_{1}

The present work illustrates a predictive method, based on graph theory, for different types of energy of subatomic particles, atoms and molecules, to be specific, the mass defect of the first thirteen elements of the periodic table, the rotational and vibrational energies of simple molecules (such as
_{2}, FH and CO) as well as the electronic energy of both atoms and molecules (conjugated alkenes). It is shown that such a diverse group of energies can be expressed as a function of few simple graph-theoretical descriptors, resulting from assigning graphs to every wave function. Since these descriptors are closely related to the topology of the graph, it makes sense to wonder about the meaning of such relation between energy and topology and suggests points of view helping to formulate novel hypotheses about this relation.

In an article published by the author a few years ago [

Now, the aim is to extend this approach to other types of energies, like the mass defect of the less massive nuclides of the first 13 elements of the periodic table (between hydrogen and aluminium), the rotational and vibrational energies of diatomic molecules, such as H 2 + , H_{2}, FH and CO, as well as the electronic energies of atoms and molecules (conjugated alkenes).

In all cases, the formalism proposed predicts accurately all these different energies, even improving the predictions made with the typical quantum approaches for such cases.

Altogether, these new results arise some interesting questions about the relation energy-topology.

In a previous paper, the author together with other colleagues proposed new ways to assign wave functions to graphs [

The relation between wave functions and graphs is implicit in Heisenberg’s matricial formulation of quantum mechanics [

It’s not difficult to see that the transformation of wave functions into graphs has a double advantage: On the one hand, unlike most predictive models, it yields with pure mathematical and simple descriptors and not physical ones. On the other hand, the same model, namely the classical model of the stationary waves, can be used to predict very different types of energies (rotational, vibrational and electronic) at different levels (subatomic particles, atoms and molecules). Furthermore, it is well known that this classical pattern of interferences show the same shape as that of the quantum models, such as particle-in-a-box, rigid rotor, harmonic oscillator, free-electrons and Hückel. All are applicable to particles, atoms and molecules.

Although the equations complexity increases along with the model complexity

Model | Wave function | Energy |
---|---|---|

Particle in a 1D-box | Ψ ( x ) = 2 / a sin ( N π x a ) | E = N 2 h 2 8 m a 2 , N = 1 , 2 , 3 , ⋯ |

1D-Rigid Rotor | Ψ ( x ) = ( 1 / 2 π ) 1 / 2 e i N ϕ | E = 2 B ( N + 1 ) , N = 1 , 2 , 3 |

Harmonic oscillator | Ψ ( x ) = ( 2 v v ! ) − 1 / 2 ( α / π ) 1 / 4 e − α x 2 / 2 H v ( α 1 / 2 x ) | E = h ν e ( N + 1 / 2 ) , N = 0 , 1 , 2 , 3 , ⋯ |

(from the particle in a box to the harmonic oscillator), the topology of the wave functions, and hence of the graphs associated, remains unaltered.

This is the key basis for this work because one unique type of graphs accounts for any of the wave functions associated to every of the physical models mentioned above.

Hence, it were assigned graphs to each level starting with a simple graph without any loop (level N = 0), and continuing by graphs containing N loops, where N is the number of the corresponding level.

Once allocated the graphs, the formalism will be applied to the prediction of the following energies:

1) Mass defect. Applied onto the less massive nuclides of the first 13 elements of the periodic table (from hydrogen to aluminium).

2) Rotational energy. Rotational energies of the molecules of H_{2}, FH and CO.

3) Vibrational energy. Vibrational energies of the hydrogen molecule-ion, H 2 + , H_{2} and CO.

4) Electronic energy. The 1s-binding energy (ionization potential) of the first 13 elements of the periodic table (from hydrogen to aluminium). Additionally, it will be also predicted the HOMO-LUMO gap of conjugated alkenes.

In the predictions of atomic properties (mass defect and binding energy) it will be considered the atomic number as the topological level of each graph. Thus hydrogen would be N = 1, helium N = 2, and so forth.

The topological indices were calculated using Dragon software [

Mass defect is defined as the mass loss produced when nuclei are formed from their constitutive particles (protons and neutrons) [

Graph | Graph’s label | N |
---|---|---|

G1 | 0 | |

G2 | 1 | |

G3 | 2 | |

G4 | 3 | |

G5 | 4 |

When correlating the mass defect for the most stable nuclides of the first thirteen elements of the periodic table, the best regression equation was:

M D ( μ u ) = − 10.47 × S C B O + 54 (1)

N d = 13 , R 2 = 0.9967 , S E = 4.79 , F = 3363

The simple correlation with the mass number (Nn + Z) was also very good:

M D ( μ u ) = − 9.60 × ( N n + Z ) + 20.104 (2)

N d = 13 , R 2 = 0.9966 , S E = 4.86 , F = 3359

where:

MD = Mass defect (in μu); SCBO = Sum of count bond orders (overall number of edges in the graph); Nn = Number of neutrons; Nn + Z = Mass number; N_{d} = Number of data; R^{2} = Coefficient of determination; SE = Standard error of estimate; F = Fisher-Snedecor parameter.

It is interesting to note that the regression with both, the atomic and the mass numbers, yielded worse fittings than SCBO (R^{2} = 0.9953, 0.9966 and 0.9967, respectively). Moreover, Equation (2) approaches the null value of hydrogen much better than Equation (3) (1.65 vs. 10.50), whereas the notable discrepancy for Be

Element | Z | Nn | Nn + Z | SCBO | Mass defect (Experimental)* | Cal. from Equation (1) | Cal. from Equation (2) |
---|---|---|---|---|---|---|---|

H | 1 | 0 | 1 | 5 | 0 | 1.6 | 10.5 |

He | 2 | 1 | 3 | 6 | −8.3 | −8.8 | −8.7 |

Li | 3 | 3 | 6 | 8 | −34.4 | −29.8 | −37.5 |

Be | 4 | 3 | 7 | 10 | −40.4 | −50.7 | −47.1 |

B | 5 | 5 | 10 | 12 | −69.5 | −71.7 | −75.9 |

C | 6 | 6 | 12 | 14 | −98.9 | −92.6 | −95.1 |

N | 7 | 7 | 14 | 16 | −112.4 | −113.5 | −114.4 |

O | 8 | 8 | 16 | 18 | −137.0 | −134.5 | −133.6 |

F | 9 | 10 | 19 | 20 | −158.7 | −155.4 | −162.4 |

Ne | 10 | 10 | 20 | 22 | −172.5 | −176.4 | −172.0 |

Na | 11 | 12 | 23 | 24 | −200.3 | −197.3 | −200.8 |

Mg | 12 | 12 | 24 | 26 | −212.8 | −218.3 | −210.4 |

Al | 13 | 14 | 27 | 28 | −241.5 | −239.2 | −239.2 |

can be explained by the exceptional fact that its more stable nuclide contains 5 and not 4 neutrons.

Equation (1) is particularly interesting as far as there are no theoretical equations for the mass defect, excepting a semi-empirical equation [

In this section the frequencies of transitions between rotational levels for three molecules, namely H_{2}, FH and CO, will be predicted.

The rotational frequencies for diatomic molecules are given by the quantum rigid and elastic rotors [

ν = 2 B × ( N + 1 ) with N = 0 , 1 , 2 , 3 , 4 , ⋯ (Rigid rotor) (3)

ν = 2 B × ( N + 1 ) − 4 D × ( N + 1 ) 3 with N = 0 , 1 , 2 , 3 , 4 , ⋯ (Elastic rotor) (4)

where ν = Rotational frequency. B = Rotational Constant. D = Centrifugal distortion constant.

The frequencies will be predicted for the spectral first lines of three different molecules: H_{2}, FH and CO in their electronic and vibrational ground states.

Following are the results.

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; R^{2} = 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; R^{2} = Coefficient of determination; SE = Standard error of estimate; F = Fisher-Snedecor parameter.

Following is

From results in

Experimental rotational frequencies (cm^{−}^{1})^{c} | Calculated from Equation (5) | Calculated from Equation (6) | Calculated from Equation (3) rigid rotor (2B = 170.4) | Calculated PCR^{2} |
---|---|---|---|---|

170.4 | 186.0 | 195.9 | 170.4 | 178.6 |

339.3 | 341.2 | 328.6 | 340.8 | 327.4 |

504.8 | 496.5 | 491.7 | 511.2 | 502.7 |

665.4 | 651.8 | 654.8 | 681.6 | 669.7 |

819.7 | 807.0 | 812.8 | 852.0 | 823.8 |

965.7 | 962.3 | 965.8 | 1022.4 | 965.5 |

1096.9 | 1117.5 | 1112.5 | 1192.8 | 1094.6 |

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; R^{2} = 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; R^{2} = Coefficient of determination; SE = Standard error of estimate; F = Fisher-Snedecor parameter.

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; R^{2} = Coefficient of determination; SE = Standard error of

Experimental frequency (cm^{−}^{1})^{d} | ∆ frequencies | Calculated from Equation (3) (B = 20.56 cm^{−}^{1}) | Calculated from Equation (7) | Calculated from Equation (8) |
---|---|---|---|---|

41.08 | 41.12 | 48.65 | 38.27 | |

82.19 | 41.11 | 82.24 | 87.40 | 74.90 |

123.15 | 40.96 | 123.36 | 126.15 | 119.90 |

164.01 | 40.85 | 164.48 | 164.90 | 164.90 |

204.62 | 40.62 | 205.6 | 203.65 | 208.51 |

244.93 | 40.31 | 246.72 | 242.40 | 250.72 |

285.01 | 40.08 | 287.84 | 281.15 | 291.19 |

324.65 | 39.64 | 328.96 | 319.90 | 330.26 |

363.93 | 39.28 | 370.08 | 358.65 | 367.93 |

402.82 | 38.89 | 411.2 | 397.40 | 404.21 |

441.13 | 38.31 | 452.32 | 436.15 | 439.80 |

478.94 | 37.81 | 493.44 | 474.90 | 474.33 |

516.20 | 37.26 | 534.56 | 513.65 | 507.82 |

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; R^{2} = Coefficient of determination; SE = Standard error of estimate; F = Fisher-Snedecor parameter.

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

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 H_{2} and HF, what is consistent with expressed in the literature [_{2}.

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.

Experimental Frequencies (cm^{−}^{1})^{e} | Calculated by Equation (9) | Calculated by Equation (10) |
---|---|---|

11.43 | 11.45 | 10.67 |

19.05 | 19.06 | 17.60 |

26.67 | 26.67 | 26.11 |

34.29 | 34.28 | 34.62 |

41.91 | 41.89 | 42.86 |

49.52 | 49.5 | 50.84 |

57.13 | 57.11 | 58.50 |

64.74 | 64.72 | 65.89 |

72.34 | 72.33 | 73.01 |

79.95 | 79.94 | 79.87 |

87.55 | 87.55 | 86.60 |

95.15 | 95.16 | 93.13 |

SUMMARY of ROTATION | |||||
---|---|---|---|---|---|

H_{2} | HF | CO | |||

Model | R^{2} | Model | R^{2} | Model | R^{2} |

N | 0.9984 | N | 0.9994 | N | 0.9999 |

PCR | 0.9980 | PCR | 0.9989 | PCR | 0.9983 |

PCR2 | 0.9996 | PCR2 | 0.9998 | PCR2 | 0.9997 |

Rigid rotor (2B = 170.4 cm^{−}^{1}) | 0.9964 | Rigid rotor (2B = 41.12 cm^{−}^{1}) | 0.9988 | Rigid rotor (2B = 11.43 cm^{−}^{1}) | 0.9999 |

Elastic rotor | 0.9985 | Elastic rotor | 0.9998 | Elastic rotor | 1 |

The energies for the different vibrational levels of diatomic molecules are given by [

E v = ( N + 1 / 2 ) h v e , N = 0 , 1 , 2 , 3 , ⋯ Harmonic oscillator (12)

E v = ( N + 1 / 2 ) h v e − ( n + 1 / 2 ) 2 h e v e X e , N = 0 , 1 , 2 , 3 , ⋯ nharmonic oscillator (13)

where ν_{e} is the (classical) fundamental vibrational frequency, X_{e} is the anharmonicity constant and h is the Planck constant.

Next are the results for the three molecules tested, namely H 2 + , H_{2} and CO.

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; R^{2} = 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

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; R^{2} = Coefficient of determination; SE = Standard error of estimate; F = Fisher-Snedecor parameter.

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; R^{2} = 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; R^{2} = 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 R^{2} is

Experimental frequencies for H 2 + (cm^{−1})^{f} | Calculated from Equation (14) | Calculated from Equation (15) |
---|---|---|

117 | 161 | 146 |

265 | 288 | 258 |

411 | 409 | 396 |

548 | 530 | 533 |

677 | 651 | 667 |

800 | 772 | 796 |

918 | 893 | 919 |

1033 | 1014 | 1039 |

1145 | 1135 | 1154 |

1257 | 1256 | 1265 |

1368 | 1377 | 1374 |

1479 | 1498 | 1479 |

1591 | 1620 | 1581 |

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).

From these results, it is to note that H_{2}, 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 H_{2}. 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; R^{2} = 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

Vibrational energy observed (cm^{−1})^{g} | Calc. from Equation (16) | Calc. from Equation (17) |
---|---|---|

4547 | 9150 | 8352 |

13,265 | 15,358 | 14,029 |

21,481 | 21,566 | 21,003 |

29,195 | 27,774 | 27,977 |

36,407 | 33,982 | 34,735 |

43,117 | 40,190 | 41,276 |

49,331 | 46,398 | 47,548 |

55,037 | 52,606 | 53,603 |

60,237 | 58,814 | 59,442 |

64,931 | 65,022 | 65,064 |

69,137 | 71,230 | 70,579 |

72,836 | 77,438 | 75,931 |

where:

ν = Vibrational frequency; PCR = Ratio between multiple path count and path count; N_{d} = Number of data; R^{2} = 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 N^{2}) is practically perfect.

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 H_{2} whereas the conventional approaches are better for larger molecules, such as CO.

That conclusion is evident just watching

Vibrational energies CO (cm^{−1})^{h} | Frequation Calc. from Equation (18) | Frequation Calc. From Equation (19) |
---|---|---|

1082 | 1238 | 1201 |

3227 | 3279 | 3009 |

5345 | 5319 | 5230 |

7438 | 7360 | 7451 |

9505 | 9400 | 9603 |

11,545 | 11,441 | 11,686 |

13,560 | 13,482 | 13,683 |

15,548 | 15,522 | 15,612 |

17,511 | 17,563 | 17,471 |

19,447 | 19,603 | 19,262 |

SUMMARY VIBRATIONAL ENERGIES | |||||
---|---|---|---|---|---|

H 2 + | H_{2} | CO | |||

Variable | R^{2} | Variable | R^{2} | Variable | R^{2} |

N | 0.9972 | N | 0.9850 | N | 0.9997 |

PCR | 0.9993 | PCR | 0.9928 | PCR | 0.9995 |

PCR^{2} | 0.9994 | PCR^{2} | 0.9998 | PCR^{2} | 0.9996 |

Harmonic oscillator | 0.9832 | Harmonic oscillator | 0.9655 | Harmonic oscillator | 0.9994 |

N^{2} | 0.9983 | N^{2} | 1,0000 | N^{2} and Anharmonic oscillator | 1,0000 |

In this section, different sorts of electronic energies for atoms and molecules will be analysed.

Electronic binding energy in atoms also usually referred to as ionization potential [

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; R^{2} = Coefficient of determination; SE = Standard error of estimate; F = Fisher-Snedecor parameter.

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 [

E = − Z 2 e 2 2 a 0 n 2 (21)

Atom | EBE-1S (eV) (observed)^{i} | Calculated from Equation (20) |
---|---|---|

H | 13.6 | 14.0 |

He | 24.6 | 25.3 |

Li | 54.7 | 57.9 |

Be | 111.5 | 111.7 |

B | 188.0 | 186.6 |

C | 284.2 | 282.7 |

N | 409.9 | 400.1 |

O | 543.1 | 538.6 |

F | 696.7 | 698.3 |

Ne | 870.2 | 879.2 |

Na | 1070.8 | 1081.4 |

Mg | 1303.0 | 1304.7 |

Al | 1559.6 | 1549.2 |

where n = Atomic level. As in this case n = 1 (1S) for all the levels, the binding energy depends on Z^{2}. Hence a good correlation with Z^{2} should be obtained, something that actually happens (R^{2} = 0.9986).

However, although with Z^{2} there is an acceptable correlation (R^{2} = 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

In this paragraph the resonance energies of conjugated alkenes will be predicted. Two different formalisms are followed: In the first the type G graphs (

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; R^{2} = Coefficient of determination; SE = Standard error of estimate; F = Fisher-Snedecor parameter.

The Balaban V index [

Alkene | Graph allocated | HLG (eV) ^{j}(Experimental) | HLG (calculated from Equation (22)) |
---|---|---|---|

ethylene | G2 | 7.52 | 7.44 |

1,3-butadiene | G3 | 5.72 | 5.73 |

1,3,5-hexatriene | G4 | 4.63 | 4.77 |

1,3,5,7-octatetraene | G5 | 4.08 | 4.15 |

1,3,5,7,9-decacapentaene | G6 | 3.72 | 3.72 |

1,3,5,7,9,11-dodecahexaene | G7 | 3.41 | 3.39 |

1,3,5,7,9,11,13-tetradecaneeptene | G8 | 3.18 | 3.15 |

1,3,5,7,9,11,13,15-esadecaneoctaene | G9 | 3.03 | 2.95 |

1,3,5,7,9,11,13,15,17-octadecanenonaene | G10 | 2.78 | 2.78 |

Although not shown here, it is important to note that this model (Equation (22)) improved both the free-electron model and the Hückel model [^{2} = 0.9958 and R^{2} = 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; R^{2} = Coefficient of determination; SE = Standard error of estimate; F = Fisher-Snedecor parameter.

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.

Alkene | Graph allocated | HLG (eV) | HLG calculated form Equation (23) |
---|---|---|---|

Ethylene | M1 | 7.52 | 7.44 |

1,3-butadiene | M2 | 5.72 | 5.65 |

1,3,5-hexatriene | M3 | 4.63 | 4.79 |

1,3,5,7-octatetraene | M4 | 4.08 | 4.22 |

1,3,5,7,9-decacapentaene | M5 | 3.72 | 3.79 |

1,3,5,7,9,11-dodecahexaene | M6 | 3.41 | 3.44 |

1,3,5,7,9,11,13-tetradecaneeptene | M7 | 3.18 | 3.15 |

1,3,5,7,9,11,13,15-esadecaneoctaene | M8 | 3.03 | 2.91 |

1,3,5,7,9,11,13,15,17-octadecanenonaene | M9 | 2.78 | 2.69 |

Graph allocated (Type G) | N (number of nodes) | SCBO | PCR | Vindex | ESpm08x | Graph allocated (Type M) | ESpm02x (for M-graphs) |
---|---|---|---|---|---|---|---|

G1 | 0 | 5 | 1 | 1.088 | 10.171 | M1 | 1.610 |

G2 | 1 | 6 | 1.105 | 1.088 | 11.094 | M2 | 2.640 |

G3 | 2 | 8 | 1.234 | 0.875 | 11.186 | M3 | 3.130 |

G4 | 3 | 10 | 1.363 | 0.755 | 11.342 | M4 | 3.470 |

G5 | 4 | 12 | 1.488 | 0.678 | 11.484 | M5 | 3.710 |

G6 | 5 | 14 | 1.609 | 0.624 | 11.608 | M6 | 3.910 |

G7 | 6 | 16 | 1.725 | 0.584 | 11.719 | M7 | 4.080 |

G8 | 7 | 18 | 1.837 | 0.553 | 11.818 | M8 | 4.220 |

G9 | 8 | 20 | 1.945 | 0.528 | 11.909 | M9 | 4.340 |

G10 | 9 | 22 | 2.049 | 0.507 | 11.992 | M10 | 4.480 |

G11 | 10 | 24 | 2.151 | 0.49 | 12.069 | M11 | 4.607 |

G12 | 11 | 26 | 2.25 | 0.475 | 12.14 | M12 | 4.724 |

G13 | 12 | 28 | 2.346 | 0.462 | 12.206 | M13 | 4.832 |

G14 | 13 | 30 | 2.440 | 0.45 | 12.269 | M14 | 4.936 |

The assignment of graphs to the energy levels of different systems (from elementary particles up to molecules) allows an excellent prediction of parameters such as masses of elementary particles, mass defects of stable nuclei, electronic energies of atoms and rotational, vibrational and electronic energies of molecules.

In particular, the wave functions associated with quantum-mechanical models, like particle in a box, rigid rotor or harmonic oscillator, are assimilated to simple graphs whose topology (number of nodes) coincides with that of the standing waves, as for instance those that appear on the strings of musical instruments. This assimilation is justified by the Heisenberg’s matricial formulation, by virtue of which every quantum object (for example the wave functions) can be associated with a matrix and this, in turn, with a graph.

For almost all cases, the formalism proposed improves the predictions of the conventional quantum-mechanical models (particle in a box, rigid rotor, harmonic oscillator, model of free electrons and Hückel), through the use of simple topological indices.

Moreover, in those cases where a very good correlation is found simply thanks to the number of nodes in the graph (N), it is demonstrated that the solution is not trivial, since it does not coincide with what expected from the aforementioned conventional quantum models.

Since such a diverse type of energy can be expressed as a function of simple topological indices which, given their mathematical nature, are not dependent on energy, formalisms such as the one proposed here, may open suggestive pathways of discussion about the relations between energy and topology.

Authors acknowledge the MINECO (Spanish Ministry of Economy, Industry and Competitivity) Project: “Desarrollo de nuevas herramientas para el control de oidios” (AGL2016-76216-C2-2-R).

The author declares no conflicts of interest regarding the publication of this paper.

Galvez, J. (2019) A Graph Theoretical Interpretation of Different Types of Energies of Elementary Particles, Atoms and Molecules. Open Journal of Physical Chemistry, 9, 33-50. https://doi.org/10.4236/ojpc.2019.92003