Hamiltonian operators superoperators

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

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. ... 

Through introduction of superoperators and a corresponding metric [13], the propagator may be represented more compactly [2, 6]. Superoperators act on field operator products, X, where the number of annihilators exceeds the number of creators by one. The identity superoperator, I, and the Hamiltonian superoperator, H, are defined by... 

H The Hamiltonian superoperator (the Hamiltonian represented as an operator acting on Hilbert Schmidt operators)... 

When considering relaxation, a Liouville space representation is typically used in which the Hamiltonian and density matrix are represented as superoperators in addition to the relaxation operator being represented as a superoperator. Once a... 

It is possible to perform more precise calculations that simultaneously account for the coherent quantum mechanical spin-state mixing and the diffusional motion of the RP. These employ the stochastic Liouville equation. Here, the spin density matrix of the RP is transformed into Liouville space and acted on by a Liouville operator (the commutator of the spin Hamiltonian and density matrix), which is then modified by a stochastic superoperator, to account for the random diffusive motion. Application to a RP and inclusion of terms for chemical reaction, W, and relaxation, R, generates the equation in the form that typically employed... 

Ernst, 1979 Chingas et al., 1981 Cavanagh et al., 1990). Similarly, the diagonalization of the mixing Hamiltonian superoperator can be facilitated if a set of basis operators is used that is completely reduced with respect to rotation, permutation, and particle number (Listerud and Drobny, 1989 Listerud et al., 1993). 

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. 

This result is the Redfield-Liouville-von Neumann equation of motion or, simply, the Redfield equation [29,30,49-53]. Here the influence of the bath is contained entirely in the Redfield relaxation tensor, 3i, which is added to the Liouville operator for the isolated subsystem to give the dissipative Redfield-Liouville superoperator (tensor) that propagates (T. Expanded in the eigenstates of the subsystem Hamiltonian, H, Eq. (9) yields a set of coupled linear differential equations for the matrix... 

We have also outlined a treatment of systems showing dissipative dynamies. These ean be treated with a natural extension of the formulation shown here. The density operator equation differs in a signifieant way when dissipation is present, because it eontains terms whieh are not derived from a Hamiltonian, and must be written as rates involving superoperators. We have shown here one such situation, with the Lindblad form of the dissipation rate, and other forms can be treated equally well with the present eombination of a PWT and a quasielassieal approximation in phase spaee. 

A. THE STOCHASTIC RELAXATION MODEL. The most general theories of magnetic relaxation in Mossbauer spectroscopy involve stochastic models see, for example. Ref. 283 for a review. A formalism using superoperators (Liouville operators) was introduced by Blume, who presented a general solution for the lineshape of radiation emitted (absorbed) by a system whose Hamiltonian jumps at random as a function of time between a finite number of possible forms that do not necessarily commute with one another. The solution can be written down in a compact form using the superoperator formalism. 

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... 

The Liouville operator = (1 lh) H embodies the dynamics of the system in thg absence of the external force, where the snperoperator H is formed nsing the system Hamiltonian H. The influeiice of the external force on the system is described by Axt(t) = (l/ )7fext(0, where the superoperator H tiO is formed from the interaction Hamiltonian HestiO describing the coupling between the external force and the system. 

As a first step, the appearance of (13.5.2) can be improved by introducing the concept of a superoperator (Banwell and Primas, 1963 Goscinski and Lukman, 1980) the Hamiltonian superopeiatoii, denoted by A, works on an arbitrary operator to generate a commutator. Thus ... 

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