Molecules uncoupled state

The main conclusion of this section is that the matrix elements of all terms in the collision Hamiltonian in the fully uncoupled space-fixed representation can be reduced to simple products of integrals of the type (8.46). Such matrix elements are very easy to evaluate numerically. The fiilly uncoupled representation is therefore very convenient for the development of the coupled channel codes for collision problems involving open-shell molecules with many angular momenta that need to be accounted for. The price for simplicity is a very large number of basis states that need to be included in the expansion of the eigenstates of the full Hamiltonian to achieve full basis set convergence (see Section 8.3.4). 

The total angular momentum basis is thus computationally more efficient, even for collision problems in external fields. There is a price to pay for this. The expressions for the matrix elements of the collision Hamiltonian for open-shell molecules in external fields become quite cumbersome in the total angular momentum basis. Consider, for example, the operator giving the interaction of an open-shell molecule in a 51 electronic state with an external magnetic field. In the uncoupled basis (8.43), the matrix of this operator is diagonal with the matrix elements equal to Mg, where is the projection of S on the magnetic field axis. In order to evaluate the matrix elements of this operator in the coupled basis, we must represent the operator 5 by spherical tensor of rank 1 (Sj = fl theorem [5]... 

In order to appreciate the size of the basis sets required for fully converged calculations, consider the interaction of the simplest radical, a molecule in a electronic state, with He. The helium atom, being structureless, does not contribute any angular momentum states to the coupled channel basis. If the molecule is treated as a rigid rotor and the hyperfine structure of the molecule is ignored, the uncoupled basis for the collision problem is comprised of the direct products NMf ) SMg) lnii), where N = is the quantum number... 

If the number of primary bonds capable of yielding a particular atom is determined by its valence state (by the number of uncoupled valence electrons corresponding to this st te), then the number of secondary bonds is determined by the valence state of this atom in a molecule and depends upon the structural features of the molecule as a whole and upon external circumstances. 

DNP is an oxidative phosphorylation uncoupler. It makes the process only about 40% efficient by uncoupling a high energy phosphate molecule from ATP and therefore turning ATP into ADP. To maintain an adequate supply of ATP, the body must step-up production. For this reason metabolism is significantly increased and an incredible amount of calories are burned. During this accelerated metabolic state, and due to the need for ATP production, most of the calories come from fatty acids (adipose/fat tissue). So little or no muscle is lost (With adequate protein intake). 

The extension to more than one dimension is rather straightforward within the time-dependent approach (Heller 1978a, 1981a,b). For simplicity we restrict the discussion to two degrees of freedom and consider the dissociation of the linear triatomic molecule ABC into A and BC(n) as outlined in Section 2.5 where n is the vibrational quantum number of the free oscillator. The Jacobi coordinates R and r are defined in Figure 2.1, Equation (2.39) gives the Hamiltonian, and the transition dipole function is assumed to be constant. The parent molecule in the ground electronic state is represented by two uncoupled harmonic oscillators with frequencies ur and ur, respectively. 

Once we have removed the terms which couple different electronic states (at least to a certain level of accuracy), we can deal with the motion in the other degrees of freedom of the molecule for each electronic state separately. The next step in the process is to consider the vibrational degree of freedom which is usually responsible for the largest energy separations within each electronic state. If we perform a suitable transformation to uncouple the different vibrational states, we obtain an effective Hamiltonian for each vibronic state. Once again, we adopt a perturbation approach. 

Each channel is defined by a unique set of quantum numbers for the target degrees of freedom. There are five such labels for each channel. They are (1) J — the total angular momentum and (2) M, its projection on an axis fixed in space. In addition there are labels (3) n for the vibrational motion of the molecule, (4) j for the molecular rotational degree of freedom, and (5) l for the atom-molecule orbital angular momentum. The equations for one set of (J,M) are uncoupled from equations for other values of (J,M). The equations for a function labeled by one value of (n,j,Z) are coupled to values of all the other functions labeled by (the same or) different values of (n,j, ). The number of coupled equations we have to solve therefore depends on the number of molecular vibration-rotation states we have to treat in the scattering dynamics at each collision energy. 

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