Method irreducible tensor

For high-dimensional matrices the time required to transform the interaction matrices from the local into the molecular-state basis set may increase considerably and the procedure loses its battle. Either the direct diagonalisa-tion of the spin Hamiltonian in the local basis set or the irreducible tensor method may be much faster. 

The principal feature of this method lies in the fact that a direct evaluation of an arbitrary matrix element is possible without knowing the explicit form of the wave function. In other words, we no longer need the coupling coefficient matrix. Only the expressions for the reduced matrix elements, the 9/-, 6j-and 3/-symbols are required. The procedure is a generalisation of the method outlined for dinuclear systems [1,2] and will be described in detail for tri-nuclear and tetranuclear clusters. 

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J. S. Griffith, The Irreducible Tensor Method for Molecular Symmetry Groups, Prentice-Hall, Englewood Cliffs, NJ, 1962, p. 20. 

The equilibrium structure is considered of flexible polymer chains within the RIS model. This model is solved by an irreducible tensor method which is somewhat different from, and simpler than, the approach of Flory and others. The results are used to compute the light scattering intensities from dilute solutions of flexible polymer chains, and the angle dependence is found... 

Silver BL (1976) Irreducible Tensor Method. Academic, New York... 

B. L. Silver, Irreducible Tensor Methods, Academic Press, New York, 1976. 

Leavitt RP (1980) Erratum An irreducible tensor method of deriving the longrange anisotropic interactions between molecules of arbitrary symmetry [J. Chem. Phys. 72, 3472 (1980)]. J Chem Phys 73 2017-2017... 

From (2.68) we see that we can add selected terms in 3/3/ to our expression for Pr in (2.51) and hence to the nuclear Hamiltonian, without altering the values of any of the physical observables. We choose these terms so that the rotational Hamiltonian has the same form as the rotational Hamiltonian of a spherical top molecule. We shall see later that with this choice for the rotational Hamiltonian, we can make use of the very powerful techniques of angular momentum theory, in particular, irreducible tensor methods, which would otherwise be denied to us. Accordingly, we modify equation (2.51) to be... 

These represent the nuclear spin Zeeman interaction, the rotational Zeeman interaction, the nuclear spin-rotation interaction, the nuclear spin-nuclear spin dipolar interaction, and the diamagnetic interactions. Using irreducible tensor methods we examine the matrix elements of each of these five terms in turn, working first in the decoupled basis set rj J, Mj /, Mi), where rj specifies all other electronic and vibrational quantum numbers this is the basis which is most appropriate for high magnetic field studies. In due course we will also calculate the matrix elements and energy levels in a ry, J, I, F, Mf) coupled basis which is appropriate for low field investigations. Most of the experimental studies involved ortho-H2 in its lowest rotational level, J = 1. If the proton nuclear spins are denoted I and /2, each with value 1 /2, ortho-H2 has total nuclear spin / equal to 1. Para-H2 has a total nuclear spin / equal to 0. 

In spite of this, there does exist a general theoretical method for dealing with just this situation of the coupling of three (or more) angular momenta. It is the irreducible tensor method of Racah (29 and ITigwer (JO). 

We now return to the two spin system and dicsuss its energies in a manner that forms a suitable basis for the irreducible tensor method. Throughout this section we shall closely follow the usage of Fano and Racak s book (29, which will henceforth be referred to as FR. 

We now consider this matter from the point of view of the irreducible tensor method. It is convenient to work in terms of the unit contra-standard tensor operators which are defined for each system i by the equation... 

The complexity of these formulae, especially of the second, is the justification of the remark made in section 3A that they could hardly be obtained by elementary manipulations and they give a good demonstration of the great power of the irreducible tensor method. 

The irreducible tensor method for molecular symmetry groups. Prentice Hall 1962. 

Griffith, J. S. The irreducible tensor method for molecular symmetric groups. London-Tokyo-Sydney-Paris Prentice Hall International 1962... 

The real version of the irreducible tensor method, related to the complex representations as mentioned, is highly useful in the ligand-field theory, as will be shown in the second part of this paper. An additional reason for this is the fact that expansion theorems concerning functions and operators achieve apt forms when the tensor method is applied to real spherical harmonics. 

The irreducible tensor method was originally developed by G. Racah in order to make possible a systematic interpretation of the spectra of atoms. In the present paper this method has been extended to irreducible sets of real functions that have the same transformation properties as the usual real spherical harmonics. Such an extension is particularly useful in the discussion of the spectra of molecules which belong to the finite point groups or to the continuous groups with axial symmetry. There are several reasons for this. 

B. L. Silver. Irreducible Tensor Methods — An Introduction for Chemists, Volume 36 of Physical Chemistry. Academic Press, New York, San Francisco, London, 1976. 

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