quantum number v, for instance, the absolute deviation increases to (∼20 cm−1 for v = 11. Such an increment in deviation could arise from the accumulation of errors. It may also indicate that the PEC in the range far away from the Franck–Condon region needs to be further improved, for example, considering multiple electron excitation. There are few experimental values of the vibrational levels for the negative ion, and the error bar is large.[12] The calculated in this paper is 1099 cm−1, which is in reasonable agreement with the experimental value of 1090 ± 20 cm−1. Excited vibrational levels for the ground state of are present in this study for the first time, which should be of value for further experimental studies.

The rotational constants Bv and Dv (v = 0–11) of these molecular systems are given in Tables 4 and 5, respectively, together with the experimental values listed for comparison. For neutral molecules, the present calculated results of Bv (v = 0–11) are in line with the experimental values with 3–4 significant digits, and are in good agreement with the results of the most recent theoretical study.[9] It is worth mentioning that our calculated Bv values are still 3 significant digits in accordance with the experimental results for vibrational levels up to v = 26 (data not shown in the table). For example, cm−1 and B26=1.01406 cm−1 of the ground-state O2 are calculated in this work, which are in good agreement with the corresponding experimental values of 1.18833 cm−1 and 1.01537 cm−1.[9] The maximum vibrational level of which the Bv is determined experimentally is v = 31, and the deviation of the present computed B31 from the experimental value[9] is −0.029 cm−1 (0.86410 cm−1 vs. 0.89248 cm . The Dv (v = 0–11) values of the neutral molecule are list in Table 4. The deviation of calculated in this work from the experimental value is in the range of 0.02–0.3 cm−1, which is much less than those of the latest calculated results (0.8–1.0 cm−1). The results of Dv for excited vibrational states, for example, v = 16, 26, and 31, are calculated with errors of 0.171 cm−1, 0.483 cm−1, and 0.765 cm−1, respectively, which are slightly less than those of the latest ab initio calculations[9] (0.837 cm−1, 0.667 cm−1, and 1.350 cm−1). The computational results for neutral O2 indicate that the present computational scheme for low-lying vibration–rotation levels is reliable.

In the system, the deviation of our computed Bv and values for v = 0–11 are in the range of cm−1 and 0.01–0.02 cm−1, respectively, which are close to the errors of recent extrapolated results with the MRCI method.[27] For v = 18, 19 levels in , the experimental values[28] of Bv are 1.3167 cm−1 and 1.2949 cm−1, respectively, which agree well with our computed values 1.3182 cm−1 and 1.2966 cm−1, respectively. The experimental values[29] of are 6.3876 ± 0.0086 cm−1 and 6.4825 ± 0.0085 cm−1, respectively, which are also in accordance with the present computed values, 6.3080 cm−1 and 6.4114 cm−1, respectively. For the v = 20 vibrational level of , our computed values of Bv and Dv (1.2746 cm−1 and 6.5212 cm−1) are also in agreement with recent experimental values[29] (1.272884(11) cm−1 and 6.596 ± 0.018 cm−1), respectively.

To the best of our knowledge, there are no experimental results for the Bv and Dv of anions. Our calculated values of Bv and Dv (v = 0–11) are listed in Tables 4 and 5, respectively. Since the same computational scheme as that for the neutral and cation is adopted for the anion, we expect that the results of low-lying vibrational and rotational spectrum information would have similar accuracy, which could help further experimental investigations on anions.

4. Conclusions

In this paper, we provide a computational scheme to obtain accurate spectroscopic constants and vibration–rotation spectra of the ground electronic states of O2 and its ions, using the explicitly correlated MRCI-F12 approach. The computed PECs of the ground states at the MRCI-F12( +Q/CVQZ +SR+SOC level, as well as the spectroscopic constants and low-lying vibration–rotation levels are in good agreement with available experimental values. We also discuss the effects of various corrections, including the Davidson correction, active space dependence, scalar relativistic correction, and spin–orbit coupling, on the accurate calculation of spectroscopic constants. The reasonable feasibility of MRCI-F12( +Q/CVQZ +SR+SOC method to calculate oxygen molecular systems is proved. The accurate results provided in the present study would be valuable for further spectroscopic measurements and for understanding the electronic states of these molecular systems.

Reference
[1] Ferguson E E Arnold F 1981 Acc. Chem. Res. 14 327
[2] Prabhakar R Siegbahn P E M Minaev B F Ågren H 2002 J. Phys. Chem. B 106 3742
[3] Zhang Y P Gan C L Song J P Yu X J Ma R Q Ge H Jiang T Lu K Q Eyler E E 2005 Chin. Phys. Lett. 22 1114
[4] Zhang Y P Cheng C H Kim J T Stanojevic J Eyler E E 2004 Phys. Rev. Lett. 92 203003
[5] Zhang Y P Gan C L Song J P Yu X J Ge H Ma R Q Li C S Lu K Q Eyler E E 2005 Chin. Phys. Lett. 22 1110
[6] Zhang Y P Song J P Gan C L Yan X A Nie Z Q Jiang T Li L Du K Zhang X C Lu K Q Eyler E E 2006 Chin. Phys. 15 2288
[7] Cheung A S C Yoshino K Esmond J R Parkinson W H 1996 J. Mol. Spectrosc. 178 66
[8] Bytautas L Ruedenberg K 2010 J. Chem. Phys. 132 074109
[9] Bytautas L Matsunaga N Ruedenberg K 2010 J. Chem. Phys. 132 074307
[10] Akahori T Morioka Y Tanaka T Yoshil H Hayaishi T Ito K 1997 J. Chem. Phys. 107 4875
[11] Prasad C V V Lacombe D Walker K Kong W Bernath P Hepburn J 1997 Mol. Phys. 91 1059
[12] Ervin K M Anusiewicz I Skurski P Simons J Lineberger W C 2003 J. Phys. Chem. A 107 8521
[13] Kerr J A Huber K P Herzberg G 1982 Anal. Chim. Acta 144 298
[14] Travers M J Cowles D C Ellison G B 1989 Chem. Phys. Lett. 164 449
[15] Liu H Shi D Sun J Zhu Z 2016 Mol. Phys. 114 3150
[16] Shiozaki T Knizia G Werner H J 2011 J. Chem. Phys. 134 034113
[17] Werner H J Knowles P J Knizia G Manby F R Schütz M 2012 WIREs Comput. Mol. Sci. 2 242
[18] Hill J G Mazumder S Peterson K A 2010 J. Chem. Phys. 132 054108
[19] Wolf A Reiher M Hess B A 2002 J. Chem. Phys. 117 9215
[20] Reiher M Wolf A 2004 J. Chem. Phys. 121 2037
[21] Reiher M Wolf A 2004 J. Chem. Phys. 121 10945
[22] Berning A Schweizer M Werner H J Knowles P J Palmieri P 2000 Mol. Phys. 98 1823
[23] Le Roy R J 2017 J. Quant. Spectrosc. Radiat Transf. 186 167
[24] Wei C Zhang X Ding D Yan B 2016 Chin. Phys. B 25 013102
[25] Li C X Guo Y C Wang B B 2017 Acta Phys. Sin. 66 103101 in Chinese
[26] Yan B Pan S Wang Z Yu J 2005 Acta Phys. Sin. 54 5618 in Chinese
[27] Zhang X Shi D Sun J Zhu Z 2011 Mol. Phys. 109 1627
[28] Zheng L Yang X Wu L Kaniki K Gao Y Liu Y Chen Y 2005 J. Mol. Spectrosc. 229 131
[29] Deng L Li C He W Zhu Y Yang X Chen Y 2012 J. Quant. Spectrosc. Radiat. Transf. 113 1547
[30] Song Y Evans M Ng C Y Hsu C W Jarvis G K 1999 J. Chem. Phys. 111 1905
[31] Lavrich D J Buntine M A Serxner D Johnson M A 1993 J. Chem. Phys. 99 5910
[32] Albritton D L Harrop W J Schmeltekopf A L Zare R N 1973 J. Mol. Spectrosc. 46 89