xDimers

Pyhton package for simulating multi-molecular emission spectra dominated by a single effective intermolecular vibrational mode. This package is accompanying a publication and is available at Zenodo.

Zenodo:

Publication:

S. Hammer et al, "Spectroscopic analysis of vibrational coupling in multi-molecular excited states", Material Horizons, 2023, 10, 221-234

https://doi.org/10.1039/D2MH00829G

Table of contents

1. Installation
2. Basic introduction

• Model and definitions
• Basic functionality
• Fitting emission data
• Examples

3. API reference guide

• xdimer
• semiclassical
• quantummechanical
• auxiliary

4. License and citation

1. Installation

Install package from PyPi with

pip install XDimer

The latest development version is available on GitHub. To install use:

python -m pip install git+https://github.com/HammerSeb/xDimer.git

The package has been tested against the latest versions of python > 3.6 .

2.Basic Introduction

This package enables the quick simulation of emission spectra from Franck-Condon vibronic transitions between the vibrational levels of two harmonic oscillators with different potential strength at different temperatures. For this purpose, it provides two ways to simulate the emission, a semi-classical one for which the final state harmonic oscillator is treated as a continous function and a full quantum-mechanical approach, for which the individual Franck-Condon factors are calculated numerically. A short introduction of the underlying physical model as well as some basic defintions are given below please refere to the related publication for an in detail description of the physical model and the mathematical definition.

2.1 Model and defintions

The emission spectra are modeled by Franck-Condon transitions between two displaced harmonic oscillators described by $$R(q) = R q^2$$

with $q$ as the generalized spatial coordinate and the

oscillator constant $R$.

which is related to the

vibrational energy quantum $E_{vib}$

and the

oscillator parameter $\alpha$

via the reduced mass as

$$R = \frac{\mu}{\hbar^2}E_{vib}^2 $$

and

$$\alpha = \frac{\mu}{\hbar^2}E_{vib} .$$

The oscillators are displaced in energy by the

energetic offset $D_e$

and along the generalized spatial coordinate by the

spatial displacement $q_e$

each with respect to the vertex of the ground state parabola.

Variable names are declared throughtout the package as follows:

gs_potential : ground state oscillator constant

xs_potential : excited state oscillator constant

gs_vib_energy : ground state vibrational energy quantum

xs_vib_energy : excited state vibrational energy quantum

gs_parameter : ground state oscillator parameter

xs_parameter : excited state oscillator parameter

q_xs : spatial displacement along generalized coordinate

e_offset : energetic offset of the oscillators

mass : reduced mass of the system in atomic units u

For the quantummechanical simulation of emission spectra the energetic broadening of the underlying lineshape function is declared as

energetic_broadening

2.2 Basic functionality

The package consists of three main parts contained in the xdimer module which is loaded by

import xdimer

This module contains the class dimer_system whicht stores the key parameters which fully define a dimer system. The functions semiclassical_emission and quantummechanical_emission take instances of the dimer_system class as an input and return the respective emission spectra.

Set up a dimer system

To create a dimer system use

dimer = xdimer.dimer_system(mass, gs, xs, q_xs, e_offset)

In the default setup_mode the parameters gs and xs define the ground state (gs) and excited state (xs) vibrational energy quantum (in eV). The parameters q_xs and e_offset are the spatial displacement $q_e$ (in Angstrom) and energetic offset $D_e$ (in eV). The mass parameter is the reduced mass $\mu$ of the dimer system in atomic units. Hence,

dimer = xdimer.dimer_system(423, 0.02, 0.025, 0.1, 1.5)

defines a dimer system with reduced mass 423 u, ground and excited state vibrational energy quantums of 20 meV and 25 meV, respecitvely, an excited state spatial displacement of 0.1 Angstrom and an energetic offset of 1.5 eV.

Calculating emission spectra

Emission spectra can be calculated by a semi-classical or quantum-mechanical approach. To calculate semi-classical emission spectra use

spectra = xdimer.semiclassical_emission(E, temp, dimer)

where E is of ndarray-type and defines the energy axis over which the emission is calculated. The temperatures for which the spectra are calcualted are given by temp either as a list of several temperatures or a single temperature value as a float. The dimer is an instance of dimer_system (dimer can also be a list, see API reference). The function semiclassical_emission returns either a dictonary, if the temperature input was given as a list of several values, or a ndarray if only one temperature value was provided. The return variable is compased as follows

For a single temperature value: emission is an array of the form

emission[0] input variable E as energy axis

emission[i] from i = 1,...,6. The emission spectrum of the i-th vibrational level of the excited state

emission[7] total emission spectrum as sum over all emissions from the first 6 vibrational levels of the excited state

For a list of temperature values temp=[T1, T2, T3, ...] emission is a dictionary with keys T1, T2, T3, ... . Each entry contains an array for a single temperature value as described above.

The quantummechanical emission is calculated by

emission = xdimer.quantummechanical_emission(E, temp, dimer)

where E is of ndarray-type and defines the energy axis over which the emission spectra are calculated. The temperatures are given by temp either as a list of several temperatures or a single temperature value as a float. The dimer is an instance of dimer_system (dimer can also be a list, see API reference). The function quantummechanical_emission returns pairs of array like return values

[spectra_full, spectra_stick] 

either as entrys in a dictonary keyed by multiple temperature values given in temp, or a single array pair if only one temperature value was given. If n vibrational levels of the excited state (0, ..., n-1) are simulated

spectra_stick is a list of ndarrays of length n. Each entry i contains the transition energies and intensities of all transitions from the i-th vibrational level of the excited state to the manifold of simulated ground state levels (default is 25). The relation between emission energy and tranisition intensity is referred to as stick spectrum of the i-th vibrational level.

The stick spectrum of the i-th vibrational level is given as

spectrum_stick[i][0] quantum number k of the vibrational level of the respective final state

spectrum_stick[i][1] photon energy of the $\ket{i} \rightarrow \ket{k}$ transition

spectrum_stick[i][2] Franck-Condon factor $|\braket{k|i}|^2$ transition weighted with a respective Boltzmann factor

spectra_full is a ndarray and contains the convolution of the stick spectra with a gaussian line shape function of energetic broadening w (specified as an optional parameter when creating a dimer_system instance) resulting in smooth emission spectra.

The smooth emission spectra are returned as

spectra_full[0]: input variable E as energy axis

spectra_full[i]: from i = 1 to the last simulated vibrational level n. The emission spectrum of the i-th vibrational level of the excited state as the sum of gaussian line shapes for each transition specified in spectra_stick[i].

spectra_full[n+1]: total emission spectrum as sum over all emissions from the simulated ;eve;s vibrational levels of the excited state

The default settings of the function simulates the first five vibrational levels of the excited states.

2.3 Fitting emission data

To use the emission functions to fit a luminescence data set the dimer input can be given as a list containing the variables for an optimization procedure.

For semiclassical_emission the list needs to be of the following form:

[gs_potential, xs_parameter, e_offset, q_xs, mass]

For quantummechanical_emission the list needs to be of the following form:

[gs_parameter, xs_parameter, e_offset, q_xs, energetic_broadening, mass]

2.4 Examples

Two examples are provided showing how to simulate spectra and use the provided functions to fit a set of temperature dependent luminescence data.

They can be called by

python -m xdimer.examples.simulating_emission

and

python -m xdimer.examples.fitting_emission_data

simulating_emission simulates a semiclassical and quantum-mechanical emission spectrum for a temperature specified by console input when running the module. The individual vibrational contributions from the excited state are unraveled and depicted color coded with the main emission spectrum. The dimer system used in the simulation is created by

dimer = xdimer.dimer_system(577.916/2, .022, .026, 0.1, 1.55, energetic_broadening=.02)

fitting_emission_spectra simulates a data set of emission data using xdimer.quantummechanical_emission for four different temperatures. It then performs a fit to the whole data set, including all temperatures. The fit does take a while (approx 90 seconds). The dimer system used to generate the data set is created by

dimer = xdimer.dimer_system(400, .027, .023, 0.08 ,1.55, energetic_broadening= .019)

3. API reference guide

API reference guide for all classes and functions available.

3.1 xdimer

Base functionality of the package containing the dimer_system class, the functions semiclassical_emission and quantummechanical_emission and the exception xDimerModeError. Import by

import xdimer

dimer_system

Class defining a dimer system containing all defining physical quantities. Instances are created via

dimer_system(mass, gs, xs, q_xs, e_offset, energtic_broadening, setup_mode)

mass, q_xs and e_offset are the reduced mass (in atomic units), the spatial displacement (in Angstrom) and the energetic offset (in eV), respectively.

gs and xs define the ground and excited state potential, respectively. See setup_mode for details.

energetic_broadening defines the line width parameter as standard deviation of the gaussian linshape of each vibronic transition calculated by quantummechanical_emission in eV. Default is 0.02.

setup_mode defines the input mode for the defintion of the ground and exited state potential by the variables gs and xs, respectively. Accepted inputs 'vib_energy' (default), 'osc_const' and 'osc_para'.

'vib_energy' (default): potentials are defined by their vibrational energy quantum $E_{vib}$ in eV.

'osc_const': potentials are defined by their oscillator constant $R$ in eV/Angstrom^2.

'osc_para': potentials are defined by their oscillator parameter $\alpha$ in 1/Angstrom^2.

If none of the above modes is used xDimerModelError is raised.

Dimer properties can be accessed by calling the respective internal variable:

reduced mass: dimer_system.mass

spatial offset: dimer_system.q_xs

energetic offset: dimer_system.e_offset

energetic broadening: dimer_system.energetic_broadening

Parameters of the harmonic potentials can be returned by properties:

Vibrational energy quantum

dimer_system.gs_vib_energy: ground state

dimer_system.xs_vib_energy: excited state

Oscillator constant

dimer_system.gs_potential: ground state

dimer_system.xs_potential: excited state

Oscillator parameter

dimer_system.gs_parameter: ground state

dimer_system.xs_parameter: excited state

semiclassical_emission

Calculates and returns a semiclassical emission spectrum for a dimer system at a given temperature or list a of temperatures. The first six vibrational levels of the excited state are taken into account.

semiclassical_emission(E, temp, dimer)

Arguments:

E (1-D ndarray): photon emission energy in eV

temp (list/Float): either list of floats or float: List of temperature values or single temperature value in Kelvin

dimer (instance dimer_system/list): either instance of dimer_system class or list of variables

  instance dimer_system: use for simulation of emission spectra from exisiting dimer system

  list: use for fitting a data set, list needs to be of the form [0: gs_potential, 1: xs_parameter, 2: e_offset, 3: q_xs, 4: mass]

COMMENT: mass should not be used as a free fit parameter but be set to the reduced mass of the dimer system

Returns:

dict/ndarray: if temp is list, dictionary (key = temp): ndarrays containing emission spectra for respective temperature temp

if temp is float: ndarray containing emission spectra for respective temperature temp

array structure: [8, size(E)]:

array[0] = E (emission energies)

array[i+1]: X-dimer emission spectrum from the i-th excited vibrational state (i in [0,...,5])

array[7] : Semi-classical X-dimer emission spectrum at temperature "temp" considering the first six vibrational levels of the excited state oscillator

Example

import numpy as np
import xdimer
# initilazises dimer system
dimer = xdimer.dimer_system(400, 0.22, 0.25, 1.5)
# Define energy axis for simulation and temperatures
E = np.linspace(1, 3, 500)
temp = [5, 50, 100, 150, 200, 250, 300]
# calculate emission spectra for given temperatures
spectra = xdimer.semiclassical_emission(E, temp, dimer)
spectrum_150K = spectra[150]

quantummechanical_emission

Calculates and returns a quantummecanical emission spectrum for a dimer system at a given temperature or list a of temperatures. Function returns stick spectra and continous spectrum as superposition of gaussian emissions with intensity and position defined by the stick spectrum

quantummechanical_emission(E, temp, dimer, simulation_parameters = [5, -.5, .5, 10000, 25])

Arguments:

E (ndarray): photon emission energy in eV

temp (list/Float): either list of floats or float: List of temperature values or single temperature value in Kelvin

dimer (instance dimer_system/list): either instance of dimer_system class or list of variables

  instance dimer_system: use for simulation of emission spectra from exisiting dimer system
  list: use for fitting a data set, list needs to be of the form [0: gs_parameter, 1: xs_parameter, 2: e_offset, 3: q_xs, 4: energetic_broadening, 5: mass]

simulation_parameters (list, optional): specifies the simulation parameters [n_sim, q_low, q_high, dq, n_gs]. First entry n_sim sets number of simulated vibrational levels of excited state. Other four parameters specify numeric evaluation of Franck-Condon factors, see quantummechanical.franck_condon_factor. Defaults to [5 ,-.5, .5, 10000, 25].

COMMENT: mass should not be used as a free fit parameter but be set to the reduced mass of the dimer system

Returns

dict/list: if temp is list, dictionary (key = temp): each entry contains list of the form [spectra_full, spectra_stick]

if temp is float: list of the form [spectra_full, spectra_stick] for respective temperature temp

spectra_full (ndarray):

[0] = E (emission energies)

[1: n_sim]: smooth emission spectrum for 0-th to (n_sim-1)-th vibrational level of excited state

[n_sim+1]: full emission spectrum as sum over all excited state transitions from 0-th to (n_sim-1)-th

spectra_stick (list of ndarrays): list of length n_sim.

Each entry contains the output of quantummechanical.franck_condon_factor as ndarray [0: final state, 1: transition energies, 2: boltzmann weighted transition intensities] for the respective excited state level, i.e. spectra_stick[0] holds 0-th vibrational level, spectra_stick[1] holds 1-st vibrational level and so forth.

Example

import numpy as np
import xdimer
# initilazises dimer system
dimer = xdimer.dimer_system(400, 0.22, 0.25, 1.5)
# Define energy axis for simulation and temperatures
E = np.linspace(1, 3, 500)
temp = [5, 50, 100, 150, 200, 250, 300]
# calculate emission spectra for given temperatures
spectra = xdimer.quantummechanical_emission(E, temp, dimer)
qm_150K = spectra[150]
spectra150K, stick_spectra150K = qm_150k[0], qm_150k[1]

Errors and Exceptions

xDimerModeError(Exception):

Subclass of Exception. Raised if instance of dimer_system class is created using an unknown setup mode.

3.2 semiclassical

Contains the functions to calculate a semi-classical emission spectrum. Can be imported by

import xdimer.semiclassical

excited_state_energy

excited_state_energy(n,vib_zero_point_energy,e_offset)

auxiliary function for semi-classical emission spectra. Calculates excited state energy of vibrational level n with respect to ground state minimum in eV. Equation (5) in publication.

Arguments:

n (int): vibrational level n

vib_zero_point_energy (Float): vibrational zero point energy in eV

e_offset (Float): energetic offset with respect to ground state minimum in eV

Returns:

Float: excited state energy of vibrational level n in eV

displacement_from_energy

displacement_from_energy(E, n, gs_potential, vib_zero_point_energy, e_offset, q_xs):

auxiliary function for semi-classical emission spectra. Inverse function of the emission energy - spatial coordinate relation. Calculates spatial displacement as function of photon energy for emission from the excited state at vibrational level n in Angstrom. Equation (S4) in electronic supplementary information.

Arguments:

E (ndarray): photon emission energy in eV

n (Int): vibrational level n

gs_potential (Float): oscillator constant $R_0$ of the ground state potential in eV/Angstrom**2

vib_zero_point_energy (Float): vibrational zero point energy in eV

e_offset (Float): energetic offset with respect to ground state minimum in eV

q_xs (Float): spatial displacement of excited state with respect to the ground state minimum in Angstrom

Returns:

ndarray: spatial displacement in Angstrom

xdimer_sc_emission

There six numbered functions from i= [0 to 5] which calculate the semi-classical emission spectrum from the i-th vibrational level of the excited state (c.f. equations (S7)-(S12) in electronic supplementary information). The emission from the ground state is disccused here exemplarily.

xdimer_sc_emission_0(E, gs_potential, xs_parameter ,vib_zero_point_energy, e_offset, q_xs)

Semi-classical X-dimer emission spectrum from the vibrational ground state

Arguments:

E (ndarray): photon emission energy in eV

gs_potential (Float): oscillator constant $R_0$ of the ground state potential in eV/Angstrom**2

xs_parameter (Float): oscillator parameter of excited state quantum-mechanical oscillator in 1/Angstrom**2 (alpha in manuscript)

vib_zero_point_energy (Float): vibrational zero point energy in eV

e_offset (Float): energetic offset with respect to ground state minimum in eV

q_xs (Float): spatial displacement of excited state with respect to the ground state minimum in Angstrom

Returns:

ndarray: size: size(E). emission intensity with respect to values of E. nan-values resulting from E-values greater than singularity are set to 0 to ensure down the line usability.

For higher vibrational levels use xdimer_sc_emission_1, xdimer_sc_emission_2, xdimer_sc_emission_3, xdimer_sc_emission_4 and xdimer_sc_emission_5.

xdimer_sc_total_emission

xdimer_sc_total_emission(E, gs_potential, xs_parameter ,vib_zero_point_energy, e_offset, q_xs, boltzmann_dist, temp)

Semi-classical X-dimer emission spectrum at temperature "temp" considering the first six vibrational levels of the excited state oscillator

Arguments:

E (ndarray): photon emission energy in eV

gs_potential (Float): oscillator constant $R_0$ of the ground state potential in eV/Angstrom**2

xs_parameter (Float): oscillator parameter of excited state quantum-mechanical oscillator in 1/Angstrom**2

vib_zero_point_energy (Float): vibrational zero point energy in eV

e_offset (Float): energetic offset with respect to ground state minimum in eV

q_xs (Float): spatial displacement of excited state with respect to the ground state minimum in Angstrom

boltzmann_dist (Dict): Boltzman distribution generated with auxiliary.boltzmann_distribution

temp (Float): Temperature in Kelvin - must be key in dictionary boltzmann_dist

Returns:

ndarray: size(E). emission intensity with respect to values of E. nan-values resulting from E-values greater than singularity are set to 0 to ensure down the line usability.

3.3 quantummechanical

Contains the functions to calculate a quantum-mechanical emission spectrum. Can be imported by

import xdimer.quantummechanical

harmonic_oscillator_wavefunction

harmonic_oscillator_wavefunction(level, spatial_coordinate, oscillator_parameter)

Auxiliary function to caluclate the wave function of harmonic oscillator functions of a given vibrational level.

Arguments:

level (Int): oscillator quantum number

spatial_coordinate (ndarray): array of spatial coordinates (in Angstrom) for which wavefunction is calculated

oscillator_parameter (type): oscillator parameter alpha in 1/Angstrom**2

Returns:

ndarray: values of the wavefuncion at given spatial coordinates

franck_condon_factors

franck_condon_factor(level, xs_parameter, gs_parameter, q_xs, e_offset, mass, q_low= -.5, q_high= .5, dq= 10000, n_gs= 25)

numerically calculates emission energies and respective Franck-Condon factors for the emission from a vibrational level (given by 'level') of an excited state oscillator to the vibrational levels of a ground state harmonical oscillator

Arguments:

level (Int): oscillator quantum number

xs_parameter (Float): excited state oscillator parameter alpha in 1/Angstrom**2

gs_parameter (Float): ground state oscillator parameter alpha in 1/Angstrom**2

q_xs (Float): spatial displacement of excited state with respect to the ground state minimum in Angstrom

e_offset (Float): energetic offset with respect to ground state minimum in eV mass (Float): mass of the dimer system in kg

Simulation parameters - optional

q_low (int, optional): lower intergration boundary of spatial coordinate. Defaults to -1.

q_high (int, optional): upper intergration boundary of spatial coordinate. Defaults to 1.

dq(int, optional): value number between lower and upper integration boundary of spatial coordinate, determines spatial resolution during integration. Defaults to 10000.

n_gs (int, optional): simulated levels of the ground state oscillator. Defaults to 25.

Returns:

ndarray: size(3, n_GS)

[0,:]: int: quantum number of respective final vibratioanl state k of ground state level for the transition n --> k

[1,:]: Floats: photon energy of transition n --> k

[2,:]: Floats: value of Frank-Condon factor of transition n --> k

3.4 auxiliary

Contains auxiliary functions. Can be imported by

import xdimer.auxiliary

osc_para_to_vib_energy

osc_para_to_vib_energy(osc_para, mass)

Transforms oscillator parameter to vibrational energy quantum.

Arguments:

osc_para (float): oscillator parameter in 1/Angstrom^2

mass (float): reduced mass in atomic units

Returns:

float: vibrational energy quantum in eV

vib_energy_to_osc_para

vib_energy_to_osc_para(vib_energy, mass)

Transforms vibrational energy quantum to oscillator parameter

Arguments:

vib_energy (float): vibrational energy quantum in eV

mass (float): reduced mass in atomic units

Returns:

float: oscillator parameter in 1/Angstrom^2

osc_const_to_vib_energy

osc_const_to_vib_energy(osc_const, mass)

Transforms oscillator constant to vibrational energy quantum.

Arguments:

osc_const (float): oscillator constant in eV/Angstrom^2

mass (float): reduced mass in atomic units

Returns:

float: vibrational energy quantum in eV

vib_energy_to_osc_const

vib_energy_to_osc_const(vib_energy, mass)

Transforms vibrational energy quantum to oscillator constant

Arguments:

vib_energy (float): vibrational energy quantum in eV

mass (float): reduced mass in atomic units

Returns:

float: oscillator constant in eV/Angstrom^2

boltzmann_distribution

boltzmann_distribution(Temp_list, vib_zero_point_energy, no_of_states= 50)

Generates a Boltzmann probability distribution for a quantum-mechanical harmonic oscillator with zero point energy "vib_zero_point_energy"

Arguments:

Temp_list (list of Floates): list of temperature values in Kelvin

vib_zero_point_energy (type): vibrational zero point energy in eV

No_of_states (int, optional): number of simulated excited states for canocial partion sum. Defaults to 50.

Returns:

dictionary:

key (Float): entries of Temp_list;

values (ndarray): (2 x no_of_states); [0]: vibrational level, [1]: corresponding occupation probability

gauss_lineshape

gauss_lineshape(x, A, w, xc):

gauss function as line shape function $$ L(x) = \frac{A}{\sqrt{2\pi\sigma}}\exp\left( - \frac{(x-x_c)^2}{2\sigma^2} \right) $$ Arguments:

x (ndarray): x-values

A (Float): area under the curve

w (Float): standard deviation

xc (Float): x center

Returns:

nadarry: function values at x-values

4. License and citation

License

Package distributed under MIT license. Copyright (c) 2022 Sebastian Hammer.

Citation

If you use this package or its contents for apublication please consider citing the zenodo archive or the corresponding publication .