Nuclear quantum phenomena beyond the Born–Oppenheimer approximation are known to play an important role in a growing number of chemical and biological processes. While there exists no unique consensus on a rigorous and efficient implementation of coupled electron–nuclear quantum dynamics, it is recognized that these problems scale exponentially with system size on classical processors and, therefore, may benefit from quantum computing implementations. Here, we introduce a methodology for the efficient quantum treatment of the electron–nuclear problem on near-term quantum computers, based upon the Nuclear–Electronic Orbital (NEO) approach. We generalize the electronic two-qubit tapering scheme to include nuclei by exploiting symmetries inherent in the NEO framework, thereby reducing the Hamiltonian dimension, number of qubits, gates, and measurements needed for calculations. We also develop parameter transfer and initialization techniques, which improve convergence behavior relative to conventional initialization. These techniques are applied to H2 and malonaldehyde for which results agree with NEO full configuration interaction and NEO complete active space configuration interaction benchmarks for ground state energy to within 10−6 hartree and entanglement entropy to within 10−4. These implementations therefore significantly reduce resource requirements for full quantum simulations of molecules on near-term quantum devices while maintaining high accuracy.
REFERENCES
1.
T. E.
Markland
and M.
Ceriotti
, “Nuclear quantum effects enter the mainstream
,” Nat. Rev. Chem.
2
(3
), 0109
(2018
).2.
W. H.
Miller
, “Classical S matrix: Numerical application to inelastic collisions
,” J. Chem. Phys.
53
, 3578
(1970
).3.
I.
Tavernelli
, U. F.
Röhrig
, and U.
Rothlisberger
, “Molecular dynamics in electronically excited states using time-dependent density functional theory
,” Mol. Phys.
103
(6–8
), 963
–981
(2005
).4.
I.
Tavernelli
, “Electronic density response of liquid water using time-dependent density functional theory
,” Phys. Rev. B
73
(9
), 094204
(2006
).5.
S. K.
Min
, F.
Agostini
, I.
Tavernelli
, and E. K. U.
Gross
, “Ab initio nonadiabatic dynamics with coupled trajectories: A rigorous approach to quantum (de)coherence
,” J. Phys. Chem. Lett.
8
(13
), 3048
–3055
(2017
).6.
T. J.
Martínez
, “Insights for light-driven molecular devices from ab initio multiple spawning excited-state dynamics of organic and biological chromophores
,” Acc. Chem. Res.
39
(2
), 119
–126
(2006
).7.
H.-D.
Meyer
, U.
Manthe
, and L. S.
Cederbaum
, “The multi-configurational time-dependent Hartree approach
,” Chem. Phys. Lett.
165
(1
), 73
–78
(1990
).8.
O.
Christiansen
, “Møller–Plesset perturbation theory for vibrational wave functions
,” J. Chem. Phys.
119
(12
), 5773
–5781
(2003
).9.
L.
Pereyaslavets
, I.
Kurnikov
, G.
Kamath
, O.
Butin
, A.
Illarionov
, I.
Leontyev
, M.
Olevanov
, M.
Levitt
, R. D.
Kornberg
, and B.
Fain
, “On the importance of accounting for nuclear quantum effects in ab initio calibrated force fields in biological simulations
,” Proc. Natl. Acad. Sci. U. S. A.
115
(36
), 8878
–8882
(2018
).10.
D. J.
Heyes
, M.
Sakuma
, S. P.
de Visser
, and N. S.
Scrutton
, “Nuclear quantum tunneling in the light-activated enzyme protochlorophyllide oxidoreductase
,” J. Biol. Chem.
284
(6
), 3762
–3767
(2009
).11.
M.
Rossi
, P.
Gasparotto
, and M.
Ceriotti
, “Anharmonic and quantum fluctuations in molecular crystals: A first-principles study of the stability of paracetamol
,” Phys. Rev. Lett.
117
(11
), 115702
(2016
).12.
A.
Vardi-Kilshtain
, N.
Nitoker
, and D. T.
Major
, “Nuclear quantum effects and kinetic isotope effects in enzyme reactions
,” Arch. Biochem. Biophys.
582
, 18
–27
(2015
).13.
S.
Raugei
and M. L.
Klein
, “Nuclear quantum effects and hydrogen bonding in liquids
,” J. Am. Chem. Soc.
125
(30
), 8992
–8993
(2003
).14.
K. H.
Kim
, H.
Pathak
, A.
Späh
, F.
Perakis
, D.
Mariedahl
, J. A.
Sellberg
, T.
Katayama
, Y.
Harada
, H.
Ogasawara
, L. G. M.
Pettersson
, and A.
Nilsson
, “Temperature-independent nuclear quantum effects on the structure of water
,” Phys. Rev. Lett.
119
(7
), 075502
(2017
).15.
B.
Pamuk
, J. M.
Soler
, R.
Ramírez
, C. P.
Herrero
, P. W.
Stephens
, P. B.
Allen
, and M.-V.
Fernández-Serra
, “Anomalous nuclear quantum effects in ice
,” Phys. Rev. Lett.
108
(19
), 193003
(2012
).16.
P. J.
Ollitrault
, A.
Baiardi
, M.
Reiher
, and I.
Tavernelli
, “Hardware efficient quantum algorithms for vibrational structure calculations
,” Chem. Sci.
11
, 6842
–6855
(2020
).17.
S. P.
Webb
, T.
Iordanov
, and S.
Hammes-Schiffer
, “Multiconfigurational nuclear-electronic orbital approach: Incorporation of nuclear quantum effects in electronic structure calculations
,” J. Chem. Phys.
117
, 4106
–4118
(2002
).18.
B.
Simmen
, E.
Mátyus
, and M.
Reiher
, “Elimination of the translational kinetic energy contamination in pre-Born–Oppenheimer calculations
,” Mol. Phys.
111
(14–15
), 2086
–2092
(2013
).19.
A.
Muolo
, A.
Baiardi
, R.
Feldmann
, and M.
Reiher
, “Nuclear-electronic all-particle density matrix renormalization group
,” J. Chem. Phys.
152
(20
), 204103
(2020
).20.
H.
Nakai
, “Simultaneous determination of nuclear and electronic wave functions without Born–Oppenheimer approximation: Ab initio NO + MO/HF theory
,” Int. J. Quantum Chem.
86
(6
), 511
–517
(2002
).21.
S.
Lee
, J.
Lee
, H.
Zhai
, et al., “Evaluating the evidence for exponential quantum advantage in ground-state quantum chemistry
,” Nat. Commun.
14
, 1952
(2023
).22.
M.
Kjaergaard
, M. E.
Schwartz
, J.
Braumüller
, P.
Krantz
, J. I.-J.
Wang
, S.
Gustavsson
, and W. D.
Oliver
, “Superconducting qubits: Current state of play
,” Annu. Rev. Condens. Matter Phys.
11
, 369
–395
(2020
).23.
H.-L.
Huang
, D.
Wu
, D.
Fan
, and X.
Zhu
, “Superconducting quantum computing: A review
,” Sci. China Inf. Sci.
63
(8
), 180501
(2020
).24.
R.
Somma
, G.
Ortiz
, J. E.
Gubernatis
, E.
Knill
, and R.
Laflamme
, “Simulating physical phenomena by quantum networks
,” Phys. Rev. A
65
, 042323
(2002
).25.
D.
Wecker
, M. B.
Hastings
, N.
Wiebe
, B. K.
Clark
, C.
Nayak
, and M.
Troyer
, “Solving strongly correlated electron models on a quantum computer
,” Phys. Rev. A
92
, 062318
(2015
).26.
B.
Bauer
, D.
Wecker
, A. J.
Millis
, M. B.
Hastings
, and M.
Troyer
, “Hybrid quantum-classical approach to correlated materials
,” Phys. Rev. X
6
, 031045
(2016
).27.
A.
Smith
, M. S.
Kim
, F.
Pollmann
, and J.
Knolle
, “Simulating quantum many-body dynamics on a current digital quantum computer
,” npj Quantum Inf.
5
(1
), 106
(2019
).28.
C.
Cade
, L.
Mineh
, A.
Montanaro
, and S.
Stanisic
, “Strategies for solving the Fermi-Hubbard model on near-term quantum computers
,” Phys. Rev. B
102
, 235122
(2020
).29.
A.
Chiesa
, F.
Tacchino
, M.
Grossi
, P.
Santini
, I.
Tavernelli
, D.
Gerace
, and S.
Carretta
, “Quantum hardware simulating four-dimensional inelastic neutron scattering
,” Nat. Phys.
15
, 455
–459
(2019
).30.
F.
Tacchino
, A.
Chiesa
, S.
Carretta
, and D.
Gerace
, “Quantum computers as universal quantum simulators: State-of-the-art and perspectives
,” Adv. Quantum Technol.
3
(3
), 1900052
(2020
).31.
P.
Suchsland
, P. K.
Barkoutsos
, I.
Tavernelli
, M. H.
Fischer
, and T.
Neupert
, “Simulating a ring-like Hubbard system with a quantum computer
,” Phys. Rev. Res.
4
, 013165
(2021
); arXiv:2104.06428.32.
A.
Miessen
, P. J.
Ollitrault
, and I.
Tavernelli
, “Quantum algorithms for quantum dynamics: A performance study on the spin-boson model
,” Phys. Rev. Res.
3
, 043212
(2021
).33.
F.
Libbi
, J.
Rizzo
, F.
Tacchino
, N.
Marzari
, and I.
Tavernelli
, “Effective calculation of the Green’s function in the time domain on near-term quantum processors
,” Phys. Rev. Res.
4
, 043038
(2022
).34.
J.
Rizzo
, F.
Libbi
, F.
Tacchino
, P. J.
Ollitrault
, N.
Marzari
, and I.
Tavernelli
, “One-particle Green’s functions from the quantum equation of motion algorithm
,” Phys. Rev. Res.
4
, 043011
(2022
).35.
E. A.
Martinez
, C. A.
Muschik
, P.
Schindler
, D.
Nigg
, A.
Erhard
, M.
Heyl
, P.
Hauke
, M.
Dalmonte
, T.
Monz
, P.
Zoller
, and R.
Blatt
, “Real-time dynamics of lattice gauge theories with a few-qubit quantum computer
,” Nature
534
(7608
), 516
–519
(2016
).36.
N.
Klco
, E. F.
Dumitrescu
, A. J.
McCaskey
, T. D.
Morris
, R. C.
Pooser
, M.
Sanz
, E.
Solano
, P.
Lougovski
, and M. J.
Savage
, “Quantum-classical computation of Schwinger model dynamics using quantum computers
,” Phys. Rev. A
98
(3
), 032331
(2018
).37.
A.
Roggero
and J.
Carlson
, “Dynamic linear response quantum algorithm
,” Phys. Rev. C
100
, 034610
(2019
).38.
S. V.
Mathis
, G.
Mazzola
, and I.
Tavernelli
, “Toward scalable simulations of lattice gauge theories on quantum computers
,” Phys. Rev. D
102
, 094501
(2020
).39.
G.
Mazzola
, S. V.
Mathis
, G.
Mazzola
, and I.
Tavernelli
, “Gauge-invariant quantum circuits for U(1) and Yang-Mills lattice gauge theories
,” Phys. Rev. Res.
3
, 043209
(2021
).40.
S. L.
Wu
, S.
Sun
, W.
Guan
, C.
Zhou
, J.
Chan
, C. L.
Cheng
, T.
Pham
, Y.
Qian
, A. Z.
Wang
, R.
Zhang
, M.
Livny
, J.
Glick
, P. K.
Barkoutsos
, S.
Woerner
, I.
Tavernelli
, F.
Carminati
, A.
Di Meglio
, A. C. Y.
Li
, J.
Lykken
, P.
Spentzouris
, S. Y.-C.
Chen
, S.
Yoo
, and T.-C.
Wei
, “Application of quantum machine learning using the quantum kernel algorithm on high energy physics analysis at the LHC
,” Phys. Rev. Res.
3
, 033221
(2021
).41.
G.
Clemente
, A.
Crippa
, and K.
Jansen
, “Strategies for the determination of the running coupling of (2 + 1)-dimensional QED with quantum computing
,” Phys. Rev. D
106
, 114511
(2022
).42.
W.
Guan
, G.
Perdue
, A.
Pesah
, M.
Schuld
, K.
Terashi
, S.
Vallecorsa
, and J.-R.
Vlimant
, “Quantum machine learning in high energy physics
,” Mach. Learn.: Sci. Technol.
2
, 011003
(2021
).43.
J.
Schuhmacher
, L.
Boggia
, V.
Belis
, E.
Puljak
, M.
Grossi
, M.
Pierini
, S.
Vallecorsa
, F.
Tacchino
, P.
Barkoutsos
, and I.
Tavernelli
, “Unravelling physics beyond the standard model with classical and quantum anomaly detection
,” arXiv:2301.10787 (2023
).44.
K.
Anna Woźniak
, V.
Belis
, E.
Puljak
, P.
Barkoutsos
, G.
Dissertori
, M.
Grossi
, M.
Pierini
, F.
Reiter
, I.
Tavernelli
, and S.
Vallecorsa
, “Quantum anomaly detection in the latent space of proton collision events at the LHC
,” arXiv:2301.10780 (2023
).45.
V.
Belis
, S.
González-Castillo
, C.
Reissel
, S.
Vallecorsa
, E. F.
Combarro
, G.
Dissertori
, and F.
Reiter
, “Higgs analysis with quantum classifiers
,” EPJ Web Conf.
251
, 03070
(2021
).46.
P. J. J.
O’Malley
, R.
Babbush
, I. D.
Kivlichan
, J.
Romero
, J. R.
McClean
, R.
Barends
, J.
Kelly
, P.
Roushan
, A.
Tranter
et al, “Scalable quantum simulation of molecular energies
,” Phys. Rev. X
6
, 031007
(2016
).47.
A.
Kandala
, A.
Mezzacapo
, K.
Temme
, M.
Takita
, M.
Brink
, J. M.
Chow
, and J. M.
Gambetta
, “Hardware-efficient variational quantum eigensolver for small molecules and quantum magnets
,” Nature
549
(7671
), 242
–246
(2017
).48.
P. K.
Barkoutsos
, J. F.
Gonthier
, I.
Sokolov
, N.
Moll
, G.
Salis
, A.
Fuhrer
, M.
Ganzhorn
, D. J.
Egger
, M.
Troyer
, A.
Mezzacapo
, S.
Filipp
, and I.
Tavernelli
, “Quantum algorithms for electronic structure calculations: Particle-hole Hamiltonian and optimized wave-function expansions
,” Phys. Rev. A
98
(2
), 022322
(2018
).49.
Y.
Cao
, J.
Romero
, J. P.
Olson
, M.
Degroote
, P. D.
Johnson
, M.
Kieferová
, I. D.
Kivlichan
, T.
Menke
, B.
Peropadre
, N. P. D.
Sawaya
et al, “Quantum chemistry in the age of quantum computing
,” Chem. Rev.
119
(19
), 10856
–10915
(2019
).50.
S.
McArdle
, S.
Endo
, A.
Aspuru-Guzik
, S. C.
Benjamin
, and X.
Yuan
, “Quantum computational chemistry
,” Rev. Mod. Phys.
92
(1
), 015003
(2020
).51.
I. O.
Sokolov
, P. K.
Barkoutsos
, P. J.
Ollitrault
, D.
Greenberg
, J.
Rice
, M.
Pistoia
, and I.
Tavernelli
, “Quantum orbital-optimized unitary coupled cluster methods in the strongly correlated regime: Can quantum algorithms outperform their classical equivalents?
,” J. Chem. Phys.
152
(12
), 124107
(2020
).52.
P. J.
Ollitrault
, A.
Kandala
, C.-F.
Chen
, P. K.
Barkoutsos
, A.
Mezzacapo
, M.
Pistoia
, S.
Sheldon
, S.
Woerner
, J. M.
Gambetta
, and I.
Tavernelli
, “Quantum equation of motion for computing molecular excitation energies on a noisy quantum processor
,” Phys. Rev. Res.
2
(4
), 043140
(2020
).53.
I. O.
Sokolov
, P. K.
Barkoutsos
, L.
Moeller
, P.
Suchsland
, G.
Mazzola
, and I.
Tavernelli
, “Microcanonical and finite-temperature ab initio molecular dynamics simulations on quantum computers
,” Phys. Rev. Res.
3
, 013125
(2021
).54.
O.
Kiss
, F.
Tacchino
, S.
Vallecorsa
, and I.
Tavernelli
, “Quantum neural networks force fields generation
,” Mach. Learn.: Sci. Technol.
3
(3
), 035004
(2022
).55.
P. K.
Barkoutsos
, F.
Gkritsis
, P. J.
Ollitrault
, I. O.
Sokolov
, S.
Woerner
, and I.
Tavernelli
, “Quantum algorithm for alchemical optimization in material design
,” Chem. Sci.
12
(12
), 4345
–4352
(2021
).56.
P. J.
Ollitrault
, A.
Miessen
, and I.
Tavernelli
, “Molecular quantum dynamics: A quantum computing perspective
,” Acc. Chem. Res.
54
(23
), 4229
–4238
(2021
).57.
M.
Rossmannek
, P. K.
Barkoutsos
, P. J.
Ollitrault
, and I.
Tavernelli
, “Quantum HF/DFT-embedding algorithms for electronic structure calculations: Scaling up to complex molecular systems
,” J. Chem. Phys.
154
(11
), 114105
(2021
).58.
H.
Ma
, M.
Govoni
, and G.
Galli
, “Quantum simulations of material on near-term quantum computers
,” Npj Comput. Mater.
6
, 85
(2020
).59.
A.
Robert
, P. K.
Barkoutsos
, S.
Woerner
, and I.
Tavernelli
, “Resource-efficient quantum algorithm for protein folding
,” npj Quantum Inf.
7
(1
), 38
(2021
).60.
S.
Mensa
, E.
Sahin
, F.
Tacchino
, P. K.
Barkoutsos
, and I.
Tavernelli
, “Quantum machine learning framework for virtual screening in drug discovery: A prospective quantum advantage
,” arXiv:2204.04017 (2022
).61.
A.
Baiardi
, M.
Christandl
, and M.
Reiher
, “Quantum computing for molecular biology
,” arXiv:2212.12220 (2022
).62.
S.
Maniscalco
, E.-M.
Borrelli
, D.
Cavalcanti
, C.
Foti
, A.
Glos
, M.
Goldsmith
, S.
Knecht
, K.
Korhonen
, J.
Malmi
, A.
Nykänen
et al, “Quantum network medicine: Rethinking medicine with network science and quantum algorithms
,” arXiv:2206.12405 (2022
).63.
I.
Kassal
, S. P.
Jordan
, P. J.
Love
, M.
Mohseni
, and A.
Aspuru-Guzik
, “Polynomial-time quantum algorithm for the simulation of chemical dynamics
,” Proc. Natl. Acad. Sci. U. S. A.
105
(48
), 18681
–18686
(2008
).64.
L.
Veis
, J.
Višňák
, H.
Nishizawa
, H.
Nakai
, and J.
Pittner
, “Quantum chemistry beyond Born–Oppenheimer approximation on a quantum computer: A simulated phase estimation study
,” Int. J. Quantum Chem.
116
, 1328
–1336
(2016
).65.
P. J.
Ollitrault
, G.
Mazzola
, and I.
Tavernelli
, “Nonadiabatic molecular quantum dynamics with quantum computers
,” Phys. Rev. Lett.
125
, 260511
(2020
).66.
F.
Pavošević
and S.
Hammes-Schiffer
, “Multicomponent unitary coupled cluster and equation-of-motion for quantum computation
,” J. Chem. Theory Comput.
17
(6
), 3252
–3258
(2021
).67.
A.
Peruzzo
, J.
McClean
, P.
Shadbolt
, M.-H.
Yung
, X.-Q.
Zhou
, P. J.
Love
, A.
Aspuru-Guzik
, and J. L.
O’Brien
, “A variational eigenvalue solver on a photonic quantum processor
,” Nat. Commun.
5
, 4213
(2014
).68.
N.
Moll
, P.
Barkoutsos
, L. S.
Bishop
, J. M.
Chow
, A.
Cross
, D. J.
Egger
, S.
Filipp
, A.
Fuhrer
, J. M.
Gambetta
, M.
Ganzhorn
, A.
Kandala
, A.
Mezzacapo
, P.
Müller
, W.
Riess
, G.
Salis
, J.
Smolin
, I.
Tavernelli
, and K.
Temme
, “Quantum optimization using variational algorithms on near-term quantum devices
,” Quantum Sci.Technol.
3
, 030503
(2018
).69.
A.
Choquette
, A.
Di Paolo
, P. K.
Barkoutsos
, D.
Sénéchal
, I.
Tavernelli
, and A.
Blais
, “Quantum-optimal-control-inspired ansatz for variational quantum algorithms
,” Phys. Rev. Res
3
, 023092
(2021
).70.
J. T.
Seeley
, M. J.
Richard
, and P. J.
Love
, “The Bravyi-Kitaev transformation for quantum computation of electronic structure
,” J. Chem. Phys.
137
(22
), 224109
(2012
).71.
P.
Suchsland
, F.
Tacchino
, M. H.
Fischer
, T.
Neupert
, P. Kl.
Barkoutsos
, and I.
Tavernelli
, “Algorithmic error mitigation scheme for current quantum processors
,” Quantum
5
, 492
(2021
).72.
J.
Preskill
, “Quantum computing in the NISQ era and beyond
,” Quantum
2
, 79
(2018
).73.
J. R.
McClean
, S.
Boixo
, V. N.
Smelyanskiy
, R.
Babbush
, and H.
Neven
, “Barren plateaus in quantum neural network training landscapes
,” Nat. Commun.
9
(1
), 4812
(2018
).74.
Qiskit, Qiskit: An open-source framework for quantum computing (
2021
); http://www.qiskit.org.75.
S.
Wang
, E.
Fontana
, M.
Cerezo
, K.
Sharma
, A.
Sone
, L.
Cincio
, and P. J.
Coles
, “Noise-induced barren plateaus in variational quantum algorithms
,” Nat. Commun.
12
(1
), 6961
(2021
).76.
M.
Skogh
, O.
Leinonen
, P.
Lolur
, and M.
Rahm
, “Accelerating variational quantum eigensolver convergence using parameter transfer
” (unpublished) (2023
).77.
G. M. J.
Barca
, C.
Bertoni
, L.
Carrington
, D.
Datta
, N.
De Silva
, J. E.
Deustua
, D. G.
Fedorov
, J. R.
Gour
, A. O.
Gunina
, E.
Guidez
, T.
Harville
, S.
Irle
, J.
Ivanic
, K.
Kowalski
, S. S.
Leang
, H.
Li
, W.
Li
, J. J.
Lutz
, I.
Magoulas
, J.
Mato
, V.
Mironov
, H.
Nakata
, B. Q.
Pham
, P.
Piecuch
, D.
Poole
, S. R.
Pruitt
, A. P.
Rendell
, L. B.
Roskop
, K.
Ruedenberg
, T.
Sattasathuchana
, M. W.
Schmidt
, J.
Shen
, L.
Slipchenko
, M.
Sosonkina
, V.
Sundriyal
, A.
Tiwari
, J. L.
Galvez Vallejo
, B.
Westheimer
, M.
Włoch
, P.
Xu
, F.
Zahariev
, and M. S.
Gordon
, “Recent developments in the general atomic and molecular electronic structure system
,” J. Chem. Phys.
152
(15
), 154102
(2020
).78.
P.
Virtanen
, R.
Gommers
, T. E.
Oliphant
, M.
Haberland
, et al, “SciPy 1.0: Fundamental algorithms for scientific computing in Python
,” Nat. Methods
17
(3
), 261
–272
(2020
).79.
M. J. D.
Powell
, “A direct search optimization method that models the objective and constraint functions by linear interpolation
,” in Advances in Optimization and Numerical Analysis
(Springer
, 1994
), pp. 51
–67
.80.
M. R.
Hestenes
and E.
Stiefel
, “Methods of conjugate gradients for solving linear systems
,” J. Res. Natl. Bur. Stand.
49
(6
), 409
(1952
).81.
J. L.
Nazareth
, “Conjugate gradient method
,” Wiley Interdiscip. Rev.: Comput. Stat.
1
(3
), 348
–353
(2009
).82.
D.
Kraft
, “Algorithm 733: TOMP–Fortran modules for optimal control calculations
,” ACM Trans. Math. Software
20
(3
), 262
–281
(1994
).83.
S.
Bravyi
, J. M.
Gambetta
, A.
Mezzacapo
, and K.
Temme
, “Tapering off qubits to simulate fermionic Hamiltonians
,” arXiv:1701.08213 (2017
).84.
R.
Ditchfield
, W. J.
Hehre
, and J. A.
Pople
, “Self-consistent molecular-orbital methods. IX. An extended Gaussian-type basis for molecular-orbital studies of organic molecules
,” J. Chem. Phys.
54
(2
), 724
–728
(1971
).85.
K. P.
Huber
and G.
Herzberg
, Molecular Spectra and Molecular Structure. IV. Constants of Diatomic Molecules
(National Institute of Standards and Technology
, 1979
).86.
T.
Iordanov
and S.
Hammes-Schiffer
, “Vibrational analysis for the nuclear-electronic orbital method
,” J. Chem. Phys.
118
(21
), 9489
–9496
(2003
).87.
A. C.
Hurley
, “The computation of floating functions and their use in force constant calculations
,” J. Comput. Chem.
9
(1
), 75
–79
(1988
).88.
M.
Tachikawa
, K.
Taneda
, and K.
Mori
, “Simultaneous optimization of GTF exponents and their centers with fully variational treatment of Hartree–Fock molecular orbital calculation
,” Int. J. Quantum Chem.
75
(4–5
), 497
–510
(1999
).89.
H.
Nakai
, K.
Sodeyama
, and M.
Hoshino
, “Non-Born–Oppenheimer theory for simultaneous determination of vibrational and electronic excited states: Ab initio NO + MO/CIS theory
,” Chem. Phys. Lett.
345
(1–2
), 118
–124
(2001
).90.
H.
Nakai
, M.
Hoshino
, K.
Miyamoto
, and S.
Hyodo
, “Elimination of translational and rotational motions in nuclear orbital plus molecular orbital theory
,” J. Chem. Phys.
122
(16
), 164101
(2005
).91.
K. R.
Brorsen
, “Quantifying multireference character in multicomponent systems with heat-bath configuration interaction
,” J. Chem. Theory Comput.
16
(4
), 2379
–2388
(2020
).92.
F.
Pavošević
, T.
Culpitt
, and S.
Hammes-Schiffer
, “Multicomponent quantum chemistry: Integrating electronic and nuclear quantum effects via the nuclear-electronic orbital method
,” Chem. Rev.
120
(9
), 4222
–4253
(2020
).93.
M.
Tachikawa
and Y.
Osamura
, “Isotope effect of hydrogen and lithium hydride molecules. Application of the dynamic extended molecular orbital method and energy component analysis
,” Theor. Chem. Acc.
104
(1
), 29
–39
(2000
).94.
W.
Kołos
and L.
Wolniewicz
, “Nonadiabatic theory for diatomic molecules and its application to the hydrogen molecule
,” Rev. Mod. Phys.
35
, 473
–483
(1963
).95.
A.
Muolo
, E.
Mátyus
, and M.
Reiher
, “as a five-body problem described with explicitly correlated Gaussian basis sets
,” J. Chem. Phys.
151
(15
), 154110
(2019
).96.
Y.
Wang
, B. J.
Braams
, J. M.
Bowman
, S.
Carter
, and D. P.
Tew
, “Full-dimensional quantum calculations of ground-state tunneling splitting of malonaldehyde using an accurate ab initio potential energy surface
,” J. Chem. Phys.
128
(22
), 224314
(2008
).97.
N. H.
List
, A. L.
Dempwolff
, A.
Dreuw
, P.
Norman
, and T. J.
Martínez
, “Probing competing relaxation pathways in malonaldehyde with transient X-ray absorption spectroscopy
,” Chem. Sci.
11
, 4180
–4193
(2020
).98.
S. L.
Baughcum
, R. W.
Duerst
, W. F.
Rowe
, Z.
Smith
, and E. B.
Wilson
, “Microwave spectroscopic study of malonaldehyde (3-hydroxy-2-propenal). 2. Structure, dipole moment, and tunneling
,” J. Am. Chem. Soc.
103
(21
), 6296
–6303
(1981
).99.
E. M.
Fluder
and J. R.
De la Vega
, “Intramolecular hydrogen tunneling in malonaldehyde
,” J. Am. Chem. Soc.
100
(17
), 5265
–5267
(1977
).100.
C.
Møller
and M. S.
Plesset
, “Note on an approximation treatment for many-electron systems
,” Phys. Rev.
46
(7
), 618
–622
(1934
).101.
Y.
Yang
, P. E.
Schneider
, T.
Culpitt
, F.
Pavošević
, and S.
Hammes-Schiffer
, “Molecular vibrational frequencies within the nuclear-electronic orbital framework
,” J. Phys. Chem. Lett.
10
(6
), 1167
–1172
(2019
).102.
Z. B.
Charles
, M.
Farber
, C. R.
Johnson
, and L.
Kennedy-Shaffer
, “The relation between the diagonal entries and the eigenvalues of a symmetric matrix, based upon the sign pattern of its off-diagonal entries
,” Linear Algebra Appl.
438
, 1427
–1445
(2013
).103.
Q.
Yu
and S.
Hammes-Schiffer
, “Nuclear-electronic orbital multistate density functional theory
,” J. Phys. Chem. Lett.
11
(23
), 10106
–10113
(2020
).104.
Ö.
Legeza
, F.
Gebhard
, and J.
Rissler
, “Entanglement production by independent quantum channels
,” Phys. Rev. B
74
, 195112
(2006
).105.
K.
Boguslawski
, P.
Tecmer
, Ö.
Legeza
, and M.
Reiher
, “Entanglement measures for single- and multireference correlation effects
,” J. Phys. Chem. Lett.
3
, 3129
–3135
(2012
).106.
R.
Feldmann
, A.
Muolo
, A.
Baiardi
, and M.
Reiher
, “Quantum proton effects from density matrix renormalization group calculations
,” J. Chem. Theory Comput.
18
(1
), 234
–250
(2022
).107.
K.
Boguslawski
and P.
Tecmer
, “Orbital entanglement in quantum chemistry
,” Int. J. Quantum Chem.
115
, 1289
–1295
(2015
).© 2023 Author(s). Published under an exclusive license by AIP Publishing.
2023
Author(s)
You do not currently have access to this content.
