Theoretical study of the ro-vibrational band of ground state for BaCl+ cation

Computational and Theoretical Chemistry 1172 (2020) 112670

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Theoretical study of the ro-vibrational band of ground state for BaCl+ cation a

c

MaoLin Yuan , Junxia Cheng , Xinlu Cheng a b c

a,b,⁎

T

Institute of Atomic and Molecular Physics, Sichuan University, Chengdu 610065, People’s Republic of China Key Laboratory of High Energy Density Physics and Technology of Ministry of Education, Sichuan University, Chengdu 610065, People’s Republic of China School of Photo-Electrical Engineering, Xi’an Technological University, Xi’an 710021, People’s Republic of China

A R T I C LE I N FO

A B S T R A C T

Keywords: Ultralcold molecule Ro-vibrational transition BaCl+

Vibrational cooling of ground state for BaCl+ had been realised with high rate by collisions with laser-cooled buffer Calcium gas; however the related properties of internal ro-vibrational levels are lack for this cation. Highly correlated ab initial methods were performed to study the only stable bound state X1Σ+ of BaCl+. Its spectroscopic parameters, potential energy curve (PEC), dipole moment curve (DMC) and vibration energy level (v = 7–0, J = 0) were obtained. Based on these spectroscopic parameters, we obtained the partition functions from T = 100 to 800 K. At T = 300 K the vibrational intensities of v = 7–0 levels and the emission line lists of 1–0, 2–1, 3–2, 4–3, 2–0, 3–1, 4–2 band were obtained. For 1–0 band the lifetime τ = 0.137 s. It is obviously that inducing transitions between vibrational level in traditional methods is difficult and the direct laser cooling technology is not suitable to cool the BaCl+ cation.

1. Introduction Diatomic cations MX+ that consist of alkaline earth metal (M) and halogen element (X) are readily trapped by external field and of the potential to be cooled to ultracold temperature. Although to govern the electronic state transitions and translational motion of molecule are easy, it is difficult to control the vibrational transitions, which makes traditional cooling methods difficult except for molecules with suitable Franc-condon factors and lifetime like MgF [1]. In recent years, BaCl+ has attracted our attentions [2–6]. Chen et al. [2] studied the spectroscopic data of BaCl+ by using the molecular-ion trap-depletion spectroscopy technique and they also reported the PECs and spectroscopic constants of X1Σ+ and A1Π with CASPT2 method. They studied its photodissociation spectrum by driving a transition from bound X1Σ+ state to the repulsive A1Π state. The rotation and translation of molecular can be cooled easily by colliding with buffer gases; however, it is impractically to quench the vibration motion for the long relaxing timescales. Hudson et al. [3] proved that the vibrational motion of BaCl+ is quenched to ground state with high-efficiency by interaction with ultracold atom calcium in sympathetic cooling experiment. They found the strong interaction between trapped BaCl+ and cold Ca atom that mainly contributed to the cooling vibration. Ab initial method CCSD(T) was performed to explained this high efficiency sympathetic cooling for the vibrational motion of Ca-BaCl+ system by Stoecklin et al. [4]. VanGundy et al. [5] experimentally researched the first three

⁎

low-energy vibrational levels of ground state of BaCl+ and they [6] extended the vibrational levels up to seven and provide improved molecular constants later. The determination of internal temperature and characterization of internal state populations distributions have been hindered for the lacking of spectroscopic data among the internal ro-vibrational band for BaCl+ [5,6] and we know little about the internal ro-vibrational levels. In this paper, we studied the PECs and DMC of X1Σ+ the only stable bound state of BaCl+. Moreover we researched its lifetime and ro-vibrational intensity of v = 7–0 levels of X1Σ+. The organizations of this paper are as following: second 2 includes two main parts: firstly, specific descriptions of the computations process and details of potential energy and dipole moment with ab initial. Secondly, using the ab initial results, we computed ro-vibrational line intensities. The relate PECs, DMC, lines intensities and lifetime were provide in Section 3. Conclusions were made in the last section. 2. Method details 2.1. Ab initial calculation The potential energy and dipole moment of BaCl+ were computed with Molpro 2009 [7] program. We choose the Pseudopotential ECP46MDF [8,9] together with its corresponding valence basis sets and AV5Z [10] basis sets for atom Ba and Cl respectively. The 5s5p6s6p

Corresponding author at: Institute of Atomic and Molecular Physics, Sichuan University, Chengdu 610065, People’s Republic of China. E-mail address: [email protected] (X. Cheng).

https://doi.org/10.1016/j.comptc.2019.112670 Received 9 October 2019; Received in revised form 7 December 2019; Accepted 7 December 2019 Available online 10 December 2019 2210-271X/ © 2019 Published by Elsevier B.V.

Computational and Theoretical Chemistry 1172 (2020) 112670

M. Yuan, et al.

Fig. 1. (a) CASSCF potential energy curves for

Σ ,

1,3 + 1,3

Π,

Δ and X1Σ+ states of BaCl+. (b) MRCI + Q potential energy curve for the X1Σ+ of BaCl+.

1,3

We also use the Av′J′v″J″ to calculate the absolute emission line intensity Iv′J′v″J″ that the rotation-vibration spectrum from upper state |v′J′〉 to lower state |v″J″〉 in the following expression:

orbitals of atom Ba and the 3s3p4s orbitals of atom Cl were considered as active to combine the molecular orbitals. The 1s2s2p orbitals of Cl are frozen. The diatomic cation BaCl+ belongs to C∞v group. However, MOLPRO program can only use C2v point group that the symmetries representations of a1, b1, b2, and a2 were used to deal with the diatomic system. As a result, for BaCl+ system with ECP46MDF basis set there are16 electrons in 13 active molecular orbitals (7a1, 3b1, 3b2, 0a2) and 10 internal electrons in 5 frozen core orbitals (3a1, 1b1, 1b2, 0a2). At first the Hartree-Fock (HF) method were conducted to get initial guess of wave function of X1Σ+ state for BaCl+ and then implemented the complete active space self consistent field (CASSCF) method to optimize the wave functions. Eventually, In order to get accurate potential energy and dipole moment the Multi-Reference Configuration Interaction (MRCI) method and Davidson correction (+Q) were also performed.

−hcE ′ ′

(2J ′ + 1) exp ⎛ KTv′ J ′ ⎞ ⎝ ⎠ ×A ′ ′ ′ Iv′J ′v′ ′J′ ′ = v′ J ′v′ J ′ 8πcv 2 ′ ′ ′ Q −1

Isum =

∑ Iv′J ′v′′J′′

(5)

Since now we need to calculate the partition function Q according to the following expression: (6)

Q = Q v × Qr

Solving the one-dimension nuclear-eletronic coupled Schrödinger equation (1) with LEVEL8.0 [11], the spectroscopic parameters were obtained, eigenfunction Ψ(v, J) and energy E(v, J) of each rotationvibration level.

ħ2 d 2ψvJ (r ) + VJ (r ) ψvJ (r ) = EvJ ψvJ (r ) 2μ dr 2

where the Qv and Qr are the vibration and rotation partition function respectively. The Qv is computed by equation (7):

Q v = 1 + e−G0 (1) hc / kT + e−G0 (2) hc / kT + ...

(1)

(7)

According to McDowell [12], for the X Σ the following equation:

1 +

where the μ, VJ(r), v and J are the reduced mass of BaCl+, the sum of a centrifugal terms and electronic potential energy obtained from MRCI + Q computation, vibration and rotation quantum numbers respectively. Using the eigenfunction from Eq. (1) combined with the dipole moment from ab initial computations, it is convenient to compute the Einstein spontaneous emission A coefficient with the following Eq. (2) from LEVEL.

A v′J ′v′ ′J′ ′ = 3.1361891 × 10−7 ×

v3

v′J ′v′ ′J ′ ′

1 ∑ A v′v′ ′ v′

′

(8)

−27

ergs s, where β = hcBe/kT, h = 6.62606896 × 10 c = 2.99792458 × 1010 cm/s, and k = 1.3806504 × 10−16 ergs/K, σ = I = 1, κ = 0. 3. Results and discussions

(2)

3.1. Potential energy and dipole moment

where Av'J'v''J'' is the Einstein A coefficient (v′J′-v″J″) with units s−1. vv′J′v″J″ is the transition frequency in units cm−1. M(r) is dipole moment in Deby and S (J′, J″) is the Hönl-London rotational intensity factor. The lifetime of each vibration is equal to the inverse of the sum Av'v'' over all lower v″:

τv ′ =

state the Qr is given in

β −β β 2 8β 2 2 Qr = σ −1I 2e 3 β −1 ⎡1 + + ....+κI −1π 2/3e 12 e−β /4ββ −1/2⎤ ⎥ ⎢ 90 2835 ⎦ ⎣

S (J ′, J ′′) 2J ′ + 1

| 〈Ψv′J ′(r )| M (r )|Ψ v′ ′J′ ′〉 |

−2

where Iv′J′v″J″ is the line intensities in (cm /(molecule cm )), Ev″J″ is the lower state energy (cm−1), T is the temperature (K), Q is the total partition function. A band intensity of v′ − v″ is the sum of Iv′J′v″J″ over all respective J′, and J″ represented as Eq. (5):

2.2. Spectroscopic constants and intensities

−

(4)

v′ J ′v′ J ′

The PECs of first exited 1,3Σ+, 1,3Π, 1,3Δ and X1Σ+ states of BaCl+ were plotted in Fig. 1(a) at CASSCF level. It is obviously to conclude that X1Σ+ is the only stable bound state and others are all nearly repulsive states with shallow potential well. For this characteristic, quenching BaCl+ to the ground electronic state is readily. Since, in this paper we further studied its ground state. The PEC of X1Σ+ at MRCI + Q level was showed in Fig. 1(b). Its spectroscopic constants computed with LEVEL8.0 and previous theoretical and experimental

(3) 2

Computational and Theoretical Chemistry 1172 (2020) 112670

M. Yuan, et al.

Table 1 Spectroscopic constants for the X1Σ+ states of BaCl+. State

Method

Re (cm−1)

Te (cm−1)

ωe (cm−1)

ωeχe (cm−1)

Be (cm−1)

De (cm−1)

Reference

X Σ

MRCI + Q MRCI + Q CASPT2 Exp Exp

2.60 2.59 2.57

0 0 0

338.79 333.80 328.30 342(2) 337.2(1.6)

0.68 0.83 1.56 1.6(4) 0.93(0.20)

0.090

38,456 39,278 39,055

This work Ref. [3] Ref. [1] Ref. [4] Ref. [5]

1 +

data were listed in Table 1. For the ground state of BaCl+ the equilibrium bond distance Re and harmonic frequency ωe are 2.60 Å and 338.79 cm−1 respectively matching well with previous experiment and theoretical results. Compared with Refs. [1,3], our ωe is closer to the experiment seen form Table 1. Using the CASTP2 method, Chen et al. obtained the anharmonic vibrational frequency ωeXe = 1.56 cm−1, which is close to the previous experimental result of VanGundy et al. However, VanGundy corrected this value: ωeXe = 0.93(0.2) cm−1 in the later experiment and this corrected value is close to our result ωeXe = 0.68 cm−1. The best available estimate of rotation constant for the X1Σ+ of BaCl+ is Be = 0.092 cm−1 from Chen et al., which is just 0.002 cm−1 greater than our result Be = 0.090 cm−1. Considering the vibrational energy of v = 0, we obtained the dissociation energy D0 = 38286 cm−1. It matches the result of Kaledin et al. [13] (D0 = 38500 ± 1100 cm−1) well. Refs. [1,3] provide its dissociation energy De = 39,055 and 39,278 cm−1 respectively, which are slightly lager than our result De = 38,456 cm−1. We also provide the vibration energy (v = 0–7, J = 0) of the ground state of BaCl+ from LEVEL and its corresponding experimental data in Table 2. Our results are slightly different from the experimental data in Table 2. There are two main reasons. Firstly, we noticed that VanGundy’s used the maximum intensity points to determine the band positions and the rotational structure was not concerned, which means the vibrational energy levels were from uncertainty ro-vibrational transitions. Concerned this our computations are acceptable. Secondly, it is must admit that for atom Ba we chosen the Pseudopotential ECP46MDF, which brings unavoidable errors. This is the first time that the DMC of X1Σ+ for BaCl+ was researched. Fig. 2 shows the DMC with two extremes values at MRCI level as R varies from 1.8 to 15 Å. When R = 3.46 Å and 5.40 Å, DM reaches its maximum and minimum value respectively. After the minimum point, DMC becomes monotonically increasing, which is accord with the cation property of BaCl+. The detailed potential energy and dipole moment results are provide in supplement material (Table 1s).

Table 2 Vibrational energy levels for the X1Σ+ of BaCl+. v

a

0 1 2 3 4 5 6 7

0 337.5 673.6 1008.2 1341.5 1673.5 2003.9 2333.2

a b

b

E (cm-1)

0.092

E (cm-1)

0 339.1 673.1 1003.4 1328.0 1660.8 1987.1 2311.0

This work vibration energy (v′ = 0–7, J′ = 0). Experiment data from Ref. [4].

Fig. 2. Dipole moment curve for the X1Σ+ of BaCl+. Table 3 Partition function for the X1Σ+ state of BaCl+.

3.2. Intensities and lifetime

T(K)

100

200

300

400

500

600

700

800

T

781.3

1700.3

2901.0

4414.0

6246.1

8399.8

10876.04

13677.1

According to Eqs. (6)–(8), partition functions of X1Σ+ for BaCl+ were obtained with the temperature varying from 100 to 800 K and the corresponding values were listed in Table 3. The ro-vibrational line lists including 1–0, 2–1, 3–2, 4–3, 2–0, 3–1, 4–2 band were showed in the Table 4S of supplement material. Table 4 shows the vibrational line intensities of v = 7–0 bands of ground state at T = 300 K, the total

Table 4 Band intensities for the X1Σ+ state of BaCl+ in cm−1/(molecule cm−2) at T = 300 K. v″

v′

1 2 3 4 5 6 7

0 1.09E−18 6.46E−21 5.81E−23 9.36E−25 1.89E−26 5.04E−28 1.78E−29

1

2

3

4

5

6

5.07E−19 3.82E−21 4.55E−23 9.20E−25 2.23E−26 5.67E−28

1.27E−19 1.52E−21 2.24E−23 5.41E−25 1.49E−26

3.39E−20 5.05E−22 8.89E−24 2.51E−25

8.52E−21 1.52E−22 3.10E−24

2.07E−21 4.32E−23

4.31E−22

3

Computational and Theoretical Chemistry 1172 (2020) 112670

M. Yuan, et al.

Fig. 3. Line intensities v = 1–0 for the X1Σ+ of BaCl+.

partition function Q = 2901.0. The intensity value is up to 1.09 × 10−18 cm−1/(molecule cm−2) for 1–0 band and it decreases fast with the increase of Δv. Seen from Table 4, it has reduced to 1.78 × 10−29 cm−1/(molecule cm−2) for 7–0 band. Fig. 3 shows the absolute line intensities of 1–0 band including 122 lines (there is 61 rotational levels for the v = 0 level of the ground state of BaCl+. Spectra of this band are strongly influenced by temperature. The wavenumber of maximum line intensity decreases at temperature varying from 100 to 300 K for the P branch opposite to the R branch seen in Fig. 3. The wavenumber of this transition are mainly in the range of 325–347 cm−1. We also evaluated the lifetime of 1–0 band for the X1Σ+ state of BaCl+ τ = 0.137 s. Even for the strongest 1–0 band, the lifetime is such a long a data, making the difficulty of governing the vibrational transitions, therefore traditional cooling technology is not applicable for the vibrational cooling for this molecular-ion.

Declaration of Competing Interest

4. Conclusions

References

Using the CASSCF method, the PECs of first exited 1,3Σ+, 1,3Π, 1,3 Δ and X1Σ+ state of BaCl+ were researched. Different from many other diatomic systems X1Σ+ is the only stable bound state for BaCl+, for which preparation its ground electronic state is convenient. Therefore we calculated the potential curve at MRCI + Q level and the dipole moment at MRCI level of X1Σ+ for BaCl+ only. Spectroscopic constants and vibrational energy levels (v = 0–7, J = 0) were obtained from the potential energy and dipole moment. Partition functions of X1Σ+ were also provided with temperature from 100 to 800 K. Our vibrational levels and spectroscopic constants are in agreement with previous experimental and theoretical results. The effects on spectra for v = 1–0 band are strongly influenced by the temperature. Band intensities among first seven vibrational levels were also obtained by summing over all the rotational levels. The lifetime τ = 0.137 s is too long to cool with traditional laser cooling way for this molecular-ion. We have to choose other cooling ways for example [3,4] collisions with ultracold calcium buffer gases and we hope our study will be helpful in the following research.

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The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements We thank the financial support from the National Natural Science Foundation of China (11774248). Appendix A. Supplementary material Supplementary data to this article can be found online at https:// doi.org/10.1016/j.comptc.2019.112670.

CRediT authorship contribution statement MaoLin Yuan: Data curation, Writing - original draft. Junxia Cheng: Investigation, Writing - review & editing. Xinlu Cheng: Software, Supervision, Funding acquisition. 4