Baird's Rule in Substituted Fulvene Derivatives: An Information-Theoretic Study on Triplet-State Aromaticity and Antiaromaticity.

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.

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 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 information-theoretic 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 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 π-electrons 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 singlet- and 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?

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/stereoselectivity[33,35] 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 SS(16,40)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 IF(16,40)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 SGBP(16,42)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 as[16,43]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 follows[16]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 ≥ 2[44]

Besides quantifying steric effect and stereoselectivity with Fisher information[32,45] 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 available in the literature.[3,6,13] However, most of these descriptors were developed based on singlet states or closed-shell 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 following[3]where n is the bond number of the considered ring, Ropt is average bond length, R is bond length with Å, and α is normalization constant, α = 257.7 for carbon–carbon. The HOMA value of <0, =0, and =1 represents antiaromatic, nonaromatic, and perfect aromatic systems, respectively. The other aromaticity index is the nucleus-independent chemical shift (NICS) derived from the electromagnetic effect because of the aromatic ring current. It is found that diamagnetic or diatropic ring current is associated with aromaticity, whereas a paramagnetic or paratropic ring current signals antiaromaticity. This difference in ring current leads to the noticeable difference in NMR chemical shifts. A more negative value of NICS is an indication of a stronger aromaticity, and a more positive value of NICS is an indication of a stronger antiaromaticity. In formulating NICS, the chemical shift is expressed as the sum of sum of partial chemical shifts arising from occupied molecular orbitals Ψ at the chosen points located at or on top of the aromatic ring[13]where is the chosen point for calculation, is the electron coordinate, = – is the relative coordinate, = × ∇ 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) 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 aswhere is an operator acting on that produces all n! permutations of atoms in the cycle.[49] A larger MCI value indicates stronger aromaticity. It should be noted that the normalized MCI was used for the comparison purpose among different cyclic sizes. The normalized MCI values were calculated as (MCI)1/n The normal MCI were calculated as (MCI)1/, where the n is the ring size,[49] but still labeled as MCI. Finally, the aromatic stabilization energy (ASE) is calculated with the same isodesmic reaction, shown in Scheme S2, as our previous work.[17] This isodesmic reaction destroys the cyclic delocalization, so positive and negative ASE refer to aromatic and antiaromatic rings, respectively.

The purpose of the present work is to calculate the above ITA quantities and aromaticity indexes for both singlet and triplet states of a series of fulvene derivatives and to compare their behaviors in correlations. Besides the desire to discuss the differences of appreciating singlet and triplet states aromaticity and antiaromaticity using these ITA quantities and aromaticity indexes, what we are interested the most is to examine the scope and applicability of Hückel’s and Baird’s rules and to see whether or not these two rules obey the same cross-correlation patterns that we previously discovered and thus to enable these properties to provide a novel understanding about the validity of these rules in different circumstances.[21,50] The answer could be affirmative based on what was available from the literature, such as works of Zilberg and Mandado.[29,30] Some reference structures (presented in Scheme S1) with explicit 4n or 4n + 2 number of electron on rings are used for understanding the relative magnitude of quantities.

Results and Discussion

Shown in Tables and 2 as illustrations are calculated values for Shannon entropy SS, and information gain IG, together with four aromaticity indexes HOMA, NICS(1), ASE, and MCI for 5MR and 7MR series. More details are enclosed in the Supporting Information. We picked 5MR and 7MR as examples because our previous studies show that these two series behaved differently in the cross-correlation between information-theoretic quantities and aromaticity indexes.[17] Results from two reference structures with the same ring size and the known aromaticity/antiaromaticity propensity are shown in the last two rows in Tables and 2 for the purpose of comparison. As can be seen from Table , for 5MR, whose total number of π electrons on its rings are influenced by the substituent group through the exocyclic double bond and clearly reflected on the aromatic indexes.[20,22] Yet, the impact from the substituent groups is not significant. This point could be witnessed by the average value of HOMA, NICS(1), and ASE indices and by the comparison with the result for the two reference structures. Compared with the reference data, the absolute value for most aromatic indices of pentafulvenes is small. The exception is outlier cases with CC–, CCH, O–, OH, and NN+. These results suggest that most pentafulvene derivatives are weak aromatic or antiaromatic, and the impact of the substituting groups on the exocyclic double bond is not significant on the ring aromaticity behavior. Switching from singlet to triplet, for the same molecule, we noticed the switch of the aromaticity indices as well. For instance, the NICS(1) value of the two cases for CCH and NN+ species is 29.70 and −12.30, respectively, for the singlet state, but those values were changed to −16.40 and 27.60, respectively, for the triplet state. This change reflects the fundamental difference from the perspective of aromaticity for the two spin states. This change is also true for the two reference structures, C5H5– and C5H5+. We should mention that this change is always true, and similar observations were reported in the literature.[12] Also shown in Table are numerical values of two information-theoretic quantities, the Shannon entropy and information gain. Their values in both the singlet and triplet states did not change significantly, consistent with the results from our previous studies.[17,44] However, as will be shown below, it is these small yet discernible fluctuations that will govern the propensity of aromaticity and antiaromaticity. On the other hand, for the triplet state results, it is found that HOMA becomes more positive and all NICS(1) values are negative, suggesting that the triplet state of 5MR is aromatic, in excellent agreement with Baird’s rule.[18,21] For the case of 5MR, for the purpose of predicting the triplet state aromaticity, the NICS index is more accurate than HOMA.[51] Again, for information-theoretic quantities, each of them fluctuated around its average value and it is its changing patterns that will dictate its usefulness and validity in predicting aromaticity and antiaromaticity.

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), 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	4.795	0.084	0.339	2.32	53.35	0.686	4.822	0.102	–0.458	–1.10	–4.33	0.669
CC–	4.937	0.114	0.212	6.38	9.68	0.615	4.987	0.141	0.358	–1.91	37.63	0.666
CCH	4.785	0.084	0.349	29.70	48.13	0.642	4.809	0.101	–0.272	–16.40	2.65	0.632
CF3	4.761	0.079	0.352	0.24	58.04	–0.477	4.786	0.096	–0.504	–0.83	–4.51	0.512
CH3	4.820	0.092	0.250	6.98	44.17	0.627	4.839	0.111	–0.222	–7.38	12.49	0.688
CMe3	4.819	0.092	0.215	7.16	46.46	0.712	4.837	0.110	–0.281	–8.23	14.21	0.713
CN	4.752	0.076	0.387	1.51	56.39	0.664	4.772	0.092	–0.408	1.39	–5.02	0.695
COCH3	4.762	0.078	0.402	–0.84	58.55	0.611	4.795	0.096	–0.508	3.66	–7.93	0.558
CONH2	4.770	0.080	0.375	–0.77	59.45	0.562	4.797	0.097	–0.491	1.85	–5.91	0.631
COO–	4.894	0.104	0.215	9.19	34.42	0.684	4.922	0.125	–0.110	–10.88	7.64	0.570
F	4.812	0.091	0.246	6.05	46.51	0.625	4.826	0.110	–0.150	–8.02	9.80	0.724
H	4.820	0.090	0.270	4.89	50.68	0.661	4.830	0.106	–0.319	–5.64	5.50	0.678
NH2	4.851	0.100	0.179	14.31	32.60	0.524	4.904	0.130	0.247	–14.84	26.97	0.416
NH3+	4.691	0.066	0.391	–5.16	61.52	0.553	4.703	0.081	–0.572	4.61	–11.67	0.516
NMe2	4.852	0.100	0.154	17.85	33.25	–0.584	4.920	0.134	0.271	–20.28	38.22	0.704
NN+	4.584	0.037	0.599	–12.30	69.64	0.667	4.631	0.058	–0.708	27.60	–42.92	0.692
NO	4.831	0.105	0.112	4.38	8.80	–0.662	4.776	0.092	–0.466	3.09	–11.19	0.645
NO2	4.729	0.072	0.435	–3.95	59.87	–0.551	4.766	0.091	–0.556	4.98	–17.14	0.501
O–	5.067	0.139	–0.067	9.85	–31.43	0.626	5.136	0.179	0.742	–12.62	71.89	0.680
OCH3	4.839	0.097	0.225	10.72	38.39	0.356	4.869	0.120	0.040	–10.96	17.56	0.645
OH	4.838	0.097	0.218	41.75	37.87	0.521	4.863	0.122	0.091	–27.88	22.15	0.615
SiMe3	4.797	0.086	0.289	5.11	50.48	0.720	4.819	0.104	–0.385	–3.86	4.28	0.716
C5H5–	5.312	0.253	0.322	69.89	–5.50	0.327	5.458	0.230	0.810	–36.10	100.84	0.659
C5H5+	4.545	–0.013	0.669	–27.08	88.92	0.612	4.545	0.004	–1.343	100.29	–99.90	0.488

Table 2

Calculated Results of Information-Theoretic Quantities and Aromatic Indexes of Triplet-State and Singlet-State Fulvene Derivatives, Exampled with SS, IG, HOMA, NICS(1), ASE, and MCI of 7MR, Respectively

	7MR, triplet state						7MR, singlet state
group	SS	IG	HOMA	NICS(1)ZZ	ASE	MCI	SS	IG	HOMA	NICS(1)ZZ	ASE	MCI
B(OH)2	4.828	0.095	0.583	34.13	31.19	0.752	4.772	0.088	0.150	11.78	20.39	0.708
CC–	4.942	0.121	0.765	35.59	54.27	–0.687	4.890	0.115	0.049	14.47	5.29	–0.613
CCH	4.818	0.094	0.633	11.08	37.02	–0.682	4.762	0.087	0.183	49.52	21.30	–0.544
CF3	4.800	0.091	0.575	31.67	27.95	–0.634	4.744	0.084	0.161	8.46	20.32	–0.678
CH3	4.839	0.100	0.633	21.44	38.57	–0.669	4.792	0.097	–0.003	10.39	10.06	–0.681
CMe3	4.830	0.099	0.619	20.79	34.90	–0.680	4.778	0.096	–0.048	4.49	14.18	0.368
CN	4.794	0.089	0.596	37.73	31.50	–0.685	4.734	0.081	0.245	9.10	25.97	–0.727
COCH3	4.808	0.092	0.562	46.77	24.37	–0.491	4.744	0.082	0.282	5.76	26.42	–0.620
CONH2	4.813	0.093	0.563	38.74	25.18	–0.575	4.751	0.084	0.229	7.59	23.40	–0.538
COO–	4.908	0.113	0.662	11.60	45.40	–0.572	4.860	0.108	–0.027	23.72	8.32	–0.683
F	4.832	0.100	0.643	16.70	39.29	–0.452	4.787	0.097	0.024	20.85	10.62	–0.680
H	4.836	0.097	0.614	20.37	38.29	–0.552	4.790	0.093	0.035	17.71	14.81	–0.618
NH2	4.877	0.110	0.713	6.40	47.64	–0.627	4.823	0.106	–0.041	15.71	9.08	–0.641
NH3+	4.744	0.080	0.549	35.64	16.75	–0.698	4.682	0.074	0.295	–1.04	23.51	–0.654
NMe2	4.873	0.111	0.718	7.41	49.45	0.578	4.799	0.101	0.005	15.69	10.35	–0.584
NN+	4.685	0.061	0.482	85.91	–9.51	0.478	4.584	0.046	0.789	–13.73	69.61	0.756
NO	4.777	0.090	0.435	45.32	27.56	–0.597	4.723	0.078	0.354	5.65	33.25	–0.635
NO2	4.788	0.087	0.533	47.01	15.08	–0.451	4.723	0.079	0.305	1.49	25.35	–0.673
O–	5.052	0.147	0.834	13.30	77.87	–0.467	5.010	0.144	0.035	24.71	–16.91	–0.721
OCH3	4.859	0.106	0.682	13.72	44.32	0.440	4.812	0.102	0.019	26.45	9.43	–0.652
OH	4.855	0.105	0.676	–20.95	43.78	–0.505	4.811	0.102	0.005	104.57	9.17	–0.646
SiMe3	4.826	0.096	0.606	26.22	35.50	0.368	4.773	0.091	0.042	9.63	16.20	–0.627
C7H7–	5.288	0.178	0.827	–35.71	102.71	0.598	5.280	0.189	0.156	81.07	–50.32	–0.436
C7H7+	4.618	0.016	0.477	9.18	–48.25	–0.386	4.588	0.017	0.984	–29.12	123.98	0.643

Similar to the 5MR, for 7MR systems, whose total number of π electrons on the conjugation ring could also change between 6 and 8 depending on the nature of the substituent group on the exocyclic bond, so, according to Hückel’s rule,[20,22] these systems in the singlet state should yield observable changes between aromaticity and antiaromaticity. The same must be true for the triplet state. Table shows the results for the same aromaticity indexes and information-theoretic quantities for both singlet and triple states. For the comparison purpose, we also listed the results for two reference structures, C7H7– and C7H7+, whose aromaticity and antiaromaticity features are known. Consistent with what we found for the 5MR case, the results from 7MR derivatives show that, generally speaking, the impact from the substituting groups is not significant, but we do see the existence of outliers, such as CC– and NN+, whose absolute value of the aromaticity indices is even larger than that of the two reference species. We also see the sign switch of the aromaticity index values in the triplet state, same as that for C7H7– and C7H7+. The result for information-theoretic quantities is similar to those for 5MR as well.

Put together, from the results shown in Tables and 2, it becomes apparent to us that aromaticity index results often provide inconsistent and sometimes contradictory predictions on both singlet-state and triplet-state aromaticity and antiaromaticity.[18,52] It is our intention that information-theoretic quantities will provide an additional dimension to appreciate aromaticity and antiaromaticity. With the new results from the latter, a better understanding about the nature of aromaticity will be achieved. This is done through the correlations between the pair of the quantities from both aromaticity index and information-theoretic quantities, whose results are shown and discussed below.

Shown in Table are the correlation coefficients from the quantities tabulated in Tables and 2. As can be seen from the table, SGBP and IG are strongly positively correlated for both 5MR and 7MR systems and for both singlet and triplet states. Also, for aromaticity indexes, HOMA is found to be reasonably conversely correlated with the two NICS indexes. In all cases (5MR/7MR, and singlet/triplet states), the correlation coefficient is negative. The correlations within the categories of aromaticity indexes and information quantities are less informative because the correlation coefficient does not change sign either across different sizes of the fulvene ring or between singlet and triplet states. What we think is more interesting and more informative is the case of cross-correlations between these two categories of properties, that is, aromaticity indexes and information-theoretic quantities. In our previous work, we have shown that the correlation coefficient of these cross-correlations changes its sign from one fulvene size to another, and the pattern of this sign changing depends on the total number of π electrons on the fulvene ring, which agrees excellently well with Hückel’s rule.[17] Now, let us see if there is a similar pattern for the triplet-state aromaticity, and, if yes, whether or not it agrees well with Baird’s rule.

Table 3

Correlation Coefficients (R) of Correlations between Information-Theoretic Quantities and Aromatic Indexes Listed in Tables and 2 (The Reference Structures are Not Included to Calculate R)

	5MR, triplet state					5MR, singlet state
	SS	IG	HOMA	NICS(1)ZZ	ASE	SS	IG	HOMA	NICS(1)ZZ	ASE
IG	0.987					0.992
HOMA	–0.921	–0.958				0.926	0.954
NICS(1)ZZ	0.457	0.483	–0.481			–0.639	–0.697	–0.681
ASE	–0.889	–0.901	0.891	–0.332		0.952	0.977	0.958	–0.734
MCI	0.062	–0.028	0.100	0.052	0.084	0.108	0.090	0.097	–0.006	0.151

For the cross-correlation between aromaticity indexes, for example, HOMA, NICS, ASE, and MCI and information-theoretic quantities, for example, Shannon entropy and information gain, as shown in Table , we found that the correlation coefficient changed sign for the singlet state from 5MR to 7MR. For example, the correlation coefficient of NICS(1) versus SS is −0.6389 for 5MR, but that is changed to +0.3871 for 7MR. (These low correlation coefficients were resulted from a few outliers.) The same is true for all other cross-correlation coefficients. This result is known, as we have reported previously.[17,44] What is new from the present study is that for the same ring-size of fulvene derivatives, the same phenomenon of the sign change is true from the singlet state to the triplet state. For instance, for 7MR, the correlation coefficient for the cross-correlation between Ss and NICS(1) is −0.5923 for the triplet state, but the same quantity is changed to +0.3871 for the singlet state. This is true for all other cross-correlations shown in Table as well. The MCI values also fluctuate insignificantly when the substituent group is changed, much to the same as information-theoretic quantities as shown in Tables and 2. As shown in Table , this quantity is not significantly correlated with any other quantity studies in this work. This result might be an indication that MCI is not a valid index to quantify aromaticity of fulvene derivatives. For the ASE index, which has been widely used in the literatures[6,28] as a reliable measure of the singlet states aromaticity and antiaromaticity,[17] for the triplet states, its reliability as a measure of aromaticity and antiaromaticity is questionable. This point can be reflected by the 7MR series, whose ASE values of both triplet and singlet states are positive, and whose ASE values in the triplet state are only modestly correlated with other aromaticity indexes. The correlations among the triplet state of other series, as shown in Table S4, are not significantly correlated either. On the basis of these results, the ASE index is not shown to be an accurate descriptor for the triplet-state aromaticity.[28]

What does this sign change mean? To be more specific, what additional information does it convey to us about aromaticity and antiaromaticity? Earlier, for the singlet state aromaticity, we did observe this phenomenon with respect to the ring-size change[17,48] and found that the pattern of these sign changes was in excellent agreement with Hückel’s 4n + 2 rule.[22] That is, for aromatic systems with the total number of π electrons equal to 2 and 6, the sign is always the same, and only for those systems whose total number of π electrons is 4, we observed the change of the signs. For the latter systems, according to Hückel’s 4n + 2 rule, they should be antiaromatic in nature. Given the fact that (i) no single aromaticity index is able to predict aromaticity and antiaromaticity correctly and (ii) different aromaticity indexes often yield inconsistent and many times contradictory results, the cross-correlation results should provide an additional dimension of knowledge to appreciate aromaticity and antiaromaticity. For the triplet-state aromaticity, as shown in Tables and 2, we also disclosed that different aromaticity indexes, for example, HOMA and NICS, do not provide robust and consistent predictions about aromaticity and antiaromaticity.[53] Plus, information-theoretic quantities alone do not tell us much about the nature and propensity of aromaticity and antiaromaticity because their numerical values only fluctuated slightly around their average value. However, combining aromaticity indexes with information-theoretic quantities and using the cross-correlations between these two categories of properties, we could unveil reasonably strong correlations with unambiguous patterns of changes.

More importantly, as can be seen from the correlation coefficient values in Table , the cross-correlations for the triplet state are opposite to those for the singlet state. That is, the value of correlation coefficients for singlet and triplet states is opposite in sign to each other. This opposite nature in correlation patterns are in consensus with the two rules governing the aromatic propensity for these two states, Hückel’s rule for the singlet state and Baird’s rule for the triplet state.[20−22,54] Our results indicate that instead of employing two separate rules through π-electron number counting to predicting aromaticity and antiaromaticity for both singlet and triplet states, we can solely utilize the cross-correlation patterns between aromaticity indexes and information-theoretic quantities to ascertain which systems are aromatic and which are antiaromatic. Unifying predictions of aromaticity and antiaromaticity with cross-correlations patterns between aromaticity indexes and information-theoretic quantities for both singlet and triplet states are one of the main results from this work.

Is this conclusion applicable to other information-theoretic quantities or other ring sizes of the fulvene derivatives? Shown in Table are values of correlation coefficients for all of the cross-correlations between four aromaticity indexes and four information-theoretic quantities for five ring sizes in both singlet and triplet states. As can be seen from the table, in singlet state, 3MR, 4MR+, and 7MR, each with 2, 2, and 6 π-electrons on the fulvene ring, respectively, have the same trend of correlation coefficients (i.e., the same sign), whereas 4MR– and 5MR each with 4 π-electrons are opposite in terms of their cross-correlation coefficients signs. For the triplet state, however, the trend is completely opposite. All positive values of correlation coefficients in the singlet state become negative in the triplet state and vice versa. This opposite nature is also reflected from the two number counting rules, Hückel’s rule and Baird’s rule, for singlet and triplet states,[20−22,54] respectively.

Table 4

Matrix of Linear Correlation Coefficient between Aromatic Indexes and Information-Theoretic Quantities for Triplet-State and Singlet-State Fulvene Derivatives (The Reference Structures are Not Included to Calculate R)

		triplet state				singlet state
series	aromatic indexes	SS	SGBP	E2	R2r	SS	SGBP	E2	R2r
3MR	HOMA	0.774	0.813	–0.937	0.788	–0.947	–0.956	0.985	–0.973
	NICS(0)	–0.864	–0.895	0.905	–0.886	0.924	0.897	–0.785	0.900
	NICS(1)	–0.726	–0.769	0.934	–0.744	0.980	0.974	–0.955	0.989
	NICS(1)ZZ	–0.450	–0.508	0.707	–0.468	0.310	0.353	–0.456	0.339
	ASE	0.922	0.938	–0.767	0.921	–0.957	–0.965	0.927	–0.965
	MCI	0.403	0.408	–0.333	0.379	–0.078	–0.068	–0.025	–0.055
4MR–	HOMA	–0.651	–0.686	0.039	–0.150	0.734	0.764	–0.575	0.468
	NICS(0)	0.576	0.481	0.513	0.684	–0.549	–0.604	0.826	–0.608
	NICS(1)	0.789	0.736	0.225	0.649	–0.433	–0.487	0.709	–0.499
	NICS(1)ZZ	0.358	0.311	0.274	0.269	–0.401	–0.464	0.598	–0.367
	ASE	–0.284	–0.231	–0.356	–0.310	–0.395	–0.338	–0.105	–0.140
	MCI	0.278	0.139	0.558	0.674	0.608	0.643	–0.523	0.295
4MR+	HOMA	–0.594	–0.549	0.461	–0.559	–0.915	–0.919	0.935	–0.919
	NICS(0)	0.176	0.120	0.064	0.109	0.795	0.712	–0.663	0.711
	NICS(1)	0.092	0.040	0.193	0.036	0.846	0.755	–0.673	0.752
	NICS(1)ZZ	0.121	0.062	0.164	0.058	0.680	0.603	–0.549	0.607
	ASE	0.336	0.314	–0.344	0.283	–0.739	–0.578	0.438	–0.569
	MCI	–0.462	–0.436	0.338	–0.423	–0.618	–0.580	0.445	–0.579
5MR	HOMA	–0.921	–0.769	0.862	–0.956	0.926	0.688	–0.982	0.951
	NICS(0)	0.926	0.646	–0.899	0.884	–0.918	–0.763	0.921	–0.953
	NICS(1)	0.949	0.699	–0.922	0.923	–0.897	–0.766	0.893	–0.931
	NICS(1)ZZ	0.457	0.770	–0.531	0.482	–0.639	–0.837	0.656	–0.698
	ASE	–0.889	–0.623	0.756	–0.899	0.952	0.671	–0.972	0.976
	MCI	0.062	–0.004	–0.096	–0.026	0.108	–0.014	–0.074	0.090
7MR	HOMA	0.898	0.903	–0.952	0.893	–0.756	–0.795	0.711	–0.799
	NICS(0)	–0.898	–0.927	0.855	–0.929	0.892	0.869	–0.889	0.866
	NICS(1)	–0.895	–0.924	0.852	–0.926	0.899	0.876	–0.894	0.874
	NICS(1)ZZ	–0.592	–0.646	0.551	–0.645	0.387	0.410	–0.402	0.405
	ASE	0.950	0.968	–0.841	0.971	–0.927	–0.948	0.896	–0.952
	MCI	–0.117	–0.122	0.019	–0.123	–0.396	–0.416	0.358	–0.410

To visualize these opposite patterns in triplet and singlet states, shown in Figure are illustrative examples using the Shannon entropy SS and the ASE aromaticity index for fulvene derivatives with three ring sizes (color-coded). The linear correlations between SS and ASE for the triplet state are exhibited in Figure a, whereas those for the singlet state are displayed in Figure b. As can be unambiguously seen, the slope of the fitted line is opposite in these two states. If the slope is positive in Figure a, it will become negative in Figure b. The opposite is true as well. This figure unquestionably demonstrates that the two states should be governed by two different rules, Hückel’s rule and Baird’s rule[20−22,54] because their slopes are opposite in sign. It is also shown that with the cross-correlation patterns between aromaticity indexes and information-theoretic quantities, we are able to distinguish them from each other, so these cross-correlations include adequate information to unify the two rules with the same physiochemical understanding. The similar thing also was reported by Mandado in several cyclic compounds with different spin states, where a separated scheme of α and β terms was proposed to uniform the 4n + 2 and 4n aromaticity for closed-/open-shell annulenes.[30]

Figure 1

Strong linear correlation between the Shannon entropy SS and the aromaticity index ASE for 3MR, 5MR, and 7MR systems. Panels (a,b) are plotted to the triplet and singlet states, respectively.

As a further illustration of the opposite trends in triplet and singlet states, shown in Figure are linear correlations for both singlet and triplet states between NICS(0) and four different information-theoretic quantities, Fisher information,[40] information gain,[16] Onicescu information energy of order 3,[43] and relative Rényi entropy of order 3.[44] As can be seen from the figure, linear correlations for the triplet state are completely opposite to those for the singlet state, suggesting, again, that the aromaticity propensity of these two states are dictated by different rules and that using these cross-correlations we should be able to adequately appreciate the propensity difference of aromaticity and antiaromaticity for fulvene derivatives and other species alike. In addition, the opposite changing tendency of aromaticity for singlet and triplet states agrees well with the results from the substituent effect. The modest correlation between various substituent constants and aromatic indexes (shown in Supporting Information Table S6) indicated that these substituent constants could provide a qualitative description about aromaticity and antiaromaticity for both singlet and triplet states, similar to the resonance analysis shown in Scheme .

Figure 2

Middle-to-good linear correlations between NICS(0) and 4 information-theoretic quantities, IF, IG, E3, and R3r for the 7MR system. Panels (a–d) are plotted with Fisher information, IF, information gain, IG, Onicescu energy of order 3, E3, and Relative Rényi entropy of order 3, R3r, respectively.

Scheme 2

Qualitative Interpretation of Aromaticity from Hückel’s and Baird’s Rules, Using 3MR, 5MR, and 7MR as Examples

Concluding Remarks

As a continuation of our recent efforts to appreciate and quantify molecular aromaticity and antiaromaticity with information-theoretic quantities, in this work, we have investigated the applicability of our earlier results to the triplet state for a series of fulvene derivatives using the cross-correlation pattern between aromaticity indexes and information-theoretic quantities as a gauge of molecular aromaticity. To that end, we made use of four aromaticity indexes and eight information-theoretic quantities and examined a total of 22 substituted fulvene derivatives. We compared the results both in singlet and triplet states. The following is what we have discovered in the present work. Predictions of molecular aromaticity and antiaromaticity for the triplet state using aromaticity indexes such as HOMA and NICS are often unreliable and inconsistent. Using information-theoretic quantities alone will not do the job either because their numerical values usually fluctuate slightly around an average value. However, if these two categories of properties are combined and cross-correlations between aromaticity indexes and information-theoretic quantities are employed, better understanding on the nature and propensity of aromaticity and antiaromaticity can be accomplished. Historically, the triplet state aromaticity is governed by Baird’s 4n rule. With cross-correlation results from this study, we have not only demonstrated the validity of this rule but also shown that Hückel’s rule for the singlet state aromaticity and Baird’s rule for the triplet state aromaticity share the same theoretical foundation because with the cross-correlation patterns between aromaticity indexes and information-theoretic quantities we are able to distinguish them from each other simultaneously.

Computational Details

Dubbed as the aromatic chameleon and examined by a few of previous studies,[19] fulvene derivatives are capricious in aromatic behaviors.[17,48] They can be either aromatic or antiaromatic as their behaviors are greatly influenced by the nature of substituting groups, ring sizes, fused rings, spin states, and coordinated ions.[17,18,31,48,55−57] We choose this system for the present study precisely for this reason. Scheme exhibits the systems studied in this work, including tria (3MR), tetra (4MR+ and 4MR– with one positive and negative charge, respectively), penta (5MR), hexa (6MR+ and 6MR–, with one positive and negative charge, respectively), and hepta (7MR) fulvene derivatives. The substituting group R was chosen from the following pool, R = H, CH3, CCH, CMe3, CN, CONH2, COCH3, CF3, CC–, COO–, F, B(OH)2, OH, OCH3, O–, NH2, NO2, NO, NMe2, NH3+, NN+, and SiMe3, with the criterion that the optimized structure should be planar, which is one of the prerequisites for aromaticity. Scheme also shows the total number of π electrons on the fulvene ring of different ring sizes. For the triplet state of fulvene derivatives, we are only concerned with the triplet state in this work. Shown in Scheme , as illustrative examples, is the resonant structure of 3MR, 5MR, and 7MR with different kinds (electron donor and acceptor) of substituents leading to different aromaticity propensities in the singlet and triplet states governed by Hückel’s and Baird’s rules,[20,21] respectively.

Scheme 1

Seven Fulvene Derivatives were Considered

Here R is B(OH)2, CC–, CCH, CF3, CH2–, CH3, CMe3, CN, COCH3, CONH2, COO–, F, H, NH–, NH2, NH3+, NMe2, NN+, NO, NO2, O–, OCH3, OH, and SiMe3. All of these structures are denoted by the ring size and net charge as well as the substituting group R. For clarity, the total number of π electrons is also shown. Each ring size is color coded.

All singlet-state and triplet-state structures listed in Scheme were optimized at the M062X/6-311++G(d,p) level of theory[58,59] with the tight SCF convergence criterion and ultrafine integration grids using Gaussian 09, Revision D.01 package.[60] The vibrational frequency was calculated to ensure the final structures obtained have no imaginary frequency. The Multiwfn 3.4.0 program developed by one of current authors was used to calculate information-theoretic quantities, NICS(1) of nonplanar rings, MCI, and HOMA based on the final optimized structure obtained. In the Multiwfn calculation,[61] the spherically averaged electron density of neutral atom at the same theoretical level was employed as reference. To perform the atomic partition, Becke’s fuzzy atom approach,[62] Bader’s zero-flux atoms-in-molecules criterion,[63] and Hirshfeld’s stockholder approach are possible.[64] As have been demonstrated earlier, these approaches yield qualitatively similar results.[65] Here, the Hirshfeld’s stockholder approach was selected to partition atoms from molecule to obtain each atomic information-theoretic quantities. The arithmetic average of atoms in the cycle was used for the description of aromaticity. A dummy atom at the geometric center of heavy atoms in the cycle was added for the NICS(0) calculation, and, meanwhile, two dummy atoms 1 Å above and 1 Å below the cyclic center were added to calculate NICS(1) and NICS(1).[13,66] The arithmetic average of the two sides was employed as the cyclic aromatic index. The unit of NICS and information-theoretic quantities is parts per million and atomic unit, respectively, whereas for HOMA and MCI are dimensionless indexes. The difference of electronic energies of the isodesmic reaction was used as the aromaticity ASE index with the unit of kJ/mol.