Electronic Structure of Fractionally Nuclear Charged Atoms

Diferentes propriedades da química de quarks são estudadas por cálculos ab initio HartreeFock em átomos com carga nuclear fracionária. Os estados fundamental e excitado de átomos de sódio com quarks ligados ao núcleo são obtidos usando cálculos de interação de configuração. Este estudo sugere que a transição eletrônica P S no sódio pode ser usada como guia para observação de quarks não confinados. A variação da energia de ligação eletrônica com a carga nuclear para a série isoeletrônica de átomos com carga nuclear fracionária A e A (A = H, Li, Na, P and Ca) foi analisada. Os cálculos apresentados sugerem, que os quarks não confinados preferem se ligar a núcleos pesados, e que o par quark-antiquark poderia ser estabilizado na presença da matéria atômica.


Introduction
Quantum chromodynamics (QCD) gives a correct phenomenological description of the strong and electroweak interactions, however experimental evidence for quarks is only indirect, being obtained from the properties of hadrons. 1 There are theoretical and experimental evidence that particles having color cannot exist as free entities, but are confined to the interior of hadrons.Although no laboratory experiment has so far liberated a quark from a nucleon, this is not a statement that unconfined quarks do not exit in a stable matter.There may be, e.g., a tiny concentration of primordial quarks that have not found their partners in the expanding Big Bang universe. 23][4][5][6][7][8][9] In these search quark experiments, it is important to know the behavior of unconfined quarks in the stable matter, that is, the so-called quark chemistry. 2By applying quantum mechanics to describe the interaction of a ±1/3e or ±2/3e electrical charge with the atomic matter, it is possible to get some useful information on this quark chemistry.In addition, there is the inherent interest to study these exotic systems, which poses new and stimulating questions in computational quantum chemistry.
The hypothesis of unconfined quarks raises the question of where they reside in the real word.0][11][12] Another possibility is that quarks will absorb baryons from ordinary matter forming quark nucleon complexes. 2In this case, atoms with fractionally nuclear charge can be formed. 2,3,5,11,12Indeed, it is argued that unconfined quarks must have large nuclear appetite, 10 which may be a reason to consider them as being associated with nuclei.The fractional quark charge in the nucleus of an atom modifies their electronic structure and an appropriated spectroscopic technique should be able to detect this change and assign transitions between quarkatom energy levels.In this paper, the 2 P P and Ca atoms indicate that quarks prefer to be bound with nucleus of heavy atoms.The variation of the binding energy with the nuclear charge and some other aspects of the quark chemistry are also discussed in results and discussions section.

Computational Details
Electronic structure calculations of fractionally nuclear charge atoms requires an appropriated choice of wavefunctions in order to take into account the peculiarities of these exotic systems.Here we use the generator coordinate method 13 to design new basis set for the atomic systems.We maintain the 6-311G set of contracted basis in internal regions and include diffuse 14 and polarization 15 functions in external regions.In sodium case the purpose is to reproduce exactly the first D line (5896 Å).Electronic correlation effects are included by performing configuration interaction (CI) calculations with single, double, triple and quadruple excitations.For calculation of electronic transitions, where we consider differences in total energy, the so-called differential correlation effects are considered.The GAUSSIAN 98 program is used in all calculations. 16

Results and Discussions
Total energies for the systems Na +2/3 , Na +1/3 , Na, Na -1/3 and Na -2/3 are given in Table 1.The fractional ionic charges in these systems reflect the quark (or antiquark) ±1/3e or ±2/3e charge added to the sodium nuclei.Each system contains 11 electrons and corresponds to a bound state.CI calculations of the 2 P 2 S electronic transition yield a wavelength of 5896 Å, which reproduces exactly the first sodium D line and is 6 Å away from the second yellow line, i.e., an error about 0.1% (as relativistic effects such as spin-orbit coupling are not included in the calculations, only one transition energy is obtained).The accurate wavelengths calculated here can be a guide in search-forquarks spectroscopic experiments.For example, the D line of the Na +1/3 system lies in the violet region (4411 Å) and the observance of this line would be indication of a +1/3-quark attached to the nucleus.
Table 1 shows that the wavelength increases by decreasing the nuclear charge, indicating that the energy levels of the same principal quantum number (3s, 3p, 3d) approach each other for the negative Na -1/3 and Na -2/3 systems.Table 1 also shows that the variation of total energy is larger for the positive quark as compared to the corresponding negative quark.This is consistent with the fact that the total binding energy varies as E = -kZ (Z is the total nuclear charge, not just the number of protons) with 2 (for Hydrogen = 2).Considering electronic binding energy for atoms from Z = 1 to 20, including Z ± 1/3 and Z ± 2/3 nuclei, we obtain = 2.39 (Figure 1).
There are other consequences of the nonlinear dependence of the binding energy with the nuclear charge.In QCD, the vacuum state can be described as the lowest energy condensate state of gluons and quark-antiquark pairs.A primary creation process occurs when a quarkantiquark pair is produced by a gluon: As the pair separates and becomes energetically favorable, another pair is created.This process goes until the hadron limit, where quarks are confined.When the mean separation between these particles exceeds the hadron limit, quarks are free.In this circumstance, if atomic matter is present we can consider formation of quark atoms according the equation: where A is a neutral atom, Aq and Aq are fractionally nuclear charged atoms with opposite ion charge.Considering the Z dependence, the reaction for the Aq and Aq formation (equation 2) will be an exothermic process.Here we are neglecting the color interaction, which overwhelms all the energies considered here.However, our purpose is to analyze only the atom-quark electronic interaction and consequences of the Z non-linear dependence of binding energy.Table 2 shows the amount of energy liberated ( E) when the quark-antiquark pair interacts with H, Li, Na, P and Ca atoms.
We can observe that increases from Hydrogen to Calcium.Then, more energy is released as the nuclear charge is increased, indicating that quarks prefer to bound with nuclei of heavy atoms, in agreement with predictions of De Rújula et al. 10 Another consequence of the negative signal of is that atomic matter stabilizes the quarkantiquark pair in relation to the condensate gluon state, which favors the hadron formation process.

Conclusions
The electronic structure of fractionally nuclear charged atoms is investigated using CI calculations.Even though more sophisticated calculation could affect the present numerical results and interpretations, our calculated 2 P 2 S electronic transition for quark-sodium atoms illustrates how usual methods of quantum chemistry can be useful in searching for unconfined quarks.In addition, the electronic energy calculations of fractionally nuclear charged atoms produce new information on the interaction of quarks with the atomic matter, not generally considered by researchers in this area.Our results suggest the possibility that unconfined quarks have preference for heavy nuclei and that quark-antiquark pairs could be stabilized in the interaction with the atomic matter.

Table 1 .
Total energy and 2 P

Table 2 .
Energy released in equation 2 for different atoms