the Static and Dynamic Description of
Photoinduced Processes in Embedded
Systems
Author:
Maximilian F. S. J. Menger
Supervisors:
Prof. Benedetta Mennucci
Prof. Leticia González
Dipartimento di Chimica e Chimica Industriale
A Thesis submitted in fulfilment of the requirements for the degree of
Doctor of Philosophy in the
Chemistry and Material Science
Titel der Dissertation / Title of the Doctoral Thesis
“Development of Multiscale Models for the Static
and Dynamic Description of Photoinduced
Processes in Embedded Systems”
verfasst von / submitted by
Maximilian F. S. J. Menger M.Sc. B.Sc.
angestrebter akademischer Grad /
in partial fulfillment of the requirements for the degree of
Doktor der Naturwissenschaften (Dr. rer. nat.)
Wien, 2018
Studienkennzahl lt. Studienblatt /
degree programme code as it appears on the student record sheet:
A 796 605 419
Dissertationsgebiet lt. Studienblatt /
field of study as it appears on the student record sheet: Chemie
Betreut von / Supervisors: Univ.-Prof. Dr. Leticia González Univ.-Prof. Dr. Benedetta Mennucci
The detailed knowledge of the processes following photoexcitation is ex-tremely important, but the understanding of the underlying mechanisms, as well as the proper description of the corresponding non-adiabatic dynamics are still real challenges for theoretical chemistry. This is true especially for biologically relevant systems, where not only the chromophores, but also the complex environment need to be included in the simulation in order to obtain a correct picture. The environment, in fact, can play an important role in the mechanisms of the photoinduced processes by tuning both the electronic and structural properties of the chromophore(s). In this thesis, we present novel computational multiscale models, based on a hybrid quantum mechanical/molecular mechanical (QM/MM) description of the system able to study such processes.
First, the theoretical formulation and implementation of excited states gradients of a polarizable QM/MM method using an induced dipole scheme are introduced within the framework of time-dependent density functional theory. Secondly, the development and the implementation of a novel exciton approach are presented and discussed, where the environment is included via an electrostatic or mechanical embedding scheme. Consistent hybrid QM/MM energies, gradients and non-adiabatic couplings were developed, allowing ab initio non-adiabatic dynamics simulations in multichromophoric systems, using the surface hopping approach.
The methods are finally applied to the study of (i) the effects of a DNA pocket on the excitation process and the corresponding excited state proper-ties of an organic dye, and (ii) the excitation energy transfer process in an orthogonal molecular dyad.
Al giorno d’oggi è di fondamentale importanza la conoscenza dettagliata dei processi seguenti una foto-eccitazione, tuttavia la piena comprensione di tali meccanismi e la precisa descrizione delle corrispondenti dinamiche non-adiabatiche rimangono una vera sfida nel campo della chimica teorica.
L’ambiente infatti, può influenzare notevolmente il meccanismo di un pro-cesso foto-indotto variando (influenzando) sia la struttura elettronica del cromoforo sia la sua conformazione geometrica. In questa tesi, presentiamo un modello computazionale multi-scale basato su una descrizione ibrida del sistema che permette di studiare questo tipo di processi combinando la teoria quanto meccanica con la meccanica molecolare. Innanzitutto sono presentate la formulazione teorica e l’implementazione dei gradienti degli stati eccitati di un metodo polarizzabile QM/MM usando uno schema di dipoli indotti nel con-testo della teoria del funzionale densità nella sua forma dipendente dal tempo. In secondo luogo viene presentato e discusso lo sviluppo e l’implementazione di un innovativo approccio per la descrizione di eccitoni. In questo modello, l’ambiente viene incluso tramite embedding scheme (uno schema ad incastro) meccanico o elettrostatico.
Energie, gradienti e accoppiamenti non-adiabatici sono stati sviluppati in maniera coerente al modello ibrido QM/MM con i quali è stato possibile condurre simulazioni non-adiabatiche ab-initio di sistemi multi-cromofori usando il metodo surface hopping. Entrambi I metodi sono stati applicati per lo studio (i) degli effetti di una tasca del DNA sul processo di eccitazione e le proprietà corrispondenti agli stati eccitati di un pigmento organico ed (ii) il processo di trasferimento energetico eccitato in una diade ortogonale.
Detalierte Kenntniss von den chemischen und physikalischen Prozessen, die einer Photoanregung folgen, ist von großer Bedeutung. Allerdings stellt so-wohl die richtige Beschreibung des zugrundeliegenden Mechanismus als auch die Simulation der entsprechenden nicht adiabatischen Dynamik eine echte Herausforderung für die theoretische Chemie dar. Dies gilt besonders im Fall von biologisch relevanten Systemen, bei denen nicht nur die Chromophore, sondern auch die molekulare Umgebung in der Simulation berücksichtigt wer-den müssen, um eine korrekte Beschreibung sicherzustellen. Die Umgebung kann hierbei eine fundamentale Rolle im Mechanismus des photoinduzierten Prozesses einnehmen, indem sie sowohl die elektronischen Eigenschaften als auch die Struktur des Chromophores (der Chromophore) grundlegend verändern kann. In dieser Arbeit stellen wir neuartige Berechnungsmethoden vor, die, basierend auf einer gemischt quantenmechanischen und molekularen Mechanik (QM/MM) Beschreibung des Systems, in der Lage sind, photodin-duzierte Prozesse zu simulieren.
Zuerst wird die theoretische Formulierung und Implementierung von Gra-dienten angeregter Zustände vorgestellt. Die Umgebung wird hierbei mit Hilfe eines polarisierbaren Kraftfeldes beschrieben und das quantenmechanische System mit zeitabhängiger Dichtefunktionaltheorie. Als nächste Methode wird ein Exziton Modell, für nicht adiabatische Moleküldynamik mit der Sur-face Hopping Methode, entwickelt. Die Umgebung wird in unserem Exziton Model ebenfalls mit einer QM/MM Methode beschrieben und die Kopplung zwischen QM Region und MM Teilsystem wird entweder mittels elektrostati-scher oder mechanielektrostati-scher Interaktion beschrieben.
Abschließend werden die entwickelten Methoden verwendet um (i) den Effekt von DNA auf die Eigenschaften der angeregten Zustände eines organi-schen Farbstoffes und (ii) den Anregunsenergieübertragungsprozess in einer orthogonalen molecularen Dyade zu beschreiben.
1 introduction 1
2 theory 5
2.1 Electronic Structure Theory . . . 5 2.2 QM/classical approaches . . . 14 2.3 Nuclear Dynamics . . . 26
3 method development 31
3.1 Molecular Properties . . . 31 3.2 Sharc . . . 37 3.3 An exciton model for Surface Hopping . . . 45
4 applications 51
4.1 DNA-DAPI: A test study for TDDFT/MMPol . . . 51 4.2 BODT4: A test study for Surface Hopping within the Exciton
Model . . . 61
5 conclusion 72
a appendix 74
a.1 AMBER/MMPol Interface and PySANDER . . . 74 a.2 SHARC/Gaussian Interface . . . 75 a.3 Gaussian External Optimizer . . . 76
I N T R O D U C T I O N
1
The accurate description of photoinduced processes in (supra)molecularsys-tems represents a real challenge for quantum chemistry. A proper description can only be provided if both excited states and structures can be modeled correctly. Although quantum chemistry has greatly benefited from the rapid progress in computer technology, which has led to tremendous improvements in computational speed and memory size, as well as, the development of more efficient algorithms, it is still not possible to use accurate wave function based ab initio methods, like coupled cluster (CC) or complete active space self-consistent field (CASSCF) approaches for medium to large molecular systems. Even with computational efficient schemes, like density functional theory (DFT), the limitations due to the system size lies within a few thousand atoms for single point calculation, not to mention molecular dynamics, where several thousand of these computations need to be performed in a single simulation. From this brief description it appears clearly that if we move from isolated (supra)molecular to complex biological systems, the hypothesis of using a fully quantum chemical description becomes unfeasible. For many bi-ologically interesting systems, however, the photoinduced process of interest is generally localized in a small region, while the by far largest part of the molecular system acts as an embedding environment. The description can thus be different for the two parts, as long as an accurate modelling of the interactions between the two parts is provided. In fact even if the environ-ment does not play an active role in the process it can affect the properties of the subsystem involved in the light-driven process and have a tremendous impact both on the underlying mechanism and the final output.
An efficient strategy to include the environment is to introduce a so-called focused, or multiscale model. The development of these methods was honored by the scientific community, in 2013, when Karplus, Levitt and Warshel got the Nobel prize in Chemistry for their work on “the development of multiscale models for complex chemical systems”. In multiscale models, instead of performing a high-level quantum mechanical description of the full system, one focuses only on an accurate description of the local region, where the process under study happens. The remaining region, the environment, is instead treated at a classical level. This can lead to a tremendous reduction in computational costs, while still capturing the important features of the processes.1–6
Over the years many flavours of multiscale models were developed and can be roughly divided into two regimes, explicit and implicit models. In explicit models the environment is included explicitly in the simulation (see Fig. 1). The most prominent family of explicit models are QM/MM (quantum mechanics/molecular mechanics) methods,2,4,6 where MM force-fields are employed to model the environment. Other explicit models are represented by subsystem schemes, like frozen density embedding,7or the Effective
Frag-Figure 1: Implicit solvation approach using continuum models for the environment (left) and explicit solvation e.g. in terms of QM/MM (right).
ment Potential formulation.8,9In implicit models, instead, the environment is described as a dielectric medium characterized through its macroscopic prop-erties, like its dielectric constant (see left panel of Fig. 1).1,3,5,10 Within this framework, the degrees of freedom of the environment disappear, reducing the dimensionality of the system dramatically.
Figure 2: Different couplings schemes to include the environment in a QM/MM manner with respect to the gas phase reference (a). In mechanical embedding (b), the environment has no influence on the QM density. In electrostatic embedding (c) the environment is included in the QM calculation in terms of fixed point charges (or multipoles) and can polarize the QM density, but does not get polarized itself. And in panel (d) the induced dipole formulation of polarizable QM/MM is seen, hereby a mutual polarization between the QM and MM region occurs.
The most crucial part in multiscale models is the coupling between the QM subsystem and the environment. In most cases, the dominant part of the interaction is represented by Coulombic interactions. In many formulation of QM/continuum models such an interaction is modeled via the use of apparent surface charges spreading on the surface of the molecular cavity.1 The two most prominent representatives of this method are the polarizable continuum model (PCM)1,10and the conductor like screening model (COSMO).11,12For hybrid QM/MM methods on the other hand, different coupling schemes are commonly used (for an overview see Fig. 2). In mechanical embedding all interactions are treated purely at the MM level of theory, while in electrostatic and polarizable embeddings, the interaction is treated with a hybrid QM/MM scheme.4,6,13 Nowadays, these hybrid QM/classical methods (QM/MM and QM/continuum) are the state-of-the-art for multiscale approaches and have been successfully employed to study a wide range of (bio)chemical prob-lems.2,4,14
For the study of multichromophoric systems, pure QM/classical methods are still not enough to treat the full system. For such cases exciton models were successfully applied to describe the electronic structure of the whole system.15–19The computationally advantage of these schemes is based on the approximation that the system can be divided into subsystems (the chro-mophores) that are weakly coupled. Hence, it is possible to compute each subsystem separately and include the interaction between the systems after-wards, via perturbation theory, reducing the overall costs of the calculation tremendously.
For a proper description of light-induced processes, however, a purely static picture is often not enough and the non-adiabatic dynamics of the system has to be taken into account. The dynamics of an isolated quantum system can be described by the time-dependent Schrödinger equation (TDSE), which defines the equations of motion of the wave function of the system Ψ(x,t)20
i ∂
∂tΨ(x,t) = H(t)Ψ(x, t) (1)
The degrees of freedom of the system are labeled with x and the (time-dependent) molecular Hamilton operator of the system is given with H. The time evolution of the system is described by this first-order differential equa-tion, that can be solved by defining an initial wave function Ψ(x,t0), followed by a propagation. Even if this equation represents a simple mathematical problem, the numerical solution is exceptionally computationally demanding and can only be carried out for small systems considering only a few degrees of freedom. This holds still true even if computational efficient algorithms, like the multi-configurational time-dependent Hartree (MCTDH) method, are used.21–24Instead of describing the system by its total wave function, mixed quantum classical approaches simplify the problem significantly by separat-ing the nuclear and electronic degrees of freedom. The nuclei are propagated as classical particles following Newton’s equation, while the electronic part of the wave function is propagated with the time-depending Schrödinger equation. Trajectory surface hopping25–28is a well-established representative
of the mixed-quantum classical dynamics approaches. Hereby, one takes advantage of the fact, that in most cases the Born-Oppenheimer approxi-mation is valid, and the nuclear motion is governed by a single electronic potential energy surface. For each trajectory, the nuclei are propagated on the potential of a single active state, but depending on the non-adiabatic coupling between the states, stochastic hops may appear, changing the active state. To approximate the nuclear wave function an ensemble of independent trajectory are used in a single simulation.
To overcome the computational limitations of (wave function based) ab initio methods we present in this Thesis our contribution to the existing methodologies. The focus of the models is an adequate description of the QM region, employing DFT and time-dependent DFT (TDDFT) as the QM method of choice, in combination of a (polarizable) MM description of the environment. To study the excited states of large multichromophoric system, we present an exciton model including electrostatic embedding designed for non-adiabatic dynamics using the well-established surface hopping approach. With the presented methods it is possible to study the excited state (dynamics) of medium to large (multi)chromophoric systems. The work is organised in three parts. In the first part we will recall the state of the art of the QM methods and their extension to dynamics utilized in this work. In the second part we present the theoretical framework of the developed multiscale models and in the final part the developed tools are applied to selected systems.
T H E O R Y
2
2.1 electronic structure theoryBy assuming valid the Born-Oppenheimer approximation, the electronic Hamiltonian of a generic molecular system can be written as:29,30
H = Vext +T + Vee (2) Vext = X A,B6=A ZAZB |RA− RB | {z } VN N −X i X A ZA |ri − RA| | {z } Vne (3) T = X i −1 2∇ 2 i (4) Vee = X i,j6=i 1 |ri− rj| (5)
whereVext is the external potential, and for atoms and molecules without external field, it contains the Coulombic interaction of the clamped nuclei with the electrons (Vne) and among themselves (VN N). The last two terms are the kinetic energy operator of the electronsT and the electron-electron interactionVee; both of them do not depend on the specific system and are only dependent on the number of electrons. Within this context, only a parametric dependence on the nuclei position is considered and an exact solution of the equation is only possible for one-electron systems, as for all other many-body effects play a fundamental role and one has to employ approximations.
The most fundamental approximation is the independent-particle approx-imation, also known as the Hartree-Fock method.31 Hereby, the N-body problem is approximated by N one-body problems. The electron-electron interaction term that coupled the one-particle equations before is now ap-proximated with a self-consistent mean-field.31
In the Hartree-Fock method the wave function is defined as an antisym-metric sum over Hartree products of one-particle wave functions (orbitals) ϕp(rq), to satisfy the physical properties of fermions. It is typically written in form of a single determinant, the so-called Slater determinant
ΨHF0 = √1 N ! ϕ1(r1) ϕ2(r1) · · · ϕk(r1) ϕ1(r2) ϕ2(r2) · · · ϕk(r2) .. . ... ... ... ϕ1(rN) ϕ2(rN) · · · ϕk(rN) (6)
Hereby, rq represents the spatial and spin coordinates of an electron. Using this ansatz for the wave function the Hartree-Fock energy can be obtained as
the expectation value of the wave function with the corresponding Hamilton-ian.
E0HF = ΨHF0 |H | ΨHF0 (7)
To find the optimal wave function within the Hartree-Fock method the vari-ational principle is employed, that states that any trial wave function will result in an higher (or equal) energy than the exact groundstate wave function (equality in energy is only valid if the trial wave function is already the exact wave function).
E[ ˜Ψ] = ˜Ψ |H | ˜Ψ ≤ hΨ0|H | Ψ0i =EHF0 (8) The optimal orbitals and therefore the Slater determinant, that leads to the lowest energy can be found by finding the stationary point of the energy with respect to the wave function. Hereby, it is imposed that the orbitals remain orthogonal throughout the minimization. With this the set of canonical Hartree-Fock equations can be obtained.31
εpϕp = h(1)|ϕpi +X i ϕi 1 r12 ϕi |ϕpi − ϕi 1 r12 ϕp |ϕii = F|ϕpi (9)
Hereby, hp denotes the all one-body operators in the electronic Hamiltonian for orbitalϕp: h(1) = −1 2∇ 2 1− X A ZA |r1− RA| (10)
The electrons move as independent particles in an averaged potential spanned by all nuclei and the other electrons. Because of this dependence of the Fock-Operator F on the orbitals and hence, on the solution of Eq. 9 it can only be solved in a self-consistent manner, meaning that one starts with an initial set of guess orbitals and inserts them into the Hartree-Fock equations, then one obtains a new, improved set of orbitals and plugs these again into Eq. 9. This continues until the change in the energy and in the HF orbitals is negligible.
Due to the averaged potential, the electronic correlation is not included. In fact, the correlation energyEc is defined as the difference of the exact ground state energy of the systemE0and its Hartree-Fock energyE0HF.
Ec =E0−E0HF (11)
Typically, the Hartree-Fock method is used as the starting point for the development of improved methods, like Møller-Plesset perturbation theory (MPn) or the complete active space self consistent field (CASSCF) method.
2.1.1 Density Functional Theory
In contrast to wave function based methods, density functional theory (DFT) tries to compute the energy and most properties of the quantum system in terms of the electronic densityρ(r).30,32The clear advantage of this procedure is that the complicated 3N (4N with spin) dimensional problem can be broken down to a 3 (4) dimensional problem, as all needed information is already contained within the electronic density.
ρ(r) = N · Z
dr2· · ·drN|Ψ(r, r2, . . . , rN)|2 (12) The main theorems for DFT were derived by Hohenberg and Kohn in their famous paper on the “inhomogenous electron gas“ in 1964.33In 1998, Kohn together with Pople received the Nobel Prize in chemistry for his contributions to the field of theoretical chemistry, especially for the development of DFT and wave function theory, respectively.
The first Hohenberg-Kohn theorem (HKI) ensures that the electronic den-sity for a non-degenerate molecular system, defines uniquely the external potential and therefore as well the electronic wave function. It is therefore possible to write the electronic wave function as a functional of the electronic density:
Ψ(r1, r2, . . . rN) = Ψ[ρ(r)](r1, r2, . . . rN) (13) Hence, the energy can be written as a functional of the electronic density, as it is a direct functional of the wave function, which is a functional of the electronic density.
E[ρ(r)] = hΨ[ρ(r)] |Vext +T + Vee| Ψ[ρ(r)]i (14) =
Z
ρ(r)Vextdr + F [ρ(r)] (15)
The first term consists of the external potential. In case of a molecule in gas phase this term includes only the interaction of the nuclei with the electronic density. The last termF [ρ(r)] represents a universal functional of the electronic density. If one is able to find a correct expression for this functional, then it is possible to solve the electronic Schrödinger equation exactly within DFT.
The second Hohenberg-Kohn theorem (HKII) ensures the existence of the variational principle for the electron densities. Analogously to Hartree-Fock, it is possible to start with a trial density and iteratively converge in a self-consistent manner to the best density within the given approximation for the corresponding system.33,34
The remaining difficulty in DFT is to find good approximations of the universal functional, as no exact solution is known. Hereby, the problem is that neither for the kinetic energy operator nor for the exchange contributions a functional form with respect to the electronic density is known. It should be noted that the kinetic energy term has the largest contribution to the energy
and hence, should be approximated accurately. The simplest approximation represents the Thomas-Fermi approximation:35,36
T [ρ(r)] = 3 10 3π
22/3 Z
drρ(r)5/3 (16)
Although it is exact for a uniform electron gas, it represents a very crude approximation for most chemically relevant systems, where the electronic density is not uniform. This holds true even if gradient corrections are introduced, like the von Weizsäcker approximation,37which is exact for one-orbital systems. T [ρ(r)] = 1 8 Z dr|∇ρ(r)|2 ρ(r) (17)
Due to this limitation orbital free DFT is rarely used for chemical computa-tions.
Instead, the Kohn-Sham formalism of DFT (KS-DFT) is largely used to approximate the kinetic energy functional.38In the following KS-DFT and DFT will be used as synonyms. KS-DFT solves in part the problem by rein-troducing a wave function like description of the system. The correlated many-body problem is hereby approximated by a uncoupled single-particle theory, analogously to the mean-field approach in Hartree-Fock.30And the real system is approximated as a fictitious system of independent particles moving in an averaged field spanned by all the other electrons and the nu-clei under the constraint that the fictitious system and the real system both share the same electronic density. For the independent particle system the electronic density can be obtained as the sum of the electronic densities of the single-particle functions (Kohn-Sham orbitals):
ρ(r) =XN i
|ϕi(r)|2 (18)
The kinetic energy of the independent particlesTscan be written in the same way as in Hartree-Fock theory in dependence of the Kohn-Sham orbitals.
Ts[ρ(r)] = N X i −1 2 ϕ i ∇2 ϕi (19) That way the main kinetic energy contribution of the system can be computed. The total energy functional is therefore given by:
E[ρ(r)] = Zρ(r)Vextdr + Ts[ρ(r)] + EH[ρ(r)] + Exc[ρ(r)] (20) EH[ρ(r)] is the Hartree energy, that can be written directly as a functional of the density: EH[ρ(r)] = 1 2 Z Z drdr0ρ(r)ρ(r0) |r − r0| (21)
All unknown parts regarding exchange and correlation contributions that map the independent particle system to the real correlated system are contained in the exchange-correlation functionalExc. It also contains the correlation energy contributions of the kinetic energy operator, as the kinetic energy of the Kohn-Sham systemTs does not give completely the kinetic energy contributions of the real system.
Tc[ρ] = T [ρ] − Ts[ρ] (22)
TheTc is rather small and is as well included into the exchange-correlation functional.
The contributions of the exchange-correlation functional to the total energy are rather small and it can be obtained either by fitting against an (empirical) data set or it is approximated on the basis of model systems or exact constraints. For the former method pioneering work has been done by Becke and the latter by Perdew. Almost all of the presently used density functionals have been influenced by their work. Nowadays a manifold of different exchange-correlation functionals exist with different strength and weaknesses and the main challenge is to choose the “right” one out of these, for the current problem.
Once the exchange-correlation functional is chosen, one can solve the Kohn-Sham system in a very similar way, as the Hartree-Fock method and obtain in analogy the Kohn-Sham equations for the independent particles.
−1 2∇ 2 i +Ve f f(r) ϕi(r) =εiϕi(r) (23)
The Kohn-Sham potentialVe f f(r) is given by: Ve f f(r) =Vext(r) + Z dr0 ρ(r0) |r − r0| + δExc[ρ(r)] δρ(r) (24)
Generally, the matrix elements of the Fock-Operator can be written in the following way:
Fpq =hpq+X i
[hpi|qii − cxhpq|iii] + fpqxc (25)
Hereby, as throughout this whole work, the common notation is used, where occupied molecular orbitals (MOs) are labeledi, j, · · ·, virtual orbitals a, b, · · · and generic orbitalp, q, · · ·. Greek letters (µ, ν, · · ·) are used as indices for atomic orbitals. The scaling parametercx is introduced to interpolate between Hartree-Fock (cx = 1,fxc = 0) and pure DFT (cx = 0). fpqxc are the matrix elements of the exchange-correlation kernel:
fpqxc = ϕp δExc[ρ(r)] δρ(r) ϕq (26) Although, KS-DFT increases the computational cost compared to orbital free DFT by several orders of magnitude, it has become the most popular
DFT approach and was successfully applied to a wide range of chemical problems. One additional reason for this, is its similarity with Hartree-Fock from a mathematical or programming point of view (although the physical assumptions are fundamentally different as DFT is formally exact). This made it relatively simple to implement KS-DFT into existing Hartree-Fock programs.
2.1.2 Time-dependent Density Functional Theory
DFT is a pure ground state method, although it can be tweaked to compute some particular excited states by imposing constraints. The probably easiest constraint one can think of is the spin multiplicity. It is possible to use a ground state method to compute the ground state energy of every accessible spin multiplicity. The excitation energy is then simply given as the energy difference between the total energies of the electronic ground state and the electronic excited state. Therefore, this type of method is in literature often called ∆−SCF method.39 Another more general approach to compute the electronically excited states of a system within the framework of DFT is its time-dependent formulation TDDFT (time-dependent density functional the-ory). Hereby, TDDFT relies on the Runge-Gross theorem,40 which represents the time-dependent analogue of the first Hohenberg-Kohn theorem. The time-dependent external potentialVext(R, r,t) is determined by the (exact) time-dependent electron density and the same holds for the electronic wave function up to a time-dependent phase factor:
Ψ(R, r,t) = Ψ[ρ(t)](t) exp[−iα(t)] (27)
The equations of motion of the time-dependent electronic wave function are of course given by the time-dependent Schrödinger equation. The second requirement for TDDFT is that the (exact) time-dependent density can be obtained by applying the variational principle to the quantum mechanical action integral. A[ρ] = t1 Z t0 dt Ψ(R, r,t) i ∂ ∂t − H(t) Ψ(R, r,t) (28)
The exact electron density can be obtained as the stationary point of the action integral and thus it has to full fill the Euler equation:
δA[ρ]
δρ = 0 (29)
Like in ground state DFT, the functional form of the exchange-correlation functional is not known and additionally the so-called memory problem of TDDFT arises, which states that the time-dependent exchange-correlation functional at timet does not only depend on the current density at the same time, but also on the history of densities ρ(r, t0 < t) of all previous times
and the initial state. Although memory-dependent functionals have been developed,41,42 the standard approach is to use the adiabatic approximation, which approximates the time-dependent exchange-correlation functional with a time-independent one. The underlying assumption is that the exchange and correlation potential changes instantaneously with every change in the electron density and therefore no memory effects appear. In practice the existing ground state exchange-correlation functionals are used as-well for time-dependent computations.
The most popular version of TDDFT is Kohn-Sham-TDDFT as well as for ground state DFT, where the motion of the independent particles of the fictitious KS system are given by the time-dependent Schrödinger equation
i∂ϕ(r, t)
∂t =F [ρ]ϕ(r, t) (30)
withF [ρ] being the Kohn-Sham operator given on the left hand site of Eq. 23. A possible approach to solve TDDFT equations is the propagation of the KS orbitals using Eq. 30. This approach is typically called real-time TDDFT and is mainly applied to study field-driven processes. However, the most common approach is the linear-response formulation of TDDFT, which computes the excited states with a first order time-dependent perturbation theory going back to the work of Casida.39 Because of that, the resulting equations are often referred to as Casida equations and form an eigenvalue problem:
A B B A XY =ω 1 0 0 −1 X Y (31) Hereby, the excitation energiesωI =EI −E0for a given excited state ΨI are obtained as the eigenvalues, while X and Y are the corresponding eigenvectors representing the particle-hole and hole-particle excitations of the system. The matrix elements of the A and B matrix are given as
Aia,jb = δijδab(εa −εi) + hij|abi + fijabxc (32)
Bia,jb = hib|aji + fibajxc (33)
withεp orbital energies of the KS orbitalp. Also the well known Dirac braket notation is used to express the exchange and Coulomb interaction terms:
hij|abi = Z Z dr1dr2ϕi(r1)ϕj(r2) 1 r12 ϕa(r1)ϕb(r2) (34)
and fijabxc being the matrix elements of the exchange-correlation kernel: fijabxc = hij | fxc|abi =
ij δ2Exc δρ(r)δρ(r0) ab (35) The main contributions to the excitation energy are the orbital energy dif-ferences in the diagonal elements of the A matrix. This can be interpreted as follows: for a system of completely independent electrons, the excitation energy to move an electron from orbitali into the virtual orbital a is simply
given by the energy difference between these two orbitals and, as long as no interaction with the other electrons appears, this gives the excitation energy. The other terms in A are correlation terms due to the interaction between the electrons.
In most cases the elements of the B matrix are quite small compared to the elements in the A matrix and a valid approximation is the Tamm-Dancoff approximation (TDA)34,43–45which neglects the B matrix completely. This simplifies the Casida equation (Eq. 31) to
AXI =ωIXI (36)
This equation is analogously to the configuration interaction singlets (CIS) equations to compute excited state based on the Hartree-Fock method, as CIS equals the TDA formulation of linear response time-dependent Hartree-Fock (TDHF). XI contains the expansion coefficients for the single excited configurations based on the Kohn-Sham single determinant.
2.1.3 Exciton Model
Even for computationally efficient methods like TDDFT, their application to the computation of excitation processes in large systems containing many chomophoric units becomes very quickly unfeasible. Cheaper, but more approximate methods are thus needed.
The excitonic Hamiltonian represents a valid strategy to overcome this limitation, and to allow a quantum-mechanical description of the excited states of large multichromophoric systems. Hereby, the Hamiltonian of the full system is expressed in the diabatic representation of locally excited states on the individual chromophores:
ˆ Hex =X α N (α) X I ΩαI|αIihαI |+ X α,β6=α N (α) X I N (β) X J VαβI J|αIihβ J | (37) In this notation the chromophores are labeled by greek lettersα, β while the specific local excited state on these chromophores are marked withI, J. The site energies ΩIα are the excitation energies of theI state on chromophore α and the excitonic couplings between different states are labeled asVαβI J.16,46
In the exciton model the computation of the full system Hamiltonian is avoided and instead individual isolated quantum chemistry calculations are performed on each of the chromophores, to obtain the site energies. Therefore, the 3N dimensional problem is reduced to M 3K-dimensional problems (with the number of atoms in the full system N = M · K in case of equally sized chromophores).M is hereby the number of chromophores andK the corresponding number of atoms. Hereby, 3K is much less than 3N , reducing the computational cost significantly.
In a general form, the coupling can be partitioned in three contributions:
The first term represents the Coulomb coupling between the transition densi-ties corresponding to the excitations in the two chromophores. The second term is the exchange contribution, accounting for the fact that the electrons are indistinguishable. And the last term is an additional contribution due to the presence of a polarizable environment. Usually this term represents a screening term of the Coulomb coupling. For “bright” singlet excited states the Coulomb coupling represents the largest term and a general approxima-tion is to neglect the exchange coupling.16 Therefore, the excitonic couplings can be approximated by the Coulomb interaction between the transition densities ˜ρ, namely Vαβcoul = Z Z dr1dr2 ˜ ρα(r1) ˜ρα(r2) |r1− r2| (39) where the integral can be solved analytically or numerically. Alternatively, one can approximate the transition densities and compute the resulting cou-pling analytically.
A very common approximation of the transition density is the point dipole approximation (PDA), valid for spatially well separated chromophores. Hereby, the transition densities are approximated by their transition dipoles µα,I. This approximation goes back more than 70 years to the pioneering work of Förster.47Within the PDA the excitonic couplings can be computed straightforwardly as:
VαβI J,(PDA) = µα,I ·µβ,J |Rαβ|3 −
3(Rαβ·µα,I)(Rαβ·µβ,J)
|Rαβ|5 (40)
whereRαβ is the vectorial distance between the transition dipoles on the chromophoresα/β.
An alternative method is the so-called transition monopole approximation (TMA). The transition density is hereby approximated in terms of point charges centered on the atoms of the chromophores. Although different fitting algorithms can be exploited to obtain these transition charges, the most used one is electrostatic potential fitting (ESP) of the transition density leading to the so-called TrESP charges.48The coupling is then given as the Coulomb interaction between these charges:
VαβI J,(T MA) =X K,L
qI
α(K)qJβ(L)
RKL (41)
withqIα(K) being the TrESP charge localized on atom K of chromophore α and obtained by ESP fitting of the transition densityραI. Both TMA and PDA approximation to the transition density are illustrated in Fig. 3.
With this approximations the matrix elements of the exciton matrix can be obtained as the results of independent quantum chemistry calculations of the individual chromophores. Once the exciton matrix is obtained, it is possible
Figure 3: a) Transition density (TrD) of the second singlet excited state on the BODIPY molecule. The corresponding transition ESP charges (TrESP) are shown in (b) and the transition dipole centered at the center of mass in (c).
to obtain the excitation energies of the total system within the exciton picture by diagonalization of the exciton model Hamiltonian matrix:
© « ω1 0 0 0 ω2 0 0 0 . .. ª ® ® ¬ = U†HexU (42)
U represents the unitary transformation matrix that diagonalizes the matrix. 2.2 qm/classical approaches
The main concept behind QM/classical methods is that many (chemical) reac-tions are spatially localized. Therefore, it is possible to restrict the quantum description of the system to the area where the process of interest occurs (the QM region) and use a simpler and computationally cheaper approach for the other part of the system (the environment) The main focus in these
multi-scale approaches is the correct modelling of the interaction between the environment and the QM system.2,4,6,49,50
The main advantage of QM/classical methods is the reduction in the degrees of freedom of the system, by describing most of the system with a simplified scheme. This reduces the computational cost significantly, while on the other hand it is possible to include the main influence of the environment on the process of interest due to an accurate modelling of the QM-classical interaction.
In literature, QM classical methods are typically divided into two main flavours, depending on the representation of the environment, either into implicit solvation schemes or explicit solvation schemes. This is illustrated in Fig. 1, where on the left implicit solvation and on the right explicit solvation is sketched. In the former case, the environment is described as a continuous medium while in explicit solvation models the individual solvent molecules are explicitly included in the simulation but they are represented in a classical way through Molecular Mechanics (MM) force fields.
Both solvation schemes can describe solvation effects on the QM region accurately through the introduction of an effective Hamiltonian which is partitioned into contributions of the QM region H0, the environment Henv(R) and the interaction Hint(r, R) between them:
He f f = H0(r) + Henv(R) + Hint(r, R) | {z }
Hemb(r,R)
(43)
where {r} and {R} indicate the degrees of freedom of the QM region and the environment, respectively. The last two terms contain all the embedding contributions (Hemb). The introduction of the effective Hamiltonian does not change anything on the strategy for solving of the time-independent Schrödinger equation, from both a formal and a numerical perspective. The most difficult part in Eq. 43 is the definition of the interaction operator Hint(r, R) and over the years several different schemes were developed to express this part.
Historically two different classes of QM/classical methods were developed, using either implicit or explicit models for representing the environment.
In the implicit formulation, also called continuum solvation model, the environment is represented as a dielectric medium. The explicit environ-mental term Henv(R) in Eq. 43 is eliminated and only the contribution of the QM region and the QM/environment interaction term remain, leading to the following effective Hamiltonian:1
He f f = H0(r) + Hint(r, R) (44)
In most continuum models, apparent surface charges on the boundary be-tween the solvent and solute are used to simulate the electrostatic interaction between the QM region and the environment. This formalism was used in many different flavours of continuum models, like the polarizable continuum model (PCM)3,10 or the conductor-like screening model (COSMO).11,12
Continuum models represent very efficient methods to simulate bulk sol-vation including the mutual polarization between the solvent and the solute, but for more heterogeneous environments they suffer from the fact that the solvent molecules are not represented explicitly and therefore directional solvation effects like hydrogen bonding cannot be described accurately. 2.2.1 QM/MM schemes
The explicit formulation of QM/classical approaches is represented by QM/MM schemes, where the environment is included explicitly in terms of molecular mechanics (MM) force-fields. Within this framework, the Hamiltonian of the MM subsystem is partitioned into bonding (typically bond stretching, angle bending, torsions and out-of-plane deformations) and non-bonding terms as follows::
H(R)env =Estr+Ebend +Etors | {z } Ebond +Evdw +Eel | {z } Enon−bond (45)
The latter consist of a van der Waals (often represented by a Lennard-Jones-type potential) and an electrostatic term. A typical non-polarizable force-field can be defined as:6,13
(46) Henv(R) = X bonds kd(d − d0)2+ X anдles kθ(θ − θ0)2+ X dihedrals kϕ[1 +cos(nϕ + δ)] + X nonbondinд ( εAB " σ AB rAB 12 − σ AB rAB 6# + 1 4πε0 qAqB rAB )
whered, θ, ϕ are used for the bond lengths, angles and torsions, while the subscript 0 indicate the corresponding equilibrium values. The force constants are labeled askd, kθ, kϕ andrAB is the distance between atoms A and B;n andδ are the torsional multiplicity and phase values. The Lennard-Jones parameters are labeled withσABandεAB, whileqAandqB represent the atomic partial charges used to represent the electrostatic interaction between any pair of MM atoms.
In order to couple this MM approach with a QM description of a part of the system, various approaches have been developed over the last years. Typically, they can be roughly divided into subtractive and additive coupling schemes.
From now on, the notation will be the following: S represent the degrees of freedom of the total system, O the degrees of freedom of the MM subsystem only and I the degrees of freedom of the QM subsystem and L those of the possible link atoms.
Subtractive QM/MM scheme make use of the fact that MM calculations are typically orders of magnitude faster than the corresponding QM calculations, even if the MM system is much larger. Therefore, it is not too costly if parts of
Figure 4: Illustration of a subtractive QM/MM scheme. The total energy of the system is obtained via the combination of three independent computations: (i) the MM energy of the total system (E(S)) (ii) the QM energy on the model system (EQ M(I + L)) and (iii) the MM energy of the model system (EM M(I + L)).
the MM calculation are done more than once. In subtractive QM/MM schemes the energy of the total system is obtained as
E(S) = EMM(S) +EQM(I + L) −EMM(I + L) (47)
whereEMM(S) is the MM energy of the total system andEQM(I + L) is the energy of the QM systemEQM(I + L). To avoid double counting, the energy of the QM subsystem is recalculated at MM level,EMM(I + O), and subtracted afterwards. Therefore the total energy can be obtained by combining the results of three (independent) calculations and an implementation into ex-isting electronic structure programs is trivial. Probably, the most common subtractive QM/MM scheme is the ONIOM method developed by Morokuma and co-workers51–56and implemented in several electronic structure codes, as e.g. Gaussian.57The main limitation of pure subtractive QM/MM schemes, is that the QM/MM interaction energy can only be computed at MM level of theory, if one wants to go beyond mechanical embedding it is therefore necessary to use a different approach.53,55,56
In additive QM/MM schemes the total energy is computed directly and no double computations are necessary, namely6
E(S) = EQM(I + L) +EMM(O) +EQM/MM(I, O) (48) In contrast to the subtractive QM/MM schemes the MM calculation is now performed on only the MM degrees of freedom and not on the total system. The last term in Eq. 48 includes QM/MM interaction implicitly, namely the bonded, van der Waals and electrostatic interactions between the QM and MM region:
EQM/MM(I, O) =EbondQM/MM +EvdWQM/MM+EeleQM/MM (49)
As the QM/MM interaction is treated explicitly, different levels of theory can be applied, which are described in the next sections.
2.2.2 QM/MM coupling schemes
The simplest implementation of QM/MM coupling schemes is named mechan-ical embedding (see Fig. 2b)). Hereby, the interaction term between the QM and MM region is computed purely at MM level of theory. The QM energy in Eq. 48 is obtained in a pure gas phase calculation and the other two terms, namely the MM energy and the QM/MM interaction energies are obtained from MM calculations.
The limitation of the mechanical embedding scheme, is that the MM sub-system has no direct influence on the electronic wave function/density of the QM subsystem and therefore all the properties of the latter are those corresponding to the isolated system.
To overcome the main shortcomings of mechanical embedding schemes, the alternative electrostatic embedding (see Fig. 2c)) was introduced.2,6 Hereby, the MM environment is included in the QM Hamiltonian in terms of fixed point charges (or multipoles).
HQM/MMele =X J qJVQM(RJ) +X α X J qJQα |Rα − RJ| (50)
whereqJ represents the MM partial charges located at spatial position RJ j (i.e. on the MM sites),VQM(RJ) is the electrostatic potential generated on those positions by the QM electronic density andQα are nuclear charges of the QM atoms.
Contrary to what happens in the mechanical embedding, the MM subsys-tem can affect (i.e. polarize) the QM density via the electrostatic interaction. We note, that at the QM-MM boundary, where the MM charges are placed in the immediate vicinity of the electronic density, the MM charges can lead to an overpolarization of the electron density. Possible approaches to avoid this problem are described in a following section.
It is straightforward to define electrostatic embedding in the additive QM/MM schemes, but also the subtractive QM/MM approaches can be gen-eralized to include electrostatic embedding. The main scheme to do that partitions the interaction into non-electronic interactions, which are treated in the standard subtractive QM/MM fashion and electronic interactions, that are included in an additive QM/MM formulation. Therefore, it is not straight-forward to differentiate between subtractive or additive QM/MM scheme anymore, as this scheme represents a combination of both.
Electrostatic embedding represents the most popular embedding scheme nowadays, due to its easy implementation into existing QM schemes and its ef-ficiency to describe environmental effects by parametrized atomic charges.2,4,6 2.2.3 Polarizable embedding
Although, electrostatic embedding schemes are able to accurately describe the electrostatic interaction between the environment and the QM region, they have a main drawback, which is the use of static charges. Therefore, the
environmental response to many chemical changes, like the redistribution of the charge density upon excitation cannot be described properly. In recent years, different groups have tried to solve this shortcoming of electrostatic embedding by using polarizable embedding schemes, that include mutual polarization between the QM region and the environment. The approaches presented so far in literature use Drude oscillators58, fluctuating charges59,60 or induced dipole formulations.61–63The last approach is will be discussed in a following section.
The first two approaches, Drude oscillators and fluctuating charges, de-scribe the QM/MM interaction in terms of the Coulombic interaction between point charges and the QM region (electronic density and electronic nuclear po-tential). This has the advantage that no new term for the QM/MM interaction need to be included into the existing QM software.
In the Drude oscillator model the polarization is described by a fixed point charge centered at the polarizable site and an attached mobile massless charged particle (Drude particle). Hereby, the sum of both charges should give the charge of the non-polarized MM atom. The two charges are connected via a harmonic spring with a predefined force constant and form an induced dipole due to the displacement of the Drude particle from the polarizable site, given by an external electric potential. Hereby, the displacement of the Drude particles depend on the strength of the external electric potential, which arises from the potential due to the nuclei of the system and its electron density. The last term also depends on the charge distribution of the environment and therefore a mutual polarization between both systems is introduced.
In the fluctuating charge scheme the MM sites bear fluctuating (non-fixed) partial charges. These partial charges can change according to the electroneg-ativity equalization, under the constrain that the total charge of the classical sites is preserved. Furthermore, additional constrains can be imposed to not only preserve the total charge of the environment, but e.g. that of specific molecules or even molecular fragments.
MMPol: An induced dipole formalism
In this work, we deal with a polarizable QM/MM scheme based on the in-duced dipole approach that will be referred to in the following as QM/MMPol. Within induced dipole based polarizable QM/MM schemes, the MM sites bear additionally to the fixed point charges (or fixed multipoles64) isotropic polarizabilities. Thanks to these polarizabilities, the MM sites can polarize in the presence of the electric field E due to the QM system and the point charges of the MM sites; as a result an induced atomic dipole appears on each polarizable MM site. The induced dipole formulation is formally very similar to the Drude oscillator scheme, with the difference that the dipoles are repre-sented as induced point dipoles due to the presence of atomic polarizabilities on the MM sites, while in the Drude oscillator approach two point charges are used to mimic the dipole and the atomic polarizabilities are replaced with a harmonic spring constant whose strength is proportional to the amount of displacement and therefore to the strength of the resulting dipole.
The polarization energy can be written as: Epol = 1
2µ
†Tµ − µ†
E (51)
The second term is the interaction term between the induced dipoles and the electric field of the system. The first term is the dipole-dipole interaction between the induced dipoles. To avoid an overestimation of this term and the well known polarization catastrophe, typically a damping scheme is ap-plied, where the most common one is the Thole’s damping. This leads to an interaction matrix T defined as:
T = © « α−1 1 T1,2 . . . T1,Npol T2,1 α−12 . . . T2,Npol .. . ... . .. ... TNpol,1 TNpol,2 . . . α−1 Npol ª ® ® ® ® ¬ (52)
The diagonal elements are the inverse of the atomic polarizability tensorsα , while the off-diagonal elements Ti,jare the damped dipole interaction tensors:
Tijαβ = − δαβ
|ri−rj|3λ3+ 3
|ri−rj|α|ri−rj|β
|ri−rj|5 λ5 (53)
The factorsλ3 andλ5 are functions of the inter-atomic distance between different dipoles and therefore the damping terms
In a variational framework, the induced dipoles are obtained to minimizes the polarization energy Epol , namely:
∂Epol
∂µ = Tµ − E = 0 (54)
This set of equations can be either solved by matrix inversion or in an iterative manner to obtain the induced dipoles as a function of the electric field of the system and therefore as a function of the electronic density. This dependence on the QM density makes the resulting Hamiltonian,HQM/MMpol nonlinear with respect to the wave function. As a matter of fact, the Hartree-Fock and KS-DFT methods show the same characteristic and a similar approach can thus be used to solve the problem, namely a self-consistent iterative approach.
After solving the QM/MMPol equations and obtaining the induced dipoles the QM-MMPol interaction energy can be divided into two contributions:
Ees = Uqes +Uees +Unes (55)
Epol = 1 2 hU
pol
en +Unepol +Ueepol +Unnpol i
(56) whereEes is the energy term due to the fixed charges andEpolthe correspond-ing one due to the induced dipoles. In both cases, we can further split each contribution into terms related the electron density (e) and the nuclei (n). In the case of the electrostatic contribution there is also a term corresponding to the interaction between MM chargesUqes, whereas in the case of the po-larization contribution, the single term has two indices as the dipoles can be induced bythe electronic density and the nuclei.
2.2.4 Analytic nuclear ground state gradient in DFT/MMPol
The first analytic gradients for polarizable QM/MM within the induced dipole formulation were formulated and presented by Caprasecca et al.65. In the following, we will use the same framework but only the QM Hamiltonian within the embedding scheme will be discussed, as the expressions of the classical terms are straightforward and do not need special attention.
Using a matrix representation in terms of the molecular orbitals (MO), the Fock-Operator in DFT (or HF) can be written as:
Fpq =hpq+X i
[hpi|qii − cxhpq|iii] + fpqxc +Vpqes +Vpqpol (57) where the scaling parametercx is introduced to interpolate between Hartree-Fock (cx = 1,fxc = 0) and pure DFT (cx = 0). For hybrid functionalscx has a value in between 0 and 1, depending on the amount of exact exchange.
The environmental effects are included by the two termsVpqes andVpqpol, representing the matrix elements of the Coulomb operatorVpqes, due to the fixed point charges and the polarization termVpqpol, due to the induced charges.
If only nuclear gradients are considered, it is convenient to switch from the MO representation to an atomic orbital (AO) representation, as the gradient with respect to a nuclear displacement of an AO orbital is zero for all AOs located on the other atoms that are not moved. In the AO representation and using Eq. 57 the ground state DFT (HF) energyE0can be written as:
E0 = X µν Dµνhµν +1 2 X µνκλ DµνDκλ(hµκ|νλi − cxhµκ|λνi) +X µν Dµνfµνxc +X µν DµνVµνes +X µνκλ DµνVµνpol[Dκλ] +Vclass (58) withVclasscontaining all density independent contributions, like the nuclear-nuclear repulsion and the interaction between the nuclei and the fixed charges, and D is the ground state density in AO representation:
Dµν =X i
nicµicνi (59)
The differentiation with respect to a generic parameter ζ gives rise to two different contributions. The first one is the Hellmann-Feynman term, representing the derivative of the matrix elements of the operators with respect toζ : EHellmann,ζ0 = X µν Dµνh ζ µν +12 X µνκλ DµνDκλ∂ζ∂ (hµκ|νλi − cxhµκ|λνi) (60) +X µν Dµν ∂ ∂ζ fµνxc + X µν Dµν ∂ ∂ζVµνes + X µνκλ DµνDκλ∂ζ∂ Vµνpol+ ∂ζ∂ Vclass wheras the second one is the density response (the Pulay force) which can be written in a compact form as:
EPulay,ζ0 = X
µν
The computation of the density response term can be avoided, as first shown by Pulay by using Eq. 59 and the Roothaan-Hall relation
FC = SCε (62)
Using this equation in combination with the orthonormality of the MO coeffi-cients, enforced by the Lagrangian constrain in the derivation of Kohn-Sham DFT (HF) the Pulay term can be rewitten as:
E0Pulay,ζ = −X
µν WµνS ζ
µν (63)
With the energy-weighted density matrix W defined as: Wµν =X
i
εicµicνi (64)
Hence, the first order derivatives of the energy with respect to a generic change in geometry can be computed without the need to compute a density derivative and only Hellmann-Feynman forces are needed.
MMPol contributions
Nuclear gradients in DFT/MMPol framework are usefully divided into electro-static termsdue to the fixed charges and polarizable terms, due to the induced dipoles.
For the electrostatic terms the derivatives with respect to an MM coordinate kζ can be simply computed as the interaction between the charge at MM site qk with the electric field due to the system at that coordinate:
∂ ∂Rk,ζVes = − X c qkqc |Rk −Rc|3 (ζk −ζc) −qk X n Zn |Rk −Rn|3 (ζk −ζn) +qkX µν Dµν µ (ζk −ζn) |ρ − Rk|3 ν = −qk Echдkζ +Enuckζ +Eelekζ (65) The last three terms represent the electric field in at MM sitek due to the other charges, the nuclei and the electronic density. In the case of derivatives with respect to QM coordinates,aζ , the pure classical charge-charge term vanishes, as it does not depend on those coordinates and only the last two remain:
∂ ∂Ra,ζV
es = −q
k Enucaζ +Eaζele
(66) The polarizable terms, can be divided into four subgroups due to the inter-action of the induced dipoles and the electric field generated of the charges, nuclei and electrons and finally, the dipole self-energy. The derivative of theses terms with respect to an MM coordinatekζ can be written n a very compact form, using the definition of the induced dipolesµ = T−1E:
∂ ∂Rk,ζVpol = −µEkζ + 1 2µ † T−1,kζµ (67)
withEkζ being the derviative of the electric field, due to the nuclei, charges, electrons and the other induced dipoles, with respect tokζ at the position of the corresponding polarizable MM sitek.
2.2.5 Excited state formulation for TDDFT
So far the QM/MM embedding schemes were only discussed for the case of ground state methods like Hartree-Fock or DFT, but it is also possible to extend these approaches to excited state methods. In this paragraph, this is done for the case of TDDFT as the QM method of choice.
Recall that for TDDFT within the linear response formulation, the eigen-value problem for the excited states is:
A B B A X Y =ω 1 0 0 −1 X Y (68) with the matrix elements of the orbital rotation A and B given in Eq. 33 and Eq. 33. For mechanical embedding the generalization is trivial, as the QM calculation is performed in gas phase and therefore no environment terms enter the QM Hamiltonian. In the case of electrostatic embedding, the interaction term due to the fixed point charges enters the Fock operator while the Casida equation has the same matrix form for A and B as in the gas phase but both the orbital energiesεp and the canonical orbitalsϕp will be changed. On the other hand, the use of polarizable embedding introduces an explicit term in the Casida equations, namely:
˜
Aia,jb = δijδab(εa −εi) + hij|abi + fijabxc + Via,jbpol (69) ˜
Bia,jb = hib|aji + fibajxc + Via,jbpol (70)
The polarization contribution Via,jbpol represents the interaction of the MM dipoles induced by the transition densityϕjϕb∗ with the electric field due to the transition densityϕiϕa∗:
Via,jbpol = −X k µindk [ϕjϕ∗b] · Z drϕi (rk − r) |rk − r|3ϕ ∗ a (71)
The resulting “dressed” Casida equations can be solved within the standard iterative procedure employed and the excitated state energies and transition densities can be computed within polarizable embedding scheme.
2.2.6 State-specific correction
In the previous section the linear-response (LR) formalism was introduced, where the induced dipoles feel the electric field of the transition density. The resulting transition energy can be written as:66
ωLRI =ω0I +RLRdyn(ρI) (72)
whereωI0is the transition energy obtained by switching off the contribution of the induced dipoles.
However, when the environment is polarizable, another energy term should be considered. This is obtained considering that, after excitation, the elec-tronic polarization of the environment will immediately relax to be in equi-librium with the electronic density of the excited state. The resulting energy term can be written as:
1 2
X
k
µindk [P∆I ]Ek∆ (73)
wherePI∆is the difference between the electron density of the excited state I and the ground state. In the framework of LR-TDDFT the excited state density can be approximately obtained in terms of the relaxed densityZ that is used as well for the computation of excited state gradients, as discussed in a following section.
The theoretical framework for this state-specific (SS) model was originally developed by Caricato et al.66 within the PCM solvation method (using the name corrected LR, cLR, approach) and later was generalized to polarizable QM/MM approaches to correct absorption energies.67–69
While the LR approach is computationally favourable for TDDFT, it can lead to an erroneous description of the environment response. This becomes obvious for dark transitions, like charge transfer states. In these cases the polarization due to the transition density is very small, although a major charge redistribution appears in the system upon excitation.
2.2.7 QM/MM boundary
In many systems it cannot be avoided to have a QM/MM boundary that cuts a covalent bond. To avoid an unphysical behaviour, the valency of the QM atom at the frontier must be saturated. Over the years, several different methods were developed to do this. For example Warshel and Levitt used in their earliest work frozen hybrid orbitals to saturate the valence of the QM atom. In that approach a single hybrid orbital is used to represent the covalent bond between QM and MM region. This hybrid orbital is kept fixed during the following computations, which can be ensured by the introduction of Lagrange multipliers. The advantage of this scheme is that the correct chemical nature of the bond is preserved, but the computations become more complicated. Therefore, this scheme is used very rarely and the main method to describe the QM/MM boundary is the link atom (LA) approach. Hereby,
Figure 5: QM/MM setup with link atom boundary. The QM region (orange) and the MM region (transparent) are shown. The link atom (green) is placed on the Q1-M1 bond at a fixed distance.
an additional atom (usually a hydrogen atom) is introduced and covalently bonded to the QM atom. As each LA creates three artificial nuclear degrees of freedom, its position is typically defined as a function of the boundary QM and MM atoms. The AMBER software package,70,71 for example, sets the QM-LA distance to a fixed predefined value (e.g. for carbon as QM atom it is 1.09 Å) along the QM-MM bond. Therefore the position of the LAR(LA) is given by:
R(LA) = R(Q1) + C(Q1) R(M1) − R(Q1)
||R(M1) − R(Q1)|| (74)
withC(Q1) being the constant Q1-LA distance depending on the QM atom andR(Q1) (R(M1)) the coordinates of the Q1 (M1) atom, respectively. One can now use the chain rule to redistribute the forces acting on the LA to the M1 and Q1 atom, which effectively removes the artificial degrees of freedom of the LA.
To avoid the overpolarization problem in electrostatic (or polarizable) em-bedding QM/MM schemes several protocols have been proposed. Most of them change the MM charges at the boundary by deleting, redistributing or smearing of the charges. One commonly used scheme is the charge elim-ination scheme Z3, default in Gaussian09. Hereby, the charges of the M1, M2, and M3 atoms (see Fig. 5) are set equal to zero. An alternative is the redistributed charge scheme by Truhlar et al.72,73, which introduces artificial MM atoms along the M1-M2 bond, in a link atom manner, and redistributes the charge of the M1 atom on the resulting artificial MM atoms.
2.3 nuclear dynamics
In the recent years many time-resolved experimental techniques were devel-oped, especially in the femto or even sub-femtosecond time-scale. Hereby, (quantum) molecular dynamics provides an essential tool for the understand-ing and interpretation of the underlyunderstand-ing physical and chemical phenomena. Several different theoretical methods were developed ranging from full quan-tum dynamics on a grid to semi-classical on-the-fly dynamics. In grid based methods, the nuclear wavepacket is propagated on precomputed potential energy surfaces of the electronic states of the system. The most common approach is the multi-configurational time-dependent Hartree (MCTDH) method developed by Meyer and Manthe and co-workers.21–24In MCTDH the wave function of the system is written as an expansion of Hartree prod-ucts. Solving the time-dependent Schrödinger equation by a variational approach with this ansatz for the wave function leads to a set of coupled equations of motion for the expansion coefficients. Although, the MCTDH method presents a computationally very efficient scheme to perform quantum molecular dynamics, it is limited to only a few (dozen) degrees of freedom, even if advanced techniques like the multilayer-MCTDH scheme74,75 are exploited.76 Another problem of these grid based methods is the fact that the potential energy surfaces and couplings need to be precomputed, which in fact adds up to the computational costs of the calculation. To include the full dimensionality of large systems in the simulation, computationally less demanding, but also more approximate methods were developed. Among them on-the-fly methods,77like the variational multiconfigurational gaussian scheme,78ab-initio multiple spawning79,80 or surface hopping.25,26 represent popular approaches. Hereby, the potential energy surfaces do not need to be computed in advance, instead only the relevant configurational space is computed on demand through out the nuclear dynamics.
2.3.1 Surface Hopping
Trajectory surface hopping (TSH)25,26is a computationally efficient scheme to study the dynamics of non-adiabatic processes, where the Born-Oppenheimer approximation breaks down for a group of electronic states.28In this approach the electronic and nuclear motions are decoupled. The electronic motion is treated fully quantum mechanically and is propagated using the time-dependent Schrödinger equation, while the nuclei follow the classical equation of motion of Newton on a single potential energy surface given by a chosen (electronic) state, the so-called active state.81,82 A similar approach is the Ehrenfest method, in which the electronic potential is described by a mean-field approach of the potential energy surfaces of all populated excited states:
Eavд = hΨ |Hele| Ψi =X i
whereEi represents the energy of thei electronic state and ciits contribution to the electronic wave function Ψ.
Although, the Ehrenfest method has been applied with great success to a number of chemical problems,83,84 it has several serious limitations due to its mean-field character.27,28 Imagine a system that moves, after passing a region with strong non-adiabatic couplings, in a region where the Born-Oppenheimer approximation is valid and the overall nuclei motion is governed by a single excited state. An average potential will not be able to describe the motion correctly and the average trajectory given by Ehrenfest will diverge from the true trajectory path. Additionally, the total wave function can have contributions of electronic states that are in the classically forbidden regions, meaning that the energy of the system is not enough to populate these states. Another severe drawback of the Ehrenfest approach is that it violates microscopic reversibility.
TSH is an approach to overcome the limitations of the Ehrenfest method85–87 and a typically surface hopping trajectory run is sketched in Fig. 6 (b-d). Instead of computing an average path, in TSH many independent surface hopping trajectories are run and averaged afterwards. The nuclei are prop-agated on a pure adiabatic state of the system. The main idea behind this approach is that, in regions were the non-adiabatic coupling is negligible (so Born-Oppenheimer approximation is valid), the nuclear motion should be governed by the potential of the corresponding Born-Oppenheimer state. To account for non-adiabatic effects, the active sate can be changed throughout the dynamics depending on the non-adiabatic coupling between the involved states. Thus, non-adiabatic transitions between different adiabatic states may occur. The expansion coefficient for the electronic wave function |ci|2should hereby correspond to the ensemble averaged number of trajectories.
The equation of motion for the coordinates of nucleusα (Rα) and massMα can be written as
Mα∂2Rα
∂t2 = −дα (76)
where the term on the right hand side is the force on the corresponding nuclei, which is given as the negative gradient of the electronic energy of the active state. The nuclear motion can be described with a single classical function R(t) that dictates the positions of each nuclei at each time step. This is what throughout this work will be referred to as a classical trajectory.
The time-dependent electronic wave function, on the other hand, is nor-mally expanded in a basis of local (adiabatic) states {Φk}:
Ψele(R, r,t) = X i
ci(t)Φi(r; R(t)) (77)
The states are typically obtained on the fly as the solution of the electronic Schrödinger equation:
Figure 6: Sketch of a surface hopping dynamics. After photoexcitation (b) the nuclei are propagated along a single excited state potential energy surface (c). In the proximity of a coupling area, with (strong) non-adiabatic couplings the active state the nuclei follow is allowed to change stochastically (d). To mimic the nuclear wavepacket an ensemble of trajectories are run (e) that should in the end reproduce the full quantum result in an approximated manner (f).
