Hamiltonian with applied fields

This is a simplified Hamiltonian that ignores the direct interaction of any nuclear spins with the applied field, B. Because of the larger coupling, Ah to most transition metal nuclei, however, it is often necessary to use second-order perturbation theory to accurately determine the isotropic parameters g and A. Consider, for example, the ESR spectrum of vanadium(iv) in acidic aqueous solution (Figure 3.1), where the species is [V0(H20)5]2+. 

The Mu spin Hamiltonian, with the exception of the nuclear terms, was first determined by Patterson et al. (1978). They found that a small muon hyperfine interaction axially symmetric about a (111) crystalline axis (see Table I for parameters) could explain both the field and orientation dependence of the precessional frequencies. Later /xSR measurements confirmed that the electron g-tensor is almost isotropic and close to that of a free electron (Blazey et al., 1986 Patterson, 1988). One of the difficulties in interpreting the early /xSR spectra on Mu had been that even in high field there can be up to eight frequencies, corresponding to the two possible values of Ms for each of the four inequivalent (111) axes. It is only when the external field is applied along a high symmetry direction that some of the centers are equivalent, thus reducing the number of frequencies. 

The approaches to this problem follow along two general lines. In the first approach, one computes derivatives of the dipole moment with respect to the applied field and relates them to the terms in the polarization expansion of equation 8. Inspection of equation 8 suggests that the second derivative of the dipole moment with respect to the field gives p. The choice of the exact form of the Hamiltonian, which incorporates the optical field and the atomic basis set, determines the accuracy of this procedure. In one popular version of this approach, the finite field method, the time dependence of the Hamiltonian is ignored for purposes of simplification and the effects of dispersion on p, therefore, cannot be accounted for. 

Sx, Sy, and Sz are the three components of the spin operator, D and E are the anisotropy constants which were determined via high-frequency electron paramagnetic resonance (D/kB 0.275 K and E/kB 0.046 K [10]), and the last term of the Hamiltonian describes the Zeeman energy associated with an applied field H. 

An applied field in the xy-plane can tune the tunnel splittings Amm. via the Sx and Sy spin operators of the Zeeman terms that do not commute with the spin Hamiltonian. This effect can be demonstrated by using the Landau-Zener method (Section 3.1). Fig. 8 presents a detailed study of the tunnel splitting A 10 at the tunnel transition between m - +10, as a function of transverse fields applied at different angles (p, defined as the azimuth angle between the anisotropy hard axis and the transverse field (Fig. 4). 

Tunneling is allowed in these half-integer (S = 9/2) spin systems because of a small transverse anisotropy Htrans,i containing Sx i and S, spin operators and transverse fields (Hx and Hy). The exact form of Htrans,i is not important in this discussion. The last term in Eq. 4 is the Zeeman energy associated with an applied field. The Mn4 units within the [Mn4J2 dimer are coupled by a weak superexchange interaction via both the six C-H—Cl pathways and the Cl—Cl approach. Thus, the Hamiltonian (H) for [Ma ] 2 is... 

An applied electric field (E) interacts with the electric dipole moment (p,e) of a polar diatomic molecule, which lies along the direction of the intemuclear axis. The applied field defines the space-fixed p = 0 direction, or Z direction, whilst the molecule-fixed q = 0 direction corresponds to the intemuclear axis. Transformation from one axis system to the other is accomplished by means of a first-rank rotation matrix, so that the interaction may be represented by the effective Hamiltonian as follows ... 

Figure 15 (a) The 4.2 K Mossbauer spectra of [(Fe(IV)=0)(TMC)(NCCH3)](0Tf)2 in acetonitrile recorded in (A) zero field and (B)a parallel field of 6.5 T. The solid line represents a spin Hamilton simulation with the parameters described in the text, (b) Mossbauer spectra of [Fe(lV)=(0)(TMCS)] recorded at temperatures and applied fields that are indicated. The solid lines represent spin Hamiltonian simulations with parameters described in the text. The spectra were simulated in the slow (at 4.2 K) and fast (at 30 K) spin fluctuation limit. The applied field was directed parallel to the observed y radiation. The doublet drawn above the topmost experimental spectrum (0 T, 4 K) represents a 7% Fe(ll) contribution from the starting complex. (From J. U. Rohde et al. (2003) Science 299 1037-1039. Reprinted with permission from AAAS)... 

Finite-field methods were first used to calculate dipole polarizabilities by Cohen and Roothaan [66]. For a fixed field strength V, the Hamiltonian potential energy term for the interaction between the electric field and ith electron is just The induced dipole moment with the applied field can be calculated from the Hartree-Fock wavefunction by integrating the dipole moment operator with the one-electron density since this satisfies the Hellmann-Feyman theorem. With the usual dipole moment expansion. 

These operators induce transitions between different spin states so that by applying an rf-field to nuclear spins in the presence of a large static magnetic field close to the Larmor frequency, the spin distribution between the energy levels is perturbed away from thermodynamic equilibrium. In pulsed NMR experiments the spin system is excited with a short if pulse near resonance and the system is measured afterwards. The total external Hamiltonian from the applied fields is then... 

The models for the control processes start with the Schrodinger equation for the molecule in interaction with a laser field that is treated either as a classical or as a quantized electromagnetic field. In Section II we describe the Floquet formalism, and we show how it can be used to establish the relation between the semiclassical model and a quantized representation that allows us to describe explicitly the exchange of photons. The molecule in interaction with the photon field is described by a time-independent Floquet Hamiltonian, which is essentially equivalent to the time-dependent semiclassical Hamiltonian. The analysis of the effect of the coupling with the field can thus be done by methods of stationary perturbation theory, instead of the time-dependent one used in the semiclassical description. In Section III we describe an approach to perturbation theory that is based on applying unitary transformations that simplify the problem. The method is an iterative construction of unitary transformations that reduce the size of the coupling terms. This procedure allows us to detect in a simple way dynamical or field induced resonances—that is, resonances that... 

To determine an effective dressed Hamiltonian characterizing a molecule excited by strong laser fields, we have to apply the standard construction of the free effective Hamiltonian (such as the Born-Oppenheimer approximation), taking into account the interaction with the field nonperturbatively (if resonances occur). This leads to four different time scales in general (i) for the motion of the electrons, (ii) for the vibrations of the nuclei, (iii) for the rotation of the nuclei, and (iv) for the frequency of the interacting field. It is well known that it is a good strategy to take into account the time scales from the fastest to the slowest one. 

It is desirable to apply fields of strong enough amplitude so that dominates all other interaction Hamiltonians except for the Zeeman interaction. The rf pulses can then be treated as infinitely short delta pulses, and the analysis of the experimental spectra becomes comparatively simple. However, arcing in the probe limits useful amplitudes to the order of 200 kHz, so that in solid-state NMR the delta-pulse approximation must be treated with care for the dipole-dipole interaction among protons, and it breaks down for the quadrupole interaction. 

Eor molecules of lower than cubic symmetry with more than one unpaired electron i.e. S > Vi), coupling to excited states through spin-orbit coupling can break the degeneracy of the states even in the absence of an applied field. This phenomenon, known as zero-field splitting, in turn will lead to deviations from the Curie law. The behaviour of an 5 = 1 system will serve as an illustration. The form of the Hamiltonian is ... 

To calculate an ESR spectrum we first of all assume the applied field to be in some direction (0, ) with respect to the crystal or molecular axes. The matrix elements of the spin Hamiltonian are loaded into a (6 x 6) Hermitian matrix (Tables 1 or 2) which is diagonalized numeri-... 

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