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1.H. Köppel, W. Domcke, and L. S. Cederbaum, Adv. Chem. Phys. 57, 59 (1984).
2.Meaning that all of the states that are treated explicitly are only weakly coupled to the remaining electronic states. The KDC Hamiltonian can be viewed as resulting from a block diagonalization of the molecular Hamiltonian projected onto the crude Born–Huang basis, in which the interacting state block is “dressed” by the weak coupling between it and the remaining states. It is this effect that builds in the “following of the nuclei” property of the wavefunctions and other slow variations with nuclear geometry. See Ref. 18 for more discussions.
3.Within the KDC model, nonzero elements of the off-diagonal term in the potential can be viewed as representing departures from the so-called crude Born–Oppenheimer approximation, which is the form of the BO approximation that is often used to establish the Franck–Condon approximation. For example, the Herzberg–Teller coupling that leads to the appearance of modes with an odd number of quanta in nonsymmetric vibrations is not a true nonadiabatic phenomenon, but does result from the term.
4.Such approach is of course problematic when conical intersections occur. Then, the surface is pathological, and a variational adiabiatic calculation of energy levels is clearly an unwise pursuit. This paper largely focuses on systems such as the ground and first excited states of BNB, where (at least the interesting regions of) the adiabatic potential energy surfaces do not coincide at any point.
5.See, for example, W. Kołos and L. Wolniewicz, J. Mol. Spectrosc. 54, 303 (1975);
5.H. Müller, R. Franke, S. Vogtner, R. Jaquet, and W. Kutzelnigg, Theor. Chem. Acc. 100, 85 (1998).
6.Some analysis along the lines of those presented here were in H. Köppel, Ph.D. thesis, TU München, 1979, and has found its way into his lecture notes entitled “Molecular photoexcitation and spectroscopy for strongly coupled potential surfaces,” especially around p. 50
6.See also H. Köppel, in Conical Intersections: Electronic Structure, Dynamics and Spectroscopy, edited by W. Domcke, D. R. Yarkony, and H. Köppel (World Scientific, Singapore, 2004), pp. 175204
6.Another example where a similar calculation was done, although the case was far less diabolical than BNB and the nonadiabatic correction was small, was R. Lefebvre and M. Garcia Sucre, Int. J. Quantum Chem. 1, 339 (1967).
7.It should be noted that the term “nonadiabatic error,” as used in this paper, has a very specific definition; that is, it refers to the difference between vibronic level spacings involving the zero-point and low-lying vibrational states, in a quite specific way in this paper.
8.J. F. Stanton J. Chem. Phys. 126, 134309 (2007) (diabatic treatment);
8.J. F. Stanton Mol. Phys. 107, 1059 (2009) (adiabatic treatment).
9.J. F. Stanton and J. Gauss, J. Chem. Phys. 101, 8938 (1994).
10.It should still be said that this is not as poor an approximation as it might seem. For example, while the asymmetric stretching force constants are very different on the two adiabatic surfaces of BNB relevant to this draft, the two diabatic electronic states involved have great qualitative similarities: they both can be viewed as arising from the closed-shell BNB anion by removal of an electron from nonbonded lone pair orbitals (out of phase for the ground state neutral, and in phase for the excited state). One would expect two such diabatic potentials to be very similar. This is essentially the justification for this frequently invoked approximation.
11.J. Almlöf and P. R. Taylor, Adv. Quantum Chem. 22, 301 (1991). The truncations used here are (designated as ANO0) and (designated as ANO1).
12.M. Musial, S. A. Kucharski, and R. J. Bartlett, J. Chem. Phys. 118, 1128 (2003). However, the calculations presented here were run with an excitation energy code, with ionized state energies obtained by means of the trick described in Ref. 19.
13.K. R. Asmis, T. R. Taylor, and D. M. Neumark, J. Chem. Phys. 111, 8838 (1999).
14.See, for example, R. G. Pearson, Symmetry Rules for Chemical Reactions. Orbital Topology and Elementary Processes (Wiley, New York, 1976);
14.E. R. Davidson and W. T. Borden, J. Phys. Chem. 87, 4783 (1983);
14.I. B. Bersuker, The Jahn-Teller Effect (Cambridge University Press, Cambridge, 2006).
15.X. Li and J. Paldus, J. Chem. Phys. 126, 224304 (2007)
15.Other high-level studies of the BNB potential include S. R. Gwaltney and M. Head-Gordon, J. Chem. Phys. 3, 4495 (2001);
15.A. Kalemos, T. H. Dunning, and A. Mavridis, J. Chem. Phys. 120, 1813 (2004);
15.Y. Liu, W. L. Zou, I. B. Bersuker, and J. E. Boggs, J. Chem. Phys. 130, 184305 (2009). None of these papers employs a diabatic treatment of the problem; however, the experimental paper (Ref. 13) contains a qualitatively correct and insightful diabatic analysis of the problem.
16.A considerably more elaborate two-dimensional parametrization of the BNB molecule uses a quartic expansion of the diabatic potential surfaces and a bilinear coupling. The parameters, which are derived from the adiabatic potential surfaces of the and states at the EOMIP-CCSDT level with the ANO1 basis set, are ; ; ; ; ; ; ; ; ; ; ; ; and (see Ref. 20 for a guide to the notation). Eigenvalues of this potential give the following values for the nominal , , , , and transition energies: 838, 2027, 3249, 4336, and . If the totally symmetric mode is excluded from the treatment, the corresponding energies are 853, 2075, 3349, 4729, and . This is the basis of the asserted and mode-mode coupling effects on the fundamental and first overtone. In passing, the former set of values agrees rather well with the experimental levels (Ref. 13) of 855, 2052, 3291 (not observed), and , while the associated adiabatic energies of 918, 2106, 3365, 4456, and are in less satisfactory agreement. It should be noted that the nonadiabatic contribution to the fundamental and first overtone (80 and ) are quite close to those obtained with the simplified one-dimensional treatment discussed in the body of this paper. The one-dimensional adiabatic values substantially overestimate the degree of anharmonicity and are in line with the calculations of Li and Paldus (871, 2089, 3393, 4802, and ).
17.W. Domcke, H. Köppel, and L. S. Cederbaum, Mol. Phys. 43, 851 (1981).
18.T. Ichino, J. Gauss, and J. F. Stanton, J. Chem. Phys. 130, 174105 (2009).
19.J. F. Stanton and J. Gauss, J. Chem. Phys. 111, 8785 (1999).
20.E. Garand, K. Klein, J. F. Stanton, J. Zhou, T. I. Yacovitch, and D. M. Neumark, J. Phys. Chem. A 114, 1374 (2010).

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The vibronic coupling model of Köuppel, Domcke, and Cederbaum in one dimension is introduced as a means to estimate the effects of electronic nonadiabaticity on the vibrational energy levels of molecules that exhibit vibronic coupling. For the BNB molecule, the nonadiabatic contribution to the nominal fundamental vibrational energy of the antisymmetric stretching mode is approximately . The surprisingly large effect for this mode, which corresponds to an adiabatic potential that is essentially flat near the minimum due to the vibronic interaction, is contrasted with another model system that also exhibits a flat potential (precisely, a vanishing quadratic force constant) but has a significantly larger gap between interacting electronic states. For the latter case, the nonadiabatic contribution to the level energies is about two orders of magnitude smaller even though the effect on the potential is qualitatively identical. A simple analysis shows that significant nonadiabatic corrections to energy levels should occur only when the affected vibrational frequency is large enough to be of comparable magnitude to the energy gap involved in the coupling. The results provide evidence that nonadiabatic corrections should be given as much weight as issues such as high-level electron correlation, relativistic corrections, etc., in quantum chemical calculations of energy levels for radicals with close-lying and strongly coupled electronic states even in cases where conical intersections are not obviously involved. The same can be said for high-accuracy thermochemical studies, as the zero-point vibrational energy of the BNB example contains a nonadiabatic contribution of approximately .


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