# AH-FC: Adiabatic Hessian Franck-Condon¶

Electronic spectra, such as absorption, emission, phosphorescence, and ionization, may contain a vibrational structure.

In the Adiabatic Hessian Franck-Condon (AH-FC) method one needs to do a frequency calculation both at the ground state as well as the excited state of interest. The model makes the following assumptions:

- 1. It employs the adiabatic (Born-Oppenheimer) approximation and treats the nuclei as moving in an effective potential defined by the electronic configuration.
- 2. It works at the Franck-Condon point and assumes that the transition occurs at the ground state equilibrium structure for absorption and at the excited state equilibrium structure for emission.
- 3. It applies the harmonic approximation to both the ground and excited state potential energy surfaces.

Franck-Condon factors can be calculated for the transition between the two electronic states, which can be done with the FCF Module, described below. These Franck-Condon factor can then be used to predict the relative intensities of absorption or emission lines in the electronic spectra. Note that the Herzberg-Teller effect is not taken into account.

In the vertical gradient Franck-Condon (VG-FC) method some extra assumptions are made compared to the AH-FC method. This approach is particularly effective for large molecules as it shows linear scaling with the number of normal modes.

- See the documentation of the VG-FC method.

## FCF module: Franck-Condon Factors¶

*fcf* is an auxiliary program which can be used to calculate Franck-Condon factors from two vibrational mode calculations [1].

*fcf* requires an ASCII input file where the user specifies the .rkf files from two vibrational mode calculations, carried out for two different electronic, spin or charge states of the same molecule.
These calculations can be either numerical or analytical. The number of vibrational quanta that have to be taken into account for both states in the evaluation of the Franck-Condon factors have to be specified.

*fcf* produces a (binary) KF file TAPE61, which can be inspected using the KF utilities. Furthermore, *fcf* writes the frequencies, vibrational displacements and electron-phonon couplings for both states too the standard output, including any error messages.

### Theory¶

Franck-Condon factors are the squares of the overlap integrals of vibrational wave functions. Given a transition between two electronic, spin or charge states, the Franck-Condon factors represent the probabilities for accompanying vibrational transitions. As such, they can be used to predict the relative intensities of absorption or emission lines in spectroscopy or excitation lines in transport measurements.

When a molecule makes a transition to another state, the equilibrium position of the nuclei changes, and this will give rise to vibrations. To determine which vibrational modes will be active, we first have to express the displacement of the nuclei in the normal modes:

Here, \(\mathbf{k}\) is the displacement vector, \(\mathbf{L}\) is the normal mode matrix, \(\mathbf{m}\) is a matrix with the mass of the nuclei on the diagonal, \(\mathbf{B}\) is the zero-order axis-switching matrix and \(\mathbf{x}_0\) is the equilibrium position of the nuclei. For free molecules, depending on symmetry constraints, the geometries of both states may have been translated and/or rotated with respect to each other. To remove the six translational and rotational degrees of freedom, we can center the equilibrium positions around the center of mass and rotate one of the states to provide maximum overlap. The latter is included with the zero-order axis-switching matrix \(\mathbf{B}\), implemented according to [2].

When we have obtained the displacement vector, it is trivial to calculate the dimensionless electron-phonon couplings. They are given by:

Here, \(\mathbf{\Gamma} = 2 \pi \mathbf{\omega} / h\) is a vector containing the reduced frequencies. [3]. The Huang-Rhys factor \(\mathbf{g}\) is related to \(\mathbf{\lambda}\) as:

The reorganization energy per mode is

When the displacement vector \(\mathbf{k}\), the reduced frequencies \(\mathbf{\Gamma}\) and \(\mathbf{\Gamma}'\), and the Duschinsky rotation matrix \(\mathbf{J} = \mathbf{L'}^T \mathbf{B}_0 \mathbf{L}\) have been obtained, the Franck-Condon factors can be calculated using the two-dimensional array method of Ruhoff and Ratner [3].

There is one Franck-Condon factor for every permutation of the vibrational quanta over both states. Since they represent transition probabilities, all Franck-Condon factors of one state which respect to one vibrational state of the other state must sum to one. Since the total number of possible vibrational quanta, and hence the total number of permutations, is infinite, in practice we will calculate the Franck-Condon factors until those sums are sufficiently close to one. Since the number of permutations rapidly increases with increasing number of vibrational quanta, it is generally possible to already stop after the sum is greater than about two thirds. The remaining one third will be distributed over so many Franck-Condon factors that their individual contributions are negligible.

In the limiting case of one vibrational mode, with the same frequency in both states, the expression for the Franck-Condon factors of transitions from the ground vibrational state to an excited vibrational state are given by the familiar expression:

### Input¶

The input for *fcf* is keyword oriented and is read from the standard input. *fcf* recognizes several keywords, but only two have to be specified to perform the calculation. All input therefore contains at least two lines of the following form:

```
$AMSBIN/fcf << eor
STATES state1 state2
QUANTA l1 l2
eor
```

Documentation for all keys of *fcf*:

```
$AMSBIN/fcf << eor
States state1 state2
Quanta l1 l2
Lambda float
Modes first last
Rotate [True | False]
Translate [True | False]
Spectrum FreqMin FreqMax NFreq
eor
```

`Lambda`

Type: Float Default value: 0.0 Description: Optional. The minimum value of the electron-phonon coupling for a mode to be taken into account in the calculation. Together with the MODES option, this provides a way to significantly reduce the total number of Franck-Condon factors. As with the MODES option, always check if the results do not change too much.

`Modes`

Type: Integer List Default value: [0, 0] Description: Optional. The first and last mode to be taken into account in the calculation. By default, all modes are taken into account. This option can be used to effectively specify and energy range for the Franck-Condon factors. When using this options, always check if the results (electron-phonon couplings, ground state to ground overlap integral, average sum of Franck-Condon factors, etc.) do not change too much.

`Quanta`

Type: Integer List Default value: [0, 0] Description: The maximum number of vibrational quanta to be taken into account for the two states. Franck-Condon factors will be calculated for every permutation of up to and including l1/l2 quanta over the vibrational modes.

`Rotate`

Type: Bool Default value: No Description: Optional, recommended to be included. Rotate the geometries to maximize the overlap of the nuclear coordinates.

`Spectrum`

Type: Float List Description: Optional. If included, the vibrational spectrum is calculated. A histogram of the spectrum is calculated for the frequency range defined by three numbers: FreqMin (minimum frequency for which the spectrum is calculated), FreqMax (maximum frequency for which the spectrum is calculated), NFreq (number of frequencies for which the spectrum is calculated). Usage example: ‘Spectrum 0.0 2.0E4 1000’

`States`

Type: String Description: The filenames of two results files of a numerical or analytical frequency calculation. The calculations must have been performed on the same molecule, i.e. the type, mass and order of occurrence of all the atoms (or fragments) has to be the same in both files.

`Translate`

Type: Bool Default value: No Description: Optional, recommended to be included. Move the center of mass of both geometries to the origin.

Only a few keys from the .rkf file are used for the calculation of the Franck-Condon factors. Disk space usage can be significantly reduced by extracting just these keys from the .rkf file before further analysis. The following shell script will extract the keys from the KF file specified by the first argument and store them in a new KF file specified by the second argument using the *cpkf* utility:

```
#!/bin/sh
cpkf $1 $2 General Molecule Vibrations
```

### Result: TAPE61¶

After a successful calculation, *fcf* produces a TAPE61 KF file. All results are stored in the Fcf section:

contents of TAPE61 |
comments |

firstmode, lastmode | the first and last vibrational mode taken into account |

lambda | the minimum value of the electron-phonon coupling |

maxl1, maxl2 | maximum level (or maximum number of vibrational quanta) in both states |

translate, rotate | whether the TRANSLATE and ROTATE options were specified in the input |

natoms | number of atoms in the molecule |

mass | atomic mass vector (m) |

xyz1, xyz2 | equilibrium geometries of both states (x_{0} and x_{0} ‘) |

b0 | zero-order axis-switching matrix matrix (B_{0} ) |

nmodes | number of vibrational modes with a non-zero frequency |

gamma1, gamma2 | reduced frequencies of both states (Γ and Γ’) |

lmat1, lmat2 | mass-weighted normal modes of both states (L and L’) |

jmat | Duschinsky rotation matrix (J) |

kvec1, kvec2 | displacement vectors for both states (k and k’, kvec1 is used for the calculation of the Franck-Condon factors) |

lambda1, lambda2 | electron-phonon couplings for both states (λ and λ’) |

maxp1, maxp2 | maximum number of permutations of maxl1/maxl2 quanta over the vibrational modes |

i0 | ground state to ground state overlap integral (I_{0,0} ) |

freq1, freq2 | frequencies of every permutation of the vibrational quanta for both states |

fcf | maxp1 by maxp2 Franck-Condon factor matrix |

fcfsum1, fcfsum2 | average sum of the Franck-Condon factors for both states |

In addition to producing a binary TAPE61 file, *fcf* also writes the frequencies, displacement vectors and electron-phonon couplings for both states to the standard output.

## FCF example absorption and fluorescence¶

In this example it is assumed that the molecule has a singlet ground state S_{0} , and the interesting excited state is the lowest singlet excited state S_{1} . First one needs to a the ground state geometry optimization, followed by a frequency calculation.
For the ground state frequency calculation we will use AMS_JOBNAME=S0.
Next one needs to do an excited state geometry optimization. Here it is assumed that the lowest singlet excited state S_{1} is of interest:

```
AMS_JOBNAME=S1_GEO $AMSBIN/ams <<eor
...
Task GeometryOptimization
Engine ADF
...
Excitation
Onlysing
Lowest 1
End
ExcitedGO
State A 1
Singlet
End
EndEngine
eor
```

To get the frequencies for this excited state, numerical frequencies need to be calculated, at the optimized geometry of the first excited state:

```
AMS_JOBNAME=S1 $AMSBIN/ams <<eor
...
LoadSystem
File S1_GEO.results/adf.rkf
End
Task SinglePoint
Properties
NormalModes True
End
Engine ADF
...
Excitation
Onlysing
Lowest 1
End
ExcitedGO
State A 1
Singlet
End
EndEngine
eor
```

Next for the absorption spectrum, we look at excitations from the lowest vibrational state of the electronic ground state to the vibrational levels of the first singlet excited state S_{1} (S_{1} \(\leftarrow\) S_{0} ), using the FCF program, which calculates the Franck-Condon factors between the vibrational modes of the two electronic states, with input

```
$AMSBIN/fcf << eor
STATES S0.results/adf.rkf S1.results/adf.rkf
QUANTA 0 5
SPECTRUM 0 10000 1001
TRANSLATE
ROTATE
eor
```

The number of vibrational quanta for the excited state should be larger in case of small molecules. See the description of FCF program for more details.

For the fluorescence spectrum, we look at excitations from the lowest vibrational state of the first singlet excited state S_{1} to the vibrational levels of the singlet ground state state S_{0} (S_{1} → S_{0} ). Input for the FCF program is in this case:

```
$AMSBIN/fcf << eor
STATES S0.results/adf.rkf S1.results/adf.rkf
QUANTA 5 0
SPECTRUM -10000 0 1001
TRANSLATE
ROTATE
eor
```

The number of vibrational quanta for the ground state should be larger in case of small molecules.

Note that the FCF program calculates the spectrum relative to the 0-0 transition. Thus one should add to spectrum calculated with FCF the difference in energy of the lowest vibrational state of the ground state S_{0} and the lowest vibrational state of the electronically singlet excited state S_{1} .

## FCF Example phosphorescence¶

In this example it is assumed that the molecule has a singlet ground state S_{0} , and the interesting excited state is the lowest triplet excited state T_{1} . Emission from a triplet state to a singlet state is spin forbidden, however, due to spin-orbit coupling such transitions may occur. In the following we assume that the geometry of the triplet excited state is not influenced much by spin-orbit coupling.

First one needs to a the ground state geometry optimization, followed by a frequency calculation, using AMS_JOBNAME=S0. Next one needs to do an excited state geometry optimization of the lowest triplet excited state, followed by a frequency calculation.

```
AMS_JOBNAME=T1 $AMSBIN/ams <<eor
...
Task GeometryOptimization
Properties
NormalModes True
End
Engine ADF
Unrestricted
SpinPolarization 2.0
EndEngine
eor
```

For the phosphorescence spectrum, we look at excitations from the lowest vibrational state of the first triplet excited state T_{1} to the vibrational levels of the singlet ground state state S_{0} (T_{1} → S_{0} ). Input for the FCF program is in this case:

```
$AMSBIN/fcf << eor
STATES S0.results/adf.rkf T1.results/adf.rkf
QUANTA 5 0
SPECTRUM -10000 0 1001
TRANSLATE
ROTATE
eor
```

The number of vibrational quanta for the ground state should be larger in case of small molecules.

Note that the FCF program calculates the spectrum relative to the 0-0 transition. Thus one should add to spectrum calculated with FCF the difference in energy of the lowest vibrational state of the ground state S_{0} and the lowest vibrational state of the electronically triplet excited state T_{1} .

Zero field splitting (ZFS) and the radiative rate constants (i.e. radiative phosphorescence lifetimes) could be calculated with spin-orbit coupled ZORA time-dependent density functional theory (ZORA-TDDFT). With the ADF engine spin-orbit coupling can be treated self-consistently (i.e. non perturbatively) during both the SCF and TDDFT parts of the computation.

An alternative to the use of the unrestricted formalism to calculate the lowest triplet excited state is to use the TDDFT formalism:

```
AMS_JOBNAME=T1_GEO $AMSBIN/ams <<eor
...
Task GeometryOptimization
Engine ADF
Excitation
Onlytrip
Lowest 1
End
ExcitedGO
State A 1
Triplet
End
EndEngine
eor
```

To get the frequencies for this excited state, numerical frequencies need to be calculated, at the optimized geometry of the first excited state.

```
AMS_JOBNAME=T1 $AMSBIN/ams <<eor
...
LoadSystem
File T1_GEO.results/adf.rkf
End
Task SinglePoint
Properties
NormalModes True
End
Engine ADF
...
Excitation
Onlytrip
Lowest 1
End
ExcitedGO
State A 1
Triplet
End
EndEngine
eor
```

References

[1] | J.S. Seldenthuis, H.S.J. van der Zant, M.A. Ratner and J.M. Thijssen, Vibrational Excitations in Weakly Coupled Single-Molecule Junctions: A Computational Analysis, ACS Nano 2, 1445 (2008) |

[2] | G.M. Sando and K.G. Spears, Ab Initio Computation of the Duschinsky Mixing of Vibrations and Nonlinear Effects, Journal of Physical Chemistry A 105, 5326 (2001) |

[3] | (1, 2) P.T. Ruhoff and M.A. Ratner, Algorithms for computing Franck-Condon overlap integrals, International Journal of Quantum Chemistry 77, 383 (2000) |