Overview of antiproton affinities for functional groups relevant in particle‐beam cancer therapy

Abstract

Background

The recent advances in the production and storing of antiprotons inspire further research regarding using an antiproton particle beam in cancer therapy.

In this work, an overview study of the chemical properties that govern the reactions between biomolecules and slowly moving antiprotons have been investigated.

Aim

The motivation for the article is to grant an improved understanding of the processes involved in antiproton particle‐beam cancer therapy as the antiproton comes to rest, in terms of the impact of molecular chemical properties on the antiproton annihilation site of the biomolecule.

Methods and results

The potential energy surfaces for an antiproton in the vicinity of common functional groups in the human body have been calculated at the CASSCF/CASPT2 level. The energies at different antiproton distances are calculated and compared between different functional groups.

A comparison of the impact on the antiproton affinity from atomic number, bond order, number of lone pairs, bond polarization due to electronegativity, and charge is presented.

Conclusion

The lone pair effect and bond polarization have a relatively large impact. This implies that even on the isolated molecular level, the chemical environment governing the trajectory of an antiproton is a complex problem. The study is applicable primarily in the ultra‐low energy collision regime but can also function as a starting point for quantum dynamical treatments.

1. INTRODUCTION

Antiparticle‐particle interaction is an exciting field of research, as we still do not know if antiparticles are the charge conjugated equivalent of their corresponding regular particle. Questions concerning the evolution of the universe, the validity of the Charge‐Parity‐Time symmetry, and differences between subatomic particles and their antiparticles are hot topics.1, 2 Recent advances in the production, synthesizing, cooling, and trapping of antimatter at CERN in order to perform high precision measurements1, 3, 4, 5, 6, 7, 8, 9 and the state‐of‐the‐art developments by the PUMA10 collaboration for transporting antiprotons have made it more practically feasible to use antiprotons in particle beams for cancer treatment at hospitals.

The interaction between antimatter and matter has been focused on model systems such as hydrogen‐antihydrogen11, 12 and antiproton‐H2,13 and the aim of this article is to investigate the interaction between an antiproton and common functional groups that govern much of the chemistry in the human body. This is done by investigating the impact of chemical properties of atoms and molecules such as the atomic number, bond order, number of lone pairs, bond polarization due to electronegativity, and charge. By understanding the relative impact of these chemical properties on the antiproton affinity at this level, more complex systems might be understood by decomposition.

Today, particle‐beam therapy uses protons to destroy cancer cells by shooting particles into the patient's body with a specified kinetic energy aimed at a specific tissue depth. The damage caused on the cells by the charged protons is at a maximum when the particles come to rest, but all nearby cells are damaged, not only cancer cells.

By using antiprotons instead of protons, the efficiency of the cell damage is higher as demonstrated in Holzscheiter et al, Sellner et al, and Shmatov14, 15, 16 since the molecules are not only affected by the electrical charge but also by strong nuclear forces between matter and antimatter as the antiproton comes to rest resulting in annihilation where mass is transformed into energy. In the annihilation process, high energy photons and short range.

secondary's and recoil ions are released that in turn destroy neighboring cancer cells. A more effective treatment means less side‐effects as the charged particles affect healthy cells along its path to the tumor, due to fewer treatments.

In 2003, the ACE collaboration in CERN started investigating the biological effects and details concerning the use of an antiproton beam,17, 18, 19, 20, 21, 22, 23 and research in this field is also under way by the QUASAR group.24, 25, 26

In this work, we investigate the antiproton affinity (the chemical environment and effect due to the impact on the electronic density and molecular structure) in the presence of functional groups in the human body. We present and analyze the potential energy surfaces (PES) for the electronic ground state of some functional groups in interaction with an antiproton.

Similar work has been carried out regarding proton affinities for amino acids in Dinadayalane et al, Bleiholder et al, and Gronert et al,27, 28, 29 proton‐molecule collisional reaction mechanisms from studying PES in Wang et al,30 and electron affinities in Driver and Jena.31 For antiproton affinities, a related topic is the capture probability of muons, since a muon has the same charge and comparable mass (about 9 times smaller) to an antiproton. In the muon studies carried out for a few selected molecules, the ionicity of the chemical bond between atoms, the atomic number, and the structure of the electronic orbits have been found to be key variables as verified by experimental data.32, 33, 34, 35, 36

However, due to the larger mass, the antiproton falls closer to the nucleus (where it eventually annihilates together with a proton) and the probability for an antiproton to be captured by an atom can thus be expected to be significantly higher, which makes it difficult to draw conclusions regarding antiproton affinities.

The chemical compounds of interest for this study are those containing H, C, N, O, and S atoms since they are found in amino acids building up proteins, and for investigating the impact of the chemical properties we also include Li due to contrasting chemical properties.

The paper continues with a brief discussion of the methods employed and the molecular structure studies. This is followed by a section discussing the results, and finally a conclusion.

2. METHODOLOGY

We have considered the interaction of an antiproton $\overline{p}$ with different organic functional groups (see Figure 1). The electronic ground states have been calculated using the CASPT2 (Complete Active Space method with 2^nd order Perturbation Theory) method37 in the MOLCAS38 program package in association with a 2^nd order Douglas‐Kroll‐Hess transformation to include scalar relativistic effects. The standard Fock operator and IPEA parameter are used, and the imaginary denominator shift is set to 0.1 for decreasing the effect of intruder states. The extended relativistic Atomic Natural Orbital basis set of valence triple zeta accuracy with polarization functions basis set (ANO‐RCC‐VTZP) is used to describe the electrons for each molecule investigated. The selection of the active space in the associated Complete Active Space, Self Consistent Field method is described below, where the antiproton is simulated by adding a negative point charge to the geometry definition and the valence electrons are placed in the active space.

Figure 1.

Molecule interactions with an antiproton investigated in this article

When the antiproton is close to a π or σ orbital, it excites the electrons to the antibonding π* or σ* orbitals and all the π, π*, σ, and σ* orbitals have been taken into account in the active space. For molecules containing an O atom, the P‐orbitals are added as the antiproton can excite electrons from a lone pair in a π or σ state into a more diffuse π* or σ* state P‐orbital (3P orbital). For the S atom, the antiproton rearranges electrons inside the same P‐orbital as the orbital is diffuse enough to avoid an excitation.

The antiproton distance $Z_{\overline{p}}$ from the molecular plane of symmetry is varied between 5 and 0 Å, and over the other coordinates such that a full map image is obtained of the PES, where the internal coordinates of the molecule are fixed in the ground state geometry for each antiproton position. Optimized internal molecular coordinates for each antiproton position would be of interest for describing a more exact antiproton trajectory, but for this comparison study we are only interested in the direction of the antiproton in terms of the relative impact of molecular properties. Once a slowly moving antiproton has entered the electronic density of an atom, it continues toward the nucleus. The geometry of the systems is depicted in Figure 2.

Figure 2.

A, Linear molecule coordinates; B, nonlinear molecule coordinates. The (X,Y,Z) coordinates of the atoms are obtained by letting the molecule reach its ground state before interaction with the antiproton, and the (X,Y) coordinates are marked as a circle with the atomic label on the XY‐plane (molecular plane of symmetry) in the PES plots, where the antiproton distance $Z_{\overline{p}}$=to the XY‐plane is given in the caption

The chemical properties investigated are not independent, making it impossible to vary only one of them to isolate the effect of one of the parameters affecting the antiproton affinity. For example, if we want to compare the effect of the atomic number and keep electronegativity fairly constant, and for the sake of the example assuming that other factors such as molecular bond is identical, we could compare the antiproton affinity for the functional groups C − H, I − H, and S − H. However, in reaching the ground state, the molecules have reordered their electronic density such that they must be described with a different type of molecular bond, affecting the effective electronegativity. The method we use to compare the relevance of non‐independent properties of the different atomic and molecular properties is to compare two molecular species for a set of effects and then try to vary just one of the effects for another set of two molecular species, and then compare the two investigations. The result is an estimation of the relative strength of the different effects.

3. RESULTS AND DISCUSSION

The following sections investigate the effect of atomic number, electronegativity, charge, number of lone pairs, and type of bond (bond order) on the interaction between a molecule and an antiproton.

3.1. Effect of atomic number and electronegativity

The effect of the atomic number is first studied from the interaction of $\overline{p}$ with H − S⁻ compared with H − O⁻ with the geometrical configuration as in Figure 2A. Since Oxygen has a greater electronegativity (≈3.44) than Sulphur (≈2.58), one might expect that the antiproton would fall toward the Oxygen nucleus earlier (at a larger distance) than the Sulphur nucleus. This is not the case, and it can be understood by realizing that due to the difference in the spatial extension of the electronic cloud, which is related to the atomic number, the antiproton will pass the electronic energy barrier sooner for the S atom and be pulled toward the nucleus earlier. The competition between the H and S, and H and O meet a critical distance $Z_{\overline{p}}^C$ where it passes the electronic energy barrier and falls toward the more positive nucleus due to the stronger attractive pull, see Figure 3. In this comparison, with a single bond between the two atoms in the molecules and with an added charge on each molecule, the impact of the atomic number is stronger than the effect of electronegativity.

Figure 3.

Two‐dimensional slices of the PES of A, H − O⁻ and B, H − S⁻ where the colored lines represent different antiproton vertical distances from the molecular plane of symmetry. The two molecules are in their ground state resulting in inter‐atomic distances to the H atom of 0.94 and 1.33 Å for oxygen and the sulphur atom, respectively. The critical distances for the two molecules are $Z_{\overline{p}}^C$(H − O⁻) ≈1.5 and $Z_{\overline{p}}^C$(H − S⁻) ≈1.9

To test the impact of charge, we compared this result with the protonated, charge‐neutral form of the above species, H2S and H2O, which exhibit the same characteristics, see Table 1. We conclude that the atomic number is a stronger effect than electronegativity for these types of single bond molecules. Due to the similarities between the results of the species H2O and H2S, we only show the two plots for H2O before and after the electronic barrier is passed, see Figure 4.

Table 1.

PES differences in a.u. where O and H, and S and H, are at placed on the X‐axis and $\overline{p}$ at Z‐values $Z_{\overline{p}}$ as in Figure 2B. For the molecules H2O and H2S, the O and S atom is at (−0.23, 0) and (−0.14, 0), with the two H atoms at (0.94 ± 1.65) and (0.96 ± 1.56) respectively. A positive ∆E corresponds to a higher energy at the heavier atom

${\mathrm{Z}}_{\overline{\mathrm{p}}}$	∆EHO−	∆EHS−	∆ EH2O	∆ EH2S
2.5	0.018	0.022	0.060	0.003
2.0	0.038	0.005	0.010	0.002
1.5	0.005	−0.048	0.014	−0.021
1.0	−0.090	−0.352	−0.040	−0.202

Figure 4.

The PES of H2O near an antiproton. After the molecule has reached the ground state, the hydrogen atoms are situated at (X,Y) = (0.497 ± 0.873) Å and the oxygen atom at (X,Y) = (0.124,0) Å. A, Before passing the electronic barrier near the heavy atom at ${\mathrm{Z}}_{\overline{\mathrm{p}}}$ = 2.0 Å; B, after passing the electronic barrier at ${\mathrm{Z}}_{\overline{\mathrm{p}}}$ = 1.5 Å

3.2. Lone pair effect

The effect of an atom with a lone pair was first studied by using the molecule H2C═NH, see Figure 5. The N atom has a larger atomic number and greater electronegativity than the C atom so the antiproton should be more attracted to N than C, but this is not the case. The difference here is that the nitrogen atom has a lone pair that increases the electronic density, repelling the antiproton.

Figure 5.

The PES for H2C=NH with an antiproton at $Z_{\overline{p}}$ = 3 Å. The atomic coordinates all lie in the XY‐plane

The comparison of the lone pair contra atomic number is interesting. Increasing the atomic number and electronegativity further but also increasing the lone pair effect, by replacing the NH group by an oxygen atom with two lone pairs, as in H2C═O and CH3―OH, the repulsion of the extra lone pair is even more pronounced, see Table 2. We conclude that the lone pair effect is stronger than both atomic number and electronegativity. Only if we add a charge on O is the lone pair effect matched, giving an indication of the relative strength of charge compared with the effect of the lone pair, atomic number, and electronegativity. This can be understood by considering that the effective number of electrons is changed due to the presence of the lone pair, which affects the spatial extension of the electronic density.

Table 2.

PES energies at $Z_{\overline{p}}$ = 2.0 Å in kcal/mol where the two heaviest atoms a1 (to the left in the notation) and a2 (to the right in the notation) are aligned along X = 0

Molecule	Bond	Ea1	Ea2	∆E
H3C―OH	Single	−115.640	−115.581	0.030
H2C═O	Double	−114.257	−114.227	0.059
H2C═NH	Double	−94.520	−94.503	0.017
H3C―O−	Single	−114.832	−114.837	−0.005
HC≡N	Triple	−93.330	−93.305	0.025
C−≡N	Triple	−92.529	−92.525	0.00

3.3. Single, double, and triple bonds

By using the organic functional groups H3C―OH and H2C═O, we compared the effect of single versus double bonds on the antiproton affinities. Similarly, we used the species H2C═NH and CH_N to compare double versus triple bonds. In these comparisons, they all have the same number of lone pairs, so this effect is neutralized. It is evident that more lone pairs give a more concentrated electronic density between the corresponding atoms, where the polarization of the bond depends on the surrounding atoms, decided in general by which atom has the higher electronegativity. Because of the polarization, it is expected that electronegativity plays a deciding role in these comparisons, and this is also the result as can be seen in Table 2. Increasing the bond order increases the polarization of the bond, and the antiproton is more attracted toward the atom with lower electronegativity. An interesting question arose if the triple bond effect between C and N, where N is more electronegative, could counter a negatively charged C atom in the molecule by comparing HC − N with C⁻ − N. It turns out that the electronic density is distributed such that the effect of charge is only minor, see Figure 6B. The PES 2D slices for C⁻ − N and HCN when the antiproton is at a distance Z = 1–3 Å away from the molecular symmetry axis X are plotted in Figure 6A.

Figure 6.

PES 2D slices for A, HCN and B, C⁻≡N for different antiproton distances $Z_{\overline{p}}$ from the linear molecule axis

3.4. Bond polarization in single bond functional groups

We have also studied H − Li in order to have an electropositive element bound by a single bond to H, see Figure 7. Interestingly, even though Li has a larger atomic number and is more electropositive than H, the antiproton is attracted to Li over H for all distances. This is explained by the effect of the polarization of the bond between H and Li, as the electronic density is shifted toward H which repel the antiproton, following the same patterns as we saw for the cases of the single, double, and triple bond.

Figure 7.

PES 2D slices for H‐Li for different antiproton distances $Z_{\overline{p}}$

4. CONCLUSIONS

We have conducted an overview study of the antiproton affinities for naturally occurring functional groups where we map the effect of the atomic number, charge, bond order, lone pair, and electropositivity.

Charge is a dominant factor regulating the antiproton attraction, but since the electronic density is polarized by the charge it is not obvious toward which atom the antiproton falls. For neutral molecules with atoms that are not particularly electropositive, the effect of the atomic number is distinct. For such atoms, we notice that the attraction of the antiproton is stronger for atoms with a larger atomic number once it has passed the repulsive electronic energy barrier, but for larger distances the antiproton is more attracted to atoms with a lower atomic number, due to the weaker electronic repulsion at a distance. However, in a molecule with a bond between an atom with a larger atomic number but less electronegative and an atom with a lower atomic number but more electronegative, toward which atom the antiproton will fall depends on the distance to the atomic nucleus, bond order and number of lone pairs, the relative electronegativity, and the relative atomic number. Here, parameters such as the number of lone pairs increase the effect of electronegativity, and the impact of relative atomic number can be negated. The strength of the impact of these parameters depends on the details of the molecular electronic distribution.

For the functional groups investigated, a rule of thumb for the relative order is charge > number of lone pairs > bond polarization due to electronegativity > bond order > atomic number for antiproton distances outside the electronic barrier. For shorter distances, the spatial extension of the electronic density, closely related to the atomic number, becomes a dominating factor. However, such an ordering can never be general since the effects are not independent, eg, the lone pair effect increases the effect of electronegativity and bond polarization.

These results and the parameters used can be compared with the formula used for calculating muon capture probability between two atoms in Yoshida et al,35 but experimental data are needed for formulating an equation for the case of antiproton capture probability and for a more detailed description of the parameters involved (in particular for the parameters regulating the ionization of the chemical bond used in Imanishi et al39).

Furthermore, we have looked for an energy minima in the PES where the minimum is not located near one of the atoms for a range of small molecules (a substantial extension of the molecules in Figure 1), but for all the molecules investigated no such molecule was found. However, this null‐result implies that a naturally occurring small molecule that can trap an antiproton for any significant amount of time is unlikely, as it will fall toward one of the atoms in the molecule, penetrate the electronic barrier, and annihilate in the nucleus.

CONFLICT OF INTEREST

The author has stated explicitly that there are no conflicts of interest in connection with this article.

Stegeby H. Overview of antiproton affinities for functional groups relevant in particle‐beam cancer therapy. Cancer Reports. 2018;1:e1128. 10.1002/cnr2.1128