Electron density physical interpretation

The hydrogen atom orbitals are functions of three variables the coordinates of the electron. Their physical interpretation is that the square of the amplitude of the wave function at any point is proportional to the probability of finding a particle at that point. Mathematically, the electron density distribution is equal to the square of the absolute value of the wave function ... 

All the information that can be known about a system in a given stationary state is contained in the state function 5 (r), particularly molecular electron densities. The use of electron density to interpret atoms in molecules, bonds and structures constitutes a bridge between the concept of state function and the physical model of matter in real space. 

In this formulation, the electron density is expressed as a linear combination of basis functions similar in mathematical form to HF orbitals. A determinant is then formed from these functions, called Kohn-Sham orbitals. It is the electron density from this determinant of orbitals that is used to compute the energy. This procedure is necessary because Fermion systems can only have electron densities that arise from an antisymmetric wave function. There has been some debate over the interpretation of Kohn-Sham orbitals. It is certain that they are not mathematically equivalent to either HF orbitals or natural orbitals from correlated calculations. However, Kohn-Sham orbitals do describe the behavior of electrons in a molecule, just as the other orbitals mentioned do. DFT orbital eigenvalues do not match the energies obtained from photoelectron spectroscopy experiments as well as HF orbital energies do. The questions still being debated are how to assign similarities and how to physically interpret the differences. 

The comparison with experiment can be made at several levels. The first, and most common, is in the comparison of derived quantities that are not directly measurable, for example, a set of average crystal coordinates or a diffusion constant. A comparison at this level is convenient in that the quantities involved describe directly the structure and dynamics of the system. However, the obtainment of these quantities, from experiment and/or simulation, may require approximation and model-dependent data analysis. For example, to obtain experimentally a set of average crystallographic coordinates, a physical model to interpret an electron density map must be imposed. To avoid these problems the comparison can be made at the level of the measured quantities themselves, such as diffraction intensities or dynamic structure factors. A comparison at this level still involves some approximation. For example, background corrections have to made in the experimental data reduction. However, fewer approximations are necessary for the structure and dynamics of the sample itself, and comparison with experiment is normally more direct. This approach requires a little more work on the part of the computer simulation team, because methods for calculating experimental intensities from simulation configurations must be developed. The comparisons made here are of experimentally measurable quantities. 

If we multiply the probability density P(x, y, z) by the number of electrons N, then we obtain the electron density distribution or electron distribution, which is denoted by p(x, y, z), which is the probability of finding an electron in an element of volume dr. When integrated over all space, p(x, y, z) gives the total number of electrons in the system, as expected. The real importance of the concept of an electron density is clear when we consider that the wave function tp has no physical meaning and cannot be measured experimentally. This is particularly true for a system with /V electrons. The wave function of such a system is a function of 3N spatial coordinates. In other words, it is a multidimensional function and as such does not exist in real three-dimensional space. On the other hand, the electron density of any atom or molecule is a measurable function that has a clear interpretation and exists in real space. 

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 phenomenon of electron pairing is a consequence of the Pauli exclusion principle. The physical consequences of this principle are made manifest through the spatial properties of the density of the Fermi hole. The Fermi hole has a simple physical interpretation - it provides a description of how the density of an electron of given spin, called the reference electron, is spread out from any given point, into the space of another same-spin electron, thereby excluding the presence of an identical amount of same-spin density. If the Fermi hole is maximally localized in some region of space all other same-spin electrons are excluded from this region and the electron is localized. For a closed-shell molecule the same result is obtained for electrons of p spin and the result is a localized a,p pair [46]. 

Our discussion of electronic structure has been in terms of band filling only. Of course, there is a lot more to know about band structures. The density of states represents only a highly simplified representation of the actual electronic structure, which ignores the three-dimensional structure of electron states in the crystal lattice. Angle-dependent photoemission gives information on this property of the electrons. The interested reader is referred to standard books on solid state physics [9,10] and photoemission [16,17]. The interpretation of photoemission and X-ray absorption spectra of catalysis-oriented questions, however, is usually done in terms of the electron density of states only. 

Thus the XC energy is the energy of interaction between the electrons and a charge distribution represented by p cM(f, r ). The question we wish to answer now is whether the expression in Equation 7.18 is just the rewriting of the XC energy in a different way or does it have a physical interpretation. We now show that it indeed has a physical interpretation the term represents the deficit in the density of electrons at r when an electron is at r. 

As mentioned in [Section 24.1], and as already demonstrated in Equation 24.39, the Fukui functions as well as the chemical hardness of an isolated system can be properly defined without invoking any change in its electron number. We define a new Fukui function called polarization Fukui function, which very much resembles the original formulation of the Fukui function but with a different physical interpretation. Because of space limitation, only a brief presentation is given here. More details will appear in a forthcoming work [33]. One assumes a potential variation <5wext(r), which induces a deformation of the density 8p(r). A normalized polarization Fukui function is defined by... 

Equations 24.75 and 24.76 generalize the Dyson equations derived previously [32], The differences between /1 and the frontier orbitals have nice and simple physical interpretations [24,32] as either the variation of the electronic density induced by effective external potentials 8vj ... 

Evidently, the LSD and GGA approximations are working, but not in the way the standard spin-density functional theory would lead us to expect. In Ref [36], a nearly-exact alternative theory, to which LSD and GGA are also approximations, is constructed, which yields an alternative physical interpretation in the absence of a strong external magnetic field. In this theory, Hf(r) and rti(r) are not the physical spin densities, but are only intermediate objects (like the Kohn-Sham orbitals or Fermi surface) used to construct two physical predictions the total electron density n(r) from... 

Abstract. Investigation of P,T-parity nonconservation (PNC) phenomena is of fundamental importance for physics. Experiments to search for PNC effects have been performed on TIE and YbF molecules and are in progress for PbO and PbF molecules. For interpretation of molecular PNC experiments it is necessary to calculate those needed molecular properties which cannot be measured. In particular, electronic densities in heavy-atom cores are required for interpretation of the measured data in terms of the P,T-odd properties of elementary particles or P,T-odd interactions between them. Reliable calculations of the core properties (PNC effect, hyperfine structure etc., which are described by the operators heavily concentrated in atomic cores or on nuclei) usually require accurate accounting for both relativistic and correlation effects in heavy-atom systems. In this paper, some basic aspects of the experimental search for PNC effects in heavy-atom molecules and the computational methods used in their electronic structure calculations are discussed. The latter include the generalized relativistic effective core potential (GRECP) approach and the methods of nonvariational and variational one-center restoration of correct shapes of four-component spinors in atomic cores after a two-component GRECP calculation of a molecule. Their efficiency is illustrated with calculations of parameters of the effective P,T-odd spin-rotational Hamiltonians in the molecules PbF, HgF, YbF, BaF, TIF, and PbO. 

Another useful physical property of the crystal is its density, which can be used to determine several useful microscopic properties, including the protein molecular weight, the proteinlwater ratio in the crystal, and the number of protein molecules in each asymmetric unit (defined later). Molecular weights from crystal density are more accurate than those from electrophoresis or most other methods (except mass spectrometry) and are not affected by dissociation or aggregation of protein molecules. The proteinlwater ratio is used to clarify electron-density maps prior to interpretation (Chapter 7). If the unit cell is symmetric (Chapter 4), it can be subdivided into two or more equivalent parts called asymmetric units (the simplest unit cell contains, or in fact is, one asymmetric unit). For interpreting electron-density maps, it is helpful to know the number of protein molecules per asymmetric unit. 

For the details and derivation of the physical interpretation we refer the reader to the original literature14,15. Since the Coulomb self-energy component of the KS electron-interaction energy functional and its derivative, the Hartree potential, are known functionals of the density, we provide in Section HA the expressions governing the interpretation of the KS exchange-correlation energy... 

The physical interpretation of the functional derivative vx(r) shows that it is comprised of a term Wx (r) representative of Pauli correlations, and a term wj (r) that constitutes part of the total correlation-kinetic contribution Wt (r). cThe exact asymptotic structure of these components in the vacuum has been determined and shown to also be image-potential-like. Although the structure of vx(r) about the surface and asymptotically in the vacuum and metal-bulk regions is comprised primarily of its Pauli component, the correlation-kinetic contribution is not insignificant for medium and low density metals. It is only for high density systems (rs < 2) that vx(r) is represented essentially by its Pauli component Wx (r). Thus, we see that the uniform electron gas result of -kF/ir for the functional derivative vx(r), which is the asymptotic metal-bulk value, is not a consequence of Pauli correlations alone as is thought to be the case. There is also a small correlation-kinetic contribution. The Pauli and correlation-kinetic contributions have now been quantified. 

---