Ground state density

Henee, there eannot be two distinet potentials Vand V that give the same ground-state p r). So, the ground-state density p r) uniquely detennines A and V, and thusy/, and therefore and q. Furthemiore, beeause T... 

Although the above discussion suggests how one might compute the ground-state energy once the ground-state density p(r) is given, one still needs to know how to obtain... 

We ve specified five excited states with NSlotes=5 (the reasons will be clear in a moment). The DensitysAll keyword tells Gaussian to perform the population analysis using all available densities the SCF (ground state) density, the Cl one-partide density, and the Cl (Cl-Singles) density. The population analyses using excited state densities will be performed for the first excited state (the default if the Root option is not included), which is the one in which we are interested. 

In the Hartree-Fock approach, the many-body wave function in form of a Slater determinant plays the key role in the theory. For instance, the Hartree-Fock equations are derived by minimization of the total energy expressed in terms of this determinantal wave function. In density functional theory (3,4), the fundamental role is taken over by an observable quantity, the electron density. An important theorem of density functional theory states that the correct ground state density, n(r), determines rigorously all electronic properties of the system, in particular its total energy. The totd energy of a system can be expressed as a functional of the density n (r) and this functional, E[n (r)], is minimized by the ground state density. 

This concludes the proof that there cannot be two different Vext that yield the same ground state electron density, or, in other words, that the ground state density uniquely specifies the external potential Vext. Using again the terminology of Section 1.2 we can simply add p0 as the property which contains the information about N, ZA, RA and summarize this as... 

One should note at this point that the ground state density uniquely determines the Hamilton operator, which characterizes all states of the system, ground and excited. Thus, all properties of all states are formally determined by the ground state density (even though we would need functionals other than J p(r) VNedr + Fhk [p], which is the functional con-... 

Stated in still other words this means that for any trial density p(r) - which satisfies the necessary boundary conditions such as p( ) - 0, J p( ) dr = N, and which is associated with some external potential Vext - the energy obtained from the functional given in equation (4-6) represents an upper bound to the true ground state energy E0. E0 results if and only if the exact ground state density is inserted into equation (4-8). The proof of the inequality (4-11) is simple since it makes use of the variational principle established for wave functions as detailed in Chapter 1. We recall that any trial density p(r) defines its own Hamiltonian H and hence its own wave function. This wave function can now be taken as the trial wave function for the Hamiltonian generated from the true external potential Vext. Thus, we arrive at... 

Let us summarize what we have shown so far. First, all properties of a system defined by an external potential Vext are determined by the ground state density. In particular the ground state energy associated with a density p is available through the functional J P(f )VNedr + I)ikIp] Second, this functional attains its minimum value with respect to all allowed densities if and only if the input density is the true ground state density, i. e for p(r) = p0(r). Of course, the applicability of this variational recipe is limited to the ground... 

In order to distinguish these orbitals from their Hartree-Fock counterparts, they are usually termed Kohn-Sham orbitals, or briefly KS orbitals. The connection of this artificial system to the one we are really interested in is now established by choosing the effective potential Vs such that the density resulting from the summation of the moduli of the squared orbitals tpj exactly equals the ground state density of our real target system of interacting electrons,... 

Thus, once we know the various contributions in equation (5-15) we have a grip on the potential Vs which we need to insert into the one-particle equations, which in turn determine the orbitals and hence the ground state density and the ground state energy by employing the energy expression (5-13). It should be noted that Veff already depends on the density (and thus on the orbitals) through the Coulomb term as shown in equation (5-13). Therefore, just like the Hartree-Fock equations (1-24), the Kohn-Sham one-electron equations (5-14) also have to be solved iteratively. 

If p is constrained to be the same as the exact ground state density p, then the orbitals will satisfy the equation... 

For the correct p(r), Ev[p eqtials the ground-state energy E. To identify the correct ground-state density p(r) consider the energy corresponding to ty1 for N particles... 

The advantage over the HF scheme is that whereas in conventional ah initio theory we must resort to costly perturbation theory or configuration interaction expansions, in DFT electron correlation is already included explicitly in the exchange-correlation functional. The key problem is instead to find an appropriate expression for xc. As stated above, when we have the correct functional we should be able to extract the exact energy, the exact ground state density, and all properties for our system. 

Among various theories of electronic structure, density functional theory (DFT) [1,2] has been the most successful one. This is because of its richness of concepts and at the same time simplicity of its implementation. The new concept that the theory introduces is that the ground-state density of an electronic system contains all the information about the Hamiltonian and therefore all the properties of the system. Further, the theory introduces a variational principle in terms of the ground-state density that leads to an equation to determine this density. Consider the expectation value (H) of the Hamiltonian (atomic units are used)... 

FIGURE 7.1 Exchange-correlation potential Vxc (in atomic units) for neon as a function of distance r (in atomic units) from the nucleus. The potential is obtained from the ground-state density by employing the ZP method. 

In Equation 9.21, T yields n and is orthogonal to the first i — 1 state of the Hamiltonian for which n0 is the ground-state density. Here, this Hamiltonian is the // in Equation 9.19. Note that instead of the ground-state electron density n0, we could use the external potential v or any ground-state Kohn-Sham orbital, etc. Thus we could use Ft[n, v]. The extension to degenerate states is studied in Section 9.4. 

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