The Formation Mechanism and Binding Energy for the Octahedral Central Structure of the He7+ Cluster

The formation mechanism for the octahedral central structure of the He7 cluster is proposed and its totalenergy curve is calculated by the method of a modified arrangement channel quantum mechanics (MACQM). The energyis a function of separation R between two nuclei at the center and an apex of the octahedral central structure. The resultof the calculation shows that the curve has a minimal energy -19.7296 a.u. At R = 2.40 α0. The binding energy of He7 with respect to He 6He was calculated to be 0.6437 a.u. This means that the cluster of He7 may be formed in thestable octahedral central structure with R = 2.40 α0.     

Formation Mechanism and Binding Energy for Icosahedral Central Structure of He13^＋ Cluster

The formation mechanism for the icosahedral central structure of the He13^ cluster is proposed and its total energy curve is calculated by the method of a Modified Arrangement Channel Quantum Mechanics. The energy is the function of separation R between two nuclei at the center and an apex of the icosahedral central structure. The result of the calculation has shown that the curve has a minimal energy -37.5765 (a.u.) at R=2.70ao. The binding energy of He13^ with respect to He^ 12He was calculated to be 1.4046 a.u. This means that the cluster of He13^ may be formed in an icosahedral central structure with strong binding energy.     

Relaxation of Small Molecules: an ab initio Study

The formation mechanism for the equilateral triangle structure of the He3^ cluster is proposed.The curve of the total energy versus the internuclear distance R for this structure has been caclulated by the method of a modified arrangement channel quantum mechanics,The result shows that the curve has a minimal -7.81373 a.u.at R=1.55 a0. The binding energy of He3^ with respect to He He^ He was calculated to be 0.1064 a.u.(about 2.89 eV).This means that the He3^ cluster may be formed in the equilateral triangle structure stable by the interaction of He^ with two helium atoms.     

Formation Mechanism and Binding Energy for Body-Centred Regular Octahedral Structure of Li7 Cluster

The formation mechanism for the body-centred regular octahedral structure of Li7 cluster is proposed. The curve of the total energy versus the separation R between the nucleus at the centre and nuclei at the apexes for this structure of Li7 has been calculated by using the method of Gou＇s modified arrangement channel quantum mechanics （MACQM）. The result shows that the curve has a minimal energy of-52.169 73 a.u. at R = 5.06ao. When R approaches infinity, the total energy of seven lithium atoms has the value of-51.996 21 a.u. So the binding energy of Li7 with respect to seven lithium atoms is 0.173 52 a.u. Therefore the binding energy per atom for Li7 is 0.024 79 a.u. or 0.674 eV, which is greater than the binding energy per atom of 0.453 eV for Li2, the binding energy per atom of 0.494 eV for Li3 and the binding energy per atom of 0.632 eV for Li5 calculated previously by us. This means that the Li7 cluster may be formed stably in a body-centred regular octahedral structure with a greater binding energy.     

The formation mechanism and the binding energy the body—centred regular tetrahedral structure of He5^＋

We propose the formation mechanism of the body-centred regular tetrahedral structure of the He5^ cluster,The total energy curve for this structure has been calculated by using a modified arrangement channel quantum mechanics method.The result shows that a minimal energy of -13.9106 a.u.occurs at a separation of 1.14a0 between the nucleus at the centre and nuclei at the apexes.Therefore we obtain the binding energy of 0.5202a .u.for this structure.This means that the He5^ cluster may be stable with a high binding energy in a body-centred regular tetrahedral structure.     

Formation Mechanism and Binding Energy for Regular Octahedral Structure of Li6 Cluster

The formation mechanism for the regular octahedral structure of Liscluster is proposed. The curve of the total energy versus the separation R between any two neighboring nuclei has been calculated by using the method of Gou＇s modified arrangement channel quantum mechanics （MACQM）. The result shows that the curve has a minimal energy of -44.736 89 a.u. at R = 5.07a0. When R approaches infinity, the total energy of six lithium atoms has the value of -44.568 17 a.u. So the binding energy of Li6 with respect to six lithium atoms is 0.1687 a.u. Therefore, the binding energy per atom for Li6 is 0.028 12 a.u., or 0.7637 eV, which is greater than the binding energy per atom of 0.453 eV for Li2 and the binding energy per atom of 0.494 eV for Li3 calculated in our previous work. This means that the Li6 cluster may be formed in a regular octahedral structure with a greater binding energy.     

Formation Mechanism and Binding Energy for Body-Centred Regular Icosahedral Structure of Li13 Cluster

The formation mechanism for the body-centred regular icosahedral structure of Li13 cluster is proposed. The curve of the total energy versus the separation R between the nucleus at the centre and nuclei at the apexes for this structure of Li13 has been calculated by using the method of Gou＇s modified arrangement channel quantum mechanics （MACQM）. The result shows that the curve has a minimal energy of-96.951 39 a.u. at R = 5.46ao. When R approaches to infinity, the total energy of thirteen lithium atoms has the value of-96.564 38 a.u. So the binding energy of Lii3 with respect to thirteen lithium atoms is 0.387 01 a.u. Therefore the binding energy per atom for Lii3 is 0.029 77 a.u. or 0.810 eV, which is greater than the binding energy per atom of 0.453 eV for Li2, 0.494 eV for Li3, 0.7878 eV for Li4. 0.632 eV for Lis, and 0.674 eV for Liv calculated by us previously. This means that the Li13 cluster may be formed stably in a body-centred regular icosahedral structure with a greater binding energy.     

Formation Mechanism and Binding Energy for Regular Tetrahedral Structure of Li4

The formation mechanism for the regular tetrahedral structure of Li4 cluster is proposed. The curve of the total energy versus the separation R between the two nuclei has been calculated by using the method of Gou‘s modified arrangement channel quantum mechanics (MACQM). The result shows that the curve has a minimal energy of-29.8279 a.u. at R=14.50 a0. When R approaches infinity the total energy of four lithium atoms has the value of-29.7121 a.u. So the binding energy of Li4 with respect to four lithium atoms is the difference of 0.1158 a.u.for the above two energy values. Therefore the binding energy per atom for Li4 is 0.029 a.u., or 0.7878 eV, which is greater than the binding energy per atom of 0.453 eV for Li2, the binding energy pcr atom of 0.494 eV for Li3 and the binding energy per atom of 0.632 eV for Li5 calculated previously by us. This means that the Li4 cluster may be formed stably in a regular tetrahedral structure of side length R=14.50 a0 with a greater binding energy.     

Formation Mechanism and Binding Energy for Regular Tetrahedral Structure of Li4

The formation mechanism for the regular tetrahedral structure of Li4 cluster is proposed. The curve of the total energy versus the separation R between the two nuclei has been calculated by using the method of Gou＇s modified arrangement channel quantum mechanics （MACQM）. The result shows that the curve has a minimal energy of-29.8279 a.u. at R = 14.50 ao. When R approaches infinity the total energy of four lithium atoms has the value of-29.7121 a.u. So the binding energy of Li4 with respect to four lithium atoms is the difference of 0.1158 a.u.for the above two energy values. Therefore the binding energy per atom for Lh is 0.020 a.u., or 0.7878 eV, which is greater than the binding energy per atom of 0.453 eV for Li2, the binding energy per atom of 0.494 eV for Lia and the binding energy per atom of 0.632 eV for Li5 calculated previously by us. This means that the Li4 cluster may be formed stably in a regular tetrahedral structure of side length R = 14.50 ao with a greater binding energy.     

Formation Mechanism and Binding Energy for Equilateral Triangle Structure of Li3 Cluster

The formation mechanism for the equilateral triangle structure of Lia cluster is proposed. The curve of the total energy versus the interatomic distance for this structure has been calculated by using the method of Gou＇s Modified Arrangement Channel Quantum Mechanics. The result shows that the curve has a minimal energy of-22.338 60 a.u at R = 5.82 ao. The total energy of Lia when R approaches co has the value of-22.284 09 a.u. This is also the total energy of three lithium atoms dissociated from Lia. The difference value of 0.0545 08 a.u. for the above two energy values is the dissociation energy of Li3 cluster, which is also its binding energy. Therefore the binding energy per lithium atom for Lia is 0.018 169 a.u. = 0.494 eV, which is greater than the binding energy of 0.453 eV per atom for Li2 calculated in a previous work. This means that the Li3 cluster may be formed in the equilateral triangle structure of side length R = 5.82ao stably with a stronger binding from the symmetrical interaction among the three lithium atoms.     

Theoretical Study on the Low-Lying Electronic States of He2, He2+, and He2++

The equilibrium geometries, potential energy curves, spectroscopic dissociation energies of the ground and low-lying electronic states of He₂, He₂⁺ and He₂⁺⁺ are calculated using symmetry adapted cluster/symmetry adapted cluster-configuration interaction (SAC/SAC-CI) method with the basis sets CC-PV5Z. The corresponding dissociation limits for all states are derived based on atomic and molecular reaction statics. The analytical potential energy functions of these states are fitted with Murrell--Sorbie potential energy function from our calculation results. The spectroscopic constants B_e, α _e, ω _e, and ω _e χ _e of these states are calculated through the relationship between spectroscopic data and analytical energy function, which are in well agreement with the experimental data. In addition, the origin of the energy barrier in the ground state X¹Σ_g⁺ of He₂⁺⁺ energy curve are explained using the avoided crossing rules of valence bond model.     

The MACQM Calculation for the BInding Energy of the Equilateral Triangle Structure of H3^— Cluster

Considering that the equilateral triangle structure of H3^- cluster can be formed from the interaction of H^- with two hydrogen atoms,a modified arrangement channel quantum mechanics method has been used to calculate the total energy curve for this structure,The result shows that the cureve has a minimal energy-1.6672 a.u.at an internuclear distance of 1.77a0,so its dissociation energy(binding energy)is D(H^- H H)=0.1395,a.u.This means that the cluster H3^- may be formed in an equilateral triangle structure with a bond length of 1.77α0.     

The Three-Pion Decays of the a1(1260)

We investigate the decay of a₁⁺ (1260)→π⁺π⁺π^- with the assumption that the a₁(1260) is dynamically generated from the coupled channel ρπ and KK^* interactions. In addition to the tree level diagrams that proceed via a₁⁺ (1260)→ρ⁰π⁺→π⁺π⁺π^-, we take into account also the final state interactions of ππ→ππ and KK→ππ. We calculate the invariant π⁺π^- mass distribution and also the total decay width of a₁⁺ (1260)→π⁺π⁺π^- as a function of the mass of a₁(1260). The calculated total decay width of a₁(1260) is significantly different from other model calculations and tied to the dynamical nature of the a₁(1260) resonance. The future experimental observations could test of model calculations and would provide vary valuable information on the relevance of the ρπ component in the a₁(1260) wave function.     

The MACQM Calculation of the Total Energy Curve for the Equilateral Triangle Structure of H3+ Cluster

Considering that the cluster H₃⁺ can be formed from the interaction of H⁺ with two hydrogen atoms, a modified arrangeLent channel quantum mechanics method has been used to calculate the total energy curve for the equilateral triangle structure of H₃⁺. The result shows that the curve has a minimal energy -1.2306 a.u. at the internuclear distance R = 1.97ao = 1.04 Å. This bond length is in good agreement with the experimental value of R = 0.98 ± 0.02 Å.     

Formation Mechanism and Binding Energy for Regular Octahedral Structure of Li6 Cluster

The formation mechanism for the regular octahedral structure of Li₆ cluster is proposed. The curve of the total energy versus the separation R between any two neighboring nuclei has been calculated by using the method of Gou's modified arrangement channel quantum mechanics (MACQM). The result shows that the curve has a minimal energy of -44.736 89 a.u. at R = 5.07a₀. When R approaches infinity, the total energy of six lithium atoms has the value of -44.568 17 a.u. So the binding energy of Li₆ with respect to six lithium atoms is 0.1687 a.u. Therefore, the binding energy per atom for Li₆ is 0.028 12 a.u., or 0.7637 eV, which is greater than the binding energy per atom of 0.453 eV for Li₂ and the binding energy per atom of 0.494 eV for Li₃ calculated in our previous work. This means that the Li₆ cluster may be formed in a regular octahedral structure with a greater binding energy.