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作为非对称多原子分子制冷的一个重要目标分子, CaSH的冷却有望打破双原子分子及线性三原子分子在激光冷却中的技术局限. 本文使用高精度的EA-EOM-CCSD (electron attachment equation-of-motion coupled cluster singles and doubles)方法, 通过cc-pVXZ/cc-pCVXZ (X = T, Q)系列基组外推至基组极限, 得到了CaSH基态和3个最低激发态精确的几何结构及基态到激发态的跃迁能. 其中, 基态
$ {\tilde{{\mathrm{X}}}}^{2}{{\mathrm{A}}}' $ 几何结构参数分别为RCaS= 2.564 Å; RSH= 1.357 Å; ∠CaSH= 91.0°; 从$ {\tilde{{\mathrm{X}}}}^{2}{{\mathrm{A}}}^{\prime} $ 到$ {\tilde{{\mathrm{A}}}}^{2}{{\mathrm{A}}}^{\prime} $ ,$ {\tilde{{\mathrm{B}}}}^{2}{{\mathrm{A}}}^{{\prime\prime} } $ 和$ {\tilde{{\mathrm{C}}}}^{2}{{\mathrm{A}}}^{\prime} $ 的垂直激发能分别为1.898, 1.945和1.966 eV, 与已有实验符合得很好. 进一步, 在3ζ级别基组上, 计算了该分子4个最低电子态的势能面, 并通过求解核运动方程给出CaS键伸缩、CaSH弯曲两个振动模的频率. 最后, 理论计算给出的$ {\tilde{{\mathrm{X}}}}^{2}{{\mathrm{A}}}^{\prime} ({\mathrm{0, 0}}, 0) $ 态到激发态$ {\tilde{{\mathrm{A}}}}^{2}{{\mathrm{A}}}^{\prime} ({\mathrm{0, 0}}, 0) $ ,$ {\tilde{{\mathrm{B}}}}^{2}{{\mathrm{A}}}^{{\prime\prime} }({\mathrm{0, 0}}, 0) $ 和$ {\tilde{{\mathrm{C}}}}^{2}{{\mathrm{A}}}^{\prime} ({\mathrm{0, 0}}, 0) $ 跃迁的Frank-Condon (FC)因子分别为0.9268, 0.9958和0.9248. 结合Frank-Condon因子和激发态寿命分析, 本文给出了可能用于CaSH冷却的光学循环, 为CaSH的激光冷却提供了理论参考.-
关键词:
- 激光冷却 /
- 激发态 /
- Franck-Condon因子 /
- 光学循环
The CaSH molecule is an important target in the field of laser cooling non-linear polyatomic molecules. Successful cooling of such molecules marks a breakthrough of the technical limitations of laser cooling diatomic and linear triatomic molecules. To identify the possible optical cycle in cooling CaSH, precise geometries of the CaSH ground state and the three lowest excited states, along with their excitation energy, are determined by utilizing the EA-EOM-CCSD (electron attachment equation-of-motion coupled cluster singles and doubles) method, in combination with energy extrapolation using cc-pVXZ/cc-pCVXZ (X = T, Q ) serial basis sets. Geometric parameters of the ground state$ {\tilde{{\mathrm{X}}}}^{2}{{\mathrm{A}}}^{\prime} $ are found to be RCaS= 2.564 Å, RSH= 1.357 Å, and∠CaSH= 91.0°. Additionally, the equilibrium geometries of three excited states are also obtained. The$ {\tilde{{\mathrm{B}}}}^{2}{{\mathrm{A}}}^{\prime\prime} $ state has a similar equilibrium structure to the ground state, while the$ {\tilde{{\mathrm{A}}}}^{2}{{\mathrm{A}}}^{\prime} $ and$ {\tilde{{\mathrm{C}}}}^{2}{{\mathrm{A}}}^{\prime} $ states exhibit significant conformer distortions. Specifically, the CaS bond of the$ {\tilde{{\mathrm{A}}}}^{2}{{\mathrm{A}}}^{\prime} $ state and$ {\tilde{{\mathrm{C}}}}^{2}{{\mathrm{A}}}^{\prime} $ state tend to contract, and the CaSH angel bends by 5° relative to the ground state. The vertical excitation energy from the ground state to$ {\tilde{{\mathrm{A}}}}^{2}{{\mathrm{A}}}^{\prime} $ ,$ {\tilde{{\mathrm{B}}}}^{2}{{\mathrm{A}}}^{\prime\prime} $ and$ {\tilde{{\mathrm{C}}}}^{2}{{\mathrm{A}}}^{\prime} $ are of 1.898, 1.945 and 1.966 eV, respectively, which are in good agreement with the previous experimental results. Moreover, the potential energy surfaces of the four lowest electronic states of CaSH are calculated by EA-EOM-CCSD with 3ζ level of basis sets. The nuclear equations of motion are solved to obtain the vibrational frequencies of the CaS bond stretching and CaSH bending. The vibrational frequencies of the (0,1,0) mode and the CaS stretching frequency of four states are 316 cm–1, 315 cm–1, 331 cm–1 and 325 cm–1, which are in close agreement with the available experimental results. The frequencies of the CaSH bending mode are presented for the first time, with the values of 357 cm–1, 396 cm–1, 384 cm–1, 411 cm–1 for the$ {\tilde{{\mathrm{X}}}}^{2}{{\mathrm{A}}}^{\prime} $ ,$ {\tilde{{\mathrm{A}}}}^{2}{{\mathrm{A}}}^{\prime} $ ,$ {\tilde{{\mathrm{B}}}}^{2}{{\mathrm{A}}}^{\prime\prime} $ and$ {\tilde{{\mathrm{C}}}}^{2}{{\mathrm{A}}}^{\prime} $ states, respectively. Theoretical calculations give the Frank-Condon factors of 0.9268, 0.9958 and 0.9248 for the$ {\tilde{{\mathrm{X}}}}^{2}{{\mathrm{A}}}^{\prime} ({\mathrm{0,0}},0) $ to$ {\tilde{{\mathrm{A}}}}^{2}{{\mathrm{A}}}^{\prime} ({\mathrm{0,0}},0) $ ,$ {\tilde{{\mathrm{B}}}}^{2}{{\mathrm{A}}}^{{{\prime} }{{\prime} }}({\mathrm{0,0}},0) $ and$ {\tilde{{\mathrm{C}}}}^{2}{{\mathrm{A}}}^{\prime} ({\mathrm{0,0}},0) $ transitions. All three excited states are the bright states with considerable oscillator strength relative to the ground state. Based on the Frank-Condon factor and lifetime of excited states, the$ {{\tilde{{\mathrm{X}}}}^{2}{{\mathrm{A}}}^{\prime} ({\mathrm{0,0}},0)\to \tilde{{\mathrm{B}}}}^{2}{{\mathrm{A}}}^{{{\prime} }{{\prime} }}({\mathrm{0,0}},0) $ transition is regarded as the main cooling cycle for the CaSH molecule. The corresponding pump light wavelength is 678 nm. By exciting the vibrational excited states (0,1,0) and (0,0,1) of the$ {\tilde{{\mathrm{X}}}}^{2}{{\mathrm{A}}}^{\prime} $ state to$ {\tilde{{\mathrm{A}}}}^{2}{{\mathrm{A}}}^{\prime} ({\mathrm{0,0}},0) $ using lasers at 666 nm and 668 nm, respectively, the optical cooling branch ratio of CaSH is expected to exceed 0.9998.-
Keywords:
- laser cooling /
- excited state /
- Franck-Condon factor /
- optical cycle
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Google Scholar
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Li Y H, Li J F, Wang L B, Zhang J L 2008 J. Henan Univ. (Nat. Sci.) 38 4
[46] Baum L, Vilas N B, Hallas C, Augenbraun B L, Raval S, Mitra D, Doyle J M 2020 Phys. Rev. Lett. 124 133201
Google Scholar
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[48] Franck-Condon factors in the harmonic approximation , Wójcik P, Gozem S, Mozhayskiy V, Krylov A I. http:// iopenshell.usc.edu/downloads [2023-5-3
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图 1 CaSH分子前线轨道图 (a) $ 15{{\mathrm{a}}}^{\prime} ; $ (b) $ 16{{\mathrm{a}}}^{\prime} ; $ (c) $ 5{{\mathrm{a}}}^{{{\prime} }{{\prime} }}; $ (d) $ 17{\mathrm{a}}^{{\prime} } $
Fig. 1. Frontier orbital diagram of the CaSH molecule: (a) $ 15{{\mathrm{a}}}^{\prime} ; $ (b) $ 16{{\mathrm{a}}}^{\prime} ; $ (c) $ 5{{\mathrm{a}}}^{{{\prime} }{{\prime} }}; $ (d) $ 17{\mathrm{a}}^{{\prime} } $ .
图 2 CaSH激光冷却光学循环示意图, 实线代表激发光, 虚线为自发辐射通道
Fig. 2. Schematic diagram of optical cycle of laser cooling of CaSH, solid lines represent the excitation light, dash lines are spontaneous emission channels.
表 1 CaSH分子基态和3个低能级激发态的平衡结构
Table 1. Equilibrium structures of the ground state and three low-lying excited states of the CaSH molecule.
State r0 (Ca-S)/Å r0 (S—H)/Å θ (Ca—S—H)/(°) Method Ref. $ {\tilde{{\mathrm{X}}}}^{2}{{\mathrm{A}}}^{\prime} $ 2.557 1.338 91.0 EA-EOM-CCSD This work 2.614 1.346 100.0 MP2 Calc.[23] 2.607 1.339 97.1 Calc.[25] 2.647 1.347 93.1 CCSD(T) Calc.[24] 2.560 1.346 96.6 Expt.[22] 2.564 1.357 91.0 Expt.[26] $ {\tilde{{\mathrm{A}}}}^{2}{{\mathrm{A}}}^{\prime} $ 2.528 1.343 86.0 EA-EOM-CCSD This work 2.513 1.346 94.6 Expt.[22] 2.517 1.357 89.1 Expt.[26] $ {\tilde{{\mathrm{B}}}}^{2}{{\mathrm{A}}}^{\prime\prime} $ 2.554 1.338 91.0 EA-EOM-CCSD This work 2.550 1.357 89.1 Expt.[26] $ {\tilde{{\mathrm{C}}}}^{2}{{\mathrm{A}}}^{\prime} $ 2.525 1.345 86.0 EA-EOM-CCSD This work 2.562 1.357 91.0 Expt.[26] 
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表 2 CaSH 基态到3个低能级激发态激发能的对比
Table 2. Comparison of excitation energies from the ground state to three low-lying excited states of the CaSH.
$ {\tilde{{\mathrm{X}}}}^{2}{{\mathrm{A}}}^{\prime} \to {\tilde{{\mathrm{A}}}}^{2}{{\mathrm{A}}}^{\prime} $ $ {\tilde{{\mathrm{X}}}}^{2}{{\mathrm{A}}}^{\prime} \to {\tilde{{\mathrm{B}}}}^{2}{{\mathrm{A}}}^{\prime\prime} $ $ {\tilde{{\mathrm{X}}}}^{2}{{\mathrm{A}}}^{\prime} \to {\tilde{{\mathrm{C}}}}^{2}{{\mathrm{A}}}^{\prime} $ EA-EOM-CCSD VEE a) 1.945 2.000 2.093 3ζ 1.915 1.966 2.016 4ζ 1.898 1.945 1.966 Basis set limit AEE b) 1.894 1.944 1.961 Basis set limit SA-CASSCF VEE 1.884 1.937 2.216 3ζ f c) 0.2628 0.2589 0.1621 Expt. 1.907 1.966 1.993 Ref. [26] 1.905 1.960 1.987 Ref. [21] Calc. 1.860 1.930 2.110 Ref. [23] VEE a) 2.000 2.060 2.330 Ref. [43] 注: a) 垂直激发能; b)绝热激发能; c) 偶极跃迁振子强度.
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表 3 CaSH分子Ca—S键伸缩和CaSH弯曲振动模式的频率(cm–1)a)
Table 3. Frequencies of the Ca—S stretching and CaSH bending vibrational modes of the CaSH molecule (cm–1) a).
$ {\tilde{{\mathrm{X}}}}^{2}{{\mathrm{A}}}^{\prime} $ $ {\tilde{{\mathrm{A}}}}^{2}{{\mathrm{A}}}^{\prime} $ $ {\tilde{{\mathrm{B}}}}^{2}{{\mathrm{A}}}^{\prime\prime} $ $ {\tilde{{\mathrm{C}}}}^{2}{{\mathrm{A}}}^{\prime} $ (0, 1, 0) 316 (326) b) 315 (318) b) 331 (320) b) 326 (312) b) (0, 0, 1) 357 396 384 411 (0, 2, 0) 634 632 661 653 (0, 1, 1) 681 724 722 742 (0, 0, 2) 704 785 760 808 (0, 3, 0) 953 949 990 981 (0, 2, 1) 1007 1051 1058 1074 (0, 1, 2) 1034 1122 1102 1145 (0, 0, 3) 1039 1167 1130 1188 (0, 4, 0) 1271 1266 1318 1308 注: a)(0, 1, 0)CaS 伸缩振动模式, (0, 0, 1)CaSH键角弯曲振动模式; b) 括号内值为实验值[23].
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表 4 CaSH分子激发态$ {\tilde{{\mathrm{A}}}}^{2}{{\mathrm{A}}}^{\prime} $, $ {\tilde{{\mathrm{B}}}}^{2}{{\mathrm{A}}}^{\prime\prime} $和$ {\tilde{{\mathrm{C}}}}^{2}{{\mathrm{A}}}^{\prime} $到基态$ {\tilde{{\mathrm{X}}}}^{2}{{\mathrm{A}}}^{\prime} $的跃迁FC因子
Table 4. The Frank-Condon factor of the excited states $ {\tilde{{\mathrm{A}}}}^{2}{{\mathrm{A}}}^{\prime} $, $ {\tilde{{\mathrm{B}}}}^{2}{{\mathrm{A}}}^{\prime\prime} $ and $ {\tilde{{\mathrm{C}}}}^{2}{{\mathrm{A}}}^{\prime} $ transition to ground state $ {\tilde{{\mathrm{X}}}}^{2}{{\mathrm{A}}}^{\prime} $ of the CaSH molecule.
$ {\tilde{{\mathrm{X}}}}^{2}{{\mathrm{A}}}^{\prime} {\to} {\tilde{{\mathrm{A}}}}^{2}{{\mathrm{A}}}^{\prime} $ $ {\tilde{{\mathrm{X}}}}^{2}{{\mathrm{A}}}^{\prime} {\to} {\tilde{{\mathrm{B}}}}^{2}{{\mathrm{A}}}^{\prime\prime} $ $ {\tilde{{\mathrm{X}}}}^{2}{{\mathrm{A}}}^{\prime} {\to} {\tilde{{\mathrm{C}}}}^{2}{{\mathrm{A}}}^{\prime} $ (0, 0, 0) →
(0, 0, 0)0.9268 0.9958 0.9248 (0, 1, 0) →
(0, 0, 0)0.0627 0.0011 0.0626 (0, 0, 1) →
(0, 0, 0)0.0094 0.0002 0.0114 (0, 1, 1) →
(0, 0, 0)0.0006 2.4×10–7 0.0008 (0, 2, 0) →
(0, 0, 0)0.0001 0.0006 2.2×10–5 (0, 0, 2) →
(0, 0, 0)6.0×10–5 0.0003 5×10–6
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[1] Balakrishnan N 2016 J. Chem. Phys. 145 150901
Google Scholar
[2] Wu Y, Bao W S, Cao S R, et al. 2021 Phys. Rev. Lett. 127 180501
Google Scholar
[3] Andreev V, Ang D G, Demille D, Doyle J M, Gabrielse G, Haefner J, Hutzler N R, Lasner Z, Meisenhelder C, O'Leary B R, Panda C D, West A D, West E P, Wu X 2018 Nature 562 355
Google Scholar
[4] Baron J, Campbell W C, DeMille D, Doyle J M, Gabrielse G, Gurevich Y V, Hess P W, Hutzler N R, Kirilov E, Kozyryev I, O’Leary B R, Panda C D, Parsons M F, Petrik E S, Spaun B, Vutha A C, West A D 2014 Science 343 269
Google Scholar
[5] Barry J F, Mccarron D J, Norrgard E B, Steinecker M H, Demille D 2014 Nature 512 286
Google Scholar
[6] Norrgard E B, McCarron D J, Steinecker M H, Tarbutt M R, DeMille D 2016 Phys. Rev. Lett. 116 063004
Google Scholar
[7] Truppe S, Williams H J, Fitch N J, Hambach M, Wall T E, Hinds E A, Sauer B E, Tarbutt M R 2017 New J. Phys. 19 022001
Google Scholar
[8] Kozyryev I, Baum L, Matsuda K, Augenbraun B L, Anderegg L, Sedlack A P, Doyle J M 2017 Phys. Rev. Lett. 118 173201
Google Scholar
[9] Augenbraun B L, Lasner Z D, Frenett A, Sawaoka H, Miller C, Steimle T C, Doyle J M 2020 New J. Phys. 22 022003
Google Scholar
[10] Baranov M A 2008 Phys. Rep. 464 71
Google Scholar
[11] Ni K K, Ospelkaus S, De Miranda M H, Hg M, Pe'Er A, Neyenhuis B, Zirbel J J, Kotochigova S, Julienne P S, Jin D S, Ye J 2008 Science 322 231
Google Scholar
[12] Shuman E S, Barry J F, DeMille D 2010 Nature 467 820
Google Scholar
[13] Hummon M T, Yeo M, Stuhl B K, Collopy A L, Xia Y, Ye J 2013 Phys. Rev. Lett. 110 143001
Google Scholar
[14] Zhelyazkova V, Cournol A, Wall T E, Matsushima A, Hudson J J, Hinds E, Tarbutt M, Sauer B 2014 Phys. Rev. A 89 053416
Google Scholar
[15] Gao Y, Wan M 2017 Phys. Chem. Chem. Phys. 19 5519
Google Scholar
[16] 陈涛, 颜波 2019 68 043701
Google Scholar
Chen T, Yan B 2019 Acta Phys. Sin. 68 043701
Google Scholar
[17] Wells N, Lane I C 2011 Phys. Chem. Chem. Phys. 13 19018
Google Scholar
[18] Li D, Fu M, Ma H, Bian W, Du Z, Chen C A 2020 Front. Chem. 8 20
[19] Li D, Cao J, Ma H, Bian W 2022 Phys. Chem. Chem. Phys. 24 10114
[20] Ivanov M V, Bangerter F H, Krylov A I 2019 Phys. Chem. Chem. Phys. 21 19447
Google Scholar
[21] Augenbraun B L, Doyle J M, Zelevinsky T, Kozyryev I 2020 Phys. Rev. X 10 031022
Google Scholar
[22] Liu L, Yang C L, Sun Z P, Wang M S, Ma X G 2021 Phys. Chem. Chem. Phys. 23 2392
Google Scholar
[23] Fernando W T M L, Ram R S, O'Brien L C, Bernath P F 1991 J. Phys. Chem. 95 2665
Google Scholar
[24] Jarman C N, Bernath P F 1993 J. Chem. Phys. 98 6697
Google Scholar
[25] Ortiz J V 1990 Chem. Phys. Lett. 169 116
Google Scholar
[26] Scurlock C T, Henderson T, Bosely S, Jung K Y, Steimle T C 1994 J. Chem. Phys. 100 5481
Google Scholar
[27] Taleb-Bendiad A, Scappini F, Amano T, Watson J K 1996 J. Chem. Phys. 104 7431
Google Scholar
[28] Sheridan P M, Dick M J, Wang J G, Bernath P F 2007 Mol. Phys. 105 569
Google Scholar
[29] Nooijen M, Bartlett R J 1995 J. Chem. Phys. 102 3629
Google Scholar
[30] Pritchard B P, Altarawy D, Didier B T, Gibson T D, Windus T L 2019 J. Chem. Inf. Model. 59 4814
Google Scholar
[31] Koput J, Peterson K A 2002 J. Chem. Phys. 116 9255
Google Scholar
[32] Woon D E, Dunning T H 1995 J. Chem. Phys. 100 4572
Google Scholar
[33] Dunning T H 1989 J. Chem. Phys. 90 1007
Google Scholar
[34] Müller T, Dallos M, Lischka H, Dubrovay Z, Szalay P G 2001 Theor. Chem. Acc. 105 227
Google Scholar
[35] Lanczos C 1950 J. Res. Nat. Bur. Stand. 45 255
Google Scholar
[36] Goldfield E M, Gray S K, Harding L B 1993 J. Chem. Phys. 99 5812
Google Scholar
[37] Light J C, Carrington Jr T 2000 Adv. Chem. Phys. 114 263
Google Scholar
[38] Lill J V, Parker G A, Light J C 1986 J. Chem. Phys. 85 900
Google Scholar
[39] Liu W J, Hong Y G, Dai D D, Li L M, Dolg M 1997 Theor. Chem. Acc. 96 75
Google Scholar
[40] Zhang Y, Suo B B, Wang Z K, Zhang N, Li Z D, Lei Y B, Zou W L, Gao J, Peng D L, Pu Z C, Sun Q M, Wang F, Ma Y T, Wang X P, Liu W J 2020 J. Chem. Phys. 152 064113
Google Scholar
[41] Yang D D, Wang F, Guo J W 2012 Chem. Phys. Lett. 531 236
Google Scholar
[42] Tu Z Y, Wang F, Li X Y 2012 J. Chem. Phys. 136 174102
Google Scholar
[43] Wang Z F, Hu S, Wang F, Guo J W 2015 J. Chem. Phys. 142 144109
Google Scholar
[44] Cao Z L, Li Z D, Wang F, Liu W J 2017 Phys. Chem. Chem. Phys. 19 3713
Google Scholar
[45] 李永红, 李俊峰, 王连宾, 张敬来 2008 河南大学学报(自然科学版) 38 4
Li Y H, Li J F, Wang L B, Zhang J L 2008 J. Henan Univ. (Nat. Sci.) 38 4
[46] Baum L, Vilas N B, Hallas C, Augenbraun B L, Raval S, Mitra D, Doyle J M 2020 Phys. Rev. Lett. 124 133201
Google Scholar
[47] Kozyryev I, Baum L, Aldridge L, Yu P, Eyler E E, Doyle J M 2018 Phys. Rev. Lett. 120 063205
Google Scholar
[48] Franck-Condon factors in the harmonic approximation , Wójcik P, Gozem S, Mozhayskiy V, Krylov A I. http:// iopenshell.usc.edu/downloads [2023-5-3
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