.. tip:: All input and output files can be downloaded :download:`here <_static/tso-basic/tso-basic.zip>`. TSO-DFT for Excited State Energies ========================================= TSO-DFT (TSO = target state optimization) is an originally developed powerful method for calculating electronic excited and diabatic states. It is a **single-determinant** method so you can study an excited or diabatic state very efficiently. For charge transfer, core, and doubly excitations, **TSO-DFT outperforms TDDFT significantly!** In this tutorial, we will describe how to compute electronic excited states for molecules. .. tip:: For details and performance of TSO-DFT, please refer to the following paper. This should also be **cited** if you use TSO-DFT in your research work. - Zhang, J.; Tang, Z.; Zhang, X.; Zhu, H.; Zhao, R.; Lu, Y.; Gao, J. `Target State Optimized Density Functional Theory for Electronic Excited and Diabatic States. `_ *J. Chem. Theory Comput.* **2023**, *19*, 1777-1789. TSO-DFT is a highly flexible method for excitation and diabatization. You can realize any electronic state with orbital subspace partition. Core Excitated State ---------------------------------- We consider the core excitation of formaldehyde. We use this as the first example because it is simple for us to understand how TSO-DFT works but also an "excellent" example where TDDFT completely fails! Ground State +++++++++++++++++++++++++++++++++++++ First, a ground state calculation is carried out using the following input: .. code-block:: :caption: hcho-gs.inp basis element H cc-pVTZ C cc-pCVTZ O cc-pCVTZ end scf charge 0 spin2p1 1 end mol C -0.000756 -0.520733 0. H 0.935697 -1.111766 0. H -0.939631 -1.107897 0. O 0.001792 0.678123 0 end task energy b3lyp end After calculation by .. code-block:: bash $ qbics hcho-gs.inp -n 4 > hcho-gs.out we get a ground state wave function ``hcho-gs.mwfn``. Let's visualize its orbitals with Multiwfn, we will find this: .. list-table:: :header-rows: 1 * - Index - Occupation - Property * - 2 - doubly occupied - 1s core orbital of C1 * - 7 - doubly occupied - π bonding orbital * - 8 - doubly occupied - n nonbonding orbital * - 9 - unoccupied - π* anti-bonding orbital .. image:: _static/tso-basic/p1.png :align: center We can also know from output file ``hcho-gs.out`` that there are totally 16 electrons and 114 basis functions, and the ground state energy is -114.55175795 Hartree. C 1s→π* Excited State +++++++++++++++++++++++++++++++++++++ Now we want to study the state when one 1s electron of carbon is excited to a π* orbital, which is MO 2 and 9, respectively. In this case, the powerful TSO-DFT should be used, with the following input: .. code-block:: :caption: hcho-c1se.inp basis element H cc-pVTZ C cc-pCVTZ O cc-pCVTZ end scf charge 0 spin2p1 1 type U # For TSO-DFT, unrestricted SCF is preferred. do_tso end scfguess type mwfn file hcho-gs.mwfn orb 16 1 1-113 : 1 3-114 orb 0 1 114 : 2 end mol C -0.000756 -0.520733 0. H 0.935697 -1.111766 0. H -0.939631 -1.107897 0. O 0.001792 0.678123 0 end task energy b3lyp end Now we will explain the key points in ``hcho-c1se.inp``: - ``do_tso`` This option in ``scf ... end`` block will turn on TSO calculations. - ``type U`` For TSO-DFT, it is preferred to using **unrestricted** SCF. - A reference state is needed. In this case, the ground state of formaldehyde, so the initial guess in ``scfguess ... end`` block should be ``type mwfn`` and ``file hcho-gs.mwfn``. .. attention:: TSO-DFT can only use 2 kind of initial guess: ``type mwfn`` and ``type fragden``. The latter will be useful in diabatic state studies. ``orb`` is the most important keyword in TSO calculations. There can be arbitrary number of ``orb``, meaning that orbitals are partitioned into several subspaces. Orbitals from different subspaces will not mix. The format of ``orb`` is .. centered:: ``orb num_electrons spin_multiplicity alpha_MO_indices : beta_MO_indices`` For example, ``orb 16 1 1-113 : 1 3-114`` means that in this subspace, there are ``16`` electrons, the spin multiplicity is ``1``. The alpha orbitals are 1,2,3,...,113, and the beta orbital are 1,3,...,114. So, ``orb 0 1 114 : 2`` defines another subspace, which has ``0`` electrons, the alpha orbitals is ``114``, and the beta orbitals is only ``2``. Why do we partition the orbitals into 2 subspaces like this? Let's see the figure below. .. image:: _static/tso-basic/p2.png :align: center The first and second subspace are rendered by black and red color, respectively. Since beta orbital ``2`` is removed, the aufbau occupation of 15 electrons will automatically skip the C 1s orbital, so the π* orbital, i.e. MO 9, is also occupied automatically. However, **the number of alpha and beta orbitals must be identical,** so one can remove the highest alpha orbital, which is ``114``. These remaining orbitals will be collected to form another subspace, none of which is occupied. In this case, a C 1s→π* excited state is successfully constructed. Let's run this calculation: .. code-block:: bash $ qbics hcho-c1sex.inp -n 4 > hcho-c1sex.out We can get the C 1s ionized state wave function ``hcho-c1sex.mwfn`` and output file ``hcho-c1sex.out``. We can find that .. code-block:: :caption: hcho-c1sex.out Molecular Orbitals ================== k = Gamma Alpha HOMO-LUMO (8-9) gap: 5.827 eV Beta HOMO-LUMO (8-9) gap: 6.141 eV Alpha Alpha Beta Beta # Occupancies Energies/Hartree Occupancies Energies/Hartree 1 1.000 -19.13049258 1.000 -19.14916538 2 1.000 -12.32577074 1.000 -1.11992044 3 1.000 -1.10653786 1.000 -0.70203028 4 1.000 -0.70362987 1.000 -0.55634086 5 1.000 -0.56613413 1.000 -0.49715471 6 1.000 -0.48311622 1.000 -0.48046027 7 1.000 -0.41628076 1.000 -0.28713576 8 1.000 -0.27184459 1.000 -0.20773869 9 0.000 -0.05770604 0.000 0.01794976 10 0.000 0.01275050 0.000 0.07049694 Final total energy: -104.06138400 Hartree From molecular orbitals, we can find that one C 1s orbital is indeed excited, and the energy is -104.06138400 Hartree. Now the C state energy of formaldehyde is: :math:`-104.06138400-(-114.55175795) = 10.490374` Hartree, i.e., 285.45 eV. For comparison: .. list-table:: :header-rows: 1 * - Method - C 1s→π* Excitation Energy * - TSO-B3LYP - 285.5 eV * - TD-B3LYP - 275.2 eV * - Experiment - 286.0 eV Obviously, TSO give excellent results! Doubly Excited States ---------------------------------- Doubly excited state means that two electrons are excited simultaneously from the ground state. Popular TDDFT implemented with adiabatic approximation **cannot** be used to study double excited states. However, this can be easily done with TSO. For formaldehyde, we consider the doubly excited state :math:`n^2\rightarrow (\pi^*)^2`, i.e. 2 electrons from MO 8 are excited to MO 9. The input file is: .. code-block:: :caption: hcho-de.inp basis element H cc-pVTZ C cc-pCVTZ O cc-pCVTZ end scf charge 0 spin2p1 1 type U # For TSO-DFT, unrestricted SCF is preferred. do_tso end scfguess type mwfn file hcho-gs.mwfn orb 16 1 1-7 9-114 : 1-7 9-114 orb 0 1 8 : 8 end mol C -0.000756 -0.520733 0. H 0.935697 -1.111766 0. H -0.939631 -1.107897 0. O 0.001792 0.678123 0 end task energy b3lyp end We can see that, `orb 16 1 1-7 9-114 : 1-7 9-114` defines a subspace that both alpha and beta MO 8 are removed, so the last alpha and beta electrons will automatically occupy MO 9. The double excitation is achieved. Run the calculation: .. code-block:: bash $ qbics hcho-de.inp -n 4 > hcho-de.out The energy is -114.15907187 Hartree, so the double excitation energy is: :math:`-114.15907187-(-114.55175795) = 0.39268` Hartree, i.e., 10.68 eV. For comparison: .. list-table:: :header-rows: 1 * - Method - :math:`n^2\rightarrow (\pi^*)^2` Excitation Energy * - TSO-B3LYP - 10.68 eV * - EOM-CC - 10.34 eV Obviously, our TSO-DFT is highly accurate! .. tip:: For a theoretical explaination of the excellent performance of TSO-DFT, please refer to the TSO paper: - Zhang, J.; Tang, Z.; Zhang, X.; Zhu, H.; Zhao, R.; Lu, Y.; Gao, J. `Target State Optimized Density Functional Theory for Electronic Excited and Diabatic States. `_ *J. Chem. Theory Comput.* **2023**, *19*, 1777-1789.