IMPRIMIR VOLTAR
A. Ciências Exatas e da Terra - 3. Física - 3. Física Atômica e Molecular
Calculation of electronic energies levels of molecular ions from the Hamilton-Jacobi equation: HeH2plus, LiH3plus and BeH4plus
Deise Tamara dos Santos Cavalcante Machado 1
Antonio Luciano A. Fonseca 1
Daniel L. Nascimento 2
Jonathas A. Campos 1
Antonio O. Cleves Nunes 1
(1. Instituto de Física, Universidade de Brasília; 2. Dept. de Matemática, IE, Universidade de Brasília)
INTRODUÇÃO:

In this work by making use of theHamilton-Jacobi equation we obtain exact value of the electronic energy levelsfor the hetero-molecular ionsHeH2+,LiH3+andBeH4+as a function ofnuclear distance. The Hylleraas’ method in association with the seriesestablished by Wind-Jaffewas applied to these molecular ionsand the results are in good agreement with those foundin literature.

METODOLOGIA:

Heteronuclear kind diatomic systems such as HeH2+, LiH3+ and BeH4+ have been studied by several methods. Among others we have  the seminumerical 2D approach [1], boundaries set in 2Z given by algebraic topological structures [2], LCAO method [3], separable Hartree-Fock-like solution [4], analytical approaches  based upon the internuclear separation [5], universal basis sets of elliptical functions [6] and treatment of the continuum radial wavefunctions for the coulomb two-centre problem [7].

            In spite of the progress brought about by QM, the Hamilton-Jacobi equation (HJE) has been used to study the atoms and molecules systems [8-12].

In the current work we obtain the exact solution of the Hamilton-Jacobi equation through eigenvalue equations. We apply this equation in association with the series established by Wind-Jaffe [13, 14] to hetero-molecular ions HeH2+, LiH3+ and BeH4+ in order to calculate the electronic energies levels for several excited states. To determine visando a fuçnetico ra a energia eletornica em funçthe exact value of the atomic energy levels we used the method developed by Hylleraas [15] in association with the series established by Wind-Jaffe [13, 14] using the symbolic calculation and Maple codes, to obtaining the electronic energy for all molecular orbitals. The results are in good agreement with those found in literature.

RESULTADOS:

Our tables we show our results and some authors results concerning the electronic energies as function of R  to the 1s sigma state for HeH2+ , the electronic energies obtained in this work and in ref. [2] are show for several R values to the 1s sigma state for LiH3+. We can see from both tables that our results are in good agreement with those found in the literature, the several molecular orbitals for the states sigma, p and d for HeH2+ molecular ion, respectively. The same is given for the LiH3+ molecular ion in the tables 6-8 and in the tables 9-11 for BeH4+ molecular ion, respectively.

 

CONCLUSÕES:

In this work we have shown that it is possible to use the HJE to obtain exact value of the electronic energy levels for the hetero-molecular ions HeH2+, LiH3+ and BeH4+ .

To determine the solutions of this eigenvalue equation we used the method developed by Hylleraas in association with the series established by Wind-Jaffe in Maple codes to obtain the electronic energy for some molecular orbitals as given in tables 1-11.

Using Eq. (32) in elliptic confocal coordinates  related  to Cartesian coordinates  through transformation given by Eq. (34) we have plotted in Fig. 1 the normalized wave function for 2p sigma orbital for HeH+2 molecule for R = 4.0 a.u. .  In  Fig. 2 we give the wave function for 3d sigma orbital for LiH+3 molecule for R = 6.2 a.u  and  in Fig. 3 the wave function for 4f sigma orbital for BeH+4 molecule is plotted for R = 9.1 a.u. is shown.

            The results presented above are in good agreement with those found in literature which makes us feel confident that those that could not be found in any other work are also reliable. In other words, we are able to determine any state and its wave function showing that the method is reliable.

Instituição de fomento: CNPq
 
Palavras-chave: Hamilton-Jacobi equation; hetero-molecular ions ; Hylleraas’ method .
Anais da 58ª Reunião Anual da SBPC - Florianópolis, SC - Julho/2006