### all known math issues fixed

parent 2a45dbf3
 ... ... @@ -78,7 +78,7 @@ TOL (MBASIS) OPTIMIZE (MBASIS) ----------------- The mixed product basis can still be quite large. In the calculation of the screened interaction, each matrix element, when represented in the basis of Coulomb eigenfunctions, is multiplied by :math:{\sqrt{v_\mu v_\nu}} with the Coulomb eigenvalues :math:{\{v_\mu\}}. This gives an opportunity for reducing the basis-set size further by introducing a Coulomb cutoff :math:{v_\mathrm{min}}. The reduced basis set is then used for the polarization function, the dielectric function, and the screened interaction. The parameter :math:{v_\mathrm{min}} can be specified after the keyword OPTIMIZE MB in three ways: first, as a "pseudo" reciprocal cutoff radius :math:{\sqrt{4\pi/v_\mathrm{min}}} (which derives from the plane-wave Coulomb eigenvalues :math:{v_\mathbf{G}=4\pi/G^2}), second, directly as the parameter :math:{v_\mathrm{min}} by using a negative real number, and, finally, as the number of basis functions that should be retained when given as an integer. The so-defined basis functions are mathematically close to plane waves. For testing purposes, one can also enforce the usage of plane waves (or rather projections onto plane waves) with the keyword OPTIMIZE PW, in which case the Coulomb matrix is known analytically. No optimization of the basis is applied, if OPTIMIZE is omitted. The mixed product basis can still be quite large. In the calculation of the screened interaction, each matrix element, when represented in the basis of Coulomb eigenfunctions, is multiplied by :math:{\sqrt{v_\mu v_\nu}} with the Coulomb eigenvalues :math:{\{v_\mu\}}. This gives an opportunity for reducing the basis-set size further by introducing a Coulomb cutoff :math:{v_\mathrm{min}}. The reduced basis set is then used for the polarization function, the dielectric function, and the screened interaction. The parameter :math:{v_\mathrm{min}} can be specified after the keyword OPTIMIZE MB in three ways: first, as a "pseudo" reciprocal cutoff radius :math:{\sqrt{4\pi/v_\mathrm{min}}} (which derives from the plane-wave Coulomb eigenvalues :math:{v_\mathbf{G}=4\pi/G^2}), second, directly as the parameter :math:{v_\mathrm{min}} by using a negative real number, and, finally, as the number of basis functions that should be retained when given as an integer. The so-defined basis functions are mathematically close to plane waves. For testing purposes, one can also enforce the usage of plane waves (or rather projections onto plane waves) with the keyword OPTIMIZE PW, in which case the Coulomb matrix is known analytically. No optimization of the basis is applied, if OPTIMIZE is omitted. +----------+-----------------------+-------------------------------------------------------------------------------------------------------------+ | Examples | OPTIMIZE MB 4.0 | Optimize the mixed product basis by removing eigenfunctions with eigenvalues below 4\pi/4.5^2. | ... ...
 ... ... @@ -104,7 +104,7 @@ CHKMISM MTACCUR ------- (*) The LAPW method relies on a partitioning of space into MT spheres and the interstitial region. The basis functions are defined differently in the two regions, interstitial plane waves in the latter and numerical functions in the spheres with radial parts :math:{u(r)}, {\dot{u}(r)=\partial u(r)/\partial\epsilon}, :math:{u^\mathrm{LO}(r)} and spherical harmonics :math:{Y_{lm}(\hat{\mathbf{r}})} The plane waves and the angular part of the MT functions can be converged straightforwardly with the reciprocal cutoff radius :math:{g_\mathrm{max}} and the maximal l quantum number :math:{l_\mathrm{max}}, respectively, whereas the radial part of the MT functions is not converged as easily. The standard LAPW basis is restricted to the functions :math:{u} and :math:{\dot{u}}. Local orbitals :math:{u^\mathrm{LO}} can be used to extend the basis set, to enable the description of semicore and high-lying conduction states. The accuracy of the radial MT basis can be analyzed with the keyword MTACCUR e1 e2 which gives the MT representation error [Phys. Rev. B 83, 081101] in the energy range between e1 and e2. (If unspecified, e1 and e2 are chosen automatically.) The results are written to the output files spex.mt.t where t is the atom type index, or spex.mt.s.t with the spin index s(=1 or 2) for spin-polarized calculations. The files contain sets of data for all l quantum numbers, which can be plotted separately with gnuplot (e.g., plot "spex.mt.1" i 3 for :math:{l=3} (*) The LAPW method relies on a partitioning of space into MT spheres and the interstitial region. The basis functions are defined differently in the two regions, interstitial plane waves in the latter and numerical functions in the spheres with radial parts :math:{u(r)}, {\dot{u}(r)=\partial u(r)/\partial\epsilon}, :math:{u^\mathrm{LO}(r)} and spherical harmonics :math:{Y_{lm}(\hat{\mathbf{r}})} The plane waves and the angular part of the MT functions can be converged straightforwardly with the reciprocal cutoff radius :math:{g_\mathrm{max}} and the maximal l quantum number :math:{l_\mathrm{max}}, respectively, whereas the radial part of the MT functions is not converged as easily. The standard LAPW basis is restricted to the functions :math:{u} and :math:{\dot{u}}. Local orbitals :math:{u^\mathrm{LO}} can be used to extend the basis set, to enable the description of semicore and high-lying conduction states. The accuracy of the radial MT basis can be analyzed with the keyword MTACCUR e1 e2 which gives the MT representation error [Phys. Rev. B 83, 081101] in the energy range between e1 and e2. (If unspecified, e1 and e2 are chosen automatically.) The results are written to the output files spex.mt.t where t is the atom type index, or spex.mt.s.t with the spin index s(=1 or 2) for spin-polarized calculations. The files contain sets of data for all l quantum numbers, which can be plotted separately with gnuplot (e.g., plot "spex.mt.1" i 3 for :math:{l=3}) +----------+------------------+------------------------------------------------------------+ | Examples | MTACCUR -1 2` | Calculate MT representation error between -1 and 2 Hartree | ... ...
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