Electrostatic Polarization Energies of Charge Carriers in Organic

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Condensed Matter, Interfaces, and Materials

Electrostatic polarization energies of charge carriers in organic molecular crystals: A comparative study with explicit state-specific atomic polarizability (SSAP)-based AMOEBA force field and implicit solvent method Tao Xu, Wenliang Wang, and Shiwei Yin J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.8b00132 • Publication Date (Web): 05 Jun 2018 Downloaded from http://pubs.acs.org on June 6, 2018

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Journal of Chemical Theory and Computation

Electrostatic polarization energies of charge carriers in organic molecular crystals: A comparative study with explicit state-specific atomic polarizability (SSAP)-based AMOEBA force field and implicit solvent method Tao Xu, Wenliang Wang and Shiwei Yin* Key Laboratory for Macromolecular Science of Shaanxi Province, School of Chemistry and Chemical Engineering, Shaanxi Normal University, Xi'an city, P. R. China, 710119 Abstract The electrostatic polarization plays an important role in determining the energy levels of charge carriers in organic solids, which is controlled by the atomic polarizability in AMOEBA polarizable force field. QTAIM-based space partitioning of molecular polarizability is utilized to uniformly parameterize the state-specific atomic polarizability (SSAP) of π-conjugated organic small molecules to avoid fitting molecular polarizability of some artificial training set. Herein, the SSAPs are applied to explicitly extrapolate the electrostatic polarization energy (Epol) of the charge-carriers of nine π-conjugated organic crystals including six p-type transfer materials oligoacenes and TIPS-substituted oligoacenes and three n-type transfer materials F-substituted oligoacenes and TCNQ. Our results demonstrate that the electrostatic polarization energy of hole carrier ( E+pol ) are smaller than that of electron carrier ( E−pol ) for p-type molecules while E+pol are larger than E−pol for n-type molecules. SSAP-based Epol values of oligoacenes behave as nearly unvaried feature with the increase of conjugation length which is similar with implicit polarizable continuum model (PCM) results, while Epol obtained from the default atomic polarizability behaves as notable decrease. Implicit PCM can correctly capture most of electrostatic polarization of ions in bulk system although it slightly underestimates the gap between electrostatic polarization of hole and electron carrier in oligoacene crystals. Our results demonstrate that this unified parameterized SSAP provides a reliable and cheap tool to estimate the energy landscape of charge carriers in 1

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condensed-phase organic solids. 1. Introduction Π-conjugated organic molecules have attracted considerable interest because of their potential applications in optoelectric devices such as organic light emitting diodes (OLEDs), organic solar cells (OSCs) and organic field effect transistors (OFETs).1-6 The conductivities of electric active materials play a very important role in controlling their optoelectric property. Due to the weak intermolecular interaction between organic molecules, the behaviors of charge-carrier transfer can be well described using the thermal-activated hopping model at high-temperature limit. Besides the molecular intrinsic charge mobility, the electronic conductivity is also strongly related to the ability of charge injection from metal electrodes into bulk solid. The charge injection is strongly related to the difference of energies between metal electrode’s Fermi level and molecular ionization potential (IP)/electron affinity (EA) in solid or interface phase. The organic molecules with suitable IP/EA values are very important to achieve the ohmic contact. Here we need to keep in mind that IP and EA are the condensed-phase values not their gas-phase values. With the assumption that charge is bound to an individual molecule, Lyons defined the polarization energy of ion (Wion) as the deviation of IP/EA in vacuum and IP/EA in condensed phase.7 To distinguish pol the electrostatic polarization energy ( Eion ) produced from induced-dipole interactions,

Bounds named this polarization energy (Wion) as apparent polarization energy.8, 9 Because of the complication of intermolecular interactions and lack of packing information in disordered systems, for a long period, the gaseous molecular IP/EA values are often used to evaluate the site energy of hole/electron carrier in the theoretical modeling of organic electron processes of organic solid, and the same Wion is usually assumed for most of molecules and even for the different locations. Recently Griffith et al utilize ultraviolet photoelectron spectroscopy (UPS)10 to measure

the

gaseous

and

solid-phase

IP

values

of

pentacene

and

tri-isopropylsilyl-ethynyl (TIPS)-substituted pentacene. The UPS data demonstrate that the big difference in apparent polarization energy of hole (Wh) between these two 2


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molecules leads to reversing the gas-phase and solid-phase IP values of TIPS-pentacene and pentacene.11 The difference of Wh is proved to result from the remarkable variation of intermolecular interactions in these two molecules, which is due to the different molecular packing structures for TIPS-pentacene and pentacene.12 Under the approximation of zero-overlap intermolecular wavefunctions, Wion in condensed phase can be described by the deviation of intermolecular interactions between the ionic and neutral molecule with its surrounding molecules. In chemistry community, Wion is also called as the variation of solvent free-energy change ( ∆∆G ) between the free-energy changes for ionic and neutral molecules transferring from vacuum into solid phase, which is shown in Fig. 1. When the ion is positive or negative charge, Wion is respectively called as the apparent polarization energies of hole (Wh) or electron (We) carriers in solid phase:

Wion = ∆∆Gion = ∆Gion − ∆Gneu

(1)

∆Gion is the free energy change of transferring an ionic molecule from vacuum into crystals. When the entropic contribution to free energy is ignored to avoid the thermally sampling processes in molecular dynamics or Monte-Carlo simulation phase space region, ∆Gion are approximately considered as the intermolecular interactions between ionic molecule and its surrounding molecules. Thus, the ∆Gion es mainly comes from the static electrostatic interaction contribution ( Eion ), electrostatic pol dis polarization contribution ( Eion ) and dispersion contribution ( Eion ): es pol dis ∆Gion = Eion + Eion + Eion

(2)

The ∆Gneu represents the free energy change for transferring a gas-phase neutral molecule into crystal, which can be similarly expressed via substituting “ion” by “neu” in eq.2. As such, the total intermolecular interactions between the center of the neutral dis molecule and its surrounding molecules. Essentially the dispersion interactions Eneu dis

and Eion are from the electrostatic higher-order contributions and are related to the 3



electron correlation, which is challenging to do the accurate evaluation. For most of conjugated molecules, the dispersion interaction is usually weaker than electrostatic dis dis and polarization energy. Thus, assuming the same values of Eneu and Eion , then the

first two terms in eq.2 are the main contributions to Wion. If the geometries of charged and neutral molecules are different, the nuclear and lattice relaxation energy (Enuc) needs to be added into the eq.2. Due to the lack of accurate full-atom model to describe intermolecular interactions for a long time, pol especially many-body electrostatic polarization ( Eion ), the study of Wion was focused

on the molecular crystals, such as oligoacenes. Previously, a whole neutral molecule in lattice was taken as a polarizable point and the charged molecule was taken as a charge point. And ∆Gion is approximately equal to interactions of charge and pol induced-dipoles ( Eion ), which is solved by self-consistent polarization field (SCPF)

procedure.8,

13

At that time, ∆Gneu is totally ignored because of zero interactions

between the neutral points. Under these assumptions, the polarization energies for electron (We) and hole (Wh) carriers in crystals have same values and are pol 8, 14, 15 approximately equal to Eion . This SCPF model capture the main contributions

of Wion from the induced-dipole interaction polarization (Epol), but it cannot explain why Wh and We for oligoacene crystals have the difference of ~ 0.5 eV in experimental data. To solve this problem, Bounds calculated the static electrostatic interactions (Ees) in eq.2 by considering charge-quadrupole interactions, the asymmetric behaviors of Wh and We could be reasonably explained.9 Although these simple models could successfully explain the polarization energy of oligoacenes crystal, these polarizable points for molecules in lattices completely neglect their structure and orientations. It is hard to be used to estimate the packing-dependent inhomogenous Wion values. Thus, submolecule-based model16 and full-atom based methods such as QM/PCM17-20, valence bond (VB)-QM21, fragmental-based QM coupled with PCM22, charge redistribution (CR) model23,

24

and microelectrostatic

(ME) model25, 26 are respectively proposed to investigate ionic polarization energy in 4


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organic crystals or heterojunction interfaces. These methods can fully or partially consider the molecular structures and packing orientations. Yet most of these methods are time-consuming and do not readily deal with the large enough molecular clusters to explicitly extrapolate the electrostatic polarization. To this end, the cheap full-atom polarizable force field (AMOEBA, Atomic Multipole Optimized Energetics for Biomolecular Applications) was firstly introduced to modeling the energetics of charge-carrier in Alq3 amorphous materials.27 Following, Brédas and his coworkers evaluated the Wion of oligoacene crystals and to rationalize the packing effect on Wh of pentacene and TIPS-pentacene crystals based on the parameterization protocol of AMOEBA developed by Ren and coworkers.12, 28, 29 Moreover, AMOEBA also is successfully applied to investigate energetics of charge carriers in donor-acceptor heterojunctions and interface complex systems.30, 31 In these solid-phase polarization studies by AMOEBA force field, state-specific atomic multipoles (AMP) derived from Stone’s GDMA program on the basis of molecular electron density were adopted.28, 32, 33 The state-specific AMPs have been demonstrated that it can well describe intermolecular electrostatic interaction.34-36 But their atomic polarizabilities are from a transferable parameter set i.e. AMOEBA, which are originally obtained by fitting some non-aromatic organic molecular polarizabilities using Thole’s modified dipole interaction model.37,

38

In order to

correctly reproduce the aromatic molecular polarizabilities, Ren and coworkers added four aromatic molecules into the training set to get a “special” atomic polarizabilities for aromatic C (1.750) and aromatic H (0.696) in 4πε0Å3 unit.28 To simplify the indication, these atomic polarizabilities are termed as the default AMOEBA parameters in this study. The default polarizabilities of aromatic C and H greatly improve the molecular polarizability tensor of benzene and polycyclic aromatics. However, the effects of conjugate length and open-shell charged state on the atomic polarizabilities are ignored by using an averaged way to get simple and transferable parameters. Actually our DFT calculations in Tab.2 and Tab.S3 demonstrate that: 1) The default polarizabilities do well produce the DFT’s neutral molecular polarizabilities of naphthalene and anthracene in Ren’s fitting compound set while 5



they notably underestimate the molecular polarizability of pentacene owning a long π-conjugated length; 2) The open-shell molecular polarizabilities of anion and cation have around 10% ~ 30% larger than that of its neutral state. Thus, using the homogenous aromatic polarizability for different conjugated-length molecules and various electron states is very unsuitable for π-conjugated systems. Moreover, the default AMOEBA force field does not provide the polarizabilities of special “aromatic” hybrid atoms such as S, N, F often used in π-conjugated molecules, which limits the application of AMOEBA in the widely π-conjugated molecular family. To address these issues, it is necessary to look for a QM-based method to parameterize static-specific atomic polarizability (SSAP). Besides the assignment atomic polarizability by fitting molecular polarizability with Thole model, many atomic partitioning schemes are proposed to get the contribution of each atom the molecular polarizability.39,

40

Stone and coworkers directly analyzed several partitioning

approaches of molecular polarizability and concluded that space-partitioned atomic polarizability would be the most efficient.41,

42

Bader and coworkers proposed a

hard-space partitioning of molecular polarizability on the basis of the quantum theory of atoms in molecules (QTAIM), which is generalized by Keith.43-45 In this paper, we apply QTAIM approach to parameterize the SSAP of π-conjugated organic small molecule. Based on SSAP parameters, the many-body pol electrostatic polarization ( Eion ) is comparably studied using the explicit extrapolate

method and implicit PCM method, respectively. To demonstrate the more general applications of this QM-based AMOEBA parameterization, we investigate polarization energy of nine organic molecular crystals, namely oligoacene crystals series including naphthalene, anthracene, tetracene and pentacene; TIPS-substituted oligoacenes including TIPS-tetracene and TIPS-pentacene; F-substituted oligoacene crystal series including perfluoronaphthalene (PFN) and perfluoropentacene (PFP); and CN-containing tetracyanoquinodimethane (TCNQ). The molecule structures are sketched in Fig.2.

6


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Fig. 1. Schematic energy diagram transferring molecule from vacuum into solid. E0g and E+g are the energies of the neutral(0) and cationic(+) molecule in gas phase, respectively; E0s and E+s are the energies of the neutral and cationic molecule in solid phase, respectively; ∆G0 and ∆G+ mean the free energy change transfer neutral and cationic molecule from vacuum into solid. IPg and IPs ionization potential in the gas and solid phases, respectively.

Fig.2. Chemical structures of oligoacenes (naphthalene, anthracene, tetracene and pentacene), F-substituted and TIPS-substituted oligoacene (PFN, PFP, TIPS-tetracene and TIPS-pentacene) and TCNQ.

7



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2. Theoretical methodology 2.1 Explicit polarizable force field method 2.1.1 AMOEBA electrostatic and polarization potentials es In AMOEBA polarizable force field, the intermolecular electrostatic interaction ( Eion )

is described by the permanent atomic multipoles (AMPs) interaction:46 es Eion =

∑∑M

T i ij

TMj

i∈ion j∈sur

(3)

where the ith AMP M iT = [qi , µi , x , µi , y , µi , z , Qi , xx , Qi , xy ,L , Qi , zz ] , includes the point charge, atomic dipole and quadrupole moments of the ith atom. Tij is AMPs interaction matrix, which represents the ith and the jth multipoles interactions. The detailed multipole interaction Tij matrix can be found in Stone's textbook.34 Eq.3 requires the

ith atoms and the jth atoms respectively belonging to the ionic molecule and its es surrounding neutral molecules. Thus, Eion describe the total electrostatic interaction

between ionic molecule and its surrounding molecules. Following the procedure proposed by Ren and coworkers, the state-specific AMPs for ionic and neutral state are derived by distributed multipole analysis (DMA)33 on the basis of its respective electron densities derived from QM calculated. pol The electrostatic polarization ( Eion ) between ionic molecule and its surrounding

molecule can be described as induced-dipole interactions: pol Eion =

1 ( µiind ⋅ Fj→i +µ ind ∑ ∑ j ⋅ Fi → j ) 2 i∈ion j∈sur

(4)

here µiind is the ith atomic induced dipole moment, which is created by atomic multipole mutual polarization. Fj→i means the ith feeling electric field produced by the

8


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jth static AMP and its induced dipole. In AMOEBA polarizable force field, the atomic induced dipole is solved by ij ij ' ind µiind ,α = α i Fi ,α = α i (∑Tα M j + ∑Tαβ µ j ', β ) { j}

(5)

{ j '}

where αi is the isotropic atomic polarizability of the ith atom and the subscript α, β means the x, y and z direction in the global cartesian coordinates. Fi ,α is the ith atomic total electric field along α direction. The Fi ,α is the summation of the feeling electric fields generated by all permanent AMPs contributions Tαij M

j

and their

ij ' ind induced-dipole contributions Tαβ µ j ',β . The atomic sets {j} and {j’} separately

controlled “direct” and “mutual” electric field.28,

47

These can give various

specifications according to chemical intuition. For our studied small conjugated molecules, we uniformly defined permanent set {j} stands for all the atoms other than atoms in the molecule containing the ith atom. And the mutual set {j’} stands for all the atoms except for the ith atom. Because of the electric field ( Fi ,α ) including induced-dipole contributions, thus eq.5 needs to be solved by the self-consistent iterative method.48 To avoid “polarization catastrophe”, the smearing function for charges is adopted

ρ=

3a exp ( −au 3 ) 4π

(6)

16 where u = Rij (αiα j ) is a function of the distance of Rij, α i and α j are the

isotropic atomic polarizabilities of the ith and jth atoms, respectively. The dimensionless damping factor (a) is set as 0.39 to controls the width of the smeared charge distribution. And Thole’s modified multipole interactions (Tα and Tαβ) for atomic pairs are applied to calculate induced-dipole moments in eq.5.37, 47

2.1.2 The extrapolation of polarization energy Ignoring thermally entropic contribution to free energy and on the basis of the 9



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expression of electrostatic and polarization energy in eqns.3-4, the free energy change transfer ion from vacuum into a cluster could be solved in eq.2. cluster es pol ∆Gion = Eion + Eion =

∑∑M

T Mj +

T i ij

i∈ion j∈sur

1 ∑ ∑ (µiind ⋅ Fj→i +µ indj ⋅ Fi→ j ) 2 i∈ion j∈sur

(7)

cluster Similarly, ∆Gneu for neutral molecule in cluster is solve by eq.7, when the ionic

AMP are substituted into neutral AMP. Here the surrounding molecules in eq.7 belong cluster to spherical molecular clusters. And the apparent polarization ( Wion ) of ion in cluster cluster can be obtained from eq.1. To explicitly solve the free energy ( ∆Gion ) and

cluster apparent polarization ( Wion ) of in crystals or infinite bulk systems, the

extrapolations are required to be done on the basis of the linear relationship of cluster cluster ∆Gion / Wion and inverse of the cube of the number of molecules in

three-dimensional clusters.24, 29 The molecular packing structures are taken from the crystal structures for naphthalene (NAPHTA06),49 anthracene (ANTCEN09),50 tetracene (TETCEN01),51 pentacene

(PENCEN04),52

TIPS-tetracene

(NIKLUQ),53

TIPS-pentacene

(VOQBIM),54 PFN (OFNAPH01),55 PFP (BEZLUO)56 and TCNQ (TCYQME)57 deposited in the Cambridge Structural Database.58, 59 The CSD identification codes are given within parentheses. Various sizes of spherical cluster are used to extrapolate the apparent polarization energy for bulk systems. The charged molecule was uniformly located at the center of the spherical clusters. All the interactions are calculated by our modified Tinker program.60

2.1.3 The state-specific AMP and polarizability parameterization Atomic multipoles (AMPs) From intermolecular interaction potentials including electrostatic and polarization contributions mentioned above, Wion in solid phase can be explicitly extrapolated. The 10


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atomic multipole (AMP) and atomic polarizability are critical parameters to calculate the intermolecular interactions. Following Ren’s parameterization procedure, the state-specific AMPs are derived by DMA analysis of electron densities of negative, positive and neutral molecule in vacuum QM calculations.28 All the electron densities of monomer in various neutral and charged states are calculated with second-order Møller−Plesset perturbation theory (MP2) coupled with 6-311G(d,p) basis set.61-63 And the monomers structures are directly from molecular crystals mentioned above. State-Specific Atomic polarizabilities (SSAPs) As we mentioned before, due to the lack of electron-state and conjugated-length dependent aromatic atomic polarizability, it is unsuitable to adopt uniformly atomic polarizability for π-conjugated molecules in the default AMOEBA. To this end, the distributed atomic polarizabilities are numerically computed on the basis of the QTAIM theory.39, 64, 65 QTAIM analysis can provide the spatial partitioning distributed atomic dipole moments in zero electric field ( µ A0 ) and electric field ( µ AF ) QM calculations. Then, the atomic polarizability tensors of A atom can be solved by following numerical gradient,

µ A,i − µ A0 ,i Fj

α A,ij = lim

F →0

2Fj

(8)

here the subscript i and j mean the x, y, z directions and F is a small applied electric field, which usually is set as 0.05 a.u. to satisfy the stabilization of numerical solution. The detailed procedure to solve the atomic polarizability can be found in Macchi’s papers.66 The isotropic atomic polarizability is solved by averaged the diagonal tensors (αA,ii)

α Aiso =

1 ∑ α A,ii 3 i

(9)

In AMOEBA force field, the isotropic molecular polarizability is expressed by interactive molecular polarizability, which is solved with Thole’s modified dipole model.37, 38 In order to reproduce QM-based molecular polarizability, we uniformly give these numerical isotropic polarizabilities a 1.4 scale. As these scaled isotropic 11



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polarizabilities are related to the electronic state of QM calculations, we define them as state-specific atomic polarizabilities (SSAPs). The interactive molecular polarizability is calculated with polarize program in Tinker software.67 The anisotropy of molecular polarizability tensor is estimated by ∆α:66

∆α =

3(α112 + α 222 + α 332 ) − (α11 + α 22 + α 33 )2 2

(10)

here α11, α22 and α33 are components of the diagonalized molecular polarizability tensors. The large ∆α relates to anisotropic molecular polarizability. QTAIM analysis is performed with AIMALL program.43 The QM calculations to numerically solve the distributed atomic polarizability are carried out at CAM-B3LYP/6-311+G(d,p) level62, 63, 68

performed on Gaussian 09 package69.

2.2 Implicit model In the implicit model, the surrounding molecular packing is totally ignored and difference of free energy (∆Gion) in eq.2 is described by electrostatic polarization pol ( Eion ) through self-consistent reaction field (SCRF) method. At this circumstance, the es dis electrostatic energy ( Eion ) and dispersion energy ( Eion ) are essentially neglected.

Although there are some state-of-art SCRF approaches including cavitation-dispersion corrections such as Truhlar and coworkers’ SMD solvation model,70 the cavitation-dispersion interaction is independent with the electronic state of the solute. Thus, the cavitation-dispersion corrected energies for ionic and neutral state will cancel each other and dispersion terms do not have an effect on the apparent polarization. In order to consider nearby electrostatic interactions, some surrounding molecules are required to be included in SCRF model, which is termed as the supermolecular solvated model (SSM). Our previous study shows that the apparent polarization energy (Wion) from SSM shows size-dependent feature due to the pol long-range Coulomb interaction.22 Therefore, to directly compare the explicit Eion

and evaluate the parameters of atomic polarizabilities, we use polarizable continuum model (PCM) to evaluate the ion and neutral molecular electrostatic polarization in 12


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solid phase.71 The isotropic dielectric constants (ε) of different organic crystals are calculated based on Clausius-Mossotti equation18, 19

ε − 1 4π ρ N α = ε +2 3 M A

(11)

where ρ, M, NA, and α are the density of material, molecular mass, Avogadro number, and the molecular isotropic polarizability, respectively. ρNA/M is the reciprocal of the molecular volume. Molecular volume in the contour of 0.001 a.u. density was solved by Monte-Carlo calculation on the basis of optimized molecules under CAM-B3LYP/6-311G(d,p) level. The sampling number of points per bohr3 in Monte-Carlo calculation is set 512000 to get accurate volume.72 Because the diffuse basis set 6-311+G(d,p) overestimates the dielectric constants compared with experimental

values

showed

in

Tab.S1,

so

ε

is

calculated

at

the

CAM-B3LYP/6-311G(d,p) level. The detailed values are listed in Tab.S1 in im Supporting Information. Thus, the difference of free-energy change ( ∆Gion ) moving

an ionic molecule from vacuum into solvent with the implicit method is expressed by: im PCM vac ∆Gion = Eion − Eion

(12)

PCM vac Eion and Eion are QM total energies of ionic molecule in PCM solvent reaction

field and in vacuum, respectively. When an isolated ion is in (within) self-consistent im reaction field (SCRF), ∆Gion is actually equivalent to electrostatic polarization pol energy ( Eion ) of the extrapolated bulk systems and can be directly compared with

im pol explicit Eion . Similarly, ∆Gneu can be solved by changing ionic into neutral electron

state for respective QM calculations to compare with explicit result.

3. Results and discussion 3.1 Comparison of electrostatic polarization using explicit and implicit approaches According to the default and state-specific atomic polarizability (SSAP), the pol electrostatic polarization energy ( Eion ) of crystal is extrapolated by the linear

13



pol relationship of electrostatic polarizations ( Eion ,cluster ) of the various clusters and

1/ 3 N , where N is the number of molecules in its clusters. Fig.3 is an example to pol extrapolate Eion of cationic and anionic molecules in naphthalene crystal. The other pol Eion values are shown in Tab.1. It can be found that the electrostatic polarizations for

ionic molecules are far larger than those of neutral ones. Comparing the implicit im pol ∆Gion and explicit Eion , we generally note that most of them are close to each other,

especially for ionic molecules, which means that the isotropic PCM model basically can capture the electrostatic polarization contribution. Through the careful analysis of the electrostatic polarizations, the following features can be attained. For herringbone stacking oligoacene molecules, explicit SSAP approach behaves almost the unchanged electrostatic polarizations (Epol) when increasing of π-conjugated length while our default parameters and Ryno’s previous studies show that Epol decrease with increasing of π-conjugated length.29 As equations 4-5 showed, the electrostatic polarization is directly related with AMP parameters and polarization atomic polarizability. For the same molecular packings, the larger AMPs would produce stronger electric field in eq.5 for its neighbor atoms and the larger atomic polarizability produce larger electrostatic polarization. Increasing conjugation length would lead to the more distributed AMPs for the charged molecules and the smaller electric field for its surrounding atoms. Thus, the more distributed AMPs combining with the same atomic polarizabilities for the default paramenters result in the decreasing Epol tendency with increases of conjugation length. For SSAP parameters, the values of atomic polarizabilities increase with increasing of conjugation length in Tab.S2. The interplay of the increasing atomic polarizability and the more distributed AMP results in nearly unvaried Epol as increasing the conjugation length. In addition, the interactive molecular polarizabilities obtained from Thole’s model are a very simple way to evaluate the atomic polarizability parameters. From Tab.2, we can note the interactive molecular polarizability from SSAP do can well reproduce the CAM-B3LYP molecular polarizability while values from default parameters 14


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underestimate molecular polarizability as conjugated length increasing. Thus, the default parameters behave different tendency of electrostatic polarization with increasing conjugated length while SSAP behaves as unvaried electrostatic polarizations. For self-consistent reaction field (SCRF) model, the electrostatic free energy is determined by electrostatic potentials (ESPs) on the reaction cavity and solvent dielectric constant. Increasing conjugation length leads to smaller ESPs on vdW surface in Tab.3 while both the area of vdW surface and dielectric constant are increased. The interplay of ESPs and dielectric constant results in that the implicit electrostatic free energy ( ∆Gim ) or electrostatic polarization (Epol) also shows nearly unvaried feature with increase of conjugated length. Thus, one may conclude that SSAPs gives better physical pictures due to better describing the molecular polarizabilities. Compared SSAP-based deviation between E+pol and E−pol , implicit PCM slightly underestimate the gap between ∆ G+im and ∆ G −im , which is possibly due to that PCM is on the basis of isotropic continuum dielectric model while real molecular crystal structures behave anisotropic and discontinuum characters. For

unsubstituted

oligoacenes

and

TIPS-oligoacenes,

Epol(SSAP)

of

TIPS-oligoacenes are smaller than those of their respective oligoacenes. These can be rationalized by the following two factors. One is larger intermolecular separations for TIPS-oligoacenes than oligoacenes due to the steric hindrance of TIPS groups. For example, the nearest center-to-center distance in pentacene crystal is 4.69 Å in Fig.S1, which is far smaller the values (7.57 Å) of TIPS-pentacene. This larger intermolecular distance results in the smaller electrostatic interaction and smaller polarization energies. The other reason is different molecular packing (herringbone for pentacene and wall-brick packing for TIPS-pentacene) resulting in different electrostatic polarization. Comparing the positive and negative electrostatic polarizations with implicit and explicit methods, we can note that: ∆G+im ≈ E +pol ( SSAP ) and ∆G−im
E+pol ( SSAP ) and ∆G−im ≈ im E −pol ( SSAP ) for oligoacenes. The different tendency of implicit ∆Gion and explicit

15



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pol Eion for oligoacenes and TIPS-substituted oligoacenes is possibly due to the various

molecular packings for two types of crystals. From crystal structures showed in Fig.S1, TIPS-substituted oligoacenes adopt 2D-like brick-wall packing structure, which makes electrons more readily polarized along the long axis of oligoacenes. Other directions of vertical long axis are hard to polarize electron. But oligoacenes adopts herringbone packing which leads to not very strong anisotropy of dielectric tensor.73 Thus, the brick-wall packing behaviors more anisotropic dielectric tensor than herringbone packing, which results in various characters for explicit Epol with isotropic-solvent based ∆G . im

Additionally, SSAPs have a similar electrostatic polarization with default AMOEA atomic polarizability for F-containing molecules, while for N-containing TCNQ, the calculations using the default parameters notably underestimates about 25% Epol compared with SSAP results. The reason for underestimation can be rationalized by the fact that the interactive molecular polarizability (21.6 in 4πε0Å3 unit) of TCNQ obtained from the default parameters in Tab.2 is around 33% smaller than CAM-B3LYP molecular polarizability (32.2 in 4πε0Å3 unit). The last but interesting feature is that E−pol in oligoacenes and TIPS-oligoacene crystals are smaller than their E+pol values while the opposite situation happens in PFN, PFP and TCNQ molecules containing the electron-withdrawing F or CN groups. Interestingly, the former six compounds, i.e. oligoacenes and TIPS-oligoacenes compounds with E−pol > E+pol , usually behave as p-type transport materials74,

75

while the last three compounds PFN, PFP and TCNQ with E−pol < E+pol behave as n-type transport materials.6, 76 This is because the larger Epol will lead to more notable electron-cloudy deformation in the surrounding molecules, which may produce bigger external reorganization during electron transfer processes. The large external reorganization energy leads to bad charge-carrier transfer property.77, 78 More detailed studies about the relation between electrostatic polarization and external reorganization are underway. 16


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In a short summary, SSAP-based explicit method and isotropic solvent PCM model provides reasonable and unvaried Epol with increasing conjugation length while the method using default parameters slightly underestimate Epol especially for oligoacenes with long chains and TCNQ containing hybrid N atoms.

Tab.1. The explicit electrostatic polarizations (Epol) are respectively extrapolated with the state-specific atomic polarizability (SSAP) and Ren suggested default atomic polarizability. The implicit electrostatic polarization energies ( ∆Gim ) are solved by eq.12. a

Default

SSAP

PCM

E0pol

E+pol

E−pol

E0pol

E+pol

E−pol

∆G0im

∆G+im

∆G−im

Naphthalene

0.04

1.14

1.35

0.05

1.14

1.36

0.06

1.25

1.31

Anthracene

0.06

1.08

1.35

0.06

1.04

1.31

0.08

1.24

1.32

Tetracene

0.09

1.04

1.32

0.09

0.94

1.24

0.11

1.24

1.34

Pentacene

0.12

1.09

1.43

0.12

0.95

1.28

0.15

1.24

1.36

TIPS-tetracene

0.22

0.85

1.20

0.23

0.87

1.23

0.11

0.93

1.08

TIPS-pentacene 0.28

0.90

1.34

0.29

0.91

1.37

0.13

0.93

1.10

PFN

0.04

1.19

0.95

0.05

1.31

1.03

0.08

1.31

0.90

PFP

0.18

1.46

1.04

0.17

1.37

0.95

0.23

1.52

1.04

TCNQ

0.27

1.82

1.07

0.18

1.37

0.87

0.35

1.89

1.27

a

All energies are in eV and scaled by -1. The subscripts 0, + and – represent the neural,

cationic and anionic molecules, respectively.

17



Fig.3. The linear relationship of E+pol and E−pol for naphthalene spherical clusters versus 1/ 3 N , where N is the number of molecules in clusters. The spherical radii of cluster range from 10 to 50 Å. The ionic molecule is positioned in center of cluster and state-specific atomic polarizabilities (SSAPs) are applied to solve the ionic electrostatic polarization of clusters based on eq.4. Black and red points mean electrostatic polarization energy of anionic and cationic molecules, respectively.

3.2 Rationalization why SSAPs well describe ionic electrostatic polarization 3.2.1 Inhomogeneous state-specific atomic polarizability From eqns.4-5 in the theoretical section, the atomic polarizabilities are key parameters to compute the electrostatic polarization. Based on QTAIM space partitioned atomic polarizability, SSAPs show inhomogeneous features for the same elements with different sites or various electron states. As an example, the numerical SSAPs of anthracene are listed in Fig.4. For the neutral C atom in anthracene, the order of atomic polarizabilities on various sites is 1C < 2C < 4C < 3C and all the SSAP for C 18


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atom are larger than default aromatic C polarizability (1.750 4πε0Å3) in AMOEBA. Since the static polarizability represents the ability of π-electron delocalization in π-conjugated molecules, the more delocalization leads to the larger polarizability. The 3C site of anthracene is directly linked with three aromatic C atoms, which means the 3C site has the largest deformation ability. Although the 4C, 2C and 1C atoms are directly connected with two aromatic C atoms, 4C locates in the center of the long molecular-axis, 1C is on the side of the long axis and 2C is in the middle of 1C and 4C. Thus, we can image the center site 4C atom has a stronger ability of delocalization, due to abundant π-electron density. Therefore, SSAPs of C atoms behaves the different values. As for the neutral H atoms, the default aromatic H atomic polarizability (0.696 4πε0Å3) is far larger than SSAP values in 1H, 2H and 3H sites, which means the fitted aromatic H overestimate atomic polarizability. Fig.4 also demonstrates that most of atomic polarizabilities in open-shell ionic state are larger than those of the closed-shell state, which indicates that open-shell electron structure more readily deforms π-electrons. Considering the effect of conjugated length showed in Fig.S2, we can note that increasing the conjugated length from naphthalene to pentacene increases the C atomic polarizabilities about 10% ~ 30%. In short, these inhomogeneous SSAPs really represent π-electron delocalization ability on various sites including the effects of conjugated length and electron state.

19



Fig.4. AMOEBA default and state-specific atomic polarizabilities of anthracene. Left side: Chemical structure of anthracene and the atom indices are labeled on it. Right side: The default polarizabilities (in black points) and inhomogeneous SSAPs of various C and H atoms in anionic (in red points), cationic (in blue points) and neutral (in green points) electron states.

3.2.2 Better description the interactive molecular polarizability with SSAPs Here keeping in mind that SSAPs is isotropic, AMOEBA uses the interactive molecular polarizability with Thole model to describe the anisotropy of molecular polarizability.46 Thus, the interactive molecular polarizability is a critical factor to evaluate electrostatic polarization. Tab.2 lists the interactive molecular polarizabilities of our studied nine molecules in neutral state. Tab.2 evidence that default AMOEBA atomic polarizability can well describe QM-based molecular polarizability of the short chain oligoacenes, such as naphthalene and anthracene, while the method using

default parameters underestimates the molecular polarizability of the longer oligoacenes and hybrid N containing TCNQ. Compared the interactive molecular polarizability obtained from SSAP and default parameters, we can find SSAP would remarkably improve the molecular polarizabilities no matter average polarizability or anisotropy. Similar situations can be found for open-shell ionic molecular polarizabilities listed in Tab.S3 in Supporting Information. Because of the better description of QM-based polarizability with SSAPs, the explicit approach using SSAPs can give better electrostatic polarization energy for these nine molecular 20


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crystals.

Tab.2. The static isotropic molecular polarizability (αiso) and its anisotropy (∆α) of neutral molecules are calculated at CAM-B3LYP/6-311+G(d,p) level. For clear comparison, the deviations of molecular polarizabilities between QM and AMOEBA force field using SSAP and the default AMOEBA parameters are listed.a CAM-B3LYP αiso ∆α 17.0 13.2 naphthalene 25.8 25.7 anthracene 36.1 42.5 tetracene 47.9 63.2 pentacene 89.1 55.1 TIPS-tetracene 101.1 64.9 TIPS-pentacene 17.8 15.3 PFN 50.0 67.1 PFP 32.2 45.0 TCNQ a 3 All polarizabilities are in 4πε0Å .

default αdiff ∆αdiff -0.3 -2.3 -2.4 -8.3 -5.7 -17.6 -10.3 -30.1 -6.0 -22.7 -11.3 -29.9 4.2 -0.9 -2.7 -26.4 -10.6 -30.9

SSAP αdiff ∆αdiff -0.3 -2.3 -0.5 -6.2 -0.9 -11.4 -1.4 -17.7 2.6 -7.9 1.8 -11.7 1.5 -2.6 2.6 -17.6 -1.1 -23.0

3.3 Molecular origin of asymmetry of ∆G+im and ∆G−im for p-type and n-type compounds Assuming the charged molecule as a simple charge-point and putting it from vacuum into the isotropic continuum dielectric, the positive and negative changes would produce the exactly equivalent electrostatic polarization energy.7, 79 However, both implicit PCM and explicit SSAP results in Tab.1 demonstrate that electrostatic polarizations of positive and negative charges are not equivalent especially for TCNQ molecules. Ignoring the molecular packings resulting in anisotropic effects, the electrostatic potential (ESP) on the molecular surface are analyzed to explore the molecular origin of various ∆G+im and ∆G−im . In the implicit continuum model, the electrostatic polarization energy essentially represents the interactions between the molecule and its apparent surface charges (ASCs) on the molecular cavity.80, 81 ASCs are obtained by solving Poisson’s equation, which is related to molecular electrostatic 21



Page 22 of 32

potential (ESP) on molecular vdW surface and solvent polarity. Usually the vdW surface is defined as 0.001 a.u. iso-surface of electron density. Those ESPs on vdW surface are widely used to describe intermolecular interactions such as nucleophilic and electrophilic sites. According to Politzer’s definition of some molecular descriptors based on ESPs,82 we respectively calculate the average of ESP ( V S ) and the standard variation (σ2) of ESP for positive and negative charged molecule on vdW surface. As ESP values of positive and negative charged molecule respectively are +/−

positive/negative number on vdW surface, we used V S

and σ +2/ − to represent the

average and variation of ESP for cation and anion.

1 m ∑ V (ri ) m i =1

(13)

2 1 m V (ri ) − VS+ / −  ∑ m i =1

(14)

+/−

VS =

σ +2/ − =

where V(ri) is ESP value on vdW surface ri site and m means the number of sampling points on surface. The descriptors are calculated by Lu developed Multiwfn 3.4.83 From the definition of Politzer’s molecular ESP descriptors eqns.13-14, the larger absolute of both V S and σ2 results in a stronger intermolecular interaction and a stronger electrostatic polarization. The detailed molecular descriptors are listed in Tab.3. We can clearly find that the former six compounds, i.e. oligoacenes and their +

−

TIPS-substituted oligoacenes, have similar | V S | and | V S | while σ −2 > σ +2 . These indict anions of these six compounds slightly behavior stronger intermolecular interactions than their positive charged molecules, which result in larger ASCs for anion on the molecular cavity than cation. So their ∆G−im are slightly larger than the ∆G+im . For the other three n-type compounds including F-substituted oligoacenes and

TCNQ, both | V S | and σ2 of positive charge are larger than negative ones, which mean that cations have stronger interactions than anions. Thus, the ∆G−im values of n-type compounds are notably smaller than positive charge polarizations ( ∆G+im ). 22


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Tab.3. The molecular ESP descriptors including ionic molecular average of surface ESP ( VS , in kcal·mol-1), their variations (σ2, in (kcal·mol-1)2) and the areas of molecular vdW surfaces (AS, in Å2). As a comparison, the deviations of electrostatic polarization (in eV) using PCM method are also listed. +

−

VS

VS

σ +2

σ −2

AS

∆G+im − ∆G−im

Naphthalene

91.6

-90.1

30.1

264.1

174.6

-0.05

Anthracene

81.4

-80.8

52.7

270.1

222.0

-0.08

Tetracene

73.9

-73.7

75.0

278.2

269.4

-0.09

Pentacene

68.1

-68.1

96.1

286.1

316.8

-0.10

TIPS-tetracene

52.9

-52.2

96.3

344.1

683.9

-0.14

TIPS-pentacene

51.3

-50.8

84.5

317.1

741.2

-0.15

PFN

90.4

-80.8

429.9

22.2

210.4

0.29

PFP

70.2

-59.6

423.5

101.8

369.9

0.43

TCNQ

86.6

-68.0

813.9

175.1

249.9

0.61

3.4 Apparent polarization energy (W) As we mentioned before, apparent polarization energy (W) is an observable value by vacuum and solid-phase photoelectron spectroscopy. According to the definition of apparent polarization energy in eq.1, we can extrapolate the apparent polarization of positive (W+) and negative (W-) charges based on the SSAP and default atomic polarizability in Tab.4. We can note that the default atomic polarizabilities produce very similar W with SSAP in naphthalene and anthracene crystals as for oligoacenes, while underestimate W values in tetracene and pentacene crystals due to the lower inductive molecular polarizabilities for the two acenes. For hybrid atom F and N containing molecules, the default atomic polarizabilities give similar W values with SSAPs in PFN and PEP crystal but notably underestimate cationic W+ values in TCNQ. For the widely studied oligoacene crystals, the deviations between the calculated W+ and W- are very similar to the experimental results, respectively. Due to 23



ignoring the nuclear contribution to apparent polarization energy in our calculations, the theoretical W values of 0.1 ~ 0.4 eV are less than the experimental results. In contrast with Ryno’s AMOEBA results in Tab.S6, the values of W+ and W- obtained the calculations using default AMOEBA parameters are slightly larger than their results. The only difference between our default and Ryno’s calculations is that Ryno adopted non-aromatic C polarizability (1.334 4πε0Å3) instead of Ren’s suggested aromatic C atomic polarizability (1.750 4πε0Å3), which results in the underestimation of the electrostatic polarization.29 Compared other theoretical W values of oligoacenes in Tab.S6, we find that SSAP and microelectrostatic (ME) method coupled with anisotropic atomic polarizability (ME0) and isotropic atomic polarizability (MEa) provide very similar change trends with the increase of conjugated-length, while both ME0 and MEa adopt uniformly neutral atomic polarizability instead of ionic and neutral molecules.73 Furthermore, in view of SSAP correct interactive molecular polarizability for neutral and ionic molecules in Tab.2, we believe that W calculated using SSAPs would provide more reasonable results. As for the n-type TCNQ molecule, SSAP-calculated W+ and W- behave far smaller than experimental values, which possible due to that there exist the factors such as impurity and defects resulting in the different packing morphologies between experimental and our used ideal TCNQ crystal. Considering the relation of W+ and W-, W+ and W- behave the following features besides TIPS-tetracene: W+ > W- for p-type molecules and W+ < Wfor n-type molecules, which is opposite to behaviors of E+pol and E−pol as we discussed before. The reverse character for W and Epol can be explained by the different sign of electrostatic interaction (Ees) between cation/anion with their surrounding molecules. More detailed electrostatic interaction of cation and anion in bulk system can be found in Tab.S5. As explicitly solving the electrostatic polarization for a very large size cluster requires a time-consuming iterative procedure and PCM method already capture main electrostatic polarization contribution, Wion can be approximately estimated by the simple combination of PCM electrostatic free energy (∆Gim) and electrostatic interaction (Ees). Tab.4 shows that PCM + Ees 24


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basically represents the characters of apparent polarization energy. Due to the isotropic continuums solvent model overestimate the electrostatic polarization (Epol), PCM + Ees slightly overestimate the gap of W+ and W-, in particular, for pentacene and TIPS-pentacene with a long π-conjugated length. Moreover, it is still a challenge for continuums dielectric-based PCM to model the electrostatic polarization of ions on the donor-acceptor interfaces or surfaces of organic solids. Thus, explicit polarizable force field still is a good tool for modeling the apparent polarization of charge carrier in complex systems.

Tab.4. Theoretical and experimental apparent polarization energy of positive (W+) and negative (W-) charge carrier in molecular crystals. SSAP means W is extrapolated by intermolecular interactions of various clusters on the basis of SSAPs and AMP. PCM + Ees means that W is estimated in eq.1 via the summation of PCM calculated ∆Gim and electrostatic interaction (Ees).a SSAP

Naphthalene Anthracene Tetracene Pentacene TIPS-tetracene TIPS-pentacene PFN PFP TCNQ a

Default

PCM + Ees

Experiment

W+

W-

W+

W-

W+

W-

W+

W-

1.33 1.29 1.33 1.39 0.76 1.21 0.86 0.84 0.59

1.09 1.04 0.95 0.91 0.86 0.51 1.12 1.27 1.82

1.33 1.24 1.23 1.25 0.77 1.22 0.97 0.76 0.23

1.09 1.00 0.87 0.77 0.88 0.53 1.19 1.19 1.72

1.42 1.42 1.50 1.51 0.95 1.33 0.94 0.86 0.59

1.03 0.99 0.95 0.82 0.86 0.45 1.03 1.22 1.95

1.72b 1.62b 1.63c 1.63b

1.10b 1.09b 0.92d 1.17b

0.70e 1.24f 2.10g 2.90b

All energies are in eV and scaled by -1. bReference 79. cReference 84, 85. dReference

86. eReference 87. fReference 88. gReference 89.

4. Conclusions In this work, we have presented a unified approach to parameterize the state-specific atomic polarizability (SSAP) of small π-conjugated molecules based on QTAIM space partitioned molecular polarizability. Different from AMOEBA using the default 25



parameters, which are obtained from fitting the polarizability of molecules belonging to a certain training set, our SSAPs more rely on the quantum chemistry computations and reduce the artificial training. The isotropic SSAP can be used to well describe the molecular polarizability tensors of studied nine molecules using Thole’s modified induced dipole model no matter in neutral state or ionic state. The SSAPs have been utilized to calculate the electrostatic polarization energy of a changed molecule transferred from vacuum into crystal. Our results show that: 1) The explicit SSAPs more reasonably represent the electrostatic polarization than implicit PCM and AMOEBA default parameters; 2) E+pol are smaller than E−pol for p-type materials while E+pol are larger than E−pol for n-type materials. The molecular origin of various electrostatic polarizations of electron and hole can be explained by ESP analysis on the molecular vdW surface. As for apparent polarization energy of electron and hole carrier, the tendency is opposite to that of electrostatic polarization due to the asymmetric sign of electrostatic interactions. The isotropic-solvent based PCM method can suitably capture the main electrostatic polarization contribution in bulk systems. But it is a challenge and open problem in particular for implicit PCM approaches to deal with electrostatic polarization for the discrete asymmetric surfaces or donor-acceptor heterojunctions. Therefore, it is necessary to develop explicit method to deal with the complex systems. The SSAPs based polarizable force field provides a useful and cheap approach to compute the charge-carries energy landscape and reorganization energy in organic solids and at their surfaces.

Associated Content Supporting information The dielectric constants, isotropic SSAPs and molecular polarizability of the molecules considered in this study, additional information on the electrostatic polarization extrapolations and the analysis of molecular ESP descriptors on vdW surface. These materials are available free of charge via the Internet at http://pubs.acs.org. 26


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Table of Contents

Abstract Graphics: QTAIM is applied to numerically parameterize state-specific atomic polarizability (SSAP) of π-conjuagted organic molecules. The electrostatic polarization energies (Epol) of charge carriers in organic molecular crystals are comparatively studied by the explicit SSAP-based AMOEBA polarizable force field and the implicit polarizable continuum model (PCM) method.

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Electrostatic Polarization Energies of Charge Carriers in Organic

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