Compton profiles momentum density

Crystal can compute a number of properties, such as Mulliken population analysis, electron density, multipoles. X-ray structure factors, electrostatic potential, band structures, Fermi contact densities, hyperfine tensors, DOS, electron momentum distribution, and Compton profiles. 

Generally, all band theoretical calculations of momentum densities are based on the local-density approximation (LDA) [1] of density functional theory (DFT) [2], The LDA-based band theory can explain qualitatively the characteristics of overall shape and fine structures of the observed Compton profiles (CPs). However, the LDA calculation yields CPs which are higher than the experimental CPs at small momenta and lower at large momenta. Furthermore, the LDA computation always produces more pronounced fine structures which originate in the Fermi surface geometry and higher momentum components than those found in the experiments [3-5]. 

Mueller, F.M. (1977) Anisotropic momentum densities from Compton profiles silicon, Phys. Rev., BIS, 3039-3044. 

Kobayasi, T., Nara, H., Timms, D.N. and Cooper, MJ. (1995) Core-orthogonalization effects on the momentum density distribution and the Compton profile of valence electrons in semiconductors, Bull. Coll. Med. Sci. Tohoku Univ., 4, 93-104. 

Nara, H., Kobayasi, T., Takegahara, K., Cooper, M.J. and Timms, D.N. (1994) Optimal number of directions in reconstructing 3D momentum densities from Compton profiles of semiconductors, Computational Materials Sci., 2, 366-374. 

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. 

On the other hand both the momentum densities and the occupation number functions were influenced by the early truncation of the B(r) function due to inadequate statistics. The effect of this influence has to be studied, which could best be done by a reconstruction using calculated Compton profiles. 

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]. 

IV. OBTAINING MOMENTUM DENSITIES A. Isotropic Compton Profiles... 

Within the impulse approximation, the gas-phase Compton profile Jo q) is related [185,186] to the isotropic momentum density by... 

In the special case that the scattering vector is parallel to one of the coordinate axes, these expressions look much simpler. For example, if q is parallel to the z axis, the directional Compton profile, expressed in Cartesian coordinates, is simply the marginal momentum density along the axis ... 

A vast number of directional Compton profiles have been measured for ionic and metallic solids, but none for free molecules. Nevertheless, several calculations of directional Compton profiles for molecules have been performed as another means of analyzing the momentum density. 

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. 

Subsequent work on graphical analysis of anisotropic momentum densities, directional Compton profiles, and their differences in diatomic molecules was reported by several groups, including Kaijser and Smith [195,316], Ramirez [317-319], Matcha, Pettit, Ramirez, and Mclntire [320-328], Leung and Brion [329,330], Simas et al. [331], Rozendaal and Baerends [332,333], Cooperand Allan [334], Anchell and Harriman [138], and Rerat et al. [335,336]. 

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. 

Podloucky, R., and J. Redinger (1984). A theoretical study of Compton scattering for magnesia. 1. Momentum density, Compton profiles and B functions. J. 

Table 22 Calculated Compton profile and Radial momentum Densities RMD (au) ... 

Ray et al first applied FSGO in Compton profile calculations. S. Bhargava and N. K. Ray used FSGO to calculate Compton Profiles (CP) for ionic system such as LiF, LiCl, NaF and NaCl. The method had been proposed by Epstein and Lipscomb. Radial momentum density for an i -type Gaussian I(p) is given by the relationship,... 

The momentum distribution of H is derived from the measured NCS TOF-spectra by standard procedures [Mayers 1994 Mayers 2004], The distribution J(y) (often called "Compton profile" [Sears 1984 Watson 1996]) is proportional to the density of protons with momentum component hy along the direction of the neutron-proton momentum transfer hq. J(y) at scattering angle 0 = 66° is shown in Fig. 13 (full line). Here hy is the H-momentum component (before collision) along the direction of momentum transfer hq. 

Experimentally, the momentum density is closely connected with the Compton profile, the spectrum of scattered radiation. Within the impulse approximation (Kilby, 1965 Eisenberger and Platzman, 1970), the directional Compton profile is given by... 

Note that all the above discussion also holds when we adopt the radial momentum density I p) or the Compton profile J(q) as a basic physical quantity instead of the three-dimensional density p(p) (Koga and Morita, 1981b, 1982a Thakkar, 1983). 

Fig. 24. Radial momentum densities l p) and isotropic Compton profiles J(q) for the o-and ir states of the H2+ system. All values in atomic units. [Reproduced from Koga et al, 1982.]...

There are two main methods for the reconstruction of 7T(p) from the directional Compton profile. In the Fourier-Hankel method [33,51], a spherical harmonic expansion of the directional Compton profile is inverted term-by-term to obtain the corresponding expansion of /T(p). In the Fourier reconstraction method [33,34], the reciprocal form factor B0) is constructed a ray at a time by Fourier transformation of the measured J(q) along that same direction. Then the electron momentum density is obtained from B( ) by using the inverse of Eq. (22). A vast number of directional Compton profiles have been measured for ionic and metallic solids, but none for free molecules. Nevertheless, several calculations of directional Compton profiles for molecules have been performed as another means of analyzing the momentum density. 

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. 

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