When correlating with N, it results:

ν ( cm − 1 ) = 155.26 × N + 185.97 (5)

N d = 7 , R 2 = 0.9984 , S E = 14.83 , F = 3068

where:

ν = Rotational frequency; N = Topological-quantum level ( N = 1 , 2 , 3 , ⋯ ); N_d = Number of data; R² = Coefficient of determination; SE = Standard error of estimate; F = Fisher-Snedecor parameter.

The regression with PCR yielded:

ν ( cm − 1 ) = 1264.26 × P C R − 1068.38 (6)

N d = 7 , R 2 = 0.9998 , S E = 5.16 , F = 25345

where:

ν = Rotational frequency; PCR = Ratio between graph’s multiple path count and path count; N_d = Number of data; R² = Coefficient of determination; SE = Standard error of estimate; F = Fisher-Snedecor parameter.

Following is Table 4, where is shown the comparison between the experimental and the predicted values, using Equations (3), (5), (6) and square PCR (equation not shown).

From results in Table 4, it can be appreciated that Equations (3) and (5) (rigid rotor and N) predict better the lower values, whereas Equation (6) (PCR) predicts much better the values from N = 3 on up.

The corresponding equations for N and PCR, were:

ν ( cm − 1 ) = 38.75 × N + 48.65 (7)

N d = 14 , R 2 = 0.9994 , S E = 1.51 , F = 77125

where:

ν = Rotational frequency; N = Topological-quantum level ( N = 1 , 2 , 3 , ⋯ ); N_d = Number of data; R² = Coefficient of determination; SE = Standard error of estimate; F = Fisher-Snedecor parameter.

ν ( cm − 1 ) = 348.85 × P C R − 310.58 (8)

N d = 14 , R 2 = 0.9989 , S E = 5.04 , F = 6931

where:

ν = Rotational frequency; PCR = Ratio between multiple path count and path count; N_d = Number of data; R² = Coefficient of determination; SE = Standard error of estimate; F = Fisher-Snedecor parameter.

Table 5 presents the comparison between the experimental and predicted values. On the second column are the values of the differences between consecutive frequencies. From them, it’s easy to see that the rigid rotor model is not followed because the differences become smaller and smaller.

The corresponding equations were:

ν ( cm − 1 ) = 7.61 × N + 11.45 (9)

N d = 12 , R 2 = 0.9999 , S E = 0.0009 , F > 100000

where:

ν = Rotational frequency; N = Topological-quantum level ( N = 1 , 2 , 3 , ⋯ ); N_d = Number of data; R² = Coefficient of determination; SE = Standard error of

estimate; F = Fisher-Snedecor parameter.

ν ( cm − 1 ) = 69.57 × P C R − 55.30 (10)

N d = 12 , R 2 = 0.9983 , S E = 1.21 , F = 4193

where:

ν = Rotational frequency; PCR = Ratio between multiple path count and path count; N_d = Number of data; R² = Coefficient of determination; SE = Standard error of estimate; F = Fisher-Snedecor parameter.

Table 6 illustrates the comparison between the experimental and the calculated values.

In this case, the regression with N is by far the best and practically coincides with that calculated using the rigid rotor equation.

Table 7 summarizes the results for all the three molecules.

Altogether it may be concluded that the excellent adjustment found for a single variable (N), does not mean that experimental results are adjustable to the rigid rotor, since in that case, as deduced from Equation (3), the slope and intercept should take the same value. This is far to be true for H₂ and HF, what is consistent with expressed in the literature [11] . On the contrary, CO fits pretty well the rigid rotor, what was to be expected considering that the centrifugal distortion must decrease as the molecular mass increases and hence that CO is less deformable than FH and H₂.

The consequence is that the formalism proposed here, according to which the quantum level is equivalent to the graph’s number of loops, remains consistent.

The hydrogen molecule-ion is an ideal molecule to study since it is the most simple molecule possible and it has one single electron, so that the possible influence of interelectronic repulsion is overcome.

The correlation with N is quite good as shown in Equation (14):

ν ( cm − 1 ) = 122.39 × N + 161.72 (14)

N d = 12 , R 2 = 0.9972 , S E = 24.51 , F = 3566

where:

ν = Vibrational frequency; N = Topological-quantum level ( N = 1 , 2 , 3 , ⋯ ); N_d = Number of data; R² = Coefficient of determination; SE = Standard error of estimate; F = Fisher-Snedecor parameter.

Note that, as mentioned above, the first value for N is here N = 1 to take into account the existence of a residual vibratory energy.

Although there is a good correlation with N, a look at Figure 1 illustrates that the dependence is not linear but quadratic. In principle, this is consistent if there is a following of the model of anharmonic oscillator (Equation (14)). But see that in Equation (14) the intercept should be zero, what doesn’t occur. Actually, the ratio between the ordinate and the slope (23/149) is too large to be consistent with Equation (14). Of course, this difference could be explained considering other effects, such as the influence of rotational over vibrational energies, but, once more, this discrepancy provide value to the graph-theoretical formalism proposed by the author.

Anyway, the adjustment with PCR is much better than the linear N:

ν ( cm − 1 ) = 1066.1 P C R − 919.61 (15)

N d = 12 , R 2 = 0.9999 , S E = 5.25 , F = 77862

where:

ν = Vibrational frequency; PCR = Ratio between multiple path count and path count; N_d = Number of data; R² = Coefficient of determination; SE = Standard error of estimate; F = Fisher-Snedecor parameter.

Table 8 illustrates the values of the vibrational frequencies of H 2 + in its

electronic ground state for the first 11 levels of vibratory energy. The values calculated from Equations (14) and (15) are also included for comparison.

In this case the aim is analysing the vibrational energy for the different levels ( N = 0 , 1 , 2 , ⋯ ).

The correlation with N is quite good as seen in Equation (16):

ν ( cm − 1 ) = 12416 × N + 18229 (16)

N d = 12 , R 2 = 0.9850 , S E = 16.62 , F = 665

where:

E_v = Vibrational energy; N = Topological-quantum level ( N = 1 , 2 , 3 , ⋯ ); N_d = Number of data; R² = Coefficient of determination; SE = Standard error of estimate; F = Fisher-Snedecor parameter.

But the correlation with PCR is much better:

ν ( cm − 1 ) = 108125 × P C R − 91423 (17)

N d = 12 , R 2 = 0.9947 , S E = 9.83 , F = 1888

where:

ν = Vibrational frequency; PCR = Ratio between multiple path count and path count; N_d = Number of data; R² = Coefficient of determination; SE = Standard error of estimate; F = Fisher-Snedecor parameter.

Similarly to H 2 + , the dependence with N is clearly quadratic, so that R² is

virtually 1 what is consistent with the well known values for Xe. but for one single variable, PCR is much better as can be seen comparing Equations (16) and (17).

Table 9 shows the experimental values of the vibration frequencies for the hydrogen molecule in its fundamental electronic state, together with the values calculated from Equations (16) and (17).

From these results, it is to note that H₂, just like H 2 + , does not fit well neither the harmonic oscillator nor the anharmonic oscillator. Therefore, the same arguments exposed for H 2 + can be repeated for H₂. Anyway, the dependence with PCR is clearly linear and better than the ones of conventional models if equations having the same number of variables are compared. It is also to be noted that other topological indices (such as ESpm02r) led to better results but PCR was left for comparative purposes.

The correlation with N is quite good as illustrated in Equation (18):

ν ( cm − 1 ) = 2040.57 × N + 1238.2 (18)

N d = 10 , R 2 = 0.9998 , S E = 105.77 , F = 30709

where:

ν = Vibrational frequency; N = Topological-quantum level ( N = 1 , 2 , 3 , ⋯ ); N_d = Number of data; R² = Coefficient of determination; SE = Standard error of estimate; F = Fisher-Snedecor parameter.

The equation for PCR IS:

ν ( cm − 1 ) = 17217 × P C R − 16016 (19)

N d = 10 , R 2 = 0.9995 , S E = 141.40 , F = 17179

where:

ν = Vibrational frequency; PCR = Ratio between multiple path count and path count; N_d = Number of data; R² = Coefficient of determination; SE = Standard error of estimate; F = Fisher-Snedecor parameter.

In this case the fitting to the harmonic oscillator continues to be poor, although the adjustment with the anharmonic (in this case matches with that of N²) is practically perfect.

Table 10 shows the comparison between the experimental values and those calculated from Equations (20) and (21).

From these overall results of vibrational frequencies, it can be concluded that the graph-theoretical model (PCR) is better for small molecules, such as H 2 + and H₂ whereas the conventional approaches are better for larger molecules, such as CO.

That conclusion is evident just watching Table 11, where it is summarized all the results.

Table 11 summarizes the results for the three molecules.

Electronic binding energy in atoms also usually referred to as ionization potential [13] is defined as the energy necessary to remove the electron from the atom. In this case we’ll correlate the binding energies for the 1s-electrons of the thirteen first elements of the periodic system.

The regresión equation with quadratic N was almost perfect:

E B E − 1 s ( eV ) = 10.60 × N 2 − 20.43 × N + 23.79 (20)

N d = 12 , R 2 = 0.9999 , S E = 4.91 , F = 43500

where:

EBE = Electronic binding energy (eV) of the first atomic level (1S); N = Topological-quantum level ( N = 1 , 2 , 3 , ⋯ ); N_d = Number of data; R² = Coefficient of determination; SE = Standard error of estimate; F = Fisher-Snedecor parameter.

Table 12 shows the values of EBEs for the 13 first elements of the periodic table, along with the values calculated from Equation (20).

Given the nature of the correlated property, it is clear that N might be identified as Z (atomic number), and since the ionization potential for electrons of a given electronic level, n, is [14] .

E = − Z 2 e 2 2 a 0 n 2 (21)

where n = Atomic level. As in this case n = 1 (1S) for all the levels, the binding energy depends on Z². Hence a good correlation with Z² should be obtained, something that actually happens (R² = 0.9986).

However, although with Z² there is an acceptable correlation (R² = 0.9986) the dependence is not proportional, as it should be according to Equation (21). This allows us to do the assignment of N as the topological level and interpret Equation (20) as a virial-like development in powers of N (see Figure 2).

In this paragraph the resonance energies of conjugated alkenes will be predicted. Two different formalisms are followed: In the first the type G graphs (Table 3) will be used to characterize the energy levels, while in the second, the conventional graph-molecular presentation (graphs type M, where M = molecular). In the M representation, each molecule is represented by a graph containing the same number of alternated loops as alternated double bonds have the molecule.

For instance, the graph allocated to ethylene is (M1): , and for butadiene (M2): , etc.

1) Formalism using graphs type G:

The best correlation was for:

H L G ( eV ) = 8.03 × V i n d e x − 1.29 (22)

N d = 9 , R 2 = 0.9979 , S E = 0.074 , F = 3392

where:

HLG = HOMO-LUMO gap (eV); Vindex = Balaban Vindex; N_d = Number of data; R² = Coefficient of determination; SE = Standard error of estimate; F = Fisher-Snedecor parameter.

The Balaban V index [15] is a typical information index which takes into account the bond multiplicity.

Table 13 shows the comparison between the observed and the calculated (Equation (22)) Values of HLG.

Although not shown here, it is important to note that this model (Equation (22)) improved both the free-electron model and the Hückel model [16] (R² = 0.9958 and R² = 0.9959, respectively).

2) Formalism using graphs type M:

The best equation was:

H L G ( eV ) = − 1.73 × E s p m 02 x + 10.23 (23)

N d = 9 , R 2 = 0.9985 , S E = 0.063 , F = 4759

where:

HLG = HOMO-LUMO gap (eV); ESpm02x = Spectral moment #2 of the edge adjacency matrix weighted by edge degrees; N_d = Number of data; R² = Coefficient of determination; SE = Standard error of estimate; F = Fisher-Snedecor parameter.

Table 14 shows the experimental and the calculated values for HLG.

Noteworthy from the outcomes that the graph theoretical approach proposed by the author, either using graphs type G or type M, improves the predictions of HLG done using conventional quantum methods, particularly the free-electrons and Hückel.

Table 15 illustrates, as a summary, the assignment of graphs and the values of the molecular graph-theoretical descriptors included in the equations of this work. The aim is ease the reader the reproduction of the results outlined throughout the work.