Many-electron systems distribution densities

A time-independent wave function is a function of the position in space (r = x,y,z) and the spin degree of freedom, which can be either up or down. The physical interpretation of the wave function is due to Max Born (25, 26), who was the first to interpret the square of its magnitude, > /(r)p, as a probability density function, or probability distribution function. This probability distribution specifies the probability of finding the particle (here, the electron) at any chosen location in space (r) in an infinitesimal volume dV= dx dy dz around r. I lu probability of finding the electron at r is given by )/(r) Id V7, which is required to integrate to unity over all space (normalization condition). A many-electron system, such as a molecule, is described by a many-electron wave function lF(r, r, l .I -.-), which when squared gives the probability den-... 

The determination of the ground state energy and the ground state electron density distribution of a many-electron system in a fixed external potential is a problem of major importance in chemistry and physics. For a given Hamiltonian and for specified boundary conditions, it is possible in principle to obtain directly numerical solutions of the Schrodinger equation. Even with current generations of computers, this is not feasible in practice for systems of large total number of electrons. Of course, a variety of alternative methods, such as self-consistent mean field theories, also exist. However, these are approximate. 

To show the validity of using Eq. (1) to compute the total electronic energy of a many-electron system, Hohenberg and Kohn, in their famous 1964 paper, presented two proofs that provided the foundation for DFT. In the first, they proved that an external potential (such as classical nuclei distributed in space) is a unique functional of the electron density (apart from a trivial additive constant). For most practical purposes, the converse is true, and the electron density of N electrons in an external potential is considered to result uniquely from that potential. Parr and Yang (1989) give an in-depth discussion of these issues, in addition to providing the staple text on DFT. We also remind the reader that a functional maps a set of functions to a set of numbers, in contrast to a function, which maps one set of numbers to another set of numbers. 

Consider the idealized system of n electrons within a cubical box of volume V throughout which there is uniformly distributed positive charge sufficient to render the system neutral. The uniform electron gas (UEG) of density p = nfV is obtained as the limit of this system as n, V -> oo. Although the UEG bears some resemblance to the electron sea in metals, its chief virtue is its simplicity. Despite being a many-electron system, it is completely defined by a single variable - its density p - and it is relatively easy to study. It is often called jellium . A detailed discussion of the properties of jellium can be found in Appendix E of Ref. 4. 

To make matters worse, the use of a uniform gas model for electron density does not enable one to carry out accurate calculations. Instead, ripples must be introduced into the uniform electron gas distribution. The way in which this has been implemented has typically been in a semiempirical manner by working backward from the known results on a particular system, usually taken to be the hehum atom. In this way, it has been possible to obtain an approximate set of functions that ako give successful approximate calculations in many other atoms and molecules. By carrying out this combination of a semiempirical approach and retreating from the pure Thomas-Fermi ideal of a uniform gas, it has actually been possible to obtain computationally better results, in many cases, than with conventional ab initio methods using orbitak and wavefunctions. ... 

The spatial distribution of the electron cloud in a molecular system is investigated quantitatively through the single-particle electron density,which has served as a basic variable in the so-called density functional theory (DFT), an approach that bypasses the many-electron wavefunction, which is the usual vehicle in conventional quantum chemistry or electronic structure theory (for a review of modem developments in quantum chemistry, see reference 7). The theoretical framework of DFT is well known for the associated conceptual simplicity as well as for the computational economy it offers. Another equally important aspect of DFT is its ability to rationalize the existing concepts in chemistry as well as to give birth to newer concepts, which has led to the important field of conceptual DFT. ... 

By contrast. Density Functional Theory (DFT) [6] methods do not attempt to solve for the ground state wavefiinction direcdy, but rather to find a universal function for a system s electron density and then to calculate each individual electron s density distribution within that system-wide density [7]. DFT dates from the mid-1960s [8] and, as with the other models, contains many approximations — as do the various different optimisation and simulation choices for each of these methods, mentioned in Sect. 6.2. 

The term T is the quantum-mechanical expectation value of the kinetic energy, and is the only term requiring knowledge of P(r, rl) for ri r, while the others have a purely classical interpretation in terms of the distribution functions for a particle and for a pair of particles respectively. These results are valid for all kinds of wavefunctions, or approximate wavefunctions, for any state of any system and because they involve the electron distribution directly it is often possible to get a useful interpretation of molecular properties in terms of the main features of the electron density, without detailed reference to the intricacies of the many-electron wavefunction. A chemical bond, for instance, arises from a concentration of electron density in the bond... 

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