Idempotent property

However, one should ask whether the ansatz Eq. (23) is a valid one, and exactly how good is the TF approximation. It is certain that for systems other than the PEG, the idempotency property in Eq. (9) satisfied by any idempotent DM1 will no longer be true for Eq. (23). Hence, the TF functional is actually not an approximation for the Ts functional, the KS idempotent KEDF. Further, Eq. (23) has the wrong asymptotic behavior for isolated finite systems as both r and r become large, where the exact DM1 goes like the product of the highest occupied molecular orbital (HOMO) of Eq. (10) at two different points rand... 

It should be clear that the three TBFWV s introduced in Eqs. (139), (143), and (148) need not to be identical proper llinctional forms have to be chosen individually. It is also cinious to note that the final forms of the OF-KEDF within the WDA and the SADA, Eqs. (137) and (147), are indifferent to the symmetrization of the exchange PCF or the DM1 and only depend on the relevant average or effective density. In fact, the functional form for g(y2) in Eq. (39) has little influence over the final form of the OF-KEDF. Hence, other functional forms can also be considered. 3 3 3 Yet, there is currently no systematic, coherent, and consistent scheme to fix the functional forms for the TBFWV and g (y2) in conjunction with the simultaneous enforcement of the idempotency property for the DM1 and the correct LR behavior. 

Unlike the WDA that enforces the idempotency property for its DM1 ansatz, the SADA trades the idempotency requirement for the correct LR behavior of the OF-KEDF ... 

Introduction of the AWF within the ADA and the SADA allows for an extra degree of freedom so that the correct LR behavior can be exactly obeyed. Then, the explicit enforcement of the idempotency property on the DM 1 should in principle determine an unique functional form for the TBFWV of the AWE We have started to work on this idea numerical results will be published elsewhere. For later reference, we call this scheme the Weighted-Average-Density Approximation (WADA)... 

The idempotent property is conserved, whereas the Hermitian one is lost. Indeed, in gener. Pi P. The k + l)th order state is defined by... 

Note The event xi (PIA fails) figures twice in the fault tree. However, it is counted only once because of the idempotent property of the binary variables x according to Eq. (9.60)]... 

This idempotent property of p allows us to project any single particle matrix into the interband (p—h) subspace... 

The implementation of the CCM in the HF LCAO method is based on the use of the idempotency property of a one-electron density matrix (DM) of crystalhne systems [100]. The underlying strategy in this case was to start from a periodic system and to make small modifications in the corresponding LCAO computer codes. It has to be mentioned that the CCM realization in PW basis seems to be impossible as the interaction region in this case can not be defined. 

Relation (6.76) provides a simple way to check the idempotency property of the DM. Therefore, it can be used to implement the CCM in HF LCAO calculations. 

Let us start with a short discussion about the kind of transformation we are seeking for. As described in the Chapter 2, the density matrix corresponding to a pure state is a projector, which satisfies the following properties (Chapter 3) p = p" and Tr(/o ) = 1. On the other hand, for a statistically mixed state, p p and Tr(p ) < 1. Now, let us look at a density operator that is obtained from a mixed state operator p by a unitary transformation, p = UpU. The question is whether this operator can or cannot be a pure state operator. The trace and idempotency properties for the transformed operator become ... 

Finally, the bronze holy grail is the WADA that concurrently enforces the correct LR behavior at the FEG limit and the exact idempotency property for any OF approximation of the DM1. Of course, the correct LR behavior is critical for any EDF to outperform its LDA counterpart. However, one should not push the limit too far with regard to higher-order response behaviors. Past numerical tests have shown that the SNDA OF-KEDF s with the DD AWF based on LR theory and the ones with the DI AWF based on QR theory perform indistinguishably from one another for bulk solids (metals and insulators alike), Hence, the... 

This idempotency property of particle number operators holds also in the more general case. Consider a many-electron multideterminantal wave function of the form ... 

The reader may easily investigate the idempotency property of Nj for this general case. It will turn out that the operator Nj is indeed idempotent for any multideterminantal wave function. 

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