Wavefunction theory

The description of electron motion and electronic states, which is at the heart of all of chemistry, is included in wavefunction theory, also referred to as self-consistent- 

We begin with a brief review of one-particle quantum mechanics [1]. An electron has spin s = and -component of spin a = (t) or — (1). 

The Hamiltonian or energy operator for one electron in the presence of an external potential u(r) is 

The energy eigenstates V a(r, r) and eigenvalues Eq, are solutions of the time-independent Schrodinger equation 

The electron-electron repulsion sums over distinct pairs of different electrons. The states of well-defined energy are the eigenstates of H  

Because electrons are fermions, the only physical solutions of (1.17) are those wavefunctions that are antisymmetric [2] under exchange of two electron labels i and j  

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In wavefunction theory an alternative way to find IEs for removal of an electron from a molecular orbital (usually the highest), is to invoke Koopmans theorem ... 

The only cases for which one might anticipate differences between DFT and wavefunction theory as regards visualization (Sections 5.5.6 and 6.3.6) are those involving orbitals as explained in Section 7.2.3.2, The Kohn-Sham equations, the orbitals of currently popular DFT methods were introduced to make the calculation of the electron density tractable, but in pure DFT theory orbitals would not exist. Thus electron density, spin density, and electrostatic potential can be visualized in Kohn-Sham DFT calculations just as in ab initio or semiempirical work. However, visualization of orbitals, so important in wavefunction work (especially the HOMO and FUMO, which in frontier orbital theory [154] strongly influence reactivity) does not seem possible in a pure DFT approach, one in which wavefunctions are not invoked. In currently popular DFT calculations one can visualize the Kohn-Sham orbitals, which are qualitatively much like wavefunction orbitals [130] (Section 7.3.5, Ionization energies and electron affinities). 

What is the essential difference between wavefunction theory and DFT What is it that, in principle anyway, makes DFT simpler than wavefunction theory ... 

Explain why a kind of molecular orbital is found in current DFT, although DFT is touted as an alternative to wavefunction theory. 

If the electron density function is mathematically and conceptually simpler than the wavefunction concept, why did DFT come later than wavefunction theory ... 

The wavefunction [1] and electron density [2] concepts came at about the same time, 1926, but the application of wavefunction theory to chemistry began in the 1920s [3], while DFT was not widely used in chemistry until the 1980s (see below). Why ... 

In Hartree-Fock theory, the simplest wavefunction theory involving an antisymmetric wavefunction, the electron repulsion energy of an N-electron system is given by... 

The first Kohn-Sham theorem tells us that it is worth looking for a way to calculate molecular properties from the electron density. The second theorem suggests that a variational approach might yield a way to calculate the energy and electron density (the electron density, in turn, could be used to calculate other properties). Recall that in wavefunction theory, the Hartree-Fock variational approach (section 5.2.3.4) led to the HF equations, which are used to calculate the energy and the wavefunction. An analogous variational approach led (1965) to the KS equations [26], the basis of current... 

In wavefunction theory an alternative way to find IBs for removal of an electron from any molecular orbital is to invoke Koopmans theorem the IE for an orbital is the negative of the orbital energy section 5.5.5. In chapters 5 and 6 both the energy difference and the Koopmans theorem methods were used to calculate some lEs (Tables 5.21 and 6.7). The problem with applying Koopmans theorem to DFT is that in DFT proper... 

These concepts, which can be analyzed quantitatively using wavefunction theory, but are often treated in connection with DFT ( perhaps because much of the underlying theory was formulated in this context [105]) will now be examined a bit more quantitatively. Consider the effect on the energy of a molecule, atom, or ion of adding electrons. Figure 7.10 shows how the energy of a fluorine cation F+ changes as one... 

The present article essentially focusses on the application of ECPs in standard molecular wavefunction theory (WFT), i.e., standard ab initio quantum chemistry using atomic and molecular orbitals as one-particle basis sets from which determinantal many-electron wavefunctions are constmcted. Applications to... 

Density functional theory provides an alternative approach to the description of the electronic structure of atoms and molecules, in principle more simple than conventional wavefunction theory, because the total energy is determined directly from the electronic density distribution p(r), through the expression... 

The basic idea underlying the development of the various density functional theory (DFT) formulations is the hope of reducing complicated, many-body problems to effective one-body problems. The earlier, most popular approaches have indeed shown that a many-body system can be dealt with statistically as a one-body system by relating the local electron density p(r) to the total average potential, y(r), felt by the electron in the many-body situation. Such treatments, in fact, produced two well-known mean-field equations i.e. the Hartree-Fock-Slater (HFS) equation [14] and the Thomas-Fermi-Dirac (TFD) equation [15], It stemmed from such formulations that to base those equations on a density theory rather than on a wavefunction theory would avoid the full solution... 

It is interesting to note that these principles may be used in the context of wavefunction theory. In this case the global hardness may be approximated by [42]... 

The difference between this Fock operator and the Hartree-Fock counterpart in Eq. (2.51) is only the exchange-correlation potential functional, Exc, which substitutes for the exchange operator in the Hartree-Eock operator. That is, in the electron-electron interaction potential, only the exchange operator is replaced with the approximate potential density functionals of the exchange interactions and electron correlations, while the remaining Coulomb operator, Jj, which is represented as the interaction of electron densities, is used as is. The point is that the electron correlations, which are incorporated as the interactions between electron configurations in wavefunction theories (see Sect. 3.3), are simply included... 

The one-to-one correspondence between electron density and effective potential, which is proven on the basis of the constrained search formulation, suggests that the effective potential can be determined directly from the electron density. Parr and coworkers developed a procedure for determining highly accurate exchange-correlation potentials from electron densities, which are calculated by high-level ab initio correlation wavefunction theories. This procedure is called the Zhao-Morrison-Parr (ZMP) method (Zhao et al. 1994). In this method, the effective potential is given by the Lagrange undetermined multiplier method with a potential. 

Perturbation theories such as the MP2 method (McWeeny 1992) (see Sect. 3.2) have been appreciated as ab initio wavefunction theories reproducing dispersion interactions with relatively short computational times. Therefore, dispersion interactions can be incorporated in the Kohn-Sham method by combining with such perturbation theories, in principle. One of the methods based on this concept is the DFT symmetry-adapted perturbation theory (DFT-SAPT), which uses Kohn-Sham orbitals to calculate the perturbation energies (Williams and Chabalowski 2001). In contrast to ab initio SAPT, in which both intermolecular and intramolecular electron correlations are calculated, only intermolecular electron correlations are calculated as a dispersion correction for the Kohn-Sham method in DFT-SAPT. Consequently, this drastically reduces the computational cost, typically by one or two orders of magnitude, compared to an ab initio SAPT calculation, with similar accuracies. 

Abstract. The Hohenberg-Kohn theorems in Inhomogeneous electron gas established a whole new perspective for the study of electronic structure theory and marked the birth of modern density functional theory (DFT). In our view, DFT and wavefunction theories complement each other. Starting with the invention of the Kohn-Sham method a fruitful synthesis of DFT and wavefunction theories took place and the most powerful computational tools currently available are combinations of both methods. The Hohenberg-Kohn theorems inspire the quest for simple density functionals of increased accuracy. We believe that the synthesis of accurate density functionals and computationally efficient wavefunction methods will continue to dominate electronic structure theory. 

Modern density functional theory (DFT) was born with the title paper and was developed in parallel with wavefunction methods. The Kohn-Sham method [6] is an example of a unification between pure DFT methods and wavefunction theory, a synthesis that has lead to some of the most powerful tools in computational quantum chemistry. 

The Kohn-Sham equations a synthesis of wavefunction theory and DFT... 

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