Density matrices natural orbitals

Chapter 4 discusses configuration interaction (Cl) and is the first of the four chapters that deal with approaches incorporating electron correlation. One-electron density matrices, natural orbitals, the multiconfiguration self-consistent-field approximation, and the generalized valence bond method are... 

We recall that the natural orbitals are the eigenfunctions of the one-electron density matrix and that the associated eigenvalues, known as the occupation numbers, represent a measure of the contributions that the different natural orbitals make to the density matrix. Natural orbitals with small occupation numbers can thus be presumed to be less important than those with larger occupations. For a discussion of natural orbitals, see Sections 1.7.1 and 2.7.1. 

This is the one-electron density matrix in orbital space for the ground state 0). For the present purpose we call it the density matrix. It is manifestly Hermitian. Therefore it is possible to find a unitary transformation U of the orbitals fi) that diagonalises it. The new orbitals a) are the natural orbitals... 

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. 

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. 

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. 

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). 

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

In the case ofthe VB wave function, the natural orbitals and occupation numbers obtained by diagonalizing the matrix representation of the VB one-density function on the atomic basis set were used in Equation (14). 

W. A. Bingel and W. Kutzelnigg, Symmetry of the first-order density matrix and its natural orbitals for linear molecules. Adv. Quantum Chem. 5, 201 (1970). 

The spin-free one-particle density matrix Fi = F is diagonal in the basis of the (spin-free) natural orbitals (NOs)... 

Pierloot, K., Dumez, B., Widmark, P.-O., Roos, B.O. Density-matrix averaged atomic natural orbital (ANO) basis-sets for correlated molecular wave-functions. IV. Medium-size basis-sets for the atoms H-Kr. Theor. Chim. Acta 1995, 90, 87-114. 

The spin density should follow from the density matrix (38), which includes the spin variables. As in (42), Qa(x x ) will be a sum of terms containing the various spinor components, summed over all spin-orbitals in the natural expansion. A typical term will be... 

It can be necessary and/or desirable to impose symmetry and equivalence restrictions on quantum chemical calculations or results beyond the single-configuration SCF level. For instance, most Cl programs generate natural orbitals (NOs) after computing the Cl wave function, by forming and diagonalizing the first-order reduced density matrix or 1-matrix p in... 

The concept of the molecular orbital is, however, not restricted to the Hartree-Fock model. Sets of orbitals can also be constructed for more complex wave functions, which include correlation effects. They can be used to obtain insight into the detailed features of the electron structure. One choice of orbitals are the natural orbitals, which are obtained by diagonalizing the spinless first-order reduced density matrix. The occupation numbers (T ) of the natural orbitals are not restricted to 2, 1, or 0. Instead they fulfill the condition ... 

The super-CI method now implies solving the corresponding secular problem and using tpq as the exponential parameters for the orbital rotations. Alternatively we can construct the first order density matrix corresponding to the wave function (4 55), diagonalize it, and use the natural orbitals as the new trial orbitals in I0>. Both methods incorporate the effects of lpq> into I0> to second order in tpq. We can therefore expect tpq to decrease in the next iteration. At convergence all t will vanish, which is equivalent to the condition ... 

The two-electron reduced density matrix is a considerably simpler quantity than the N-electron wavefunction and again, if the A -representability problem could be solved in a simple and systematic manner the two-matrix would offer possibilities for accurate treatment of very large systems. The natural expansion may be compared in form to the expansion of the electron density in terms of Kohn-Sham spin orbitals and it raises the question of the connection between the spin orbital space and the -electron space when working with reduced quantities, such as density matrices and the electron density. 

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