3.1 Calculation of the adiabatic potential
We used a quantum chemical calculation software named “Gaussian 09” (G09) [29] to calculate the adiabatic energies for the systems of an antiproton-ion, an atom, and an ion. The coupled-cluster method using single and double substitution from Hartree-Fock determinant (CCSD(T)) method is used in our calculation. We choose AUG-cc-pVQZ basis set (Dunning’s correlation consistent basis set with quadruple-zeta). In our model, it is important to include the polarization functions and diffusion functions, because the antiproton has a negative charge which can polarize and diffuse the atomic/ionic electron cloud. AUG-cc-pVQZ basis set includes the polarization functions, and the “AUG-“prefix represents adding the diffusion functions.
We used the keyword “charge” to include the existence of antiproton. The keyword
“charge” allows us to put a point charge in the calculation. Antiproton has a negative charge and the mass is assumed infinity in quantum chemistry calculation (or adiabatic
approximation), so we just put a negative charge at the position of antiproton. The existence of electrons is strongly prohibited in the vicinity of the antiproton because of the Coulomb repulsion between negatively charged particles. In order to include this effect efficiently, we added some basis functions centered on the position of the antiproton. Because the practical limitation of the smallest distance between particles in G09, we need the keyword “GFInput”
to input the basis by listing the basis set. After writing an input file, we can use G09 to get the adiabatic energies.
If the target atom is an alkali atom, we found that the adiabatic potential can be nicely approximated by the following function:
(3-1)
Here, Z is the atomic number, represents the potential at , which is equal to the total energy of the atom with atomic number , and is the potential at
, which equals the total energy of the alkali ion. In the asymptotic region, the function Eq. (3-1) behaves as a Coulomb-like function ( ), which is the correct asymptotic behavior of the adiabatic potential. The Eq. (3-1) are compared with the numerically obtained adiabatic potentials by taking , , as examples. The Figure 3-1 show that the simple model works nicely for all three atoms.
Figure 3-1 Adiabatic potential energies of antiproton + Atomic ion system. (a). The adiabatic potential energies of , given by Eq.
(3-1) (green line), calculated using G09 (blue cross), and obtained by Ahlrichs et. al. in Ref. [30]. All three results agree well.
(b). The adiabatic potential energies of , given by Eq.
(3-1) (green line), calculated using G09 (blue cross).
(c). Same as (b), for .
(C)
3.2 The atomic binding energy and the reduced mass
In our model, not only the adiabatic potential of system, but the atomic binding energy of is necessary. We used G09 program to calculate the atom energies. We calculate the binding energy of an atom assuming the spherical symmetry of the system, but the adiabatic potential of is calculated assuming the axial symmetry along the
antiproton-atom axis. There may be a numerical discrepancy between these two different calculations. We tested some atoms to make sure that these two calculations are consistent.
The comparison of ionization potentials obtained by various theoretical methods and experiments are shown in the next chapter.
The reduce mass is derived by the formula
(3-2)
, , and are the number of proton, neutron, and electron respectively. , , , and are the mass of proton, neutron, electron, and antiproton respectively.
We choose these particle numbers from the one of the isotope having the largest relative abundance in the earth.
3.3 Cubic spline
The method how to obtain is introduced in the previous chapter. This method requires finding the minima of the effective adiabatic energy. But we can only obtain the adiabatic potential at discretized distances , so we employed an interpolation method cubic spline to obtain the minimum of the adiabatic energy. The cubic spline is a smooth third-order polynomial function that is piecewise-defined. Cubic spline has these properties: (1)
Continuous and smooth, (2) Differentiable, (3) The curve does not turn over, (4) the value is exact at the control points. By using cubic spline, it is easy to find the extreme value of a series because it is differentiable. If the spacing of the control points is small enough, the interpolation result is quite accurate.
When utilizing the cubic spline method to interpolate the values of the adiabatic energy calculated by G09, there is an important thing that is worth to be mentioned. The output file of G09 is the total adiabatic energy which is the electronic adiabatic energy plus the nuclear charge potential. The nuclear-charge potential is given by
(3-3)
This potential is proportional to , but the cubic spline is given by
There is no term, so the interpolation is not good for the total adiabatic energy. Thus, we utilized the cubic spline method to interpolate the electronic adiabatic energy, but not the total adiabatic energy.
3.4 How to obtain the parameters and
We obtain the parameters and from the effective adiabatic potential
(Eq. (2-17)) by solving the conditions Eq. (2-18) and Eq. (2-19). Here we summarize the procedure we followed to obtain the parameters.
(1) The adiabatic potentials of ion-antiprotonic system are calculated by G09 at discrete points.
(2) Obtain the electronic adiabatic energy by subtracting the nuclear-charge Coulomb potential from the adiabatic potential.
(3) Use the cubic spline interpolation method to have a piecewise polynomial functions of electronic adiabatic energy.
(4) Add the nuclear-charge Coulomb potential to (3), and obtain the adiabatic energy in a polynomial form.
(5) Evaluate the effective adiabatic energy for a given angular momentum.
(6) Find the minimum of the effective adiabatic energy.
(7) If the minimum is larger/smaller than the total ionization energy, go back to step (6) and increase/decrease the angular momentum.
(8) If the minimum of the effective adiabatic potential is equal to the atomic binding energy, the position of the minimum is and the angular momentum is . is
approximated by Eq. (2-19).