Homogeneous Molecular Systems are Positively Cooperative, but

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Homogeneous Molecular Systems are Positively Cooperative but Charged Molecular Systems are Negatively Cooperative Chunying Rong, Dongbo Zhao, Tianjing Zhou, Siyuan Liu, Donghai Yu, and Shubin Liu J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.9b00639 • Publication Date (Web): 27 Mar 2019 Downloaded from http://pubs.acs.org on March 28, 2019

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The Journal of Physical Chemistry Letters

Homogeneous Molecular Systems are Positively Cooperative but Charged Molecular Systems are Negatively Cooperative

Chunying Rong,† Dongbo Zhao,‡ Tianjing Zhou,† Siyuan Liu,† Donghai Yu,† and Shubin Liu‖,* †College

of Chemistry and Chemical Engineering, Hunan Normal University, Changsha Hunan 410081, People’s Republic of China

‡School

of Chemistry and Chemical Engineering, Nanjing University, Nanjing 210093, People’s Republic of China

‖Research

Computing Center, University of North Carolina, Chapel Hill, North Carolina 27599-3420, USA

Abstract Molecular systems bounded together through non-covalent interactions are ubiquitous in nature, most of which are involved in life essential processes, yet little is known about the principles governing their structure, stability and function. Cooperativity as one of the intrinsic properties in these systems plays a key role. In this work, based on our recent quantification scheme of the cooperativity effect, we present a general pattern to identify which systems are positively cooperative and which are negatively cooperativity. We show that cooperativity in homogenous molecular systems is positive, but that in charged molecular systems is negative. We also employ analytical tools from energetics and information perspectives to appreciate the origin of the cooperativity effect. We find that positive cooperativity is dominated by the exchange-correlation interaction and steric effect, whereas negative cooperativity is 1 ACS Paragon Plus Environment

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governed by the electrostatic interaction. Our results should have strong implications for better understanding molecular recognition, protein folding, signal transduction, allosteric regulation, and other processes.

TOC Graphic

Molecular systems bound together through non-covalent interactions are of paramount importance in supramolecular chemistry and biological sciences.1–8 All life essential systems including proteins and DNA are outstanding examples of such molecular systems. Depending on the nature of their ingredients, these systems can be homogeneous and heterogeneous (e.g., absorbed on surfaces), or neutral and charged.2,3 The non-covalent interactions involved in these systems are weak van der Waals interactions, hydrogen bonding, Coulombic interactions, etc. It is often true that in these systems, a number, if not all, of these non-covalent interactions are simultaneously coexistent. One intrinsic feature of such molecular systems is cooperativity.4–8 Arising from the competition and compromise among two or more interactions, cooperativity is the effect enabling the components of a system to behave non-additively. Cooperativity can be either positive or negative. A positive cooperativity empowers the molecular system, for example, to increase the affinity upon binding of

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another component, whereas negative cooperativity is otherwise. Cooperativity plays an essential role in phenomena like conformational changes, ligand-binding interactions, enzymatic reactions, and many other biological processes involved molecular recognition, protein folding, signal transduction, and allosteric regulation. From the perspective of the macroscopic picture, cooperativity is closely related to synergetics and emergence of new physio-chemical properties. Though the significance of cooperativity had been widely appreciated and there have also been numerous previous studies in the literature,1,9–18 there are still tremendous controversies about how cooperativity should be quantified, where the effect of cooperativity is originated from, and whether or not there is any reliable approach to predict positive or negative cooperativity.19–21 This Letter is our latest effort to address these questions. Very recently,22,23 for the molecular systems consisting of multiple copies of a building block, we quantified cooperativity, , as the negative of the interaction energy change per building block change,  = – ( En / n), where En is the interaction energy per building block and n is the number of the building blocks within the system. A positive cooperativity index  indicates that the interaction energy per unit becomes larger and thus the interaction becomes more attractive when an extra building block is added into the system. Otherwise, if  is negative, it means that adding an additional building block into a system leads to a smaller En value, and thus the system becomes less stable because the interaction is more repulsive. This system is, therefore, negatively cooperative. A value of  = 0 shows that the system is non cooperative. We applied this quantification approach to a few molecular systems and discovered that these systems can be either positive or negative cooperativity.22,23 For example, water clusters are positively cooperative but protonated water clusters are negatively cooperative. Employing the total energy decomposition schemes and information-theoretic approach,24–26 we demonstrated that that the interactions governing the existence and validity of the cooperativity effect is rather complicated. Different systems may have different origins of cooperativity from the viewpoint of energetics and 3 ACS Paragon Plus Environment


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information. Notice that the total energy partition schemes employed in this work substantially differ from those in the literature,27–34 where partitioning the interaction energy is their main focus. While it is true that there exists no single rule about cooperativity to govern all circumstances, it is possible that for the same category of molecular systems one is able to obtain the same pattern or trend. In this work, we report the finding that for homogenous molecular systems, cooperativity is positive, but for charged molecular systems, cooperativity turns negative. We also present analyses using energy decomposition and information-theoretic approaches, which provide in-depth insights on the nature and origin of the cooperativity effect for these two categories of molecular systems. We found that for the homogenous molecular system, contributions from the exchange-correlation interaction and steric effect play the dominant role in making it positive cooperativity, whereas for the charged molecular systems it is the electrostatic interaction that dictates, turning the system to be negatively cooperative. These findings should shed new light on better understanding about weak interactions in protein folding, enzymatic interactions, allosteric regulation, etc., where interactions involved in homogenous and charged systems are ubiquitous. In this Letter, we consider additional four homogeneous molecular systems and two charged molecular systems. The four homogeneous systems include hydrogen fluoride cluster (HF)n, carbon dioxide cluster (CO2)n, dichlorine embedded in argon cluster Cl2Arn, and ammonia embedded in water cluster NH3(H2O)n, all with n=1-20. The two charged systems are lithium cation in water cluster Li+(H2O)n and fluoride anion in water cluster F– (H2O)n, also with n=1-20. These molecular systems are in addition to eight other homogeneous and charged systems that we have already examined in our previous studies,22,23 such as water cluster (H2O)n, argon cluster Arn, zinc atom cluster Znn, glycine alpha helix (NH2)(Gly)n(COOH), protonated water cluster H3+O (H2O)n, and argon cluster embedded by one lithium cation Li+Arn, and argon cluster with one fluorine anion F–Arn, all with n = 1-20. Figure 1 shows the optimized structure for the six molecular systems studied in this work for n=20. Computational details 4 ACS Paragon Plus Environment

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together with the Cartesian coordinates of all optimized structyres are provided in the Supplementary Information (SI). It can be seen that their structures and 3D shapes are enormously different. This is because the dominant interaction involved in these systems is markedly different. For example, Cl2Arn is dominated by van der Waals interactions, (HF)n dictated by hydrogen bonding, and Li+(H2O)n governed by both hydrogen bonding and Coulombic interactions. The intrinsic cooperativity property of a molecular system can be illustrated by the cooperativity profile, the change of the interaction energy per building block as a function of the number of building blocks. If the profile goes downward as the number of block increases (i.e., increasing the interaction energy per building block with a larger number of building blocks), it is positive cooperative; if the cooperativity profile is upwards, negative cooperation is present; and if horizontally it is a straight line, there exists non cooperativity in the system. Shown in Figure 2 are cooperativity profiles for the six systems investigated in this work. It can be seen that the four homogeneous systems, (HF)n, (CO2)n, Cl2Arn, and NH3(H2O)n with n=1-20, their cooperative profile goes upwards, so these systems are positively cooperative. On the other hand, for the two charged systems, Li+(H2O)n and F–(H2O)n with n=1-20, their profile goes downwards, so they are negatively cooperative. These results are consistent with our previous observations for three other homogeneous molecular systems, (H2O)n, Arn, and Znn, which were found to be positively cooperative, and three other charged molecular systems, H3+O (H2O)n, Li+Arn, and F–Arn, which were showed to be negatively cooperative. In Table 1, we listed the interaction energy change per building block as a function of the number of building blocks for the six clusters studied in this work. The last row tabulates the mean absolute percentage of the interaction energy change from n=2 to n=12. It can be seen from the Table that the effect of cooperativity is strongly size dependent. For the first 10 cluster sizes, the effect could be in the range from 4% to 20%. Put together, our available results point to the conclusion that homogeneous molecular systems are positively cooperative but charged molecular systems are negatively cooperative. 5 ACS Paragon Plus Environment


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The next question to address is where the effect of positive and negative cooperativity originates from. To that end, we resort to the two total energy partition schemes and eight quantities from the information-theoretic approach (ITA) that we recently developed. Details of their formulations and applications are available elsewhere.24,35–52 In a nutshell, the total energy change E can be partitioned in terms of the contributions either from the combination of the kinetic TS, electrostatic Ee and exchangecorrelation energy Exc changes, or from the sum of the steric ES, electrostatic Ee, and fermionic quantum Eq effects. On the other hand, ITA provides numerous simple density functionals to describe different aspects of the electron density distribution. According to the basic theorems of density functional theory (DFT),53 the electron density should have contained all the necessary information to determine everything for molecular systems in the ground state, including structure, bonding, and reactivity. These simple density functionals from ITA have shown to be informative in this regard. Shown in Table 2 are correlation coefficients (R) of the interaction energy change per building block, En, for the six cluster models (with n=1-20) studied in this work with five energetic components TS, Exc, Ee, ES, and Eq, three thermochemistry changes, H, S, and G, and eight ITA quantities, Shannon entropy SS, Fisher information IF, Ghosh-Berkowitz-Parr entropy SGBP, information gain IG, the second-order of Rényi entropy R2, the third-order of Rényi entropy R3, the second-order relative Rényi entropy rR2, and the third-order relative Rényi entropy rR3. From the results in Table 2, following three points can be clearly observed. At first, from the energetics viewpoint, for the systems with positive cooperativity, it is the contribution from the exchange-correlation interaction Exc and steric effect ES that most strongly correlates with the interaction energy per building block En, indicating that these energy components are responsible for the validity of the cooperativity effect in these systems. This result can be understood in this manner. As a homogeneous system increases its size with more building blocks added to it, the system becomes larger and more bulky, and more exchange-correlation interactions are involved among the components, so ES and Exc play the dominant roles. Notice that these two 6 ACS Paragon Plus Environment

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components are from two different total energy partition schemes. For the systems with negative cooperativity, as shown in Table 2, it is the electrostatic component Ee shared by both energy partition schemes that governs the validity of the cooperativity effect. This result makes a lot sense because in charged systems, the electrostatic interaction should be more important than others. Secondly, from the thermochemistry point of view, the enthalpy change H strongly correlates with the interaction energy per building block En, no matter whether the system is positively or negatively cooperative, suggesting that it does not matter to use the interaction energy or enthalpy to quantify cooperativity. Also, same as what we’ve observed previously, for systems with positive cooperativity, the entropy change S is strongly positively correlated with En, but for systems with negative cooperativity, the correlation turns negative. No pattern is seen for the Gibbs free energy change G. Lastly, from the information-theoretic perspective, information gain IG is found to be positively correlated to En for all systems. For positively cooperative systems, Shannon entropy SS, Fisher information IF, and Ghosh-Berkowitz-Parr entropy SGBP are found to be strongly and positively correlated with the interaction energy per building block En, but for systems with negative cooperativity, it is the second-order and third-order of Rényi entropy, R2 and R3, that are found to be strongly and positively correlated with En. Recall that these different ITA quantities have different meanings, and they can be employed to gauge different aspects of the electron density distribution. For example, Shannon entropy is a measure of the electron density delocalization, whereas Fisher information, differing from the steric energy by merely a factor of 1/8, is a gauge of the electron density location. Other entropy quantities are variants of the Shannon entropy. Our results in Table 2 unambiguously exhibited that systems with positive and negative cooperativity possess unique characteristics, with which one is able to appreciate the origin of the cooperativity effect. To address the concern that BSSE (basis set superposition error)54,55 might play a role in quantifying cooperativity, an illustrative examples, shown in Figure 3 are profiles of the BSSE-corrected interaction energy per building block for (CO2)n, Cl2Arn, and Li+(H2O)n clusters with n=1-20. Comparing with 7 ACS Paragon Plus Environment


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their counterparts in Figure 2 without BSSE correction, no qualitative difference is discernible, suggesting the basis set effect is not a concern in this case. In summary, in this Letter, we have presented a general pattern to distinguish positively cooperative molecular systems from negatively cooperative molecular systems. Based on the results from this work, together with those from our earlier studies, it is shown that homogenous molecular systems are positively cooperative, whereas charged molecular systems are negatively cooperative. This pattern has been further examined by employing analytical tools to dissect contributions from energetics and information to appreciate the origin. We found that positive cooperativity is dominant by the exchangecorrelation interaction and steric effect, but the negative cooperativity is dictated by the electrostatic interaction. From the information-theoretic perspective, these two categories of systems also demonstrated vastly different behaviors. Our results from this work should have strong implications for better understanding processes such as molecular recognition, protein folding, signal transduction, allosteric regulation, and so on.

AUTHOR INFORMATION Corresponding Author *E-mail: [email protected]. Tel: (919)962-4032 (O) ORCID Dongbo Zhao: 0000-0002-0927-4361 Donghai Yu: 0000-0002-3119-7999 Shubin Liu: 0000-0001-9331-0427 Notes The authors declare no competing financial interest.

ACKNOWLEDGEMENTS 8 ACS Paragon Plus Environment

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CYR and SBL acknowledge support from the National Natural Science Foundation of China (No. 21503076) and Hunan Provincial Natural Science Foundation of China (Grant No. 2017JJ3201).

References (1) Mahadevi, A. S.; Sastry, G. N. Cooperativity in Noncovalent Interactions. Chem. Rev. 2016, 116, 2775– 2825. (2) Molina, P.; Zapata, F.; Caballero, A. Anion Recognition Strategies Based on Combined Noncovalent Interactions. Chem. Rev. 2017, 117, 9907–9972. (3) Rodgers, M. T.; Armentrout, P. B. Cationic Noncovalent Interactions: Energetics and Periodic Trends. Chem. Rev. 2016, 116, 5642–5687. (4) Shank, E. A.; Cecconi, C.; Dill, J. W.; Marqusee, S.; Bustamante, C. The folding cooperativity of a protein is controlled by its chain topology. Nature 2010, 465, 637–640. (5) Motlagh, N. H.; Wrabl, J. O.; Li, J.; Hilser, V. J. The ensemble nature of allostery. Nature 2014, 508, 331– 339. (6) Williamson, J. R. Cooperativity in macromolecular assembly. Nat. Chem. Biol. 2008, 4, 458–465. (7) Whitty, A. Cooperativity and biological complexity. Nat. Chem. Biol. 2008, 4, 435–439. (8) Li, Y.; Wang, Y.; Huang, G.; Gao, J. Cooperativity Principles in Self-Assembled Nanomedicine. Chem. Rev. 2018, 118, 5359–5391. (9) Xantheas, S. Cooperativity and hydrogen bonding network in water clusters. Chem. Phys. 2000, 258, 225–231. (10) Wu, Y.; Zhao, Y. A theoretical study on the origin of cooperativity in the formation of 310- and α-helices. J. Am. Chem. Soc. 2001, 123, 5313–5319. (11) Ercolani, G. Assessment of cooperativity in self-assembly. J. Am. Chem. Soc. 2003, 125, 16097–16103. (12) Kobko, N.; Dannenberg, J. Cooperativity in Amide Hydrogen Bonding Chains. Relation between Energy, Position, and H-Bond Chain Length in Peptide and Protein Folding Models. J. Phys. Chem. A 2003, 107, 10389–10395. (13) Kar, T.; Scheiner, S. Comparison of Cooperativity in CH···O and OH···O Hydrogen Bonds. J. Phys. Chem. A 2004, 108, 9161–9168. (14) Albrecht, L.; Chowdhury, S.; Boyd, R. J. Hydrogen Bond Cooperativity in Water Hexamers: Atomic Energy Perspective of Local Stabilities. J. Phys. Chem. A 2013, 117, 10790–10799. (15) Chen, Y.; Dannenberg, J. Cooperative 4-Pyridone H-Bonds with Extraordinary Stability. A DFT Molecular Orbital Study. J. Am. Chem. Soc. 2006, 128, 8100–8101.



Page 10 of 17

(16) Nochebuena, J.; Ireta, J. On cooperative effects and aggregation of GNNQQNY and NNQQNY peptides. J. Chem. Phys. 2015, 143, 135103. (17) Guevara-Vela, J. M.; Romero-Montalvo, E.; Gomez, V. A. M.; Chavez-Calvillo, R.; Garcia-Revilla, M.; Francisco, E.; Pendas, A. M.; Rocha-Rinza, T. Hydrogen bond cooperativity and anticooperativity within the water hexamer. Phys. Chem. Chem. Phys. 2016, 18, 19557–19566. (18) Nochebuena, J.; Cuautli, C.; Ireta, J. Origin of cooperativity in hydrogen bonding. Phys. Chem. Chem. Phys. 2017, 19, 15256–15263. (19) Hunter, C. A.; Anderson, H. L. What is Cooperativity? Angew. Chem., Int. Ed. 2009, 48, 7488–7499. (20) Martini, J. W. R.; Diambra, L.; Habeck, M. Cooperative binding: a multiple personality. J. Math. Biol. 2016, 72, 1747–1774. (21) Stefan, M. I. Cooperativity: a competition of definitions. J. Math. Biol. 2017, 74, 1679–1681. (22) Rong, C.; Zhao, D.; Yu, D.; Liu, S. Quantification and origin of cooperativity: insights from density functional reactivity theory. Phys. Chem. Chem. Phys. 2018, 20, 17990–17998. (23) Zhou, T.; Liu, S.; Yu, D.; Zhao, D.; Rong, C.; Liu, S. On the negative cooperativity of argon clusters containing one lithium cation or fluorine anion. Chem. Phys. Lett. 2019, 716, 192–198. (24) Liu, S. Steric effect: a quantitative description from density functional theory. J. Chem. Phys. 2007, 126, 244103. (25) Liu, S. Conceptual Density Functional Theory and Some Recent Developments. Acta Phys.–Chim. Sin. 2009, 25, 590–600. (26) Liu, S. Information-Theoretic Approach in Density Functional Reactivity Theory. Acta Phys.–Chim. Sin. 2016, 32, 98–118. (27) Kitaura, K.; Morokuma, K. A new energy decomposition scheme for molecular interactions within the Hartree-Fock approximation. Int. J. Quantum Chem. 1976, 2, 325–331. (28) Jeziorski, B.; Moszynski, R.; Szalewicz, K. Perturbation Theory Approach to Intermolecular Potential Energy Surfaces of van der Waals Complexes. Chem. Rev. 1994, 94, 1887−1930. (29) Wu, Q.; Ayers, P. W.; Zhang, Y. Density-based energy decomposition analysis for intermolecular interactions with variationally determined intermediate state energies. J. Chem. Phys. 2009, 131, 164112. (30) Hesselmann, A.; Jansen, G.; Schutz, M. Density-functional theory-symmetry-adapted intermolecular perturbation theory with density fitting: A new efficient method to study intermolecular interaction energies. J. Chem. Phys. 2005, 122, 014103. (31) Fedorov, D. G.; Kitaura, K. Pair interaction energy decomposition analysis. J. Comput. Chem. 2007, 28, 222−237. (32) Su, P.; Li, H. Energy decomposition analysis of covalent bonds and intermolecular interactions. J. Chem. Phys. 2009, 131, 014102.


Page 11 of 17 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 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(33) Horn, P. R.; Mao, Y.; Head-Gordon, M. Probing non-covalent interactions with a second generation energy decomposition analysis using absolutely localized molecular orbitals. Phys. Chem. Chem. Phys. 2016, 18, 23067−23079. (34) Bickelhaupt, F. M.; Baerends, E. J. Kohn-Sham Density Functional Theory: Predicting and Understanding Chemistry. In Reviews in Computational Chemistry; Lipkowitz, K. B., Boyd, D. B., Eds.; Wiley: New York, 2000; Vol. 15, pp 1−86. (35) Shannon, C. E. A mathematical theory of communications. Bell Syst. Tech. J. 1948, 27, 379–423. (36) Fisher, R. A. Theory of Statistical Estimation. Proc. Cambridge Philos. Soc. 1925, 22, 700–725. (37) Ghosh, S. K.; Berkowitz, M.; Parr, R. G. Transcription of ground-state density-functional theory into a local thermodynamics. Proc. Natl. Acad. Sci. U.S.A. 1984, 81, 8028−8031. (38) Rong, C.; Lu, T.; Chattaraj, P. K.; Liu, S. On the relationship among Ghosh-Berkowitz-Parr entropy, Shannon entropy and Fisher information. Indian J. Chem. 2014, 53, 970–977. (39) Kullback, S. Information Theory and Statistics, Dover, Mineola, NY, 1997. (40) Liu, S.; Rong, C.; Wu, Z.; Lu, T. Rényi Entropy, Tsallis Entropy and Onicescu Information Energy in Density Functional Reactivity Theory. Acta Phys.–Chim. Sin. 2015, 31, 2057–2063. (41) Zhao, D.; Liu, S.; Rong, C.; Zhong, A.; Liu, S. Toward Understanding the Isomeric Stability of Fullerenes with Density Functional Theory and the Information-Theoretic Approach. ACS Omega, 2018, 3, 17986– 17990. (42) Rong, C.; Lu, T.; Ayers, P. W.; Chattaraj, P. K.; Liu, S. Scaling properties of information-theoretic quantities in density functional theory. Phys. Chem. Chem. Phys. 2015, 17, 4977–4988. (43) Liu S. Origin and Nature of Bond Rotation Barriers: A Unified View. J. Phys. Chem. A 2013, 117, 962– 965. (44) Zhao, D.; Rong, C.; Jenkins, S.; Kirk, S. R.; Yin, D.; Liu, S. Origin of the cis-Effect: a Density Functional Theory Study of Doubly Substituted Ethylenes. Acta Phys.–Chim. Sin. 2013, 29, 43–54. (45) Liu, S.; Govind, N. Toward understanding the nature of internal rotation barriers with a new energy partition scheme: ethane and n-butane. J. Phys. Chem. A 2008, 112, 6690–6699. (46) Liu, S.; Govind, N.; Pedersen, L. G. Exploring the origin of the internal rotational barrier for molecules with one rotatable dihedral angle. J. Chem. Phys. 2008, 129, 094104. (47) Torrent-Sucarrat, M.; Liu, S.; DeProft, F. Steric effect: partitioning in atomic and functional group contributions. J. Phys. Chem. A 2009, 113, 3698–3702. (48) Liu, S.; Hu, H.; Pedersen, L. G. Steric, quantum, and electrostatic effects on SN2 reaction barriers in gas phase. J. Phys. Chem. A 2010, 114, 5913–5918. (49) Ess, D. H.; Liu, S.; DeProft, F. Density functional steric analysis of linear and branched alkanes. J. Phys. Chem. A 2010, 114, 12952–12957.



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(50) Tsirelson, V. G.; Stash, A. I.; Liu, S. Quantifying steric effect with experimental electron density. J. Chem. Phys. 2010, 133, 114110. (51) Huang, Y.; Zhong, A.; Yang, Q.; Liu, S. Origin of anomeric effect: A density functional steric analysis. J. Chem. Phys. 2011, 134, 084103. (52) Liu, S. On the relationship between densities of Shannon entropy and Fisher information for atoms and molecules. J. Chem. Phys. 2007, 126, 191107. (53) Parr, R. G.; Yang, W. Density Functional Theory of Atoms and Molecules; Oxford University n

2 3 4 5 6 7 8 9 10 11 12

(HF)n

-2.51 -5.26 -7.28 -8.04 -8.27 -8.36 -8.42 -8.48 -8.50 -8.53 -8.54

%

109.2 38.5 10.4 2.9 1.2 0.7 0.6 0.3 0.4 0.1

Avg. 16.4 Press:Oxford, U.K., 1989.

(CO2)n

-0.84 -1.55 -2.32 -2.78 -3.17 -3.42 -3.60 -3.82 -3.92 -4.11 -4.24

%

85.2 49.7 19.9 14.0 8.0 5.1 6.1 2.7 4.8 3.3 19.9

Cl2Arn

-0.87 -0.92 -0.96 -1.05 -1.11 -1.15 -1.20 -1.22 -1.31 -1.33 -1.31

%

6.4 4.0 9.4 5.3 4.1 4.3 1.9 7.1 1.9 -2.0

NH3(H2O)n

-9.51 -10.37 -10.04 -10.12 -10.51 -11.53 -10.59 -11.32 -11.22 -11.33 -11.45

4.6

%

9.1 -2.5 0.1 3.9 9.8 -6.8 5.3 -0.9 0.9 1.1 4.0

Li+(H2O)n

-34.81 -31.30 -28.23 -26.11 -24.58 -23.37 -22.88 -22.16 -21.56 -20.85 -20.78

%

-10.1 -9.8 -7.5 -5.8 -5.0 -2.1 -3.2 -2.7 -3.3 -0.3

F–(H2O)n

-22.78 -22.31 -20.43 -19.44 -18.74 -17.84 -17.22 -16.68 -16.34 -15.91 -16.00

%

-2.0 -8.4 -4.8 -3.6 -4.8 -3.5 -3.1 -2.0 -2.6 0.6

5.0

3.5

(54) Boys, S. F.; Bernardi, F. Calculation of Small Molecular Interactions by Differences of Separate Total Energies – Some Procedures with Reduced Errors. Mol. Phys. 1970, 19, 553–566. (55) Simon, S.; Duran, M.; Dannenberg, J. J. How does basis set superposition error change the potential surfaces for hydrogen bonded dimers? J. Chem. Phys. 1996, 105, 11024–11031.

Table 1. The Interaction Energy per Building Block and Its Relative Change Percentage for the Six Clusters Studied in This Work. The Last Row Tabulates the Mean Absolute Percentage of the Change from n=2 to n=12.


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Table 2. The Correlation Coefficient (R) of the Interaction Energy per Building Block, En, for the Six Cluster Models (n=1-20) Studied in this Work with Five Energetic Components (TS, Exc, Ee, ES, and Eq), Three Thermodynamic (H, S, and G), Eight Information-Theoretic Quantities (SS, IF, SGBP, IG, R2, R3, rR2, and rR3). (HF)n

(CO2)n

Cl2 Arn

NH3(H2O)n

F– (H2O)n

Li+ (H2O)n

Ts

-0.591

0.998

-0.025

-0.946

-0.981

-0.922

Exc

0.998

0.999

0.995

0.994

0.762

-0.944

Ee

0.764

-0.999

-0.823

0.945

0.990

0.974

Es

0.994

0.999

0.987

0.981

-0.647

-0.936

Eq

-0.994

-0.999

-0.978

-0.988

0.467

0.937

G

0.660

-0.908

-0.950

0.975

0.994

1.000

H

1.000

1.000

0.953

0.999

1.000

1.000

S

0.947

0.986

0.983

0.904

-0.747

-0.994

Ss

0.985

0.999

0.995

0.974

0.484

-0.962

IF

0.994

0.999

0.987

0.980

-0.647

-0.936

SGBP

0.980

0.999

0.992

0.951

-0.068

-0.977

IG

0.994

0.985

0.858

0.951

0.981

0.999

R2

-0.938

-0.983

-0.986

-0.925

0.993

0.996

R3

-0.947

-0.979

-0.984

-0.923

0.992

0.996

rR2

-0.921

-0.988

-0.987

-0.927

-0.769

0.995

rR3

-0.923

-0.984

-0.927

0.513

-0.972

0.995


Page 15 of 17 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 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure 1. Illustrative examples of the optimized structure for six cluster models with n=20 studied in this work. The computational method for these systems is different because of these different nature of interactions. Computational details are available in SI and elsewhere.22,23



Page 16 of 17

Figure 2. Cooperativity profiles of the six cluster models studied in this work.


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Figure 3. Profiles of the BSSE (basis set superposition error) corrected interaction energy per building block for (CO2)n, Cl2Arn, and Li+(H2O)n (n=1-20) clusters.


Homogeneous Molecular Systems are Positively Cooperative, but

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