Atomic momentum density

The spherically symmetric atomic momentum densities, in contrast, exhibit monotonic as well as nonmonotonic behavior even in their ground states. Further, it was... 

FIGURE 5.3 Monotonic and nonmonotonic atomic momentum densities for He (dotted) and Ne are portrayed. 

At this point, one may wonder why there is an interest in the atomic momentum densities and their nature and what sort of information does one derive from them. In a system in which all orientations are equally probable, the full three-dimensional (3D) momentum density is not experimentally measurable, but its spherical average is. The moments of the atomic momentum density distributions are of experimental significance. The moments and the spherically averaged momentum densities are defined in the equations below. 

Spin densities determine many properties of radical species, and have an important effect on the chemical reactivity within the family of the most reactive substances containing free radicals. Momentum densities represent an alternative description of a microscopic many-particle system with emphasis placed on aspects different from those in the more conventional position space particle density model. In particular, momentum densities provide a description of molecules that, in some sense, turns the usual position space electron density model inside out , by reversing the relative emphasis of the peripheral and core regions of atomic neighborhoods. 

Figure 1. The electron momentum density for atomic hydrogen measured by EMS for the indicated energies compared with the square of Schrodinger wave function (solid curve) [4].

The outer most levels in C60 are due to rc orbitals . These are formed by 2p electrons which have their orbitals oriented along the radius of the molecule. The different environment inside and outside the spherical molecule causes the double-peaked structure in the momentum densities. In graphite the n band is formed by 2p orbitals oriented perpendicular to the sheets of carbon atoms. Using single-crystal graphite films we have a unique opportunity to study the effects of the orientation of these 2p orbitals in detail. 

In this chapter we will have a closer look at the methods of the reconstruction of the momentum densities and the occupation number densities for the case of CuAl alloys. An analogous reconstruction was successfully performed for LiMg alloys by Stutz etal. in 1995 [3], It was found that the shape of the Fermi surface changed and its included volume grew with Mg concentration. Finally the Fermi surface came into contact with the boundary of the first Brillouin zone in the [110] direction. Similar changes of the shape and the included volume of the Fermi surface can be expected for CuAl [4], although the higher atomic number of Cu compared to that of Li leads to problems with the reconstruction, which will be examined. 

The chemistry of momentum densities becomes more interesting when one goes over to molecules. There is an inherent stabilization when atoms bond together to form molecules. One of the first chemical interpretations of the EMD was presented by Coulson [11]. The simplest valence bond (VB) and molecular orbital (MO) wave functions for H2 molecule given in the following equations were used for this purpose ... 

C. Gatti and A. Famulari Interaction Energies and Densities. A Quantum Theory of Atom in Molecules insight on the Effect of Basis Set Superposition Error Removal , P.G. Mezey and B. Rohertson (Eds.), Understanding Chemical Reactivity Electron, Spin and Momentum Densities and Chemical Reactivity, Vol. 2, Kluwerhook series (1999). In press. 

THE MOMENTUM DENSITY PERSPECTIVE OE THE ELECTRONIC STRUCTURE OF ATOMS AND MOLECULES... 

Several review articles on the theoretical aspects of electron momentum densities of atoms and molecules were written in the 1970s by Benesch and Smith [9], Epstein [10,11], Mendelsohn and Smith [12], Epstein and Tanner [13], Lindner [14], and Kaijser and Smith [15]. Since that time (e,2e) spectroscopy and the momentum densities of Dyson orbitals have been reviewed very often [16-28]. However, to my knowledge, a review article on molecular electron momentum densities has not been written recently apart from one [29] devoted solely to the zero-momentum critical point. The purpose of this chapter is to survey what is known about the electron momentum density of atoms and molecules, and to provide an extensive, but not exhaustive, bibliography that should be sufficient to give a head start to a nonspecialist who wishes to enter the field. 

Section III. Methods for obtaining momentum densities, both experimental and computational, are reviewed in Section IV. Only a sample of representative work on the electron momentum densities of atoms and molecules is summarized in Sections V and VI because the topic is now too vast for comprehensive coverage. Electron momentum densities in solids and other condensed phases are not considered at all. The literature on electron momentum spectroscopy and Dyson orbital momentum densities is not surveyed, either. Hartree atomic units are used throughout. 

For 5-state atoms, the momentum density is spherically symmetric and therefore n( p) = IIo(p). Nonspherical II( p) can occur in atomic states of higher angular momentum. However, these states are degenerate, and an ensemble average is required. This is equivalent to a spherical average for any one of the states in the degenerate manifold. The spherically averaged momentum density IIo(p) is also of interest for molecules because it is a quantity that can be obtained from gas-phase experiments. 

A fair amount of effort has been devoted to the manifestations of atomic shell structure in the momentum density and related functions [210-215]. The number of maxima observed in I p) varies from one to four, with 35 and 48 atoms exhibiting two and three local maxima [215]. No maxima in I p) corresponding to the most corelike shells are found in heavy atoms. Correlations have been found between 1 jpmax, where p ax is the location of the innermost maximum of I p), and the relative size of the atom [215]. 

Figure 5.5. Types of electron momentum densities no(p) >n atoms. Solid lines are used for the total density, whereas dotted lines and crosses indicate the contribution from the outermost s and p orbitals, respectively. Top left a type I density for the potassium atom. Top right a typical type II density for the argon atom. Bottom left a typical type III density for the silver atom. Bottom right a closeup of rio(p) for the silver atom showing the minimum and secondary maximum. Adapted from Thakkar [29].

Type I momentum densities, characteristic of He, N, Mn, all atoms from groups 1-6, 13, and 14 except Ge, and the lanthanides and actinides, are unimodal and have a maximum at p = 0. A density of this type is shown in Figure 5.5 for the potassium atom. The maximum at p = 0 comes mainly from... 

Hartree-Fock calculations of the three leading coefficients in the MacLaurin expansion, Eq. (5.40), have been made [187,232] for all atoms in the periodic table. The calculations [187] showed that 93% of rio(O) comes from the outermost s orbital, and that IIo(O) behaves as a measure of atomic size. Similarly, 95% of IIq(O) comes from the outermost s and p orbitals. The sign of IIq(O) depends on the relative number of electrons in the outermost s and p orbitals, which make negative and positive contributions, respectively. Clearly, the coefficients of the MacLaurin expansion are excellent probes of the valence orbitals. The curvature riQ(O) is a surprisingly powerful predictor of the global behavior of IIo(p). A positive IIq(O) indicates a type 11 momentum density, whereas a negative rio(O) indicates that IIo(O) is of either type 1 or 111 [187,230]. MacDougall has speculated on the connection between IIq(O) and superconductivity [233]. 

Duncanson and Coulson [242,243] carried out early work on atoms. Since then, the momentum densities of aU the atoms in the periodic table have been studied within the framework of the Hartree-Fock model, and for some smaller atoms with electron-correlated wavefunctions. There have been several tabulations of Jo q), and asymptotic expansion coefficients for atoms [187,244—251] with Hartree-Fock-Roothaan wavefunctions. These tables have been superseded by purely numerical Hartree-Fock calculations that do not depend on basis sets [232,235,252,253]. There have also been several reports of electron-correlated calculations of momentum densities, Compton profiles, and momentum moments for He [236,240,254-257], Li [197,237,240,258], Be [238,240,258, 259], B through F [240,258,260], Ne [239,240,258,261], and Na through Ar [258]. Schmider et al. [262] studied the spin momentum density in the lithium atom. A review of Mendelsohn and Smith [12] remains a good source of information on comparison of the Compton profiles of the rare-gas atoms with experiment, and on relativistic effects. 

The Momentum Density Perspective of the Electronic Structure of Atoms and Molecules By Ajit J. Thakkar... 

---