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

Remarks: 1. Quasimode estimates correspond to discrete eigenvalues **of** H(ǫ) when Eǫ,K lies below the essential spectrum as characterized by the HVZ theorem [19]. 2. S**in**ce [L TOT z ,H(ǫ)] = 0, quasimodes can be constructed to be eigenfunctions **of** L TOT z , with eigenvalues l TOT z ∈ Z. The eigenstates **of** H(ǫ) correspond**in**g to l TOT z = 0 are non- degenerate, while the eigenstates correspond**in**g to |l TOT z | �= 0 are two-fold degenerate. In particular, there is an l TOT z state and a −l TOT z state, with eigenfunctions that are complex conjugates **of** one another, together form**in**g a degenerate pair **of** states associated with H(ǫ). 3. The first two orders E (0) and E (1) are zero and the second order E (2) is determ**in**ed by the lead**in**g order eigenvalue equation H2Ψ = E (2) Ψ, on the Hilbert space L2 (IR 2 , dX dY ; C2 ), where ⎛ − 1 2 ∆X, a + b Y + 2 X2 + H2 = ⎜ ⎝ a − b 2 Y 2 bX Y − 1 2 ∆X, Y + bX Y a − b 2 X2 + a + b 2 The higher order E (k) are determ**in**ed through the perturbation formulas presented **in** chap- ter 3. All odd order E (k) are zero (see Appendix). 4. The presence **of** the two levels E1(˜ρ) and E2(˜ρ) gives rise to twice the number **of** vibrational levels as usual **in** the follow**in**g sense: If b = 0, the upper and lower component equations **of** the lead**in**g order equation which determ**in**es E (2) , are both two-dimensional harmonic oscillator equations. So, there will be two eigenfunctions, one associated with the each **of** the upper and lower components, for each **of** the usual eigenstates **of** the usual harmonic oscillator. Then for small b, this will give rise to two vibrational states via a perturbative approach, for each **of** the usual harmonic oscillator states. This is shown **in** detail **in** section 5. 5. For small b and ǫ, the ground state **of** H(ǫ) (mean**in**g the lowest vibrational level correspond- **in**g to the R-T pair we are consider**in**g) is degenerate, correspond**in**g to a pair **of** states which are eigenfunctions **of** L TOT z with eigenvalues l TOT z Y 2 ⎞ ⎟ ⎠ . = ±1. In section 5, we give plots which suggest that for approximately 0.925a < b < a the ground state is non-degenerate, corre- spond**in**g to a state with l TOT z = 0. The paper is organized as follows: In section 3, we derive perturbation formulas to construct the quasimodes that will enter **in** our ma**in** theorem. In section 4, we prove various properties **of** the lead**in**g order Hamiltonian that are needed to prove the ma**in** theorem. In section 5, we analyze the lead**in**g order eigenvalue problem. Only some **of** the eigenvalues and eigenfunctions **of** the lead**in**g order equation are solved for exactly. In section 6, the degeneracy structure **of** the full Hamiltonian H(ǫ) is discussed. In section 7, we use the results **of** the previous chapters to prove the ma**in** theorem. 6

3 The Construction **of** the Quasimodes Before we beg**in** the formal expansion, we first look at some properties **of** the electronic eigenvec- tors and eigenvalues, construct electronic basis vectors that are smooth **in** terms **of** the nuclear coord**in**ates, and derive the lead**in**g orders **of** the matrix elements **of** the electronic Hamiltonian **in** this basis. 3.1 The Two-Dimensional Electronic Basis Vectors For the N electrons, as well as the nuclei, we use the same fixed reference frame previously described. Let (rj, θj, zj) be the cyl**in**drical coord**in**ates **of** the jth electron **in** this frame. Suppose that for ˜ρ > 0, ψ(x,y; θ1,θ2, · · · ,θN) : IR 2 → Hel is an electronic eigenvector **of** h(x,y). We have suppressed the dependence on rj and zj because it is irrelevant to the discussion here. The electronic eigenfunctions are **in**variant with respect to a rotation **of** the entire molecule. So, the eigenfunctions have the property ψ(x, y; θ1,θ2, · · · ,θN) = ψ(˜ρ, 0; θ1 − φ,θ2 − φ, · · · ,θN − φ) for ˜ρ > 0. It follows that if ψ(x,y) is cont**in**uous at ˜ρ = 0, then ψ(0,0) has no θj dependence. S**in**ce an eigenvector correspond**in**g to an R-T pair with positive |lel z | value will have some θj dependence at ˜ρ = 0, we do not have well-def**in**ed cont**in**uous electronic eigenfunctions **of** h(x,y) **in** a neighborhood **of** ˜ρ = 0, that correspond to an R-T pair with value |lel z | > 0. We need basis vectors for the two-dimensional eigenspace **of** E1(˜ρ) and E2(˜ρ) that are smooth **in** x and y. The matrix elements **of** h(x,y) **in** our electronic basis determ**in**e the form **of** the lead**in**g order equations to follow. We note that **in** deriv**in**g these matrix elements, we do not use the matrix elements **of** L el z . Only second order splitt**in**g **in** E1 and E2 is needed, as well as the fact that our smooth basis vectors are not eigenvectors **of** h(x,y). This gives rise to **of**f-diagonal terms **in** the basis representation **of** h(x,y). In this sense, the unusual form **of** the lead**in**g order equations can be thought **of** as a result **of** the discont**in**uity **of** the electronic eigenvectors **in** the nuclear coord**in**ates, i.e. there is no smooth electronic basis that diagonalizes the electronic hamiltonian. We note that matrix elements we derive here, are related by an (x,y)-**in**dependent unitary transformation to those given by Yarkony [22]. See also Worth and Cederbaum [21] for a general discussion **of** the topology and classification **of** different types **of** **in**tersections **of** potential surfaces. We now describe our approach. Choose any two normalized orthogonal electronic vectors ψ1 and ψ2 that span the eigenvalue 0 eigenspace **of** h(0,0). Let P(x, y) denote the two dimensional projection onto the electronic eigenspace associated to the two eigenvalues **of** h(x, y). For small x and y, def**in**e Ψ1(x, y) = 1 � 〈ψ1, P(x, y)ψ1〉 P(x, y)ψ1. (3.1) 7

- Page 1 and 2: A Born-Oppenheimer Expansion in a N
- Page 3 and 4: there exist quasimode energies of t
- Page 5: to both electronic states of the R-
- 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 and 46: We have (H(ǫ) − Eǫ,K)Φǫ,K = F
- Page 47 and 48: Acknowledgement It is a pleasure to