The effective Hamiltonian

The propagator in Eq. 15 can also be written in terms of the ladder operators. However, before doing so we insert the unit operator U 0)U(0) and obtain 

This demonstrates again the special form of the propagator, a doubly periodic operator times an unitary operator defined by a time independent effective Hamiltonian. 

The inverse transformation of operators from Fourier space to Hilbert space can be performed straightforwardly. Using the fact that all operators (except the number operators in the Hamiltonian) satisfy the diagonal property 

At this point we have expressed the Hamiltonian, the density operator and the evolution operator in Fourier space. We have introduced an effective Hamiltonian, defined in the Hilbert space of the same dimension 2 as the total time-dependent Hamiltonian itself, and we have shown how to transform operators between the two representations. The definition of the effective Hamiltonian enables us to predict the overall evolution of the spin system, despite the fact that we can not find time-points for synchronous detection, f, where Uint f) = exp —iWe//t In actual experiments the time dependent signals are monitored and after Fourier transformation they result in frequency sideband 

According to the second conjecture above, we should replace vowith(l/2)(v + Vn ) when calculating the probability P /. Inserting this in eq. (8.60), we obtain 

This condition is fulfilled for thermal velocities and phonon frequencies less than 0.2-0.5 lO sec These frequencies correspond to Debye temperatures of 2-400 K, i.e, the above expression is valid for the low-frequency modes, which also are those responsible for most of the energy accoimnodation at low collision energies. 

Equation (8.62) shows that the symmetrization decreases the excitation and increases the de-excitation probabilities as compared to the original expression (8.60). 

It is now relatively straightforward to extend this forced oscillator model for a single oscillator to the situation in a solid where we have a distribution of oscillators. Omitting the zero-point energy of the crystal we can write the total hamiltonian as 

Note that we have furthermore dropped the two-quantum operators and Ok dkt. Thus two quantum transitions will take place through successive operations with the one-quantum operators Ok and d. This approximation is introduced in order to obtain a solvable algebra for the operators. Such an algebra should be closed with respect to commutations. The above hamiltonian fulfills this condition. 

The choice of the effective Hamiltonian is often far from straightforward indeed we have devoted a whole chapter to this subject (chapter 7). In this section we give a gentle introduction to the problems involved, and show that the definition of a particular molecular parameter is not always simple. The problem we face is not difficult to understand. We are usually concerned with the sub-structure of one or two rotational levels at most, and we aim to determine the values of the important parameters relating to those levels. However, these parameters may involve the participation of other vibrational and electronic states. We do not want an effective Hamiltonian which refers to other electronic states explicitly, because it would be very large, cumbersome and essentially unusable. We want to analyse our spectrum with an effective Hamiltonian involving only the quantum numbers that arise directly in the spectrum. The effects of all other states, and their quantum numbers, are to be absorbed into the definition and values of the molecular parameters . The way in which we do this is outlined briefly here, and thoroughly in chapter 7. 

The development of the effective Hamiltonian has been due to many authors. In condensed phase electron spin magnetic resonance the so-called spin Hamiltonian [20,21] is an example of an effective Hamiltonian, as is the nuclear spin Hamiltonian [22] used in liquid phase nuclear magnetic resonance. In gas phase studies, the first investigation of a free radical by microwave spectroscopy [23] introduced the ideas of the effective Hamiltonian, as also did the first microwave magnetic resonance study [24], Miller [25] was one of the first to develop the more formal aspects of the subject, particularly so far as gas phase studies are concerned, and Carrington, Levy and Miller [26] have reviewed the theory of microwave magnetic resonance, and the use of the effective Hamiltonian. 

As a simple introduction to the subject [27], let us consider the four angular momentum vectors illustrated in figure 1.9. They are as follows  

Each angular momentum can interact with the other three, and figure 1.9 draws attention to the following pairwise interactions  

Many different procedures for reducing the complete Hamiltonian to a suitable effective Hamiltonian have been devised. These are reviewed in detail in chapter 7 we will see that the methods involve different forms of perturbation theory [29]. 

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In the presence of a phase factor, the momentum operator (P), which is expressed in hyperspherical coordinates, should be replaced [53,54] by (P — h. /r ) where VB creates the vector potential in order to define the effective Hamiltonian (see Appendix C). It is important to note that the angle entering the vector potential is shictly only identical to the hyperangle <]> for an A3 system. 

We introduce the dimensionless bending coordinates qr = t/XrPr anti qc = tAcPc ith Xt = (kT -r) = PrOir, Xc = sJ kcPc) = Pc nc. where cor and fOc are the harmonic frequencies for pure trans- and cis-bending vibrations, respectively. After integrating over 0, we obtain the effective Hamiltonian H = Ho + H, which is employed in the perturbative handling of the R-T effect and the spin-orbit coupling. Its zeroth-order pait is of the foim... 

In another promising method, based on the effective Hamiltonian theory used in quantum chemistry [19], the protein is divided into blocks that comprise one or more residues. The Hessian is then projected into the subspace defined by the rigid-body motions of these blocks. The resulting low frequency modes are then perturbed by the higher... 

For QM-MM methods it is assumed that the effective Hamiltonian can be partitioned into quantum and classical components by writing [9]... 

The special case where only rotators are present, Np = 0, is of particular interest for the analysis of molecular crystals and will be studied below. Here we note that in the other limit, where only spherical particles are present, Vf = 0, and where only symmetrical box elongations are considered with boxes of side length S, the corresponding measure in the partition function (X Qxp[—/3Ep S, r )], involving the random variable S, can be simplified considerably, resulting in the effective Hamiltonian... 

We assume that exploring all possible forms for the fields corresponds to exploring the overall usual phase space. To determine the partition function Z the contributions from all the p+ r) and P- r) distributions are summed up with a statistical weight, dependent on p+ r) and p (r), put in the form analogous to the Boltzmann factor exp[—p (F)]], where the effective Hamiltonian p (F)] is a functional of the fields. The... 

This expression has a formal character and has to be complemented with a prescription for its evaluation. A priori, we can vary the values of the fields independently at each point in space and then we deal with uncountably many degrees of freedom in the system, in contrast with the usual statistical thermodynamics as seen above. Another difference with the standard statistical mechanics is that the effective Hamiltonian has to be created from the basic phenomena that we want to investigate. However, a description in terms of fields seems quite natural since the average of fields gives us the actual distributions of particles at the interface, which are precisely the quantities that we want to calculate. In a field-theoretical approach we are closer to the problem under consideration than in the standard approach and then we may expect that a simple Hamiltonian is sufficient to retain the main features of the charged interface. A priori, we have no insurance that it... 

Let us underline some similarities and differences between a field theory (FT) and a density functional theory (DFT). First, note that for either FT or DFT the standard microscopic-level Hamiltonian is not the relevant quantity. The DFT is based on the existence of a unique functional of ionic densities H[p+(F), p (F)] such that the grand potential Q, of the studied system is the minimum value of the functional Q relative to any variation of the densities, and then the trial density distributions for which the minimum is achieved are the average equihbrium distributions. Only some schemes of approximations exist in order to determine Q. In contrast to FT no functional integrations are involved in the calculations. In FT we construct the effective Hamiltonian p f)] which never reduces to a thermo-... 

We may give a meaning to the coupling constants in by considering the MFA for the effective Hamiltonian + - ideai t 1oc pj-om... 

In this part we consider the effective hamiltonian H coul ideal loc nonloc simplified fOHU =... 

This prescription transforms the effective Hamiltonian to a tridiagonal form and thus leads directly to a continued fraction representation for the configuration averaged Green function matrix element = [G i]at,. This algorithm is usually continued... 

Molecules and Clusters. The local nature of the effective Hamiltonian in the LDF equations makes it possible to solve the LDF equations for molecular systems by a numerical LCAO approach (16,17). In this approach (17), the atomic basis functions are constructed numerically for free atoms and ions and tabulated on a numerical grid. By construction, the molecular basis becomes exact as the system dissociates into its atoms. The effective potential is given on the same numerical grid as the basis functions. The matrix elements of the effective LDF Hamiltonian in the atomic basis are given by... 

The real wave packet (RWP) method, developed by Gray and Bahnt-Kuiti [ 1], is an approach for obtaining accurate quantum dynamics information. Unlike most wave packet methods [2] it utilizes only the real part of the generally complex-valued, time-evolving wave packet, and the effective Hamiltonian operator generating the dynamics is a certain function of the actual Hamiltonian operator of interest. Time steps in the RWP method are accomphshed by a simple three-term Chebyshev... 

It is also of interest to study the "inverse" problem. If something is known about the symmetry properties of the density or the (first order) density matrix, what can be said about the symmetry properties of the corresponding wave functions In a one electron problem the effective Hamiltonian is constructed either from the density [in density functional theories] or from the full first order density matrix [in Hartree-Fock type theories]. If the density or density matrix is invariant under all the operations of a space CToup, the effective one electron Hamiltonian commutes with all those elements. Consequently the eigenfunctions of the Hamiltonian transform under these operations according to the irreducible representations of the space group. We have a scheme which is selfconsistent with respect to symmetty. 

Electron correlation was treated by the CIPSI multi-reference perturbation algorithm ([24,25] and refs, therein). The Quasi Degenerate Perturbation Theory (QDPT) version of the method was employed, with symmetrisation of the effective hamiltonian [26], and the Maller-Plesset baricentric (MPB) partition of the C.I. hamiltonian. 

Figure 4-2. Computed potential energy surface from (A) ab initio valence-bond self-consistent field (VB-SCF) and (B) the effective Hamiltonian molecular-orbital and valence-bond (EH-MOVB) methods for the S 2 reaction between HS- and CH3CI...

Wesolowski and Warshel197 introduced a DFT based approach in which all short-range terms in the effective Hamiltonian (Eq. 4.25) were derived entirely from density functional theory and were involved in the construction of the Fock matrix195 196. In this approach, the H croEnv is expressed using explicit functionals of the electron density ... 

According to the effective Hamiltonian method, the energy Ea is complex, that is,... 

To show the importance of the damping operator, we apply the effective Hamiltonian to optical absorption. In this case, we have... 

In the previous section we discussed the effective Hamiltonian method a main feature of this method is that it results in the appearance of damping operator T in the Liouville equation. However, the damping operator is introduced in an ad hoc manner. In this section we shall show that the damping operator results from the interaction between the system and heat bath. 

C. The Effective Hamiltonians of the Slow Mode in Different Representations... 

The Standard Spectral Density Within the Effective Hamiltonian Procedure... 

Passing to the Boson operators by aid of Table II, and after neglecting the zero-point-energy of the fast mode, we obtain a quantum representation we shall name I, in which the effective Hamiltonians of the slow mode corresponding respectively to the ground and first excited states of the fast mode are... 

The eigenvalue equation of the representation of the effective Hamiltonian operators (28) in the base of the number occupation operator of the slow mode is characterized by the equation... 

On the other hand, one may perform on the effective Hamiltonian in representation II the canonical transformations ... 

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