Hamiltonian 2-center

For vibrational frequencies, one needs the derivatives of the energy E with respect to deformation of the bond lengths and angles of the molecule, so V is the sum of all changes in the electronic Hamiltonian that arise from displacements 5Ra of the atomic centers... 

For non-linear molecules, when treated as rigid (i.e., having fixed bond lengths, usually taken to be the equilibrium values or some vibrationally averaged values), the rotational Hamiltonian can be written in terms of rotation about three axes. If these axes (X,Y,Z) are located at the center of mass of the molecule but fixed in space such that they do not move with the molecule, then the rotational Hamiltonian can be expressed as ... 

In the DC-biased structures considered here, the dynamics are dominated by electronic states in the conduction band [1]. A simplified version of the theory assumes that the excitation occurs only at zone center. This reduces the problem to an n-level system (where n is approximately equal to the number of wells in the structure), which can be solved using conventional first-order perturbation theory and wave-packet methods. A more advanced version of the theory includes all of the hole states and electron states subsumed by the bandwidth of the excitation laser, as well as the perpendicular k states. In this case, a density-matrix picture must be used, which requires a solution of the time-dependent Liouville equation. Substituting the Hamiltonian into the Liouville equation leads to a modified version of the optical Bloch equations [13,15]. These equations can be solved readily, if the k states are not coupled (i.e., in the absence of Coulomb interactions). 

The simplest iron-sulfur centers, which were first discovered in ru-bredoxins, consist of one iron ion coordinated by a distorted tetrahedron of cysteinyl sulfur atoms. This environment provides a weak ligand field giving a spin equal to and 2 when the ion is Fe(III) and Fe(II), respectively. It also determines the splitting of the ground spin manifold, and consequently the characteristics of the EPR spectrum. This splitting is generally described in the framework of the spin Hamiltonian ... 

A priori, one might have expected a [3Fe-4S] center to give a particularly simple EPR spectrum. Contrary to what was suggested in Ref. (13), the electronic structure of this cluster, which possess three ferric sites, is not liable to be complicated by valence delocalization phenomena, so that the intersite interactions can be described by the Heisenberg Hamiltonian ... 

With the knowledge of g, we can estimate the inverse mean free path of a phonon with frequency co. As done originally within the TLS model, the quantum dynamics of the two lowest energies of each tunneling center are described by the Hamiltonian //tls = gcTz/2 + Aa /2. This expression, together with Eqs. (15) and (17), is a complete (approximate) Hamiltonian of... 

The ESR measurements were made at RT or 77 K on a Varian E-9 spectrometer (X-band), equipped with an on-line computer for data analysis. Spin-Hamiltonian parameters (g and A values) were obtained from calculated spectra using the program SIM14 A [26]. The absolute concentration of the paramagnetic species was determined from the integrated area of the spectra. Values of g were determined using as reference the sharp peak at g = 2.0008 of the E i center (marked with an asterisk in Fig. 3) the center was formed by UV irradiation of the silica dewar used as sample holder. 

From the given Hamiltonian, adiabatic potential energy surfaces for the reaction can be calculated numerically [Santos and Schmickler 2007a, b, c Santos and Schmickler 2006] they depend on the solvent coordinate q and the bond distance r, measured with respect to its equilibrium value. A typical example is shown in Fig. 2.16a (Plate 2.4) it refers to a reduction reaction at the equilibrium potential in the absence of a J-band (A = 0). The stable molecule correspond to the valley centered at g = 0, r = 0, and the two separated ions correspond to the trough seen for larger r and centered at q = 2. The two regions are separated by an activation barrier, which the system has to overcome. 

Orbitally degenerate grormd states, in general, cannot be treated in the spin-Hamiltonian approach. In this case, SOC has to be evaluated explicitly on an extended basis of spin-orbit functions. However, in coordination chemistry and bioinorganic chemistry, this is only of marginal importance, because the metal centers of... 

We next apply these classical relationships to the rigid diatomic molecule. Since the molecule is rotating freely about its center of mass, the potential energy is zero and the classical-mechanical Hamiltonian function H is just the kinetic energy of the two particles,... 

We see that the kinetie energy contribution to the Hamiltonian is the sum of two parts, the kinetic energy due to the translational motion of the center of mass of the system as a whole and the kinetic energy due to the relative motion of the two particles. Since the potential energy F(r) is assumed to be a function only of the relative position coordinate r, the motion of the center of mass of the system is unaffected by the potential energy. 

For time-dependent Hamiltonian systems we chose in Section IVB to use a normal form that decouples the reactive mode from the bath modes, but does not attempt a decoupling of the bath modes. This procedure is always safe, but in many cases it will be overly cautious. If it is relaxed, the dynamics within the center manifold is also transformed into a (suitably defined) normal form. This opens the possibility to study the dynamics within the TS itself, as has been done in the autonomous case, for example in Ref. 107. One can then try to identify structures in the TS that promote or inhibit the transport from the reactant to the product side. 

Equation (106) shows that the interaction of the proton with the motion of the center of mass, described by the terms proportional to fx, is formally of the same form as the interaction with the medium atoms, and the first three terms in the Hamiltonian in Eq. (106) are equivalent to addition of one more degree of freedom to the vibrational subsystem. Thus, this problem does not differ from that for the process of tunnel transfer of the particles stimulated by the vibrations which were discussed in Section IV. So we may use directly the expressions obtained previously with substitution of the appropriate parameters. 

For relatively simple systems of high symmetry (or for systems assumed to be simple) the system spin Hamiltonian parameters are readily relatable to those of the individual centers, for example,... 

---