Density matrix first-order reduced

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... 

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]. 

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. 

Density matrices, in particular, the so-called first- and second-order reduced density matrices, are important quantities in the theoretical description of electronic structures because they contain all the essential information of the system under study. Given a set of orthonormal MOs, we define the first-order reduced density matrix D with matrix elements as the expectation value of the excitation operator E = aLa, -I- with respect to some electronic wave function Fgi,... 

He goes on to show that for the description of the energy it is sufficient to know the second-order reduced density matrix F(jc, jc2 i 2)- This was a favorite subject when he was lecturing and led to speculations about the possibility to compute the second-order density matrix directly, and discussions of the so called A/ -represent-ability problem. In spite of several attempts, this way of attacking the quantum chemical many-particle problem has so far been unsuccessful. Of special interest was the first-order reduced density matrix, 7( 1 jC ), which when expanded in a complete one-electron basis, ij/, is obtained as... 

The orbitals that diagonalize the first-order reduced density matrix are the natural spin orbitals x-... 

The probability density of electrons p x) in a quantum-mechanical system is given by the diagonal element of the first-order reduced-density matrix, (the superscript s indicates... 

Within the conventional density-functional formalism one seeks an expression for the total electronic energy as a functional of the electron density of Eq. (16). The electron density of Eq. (16) can be considered the diagonal elements of the first-order reduced density matrix... 

In order to define natural orbitals, we now consider the first-order reduced density matrix of an iV-electron system. Given a normalized wave function, O, then 0(xi,..., x y), Xjy) dx dxjy is the probability... 

The first-order reduced density matrix of the exact wave function also has a number of weakly occupied orbitals for which i > N. These orbitals never appear in a single Slater determinant method, but are important for a correct description of the correlation hole. 

The generalized fractional occupation Up is related to diagonal matrix elements of the first-order reduced density matrix constructed in natural... 

Finally, it should be stressed that this full energetic decomposition, coined interacting quantum atoms (IQA) by Pendas and coworkers (who have applied it to very different systems, proving its versatihty [81-87]), is in principle exact. However, as it is grounded on the partition of the first-order reduced density matrix (RDM) for the kinetic part and of the second-order RDM to decompose the total electron-electron repulsion energy, it should a priori be used only in conjunction with (post Hartree-Fock) wavefunction theory. We will mark this fact by exphcitly writing these energetic terms as functionals of the wavefunction E = E[ j/]). 

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