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:
It employs the adiabatic (Born-Oppenheimer) approximation and treats the nuclei as moving in an effective potential defined by the electronic configuration.
It works at the Franck-Condon (FC) point and assumes that the transition occurs at the ground state equilibrium structure for absorption and at the excited state equilibrium structure for emission.
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. The program assumes that the starting vibrational wavefunction is in the ground state, as is the case for most spectroscopic experiments performed at room temperature.
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.
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.
In AMS2022 the algorithm for the calculation of the Franck-Condon factors has been changed in order to improve the precision, increase the speed, and lower the memory requirements.
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:
Algorithm¶
The algorithm for the calculation of the FC integrals is based on the work by Santoro et al. 4 which is here summarized. There are in principle infinite FC factors that should be computed in order to achieve 100% convergence in the spectrum. This number is reduced by dividing the vibrational into classes.
A class is defined based on the number of simultaneously excited vibrational modes, thus a class 3 state has exactly 3 modes which are simultaneously excited, while all the remaining modes have zero quanta. The fcf program first evaluates the class 1 e 2 FC up to a fairly large vibrational quantum number and uses these results to infer how much each mode should be excited for the calculation of the integrals involving states of higher classes. We refer to the original paper for the details of the algorithm.
The convergence of the results after each class can be estimated by summing all FC factors, which should converge to 1 in the limit of a complete set. The program is terminated when the maximum allowed class is reached, when there is little gain from one class to the next, or when a sufficient convergence criterion is met. It should be noted that if one is interested in the overall shape of the spectrum then in most cases it isn’t necessary to reach a convergence that is close to 100%.
Input¶
The input for fcf is keyword-oriented and is read from the standard input. fcf recognizes several keywords, but only the states have to be specified to perform the calculation. All input therefore contains at least the following lines:
$AMSBIN/fcf << eor
STATE1 state1
STATE2 state2
eor
The spectrum is always calculated from the ground vibrational state belonging to the potential energy surface specified by state1 to the vibronic states belonging to state2. Documentation for all keys of fcf:
$AMSBIN/fcf << eor
State1 state1
State2 state2
Lambda float
Modes first last
Rotate [True | False]
Translate [True | False]
NumericalQuality ["Basic" | "Normal" | "Good" | "Excellent"]
Prescreening
Class1 integer
Class2 integer
MaxClass integer
MaxFCI integer
End
Spectrum FreqMin FreqMax NFreq
LineShape ["Stick" | "Gaussian" | "Lorentzian" ]
SpcMin float
SpcMax float
SpcLen integer
HWHM float
End
Printing
Verbose
FCFThresh float
End
eor
Convergence
- Type
Float
- Description
Minimum fraction of the FC factors sum to be calculated
Lambda
- Type
Float
- Default value
0.0
- GUI name
Minimum coupling
- 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.
NumericalQuality
- Type
Multiple Choice
- Default value
Normal
- Options
[0-0, Basic, Normal, Good, Excellent]
- Description
Set the quality of several technical aspects of an FC calculation, including details for the prescreening, the amount of computed integrals, the maximum integral class evaluated, as well as the convergence criteria.
Printing
- Type
Block
- Description
Optional. Printing Options.
FCFThresh
- Type
Float
- Default value
0.01
- GUI name
FC factor threshold
- Description
The minimum value for the printing of a FC factor.
Verbose
- Type
Bool
- Default value
No
- GUI name
Verbose printing
- Description
Increase output verbosity.
Rotate
- Type
Bool
- Default value
Yes
- Description
Recommended to be used. Rotate the geometries to maximize the overlap of the nuclear coordinates.
Spectrum
- Type
Block
- Description
Optional. Controls the details of the spectrum calculation
HWHM
- Type
Float
- Default value
100.0
- Description
Half-Width at Half-Maximum for the spectral lineshape function
LineShape
- Type
Multiple Choice
- Default value
Stick
- Options
[Stick, Gaussian, Lorentzian]
- Description
Type of lineshape function
SpcLen
- Type
Integer
- Default value
2001
- Description
Number of points in the spectrum
SpcMax
- Type
Float
- Default value
9500.0
- Description
Maximum absolute energy difference between the states relative to the 0-0 transition
SpcMin
- Type
Float
- Default value
-500.0
- Description
Minimum absolute energy difference between the states relative to the 0-0 transition
Type
- Type
Multiple Choice
- Default value
Absorption
- Options
[Absorption, Emission]
- Description
Type of spectrum
State1
- Type
String
- Description
Filename of the result files of a numerical or analytical frequency calculation for the first (initial) state.
State2
- Type
String
- Description
Filename of the result files of a numerical or analytical frequency calculation for the second (final) state.
Translate
- Type
Bool
- Default value
Yes
- Description
Recommended to be used. 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 |
lambda |
the minimum value of the electron-phonon coupling |
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’) |
i0 |
ground state to ground state overlap integral (I_{0,0} ) |
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
STATE1 S0.results/adf.rkf
STATE2 S1.results/adf.rkf
TRANSLATE
ROTATE
eor
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
STATE1 S1.results/adf.rkf
STATE2 S0.results/adf.rkf
TRANSLATE
ROTATE
eor
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
STATE1 T1.results/adf.rkf
STATE2 S0.results/adf.rkf
TRANSLATE
ROTATE
eor
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
Density of States computed with AH-FC method¶
The FCF module can also be used to evaluate the density of states rho that enters the expression of the rate constant for a number of transfer phenomena based on Fermi’s Golden Rule:
To calculate the transfer rate from a donor molecule to an acceptor molecule it is sufficient to employ the FCF code as described above to evaluate the emission spectrum of the donor and the absorption spectrum of the acceptor. In fact, it can be shown 5 that the expression for the density of states can be written as:
Where F denotes the Franck-Condon spectrum calculated by the FCF code, specifically the absorption for the acceptor and the emission for the donor.
The convolution may be easily calculated by using the PLAMS library.
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)
- 4
F. Santoro, R. Improta, A. Lami, J. Bloino, V. Barone, Effective method to compute Franck-Condon integrals for optical spectra of large molecules in solution, J. Chem. Phys. 126, 084509 (2007)
- 5
X. de Vries High energy acceptor states strongly enhance exciton transfer between metal organic phosphorescent dyes, Nature Communications 11, 1292 (2020)