Idempotent density matrices

In view of these results, entropy can be used as a measure of dispersions. Pure states (idempotent density matrices) have zero entropy, whereas stable equilibrium states have an entropy larger than that of any other state with the same values of energy, parameters, and number of particles. 

Frishbeig, C., Massa, L. J. (1981). Idempotent density matrices for correlated systems from x-ray-diffraction structure factors. Phys. Rev. B 24, 7018-7024. 

Within the framework of the SALDA method,restoring idem-potency of density matrices at displaced nuclear geometries involves the parent molecules, where for small geometry variations a new, approximate, but exactly idempotent density matrix, as well as the associated, improved fragment density matrices can be computed using the above method. 

The breadth "of dispersion is measured by entropy, which ranges in value from zero for a pure state (idempotent density matrix) of a given energy, to a maximum for the stable equilibrium (thermodynamic equilibrium) mixed state of the same energy. 

The field Fs(r) depends only on the density p(r) and the idempotent density matrix y,(r,r ) through the field z(r [yj). The field. elr) does not appear in the expression for Fs(r) because there is no electron-interaction operator in the... 

This purification transformation of McWeenyallows one to create, out of a nearly idempotent density matrix P a more idempotent matrix P. The method converges quadratically toward the fully idempotent matrix. The function x (x) = 3x 2x (for S = 1) is shown in Figure 16 and possesses... 

The macromolecular density matrix built from such displaced local fragment density matrices does not necessarily fulfill the idempotency condition that is one condition involved in charge conservation. It is possible, however, to ensure idempotency for a macromolecular density matrix subject to small deformations of the nuclear arrangements by a relatively simple algorithm, based on the Lowdin transform-inverse Lowdin transform technique. 

This new, approximate macromolecular density matrix (q K ), K [A]) for the new, slightly distorted nuclear geometry K1 is also idempotent with respect to multiplication involving the actual new overlap matrix S(K... 

The origins of density functional theory (DFT) are to be found in the statistical theory of atoms proposed independently by Thomas in 1926 [1] and Fermi in 1928 [2]. The inclusion of exchange in this theory was proposed by Dirac in 1930 [3]. In his paper, Dirac introduced the idempotent first-order density matrix which now carries his name and is the result of a total wave function which is approximated by a single Slater determinant. The total energy underlying the Thomas-Fermi-Dirac (TFD) theory can be written (see, e.g. March [4], [5]) as... 

The concept of purification is well known in the linear-scaling literature for one-particle theories like Hartree-Fock and density functional theory, where it denotes the iterative process by which an arbitrary one-particle density matrix is projected onto an idempotent 1-RDM [2,59-61]. An RDM is said to be pure A-representable if it arises from the integration of an Al-particle density matrix T T, where T (the preimage) is an Al-particle wavefiinction [3-5]. Any idempotent 1-RDM is N-representable with a unique Slater-determinant preimage. Within the linear-scaling literature the 1-RDM may be directly computed with unconstrained optimization, where iterative purification imposes the A-representabUity conditions [59-61]. Recently, we have shown that these methods for computing the 1 -RDM directly... 

The n-particle density matrix of an w-particle state is pure-state n-representable if—for unit trace—it is idempotent. Since we normalize y as... 

The N particle representable density matrix T is an idempotent projector... 

Though we can compare electron densities directly, there is often a need for more condensed information. The missing link in the experimental sequence are the steps from the electron density to the one-particle density matrix f(1,1 ) to the wavefunction. Essentially the difficulty is that the wavefunction is a function of the 3n space coordinates of the electrons (and the n spin coordinates), while the electron density is only a three-dimensional function. Drastic assumptions must be introduced, such as the description of the molecular orbitals by a limited basis set, and the representation of the density by a single Slater-determinant, in which case the idempotency constraint reduces the number of unknowns... 

The same Lowdin-type orthonormalization-deorthonormalization method can also be applied for the restoration of the idempotency of geometry-dependent macromolecular density matrices within the context of the recently introduced ADMA macromolecular density matrix technique, reviewed in the following text. 

According to the assumption we have made the change in the density matrix, ARX, due to the coulombic interaction between fragments will be more or less localized. It is tempting to set ARX = XL. By doing that, however, one is forced [8] to split off the local space from the remainder of the system to satisfy the idempotency condition. This results in an ordinary cluster model which does not allow electron transfer to or from the surroundings and, as we will see in Sect. 5, is unsuitable for our purposes. In order to properly embed the cluster we take advantage of the fact that the sum of the occupied and unoccupied molecular orbital (MO) spaces is identical to the total AO space. So, instead of ARX = XL, we write... 

It is customary to represent the probabilities of a quantum-mechanical or "actual" state by a wave function or, equivalently, by a density matrix that is pure or idempotent, i.e., by a matrix that, in diagonal form, has one element equal to unity and all others equal to zero. 

Yet another technique is to minimize the Kohn-Sham energy as a functional of the density matrix, Eq. (65). In the first method of this type, due to Li et al. [47], the ground state of the system was only a local minimum of the energy functional. To surmount this difficulty, one may choose to explicitly impose the idempotency constraint on the density matrix, Eq. (71), with a Lagrange multiplier. For exam-... 

The simplest case is that the Cl expansion coefficients are fixed and not effected by relativity. Then one can essentially use the same formalism as in section 7.5, except that the one-particle density matrix is no longer idempotent and that the two-particle density matrix is not simply related to the one-particle density matrix. 

Where the non-correlated energy E c is of the same form as the Hartree-Fock energy, but with 7 the exact one-particle density matrix, which is usually not idempotent. This concept has not yet been applied in practice. 

This result contains the idempotency constraint on the density matrix. 

Since the integral over cIt2 is nothing more than the idempotency condition on the one-particle density matrix ... 

The Fermi hole in turn is defined in terms of the idempotent Dirac density matrix ys(r, r ) of Eq. (36) where the orbitals < j(x) are solutions of the Har-... 

The field z(r [y]) thus defined is for the interacting system since the tensor involves the density matrix y(r,r ) of Eq. (6). With the field z(r [yj) derived similarly from the tensor tajj(r [ys])) written in terms of the idempotent Dirac density matrix > s( F) of Kohn-Sham theory, the field Z, (r) is then defined as... 

---