Spectroscopy irreducible

Equation (4.15) would be extremely onerous to evaluate by explicit treatment of the nucleons as a many-particle system. However, in Mossbauer spectroscopy, we are dealing with eigenstates of the nucleus that are characterized by the total angular momentum with quantum number 7. Fortunately, the electric quadrupole interaction can be readily expressed in terms of this momentum 7, which is called the nuclear spin other properties of the nucleus need not to be considered. This is possible because the transformational properties of the quadrupole moment, which is an irreducible 2nd rank tensor, make it possible to use Clebsch-Gordon coefficients and the Wigner-Eckart theorem to replace the awkward operators 3x,xy—(5,yr (in spatial coordinates) by angular momentum operators of the total... 

In molecular spectroscopy the term symmetry species is usually substituted for the rather unwieldy expression irreducible representation Furthermore, the word degeneracy, as employed in spectroscopy, is often used to specify their dimension. 

Thus, any representation Tcan be expressed as a function of its irreducible representations Pi. This operation is written as P = S a, Pi, where a, indicates the number of times that Pi appears in the reduction. In group theory, it is said that the reducible representation P is reduced into its Pi irreducible representations. The reduction operation is the key point for applying group theory in spectroscopy. To perform a reduction, we need to use the so-called character tables. 

Ternary virial coefficients consist of two parts. One part is defined in terms of the pairwise additive functions of the interaction, the other by the irreducible ones. Since the latter are not well known, a vast field of study, nearly untouched, is being opened up by collision-induced spectroscopy. 

A fundamental role is played in theoretical atomic spectroscopy by the Wigner-Eckart theorem, the utilization of which allows one to find the dependence of any matrix element of an arbitrary irreducible tensorial operator on projection parameters,... 

In practice, the transformation of any operator to irreducible form means in atomic spectroscopy that we employ the spherical coordinate system (Fig. 5.1), present all quantities in the form of tensors of corresponding ranks (scalar is a zero rank tensor, vector is a tensor of the first rank, etc.) and further on express them, depending on the particular form of the operator, in terms of various functions of radial variable, the angular momentum operator L(1), spherical functions (2.13), as well as the Clebsch-Gordan and 3n -coefficients. Below we shall illustrate this procedure by the examples of operators (1.16) and (2.1). Formulas (1.15), (1.18)—(1.22) present concrete expressions for each term of Eq. (1.16). It is convenient to divide all operators (1.15), (1.18)—(1.22) into two groups. The first group is composed of one-electron operators (1.18), the first two... 

Molecular vibrations, as detected in infrared and Raman spectroscopy, provide useful information on the geometric and electronic structures of a molecule. As mentioned earlier, each vibrational wavefunction of a molecule must have the symmetry of an irreducible representation of that molecule s point group. Hence the vibrational motion of a molecule is another topic that may be fruitfully treated by group theory. 

The considerations on the symmetries of the ground and excited states and the above conditions lead to the selection rule for infrared spectroscopy A fundamental vibration will be infrared active if the corresponding normal mode belongs to the same irreducible representation as one or more of the Cartesian coordinates. 

Character tables, which can be found in several vibrational spectroscopy books, allow the determination, for any molecular point group, of the species (or irreducible representations) in relation to the symmetry elements typical of that group. As further cited below, the classification in terms of a particular symmetry species determines the activity (IR activity, Raman activity, both IR and Raman activity or inactivity) of any mode. 

One of our main motivations for pursuing the development of a density functional response theory for open-shell systems has been to calculate spln-Hamiltonian parameters which are fundamental to experimental magnetic resonance spectroscopy. It is only within the context of a state with well-defined spin we can speak of effective spin Hamiltonians. The relationship between microscopic and effective Hamiltonians rely on the Wigner-Eckart theorem for tensor operators of a specific rank and states which transform according to their irreducible representations [45]. 

Keywords Nonlinear spectroscopy hyperpolarizability carbon nanotubes irreducible representations... 

The symmetry of an isolated atom is that of the full rotation group R+ (3), whose irreducible representations (IRs) are D where j is an integer or half an odd integer. An application of the fundamental matrix element theorem [22] tells that the matrix element (5.1) is non-zero only if the IR DW of Wi is included in the direct product x of the IRs of ra and < f. The components of the electric dipole transform like the components of a polar vector, under the IR l)(V) of R+(3). Thus, when the initial and final atomic states are characterized by angular momenta Ji and J2, respectively, the electric dipole matrix element (5.1) is non-zero only if D(Jl) is contained in Dx D(j 2 ) = D(J2+1) + T)(J2) + )(J2-i) for j2 > 1 This condition is met for = J2 + 1, J2, or J2 — 1. However, it can be seen that a transition between two states with the same value of J is allowed only for J 0 as DW x D= D( D(°) is the unit IR of R+(3)). For a hydrogen-like centre, when an atomic state is defined by an orbital quantum number , this can be reduced to the Laporte selection rule A = 1. This is of course formal, as it will be shown that an impurity state is the weighted sum of different atomic-like states with different values of but with the same parity P = ( —1) These states are represented by an atomic spectroscopy notation, with lower case letters for the values of (0, 1, 2, 3, 4, 5, etc. correspond to s, p, d, f, g, h, etc.). The impurity states with P = 1 and -1 are called even- and odd-parity states, respectively. For the one-valley EM donor states, this quasi-atomic selection rule determines that the parity-allowed transitions from Is states are towards np (n > 2), n/ (n > 4), nh (n > 6), or nj (n > 8) states. For the acceptor states in cubic semiconductors, the even- and odd-parity states labelled by the double IRs T of Oh or Td are indexed by + or respectively, and the parity-allowed transition take place between Ti+ and... 

The far-right column of the character table lists the quadratic functions (or functions involving the quadratic terms of x, y, z, xy, xz, or yz) associated with each irreducible representation that are helpful for both IR and Raman spectroscopy, as discussed later in this chapter. 

Any textbook on application of group theory in molecular spectroscopy contains tables of characters of irreducible representations, which correspond to various symmetry groups of molecules. 

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