Hamiltonian operators dynamical symmetries

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... 

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... 

Dynamic symmetries for chain (II) correspond to an expansion of the Hamiltonian in terms of invariant operators of 0(2). The linear invariant is... 

Consider first chain (I). A dynamical symmetry corresponding to this route implies that the Hamiltonian operator contains only invariant operators of the chain,... 

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

If the Hamiltonian now contains the Casimir operators of both G, and G[, which do not commute, then the labels of neither provide good quantum numbers. Of course, in general such a Hamiltonian has to be diagonalized numerically. In this way one can proceed to break the dynamical symmetries in a progressive fashion. In (61) all the quantum numbers of G, up to G remain good. If we add another subalgebra beside Gz only those quantum numbers provided by G, on will be conserved, etc. In applications, the different chains are found to correspond to different limiting cases such as the normal versus the local mode limits for coupled stretch vibrations (99). 

The main purpose of this section is to illustrate some specific dynamical symmetries suitable for molecular Hamiltonian operators. In this section we bridge groups, algebras, and related abstract arguments on one side and rovibrational modes, infrared, Raman spectra, and related physical subjects on the other. 

To reiterate, we prefer to describe the one-dimensional model first because of its mathematical simplicity in comparison to the three-dimensional model. From a strictly historical point of view, the situation is slightly more involved. The vibron model was officially introduced in 1981 by lachello [26]. In his work one can find the fundamental idea of the dynamical symmetry, based on U(4), for realizing an algebraic version of the three-dimensional Hamiltonian operator of a single diatomic molecule. After this work, many other realizations followed (see specific... 

Before providing more detail on this procedure, we prefer to outline the strategy specifically adopted in the U(4) problem. By analogy with the U(2) case, we need to perform two distinct steps (1) to identify the subalgebra chains of U(4) closing in SO(3), and (2) to construct the Hamiltonian operator in a dynamical symmetry sense. The first step is a strictly Lie algebraic question, which can be solved by making explicit use... 

From the molecular point of view, the dynamical symmetry based on chain (b) of Eq. (2.104) definitely deserves more attention. If we start from eigenvalues (2.111) and then introduce the vibrational quantum number v defined as in Eq. (2.67), we can write the eigenvalues of the Hamiltonian operator (2.110) in the following form ... 

Finally, in physical situations characterized by potential energy functions intermediate between purely rigid and nonrigid rovibrators, one should consider more complex algebraic treatments in which both U(3) and 0(4) invariant operators are included. Consequently, the Hamiltonian operator can no longer be diagonal in the chosen algebraic basis (related to either one or the other of the two dynamical symmetries). However, matrix elements for any operator of interest have already been explicitly computed in analytical form [35]. 

The next step is the construction of Hamiltonian operators in the dynamical symmetry framework. The general procedure is to restrict the expansion (3.6) to invariant operators of the subalgebra chains, thus leading to two distinct models ... 

By restricting our interest to pure vibrational modes, the dynamical symmetry Hamiltonian operator (4.23) (excluding the Majorana operator... 

Let us consider the simple example where the algebraic Hamiltonian operator is of the dynamic symmetry type U(2)DU(1). Thus we have... 

The second example is a U(2) D 0(2) dynamic symmetry Hamiltonian operator. 

The transformation element is completely independent of M, as is the Hamiltonian operator in Eq. (5.18). This demonstrates that the reaction probabiHties are independent of M. Formally they must be, since our choice of SF z-axis orientation should not affect the dynamics. The first factor in Eq. (5.38) is the usual DVR-FBR transformation element [39]. The second factor is the frame transformation. The remaining factors demonstrate the relationship between the symmetry of the BF rotational state and the SF rotational state. In particular, since the SF rotational state has both definite exchange and inversion symmetry, it projects onto the block of the Hamiltonian... 

The reference (zeroth-order) function in the CASPT2 method is a predetermined CASSCF wave function. The coefficients in the CAS function are thus fixed and are not affected by the perturbation operator. This choice of the reference function often works well when the other solutions to the CAS Hamiltonian are well separated in energy, but there may be a problem when two or more electronic states of the same symmetry are close in energy. Such situations are common for excited states. One can then expect the dynamic correlation to also affect the reference function. This problem can be handled by extending the perturbation treatment to include electronic states that are close in energy. This extension, called the Multi-State CASPT2 method, has been implemented by Finley and coworkers.24 We will briefly summarize the main aspects of the Multi-State CASPT2 method. 

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