Accurate calculations on the 22 electronic states and 54 spin-orbit states of the O2 molecule: Potential energy curves, spectroscopic parameters and spin-orbit coupling

Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy 124 (2014) 216–229

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Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy journal homepage: www.elsevier.com/locate/saa

Accurate calculations on the 22 electronic states and 54 spin-orbit states of the O2 molecule: Potential energy curves, spectroscopic parameters and spin-orbit coupling Hui Liu a,b, Deheng Shi a,⇑, Jinfeng Sun a, Zunlue Zhu a, Zhang Shulin a a b

College of Physics and Electronic Engineering, Henan Normal University, Xinxiang 453007, China College of Physics and Electronic Engineering, Xinyang Normal University, Xinyang 464000, China

h i g h l i g h t s

g r a p h i c a l a b s t r a c t

 PECs are extrapolated to the CBS

The PECs of 54 X states generated from the 22 electronic states of O2 molecule are studied. Of the 22 03 3 þ 3  3 1 1 þ 1  1 1 þ 5 3 3  5  1  states, the X3 R g , A Du , A Ru , B Ru , C Pg , a Dg , b Rg , c Ru , d Pg , f Ru , 1 Pg , 1 Pu , 2 Rg , 1 Ru , 2 Ru 1 5 þ 1 5 5 and 21Dg are found to be bound, whereas the 15 Rþ , 2 R , 1 P , 1 D , 1 P and 2 P are found to be u g u u g g 1 3 3 5 03 repulsive ones. The B3 R u and d Pg states possess the double well. And the 1 Pu, C Pg, A Du, 1 Dg and 25 R þ g states are the inverted ones. The PECs are calculated by the CASSCF method followed by the icMRCI approach. Core-valence correlation and scalar relativistic corrections are included. All the PECs are extrapolated to the CBS limit. The vibrational properties are discussed for the 15 Pg , 13 Pu , d1 Pg , 15 R u and B3 R u states. The SO coupling effect is accounted for. The spectroscopic parameters are evaluated. The present spectroscopic parameters can be expected to be reliably predicted ones. The effect of SO coupling on the spectroscopic parameters are small almost for all the electronic states except for the 3 5 15 R  u ; 1 Pg and 1 Pu.

limit.  Convergence of the present calculations is found.  Effect of SO coupling on the spectroscopic parameters is evaluated.  Spectroscopic parameters of 16 electronic states and 35 X states are obtained.  Effect of core-valence correlation and scalar relativistic corrections is included.

9 6 7

Potential energy /Hartree

-150.0

O(1Dg) + O(1Sg) 11 10

-150.1

O(3Pg) + O(1Dg)

8 -150.2

4 3

O(1Dg) + O(1Dg)

5

O(3Pg) + O(3Pg)

-150.3

2 1

-150.4 0.1

0.2

0.3

0.4

Internuclear separation /nm PECs of 11 Σ states of O2 molecule 1-X3Σg-; 2-b1Σg+; 3-c1Σu-; 4-A3Σu+; 5-15Σu-; 5 + 6-1 Σg ; 7-25Σg+; 8-B3Σu-; 9-23Σg-; 10-21Σu-; 11-f1Σu+

a r t i c l e

i n f o

Article history: Received 26 November 2013 Received in revised form 23 December 2013 Accepted 5 January 2014 Available online 10 January 2014 Keywords: Spin-orbit coupling Potential energy curve Spectroscopic parameter

a b s t r a c t The potential energy curves (PECs) of 54 spin-orbit states generated from the 22 electronic states of O2 molecule are investigated for the first time for internuclear separations from about 0.1 to 1.0 nm. Of 03 3 þ 3  3 1 1 þ 1  1 1 þ 5 3 the 22 electronic states, the X3 R g , A Du , A Ru , B Ru , C Pg , a Dg , b Rg , c Ru , d Pg , f Ru , 1 Pg , 1 Pu , 1 5  1  5 þ 5 þ 1 5 5 23 R  g , 1 Ru , 2 Ru and 2 Dg are found to be bound, whereas the 1 Rg , 2 Rg , 1 Pu , 1 Dg , 1 Pu and 1 3 21Pu are found to be repulsive ones. The B3 R u and d Pg states possess the double well. And the 1 Pu, 3 5 03 5 þ C Pg, A Du, 1 Dg and 2 Rg states are the inverted ones when the spin-orbit coupling is included. The PEC calculations are done by the complete active space self-consistent field (CASSCF) method, which is followed by the internally contracted multireference configuration interaction (icMRCI) approach with the Davidson correction. Core-valence correlation and scalar relativistic corrections are taken into account. The convergence of present calculations is evaluated with respect to the basis set and level of

⇑ Corresponding author. Tel./fax: +86 376 6393178. E-mail address: [email protected] (D. Shi). 1386-1425/$ - see front matter Ó 2014 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.saa.2014.01.003

H. Liu et al. / Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy 124 (2014) 216–229 Relativistic correction Core-valence correlation correction

217

theory. The vibrational properties are discussed for the 15Pg, 13Pu, d1Pg and 15 R u states and for the second well of the B3 R u state. The spin-orbit coupling effect is accounted for by the state interaction method with the Breit-Pauli Hamiltonian. The PECs of all the electronic states and spin-orbit states are extrapolated to the complete basis set limit. The spectroscopic parameters are obtained, and compared with available experimental and other theoretical results. Analyses demonstrate that the spectroscopic parameters reported here can be expected to be reliably predicted ones. The conclusion is obtained that the effect of spin-orbit coupling on the spectroscopic parameters are small almost for all the electronic states 3 5 involved in this paper except for the 15 R u , 1 Pg and 1 Pu. Ó 2014 Elsevier B.V. All rights reserved.

Introduction O2 molecule is of great significance in human health, combustion, biological, and atmospheric chemistry as well as in many other processes. It is the second most abundant molecule in the earth’s atmosphere, and plays an extremely important role in terrestrial chemistry. Detailed knowledge of its potential energy curves (PECs) in various electronic states is frequently needed to help in understanding the complex oxygen spectrum, collision processes between oxygen atoms as well as mechanisms for many chemical reactions involving the O2 molecule, to calculate transport properties, and to model the energy flow in the atmosphere of Earth and Venus. For this reason, a great amount of experimental [1–15] and theoretical [16–37] work has been performed in the past several decades so as to determine its accurate PECs and evaluate its various spectroscopic properties. In experiment, a large number of measurements [1–15] have been made in order to determine its spectroscopy and relevant properties. In 1972, from the experimental aspect, Krupenie [38] performed a critical review of the observed spectroscopic data on the O2 molecule as of that time. In 1979, Huber and Herzberg [39] summarized the experimental spectroscopic parameters prior to 1979. Since 1979, there have still been a number of experiments [9–15], in which the spectroscopic parameters and some relevant properties were extracted for many electronic states. To our surprise, when we summarize the measurements, we find the following. (1) The spectroscopic parameters, in particular for the Te, Re and xe, are accurately determined only for several electronic states 3 þ 3  1 1 þ 1  such as X3 R g , A Ru , B Ru , a Dg , b Rg and c Ru , though a large number of electronic states have been observed experimentally. And (2) no spectroscopic parameters of any X states [Here, the X state refers to the spin-orbit (SO) state] have been calculated for the electronic states involved in this paper except for the X3 R g;0þ and X3 R g;1 . In theory, the first ab initio spectroscopic calculations of O2 molecule were done by Meckler [16] in 1953. Meckler [16] employed the configuration interaction (CI) method with a minimum Gaussian atomic-orbital basis set to determine the PECs of the 9 low-ly1 þ ing 3 R g and 12 low-lying Rg electronic states. Kotani et al. [17] in 1958 performed the second ab initio spectroscopic calculations for the 9 low-lying electronic states using the CI approach and a minimum basis set of Slater-type orbitals (STO’s). Schaefer and Harris [18] in 1968 calculated the PECs of 62 low-lying electronic states by the valence CI (VCI) method and a minimum basis set of STO’s. In 1972, from the theoretical aspect, Krupenie [38] made a critical review of the predicted spectroscopic data on the O2 molecule prior to that time. Since 1972, there have still been many theoretical investigations [19–37] about the PECs and the spectroscopic parameters of O2 molecule. Of these investigations, only few calculations performed in recent years have obtained high-quality spectroscopic results. For example, in 2010, Bytautas et al. [36] obtained a very

accurate ab initio PEC of ground state. In their work, all valence shell electron correlations were calculated at the complete basis set (CBS) limit. The ground-state dissociation energy and equilibrium position they derived were 42030.05 cm1 and 0.12075 nm, which deviated from the corresponding measurements only by 25.35 cm1 [14] and 0.00001 nm [39], respectively. However, they [36] only evaluated the ground-state spectroscopic parameters. In 2010, Bytautas and Ruedenberg [37] performed the non-relativistic full CI valence correlation by the correlation energy extrapolation. They evaluated the ground-state dissociation energy and internuclear equilibrium separation, which also agree well with the available experimental ones [14,39]. Summarizing these theoretical results, on the whole, we find the following. (1) The accurate spectroscopic calculations concentrate only on few electronic states, in particular on the ground state, though a large number of electronic states have been investigated in the past several decades. And (2) to this day, no SO coupling effect has been involved for the X states in the literature except for the ground state, though the SO coupling effect may produce important influences on the spectroscopic properties, especially for the electronic states possessing the very shallow potential well. The aim of the present work is to determine the accurate spectroscopic parameters of O2 molecule. Firstly, extensive ab initio calculations on the PECs of 22 electronic states will be made over a wide internuclear separation. In order to evaluate the spectroscopic parameters of O2 molecule as accurately as possible, on the one hand, core-valence correlation and scalar relativistic corrections are taken into account; on the other hand, the extrapolation to the CBS limit is made so that the residual errors behind the basis sets can be eliminated. Secondly, the effect of SO coupling on the PECs will be introduced into the calculations since few PECs have been calculated for the X states up to now. Analyses demonstrate that the present spectroscopic results can achieve very high quality. In the next section, we will briefly describe the theory and method used in this paper. In Results and discussion, the PECs of h 03 3 þ 3  3 1 1 þ 1 22 electronic states X3 R g ; A Du ; A Ru ; B Ru ; C Pg ; a Dg ; b Rg ; c

Ru ; d1 Pg ; f 1 Rþu ; 15 Pg ; 13 Pu ; 23 Rg ; 15 Ru ; 21 Ru ; 21 Dg ; 15 Rþg ; 25 Rþg ; 11 Pu ; 15 Dg ; 15 Pu ; and 21 Pu  are calculated for internuclear separations from about 0.1 to 1.0 nm. The PECs of 54 X states generated from the 22 states are studied for the first time over the same internuclear separations. The PEC calculations are made using the complete active space self-consistent field (CASSCF) method, which is followed by the internally contracted multireference CI (icMRCI) approach [40,41] with the Davidson correction (icMRCI + Q) [42,43]. The SO coupling effect is accounted for by the state interaction method with the Breit-Pauli Hamiltonian. The effect of core-valence correlation and scalar relativistic corrections on the PECs is included. The PECs are extrapolated to the CBS limit. The spectroscopic parameters are calculated for all the bound states involved in the present paper, and compared with those available in the literature. Concluding remarks are given in Conclusion.

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49660.45

33792.58 33531.52

49255.07 1 1 1 þ 1 1 Rþ g ð6Þ; Pg ð4Þ; Dg ð4Þ; Ru ð1Þ; Pu ð4Þ; Du ð2Þ

Oð Dg Þ þ Oð Sg Þ

1 1 1

3

Oð Dg Þ þ Oð Dg Þ

Oð3 Pg Þ þ Oð1 Sg Þ

3 3  3 R u ð3Þ; Pu ð5Þ; Rg ð4Þ; Pg ð5Þ

15867.86

0.0

31735.72

15723.45

31466.06 1 þ 1 þ 1  1  1 1 1 1 1 1 1 1 1 1 Rþ g ð3Þ; Rg ð4Þ; Rg ð5Þ; Ru ð2Þ; Ru ð3Þ; Pg ð2Þ; Pg ð3Þ; Pu ð2Þ; Pu ð3Þ; Dg ð2Þ; Dg ð3Þ; Du ð1Þ; Ug ð1Þ; Uu ð1Þ; Cg ð1Þ 1 1

Oð Pg Þ þ Oð Dg Þ

1

3 þ 3  3  3  3  3 3 3 3 3 3 3 3 3 3 3 3 Rþ g ð1Þ; Ru ð3Þ; Rg ð2Þ; Rg ð3Þ; Ru ð2Þ; Ru ð2Þ; Pg ð2Þ; Pg ð3Þ; Pg ð4Þ; Pu ð2Þ; Pu ð3Þ; Pu ð4Þ; Dg ð1Þ; Dg ð2Þ; Du ð2Þ; Ug ð1Þ; Du ð3Þ; Uu ð1Þ 3 1 3

This work

0.0 1 þ 1  1 1 1 3  3 þ 3 þ 3 3 3 5 þ 5 þ 5  5 5 5 Rþ g ð1Þ; Rg ð2Þ; Ru ð1Þ; Pg ð1Þ; Pu ð1Þ; Dg ð1Þ; Rg ð1Þ; Ru ð1Þ; Ru ð2Þ; Pg ð1Þ; Pu ð1Þ; Du ð1Þ; Rg ð1Þ; Rg ð2Þ; Ru ð1Þ; Pg ð1Þ; Pu ð1Þ; Dg ð1Þ 1

Oð3 Pg Þ þ Oð3 Pg Þ

Electronic states Dissociation channel

Table 1 Dissociation relationships of a few electronic states of O2 molecule determined by the icMRCI + Q/CV + DK + 56 calculations.

Relative energy (cm1)

Exp. [44,45]

Theory and method To find out the dissociation channels of the electronic states involved here, we first deduce all the electronic states resulting from the lowest five dissociation channels of the O2 molecule. These states together with their corresponding dissociation channels are collected in Table 1. Among the twenty-two electronic states involved here, sixteen 1 þ 1  states [11 Rþ 11 R 11 Pg ðd1 Pg Þ, 11 Pu ð11 Pu Þ, g ðb Rg Þ, u ðc Ru Þ, 1 1 3  3  3 þ 3 þ 1 Dg ða Dg Þ, 1 Rg ðX Rg Þ, 1 Ru ðA Ru Þ, 13 Pg ðC3 Pg Þ, 13 Pu ; 5 þ 5  5 5 5 13 Du ðA0 3 Du Þ, 15 Rþ , 2 R , 1 R , 1 P , 1 P and 1 D ] are attributed g u g g g u to the first dissociation channel, O(3Pg) + O(3Pg). Two states 3  3  ½23 R g and 1 Ru ðB Ru Þ are attributed to the second dissociation 1 1 channel, O(3Pg) + O(1Dg). Three states ½21 R u ; 2 Pu and 2 Dg  are 1 1 attributed to the third dissociation channel, O( Dg) + O( Dg). And 1 þ one state ½11 Rþ u ðf Ru Þ is attributed to the fifth dissociation chan1 1 nel, Oð Dg Þ þ Oð Sg Þ. All the PECs are calculated by the CASSCF method, which is followed by the icMRCI approach. Here, the CASSCF is used as the reference wave function for the icMRCI calculations. The basis set used to calculate all the PECs is the aug-cc-pV6Z (AV6Z) [46]. All the PECs are calculated with the MOLPRO 2010.1 program package [47]. To determine accurately all the PECs, the point spacing interval used here is 0.02 nm for each state, except near the equilibrium separation where the point spacing is 0.002 nm. Here, the smaller step is adopted near the equilibrium position of each state so that the properties of each PEC can be displayed more clearly. The O2 molecule belongs to the D1h symmetry. To make the PEC calculations in the MOLPRO 2010.1 program package, we must substitute the D1h symmetry with the D2h point group. As we know, the D2h point group has eight irreducible representations, Ag, B3u, B2u, B1g, B1u, B2g, B3g and Au. The corresponding symmetry operations for the D1h ? D2h are the Rþg ! Ag ; Rg ! B1g ; Pg ! B2g þ B3g ; Dg ! Ag þ B1g ; Rþu ! B1u ; Ru ! Au ; Pu ! B2u þ B3u and Du ? Au + B1u, respectively. The orbitals are optimized by the CASSCF method. The state-averaged technique is employed in the CASSCF calculations. In the CASSCF and subsequent icMRCI calculations, eight valence molecular orbitals (MOs) are put into the active space, including two ag, one b3u, one b2u, two b1u, one b2g and one b3g symmetry MOs. The six valence electrons in the 2s2p shell of oxygen atom are placed in the active space, which consists of full valence space. That is, twelve valence electrons in the O2 molecule are distributed into eight orbitals (2–3rg, 2–3ru, 1pu and 1pg) in the calculations. The energy ordering of the eight valence MOs is 2rg 2ru3rg1pu1pg3ru. As a result, this active space is referred to as CAS (12, 8). The four inner electrons in the O2 molecule are put into the closed-shell orbitals, including one ag and one b1u symmetry MOs, which correspond to the 1rg and 1ru MOs in the molecule. In addition, the four inner electrons in the molecule are used as core electrons for the corevalence correlation calculations, whereas they are frozen when we perform the frozen-core calculations [48]. In the present calculations, the number of external orbitals reaches 108, including 22ag, 13b3u, 13b2u, 6b1g, 22b1u, 13b2g, 13b3g and 6au symmetry MOs. When we use the ten MOs (3ag, 1b3u, 1b2u, 3b1u, 1b2g and 1b3g) to make the PEC calculations, we find that each PEC is smooth over the present internuclear separation range, and each PEC is convergent. It means that the two atomic oxygen fragments are completely separated at 1.0 nm. The convergence of each PEC clarifies that the dissociation energy can be determined by the difference between the total energy of O2 molecule at the equilibrium position (which is obtained by fitting) and the energy sum of two atomic fragments at 1.0 nm, or is obtained by the difference between the total energy of O2 molecule at the equilibrium position

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and the total energy at the highest barrier when the energy at the highest barrier is higher than that at the dissociation limit. In this work, core-valence correlation correction is included with an aug-cc-pCV5Z basis set [48] at the level of icMRCI + Q theory, and its contribution is denoted as CV. Scalar relativistic correction is taken into account by the third-order Douglas-Kroll Hamiltonian approximation (DKH3) at the level of a cc-pV5Z basis set [49] by both taking and not taking into account the scalar relativistic effect. In detail, the cc-pV5Z-DK basis set with the DKH3 approximation and the cc-pV5Z basis set with no DKH3 approximation are employed to calculate the scalar relativistic correction contributions. The difference between the two energies produces the scalar relativistic correction result, and is denoted as DK. Core-valence correlation and scalar relativistic corrections are applied across the entire PEC of each state. The SO coupling effect is included into the PEC calculations by the state interaction method with the Breit-Pauli Hamiltonian [50], which has been implemented in MOLPRO 2010.1 program package [47] by incorporating the most important two-electron contributions of SO operator [51]. Two basis sets, aug-cc-pV5Z (AV5Z) and AV6Z [46,52], are used for the extrapolation scheme (denoted as 56). The extrapolation formula is written as [53]

DEtotal;1 ¼

DEtotal;nþ1 ðn þ 1Þ3  DEtotal;n n3 ðn þ 1Þ3  n3

ð1Þ

:

Here, DEtotal, 1 is the total energy extrapolated to the CBS limit, and DEtotal, n and DEtotal, n+1 are the total energies obtained by the basis sets, aug-cc-pVnZ (AVnZ) and aug-cc-pV(n + 1)Z [AV(n + 1)Z]. From the PECs obtained here, the spectroscopic parameters, including the excitation energy term Te referred to the ground state, harmonic frequency xe, equilibrium internuclear separation Re, first- and second-order anharmonic constants xexe and xeye, vibration coupling constant ae, rotational constant Be and dissociation energy De, are evaluated. To determine accurately the spectroscopic parameters, all the PECs are fitted to an analytical form by cubic splines so that the corresponding rovibrational Schrödinger equation can be conveniently solved. In this paper, we solve the rovibrational Schrödinger equation by Numerov’s method [54]. That is, the rovibrational constants are first obtained in a direct forward manner from the analytic potential by solving the rovibrational Schrödinger equation, and the spectroscopic parameters

are evaluated by fitting the first ten vibrational levels whenever available. Results and discussion High-quality ab initio calculations must be convergent with respect to the basis set and level of theory. Otherwise, the spectroscopic results determined by the calculations are insignificant because of low accuracy and poor reliability. For some electronic states involved here, no accurate measurements can be available, and the available theoretical spectroscopic parameters sometimes greatly differ from each other. To verify the rationality of spectroscopic parameters obtained in this paper, and to estimate the resident errors behind the present calculations, we must discuss the convergent behavior of present calculations with respect to the basis set and level of theory. Due to length limitation, we only 1  03 collect the Te, Re, xe and De results of the b1 Rþ g , c Ru and A Du states in Table 2, which are obtained by the aug-cc-pVTZ (AVTZ), aug-cc-pVQZ (AVQZ), AV5Z and AV6Z basis sets and the extrapolation to the CBS limit. It should be pointed out that the three states used for the present discussion are optionally selected. As demonstrated in Table 2, we can clearly see that Te converges toward the CBS limit when we systematically improve the quality of correlation-consistent basis sets. For the b1 Rþ g state, the basis sets from the AVTZ to the AVQZ, from the AVQZ to the AV5Z and from the AV5Z to the AV6Z lower Te by 102.27, 66.50 and 24.80 cm1 for the icMRCI, and lower Te by 109.30, 31.44 and 25.68 cm1 for the icMRCI + Q calculations, respectively. It reminds us that Te of the b1 Rþ g state may be converged near 30 cm1 with respect to the basis set. It has been proved by the extrapolation of basis sets to the CBS limit. From Table 2, we can 1 see that Te of the b1 Rþ g state is lowered by 34.24 and 35.33 cm for the icMRCI and icMRCI + Q calculations, respectively, when we extrapolate the AVnZ basis set to the aug-cc-pV1Z (AV1Z). Similar to the b1 Rþ g state, with the quality of basis sets improved, 1 Te of the c1 R u state are raised by 580.07, 91.49 and 79.92 cm at the icMRCI, and raised by 595.66, 334.89 and 82.33 cm1 at the icMRCI + Q level of theory; Te of the A0 3Du state are increased by 531.57, 75.03 and 72.67 cm1 at the icMRCI, and raised by 545.40, 317.11 and 75.09 cm1 at the icMRCI + Q level of theory, 03 respectively. It suggests us that Te of the b1 Rþ g and A Du states may be converged at about 80 cm1 with respect to the basis set.

Table 2 1  03 Convergence observations of spectroscopic parameters with respect to the basis set and level of theory for the b1 Rþ g ; c Ru and A Du states. icMRCI

icMRCI + Q

Te (cm1)

Re (nm)

xe (cm1)

De (eV)

Te (cm1)

Re (nm)

xe (cm1)

De (eV)

AVTZ AVQZ AV5Z AV6Z 56

12806.56 12704.29 12637.79 12612.99 12578.75

0.12349 0.12289 0.12276 0.12270 0.12262

1405.19 1430.81 1434.16 1438.86 1442.60

3.3724 3.5013 3.5280 3.5459 3.5706

13274.76 13165.46 13134.02 13108.34 13073.01

0.12370 0.12307 0.12294 0.12287 0.12279

1394.67 1421.20 1425.84 1429.68 1433.59

3.4070 3.5498 3.5517 3.5712 3.5980

c1 R u AVTZ AVQZ AV5Z AV6Z 56

32152.16 32732.23 32823.72 32903.64 32985.94

0.15283 0.15192 0.15168 0.15151 0.15135

759.78 773.15 780.74 783.45 787.14

0.9773 1.0215 1.0003 1.0078 1.0182

32032.76 32628.42 32963.31 33045.64 33131.45

0.15286 0.15193 0.15168 0.15150 0.15133

773.83 787.86 794.72 798.54 802.42

1.0587 1.1105 1.0869 1.0954 1.1070

A03 Du AVTZ AVQZ AV5Z AV6Z 56

33854.62 34386.19 34461.22 34533.89 34606.32

0.15234 0.15147 0.15120 0.15109 0.15094

785.47 798.75 804.16 808.76 812.31

0.7409 0.7935 0.7972 0.8055 0.8169

33802.82 34348.22 34665.33 34740.42 34816.14

0.15250 0.15160 0.15132 0.15121 0.15105

795.36 809.66 816.39 820.10 823.80

0.8099 0.8701 0.8831 0.8923 0.9051

b1 Rþ g

H. Liu et al. / Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy 124 (2014) 216–229

Spectroscopic parameters of the 16 bound electronic states To show clearly the relationships of all the PECs, we depict them in Figs. 1 and 2. To demonstrate the details of the PEC of each state,

9 6 7

O(1Dg) + O(1Sg)

-150.0

Potential energy /Hartree

When we extrapolate the AVnZ to the AV1Z basis set, Te of the b1 Rþ g and A0 3Du states are raised by 82.30 and 72.43 cm1 for the icMRCI, 1 and raised by 85.81 and 75.72 cm for the icMRCI + Q calculations, respectively. As a conclusion, we think that Te of these states is convergent with respect to the basis set at the present level of theory. Then we discuss the convergent behavior of the Re. With the quality of basis sets raised, Re of the b1 Rþ g state are shortened by 0.00060, 0.00013 and 0.00006 nm for the icMRCI, and by 0.00063, 0.00013 and 0.00007 nm for the icMRCI + Q calculations, respectively. As a result, we think that Re of this state should converge rapidly with respect to the basis set. As clearly seen in Table 2, Re of the b1 Rþ g state converge within 0.00008 nm with respect to the basis set for both the icMRCI and icMRCI + Q calculations when the basis sets are extrapolated to the CBS limit. Sim03 ilarly, we have also found that Re of the c1 R u and A Du electronic states converge at about 0.00017 and 0.00016 nm, respectively, when we extrapolate the basis sets to the CBS limit. The xe still converges rapidly with respect to the basis set, whether for the icMRCI or the icMRCI + Q calculations. For example, for the icMRCI + Q calculations, when the basis sets are changed from the AVTZ to the AVQZ, from the AVQZ to the AV5Z and from the AV5Z to the AV6Z, xe are raised by 26.53, 4.64 and 1 3.84 cm1 for the b1 Rþ for g , raised by 14.03, 6.86 and 3.82 cm 1 03 the c1 R , and raised by 14.30, 6.73 and 3.71 cm for the A Du u state, respectively. These results suggest that the xe may be converged at about 3.8 cm1. According to Table 2, we can clearly find that the xe are converged at about 3.91, 3.88 and 3.70 cm1 for the 1  03 b1 Rþ g ; c Ru and A Du states, respectively, when we extrapolate the basis sets from the AVnZ to the AV1Z. In conclusion, the xe of the three states have excellent convergent behavior. Now we investigate the convergent behavior of present calculations with respect to the level of theory. For the b1 Rþ g state, the differences of the Te between the icMRCI and the icMRCI + Q method are 468.20, 461.17, 496.23, 495.35 and 494.26 cm1 at the AVTZ, AVQZ, AV5Z and AV6Z basis sets and the CBS limit, respectively. These results tell us that the triplet and quadruple excitations in the present calculations lower Te by less than 495.4 cm1. It reminds us that the contribution to the Te of b1 Rþ g state by the corrections higher than quadruple excitations can be estimated by less 03 than 247.7 cm1. For the c1 R u and A Du states, similar calculations tell us that the triplet and quadruple excitations in the present calculations lower Te by less than 145.6 and 209.8 cm1, and the contribution to the Te by the corrections higher than quadruple excitations can be estimated by less than 72.8 and 104.9 cm1, respectively. From these calculations, we can see that the contribution to the Te of the b1 Rþ g state by the corrections higher than quadruple excitations is very large. Summarizing the discussion made here, we can draw two conclusions. One is that the present icMRCI + Q calculations have excellent convergent behavior for the electronic states collected in Table 2. The other is that the contribution to the Te by the corrections higher than quadruple excitations may be large for some electronic states. In addition to the above three electronic states, we have investigated the convergent behavior with respect to the basis set and level of theory for other thirteen bound states of O2 molecule involved here. Similar conclusion can be gained. In conclusion, we think that the present calculations are convergent with respect to the basis set and present level of theory. According to the above discussion, we will use the PECs obtained by the icMRCI + Q/56 + CV + DK calculations to perform the following spectroscopic calculations.

11 10

-150.1

O(1Dg) + O(1Dg) O(3Pg) + O(1Dg)

8 -150.2

O(3Pg) + O(3Pg)

5

4 3 -150.3

2 1

-150.4 0.1

0.2

0.3

0.4

Internuclear separation /nm 1 þ 1  3 þ Fig. 1. PECs of the 11 states of O2 molecule 1 – X3 R g ; 2 – b Rg ; 3 – c Ru ; 4 – A Ru ; 5 þ 5 þ 3  3  1  1 þ 5 – 15 R ; 6 – 1 R ; 7 – 2 R ; 8 – B R ; 9 – 2 R ; 10 – 2 R ; 11 – f R . u g g u g u u

Potential energy /Hartree

220

-150.0

11

6 7 9

-150.1

O(1Dg) + O(1Dg)

10 8

5

O(3Pg) + O(1Dg)

4

-150.2

2

3

-150.3

1 0.1

0.2

0.3

0.4

Internuclear separation /nm Fig. 2. PECs of the 11 states of O2 molecule 1 – a1 Dg ; 2 – A0 3 Du ; 3 – 15 Pg ; 4 – 13 Pu ; 5 – C3 Pg ; 6 – 11 Pu , 7 – d1 Pg , 8 – 15 Pu ; 9 – 15 Dg ; 10 – 21 Dg ; 11 – 21 Pu .

we depict them only within a small internuclear separation range. From Figs. 1 and 2, we find the following. The C3 Pg electronic state has an obvious barrier, whereas it only possesses the single well. The d1 Pg and B3 R u electronic states possess the double well. 5 þ 1 5 5 1 And the 15 Rþ ; 2 R g g ; 1 Pu ; 1 Dg ; 1 Pu and 2 Pu are the repulsive states. It should be pointed out that the PECs of the 15 Rþ g and 25 Rþ g states are almost overlapped together in Fig. 1. For the sake of length limitation, we do not depict each of them in a separate figure. In Fig. 2, the PECs of the 15 Pu and 15 Dg electronic states cannot be distinguished from each other. For the same reason, we either do not show each of them in a separate figure. The present calculations are involved with the four dissociation channels. For comparison with the measurements [44,45], we determine the energy separation between each higher dissociation channel and the lowest one by the icMRCI + Q/56 + CV + DK calculations, and collect them in Table 1. As seen in Table 1, the present energy separation between the Oð3 Pg Þ þ Oð1 Dg Þ dissociation channel and the lowest one is 15723.45 cm1 which agrees well with the corresponding measurements of 15867.86 cm1 [44,45]. The present energy separation between the Oð1 Dg Þ þ Oð1 Dg Þ dissociation channel and the lowest one is 31466.06 cm1, which is in fair agreement with the measurements of 31735.72 cm1 [44,45]. The present energy separation between the Oð1 Dg Þ þ Oð1 Sg Þ dissociation channel and the lowest one is 49255.07 cm1, which deviates from the corresponding measurements of 49660.45 cm1 only by 405.38 cm1. These comparisons show that the electronic states

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Table 3 Comparison of the spectroscopic parameters obtained by the icMRCI + Q/56 + CV + DK calculations with experimental and other theoretical results for the 03 1 1 þ 1  3 þ 3  X3 R g ; a Dg ; b Rg ; c Ru ; A Du ; A Ru and B Ru states of the O2 molecule. De (eV)

Te (cm1)

Re (nm)

xe (cm1)

xexe (cm1)

102xeye (cm1)

Be (cm1)

102ae (cm1)

R g

5.2203

0.0

0.12068

1581.61

10.039

5.0175

1.4376

1.2539

Exp.[14] Exp. [39] Cal. [28]a Cal. [29]b Cal. [32] Cal. [34]c Cal. [35] Cal. [36] Cal. [37]

5.2142 5.2132 5.0606 5.0996 5.2111 – – 5.2111 5.2111

0.0 0.0 0.0 0.0

0.12075 0.12074 0.12076

1580.19 1604 1601

11.98 10.8

–

1.4456

1.59

0.0 0.0 0.0 0.0

– 0.1207 0.12075 0.12075

1580 1645

12.2

–

1.4191

1.58

a1 Dg Exp. [38] Exp. [39] Cal. [24]f Cal. [30]g

4.2258

7776.43

0.12147

1491.07

8.1245

19.251

1.3814

0.4238

4.2316 – 5.723 4.2690

7918.11 7918.1 – 7769

0.12157 0.12156 0.1200 0.12179

(1509.3)d [1483.5]e 1662.2 1512

(12.9) (12.9) 19.87

– – –

1.4263 1.4264 1.40

1.71 1.71

b1 Rþ g

X

a b c d e f g h i j

3

3.6058

13099.92

0.12258

1438.65

12.723

10.538

1.4030

1.8018

Exp. [38] Cal. [30]g

3.5772 3.6129

13195.31 12864

0.12268 0.12291

(1432.67) 1434

13.934

1.43

1.4005

1.8169

c1 R u Exp. [38] Cal. [18] Cal. [20] Cal. [21]h Cal. [30]g

1.1004

33220.12

0.15109

804.29

13.102

39.212

0.9233

1.6899

0.8665 0.90 1.062 1.425 1.1452

33058.4 – 31358.82 – 31924

0.15174 0.156 0.1555 0.1520 0.15175

794.29 920 759.8 794.2 791

12.736 27 12.25 17.01

24.44 – – –

0.9155 0.87 0.872 0.912

1.391 1.7 1.46 1.96

A03 Du Exp. [38] Exp. [39] Cal. [21]h Cal. [24]i Cal. [30]g

0.8939 0.9066 – 1.216 0.929 0.9451

34913.80 34735 (34690) – – 34632

0.15078 – (0.148) 0.1515 0.1536 0.15152

826.44 (750) (850) 820.5 753.8 810

12.594 (14) (20) 16.71 9.08

58.940

0.9271

1.6049

– – –

(0.96) 0.918 0.856

(2.62) 1.84

A3 Rþ u Exp. [38] Cal. [21]h Cal. [30]g

0.8128

35450.42

0.15151

810.13

12.419

68.473

0.9181

1.6122

0.8244 1.140 0.8659

35398.70 – 35271

0.15215 0.1523 0.15224

799.08 802.5 795

12.16 16.61

55.0 –

0.908

1.87

B3 R u 1st well Exp. [9] Exp. [38] Exp. [39] Exp. [12]j Cal. [20]

0.8417 – 1.0069 – – 1.136

51028.73 49005 49794.33 49793.28 48534 49030.42

0.15978 – 0.16043 0.16042 – 0.1627

723.60 709.06 709.06 709.31 728.0 724.9

10.756 10.61 10.614 10.65 12.33 7.04

74.742 5.92 5.9212 – 26.1 –

0.8256

1.3596

0.9105 0.8190

1.416 1.206

0.791

0.77

2nd well

0.0632

57345.21

0.23240

186.88

19.921

28.205

0.3838

4.0519

CCSDT/cc-pVQZ. CCSDT/aug-cc-pVQZ. p-RBXCISD(T). Results in squares are uncertain data. results in brackets are those refering to t = 0 or the lowest observed level. Ht (2rd). MR-AQCC/(TQ). Ht . Ht (3rd). in Ar.

involved in this paper can dissociate into the atomic fragments in a proper form. Employing the PECs determined by the icMRCI + Q/56 + CV + DK calculations, we have evaluated the spectroscopic parameters of all the bound states involved here by the theoretical approach outlined in Theory and method. For convenience of discussion, we divide the present bound states into two categories. One is the X3 R g, 03 1  3 þ 3  a1 Dg , b1 Rþ , c R , A D , A R and B R , which are the states posu g u u u sessing the accurate experimental spectroscopic parameters [2,3,5,6,9,14,38,39]. The other is the 15 Pg , 13 Pu , C3 Pg , 23 R g, 1  1 1 þ d1 Pg , 15 R u , 2 Ru , 2 Dg and f Ru , which are the states possessing the inaccurate measurements or no measurements so far. Here, 1 1 þ we collect the spectroscopic parameters of X3 R g , a Dg , b Rg ,

03 3 þ 3  c1 R u , A Du , A Ru and B Ru states in Table 3, and tabulate the spec1 5  1  troscopic parameters of 15 Pg , 13 Pu , C3 Pg , 23 R g , d P g , 1 Ru , 2 Ru , 1 1 þ 2 Dg and f Ru states in Table 4. For convenience of discussion, the electronic configurations of the 16 bound states as obtained from the icMRCI wavefunctions near the equilibrium positions are collected in Table I in the Supplementary material.

03 1 1 þ 1  3 þ 3  X3 R g ; a Dg ; b Rg ; c Ru ; A Du ; A Ru and B Ru states For the sake of length limitation, we only tabulate some selected measurements [9,12,14,38,39] in Table 3 for convenience of discussion. In addition, we also only collect some selected theoretical results [18,20,21,24,28–30,32,34–37] in Table 3, which quality can be comparable with that of the present spectroscopic parameters.

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Table 4 Comparison of the spectroscopic parameters obtained by the icMRCI + Q/56 + CV + DK calculations with experimental and other theoretical results for the 1 5  1  1 1 þ 15 Pg ; 13 Pu ; C3 Pg ; 23 R g ; d Pg ; 1 Ru ; 2 Ru ; 2 Dg and f Ru states of the O2 molecule. De (eV)

Te (cm1)

Re (nm)

xe (cm1)

xexe (cm1)

102xeye (cm1)

Be (cm1)

1 Pg

0.0883

41595.22

0.22369

157.41

11.872

56.031

0.4220

3.0349

13 Pu C3 Pg Exp. [39] Cal. [30]c

0.0200 1.4845

41931.95 54770.33

0.30052 0.14775

53.468 696.71

6.6939 13.604

106.89 1769.8

0.2174 0.9637

0.2678 2.3527

– –

(65530)a 54079

– 0.14845

[1840]b 681

5

23 R g

0.3423

55078.04

0.21044

468.95

12.930

103.84

0.4760

0.30 0.511 0.807

– 54031.06 –

0.200 0.2069 0.2136

872 537.0 455.8

18 13.73 7.41

– – –

0.53 0.490 0.461

0.0905 – 0.84 1.111 1.364 –

65516.91 (69180)a – 67839.27 – 65040

0.14487 – 0.157 0.1620 0.1620 0.14610

829.75 [1860]b 2559 1626.4 2426.0 810

664.67

15475

3.3954

65 163.67 320.39

– – –

0.85 0.819 0.796

21 R u

0.0159 0.0051 0.6583

41962.01 42097.21 68232.91

0.32032 0.32890 0.16095

54.267 30.855 685.65

6.4377 8.8605 13.044

7.1410 83.011 96.748

0.2052 0.2006 0.8135

21 Dg Cal. [18] Cal. [20] Cal. [21]d

0.2646

71430.21

0.20052

404.28

16.767

65.133

0.5245

1.6669

0.14 0.437 0.715

– 72839.90 –

0.188 0.1911 0.2030

597 499.5 408.2

36 13.51 13.92

– – –

0.60 0.547 0.510

1.5 1.02 1.15

f 1 Rþ u Cal. Cal. Cal. Cal. Cal.

1.2083

81607.47

0.15984

750.85

10.186

38.606

0.8249

1.2567

0.74 1.653 1.639 2.514 –

– 84123.59 – – 54805.29

0.166 0.1611 0.1622 0.1543 0.1467

625 792.4 715.1 782.3 931.1

1 7.71 6.23 18.68

– – – –

0.76 0.811 0.800 0.849

2.2 0.92 0.12

Cal. [18] Cal. [20] Cal. [21]d d1 Pg 1st well Exp. [39] Cal. [18] Cal. [20] Cal. [21]d Cal. [30]c 2nd well 15 R u

a b c d e

102ae (cm1)

[18] [20] [21]d [24]e [33]

1.0007 0.9 0.45 0.57 532.71 0.5 3.89 0.98 2.1403 5.9015 1.5833

Results in squares are uncertain data. Results in brackets are those refering to t = 0 or the lowest observed level. MR-AQCC/(TQ). Ht . Ht (2rd).

Now we investigate the multiconfiguration characterizations of these states collected in Table 3. According to Table I in the Supple1 1 þ mental material, we can clearly see that the X3 R g , a Dg ; and b Rg states have poor multireference characterizations near the internu03 3 þ clear equilibrium positions. The c1 R u , A Ru and A Du states and the first well of B3 R state possess the multireference characterizau tions near the internuclear equilibrium positions when the spin orientation of electrons is taken into account, whereas their multireference characterizations become poor when we ignore the spin orientation of electrons. As clearly demonstrated in Table I, only the spin orientation of electrons in the 1pu and1pg orbitals is different for each main valence configuration of each state 03 3 þ 3  (c1 R u , A Ru , A Du and the first well of B Ru ). Of the seven states collected in Table 3, only the second well of the B3 R u state possess the multireference characterizations around the equilibrium position whether the spin orientation of electrons is ignored or not. From these configurations, we can instantly find out how the electronic transition occurs from one state to another. As seen in Table I in the Supplemental material, the ground state is essentially represented by one main valence configuration, 0 2rgab 2ruab 3rgab 1puabab 1paa g 3ru , around the equilibrium separation. abab 3r0 ; Other valence configurations, 2rgab 2ruab 3rgab 1paa u 1pg u a b a b a a ba aa b b a b a b a aa b 2rg 2ru 3rg 1pu 1pg 3ru and 2rg 2ru 3rg 1pu 1pag ba 3rbu , þ are indeed negligible. The b1 Rg state can also be basically represented by one main valence configuration, 2rgab 2ruab 3rgab 1pau bab 1pgab 3r0u , around the equilibrium separation. Other valence configurations of this state can be dismissed. Here, we can clearly see that the main valence configuration of the X3 R g state is different

from that of the b1 Rþ g state only by the spin orientation of electrons 1 þ in the 1pg orbitals. The transitions between the X3 R g and the b Rg state have been observed in experiment early forty years ago. The 1 present Te of the b1 Rþ . Albritton et al. [6] g state is 13099.92 cm 1 þ determined the Te of the b Rg state as 13195.31 cm1. The difference between them is 95.39 cm1. Calculations have obtained that the contribution to the Te of the b1 Rþ g state by the corrections higher than quadruple excitations may amount to 247.7 cm1. If the energy corrections contributed by higher than quadruple excitations are included into the present calculations, the difference of the Te between our result and the measurements may be reduced further. When we ignore the spin orientation of electrons in the 1pg and 1pu orbitals, the c1 R u state can be essentially expressed by one main valence electronic configuration, 2r2g 2r2u 3r2g 1p3u 1p3g 3r0u , around the internuclear equilibrium position. From this point, 1  the electronic transition between the X3 R g and the c Ru state can be regarded as arising from the 1pu ? 1pg electron promotion. As shown in Table 3, the present prediction of energy separation 1 1  between the X3 R , g and the c Ru state is equal to 33220.12 cm 1 which deviates from the measurements of 33058.4 cm [38] by 161.72 cm1. As pointed out above, the contribution to the Te of the c1 R u state by the corrections higher than quadruple excitations can reach 72.8 cm1. When such corrections are included, the difference of the Te between the present result and the experimental one may be minished further. Of the seven electronic states, only the B3 R u possesses the double well. From Fig. 1, we can see that the energy of the first well at the internuclear equilibrium position is obviously lower than that

H. Liu et al. / Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy 124 (2014) 216–229

at the dissociation limit. The depth of the first well is 6789.01 cm1. The number of its vibrational states is 10, and the corresponding vibrational levels are 293.71, 894.41, 1514.23, 2154.83, 2817.88, 3505.05, 4218.01, 4958.43, 5727.97 and 6528.31 cm1 for t = 0–9, respectively. According to these, we think that the first well should be stable and is not hard to be observed. In fact, several experiments have detected the spectroscopic transitions of the first well [9,12]. As for the second well of the B3 R u state, the energy at the internuclear equilibrium position is slightly lower than that at the dissociation limit. The depth of the second well is only 742.48 cm1. To our surprise, it possesses the 15 vibrational states, which vibrational levels are 82.83, 220.99, 325.95, 401.92, 453.10, 483.68, 497.88, 499.88, 493.88, 484.09, 474.70, 469.91, 473.92, 490.94 and 525.15 cm1 for t = 0–14, respectively. Thus, the second well of the B3 Ru state can be observed, though it has not been found in experiment till now. We expect that some measurements will be made in the near future so that the spectroscopic properties proposed here can be validated. The detailed calculations tell us that the barrier between the two wells of the B3 R u state is at about 0.2050 nm. The further studies point out that this barrier is formed by the avoided crossing 3  between the B3 R u and the E Ru state. This avoided crossing was also predicted by Wang et al. [55], though they did not report the double well of the B3 R u state. A large number of calculations [16–37] have been made to determine the spectroscopic properties of these states, in particular for the ground state. By careful comparison, we have found that few prior theoretical spectroscopic parameters are closer to the measurements than the present ones. For example, for the X3 R g state, only the xe value calculated by Nooijen and Le Roy [34] in 2006 is closer to the measurements [39] than the present one. In recent several years, at least three groups of calculations [34,36,37] are concentrated to determine the De and Re by the complicated approaches, for which the core-valence correlation and scalar relativistic corrections and the extrapolation to the CBS limit are included. Their De and Re results [34,36,37] are only slightly closer to the measurements [39] than the present ones. For the a1 Dg and b1 Rþ g states, no other theoretical De, Te and Re results are closer to the measurements [38] than the present ones. However, no accurate experimental xe can be available to this day for these two states. Therefore, we can not make any direct comparison between theory and experiment. For the A3 Rþ u state, no other theoretical De are closer to the measurements [38] than the present one, and only the theoretical Te obtained by Müller et al. [30] in 2001 is slightly closer to the measurements [38] than the present result. For the A3 Rþ u state, no other theoretical De and Te are closer to the measurements [38] than the present ones, and only the Re and xe obtained by Takada and Freed [21] in 1984 and Müller et al. [30] in 2001 are closer to the measurements [38] than the present values. 3  As for the c1 R u state and the first well of the B Ru state, the present spectroscopic parameters are still encouraging when compared with other theoretical results [18,20,21,24,30,33]. For the X3 R g state, the present De of 5.2203 eV agrees favorably with the recent measurements of 5.2142 eV [14], and the difference between them is only 0.0061 eV (0.12%). The present Re of 0.12068 nm is in excellent agreement with the measurements of 0.12075 nm [39], and the difference between them is also only 0.00007 nm (0.06%). As for the xe, the present result is 1581.61 cm1, which deviates from the measurements of 1580.19 cm1 [39] only by 1.42 cm1 (0.09%). For the a1 Dg state, the present De of 4.2258 eV is in line with the measurements of 4.2316 eV [38], and the difference between them is only 0.0058 eV (0.14%). the present Re is 0.12147 nm, which deviates from the measurements of 0.12157 nm [38] only by 0.00010 nm (0.94%). The deviation of present Te from the measurements [38] reaches 141.68 cm1 (1.79%), which is slightly large. Further

223

analyses tell us that such deviation may come from the correction contribution higher than quadruple excitations in the present calculations. For the b1 Rþ g state, the present De is 3.6058 eV, which agrees favorably with the recent measurements of 3.5772 eV [38], and the difference between them is 0.0286 eV (0.80%). The present Te is 13099.92 cm1, which deviates from the measurements of 13195.31 cm1 [38] by 95.39 cm1 (0.72%). And the present Re is 0.12258 nm, which deviates from the measurements of 0.12268 nm [38] by only 0.00010 nm (0.08%). Obviously, the comparison is encouraging. When we compare the present spectroscopic parameters of other states collected in Table 3 with their corresponding measurements, fair agreement can still be found. On the whole, these comparisons indicate that the present spectroscopic results obtained by the icMRCI + Q/56 + CV + DK calculations should be high quality. We think that the reasons for achieving such high-quality spectroscopic parameters may be several aspects. Two main reasons are as follows. One is that the corevalence correlation and scalar relativistic corrections are included into the present calculations. The other is that the residual errors behind the basis sets are eliminated by the total-energy extrapolation to the CBS limit. 1 5  1  1 1 þ 15 Pg ; 13 Pu ; C3 Pg ; 23 R g ; d Pg ; 1 Ru ; 2 Ru ; 2 Dg and f Ru states 1 5  1  1 1 þ For the 15 Pg , 13 Pu , C3 Pg , 23 R , d P , 1 R , 2 R g g u u , 2 Dg and f Ru electronic states, no accurate measurements or even no measurements can be available in the literature, though several groups of calculations [18,20,21,24,30,33] have been made in the past several decades. We collect the present spectroscopic parameters determined by the icMRCI + Q/56 + CV + DK calculations in combination with the theoretical results [18,20,21,24,30,33] and measurements [39] in Table 4 for discussion. We evaluate the multireference characterizations of these electronic states. According to Table I in the Supplemental material, near the equilibrium positions, the C3 Pg state and the first well of the d1 Pg state can be essentially represented by only one main 1 valence electronic configuration, and the 15 Pg , 15 R u and 2 Dg states can be basically represented by two main valence configurations, whether the spin orientation of electrons is taken into ac1  1 þ count or not. Only the 13 Pu , 23 R g , 2 Ru and 2 Ru states and the 1 3  second wells of the d Pg and B Ru states possess the obvious multireference characterizations around the internuclear equilibrium positions whether the spin orientation of electrons is included or not. For example, as seen in Table I in the Supplemental material, the 13 Pu state is represented by six main valence electronic configaba b ab ab a abb aa urations, 2rgab 2ruab 3rgab 1paa u 1pg 3ru ; 2rg 2ru 3rg 1pu 1pg a b a b a b a a ba ba a b a b a b a b a bab a a 3ru ; 2rg 2ru 3rg 1pu 1pg 3ru ; 2rg 2ru 3rg 1pu 1pg 3ru ; 2 b a ab ab a aab ab ab rgab 2ruab 3rgab 1puab 1paa g 3ru and 2rg 2ru 3rg 1pu 1pg 3ru . From these configurations, it is easy to find out how the electronic transition occurs from one state to another. Of these nine electronic states, the C3 Pg possesses an obvious barrier, which is at about 0.1650 nm. The depth of this well is about 11974.10 cm1. Thus, this state should be stable and can be observed in experiment, though the energy of this well at the internuclear equilibrium position is greatly higher than that at the dissociation limit. The earlier spectroscopic experiments have observed this electronic state, though its spectroscopic parameters obtained there is not accurate. We expect that the present spectroscopic results can provide some useful guidelines for future measurements of this state. Each of the 15 Pg , 13 Pu and 15 R u states possesses one shallow well. We can not clearly see these wells on the corresponding PECs because these wells are too shallow. Detailed PECs tell us that the energy of their respective well at the equilibrium position is lower than that at the dissociation limit. For the 15 Pg state, the depth of its well is only 712.19 cm1. However, this electronic state possesses eleven vibrational states, which vibrational levels are

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76.17, 212.86, 329.73, 428.04, 509.05, 574.00, 624.17, 660.79, 685.14, 698.46 and 702.02 cm1 for t = 0–10, respectively. For the 13 Pu state, the depth of its well is only 161.31 cm1. To our surprise, it possesses seven vibrational states, which vibrational levels are 26.60, 71.62, 106.34, 131.49, 147.82, 156.05 and 156.92 cm1 for t = 0–6, respectively. To validate this point, we have calculated the vibrational properties of this electronic state with great care from the different theoretical aspects. The conclusion is that the vibrational levels obtained here are slightly different from these when the core-valence correlation correction is excluded. As for the 15 R state, the well depth is only u 41.13 cm1. Even for such a shallow well, the vibrational states are found to be three, and the corresponding vibrational levels are found to be 13.32, 29.15 and 34.73, for t = 0–2, respectively. 3 To some extent, it is difficult to observe the 15 R u and 1 Pu states 3 in experiment, though both the 15 R and 1 P states possess sevu u eral vibrational states. Chiu et al. [56] in 1992 predicted that the 13 Pu was a replusive state. Whereas in the present work, we find that the 13 Pu is a weakly bound state whether the core-valence correlation and sclalar relativistic corrections are included or not. We have validated this point from the different theoretical aspects. 1 Similar to the B3 R u , the d Pg is another state, which possesses the double well. Different from the B3 R u , both the two well of the d1 Pg are very shallow. In detail, their depths are 729.93 and 128.24 cm1 for the first and second wells, respectively. As demonstrated in Fig. 2, the energy of the first well at the internuclear equilibrium pisition is greatly higher than that at the dissociation limit. Calculations have found that the first well possesses only three vibrational states, which vibrational levels are 268.05, 271.40 and 338.17 cm1 for t = 0–2, respectively. Form this point, we think that the first well is not stable and is difficult to be observed in experiment. The energy of the second well at the internuclear equilibrium pisition is slightly lower than that at the dissociation limit. To our surprise, the second well possesses six vibrational states. Calculations have determined that the vibrational levels of the second well are 26.09, 66.64, 94.10, 111.40, 121.41 and 127.06 cm1 for t = 0–5, respectively. Because the second well is very shallow, we need to make great effort if we want to observe it in experiment. The spectroscopic parameters tabulated in Table 4 should be reliable. The reasons may be several aspects. Three main reasons are as follows. The first one is that the present calculations are convergent with respect to the basis set and the present level of theory, as discussed above. The second one is that the two main corrections, core-valence correlation and scalar relativistic corrections, are included into the present calculations. And the last one is

that the residual errors behind the basis sets are eliminated by the extrapolation to the CBS limit. On the one hand, as pointed out above, the PECs of all the states involved here are calculated by the same method; on the other hand, the spectroscopic parameters of all these states are evaluated by the same approach. As dis1 cussed above, the spectroscopic parameters of the X3 R g , a Dg , 03 1 þ 1  3 þ 3  b Rg , c Ru , A Du , A Ru and B Ru states collected in Table 3 are in fair agreement with available measurements [9,12,14,38,39]. In addition, as discussed in the following section, we can still see that the energy splitting in the X3 R g state also agrees well with the available experimental result [7]. According to these, we think that the spectroscopic parameters of the nine electronic states collected in Table 4 can be expected to be reliably predicted ones. These spectroscopic parameters can be good references for the future laboratory research and theoretical work. Spectroscopic parameters of the 35 X bound states With the SO coupling included, the ground state of the O atom splits into three components, 3 P2 , 3 P1 and 3 P0 , and its first and second excited states 1 Dg and 1 Sg do not split. Thus, the first dissociation channel of the O2 molecule splits into six asymptotes. The second dissociation channel splits into three asymptotes. And the third, fourth and fifth dissociation channels do not split, as collected in Table 5. In the present work, the SO coupling effect is accounted for by the state interaction approach with the Breit-Pauli Hamiltonian. The calculations are made at the level of icMRCI + Q theory using the all- electron aug-cc-pCV5Z basis set over the internuclear separations from about 0.1 to 1.0 nm. From that, the PECs of 54 X states generated from the 22 electronic states are determined. It should be pointed out that all the SO coupling calculations are made across the entire PEC of each X state. The all-electron augcc-pCV5Z basis set with the Breit-Pauli Hamiltonian and the allelectron aug-cc-pCV5Z basis set with no Breit-Pauli Hamiltonian are used to calculate the SO coupling contribution. The difference between the two energies generates the SO coupling result. Adding the SO coupling splitting energies to the icMRCI + Q/56 + CV + DK results, we can obtain the final PEC of each X state at the icMRCI + Q/ 56 + CV + DK + SO level of theory. We have calculated the energy separation relative to the lowest dissociation channel for each higher dissociation asymptote. These results are tabulated in Table 5. In addition, Table 5 also collects all the possible X states, which are attributed to the first five dissociation channels of the O2 molecule. For convenience of discussion, we also tabulate the available experimental energy separations [57] between each of

Table 5 Dissociation relationships of possible X states of O2 molecule obtained by the icMRCI + Q/CV + DK + 56 + SO calculations. Atomic states

Possible X states

O(3P2) + O(3P2)

þ þ   4g, 3g, 3u, 2g, 2g, 2u, 1g, 1g, 1u, 1u, 0þ g , 0g , 0g , 0u , 0u

O(3P2) + O(3P1)

Relative energy (cm1) This work

Exp. [57]

0.0

0.0

  þ   3g, 2g, 2g, 1g, 1g, 1g, 0þ g , 0g , 0g , 3u, 2u, 2u, 1u, 1u, 1u, 0u , 0u , 0u

144.84

158.27

O(3P2) + O(3P0)

þ 2 g, 1 g, 0 þ g , 0u , 1u, 2u

232.89

226.98

O(3P1) + O(3P1)

þ  2 g, 1 g, 0 þ g , 0g , 0u , 1u

315.87

316.53

O(3P1) + O(3P0)

 1 g, 0  g , 0u , 1u

380.13

385.24

O(3P0) + O(3P0)

0þ g

465.78

453.95

O( P2) + O( D2)

þ þ   4g, 3g, 3u, 2g, 2g, 2u, 1g, 1g, 1u, 1u, 0þ g , 0g , 0g , 0u , 0u

15866.25

15867.86

O(3P1) + O(1D2)

  þ   3g, 2g, 2g, 1g, 1g, 1g, 0þ g , 0g , 0g , 3u, 2u, 2u, 1u, 1u, 1u, 0u , 0u , 0u

15947.07

16026.13

O(3P0) + O(1D2)

þ 2 g, 1 g, 0 þ g , 0u , 1u, 2u

16002.70

16094.84

O(1D2) + O(1D2)

þ þ   4g, 3g, 3u, 2g, 2g, 2u, 1g, 1g, 1u, 1u, 0þ g , 0g , 0g , 0u , 0u

31674.36

31735.72

O(3P2) + O(1S0)

þ 2 g, 1 g, 0 þ g , 0u , 1u, 2u

33597.65

33792.58

þ 2 g, 1 g, 0 þ g , 0u , 1u, 2u

49463.90

49660.45

3

1

1

1

O( D2) + O( S0)

225

H. Liu et al. / Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy 124 (2014) 216–229

 g;1 ,

15 Pu;0 , X3 R 03

5

15 Pg;1 , 13 Pu;1 , 13 Pu;2 , 15 Pu;1 , 15 R

þ g;2 ,

25 R

þ g;2 ,

5

A Du;3 , 1 Dg;3 and 1 Dg;4 ) are contributed to the lowest dissociation asymptote, Oð3 P2 Þ þ Oð3 P2 Þ. The 17 X states ð15 Pg;0 ; 15 Pg;0þ ; 03 03 5 3 þ 5 þ 5 þ 5 5 15 R u;0 ; 1 Pu;0þ ; A Ru;0 ; 1 Rg;1 ; 2 Rg;1 ; 1 Ru ;1 ; A Du;1 ; A Du;2 ; 1 Pu;1 ; 15 Pu;2 ; 15 Pu;3 ; C3 Pg;2 ; 15 Pg;1 ; 15 Pg;2 and 15 Pg;3 Þ are contributed to the Oð3 P2 Þ þ Oð3 P1 Þ dissociation asymptote. The 6 X states   1 5  13 Pu;0þ ; C3 Pg;0þ ; C3 Pg;1 ; A3 Rþ are contributed u;1 ; a Dg;2 and 1 Ru;2 to the dissociation asymptote, Oð3 P2 Þ þ Oð3 P0 Þ. The 4 X states   5 5 5 correlate to the Oð3 P1 Þ þ Oð3 P1 Þ 25 Rþ g;0þ ; 1 Dg;0þ ; 1 Dg;1 and 1 Dg;2 3

3

1

dissociation asymptote. The 4 X states ð1 Pu;0 ; C Pg;0 ; 1 Pu;1 and d1 Pg;1 Þ correlate to the Oð3 P1 Þ þ Oð3 P0 Þ dissociation asymptote.   3  3  3  correlate to the The 4 X states 23 R g;0þ ; B Ru;0þ ; B Ru;1 and 2 Rg;1

7 9 8 10 12

-150.0

Potential energy /Hartree

the higher dissociation asymptote and the lowest one. As clearly seen in Table 5, the energy separation between each higher dissociation asymptote and the lowest one is in excellent agreement with the measurements [57]. The present 54 X states correlate to the eight dissociation asymp1 þ 5 þ 1  totes. In detail, the 15 X states (X3 R g;0þ , b Rg;0þ , 1 Rg;0þ , c Ru;0 ,

O(1D2) +O(1S0)

11 -150.1

6

O(3P1) +O(1D2)

5

O(3P1) +O(3P1)

4

-150.2

3 -150.3

O(3P2) +O(3P0) O(3P2) + O(3P1) O(3P2) + O(3P2)

2 1

-150.4

0.15

0.30

0.45

0.60

Internuclear separation /nm 1 þ 5 Fig. 4. PECs of twelve X states for X = 0+. 1 – X3 R g;0þ ; 2 – b Rg;0þ ; 3 – 1 Pg;0þ ; 4 – 3 5 þ 5 5 þ 5 13 Pu;0þ ; 5 – B3 R u;0þ ; 6 – C Pg;0þ ; 7 – 1 Rg;0þ ; 8 – 1 Dg;0þ ; 9 – 2 Rg;0þ ; 10 – 1 Pu;0þ ; 1 þ 11 – 23 R g;0þ ; 12 – f Ru;0þ .

Oð3 P1 Þ þ Oð1 D2 Þ dissociation asymptote. The 3 X states (21 R u;0 ,

Oð1 D2 Þ þ Oð1 S0 Þ dissociation asymptote. The PECs of 53 X states are depicted in Figs. 3–8, and their respective dissociation asymptotes are labeled in the same figure. As seen in these figures, the PECs of the 5 5 þ 5 15 Rþ g;0þ ; 1 Dg;0þ ; 2 Rg;0þ and 1 Pu;0þ states in Fig. 4, the PECs of the 5 5 A0 3 Du;1 and A3 Rþ u;1 states and the PECs of the 1 Pu;1 and 1 Pu;1

states in Fig. 5, the PECs of the 15 Pg;1 and 15 Pg;1 states and the 5 þ 5 PECs of the 15 Rþ g;1 ; 2 Rg;1 and 1 Dg;1 states in Fig. 6 are almost over-

-150.01

Potential energy /Hartree

21 Pu;1 and 21 Dg;2 ) are contributed to the Oð1 D2 Þ þ Oð1 D2 Þ dissocia  is contributed to the tion asymptote. And only 1 X state f 1 Rþ u;0þ

9

Potential energy /Hartree

8 -150.12

3 1

O(3P2) + O(3P2)

2

0.1

0.2

0.3

0.4

Internuclear separation /nm 3 Fig. 5. PECs of eight X states for X = ± 1u. 1 – A0 3 Du;1 ; 2 – A3 Rþ u;1 ; 3 – 1 Pu;1 ; 4 – 1 3  5 5 1 15 R ; 5 – 1 P ; 6 – B R ; 7 – 1 P ; 8 – 1 P ; 9 – 2 P . u;1 u;1 u;1 u;1 u;1 u;1

6 7 8

-150.02

5

O(3P1) + O(1D2)

9

4

-150.15

2

O(3P1) + O(3P0) O(3P1) + O(3P1)

3

O(3P2) + O(3P0) O(3P2) + O(3P1) O(3P2) + O(3P2)

-150.28

6

1 -150.41

5

0.1

7 3

-150.18

O(3P1) +O(1D2) O(3P1) + O(3P0) O(3P2) + O(3P0) O(3P2) + O(3P1)

4

6

-150.15

-150.22

Potential energy /Hartree

O(1D2)+ O(1D2)

-150.06

O(1D2) + O(1D2)

78

-150.08

5 5 5 þ lapped together. The PECs of the 15 Rþ g;2 ; 1 Pu;2 ; 1 Dg;2 and 2 Rg;2

states in Fig. 7 and the PECs of the 15 Dg;3 and 15 Pu;3 states in Fig. 8 are also almost superposed. For the sake of length limitation, we do not demonstrate each of them in a separate figure. It should be pointed out that only one X = 4 state, 15 Dg;4 , is involed in the present paper. The PEC of this X state is not depicted in a separate figure also due to length limitation. Similar to Figs. 1 and 2, we show the PECs of these X states only over a small internuclear separation range so that some details can be clearly seen. There are 35 X bound states altogether among the present 54 X states. Using the PECs obtained by the icMRCI + Q/56 + CV + DK + SO calculations, we have evaluated the

5

3

O( P1) +O( P0) O(3P2) + O(3P1) O(3P2) + O(3P2)

4

0.2

0.3

0.4

0.5

Internuclear separation /nm 5 5 Fig. 6. PECs of eight X states for X = ±1g. 1 – X3 R g;1 ; 2 – 1 Pg;1 ; 3 – 1 Pg;1 ; 4 – 5 þ 5 3  C3 Pg;1 ; 5 – d1 Pg;1 ; 6 – 15 Rþ g;1 ; 7 – 2 Rg;1 ; 8 – 1 Dg;1 ; 9 – 2 Rg;1 .

3 -150.24

2 1 0.15

0.30

0.45

0.60

Internuclear separation /nm 3 þ 5 Fig. 3. PECs of eight X states for X = 0. 1 – c1 R u;0 ; 2 – A Ru;0 ; 3 – 1 Pg;0 ; 4 – 3 5 1  13 Pu;0 ; 5 – 15 R u;0 ; 6 – C Pg;0 ; 7 – 1 Pu;0 ; 8 – 2 Ru;0 .

spectroscopic parameters (Te, Re, xe and De) of these X bound states by the method given in Theory and method. The spectroscopic parameters are collected in Tables 6–8 together with available measurements [7]. Meanwhile, the electronic state compositions of each X state near the equilibrium position are also tabulated in Tables 6–8. It should be pointed out that the

226

H. Liu et al. / Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy 124 (2014) 216–229

Potential energy /Hartree

-150.04

-150.15

O(1D2) + O(1D2)

11

7 8

9 6 10

5

O(3P1) + O(3P1) O(3P2) + O(3P0)

4 2

3

O(3P2) + O(3P1) O(3P2) + O(3P2)

-150.26

-150.37 0.08

5 þ 1 5 5 1 15 Rþ g ; 2 Rg ; 1 Pu ; 1 Dg ; 1 Pu and 2 Pu are still the repulsive states with the SO coupling included. Of these repulsive electronic states, the 15 Dg and 25 Rþ g are found to be the inverted ones. We divide the present 35 X bound states into three categories according to their symmetries for convenience of discussion. The first group is the fifteen X states, which are generated from the nine R states, X3 Rg ; b1 Rþg ; c1 Ru ; A3 Rþu ; 15 Ru ; B3 Ru ;23 Rg ; 21 Ru and f 1 Rþu . The second group is the fifteen X states, which are yielded by the four P states, 15 Pg ; 13 Pu ; C3 Pg and d1 Pg . And the last group is the five X states, which come from the three D states, a1 Dg ; A0 3 Du and 21 Dg .

1 0.16

0.24

0.32

0.40

0.48

Internuclear separation /nm Fig. 7. PECs of eleven X states for X = 2. 1 – a1 Dg;2 ; 2 – A0 3 Du;2 ; 3 – 15 Pg;2 ; 4 – 5 þ 5 5 5 þ 13 Pu;2 ; 5 – C3 Pg;2 ; 6 – 15 R u;2 ; 7 – 1 Rg;2 ; 8 – 1 Pu;2 ; 9 – 1 Dg;2 ; 10 – 2 Rg;2 ; 11 – 21 Dg;2 .

Potential energy /Hartree

-150.05

-150.10

34 -150.15

O(3P2) + O(3P1) -150.20

2

O(3P2) + O(3P2)

1 -150.25 0.1

0.2

0.3

0.4

0.5

0.6

Internuclear separation /nm Fig. 8. PECs of four X states for X = 3. 1 – A0 3 Du;3 ; 2 – 15 Pg;3 ; 3 – 15 Dg;3 ; 4 – 15 Pu;3 .

Fifteen X states generated from the nine R electronic states Table 6 tabulates the spectroscopic parameters of the fifteen X 1 þ 1  states, which are generated from the X3 R g ; b Rg ; c Ru ; 3 þ 5  3  3  1  1 þ A Ru ; 1 Ru ; B Ru ; 2 Rg ; 2 Ru and f Ru states. At the same time, Table 6 also collects the electronic state compositions of each X state near the equilibrium position for convenience of discussion. For the 1  1  1 þ b1 Rþ g ; c Ru ; 2 Ru and f Ru states, both K and R equal 0. That is, the four states do not split with the SO coupling included. For the 3 þ 3  3  X3 R g ; A Ru ; B Ru and 2 Rg states, K is 0 and R equals 1. Accordingly, each 3R state produces two X components. And for the 15 R u state, K is 0 and R equals 2. Accordingly, it generates three X components, as collected in Table 6. 3  The electronic state composition of the X3 R g;0þ and X Rg;1 X components is pure around the equilibrium position. As seen in Ta3  bles 3 and 6, the Re of the X3 R g is the same as that of X Rg;0þ and 3  X Rg;1 states. The largest differences of the xe and De between the 1 X3 R and 0.0268 eV, g and the corresponding X states are 0.58 cm respectively. And the energy splitting in the X3 R g state is only 2.19 cm1. Therefore, the effect of SO coupling on the spectroscopic parameters of the X3 R g state is small. In addition, the experimental 3  energy separation between the X3 R g;0þ and X Rg;1 X states was 1 determined to be about 2 cm [7]. The present energy splitting compares well with this result. 1  1  For the b1 Rþ g ; c Ru and 2 Ru states, the electronic state compositions of their respective X states are almost pure around the equilibrium position. Therefore, almost no effect of SO coupling on the spectroscopic parameters is shown. The Te of each X component is

Table 6 Spectroscopic parameters obtained by the icMRCI + Q/56 + CV + DK + SO calculations for the fifteen X states generated from the nine R electronic states of the O2 molecule. Te (cm1)

Re (nm)

xe (cm1)

De (eV)

State compositions near the Re (%)

0.0

0.12068

1581.03

5.1938

X3 R g (100.00)

2.19

0.12068

1581.18

5.1935

X3 R g (100.00)

b1 Rþ g;0þ

2 13102.12

0.12258

1438.65

3.6058

1 þ b1 Rþ g (99.98), 2 Rg (0.02)

c1 R u;0

33222.31

0.15109

804.29

1.1004

c1 R u (100.00)

A Rþ u;0 A3 Rþ u;1 15 R u;0 15 R u;1 15 R u;2

35452.62

0.15151

810.13

0.8278

A3 Rþ u (100.00)

35462.27

0.15151

810.05

0.8318

A3 Rþ u (100.00)

42175.12

0.34177

36.00

0.0071

5 þ l5 R u (99.99), l Ru (0.01)

42183.46

0.34426

38.28

0.0074

l5 R u (100.00)

42203.22

0.31634

79.82

0.0095

l5 R u (100.00)

1st well

51029.29

0.15978

723.60

2.5946

B3 R u (100.00)

2nd well

57346.10

0.23240

186.72

0.0586

B3 R u (100.00)

1st well

51030.92

0.15978

723.63

2.5946

B3 R u (100.00)

2nd well

57347.62

0.23240

186.78

0.0587

B3 R u (100.00)

23 R g;0þ

55079.85

0.21044

468.91

0.3366

23 R g (100.00)

23 R g;1

55080.01

0.21044

468.91

0.3364

23 R g (100.00)

2 R u;0 f 1 Rþ u;0þ

68235.10

0.16095

685.65

0.6583

21 R u (99.99)

81609.67

0.15984

750.85

1.2083

03 f 3 Rþ u (51.33), A Du (48.67)

X R g;0þ X3 R g;1 3

Exp. [7]

3

B3 R u;0þ

B

3

1

R u;0þ

H. Liu et al. / Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy 124 (2014) 216–229

227

Table 7 Spectroscopic parameters obtained by the icMRCI + Q/56 + CV + DK + SO calculations for the fifteen X states generated from the four P electronic states of the O2 molecule. Te (cm1)

Re (nm)

xe (cm1)

De (eV)

State compositions near the Re (%)

1 Pg,1 15Pg,0 15Pg,0+ 15Pg,1 15Pg,2 15Pg,3 13Pu,2 13Pu,1 13Pu,0+ 13Pu,0 C3Pg,2 C3Pg,1 C3Pg,0+ C3Pg,0

41451.52 41521.31 41561.91 41588.25 41611.07 41615.70 41903.63 41913.73 41979.35 41979.53 54684.30 54860.76 54991.84 54992.00

0.20222 0.20838 0.20737 0.20547 0.20479 0.20479 0.28826 0.29309 0.34684 0.34684 0.14774 0.14776 0.14775 0.14775

140.86 368.87 448.60 553.19 671.64 671.64 54.908 80.324 49.670 49.670 697.92 697.82 697.86 697.88

0.0730 0.0719 0.0645 0.0645 0.0758 0.0753 0.0059 0.0108 0.0107 0.0107 1.4686 1.4996 1.4843 1.4845

15Pg (100.00) 15Pg (50.13), 23Pg (49.85) 15Pg (100.00) 15Pg (50.06), 23Pg (49.78) 15Pg (100.00) 15Pg (100.00) 13Pu (93.69), 11Pu (3.84), 15Pu (2.47) 13Pu (100.00) 13Pu (94.37), 15Pu (3.64) 13Pu (94.37), 15Pu (3.64) 13Pg (50.04), 23Pg (49.95) 13Pg (50.04), 23Pg (49.95) 13Pg (50.04), 23Pg (49.95) 13Pg (50.04), 23Pg (49.95)

d1Pg,1 1st well 2nd well

65519.10 41964.21

0.14487 0.32032

829.75 54.267

0.0905 0.0159

d1Pg (100.00) d1Pg (50.04), 21Pg (49.95)

5

Table 8 Spectroscopic parameters obtained by the icMRCI + Q/56 + CV + DK + SO calculations for the five X states generated from the three D electronic states of the O2 molecule.

1

a Dg,2 A0 3Du,3 A0 3Du,2 A0 3Du,1 21Dg,2

Te (cm1)

Re (nm)

xe (cm1)

De (eV)

State compositions near the Re (%)

7778.62 34889.22 34891.64 35036.49 71432.41

0.12147 0.15078 0.15081 0.15081 0.20052

1491.07 826.44 824.31 824.76 404.28

4.2258 0.8937 0.8844 0.8664 0.2646

a1Dg (100.00) A0 3Du (100.00) A0 3Du (100.00) A0 3Du (100.00) 3 þ 21 R g (99.62), l Rg (0.38)

larger than that of corresponding electronic state by about 2 cm1. The reason is that the energy of X ground state at the internuclear equilibrium position is lowered by about 2 cm1. As demonstrated in Tables 4 and 6, there is still almost no effect of SO coupling on the spectroscopic parameters of the f 1 Rþ u state, though the electronic state compositions of the f 1 Rþ u;0þ X component strongly mix with the A0 3 Du state. 1  1  Similar to the b1 Rþ g ; c Ru and 2 Ru , the electronic state compo3  sition of each X component yielded from the A3 Rþ u ; B Ru and 23 R states keeps pure around the internuclear equilibrium g position. As a result, almost no effect of SO coupling on the Re, xe and De can be seen for the three electronic states, and the effect of SO coupling on the Te of each electronic state is also very small. 5  The Te of the 15 R u;2 state is larger than that of the 1 Ru;0 state. 5  Therefore, the 1 Ru is a regular electronic state. Very obvious effect of SO coupling on the spectroscopic parameters can be seen for the 15 R u state, though the electronic state composition of each X component is pure. The main reason may be that the potential well of this electronic state is very shallow. Vrey small correction generated by the SO coupling effect can bring about the obvious change of the PEC shape. As a result, obvious effect of SO coupling on the spectroscopic parameters is demonstrated. In conclusion, we think that the effect of SO coupling on the spectroscopic parameters is very small for all the R electronic states except for the 15 R u. Fifteen X states generated from the four P electronic states Table 7 collects the spectroscopic parameters of the fifteen X states generated from the 15 Pg ; 13 Pu ; C3 Pg and d1 Pg states. At the same time, Table 7 also tabulates the electronic state compositions of each X state near the equilibrium position. For the d1P state, X is only 1. It means that only one X state is generated with the SO coupling included. As seen in Table 7, around the equilibrium position, the electronic state composition of d1 Pg;1 X

component is pure for the first well, but those strongly mix with the 21 Pg state for the second well. By comparison between Tables 4 and 7, we can instantly find out that no effect of SO coupling on the spectroscopic parameters is demonstrated. The state composition of 15 Pg;1 ; 15 Pg;0þ ; 15 Pg;2 and 15 Pg;3 X components keeps pure, but those of the 15 Pg;0 and 15 Pg;1 X components strongly mix with the 23 Pg state. Similar to the 5 15 R u , the well of the 1 Pg state is very shallow. As a result, the shape of each X state is very sensitive to the effect brought about by the SO coupling. For this reason, the effect of SO coupling on the spectroscopic parameters of 15 Pg state is very profound. For example, the deviations of the xe from that of the 15 Pg state reach 16.55, 211.46, 291.19, 395.78, 514.23 and 514.23 cm1, and the deviations of the Re from that of the 15 Pg state amount to 0.02143, 0.01527, 0.01628, 0.01818, 0.01886 and 0.01886 nm for the 15 Pg;1 ; 15 Pg;0 ; 15 Pg;0þ ; 15 Pg;1 ; 15 Pg;2 and 15 Pg;3 X components, respectively. The energy separations between the two neighboring X states from the 15 Pg;1 to the 15 Pg;3 are 69.79, 40.60, 26.34, 22.82 and 4.63 cm1, respectively. As demonstrated in Table 7, Te of the 15 Pg;1 X component is smaller than, and Te of the 15 Pg;3 X component is larger than that of the 15 Pg state. Therefore, the 15 Pg is a regular state with the SO coupling included. The state composition of the 13 Pu;1 component is pure, whereas those of the 13 Pu;2 ; 13 Pu;0þ and 13 Pu;0 components slightly mix with other electronic states. Similar to the 15 R u, the potential well of the 13 Pu state is also shallow. To some extent, for such a shallow well, very small effect of SO coupling may bring about the obvious change of PEC shape. It is the reason why the spectroscopic parameters of each X component are greatly changed when compared with those of the 13 Pu state. As seen in Table 7, Te of the 13 Pu;2 state is the smallest, and Te of the 13 Pu;0 state is the largest among Te of the four X states. As a result, the 13 Pu is an inverted electronic state with the SO coupling included.

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The well of the C3 Pg state is very deep. For this reason, it is impossible that small effect of SO coupling can bring about the obvious change of its PEC shape. That is, the effect of SO coupling on the spectroscopic parameters of the C3 Pg must be greatly smal5 3 ler than that of the 15 R u ; 1 Pg and 1 Pu states. It can be clearly seen in Table 7. In detail, the largest deviations of the Re and xe of these four X states from those of the C3 Pg state are 0.00001 nm and 1.21 cm1, respectively. The energy separation between the C3 Pg;2 and the C3 Pg;1 state amounts to 176.46 cm1. The energy separation between the C3 Pg;1 and C3 Pg;0þ states reaches 131.08 cm1. But the energy separation between the C3 Pg;0þ and the C3 Pg;0 state is only 0.16 cm1, though the state compositions of each X component strongly mix with the 23 Pg state. As seen in Table 7, Te of the C3 Pg;2 is smaller, whereas Te of the C3 Pg;0 is larger than that of the C3 Pg state. Therefore, the C3 Pg is an inverted electronic state with the SO coupling added. In conclusion, (1) only the 13 Pu and C3 Pg are the inverted states among these four P electronic states with the SO coupling included; (2) on the whole, the effect of SO coupling on the spectroscopic parameters of C3 Pg state is small. The effect of SO coupling on the spectroscopic parameters of 15 Pg and 13 Pu states is very obvious. And almost no effect on the spectroscopic parameters of d1 Pg state can be seen.

B3 R u state are found to be the weakly bound ones. And the 13 Pu ; C3 Pg ; A0 3 Du ; 15 Dg and 25 Rþ g are found to be the inverted ones when the SO coupling is included. The SO coupling effect is accounted for by the Breit-Pauli Hamiltonian with the all-electron aug-cc-pCV5Z basis set. To determine more reliable spectroscopic results, the effect of core-valence correlation and scalar relativistic corrections on the PECs is taken into account. Scalar relativistic correction is calculated using the DKH3 approximation at the level of a cc-pV5Z basis set. Core-valence correlation correction is included with the aug-cc-pCV5Z basis set. The spectroscopic parameters have been evaluated for the 16 electronic bound states and 35 X bound states, and compared in detail with those available in the literature. Excellent agreement has been found between the present results and the available experimental ones. On the whole, the effect of SO coupling on the spectroscopic parameters is not obvious except for few electronic states such as the 5 3 15 R u ; 1 Pg and 1 Pu . The analyses demonstrate that the spectroscopic parameters of 16 electronic bound states and 35 X bound states reported in this paper can be expected to be reliably predicted ones, and can be good references for the future laboratory research and theoretical work.

Five X states generated from the three D electronic states Table 8 collects the spectroscopic parameters of five X states generated from the a1 Dg ; A0 3 Du and 21 Dg electronic states. Similar to Tables 6 and 7, in Table 8, we still tabulate the state compositions of each X state around the equilibrium position for convenience of discussion. Obviously, the electronic state compositions of each X component generated from the a1 Dg ; A0 3 Du and 21 Dg states almost keep pure around the internuclear equilibrium position. For the A0 3 Du state, Te of the A0 3 Du;3 is smaller than, whereas Te of the A0 3 Du;1 X component is larger than that of the A0 3 Du state. Thus, the A0 3 Du is an inverted electronic state with the SO coupling included. Because the electronic state composition of each X component is pure near the equilibrium position, the effect of SO coupling on the spectroscopic parameters is not obvious. This can be clearly seen by comparison of the Re and xe results collected in Tables 3 and 8. In detail, for the A0 3 Du;3 , A0 3 Du;2 and A0 3 Du;1 states, the largest deviation of the Re from that of the A0 3 Du state is 0.00003 nm, and the largest deviation of the xe from that of the A0 3 Du state is 1.68 cm1. Obviously, all the deviations are indeed small. Only one X component is generated for each of the a1 Dg and 21 Dg state when the SO coupling is included. As seen in Table 8, the electronic state compositions of the a1 Dg;2 and 21 Dg;2 X components are almost pure. For this reason, very small effect of SO coupling on their spectroscopic parameters is demonstrated. As a conclusion, (1) the A0 3 Du is the inverted electronic state; and (2) the effect of SO coupling on the spectroscopic parameters of the A0 3 Du ; a1 Dg and 21 Dg states is very small.

This work was sponsored by the National Natural Science Foundation of China under Grant Nos. 11274097 and 61177092, the Natural Science Foundation of Education Bureau of Henan Province in China under Grant No. 2010B140013 and the Program for Science & Technology of Henan Province in China under Grant No. 122300410303.

Conclusions In this paper, the PECs of 22 electronic states and 54 X states have been studied for internuclear separations from about 0.1 to 1.0 nm using the CASSCF method, which is followed by the icMRCI 03 approach with the Davidson correction. The X3 R g ; A Du ; 3  3 1 1 þ 1  1 1 þ 5 3 3  A3 Rþ ; B R ; C P ; a D ; b R ; c R ; d P ; f R ; 1 P ; 1 P ; 2 Rg ; g g g g u u u g u u 1  1 15 R ; 2 R and 2 D are found to be bound states, whereas the g u u 5 þ 1 5 5 1 15 Rþ g ; 2 Rg ; 1 Pu ; 1 Dg ; 1 Pu and 2 Pu are found to be repulsive 1 ones. The B3 R u and d Pg are found to possess the double well. The 15 Pg ; 13 Pu ; d1 Pg and 15 R u states and the second well of

Acknowledgments

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