1. Remarks on the phase change of the Jahn-Teller electronic wave function upon going once around the conical intersection in vibrational coordinate space Jon T. Hougen Sensor Science Division, NIST, Gaithersburg, MD, USA Outline of talk 1. Short introduction, using words. 2. Long explanation, using equations. 1

2. We know from an early paper by Longuet-Higgins, Öpik, Pryce, and Sack in Proc. Roy. Soc. 244A (1958) 1-16 that: The Born-Oppenheimer fixed-nucleus electronic wavefunction in the Jahn-Teller problem transforms into its negative if the nuclear coordinates are taken once around the conical intersection, provided that this electronic wavefunction  (q e ;Q v ) is chosen to be real. Various aspects of this statement are the subject of today’s talk. In particular, what happens when we relax the requirement that  (q e ;Q v ) be real? 2

3. The answer was given 30 years ago by C.A. Mead, D. G. Truhlar, JCP 70 (1979) 2284-2296: We can make the electronic wavefunction transform in many different ways. In particular we can make it transform into itself when the nuclear coordinates are taken once around the conical intersection. One conclusion (at least for me): These transformation properties are not as fundamental as they are sometimes thought to be, because they can be varied at will within the usual confines of quantum mechanical algebra. Now look at some wave functions to try to understand this in more detail. 3

4. Specialize today’s discussion to: XY 3 molecule with C 3v symmetry Doubly degenerate electronic state ( e E) Doubly degenerate vibrational state ( v E) First-order Jahn-Teller interaction term Just do a quick review of the problem above in 5 slides before discussing Berry phase implications, from M.V. Berry, Proc. Roy. Soc. A392 (1984) 45-57. 4

5. 5 Hamiltonian: H = H e + H v + H ev H e = E e = constant energy of electronic state H v = (1/2m)(P x 2 + P y 2 ) + (½)k(Q x 2 + Q y 2 ) = doubly degenerate harmonic oscillator H ev = k JT (Q + e +2i  + Q - e -2i  ) = first-order Jahn-Teller term for C 3v Basis set:  e (q e )  v (Q,  ) = |  =  1  |v =1, =  1   e +i  e +i 

6. 6 To get the electronic eigenfunctions for fixed nuclei, take matrix elements of H in the electronic basis set |  =  1  only (  d  only !) |  = -1  |  = +1   = -1| E e + (½)kQ 2 k JT Qe -i   = +1| k JT Qe +i  E e + (½)kQ 2 The eigenvalues of this 2x2 matrix are not of interest today, but they give the well known pair of potential surfaces with a conical intersection, shown in the next slide.

7. 7 From Longuet-Higgins, Öpik, Pryce, and Sack, Proc. Roy. Soc. 244A (1958) 1-16:

8. 8 The eigenvectors are of interest today. Look again at the 2x2 matrix. |  = -1  |  = +1  -1 +1  = -1| E e + (½)kQ 2 k JT Qe -i   0 e -i   = +1| k JT Qe +i  E e + (½)kQ 2 e +i  0 +1 +e +i  +1 -e +i  E + E - +e -i  /2 +e +i  /2 +e -i  /2 -e +i  /2 E + E - Complex Real  (  +2  ) = +  (  )  (  +2  ) = -  (  )

9. 9 Eigenfunctions = coefficient*electronic basis function Complex Real (1/2  ) [e -i  + e +i  e +i  ] (1/2  ) [e -i  e -i  /2 + e +i  e +i  /2 ] (1/2  ) [e -i  - e +i  e +i  ] (i/2  ) [e -i  e -i  /2 - e +i  e +i  /2 ] Transformation properties of these complex eigenfunctions under PI operations are “normal” for C 3v. (123) [e -i  + e +i  e +i  ] = e +2  i/3 [e -i  + e +i  e +i  ] (123) [e -i  - e +i  e +i  ] = e +2  i/3 [e -i  - e +i  e +i  ] but are not normal for the real eigenfunctions (123) [e -i  e -i  /2 + e +i  e +i  /2 ] = e +i  [e -i  e -i  /2 + e +i  e +i  /2 ] and cannot be treated with ordinary PI group.

10. How does this all fit with the paper by M.V. Berry in Proc. Roy. Soc. A392 (1984) 45-57 on Quantal phase factors accompanying adiabatic changes, from which the term “Berry phase” comes? First, what does Berry actually do in his paper? Summarize in 4 points on 2 slides the vector calculus presented in his section 2 on General Formula for Phase Factor We need to look at these mathematical details to understand what (in my opinion) is going wrong when Berry’s paper is applied to the JT problem. 10

11. 1. He defines, for a given Born-Oppenheimer electronic state  e (q e ;Q x,Q y ), a vector u(Q x,Q y ) in vibrational coordinate space u(Q x,Q y ) =   e (q e ;Q x,Q y )* [  Qx,Qy  e (q e ;Q x,Q y )] dq e 2  =  0  e (  ;Q x,Q y )* [  Qx,Qy  e (  ;Q x,Q y )] d  Here we use a 2-D vibrational space for the v E state. 2. He defines the “geometrical phase change”  (C) for  e (q e ;Q x,Q y ) by a line integral around a closed circuit C  (C)  i  C u(Q x,Q y )  dR = i  C u(Q x,Q y )  (i dQ x + j dQ y ) 11

12. 3. He uses Stokes’ (3-D) or Green’s (2-D) theorem, to equate the line integral to a surface integral  C u(Q x,Q y )  dR =  S (  u)  dS =  S (  u)  k dQ x dQ y 4. He uses the surface integral to conclude that his treatment gives a unique value for  (C) for any choice of (differentiable) phase factor exp[i f(Q x,Q y )] for the Born-Oppenheimer electronic wave function  e (q e ;Q x,Q y ). 12

13. But there is some trouble here, because for today’s real and complex electronic wavefunctions the line integral definition of  (C), where C is a closed curve enclosing the conical intersection, gives  (C) = 0 for real (LH’s) BO electronic wavefunction  (C) =  for complex BO electronic wavefunction. which is not a unique value independent of phase factor. On the other hand, the surface integral definition gives  (C) = 0 for both the real and complex wavefunction, which is a unique value independent of phase factor. 13

14. The mathematical problem is as follows (in my opinion): We do NOT USE u(Q x,Q y ) in Cartesian coordinates, BUT instead use u(Q,  ) in polar coordinates, where neither  nor  is defined at Q = 0, i.e., neither  nor  is defined at the conical intersection. Only   arctan(Q y /Q x ) occurs in the phase factors. For f(  ) phase factors, the requirements for differentiability of the phase factor that are necessary for the application of Stokes’ or Green’s theorem (mentioned only once by Berry, & only in parentheses) are not (in my opinion) satisfied. Conclusion of this talk (controversial): Berry’s formalism is not applicable to this Jahn-Teller problem. 14

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16. 16 It is possible to redo the Longuet-Higgins et al. calculation of the vibronic energy levels at the bottom of a deep Jahn-Teller moat, using only integer quantum numbers, and only ordinary C 3v group theory, but we have no time for that today, so just give the form of the starting wavefunction in the lower well.  ev =  e (q e ;Q v )  v (Q) = [e -i  - e +i  e +i  ] e [e +i   (Q)] v Important point: is an integer, not a half-integer, in this calculation.

17. Today, for all values of the two vibrational coordinates of the degenerate vibration, we seek: 1. Potential surface V 2. Electronic wavefunctions e  and energy levels Usually, for high-resolution work, we seek: Vibronic eigenfunctions ev  and energy levels or, Rovibronic eigenfunctions evr  and energy levels, To compare with experiment (but not today). 17