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

= =: ∞� ǫ k ⎛ ⎝ k=0 ∞� ǫ k Φk, k=0 k� j=0 � Ψ (j) 1 (X, Y )f(k−j) (X, Y ) + Ψ (j) 2 (X, Y )g(k−j) � (X, Y ) + ψ (k) ⊥ ⎞ (X, Y ) ⎠ , where {Ψ1(ǫX, ǫY ), Ψ2(ǫX, ǫY )} is the electronic eigenfunction basis. The f (k) and g (k) have no electronic dependence (they are scalar functions) and ψ (k) ⊥ has both electronic and nuclear dependence. ⎛ Recall that for |l| �= 0, if ⎝ f[|l|] (r) g [|l|] ⎞ ⎠ is an eigenfunction **of** H (r) [|l|] UZ with eigenvalue E, then ⎛ ⎝ f[−|l|] (r) g [−|l|] (r) ⎞ ⎠ = ⎛ ⎝ g[|l|] (r) f [|l|] (r) ⎞ ⎠ is an eigenfunction **of** H [−|l|] UZ have two-fold degenerate eigenfunctions **of** H2 **of** the form (recall r = a 1/4 ρ) and ⎛ ⎝ F (0) G (0) ⎞ ⎠ =: ⎛ U Z ⎝ ei|l|φ f [|l|] ⎞ (r) ⎠ = 1 √ 2 ⎛ U Z ⎝ e−i|l|φ g [|l|] ⎞ (r) ⎠ = e −i|l|φ f [|l|] (r) = ⎛ e i|l|φ g [|l|] (r) ⎝ ei(|l|−1)φ f [|l|] (r) + e i(|l|+1)φ g [|l|] (r) 1 √ 2 ⎛ ⎝ i � e i(|l|−1)φ f [|l|] (r) − e i(|l|+1)φ g [|l|] (r) � ⎛ ⎝ F (0) G (0) with eigenvalue E. So if |l| �= 0, we ⎞ e −i(|l|−1)φ f [|l|] (r) + e −i(|l|+1)φ g [|l|] (r) −i � e −i(|l|−1)φ f [|l|] (r) − e −i(|l|+1)φ g [|l|] (r) � ⎞ ⎠ (6.1) ⎠ (6.2) By tak**in**g appropriate l**in**ear comb**in**ations, these degenerate zeroth order functions lead to two orthogonal quasimodes us**in**g the perturbation formulas **of** chapter 3, possibly degenerate (no splitt**in**g) or non-degenerate (splitt**in**g). We adopt the follow**in**g nomenclature: We refer to the eigenfunctions **of** H2 that arise from the eigenfunctions **of** H [|l|] UZ , where |l| �= 0, as +|l| states. We refer to the eigenfunctions **of** H2 that arise from the eigenfunctions **of** H [−|l|] UZ **of** H2 that arise from the eigenfunctions **of** H [|l|=0] U Theorem 6.1. Let L TOT z 0 < ˜ b < 1. Then: , where |l| �= 0, as −|l| states. We refer to the eigenfunctions as |l| = 0 states. be the operator **of** total angular momentum around the z-axis and 40 ⎞ ⎠

1. For l �= 0, each +|l| state generates a quasimode ΦA ǫ **of** H(ǫ) that satisfies LTOT z ΦA ǫ = |l|ΦA ǫ . The correspond**in**g degenerate −|l| state generates a quasimode ΦB ǫ that satisfies ΦB ǫ = Φ A ǫ and LTOT z ΦB ǫ = −|l|ΦBǫ . The ΦAǫ and ΦBǫ quasimodes are orthogonal, and asymptotic to two-fold degenerate eigenfunctions **of** H(ǫ). We see that l**in**ear comb**in**ations **of** the these two-fold degenerate ±|l| states also generate valid quasimodes. 2. Each |l| = 0 state generates a quasimode that is asymptotic to a non-degenerate eigenfunction **of** H(ǫ). In either case, the zeroth order **of** the electronic eigenfunction basis vectors Ψ1(0,0) and Ψ2(0,0) are l**in**ear comb**in**ations **of** eigenfunctions **of** L el z with eigenvalues ±1. Remark: The physical mean**in**g **of** l is now apparent. It corresponds to the total angular momentum about the z-axis **of** the wave function be**in**g approximated. From the pro**of** to follow, it will be clear that the zeroth order Φ0 **of** a quasimode, can be constructed to satisfy L TOT z In this case it is a l**in**ear comb**in**ation **of** two states **of** the form where Pro**of**: ΞlTOT z −1(�rnuc) ˜ Ψ+(�rel) and ΞlTOT z +1(�rnuc) ˜ Ψ−(�rel), L el z ˜ Ψ+ = ˜ Ψ+, L el z ˜ Ψ− = − ˜ Ψ−, L nuc z ΞlTOT z −1 = (l TOT z − 1)Ξ l TOT z −1, L nuc z Ξ l TOT z +1 = (l TOT z Φ0 = l TOT z + 1)Ξl TOT z +1. S**in**ce [H(ǫ), L TOT z ] = 0, we know that the true eigenfunctions Ψ(ǫ) **of** H(ǫ) can be constructed to satisfy L TOT z implies that L TOT z Ψ(ǫ) = l TOT z Ψ(ǫ) = − l TOT z Ψ(ǫ), at each ǫ **in** a neighborhood **of** 0, for some l TOT z Ψ(ǫ), s**in**ce L TOT z Ψ(ǫ) = − L TOT z Φ0. ∈ Z. This Ψ(ǫ). We can therefore arrange so that the asymptotic series Φǫ = �∞ k=0 ǫkΦk satisfies LTOT z Φǫ = lTOT z Φǫ at each order **of** ǫ. We then know that each order Φk **of** the quasimode, and its complex conjugate, are eigenfunctions **of** L TOT z We now separate **in**to two cases: Case 1: |l| �= 0 with eigenvalues l TOT z and −l TOT z respectively. In this case, we have degenerate zeroth order states **of** the form **in** (6.1) and (6.2). Regard- less **of** whether � splitt**in**g occurs, assume that we depart from zeroth order with a correct l**in**ear f (0) comb**in**ation g (0) � � F (0) = α G (0) � � F (0) + β G (0) � , so that this leads to a valid quasimode, which satisfies L TOT z Φǫ = l TOT z Φǫ. Then the zeroth order Φ0 must also be an eigenfunction **of** 41

- 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: 4 3.5 3 2.5 2 1.5 1 0.5 0 0 0.1 0.2
- 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