Quantitative prediction of optical absorption in molecular solids from

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Quantum Electronic Structure

Quantitative prediction of optical absorption in molecular solids from an optimally-tuned screened range-separated hybrid functional Arun K. Manna, Sivan Refaely-Abramson, Anthony Martin Reilly, Alexandre Tkatchenko, Jeffrey B. Neaton, and Leeor Kronik J. Chem. Theory Comput., Just Accepted Manuscript • DOI: 10.1021/acs.jctc.7b01058 • Publication Date (Web): 04 May 2018 Downloaded from http://pubs.acs.org on May 15, 2018

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Journal of Chemical Theory and Computation

Quantitative predi tion of opti al absorption in mole ular solids from an optimally-tuned s reened range-separated hybrid fun tional Arun K. Manna,† Sivan Refaely-Abramson,† Anthony M. Reilly,¶ Alexandre Tkat henko,§ Jerey B. Neaton,‡ and Leeor Kronik∗ † ,

†Department

of Materials and Interfa es, Weizmann Institute of S ien e, Rehovoth 76100, Israel

‡Department

of Physi s, University of California Berkeley, Berkeley, California 94720, USA

¶S hool

of Chemi al S ien es, Dublin City University, Glasnevin, Dublin 9, Ireland

§Physi s

and Materials S ien e Resear h Unit, University of Luxembourg, L-1511, Luxembourg

kMole ular

Foundry, Lawren e Berkeley National Laboratory, Berkeley, California 94720, USA

E-mail: leeor.kronikweizmann.a .il

Abstra t We show that fundamental gaps and opti al spe tra of mole ular solids an be predi ted quantitatively and non-empiri ally within the framework of time-dependent density fun tional theory (TDDFT), using the re ently-developed optimally-tuned s reened range-separated hybrid (OT-SRSH) approa h. In this s heme, the ele troni stru ture of the gas-phase mole ule is determined by optimal tuning of the range-separation pa1

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Journal of Chemical Theory and Computation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

rameter in a range-separated hybrid fun tional. S reening and polarization in the solidstate are taken into a

ount by adding long-range diele tri s reening to the fun tional form, with the modied fun tional used to perform self- onsistent periodi -boundary

al ulations for the rystalline solid. We provide a omprehensive ben hmark for the a

ura y of our approa h by onsidering the X23 set of mole ular solids and omparing results obtained from TDDFT with those obtained from many-body perturbation theory in the GW-BSE approximation. We additionally ompare results obtained from diele tri s reening omputed within the random-phase approximation to those obtained from the omputationally more e ient many-body dispersion approa h and nd that this inuen es the fundamental gap but has little ee t on the opti al spe tra. Our approa h is therefore robust and an be used for studies of mole ular solids that are typi ally beyond the rea h of omputationally more intensive methods.

Introdu tion The ele troni and opti al properties of mole ular solids have re ently attra ted signi ant attention, primarily in the ontext of optoele troni devi es based on small mole ules (see, e.g., Refs. 1-4). In parti ular, there is on-going interest in identifying small-gap organi mole ules for high-performan e, low- ost, or enhan ed-stability optoele troni devi es based on solids opmrised of these mole ules (see, e.g., Refs. 5-9 for some re ent overviews). Theory

an and should play an important role in su h investigations, as it an larify the properties of existing mole ular solids and point out promising new ones.

1013

Ele troni properties, su h as the band stru ture and in parti ular the transport gap, and opti al properties, su h as opti al absorption in general and the opti al gap in parti ular, are ex ited-state properties. For inorgani solids, these properties have long been al ulated using many-body perturbation theory (MBPT). solved using Hedin's GW approximation,

W

17

1416

where

G

is the dynami ally s reened Coulomb intera tion.

2

In MBPT, Dyson's equation is often is the one-parti le Green fun tion and

14,18

The Bethe-Salpeter equation (BSE)


Page 2 of 31

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for the two-parti le Green fun tion is then solved approximately to predi t opti al properties.

14,19,20

In re ent years, the GW-BSE approa h has been in reasingly applied to mole ular

solids, yielding many important insights (see Ref. 13 for a re ent overview). Unfortunately, su h GW-BSE al ulations an be quite ompli ated and omputationally intensive, limiting our ability to use them routinely, espe ially in the ontext of high-throughput al ulations for new materials. Density fun tional theory (DFT), in both its time-independent

21,22

and time-dependent

2326

forms, suitable for ground and ex ited state properties, respe tively, is mu h more omputationally e ient. However, ommon approximations to time-dependent DFT (TDDFT) are known to fail in the solid-state limit.

14,27

For mole ular solids in parti ular, key quantities,

su h as the transport gap, the opti al gap, and the ex iton binding energy (i.e., the dieren e between the two gaps), are often in qualitative or gross quantitative error.

13

Re ently, Refaely-Abramson et al. have suggested the optimally-tuned s reened rangeseparated hybrid (OT-SRSH) fun tional as a means for quantitative DFT-based predi tion of ex ited-state properties in mole ular solids.

28,29

In this approa h, one rst om-

putes the underlying gas-phase mole ule using an asymptoti ally orre t range-separated hybrid (RSH) fun tional, in whi h an optimal range-separation parameter is determined nonempiri ally,

3033

based on satisfa tion of the ionization potential theorem. One then uses the

same range-separation parameter in the solid-state environment, while a

ounting expli itly for solid-state polarization by s reening the asymptoti potential with a non-empiri al diele tri onstant.

28,29

While preliminary results obtained with the OT-SRSH method have shown ex ellent agreement with GW-BSE data, two important questions remain. First, results have been reported to-date only for a few mole ular solids - penta ene, quina ridone,

34

28,29

benzene, C60 ,,

28

and

(an air-stable penta ene derivative), with the opti al absorption spe trum

omputed only for penta ene. The validity of the OT-SRSH approa h a ross a wider range of mole ular rystals, espe ially as far as opti al properties are on erned, is therefore in

3



need of demonstration.

Page 4 of 31

Se ond, previous OT-SRSH al ulations have used the diele tri

onstant obtained within the random-phase approximation (RPA) for fa ilitating omparison to MBPT data. However, this step an itself be expensive and it remains to be seen whether su iently a

urate results an be obtained from simpler methods for determining the diele tri onstant. In this arti le, we address both questions by assessing the a

ura y of the OT-SRSH approa h for transport gaps and opti al absorption spe tra a ross the X23 set of mole ular solids.

35,36

This set omprises rystals based on small- to medium-sized organi mole ules,

possessing a variety of weak inter-mole ular intera tions and dierent degrees of solid-state polarization.

It therefore provides a stri t ben hmark for the OT-SRSH approa h.

We

further ompare results obtained using an RPA-based diele tri onstant with those obtained using the many-body dispersion (MBD) method.

37

Within the RPA, we nd our OT-SRSH

results to be in very good agreement with those obtained from GW for quasi-parti le gaps and from GW-BSE for the opti al spe trum. We further nd that using MBD-based diele tri s reening results in larger deviations for quasi-parti le gaps but has an essentially negligible ee t on the opti al absorption, allowing for an inexpensive yet non-empiri al predi tion of opti al properties.

Theoreti al and Computational Approa h Optimally-tuned range-separated hybrid fun tionals In the range-separated hybrid (RSH) method, the Coulomb intera tion is range-split. Here, we use the range-separation s heme suggested by Yanai et al.,

38

whi h is based on the identity

1 α + βerf(γr) 1 − [α + βerf(γr)] = + , r r r

4


(1)

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where

r

is the inter-ele tron oordinate and

α, β , γ

are parameters. The full

1/r

repulsion

is used for the Hartree and orrelation terms, but the two terms on the right-hand side of Eq. (1) are treated dierently in the omputation of the ex hange term. The rst term is treated using exa t (i.e., Fo k) ex hange, whereas the se ond term is treated using lo al or semi-lo al ex hange. This leads to the following expression for the ex hange- orrelation energy,

Exc : RSH SR SR LR Exc = αExx + (1 − α)EDFAx + (α + β)Exx LR + (1 − α − β)EDFAx + EDFAc ,

(2)

where the super-s ripts `SR' and `LR' denote short-range and long-range ontributions, respe tively, and the subs ripts `xx', `DFAx' and `DFA ' denote Fo k-like exa t ex hange, approximate (semi-)lo al ex hange, and approximate (semi-)lo al orrelation, respe tively. Equation (2) reveals that the parameter short range (r

→ 0)

γ

di tates the amount of Fo k-like ex hange in the

and the parameter sum

hange in the long range (r fun tion, with

α

→ ∞).

39

α+β

determines the amount of Fo k-like ex-

The two limits are smoothly interpolated using the error

being the range-separation parameter, i.e.,

1/γ

orresponds to a typi al

length denoting the transition from SR to LR. In order to turn Eq. (2) into a pra ti al fun tional, one needs to hoose the approximate (semi-)lo al ex hange- orrelation fun tional and set the parameters

α, β , and γ .

To pro eed

without introdu ing empiri ism, typi al hoi es for the (semi)-lo al fun tional would be the lo al density approximation, LDA,

40

or the Perdew-Burke-Ernzerhof (PBE)

41

form of the

generalized-gradient approximation (GGA). In some ases, the fra tion of SR Fo k ex hange,

α,

an be determined from rst-prin iples based on satisfa tion of pie ewise linearity in

fra tional DFT,

39,42

but this is not always possible.

43

For a wide variety of organi mole ules

a universal value of 0.2 has been found to be useful (see, e.g., Refs. 28,39,4244). This value is used in this work throughout. For any hoi e of

5

α,

the ondition


α + β = 1 guarantess

100%


Page 6 of 31

of LR Fo k ex hange and therefore the orre t asymptoti potential in the gas phase. In many popular RSH fun tionals, the range-separation parameter, value based on tting against an appropriate data set. s heme,

γ

31,38,4446

30,39,42

γ , is given a universal

In the optimal tuning (OT)

is system-dependent, but still hosen non-empiri ally. For gas-phase systems, it is

obtained by satisfying the ionization potential (IP) theorem,

4750

whi h states that for the

exa t fun tional the energy of the highest o

upied mole ular orbital (HOMO) is equal and opposite to the ionization potential, i.e.,

IP = −ǫHOM O .

Often, this ondition is demanded

simultaneously for the system in both its neutral and anioni state (where the ionization potential orresponds to the ele tron anity of the neutral).

30,31,51

For any hoi e of

α,

the

optimal tuning s heme then involves the minimization of a target fun tion, J(γ; α), dened by

X

J 2 (γ; α) =

2 [IP γ;α (i) + εγ;α H (i)] .

(3)

i=N,N +1 Minimizing this target fun tion has been shown to be equivalent to enfor ing pie ewise linearity,

33,34,42,5254

resulting in an a

urate predi tion of the ionization potential and the

ele tron anity dire tly from the energy levels of the highest o

upied and lowest uno

upied orbitals, respe tively.

31,33,55

Note that use of Eq. (3) does not require any external referen e

value for the IP, e.g. from experiment or from wave-fun tion based al ulation. Instead, Eq. (3) in an internal self- onsisten y ondition between the IP and the HOMO energy. Refaely-Abramson et al.

have suggested that the OT-RSH s heme an be extended

screened

to mole ular solids by using a approa h one rst sele ts

α

and

γ

range-separated hybrid (SRSH).

where

ǫ

tensor. Therefore instead of

Briey, in this

for the gas-phase mole ule as des ribed above. One then

notes that in the gas-phase the asymptoti potential is

−1/(ǫr),

28

−1/r

but in the solid state it is

is the s alar diele tri onstant, i.e., the averaged tra e of the diele tri

β

is re-adjusted to ree t this s reening, by demanding that

α+β = 1

as in the gas-phase.

applied to the mole ular solid.

13,28,29,34

α + β = 1/ǫ

The resulting s reened RSH fun tional is then

Note that this pro edure assumes that the ee t of

anisotropy in the diele tri tensor is negligible and that intra-mole ular diele tri s reening

6


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an be negle ted during the gas-phase tuning.

56

The X23 set of mole ular solids For evaluating the a

ura y of the above approa h in a systemati manner, we onsider quasi-parti le (QP) gaps and opti al absorption spe tra for the set of 23 non- ovalently bound mole ular solids, known as the X23 set.

35

A s hemati diagram displaying all mole -

ular entities onsidered in this set is given in Figure 1. The mole ules used are small- to medium-sized organi mole ules that an be grouped into four subsets based on their hemi al identity: Open- y li aliphati mole ules ( arbon dioxide, ammonia, a eti a id, su

ini a id, yanamide, ethyl arbamate, oxali a id (in both

α

and

β

polymorphs), urea, and for-

mamide), y li aliphati mole ules (adamantane, hexamine, trioxane, and 1,4- y lohexanedi-one), y li aromati mole ules (benzene, naphthalene, and anthra ene), and hetero- y li aromati mole ules ( ytosine, ura il, triazine, imidazole, pyrazine, and pyrazole). In rystalline solid form, these mole ules are weakly bound, typi ally through H-bonding,

π−π

sta king, van der Waals intera tions, et .

Figure 1: (Color online) Chemi al stru tures of all organi mole ules present in the X23 mole ular rystal set.

7



Page 8 of 31

Computational Details All gas-phase OT-RSH al ulations presented in this work were based on the LRC-ω PBE0 RSH fun tional,

44

whi h is based on Eq. 2 with

α=0.2

and PBE orrelation and short-

range ex hange omponents, as implemented in the Q-CHEM ode (version 4.3), the range-separated parameter

γ

57

but with

optimally-tuned per system, rather than xed to its default

value. Optimization pro eeded via minimization of the target fun tion

J

given in Eq. (3), i.e.,

both neutral and anion forms were onsidered, ex ept for mole ules exhibiting an unbound LUMO, where only the neutral form was onsidered. Optimal values of the range-separated parameter, for all gas-phase mole ules used in the X23 set, are given in Table S1 of the Supplementary Material. The all-ele tron

-pVTZ basis set

58

was used throughout for all

atoms. All solid-state OT-SRSH al ulations were arried out using a modied version of PARATEC (revision 499),

59

a pseudopotential-planewave ode. Here, short-range LDA ex hange was

used, together with LDA orrelation, again with

α=0.2

throughout. Dieren es in tuning

based on LDA or PBE were found to be insigni ant. See Refs. 28,29 for more implementation details. LDA-based Troullier-Martins from the ABINIT website,

61

60

norm- onserving pseudopotentials, adapted

were used for all atoms. An energy uto of 816 eV was used

throughout. The number of bands used, as well as details of oarse and ne k-grid meshes used to onstru t the OT-SRSH wavefun tions, are given in Table S2 of the Supplementary Material. Two dierent methods were used to evaluate the s alar diele tri onstant needed for the determination of

β

in the solid-state al ulations. In one, we used the ran-

dom phase approximation (RPA), as used in the G0 W0 al ulations elaborated below. In the other, we used the framework of the many-body dispersion method, as follows. The diele tri onstant has been al ulated using the Clausius-Mossotti (CM) equation. The required polarizabilities used for evaluating the CM equation were obtained as a unit- ell sum of atomi polarizabilities that in orporate lo al hybridization ee ts, as well as ele trodynami s reening, as des ribed in Ref.

62

All al ulations of

8

ǫMBD

employed the same omputational


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proto ol and optimized geometries as in the original X23 ben hmark.

35

For omparison purposes, all mole ular solids were also omputed using a standard oneshot perturbative

G0 W0

al ulation,

13,18

based on the DFT eigenvalues and eigenve tors

obtained from an LDA al ulation within PARATEC. We used a generalized plasmon pole model,

18

implemented within the BerkeleyGW pa kage (trunk version, revision 6539).

63

This approa h has previously been established as a quantitatively useful tool for the study of mole ular solids (see Ref. 13, and referen es therein).

The diele tri fun tion and the

self-energy were omputed using a large number of uno

upied states, as listed in Table S3 of the Supplementary Material. Opti al spe tra in the solid state were omputed using TDDFT with the LDA and the OT-SRSH fun tional, as well as with the Bethe-Salpeter equation (BSE) based on the

G0 W0

output. Both TDDFT and BSE al ulations were performed using the BerkeleyGW pa kage,

63

modied to in lude TDDFT, with in ident light polarization averaged over the main

unit- ell axes. The kernel was al ulated on a oarse wavefun tion grid, then interpolated to a ne grid using the interpolation s heme suggested by Rohlng and Louie,

64

ex ept

for a few rystals with a very large unit ells, for whi h only a oarse grid was used. We used a slightly shifted grid to generate the transition matrix elements in the diele tri fun tion, using a velo ity operator to approximate an in ident light along a spe i dire tion. Grid shift dire tions were set along the

a, b,

and

c

64

unit- ell axes. The number of o

upied

and uno

upied states used to onstru t the kernel matrix is provided in Table S4 of the Supplementary Material.

Results and Dis ussion Fundamental gap We begin our ben hmark evaluation by omparing the fundamental gaps omputed using OT-SRSH and

G0 W0

methods for all solids in the X23 set. For onsisten y, all OT-SRSH

9



Page 10 of 31

results presented are based on the diele tri onstant obtained within RPA, ex ept in the last part of this se tion, where omparison with MBD-based results is expli itly made. With both OT-SRSH and GW, we omputed the fundamental gap as the energy dieren e between the highest o

upied and lowest uno

upied state at the

Γ point of the Brillouin zone,

as the

inter-mole ular orbital hybridization and therefore band dispersion are very small. OT-SRSH and GW omputed fundamental gaps are given in Table 1, where they are additionally ompared to LDA- omputed gaps and to gas-phase OT-RSH gaps. As expe ted, the gas-phase gaps are substantially larger than the solid-state ones (by as little as 0.8 eV and as mu h as 4.8 eV). This ree ts the well-known phenomena of polarization-indu ed gaprenormalization in mole ular solids,

65

whi h is learly aptured in the OT-SRSH s heme

but is known to be absent in standard fun tionals.

13,66

28

It is readily observed that the OT-

SRSH gaps agree very well indeed with the GW ones.

The deviation between the gaps

omputed with GW and OT-SRSH is summarized graphi ally in Fig. 2.

The dieren es

are usually 0.2 eV at most, with a mean absolute deviation of only 0.15 eV. Only two solids (hexamine and imidazole) exhibited a somewhat larger deviation of 0.3 eV and only one solid (pyrazole) exhibits a larger deviation of 0.5 eV. Not surprisingly, these gaps are substantially larger than those obtained with LDA, whi h is well-known to underestimate fundamental gaps in general.

14,18

The above results establish diele tri s reening as key to a

urate treatment of mole ular solids. This observation suggests several additional omments. First, there is growing re ent interest in solid-state s reening as an ingredient in the onstru tion of density fun tionals in general (see, e.g., 29,6771). Spe i ally for mole ular solids, there is growing interest in embedding a range-separated hybrid mole ular al ulation within a polarizable ontinuum model (PCM) to mimi solid-state ee ts (see, e.g., 56,7274 and referen es therein). We note that the optimal tuning of

γ

an be performed within the PCM also in the absen e

of a s reening term in the fun tional, i.e., with the full asymptoti potential.

While this

too leads to improved predi tions for fundamental gaps, it usually omes at the ost of

10


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a greatly redu ed range-separation parameter, whi h may ae t other system properties, and does not ontain a full physi al des ription of diele tri s reening. dis ussion, see Refs. 56,74,75.

For a omplete

Additionally, it is important to note that in lusion of a

diele tri onstant renormalizes the fundamental gap even in a gas-phase al ulation.

56,74

This is still a manifestation of bulk polarization, as the value of the diele tri onstant must be supplied by a bulk al ulation or a PCM-based al ulation.

Beyond apturing

polarization, our approa h, whi h is based on a full solid-state al ulation, also aptures inter-mole ular dispersion ee ts.

28

Table 1: Fundamental gaps of the X23 set of mole ular solids (in eV), al ulated using LDA, OT-SRSH and G0 W0 , additionally ompared to gas-phase fundamental gaps al ulated using OT-RSH. Mole ular Solid

Fundamental Gap (Eg )

LDA

OT-RSH

OT-SRSH

G0 W 0

Carbon dioxide Ammonia Cyanamide Formamide Urea Ethyl arbamate A eti a id Oxali a id (α) Oxali a id (β ) Su

ini a id

6.4 4.3 4.6 4.9 4.8 5.6 5.1 3.2 3.5 5.2

13.9 10.7 10.8 10.7 10.0 10.3 10.8 11.5 11.5 10.7

11.2 7.9 8.0 8.8 8.0 9.0 9.1 6.7 7.3 9.1

11.2 7.7 8.0 8.8 7.9 8.8 9.3 6.9 7.5 9.1

Adamantane Hexamine Trioxane 1,4-Cy lohexane-di-one

4.8 5.0 5.9 3.5

9.7 8.3 10.7 10.0

7.5 7.5 9.7 7.0

7.6 7.8 9.5 7.0

Benzene Naphthalene Anthra ene

4.3 3.2 2.1

9.3 8.1 6.8

6.8 5.2 3.9

6.9 5.4 4.1

Pyrazine Triazine Pyrazole Imidazole Ura il Cytosine

2.8 3.0 4.8 4.8 3.4 3.4

10.0 10.6 9.4 8.9 9.5 8.8

6.2 6.2 7.6 7.6 6.4 6.1

6.0 6.3 8.1 7.9 6.4 6.1

Opti al Absorption Spe tra We next ompare the opti al absorption spe tra al ulated using TD-OT-SRSH with those obtained from the G0 W0 -BSE methods. As we are interested in allowed opti al transitions, 11



Page 12 of 31

Figure 2: (Color online) Absolute deviations and mean absolute deviation (MAD) in the

al ulated quasi-parti le gap between the OT-SRSH and G0 W0 for the X23 set of mole ular solids. For OT-SRSH al ulations, the G0 W0 omputed RPA ma ros opi diele tri onstant RPA (ǫ ) is used.

we only onsider singlet ex itations. The omplete set of opti al absorption spe tra is shown in Figs. 3 and 4, for solids based on aliphati and aromati mole ules, respe tively. In addition, the energy of the lowest singlet ex itation and the position of lowest-lying main opti al absorption peak are provided in Table 2, along with experimental values for omparison, where available. Additional omparison between lowest solid-state OT-SRSH and gas-phase OT-RSH energies an be found in Table S5 of the SI. Generally, the opti al gap is not as strongly ae ted by the transition from the gas-phase to the solid-state as the fundamental gap, whi h is expe ted given that the opti al gap orresponds to a neutral ex itation that is less sensitive to diele tri s reening (an issue we return to below). However, the omplete opti al spe trum may be very dierent, as demonstrated for ammonia in Fig. S1 of the SI. Therefore the following dis ussion fo uses on the solid-state results. It is readily observed from Figs. 3 and 4 that absorption spe tra omputed with the TD-OT-SRSH approa h do indeed agree well with those omputed using G0 W0 -BSE, a ross the board, over a range of several eV. Spe i ally, the position of intense peaks, found by the two approa hes, typi ally agrees within of

∼0.1

∼0.2-0.3

eV, whi h is ex ellent given an a

ura y

eV at best for either approa h separately. This observation is quantied in Fig. 5,

whi h shows deviations between peak positions using TD-OT-SRSH and GW-BSE, for the lowest-energy peak (top) and for all peaks shown in Figs. 3 and 4 (bottom).

The mean

absolute deviation is only 0.2 eV for either the lowest-energy peak or all shown peaks, with

12


Page 13 of 31 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure 3: (Color online) The imaginary part of the diele tri fun tion of aliphati -mole ule13 based solids in the X23 data set,ACS

al ulated using G0 W0 /BSE (bla k, solid lines) and TDParagon Plus Environment OT-SRSH (red, dashed lines).


Figure 4: (Color online) The imaginary part of the diele tri fun tion of aromati -mole ulebased solids mole ules in the X23 data set, al ulated using G0 W0 /BSE (bla k, solid lines) and TD-OT-SRSH (red, dashed lines).

14


Page 14 of 31

Page 15 of 31 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Table 2: Energy of the lowest singlet ex itation and position of lowest-lying main opti al absorption peak, al ulated using TD-OT-SRSH and the G0 W0 -BSE for the X23 set of mole ular solids, and additionally ompared to experimental values (where available). All energies are in eV. Mole ular Solid

First Ex ited State (S1 ) Energy TD-LDA TD-SRSH G0 W0 -BSE

Opti al Peak Position

Expt.

TD-LDA

TD-SRSH

G0 W0 -BSE

Carbon dioxide Ammonia Cyanamide Formamide Urea Ethyl arbamate A eti a id Oxali a id (α) Oxali a id (β ) Su

ini a id

6.4 4.3 4.6 4.5 4.8 5.6 5.1 3.2 3.4 5.0

8.9 6.7 6.0 5.9 7.1 7.3 5.8 4.7 4.8 6.2

8.3 6.6 5.4 5.4 6.5 6.5 5.6 4.4 4.5 5.7

8.9 76 6.6 77 6.2 78 -

7.2 4.6 5.6 5.3 6.4 7.2 5.5 3.7 3.7 5.4

10.7 7.1 7.3 7.5 7.4 7.7 8.2 6.0 6.1 8.7

10.8 7.1 7.1 7.5 7.1 7.5 8.3 6.3 6.3 8.6

Adamantane Hexamine Trioxane 1,4-Cy lohexane-di-one

4.8 4.7 5.8 3.2

6.8 6.0 8.2 4.5

6.8 6.2 7.9 3.9

6.5 79 -

6.8 5.4 6.3 3.9

8.2 6.6 8.3 6.2

8.5 6.9 8.1 6.3

Benzene Naphthalene Anthra ene

4.3 3.1 2.0

5.4 4.2 2.9

4.9 4.1 3.2

4.7 80 3.9 81 3.1 82

4.9 3.3 2.1

6.3 4.7 3.0

6.5 4.4 3.2

Pyrazine Triazine Pyrazole Imidazole Ura il Cytosine

2.7 2.8 4.8 4.5 3.4 3.4

4.0 4.4 6.7 6.3 4.6 4.7

3.5 3.9 6.5 6.3 4.9 4.9

3.8 83 3.7 84 4.5 85 4.4 86

3.0 3.4 5.5 5.0 3.8 3.8

4.0 4.6 6.8 6.4 5.1 5.0

3.5 4.2 6.5 6.3 4.9 4.9

deviations rarely ex eeding 0.3 eV. As expe ted based on known short omings of TDLDA,

13,14,27

the TD-OT-SRSH data

oer an improvement over TDLDA data that is not only quantitative but also qualitative. This is demonstrated in Fig. 6 for two representative ases - the ammonia and ura il solids. For ammonia, TDLDA produ es an extended spurious absorption tail, starting at

∼4.6 eV,

whereas both TD-OT-SRSH and GW-BSE predi t a sharp onset of absorption, at

∼7.1

The same phenomenon has been previously observed for a non-mole ular solid - LiF. ura il, TDLDA (as well as TDPBE

87

) produ es a spurious peak at

29

eV. For

∼3.8 eV. Further analysis

(not shown for brevity) reveals that this peak results from LDA mis-ordering of the HOMO and HOMO-1 orbitals, whi h is remedied by the OT-SRSH al ulation and removes the false peak.

15



Figure 5: (Color online) Absolute deviations between the TD-OT-SRSH and G0 W0 -BSE

omputed peak positions in the opti al spe tra of the X23 mole ular solid set. Top: lowest peak position. Bottom: Mean absolute deviations of all peaks shown in Fig. 3 or 4.

Figure 6: (Color online) The imaginary part of the diele tri fun tion of the ammonia and ura il mole ular solids, al ulated using G0 W0 /BSE (bla k, solid lines), TD-OT-SRSH (red, dashed lines), and TDLDA (dotted, blue lines).

16


Page 16 of 31

Page 17 of 31 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


In many of the mole ular solids, the lowest singlet ex itation possesses a small matrix element and ontributes little to the opti al spe trum. This is ree ted in Table 2, where the energy of the rst-ex ited state is often predi ted to be quite dierent from the energy of the lowest-lying absorption peak, using either TD-OT-SRSH or GW-BSE. We found that larger dieren es between the two methods often, but not always, arise for these low-absorption ex itations, despite the ex ellent agreement in predi tions of the fundamental gap and the high-absorption ex itations. As an example, for ammonia the lowest-energy singlet ex itation is predi ted to be 6.7 or 6.6 eV using TD-OT-SRSH or GW-BSE, respe tively, with both values in ex ellent agreement with the experimental values of 6.6 eV.

77

A similar pi ture

emerges for adamantane. But for urea or 1,4- y lohexane-di-one, the dieren e between the two predi tions is a mu h larger and entirely non-negligible 0.6 eV. We note that these lowestlying transitions often involve transitions between highly lo alized orbitals, whi h an exhibit large self-intera tion errors. Therefore the dis repan y may be due to remaining issues in the TDDFT al ulation, but may well be also due to LDA being an insu ient starting point for the GW-BSE al ulation.

8890

We note in passing that starting point issues and

TD-OT-SRSH ina

ura ies an be observed also in the ontext of delo alized orbitals, e.g., for benzene and oligoa ene mole ules and solids,

29,43,9092

but this is an issue separate from

the opti al absorption dieren es seen here. Finally, we note that Table 2 and Table S5 additionally show that in some ase absorption peak positions revealed in the solid-state an dier meaningfully from those obtained in the gas-phase with TD-OTRSH. This ree ts the fa t that owing to orbital lo alization or delo alization, band dispersion, and possible harge transfer, the nature of the solid-state ex iton an be very dierent from that of the gas-phase one.

13,93,94

Ee t of the diele tri onstant To fa ilitate omparison to GW-BSE, whi h relies on evaluation of the diele tri fun tion using the random-phase approximation (RPA), all OT-SRSH and TD-OT-SRSH results re-

17



ported above were obtained using

ǫRPA .

Page 18 of 31

Here, we explore the ee t of basing the al ula-

tion on an evaluation of the diele tri onstant using the inexpensive many-body dispersion (MBD) method.

37

A omparison of diele tri onstants and quasi-parti le gaps obtained

from using (the averaged tra e of ) material. We nd that

ǫRPA

ǫMBD ≥ ǫRPA

and

ǫMBD

is given in Table S6 of the supplementary

throughout the X23 set, possibly be ause it is omputed

based on PBE, whi h tends to overestimate polarizabilities. Therefore, the MBD- omputed fundamental gaps are generally smaller than RPA- omputed ones. While the dieren e is often small, it an be substantial - as mu h as 0.8 eV for su

ini a id and 0.7 eV for ura il. It would be interesting for future work to examine whether a self- onsistent

ǫ value, obtained

from MBD al ulations based on the OT-SRSH results, would result in improved agreement. While the dieren es in the diele tri onstant do ae t the fundamental gap, their ee t on the opti al spe tra is mu h smaller. This is reasonable, as the fundamental gap ree ts

harged ex itations, whereas opti al ex itations are neutral.

The ee t of the diele tri

onstant on the TD-OT-SRSH absorption spe tra is demonstrated in Figure 7, for the ase of the a ene-based mole ular solids. Clearly, the ee t of

∼0.1

ǫ

is marginal (e.g., dieren es of

eV at most in the absorption peak position for the benzene solid). The ee t on the

lowest singlet-ex itation energy is equally small. Therefore, using MBD diele tri onstants leads to an inexpensive and predi tive al ulation of opti al spe tra in mole ular solids.

Figure 7: (Color online) The imaginary part of the diele tri fun tion for the aromati a ene mole ular solids (benzene, naphthalene and anthra ene), al ulated using G0 W0 /BSE (bla k, RPA solid lines), TD-OT-SRSH based on ǫ (red, dashed lines)℄, and TD-OT-SRSH based on MBD ǫ (blue, dash-dotted lines). The dierent ǫ values are denoted in the gure.

18


Page 19 of 31 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Con lusions In on lusion, we have omputed fundamental gaps and opti al spe tra for the entire X23 ben hmark set of mole ular solids using the re ently developed optimally-tuned s reened range-separated hybrid fun tional approa h. In this two-stage approa h, optimal tuning of a range-separated hybrid fun tional is rst used for an a

urate and predi tive al ulation of the gas-phase ele troni stru ture. Diele tri s reening is then built into the fun tional to obtain a self- onsistent predi tion for the solid-state ele troni stru ture and opti al properties. The obtained results have been ompared to many-body perturbation theory al ulations within the GW-BSE approa h.

Agreement has been found to be very good to ex ellent

throughout, with somewhat larger dieren es possible for opti ally dark singlet ex itations that do not ae t the opti al spe trum.

Furthermore, we have shown that inexpensive

evaluation of the diele tri onstant using many-body dispersion is su ient for obtaining a

urate opti al spe tra, opening the door to low- ost, fully predi tive al ulation of opti al spe tra in mole ular solids.

Supporting Information The following Supporting Information is available free of harge on the ACS Publi ation website: Gas-phase optimal range-separation parameters (γ ); oarse and ne k-grid meshes used for the LDA, OT-SRSH and G0 W0 al ulations; Number of uno

upied ele troni bands used to onstru t the diele tri fun tion and self-energy in the G0 W0 al ulations; Number of o

upied and uno

upied states used to onstru t the BSE and TDDFT kernel matrix; RPA and MBD diele tri onstants and their orresponding OT-SRSH gaps.

A knowledgments AKM gratefully a knowledges the VATAT postdo toral fellowship awarded by Government of Israel and the Weizmann Institute of S ien e for resear h support. LK a knowledges support of the US Air For e and the NSF-BSF program. Portions of this work were supported by

19



Page 20 of 31

the National S ien e Foundation under Grant No. DMR-1708892. Work at the Mole ular Foundry was supported by the O e of S ien e, O e of Basi Energy S ien es, of the U.S. Department of Energy under Contra t No. DE-AC02-05CH11231. The authors thank NERSC for providing omputational fa ilities.

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