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Article Cite This: ACS Omega 2018, 3, 18370−18379
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Baird’s Rule in Substituted Fulvene Derivatives: An InformationTheoretic Study on Triplet-State Aromaticity and Antiaromaticity Donghai Yu,†,‡ Chunying Rong,*,† Tian Lu,§ Frank De Proft,*,‡ and Shubin Liu*,∥ †
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Key Laboratory of Chemical Biology and Traditional Chinese Medicine Research (Ministry of Education of China), College of Chemistry and Chemical Engineering, Hunan Normal University, Changsha 410081, China ‡ Research Group of General Chemistry (ALGC), Vrije Universiteit Brussel (VUB), Pleinlaan 2, 1050 Brussels, Belgium § Beijing Kein Research Center for Natural Sciences, Beijing 100022, China ∥ Research Computing Center, University of North Carolina, Chapel Hill, North Carolina 27599-3420, United States S Supporting Information *
ABSTRACT: Originated from the cyclic delocalization of electrons resulting in extra stability and instability, aromaticity and antiaromaticity are important chemical concepts whose appreciation and quantification are still much of recent interest in the literature. Employing information-theoretic quantities can provide us with more insights and better understanding about them, as we have previously demonstrated. In this work, we examine the triplet-state aromaticity and antiaromaticity, which are governed by Baird’s 4n rule, instead of Hückel’s 4n + 2 rule for the singlet state. To this end, we have made use of 4 different aromaticity indexes and 8 information-theoretic quantities, examined a total of 22 substituted fulvene derivatives, and compared the results both in singlet and triplet states. It is found that cross-correlations of these two categories of molecular property descriptors enable us to better understand the nature and propensity of aromaticity and antiaromaticity for the triplet state. Our results have not only demonstrated the existence and validity of Baird’s rule but also shown that Hückel’s rule and Baird’s rule indeed share the same theoretical foundation because with these cross-correlation patterns we are able to distinguish them from each other simultaneously in both singlet and triplet states. Our results should provide new insights into the nature of aromaticity and antiaromaticity in the triplet state and pave the road toward new ways to quantify this pair of important chemical concepts.
1. INTRODUCTION Even though theoretical and computational chemistry has been well established nowadays from the perspective of both accuracy and complexity, how to quantify chemical concepts widely employed in the literature and textbooks to appreciate molecular structure, stability, and reactivity properties is a still unresolved task.1,2 The concept of aromaticity or antiaromaticity is such an example.3−5 It has to do with the extra stability or instability of a cyclic yet planar structure because of the additional delocalization of electrons,6 either π or σ, leading to the redistribution of the electron density of the system.7−10 The controversy about aromaticity lies in the fact that many different categories of aromaticity have been unveiled in the literature,3,11,12 lots of descriptors to characterize it have been proposed as well,13 but there exists no single descriptor that can be used as the general quantitative measure.14 The culprit is the fact that these descriptors are often measures of one property only from its many manifestations of the phenomenon, such as energetics,6 geometry,3 ring current,15 and so forth, which are the consequences of the aromaticity © 2018 American Chemical Society
phenomenon, not its root cause. We need to adopt an approach to examine the concept of aromaticity from a more general viewpoint which should cover various aspects of aromaticity of different structures, subsets, and spin states. The information-theoretic approach (ITA) from density functional reactivity theory (DFRT) is believed to be such a framework.16 Using substituted fulvene derivatives in the singlet state, in our previous study,17 we have demonstrated the feasibility of applying ITA to study aromaticity and antiaromaticity from a completely different perspective.17 In that work, we employed a four-dimensional approach, with 5 fulvene ring sizes, 24 substitution groups, 4 aromaticity indexes, and 8 informationtheoretic quantities. We chose fulvene derivatives because their aromaticity character is influenced by the nature of the substituting group, and these model systems were widely used in the literature.17−19 A large wealth of data stemmed from our Received: October 19, 2018 Accepted: December 14, 2018 Published: December 26, 2018 18370
DOI: 10.1021/acsomega.8b02881 ACS Omega 2018, 3, 18370−18379
ACS Omega
Article
previous study enabled us to provide novel insights into aromaticity and antiaromaticity. The most important finding from that work is that there exist two completely opposite yet strong correlations, one positive and the other negative, between information-theoretic quantities and aromaticity indexes. These two groups of cross-correlations between these two sets of properties (aromaticity indexes and ITA quantities) for fulvene derivatives with different ring sizes happen to have a total number of 4n + 2 and 4n π electrons, where n is a whole number, on their fulvene ring, respectively, which agrees remarkably well with Hückel’s rule of aromaticity and antiaromaticity.20,21 For triplet states, it is known that Hückel’s 4n + 2 rule is no longer valid in governing the propensity of aromaticity. It is instead replaced by Baird’s rule,21,22 which dictates that the lowest triplet state of a ring structure is aromatic only when it has 4n π-electrons. To prove it, Baird developed a theoretical argument from the perturbation molecular orbital theory,21 whose validity has since been confirmed by many more accurate studies.23−28 Meanwhile, Zilberg et al. proposed that the 4n aromatic rule of triplet states can be simply regarded as 4n − 2 Hückel aromatic cycles plus two nonbonding πelectrons of the same spin.29 Mandado also interpreted 4n triplet state and 4n + 2 singlet state rules using 2n + 1 πα/2n − 1 πβ orbitals and 2n + 1 πα/2n + 1 πβ orbitals, respectively.30 It remains to be seen, however, that among the many descriptors proposed for the singlet state aromaticity if they are still applicable for the triplet state and whether or not there exists the same perplexed situation as for the singlet state aromaticity and antiaromaticity as we have demonstrated previously.17 In this work, continuing to use substituted fulvenes as illustrative examples,17,18,31 we will address these issues. In specific, we seek answers for the following three questions. (i) Are the conventional aromaticity descriptors able to characterize the triplet state aromaticity and antiaromaticity? (ii) Does the ITA provide any new physiochemical insight into aromaticity, similar to what we have observed in the singlet state? (iii) More importantly, is it possible to employ the ITA quantities to provide a common theoretic foundation for both singletand triplet-state aromaticity and antiaromaticity, and henceforth we can appreciate both Hückel’s 4n + 2 and Baird’s 4n rules with the same theoretic understanding?
which is a local functional of the electron density and a measure of the spatial delocalization of the electron density. The second quantity is Fisher information IF16,40 2
IF = −
∫ ρ(r)ln
ρ(r)dτ
dτ
(2)
which is a functional of both the electron density and its gradient |∇ρ| and is a gauge of the sharpness or localization of the electron density distribution. That is, for uniform electron gas, where |∇ρ| = 0, IF = 0. The larger the density gradient |∇ρ|, the larger the Fisher information IF is. Fisher information is closely related to the Weizsäcker kinetic energy,28 which has recently been used to quantify the steric effect and stereoselective properties.41 Other information-theoretic quantities are the Ghosh− Berkowitz−Parr entropy SGBP16,42 ÅÄÅ ÑÉ Å t(r;ρ(r)) ÑÑÑ 3 ÑÑdτ SGBP = − kρ(r)ÅÅÅÅc + ln ÅÅÇ 2 t TF(r;ρ(r)) ÑÑÑÖ (3)
∫
where t(r,ρ) is the kinetic energy density, tTF(r;ρ) is the Thomas−Fermi kinetic energy density, and c and k are constants. SGBP was resulted from transcribing the singlet-state density functional theory into a local thermodynamics through a phase-space distribution function. The Onicescu information energy of order n is defined as16,43 En =
1 n−1
∫ ρn (r)dτ
(4)
with n ≥ 2. Onicescu introduced this quantity to define a finer measure of dispersion distribution than that of Shannon entropy, which is closely related to Rényi entropy and Tsallis entropy. The relative Shannon entropy, also called information gain, Kullback−Leibler divergence, or information divergence, has been shown to be an effective descriptor to determine electrophilicity, nucleophilicity, and regioselectivity,28−31,37 whose definition is as follows16 IG =
∫ ρ(r)ln ρρ((rr)) dτ 0
(5)
where ρ0(r) is the reference state density satisfying the same normalization condition as ρ(r). This reference density can be from the same molecule with different conformations or from the reactant of a chemical reaction when the transition state is investigated. Related to the information gain is the relative Rényi entropy of order n with n ≥ 244
2. THEORETICAL FRAMEWORK The key idea of DFRT is to employ simple density functionals to appreciate and quantify chemical properties for a molecular system.16 This idea originates from the basic theorems of DFT, which dictate that all structure, bonding, reactivity, and other properties of a molecular system in the singlet state should be determined by the information contained in the electron density, ρ(r).1 Steric effect,32 electrophilicity/nucleophilicity,33−35 acidity/basicity,36−39 and regioselectivity/stereoselectivity33,35 are a few recent examples of chemical concepts quantified by simple density functionals. On the other hand, a list of quantities from information theory is well known to be simple density functionals. Whether or not they can be employed to appreciate molecular properties and chemical concepts is a recent research interest in the literature.16 These information-theoretic quantities include Shannon entropy SS16,40 SS = −
∫ |∇ρρ((rr))|
R nr =
1 n−1
∫
ρn (r) dτ ρ0 n − 1(r)
(6)
Besides quantifying steric effect and stereoselectivity with Fisher information32,45 and electrophilicity, nucleophilicity, and regioselectivity with the relative Shannon entropy,33 these quantities are additionally applied to accurately predict molecular acidity,36 highest occupied molecular orbital/lowest unoccupied molecular orbital gap,46 and so forth. We have also applied these quantities to appreciate chemical phenomena in a number of other systems, including aromaticity in the singlet state.17,47,48 A review article on this subject has recently been appeared.16 Meanwhile, to quantitatively determine aromaticity and antiaromaticity, numerous descriptors using electronic properties such as energetics, geometry, ring current, and so forth are
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DOI: 10.1021/acsomega.8b02881 ACS Omega 2018, 3, 18370−18379
ACS Omega
Article
Table 1. Calculated Results of Information-Theoretic Quantities and Aromatic Indexes of Triplet-State and Singlet-State Fulvene Derivatives, Exampled with SS, IG, HOMA, NICS(1)ZZ, ASE, and MCI of 5MR, Respectively 5MR, triplet state
5MR, singlet state
group
SS
IG
HOMA
NICS(1)ZZ
ASE
MCI
SS
IG
HOMA
NICS(1)ZZ
ASE
MCI
B(OH)2 CC− CCH CF3 CH3 CMe3 CN COCH3 CONH2 COO− F H NH2 NH3+ NMe2 NN+ NO NO2 O− OCH3 OH SiMe3 C5H5− C5H5+
4.795 4.937 4.785 4.761 4.820 4.819 4.752 4.762 4.770 4.894 4.812 4.820 4.851 4.691 4.852 4.584 4.831 4.729 5.067 4.839 4.838 4.797 5.312 4.545
0.084 0.114 0.084 0.079 0.092 0.092 0.076 0.078 0.080 0.104 0.091 0.090 0.100 0.066 0.100 0.037 0.105 0.072 0.139 0.097 0.097 0.086 0.253 −0.013
0.339 0.212 0.349 0.352 0.250 0.215 0.387 0.402 0.375 0.215 0.246 0.270 0.179 0.391 0.154 0.599 0.112 0.435 −0.067 0.225 0.218 0.289 0.322 0.669
2.32 6.38 29.70 0.24 6.98 7.16 1.51 −0.84 −0.77 9.19 6.05 4.89 14.31 −5.16 17.85 −12.30 4.38 −3.95 9.85 10.72 41.75 5.11 69.89 −27.08
53.35 9.68 48.13 58.04 44.17 46.46 56.39 58.55 59.45 34.42 46.51 50.68 32.60 61.52 33.25 69.64 8.80 59.87 −31.43 38.39 37.87 50.48 −5.50 88.92
0.686 0.615 0.642 −0.477 0.627 0.712 0.664 0.611 0.562 0.684 0.625 0.661 0.524 0.553 −0.584 0.667 −0.662 −0.551 0.626 0.356 0.521 0.720 0.327 0.612
4.822 4.987 4.809 4.786 4.839 4.837 4.772 4.795 4.797 4.922 4.826 4.830 4.904 4.703 4.920 4.631 4.776 4.766 5.136 4.869 4.863 4.819 5.458 4.545
0.102 0.141 0.101 0.096 0.111 0.110 0.092 0.096 0.097 0.125 0.110 0.106 0.130 0.081 0.134 0.058 0.092 0.091 0.179 0.120 0.122 0.104 0.230 0.004
−0.458 0.358 −0.272 −0.504 −0.222 −0.281 −0.408 −0.508 −0.491 −0.110 −0.150 −0.319 0.247 −0.572 0.271 −0.708 −0.466 −0.556 0.742 0.040 0.091 −0.385 0.810 −1.343
−1.10 −1.91 −16.40 −0.83 −7.38 −8.23 1.39 3.66 1.85 −10.88 −8.02 −5.64 −14.84 4.61 −20.28 27.60 3.09 4.98 −12.62 −10.96 −27.88 −3.86 −36.10 100.29
−4.33 37.63 2.65 −4.51 12.49 14.21 −5.02 −7.93 −5.91 7.64 9.80 5.50 26.97 −11.67 38.22 −42.92 −11.19 −17.14 71.89 17.56 22.15 4.28 100.84 −99.90
0.669 0.666 0.632 0.512 0.688 0.713 0.695 0.558 0.631 0.570 0.724 0.678 0.416 0.516 0.704 0.692 0.645 0.501 0.680 0.645 0.615 0.716 0.659 0.488
available in the literature.3,6,13 However, most of these descriptors were developed based on singlet states or closedshell structures. Here, we consider only four representative descriptors for the two states. The harmonic oscillator model of aromaticity (HOMA) index is a geometrical measurement of the equalization of chemical bonds on a conjugate ring. Its definition is the following3
occ
NICS =
α ∑ (R opt − R i)2 n i=1
Ψk 0
occ
−
2 ∑ c k
Ψk 0
rrN I − rN ⊗ r |r − RN |3 (L N ) |r − RN |3
Ψk 0
Ψk1 (8)
where RN is the chosen point for calculation, r is the electron coordinate, rN = r − RN is the relative coordinate, LN = rN × ∇ is the angular momentum, I is the unit matrix, and the first and second term on the right side are diamagnetic and paramagnetic contributions, respectively. Three popular NICS values, NICS(0), NICS(1), and NICS(1)ZZ considered in this work stand for the NICS value at the center of the aromatic carbon ring, 1 Å above the ring center, and the ZZ component of NICS(1), respectively.13 The multicenter index (MCI) was also used to measure the aromaticity from electronic aspect, which is defined as
n
HOMA = 1 −
1 ∑ 2c 2 k
(7)
where n is the bond number of the considered ring, Ropt is average bond length, Ri is bond length with Å, and α is normalization constant, α = 257.7 for carbon−carbon. The HOMA value of
