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Polyspherical Description of a N-atom system

Polyspherical Description of a N-atom system. Christophe Iung LSDSMS, UMR 5636 Université Montpellier II e-mail : iung@univ-montp2.fr Collaboration avec Dr. Fabien Gatti , Dr. Fabienne Ribeiro et G. Pasin Pr. Claude Leforestier (Montpellier) Pr. Xavier Chapuisat et Pr. André Nauts

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Polyspherical Description of a N-atom system

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  1. Polyspherical Description ofa N-atom system Christophe Iung LSDSMS, UMR 5636 Université Montpellier II e-mail : iung@univ-montp2.fr Collaboration avec Dr. Fabien Gatti, Dr. Fabienne Ribeiro et G. Pasin Pr. Claude Leforestier (Montpellier) Pr. Xavier Chapuisat et Pr. André Nauts (Orsay et Louvain La Neuve)

  2. (H2O)n FIT of a Potential Energy Surface (PES) to describe the water solvent EXAMPLES OF SYSTEMS H nnCH F F F CF3H IntramolecularEnergytransfer in an excited System : Dynamical Behaviour of an Excited system : Is it ergodic or selective?

  3. Schrödinger ro-vibrational Equation 1- Born-Oppenheimer Approximation : ===> The Potential energy surface V can be expressed in terms of (3N-6) internal coordinates that describe the deformation of the molecular system 2- A Body-Fixed Frame (BF) has to be defined : Tc = Tc(G : 3 coordinates) + Tc(rotation-vibration:3N-3 coordinates) 3- Ro-Vibrationnal Schrödinger Equation : an eigenvalue equation H|Y> = (Tc+V) |Y(3N-3) internal coordinates> = Ero-vibrationnal|Y>

  4. Problem to be Solved 1- Choice of the set of coordinates adopted to describe the system : A crucial Choice 2- Expression of the Kinetic Energy Operator (KEO) Tc 3- Calculation and Fit of the Potential energy Surface (PES), V, a function of the 3N-6 internal nuclear coordinates. 4- Definition of a working basis set in which the Hamiltonian is diagonalized, this basis should contain 150000 states, for instance. • 5- Schrödinger Equation to be solved • -Pertubative Methods (CVPT...) • - Variational method (VSCF, MCSCF, Lanczos, Davidson,...) 6- Comparison between the calculated and experimental spectrum

  5. Curvilinear Large amplitude motions Rectilinear Low energy spectrum DY1 DY2 Dq DZ1 DZ2 Very Simple Expression of the Kinetic Energy Operator Basis of the traditional Spectroscopy More Intricate expression of the KEO 1-Choice of the set of coordinates

  6. P = - i , h x ¶ x i i 2- Expression of the KEO (Tc) * We need an exact expression of the KEO adapted to the numerical methods used to solve the Schrödinger equation. * We have to know how to act this operator on vectors of the working basis set. with

  7. 3- Analytical Expression of the PES calculated on a grid (of few thousands points). (Fit of this function) Potential Energy Surface Coordinate 1 Coordinates 2

  8. Outlines of the talk 1- KEO Expression 2.1 : Historical Expressions of the KEO 2.2 : More Recent (1990-2005) Strategies that provide KEO operator 2- Polyspherical Parametrization of a N-atom System (IJQC review paper on the web) 2.1 : Principle 2.2 : Application to the study of large amplitude motion 2.3 : Application to highly excited semi-rigid systems : Jacobi Wilson Method • 3- Direct Methods that solve the Schrödinger Equation • 3.1 : Lanczos Method • 3.2 : Block Davidson Method 4- Application to HFCO

  9. 1- Some Famous References • B. Podolsky, Phys. Rev. 32,812 (1928) • E.C. Kemble “The fundamental Principles of Quantum Mechanics” • Mc GrawHill, 1937 • E.B. Wilson, J.C. Decius, P.C. Cross “Molecula Vibrations” • McGrawHill, 1955 • H.M. Pickett, J. Chem. Phys, 56, 1715 (1971) • Nauts et X. Chapuisat, Mol. Phys., 55, 1287 1985 • N.C. Handy, Mol. Phys., 61, 207 (1987) • X.G. Wang, E.L. Sibert et M.S Child, Mol. Phys., 98, 317 (2000)

  10. q =J-1xpx=t(J-1)pq Quantum Expression of KEO for J=0 in the Euclidean Normalization 2Tc = (tpx)+px where pxi is the conjugate momentum associated with the mass-ponderated coordinates If a new set of curvilinear coordinates qi (i=1,…,3n-6) is introduced where J is the matrix which relies the cartesian coordinates to the new set of coordinates qi The determinant of J is the Jacobian of the transformation denoted byJ dtEuclide = dx1 dx2… dx3N-6= J dq1 dq2… dq3N-6

  11. Tc expression of the KEO for J=0 in Euclidian normalization If 2Tc = (tpx)+px and px=t(J-1)pq 2Tc = (tpq)+J-1t(J-1)pq 2Tc = (tpq)+gpq  det(g)=J-2 What is the adjoint of pqi ? It depends on the normalisation chosen In an Euclidean Normalization (pqi)+ = J-1 pqi Jwhere J est the Jacobian 2Tc = J-1 tpq Jgpq

  12. Démonstration de (pq)+=J-1 pq Jen normalisation euclidienne Définition de l’adjoint de pqi ? <(pqi)+j| f>= < j| pqi f> Or ... pq (J j f) dq1 dq2… dq3n-6 = 0 si (J j f)s ’annule sur les bornes d ’intégration ... pq (J j* f) dq1 … = ... pq (J j*) f dq1…dq3n-6 + ... J j*pq (f)dq1…dq3n-6 d ’où... (J-1pq J j)* f Jdq1… dq3n-6= ...j*pq (f)Jdq1... dq3n-6 =0 ... (J-1pq J j)* f dtEuclide= ...j*pq (f)dtEuclide =0 d ’où (pq)+ = J-1 pq J

  13. Other way to find 2Tc = J-1 tpq Jg pq Let use the expression of the Laplacian in spherical coordinates : 2Tc = -h/2pD, This expression can be re-expressed by 2Tc = (tpq)+g pq

  14. Quantum Expression of Tc for J=0 in Wilson Normalization dtWilson =dq1 dq2… dq3n-6 This normalization can be helpful to calculate some integrals. (jEuclide)*ÂEuclide fEuclidedtEuclide= (jWilson)*ÂWilson fWilsondtWilson (jEuclide)*ÂEuclide fEuclideJdq1 … dq3n-6= (jWilson)*ÂWilson fWilsondq1 … dq3n-6 (J0.5jEu)* (J0.5ÂEuJ-0.5) (J0.5 fEu)dtWilson= (jWilson)*ÂWilson fWilsondtWilson (jWilson)* ÂWilson fWilson 2TcW = J0.5TcEuJ-0.5 = J0.5J-1tpq Jg pqJ-0.5= J-0.5tpqJg pq J-0.5 2 TcWilson = J-0.5tpq Jg pq J-0.5

  15. 2 TcWilson = J-0.5tpq Jg pq J-0.5 OR 2TcEuclide= J-1tpq Jg pq Curvilinear Description J, g depend on q Rectilinear Description J, g do not depend on q 2Tc =tpq g pq No problem for Tc but problem for the fit of V and for the physical meaning of q Problem of no-commutation More Intricate expression To find and to act on a basis But easy fit of V et better physical meaning of q

  16. Different strategies developed : Application of the Chain Rule Handy et coll. (Mol. Phys., 61, 207 (1987)) Starting with the expression with cartesean coordinates : 2Tc = (tpx)+px The chain rule is acted (with the kelp of symbolic calculation) and provides : 2 Tc = S gkl pk pl + S hk pk in Euclidean Normalization Other normailization can be used… But it results more intricate expression of the KEO Tc

  17. Other formulation : Pickett expression: JCP, 56,1715 (1972) Starting from 2 TcWilson = J-0.5tpq J g pq J-0.5 One can find 2 TcWilson = tpq g pq + V’ V’ « extrapotential term » that depends on the masses. It can be treated with the potential This formulation has be exploited by E.L. Sibert et coll. in his CVPT perturbative formulation: J. Chem. Phys., 90, 2672 (1989)

  18. Ideal features of a KEO expression 1- Compact Expression of the KEO : larger is the number of terms, larger is the CPU time 2- Use of a set of coordinates adapted to describe the motion of atoms in order *to reduce the coupling between these coordinates * to define a working basis set such that the Hamiltonian matrix is sparse 3- The numerical action of the KEO must be possible and not too much CPU time consuming 4- The expression should be general and should allow to treat a large variety of systems

  19. 2- Polyspherical Parametrization The N-atom system is parametrized by (N-1) vectors described by their Spherical Coordinates ((Ri,i, i), i=1,...,N-1) The General Expression of the KEO is given in terms of either 1- the kinetic momenta associated to the vectors And the (N-1) radial conjugate momenta pRi ===>adapted to the description of large amplitude motion OR 2- the momenta conjugated with the polyspherical coordinates ((Ri,i, i),i=1,...,N-1) ===> adapted to the description of highly excited semi-rigid systems

  20. Development of this parametrization First description of its interest : X. Chapuisat et C. Iung , Phys. Rev. A,45, 6217 (1992) Review papers : F. Gatti et C. Iung,J. Theo. Comp. Chem.,2 ,507 (2003) et C. Iung et F. Gatti, IJQC (sous presse) Orthogonal Vectors : F. Gatti, C. Iung,X. Chapuisat JCP, 108, 8804 (1998), and 108, 8821 (1998) M. Mladenovic, JCP, 112, 112 (2000) NH3 Spectroscopy : F. Gatti et al , JCP, 111, 7236, (1999) and 111, 7236, (1999) Non Orthogonal Vectors : C. Iung, F. Gatti, C. Munoz, PCCP, 1, 3377 (1999) M. Mladenovic, JCP, 112, 1082 (2000);113,10524(2000) Semi-Rigid Molecules : C. Leforestier, F. Ribeiro, C. Iung 114,2099 (2001) F. Gatti, C. Munoz and C.Iung : JCP, 114, 8821 (2001) X. Wang, E.L.Sibert and M. Child : Mol. Phys, 98, 317(2000) H.G Yu, JCP,117, 2020 (2002);117,8190(2002) HF trimer : L.S. Costa et D.C. Clary, JCP, 117,7512 (2002) Diatom-diatom collision : E.M. Goldfield,S.K. Gray, JCP, 117,1604(2002) S.Y. Lin and H. Guo, JCP, 117, 5183(2002)

  21. “ORTHOGONAL” SET OF VECTORS H H C O C O BF Gz F H Jacobi Vectors Radau Vectors Polyspherical Coordinates : R3, R2, R1, 1, et out-of-plane dihedral angle)

  22. Non Orthogonal Set of Vectors H C O BF Gz H Valence Vectors Polyspherical Description : R3, R2, R1, 1,  and  - M matrix determination M (Trivial) - Dramatic Increase of term number… CPU can dramatically increase

  23. Determination de la Matrice M Any set of vectors can be related to a set of Jacobi vectors : La Matrice M est une matrice très facile à déterminer et dépendant des masses Elle permet de généraliser les résultats obtenus avec les vecteurs orthogonaux

  24. Developed expressions of the KEO *kinetic momentum Li associated with Ri and the radial momenta *Conjugate radial and angular momenta ¶ ¶ ¶ P = - i , P = - i , P = - i h h h j R J ¶ ¶ J ¶ j R i i i i i i Obtained by the substitution of the angular momentum By using { A BF (Body Fixed) frame has to be defined to introduce the total angular momentum (full rotation) vector J

  25. Choix du Body Fixed The (Gz)BF is chosen parallel to RN-1; LN-1is substituted by This requires 2 Euler rotations () The last Euler rotation ( can be chosen by the user In general, RN-2 is taken parallel to the plane (Gxz)BF But other choice can be done : N atoms = 3N-3 degrees of freedom • Kinetic MomentaLi (i-1,...,N-2) • (2N-5) angles (N-1 N-2 N-1) • the (N-1) radial conjugate momenta • the full rotation J (3 angles) *(3N-6) conjugate momenta (N-1 N-2 N-1) *the full rotation J (3 angles)

  26. By taking into account the fact that RN-1 and RN-2 are linked to the BF frame (problem of no-commutation of the operator that depends on vectors RN-1 and RN-2 ) It results in general expression of the KEO

  27. with One finds that :

  28. The problem of no-commutation are such that

  29. KEO developed expression for a system described by a set of (N-1) orthogonal vectors KEO developed expression for a system described by a set of (N-1) orthogonal vectors

  30. General Expression of Tc in terms of the conjugate momenta Associated with the polyspherical coordinates Expression used to study semi-rigid systems F. Gatti, C. Munoz, C. Iung, JCP, 114, 8821 (2001)

  31. The expression of the KEO are known… How can we use them for instance for semi-rigids systems ? 1- Orthogonal Coordinates provides rather simple expression of KEO… However, these coordinates does not necessary describe a real deformation of the system 2- Interesting coordinates, such valence coordinates, are not ‘orthogonal’ The KEO expression is intricate Two sets of coordinates can be used… This is the idea of the Jacobi-Wilson Method

  32. and where Definition of “Curvilinear Normal Coordinates”,Qi, In terms of polyspherical coordinates qj : Normal Modes Defined in terms of Polyspherical coordinates Polyspherical Coordinates Pqis substituted by(tL)PQin Tc Advantages : Simplicity of Tc in terms of polyspherical coordinates Physical Interest of the Normal Modes Jacobi-Wilson Method (C. Leforestier, A. Viel, C. Munoz, F. Gatti and C. Iung, JCP, 114, 2099 (2001))

  33. H • Description Polyspherique • Simple Expression of the KEO T O C JACOBI F Jacobi Vector • Normal Mode Coordinates : • Definition of a working basis set : WILSON DIAGONALIZATION JACOBI-WILSON STRATEGY Application to HFCO et H2CO Up to10000 cm-1

  34. Improvement of the zero-order basis set On can take into account to the diagonal anharmonicity:

  35. H Matrix calculation semi-analytical estimation of its action pseudo spectral scheme used Spectral Representation : Grid Representation:

  36. Ideal features of a method that provides eigenstates and eigenvalues which can be located in a dense part of the spectrum • Application to a large variety of systems ; • Use of huge basis set ; • Obtention of eigenvalues and eigenstates; • Control of the accuracy of the results ; • Small CPU time, Small memory requirement ;; • Easy to use and to adopt ; • Specific Calculation of energies in a given part of the spectrum ;

  37. LANCZOS METHOD Iterative Construction of the Krylov subspace generated by {un, n=O,N} : 1- Initialization : A first guess vectoru0 is chosen 2- Propagation : The following vector un+1 is calculated bn+1 un+1 = (H – an) un – bn un-1 with an = <un|H|un> bn+1= <un+1|H|un>

  38. é ù a b é ù 0 ê ú 0 1 ê ú ê ú H b a b ê ú 1 1 2 ê ú Þ ê ú 0 ê ú ê ú ê ú bN a ë û ê ú N ë û < dim N dim B dim B LANCZOS FEATURES • Lanczos Method: • Avoid the determination of the full H matrix. • The convergence is slower when the state density increases

  39. Ouverture de Fenêtres en énergie E0 DIAGONALISATION DE H Méthode de Lanczos Diagonalisation directe

  40. One has to open some window energy • Spectral Transform . Lanczos applied to G=(Eref°-H)-1. or exp(-a(H-Eref)2)

  41. Modified Block-Davidson Algorithm to calculate a set of b coupled eigenstates Method based on one parameter : e which sets the accuracy F. Ribeiro, C. Iung, C. Leforestier JCP in press C. Iung and F. Ribeiro JCP in press

  42. E°1 P° 0 0 0 E°i Eanh1 0 Q 0 0 Eanhq Prediagonalization step in order to reduce the off-diagonal terms The working basis set Banhis divided into : Banh= P°Q . Where P° contains the zero-order states which play a significant rôle in the calculation performed : H is diagonalized in P°, et this new basis set {u°i ,E°i} is used during the Davidson scheme H°=

  43. Determination of the Block of states We can defined the block of states using the second-order perturbation States such that are retained un a given block :

  44. APPLICATION TO HFC0 • Faible barrière de dissociation (14000 cm-1) • HFCO HF + CO • Mode de déplacement hors du plan très découplé à haute énergie. • Forte densité d ’états

  45. Excitation of the Out-of-plane mode C O H F Selectivity of the energy transfer in HFCO whose out-of-plane mode is excited 6 modes 2981 cm-1 : CH stretch 1837cm-1 : C=0 stretch 1347 cm-1 : HCO bend 1065 cm-1 : CF stretch 662 cm-1 : FCO bend 1011 cm-1 In-plane modes Out of plane mode Moore et coll. have studied the highly excited out-of-plane overtones (nnout-of-plane, n=14,…,20) : they predict the localization of energy in these states How can we understand that a highly excited state can be localized in one mode while the state density is large for Eexc=14000-20000cm-1?

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