Equations, mathematical dipole moment

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. 

For the respective quantum mechanical description of a molecule in a stationary state, a few additional aspects need to be addressed. First, the system state is characterized by a wavefunction VP, and system properties, such as the total energy or dipole moment, are calculated through integration of VP with the relevant operator in a distinct way. Note that an operator is simply an instruction to do some mathematical operation such as multiplication or differentiation, and generally (but not always) the order in which such calculations are performed affects the final result. Second, the wavefunctions V obey the Schrodinger equation ... 

We will derive the mathematical expressions for monopole and dipole fields in the next section, but based on those results, we can give a physical interpretation of the source terms in each of the integrals on the right-hand side of Equation 20.9. In the first, we note that —V / is a volume source density, akin to charge density in electrostatics. In the second integral of Equation 20.9, J behaves with the dimensions of dipole moment per unit volume. This confirms an assertion, above, that / has a dual interpretation as a current density, as originally defined in Equation 20.1, or a volume dipole density, as can be inferred from Equation 20.9 in either case, its dimension are mA/cm = mA cm/cm. ... 

Purely mathematical restrictions arise from die appearance of a large number of parameters in the equations, as already discussed in Section 3.4. Hylden and Overend [165] have applied the charge flow model in analyzing die high derivatives of the dipole moment for a number of oxygen containing triatomics. 

The terms in equation (4) are generally referred to as the orbital-dipolar interaction (o) between the orbital magnetic fields of the electrons and the nuclear spin dipole, the spin-dipolar interaction (D) between the spin magnetic moments of the electrons and nucleus and the Fermi contact interaction (c) between the electron and nuclear spins, respectively. Discussion of the mathematical forms of each of these three terms appears elsewhere. (3-9)... 

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