The Bethe–Salpeter equation (BSE) formalism is a computationally affordable method for the calculation of accurate optical excitation energies in molecular systems. Similar to the ubiquitous adiabatic approximation of time-dependent density-functional theory, the static approximation, which substitutes a dynamical (i.e., frequency-dependent) kernel by its static limit, is usually enforced in most implementations of the BSE formalism. Here, going beyond the static approximation, we compute the dynamical correction of the electron–hole screening for molecular excitation energies, thanks to a renormalized first-order perturbative correction to the static BSE excitation energies. The present dynamical correction goes beyond the plasmon-pole approximation as the dynamical screening of the Coulomb interaction is computed exactly within the random-phase approximation. Our calculations are benchmarked against high-level (coupled-cluster) calculations, allowing one to assess the clear improvement brought by the dynamical correction for both singlet and triplet optical transitions.

1.
E. E.
Salpeter
and
H. A.
Bethe
,
Phys. Rev.
84
,
1232
(
1951
).
2.
G.
Strinati
,
Riv. Nuovo Cimento
11
,
1
(
1988
).
4.
D.
Golze
,
M.
Dvorak
, and
P.
Rinke
,
Front. Chem.
7
,
377
(
2019
).
5.
G.
Onida
,
L.
Reining
, and
A.
Rubio
,
Rev. Mod. Phys.
74
,
601
(
2002
).
6.
R. M.
Martin
,
L.
Reining
, and
D. M.
Ceperley
,
Interacting Electrons: Theory and Computational Approaches
(
Cambridge University Press
,
2016
).
7.
E.
Runge
and
E. K. U.
Gross
,
Phys. Rev. Lett.
52
,
997
(
1984
).
8.
M. E.
Casida
,
Time-dependent Density Functional Response Theory for Molecules
(
World Scientific
,
Singapore
,
1995
), pp.
155
192
.
9.
P.
Hohenberg
and
W.
Kohn
,
Phys. Rev.
136
,
B864
(
1964
).
10.
W.
Kohn
and
L. J.
Sham
,
Phys. Rev.
140
,
A1133
(
1965
).
11.
P.
Boulanger
,
D.
Jacquemin
,
I.
Duchemin
, and
X.
Blase
,
J. Chem. Theory Comput.
10
,
1212
(
2014
).
12.
D.
Jacquemin
,
I.
Duchemin
, and
X.
Blase
,
J. Chem. Theory Comput.
11
,
3290
(
2015
).
13.
F.
Bruneval
,
S. M.
Hamed
, and
J. B.
Neaton
,
J. Chem. Phys.
142
,
244101
(
2015
).
14.
D.
Jacquemin
,
I.
Duchemin
, and
X.
Blase
,
J. Chem. Theory Comput.
11
,
5340
(
2015
).
15.
D.
Hirose
,
Y.
Noguchi
, and
O.
Sugino
,
Phys. Rev. B
91
,
205111
(
2015
).
16.
D.
Jacquemin
,
I.
Duchemin
, and
X.
Blase
,
J. Phys. Chem. Lett.
8
,
1524
(
2017
).
17.
D.
Jacquemin
,
I.
Duchemin
,
A.
Blondel
, and
X.
Blase
,
J. Chem. Theory Comput.
13
,
767
(
2017
).
18.
T.
Rangel
,
S. M.
Hamed
,
F.
Bruneval
, and
J. B.
Neaton
,
J. Chem. Phys.
146
,
194108
(
2017
).
19.
K.
Krause
and
W.
Klopper
,
J. Comput. Chem.
38
,
383
(
2017
).
20.
X.
Gui
,
C.
Holzer
, and
W.
Klopper
,
J. Chem. Theory Comput.
14
,
2127
(
2018
).
21.
C.
Liu
,
J.
Kloppenburg
,
Y.
Yao
,
X.
Ren
,
H.
Appel
,
Y.
Kanai
, and
V.
Blum
,
J. Chem. Phys.
152
,
044105
(
2020
).
22.
X.
Blase
,
I.
Duchemin
, and
D.
Jacquemin
,
Chem. Soc. Rev.
47
,
1022
(
2018
).
23.
M.
Boggio-Pasqua
,
M. J.
Bearpark
, and
M. A.
Robb
,
J. Org. Chem.
72
,
4497
(
2007
).
24.
N.
Helbig
,
J. I.
Fuks
,
I. V.
Tokatly
,
H.
Appel
,
E. K. U.
Gross
, and
A.
Rubio
,
Chem. Phys.
391
,
1
(
2011
).
25.
P.
Elliott
,
S.
Goldson
,
C.
Canahui
, and
N. T.
Maitra
,
Chem. Phys.
391
,
110
(
2011
).
26.
M. E.
Casida
,
J. Chem. Phys.
122
,
054111
(
2005
).
27.
P.
Romaniello
,
S.
Guyot
, and
L.
Reining
,
J. Chem. Phys.
131
,
154111
(
2009
).
28.
M.
Huix-Rotllant
,
A.
Ipatov
,
A.
Rubio
, and
M. E.
Casida
,
Chem. Phys.
391
,
120
(
2011
).
29.
P.-F.
Loos
,
A.
Scemama
,
M.
Boggio-Pasqua
, and
D.
Jacquemin
,
J. Chem. Theory Comput.
16
,
3720
(
2020
).
30.
P.-F.
Loos
,
A.
Scemama
,
A.
Blondel
,
Y.
Garniron
,
M.
Caffarel
, and
D.
Jacquemin
,
J. Chem. Theory Comput.
14
,
4360
(
2018
).
31.
P.-F.
Loos
,
M.
Boggio-Pasqua
,
A.
Scemama
,
M.
Caffarel
, and
D.
Jacquemin
,
J. Chem. Theory Comput.
15
,
1939
(
2019
).
32.
P.-F.
Loos
,
F.
Lipparini
,
M.
Boggio-Pasqua
,
A.
Scemama
, and
D.
Jacquemin
,
J. Chem. Theory Comput.
16
,
1711
(
2020
).
33.
M.
Olivucci
,
Computational Photochemistry
(
Elsevier Science
,
Amsterdam; Boston, MA; Paris
,
2010
).
34.
M. A.
Robb
,
M.
Garavelli
,
M.
Olivucci
, and
F.
Bernardi
, “
A computational strategy for organic photochemistry
,” in
Reviews in Computational Chemistry
, edited by
K. B.
Lipkowitz
and
D. B.
Boyd
(
John Wiley & Sons, Inc.
,
Hoboken, NJ, USA
,
2007
), pp.
87
146
.
35.
M.
Manathunga
,
X.
Yang
,
H. L.
Luk
,
S.
Gozem
,
L. M.
Frutos
,
A.
Valentini
,
N.
Ferrè
, and
M.
Olivucci
,
J. Chem. Theory Comput.
12
,
839
(
2016
).
36.
N. T.
Maitra
,
F.
Zhang
,
R. J.
Cave
, and
K.
Burke
,
J. Chem. Phys.
120
,
5932
(
2004
).
37.
R. J.
Cave
,
F.
Zhang
,
N. T.
Maitra
, and
K.
Burke
,
Chem. Phys. Lett.
389
,
39
(
2004
).
38.
B.
Saha
,
M.
Ehara
, and
H.
Nakatsuji
,
J. Chem. Phys.
125
,
014316
(
2006
).
39.
M. A.
Watson
and
G. K.-L.
Chan
,
J. Chem. Theory Comput.
8
,
4013
(
2012
).
40.
Y.
Shu
and
D. G.
Truhlar
,
J. Am. Chem. Soc.
139
,
13770
(
2017
).
41.
G. M. J.
Barca
,
A. T. B.
Gilbert
, and
P. M. W.
Gill
,
J. Chem. Theory Comput.
14
,
1501
(
2018
).
42.
G. M. J.
Barca
,
A. T. B.
Gilbert
, and
P. M. W.
Gill
,
J. Chem. Theory Comput.
14
,
9
(
2018
).
43.
M.
Rohlfing
and
S. G.
Louie
,
Phys. Rev. B
62
,
4927
(
2000
).
44.
F.
Sottile
,
V.
Olevano
, and
L.
Reining
,
Phys. Rev. Lett.
91
,
056402
(
2003
).
45.
P.
Myöhänen
,
A.
Stan
,
G.
Stefanucci
, and
R.
van Leeuwen
,
Europhys. Lett.
84
,
67001
(
2008
).
46.
Y.
Ma
,
M.
Rohlfing
, and
C.
Molteni
,
Phys. Rev. B
80
,
241405
(
2009
).
47.
Y.
Ma
,
M.
Rohlfing
, and
C.
Molteni
,
J. Chem. Theory Comput.
6
,
257
(
2009
).
48.
P.
Romaniello
,
D.
Sangalli
,
J. A.
Berger
,
F.
Sottile
,
L. G.
Molinari
,
L.
Reining
, and
G.
Onida
,
J. Chem. Phys.
130
,
044108
(
2009
).
49.
D.
Sangalli
,
P.
Romaniello
,
G.
Onida
, and
A.
Marini
,
J. Chem. Phys.
134
,
034115
(
2011
).
50.
N.
Säkkinen
,
M.
Manninen
, and
R.
van Leeuwen
,
New J. Phys.
14
,
013032
(
2012
).
51.
D.
Zhang
,
S. N.
Steinmann
, and
W.
Yang
,
J. Chem. Phys.
139
,
154109
(
2013
).
52.
E.
Rebolini
and
J.
Toulouse
,
J. Chem. Phys.
144
,
094107
(
2016
).
53.
V.
Olevano
,
J.
Toulouse
, and
P.
Schuck
,
J. Chem. Phys.
150
,
084112
(
2019
).
54.
T.
Lettmann
and
M.
Rohlfing
,
J. Chem. Theory Comput.
15
,
4547
(
2019
).
55.
G.
Strinati
,
Phys. Rev. Lett.
49
,
1519
(
1982
).
56.
G.
Strinati
,
Phys. Rev. B
29
,
5718
(
1984
).
57.
B.
Baumeier
,
D.
Andrienko
,
Y.
Ma
, and
M.
Rohlfing
,
J. Chem. Theory Comput.
8
,
997
(
2012
).
58.
E.
Rebolini
, “
Range-separated density-functional theory for molecular excitation energies
,” Ph.D. thesis,
Université Pierre et Marie Curie—Paris VI
,
2014
.
59.
J.
Authier
and
P.-F.
Loos
, “
Dynamical kernels for optical excitations
,”
J. Chem. Phys.
(submitted) (
2020
); arXiv:2008.13143 [physics.chem-ph].
60.
J.
Wambach
,
Rep. Prog. Phys.
51
,
989
(
1988
).
61.
N. T.
Maitra
, “
Memory: History, initial-state dependence, and double-excitations
,” in
Fundamentals of Time-dependent Density Functional Theory
, edited by
M. A.
Marques
,
N. T.
Maitra
,
F. M.
Nogueira
,
E.
Gross
, and
A.
Rubio
(
Springer Berlin Heidelberg
,
Berlin, Heidelberg
,
2012
), Vol. 837, pp.
167
184
.
62.
M. E.
Casida
and
M.
Huix-Rotllant
, “
Many-body perturbation theory (MBPT) and time-dependent density-functional theory (TD-DFT): MBPT insights about what is missing in, and corrections to, the TD-DFT adiabatic approximation
,” in , edited by
N.
Ferré
,
M.
Filatov
, and
M.
Huix-Rotllant
(
Springer International Publishing
,
Cham
,
2016
), pp.
1
60
.
63.
M. S.
Hybertsen
and
S. G.
Louie
,
Phys. Rev. Lett.
55
,
1418
(
1985
).
64.
M. S.
Hybertsen
and
S. G.
Louie
,
Phys. Rev. B
34
,
5390
(
1986
).
65.
M.
Shishkin
and
G.
Kresse
,
Phys. Rev. B
75
,
235102
(
2007
).
66.
X.
Blase
and
C.
Attaccalite
,
Appl. Phys. Lett.
99
,
171909
(
2011
).
67.
C.
Faber
,
C.
Attaccalite
,
V.
Olevano
,
E.
Runge
, and
X.
Blase
,
Phys. Rev. B
83
,
115123
(
2011
).
68.
T.
Rangel
,
S. M.
Hamed
,
F.
Bruneval
, and
J. B.
Neaton
,
J. Chem. Theory Comput.
12
,
2834
(
2016
).
69.
F.
Kaplan
,
M. E.
Harding
,
C.
Seiler
,
F.
Weigend
,
F.
Evers
, and
M. J.
van Setten
,
J. Chem. Theory Comput.
12
,
2528
(
2016
).
70.
I.
Duchemin
and
X.
Blase
,
J. Chem. Phys.
150
,
174120
(
2019
).
71.
I.
Duchemin
and
X.
Blase
,
J. Chem. Theory Comput.
16
,
1742
(
2020
).
72.
M. J.
van Setten
,
F.
Weigend
, and
F.
Evers
,
J. Chem. Theory Comput.
9
,
232
(
2013
).
73.
P.-F.
Loos
,
P.
Romaniello
, and
J. A.
Berger
,
J. Chem. Theory Comput.
14
,
3071
(
2018
).
74.
M.
Véril
,
P.
Romaniello
,
J. A.
Berger
, and
P.-F.
Loos
,
J. Chem. Theory Comput.
14
,
5220
(
2018
).
75.
M.
Head-Gordon
,
R. J.
Rico
,
M.
Oumi
, and
T. J.
Lee
,
Chem. Phys. Lett.
219
,
21
29
(
1994
).
76.
M.
Head-Gordon
,
D.
Maurice
, and
M.
Oumi
,
Chem. Phys. Lett.
246
,
114
(
1995
).
77.
A. B.
Trofimov
and
J.
Schirmer
,
Chem. Phys.
214
,
153
(
1997
).
78.
A.
Dreuw
and
M.
Wormit
,
Wiley Interdiscip. Rev.: Comput. Mol. Sci.
5
,
82
(
2015
).
79.
O.
Christiansen
,
H.
Koch
, and
P.
Jørgensen
,
Chem. Phys. Lett.
243
,
409
(
1995
).
80.
G. D.
Purvis
 III
and
R. J.
Bartlett
,
J. Chem. Phys.
76
,
1910
(
1982
).
81.
O.
Christiansen
,
H.
Koch
, and
P.
Jørgensen
,
J. Chem. Phys.
103
,
7429
(
1995
).
82.
P. F.
Loos
, QuAcK: A software for emerging quantum electronic structure methods,
2019
, https://github.com/pfloos/QuAcK.
83.
P.-F.
Loos
and
D.
Jacquemin
,
J. Phys. Chem. Lett.
11
,
974
(
2020
).
84.
P.-F.
Loos
,
N.
Galland
, and
D.
Jacquemin
,
J. Phys. Chem. Lett.
9
,
4646
(
2018
).
85.
P. F.
Loos
and
D.
Jacquemin
,
ChemPhotoChem
3
,
684
(
2019
).
86.
M.
Boggio-Pasqua
,
M. J.
Bearpark
,
M.
Klene
, and
M. A.
Robb
,
J. Chem. Phys.
120
,
7849
(
2004
).
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