Liouville—von Neumann equation

Altematively, in the case of incoherent (e.g. statistical) initial conditions, the density matrix operator P(t) I 1>(0) (v(01 at time t can be obtained as the solution of the Liouville-von Neumann equation ... 

A formulation of electronic rearrangement in quantum molecular dynamics has been based on the Liouville-von Neumann equation for the density matrix. Introducing an eikonal representation, it naturally leads to a general treatment where Hamiltonian equations for nuclear motions are coupled to the electronic density matrix equations, in a formally exact theory. Expectation values of molecular operators can be obtained from integrations over initial conditions. 

Ignoring relaxation, the spin evolution is governed by the Liouville-von-Neumann equation... 

Prior to an effective Hamiltonian analysis it is, in order to get this converging to the lowest orders, typical to remove the dominant rf irradiation from the description by transforming the internal Hamiltonian into the interaction frame of the rf irradiation. This procedure is well established and also used in the most simple description of NMR experiments by transforming the Hamiltonian into the rotating frame of the Zeeman interaction (the so-called Zeeman interaction frame). In the Zeeman interaction frame the time-modulations of the rf terms are removed and the internal Hamiltonian is truncated to form the secular high-field approximated Hamiltonian - all facilitating solution of the Liouville-von-Neumann equation in (1) and (2). The transformation into the rf interaction frame is given by... 

By analogy the propagation of a density matrix, which corresponds to the solution of the Liouville-von Neumann equation 231... 

Dependent Liouville-von Neumann Equation Dissipative Evolution. 

Another important hierarchy of equations is obtained by applying the (MCM) to the matrix representation of the Liouville-von Neumann Equation LVNE) [12,13]. In this way the p-order Contracted Liouville-von Neumann Elquation (p-CLVNE) is obtained [4]. It will be shown here that The structure of a particular p-CSE, that involves the higher order CSE s for a given state, can be replaced by an equivalent set of equations, 1-CSE and 1-CLVNE, but for the whole spectrum, i. e. involving all the states. 

The evolution of the molecule is described by the generalized Liouville-von Neumann equation [35, 36]... 

The powerful mathematical tools of linear algebra and superoperators in Li-ouville space can be used to proceed from the identification of molecular phenomena, to modelling and calculation of physical properties to interpret or predict experimental results. The present overview of our work shows a possible approach to the dissipative dynamics of a many-atom system undergoing localized electronic transitions. The density operator and its Liouville-von Neumann equation play a central role in its mathematical treatments. 

In an NMR experiment, the energy of the lattice is practically constant (the lattice has a large heat capacity). It is therefore assumed that the lattice is always in a state of thermodynamic equilibrium. Thus, it is possible to use a semi-classical description of its interactions with the spin system. Within this approach, the Liouville-von Neumann equation of motion for a spin system is given by ... 

To describe consistently cotunneling, level broadening and higher-order (in tunneling) processes, more sophisticated methods to calculate the reduced density matrix were developed, based on the Liouville - von Neumann equation [186-193] or real-time diagrammatic technique [194-201]. Different approaches were reviewed recently in Ref. [202]. 

We consider also the other method, based on the equations of motion for operators. From Liouville - von Neumann equation we find (all c-operators are Heisenberg operators in the formula below, (t) is omitted for shortness)... 

Having established the Hamiltonian, the next step is to derive an equation of motion for the density matrix. One can start with the Liouville-von Neumann equation for the complete system-plus-bath density matrix a(t)... 

By performing a partial Wigner transform with respect to the coordinates of the environment, we obtain a classical-like phase space representation of those degrees of freedom. The subsystem coordinate operators are left untransformed, thus, retaining the operator character of the density matrix and Hamiltonian in the subsystem Hilbert space [4]. In order to take the partial Wigner transform of Eq. (1) explicitly, we express the Liouville-von Neumann equation in the Q representation,... 

Let us briefly discuss, as an example of inconsistency, the case of the quantum-classical Liouville representation [19]. The starting point of such a derivation is the Liouville-von Neumann equation of motion for the evolution... 

Abstract Conventional dynamic NMR spectrum simulation methods are based on the solution of the phenomenologically extended Liouville-von Neumann equation of spin systems in Liouville space. In this work, we show an alternative method in which the... 

So to obtain expectation values relevant to any particular experiment one needs an estimate of the density matrix at the time of measurement. For an NMR experiment, this typically requires the ability to estimate the time evolution of the density matrix for the pulse sequence used for the experiment. The time dependent differential equation that describes the time evolution of the density matrix, known as the Liouville-von Neumann equation is given by... 

Simulation packages such as GAMMA take advantage of the fact that evolution of the density matrix under the Liouville-von Neumann equation is well approximated by a small number of easily applied transformations of the density matrix, namely free evolution can be represented by a simple unitary transformation and application of ideal RF pulses can be represented by a simple rotation. Real RF pulses can be effectively modelled as a succession of ideal RF pulses. The beauty of this method is that fairly complex, realistic effects, such as evolution of coupled spin systems through complex pulses, can be modelled by a straightforward combination of these simple building blocks. 

The calculations presented here are based on the density operator formalism using the Liouville-von-Neumann equation and the theoretical approach is confined to quadrupolar nuclei subjected to EFG as well as CSA-interactions. Following the approach of Barbara et al.,20 the Hamiltonian for an N-site jump may be written as... 

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