Title Combined first-principles calculation and neural

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Combined first-principles calculation and neural-network correction approach for heat of formation

Hu, LH; Wang, XJ; Wong, LH; Chen, G

Journal of Chemical Physics, 2003, v. 119 n. 22, p. 11501-11507

2003

http://hdl.handle.net/10722/42048

Journal of Chemical Physics. Copyright © American Institute of Physics.

JOURNAL OF CHEMICAL PHYSICS

VOLUME 119, NUMBER 22

8 DECEMBER 2003

COMMUNICATIONS Combined first-principles calculation and neural-network correction approach for heat of formation LiHong Hu, XiuJun Wang, LaiHo Wong, and GuanHua Chen Department of Chemistry, The University of Hong Kong, Hong Kong, China

共Received 17 March 2003; accepted 14 October 2003兲 Despite their success, the results of first-principles quantum mechanical calculations contain inherent numerical errors caused by various intrinsic approximations. We propose here a neural-network-based algorithm to greatly reduce these inherent errors. As a demonstration, this combined quantum mechanical calculation and neural-network correction approach is applied to the evaluation of standard heat of formation ⌬ fH 両 for 180 small- to medium-sized organic molecules at 298 K. A dramatic reduction of numerical errors is clearly shown with systematic deviation being eliminated. For example, the root-mean-square deviation of the calculated ⌬ fH 両 for the 180 molecules is reduced from 21.4 to 3.1 kcal mol⫺1 for B3LYP/6-311⫹G(d,p) and from 12.0 to 3.3 kcal mol⫺1 for B3LYP/6-311⫹G(3d f ,2p) before and after the neural-network correction. © 2003 American Institute of Physics. 关DOI: 10.1063/1.1630951兴

One of the Holy Grails of computational science is to quantitatively predict properties of matter prior to experiments. Despite the fact that the first-principles quantum mechanical calculation1,2 has become an indispensable research tool and experimentalists have been increasingly relying on computational results to interpret their experimental findings, the practically used numerical methods by far are often not accurate enough in particular for complex systems. This limitation is caused by the inherent approximations adopted in the first-principles methods. Because of computational cost, electron correlation has always been a difficult obstacle for first-principles calculations. Finite basis sets chosen in practical computations are not able to cover entire physical space and this inadequacy introduces also inherent computational errors. Effective core potential is frequently used to approximate the relativistic effects, resulting inevitably in errors for systems that contain heavy atoms. The accuracy of a density functional theory 共DFT兲 calculation is mainly determined by the exchange-correlation 共XC兲 functional being employed,1 whose exact form is however unknown. Nevertheless, the results of first-principles quantum mechanical calculation can capture the essence of physics. For instance, the calculated results, despite that their absolute values may agree poorly with measurements, are usually of the same tendency among different molecules as their experimental counterparts. The quantitative discrepancy between the calculated and experimental results depends predominantly on the property of primary interest and, to a less extent, also on other related properties of the material. There exists thus a sort of quantitative relation between the calculated and experimental results, as the aforementioned approximations to a large extent contribute to the systematic errors of specified first-principles methods. Can we develop general ways to eliminate the systematic computational errors and further to quantify the accuracies of numerical 0021-9606/2003/119(22)/11501/7/$20.00

methods used? It has been proven an extremely difficult task to determine the calculation errors from the first-principles. Alternatives must be sought. We propose here a neural-network-based algorithm to determine the quantitative relationship between the experimental data and the first-principles calculation results. The determined relation will subsequently be used to eliminate the systematic deviations of the calculated results and thus reduce the numerical uncertainties. Since its beginning in the late fifties, neural networks has been applied to various engineering problems, such as robotics, pattern recognition, speech, etc.3,4 As the first application of neural networks to quantum mechanical calculations of molecules, we choose the standard heat of formation ⌬ fH 両 at 298.15 K as the property of interest. A total of 180 small- or medium-sized organic molecules, whose ⌬ fH 両 values are well documented in Refs. 5–7, are selected to test our proposed approach. The three tabulated values of ⌬ fH 両 in the three references differ less than 1.0 kcal mol⫺1 for any one of the 180 molecules. The uncertainties of all ⌬ fH 両 values are less than 1.0 kcal mol⫺1 in Refs. 5–7. These selected molecules contain elements such as H, C, N, O, F, Si, S, Cl, and Br. The heaviest molecule contains 14 heavy atoms, and the largest has 32 atoms. We divide these molecules randomly into the training set 共150 molecules兲 and the testing set 共30 molecules兲. The geometries of 180 molecules are optimized via B3LYP/6-311 ⫹G(d,p) 8 calculations, and the zero point energies 共ZPEs兲 are calculated at the same level. The enthalpy of each molecule is calculated at both B3LYP/6-311⫹G(d,p) and B3LYP/6-311⫹G(3d f ,2p). 8 B3LYP/6-311⫹G(3d f ,2p) employs a larger basis set than B3LYP/6-311⫹G(d,p). The unscaled B3LYP/6-311⫹G(d,p) ZPE is employed in the ⌬ fH 両 calculations. The strategies in Ref. 9 are adopted to calculate ⌬ fH 両 . The calculated ⌬ fH 両 s for B3LYP/6-311

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© 2003 American Institute of Physics

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FIG. 1. Experimental ⌬ fH 両 vs the calculated ⌬ fH 両 for all 180 compounds. 共a兲 and 共b兲 are the comparisons of the experimental ⌬ fH 両 s to their raw B3LYP/6-311⫹G(d,p) and B3LYP/6-311⫹G(3d f ,2p) results, respectively. 共c兲 and 共d兲 are the comparisons of the experimental ⌬ fH 両 s to the neural-network corrected B3LYP/ 6-311⫹G(d,p) and B3LYP/6-311 ⫹G(3d f ,2p) ⌬ fH 両 s, respectively. In 共c兲 and 共d兲, the triangles are for the training set and the crosses for the testing set. The correlation coefficients of the linear fits are 0.998 and 0.994 in 共c兲 and 共d兲, respectively. Insets are the histograms for the differences between the experimental and calculated ⌬ fH 両 s 共before and after the neuralnetwork corrections兲. All values are in the units of kcal mol⫺1 .

⫹G(d,p) and B3LYP/6-311⫹G(3d f ,2p) are compared to their experimental data in Figs. 1共a兲 and 1共b兲. The horizontal coordinates are the raw calculated data and the vertical coordinates are the experimental values. The dashed lines are where the vertical and horizontal coordinates are equal, i.e., where the B3LYP calculations and experiments would have the perfect match. The raw calculation values are mostly below the dashed line, i.e., most raw ⌬ fH 両 s are larger than the experimental data. In other words, there are systematic deviations for B3LYP ⌬ fH 両 . Compared to the experimental measurements, the root-mean-square 共RMS兲 deviations for ⌬ fH 両 are 21.4 and 12.0 kcal mol⫺1 for B3LYP/6-311 ⫹G(d,p) and B3LYP/6-311⫹G(3d f ,2p) calculations, respectively. In Table I we compare the B3LYP and experimental ⌬ fH 両 s for 180 molecules with first seven small molecules reported in Ref. 9. Overall, B3LYP/6-311 ⫹G(3d f ,2p) calculations yield better agreements with the experiments than B3LYP/6-311⫹G(d, p). In particular, for small molecules with few heavy elements B3LYP/6-311 ⫹G(3dp,2p) calculations result in very small deviations from the experiments. For instance, the ⌬ fH 両 deviations for CH4 and CS2 are only ⫺0.3 and 0.4 kcal mol⫺1 , respectively. The results in Ref. 9 are slightly better than the B3LYP/6-311⫹G(3d f ,2p) results for small molecules, and this is because the ZPEs in Ref. 9 are scaled by an empirical factor 0.98. For large molecules, both B3LYP/6-311 ⫹G(d, p) and B3LYP/6-311⫹G(3d f ,2p) calculations yield quite large deviations from their experimental counterparts. To improve the comparison with the experiment, different empirical scaling factors have been employed for large molecules. Our neural network adopts a three-layer architecture which has an input layer consisting of inputs from the physical descriptors and a bias, a hidden layer containing a number of hidden neurons, and an output layer that outputs the corrected values for ⌬ fH 両 共see Fig. 2兲. The number of hid-

den neurons is to be determined. The most important issue is to select the proper physical descriptors of our molecules, which are to be used as the input for our neural network. The calculated ⌬ fH 両 contains the essence of exact ⌬ fH 両 , and is thus an obvious choice of the primary descriptor for correcting ⌬ fH 両 . We observe that the size of a molecule affects the accuracies of calculations. We plot the difference between the calculated ⌬ fH 両 and the measured ⌬ fH 両 versus the number of atoms N t in Fig. 3. Roughly speaking, the more atoms a molecule has, the larger the deviation. The deviation is approximately proportional to N t , although the proportionality is by no means strict. This is consistent with the general observations of others in the field.9 N t of a molecule is thus chosen as the second descriptor for the molecule. ZPE is an important parameter in calculating ⌬ fH 両 . Its calculated value is often rescaled in evaluating ⌬ fH 両 , 9 and it is thus

FIG. 2. Structure of our neural network.

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J. Chem. Phys., Vol. 119, No. 22, 8 December 2003

Calculated heat of formation

FIG. 3. Deviations 共theory–expt.兲 vs the number of atoms 共a兲 for B3LYP/6-311⫹G(d,p); 共b兲 for B3LYP/6-311⫹G(3d f ,2p).

taken as the third physical descriptor. Finally, the number of double bonds, N db , is selected as the fourth and last descriptor to reflect the chemical structure of a molecule. To ensure the quality of our neural network, a cross-validation procedure is employed to determine our neural network including its structure and weights.10 We randomly divide further 150 training molecules into five subsets of equal size. Four of them are used to train the neural network, and the fifth to validate its predictions. This procedure is repeated five times in rotation. The number of neurons in the hidden layer is varied from 1 to 10 to decide the optimal structure of our neural network. We find that the hidden layer containing two neurons yields the best overall results. Therefore, the 5-2-1 structure is adopted for our neural network as depicted in Fig. 2. The input values at the input layer, x 1 , x 2 , x 3 , x 4 , and x 5 , are scaled ⌬ fH 両 , N t , ZPE, N db , and bias, respectively. The bias x 5 is set to 1. The weights 兵 Wx i j 其 s connect the input neurons 兵 x i 其 and the hidden neurons y 1 and y 2 , and 兵 Wy j 其 s connect the hidden neurons and the output Z which is the scaled ⌬ fH 両 upon neural-network correction. The output Z is related to the input 兵 x i 其 as Z⫽

兺

j⫽1,2

Wy j Sig

冉兺

i⫽1,5

冊

Wx i j x i ,

共1兲

where Sig( v )⫽ 1/关 1⫹exp(⫺␣v)兴 and ␣ is a parameter that controls the switch steepness of Sigmoidal function Sig( v ). An error back-propagation learning procedure4 is used to optimize the values of Wx i j and Wy j (i⫽1,2,3,4,5 and j ⫽1,2). In Figs. 1共c兲 and 1共d兲 we plot the comparison be-

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tween the neural-network corrected ⌬ fH 両 s and their experimental values 共with the vertical coordinates for the experimental values and the horizontal coordinates for the calculated ⌬ fH 両 s). The triangles belong to the training set and the crosses belong to the testing set. Compared to the raw calculated results, the neural-network corrected values are much closer to the experimental values for both training and testing sets. More importantly, the systematic deviations for ⌬ fH 両 in Figs. 1共a兲 and 1共b兲 are eliminated, and the resulting numerical deviations are reduced substantially. This can be further demonstrated by the error analysis performed for the raw and neural-network corrected ⌬ fH 両 s of all 180 molecules. In the insets of Fig. 1, we plot the histograms for the deviations 共from the experiments兲 of the raw B3LYP ⌬ fH 両 s and their neural-network corrected values. Obviously, the raw calculated ⌬ fH 両 s have large systematic deviations: 21.4 and 12.0 kcal mol⫺1 for ⌬ fH 両 s at B3LYP/6-311 ⫹G(d,p) and B3LYP/6-311⫹G(3d f ,2p), respectively. On the contrary, the neural-network corrected ⌬ fH 両 s have virtually no systematic deviations. Moreover, the remaining numerical deviations are much smaller. Upon the neuralnetwork corrections, the RMS deviations of ⌬ fH 両 s are reduced from 21.4 to 3.1 kcal mol⫺1 and 12.0 to 3.3 kcal mol⫺1 for B3LYP/6-311⫹G(d,p) and B3LYP/6-311⫹G(3d f ,2p), respectively. Note that the error distributions after the neural-network correction are of approximate Gaussian distributions 关see Figs. 1共c兲 and 1共d兲兴. Although the raw B3LYP/6-311⫹G(d,p) results have much larger deviations than those of B3LYP/6-311⫹G(3d f ,2p), the neural-network corrected values of both calculations have deviations of the same magnitude. This implies that it may be sufficient to employ the smaller basis set 6-311 ⫹G(d,p) in our combined DFT calculation and neuralnetwork correction 共or DFT-NEURON兲 approach. The neural-network-based algorithm can correct easily the deficiency of a small basis set. Therefore, the DFT-NEURON approach can potentially be applied to much larger systems. In Table I we distinguish the molecules of the testing sets. The deviations of large molecules are of the same magnitude as those of small molecules. Unlike other quantum mechanical calculations that usually yield worse results for larger molecules than for small ones, the DFT-NEURON approach does not discriminate against the large molecules. In Table II we list the values of 兵 Wx i j 其 and 兵 Wy j 其 of the two neural networks for correcting ⌬ fH 両 of B3LYP/6-311 ⫹G(d,p) and B3LYP/6-311⫹G(3d f ,2p) calculations. Analysis of our neural network reveals that the weights connecting the input for ⌬ fH 両 have the dominant contribution in all cases. This confirms our fundamental assumption that the calculated ⌬ fH 両 captures the essential value of exact ⌬ fH 両 . The bias contributes significantly to the correction of systematic deviations in the raw calculated data, and always has the second largest weights for all cases. The input for the second physical descriptor, N t , has also large weights in all cases, in particular for 6-311⫹G(d,p). This is because the raw ⌬ fH 両 deviations are roughly proportional to N t as shown in Fig. 3, which confirms the importance of N t as a significant descriptor of our neural network. ZPE has been often rescaled to account for the discrepancies between cal-

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TABLE I. Experimental and calculated ⌬ fH 両 共298 K兲 for 180 compounds 共all data are in the units of kcal mol⫺1 ). ⌬ fH 両 共298 K兲 Formula CF2 O CH2 Cl2 CH2 F2 g CH4 CH4 Og CS2 C6 H6 g CBrCl3 CBrF3 g CClF3 g,h CClNg CCl2 O CF4 CHCl3 CHF3 CH2 O2 CH3 Br CH3 NO2 CH3 NO2 CH4 Sg,h CH5 N COS C2 H2 C2 H2 Cl2 g C2 H2 F2 C2 H2 O4 C2 H3 Br C2 H3 ClOh C2 H3 ClO2 C2 H3 Cl3 g C2 H3 F C2 H4 g C2 H4 Br2 C2 H4 Cl2 C2 H4 Cl2 C2 H4 F2 C2 H4 O C2 H4 O2 C2 H4 S C2 H5 Br C2 H5 Cl C2 H5 N C2 H5 NOg,h C2 H5 NO2 C2 H5 NO3 C2 H6 C2 H6 O C2 H6 S C2 H7 N C2 H7 Nh C2 H8 N2 C2 N2 C3 H3 NO C3 H4 g C3 H4 h C3 H4 O3 C3 H5 Cl3 C3 H6 g C3 H6 g C3 H6 Br2 C3 H6 Cl2 h C3 H6 O C3 H6 O2 h C3 H6 O2 g C3 H6 Sh

Name carbonyl fluoride dichloromethane difluoromethane methane methanol carbon disulfide benzene bromotrichloromethane bromotrifluoromethane chlorotrifluoromethane cyanogen chloride phosgene carbon tetrafluoride chloroform trifluoromethane formic acid methyl bromide nitromethane methyl nitrite methyl mercaptan methylamine carbonyl sulfide acetylene 1,1-dichloroethylene 1,1-difluoroethylene oxalic acid vinyl bromide acetyl chloride chloroacetic acid 1,1,1-trichloroethane vinyl fluoride ethylene 1,2-dibromoethane 1,1-dichloroethane 1,2-dichloroethane 1,1-difluoroethane ethylene oxide acetic acid thiacyclopropane bromoethane ethyl chloride ethyleneimine acetamide nitroethane ethyl nitrate ethane dimethyl ether dimethyl sulfide dimethylamine ethylamine ethylenediamine cyanogen oxazole methylacetylene propadiene ethylene carbonate 1,2,3-trichloropropane cyclopropane propylene 1,2-dibromopropane 1,2-dichloropropane acetone methyl acetate propionic acid thiacyclobutane

Expt.a

DFT1b

NN1c

DFT2d

NN2e

DFT3f

⫺152.7 ⫺22.8 ⫺108.2 ⫺17.9 ⫺48.1 28.0 19.8 ⫺9.3 ⫺155.1 ⫺169.2 33.0 ⫺52.3 ⫺223.0 ⫺24.2 ⫺166.7 ⫺90.5 ⫺9.0 ⫺17.9 ⫺15.3 ⫺5.5 ⫺5.5 ⫺33.1 54.2 0.6 ⫺80.5 ⫺173.0 18.7 ⫺58.3 ⫺104.3 ⫺34.0 ⫺33.2 12.5 ⫺9.3 ⫺31.0 ⫺31.0 ⫺118.0 ⫺12.6 ⫺103.9 19.7 ⫺15.3 ⫺26.7 29.5 ⫺57.0 ⫺24.2 ⫺36.8 ⫺20.2 ⫺44.0 ⫺9.0 ⫺4.5 ⫺11.0 ⫺4.1 73.8 ⫺3.7 44.3 45.9 ⫺121.2 ⫺44.4 12.7 4.9 ⫺17.4 ⫺39.6 ⫺52.0 ⫺98.0 ⫺108.4 14.6

⫺132.9 ⫺12.2 ⫺100.1 ⫺16.6 ⫺42.9 36.5 36.2 11.3 ⫺140.4 ⫺150.9 40.3 ⫺40.4 ⫺203.2 ⫺7.4 ⫺153.2 ⫺82.9 ⫺6.8 ⫺11.0 ⫺7.9 1.4 ⫺3.4 ⫺25.8 59.8 15.0 ⫺73.5 ⫺152.7 22.6 ⫺47.5 ⫺97.3 ⫺12.7 ⫺27.7 16.3 ⫺0.0 ⫺17.6 ⫺18.2 ⫺109.3 ⫺3.6 ⫺91.9 30.3 ⫺9.8 ⫺18.1 36.8 ⫺49.6 ⫺14.4 ⫺24.2 ⫺15.9 ⫺36.3 1.9 0.9 ⫺6.3 3.4 78.4 8.9 51.5 49.4 ⫺101.4 ⫺17.5 21.6 11.9 ⫺5.2 ⫺20.4 ⫺41.6 ⫺83.9 ⫺91.8 31.1

⫺146.0 ⫺19.2 ⫺107.2 ⫺16.7 ⫺46.2 31.2 21.1 ⫺5.3 ⫺156.9 ⫺168.0 34.3 ⫺49.8 ⫺218.6 ⫺18.8 ⫺167.6 ⫺89.2 ⫺10.3 ⫺20.1 ⫺17.8 ⫺4.0 ⫺6.5 ⫺29.4 54.2 4.4 ⫺83.9 ⫺177.3 15.6 ⫺58.3 ⫺112.6 ⫺29.4 ⫺33.8 13.5 ⫺12.4 ⫺29.7 ⫺29.9 ⫺121.9 ⫺9.8 ⫺103.8 22.4 ⫺18.0 ⫺26.0 29.4 ⫺60.5 ⫺28.8 ⫺42.3 ⫺20.2 ⫺44.5 ⫺7.9 ⫺7.2 ⫺14.2 ⫺8.2 65.7 ⫺1.6 42.5 41.5 ⫺119.3 ⫺38.3 14.1 4.4 ⫺22.5 ⫺37.2 ⫺53.6 ⫺100.3 ⫺108.3 18.8

⫺144.2 ⫺17.8 ⫺107.5 ⫺18.2 ⫺47.2 28.4 26.7 1.2 ⫺153.4 ⫺165.0 32.7 ⫺49.2 ⫺218.9 ⫺15.6 ⫺164.5 ⫺89.6 ⫺9.2 ⫺20.5 ⫺17.4 ⫺3.3 ⫺7.3 ⫺34.0 56.7 6.8 ⫺83.5 ⫺166.5 18.5 ⫺55.0 ⫺97.3 ⫺22.0 ⫺34.0 13.1 ⫺4.3 ⫺24.3 ⫺24.6 ⫺117.9 ⫺10.3 ⫺100.1 23.9 ⫺13.2 ⫺22.6 30.5 ⫺57.5 ⫺25.1 ⫺38.4 ⫺18.8 ⫺42.6 ⫺4.6 ⫺4.5 ⫺11.4 ⫺4.0 70.7 ⫺2.1 46.8 44.4 ⫺115.9 ⫺27.2 16.7 7.2 ⫺10.5 ⫺28.1 ⫺48.5 ⫺94.1 ⫺101.3 23.9

⫺146.2 ⫺17.9 ⫺107.5 ⫺16.8 ⫺47.3 31.1 21.1 ⫺1.1 ⫺156.7 ⫺169.1 33.8 ⫺48.6 ⫺220.1 ⫺16.7 ⫺168.2 ⫺89.2 ⫺8.4 ⫺22.0 ⫺19.1 ⫺3.7 ⫺8.0 ⫺30.6 55.7 5.8 ⫺84.7 ⫺176.9 18.1 ⫺57.0 ⫺101.2 ⫺26.7 ⫺34.1 13.7 ⫺7.8 ⫺27.9 ⫺28.1 ⫺121.7 ⫺11.7 ⫺103.4 21.8 ⫺15.9 ⫺25.2 27.7 ⫺61.1 ⫺30.7 ⫺45.1 ⫺20.5 ⫺46.2 ⫺8.3 ⫺8.6 ⫺15.5 ⫺10.5 66.9 ⫺3.9 43.7 42.9 ⫺122.7 ⫺35.1 13.4 4.6 ⫺17.5 ⫺35.1 ⫺53.2 ⫺100.8 ⫺108.1 18.6

⫺143.6 ⫺18.2 ⫺107.7 ⫺19.5 ⫺48.1 28.2 24.2 – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – –

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J. Chem. Phys., Vol. 119, No. 22, 8 December 2003

Calculated heat of formation

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DFT3f

TABLE I 共Continued兲. ⌬ fH 両 共298 K兲 Formula C3 H7 Br C3 H7 Brh C3 H7 Cl C3 H7 Cl C3 H7 F C3 H7 NOg C3 H7 NO2 C3 H7 NO2 C3 H7 NO3 C3 H7 NO3 C3 H8 h C3 H8 O C3 H8 S C3 H8 S C3 H8 S C3 H9 N C3 H9 N C3 H9 N C3 H10N2 h C4 H4 N2 C4 H6 C4 H6 Oh C4 H8 h C4 H8 O C4 H8 O2 C4 H9 Br C4 H9 Cl C4 H10O C4 H10O2 h C4 H10S C4 H10S C4 H11N C5 H5 N C5 H6 S C5 H8 C5 H8 O2 h C5 H10h C5 H10 C5 H10 C5 H10 C5 H10 C5 H10 C5 H10h C5 H10O C5 H10O C5 H10O2 g C5 H10Sh C5 H10Sg,h C5 H11Br C5 H11Cl C5 H11N C5 H12 C5 H12 C5 H12O C5 H12Og,h C5 H12O C5 H12O C5 H12O C5 H12O C5 H12O4 C5 H12Sg C5 H12S C 6 F6 C6 H4 Cl2 C6 H4 F2 g,h

Name 1-bromopropane 2-bromopropane isopropyl chloride n-propyl chloride 1-fluoropropane N,N-dimethylformamide 1-nitropropane 2-nitropropane propyl nitrate isopropyl nitrate propane methyl ethyl ether n-propyl mercaptan isopropyl mercaptan ethyl methyl sulfide n-propylamine isopropylamine trimethylamine 1,2-propanediamine succinonitrile 1,2-butadiene divinyl ether 1-butene isobutyraldehyde ethyl acetate 1-bromobutane tert-butyl chloride sec-butanol 1,4-butanediol isobutyl mercaptan methyl propyl sulfide tert-butylamine pyridine 2-methylthiophene trans-1,3-pentadiene acetylacetone cyclopentane 2-methyl-1-butene 2-methyl-2-butene 3-methyl-1-butene 1-pentene cis-2-pentene trans-2-pentene 2-pentanone valeraldehyde valeric acid thiacyclohexane cyclopentanethiol 1-bromopentane 1-chloropentane piperidine isopentane n-pentane 2-methyl-1-butanol 3-methyl-1-butanol 3-methyl-2-butanol 2-pentanol 3-pentanol ethyl propyl ether pentaerythritol n-pentyl mercaptan butyl methyl sulfide hexafluorobenzene m-dichlorobenzene p-difluorobenzene

Expt.a

DFT1b

NN1c

DFT2d

NN2e

⫺21.0 ⫺23.2 ⫺35.0 ⫺31.1 ⫺67.2 ⫺45.8 ⫺29.8 ⫺33.5 ⫺41.6 ⫺45.6 ⫺24.8 ⫺51.7 ⫺16.2 ⫺18.2 ⫺14.2 ⫺17.3 ⫺20.0 ⫺5.7 ⫺12.8 50.1 38.8 ⫺3.3 ⫺0.0 ⫺51.5 ⫺105.9 ⫺25.6 ⫺43.8 ⫺69.9 ⫺102.0 ⫺23.2 ⫺19.5 ⫺28.6 33.5 20.0 18.6 ⫺90.8 ⫺18.5 ⫺8.7 ⫺10.2 ⫺6.9 ⫺5.0 ⫺6.7 ⫺7.6 ⫺61.8 ⫺54.4 ⫺117.2 ⫺15.1 ⫺11.5 ⫺30.9 ⫺41.8 ⫺11.7 ⫺36.9 ⫺35.0 ⫺72.2 ⫺72.2 ⫺75.1 ⫺75.0 ⫺75.7 ⫺65.1 ⫺185.6 ⫺25.9 ⫺24.4 ⫺228.6 6.3 ⫺73.4

⫺10.7 ⫺12.8 ⫺21.7 ⫺18.9 ⫺58.9 ⫺36.2 ⫺15.0 ⫺17.9 ⫺24.9 ⫺28.1 ⫺16.8 ⫺40.6 ⫺1.6 ⫺2.7 0.7 ⫺7.0 ⫺9.7 3.4 0.4 63.5 46.5 10.0 11.3 ⫺35.6 ⫺88.0 ⫺11.4 ⫺24.6 ⫺52.4 ⫺79.8 ⫺2.4 ⫺0.1 ⫺12.1 46.2 44.3 31.7 ⫺66.6 0.9 7.9 5.9 10.6 10.4 9.0 7.5 ⫺43.8 ⫺35.3 ⫺94.4 11.4 14.7 ⫺12.2 ⫺20.5 9.7 ⫺18.4 ⫺18.4 ⫺49.2 ⫺48.2 ⫺52.1 ⫺52.0 ⫺49.8 ⫺45.4 ⫺143.7 ⫺3.2 ⫺0.9 ⫺194.8 32.8 ⫺51.3

⫺23.7 ⫺26.1 ⫺34.7 ⫺31.7 ⫺71.5 ⫺51.5 ⫺33.7 ⫺36.8 ⫺47.7 ⫺51.3 ⫺25.9 ⫺53.7 ⫺16.5 ⫺17.8 ⫺13.8 ⫺19.7 ⫺22.7 ⫺9.8 ⫺16.1 44.5 34.4 ⫺4.5 ⫺0.8 ⫺52.3 ⫺109.4 ⫺29.3 ⫺42.9 ⫺70.5 ⫺101.3 ⫺22.3 ⫺19.4 ⫺30.4 31.2 25.6 16.0 ⫺91.4 ⫺14.9 ⫺9.2 ⫺11.6 ⫺6.5 ⫺6.5 ⫺8.1 ⫺9.7 ⫺65.5 ⫺56.6 ⫺120.6 ⫺9.2 ⫺7.3 ⫺35.0 ⫺43.0 ⫺9.6 ⫺37.4 ⫺37.2 ⫺71.9 ⫺70.7 ⫺75.1 ⫺74.7 ⫺72.6 ⫺68.2 ⫺179.6 ⫺27.8 ⫺25.1 ⫺230.3 9.3 ⫺74.8

⫺15.4 ⫺17.5 ⫺27.5 ⫺24.8 ⫺65.6 ⫺45.9 ⫺27.0 ⫺29.6 ⫺40.3 ⫺43.4 ⫺21.0 ⫺48.1 ⫺8.7 ⫺9.7 ⫺7.0 ⫺13.4 ⫺16.2 ⫺3.6 ⫺8.0 53.2 40.1 ⫺0.5 5.3 ⫺43.7 ⫺99.4 ⫺17.5 ⫺31.7 ⫺60.3 ⫺90.3 ⫺10.7 ⫺9.1 ⫺19.8 35.6 31.7 23.7 ⫺78.8 ⫺5.8 0.5 ⫺1.7 3.3 3.1 1.6 0.1 ⫺53.3 ⫺44.7 ⫺106.5 1.8 5.2 ⫺19.6 ⫺29.0 0.8 ⫺25.3 ⫺25.2 ⫺58.4 ⫺57.4 ⫺61.4 ⫺61.1 ⫺59.0 ⫺55.4 ⫺159.3 ⫺12.9 ⫺11.2 ⫺225.2 18.5 ⫺67.1

⫺21.5 ⫺23.6 ⫺33.6 ⫺30.8 ⫺71.7 ⫺52.8 ⫺35.3 ⫺38.1 ⫺50.5 ⫺53.7 ⫺26.2 ⫺55.1 ⫺16.0 ⫺17.0 ⫺14.1 ⫺21.0 ⫺23.8 ⫺11.2 ⫺17.9 45.3 35.7 ⫺5.6 ⫺0.6 ⫺51.7 ⫺109.8 ⫺27.0 ⫺41.3 ⫺70.7 ⫺102.1 ⫺21.4 ⫺19.6 ⫺31.0 30.6 24.6 16.7 ⫺90.2 ⫺15.5 ⫺8.9 ⫺11.2 ⫺6.1 ⫺6.2 ⫺7.7 ⫺9.3 ⫺64.8 ⫺56.1 ⫺120.6 ⫺9.9 ⫺6.7 ⫺32.5 ⫺41.9 ⫺11.3 ⫺37.4 ⫺37.3 ⫺72.2 ⫺71.1 ⫺75.2 ⫺74.9 ⫺72.8 ⫺69.2 ⫺181.2 ⫺27.0 ⫺25.2 ⫺232.6 11.1 ⫺75.3

– – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – –

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Hu et al.

J. Chem. Phys., Vol. 119, No. 22, 8 December 2003

TABLE I 共Continued兲. ⌬ fH 両 共298 K兲 Formula C6 H5 Cl C6 H5 F C6 H5 NO2 C6 H6 N2 O2 C6 H6 O C6 H6 O2 C6 H7 N C6 H8 N2 C6 H10g C6 H10 C6 H10O3 C6 H11NOg C6 H12 C6 H12O C6 H12Og C6 H14g,h C6 H14S C7 H5 N C7 H6 O C7 H8 g C7 H8 O C7 H9 N C7 H14 C7 H15Br C7 H16h C7 H16 C7 H16S C8 H6 O4 C8 H8 O C8 H10h C8 H10Og C8 H16 C8 H16h C8 H16h C8 H18 C8 H18 C8 H18 C8 H18g C8 H18Og C8 H18S2 C9 H12 C9 H12g C9 H18Oh C9 H20h C9 H20 C10H14 C10H14 C10H18O4 C10H20O2 C12H10

Name monochlorobenzene fluorobenzene nitrobenzene m-nitroaniline phenol 1,3-benzenediol 2-methylpyridine adiponitrile 1-methylcyclopentene 1,5-hexadiene propionic anhydride e-caprolactam trans-3-hexene butyl vinyl ether 3-hexanone 3-methylpentane methyl pentyl sulfide benzonitrile benzaldehyde toluene o-cresol 2,6-dimethylpyridine cis-1,2-dimethylcyclopentane 1-bromoheptane 3,3-dimethylpentane 2,2,3-trimethylbutane n-heptyl mercaptan terephthalic acid acetophenone o-xylene 3,4-xylenol cis-1,2-dimethylcyclohexane trans-1,4-dimethylcyclohexane 2,4,4-trimethyl-2-pentene 2,3-dimethylhexane 3-ethylhexane 4-methylheptane 2,3,4-trimethylpentane 2-ethyl-1-hexanol dibutyl disulfide m-ethyltoluene 1,2,3-trimethylbenzene diisobutyl ketone 3,3-diethylpentane 2,2,3,4-tetramethylpentane sec-butylbenzene isobutylbenzene sebacic acid n-decanoic acid acenaphthene

Expt.a

DFT1b

NN1c

DFT2d

NN2e

12.4 ⫺27.9 16.2 14.0 ⫺23.0 ⫺65.7 23.6 35.7 ⫺1.3 20.1 ⫺149.7 ⫺58.8 ⫺13.0 ⫺43.7 ⫺66.4 ⫺41.0 ⫺29.3 52.3 ⫺8.8 11.9 ⫺30.7 14.0 ⫺31.0 ⫺40.2 ⫺48.2 ⫺48.9 ⫺35.8 ⫺171.6 ⫺20.7 4.5 ⫺37.4 ⫺41.1 ⫺44.1 ⫺25.1 ⫺51.1 ⫺50.4 ⫺50.7 ⫺52.0 ⫺87.3 ⫺37.9 ⫺0.5 ⫺2.3 ⫺85.5 ⫺55.4 ⫺56.6 ⫺4.0 ⫺4.9 ⫺220.3 ⫺142.0 37.0

34.0 ⫺8.3 37.1 38.2 ⫺1.5 ⫺39.3 40.7 59.3 20.9 38.7 ⫺116.5 ⫺32.1 6.9 ⫺18.9 ⫺43.8 ⫺16.1 ⫺1.7 72.1 13.2 32.9 ⫺5.0 35.2 ⫺1.0 ⫺13.8 ⫺17.4 ⫺17.4 ⫺4.1 ⫺121.9 7.0 30.3 ⫺7.1 ⫺5.7 ⫺4.7 6.8 ⫺17.7 ⫺16.6 ⫺19.4 ⫺14.0 ⫺47.7 7.5 29.3 29.1 ⫺45.4 ⫺11.2 ⫺14.1 31.5 30.4 ⫺163.4 ⫺92.9 79.0

14.8 ⫺26.8 12.1 8.8 ⫺20.3 ⫺63.1 20.8 31.7 1.4 18.2 ⫺153.0 ⫺58.4 ⫺15.1 ⫺44.9 ⫺70.2 ⫺39.8 ⫺30.7 48.4 ⫺8.6 12.7 ⫺29.1 10.2 ⫺26.9 ⫺46.2 ⫺46.2 ⫺46.6 ⫺38.4 ⫺169.0 ⫺19.9 5.1 ⫺36.7 ⫺36.1 ⫺35.0 ⫺25.4 ⫺51.3 ⫺50.0 ⫺52.8 ⫺47.6 ⫺84.4 ⫺36.2 ⫺0.8 ⫺1.3 ⫺86.3 ⫺49.4 ⫺52.8 ⫺3.1 ⫺4.2 ⫺218.8 ⫺142.2 41.6

22.1 ⫺21.0 19.6 18.0 ⫺14.0 ⫺54.8 28.7 46.4 12.3 29.6 ⫺133.6 ⫺45.0 ⫺1.8 ⫺31.1 ⫺54.6 ⫺24.2 ⫺13.3 58.7 ⫺0.3 22.0 ⫺18.8 21.9 ⫺10.3 ⫺23.8 ⫺26.7 ⫺26.8 ⫺16.5 ⫺144.6 ⫺7.8 17.9 ⫺22.4 ⫺16.1 ⫺15.0 ⫺4.5 ⫺28.5 ⫺27.3 ⫺30.1 ⫺24.7 ⫺60.8 ⫺11.9 15.6 15.3 ⫺60.0 ⫺23.0 ⫺26.1 16.7 15.5 ⫺188.6 ⫺112.1 60.5

15.7 ⫺27.3 11.0 6.9 ⫺21.1 ⫺64.5 20.5 32.6 1.6 19.1 ⫺153.9 ⫺59.9 ⫺14.6 ⫺45.9 ⫺69.5 ⫺39.7 ⫺30.7 49.2 ⫺8.1 13.1 ⫺29.5 10.3 ⫺27.0 ⫺43.6 ⫺45.7 ⫺45.9 ⫺37.5 ⫺169.6 ⫺19.3 5.6 ⫺36.8 ⫺36.2 ⫺35.1 ⫺24.3 ⫺50.9 ⫺49.7 ⫺52.5 ⫺47.1 ⫺84.6 ⫺38.2 ⫺0.1 ⫺0.5 ⫺85.3 ⫺48.8 ⫺52.0 ⫺2.4 ⫺3.5 ⫺221.4 ⫺144.6 42.7

DFT3f – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – –

a

The experimental data were taken from Ref. 5. The calculated ⌬ fH 両 by using B3LPY/6-311⫹G(d,p) geometries, zero point energies and enthalpies. c The calculated ⌬ fH 両 by B3LYP/6-311⫹G(d,p)-neural networks approach. d The calculated ⌬ fH 両 by using the 6-311⫹G(d,p) geometries and zero point energies, and recalculated enthalpies by 6-311⫹G(3d f ,2p) basis. e The calculated ⌬ fH 両 by B3LYP/6-311⫹G(3d f ,2p)-neural networks approach. f The ⌬ fH 両 s were taken from Ref. 9, where the zero point energies were corrected by a scale factor 0.98. g The molecule belongs to the testing set in NN1 calculation. h The molecule belongs to the testing set in NN2 calculation. b

culations and experiments,9 and it is thus expected to have large weights. This is indeed the case, especially when the smaller basis set 6-311⫹G(d, p) is adopted in the calculations. In all cases the number of double bonds, N db , has the smallest but non-negligible weights.

Our DFT-NEURON approach has a RMS deviation of ⬃3 kcal mol⫺1 for the 180 small- to medium-sized organic molecules. The physical descriptors adopted in our neural network, the raw calculated ⌬ fH 両 , the number of atoms N t , the number of double bonds N db , and the ZPE are quite

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J. Chem. Phys., Vol. 119, No. 22, 8 December 2003

Calculated heat of formation

TABLE II. Weights of DFT-neural networks for ⌬ fH 両 . NN1a

a

NN2b

Weights

y1

y2

y1

y2

Wx1 j Wx2 j Wx3 j Wx4 j Wx5 j Wy j

⫺2.48 0.25 0.01 0.26 0.41 ⫺0.21

0.79 ⫺0.50 0.38 0.01 ⫺0.49 1.41

0.85 ⫺0.18 0.09 0.01 ⫺0.56 1.43

⫺2.77 0.09 0.20 0.20 0.59 ⫺0.20

NN1 refers B3LYP/6-311⫹G(d,p)-neural networks approach. NN2 refers B3LYP/6-311⫹G(3d f ,2p)-neural networks approach.

b

general, and are not limited to special properties of organic molecules. The DFT-NEURON approach developed here is expected to yield a RMS deviation of ⬃3 kcal mol⫺1 for ⌬ fH 両 s of any small- to medium-sized organic molecules. G2 共Ref. 9兲 and G3 共Ref. 11兲 methods result in more accurate ⌬ fH 両 for small molecules. Like the DFT-NEURON approach, G2 and G3 methods have empirical fittings to the high order electron correlations and ZPEs, and are thus not entirely ab initio. Our approach is much more efficient and can be applied to much larger systems. To improve the accuracy for the DFT-NEURON approach, we need more and better experimental data, and possibly, more and better physical descriptors for the molecules. Besides ⌬ fH 両 , the DFT-NEURON approach can be generalized to calculate other properties such as Gibbs free energy, ionization energy, dissociation energy, absorption frequency, band gap, etc. The raw first-principles property of interest contains the essence of its exact value, and is thus always the primary descriptor. As the raw calculation error accumulates with increasing molecular size, the number of atoms N t should thus be selected for other DFT-NEURON calculations. Additional physical descriptors should be chosen according to their relations to the property of interest and to the physical and chemical characteristics of the compounds. Others have used neural networks to determine the quantitative relationship between the experimental measured thermodynamic properties and the structure parameters of the molecules.10 We distinguish our work from them by utilizing specifically the firstprinciples methods and with the objective to improve quantum mechanical results. Since the first-principles calculations capture readily the essences of the properties of interest, our approach is more reliable and covers much a wider range of molecules or compounds.

11507

To summarize, we have developed a promising new approach to improve the results of first-principles quantum mechanical calculations and to calibrate their uncertainties. The accuracy of DFT-NEURON approach can be systematically improved as more and better experimental data are available. As the systematic deviations caused by small basis sets and less sophisticated methods adopted in the calculations can be easily corrected by neural networks, the requirements on first-principles calculations are modest. Our approach is thus highly efficient compared to much more sophisticated firstprinciples methods of similar accuracy, and more importantly, is expected to be applied to much larger systems. The combined first-principles calculation and neural-network correction approach developed in this work is potentially a powerful tool in computational physics and chemistry, and may open the possibility for first-principles methods to be employed practically as predictive tools in materials research and design. We thank Professor YiJing Yan for extensive discussion on the subject and generous help in manuscript preparation. Support from the Hong Kong Research Grant Council 共RGC兲 and the Committee for Research and Conference Grants 共CRCG兲 of the University of Hong Kong is gratefully acknowledged.

1

R. G. Parr and W. Yang, Density-Functional Theory of Atoms and Molecules 共Oxford University Press, New York, 1989兲, and references therein. 2 H. F. Schaefer III, Methods of Electronic Structure Theory 共Plenum, New York and London, 1977兲, and references therein. 3 B. D. Ripley, Pattern Recognition and Neural Networks 共Cambridge University Press, New York, 1996兲. 4 D. E. Rumelhart, G. E. Hinton, and R. J. Williams, Nature 共London兲 323, 533 共1986兲. 5 C. L. Yaws, Chemical Properties Handbook 共McGraw–Hill, New York, 1999兲. 6 D. R. Lide, CRC Handbook of Chemistry and Physics, 3rd electronic ed. 共CRC, Boca Raton, FL, 2000兲. 7 J. B. Pedley, R. D. Naylor, and S. P. Kirby, Thermochemical Data of Organic Compounds, 2nd ed. 共Chapman and Hall, New York, 1986兲. 8 M. J. Frisch, G. W. Trucks, H. B. Schlegel et al., GAUSSIAN 98, Revision A.11.3, Gaussian, Inc., Pittsburgh, PA, 2002. 9 L. A. Curtiss, K. Raghavachari, P. C. Redfern, and J. A. Pople, J. Chem. Phys. 106, 1063 共1997兲. 10 X. Yao, X. Zhang, R. Zhang, M. Liu, Z. Hu, and B. Fan, Comput. Chem. 25, 475 共2001兲. 11 L. A. Curtiss, K. Raghavachari, P. C. Redfern, V. Rassolov, and J. A. Pople, J. Chem. Phys. 109, 7764 共1998兲.

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