Diagonal element quantization

The orbital representation is not used in most of the recent work on computational methods based on diagonal elements of density matrices. This is partly for historical reasons—most of the work has been done by people trained in density functional theory—and partly this is because most of the available kinetic energy functionals are known only in first-quantized form. For example, the popular generalized Weisacker functional [2, 7-11],... 

For the geometries considered in this chapter (see Figs. 2.5 and 2.6), which are peculiar in the sense that no external fields are applied and the quantization axis z is chosen along the symmetry axis of excitation, only diagonal elements of the excited state /mm and ground state ip density matrix appear. The connection between /mm and in analogy with the classical expression (3.16), may be written in the form... 

At microwave frequencies the first term in Eq. (18-1) predominates allowing the approximation S is quantized along the unit vector k, where k = Hg/ H g. With this approximation off-diagonal elements resulting from components of S perpendicular to H0 can be neglected. Since there are no terms in Eq. (18-1) connecting different nuclear spins, it follows that the spectrum and intensity pattern for each nucleus can be analyzed independently of the other nuclei. Equation (18-1) may then be written as... 

These experiments are in fact entirely analagous to the Hanle effect or zero-field level-crossing experiments involving excited atoms discussed in Chapter 15. The coherent polarization of the pumping light referred to the quantization axis Oz in Fig.17.12 prepares the atoms in a coherent superposition of ground-state Zeeman sub-levels. The ensemble density matrix now has finite off-diagonal elements... 

Calculating the expectation values of (2.7.29) and (2.7.30) in first quantization, we find that these physical densities correspond to the diagonal elements of the first- and second-order reduced density matrices (1.7.37) and (1.7.43) integrated over the spin coordinates... 

The method presented enables expressions to be found for submatrix elements of irreducible tensorial products of second-quantization operators for configurations of any complexity. This method provides a unified approach both to diagonal and non-diagonal (relative to the configuration) matrix elements of operators of physical quantities. 

Considering a given region. A, in configuration space we define those x-matrices, that yield diagonal D-matrices in A, as being quantized in that region. The concept of quantization becomes clear when we consider a 2x2 x-matrix which is presented in terms of one non-zero matrix element ... 

The proportionality constant k is usually assigned a value of approximately 1.75. The overlap matrix elements Sy are calculated with respect to a set of two component basis functions with lsjm) quantization. The radial parts were chosen to be one or two Slater functions yielding (r ) (k=-l,0,1,2) expectation values as close as possible (Lohr and Jia 1986) to the Dirac-Hartree-Fock or Hartree-Fock results tabulated by Desclaux (1973) for the relativistic and nonrelativistic case, respectively. The diagonal Hamiltonian matrix elements Hu were set equal to the corresponding orbital energies from Desclaux s tables. Due to the use of a two-component lsjm) basis set the matrices H and S are generally complex and of dimension 2nx2n, when is the number of spatial orbitals. 

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