.. tip::
    All input files can be downloaded: :download:`Files <http://www.qbics.info/input_library/tutorials-msdft3.tar.gz>`.

.. tip::
    For more information of this section, please refer to these pages:

    - :doc:`./tso1`
    - :doc:`./msdft1`
    - :doc:`./msdft2`
    - :doc:`./msdft4`
    - :doc:`../keywords/nosi`
    - :doc:`../keywords/msdft`

MSDFT (3): Double Excitations
================================================================================

.. contents::
   :local:

This tutorial will lead you step by step to study excited states using Multi-State Density Functional Theory (MSDFT).

MSDFT is a powerful method for studying excited states. It optimizes excited states using TSO-DFT method, so it is **free of orbital relaxation problem** like in TDDFT. In this section, you will see that MSDFT can give **much more reasonable** results than TDDFT for excited states.

In :doc:`tso1`, we have introduced how to use TSO-DFT to study excited states. In :doc:`msdft1`, we have introduced how to use MSDFT to study excitations generally. In :doc:`msdft2`, we have introduced how to use MSDFT to study core excitations. In this tutorial, we will introduce how to use MSDFT to study double excitations. We strongly recommend you to read these two tutorials before reading this tutorial.

Actually, MSDFT can be considered as TSO+NOSI. MSDFT is a framework that automaitcally performs TSO-DFT and NOSI for required excited states. Of course, you can also perform TSO-DFT and NOSI separately for special purposes.

Example: Double Excitations of Glyoxal
----------------------------------------------------------

Below is the input file for calculating double excitations of glyoxal:

.. code-block::
    :caption: msdft-3.inp
    :linenos:

    basis
        cc-pVTZ
    end
    
    scf
        charge  0
        spin2p1 1
        type    U
    end
    
    mol
        C  0.642211002  0.401329163 0.
        C -0.642211002 -0.401329163 0.
        O  1.722902738 -0.139984242 0.
        O -1.722902738  0.139984242 0.
        H  0.508726009  1.491661991 0.
        H -0.508726009 -1.491661991 0.
    end
    
    msdft
        double_ex 15 : 16 17
    end
    
    task
        msdft b3lyp
    end
    
- ``msdft...end`` indicates the MSDFT calculation. Here, ``double_ex 15 : 16 17`` means we want to study the double excitations from orbitals 15 to orbitals 16 and 17. Explicitly, we want to study the following double excitations:

  - 15 (HOMO) → 16 (LUMO)
  - 15 (HOMO) → 17 (LUMO+1)

In the output file, we can find the following information:

Below is the input file for calculating double excitations of glyoxal:

.. code-block::
    :caption: msdft-3.out
    :linenos:
    :emphasize-lines: 6,7,18,19

    ---- NOSI Results ----
    ======================
       State   NOSI Energies  Excited Energy       Osc. Str.        DX        DY        DZ
                   (Hartree)            (eV)                    (a.u.)    (a.u.)    (a.u.)
           0   -227.91006738      0.00000000      0.00000000  -0.00002  -0.00001   0.00000
           1   -227.72694413      4.98278376      0.00000000   0.00000   0.00000  -0.00000
           2   -227.36327349     14.87826196      0.00000000  -0.00001  -0.00000   0.00000
    
    ---- NOSI State Identification (Coefficients) ----
    ==================================================
    State |0> = -0.999 |msdft-3-gs.mwfn>
    State |1> = -0.952 |msdft-3-15-to-16-de.mwfn> +0.303 |msdft-3-15-to-17-de.mwfn>
    State |2> = +0.302 |msdft-3-15-to-16-de.mwfn> +0.953 |msdft-3-15-to-17-de.mwfn>
    
    ---- NOSI State Identification (Weights) ----
    =============================================
    State |0> = 0.997 |msdft-3-gs.mwfn>
    State |1> = 0.906 |msdft-3-15-to-16-de.mwfn> 0.092 |msdft-3-15-to-17-de.mwfn>
    State |2> = 0.091 |msdft-3-15-to-16-de.mwfn> 0.908 |msdft-3-15-to-17-de.mwfn>
    

From the highlighted lines, for state 1 (2), the weight of double excitation from orbitals 15 to orbitals 16 (17) is 0.906 (0.908), so state 1 (2) can be identified as the double excitation from orbitals 15 to orbitals 16 (17). For (15)\ :sup:`2` → (16)\ :sup:`2`, i.e. (HOMO)\ :sup:`2` → (LUMO)\ :sup:`2`, the excited energy is 4.98 eV. The theoretical best estimate for this excitation is 5.54 eV(J. Chem. Theory Comput. 2019, 15, 1939). Interestingly, if we only use TSO-DFT to study this excitation, the excited energy is 5.86 eV (see Line 1020 in ``msdft-3.out``).

Summarizing the results, we can list them below:

.. list-table::
      
   * - State
     - TSO-DFT
     - MSDFT
     - Theoretical Best Estimate
   * - (15)\ :sup:`2` → (16)\ :sup:`2`
     - 5.86 eV
     - 4.98 eV
     - 5.54 eV
   * - (15)\ :sup:`2` → (17)\ :sup:`2`
     - 13.96 eV
     - 14.88 eV
     - N/A