Matrix idempotent

The density matrix idempotency relations described here may be easily extended to the unrestricted Hartree-Fock (UHF) method when the orbitals for a and / spins are treated independently. 

This eonstruetion in whieh a veetor is used to form a matrix v(i)Xv(i) is ealled an "outer produet". The projeetion matrix thus formed ean be shown to be idempotent. whieh means that the result of applying it twiee (or more times) is identieal to the result of applying it onee P P = P. This property is straightforward to demonstrate. Let us eonsider... 

It should be emphasized that not all normalizable hermitean matrices r(x x 2. . . x xlx2. . . xp) having the correct antisymmetry property are necessarily strict density matrices, i.e., are derivable from a wave function W. For instance, for p — N, it is a necessary and sufficient condition that the matrix JT is idempotent, so that r2 = r, Tr (JH) = 1. This means that the F-space goes conceptually outside the -space, which it fully contains. The relation IV. 5 has apparently a meaning within the entire jT-space, independent of whether T is connected with a wave function or not. The question is only which restrictions one has to impose on r in order to secure the validity of the inequality... 

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. 

The inverse Lowdin transform constructed for the above idempotent matrix S(AT)i/2P(c AT), K)S(K) n- given with respect to the actual new, macromolecular overlap matrix S(K1), is expressed as... 

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... 

A projector is another case of a symmetric matrix. Since it is idempotent, its eigenvalues must be either 1 or 0. Indeed, idempotence relative to eigenvectors , implies... 

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... 

In illustration, March and Young [55] constructed an idempotent 1-matrix which, for example, for the one-dimensional case with N odd N — 2m + 1, for m integer) is given by ... 

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 quantity Pp is a projection tensor, which reduces to the identity 8p in the case of an invertible mobility matrix, and which is always idempotent, since... 

The first two terms on the r.h.s. of equation (10) coincide with those obtained when evaluating the 2-RDM corresponding to a simple Slater determinant. Therefore, the only way in which these two terms may describe correlation effects is through the 1-RDM itself, when the state is a correlated one. Note that in such a case the 1-RDM will not be an idempotent matrix. 

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

It is easy to see that in general the new R matrix created from equation (33) will not be accurately idempotent (i.e. orthogonality will be lost), and that therefore one will not be able to use this matrix at the next iteration without correcting it for this defect. However, as the minimum is approached the new R matrix will become more and more accurately idempotent. 

In 1970 Fletcher5 observed that slow convergence was a characteristic of the steepest-descent method and that a more modem method would probably work better. However, he noted that the scheme for incorporating constraints used by McWeeny was unsuitable for most modem methods, since modem methods often needed information from the previous cycle and this information would be misleading if it had been necessary, after the previous cycle, to restore idempotency. Fletcher therefore suggested the method we have already referred to in equation (15) leading to equation (17). If we denote the matrix (F+SF) by A, it can easily be seen from equation (17) that, to first order,... 

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... 

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 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. 

In mathematics, matrices having these properties (idempotency, mutual exclusivity, completeness3) are called projectors. In fact, acting on matrix C of Equation (1.21)... 

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... 

At the heart of SCEP theory lie the matrices, Cp, that give the coefficients for excitation of electrons from a given pair of occupied MOs (P = ij) to an arbitrary pair of AOs p, q. In the spin-unrestricted treatment i and j refer to spinorbitals with i > j. The use of a matrix formulation in an AO basis leads to a perfect fit between SCEP theory and either the KS/ or HF/ LSA method. In treating pair correlations the idempotency condition on R is replaced by a strong orthogonality requirement on the double excitations — namely, CpSR = 0 = RSCp or, equivalently,... 

In the calculations of the matrix elements concerned, three points are to be taken into account (i) the operator (projector) A is idempotent (ii) the Hamiltonian H is formulated according to the principles of quantum mechanics and as such it commutes with A as well as with the individual spin-space permutations (iii) the spin-dependent... 

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. 

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