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Bond Energy Density Analysis Combined with Informatics Technique Hiromi Nakai, Junji Seino, and Kairi Nakamura J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.9b04030 • Publication Date (Web): 19 Aug 2019 Downloaded from pubs.acs.org on August 23, 2019

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The Journal of Physical Chemistry

Bond Energy Density Analysis Combined with Informatics Technique Hiromi Nakai1,2,3,*, Junji Seino2,4, Kairi Nakamura1

1 Department

of Chemistry and Biochemistry, School of Advanced Science and Engineering, Waseda

University, Tokyo169-8555, Japan 2 Waseda

Research Institute for Science and Engineering, Waseda University, Tokyo 169-8555,

Japan 3 ESICB,

Kyoto University, Kyotodaigaku-Katsura, Nishigyoku, Kyoto 615-8520, Japan

4 PRESTO,

Japan Science and Technology Agency, 4-1-8 Honcho, Kawaguchi, Saitama, 332-0012,

Japan

*Corresponding author. Email address: [email protected] (Hiromi Nakai) Fax: +81 3 3205 2504

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ABSTRACT Bond energy density analysis, a two-body energy decomposition scheme, was extended by revisiting the constraint conditions and using the informatics technique. The present scheme can evaluate the bond energies (BEs) for all interatomic pairs including both strong chemical bonds and weak through-space/bond interactions, and bond dissociation energies (BDEs) constructed from BEs. The newly derived formula, presented in the form of the system of linear equations, tends to result in the overfitting problem owing to the small components originating from the weak through-space/bond interactions. Hence, we adopt the least absolute shrinkage and selection operator technique. Numerical assessments of the present scheme were performed for C−C and C−H bonds in typical hydrocarbons as well as 44 chemical bonds, i.e., covalent and ionic bonds, in 33 small molecules involving secondand third-row atoms. The statistics for the BDE estimation confirms the accuracy of the present scheme.


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1. INTRODUCTION Chemical bonds are crucial in chemistry. The rearrangements of chemical bonds are the essence of chemical reactions. Thus, the reactivity of a molecule is directly related with the strength of chemical bonds that might be evaluated quantitatively by the bond dissociation energies (BDEs). The BDE can be observed by experimental techniques and/or estimated by theoretical calculations. In fact, many experimental BDEs are summarized in handbooks.1,2 When a molecule is dissociated into two distinct fragments, i.e., X and Y, by breaking a specific chemical bond, i.e., A−B, the BDE is defined as AB EBDE  E tot  E X  E Y .

(1)

Here, Etot, EX, and EY are the energies of the molecule, fragment X, and fragment Y, respectively. In the dissociation process in experiments, various effects such as structural relaxation and vibrational mode change are involved. Meanwhile, the theoretical method can include and exclude such effects in calculating Etot, EX, and EY. It is noteworthy, however, that the BDE cannot be defined when the distinct fragments are not produced by breaking the specific chemical bond, for example, the ring structure. Chan and Radom3 carefully investigated the accuracies of density functional theory (DFT) calculations and composite protocols such as high-level W1X-24 and G4(MP2)-6X5 to estimate the BDEs. A comprehensive benchmark set of accurate theoretical BDEs, termed the BDE261 set,6 was presented for the numerical assessments. Henceforth, the DFT and high-level methods were examined to evaluate the BDEs of several types of chemical bonds such as C−X (X = H, C, N, O, F, S, and Br),6-8 -3ACS Paragon Plus Environment


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RX−H (X = CH2, NH, O, PH, and S; R = H, H3C, H2N, HO, F, H2P, HS, and Cl),9 and S−X (X = F and Cl).10,11 Recently, the machine-learning technique was introduced for the efficient prediction of a large amount of BDEs of organic molecules.12 An alternative approach for estimating the strength of chemical bonds is using the two-body energy decomposition scheme, in which the total energy of a molecule is divided into one- and twobody components: namely, the intra-atomic components within an atom A, EAA, and interatomic components between atoms A and B, i.e., EAB.

E tot   E AA   A

E

A B   A

AB

.

(2)

The opposite sign of the two-body component, (−EAB), directly corresponds to the bond energy (BE). Furthermore, the proper summation of BEs can estimate the BDE without the structural relaxation and vibrational mode change as follows: AB EBDE    E AB  

  E     E  . CB

C  A , C X

AD

(3)

D  B , DY

Eq. (3) indicates that the first term on the right-hand side is the leading term but the other interactions between fragments X and Y are involved in the BDEs. The advantage of adopting the two-body energy decomposition scheme is the applicability to the case where the BDE cannot be defined as mentioned above. In addition, the scheme provides BEs for all chemical bonds in a molecule by a single quantum chemical calculation, and it can thus easily obtain a large amount of data. In fact, a recent study adopted such data as a reference for machine learning to predict BEs quickly.13 The development of the energy decomposition scheme might have started with semi-empirical -4ACS Paragon Plus Environment

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calculations in the earlier work of Pople and cowokers.14 Subsequently, the scheme was developed to the extended Hückel method,15-19 Hartree–Fock (HF) method,20,21 post-HF methods,22 and DFT.23 In the scheme, different definitions of one- and two-body energy components were proposed because the arbitrariness was inevitable for the energy partitioning. At the HF level, Kollmar proposed one of the starting formulas in the energy decomposition scheme.21 Ichikawa and Yoshida modified the Coulomb interaction in Kollmar’s scheme because of the overestimation of BEs.24 Mayer slightly modified the decomposition of the HF exchange energy term.25 Sato and Sakaki proposed a combination between the Ichikawa–Yoshida and Mayer schemes to improve intermolecular interactions between ammonia and water.26 The authors’ group, i.e., Nakai and Kikuchi, first introduced a fixed parameter to decrease the interatomic contributions of HF exchange and kinetic terms.27 Vyboishchikov introduced another parameter only to the interatomic kinetic terms, which was determined to reproduce the atomization energy.28 The authors’ group, i.e., Nakai, Ohashi, Imamura, and Kikuchi, further introduced an adjustable parameter depending on the atomic forces,29 to provide reasonable BEs for both single and multiple bonds. The authors’ group uses the terminology of bond energy density analysis (Bond-EDA) for the two-body energy decomposition scheme, because the scheme is regarded as the extension of a one-body energy decomposition scheme proposed by one of the authors, named as energy density analysis (EDA).30-41 The Bond-EDA scheme was applied to the analyses in various chemical phenomena such as the Diels–Alder reaction,42 hexacoordinate hypervalent carbon compounds,43 and effective exchange integrals for radical dimers.44 Despite the long history and various improvements -5ACS Paragon Plus Environment


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of the two-body decomposition scheme, a problem still exists for simultaneously evaluating the BEs of various types of chemical bonds such as strong covalent and ionic bonds and weak throughspace/bond interactions. Hence, an extension of the Bond-EDA scheme is proposed herein by revisiting the constraint conditions and adopting the informatics technique. The organization of this article is as follows. Section 2 provides the theoretical aspects including a brief summary of the conventional Bond-EDA technique and the present modification. Section 3 summarizes the computational details. Section 4 describes the numerical assessments of the present scheme compared with the previous ones. Concluding remarks are summarized in Sec. 5.


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2. THEORY This section reviews the theoretical aspects of the conventional Bond-EDA methods at the ab initio HF/DFT level of theory and provides its modification. The total energy E can be written as

E  TS  ENN  ENe  Eee  EHFx  EDFTxc ,

(4)

where TS, ENN, ENe, Eee, EHFx, and EDFTxc represent the kinetic, nucleus–nucleus repulsion, nucleus– electron attraction, electron–electron repulsion, HF exchange, DFT exchange-correlation energies, respectively. In the Bond-EDA, the total energy is partitioned into the intra- and interatomic components as given in Eq. (2). Here, the explicit formulas of the intra- and interatomic components are given by AA AA AA AB AB E AA  TSAA  ENe  EeeAA  EHFx  EDFTxc    AB TSAB  EHFx  EDFTxc ,

(5)

AB AB AB AB  EeeAB  ENN  1   AB TSAB  EHFx  EDFTxc E AB  ENe .

(6)

A B

and

The minus value of EAB corresponds to the BE, as mentioned in the introduction section. Here, αAB is the parameter to correct the energy decomposition between the intra- and interatomic contributions. The explicit formulas of the individual energy components are defined as follows:

 1  TSAA    D    dr * (r ) 2  (r )  ,  2   A   A

(7)

 Z A  AA , ENe   dr   A (r ) A   r  R  

(8)



AA ee

E

  A  r   A  r   1   drdr   ,  r  r 2  

AA  EHFx

 A   A

AB ENN 

(9)

1   D     D , 4   A   A 

AA EDFTxc    D

  dr

xc

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(10)



(r )  * (r )  (r ) ,

(11)

Z AZ B , R A  RB

(12)

 1  TSAB  2   D    dr * (r ) 2  (r )  ,  2  A  B

(13)

  Z A  Z B  AB A   , ENe d ( )    dr   B (r )  r r   A  B   r  R r  R    

(14)

AB ee

E

  A  r   B  r     drdr   ,  r  r  

AB  EHFx

(15)

1   D     D , 2 A  B 

(16)

and AB EDFTxc  2   D

A  B

  dr

xc



(r )  * (r )  (r ) ,

(17)

with

 A  r     D    r    r  .

(18)

 A 

Here, ZA is the nuclear charge, RA is the nuclear coordinate, r is the electronic coordinate, {} are the atomic orbitals (AOs), D is the one-electron density matrix on the AO basis, and xc(r) is the exchange-correlation energy density per electron. The general formulas of Eqs. (5) and (6) can describe some of the different energy decomposition schemes by changing the value of αAB. When αAB is set to zero, the expressions of Eqs. (5) and (6) -8ACS Paragon Plus Environment

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correspond to Mayer’s scheme (M03).25 This expression implies that the energy is decomposed based on the definition on the AO basis in HF/DFT. The scheme tends to underestimate the interatomic components EAB. Thus, in the Nakai–Kikuchi scheme (NK05),27 αAB is set to 0.5. The Nakai–Ohashi– Imamura–Kikuchi scheme (NOIK11)29 determines αAB by introducing the constraint condition for each bond independently to reasonably estimate the BEs of multiple bonds. Next, we revisit the constraint condition. At the optimized geometry, the following relationship is established:

E tot  0, R AB

(19)

where RAB is the distance between atoms A and B. Using the two-body decomposition formula in Eq. (2),

   E CC    E CD   0 .  AB   R  C C D C  

(20)

When assuming the leading term of the left-hand side in Eq. (20), we obtain

E AB  0. R AB

(21)

Eq. (21) is the constraint condition adopted in NOIK11. Using Eq. (6) and assuming that the parameter αAB is independent of RAB, the following equation is obtained:   AB AB AB  1   AB   EDFTxc E AB  EeeAB  ENN T AB  EHFx    0. AB  Ne AB  R R

(22)

From Eq. (22), the parameter αAB can be uniquely determined. The present study proposes to use Eq. (20) itself as the constraint condition. We introduce the following assumption: -9ACS Paragon Plus Environment


E CC  0. R AB

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(23)

This condition is related to the assumption by Vyboishchikov.28 Finally, we derive a system of linear equations to determine the parameter set {αAB} as follows:

  CD CD CD E CD  EeeCD  ENN    1   CD  AB T CD  EHFx  EDFTxc    0. AB  Ne R D   C  R C D C 

 C

(24)

Because the number of parameters is the same as the number of equations, namely, the number of interatomic pairs, the parameter set {αAB} can be obtained. Although Eq. (24) is established only at the optimized geometry, more general formulas are given using the energy gradients,   E tot CD CD CD CD CD CD CD .         E E E 1 T E E  R AB    AB  Ne ee NN    HFx DFTxc  R AB C D   C  R C D C 

(25)

Let us consider the behaviors of Eqs. (24) and (25). The interactions between the interatomic pairs involve strong chemical bonds, weak through-space/bond interactions, and even non-interactions. For such weak and/or non-interactions, the energy components and their gradients might exhibit negligible values. This situation may cause overfitting in solving the system of linear equations. In the present study, we adopt the least absolute shrinkage and selection operator (LASSO) technique instead of the direct solution in the system of linear equations because the LASSO can neglect coefficients with small contributions.

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3. COMPUTATIONAL DETAILS This subsection summarizes the computational methods for the energy decomposition schemes. All quantum chemical calculations were performed at the B3LYP/6-311G** level using the modified version of the GAMESS program45 that include the calculations of geometry optimizations, total energies, and all energy components. The present scheme utilizes Eqs. (5) and (6) for the energy decomposition formalisms. The system of linear equations of Eq. (24) were solved by two methods, namely, the direct evaluation in the linear equation (LE) and LASSO. For comparison, three conventional energy decomposition schemes, i.e., M03, NK05, and NOIK11, were examined. The gradients of the total energy and energy components appearing in the NOIK11, LE, and LASSO formulas were evaluated numerically by changing the distance around the equilibrium bond distance by 0.0001 Å. A regularization term as a hyperparameter in the LASSO was set to 0.005, and was manually determined to minimize the statistical deviations in the test set described in Sec. 4.3.

4. RESULTS AND DISCUSSION 4.1. BEs for Typical Hydrocarbons This subsection focuses on the accuracies of the present energy decomposition scheme to evaluate the BEs for typical hydrocarbons including C2H6, C2H4, C2H2, and C6H6. These molecules exhibit different types of C−C bonds, namely, single, double, triple, and aromatic bonds, respectively. Table 1 shows the results of the LASSO as well as those for M03, NK05, NOIK11, and LE. The structures - 11 ACS Paragon Plus Environment


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and atomic labeling in Table 1 are illustrated in Fig. 1. We next discuss the BEs estimated for the C−C and C−H covalent bonds. Except for M03, the C−C BEs are in the order of C2H6, C6H6, C2H4, and C2H2. This trend is reasonable if we consider the C−C bond orders in these molecules. The M03 scheme typically underestimates the C−C BEs. The degree of underestimation increases with the bond order. In fact, the C−C BE for the triple bond in C2H2 is estimated to be 70.9 kcal/mol, i.e., smaller than that for the single bond in C2H6 of 89.5 kcal/mol. The NK05 scheme also underestimates the BEs for the double bond in C2H4 and the triple bond in C2H2. The trends above in M03 and NK05 are due to an unfavorable behavior of the estimated BEs for the C−C bond elongation, which was well analyzed and discussed in Ref. 29. The NOIK11, LE, and LASSO schemes present comparatively similar values of the C−C BEs. Concerning the C−H BEs, a typical underestimation appears in M03 in comparison with the other schemes. NK05 tends to slightly underestimate the C−H BEs. NOIK11, LE, and LASSO exhibit similar behavior in estimating the C−H BEs, as in the case of the C−C BEs. We next compare the weak through-space/bond interactions. M03 and NK05 present many negative BEs, which correspond to repulsive interactions. Some of them exhibit absolute values larger than 10 kcal/mol that appear unrealistic for the weak interaction. NOIK11 and LASSO present more reasonable values, except for CL…HR1 in C2H4 by NOIK11. Although NOIK11 and LASSO tend to estimate the negative and positive BEs, respectively, it is difficult to understand which is more reliable from the viewpoint of BE. This will be discussed later. It is noteworthy that LE, which adopts the same - 12 ACS Paragon Plus Environment

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equation, i.e., Eq. (24), as the LASSO, results in an unphysical behavior owing to the overfitting problem. For example, the BEs of HL1…HL2 in C2H4 and HL…HR in C2H2 are estimated to be −57.2 and −49.0 kcal/mol, respectively. Therefore, the LASSO technique is essential to reasonably reproduce BEs by solving the linear equations of Eq. (24) or (25). The parameter set {αAB} in Eqs. (5) and (6) are summarized in Table 2. The parameters of M03 and NK05 are typically constant, i.e., 0.0 and 0.5, respectively. For covalent bonds, NOIK11 presents parameters that are comparatively close to those of the LASSO. For example, the parameters for {CL−CR, CL−HL1} in C2H6 are {0.70, 0.83} for NOIK11 and {0.73, 0.77} for LASSO. This indicates that the assumption in Eq. (21) applies well for covalent bonds. Furthermore, the parameters increase when the bond orders increase except for C6H6: the parameters for C−C in the LASSO are 0.73 for C2H6, 0.81 for C2H4, 0.95 for C2H2, and 0.90 for C6H6. This implies that the large amount of kinetic, HF exchange, and DFT exchange-correlation energy components are reassigned from interatomic to intra-atomic terms, especially for multiple bonds, through the parameters in Eq. (5). For the weak through-space/bond interactions, NOIK11 presents values larger than 1.0, which are unphysical. This indicates that the assumption in Eq. (21) is not adequate for such interactions. The situation becomes worse in LE owing to the overfitting problem. For example, negative value of −2.27 appears for HL1…HL2 in C2H4, of which the BE is estimated to be −57.2 kcal/mol, as shown in Table 1. The largest positive value is 40.83 for HL…HR in C2H2, of which the BE is −49.0 kcal/mol. Such overfitting problems can be solved by adopting the LASSO technique. In fact, all parameters obtained - 13 ACS Paragon Plus Environment


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by the LASSO are within the range of 0.00 and 1.00, which should satisfy. Especially, the parameters for the weak through-space/bond interactions become 1.00 in the LASSO. This indicates that the LASSO technique can automatically judge the covalent bonds or not by neglecting coefficients with small contributions. Actually, the solution of the system of linear equations only for the covalent bonds provides the similar results to the LASSO as shown in Supporting Information (SI). Then, the longrange Coulomb interactions for nucleus-nucleus, nucleus-electron, and electron-electron are dominant in the weak through-space/bond interactions, which are independent of the parameters. Therefore, the LASSO technique can be applicable to systems with various long-range interactions such as large molecules.

4.2. BDE for Typical Hydrocarbons This subsection investigates the BDEs estimated from BEs by the energy decomposition schemes such as M03, NK05, NOIK11, and LASSO. The relationship between BDE and BE is given in Eq. (3). Table 3 shows the BEs and BDEs of C−C and C−H covalent bonds in typical hydrocarbons C2H6, C2H4, and C2H2. For comparison, the experimental and quantum chemical BDEs, as defined in Eq. (1), are listed in the table. As mentioned in the introduction section, the experimental BDEs involves various effects such as structural relaxation and vibrational mode change in dissociating a molecule into fragments. Meanwhile, the BDEs estimated from the BEs do not include such effects. Thus, we estimated two types of quantum chemical estimations of BDEs, namely, with and without the structural - 14 ACS Paragon Plus Environment

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relaxation. The former might be closer to the experimental BDEs and the latter to the energy decomposition results. The differences in BDEs between the energy decomposition results and the quantum chemical ones without structural relaxation are also tabulated in Table 3. As a general trend shown in M03, NK05, and LASSO, the BDE decreases from the corresponding BE. This implies that the other through-space/bond interactions, i.e., the second and third terms on the right-hand side in Eq. (3), provide negative values in total, namely, work as repulsive interactions. On the contrary, a trend appears in NOIK11 where the BDE increases from the corresponding BE owing to the opposite reason to the M03, NK05, and LASSO cases. It is noteworthy that the changes from the BEs to BDEs are the smallest in the present LASSO case. This indicates that the throughspace/bond interactions are reasonably estimated to be small in the LASSO scheme. We next discuss the quantum chemical and experimental BDEs. It is obvious that the difference in BDEs between the quantum chemical calculations at the fixed and relaxed structures shows the structural relaxation effect in the dissociation process. The difference between the quantum chemical BDEs at fixed structures and the experimental BDEs originated from the effects associated with the dissociation except for the structural relaxation and from the accuracy of the adopted computational level itself. We herein compare the BDEs estimated from the BEs by the energy decomposition schemes with the quantum chemical BDEs in the fixed structure. The deviations are shown in parentheses in Table 3. In the cases of M03 and NK05, the deviations are large negative values, especially for the C−C - 15 ACS Paragon Plus Environment


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double bond in C2H4 and the C−C triple bond in C2H2. The primary reason is the underestimation of the BEs for such bonds, which are the leading term in estimating the BDEs in Eq. (3), as discussed in the previous subsection. Furthermore, the unreliable estimation of the through-space/bond interactions, which correspond to the second and third terms in Eq. (3), contributes to the large deviations. The NOIK11 scheme might improve completely, primarily owing to the reasonable estimation of BEs. However, the unphysical estimation of the through-space/bond interactions results in the overestimation of the BDEs. In particular, the BDE for the C−C double bond in C2H4 is overestimated by 83.6 kcal/mol, while the difference between the estimated BE and quantum chemical BDE is 8.5 kcal/mol. In the present LASSO scheme, the deviations are typically small, that is, the maximum deviation is 19.8 kcal/mol for the BDE of the C−C triple bond in C2H2. Consequently, the reasonable description of BEs for both the strong covalent bonds and weak through-space/bond interactions is essential for the accurate estimation of the BDEs from the energy decomposition scheme.

4.3. Statistics of BDE Estimation This subsection presents the statistical analysis of the estimation of BDEs from the BEs by the energy decomposition schemes for 44 chemical bonds in 33 covalent and ionic molecules involving the second- and third-row atoms, which are summarized in Table 4. Figure 2 shows the correlation between the BDEs obtained by the quantum chemical calculations without structural relaxation and those by the energy decomposition schemes, i.e., (a) M03, (b) NK05, (c) NOIK11, and (d) LASSO. - 16 ACS Paragon Plus Environment

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The gray circle and purple square correspond to the results of covalent and ionic bonds, respectively. The statistical results including the mean absolute deviation (MAD), maximum deviation (MaxD), and R2 with zero for intercept are summarized in Table 5. The numerical data corresponding to Figure 2 are summarized in SI. Figure 2(a) and the statistical data in Table 5 indicate no correlation between the BDEs estimated by M03 and those by the quantum chemical calculations. Particularly, the large values of BDEs cannot be reproduced by the M03 scheme, as discussed in the previous subsections 4.1 and 4.2. The statistical behaviors of NK05 and NOIK11 appear to be similar; for example, the R2 values are 0.62 and 0.66, respectively. The NOIK11 scheme shown in Figure 2(c) tends to overestimate the BDEs compared with the quantum chemical values, although the NK05 scheme in Figure 2(b) slightly underestimates them. The present LASSO scheme in Figure 2(d) demonstrates a remarkably good performance in evaluating the BDEs. The MAD of the LASSO results is less than half of those of the NK05 and NOIK11. The ratios of MaxD are small. The R2 value of LASSO becomes 0.91, thus indicating that the present LASSO scheme can well reproduce the quantum chemical BDEs. The largest deviations are shown in C−C for C2F2 as shown in SI. The results for pairs associated with many atoms with high electronegativity such as oxygen and fluorine can require the careful handling. The statistical analysis confirms the high accuracy of the energy decomposition scheme using the LASSO technique developed in this study.



Page 18 of 42

5. CONCLUSION So far, several researchers have proposed energy decomposition schemes with different formulae and performance, which partition the total energy obtained by a quantum chemical calculation into one- and two-body contributions, i.e. atomic and bond energies. This is because such partitioning involves arbitrariness, just like the partitioning of the total charge into atomic and/or bond charges in the electron density analysis. The purpose of this study was the improvement of energy decomposition scheme to adequately reproducing the BDEs obtained by the standard quantum chemical calculation. This study extended the Bond-EDA scheme, one of the two-body energy decomposition schemes, by revisiting the constraint condition and adopting the informatics technique. The present scheme evaluated the BEs for all interatomic pairs involving both strong chemical bonds and weak throughspace/bond interactions. The adjustable parameters were determined by solving the linear equations that applied the gradients of energy components obtained by quantum chemical calculations such as HF and DFT. In particular, to avoid the overfitting problem originating from weak and/or negligible interactions, we adopted the LASSO technique. The BDEs, which could be evaluated from the BEs by the energy decomposition schemes, could be compared with the experimental and theoretical BDEs. The numerical results confirmed that the present LASSO scheme could accurately reproduce the theoretical BDEs, for which the reliable description of BEs for both strong chemical bonds and weak through-space/bond interactions was found to be essential. Furthermore, the energy decomposition scheme could evaluate the BDEs simultaneously without dissociating the corresponding chemical - 18 ACS Paragon Plus Environment

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bonds, namely, although the standard quantum chemical estimation of the BDEs required the calculations of all fragments. Therefore, the scheme proposed here might be a promising analysis tool for both small molecules and complicated systems.



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ASSOCIATED CONTENT Supporting Information. The Supporting Information is available free of charge on the ACS Publications website at DOI: *******. The lists of the BEs and parameters for hydrocarbons, which were evaluated by solving the system of linear equations only for covalent bonds, the lists of the BDEs, BEs, and parameters for 44 chemical bonds in 33 molecules, and the list of BDEs for O22+ and F22+ are included.

ACKNOWLEDGMENTS This paper is dedicated to Professor Leo Radom on the occasion of his 75th birthday. Some of the present calculations were performed at the Research Center for Computational Science (RCCS), Okazaki Research Facilities, National Institutes of Natural Sciences (NINS). Author J.S. is grateful for the PRESTO program, “Advanced Materials Informatics through Comprehensive Integration among Theoretical, Experimental, Computational, and Data-Centric Sciences,” sponsored by the Japan Science and Technology Agency (JST).


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FIGURE CAPTIONS

Figure 1. Structures and atomic labelings of (a) C2H6, (b) C2H4, and (c) C2H2.

Figure. 2. Correlation between the BDES for 44 chemical bonds in 33 covalent and ionic molecules obtained by the quantum chemical calculations without structural relaxation and those by the energy decomposition schemes, i.e., (a) M03, (b) NK05, (c) NOIK11, and (d) LASSO. The gray circle and purple square correspond to results of covalent and ionic bonds, respectively.



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Table 1. BEs (kcal/mol) in C2H6, C2H4, C2H2, and C6H6 estimated by M03, NK05, NOIK11, LE, and LASSO. Molecule

Bond*1

C2H6

CL−CR

89.5

108.3

104.5

125.2

117.1

CL−HL1

97.2

115.4

121.0

124.5

125.1

CL…HR2

−12.0

−6.7

6.0

10.5

−1.4

HL1…HL2

−15.4

−10.4

3.3

−13.7

−5.4

HL1…HR1

4.2

1.8

1.5

−17.3

−0.7

HL1…HR2

−4.6

−3.2

1.8

−5.1

−1.8

CL−CR

113.5

169.8

192.6

220.7

204.1

CL−HL1

100.3

117.2

124.0

127.1

126.1

CL…HR1

−15.6

−9.5

18.8

0.8

−3.4

HL1…HL2

−22.2

−14.5

9.6

−57.2

−6.8

HL1…HR1

4.2

1.8

−0.3

−9.5

−0.5

HL1…HR2

−7.9

−5.2

1.1

−5.3

−2.5

CL−CR

70.9

170.3

261.0

269.5

260.1

CL−HL

117.9

128.7

133.9

135.1

134.0

CL…HR

−24.4

−13.8

3.3

−7.6

−3.2

HL…HR

−0.2

−0.8

−2.2

−49.0

−1.4

C−C

77.2

125.2

147.6

189.7

164.7

C−H

108.0

122.6

126.7

128.7

129.5

C2H4

C2H2

C6H6

M03

NK05

NOIK11

LE

LASSO

*1 A−B and A…B mean covalent bond and through-space/bond interaction, respectively.


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Table 2. Parameter set {αAB} in Eqs. (5) and (6) for M03, NK05, NOIK11, LE, and LASSO. Molecule

Bond*1

C2H6

CL−CR

0.00

0.50

0.70

0.95

0.73

CL−HL1

0.00

0.50

0.83

0.75

0.77

CL…HR1

0.00

0.50

1.35

2.11

1.00

HL1…HL2

0.00

0.50

1.43

0.18

1.00

HL1…HR1

0.00

0.50

0.77

4.43

1.00

HL1…HR2

0.00

0.50

1.64

−0.17

1.00

CL−CR

0.00

0.50

0.85

0.95

0.81

CL−HL1

0.00

0.50

0.85

0.79

0.76

CL…HR1

0.00

0.50

1.90

1.34

1.00

HL1…HL2

0.00

0.50

1.53

−2.27

1.00

HL1…HR1

0.00

0.50

0.98

2.88

1.00

HL1…HR2

0.00

0.50

1.34

0.48

1.00

CL−CR

0.00

0.50

0.98

1.00

0.95

CL−HL

0.00

0.50

0.87

0.80

0.75

CL…HR

0.00

0.50

1.15

0.79

1.00

HL…HR

0.00

0.50

1.34

40.83

1.00

C−C

0.00

0.50

0.87

1.17

0.90

C−H

0.00

0.50

0.82

0.71

0.72

C2H4

C2H2

C6H6

M03

NK05

NOIK11

LE

LASSO

*1 A−B and A…B mean covalent bond and through-space/bond interaction, respectively.


The Journal of Physical Chemistry 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46

Page 30 of 42

Table 3. BDEs (kcal/mol) of C−H and C−C covalent bonds in typical hydrocarbons, C2H6, C2H4, and C2H2. The BDEs are estimated from BEs by the energy decomposition schemes of M03, NK05, NOIK11, and LASSO. Experimental and quantum chemical BDEs, as defined in Eq. (1), are also listed. Molecule Bond C2H6 C2H4 C2H2

Type

M03 BE

NK05

BDE*1

NOIK11

BDE*1

BE

BE

QC*2

LASSO

BDE*1

BE

BDE*1

Fix

Relax

Exptl.

C−C

Single

89.5

17.8

(−92.9)

108.3

68.4 (−42.3)

104.5

140.6

(29.9)

117.1

109.1

(1.6)

110.7

102.9

87.7

C−H

Single

97.2

61.4

(−52.2)

115.4

90.0 (−23.6)

121.0

132.6

(19.0)

125.1

108.8

(−4.8)

113.6

106.7

99.4

C−C

Double

113.5

51.0 (−133.1)

169.8

131.8 (−52.3)

192.6

267.7

(83.6)

204.1

190.8

(6.7)

184.1

178.0

172.1

C−H

Single

100.3

58.8

(−56.9)

117.2

89.9 (−25.8)

124.0

153.3

(37.6)

126.1

113.0

(−2.7)

115.7

115.7

99.9

C−C

Triple

70.9

22.1 (−212.0)

170.3

142.8 (−91.4)

261.0

267.6

(33.5)

260.1

253.9

(19.8)

234.1

231.7

229.0

C−H

Single

117.9

93.3

128.7

114.1 (−25.5)

133.9

135.0

(−4.6)

134.0

129.4 (−10.2)

139.6

139.6

103.9

(−46.3)

*1 Deviations from the quantum chemical BDEs at the fixed structure are shown in parentheses. *2 Quantum chemical evaluation of BDEs at fixed and relaxed structures for fragments are tabulated.


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Table 4. 44 Chemical bonds in 33 molecules used in the article for numerical assessments. Entry

Molecule

Bond

Entry

Molecule

Bond

#1

CH

C−H

#23

HCO

C−O

#2

CH2

C−H

#24

CH3OH

C−H

#3

CH3

C−H

#25

C−O

#4

CH4

C−H

#26

O−H

#5

C2H2

C−C

#27

HF

H−F

C−H

#28

SiH2

Si−H

C−C

#29

SiH3

Si−H

C−H

#30

SiH4

Si−H

C−C

#31

PH2

P−H

C−H

#32

PH3

P−H

C−C

#33

H2S

S−H

C−F

#34

CH3SH

C−H

#6 #7

C2H4

#8 #9

C2H6

#10 #11

C2F4

#12 #13

NH

N−H

#35

C−S

#14

NH2

N−H

#36

S−H

#15

NH3

N−H

#37

HCl

H−Cl

#16

HCN

H−C

#38

CO2

C−O

C−N

#39

SO2

S−O

#17 #18

OH

O−H

#40

LiF

Li−F

#19

H2O

O−H

#41

LiCl

Li−Cl

#20

H2O2

O−O

#42

NaCl

Na−Cl

O−H

#43

MgCl2

Mg−Cl

H−C

#44

CaCl2

Ca−Cl

#21 #22

HCO



Page 32 of 42

Table 5. Statistical MAD, MaxD (kcal/mol), and R2 in 44 chemical bonds obtained by M03, NK05, NOIK11, and LASSO from the quantum chemical evaluation of BDEs.. Method

M03

NK05

NOIK11

LASSO

MAD MaxD

61.7 212.0

19.7 91.4

85.6 2269.8

10.9 47.8

R2

−0.14

0.76

−0.03

0.89


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Figure 1. Structures and atomic labelings of (a) C2H6, (b) C2H4, and (c) C2H2.



Page 34 of 42

Figure. 2. Correlation between the BDES for 44 chemical bonds in 33 covalent and ionic molecules obtained by the quantum chemical calculations without structural relaxation and those by the energy decomposition schemes, i.e., (a) M03, (b) NK05, (c) NOIK11, and (d) LASSO. The gray circle and purple square correspond to results of covalent and ionic bonds, respectively.


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Bond Energy Density Analysis Combined with ... - ACS Publications

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