Reduced-density-matrix formalism

As mentioned in the introduction, only few vibrational degrees of freedom are usually actively involved in the ultrafast dynamics at the conical intersection. To account for the effect of energy transfer from the active modes to the many inactive modes of the polyatomic molecule or a condensed-phase environment, the reduced-density-matrix formalism may... 

From a quantum mechanical perspective, the transient vibrational relaxation dynamics is often studied within a reduced density matrix formalism. The dissipative dynamics of the so-called open system is described by a Liouville von Neumann equation that includes implicitly the effects of the surroundings on the dynamics... 

Husimi, K., Proc. Phys.-Math. Soc. Japan 22, 264, "Some formal properties of the density matrix." Introduction of the concept of reduced density matrix. Statistical-mechanical treatment of the Hartree-Fock approximation at an arbitrary temperature and an alternative method of obtaining the reduced density matrices are discussed. 

It can be seen that, in the average density matrix formalism which is based on spin system model, the scalar couplings and the exchange processes are handled simultaneously. Thus they cannot be separated and a larger atomic basis (spin system) is required for their description. Meanwhile, the Monte Carlo method based on spin sets separates the two interactions, and thus spin systems can be reduced to smaller spin sets. 

From the physical point of view, we are representing a many-electron state by an antisymmetrized product of one-electron states. The density matrix formalism [4,11-13] allows one to analyse in the same footing calculations resulting from different levels of approximation. The density matrix is called reduced when is formed from a pure state ... 

As described in Sec. 3.1, each Hartree-Fock iteration involves the construction of the Fock matrix for a given density matrix, followed by the diagonalization of the Fock matrix to generate a set of improved spin orbitals and thus an improved density matrix. Formally, the construction of the Fock matrix requires a number of operations proportional to K4, where K is the number of atoms (because the number of two-electron integrals scales as Al4). For large systems, however, this quartic scaling with K (i.e., with system size) can be reduced to linear by special techniques, as will now be discussed. 

In fact in this resides the power of the density matrix formalism reducing a many-body problem to the single particle density matrix, abstracted from the single Slater determinant of Eq. (4.190) called also as Fock-Dirac matrix... 

Within the conventional density-functional formalism one seeks an expression for the total electronic energy as a functional of the electron density of Eq. (16). The electron density of Eq. (16) can be considered the diagonal elements of the first-order reduced density matrix... 

The term decoherence describes the process by which the off-diagonal elements of the reduced density matrix tend to zero when evolving with time. Our objective is to reach an understanding of the molecular mechanisms governing decoherence with an atomic resolution. In addition we wish to be in a position to treat systems consisting of tens to thousands of atoms since the brute force simulation of the time evolution of p t) by the Liouville-von Neuman equation (p (i) = ih [H, p ]), the equivalent of the TDSE in the density matrix formalism, is out of question for such molecular systems. 

Note that the term open system refers here to exchange of energy and phase with the environment, as the number of particles is conserved throughout. The reduced density matrix p t) evolves coherently under the influence of the nuclear Hamiltonian, Hnuc, and the non-adiahatic effects enter the equation via the dissipative Liouvillian superoperator jSfn- The latter is also termed memory kernel , as it contains information about the entire history of the environmental evolution and its interaction with the system. The definition of the memory kernel is by no means unique nor straightforward. One possible solution is to start from the microscopic Hamiltonian of the total system, eqn (1). Using the projector formalism, it is possible to separate the evolution of the system, i.e., the... 

Open systems are generally described within the density matrix formahsm [27-30,41,42]. The reduced density matrix Ps t) associated with the system is obtained by tracing over the bath coordinates. In the Nakajima-Zwanzig formalism [43], Ps t) is solution of a reduced equation containing a memory which depends on the whole history of the global system-bath... 

A quantum system of N particles may also be interpreted as a system of (r — N) holes, where r is the rank of the one-particle basis set. The complementary nature of these two perspectives is known as the particle-hole duality [13, 44, 45]. Even though we treated only the iV-representability for the particles in the formal solution, any p-hole RDM must also be derivable from an (r — A)-hole density matrix. While the development of the formal solution in the literature only considers the particle reduced Hamiltonian, both the particle and the hole representations for the reduced Hamiltonian are critical in the practical solution of N-representability problem for the 1-RDM [6, 7]. The hole definitions for the sets and are analogous to the definitions for particles except that the number (r — N) of holes is substituted for the number of particles. In defining the hole RDMs, we assume that the rank r of the one-particle basis set is finite, which is reasonable for practical calculations, but the case of infinite r may be considered through the limiting process as r —> oo. 

We present in Section 2 the formalism giving the equations for the reduced density operator and for competing instantaneous and delayed dissipation. Section 3 presents matrix equations in a form suitable for numerical work, and the details of the numerical procedure used to solve the integrodiffer-ential equations with the two types of dissipative processes. In Section 4 on applications to adsorbates, results are shown for quantum state populations versus time for the dissipative dynamics of CO/Cu(001). The fast electronic relaxation to the ground electronic state is shown first without the slow relaxation of the frustrated translation mode of CO vibrations, for comparison with previous work, and this is followed by results with both fast and slow relaxation. In Section 5 we comment on the general conclusions that can be reached in problems involving both vibrational and electronic relaxation at surfaces. 

Based on the TD-HEDT, Chen et al. have proposed and developed a practical first-principles formalism for open electronic systems and implemented it to simulate transient currents through electronic devices [42, 60]. The resulting first-principles formalism for open system starts from a closed equation of motion (EOM) for the Kohn-Sham (KS) reduced single-electron density matrix (RSDM) of the entire system, and reduces to the following Liouville-von Neumann equation by projecting out the electronic degrees of freedom of the electrodes ... 

Formally, the theory can be extended beyond the Golden Rule limit by considering higher order terms in the perturbation expansion of the rate in V. Starting with the Liouville equation for the density matrix, one can derive reduced equations of motion for the state populations (generalized master equations), formally exact for arbitrary coupling [88, 99, 295, 296],... 

However a quantitative analysis of the absorption bands requires the formalism of the stochastic theory, outlined in Section 10.2, which is able to connect the measured solvent shifts and inhomogeneous bandwidths to two microscopic parameters of the system, namely the respective number densities p of matrix units aroimd a dye molecule and the depths e of the dye-matrix interaction potentials. While for polymer matrices the stochastic theory was to be used to determine geometric parameters [4], they are already known for our rare gas model systems from independent investigations. This enabled us to reduce the number of fit factors and calculate the p and values as listed in Tab. 10.3. 

Hence, the results stated above have shown that analysis of structural properties for epoxy polymers, which are considered as natural nanocomposites, can be carried out within the frameworks of multifractal formalism in its most simple variant. The structure adaptability resource is reduced as the crosslinking density increases and is defined by the relative fraction of the loosely packed matrix. The properties of epoxy polymers are a function of their structure adaptability. 

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