:orphan: 1s core-ionization of N2 by TD-DFT ================================== Introduction ------------ We want to study the excitation of a 1s electron of the :math:`N_2` molecule to an empty orbital. More precisely we shall look at the excitation of an electron from the bonding :math:`1s\sigma_g` or anti-bonding :math:`1s\sigma_u` -orbitals to the vacant :math:`2p\pi_g` or :math:`2p\sigma_u` orbitals (see MO-diagram below). .. image:: N2_MOdiagram.jpg :scale: 40 :align: center :alt: N2_MOdiagram Note that this diagram does not take spin-orbit into account, but we shall consider this interaction later on. Let us first consider the possible final states. One electron leaves from one of four spin-orbitals and enters one of six spin-orbitals. This gives 24 determinants which translates into the following states: +-----------------------------------+------------------------+ | Configuration | States | +===================================+========================+ | :math:`1s\sigma_g^{-1}2p\pi_g` | :math:`^{1,3}\Pi_g` | +-----------------------------------+------------------------+ | :math:`1s\sigma_g^{-1}2p\sigma_u` | :math:`^{1,3}\Sigma_u` | +-----------------------------------+------------------------+ | :math:`1s\sigma_u^{-1}2p\pi_g` | :math:`^{1,3}\Pi_u` | +-----------------------------------+------------------------+ | :math:`1s\sigma_u^{-1}2p\sigma_u` | :math:`^{1,3}\Sigma_g` | +-----------------------------------+------------------------+ Spin-orbit free calculation --------------------------- Preparing the input files ~~~~~~~~~~~~~~~~~~~~~~~~~ We employ the following molecular input file `N2.mol` .. literalinclude:: N2.mol Here we do not provide any symmetry information, meaning that we ask DIRAC to detect it. DIRAC will find that the full group is :math:`D_{\infty h}`. With spin-orbit coupling DIRAC will then activate linear supersymmetry, but in the spin-orbit free case it will simply use the highest Abelian single point group, that is :math:`D_{2h}`. For the final states DIRAC will employ the *total* symmetry, that is the combined spin and spatial symmetry. Here we shall keep in mind that the singlet spin function is totally symmetric (:math:`A_g`), whereas the triplet spin functions transform as rotations. We know the triplet functions as: .. math:: T_{-1} = \alpha_1\alpha_2;\quad T_0=\frac{1}{\sqrt{2}}\left(\alpha_1\beta_2-\beta_1\alpha_2\right);\quad T_{+1}=\beta_1\beta_2 but for our purposes it will be more convenient to form the combinations: .. math:: T_x=\frac{1}{\sqrt{2}}\left(T_{-1}-T_{+1}\right);\quad T_y=\frac{i}{\sqrt{2}}\left(T_{-1}+T_{+1}\right);\quad T_z=T_0 which transform as rotations :math:`R_x\left(B_{3g}\right)` , :math:`R_y\left(B_{2g}\right)` and :math:`R_z\left(B_{1g}\right)` , respectively. We can now set up the following correlation of states: +------------------------+--------------------------------+------------------------+----------------------------------------------------------------------------+ | State | Spin | Spatial | Spin :math:`\otimes` Spatial | +========================+================================+========================+============================================================================+ | :math:`^1\Sigma_g` | :math:`A_g` | :math:`A_g` | :math:`A_g` | +------------------------+--------------------------------+------------------------+----------------------------------------------------------------------------+ | :math:`^1\Sigma_u` | :math:`A_g` | :math:`B_{1u}` | :math:`B_{1u}` | +------------------------+--------------------------------+------------------------+----------------------------------------------------------------------------+ | :math:`^1\Pi_{x,y;g}` | :math:`A_g` | :math:`B_{3g}, B_{2g}` | :math:`B_{3g}, B_{2g}` | +------------------------+--------------------------------+------------------------+----------------------------------------------------------------------------+ | :math:`^1\Pi_{x,y;u}` | :math:`A_g` | :math:`B_{3u}, B_{2u}` | :math:`B_{3u}, B_{2u}` | +------------------------+--------------------------------+------------------------+----------------------------------------------------------------------------+ | :math:`^3\Sigma_g` | :math:`B_{3g}, B_{2g}, B_{1g}` | :math:`A_g` | :math:`B_{3g}, B_{2g}, B_{1g}` | +------------------------+--------------------------------+------------------------+----------------------------------------------------------------------------+ | :math:`^3\Sigma_u` | :math:`B_{3g}, B_{2g}, B_{1g}` | :math:`B_{1u}` | :math:`B_{2u}, B_{3u}, A_{u}` | +------------------------+--------------------------------+------------------------+----------------------------------------------------------------------------+ | :math:`^3\Pi_{x,y;g}` | :math:`B_{3g}, B_{2g}, B_{1g}` | :math:`B_{3g}, B_{2g}` | :math:`\left(A_g, B_{1g}, B_{2g}\right), \left(B_{1g}, A_g, B_{3g}\right)` | +------------------------+--------------------------------+------------------------+----------------------------------------------------------------------------+ | :math:`^3\Pi_{x,y;u}` | :math:`B_{3g}, B_{2g}, B_{1g}` | :math:`B_{3u}, B_{2u}` | :math:`\left(A_u, B_{1u}, B_{2u}\right), \left(B_{1u}, A_u, B_{3u}\right)` | +------------------------+--------------------------------+------------------------+----------------------------------------------------------------------------+ Counting total symmetries we find the 24 microstates are evenly distributed amongst the eight irreps of :math:`D_{2h}`: +----------------+-------------------------------------------------------------+-----------------------+ | Irrep | Core-ionized state | | +================+=============================================================+=======================+ | :math:`A_g` | :math:`^1\Sigma_g, ^{3(x)}\Pi_{x;g}, ^{3(y)}\Pi_{y;g}` | :math:`x^2, y^2, z^2` | +----------------+-------------------------------------------------------------+-----------------------+ | :math:`B_{3u}` | :math:`^{1}\Pi_{x;u}, ^{3(y)}\Sigma_u, ^{3(z)}\Pi_{y;u}` | :math:`x` | +----------------+-------------------------------------------------------------+-----------------------+ | :math:`B_{2u}` | :math:`^{1}\Pi_{y;u}, ^{3(x)}\Sigma_u, ^{3(z)}\Pi_{x;u}` | :math:`y` | +----------------+-------------------------------------------------------------+-----------------------+ | :math:`B_{1g}` | :math:`^{3(z)}\Sigma_g, ^{3(y)}\Pi_{x;g}, ^{3(x)}\Pi_{y;g}` | :math:`xy` | +----------------+-------------------------------------------------------------+-----------------------+ | :math:`B_{1u}` | :math:`^1\Sigma_u, ^{3(y)}\Pi_{x;u}, ^{3(x)}\Pi_{y;u}` | :math:`z` | +----------------+-------------------------------------------------------------+-----------------------+ | :math:`B_{2g}` | :math:`^{1}\Pi_{y;g}, ^{3(y)}\Sigma_g, ^{3(z)}\Pi_{x;g}` | :math:`xz` | +----------------+-------------------------------------------------------------+-----------------------+ | :math:`B_{3g}` | :math:`^{1}\Pi_{x;g}, ^{3(x)}\Sigma_g, ^{3(z)}\Pi_{y;g}` | :math:`yz` | +----------------+-------------------------------------------------------------+-----------------------+ | :math:`A_{u}` | :math:`^{3(z)}\Sigma_u, ^{3(x)}\Pi_{x;u}, ^{3(y)}\Pi_{y;u}` | :math:`xyz` | +----------------+-------------------------------------------------------------+-----------------------+ From these considerations we now set up the following menu file for our calculation .. literalinclude:: N2spf.inp In the :ref:`*SCF` section we give the electron occupation of :math:`N_2`: 6 and 8 electrons in *gerade* and *ungerade* orbitals, respectively. We also ask for a Mulliken population analysis (:ref:`ANALYZE_.MULPOP`) for the occupied orbitals and the orbitals involved in the core excitation. Let us now look at how we set up the calculation of excitation energies under `*EXCITATION ENERGIES`. We have seen that there are three excitations per boson irrep. Note that the numbering of irreps follow what you for instance find in the :math:`D_{2h}` direct product table in the output:: | | Ag B3u B2u B1g B1u B2g B3g Au -----+---------------------------------------- Ag | Ag B3u B2u B1g B1u B2g B3g Au B3u | B3u Ag B1g B2u B2g B1u Au B3g B2u | B2u B1g Ag B3u B3g Au B1u B2g B1g | B1g B2u B3u Ag Au B3g B2g B1u B1u | B1u B2g B3g Au Ag B3u B2u B1g B2g | B2g B1u Au B3g B3u Ag B1g B2u B3g | B3g Au B1u B2g B2u B1g Ag B3u Au | Au B3g B2g B1u B1g B2u B3u Ag Note also that we skip excitations in :math:`B_{2u}` and :math:`B_{2g}`, since they are related by symmetry to the excitations of :math:`B_{3u}` and :math:`B_{3g}`, respectively. If nothing further is specified the excitation energies are calculated by a "bottoms-up" approach and so we will get valence excitations only, since the core-excitations are much higher in energy. We therefore restrict the excitations to the occupied :math:`1s\sigma_g` and :math:`1s\sigma_u` orbitals. We furthermore ask for transition moments to be calculated with respect to the component of the dipole moment operator. These will be non-zero only for excitations in irreps :math:`B_{3u}, B_{2u}` and :math:`B_{1u}`. Finally we ask for analysis of what orbitals contribute to the various excitations. For this the Mulliken population analysis may come in handy as reference. Looking at the output ~~~~~~~~~~~~~~~~~~~~~ After running the calculation, let us now look at the output. The following excitation energies were calculated .. literalinclude:: N2spf_exc.txt DIRAC assumes that excitation energies that are within :math:`10^{-9}\ E_h` of each other come from the same degenerate state. This threshold is somewhat arbitrary and we shall see that DIRAc is not always correct. There is sufficient symmetry in the calculation (symmetry distinct rotation) to allow DIRAC to pinpoint the symmetry of the core-ionized state and we therefore find the following distribution .. literalinclude:: N2spf_exc2.txt The first and second block refers to singlet and triplet states, respectively. Based on the discussion in the preceeding section we see that levels 1, 2 and 3 all come from a :math:`^3\Pi_u` which in :math:`D_{2h}` splits into :math:`^3B_{2u}` and :math:`^3B_{3u}`. After careful inspection we can set up the following table +-------+-----------------+--------------------+ | Level | eigenvalue (eV) | | +=======+=================+====================+ | 0 | 0.000 | :math:`^1\Sigma_g` | +-------+-----------------+--------------------+ | 1,2,3 | 388.780 | :math:`^3\Pi_u` | +-------+-----------------+--------------------+ | 4,5,6 | 388.831 | :math:`^3\Pi_g` | +-------+-----------------+--------------------+ | 7 | 389.916 | :math:`^1\Pi_u` | +-------+-----------------+--------------------+ | 8 | 389.936 | :math:`^1\Pi_g` | +-------+-----------------+--------------------+ | 9,10 | 399.834 | :math:`^3\Sigma_g` | +-------+-----------------+--------------------+ | 11,12 | 399.876 | :math:`^3\Sigma_u` | +-------+-----------------+--------------------+ | 13 | 400.519 | :math:`^1\Sigma_g` | +-------+-----------------+--------------------+ | 14 | 400.568 | :math:`^1\Sigma_u` | +-------+-----------------+--------------------+ Looking further down in the output we find dominant inactive and virtual orbitals. Restricting attention to :math:`B_{3u}` total symmetry we find that the first excited state :math:`^3\Pi_u`, at 388.78 eV, is dominated by the excitation `1(i:E1u) ---> 4(v:E1g)`, which, as can be inferred from the Mulliken population analysis, corresponds to :math:`1s\sigma_u \rightarrow 2p\pi_{y;g}`. The second excited state :math:`^1\Pi_u`, at 389.91 eV, corresponds to `1(i:E1u) ---> 5(v:E1g)` (:math:`1s\sigma_u \rightarrow 2p\pi_{x;g}`), whereas the third excited state :math:`^3\Sigma_u`, at 399.88 eV, is dominated by `1(i:E1g) ---> 5(v:E1u)` (:math:`1s\sigma_g \rightarrow 2p\sigma_u`). Within the electric dipole approximation only singlet states get oscillator strengths. In the output we find .. literalinclude:: N2spf_osc.txt showing intensity to the :math:`^1\Pi_u` and :math:`^1\Sigma_u` states. Including spin-orbit -------------------- Spin-orbit is included by simply commenting out the keyword :ref:`HAMILTONIAN_.SPINFREE` in the input above:: **HAMILTONIAN !.SPINFREE This leads to the following states: .. literalinclude:: N2so_exc.txt with the following distribution on linear symmetries: .. literalinclude:: N2so_exc2.txt Comparing with the preceeding section we see the following spin-orbit decomposition of the :math:\Lambda-S` states: +-------+-----------------+----------------------+------------------------------------------------------------------------------------------------+ | Level | eigenvalue (eV) | | | +=======+=================+======================+================================================================================================+ | 0 | 0.000 | :math:`^1\Sigma^+_g` | :math:`0_g^+` (0.000) | +-------+-----------------+----------------------+------------------------------------------------------------------------------------------------+ | 1,2,3 | 388.780 | :math:`^3\Pi_u` | :math:`0^+_u` (388.771), :math:`0^-_u` (388.771), :math:`1_u` (388.780), :math:`2_u` (388.788) | +-------+-----------------+----------------------+------------------------------------------------------------------------------------------------+ | 4,5,6 | 388.831 | :math:`^3\Pi_g` | :math:`0^+_g` (388.821), :math:`0^-_g` (388.822), :math:`1_g` (388.830), :math:`2_g` (388.839) | +-------+-----------------+----------------------+------------------------------------------------------------------------------------------------+ | 7 | 389.916 | :math:`^1\Pi_u` | :math:`1_u` (389.916) | +-------+-----------------+----------------------+------------------------------------------------------------------------------------------------+ | 8 | 389.936 | :math:`^1\Pi_g` | :math:`1_g` (389.936) | +-------+-----------------+----------------------+------------------------------------------------------------------------------------------------+ | 9,10 | 399.834 | :math:`^3\Sigma^+_g` | :math:`0^-_g` (399.834), :math:`1_g` (399.834) | +-------+-----------------+----------------------+------------------------------------------------------------------------------------------------+ | 11,12 | 399.876 | :math:`^3\Sigma^+_u` | :math:`0^-_u` (399.876), :math:`1_u` (399.876) | +-------+-----------------+----------------------+------------------------------------------------------------------------------------------------+ | 13 | 400.519 | :math:`^1\Sigma^+_g` | :math:`0^+_g` (400.519) | +-------+-----------------+----------------------+------------------------------------------------------------------------------------------------+ | 14 | 400.568 | :math:`^1\Sigma^+_u` | :math:`0^+_u` (400.568) | +-------+-----------------+----------------------+------------------------------------------------------------------------------------------------+ Note that the energies are given relative to the lowest level, that is, the ground state and that it is somewhat stabilized by spin-orbit coupling. The effect of spin-orbit coupling shows up in the oscillator strengths: .. literalinclude:: N2so_osc.txt What we see is the :math:`0_u^+` and :math:`1_u` components of the :math:`^3\Pi_u` state stealing intensity from the singlet states. This change is not very spectacular since the nitrogen molecule is composed of light atoms for which relativistic effects are not very strong. We can mimic a more strongly relativistic system by reducing the speed of light to e.g. 20 a.u.:: **GENERAL .CVALUE 20.0D0 We now see .. literalinclude:: N2so20_osc.txt Simulating the core-excitation spectrum using complex response -------------------------------------------------------------- `Gas Phase Core Excitation Database `_