The angular momentum of a molecule

Angular momentum and transition dipole moment 1.2 The angular momentum of a molecule 

The angular momentum of a molecule forms the basic object of our discussion. We will accordingly attempt to remind the reader briefly of which components it consists and how they combine. For details we refer the reader to well-known books, such as [194, 218, 294, 402]. 

The fundamental problem in the case of a rotating molecule (as a rule, for simplicity s sake, we will consider a diatomic or a linear polyatomic one) is that of the interaction between the electronic motion and rotation of the nuclei. For better clarity and in conformity with the style of our further presentation, we will apply the vector model approach. 

In the more general case S 0 and the molecular angular momenta can be coupled in various ways. It is of primary importance to ascertain to what extent the interaction of the spin momentum S with the orbital momentum L is comparable to the rotation of the molecule, as well as to the interaction of each of the momenta L and S with the internuclear axis. An attempt to establish a hierarchy of interactions yields a number of possible, certainly idealized, coupling cases between angular momenta, first considered by Hund and known as Hund s coupling cases. Here we will discuss the three basic (out of five) cases of coupling of momenta in a linear molecule. 

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For a diatomic molecule, the orbital motion of the two nuclei about each other (the rotational motion) satisfies exactly the same equations, except that the upper limit imposed on / by the principal quantum number n does not apply. By convention, we write the angular momentum of a molecule as J, to distinguish it from the atomic angular momentum L ... 

Gordon s model assumes that molecules in a liquid are undergoing collision-interrupted free rotation. A collision is defined as an event which changes the angular momentum of a molecule. It is furthermore assumed that (a) collisions are of zero duration, (b) collisions change the molecule s rotational velocity but do not change its orientation, (c) successive hard-core collisions are uncorrelated that is, the in-... 

The dynamic state is defined by the values of certain observables associated with orbilal and spin motions of the electrons and with vibration and rotation of [lie nuclei, and also by symmetry properties of the corresponding stationary-state wave functions. Except when heavy nuclei ate present, the total electron spin angular momentum of a molecule is separately conserved with magnitude Sh. and molecular slates are classified as singlet, doublet, triplet., . according to the value of the multiplicity (25 + I). This is shown by a prefix superscript lo the term symbol, as in atoms. 

Let the distribution Pb(0, excited state (see Fig. 4.1(6)) arise under the action of light in the form of a short pulse (6-pulse). The magnetic field B creates precession around B not only of the angular momentum of a separate molecule, but also of the distribution over the ensemble. Since, however, all momenta J are in precession with one and the same angular velocity ujj (4.2), their mutual positions with respect to each other remain the same. Hence, the whole rose of vectors J is in precession as a single entity, which means that the... 

Thus, we come to the important conclusion that the angular momentum of a particle moving at constant speed in a circle can only have values which are multiples of % This quantization in units of was first postulated by Niels Bohr in 1913 for the movement of an electron around the hydrogen atom. Later, we shall see that it applies quite generally to the movement of electrons in atoms and molecules. 

The molecule also has angular momentum, which you would expect it to have because it is rotating. The quantum number / is used to define the total angular momentum of the molecule rotating in three dimensions. The total angular momentum of a molecule is given by the same eigenvalue equation from three-dimensional rotational motion ... 

Fig. 2. A schematic diagram illustrating how a time delay, r, permits the product molecule of an A + BC reaction to rotate into the forward scattering direction. The frequency u) of the rotating complex is set by the angular momentum of the collision, J, and hence by the impact parameter, b.

The mechanical modes whereby molecules may absorb and store energy are described by quadratic terms. For translational kinetic energy it involves the square of the linear momentum (E = p2/2m), for rotational motion it is the square of angular momentum (E = L2121) and for vibrating bodies there are both kinetic and potential energy (kx2/2) terms. The equipartition principle states that the total energy of a molecule is evenly distributed over all available quadratic modes. 

In summary, the molecular orbitals of a linear molecule can be labeled by their m quantum number, which plays the same role as the point group labels did for non-linear polyatomic molecules, and which gives the eigenvalue of the angular momentum of the orbital about the molecule s symmetry axis. Because the kinetic energy part of the... 

Although nitric oxide has an unpaired electron, it is difficult to detect directly by electron paramagnetic resonance. In addition to the low concenttation of nitric oxide in vivo, the angular momentum of the unpaired electron can couple with the angular momentum of the nitric oxide molecule to obscure the paramagnetic properties of nitric oxide (Jones, 1973). However, nitric oxide can be detected as a complex with heme groups, which has been used to show the... 

The magnetic properties of a molecule or ion result primarily from the interaction of the angular momentum and spin of the electrons and nuclei with the magnetic field. To deal with any quantum-mechanical theory of magnetic properties, we need to know some basic properties of the angular-momentum operators and their eigenfunctions. 

Two electrons with opposite spins in the same orbital are described as paired. When extended to molecules, the exclusion principle allows us to understand the pairing of electrons in covalent bonds. The net spin angular momentum of a pair of electrons is zero. If not all electrons are paired in a molecule or solid, magnetic properties arise, as happens with many compounds of transition metals. 

The rotational quantum number for a diatomic molecule (to which the symbol J is usually given) is often visualized as the number of units of angular momentum of a dumbbell rotating end over end. This picture enables us to describe some of the events which occur in reaction kinetics and is often helpful in obtaining orders of... 

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