Reduced density operator

Factoring the Energy Functional through the First Order Reduced Density Operator. 

An appealing way to apply the constraint expressed in Eq. (3.14) is to make connection with Natural Orbitals (31), in particular, to express p as a functional of the occupation numbers, n, and Natural General Spin Ckbitals (NGSO s), yr,, of the First Order Reduced Density Operator (FORDO) associated with the N-particle state appearing in the energy expression Eq. (3.8). In order to introduce the variables n and yr, in a well-defined manner, the... 

The of First Orda Reduced Density Operators (FORDO s)... 

The first-order reduced density operator y can be defined in terms of its kernel function37... 

Reduced density operators were first introduced by K. Husimi (Proc. Phys. Math. Soc. Japan 22 [1940], 264) to describe subsets of the IV-electron distribution (first-order for one-particle distributions, second-order for pair distributions, etc.) and are obtained from the full Mh-order (von Neumann) density operator electronic coordinates see, e.g., E. R. Davidson, Reduced Density Matrices in Quantum Chemistry (New York, Academic Press, 1976) and note 31. 

For a system of N identical fermions in a state ij/ there is associated a reduced density matrix (RDM) of order p for each integer p, 1 Hermitian operator DP, which we call a reduced density operator (RDO) acting on a space of antisymmetric functions of p particles. The case p = 2 is of particular interest for chemists and physicists who seldom consider... 

Reduced Density Operators Conference, Kingston, June 20-22, 1974. Sponsored by National Research Council of Canada Queen s University. Organizer R. M. Erdahl. Proceedings R. M. Erdahl, editor. Reduced Density Operators with Applications to Physical and Chemical Systems—II, Queen s Papers in Pure and Applied Mathematics No. 40 (1974), 234 pp. 

M. Rosina, (a) Direct variational calculation of the two-body density matrix (b) On the unique representation of the two-body density matrices corresponding to the AGP wave function (c) The characterization of the exposed points of a convex set bounded by matrix nonnegativity conditions (d) Hermitian operator method for calculations within the particle-hole space in Reduced Density Operators with Applications to Physical and Chemical Systems—II (R. M. Erdahl, ed.), Queen s Papers in Pure and Applied Mathematics No. 40, Queen s University, Kingston, Ontario, 1974, (a) p. 40, (b) p. 50, (c) p. 57, (d) p. 126. 

R. M. Erdahl (ed.), Reduced Density Operators with Applications to Physical and Chemical Systems II, Queen s University, Kingston, Ontario, 1974. 

A. J. Coleman, Reduced density operators and N-particle problem. Int J. Quantum Chem. 13, 67 (1978). 

The second-order reduced density matrix in geminal basis is expressed by the parameters of the wave function [6-9]. The second-order reduced density matrix (3) is the kernel of the second-order reduced density operator. Quantities 0 are matrix elements of the second-order reduced density operator in the basis of geminals. In spite of this, the expression element of density matrix is usual. In this sense, in the followings 0 is called as element of second-order reduced density matrix. 

Extensions from the preceding ideal, isolated systems to others that are coupled to an environment are quite demanding and nontrivial [32] because the IR femtosecond/picosecond laser pulse has to achieve the selective vibrational transition (9) in competition against nonselective processes such as dissipation. For simulations, we employ the equation of motion for the reduced-density operator... 

A similar equation is obtained for the reduced density operator T8 = trp[r] for the s-region. The right-hand side can not be written in general as a commutator between a hamiltonian and a density operator, so that in this case... 

Previously we have considered the reduced density matrices and the equations of motion. These quantities are the representation of the reduced density operators (Bogolyubov,6 Gurov7) defined by... 

The equation of motion for the reduced density operator (quantum master equation) takes the form [40]... 

A version of the reduced density operator to be used in the mixed quantum classical description may be obtained if we replace p(t) by the pure state... 

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. 

The present reduced density operator treatment allows for a general description of fluctuation and dissipation phenomena in an extended atomic system displaying both fast and slow motions, for a general case where the medium is evolving over time. It involves transient time-correlation functions of an active medium where its density operator depends on time. The treatment is based on a partition of the total system into coupled primary and secondary regions each with both electronic and atomic degrees of freedom, and can therefore be applied to many-atom systems as they arise in adsorbates or biomolecular systems. 

Of course, the reduced-density operator of the driven damped quantum harmonic oscillator at time t is the partial trace over the thermal bath of the full density operator ... 

Next, the reduced density operator at time t may be obtained by performing the partial trace tre over the thermal bath ... 

The fact that we can express the propagator in this form allows us also to expand the reduced density operator in the RF interaction frame as... 

Starting from the gap Hamiltonian (33) and the interactions with the reservoirs (34) and (35), eliminating the reservoir degrees of freedom within the standard Bom-Markov approach we derive a master equation for the reduced density operator of the atomic ensemble. The calculation is lengthy but straight forward. Disregarding level shifts caused by the bath interaction we find for the populations in states cj ) the following density matrix equations in the interaction picture ... 

Decoherence theory states that the entanglement in a system is broken only when the environment vectors, to which it is coupled, form an orthogonal set. This can be expressed as a condition on the reduced density operator ps> corresponding to the product wavefunction J>) = afc) E, i)(Sk, particle system (with orthonormal basis vectors ) ) coupled to an environment E (basis vectors E,i) ),... 

Let us first assume that the reduced density operator p(0) can be chosen to be diagonal in the preferred -representation (which corresponds to the usual random phase approximation at t = 0). Then each term of Eq. (29) is of the form... 

Time evolution equations for reduced density operators ... 

We have already encountered the projection operator formalism in Appendix 9A, where an apphcation to the simplest system-bath problem—a single level interacting with a continuum, was demonstrated. This formalism is general can be applied in different ways and flavors. In general, a projection operator (or projector) P is defined with respect to a certain sub-space whose choice is dictated by the physical problem. By definition it should satisfy the relationship = P (operators that satisfy this relationship are called idempotent), but other than that can be chosen to suit our physical intuition or mathematical approach. For problems involving a system interacting with its equilibrium thermal environment a particularly convenient choice is the thermal projector. An operator that projects the total system-bath density operator on a product of the system s reduced density operator and the... 

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