Projections, vibration-rotation Hamiltonians

First, as the molecule on which the chromophore sits rotates, this projection will change. Second, the magnitude of the transition dipole may depend on bath coordinates, which in analogy with gas-phase spectroscopy is called a non-Condon effect For water, as we will see, this latter dependence is very important [13, 14]. In principle there are off-diagonal terms in the Hamiltonian in this truncated two-state Hilbert space, which depend on the bath coordinates and which lead to vibrational energy relaxation [4]. In practice it is usually too difficult to treat both the spectral diffusion and vibrational relaxation problems at the same time, and so one usually adds the effects of this relaxation phenomenologically, and the lifetime 7j can either be calculated separately or determined from experiment. Within this approach the line shape can be written as [92 94]... 

Here, Q is the projector on the bound subspace and P projects onto the open, continuum channels. The intramolecular coupling is written as V+ U so that, as before, U is any additional coupling brough about by external perturbations. The equation H = Hq + V+U, where Ho is the zero-order Hamiltonian of the Rydberg electron and so includes only the central part of the potential due to the core plus the motion (vibration, rotation) of the core, uncoupled to the electron. The perturbations V + U can act within the bound subspace, as the operator Q(V+l/)Q is not necessarily diagonal and is the cause of any intramolecular dynamics even in the absence of coupling to the continuum. The intramolecular terms can also couple the bound and dissociative states. 

After the separation of the kinetic energy operator due to the center-of-mass motion from the Hamiltonian, the Hamiltonian describes the internal motions of electrons and nuclei in the system. These in the BO approximation can be separated into the vibrational and rotational motions of the nuclear frame of the molecule and the electronic motion that only parametrically depends on the instantenous positions of the nuclei. When the BO approximation is removed, the electronic and nuclear motions become coupled and the only good quantum numbers, which can be used to quantize the stationary states of the system, are the principle quantum number, the quantum number quantizing the square of the total (nuclear and electronic) squared angular momentum, and the quantum number quantizing the projection of the total angular momentum vector on a selected direction (usually the z axis). The separation of different rotational states is an important feamre that can considerably simplify the calculations. 

In this equation Av, Bv, and Cv are the rotational constants and La, Lb, and Lc are the projections of the rotational angular momentum on the principal inertial axes of the molecule. The Hamiltonian in Eq. (1) is often referred to as a rigid-rotor Hamiltonian, even though significant vibrational effects appear in the rotational constants. To good approximation... 

Thus the asymptotic quantum states are labeled by the vibrational and rotational quantum numbers, whereas the projection quantum number is treated classically. We have in the above Hamiltonian indicated that it depends upon time through the classical variables. Thus the quantum mechanical problem consists of propagating the solution to the TDSE... 

To form a microcanonical ensemble for the total Hamiltonian, H = HTib + Hrot, orthant sampling may be used for energy E = H. A (2n + 3)-dimen-sional random unit vector is chosen and projected onto the semiaxes for jx, jy, and jz [e.g., the semiaxis for jx is (2fx )1/2] as well as the semiaxes for Q and P. Since rotation has one squared-term in the total energy expression, whereas vibration has two, the average energy in a rotational degree of freedom will be one-half of that in a vibrational degree of freedom. 

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