Ensemble potentials exchange potential

Xa ensemble exchange potential has been proposed [15]. Accurate ensemble exchange potentials have been calculated as a function of the radial distance from the Hartree-Fock ensemble electron density [16]. 

As we have seen in the previous sections the currently existing exchange-correlation potentials do not always perform well for ensemble-states. Recently, a simple Xa ensemble potential has been proposed [15]. In this section this potential is discussed and results for other atoms are presented. The form of the Xa ensemble exchange potential is... 

In a previous paper [16] the ensemble exchange potentials are studied for the following multiplets D, S for the atoms O and S and and... 

This figure shows a shell structure. For the ground state the shell structure has already been demonstrated [37]. Though the ensemble exchange potentials are different from the ground state one, the difference is not too much and the factors a show a very similar behaviour. 

The fact that the exact exchange potential has similar behaviour for the ensemble of multiplets suggests that approximations might also be similar. Probably, a small change in the ground-state exchange functionals might lead... 

Fig.6. presents the factors a as functions of the density. The shell structure can be clearly seen, too. If the ensemble exchange potential were a unique function of the ensemble density alone (for different ensembles) the curves a would be exactly the same. However, we can also see that the ensemble exchange potential is not the same function of the ensemble density. Though the curves are very close together, they are not exactly the same. So the ensemble exchange potential has a different dependence on the ensemble densitites for different ensembles. 

It was emphasized in the theory of Gross, Oliveira and Kohn [7] that the ensemble exchange potential depends on w. In a previous paper [16] it was shown that the ensemble exchange factor for multiplets is different for different values of w. 

The canonical ensemble corresponds to a system of fixed and V, able to exchange energy with a thennal bath at temperature T, which represents the effects of the surroundings. The thennodynamic potential is the Helmholtz free energy, and it is related to the partition fiinction follows ... 

The grand canonical ensemble corresponds to a system whose number of particles and energy can fluctuate, in exchange with its surroundings at specified p VT. The relevant themiodynamic quantity is the grand potential n = A - p A. The configurational distribution is conveniently written... 

Another example of phase transitions in two-dimensional systems with purely repulsive interaction is a system of hard discs (of diameter d) with particles of type A and particles of type B in volume V and interaction potential U U ri2) = oo for < 4,51 and zero otherwise, is the distance of two particles, j l, A, B] are their species and = d B = d, AB = d A- A/2). The total number of particles N = N A- Nb and the total volume V is fixed and thus the average density p = p d = Nd /V. Due to the additional repulsion between A and B type particles one can expect a phase separation into an -rich and a 5-rich fluid phase for large values of A > Ac. In a Gibbs ensemble Monte Carlo (GEMC) [192] simulation a system is simulated in two boxes with periodic boundary conditions, particles can be exchanged between the boxes and the volume of both boxes can... 

The physical system and the scaled system are related by Eq. (6). The scaling of velocities permits the exchange of heat between the simulated and the external heat reservoir. The equations of motion for the N atoms and the scaling factor g are solved numerically and averages calculated from the trajectory. The special choice (/+ I) kTo In g for the potential energy associated with g guarantees that the averages of equilibrium quantities calculated from the MD trajectory are the same as those in the canonical ensemble. 

In section 2 the theory of ensembles is reviewed. Section 3 summarizes the parameter-free theory of G par[ll]. The self-consistently determined ensemble a parameters of the ensemble Xa potential are presented. In section 4 spin-polarized calculations using several ground-state exchange-correlation potentials are discussed. In section 5 the w dependence of the ensemble a parameters is studied. It is emphasized that the excitation energy can not generally be calculated as a difference of the one-electron energies. The additional term should also be determined. Section 6 presents accurate... 

Ami occupation numbers. The ensemble exchange-correlation potential... 

To solve the Kohn-Sham equations (13) one needs the ensemble exchange-correlation potential. In the following sections several approximate forms are discussed. 

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