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. 2022 Dec 15;12(55):35950–35958. doi: 10.1039/d2ra07613f

Achieving vibrational energies of diatomic systems with high quality by machine learning improved DFT method

Zhangzhang Yang 1, Zhitao Wan 1, Li Liu 1, Jia Fu 1,✉, Qunchao Fan 1,✉, Feng Xie 2, Yi Zhang 3, Jie Ma 4
PMCID: PMC9753899  PMID: 36545113

Abstract

When using ab initio methods to obtain high-quality quantum behavior of molecules, it often involves a lot of trial-and-error work in algorithm design and parameter selection, which requires enormous time and computational resource costs. In the study of vibrational energies of diatomic molecules, we found that starting from a low-precision DFT model and then correcting the errors using the high-dimensional function modeling capabilities of machine learning, one can considerably reduce the computational burden and improve the prediction accuracy. Data-driven machine learning is able to capture subtle physical information that is missing from DFT approaches. The results of 12C16O, 24MgO and Na35Cl show that, compared with CCSD(T)/cc-pV5Z calculation, this work improves the prediction accuracy by more than one order of magnitude, and reduces the computation cost by more than one order of magnitude.

By systematically correcting the calculation errors through machine learning, the accuracy of the diatomic vibrational energy prediction based on typical DFT methods has been improved by order of magnitude.graphic file with name d2ra07613f-ga.jpg

1. Introduction

Diatomic molecules and their corresponding energy spectra are widely used in astrophysics, ultracold molecules, fundamental physical constants and thus on.1–5 Various experimental techniques have been developed for high-precision spectra measurement such as velocity modulation laser spectroscopy (VMS),6 noise immune cavity enhanced optical heterodyne molecular spectroscopy (NICE-OHMS),7 laser-induced breakdown spectroscopy (LIBS)8 and so forth. However, limited by experimental conditions, generally only part of the energy levels corresponding to lower quantum numbers can be measured accurately. In the theoretical side, there are two main options: (1) ab initio methods based on the principles of quantum mechanics, such as Hartree Fock and its extension, DFT, etc.9–13 The post-Hartree Fock methods like multireference configuration interaction methods have decent performance in accuracy, whereas the steep computation cost and lengthy time make them limited to small systems.12,14,15 The DFT method is compromised in accuracy, but it has rapid calculation speed, which makes it the first choice for the calculation of large systems.16–18 In order to improve the performance of DFT, both general and accurate exchange-correlation functionals and basis set are required,16,18–20 which is still a challenge to us.18,21,22 (2) Data driven algorithms, such as empirical potential energy function and direct parameter formulas for energy levels.23,24 The accuracy of data driven algorithms are higher than ab initio methods except that they are only applicable to some molecular systems which have superior-quality experimental data.

Recently, the machine learning algorithm has made many achievements in spectroscopy study.25–27 In this work, machine learning algorithm was applied to improve the performance of DFT in the study of diatomic vibrational spectrum. To obtain the best prediction, three widely used machine learning regression algorithms were tested with H35Cl as an example. Then, the best performing algorithm was used to predict the vibrational energy levels of 12C16O, 24MgO, and Na35Cl. Finally, the spectral quality of many systems (24MgO, HF, N2, H35Cl, Na35Cl, 12C16O, BeH, and SiN et al.) were greatly improved.

2. Theory and method

2.1. DFT for vibrational energies

For the purpose of obtain the vibrational energy spectrum of diatomic molecules systems, it is inevitable to solve the Schrodinger equation (r and R denote the electronic and the nuclear coordinate, respectively),

ĤΨ(r, R) = EΨ(r, R), 1

where Ĥ, E and Ψ(r, R) represent the Hamiltonian of the system, total energy and wave function, respectively. It is worth noting that the wave function must be first order continuous differentiable, square integrable, and so on, which makes the partial solutions of the Schrodinger equation called mathematical but nonphysical.28 In other words, the wave function is the result of approximation. Then, when studying the radial motion of the binuclear system, taking nonrelativistic approximation and Born–Oppenheimer approximation (BOA) into consider, eqn (1) can be simplified to

2.1. 2

where ℏ = h/2π (h is the Planck constant), μ is the reduced mass of two nucleus, J is total angular momentum quantum number, r is internuclear distance, and V(r) corresponds to the electrostatic interaction of all the particles.23 One problem is that the exchange–correlation energy has no universal accurate form,29 which introduces error to results. Finally, the ro-vibrational energy level EνJ can be expressed as

EνJ = G(v) + Fν(J), 3

and

2.1. 4
2.1. 5

While J = 0, EνJ degenerates to Gν, namely vibrational energy level.30

A conclusion is drawn that obtaining the vibrational energy levels of molecules, a large quantity of approximations need to be included, and various approximations affect each other. Error is often unavoidable and difficult to predict in advance.

2.2. Combining machine learning algorithm and DFT

2.2.1. Machine learning algorithm

Among a variety of machine learning (ML) regression algorithms, artificial neural network (ANN),31,32 random forest (RF)33,34 and extreme gradient boosting (XGBoost)35 algorithms were widely used and usually found successful. All three algorithms have been tested in this work for vibrational energy prediction. The results were compared in the Fig. 1. The absolute error is the predicted values minus the experimental values. Clearly, ANN performs the best. Thus, it is the algorithm used in this work.

Fig. 1. Absolute error of H35Cl in ML algorithms and DFT methods.

Fig. 1

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An ANN consists of an input layer of neurons, followed by many hidden layers (two, three or more layers are all fine), and a final layer of output neurons. Neurons are connected by weights Vij. Given the input, xj, the output, hi, of neuron i is,

2.2.1. 6

where σ(*) is called activation function, N is the number of input neurons, and Thidi is the threshold term of the neurons.31 It is worth noting that activation function not only introduces nonlinearity into the neural network, but also constrains the value of neurons to prevent the ANN from being paralyzed by divergent neurons. And a common example of the activation function is the sigmoid function,31 defined as

2.2.1. 7

The architecture is shown in Fig. 2. When ANN is used to establish a high-dimensional functional relationship between input and output variables, the data samples will be divided into three groups, namely training set, validation set, and test set. For convenience, the training set and the validation set are also called sub-sample set uniformly. Similar to how humans learn through feedback, neural networks obtain training errors through their performance on the training set. Then, the weights between the connected neurons are adjusted for learning, which reduces the training error. The performance of ANN on the validation set is tracked during the learning process. And the one that performs the best is selected as the chosen model. Finally, the test set is used to determine the performance level of the ANN.31

Fig. 2. Architecture of an ANN.

Fig. 2

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2.2.2. Prediction of vibrational energies

By analyzing the error data between DFT and experimental results, a definite and clear trend is found. It is illustrated in Fig. 3 (take B3PW91/def2-QZVP as an example) that higher vibrational quantum number means greater error, and the error trajectories of different molecules are similar. The absolute error here is the theoretical vibrational energy minus the experiment value. This trend law with abundant details can be learned by ML method. Therefore, after getting the theoretical value Eabv of the DFT methods and the corresponding experimental results, the systematic error Esysv of DFT methods can be obtained through ANN. Ultimately, the predicted vibrational energy Eν is defined as

Ev(α) = Eabv + Esysv(α), 8

where α represents the kind of diatomic molecular system. It's worth noting that Esysv(α) is an error function associated with molecules. So that it is not a fixed constant.

Fig. 3. Systematic error of B3PW91/def2-QZVP in vibrational energy levels prediction.

Fig. 3

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3. Results and discussion

3.1. Obtain initial sample set

The potential energy curves of 39 molecules, such as H2, 12C16O, ClF, et al., were obtained by Gaussian 09 (ref. 36) with B3PW91/def2-XVP (where X = QZ, TZ or S). Then Eabv was obtained by solving eqn (2) by LEVEL.37 The corresponding data of partial experimental vibration energy levels38–59 are displayed in Table 1 as the initial sample set. Finally, the predicted vibrational energy and relative deviation δ of the molecules are obtained through ANN. And the expression of δ is:

3.1. 9

where Inline graphic represents the theoretical energy (the value of DFT or ANN), Eν represents the experimental value. The input characteristic variables are shown below:

Partial experimental vibrational energy levels of molecules in the ground state (cm−1).

ν BH 12C16O 12C17O 12C18O 13C16O 13C17O 13C18O 14C16O
0 1171.08 1081.77 1068.03 1055.71 1057.72 1043.66 1031.05 1036.74
1 3440.30 3225.04 3184.32 3147.84 3153.79 3112.11 3074.74 3091.61
2 5614.11 5341.83 5274.81 5214.74 5224.54 5155.92 5094.38 5122.15
3 7694.67 7432.21 7339.54 7256.48 7270.04 7175.14 7090.03 7128.44
4 9684.16 9496.24 9378.60 9273.14 9290.35 9169.83 9061.73 9110.52
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ν 14C17O 14C18O 24Mg16O 25Mg16O 26Mg16O SO AlO BeH
0 1022.39 1009.51 391.14 388.01 385.10 576.94 488.00 1021.30
1 3049.05 3010.88 1165.88 1156.60 1147.99 1740.42 1453.40 3008.15
2 5052.07 4989.20 1930.32 1915.05 1900.90 2916.75 2404.76 4921.11
3 7031.50 6944.51 2684.16 2663.08 2643.55 4105.98 3342.12 6760.67
4 8987.38 8876.87 3427.11 3400.42 3375.67 5308.16 4265.42 8527.35
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ν BeD BeT BF ClF H35Cl H37Cl D35Cl D37Cl
0 759.86 649.25 742.00 390.50 1483.88 1482.76 1066.60 1065.04
1 2248.84 1925.35 2208.00 1164.00 4369.86 4366.64 3157.66 3153.10
2 3697.03 3171.70 3651.00 1927.60 7151.86 7146.69 5195.04 5187.63
3 5104.62 4388.42 5072.00 2681.20 9830.66 9823.69 7179.05 7168.96
4 6471.83 5575.64 6470.00 3424.50 12 406.70 12 398.10 9109.98 9097.36
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ν HF DF H2 HBr MgH N2 Na35Cl Na37Cl
0 2050.77 1490.30 2170.88 1314.65 739.11 1175.77 181.90 179.94
1 6012.19 4396.97 6332.02 3873.57 2171.09 3505.69 543.05 537.24
2 9801.57 7212.12 10 257.99 6341.99 3539.79 5806.93 900.70 891.11
3 13 423.60 9937.66 13 953.23 8719.91 4841.14 8079.47 1254.89 1241.59
4 16 882.40 12 575.30 17 421.24 11 007.00 6070.50 10 323.30 1605.65 1588.71
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ν NaLi O2 SiC SiCl SiN SiO
0 127.81 787.14 475.47 267.25 574.0616 619.20
1 384.08 2343.47 1416.67 798.54 1712.46 1848.90
2 631.07 3876.15 2344.87 1325.50 2837.85 3066.50
3 877.74 5385.51 3260.07 1847.50 3950.20 4272.30
4 1121.07 6871.86 4162.27 2365.50 5049.46 5466.10
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(1) Vibrational energy of B3PW91/def2-QZVP: EQZv;

(2) Vibrational energy of B3PW91/def2-TZVP: ETZv;

(3) Vibrational energy of B3PW91/def2-SVP: ESv.

The output variable is written as EBv. A case in point is 24MgO in Table 2. The task of the ANN is to learn the correct mapping relationship between input characteristic variables and system deviation Esysv.

The input characteristic variables of 24MgO (cm−1).

E QZ E TZ E S
380.703 385.124 379.880
1203.812 1145.498 1141.218
2016.469 1894.550 1890.964
2818.615 2632.317 2629.060
3610.286 3358.877 3355.515
4391.528 4074.295 4070.334
5162.374 4778.626 4773.501
5922.839 5471.926 5465.001
6672.946 6154.270 6144.839
7412.729 6825.742 6813.034
8142.228 7486.417 7469.602
8861.481 8136.358 8114.558
9570.527 8775.622 8747.914
10 269.395 9404.272 9369.688
10 958.118 10 022.374 9979.911
11 636.728 10 630.005 10 578.627
12 305.264 11 227.241 11 165.886
12 963.767 11 814.158 11 741.743
13 612.285 12 390.825 12 306.256
14 250.864 12 957.311 12 859.483
14 879.549 13 513.684 13 401.489
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3.2. Prediction results of vibrational energies

The initial sample set (39 molecules in all) was assigned to the sub-sample set (36 molecules) and the test set (3 molecules). After plenty of parameter set tests, and considering the calculation time and accuracy comprehensively, the most balanced set of parameters was chosen as the training parameters, which are listed in Table 3. The relative deviation (see eqn (9)) is used to measure the performance of ANN. For the test set consists of Na35Cl, 24MgO and 12C16O, the average relative deviation of the final ANN model is 0.42%, the maximum relative deviation is 4.65%, the minimum relative deviation is 0.0000099%. On the sub-sample set, the average relative deviation is 1.10%, the maximum relative deviation is 15.92%, the minimum relative deviation is 0.0000038%. The result shows consistent performance on the training and sub-sample set, which means that the learned model is reliable.

Main parameter settings of ANNa.

Hidden layer 1 Hidden layer 2 trainRatio valRatio Train function Error function Divide function
30 20 0.85 0.15 trainbr mse Dividerand
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a

trainRatio = the number of train set/the number of subsample set. valRatio = the number of validation set/the number of subsample set.

3.3. Comparison and analysis

The comparison with CCSD(T)/cc-pV5Z results are listed in Table 4. As shown in Tables 2 and 4, it can be found that ANN can effectively improve the performance of B3PW91/def2-XVP, even better than the more complex ab initio method (CCSD(T)/cc-pV5Z). In detail, the error of ab initial methods increases significantly at high energy levels and easily exceeds 100 cm−1, not to mention the maximum error of B3PW91/def2-QZVP exceeds 1000 cm−1. However, the maximum error of ANN does not exceed 70 cm−1, and the minimum error is only 0.006 cm−1. In addition, the error of the current method is smaller than that of CCSD(T) at each vibrational energy level.

Prediction of vibrational energies of 12C16O/24MgO/Na35Cl (cm−1)a.

ν 12C16O 24MgO Na35Cl
E v E B − Ev E 5 − Ev E v E B − Ev E 5 − Ev E v E B − Ev E 5 − Ev
0 1081.77 20.430 −41.597 391.14 −7.408 −24.315 181.90 −8.463 −5.210
1 3225.04 40.073 −47.228 1165.88 23.542 −67.207 543.05 −8.877 −13.490
2 5341.83 50.975 −52.550 1930.32 37.748 −105.760 900.70 −9.328 −21.736
3 7432.21 16.260 −57.526 2684.16 39.576 −139.913 1254.89 −9.697 −29.886
4 9496.24 −48.621 −62.174 3427.11 34.611 −169.528 1605.650 −9.806 −37.961
5 11 533.99 −21.454 −66.434 4158.89 28.412 −194.442 1953.01 −9.425 −45.949
6 13 545.54 66.471 −70.264 4879.21 24.892 −214.454 2297.02 −8.353 −53.853
7 15 530.95 21.302 −73.663 5587.77 24.813 −229.336 2637.69 −6.375 −61.666
8 17 490.31 29.300 −76.624 6284.28 25.498 −238.849 2975.08 −3.382 −69.383
9 19 423.68 18.285 −79.126 6968.45 22.793 −242.728 3309.20 0.641 −77.004
10 21 331.14 −32.063 −81.149 7640.00 15.022 −240.705 3640.09 5.541 −84.527
11 23 212.78 −44.368 −82.679 8298.63 5.976 −232.502 3967.80 10.937 −91.957
12 25 068.67 −13.295 −83.713 8944.06 2.979 −217.836 4292.33 16.183 −99.282
13 26 898.89 −44.542 −84.264 9575.98 9.678 −196.420 4613.74 20.281 −106.509
14 28 703.54 −10.990 −84.316 10 194.12 20.080 −167.961 4932.05 21.831 −113.634
15 30 482.68 12.240 −83.849 10 798.17 22.109 −132.166 5247.30 19.027 −120.651
16 32 236.41 4.691 −82.890 11 387.86 10.540 −88.739 5559.51 9.710 −127.571
17 33 964.80 1.2907 −81.508 11 962.89 −3.230 −37.381 5868.71 −8.600 −134.390
18 35 667.96 5.946 −79.757 12 522.96 −1.051 22.204 6174.94 −38.528 −141.106
19 37 345.95 −1.119 −77.713 13 067.80 8.500 90.311
20 38 998.86 7.838 −75.525 13 597.10 −47.268 167.237
21 40 626.79 3.069 −73.411
22 42 229.80 0.363 −71.673
23 43 807.99 −1.943 −70.756
24 45 361.43 −0.141 −71.309
25 46 890.20 −0.872 −74.258
26 48 394.02 −2.294 −80.588
27 49 874.02 −0.886 −93.253
28 51 329.22 3.742 −113.948
29 52 760.00 5.960 −146.852
30 54 166.50 1.847 −196.933
31 55 548.70 −1.806 −268.367
32 56 906.67 −0.006 −360.419
33 58 240.46 −0.339 −466.403
34 59 550.10 −0.563 −583.000
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a

E ν represents the experimental vibrational energy. EBν represents the vibrational energy predicted by machine learning method. E5ν represents the vibrational energy of CCSD(T)/cc-pV5Z.

In order to further illustrate the reliability of the current method, many more diatomic molecules have been studied and compared with CCSD(T)/cc-pV5Z. Some are shown in Fig. 4, the height of red pillar is the average relative deviation of CCSD(T)/cc-pV5Z and the height of blue pilar is the average relative deviation of ANN. It shows that the improvement in prediction introduced by ANN over the ab initio method is better than that obtained by expanding the basis set.

Fig. 4. Comparison of the average relative deviation of vibrational energy levels between ANN and CCSD(T)/cc-pV5Z.

Fig. 4

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It should also be emphasized that this work also considerably reduces the computational cost. Taking Na35Cl for an example, it takes more than 40 hours to obtain the results of CCSD(T)/cc-pV5Z, compared to less than one hour for the current method, which includes preparing DFT data and executing the ANN algorithm.

4. Conclusion

In this work, a general method is presented to obtain vibrational spectra of diatomic molecules of high quality by starting from conventional DFT calculations and modifying them with artificial neural network models. This approach provides a different path to improve DFT results without introducing sophisticated models (such as specific hybrid functionals) and large basis sets. Compared with the results of CCSD(T)/cc-pV5Z, the current work reduces the vibrational energies prediction error for diatomic systems from hundreds to dozens, even to tenths, and takes less than a tenth of the time. Since the strategy employed in this paper is a general data-driven approach, it can be easily extended to calculations of other molecular properties. For example, current DFT calculations of fluorescence spectra of macromolecular systems can be easily exceed 1000 cm−1.60,61 In future work, it is expected that the fluorescence spectral prediction capability of DFT can be improved by building a fluorescence spectral data set and adopting a correction method similar to that used in this work. There are several keys that should be attention: (1) collect accurate experimental (or computational) data of macromolecular system properties to establish a data set; (2) from simple to complex, try a variety of DFT methods for these properties, so that the calculation error on the data set can show a certain trend (similar to Fig. 3); (3) build a high-dimensional function through ANN and learn the rule of calculation error; (4) combine DFT and ANN error model to achieve higher prediction quality.

Conflicts of interest

There are no conflicts to declare

Supplementary Material

Acknowledgments

This research is 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), the Open Foundation of Key Laboratory of Advanced Reactor Engineering and Safety (Grant No. ares-2019-01) and the Open Re-search Fund Program of the Collaborative Innovation Center of Extreme Optics (Grant No. KF2020003).

Notes and references

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