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Kinetic Energy Operator in curvilinear coordinates: numerical approach

Kinetic Energy Operator in curvilinear coordinates: numerical approach. A. Nauts Y. Justum M. Desouter-Lecomte L. Bomble (PhD). Spectroscopy, floppy systems. contr ol quantum gates. Why we need curvilinear coordinates.

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Kinetic Energy Operator in curvilinear coordinates: numerical approach

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  1. 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

  2. Why we need curvilinear coordinates In quantum dynamics, the calculations are easier with curvilinear coordinates: • The center of mass is separable: XCM=[XCM, YCM, ZCM] • The overall rotation is well-described by means of 3 Euler angles:q=[q,j,c] • The torsion of a chemical fragment (ex: Methyl) can be described by one coordinates: a dihedral angle, f. • .... • All 3N curvilinear coordinates will be noted: q=[qi] La Grande Motte, February 2008

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

  4. 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 f2(q) and f1(q) on a grid. Ex: Numerical integrations La Grande Motte, February 2008

  5. Example: For the 1D-Kinetic energy operator of methanol: One active (dynamical) coordinate, f, and 11 frozen coordinates G(f)=A/B A = -6 RCH2 (MCH3 MOH RCO2 - 2 MH RCO (3 MOH RCH Cos(aHCO) + MCH3 RCOH Cos(aHOC)) + MH (3 MCHO RCH2 Cos(aHCO)2 + 6 MH RCH RCOH Cos(aHCO) Cos(aHOC) + MCH3O RCOH2 Cos(aHOC)2)) Sin(aHCO)2 - 9 MH MCH3OH RCH 4 Sin(aHCO)4 - RCOH2 (2 MCH3 MO RCO2 - 12 MH MO RCH RCO Cos(aHCO) + 6 MH MCO RCH2 Cos(aHCO)2 + 3 MH MCH3O RCH2 Sin(aHCO)2) Sin(aHOC)2 B = 6 MH RCH2 RCOH2 Sin(aHCO)2 (2 MCH3 MO RCO2 -12 MH MO RCH RCO Cos(aHCO) + 6 MH MCO RCH2 Cos(aHCO)2 + 3 MH MCH3O RCH2 Sin(aHCO)2) Sin(aHOC)2 La Grande Motte, February 2008

  6. 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 Tdifficult to obtain. For numerical applications the analytical expressions may not be needed, but only the values of the functions f2(q) and f1(q) on a grid. Ex: Numerical integrations La Grande Motte, February 2008

  7. Objective of Tnum[1] To calculate -numerically and exactly -for molecular systems of any size the functions f2(q) and f1(q) for a given value of q using a Z-matrix definition of the curvilinear coordinates. Similar numerical procedures 1-2active coordinate(s) (inversion of ammonia[2], ring puckering[3,4], torsion[5]) 6active 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

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

  9. Especially developed for MCTDH Cartesian coordinates in BF Z-matrix O1 O2 O1 R H1 O1 R1 O2 a1 H1 O2 R2 O1 a2H1phi Cartesian construction with the vectors in any order. => Polyspherical, Jacobi vectors.... bunch of vectors Analytical expression La Grande Motte, February 2008

  10. Further transformations: Qzmat => Qdyn Qzmat: Coordinates associated with the Z-matrix or the bunch of vectors Qdyn: Coordinates used in the dynamic (active, inactive) Transformations Identity Linear combinations Polar transformations La Grande Motte, February 2008

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

  12. Test : H-CN (Jacobi) Z-matrix : C X C (1-m)R N X mRC 180,0 H XrC aN 0,0 Normalization : r(x,R,r)=1 with x=cos(a) constants m=MC/MCN Spectrum of HCN/CNH (diagonalization) : adiabatic constraint (1d : a) Spectrum of HCN/CNH (WP)[3] : [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

  13. Use of Tnum: To set up or to check analytical kinetic energy operators: Ethene with constraints, W2H+ (MCTDH) Spectroscopy: MethylPropanal, Methanol (1+11D), Fluoroproprene, Ammonia (6D) 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

  14. 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 Incompatibility between Tnum and MCTDH! Can be easily used with CDVR!! What can be done? -Use a fitting procedure -Taylor expansion of G La Grande Motte, February 2008

  15. Taylor expansion of G around Q0 This form can be used with MCTDH -The calculation of the derivatives of G (up to the second order) is already implemented in Tnum. This approach is well known: R. Wallace, CP, 11 p189 1975: H2O, CH of benzene (Zero order) E. L Sibert III, W. P. Reinhardt, J. T. Hynes, JCP, 81, p1115 1984: CH in benzene (first order) L. Halonen, T. Carrington Jr JCP 88 p4171 1988: H2X (third order) L Lespade, S. Robin D. Cavagnat, JPC, 97, p6134, 1993: Cyclohexene (second order) .... La Grande Motte, February 2008

  16. Taylor expansion of G around Q0 The Taylor expansion may give non-hermitian KEO ! Ex: AB-C with Jacobi coordinates : The Taylor expansion (2d order) 2d order in x and R. Non-hermitian with Legendre polynomial Always Hermitian with the sine and the HO basis-sets. La Grande Motte, February 2008

  17. Example: H2O in valence coordinates ------------------------------------------------- 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 .... &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 defined constraints (here no constraint) reference geometry, Q0 Number of terms for (H2O)2H+ (D2d): 29,92,279

  18. Advantages/drawback ofTnum • + 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

  19. Thank you

  20. Curvilinear coordinates: Z-matrix atom 3 : distance d3 from atom 2 and angle a3 between atoms 1, 2 and 3 atom 2 : distance d2 from atom 1 atom 1: at origin La Grande Motte, February 2008

  21. Curvilinear coordinates: Z-matrix atom 3 : distance d3 from atom 2 and angle a3 between atoms 1, 2 and 3 atom 2 : distance d2 from atom 1 atom 1: at origin La Grande Motte, February 2008

  22. Curvilinear coordinates: Z-matrix d2=2. d3=2. a3=120° Qk= d3 atom 3 : distance d3 from atom 2 and angle a3 between atoms 1, 2 and 3 atom 2 : distance d2 from atom 1 atom 1: at origin La Grande Motte, February 2008

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