electronic
variablesOn the possible existence of non—Born-Oppenheimer hydrogen excited states
P. Accomazzi
Polimeri Europa s.p.a., Istituto G. Donegani, 28100 Novara, Italy
Email: paolo.accomazzi@polimerieuropa.com
In this work we try to interpret experimental evidence for heliumlike hydrogen spectra in terms of binuclear atoms, following a model proposed to account for a wide range of anomalous phenomena related to low energy nuclear reactions. The original idea of binuclear atom, a hypothetical metastable configuration characterized by nuclear motion with non-negligible kinetic energy, is further analyzed taking into consideration hydrogen electronic excited states. As a result it is shown that H-H atomic impacts, where one of the two H atoms is in an excited electronic state, may give rise to situations that can not be handled within the framework of the Born-Oppenheimer approximation – that leaves open the possibility of a new kind of metastable bound states.
In Condensed Matter Nuclear Science one of the principal problems to be faced is the overcoming of the Coulomb barrier between nuclei, a question that from a theoretical point of view severely challenges experimental claims of low energy nuclear reactions. In 1992 Cerofolini proposed a model, the binuclear atom, that supposed the existence of excited states, normally accessible at energies available during chemical reactions, electrochemical experiments and ion implantation tests, in which nuclei are, in a counterintuitive manner, at a shorter distance than in normal ground state [1]. Such metastable states should have been characterized by a mean life long enough to allow, in particular conditions, for nuclear reactions [2]. In fact shorter nucleus-nucleus distance would increase the nuclear reaction rates by orders of magnitude, due to the sharp decay of nuclear wavefunctions.
The principle of the binuclear atom idea is simple: being hydrogenlike electronic energy a function of the square of the atomic charge, the electronic energy of two nuclei held at a very short distance is much lower than the sum of the electronic energies when atoms are at a valence distance due to the fact that in former case electrons experience a localized nuclear charge which is the sum of the charge of the two nuclei. The internuclear Coulomb repulsion can thus be counterbalanced by this stabilizing electronic energy contribution keeping the total energy of such configuration relatively low. This could possibly give rise to a metastable configuration at accessible energy with two nuclei very close in space.
In this paper an extensive analysis is carried out on the possible existence of a hydrogen binuclear configuration which would have electrons held in a metastable state at energy near, but slightly higher than, that of helium atom. Because nuclear kinetic energy as well as nuclear motion are explicitly taken into account, the analysis must be carried out outside of the framework of the Born-Oppenheimer approximation.
Atomic units are used throughout this work.
electronic
variables
nuclear
variables
total
wavefunction
electronic
wavefunction
nuclear
wavefunction
nuclear
mass
electronic
kinetic operator
nuclear
kinetic operator
As we are concerned with the hydrogen molecule, let’s write the total Hamiltonian as a sum of two contributions, the electronic Hamiltonian Hel and the nuclear part, composed from the nuclear kinetic energy and the internuclear coulomb repulsion term:
The Born-Oppenheimer approximation (BO in short) is based on the following observation: “If a physical system has variables that change slowly and others that change rapidly, the behavior of the fast variables should not be significantly influenced by the rates of change of the slow variables” [3]. The usual study of a molecular system in the BO framework is normally done as follows: the electronic Schrödinger equation is firstly solved with nuclei considered as classical pointlike charges, which means the nuclear kinetic energy is neglected in first approximation. Once determined, the electronic wavefunction is used as effective potential for nuclear motion [4].
Let’s write the total wavefunction of our system as a sum over several BO states:
where the electronic wavefunction is a solution of the electronic eigenvalues equation
.
If we introduce expression (1) into the eigenvalue Schrödinger equation for the total wavefunction and integrate over electronic variables, we obtain a system of coupled equations for the nuclear components [5]:
where the quantities
and
are the non-adiabatic coupling terms between the adiabatic electronic states n and m.
The normal route is to observe that the coupling terms, in case of non degeneracy of electronic states, are exceedingly small and negligible for two reasons: [5]
electronic wavefunctions vary very slowly over a space of the order of the Bohr radius, the value of V’ and V” should consequently be small;
nuclear mass appears at the denominator of the coupling quantity - see second member of eq. (2)
The equations for nuclear motion are thus decoupled and written in the following way:
The only exception arises when two or more electronic states become degenerate or near degenerate in the energy eigenvalue. This may be better understood considering the form taken by the non adiabatic coupling terms after the use of the Hellmann-Feynman theorem [6-7], giving for V’ and V” the following expressions [8]:
This is the case of the first and second excited states for the singlet electronic wavefunction of the hydrogen molecule, as can be seen in the following diagram:
Fig. 1 - Electronic energies for the hydrogen molecule, ground, first and second excited states, singlet case. Atomic units.
Recalling eq. (2) for the coupled nuclear motion in case of degeneracy of electronic wavefunction, a formally correct solution for nuclear equation near R=0 would be the following system of coupled equations:
Expression (8) represents a system of equations for the nuclear wavefunctions coupled through the electronic terms. Recall (eqs. 6 and 7) that the coupling terms V’ and V” are not limited in R=0. If a particular solution exists for this system which is not an ordinary BO solution, it should be located near R=0 where the coupling strength is maximum. This kind of solution could represent a possible metastable state for the hydrogen molecule. The peculiarity of the kind of state thus described would be:
nuclear motion coupled with electronic motion;
heliumlike electronic energies.
H-H atomic impacts with kinetic energy sufficient to bring nuclei at a distance of the order of a few tenth of the Bohr radius, and where one of the H atoms is in an excited electronic state, may give rise to a situation that can be solved only outside the domain of BO approximation – see eq. (8). If a particular solution exists for this system, it would describe a heliumlike metastable state for the hydrogen molecule characterized by nuclear motion coupled with electronic motion [1]. This is a possible conventional explanation for heliumlike hydrogen spectra observed in helium – hydrogen plasmas by Mills and coworkers [9].
G. F. Cerofolini, Can binuclear atoms be formed in head-on impacts at moderate energy?, J. Phys. Chem. 96, 3298 (1992)
G. F. Cerofolini and A. Foglio Para, Can binuclear atoms solve the cold fusion puzzle?, Fusion Technol. 23, 98 (1993)
E. Teller, H. L. Sahlin, General remarks on Electronic Structure, in Physical Chemistry an advanced treatise edited by Henry Eyring, Academic Press, New York/London (1970) p. 3
M. Born, K. Huang, Quantum mechanical foundation in Dynamical theory of Crystal Lattices, Oxford Clarendon Press, (1954) p. 166
L. D. Landau, E. M. Lifsits, in Meccanica quantistica - teoria non relativistica, Editori Riuniti - Edizioni Mir, (1976) p. 379
H.Hellmann, in Einführung in die Quantumchemie (Deuticke, Leipzig, 1937), p. 285
R.P. Feynman, Phys Rev. 56, 340 (1939)
M. S. Child, Molecular collision Theory, Academic Press New York/London, (1974) p. 89
R. Mills, P. Ray, Extreme ultraviolet spectroscopy of helium-hydrogen plasma, J. Phys. D: Appl. Phys. 36 (2003) 1535-1542