Encoding the complex features of an energy landscape is a challenging task, and often, chemists pursue the most salient features (minima and barriers) along a highly reduced space, i.e., two- or three-dimensions. Even though disconnectivity graphs or merge trees summarize the connectivity of the local minima of an energy landscape via the lowest-barrier pathways, there is much information to be gained by also considering the topology of each connected component at different energy thresholds (or sublevelsets). We propose sublevelset persistent homology as an appropriate tool for this purpose. Our computations on the configuration phase space of n-alkanes from butane to octane allow us to conjecture, and then prove, a complete characterization of the sublevelset persistent homology of the alkane CmH2m+2 Potential Energy Landscapes (PELs), for all m, in all homological dimensions. We further compare both the analytical configurational PELs and sampled data from molecular dynamics simulation using the united and all-atom descriptions of the intramolecular interactions. In turn, this supports the application of distance metrics to quantify sampling fidelity and lays the foundation for future work regarding new metrics that quantify differences between the topological features of high-dimensional energy landscapes.

1.
M.
Born
and
R.
Oppenheimer
,
Ann. Phys.
84
,
457
(
1927
).
2.
D.
Wales
,
Energy Landscapes: Applications to Clusters, Biomolecules and Glasses
(
Cambridge University Press
,
2003
).
3.
F. H.
Stillinger
,
Energy Landscapes, Inherent Structures, and Condensed-Matter Phenomena
(
Princeton University Press
,
2016
).
4.
B.
Denzel
and
F. F.
Seelig
,
Z. Naturforsch. A
33
,
1341
(
1978
).
5.
R. Q.
Topper
, in
Reviews in Computational Chemistry
, edited by
K. B.
Lipkowitz
and
D. B.
Boyd
(
John Wiley and Sons
,
Hoboken, NJ
,
1997
), Chap. 3, pp.
101
169
.
6.
R. G.
Mullen
,
J.-E.
Shea
, and
B.
Peters
,
J. Chem. Theory Comput.
10
,
659
(
2014
).
7.
F.
Pietrucci
and
W.
Andreoni
,
Phys. Rev. Lett.
107
,
085504
(
2011
).
8.
T.
Zhou
,
E.
Martinez-Baez
,
G.
Schenter
, and
A. E.
Clark
,
J. Chem. Phys.
150
,
134102
(
2019
).
9.
H. J.
Kulik
,
Wiley Interdiscip. Rev.: Comput. Mol. Sci.
10
,
e1439
(
2020
).
10.
K. T.
Butler
,
D. W.
Davies
,
H.
Cartwright
,
O.
Isayev
, and
A.
Walsh
,
Nature
559
,
547
(
2018
).
11.
O. M.
Becker
and
M.
Karplus
,
J. Chem. Phys.
106
,
1495
(
1997
).
12.
M.
Li
,
M.
Duan
,
J.
Fan
,
L.
Han
, and
S.
Huo
,
J. Chem. Phys.
139
,
185101
(
2013
).
13.
D. J.
Wales
,
Philos. Trans. R. Soc. A
363
,
357
377
(
2005
).
14.
F.
Cazals
,
T.
Dreyfus
,
D.
Mazauric
,
C.-A.
Roth
, and
C. H.
Robert
,
J. Comput. Chem.
36
,
1213
(
2015
).
15.
L. C.
Smeeton
,
M. T.
Oakley
, and
R. L.
Johnston
,
J. Comput. Chem.
35
,
1481
1490
(
2014
).
16.
K.
Beketayev
,
G. H.
Weber
,
M.
Haranczyk
,
P.-T.
Bremer
,
M.
Hlawitschka
, and
B.
Hamann
, in
Computer Graphics Forum
(
John Wiley & Sons Ltd.
,
2011
), Vol. 30, pp.
663
672
.
17.
G.
Reeb
,
C. R. Acad. Sci., Paris
222
,
847
(
1946
).
18.
J. M.
Carr
,
D.
Mazauric
,
F.
Cazals
, and
D. J.
Wales
,
J. Chem. Phys.
144
,
054109
(
2016
).
19.
H.
Edelsbrunner
,
D.
Letscher
, and
A.
Zomorodian
, in
Proceedings 41st Annual Symposium on Foundations of Computer Science
(
IEEE
,
2000
), pp.
454
463
.
20.
H.
Edelsbrunner
and
E. P.
Mücke
,
ACM Trans. Graphics
13
,
43
(
1994
).
21.
H.
Edelsbrunner
and
P.
Koehl
,
Comb. Comput. Geom.
52
,
243
(
2005
).
22.
H.-W.
Chang
,
S.
Bacallado
,
V. S.
Pande
, and
G. E.
Carlsson
,
PLoS One
8
,
e58699
(
2013
).
23.
G. G.
Henkelman
,
G.
Jóhannesson
, and
H.
Jónsson
,
Theoretical Methods in Condensed Phase Chemistry
(
Springer Netherlands
,
2002
), pp.
269
302
.
24.
H.
Adams
,
A.
Atanasov
, and
G.
Carlsson
,
Topol. Methods Nonlinear Anal.
45
,
247
(
2015
).
25.
K.
Xia
,
X.
Feng
,
Y.
Tong
, and
G.-W.
Wei
,
J. Comput. Chem.
36
,
408
(
2015
).
26.
K.
Xia
and
G.-W.
Wei
,
Int. J. Numer. Methods Biomed. Eng.
30
,
814
(
2014
).
27.
J.
Townsend
,
C. P.
Micucci
,
J. H.
Hymel
,
V.
Maroulas
, and
K. D.
Vogiatzis
,
Nat. Commun.
11
,
3230
(
2020
).
28.
I.
Membrillo-Solis
,
M.
Pirashvili
,
L.
Steinberg
,
J.
Brodzki
, and
J. G.
Frey
, arXiv:1907.07770 (
2019
).
29.
S.
Martin
,
A.
Thompson
,
E. A.
Coutsias
, and
J.-P.
Watson
,
J. Chem. Phys.
132
,
234115
(
2010
).
30.
A.
Baeyer
,
Ber. Dtsch. Chem. Ges.
10
,
1286
(
1877
).
31.
E.
Badea
,
G.
Della Gatta
,
D.
D’Angelo
,
B.
Brunetti
, and
Z.
Rečková
,
J. Chem. Thermodyn.
38
,
1546
(
2006
).
32.
M. K.
Mishra
,
S.
Varughese
,
U.
Ramamurty
, and
G. R.
Desiraju
,
J. Am. Chem. Soc.
135
,
8121
(
2013
).
33.
R.
Boese
,
H.-C.
Weiss
, and
D.
Bläser
,
Angew. Chem., Int. Ed.
38
,
988
(
1999
).
34.
K.
Yang
,
Z.
Cai
,
A.
Jaiswal
,
M.
Tyagi
,
J. S.
Moore
, and
Y.
Zhang
,
Angew. Chem., Int. Ed.
55
,
14090
(
2016
).
35.
G.
Carlsson
,
Bull. Am. Math. Soc.
46
,
255
(
2009
).
36.
H.
Edelsbrunner
and
J. L.
Harer
,
Computational Topology: An Introduction
(
American Mathematical Society
,
Providence
,
2010
).
37.
H.
Adams
,
S.
Chepushtanova
,
T.
Emerson
,
E.
Hanson
,
M.
Kirby
,
F.
Motta
,
R.
Neville
,
C.
Peterson
,
P.
Shipman
, and
L.
Ziegelmeier
,
J. Mach. Learn. Res.
18
,
1
(
2017
).
38.
N.
Milosavljević
,
D.
Morozov
, and
P.
Skraba
, in
Proceedings of the Twenty-Seventh Annual Symposium on Computational Geometry
(
Association for Computing Machinery
,
2011
), pp.
216
225
.
39.
R.
Pradhan
and
U.
Lourderaj
,
Phys. Chem. Chem. Phys.
19
,
27468
(
2017
).
40.
O.
Dmitrenko
and
R. D.
Bach
,
J. Phys. Chem. A
108
,
6886
(
2004
).
41.
S. P.
de Visser
,
F.
Ogliaro
, and
S.
Shaik
,
Chem. Commun.
2001
,
2322
.
42.
R.
Pradhan
and
U.
Lourderaj
,
Phys. Chem. Chem. Phys.
21
,
12837
(
2019
).
43.
D.
Cohen-Steiner
,
H.
Edelsbrunner
, and
J.
Harer
,
Discrete Comput. Geom.
37
,
103
(
2007
).
44.
A.
Banyaga
and
D.
Hurtubise
,
Lectures on Morse Homology
, Kluwer Texts in the Mathematical Sciences (
Kluwer Academic Publishers Group
,
Dordrecht
,
2004
), Vol. 29, p.
x+324
.
45.
W. L.
Jorgensen
and
J.
Tirado-Rives
,
J. Am. Chem. Soc.
110
,
1657
(
1988
).
46.
B.
Chen
,
M. G.
Martin
, and
J. I.
Siepmann
,
J. Phys. Chem. B
102
,
2578
(
1998
).
47.
J.-P.
Ryckaert
,
G.
Ciccotti
, and
H. J. C.
Berendsen
,
J. Comput. Phys.
23
,
327
(
1977
).
48.
M. J.
Abraham
,
T.
Murtola
,
R.
Schulz
,
S.
Páll
,
J. C.
Smith
,
B.
Hess
, and
E.
Lindahl
,
SoftwareX
1-2
,
19
(
2015
).
49.
C.
Maria
,
J.-D.
Boissonnat
,
M.
Glisse
, and
M.
Yvinec
, in
International Congress on Mathematical Software
(
Springer
,
2014
), pp.
167
174
.
50.
J.
Mirth
,
J.
Bush
,
M.
Heim
, and
H.
Adams
, deltapersistence,
2020
, https://gitlab.com/delta-topology-public/deltapersistence.
51.
V.
de Silva
and
G. E.
Carlsson
, in
Eurographics Symposium on Point-Based Graphics
(
Eurographics Association
,
2004
), Vol. 4, p.
157
.
52.
J.-D.
Boissonnat
,
R.
Dyer
, and
A.
Ghosh
,
Found. Comput. Math.
18
,
399
(
2018
).
53.
J.-D.
Boissonnat
,
R.
Dyer
,
A.
Ghosh
, and
N.
Martynchuk
,
Discrete Comput. Geom.
59
,
226
(
2018
).
54.
G.
Leibon
and
D.
Letscher
, in
Proceedings of the Sixteenth Annual Symposium on Computational Geometry
(
Association for Computing Machinery
,
2000
), pp.
341
349
.
55.
Y.
Rubner
,
C.
Tomasi
, and
L. J.
Guibas
,
Int. J. Comput. Vision
40
,
99
(
2000
).
56.
D.
Cohen-Steiner
,
H.
Edelsbrunner
,
J.
Harer
, and
Y.
Mileyko
,
Found. Comput. Math.
10
,
127
(
2010
).
57.
J.
Milnor
,
Morse Theory
(
Princeton University Press
,
2016
), Vol. 51.
58.
H.
Gakhar
and
J. A.
Perea
, arXiv:1910.05656 (
2019
).
59.
J.
Towns
,
T.
Cockerill
,
M.
Dahan
,
I.
Foster
,
K.
Gaither
,
A.
Grimshaw
,
V.
Hazlewood
,
S.
Lathrop
,
D.
Lifka
,
G. D.
Peterson
,
R.
Roskies
,
J. R.
Scott
, and
N.
Wilkins-Diehr
,
Comput. Sci. Eng.
16
,
62
(
2014
).
60.
M.
Pierce
,
S.
Marru
,
E.
Abeysinghe
,
S.
Pamidighantam
,
M.
Christie
, and
D.
Wannipurage
, in
Proceedings of the Practice and Experience on Advanced Research Computing
(
Association for Computing Machinery
,
2018
), pp.
1
4
.
61.
M.
Pierce
,
S.
Marru
,
L.
Gunathilake
,
T. A.
Kanewala
,
R.
Singh
,
S.
Wijeratne
,
C.
Wimalasena
,
C.
Herath
,
E.
Chinthaka
,
C.
Mattmann
 et al, in
2014 6th International Workshop on Science Gateways
(
IEEE
,
2014
), pp.
48
54
.
62.
J.
Jost
,
Riemannian Geometry and Geometric Analysis
(
Springer
,
2008
), Vol. 42005.
63.
A.
Hatcher
,
Algebraic Topology
(
Cambridge University Press
,
Cambridge
,
2002
).
64.
J.
Tierney
,
G.
Favlier
, and
J. A.
Levine
,
IEEE Trans. Visualization Comput. Graphics
24
,
832
(
2018
).
65.
H.
Jordan
and
M.
Pflaum
, TTKAlkanes,
2020
, https://gitlab.com/delta-topology-public/ttkalkanes.
You do not currently have access to this content.