Operator Casimir

Thus the Casimir operator of SO(3) is the familiar square of the angular momentum (a constant of the motion when the Hamiltonian is invariant under rotation). One can show that SO(3) has only one Casimir operator, and it is thus an algebra of rank one. Multiplication of C by a constant a, which obviously satisfies (2.7), does not count as an independent Casimir operator, nor do powers of C (i.e., C2,...) count. Casimir operators can be constructed directly from the algebra. This construction has been done for the large majority of algebras used in physics. 

The eigenvalues / have been evaluated for any Casimir operator of any Lie algebra, and a summary of the results is given in Appendix A. Using the expressions of the appendix, we find, for example, that the eigenvalues of the Casimir operator of SO(3), J2, in the representation 1/ > is... 

The Hamiltonian (2.23) represents the general expansion in terms of the elements Gap, and it corresponds to a Schrodinger equation with a generic potential. In some special cases, one does not have in Eq. (2.23) generic coefficients e ap, apY8 but only those combinations that can be written as invariant Casimir operators of G and its subalgebras, GdG dG"D ", This situation... 

Dynamic symmetry corresponds to an expansion of the Hamiltonian in terms of Casimir operators. The Casimir operator of U(2) plays no role, since it is a given number within a given representation of U(2) and thus can be reabsorbed in a constant term E(). The algebra U(l) has a linear invariant... 

It is convenient to take the U(2) z> 0(2) symmetry of the preceding section as the starting point for approximations. Since it is unnecessary to carry the index z or x, the wave functions can be written simply as IN, m >. Denoting by C2 the Casimir operator of 0(2) with eigenvalues... 

The eigenvalue problem for the Hamiltonian H [Eq. (2.92)] can be solved in closed form whenever H does not contain all the elements but only a subset of them, the invariant or Casimir operators. For three-dimensional problems there are two such situations corresponding to the two chains discussed in the preceding sections. We begin with chain (I). Restricting oneself only to terms up to quadratic in the elements of the algebra, one can write the most general Hamiltonian with dynamic symmetry (I) as... 

It has become customary to call the Casimir operator of U12(2), Majorana operator since it was introduced by Majorana in the 1930 s within the context of other problems,... 

We have stated several times that whenever the Hamiltonian can be written in terms of invariant (Casimir) operators of a chain, its eigenvalue problem can be solved analytically. This method can be applied to the construction of both local and normal Hamiltonians. For local Hamiltonians, one writes H in terms of Casimir invariants of Eq. (4.43). 

Note that there is a duality that stems from the two different ways one can view the Hamiltonian (4.67) (Lehmann, 1983 Levine and Kinsey, 1986). As written, the Majorana operator serves to couple the local-mode states. But the Majorana operator is [cf. Eq. (4.66)] the Casimir operator of U(4) and is a leading contributor to the Hamiltonian, Eq. (4.56) describing the exact normal-... 

Using only terms linear in the Casimir operators. 

The following notation has been introduced in Eq. (4.92) As denote coefficients of terms linear in the Casimir operators, A.s denote coefficients of terms linear in the Majorana operators, Xs denote coefficients of terms quadratic in the Casimir operators, Ks denote coefficients of terms containing the product of one Casimir and one Majorana operator, and Zs denote coefficients of terms quadratic in the Majorana operators. This notation is introduced here to establish a uniform notation that is similar to that of the Dunham expansion, where (Os denote terms linear in the vibrational quantum numbers, jcs denote terms that are quadratic in the vibrational quantum numbers and y s terms which are cubic in the quantum numbers (see Table 0.1). Results showing the improved fit using terms bilinear in the Casimir operators are given in Table 4.8. Terms quadratic in the Majorana operators, Z coefficients, have not been used so far. A computer code, prepared by Oss, Manini, and Lemus Casillas (1993), for diagonalizing the Hamiltonian is available.2... 

In the local basis, all terms involving Casimir operators are diagonal. For example, the term XuCf has an expectation value of... 

Using only terms linear in the Casimir operators. c Using all the terms bilinear in the Casimir operators in Eq. (4.92). 

Adapted from Iachello and Oss (1990). Terms both linear and bilinear in the Casimir operators in Eq. (4.96) have been used in the fit. See Appendix C. States are designated both by normal-mode quantum numbers and by localmode quantum numbers. 

The Casimir operators, C, and Cy have been defined in Eq. (5.12). The operators Cjjk are given by... 

The operator Cy is diagonal and the vibrational quantum numbers v, have been used instead of /w,. In practical calculations, it is sometime convenient to subtract from Cy a contribution that can be absorbed in the Casimir operators of the individual modes i and j, thus considering an operator Cy whose matrix elements are... 

Situations in which the Hamiltonian does not have a dynamic symmetry (i.e., it contains Casimir operators of both chains), as, for example,... 

In order to obtain spectroscopic accuracy it was found necessary, as discussed in Chapter 4, to introduce higher-order terms, for example, products and powers of Casimir operators. These higher-order terms can be dealt with easily within the approach discussed here. For example, a Hamiltonian of the type... 

For any Lie algebra, one can construct a set of operators, called invariant or Casimir operators, C, such that... 

A Casimir operator containing p operators X, is called of order p. Only unitary algebras U(n) have linear Casimir operators. All other algebras have Casimir... 

Another ingredient one needs in the application of algebraic methods to problems in physics and chemistry is the eigenvalues of Casimir operators in the representations of Section A.8. The known solution is given in Table A.5. 

Table A.5 gives the eigenvalues of the Casimir operator of SO(3) in the representation J as...

Table A.5 Eigenvalues of some Casimir operators of Lie algebras...

This appendix provides a summary of the functional form of the algebraic Hamiltonian used in the text for tri- and tetratomic molecules. Values of the parameters are reported both for a low-order realistic representation of the spectrum and for accurate fits using terms quadratic in the Casimir operators. 

Many of these coefficients are not used in actual fits. The actual Amat-Nielsen parameters are denoted by %12>12, 13,13, 23,23- In some of the fits, for convergence reasons, the Casimir operators C12, C13, C23, C123 are divided by their respective Ns, that is, the operators... 

---