Toward Understanding the Isomeric Stability of

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Article Cite This: ACS Omega 2018, 3, 17986−17990

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Toward Understanding the Isomeric Stability of Fullerenes with Density Functional Theory and the Information-Theoretic Approach Dongbo Zhao,†,⊥ Siyuan Liu,‡,⊥ Chunying Rong,‡ Aiguo Zhong,§ and Shubin Liu*,∥ †

School of Chemistry and Chemical Engineering, Nanjing University, Nanjing 210023, People’s Republic of China College of Chemistry and Chemical Engineering, Hunan Normal University, Changsha 410081, People’s Republic of China § School of Pharmaceutical and Chemical Engineering, Taizhou University, Linhai 318000, Zhejiang, People’s Republic of China ∥ Research Computing Center, University of North Carolina, Chapel Hill, North Carolina 27599-3420, United States Downloaded via 181.214.179.20 on December 22, 2018 at 01:04:25 (UTC). See https://pubs.acs.org/sharingguidelines for options on how to legitimately share published articles.

‡

S Supporting Information *

ABSTRACT: For a given size of one fullerene molecule, there could exist many different isomers and their energy landscape is remarkably complex. To have a better understanding of the nature and origin of their isomeric stability, as a continuation of our previous endeavors, we systematically dissect the molecular stability of four fullerene systems, C44, C48, C52, and C60, with a total of 2547 structures, using density functional theory and the informationtheoretic approach. The total energy decomposition analysis is beneficial to understand the origin and nature of isomeric stability. Our results showcase that the electrostatic potential is the dominant factor contributing to the isomeric stability of these fullerenes, and other contributions such as steric and quantum effects play minor but indispensable roles. This study also finds that the origin of the isomeric stability of these species is due to the spatial delocalization of the electron density. Our work should provide novel insights into the isomeric stability of fullerene molecules, which have found tremendous applications in solar-energy studies and nanomaterial sciences.

1. INTRODUCTION Since their first theoretical prediction in 1970,1 and then experimental verification in 1985,2 fullerenes have served as the starting point of numerous studies for a wide variety of other carbon structures,3,4 among which are carbon nanotubes,5 graphene,6,7 carbon nanodots,8 schwarzites,9 and carbon nanocombs.10 Nowadays, carbon nanomaterials are promising for many applications, such as electrical energy storage, photoconversion, biological imaging, gas storage and separation, or nanomedical therapy.10−14 Yet, what dictates the origin of the isomeric stability of fullerenes is still an unresolved and ongoing research task. For example, why does the most stable isomer of C60 possess the Ih symmetry? Previously, both steric hindrance and hyperconjugation effects have been employed to study the origin and nature of isomeric stability. However, even for ethane, no consensus is available on which of these effects is more significant.15−21 Recently, we tackled the issue in a different manner in terms of physiochemical insights such as steric effect.22 In addition, several external factors might influence the molecular stability of the different isomers of fullerenes. For instance, some nonstable isomers are stabilized when some molecular fragments are included in the cavity.23 In this work, we employ two different energy decomposition schemes from density functional reactivity theory (DFRT),22 © 2018 American Chemical Society

to distinguish the core factors dominating the relative isomeric stability of fullerenes (C44, C48, C52, and C60) as shown in Figure 1. Our results reveal that the electrostatic potential plays a key role whereas the exchange−correlation contribution and steric hindrance play minor roles in determining the relative isomeric stability. In addition, some commonly used densitybased quantities from the information-theoretic approach (ITA), such as Shannon entropy, Fisher information, Onicescu information energy, and Ghosh−Berkowitz−Parr (GBP) entropy, are also employed to gain insights into their relative stability. Our results show that there exists a strong linear relationship between the total energy difference and the Shannon entropy difference among isomers. The remaining part of this article is ordered in the following manner. In Section 2, the main results and discussion are presented, followed by the Concluding Remarks to summarize the main findings in Section 3. In Section 4, the two energy decomposition schemes are outlined, followed by the introduction of a few well-known information-theoretic quantities. Computational details are provided Section 5. Received: October 8, 2018 Accepted: December 10, 2018 Published: December 21, 2018 17986

DOI: 10.1021/acsomega.8b02702 ACS Omega 2018, 3, 17986−17990

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Figure 3. Strong linear relationships between the total energy difference and its components for C44, C48, C52, and C60 under study.

between the total energy difference ΔE and its two-component fitting results ΔE = 0.572ΔEe + 0.130ΔExc + 0.003

(1)

and ΔE = 0.627ΔEe − 0.006ΔEs + 0.004

(2)

It is unambiguous that from eqs 1 and 2 that electrostatic potential is the predominant energy component as showcased by the coefficients. In conventional Kohn−Sham density functional theory (KS-DFT) scheme, both Ee and Exc make positive contributions to the total energy difference in a synchronous way as shown in eq 1. However, in eq 2, compared with that from Ee, the contribution from the steric effect, Es, makes negative contributions to the relative stability of fullerene buckyballs. Taken together, we have verified that electrostatic potential plays a key role in determining the relative isomeric stability of fullerene buckyballs. In the meantime, exchange−correlation potential plays a minor role. Moreover, the impact resulted from the steric effect is minimal. 2.2. Total Energy and Information Quantities. Figure 4 shows the strong linear relationship between the total energy

Figure 1. Schematic representations of fullerene buckyballs: (a) C44, (b) C48, (c) C52, and (d) C60.

2. RESULTS AND DISCUSSION 2.1. Total Energy and Its Components. One basic question we aim to answer in this work is whether there is an energetic component that is responsible for the isomeric stability of fullerene buckyballs. Figure 2 shows strong linear

Figure 2. Strong linear relationships between the total energy difference (ΔE) and its components: (a) ΔE = 0.626ΔEe + 0.004; (b) ΔE = 0.626ΔExc − 0.004.

correlations between the total energy difference ΔE and its components: electrostatic interaction ΔEe (Figure 2a) and exchange−correlation ΔExc (Figure 2b). The corresponding correlation coefficients R2 are larger than 0.99, indicating that electrostatic and exchange−correlation interactions both positively contribute to the relative stability of fullerene conformers. In addition, a strong linear relationship is observed between ΔExc and ΔEe (results not shown). Its implication is straightforward that electrostatic effects may be made use of in constructing better exchange−correlation energy functionals. This idea was first proposed by Fermi and Amaldi in 1934.24 Furthermore, we aim to answer the following question: Are the two energy components equivalently important? To further pin down this point, we employ a binary fit strategy. Figure 3 displays the strong linear regression equations

Figure 4. Strong linear correlation between the total energy difference (ΔE) and the Shannon entropy difference (ΔSS).

difference ΔE and the Shannon entropy difference ΔSS with the correlation coefficient R2 equal to 0.983. ITA quantities have their own physical meaning and gauge different aspects of the electronic properties as shown in eqs 8−15. Although there is no significantly strong correlation between ΔE and many of the ITA quantities (as shown in Table S1), shown in Figure 4, the strong linear correlation between the total energy difference (ΔE) and the Shannon entropy ΔSS suggests that spatial delocalization of the electron density gauged by the Shannon entropy is the origin of the isomeric stability of these fullerene molecules. This result provides us a novel insight into this complicated phenomenon of isomeric stability from the 17987


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tackle the issue of how to pin down the nature and origin of the isomeric stability of fullerene molecules. Through the calculations and analyses of the four fullerene systems, C44, C48, C52, and C60 with a total of 2547 isomeric structures, we found that the electrostatic interaction is the single largest contributor to the isomeric stability of these fullerene species. This result is consistent with our earlier results from other systems. Also, other energetic components such as steric effect and exchange−correlation interaction play a minor but indispensable role in determining the isomeric stability of these compounds. Finally, the strong linear correlation between the total energy difference and the Shannon entropy difference indicates that the isomeric stability is dictated by the delocalization of the electron density in these fullerene systems. We mention in passing that the question whether a maximum entropy principle can complement the usual minimum energy principle of stability deserves our further attention. It is still an open question, and its universality should be amenable to more systematic studies in future. This is what we do, and its results will be presented elsewhere.

perspective of information theory. As a piece of evidence to support this point, Figure S1 shows the reasonable correlation between the total energy difference (ΔE) and the Rydberg non-Lewis contribution from natural bond orbital (NBO) analysis at the M06-2X/Def2-TZVP level of theory for all of the isomers of C44, C48, and C52 fullerenes. The Rydberg nonLewis contribution from NBO analysis is a measure of the deviation from the traditional Lewis structure and thus could be used as a gauge of the electron delocalization. 2.3. Correlations among ITA and Energetic Quantities. It is well known that ITA and energetic quantities can be strongly intercorrelated.25 These different but strongly correlated relationships provide effective measurements about the electron density distribution of the system; thus, they attribute useful and novel insights into the nature and origin of various physicochemical phenomena including the isomeric stability of different fullerene buckyballs. As an illustrative example, Figure 5 exhibits two strong linear correlations of the

4. METHODOLOGY 4.1. Energy Decomposition Schemes in DFT. In Kohn−Sham DFT,21 the total energy of a given molecular system is formulated as E[ρ] = Ts[ρ] + Ee[ρ] + Exc[ρ]

where Ts, Ee, and Exc are the noninteracting kinetic, electrostatic, and exchange−correlation energies, respectively. The electrostatic energy Ee includes three independent components: the nuclear−electron attraction, Vne; the classical interelectron Coulombic repulsion, J; and the nuclear−nuclear repulsion, Vnn. The last term Exc consists of exchange (Ex) and correlation (Ec) components. Recently, to evaluate the steric effect within DFT, one of us has proposed a novel total energy decomposition scheme

Figure 5. Strong linear correlations between information-theoretic quantities for C44, C48, C52, and C60 studied in this work.

relative Rényi entropy of order 2 with two other ITA quantities for all of the conformers of C44, C48, C52, and C60 studied in this work. As can be seen from the figure, Rr2 is positively correlated with Rr3 and IG, each with R2 > 0.95. On the other hand, Table 1 shows the correlation coefficient (R) between

E[ρ] = Es[ρ] + Ee[ρ] + Eq [ρ]

ΔE

ΔTs

1 0.371 0.998 0.993 0.497 −0.075

1 0.381 0.433 0.663 0.661

ΔEe

1 0.994 0.500 −0.067

ΔExc

ΔEs

ΔEq

Es = TW[ρ] = 1 0.556 −0.028

1 0.115

(4)

where Ee is the same electrostatic energy in eq 3, Es stands for the steric effect, and Eq signifies the contribution originating from Fermionic quantum effect (due to the exchange− correlation interaction). In eq 4, the energy component associated with the steric effect Es has been proved to be simply the Weizsäcker kinetic energy

Table 1. Correlation Coefficients R between the Total Energy Difference and Its Componentsa ΔE ΔTs ΔEe ΔExc ΔEs ΔEq

(3)

1 8

2

∫

|∇ρ(r)| dr ρ(r)

(5)

where ρ(r) and ∇ρ(r) are the electron density and its gradient, respectively. After combining eqs 3 and 4, one has the definition of Eq

1

a

Correlation coefficients among the energy components themselves are also shown.

Eq = Exc[ρ] + Ts[ρ] − Es[ρ]

(6)

This new formulation has its own distinct physical meaning with a corresponding physical state.22 It has been applied to a number of molecular systems and phenomena,22,26−34 whose results are consistent with our chemical intuition and conventional wisdom. 4.2. Information-Theoretic Quantities in DFT. Shannon entropy SS is a measure of the spatial delocalization of the electron density, and Fisher information IF measures the sharpness or localization of the same. They are defined as eqs 7 and 8, respectively

the total energy difference and five components. Besides the strong correlation of ΔE with ΔEe and ΔExc, we also found that there exists a strong linear correlation between ΔEe and ΔExc, which is unique in these systems.

3. CONCLUDING REMARKS In summary, using two different total energy partition schemes from density functional theory together with the quantities from the information-theoretic approach, in this work, we 17988


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∫ ρ(r) ln ρ(r) dr

frequency calculations. All DFT calculations were carried out using the Gaussian 0941 package version D01 with tight selfconsistent field convergence criteria and ultrafine integration grids to rule out numerical instabilities. The molecular wave functions were obtained at the M06-2X/Def2-TZVP42,43 level as well as the energetic components as obtained with the Gaussian keyword IOp(5/33 = 1). All ITA quantities mentioned above were evaluated using the Multiwfn 3.3.9 program,44 with the Gaussian checkpoint file as the input file. The steric energy component was obtained via Fisher information, which differs from the Weizsäcker kinetic energy by a factor of 8. In addition, we have taken the isomer with the lowest energy as a reference for each series of C44, C48, C52, and C60. Accordingly, the total energy difference should always be positive. Put all together, we have a total of 2547 structures for these 4 fullerene systems. When plotting a correlation, we put all these 2547 data points together, indicating that the relationships should be universal to all of these species.

(7)

2

IF =

∫ |∇ρρ((rr))|

dr

(8) 35

Earlier, we have proven that for atoms and molecules, IF has an equivalent expression (eq 9) in terms of the Laplacian of the electron density, ∇2ρ(r) I′F = −

∫ [∇2 ρ(r)] ln ρ(r) dr

(9)

Ghosh−Berkowitz−Parr (GBP) entropy ÄÅ ÉÑ ÅÅ t(r; ρ) ÑÑÑ 3 Å Å ÑÑdr SGBP = kρ(r)ÅÅc + ln ÅÅÇ t TF(r; ρ) ÑÑÑÖ 2

SGBP36

∫

(10)

where t(r; ρ) is the kinetic energy density, which is related to the total kinetic energy Ts via

∫ t(r; ρ)dr = Ts

■

(11)

ASSOCIATED CONTENT

S Supporting Information *

tTF(r; ρ) is the Thomas−Fermi kinetic energy density given by (12)

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acsomega.8b02702.

with K as the Boltzmann constant (set to be unity for convenience in this work), c = (5/3) + ln(4πcK/3), and cK = (3/10)(3π2)2/3, the specific form of the local kinetic energy

Optimized coordinates of C44 (89), C48 (199), and C52 (437) in xyz format; correlations between DFRT quantities and total energy difference (PDF)

t TF(r; ρ) = cK ρ

t(r; ρ) = ∑i

5/3

1 ∇ ρi ·∇ ρi 8 ρi

(r)

■

1

− 8 ∇2 ρ.

More recently, additional ITA quantities have been introduced as new reactivity descriptors in DFRT. One example is Onicescu information energy of order n En =

1 n−1

∫ ρn (r)dr

relative Rényi entropy of order n ÅÄÅ ÑÉ ÅÅ ρn (r) ÑÑÑÑ 1 r Å Rn = lnÅ drÑ n − 1 ÅÅÅÅ ρ0n − 1(r) ÑÑÑÑ Ç Ö

∫

Corresponding Author

*E-mail: [email protected]. ORCID

Shubin Liu: 0000-0001-9331-0427

(13)

Author Contributions ⊥

∫ ρ(r) ln ρρ((rr)) dr 0

D.Z. and S.L. contributed equally to this work.

Notes

The authors declare no competing financial interest.

■

(14)

ACKNOWLEDGMENTS S.B.L. and C.R. acknowledge support from the National Natural Science Foundation of China (No. 21503076) and Hunan Provincial Natural Science Foundation of China (Grant 2017JJ3201).

and information gain (also called Kullback−Leibler divergence) IG is given in eq 15 IG =

AUTHOR INFORMATION

(15)

■

where ρ0(r) is the reference-state density satisfying the same normalization condition as ρ(r).

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