Wavefunction and Density Functional Theories

The stationary states of electrons in molecular systems satisfy the time-independent Schrodinger equation  

The stationary-state Schrodinger equation also implies that the local energy [q] = H(q) lf/(q)/ / (q). is equalized for the exact eigenfunction of the hamiltonian, throughout the whole (spin-position) configuration space q of N electrons, at the exact eigenvalue, e.g., 

As demonstrated by Hohenberg-Kohn [1], the external potential is uniquely determined, up to an additive constant, by the ground-state electron density  

It follows from the first HK theorem that the non-degenerate ground-state is also uniquely determined by its electron density 1P0 = P0[p0]. Thus, p0(r) represents the alternative, exact specification of the molecular quantum-mechanical ground-state. In other words, there is a unique mapping between P,l and p0, P0 - po, so that both functions carry exactly all the information about the quantum-mechanical state of the N electron system. 

The fact that po identifies the molecular Hamiltonian should not come as a surprise. Indeed, the nuclear cusps [116] of the electron density in an atom, molecule or solid, in the neighborhood of the atomic nuclei, necessary to avoid divergences in // P of the Schrodinger equation for r— Ra, i.e., r,s = Ir — Rj = lral — 0, 

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Traditional wavefunction and density functional theories are applicable to large systems, including nanomaterials however, their inqilementation often involves different algorithms. These include the various linear scaling methods, hybrid (often referred to as QM/MM) mediods, and sparse matrix mediods. 

Two commonly used approximations are the Hartree-Fock approach and density-functional theory (DFT). The Hartree-Fock approach approximates the exact solution of the Schrodinger equation using a series of equations that describe the wavefunc-tions of each individual electron. If these equations are solved explicitly during the calculation, the method is known as ab initio Hartree-Fock. The less expensive (i.e., less time-consuming) semi-empirical methods use preselected parameters for some of the integrals. DFT, on the other hand, uses the electronic density as the basic quantity, instead of a many-body electronic wavefunction. The advantage of this is that the density is a function of only three variables (instead of 3N variables), and is simpler to deal with both in concept and in practice. 

For anything but the most trivial systems, it is not possible to solve the electronic Schrodinger equation exactly, and approximate techniques must instead be used. There exist a variety of approximate methods, including Hartree-Fock (HF) theory, single- and multireference correlated ab initio methods, semiempirical methods, and density functional theory. We discuss each of these in turn. In Hartree-Fock theory, the many-electron wavefunction vF(r1, r2,..., r ) is approximated as an antisymmetrized product of one-electron wavefunctions, ifijfi) x Pauli principle. This antisymmetrized product is known as a Slater determinant. 

An important conceptual, or even philosophical, difference between the orbital/wavefunction methods and the density functional methods is that, at least in principle, the density functional methods do not appeal to orbitals. In the former case the theoretical entities are completely unobservable whereas electron density invoked by density functional theories is a genuine observable. Experiments to observe electron densities have been routinely conducted since the development of X-ray and other diffraction techniques (Coppens, 2001).18... 

A computer program for the theoretical determination of electric polarizabilities and hyperpolarizabilitieshas been implemented at the ab initio level using a computational scheme based on CHF perturbation theory [7-11]. Zero-order SCF, and first-and second-order CHF equations are solved to obtain the corresponding perturbed wavefunctions and density matrices, exploiting the entire molecular symmetry to reduce the number of matrix element which are to be stored in, and processed by, computer. Then a /j, and iap-iS tensors are evaluated. This method has been applied to evaluate the second hyperpolarizability of benzene using extended basis sets of Gaussian functions, see Sec. VI. 

The usual way chemistry handles electrons is through a quantum-mechanical treatment in the frozen-nuclei approximation, often incorrectly referred to as the Born-Oppenheimer approximation. A description of the electrons involves either a wavefunction ( traditional quantum chemistry) or an electron density representation (density functional theory, DFT). Relativistic quantum chemistry has remained a specialist field and in most calculations of practical... 

The density functional theory (DFT) [32] represents the major alternative to methods based on the Hartree-Fock formalism. In DFT, the focus is not in the wavefunction, but in the electron density. The total energy of an n-electron system can in all generality be expressed as a summation of four terms (equation 4). The first three terms, making reference to the noninteracting kinetic energy, the electron-nucleus Coulomb attraction and the electron-electron Coulomb repulsion, can be computed in a straightforward way. The practical problem of this method is the calculation of the fourth term Exc, the exchange-correlation term, for which the exact expression is not known. 

Spin-coupled theory has been used to smdy the changes that occur in the electronic wavefunction as a system moves along the intrinsic reaction coordinate for the case of the conrotatory and disrotatory pathways in the electrocyclization of cyclobutene to c/x-butadiene. Against intuitive expectations, conrotatory opening of cyclobutenes was found to be promoted by pressure. Ab initio MO and density functional calculations have indicated that the ring opening of the cyclobutene... 

Space-coordinate density transformations have been used by a number of authors in various contexts related to density functional theory [26,27, 53-64, 85-87]. As the free-electron gas wavefunction is expressed in terms of plane waves associated with a constant density, these transformations were introduced by Macke in 1955 for the purpose of producing modified plane waves that incorporate the density as a variable. In this manner, the density could be then be regarded as the variational object [53, 54]. Thus, explicitly a set of plane waves (defined in the volume V in and having uniform density po = N/V) ... 

In the constructive approach to density functional theory reviewed here, the extension to excited states is straightforward, provided that the orbit-generating wavefunction s, )eC J satisfies both wavefunction- and Hamiltonian... 

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