Hamiltonian rotational operator

Here it is taken into account that density matrix p, being a scalar, commutates with any rotation operator, and diq defined in Eq. (7.51) is used. After an analogous transformation, in master equation (7.51) there remains the Hamiltonian, which does not depend on e ... 

It is important to note that the Hamiltonian (2.120) contains the terms which produce both the adiabatic and non-adiabatic effects. In chapter 7 we shall show how the total Hamiltonian can be reduced to an effective Hamiltonian which operates only in the rotational subspace of a single vibronic state, the non-adiabatic effects being treated by perturbation theory and incorporated into the molecular parameters which define the effective Hamiltonian. Almost for the first time in this book, this introduces an extremely important concept and tool, outlined in chapter 1, the effective Hamiltonian. Observed spectra are analysed in terms of an appropriate effective Hamiltonian, and this process leads to the determination of the values of what are best called molecular parameters . An alternative terminology of molecular constants , often used, seems less appropriate. The quantitative interpretation of the molecular parameters is the link between experiment and electronic structure. 

The inversion operation i which leads to the g/u classification of the electronic states is not a true symmetry operation because it does not commute with the Fermi contact hyperfine Hamiltonian. The operator i acts within the molecule-fixed axis system on electron orbital and vibrational coordinates only. It does not affect electron or nuclear spin coordinates and therefore cannot be used to classify the total wave function of the molecule. Since g and u are not exact labels, it was realised by Bunker and Moss [265] that electric dipole pure rotational transitions were possible in ll], the g/u symmetry breaking (and simultaneous ortho-para mixing) being relatively large for levels very close to the dissociation asymptote. The electric dipole transition moment for the 19,1 19,0 rotational transition in the ground electronic state was calculated... 

The complete set of the molecular conversion operations which commute with such an Hamiltonian operator (18) will contain overall rotation operations describing the molecule rotating as a whole, and internal motion operations describing molecular moieties moving with respect to the rest of the molecule. Such a group is called the full Non-Rigid Molecule Group (full NRG). 

Using second-order perturbation theory and evaluating the matrix elements of the vibrational and rotational operators occuring in our simplified Hamiltonian... 

Next consider two rotation operators, T and T ,., that satisfy Eq. (242). Substitution of Eq. (242) into the expression for the blocks of the Hessian matrix, under the assumption that the current CSF expansion coefficients are an eigenvector of the current Hamiltonian matrix, gives... 

The diatomic molecule Hamiltonian has in it a potential energy term that transforms like z. The presence of this term means that the Hamiltonian remains invariant under those transformations for which Rz = z. Only one of the infinitesimal rotation operators Z, I , Iz can generate irreducible representations of the group (C , or As before,... 

The theoretical basis for such features can be obtained by analysis of the time evolution in the rotating frame under action of the Hamiltonian (2.9.8). This requires the diagonaliza-tion of such Hamiltonian and the solution of the Liouville-von Neumann equation (Equation (2.5.3)). Usually, the results are properly described using the fictitious spin-1/2 formalism or the single-transition operator approach [18,20,21]. As example, we give below the matrix representation of the rotation operator corresponding to the selective excitation of the central transition in the case / = 3/2 for a r/2 pulse [22] ... 

When angular momentum M(apy) is governed by Hamiltonian H, this is transformed with a rotation operator to M (aPy), such that... 

When the Hamiltonian is defined as the Zeeman interaction between an angular momentum with direction cosines cos, costi , and cosQ, and magnetic field, which is aligned, a rotation operator is expressed, assuming... 

As indicated earlier, the commutator of with any of the angular momentum component operators happens to be zero, which means that also commutes with L, L, and L. The component operators, though, do not commute with each other. Therefore, the largest set of mutually commuting operators for the rigid rotator problem consists of three, the Hamiltonian, the operator, and any one of the component operators. 

At this point the reader may feel that we have done little in the way of explaining molecular synnnetry. All we have done is to state basic results, nonnally treated in introductory courses on quantum mechanics, connected with the fact that it is possible to find a complete set of simultaneous eigenfiinctions for two or more commuting operators. However, as we shall see in section Al.4.3.2. the fact that the molecular Hamiltonian //coimmites with and F is intimately coimected to the fact that //commutes with (or, equivalently, is invariant to) any rotation of the molecule about a space-fixed axis passing tlirough the centre of mass of the molecule. As stated above, an operation that leaves the Hamiltonian invariant is a symmetry operation of the Hamiltonian. The infinite set of all possible rotations of the... 

A simple, non-selective pulse starts the experiment. This rotates the equilibrium z magnetization onto the v axis. Note that neither the equilibrium state nor the effect of the pulse depend on the dynamics or the details of the spin Hamiltonian (chemical shifts and coupling constants). The equilibrium density matrix is proportional to F. After the pulse the density matrix is therefore given by and it will evolve as in equation (B2.4.27). If (B2.4.28) is substituted into (B2.4.30), the NMR signal as a fimction of time t, is given by (B2.4.32). In this equation there is a distinction between the sum of the operators weighted by the equilibrium populations, F, from the unweighted sum, 7. The detector sees each spin (but not each coherence ) equally well. 

A mapping is said to be symplectic or canonical if it preserves the differential form dp A dq which defines the symplectic structure in the phase space. Differential forms provide a geometric interpretation of symplectic-ness in terms of conservation of areas which follows from Liouville s theorem [14]. In one-degree-of-freedom example symplecticness is the preservation of oriented area. An example is the harmonic oscillator where the t-flow is just a rigid rotation and the area is preserved. The area-preserving character of the solution operator holds only for Hamiltonian systems. In more then one-degree-of-freedom examples the preservation of area is symplecticness rather than preservation of volume [5]. 

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