Diabatic versus Adiabatic Calculations of Torsion-Vibration Interactions Jon T. Hougen Senior Science Division, NIST, Gaithersburg, MD , USA Material is taken from a paper (almost finished) by Li-Hong Xu, Jon T. Hougen, R. M. Lees: On the physical interpretation of ab-initio normal-mode coordinates for the three C-H stretching vibrations of methanol along the internal rotation path Next talk: Li-Hong Xu on other material from this paper. 1

Outline ~n slides Topic 7 Theoretical concepts used in this talk 3 of 20 Example of diabatic calculation 3 of 20 Example of adiabatic calculation 1 Concluding remarks 2

Adiabatic = Impassable in Greek Adiabatic = No transfer of heat in thermodynamics Non-adiabatic = Transfer of heat occurs Adiabatic is used in classical mechanics when a system contains slow and fast motions, and the fast motion adjusts “instantaneously” to the slow one. Adiabatic and Diabatic are used when discussing quantum mechanical systems that contain slow motions with small energy quanta and fast motions with large energy quanta. 3

4 Adiabatic and Diabatic diatomic potential curves Fast motion = electrons Slow motion = vibration Adiabatic curves (ab initio) Slow motion is really slow No jump between curves Diabatic Curves Slow motion is a little fast (Like a car on a curve) System can’t make the turn, and jumps between curves

The molecular problem for today’s talk: Fast SAV’s = three C-H stretches of CH 3 OH, normally called 3, 2, 9, with vibrational local-mode C-H i=4,5,6 coordinates  r 4,  r 5,  r 6 or symmetry coordinates S 3, S 2, S 9 or normal coordinates Q 3, Q 2, Q 9 Slow LAM = CH 3 internal rotation =  Diatomic  Methanol  elec (r)  Q 3 (  ), Q 2 (  ), Q 9 (  ) V elec (r)  3 (  ), 2 (  ), 9 (  ), 5

Now look at some key theoretical differences between a diabatic and an adiabatic calculation of “inverted torsional splittings” (A state above E state) in 2 and 9 of methanol. Some papers on regular and inverted torsional splittings in methanol from other groups often at this meeting are: X. Wang, D.S. Perry, J. Chem. Phys. 109 (1998) B. Fehrensen, D. Luckhaus, M. Quack, M. Willeke, T.R. Rizzo, J. Chem. Phys. 119 (2003) E.L. Sibert III, J. Castillo-Chará, J. Chem. Phys. 122 (2005) – J.M. Bowman, X. Huang, N.C. Handy, S. Carter, J. Phys. Chem. A 111 (2007)

Why say that the fast-slow interaction terms are in V for a diabatic calculation (H = T + V), but are in T for an adiabatic calculation? Why say that direct use of the numerical output from quantum chemical packages forces us to do an adiabatic calculation, but indirect use permits a diabatic calculation? Are adiabatic or diabatic calculations: easier and faster to carry out and understand? more accurate under the necessary approximations? 7 F. T. Smith, Phys. Rev. 179 (1969) Diabatic and Adiabatic Representations for Atomic Collision Problems

Diabatic Calculation H = (T fast + V fast ) + (T slow + V slow ) + V fast-slow H = (T CH + V CH ) + (T tors + V tors ) + V CH-tors Born-Oppenheimer approximation: First solve separately the fast and slow parts of H – V CH-tors = (T CH + V CH ) + (T tors + V tors ) to obtain  CH (S 2,S 9 )  tors (  ) and E CH + E tors. Then solve H in the basis  CH (S 2,S 9 )  tors (  ), treating V CH-tors as a perturbation that mixes CH stretches & torsion, to get final  ’s and E’s. 8

Adiabatic Calculation H = (T fast + V fast ) + (T slow + V slow ) + V fast-slow Use B-O approximation again, to solve the same (T slow + V slow ) problem, but solve a different fast part (as Q. Chem. packages do), H – T tors = (T CH + V CH ) + V tors + V CH-tors to obtain  CH (S 2,S 9 ;  ) and E CH (  ). Then use the basis set  CH (S 2,S 9 ;  )  tors (  ), to solve H, where  /   in T tors mixes different  CH (S 2,S 9 ;  ) fcns, to obtain final  ’s and E’s. 9

The four terms in the Diabatic Hamiltonian (no rotation, J=0): H = (T CH + V CH ) + (T tors + V tors ) + V CH-tors = (1/2m)(P x 2 + P y 2 ) + (½)k E (S x 2 + S y 2 ) + FJ  2 + (½)V 3 (1 - cos3  ) + (¼)k 1 [e +i  (S x - iS y ) 2 + e -i  (S x + iS y ) 2 ] + (¼)k 2 [e -2i  (S x - iS y ) 2 + e +2i  (S x + iS y ) 2 ] = Doubly degenerate Harmonic oscillator + Ordinary methyl torsion Hamiltonian + Jahn-Teller-like torsion-vibration term + Renner-Teller-like torsion-vibration term 10

The Diabatic basis set (no rotation, J=0) and selection rules for the four terms in H:  = |v E =1, E  HO |e +im   tors (1/2m)(P x 2 + P y 2 ) + (½)k E (S x 2 + S y 2 )  v E =  E =  m = 0 FJ  2 + (½)V 3 (1 - cos3  )  v E =  E = 0;  m = 0,  3 (¼)k 1 [e +i  (S x - iS y ) 2 + e -i  (S x + iS y ) 2 ] “  v E = 0”;  E =  2;  m =  1 (Jahn-Teller if , m  ) (¼)k 2 [e -2i  (S x - iS y ) 2 + e +2i  (S x + iS y ) 2 ] “  v E = 0”;  E =  m =  2 (Renner-Teller if , m  ) 11

The Diabatic results (no rotation, J=0) E( vt A 1,2 ) = E vE=1 + E( t E)  (½)(k 1 +k 2 )   E /k E E( vt E) = E vE=1 + (½)[E( t E) + E( t A 1 )]  (½){[E( t E) - E( t A 1 )] 2 + (k 1 +k 2 ) 2 (   E /k E ) 2 } 1/2  E vE=1 + (½)[E( t E) + E( t A 1 )]  (½)(k 1 +k 2 )   E /k E The torsional splitting in v E = 1 becomes E( vt A 1,2 ) - E( vt E) = (½)[E( t E)  E( t A 1 )] This is true for Berry phase = -1 (k 2 = 0, only J-T), or Berry phase = +1 (k 1 = 0, only R-T). 12

The two terms in the Adiabatic Hamiltonian (no rotation, J=0): H = (T CH + V CH ) + (T tors + V tors ) = (1/2m)(P P 9 2 ) + (½)k 2 Q (½)k 9 Q F(J   p v (  ) ) 2 + (½)V 3 (1 - cos3  ) = Two non-degenerate Harmonic oscillators V CH-tors has been “diagonalized out” of H Methyl torsion Hamiltonian now has torsion-vibration Coriolis interaction term -2Fp v (  ) J  = -2F   (Q 2 P 9 – Q 9 P 2 )J  13

The Adiabatic basis set (no rotation, J=0) and selection rules for the terms in H:  = |v 2  HO |v 9  HO |e +im   tors (1/2m)P (½)k 2 Q (1/2m)P (½)k 9 Q 9 2  v 2 =  v 9 =  m = 0 F(J   p v (  ) ) 2 + (½)V 3 (1 - cos3  )  v 2 =  v 9 = 0;  m = 0,  3 for J  and cos3   v 2 = -  v 9 =  1;  m = 0 for p v (  ) J  14

Adiabatic results = same as diabatic results E( vt A 1 ) = (3/2)   2 + (1/2)   9 + E( t E) E( vt A 2 ) = (1/2)   2 + (3/2)   9 + E( t E) E( vt E) = (3/2)   2 + (1/2)   9 + (½)[E( t E) + E( t A 1 )] E( vt E) = (1/2)   2 + (3/2)   9 + (½)[E( t E) + E( t A 1 )] The torsional splitting again becomes E( vt A 1,2 ) - E( vt E) = (½)[E( t E)  E( t A 1 )] This is true for Berry phase = -1 (k 2 = 0,   = +1/2), or Berry phase = +1 (k 1 = 0,   = -1). 15

Present results For two simplified models ( = constant, L(  ) = sine or cosine,  = constant) that permit an algebraic solution, it can be shown that a diabatic calculation gives the same inverted A-E torsional splittings in 2 and 9 as an adiabatic calculation does. Future work Try to figure out how to do the adiabatic calculation numerically so that the Gaussian numerical output can be used directly. The main problem is how to deal with derivatives of the very rapidly varying functions (  ), L(  ), and  (  ), as shown in next talk. 16

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Group theory uses 3 groups C s = point group of CH 3 OH at equilibrium G 6 = permutation-inversion group of CH 3 OH C 3v = G 6 = point group of CH 3 F (CH 3 -O-H) Symmetry of vibrational coordinates S and Q A’ A’ A” S and Q in C s point group A 1 E S in G 6 PI group, like (A E) S and Q in C 3v point group of CH 3 F A 1 A 1 A 2 S and Q in G 6 PI group 18

Slow versus Fast motions in molecules Small versus Large energy quanta in molecules  E vib ~ 10 2 cm -1 <<  E elec ~ 10 4 cm -1  E rot ~ 10 0 cm -1 <<  E vib ~ 10 2 cm -1  E LAM ~ 10 2 cm -1 <<  E SAV ~ 10 3 cm -1  E tors ~ 10 2 cm -1 <<  E CH stretch ~ 10 3 cm -1 Small-amplitude vibrations (SAVs) are fast and have large quanta. Large-amplitude motions (LAMs) are slow and have small quanta. 19