Hamiltonian superoperator

According to equation 15, eigenvalues of the superoperator Hamiltonian matrix, H, are poles (electron binding energies) of the electron propagator. Several renormalized methods can be defined in terms of approximate H matrices. The... 

The matrix Hfj would be the transpose of Hf, if it were Hermitian. The Hermiticity of the superoperator Hamiltonian has been a concern since the beginnings of the electron propagator theory (46,129). For a Hermitian spin ftee Hamiltonian (// ) the following relation can be written describing the Hermiticity problem,... 

The matrix H 3 can be obtained fiom the general ex)x ession for the superoperator Hamiltonian matrix H33 recently derived (126). For the non-diagonal one-electron Hamiltonian it can be written as... 

P. O. Lowdin. On operators, superoperators, hamiltonians and liouvil-lians. Intern. J. Quantum Chem., QCS 16 485, 1982. 

For many of the commonly used renormalized methods, such as 2ph-TDA, NR2, and ADC(3), the operator space spans the h, p, 2hp, and 2ph subspaces [7,22]. Reference states are built from Hartree-Fock determinantal wavefunctions plus perturbative corrections. The resulting expressions for various blocks of the superoperator Hamiltonian matrix may be evaluated through a given order in the fluctuation potential. 

Earlier in this chapter, we noted that the question of the hermiticity of f) T ) had to be examined in individual cases (i.e., it was not automatically valid). When a perturbation expansion is used to determine the reference slate, we may more explicitly state the conditions under which the matrix is hermitian by examining the difference between the (k/)th and the complex conjugate of the (/k)th element of the superoperator Hamiltonian. When this difference... 

Such a treatment can, with advantage, be expressed in terms of the superoperators introduced in Eq. (4.19) and in terms of a basis of field operators. The basis of fermion-like operators Xj = a, aj[aja, ,a aja, a ap, - is chosen, such that the electron field operators correspond to the SCF spin orbitals. The field operator space supports a scalar product (XjlXj) = ([A , X,]+) = Tr /9[Xl,Xj]+, where p is the density operator defined in Eq. (4.33). The superoperator identity and the superoperator hamiltonian operate on this space of fermion-like field operators and, in particular, Xi HXj) = [x/, [H,Xj - J. ) = Tt p[xI[H,X ] U. ... 

Perturbation theory starts with a partitioning of the hamiltonian, and thus of the superoperator hamiltonian, into an unperturbed part and a perturbation ... 

The state employed in the definition of the superoperator binary product is often called the reference state and need not be the ground state of the system. The transformations working on the vectors in this vector space of operators, i.e. the ( erators, are called superoperators and are here denoted with a wide hat as, e.g. in O. Commonly, only the superoperator Hamiltonian and the superoperator identity operator I are used, which are defined as... 

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