Spin propagator representation

Second, the mapping approach to nonadiabatic quantum dynamics is reviewed in Sections VI-VII. Based on an exact quantum-mechanical formulation, this approach allows us in several aspects to go beyond the scope of standard mixed quantum-classical methods. In particular, we study the classical phase space of a nonadiabatic system (including the discussion of vibronic periodic orbits) and the semiclassical description of nonadiabatic quantum mechanics via initial-value representations of the semiclassical propagator. The semiclassical spin-coherent state method and its close relation to the mapping approach is discussed in Section IX. Section X summarizes our results and concludes with some general remarks. 

The path integral representation of the spin-coherent state propagator is formally given by [139]... 

Within the theoretical framework of time-dependent Hartree-Fock theory, Suzuki has proposed an initial-value representation for a spin-coherent state propagator [286]. When we adopt a two-level system with quantum Hamiltonian H, this propagator reads... 

Another possibility to introduce a semiclasscial initial value representation for the spin-coherent state propagator is to exploit the close relation between Schwinger s representation of a spin system and the spin-coherent state theory [100, 133-135]. To illustrate this approach, we consider an electronic two-level system coupled to Vvib nuclear DoF. Within the mapping approach the semiclassical propagator for this system is given by... 

Figure 11 Simulation of the fid of a spin set. (A) Individual density matrix is calculated at each exchange point. (B) Eigencoherence representation of the density matrix is propagated from the beginning of the time slice for each detection point.

The limit with respect to rj is taken because of integration techniques required in a Fourier transform from the time-dependent representation. Indices r and s refer to general, orthonormal spin-orbitals, r x) and os(x), respectively, where x is a space-spin coordinate. Matrix elements of the corresponding field operators, al and as/ depend on the N-electron reference state, N), and final states with N 1 electrons, labeled by the indices m and n. The propagator matrix is energy-dependent poles occur when E equals a negative VDE, Eq(N) — En(N — 1), or a negative VAE, Em(N +1) — Eq(N). 

It follows from Eq. (25) that any periodicity (/) in the propagator should give rise to a coherence peak, that is, to a local maximum for q = ybg = Iti/I in the representation of the NMR spin echo intensity versus the intensity of the field gradient pulses (78 ). Such behavior has indeed been observed in recent PEG NMR studies of water diffusion through the free space within an array of loosely packed monodisperse polysterene spheres [78],... 

A full relativistic theory for coupling tensors within the polarization propagator approach at the RPA level was presented as a generalization of the nonrelativistic theory. Relativistic calculations using the PP formalism have three requirements, namely (i) all operators representing perturbations must be given in relativistic form (ii) the zeroth-order Hamiltonian must be the Dirac-Coulomb-Breit Hamiltonian, /foBC, or some approximation to it and (iii) the electronic states must be relativistic spin-orbitals within the particle-hole or normal ordered representation. Aucar and Oddershede used the particle-hole Dirac-Coulomb-Breit Hamiltonian in the no-pair approach as a starting point, Eq. (18),... 

Reff = observed dipolar coupling constant t = time T20 = spin term in the spherical tensor representation of the dipolar Hamiltonian = zero-quantum relaxation time constant U = propagator = magne-togyric ratio of spin / A/ = anisotropy of the indirect spin-spin interaction 0 = angle between the applied field and the internuclear vector A = dephasing parameter /Uq = permeability of free space Vj. = rotor frequency in Hz 1/, = isotropic resonant frequen-... 

This equation reveals an important connection between propagators and density matrices, since in Chapter S (p. 132) we noted that p , a density-matrix element in the discrete representation provided by a set of spin-orbitals V r(x) > could be expressed quite generally as an expectation value of ajar and wc let t— 0 from below, the first term in the braces, in (13.3.2), will be zero while the second will yield (ajar)- Thus we find... 

Spectral representation of a propagator, 257 Sphere optimization, 331 Spin contamination, 113, 189 Spin functions, 58... 

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