:orphan: Core ionization in the CO and N2 molecules ========================================== Introduction ============ We would like to study K-edge energies of CO and N2 molecules. Carbon monoxide =============== A first approximation to the O1s and C1s ionization energies is provided by Koopmans' theorem. Starting from the molecular input file CO.mol .. literalinclude:: CO.mol and the menu file CO.inp .. literalinclude:: CO.inp we first run a Hartree-Fock calculation of the ground electronic state of carbon monoxide:: pam --inp=CO --mol=CO --get "DFCOEF=cf.CO" We find that :math:-\varepsilon (O1s)=20.6838\,E_h=562.8 eV and :math:-\varepsilon (C1s)=11.3672 \,E_h=309.3 eV. This is significantly off the values of **542.1 eV** and **295.9 eV**, respectively, reported by Kai Siegbahn and co-workers :cite:Siegbahn1969 . Koopmans' theorem work quite well for valence excitations because the two major sources, that is, i) lack of electron correlation and ii) lack of orbital relaxation, typically are of about the same magnitude, but of opposite sign and hence tend to cancel out. This is not the case for core ionization, where orbital relaxation is very much more important than electron correlation. To introduce orbital relaxation of the core-ionized state we carry out a :math:\Delta SCF calculation, as pioneered by Paul Bagus and Henry F. Schaefer :cite:Bagus_JCP1971. In order to calculate the O1s ionized system we set up the menu file CO_O1s.inp .. literalinclude:: CO_O1s.inp and use the command:: pam --inp=CO_O1s --mol=CO --put "cf.CO=DFCOEF" It is seen that we start from the coefficients of the neutral CO molecule. We then converge to the core-ionized system using reordering and overlap selection: In the input we have specified twelve electrons in six inactive (closed) orbitals, followed by a single electron in an active (open) orbital. We want the O1s to be the active orbital, but this is not achieved automatically since DIRAC will normally order orbitals according to their energy. We therefore start be reordering the orbitals such that the O1s orbital from the previous calculation on the neutral system comes out on top of the occupied orbitals. However, this is not enough to converge to the desired state since after the first diagonalization DIRAC will again by default order orbitals according to their energy. This is why we use *overlap selection*, that is, we ask DIRAC to rather order orbitals according to their overlap with some reference orbitals. By default (dynamic overlap selection) this will be the orbitals from the previous iteration. However, in this case we activate *non-dynamic* overlap selection, which means that we order orbitals according to their overlap with the starting orbitals. The analogous menu file for C1s ionization is CO_C1s.inp .. literalinclude:: CO_C1s.inp which puts the C1s on top of the occupied orbitals, and then use the command:: pam --inp=CO_C1s --mol=CO --put "cf.CO=DFCOEF" The HF energies of the neutral, the O1s-ionized and the C1s-ionized systems are -112.857912 :math:E_h, -92.939954 :math:E_h and -101.933919 :math:E_h, respectively, from which we calculate IP(O1s) = 19.9180 :math:E_h = 542.0 eV and IP(O1s) = 10.9240 :math:E_h = 297.3 eV, which agrees remarkably well with the experimental values, in particular for the oxygen K-edge. .. note:: Overlap selection is nowadays marketed hard as MOM (Maximum Orbital Method, see :cite:Gilbert_JPCA2008), but this method has been included in DIRAC for at least two decades and goes back to the pioneering work of Paul Bagus _ It was used in :cite:Bagus_JCP1971, but not reported explicitly. However, it is for instance documented in the 1975 manual of the ALCHEMY program _ (On pdf page 15 you find a description of keyword MOORDR using a "maximum overlap criterion"). Nitrogen molecule ================= We now switch to the isoelectronic nitrogen molecule. Starting from the molecular input file N2.mol .. literalinclude:: N2.mol and the menu file N2.inp .. literalinclude:: N2.inp we run a Hartree-Fock calculation of the ground electronic state of the nitrogen molecule:: pam --inp=N2 --mol=N2 --get "DFCOEF=cf.N2 DFACMO=ac.N2" (note that we are also getting the :math:C_1 coefficients; we shall come back to that). We find that :math:-\varepsilon (1s\sigma_g)=15.6942\,E_h=427.1 eV and :math:-\varepsilon (1s\sigma_u)=15.6907 \,E_h=427.0 eV. This is significantly off the value of **409.9 eV** reported by Kai Siegbahn and co-workers :cite:Siegbahn1969 ; the :math:1s\sigma_g and :math:1s\sigma_u are indistinguishable in the X-ray PES study (see also :cite:Wight_JEPRP1972). We therefore proceed to :math:\Delta SCF calculations: for the :math:1s\sigma_g ionization we use the menu file N2_Kg.inp .. literalinclude:: N2_Kg.inp and the command:: pam --inp=N2_Kg --mol=N2 --put "cf.N2=DFCOEF" and equivalent menu file N2_Ku.inp for the :math:1s\sigma_u ionization: .. literalinclude:: N2_Ku.inp The HF energies of the neutral, the :math:1s\sigma_g -ionized and the :math:1s\sigma_u -ionized systems are -109.051015 :math:E_h, -93.626378 :math:E_h and -93.630989 :math:E_h, respectively, from which we calculate IP(:math:1s\sigma_g) = 15.4246 :math:E_h = 419.7 eV and IP(:math:1s\sigma_u) = 15.4200 :math:E_h = 419.6 eV. However, *this is still far from the experimental value* of 409.9 eV ! Bagus and Schaefer found a similar discrepancy in their study of the :math:O_2^+ molecule :cite:Bagus_JCP1972 and found that results improved dramatically by *localization* of the core hole, so let us try this. Localization breaks molecular symmetry, so we go down to :math:C_1 using the molecule input file N2_C1.mol .. literalinclude:: N2_C1.mol This is the reason why we saved the :math:C_1 coefficients in the initial calculation ! We next set up the menu file N2loc.inp .. literalinclude:: N2loc.inp for Pipek-Mezey localization. Note that we select only the a little trick: We select only the :math:1s\sigma_g and :math:1s\sigma_u orbitals for localization. We now localize using the command:: pam --inp=N2loc --mol=N2_C1 --put "ac.N2=DFCOEF" --get "DFCOEF=ac.N2loc" We calculate the system with a localized core hole using the menu file N2_K.inp .. literalinclude:: N2_K.inp and the command:: pam --inp=N2_K --mol=N2_C1 --put "ac.N2loc=DFCOEF" In fact, we can do slightly better because we can go up to :math:C_{2v} symmetry using that DIRAC allows the import of coefficients in :math:C_1 symmetry from the unformatted file DFACMO to current symmetry. This assumes that the current symmetry is lower than the symmetry used for obtaining the original coefficients, which is the case here. We now calculate the system with a localized core hole using the menu file N2_KC2v.inp .. literalinclude:: N2_KC2v.inp the molecule input file N2_C2v.mol .. literalinclude:: N2_C2v.mol and the command:: pam --inp=N2_KC2v --mol=N2_C2v --put "ac.N2loc=DFACMO" The calculated energy of the core ionized system is -93.971190 :math:E_h and we now obtain an ionization energy of 15.0978 :math:E_h = 410.3 eV, in much better agreement with experiment. The price we had to pay was to break symmetry. It is possible to keep symmetry, but now at the price of going to more complex wave functions, as shown by Agren, Bagus and Roos :cite:Agren_CPL1981.