Potential Energy Curves, Spectroscopic Parameters, and SpinâOrbit

Article pubs.acs.org/JPCA

Potential Energy Curves, Spectroscopic Parameters, and Spin−Orbit Coupling: A Theoretical Study on 24 Λ‑S and 54 Ω States of C2+ Cation Deheng Shi,* Xianghong Niu, Jinfeng Sun, and Zunlue Zhu College of Physics and Electronic Engineering, Henan Normal University, Xinxiang 453007, China

ABSTRACT: The potential energy curves (PECs) of 24 Λ-S states and 54 Ω states of the C2+ cation are studied in detail using an ab initio quantum chemical method. All the PEC calculations are made for internuclear separations from 0.09 to 1.11 nm by the complete active space self-consistent field method, which is followed by the internally contracted multireference configuration interaction approach with the Davidson modification (MRCI+Q). All the Λ-S states involved dissociate into the first dissociation limit, C(3Pg) + C+(2Pu), of C2+ cation, of which only the 22Σg− and 24Σg− are repulsive. The spin−orbit (SO) coupling effect is accounted for by the Breit−Pauli Hamiltonian with an aug-cc-pCVTZ basis set. To improve the quality of PECs, core−valence correlation and scalar relativistic corrections are included. Core−valence correlation corrections are taken into account with an aug-cc-pCVTZ basis set. Scalar relativistic correction calculations are done by the third-order Douglas−Kroll Hamiltonian approximation with the cc-pVQZ basis set. All the PECs are extrapolated to the complete basis set limit. The convergence observations of present calculations are made, and the convergent behavior is discussed with respect to the basis set and level of theory. With the PECs obtained by the MRCI+Q/CV+DK+56 calculations, the spectroscopic parameters of 22 Λ-S bound states of C2+ cation are evaluated by fitting the first ten vibrational levels, which are obtained by solving the rovibrational Schrödinger equation using Numerov’s method. In addition, the spectroscopic parameters of 51 Ω bound states generated from these Λ-S bound states are also obtained. The spectroscopic parameters are compared with those reported in the literature. Excellent agreement with available measurements is found. It is expected that the spectroscopic parameters of Λ-S and Ω states reported here are reliable predicted ones.

1. INTRODUCTION

parameters of the cation is sparse, whether in experiment or in theory. Historically, in 1959, Drowart et al.5 was the first to observe the C2+ cation in a mass spectrometric study of carbon vapor. In 1972, Meinel6 made his first spectra observations in a flash discharge. In 1984, O’Keefe et al.7 performed the first observations of electronic transitions in the C2+ by highresolution translational energy spectroscopy. Subsequently in 1987, Forney et al.8 identified the B4Σu− ← X4Σg− electronic transition of the cation in a 5 K neon matrix by laser-induced fluorescence (LIF) spectrum. Rasanen et al.9 carried out the matrix studies by laser vaporization of graphite and observed a

+

The C2 cation is of considerable importance for astrophysical processes, interstellar chemistry, and combustion as well as some plasma reactions.1 On the one hand, the C2+ cation may be involved with the evolution of stellar and interstellar media in view of the abundance of C2 and carbon-containing species in many stars. It has been found that the C2+ cation is an important constituent in extraterrestrial environments. For example, it has been detected by the mass-spectroscopic sampling in comets Halley and Giacobini-Zinner;2 on the other hand, the cation has been incorporated in ion−molecule reactions such as the interstellar gas phase production of highly complex hydrocarbons in interstellar clouds.3,4 As we know, the spectroscopic properties of the cation have extensive applications in these fields. However, the accurate spectroscopic © 2013 American Chemical Society

Received: December 22, 2012 Revised: February 4, 2013 Published: February 7, 2013 2020

dx.doi.org/10.1021/jp312670v | J. Phys. Chem. A 2013, 117, 2020−2034

The Journal of Physical Chemistry A

Article

new spectrum, which was characterized by Te = 19 732 cm−1 and assigned to the B4Σu− electronic state of C2+ cation. Rösslein et al.10 recorded the B4Σu−−X4Σg− laser excitation spectrum of C2+ cation and identified some transitions in their experiment. In 1988, Maier and Rösslein11 studied the B4Σu−− X4Σg− electronic spectrum of C2+ in the gas phase by LIF spectrum. In 1989, Carré et al.12 observed a high-resolution spectrum of the B4Σu−−X4Σg− electronic system of the C2+ by fast-ion beam-laser spectroscopy. In 1990, Celii and Maier13 detected the rotationally resolved B4Σu−−X4Σg− transitions using the stimulated emission pumping technique and characterized the vibrationally excited levels (υ = 4−6) of the ground state. In 1993, Zackrisson and Royen14 obtained the (1,0) band spectrum of the B4Σu−−X4Σg− system by the velocity modulation laser absorption technique. In 1995, Boudjarane et al.15 studied the B4Σu−−X4Σg− system by highresolution LIF spectrum and observed the high-resolution rotational transitions of the (1,1) band. In 1998, Orden and Saykally16 reviewed the experimental and theoretical work about the spectroscopic properties of C2+ cation that far. Recently, in 2004, Tarsitano et al.17 obtained high-resolution spectroscopy of the 22Πu ← X4Σg− forbidden transitions of the C2+ using the velocity modulation technique in conjunction with heterodyne detection. Some spectroscopic and molecular properties have been extracted from these experiments.5−17 Summarizing these measurements, we find that (1) few spectroscopic parameters of Ω states have been reported for the Λ-S states involved in this article except for the 22Πu, though the spin−orbit (SO) coupling effect has been studied by two groups of experiments;14,15 and (2) the spectroscopic parameters, in particular for the Te, Re, and ωe, are accurately determined only for the X4Σg− and B4Σu− Λ-S states in experiment, though a number of spectroscopic measurements5−17 have been done in the past several decades. In theory, the first ab initio calculations on the C2+ cation were made by Verhaegen18 in 1968. Verhaegen18 used the linear combination atomic orbital-molecular orbital-self-consistent field (LCAO-MO-SCF) method and double-ζ plus dpolarization Slater-type orbital (STO) basis set to determine the potential energy curves (PECs) of nine electronic states. In 1971, Lathan et al.19 used the SCF method and STO-3G and 431G basis sets for further calculations. In 1974, Cade et al.20 made the Hartree−Fock (HF) calculations using a very large STO basis set for the study of two 2Π Λ-S states at fixed internuclear separation. In 1975, Ć ársky et al.21 carried out the semiempirical intermediate neglect of differential overlapconfiguration interaction (INDO-CI) calculations and confirmed that the 4Σg− is the ground state of C2+. Some spectroscopic parameters were extracted in these calculations.18−21 As shown in these studies,18−21 obvious deviations of the spectroscopic parameters from the measurements8−11,13 can be seen. From then on, there have been still a number of ab initio calculations22−31 in which the PECs of various Λ-S states were calculated and a number of spectroscopic parameters were evaluated. Summarizing these calculations,18−31 we find that (1) few spectroscopic parameters achieve high quality, though a large number of Λ-S states have been studied; and (2) few SO coupling splittings have been included into the Λ-S states, though the SO coupling effect may yield important influences on the spectroscopic results, in particular for the Te. For high-quality quantum chemistry calculations, core− valence correlation and scalar relativistic corrections must be taken into account to predict accurately the spectroscopic

parameters since the two corrections may yield important effects on the spectroscopic parameters even for small molecules such as C2+. On the one hand, when we summarize the spectroscopic properties in the literature, we find that no calculations have included the core−valence correlation and scalar relativistic effects; on the other hand, we only find one group of theoretical work29 in which the SO coupling calculations were involved by using the Breit−Pauli Hamiltonian. Therefore, to improve the quality of spectroscopic parameters of these Λ-S states of C2+ cation, more theoretical work should be done. The aim of the present work is to extend the spectroscopic knowledge of C2+ cation. On the one hand, the effect on the PECs by the SO coupling will be introduced into the calculations since few SO coupling calculations have been performed for Ω states,29 though the SO coupling effect has been observed by two groups of experiments;14,15 on the other hand, extensive ab initio calculations of PECs will be made in which the core−valence correlation and scalar relativistic corrections are included so that the spectroscopic parameters of the cation can be determined as accurately as possible. In the next section, we will briefly describe the theory and method used in this work. In section 3, the PECs of 22 bound (X4Σg−, 12Πu, 14Πg, 12Δg, 12Σg−, 12Σg+, 22Πu, B4Σu−, 12Πg, 12Σu+, 14Δu, 14Σu+, 12Σu−, 12Δu, 22Πg, 24Πg, 14Πu, 22Σu−, 24Σu−, 14Σg+, 24Πu, and 14Δg) and two repulsive (22Σg− and 24Σg−) ΛS states are studied for internuclear separations from 0.09 to 1.10 nm. The PECs of 54 Ω states generated from all the Λ-S states are studied for the first time over a large internuclear separation. The PEC calculations are carried out using the complete active space self-consistent field (CASSCF) method, which is followed by the internally contracted multireference configuration interaction (MRCI) approach32,33 with the Davidson modification (MRCI+Q).34,35 The effect on the PECs by the core−valence correlation and scalar relativistic corrections is included. The SO coupling effect is accounted for. The spectroscopic parameters are calculated for the 22 Λ-S bound states and the 51 Ω bound states. The spectroscopic parameters are compared with those available in the literature. Concluding remarks are given in section 4.

2. THEORY AND METHOD In this article, we will study these electronic states, which dissociate into the first dissociation limit, C(3Pg) + C+(2Pu), of C2+ cation. Foremost, we must find out all these electronic states. As we know, the C2+ cation belongs to D∞h molecular symmetry. The ground states of C atom and C+ cation are 3Pg and 2Pu, respectively. The representations 3Pg and 2Pu of the atomic group are resolved into those of D∞h 3 − Σg

+

⊕ 3 Π g and 2Σ u ⊕ 2Π u

(1)

and their direct product −

+

(3Σ g ⊕ 3 Π g) ⊗ (2Σ u ⊕ 2Π u) =

+ 2,4 − Σ u (2), 2,4 Σ u , 2,4 Π u(2), 2,4Δu

(2)

Because of the exchange symmetry of the excited energy in the C2+ cation, eq 2 has the following additional electronic states: + 2,4 − Σ g (2), 2,4 Σ g , 2,4 Π g(2), 2,4Δg

2021

(3)



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Ag, two doublet B3u, two doublet B2u, three doublet B1g, two doublet B1u, two doublet B2g, two doublet B3g, three doublet Au, two quartet Ag, three quartet B1g, two quartet B1u, two quartet B2g, two quartet B3g, two quartet B3u, two quartet B2u, and three quartet Au states, respectively. In the CASSCF and subsequent internally contracted MRCI and MRCI+Q 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 valence electrons in the 2s2p shell of C are placed in the active space, which consists of full valence space. That is, seven electrons of C2+ cation are distributed into eight orbitals (2−3σg, 2−3σu, 1πu, and 1πg). The energy ordering of the eight valence MOs is 2σg1πu2σu3σg1πg3σu. As a result, this active space is referred to as CAS (7,8). The rest of the four electrons in C2+ are put into the closed-shell orbitals, including one ag and one b1u symmetry MOs, which correspond to the 1σg and 1σu MOs in the cation. In addition, the two electrons in the 1s shell of each C are used as core electrons for the present core−valence correlation correction calculations. That is, when we make the core−valence correlation calculations,40,41 all the electrons in the C2+ cation are correlated. When we perform the frozen-core calculations, the two electrons in the 1s orbital of each C are frozen, which correspond to the 1σg and 1σu MOs in the C2+. When we use the ten MOs (3ag, 1b3u, 1b2u, 3b1u, 1b2g, and 1b3g) to make the PEC calculations, we find that the each PEC is smooth over the present internuclear separation range, and each PEC is convergent. To determine accurately the PECs of all the Λ-S and Ω states, the point spacing intervals used here are 0.02 nm for each state, except near the equilibrium internuclear position where the point spacing is 0.002 nm. Here, the smaller step is adopted around the equilibrium separation of each state so that the properties of each PEC can be displayed more clearly. To improve the quality of spectroscopic parameters, we include the core−valence correlation and scalar relativistic corrections into the PEC calculations. Core−valence correlation correction is included with an aug-cc-pCVTZ (ACVTZ) basis set.40,41 Scalar relativistic effect is taken into account using the third-order Douglas−Kroll Hamiltonian (DKH3) approximation42−44 with the cc-pVQZ-DK basis set.45 That is, The ACVTZ basis set with the core−valence correlations and the ACVTZ basis set within the frozen-core approximation are used for the core−valence correlation contribution calculations. The difference between the two energies yields the core− valence correlation contribution (denoted as CV). The ccpVQZ-DK basis set with the DKH3 approximation and the ccpVQZ basis without the DKH3 approximation are employed for the scalar relativistic correction calculations. The difference between the two energies produces the scalar relativistic contribution (denoted as DK). We use two successive correlation-consistent basis sets, augcc-pV5Z (AV5Z) and aug-cc-pV6Z (AV6Z),46 for the totalenergy extrapolation scheme (denoted as 56). For two successive correlation-consistent basis sets, the total-energy extrapolation formula is written as47

Then, we obtain all the electronic states, which dissociate into the C(3Pg) + C+(2Pu) dissociation limit of C2+ cation + − + 2,4 − Σ u (2), 2,4 Σ u , 2,4 Π u(2), 2,4Δu , 2,4 Σ g (2), 2,4 Σ g

,

2,4

Π g(2),

2,4

Δg

(4)

As seen in eq 4, there are 24 electronic states together. That is, these 24 Λ-S states of C2+ cation dissociate into its first dissociation limit. Now, we briefly describe the method used for the SO coupling calculations. For a given molecular system, the whole Hamiltonian can be expressed as el so (5) Ĥ = Ĥ + Ĥ el ̂ where H is the spin-free Hamiltonian, which can be expressed as the nonrelativistic Schrödinger operator plus Douglas−Kroll Hamiltonian. Ĥ so is the SO part of Breit−Pauli Hamiltonian. The Breit−Pauli SO operator Ĥ so can be expressed as36−38 so Ĥ =

⎡ Z e2 1 ⎢ ∑ ∑ α 3 Iiα̂ ·Sî 2 2⎢ 2μ c ⎣ i α riα −

∑∑ i

j

⎤ e2 ̂ ̂ ⎥ ̂ I · ( S + 2 S ) ij i j ⎥⎦ rij 3

(6)

In eq 2, the first double sum is known as the one-electron and the second as the two-electron operators, respectively. I ̂ and Ŝ are space and spin angular momentum operators Iiα̂ = (ri − R α) × Pi

(7)

Iiĵ = (ri − rj) × Pi

(8)

where Roman and Greek subscripts refer to electrons and nuclei, respectively. The other symbols have their usual meanings. For example, Zα and e are nuclear and electronic charges, respectively. Pi is the momentum operator. riα is the distance between the ith electron and the αth nuclei, and rij is the distance between the ith and jth electron. Berning et al.36 thought that the Breit−Pauli operator can be well approximated by an effective one-electron Fock operator. Using the effective one-electron Fock operator, Berning et al.36 have incorporated the most important two-electron contributions of SO operator and presented an efficient method for the calculations of Breit−Pauli SO matrix elements for the internally contracted MRCI wave function, which has been implemented in the MOLPRO 2010.1 program package.39 Because of a limitation of the MOLPRO program package,39 we must treat D∞h molecules in the D2h subgroup. As noted above, the C2+ cation belongs to D∞h molecular symmetry. Therefore, to make the present calculations in the MOLPRO 2010.1 program package, we must substitute the D∞h symmetry with the D2h point group, which can be made by orienting the C2+ along the Z axis. There are eight irreducible representations, Ag, B3u, B2u, B1g, B1u, B2g, B3g, and Au, in the D2h point group. The corresponding symmetry operations for both D∞h and D2h are Σg+ → Ag, Σg− → B1g, Πg → B2g + B3g, Δg → Ag + B1g, Σu+ → B1u, Σu− → Au, Πu → B2u + B3u, and Δu → Au + B1u, respectively. The orbitals are optimized using the CASSCF approach. The state-averaged technique is employed in the CASSCF calculations only for the 36 excited states, for which the same weight factor is used. The 36 states are two doublet

ΔEtotal, ∞ =

ΔEtotal, n + 1(n + 1)3 − ΔEtotal, nn (n + 1)3 − n3

3

(9)

Here, ΔEtotal,∞ is the total energy extrapolated to the complete basis set (CBS) limit, and ΔEtotal,n and ΔEtotal n+1 are the total 2022



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Table 1. Spectroscopic Parameters of 12Σg+ and B4Σu− Λ-S States: Convergence Observations of Basis Set and Level of Theory MRCI 12Σg+ AVTZ AVQZ AV5Z AV6Z 56 B4Σu− AVTZ AVQZ AV5Z AV6Z 56

MRCI+Q

Te (cm−1)

Re (nm)

ωe (cm−1)

ωexe (cm−1)

Te (cm−1)

Re (nm)

ωe (cm−1)

ωexe (cm−1)

13811.54 13714.31 13669.76 13642.10 13612.48

0.14591 0.14523 0.14507 0.14501 0.14494

1106.46 1120.46 1124.25 1126.07 1128.54

13.4521 12.7421 12.5923 12.5304 12.4453

13666.90 13563.09 13517.00 13486.72 13456.21

0.14599 0.14529 0.14512 0.14507 0.14499

1100.23 1115.80 1119.93 1121.88 1124.53

13.9974 14.0265 14.0064 13.9846 13.9510

20006.43 19841.17 19781.91 19757.76 19736.70

0.13595 0.13542 0.13530 0.13526 0.13521

1480.86 1492.41 1496.30 1495.85 1497.64

11.8057 11.8580 11.9026 11.2212 11.1937

20131.75 19964.73 19905.03 19879.79 19858.28

0.13592 0.13539 0.13526 0.13522 0.13517

1477.88 1489.29 1493.11 1490.12 1491.94

12.5976 12.4359 12.4238 10.7118 10.6806

the basis set. As shown in Table 1, it has been verified by the extrapolation to the CBS limit. In fact, when we extrapolate the AVnZ basis set to the aug-cc-pV∞Z (AV∞Z), the Te of the 12Σg+ and B4Σu− Λ-S states are lowered by 29.62 and 21.06 cm−1 for the internally contracted MRCI calculations and lowered by 30.51 and 21.51 cm−1 at the internally contracted MRCI+Q level of theory, respectively. In conclusion, we think that the Te results of the 12Σg+ and B4Σu− states are convergent with respect to the basis set at the present level of theory. For the 12Σg+ Λ-S state, as demonstrated in Table 1, the differences of the Te between the internally contracted MRCI and MRCI+Q approach are 144.64, 151.22, 152.76, 155.38, and 156.27 cm−1 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 the Te of 12Σg+ Λ-S state by less than 157 cm−1. If we use the error bar estimation approach employed in refs 49 and 50, the contribution to the Te by the corrections higher than quadruple excitations can be estimated by less than 78.5 cm−1. For the B4Σu− Λ-S state, according to Table 1, the differences of the Te between the internally contracted MRCI and MRCI +Q method are 125.32, 123.56, 123.12, 122.03, and 121.58 cm−1 at the AVTZ, AVQZ, AV5Z, and AV6Z basis sets and the CBS limit, respectively. These remind us that the contribution to the Te by the triplet and quadruple excitations is less than 124 cm−1. Similar to the 12Σg+, by simple calculations, we find that the Te error of the B4Σu− Λ-S state brought about by the corrections higher than quadruple excitations may be smaller than 62 cm−1. These results demonstrate that the present MRCI+Q approach has excellent convergent behavior for these two Λ-S states, 12Σg+ and B4Σu−. In addition to the 12Σg+ and B4Σu− Λ-S states, we have investigated the convergent behavior with respect to the basis set and level of theory for the other 20 Λ-S bound states involved in this article, and a similar conclusion can still be gained. In conclusion, we think that the present calculations are convergent with respect to the basis set and level of theory. Consequently, we use the PECs obtained by the MRCI+Q approach and extrapolation to the CBS limit for the following calculations. As pointed out in section 2, the combination of ground-state C(3Pg) atom and ground-state C+(2Pu) cation can yield 24 Λ-S states. They are 12 doublet and 12 quartet Λ-S states. By the careful PEC calculations over a wide internuclear separation, we have found that 22 of them are bound, whereas two of them (24Σg− and 22Σg−) are repulsive states. To demonstrate clearly

energies obtained by the basis sets, aug-cc-pVnZ (AVnZ) and aug-cc-pV(n+1)Z [AV(n+1)Z], respectively. From the PECs of 22 Λ-S and 51 Ω bound states obtained here, the spectroscopic parameters, including the dissociation energy De, excitation energy term Te referred to the ground state, equilibrium internuclear separation Re, harmonic frequency ωe, first- and second-order anharmonic constants ωexe and ωeye, rotational constant Be, and vibration coupling constant αe, are evaluated. To determine accurately the spectroscopic parameters, the PECs of all the Λ-S and Ω states of the C2+ are fitted to an analytical form by cubic splines so that the corresponding rovibrational Schrödinger equation can be conveniently solved. In this article, we solve the rovibrational Schrödinger equation by Numerov’s method.48 That is, the rovibrational constants are first determined in a direct forward manner from the analytic potential by solving the rovibrational Schrö d inger equation, and then the spectroscopic parameters are evaluated by fitting the first ten vibrational levels obtained here.

3. RESULTS AND DISCUSSION Now, we first discuss the convergent behavior of present calculations with respect to the basis set and level of theory. For the sake of length limitation and for convenient discussion, here we only collect the Te, Re, ωe and ωexe results of 12Σg+ and B4Σu− Λ-S states obtained by the internally contracted MRCI and MRCI+Q approaches in combination with the correlationconsistent basis sets from the AVTZ to AV6Z and the extrapolation to the CBS limit by the AV5Z and AV6Z basis sets in Table 1. It should be pointed out that the two Λ-S states selected for the present discussion are optimal. As demonstrated in Table 1, we can see that the Te values converge toward the CBS limit when systematically increasing the quality of correlation-consistent basis sets. For the 12Σg+ ΛS state, the basis sets from the AVTZ to AVQZ, from the AVQZ to AV5Z, and from the AV5Z to AV6Z lower the Te by 97.23, 44.55, and 27.66 cm−1 at the internally contracted MRCI and lower the Te by 103.81, 46.09, and 30.28 cm−1 at the internally contracted MRCI+Q level of theory, respectively. For the B4Σu− Λ-S state, the basis sets from the AVTZ to AVQZ, from the AVQZ to AV5Z ,and from the AV5Z to AV6Z lower the Te by 165.26, 59.26, and 24.15 cm−1 at the internally contracted MRCI and lower the Te by 167.02, 59.70, and 25.24 cm−1 at the internally contracted MRCI+Q level of theory, respectively. These results suggest that the Te of the two Λ-S states may converge to about within 30 cm−1 with respect to 2023



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3.95 cm−1 for the 12Σg− and 12Δg Λ-S states, respectively. According to these, we can say that the effect on the Te by the two corrections is sometimes obvious, and the contribution to the Te by the core−valence correlation correction is more important than that by the scalar relativistic correction. (2) For the Re, with only the core−valence correlation correction added, the Re values are shortened by 0.00034, 0.00034, 0.00042, and 0.00045 nm for the X4Σg−, B4Σu−, 12Σg−, and 12Δg Λ-S states, respectively. With only the scalar relativistic correction included, the shifts of the Re are within 0.00004 nm for the four Λ-S states. Therefore, the effect on the Re by the core−valence correlation correction is pronounced, whereas the effect on the Re by the scalar relativistic correction is very small. By the way, the core−valence correlation correction makes the Re shorter for all the Λ-S states involved here. The reason may be that the core−valence correlation correction lowers the total energy of each Λ-S state, and the lowered total energy can make the C2+ cation become slightly steadier. As a result, the core−valence correlation correction more or less shortens the equilibrium separation of each Λ-S state. (3) For the ωe, its values are raised by 12.77, 14.95, 15.19, and 16.00 cm−1 with only the core−valence correlation correction included, and shifts of the ωe are within 0.91 cm−1 for the four Λ-S states collected in Table 3 with only the scalar relativistic correction added. Here, the contribution to the ωe by the core−valence correlation correction is greatly larger than that by the scalar relativistic correction. When we analyze the Te, Re, and ωe results of other Λ-S states involved in this article, similar conclusion can still be gained. In conclusion, we can say that the effect on the spectroscopic parameters by the core− valence correlation correction is more pronounced than that by the scalar relativistic correction. Meanwhile, their effect on the spectroscopic parameters, in particular for the Te, can not be dismissed for high-quality quantum chemistry calculations. As shown in Table 3, the Re value of the X4Σg− Λ-S state is longer than that of the B4Σu−. We think that such situation may be caused by the different electronic distributions of the two ΛS states. In other words, the electronic distributions of the B4Σu− may make the structure of this Λ-S state more compact when compared with those of the X4Σg− Λ-S state. As a result, the Re of the B4Σu− is shorter than that of the X4Σg− state. 3.1. Spectroscopic Parameters of 22 Λ-S Bound States. The 22 Λ-S bound states are divided into three categories for convenient discussion. The first is the X4Σg− and B4Σu−, for which accurate spectroscopic measurements are available in the literature.10,11,13 The second is the 12Πu, 14Πg, 12Δg, 12Σg−, 12Σg+, 22Πu, 12Πg, 14Δu, 14Σu+, 12Σu−, 22Πg, 22Σu−, 24Σu−, 14Σg+, 24Πu, and 14Δg, for which only one potential well exists on each PEC and no accurate experimental spectroscopic parameters exist elsewhere. The last is the 12Σu+, 12Δu, 24Πg, and 14Πu, for which two potential wells can be found on each PEC. 3.1.1. X4Σg− and B4Σu− Λ-S States. With these PECs determined by the MRCI+Q/CV+DK+56 calculations, we have evaluated the spectroscopic parameters of X4Σg− and B4Σu− Λ-S states by the theoretical method described in section 2. These spectroscopic parameters are collected in Table 4 together with available theoretical ones18,22−28,30,31 for comparison. To our knowledge, at least five groups of spectroscopic measurements exist in the literature8−11,13 for the X4Σg− and B4Σu− Λ-S states. Of these experiments, two groups9,10 are performed in neon matrix. Therefore, we only gather the measurements reported in refs 10, 11, and 13 in Table 4 for comparison.

the dissociation relationships of these states, we depict the PECs obtained by the MRCI+Q/CV+DK+56 calculations for internuclear separations from 0.09 to 1.10 nm in Figures 1 and 2. In addition, the important electronic configurations as obtained from the MRCI wave functions around their equilibrium positions are collected in Table 2.

Figure 1. PECs of 12 quartet Λ-S states of the C2+ cation: 1, X4Σg−; 2, 14Πg; 3, B4Σu−; 4, 14Δu; 5, 14Σu+; 6, 24Πg; 7, 14Πu; 8, 24Σu−; 9, 14Σg+; 10, 24Πu; 11, 14Δg; 12, 24Σg−.

Figure 2. PECs of 12 doublet Λ-S states of the C2+ cation: 1, 12Πu; 2, 12Δg; 3, 12Σg−; 4, 12Σg+; 5, 22Πu; 6, 12Πg; 7, 12Σu+; 8, 12Δu; 9, 12Σu−; 10, 22Πg; 11, 22Σu−; 12, 22Σg−.

Now, we discuss the effect on the spectroscopic parameters by the core−valence correlation and scalar relativistic corrections. To simplify the discussion, here we only take the X4Σg−, B4Σu−, 12Σg−, and 12Δg Λ-S states as examples at the level of MRCI+Q theory. It should be pointed out that the four Λ-S states selected for discussion are optimal. For convenient discussion, Table 3 collects the De, Te, Re, and ωe results when the core−valence correlation and/or scalar relativistic corrections are included. Because of length limitation, here we only study in detail the effect on the Te, Re, and ωe by the two corrections. As shown in Table 3, (1) for the Te, with only the core−valence correlation correction added, its values are lowered by 161.09 cm−1 for the B4Σu− and raised by 70.23 and 50.48 cm−1 for the 12Σg− and 12Δg Λ-S states, respectively. With only the scalar relativistic correction included, its values are raised by 70.89 cm−1 for the B4Σu− and lowered by 7.90 and 2024



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Table 2. Important Electronic Configurations of 22 Λ-S Bound States of C2+ Cation near the Internuclear Equilibrium Separationsa Λ-S state X4Σg− 12Πu 14Πg 12Δg 12Σg− 12Σg+ 22Πu B4Σu− 12Πg

14Δu 14Σu+ 12Σu−

22Πg

22Σu− 24Σu− 14Σg+ 24Πu

14Δg 12Σu+ (1st well) (2st well) 12Δu (1st well) (2st well) 24Πg (1st well) (2st well) 14Πu (1st well)

(2st well) a

dominant configurations ...2σαβg3σαg1πααu2σαβu3σ0u1π0g (0.859); ...2σαβg3σαβg1παu2σαu3σ0u1παg (0.068); ...2σαβg3σ0g1παβαu2σαβu3σ0u1π0g (0.709); ...2σαβg3σαg1πααu2σβu3σ0u1πβg (0.024); ...2σαβg3σαg1παβαu2σαu3σ0u1π0g (0.880); ...2σαβg3σαβg1πααu2σ0u3σ0u1παg (0.015); ...2σαβg3σαg1παβu2σαβu3σ0u1π0g (0.864); ...2σαβg3σαg1π0u2σαβu3σ0u1παβg (0.024). ...2σαβg3σαg1παβu2σαβu3σ0u1π0g (0.876); ...2σαβg3σαg1π0u2σαβu3σ0u1παβg (0.016); ...2σαβg3σαg1παβu2σαβu3σ0u1π0g (0.822); ...2σαβg3σαg1π0u2σαβu3σ0u1παβg (0.043); ...2σαβg3σαβg1παu2σαβu3σ0u1π0g (0.629); ...2σαβg3σαβg1παβαu2σ0u3σ0u1π0g (0.055); ...2σαβg3σαβg1πααu2σαu3σ0u1π0g (0.772); ...2σαβg3σαg1παβαu2σ0u3σ0u1παg (0.015). ...2σαβg3σαg1παββu2σαu3σ0u1π0g (0.675); ...2σαβg3σ0g1παβu2σαβu3σ0u1παg (0.049); ...2σ2g3σ0g1πααu2σ2u3σ0u1πβg (0.016). ...2σαβg3σαg1παu2σαβu3σ0u1παg (0.913); ...2σαβg3σαg1παu2σαβu3σ0u1παg (0.912); ...2σαβg3σαg1παu2σαβu3σ0u1παg (0.188); ...2σαβg3σαβg1παβu2σαu3σ0u1π0g (0.052); ...2σαβg3σ0g1παβu2σαβu3σαu1π0g (0.019). ...2σαβg3σ0g1παβu2σαβu3σ0u1παg (0.333); ...2σαβg3σ0g1πααu2σαβu3σ0u1πβg (0.187); ...2σαβg3σαg1παβαu2σαu3σ0u1π0g (0.032). ...2σαβg3σαβg1παβu2σαu3σ0u1π0g (0.324); ...2σαβg3σαg1πβu2σαβu3σ0u1παg (0.135); ...2σαβg3σαg1παu2σαβu3σ0u1παg (0.179); ...2σαβg3σ0g1πααu2σαβu3σαu1π0g (0.373); ...2σαβg3σ0g1παβαu2σαu3σ0u1παg (0.781); ...2σαβg3σ0g1παu2σαu3σ0u1παβαg (0.056); ...2σαβg3σαg1παβu2σαu3σ0u1παg (0.025); ...2σαβg3σαg1πααu2σαu3σ0u1πβg (0.124); ...2σαg3σαg1παβαu2σαβu3σ0u1π0g (0.065); ...2σαβg3σαg1π0u2σαβu3σ0u1πααg (0.384); ...2σαg3σαβg1πααu2σαβu3σ0u1π0g (0.090); ...2σαβg3σ0g1παβαβu2σαu3σ0u1π0g (0.762); ...2σαβg3σαg1παβαu2σ0u3σ0u1πβg (0.052); ...2σαβg3σαg1πβu2σαβu3σ0u1παg (0.632); ...2σαβg3σαβg1παβu2σαu3σ0u1π0g (0.052); ...2σαβg3σαβg1παβu2σαu3σ0u1π0g (0.774); ...2σαβg3σαβg1π0u2σαu3σ0u1παβg (0.012). ...2σαβg3σαg1πβu2σαβu3σ0u1παg (0.625); ...2σαβg3σαβg1παβu2σαu3σ0u1π0g (0.049); ...2σαβg3σαg1παβαu2σαu3σ0u1π0g (0.166); ...2σαβg3σαβg1πααu2σ0u3σ0u1παg (0.044). ...2σαβg3σαg1παu2σαβu3σαu1π0g (0.897); ...2σαβg3σαg1παβu2σαu3σ0u1παg (0.345); ...2σαβg3σαg1πααu2σαu3σ0u1πβg (0.134); ...2σαg3σαg1παβαu2σαβu3σ0u1π0g (0.034); ...2σαβg3σαg1π0u2σαβu3σαu1παg (0.932);

...2σαβg3σ0g1παβαu2σαu3σ0u1παg (0.013); ...2σαβg3σαg1πααu2σ0u3σ0u1παβg (0.016). ...2σαβg3σαβg1παβαu2σαu3σ0u1π0g (0.126); ...2σαβg3σ0g1παu2σαβu3σ0u1παβg (0.024). ...2σαβg3σ0g1πααu2σαβu3σ0u1παg (0.029); ...2σαβg3σαg1παu2σαu3σ0u1παβg (0.027). ...2σαβg3σαβg1παu2σβu3σ0u1παg (0.027); ...2σαβg3σαβg1παu2σβu3σ0u1παg (0.025); ...2σαβg3σαg1πααu2σβu3σβu1π0g (0.012). ...2σαβg3σαg1π4u2σ0u3σ0u1π0g (0.039); ...2σαβg3σαβg1παu2σβu3σ0u1παg (0.023). ...2σαβg3σ0g1παβαu2σαβu3σ0u1π0g (0.155); ...2σαβg3σ0g1παu2σαβu3σ0u1παβg (0.023). ...2σαβg3σαg1παu2σαβu3σ0u1παg (0.155); ...2σαβg3σαg1παβαu2σβu3σ0u1π0g (0.117); ...2σ2g3σ0g1παβu2σαβu3σ0u1παg (0.027); ...2σαβg3σαg1πβu2σαu3σαu1παg (0.015); ...2σαβg3σαg1πβu2σαu3σαu1παg (0.015). ...2σαβg3σαg1πβu2σαβu3σ0u1παg (0.632); ...2σαβg3σαg1παu2σβu3σβu1παg (0.022); ...2σαβg3σ0g1παβu2σαβu3σ0u1παg (0.202); ...2σαβg3σ0g1π0u2σαβu3σ0u1παβαg (0.081); ...2σαβg3σαg1παu2σαβu3σ0u1παg (0.396); ...2σαβg3σαg1παβαu2σ0u3σ0u1παg (0.013). ...2σαβg3σαβg1πααu2σαu3σ0u1π0g (0.238); ...2σαβg3σαβg1π0u2σαu3σ0u1πααg (0.099); ...2σαβg3σαβg1παu2σαu3σ0u1παg (0.038); ...2σαβg3σαg1πααu2σ0u3σ0u1παβg (0.032); ...2σαβg3σαg1παβu2σαu3σ0u1παg (0.544); ...2σαβg3σ0g1παu2σαβu3σ0u1πααg (0.114); ...2σαβg3σαg1π0u2σαu3σ0u1παβαg (0.037). ...2σαβg3σαβg1παu2σαu3σ0u1παg (0.387); ...2σαβg3σαg1πααu2σβu3σ0u1π0g (0.032). ...2σαβg3σαβg1παβu2σαu3σ0u1π0g (0.063); ...2σαβg3σ0g1παβu2σαu3σ0u1παβg (0.032). ...2σαβg3σαg1παu2σαβu3σ0u1παg (0.188); ...2σαβg3σαg1παu2σβu3σβu1παg (0.022). ...2σαβg3σαg1πβu2σαβu3σ0u1παg (0.132); ...2σαβg3σαg1παu2σαβu3σ0u1παg (0.194); ...2σαβg3σαg1παu2σβu3σβu1παg (0.022). ...2σαβg3σ0g1πααu2σαβu3σ0u1παg (0.642); ...2σαβg3σ0g1πααu2σαβu3σ0u1παg (0.024). ...2σαβg3σαg1παβu2σαu3σ0u1παg (0.303); ...2σαβg3σ0g1παu2σαβu3σ0u1πααg (0.077); ...2σαβg3σαg1π0u2σαu3σ0u1παβαg (0.023). ...2σαg3σαg1παu2σβu3σαu1παβg (0.016).

Values in parentheses are the coefficients squared of CSF associated with the electronic configuration.

insignificant. The B4 Σ u − Λ-S state can be essentially represented by two main configurations, (core)2σαβg3σαβg1πααu 2σαu3σ0u1π0g (0.772) and (core)2σαβg3σαg1παu2σαβu3σ0u1παg (0.155). The configuration (core)2σαβg3σαg1παβαu2σ0u3σ0u1παg

As seen in Table 2, near the internuclear equilibrium separation, the ground state can be basically represented by the configuration, (core)2σαβg3σαg1πααu2σαβu3σ0u1π0g (0.859). The other three configurations collected in Table 2 are indeed 2025



Article

Table 3. Effect on the Spectroscopic Parameters of X4Σg−, B4Σu−, 12Σg−, and 12Δg Λ-S States by the Core−Valence Correlation and/or Scalar Relativistic Corrections De (eV)

Te (cm−1)

Re (nm)

ωe (cm−1)

De (eV)

−

Te (cm−1)

Re (nm)

ωe (cm−1)

19879.79 19718.70 19950.68 19789.15

0.13522 0.13488 0.13518 0.13484

1490.12 1505.07 1490.70 1505.84

9927.28 9977.76 9923.33 9973.59

0.14385 0.14340 0.14385 0.14340

1215.51 1231.51 1214.60 1230.64

Σu−

X Σg MRCI+Q CV DK CV+DK 12Σg− MRCI+Q CV DK CV+DK 4

4

5.6324 5.7017 5.6278 5.6970

0.0 0.0 0.0 0.0

0.14103 0.14069 0.14102 0.14069

1341.88 1354.65 1341.23 1354.02

4.1222 4.1838 4.1185 4.1801

12151.65 12221.88 12143.75 12213.98

0.14302 0.14260 0.14301 0.14260

1245.33 1260.52 1244.83 1260.09

B 3.1668 3.2562 3.1534 3.2428 12Δg 4.3978 4.4618 4.3936 4.4576

Table 4. Comparison of Spectroscopic Parameters of X4Σg− and B4Σu− Λ-S States Obtained by the MRCI+Q/CV+DK+56 Calculations with Experimental and Other Theoretical Results De (eV)

Te (cm−1)

Re (nm)

ωe (cm−1)

ωexe (cm−1)

102ωeye (cm−1)

Be (cm−1)

102αe (cm−1)

X Σg exptl10 exptl11 exptl13 calcd18 calcd22 calcd23

5.7080

0.14064 0.14083a 0.14034

41.728

1.42050 1.417b 1.4266 1.4258

1.5058 1.76 1.762

5.53 5.62 ∼6.00 5.64 5.66 3.46 5.50 6.0936

1355.39 1351.5 1351.21 1351.7 1560 1360 1390 1335 1339 1330 1341 1369g 1368h

11.396 12.1 12.06 12.18

calcd24

0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 19767.42 19652.9 0.13475 19652.2 16695.70 20163.85 18631.40 19921.89 20069 19959

1.438 1.410 1.403 1.407

1.8c 1.7d 1.55e 1.65f

1.3935

1.728 1.5772

12.69

1.54400 1.437b 1.711 1.5465

1.704

10 11

1.557 1.537

1.5c 1.2d

4

−

calcd25 calcd26 calcd27 calcd28 calcd30 calcd31 B4Σu− exptl10 exptl11 exptl13 calcd18 calcd22 calcd23 calcd26 a

3.2565 19652.2

3.16 3.19

0.142 0.14290 0.1398 0.1411 0.1415 0.1413 0.14129 0.1402 0.1403 0.1304i 0.14129 0.14142 0.14658k 0.13479 0.1352a 1508.1 0.1340 0.13547 0.1343 0.1352 0.1343 0.1343

1393j 1358.17 1507.64 1507.0 12.69 1508.1 1560 1470 1579 1507 1553g 1547h

12 12 12.7 14.0

16 22

12.15 11.073 12.3

14.653 1.5474

r0 result. bB0 result. cMCSCF. dMCSCF-SCEP. eCAS. fCCI. gCCSD(T)/cc-pVTZ. hCCSD(T)/cc-pVQZ. iSDCI. jBasis set III. kVCI.

measurements10,11,13 than the present one. For the B4Σu− Λ-S state, no other theoretical Te18,22,23,28 is closer to the measurements10,11,13 than the present one, and no other theoretical Re18,22,23,28 is superior to the present one in quality when compared with the measurements.11 As shown in Table 4, the deviation of the present Re from the experimental one10 is slightly large. We should notice that the value reported by Rösslein et al.10 is r0, not Re. As for the ωe, only the result of 1507 cm−1 calculated by Rosmus et al.23 is equivalent to the present one in quality when compared with the measurements.10,11,13 As seen in Table 4, on the whole, the present spectroscopic results are in excellent agreement with the measurements.10,11,13 For example, for the X4Σg− Λ-S state, the deviations of the present Re, ωe, and Be values from those

(0.015) is unimportant. From this point, we think that the multiconfiguration character of the B4Σu− Λ-S state is more obvious than that of the ground state. The electronic transition X4Σg−−B4Σu− can be viewed as arising from the 2σu → 3σg electron promotion. As shown in Table 4, the present theoretical prediction for the energy separation of 19767.42 cm−1 between the X4Σg− and B4Σu− Λ-S states is in fair agreement with the measurements of 19652.9 and 19652.2 cm−1.10,11,13 A number of calculations18,22−28,30,31 have reported the spectroscopic parameters of the C2+ in the past several decades. As demonstrated in Table 4, for the X4Σg− Λ-S state, only the Re result calculated by Watts and Bartlett26 is equivalent to the present one in quality when compared with the measurements.10,11 No other theoretical ωe18,22−28,30,31 is closer to the 2026



Article

Table 5. Comparison of Spectroscopic Parameters of 16 Λ-S states Obtained by the MRCI+Q/CV+DK+56 Calculations with Other Theoretical Spectroscopic Results De (eV)

Te (cm−1)

Re (nm)

ωe (cm−1)

ωexe (cm−1)

102ωeye (cm−1)

Be (cm−1)

102αe (cm−1)

1 Πu calcd18 calcd22 calcd23

5.1466

1605.74 [1280]a 1630 1717 1624 1610 1590 1643f 1632g 1651h

110.46

1.65217

1.8793

4.73 4.92 5.08 5.12 4.81

0.13039 0.1312 0.13229 0.1283 0.1303 0.1311 0.1309 0.1300 0.1300 0.13124 0.13600i 0.12542 [0.1205]a 0.12700 0.1251 0.1259 0.1254 0.1253 0.13124i 0.14333 0.1423 0.14394 0.1438 0.1445 0.15082i 0.14254 0.1428 0.14341 0.1446 0.1434 0.14923i 0.14450 0.1437 0.14605 0.1449 0.1455 0.15420i 0.15079 0.15029 0.12621 [0.1222]a 0.12753 0.1200 0.1252 0.19115

12.924

calcd24

4590.09 5887.84 6775.05 10565.86 6371.78 4758.67 5465.88 4535 4399 5565.22

17 14 9.94 7.43

92 95

1.705 1.654 1.638 1.641

2.0b 1.9c 2.21d 2.31e

1890.08 [1400]a 1840 1954 1874 1924f 1911g

13.321

3.1389

1.78610

1.6580

1.795 1.773

1.5b 1.6c

1233.04 1560 1230 1224 1193

14.606

1.36765

1.9786

1.359 1.346

1.9b 2.0c

1262.07 1530 1265 1196 1228

15.927

1.38301

2.0016

1.343 1.366

1.6b 1.8c

1138.86 [1500]a 1090 1167 1103

14.212

1.34589

2.0203

1.338 1.327

2.1b 2.0c

1395.42 1360 1777.63 [1300]a 1660 2227 1859 558.301 0.19474 548.834

11.413

190.89

1.23572

1.9390

20.447

117.29

1.76374

2.2549

40 19 5.1553

10.277

1.949 1.792 0.76884

3.8b 2.1c 1.0659

5.0148

7.6797

0.75806

1.0479

535.768

2.9007

45.328

0.78866

1.7562

455.750 1595.81 771.523 1188.83 608.740 629.830

3.7796 24.617 3.7807 16.528 12.722 36.220

12.717 567.11 118.18 182.91 893.43 9.3665

0.57285 1.20096 0.78064 1.37376 1.13994 1.06622

0.6208 28.761 0.4557 2.2199 0.9387 5.1269

2

calcd26 calcd28 calcd31 14Πg calcd18 calcd22 calcd23 calcd26 calcd31 12Δg calcd18 calcd22 calcd23 calcd31 12Σg− calcd18 calcd22 calcd23 calcd31 12Σg+ calcd18 calcd22 calcd23 calcd31 22Πu calcd22 12Πg calcd18 calcd22 calcd23 14Δu calcd22 14Σu+ calcd22 12Σu− calcd22 22Πg 22Σu− 24Σu− 14Σg+ 24Πu 14Δg

a

4.5448

4.48 4.52 4.4724

4.1954

4.0532

3.8283

9597.85 5565.22 11049.79 5323.26 9678.65 9392 9206 9943.30 4194.08 11775.69 12743.55 11049.79 12179.74 10727.17 14034.04 15163.22 13066.18 13505.81 13630.76 15001.90 16292.39 14195.35

2.2074

15366.74 16453.70 22618.62 [6613.74]a 24761.21 35246.41 26051.70 28425.26

2.1539

28859.38

1.9659

30174.91

1.1987 0.8791 0.6411 0.6512 0.9320 0.3246

36448.37 49951.99 55137.07 59650.79 61898.65 64455.31

2.9294

0.19251 0.19738 0.18754 0.19103 0.22147 0.15990 0.18967 0.14298 0.15608 0.16280

11 13

10.037

13 14 50.267

11 12 8.6371

17 11

Values in brackets are not accurate. bMCSCF. cMCSCF-SCEP. dCAS. eCCI. fCCSD(T)/cc-pVTZ. gCCSD(T)/cc-pVQZ. hBasis set III. iVCI. 2027



Article

2σαβg3σ0g1παβu2σαβu3σ0u1παg (0.333), (core)2σ α β g 3σ 0 g 1π α β u 2σ α β u 3σ 0 u 1π α g (0.202), and (core) 2σαβg3σ0g1πααu2σαβu3σ0u1πβg (0.187) configurations. From these configurations, we can instantly find out how the electronic transition happens from one Λ-S state to another. Among the 16 Λ-S states, 11 have no potential barriers. They are 12Πu, 14Πg, 12Δg, 12Σg−, 12Σg+, 22Πu, 12Πg, 14Δu, 14Σu+, 12Σu−, and 22Πg, respectively. As shown in Figures 1 and 2, we see that all the 11 Λ-S states have deep potential wells. No accurate spectroscopic parameters can be found in the literature so far, though several groups of theoretical studies18,22−24,26,28,31 have reported their spectroscopic properties. From Table 5, one important feature can be observed. That is, the 14Δu, 14Σu+, 12Σu−, 22Πg, and 24Σu− Λ-S states have very large Re and very small ωe when compared with those of the ground state. As seen in Table 5, only one group of theoretical Re can be found in the literature,22 which are obviously larger than the present ones. From Table 4, we can instantly find out that the ground-state Re calculated by Petrongolo et al.22 is longer than the present one by 0.00226 nm. For these five Λ-S states (14Δu, 14Σu+, 12Σu−, 22Πg, and 24Σu−), no other theoretical ωe can be found in the literature. According to the shape of PECs shown in Figures 1 and 2, a barrier occurs for each of these five Λ-S states, 22Σu−, 24Σu−, 14Σg+, 24Πu, and 14Δg. In detail, (1) for the 22Σu− Λ-S state, the depth of potential well is 7139.73 cm−1. The barrier on the PEC is at 0.18444 nm, and its height is 11021.44 cm−1. (2) For the 24Σu− Λ-S state, the barrier on the PEC appears around the internuclear separation, 0.23974 nm. The depth of potential well and the height of barrier are 5290.22 and 14344.38 cm−1, respectively. As seen in Figure 2, the energies at the potential well are higher than those at the dissociation limit for the two Λ-S states, 22Σu− and 24Σu−. (3) For the 14Σg+ and 24Πu Λ-S states, the potential barriers come out near 0.19199 and 0.21431 nm. The depths of potential wells are 6254.15 and 7554.98 cm−1, and the heights of the barriers are 19719.57 and 22170.64 cm−1, respectively. Similar to 22Σu− and 24Σu−, the energies at the potential wells of the 14Σg+ and 24Πu Λ-S states are also higher than those at the dissociation limit. (4) For the 14Δg Λ-S state, the depth of potential well is only 1095.18 cm−1. The barrier on the PEC is at 0.19515 nm, and its height equals 19549.71 cm−1. By calculations, we find that the 14Δg ΛS state only possesses five vibrational levels. Therefore, this Λ-S state should be unstable and would be hard to observe. In addition, the total number of vibrational levels is 31, 14, 14, and 33 for the 22Σu−, 24Σu−, 14Σg+, and 24Πu Λ-S states, respectively. As a result, the 14Δg Λ-S state is hard to observe when compared with the 22Σu−, 24Σu−, 14Σg+, and 24Πu Λ-S states. The reasons to form these barriers have several aspects. One of the most important aspects is the avoided crossings between the two Λ-S states with the same symmetry and spin. As for the barriers of the present five Λ-S states, we think that they may mainly come from the avoided crossings of Λ-S states with the same symmetry and spin since these barriers are not far from the equilibrium positions. Of course, to verify this point, further calculations on the PECs with the same symmetry and spin are required. For the reason, we will study these PECs in the near future, which dissociate into the second and third dissociation limits. 3.1.3. Four Λ-S States with Two Potential Wells. With the PECs obtained by the MRCI+Q/CV+DK+56 calculations, we have also determined the spectroscopic parameters of the

reported by Maier and Rösslein11 are only 0.0003 nm (0.21%), 4.18 cm−1 (0.31%) ,and 0.0061 cm−1 (0.43%), respectively. For the B4Σu− Λ-S state, the deviations of the present Te, Re, ωe, and Be results from the measurements11 are only 115.22 cm−1 (0.59%), 0.00004 nm (0.03%), 0.46 cm−1 (0.03%), and 0.0034 cm−1 (0.22%), respectively. These comparisons indicate that the spectroscopic parameters determined by the MRCI+Q/CV +DK+56 calculations are of high quality. The reasons why such high-quality spectroscopic parameters can be achieved may be several aspects. Two main reasons are as follows. One is that the core−valence 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 totalenergy extrapolation to the CBS limit. Nevertheless, the residual error in the present calculations still exists. For example, the difference of Te between the present result and the measurements11 for the B4Σu− Λ-S state is up to 115.22 cm−1. The reasons may be several aspects. Two main reasons are as follows. One is the contribution to the Te by the corrections higher than quadruple excitations in the MRCI approach. As discussed above, such contribution can be up to 62 cm−1 for the B4Σu− Λ-S state. The other is the truncation of basis sets. The truncation error tends to zero when the basis set tends to infinity. Here, we eliminate the truncation error by extrapolating the potential energy to the CBS limit. The total energy obtained by the MRCI calculations consists of two parts. One is the reference energy. The other is the electron-correlation energy. The convergence behavior of reference energy is somewhat faster than that of electroncorrelation energy. Therefore, to obtain highly accurate results, the reference energy should be extrapolated separately. However, here we use the same exponent in both cases and extrapolate the total energies to the CBS limit due to lack of a well-founded way to evaluate the separate exponents for the reference- and electron-correlation energies up to date. The same exponent in reference- and correlation-energy extrapolation schemes somewhat slows down the convergence of the reference energy. We think that the same exponent in both cases may be responsible for some residual errors of the Te for the B4Σu− Λ-S state. 3.1.2. Sixteen Λ-S States with One Potential Well. With the PECs obtained by the MRCI+Q/CV+DK+56 calculations, we have also evaluated the spectroscopic parameters of 12Πu, 14Πg, 12Δg, 12Σg−, 12Σg+, 22Πu, 12Πg, 14Δu, 14Σu+, 12Σu−, 22Πg, 22Σu−, 24Σu−, 14Σg+, 24Πu, and 14Δg Λ-S states by the theoretical method described in section 2, for which only one potential well exists on each PEC and no accurate measurements can be found in the literature so far. These spectroscopic results are gathered in Table 5 together with available theoretical ones18,22−24,26,28,31 for comparison. For the 16 Λ-S states collected in Table 5, the multiconfiguration character can be clearly seen in Table 2 for several Λ-S states, in particular for the 12Πu, 22Πu, 12Πg, 22Πg, 22Σu−, 24Σu−, 22Πu, and 14Δg. For example, the 22Σu− Λ-S state can be essentially characterized by three configurations, (core)2σ α β g 3σ α g 1π α u 2σ α β u 3σ 0 u 1π α g (0.396), (core) 2σ α β g 3σ α β g 1π α β u 2σ α u 3σ 0 u 1π 0 g (0.324), and (core)2σαβg3σαg1πβu2σαβu3σ0u1παg (0.135). The 24Σu− Λ-S state can be basically represented by three configurations, (core)2σαβg3σ0g1πααu 2σαβu3σαu1π0g (0.373), (core)2σαβg3σαβg1πααu2σαu3σ0u1π0g (0.238), and (core)2σαβg3σαg1παu 2σαβu3σ0u1παg (0.179), and the 22Πg Λ-S state can also be essentially represented by the (core) 2028



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Table 6. Comparison of Spectroscopic Parameters of Four Λ-S States Obtained by the MRCI+Q/CV+DK+56 Calculations with Other Theoretical Spectroscopic Results 1 Σu 1st well 2nd well calcd18 calcd22 calcd23 2

12Δu 1st well 2nd well calcd22 24Πg 1st well 2nd well calcd22 14Πu 1st well 2nd well calcd30 a

De (eV)

Te (cm−1)

Re (nm)

ωe (cm−1)

ωexe (cm−1)

102ωeye (cm−1)

Be (cm−1)

102αe (cm−1)

1.7925 1.0862

26797.85 37449.61 [19115.33]a 30810.37 35327.07 29358.57

0.12168 0.19587 0.1219 0.12277 0.1210 0.1214

1932.56 458.886 1780 2330 2009 1977

16.761 4.0389

139.03 97.966

1.89716 0.76115

2.0711 8.6687

8 16

1.918 1.906

1.1b 2.1c

0.4851 1.1411

34221.58 36993.11

0.13574 0.19349 0.13706

1394.22 448.389

23.081 2.1603

499.31 419.52

1.52487 0.96313

2.8351 0.5960

1.9548 0.3055

40271.62 43687.74

0.15397 0.28686 0.15346

1244.47 474.807

5.5387 161.01

69.030 2958.8

1.18481 0.34219

0.5580 5.4329

1.1230 0.0763 7.6274

51479.75 46659.65

0.14795 0.40489 0.12637

1078.65 62.5002 1914.41

18.450 3.6366 12.6772

769.80 11.086

1.28146 0.17135 1.7452

2.0284 1.0342 1.259

+

Values in brackets are not accurate. bMCSCF. cMCSCF-SCEP.

12Σu+, 12Δu, 24Πg, and 14Πu Λ-S states by the theoretical method given in Section 2, for which two potential wells can be found on each PEC. To our knowledge, for the four Λ-S states, no available spectroscopic measurements can be found in the literature thus far, and only four groups of theoretical work18,22,23,30 have reported their spectroscopic results. The present spectroscopic parameters are collected in Table 6 together with these available theoretical ones18,22,23,30 for comparison. As clearly seen in Figures 1 and 2, the energy at the potential well of the 14Πu Λ-S state is higher than, whereas the energies at the potential wells of 12Σu+, 12Δu, and 24Πg Λ-S states are lower than those at the dissociation limit. From this point, the 14Πu Λ-S state is harder to observe than the 12Σu+, 12Δu, and 24Πg Λ-S states. The present calculations have found that each potential well of these four Λ-S states possesses at least more than ten vibrational levels. For example, even for the 14Πu Λ-S state for which the total number of the vibrational states is the fewest, 13 vibrational levels can be found for each potential well. The past theoretical investigations18,22,23,30 did not report that the four Λ-S states have double potential wells on their PECs. The main reason may be that the past PEC calculations are performed only within a relative narrow internuclear separation range. In addition, each of these four Λ-S states only has one barrier, though each of them possesses the double wells. Similar to the 22Σu−, 24Σu−, 14Σg+, 24Πu, and 14Δg Λ-S states, we think that these barriers might mainly originate from the avoided crossings of Λ-S states with the same symmetry and spin since these barriers are not far from the equilibrium separations. In the process of the avoided crossings, the double wells are produced. No accurate experimental spectroscopic parameters have been reported for the 22 Λ-S bound states except for the X4Σg− and B4Σu−. On the one hand, the PECs of all the 22 Λ-S bound states are calculated by the same approach; on the other hand, all the spectroscopic parameters given in Tables 4−6 are determined by the same method. Because the spectroscopic parameters of the X4Σg− and B4Σu− Λ-S states are in fair

agreement with the measurements,10,11,13 we think, with reason, that spectroscopic parameters collected in Tables 5 and 6 are expected to be reliable predicted ones. We believe that the spectroscopic results collected in Tables 5 and 6 can be good references for future laboratory research. Up to now, all the calculations and measurements about the C2+ cation have been focused on the Λ-S states, which dissociate into the first dissociation limit. As shown in Table 5, the Te results of some Λ-S states (such as 24Πu and 14Δg) are very high. Therefore, we cannot exclude that there are possibilities of the Rydberg states lower in energy than these higher-lying Λ-S states. To clarify this point, in the near future, we will perform the calculations on the PECs of these Λ-S states, which dissociate into the second and third dissociation limits of the cation. 3.2. Spectroscopic Parameters of 51 Ω Bound States. The SO coupling effect is introduced into the present calculations by the Breit−Pauli SO operation. From that, the PECs of 54 Ω states generated from the 24 Λ-S states are calculated. For convenient discussion, we collect the dissociation relationships for the possible Ω states in Table 7. In addition, we also collect the energy separations relative to the lowest dissociation channel of Ω states calculated at the MRCI Table 7. Dissociation Relationships of Possible Ω States of C2+ Cation Obtained by the MRCI+Q/CV+DK+56+SO Calculations relative energy (cm−1) Ω states

atomic state (C + C+) 3

P0 P1 3 P2 3 P0 3 P1 3 P2 3

2029

+ + + + + +

2

P1/2 P1/2 2 P1/2 2 P3/2 2 P3/2 2 P3/2 2

1/2(2) 1/2(4), 1/2(4), 1/2(2), 1/2(6), 1/2(8),

3/2(2) 3/2(4), 5/2(2) 3/2(2) 3/2(4), 5/2(2) 3/2(6), 5/2(4), 7/2(2)

calcd

exptl51

0.0 16.65 46.64 68.31 83.55 109.56

0.0 16.4 43.5 64.0 80.4 107.5



Article

+Q/CV+DK+56 level of theory and their corresponding experimental ones51 in Table 7 for comparison. It should be pointed out that the all-electron ACVTZ basis set is used to calculate the SO coupling splitting in this article. As shown in Table 7, the relative energies calculated here agree favorably with the measurements.51 For example, the present energy difference of 16.65 cm−1 between the C(3P1) + C+(2P1/2) and the lowest dissociation channel is in excellent agreement with the measurements of 16.4 cm−1.51 The present energy difference of 46.64 cm−1 between the C(3P2) + C+(2P1/2) and the lowest dissociation channel is very close to the measurements of 43.5 cm−1,51 and the present energy difference of 68.31 cm−1 between the C(3P0) + C+(2P3/2) and the lowest dissociation channel is also in line with the measurements of 64.0 cm−1.51 The SO coupling effect causes different Λ-S states that have common Ω states to recombine. The dissociation limits of the present 54 Ω states generated from the 24 Λ-S states are the C(3P0) + C+(2P1/2) for the X4Σg,1/2− and 22Σu,1/2− Ω states, the C(3P1) + C+(2P1/2) for the X4Σg,3/2−, 12Δg,3/2, 12Σu,1/2−, 24Σu,1/2−, 12Σg,1/2−, and 22Σg,1/2− Ω states, the C(3P2) + C+(2P1/2) for the 14Πu,5/2, 12Δg,5/2, 24Σu,3/2−, 24Σg,3/2−, 14Δg,3/2, 14Πu,3/2, 14Πu,1/2, 14Δg,1/2, 24Σg,1/2−, and B4Σu,1/2− Ω states, the C(3P0) + C+(2P3/2) for the B4Σu,3/2−, 22Πg,3/2, 22Πg,1/2, and 14Πu,−1/2 Ω states, the C(3P1) + C+(2P3/2) for the 24Πu,5/2, 14Δg,5/2, 22Πu,3/2, 12Δu,3/2, 12Πg,3/2, 24Πg,3/2, 12Σg,1/2+, 22Πu,1/2, 12Σ+u,1/2, 12Πg,1/2, 24Πg,1/2, and 24Πg,−1/2 Ω states, and the C(3P2) + C+(2P3/2) for the 14Δu,7/2, 14Δg,7/2, 12Δu,5/2, 14Δu,5/2, 14Πg,5/2, 24Πg,5/2, 14Σ+g,3/2, 12Πu,3/2, 24Πu,3/2, 14Δu,3/2, 14Σ+u,3/2, 14Πg,3/2, 14Σ+g,1/2, 12Πu,1/2, 24Πu,1/2, 24Πu,−1/2, 14Δu,1/2, 14Σ+u,1/2, 14Πg,1/2, and 14Πg,−1/2 Ω states, respectively. That is, when the SO coupling effect is included in the calculations, on the one hand, the lowest dissociation limit, C(3Pg) + C+(2Pu), of C2+ cation splits into six asymptotes [C(3P0) + C+(2P1/2), C(3P1) + C+(2P1/2), C(3P2) + C+(2P1/2), C(3P0) + C+(2P3/2), C(3P1) + C+(2P3/2), and C(3P2) + C+(2P3/2)], of which the C(3P0) + C+(2P1/2) is the lowest one; on the other hand, these asymptotes correlate with 54 Ω states with Ω = 1/2 (−1/2), 3/2, 5/2, and 7/2, respectively. The PECs of 54 Ω states are depicted, and their corresponding dissociation limits are labeled in Figures 3−7, respectively. The spectroscopic parameters (De, Te, Re, ωe, and ωexe) of 51 Ω bound states generated from the 22 Λ-S bound states are collected in Table 8. At the same time, the dominant Λ-S state compositions of each Ω state around the internuclear equilibrium separation are also tabulated in Table 8 for discussion. For convenient discussion, we divide the 51 Ω bound states into three groups. One is the 15 Ω states generated from the ten Σ Λ-S bound states. One is the 24 Ω states generated from the eight Π Λ-S bound states, and one is the 12 Ω states generated from the four Δ Λ-S bound states. Now, we first discuss the effect on the spectroscopic parameters of ten Σ Λ-S bound states by the SO coupling. No experimental spectroscopic parameters can be found in the literature for any Ω states generated from these Σ Λ-S states. As we know, with the SO coupling included, each 2Σ Λ-S state does not split, whereas each 4Σ Λ-S state splits into two Ω components, 4Σ3/2 and 4Σ1/2. For the present 15 Ω states generated from the five 2Σ and five 4Σ Λ-S states, as seen in Table 8, the dominant Λ-S state composition remains almost pure at the internuclear equilibrium separations for each Ω state except for the X4Σg,1/2−, X4Σg,3/2−, 12Σu,1/2−, 14Σg,1/2+ and

Figure 3. PECs of 22 Ω states of C2+ cation: 1, X4Σg,1/2−; 2, 12Πu,1/2; 3, 14Πg,1/2; 4, 12Σg, 1/2−; 5, 12Σ+g,1/2; 6, 22Πu,1/2; 7, B4Σu,1/2−; 8, 12Πg,1/2; 9, 12Σ+u,1/2; 10, 14Δu,1/2; 11, 14Σ+u,1/2; 12, 12Σu,1/2−; 13, 22Πg,1/2; 14, 24Πg,1/2; 15, 22Σu,1/2−; 16, 14Πu,1/2; 17, 24Σu,1/2−; 18, 14Σg,1/2+; 19, 24Πu,1/2; 20, 14Δg,1/2; 21, 24Σg,1/2−; 22, 22Σg,1/2−.

Figure 4. PECs of 18 Ω states of C2+ cation: 1, X4Σg,3/2−; 2, 12Πu,3/2; 3, 14Πg,3/2; 4, 12Δg,3/2; 5, 22Πu,3/2; 6, B4Σu,3/2−; 7, 12Πg,3/2; 8, 14Δu,3/2; 9, 14Σu,3/2+; 10, 12Δu,3/2; 11, 22Πg,3/2; 12, 24Πg,3/2; 13, 14Πu,3/2; 14, 24Σu,3/2−; 15, 14Σg,3/2+; 16, 24Πu,3/2; 17, 14Δg,3/2; 18, 24Σg,3/2−.

14Σg,3/2+ Ω states as well as the 12Σu,1/2+ Ω state at the second potential well. On the whole, the effect on the spectroscopic parameters of these ten Σ Λ-S states by the SO coupling is small. Because of length limitation, here we only take four Λ-S states as examples for discussion. For the X4Σg− Λ-S state, with the SO coupling included, the X 4 Σ g,1/2 − and X 4 Σ g,3/2 − compositions mix with the 14Πg Λ-S state. As a result, the energy splitting of ground state is 1.50 cm−1, which is larger than those of other Σ Λ-S states with almost pure dominant ΛS state composition. As seen in Table 8, the ωe separation between the X4Σg,1/2− and X4Σg,3/2− Ω states is only 0.15 cm−1, and no change can be observed about their Re. For the 14Σg,1/2+ and 14Σg,3/2+ Ω states, their dominant compositions around the equilibrium internuclear separations mix with the 14Δg Λ-S state. The energy separation between the two Ω states is 1.56 2030



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mixing with other Λ-S states. For example, for the 14Σu,1/2+ and 14Σu,3/2+ Ω states, the difference of the Te between the two Ω states is 0.66 cm−1. No effect on the Re by the SO coupling can be observed for this state. For the B4Σu− Λ-S state, the separation of the Te between the two Ω states is only 0.66 cm−1. On the whole, the energy splitting is tiny for these Σ Λ-S states. Meanwhile, the effect on their Re and ωe is also small. Then we investigate the influences on the spectroscopic parameters of eight Π Λ-S states by the SO coupling, here including four 2Π and four 4Π Λ-S states. With the SO coupling included, each 2Π Λ-S state splits into two Ω states, 4Π3/2 and 4 Π1/2; and each 4Π Λ-S state splits into four Ω components, 4 Π5/2, 4Π3/2, 4Π1/2, and 4Π−1/2. The dominant Λ-S state compositions of all the 2Π and 4Π Ω states are pure except for the 12Πu,3/2 and 12Πu,1/2 near the equilibrium internuclear separation. As tabulated in Table 8, the dominant compositions of 12Πu,3/2 and 12Πu,1/2 Ω states mix with the 14Σu− Λ-S state. As a result, the two Ω states are separated by 31.61 cm−1, though the separation of their ωe is only 0.16 cm−1 and no effect on their Re can be observed. As seen in Table 8, the Te of the 22Πg,3/2 is lower, and the Te of the 22Πg,1/2 is higher than that of the 22Πg Λ-S state by 16.73 and 18.16 cm−1, respectively. Therefore, the two Ω states are separated by 34.89 cm−1, though their composition is pure around the equilibrium internuclear separation. In experiment,14,15 only the SO coupling constant of the 22Πu Λ-S state has been reported. For the present calculations, the two Ω states generated from the 22Πu are separated by 20.41 cm−1, which compares well with the measurements of 16.6414 and 15.4 cm−1.15 In addition, the two Ω states are separated by 0.00002 nm and 3.43 cm−1 for the Re and ωe, respectively. Similar to the 22Πu, with the SO coupling included for the 12Πg Λ-S state, the energy splitting is about 7.68 cm−1, whereas the separation of their ωe is only 0.20 cm−1 and no effect on the Re can be found. In conclusion, Table 8 demonstrates that the effect on the energy splitting of Π Λ-S states by the SO coupling is more pronounced than that of Σ Λ-S states, whereas the effect on the Re and ωe by the SO coupling is not obvious, whether for the Σ or Π Λ-S states. As for the four Δ Λ-S states, 12 Ω states can be yielded with the SO coupling included. For the 14Δg,1/2, 14Δg,3/2, 14Δg,5/2 and 14Δg,7/2 Ω states, their dominant compositions strongly mix with the 14Σg− Λ-S state in the internuclear equilibrium region. The energy separations between the 14Δg,1/2 and 14Δg,3/2 Ω states and between the 14Δg,5/2 and 14Δg,7/2 Ω states are only 5.04 and 5.05 cm−1, respectively, but the 14Δg,3/2 and 14Δg,5/2 Ω states are separated by 158.46 cm−1. The Re and ωe of the 14Δg,1/2 and 14Δg,3/2 Ω states change marginally, whereas those of the 14Δg,5/2 and 14Δg,7/2 shift dramatically. As for the 14Δg ΛS state, the 14Δu,1/2 and 14Δu,3/2 Ω states and the 14Δu,5/2 and 14Δu,7/2 Ω components are separated by 11.84 and 11.86 cm−1, respectively. The separation between the 14Δu,3/2 and 14Δu,5/2 Ω states is 67.15 cm−1, which is obviously smaller that of the 14Δg,3/2 and 14Δg,5/2 components. In addition, the Re and ωe of these four components change inappreciably. According to the discussion, we can say that the effect on the 14Δg by the SO coupling is more pronounced than that of the 14Δu Λ-S state. For the four Ω states generated from 12Δg and 12Δu Λ-S states, their dominant Λ-S state compositions near the equilibrium internuclear separation mix with 14Πg or 14Σ+u. As a result, the energy separation between the 12Δg,5/2 and 12Δg,3/2 Ω states is 30.95 cm−1, and the energy separations between the 12Δu,5/2 and 12Δu,3/2 Ω states are 99.20 and 89.11 cm−1 for the first and second potential wells, respectively.

Figure 5. PECs of eight Ω states of C2+ cation: 1, 14Πg,5/2; 2, 12Δg,5/2; 3, 14Δu,5/2; 4, 12Δu,5/2; 5, 24Πg,5/2; 6, 14Πu,5/2; 7, 24Πu,5/2; 8, 14Δg,5/2.

Figure 6. PECs of four Ω states of C2+ cation: 1, 14Πg,−1/2; 2, 24Πg,−1/2; 3, 14Πu,−1/2; 4, 24Πu,−1/2.

Figure 7. PECs of two Ω states of C2+ cation: 1, 14Δu,7/2; 2, 14Δg,7/2.

cm−1. In conclusion, for the Ω states with almost pure dominant Λ-S state composition, we can see that their energy separations are obviously small when compared with those 2031



Article

Table 8. Spectroscopic Parameters of 51 Ω States Generated from the 22 Λ-S Bound States of C2+ Cation Obtained by the MRCI+Q/CV+DK+56 +SO Calculations −

X Σg,1/2 X4Σg,3/2− 12Πu,3/2 12Πu,1/2 14Πg,5/2 14Πg,3/2 14Πg,1/2 14Πg,−1/2 12Δg,5/2 12Δg,3/2 12Σg,1/2− 12Σg,1/2+ 22Πu,3/2 22Πu,1/2 B4Σu,3/2− B4Σu, 1/2− 12Πg,1/2 12Πg,3/2 12Σu,1/2+ 1st well 2nd well 14Δu,1/2 14Δu,3/2 14Δu,5/2 14Δu,7/2 14Σu, 1/2+ 14Σu,3/2+ 12Σu, 1/2− 12Δu,3/2 1st well 2nd well 12Δu, 5/2 1st well 2nd well 22Πg,3/2 22Πg,1/2 24Πg,,−1/2 1st well 2nd well 24Πg,1/2 1st well 2nd well 24Πg,3/2 1st well 2nd well 24Πg,5/2 1st well 2nd well 14Πu,5/2 1st well 2nd well 14Πu,3/2 1st well 2nd well 14Πu,1/2 1st well 2nd well 14Πu,,−1/2 1st well 2nd well 4

De (eV)

Te (cm−1)

Re (nm)

ωe (cm−1)

ωexe (cm−1)

dominant Λ-S states at Re (%)

5.7178 5.7160 5.1479 5.1416 4.5464 4.5466 4.5464 4.5452 4.4720 4.4683 4.1949 4.0519 3.8289 3.8269 3.2543 3.2534 2.9287 2.9283

0.0 1.50 4574.89 4606.5 9587.69 9598.23 9605.03 9616.01 9943.68 9974.63 12180.13 13506.63 15357.24 15377.65 19773.29 19773.95 22615.49 22623.17

0.14064 0.14064 0.13039 0.13039 0.12490 0.12486 0.12538 0.12538 0.14333 0.14334 0.14254 0.14449 0.15078 0.15080 0.13478 0.13479 0.12621 0.12621

1355.83 1355.68 1605.82 1605.66 1778.74 1773.70 1876.62 1876.67 1233.12 1232.11 1261.97 1139.15 1397.16 1393.73 1507.62 1506.87 1777.33 1777.53

12.271 12.218 12.922 12.928 38.434 40.503 6.9719 7.0427 14.675 14.432 15.886 14.255 11.262 11.527 10.518 10.555 20.265 20.466

X4Σg− (95.77), 14Πg (4.23) X4Σg− (96.16), 14Πg (3.84) 12Πu (98.23), 14Σu− (1.77) 12Πu (97.92), 14Σu− (2.08) 14Πg (100) 14Πg (100) 14Πg (100) 14Πg (100) 12Δg (96.29), 14Πg (3.71) 12Δg (96.28), 14Πg (3.72) 12Σg− (99.98), 12Σ+g (0.02) 12Σ+g (99.97), 12Σg− (0.03) 22Πu (100) 22Πu (100) B4Σu− (99.16), 22Πu (0.84) B4Σu− (99.53), 22Πu (0.46) 12Πg (100) 12Πg (100)

1.7926 1.0867 2.2173 2.2162 2.2079 2.2053 2.1550 2.1535 1.9640

26798.45 37450.22 28348.82 28360.68 28427.83 28439.69 28859.32 28859.98 30176.17

0.12168 0.19587 0.19114 0.19114 0.19115 0.19117 0.19251 0.19251 0.18755

1932.54 459.15 557.40 557.26 558.27 558.14 548.82 548.86 535.74

16.719 4.5347 5.2802 5.2865 5.1576 6.7354 5.0096 5.0283 2.8725

12Σ+u (100) 12Σ+u (93.75), 12Σu− (6.24) 14Δg (99.99), 14Σg+ (0.01) 14Δg (99.98), 14Σg+ (0.02) 14Δg (99.62), 24Σg− (0.38) 14Δg (96.85), 24Σg− (3.12) 14Σu+ (99.96), 12Δu (0.04) 14Σu+ (99.97), 12Δu (0.03) 12Σu− (93.75), 12Σu+ (6.25)

0.4845 1.1528

34122.98 36910.09

0.13575 0.19351

1388.11 449.64

22.009 2.4433

12Δu (96.23), 14Σ+u (3.76) 12Δu (97.63), 14Σ+u (2.36)

0.4853 1.1418 1.1958 1.1907

34222.18 36999.20 36431.64 36466.53

0.13576 0.19349 0.22145 0.22146

1391.88 448.60 455.82 455.04

16.625 1.5643 3.8369 3.9368

12Δu 12Δu 22Πg 22Πg

1.9552 0.3167

40257.96 43709.41

0.15395 0.28660

1245.61 274.55

5.5536 33.750

24Πg (100) 24Πg (100)

1.9549 0.3117

40267.39 43718.19

0.15396 0.28664

1244.83 273.20

5.5525 32.400

24Πg (100) 24Πg (100)

1.9546 0.3143

40276.83 43727.19

0.15398 0.28667

1244.07 272.44

5.5439 31.804

24Πg (100) 24Πg (100)

1.9543 0.3130

40286.49 43736.19

0.15399 0.28671

1243.32 271.73

5.5271 31.262

24Πg (100) 24Πg (100)

1.1142 0.0764

51478.6 46647.52

0.14795 0.40485

1078.00 62.53

17.830 3.6327

14Πu (100) 14Πu (100)

1.1201 0.0763

51479.91 46655.86

0.14795 0.40487

1078.09 62.51

17.671 3.6316

14Πu (100) 14Πu (100)

1.1260 0.0763

51481.01 46664.42

0.14795 0.40490

1078.41 62.49

17.633 3.6315

14Πu (100) 14Πu (100)

1.1319 0.0763

51482.33 46672.98

0.14795 0.40492

1078.51 62.47

17.482 3.6301

14Πu (100) 14Πu (100)

2032

(95.21), 14Σ+u (4.79) (96.60), 14Σ+u (3.40) (100) (100)



Article

Table 8. continued −

2 Σu,1/2 24Σu,1/2− 24Σu,3/2− 14Σg,1/2+ 14Σg,3/2+ 24Πu,5/2 24Πu,3/2 24Πu,1/2 24Πu,,−1/2 14Δg,1/2 14Δg,3/2 14Δg,5/2 14Δg,7/2 2

De (eV)

Te (cm−1)

Re (nm)

ωe (cm−1)

ωexe (cm−1)

dominant Λ-S states at Re (%)

0.8791 0.6411 0.6411 0.6543 0.6528 0.9315 0.9318 0.9320 0.9324 0.3198 0.3239 0.2717 0.2758

49952.59 55136.68 55137.39 59649.63 59651.09 61893.33 61897.28 61901.23 61904.96 64450.21 64455.25 64613.71 64618.76

0.15990 0.18967 0.18967 0.14298 0.14298 0.15600 0.15605 0.15610 0.15615 0.16282 0.16281 0.16307 0.16305

1595.95 771.53 771.52 1188.82 1188.84 610.94 609.48 608.03 606.59 629.39 629.77 616.63 617.01

24.999 3.7827 3.7803 16.537 16.535 11.461 12.301 13.128 13.942 36.351 36.234 37.500 37.395

22Σu− (100) 24Σu− (100) 24Σu− (100) 14Σg+ (95.26), 14Δg (4.45) 14Σg+ (97.25), 14Δg (2.73) 24Πu (100) 24Πu (100) 24Πu (100) 24Πu (100) 14Δu (97.31), 14Σg− (2.69) 14Δu (84.9), 14Σg− (15.1) 14Δu (89.75), 14Σg− (10.24) 14Δu (96.23), 14Σg− (3.76)

all the Λ-S and Ω bound states reported here can be expected to be reliable predicted ones.

Unfortunately, to our knowledge, no measurements can be available in the literature for the Ω states involved in this article except for the energy splitting of the 22Πg Λ-S state. Therefore, we cannot make any direct comparison between theory and experiment. On the one hand, the PECs of all the Λ-S and Ω states involved are calculated by the same approach; on the other hand, the spectroscopic parameters of all the Λ-S and Ω states involved are evaluated by the same methods. As discussed above, excellent agreement has been found between the present results and the available experimental data, whether for the Λ-S or the Ω states. According to these analyses, we believe, with reason, that the spectroscopic parameters of 51 Ω bound states collected in Table 8 should be reliable predicted ones.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] or [email protected]. Tel/Fax: 86-376 -6393178. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This work is sponsored by the National Natural Science Foundation of China under Grant No. 11274097, the Program for Science and Technology of Henan Province in China under Grant No. 122300410303, and the Program for Science and Technology Innovation Talents in Universities of Henan Province in China under Grant No. 2008HASTIT008.

4. CONCLUSIONS In this article, the PECs of 24 Λ-S states of C2+ cation and 54 Ω states generated from these Λ-S states have been investigated for internuclear separations from 0.09 to 1.11 nm using the CASSCF method, which is followed by the internally contracted MRCI and MRCI+Q approaches. In the calculations, the eight outermost MOs are put into the active space, which correspond to the 2−3σg, 2−3σu, 1πu, and 1πg MOs in the C2+ cation, and the two inner MOs are put into the closedshell orbitals, which correspond to the 1σg and 1σu orbitals in the C2+ cation. The SO coupling effect is included by the Breit−Pauli Hamiltonian using the ACVTZ basis set. To obtain more reliable results, the effect on the PECs by the core− valence correlation and relativistic corrections is taken into account. Relativistic corrections are incorporated using the DKH3 approximation with the cc-pVQZ-DK basis set. Core− valence correlation correction calculations are made using the ACVTZ basis set. With these PECs obtained by the MRCI+Q/ CV+DK+56 calculations, the spectroscopic parameters of 22 ΛS bound states have been calculated by fitting the first ten vibrational levels, which are determined by solving the rovibrational Schrö dinger equation using the Numerov’s method. These results have been compared in detail with those reported in the literature. Excellent agreement has been found between the present spectroscopic results and the experimental ones. With the PECs obtained by the MRCI+Q/ CV+DK+56+SO calculations, the spectroscopic parameters of 51 Ω bound states have been evaluated. The energy separation of the 22Πu Λ-S state is determined as 20.41 cm−1, which compares well with the measurements of 16.64 and 15.4 cm−1. The analyses demonstrate that the spectroscopic parameters of

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REFERENCES

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Potential Energy Curves, Spectroscopic Parameters, and SpinâOrbit

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