Commun. Theor. Phys. Beijing, China) 42 2004) pp. 763 767 c International Academic Publishers Vol. 42, No. 5, November 5, 2004 Formation Mechanism and Binding Energy for Icosahedral Central Structure of He 3 Cluster ZHANG Jian-Ping, 2 GOU Qing-Quan,, and LI Ping Institute of Atomic and Molecular Physics, Sichuan University, Chengdu 60065, China 2 Department of Physics, Leshan Teacher s College, Leshan 64000, China Received March 30, 2004) Abstract The formation mechanism for the icosahedral central structure of the He 3 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 = 2.70a 0. The binding energy of He 3 with respect to He 2He was calculated to be.4046 a.u. This means that the cluster of He 3 may be formed in an icosahedral central structure with strong binding energy. PACS numbers: 36.40.-c Key words: He 3 cluster, binding energy, icosahedral central structure Introduction In recent years, ionic clusters of rare gas species are of considerable interest. [ 5] Nobuo Kobayashi et al. [] have observed stable He 4 cluster produced by the interaction of He 2 with twelve helium atoms in cooled helium gas. They thought that if He 2 remains and forms a nucleus in the center, and the twelve helium atoms may be weakly bounded by polarization force, and then the stability of He 4 is possible to be explained by the closed shell of the icosahedral structure centered by ion He 2. However, the binding energy, for the icosahedral structure centered with He 2 formed by weak polarization force, is very small and the stability of this structure is weak. But we may predict that, if the He 2 at the center is replaced by helium ion He, the new cluster He 3 formed in icosahedal structure centered by He may be greatly more stable than the cluster He 4 formed in icosahedal structure centered by He 2. In order to verify the prediction, in this paper we report the formation mechanism and the calculation for the total energy curve of this structure of the He 3 cluster by using Gou s modified Method of Arrangement Channel Quantum Mechanics MACQM). [6,7] This method has been used by us to calculate successfully the binding energies and some symmetric structures of H n, H n, H n n = 3, 5, 7, 9, 3) [6 0] and He n n = 2, 3, 5, 7). [2 4] 2 Formation Mechanism We think that the cluster He 3 can be formed in the icosahedral central structure by the interaction of He at the center A 3 with twelve neighbor helium atoms at apices A, A 2,..., A,..., and A 2, as shown in Fig.. Their electrons are denoted respectively by e 2 with positive spin and by e 2 with negative spin for the helium atoms situated at A =, 2, 3..., 2), where e 2 are localized around the nuclei at the apex A respectively and e 25 is also localized around the helium ion nuclei situated at center A 3. At a certain instantaneous time, the electron e 2 of the helium atom He e, e 2 ) at the A may interact with the helium ion He e 25 ) at the center A 3 and form bond A A 3 [He e ) e 2 He e 25 )] by the resonance of e 2 between He e ) at A and He e 25 ) at the center A 3. In the meantime, all the electrons around the nuclei at A 2,..., A,..., and A 2, are localized. Fig. The structure of the He 3 cluster. In the same way the twelve bonds A A 3 [He e 2 ) e 2 He e 25 )] =, 2, 3,..., 2) can be formed sequentially and instantaneous with equal opportunities The project supported by National Natural Science Foundation of China under Grant No. 9974027, and the Science Foundation of Education Committee of Sichuan Province of China under Grant No. 0LB04 Correspondence author
764 ZHANG Jian-Ping, GOU Qing-Quan, and LI Ping Vol. 42 as shown in Fig. 2. The resonance of the twelve bonds around the center A 3 makes the He at the center and the twelve He atoms at the apexes bounded together to form the icosahedral structure by the interaction of He with twelve neighboring He atoms according to the twelve channels: where and δ =, 2,..., 2. A A 3 [He e 2 ) e 2 He e 25 )] 2 δ A δ [Hee 2δ, e 2δ )], ) Fig. 2 Formation of twelve bound A A 3 =, 2, 3,..., 2). 3 Method for Calculation of Total Energy According to the MACQM method, [6,7] the twelve channels as shown in Fig. 2 and denoted by Eq. ) are selected for the calculation of the total energy of the system. The wave function for the bond A A 3 in the first channel is made from the linear combination of wave function A e, e 2 ) for helium atom at A multiplying wave function A 3 e 25 ) for helium ion at A 3 and A e 2 ) A 3 e 2, e 25 ). The total wave function for the whole system with the first channel is ψ = N [A e, e 2 )A 3 e 25 ) A e )A 3 e 2, e 25 )]A 2 e 3, e 4 )A 3 e 5, e 6 )... A 2 e 23, e 24 ). In the same way, we may write down the other channel wave functions. The twelve channel wave functions are denoted by ψ = N [A e 2, e 2 )A 3 e 25 ) A e 2 )A 3 e 2, e 25 )] 2 δ A δ e 2δ, e 2δ ), =, 2,..., 2, δ =, 2,..., 2, 2) where N is the normalization factor, and A e 2 ) is ground state wave functions of hydrogen-like ion of helium ions, A e 2, e 2 ) and A 3 e 25, e 2 ) are the ground state wave functions of helium atoms. They have the
No. 5 Formation Mechanism and Binding Energy for Icosahedral Central Structure of He 3 Cluster 765 forms of µ 3 ) /2 µ 3 ) /2 A e ) = exp µra,), A e 2 ) = exp µra,2 ), 2a) π π A e, e 2 ) = µ 3 π exp[ µ r A, r A,2)], A e 2, e 2 ) = µ 3 π exp[ µ r A,2 r A,2 )], where µ =.6875 is the effective nuclear charge for electrons of He determined by variation method and µ = 2.0 is the nuclear charge for the electron of He. The expressions in the brackets of Eq. 2) denote the wave functions of the instantaneous single electron bonds. Under the Born Oppenheimer approximation, the Schrödinger Hamiltonian Ĥ in a.u.) is Ĥ = i= 2 2 i 3 γ i Z 0 r Aγ,i 24 i j>i r i,j 2 µ 3 γ>µ 2b) Z 2 0 R Aµ,A γ, 3) where i or j is used to stand for electrons, A µ and A γ represent any two of the A, A 2,..., and A M nuclei, R Aµ,Aγ is the separation between µ-th and γ-th among helium nuclei, Z 0 = 2 is the nuclear charge of helium nuclei He 2. The partitioning Hamiltonian Ĥ) and the interaction partial ˆV ) for -th channel are Ĥ ) = i= [ 2 i= V ) = Ĥ Ĥ) = 2 j=2 { 2 [ 2 2 i z 0 i= r 2i,2i i r Ai,2i )] r 25,2 r 25,2 2 j>i ] r Ai,2i r AM,25 r 2i ),2j ) z 2 0 R AM,A r 2i ),2j ) r,2j ) r,2j r 2,2j ) r 2,2j Then the Hamiltonian of the system can be expressed as 2 i= i r AM,2 } r AM,2 r A,25, 4) ) r 2i,2j ) r 2i,2j ) r 2i ),25 r 2i,25 r,25 r 2,25. 5) Ĥ = Ĥ) ˆV ), =, 2, 3,..., 2, 6) while the Schrödinger equation for the whole system can be ĤΨ = EΨ, where Ψ = 2 β= C βψ β, all Ψ β β =, 2, 3,..., 2) satisfying Ĥβ) Ψ β = E β Ψ β are defined by the set of equations E E)C Ψ = and so we can be re-expressed explicitly the secular equation form as E E)C = 2 i= 2 i= W β ˆV β) C β Ψ β = 0 β =, 2, 3,..., 2), 7) W β V β) β C β = 0 β =, 2, 3,..., 2), 8) where E = Ψ Ĥ) Ψ, V β β = Ψ ˆV β) Ψ β, W β is an element of the channel coupling matrix W satisfying 2 = W β = and 2 β= W β =, and can be chosen as 2 = W = V 2 = V 2 β= V β β 2 = V = β), β V β β W β = V β 2 = V 2 β= V β β 2 = V β β). 9) Using the procedures of the MACQM method as in Refs. [4], [6], and [7] to solve the secular equation 8), the energy curve ER) for He 3 cluster can be obtained.
766 ZHANG Jian-Ping, GOU Qing-Quan, and LI Ping Vol. 42 According to the spatial symmetry of the icosahedral central structure of the He 3 cluster, we can prove the following relations ε = ψ Ĥ) ψ dτ = H ) = H δ) δδ, 0) x = ψ ˆV ) ψ dτ = V ) = V δ) δδ, ) y = ψ ˆV 2) ψ 2 dτ = V 2) 2 = V 3) 3 = V 4) 4 = V 5) 5 = V 6) 6 = V 3) 23 = V 6) 26 = V 7) 27 = V ) 2 = V 4) 34 = V 0) 3.0 = V ) 3 = V 5) 45 = V 0) 40 = V 6) 56 = V 8) 58 = V 9) 59 = V 7) 67 = V 8) 68 = V 8) 78 = V ) 7 = V 2) 72 = V 9) 89 = V 2) 82 = V 0) 90 = V 2) 92 = V ) 0 = V 2) 02 = V 2) 2, 2) z = ψ ˆV 7) ψ 7 dτ = V 7) 7 = V 8) 8 = V 9) 9 = V 0) 0 = V ) = V 4) 24 = V 5) 25 = V 8) 28 = V 0) 20 = V 2) 22 = V 5) 35 = V 6) 36 = V 7) 37 = V 9) 39 = V 2) 32 = V 6) 46 = V 8) 48 = V ) 4 = V 2) 42 = V 7) 57 = V 0) 50 = V 2) 52 = V 9) 69 = V ) 6 = V 2) 62 = V 9) 79 = V 0) 70 = V 0) 80 = V ) 8 = V ) 9, 3) u = ψ ˆV 2) ψ 2 dτ = V 2) 2 = V 8) 28 = V 9) 39 = V 7) 47 = V ) 5 = V 0) 60, 4) V δ) δ = V ) δ. 5) Inserting Eqs. ), 2), 4), and 5) into Eqs. 0) 5), by simplification the detailed expressions of ε, x, y, z, and u can be obtained. Each expression contains a large number of all kinds of integrals, such as overlap integrals, the Coulomb integrals with two centers between electron and nuclei, the electron and nuclei attractive integrals with three centers and the repulsive integrals with four centers between electrons. The integrals with three and four centers can be calculated numerically with the STO 4G basis sets. The other integrals can be computed analytically in ellipse coordinates. By using the MACQM, we can construct the secular equation for the energy of the icosahedral central structure of the He 3 cluster, i.e. the determinant with 2 orders is equal to zero. 4 Results and Discussions After computing all the above integrals, the Jacobi method is used to resolve the eigenvalue of the secular equation for the cluster He 3, and we have obtained the energy curve which varies with R R is the distance of two nuclei between at the center and at any peripheral apex), as shown in Fig. 3 and Table. The potential curve for He 3 has a minimal energy 37.5765 a.u. at R = 2.70 a 0. Hence, when R approaches, namely the He 3 cluster may be fully dissociated into one helium ion and twelve helium atoms He 3 He 2He), the dissociation energy of the He 3 cluster is.4046 a.u. Namely, the dissociation energy D 0 = EHe 3 ) [EHe ) 2EHe)], where EHe ) is the ground state energy of hydrogen-like ion with effective nuclear charge µ = 2.0 for electron and EHe) is the ground state energy of helium atom with effective nuclear charge µ =.6875 for electrons, which is determined by variation method. This means that the He 3 cluster may be formed in the icosahedral central structure of center radius length R = 2.70 a 0 with strong binding energy. This cluster may be produced by the interaction of He with twelve neighbouring helium atoms in cooled helium gas. Fig. 3 Total energy curve of He 3 versus inter-nuclear distance R.
No. 5 Formation Mechanism and Binding Energy for Icosahedral Central Structure of He 3 Cluster 767 Table Energy values varying with R for the icosahedral central structure of the He 3 cluster. Ra 0 ) Ea.u.) Ra 0 ) Ea.u.).5 36.5358 2.8 37.5738.6 36.7850 2.9 37.5665.7 36.9799 3.0 37.5554.8 37.333 3. 37.54.9 37.3038 3.2 37.5243 2.0 37.3480 3.3 37.5057 2. 37.4207 3.4 37.4855 2.2 37.4759 3.5 37.464 2.3 37.565 3.6 37.4420 2.4 37.5450 3.7 37.492 2.5 37.5633 4.0 37.3498 2.6 37.5734 5.0 37.355 2.7 37.5765 6.0 36.9692 References [] Nobuo Kobayashi and Yozaburo Kaneko, Proceedings of the Second China-Japan Joint Seminar on Atomic and Molecular Physics 988) p. 24. [2] Xiong Yong, Li Ping, and Zhang Jian-Ping, Chin. J. At. Mol. Phys. 9 2003) 597. [3] Gou Qing-Quan, Zhang Jian-Ping, and Li Ping, Commun. Theor. Phys. Beijing, China) 38 2002) 597. [4] Li Ping, Xiong Yong, Gou Qing-Quan, and Zhang Jian- Ping, Chin. Phys. 2002) 08. [5] H. Tanuma, J.H. Sanderson, and N. Kobayashi, J. Phys. Soc. Jpn. 68 999) 2570. [6] Gou Qing-Quan, Zhou Qi-Nian, and Liu Jin-Chao, Chin. J. At. Mol. Phys. 5 988) 78. [7] Gou Qing-Quan, Zhang Xian-Zhou, and Liu Jin-Chao, Chin. J. At. Mol. Phys. 6 989) 995. [8] Li Ping, Yang Si-Qing, and Gou Qing-Quan, Chin. J. At. Mol. Phys. 994) 58. [9] Li Ping, Yin Hua, Gou Qing-Quan, and Miao Jin-Wei, Commun. Theor. Phys. Beijing China) 32 999) 467. [0] Li Ping, Gou Qing-Quan, Zhang Jian-Ping, and Miao Jin- Wei, Commun. Theor. Phys Beijing China) 35 200) 327.