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

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

which gives rise to ⎛

which gives rise to ⎛ ⎝ h11(x, y) h12(x, y) h21(x, y) h22(x, y) ⎞ ⎠ = ⎛ ⎜ ⎝ a + b 2 0 ˜ρ 2 0 a − b 2 We do not consider this case. Aside from being uninteresting, it implies that the basis vectors are the eigenfunctions of h(x, y) (at least to leading order). We assume that the off diagonal terms in (3.3) are bxy, since the −bxy case is related by the trivial change of coordinates y ↦→ −y. 3.3 The Formal Expansion To construct the quasimodes in theorem 2.1, we introduce the scaled variables (X,Y ) = (x/ǫ,y/ǫ). The intuition of the Born-Oppenheimer approximation suggests that the adiabatic effects will occur on the (x,y) = (ǫX,ǫY ) scale, whereas the semi-classical motion of the nuclei is determined on the (X,Y ) scale. In terms of the (X,Y ) variables, the Hamiltonian in (2.2) is H(ǫ) = − ǫ2 2 ∆X, Y + h(ǫX, ǫY ). We define H to be the Hilbert space L 2 (IR 2 , dX dY ; C 2 ) and we denote the inner product on this space by 〈 ·, · 〉H. We seek solutions to H(ǫ) Ψ(ǫ, X, Y ) = E(ǫ) Ψ(ǫ, X, Y ). The wave function Ψ(ǫ, X, Y ) can be written in terms of the orthonormal basis functions {Ψ1(x, y), Ψ2(x, y) } from (3.1) and (3.2) as Ψ(ǫ, X, Y ) = f(ǫ, X, Y ) Ψ1(ǫX, ǫY ) + g(ǫ, X, Y ) Ψ2(ǫX, ǫY ) + ψ⊥(ǫ, X, Y ), (3.11) where 〈 ψ⊥, Ψi 〉el = 0. Substituting (3.11) in H(ǫ) Ψ(ǫ, X, Y ) = E(ǫ) Ψ(ǫ, X, Y ) gives three equations; one along Ψ1, one along Ψ2, and one in span{Ψ1, Ψ2} ⊥ . We denote the projection on span{Ψ1, Ψ2} ⊥ by P⊥. Along Ψ1: − ǫ2 2 ∆X, Y f + h11 f + h12 g − ǫ2 2 〈 Ψ1, ∆X, Y ψ⊥ 〉el − ǫ4 2 f 〈 Ψ1,∆x, y Ψ1 〉el − ǫ4 2 g 〈 Ψ1,∆x, y Ψ2 〉el − ǫ 3 � ∂g ∂X ∂Ψ2 〈 Ψ1, ∂x 〉el + ∂g ∂Y ˜ρ 2 ⎞ ⎟ ⎠ . 〈 Ψ1, ∂Ψ2 ∂y 〉el = E(ǫ) f. (3.12) Above we have used that 〈 Ψi, ∂Ψi ∂x 〉el = 0, which we know from normalization and the fact that the electronic basis vectors were chosen real. Along Ψ2 we get a similar equation with 10 �

f ↔ g, Ψ1 ↔ Ψ2, h11 ↔ h22, h12 ↔ h21. In span{Ψ1, Ψ2} ⊥ : − ǫ2 2 P⊥ [ ∆X, Y ψ⊥ ] + (h P⊥) ψ⊥ − ǫ4 2 f P⊥ [ ∆x, y Ψ1 ] − ǫ4 − ǫ 3 � ∂f ∂X P⊥ � � ∂Ψ1 + ∂x ∂f ∂Y P⊥ � � ∂Ψ1 + ∂y ∂g ∂X P⊥ � � ∂Ψ2 + ∂x ∂g ∂Y P⊥ 2 g P⊥ [ ∆x, y Ψ2 ] � ∂Ψ2 ∂y = E(ǫ) ψ⊥. (3.13) We adopt the following notation for simplicity: Tij(x, y) = 〈 Ψi,∆x, y Ψj 〉el, Aij(x, y) = Bij(x, y) = � � Ψi, ∂Ψj � , ∂x el Ψi, ∂Ψj ∂y We have identities involving these quantities since {Ψ1, Ψ2 } are orthonormal and real valued. For instance we know the diagonal elements of A and B are zero and A12 = −A21, B12 = −B21. Now we expand all functions and operators with ǫ dependence. For example, f(ǫ, X, Y ) = ∞� k=0 ǫ k f (k) (X, Y ) . For functions and operators with exclusively (x, y) dependence, we know the form of the expansions. For example, Ψ1(x, y) = Ψ1(ǫX, ǫY ) = Ψ (k) 1 (X, Y ) = ∞� k=2 + + + ǫ k ∞� k=2 ∞� k=4 ∞� k=3 k� j=0 � − 1 � 2 ǫ k ǫ k ǫ k k� j=2 k� j=4 k� j=3 1 j!(k − j)! ∆X, Y f (k−2) + � − 1 � 2 � . el ∞� k=0 � � ǫ k Ψ (k) 1 (X, Y ), where ∂ k Ψ1 ∂x j ∂y k−j (0,0) Xk Y k−j . Equations (3.12) and (3.13) become: ∞� k=0 ǫ k k� j=0 〈 Ψ (j−2) 1 , ∆X, Y ψ (k−j) ⊥ 〉el � − 1 � � T 2 (j−4) 11 f (k−j) + T (j−4) � (k−j) 12 g � − A (j−3) 12 ∂ ∂X − B(j−3) 12 ∂ ∂Y 11 � h (j) 11 f(k−j) + h (j) � 12 g(k−j) � g (k−j) = ∞� k=0 ǫ k k� j=0 E (j) f (k−j) (3.14)

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