Symmetric states field equations

In the triplet model the spin polarization is with respect to the internal molecular states, TjJ>, Ty>, and T > of the triplet and evolves with time according to the time-dependent Schrodinger equation into a spin polarization with respect to the electron spin Zeeman levels Ti>, Tq>, and T i> in an external magnetic field Bq. Consider a simple case of axially symmetric zero-field splitting (i.e., D y 0 and E = 0 D and E are the usual zero-field parameters). Tx>, [Ty>, and TZ> are the eigenstates of the zero-field interaction Hzfs, where Z is the major principal axis. The initial polarization arising from the population differences among these states can be expressed as... 

This factorization amounts to the statement thatEq. (2.25) breaks down into two separate linear systems, one for the determination of a orbitals, and the other for n orbitals. In the Hartree-Fock scheme, self-consistent field equations (SCF equations 2.25) have as solutions symmetry-adapted functions (i.e. in the case of planar unsaturated molecules symmetric or antisymmetric functions with respect ot the molecular plane), at least for closed-shell ground states iM8,20,2i) ... 

After transforming to the collective state basis, the master equation (31) leads to a closed system of 15 equations of motion for the density matrix elements [46]. However, for a specifically chosen geometry for the driving field, namely, that the field is propagated perpendicularly to the atomic axis (k rn = 0), the system of equations decouples into 9 equations for symmetric and 6 equations for antisymmetric combinations of the density matrix elements [45-50]. In this case, we can solve the system analytically, and find that the steady-state values of the populations are [45,46]... 

One might speculate whether the present picture would predict or even allow gravitational waves. It is evident that the present microscopic model will bring up the non-participation of gravitational waves. This is not at all surprising, since it is supported by Birkoff s theorem [17] that states that any spherically symmetric solution of the vacuum field equations must be stationary and asymptotically flat... 

The Entrance Velocity Field For an isothermal, steady state, incompressible flow of a Newtonian fluid being symmetrical in the azimuthal direction, the governing equations are the Navier-Stokes equations and the steady state continuity equation. In dimensionless form the equations are ... 

Detailed balance is a chemical application of the more general principle of microscopic reversibility, which has its basis in the mathematical conclusion that the equations of motion are symmetric under time reversal. Thus, any particle trajectory in the time period t = 0 to / = ti undergoes a reversal in the time period t = —ti to t = 0, and the particle retraces its trajectoiy. In the field of chemical kinetics, this principle is sometimes stated in these equivalent forms ... 

If kinetic processes on catalytic surfaces in S02 oxidation are assumed to be at steady state, temperature and concentration fields in a radially symmetrical, adiabatic catalyst bed are described by the equations collected in Table IX for the reactor space 0 and time I > 0 (Matros, 1989 ... 

This equation is to be compared with the second equation (8) In the absence of the field, g = 0, the two equations become identical and predict a symmetric bifurcation from the trivial state 3 = 0 ... 

Only spatially degenerate states exhibit a first-order zero-field splitting. This condition restricts the phenomenon to atoms, diatomics, and highly symmetric polyatomic molecules. For a comparison with experiment, computed matrix elements of one or the other microscopic spin-orbit Hamiltonian have to be equated with those of a phenomenological operator. One has to be aware of the fact, however, that experimentally determined parameters are effective ones and may contain second-order contributions. Second-order SOC may be large, particularly in heavy element compounds. As discussed in the next section, it is not always distinguishable from first-order effects. 

When the two dipoles occupy symmetrical positions in the cell, as for instance in the case of the anthracene crystal, (1.70) may be further simplified by introducing symmetric and antisymmetric states for all directions of K, with respect to the assumed symmetry. Then (1.70) reduces to 2 x 2 determinants for the two (symmetric and antisymmetric) transitions. The solution of (1.70) leads to four values of co for each wave vector K, i.e. to four excitonic branches. In general, the crystal field is assumed weak compared to intramolecular forces, so that coupling between excitonic branches may be neglected. To a first approximation, each of the excitonic branches, symmetric and antisymmetric, is given by the equation... 

We now specialize the discussion to the ligand field theory situation and define the orthonormal set of spin-orbitals we shall use in the determinantal expansion of the many-electron functions Vyy for the groups M and L. First we suppose that we have a set of k orbitals describing the one-electron states in the metal atom these will be orthonormal solutions of a Schrodinger equation for a spherically symmetric potential, V<,(r), which may be thought of as the average potential about the metal atom which an electron experiences ... 

When a nucleus with a quadrupole moment is situated in an inhomogeneous field that is not axially symmetric (i.e., in 0), the energies of the various quadrupole levels are no longer given by Eq. (2.40). For I = V2, the following equations can be derived for the energies of the two states ... 

An alternative way of viewing the process of the reduction of the dressed trapping state to the state a) in a very strong field is to analyze the equations of motion for the density matrix elements in terms of the symmetric and antisymmetric superpositions of the atomic excited states... 

In these equations the position of the molecule is described by the vector R the wavevectors of the two beams of modes r2 and are k2 and k3 respectively, with ( 2) and (q3) the corresponding mean photon numbers (mode occupancies) and is a unit vector describing the polarization state of mode rn. In deriving Eqs. (120) and (121), the state vectors describing the radiation fields have been assumed to be coherent laser states, and so, for example, (<72) = (oc n a(2 ), where a ) is the coherent state representing mode 2 and h is the number operator a similar expression may be written for (<73). Also, the molecular parameters apparent in Eqs. (120) and (121) are the components of the transition dipole, p °, and the index-symmetric second-order molecular transition tensor,... 

The system of equations (1.8) is based on the central field approximation, and therefore its application to real atoms is entirely dependent on the existence of closed shells, which restore spherical symmetry in each successive row of the periodic table. For spherically symmetric atoms with closed shells, the Hartree-Fock equations do not involve neglecting noncentral electrostatic interactions and are therefore said to apply exactly. This does not mean that they are expected to yield exact values for the experimental energies, but merely that they will apply better than for atoms which are not centrally symmetric. One should bear in mind that, in any real atom, there are many excited configurations, which mix in even with the ground state and which are not spherically symmetric. Even if one could include all of them in a Hartree-Fock multiconfigura-tional calculation, they would not be exactly represented. Consequently, there is no such thing as an exact solution for any many-electron atom, even under the most favourable assumptions of spherical symmetry. 

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