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

Let P1(x, y) denote the orthogonal projection onto this vector, i.e., Next, def**in**e and P1(x, y) = |Ψ1(x, y) 〉 〈Ψ1(x, y) |. χ(x, y) = ( 1 − P1(x, y)) P(x, y) ψ2, Ψ2(x, y) = 1 � 〈χ(x, y), χ(x, y) 〉 χ(x, y). (3.2) Then {Ψ1(x, y), Ψ2(x, y) } is an orthonormal basis for the range **of** P(x, y). From the formula [19] P(x,y) = 1 2πi � C (λ − h(x,y)) −1 dλ, where C is a closed path **in** the complex plane encircl**in**g E1(˜ρ) and E2(˜ρ) but no other spectrum **of** h(x,y), we see that these vectors are smooth **in** x and y, s**in**ce we have assumed that the potentials are smooth and hence the resolvent **of** h(x,y) is as well (recall we are only work**in**g **in** a neighborhood **of** the orig**in** (x,y) = (0,0)). Note that we can arrange for these vectors to be real, which we assume has been done. 3.2 The Matrix Elements **of** the Electronic Hamiltonian The span **of** {Ψ1(x, y), Ψ2(x, y) } is an **in**variant subspace for h(x, y). Us**in**g coord**in**ates **in** this basis, the restriction **of** h(x, y) to this subspace is unitarily equivalent to the real symmetric matrix ⎛ where ⎝ h11(x, y) h12(x, y) h21(x, y) h22(x, y) ⎞ ⎠ , hjk(x,y) = 〈Ψj(x, y), h(x, y)Ψk(x, y) 〉. Aga**in**, s**in**ce we have smooth potentials, hij(x,y) can be expanded **in** powers **of** x and y. S**in**ce we assume the degeneracy splits at second order, the eigenvalues **of** this matrix are E1(x, y) = a+b 2 (x2 +y2 ) + O(˜ρ 4 a − b ) and E2(x, y) = 2 (x2 +y2 ) + O(˜ρ 4 ). Us**in**g these expressions for the eigenvalues we show that up to an (x,y)-**in**dependent unitary transformation, this matrix is ⎛ a + b 2 x 2 + a − b 2 y 2 ± bxy ⎜ ⎝ ± bxy a − b x 2 2 a + b + y 2 2 ⎟ ⎠. (3.3) To show this, we consider a traceless, real symmetric matrix ⎛ ˜h11(x, y) h12(x, ˜ y) ⎝ ˜h21(x, y) −˜ ⎞ ⎠ , h11(x, y) (3.4) 8 ⎞

with eigenvalues ˜ E±(x, y) = ± ˜ρ 2 + O(˜ρ 4 ). The form **in** (3.3) will follow from the analysis below. Us**in**g (3.4), we have the characteristic equation ˜E 2 ± + O(˜ρ6 ) = ˜ h 2 11 + ˜h 2 12 . (3.5) By expand**in**g **in** powers **of** x and y and equat**in**g orders **in** the above equation, it can be easily shown that the constant and l**in**ear terms **of** ˜ h11 and ˜ h12 must vanish. We then write, ⎛ ⎝ ˜h11(x, y) ˜ h12(x, y) ˜h21(x, y) − ˜ h11(x, y) ⎞ ⎠ = Ax 2 + B y 2 + C xy + O(˜ρ 3 ), (3.6) where A, B, and C are traceless 2 by 2 matrices with constant entries. We can apply a constant unitary transformation to (3.6) that diagonalizes A, which we assume has been done. An obvious consequence **of** (3.5) and ˜ E±(x, y) = ± ˜ρ 2 + O(˜ρ 4 ) is that if A is diagonal, it must be We let B = ⎛ ⎝ b11 b12 b12 −b11 A = ⎞ ⎛ ⎝ 1 0 0 −1 ⎞ ⎠ . ⎠ , and C = ⎛ ⎝ c11 c12 c12 −c11 and use (3.5) with ˜ E±(x, y) = ± ˜ρ 2 + O(˜ρ 4 ) to solve for bij and cij by equat**in**g the powers **of** x and y. This gives us four equations, the first equation comes from the y 4 coefficients, the second comes from the x 2 y 2 coefficients, etc. These equations have 3 solutions. Two **of** the solutions are which give ⎛ ⎝ y 4 : 1 = b 2 11 + b2 12 (3.7) x 2 y 2 : 2 = 2b11 + c 2 11 + c2 12 (3.8) xy 3 : 0 = 2(b11c11 + b12c12) (3.9) x 3 y : 0 = 2c11 (3.10) (b11,b12,c11,c12) = (−1,0,0, ±2), ˜h11(x, y) ˜ h12(x, y) ˜h21(x, y) − ˜ h11(x, y) ⎞ ⎠ = ⎛ ⎝ x2 − y 2 ± 2xy ± 2xy −(x 2 − y 2 ) These solutions give rise to (3.3). The only other solution **of** equations (3.7)-(3.10) is (b11,b12,c11,c12) = (1,0,0,0), 9 ⎞ ⎠, ⎞ ⎠ .

- 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: 3 The Construction of the Quasimode
- 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