First-order reduced density

Factoring the Energy Functional through the First Order Reduced Density Operator. 

An appealing way to apply the constraint expressed in Eq. (3.14) is to make connection with Natural Orbitals (31), in particular, to express p as a functional of the occupation numbers, n, and Natural General Spin Ckbitals (NGSO s), yr,, of the First Order Reduced Density Operator (FORDO) associated with the N-particle state appearing in the energy expression Eq. (3.8). In order to introduce the variables n and yr, in a well-defined manner, the... 

The first-order reduced density operator y can be defined in terms of its kernel function37... 

Here T(r r ) signifies what is termed as the first-order reduced density matrix, which is defined as... 

If electronic spin is not a focus of attention, then the spin-traced versions of these density matrices can be used. The r-space, spin-traced, first-order, reduced-density matrix is... 

The many-body problem is reformiilated here by using a system of equations involving only first order Reduced Density Matrices. These matrices correspond to all the states of the spectrum of the system and to the transitions among the different states. Some results concerning the correlation effects are also reported here. 

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 ... 

Density matrices of the state functions provide a compact graphical representation of important microscopic features for second order nonlinear optical processes. The transition moment y is expressed in terms of the transition density matrix p jji(r,r ) by nn " /j, ptr Pjj t(r,r )dr and the dipole moment difference Ay by the difference density function p - p between the excited and ground state functions = -e / r( p -p )dr where p is the first order reduced density matrix. 

This can be generalized to the first-order reduced density matrix -y(xi, Xj ) that depends on two continuous variables. 

Davidson suggested that the wavefunction be projected onto a set of orbitals that have intuitive significance. These orbitals are a minimum set of atomic orbitals that provide the best least-squares fit of the first-order reduced-density matrix. Roby expanded on this idea by projecting onto the wavefunction of the isolated atom. One then uses the general Mulliken idea of counting the number of electrons in each of these projected orbitals that reside on a given atom to obtain the gross atomic population. 

If we are interested only in properties that can be expressed in terms of one-electron operators, then it is sufficient to work with the first-order reduced density matrix rather than the A-electron wave function [23-27]. 

Fig. 19.1 provides a concise summary of these relationships. A more elaborate figure that adds the connections to the Wigner [38,39] and Moyal [40] mixed position-momentum representations of the first-order reduced density matrix can be found in an article that also works out all these functions in closed form for a simple harmonic model of the helium atom [41]. 

What is needed for a correct computation of momentum-space properties from DPT is an accurate functional for approximating the exact first-order reduced density matrix r f f ), or failing that, good functionals for each of the p-space properties of interest. Of course, a sufficiently good functional for (p ) would obviate the necessity of using Kohn-Sham orbitals and enable the formulation of an orbital-free DFT. Unfortunately, a kinetic energy functional sufficiently accurate for chemical purposes remains an elusive goal [118,119]. 

The concept of the molecular orbital and their occupation is, however, not restricted to the HF model. It has much wider relevance and is applicable also for more accurate wave functions. For each wave function we can form the first-order reduced density matrix. This matrix is Hermitian and can be diagonalized. The basis for this diagonal form of the density matrix are the Natural Orbitals first introduced in quantum chemistry by Per-Olof Lowdin [4]. 

Natural resonance theory (NRT) is an optimal ab initio realization of the resonance weighting concepts expressed in Equation 7.10 and Equation 7.11. The necessary and sufficient condition that Equation 7.10 be satisfied for all possible density-related (one-electron) properties p is that the first-order reduced density operator be expressible as such a weighted average of localized density operators t... 

Diagonalization of F with the constraint that the first-order reduced density matrix 7 (the one-matrix) satisfies 7sr = (ar s) = (nr)Ssr with occupation numbers (n ) = 0 or 1 i.e., Tr y = N and 7 = 7) is done iteratively and converges to a single determinantal SCF approximation for the iV-electron ground state corresponding to the appropriate set of occupation numbers. 

Methods using the first-order reduced density matrix as variable can be chosen to strictly enforce the A-representability of yi, and employ the exact energy functional for all the terms except the correlation energy. The latter, however, inherently depends on 72, which in this approach must be approximated as a function of yi. One can argue that Hartree-Fock belongs to this class of methods, with imphcit neglect of the electron correlation. 

Alternative approaches to the many-electron problem, working in real space rather than in Hilbert space and with the electron density playing the major role, are provided by Bader s atoms in molecule [11, 12], which partitions the molecular space into basins associated with each atom and density-functional methods [3,13]. These latter are based on a modified Kohn-Sham form of the one-electron effective Hamiltonian, differing from the Hartree-Fock operator for the inclusion of a correlation potential. In these methods, it is possible to mimic correlated natural orbitals, as eigenvectors of the first-order reduced density operator, directly... 

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