Kohn-Sham equations defined

Solve the Kohn-Sham equations defined using the trial electron density to find the single-particle wave functions, i i,(r). 

Note that in all current implementations of TDDFT the so-called adiabatic approximation is employed. Here, the time-dependent exchange-correlation potential that occurs in the corresponding time-dependent Kohn-Sham equations and which is rigorously defined as the functional derivative of the exchange-correlation action Axc[p] with respect to the time-dependent electron-density is approximated as the functional derivative of the standard, time-independent Exc with respect to the charge density at time t, i. e.,... 

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

If you have a vague sense that there is something circular about our discussion of the Kohn-Sham equations you are exactly right. To solve the Kohn-Sham equations, we need to define the Hartree potential, and to define the Hartree potential we need to know the electron density. But to find the electron density, we must know the single-electron wave functions, and to know these wave functions we must solve the Kohn-Sham equations. To break this circle, the problem is usually treated in an iterative way as outlined in the following algorithm 1 2 3... 

We have skipped over a whole series of important details in this process (How close do the two electron densities have to be before we consider them to be the same What is a good way to update the trial electron density How should we define the initial density ), but you should be able to see how this iterative method can lead to a solution of the Kohn-Sham equations that is self-consistent. 

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 ... 

This approximation uses only the local density to define the approximate exchange-correlation functional, so it is called the local density approximation (LDA). The LDA gives us a way to completely define the Kohn-Sham equations, but it is crucial to remember that the results from these equations do not exactly solve the true Schrodinger equation because we are not using the true exchange-correlation functional. 

We now have all the pieces in place to perform an HF calculation—a basis set in which the individual spin orbitals are expanded, the equations that the spin orbitals must satisfy, and a prescription for forming the final wave function once the spin orbitals are known. But there is one crucial complication left to deal with one that also appeared when we discussed the Kohn-Sham equations in Section 1.4. To find the spin orbitals we must solve the singleelectron equations. To define the Hartree potential in the single-electron equations, we must know the electron density. But to know the electron density, we must define the electron wave function, which is found using the individual spin orbitals To break this circle, an HF calculation is an iterative procedure that can be outlined as follows ... 

As we discussed in Chapter 1, the main aim of a DFT calculation is to find the electron density that corresponds to the ground-state configuration of the system, p(r). The electron density is defined in terms of the solutions to the Kohn-Sham equations, i /-(r), by... 

In the remainder of this section, we give a brief overview of some of the functionals that are most widely used in plane-wave DFT calculations by examining each of the different approaches identified in Fig. 10.2 in turn. The simplest approximation to the true Kohn-Sham functional is the local density approximation (LDA). In the LDA, the local exchange-correlation potential in the Kohn-Sham equations [Eq. (1.5)] is defined as the exchange potential for the spatially uniform electron gas with the same density as the local electron density ... 

To[ (r)] defines the kinetic energy of a non-interaoting electron gas which requires the density />(r). (r) is the classical (direct) Coulomb interaction which is equivalent to the Hartree potential. Using the constraints that the total number of electrons are conserved, Kohn and Sham [20] reduced the many electron Schrodinger equations to a set of one-electron equations known as the Kohn-Sham equations... 

The main challenge, after having defined the Kohn-Sham equations, is to solve them in an accurate way where the main problem is to find an accurate description of the multi-centre Coulomb potential. The method used is, to a large extent, determined by the character of the atoms involved in the bonding. The electronic structure of alkali and noble metals is for example rather simple... 

C. Density functional theory Density functional theory (DFT) is the third alternative quantum mechanics method for obtaining chemical structures and their associated energies.Unlike the other two approaches, however, DFT avoids working with the many-electron wavefunction. DFT focuses on the direct use of electron densities P(r), which are included in the fundamental mathematical formulations, the Kohn-Sham equations, which define the basis for this method. Unlike Hartree-Fock methods of ab initio theory, DFT explicitly takes electron correlation into account. This means that DFT should give results comparable to the standard ab initio correlation models, such as second order M(j)ller-Plesset (MP2) theory. 

The Kohn-Sham equations would indeed be equivalent to the Schrodinger equation if the universal exchange-correlation functional were known. Since it has not yet been discovered, practical forms of Kohn-Sham theory are (like Hartree-Fock theory) well-defined approximations to Schrodinger theory. 

This is our first basic relation. In order to derive the Kohn-Sham equations we define the following energy functional for a system of noninteracting particles with external potential vs and with ground state wavefunction I f fvJ) ... 

These equations can be written more explicitly in terms of Kohn-Sham orbitals defined to be the eigenstates of the single particle equation... 

For notational simplicity, this is generally simplified further by defining the electrostatic potential, vj [p r] = and the exchange-correlation potential, vxc [p r] =. Substitution of these results into Eq. (37) yields the celebrated Kohn-Sham equations [15]... 

This says that /(r) is the functional derivative (section 7.2.3.2, The Kohn-Sham equations) of the chemical potential with respect to the external potential (i.e. the potential caused by the nuclear framework), at constant electron number and that it is also the derivative of the electron density with respect to electron number at constant external potential. The second equality shows /(r) to be the sensitivity of p(r) to a change in N, at constant geometry. A change in electron density should be primarily electron withdrawal from or addition to the HOMO or LUMO, the frontier orbitals of Fukui [114], hence the name bestowed on the function by Parr and Yang. Since p(r) varies from point to point in a molecule, so does the Fukui function. Parr and Yang argue that a large value of fix) at a site favors reactivity at that site, but to apply the concept to specific reactions they define three Fukui functions ( condensed Fukui functions [80]) ... 

We therefore arrive at the Kohn-Sham equation for the optimum orbitals which define the exact one-electron density ... 

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