General Approach to Coupled Reactive Smoluchowski Equations: Integration and Application of DVR and Generalized Coordinate Methods to Diffusive Problems

Abstract

A new and more general approach to diffusion problems with the inclusion of reactivity among different coupled diffusional states is rationalized and presented. The integration of our previous developments in such field [Phys. Chem. Chem. Phys., 2015, 17, 17362-17374; J. Chem. Theory Comput., 2016, 12, 3482-3490] are implemented in a software package tool allowing the generic user to set up and run diffusional calculations with very low efforts. We show the applicability of the whole framework to a generic diffusional case of chemical interest that is the study case of (N, N-dimethylamino) benzonitrile (DMABN) fluorescence, whose excited state undergoes twisted intramolecular charge transfer (TICT) relaxation. The population dynamics of the excited state coupled to the ground state is followed, and fluorescence decay spectrum is calculated. The theoretical and numerical background here presented is robust and general enough to complement a wide number of diffusional problems of current interest.

1. Introduction

The chemist is often faced with several microscopic processes that are diffusive in nature, for example solution phase chemical processes, biochemical kinetics or configurational dynamics of fluorophores in liquid solution. For such microscopic systems in solution, apart from pure diffusion, reactive phenomena involving one or more species can also be included. As an example, problems such as recombination of isolated ion pairs,¹ fluorescence quenching,^2–4 and intramolecular electron transfer^5–7 have been considered in the past. The picture of the whole process must include the coupling between the reactant species and in this view the description and theoretical modeling of such complicated systems become difficult and in many cases is not trivial.

The main problems that emerge are the selection of reliable tools from the theoretical chemistry and how they can be used in order to properly model the reactivity; in particular how to face and include the coupling of the different reactant species to diffusion and between them. This problem has been faced in the past by many scientific groups using chemical kinetics and in many cases the most robust theoretical tools from stochastic processes field.^8,9 Here we take into consideration this last branch that has paved the road to a proper modeling of several microscopic processes which are diffusive in nature, adopting the Smoluchowski or more complicated Fokker-Planck equations.

For such systems diffusion can be thought of as occurring along one or more relevant coordinates. Then, reactivity can be taken into account by adding a coordinate dependent rate constant to the equation that is coupled to diffusion. Just to mention, in the last four decades, several works that have used this approach have adopted the so called reactive Smoluchowski equation; e.g. the pioneering works of Agmon and Hopfield on geminate recombination in excited-state proton transfer reactions,^10–12 the diffusive dynamics of CO binding to heme proteins,^13,14 the barrierless isomerization in solution,¹ the twisted-intramolecular charge transfer (TICT) from singlet excited state,^4,15 etc.

By the way, all these problems still lack a general and solid approach. In many cases the coupling that describes the population exchange between different diffusional states is absent and/or not directly explicit. In fact, the chemist is often faced with such diffusional problems and in the presence of reversible reactions, two or more coupled Smoluchowski equations are needed in order to describe completely the time evolution of the probability density. From this information, after proper modeling, observables can be retrieved; just to mention, computed time resolved emission spectra, rates, binding constants and survival probabilities.

In this work we have proposed a general approach to a system of coupled reactive Smoluchowski equations capable to properly describe reactive systems in solution. We have set down the main equations that govern the microscopical evolution of the system where we include the possibility of population exchange or disappearance; calculating and considering respectively, coordinate dependent rate constants and sink contributions. We have merged in a modular code all our previous developments on the solution of the one-dimensional Smoluchowski equation and molecular diffusion field. These are respectively, the robust numerical approach rooted in the discrete variable representation (DVR) in order to solve the equation¹⁶ and the variable diffusion tensor along a one-dimensional generalized coordinate considered as a diffusive process.¹⁷

We briefly recall that the advantage in using DVR, instead of classical numerical methods such as finite difference methods and/or spectral methods, is the higher performances in terms of rapid eigenvalues convergence with the use of few grid points.

While the advantage in using a generalized coordinate is to treat diffusive problems involving motion along more general paths, this motion is parameterized by the generalized coordinate that lacks an absolute analytical expression, but is rather described only by a nonlinear combination of local (curvilinear) coordinates.

The final goal is to have a reliable framework with which it is possible to set up and solve two or more coupled Smoluchowski equations along a generalized coordinate with reactive and/or sink coordinate dependent contributions. In this view we propose a novel and reliable machinery different from the previous one proposed by Krissinel’ and Agmon,¹ surpassing the problems found in the past and integrating the combined effect of our two novel approaches. The code is embedded in the previously implemented framework for the diffusion tensor, in a development version of the Gaussian code.¹⁸

A specific test case is used in order to verify the reliability of the code. Then we apply our integrated tool to the study case of TICT state fluorescence of (N, N-dimethylamino)benzonitrile (DMABN) molecule in water.

The paper continues as follows: in the Section “Theory” we provide the mathematical background underlying the implemented code for an arbitrary number of coupled diffusive and reactive states; in the Sections “Test-case” and “Application” we give some details about the reliability of the implemented code and the application of the same to a specific case of chemical interest. Finally, the Section “Conclusions” is dedicated to some discussion and perspectives.

2. Theory

Coupled reactive Smoluchowski equations

We consider n coupled states upon which diffusion and reaction take place along the same generalized coordinate x. The reactive one-dimensional Smoluchowski equation for the i-th state reads

$$\frac{\partial p_i(x,t)}{\partial t}=-{\Gamma }_i{p}_i(x,t)-\left[s_i(x)+{\displaystyle \sum _{j\ne i}^n{k}_{ij}(x)}\right]p_i(x,t)+{\displaystyle \sum _{j\ne i}^n{k}_{ji}(x)}{p}_j(x,t)$$ (1)

where p_i(x, t) is the probability density of finding a value of x at time t. Γ_i is the diffusion operator for the i-th state that reads

$${\Gamma }_i=-\frac{\partial }{\partial x}{D}_i(x)p_{i,eq}(x)\frac{\partial }{\partial x}{p}_{i,eq}^{-1}(x)$$ (2)

This operator, under the potential V_i(x) and diffusion function D_i(x) of the i-th state, regulates the population relaxation till the stationary limit lim_t→+∞ p_i(x, t) = p_i,eq(x), with

$$p_{i,eq}(x)=\frac{e^{-\beta V_i(x)}}{Z_i}$$ (3)

the Boltzmann distribution, Z_i = ∫ dx e^−βV_i(x) the partition function and β = 1/k_BT the Boltzmann factor at temperature T. Apart from stationarity, we assume that x describes a Markov type process,^8,9 i.e., in order to know the actual state of the system it is necessary to know the state of the system just at a previous time. Substantially, we are considering that x dynamics has no memory effects. We underline that in the entire development here presented we are assuming a priori that x has a Markovian behavior. It is generic users’ task to verify the Markov behavior of the chosen/computed generalized coordinate.¹⁹

The term k_ij(x) physically represents the rate at which the population of the i-th state at a specific value of x migrates towards the j-th state, while the sink term s_i(x) physically represents the rate of population depletion from the i-th state at a specific value of x. The combined effect of potential and diffusion governs the population dynamics starting from an initial distribution and widening it till the equilibrium state. But in the presence of reactive terms that are respectively, the reaction rate k_ij(x) from level i to level j and the sink term, s_i(x) ≡ k_ii(x), the stationary limit is no longer guaranteed.

Computation of generalized coordinate

A straightforward strategy for calculating the generalized coordinate is to perform a relaxed potential energy scan along a predefined internal coordinate and obtain the generalized coordinate as the distance between mass-weighted Cartesian coordinates of the successive geometries obtained by the scan. The angular momentum between the pairs of consecutive structures should be minimized to minimize the coupling between the rotations of the system and the generalized coordinate. For this we have used an approach based on the Intrinsic Reaction Coordinate (IRC).^20,21 A closed curve representing the minimum energy path in the mass-weighted Cartesian space C of the motion along the internal rotation, is generated.²² The space C contains the coordinates of a point c which are the elements of the set c_i : i = [1, 2, … , 3N]. c is defined as a function of x, the parameter corresponding to the generalized coordinate; in the following way

$$\frac{d\mathbf{\text{c}}(x)}{dx}=\frac{\mathbf{\text{g}}}{\sqrt{{\mathbf{\text{g}}}^{†}\mathbf{\text{g}}}}$$ (4)

Here g is the energy gradient in mass-weighted Cartesian coordinates. Each value of x is associated with a point c in C by a map a(x), according to the Reaction Path Hamiltonian (RPH) of Miller and coworkers²³

$${\mathbf{\text{c}}}_i={\mathbf{\text{a}}}_i(x)+{\displaystyle \sum _{k=1}^{3N-7}{\mathbf{\text{L}}}_{i,k}(x)Q_k}$$ (5)

Here we have taken into consideration the fact that eqn. (4) has seven solutions which are zero: six corresponding to translations and external rotations and one related to the curve as the energy along the internal rotation is expected to vary slowly. In eqn. (5), L_k(x) contains the eigenvectors corresponding to the non-zero eigenvalue of the Hamiltonian and Q_k are the vibrational normal modes.

The rototranslational displacements between consecutive geometries obtained from the scan are minimized by imposing the Eckart orientation. Finally, the map a(x) is calculated as the distance between two consecutive points in C in terms of mass-weighted coordinates

$$|dx{|}^2=|{\mathbf{\text{a}}}_{i+1}(x)-{\mathbf{\text{a}}}_i(x){|}^2=|{\mathbf{\text{c}}}_{i+1}-{\mathbf{\text{c}}}_i{|}^2$$ (6)

Considering that two consecutive points (i.e. two consecutive geometries) in the configurational space C are close enough to neglect the vibrational displacement (i.e. the second part in RHS of eqn. (5)).

The reduced mass has been taken to be unity and the information about kinematic coupling is stored as the discrepancy between the curve x, parameterized by the map a(x) in a point to point basis and the corresponding internal coordinate along which the scan is performed.

Variable diffusion tensor and Discrete Variable Representation

In the following we recall and apply the main ideas presented in our previous works^16,17 to the system of coupled reactive Smoluchowski equations. The theory that underlies the diffusion tensor construction relates the set of forces and velocities of a system of N atoms considered with and without constraints, respectively, the system is constituted of independent material points and the system is constituted by atoms linked by bonds. For these two different views a set of Cartesian velocities and forces for the unconstrained system and a set of external velocities and forces for the constrained system are considered. A laboratory frame (LF) and a molecular frame (MF) (fixed in the Eckart orientation on the molecular center of mass) are chosen and after some classical mechanics passages^17,24 we can express the Cartesian velocities v_i of the i-th atom for the unconstrained system as

$${\mathbf{\text{v}}}_i=\mathit{\text{υ}}+{\mathbf{\text{E}}}^{\text{tr}}(\Omega )\cdot \left(\omega \times {\mathbf{\text{c}}}_i+\frac{\partial {\mathbf{\text{c}}}_i}{\partial x}\dot{x}\right)$$ (7)

where υ, ω, ẋ are the set of external velocities, respectively the translational velocity, the angular velocity and the generalized coordinate moment. E(Ω) is the Euler matrix dependent on the set of Euler angles Ω that brings the LF into the MF and c_i are the Cartesian coordinates expressed in the MF of the i-th atom. It follows that the set of Cartesian and external velocities are related by a pure geometrical matrix, B

$$\left(\begin{array}{c}{\mathbf{\text{v}}}_1\\ ⋮\\ {\mathbf{\text{v}}}_i\\ ⋮\\ {\mathbf{\text{v}}}_N\end{array}\right)=\underset{\text{B}}{\underbrace{\left(\begin{array}{ccc}{\mathbf{1}}_3& -{\mathbf{\text{E}}}^{\text{tr}}(\mathbf{\Omega }){\mathbf{\text{c}}}_1^{\times }& {\mathbf{\text{E}}}^{\text{tr}}(\mathbf{\Omega })\frac{\partial {\mathbf{\text{c}}}_1}{\partial x}\\ ⋮& ⋮& ⋮\\ {\mathbf{1}}_3& -{\mathbf{\text{E}}}^{\text{tr}}(\mathbf{\Omega }){\mathbf{\text{c}}}_i^{\times }& {\mathbf{\text{E}}}^{\text{tr}}(\mathbf{\Omega })\frac{\partial {\mathbf{\text{c}}}_i}{\partial x}\\ ⋮& ⋮& ⋮\\ {\mathbf{1}}_3& -{\mathbf{\text{E}}}^{\text{tr}}(\mathbf{\Omega }){\mathbf{\text{c}}}_N^{\times }& {\mathbf{\text{E}}}^{\text{tr}}(\mathbf{\Omega })\frac{\partial {\mathbf{\text{c}}}_N}{\partial x}\end{array}\right)}}\left(\begin{array}{c}\mathit{\upsilon }\\ \mathit{\omega }\\ \dot{x}\end{array}\right)$$ (8)

where ${\mathbf{\text{c}}}_i^{\times }$ is the matrix representation of the cross vector product with elements ${\left({\mathbf{\text{c}}}_i^{\times }\right)}_{mn}=\sum {\epsilon }_{mln}{\left({\mathbf{\text{c}}}_i\right)}_l$ and with ε_mln the Levi-Civita symbol. The construction of the generalized coordinate from the variation of the molecular geometries visited, for example, with a relaxed potential energy surface scan, vide infra, and the derivatives of the various geometries with respect to x, allows the B matrix construction.

After B is constructed, the friction tensor for the constrained system ξ is calculated as

$$\mathit{\xi }={\mathbf{\text{B}}}^{\text{tr}}\Xi \mathbf{\text{B}}$$ (9)

where Ξ is the friction tensor for the unconstrained system. The choice of Ξ depends on the specific model adopted, typically, the model of non-interacting beads, Ξ = Ξ₀ 1_N, or more refined models that take in consideration hydrodynamical interactions, like the Oseen^25,26 or the Rotne-Prager²⁷ models.

The diffusion tensor D is finally calculated from the friction tensor ξ using Einstein relation

$$\mathbf{\text{D}}=k_{B}T{\mathit{\xi }}^{-1}=\left(\begin{array}{lll}{\mathbf{\text{D}}}_{TT}\hfill & {\mathbf{\text{D}}}_{TR}\hfill & {\mathbf{\text{D}}}_{TG}\hfill \\ {\mathbf{\text{D}}}_{RT}\hfill & {\mathbf{\text{D}}}_{RR}\hfill & {\mathbf{\text{D}}}_{RG}\hfill \\ {\mathbf{\text{D}}}_{GT}\hfill & {\mathbf{\text{D}}}_{GR}\hfill & {\mathbf{\text{D}}}_{GG}\hfill \end{array}\right)$$ (10)

where the subscripts T, R, G stand, respectively, for translational, rotational and generalized contributions. The diffusion tensor along the generalized coordinate is retrieved from the generalized term D_GG that is a scalar function of x, since we are dealing with just one generalized coordinate. All the technical details about the diffusion tensor model here omitted for sake of brevity can be found in our previous work, ref. 17, to which the interested reader is addressed.

The application of DVR to the system of coupled reactive Smoluchowski equations can be summarized in the following main steps (the main theory and equations underlying the DVR approach are reported in Appendix A):

1. Using eqn. (3) we express Γ_i in a more convenient way, allowing the evaluation of large values of V_i(x) with no numerical problems

   $$\begin{array}{l}{\Gamma }_i=D_i(x)\left[\frac{{\partial }^2}{\partial x^2}+\frac{\partial }{\partial x}\frac{V_i(x)}{k_{B}T}\frac{\partial }{\partial x}+\frac{{\partial }^2}{\partial x^2}\frac{V_i(x)}{k_{B}T}\right]p_i(x,t)\\ +\frac{\partial }{\partial x}{D}_i(x)\frac{\partial }{\partial x}{p}_i(x,t)+\frac{\partial }{\partial x}{D}_i(x)\frac{\partial }{\partial x}\frac{V_i(x)}{k_{B}T}\end{array}$$ (11)
2. We express the operators acting on p_i(x, t) and p_j(x, t) in the DVR matrix form by using product approximation, making use of approximate resolution of the identity and recalling the DVR matrix representation of first and second derivative operators (respectively D^(1)DVR and D^(2)DVR) for bounded and periodic potentials see eqns. (21-26) in Appendix A,

   $$\begin{array}{l}{\mathbf{\text{M}}}_i^{\text{DVR}}={\mathbf{\text{D}}}_i^{\text{DVR}}[{\mathbf{\text{D}}}^{(2)\text{DVR}}+\beta {\mathbf{\text{V}}}_i^{\prime \text{DVR}}{\mathbf{\text{D}}}^{(1)\text{DVR}}+\beta {\mathbf{\text{V}}}_i^{″\text{DVR}}]\\ +{\mathbf{\text{D}}}_i^{\prime \text{DVR}}{\mathbf{\text{D}}}^{(1)\text{DVR}}+\beta {\mathbf{\text{D}}}_i^{\prime \text{DVR}}{\mathbf{\text{V}}}_i^{\prime \text{DVR}}\\ -\left[{\mathbf{\text{s}}}_i^{\text{DVR}}+{\displaystyle \sum _{j\ne i}^n{\mathbf{\text{k}}}_{ij}^{\text{DVR}}}\right]\end{array}$$ (12)
3. Choosing the same DVR grid points {x_α} for the generalized coordinate x we construct the total DVR matrix M^DVR;

   $$\frac{\partial }{\partial t}\left(\begin{array}{c}{\mathbf{\text{p}}}_1\\ ⋮\\ {\mathbf{\text{p}}}_i\\ ⋮\\ {\mathbf{\text{p}}}_n\end{array}\right)=-\underset{{\mathbf{\text{M}}}^{\text{DVR}}}{\underbrace{\left(\begin{array}{ccccc}{\mathbf{\text{M}}}_1^{\text{DVR}}& \cdots & -{\mathbf{\text{k}}}_{i1}^{\text{DVR}}& \cdots & -{\mathbf{\text{k}}}_{n1}^{\text{DVR}}\\ ⋮& & ⋮& & ⋮\\ -{\mathbf{\text{k}}}_{1i}^{\text{DVR}}& \cdots & {\mathbf{\text{M}}}_i^{\text{DVR}}& \cdots & -{\mathbf{\text{k}}}_{ni}^{\text{DVR}}\\ ⋮& & ⋮& & ⋮\\ -{\mathbf{\text{k}}}_{1n}^{\text{DVR}}& \cdots & -{\mathbf{\text{k}}}_{in}^{\text{DVR}}& \cdots & {\mathbf{\text{M}}}_n^{\text{DVR}}\end{array}\right)}}\left(\begin{array}{c}{\mathbf{\text{p}}}_1\\ ⋮\\ {\mathbf{\text{p}}}_i\\ ⋮\\ {\mathbf{\text{p}}}_n\end{array}\right)$$ (13)

4. Eigenvalues and eigenfunctions are then calculated from the numerical diagonalization of M^DVR. Given the initial distributions p_i(x, 0), the profiles p_i(x, t) are built expanding upon DVR functions calculated at DVR points {x_α}.

Code details

The code has been written in a modular fashion inside the previously implemented framework for the diffusion tensor, in the same development version of the Gaussian code.¹⁸ After the user has chosen the system of interest, he/she must specify the number n of coupled states upon which the population relaxes under the combined effect of diffusion and reaction. Also he/she must specify the number α of DVR grid points {x_α} and the endpoints [a, b] of the generalized coordinate domain. Finally reactivity informations, i.e. all s_i(x) and k_ij(x) functions, must be given as input numerical files.

Supplementary input informations, such as the number of time steps upon which the profile is calculated, the initial and final time of propagation, hydrodynamical parameters for the diffusion function calculation must be given.

From the relaxed or rigid scan calculation (respectively, the molecular geometry is optimized at each scan step, the molecular geometry is not optimized at each scan step) the generalized coordinate x is calculated and the energetics V_i(x) and the diffusion functions D_i(x) are automatically retrieved from the program. The code, thanks to a built-in FORTRAN interpreter, can also retrieve these informations from numerical input files prepared from the output of other computational chemistry tools.

Since M^DVR is not hermitian and computations with complex numbers are required, all the linear algebra calculations are carried out using standard LAPACK and BLAS routines,²⁸ optimized for such goal.

We are currently trying to merge the calculation of rate constants and sink terms, still not available in Gaussian, inside our framework (vide infra). Also many efforts are being done in order to ease the tedious input stage via a Guided User Interface (GUI) in the framework of the Virtual Multi-frequency Spectrometer (VMS) project; a project currently under active development in our group.^29–31

3. Test-case

In this section we show the reliability of the program by comparing the results obtained for a specific case where analytical solution is available. In particular this is the case for one state with a parabolic potential V(x) = c₁x², constant diffusion coefficient D, and a parabolic sink term s(x) = c₂x² for a chosen normalized initial distribution (here we have chosen p(0, t) ≡ p_eq(x)). For this case the survival probability Q(t) defined as

$$Q(t)={\displaystyle {\int }_a^{b}p(x,t)dx}$$ (14)

is analytical, and it reads³²

$$Q\left(t\right)=\frac{\sqrt{8\gamma }{e}^{-D{c}_1\left(2-\gamma \right)t/\gamma }}{\sqrt{{\left(2+\gamma \right)}^2-{\left(2-\gamma \right)}^2{e}^{-4D{c}_{1}t/\gamma }}}$$ (15)

where γ is defined as

$$\gamma =\frac{2}{\sqrt{1+\frac{c_2}{D{c}_1^2}}}$$ (16)

The survival probability gives information about the cumulative probability of not reacting till time t.

From now on we express the potential V(x) in k_BT units (multiplied by β) for a constant temperature of 300 K.

We have chosen c₁ = 0.25, c₂ = 0.5, D = 10⁻⁶ s⁻¹ and the number of DVR grid points is α = 100, in the domain [−4.0, 4.0]. In Figure 1 we compare the resulting profile of Q(t) and the analytical one, eqn. (15). The two profiles are identical, confirming the consistency of our approach. The agreement has been verified also for steeper potential (higher values of c₁), different diffusion and sink magnitudes and diffusional space domains. Here we have shown the limit in which both diffusion and reaction take place, that is one of the most problematic case where space-dependent sink term is present. The reliability of the method in the case of free diffusion and/or no reactivity has been shown in our previous work, ref.,¹⁶ guaranteeing the performance also in other simpler limits.

Figure 1.

Plot of the survival probability for the test-case. The black line is the calculated survival probability with our code, while the red line is the analytical one, eqn. (15). The physical and diffusional parameters are reported in the text.

Application

In this section we apply the whole framework presented till here to a specific case-study of chemical interest. We focused on the prototype TICT system of (N, N-dimethylamino)benzonitrile (DMABN) molecule depicted in Figure 2, studied with similar approaches in the past by Polimeno and coworkers, ref.15 This system and in general TICT systems are characterized by a high dipole moment in the excited singlet state S₁ when excited from the ground state S₀ minimum. The system then undergoes relaxation via the intramolecular rotation of the dimethyl group along the C–N bond (dihedral angle θ in Figure 2) reaching a potential minimum. Inversely, in this situation the dipole reaches its maximum value, and this state is known as TICT state. In the singlet excited state and in particular in the twisted situation, charge-transfer from S₁ occurs via a deactivation decay mechanism (both radiative and non-radiative in nature).

Figure 2.

Panel a; sketch of the DMABN molecule depicting the optimized ground state geometry calculated at the level of CAM-B3LYP/6-31+G* in water. Panel b; starting conformation for the scan along the highlighted internal coordinate θ.

Inspired by the previous work of Pedone and coworkers, ref.,⁴ our objective is to determine the excited state dynamics of the population density p_ES(x, t) from which it is possible to calculate the time resolved emission spectra. By using our established modular framework we performed a relaxed scan calculation for both ground and excited states along the θ internal coordinate from which the generalized coordinate x, the ground V_GS(x) and excited state V_ES(x) potential energies and the diffusion functions, D_GS(x), D_ES(x) are calculated. Finally using quantum mechanical calculations (vide infra) we have calculated the reactive rate constant from the excited state to the ground state, k_ES,GS(x). We neglected the possibility of population disappearance and so we didn’t consider the sink term s_ES(x). Also, since our focus is on the excited state dynamics and since the laser pulse is not continued after the instantaneous irradiation, the rate constant from the ground state to the excited state, k_GS,ES(x) is assumed to be zero.

S₀ and S₁ energetic and diffusion properties have been calculated by means of density functional theory (DFT) and time dependent (TD)-DFT³³ relaxed scan computations done with a development version of Gaussian,¹⁸ employing hybrid exchange-correlation functional CAM-B3LYP³⁴ and using 6-31+G* basis set. We chose the long-range corrected CAM-B3LYP functional since it is known that B3LYP functional³⁵ overestimates the energies of charge transfer states, for example as seen in the work of Pedone and co-workers for coumarin derivatives. Bulk solvent effects of water have been included by means of the polarizable continuum model (PCM).³⁶

In panel a of Figure 3 we report V_GS(x) and V_ES(x) profiles, while in panel b we report the oscillator strength for the S₀ → S₁ transition as a function of the generalized coordinate x. The ground state has a minimum at θ = 0°, while at the same angle, the excited state has a maximum, corresponding to the locally excited (LE) state. Specular to this, the situation is inverted at θ = 90°, where the excited state has now a minimum that is the TICT state, separated from the LE state by 9.018 k_BT units. Just to have an idea, this is the order of magnitude of rotation barriers in common alkanes, e.g. n-butane,.^17,37 The oscillator strength is widely affected by the twist of the angle θ reaching its minimum at θ = 90°; this suggests that at room temperature, the TICT state should not be populated.

Figure 3.

Panel a; calculated potential energies of S₀ (black line) and S₁ (red line) states, of DMABN from the relaxed scan in water. Panel b, oscillator strength plot for the S₀ → S₁ transition. In the upper axis is shown the corresponding internal coordinate θ along which the scan is performed. (T = 300K) As expected, the oscillator strength falls from the LE to the TICT state, which is dark.

Figure 4a shows the diffusion function profiles along the generalized coordinate, for the two states. Since there is no direct correlation between energetics and diffusion, providing quantitative inferences about the profiles’ shape is not trivial. Qualitatively, we can state that they reflect the geometry change of the system step by step during the relaxed scan. In Figure 4b we see the plot of the dipole moment μ in the excited state; as expected, the LE state has a lower dipole moment around 14.6 D that increases till its maximum (19.8 D) in the TICT state. The dipole moment at the ground state minimum is 10.4 D.

Figure 4.

Panel a; diffusion function along the generalized coordinate for both the ground and excited states. Panel b; variation of the dipole moment of the S₁ state along the generalized coordinate. In the upper axis is shown the corresponding internal coordinate θ along which the scan is performed. (T = 300K) The dipole moment increases by approximately 5 D from the LE to the TICT state.

The rate constant term k_ES,GS(x) accounts for the continuous depletion of the excited state population due to radiative and non-radiative decay.

$$k_{\text{ES},\text{GS}}(x)=k_{\text{ES},\text{GS}}^{\text{r}}(x)+k_{\text{ES},\text{GS}}^{\text{nr}}(x)$$ (17)

These two separated contributions have been calculated using Fermi’s golden rule (main theory and equations here used are reported in Appendix B). In Figure 5 we show the computed profiles of the radiative and non-radiative rate constants $(k_{\text{ES},\text{GS}}^{\text{r}}(x)$ and $k_{\text{ES},\text{GS}}^{\text{nr}}(x))$ as a function of the generalized coordinate. The non-radiative contribution dominates over the radiative one by a factor around 72. This compares fairly well to the ratio of 30 between the two for DMABN in n-butyl chloride at 150 K.¹⁵ As expected, the emission rate is near zero at the charge transfer state conformation, because of orbital symmetry. Symmetrically, the non-radiative rate is lower at the LE state and reach a maximum around the TICT state.

Figure 5.

Panel a; radiative rate constant along the generalized coordinate; $k_{\text{ES,GS}}^{\text{r}}\left(x\right)$ falls sharply from the LE to the TICT state. Panel b; non-radiative rate constant along the generalized coordinate; $k_{\text{ES,GS}}^{\text{nr}}\left(x\right)$ increases from LE to TICT state which is predominantly a dark state. In the upper axis is shown the corresponding internal coordinate θ along which the scan is performed.

After we have calculated all the above mentioned energetic, diffusion and reaction input parameters, we followed the excited state population dynamics p_ES(x, t) by means of eqn. (1). Just before the radiation pulse excitation, the ground state population is assumed to be at equilibrium, and after the laser pulse excitation the excited state population resembles the ground state one. Consequently, as initial population we have chosen

$$p_{\text{ES}}(0,t)=p_{eq,\text{GS}}(x)$$ (18)

In other words, the ground state energetics provides the initial distribution function on the excited state, after photon excitation.

In Figure 6 we plot the calculated probability density profile for the excited state, p_ES(x, t) as a function of the generalized coordinate x and time t, while in panel b we report the same computed at several times. The initial population quickly diffuses towards the TICT state and after 15 ps it is redistributed at the minimum; at the same time, it is disappearing due to the combined effect of reactive contribution and after 50 ps it relaxes back completely to the ground state. The migration to the TICT dark state (radiative contribution is approximately zero in this conformation) decreases the fluorescent population. The rate of this decay is affected both by diffusion and reaction.

Figure 6.

Panel a; calculated probability density profile along the generalized coordinate and along time. Panel b; The same probability density profile extrapolated at several times; in the upper axis is shown the corresponding normalized coordinate z used in the time resolved emission spectra calculation (see Appendix C).

In order to better state this effect we have calculated the time resolved emission spectrum of DMABN molecule using p_ES(x, t) using the relations reported in Appendix C. In panel a of Figure 7 we report the calculated time resolved fluorescence spectra as a function of the frequency ν and time t, while in panel b we report the same computed at several times. The parameters used for the line shape function, g are the asymmetry parameter γ = −0.4 and the bandwidth Δ = 3300 cm⁻¹, while the electronic transition moment, M(z) is collected from the Gaussian scan calculation output. It can be seen that the peak position shifts to red and the fluorescence intensity gradually decreases till a total loss of fluorescence emission occurs. In Figure 8 we have collected the peak intensities as a function of time (black dots and line) and fitted the peak intensity decay with a mono-exponential function (red line) in order to retrieve an approximate life time; τ = 12.98 ps. This value is in good agreement with the measured life time of 15 ps of DMABN in water.³⁸

Figure 7.

Panel a; calculated time resolved emission spectra. Panel b; The same spectra extrapolated at several times.

Figure 8.

Calculated peak intensities of the DMABN time resolved spectrum along time (black dots and line) and mono-exponential fit (red line).

We also performed a similar set of calculations in n-hexane solution to determine the variation of the radiative and non-radiative rates with the rotation of the dimethyl group along the C–N bond. Hexane, being a non-polar solvent; results in a more pronounced character of the LE state and therefore, the relative variation of the radiative and non-radiative rates between the LE and TICT states is different from that in water. For example, we see that the dipole moments of both states are lower by about 4 to 5 D in hexane, compared to those in the highly polar solvent water. The transition dipole moment shows an initial increase corresponding to the LE state and then from θ ~ 30°, starts falling steadily to zero, as expected for the TICT state. This leads to an initial increase in the rate of radiative decay (in the order of 10⁷ s⁻¹), followed by a steady decay to zero. This agrees very well to the already-mentioned fact that the TICT state is dark.

On the other hand, the rate of non-radiative decay shows a slow increase till θ = 30°, from around 7.6 × 10⁸ s⁻¹ to 1.7 × 10⁹ s⁻¹, and then shows a more pronounced increase till 3.2 × 10⁹ s⁻¹ corresponding to the dark TICT state. The data agree fairly well to previously reported data for DMABN in n-butyl chloride at 150 K, as also observed before for our simulations performed in water.¹⁵

4. Conclusions

We have presented in this work a general approach for the solution of coupled reactive one-dimensional Smoluchowski equations. This updated approach embedded in the Gaussian framework covers a wide class of chemical problems and overcomes the limitations of previous models. It allows for a generic treatment of diffusion coupled to reactivity coupled also to different possible diffusional states. The reliability of the solution is guaranteed by the consolidated numerical method of DVR presented in a past work of us. The inclusion of a generalized coordinate along which the system is evolving across more general paths give access to more detailed informations.

We have shown the robustness of our implementation with a test case, by comparing the calculated survival probability with an analytical one for a specific diffusional problem. Then we have considered a concrete example of chemical interest that is the fluorescence intensity decay for a TICT molecule. The computed evolution of the probability density of the excited state coupled to the ground state with a specific rate constant gives access to the spectra calculation. The modularity of the implemented code allows the integration of scan calculations with the computation of the generalized coordinate and the diffusion tensor along the same, reducing the input stage efforts for the generic user. If needed, these informations can also be given manually.

A recent diabatization approach based on the dipole moments of the electronic states and the corresponding transition dipole moments was used to compute the rates of radiative and non-radiative transitions for the depletion of the excited state population. A definitive implementation of such theory in the above mentioned framework is currently under work in our group.

There are encouraging perspectives about the extension of the approach to more than one internal generalized coordinate leading to multidimensional coupled reactive Smoluchowski equations; this will be the main objective of our future investigations. Also a more versatile graphical input format interface, using the VMS software, is currently under implementation. It will facilitate the creation of complicated input files for diffusional computations and the understanding of difficult output files, allowing direct visualization of the results. The modularity of the software constructed as an integrated environment for electronic and magnetic spectroscopies will function as a base for future comparisons between collected experimental results and theoretical ones. The embedded scientific data visualizer for the analysis and processing of data will then ease the interpretation of results with the aid of different graphical tools.

In conclusion, apart from the further improvements and developments mentioned above, we think that, we already have at our disposal a quite powerful “black-box” machinery allowing us to complement experimental and theoretical studies for a number of diffusional problems of fundamental and applicative interests.