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

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

= =: ∞� ǫ k ⎛ ⎝

= =: ∞� 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 taking appropriate linear combinations, these degenerate zeroth order functions lead to two orthogonal quasimodes using the perturbation formulas of chapter 3, possibly degenerate (no splitting) or non-degenerate (splitting). We adopt the following 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 corresponding 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 linear combinations 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 linear combinations of eigenfunctions of L el z with eigenvalues ±1. Remark: The physical meaning of l is now apparent. It corresponds to the total angular momentum about the z-axis of the wave function being approximated. From the proof 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 linear combination of two states of the form where Proof: 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. Since [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), since 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 into 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 � splitting occurs, assume that we depart from zeroth order with a correct linear f (0) combination 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

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