DOI: 10.1063/1.3671386. Quantum entanglement between electronic and vibrational degrees of freedom in molecules

Quantum entanglement between electronic and vibrational degrees of freedom in molecules

10.1063/1.3671386

Crossref journal-article

AIP Publishing

The Journal of Chemical Physics (317)

Abstract

We consider the quantum entanglement of the electronic and vibrational degrees of freedom in molecules with tendencies towards double welled potentials. In these bipartite systems, the von Neumann entropy of the reduced density matrix is used to quantify the electron-vibration entanglement for the lowest two vibronic wavefunctions obtained from a model Hamiltonian based on coupled harmonic diabatic potential-energy surfaces. Significant entanglement is found only in the region in which the ground vibronic state contains a density profile that is bimodal (i.e., contains two separate local maxima). However, in this region two distinct types of density and entanglement profiles are found: one type arises purely from the degeneracy of energy levels in the two potential wells and is destroyed by slight asymmetry, while the other arises through strong interactions between the diabatic levels of each well and is relatively insensitive to asymmetry. These two distinct types are termed fragile degeneracy-induced entanglement and persistent entanglement, respectively. Six classic molecular systems describable by two diabatic states are considered: ammonia, benzene, BNB, pyridine excited triplet states, the Creutz-Taube ion, and the radical cation of the “special pair” of chlorophylls involved in photosynthesis. These chemically diverse systems are all treated using the same general formalism and the nature of the entanglement that they embody is elucidated.

Bibliography

McKemmish, L. K., McKenzie, R. H., Hush, N. S., & Reimers, J. R. (2011). Quantum entanglement between electronic and vibrational degrees of freedom in molecules. The Journal of Chemical Physics, 135(24).

Authors 4

Laura K. McKemmish (first)
Ross H. McKenzie (additional)
Noel S. Hush (additional)
Jeffrey R. Reimers (additional)

References 105 Referenced 57

10.1016/S1381-1169(03)00034-7 / J. Mol. Catal. A (2003)
{'key': '2023080407445799100_c1b', 'first-page': '505', 'volume': '2', 'year': '1935', 'journal-title': 'Transl. Acta Physicochim. URSS'} / Transl. Acta Physicochim. URSS (1935)
10.1002/pol.1953.120110401 / J. Polym. Sci. (1953)
{'key': '2023080407445799100_c3', 'first-page': '423', 'volume': '19', 'year': '1957', 'journal-title': 'Trans. N. Y. Acad. Sci.'} / Trans. N. Y. Acad. Sci. (1957)
10.1063/1.1744305 / J. Chem. Phys. (1958)
10.1039/df9602900021 / Discuss. Faraday Soc. (1960)
10.1039/tf9615700557 / Trans. Faraday Soc. (1961)
10.1016/0304-4173(85)90014-x / Biochim. Biophys. Acta (1985)
10.1021/ba-1990-0226 / Electron Transfer in Biology and the Solid State by Johnson (1989)
10.1142/3347 / Electron and Ion Transfer in Condensed Media by Kornyshev (1997)
10.1021/ba-1997-0253 / Electron Transfer Reactions by Isied (1997)
{'volume-title': 'Electron Transfer in Chemistry and Biology', 'year': '1999', 'author': 'Kutnetsov', 'key': '2023080407445799100_c11'} / Electron Transfer in Chemistry and Biology by Kutnetsov (1999)
10.1002/9783527618248 / Electron Transfer in Chemistry. Principles, Theories, Methods and Techniques by Balzani (2001)
10.1146/annurev.physchem.52.1.681 / Annu. Rev. Phys. Chem. (2001)
10.1039/b106962b / Chem. Soc. Rev. (2002)
10.1021/jp0124490 / J. Phys. Chem. A (2002)
10.1021/ic060847h / Inorg. Chem. (2006)
10.1016/S0065-3160(06)41003-0 / Adv. Phys. Org. Chem. (2006)
10.1021/cr050140x / Chem. Rev. (2007)
10.1016/S0009-2614(97)00606-4 / Chem. Phys. Lett. (1997)
10.1103/PhysRevLett.103.023202 / Phys. Rev. Lett. (2009)
10.1063/1.1681573 / J. Chem. Phys. (1974)
10.1016/0301-0104(81)80198-X / Chem. Phys. (1981)
10.1002/9780470142813.ch2 / Adv. Chem. Phys. (2007)
{'volume-title': 'The Jahn-Teller Effect in Molecules and Crystals', 'year': '1972', 'key': '2023080407445799100_c24'} / The Jahn-Teller Effect in Molecules and Crystals (1972)
{'volume-title': 'Vibronic Coupling', 'year': '1984', 'key': '2023080407445799100_c25'} / Vibronic Coupling (1984)
10.1021/cr0004411 / Chem. Rev. (2001)
{'volume-title': 'Quantum Computation and Quantum Information', 'year': '2000', 'key': '2023080407445799100_c27'} / Quantum Computation and Quantum Information (2000)
10.1103/PhysRevA.72.022309 / Phys. Rev. A (2005)
10.1103/PhysRevA.71.034302 / Phys. Rev. A (2005)
10.1103/RevModPhys.81.865 / Rev. Mod. Phys. (2009)
10.1103/PhysRevA.70.022303 / Phys. Rev. A (2004)
10.1063/1.3505217 / J. Chem. Phys. (2010)
10.1016/0301-0104(75)87049-2 / Chem. Phys. (1975)
10.1021/ja01042a072 / J. Am. Chem. Soc. (1969)
10.1098/rsta.2007.2136 / Philos. Trans. R. Soc. A (2008)
10.1038/320615a0 / Nature (London) (1986)
10.1016/S0065-2792(08)60179-X / Adv. Inorg. Chem. Radiochem. (1967)
10.1098/rsta.2007.2135 / Philos. Trans. R. Soc. A (2008)
10.1021/jp000962s / J. Phys. Chem. A (2000)
10.1063/1.1589742 / J. Chem. Phys. (2003)
10.1002/SERIES2229 / Prog. Inorg. Chem. (1967)
10.1103/PhysRevA.78.033833 / Phys. Rev. A (2008)
10.1103/PhysRevA.81.043805 / Phys. Rev. A (2010)
{'key': '2023080407445799100_c44', 'first-page': '579', 'volume': '24', 'year': '2007', 'journal-title': 'Liangzi Dianzi Xuebao'} / Liangzi Dianzi Xuebao (2007)
10.1088/1674-1056/18/11/046 / Chin. Phys. B (2009)
10.1088/1674-1056/18/12/044 / Chin. Phys. B (2009)
10.1038/nphys1730 / Nat. Phys. (2010)
10.1039/tf9565201163 / Trans. Faraday Soc. (1956)
10.1021/ja00732a006 / J. Am. Chem. Soc. (1971)
10.1007/BF00549666 / Theor. Chim. Acta (1984)
10.1016/0009-2614(75)85236-5 / Chem. Phys. Lett. (1975)
10.1063/1.432571 / J. Chem. Phys. (1976)
10.1063/1.440886 / J. Chem. Phys. (1981)
10.1016/0301-0104(77)89052-6 / Chem. Phys. (1977)
10.1021/jp064885y / J. Phys. Chem. A (2006)
See supplementary material at http://dx.doi.org/10.1063/1.3671386 for the derivation of Eq. (15) and a description of the 3-state and 5-state extended vibronic coupling models for closed-shell reactions, their reduction to effective 2-state models, parameter determinations for ammonia and benzene based on experimental data, and the differences between the entanglements manifest by the full models and their effective 2-state approximations.
10.1063/1.1695698 / J. Chem. Phys. (1965)
10.1002/andp.19273892002 / Ann. Phys. (1927)
10.1103/PhysRevA.73.032346 / Phys. Rev. A (2006)
10.1103/PhysRevLett.89.247902 / Phys. Rev. Lett. (2002)
10.1038/nature03350 / Nature (London) (2005)
10.1103/PhysRevA.65.034303 / Phys. Rev. A (2002)
10.1103/PhysRevA.73.012109 / Phys. Rev. A (2006)
10.1063/1.2236111 / J. Chem. Phys. (2006)
10.1016/j.physleta.2007.03.018 / Phys. Lett. A (2007)
10.1103/PhysRevA.75.022315 / Phys. Rev. A (2007)
10.1103/PhysRevA.78.042323 / Phys. Rev. A (2008)
{'volume-title': 'Molecular Vibrations: The Theory of Infrared and Raman Vibrational Spectra', 'year': '1955', 'key': '2023080407445799100_c68'} / Molecular Vibrations: The Theory of Infrared and Raman Vibrational Spectra (1955)
10.1016/j.chemphys.2003.12.010 / Chem. Phys. (2004)
10.1021/cr990413m / Chem. Rev. (2001)
10.1098/rspa.1937.0142 / Proc. R. Soc. A (1937)
{'volume-title': 'The Jahn-Teller Effect and Vibronic Interactions in Modern Chemistry', 'year': '1984', 'key': '2023080407445799100_c72'} / The Jahn-Teller Effect and Vibronic Interactions in Modern Chemistry (1984)
10.1103/PhysRevA.68.034301 / Phys. Rev. A (2003)
10.1103/PhysRevLett.98.220401 / Phys. Rev. Lett. (2007)
10.1088/0953-8984/18/32/015 / J. Phys.: Condens. Matter (2006)
10.1088/0953-8984/20/23/235203 / J. Phys.: Condens. Matter (2008)
10.1016/j.physleta.2005.06.064 / Phys. Lett. A (2005)
10.1103/PhysRevLett.92.073602 / Phys. Rev. Lett. (2004)
10.1103/PhysRevB.69.113203 / Phys. Rev. B (2004)
10.1103/PhysRevA.71.062302 / Phys. Rev. A (2005)
10.1016/j.physleta.2006.01.028 / Phys. Lett. A (2006)
10.1140/epjd/e2006-00103-6 / Eur. Phys. J. D (2006)
10.1103/PhysRevE.77.051102 / Phys. Rev. E (2008)
10.1103/PhysRevA.78.032523 / Phys. Rev. A (2008)
10.1103/PhysRevE.77.026213 / Phys. Rev. E (2008)
10.1103/PhysRevA.77.063815 / Phys. Rev. A (2008)
10.1016/j.physleta.2008.06.080 / Phys. Lett. A (2008)
10.1103/PhysRevA.79.022308 / Phys. Rev. A (2009)
10.1088/1751-8113/42/7/075301 / J. Phys. A (2009)
10.1103/PhysRevB.80.214301 / Phys. Rev. B (2009)
10.1103/PhysRevA.82.025802 / Phys. Rev. A (2010)
10.1103/PhysRev.47.777 / Phys. Rev. (1935)
{'volume-title': 'Quantum Theory', 'year': '1951', 'key': '2023080407445799100_c93'} / Quantum Theory (1951)
10.1103/PhysRev.108.1070 / Phys. Rev. (1957)
10.1103/PhysRev.97.1509 / Phys. Rev. (1955)
10.1021/ja00800a064 / J. Am. Chem. Soc. (1973)
10.1063/1.1701290 / J. Chem. Phys. (1962)
10.1016/0301-0104(78)85059-9 / Chem. Phys. (1978)
10.1016/0301-0104(78)85058-7 / Chem. Phys. (1978)
10.1016/0301-0104(79)85216-7 / Chem. Phys. (1979)
10.1016/0301-0104(89)87167-8 / Chem. Phys. (1989)
10.1063/1.457771 / J. Chem. Phys. (1990)
10.1063/1.439563 / J. Chem. Phys. (1980)
10.1063/1.1569910 / J. Chem. Phys. (2003)

Dates

Type	When
Created	13 years, 7 months ago (Jan. 3, 2012, 4:08 p.m.)
Deposited	2 years ago (Aug. 4, 2023, 3:47 a.m.)
Indexed	3 weeks, 6 days ago (July 30, 2025, 6:57 a.m.)
Issued	13 years, 7 months ago (Dec. 28, 2011)
Published	13 years, 7 months ago (Dec. 28, 2011)
Published Online	13 years, 7 months ago (Dec. 30, 2011)
Published Print	13 years, 7 months ago (Dec. 28, 2011)

Funders 0

None

BibTeX

@article{McKemmish_2011, title={Quantum entanglement between electronic and vibrational degrees of freedom in molecules}, volume={135}, ISSN={1089-7690}, url={http://dx.doi.org/10.1063/1.3671386}, DOI={10.1063/1.3671386}, number={24}, journal={The Journal of Chemical Physics}, publisher={AIP Publishing}, author={McKemmish, Laura K. and McKenzie, Ross H. and Hush, Noel S. and Reimers, Jeffrey R.}, year={2011}, month=dec }

JSON

{
  "indexed": {
    "date-parts": [
      [
        2025,
        7,
        30
      ]
    ],
    "date-time": "2025-07-30T10:57:09Z",
    "timestamp": 1753873029880,
    "version": "3.41.2"
  },
  "reference-count": 105,
  "publisher": "AIP Publishing",
  "issue": "24",
  "content-domain": {
    "domain": [
      "pubs.aip.org"
    ],
    "crossmark-restriction": true
  },
  "published-print": {
    "date-parts": [
      [
        2011,
        12,
        28
      ]
    ]
  },
  "abstract": "<jats:p>We consider the quantum entanglement of the electronic and vibrational degrees of freedom in molecules with tendencies towards double welled potentials. In these bipartite systems, the von Neumann entropy of the reduced density matrix is used to quantify the electron-vibration entanglement for the lowest two vibronic wavefunctions obtained from a model Hamiltonian based on coupled harmonic diabatic potential-energy surfaces. Significant entanglement is found only in the region in which the ground vibronic state contains a density profile that is bimodal (i.e., contains two separate local maxima). However, in this region two distinct types of density and entanglement profiles are found: one type arises purely from the degeneracy of energy levels in the two potential wells and is destroyed by slight asymmetry, while the other arises through strong interactions between the diabatic levels of each well and is relatively insensitive to asymmetry. These two distinct types are termed fragile degeneracy-induced entanglement and persistent entanglement, respectively. Six classic molecular systems describable by two diabatic states are considered: ammonia, benzene, BNB, pyridine excited triplet states, the Creutz-Taube ion, and the radical cation of the \u201cspecial pair\u201d of chlorophylls involved in photosynthesis. These chemically diverse systems are all treated using the same general formalism and the nature of the entanglement that they embody is elucidated.</jats:p>",
  "DOI": "10.1063/1.3671386",
  "type": "journal-article",
  "created": {
    "date-parts": [
      [
        2012,
        1,
        3
      ]
    ],
    "date-time": "2012-01-03T21:08:40Z",
    "timestamp": 1325624920000
  },
  "update-policy": "https://doi.org/10.1063/aip-crossmark-policy-page",
  "source": "Crossref",
  "is-referenced-by-count": 57,
  "title": "Quantum entanglement between electronic and vibrational degrees of freedom in molecules",
  "prefix": "10.1063",
  "volume": "135",
  "author": [
    {
      "given": "Laura K.",
      "family": "McKemmish",
      "sequence": "first",
      "affiliation": [
        {
          "name": "The University of Sydney 1 School of Chemistry, , NSW 2006, Australia"
        }
      ]
    },
    {
      "given": "Ross H.",
      "family": "McKenzie",
      "sequence": "additional",
      "affiliation": [
        {
          "name": "The University of Queensland 2 School of Mathematics and Physics, , QLD 4072, Australia"
        }
      ]
    },
    {
      "given": "Noel S.",
      "family": "Hush",
      "sequence": "additional",
      "affiliation": [
        {
          "name": "The University of Sydney 3 School of Molecular Biosciences and School of Chemistry, , NSW 2006, Australia"
        }
      ]
    },
    {
      "given": "Jeffrey R.",
      "family": "Reimers",
      "sequence": "additional",
      "affiliation": [
        {
          "name": "The University of Sydney 1 School of Chemistry, , NSW 2006, Australia"
        }
      ]
    }
  ],
  "member": "317",
  "published-online": {
    "date-parts": [
      [
        2011,
        12,
        30
      ]
    ]
  },
  "reference": [
    {
      "key": "2023080407445799100_c1a",
      "doi-asserted-by": "publisher",
      "first-page": "185",
      "DOI": "10.1016/S1381-1169(03)00034-7",
      "volume": "199",
      "year": "2003",
      "journal-title": "J. Mol. Catal. A"
    },
    {
      "key": "2023080407445799100_c1b",
      "first-page": "505",
      "volume": "2",
      "year": "1935",
      "journal-title": "Transl. Acta Physicochim. URSS"
    },
    {
      "key": "2023080407445799100_c2",
      "doi-asserted-by": "publisher",
      "first-page": "289",
      "DOI": "10.1002/pol.1953.120110401",
      "volume": "11",
      "year": "1953",
      "journal-title": "J. Polym. Sci."
    },
    {
      "key": "2023080407445799100_c3",
      "first-page": "423",
      "volume": "19",
      "year": "1957",
      "journal-title": "Trans. N. Y. Acad. Sci."
    },
    {
      "key": "2023080407445799100_c4",
      "doi-asserted-by": "publisher",
      "first-page": "962",
      "DOI": "10.1063/1.1744305",
      "volume": "28",
      "year": "1958",
      "journal-title": "J. Chem. Phys."
    },
    {
      "key": "2023080407445799100_c5",
      "doi-asserted-by": "publisher",
      "first-page": "21",
      "DOI": "10.1039/df9602900021",
      "year": "1960",
      "journal-title": "Discuss. Faraday Soc."
    },
    {
      "key": "2023080407445799100_c6",
      "doi-asserted-by": "publisher",
      "first-page": "577",
      "DOI": "10.1039/tf9615700557",
      "volume": "57",
      "year": "1961",
      "journal-title": "Trans. Faraday Soc."
    },
    {
      "key": "2023080407445799100_c7",
      "doi-asserted-by": "publisher",
      "first-page": "265",
      "DOI": "10.1016/0304-4173(85)90014-x",
      "volume": "811",
      "year": "1985",
      "journal-title": "Biochim. Biophys. Acta"
    },
    {
      "key": "2023080407445799100_c8",
      "doi-asserted-by": "crossref",
      "DOI": "10.1021/ba-1990-0226",
      "volume-title": "Electron Transfer in Biology and the Solid State",
      "author": "Johnson",
      "year": "1989"
    },
    {
      "key": "2023080407445799100_c9",
      "doi-asserted-by": "crossref",
      "DOI": "10.1142/3347",
      "volume-title": "Electron and Ion Transfer in Condensed Media",
      "author": "Kornyshev",
      "year": "1997"
    },
    {
      "key": "2023080407445799100_c10",
      "doi-asserted-by": "crossref",
      "DOI": "10.1021/ba-1997-0253",
      "volume-title": "Electron Transfer Reactions",
      "author": "Isied",
      "year": "1997"
    },
    {
      "volume-title": "Electron Transfer in Chemistry and Biology",
      "year": "1999",
      "author": "Kutnetsov",
      "key": "2023080407445799100_c11"
    },
    {
      "key": "2023080407445799100_c12",
      "doi-asserted-by": "crossref",
      "DOI": "10.1002/9783527618248",
      "volume-title": "Electron Transfer in Chemistry. Principles, Theories, Methods and Techniques",
      "author": "Balzani",
      "year": "2001"
    },
    {
      "key": "2023080407445799100_c13",
      "doi-asserted-by": "publisher",
      "first-page": "681",
      "DOI": "10.1146/annurev.physchem.52.1.681",
      "volume": "52",
      "year": "2001",
      "journal-title": "Annu. Rev. Phys. Chem."
    },
    {
      "key": "2023080407445799100_c14",
      "doi-asserted-by": "publisher",
      "first-page": "22",
      "DOI": "10.1039/b106962b",
      "volume": "31",
      "year": "2002",
      "journal-title": "Chem. Soc. Rev."
    },
    {
      "key": "2023080407445799100_c15",
      "doi-asserted-by": "publisher",
      "first-page": "1131",
      "DOI": "10.1021/jp0124490",
      "volume": "106",
      "year": "2002",
      "journal-title": "J. Phys. Chem. A"
    },
    {
      "key": "2023080407445799100_c16",
      "doi-asserted-by": "publisher",
      "first-page": "6556",
      "DOI": "10.1021/ic060847h",
      "volume": "45",
      "year": "2006",
      "journal-title": "Inorg. Chem."
    },
    {
      "key": "2023080407445799100_c17",
      "doi-asserted-by": "publisher",
      "first-page": "183",
      "DOI": "10.1016/S0065-3160(06)41003-0",
      "volume": "41",
      "year": "2006",
      "journal-title": "Adv. Phys. Org. Chem."
    },
    {
      "key": "2023080407445799100_c18",
      "doi-asserted-by": "publisher",
      "first-page": "926",
      "DOI": "10.1021/cr050140x",
      "volume": "107",
      "year": "2007",
      "journal-title": "Chem. Rev."
    },
    {
      "key": "2023080407445799100_c19",
      "doi-asserted-by": "publisher",
      "first-page": "407",
      "DOI": "10.1016/S0009-2614(97)00606-4",
      "volume": "273",
      "year": "1997",
      "journal-title": "Chem. Phys. Lett."
    },
    {
      "key": "2023080407445799100_c20",
      "doi-asserted-by": "publisher",
      "first-page": "023202",
      "DOI": "10.1103/PhysRevLett.103.023202",
      "volume": "103",
      "year": "2009",
      "journal-title": "Phys. Rev. Lett."
    },
    {
      "key": "2023080407445799100_c21",
      "doi-asserted-by": "publisher",
      "first-page": "3558",
      "DOI": "10.1063/1.1681573",
      "volume": "60",
      "year": "1974",
      "journal-title": "J. Chem. Phys."
    },
    {
      "key": "2023080407445799100_c22",
      "doi-asserted-by": "publisher",
      "first-page": "187",
      "DOI": "10.1016/0301-0104(81)80198-X",
      "volume": "62",
      "year": "1981",
      "journal-title": "Chem. Phys."
    },
    {
      "key": "2023080407445799100_c23",
      "doi-asserted-by": "publisher",
      "first-page": "59",
      "DOI": "10.1002/9780470142813.ch2",
      "volume": "57",
      "year": "2007",
      "journal-title": "Adv. Chem. Phys."
    },
    {
      "volume-title": "The Jahn-Teller Effect in Molecules and Crystals",
      "year": "1972",
      "key": "2023080407445799100_c24"
    },
    {
      "volume-title": "Vibronic Coupling",
      "year": "1984",
      "key": "2023080407445799100_c25"
    },
    {
      "key": "2023080407445799100_c26",
      "doi-asserted-by": "publisher",
      "first-page": "1067",
      "DOI": "10.1021/cr0004411",
      "volume": "101",
      "year": "2001",
      "journal-title": "Chem. Rev."
    },
    {
      "volume-title": "Quantum Computation and Quantum Information",
      "year": "2000",
      "key": "2023080407445799100_c27"
    },
    {
      "key": "2023080407445799100_c28",
      "doi-asserted-by": "publisher",
      "first-page": "022309",
      "DOI": "10.1103/PhysRevA.72.022309",
      "volume": "72",
      "year": "2005",
      "journal-title": "Phys. Rev. A"
    },
    {
      "key": "2023080407445799100_c29",
      "doi-asserted-by": "publisher",
      "first-page": "034302",
      "DOI": "10.1103/PhysRevA.71.034302",
      "volume": "71",
      "year": "2005",
      "journal-title": "Phys. Rev. A"
    },
    {
      "key": "2023080407445799100_c30",
      "doi-asserted-by": "publisher",
      "first-page": "865",
      "DOI": "10.1103/RevModPhys.81.865",
      "volume": "81",
      "year": "2009",
      "journal-title": "Rev. Mod. Phys."
    },
    {
      "key": "2023080407445799100_c31",
      "doi-asserted-by": "publisher",
      "first-page": "022303",
      "DOI": "10.1103/PhysRevA.70.022303",
      "volume": "70",
      "year": "2004",
      "journal-title": "Phys. Rev. A"
    },
    {
      "key": "2023080407445799100_c32",
      "doi-asserted-by": "publisher",
      "first-page": "174309",
      "DOI": "10.1063/1.3505217",
      "volume": "133",
      "year": "2010",
      "journal-title": "J. Chem. Phys."
    },
    {
      "key": "2023080407445799100_c33",
      "doi-asserted-by": "publisher",
      "first-page": "361",
      "DOI": "10.1016/0301-0104(75)87049-2",
      "volume": "10",
      "year": "1975",
      "journal-title": "Chem. Phys."
    },
    {
      "key": "2023080407445799100_c34",
      "doi-asserted-by": "publisher",
      "first-page": "3988",
      "DOI": "10.1021/ja01042a072",
      "volume": "91",
      "year": "1969",
      "journal-title": "J. Am. Chem. Soc."
    },
    {
      "key": "2023080407445799100_c35",
      "doi-asserted-by": "publisher",
      "first-page": "15",
      "DOI": "10.1098/rsta.2007.2136",
      "volume": "366",
      "year": "2008",
      "journal-title": "Philos. Trans. R. Soc. A"
    },
    {
      "key": "2023080407445799100_c36",
      "doi-asserted-by": "publisher",
      "first-page": "615",
      "DOI": "10.1038/320615a0",
      "volume": "320",
      "year": "1986",
      "journal-title": "Nature (London)"
    },
    {
      "key": "2023080407445799100_c37",
      "doi-asserted-by": "publisher",
      "first-page": "247",
      "DOI": "10.1016/S0065-2792(08)60179-X",
      "volume": "10",
      "year": "1967",
      "journal-title": "Adv. Inorg. Chem. Radiochem."
    },
    {
      "key": "2023080407445799100_c38",
      "doi-asserted-by": "publisher",
      "first-page": "5",
      "DOI": "10.1098/rsta.2007.2135",
      "volume": "366",
      "year": "2008",
      "journal-title": "Philos. Trans. R. Soc. A"
    },
    {
      "key": "2023080407445799100_c39",
      "doi-asserted-by": "publisher",
      "first-page": "8389",
      "DOI": "10.1021/jp000962s",
      "volume": "104",
      "year": "2000",
      "journal-title": "J. Phys. Chem. A"
    },
    {
      "key": "2023080407445799100_c40",
      "doi-asserted-by": "publisher",
      "first-page": "3262",
      "DOI": "10.1063/1.1589742",
      "volume": "119",
      "year": "2003",
      "journal-title": "J. Chem. Phys."
    },
    {
      "key": "2023080407445799100_c41",
      "doi-asserted-by": "publisher",
      "first-page": "391",
      "DOI": "10.1002/SERIES2229",
      "volume": "8",
      "year": "1967",
      "journal-title": "Prog. Inorg. Chem."
    },
    {
      "key": "2023080407445799100_c42",
      "doi-asserted-by": "publisher",
      "first-page": "033833",
      "DOI": "10.1103/PhysRevA.78.033833",
      "volume": "78",
      "year": "2008",
      "journal-title": "Phys. Rev. A"
    },
    {
      "key": "2023080407445799100_c43",
      "doi-asserted-by": "publisher",
      "first-page": "043805",
      "DOI": "10.1103/PhysRevA.81.043805",
      "volume": "81",
      "year": "2010",
      "journal-title": "Phys. Rev. A"
    },
    {
      "key": "2023080407445799100_c44",
      "first-page": "579",
      "volume": "24",
      "year": "2007",
      "journal-title": "Liangzi Dianzi Xuebao"
    },
    {
      "key": "2023080407445799100_c45",
      "doi-asserted-by": "publisher",
      "first-page": "4893",
      "DOI": "10.1088/1674-1056/18/11/046",
      "volume": "18",
      "year": "2009",
      "journal-title": "Chin. Phys. B"
    },
    {
      "key": "2023080407445799100_c46",
      "doi-asserted-by": "publisher",
      "first-page": "4683",
      "DOI": "10.1088/1674-1056/18/12/044",
      "volume": "18",
      "year": "2009",
      "journal-title": "Chin. Phys. B"
    },
    {
      "key": "2023080407445799100_c47",
      "doi-asserted-by": "publisher",
      "first-page": "772",
      "DOI": "10.1038/nphys1730",
      "volume": "6",
      "year": "2010",
      "journal-title": "Nat. Phys."
    },
    {
      "key": "2023080407445799100_c48",
      "doi-asserted-by": "publisher",
      "first-page": "1163",
      "DOI": "10.1039/tf9565201163",
      "volume": "52",
      "year": "1956",
      "journal-title": "Trans. Faraday Soc."
    },
    {
      "key": "2023080407445799100_c49",
      "doi-asserted-by": "publisher",
      "first-page": "593",
      "DOI": "10.1021/ja00732a006",
      "volume": "93",
      "year": "1971",
      "journal-title": "J. Am. Chem. Soc."
    },
    {
      "key": "2023080407445799100_c50",
      "doi-asserted-by": "publisher",
      "first-page": "161",
      "DOI": "10.1007/BF00549666",
      "volume": "66",
      "year": "1984",
      "journal-title": "Theor. Chim. Acta"
    },
    {
      "key": "2023080407445799100_c51",
      "doi-asserted-by": "publisher",
      "first-page": "541",
      "DOI": "10.1016/0009-2614(75)85236-5",
      "volume": "32",
      "year": "1975",
      "journal-title": "Chem. Phys. Lett."
    },
    {
      "key": "2023080407445799100_c52",
      "doi-asserted-by": "publisher",
      "first-page": "3077",
      "DOI": "10.1063/1.432571",
      "volume": "64",
      "year": "1976",
      "journal-title": "J. Chem. Phys."
    },
    {
      "key": "2023080407445799100_c53",
      "doi-asserted-by": "publisher",
      "first-page": "273",
      "DOI": "10.1063/1.440886",
      "volume": "74",
      "year": "1981",
      "journal-title": "J. Chem. Phys."
    },
    {
      "key": "2023080407445799100_c54",
      "doi-asserted-by": "publisher",
      "first-page": "141",
      "DOI": "10.1016/0301-0104(77)89052-6",
      "volume": "23",
      "year": "1977",
      "journal-title": "Chem. Phys."
    },
    {
      "key": "2023080407445799100_c55",
      "doi-asserted-by": "publisher",
      "first-page": "11219",
      "DOI": "10.1021/jp064885y",
      "volume": "110",
      "year": "2006",
      "journal-title": "J. Phys. Chem. A"
    },
    {
      "key": "2023080407445799100_c56",
      "unstructured": "See supplementary material at http://dx.doi.org/10.1063/1.3671386 for the derivation of Eq. (15) and a description of the 3-state and 5-state extended vibronic coupling models for closed-shell reactions, their reduction to effective 2-state models, parameter determinations for ammonia and benzene based on experimental data, and the differences between the entanglements manifest by the full models and their effective 2-state approximations."
    },
    {
      "key": "2023080407445799100_c57",
      "doi-asserted-by": "publisher",
      "first-page": "351",
      "DOI": "10.1063/1.1695698",
      "volume": "42",
      "year": "1965",
      "journal-title": "J. Chem. Phys."
    },
    {
      "key": "2023080407445799100_c58",
      "doi-asserted-by": "publisher",
      "first-page": "457",
      "DOI": "10.1002/andp.19273892002",
      "volume": "84",
      "year": "1927",
      "journal-title": "Ann. Phys."
    },
    {
      "key": "2023080407445799100_c59",
      "doi-asserted-by": "publisher",
      "first-page": "032346",
      "DOI": "10.1103/PhysRevA.73.032346",
      "volume": "73",
      "year": "2006",
      "journal-title": "Phys. Rev. A"
    },
    {
      "key": "2023080407445799100_c60",
      "doi-asserted-by": "publisher",
      "first-page": "247902",
      "DOI": "10.1103/PhysRevLett.89.247902",
      "volume": "89",
      "year": "2002",
      "journal-title": "Phys. Rev. Lett."
    },
    {
      "key": "2023080407445799100_c61",
      "doi-asserted-by": "publisher",
      "first-page": "39",
      "DOI": "10.1038/nature03350",
      "volume": "434",
      "year": "2005",
      "journal-title": "Nature (London)"
    },
    {
      "key": "2023080407445799100_c62",
      "doi-asserted-by": "publisher",
      "first-page": "034303",
      "DOI": "10.1103/PhysRevA.65.034303",
      "volume": "65",
      "year": "2002",
      "journal-title": "Phys. Rev. A"
    },
    {
      "key": "2023080407445799100_c63",
      "doi-asserted-by": "publisher",
      "first-page": "012109",
      "DOI": "10.1103/PhysRevA.73.012109",
      "volume": "73",
      "year": "2006",
      "journal-title": "Phys. Rev. A"
    },
    {
      "key": "2023080407445799100_c64",
      "doi-asserted-by": "publisher",
      "first-page": "064102",
      "DOI": "10.1063/1.2236111",
      "volume": "125",
      "year": "2006",
      "journal-title": "J. Chem. Phys."
    },
    {
      "key": "2023080407445799100_c65",
      "doi-asserted-by": "publisher",
      "first-page": "345",
      "DOI": "10.1016/j.physleta.2007.03.018",
      "volume": "367",
      "year": "2007",
      "journal-title": "Phys. Lett. A"
    },
    {
      "key": "2023080407445799100_c66",
      "doi-asserted-by": "publisher",
      "first-page": "022315",
      "DOI": "10.1103/PhysRevA.75.022315",
      "volume": "75",
      "year": "2007",
      "journal-title": "Phys. Rev. A"
    },
    {
      "key": "2023080407445799100_c67",
      "doi-asserted-by": "publisher",
      "first-page": "042323",
      "DOI": "10.1103/PhysRevA.78.042323",
      "volume": "78",
      "year": "2008",
      "journal-title": "Phys. Rev. A"
    },
    {
      "volume-title": "Molecular Vibrations: The Theory of Infrared and Raman Vibrational Spectra",
      "year": "1955",
      "key": "2023080407445799100_c68"
    },
    {
      "key": "2023080407445799100_c69",
      "doi-asserted-by": "publisher",
      "first-page": "79",
      "DOI": "10.1016/j.chemphys.2003.12.010",
      "volume": "299",
      "year": "2004",
      "journal-title": "Chem. Phys."
    },
    {
      "key": "2023080407445799100_c70",
      "doi-asserted-by": "publisher",
      "first-page": "2655",
      "DOI": "10.1021/cr990413m",
      "volume": "101",
      "year": "2001",
      "journal-title": "Chem. Rev."
    },
    {
      "key": "2023080407445799100_c71",
      "doi-asserted-by": "publisher",
      "first-page": "220",
      "DOI": "10.1098/rspa.1937.0142",
      "volume": "161",
      "year": "1937",
      "journal-title": "Proc. R. Soc. A"
    },
    {
      "volume-title": "The Jahn-Teller Effect and Vibronic Interactions in Modern Chemistry",
      "year": "1984",
      "key": "2023080407445799100_c72"
    },
    {
      "key": "2023080407445799100_c73",
      "doi-asserted-by": "publisher",
      "first-page": "034301",
      "DOI": "10.1103/PhysRevA.68.034301",
      "volume": "68",
      "year": "2003",
      "journal-title": "Phys. Rev. A"
    },
    {
      "key": "2023080407445799100_c74",
      "doi-asserted-by": "publisher",
      "first-page": "220401",
      "DOI": "10.1103/PhysRevLett.98.220401",
      "volume": "98",
      "year": "2007",
      "journal-title": "Phys. Rev. Lett."
    },
    {
      "key": "2023080407445799100_c75",
      "doi-asserted-by": "publisher",
      "first-page": "7669",
      "DOI": "10.1088/0953-8984/18/32/015",
      "volume": "18",
      "year": "2006",
      "journal-title": "J. Phys.: Condens. Matter"
    },
    {
      "key": "2023080407445799100_c76",
      "doi-asserted-by": "publisher",
      "first-page": "235203",
      "DOI": "10.1088/0953-8984/20/23/235203",
      "volume": "20",
      "year": "2008",
      "journal-title": "J. Phys.: Condens. Matter"
    },
    {
      "key": "2023080407445799100_c77",
      "doi-asserted-by": "publisher",
      "first-page": "164",
      "DOI": "10.1016/j.physleta.2005.06.064",
      "volume": "344",
      "year": "2005",
      "journal-title": "Phys. Lett. A"
    },
    {
      "key": "2023080407445799100_c78",
      "doi-asserted-by": "publisher",
      "first-page": "073602",
      "DOI": "10.1103/PhysRevLett.92.073602",
      "volume": "92",
      "year": "2004",
      "journal-title": "Phys. Rev. Lett."
    },
    {
      "key": "2023080407445799100_c79",
      "doi-asserted-by": "publisher",
      "first-page": "113203",
      "DOI": "10.1103/PhysRevB.69.113203",
      "volume": "69",
      "year": "2004",
      "journal-title": "Phys. Rev. B"
    },
    {
      "key": "2023080407445799100_c80",
      "doi-asserted-by": "publisher",
      "first-page": "062302",
      "DOI": "10.1103/PhysRevA.71.062302",
      "volume": "71",
      "year": "2005",
      "journal-title": "Phys. Rev. A"
    },
    {
      "key": "2023080407445799100_c81",
      "doi-asserted-by": "publisher",
      "first-page": "60",
      "DOI": "10.1016/j.physleta.2006.01.028",
      "volume": "354",
      "year": "2006",
      "journal-title": "Phys. Lett. A"
    },
    {
      "key": "2023080407445799100_c82",
      "doi-asserted-by": "publisher",
      "first-page": "277",
      "DOI": "10.1140/epjd/e2006-00103-6",
      "volume": "39",
      "year": "2006",
      "journal-title": "Eur. Phys. J. D"
    },
    {
      "key": "2023080407445799100_c83",
      "doi-asserted-by": "publisher",
      "first-page": "051102",
      "DOI": "10.1103/PhysRevE.77.051102",
      "volume": "77",
      "year": "2008",
      "journal-title": "Phys. Rev. E"
    },
    {
      "key": "2023080407445799100_c84",
      "doi-asserted-by": "publisher",
      "first-page": "032523",
      "DOI": "10.1103/PhysRevA.78.032523",
      "volume": "78",
      "year": "2008",
      "journal-title": "Phys. Rev. A"
    },
    {
      "key": "2023080407445799100_c85",
      "doi-asserted-by": "publisher",
      "first-page": "026213",
      "DOI": "10.1103/PhysRevE.77.026213",
      "volume": "77",
      "year": "2008",
      "journal-title": "Phys. Rev. E"
    },
    {
      "key": "2023080407445799100_c86",
      "doi-asserted-by": "publisher",
      "first-page": "063815",
      "DOI": "10.1103/PhysRevA.77.063815",
      "volume": "77",
      "year": "2008",
      "journal-title": "Phys. Rev. A"
    },
    {
      "key": "2023080407445799100_c87",
      "doi-asserted-by": "publisher",
      "first-page": "5564",
      "DOI": "10.1016/j.physleta.2008.06.080",
      "volume": "372",
      "year": "2008",
      "journal-title": "Phys. Lett. A"
    },
    {
      "key": "2023080407445799100_c88",
      "doi-asserted-by": "publisher",
      "first-page": "022308",
      "DOI": "10.1103/PhysRevA.79.022308",
      "volume": "79",
      "year": "2009",
      "journal-title": "Phys. Rev. A"
    },
    {
      "key": "2023080407445799100_c89",
      "doi-asserted-by": "publisher",
      "first-page": "075301",
      "DOI": "10.1088/1751-8113/42/7/075301",
      "volume": "42",
      "year": "2009",
      "journal-title": "J. Phys. A"
    },
    {
      "key": "2023080407445799100_c90",
      "doi-asserted-by": "publisher",
      "first-page": "214301",
      "DOI": "10.1103/PhysRevB.80.214301",
      "volume": "80",
      "year": "2009",
      "journal-title": "Phys. Rev. B"
    },
    {
      "key": "2023080407445799100_c91",
      "doi-asserted-by": "publisher",
      "first-page": "025802",
      "DOI": "10.1103/PhysRevA.82.025802",
      "volume": "82",
      "year": "2010",
      "journal-title": "Phys. Rev. A"
    },
    {
      "key": "2023080407445799100_c92",
      "doi-asserted-by": "publisher",
      "first-page": "777",
      "DOI": "10.1103/PhysRev.47.777",
      "volume": "47",
      "year": "1935",
      "journal-title": "Phys. Rev."
    },
    {
      "volume-title": "Quantum Theory",
      "year": "1951",
      "key": "2023080407445799100_c93"
    },
    {
      "key": "2023080407445799100_c94",
      "doi-asserted-by": "publisher",
      "first-page": "1070",
      "DOI": "10.1103/PhysRev.108.1070",
      "volume": "108",
      "year": "1957",
      "journal-title": "Phys. Rev."
    },
    {
      "key": "2023080407445799100_c95",
      "doi-asserted-by": "publisher",
      "first-page": "1509",
      "DOI": "10.1103/PhysRev.97.1509",
      "volume": "97",
      "year": "1955",
      "journal-title": "Phys. Rev."
    },
    {
      "key": "2023080407445799100_c96",
      "doi-asserted-by": "publisher",
      "first-page": "6472",
      "DOI": "10.1021/ja00800a064",
      "volume": "95",
      "year": "1973",
      "journal-title": "J. Am. Chem. Soc."
    },
    {
      "key": "2023080407445799100_c97",
      "doi-asserted-by": "publisher",
      "first-page": "1914",
      "DOI": "10.1063/1.1701290",
      "volume": "36",
      "year": "1962",
      "journal-title": "J. Chem. Phys."
    },
    {
      "key": "2023080407445799100_c98",
      "doi-asserted-by": "publisher",
      "first-page": "43",
      "DOI": "10.1016/0301-0104(78)85059-9",
      "volume": "29",
      "year": "1978",
      "journal-title": "Chem. Phys."
    },
    {
      "key": "2023080407445799100_c99",
      "doi-asserted-by": "publisher",
      "first-page": "31",
      "DOI": "10.1016/0301-0104(78)85058-7",
      "volume": "29",
      "year": "1978",
      "journal-title": "Chem. Phys."
    },
    {
      "key": "2023080407445799100_c100",
      "doi-asserted-by": "publisher",
      "first-page": "315",
      "DOI": "10.1016/0301-0104(79)85216-7",
      "volume": "44",
      "year": "1979",
      "journal-title": "Chem. Phys."
    },
    {
      "key": "2023080407445799100_c101",
      "doi-asserted-by": "publisher",
      "first-page": "323",
      "DOI": "10.1016/0301-0104(89)87167-8",
      "volume": "134",
      "year": "1989",
      "journal-title": "Chem. Phys."
    },
    {
      "key": "2023080407445799100_c102",
      "doi-asserted-by": "publisher",
      "first-page": "4115",
      "DOI": "10.1063/1.457771",
      "volume": "92",
      "year": "1990",
      "journal-title": "J. Chem. Phys."
    },
    {
      "key": "2023080407445799100_c103",
      "doi-asserted-by": "publisher",
      "first-page": "3263",
      "DOI": "10.1063/1.439563",
      "volume": "72",
      "year": "1980",
      "journal-title": "J. Chem. Phys."
    },
    {
      "key": "2023080407445799100_c104",
      "doi-asserted-by": "publisher",
      "first-page": "3249",
      "DOI": "10.1063/1.1569910",
      "volume": "119",
      "year": "2003",
      "journal-title": "J. Chem. Phys."
    }
  ],
  "container-title": "The Journal of Chemical Physics",
  "original-title": [],
  "language": "en",
  "link": [
    {
      "URL": "https://pubs.aip.org/aip/jcp/article-pdf/doi/10.1063/1.3671386/14111731/244110_1_online.pdf",
      "content-type": "application/pdf",
      "content-version": "vor",
      "intended-application": "syndication"
    },
    {
      "URL": "https://pubs.aip.org/aip/jcp/article-pdf/doi/10.1063/1.3671386/14111731/244110_1_online.pdf",
      "content-type": "unspecified",
      "content-version": "vor",
      "intended-application": "similarity-checking"
    }
  ],
  "deposited": {
    "date-parts": [
      [
        2023,
        8,
        4
      ]
    ],
    "date-time": "2023-08-04T07:47:29Z",
    "timestamp": 1691135249000
  },
  "score": 1,
  "resource": {
    "primary": {
      "URL": "https://pubs.aip.org/jcp/article/135/24/244110/983379/Quantum-entanglement-between-electronic-and"
    }
  },
  "subtitle": [],
  "short-title": [],
  "issued": {
    "date-parts": [
      [
        2011,
        12,
        28
      ]
    ]
  },
  "references-count": 105,
  "journal-issue": {
    "issue": "24",
    "published-print": {
      "date-parts": [
        [
          2011,
          12,
          28
        ]
      ]
    }
  },
  "URL": "http://dx.doi.org/10.1063/1.3671386",
  "relation": {},
  "ISSN": [
    "0021-9606",
    "1089-7690"
  ],
  "subject": [],
  "published-other": {
    "date-parts": [
      [
        2011,
        12,
        28
      ]
    ]
  },
  "published": {
    "date-parts": [
      [
        2011,
        12,
        28
      ]
    ]
  },
  "article-number": "244110"
}