Orbital-density matrices

We consider the expectation value of a Hermitian operator of singlet spin symmetry 

The function may be an eigenfunction also of the total-spin operator, although this property is not assumed in the development of this section. 

Evaluation of the expectation value of with respect to the reference state (2.7.2) gives the expression 

These expressions should be compared with (1.7.5) and (1.7.6) in the spin-orbital basis. Expanding the orbital densities in the spin-orbital densities following the definitions of the excitation operators (2.2.7) and (2.2.16), we obtain 

The loss of the permutational symmetries (upper-case indices used for spin orbitals) 

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Total Valence Shell Electron Density, and d-Orbital Density Matrices of Phosphorus in... 

A. Definition of Orbitals, Density Matrices and Integral Matrices. 

The reduction coefficients q and Q are independent of both the Cl coefficients Ajf and the orbitals, and are determined by the form of the many-electron wavefunction. For simple wavefunctions they are very simple e.g. in the closed-shell case, the configuration subscripts can be omitted, and = 2dij, — 23ii,3j,. It is useful to introduce the orbital density matrices yij and rij i as the weighted sum of the reduction coefficients... 

Forms a stack of orbital density matrices plus an empty shell matrix. 

The eigenvalues of the one- and two-orbital density matrices, w. and w .y, respectively, can then be employed to determine the von Neumann entropy of each spatial orbital. This yields the so-called single-orbital entropy [96],... 

From the orbital density matrices considered in Section 2.7.1, we may calculate expectation values of singlet operators. For triplet operators such as the Fermi contact operator, a different set of density matrices is needed. Consider the evaluation of the expectation value for a one-electron triplet operator of the general form... 

Ochsenfeld C, Head-Gordon M. A reformulation of the coupled perturbed self-consistent field equations entirely within a local atomic orbital density matrix-based scheme. Chem Phys Lett 1997 270 399 405. 

To be more specific, we restate the approach [60] for the full (spin orbital) density matrix D what is not principal for our purposes. Then the approach is in fact assumed that the special density matrix, associated with effectively unpaired... 

Optimization of the Atomic-orbital Density Matrix Using the Conjugate-gradient Method with a Multilevel Preconditioner. 

The one-electron density matrix is positive semidefinite since its elements are either trivially equal to zero or inner products of states in the subspace F(M, N — 1). The diagonal elements of the spin-orbital density matrix are the expectation values of the occupation-number operators (1.3.1) in F(M, N) and are referred to as the occupation numbers Top of the electronic state ... 

For a state consisting of a single ON vector, the one-electron spin-orbital density matrix has a simple diagonal structure ... 

The main drawback of the chister-m-chister methods is that the embedding operators are derived from a wavefunction that does not reflect the proper periodicity of the crystal a two-dimensionally infinite wavefiinction/density with a proper band structure would be preferable. Indeed, Rosch and co-workers pointed out recently a series of problems with such chister-m-chister embedding approaches. These include the lack of marked improvement of the results over finite clusters of the same size, problems with the orbital space partitioning such that charge conservation is violated, spurious mixing of virtual orbitals into the density matrix [170], the inlierent delocalized nature of metallic orbitals [171], etc. 

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]. 

If you define a density matrix R by summing over all occupied molecular orbitals ... 

N() -e that the summations are over the N/2 occupied orbitals. Other properties can be cali ulated from the density matrix for example, the electronic energy is ... 

The calculation proceeds as illustrated in Table 2.2, which shows the variation in the coefficients of the atomic orbitals in the lowest-energy wavefunction and the energy for the first four SCF iterations. The energy is converged to six decimal places after six iterations and the charge density matrix after nine iterations. 

We have extended the linear combination of Gaussian-type orbitals local-density functional approach to calculate the total energies and electronic structures of helical chain polymers[35]. This method was originally developed for molecular systems[36-40], and extended to two-dimensionally periodic sys-tems[41,42] and chain polymers[34j. The one-electron wavefunctions here are constructed from a linear combination of Bloch functions c>>, which are in turn constructed from a linear combination of nuclear-centered Gaussian-type orbitals Xylr) (in ihis case, products of Gaussians and the real solid spherical harmonics). The one-electron density matrix is given by... 

Both the Fock matrix—through the density matrix—and the orbitals depend on the molecular orbital expansion coefficients. Thus, Equation 31 is not linear and must be solved iteratively. The procedure which does so is called the Self-Consistent Field... 

Form an initial guess for the molecular orbital coefficients, and construct the density matrix. 

When working with atomic orbitals, it is usual to write the electron density in terms of a certain matrix called (not surprisingly) the electron density matrix. For the simple dihydrogen VB wavefunction, we have... 

In the electronic ground state, the six rr-electrons occupy the three lowest-energy orbitals (the first three from Table 7.2). The Huckel rr-electron charge density matrix is... 

Anotiier way of justifying the use of eq. (6.4) for calculating tire kinetic energy is by reference to natural orbitals (eigenvectors of the density matrix. Section 9.5). The exact kinetic energy can be calculated from the natural orbitals (NO) arising from tire exact density matrix. 

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. 

Since the exact density matrix is not known, the (approximate) density is written in terms of a set of auxiliary one-electron functions, orbitals, as... 

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. 

The concept of natural orbitals may be used for distributing electrons into atomic and molecular orbitals, and thereby for deriving atomic charges and molecular bonds. The idea in the Natural Atomic Orbital (NAO) and Natural Bond Orbital (NBO) analysis developed by F. Weinholt and co-workers " is to use the one-electron density matrix for defining the shape of the atomic orbitals in the molecular environment, and derive molecular bonds from electron density between atoms. 

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