Open density matrix

Dead time A very short delay introduced before the start of acquisition that allows the transmitter gate to close and the receiver gate to open. Density matrix A description of the state of nuclei in quantum mechanical terms. 

Although quantitative calculation of the accurate PESs remains a difficult task (see Section 20.1.2), the two-state-model describes the essential reaction dynamic process and is useful for a qualitative understanding. When the reaction coordinate is set to the adsorbate-surface distance (one-dimension), the two-state-model is called the Menzel-Gomer-Redhead [49] and/or Antoniewicz [50] model. We refer to them as the MGR models. The MGR models are often used successfully to analyze photodesorption on metal surfaces by assuming a short residential time on the excited PES. There are several methods to simulate the quantum dynamics of the MGR models, for example, stochastic wavepacket [51], open density matrix methods [52], and so on. 

It is clear that the strong form of the QCT is impossible to obtain from either the isolated or open evolution equations for the density matrix or Wigner function. For a generic dynamical system, a localized initial distribution tends to distribute itself over phase space and either continue to evolve in complicated ways (isolated system) or asymptote to an equilibrium state (open system) - whether classically or quantum mechanically. In the case of conditioned evolution, however, the distribution can be localized due to the information gained from the measurement. In order to quantify how this happ ens, let us first apply a cumulant expansion to the (fine-grained) conditioned classical evolution (5), resulting in the equations for the centroids (x = (t), P= (P ,... 

The calculation of the indices requires the overlap matrix S of atomic orbitals and the first-order density (or population) matrix P (in open-shell systems in addition the spin density matrix Ps). The summations refer to all atomic orbitals /jl centered on atom A, etc. These matrices are all computed during the Hartree-Fock iteration that determines the molecular orbitals. As a result, the three indices can be obtained... 

J. R. Hammond and D. A. Mazziotti, Variational reduced-density-matrix calculations on radicals an alternative approach to open-shell ab initio quantum chemistry. Phys. Rev. A 73, 012509 (2006). 

M. Nakata, M. Ehara, K. Yasuda, and H. Nakatsuji, Direct determination of second-order density matrix open-shell system and excited state. J. Chem. Phys. 112, 8772 (2000). 

This volume in Advances in Chemical Physics provides a broad yet detailed survey of the recent advances and applications of reduced-density-matrix mechanics in chemistry and physics. With advances in theory and optimization, Coulson s challenge for the direct calculation of the 2-RDM has been answered. While significant progress has been made, as evident from the many contributions to this book, there remain many open questions and exciting opportunities for further development of 2-RDM methods and applications. It is the hope of the editor and the contributors that this book will serve as a guide for many further advenmres and advancements in RDM mechanics. 

Separability can be exploited even with admission of relativistic effects, by using the standard density matrix formalism with a simple extension to admit 4-component Dirac spin-orbitals this opens up the possibility of performing ab initio calculations, with extensive d, on systems containing heavy atoms. 

The question, what conditions are to be fulfilled by a density matrix to be the image of a wave function, that is, to describe a real physical system is opened till today. The contracted Schrodinger-equations derived for different order reduced density matrices by H. Nakatsui [1] give opportunity to determine density matrices by a non-variational way. The equations contain density matrices of different order, and the relationships needed for the exact solutions are not yet known in spite of the intensive research activity [2,3]. Recently perturbation theory corrections were published for correcting the error of the energy obtained by minimizing the density matrix directly applying the known conditions of N-representability [4], and... 

To this end, we resort to a novel general approach to the control of arbitrary multidimensional quantum operations in open systems described by the reduced density matrix p(t) if the desired operation is disturbed by linear couplings to a bath, via operators S B (where S is the traceless system operator and B is the bath operator), one can choose controls to maximize the operation fidelity according to the following recipe, which holds to second order in the system-bath coupling (i) The control (modulation) transforms the system-bath coupling operators to the time-dependent form S t) (S) B(t) in the interaction picture, via the rotation matrix e,(t) a set of time-dependent coefficients in the operator basis, (Pauli matrices in the case of a qubit), such that ... 

We have previously defined the one-electron spin-density matrix in the context of standard HF methodology (Eq. (6.9)), which includes semiempirical methods and both the UHF and ROHF implementations of Hartree-Fock for open-shell systems. In addition, it is well defined at the MP2, CISD, and DFT levels of theory, which permits straightforward computation of h.f.s. values at many levels of theory. Note that if the one-electron density matrix is not readily calculable, the finite-field methodology outlined in the last section allows evaluation of the Fermi contact integral by an appropriate perturbation of the quantum mechanical Hamiltonian. 

We now connect the analysis given above with the equation of motion displayed in Eq. (5.5). That equation of motion follows from subdivision of a system into an open subsystem S and a complementary reservoir R. When the coupling between S and R is weak, the evolution of the open system 5, due to the internal dynamics of 5 and the interaction with the reservoir R, can be described in density matrix form by Eq. (5.5). Now writing... 

We emphasize that the density matrix calculated from Eq. (6) is equivalent to that from Eq. (4), but Eq. (6) is much easier to compute for open systems. To see why this is so, let us consider zero temperature and assume ftL — ftR = eV], > 0. Then, in the energy range -oo < E < pR the Fermi functions = fR = 1. Because the Fermi functions are equal, no information about the non-equilibrium statistics exists and the NEGF must reduce to the equilibrium Green s function GR. In the range pR < E < pR, fL 7 fR and NEGF must be used in Eq. (6). A more careful mathematical manipulation shows that this is indeed true [30], and Eq. (6) can be written as a sum of two terms ... 

The vector model is a way of visualizing the NMR phenomenon that includes some of the requirements of quantum mechanics while retaining a simple visual model. We will jump back and forth between a classical spinning top model and a quantum energy diagram with populations (filled and open circles) whenever it is convenient. The vector model explains many simple NMR experiments, but to understand more complex phenomena one must use the product operator (Chapter 7) or density matrix (Chapter 10) formalism. We will see how these more abstract and mathematical models grow naturally from a solid understanding of the vector model. 

Lathiotakis NN, Helbig N, Gross EKU (2005) Open shells in reduced-density-matrix-functional theory, Phys Rev A, 72 030501... 

W. Huisinga, et al, Faber and Newton polynomial integrators for open-system density matrix propagation, /. Chem. Phys. 110 (12) (1999) 5538-5547. 

Generalization to open-shell systems does not represent any specific problem of ab initio calculations and therefore it will not be treated here. In eqn, (3.3), denotes the matrix element of the one-electron part of Hamiltonian, is the element of the density matrix... 

The extensive experience with Gaussian-type basis sets shows that basis set sequences that increase rapidly in accuracy can be constructed in a systematic way [9]. At the same time, a compact description of the wave functions is maintained, and this opens the way for efficient methods to solve for the self consistent field (SCF) equations. Furthermore, as Gaussian functions are localized, the representations of the KS, overlap and density matrix in this basis become sparse with increasing system size [11]. This eventually allows for solving the KS equations using computational resources that scale linearly with system size. 

The integral-driven procedure indicated above is practicable only if the elements of the two-particle density matrix can be rapidly accessed. In the closed-shell Hartree-Fock case, the two-particle density matrix can be easily constructed from the one-particle density. The situation is similar for open-shell and small multiconfigurational SCF wavefunctions the two-particle density matrix can be built up from a few compact matrices. In most open-shell Hartree Fock theories (Roothaan, 1960), the energy expression (Eq. (23))... 

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