Moment calculations, 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 These tables have been superseded by purely numerical Hartree-Fock calculations that do not depend on basis sets [232,235,252,253].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. 

Other calculations tested using this molecule include two-dimensional, fully numerical solutions of the molecular Dirac equation and LCAO Hartree-Fock-Slater wave functions [6,7] local density approximations to the moment of momentum with Hartree-Fock-Roothaan wave functions [8] and the effect on bond formation in momentum space [9]. Also available are the effects of information theory basis set quality on LCAO-SCF-MO calculations [10,11] density function theory applied to Hartree-Fock wave functions [11] higher-order energies in... 

The results in (92)-(96) allow us to calculate the moments of (91) up to second order. Consistency with the mass density, momentum density and Euler sdess for a given p and u, uniquely determines the equilibrium distribution. 

Here, 2 is an operator appropriate to the quantity of interest, ij/ is the wave function for the system being studied, and a is the result of an experimental measurement. Experimental quantities include the energy, the charge density, the momentum, the dipole moment, etc. The molecular orbitals and their energies are not experimental quantities and the wave function for a system need not be expressed in terms of molecular orbitals. In fact, the wave functions themselves would not be necessary if there were a direct way in which to calculate the electron density distribution for a molecule since all of the properties may be derived from the density. ... 

In practice, using currently available exchange and correlation potentials, this path leads to results [113] worse than those obtained with the Hartree-Fock method. This is illustrated for momentum moments in Table 19.2 which shows median absolute percent errors of (p ) for 78 molecules relative to those computed by an approximate singles and doubles coupled-cluster method often called QCISD [114,115]. The molecules are mostly polyatomic, and contain H, C, N, O, and F atoms. The correlation-consistent cc-pVTZ basis set [110] was used for these computations. Table 19.2 shows the median errors for the Hartree-Fock method, for second-order Mpller-Plesset permrbation theory (MP2), and for DFT calculations done with the B3LYP hybrid density functional... 

The calculations include, as said previously, overlaps, conditional probability distributions of the electron probability densities, and these observables oscillator strengths, quadrupole moments (for states with total angular momentum quantum numbers of 1 or more) and expectation values (pi p2)/( pi i>2 )- (Distributions of this last quantity have also been computed, in preparation for two-electron ionization experiments by electron impact, but are not reported here.) We can proceed to summarize these indicators and then examine them and ask how well each model performs. 

The density-matrix-spectral-moments algorithm (DSMA) °° is an approximate scheme for solving the TDHF equations that allows us to calculate from the source ( W) by solving eq A18 without a direct diagonalization of L. This is accomplished by computing the set of electronic oscillators that dominate the expansion of Without loss of generality, we can take rj itj to be real and express it in terms of our momentum variables as - °°... 

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