Density matrix special cases

In the usual preparatioii-evohition-detection paradigm, neither the preparation nor the detection depend on the details of the Hamiltonian, except hi special cases. Starthig from equilibrium, a hard pulse gives a density matrix that is just proportional to F. The detector picks up only the unweighted sum of the spin operators,... 

Population analysis with semi-empirical methods requires a special comment. These methods normally employ the ZDO approximation, i.e. the overlap S is a unit matrix. The population analysis can therefore be performed directly on the density matrix. In some cases, however, a Mulliken population analysis is performed with DS, which requires an explicit calculation of the S matrix. 

Approximations have been reviewed in the case of short deBroglie wavelengths for the nuclei to derive coupled quantal-semiclassical computational procedures, by choosing different types of many-electron wavefunctions. Time-dependent Hartree-Fock and time-dependent multiconfiguration Hartree-Fock formulations are possible, and lead to the Eik/TDHF and Eik/TDMCHF approximations, respectively. More generally, these can be considered special cases of an Eik/TDDM approach, in terms of a general density matrix for many-electron systems. 

There are two ways to fix this problem. First, one can attempt to derive N-representability conditions for the g-density in the spatial representation. This seems hard to do, although one constraint (basically a special case of the G condition for the density matrix) of this type is known, see Eq. (77). Deriving additional constraints is a priority for future work. 

It is instructive to compute the gradient (4 42) for the special case of a closed shell Hartree-Fock wave function. Let us denote the occupied orbitals with labels ij,k,l... The only non-vanishing first and second order density matrix elements are in this case ... 

The use of this expression for a variational determination of T is a complex problem because of the /V-representability requirement [15, 16, 17], Nevertheless, there is a renewed interest in this problem and a number of methods, including so called cumulant-based approximations [18, 19] are being put forth as solutions to the representability problem. Although some advances can be obtained for special cases there appears to be no systematic scheme of approximating the density matrix with a well-defined measure of the N-representability error. Obviously, the variational determination of density matrices that are not guaranteed to correspond to an antisymmetric electronic wavefunction can lead to non-physical results. 

We have written the operator Fl(x) as a function of the combined space-spin coordinates X, because while the spin summations can be carried out in Jl(x) before calculating matrix elements, Kl(x) may connect spin-orbitals that are off-diagonal in the spin wavefunctions however in the special case of the density matrix p (xi, Xa) arising from a wavefunction that is a spin singlet (5 = 0) one can show that must also be diagonal. This leads to a useful simplification here since we can usually assume this property for Wlo, and it means that Vl(x) reduces to a (non-local) function of the space variable r only we can therefore consistently parameterize the matrix elements for the whole potential, (/bI Vl(x) j) without having to decompose them into different spin combinations for the Coulomb and exchange potentials. 

That Eq. (44) and Eq. (34) are different is due to the fact that GHF exchange is quadratic in the first-order density matrix, whereas Slater s approximation depends on the 4/3 power of the density. Only for the special case of a single determinantal wavefunction that has constant direction of S (i.e. for conventional but not generalized single determinants) can we regroup terms in Eq. (43) to yield Eq. (34). For this case N and S can be simplified as... 

In the special case of the initial states whose density matrix is diagonal in the Fock basis, so that (b tbk) = v 8nk and (bnbk) = 0 (e.g., the Fock or thermal states v is the mean number of quanta in the rath mode), the sum in (179) is proportional to the initial total energy 6" in all the modes (above the Casimir level) ... 

So far, biexponential relaxation has been the only aspect of the behaviour of quadrupolar nuclei which has distinguished them from the classical model of nuclear magnetism. This is because all the density matrices generated have contained only single-quantum elements, which behave classically in response to rf pulses. The order of multiple-quantum coherence for a density matrix element ernm is n — m for the special case of a single spin. The coherence pattern for a spin- density matrix is thus,... 

In the special case that the wave function T is a Slater determinant, i.e., the wave function of N noninteracting fermions, the single-particle density matrix can be written in terms of the orbitals comprising the determinant,... 

In the usual preparation-evolution-detection paradigm, neither the preparation nor the detection depend on the details of the Hamiltonian, except in special cases. Starting from equilibrium, a hard pulse gives a density matrix that is just proportional to F. The detector picks up only the unweighted sum of the spin operators, /. It is only during an evolution (perhaps between sampling points in an FID) that these totals need be divided amongst the various lines in the spectrum. Therefore, one of the factors in the transition probability represents the conversion from preparation to evolution the other factor represents the conversion back from evolution to detection. 

For a single Slater determinant all information is contained in its one particle density matrix [96]. For this special case we use q instead of 7. 

The density matrices play a special role in statistical mechanics but also in any situation in which we possess incomplete information about some general system. In that case, the system will not be in a pure state and will thus not be represented by a wavefunction instead, the mixed state must be represented by a density matrix. The density matrix will then refer to an ensemble of identical systems of which a fractional number wk are in the definite state and the ensemble density matrix will be... 

Equation (8) introduces notation, in which a labels the specific matrix element of rank R, SR(jij ) is a single-particle matrix element for the transition j-, - jf in the impulse approximation, and the quenching factor qJUiJf) corrects SrO jV) for the finite size of the model space and some effects of the nuclear medium so that the effective value of Spijijf) is qa jijf)SR jijf) = SKy ijf, eff). The DR jij[) are the one-body-transition densities that are the result of the shell-model calculations. For the special case of the axial-charge matrix element Mj the defining equation has the emedjijf) of Table 1 incorporated into the sum, i.e.. 

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