Exact ground state

Note that h is simply the diagonal matrix of zeroth-order eigenvalues In the following, it will be assumed that the zeroth-order eigenfunction a reasonably good approximation to the exact ground-state wavefiinction (meaning that Xfi , and h and v will be written in the compact representations... 

The exact ground-state eigenvalue and corresponding eigenvector... 

Nevertheless, equation (A 1.1.145) fonns the basis for the approximate diagonalization procedure provided by perturbation theory. To proceed, the exact ground-state eigenvalue and correspondmg eigenvector are written as the sums... 

This proof shows that any approximate wave function will have an energy above or equal to the exact ground-state energy. There is a related theorem, known as MacDonald s Theorem, which states that the nth root of a set of secular equations (e.g. a Cl matrix) is an upper limit to the n — l)th excited exact state, within the given symmetry subclass. In other words, the lowest root obtained by diagonalizing a Cl matrix is an upper limit to the lowest exact wave functions, the 2nd root is an upper limit to the exact energy of the first excited state, the 3rd root is an upper limit to the exact second excited state and so on. 

In this section, we compare our results for the magnetic moments of Ni, Fe and Co, n< 55, clusters with available experimental data. We point out that the determination of the exact ground state of these clusters is a very difficult task because these clusters exhibit a number of various spin states with energies lying very close to the ground state and within the range of both the calculational and the experimental errors. 

The exact ground-state energy Ei is shown in equation (2.39) to be 7i h /2ma. Thus, we have... 

The reason why we obtain the exact ground-state energy in this simple example is that the trial function 0 has the same mathematical form as the exact ground-state eigenfunction, given by equation (4.39). When the parameter c is evaluated to give a minimum value for S , the function 0 becomes identical to the exact eigenfunction. 

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... 

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

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. 

Density functional theory (DFT) uses the electron density p(r) as the basic source of information of an atomic or molecular system instead of the many-electron wave function T [1-7]. The theory is based on the Hohenberg-Kohn theorems, which establish the one-to-one correspondence between the ground state electron density of the system and the external potential v(r) (for an isolated system, this is the potential due to the nuclei) [6]. The electron density uniquely determines the number of electrons N of the system [6]. These theorems also provide a variational principle, stating that the exact ground state electron density minimizes the exact energy functional F[p(r)]. 

Using the fact that the energy is linear with respect to the number of electrons and Janak s theorem [31], the orbital energies of the N—n and N+n electron system become equal to the exact ground state vertical ionization energy and electron affinity, respectively ... 

The meaning of adiabatic correction is that the addition of C to the ground state energy calculated with Roo should yield a value equal to (close to) the exact ground state energy of the hydrogen atom... 

In the united-atom limit, R = 0, the positive root is k = 2, which corresponds to the exact ground-state energy ofthe He ion (in Hartrees) ... 

The exact ground state energy will be given by perturbation expansion (up to the second order)... 

The Exact Ground-State Problem Posed as the Hartree-Fock... 

Fig. 6. Many-to-one correspondance between wavefunctions in and one-particle densities is the Hohenberg-Kohn orbit, i.e., the orbit that contains the exact ground-state wavefunction...

Orbit is reached by optimization of the energy density functional through inter-orbit jumping. This process, which is illustrated in Fig. 7 by means of a sequence of arrows, is discussed in detail below. The inter-orbit jumping process is repeated until one finally reaches orbit This is the orbit where, by definition, one finds the exact ground state wavefunction that satisfies the Schrodinger equation = El Pl. For this reason, we call this the... 

Hohenberg-Kohn orbit. Clearly, within the application of local-scaling transformations to any initial wavefunction leads to the exact ground-state wavefunction as well as to the exact ground-state density. 

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