H ₂⁺ molecule - the basic element of the chemical world. It is an experimental fact that two protons and an electron may join together and form a stable molecular system with well-defined properties. This compound object, bombarded by a photon, electron or other particle may split into two parts: into a neutral hydrogen atom and a free proton, see Fig. 1.

The process in which molecules are split into smaller elements is called dissociation. Dissociation is an endothermic process and some energy is needed to initiate the process. The minimal energy, which is needed, for the process to occur, is called dissociation energy. Dissociation energy is a key parameter, measured experimentally, describing the molecule and molecular kinetics in general. The energy, which is needed to split the molecular hydrogen ion into a neutral atom and a free proton, is quite large and has the value:

 

Since at the threshold, kinetic energies of the dissociation products are equal to zero, we can write the following energy balance for the reaction:

(1a)

On the left side of this equation stands the sum of internal energy of the molecular ion H ₂⁺ and the dissociation energy U_diss. On the right side stands the ionization potential of the hydrogen atom I _H. The internal energy of the molecule containing three particles may be split into two parts: the first one is the electrostatic energy of two protons situated at a distance 2X, see Fig. 2, and the second one is the binding energy, W, of the electron moving in the electric field of two protons:

(1b)

Taking into account the above, and introducing the dimensionless parameter defined in the following way:

(2)

one obtains, assuming that instead of two protons there are two bare nuclei with the electric charge Ze, the relation:

(3)

where a₀ is Bohr radius. The dimensionless quantity w, which contains key information about the energy state of the electron bound in the molecule we will call molecular parameter (M.G. Phys. Lett. 123 (1987) 170). Since the ionization potential of the hydrogen atom is known, I _H = 13.6 eV , the distance between two protons is known, X ~ a₀, and the dissociation energy of the molecule is known, U_diss = 2.79 eV, the key parameter describing the molecule can be calculated.

Introducing the known values of U _diss and X into Eq. (3) one at Z = 1 obtains:

(4)

It is a chief task of the theory to derive this quantity containing key information about the considered molecular ion from some basic postulates of the theory. Since the molecular parameter w is directly related to the electrostatic and kinetic energy of the electron, one can suppose that Newton's equation of motion and Coulomb's interaction law are satisfactory to solve the problem. Unfortunately, the first attempts to solve the problem within classical dynamics were not crowned as a success as the Bohr circular model of the molecule had evidently wrong properties, wrong value of the molecular parameter and lack of stability. Although the Bohr model of the molecule was evidently wrong, it is worthwhile to recall his considerations to present essential aspects of the problem.

An academic problem of molecular chemistry. It is well known that the electronic structure of the atom plays a key role in formation of molecules. It was a circular model of the atom, which formed the basis of Bohr's historical considerations on the model of H₂⁺ molecule. Bohr assumed that the binding electron stays on a circular orbit in the central plane situated between protons (between nuclei). Taking into account that the mass of the electron is by three orders of magnitude smaller than the mass of the proton he could neglect slow motion of heavy protons and perform analysis assuming than those remain at rest, see Fig. 3.

The nuclei can stay at rest, if:

 

where the left side of the equation represents repulsive interaction of protons, while the right side of the equation represents the attraction of protons by a negatively charged electron circulating in a central plane. On the other hand, in the case of the electron staying on a circular orbit, the centrifugal force and radial component of the Coulomb force are equal to each other and we have:

 

At the assumption that protons are at rest, the sum of kinetic energy and potential energy of the electron is a constant quantity. Thus,

 

After elimination v and R, we obtain the value of the molecular parameter w describing dynamic equilibrium of the considered molecular system:

 

In a particular case, that is at Z = 1, we have

 

w = 0.440 .

The comparison of the obtained result with experiment shows, as the experimental value is as high as w = 0.553, that the presented model of the molecule cannot represent a physical reality. reality.

Free-fall molecular oscillator - an electron on the way between two nuclei. With the discovery of the free-fall atomic model I did try to solve the enigma assuming that the electron in the H₂⁺ molecule moves along the shortest path from the one nucleus to the other (M.G. Phys. Rev. Lett. 217 (1994) 43). I assumed that in the close vicinity of the nucleus the electron is back scattered, like a ball from a massive wall, by some kind of a short range conservative force d F (it was an analogy to a short range force related to a spin magnetic field of the electron), see Fig. 4.

In view of the large difference in the mass of the electron and the proton, this formally three body problem, one could reduce to a two body problem with two differential equations; the one describing fast oscillatory motion of the electron between two fixed centres of force and the other describing relatively slow translations of the heavy centres.

Thus, during short intervals of time - that is during one period of motion of the electron, one can assume that the coordinates of the nuclei +X and -X are constant quantities and the energy of the electron is conserved:

(5)

where x is the coordinate, v is the velocity and W is the binding energy of the electron. The above equation for X = const can be easily integrated and the relation between t and x can be given in analytical form. Performing the integration along the entire closed orbit of the electron one obtains the interval of time needed to complete a round trip between the considered nuclei:

(6)

where

(7)

and E and K are elliptic integrals of the first and the second kind of the argument

(8)

If the motion of the electron is known, one can calculate the effective interaction between the electron and each of the protons. Thus

(9)

where d P is the transfer of momentum to the nucleus in the back scattering phase of motion. This is the force which determines slow translations of the nuclei. With the help of the momentum conservation law singularity in the integral (9) can be eliminated and we obtain two identical equations of motion for each of the nuclei:

(10)

where v ₀ is velocity of the electron at x = 0 (in a saddle point of the potential of the two nuclei). At this point it is worthy to note that motion of heavy nuclei is determined by three qualitatively different phenomena. These are: electrostatic interaction between nuclei, "kinetic" pressure of the electron back-scattered from the nucleus, and electrostatic attraction of the nuclei by a negatively charged electron moving between slowly moving nuclei. Averaging with respect to a fast variable x(t) the equation of motion for the nuclei assumes the form:

(11)

It follows from the above that the nuclei may stay at rest at a single, precisely determined value of the molecular parameter w. This is the case when:

(12)

w = w ₀ = 0.907 (for Z = 1) .

The obtained result shows that the considered free-fall molecular oscillator cannot represent a model of the H₂⁺ molecular ion. In the next lecture I will show that ff-molecular oscillator quite accurately describes the motion of binding electrons in a broad class of a crystal lattice of a solid body.

A classical three-body problem - a planar case. The lack of quick success in solving the enigma was not a convincing argument that the problem of the H₂⁺ molecule within a classical three-body problem cannot be solved. Not giving up, it was necessary to look deeply into a difficult three-body problem, which in the past was intensely investigated by such giants of mathematics, as Gauss, Euler and Lagrange.

Formally the Lagrangian describing our problem has the form:

(13)

where indices a , b and e represent two nuclei and the electron. Fortunately, in view of the large difference in the mass of the electron and proton, this fundamental problem of physics could be quite effectively investigated on the basis of a perturbation calculus starting with a well-known solution of the two-fixed center problem.

Like in the case of the free-fall molecular oscillator considered above, the analysis was based on two equations. The first one, describing motion of the electron in the electric field of the fixed charged centers:

(14)

and the other, describing slow translations of the two nuclei at the time averaged interaction with the electron staying on a closed orbit with slowly changing parameters:

(15)

where r(t) is a known solution of a two fixed center problem. Taking into account that the molecular ion H₂⁺ is deprived of angular momentum, which is a result of the fact that the angular momentum of a main dissociation product - that is of the hydrogen atom - is equal to zero, a solution of the two fixed center problem assumes a simple form. In elliptic coordinates l, m it has a following form:

(16)

where S is Hamilton-Jacobi function of the problem, d is the distance between nuclei, E and C are two integration constants and w is the above defined molecular parameter. The integrals describing the motion of the electron can be effectively calculated and the trajectory equation with help of elliptic Jacobi functions sn(u, k _l(w)) and dn((u, k _m(w)) can be written explicitly. The motion is periodic, orbits are closed, if periods T l and T m of Jacobi functions sn((u, k _l(w)) and dn((u, k _m(w)) are related in a following way

(18)

n _l T _l ( k ) = n _m T _m ( k )  _.

where n _l and n _m are integer numbers defining the form of a closed orbit and a detailed shape of the orbit is determined by the value of the molecular parameter w (M.G. INR report No. 810/XVIII/PP, 1967). When the shape of the orbit is known, one can calculate the time-averaged interaction of the electron with a given nucleus. It is important to note that for a given orbit there is a single value of the molecular parameter w , at which the nuclei can stay at rest.

Eureka! H ₂⁺ molecule finally deciphered. In fact, there is an infinite number of closed orbits corresponding to various combination of integer numbers integers n _l and n _m with a precisely defined value of the molecular parameter w. One can suppose that ground state orbit should be relatively simple, corresponding to small values of integers n _l and n _m Investigations of the possible combinations of integers n _l and n _m shows that there are two kinds of closed orbits. Orbits of the first kind, when the electron remains at all times at a large distance from both nuclei, we will call satellite orbits. A particular case of the satellite orbit is shown in Fig. 5.

Orbits of the second kind, when the electron starts and comes back to the same nucleus, we will call singular orbits. Two particular singular orbits are shown in Fig. 6.

The answer to the question, which among closed orbits may represent the considered molecule, is hidden in the value of the molecular parameter w. Examination of a reasonably large number of orbits shows that the only orbit with the value of the molecular parameter exactly equal to the value observed in the experiment does exist. (M.G. "Sprawa Atomu" p. 135, Homo-Sapiens Warszawa 2002r). This is the satellite orbit shown in figure 5. Its animated picture is shown below.

Singular orbits and spin of the electron. Although the found free-fall satellite orbit shown above solved in principle our problem and the electronic structure of H₂⁺ molecule was deciphered, there was a question about the role of singular orbits in molecular chemistry when a short range strong spin magnetic field determines the back scattering phase of the electron from the nucleus. In this case dynamic equilibrium may be reached at few reflections of the electron from the nucleus, however, radial asymptotes of singular orbits of two fixed center problem must fit to radial asymptotes of the free-fall model of the atom, which we discussed in lecture 5. Thus, in principle, radial asymptotes of singular orbits must fit to the angles shown in Fig. 8.

One can be highly surprised, but the two simplest singular orbits as shown in Fig. 7 can together form a spatial orbit with a value of the molecular parameter w just equal to the experimental value.

The obtained result shows that the external shell of the atom, depending on the environment, may have a different form. Depending on the environment, electron orbits in various molecules may have quite different forms, but there exist a limited number of closed orbits which play a decisive role in chemical bonding and are directly related with basic orbits of a two fixed center problem, two of them are shown in Fig. 10.

In the next lecture I will show that one of the orbits shown in Fig. 10 forms the basis for the description of a metallic bond. Few examples of closed orbits of the two fixed center problem shown above are a beginning of the way to describe a rich world of chemistry.

Closing the lecture it is worthy to note that there is, moreover, a whole class of two electron orbits similar to that shown above and joined together by a spin magnetic field, like in the case of the helium atom. How atoms can join together by Van der Walls forces one can learn from appendix to lecture 5 on electronic structure of the helium atom. Some aspects of the problem will be described in the next lecture.

H ₂⁺ molecule in quantum mechanics. Without going into a general discussion on the validity of the entire quantum formalism, some elements of which the reader can find in previous lectures, I will briefly present the essence of the quantum formalism. In the considered case, a Schrödinger equation, which has its origin in a classical energy conservation law for a point like electron

 

has the form:

(19)

According to quantum mechanics the problem will be solved if one invents the wave-function satisfying Schrödinger equation and, which introduced into the relation

(20)

will give the value of the binding energy W ( W = - E ) in agreement with the experiment.

Already at the start, one can be surprised that in quantum mechanics, to solve the problem of molecule, Schrödinger equation for the nuclei is not needed - an equivalent of classical equation (15). Thus, equation (19), which may be considered as an equivalent of a classical two fixed center problem, may describe a part of the problem only. A distance between the nuclei in quantum formalism is fixed and a completely new procedure is needed to describe oscillations of the nuclei.

Keeping in mind that a distance R plays the role of fitting parameter, let us trace out step by step the way to agreement of the theory with the experiment.

The first step of the fitting procedure: it is assumed that the wave-function is a sum of atomic wave-functions for bonding and anti-bonding states,

(21)

This assumption yields the following relation for the energy of the electron E as a function of a distance r between the nuclei ( r = R /a₀):

(22)

This function has at r = 2.157 a minimum, which is equal to 1.76 eV. Thus, for the molecular parameter w theory gives the value 0.640 - the experimental value is as high as 0.553.

In one of the text-books on quantum chemistry, this evidently bad result, bears the following comment :"the presented analysis quite well describes basic properties of the molecular ion H₂⁺" (H.Haken, H.Ch.Wolf, Molec. Phys. and Elem. of Quant. Chem. p. 45-51, Springer-Verlag Berlin, Heidelberg, 1995).
The second step of the fitting procedure: the trial wave-function improved by the variational parameter a

(23)

Really, having at one’s disposal one free parameter more one could improve the final result. Now the molecular parameter calculated with the improved wave-function has the value: 0.610.

The third step of the fitting procedure: a trial wave-function needs "a correction" factor l. It is argued: that due to polarisability initially spherical wave functions of the hydrogen atom must have greater density in the central part between two nuclei. Now, the molecular wave-function had two fitting parameters: a and l:

(24)

A "polarized" wave function improved the situation appreciably - now, the molecular parameter w was not much different from the experimental value: 0.526.

The fourth and final step of the fitting procedure: a trial wave-function corrected for the second time.

(25)

Finally the theory achieved ideal agreement with measurements (although there should be a small difference, by 0.2%, representing oscillation energy of the nuclei). It is a pity but, a satisfaction with the achieved agreement is spoiled by the fact that the excellent agreement between the theory and the experiment may be achieved with many other wave-functions and it is not known, which one among "good" molecular wave-functions is appropriate for a real situation in the considered molecule? Which among "good" wave functions correctly describes, therefore, the density of the electric charge in the considered H₂⁺ molecule? And it is not only an academic problem.