Wave operator, effective Hamiltonians

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. 

From Adiabatic to Effective Hamiltonian Matrices Through the Wave Operator Procedure... 

Figure 5. Illustration of the equivalence between the spectral densities obtained within the adiabatic approximation and those resulting from the effective Hamiltonian procedure, using the wave operator. Common parameters a0 = 0.4, co0 = 3000cm-1, C0Oo = 150cm-1, y = 30cm-1, and T = 300 K.

Durand P (1983) Direct determination of effective Hamiltonians by wave-operator methods. I. General formalism. Phys Rev A 28 3184... 

Note that all the above expressions characterize the effective Hamiltonian formalism per se, and are independent of a particular form of the wave operator U. Indeed, this formalism can be exploited directly, without any cluster Ansatz for the wave operator U (see Ref. [75]). We also see that by relying on the intermediate normalization, we can easily carry out the SU-Ansatz-based cluster analysis We only have to transform the relevant set of states into the form given by Eq. (16) and employ the SU CC Ansatz,... 

D. Maynau, P. Durand, J. P. Duadey, and J. P. Malrieu, Phys. Rep. A, 28, 3193 (1983). Direct Determination of Effective-Hamiltonians by Wave-Operator Methods. 2. Application to Effective-Spin Interactions in -Electron Systems. P. Durand and J. P. Malrieu, in Advances in Chemical Physics (Ah Initio Methods in Quantum Chemistry—I), K. P. Lawley, Ed., Wiley, New York, 1987, Vol. 67, pp. 321-412. Effective Hamiltonians and Pseudo-Operators as Tools for Rigorous Modelling. 

The coupled Schrodinger equations can be projected onto the fa fa subspace by Feshbach partitioning, giving an equation for the coefficient function Xd(q) in the component faxdiq) of the total wave function. The effective Hamiltonian in this equation is tn + Vd(q) + Vopt, which contains an optical potential that is nonlocal in the <7-space. This operator is defined by its kernel in the fa - fa subspace,... 

Wave function of electrons in quantum R-system Ap satisfies the Schrodinger equation with the effective Hamiltonian iTff eq. (1.246), which is obtained by averaging the interaction operators in eq. (1.232) over the ground state of the M-system, i.e. over Ap, and acts on the quantum numbers (variables) of electrons in the R-system. 

We now study the disordered effective hamiltonian (4.4). Since a direct diagonalization of (4.4) is too hard, we shall have to use approximations which are conveniently expressed in the resolvent (or Green s function) formalism. The translation-invariant K sum in HeU is restricted to the optical wave vectors only (for K oj/c, RK / K 0I)- Therefore, it is possible to restrict the problem to this small part of the Brillouin zone using the projector operator... 

As for the QM/MM description also for PCM, non-electrostatic (or van der Walls) terms can be added to the Vent operator in this case, besides the dispersion and repulsion terms, a new term has to be considered, namely the energy required to build a cavity of the proper shape and dimension in the continuum dielectric. This further continuum-specific term is generally indicated as cavitation. Generally all the non-electrostatic terms are expressed using empirical expressions and thus their effect is only on the energy and not on the solute wave function. As a matter of fact, dispersion and repulsion effects can be (and have been) described at a PCM-QM level and included in the solute-effective Hamiltonian 7/eff as two new operators modifying the SCRF scheme. Their definition can be found in Ref. [17] while a recent systematic comparison of these contributions determined either using the QM or the classical methods is reported in Ref. [18]... 

If we now consider the generalized Silverstone-Sinanoglu strategy discussed above, then the most natural way to proceed is by way of an effective hamiltonian formalism. We introduce a single (state-universal) wave-operator O, whose action on 4 s produce the functions 4>k, defined by eq. (6.1.1) or (6.1.7), and write Schrodinger equations for 4[Pg.327]

The OOA, also known as Kugel-Khomskii approach, is based on the partitioning of a coupled electron-phonon system into an electron spin-orbital system and crystal lattice vibrations. Correspondingly, Hilbert space of vibronic wave functions is partitioned into two subspaces, spin-orbital electron states and crystal-lattice phonon states. A similar partitioning procedure has been applied in many areas of atomic, molecular, and nuclear physics with widespread success. It s most important advantage is the limited (finite) manifold of orbital and spin electron states in which the effective Hamiltonian operates. For the complex problem of cooperative JT effect, this partitioning simplifies its solution a lot. 

We now determine particular classes of commutation relations that are, indeed, conserved upon transformation to state-independent effective operators. The proof of (4.1) demonstrates that the preservation of [A, B] by definition A requires the existence of a relation between K, K, or both and one or both of the true operators A or B. Likewise, there must be a relation between the appropriate wave operator, the inverse mapping operator, or both, and A, B, or both for other state-independent effective operator definitions to conserve [A, B]. All mapping operators depend on the spaces and fl. Although the model space is often specified by selecting eigenfunctions of a zeroth order Hamiltonian, it may, in principle, be arbitrarily defined. On the other hand, the space fl necessarily depends on H. Therefore, the existence of a relation between mapping operators and A, B, or both, implies a relation between H and A, B, or both. 

Here g is the projector to the model space of nonrelativistic wave functions. What we have done in two steps, can also be done in one step. By means of the operator U we switch from the Dirac operator D — mc to the effective Hamiltonian L in the model space... 

Once we know the wave operator through Eqs. (13) and (14), we are ready to evaluate the matrix elements of the effective Hamiltonian in the basis of reference configurations because the effective Hamiltonian is given by the following relationship (see, for example. Ref. [55])... 

Before considering approximations for the wave operator, we will establish an important equation. We will show that the operator HQ, can be expressed in terms of the effective Hamiltonian. First, we note that... 

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