Normalized irreducible tensorial matrix expansions applied to an effective sp‐type Hamiltonian for the PES of water

Abstract

Any matrix can be expanded on a basis of su(2) normalized irreducible tensorial matrices, NITM, defined in terms of 3-j symbols or coupling coefficients of su(2). The NITM transform under rotations according to Wigner's matrices. If one dimension of an NITM is odd and the other even, the tensor has half-integer rank. A simple NITM basis consists of all NITM having the same numbers of rows and columns as the expanded matrix. A compound NITM basis consists of two or more simple bases, each spanning a corresponding block in the expanded matrix. The choice of NITM basis for expanding an effective Hamiltonian matrix is a crucial step in formulating a model. To illustrate the use of a compound NITM basis, including nonsquare NITM, an effective sp-type overlap-free superposition Hamiltonian is constructed and applied to the photoelectron ionization potential spectrum of water.

Symmetry-Adapted Bases of Matrix Spaces Applied to Quantum Chemistry

“…1. When both representations are irreducible, (4.3) is a simple basis of M(ω 1 ×ω 2 ) discussed previously. , Using (3.12) for semisimple groups with coupling frequencies of zero or one, elements of the NITM are given by where φ is an appropriate phase factor, V is Griffith's V coefficient for finite groups or a 3-j symbol for SU(2), and r̂ 1 is an appropriate contragredient index.…”

“…A notable exception is the mixing of the |2s N 〉 and |2p zN 〉 orbitals, absent in the overlap paradigm. With this term included [ H ] N , N fits the superposition model. , …”

“…These bases consist of generalized normalized irreducible tensorial matrices (GNITM). Unlike previously defined normalized irreducible tensorial matrices NITM, − the GNITM cannot generally be expressed in terms of tabulated V coefficients , or 3j symbols − but are obtained by symmetry-adaptation , of the relevant matrix basis. The GNITM can be used to construct the most general Hamiltonian matrix for a physical system having a particular symmetry group, thereby clarifying the fundamental parameterization of the problem.…”

Normalized irreducible tensorial matrices and the Wigner–Eckart theorem for unitary groups: A superposition hamiltonian constructed from octahedral NITM

“…(1.1) shown to be or-(1. 2) and, by the Wigner-Eckart theorem [6], to transform under elements of SU (2) according to the Wigner rotation matrices [7] It was further shown that the set of NITM { [~~) ] ( " ' ) , k = ljj'l,. .. , j + j ' ; 4 = -k, .. .…”

“…2) and, by the Wigner-Eckart theorem [6], to transform under elements of SU (2) according to the Wigner rotation matrices [7] 0 1992 John Wiley & Sons, Inc.…”