Multiscale approaches that partition the system into an active site (where the electronic process under study occurs) and a remaining region, the environment, have proven to be good strategies for the computation of electronic excitations in complex systems. In this work the implementation of a polarizable QM/MM scheme for the computation of excited state gradients is presented and are applied to a test case

Caprasecca, Stefano

Menger, Maximilian F.S.J.

Mennucci, Benedetta

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1Excited state gradients within a polarizableQM/MM formulationMaximilian F.S.J. Menger*, Stefano Caprasecca, and Benedetta MennucciDipartimento di Chimica e Chimica Industriale, University of Pisa*m.menger@studenti.unipi.itAbstract—Multiscale approaches that partition the system into an active site (where the electronic process under study occurs) and aremaining region, the environment, have proven to be good strategies for the computation of electronic excitations in complex systems.In this work the implementation of a polarizable QM/MM scheme for the computation of excited state gradients is presented and areapplied to a test case.F1 INTRODUCTIONIn standard QM/MM models, the environment is de-scribed in terms of fixed charges (or multipoles), that re-main unchanged during the calculation. In our polarizableQM/MM approach (QM/MMPol), on the other hand, po-larizablities on the MM sites are included additionally (seeFig. 1). Therefore the environment is allowed to respond tothe electron density of the QM subsystem or its changesupon excitation by polarizing. The polarization of the envi-ronment is represented in terms of classical dipoles, inducedby the QM electric field Fp. The dipoles are obtained byFp = (Dµ)pwhere D is the MMPol matrix, that only depends on thegeometrical parameters of the MM sites and the atomicpolarizabilities.The effective interaction energy between QM and MMregion of the system is given by:Ees = Uesq + Uesn + UeseEpol =12[Upolen + Upolne + Upolee + Upolnn]Hereby is Uesi the electrostatic interaction energy betweenMM charges and the potential by the other MM charges(i = q), the nuclei (i = n) and the electron density (i = e).Epol denotes the interaction energy between the induceddipoles and the electric fields generated by charges, nuclei,electrons and the other dipoles.Fig. 1. Schematic representation of the polarizable QM/MM approach.2 POLARIZABLE TDDFT/MM GRADIENTSThe energy derivative of a system represented by its wave-function Ψ and its corresponding Hamiltonian H with re-spect to a generic parameter λ can in general be written as:Eλ =〈Ψ∣∣∣∣∂H∂λ∣∣∣∣Ψ〉+ 2〈∂Ψ∂λ |H|Ψ〉(1)The first term is the so-called Hellmann Feynman force,while the second term is the Pulay force. In the case of apurely variational energy functional the Pulay force van-ishes. This is also desirable in the case for non variationalenergy functionals, as the computation of the molecular or-bital response with respect to the perturbation is very costly.A better ansatz is to introduce a so-called relaxed densityto account for the wavefunction response. This avoids thecomputation of the perturbed basis.In our approach for the analytic gradients of the po-larizable TDDFT/MM approach, we use an ansatz thatwas already applied by Furche and Ahlrichs [1] to pureTDDFT and later expanded by Scalmani et al. [2] on TDDFTgradients in combination with the polarizable continuummodel [3] (PCM). The starting point for this approach is afully variational Lagrangian L, defined as following:L[X,Y,Ω,C,Z,W] = G[X,Y,Ω] +∑iaZiaFia−∑pq,p≤qWpq(Spq − δpq)The derivation of analytic gradients from L is straightfor-ward and the resulting Z-Vector method is computationaladvantageous with respect to others.The equation for the analytic derivatives can be writtenin a compact form, by expressing it in the AO basis:Ωξ =∑µνhξµνP∆µν +∑µνSξµνWµν +∑µνκλ〈µν|κλ〉ξΓµνκλ+ωxc,ξ + ωMMPol,ξThe exponent ξ denotes derivatives with respect to a generalvariable. P∆ and Γ are the one- and two-particle den-sity matrices. ωxc,ξ is the derivative of the DFT exchange-4th BSC Severo Ochoa Doctoral Symposium912correlation energy and ωMMPol,ξ of the QM-MMPol interac-tion energy. The exact expressions and a detailed discussioncan be found in the articles of Furche [1] and Scalmani [2]for the case of pure DFT and DFT/PCM.3 IMPLEMENTATION AND APPLICATION3.1 ImplementationThe analytic gradients are implemented in a local mod-ified version of Gaussian 09 [4]. Analytical Forces forQM and MM sites in excited state calculations at (TDA)-TDDFT/TDHF level of theory for a polarizable embeddingare available within the ONIOM scheme. The implementa-tion is an extension of the work of Caprasecca et al. [5], whoimplemented the corresponding ground state QM/MMPolgradients.3.2 ApplicationWe applied the model to study the intercalation ofthe doubly protonated fluorescent stain 4’,6-diamidino-2-phenylindole (DAPI, shown in Fig. 2) within two pairsof CG DNA bases. DAPI is a commonly used fluorescentmarker that can interact with DNA both by minor groovebinding and intercalation. In this test study we will onlyfocus on the latter. The initial structure of the systemFig. 2. Structure of the doubly protonated 4’,6-diamidino-2-phenylindole(DAPI).was taken from the article of Biancardi et al. [6] and theconformer 1 of the intercalated structures was used. Bothground state (GS) and excited state (ES) geometry opti-mizations were performed and the QM method of choicewas DFT/TDA-TDDFT. To assess the effect of polarizationon the optimized structures, three different models wereintroduced: (i) a full-QM model, where both DAPI andDNA are treated at QM level (f-QM); (ii) a non polarizableQM/MM model, where the DAPI is treated at QM level andthe DNA is treated at MM level; (iii) a polarizable QM/MMmodel (QM/MMPol), where the DNA is now representedas a polarizable environment. Computations of the isolatedDAPI molecule were performed (g-QM), additionally. In allcalculations the DFT functional m06-2X and the 6-31G(d)basis set was used for the DAPI molecule, while 6-31G wasused on the DNA in the f-QM calculations. The excitedstate of interest is the lowest state localized on the DAPImolecule, characterized by a HOMO-LUMO transition inthe g-QM model. This state corresponds to the first excitedstate in the QM/MM and QM/MMPol approaches and thethird for f-QM, as two charge transfer state appear at lowerenergy.To assess the structural changes induced by the environ-ment on ground and excited state geometries, as well as thestructural rearrangement upon excitation, eleven internalbonds (marked in red in Fig. 2) were chosen. From theseit is possible to define the bond length alternation (BLA),which is often used as a measure of conjugation. The bonddifference between GS and ES calculations are shown inFig. 3. It is evident from the alternating pattern of the first8 bond length that, upon excitation, the single (double)bonds tend to become shorter (longer), with an increasein conjugation as consequence. The comparison betweendifferent models for the inclusion of the environment donot show significant differences. In several cases, however,the optimization carried out with the newly implementedQM/MMPol model is closer to the more expensive f-QMoptimizations than the other models. This seems to point outthat the inclusion of the polarisation is particularly useful indescribing accurately the environment response to electrondensity redistribution.Fig. 3. Difference in bond length of eleven internal bonds and thecorresponding BLA in the DAPI molecule (marked in Fig. 2) betweenES and GS optimized structures at different levels of theory.ACKNOWLEDGMENTSFinancial support of the ITN-EJD-TCCM PhD Fellowshipas part of the Marie Sklodowska-Curie Actions is greatlyacknowledged.REFERENCES[1] F. Furche, R. Ahlrichs: ”Adiabatic time-dependent density functionalmethods for excited state properties”, J. Chem. Phys. 2002, 117, 7433;Erratum J. Chem. Phys. 2004, 121, 12772.[2] G. Scalmani, M. J. Frisch, B. Mennucci, J. Tomasi, R. Cammi, V.Barone: ”Excited States in the Gas Phase and in Solution. Theory andApplication of a Time-Dependent DFT Polarizable Continuum Model”, J.Chem. Phys. 2006, 124, 094107.[3] J. Tomasi, B. Mennucci and R. Cammi: ”Quantum Mechanical Con-tinuum Solvation Models”, Chem. Rev. 2005, 105, 2999.[4] M. J. Frisch et al.: Gaussian 09, Gaussian Inc. Wallingford CT 2009.[5] S. Caprasecca, S. Jurinovich, L. Viani, C. Curutchet, B. Mennucci:”Geometry Optimization in Polarizable QM/MM Models: The InducedDipole Formulation”, J. Chem. Theory Comput. 2014 , 10, 1588.[6] A. Biancardi, T. Biver, F. Secco and B. Mennucci: ”An investigationof the photophysical properties of minor groove bound and intercalatedDAPI through quantum-mechanical and spectroscopy tools”, Phys. Chem.Chem. Phys. 2013, 15, 4596.[7] J. Wang, P. Cieplak, J. Li, T. Hou, R. Luo, Y. Duan: ”Development ofPolarizable Models for Molecular Mechanical Calculations I: Parameteri-zation of Atomic Polarizability” J. Phys. Chem. B 2011, 115, 3091.[8] M. Caricato, B. Mennucci, J. Tomasi, F. Ingrosso, R. Cammi, S. Corni,G. Scalmani: ”Formation and relaxation of excited states in solution: Anew time dependent polarizable continuummodel based on time dependentdensity functional theory” J. Chem. Phys. 2006, 124, 124520.4th BSC Severo Ochoa Doctoral Symposium923Maximilian F.S.J. Menger Bachelor (2013) and Master degree (2015)in Chemistry from the Ruprecht-Karls University Heidelberg with adissertation on ultrafast electrondynamics in phenylalanine, under thesupervision of PD Alexander Kuleff. Currently, he works as a PhDstudent within the EJD-ITN-TCCM Marie Skłodowska-Curie actions inthe groups of Prof. Benedetta Mennucci (University of Pisa) and Prof.Leticia Gonza´lez (University of Vienna), with a project on The mod-elling of light-harvesting processes in multichromophoric systems. Hisresearch interest include QM/classical models as well as diabatic andnon adiabatic molecular dynamics.4th BSC Severo Ochoa Doctoral Symposium93

Excited state gradients within a polarizable QM/MM formulation

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