The Electron Density

The ultimate goal of most quantum chemical approaches is the - approximate - solution of the time-independent, non-relativistic Schrodinger equation 

A and B run over the M nuclei while i and j denote the N electrons in the system. The first two terms describe the kinetic energy of the electrons and nuclei respectively, where the Laplacian operator V2 is defined as a sum of differential operators (in cartesian coordinates) 

All equations given in this text appear in a very compact form, without any fundamental physical constants. We achieve this by employing the so-called system of atomic units, which is particularly adapted for working with atoms and molecules. In this system, physical quantities are expressed as multiples of fundamental constants and, if necessary, as combinations of such constants. The mass of an electron, me, the modulus of its charge, lei, Planck s constant h divided by lit, h, and 4jt 0, the permittivity of the vacuum, are all set to unity. Mass, charge, action etc. are then expressed as multiples of these constants, which can therefore be dropped from all equations. The definitions of atomic units used in this book and their relations to the corresponding SI units are summarized in Table 1-1. 

Quantity Atomic unit Value in SI units Symbol (name) 

Note that the unit of energy, 1 hartree, corresponds to twice the ionization energy of a hydrogen atom, or, equivalently, that the exact total energy of an H atom equals -0.5 Eh. Thus, 1 hartree corresponds to 27.211 eV or 627.51 kcal/mol.2 

Although measurement of, in many cases, the hundreds of thousands of X-ray reflections that are necessary to compute the Fourier synthesis of a macromolecule has in the past been an extremely time-consuming endeavor, this has, in recent years, become a far less arduous task. This is due to the advent of very rapid data collection devices based on area detectors, 

FIGURE 1.15 Electron density from a monoclinic unit cell of the Gene 5 DNA Unwinding Protein crystal, lying between y = 0.125 and y = 0.250, is projected onto a single plane. Superimposed upon this electron density is a portion of the atomic model of the Gene 5 Protein. Electron density planes, like those shown here, are the images obtained directly by X-ray diffraction from computed Fourier 

FIGURE 1.16 A more sophisticated presentation of electron density, in virtual three dimensions, is possible using computer graphics. In this stereo diagram, two tyrosines separated by a valine residue are superimposed upon their density in a 1.8 A resolution electron density map of the serine protease from peniciUium cyclopium. 

Obviously, this contribution is independent of k if the potential is local, and the matrix elements is just the Fourier transform of the valence potential. 

The Coulomb contribution to this Fourier transform is as obtained from the Poisson equation. If one introduces the effects of the 

Of course, as discussed earlier, the determination of G q) remains a still unsolved many-body problem. In fact, it is quite questionable whether the exchange and correlation hole can be describe in such a simple way by a local function. But in practice, one can proceed by using the best known approximations. 

One should keep in mind that the calculation of the density has nof to start from the pseudo wave function, but from the wave function  

Since the zero order pseudo wave functions are plane waves, each one of these zero order terms contains a uniform contribution I/O to the density. This gives a uniform distribution outside the core region  

In this chapter we make first contact with the electron density. We will discuss some of its properties and then extend our discussion to the closely related concept of the pair density. We will recognize that the latter contains all information needed to describe the exchange and correlation effects in atoms and molecules. An appealing avenue to visualize and understand these effects is provided by the concept of the exchange-correlation hole which emerges naturally from the pair density. This important concept, which will be of great use in later parts of this book, will finally be used to discuss from a different point of view why the restricted Hartiee-Fock approach so badly fails to correctly describe the dissociation of the hydrogen molecule. 

The probability interpretation from equation (1 -7) of the wave function leads directly to the central quantity of this book, the electron density p(r). It is defined as the following multiple integral over the spin coordinates of all electrons and over all but one of the spatial 

Unlike the wave function, the electron density is an observable and can be measured experimentally, e. g. by X-ray diffraction. One of its important features is that at any position of an atom, p(r) exhibits a maximum with a finite value, due to the attractive force exerted by the positive charge of the nuclei. However, at these positions the gradient of the density has a discontinuity and a cusp results. This cusp is a consequence of the singularity Z a 

As computational facilities improve, electronic wavefunctions tend to become more and more complicated. A configuration interaction (Cl) calculation on a medium-sized molecule might be a linear combination of a million Slater determinants, and it is very easy to lose sight of the chemistry and the chemical intuition , to say nothing of the visualization of the results. Such wavefunctions seem to give no simple physical picture of the electron distribution, and so we must seek to find ways of extracting information that is chemically useful. 

In Chapter 3, I showed you how to write a simple LCAO wavefunction for the electronic ground state of the hydrogen molecule-ion, H2  

Sab is the overlap integral between atomic orbitals Isa and Isb, and the factor 1/V2(1 + Sab) is often called a normalization coefficient or the normalizing factor. It is introduced to make sure that 

I have included the modulus bars in IV (r)p because wavefunctions can be complex quantities. For most of this and subsequent chapters, I will assume that we are dealing with real wavefunctions. 

The total electronic wavefunction is the product of a spatial part and a spin part it is it(r) times a(s) or /3(s) for this one-electron molecule. There are thus two different quantum states having the same spatial part i/r(r). In the absence of a magnetic field, these are degenerate. 

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Another usefiil quantity is defining the electronic structure of a solid is the electronic density of states. In general the density of states can be defined as... 

The value of at zero temperature can be estimated from the electron density ( equation Al.3.26). Typical values of the Femii energy range from about 1.6 eV for Cs to 14.1 eV for Be. In temis of temperature (Jp = p//r), the range is approxunately 2000-16,000 K. As a consequence, the Femii energy is a very weak ftuiction of temperature under ambient conditions. The electronic contribution to the heat capacity, C, can be detemiined from... 

Kim Y S, Kim S K and Lee W D 1981 Dependence of the closed-shell repulsive interaction on the overlap of the electron densities Chem. Phys. Lett. 80 574... 

The reason that relaxation occurs can be understood in tenus of the free electron character of a metal. Because the electrons are free, they are relatively uuperturbed by the periodic ion cores. Thus, the electron density is homogeneous... 

In many materials, the relaxations between the layers oscillate. For example, if the first-to-second layer spacing is reduced by a few percent, the second-to-third layer spacing would be increased, but by a smaller amount, as illustrated in figure Al,7,31b). These oscillatory relaxations have been measured with FEED [4, 5] and ion scattering [6, 7] to extend to at least the fifth atomic layer into the material. The oscillatory nature of the relaxations results from oscillations in the electron density perpendicular to the surface, which are called Eriedel oscillations [8]. The Eriedel oscillations arise from Eenni-Dirac statistics and impart oscillatory forces to the ion cores. 

Above approximately 80 km, the prominent bulge in electron concentration is called the ionosphere. In this region ions are created from UV photoionization of the major constituents—O, NO, N2 and O2. The ionosphere has a profound effect on radio conmumications since electrons reflect radio waves with the same frequency as the plasma frequency, f = 8.98 x where 11 is the electron density in [147]. The... 

As the number of atoms in the asyimnetric unit increases, the solution of a structure by any of these phase-independent methods becomes more difficult, and by 1950 a PhD thesis could be based on a single crystal structure. At about that time, however, several groups observed that the fact that the electron density must be non-negative everywhere could be exploited to place restrictions on possible phases. The first use of this fact was by D Marker and J S Kasper [24], but their relations were special cases of more general relations introduced by J Karle and H Hauptman [25]. Denoting by A. the set of indices h., k., /., the Karle-Hauptman condition states that all matrices of the fonu... 

In order to understand the tendency to fomi a dipole layer at the surface, imagine a solid that has been cleaved to expose a surface. If the truncated electron distribution originally present within the sample does not relax, this produces a steplike change in the electron density at the newly created surface (figme B1.26.19(A)). 

Since the electron density p(x) oc /(v)p, where /(v) is die electron wavefiinction, this implies that the electron wavefiinction varies in a similarly step-wise fashion at the interface. This indicates that d i //dx, where s indicates that the derivative is evaluated at the surface, becomes infinite. Since the electron kinetic... 

Figure Bl.26.19. The variation of the electron density (A) from an iimelaxed surface and (B) showing the smoothing of the electron density to lower the kinetic density.

The polarization fiinctions are essential in strained ring compounds because they provide the angular flexibility needed to direct the electron density mto the regions between the bonded atoms. 

This gives the total energy, which is also the kinetic energy in this case because the potential energy is zero within the box , m tenns of the electron density p x,y,z) = (NIL ). It therefore may be plausible to express kinetic energies in tenns of electron densities p(r), but it is by no means clear how to do so for real atoms and molecules with electron-nuclear and electron-electron interactions operative. 

The so-called orbital-free DFT teclmique, which aims to directly calculate the electron density for which the... 

In DFT, the electronic density rather than the wavefiinction is tire basic variable. Flohenberg and Kohn showed [24] that all the observable ground-state properties of a system of interacting electrons moving in an external potential are uniquely dependent on the charge density p(r) that minimizes the system s total... 

Figure B3.2.10. Contour plot of the electron density obtained by an orbital-free Hohenberg-Kolnr teclmique [98], The figure shows a vacancy in bulk aluminium in a 256-site cell containing 255 A1 atoms and one empty site, the vacancy. Dark areas represent low electron density and light areas represent high electron density. A Kolm-Sham calculation for a cell of this size would be prohibitively expensive. Calculations on smaller cell sizes using both techniques yielded densities that were practically identical.

The electron density, pj, of the embedded cluster/adsorbate atoms is calculated using quantum chemistry methods (HF, PT, multireference SCF, or Cl). The initial step in this iterative procedure sets to zero,... 

Wang L-W and Teter M P 1992 Kinetic-energy functional of the electron density Phys. Rev. B 45 13 196-220... 

For each configuration of the nuclei, minimization of tlie total energy with respect to the electron density yields the instantaneous value of a potential energy fiinction V(/ ), and the corresponding forces on the nuclei. In principle,... 

E is tire density of states between E and E + AE. A simpler way of calculating n is to represent all tire electron states in tire CB by an effective density of states at tire energy E (band edge). The electron density is tlien simply n = NJ (Ef. 

Besides the expressions for a surface derived from the van der Waals surface (see also the CPK model in Section 2.11.2.4), another model has been established to generate molecular surfaces. It is based on the molecular distribution of electronic density. The definition of a Limiting value of the electronic density, the so-called isovalue, results in a boundary layer (isoplane) [187]. Each point on this surface has an identical electronic density value. A typical standard value is about 0.002 au (atomic unit) to represent electronic density surfaces. 

Molecular orbitals were one of the first molecular features that could be visualized with simple graphical hardware. The reason for this early representation is found in the complex theory of quantum chemistry. Basically, a structure is more attractive and easier to understand when orbitals are displayed, rather than numerical orbital coefficients. The molecular orbitals, calculated by semi-empirical or ab initio quantum mechanical methods, are represented by isosurfaces, corresponding to the electron density surfeces Figure 2-125a). 

Note that the sums of the squares of the coefficients in a given MO must equal 1 (e.g., 0.3717 + 0.6015 + 0.3717 + 0.6015 = 1.0 for Pi) because each of the AOs represents a probability distribution of finding the electron at a given point in space. The total probability of finding an electron in all space for an MO must be unity, exactly as for its constituent AOs. We now can see that the LCAO approximation is only one of many possibilities to describe the electron density (= probability) for MOs. We do not have to express the electron density as a linear combination of the electron densities of AOs centered at the atoms. We could also... 

Although split valence basis sets give far better results than minimal ones, they still have systematic weaknesses, such as a poor description of three-inembered rings, This results from their inability to polarize the electron density to one side of an atom. Consider the /T-bond shown in Figure 7-23. 

The solution is therefore to add basis functions that allow a shift of the electron density without moving the centers of the basis functions away from their nuclei. 

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