Wave operator methods

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

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. 

For the third overtone, the wave operator method diverges after three iterations. For this reason, the results obtained after either three or four iterations are provided. 

Mintzer, David, 1 Mitropolsky, Y. A361,362 Mixed groups, 727 Modality of distribution, 123 Models in operations research, 251 Modification, method of, 67 Molecular chaos, assumption of, 17 Miller wave operator, 600 Moment generating function, 269 Moment, 119 nth central, 120... 

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... 

The RWP method also has features in common with several other accurate, iterative approaches to quantum dynamics, most notably Mandelshtam and Taylor s damped Chebyshev expansion of the time-independent Green s operator [4], Kouri and co-workers time-independent wave packet method [5], and Chen and Guo s Chebyshev propagator [6]. Kroes and Neuhauser also implemented damped Chebyshev iterations in the time-independent wave packet context for a challenging surface scattering calculation [7]. The main strength of the RWP method is that it is derived explicitly within the framework of time-dependent quantum mechanics and allows one to make connections or interpretations that might not be as evident with the other approaches. For example, as will be shown in Section IIB, it is possible to relate the basic iteration step to an actual physical time step. 

The primary characteristic of WT that distinguishes it from DFT, is that two-electron operators are treated explicitly. However, except for a few methods that attempt to use explicit two electron operator r12 = r,-r2 ) terms in the wave function, [4] the vast majority of wave function methods attempt to describe the innate correlation effects ultimately in terms of products of basis functions, fcP(l)Xq(2) - %P(2)%q(l)], where (1) indicates the space (r2) and spin (a) coordinates of electron one (together... 

M. Rosina, (a) Direct variational calculation of the two-body density matrix (b) On the unique representation of the two-body density matrices corresponding to the AGP wave function (c) The characterization of the exposed points of a convex set bounded by matrix nonnegativity conditions (d) Hermitian operator method for calculations within the particle-hole space in Reduced Density Operators with Applications to Physical and Chemical Systems—II (R. M. Erdahl, ed.), Queen s Papers in Pure and Applied Mathematics No. 40, Queen s University, Kingston, Ontario, 1974, (a) p. 40, (b) p. 50, (c) p. 57, (d) p. 126. 

D.A. Micha, E. Brandas, Variational Methods in the Wave Operator Formalism. A Unified Treatment for Bound and Quasi-Bound Electronic and Molecular States, J. Chem. Phys. 55 (1971) 4792. 

Beyond the systems and applications described in this chapter, projection-operator methods can, for example, be used to study the dynamics near glass transitions [77] and the propagation of wave functions in systems with non-resonant transitions. The latter application has recently been analyzed in connection with the decomposition of the spectral density [78] showing the wide range of applicability of the proposed schemes. 

The theory of the multi-vibrational electron transitions based on the adiabatic representation for the wave functions of the initial and final states is the subject of this chapter. Then, the matrix element for radiationless multi-vibrational electron transition is the product of the electron matrix element and the nuclear element. The presented theory is devoted to the calculation of the nuclear component of the transition probability. The different calculation methods developed in the pioneer works of S.I. Pekar, Huang Kun and A. Rhys, M. Lax, R. Kubo and Y. Toyozawa will be described including the operator method, the method of the moments, and density matrix method. In the description of the high-temperature limit of the general formula for the rate constant, specifically Marcus s formula, the concept of reorganization energy is introduced. The application of the theory to electron transfer reactions in polar media is described. Finally, the adiabatic transitions are discussed. 

Haque and Mukherjee/126/ developed a Fock-space method for generating hermitian Hg in which the cluster wave-operator ft is not unitary, following a suggestion of Jorgensen/127/, who chose ft to preserve the norm of the functions in the model space. Thus ft satisfies... 

The time evolution operator exp(—/HAf/ft) acting on ( ) propagates the wave function forward in time. A number of propagation methods have been developed and we will briefly describe the following the split operator method [91,94,95], the Lanzcos method [96] and the polynomial methods such as Chebychev [93,97], Newtonian [98], Faber [99] and Hermite [100,101]. A classical comparison between the three first mentioned methods was done by Leforestier et al. [102]. 

A formal similarity arises because both the scattering matrix in quantum electrodynamics and the wave operator in a full coupled-cluster method [cf. Equations (1) and (4)] are exponential operators. In Bogo-liubov s axiomatic formulation of the scattering matrix,29... 

In the LDA, Adolph and Bechstedt [157,158] adopted the approach of Aspnes [116] with a plane-wave-pseudopotential method to determine the dynamic x of the usual IB V semiconductors as well as of SiC polytypes. They emphasized (i) the difficulty to obtain converged Brillouin zone integration and (ii) the relatively good quality of the scissors operator for including quasiparticle effects (from a comparison with the GW approximation, which takes into account wave-vector- and band-dependent shifts). Another implementation of the SOS x —2 ffi, ffi) expressions at the independent-particle level was carried out by Raskheev et al. [159] by using the linearized muffin-tin orbital (LMTO) method in the atomic sphere approximation. They considered... 

Each coupled spin level in the zero-field or correlation diagram had energy E(Si2, S34, S) for each wave function Si2, S34, S, M). The coupling scheme adopted was S12 = Sj + S2 S34 = S3 + S4 S = S12 + S34. Since the matrix of Eq. (14) cannot be diagonalized, matrix elements were worked out by tensor operator methods (50, 68). 

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