Momentum space Compton profile

To provide experimental information about (k) from Compton profile measurements, we performed Compton measurements on Li using an experimental setup described elsewhere [21], The momentum space resolution obtained was APz = 0.12a.u.,... 

Using the valence profiles of the 10 measured directions per sample it is now possible to reconstruct as a first step the Ml three-dimensional momentum space density. According to the Fourier Bessel method [8] one starts with the calculation of the Fourier transform of the Compton profiles which is the reciprocal form factor B(z) in the direction of the scattering vector q. The Ml B(r) function is then expanded in terms of cubic lattice harmonics up to the 12th order, which is to take into account the first 6 terms in the series expansion. These expansion coefficients can be determined by a least square fit to the 10 experimental B(z) curves. Then the inverse Fourier transform of the expanded B(r) function corresponds to a series expansion of the momentum density, whose coefficients can be calculated from the coefficients of the B(r) expansion. 

Local-scaling transformations have been employed [39] in order to obtain a one-particle density in position space from the one-particle density in momentum space, and vice versa. This problem arises from example when we have a y(p), obtained from experimental Compton profiles, and wish to calculate the corresponding p(r) [98]. 

B( ) is variously called the reciprocal form factor, the p-space form factor, and the internally folded density. B(s) is the basis of a method for reconstructing momentum densities from experimental data [145,146], and it is useful for the r-space analysis of Compton profiles [147-151]. The B(s) function probably first arose in an examination of the connection between form factors and the electron momentum density [129]. The B f) function has been rediscovered by Howard et al. [152]. 

Bond additivity [377] has been studied with localized molecular orbitals by Epstein [308], Smith et al. [378,379], and Measures et al. [380], and with phenomenological analysis of theoretical Compton profiles by Hirst and Liebmann [381-383], Reed et al. [384], and by Cade et al. [385]. Hybridization has been examined from a momentum space viewpoint by Cooper et al. [386] and Clark et al. [387]. 

Finally, (for atoms), the momentum densities corresponding to hybrid orbitals exhibit a few basic extremal features close to the origin. These depend on the weight that is given to s, p and d contributions, and they determine the basic look of the density. Outwardly, momentum-space hybrids share one feature with a related experimental quantity, the Compton profile they all look alike. On closer inspection, however, there are a variety of complex features, mainly arising from the nodal structure of the orbitals. Apart from the obvious use of hybrids in position space for the description of bond situations, there is another feature that has always captured the interest of scientists and laymen their intricate structure. This feature is less apparent in momentum space, but it is still present. If nothing else, its enjoyment makes a close look at these entities worthwhile. 

In the early 1940s, an investigation of chemical bonding from the momentum-space viewpoint was initiated by Coulson and Duncanson (Coulson, 1941a,b Duncanson, 1941, 1943 Coulson and Duncanson, 1941,1942 Duncanson and Coulson, 1941) based on the Fourier transformation of the position wave function. [They also gave a systematic analysis of the momentum distributions and the Compton profiles of atoms (Duncanson and Coulson, 1944, 1945, 1948).] They first clarified the momentum-space properties of the fundamental two-center MO and VB wave functions, which may be outlined as follows. 

The Fourier transforms of Eq. (51) can be performed in closed form for most commonly used basis sets. Moreover, formulas and techniques for the computation of the spherically averaged momentum density, isotropic and directional Compton profiles, and momentum moments have been worked out for both Gaussian- and Slater-type basis sets. Older work on the methods and formulas has been summarized in a review article by Kaijser and Smith [79]. A bibliography of more recent methodological work can be found in another review article [11]. Advantages and disadvantages of various types of basis sets, including many unconventional ones, have been analyzed from a momentum-space perspective [80-82]. Section 19.7 describes several illustrative computations chosen primarily from my own work for convenience. 

In quantum chemistry, the state of a physical system is usually described by a wave function in the position space. However, it is also well known that a wave function in the momentum space can provide complementary information for electronic structure of atoms or molecules [1]. The momentum-space wave function is especially useful to analyse the experimental results of scattering problems, such as Compton profiles [2] and e,2e) measurements [3]. Recently it is also applied to study quantum similarity in atoms and molecules [4]. In the present work, we focus our attention on the inner-shell ionization processes of atoms by charged-particle impact and study how the electron momentum distribution affects on the inner-shell ionization cross sections. 

Three functions may be computed that have the same information content but different use in the discussion of theoretical and experimental results [23] the electron momentum density itself (EMD) p p) the Compton profile (CP) function J(p) the autocorrelation function, or reciprocal space form factor, B r). 

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