Single-particle density operator

Thus the binary density operator is given only by the single-particle density operator Fl, which depends on the earlier time t0. Therefore, retardation effects appear. In order to eliminate this time we use the formal solution (1.30) for F, ... 

It turns out that it is impossible now to get a closed equation for the single-particle density operator alone. We also have to consider a kinetic... 

Because of the interaction between the particles on the right-hand side, the pair operators C and the operator of the single particle density pa occur. Using relation (4.19) we find... 

It is worthwhile mentioning at this point that all properties of a subsystem defined in real space, including its energy, necessarily require the definition of corresponding three-dimensional density distribution functions. Thus, all the properties of an atom in a molecule are determined by averages over effective single-particle densities or dressed operators and the one-electron picture is an appropriate on ] [y)... 

The usefulness of the single-particle density matrix becomes apparent when we consider how one would calculate the expectation value of a multiplicative single-particle operator A = Y, a i) (such as the potential V =... 

Using this expression as the single-particle operator o(r), and combining it with the expression for the single-particle density matrix above, we obtain for... 

This expression involves exclusively matrix elements of the single-particle operators o(r) and the single-particle density matrix y(r, rO in the single-particle states (p (r), which is very convenient for actual calculations of physical properties (see, for example, the discussion of the dielectric function in chapter 6). 

As discussed in [11], the single particle density function can be calculated by evaluating the expectation value of the 5(r) operator,... 

In order to have an expression for the mixed-system exchange energy (192) in terms of the single-particle KS orbitals we need such an expression for the ensemble DM - the diagonal of pP. When the ensemble density operator is characterized by the following decompc ition into its pure-state contributions... 

Now we want to generalize the kinetic equation for free (unbound) particles that is, we want to derive a kinetic equation for free particles that takes into account collisions between free and bound particles as well. For this purpose it is necessary to determine the binary density operator, occurring in the collision integral of the single-particle kinetic equation, at least in the three-particle collision approximation. An approximation of such type was given in Section II.2 for systems without bound states. Thus we have to generalize, for example, the approximation for/12 given by Eq. (2.40), to systems with bound states. 

The averages of the operators in Eqs. (190) and (194) are the elements of the density matrix in the single-particle space... 

After having described the expression for the rate constant within the framework of classical mechanics, we turn now to the quantum mechanical version. We consider first the definition of a flux operator in quantum mechanics.2 To that end, the flux density operator (for a single particle of mass to) is defined by... 

Because field quantization falls outside the scope of the present text, the discussion here has been limited to properties of classical fields that follow from Lorentz and general nonabelian gauge invariance of the Lagrangian densities. Treating the interacting fermion field as a classical field allows derivation of symmetry properties and of conservation laws, but is necessarily restricted to a theory of an isolated single particle. When this is extended by field quantization, so that the field amplitude rjr becomes a sum of fermion annihilation operators, the theory becomes applicable to the real world of many fermions and of physical antiparticles, while many qualitative implications of classical gauge field theory remain valid. 

The result is multiplied by JV, the total number of electrons, in the definition of an atomic property. The reader is reminded that the mode of integration indicated by N dx [l/ ijy as used in this definition of an atomic average is the same as that employed in the definition of the electronic charge density, p r) (eqns (1.3) and (1.4)). From this point on the subscript T will be dropped from the coordinates of the electron whose coordinates are integrated only over 2 and all single-particle, unlabelled coordinates and operators will refer to this electron. 

Because the many-particle Lagrangian density L° reduces to a sum of singleparticle operators, one may define an effective single-particle Lagrangian density iP°(r, t) by the usual recipe of summing over all spins and integrating over the coordinates of all the electrons but one, a step which expresses the result in terms of the charge density p(r, t) as... 

From the extremum condition that AAva r should be zero for arbitrary variations hP and thus for arbitrary changes Ap/l1 in the density operator, we derive the expression for the optimized single-particle Hamiltonian... 

Three known facts are essentially important in the development of a divide-and-conquer strategy. First, the KS Hamiltonian is a single particle operator that depends only on the total density, not on individual orbitals. This enables one to project the energy density in real space in the same manner in which one projects the density (see below). Second, any complete basis set can solve the KS equation exactly no matter where the centers of the basis functions are. Thus, one has the freedom to select the centers. It is well known that for a finite basis set the basis functions can be tailored to better represent wavefunctions, and thus the density, of a particular region. The inclusion of basis functions at the midpoint of a chemical bond is the best known example. Finally, the atomic centered basis functions used in almost all quantum chemistry computations decay exponentially. Hence both the density and the energy density contributed by atomic centered basis functions also decrease rapidly. All these... 

This equation follows immediately from comparing (8) with (46). We can define an alternative density operator, n, by requiring that the same equation must also be obtained by substituting n(r) into Eq. (51), which holds for any single-particle operator. This requirement implies that n(r) = 4(r — ry).24... 

S OLID-LIQUID FLOWS are encountered in a variety of applications ranging from food to mining industries (I). Unlike single fluid flow in pipes, slurry flow in pipelines is complex. The complexity of these flows has necessitated the use of empirical equations in the design of slurry handling equipment, often leading to expensive systems. This complexity depends on the physical properties of the solid particles, for example, particle density, shape, and mean diameter. It also depends on the viscosity and density of the carrier fluid and, finally, on the operating con-... 

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