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BH分子8个Λ-S态和23个Ω态光谱性质的理论研究

邢伟 李胜周 孙金锋 李文涛 朱遵略 刘锋

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BH分子8个Λ-S态和23个Ω态光谱性质的理论研究

邢伟, 李胜周, 孙金锋, 李文涛, 朱遵略, 刘锋

Theoretical study on spectroscopic properties of 8 Λ-S and 23 Ω states for BH molecule

Xing Wei, Li Sheng–Zhou, Sun Jin–Feng, Li Wen–Tao, Zhu Zun–Lüe, Liu Feng
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  • 本文利用内收缩多参考组态相互作用方法计算了BH分子8个低电子态(X1Σ+, a3Π, A1Π, b3Σ, 23Π, 13Σ+, 15Σ和15Π)和在自旋-轨道耦合效应下所产生的23个Ω态的势能曲线、以及${\rm{X}}{}^1\Sigma _{{0^ + }}^ +$, ${{\rm{a}}^3}{\Pi _{{0^ + }}}$, ${{\rm{a}}^3}{\Pi _1}$, ${{\rm{a}}^3}{\Pi _2}$${{\rm{A}}^1}{\Pi _1}$态之间6对跃迁的跃迁偶极矩. 为了获得精确的势能曲线, 计算中修正了单双电子激发、核价相关效应、相对论效应和基组截断带来的误差. 获得的BH分子的光谱和跃迁数据与现有的理论值和实验值符合得很好. 计算结果表明: BH分子的A1Π1(υ' = 0 – 2, J' = 1, +) →$ {\text{X}}{}^1\Sigma _{{0^ + }}^ + $(υ′′ = 0 – 2, J ′′ = 1, –)跃迁具有较大的爱因斯坦A系数和加权的吸收振子强度、高度对角化分布的振动分支比, A1Π1态具有较短的辐射寿命. 另外, ${{\rm{a}}^3}{\Pi _{{0^ + }}}$和a3Π1态对A1Π1(υ' = 0) ↔ $ {\rm X}^1\Sigma _{{0^ + }}^ + $(υ′′ = 0)循环跃迁的影响可以忽略. 因此, 基于A1Π1(υ'= 0—1, J ′ = 1, +) ↔ $ {\rm X}^1\Sigma _{{0^ + }}^ + $(υ′′ = 0—3, J′′ = 1, –)循环跃迁, 我们提出了用一束主冷却激光(λ00 = 432.45 nm)和两束再泵浦激光(λ10 = 479.67 nm和λ21 = 481.40 nm)冷却BH分子的方案, 并评价了冷却效果.
    In this work, the potential energy curves of eight low electronic states (X1Σ+, a3Π, A1Π, b3Σ-, 23Π, 13Σ+, 15Σ-, and 15Π) and twenty-three Ω states of BH molecule, and the transition dipole moments among the $ {\text{X}}{}^{\text{1}}{\Sigma}_{{{\text{0}}^ + }}^ + $, $ {{\text{a}}^{\text{3}}}{\Pi_{{{\text{0}}^ + }}} $, a3Π1, a3Π2, and A1Π1 states are calculated by using the internally contracted multireference configuration interaction (icMRCI) method. In order to obtain the accurate potential energy curve, the errors caused by single and double electron excitation, core-valence correlation effects, relativistic effects and basis set truncation are corrected. The spectral and transition data of BH molecule are in good agreement with the available theoretical and experimental data. The calculation results show that the A1Π1(υ′ = 0-2, J′ = 1, +) →$ {\text{X}}{}^{\text{1}}{\Sigma}_{{{\text{0}}^ + }}^ + $(υ′′ = 0-2, J′′ = 1, –) transition has large Einstein A-coefficient, weighted absorption oscillator strength, and highly diagonal vibrational branching ratio Rυ′υ′′, and the excited state A1Π1(υ′ = 0, 1) have short spontaneous radiation lifetimes. Moreover, the effects of $ {{\text{a}}^{\text{3}}}{\Pi_{{{\text{0}}^ + }}} $and a3Π1 states on A1Π1(υ′ = 0) ↔ $ {\text{X}}{}^{\text{1}}{\Sigma}_{{{\text{0}}^ + }}^ + $(υ′′ = 0) cycle transition can be ignored. Therefore, according to the A1Π1(υ′ = 0-1, J′ = 1, +) ↔ $ {\text{X}}{}^{\text{1}}{\Sigma}_{{{\text{0}}^ + }}^ + $(υ′′ = 0-3, J′′ = 1, –) cycle transition, we propose to apply one main cooling laser (λ00 = 432.45 nm) and two repumping lasers (λ10 = 479.67 nm and λ21 = 481.40 nm) to laser cooling BH molecules, and evaluation of the cooling effect.
      通信作者: 邢伟, wei19820403@163.com
    • 基金项目: 国家自然科学基金(批准号: 61275132, 11274097)、河南省自然科学基金(批准号: 212300410233)、河南省高等学校重点科研项目(批准号: 21A140023)和信阳师范学院南湖学者奖励计划青年项目资助的课题.
      Corresponding author: Xing Wei, wei19820403@163.com
    • Funds: Project supported by the National Natural Science Foundation of China (Grant No. 61275132, 11274097), the Natural Science Foundation of Henan Province, China (Grant No. 212300410233), the Key Scientific Research Prgoram of Higher Education of Henan Province, China (Grant No. 21A140023), and the Nanhu Scholars Program for Young Scholars of XYNU, China.
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  • 图 1  BH分子8个Λ-S态的势能曲线

    Fig. 1.  Potential energy curves of 8Λ-S states of the BH molecule.

    图 2  BH分子23个Ω态的势能曲线

    Fig. 2.  Potential energy curves of 23 Ω states of the BH molecule.

    图 3  BH分子6对跃迁的跃迁偶极矩曲线

    Fig. 3.  Curves of the transition dipole moments versus internuclear separation of six-pair states of the BH molecule.

    图 4  利用A1Π1(υ′) ↔$ {\text{X}}{}^{\text{1}}{\Sigma}_{{{\text{0}}^ + }}^ + $(υ′′)跃迁进行激光冷却BH分子的方案. 虚线表示A1Π1(υ′ = 0, 1) →$ {\text{X}}{}^{\text{1}}{\Sigma}_{{{\text{0}}^ + }}^ + $(υ′′ = 0 –3)跃迁的自发辐射振动分支比(Rυ′υ′′). 红色实线表示激光驱动$ {\text{X}}{}^{\text{1}}{\Sigma}_{{{\text{0}}^ + }}^ + $(υ′′) → A1Π1 (υ′)跃迁

    Fig. 4.  The proposed laser cooling scheme for the BH using A1Π1(υ′) ↔$ {\text{X}}{}^{\text{1}}{\Sigma}_{{{\text{0}}^ + }}^ + $(υ′′) transition. The dotted line indicate the spontaneous radiation vibrational branching ratio (Rυ′υ′′) of A1Π1(υ′ = 0, 1) →$ {\text{X}}{}^{\text{1}}{\Sigma}_{{{\text{0}}^ + }}^ + $(υ′′ = 0 – 3) transition. The red solid line indicate the wavelength (λυ′′υ′) at which the laser drives the $ {\text{X}}{}^{\text{1}}{\Sigma}_{{{\text{0}}^ + }}^ + $(υ′′) →A1Π1 (υ′). transition.

    表 1  BH分子前两个离解极限产生的8个Λ-S态的离解关系

    Table 1.  Dissociation relationships of the 8 Λ –S states generated from the first two dissociation asymptotes of the BH molecule

    离解极限Λ-S态能级a/cm–1
    本文实验[30]理论[31]
    B(2Pu) + H(2Sg)X1Σ+, a3Π, A1Π, 13Σ+0.000.000.00
    B(4Pg) + H(2Sg)b3Σ, 15Σ, 23Π, 15Π28907.6628644.99+xb28932.70
    a, 4Pg态能级为4P1/2, 4P3/24P5/2能级的算术平均值减去2P3/22P1/2能级的算术平均值; b, 4P5/2能级外推值的不确定度.
    下载: 导出CSV

    表 2  icMRCI + Q/56 + CV + SR理论水平上BH分子7个Λ-S态的光谱常数

    Table 2.  Spectroscopic parameters of the 7 Λ-S states of BH at level of icMRCI + Q/56 + CV + SR.

    Λ-S态来源Te/cm–1Re/nmωe/cm–1ωexe/cm–1Be/cm–1αe/(102 cm–1)De/eV
    X1Σ+本文00.122952367.2848.778212.039537.09853.7137
    实验[8]00.123222366.7349.338412.025542.1516
    实验[10]02366.7349.339812.025842.1565
    实验[12]02364.6647.709812.025742.15913.6476±0.0037a
    实验[13]00.123222366.7349.340512.025542.1450
    理论[5]00.122902352.044.012.0863.6863
    理论[15]00.12301237946.7912.073.70
    理论[16]00.12312237812.0553.578b
    理论[17]00.1230235948.841.83.6773
    理论[18]00.123272368.4850.695712.11043.053.6580
    理论[19]00.123003.6751
    理论[20]00.122932365.6947.231012.080141.63.6851
    a3Π本文10944.320.118992625.9759.417712.891941.64042.3507
    实验[14]xc0.119002625.1455.784012.893141.56102.3867
    理论[5]10645.00.119002961.0109.612.9042.3806
    理论[15]0.11913265362.7012.872.38
    理论[17]105830.11900262560.445.52.3677
    理论[18]9557.670.119252598.9846.630012.940042.532.3135
    A1Π本文23203.520.122232253.2836.831011.834311.62540.8368
    实验[10]23135.440.12195d2251.4656.572512.2003553.76700.697d
    实验[12]23105.102342.41127.761812.1998653.67360.7786±0.0037a
    理论[5]22997.900.122102404.60147.312.27950.9098
    理论[15]0.122132320136.512.240.71
    理论[16]230610.12235229012.200.73b
    理论[17]231440.12222341129.685.10.8109
    理论[18]22260.890.122672280.2693.623312.22960.830.7536
    理论[19]23099.840.122122343.96128.17812.283674.00.8938
    b3Σ本文38238.630.121642440.8954.447712.250833.67122.5959
    实验[14]xc+27152.750.1216252438.1055.56212.342643.0872.5987
    理论[15]0.12256234548.4512.162.54
    理论[17]377080.1217243057.345.92.5845
    理论[18]36859.520.121992428.3355.40912.28444.312.5403
    23Π本文50730.460.192151273.8920.78964.944713.09571.0467
    理论[15]0.19338142557.044.881.04
    理论[17]502160.1931129538.69.91.0321
    13Σ+本文51738.070.125920.0031
    理论[17]516880.123
    15Σ本文58295.540.16981634.868167.6766.51936192.6410.1093
    理论[17]576740.170152887.3153.20.1084
    a, 文献[11]中的值; b, D0值; c, x表示a3Π态相对于X1Σ+态的Te值; d, 文献[7]中的值.
    下载: 导出CSV

    表 3  BH分子23个Ω态的离解关系

    Table 3.  Dissociation relationships of the 23 Ω states of the BH molecule.

    原子态(B + H)Ω态能级/cm–1
    本文实验[30]
    B(2P1/2) + H(2S1/2)0, 0+, 10.000.00
    B(2P3/2) + H(2S1/2)2, 1(2), 0+, 014.57215.287
    B(4P1/2) + H(2S1/2)0, 0+, 128910.6328647.43+x a
    B(4P3/2) + H(2S1/2)2, 1(2), 0+, 028914.6728652.07+x a
    B(4P5/2) + H(2S1/2)3, 2(2), 1(2), 0+, 028921.4128658.40+x a
    a, 4P5/2能级外推值的不确定度.
    下载: 导出CSV

    表 4  利用icMRCI + Q/56 + CV + SR + SOC理论计算获得的17个Ω态的光谱常数

    Table 4.  Spectroscopic parameters obtained by the icMRCI + Q/56 + CV + SR + SOC calculations for the 17 Ω states.

    Ω态Te/cm–1Re/nmωe/cm–1ωexe/cm–1Be/cm–1102αe/cm–1De/eVRe附近主要的Λ–S态/%
    $ {\text{X}}{}^{\text{1}}{\Sigma}_{{{\text{0}}^ + }}^ + $00.122952367.2848.778312.039537.09853.7138X1Σ+ (100.00)
    ${\text{a} }{}^{\text{3} }{\Pi_{ { {\text{0} }^{{ - } } } }}$10940.360.118992625.9359.416512.891841.64262.3506a3Π (100.00)
    $ {\text{a}}{}^{\text{3}}{\Pi_{{{\text{0}}^ + }}} $10940.370.118992625.9359.419212.891841.64242.3506a3Π (100.00)
    a3Π110944.320.118992625.9759.414012.891941.64032.3513a3Π (100.00)
    a3Π210948.490.118992626.0159.413112.891941.63842.3509a3Π (100.00)
    A1Π123203.520.122232253.2836.831711.833811.70340.9051A1Π (100.00)
    (3)0+第一势阱38244.330.121632438.0844.728112.292538.30411.5501b3Σ (100.00)
    (3)0+第二势阱50725.860.192130.002623Π (100.00)
    (3)138244.350.121632447.6954.293412.316737.61490.8995b3Σ (100.00)
    (4)145758.490.164964850.091293.006.739524.021681.662913Σ+ (100.00)
    ${\text{2} }{}^{\text{3} }{\Pi_{ { {\text{0} }^{ { - } } } } }$50726.070.192141274.0020.84504.944703.096111.046623Π (100.00)
    (4)0+50726.510.188602344.5477.797616.561629.69761.0478b3Σ (99.82), 23Π (0.18)
    (5)150728.050.188632531.33412.6715.2041561.47521.0476b3Σ (99.92), 23Π (0.08)
    23Π250734.850.192151273.8620.79034.944713.095201.046723Π (100.00)
    ${\text{1} }{}^{\text{3} }{\Sigma}_{ { {\text{0} }^{{ - } } } }^ +$51738.080.125920.003113Σ+ (100.00)
    ${\text{1} }{}^{\text{5} }{\Sigma}_{ { {\text{0} }^{{ - } } } }^{{ - } }$58295.530.16981634.857167.6536.51872192.4710.109615Σ (100.00)
    ${\text{1} }{}^{\text{5} }{\Sigma}_{\text{2} }^{{ - } }$58295.550.16981634.862167.6606.51884192.5020.109615Σ (100.00)
    ${\text{1} }{}^{\text{5} }{\Sigma}_{\text{2} }^{{ - } }$58295.570.16981634.867167.6696.51902192.5500.109515Σ (100.00)
    下载: 导出CSV

    表 5  A1Π1(υ′, J′ = 1, +) → $ {\text{X}}{}^{\text{1}}{\Sigma}_{{{\text{0}}^ + }}^ + $(υ′′, J′′ = 1, –)跃迁的跃迁波数(${\tilde v} $)、爱因斯坦A系数(Aυ′υ′′)、振动分支比(Rυ′υ′′)、波长(λυ′υ′′)、加权的吸收振子强度(gfυ′υ′′)

    Table 5.  The transition wavenumber (${\tilde v} $), Einstein A-coefficients (Aυ′υ′′), vibrational branching ratios (Rυ′υ′′), wavelength (λυ′υ′′), and weighted absorption oscillator strengths (gfυ′υ′′) for the A1Π1(υ′, J′ = 1, +) → $ {\text{X}}{}^{\text{1}}{\Sigma}_{{{\text{0}}^ + }}^ + $(υ′′, J′′ = 1, –) transitions.

    υ′–υ${\tilde v}/$cm–1Aυ′υ′′/sRυ′υ′′λυ′υ′′/nmgfυ′υ′′ υ′–υ${\tilde v} $/cm–1Aυ′υ′′/sRυ′υ′′λυ′υ′′/nmgfυ′υ′′
    0-023140.447.98×1060.9912432.450.0067 1-025243.779.61×1040.0138396.426.78×10–4
    0-120862.556.67×1040.0083479.676.89×10–41-122965.886.80×1060.9777435.740.0580
    0-218684.363.86×1034.79×10–4535.594.97×10–51-220787.694.72×1040.0068481.404.91×10–4
    0-316602.974.43×1015.50×10–6602.737.22×10–71-318706.301.13×1040.0016534.961.45×10–4
    0-414612.081.752.17×10–7684.853.69×10–81-416715.407.15×1011.03×10–5598.681.15×10–6
    2-027090.981.76×1033.10×10–4369.391.08×10–53-028588.601.08×1032.66×10–4350.045.95×10–6
    2-124813.094.31×1050.0759403.300.00323-126310.721.89×1034.66×10–4380.341.23×10–5
    2-222634.905.22×1060.9192442.110.04583-224132.531.16×1060.2858414.670.0090
    2-320553.511.38×1032.43×10–4486.881.47×10–53-322051.132.80×1060.6887453.810.0259
    2-418562.622.46×1040.0043539.103.21×10–43-420060.244.31×1040.0106498.854.82×10–4
    2-516658.641.50×1012.64×10–6600.722.43×10–73-518156.265.40×1040.1330551.177.37×10–4
    下载: 导出CSV

    表 7  a3Π1(υ′, J′ = 1, +) →$ {\text{X}}{}^{\text{1}}{\Sigma}_{{{\text{0}}^ + }}^ + $(υ′′, J′′ = 1, –)跃迁的跃迁波数(${\tilde v} $)、爱因斯坦A系数(Aυ′υ′′)、振动分支比(Rυ′υ′′)、波长(λυ′υ′′)、加权的吸收振子强度(gfυ′υ′′)

    Table 7.  The transition wavenumber (${\tilde v} $), Einstein A-coefficients(Aυ′υ′′), vibrational branching ratios (Rυ′υ′′), wavelength (λυ′υ′′), and weighted absorption oscillator strengths (gfυ′υ′′) for the a3Π1(υ′, J′ = 1, +) →$ {\text{X}}{}^{\text{1}}{\Sigma}_{{{\text{0}}^ + }}^ + $(υ′′, J′′ = 1, –).

    υ′–υ${\tilde v} $/cm–1Aυ′υ′′/sRυ′υ′′λυ′υ′′/nmgfυ′υ′′ υ′–υ${\tilde v} $/cm–1Aυ′υ′′/sRυ′υ′′λυ′υ′′/nmgfυ′υ′′
    0-011039.580.12780.9615906.484.72×10–9 1-013546.430.00810.0607738.731.98×10–10
    0-18761.690.00500.03741142.142.91×10–101-111268.540.11480.8613888.064.07×10–9
    0-26583.501.48×10–40.00111520.031.54×10–111-29090.350.00990.07391100.855.36×10–10
    0-34502.113.00×10–62.25×10–52222.766.65×10–131-37008.965.33×10–40.00401427.764.88×10–11
    0-42511.213.32×10–82.50×10–73984.982.37×10–141-45018.071.86×10–51.39×10–41994.223.32×10–12
    2-015928.882.81×10–52.12×10–4628.244.98×10–133-018182.601.94×10–61.47×10–5550.372.63×10–14
    2-113651.000.01520.1142733.073.66×10–103-115904.715.54×10–54.22×10–4629.199.84×10–13
    2-211472.810.10230.7703872.253.49×10–93-213726.520.02110.1607729.045.04×10–10
    2-39391.410.01410.10591065.567.17×10–103-311645.130.09120.6945859.343.03×10–9
    2-47400.520.00120.00891352.229.71×10–113-49654.230.01680.12771036.558.09×10–10
    下载: 导出CSV

    表 6  $ {{\text{a}}^{\text{3}}}{\Pi_{{{\text{0}}^ + }}} $(υ′, J′ = 0, + ) →$ {\text{X}}{}^{\text{1}}{\Sigma}_{{{\text{0}}^ + }}^ + $(υ′′, J′′ = 1, –)跃迁的跃迁波数(${\tilde v} $)、爱因斯坦A系数(Aυ′υ′′)、振动分支比(Rυ′υ′′)、波长(λυ′υ′′)、加权的吸收振子强度(gfυ′υ′′)

    Table 6.  The transition wavenumber(${\tilde v} $), Einstein A-coefficients(Aυ′υ′′), vibrational branching ratios (Rυ′υ′′), wavelength (λυ′υ′′), and weighted absorption oscillator strengths (gfυ′υ′′) for the $ {{\text{a}}^{\text{3}}}{\Pi_{{{\text{0}}^ + }}} $(υ′, J′ = 0, + ) →$ {\text{X}}{}^{\text{1}}{\Sigma}_{{{\text{0}}^ + }}^ + $(υ′′, J′′ = 1, –) transitions.

    υ′–υ${\tilde v} $/cm–1Aυ′υ′′/sRυ′υ′′λυ′υ′′/nmgfυ′υ′′ υ′–υ${\tilde v} $/cm–1Aυ′υ′′/sRυ′υ′′λυ′υ′′/nmgfυ′υ′′
    0-011039.100.18780.8913906.522.31×10–9 1-013546.355.67×10–40.0030738.734.63×10–12
    0-18761.220.02160.10271142.214.22×10–101-111268.460.14410.7666888.061.70×10–9
    0-26583.030.00120.00581520.144.24×10–111-29090.270.03910.20821100.867.10×10–10
    0-34501.634.60×10–52.18×10–42223.003.40×10–121-37008.880.00390.02091427.781.20×10–10
    0-42510.749.71×10–74.61×10–63985.722.31×10–131-45017.982.33×10–40.00121994.251.39×10–11
    2-015929.213.36×10–40.0020628.221.98×10–123-018183.371.76×10–61.12×10–5550.347.97×10–15
    2-113651.320.00230.0142733.051.84×10–113-115905.489.50×10–40.0061629.165.63×10–12
    2-211473.130.10810.6354872.221.23×10–93-213727.290.00500.0320728.993.99×10–11
    2-39391.740.05070.29821065.528.62×10–103-311645.900.08170.5216859.289.03×10–10
    2-47400.840.00800.04681352.162.18×10–103-49655.010.05420.34641036.478.72×10–10
    下载: 导出CSV

    表 8  A1Π1(υ′, J′ = 1, +), $ {{\text{a}}^{\text{3}}}{\Pi _{{0^ + }}} $(υ′, J′ = 0, + )和a3Π1(υ′, J′ = 1, +)态的辐射寿命(τυ)

    Table 8.  Spontaneous radiative lifetimes(τυ′) for the A1Π1(υ′, J′ = 1, +), $ {{\text{a}}^{\text{3}}}{\Pi _{{0^ + }}} $(υ′, J′ = 0, +)和a3Π1(υ′, J′ = 1, +) transitions

    υ$ {\text{a}}{}^{\text{3}}{\Pi_{{{\text{0}}^ + }}} $/s a3Π1/s A1Π1/ns
    总和/ns$ {\text{A}}{}^{\text{1}}{\Pi_{\text{1}}}{\text{ - }}{{\text{X}}^{\text{1}}}{\Sigma}_{{{\text{0}}^ + }}^ + $/ns$ {\text{A}}{}^{\text{1}}{\Pi_{\text{1}}} $-$ {\text{a}}{}^{\text{3}}{\Pi_{{{\text{0}}^ + }}} $/sA1Π1– a3Π1/sA1Π1– a3Π2/s
    04.75 7.52 124.18124.182.71111.48177.04
    15.327.50143.86143.863.0390.08116.30
    25.887.53176.12176.123.5883.05192.36
    36.397.61246.20246.204.7793.55255.19
    46.857.78
    57.278.09
    下载: 导出CSV
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出版历程
  • 收稿日期:  2022-01-07
  • 修回日期:  2022-02-07
  • 上网日期:  2022-02-15
  • 刊出日期:  2022-05-20

BH分子8个Λ-S态和23个Ω态光谱性质的理论研究

  • 1. 信阳师范学院物理电子工程学院, 信阳 464000
  • 2. 河南师范大学物理学院, 新乡 453000
  • 3. 潍坊科技学院, 寿光 262700
  • 通信作者: 邢伟, wei19820403@163.com
    基金项目: 国家自然科学基金(批准号: 61275132, 11274097)、河南省自然科学基金(批准号: 212300410233)、河南省高等学校重点科研项目(批准号: 21A140023)和信阳师范学院南湖学者奖励计划青年项目资助的课题.

摘要: 本文利用内收缩多参考组态相互作用方法计算了BH分子8个低电子态(X1Σ+, a3Π, A1Π, b3Σ, 23Π, 13Σ+, 15Σ和15Π)和在自旋-轨道耦合效应下所产生的23个Ω态的势能曲线、以及${\rm{X}}{}^1\Sigma _{{0^ + }}^ +$, ${{\rm{a}}^3}{\Pi _{{0^ + }}}$, ${{\rm{a}}^3}{\Pi _1}$, ${{\rm{a}}^3}{\Pi _2}$${{\rm{A}}^1}{\Pi _1}$态之间6对跃迁的跃迁偶极矩. 为了获得精确的势能曲线, 计算中修正了单双电子激发、核价相关效应、相对论效应和基组截断带来的误差. 获得的BH分子的光谱和跃迁数据与现有的理论值和实验值符合得很好. 计算结果表明: BH分子的A1Π1(υ' = 0 – 2, J' = 1, +) →$ {\text{X}}{}^1\Sigma _{{0^ + }}^ + $(υ′′ = 0 – 2, J ′′ = 1, –)跃迁具有较大的爱因斯坦A系数和加权的吸收振子强度、高度对角化分布的振动分支比, A1Π1态具有较短的辐射寿命. 另外, ${{\rm{a}}^3}{\Pi _{{0^ + }}}$和a3Π1态对A1Π1(υ' = 0) ↔ $ {\rm X}^1\Sigma _{{0^ + }}^ + $(υ′′ = 0)循环跃迁的影响可以忽略. 因此, 基于A1Π1(υ'= 0—1, J ′ = 1, +) ↔ $ {\rm X}^1\Sigma _{{0^ + }}^ + $(υ′′ = 0—3, J′′ = 1, –)循环跃迁, 我们提出了用一束主冷却激光(λ00 = 432.45 nm)和两束再泵浦激光(λ10 = 479.67 nm和λ21 = 481.40 nm)冷却BH分子的方案, 并评价了冷却效果.

English Abstract

    • BH分子在天体物理[1-3]和激光冷却分子[4,5]中起着重要作用. 获得BH分子精确的光谱和跃迁数据对识别太阳光球和太阳黑子中的BH分子、分析激光冷却BH分子的可行性和构建激光冷却方案至关重要.

      实验上随着光谱技术的发展, 科学家们利用气相光谱技术从微波到紫外区域对BH分子进行高分辨的电子和振动-转动光谱研究[6-14]. 例如, John等[6]测量了近紫外区域A1Π ↔ X1Σ+跃迁的0-0, 1-0, 1-1, 2-1, 2-2, 3-2和3-3振转带. Luh和Stwalley[7]利用实验[6]中的光谱常数和RKR方法构建了X1Σ+, A1Π和B1Σ+态的势能曲线. Pianalto等[8]利用傅里叶变换光谱仪记录了X1Σ+态1-0, 2-1和3-2振转带的红外发射光谱. Douglass等[9]利用激光诱导荧光(LIF)技术观察到A1Π → X1Σ+跃迁的0-1和1-2振转带. Fernando和Bernath[10]利用傅里叶变换光谱记录了433 nm附近A1Π → X1Σ+跃迁的0-0, 1-1和2-2振转带. Persico[11]对A1Π态的各种衰减通道进行了相对完整的研究, 并推导出X1Σ+态离解能De的最佳实验值. Clark等[12]利用光子共振增强的多光子电离光谱观察了A1Π → X1Σ+跃迁的2-0振转带. Shayesteh和Ghazizadeh[13]利用获得的X1Σ+态的光谱数据并结合同位素拟合, 报道了X1Σ+态的Dunham系数. Brazier[14]利用发射光谱对b3Σ → a3Π跃迁进行了研究. 这些实验集中于研究该分子X1Σ+, A1П, a3П和b3Σ态的光谱性质, 报道了这4个电子态精确的光谱常数和分子常数、A1П → X1Σ+跃迁的部分数据(Franck-Condon因子、爱因斯坦A系数Aυ′υ′′和A1П态的辐射寿命τυ′); 但未报道考虑自旋-轨道耦合(SOC)后Ω 态的光谱和跃迁数据.

      近年来, 随着从头计算方法的快速发展, 人们对BH分子基态和激发态电子结构进行了高精度的理论研究[5,15-20]. Petsalakis和Theodorakopoulos[15,16],Miliordos和Mavridis[17]以及王新强等[18]利用多参考组态相互作用方法(MRCI)结合大的相关一致基组计算了BH分子一些Λ-S电子态的势能曲线, 并获得了这些电子态的光谱常数. Koput[19]采用多参考平均耦合对泛函(MR-ACPF)方法, 结合相关一致核价基确定了X1Σ+态的势能曲线, 为了获得可靠的光谱常数, 在计算中包含高阶电子相关、标量相对论(SR)效应、绝热和非绝热效应修正. Yan和Yan[20]纳入SR效应, 采用考虑Davidson修正(+Q)的显关联MRCI(MRCI-F12 + Q + SR)方法对X1Σ+和A1Π态的电子结构进行了高精度的研究, 并报道了势能曲线、光谱常数、振动能级ΔGυ、惯性转动常数Bυ和离心畸变常数Dυ, A1Π → X1Σ+跃迁数据(Franck-Condon因子、跃迁能量和A1Π态的τυ′). Gao和Gao[5]基于光谱和跃迁特性研究了激光冷却BH分子的可行性, 得到了A1Π → X1Σ+跃迁高度对角化的Franck-Condon因子. 然而, a3Π → X1Σ+是自旋禁阻跃迁, 只有在考虑SOC效应后, $ {{\rm a}^3}{\Pi _{{0^ + }}} $和a3Π1态到$ {\rm X}^1\Sigma _{{0^ + }}^ + $态的跃迁才可以发生; 在他们的研究中未涉及$ {{\rm a}^3}{\Pi _{{0^ + }}} $和a3Π1态对A1Π1$ {\rm X}^1\Sigma _{{0^ + }}^ + $光学循环的影响. 此外, 他们在构建BH分子电子态的势能曲线时没有考虑基组截断误差和相对论(SR和SOC)效应的影响. 因此, 本文纳入SR和SOC效应、核价相关效应(CV)和外推势能到完全基组(CBS)极限对BH分子的光谱和跃迁特性进行深入的研究.

    • H原子第一激发态(2Pu)与基态(2Sg)的能级间隔大于B原子的第一激发态(4Pg)与相应基态(2Pu)的能级间隔. 因此, BH分子前两个离解极限是B(2Pu) + H(2Sg)和B(4Pg) + H(2Sg). 利用 Wigner-Witmer定则, 推算出这两个离解极限产生8个Λ-S态(X1Σ+, a3Π, A1Π, b3Σ, 23Π, 13Σ+, 15Σ和15Π), 在SOC效应的作用下, 这8个Λ-S态将产生23个Ω态. 为了探讨电子态之间的相互作用对光谱和跃迁特性的影响, 我们对8个Λ-S态和23个Ω态的电子结构进行了研究. 本文在MOLPRO 2010.1程序包[21]C2v点群中计算BH分子8个Λ-S态、23个Ω态的势能曲线和电子态之间的跃迁偶极距. 在0.06322—1.04322 nm的核间距内, 首先基于Hartree-Fock(HF SCF)方法处理基态(X1Σ+)的电子波函数, 为了描述原子轨道, 两个原子都使用相关一致基组aug-cc-pV6Z(AV6Z)[22]. 然后利用态平均的完全活性空间自洽场(SA-CASSCF)和内收缩MRCI(icMRCI)方法分别处理静态电子相关和动态电子相关. 活性空间包括所有的价轨道(B原子的2s2p轨道和H原子的1s轨道)和B原子3s轨道, 即4个电子在6个分子轨道上. 此外, SA–CASSCF和icMRCI方法用于计算激发态(a3Π, A1Π, b3Σ, 23Π, 13Σ+, 15Σ和15Π)的电子结构. 由于icMRCI方法仅考虑了单双电子激发, 本文使用Davidson修正(+Q)[23]估计三阶和四阶电子激发对相关能的贡献. 每个电子态的核间距间隔为0.02 nm, 为了显示势能曲线的细节信息, 在0.10322—0.20122 nm范围内, 核间距间隔为0.002 nm.

      为了获得8个电子态精确的势能曲线, 在上述计算的基础上, 本文考虑了CV效应、SR效应并外推势能到CBS极限. 具体处理方法为: 在icMRCI + Q理论水平上使用cc-pCVTZ (CVTZ)基组[24]计算CV. CV贡献的势能为: $\Delta {E_{\rm corr}} = {E_{\rm corr}} (\text{all-electron}) - $$ E_{\rm corr}(\text{frozen-core})$, 其中$E_{\rm corr}(\text{all-electron})$为BH分子的6个电子都参与计算获得的相关能, $E_{\rm corr}(\text{frozen-core})$为所有价轨道上的电子参与计算获得的相关能; 在icMRCI + Q理论水平上利用包含三阶Douglas-Kroll-Hess (DKH3)[25]近似的cc-pV5Z-DK基组[26]计算SR效应. 为了消除AV6Z存在的基组截断误差, 本文在icMRCI + Q方法结合AV6Z和AV5Z基组获得的势能曲线基础上, 利用Oyeyemi等[27]提出的外推公式将这8个电子态的势能曲线外推至CBS极限, 表示为icMRCI + Q/56; 将CV和SR贡献的势能加到icMRCI + Q/56势能里, 便得到icMRCI + Q/56 + CV + SR理论水平上8个Λ-S态的势能曲线, 如图1所示.

      图  1  BH分子8个Λ-S态的势能曲线

      Figure 1.  Potential energy curves of 8Λ-S states of the BH molecule.

      本文利用icMRCI + Q方法结合全电子CVTZ基组进行SOC计算. 采用带Breit-Pauli SOC算符[28] (HSO)和不带HSO的全电子CVTZ基组来计算势能. 这两种能量的差值即为SOC效应对总能量的贡献. 将SOC效应贡献的能量加到icMRCI + Q/56 + CV + SR结果的势能中, 便得到icMRCI + Q/56 + CV + SR + SOC理论水平上23个Ω态精确的势能曲线, 如图2所示.

      图  2  BH分子23个Ω态的势能曲线

      Figure 2.  Potential energy curves of 23 Ω states of the BH molecule.

      基于上述势能曲线, 利用LEVEL 8.2 程序[29]求解核运动的振转 Schrödinger方程, 获得 7个束缚Λ-S态(X1Σ+, a3Π, A1Π, b3Σ, 23Π, 13Σ+和15Σ)以及17个束缚和准束缚Ω态的光谱常数(Te, Re, ωe, ωexe, Be, αeDe)和分子常数; 然后, 基于$ X^1\Sigma _{{0^ + }}^ + $, $ {a^3}{\Pi _{{0^ + }}} $, $ {a^3}{\Pi _1} $, $ {a^3}{\Pi _2} $$ {{\text{A}}^1}{\Pi _1} $态的势能曲线和icMRCI/AV6Z + SOC理论水平的跃迁偶极距, 获得这5个Ω态之间跃迁的 Franck-Condon因子和Aυ′υ′′. 由于振动分支比(Rυ′υ′′)决定了不同电子振动态之间跃迁光子损失路径的相对强度、振子强度决定了跃迁体系吸收或发射的能力、并且天文学家通常使用吸收振子强度 ($ {f_{\upsilon 'J' \leftarrow \upsilon ''J''}} $)和加权的吸收振子强度$ g{f_{\upsilon 'J' \leftarrow \upsilon ''J''}} $, 所以将计算这5个Ω态之间跃迁的$ {R_{\upsilon '\upsilon ''}} $, $ {f_{\upsilon 'J' \leftarrow \upsilon ''J''}} $$ g{f_{\upsilon 'J' \leftarrow \upsilon ''J''}} $. 最后, 计算激发Ω态($ {{\rm a}^3}{\Pi _{{0^ + }}} $, a3Π1和A1Π1)的τυ′以及A1Π1 (υ′ = 0, J′ = 1, +) ↔ $ {\rm X}^1\Sigma _{{0^ + }}^ + $(υ′′ = 0, J′′ = 1, –)跃迁的多普勒温度(TDoppler)和回弹温度(TRecoil ).

    • icMRCI + Q/56 + CV + SR理论水平上计算的Λ-S态的离解关系列于表1. 由表1可知, 本文结果与实验估计值[30]和理论[31]符合非常好, 因此本文利用的方法很好地描述了BH分子的解离情况.

      离解极限Λ-S态能级a/cm–1
      本文实验[30]理论[31]
      B(2Pu) + H(2Sg)X1Σ+, a3Π, A1Π, 13Σ+0.000.000.00
      B(4Pg) + H(2Sg)b3Σ, 15Σ, 23Π, 15Π28907.6628644.99+xb28932.70
      a, 4Pg态能级为4P1/2, 4P3/24P5/2能级的算术平均值减去2P3/22P1/2能级的算术平均值; b, 4P5/2能级外推值的不确定度.

      表 1  BH分子前两个离解极限产生的8个Λ-S态的离解关系

      Table 1.  Dissociation relationships of the 8 Λ –S states generated from the first two dissociation asymptotes of the BH molecule

      图1可知, X1Σ+, a3Π, A1Π, 13Σ+, b3Σ, 23Π和15Σ为束缚态, 15Π态为排斥态. 为方便讨论, 表2列出了本文计算的7个束缚Λ-S态光谱常数、挑选的实验值[7,8,10-14]和其它理论值[5,15-20].

      Λ-S态来源Te/cm–1Re/nmωe/cm–1ωexe/cm–1Be/cm–1αe/(102 cm–1)De/eV
      X1Σ+本文00.122952367.2848.778212.039537.09853.7137
      实验[8]00.123222366.7349.338412.025542.1516
      实验[10]02366.7349.339812.025842.1565
      实验[12]02364.6647.709812.025742.15913.6476±0.0037a
      实验[13]00.123222366.7349.340512.025542.1450
      理论[5]00.122902352.044.012.0863.6863
      理论[15]00.12301237946.7912.073.70
      理论[16]00.12312237812.0553.578b
      理论[17]00.1230235948.841.83.6773
      理论[18]00.123272368.4850.695712.11043.053.6580
      理论[19]00.123003.6751
      理论[20]00.122932365.6947.231012.080141.63.6851
      a3Π本文10944.320.118992625.9759.417712.891941.64042.3507
      实验[14]xc0.119002625.1455.784012.893141.56102.3867
      理论[5]10645.00.119002961.0109.612.9042.3806
      理论[15]0.11913265362.7012.872.38
      理论[17]105830.11900262560.445.52.3677
      理论[18]9557.670.119252598.9846.630012.940042.532.3135
      A1Π本文23203.520.122232253.2836.831011.834311.62540.8368
      实验[10]23135.440.12195d2251.4656.572512.2003553.76700.697d
      实验[12]23105.102342.41127.761812.1998653.67360.7786±0.0037a
      理论[5]22997.900.122102404.60147.312.27950.9098
      理论[15]0.122132320136.512.240.71
      理论[16]230610.12235229012.200.73b
      理论[17]231440.12222341129.685.10.8109
      理论[18]22260.890.122672280.2693.623312.22960.830.7536
      理论[19]23099.840.122122343.96128.17812.283674.00.8938
      b3Σ本文38238.630.121642440.8954.447712.250833.67122.5959
      实验[14]xc+27152.750.1216252438.1055.56212.342643.0872.5987
      理论[15]0.12256234548.4512.162.54
      理论[17]377080.1217243057.345.92.5845
      理论[18]36859.520.121992428.3355.40912.28444.312.5403
      23Π本文50730.460.192151273.8920.78964.944713.09571.0467
      理论[15]0.19338142557.044.881.04
      理论[17]502160.1931129538.69.91.0321
      13Σ+本文51738.070.125920.0031
      理论[17]516880.123
      15Σ本文58295.540.16981634.868167.6766.51936192.6410.1093
      理论[17]576740.170152887.3153.20.1084
      a, 文献[11]中的值; b, D0值; c, x表示a3Π态相对于X1Σ+态的Te值; d, 文献[7]中的值.

      表 2  icMRCI + Q/56 + CV + SR理论水平上BH分子7个Λ-S态的光谱常数

      Table 2.  Spectroscopic parameters of the 7 Λ-S states of BH at level of icMRCI + Q/56 + CV + SR.

      X1Σ+态在Re处的主要电子组态为1σ222000(0.8995), 小括号里为组态波函系数的平方. 它的势阱深度为29954.56 cm–1, 包含22个振动态. 由表2可知, 本文计算的 Re, ωe, BeDe与实验值[8,10,12,13]吻合, 它们与实验值[8,10,12,13]的最大偏离分别为0.00027 nm (0.219%), 2.62 cm–1 (0.1108%), 0.014 cm–1 (0.1164%)和0.0661 eV (1.8122%); 仅文献[15-19]中的Re值和文献[5,15,17-20]中的De值分别比本文的计算值稍微接近实验值[8,13]和实验值[11].

      第一激发态a3Π和第二激发态A1Π通过3σ → 1π的单电子激发形成, 它们在各自Re处的主要电子组态分别为1σ221100(0.9413)和1σ221100 (0.9129). a3Π态的势阱深度为18961.07 cm–1, 包含12个振动态. A1Π态在R = 0. 21289 nm附近出现势垒, 势垒顶部的势能高于无穷远处的势能, 势阱深度为7342.74 cm–1, 包含4个振动态, 这与实验[12]和理论[16]的结论相同. 由表2知, 本文计算的这两个态的光谱常数与实验值[7,10-12,14]吻合很好.

      第三激发态b3Σ态通过3σ → 1π的双电子激发形成, 其在Re处的主要电子组态为1σ220200(0.8939). b3Σ态的势阱深度为20938.32 cm–1, 包含14个振动态, 它与13Σ+态的排斥部分在R = 0.16467 nm处交叉, 计算表明b3Σ态的预解离始于υ′ = 2, J' = 11能级.

      13Σ+态由3σ → 4σ的单电子激发形成, 其在Re处的主要电子组态为2σ21010(0.8001), 并在R = 0.13122 nm附近出现势垒, 势垒高于无穷远处, 局域势阱深度为24.80 cm–1, 不包含任何振动态. 本文的结论与Miliordos和Mavridis [17]的相同.

      23Π态具有明显的多参考特征, 其在Re处主要价电子组态为1σ212100(0.6130)和1σ220110(0.2370). 因此, 从a3Π态到23Π态的主要电子跃迁是2σ → 3σ和3σ → 4σ. 23Π态与b3Σ态在R = 0.18920 nm处交叉, 交叉点位于23Π态的Re附近, 23Π(υ′ 0)能级将受到b3Σ(υ′ 6)能级的微扰, 这解释了实验上未报道23Π态光谱的原因.

      弱束缚态15Σ通过2σ → 1π和3σ → 1π的双电子激发形成, 其在Re处的主要价电子组态为1σ211200(0.9645). 它的势阱深度为882.07 cm–1, 仅包含3个振动态. 2σ → 1π和3σ → 4σ的双电子激发形成排斥态15Π.

    • SOC效应使B原子的基态2Pu和第一激发态4Pg分别分裂成2P1/22P3/2组分以及4P1/2, 4P3/24P5/2组分. 因此, 前两个离解极限B(2Pu)+H(2Sg)和B(4Pg) + H(2Sg)分裂成5条离解极限, 即B(2P1/2) + H(2S1/2), B(2P3/2) + H(2S1/2), B(4P1/2) + H(2S1/2), B(4P3/2) + H(2S1/2)和B(4P5/2) + H(2S1/2). 表3中列入了这5个离解极限的能量间隔及它们所产生的23个Ω态.

      原子态(B + H)Ω态能级/cm–1
      本文实验[30]
      B(2P1/2) + H(2S1/2)0, 0+, 10.000.00
      B(2P3/2) + H(2S1/2)2, 1(2), 0+, 014.57215.287
      B(4P1/2) + H(2S1/2)0, 0+, 128910.6328647.43+x a
      B(4P3/2) + H(2S1/2)2, 1(2), 0+, 028914.6728652.07+x a
      B(4P5/2) + H(2S1/2)3, 2(2), 1(2), 0+, 028921.4128658.40+x a
      a, 4P5/2能级外推值的不确定度.

      表 3  BH分子23个Ω态的离解关系

      Table 3.  Dissociation relationships of the 23 Ω states of the BH molecule.

      表3可知, 本文利用icMRCI + Q/56+CV+ SR+SOC计算的B原子2P3/2 2P1/2, 4P1/2 2P3/2, 4P3/2 4P1/24P5/2 4P3/2的能量间隔分别与相应实验值[30]的差别仅为0.715 cm–1, 263.915 - x cm–1, 0.600 cm–1和0.410 cm–1. 17个束缚和准束缚Ω态的光谱常数见表4.

      Ω态Te/cm–1Re/nmωe/cm–1ωexe/cm–1Be/cm–1102αe/cm–1De/eVRe附近主要的Λ–S态/%
      $ {\text{X}}{}^{\text{1}}{\Sigma}_{{{\text{0}}^ + }}^ + $00.122952367.2848.778312.039537.09853.7138X1Σ+ (100.00)
      ${\text{a} }{}^{\text{3} }{\Pi_{ { {\text{0} }^{{ - } } } }}$10940.360.118992625.9359.416512.891841.64262.3506a3Π (100.00)
      $ {\text{a}}{}^{\text{3}}{\Pi_{{{\text{0}}^ + }}} $10940.370.118992625.9359.419212.891841.64242.3506a3Π (100.00)
      a3Π110944.320.118992625.9759.414012.891941.64032.3513a3Π (100.00)
      a3Π210948.490.118992626.0159.413112.891941.63842.3509a3Π (100.00)
      A1Π123203.520.122232253.2836.831711.833811.70340.9051A1Π (100.00)
      (3)0+第一势阱38244.330.121632438.0844.728112.292538.30411.5501b3Σ (100.00)
      (3)0+第二势阱50725.860.192130.002623Π (100.00)
      (3)138244.350.121632447.6954.293412.316737.61490.8995b3Σ (100.00)
      (4)145758.490.164964850.091293.006.739524.021681.662913Σ+ (100.00)
      ${\text{2} }{}^{\text{3} }{\Pi_{ { {\text{0} }^{ { - } } } } }$50726.070.192141274.0020.84504.944703.096111.046623Π (100.00)
      (4)0+50726.510.188602344.5477.797616.561629.69761.0478b3Σ (99.82), 23Π (0.18)
      (5)150728.050.188632531.33412.6715.2041561.47521.0476b3Σ (99.92), 23Π (0.08)
      23Π250734.850.192151273.8620.79034.944713.095201.046723Π (100.00)
      ${\text{1} }{}^{\text{3} }{\Sigma}_{ { {\text{0} }^{{ - } } } }^ +$51738.080.125920.003113Σ+ (100.00)
      ${\text{1} }{}^{\text{5} }{\Sigma}_{ { {\text{0} }^{{ - } } } }^{{ - } }$58295.530.16981634.857167.6536.51872192.4710.109615Σ (100.00)
      ${\text{1} }{}^{\text{5} }{\Sigma}_{\text{2} }^{{ - } }$58295.550.16981634.862167.6606.51884192.5020.109615Σ (100.00)
      ${\text{1} }{}^{\text{5} }{\Sigma}_{\text{2} }^{{ - } }$58295.570.16981634.867167.6696.51902192.5500.109515Σ (100.00)

      表 4  利用icMRCI + Q/56 + CV + SR + SOC理论计算获得的17个Ω态的光谱常数

      Table 4.  Spectroscopic parameters obtained by the icMRCI + Q/56 + CV + SR + SOC calculations for the 17 Ω states.

      图1可知, X1Σ+, a3Π, A1Π和15Σ态势能曲线不与其它电子态的势能曲线交叉, 考虑SOC效应后, 这4个Λ-S态所产生的9个Ω态($ X^1 \Sigma _{{0^ + }}^ + $, $ {{\text{a}}^{3}}{{\Pi}_{{0^{ - }}}} $, $ {{\text{a}}^3}{{\Pi}_{{0^ + }}} $, a3Π1, a3Π2, A1Π1, $ 1^5\Sigma_{{0^{-}}}^{-} $, $ 1^5\Sigma _1^ - $$ 1^5{\Sigma}_2^{-} $)的光谱常数与相应Λ-S态的光谱常数差别很小, 并且这9个Ω态在各自Re处波函的Λ-S成分全部来自相应的Λ-S态. 当R < 0.44322 nm时, ${a^3}{\Pi _{{0^{-}}}}$, $ {a^3}{\Pi _{{0^ + }}} $, {\rm a}3Π1和{\rm a}3Π2的势能依次增加. 由表4可知, 在各自Re处, $ {{\rm a}^3}{\Pi _{{0^ + }}} $${\rm a}^3\Pi_{0^{-}}$的分裂能为0.01 cm–1; a3Π1${{\text{a}}^3}{{\Pi}_{{0^ + }}}$和a3Π2 – a3Π1的分裂能分别为3.95 cm–1和4.17 cm–1, 它们与Brazier[14]报道的a3Π态的SOC分裂能4.3878 cm–1吻合地很好. 在R = 0.44322 nm附近, $ {{\rm a}^3}{\Pi _{{0^ + }}} $态势能曲线与a3Π1态势能曲线交叉, 当R > 0.44322 nm时, $ {{\rm a}^3}{\Pi _{{0^ + }}} $的势能大于a3Π1态的势能, 这导致$ {{\text{a}}^3}{{\Pi}_{{0^ + }}} $和a3Π1态的离解极限分别为B(2P3/2) + H(2S1/2)和B(2P1/2) + H(2S1/2).

      在–25.084165至–25.060894 Hartree的能量范围内, b3Σ态的势能曲线与13Σ+和23Π态的势能曲线交叉; 考虑SOC效应后, 这3个Λ-S态分裂出的Ω = 0+和1的成分出现避免交叉, 这导致(3)0+, (4)0+, (3)1, (4)1和(5)1态势能曲线的形状和相应Λ-S态势能曲线的形状不同, 并且这5个Ω态出现了一些局域势阱; 因此, 这5个Ω态的光谱常数也有很大的变化.

      13Σ+分裂出的$ 1^3{\Sigma}_{{0^{-}}}^ + $态、23Π态分裂出的$ {2^3}{\Pi _{{0^{-}}}} $和23Π2与其它Ω态没有避免交叉. 由表2表4可知, 它们的光谱常数与相应Λ-S态的光谱常数的差别也很小.

      SOC效应使排斥态$ 1^5\Pi $态分裂为$ 1^5{\Pi _{ - 1}} $, $ 1^5{{\Pi}_{{0^{-}}}} $, $ 1^5{\Pi _{{0^ + }}} $, $ 1^5{\Pi _1} $, $ 1^5{{\Pi}_2} $$ 1^5{{\Pi}_3} $6个Ω态.

    • 由3.2的讨论可知, 最低的5个Ω态($ {\rm X}^1\Sigma _{{0^ + }}^ + $, $ {{\text{a}}^3}{{\Pi}_{{0^ + }}} $, a3Π1, a3Π2和A1Π1)受到其他态的微扰较小, 基于电偶极跃迁的角动量选择定则ΔJ = J′J′′ = 0, ±1和宇称选择定则: + ↔ –(+, –是电子态的转动能级的宇称), 利用icMRCI/AV6Z + SOC方法计算了它们之间6对跃迁[A1Π1(υ′, J′ = 1, +) → $ {\text{X}}^1{\Sigma}_{{0^ + }}^ + $(υ′′, J′′ = 1, –), A1Π1(υ′, J′ = 1, +) → a3Π2(υ′′, J′′ = 2, –), A1Π1(υ′, J′ = 1, +) → $ {{\text{a}}^3}{{\Pi}_{{0^ + }}} $(υ′′, J′′ = 1, –), A1Π1(υ′, J′ = 1, +) → a3Π1(υ′′, J′′ = 1, –), $ {{\text{a}}^3}{{\Pi}_{{0^ + }}} $(υ′, J′ = 0, +) → $ {\text{X}}^1{\Sigma}_{{0^ + }}^ + $(υ′′, J′′ = 1, –)和a3Π1(υ′, J′ = 1, +) → $ {\text{X}}^1{\Sigma}_{{0^ + }}^ + $(υ′′, J′′ = 1, –)]的跃迁偶极矩, 如图3所示. 为了分析太阳光球和太阳黑子光谱、建立它们的物理模型和探讨激光冷却BH分子的可行性, 本文基于上述的势能曲线和跃迁偶极矩, 借助于LEVEL 8.2程序[29]获得了这6对跃迁的Aυ′υ′′, 波长λυ′υ′′, Rυ′υ′′, $ {f_{\upsilon 'J' \leftarrow \upsilon ''J''}} $$ g{f_{\upsilon 'J' \leftarrow \upsilon ''J''}} $, 如表57附录A表A1A3所列; 并计算了A1Π1(υ′, J′ = 1, +), $ {{\text{a}}^3}{{\Pi}_{{0^ + }}} $(υ′, J′ = 0, + )和a3Π1(υ′, J′ = 1, +)态的τυ′, 见表8.

      图  3  BH分子6对跃迁的跃迁偶极矩曲线

      Figure 3.  Curves of the transition dipole moments versus internuclear separation of six-pair states of the BH molecule.

      υ′–υ${\tilde v}/$cm–1Aυ′υ′′/sRυ′υ′′λυ′υ′′/nmgfυ′υ′′ υ′–υ${\tilde v} $/cm–1Aυ′υ′′/sRυ′υ′′λυ′υ′′/nmgfυ′υ′′
      0-023140.447.98×1060.9912432.450.0067 1-025243.779.61×1040.0138396.426.78×10–4
      0-120862.556.67×1040.0083479.676.89×10–41-122965.886.80×1060.9777435.740.0580
      0-218684.363.86×1034.79×10–4535.594.97×10–51-220787.694.72×1040.0068481.404.91×10–4
      0-316602.974.43×1015.50×10–6602.737.22×10–71-318706.301.13×1040.0016534.961.45×10–4
      0-414612.081.752.17×10–7684.853.69×10–81-416715.407.15×1011.03×10–5598.681.15×10–6
      2-027090.981.76×1033.10×10–4369.391.08×10–53-028588.601.08×1032.66×10–4350.045.95×10–6
      2-124813.094.31×1050.0759403.300.00323-126310.721.89×1034.66×10–4380.341.23×10–5
      2-222634.905.22×1060.9192442.110.04583-224132.531.16×1060.2858414.670.0090
      2-320553.511.38×1032.43×10–4486.881.47×10–53-322051.132.80×1060.6887453.810.0259
      2-418562.622.46×1040.0043539.103.21×10–43-420060.244.31×1040.0106498.854.82×10–4
      2-516658.641.50×1012.64×10–6600.722.43×10–73-518156.265.40×1040.1330551.177.37×10–4

      表 5  A1Π1(υ′, J′ = 1, +) → $ {\text{X}}{}^{\text{1}}{\Sigma}_{{{\text{0}}^ + }}^ + $(υ′′, J′′ = 1, –)跃迁的跃迁波数(${\tilde v} $)、爱因斯坦A系数(Aυ′υ′′)、振动分支比(Rυ′υ′′)、波长(λυ′υ′′)、加权的吸收振子强度(gfυ′υ′′)

      Table 5.  The transition wavenumber (${\tilde v} $), Einstein A-coefficients (Aυ′υ′′), vibrational branching ratios (Rυ′υ′′), wavelength (λυ′υ′′), and weighted absorption oscillator strengths (gfυ′υ′′) for the A1Π1(υ′, J′ = 1, +) → $ {\text{X}}{}^{\text{1}}{\Sigma}_{{{\text{0}}^ + }}^ + $(υ′′, J′′ = 1, –) transitions.

      υ′–υ${\tilde v} $/cm–1Aυ′υ′′/sRυ′υ′′λυ′υ′′/nmgfυ′υ′′ υ′–υ${\tilde v} $/cm–1Aυ′υ′′/sRυ′υ′′λυ′υ′′/nmgfυ′υ′′
      0-011039.580.12780.9615906.484.72×10–9 1-013546.430.00810.0607738.731.98×10–10
      0-18761.690.00500.03741142.142.91×10–101-111268.540.11480.8613888.064.07×10–9
      0-26583.501.48×10–40.00111520.031.54×10–111-29090.350.00990.07391100.855.36×10–10
      0-34502.113.00×10–62.25×10–52222.766.65×10–131-37008.965.33×10–40.00401427.764.88×10–11
      0-42511.213.32×10–82.50×10–73984.982.37×10–141-45018.071.86×10–51.39×10–41994.223.32×10–12
      2-015928.882.81×10–52.12×10–4628.244.98×10–133-018182.601.94×10–61.47×10–5550.372.63×10–14
      2-113651.000.01520.1142733.073.66×10–103-115904.715.54×10–54.22×10–4629.199.84×10–13
      2-211472.810.10230.7703872.253.49×10–93-213726.520.02110.1607729.045.04×10–10
      2-39391.410.01410.10591065.567.17×10–103-311645.130.09120.6945859.343.03×10–9
      2-47400.520.00120.00891352.229.71×10–113-49654.230.01680.12771036.558.09×10–10

      表 7  a3Π1(υ′, J′ = 1, +) →$ {\text{X}}{}^{\text{1}}{\Sigma}_{{{\text{0}}^ + }}^ + $(υ′′, J′′ = 1, –)跃迁的跃迁波数(${\tilde v} $)、爱因斯坦A系数(Aυ′υ′′)、振动分支比(Rυ′υ′′)、波长(λυ′υ′′)、加权的吸收振子强度(gfυ′υ′′)

      Table 7.  The transition wavenumber (${\tilde v} $), Einstein A-coefficients(Aυ′υ′′), vibrational branching ratios (Rυ′υ′′), wavelength (λυ′υ′′), and weighted absorption oscillator strengths (gfυ′υ′′) for the a3Π1(υ′, J′ = 1, +) →$ {\text{X}}{}^{\text{1}}{\Sigma}_{{{\text{0}}^ + }}^ + $(υ′′, J′′ = 1, –).

      分别利用下面的公式计算$ {R_{\upsilon '\upsilon ''}} $$ {f_{\upsilon 'J' \leftarrow \upsilon ''J''}} $$ g{f_{\upsilon 'J' \leftarrow \upsilon ''J''}} $$ {\tau _{\upsilon '}} $:

      $ {R_{\upsilon '\upsilon ''}} = {A_{\upsilon '\upsilon ''}}\Big/\sum\limits_{\upsilon ''} {{A_{\upsilon '\upsilon ''}}} \text{, } $

      $\begin{split} & {f_{\upsilon 'J' \leftarrow \upsilon ''J''}} \\ =\;& 1.4991938\frac{1}{{{{\tilde u }^2}}}\frac{{2J' + 1}}{{2J'' + 1}}{A_{\upsilon 'J' \to \upsilon ''J''}}, \end{split}$

      $ g{f_{\upsilon 'J' \leftarrow \upsilon ''J''}} = \left( {2J'' + 1} \right){f_{\upsilon 'J' \leftarrow \upsilon ''J''}} \text{, } $

      $ {\tau _{\upsilon '}} = 1/{A_{\upsilon '}} = 1/\sum\nolimits_i {{A_{i\upsilon '}}} . $

      (1)式—(4)式中, $ \tilde u $为跃迁波数、单位为cm–1; g为低能级的统计权重; Ai, υ′是从上能级υ′发射的第i个系统的总爱因斯坦A系数.

      表5知, A1Π1(υ′, +) → $ {\text{X}}^1{\Sigma}_{{0^ + }}^ + $(υ′′, –)跃迁的19条振转带(0-0, 0-1, 0-2, 1-0, 1-1, 1-2, 1-3, 2-0, 2-1, 2-2, 2-3, 2-4, 3-0, 3-1, 3-2, 3-3, 3-4, 3-5和3-6)的Q(1)支具有较大的Aυ′υ′′$ g{f_{\upsilon 'J' \leftarrow \upsilon ''J''}} $, 这表明这19条振转带的跃迁强度比较强. 因此0-2, 1-3, 2-3, 2-4, 3-0, 3-1, 3-4, 3-5和3-6也应是潜在的可观察振转带. 本文计算的0-0, 1-0, 1-1, 2-0, 2-1, 2-2, 3-2和3-3的${\tilde v} $稍微大于相应的实验值[6,10,12], 它们与实验值[6,10,12]的最大偏差仅为66.324 cm–1 (0.287%), 84.256 cm–1 (0.335%), 74.807 cm–1 (0.327%), 101.920 cm–1 (0.378%), 92.492 cm–1 (0.374%), 87.300 cm–1 (0.387%), 106.62 cm–1 (0.444%)和104.91 cm–1 (0.478%); A1Π1 (υ′, J′ = 1, +) → $ {\text{X}}^1{\Sigma}_{{0^ + }}^ + $(υ′′, J′′ = 1, –)跃迁具有高度对角化的Rυ′υ′′ (R00 = 0.9912, R11 = 0.9777和R22 = 0.9192), 并且本文获得的0-0, 0-1, 0-2和0-3振转带的Rυ′υ′′分别与Hendricks等[4]报道的相应值吻合. 另外, 我们计算的A1Π1(υ′ = 0, J′ = 1, +) → $ {\text{X}}^1{\Sigma}_{{0^ + }}^ + $(υ′′ = 0, J′′ = 1, –)的τ00为125.28 ns, 对应的光子散射速率(7.98 × 106 s–1)符合快速激光冷却的要求(105—108 s–1)[32]. 基于获得的Rυ′υ′′$ {\rm X}^1\Sigma _{{0^ + }}^ + $(υ′′ = 0 – 3, J′′ = 1, –) → A1Π1 (υ′ = 0 – 1, J′ = 1, +)跃迁的主冷却激光波长和再泵浦激光波长, 构建了一个需要三束激光的振动-转动态的闭合冷却方案, 如图4所示. 由图4可知, 需要一束波长λ00 = 432.45 nm的主激光驱动$ {\text{X}}^1{\Sigma}_{{0^ + }}^ + $(υ′′ = 0, J′′ = 1, –) → A1Π1(υ′ = 0, J′ = 1, +)跃迁, 为了增强冷却效果, 我们增加了两束波长为λ10 = 479.67 nm 和λ21 = 481.40 nm的再泵浦激光分别驱动$ {\text{X}}^1{\Sigma}_{{0^ + }}^ + $(υ′′ = 1, J′′ = 1, –) → A1Π1(υ′ = 0, J′ = 1, +)和$ {\text{X}}^1{\Sigma}_{{0^ + }}^ + $(υ′′ = 2, J′′ = 1, –) → A1Π1(υ′ = 1, J′ = 1, +)跃迁. 可见, 三束激光的波长都在可见光的范围, 可以用价格低廉的半导体激光器获得这三束激光. 此外, 本文计算的激光波长仅分别比实验[9]中的相应值小0.75 nm(0.173%), 0.73 nm(0.152%), 1.00 nm (0.207%). 在冷却循环中, 每个BH分子至少散射1.58 × 105个光子才会有一个光子损失在$ {\text{X}}^1{\Sigma}_{{0^ + }}^ + $υ′′ ≥ 3能级.

      图  4  利用A1Π1(υ′) ↔$ {\text{X}}{}^{\text{1}}{\Sigma}_{{{\text{0}}^ + }}^ + $(υ′′)跃迁进行激光冷却BH分子的方案. 虚线表示A1Π1(υ′ = 0, 1) →$ {\text{X}}{}^{\text{1}}{\Sigma}_{{{\text{0}}^ + }}^ + $(υ′′ = 0 –3)跃迁的自发辐射振动分支比(Rυ′υ′′). 红色实线表示激光驱动$ {\text{X}}{}^{\text{1}}{\Sigma}_{{{\text{0}}^ + }}^ + $(υ′′) → A1Π1 (υ′)跃迁

      Figure 4.  The proposed laser cooling scheme for the BH using A1Π1(υ′) ↔$ {\text{X}}{}^{\text{1}}{\Sigma}_{{{\text{0}}^ + }}^ + $(υ′′) transition. The dotted line indicate the spontaneous radiation vibrational branching ratio (Rυ′υ′′) of A1Π1(υ′ = 0, 1) →$ {\text{X}}{}^{\text{1}}{\Sigma}_{{{\text{0}}^ + }}^ + $(υ′′ = 0 – 3) transition. The red solid line indicate the wavelength (λυ′′υ′) at which the laser drives the $ {\text{X}}{}^{\text{1}}{\Sigma}_{{{\text{0}}^ + }}^ + $(υ′′) →A1Π1 (υ′). transition.

      表6表7可知, $ {{\text{a}}^3}{{\Pi}_{{0^ + }}} $(υ′, J′ = 0, + ) → $ {\text{X}}^1{\Sigma}_{{0^ + }}^ + $(υ′′, J′′ = 1, –)和a3Π1(υ′, J′ = 1, +) → $ {\text{X}}^1{\Sigma}_{{0^ + }}^ + $(υ′′, J′′ = 1, –)跃迁的Rυ′υ′′也具有对角化, 但是这两对跃迁的Aυ′υ′′$ g{f_{\upsilon 'J' \leftarrow \upsilon ''J''}} $很小, τυ′太长[τ0($ {{\text{a}}^3}{{\Pi}_{{0^ + }}} $) = 4.75 s和τ0(a3Π1) = 7.52 s], 不满足Di Rosa准则[32]. 因此, 这两对跃迁不能用于激光冷却BH分子. 但$ {{\text{a}}^3}{{\Pi}_{{0^ + }}} $和a3Π1态是A1Π1(υ′, J′ = 1, +) ↔ $ {\rm X}^1\Sigma _{{0^ + }}^ + $(υ′′, J′′ = 1, –)光学循环的中间态, 本文计算获得A1Π1(υ′, J′ = 1, +) → $ {\text{X}}^1{\Sigma}_{{0^ + }}^ + $(υ′′, J′′ = 1, –), A1Π1(υ′, J′ = 1, +) → $ {{\text{a}}^3}{{\Pi}_{{0^ + }}} $(υ′′, J′′ = 1, –)和A1Π1(υ′, J′ = 1, +) → a3Π1(υ′′, J′′ = 1, –)的总Aυ′υ′′分别为$ {\gamma _\Sigma } $= 8.05×106 s–1, $ {\gamma _1} $= 0.3686 s–1$ {\gamma _2} $= 0.0090 s–1, 因此A1Π1(υ′, J′ = 1, +) → $ {{\text{a}}^3}{{\Pi}_{{0^ + }}} $(υ′′, J′′ = 1, –)和A1Π1(υ′, J′ = 1, +) →a3Π1(υ′′, J′′ = 1, –)的振动分支损失比分别为$ {\eta _1} = {\gamma _1}/{\gamma _\Sigma } $< 4.6 × 10–8$ {\eta _2} = {\gamma _2}/{\gamma _\Sigma } $< 1.2 × 10–9. 这两个值远小于YO的实验值4.0 × 10–4[33], 这表明$ {{\text{a}}^3}{{\Pi}_{{0^ + }}} $和a3Π1态对A1Π1(υ′, J′ = 1, +) ↔ $ {\text{X}}^1{\Sigma}_{{0^ + }}^ + $(υ′′, J′′ = 1, –)的影响可以忽略.

      υ′–υ${\tilde v} $/cm–1Aυ′υ′′/sRυ′υ′′λυ′υ′′/nmgfυ′υ′′ υ′–υ${\tilde v} $/cm–1Aυ′υ′′/sRυ′υ′′λυ′υ′′/nmgfυ′υ′′
      0-011039.100.18780.8913906.522.31×10–9 1-013546.355.67×10–40.0030738.734.63×10–12
      0-18761.220.02160.10271142.214.22×10–101-111268.460.14410.7666888.061.70×10–9
      0-26583.030.00120.00581520.144.24×10–111-29090.270.03910.20821100.867.10×10–10
      0-34501.634.60×10–52.18×10–42223.003.40×10–121-37008.880.00390.02091427.781.20×10–10
      0-42510.749.71×10–74.61×10–63985.722.31×10–131-45017.982.33×10–40.00121994.251.39×10–11
      2-015929.213.36×10–40.0020628.221.98×10–123-018183.371.76×10–61.12×10–5550.347.97×10–15
      2-113651.320.00230.0142733.051.84×10–113-115905.489.50×10–40.0061629.165.63×10–12
      2-211473.130.10810.6354872.221.23×10–93-213727.290.00500.0320728.993.99×10–11
      2-39391.740.05070.29821065.528.62×10–103-311645.900.08170.5216859.289.03×10–10
      2-47400.840.00800.04681352.162.18×10–103-49655.010.05420.34641036.478.72×10–10

      表 6  $ {{\text{a}}^{\text{3}}}{\Pi_{{{\text{0}}^ + }}} $(υ′, J′ = 0, + ) →$ {\text{X}}{}^{\text{1}}{\Sigma}_{{{\text{0}}^ + }}^ + $(υ′′, J′′ = 1, –)跃迁的跃迁波数(${\tilde v} $)、爱因斯坦A系数(Aυ′υ′′)、振动分支比(Rυ′υ′′)、波长(λυ′υ′′)、加权的吸收振子强度(gfυ′υ′′)

      Table 6.  The transition wavenumber(${\tilde v} $), Einstein A-coefficients(Aυ′υ′′), vibrational branching ratios (Rυ′υ′′), wavelength (λυ′υ′′), and weighted absorption oscillator strengths (gfυ′υ′′) for the $ {{\text{a}}^{\text{3}}}{\Pi_{{{\text{0}}^ + }}} $(υ′, J′ = 0, + ) →$ {\text{X}}{}^{\text{1}}{\Sigma}_{{{\text{0}}^ + }}^ + $(υ′′, J′′ = 1, –) transitions.

      υ$ {\text{a}}{}^{\text{3}}{\Pi_{{{\text{0}}^ + }}} $/s a3Π1/s A1Π1/ns
      总和/ns$ {\text{A}}{}^{\text{1}}{\Pi_{\text{1}}}{\text{ - }}{{\text{X}}^{\text{1}}}{\Sigma}_{{{\text{0}}^ + }}^ + $/ns$ {\text{A}}{}^{\text{1}}{\Pi_{\text{1}}} $-$ {\text{a}}{}^{\text{3}}{\Pi_{{{\text{0}}^ + }}} $/sA1Π1– a3Π1/sA1Π1– a3Π2/s
      04.75 7.52 124.18124.182.71111.48177.04
      15.327.50143.86143.863.0390.08116.30
      25.887.53176.12176.123.5883.05192.36
      36.397.61246.20246.204.7793.55255.19
      46.857.78
      57.278.09

      表 8  A1Π1(υ′, J′ = 1, +), $ {{\text{a}}^{\text{3}}}{\Pi _{{0^ + }}} $(υ′, J′ = 0, + )和a3Π1(υ′, J′ = 1, +)态的辐射寿命(τυ)

      Table 8.  Spontaneous radiative lifetimes(τυ′) for the A1Π1(υ′, J′ = 1, +), $ {{\text{a}}^{\text{3}}}{\Pi _{{0^ + }}} $(υ′, J′ = 0, +)和a3Π1(υ′, J′ = 1, +) transitions

      Douglass等[9]报道了A1Π(υ′ = 0 – 2, J′ = 3, +)的τυ′分别为(127 ± 10), (146 ± 12), (172 ± 14) ns, 本文计算的相应值仅比实验值小2.42, 1.50和5.38 ns, 可见它们符合得很好. 因此, 本文计算的A1Π1(υ′ = 0 – 3, J′ = 1, +)的τυ′也应是精确的, 另外本文计算的A1Π1(υ′ = 0 - 3, J′ = 1, +) → $ {\text{X}}^1{\Sigma}_{{0^ + }}^ + $(υ′′, J′′ = 1, –)的辐射宽度Γr[Γr = (2πcτυ′)–1]分别为4.27 × 10–5, 3.69 × 10–5, 3.01 × 10–5和2.16 × 10–5 cm–1, 这些结果表明可以快速激光冷却BH分子.

      为了评价激光冷却效果, 基于A1Π1态的τ00 = 125.28 ns, TDoppler = h/(4πkBτυ′)、λ00 = 432.45 nm和TRecoil = h2/(mkBλ2), 我们计算了主冷却循环A1Π1(υ′ = 0, J′ = 1, +) ↔$ {\text{X}}^1{\Sigma}_{{0^ + }}^ + $(υ′′ = 0, J′′ = 1, –)的TDoppler = 30.48 μK和TRecoil = 8.52 μK, 这意味本文所提出的冷却方案可以将BH分子冷却到微开尔文的温度.

    • 本文利用icMRCI + Q方法连同AV5Z和AV6Z基组获得了BH分子8个Λ-S态和23个Ω态的势能曲线, 计算中修正了SR和SOC效应、CV效应和基组截断带来的误差. 利用icMRCI方法和AV6Z基组计算了最低的5个Ω态($ {\text{X}}^1{\Sigma}_{{0^ + }}^ + $, $ {{\text{a}}^3}{{\Pi}_{{0^ + }}} $, a3Π1, a3Π2和A1Π1)之间的跃迁偶极矩. 并且本文获得的光谱常数与现有的实验值符合得很好. 基于上述的势能曲线和跃迁偶极矩, 获得了A1Π1(υ′ = 0 – 2, J′ = 1, +) → $ {\text{X}}^1{\Sigma}_{{0^ + }}^ + $(υ′′ = 0 – 2, J′′ = 1, –)跃迁较大的Aυ′υ′′$ g{f_{\upsilon 'J' \leftarrow \upsilon ''J''}} $、高度对角化分布的Rυ′υ′′、A1Π1态较短的τυ′, 这些条件可以保证快速高效的激光冷却BH分子. 计算表明: A1Π1(υ′, J′ = 1, +) → $ {{\text{a}}^3}{{\Pi}_{{0^ + }}} $(υ′′, J′′ = 1, –)和A1Π1(υ′, J′ = 1, +) → a3Π1(υ′′, J′′ = 1, –)的振动分支损失比都很小, 可以忽略不计. 利用A1Π1 (υ′ = 0 – 1, J′ = 1, +) ↔ $ {\text{X}}^1{\Sigma}_{{0^ + }}^ + $(υ′′ = 0 –3, J′′ = 1, –)跃迁激光冷却BH分子所需的3束泵浦激光在可见光范围(主泵激光λ00 = 432.45 nm, 2束再泵浦激光λ10 = 479.67 nm和λ21 = 481.40 nm). 此外, 本文预测的A1Π1(υ′ = 0, J′ = 1, +) ↔$ {\text{X}}^1{\Sigma}_{{0^ + }}^ + $(υ′′ = 0, J′′ = 1, –)的TDopplerTrecoil分别为30.48和8.52 μK. 这些结果表明BH是潜在的激光冷却候选分子, 并且可以达到微开尔文的冷却温度.

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