Theoretical Study of Vibronic Spectra of Molecule Systems Generated by Photo- and Electronic Excitations
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Theoretical Study of Vibronic Spectra of Molecule Systems Generated by Photo- and Electronic Excitations Ce Song Licentiate Thesis in Biotechnology School of Engineering Sciences in Chemistry, Biotechnology and Health Royal Institute of Technology Stockholm, Sweden 2022
Theoretical Study of Vibronic Spectra of Molecule Systems Generated by Photo- and Electronic Excitations Licentiate Thesis in Biotechnology School of Engineering Sciences in Chemistry, Biotechnology and Health Royal Institute of Technology (KTH) © Ce Song, 2022 ISBN 978-91-8040-149-4 TRITA-CBH-FOU 2022:15 Printed by Universitetsservice US-AB Stockholm, Sweden, 2022
Abstract Spectra represent fingerprints of molecules, which contain unique information about their properties. Through analyzing the spectral data, one can reveal the molecules’ energy level alignments, identify their species and geometric structures, and explore relevant chemical processes and microscopic mechanisms. Currently, spectroscopy is one of the main means for human beings to enter the mysterious world of molecules and hear their stories. However, interpreting molecular spectra is not a straightforward process, because the occurrence of spectra involves complex interactions between molecules and external stimuli. Theoretical simulations based on quantum chemistry play an indispensable role in this regard, which makes developing and applying related computational software become very important. This thesis focuses on the theoretical simulations of two types of molecular spectra, namely the vibrationally resolved optical spectra and the inelastic electron tunneling spectra (IETS). The former involves the transitions of electrons between a molecule’s ground state and its excited states with the involvement of molecular vibrations, and the latter comes from the excitations of a molecule’s vibrational states within its electronic ground state by inelastic tunneling electrons across a molecular junction. By performing time-dependent density functional theory calculations as well as applying the DynaVib code, I have systematically investigated the optical absorption properties of two types of functional molecules, i.e., naphthalenediimide cyclophane (NDIC) derivatives and fused porphyrin derivatives, which have been proposed as building blocks for future single- molecule optoelectronic devices. Based on the Franck-Condon (FC) principle, the simulations well explain the energy shifts induced by chemical substitutions in the first intense absorption bands of the three NDIC derivatives, and nicely reproduce the vibrational features of their first two bands. Furthermore, by using three different exchange-correlation functionals (i.e., the conventional functional B3LYP and two long-range corrected functionals CAM-B3LYP and ωB97XD), it is found that long-range corrections are very important for the description of the spectral features owing to the strong charge transfer in the related excited states. By taking into account both the FC and the non-FC Herzberg-Teller (HT) contributions, the experimentally measured electroluminescence spectrum of a single fused 5,15-(diphenyl)-10,20-(dibromo)porphyrin (fused-H2 P) molecule is nicely reproduced by the simulations. It is found that the FC contribution also dominates the emission of the molecule, while the HT terms mainly contribute to
vi the low-energy tail of the spectrum. The vibrational fine structures as observed in the experiments are unambiguously assigned based on the simulation results. In terms of the development of computational software, I have developed a Windows version for the QCME package − an efficient package to perform first principles calculations of electron transport through molecules such as simulating the IETS. The implementation has been achieved by using the C# language and the Windows Presentation Foundation (WPF) user interface framework. The Windows version of QCME exhibits compatibility, stability, scalability, and strong operability. It has a beautiful interface, is easy to learn and to use, and has improved human-computer interactions. Such an approach of the implementation can be also extended to other quantum chemistry packages.
Sammanfattning Spektra representerar molekylära fingeravtryck som innehåller unik information om deras egenskaper. Genom att analysera spektraldata så kan man avslöja molekylernas energinivåjusteringar, identifiera deras art och geometriska strukturer, och utforska relevanta kemiska processer och mikroskopiska mekanismer. För närvarande är spektroskopi ett av de främsta sätten för människor att ge sig in i den mystiska världen av molekyler och ta del av deras berättelser. Det är dock inte en enkel process att tolka molekylära spektra, eftersom förekomsten av spektra involverar komplexa interaktioner mellan molekyler och externa stimuli. Teoretiska simulationer baserat på kvantkemi spelar en oumbärlig roll i detta avseende, vilket gör att utveckling och tillämpning av relaterad beräkningsprogramvara blir väldigt viktigt. Denna avhandling fokuserar på de teoretiska simulationerna hos två typer av molekylära spektra, nämligen den vibrationsupplösta optiska spektra och den oelastiska elektrontunnelspektra (IETS). Den föregående involverar elektronövergångar mellan en molekyls grundtillstånd och dess exciterade tillstånd med involvering av molekylära vibrationer, och den senare uppstår från excitationerna av en molekyls vibrationstillstånd inom dess elektroniska grundtillstånd genom oelastisk tunnling av elektroner över en molekylövergång. Genom att utföra tidsberoende densitetsfunktionella teoriberäkningar samt tillämpa DynaVib koden så har vi systematiskt undersökt de optiska absorptionsegenskaperna hos två typer av funktionella molekyler, d.v.s., naphthalenediimide cyclophane (NDIC) derivat och fusionerade porphyrin derivat, varav dessa har föreslagits som byggstenar för framtida enmolekylära optoelektroniska enheter. Baserat på Franck-Condon (FC) principen så har simulationerna förklarat energiskiftena inducerade av kemiska substitutioner i de första intensiva absorptionsbanden hos de tre NDIC-derivaten väl, och återger vibrationsegenskaperna hos deras två första band på ett snyggt sätt. Vidare, genom att använda tre olika utbytes korrelationsfunktioner d.v.s., den konventionella funktionella B3LYP och två långdistans korrigerade funktioner CAM-B3LYP and ωB97XD), så har man upptäckt att långdistanskorrigeringar är väldigt viktiga för beskrivningen av de spektrala egenskaperna på grund av den starka laddningsöverföringen i de relaterade exciterade tillstånden. Genom att ta hänsyn till både FC och icke-FC Herzberg-Teller (HT) bidrag så kan det experimentellt uppmätta elektroluminescens spektrumet hos en enstaka fuserad 5,15-(difenyl)-10,20-(dibrom)porfyrin (fused-H2 P) molekylen återges på ett snyggt sätt av simulationerna. Det visar sig att FC-bidraget också dominerar emissionen
viii av molekylen, medan HT-termerna huvudsakligen bidrar till lågenergi-änden av spektrumet. Vibrations finstrukturerna som observerats i experimenten är tilldelat otvetydig baserat på simuleringsresultaten. Angående utvecklingen av beräkningsprogramvara så har jag utvecklat en Windows-version för QCME-paketet - ett effektivt paket att utföra de första princip beräknelserna av elektrontransport genom molekyler såsom IETS simulation. Implementeringen har uppnåtts genom att använda språket C# och användargränssnittet Windows Presentation Foundation (WPF). Windows- versionen av QCME uppvisar kompatibilitet, stabilitet, skalbarhet, och stark funktionsduglighet. Den har ett vackert gränssnitt, är lätt att lära sig att använda, och har förbättrad human-computer interaktioner. Ett sådant tillvägagångssätt för implementeringen kan även utvidgas till andra kvantkemi-paket.
The works presented in this thesis were carried out at the Division of Theoretical Chemistry and Biology, School of Engineering Sciences in Chemistry, Biotechnology and Health, Royal Institute of Technology (KTH), Sweden and at the Hefei National Laboratory for Physical Sciences at the Microscale, University of Science and Technology of China, Hefei, China. List of papers included in the thesis Paper 1. Theoretical simulations for vibrationally-resolved absorption spectra of naphthalenediimide cyclophane derivatives, Ce Song, Li Li, Sai Duan, Yi Luo, Guangjun Tian, Spectrochimica Acta Part A: Molecular and Biomolecular Spectroscopy 2017, 183, 339-347. Paper 2. First-principles study on vibrationally resolved fluorescence of fused 5,15- (diphenyl)-10,20-(dibromo)porphyrin molecule, Feifei Qiu, Ce Song, Li Li, Yong Wei, Guangjun Tian, The Journal of Chemical Physics 2018, 149, 074312. Comments on my contribution to the papers included All the papers are the results of a teamwork. I am responsible for the computations and a large part of the writing in paper I. I have contributed to the discussions and computations in paper II. List of papers not included in this thesis Paper 1. Theoretical studies of the structure and properties of anticancer drug taxol, Meiling Zhang, Ce Song, Zhi Yao, Qiang Ji, Current Organic Chemistry 2012, 16, 2321-2331. Paper 2. Comprehensive theoretical studies on the reaction of 1-bromo-3,3,3- trifluoropropene with OH free radicals, Meiling Zhang, Ce Song, Yan Tian, Molecules 2013, 18, 7873-7885.
x Paper 3. A multiphysics fully coupled modeling tool for the design and operation analysis of planar solid oxide fuel cell stacks, Ang Li, Ce Song, Zijing Lin, Applied Energy 2017, 190, 1234-1244. Paper 4. Structural information-based method for the efficient and reliable prediction of oligopeptide conformations, Xiao Ru, Ce Song, and Zijing Lin, The Journal of Physical Chemistry B 2017, 121, 2525-2533. Paper 5. Conformers, properties, and docking mechanism of the anticancer drug docetaxel: DFT and molecular dynamics studies, Chuancai Sun, Lijuan Zhu, Chao Zhang, Ce Song, Cuihong Wang, Meiling Zhang, Yaoming Xie, Henry F. Schaefer III, Journal of Computational Chemistry 2018, 39, 889-900. Paper 6. Binding modes of cabazitaxel with the different human β-tubulin isotypes: DFT and MD studies, Lijuan Zhu, Chao Zhang, Xudong L, Ce Song, Cuihong Wang, Meiling Zhang, Yaoming Xie, Henry F. Schaefer III, Journal of Molecular Modeling 2020, 26, 162.
Acknowledgments It is a great honor to express my acknowledgment to all people who helped me during my study in Sweden. First, I would like to express my deepest gratitude to my supervisor, Prof. Yi Luo, for his professional guidance, continued encouragement, great help, and strong support. Prof. Luo provides me with an excellent atmosphere for research. His rigorous spirit in science and keen grasp of research directions have benefited me a lot. The discussion with Prof. Luo has always been an inspired and enjoyable experience. I firmly believe that the influence of Prof. Luo on my academic growth will be long-lasting and continuous. I also want to thank my co-supervisor, Prof. Yaoquan Tu. I am very grateful for his valuable advice and warm-hearted help. Even though a long time has passed, I still remember what Prof. Tu taught me that scientific research cannot be achieved overnight and it takes time to get to know the little things. The conversation with Prof. Tu always makes me feel rewarded. I am very grateful to Prof. Guangjun Tian, who has helped me a lot in performing theoretical simulations. Guangjun has not only taught me many specific skills in spectral simulations but also provided me with valuable advice on how to carry out a scientific project. I am also grateful to guangjun for carefully reading my thesis and giving very good suggestions for improvement. Many thanks to Profs. Hans Ågren, Faris Gelmukhanov, Olav Vahtras, Mårten Ahlquist, and Zilvinas Rinkevicius for their kind discussion and help. Many thanks to my friends and colleagues Sai Duan, Peng Cui, Li Gao, Xin Li, Hao Ren, Qiang Fu, Lijun Liang, Wei Hu, Xinrui Cao, Ying Wang, Liqin Xue, Jiachen Li, Zhengzhong Kang, Bogdan Frecus, Yan Wang, Li Li, Ignat Harczuk, Weijie Hua, Yuejie Ai, Xiao Cheng, Hongbao Li, Xiaofei Li, Ke-Yan Lian, Vinicius Vaz da Cruz, Guangping Zhang, Zhen Xie, Yong Ma, Xiuneng Song, Yongfei Ji, Li-Li Lin, Xing Chen, Junfeng Li, and Lu Sun for their help and all the time we shared. Last but not least, I would like to give my sincere gratitude to my wife Yongjin and my son Chenyu, for their unselfish love and delighted time we shared in the past, now, and in future.
Contents 1 Introduction 1 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Vibrational spectroscopy . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 DynaVib and QCME . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2 Simulation of Vibrationally Resolved Optical Spectra 7 2.1 Theoretical background . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Spectral simulations of NDIC derivatives . . . . . . . . . . . . . . 13 2.3 Spectral simulations of fused-H2 P molecule . . . . . . . . . . . . . 18 2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3 Simulation of Inelastic Electron Tunneling Spectroscopy 25 3.1 Theoretical background . . . . . . . . . . . . . . . . . . . . . . . . 26 3.2 Object-oriented QCME software . . . . . . . . . . . . . . . . . . . 29 3.2.1 Development environment . . . . . . . . . . . . . . . . . . 29 3.2.2 Description of the QCME calculation on Linux . . . . . . . 30 3.2.3 Design and implementation of QCME on Windows . . . . 30 3.2.4 Extension of the graphic interface to other packages . . . . 35 3.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 4 Conclusions and Future Outlook 39 4.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 4.2 Future outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 References 41 xiii
Chapter 1 Introduction 1.1 Background The inclusive and sustainable development of industrialization (ISID) plays a vital role in improving the living standards of all people worldwide and in overcoming the environmental and energy challenges faced by modern society. [1] Large-scale, integrated, and continuous industrial production is the key driving force for solving the poverty problem and promoting common prosperity. An indispensable guarantee for the industrialization processes is the accurate and convenient examination of products. For this purpose, detection technologies based on spectroscopy have been employed more and more widely. Structural identification is one of the most vital issue in chemistry, since once the information of a chemical structure is obtained, all corresponding physical, chemical and biological properties of the compound can be determined. The identification of structures thus provides a unique, long-lasting and clear way of expressing chemical compounds. [2] The spectral analysis developed by German chemist Bunsen and physicist Kirchhoff in 1859 [3] is one of the effective approaches for structural identification. As each structure of a substance has its own specific spectral characteristics, the relationships established between the structures and the spectra constitute the basis for identifying structural information according to the spectra. According to the mechanism of generation, a spectrum can be classified into emission spectrum, absorption spectrum, and scattering spectrum. Based on the corresponding wavelength range, it can be divided into X-ray spectrum, ultraviolet-visible spectrum, infrared spectrum, and Raman spectrum, etc. Among them, the characteristics of vibrational spectra have significant correlations 1
2 1 Introduction with the configurations of molecules. Besides, even in the electronic spectra of molecules, vibrational features can also be captured (Figure 1.1). Thus, vibrational spectra play an important role in detecting the structures and properties of molecules. Figure 1.1 Schematic diagram of vibration features in electronic spectrs 1.2 Vibrational spectroscopy Molecular vibration, one of the intrinsic properties of molecules, corresponds to the periodic back and forth movements of constituent atoms. The generated vibrational spectrum originates from transitions between different vibrational levels in the same electronic state. Vibrational spectrum constitutes an important branch of spectral analysis. Infrared spectrum, Raman spectrum, and sum- frequency generation, to name a few, belong to this category. Infrared spectroscopy is widely used in qualitative studies of mixed systems. This method, however, suffers from the shortcoming of cluttered signals, since signals from the background cannot be avoided. External environment and the solvents may contaminate the signal of the target molecule, making it difficult to distinguish and strip the desired information. Raman spectroscopy can complement with the infrared spectrum in identifying specific structural features or characteristic groups, but the weak signal intensity and the low sensitivity had limited its application for a long time. Fortunately, Raman spectroscopy has regained the favor of researchers thanks to the advancement of the optical technology, and two new experimental methods are thus developed, i.e., the surface-enhanced Raman spectroscopy and the resonant Raman spectroscopy. The former utilizes surface plasmon and the corresponding signals are greatly affected
1.3 DynaVib and QCME 3 by the substrate, while lasers are required in the latter for selectively exciting vibrations to increase the intensities of the spectra. As another means of the vibrational spectrum analysis, sum frequency spectroscopy is often used to reveal the microscopic information of the interface, being a powerful and versatile tool in surface science. This approach has its shortcomings too, like the different spectral features appearing under different polarization combinations, for example. When studying the adsorption and emission spectra that involve transitions between different electronic states, vibrational fine structures are the key information in interpreting the high-resolution spectra. Theoretical simulations of the vibrational fine structural features play an essential role in understanding the geometric and electronic properties of the target molecules. From the calculated vibrational fine structures, one can not only compare them with the vibrational-resolved absorption/emission spectra from the experiments for substance identification but can also reveal detailed mechanisms about the molecules at the microscales. It thus provides a practical way to assist the experimental analysis and provide further information that cannot be easily obtained merely from experiments. The vibrational-resolved absorption/emission spectroscopy is considered as one of the most commonly used methods in spectra analysis. With the rapid development of molecular electronics, much effort has been devoted to the fabrication, measurement and artificial control of molecular devices. [4–9] Here, precisely tuning the contacting and bonding patterns between functional molecules and electrodes is greatly important, since a very small change in the molecular configuration may significantly affect the performance of the constructed device. However, identifying the structural feature is challenging at the atomic scales. In this context, the inelastic electron tunneling spectrum (IETS) is extremely useful in identifying configurations of molecular devices, especially via systematic comparison between experimental measurements and theoretical simulations. 1.3 DynaVib and QCME DynaVib [10] and QCME [11] are two packages for the simulations of vibration- related spectra. Based on the harmonic approximation, the DynaVib code can be used to simulate vibrationally-resolved optical spectra of polyatomic molecules. QCME is an efficient tool for simulating elastic and inelastic charge transport
4 1 Introduction properties of molecular junctions. These two packages have been employed in studying a wide range of molecular systems and have been demonstrated to work very well. However, there is still room for improvement when using these two packages. To be specific, for DynaVib, it is not easy to achieve convergence of the simulated spectra when the time-independent method is used. To address this issue, we have developed a revised version of DynaVib in which a more efficient time- dependent method is implemented. We have also considered the contribution of the Herzberg-Teller (HT) part in the new version to further enhance the accuracy of the simulations. As a demonstration, we have simulated the vibrationally-resolved absorption spectra of the naphthalenediimide cyclophane (NDIC) molecules and the electroluminescence spectra of fused 5,15-(diphenyl)- 10,20-(dibromo)porphyrin (fused-H2 P) molecule by using the newly developed package. Developing a graphical user interface (GUI) for a package will be great helpful for users to conveniently use the corresponding tool. Here, we have put our attempt and effort in the QCME package. Concretely, we have developed an easy-to-use interface for QCME under the Windows system and transplanted the QCME code to the same operating system. The Windows version of QCME possesses the advantages of compatibility, stability, scalability, beautiful interface, easy to learn, easy to use, and strong operability for improved human-computer interactions. 1.4 Summary The aim of this thesis is to illustrate the importance of the two types of vibrational spectra, the vibrationally-resolved optical spectra and the IET spectra, in identifying molecular configurations. Through the development and transplantation of the corresponding codes and the investigation of the underlying mechanisms involving specific molecular systems, the applications of the two types of spectra are discussed. By developing a graphical user interface for the simulation code, it is convenient to implement and transplant some typical quantum chemistry software from Linux to the Windows operating system. The contents of the following chapters are organized as follows. In Chapter 2, theoretical backgrounds and applications of DynaVib are provided. In this chapter, the time-dependent method and the Herzberg-Teller
1.4 Summary 5 part will be discussed. The implementation of the code will also be introduced, followed by an application to the NDIC derivatives and the fused-H2 P molecule. In Chapter 3, I will give the theoretical backgrounds of IETS and introduce how the QCME code is developed and transplanted to the Windows operating system using the C# language and the Windows Presentation Foundation (WPF) user interface framework. I will also present how to use the Windows version of QCME. Chapter 4 includes conclusion and future outlook. We think that the resonant Raman scattering could be a new feature of the DynaVib software. A portable graphical user interface of DynaVib, like a mobile app for example, is considered and could be further developed.
Chapter 2 Simulation of Vibrationally Resolved Optical Spectra Vibrationally-resolved absorption and emission spectra of molecules can be simulated by using the DynaVib code. [10] Especially, both the time-independent sum-over-state method and the time-dependent eigenstate free method have been implemented, making DynaVib an efficient tool to simulate the spectroscopic properties of molecules. In this chapter, I will first introduce the theoretical background, and then present the applications of the DynaVib code, in investigating the optical absorption properties of a series of naphthalenediimide cyclophane (NDIC) derivatives and the optical electroluminescence properties of the fused 5,15-(diphenyl)-10,20-(dibromo)porphyrin (fused-H2 P) molecule. The NDIC molecules and the fused-H2 P molecule are suitable research objects in scanning tunneling microscope based single-molecule optical characterizations and have been demonstrated to exhibit interesting optical responses. [12,13] Detailed theoretical analysis on the corresponding optical properties of such systems could be beneficial for the design of single-molecule optoelectronic devices. The large sizes and the interesting optical properties of the two types of molecules also enable us to check the accuracy and the practicability of the DynaVib code. 2.1 Theoretical background In the simulations of the vibrationally-resolved absorption and emission spectra with DynaVib, the key is to obtain the vibrational profiles, which can be calculated either by directly computing the vibrational integrals or by evaluating the time- evolution of the appropriate time-correlation functions. In this section, I will 7
8 2 Simulation of Vibrationally Resolved Optical Spectra illustrate the processes by using one-photon absorption as an example. Other types of optical processes, like the one-photon emission and the multi-photon nonlinear processes, can be treated in a similar way. The incident frequency-dependent absorption intensities of an one-photon absorption process can be written as Eq. 2.1 [14–17] 4π 2 ω X γ σ (ω) = P (i, T ) |⟨ψi |µ̂| ψf ⟩|2 , (2.1) 3c i,f (ω − ωif )2 + γ 2 where ω represents the frequency of the incident light and c is the speed of light. |ψi ⟩ and |ψf ⟩ are the wave functions of the initial and the final states, respectively. ωif is the energy difference between the above two states. P (i, T ) represents the temperature (T ) dependent Boltzmann population of the initial state |ψi ⟩. Here, the line shape broadening is described by a Lorentzian function with γ being the half-width at half-maximum (HWHM). In practice, the broadening can also be exerted by using other types of functions such as the Gaussian or the Voigt schemes, depending on the actual requirements. The essential part of Eq. 2.1 is to calculate the transition dipole moment ⟨ψi |µ̂| ψf ⟩ with µ̂ being the dipole operator. For practical calculations, the Born-Oppenheimer (BO) approximation and the harmonic approximation need to be employed to obtain the vibrational transition dipole moment. According to the BO approximation, the electronic and the nuclear parts of the wave functions can be separated and thus, we have |ψi ⟩ = |ψie ⟩ |ψiv ⟩ and |ψf ⟩ = ψfe ψfv . Here, the superscripts e and v represent the electronic and the nuclear part, respectively. Accordingly, the transition dipole moment can be re-written as Eq. 2.2 ⟨ψi |µ̂| ψf ⟩ = ψiv ψie |µ̂| ψfe ψfv , (2.2) where µeif = ψie |µ̂| ψfe is the electronic part of the transition dipole moment which is dependent on the nuclear coordinates. However, the nuclear coordinates dependence of µeif makes it inefficient to directly calculate the integrals on the right side of Eq. 2.2. To address this issue, it is helpful to expand µeif into a Taylor series at the equilibrium position (Q0 ) of the initial state |ψi ⟩: N X ∂µeif µeif = µeif (Q0 ) + Qk + · · · · · · , (2.3) k=1 ∂Qk where N is the number of the vibration modes. It is worth noting that the well- known Condon approximation is equivalent to merely keeping the first term on
2.1 Theoretical background 9 the right side of Eq. 2.3. In this case, the electronic transition dipole moment is assumed to be independent of the nuclear coordinates. By substituting Eq. 2.2 into Eq. 2.3, we can get the following expression N X ∂µeif ⟨ψi |µ̂| ψf ⟩ = µeif (Q0 ) ψiv | ψfv + ψiv |Qk | ψfv + · · · · · · . (2.4) k=1 ∂Qk Here, the first term on the right side is the Franck-Condon (FC) term, while the second one is the Herzberg-Teller (HT) term. In most cases, these two terms are adequate in the calculations and thus we will neglect higher order terms in the following discussions. After substituting Eq. 2.4 into Eq. 2.1, we can obtain the expression for the one-photon absorption process considering both the FC and the HT terms as N 2 X ∂µeif |⟨ψi |µ̂| ψf ⟩|2 = µeif (Q0 )2 ψiv | ψfv + µeif (Q0 ) ψiv | ψfv ψiv |Qk | ψfv k=1 ∂Qk N N X X ∂µeif ∂µeif + ψiv |Qk | ψfv ψiv |Ql | ψfv . (2.5) k=1 l=1 ∂Qk ∂Ql From the right hand side of Eq. 2.5, it can be found that there are three parts in the absorption spectrum: The first one only contains the FC integral; the second term includes both the FC and the HT integrals (often referred to as the FC/HT part); and the third term only contains the HT integrals (often referred to as the HT part). Imitating the form of Eq. 2.5, Eq. 2.1 can be rewritten as δ (ω) = δ F C (ω) + δ F C/HT (ω) + δ HT (ω) , (2.6) where 4π 2 ω X 2 γ δ F C (ω) = P (i, T ) µeif (Q0 )2 ψiv | ψfv , (2.7) 3c i,f (ω − ωif )2 + γ 2 N F C/HT 4π 2 ω X X e ∂µeif v v γ δ (ω) = P (i, T ) µif (Q0 ) ψ |ψ ψiv |Qk | ψfv , 3c i,f k=1 ∂Qk i f (ω − ωif )2 + γ 2 (2.8) and
10 2 Simulation of Vibrationally Resolved Optical Spectra N X N 4π 2 ω X X ∂µeif ∂µeif v γ δ HT (ω) = P (i, T ) ψi |Qk | ψfv ψiv |Ql | ψfv . 3c i,f k=1 l=1 ∂Q k ∂Q l (ω − ωif )2 + γ 2 (2.9) Eq. 2.6 is the final form for calculating the line intensities within the framework of the harmonic approximation. As aforementioned, there are two approaches, the time-independent and the time-dependent ones, that can be used for the calculation. Considering that detailed discussions have been given in many previous works [18] regarding the time-independent method, here we mainly focus on the time-dependent approach. In contrast to the time-independent method, in which one needs to calculate the vibrational integrals and perform sum-over-state among a large number of involved vibrational transitions, in the time-dependent method, one converts the sum-over-state in Eq. 2.6 into the Fourier integrals of the corresponding dipole correlation functions. In this way, heavy computations of the vibrational integrals in the former method can be avoided. Upon the Fourier transformation, Eq. 2.6 can be written as [16] Z ∞ FC 4π 2 ω e 2 δ (ω) = µ (Q0 ) dtexp[i (ω − ωif ) t]δ F C (t) exp(−γt), (2.10) 3c if 0 N ∂µeif Z ∞ F C/HT 4π 2 ω X e δ (ω) = µ (Q0 ) Re dtexp[i (ω − ωif ) t − γt]δ F C/HT (t), 3c k=1 if ∂Qk 0 (2.11) and N N 4π 2 ω X X ∂µeif ∂µeif Z ∞ HT δ (ω) = Re dtexp[i (ω − ωif ) t − γt]δ HT (t). (2.12) 3c k=1 l=1 ∂Qk ∂Ql 0 Here, δ F C , δ F C/HT , and δ HT are the Franck-Conon, the Franck- Condon/Herzberg-Teller, and the Herzberg-Teller parts of the absorption cross section, respectively. An analytical form for the δ F C (t) term has been provided by Yan and Mukamel [19] as δ F C (t) = |ψ (t)|−1/2 exp DT f (t) D , (2.13) where 1 ψ (t) = (C+ S ′ A− + C− SA+ ) (C+ SA− + C− S ′ A+ ) , (2.14) 4
2.1 Theoretical background 11 and −1 f (t) = −S ′ A− (C+ S ′ A− + C− SA+ ) C− , (2.15) with A± = (n̄ + 1) ± n̄exp(iω ′ t), C± = 1 ± exp(iωt), −1 ℏω ′ n̄ = exp( )−1 , kT −1/2 ωi J −1 Sij = , ωj′ ij Di = (ωi )1/2 −J −1 K i , and T S ′ = S −1 . Here, J and K are the Duschinsky rotation matrix and the displacement vector, respectively. ω ′ and ω are the vibrational frequencies of the initial state and the final state, respectively. Shuai et al. formulated the analytical expressions for δ F C/HT and δ HT . [16] In F C/HT their method, an auxiliary column matrix Hk is defined to obtain δ F C/HT (t), as in the formula below: F C/HT Hk = [01 . . . 1k . . . 02N ]T1×2N . (2.16) What follows, δ F C/HT (t) can be expressed as n o F C/HT T −1 δ F C/HT (t) = −δ F C (t) (Hk ) L F . (2.17) For the HT term, a square matrix GHT kl is introduced: 011 012 · · · 01N +1 ··· 021 022 · · · 02N +1 · · · GHT kl · · · = ··· ··· ··· · · · . (2.18) 0k1 0k2 · · · 1kN +1 · · · ··· ··· ··· ··· ···
12 2 Simulation of Vibrationally Resolved Optical Spectra Thus, the HT term can be rewritten as n T HT −1 o −1 −1 δ HT (t) = δ F C (t) iℏTr GHT kl L + L F Gkl L F , (2.19) where " # B −A L= , (2.20) −A B 2N ×2N and T F = K T EJ K T EJ 1×2N , (2.21) with A = af + J T ai J, B = bf + J T bi J, and E = b i − ai . Here, ai,f is the diagonal matrix for the initial electronic states: ωi,f k ai,f k (τ ) = , (2.22) sin(ℏωi,f k τi,f ) and bi,f is that for the final electronic states: ωi,f k bi,f k (τ ) = . (2.23) tan(ℏωi,f k τi,f ) The above approach has been implemented in DynaVib, [10] which enables us to simulate vibrationally-resolved absorption and emission spectra very efficiently. In the following, I will apply DynaVib to investigate the experimentally observed changes of the optical absorption properties of NDIC derivatives induced by chemical substitutions and the vibrational fine structure in the electroluminescence spectra of the fused-H2 P molecule.
2.2 Spectral simulations of NDIC derivatives 13 2.2 Spectral simulations of NDIC derivatives Cyclophanes are hydrocarbons composed of one or more interconnected aromatic units. Due to their special chemical structures and wide applications, cyclophanes have attracted the attention of many researchers. [20] NDIC molecule belongs to a type of special cyclophane compounds consisting of two NDI molecules in a face-to-face pattern. NDIC derivatives have also been synthesized via group substitutions for various research purposes. [21–24] For example, Gabutti et al. synthesized three new NDIC derivatives to perform scanning tunneling microscope (STM) induced fluorescence measurements. [25,26] In the STM experiments, the use of such double-layer NDIC derivatives eliminates the need of spacers since one of the two NDI chromophores can be used to separate the other from the STM substrates. This self-decoupling property made the NDIC type of molecules quite appealing for STM luminescence studies which has attracted the attention of many researchers. [25,27,28] The core-substituted NDIC derivatives exhibit rich photo-absorption and emission properties that are dependent on the type of the substituent groups. Gabutti et al. experimentally studied three NDIC derivatives of NDIC- OMe, NDIC-StBu, NDIC-N(CH2 )5 , which were produced via substitutions by dimethoxy, tert-butylsulfanyl and dipiperidinyl, respectively. [26] It was found that upon the substitution, the first (low-energy) absorption band exhibits a significant redshift, while there is almost no change in the position of the second band. In this way, they had achieved chemically tunable Förster resonance energy transfer (FRET) within the three NDIC derivatives. [26] We have theoretically investigated the geometric and optical properties of the three derivatives. Our simulations reproduced the vibrationally-resolved absorption spectra of the three molecules. The accuracy of the methods and the DynaVib implementation have been well verified. This theoretical work may help to shed light on the behaviors of the NDIC derivatives in the STM induced luminescence experiments as well as in the corresponding energy transfer processes. First principles calculations were performed to simulate the vibrationally- resolved absorption spectra of the three NDIC derivatives, NDIC-OMe, NDIC- StBu and NDIC-N(CH2 )5 , for a comparison with the experiments. [26] According to the crystal structure of the NDIC molecule, [25] the two planes of the NDI components are not parallel but exhibit a certain inclination. For NDIC-StBu, [26] by contrast, the two planes are almost parallel. Thus, we constructed a tilted
14 2 Simulation of Vibrationally Resolved Optical Spectra Figure 2.1 Optimized non-planar (left) and planar (right) structures of NDIC-OMe © (a), NDIC-StBu (b), and NDIC-N(CH2)5 (c). Reused with permission from ref. 34. Copyright 2017 Elsevier. structure (called non-planar structure) and an almost parallel structure (called planar structure) by replacing the corresponding H atoms and the -StBu groups, respectively. In Figure 2.1, we present the structures of all the three NDIC derivatives. The geometries of these molecules at their respective ground states were optimized within the framework of density functional theory (DFT) by employing three exchange correlation functionals B3LYP, [29–31] CAM-B3LYP, [32] and ωB97XD. [33] The 6-31G(d) basis set was used to expand the wave functions. We have confirmed that the optimized structures correspond to local minima on the potential energy surfaces since no imaginary frequency was found in the vibrational analysis. Since it is computationally very expensive to perform vibrational analysis on the excited states of large molecules, the linear coupling model (LCM) [35,36] was employed to simulate the vibrationally-resolved spectra of the NDIC derivatives. By using the LCM model, one can efficiently obtain the vibrationally-resolved absorption and emission spectra of large-sized molecules. [36–39] LCM neglects the Duschinsky mode mixing effect and assumes the potential energy surfaces in the excited states have the same curvature as those in the ground state and they differ
2.2 Spectral simulations of NDIC derivatives 15 with each other only by a shift of the equilibrium positions (the displacement vector K). Within the harmonic approximation, one can obtain the displacement vectors from the excited state potential energy gradient at the stable geometries of the ground states and thus, avoid the computationally demanding vibrational analysis on the excited states. In practical calculations, only the forces, oscillator strengths and the excitation energies of the excited state are required which we obtain with the time-dependent density functional theory (TDDFT). The polarizable continuum model (PCM) was used to implicitly consider the solvent effects. The first-principles calculations were carried out by using the Gaussian 09 software package, [40] while the spectral simulations were performed with the DynaVib code. [10] The optimized structures of the three NDIC derivatives both in vacuum and in the dichloromethane solvent were obtained by using different DFT functionals, with the corresponding results shown in Table 2.1. One can see that for all the three derivatives, the non-planar structures are more stable than the planar ones. Both the long-range correction and the empirical dispersion have a big effect on the calculated results. Table 2.1 Relative energies (eV) of the three NDIC molecules obtained using different © functionals. ∆E is the energy difference (eV). Reproduced with permission from ref. 34. Copyright 2017 Elsevier. We then calculated the excitation energies and the corresponding oscillator strengths for the first three excited states of the non-planar structures of the NDIC derivatives by using the TD-DFT method, with the results shown in Table 2.2. One can see that the excitation energies obtained with the B3LYP functional
16 2 Simulation of Vibrationally Resolved Optical Spectra Table 2.2 Vertical excitation energies and oscillator strengths (in the parentheses) © of the three NDIC molecules (non-planar) obtained using different functionals. Reproduced with permission from ref. 34. Copyright 2017 Elsevier. Figure 2.2 Vibrationally-resolved absorption spectra of the three NDIC derivatives. From top to bottm: NDIC-OMe, NDIC-StBu, NDIC-N(CH2 )5 . The absorption spectra in the low energy band (band I), the high energy band (band II) and the total absorption spectra were shown from left to right. The theoretical spectra were shifted to match the experimental ones with values indicated in the figure. For the combined spectra, © the energy shift is the same as those in band II. Reused with permission from ref. 34. Copyright 2017 Elsevier.
2.2 Spectral simulations of NDIC derivatives 17 are significantly lower than the corresponding results obtained from the other two functionals. Moreover, the first excited states predicted by B3LYP are only weakly allowed with oscillator strengths, much smaller than those obtained with the other functionals where the long-range correction were included. In the experiment, it has been shown that all the three molecules have strong absorption and emission at the low energy regime, [26] which strongly indicates that the first excited states are allowed transitions. Therefore, we applied the CAM-B3LYP and ωB97XD functionals in the simulations of the vibrationally- resolved absorption spectra. According to the calculation results, it was found that for all the three molecules, NDIC-OMe, NDIC-StBu, and NDIC-N(CH2 )5 , their first excited states exhibit significant redshifts while the positions of the second absorption band are almost unchanged. By substituting different groups, a chemically tunable FRET was achieved. [34] Our theoretical simulations nicely reproduced the observed phenomenon in the experiment. [25,26] The vibrationally-resolved absorption spectra of the three NDIC derivatives were simulated by using the CAM-B3LYP and the ωB97XD functionals. Since the first two absorption bands are both strongly allowed transitions, only the FC contributions were considered in the simulations. In Figure 2.2, we present the results for the non-planar structures using the CAM-B3LYP functional. One can see that the vibrational features of the absorption spectra (the left and the middle panels), especially those in the first absorption band, are very similar to the experimental results. There are some differences in the peaks of the first and the second absorption bands, which may be caused by the core substitution effect. From the right panel of Figure 2.2, it can be found that differences still appear between the total simulated absorption spectra and the corresponding experimentally measured ones. Here, the red shift of the first absorption band is underestimated with the intensity overestimated, and the intensity of the second absorption band is underestimated. We have investigated the influence of the basis set on the simulated spectra, and the corresponding results are shown in Figure 2.3. One can see that changing the size of the basis sets can slightly affect the excitation energy. The changes to the energy shift between the two absorption bands, however, are not obvious. Moreover, adding diffuse functions in the basis set basically does not cause obvious changes in the simulated spectra. This indicates that changing basis sets could not improve the comparison between the simulations and the experiments. Advanced quantum chemistry methodologies and more accurate solvent models could be needed to further improve the quality of the simulated vibrationally-resolved absorption spectra.
18 2 Simulation of Vibrationally Resolved Optical Spectra © Figure 2.3 Simulated absorption spectra of the three NDIC derivatives obtained using different basis sets. Reused with permission from ref. 34. Copyright 2017 Elsevier. In short, using the DynaVib code, we have simulated the vibrationally- resolved spectral profiles at the Franck-Condon level for the first two intense absorption bands of three NDIC derivatives. Good agreements between the simulated spectra and the experimental ones are obtained, from which we have reproduced the vibrational fines structures and explained the electronic origins of the substitution induced energy shifts observed in the experiments. These results demonstrate the practicality of the time-dependent method implemented in the DynaVib code in simulating the vibrationally-resolved optical properties of large sized molecules. 2.3 Spectral simulations of fused-H2P molecule Porphyrin and its derivatives have a wide range of applications due to their unique geometrical and optical properties. Especially, porphyrin molecules have become the most commonly studied model systems in STM based single-molecule optical measurements. By combining the advantages of the ultra-high spatial resolution from STM and the chemical information recognition from optical
2.3 Spectral simulations of fused-H2 P molecule 19 characterizations, such type of measurements have significantly enriched our understanding at the single-molecule level in many fundamental physical and chemical processes. Meanwhile, the experimental advances also call for more theoretical efforts in detailed analysis of measured single-molecule spectra, which plays a key role in uncovering the experimental phenomena. One interesting example is to analyze the STM induced single-molecule electroluminesence spectra of the fused-H2 P molecule as measured by Chong et al. [13] The obtained emission spectra have ultra-narrow peaks with a full-width- at-half-maximum down to 2.5 meV and exhibiting rich vibronic features. These results make fused-H2 P an excellent system for spectral simulations because, on one hand, the experiments were performed using a single-molecule decoupled from the environment, which is ideal for comparison with theoretical simulations that employ free molecular models. On the other hand, the rich vibrational features also serve as a valuable reference for testing different vibrational models. Especially, unlike the NDIC molecules where the spectra were dominated by the Franck-Condon parts, the Herzberg-Teller contributions are expected to play an important role in the emission spectra of porphyrin derivatives and therefore, should be taken into account in the spectral simulations. In this section, I will present our work on the theoretical simulations of the vibrationally-resolved emission spectra of the fused-H2 P molecule. Similar to the study of NDIC molecules, we also performed DFT/TDDFT calculations to obtain the simulated spectra of fused-H2 P. Thanks to the relatively smaller size of the fused-H2 P molecule (shown in Figure 2.4), both the ground and the first excited states were optimized. Frequency analysis were also performed for the both two states to confirm that stable geometries had been obtained. Three density functionals, B3LYP, [29–31] ωB97X-D, [33] and M06-2X [41] were used in the calculations. Our test calculations showed that increasing the size of the basis set (6-31G(d,p) in the current simulations) did not bring a significant improvement in the simulated emission spectra. In the spectral simulations, both the FC and the HT parts were included. The Duschinsky mode mixing effect was also considered. The derivatives of the transition electronic dipole moment with respect to the vibrational normal modes were obtained from the analytical nuclear derivatives that can be calculated using the Gaussian 16 [40] software after the vibrational calculations at the excited state [42] . Such the analytical derivatives eliminated possible errors that may appear in numerical differential approach and thus, further improved the accuracy of the HT part of the spectra. All the spectra were simulated using the time-
20 2 Simulation of Vibrationally Resolved Optical Spectra © Figure 2.4 Geometrical structure of the fused-H2 P molecule. Reused with permission from ref. 42. Copyright 2018 AIP Publishing. independent method as implemented in the DynaVib software. Such a method was efficient in simulating the fused-H2 P molecule, which allowed a convenient assignment to the fine structures of the spectra. In Figure 2.5, we compare the fluorescence spectra of the fused-H2 P molecule simulated using the three density functionals with the electroluminescence spectra of the molecule reported by Chong et al. [13] . It can be immediately noticed that the three functionals give quite similar spectral profiles, indicating the simple valence transition feature of the S1 → S0 transition. By comparing the simulated and the measured spectra, it can be found that the most obvious discrepancy is the relative intensity of the vibrational peaks. For example, the intensities of the peaks located at about 1.33 eV and 1.46 eV are overestimated in the simulated spectra. However, all the experimentally observed vibrational structures in the electroluminescence spectrum have been nicely reproduced in the simulated spectra, as indicated by the vertical dashed lines. The good agreement between the simulated and the experimental spectra indicates that the applied method is sufficient for the description of the emission process of the fused-H2 P molecule. An interesting aspect of the porphyrin derivatives is that the HT parts could also have significant contributions to the optical properties of the molecules. For example, the optical absorption and emission spectra of the simplest porphyrin derivatives, porphine (H2 P), were dominated by the HT terms. [43,44] To examine the HT effect on the emission spectra of the fused-H2 P molecule, we have taken the simulated spectra using ωB97X-D as the example and compared the total spectrum with those obtained by considering only the FC or the HT contributions. The results are demonstrated in Figure 2.6. It can be found that, similar to the case of the NDIC molecule, the fluorescence of the fused-H2 P molecule is
2.3 Spectral simulations of fused-H2 P molecule 21 Figure 2.5 Vibrationally-resolved fluorescence spectra of the fused-H2 P molecule as simulated with B3LYP (red line), ω B97X-D (green line) and M06-2X (blue line). The experimental spectrum (black line) from ref. 13 was also shown for comparison. The © simulate spectra were blue shifted by 0.27 eV, 0.10 eV, and 0.09 eV for the three functionals, respectively. Reused with permission from ref. 42. Copyright 2018 AIP Publishing. also dominated by the FC part. The HT part only contributes to the low- energy tail of the spectrum. Such an interesting change can be attributed to the geometrical changes from free H2 P type to the fused configuration in fused- H2 P. In fact, symmetry analysis shows that the S1 → S0 transition is a dipole allowed Bu → Ag transition with major contributions coming from the lowest unoccupied molecular orbital (LUMO) to the highest occupied molecular orbital (HOMO), as shown in the inset of Figure 2.6. This also confirms the valence nature of the emission transition and further explains the good performance of all the three density functionals. The good agreement between the simulated and the measured spectra enables us to make a detailed assignment to the vibrational fine structures in the emission spectra of the fused-H2 P molecule. Figure 2.7 (a) shows such a detailed plot where we have included both the simulated (red) and the measured (black) spectra together with the bar spectrum of the FCHT factors as computed by the time- independent method using the DynaVib software. It can be found that the most intense vibrational peaks are the 0-1 vibrational transitions of different modes. The lack of high order transition in the emission spectra can be attributed to the relatively small FC activity of the totally symmetric modes of the fused-H2 P molecule. The broad peaks in the emission spectra, such as the one around 1.4 eV, are a collective contribution from several nearby transitions. It is interesting to see that the simulated spectra actually capture such detailed features quite nicely,
22 2 Simulation of Vibrationally Resolved Optical Spectra Figure 2.6 Detailed analysis of the fluorescence spectra of the fused-H2 P molecule as obtained with the ω B97X-D functional. Black line is the total spectrum containing both the FC (red) and HT (blue) contributions. The electronic transition between LUMO © and HOMO, which dominates the emission process, is shown as the inset. Reused with permission from ref. 42. Copyright 2018 AIP Publishing. demonstrating the accuracy of the applied method. The six most active modes corresponding to the intense vibrational peaks in the emission spectra can be found in Figure 2.7 (b). It can be seen that all the six modes are in-plane modes mainly involving atoms in the porphyrin chromophore. This is also part of the reasons for the good agreement between the simulated fluorescence spectra and the experimentally measured electroluminescence spectrum for which the fused-H2 P molecule was decoupled from the metallic electrodes through the terthiophene side chains. This was further verified in our calculations, where it was found that the inclusion of the terthiophene side chain have negligible influences on the simulated emission spectra [42] . In this section, I have presented our simulations on the optical emission spectra of the fused-H2 P molecule. It is found that the three used density functionals (B3LYP, ωB97Xd, and M06-2X) give quite similar spectral profiles while the ωB97Xd and M06-2X functionals improve the obtained excitation energy by 0.15 eV. Test calculations with different basis set indicates that the standard 6-31G(d,p) basis set is sufficient to describe the optical properties of the fused-H2 P molecule. The simulated spectra are in good agreement with the experimentally measured single-molecule electroluminescence spectra of the molecule, which enables us to a give detailed assignment to the vibrational origins of the fine spectral structures. It has also been found that the high energy part of
2.4 Summary 23 Figure 2.7 (a) Vibrational assignment of the fluorescence spectra of the fused-H2 P molecule. Black and red lines are the experimental spectrum and the simulated spectrum, respectively. The blue bar spectrum is the calculated FCHT factors. (b) © Active vibrational modes for the main emission peaks in the fluorescence spectra. Reused with permission from ref. 42. Copyright 2018 AIP Publishing. the emission spectra of the molecule is dominated by the 0-1 FC transitions and the HT part only contributes to the low energy tail of the spectra. 2.4 Summary This chapter is devoted to the theoretical simulations of vibrationally-resolved molecular optical spectra. The theoretical background, especially the theory of the time-dependent method for the simulation of the vibrationally-resolved electronic spectra is presented. This method, which has been implemented in the DynaVib software developed by our group, is then applied for two types of functional molecules, namely NDIC derivatives and fused-H2 P molecule. The simulated spectra are used to explain some interesting experimental features reported for the two types of molecules, which is not only helpful for our understanding of the
24 2 Simulation of Vibrationally Resolved Optical Spectra experimental results but also facilitates the future application of such molecules in the design of single-molecule optoelectronic devices.
Chapter 3 Simulation of Inelastic Electron Tunneling Spectroscopy With the aim of building ultra-small electronic devices, molecular electronics has attracted tremendous research interests in the past two decades. Great advancements have been achieved in the experimental realization of functional molecular devices like switches and rectifiers. [8,45–47] However, since the measured electronic signals in the current-voltage or the conductance-voltage characteristics lack chemical information, it is not straightforward to determine whether a molecule really exists, and if so, how the electron transport properties across the junction are affected by the behavior of this molecule. Inelastic electron tunneling spectroscopy (IETS), [7] which corresponds to the second derivative of the current with respect to the bias voltage, is an ideal tool to address the above issue. Generally, there are two types of charge transport processes in molecular junctions: the elastic tunneling and the inelastic tunneling (Figure 3.1). Since the inelastic tunneling can be triggered via vibrational excitations of molecules within the junctions due to the coupling between the tunneling electrons and the molecular vibrational motions, the vibrational properties of the sandwiched molecules can be reflected by IETS. Compared with conventional vibrational spectroscopic techniques, such as infrared absorption or Raman scattering, IETS has different selection rules. [48] This makes IETS be an promising characterization tool for capturing vibration modes that are invisible in conventional vibration spectra. [49] Intriguingly, IETS measurements can be conducted in real space at the single-molecular scale, empowering IETS the ability to provide the real space distribution of vibrational modes with sub-angstrom resolutions. The single-molecular scale measurement 25
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