Kohn functional

Note in particular that the exchange-correlation functional that emCTges here does not involve the kinetic energy. From the perspective of the DFT literature, (3.16) is a formulation of the Hohenberg-Kohn functional that is constructed to ensure that the functional derivatives required for variational minimization actually exist. We return to these issues in Sect. 3.3. Also note that in the time-dependent case the external potential V(r, )is often considered to be explicitly... 

Collecting the system independent parts into a new quantity, the Hohenberg-Kohn functional FHK[p0], we arrive at... 

Writing this matrix as AX = —B, where X is the column matrix of the unknown linear parameters, then X = —A 1B and the stationary value of tan f/v is obtained by substituting the elements of X back into the Kohn functional, equation (3.37). The optimum values of the non-linear variational parameters in the trial function are determined by repeating the entire calculation for a range of values of each non-linear parameter, thereby identifying the values which yield a maximum value of tan ryv. Further details of this procedure were given by Armour and Humberston (1991). 

The scattering length can be calculated using the Kohn variational method in a similar manner to that employed for the phase shift, but the Kohn functional then becomes... 

Among the most detailed and accurate investigations of positronium formation in the Ore gap are those of Humberston (1982, 1984) and Brown and Humberston (1984, 1985), who used an extension of the Kohn variational method described previously, see section 3.2, to two open channels. The single-channel Kohn functional, equation (3.37), is now replaced by the following stationary functional for the K-matrix ... 

The Levy construction [222] can be used to prove Hohenberg-Kohn theorems for the ground state of any such theory. It should be noted that any explicit model of the Hohenberg-Kohn functional F[p] implies a corresponding orbital functional theory. The relevant density function p(r) is that constructed from an OFT ground state. This has the orbital decomposition , as postulated by Kohn and Sham [205]. Unlike the density p,, for an exact A-electron wave function T, which cannot be determined for most systems of interest, the OFT ground-state density function is constructed from explicit solutions of the orbital Euler-Lagrange equations, and the theory is self-contained. 

The derivation given above of the stationary Kohn functional [ K] depends on logic that is not changed if the functions Fo and l< of Eq. (8.5) are replaced in each channel by any functions for which the Wronskian condition mm — m 0 = l is satisfied [245, 191]. The complex Kohn method [244, 237, 440] exploits this fact by defining continuum basis functions consistent with the canonical form cv() = I.a = T, where T is the complex-symmetric multichannel transition matrix. These continuum basis functions have the asymptotic forms... 

Here tfo is the ground state wave function and Tno is any other wave function yielding the same density. We recognize this as the Hohenberg-Kohn functional, Fhk- It turns out that the ground state wave function of density n(r) can be defined as the wave function which yields n(r) and minimizes the Hohenberg-Kohn functional. 

In particular we can thus define the Hohenberg-Kohn functional HK on the set A as... 

Let us discuss some properties of LL- The functional Lll is an extension of the Hohenberg-Kohn functional /-nK. which is defined on A, to the larger set S, i.e.,... 

The core problem is finding adequate approximations for the Hohenberg-Kohn functional [cf. Eq. (21)]... 

Graupe, D. and K.H. Kohn, Functional Electrical Stimulation for Ambulation by Paraplegics. 1994, Malabar, FL Krieger Publishing Co. 

The Hohenberg-Kohn functional E ipJ attains minimum Eq = F [poJ for the ideal density distribution. Now our job will be to find out what mathematical form the functional could have. And here we meet the basic problem of the DFT method nobody has so far been able to give such a formula. The best that has been achieved to date are some approximations. These approximations, however, are so good that they begin to supply results that satisfy chemists. 

By the way in which the local sensitivity descriptors were introduced, it can de seen the importance of the universal Honenbeig-Kohn functional F p]. It still can be developed a method which avoid this dependency (Garza Robles, 1993), having in mind that an exact analytic expression of this functional is not (yet) known. This inconvenient can be transpose in the unrecognizing the exact functional relation between the electronic density p r) and the global potential w(r), and this new inconvenience can be eluded by considering the translational invariance of the external potential applied to the electrons system, respectively of the density of them in atomic or molecular system. The explicit bound between those types of variations (of the global potential and of the electronic density), for an open system can took the forms ... 

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