Molecules moments

Molecules Moment of inertia Molecules Moment of inertia ... 

The influence of rotational degrees of freedom during the vibrational predissociation process is the most difficult to model simply. In the spirit of this presentation we have used the simplest possible treatment by replacing the reduced mass n for the translational motion in eq. 6 by I/r for the rotational motion. This substitution, first used by Moore to relate the transfer of collision pair vibrational relaxation to rotational motions, involves I, the vibrating molecule moment of inertia, and r the distance between its center of mass and the vibrating atom. With CH for example, r becomes just the C-H bond length. For a diatomic molecule we... 

Although a diatomic molecule can produce only one vibration, this number increases with the number of atoms making up the molecule. For a molecule of N atoms, 3N-6 vibrations are possible. That corresponds to 3N degrees of freedom from which are subtracted 3 translational movements and 3 rotational movements for the overall molecule for which the energy is not quantified and corresponds to thermal energy. In reality, this number is most often reduced because of symmetry. Additionally, for a vibration to be active in the infrared, it must be accompanied by a variation in the molecule s dipole moment. 

We have two interaction potential energies between uncharged molecules that vary with distance to the minus sixth power as found in the Lennard-Jones potential. Thus far, none of these interactions accounts for the general attraction between atoms and molecules that are neither charged nor possess a dipole moment. After all, CO and Nj are similarly sized, and have roughly comparable heats of vaporization and hence molecular attraction, although only the former has a dipole moment. 

The rotational energy of a rigid molecule is given by 7(7 + l)h /S-n- IkT, where 7 is the quantum number and 7 is the moment of inertia, but if the energy level spacing is small compared to kT, integration can replace summation in the evaluation of Q t, which becomes... 

Equation XVI-21 provides for the general case of a molecule having n independent ways of rotation and a moment of inertia 7 that, for an asymmetric molecule, is the (geometric) mean of the principal moments. The quantity a is the symmetry number, or the number of indistinguishable positions into which the molecule can be turned by rotations. The rotational energy and entropy are [66,67]... 

The long-range interactions between a pair of molecules are detemiined by electric multipole moments and polarizabilities of the individual molecules. MuJtipoJe moments are measures that describe the non-sphericity of the charge distribution of a molecule. The zeroth-order moment is the total charge of the molecule Q = Yfi- where q- is the charge of particle and the sum is over all electrons and nuclei in tlie molecule. The first-order moment is the dipole moment vector with Cartesian components given by... 

Therefore, it has at most five independent components, and fewer if the molecule has some synnnetry. Syimnetric top molecules have only one independent component of 0, and, in such cases, the axial component is often referred to as the quadnipole moment. A quadnipolar distribution can be created from four charges of the same magnitude, two positive and two negative, by arranging them m the fonn of two dipole moments parallel to each other but pointing in opposite directions. Centro-syimnetric molecules, like CO2, have a zero dipole moment but a non-zero quadnipole moment. 

The multipole moment of rank n is sometimes called the 2"-pole moment. The first non-zero multipole moment of a molecule is origin independent but the higher-order ones depend on the choice of origin. Quadnipole moments are difficult to measure and experimental data are scarce [17, 18 and 19]. The octopole and hexadecapole moments have been measured only for a few highly syimnetric molecules whose lower multipole moments vanish. Ab initio calculations are probably the most reliable way to obtain quadnipole and higher multipole moments [20, 21 and 22]. 

The charge redistribution that occurs when a molecule is exposed to an electric field is characterized by a set of constants called polarizabilities. In a imifonn electric field F, a component of the dipole moment is... 

The dipole polarizability tensor characterizes the lowest-order dipole moment induced by a unifonu field. The a tensor is syimnetric and has no more than six independent components, less if tire molecule has some synnnetry. The scalar or mean dipole polarizability... 

The electrostatic potential generated by a molecule A at a distant point B can be expanded m inverse powers of the distance r between B and the centre of mass (CM) of A. This series is called the multipole expansion because the coefficients can be expressed in temis of the multipole moments of the molecule. With this expansion in hand, it is... 

The leading tenn in the electrostatic interaction between the dipole moment of molecule A and the axial quadnipole moment of a linear, spherical or synunetric top B is... 

Consider the interaction of a neutral, dipolar molecule A with a neutral, S-state atom B. There are no electrostatic interactions because all the miiltipole moments of the atom are zero. However, the electric field of A distorts the charge distribution of B and induces miiltipole moments in B. The leading induction tenn is the interaction between the pennanent dipole moment of A and the dipole moment induced in B. The latter can be expressed in tenns of the polarizability of B, see equation (Al.S.g). and the dipole-mduced-dipole interaction is given by... 

If molecule A is a linear, spherical or synnnetric top that has a zero dipole moment like benzene, then the leading induction tenn is the quadnipole-mduced-dipole interaction... 

Note the r dependence of these tenns the charge-indiiced-dipole interaction varies as r, the dipole-indiiced-dipole as and the quadnipole-mduced-dipole as In general, the interaction between a pennanent 2 -pole moment and an induced I -pole moment varies as + L + l) gQ enough r, only the leading tenn is important, with higher tenns increasing in importance as r decreases. The induction forces are clearly nonadditive because a third molecule will induce another set of miiltipole moments in tlie first two, and these will then interact. Induction forces are almost never dominant since dispersion is usually more important. 

Many methods for the evaluation of from equation ( Al.5.20) use moments of the dipole oscillator strength distribution (DOSD) defined, for molecule A, by... 

These moments are related to many physical properties. The Thomas-Kulm-Reiche sum rule says that. S (0) equals the number of electrons in the molecule. Other sum rules [36] relate S(2),, S (1) and. S (-l) to ground state expectation values. The mean static dipole polarizability is md = e-S(-2)/m,.J Q Cauchy expansion... 

Bundgen P, Grein F and Thakkar A J 1995 Dipole and quadrupole moments of small molecules. An ab initio study using perturbatively corrected, multi-reference, configuration interaction wavefunctions J. Mol. Struct. (Theochem) 334 7... 

The microscopic origin of x and hence of Pis the non-unifonnity of the charge distribution in the medium. To lowest order this is given by the dipole moment, which in turn can be related to the dipole moments of the component molecules in the sample. Thus, on a microscopic quantum mechanical level we have the relation... 

It turns out that there is another branch of mathematics, closely related to tire calculus of variations, although historically the two fields grew up somewhat separately, known as optimal control theory (OCT). Although the boundary between these two fields is somewhat blurred, in practice one may view optimal control theory as the application of the calculus of variations to problems with differential equation constraints. OCT is used in chemical, electrical, and aeronautical engineering where the differential equation constraints may be chemical kinetic equations, electrical circuit equations, the Navier-Stokes equations for air flow, or Newton s equations. In our case, the differential equation constraint is the TDSE in the presence of the control, which is the electric field interacting with the dipole (pemianent or transition dipole moment) of the molecule [53, 54, 55 and 56]. From the point of view of control theory, this application presents many new features relative to conventional applications perhaps most interesting mathematically is the admission of a complex state variable and a complex control conceptually, the application of control teclmiques to steer the microscopic equations of motion is both a novel and potentially very important new direction. 

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