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Calculated and experimental data of the hydrogen molecule

In chapter 4.4 of the book, "How chemical bonds form and chemical reactions proceed," we tried to explain the divergences between the model calculation and the experiment. The greatest divergence between calculation and experiment (20%) was observed when comparing the calculative and experimental data on the length of bonds in the hydrogen molecule.

The length of bonds in the hydrogen molecule given by the principal, most respected reference book ("Hand Book of Chemistry and Physics", ed. 85, p.21) is 0.7414Å and, correspondingly, the covalent radius of hydrogen is 0, 3707 Å. The value of the covalent radius of the hydrogen atom in the hydrogen molecule has concerned the progressive chemical community for over seventy years. The cause of this agitation is the fact that for all compounds in which the atoms are bound by the covalent bond A - B, the length of the covalent radius of the atom A is practically the same and is equal to its covalent radius in the two atom molecule A - A.

The only exception is the hydrogen atom. The length of its covalent radius in all other compounds, HF, HCL, HBr, HJ, H20, H2S, H2Se, PH3, AsH3, SbH3, CH4, C2H4, C3H3, C6H6, HCN, SiH4, GeH4, SnH4, with the exception of the hydrogen molecule, varies within the interval of 0.28 - 0. 32 Å. Within the bonds A - H, where A differs in ionization energy from the hydrogen atom by less than 1eB, the variation interval is 0.29 - 0.31 Å. It should be added that, in the afore-mentioned compounds, the ionization potential of the atom which the hydrogen atom and the covalent radius are bound to varies in the interval of 7,3 (Sn) - 17,4(F) eV and 0,64(F) -- 141(Sb) Å. For comparison, the length of the covalent radius of the hydrogen atom in all the compounds studied differs from its length in diamond 0.771 Å by less than 0.011 Å. No well-grounded explanation of this anomaly of the covalent radius of the hydrogen atom - not even a phenomenological one - has been yet proposed in the seventy years of general interest in this paradox.

As was mentioned above (see 4.4.1), the model calculations gave the value of the electron energy of the hydrogen molecule with a divergence of ±3% from the experimentally obtained value.

In the course of the calculations, besides quanitfying electron energy, we found that the bond length of the hydrogen molecule re was 0,582 Å. That is, the calculated value of the covalent radius of the hydrogen atom 0,291 was within the limits of its variations, from 0,29 - 0.31 Å, for all the studied covalent compounds of hydrogen. It should be added that bond length in the hydrogen molecule was calculated before the correction of -5.5% inserted into the calculation of the molecule energy. Because of this amendment, the molecule's energy reduces and, correspondingly, the calculated bond length must increase, though the limit of the divergence cannot exceed +5.5%. So, the upper limit of the calculated length of the covalent radius of the hydrogen atom is 0.307 Å, which is also within the limits of 0.29 - 0.31 Å.

The correspondence with the experiment of the principal parameters of the hydrogen molecule (the energy and covalent radius of hydrogen) confirmed the correctness of the model but raised a question about the circumstances that led to the incorrect experimental data on the bond length in the hydrogen molecule being experimentally obtained and subsequently included in reference books.  

According to "Introduction to Quantum Mechanics with Applications to Chemistry," by L.Pauling and E.Bright (p. 349), starting in 1927, when calculations on  the hydrogen molecule were made by Heitler and London, and during the following six years, six more quantum-mechanic calculations were carried out. The bond energy and bond length of the hydrogen molecule were determined using these calculations.  At this time, according to the experimental data, the bond energy in the hydrogen molecule (De) was 4.72 eV. The calculated value increased with the time in the following order: 3.14 (W.Heitler and F.London, 1927), 3.47 (W.Heitler, F.London, Wang), 3.76 (Wang), 4.00 (Weinbaum (ionic), 4.02 (Rosen, 1931), 4.10 (Weinbaum, 1933), 4.722 (James and Coolidge, 1933).

TABLE    

APPROXIMATE TREATMENTS 
OF THE NORMAL HYDROGEN MOLECULE

Result   De,   eV  re, Å
Heitler-London-Suguira .......  3.14   0.80
Molecular-orbital treatment.... 3.47   0.73
Wang .............................. 3.76  0.76
Weinbaum (ionic)...............  4.00   0.77
Rosen (polarization).............. 4.02    0.77
Weinbaum (ionic-polarization)... 4.10       
James-Coolidge..................... 4.722     0.74
Experiment........................ 4.72     0.739

The history of the issue is as follows. Anybody can make an error, especially while processing the electron specters of molecules, with hundreds of lines from which to select in the spectral data of the experiment. The probability of an error rises sharply when an experiment is carried out after a theoretical calculation, and in the course of the experiment its inaccuracy is intensified by the method of the experiment and the processing of its results is not evaluated. Earlier, we showed that, within the framework of quantum theory, an unlimited number of unevaluated suppositions during calculation is permitted. Their number is always more than enough for increased proximity of the results of the calculation and the results of the experiment. For example, it was shown in Hertsberg, Kolos and Volnevich's case that, at least in the case of spectral definition of the bond energy in the hydrogen molecule, it is possible to obtain the correspondence of the experimental results with the calculated results up to the fourth significant figure.

The calculated energy of dissociation is 36117,4 cm -1. The experimental value (according toHertsberg's experiment, carried out before 1968) is - 36113,6 ± 0,3 cm -1.

In regular scientific work, such a coincidence of the experimental and calculated data proves that the amount of supposition permitted in the course of the quantum-mechanical calculations and in the course of processing the spectral experiments is more than enough for practically unlimited increase of proximity between the calculated and experimental results. We have not been able to think of any other reasonable argument to distract readers from this evident conclusion. On the other hand, it is hard for us to accept the legend that has accompanied this story for fifty years now. According to this legend, Kolos and Hertsberg were discouraged. What do you think discouraged them? We are certain that you will never guess. According to the legend, for a whole year Kolos was upset because the calculated value of the dissociation energy was lower (only in the fifth sign, but still lower) than the experimental one. One of his reasons for distress, if there really were any, was the evidence that he had overdone his adjustments of calculations. It was considered good form in quantum theory and its philosophy to end up with a calculated value of the energy lower than the experimental value (towards which it was being adjusted). 

According to the textbooks (particularly K.B. Yatsimirsky and B.K. Yatsimirsky, Chemical Bond,1975, p.145), in 1969 Hertsberg carried out very difficult but accurate measurements of the dissociation energy of the hydrogen molecule and discovered that its dissociation energy is equal to 36117,3 +- 1 cm -1. Such an unthinkable correspondence of the experiment with the theory proves the unlimited adjusted capabilities of the spectral experimental data towards the results of the calculation.

In the same book, the Yatsimirskys describe the history of the experiments and calculations of the hydrogen molecule that occurred before Kolos' calculations and Hertsberg's experiments.

... In order to improve the result (4.02 eV, see  the Table) in the method BC, it is necessary to exceed the limits of the description of the pure covalent bond and take into account the ionic component. For the further improvement of the molecular orbital (MO) method, the interaction of various MO, called configurative interaction, must be taken into account.  

The value 4.12 eV (BC with the ionic members or MO with the configurative interaction) is significantly less than the experimental value.

Besides the left - right correlation, the azimuth and radial correlations should also be taken into account. James and Coolidge took these factors into consideration and obtained the result 4.72eV, with the difference of 0.027 eV from the experimental one. Kolos and Rutan carried out a still more perfect calculation (1960) and obtained Ediss. = 4.7467 ev, with the experimental value of Ediss. = 4.7466 +- 0.0007 e.

The similar resultis observed in the case that we considered in detail above.   

In 1927, Hory (Hory Zeit.f.Phys., 44, 834, 1927) published the results of the experimental definition of the bond length in the hydrogen molecule carried out by the spectral method. According to his results, the bond length was 0.75 Å. The calculated value at that period of time was 0.73 -0.8 Å.

In 1928, Wang (Wang Phys. Rev., 31, 579; 1928) published the results of his quantum - mechanical calculation, in which he defined the moment of rotation and bond length in the hydrogen molecule. These values were 0.4565 x 10-40 and 0.76 Å, respectively.

In January, 1930, H. Hyman and R. Jeppesen (H. Hyman, R.Jeppesen, Nature, 1930, p.462) published their results of the experimental definition of the moment of rotation and bond length in the hydrogen molecule. The experimental method is practically identical to Hory's, but the processing of the experimental data is different. According to the results of the work, the moment of rotation and bond length in the hydrogen molecule are 0.459 x 10-40 and 0.7412 x 10-8, respectively. The article ends with the following:

"In this connection we should like to emphasize that Hory's published I  = 0.467 x 10-40 refers to the true state of zero vibration (v = - 1 / 2) and would now be denoted Ie. Nearly everyone has quoted and used Hory's Ie value as though it were J0.  Our own value of Be is given by the constant term in our equation (60.587), and this leads to Ie = 0.4565 x 10-40 and re = 0.7412 x 10-8.  It is interesting to note that Wang (Phys, Rev., 31, 579; 1928) obtained Ie = 0.459 x 10-40 from a theoretical wave mechanics calculation. We believe that the probable error in the values of B1 to .B9, as given by the above equation, is 0.1 percent or less. As far as the uncertainty of the extrapolation is concerned, the probable error in B0 and Be is not more than 0.2 percent. It is, however, shown in the following letter by Birge and Jeppensen that B0 is definitely perturbed, so our value is not correct. Whether B is correct cannot be tested, since this constant refers to a molecular state which does not exist."

After the experiments by H. Hyman and R. Jeppensen (H. Hyman, R. Jeppensen, Nature, 1930, p. 462) - that is, since 1930 - no experiments on determining the bond length in the hydrogen molecule have been run. Experiments in which the bond length of hydrogen with other atoms was determined were carried out twenty years later. The 1930s - 1950s were the years of fascination with quantum mechanics. The period of the critical approach to quantum theory came only in the '90s. Noting the given dates, it is clear why the length of the bond in the hydrogen molecule has been put into reference books and why the issue of the covalent radius of the hydrogen atom has remained unsolved until our current work.

The coincidence of the covalent radius value of the hydrogen atom, calculated according to the model, with the covalent radius of hydrogen in its compounds with other atoms is additional independent evidence of the correctness of the hydrogen molecule model.

Having summarized all comparisons of the model's calculated and experimental data, we obtain the following results: 

The electron energy of the hydrogen molecule is

Calculated  (Enet ) 2800 kJ/moles                  Experimental   2800 +- 83 kJ/moles

The bond length in the hydrogen molecule is

Calculated  0.61Å                                         Experimental  0.58 -0.62Å