Chapter 17
Exact Factorization of the Electron–Nuclear Wave Function: Theory and Applications
Federica Agostini,
E. K. U. Gross,
Federica Agostini
Universitè Paris-Saclay, CNRS, Institut de Chimie Physique UMR8000, 91405, Orsay, France
Search for more papers by this authorE. K. U. Gross
Fritz Haber Center for Molecular Dynamics, Institute of Chemistry, The Hebrew University of Jerusalem, Jerusalem, 91904 Israel
Search for more papers by this authorFederica Agostini,
E. K. U. Gross,
Federica Agostini
Universitè Paris-Saclay, CNRS, Institut de Chimie Physique UMR8000, 91405, Orsay, France
Search for more papers by this authorE. K. U. Gross
Fritz Haber Center for Molecular Dynamics, Institute of Chemistry, The Hebrew University of Jerusalem, Jerusalem, 91904 Israel
Search for more papers by this authorBook Editor(s):Leticia González,
Roland Lindh,
Leticia González
Institute of Theoretical Chemistry, Faculty of Chemistry, University of Vienna, Austria
Search for more papers by this authorRoland Lindh
Department of Chemistry – BMC, Uppsala University, Sweden
Search for more papers by this authorSummary
In this Chapter we review the exact factorization of the electron-nuclear wave function. The molecular wave function, solution of a time-dependent Schröodinger equation, is factored into a nuclear wave function and an electronic wave function with parametric dependence on nuclear configuration. This factorization resembles the (approximate) adiabatic product of a single Born-Oppenheimer state and a time-dependent nuclear wave packet, but it introduces a fundamental difference: both terms of the product are explicitly time-dependent.
Such feature introduces new concepts of time-dependent vector potential and time-dependent potential energy surface that allow for the treatment of nonadiabatic dynamics, thus of dynamics beyond the Born-Oppenheimer approximation. The theoretical framework of the exact factorization is presented, also in connection to the more standard Born-Huang (still exact) representation of the molecular wave function. A trajectory-based approach to nonadiabatic dynamics is derived from the exact factorization. A discussion on the connection between the molecular Berry phase and the corresponding quantity arising from the exact factorization is briefly discussed.
References
- Worth, G. and Lasorne, B. (2020). Gaussian Wave packets and the DD-vMCG Approach. Wiley.
- Curchod, B.F.E. (2020). Full and Ab Initio Multiple Spawning. Wiley.
- Kirrander, A. and Vacher, M. (2020). Ehrenfest Methods for Electron and Nuclear Dynamics . Wiley.
- Mai, S., Marquetand, P., and Gonzalez, L. (2020). Surface-Hopping Molecular Dynamics. Wiley.
- Albareda, G. and Tavernelli, I. (2020). Bohmian Approaches to Nonadiabatic Molecular Dynamics. Wiley.
- Conte, R. and Ceotto, M. (2020). Semiclassical Molecular Dynamics for Spectroscopic Calculations. Wiley.
- Saller, M.A.C., Runeson, J.E., and Richardson, J.O. (2020). Path Integral Approaches to Nonadiabatic Dynamics. Wiley.
-
Born, M. and Oppenheimer, R.J. (1927). Zur Quantentheorie der Molekeln.
Ann. Phys.
389
: 457–484.
10.1002/andp.19273892002 Google Scholar
- Cafiero, M., Bubin, S., and Adamowicz, L. (2003). Non-Born-Oppenheimer calculations of atoms and molecules. Phys. Chem. Chem. Phys. 5 : 1491–1501.
- Bubin, S., Pavanello, M., Tung, W.C. et al. (2013). Born-Oppenheimer and Non-Born-Oppenheimer, atomic and molecular calculations with explicitly correlated gaussians. Chem. Rev. 113 : 36–79.
- Mátyus, E. (2018). Pre-Born-Oppenheimer molecular structure theory. Mol. Phys. 117 : 590–609.
- Tully, J.C. (1990). Molecular dynamics with electronic transitions. J. Chem. Phys. 93 : 1061–1071.
- Tully, J.C. (1998). Mixed quantum classical dynamics. Faraday Discuss. 110 : 407–419.
- Ben-Nun, M. and Martínez, T.J. (1998). Nonadiabatic molecular dynamics: validation of the multiple spawning method for a multidimensional problem. J. Chem. Phys. 108 : 7244–7257.
- Michl, J. and Bonačić-Koutecký, V. (1990). Electronic Aspects of Organic Photochemistry. New York: Wiley.
- Michl, J. (1972). Photochemical reactions of large molecules. I. A simple physical model of photochemical reactivity. Mol. Photochem. 4 : 243–257.
- Mead, C.A. and Truhlar, D.G. (1979). On the determination of Born–Oppenheimer nuclear motion wave functions including complications due to conical intersections and identical nuclei. J. Chem. Phys. 70 (5): 2284–2296.
- Berry, M.V. and Wilkinson, M. (1984). Diabolical points in the spectra of triangles. Proc. Roy. Soc. London Ser. A. 392 (1802): 15–43.
- Berry, M.V. (1984). Quantal phase factors accompanying adiabatic changes. Proc. Roy. Soc. London Ser. A. 392 (1802): 45–57.
- Yarkony, D.R. (1996). Diabolical conical intersections. Rev. Mod. Phys. 68 (4): 985–1013.
- Bernardi, F., Olivucci, M., and Robb, M.A. (1996). Potential energy surface crossings in organic photochemistry. Chem. Soc. Rev. 25 (5): 321–328.
- Yarkony, D.R. (2001). Conical intersections: the new conventional wisdom. J. Phys. Chem. 105 : 6277–6293.
-
W. Domcke, D. Yarkony, and H. Köppel (eds.) (2004). Conical Intersections: Electronic Structure, Dynamics & Spectroscopy 15. World Scientific Pub Co Inc.
10.1142/5406 Google Scholar
- Worth, G.A. and Cederbaum, L.S. (2004). Beyond Born-Oppenheimer: molecular dynamics through a conical intersection. Annu. Rev. Phys. Chem. 55 : 127–158.
-
Baer, M. (2006). Beyond Born-Oppenheimer: Electronic Nonadiabatic Coupling Terms and Conical Intersections. Wiley.
10.1002/0471780081 Google Scholar
- Levine, B.G. and Martínez, T.J. (2007). Isomerization through conical intersections. Annu. Rev. Phys. Chem. 58 : 613–634.
- Martinez, T.J. (2010). Physical chemistry: seaming is believing. Nature 467 (7314): 412–413.
- Matsika, S. and Krause, P. (2011). Nonadiabatic events and conical intersections. Annu. Rev. Phys. Chem. 62 : 621–643.
- Domcke, W. and Yarkony, D.R. (2012). Role of conical intersections in molecular spectroscopy and photoinduced chemical dynamics. Annu. Rev. Phys. Chem. 63 : 325–352.
- Domcke, W., Yarkony, D., and Köppel, H. (2012). Conical Intersections: Theory, Computation and Experiment 17. World Scientific Pub Co Inc.
- Malhado, J.P., Bearpark, M.J., and Hynes, J.T. (2014). Non-adiabatic dynamics close to conical intersections and the surface hopping perspective. Front. Chem. 2 : 97.
- Zhu, X. and Yarkony, D.R. (2016). Non-adiabaticity: the importance of conical intersections. Mol. Phys. 114 (13): 1983–2013.
- Xie, C., Malbon, C.L., Yarkony, D.R. et al. (2018). Signatures of a conical intersection in adiabatic dissociation on the ground electronic state. J. Am. Chem. Soc. : 1986–1989.
-
Agostini, F., Curchod, B.F.E., Vuilleumier, R. et al. (2018). TDDFT and quantum-classical dynamics: a universal tool describing the dynamics of matter. In: Handbook of Materials Modeling (eds. W. Andreoni and S. Yip), 1–47. Netherlands: Springer.
10.1007/978-3-319-42913-7_43-1 Google Scholar
- Abedi, A., Maitra, N.T., and Gross, E.K.U. (2010). Exact factorization of the time-dependent electron–nuclear wave function. Phys. Rev. Lett. 105 (12): 123002.
- Abedi, A., Maitra, N.T., and Gross, E.K.U. (2012). Correlated electron–nuclear dynamics: exact factorization of the molecular wave-function. J. Chem. Phys. 137 (22): 22A530.
-
Agostini, F. and Curchod, B.F.E. (2019). Different flavors of non-adiabatic molecular dynamics.
WIREs Comput. Mol. Sci.
9
: e1417.
10.1002/wcms.1417 Google Scholar
- Agostini, F., Tavernelli, I., and Ciccotti, G. (2018). Nuclear quantum effects in electronic (non)adiabatic dynamics. Eur. Phys. J. B. 91 : 139.
- Alonso, J.L., Clemente-Gallardo, J., Echeniche-Robba, P., and Jover-Galtier, J.A. (2013). Comment on Correlated electron–nuclear dynamics: exact factorization of the molecular wave-function. J. Chem. Phys. 139 : 087101.
- Abedi, A., Maitra, N.T., and Gross, E.K.U. (2013). Reply to comment on Correlated electron–nuclear dynamics: exact factorization of the molecular wave-function. J. Chem. Phys. 139 (8): 087102.
- Abedi, A., Agostini, F., Suzuki, Y., and Gross, E.K.U. (2013). Dynamical steps that bridge piecewise adiabatic shapes in the exact time-dependent potential energy surface. Phys. Rev. Lett. 110 (26): 263001.
- Agostini, F., Abedi, A., Suzuki, Y., and Gross, E.K.U. (2013). Mixed quantum-classical dynamics on the exact time-dependent potential energy surfaces: a novel perspective on non-adiabatic processes. Mol. Phys. 111 (22-23): 3625–3640.
- Agostini, F., Min, S.K., and Gross, E.K.U. (2015). Semiclassical analysis of the electron–nuclear coupling in electronic non-adiabatic processes. Ann. Phys. 527 (9-10): 546–555.
- Agostini, F., Abedi, A., Suzuki, Y. et al. (2015). The exact forces on classical nuclei in non-adiabatic charge transfer. J. Chem. Phys. 142 (8): 084303.
- Curchod, B.F.E., Agostini, F., and Gross, E.K.U. (2016). An exact factorization perspective on quantum interferences in non-adiabatic dynamics. J. Chem. Phys. 145 : 034103.
- Curchod, B.F.E. and Agostini, F. (2017). On the dynamics through a conical intersection. J. Phys. Chem. Lett. 8 : 831–837.
- Agostini, F. and Curchod, B.F.E. (2018). When the exact factorization meets conical intersections. Eur. Phys. J. B. 91 : 141.
- Eich, F.G. and Agostini, F. (2016). The adiabatic limit of the exact factorization of the electron–nuclear wave function. J. Chem. Phys. 145 : 054110.
- Scherrer, A., Agostini, F., Sebastiani, D. et al. (2015). Nuclear velocity perturbation theory for vibrational circular dichroism: an approach based on the exact factorization of the electron–nuclear wave function. J. Chem. Phys. 143 (7): 074106.
- Scherrer, A., Agostini, F., Sebastiani, D. et al. (2017). On the mass of atoms in molecules: beyond the Born-Oppenheimer approximation. Phys. Rev. X 7 : 031035.
- Schild, A., Agostini, F., and Gross, E.K.U. (2016). Electronic flux density beyond the born-oppenheimer approximation. J. Phys. Chem. A 120 : 3316.
- deCarvalho, F.F., Bouduban, M.E.F., Curchod, B.F.E., and Tavernelli, I. (2014). Nonadiabatic molecular dynamics based on trajectories. Entropy 16 (1): 62–85.
- Suzuki, Y. and Watanabe, K. (2016). Bohmian mechanics in the exact factorization of electron–nuclear wave functions. Phys. Rev. A 94 : 032517.
- Suzuki, Y., Abedi, A., Maitra, N.T., and Gross, E.K.U. (2015). Laser-induced electron localization in : mixed quantum-classical dynamics based on the exact time-dependent potential energy surface. Phys. Chem. Chem. Phys. 17 : 29271–29280.
- Ha, J.-K., Lee, I.S., and Min, S.K. (2018). Surface hopping dynamics beyond non-adiabatic couplings for quantum coherence. J. Phys. Chem. Lett. 9 : 1097–1104.
- Min, S.K., Agostini, F., Tavernelli, I., and Gross, E.K.U. (2017). Ab initio non-adiabatic dynamics with coupled trajectories: a rigorous approach to quantum (de)coherence. J. Phys. Chem. Lett. 8 : 3048–3055.
- Agostini, F., Min, S.K., Abedi, A., and Gross, E.K.U. (2016). Quantum-classical non-adiabatic dynamics: coupled- vs. independent-trajectory methods. J. Chem. Theory Comput. 12 (5): 2127–2143.
- Agostini, F., Abedi, A., and Gross, E.K.U. (2014). Classical nuclear motion coupled to electronic non-adiabatic transitions. J. Chem. Phys. 141 (21): 214101.
-
Abedi, A., Agostini, F., and Gross, E.K.U. (2014). Mixed quantum-classical dynamics from the exact decomposition of electron–nuclear motion.
Europhys. Lett.
106
(3): 33001.
10.1209/0295-5075/106/33001 Google Scholar
- Min, S.K., Agostini, F., and Gross, E.K.U. (2015). Coupled-trajectory quantum-classical approach to electronic decoherence in non-adiabatic processes. Phys. Rev. Lett. 115 (7): 073001.
- Agostini, F. (2018). An exact-factorization perspective on quantum-classical approaches to excited-state dynamics. Eur. Phys. J. B. 91 : 143.
- Curchod, B.F.E., Agostini, F., and Tavernelli, I. (2018). CT-MQC – A coupled-trajectory mixed quantum/classical method including non-adiabatic quantum coherence effects. Eur. Phys. J. B. 91 : 168.
- Gossel, G., Agostini, F., and Maitra, N.T. (2018). Coupled-trajectory mixed quantum-classical algorithm: a deconstruction. J. Chem. Theory Comput. 14 : 4513–4529.
- Requist, R. and Gross, E.K.U. (2016). Exact factorization-based density functional theory of electrons and nuclei. Phys. Rev. Lett. 117 : 193001.
- Li, C., Requist, R., and Gross, E.K.U. (2018). Density functional theory of electron transfer beyond the Born–Oppenheimer approximation: case study of LiF. J. Chem. Phys. 148 : 084110.
- Requist, R., Proetto, C.R., and Gross, E.K.U. (2019). Exact factorization-based density functional theory of electron-phonon systems. Phys. Rev. B 99 : 165136.
- Khosravi, E., Abedi, A., Rubio, A., and Maitra, N.T. (2017). Electronic non-adiabatic dynamics in enhanced ionization of isotopologues of hydrogen molecular ions from the exact factorization perspective. Phys. Chem. Chem. Phys. 19 : 8269–8281.
- Khosravi, E., Abedi, A., and Maitra, N.T. (2015). Exact potential driving the electron dynamics in enhanced ionization of . Phys. Rev. Lett. 115 : 263002.
- Suzuki, Y., Abedi, A., Maitra, N.T. et al. (2014). Electronic Schrödinger equation with nonclassical nuclei. Phys. Rev. A 89 (4): 040501(R).
- Schild, A. and Gross, E.K.U. (2017). Exact single-electron approach to the dynamics of molecules in strong laser fields. Phys. Rev. Lett. 118 : 163202.
- Hoffmann, N.M., Appel, H., Rubio, A., and Maitra, N. (2018). Light-matter interactions via the exact factorization approach. Eur. Phys. J. B. 91 : 180.
- Abedi, A., Khosravi, E., and Tokatly, I. (2018). Shedding light on correlated electronphoton states using the exact factorization. Eur. Phys. J. B. 91 : 194.
- Jeck, T., Sutcliffe, B.T., and Woolley, R.G. (2015). On factorization of molecular wave functions. J. Phys. A Math. Theor. 48 : 445201.
- Parashar, S., Sajeev, Y., and Ghosh, S.K. (2015). Towards an exact factorization of the molecular wave function. Mol. Phys. 113 (19-20): 3067–3072.
- Bishop, D.M. and Cheung, L.M. (1977). Conditional probability amplitudes and the variational principle. Chem. Phys. Lett. 50 (1): 172–174.
- Hunter, G. and Tai, C.C. (1982). Variational marginal amplitudes. Int. J. Quantum Chem. 21 : 1041–1050.
- Hunter, G. (1981). Nodeless wave functions and spiky potentials. Int. J. Quantum Chem. 19 : 755–761.
- Hunter, G. (1975). Conditional probability amplitudes in wave mechanics. Int. J. Quantum Chem. 9 : 237–242.
- Lefebvre, R. (2015). Perturbations in vibrational diatomic spectra: factorization of the molecular wave function. J. Chem. Phys. 142 (7): 074106.
- Lefebvre, R. (2015). Factorized molecular wave functions: analysis of the nuclear factor. J. Chem. Phys. 142 (21): 214105.
- Cederbaum, L.S. (2015). The exact wave function of interacting N degrees of freedom as a product of N single-degree-of-freedom wave functions. Chem. Phys. 457 (22): 129.
- Cederbaum, L.S. (2013). The exact molecular wave function as a product of an electronic and a nuclear wave function. J. Chem. Phys. 138 (22): 224110.
- Chiang, Y.-C., Klaiman, S., Otto, F., and Cederbaum, L.S. (2014). The exact wave function factorization of a vibronic coupling system. J. Chem. Phys. 140 (5): 054104.
- Gidopoulos, N.I. and Gross, E.K.U. (2014). Electronic non-adiabatic states: towards a density functional theory beyond the Born-Oppenheimer approximation. Phil. Trans. R. Soc. A 372 : 20130059.
- Gidopoulos, N.I. and Gross, E.K.U (2005). Electronic non-adiabatic states. arXiv:cond-mat/0502433.
- Requist, R., Proetto, C.R., and Gross, E.K.U. (2017). Asymptotic analysis of the Berry curvature in the E ⨂ e Jahn-Teller model. Phys. Rev. A 96 : 062503.
- Min, S.K., Abedi, A., Kim, K.S., and Gross, E.K.U. (2014). Is the molecular berry phase an artefact of the Born-Oppenheimer approximation? Phys. Rev. Lett. 113 (26): 263004.
- Requist, R., Tandetzky, F., and Gross, E.K.U. (2016). Molecular geometric phase from the exact electron–nuclear factorization. Phys. Rev. A 93 : 042108.
- Shin, S. and Metiu, H. (1995). Nonadiabatic effects on the charge transfer rate constant: a numerical study of a simple model system. J. Chem. Phys. 102 : 9285–9295.
- Feit, M.D., Fleck, F.A., and Steiger, A. (1982). Solution of the Schrödinger equation by a spectral method. J. Comput. Phys. 47 (3): 412–433.
- Gossel, G.H., Lacombe, L., and Maitra, N.T. (2019). On the numerical solution of the exact factorization equations. J. Chem. Phys. 150 : 154112.
- Agostini, F., Gross, E.K.U., and Curchod, B.F.E. (2019). Electron-nuclear entanglement in the time-dependent molecular wave function. Comput. Theo. Chem. : 99, 1151–106.
- Hader, K., Albert, J., Gross, E.K.U., and Engel, V. (2017). Electron-nuclear wave-packet dynamics through a conical intersection. J. Chem. Phys. 146: 074304.
- CPMD, http://www.cpmd.org/, Copyright IBM Corp 1990-2015, Copyright MPI für Festkörperforschung Stuttgart 1997–2001.
- Engel, G.S., Calhoun, T.R., Read, E.L. et al. (2007). Evidence for wavelike energy transfer through quantum coherence in photosynthetic systems. Nature 446 : 782–786.
- Ishizaki, A. and Fleming, G.R. (2009). Theoretical examination of quantum coherence in a photosynthetic system at physiological temperature. Proc. Natl. Acad. Sci. USA 106 : 17255–17260.
- Scholes, G.D. (2010). Quantum coherent electronic energy transfer: did nature think of it first? J. Phys. Chem. Lett. 1 : 2–8.
- Collini, E., Wong, C.Y., Wilk, K.E. et al. (2010). Coherently wired light-harvesting in photosynthetic marine algae at ambient temperature. Nature 463 : 644–647.
- Panitchayangkoon, G., Hayes, D., Fransted, K.A. et al. (2010). Long-lived quantum coherence in photosynthetic complexes at physiological temperature. Proc. Natl. Acad. Sci. U.S.A. 107 : 12766–12770.
- Turner, D.B., Wilk, K.E., Curmi, P.M.G., and Scholes, G.D. (2011). Comparison of electronic and vibrational coherence measured by two-dimensional electronic spectroscopy. J. Phys. Chem. Lett. 2 : 1904–1911.
- Huo, P. and Coker, D.F. (2011). Theoretical study of coherent excitation energy transfer in cryptophyte phycocyanin 645 at physiological temperature. J. Phys. Chem. Lett. 2 : 825–833.
- Strümpfer, J., Sener, M., and Schulten, K. (2012). How quantum coherence assists photosynthetic light-harvesting. J. Phys. Chem. Lett. 3: 536–542.
- Dorfman, K.E., Voronine, D.V., Mukamel, S., and Scully, M.O. (2013). Photosynthetic reaction center as a quantum heat engine. Proc. Natl. Acad. Sci. U.S.A. 110 : 2746–2751.
- Rozzi, C.A., Falke, S.M., Spallanzani, N. et al. (2013). Quantum coherence controls the charge separation in a prototypical artificial light-harvesting system. Nature Comm. 4 : 1602.
- Akimov, A.V. and Prezhdo, O.V. (2013). Persistent electronic coherence despite rapid loss of electron–nuclear correlation. J. Phys. Chem. Lett. 4 : 3857–3864.
- Landry, B.R. and Subotnik, J.E. (2014). Quantifying the lifetime of triplet energy transfer processes in organic chromophores: a case study of 4-2-(Naphthylmethyl)benzaldehyde. J. Chem. Theory Comput. 10: 4253–4263.
- Curutchet, C. and Mennucci, B. (2017). Quantum chemical studies of light harvesting. Chem. Rev. 117 : 294–343.
- Schwartz, B.J., Bittner, E.R., Prezhdo, O.V., and Rossky, P.J. (1996). Quantum decoherence and the isotope effect in condensed phase non-adiabatic molecular dynamics simulations. J. Chem. Phys. 104 : 5942–5955.
- Fang, J.-T. and Hammes-Schiffer, S. (1999). Improvement of the internal consistency in trajectory surface hopping. J. Phys. Chem. A 103 : 9399–9407.
- Granucci, G. and Persico, M. (2007). Critical appraisal of the fewest switches algorithm for surface hopping. J. Chem. Phys. 126 : 134114.
- Shenvi, N., Subotnik, J.E., and Yang, W. (2011). Simultaneous-trajectory surface hopping: a parameter-free algorithm for implementing decoherence in non-adiabatic dynamics. J. Chem. Phys. 134 : 144102.
- Shenvi, N., Subotnik, J.E., and Yang, W. (2011). Phase-corrected surface hopping: correcting the phase evolution of the electronic wave function. J. Chem. Phys. 13 : 024101.
- Shenvi, N. and Yang, W. (2012). Achieving partial decoherence in surface hopping through phase correction. J. Chem. Phys. 137 : 22A528.
- Subotnik, J.E. and Shenvi, N. (2011). A new approach to decoherence and momentum rescaling in the surface hopping algorithm. J. Chem. Phys. 134 : 024105.
- Subotnik, J.E. and Shenvi, N. (2011). Decoherence and surface hopping: when can averaging over initial conditions help capture the effects of wave packet separation? J. Chem. Phys. 134 : 244114.
- Perdew, J.P., Burke, K., and Ernzerhof, M. (1996). Generalized gradient approximation made simple. Phys. Rev. Lett. 77 : 3865–3868.
- Runge, E. and Gross, E.K.U. (1984). Density-functional theory for time-dependent systems. Phys. Rev. Lett. 52 : 997–1000.
- Petersilka, M., Gossmann, U.J., and Gross, E.K.U. (1996). Excitation energies from time-dependent density-functional theory. Phys. Rev. Lett. 76 : 1212–1215.
-
Casida, M.E. (1995). Time-dependent density-functional response theory for molecules. In: Recent Advances in Density Functional Methods (ed. D.P. Chong), 155–192. Singapore: World Scientific.
10.1142/9789812830586_0005 Google Scholar
- Tamm, I. (1945). Relativistic interaction of elementary particles. J. Phys. (USSR) (9): 449.
- Dancoff, S.M. (1950). Non-adiabatic meson theory of nuclear forces. Phys. Rev. 78 : 382–385.
- Kleinman, L. and Bylander, D.M. (1982). Efficacious form for model pseudopotentials. Phys. Rev. Lett. 48 : 1425–1428.
- Agostini, F., Min, S.K., Tavernelli, I., and Gossel, G.H. (2018). CTMQC. https://e-cam.readthedocs.io/en/latest/Quantum-Dynamics-Modules/modules/CTMQC/readme.html,
- Longuet-Higgins, H.C. (1961). Some recent developments in the theory of molecular energy levels. In: Advances in Spectroscopy, vol. II (ed. H.W. Thompson), 429–472. Interscience Publishers, Ltd.
- Berry, M.V. (1984). Quantal phase factors accompanying adiabatic changes. Proc. R. Soc. Lond. A 392 : 45–57.
- Yarkony, D.R. (2012). Nonadiabatic quantum chemistry: past, present and future. Chem. Rev. 112 (1): 481–498.
- Engelman, R. (1972). The Jahn-Teller Effect in Molecules and Crystals. New York: Wiley-Interscience.
- Ryabinkin, G. and Izmaylov, A.F. (2013). Geometric phase effects in dynamics near conical intersections: symmetry breaking and spatial localization. Phys. Rev. Lett. 111 : 220406.
- Ryabinkin, I.G., Joubert-Doriol, L., and Izmaylov, A.F. (2017). Geometric phase effects in non-adiabatic dynamics near conical intersections. Acc. Chem. Res. 50 : 1785–1793.
- Halász, G.J., Vibók, Á., Baer, R., and Baer, M. (2007). Conical intersections induced by the Renner effect in polyatomic molecules. J. Phys. A Math. Theor. 40 (15): F267–F272.
- Baer, M. (1975). Adiabatic and diabatic representations for atom-molecule collisions: treatment of the collinear arrangement. Chem. Phys. Lett. 35 : 112–118.
- Baer, M. and Engelman, R. (1992). A study of the diabatic electronic representation within the Born–Oppenheimer approximation. Mol. Phys. 75 : 293–303.
- Domcke, W., Köppel, H., and Cederbaum, L. (1981). Spectroscopic effects of conical intersections of molecular potential energy surfaces. Mol. Phys. 43 : 851–875.
- Baer, M. (2002). Introduction to the theory of electronic non-adiabatic coupling terms in molecular systems. Phys. Rep. 358 : 75–142.
- Hunter, G. (1974). Conditional probability amplitude analysis of coupled harmonic oscillators. Int. J. Quantum Chem. 8 : 413–420.
- Berry, M.V. (1989). The quantum phase, five years after. In: Geometric Phases in Physics (eds. A. Shapere and F. Wilczek), 7–28. World Scientific.
