Atoms radial factor

The radial factors of the hydrogen-like atom total wave functions ip r, 0, tp) are related to the functions Sni(p) by equation (6.23). Thus, we have... 

T(S) is the Debye-WaUer factor introduced in (2). The atomic form factors are typically calculated from the spherically averaged electrcai density of an atom in isolation [24], and therefore they do not contain any information on the polarization induced by the chemical bonding or by the interaction with electric field generated by other atoms or molecules in the crystal. This approximation is usually employed for routine crystal stmcture solutions and refinements, where the only variables of a least square refinement are the positions of the atoms and the parameters describing the atomic displacements. For more accurate studies, intended to determine with precisicai the electron density distribution, this procedure is not sufficient and the atomic form factors must be modeled more accurately, including angular and radial flexibihty (Sect. 4.2). 

In order to test such an application we have calculated the spin and charge structure factors from a theoretical wave function of the iron(III)hexaaquo ion by Newton and coworkers ( ). This wave function is of double zeta quality and assumes a frozen core. Since the distribution of the a and the B electrons over the components of the split basis set is different, the calculation goes beyond the RHF approximation. A crystal was simulated by placing the complex ion in a lOxIOxlOA cubic unit cell. Atomic scattering factors appropriate for the radial dependence of the Gaussian basis set were calculated and used in the analysis. 

For Eq. (11) S is the Bragg vector S = 2ttH, IT is the row vector (htk,l) and the scalar S - S = 4ir sin 0/A. The index / covers the N atoms in the unit cell. The atomic scattering factor f (S) is the Fourier-Bessel transform of the electronic, radial density function of the isolated atom. This density function is usually derived from a spin-restricted Hartree-Fock wave function for the atom in its ground state. The structure fac-... 

Electron-electron interaction is significant in studies where spin-orbital degeneracy, or near degeneracy, is present. The atomic d-orbitals have a common radial factor and their 15 distinct products give rise to 120 density-density integrals. These are expressed in terms of three basic ones, F0, F2, and F4 in the Slater-Condon formulation. The 100 spin-orbital densities are linear combination of the orbital ones and the 100 by 100 interaction integral matrix has a rank of 15 and is expressed by the Slater-Condon parameters. 

The number of electrons in the system is denoted by q. This conclusion depends on the assumption that the radial factor of the atomic orbitals is the same and will be void when a more general model is invoked. 

The liquid structure factor of CCI4 and its derivatives with respect to temperature at fixed pressure or fixed volume, needed by eq. (2), were evaluated by Molecular Dynamics (MD) simulations. We have used the OPLS model for tetrachloromethane [9] In this model, the CCI4 molecules are described as rigid tetrahedra (dc-ci = 1 -769 A) and the intermolecular potentials are atom centered 6-12 Lennard-Jones potentials plus the coulombic interaction with partial charges on C and Cl. We performed NVT simulations with 512 molecules for about 1 ns each. The different x-ray structure factors were obtained from the accumulated partial radial distribution functions [10], using the atomic form-factors from the DABAX database [11]. In order to estimate the partial derivatives of the structure factor, we have used finite differences we considered two different temperatures, Ti = 300 K and T2 = 328 K, and two molar volumes, Vi = 97.3 cm mol and V2 = 100.65 cm mol which are the molar volumes along the liquid-vapor coexistence line for the two temperatures Tj and Tz respectively [12]. Three simulations were then run for the temperature and molar volume conditions (TiiVi), T2,V )... 

Atomic units have been used, and Fi is the integro-differential operator of the Hartree-Fock equations that determines the radial factors P i. The usual way of making stationary an energy functional that includes the orthonormality constraints of these radial factors introduces undetermined multipliers e(nl). As the definition of Fi introduces the same potential for all Pni, there is no need for off-diagonal undetermined multipliers, and the variation of the energy functional with the constraints yields the equation... 

Ground-State Wave Function and Energy. For the ground state of the hydrogenlike atom, we have = 1, / = 0, and m = 0. The radial factor (6.100) is... 

TABLE 6.1 Radial Factors in the Hydrogenlike-Atom Wave Functions... 

FIGURE 6.8 Graphs of the radial factor in the hydrogen-atom (Z = 1) wave functions.The same scale is used in all graphs. (In some texts, these functions are not properly drawn to scale.)... 

Apply the particle-in-a-box basis functions to the radial equation for the hydrogen atom for the / = 0 states. Recall that in Section 6.9, we expressed the radial factor in the H-atom wave function as R r) = r F r), where F(r) = 0 at r = 0. The variation function in this problem will have the form = r F r)Yf 6, ) take the dimensionless function F,(r,) to be a linear combination of 28 pib basis functions, where the box goes from r, = 0 to 27, where r, is defined in Section 6.9. Work out the proper forms for the integrals //y and 5. Find the estimates for the lowest three / = 0 energies. For the ground-state variation function, how many pib basis functions appear with a coefficient greater than 0.1 ... 

To simplify matters somewhat, we approximate the best possible atomic orbitals with orbitals that are the product of a radial factor and a spherical harmonic ... 

Originally, Hartree-Fock atomic calculations were done by using numerical methods to solve the Hartree-Fock differential equations (11.12), and the resulting orbitals were given as tables of the radial functions for various values of r. [The Numerov method (Sections 4.4 and 6.9) can be used to solve the radial Hartree-Fock equations for the radial factors in the Hartree-Fock orbiteds the angular factors are spherical harmonics. See D. R. Heu tree, The Calculation of Atomic Structures, Wiley, 1957 C. Froese Fischer, The Hartree-Fock Method for Atoms, Wiley, 1977.]... 

For a many-electron atom, the self-consistent-field (SCF) method is used to construct an approximate wave function as a Slater determinant of (one-electron) spin-orbitals. The one-electron spatial part of a spin-orbital is an atomic orbital (AO). We took each AO as a product of a spherical harmonic and a radial factor. As an initial approximation to the radial factors, we can use hydrogenlike radial functions with effective nuclear charges. 

The following table lists the cation radii in LiF, NaCl, and KCl, the distances of the p (r) minima from the centers of the cations, and the ionic radii rg which we deduced from the radial electron density distributions obtained by approximating the atomic scattering factors with smooth curves ... 

The method used to calculate rp was similar to that described earlier [19, 20]. The experimental values of the atomic scattering factors were approximated by f(s) curves calculated on the assumption that the atoms were neutral. Having found f(s), we determined tiie radial distribution functions in accordance with Eq. (5). These functions were employed to find the number of electrons outside a sphere of radius r ... 

K and K2 are normalization constants and Z[ and adjustable constants. Note that the radial part of the 2s AO differs from that of the 2s orbital of a one-electron atom the factor (2 - Z2jr/flo) has been replaced by (Z r/ao)-... 

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