Interaction-free Hamiltonian

The quantum dynamics under the control-free Hamiltonian Hq is assumed to be incapable of producing the desired evolution - Thus, a suitable control interaction Vc(0 is introduced... 

Prepared State. Here the Hamiltonian H is the time-independent molecular Hamiltonian. Both H0 and T are time independent. The initial prepared state is an eigenket to H0 and thus is nonstationary with respect to H = H0 + T. One example is provided by considering H0 as the spin-free Hamiltonian 77sp and the perturbation T as a spin interaction. A second example is provided by considering H0 as the spin-free Born-Oppenheimer Hamiltonian and T as a spin-free nonadiabatic perturbation. In the first example spin-free symmetry is not conserved but double-point group symmetry may be. In the second example point-group symmetry is not conserved, but spin-free symmetry is. The initial prepared state arises from some other time-dependent process as, for example, radiative absorption which occurs at a rate very much faster than the rate at which our prepared state evolves. Mechanisms for radiationless transitions in excited benzene may involve such prepared states, as is discussed in Section XI. 

There are point-group selection rules in the presence of spin interactions.73,115117 172 We recall that a spin-free Hamiltonian //SF(Qeq) for a rigid nuclear framework Qeq has a point group SF which acts on electronic spatial coordinates, and that... 

We see that it is the interaction potential E(R, r) that couples the motion of the A atbnTto the motion of the BC diatomic. In its absence the two free fragments A and BG described by the free Hamiltonian... 

Before continuing our discussion of gauge-origin dependence, we note that the substitution of Eq. 70 in the spin-free Hamiltonian Eq. 69 followed by expansion does not lead to the expression Eq. 52. To account for the missing Zeeman spin interaction, we must first replace the nonrelativistic spin-0 Hamiltonian Eq. 69 with a nonrelativistic spin- Hamiltonian, which for a one-electron system is given by... 

This result may be re-expressed in terms of H°, the usual field-free Hamiltonian, and H an interaction Hamiltonian, as... 

Here // < is the Hamiltonian for the radiation field in vacuo, flmo the field-free Hamiltonian for molecule , and //m( is a term representing molecular interaction with the radiation. It is worth emphasising that the basic simplicity of Eq. (1) specifically results from adoption of the multipolar form of light-matter interaction. This is based on a well-known canonical transformation from the minimal-coupling interaction [17-21]. The procedure results in precise cancellation from the system Hamiltonian of all Coulombic terms, save those intrinsic to the Hamiltonian operators for the component molecules hence no terms involving intermolecular interactions appear in Eq. (1). 

In the three mentioned spin-free Hamiltonians, the electron-electron interaction is described by the nonrelativistic Coulumb interaction,... 

These states are formed inside the continuous spectra of the total Hamiltonian and are responsible for phenomena such as resonances in electron scattering from atoms or molecules, autoionization, predissociation, etc. Furthermore, in this work we also consider as unstable states those states that are constructs of the time-independent theory of the interaction of an atom (molecule) with an external field which is either static or periodic, in which case the effect of the interaction is obtained as an average over a cycle. In this framework, the "atom - - field" state is inside the continuous (ionization or dissociation) spectrum, and so certain features of the problem resemble those of the unstable states of the field-free Hamiltonian. The probability of decay of these field-induced unstable states corresponds either to tunneling or to ionization-dissociation by absorption of one or more photons. 

Regardless of the sizes of the spaces, we can formulate a number of possible ways in which the effective Hamiltonian including spin-orbit interaction can be constructed. In all of these, the first step remains the solution of the eigenvalue problem in the space S for the spin-free Hamiltonian,... 

Even the excited states of a single atom are embedded in a continuum of other states. As discussed in Section 3.2.3 this continuum corresponds to the states of the radiation field sitting on lower atomic states. Casting that discussion in our present notation we have (cf Eqs (3.21)-(3.24)). o = +Hk,H =. o +. mr, where. m and. r are the Hamiltonians of the molecule and of the free radiation field, respectively, and. mr is their mutual interaction. The Hamiltonian. r was shown to represent a collection of modes—degrees of freedom that are characterized by a frequency cu, a polarization vector [Pg.314]

In the previous section we saw on an example the main steps of a standard statistical mechanical description of an interface. First, we introduce a Hamiltonian describing the interaction between particles. In principle this Hamiltonian is known from the model introduced at a microscopic level. Then we calculate the free energy and the interfacial structure via some approximations. In principle, this approach requires us to explore the overall phase space which is a manifold of dimension 6N equal to the number of degrees of freedom for the total number of particles, N, in the system. 

In a seminal paper [4], Su, Schrieffer, and Heeger discussed the energetics of solitons using a simple model of free electrons interacting with the lattice described by the Hamiltonian ... 

Now knowing how to evaluate solvation-free energies, we are ready to explore the effect of the solvent on the potential surface of the reacting solute atoms. Adapting the EVB approach we can describe the reaction by including the solute-solvent interaction in the diagonal elements of the solute Hamiltonian, using... 

The frustration effects are implicit in many physical systems, as different as spin glass magnets, adsorbed monomolecular films and liquid crystals [32, 54, 55], In the case of polar mesogens the dipolar frustrations may be modelled by a spin system on a triangular lattice (Fig, 5), The corresponding Hamiltonian consists of a two particle dipolar potential that has competing parallel dipole and antiparallel dipole interactions [321, The system is analyzed in terms of dimers and trimers of dipoles. When the dipolar forces between two of them cancel, the third dipole experiences no overall interaction. It is free to permeate out of the layer, thus frustrating smectic order. 

This treatment differs from the usual approach to molecule-radiation interaction through the inclusion of the contribution from the electric field from the beginning and by not treating it as a perturbation to the field free situation. The notation 7/ei(r R, e(f)) makes the parametric dependence of the electronic Hamiltonian on the nuclear coordinates and on the electric field explicit. 

A possible reason that the problem of C < 0 did not receive much attention was the assertion [15] (BLH) that such an anomaly was forbidden. The proof was based on the statistical mechanical analysis of the primitive model of electrolytes between two oppositely charged planes, cr and —a. It was noticed in Ref. 10 that the BLH analysis missed a very simple contribution to the Hamiltonian, direct interaction between the charged walls, ItzLct (L is the distance between the walls). With proper choice of the Hamiltonian the condition on the capacitance would be C > 27re/L. It simply means that due to ionic shielding of the electric field, the capacitance exceeded its geometrical value corresponding to the electrolyte-free dielectric gap. 

An alternative approach to calculating the free energy of solvation is to carry out simulations corresponding to the two vertical arrows in the thermodynamic cycle in Fig. 2.6. The transformation to nothing should not be taken literally -this means that the perturbed Hamiltonian contains not only terms responsible for solute-solvent interactions - viz. for the right vertical arrow - but also all the terms that involve intramolecular interactions in the solute. If they vanish, the solvent is reduced to a collection of noninteracting atoms. In this sense, it disappears or is annihilated from both the solution and the gas phase. For this reason, the corresponding computational scheme is called double annihilation. Calculations of... 

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