Application of the Extended HOMED ( Harmonic Oscillator Model of Aromaticity ) Index to Simple and Tautomeric Five-Membered Heteroaromatic Cycles with C , N , O , P , and S Atoms

Abstract: The geometry-based HOMA (Harmonic Oscillator Model of Aromaticity) descriptor, based on the reference compounds of different delocalizations of nand π-electrons, can be applied to molecules possessing analogous bonds, e.g., only CC, only CN, only CO, etc. For compounds with different heteroatoms and a different number of CC, CX, XX, and XY bonds, its application leads to some discrepancies. For this reason, the structural descriptor was modified and the HOMED (Harmonic Oscillator Model of Electron Delocalization) index defined. In 2010, the HOMED index was parameterized for compounds with C, N and O atoms. For parametrization, the reference molecules of similar delocalizations of nand π-electrons were employed. In this paper, the HOMED index was extended to compounds containing the CP, CS, NN, NP, PP, NO, NS, PO, and PS bonds. For geometrical optimization of all reference molecules and of all investigated heterocompounds, the same quantum–chemical method {B3LYP/6-311+G(d,p)} was used to eliminate errors of the HOMED estimation. For some tautomeric systems, the Gn methods were also employed to confirm tautomeric preferences. The extended HOMED index was applied to five-membered heterocycles, simple furan and thiophene, and their N and P derivatives as well as for tautomeric pyrrole and phosphole and their N and P derivatives. The effects of additional heteroatom(s) in the ring on the HOMED values for furan are parallel to those for thiophene. For pyrroles, aromaticity dictates the tautomeric preferences. An additional N atom in the ring only slightly affects the HOMED values for the favored and well delocalized NH tautomers. Significant changes take place for their rare CH forms. When intramolecular proton-transfer is considered for phosphole and its P derivatives, the PH tautomers seem to be favored only for 1,2,3-triphosphole/1,2,5-triphosphole and for 1,2,3,5-tetraphosphole. For other phospholes, the CH forms have smaller Gibbs energies than the PH isomers. For phosphazoles, the labile proton in the favored form is linked to the N atom. The PH forms have smaller HOMED indices than the NH tautomers but higher than the CH ones.


Introduction
For a quantitative description of aromaticity in homo-and heteroaromatic compounds, Kruszewski andKrygowski proposed in 1972-1974 the Harmonic Oscillator Model of Aromaticity (HOMA) index [1][2][3][4].This criterion of aromatic character is directly based on molecular structures (bond lengths) of π-electron systems.Twenty years later, taking into account the concept of resonance coordinate proposed by Jug and Koester [5], Krygowski reformulated the original HOMA index [6].
Looking for a more appropriate structural descriptor of electron delocalization in any conjugated heterocompound, we returned to the original HOMA idea.In 2006, a new parametrization was signaled for HOMA [20].To distinguish the reformulated HOMA index that can be used for hydrocarbons and also for some heteroaromatic π-electron molecules, a name 'Harmonic Oscillator Model of Electron Delocalization' and an abbreviation 'HOMED' was suggested for the new parameter [20].However, it can also be named HOMA.In 2010, the HOMED index was parametrized for molecules with C, N and O atoms [21].This index describes π-electron delocalization in conjugated cyclic and acyclic homoand heterosystems [21][22][23][24].It properly measures electron delocalization in aromatic systems, medium π-π and n-π conjugated compounds, as well as weak σ-π hyperconjugated molecules.In 2012, the HOHMED (Harmonic Oscillator Model of Heterocyclic Electron Delocalization) descriptor was applied to 16 heterocompounds [25].For parametrization, authors took the idea of the original HOMA [1][2][3][4] and HOMED indices [21] and the statistical experimental bond lengths for C-C, C-X, X-Y, C=C, C=X, and X=Y.Both the HOMED and HOHMED procedures predict the values close to unity (>0.65) for furan, oxazole, pyrrole, imidazole, etc. [12,21,25].
In the literature, one can also find other structural as well as electronic, energetic, magnetic, and spectroscopic descriptors of aromaticity [7,8,12,13,.Note that geometry-based indices can be applied for entire conjugated systems and also for selected fragment(s) in these systems [12,21].Since the same resonance phenomenon operates in aromatic compounds and in other cyclic and acyclic homo-and hetero-conjugated systems [56][57][58][59][60], various parameters quantitatively describing π-electron delocalization in a series of weakly, moderately and strongly conjugated compounds can be mutually interrelated [34,[45][46][47][48][49][50][52][53][54][55].However, the situation is more complex, and a hypothesis of the multidimensional character of aromaticity has been formulated [7,12].Nevertheless, in some cases, a lack of general correlations between various descriptors of aromaticity needs some explanation.Recently [12,21], it has been signaled that values of various descriptors depend on the reference systems.Hence, it is necessary to analyze first the similarities and differences for descriptors based on the same property, and in the case of a lack of correlation between them, it is important to clarify the reason(s).
In this paper, the HOMED index was extended to π-electron heterocompounds possessing the CP, CS, NN, NP, PP, NO, NS, PO, and PS bonds.For investigations, the same quantum-chemical method {B3LYP/6-311+G(d,p) [61][62][63][64]} was employed as previously described for π-electron molecules with C, N and O atoms [21].The procedures and parametrizations of the HOMED descriptor were compared to those of the rHOMA index, and some similarities and differences were analyzed.This analysis gives the possibility to indicate the most important discrepancies between the rHOMA and HOMED values for compounds containing various numbers of CC, CX, XX, and XY bonds and to explain their reasons.It also makes it possible to formulate some limits of rHOMA application.The extended HOMED index was employed to five-membered heterocycles, simple furan, thiophene, and their N-and P-derivatives as well as tautomeric pyrrole, phosphole, their N-and P-derivatives, and phosphazoles.The effects of additional heteroatom(s) in the ring on the HOMED values were discussed.Tautomeric conversions for azoles, phospholes and phosphazoles were analyzed.For selected tautomeric systems, the Gn levels of theory [65] were also applied to confirm the stability of favored tautomers.The effects of planarity and aromaticity of the ring on tautomeric equilibria were considered.

HOMED Procedure
The procedures of structural descriptors, HOMED [21], original HOMA [1][2][3][4], and reformulated HOMA [6] are analogous.The HOMED and HOMA indices can be estimated according to Equation (1), where α are the normalization constants for CC, CX, XX, and XY bonds (X, Y = N, P, O, S, etc.); R o are the appropriate optimum CC, CX, XX, and XY bond lengths for completely delocalized compounds with only CC, only CX, only XX, and only XY bonds; R x are the experimental or calculated bond lengths in a studied system; and n is the number of CC, CX, XX, and/or XY bonds taken into account.In the original HOMA procedure, the normalization α constant for aromatic hydrocarbons is equal to 98.89 [1][2][3].This value is a consequence of the following condition: HOMA = 0 for the structure of benzene, proposed by Kekulé, with hypothetical pure C-C and pure C=C bonds.For aromatic benzene, HOMA = 1.The same factor of 98.89 was employed for conjugated heterocycles [4].Considering the harmonic oscillator method of optimization, the optimum bond lengths in HOMA could be calculated using Equation (2), where R s and R d are the single and double bond lengths, respectively, in selected reference compounds, and ω (assumed to be close to 2 for different types of bonds) is the ratio of stretching force constants for pure double and pure single bonds [1][2][3][4]6].

HOaMA or HOMED
In the reformulated HOMA procedure [6], the normalization α constants are not the same for the CC, CX, XX, and XY bonds.They can be calculated using Equation (3).This equation can also be applied in the HOMED procedure [21] for molecules possessing an even number of bonds (2i).However, equations ( 4) and ( 5) can be employed for molecules with an odd number of bonds (2i + 1), i.e., i double bonds and (i + 1) single bonds, and (i + 1) double bonds and i single bonds, respectively.Equations ( 3)- (5) fulfill the following condition: rHOMA = 1 and HOMED = 1 for Symmetry 2019, 11, 146 4 of 20 completely delocalized compounds with R x = R o , like for original HOMA.For molecules possessing the calculated single and double bond lengths close to R s and R d , respectively, rHOMA and HOMED indices are close to zero [6,21].Since differently delocalized molecules were taken for the reference bond lengths in both descriptors, zero in the rHOMA and HOMED scales does not correspond to the same molecule (see subchapters 3.2 and 3.3).

HOMED Parametrization
Based on parametrization of the original HOMA index [1][2][3][4], the simplest compounds of analogous delocalization of n-and π-electrons were chosen for R s , R d and R o in the HOMED procedure [21].Their formulae and selected bond lengths are given in Table 1.This choice leads to the normalization α constant (α 2i and α 2i + 1 ) values for the CC, CX, XX, and XY bonds between 40 and 100 (Table 2).The α values in HOMED are not larger than the factor of 98.89 proposed in the original HOMA procedure [1][2][3][4].applied in the HOMED procedure [21] for molecules possessing an even number of bonds (2i).However, equations ( 4) and ( 5) can be employed for molecules with an odd number of bonds (2i + 1), i.e., i double bonds and (i + 1) single bonds, and (i + 1) double bonds and i single bonds, respectively.Equations ( 3)-( 5) fulfill the following condition: rHOMA = 1 and HOMED = 1 for completely delocalized compounds with Rx = Ro, like for original HOMA.For molecules possessing the calculated single and double bond lengths close to Rs and Rd, respectively, rHOMA and HOMED indices are close to zero [6,21].Since differently delocalized molecules were taken for the reference bond lengths in both descriptors, zero in the rHOMA and HOMED scales does not correspond to the same molecule (see subchapters 3.2 and 3.3).

HOMED Parametrization
Based on parametrization of the original HOMA index [1][2][3][4], the simplest compounds of analogous delocalization of n-and π-electrons were chosen for Rs, Rd and Ro in the HOMED procedure [21].Their formulae and selected bond lengths are given in Table 1.This choice leads to the normalization  constant (2i and 2i + 1) values for the CC, CX, XX, and XY bonds between 40 and 100 (Table 2).The  values in HOMED are not larger than the factor of 98.89 proposed in the original HOMA procedure [1][2][3][4].
The reference compounds possessing different delocalizations of n-and π-electrons and applied for the Rs and Rd bond lengths in the rHOMA procedure [6] cause considerably greater variations in the normalization  constant values than those in original HOMA and HOMED.For example, the applied in the HOMED procedure [21] for molecules possessing an even number of bonds (2i).However, equations ( 4) and ( 5) can be employed for molecules with an odd number of bonds (2i + 1), i.e., i double bonds and (i + 1) single bonds, and (i + 1) double bonds and i single bonds, respectively.Equations ( 3)-( 5) fulfill the following condition: rHOMA = 1 and HOMED = 1 for completely delocalized compounds with Rx = Ro, like for original HOMA.For molecules possessing the calculated single and double bond lengths close to Rs and Rd, respectively, rHOMA and HOMED indices are close to zero [6,21].Since differently delocalized molecules were taken for the reference bond lengths in both descriptors, zero in the rHOMA and HOMED scales does not correspond to the same molecule (see subchapters 3.2 and 3.3).

HOMED Parametrization
Based on parametrization of the original HOMA index [1][2][3][4], the simplest compounds of analogous delocalization of n-and π-electrons were chosen for Rs, Rd and Ro in the HOMED procedure [21].Their formulae and selected bond lengths are given in Table 1.This choice leads to the normalization  constant (2i and 2i + 1) values for the CC, CX, XX, and XY bonds between 40 and 100 (Table 2).The  values in HOMED are not larger than the factor of 98.89 proposed in the original HOMA procedure [1][2][3][4].
The reference compounds possessing different delocalizations of n-and π-electrons and applied for the Rs and Rd bond lengths in the rHOMA procedure [6] cause considerably greater variations in the normalization  constant values than those in original HOMA and HOMED.For example, the applied in the HOMED procedure [21] for molecules possessing an even number of bonds (2i).However, equations ( 4) and ( 5) can be employed for molecules with an odd number of bonds (2i + 1), i.e., i double bonds and (i + 1) single bonds, and (i + 1) double bonds and i single bonds, respectively.Equations ( 3)-( 5) fulfill the following condition: rHOMA = 1 and HOMED = 1 for completely delocalized compounds with Rx = Ro, like for original HOMA.For molecules possessing the calculated single and double bond lengths close to Rs and Rd, respectively, rHOMA and HOMED indices are close to zero [6,21].Since differently delocalized molecules were taken for the reference bond lengths in both descriptors, zero in the rHOMA and HOMED scales does not correspond to the same molecule (see subchapters 3.2 and 3.3).

HOMED Parametrization
Based on parametrization of the original HOMA index [1][2][3][4], the simplest compounds of analogous delocalization of n-and π-electrons were chosen for Rs, Rd and Ro in the HOMED procedure [21].Their formulae and selected bond lengths are given in Table 1.This choice leads to the normalization  constant (2i and 2i + 1) values for the CC, CX, XX, and XY bonds between 40 and 100 (Table 2).The  values in HOMED are not larger than the factor of 98.89 proposed in the original HOMA procedure [1][2][3][4].
The reference compounds possessing different delocalizations of n-and π-electrons and applied for the Rs and Rd bond lengths in the rHOMA procedure [6] cause considerably greater variations in the normalization  constant values than those in original HOMA and HOMED.For example, the applied in the HOMED procedure [21] for molecules possessing an even number of bonds (2i).However, equations ( 4) and ( 5) can be employed for molecules with an odd number of bonds (2i + 1), i.e., i double bonds and (i + 1) single bonds, and (i + 1) double bonds and i single bonds, respectively.Equations ( 3)-( 5) fulfill the following condition: rHOMA = 1 and HOMED = 1 for completely delocalized compounds with Rx = Ro, like for original HOMA.For molecules possessing the calculated single and double bond lengths close to Rs and Rd, respectively, rHOMA and HOMED indices are close to zero [6,21].Since differently delocalized molecules were taken for the reference bond lengths in both descriptors, zero in the rHOMA and HOMED scales does not correspond to the same molecule (see subchapters 3.2 and 3.3).

HOMED Parametrization
Based on parametrization of the original HOMA index [1][2][3][4], the simplest compounds of analogous delocalization of n-and π-electrons were chosen for Rs, Rd and Ro in the HOMED procedure [21].Their formulae and selected bond lengths are given in Table 1.This choice leads to the normalization  constant (2i and 2i + 1) values for the CC, CX, XX, and XY bonds between 40 and 100 (Table 2).The  values in HOMED are not larger than the factor of 98.89 proposed in the original HOMA procedure [1][2][3][4].Taken from Ref. [21]. 2 This work. 3For planar structure taken from Ref. [66]. 4 For planar structure.
The reference compounds possessing different delocalizations of n-and π-electrons and applied for the Rs and Rd bond lengths in the rHOMA procedure [6] cause considerably greater variations in the normalization  constant values than those in original HOMA and HOMED.For example, the applied in the HOMED procedure [21] for molecules possessing an even number of bonds (2i).However, equations ( 4) and ( 5) can be employed for molecules with an odd number of bonds (2i + 1), i.e., i double bonds and (i + 1) single bonds, and (i + 1) double bonds and i single bonds, respectively.Equations ( 3)-( 5) fulfill the following condition: rHOMA = 1 and HOMED = 1 for completely delocalized compounds with Rx = Ro, like for original HOMA.For molecules possessing the calculated single and double bond lengths close to Rs and Rd, respectively, rHOMA and HOMED indices are close to zero [6,21].Since differently delocalized molecules were taken for the reference bond lengths in both descriptors, zero in the rHOMA and HOMED scales does not correspond to the same molecule (see subchapters 3.2 and 3.3).

HOMED Parametrization
Based on parametrization of the original HOMA index [1][2][3][4], the simplest compounds of analogous delocalization of n-and π-electrons were chosen for Rs, Rd and Ro in the HOMED procedure [21].Their formulae and selected bond lengths are given in Table 1.This choice leads to the normalization  constant (2i and 2i + 1) values for the CC, CX, XX, and XY bonds between 40 and 100 (Table 2).The  values in HOMED are not larger than the factor of 98.89 proposed in the original HOMA procedure [1][2][3][4].Taken from Ref. [21]. 2 This work. 3For planar structure taken from Ref. [66]. 4 For planar structure.
The reference compounds possessing different delocalizations of n-and π-electrons and applied for the Rs and Rd bond lengths in the rHOMA procedure [6] cause considerably greater variations in the normalization  constant values than those in original HOMA and HOMED.For example, the applied in the HOMED procedure [21] for molecules possessing an even number of bonds (2i).However, equations ( 4) and ( 5) can be employed for molecules with an odd number of bonds (2i + 1), i.e., i double bonds and (i + 1) single bonds, and (i + 1) double bonds and i single bonds, respectively.Equations ( 3)-( 5) fulfill the following condition: rHOMA = 1 and HOMED = 1 for completely delocalized compounds with Rx = Ro, like for original HOMA.For molecules possessing the calculated single and double bond lengths close to Rs and Rd, respectively, rHOMA and HOMED indices are close to zero [6,21].Since differently delocalized molecules were taken for the reference bond lengths in both descriptors, zero in the rHOMA and HOMED scales does not correspond to the same molecule (see subchapters 3.2 and 3.3).

HOMED Parametrization
Based on parametrization of the original HOMA index [1][2][3][4], the simplest compounds of analogous delocalization of n-and π-electrons were chosen for Rs, Rd and Ro in the HOMED procedure [21].Their formulae and selected bond lengths are given in Table 1.This choice leads to the normalization  constant (2i and 2i + 1) values for the CC, CX, XX, and XY bonds between 40 and 100 (Table 2).The  values in HOMED are not larger than the factor of 98.89 proposed in the original HOMA procedure [1][2][3][4].Taken from Ref. [21]. 2 This work. 3For planar structure taken from Ref. [66]. 4 For planar structure.
The reference compounds possessing different delocalizations of n-and π-electrons and applied for the Rs and Rd bond lengths in the rHOMA procedure [6] cause considerably greater variations in the normalization  constant values than those in original HOMA and HOMED.For example, the applied in the HOMED procedure [21] for molecules possessing an even number of bonds (2i).However, equations ( 4) and ( 5) can be employed for molecules with an odd number of bonds (2i + 1), i.e., i double bonds and (i + 1) single bonds, and (i + 1) double bonds and i single bonds, respectively.Equations ( 3)-( 5) fulfill the following condition: rHOMA = 1 and HOMED = 1 for completely delocalized compounds with Rx = Ro, like for original HOMA.For molecules possessing the calculated single and double bond lengths close to Rs and Rd, respectively, rHOMA and HOMED indices are close to zero [6,21].Since differently delocalized molecules were taken for the reference bond lengths in both descriptors, zero in the rHOMA and HOMED scales does not correspond to the same molecule (see subchapters 3.2 and 3.3).

HOMED Parametrization
Based on parametrization of the original HOMA index [1][2][3][4], the simplest compounds of analogous delocalization of n-and π-electrons were chosen for Rs, Rd and Ro in the HOMED procedure [21].Their formulae and selected bond lengths are given in Table 1.This choice leads to the normalization  constant (2i and 2i + 1) values for the CC, CX, XX, and XY bonds between 40 and 100 (Table 2).The  values in HOMED are not larger than the factor of 98.89 proposed in the original HOMA procedure [1][2][3][4].Taken from Ref. [21]. 2 This work. 3For planar structure taken from Ref. [66]. 4 For planar structure.
The reference compounds possessing different delocalizations of n-and π-electrons and applied for the Rs and Rd bond lengths in the rHOMA procedure [6] cause considerably greater variations in the normalization  constant values than those in original HOMA and HOMED.For example, the applied in the HOMED procedure [21] for molecules possessing an even number of bonds (2i).However, equations ( 4) and ( 5) can be employed for molecules with an odd number of bonds (2i + 1), i.e., i double bonds and (i + 1) single bonds, and (i + 1) double bonds and i single bonds, respectively.Equations ( 3)-( 5) fulfill the following condition: rHOMA = 1 and HOMED = 1 for completely delocalized compounds with Rx = Ro, like for original HOMA.For molecules possessing the calculated single and double bond lengths close to Rs and Rd, respectively, rHOMA and HOMED indices are close to zero [6,21].Since differently delocalized molecules were taken for the reference bond lengths in both descriptors, zero in the rHOMA and HOMED scales does not correspond to the same molecule (see subchapters 3.2 and 3.3).

HOMED Parametrization
Based on parametrization of the original HOMA index [1][2][3][4], the simplest compounds of analogous delocalization of n-and π-electrons were chosen for Rs, Rd and Ro in the HOMED procedure [21].Their formulae and selected bond lengths are given in Table 1.This choice leads to the normalization  constant (2i and 2i + 1) values for the CC, CX, XX, and XY bonds between 40 and 100 (Table 2).The  values in HOMED are not larger than the factor of 98.89 proposed in the original HOMA procedure [1][2][3][4].Taken from Ref. [21]. 2 This work. 3For planar structure taken from Ref. [66]. 4 For planar structure.
The reference compounds possessing different delocalizations of n-and π-electrons and applied for the Rs and Rd bond lengths in the rHOMA procedure [6] cause considerably greater variations in the normalization  constant values than those in original HOMA and HOMED.For example, the applied in the HOMED procedure [21] for molecules possessing an even number of bonds (2i).However, equations ( 4) and ( 5) can be employed for molecules with an odd number of bonds (2i + 1), i.e., i double bonds and (i + 1) single bonds, and (i + 1) double bonds and i single bonds, respectively.Equations ( 3)-( 5) fulfill the following condition: rHOMA = 1 and HOMED = 1 for completely delocalized compounds with Rx = Ro, like for original HOMA.For molecules possessing the calculated single and double bond lengths close to Rs and Rd, respectively, rHOMA and HOMED indices are close to zero [6,21].Since differently delocalized molecules were taken for the reference bond lengths in both descriptors, zero in the rHOMA and HOMED scales does not correspond to the same molecule (see subchapters 3.2 and 3.3).

HOMED Parametrization
Based on parametrization of the original HOMA index [1][2][3][4], the simplest compounds of analogous delocalization of n-and π-electrons were chosen for Rs, Rd and Ro in the HOMED procedure [21].Their formulae and selected bond lengths are given in Table 1.This choice leads to the normalization  constant (2i and 2i + 1) values for the CC, CX, XX, and XY bonds between 40 and 100 (Table 2).The  values in HOMED are not larger than the factor of 98.89 proposed in the original HOMA procedure [1][2][3][4].Taken from Ref. [21]. 2 This work. 3For planar structure taken from Ref. [66]. 4 For planar structure.
The reference compounds possessing different delocalizations of n-and π-electrons and applied for the Rs and Rd bond lengths in the rHOMA procedure [6] cause considerably greater variations in the normalization  constant values than those in original HOMA and HOMED.For example, the applied in the HOMED procedure [21] for molecules possessing an even number of bonds (2i).However, equations ( 4) and ( 5) can be employed for molecules with an odd number of bonds (2i + 1), i.e., i double bonds and (i + 1) single bonds, and (i + 1) double bonds and i single bonds, respectively.Equations ( 3)-( 5) fulfill the following condition: rHOMA = 1 and HOMED = 1 for completely delocalized compounds with Rx = Ro, like for original HOMA.For molecules possessing the calculated single and double bond lengths close to Rs and Rd, respectively, rHOMA and HOMED indices are close to zero [6,21].Since differently delocalized molecules were taken for the reference bond lengths in both descriptors, zero in the rHOMA and HOMED scales does not correspond to the same molecule (see subchapters 3.2 and 3.3).

HOMED Parametrization
Based on parametrization of the original HOMA index [1][2][3][4], the simplest compounds of analogous delocalization of n-and π-electrons were chosen for Rs, Rd and Ro in the HOMED procedure [21].Their formulae and selected bond lengths are given in Table 1.This choice leads to the normalization  constant (2i and 2i + 1) values for the CC, CX, XX, and XY bonds between 40 and 100 (Table 2).The  values in HOMED are not larger than the factor of 98.89 proposed in the original HOMA procedure [1][2][3][4].Taken from Ref. [21]. 2 This work. 3For planar structure taken from Ref. [66]. 4 For planar structure.
The reference compounds possessing different delocalizations of n-and π-electrons and applied for the Rs and Rd bond lengths in the rHOMA procedure [6] cause considerably greater variations in the normalization  constant values than those in original HOMA and HOMED.For example, the applied in the HOMED procedure [21] for molecules possessing an even number of bonds (2i).However, equations ( 4) and ( 5) can be employed for molecules with an odd number of bonds (2i + 1), i.e., i double bonds and (i + 1) single bonds, and (i + 1) double bonds and i single bonds, respectively.Equations ( 3)-( 5) fulfill the following condition: rHOMA = 1 and HOMED = 1 for completely delocalized compounds with Rx = Ro, like for original HOMA.For molecules possessing the calculated single and double bond lengths close to Rs and Rd, respectively, rHOMA and HOMED indices are close to zero [6,21].Since differently delocalized molecules were taken for the reference bond lengths in both descriptors, zero in the rHOMA and HOMED scales does not correspond to the same molecule (see subchapters 3.2 and 3.3).

HOMED Parametrization
Based on parametrization of the original HOMA index [1][2][3][4], the simplest compounds of analogous delocalization of n-and π-electrons were chosen for Rs, Rd and Ro in the HOMED procedure [21].Their formulae and selected bond lengths are given in Table 1.This choice leads to the normalization  constant (2i and 2i + 1) values for the CC, CX, XX, and XY bonds between 40 and 100 (Table 2).The  values in HOMED are not larger than the factor of 98.89 proposed in the original HOMA procedure [1][2][3][4].Taken from Ref. [21]. 2 This work. 3For planar structure taken from Ref. [66]. 4 For planar structure.
The reference compounds possessing different delocalizations of n-and π-electrons and applied for the Rs and Rd bond lengths in the rHOMA procedure [6] cause considerably greater variations in the normalization  constant values than those in original HOMA and HOMED.For example, the applied in the HOMED procedure [21] for molecules possessing an even number of bonds (2i).However, equations ( 4) and ( 5) can be employed for molecules with an odd number of bonds (2i + 1), i.e., i double bonds and (i + 1) single bonds, and (i + 1) double bonds and i single bonds, respectively.Equations ( 3)-( 5) fulfill the following condition: rHOMA = 1 and HOMED = 1 for completely delocalized compounds with Rx = Ro, like for original HOMA.For molecules possessing the calculated single and double bond lengths close to Rs and Rd, respectively, rHOMA and HOMED indices are close to zero [6,21].Since differently delocalized molecules were taken for the reference bond lengths in both descriptors, zero in the rHOMA and HOMED scales does not correspond to the same molecule (see subchapters 3.2 and 3.3).

HOMED Parametrization
Based on parametrization of the original HOMA index [1][2][3][4], the simplest compounds of analogous delocalization of n-and π-electrons were chosen for Rs, Rd and Ro in the HOMED procedure [21].Their formulae and selected bond lengths are given in Table 1.This choice leads to the normalization  constant (2i and 2i + 1) values for the CC, CX, XX, and XY bonds between 40 and 100 (Table 2).The  values in HOMED are not larger than the factor of 98.89 proposed in the original HOMA procedure [1][2][3][4].
The reference compounds possessing different delocalizations of n-and π-electrons and applied for the Rs and Rd bond lengths in the rHOMA procedure [6] cause considerably greater variations in the normalization  constant values than those in original HOMA and HOMED.For example, the applied in the HOMED procedure [21] for molecules possessing an even number of bonds (2i).However, equations ( 4) and ( 5) can be employed for molecules with an odd number of bonds (2i + 1), i.e., i double bonds and (i + 1) single bonds, and (i + 1) double bonds and i single bonds, respectively.Equations ( 3)-( 5) fulfill the following condition: rHOMA = 1 and HOMED = 1 for completely delocalized compounds with Rx = Ro, like for original HOMA.For molecules possessing the calculated single and double bond lengths close to Rs and Rd, respectively, rHOMA and HOMED indices are close to zero [6,21].Since differently delocalized molecules were taken for the reference bond lengths in both descriptors, zero in the rHOMA and HOMED scales does not correspond to the same molecule (see subchapters 3.2 and 3.3).

HOMED Parametrization
Based on parametrization of the original HOMA index [1][2][3][4], the simplest compounds of analogous delocalization of n-and π-electrons were chosen for Rs, Rd and Ro in the HOMED procedure [21].Their formulae and selected bond lengths are given in Table 1.This choice leads to the normalization  constant (2i and 2i + 1) values for the CC, CX, XX, and XY bonds between 40 and 100 (Table 2).The  values in HOMED are not larger than the factor of 98.89 proposed in the original HOMA procedure [1][2][3][4].
The reference compounds possessing different delocalizations of n-and π-electrons and applied for the Rs and Rd bond lengths in the rHOMA procedure [6] cause considerably greater variations in the normalization  constant values than those in original HOMA and HOMED.For example, the applied in the HOMED procedure [21] for molecules possessing an even number of bonds (2i).However, equations ( 4) and ( 5) can be employed for molecules with an odd number of bonds (2i + 1), i.e., i double bonds and (i + 1) single bonds, and (i + 1) double bonds and i single bonds, respectively.Equations ( 3)-( 5) fulfill the following condition: rHOMA = 1 and HOMED = 1 for completely delocalized compounds with Rx = Ro, like for original HOMA.For molecules possessing the calculated single and double bond lengths close to Rs and Rd, respectively, rHOMA and HOMED indices are close to zero [6,21].Since differently delocalized molecules were taken for the reference bond lengths in both descriptors, zero in the rHOMA and HOMED scales does not correspond to the same molecule (see subchapters 3.2 and 3.3).

HOMED Parametrization
Based on parametrization of the original HOMA index [1][2][3][4], the simplest compounds of analogous delocalization of n-and π-electrons were chosen for Rs, Rd and Ro in the HOMED procedure [21].Their formulae and selected bond lengths are given in Table 1.This choice leads to the normalization  constant (2i and 2i + 1) values for the CC, CX, XX, and XY bonds between 40 and 100 (Table 2).The  values in HOMED are not larger than the factor of 98.89 proposed in the original HOMA procedure [1][2][3][4].
The reference compounds possessing different delocalizations of n-and π-electrons and applied for the Rs and Rd bond lengths in the rHOMA procedure [6] cause considerably greater variations in the normalization  constant values than those in original HOMA and HOMED.For example, the applied in the HOMED procedure [21] for molecules possessing an even number of bonds (2i).However, equations ( 4) and ( 5) can be employed for molecules with an odd number of bonds (2i + 1), i.e., i double bonds and (i + 1) single bonds, and (i + 1) double bonds and i single bonds, respectively.Equations ( 3)-( 5) fulfill the following condition: rHOMA = 1 and HOMED = 1 for completely delocalized compounds with Rx = Ro, like for original HOMA.For molecules possessing the calculated single and double bond lengths close to Rs and Rd, respectively, rHOMA and HOMED indices are close to zero [6,21].Since differently delocalized molecules were taken for the reference bond lengths in both descriptors, zero in the rHOMA and HOMED scales does not correspond to the same molecule (see subchapters 3.2 and 3.3).

HOMED Parametrization
Based on parametrization of the original HOMA index [1][2][3][4], the simplest compounds of analogous delocalization of n-and π-electrons were chosen for Rs, Rd and Ro in the HOMED procedure [21].Their formulae and selected bond lengths are given in Table 1.This choice leads to the normalization  constant (2i and 2i + 1) values for the CC, CX, XX, and XY bonds between 40 and 100 (Table 2).The  values in HOMED are not larger than the factor of 98.89 proposed in the original HOMA procedure [1][2][3][4].
The reference compounds possessing different delocalizations of n-and π-electrons and applied for the Rs and Rd bond lengths in the rHOMA procedure [6] cause considerably greater variations in the normalization  constant values than those in original HOMA and HOMED.For example, the applied in the HOMED procedure [21] for molecules possessing an even number of bonds (2i).However, equations ( 4) and ( 5) can be employed for molecules with an odd number of bonds (2i + 1), i.e., i double bonds and (i + 1) single bonds, and (i + 1) double bonds and i single bonds, respectively.Equations ( 3)-( 5) fulfill the following condition: rHOMA = 1 and HOMED = 1 for completely delocalized compounds with Rx = Ro, like for original HOMA.For molecules possessing the calculated single and double bond lengths close to Rs and Rd, respectively, rHOMA and HOMED indices are close to zero [6,21].Since differently delocalized molecules were taken for the reference bond lengths in both descriptors, zero in the rHOMA and HOMED scales does not correspond to the same molecule (see subchapters 3.2 and 3.3).

HOMED Parametrization
Based on parametrization of the original HOMA index [1][2][3][4], the simplest compounds of analogous delocalization of n-and π-electrons were chosen for Rs, Rd and Ro in the HOMED procedure [21].Their formulae and selected bond lengths are given in Table 1.This choice leads to the normalization  constant (2i and 2i + 1) values for the CC, CX, XX, and XY bonds between 40 and 100 (Table 2).The  values in HOMED are not larger than the factor of 98.89 proposed in the original HOMA procedure [1][2][3][4].
The reference compounds possessing different delocalizations of n-and π-electrons and applied for the Rs and Rd bond lengths in the rHOMA procedure [6] cause considerably greater variations in the normalization  constant values than those in original HOMA and HOMED.For example, the applied in the HOMED procedure [21] for molecules possessing an even number of bonds (2i).However, equations ( 4) and ( 5) can be employed for molecules with an odd number of bonds (2i + 1), i.e., i double bonds and (i + 1) single bonds, and (i + 1) double bonds and i single bonds, respectively.Equations ( 3)-( 5) fulfill the following condition: rHOMA = 1 and HOMED = 1 for completely delocalized compounds with Rx = Ro, like for original HOMA.For molecules possessing the calculated single and double bond lengths close to Rs and Rd, respectively, rHOMA and HOMED indices are close to zero [6,21].Since differently delocalized molecules were taken for the reference bond lengths in both descriptors, zero in the rHOMA and HOMED scales does not correspond to the same molecule (see subchapters 3.2 and 3.3).

HOMED Parametrization
Based on parametrization of the original HOMA index [1][2][3][4], the simplest compounds of analogous delocalization of n-and π-electrons were chosen for Rs, Rd and Ro in the HOMED procedure [21].Their formulae and selected bond lengths are given in Table 1.This choice leads to the normalization  constant (2i and 2i + 1) values for the CC, CX, XX, and XY bonds between 40 and 100 (Table 2).The  values in HOMED are not larger than the factor of 98.89 proposed in the original HOMA procedure [1][2][3][4].
The reference compounds possessing different delocalizations of n-and π-electrons and applied for the Rs and Rd bond lengths in the rHOMA procedure [6] cause considerably greater variations in the normalization  constant values than those in original HOMA and HOMED.For example, the applied in the HOMED procedure [21] for molecules possessing an even number of bonds (2i).However, equations ( 4) and ( 5) can be employed for molecules with an odd number of bonds (2i + 1), i.e., i double bonds and (i + 1) single bonds, and (i + 1) double bonds and i single bonds, respectively.Equations ( 3)-( 5) fulfill the following condition: rHOMA = 1 and HOMED = 1 for completely delocalized compounds with Rx = Ro, like for original HOMA.For molecules possessing the calculated single and double bond lengths close to Rs and Rd, respectively, rHOMA and HOMED indices are close to zero [6,21].Since differently delocalized molecules were taken for the reference bond lengths in both descriptors, zero in the rHOMA and HOMED scales does not correspond to the same molecule (see subchapters 3.2 and 3.3).

HOMED Parametrization
Based on parametrization of the original HOMA index [1][2][3][4], the simplest compounds of analogous delocalization of n-and π-electrons were chosen for Rs, Rd and Ro in the HOMED procedure [21].Their formulae and selected bond lengths are given in Table 1.This choice leads to the normalization  constant (2i and 2i + 1) values for the CC, CX, XX, and XY bonds between 40 and 100 (Table 2).The  values in HOMED are not larger than the factor of 98.89 proposed in the original HOMA procedure [1][2][3][4].
The reference compounds possessing different delocalizations of n-and π-electrons and applied for the Rs and Rd bond lengths in the rHOMA procedure [6] cause considerably greater variations in the normalization  constant values than those in original HOMA and HOMED.For example, the applied in the HOMED procedure [21] for molecules possessing an even number of bonds (2i).However, equations ( 4) and ( 5) can be employed for molecules with an odd number of bonds (2i + 1), i.e., i double bonds and (i + 1) single bonds, and (i + 1) double bonds and i single bonds, respectively.Equations ( 3)-( 5) fulfill the following condition: rHOMA = 1 and HOMED = 1 for completely delocalized compounds with Rx = Ro, like for original HOMA.For molecules possessing the calculated single and double bond lengths close to Rs and Rd, respectively, rHOMA and HOMED indices are close to zero [6,21].Since differently delocalized molecules were taken for the reference bond lengths in both descriptors, zero in the rHOMA and HOMED scales does not correspond to the same molecule (see subchapters 3.2 and 3.3).

HOMED Parametrization
Based on parametrization of the original HOMA index [1][2][3][4], the simplest compounds of analogous delocalization of n-and π-electrons were chosen for Rs, Rd and Ro in the HOMED procedure [21].Their formulae and selected bond lengths are given in Table 1.This choice leads to the normalization  constant (2i and 2i + 1) values for the CC, CX, XX, and XY bonds between 40 and 100 (Table 2).The  values in HOMED are not larger than the factor of 98.89 proposed in the original HOMA procedure [1][2][3][4].
The reference compounds possessing different delocalizations of n-and π-electrons and applied for the Rs and Rd bond lengths in the rHOMA procedure [6] cause considerably greater variations in the normalization  constant values than those in original HOMA and HOMED.For example, the applied in the HOMED procedure [21] for molecules possessing an even number of bonds (2i).However, equations ( 4) and ( 5) can be employed for molecules with an odd number of bonds (2i + 1), i.e., i double bonds and (i + 1) single bonds, and (i + 1) double bonds and i single bonds, respectively.Equations ( 3)-( 5) fulfill the following condition: rHOMA = 1 and HOMED = 1 for completely delocalized compounds with Rx = Ro, like for original HOMA.For molecules possessing the calculated single and double bond lengths close to Rs and Rd, respectively, rHOMA and HOMED indices are close to zero [6,21].Since differently delocalized molecules were taken for the reference bond lengths in both descriptors, zero in the rHOMA and HOMED scales does not correspond to the same molecule (see subchapters 3.2 and 3.3).

HOMED Parametrization
Based on parametrization of the original HOMA index [1][2][3][4], the simplest compounds of analogous delocalization of n-and π-electrons were chosen for Rs, Rd and Ro in the HOMED procedure [21].Their formulae and selected bond lengths are given in Table 1.This choice leads to the normalization  constant (2i and 2i + 1) values for the CC, CX, XX, and XY bonds between 40 and 100 (Table 2).The  values in HOMED are not larger than the factor of 98.89 proposed in the original HOMA procedure [1][2][3][4].
The reference compounds possessing different delocalizations of n-and π-electrons and applied for the Rs and Rd bond lengths in the rHOMA procedure [6] cause considerably greater variations in the normalization  constant values than those in original HOMA and HOMED.For example, the 2.0496 2 1 Taken from Ref. [21]. 2 This work. 3For planar structure taken from Ref. [66]. 4 For planar structure.
The use of monomeric formic acid (HO−CH=O, moderately delocalized by n-π conjugation) for the reference C−O and C=O bond lengths also gives an exceptionally high α(CO) value (157.38).Its application together with the extreme α(CC) value for O-containing heterocompounds can only explain why the rHOMA values are close to zero [7,8,12,13,21].In some cases, the rHOMA indices for compounds containing C and O atoms are even negative [21,23].The reference molecules chosen for CP (CH 3 -P=CH 2 ), NN {(CH 3 ) 2 N-N=C(CH 3 ) 2 and CH 3 -N=N-CH 3 }, and NO (CH 3 -O-N=O) bonds are not analogously delocalized.Consequently, their α values in the rHOMA procedure differ from 57.21 (for NO) to 130.33 (for NN).
Quite a different situation occurs for the reference C−N and C=N bond lengths.In this case, the application of the simplest molecules (CH 3 -NH 2 and CH 2 =NH) in the rHOMA procedure, analogously as in the original HOMA and HOMED indices, explains both the low α(CN) value (93.52) and the large rHOMA indices (>0.7) for cyclic azines and azoles [7,8,12,13,21].The same is true for the reference C−S and C=S bond lengths for which the simplest molecules (CH 3 -S-CH 3 and CH 2 =S) used in rHOMA are not very different from those (CH 3 -SH and CH 2 =S) employed in HOMED.Consequently, the α(CS) value is also low (94.09) and the rHOMA indices are also large (>0.7) for thiophene and thioazoles [7,8,12,21].For less delocalized N and/or S compounds, rHOMA is close to zero [21].
The other discrepancy in the HOMA and HOMED parametrization results from the use of different methods for R o calculation.This parameter can be estimated from Equation (2) in the original and reformulated HOMA procedure, whereas in the HOMED procedure, R o corresponds to the intrinsic bond length of the fully delocalized system optimized by using the same theoretical method {B3LYP/6-311+G(d,p)} as that applied for the reference molecules (Table 1).The differences between the reference R o bond lengths slightly affect the HOMA and HOMED values for heterocycles.
The use of the experimental (X-ray) and computational {DFT(B3LYP)/6-311+G(d,p)} reference R s and R d bond lengths in the original HOMA and HOMED procedures, respectively, is an additional Symmetry 2019, 11, 146 6 of 20 difference between the two descriptors.It should be mentioned here that the application of the HOMA procedure to experimental and computational structures of investigated heterocompounds does not lead to the same HOMA values.For some derivatives (e.g., furan and oxazoles), differences are even larger than 0.2 units [12].For this reason, it is important to use the same method for determination of the reference (R s , R d , and R o ) and running (R x ) bond lengths [21,68].The use of the same method for all bond lengths gives the possibility to minimize errors in the structural-descriptor estimation (<0.05 for HOMED) [21].
Symmetry 2019, 11, x FOR PEER REVIEW 6 of 21 {B3LYP/6-311+G(d,p)} as that applied for the reference molecules (Table 1).The differences between the reference Ro bond lengths slightly affect the HOMA and HOMED values for heterocycles.The use of the experimental (X-ray) and computational {DFT(B3LYP)/6-311+G(d,p)} reference Rs and Rd bond lengths in the original HOMA and HOMED procedures, respectively, is an additional difference between the two descriptors.It should be mentioned here that the application of the HOMA procedure to experimental and computational structures of investigated heterocompounds does not lead to the same HOMA values.For some derivatives (e.g., furan and oxazoles), differences are even larger than 0.2 units [12].For this reason, it is important to use the same method for determination of the reference (Rs, Rd, and Ro) and running (Rx) bond lengths [21,68].The use of the same method for all bond lengths gives the possibility to minimize errors in the structural-descriptor estimation (< 0.05 for HOMED) [21].

Differences in the HOMED and rHOMA Scales
Since the reference molecules used for Rs and Rd in the rHOMA procedure are differently delocalized, the normalization  constants vary from ca. 60 to 260 [6][7][8][12][13][14].Consequently, the rHOMA and HOMED scales are not analogous (Figure 1) [21][22][23].For completely delocalized πelectron compounds, e.g., benzene and its heteroanalogs, rHOMA = 1 and HOMED = 1 [6][7][8][12][13][14]20,21,69,70].However, rHOMA and HOMED have different values for Kekulé benzene (cyclohexatriene) with pure CC and C=C bonds.When we assume that the CC and C=C bond lengths in cyclohexatriene are equal to those in 1,3-butadiene (reference compound chosen in the rHOMA procedure), rHOMA = 0 [6][7][8], while HOMED = 0.694 [21,22].When we take the CC and C=C bond lengths in Kekulé benzene to be equal to those in ethane and ethene, respectively (reference compounds in the HOMA and HOMED procedures), HOMED = 0, while rHOMA = −2.023[21,22].Lack of analogy in the HOMED and rHOMA scales also takes place for heterocompounds possessing CP, CO, or NN bonds for which the selected reference molecules are also differently delocalized, and zero in the HOMED and rHOMA scales does not correspond to the same heterocompound.The two geometry-based indices do not have the identical values for the reference molecules employed in the rHOMA procedure for CO and CP bonds.For example, rHOMA = 0 and HOMED = 0.569 for monomeric formic acid, HO-CHO (n-π conjugated) is used for CO bonds.For H2CP-CH3 (σ-π hiperconjugated) applied for CP bonds, rHOMA = 0 and HOMED = 0.059.There are also some differences for triazine HNN-NH2 (n-π conjugated), for which rHOMA = 0.699 and HOMED = 0.720, indicating that zero in both scales for π-electron systems with NN bonds can refer to various molecules.
For compounds containing CN bonds, the situation is quite different.For hypothetical 1,3,5triazine with pure CN and C=N bonds (aza derivative of Kekulé benzene), rHOMA = 0 and HOMED = 0.In this case, the same reference compounds, methylamine and methylimine, were used for CN and C=N bonds, respectively.Very similar values of both descriptors can also be observed for compounds containing CS bonds.In this case, almost analogously delocalized molecules were used for the reference Rs and Rd bonds, CH3-S-CH3 and CH2S in rHOMA, and CH3-SH and CH2S in HOMED.

From Linear Tendencies to Scatter Plots Between rHOMA and HOMED Indices
All discrepancies between the rHOMA and HOMED parametrizations discussed above affect the rHOMA vs HOMED relations for simple π-electron compounds containing heteroatoms.They Lack of analogy in the HOMED and rHOMA scales also takes place for heterocompounds possessing CP, CO, or NN bonds for which the selected reference molecules are also differently delocalized, and zero in the HOMED and rHOMA scales does not correspond to the same heterocompound.The two geometry-based indices do not have the identical values for the reference molecules employed in the rHOMA procedure for CO and CP bonds.For example, rHOMA = 0 and HOMED = 0.569 for monomeric formic acid, HO-CH=O (n-π conjugated) is used for CO bonds.For H 2 C=P-CH 3 (σ-π hiperconjugated) applied for CP bonds, rHOMA = 0 and HOMED = 0.059.There are also some differences for triazine HN=N-NH 2 (n-π conjugated), for which rHOMA = 0.699 and HOMED = 0.720, indicating that zero in both scales for π-electron systems with NN bonds can refer to various molecules.
For compounds containing CN bonds, the situation is quite different.For hypothetical 1,3,5-triazine with pure C−N and C=N bonds (aza derivative of Kekulé benzene), rHOMA = 0 and HOMED = 0.In this case, the same reference compounds, methylamine and methylimine, were used for C−N and C=N bonds, respectively.Very similar values of both descriptors can also be observed for compounds containing CS bonds.In this case, almost analogously delocalized molecules were used for the reference R s and R d bonds, CH 3 -S-CH 3 and CH 2 =S in rHOMA, and CH 3 -SH and CH 2 =S in HOMED.

From Linear Tendencies to Scatter Plots Between rHOMA and HOMED Indices
All discrepancies between the rHOMA and HOMED parametrizations discussed above affect the rHOMA vs. HOMED relations for simple π-electron compounds containing heteroatoms.They cannot be ignored even for long series, because the relation between the two parameters strongly depends not only on the type of atoms present in heterocompounds but also on their number.The rHOMA and Symmetry 2019, 11, 146 7 of 20 HOMED indices are almost parallel for compounds containing only one type of bond, e.g., only CC, only CX, or only XY. Figure 2a shows linear tendencies between the rHOMA and HOMED indices estimated for three series of cyclic and acyclic π-electron compounds considered in our previous article [21] such as hydrocarbons containing only CC bonds, heterocompounds containing only CN bonds, and derivatives containing only CO bonds.The slopes of regression lines are different for each series because the normalization α constants for CC, CN and CO bonds are not the same in the rHOMA and HOMED procedures (Table 2).Generally, the slopes of regression lines are close to 1, 2 and 3 for the series 'CN', 'CO' and 'CC', similar to the α(rHOMA)/α(HOMED) ratio for the CN, CO and CC bonds.Some deviations of points from the linear relationships in Figure 2a  On the basis of these linear tendencies found for compounds possessing only one type of bond and regarding their different slopes, we can conclude that the character of the rHOMA vs HOMED plots for conjugated heterocompounds containing various CC, CX, and XY bonds can be different.They can be almost linear, parabolic, or scatter.For example, a scatter plot exists for a series of cyclic and acyclic conjugated heterosystems containing a different number of CC and CN bonds (Figure 2b).The points are placed between the lines corresponding to the two series of compounds containing only CN and only CC bonds.The great difference between the slopes of the 'CC' and 'CN' lines leads to a large scatter plot for a series of conjugated systems containing CC and CN bonds.For three other series of compounds ('CC & CO', 'CN & CO', and 'CC & CN & CO'), their data points are not so scattered as those for the 'CC & CN' series, and in some cases the rHOMA vs HOMED relations can be considered as linear.Nevertheless, the data points for series 'CC & CO', 'CN & CO', and 'CC & CN & CO', are placed between the lines found for the corresponding series of compounds possessing only one type of bond.The rHOMA and HOMED values for all heterocompounds considered in Figure 2 were taken from Ref. [21].
Even for isomers of the same compound possessing the same number of CC, CX, and XY bonds, the relation between the rHOMA and HOMED indices cannot be linear.For example, bicyclic purine contains five C atoms and four N atoms.It also possesses one labile proton that can be moved between nine conjugated sites according to amine-enamine and amine-imine tautomeric conversions.Nine tautomers are thus possible for bicyclic purine.All of them contain the same number of CC and CN bonds in the imidazole and pyrimidine rings and also in the entire purine molecule.However, the rHOMA vs HOMED plots, found for the imidazole and pyrimidine rings and bicyclic purine, are analogous to that presented in Figure 2b for the series 'CC & CN' [71].On the basis of these linear tendencies found for compounds possessing only one type of bond and regarding their different slopes, we can conclude that the character of the rHOMA vs. HOMED plots for conjugated heterocompounds containing various CC, CX, and XY bonds can be different.They can be almost linear, parabolic, or scatter.For example, a scatter plot exists for a series of cyclic and acyclic conjugated heterosystems containing a different number of CC and CN bonds (Figure 2b).The points are placed between the lines corresponding to the two series of compounds containing only CN and only CC bonds.The great difference between the slopes of the 'CC' and 'CN' lines leads to a large scatter plot for a series of conjugated systems containing CC and CN bonds.For three other series of compounds ('CC & CO', 'CN & CO', and 'CC & CN & CO'), their data points are not so scattered as those for the 'CC & CN' series, and in some cases the rHOMA vs. HOMED relations can be considered as linear.Nevertheless, the data points for series 'CC & CO', 'CN & CO', and 'CC & CN & CO', are placed between the lines found for the corresponding series of compounds possessing only one type of bond.The rHOMA and HOMED values for all heterocompounds considered in Figure 2 were taken from Ref. [21].
Even for isomers of the same compound possessing the same number of CC, CX, and XY bonds, the relation between the rHOMA and HOMED indices cannot be linear.For example, bicyclic purine contains five C atoms and four N atoms.It also possesses one labile proton that can be moved between nine conjugated sites according to amine-enamine and amine-imine tautomeric conversions.Nine tautomers are thus possible for bicyclic purine.All of them contain the same number of CC and CN bonds in the imidazole and pyrimidine rings and also in the entire purine molecule.However, the rHOMA vs. HOMED plots, found for the imidazole and pyrimidine rings and bicyclic purine, are analogous to that presented in Figure 2b for the series 'CC & CN' [71].Pyrimidine bases (uracil, cytosine, and isocytosine) possess the same number of CC and CN bonds in the pyrimidine ring and two different groups (-NH 2 /=NH and/or -OH/=O) at 2-and 4-positions.They contain two labile protons that can be moved between five conjugated sites according to different types of prototropic conversions (amide-iminol, amine-imine, enamine-imine, and/or keto-enol).All possible rearrangements lead to nine prototropic tautomers, which are analogous for pyrimidine bases [23,72,73].For tautomers with the -OH and =NH substituents, additional isomerism takes place, rotational isomerism for -OH and geometrical isomerism for =NH.Hence, 18 isomers can be distinguished for uracil [72] and 21 isomers are possible for cytosine [73] and isocytosine [23].Only for aromatic isomers with structural descriptors larger than 0.7 (Figure 3), the rHOMA vs. HOMED relations seem to be linear for the six-membered ring (rHOMA6 vs. HOMED6) and for the complete molecule including exo substituents (rHOMA8 vs. HOMED8).For medium (π-π and n-π cross conjugated) and weak (π-π, n-π, and σ-π cross conjugated) delocalized isomers, variations of the rHOMA values are larger than those of the HOMED ones.This leads to non-linear rHOMA vs. HOMED relations for isocytosine, cytosine and uracil isomers.The relations seem to have a parabolic character.The points for pyrimidine bases are placed between two lines corresponding to compounds containing only CC and only CN bonds.Pyrimidine bases (uracil, cytosine, and isocytosine) possess the same number of CC and CN bonds in the pyrimidine ring and two different groups (NH2/NH and/or OH/O) at 2-and 4positions.They contain two labile protons that can be moved between five conjugated sites according to different types of prototropic conversions (amide-iminol, amine-imine, enamine-imine, and/or keto-enol).All possible rearrangements lead to nine prototropic tautomers, which are analogous for pyrimidine bases [23,72,73].For tautomers with the OH and NH substituents, additional isomerism takes place, rotational isomerism for OH and geometrical isomerism for NH.Hence, 18 isomers can be distinguished for uracil [72] and 21 isomers are possible for cytosine [73] and isocytosine [23].Only for aromatic isomers with structural descriptors larger than 0.7 (Figure 3), the rHOMA vs HOMED relations seem to be linear for the six-membered ring (rHOMA6 vs HOMED6) and for the complete molecule including exo substituents (rHOMA8 vs HOMED8).For medium (ππ and n-π cross conjugated) and weak (π-π, n-π, and σ-π cross conjugated) delocalized isomers, variations of the rHOMA values are larger than those of the HOMED ones.This leads to non-linear rHOMA vs HOMED relations for isocytosine, cytosine and uracil isomers.The relations seem to have a parabolic character.The points for pyrimidine bases are placed between two lines corresponding to compounds containing only CC and only CN bonds.Figures 2 and 3 confirm the most important discrepancies between the two structural indices observed for heterocompounds.The choice of the reference compounds with different delocalizations of n-and π-electrons for the CC and C=C, CX and C=X, XX and X=X, XY and X=Y bonds can be considered as the main reason for differences between the rHOMA and HOMED values, and consequently, non-linear rHOMA vs HOMED relations found for heterocompounds [21][22][23][71][72][73].Although the HOMA idea is based on the theory of resonance and equalization of bond lengths in completely delocalized π-electron systems, the rHOMA parametrization seems to be inappropriate for compounds containing heteroatoms.The rHOMA parametrization can be considered as one of the principal reasons of artificial relations between rHOMA and other descriptors of aromaticity.The rHOMA index can be applied for non-tautomeric compounds containing only CC, only CX, only XX, or only XY bonds, e.g., hydrocarbons [69,70].
The choice of the experimental Rs and Rd bond lengths for the rHOMA parametrization, and the rHOMA application to the theoretical structures of π-electron systems is a secondary factor that causes significant differences between rHOMAs even for the same compound optimized at different levels of theory.These discrepancies have been confirmed by Andrzejak et al. [68], who compared the rHOMA indices for hydrocarbons studied by using different theoretical methods for optimization of their geometries.In the case of the HOMED procedure, when the same method is applied for the calculation of Rs, Rd, Ro in the reference compounds and Rx in the investigated π-electron systems, computational errors cancel out.Consequently, differences between the HOMED values are lower than 0.05 [21,74].
Figures 2 and 3 confirm the most important discrepancies between the two structural indices observed for heterocompounds.The choice of the reference compounds with different delocalizations of n-and π-electrons for the C-C and C=C, C-X and C=X, X-X and X=X, X-Y and X=Y bonds can be considered as the main reason for differences between the rHOMA and HOMED values, and consequently, non-linear rHOMA vs. HOMED relations found for heterocompounds [21][22][23][71][72][73].Although the HOMA idea is based on the theory of resonance and equalization of bond lengths in completely delocalized π-electron systems, the rHOMA parametrization seems to be inappropriate for compounds containing heteroatoms.The rHOMA parametrization can be considered as one of the principal reasons of artificial relations between rHOMA and other descriptors of aromaticity.The rHOMA index can be applied for non-tautomeric compounds containing only CC, only CX, only XX, or only XY bonds, e.g., hydrocarbons [69,70].
The choice of the experimental R s and R d bond lengths for the rHOMA parametrization, and the rHOMA application to the theoretical structures of π-electron systems is a secondary factor that causes significant differences between rHOMAs even for the same compound optimized at different levels of theory.These discrepancies have been confirmed by Andrzejak et al. [68], who compared the rHOMA indices for hydrocarbons studied by using different theoretical methods for optimization of their geometries.In the case of the HOMED procedure, when the same method is applied for the calculation of R s , R d , R o in the reference compounds and R x in the investigated π-electron systems, Symmetry 2019, 11, 146 9 of 20 computational errors cancel out.Consequently, differences between the HOMED values are lower than 0.05 [21,74].

HOMED Application to Furan, Thiophene and Their N-and P-Derivatives
Furan and thiophene belong to the family of simple five-membered heterocycles that display an aromatic character [4,21,25,26,[36][37][38][39][40][41][42][43][44][55][56][57].Similar to benzene, they possess six delocalized electrons, four π-electrons and two n-electrons in the ring, and fulfill the H űckel rule (4n + 2).Both compounds have planar structures.Hence, their geometric descriptors should not be very different than those of benzene.The presence of additional N (or P) atoms in the ring can cause similar effects.Lone pairs of electrons at additional N (or P) atoms are outside of the ring and do not participate in delocalization of the six ring electrons.
However, the rHOMA values reported for furan are between 0.0 and 0.3 and those for thiophene are between 0.7 and 0.9 [12].Additional N atoms increase the rHOMA indices for furan derivatives (even to 0.7 for 2,5-diazafuran [75]), while they vary very little for thiophene.These variations do not describe well the N-effects on electron delocalization in furan and thiophene derivatives observed for other descriptors of aromaticity.For example, linear relationships (Figure 4) exist between the magnetic descriptors and nucleus-independent chemical shift {NICS(1)} indices, and between the energetic parameters and aromatic stabilization energy (ASE) indices, both being estimated earlier for N-derivatives of furan and thiophene [75].The correlation coefficients (0.961 and 0.977, respectively) are close to unity.The slopes of these lines (0.792 and 0.721, respectively) are also not very different from unity.

HOMED Application to Furan, Thiophene and Their N-and P-Derivatives
Furan and thiophene belong to the family of simple five-membered heterocycles that display an aromatic character [4,21,25,26,[36][37][38][39][40][41][42][43][44][55][56][57].Similar to benzene, they possess six delocalized electrons, four π-electrons and two n-electrons in the ring, and fulfill the Hűckel rule (4n + 2).Both compounds have planar structures.Hence, their geometric descriptors should not be very different than those of benzene.The presence of additional N (or P) atoms in the ring can cause similar effects.Lone pairs of electrons at additional N (or P) atoms are outside of the ring and do not participate in delocalization of the six ring electrons.
However, the rHOMA values reported for furan are between 0.0 and 0.3 and those for thiophene are between 0.7 and 0.9 [12].Additional N atoms increase the rHOMA indices for furan derivatives (even to 0.7 for 2,5-diazafuran [75]), while they vary very little for thiophene.These variations do not describe well the N-effects on electron delocalization in furan and thiophene derivatives observed for other descriptors of aromaticity.For example, linear relationships (Figure 4) exist between the magnetic descriptors and nucleus-independent chemical shift {NICS(1)} indices, and between the energetic parameters and aromatic stabilization energy (ASE) indices, both being estimated earlier for N-derivatives of furan and thiophene [75].The correlation coefficients (0.961 and 0.977, respectively) are close to unity.The slopes of these lines (0.792 and 0.721, respectively) are also not very different from unity.The situation seems to be analogous for the HOMED indices.Preliminary HOMED estimations made in 2010 for furan (0.749) and oxazole (0.702) [21] showed evidently that the HOMED procedure, based on the original HOMA one, gives a values typical for five-membered aromatic systems.For this reason, the HOMED index has been extended here to π-electron compounds containing CS, NN, NO, NS, CP, PO, and PS bonds, and applied to isolated furan and thiophene and their derivatives substituted at 2-, 3-, 2,3-, 2,4-, 2,5-, 3,4-, 2,3,4-, 2,3,5-, and 2,3,4,5-positions by N or P atoms.The estimated HOMED indices for furans and thiophenes are listed in Table 3.Generally, some linear tendencies exist between the HOMED values estimated for furan and thiophene and their N and P derivatives (Figure 5).Effects of N and P atoms in furan are analogous to those in thiophene.The slopes of regression lines (1.003 and 0.282 for N and P effects, respectively) are only different.3.
The situation seems to be analogous for the HOMED indices.Preliminary HOMED estimations made in 2010 for furan (0.749) and oxazole (0.702) [21] showed evidently that the HOMED procedure, based on the original HOMA one, gives a values typical for five-membered aromatic systems.For this reason, the HOMED index has been extended here to π-electron compounds containing CS, NN, NO, NS, CP, PO, and PS bonds, and applied to isolated furan and thiophene and their derivatives substituted at 2-, 3-, 2,3-, 2,4-, 2,5-, 3,4-, 2,3,4-, 2,3,5-, and 2,3,4,5-positions by N or P atoms.The estimated HOMED indices for furans and thiophenes are listed in Table 3.Generally, some linear tendencies exist between the HOMED values estimated for furan and thiophene and their N and P derivatives (Figure 5).Effects of N and P atoms in furan are analogous to those in thiophene.The slopes of regression lines (1.003 and 0.282 for N and P effects, respectively) are only different.3. Derivatives of furan possess lower HOMED values than those of thiophene.This is in good agreement with the structural chemistry on heterocyclic compounds that furan is less aromatic than thiophene.Furan and thiophene also have lower HOMED values than benzene (1.000 by definition).Heteroatoms O and S are more electronegative than C atom, consequently, electron delocalization in benzene is stronger than that in furan and thiophene.Moreover, in benzene, six electrons are π-π conjugated, while in furan and thiophene four π-electrons are cross conjugated with n-electrons.3. Derivatives of furan possess lower HOMED values than those of thiophene.This is in good agreement with the structural chemistry on heterocyclic compounds that furan is less aromatic than thiophene.Furan and thiophene also have lower HOMED values than benzene (1.000 by definition).Heteroatoms O and S are more electronegative than C atom, consequently, electron delocalization in benzene is stronger than that in furan and thiophene.Moreover, in benzene, six electrons are π-π conjugated, while in furan and thiophene four π-electrons are cross conjugated with n-electrons.Derivatives of furan possess lower HOMED values than those agreement with the structural chemistry on heterocyclic compounds t thiophene.Furan and thiophene also have lower HOMED values than Heteroatoms O and S are more electronegative than C atom, conseque benzene is stronger than that in furan and thiophene.Moreover, in b conjugated, while in furan and thiophene four π-electrons are cross Derivatives of furan possess lower HOMED values than those of thiophene.This is in good agreement with the structural chemistry on heterocyclic compounds that furan is less aromatic than thiophene.Furan and thiophene also have lower HOMED values than benzene (1.000 by definition).Heteroatoms O and S are more electronegative than C atom, consequently, electron delocalization in benzene is stronger than that in furan and thiophene.Moreover, in benzene, six electrons are π-π conjugated, while in furan and thiophene four π-electrons are cross conjugated with n-electrons.Derivatives of furan possess lower HOMED values than those of thiophene.This is in good agreement with the structural chemistry on heterocyclic compounds that furan is less aromatic than thiophene.Furan and thiophene also have lower HOMED values than benzene (1.000 by definition).Heteroatoms O and S are more electronegative than C atom, consequently, electron delocalization in benzene is stronger than that in furan and thiophene.Moreover, in benzene, six electrons are π-π conjugated, while in furan and thiophene four π-electrons are cross conjugated with n-electrons.Derivatives of furan possess lower HOMED values than those of thiophene.This is in good agreement with the structural chemistry on heterocyclic compounds that furan is less aromatic than thiophene.Furan and thiophene also have lower HOMED values than benzene (1.000 by definition).Heteroatoms O and S are more electronegative than C atom, consequently, electron delocalization in benzene is stronger than that in furan and thiophene.Moreover, in benzene, six electrons are π-π conjugated, while in furan and thiophene four π-electrons are cross conjugated with n-electrons.Derivatives of furan possess lower HOMED values than those of thiophene.This is in good agreement with the structural chemistry on heterocyclic compounds that furan is less aromatic than thiophene.Furan and thiophene also have lower HOMED values than benzene (1.000 by definition).Heteroatoms O and S are more electronegative than C atom, consequently, electron delocalization in benzene is stronger than that in furan and thiophene.Moreover, in benzene, six electrons are π-π conjugated, while in furan and thiophene four π-electrons are cross conjugated with n-electrons.Derivatives of furan possess lower HOMED values than those of thiophene.This is in good agreement with the structural chemistry on heterocyclic compounds that furan is less aromatic than thiophene.Furan and thiophene also have lower HOMED values than benzene (1.000 by definition).Heteroatoms O and S are more electronegative than C atom, consequently, electron delocalization in benzene is stronger than that in furan and thiophene.Moreover, in benzene, six electrons are π-π conjugated, while in furan and thiophene four π-electrons are cross conjugated with n-electrons.Derivatives of furan possess lower HOMED values than those of thiophene.This is in good agreement with the structural chemistry on heterocyclic compounds that furan is less aromatic than thiophene.Furan and thiophene also have lower HOMED values than benzene (1.000 by definition).Heteroatoms O and S are more electronegative than C atom, consequently, electron delocalization in benzene is stronger than that in furan and thiophene.Moreover, in benzene, six electrons are π-π conjugated, while in furan and thiophene four π-electrons are cross conjugated with n-electrons.Derivatives of furan possess lower HOMED values than those of thiophene.This is in good agreement with the structural chemistry on heterocyclic compounds that furan is less aromatic than thiophene.Furan and thiophene also have lower HOMED values than benzene (1.000 by definition).Heteroatoms O and S are more electronegative than C atom, consequently, electron delocalization in benzene is stronger than that in furan and thiophene.Moreover, in benzene, six electrons are π-π conjugated, while in furan and thiophene four π-electrons are cross conjugated with n-electrons.Derivatives of furan possess lower HOMED values than those of thiophene.This is in good agreement with the structural chemistry on heterocyclic compounds that furan is less aromatic than thiophene.Furan and thiophene also have lower HOMED values than benzene (1.000 by definition).Heteroatoms O and S are more electronegative than C atom, consequently, electron delocalization in benzene is stronger than that in furan and thiophene.Moreover, in benzene, six electrons are π-π conjugated, while in furan and thiophene four π-electrons are cross conjugated with n-electrons.Derivatives of furan possess lower HOMED values than those of thiophene.This is in good agreement with the structural chemistry on heterocyclic compounds that furan is less aromatic than thiophene.Furan and thiophene also have lower HOMED values than benzene (1.000 by definition).Heteroatoms O and S are more electronegative than C atom, consequently, electron delocalization in benzene is stronger than that in furan and thiophene.Moreover, in benzene, six electrons are π-π conjugated, while in furan and thiophene four π-electrons are cross conjugated with n-electrons.Derivatives of furan possess lower HOMED values than those of thiophene.This is in good agreement with the structural chemistry on heterocyclic compounds that furan is less aromatic than thiophene.Furan and thiophene also have lower HOMED values than benzene (1.000 by definition).Heteroatoms O and S are more electronegative than C atom, consequently, electron delocalization in benzene is stronger than that in furan and thiophene.Moreover, in benzene, six electrons are π-π conjugated, while in furan and thiophene four π-electrons are cross conjugated with n-electrons.Derivatives of furan possess lower HOMED values than those of thiophene.This is in good agreement with the structural chemistry on heterocyclic compounds that furan is less aromatic than thiophene.Furan and thiophene also have lower HOMED values than benzene (1.000 by definition).Heteroatoms O and S are more electronegative than C atom, consequently, electron delocalization in benzene is stronger than that in furan and thiophene.Moreover, in benzene, six electrons are π-π conjugated, while in furan and thiophene four π-electrons are cross conjugated with n-electrons.Derivatives of furan possess lower HOMED values than those of thiophene.This is in good agreement with the structural chemistry on heterocyclic compounds that furan is less aromatic than thiophene.Furan and thiophene also have lower HOMED values than benzene (1.000 by definition).Heteroatoms O and S are more electronegative than C atom, consequently, electron delocalization in benzene is stronger than that in furan and thiophene.Moreover, in benzene, six electrons are π-π conjugated, while in furan and thiophene four π-electrons are cross conjugated with n-electrons.Derivatives of furan possess lower HOMED values than those of thiophene.This is in good agreement with the structural chemistry on heterocyclic compounds that furan is less aromatic than thiophene.Furan and thiophene also have lower HOMED values than benzene (1.000 by definition).Heteroatoms O and S are more electronegative than C atom, consequently, electron delocalization in benzene is stronger than that in furan and thiophene.Moreover, in benzene, six electrons are π-π conjugated, while in furan and thiophene four π-electrons are cross conjugated with n-electrons.Derivatives of furan possess lower HOMED values than those of thiophene.This is in good agreement with the structural chemistry on heterocyclic compounds that furan is less aromatic than thiophene.Furan and thiophene also have lower HOMED values than benzene (1.000 by definition).Heteroatoms O and S are more electronegative than C atom, consequently, electron delocalization in benzene is stronger than that in furan and thiophene.Moreover, in benzene, six electrons are π-π conjugated, while in furan and thiophene four π-electrons are cross conjugated with n-electrons.Derivatives of furan possess lower HOMED values than those of thiophene.This is in good agreement with the structural chemistry on heterocyclic compounds that furan is less aromatic than thiophene.Furan and thiophene also have lower HOMED values than benzene (1.000 by definition).Heteroatoms O and S are more electronegative than C atom, consequently, electron delocalization in benzene is stronger than that in furan and thiophene.Moreover, in benzene, six electrons are π-π conjugated, while in furan and thiophene four π-electrons are cross conjugated with n-electrons.Derivatives of furan possess lower HOMED values than those of thiophene.This is in good agreement with the structural chemistry on heterocyclic compounds that furan is less aromatic than thiophene.Furan and thiophene also have lower HOMED values than benzene (1.000 by definition).Heteroatoms O and S are more electronegative than C atom, consequently, electron delocalization in benzene is stronger than that in furan and thiophene.Moreover, in benzene, six electrons are π-π conjugated, while in furan and thiophene four π-electrons are cross conjugated with n-electrons.Derivatives of furan possess lower HOMED values than those of thiophene.This is in good agreement with the structural chemistry on heterocyclic compounds that furan is less aromatic than thiophene.Furan and thiophene also have lower HOMED values than benzene (1.000 by definition).Heteroatoms O and S are more electronegative than C atom, consequently, electron delocalization in benzene is stronger than that in furan and thiophene.Moreover, in benzene, six electrons are π-π conjugated, while in furan and thiophene four π-electrons are cross conjugated with n-electrons.Derivatives of furan possess lower HOMED values than those of thiophene.This is in good agreement with the structural chemistry on heterocyclic compounds that furan is less aromatic than thiophene.Furan and thiophene also have lower HOMED values than benzene (1.000 by definition).Heteroatoms O and S are more electronegative than C atom, consequently, electron delocalization in benzene is stronger than that in furan and thiophene.Moreover, in benzene, six electrons are π-π conjugated, while in furan and thiophene four π-electrons are cross conjugated with n-electrons.Derivatives of furan possess lower HOMED values than those of thiophene.This is in good agreement with the structural chemistry on heterocyclic compounds that furan is less aromatic than thiophene.Furan and thiophene also have lower HOMED values than benzene (1.000 by definition).Heteroatoms O and S are more electronegative than C atom, consequently, electron delocalization in benzene is stronger than that in furan and thiophene.Moreover, in benzene, six electrons are π-π conjugated, while in furan and thiophene four π-electrons are cross conjugated with n-electrons.Derivatives of furan possess lower HOMED values than those of thiophene.This is in good agreement with the structural chemistry on heterocyclic compounds that furan is less aromatic than thiophene.Furan and thiophene also have lower HOMED values than benzene (1.000 by definition).Heteroatoms O and S are more electronegative than C atom, consequently, electron delocalization in benzene is stronger than that in furan and thiophene.Moreover, in benzene, six electrons are π-π conjugated, while in furan and thiophene four π-electrons are cross conjugated with n-electrons.Derivatives of furan possess lower HOMED values than those of thiophene.This is in good agreement with the structural chemistry on heterocyclic compounds that furan is less aromatic than thiophene.Furan and thiophene also have lower HOMED values than benzene (1.000 by definition).Heteroatoms O and S are more electronegative than C atom, consequently, electron delocalization in benzene is stronger than that in furan and thiophene.Moreover, in benzene, six electrons are π-π conjugated, while in furan and thiophene four π-electrons are cross conjugated with n-electrons.Derivatives of furan possess lower HOMED values than those of thiophene.This is in good agreement with the structural chemistry on heterocyclic compounds that furan is less aromatic than thiophene.Furan and thiophene also have lower HOMED values than benzene (1.000 by definition).Heteroatoms O and S are more electronegative than C atom, consequently, electron delocalization in benzene is stronger than that in furan and thiophene.Moreover, in benzene, six electrons are π-π conjugated, while in furan and thiophene four π-electrons are cross conjugated with n-electrons.Derivatives of furan possess lower HOMED values than those of thiophene.This is in good agreement with the structural chemistry on heterocyclic compounds that furan is less aromatic than thiophene.Furan and thiophene also have lower HOMED values than benzene (1.000 by definition).Heteroatoms O and S are more electronegative than C atom, consequently, electron delocalization in benzene is stronger than that in furan and thiophene.Moreover, in benzene, six electrons are π-π conjugated, while in furan and thiophene four π-electrons are cross conjugated with n-electrons.Derivatives of furan possess lower HOMED values than those of thiophene.This is in good agreement with the structural chemistry on heterocyclic compounds that furan is less aromatic than thiophene.Furan and thiophene also have lower HOMED values than benzene (1.000 by definition).Heteroatoms O and S are more electronegative than C atom, consequently, electron delocalization in benzene is stronger than that in furan and thiophene.Moreover, in benzene, six electrons are π-π conjugated, while in furan and thiophene four π-electrons are cross conjugated with n-electrons.Derivatives of furan possess lower HOMED values than those of thiophene.This is in good agreement with the structural chemistry on heterocyclic compounds that furan is less aromatic than thiophene.Furan and thiophene also have lower HOMED values than benzene (1.000 by definition).Heteroatoms O and S are more electronegative than C atom, consequently, electron delocalization in benzene is stronger than that in furan and thiophene.Moreover, in benzene, six electrons are π-π conjugated, while in furan and thiophene four π-electrons are cross conjugated with n-electrons.Derivatives of furan possess lower HOMED values than those of thiophene.This is in good agreement with the structural chemistry on heterocyclic compounds that furan is less aromatic than thiophene.Furan and thiophene also have lower HOMED values than benzene (1.000 by definition).Heteroatoms O and S are more electronegative than C atom, consequently, electron delocalization in benzene is stronger than that in furan and thiophene.Moreover, in benzene, six electrons are π-π conjugated, while in furan and thiophene four π-electrons are cross conjugated with n-electrons.Derivatives of furan possess lower HOMED values than those of thiophene.This is in good agreement with the structural chemistry on heterocyclic compounds that furan is less aromatic than thiophene.Furan and thiophene also have lower HOMED values than benzene (1.000 by definition).Heteroatoms O and S are more electronegative than C atom, consequently, electron delocalization in benzene is stronger than that in furan and thiophene.Moreover, in benzene, six electrons are π-π conjugated, while in furan and thiophene four π-electrons are cross conjugated with n-electrons.Derivatives of furan possess lower HOMED values than those of thiophene.This is in good agreement with the structural chemistry on heterocyclic compounds that furan is less aromatic than thiophene.Furan and thiophene also have lower HOMED values than benzene (1.000 by definition).Heteroatoms O and S are more electronegative than C atom, consequently, electron delocalization in benzene is stronger than that in furan and thiophene.Moreover, in benzene, six electrons are π-π conjugated, while in furan and thiophene four π-electrons are cross conjugated with n-electrons.Derivatives of furan possess lower HOMED values than those of thiophene.This is in good agreement with the structural chemistry on heterocyclic compounds that furan is less aromatic than thiophene.Furan and thiophene also have lower HOMED values than benzene (1.000 by definition).Heteroatoms O and S are more electronegative than C atom, consequently, electron delocalization in benzene is stronger than that in furan and thiophene.Moreover, in benzene, six electrons are π-π conjugated, while in furan and thiophene four π-electrons are cross conjugated with n-electrons.Derivatives of furan possess lower HOMED values than those of thiophene.This is in good agreement with the structural chemistry on heterocyclic compounds that furan is less aromatic than thiophene.Furan and thiophene also have lower HOMED values than benzene (1.000 by definition).Heteroatoms O and S are more electronegative than C atom, consequently, electron delocalization in benzene is stronger than that in furan and thiophene.Moreover, in benzene, six electrons are π-π conjugated, while in furan and thiophene four π-electrons are cross conjugated with n-electrons.Derivatives of furan possess lower HOMED values than those of thiophene.This is in good agreement with the structural chemistry on heterocyclic compounds that furan is less aromatic than thiophene.Furan and thiophene also have lower HOMED values than benzene (1.000 by definition).Heteroatoms O and S are more electronegative than C atom, consequently, electron delocalization in benzene is stronger than that in furan and thiophene.Moreover, in benzene, six electrons are π-π conjugated, while in furan and thiophene four π-electrons are cross conjugated with n-electrons.Derivatives of furan possess lower HOMED values than those of thiophene.This is in good agreement with the structural chemistry on heterocyclic compounds that furan is less aromatic than thiophene.Furan and thiophene also have lower HOMED values than benzene (1.000 by definition).Heteroatoms O and S are more electronegative than C atom, consequently, electron delocalization in benzene is stronger than that in furan and thiophene.Moreover, in benzene, six electrons are π-π conjugated, while in furan and thiophene four π-electrons are cross conjugated with n-electrons.Derivatives of furan possess lower HOMED values than those of thiophene.This is in good agreement with the structural chemistry on heterocyclic compounds that furan is less aromatic than thiophene.Furan and thiophene also have lower HOMED values than benzene (1.000 by definition).Heteroatoms O and S are more electronegative than C atom, consequently, electron delocalization in benzene is stronger than that in furan and thiophene.Moreover, in benzene, six electrons are π-π conjugated, while in furan and thiophene four π-electrons are cross conjugated with n-electrons.Derivatives of furan possess lower HOMED values than those of thiophene.This is in good agreement with the structural chemistry on heterocyclic compounds that furan is less aromatic than thiophene.Furan and thiophene also have lower HOMED values than benzene (1.000 by definition).Heteroatoms O and S are more electronegative than C atom, consequently, electron delocalization in benzene is stronger than that in furan and thiophene.Moreover, in benzene, six electrons are π-π conjugated, while in furan and thiophene four π-electrons are cross conjugated with n-electrons.Derivatives of furan possess lower HOMED values than those of thiophene.This is in good agreement with the structural chemistry on heterocyclic compounds that furan is less aromatic than thiophene.Furan and thiophene also have lower HOMED values than benzene (1.000 by definition).Heteroatoms O and S are more electronegative than C atom, consequently, electron delocalization in benzene is stronger than that in furan and thiophene.Moreover, in benzene, six electrons are π-π conjugated, while in furan and thiophene four π-electrons are cross conjugated with n-electrons.Derivatives of furan possess lower HOMED values than those of thiophene.This is in good agreement with the structural chemistry on heterocyclic compounds that furan is less aromatic than thiophene.Furan and thiophene also have lower HOMED values than benzene (1.000 by definition).Heteroatoms O and S are more electronegative than C atom, consequently, electron delocalization in benzene is stronger than that in furan and thiophene.Moreover, in benzene, six electrons are π-π conjugated, while in furan and thiophene four π-electrons are cross conjugated with n-electrons.Derivatives of furan possess lower HOMED values than those of thiophene.This is in good agreement with the structural chemistry on heterocyclic compounds that furan is less aromatic than thiophene.Furan and thiophene also have lower HOMED values than benzene (1.000 by definition).Heteroatoms O and S are more electronegative than C atom, consequently, electron delocalization in benzene is stronger than that in furan and thiophene.Moreover, in benzene, six electrons are π-π conjugated, while in furan and thiophene four π-electrons are cross conjugated with n-electrons.According to the rules of resonance theory [56,57], we can propose many Lewis structures without separation of the charge for the resonance hybrid of benzene, while for those of furan and thiophene we can write many Lewis structures with separation of the charge.Contribution of the later structures in the resonance hybrid of five-membered heterocycles is usually smaller than that of the former ones, for which some localization of π-electrons takes place.

HOMED Application to Tautomeric Azoles and Phospholes
It is well recognized that pyrrole and its N derivatives exhibit prototropic tautomerism [9,21,22,57,[76][77][78][79][80][81][82][83][84][85][86][87][88].The labile proton can move from one site to the other according to 1,3-or 1,5-shift.Since proton transfer takes place between conjugated sites, it is always accompanied by a rearrangement of π-electrons.Hence, prototropic tautomers are never zwitterions.For azoles, five tautomers are possible with the labile proton at N (NH tautomers) or C atom (CH tautomers).Analogous proton transfer (Scheme 1) and analogous five tautomers (PH and CH) can be considered for phosphole and its P derivatives [89].Depending on the position of additional N atoms in azoles and P atom in phospholes, some tautomers have the same structures.For example, tautomers with the labile proton at C2 and C3 possess the same structures as those with the labile proton at C5 and C4, respectively, for pyrrole and phosphole.According to the rules of resonance theory [56,57], we can propose many Lewis structures without separation of the charge for the resonance hybrid of benzene, while for those of furan and thiophene we can write many Lewis structures with separation of the charge.Contribution of the later structures in the resonance hybrid of five-membered heterocycles is usually smaller than that of the former ones, for which some localization of π-electrons takes place.
The structural HOMED indices (Table 4) determined here for azoles confirm the general trend of their aromaticity.The HOMED values are exceptionally large (0.9) for the well delocalized and planar NH tautomers, whereas they are very low (0.4) for the rare CH forms with C-sp 3 atom.The CH tautomer of tautomeric 2,3,4-triazapyrrole/2,3,5-triazapyrrole is not stable at the DFT level.Its five-membered ring opens during structure optimization.The calculated relative Gibbs energies of the CH tautomers (G  15 kcal•mol −1 ) are in good agreement with earlier observations.The CH isomers can be neglected for neutral azoles which prefer the planar NH forms.Interestingly, the relative Gibbs energies estimated for the NH and CH tautomers of azoles are almost parallel to the structural HOMED indices.
Figure 6 shows that some simple linear relationship can be distinguished for azoles between the HOMED descriptors that measure the delocalization of n-and π-electrons and relative Gibbs energies (G) that refer to tautomeric conversions.Analogous linear tendencies between energetic and geometric parameters of NH and CH tautomers have been reported for other tautomeric heterocompounds possessing CC and CN bonds, e.g., 2-and 4-aminopyridines, 2-and 4-Scheme 1. Proton transfer for five prototropic tautomers of azoles and phospholes (X = N or P).
The NH tautomers of azoles have planar structures.
The structural HOMED indices (Table 4) determined here for azoles confirm the general trend of their aromaticity.The HOMED values are exceptionally large (≥0.9) for the well delocalized and planar NH tautomers, whereas they are very low (≤0.4) for the rare CH forms with C-sp 3 atom.The CH tautomer of tautomeric 2,3,4-triazapyrrole/2,3,5-triazapyrrole is not stable at the DFT level.Its five-membered ring opens during structure optimization.The calculated relative Gibbs energies of the CH tautomers (∆G ≥ 15 kcal•mol −1 ) are in good agreement with earlier observations.The CH isomers can be neglected for neutral azoles which prefer the planar NH forms.Interestingly, the relative Gibbs energies estimated for the NH and CH tautomers of azoles are almost parallel to the structural HOMED indices.
DFT calculations performed here for phosphole and its P derivatives confirm the literature data.Only P5H has a planar structure.Other phospholes contain the pyramidal PH group.Generally, the HOMED values estimated for the PH tautomers of phospholes (Table 5) are smaller than those for the NH tautomers of azoles (Table 4).A different situation occurs for the CH forms.Their HOMED values for phospholes are higher than those for azoles.Consequently, there is no simple relation between the structural descriptors of phospholes and azoles (Figure 7).  2 Estimated at the G2 level. 3Obtained at the G2MP2 level. 4Found at the G3 level.
DFT calculations performed here for phosphole and its P derivatives confirm the literature data.Only P5H has a planar structure.Other phospholes contain the pyramidal PH group.Generally, the HOMED values estimated for the PH tautomers of phospholes (Table 5) are smaller than those for the NH tautomers of azoles (Table 4).A different situation occurs for the CH forms.Their HOMED values for phospholes are higher than those for azoles.Consequently, there is no simple relation between the structural descriptors of phospholes and azoles (Figure 7).  2 Estimated at the G2 level. 3Obtained at the G2MP2 level. 4Found at the G3 level.
DFT calculations performed here for phosphole and its P derivatives confirm the literature data.Only P5H has a planar structure.Other phospholes contain the pyramidal PH group.Generally, the HOMED values estimated for the PH tautomers of phospholes (Table 5) are smaller than those for the NH tautomers of azoles (Table 4).A different situation occurs for the CH forms.Their HOMED values for phospholes are higher than those for azoles.Consequently, there is no simple relation between the structural descriptors of phospholes and azoles (Figure 7).  2 Estimated at the G2 level. 3Obtained at the G2MP2 level. 4Found at the G3 level.
DFT calculations performed here for phosphole and its P derivatives confirm the literature data.Only P5H has a planar structure.Other phospholes contain the pyramidal PH group.Generally, the HOMED values estimated for the PH tautomers of phospholes (Table 5) are smaller than those for the NH tautomers of azoles (Table 4).A different situation occurs for the CH forms.Their HOMED values for phospholes are higher than those for azoles.Consequently, there is no simple relation between the structural descriptors of phospholes and azoles (Figure 7).  2 Estimated at the G2 level. 3Obtained at the G2MP2 level. 4Found at the G3 level.
DFT calculations performed here for phosphole and its P derivatives confirm the literature data.Only P5H has a planar structure.Other phospholes contain the pyramidal PH group.Generally, the HOMED values estimated for the PH tautomers of phospholes (Table 5) are smaller than those for the NH tautomers of azoles (Table 4).A different situation occurs for the CH forms.Their HOMED values for phospholes are higher than those for azoles.Consequently, there is no simple relation between the structural descriptors of phospholes and azoles (Figure 7).  2 Estimated at the G2 level. 3Obtained at the G2MP2 level. 4Found at the G3 level.
DFT calculations performed here for phosphole and its P derivatives confirm the literature data.Only P5H has a planar structure.Other phospholes contain the pyramidal PH group.Generally, the HOMED values estimated for the PH tautomers of phospholes (Table 5) are smaller than those for the NH tautomers of azoles (Table 4).A different situation occurs for the CH forms.Their HOMED values for phospholes are higher than those for azoles.Consequently, there is no simple relation between the structural descriptors of phospholes and azoles (Figure 7).  2 Estimated at the G2 level. 3Obtained at the G2MP2 level. 4Found at the G3 level.
DFT calculations performed here for phosphole and its P derivatives confirm the literature data.Only P5H has a planar structure.Other phospholes contain the pyramidal PH group.Generally, the HOMED values estimated for the PH tautomers of phospholes (Table 5) are smaller than those for the NH tautomers of azoles (Table 4).A different situation occurs for the CH forms.Their HOMED values for phospholes are higher than those for azoles.Consequently, there is no simple relation between the structural descriptors of phospholes and azoles (Figure 7).  2 Estimated at the G2 level. 3Obtained at the G2MP2 level. 4Found at the G3 level.
DFT calculations performed here for phosphole and its P derivatives confirm the literature data.Only P5H has a planar structure.Other phospholes contain the pyramidal PH group.Generally, the HOMED values estimated for the PH tautomers of phospholes (Table 5) are smaller than those for the NH tautomers of azoles (Table 4).A different situation occurs for the CH forms.Their HOMED values for phospholes are higher than those for azoles.Consequently, there is no simple relation between the structural descriptors of phospholes and azoles (Figure 7).  2 Estimated at the G2 level. 3Obtained at the G2MP2 level. 4Found at the G3 level.
DFT calculations performed here for phosphole and its P derivatives confirm the literature data.Only P5H has a planar structure.Other phospholes contain the pyramidal PH group.Generally, the HOMED values estimated for the PH tautomers of phospholes (Table 5) are smaller than those for the NH tautomers of azoles (Table 4).A different situation occurs for the CH forms.Their HOMED values for phospholes are higher than those for azoles.Consequently, there is no simple relation between the structural descriptors of phospholes and azoles (Figure 7).  2 Estimated at the G2 level. 3Obtained at the G2MP2 level. 4Found at the G3 level.
DFT calculations performed here for phosphole and its P derivatives confirm the literature data.Only P5H has a planar structure.Other phospholes contain the pyramidal PH group.Generally, the HOMED values estimated for the PH tautomers of phospholes (Table 5) are smaller than those for the NH tautomers of azoles (Table 4).A different situation occurs for the CH forms.Their HOMED values for phospholes are higher than those for azoles.Consequently, there is no simple relation between the structural descriptors of phospholes and azoles (Figure 7).  2 Estimated at the G2 level. 3Obtained at the G2MP2 level. 4Found at the G3 level.
DFT calculations performed here for phosphole and its P derivatives confirm the literature data.Only P5H has a planar structure.Other phospholes contain the pyramidal PH group.Generally, the HOMED values estimated for the PH tautomers of phospholes (Table 5) are smaller than those for the NH tautomers of azoles (Table 4).A different situation occurs for the CH forms.Their HOMED values for phospholes are higher than those for azoles.Consequently, there is no simple relation between the structural descriptors of phospholes and azoles (Figure 7).  2 Estimated at the G2 level. 3Obtained at the G2MP2 level. 4Found at the G3 level.
DFT calculations performed here for phosphole and its P derivatives confirm the literature data.Only P5H has a planar structure.Other phospholes contain the pyramidal PH group.Generally, the HOMED values estimated for the PH tautomers of phospholes (Table 5) are smaller than those for the NH tautomers of azoles (Table 4).A different situation occurs for the CH forms.Their HOMED values for phospholes are higher than those for azoles.Consequently, there is no simple relation between the structural descriptors of phospholes and azoles (Figure 7).  2 Estimated at the G2 level. 3Obtained at the G2MP2 level. 4Found at the G3 level.
DFT calculations performed here for phosphole and its P derivatives confirm the literature data.Only P5H has a planar structure.Other phospholes contain the pyramidal PH group.Generally, the HOMED values estimated for the PH tautomers of phospholes (Table 5) are smaller than those for the NH tautomers of azoles (Table 4).A different situation occurs for the CH forms.Their HOMED values for phospholes are higher than those for azoles.Consequently, there is no simple relation between the structural descriptors of phospholes and azoles (Figure 7).  2 Estimated at the G2 level. 3Obtained at the G2MP2 level. 4Found at the G3 level.
DFT calculations performed here for phosphole and its P derivatives confirm the literature data.Only P5H has a planar structure.Other phospholes contain the pyramidal PH group.Generally, the HOMED values estimated for the PH tautomers of phospholes (Table 5) are smaller than those for the NH tautomers of azoles (Table 4).A different situation occurs for the CH forms.Their HOMED values for phospholes are higher than those for azoles.Consequently, there is no simple relation between the structural descriptors of phospholes and azoles (Figure 7).  2 Estimated at the G2 level. 3Obtained at the G2MP2 level. 4Found at the G3 level.
DFT calculations performed here for phosphole and its P derivatives confirm the literature data.Only P5H has a planar structure.Other phospholes contain the pyramidal PH group.Generally, the HOMED values estimated for the PH tautomers of phospholes (Table 5) are smaller than those for the NH tautomers of azoles (Table 4).A different situation occurs for the CH forms.Their HOMED values for phospholes are higher than those for azoles.Consequently, there is no simple relation between the structural descriptors of phospholes and azoles (Figure 7).  2 Estimated at the G2 level. 3Obtained at the G2MP2 level. 4Found at the G3 level.
DFT calculations performed here for phosphole and its P derivatives confirm the literature data.Only P5H has a planar structure.Other phospholes contain the pyramidal PH group.Generally, the HOMED values estimated for the PH tautomers of phospholes (Table 5) are smaller than those for the NH tautomers of azoles (Table 4).A different situation occurs for the CH forms.Their HOMED values for phospholes are higher than those for azoles.Consequently, there is no simple relation between the structural descriptors of phospholes and azoles (Figure 7).  2 Estimated at the G2 level. 3Obtained at the G2MP2 level. 4Found at the G3 level.
DFT calculations performed here for phosphole and its P derivatives confirm the literature data.Only P5H has a planar structure.Other phospholes contain the pyramidal PH group.Generally, the HOMED values estimated for the PH tautomers of phospholes (Table 5) are smaller than those for the NH tautomers of azoles (Table 4).A different situation occurs for the CH forms.Their HOMED values for phospholes are higher than those for azoles.Consequently, there is no simple relation between the structural descriptors of phospholes and azoles (Figure 7).  2 Estimated at the G2 level. 3Obtained at the G2MP2 level. 4Found at the G3 level.
DFT calculations performed here for phosphole and its P derivatives confirm the literature data.Only P5H has a planar structure.Other phospholes contain the pyramidal PH group.Generally, the HOMED values estimated for the PH tautomers of phospholes (Table 5) are smaller than those for the NH tautomers of azoles (Table 4).A different situation occurs for the CH forms.Their HOMED values for phospholes are higher than those for azoles.Consequently, there is no simple relation between the structural descriptors of phospholes and azoles (Figure 7).  2 Estimated at the G2 level. 3Obtained at the G2MP2 level. 4Found at the G3 level.
Note that the PH tautomers can be detected only in the tautomeric mixtures of 1,2,3triphosphole/1,2,5-triphosphole and 1,2,3,4-tetraphosphole/1,2,3,5-tetraphosphole.Their relative Phosphorus with H atom in five-membered phospholes is pyramidal, and the bond-angles sum about P is smaller than 360°.Hence, the electron lone pair of PH cannot interact with the ring π-electrons.Various types of aromaticity descriptors estimated for phosphole and for its P derivatives have smaller values than those for azoles [30,[98][99][100][101][102][103][104].However, planar mono-, di-, tri-, and tetraphospholes (transition states-first order saddle points) and also planar P5H display an aromatic character analogous to that of the corresponding azoles [99][100][101][102][103].Note that P substitution in phospholes reduces the pyramidal character of phosphorus, and P-substituted phosphole becomes aromatic [98,103,104,[112][113][114].DFT calculations performed here for phosphole and its P derivatives confirm the literature data.Only P5H has a planar structure.Other phospholes contain the pyramidal PH group.Generally, the HOMED values estimated for the PH tautomers of phospholes (Table 5) are smaller than those for the NH tautomers of azoles (Table 4).A different situation occurs for the CH forms.Their HOMED values for phospholes are higher than those for azoles.Consequently, there is no simple relation between the structural descriptors of phospholes and azoles (Figure 7).  2 Estimated at the G2 level. 3Obtained at the G2MP2 level. 4Found at the G3 level.
Note that the PH tautomers can be detected only in the tautomeric mixtures of 1,2,3-triphosphole/1,2,5-triphosphole and 1,2,3,4-tetraphosphole/1,2,3,5-tetraphosphole.Their relative Gibbs energies are close to zero.Additional calculations performed at the Gn levels prove that the CH tautomers are favored for other P derivatives (except for the derivative with five P atoms) (Table 5).The tautomeric preferences for phospholes are different from those for azoles.Delocalization of n-and π-electrons does not determine the tautomeric preferences for phospholes.

Figure 1 .
Figure 1.Comparison of the HOMED and rHOMA scales for π-electron delocalized compounds possessing CC, CN, CP, CO, CS, or NN bonds.

Figure 1 .
Figure 1.Comparison of the HOMED and rHOMA scales for π-electron delocalized compounds possessing CC, CN, CP, CO, CS, or NN bonds.

Figure 2 .
Figure 2. Linear tendencies between the rHOMA and HOMED indices estimated for three series of compounds containing only CC, only CN, and only CO bonds (a) and scatter plots between the rHOMA and HOMED indices estimated for four series of compounds containing only CC and CN, only CC and CO, only CN and CO, and only CC, CN, and CO bonds (b).The rHOMA and HOMED values are taken from Ref. [21].

Figure 2 .
Figure 2. Linear tendencies between the rHOMA and HOMED indices estimated for three series of compounds containing only CC, only CN, and only CO bonds (a) and scatter plots between the rHOMA and HOMED indices estimated for four series of compounds containing only CC and CN, only CC and CO, only CN and CO, and only CC, CN, and CO bonds (b).The rHOMA and HOMED values are taken from Ref. [21].

Figure 3 .
Figure 3. Plots between the rHOMA and HOMED indices estimated for isocytosine, cytosine and uracil: for the six-membered ring (a) and for the complete molecule of pyrimidine base (b).Data taken from Ref. [23].

Figure 4 .
Figure 4. Linear relations between magnetic NICS(1) indices (a) and between energetic ASE indices (b) estimated for N derivatives of furan and thiophene.NICS(1) and ASE are taken from Ref. [75].NICS(1)-ONn and ASE-ONn correspond to furan and its N-derivatives and NICS(1)-SNn and ASE-SNn refer to thiophene and its N derivatives.

Figure 4 .Figure 5 .Table 3 .Figure 5 .
Figure 4. Linear relations between magnetic NICS(1) indices (a) and between energetic ASE indices (b) estimated for N derivatives of furan and thiophene.NICS(1) and ASE are taken from Ref. [75].NICS(1)-ON n and ASE-ON n correspond to furan and its N-derivatives and NICS(1)-SN n and ASE-SN n refer to thiophene and its N derivatives.Symmetry 2019, 11, x FOR PEER REVIEW 10 of 21

Table 3 .Figure 5 .
Figure 5. Linear relations between the structural HOMED indices for N derivatives (a) and P derivatives (b) of furan and thiophene.HOMED-ONn and HOMED-OPn correspond to furans and HOMED-SNn and HOMED-SPn refer to thiophenes.Data are taken from Table3.

0.749 1 Figure 5 .
Figure 5. Linear relations between the structural HOMED indices for N derivatives (a) and P derivatives (b) of furan and thiophene.HOMED-ONn and HOMED-OPn correspond to furans and HOMED-SNn and HOMED-SPn refer to thiophenes.Data are taken from Table3.

Figure 5 . 3 .
Figure 5. Linear relations between the structural HOMED indices fo derivatives (b) of furan and thiophene.HOMED-ONn and HOMED-OP HOMED-SNn and HOMED-SPn refer to thiophenes.Data are taken from

Figure 5 .Table 3 .
Figure 5. Linear relations between the structural HOMED indices for N derivatives (a) and P derivatives (b) of furan and thiophene.HOMED-ONn and HOMED-OPn correspond to furans and HOMED-SNn and HOMED-SPn refer to thiophenes.Data are taken from 3.

Figure 5 .Table 3 .
Figure 5. Linear relations between the structural HOMED indices for N derivatives (a) and P derivatives (b) of furan and thiophene.HOMED-ONn and HOMED-OPn correspond to furans and HOMED-SNn and HOMED-SPn refer to thiophenes.Data are taken from Table3.

Figure 5 .Table 3 .
Figure 5. Linear relations between the structural HOMED indices for N derivatives (a) and P derivatives (b) of furan and thiophene.HOMED-ONn and HOMED-OPn correspond to furans and HOMED-SNn and HOMED-SPn refer to thiophenes.Data are taken from Table3.

Figure 5 .Table 3 .
Figure 5. Linear relations between the structural HOMED indices for N derivatives (a) and P derivatives (b) of furan and thiophene.HOMED-ONn and HOMED-OPn correspond to furans and HOMED-SNn and HOMED-SPn refer to thiophenes.Data are taken from Table3.

Figure 5 .Table 3 .
Figure 5. Linear relations between the structural HOMED indices for N derivatives (a) and P derivatives (b) of furan and thiophene.HOMED-ONn and HOMED-OPn correspond to furans and HOMED-SNn and HOMED-SPn refer to thiophenes.Data are taken from Table3.

Figure 5 .Table 3 .
Figure 5. Linear relations between the structural HOMED indices for N derivatives (a) and P derivatives (b) of furan and thiophene.HOMED-ONn and HOMED-OPn correspond to furans and HOMED-SNn and HOMED-SPn refer to thiophenes.Data are taken from Table3.

Figure 5 .Table 3 .
Figure 5. Linear relations between the structural HOMED indices for N derivatives (a) and P derivatives (b) of furan and thiophene.HOMED-ONn and HOMED-OPn correspond to furans and HOMED-SNn and HOMED-SPn refer to thiophenes.Data are taken from Table3.

Figure 5 .Table 3 .
Figure 5. Linear relations between the structural HOMED indices for N derivatives (a) and P derivatives (b) of furan and thiophene.HOMED-ONn and HOMED-OPn correspond to furans and HOMED-SNn and HOMED-SPn refer to thiophenes.Data are taken from Table3.

Figure 5 .Table 3 .
Figure 5. Linear relations between the structural HOMED indices for N derivatives (a) and P derivatives (b) of furan and thiophene.HOMED-ONn and HOMED-OPn correspond to furans and HOMED-SNn and HOMED-SPn refer to thiophenes.Data are taken from Table3.

Figure 5 .Table 3 .
Figure 5. Linear relations between the structural HOMED indices for N derivatives (a) and P derivatives (b) of furan and thiophene.HOMED-ONn and HOMED-OPn correspond to furans and HOMED-SNn and HOMED-SPn refer to thiophenes.Data are taken from Table3.

Figure 5 .Table 3 .
Figure 5. Linear relations between the structural HOMED indices for N derivatives (a) and P derivatives (b) of furan and thiophene.HOMED-ONn and HOMED-OPn correspond to furans and HOMED-SNn and HOMED-SPn refer to thiophenes.Data are taken from Table3.

Figure 5 .Table 3 .
Figure 5. Linear relations between the structural HOMED indices for N derivatives (a) and P derivatives (b) of furan and thiophene.HOMED-ONn and HOMED-OPn correspond to furans and HOMED-SNn and HOMED-SPn refer to thiophenes.Data are taken from Table3.

Figure 5 .Table 3 .
Figure 5. Linear relations between the structural HOMED indices for N derivatives (a) and P derivatives (b) of furan and thiophene.HOMED-ONn and HOMED-OPn correspond to furans and HOMED-SNn and HOMED-SPn refer to thiophenes.Data are taken from Table3.

Figure 5 .Table 3 .
Figure 5. Linear relations between the structural HOMED indices for N derivatives (a) and P derivatives (b) of furan and thiophene.HOMED-ONn and HOMED-OPn correspond to furans and HOMED-SNn and HOMED-SPn refer to thiophenes.Data are taken from Table3.

Figure 5 .Table 3 .
Figure 5. Linear relations between the structural HOMED indices for N derivatives (a) and P derivatives (b) of furan and thiophene.HOMED-ONn and HOMED-OPn correspond to furans and HOMED-SNn and HOMED-SPn refer to thiophenes.Data are taken from Table3.

Figure 5 .Table 3 .
Figure 5. Linear relations between the structural HOMED indices for N derivatives (a) and P derivatives (b) of furan and thiophene.HOMED-ONn and HOMED-OPn correspond to furans and HOMED-SNn and HOMED-SPn refer to thiophenes.Data are taken from Table3.

Figure 5 .Table 3 .
Figure 5. Linear relations between the structural HOMED indices for N derivatives (a) and P derivatives (b) of furan and thiophene.HOMED-ONn and HOMED-OPn correspond to furans and HOMED-SNn and HOMED-SPn refer to thiophenes.Data are taken from Table3.

Figure 5 .Table 3 .
Figure 5. Linear relations between the structural HOMED indices for N derivatives (a) and P derivatives (b) of furan and thiophene.HOMED-ONn and HOMED-OPn correspond to furans and HOMED-SNn and HOMED-SPn refer to thiophenes.Data are taken from Table3.

Figure 5 .Table 3 .
Figure 5. Linear relations between the structural HOMED indices for N derivatives (a) and P derivatives (b) of furan and thiophene.HOMED-ONn and HOMED-OPn correspond to furans and HOMED-SNn and HOMED-SPn refer to thiophenes.Data are taken from Table3.

Figure 5 .Table 3 .
Figure 5. Linear relations between the structural HOMED indices for N derivatives (a) and P derivatives (b) of furan and thiophene.HOMED-ONn and HOMED-OPn correspond to furans and HOMED-SNn and HOMED-SPn refer to thiophenes.Data are taken from Table3.

Figure 5 .Table 3 .
Figure 5. Linear relations between the structural HOMED indices for N derivatives (a) and P derivatives (b) of furan and thiophene.HOMED-ONn and HOMED-OPn correspond to furans and HOMED-SNn and HOMED-SPn refer to thiophenes.Data are taken from Table3.

Figure 5 .Table 3 .
Figure 5. Linear relations between the structural HOMED indices for N derivatives (a) and P derivatives (b) of furan and thiophene.HOMED-ONn and HOMED-OPn correspond to furans and HOMED-SNn and HOMED-SPn refer to thiophenes.Data are taken from Table3.

Figure 5 .Table 3 .
Figure 5. Linear relations between the structural HOMED indices for N derivatives (a) and P derivatives (b) of furan and thiophene.HOMED-ONn and HOMED-OPn correspond to furans and HOMED-SNn and HOMED-SPn refer to thiophenes.Data are taken from Table3.

Figure 5 .Table 3 .
Figure 5. Linear relations between the structural HOMED indices for N derivatives (a) and P derivatives (b) of furan and thiophene.HOMED-ONn and HOMED-OPn correspond to furans and HOMED-SNn and HOMED-SPn refer to thiophenes.Data are taken from Table3.

Figure 5 .Table 3 .
Figure 5. Linear relations between the structural HOMED indices for N derivatives (a) and P derivatives (b) of furan and thiophene.HOMED-ONn and HOMED-OPn correspond to furans and HOMED-SNn and HOMED-SPn refer to thiophenes.Data are taken from Table3.

Figure 5 .Table 3 .
Figure 5. Linear relations between the structural HOMED indices for N derivatives (a) and P derivatives (b) of furan and thiophene.HOMED-ONn and HOMED-OPn correspond to furans and HOMED-SNn and HOMED-SPn refer to thiophenes.Data are taken from Table3.

Figure 5 .Table 3 .
Figure 5. Linear relations between the structural HOMED indices for N derivatives (a) and P derivatives (b) of furan and thiophene.HOMED-ONn and HOMED-OPn correspond to furans and HOMED-SNn and HOMED-SPn refer to thiophenes.Data are taken from Table3.

Figure 5 .Table 3 .
Figure 5. Linear relations between the structural HOMED indices for N derivatives (a) and P derivatives (b) of furan and thiophene.HOMED-ONn and HOMED-OPn correspond to furans and HOMED-SNn and HOMED-SPn refer to thiophenes.Data are taken from Table3.

Figure 5 .Table 3 .
Figure 5. Linear relations between the structural HOMED indices for N derivatives (a) and P derivatives (b) of furan and thiophene.HOMED-ONn and HOMED-OPn correspond to furans and HOMED-SNn and HOMED-SPn refer to thiophenes.Data are taken from Table3.

Figure 5 .Table 3 .
Figure 5. Linear relations between the structural HOMED indices for N derivatives (a) and P derivatives (b) of furan and thiophene.HOMED-ONn and HOMED-OPn correspond to furans and HOMED-SNn and HOMED-SPn refer to thiophenes.Data are taken from Table3.

Figure 5 .Table 3 .
Figure 5. Linear relations between the structural HOMED indices for N derivatives (a) and P derivatives (b) of furan and thiophene.HOMED-ONn and HOMED-OPn correspond to furans and HOMED-SNn and HOMED-SPn refer to thiophenes.Data are taken from Table3.

Figure 5 .Table 3 .
Figure 5. Linear relations between the structural HOMED indices for N derivatives (a) and P derivatives (b) of furan and thiophene.HOMED-ONn and HOMED-OPn correspond to furans and HOMED-SNn and HOMED-SPn refer to thiophenes.Data are taken from Table3.

Figure 5 .Table 3 .
Figure 5. Linear relations between the structural HOMED indices for N derivatives (a) and P derivatives (b) of furan and thiophene.HOMED-ONn and HOMED-OPn correspond to furans and HOMED-SNn and HOMED-SPn refer to thiophenes.Data are taken from Table3.

Figure 5 .Table 3 .
Figure 5. Linear relations between the structural HOMED indices for N derivatives (a) and P derivatives (b) of furan and thiophene.HOMED-ONn and HOMED-OPn correspond to furans and HOMED-SNn and HOMED-SPn refer to thiophenes.Data are taken from Table3.

Figure 5 .Table 3 .
Figure 5. Linear relations between the structural HOMED indices for N derivatives (a) and P derivatives (b) of furan and thiophene.HOMED-ONn and HOMED-OPn correspond to furans and HOMED-SNn and HOMED-SPn refer to thiophenes.Data are taken from Table3.

Figure 6 .
Figure 6.Plot between the relative Gibbs energies (G) and HOMED indices for possible tautomers of azoles.Data taken from Table4.

Figure 6 .
Figure 6.Plot between the relative Gibbs energies (G) and HOMED indices for possible tautomers of azoles.Data taken from Table4.

Figure 6 .
Figure 6.Plot between the relative Gibbs energies (G) and HOMED indices for possible tautomers of azoles.Data taken from Table4.

Figure 6 .
Figure 6.Plot between the relative Gibbs energies (G) and HOMED indices for possible tautomers of azoles.Data taken from Table4.

Figure 6 .
Figure 6.Plot between the relative Gibbs energies (G) and HOMED indices for possible tautomers of azoles.Data taken from Table4.

Figure 6 .
Figure 6.Plot between the relative Gibbs energies (G) and HOMED indices for possible tautomers of azoles.Data taken from Table4.

Figure 6 .
Figure 6.Plot between the relative Gibbs energies (G) and HOMED indices for possible tautomers of azoles.Data taken from Table4.

Figure 6 .
Figure 6.Plot between the relative Gibbs energies (G) and HOMED indices for possible tautomers of azoles.Data taken from Table4.

Figure 6 .
Figure 6.Plot between the relative Gibbs energies (G) and HOMED indices for possible tautomers of azoles.Data taken from Table4.

Figure 6 .
Figure 6.Plot between the relative Gibbs energies (G) and HOMED indices for possible tautomers of azoles.Data taken from Table4.

Figure 6 .
Figure 6.Plot between the relative Gibbs energies (G) and HOMED indices for possible tautomers of azoles.Data taken from Table4.

Figure 6 .
Figure 6.Plot between the relative Gibbs energies (G) and HOMED indices for possible tautomers of azoles.Data taken from Table4.

Figure 6 .1
Figure 6.Plot between the relative Gibbs energies (G) and HOMED indices for possible tautomers of azoles.Data taken from Table4.

Figure 6 .
Figure 6.Plot between the relative Gibbs energies (G) and HOMED indices for possible tautomers of azoles.Data taken from Table4.

Figure 6 .
Figure 6.Plot between the relative Gibbs energies (∆G) and HOMED indices for possible tautomers of azoles.Data taken from Table4.

Table 1 .
The reference molecules and the DFT-calculated R s , R d , and R o bond lengths (in Å) applied in the HOMED procedure.

Table 1 .
The reference molecules and the DFT-calculated Rs, Rd, and Ro bond lengths (in Å ) applied in the HOMED procedure.

Table 1 .
The reference molecules and the DFT-calculated Rs, Rd, and Ro bond lengths (in Å ) applied in the HOMED procedure.

Table 1 .
The reference molecules and the DFT-calculated Rs, Rd, and Ro bond lengths (in Å ) applied in the HOMED procedure.

Table 1 .
The reference molecules and the DFT-calculated Rs, Rd, and Ro bond lengths (in Å ) applied in the HOMED procedure.

Table 1 .
The reference molecules and the DFT-calculated Rs, Rd, and Ro bond lengths (in Å ) applied in the HOMED procedure.

Table 1 .
The reference molecules and the DFT-calculated Rs, Rd, and Ro bond lengths (in Å ) applied in the HOMED procedure.

Table 1 .
The reference molecules and the DFT-calculated Rs, Rd, and Ro bond lengths (in Å ) applied in the HOMED procedure.

Table 1 .
The reference molecules and the DFT-calculated Rs, Rd, and Ro bond lengths (in Å ) applied in the HOMED procedure.

Table 1 .
The reference molecules and the DFT-calculated Rs, Rd, and Ro bond lengths (in Å ) applied in the HOMED procedure.

Table 1 .
The reference molecules and the DFT-calculated Rs, Rd, and Ro bond lengths (in Å ) applied in the HOMED procedure.

Table 1 .
The reference molecules and the DFT-calculated Rs, Rd, and Ro bond lengths (in Å ) applied in the HOMED procedure.

Table 1 .
The reference molecules and the DFT-calculated Rs, Rd, and Ro bond lengths (in Å ) applied in the HOMED procedure.

Table 1 .
The reference molecules and the DFT-calculated Rs, Rd, and Ro bond lengths (in Å ) applied in the HOMED procedure.

Table 1 .
The reference molecules and the DFT-calculated Rs, Rd, and Ro bond lengths (in Å ) applied in the HOMED procedure.

Table 1 .
The reference molecules and the DFT-calculated Rs, Rd, and Ro bond lengths (in Å ) applied in the HOMED procedure.

Table 1 .
The reference molecules and the DFT-calculated Rs, Rd, and Ro bond lengths (in Å ) applied in the HOMED procedure.

Table 1 .
The reference molecules and the DFT-calculated Rs, Rd, and Ro bond lengths (in Å ) applied in the HOMED procedure.

Table 1 .
The reference molecules and the DFT-calculated Rs, Rd, and Ro bond lengths (in Å ) applied in the HOMED procedure.

Table 1 .
The reference molecules and the DFT-calculated Rs, Rd, and Ro bond lengths (in Å ) applied in the HOMED procedure.

Table 1 .
The reference molecules and the DFT-calculated Rs, Rd, and Ro bond lengths (in Å ) applied in the HOMED procedure.

Table 2 .
Comparison of the normalization constants (α) for CC, CX, XX, and XY bonds applied in the rHOMA and HOMED procedures.

Table 3 .
Structures of N and P derivatives of furan and thiophene and their HOMED indices estimated for the DFT structures.

Table 3 .
Structures of N and P derivatives of furan and thiophene and their HOMED indices estimated for the DFT structures.

Table 4 .
Structures of azoles, their DFT-calculated relative Gibbs energies (G in kcal•mol −1 ) and HOMED indices estimated for DFT structures.

Table 4 .
Structures of azoles, their DFT-calculated relative Gibbs energies (G in kcal•mol −1 ) and HOMED indices estimated for DFT structures.

Table 4 .
Structures of azoles, their DFT-calculated relative Gibbs energies (G in kcal•mol −1 ) and HOMED indices estimated for DFT structures.

Table 4 .
Structures of azoles, their DFT-calculated relative Gibbs energies (G in kcal•mol −1 ) and HOMED indices estimated for DFT structures.

Table 4 .
Structures of azoles, their DFT-calculated relative Gibbs energies (G in kcal•mol −1 ) and HOMED indices estimated for DFT structures.

Table 4 .
Structures of azoles, their DFT-calculated relative Gibbs energies (G in kcal•mol −1 ) and HOMED indices estimated for DFT structures.

Table 4 .
Structures of azoles, their DFT-calculated relative Gibbs energies (G in kcal•mol −1 ) and HOMED indices estimated for DFT structures.

Table 4 .
Structures of azoles, their DFT-calculated relative Gibbs energies (G in kcal•mol −1 ) and HOMED indices estimated for DFT structures.

Table 4 .
Structures of azoles, their DFT-calculated relative Gibbs energies (G in kcal•mol −1 ) and HOMED indices estimated for DFT structures.

Table 4 .
Structures of azoles, their DFT-calculated relative Gibbs energies (G in kcal•mol −1 ) and HOMED indices estimated for DFT structures.

Table 4 .
Structures of azoles, their DFT-calculated relative Gibbs energies (G in kcal•mol −1 ) and HOMED indices estimated for DFT structures.

Table 4 .
Structures of azoles, their DFT-calculated relative Gibbs energies (G in kcal•mol −1 ) and HOMED indices estimated for DFT structures.

Table 4 .
Structures of azoles, their DFT-calculated relative Gibbs energies (G in kcal•mol −1 ) and HOMED indices estimated for DFT structures.