Combining machine learning algorithm to improve  prediction performance of ab initio method for vibrational energy spectra of HF/HBr/H<sup>35</sup>Cl/Na<sup>35</sup>Cl

Yang Zhang-Zhang; Liu Li; Wan Zhi-Tao; Fu Jia; Fan Qun-Chao; Xie Feng; Zhang Yi; Ma Jie

doi:10.7498/aps.72.20221953

Abstract

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.

Keywords:

Authors and contacts

1.
School of Science, Key Laboratory of High Performance Scientific Computation, Xihua University, Chengdu, 610039, China
2.
Institute of Nuclear and New Energy Technology, Collaborative Innovation Center of Advanced Nuclear Energy Technology, Key Laboratory of Advanced Reactor Engineering and Safety of Ministry of Education, Tsinghua University, Beijing, 100084, China
3.
College of Advanced Interdisciplinary Studies, National University of Defense Technology, Changsha 410073, China
4.
State Key Laboratory of Quantum Optics and Quantum Optics Devices, Laser Spectroscopy Laboratory, College of Physics and Electronics Engineering, Shanxi University, Taiyuan 030006, China

Corresponding author: Fu Jia, fujiayouxiang@126.com ; Fan Qun-Chao, fanqunchao@sina.com ;

Funds: Project supported by the National Natural Science Foundation of China (Grant No. 11904295), the Program of Science and Technology of Sichuan Province of China (Grant No. 2021ZYD0050), and the Open Research Found Program of Collaborative Innovation Center of Extreme Optics (Grant No. KF2020003).

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References

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图 1 RFRA结构图

Figure 1. Architecture of RFRA.

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图 2 CCSD(T)/cc-pVQZ的误差趋势图

Figure 2. Error trend figure of CCSD(T)/cc-pVQZ.

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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

Figure 3. Comparison of the errors of the new method and CCSD(T)/cc-pV5 Z: (a) HF; (b) HBr; (c) H³⁵Cl; (d) Na³⁵Cl.

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图 4 新方法与CCSD(T)/cc-pV5 Z的平均相对误差对比

Figure 4. Comparison of the average relative errors of the new method and CCSD(T)/cc-pV5 Z.

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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}} $	目标变量$\hat Y:{E_\nu }$	文献[59]	$E_\nu ^{{\text{this work}}}$
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
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Combining machine learning algorithm to improve prediction performance of ab initio method for vibrational energy spectra of HF/HBr/H³⁵Cl/Na³⁵Cl

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Corresponding author: Fu Jia, fujiayouxiang@126.com ; Fan Qun-Chao, fanqunchao@sina.com ;

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Combining machine learning algorithm to improve prediction performance of ab initio method for vibrational energy spectra of HF/HBr/H35Cl/Na35Cl

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Combining machine learning algorithm to improve prediction performance of ab initio method for vibrational energy spectra of HF/HBr/H³⁵Cl/Na³⁵Cl