Density matrices matrix elements

It is therefore clear that particle number conservation considerations are not sufficient to determine S q, q) at very small but finite q. In the case of broken translational symmetry, as certainly occurs in the vicinity of a surface, the perfect screening of density fluctuation matrix elements, which is characteristic of homogeneous systems, does not hold due to nonconservation of momentum, and the small q limit of S(q, q) is nonuniversal even in the zero temperature case. 

P is the so-called density matrix, the elements of which involve a product of two molecular orbital coefficients summed over all occupied molecular orbitals. ... 

It is seen from (60)-(61) that there are two alternative ways to calculate the density variation i) through the transition density and matrix elements of Qfc-operator and ii) through the ground state density. The second way is the most simple. It becomes possible because, in atomic clusters, Vres has no T-odd Ffc-operators and thus the commutator of Qk with the full Hamiltonian is reduced to the commutator with the kinetic energy term only ... 

By means of (3 20) and (3 21) we can write down the second order reduced density matrix with elements Piikl for the wave function (3 15) as... 

Besides displaying only linear local convergence, another problem with the SCI method is that the construction of the H matrix requires either the explicit construction of the single excitation states or the evaluation of four-particle density matrix type elements of the form <0 In an... 

Analogically to the representation of the wave-function in structural terms, there is a way to separate (hyper)polarizabilities into the individual contributions from individual atoms. A method for such separation was developed by Bredas [15, 16] and is called the real-space finite-field method. The approach can be easily implemented for a post-Hartree-Fock method in the r-electron approximation due to the simplicity of e calculation of the one-electron reduced density matrix (RDMl) elements. In our calculations we are using a simple munerical-derivative two-points formula for RDMl matrix elements (Z ) [88] (see also [48]) ... 

The spinless one-electron density matrix (DM) elements are defined in the LCAO approximation as... 

The one-electron density matrix the elements of which are defined as... 

The preceding derivation of an expression for l/Tj, equation (29), was for a simple case with two eigenstates and only one interaction V. A more general case will involve further simultaneous equations, and relaxation times that depend on more than one integral of type Such cases can often be made more tractable by replacing expression (19) with a density matrix, whose elements are the averaged products C C . In this representation the wave equation converts to an equation involving this matrix. Its solution was first studied by Kubo and Tomita, and then developed by Redfield and many others. Their approach is essential in complex... 

Under the assumption that the matrix elements can be treated as constants, they can be factored out of the integral. This is a good approximation for most crystals. By comparison with equation Al.3.84. it is possible to define a fiinction similar to the density of states. In this case, since both valence and conduction band states are included, the fiinction is called the joint density of states ... 

It is more convenient to re-express this equation in Liouville space [8, 9 and 10], in which the density matrix becomes a vector, and the commutator with the Hamiltonian becomes the Liouville superoperator. In tliis fomuilation, the lines in the spectrum are some of the elements of the density matrix vector, and what happens to them is described by the superoperator matrix, equation (B2.4.25) becomes (B2.4.26). 

The xy magnetizations can also be complicated. Eor n weakly coupled spins, there can be n 2" lines in the spectrum and a strongly coupled spin system can have up to (2n )/((n-l) (n+l) ) transitions. Because of small couplings, and because some lines are weak combination lines, it is rare to be able to observe all possible lines. It is important to maintain the distinction between mathematical and practical relationships for the density matrix elements. 

The oharge- and exohange-density matrix elements in the AO basis are ... 

Frequent approximations made in TB teclmiques in the name of achieving a fast method are the use of a minimal basis set, the lack of a self-consistent charge density, the fitting of matrix elements of the potential. 

It is helpful to simplify Equation (2.157) by introducing the charge density matrix, F, whose elements are defined as ... 

Living calculated the integrals, we are now ready to start the SCF calculation. To formulate the Fock mahix it is necessary to have an initial guess of the density matrix, P. The simplest approach is to use the null matrix in which all elements are zero. In this initial step the Fock nulrix F is therefore equal to... 

Mulliken population analysis is a trivial calculation to perform once a self-consistent field has been established and the elements of the density matrix have been determined. 

We have the makings of an iterative computer method. Start by assuming values for the matr ix elements and calculate electron densities (charge densities and bond orders). Modify the matr ix elements according to the results of the electron density calculations, rediagonalize using the new matrix elements to get new densities, and so on. When the results of one iteration are not different from those of the last by more than some specified small amount, the results are self-consistent. 

The two-eleetron density matrix elements are given in similar fashion ... 

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