Study of the stretching modes of the arsine molecule

Vibrational modes of the stibine molecule

“…We show that the accuracy of our model is better than in a previous analysis [5]. In [23] we have proposed an algebraic treatment of the vibrational stretching modes for arsine molecule AsH 3 which is characterized by the condition m 1 (A 1 ) ' m 3 (E), namely U ð4Þ ' U ð3Þ ' Kð3Þ ' Sð3Þ % C 3v . In chain (1), K (3) is the semi-continuous group defined as the semidirect product of the group A (3) (formed by the diagonal unitary matrices) and the group of permutations S (3)…”

“…In (25), [C s ] represents the dimension of irrep C s . The matrix D Cs ðRÞ of the oriented irreps are given in [23]. As usual, in the G-Z representation, n i represents the eigenvalue of the number operator E ii = N i associated with the ith bond (bond XY i of Fig.…”

“…As usual, in the G-Z representation, n i represents the eigenvalue of the number operator E ii = N i associated with the ith bond (bond XY i of Fig. 2 of [23]) of the molecule. n s ¼ P 3 i¼1 N i is the first order invariant of U (3) (up to an additive constant), the eigenvalues of which are bounded condition 0 6 n s 6 N s .…”

“…Since we have discussed the irreps of K (3), its invariant operators and the isomorphism between S (3) and C 3v in [23], we do not repeat that discussion here.…”

Some considerations on the description of vibrational excitations in terms of U(ν+1) unitary groups

“…The description of vibrational excitations of a set of equivalent oscillators in terms of u (m + 1) algebras was first developed by Michelot and Moret-Bailly [15], and later on was further analyzed to include the local subgroup K (m) in order to introduce the most important local interactions as a part of an expansion of the Hamiltonian in terms of Casimir operators [16]. This approach has been applied to several molecular systems, for instance tetrahedral [17] and pyramidal molecules [18,19], and is based on the methodology of algebraic techniques where a chain of groups provides the basis as well as the interactions of the Hamiltonian through the Casimir operator associated with different chains [5].…”