1. La Grande Motte, February 2008 Kinetic Energy Operator in curvilinear coordinates: numerical approach A. Nauts Y. Justum M. Desouter-Lecomte L. Bomble (PhD) Spectroscopy, floppy systems control quantum gates

2. La Grande Motte, February 2008 Why we need curvilinear coordinates In quantum dynamics, the calculations are easier with curvilinear coordinates: -The center of mass is separable: X CM =[X CM, Y CM, Z CM ] -The overall rotation is well-described by means of 3 Euler angles:  -The torsion of a chemical fragment (ex: Methyl) can be described by one coordinates: a dihedral angle, . -.... -All 3N curvilinear coordinates will be noted: q=[q i ]

3. La Grande Motte, February 2008 Kinetic energy operator in curvilinear coordinates where and Contravariant conponents of metric tensor extrapotential term (function of J and  )

4. La Grande Motte, February 2008 Where is the problem? With few degrees of freedom, analytical expressions are easily determined: up to 3-4 atom systems or with (quasi) orthogonal coordinates: Jacobi For larger molecular systems, the analytical expression of T is difficult to obtain. For numerical applications the analytical expressions may not be needed, but only the values of the functions f 2 (q) and f 1 (q) on a grid. Ex: Numerical integrations

5. La Grande Motte, February 2008 Example: For the 1D-Kinetic energy operator of methanol: One active (dynamical) coordinate, , and 11 frozen coordinates G(  )=A/B A = -6 R CH 2 (M CH3 M OH R CO 2 - 2 M H R CO (3 M OH R CH Cos(a HCO ) + M CH3 R COH Cos(a HOC )) + M H (3 M CHO R CH 2 Cos(a HCO ) 2 + 6 M H R CH R COH Cos(a HCO ) Cos(a HOC ) + M CH3O R COH 2 Cos(a HOC ) 2 )) Sin(a HCO ) 2 - 9 M H M CH3OH R CH 4 Sin(a HCO ) 4 - R COH 2 (2 M CH3 M O R CO 2 - 12 M H M O R CH R CO Cos(a HCO ) + 6 M H M CO R CH 2 Cos(a HCO ) 2 + 3 M H M CH3O R CH 2 Sin(a HCO ) 2 ) Sin(a HOC ) 2 B = 6 M H R CH 2 R COH 2 Sin(a HCO ) 2 (2 M CH3 M O R CO 2 -12 M H M O R CH R CO Cos(a HCO ) + 6 M H M CO R CH 2 Cos(a HCO ) 2 + 3 M H M CH3O R CH 2 Sin(a HCO ) 2 ) Sin(a HOC ) 2

6. La Grande Motte, February 2008 Where is the problem? With few degrees of freedom, analytical expressions are easily determined: up to 3-4 atom systems or with (quasi) orthogonal coordinates: Jacobi For larger molecular systems, the analytical expression of T difficult to obtain. For numerical applications the analytical expressions may not be needed, but only the values of the functions f 2 (q) and f 1 (q) on a grid. Ex: Numerical integrations

7. La Grande Motte, February 2008 Objective of Tnum [1] To calculate -numerically and exactly -for molecular systems of any size the functions f 2 (q) and f 1 (q) for a given value of q using a Z-matrix definition of the curvilinear coordinates. Similar numerical procedures 1-2 active coordinate(s) (inversion of ammonia [2], ring puckering [3,4], torsion [5] ) 6 active coordinates (inversion of ammonia…) [6] B-matrix used to calculate the gradient and hessian in internal coordinates [1] D. Lauvergnat et al., JCP 2002, 116, p8560 [2] D. J. Rush et al., JPC A 1997, 101, p3143 [3] J. R. Durig et al.,JPC 1994, 98, p9202 [4] S. Sakurai et al.,JCP 1998, 108, p3537 [5] M. L. Senent, CPL 1998, 296, p299 [6] D. Luckhaus, JCP 2000, 113, p1329

8. La Grande Motte, February 2008 IIIb Jacobian and its derivatives (Numerical) Calculation of T I Mass-weighted cartesian coordinates in terms of the curvilinear ones II g matrix and its derivatives IV Kinetic energy operator IIIa G matrix and its derivatives

9. La Grande Motte, February 2008 Cartesian coordinates in BF O1 O2 O1 R H1 O1  R1 O2 a1 H1 O2  R2 O1 a2 H1 phi Z-matrix Cartesian construction with the vectors in any order. => Polyspherical, Jacobi vectors.... Analytical expression bunch of vectors Especially developed for MCTDH

10. La Grande Motte, February 2008 Further transformations: Q zmat => Q dyn Q zmat : Coordinates associated with the Z-matrix or the bunch of vectors Q dyn : Coordinates used in the dynamic (active, inactive) Transformations IdentityLinear combinationsPolar transformations

11. La Grande Motte, February 2008 Rigid constraints:Q inact = Cte Flexible constraints:Q inact = Q inact (Q act ) Rigid or flexible constraints Q dyn is split in active and inactive coordinates. The inactive coordinates are not used in the dynamics.

12. Test : H-CN (Jacobi) Spectrum of HCN/CNH (diagonalization) : [1] F. Gatti, Y. Justum, M. Menou, A.Nauts et X. Chapuisat J. Mol. Spectroscp. 1997, 181, p403. [2] D. Lauvergnat, A.Nauts, Y. Justum et X. Chapuisat, JCP 2001, 114, p6592. [3] D. Lauvergnat, Y. Justum, M. Desouter-Lecomte et X. Chapuisat, Theochem 2001 Spectrum of HCN/CNH (WP) [3] : Z-matrix : C X C (1-  )R N X  R C 180,0 H X r C  N 0,0 Normalization :  (x,R,r)=1 with x=cos(  ) constants  =M C /M CN adiabatic constraint (1d :  )

13. La Grande Motte, February 2008 Use of Tnum: To set up or to check analytical kinetic energy operators: Ethene with constraints, W 2 H + (MCTDH) Spectroscopy: MethylPropanalMethylPropanal, Methanol (1+11D), Fluoroproprene, Ammonia (6D)MethanolFluoropropreneAmmonia Implementation in pvscf with D. Benoit and Y. Scribano Propagation: WP: Optimal control and quantum gates (4D) Single Gaussian WP (60D) and classical trajectories Other groups: Double proton transfer by Harke, JPC A 110, p13014, 2006 WP on 1,3-dibromopropane by R. Brogaard in Denmark

14. La Grande Motte, February 2008 Tnum and MCTDH -In Tnum, the elements of the G tensor can be known only on the full dimensionality grid! => We do not know whether one element is zero or a constant or a function of only 3 variables. - In MCTDH, the KEO has to be given as a sum of "single" mode products What can be done? -Use a fitting procedure -Taylor expansion of G Incompatibility between Tnum and MCTDH! Can be easily used with CDVR!!

15. La Grande Motte, February 2008 Taylor expansion of G around Q 0 -The calculation of the derivatives of G (up to the second order) is already implemented in Tnum. This form can be used with MCTDH This approach is well known: 1.R. Wallace, CP, 11 p189 1975: H 2 O, CH of benzene (Zero order) 2.E. L Sibert III, W. P. Reinhardt, J. T. Hynes, JCP, 81, p1115 1984: CH in benzene (first order) 3.L. Halonen, T. Carrington Jr JCP 88 p4171 1988: H 2 X (third order) 4.L Lespade, S. Robin D. Cavagnat, JPC, 97, p6134, 1993: Cyclohexene (second order)....

16. La Grande Motte, February 2008 Taylor expansion of G around Q 0 The Taylor expansion may give non-hermitian KEO ! Ex: AB-C with Jacobi coordinates : The Taylor expansion (2 d order) 2 d order in x and R. Non-hermitian with Legendre polynomial Always Hermitian with the sine and the HO basis-sets.

17. Example: H 2 O in valence coordinates &geom zmat=T nat=3 / 16. 1. 1 1. 1 2 1 1 1 &niveau nrho=1 read_nameQ=t / r1OH 1. r2OH 1. a 1.6 Z-matrix reference geometry, Q 0 ------------------------------------------------- HAMILTONIAN-SECTION modes | r1OH | r2OH | a ------------------------------------------------ # Zero order part: -1/2*G^ij(Qref) ------------------------------------------------- -0.53125000000000011 |1 dq^2 -0.53125000000000000 |2 dq^2 -1.0643249701438311 |3 dq^2 1.82497014383055053E-003 |1 dq |2 dq 6.24733501900941249E-002 |1 dq |3 dq 6.24733501900940971E-002 |2 dq |3 dq ------------------------------------------------- # First order part: # -1/2*dG^ij/dDQk d./dQi * DQk * d./dQj ------------------------------------------------- 1.0643249701438311 |3 dq^2 |1 q 1.0643249701438320 |3 dq^2 |2 q -6.24733501900941179E-002 |3 dq*q*dq.... defined constraints (here no constraint) Number of terms for (H 2 O) 2 H + (D 2 d): 29,92,279

18. La Grande Motte, February 2008 Advantages/drawback of Tnum + A numerical and exact representation of T is possible including constrained model. + Can deal with very large systems (tested up to 60 degrees of freedom). + Implemented for: Wave packets propagation Time independent methods Classical Trajectories (Hamilton) - Hard to find a basis set well adapted to T (with a diagonal representation).

20. La Grande Motte, February 2008 Curvilinear coordinates: Z-matrix atom 2 : distance d 2 from atom 1 atom 1: at origin atom 3 : distance d 3 from atom 2 and angle a 3 between atoms 1, 2 and 3

21. La Grande Motte, February 2008 Curvilinear coordinates: Z-matrix atom 2 : distance d 2 from atom 1 atom 1: at origin atom 3 : distance d 3 from atom 2 and angle a 3 between atoms 1, 2 and 3

22. La Grande Motte, February 2008 Curvilinear coordinates: Z-matrix atom 2 : distance d 2 from atom 1 atom 1: at origin atom 3 : distance d 3 from atom 2 and angle a 3 between atoms 1, 2 and 3 d 2 =2. d 3 =2. a 3 =120° Q k = d 3