Number density matrices

The following relation and its inverse hold between the elements of the number density matrices in momentum and position space [21] ... 

The electron density is the diagonal element of the number density matrix N(r,r ), i.e the first order redueed density matrix after integration over the spin coordinates, ... 

A transformation of the number density matrix N under a space group operation means that both variables are transformed ... 

If the elements of the number density matrix in position space are invariant under all operations of the space group, i.e. if... 

We notice that neither the momentum distribution nor the reciprocal form factor seems to carry any information about the translational part of the space group. The non diagonal elements of the number density matrix in momentum space, on the other hand, transform under the elements of the space group in a way which brings in the translational parts explicitly. 

The number density matrix for a crystal with translation symmetry can be written in terms of its natural orbitals [23, 24], as... 

Combining the inverses of (III. 14) and (III. 16) we get the natural expansion for a general element of the number density matrix in momentum space ... 

Here the component of the number density matrix associated with the wave vector k is thus... 

From the point of view of interpretation it is preferable to write the orbital form of the Fock-Dirac matrix in terms of the number density matrix... 

Following Fukutome we can therefore use the subgroup structure of G to classify the different types of GHF solutions that are possible, with respect to the properties of the number density matrix N(r, r ) and the spin density matrix vector S(r, r ). With the trivial subgroups there are eight subgroups of G, which are denoted as indicated in Table I. Each such subgroup corresponds to a class of GHF solutions, with properties summed up in Table II. 

If N(r, r ) or S(r, f ) has a nonvanishing imaginary part there is a corresponding current density. The charge-current wave (CCW) class, for example, has a complex number density matrix, whereas the axial spin-current wave (ASCW) class has a purely imaginary spin density matrix. 

Flead and Silva used occupation numbers obtained from a periodic FIF density matrix for the substrate to define localized orbitals in the chemisorption region, which then defines a cluster subspace on which to carry out FIF calculations [181]. Contributions from the surroundings also only come from the bare slab, as in the Green s matrix approach. Increases in computational power and improvements in minimization teclmiques have made it easier to obtain the electronic properties of adsorbates by supercell slab teclmiques, leading to the Green s fiinction methods becommg less popular [182]. 

A number of procedures have been proposed to map a wave function onto a function that has the form of a phase-space distribution. Of these, the oldest and best known is the Wigner function [137,138]. (See [139] for an exposition using Louiville space.) For a review of this, and other distributions, see [140]. The quantum mechanical density matrix is a matrix representation of the density operator... 

Thus, HyperChem occasionally uses a three-point interpolation of the density matrix to accelerate the convergence of quantum mechanics calculations when the number of iterations is exactly divisible by three and certain criteria are met by the density matrices. The interpolated density matrix is then used to form the Fock matrix used by the next iteration. This method usually accelerates convergent calculations. However, interpolation with the MINDO/3, MNDO, AMI, and PM3 methods can fail on systems that have a significant charge buildup. 

The orbital occupation numbers n, (eigenvalues of the density matrix) will be between 0 and 1, corresponding to the number of electrons in the orbital. Note that the representation of the exact density normally will require an infinite number of natural orbitals. The first N occupation numbers N being the total number of electrons in the system) will noraially be close to 1, and tire remaining close to 0. 

The sum of the product of MO coefficients and the occupation numbers is the density matrix defined in Section 3.5 (eq. (3.51)). The sum over the product of the density and overlap matrix elements is the number of electrons. 

The original definition of natural orbitals was in terms of the density matrix from a full Cl wave function, i.e. the best possible for a given basis set. In that case the natural orbitals have the significance that they provide the fastest convergence. In order to obtain the lowest energy for a Cl expansion using only a limited set of orbitals, the natural orbitals with the largest occupation numbers should be used. 

NAOs for an atomic block in the density matrix which have occupation numbers very close to 2 (say >1.999) are identified as core orbitals. Their contributions to the density matrix are removed. 

Each pair of atoms (AB, AC, BC,...) is now considered, and the two-by-two subblocks of the density matrix (with the core and lone pair contributions removed) are diagonalized. Natural bond orbitals are identified as eigenvectors which have large eigenvalues (occupation numbers larger than say 1.90). 

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