Tip

All input and output files can be downloaded `here`

.

# 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:

```
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

```
$ 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:

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 |

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:

```
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

`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.

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:

```
$ 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

```
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: \(-104.06138400-(-114.55175795) = 10.490374\) Hartree, i.e., 285.45 eV. For comparison:

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 \(n^2\rightarrow (\pi^*)^2\), i.e. 2 electrons from MO 8 are excited to MO 9. The input file is:

```
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:

```
$ qbics hcho-de.inp -n 4 > hcho-de.out
```

The energy is -114.15907187 Hartree, so the double excitation energy is: \(-114.15907187-(-114.55175795) = 0.39268\) Hartree, i.e., 10.68 eV. For comparison:

Method |
\(n^2\rightarrow (\pi^*)^2\) 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.