Baer M., Billing G.D. (eds.) - The role of degenerate states in chemistry (Adv.Chem.Phys. special issue, Wiley, 2002)

III.FUNCTIONAL FORM OF THE ENERGY

Since the potential matrix W is invariant and restricted to E space, it has the form

where WA1 , and so on, are functions of the nuclear coordinates transforming under C3v as indicated by their subscripts. On the other hand W must be Hermitian, and in our case can be real, from which it follows that WA2 ¼ 0. The energies that reduce to the degenerate pair at the reference configuration are then just the eigenvalues of W:

To all orders, the general form for functions with transformation properties uA1 , and so on, may be shown [13] to be

where fi ði ¼ 1; 2; 3; 4Þ are functions formally representable as a double power series in their arguments other than z, with the coefficients being constant or functions of z. From Eq. (E.37)–(E.42), one then obtains

where f ¼ f ðz; r2; r3 cosð3jÞÞ, g ¼ gðz; r2; r3 cosð3jÞÞ, and w are formally analytic functions of their arguments. These equations define the correct behavior of the potential energy in the vicinity of the conical intersection, and hence may be valuable in delineating fitting forms, as it was the case for the H3 DMBE potential energy surface [82].

Acknowledgments

We thank Dr. Alexander Alijah for helpful discussions. This work was supported by Fundac¸~ao para a Cieˆncia e Tecnologia, Portugal.

References

1.A. J. C. Varandas and H. G. Yu, Chem. Phys. Lett. 259, 336 (1996).

2.A. J. C. Varandas and H. G. Yu, J. Chem. Soc. Faraday Trans. 93, 819 (1997).

3.S. Mahapatra and H. Ko¨ppel, J. Chem. Phys. 109, 1721 (1998).

4.A. J. C. Varandas and Z. R. Xu, J. Chem. Phys. 112, 2121 (2000).

5.Z. R. Xu, M. Baer, and A. J. C. Varandas, J. Chem. Phys. 112, 2746 (2000).

6.Z. R. Xu and A. J. C. Varandas, Int. J. Quantum Chem. 80, 454 (2000).

7.Z. R. Xu and A. J. C. Varandas, Int. J. Quantum Chem. 83, 279 (2001).

8.Z. R. Xu and A. J. C. Varandas, J. Phys. Chem. 105, 2246 (2001).

9.W. H. Gerber and E. Schumacher, J. Chem. Phys. 69, 1692 (1978).

10.M. Mayer and L. S. Cederbaum, J. Chem. Phys. 105, 4938 (1996).

11.A. J. C. Varandas, H. G. Yu, and Z. R. Xu, Mol. Phys. 96, 1193 (1999).

12.A. J. C. Varandas and Z. R. Xu, Int. J. Quantum Chem. 75, 89 (1999).

13.T. C. Thompson and C. A. Mead, J. Chem. Phys. 82, 2408 (1985).

14.B. K. Kendrick, Phys. Rev. Lett. 79, 2431 (1997).

15.D. H. Zhang and J. Z. H. Zhang, J. Chem. Phys. 101, 3671 (1994).

16.V. A. Mandelshtam, T. P. Grozdanov, and H. S. Taylor, J. Chem. Phys. 103, 10074 (1995).

17.A. J. Dobbyn, M. Stumpf, H.-M. Keller, W. L. Hase, and R. Shinke, J. Chem. Phys. 103, 9947 (1995).

18.B. K. Kendrick and R. T Pack, J. Chem. Phys. 106, 3519 (1997).

19.C. A. Mead, Chem. Phys. 49, 23 (1980).

20.T. C. Thompson, D. G. Truhlar, and C. A. Mead, J. Chem. Phys. 82, 2392 (1985).

21.M. V. Berry, Proc. R. Soc. London, Ser. A 392, 45 (1984).

22.C. A. Mead, Rev. Mod. Phys. 64, 51 (1992).

23.D. R. Yarkony, Rev. Mod. Phys. 68, 985 (1996).

24.D. R. Yarkony, J. Phys. Chem. 100, 18612 (1996).

25.M. Baer, J. Chem. Phys. 107, 2694 (1997).

26.D. Neuhauser, J. Chem. Phys. 100, 9272 (1994).

27.H. G. Yu and S. C. Smith, Ber. Bunsenges. Phys. Chem. 101, 400 (1997).

28.G. Herzberg, Molecular Spectra and Molecular Structure. III. Electronic Spectra and Electronic Structure of Polyatomic Molecules, van Nostrand, New York, 1966.

29.D. M. Dennison, Rev. Mod. Phys. 3, 280 (1931).

30.E. P. Wigner, Group Theory and Its Applications to the Quantum Mechanics of Atomic Spectra, Academic Press, New York, 1959.

31.R. C. Whitten and F. T. Smith, J. Math. Phys. 9, 1103 (1968).

32.B. R. Johnson, J. Chem. Phys. 73, 5051 (1980).

33.B. R. Johnson, J. Chem. Phys. 79, 1906 (1983).

34.B. R. Johnson, J. Chem. Phys. 79, 1916 (1983).

35.J. Zu´n˜iga, A. Bastida, and A. Requena, J. Chem. Soc. Faraday Trans. 93, 1681 (1997).

36.D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii, Quantum theory of angular momentum, World Scientific, Singapore, 1988.

37.A. Kiselev, Can. J. Phys. 56, 615 (1978).

38.M. Born and J. R. Oppenheimer, Ann. Phys. 84, 457 (1927).

39.M. Born and K. Huang, Dynamical Theory of Crystal Lattices, Oxford, London, 1954.

40.E. E. Nikitin and L. Zulicke, Lecture Notes in Chemistry, Theory of Elementary Processes, Vol. 8, Springer-Verlag, Heidelberg, 1978.

41.I. B. Bersuker and V. Z. Polinger, Vibronic Interactions in Molecules and Crystals, SpringerVerlag, Berlin, 1989.

42.H. C. Longuet-Higgins, Adv. Spectrosc. 2, 429 (1961).

43.G. Herzberg and H. C. Longuet-Higgins, Faraday Discuss. Chem. Soc. 35, 77 (1963).

44.H. C. Longuet-Higgins, Proc. R. Soc., London Ser. A 344, 147 (1975).

45.B. K. Kendrick and C. A. Mead, J. Chem. Phys. 102, 4160 (1995).

46.C. A. Mead and D. G. Truhlar, J. Chem. Phys. 70, 2284 (1979).

47.C. A. Mead, J. Chem. Phys. 72, 3839 (1980).

48.Y.-S. M. Wu, B. Lepetit, and A. Kuppermann, Chem. Phys. Lett. 186, 319 (1991).

49.Y.-S. M. Wu and A. Kuppermann, Chem. Phys. Lett. 201, 178 (1993).

50.A. Kuppermann and Y.-S. M. Wu, Chem. Phys. Lett. 213, 636 (1993).

51.G. D. Billing and N. Markovic´, J. Chem. Phys. 99, 2674 (1993).

52.B. K. Kendrick and R. T. Pack, J. Chem. Phys. 104, 7475 (1996).

53.M. Baer and R. Englman, Chem. Phys. Lett. 265, 105 (1997).

54.E. Merzbacher, Quantum Mechanics, John Wiley & Sons, Inc., New York, 1970.

55.B. G. Levich, V. A. Myanlin, and Y. A. Vdovin, Theoretical Physics. An Advanced Text, NorthHolland, Amsterdam, 1973, p. 418.

56.I. N. Levine, Molecular Spectroscopy, John Wiley & Sons, Inc., New York, 1975.

57.J. N. Murrell, S. Carter, S. C. Farantos, P. Huxley, and A. J. C. Varandas, Molecular Potential Energy Functions, John Wiley & Sons, Inc., Chichester, 1984.

58.A. J. C. Varandas, Adv. Chem. Phys. 74, 255 (1988).

59.G. C. Hancock, C. A. Mead, D. G. Truhlar, and A. J. C. Varandas, J. Chem. Phys. 91, 3492 (1989).

60.A. J. C. Varandas, in Reaction and Molecular Dynamics, A. Lagana´ and A. Riganelli, eds., Springer, Berlin, 2000, Vol. 75 of Lecture Notes in Chemistry, p. 33.

61.A. J. C. Varandas, Int. J. Quantum Chem. 32, 563 (1987).

62.F. London, Z. Electrochem. 35, 552 (1929).

63.H. Eyring and M. Polanyi, Z. Phys. Chem. Abt. 12, 279 (1931).

64.S. Sato, J. Chem. Phys. 23, 592 (1955).

65.S. Sato, J. Chem. Phys. 23, 2465 (1955).

66.J. N. Murrell and A. J. C. Varandas, Mol. Phys. 57, 415 (1986).

67.V. M. F. Morais, A. J. C. Varandas, and A. A. C. C. Pais, Mol. Phys. 58, 285 (1986).

68.A. J. C. Varandas, Chem. Phys. Lett. 138, 455 (1987).

69.A. J. C. Varandas and A. A. C. C. Pais, J. Chem. Soc. Faraday Trans. 89, 1511 (1993).

70.A. A. C. C. Pais, R. F. Nalewajski, and A. J. C. Varandas, J. Chem. Soc. Faraday Trans. 90, 1381 (1994).

71.A. J. C. Varandas and A. I. Voronin, Mol. Phys. 95, 497 (1995).

72.H. A. Jahn and E. Teller, Proc. R. Soc. London, Ser. A 161, 220 (1937).

73.W. Moffitt and A. D. Liehr, Phys. Rev. 106, 1195 (1957).

74.A. J. C. Varandas, J. Tennyson, and J. N. Murrell, Chem. Phys. Lett. 61, 431 (1979).

75.M. L. Mehta, Random Matrices and the Statistical Theory of Energy Levels, Academic Press, New York, 1967.

76.C. A. Mead, J. Chem. Phys. 78, 807 (1983).

77.A. J. C. Varandas and Z. R. Xu, Chem. Phys. Lett. 316, 248 (2000).

78.J.-P. Wolf, G. Delacre´taz, and L. Wo¨ste, Phys. Rev. Lett. 63, 1946 (1989).

79.H. A. Jahn, Proc. R. Soc. London Ser. A 164, 117 (1938).

80.A. J. C. Varandas and V. M. F. Morais, Mol. Phys. 47, 1241 (1982).

81.R. Abrol, A. Shaw, A. Kuppermann, and D. R. Yarkony, J. Chem. Phys. 115, 4640 (2001).

82.A. J. C. Varandas, F. B. Brown, C. A. Mead, D. G. Truhlar, and N. C. Blais, J. Chem. Phys. 86, 6258 (1987).

83.A. Kuppermann and Y.-S. M. Wu, Chem. Phys. Lett. 205, 577 (1993).

84.A. Kuppermann, Chem. Phys. Lett. 32, 374 (1975).

85.M. Baer, Chem. Phys. Lett. 35, 112 (1975).

86.M. Baer, Mol. Phys. 40, 1011 (1980).

87.A. J. C. Varandas, J. Chem. Phys. 107, 867 (1997).

88.A. J. C. Varandas, A. I. Voronin, and P. J. S. B. Caridade, J. Chem. Phys. 108, 7623 (1998).

89.R. K. Preston and J. C. Tully, J. Chem. Phys. 54, 4297 (1971).

90.D. Grimbert, B. Lassier-Govers, and V. Sidis, Chem. Phys. 124, 187 (1988).

91.F. Gianturco, A. Palma, and F. Schnider, Chem. Phys. 137, 177 (1989).

92.B. K. Kendrick, C. A. Mead, and D. G. Truhlar, J. Chem. Phys. 110, 7594 (1999).

93.E. P. Wigner, J. Math. Phys. 1, 409 (1960).

94.R. N. Porter, R. M. Stevens, and M. Karplus, J. Chem. Phys. 49, 5163 (1968).

95.S. Califano, Vibrational States, John Wiley & Sons, Inc., London, 1976.

A.Wigner Rotation Matrices

B.The Adiabatic-to-Diabatic Transformation Matrix and the Wigner d j Matrix XIII. Curl Condition Revisited: Introduction of the Yang–Mills Field

A.The Non-Adiabatic Coupling Term as a Vector Potential

B.The Pseudomagnetic Field and the Curl Equation

C.Conclusions

XIV. A Theoretic-Numeric Approach to Calculate the Electronic

Non-Adiabatic Coupling Terms

A.The Treatment of the Two-State System in a Plane

1.The Solution for a Single Conical Intersection

2.The Solution for a Distribution of Conical Intersections

B.The Treatment of the Three-State System in a Plane

XV. Studies of Specific Systems

A.The Study of Real Two-State Molecular Systems

1.The H3-System and Its Isotopic Analogues

2.The C2H-Molecule: The Study of the (1,2) and the (2,3) Conical Intersections

B.The Study of a Real Three-State Molecular System: Strongly Coupled (2,3)

and (3,4) Conical Intersections XVI. Summary and Conclusions

Appendix A: The Jahn–Teller Model and the Longuet–Higgins Phase

Appendix B: The Sufficient Conditions for Having an Analytic Adiabatic-to-Diabatic Transformation Matrix

I. Orthogonality II. Analyticity

Appendix C: On the Single/Multivaluedness of the Adiabatic-to-Diabatic Transformation Matrix

Appendix D: The Diabatic Representation

Appendix E: A Numerical Study of a Three-State Model

Appendix F: The Treatment of a Conical Intersection Removed from the Origin of Coordinates

Acknowledgments References

I.INTRODUCTION

Electronic non-adiabatic effects are an outcome of the Born–Oppenheimer– Huang treatment and as such are a result of the distinction between the fast moving electrons and the slow moving nuclei [1,2]. The non-adiabatic coupling terms, together with the adiabatic potential energy surfaces (which are also an outcome of this treatment) form the origin for the driving forces that govern the

motion of the atoms in the molecular system. Still they differ from the potential energy surfaces because they are, as we shall show, derivative coupling and as such they are vectors, in contrast to adiabatic potential energy surfaces, which are scalars.

The ordinary way to get acquainted with objects like the non-adiabatic coupling terms is to derive them from first principles, via ab initio calculations [4–6], and study their spatial structure—somewhat reminiscent of the way potential energy surfaces are studied. However, this approach is not satisfactory because the non-adiabatic coupling terms are frequently singular (in addition to being vectors), and therefore theoretical means should be applied in order to understand their role in molecular physics. During the last decade, we followed both courses but our main interest was directed toward studying their physical– mathematical features [7–13]. In this process, we revealed (1) the necessity to form sub-Hilbert spaces [9,10] in the region of interest in configuration space and (2) the fact that the non-adiabatic coupling matrix has to be quantized for this sub-space [7–10].

In the late 1950s and the beginning of the 1960s Longuet-Higgins and colleagues [14–17] discovered one of the more interesting features in molecular physics related to the Born–Oppenheimer–Huang electronic adiabatic eigenfunctions (which are parametrically dependent on the nuclear coordinates). They found that these functions, when surrounding a point of degeneracy in configuration space, may acquire a phase that leads to a flip of sign of these functions. Later, this feature was explicitly demonstrated by Herzberg and Longuet-Higgins [16] for the Jahn–Teller conical intersection model [18–29] (see also Appendix A). This interesting observation implies that if a molecular system possesses a conical intersection at a point in configuration space the relevant electronic eigenfunctions that are parametrically dependent on the nuclear coordinates, are multivalued (this finding was later confirmed for a real case following ab initio calculations [30]). No hints were given to the fact that this phenomenon is associated with the electronic non-adiabatic coupling terms. In this chapter, we not only discuss this connection but also extend the two-state case to the multistate cases.

In molecular physics, one distinguishes between (1) the adiabatic framework that is characterized by the adiabatic surfaces and the above-mentioned nonadiabatic coupling terms [31–46] and (2) the diabatic framework that is characterized by the smoothly behaving potential couplings (and the nonexistence of non-adiabatic couplings) [31–53]. The adiabatic framework is in most cases inconvenient for treating the nuclear Schro¨dinger equation because of two reasons. (1) The non-adiabatic coupling terms are usually spiky [3,54] and frequently singular [3,36,55,56] so that any numerical recipe for solving this equation becomes unstable. (2) The singular feature of the non-adiabatic coupling terms dictates certain boundary conditions that may not be easily implemented for deriving the solution within this framework [56]. Therefore,

transforming to the diabatic framework, to be termed diabatization, is essentially enforced when treating the multistate problem as created by the Born– Oppenheimer–Huang approach.

The diabatization can be carried out in various ways—and we discuss some of them here—but the way recommended in the present composition is based on the following two-step process: (1) Forming the Schro¨dinger equation within the adiabatic framework (which also includes the non-adiabatic coupling terms) and (2) employing a unitary transformation that eliminates these terms from the adiabatic Schro¨dinger equation and replacing them with the relevant potential coupling terms [34–36,57]. This two-step process creates, as will be shown, an opportunity to study the features of the non-adiabatic coupling terms and the results of this study constitute the main subject of this chapter.

The theoretical foundations for this study were laid in a 1975 publication [34] in which this transformation matrix, hence to be termed the adiabatic-to- diabatic transformation matrix, was shown to be a solution of an integral equation defined along a given contour. In what follows, this equation will be termed as a line integral. The line integral reduces, for the two-state case, to an ordinary integral over the corresponding non-adiabatic coupling term, and yields the adiabatic-to-diabatic transformation angle [34–36]. In addition, the sufficient conditions that guarantee the existence and the single-valuedness of the integral-equation solution (along a contour in a given region in configuration space) were derived. In this context, it was shown that these conditions, hence termed the curl conditions, are fulfilled by the system of Born–Oppenheimer–Huang eigenfunctions that span a full-Hilbert space [34], and sometimes, under certain conditions, also span a sub-Hilbert space [8–10].

These two findings form a connection of the theory of the electronic nonadiabatic coupling terms with the Yang–Mills isotopic gauge transformation [58,59]. The existence of the curl conditions may lead to nonzero Yang–Mills fields as will be proposed in Section XIV. Still, it is important to emphasize that the curl condition as it emerges from our theory and the Yang–Mills field that is a quantum mechanical extension of the classical electromagnetic theory are far from being identical or of the same origin.

In 1992, Baer and Englman [55] suggested that Berry’s topological phase [60–62], as derived for molecular systems, and likewise the Longuet-Higgins phase [14–17], should be related to the adiabatic-to-diabatic transformation angle as calculated for a two-state system [56] (see also [63]). Whereas the Baer–Englman suggestion was based on a study of the Jahn–Teller conical intersection model, it was later supported by other studies [11,12,64–75]. In particular, it can be shown that these two angles are related by comparing the ‘‘extended’’ Born–Oppenheimer approximation, once expressed in terms of the gradient of the Longuet-Higgins phase (see Appendix A) and once in terms of the two-state non-adiabatic coupling term [75].