Effective Hamiltonian Hermitian

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,... 

Effective Operators and Classification of Mapping Operators Effective Operators Generated by Norm-Preserving Mappings Effective Operators Generated by Non-Norm-Preserving Mappings which Produce a Non-Hermitian Effective Hamiltonian... 

Appendix B Non-Norm Preserving Mappings that Produce a Hermitian Effective Hamiltonian Appendix C Proof of Theorem I... 

Norm-preserving mappings these necessarily generate a Hermitian effective Hamiltonian ... 

Norm-preserving mappings are denoted by K, fC) and, as discussed in Section II.B, generate a Hermitian effective Hamiltonian K HK = i. The orthonormalized model eigenfunctions of H are written as a)o and the corresponding true eigenfunctions are designated by I Pa). Thus, Eq. (2.2) specializes to... 

The unity normalized right eigenvectors of the non-Hermitian effective Hamiltonian LHK = H are denoted by a )o- Because condition (2.18) is not fulfilled, the corresponding true eigenvectors are not unity normed. As is customary [6-9], the are taken to be mutually... 

Mappings which Produce a Hermitian Effective Hamiltonian... 

The simple projection relation between the right model eigenfunctions of Hg and their true counterparts is an appealing aspect of Bloch s formalism. However, the non-Hermiticity of the resulting effective Hamiltonian represents a strong drawback, as discussed in Section VII. This has led many, beginning with des Cloizeaux [7], to derive Hermitian effective Hamiltonians, des Cloizeaux s method transforms the lag)ol not the... 

Section II.D shows how the mappings K, L) generate the non-Hermitian effective Hamiltonian H = LHK corresponding to H,... 

APPENDIX B NON-NORM-PRESERVING MAPPINGS THAT PRODUCE A HERMITIAN EFFECTIVE HAMILTONIAN... 

Both VU and SU MR CC methods employ the effective Hamiltonian formalism the relevant cluster amplimdes are obtained by solving Bloch equations and the (in principle exact) energies result as eigenvalues of a non-Hermitian effective Hamiltonian that is defined on a finite-dimensional model space Mq. An essential feature characterizing this formalism is the so-called intermediate or Bloch normalization of the projected target space wave functions I f, ) with respect to the corresponding model space configurations 1, ), namely = 8 (for details, see, e.g. Refs. [172,174]). 

E) can be easily obtained from Eq. (25) by diagonalization of a smalldimensional non-Hermitian effective Hamiltonian. This is not true if the effective Hamiltonian has the form of a Jordan block (see p. 475 in Ref. [2]). A simple example of Jordan structure is the matrix representation (30) in Section 2.2.2 when = 4A. In this nonphysical case the two eigenvalues coalesce at the value -2iA. 

E and are the energy and the width of the useful part of the continuum (doorway state) [22, 33]. The two-dimensional non-Hermitian effective Hamiltonian (30) is the simplest matrix representation linking the microscopic level characterized by the complex energy E — iFc/2 to the macroscopic level of interest (the resonance). In Eq. (30), the energy of the resonance El is real. We will see below that if the resonance is weakly coupled to the microscopic level (AE F ), the complex part of energy can be uncovered by... 

This is the recursive form of the generalized Bloch equation. In a similar way, we can separate the effective Hamiltonian and effective interaction (15) due to the powers of V. Despite the fact, that the effective Hamiltonian (15) is not hermitian in intermediate normalization, we can diagonalize the corresponding (Hamiltonian) matrix and shall obtain (always) real energies, as they represent the exact energies of fhe system. This property is satisfied for each order independently. 

The Gram-Schmidt procedure provides a simpler ordu onalization scheme that leads to a hermitian effective Hamiltonian. Since and we already orthogonal to the other projections, we only have to worry about and This means that the coefficients of 4 are defined by that is, if I l = a ( afe + 6a ) -I- p ( aa -P bb )... 

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