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A Born-Oppenheimer Expansion in a Neighborhood of a Renner ...

where Cm, Dm, Cl,j, Dl,j < ∞. We see that this term is **in**deed **of** order greater or equal to O(ǫ K+2 ). All **of** the terms **of** χ1 and χ⊥ can be handled **in** a similar fashion us**in**g the results **of** theorem 4.9. The term **in**volv**in**g the derivatives **of** F **in** equation (7.1) are handled us**in**g theorem 4.9 as well. The derivatives **of** F are supported away from the orig**in** and the terms **of** Ψǫ,K are exponentially decay**in**g. We consider the terms **in**volv**in**g derivatives with respect to X. Let M be larger than sup IR 2 | ∂2 F ∂x2 | and supIR2 | ∂F ∂x ��� � �� �� ∂2 �� , F(ǫρ) ∂X2 Ψǫ,K ≤ M e −γ√ 1+R 2 /ǫ 2 ≤ O(ǫ ∞ ). |. Then, us**in**g theorem 4.9, one can easily show that �� �� �� �� Hnuc⊗Hel � ǫ 2 �� �� � �e γ〈x〉 Ψǫ,K The conclusion **of** the theorem follows. � Appendix �� �� �� Hnuc⊗Hel �� �� + 2ǫ �� �� eγ〈x〉 ∂Ψǫ,K ∂X �� � �� �� �� Hnuc⊗Hel We now argue that the odd terms **in** the E(ǫ) series must be zero. See [9] for a detailed pro**of** which utilizes the perturbation formulas derived **in** chapter 3. The Hamiltonian **of** **in**terest **in** terms **of** the scaled nuclear coord**in**ates (X,Y ) = (x/ǫ,y/ǫ) is given by H(ǫ) = −ǫ2 2 ∆X,Y + h(ǫX,ǫY ), where h(x,y) is the electronic hamiltonian that also conta**in**s the nuclear repulsion terms. If E(ǫ) is an eigenvalue **of** H(ǫ), then E(−ǫ) is an eigenvalue **of** H(−ǫ) = −ǫ2 2 ∆X,Y + h(−ǫX, −ǫY ). Under the unitary change ˜ X = −X, ˜ Y = −Y , we see that H(−ǫ) becomes H(−ǫ) = −ǫ2 2 ∆ ˜ X,˜Y + h(ǫ ˜ X,ǫ ˜ Y ). It is clear that H(ǫ) and H(−ǫ) share the same eigenvalues. This does not immediately imply E(ǫ) = E(−ǫ), s**in**ce there could be a pair **of** eigenvalues related by EA(ǫ) = EB(−ǫ). However, this would imply that E (2) A = E(2) B , and then theorem 6.1 implies EA(ǫ) = EB(ǫ). Therefore, E(ǫ) = E(−ǫ), and as a result the odd terms **in** the expansion must vanish. 46

Acknowledgement It is a pleasure to thank Pr**of**essor George A. Hagedorn for his advise and many useful comments. This research was supported **in** part by National Science Foundation Grant DMS–0600944 while at Virg**in**ia Polytechnic Institute and State University, and also by the Institute for Mathematics and its Applications at the University **of** M**in**nesota, with funds provided by the National Science Foundation. References [1] M. **Born** and R. **Oppenheimer**, Zur Quantentheorie der Molekeln. Ann. Phys. (Leipzig) 84, 457-484 (1927). [2] J. M. Brown and F. Jørgensen, **in** Advances **in** Chemical Physics, edited by I. Prigog**in**e and S. A. Rice, John Wiley & Sons, Inc., New York, 1983, Vol. 52, p. 117. [3] J.-M. Combes and R. Seiler, **in** Quantum Dynamics **of** Molecules: The New Experimental Challenge to Theorists, edited by R. G. Wooley, NATO Advanced Study Institutes Series, Series B, Physics v. 57, Plenum Press, New York, 1980, pages 435-482. [4] J.-M. Combes, P. Duclos, and R. Seiler, **in** Rigorous Atomic and Molecular Physics, edited by G. Velo and A. Wightman, Plenum Press, New York, 1981, pages 185-212. [5] G. A. Hagedorn, Ann. Inst. H. Po**in**caré Sect. A. 47, 1-16 (1987). [6] G. A. Hagedorn, Commun. Math. Phys., 116, 23-44 (1988). [7] G. A. Hagedorn and J. H. Toloza, Int. J. Quantum Chem. 105, 463-477 (2005). [8] G. A. Hagedorn and A. Joye, Commun. Math. Phys. 274, 691-715 (2007). [9] M. S. Herman, **Born**-**Oppenheimer** Corrections Near a **Renner**-Teller Cross**in**g, Ph. D. Thesis, Virg**in**ia Polytechnic Institute and State University. [10] G. Herzberg, E. Teller, Z. Phys. Chemie B21, 410 (1933). [11] G. Herzberg, Electronic Spectra **of** Polyatomic Molecules (Pr**in**ceton, New Jersey, D. Van Nostrand Company, Inc., 1966). [12] P. Jensen, G. Osmann, and P. R. Bunker, **in** Computational Molecular Spectroscopy, edited by P. Jensen and P. R. Bunker, John Wiley & Sons, Inc., New York 2000, p. 485. [13] M. Kle**in**, A. Mart**in**ez, R. Seiler, and X. Wang, Commun. Math. Phys. 143, 607-639 (1992). [14] T. J. Lee, D. J. Fox, H. F. Schaefer III, R. M. Pitzer, J. Chem. Phys., 81, 356-361 (1984). [15] A. Messiah, Quantum Mechanics (New York, John Wiley & Sons, Inc., 1958). 47

- Page 1 and 2: A Born-Oppenheimer Expansion in a N
- Page 3 and 4: there exist quasimode energies of t
- Page 5 and 6: to both electronic states of the R-
- Page 7 and 8: 3 The Construction of the Quasimode
- Page 9 and 10: with eigenvalues ˜ E±(x, y) = ±
- Page 11 and 12: f ↔ g, Ψ1 ↔ Ψ2, h11 ↔ h22,
- Page 13 and 14: So we get E (1) = 0 and ψ (1) ⊥
- Page 15 and 16: and ⎛ Using what we know through
- Page 17 and 18: and ⎛ ⎝ f(k−2) g (k−2) ⎞
- Page 19 and 20: This easily follows from � �
- Page 21 and 22: H ± 1 = W −1 H ± 0 W = ⎛ ⎜
- Page 23 and 24: so that H2 = H0 + V . Note that for
- Page 25 and 26: Let β > 0. Then, � e 2γ〈x〉
- Page 27 and 28: � Proposition 4.7. Let Ψ = ≥ |
- Page 29 and 30: � = IR n \BS(t) � ≥ IR n \BS(
- Page 31 and 32: form (see Theorem XII.5 in [19])
- Page 33 and 34: Let r = a1/4 ρ, ˜b = b , and a HU
- Page 35 and 36: Denote the eigenfunctions of H [±|
- Page 37 and 38: for the larger values of |l|. The s
- Page 39 and 40: 4 3.5 3 2.5 2 1.5 1 0.5 0 0 0.1 0.2
- Page 41 and 42: 1. For l �= 0, each +|l| state ge
- Page 43 and 44: Φ A ǫ is also asymptotic to an ei
- Page 45: We have (H(ǫ) − Eǫ,K)Φǫ,K = F