Polyspherical Description of a N-atom system Christophe Iung LSDSMS, UMR 5636 Université Montpellier II 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)

H F F F n CH CF3HCF3H Intramolecular Energy transfer in an excited System : Dynamical Behaviour of an Excited system : Is it ergodic or selective? FIT of a Potential Energy Surface (PES) to describe the water solvent (H 2 O) n EXAMPLES OF SYSTEMS

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 : T c = T c (G : 3 coordinates) + T c (rotation-vibration:3N-3 coordinates) Schrödinger ro-vibrational Equation 3- Ro-Vibrationnal Schrödinger Equation : an eigenvalue equation H |  > = ( T c + V ) |   ) internal coordinates > = E ro-vibrationnal |  >

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

Rectilinear Low energy spectrum Very Simple Expression of the Kinetic Energy Operator Basis of the traditional Spectroscopy  Y1  Z1  Z2  Y2 Curvilinear Large amplitude motions More Intricate expression of the KEO  1-Choice of the set of coordinates

* 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. 2- Expression of the KEO (T c ) with  P i x = -i    x i,

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

1- KEO Expression 2.1 : Historical Expressions of the KEO 2.2 : More Recent ( ) Strategies that provide KEO operator 3- Direct Methods that solve the Schrödinger Equation 3.1 : Lanczos Method 3.2 : Block Davidson Method Outlines of the talk 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 4- Application to HFCO

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) A.Nauts et X. Chapuisat, Mol. Phys., 55, N.C. Handy, Mol. Phys., 61, 207 (1987) X.G. Wang, E.L. Sibert et M.S Child, Mol. Phys., 98, 317 (2000)

Quantum Expression of KEO for J=0 in the Euclidean Normalization 2T c = ( t p x ) + p x where p xi is the conjugate momentum associated with the mass-ponderated coordinates If a new set of curvilinear coordinates q i (i=1,…,3n-6) is introduced where J is the matrix which relies the cartesian coordinates to the new set of coordinates q i The determinant of J is the Jacobian of the transformation denoted by J d  Euclide = dx 1 dx 2 … dx 3N-6 = J dq 1 dq 2 … dq 3N-6 q = J -1 x  p x = t (J -1 ) p q

T c expression of the KEO for J=0 in Euclidian normalization If 2T c = ( t p x ) + p x and p x = t (J -1 )p q 2T c = ( t p q ) + J -1 t (J -1 ) p q 2T c = ( t p q ) + g p q  det(g)= J -2  2T c = J -1 t p q J g p q What is the adjoint of p qi ? It depends on the normalisation chosen In an Euclidean Normalization (p qi ) + = J -1 p qi J where J est the Jacobian

Démonstration de (p q ) + =J -1 p q J en normalisation euclidienne Définition de l’adjoint de p qi ? = Or... p q (J  dq 1 dq 2 … dq 3n-6   si (J  s ’annule sur les bornes d ’intégration... p q (J    dq 1 … ... p q (J    dq 1 …dq 3n-6 ... J    p q (  dq 1 …dq 3n-6 d ’où... (J -1 p q J    J  dq 1 … dq 3n-6 ...    p q (  J  dq 1... dq 3n-6 ... (J -1 p q J    d  Euclide ...    p q (  d  Euclide  d ’où (p q ) + = J -1 p q J

Let use the expression of the Laplacian in spherical coordinates : 2T c = -h/2   This expression can be re-expressed by Other way to find 2T c = J -1 t p q J g p q 2T c = ( t p q ) + g p q

This normalization can be helpful to calculate some integrals. (  Euclide ) *  Â Euclide  Euclide  d  Euclide  (  Wilson ) *  Â Wilson  Wilson  d  Wilson (  Euclide ) *  Â Euclide  Euclide  J  dq 1 … dq 3n-6  (  Wilson ) *  Â Wilson  Wilson  dq 1 … dq 3n-6 (J 0.5  Eu ) * (J 0.5  Â Eu J -0.5 )  J 0.5  Eu )  d  Wilson  (  Wilson)*  Â Wilson  Wilson  d  Wilson (  Wilson ) *  Â Wilson  Wilson 2T c W = J 0.5  T c Eu J -0.5 = J 0.5  J -1 t p q J g p q J -0.5   J -0.5t p q J g p q J -0.5 Quantum Expression of T c for J=0 in Wilson Normalization in Wilson Normalization d  Wilson =dq 1 dq 2 … dq 3n-6 2 T c Wilson  J -0.5 t p q J g p q J -0.5

Rectilinear Description J, g do not depend on q 2 T c Wilson  J -0.5 t p q J g p q J -0.5 OR 2T c Euclide = J -1 t p q J g p q Curvilinear Description J, g depend on q 2T c = t p q g p q No problem for T c 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

Different strategies developed : Application of the Chain Rule Handy et coll. (Mol. Phys., 61, 207 (1987)) Starting with the expression with cartesean coordinates : 2T c = ( t p x ) + p x The chain rule is acted (with the kelp of symbolic calculation) and provides : 2 T c =  g kl p k p l +  h k p k in Euclidean Normalization Other normailization can be used… But it results more intricate expression of the KEO T c

Other formulation : Pickett expression: JCP, 56,1715 (1972) Starting from 2 T c Wilson = J -0.5 t p q J g p q J -0.5 One can find 2 T c Wilson = t p q g p q + 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)

Ideal features of a KEO expression  Compact Expression of the KEO : larger is the number of terms, larger is the CPU time  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  The numerical action of the KEO must be possible and not too much CPU time consuming  The expression should be general and should allow to treat a large variety of systems

2- Polyspherical Parametrization The N-atom system is parametrized by (N-1) vectors described by their Spherical Coordinates ((R i, 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 p Ri ===> adapted to the description of large amplitude motion OR 2- the momenta conjugated with the polyspherical coordinates ((R i, i,  i ),i=1,...,N-1) ===> adapted to the description of highly excited semi-rigid systems

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) NH 3 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)

“ORTHOGONAL” SET OF VECTORS Polyspherical Coordinates : R 3, R 2, R 1,  1,   et   out-of-plane dihedral angle) BF Gz H H Jacobi VectorsRadau Vectors O O C C FH

Non Orthogonal Set of Vectors Polyspherical Description : R 3, R 2, R 1,  1,   and  - M matrix determination M (Trivial) - Dramatic Increase of term number… CPU can dramatically increase Valence Vectors BF Gz OC H H

 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

Developed expressions of the KEO *kinetic momentum L i associated with R i and the radial momenta *Conjugate radial and angular momenta By using Obtained by the substitution of the angular momentum A BF (Body Fixed) frame has to be defined to introduce the total angular momentum (full rotation) vector J {  P i R = -i    R i, P i = -i    i, P i  = -i    i

Choix du Body Fixed The (Gz) BF is chosen parallel to R N-1 ; L N-1 is substituted by This requires 2 Euler rotations (  ) The last Euler rotation (   can be chosen by the user In general, R N-2 is taken parallel to the plane (Gxz) BF But other choice can be done : N atoms =  3N-3 degrees of freedom Kinetic Momenta L i (i-1,...,N-2) (2N-5) angles        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)

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

with One finds that :

The problem of no-commutation are such that

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

General Expression of T c 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)

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

Definition of “Curvilinear Normal Coordinates”,Q i, In terms of polyspherical coordinates q j : where and Polyspherical Coordinates Normal Modes Defined in terms of Polyspherical coordinates Advantages : Simplicity of T c 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)) P q is substituted by ( t L) P Q in T c

JACOBI-WILSON STRATEGY H Jacobi Vector O C F Description Polyspherique Simple Expression of the KEO T JACOBI Normal Mode Coordinates : Definition of a working basis set : WILSON DIAGONALIZATION Application to HFCO et H 2 CO Up to10000 cm -1

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

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

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 ;

Iterative Construction of the Krylov subspace generated by {u n, n=O,N} : 1- Initialization : A first guess vector u 0 is chosen 2- Propagation : The following vector u n+1 is calculated  n+1 u n+1 = (H –  n ) u n –  n u n-1 with  n =  n+1 = LANCZOS METHOD

Lanczos Method: 0    BN B N H dim                             LANCZOS FEATURES Avoid the determination of the full H matrix. The convergence is slower when the state density increases

DIAGONALISATION DE H Diagonalisation directe Méthode de Lanczos Ouverture de Fenêtres en énergie E0E0

Spectral Transform. Lanczos applied to G=(E ref °-H) -1. or exp(-  (H-E ref ) 2 ) One has to open some window energy

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

The working basis set B anh is divided into : B anh = 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 E° 1 E° i E anh 1 E anh q P° Q H°= Prediagonalization step in order to reduce the off-diagonal terms

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

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 APPLICATION TO HFC0

Selectivity of the energy transfer in HFCO whose out-of-plane mode is excited Moore et coll. have studied the highly excited out-of-plane overtones (n out-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 E exc = cm -1 ? 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 Out of plane mode In-plane modes C F O H Excitation of the Out-of-plane mode

Between 1 and 60 Davidson iterations are required to calculate each state.  E is not a correct indicator of the convergence Davidson Iteration Number -État 60 -État 120 -État 180 Lanczos CONVERGENCE OF THE DAVIDSON SCHEME Nombre d ’itérations Davidson Error On the Energy (cm -1 )

I : IIq (M) II max < 10 cm -1 Pour  E < 0.01 cm -1 IIq (M) II max < 50 cm -1 Pour  E < 0.5 cm -1 Convergence criterion Davidson Iteration Number Erreur (cm -1 ) IIq (M) II constitue an excellent Indicator of the convergence And the accuracy of the eigenenergy and eigenstate.

Numerical Cost Nombre d ’actions de H Energie d ’excitation (cm -1 )

Determination of an Active Space specificaly built to study a given state Example : state |10 6 > in HFCO ( State n° 1774, in a 100,000 state primitive basis set Caracterized by v max (in plane)=8 and E max =32,000cm -1 ) We begin with a Davidson calculation performed on |10 6> (0) in a 7,000 state basis set defined by vmax(in plane)=4 and Emax=24,000cm-1 The Davidson scheme in this small basis set converges and provides an estimation of the eigenstate studied: |10 6> (1) The largest |v 1,…,v 6 >°contributions of this estimated eigenstate |10 6> (1) are retained in a small (368 states) « active space » P o used in the calculation in the large (100,000 states) basis set

Application to the calculation of highly Excited overtones in HFCO State 10 6 : State n°1700 Similar coefficients obtained for different overtones : The nature of the coupling is identical for these overtones The CH stretch is the more coupled mode CH stretch HCO bend

Main features of this new Prediagonalized Block-Davidson Scheme Prediagonalized Block-Davidson Scheme It can be coupled to any method which can provide the action of the Hamiltonian on a vector Huge basis set can be used (more than states) Calculation of the eigenstates and eigenenergies The accuracy of the results can be controled (with ||q M ||) Low memory cost Faster and more efficient than Lanczos Very easy to use because it depends only on one parameter  It is adapted to calculate a series of coupled states

Conclusions The development of a general method to calculate high excited ro-vibrational state is crucial Different approachs have to be exploited : The Jacobi-Wilson method coupled with the Davidson algorithm presents interesting advantages. It allows the specific calculation of eigenstates associated with highly excited states It can be improved by using a MC-SCF or SCF treatment However a lot of work has to be done… improve the fit of V, use a fit of the KEO in order to reduce the CPU time…

3 – MCTDH Method (Time Dependant method) A fit of the global PES has to be performed : A fit of the global PES has to be performed : A factorized form of H is required Spectrum calculation By Fourrier transform of the survival probability By filtered diagonalization References : H-D Meyer, U. Manthe and L. Cederbaum, Chem. Phys. Lett. 165, 73 (1990) M. Beck, A. Jaeckle, G. Worth and H-D Meyer, Physics Reports, 324,1 (2000) C. Iung, F. Gatti and H-D Meyer, J. Chem. Phys., 120, 6992 (2004)

The MCTDH Approach Primitive Basis set : { |v 1,…,v 9 >°,v i =0,1,…,v i max } The MCTDH « Active Space » is generated by the configurations : {  i 1 (1) (Q 1,t).. …  i 9 (2) (Q 9, t); i j =0,1,…,i j max ; j=1,…,9} : (time-dependant functions which are adapted to the location of the wave-packet describing the system) It is efficient if i j max < v i max

Time Dependent Coefficient and Functions to estimate Time dependent coefficient to optimize Projection on the Space generated by Functions  j (  ) (Q k ) Time Dependent 3D functions to optimize Density Matrix Mean Field Hamiltonian

Application to the calculation of highly excited overtones in HFCO (|n 6 >) Spectrum obtained with filtered Diagonalization

Fraction of Energy in the different Normal Modes The CH stretch is an energy reservoir