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

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 ]

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

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

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 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 ) M H R CH R COH Cos(a HCO ) Cos(a HOC ) + M CH3O R COH 2 Cos(a HOC ) 2 )) Sin(a HCO ) M H M CH3OH R CH 4 Sin(a HCO ) 4 - R COH 2 (2 M CH3 M O R CO M H M O R CH R CO Cos(a HCO ) + 6 M H M CO R CH 2 Cos(a HCO ) 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 M H M O R CH R CO Cos(a HCO ) + 6 M H M CO R CH 2 Cos(a HCO ) M H M CH3O R CH 2 Sin(a HCO ) 2 ) Sin(a HOC ) 2

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

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

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

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

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

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.

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 :  )

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

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!!

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 p : H 2 O, CH of benzene (Zero order) 2.E. L Sibert III, W. P. Reinhardt, J. T. Hynes, JCP, 81, p : CH in benzene (first order) 3.L. Halonen, T. Carrington Jr JCP 88 p : H 2 X (third order) 4.L Lespade, S. Robin D. Cavagnat, JPC, 97, p6134, 1993: Cyclohexene (second order)....

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.

Example: H 2 O in valence coordinates &geom zmat=T nat=3 / &niveau nrho=1 read_nameQ=t / r1OH 1. r2OH 1. a 1.6 Z-matrix reference geometry, Q HAMILTONIAN-SECTION modes | r1OH | r2OH | a # Zero order part: -1/2*G^ij(Qref) |1 dq^ |2 dq^ |3 dq^ E-003 |1 dq |2 dq E-002 |1 dq |3 dq E-002 |2 dq |3 dq # First order part: # -1/2*dG^ij/dDQk d./dQi * DQk * d./dQj |3 dq^2 |1 q |3 dq^2 |2 q E-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

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).

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

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

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