Conventional wave function-based quantum

It is instructive to compare the Hohenberg-Kohn theorems to their counterparts in conventional wave function-based quantum theory. The wave function provides a... 

Relativistic quantum chemistry is currently an active area of research (see, for example, the review volume edited by Wilson [102]), although most of the work is beyond the scope of this course. Much of the effort is based on Dirac s relativistic formulation of the Schrodinger equation this results in wave functions that have four components rather than the single component we conventionally think of. As a consequence the mathematical and computational complications are substantial. Nevertheless, it is very useful to have programs for Dirac-Fock (the relativistic analogue of Hartree-Fock) calculations available, as they can provide calibration comparisons for more approximate treatments. We have developed such a program and used it for this purpose [103]. 

The mathematical term functional, which is akin to function, is explained in Section 7.2.3.1. To the chemist, the main advantage of DFT is that in about the same time needed for an HF calculation one can often obtain results of about the same quality as from MP2 calculations (cf. e.g. Sections 5.5.1 and 5.5.2). Chemical applications of DFT are but one aspect of an ambitious project to recast conventional quantum mechanics, i.e. wave mechanics, in a form in which the electron density, and only the electron density, plays the key role [5]. It is noteworthy that the 1998 Nobel Prize in chemistry was awarded to John Pople (Section 5.3.3), largely for his role in developing practical wavefunction-based methods, and Walter Kohn,1 for the development of density functional methods [6]. The wave-function is the quantum mechanical analogue of the analytically intractable multibody problem (n-body problem) in astronomy [7], and indeed electron-electron interaction, electron correlation, is at the heart of the major problems encountered in... 

The FC method is completely different from the conventional quanmm chemistry. In the state-of-the-art quantum chemistry, one first defines Harfree-Fock orbitals based on the initially chosen basis set and then expands many-electron correlated wave functions by means of the Hartree-Fock orbitals. In this approach, any theory lies between the Hartree-Fock and the full Cl and so, tiie full Cl is a goal of this type of the theory. However, the full Cl cannof be tiie exacf solution of the Schrodinger equation because of the incompleteness of the basis set first introduced. When we use numerical Hartree-Fock that is free from the basis set, the full Cl becomes infinite expansion that cannot be handled in principle. 

Nowadays such quantum calculations are usually based on density junctional formalism [17], in which the electronic density, and not the wave functions, is used as the basic variable. This formalism results in much faster programs, so that larger systems can be treated than that with the conventional approaches. 

So far, we focused on conventional quantum chemical approaches that approximate the FCI wave function by truncating the complete N-particle Hilbert space based on predefined configuration selection procedures. In a different approach, the number of independent Cf coefficients can be reduced without pruning the FCI space. This is equivalent to seeking a more efficient parameterization of the wave function expansion, where the Cl coeflBcients are approximated by a smaller set of variational parameters that allow for an optimal representation of the quantum state of interest. Different approaches, which we will call modern solely to distinguish them from the standard quantum chemical methods, have emerged from solid-state physics. 

---