The Exchange Potential

In order to give you some background to Slater s Xa method, I would like to describe some very simple models that were used many years ago in order to understand the behaviour of electrons in metallic conductors. 

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Just to remind you, the electron density and therefore the exchange potential are both scalar fields they vary depending on the position in space r. We often refer to models that make use of such exchange potentials as local density models. The disagreement between Slater s and Dirac s numerical coefficients was quickly resolved, and authors began to write the exchange potential as... 

Becke proposed a widely used correction (B or B88) to tire LSDA exchange energy, which has the correct — asymptotic behaviour for the energy density (but not for the exchange potential). ... 

In Eq. (2.30), F is the Fock operator and Hcore is the Hamiltonian describing the motion of an electron in the field of the spatially fixed atomic nuclei. The operators and K. are operators that introduce the effects of electrons in the other occupied MOs. Hence, when i = j, J( (= K.) is the potential from the other electron that occupies the same MO, i ff IC is termed the exchange potential and does not have a simple functional form as it describes the effect of wavefunction asymmetry on the correlation of electrons with identical spin. Although simple in form, Eq. (2.29) (which is obtained after relatively complex mathematical analysis) represents a system of differential equations that are impractical to solve for systems of any interest to biochemists. Furthermore, the orbital solutions do not allow a simple association of molecular properties with individual atoms, which is the model most useful to experimental chemists and biochemists. A series of soluble linear equations, however, can be derived by assuming that the MOs can be expressed as a linear combination of atomic orbitals (LCAO)44 ... 

The exchange potential of Equation 7.31 is called the Slater potential [12], because it was Slater who had proposed [18] that the nonlocal exchange potential of HF theory can be replaced by the potential... 

The relationship between the exchange potential of DFT and the corresponding energy functional is established through the virial theorem. The two are related via the following relationship derived by Levy and Perdew [23]... 

Given in Table 7.1 are the results [24] of the total energy of some atoms obtained by solving the Kohn-Sham equation self-consistently with the exchange potential Wx within the central field approximation. The energy is obtained from Equation 7.10... 

The shape function had a role in theoretical chemistry and physics long before it was named by Parr and Bartolotti. For example, in x-ray measurements of the electron density, what one actually measures is the shape function—the relative abundance of electrons at different locations in the molecule. Determining the actual electron density requires calibration to a standard with known electron density. On the theoretical side, the shape function appears early in the history of Thomas-Fermi theory. For example, the Majorana-Fermi-Amaldi approximation to the exchange potential is just [3,4]... 

However, there is no explicit expression known for calculating in practice [e.g. in terms of occupied (f>iix) or n(r)] the exchange potential defined formally as... 

Taking into account the fact established in the previous section that Eg " is close to Eg = E, and, on the other hand, that an exact expression for the exchange potential t , = SEJdn is not known, one may wish to modify the HF method in such a way that it would lead to the exact GS energy and density (Baroni and Tuncel [15]). This can be easily obtained from Eq. (68) rewritten as... 

Because v(r) is given and t)es( ) easily calculable from the known density, Eq. (34), the KS exchange potential u,(r) is readily available from the OP solution - ags( )- It should be noticed, however, that this exchange potential v (r [ngs]) differs slightly from the exchange potential Vx(r [nos]) occurring in the exact GS problem solved by the KS method [see Eqs. (50), (51) and (56)], because the densities ngs(r) and nGs(r) are slightly different. 

The long-range asymptotic form of the exchange potential, a constituent of the exchange-correlation potential (56), will be discussed in this and the remaining subsections. Since the known derivations of this form are very complicated and involve an analysis of an integral equation of the OP method (see, e.g. [17], [27],... 

The line-integral expression (189), for the exchange-correlation potential of the HF-KS approach in Sect. 2.4, offers an interesting way to reconstruct the exchange potential for a given system, from the known HF solution for this system. Being alternative to schemes discussed in Sect. 3, this method provides an expression for the exchange potential solely in terms of the HF orbitals in the form... 

Long-range asymptotic properties of KS orbitals were recalled in Sect. 4, as dictated by asymptotic properties of all potentials of KS equations. A new, simple and direct method was applied to obtain the asymptotic form of the exchange potential. For pure-state systems the known results were confirmed in Sects. 4.2 and 4.3, while for mixed-state systems a new, exact result was obtained in Sect. 5.6. 

The above equations are especially suitable for obtaining the exchange potential, as this potential is implicitly dependent on the density but explicitly on the Kohn-Sham orbitals. The above equations are, furthermore, very useful... 

From the above we can expect that, if the functional derivatives of the GGA functionals give a good representation of the properties of the exchange-correlation hole, the near-degeneracy correlation properties responsible for the bond midpoint peak will show up in the exchange potentials rather than the correlation potentials. 

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