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Halides play an important role in atmospheric chemistry, corrosion of steel, and also in controlling the abundance of O3. Moreover high-precision vibrational energy spectra contain a large amount of quantum information of molecular system and are basic data for people to understand and manipulate molecules. At present, ab-initio methods have achieved many calculation results of the potential energy surfaces and corresponding vibrational energy of molecules, but they still face challenges in terms of accuracy and computational cost. Recently, data-driven machine learning methods have demonstrated very strong capability of extracting high-dimensional functional relationships from massive data and have been widely used in spectrum studies. Therefore, a theoretical approach to combining ab-initio method and machine learning algorithm is presented here to predict the vibrational energy of diatomic systems, which improves the accuracy and simultaneously reduces the computational cost. Firstly, the vibrational energy levels of 42 diatomic molecules are obtained by using different CCSD(T) methods to calculate the configurations from simple to complex and the corresponding experimental results are also collected. A machine learning algorithm is then used to learn the difference between the CCSD(T) method calculated vibrational results and the experimental vibrational results, and a high-dimensional error function is finally constructed to improve the original CCSD(T) computational accuracy. The results for HF, HBr, H35Cl and Na35Cl (they did not appear in the training set) and other halogen molecules show that compared with the CCSD(T)/cc-pV5Z calculation method alone, the present method reduces the prediction error by more than 50% and the computational cost by nearly one order of magnitude. It is worth noting that the method proposed in this paper is not only limited to the energy level prediction of diatomic systems, but also applicable in other fields where data can be obtained by ab initio methods and experimental methods simultaneously, such as the energy spectrum properties of macromolecular systems.
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Keywords:
- vibrational spectrum /
- ab initio /
- machine learning /
- halogen molecules
[1] Ye Y W, Jiang Z L, Zou Y J, Chen H, Guo S D, Yang Q M, Chen L Y 2020 J. Mater. Sci. Technol. 43 144
Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
[19] 黎威, 龙连春, 刘静毅, 杨洋 2022 物理学报 71 060202
Google Scholar
Li W, Long L C, Liu J Y, Yang Y 2022 Acta Phys. Sin. 71 060202
Google Scholar
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Google Scholar
[21] Le Roy R J. 2017 J. Quant. Spectrosc. Ra. 186 167
Google Scholar
[22] Goodfellow I, Bengio Y, Courville A 2016 Deep Learning (Cambridge: MIT Press) p351
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Google Scholar
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Google Scholar
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图 3 新方法的预测能级误差与CCSD(T)/cc-pV5 Z的误差对比 (a) HF; (b) HBr; (c) H35Cl; (d) Na35Cl
Fig. 3. Comparison of the errors of the new method and CCSD(T)/cc-pV5 Z: (a) HF; (b) HBr; (c) H35Cl; (d) Na35Cl.
图 4 新方法与CCSD(T)/cc-pV5 Z的平均相对误差对比
Fig. 4. Comparison of the average relative errors of the new method and CCSD(T)/cc-pV5 Z.
表 1 基态分子的部分实验振动能级 (cm–1)
Table 1. Partial experimental vibrational energy levels of molecules in the ground state (cm–1).
$ \nu $ H2 HF DF H35Cl LiH Li2 0 2179.69 2050.77 1490.30 1483.88 705.08 175.032 1 6341.75 6012.19 4396.97 4369.86 2078.45 521.26 2 10268.40 9801.57 7212.12 7151.86 3406.08 862.26 3 13694.54 13423.60 9937.66 9830.66 4688.81 1198.00 4 17433.22 16882.45 12575.33 12406.71 5927.55 1528.41 ${\text{H}}{}^{{\text{79}}}{\text{Br}}$ N2 BF ClF NO CN 0 1314.65 1175.77 742.00 191.77 948.50 1031.20 1 3873.57 3505.69 2208.00 573.06 2824.50 3073.60 2 6341.99 5806.93 3651.00 951.35 4627.30 5089.70 3 8719.91 8079.47 5072.00 1326.60 6491.90 7079.50 4 11007.01 10323.28 6470.00 1698.90 8283.50 9042.80 Br2 BCl CP CS SiCl O2 0 162.38 416.83 618.19 709.00 267.25 664.69 1 485.53 1242.63 1844.32 2117.00 798.54 2326.53 2 806.51 2058.80 3056.76 3515.00 1325.50 3968.09 3 1125.29 2865.49 4255.53 4902.00 1847.50 5589.60 4 1441.88 3662.85 5440.62 6278.00 2365.50 7191.32 12C16O 13C16O 14C16O 12C17O 13C17O 14C17O 0 1081.77 1057.72 1036.74 1068.03 1043.66 1022.39 1 3225.04 3153.79 3091.61 3184.32 3112.11 3049.05 2 6341.83 5224.54 5122.15 5274.81 5155.92 5052.07 3 7432.21 7270.04 7128.44 7339.54 7175.14 7031.50 4 9496.24 9290.35 9110.52 9378.60 9169.83 8987.38 SO SiC SiN 24Mg16O Na35Cl NaLi 0 576.94 475.47 574.06 391.14 181.90 127.83 1 1740.42 1416.67 1712.46 1165.88 543.05 381.22 2 2916.75 2344.87 2837.85 1930.32 900.70 631.10 3 4105.98 3260.07 3950.20 2684.16 1254.89 877.79 4 5308.16 4162.27 5049.47 3427.11 1605.65 1121.12 MgH AlO AlF Al Cl SiO BH 0 739.11 488.00 394.60 246.60 619.20 1171.06 1 2171.09 1453.40 1180.00 734.10 1848.90 3440.30 2 3539.79 2404.76 1956.10 1217.30 3066.50 5614.11 3 4841.14 3342.19 2722.30 1698.00 4272.30 7694.16 4 6070.50 4265.42 3480.20 2175.20 5466.10 9684.16 
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表 2 Li2的输入数据与预测数据 (cm–1)
Table 2. Input data and predicted data of Li2 (cm–1).
输入变量$X$ 目标变量$\hat Y:{E_\nu }$ 文献[59] $E_\nu ^{{\text{this work}}}$ $ E_\nu ^{\text{T}} $ $ E_\nu ^{\text{Q}} $ 171.702 172.1623 175.03 174.059 178.903 511.410 513.631 521.26 518.255 531.853 846.059 849.982 862.26 857.832 878.969 1175.599 1181.177 1198.00 1191.397 1173.096 1499.987 1507.169 1528.41 1519.330 1496.133 1819.161 1827.906 1853.46 1841.991 1906.993 2133.063 2143.328 2173.07 2159.852 2106.362 2441.620 2453.368 2487.19 2471.991 2438.853 2744.753 2757.954 2795.74 2778.334 2693.961 3042.368 3057.002 3090.64 3072.703 3102.89 3334.377 3350.421 3395.80 3373.920 3369.871 3620.669 3638.108 3687.11 3663.159 3630.759 3901.126 3919.951 3972.43 3946.618 4009.308 4175.619 4195.927 4251.73 4223.997 4297.662 4444.003 4465.597 4524.78 4495.270 4487.998 4706.126 4729.113 4791.43 4760.610 4530.406 4961.805 4986.210 5051.53 5018.559 4962.152 5210.858 5236.706 5304.93 5270.268 5225.061 5453.074 5480.402 5551.4 5514.831 5506.045 5688.224 5717.081 6790.71 5753.539 5701.189 5916.059 5946.503 6022.66 5983.959 6022.425 注: 文献[59]的数据和$E_\nu ^{{\text{this work}}}$不作为输入数据.
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表 3 HF/HBr/H35Cl/Na35Cl的预测振动能级( cm–1)
Table 3. Predicted vibrational energy levels of HF/HBr/H35Cl/Na35Cl (cm–1).
HF$ \nu $ ${E_\nu }$ ${\delta _{{\text{this work}}}}$ $E_\nu ^{{\text{this work}}}$ $\delta _\nu ^{\text{Q}}$ $E_\nu ^{\text{Q}} $ 0 2050.771 0.04093 2134.714 0.01200 2075.378 1 6012.194 0.05485 6341.983 0.01562 6106.099 2 9801.566 0.03044 10099.884 0.01694 9967.644 3 13423.603 0.001166 13407.950 0.01797 13664.780 4 16882.448 0.004083 16951.383 0.01893 17202.024 5 20181.824 0.0005710 20193.347 0.01990 20583.515 6 23324.620 0.008118 23135.267 0.02093 23812.920 7 26313.146 0.01637 25882.423 0.02205 26893.357 8 29148.927 0.01435 28730.690 0.02327 29827.322 9 31832.367 0.01963 31207.435 0.02464 32616.648 10 34362.909 0.004397 34211.806 0.02618 35262.441 11 36738.405 0.01714 36108.527 0.02794 37764.995 12 38954.943 0.01769 38265.658 0.03000 40123.716 13 41006.593 0.01487 40396.799 0.03244 42337.026 14 42884.443 0.01483 42248.544 0.03539 44402.227 15 44576.005 0.01264 44012.508 0.03902 46315.354 16 46064.207 0.009428 45629.921 0.04357 48071.083 17 47325.663 0.006234 47030.644 0.04938 49662.648 18 48328.541 0.0003463 48311.807 0.05697 51081.932 19 49026.508 0.005459 49294.146 0.06717 52319.807 HBr$ \nu $ ${E_\nu }$ ${\delta _{{\text{this work}}}}$ $E_\nu ^{{\text{this work}}}$ $\delta _\nu ^{\text{Q}}$ $E_\nu ^{\text{Q}}$ 0 1314.653 0.02080 1341.993 0.01314 1331.929 1 3873.566 0.06601 4129.243 0.01762 3941.805 2 6341.990 0.01130 6413.868 0.01935 6464.682 3 8719.913 0.01104 8816.207 0.02093 8902.436 4 11007.012 0.01316 11151.888 0.02259 11255.684 5 13202.585 0.01387 13385.700 0.02437 13524.341 6 15305.471 0.01795 15580.146 0.02630 15707.958 7 17313.970 0.002268 17274.704 0.02841 17805.806 H35Cl $ \nu $ ${E_\nu }$ ${\delta _{{\text{this work}}}}$ $E_\nu ^{{\text{this work}}}$ $\delta _\nu ^{\text{Q}}$ $E_\nu ^{\text{Q}}$ 0 1483.881 0.01052 1468.276 0.01050 1468.294 1 4369.857 0.01009 4413.952 0.003299 4384.273 2 7151.864 0.009209 7217.724 0.006637 7199.330 3 9830.658 0.01910 10018.462 0.008566 9914.869 4 12406.710 0.004748 12465.616 0.01028 12534.187 5 14880.156 0.002706 14839.892 0.01250 15066.200 6 17250.746 0.002461 17208.284 0.01563 17520.446 7 19517.778 0.003098 19457.320 0.01906 19889.759 8 21680.003 0.01529 21348.516 0.02231 22163.716 9 23735.517 0.01306 23425.579 0.02561 24343.381 10 25681.608 0.005119 25813.072 0.02871 26418.895 11 27514.609 0.009392 27256.180 0.03154 28382.336 12 29229.647 9.641 E-05 29226.829 0.03378 30217.072 13 30820.291 0.008591 30555.509 0.03507 31901.224 14 32278.144 0.004052 32408.930 0.03476 33400.117 Na35Cl $ \nu $ ${E_\nu }$ ${\delta _{{\text{this work}}}}$ $E_\nu ^{{\text{this work}}}$ $\delta _\nu ^{\text{Q}}$ $E_\nu ^{\text{Q}}$ 0 181.899 0.008954 180.270 0.04204 174.252 1 543.050 0.01550 534.633 0.03299 525.138 2 900.702 0.007215 894.204 0.03107 872.718 3 1254.890 0.01144 1240.538 0.03014 1217.063 4 1605.649 0.005573 1596.700 0.02955 1558.204 5 1953.013 0.008120 1937.155 0.02912 1896.150 6 2297.016 0.005233 2284.995 0.02877 2230.937 7 2637.692 0.001172 2640.784 0.02847 2562.592 8 2975.075 0.01092 2942.574 0.02821 2891.142 9 3309.198 0.001195 3305.243 0.02798 3216.614 10 3640.093 0.01084 3600.653 0.02776 3539.057 11 3967.795 0.001966 3959.992 0.02755 3858.485 12 4292.334 0.002171 4283.017 0.02735 4174.918
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表 3 （续） HF/HBr/H35Cl/Na35Cl的预测振动能级( cm–1)
Table 3 （续）. Predicted vibrational energy levels of HF/HBr/H35Cl/Na35Cl (cm–1).
Na35Cl $ \nu $ ${E_\nu }$ ${\delta _{{\text{this work}}}}$ $E_\nu ^{{\text{this work}}}$ $\delta _\nu ^{\text{Q}}$ $E_\nu ^{\text{Q}}$ 13 4613.743 0.01077 4564.070 0.02717 4488.398 14 4932.054 0.002617 4919.147 0.02699 4798.950 15 5247.298 0.01274 5180.428 0.02681 5106.596 16 5559.506 0.009977 5504.042 0.02665 5411.364 17 5868.710 0.02119 5744.339 0.02648 5713.289 18 6174.940 0.003747 6198.078 0.02632 6012.398 19 6478.226 0.007659 6428.612 0.02617 6308.719 20 6778.599 0.0008396 6772.914 0.02601 6602.272
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[1] Ye Y W, Jiang Z L, Zou Y J, Chen H, Guo S D, Yang Q M, Chen L Y 2020 J. Mater. Sci. Technol. 43 144
Google Scholar
[2] Wick C D 2017 J. Chem. Phys. 147 161703
Google Scholar
[3] Devlin J P, Farnik M, Suhm M A, Buch V 2005 J. Phys. Chem. A. 109 955
Google Scholar
[4] Delval C, Fluckiger B, Rossi M J 2003 Atmos. Chem. Phys. 3 1131
Google Scholar
[5] Barone S B, Zondlo M A, Tolbert M A 1999 J. Phys. Chem. A. 103 9717
Google Scholar
[6] Smart R S C, Sheppard N 1971 Proc. R. Soc. Lond. A. 320 417
Google Scholar
[7] Blass P M, Jackson R C, Polanyi J C, Weiss H 1991 J. Chem. Phys. 94 7003
Google Scholar
[8] Giorgi J B, Kuhnemuth R, Polanyi J C, Wang J X 1997 J. Chem. Phys. 106 3129
Google Scholar
[9] Carvalho A, Hancock G, Saunders M, 2006 Phys. Chem. Chem. Phys. 8 4337
Google Scholar
[10] Sun Q 2012 Vib. Spectros. 62 110
Google Scholar
[11] Weiss P S, Mestdagh J M, Covinsky M H, Balko B A, Lee Y T 1988 Chem. Phys. 126 93
Google Scholar
[12] Rubio L, Samoudi B, Santos M, Diaz L 2012 J. Photoch. Photobio. A. 237 1
Google Scholar
[13] Reiser C, Lussier F M, Jensen C C, Steinfeld J I 1979 J. Am. Chem. Soc. 101 350
Google Scholar
[14] Rauhut G, Knizia G, Werner H J 2009 J. Chem. Phys. 130 054105
Google Scholar
[15] Neff M, Hrenar T, Oschetzki D, Rauhut G 2011 J. Chem. Phys. 134 064105
Google Scholar
[16] Kowalski K, Piecuch P 2000 J. Chem. Phys. 113 18
Google Scholar
[17] 张瑶, 张云波, 陈立 2021 物理学报 70 168702
Google Scholar
Zhang Y, Zhang Y B, Chen L 2021 Acta Phys. Sin. 70 168702
Google Scholar
[18] 南虎, 麻晓晶, 赵海博, 汤少杰, 刘卫华, 王大威, 贾春林 2021 物理学报 70 076803
Google Scholar
Nan H, Ma X J, Zhao H B, Tang S J, Liu W H, Wang D W, Jia C L 2021 Acta Phys. Sin. 70 076803
Google Scholar
[19] 黎威, 龙连春, 刘静毅, 杨洋 2022 物理学报 71 060202
Google Scholar
Li W, Long L C, Liu J Y, Yang Y 2022 Acta Phys. Sin. 71 060202
Google Scholar
[20] Řezáč J, Šimová L, Hobza P 2013 J. Chem. Theory Comput. 9 364
Google Scholar
[21] Le Roy R J. 2017 J. Quant. Spectrosc. Ra. 186 167
Google Scholar
[22] Goodfellow I, Bengio Y, Courville A 2016 Deep Learning (Cambridge: MIT Press) p351
[23] Daszykowski M, Kaczmarek K, Heyden Y V, Walczak B 2007 Chemom. Intell. Lab. Syst. 85 203
Google Scholar
[24] Li Y, Zou C F, Maitane B, et al. 2018 Appl. Energ. 232 197
Google Scholar
[25] Abdel-Rahman E M, Ahmed F B, Ismail R 2013 Int. J. Remote Sens. 34 712
Google Scholar
[26] Breiman L 2000 Some Infinity Theory for Predictor Ensembles Technical Report 579 Statistics Dept. UCB
[27] Cutler A, Zhao G 2001 Comput. Sci. Stat. 33 90
[28] Yali A, Donald G 1997 Neural Comput. 9 1545
Google Scholar
[29] Leo B 2001 Mach. Learn. 45 5
Google Scholar
[30] Dietterich T G 2000 Mach. Learn. 40 139
Google Scholar
[31] Biau G, Devroye L 2010 J. Multivariate Anal. 101 2499
Google Scholar
[32] Pokluda J, Cerny M, Sob M, Umeno Y 2015 Prog. Mater. 73 127
Google Scholar
[33] Frisch M J, Trucks G W, Schlegel H B, et al. J 2016 Gaussian 09 (Revision A. 02) Gaussian Inc
[34] Ram R S, Dulick M, Guo B, Zhang K Q, Bernath P F 1997 J. Mol. Spectrosc. 183 360
Google Scholar
[35] Saksena M D, Deo M N, Sunanda K, Behere S H, Londhe C T 2008 J. Mol. Spectrosc. 247 47
Google Scholar
[36] Shayesteh A, Henderson R D E, Le Roy R J 2007 J. Phys. Chem. A 111 1249
Google Scholar
[37] Coxon J A, Hajigeorgiou P G 1990 J. Mol. Spectrosc. 142 254
Google Scholar
[38] Coxon J A, Hajigeorgiou P G 2015 J. Quant. Spectrosc. Radiat. Transf. 151 133
Google Scholar
[39] Blom C E, Hedderich H G, Lovas F J, Suenram R D, Maki A G 1992 J. Mol. Spectrosc. 152 109
Google Scholar
[40] Edwards S, Roncin J Y, Launay F, Rostas F 1993 J. Mol. Spectrosc. 162 257
Google Scholar
[41] Slanger T G, Cosby P C. 1988 J. Phys. Chem. 92 267
Google Scholar
[42] Le Roy R J, Appadoo D R T, Colin R, Bernath P F 2006 J. Mol. Spectrosc. 236 178
Google Scholar
[43] Reddy R R, Rao T V R, Viswanath R, Viswanath R 1992 Astrophys. Space Sci. 189 29
Google Scholar
[44] Speth R S, Braatz C, Tiemann E 1998 J. Mol. Spectrosc. 192 69
Google Scholar
[45] Reddy R R, Nazeer A Y, Rama G K, Azeem P A, Anjaneyulu S 1998 Astrophys. Space Sci. 262 223
Google Scholar
[46] Venkataramanaiah M, Lakshman S V J 1981 J. Quant. Spectrosc. Radiat. Transfer 26 11
Google Scholar
[47] Cai Z L, Martin J M L, François J P, Gijbels R 1996 Chem. Phys. Lett. 252 398
Google Scholar
[48] Botschwina P 1986 J. Mole. Spectrosc. 118 76
Google Scholar
[49] Dabrowski I 1984 Can. J. Phys. 62 1639
Google Scholar
[50] Fellows C E 1991 J. Chem. Phys. 94 5855
Google Scholar
[51] Coxon J A, Hajigeorgiou P G 2000 J. Mol. Spectrosco. 203 49
Google Scholar
[52] Focsa C, Li H, Bernath P H 2000 J. Mol. Spectrosco. 200 104
Google Scholar
[53] Fallon R J, Vanderslice J T, Cloney R D 1962 J. Chem. Phys. 37 1097
Google Scholar
[54] Reddy R R, Ahammed Y N, Basha D B, Narasimhulu K, Reddy S S, Gopa K R 2006 J. Quant. Spectrosco Radiat. Transf. 97 344
Google Scholar
[55] Barakat B, Bacis R, Carrot F, Churassy S, Crozet P, Martin F, Verges J 1986 Chem. Phys. 102 215
Google Scholar
[56] Peterson K A, Woods R C 1987 J. Chem. Phys. 87 4409
Google Scholar
[57] Clyne M A A, McDermid I S 1976 J. Chem. Soc. 72 2242
Google Scholar
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