Applications of Quantum Density Functional Theory

The electronic case, which has been treated in all previous chapters of this book is essentially based on the Hohenberg-Kohn theorems [1] and the Kohn-Sham procedure [2]. This is the most modern use of DFT, to solve the Schrodinger equation of an electronic system, and it is the most precise first principles method for the calculation of energies of chemical interest. Although there have been, for more than 100 years, other series of developments that also use the electronic density, we will avoid using DFT to 

Therefore, using the time-independent, nonrelativistic, electronic Hamiltonian, the total energy can be written as, 

Another important development, treated in chapter 3 of this book, is the local-scaling transformation version of DFT, in which a one-to-one correspondence is established between the energy as a functional of a wavefunction in an orbit in Hilbert space and the energy as a functional of the one-particle density. Although apparently wavefunction-dependent (as an arbitrary wavefunction is required to generate an orbit), this is a true density functional theory, in which the energy density functional depends upon the local-scaling transformation function f([p] r), which is an implicit function of the one-particle density. 

In order to have a more complete picture of the many-body problem for more general or complicated cases that DFT could help to treat, it is necessary to make a correspondence with the use of many-body perturbation theory. Under this wider classification of perturbation theory are included all the methods that treat electron correlation beyond the Hartree-Fock level, including configuration interaction, coupled cluster, etc. This perturbational approach has traditionally been known as second quantization, and its power for some applications can be seen when dealing with problems beyond the standard quantum mechanics. 

The Hamiltonian operator, using second quantization, can be written as 

---