Canonical effective Hamiltonian

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

Note that in contrast to a general similarity transformation (e.g., as found in the usual coupled-cluster theory) the canonical transformation produces a Hermitian effective Hamiltonian, which is computationally very convenient. When U is expressed in exponential form, the effective Hamiltonian can be constructed termwise via the formally infinite Baker-Campbell-Hausdorff (BCH) expansion,... 

It may be of interest to remove the driven term a°(at + a)/2, which is common to all the effective Hamiltonians (40). For this purpose, we perform the following canonical transformation ... 

Consider the effective Hamiltonian (47) of a driven quantum harmonic oscillator. Since it is not diagonal, it may be suitable to diagonalize it with the aid of a canonical transformation that will affect it or its equivalent form (50), but not that of (46) or its equivalent expression (49), which is yet to be diagonal [13]. 

For this purpose, consider selective canonical transformations leading to a new quantum representation that we name /// in order to diagonalize the effective Hamiltonian corresponding to k = 1, without affecting that corresponding to k = 0. This may be performed on the different effective operators B dealing with k = 0,1 with the aid of... 

Equation (5-9) can be regarded as a canonical (Lowdin) orthonormalisation of the set of vectors 0 , or equivalently as a polar decomposition of the operator 3l (J0rgensen 9) Thus the Schrodinger equation for the n-electron Hamiltonian, H, Eq. (2-2), can always be formally transformed to the eigenvalue problem (5-10 a) for the effective Hamiltonian,, acting in the subspace S sparmed by a finite set of orthonormal vectors 0 the ligand field Hamiltonian (1-5) must therefore be an approximation to this object. 

Continuum effective Hamiltonian needs a definition of the electronic charge distribution pMe. All quantum methods giving this quantity can be used, whereas other methods must be suitably modified. Quantum methods are not limited to those based on a canonical molecular orbital formulation. Valence Bond (VB) and related methods may be employed. The interpretation of reaction mechanisms in the gas phase greatly benefits by the shift from one description to another (e.g. from MO to VB). The same techniques can be applied to continuum effective Hamiltonians. We only mention this point here, which would deserve a more detailed discussion. 

First-Principles Approach to Guinier-Preston Zones. We have already seen that the combination of first-principles calculations with Monte Carlo methods is a powerful synthesis which allows for the accurate analysis of structural questions. In chap. 6 we noted that with effective Hamiltonians deduced from a lower-level microscopic analysis it is possible to explore the systematics of phase diagrams with an accuracy that mimics that of the host microscopic model. An even more challenging set of related questions concern the emergence of microstructure in two-phase systems. An age-old question of this type hinted at in the previous chapter is the development of precipitates in alloys, with the canonical example being that of the Al-Cu system. 

Some properties of the Fock space transformations W and effective Hamiltonians h and, thus, of the resulting h, appear to differ from those obtciined by Hilbert space transformations. For example, their canonical unitary W is not separable and yields an h and, thus, an h with disconnected diagrams on each degenerate subspace. However, the analogous U of Eq. (5.13) may be shown to be separable 71), and the resulting He on each complete subspace flo is fully linked , as proven by Brandow [8]. These differences are not explained. 

An effective Hamiltonian of file electron subsystem can be constructed with the displaced phonon operator method (Elliott et al. 1972, Young 1975) or the method of canonical transformation (Mutscheller and Wagner 1986) analoguous results are given by a perturbation method in the second order in the electron-qrhonon interaction (13) (Baker 1971). 

Two neighbor molecules interact via van der Waals and hydrogen bonding of energies given, respectively, by parameters e and e + y. The model is described by the following effective Hamiltonian, in the grand-canonical ensemble ... 

In Sections 10.3 and 10.4, we introduced the Fock operator as an effective Hamiltonian for the calculation of Hartree-Fock orbitals (the canonical orbitals) and orbital energies by the repeated diagonalization of the Fock matrix. In the present section, we consider in more detail the properties of the canonical orbitals and the associated orbitals energies - the eigenfunctions and eigenvalues, respectively, of the Fock operator. In particular, we shall introduce Koopmans theorem and identify the ionization potentials and electron affinities of a closed-shell system with the negative energies of the canonical orbitals. 

The ACSE has important connections to other approaches to electronic structure including (i) variational methods that calculate the 2-RDM directly [36-39] and (ii) wavefunction methods that employ a two-body unitary transformation including canonical diagonalization [22, 29, 30], the effective valence Hamiltonian method [31, 32], and unitary coupled cluster [33-35]. A 2-RDM that is representable by an ensemble of V-particle states is said to be ensemble V-representable, while a 2-RDM that is representable by a single V-particle state is said to be pure V-representable. The variational method, within the accuracy of the V-representabihty conditions, constrains the 2-RDM to be ensemble N-representable while the ACSE, within the accuracy of 3-RDM reconstruction, constrains the 2-RDM to be pure V-representable. The ACSE and variational methods, therefore, may be viewed as complementary methods that provide approximate solutions to, respectively, the pure and ensemble V-representabihty problems. 

When the orbital functions are chosen as d-like or/-like, the axial symmetry of the molecule again leads to factorize the Renner-Teller Hamiltonian in a double chain. In effect the total angular momentum Jz remains a constant of motion and can be put in a diagonal form by means of the same type of canonical transformation, so we have... 

Thus, the fractional equilibrium state (99) can be considered as a consequence of anomalous transport of phase points in the phase space resulting in the anomalous continuity equation (104). Note that the usual form of the evolution (93) is a direct consequence of the canonical Hamiltonian form of the microscopic equations of motion. Thus, the evolution of (105) implies that the microscopic equations of motion are not canonical. The actual form of these equations has not yet been investigated. However, there are strong indications that dissipative effects on the microscopic level become important. 

The most general approach to the formation of the effective operator of the intersite virtual phonon exchange interaction is based on the canonical shift transformation of the Hamiltonian... 

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