Idempotent

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... 

In the table, the rules of commutation, association and distribution are the same for both algebras. Idempotency, unique to Boolean algebra, relates to redundancy. Having "A and A is the same as only having A A or A is superfluous and equals A. Complementation is introduced in the next rule. The universe is represented by 1 . Completeness includes every thing in this world and not in this world, hence A +A = 1 where not A is A 1 -A which is the meaning of complementation. With this understanding, it is impossible to be both A and not A. Similarly A or not A is complete (the universe). Under unity, A is included in the universe (1) so A-f 1 = 1 For this... 

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... 

Introducing the HPHF wave-function expression (1) in (3), and taking into account the idempotency of operator A(s),the following equation may be obtained ... 

The approximate transferability of fuzzy fragment density matrices, and the associated technical, computational aspects of the idempotency constraints of assembled density matrices, as well as the conditions for adjustability and additivity of fragment density matrices are discussed in Section 4, whereas in Section 5, an algorithm for small deformations of electron densities are reviewed. The Summary in Section 6 is followed by an extensive list of relevant references. 

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... 

Including the obvious ones—associative, commutative, and idempotent—and many not-so-obvi-ous ones that help factor fundamental commonality in structure and behavior. 

This means that the operators are mutually exclusive and that the operator is idempotent. Nevertheless, in three-dimensional Cartesian space the atoms do overlap, often even to a large extent. So they have no boundaries. 

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 diagonal e-operators, efi, are special cases of what are called "primitive idempotents , which we now proceed to define. An operator p 6 U is called "idempotent if it satisfies... 

If we imagine an idempotent p expressed in terms of the e-operators, we see immediately that (21) must be satisfied separately for the parts belonging to each irreducible representation hence, we can treat the parts independently in deducing consequences from (21). Consider, then, an idempotent pM, belonging to the representation fW. We define a />-adapted basis for I M as follows ... 

A primitive idempotent is now defined as an idempotent belonging to a single irreducible representation, and with h — 1. A primitive idem-potent p, in the -adapted basis, has the form... 

The right-hand side of (28) is a primitive idempotent (cf. (25)), and can be a multiple of q only if q is itself a primitive idempotent. We have thus proved... 

Theorem 3 If e and ez are two primitive idempotents, a necessary and sufficient condition for their belonging to the same IR is the existence of an x e U such that 0. 

Theorem 1 The operator Y defined by (39) is, apart from a multiplicative constant, a primitive idempotent of <3n. Y operators belonging to the same Young diagram belong to the same irreducible representation, while those belonging to different diagrams belong to different representations. 

We will also make some use of the fact that Y —QP is also a constant times a primitive idempotent on the same IR as Y. That Y is a constant times a primitive idempotent can be proved analogously to Theorem 1. That it belongs to the same IR as Y follows from... 

Thus, symmetry projection need only be performed on the ket. Typically, projection operators are Hermitian and essentially idempotent cx in any... 

Symmetry adaptation ofVB wavefunctions Defining the (idempotent) projection operator... 

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... 

Given the inequalities in Eq. (32), these trace expressions make it clear that trA2 N and trA3 N, even as N 00, and furthermore they demonstrate that the normalization of the p-RDMC depends on the system in question. (In particular, the traces depend on how far D deviates from idempotency.) A few absolute bounds can be derived, such as... 

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