Derivation of the Kohn-Sham equations

Now we will make a variation of E, i.e. we will find the linear effect of changing E due to a variation of the spinorbitals (and therefore also of the density). We make a spinorbital variation denoted by 8 j i (as before, p. 336, it is justified to vary either or / , the result is the same, we choose, therefore, 84 ) and see what effect it will have on E keeping only the linear term. We have (see eq. (11.4)), 

We insert the right-hand sides of the above expressions into E, and identify the variation, Le. the linear part of the change of E. The variations of the individual terms off look like (note, see p. 334, that the symbol ( ) stands for an integral over space coordinates and a summation over the spin coordinates)  

Finally, we come to the variation of E c, i.e. 8Exc- We are in a quite difficult situation, because we do not know the mathematical dependence of the functional Exc on p, and therefore also on 8(f . Nevertheless, we somehow have to get the linear part of Exc, i.e. the variation. 

A change of functional F[f] (due to f f + 8f) contains a part linear in 8f denoted by 8F plus some higher powers of 8f denoted by 0((8/) ) 

The 8F is defined through th functional derivative (Fig. 11.7) of F with respect functional 

Since Ts is now written as an orbital functional one cannot directly minimize Eq. (55) with respect to n. Instead, one commonly employs a scheme suggested by Kohn and Sham [60] for performing the minimization indirectly. This scheme starts by writing the minimization as 

As a consequence of Eq. (27), 8V/8n = v(r), the external potential the electrons move in.33 The term SUn/Sn simply yields the Hartree potential, 

33This potential is called external because it is external to the electron system and not generated self-consistently from the electron-electron interaction, as vh and vxc. It comprises the lattice potential and any additional truly external field applied to the system as a whole. 

Consider now a system of noninteracting particles moving in the potential us(r). For this system the minimization condition is simply 

Consequently, one can calculate the density of the interacting (many-body) system in potential u(r), described by a many-body Schrodinger equation of the form (2), by solving the equations of a noninteracting (single-body) 

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The exchange-correlation potential is the source of both the strengths and the weaknesses of the DF approach. In HF theory, the analytical form of the term equivalent to Vxc, the exchange potential, arises directly during the derivation of the equations, but it depends upon the one-particle density matrix, making it expensive to calculate. In DF theory the analytical form of Vxc must be put into the calculations because it does not come from the derivation of the Kohn-Sham equations. Thus, it is possible to choose forms for Vxc that depend only upon the density and its derivatives and which are cheap to calculate (the so-called local and non-local density approximations). The Vxc factor can also be chosen to account for some of the correlation between the electrons, in contrast to HF methods for which additional calculations must be made. The drawback is that there does not appear to be any systematic way of improving the potential. Indeed, many such terms have been proposed. 

Note that the most common constraints in Euler-Lagrange equation take the form C[n] = 0, where C[n] is some density functional. For instance, the constraint C[n] = f n r)dr - N = 0 is used in the derivation of the Kohn-Sham equations. Additional constraints expressed as C[n] =0 are also used in some computational schemes such as the procedure to generate diabatic electronic states for the evaluation of the rate of the electron-transfer reaction [7],... 

The Hohenberg-Kohn theorems find a very important application in the derivation of the Kohn-Sham equations, in which the problem of approximating the noninteracting kinetic energy (Ts) is eliminated by introducing single-particle orbitals 9,. The exact electron density is written as the electron density of a Slater determinant,... 

The last term, the xc potential, comprises all the non-trivial many-body effects. In ordinary DFT, is normally written as a functional derivative of the xc energy. This follows from a variational derivation of the Kohn-Sham equations starting from the total energy. It is not straightforward to extend this formulation to the time-dependent case due to a problem related to causality [11,2]. The problem was solved by van Leeuwen in 1998, by using the Keldish formalism to define a new action functional, A [12]. The time-dependent xc potential can then be written as the functional derivative of the xc part of A,... 

In the derivation of the Kohn-Sham equations we have hidden a number of difficulties in the exchange-correlation potential, (r). Indeed, the success of DFT depends on finding an accurate and convenient form of this potential. There is an extensive literature discussing the merits of various potentials, and good accounts of these may be found elsewhere (Koch and Holthausen 2001). Here, we restrict the discussion to the local density approximation (LDA), because it provides a link to another approximation that has been used extensively in relativistic atomic and molecular calculations and which predates the Kohn-Sham equations. 

The calculation itself is somewhat lengthy, since it involves second derivatives of the Kohn-Sham functional with respect to the orbitals, and does not provide much insight into the physics of the problem. We therefore refer the interested reader to related references [13, 91]. The final stationarity equation reads ... 

From the constraint at Eq. (78) it follows that the functional derivatives of ys, must contain delta functions in order to cancel the delta functions in the second part of the above equation for ri equal to iz- As yj, depends explicitly on the orbitals we therefore have to calculate the functional derivative of the Kohn Sham orbitals with respect to the density which is given by [66]... 

To. solve the Kohn-Sham equations a self-consistent approach is taken. An initial guess of the density is fed into Equation (3.47) from which a set of orbitals can be derived, leading to an improved value for the density, which is then used in the second iteration, and so on until convergence is achieved. 

While the hydrodynamical scheme mentioned above involves the density quantities directly, an alternative second scheme based on their orbital partitioning along the lines of the Kohn-Sham [4] version of time-independent DFT has been derived by Ghosh and Dhara [14]. In this scheme, one obtains the exact densities p(r, t) and j(r, t) from the TD orbitals (///,(r, t) obtained by solving the effective one-particle TD Schrodinger-like equations given by... 

In the Kohn-Sham equation above, the Coulomb potential and the XC potential are obtained from their energy counterparts by taking the functional derivative of the latter with respect to the density. Thus... 

As is evident from the above, both the physics invoked to derive the potential of Equation 7.31 and the numerical results presented show that Wx gives an accurate exchange potential for the excited states. When the proposal was initially made, there was no mathematical proof of the existence of a Kohn-Sham equation for excited states. It is only during the past few years that DFT of excited states [34-37], akin to its ground-state counterpart, is being developed. 

Kohn-Sham equations is rather complicated for an arbitrarily selected set of weighting factors, and has to be derived separately for every different case of interest. For a spherically symmetric external potential and equal weighting factors, however, the Kohn-Sham equations have a very simple form, as shown in Ref. [72], In this case the noninteracting kinetic energy is given by... 

Within any sphere, the exchange-correlation potential in the Kohn-Sham equation is defined to be the conventional functional derivative of Ex c-... 

Finally we mention some basic relations which are essential in the discussion of explicitly orbital dependent functionals. Examples of such functionals are the Kohn-Sham kinetic energy and the exchange energy which are dependent on the density due to the fact that the Kohn-Sham orbitals are uniquely determined by the density. The functional dependence of the Kohn-Sham orbitals on the density is not explicitly known. However one can still obtain the functional derivative of orbital dependent functionals as a solution to an integral equation. Suppose we have an explicit orbital dependent approximation for in terms of the Kohn-Sham orbitals then... 

In fact, the true form of the exchange-correlation functional whose existence is guaranteed by the Hohenberg-Kohn theorem is simply not known. Fortunately, there is one case where this functional can be derived exactly the uniform electron gas. In this situation, the electron density is constant at all points in space that is, n(r) = constant. This situation may appear to be of limited value in any real material since it is variations in electron density that define chemical bonds and generally make materials interesting. But the uniform electron gas provides a practical way to actually use the Kohn-Sham equations. To do this, we set the exchange-correlation potential at each position to be the known exchange-correlation potential from the uniform electron gas at the electron density observed at that position ... 

In comparing Eq. (13) to the Kohn-Sham equations Eq. (3) one concludes that E(.r, x E), since it is derived from exact many-electron theory [22], is the exact Coulomb (direct) plus exchange-correlation potential. It is non-local and also energy-dependent. In view of this it is hard to see how the various forms of constructed local exchange correlation potentials that are in use today can ever capture the full details of the correlation problem. 

The KS equations are obtained by differentiating the energy with respect to the KS molecular orbitals, analogously to the derivation of the Hartree-Fock equations, where differentiation is with respect to wavefunction molecular orbitals (Section 5.2.3.4). We use the fact that the electron density distribution of the reference system, which is by decree exactly the same as that of the ground state of our real system (see the definition at the beginning of the discussion of the Kohn-Sham energy), is given by (reference [9])... 

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