We present in detail and validate an effective Monte Carlo approach for the calculation of the nuclear vibrational densities via integration of molecular eigenfunctions that we have preliminary employed to calculate the densities of the ground and the excited OH stretch vibrational states in the protonated glycine molecule [Aieta et al., Nat Commun 11, 4348 (2020)]. Here, we first validate and discuss in detail the features of the method on a benchmark water molecule. Then, we apply it to calculate on-the-fly the ab initio anharmonic nuclear densities in the correspondence of the fundamental transitions of NH and CH stretches in protonated glycine. We show how we can gain both qualitative and quantitative physical insight by inspection of different one-nucleus densities and assign a character to spectroscopic absorption peaks using the expansion of vibrational states in terms of harmonic basis functions. The visualization of the nuclear vibrations in a purely quantum picture allows us to observe and quantify the effects of anharmonicity on the molecular structure, also to exploit the effect of IR excitations on specific bonds or functional groups, beyond the harmonic approximation. We also calculate the quantum probability distribution of bond lengths, angles, and dihedrals of the molecule. Notably, we observe how in the case of one type of fundamental NH stretching, the typical harmonic nodal pattern is absent in the anharmonic distribution.

1.
R.
Fausto
,
G. O.
Ildiz
,
E. M.
Bras
, and
B. A.
Nogueira
,
Molecular Spectroscopy-Experiment and Theory
(
Springer
,
2019
), pp.
199
222
.
2.
D.
Gerlich
,
J. Chin. Chem. Soc.
65
,
637
(
2018
).
3.
J.
Roithová
,
A.
Gray
,
E.
Andris
,
J.
Jašík
, and
D.
Gerlich
,
Acc. Chem. Res.
49
,
223
(
2016
).
4.
A. P.
Cismesia
,
L. F.
Tesler
,
M. R.
Bell
,
L. S.
Bailey
, and
N. C.
Polfer
,
J. Mass Spectrom.
52
,
720
(
2017
).
5.
A. B.
Wolk
,
C. M.
Leavitt
,
E.
Garand
, and
M. A.
Johnson
,
Acc. Chem. Res.
47
,
202
(
2013
).
6.
K. R.
Asmis
,
A.
Fielicke
,
G.
von Helden
, and
G.
Meijer
,
Chem. Phys. Solids Their Surf.
12
,
327
(
2007
).
7.
8.
T. R.
Rizzo
and
O. V.
Boyarkin
,
Gas-Phase IR Spectroscopy and Structure of Biological Molecules
(
Springer
,
2014
), pp.
43
97
.
9.
H.
Schwarz
and
K. R.
Asmis
,
Chem. - Eur. J.
25
,
2112
(
2019
).
10.
11.
M. A.
Czarnecki
,
Y.
Morisawa
,
Y.
Futami
, and
Y.
Ozaki
,
Chem. Rev.
115
,
9707
(
2015
).
12.
J. M.
Voss
,
K. C.
Fischer
, and
E.
Garand
,
J. Phys. Chem. Lett.
9
,
2246
(
2018
).
13.
K. R.
Asmis
and
D. M.
Neumark
,
Acc. Chem. Res.
45
,
43
(
2011
).
14.
M. F.
Bush
,
R. J.
Saykally
, and
E. R.
Williams
,
J. Am. Chem. Soc.
129
,
2220
(
2007
).
15.
M. Y.
Choi
and
R. E.
Miller
,
J. Am. Chem. Soc.
128
,
7320
(
2006
).
16.
F.
Gabas
,
G.
Di Liberto
,
R.
Conte
, and
M.
Ceotto
,
Chem. Sci.
9
,
7894
(
2018
).
17.
B. C.
Stipe
,
M. A.
Rezaei
, and
W.
Ho
,
Science
280
,
1732
(
1998
).
18.
S.
Duan
,
G.
Tian
, and
Y.
Luo
,
Angew. Chem., Int. Ed.
55
,
1041
(
2016
).
19.
P.
Liu
,
D. V.
Chulhai
, and
L.
Jensen
,
ACS Nano
11
,
5094
(
2017
).
20.
J.
Lee
,
K. T.
Crampton
,
N.
Tallarida
, and
V. A.
Apkarian
,
Nature
568
,
78
(
2019
).
21.
A. M.
Carrascosa
,
T.
Northey
, and
A.
Kirrander
,
Phys. Chem. Chem. Phys.
19
,
7853
(
2017
).
22.
F.
Teixeira
and
M. N. D. S.
Cordeiro
,
J. Chem. Theory Comput.
15
,
456
(
2018
).
23.
J. M.
Bowman
,
Acc. Chem. Res.
19
,
202
(
1986
).
24.
R.
Gerber
and
M. A.
Ratner
,
Adv. Chem. Phys.
70
,
97
(
1988
).
25.
I.
Kosztin
,
B.
Faber
, and
K.
Schulten
,
Am. J. Phys.
64
,
633
(
1996
).
26.
O.
Christiansen
,
Phys. Chem. Chem. Phys.
9
,
2942
(
2007
).
27.
O.
Christiansen
,
Phys. Chem. Chem. Phys.
14
,
6672
(
2012
).
28.
J. S.
Mancini
and
J. M.
Bowman
,
J. Phys. Chem. Lett.
5
,
2247
(
2014
).
29.
A. B.
McCoy
,
Int. Rev. Phys. Chem.
25
,
77
(
2006
).
30.
J. M.
Bowman
,
T.
Carrington
, and
H.-D.
Meyer
,
Mol. Phys.
106
,
2145
(
2008
).
31.
E.
Mátyus
,
J.
Šimunek
, and
A. G.
Császár
,
J. Chem. Phys.
131
,
074106
(
2009
).
32.
K.
Ruud
,
P.-O.
Åstrand
, and
P. R.
Taylor
,
J. Chem. Phys.
112
,
2668
(
2000
).
33.
P.-O.
Åstrand
,
K.
Ruud
, and
P. R.
Taylor
,
J. Chem. Phys.
112
,
2655
(
2000
).
34.
O.
Vendrell
,
F.
Gatti
, and
H.-D.
Meyer
,
J. Chem. Phys.
127
,
184303
(
2007
).
35.
M.
Ceotto
,
S.
Valleau
,
G. F.
Tantardini
, and
A.
Aspuru-Guzik
,
J. Chem. Phys.
134
,
234103
(
2011
).
37.
T.
Culpitt
,
Y.
Yang
,
F.
Pavošević
,
Z.
Tao
, and
S.
Hammes-Schiffer
,
J. Chem. Phys.
150
,
201101
(
2019
).
38.
C.
Aieta
,
M.
Micciarelli
,
G.
Bertaina
, and
M.
Ceotto
,
Nat. Commun.
11
,
4348
(
2020
).
39.
M.
Micciarelli
,
R.
Conte
,
J.
Suarez
, and
M.
Ceotto
,
J. Chem. Phys.
149
,
064115
(
2018
).
40.
W.
Humphrey
,
A.
Dalke
, and
K.
Schulten
,
J. Mol. Graphics
14
,
33
(
1996
).
41.
A. L.
Kaledin
and
W. H.
Miller
,
J. Chem. Phys.
118
,
7174
(
2003
).
42.
A. L.
Kaledin
and
W. H.
Miller
,
J. Chem. Phys.
119
,
3078
(
2003
).
43.
M.
Ceotto
,
S.
Atahan
,
G. F.
Tantardini
, and
A.
Aspuru-Guzik
,
J. Chem. Phys.
130
,
234113
(
2009
).
44.
M.
Ceotto
,
S.
Atahan
,
S.
Shim
,
G. F.
Tantardini
, and
A.
Aspuru-Guzik
,
Phys. Chem. Chem. Phys.
11
,
3861
(
2009
).
45.
M.
Ceotto
,
D.
Dell’Angelo
, and
G. F.
Tantardini
,
J. Chem. Phys.
133
,
054701
(
2010
).
46.
M.
Ceotto
,
G. F.
Tantardini
, and
A.
Aspuru-Guzik
,
J. Chem. Phys.
135
,
214108
(
2011
).
47.
R.
Conte
,
A.
Aspuru-Guzik
, and
M.
Ceotto
,
J. Phys. Chem. Lett.
4
,
3407
(
2013
).
48.
D.
Tamascelli
,
F. S.
Dambrosio
,
R.
Conte
, and
M.
Ceotto
,
J. Chem. Phys.
140
,
174109
(
2014
).
49.
R. P.
Feynman
and
A. R.
Hibbs
,
Quantum Mechanics and Path Integrals
(
McGraw-Hill
,
1965
).
50.
J.
Liu
,
W. H.
Miller
,
G. S.
Fanourgakis
,
S. S.
Xantheas
,
S.
Imoto
, and
S.
Saito
,
J. Chem. Phys.
135
,
244503
(
2011
).
51.
X.
Liu
and
J.
Liu
,
Mol. Phys.
116
,
755
(
2018
).
52.
N. V.
Golubev
,
T.
Begusic
, and
J.
Vanicek
,
Phys. Rev. Lett.
125
,
083001
(
2020
).
53.
M.
Micciarelli
,
F.
Gabas
,
R.
Conte
, and
M.
Ceotto
,
J. Chem. Phys.
150
,
184113
(
2019
).
54.
T.
Begusic
and
J.
Vanicek
,
J. Chem. Phys.
153
,
024105
(
2020
).
55.
M.
Ceotto
,
G.
Di Liberto
, and
R.
Conte
,
Phys. Rev. Lett.
119
,
010401
(
2017
).
56.
F.
Gabas
,
R.
Conte
, and
M.
Ceotto
,
J. Chem. Theory Comput.
13
,
2378
(
2017
).
57.
G.
Di Liberto
,
R.
Conte
, and
M.
Ceotto
,
J. Chem. Phys.
148
,
014307
(
2018
).
58.
G.
Di Liberto
,
R.
Conte
, and
M.
Ceotto
,
J. Chem. Phys.
148
,
104302
(
2018
).
59.
X.
Ma
,
G.
Di Liberto
,
R.
Conte
,
W. L.
Hase
, and
M.
Ceotto
,
J. Chem. Phys.
149
,
164113
(
2018
).
60.
M.
Buchholz
,
F.
Grossmann
, and
M.
Ceotto
,
J. Chem. Phys.
148
,
114107
(
2018
).
61.
M.
Buchholz
,
F.
Grossmann
, and
M.
Ceotto
,
J. Chem. Phys.
147
,
164110
(
2017
).
62.
M.
Buchholz
,
F.
Grossmann
, and
M.
Ceotto
,
J. Chem. Phys.
144
,
094102
(
2016
).
63.
F.
Gabas
,
G.
Di Liberto
, and
M.
Ceotto
,
J. Chem. Phys.
150
,
224107
(
2019
).
64.
F.
Gabas
,
R.
Conte
, and
M.
Ceotto
,
J. Chem. Theory Comput.
16
,
3476
(
2020
).
65.
G.
Bertaina
,
G.
Di Liberto
, and
M.
Ceotto
,
J. Chem. Phys.
151
,
114307
(
2019
).
66.
M.
Cazzaniga
,
M.
Micciarelli
,
F.
Moriggi
,
A.
Mahmoud
,
F.
Gabas
, and
M.
Ceotto
,
J. Chem. Phys.
152
,
104104
(
2020
).
67.
R.
Pérez de Tudela
and
D.
Marx
,
J. Phys. Chem. Lett.
7
,
5137
(
2016
).
68.
E. B.
Wilson
,
J. C.
Decius
, and
P. C.
Cross
,
Molecular Vibrations: The Theory of Infrared and Raman Vibrational Spectra
(
Courier Corporation
,
1980
).
69.
G. E. P.
Box
and
M. E.
Muller
,
Ann. Math. Stat.
29
,
610
(
1958
).
70.
R. G.
Parr
and
W.
Yang
,
Annu. Rev. Phys. Chem.
46
,
701
(
1995
).
73.
E. J.
Heller
,
J. Chem. Phys.
75
,
2923
(
1981
).
74.
E. J.
Heller
,
Acc. Chem. Res.
14
,
368
(
1981
).
75.
E. J.
Heller
,
J. Chem. Phys.
94
,
2723
(
1991
).
76.
W. H.
Miller
,
J. Chem. Phys.
53
,
1949
(
1970
).
77.
M. F.
Herman
and
E.
Kluk
,
Chem. Phys.
91
,
27
(
1984
).
78.
K. G.
Kay
,
J. Chem. Phys.
100
,
4377
(
1994
).
79.
K. G.
Kay
,
J. Chem. Phys.
101
,
2250
(
1994
).
80.
K. G.
Kay
,
J. Chem. Phys.
100
,
4432
(
1994
).
81.
S. V.
Antipov
,
Z.
Ye
, and
N.
Ananth
,
J. Chem. Phys.
142
,
184102
(
2015
).
82.
M. S.
Church
,
S. V.
Antipov
, and
N.
Ananth
,
J. Chem. Phys.
146
,
234104
(
2017
).
83.
M.
Wehrle
,
M.
Šulc
, and
J.
Vaníček
,
J. Chem. Phys.
140
,
244114
(
2014
).
84.
M.
Wehrle
,
S.
Oberli
, and
J.
Vaníček
,
J. Phys. Chem. A
119
,
5685
(
2015
).
85.
M. L.
Brewer
,
J. S.
Hulme
, and
D. E.
Manolopoulos
,
J. Chem. Phys.
106
,
4832
(
1997
).
86.
Y.
Zhuang
,
M. R.
Siebert
,
W. L.
Hase
,
K. G.
Kay
, and
M.
Ceotto
,
J. Chem. Theory Comput.
9
,
54
(
2013
).
87.
M.
Ceotto
,
Y.
Zhuang
, and
W. L.
Hase
,
J. Chem. Phys.
138
,
054116
(
2013
).
88.
R.
Conte
,
F.
Gabas
,
G.
Botti
,
Y.
Zhuang
, and
M.
Ceotto
,
J. Chem. Phys.
150
,
244118
(
2019
).
89.
S.
Dressler
and
W.
Thiel
,
Chem. Phys. Lett.
273
,
71
(
1997
).
90.
M.
Valiev
,
E. J.
Bylaska
,
N.
Govind
,
K.
Kowalski
,
T. P.
Straatsma
,
H. J. J.
Van Dam
,
D.
Wang
,
J.
Nieplocha
,
E.
Apra
,
T. L.
Windus
, and
W. A.
de Jong
,
Comput. Phys. Commun.
181
,
1477
(
2010
).
91.
K.
Zhang
and
A.
Chung-Phillips
,
J. Comput. Chem.
19
,
1862
(
1998
).
92.
C.
Aieta
,
G.
Bertaina
,
M.
Micciarelli
, and
M.
Ceotto
(
2020
). “
Semiclassical nuclear density
,” Zenodo.
93.
R.
Conte
,
G.
Botti
, and
M.
Ceotto
,
Vib. Spectrosc.
106
,
103015
(
2020
).

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