Polyspherical Description of a N-atom system

1
Polyspherical Description ofa N-atom system

• Christophe Iung
• LSDSMS, UMR 5636
• Université Montpellier II
• e-mail iung_at_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
(No Transcript)
3
EXAMPLES OF SYSTEMS
H
nnCH
F
F
F
CF3H
Intramolecular Energy transfer in an excited
System Dynamical Behaviour of an Excited system
Is it ergodic or selective?
4
Schrödinger ro-vibrational Equation
1- Born-Oppenheimer Approximation gt 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-vibration3N-3 coordinates)
3- Ro-Vibrationnal Schrödinger Equation an
eigenvalue equation H Ygt (TcV) Y(3N-3)
internal coordinatesgt Ero-vibrationnalYgt
5
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
6
1-Choice of the set of coordinates
7
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
8
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
9
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
10
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)

11
Quantum Expression of KEO for J0 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 (i1,,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 by J
dtEuclide dx1 dx2 dx3N-6 J dq1 dq2 dq3N-6
12
Tc expression of the KEO for J0 in Euclidian
normalization
If 2Tc (tpx) px and
pxt(J-1)pq 2Tc (tpq) J-1 t(J-1) pq
2Tc (tpq) g pq ? det(g)J-2

What is the adjoint of pqi ? It depends on
the normalisation chosen In an Euclidean
Normalization (pqi) J-1 pqi J where J est
the Jacobian
? 2Tc J-1 tpq J g pq
13
Démonstration de (pq)J-1 pq J en normalisation
euclidienne
Définition de ladjoint de pqi ? lt(pqi) j f gt
lt j pqi f gt
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 dq1dq3n-6 ... J j pq ( f)
dq1dq3n-6 d où ... (J-1pq J j) f J
dq1 dq3n-6 ... j pq ( f) J dq1... dq3n-6
0 ... (J-1pq J j) f
dtEuclide ... j pq ( f) dtEuclide 0
d où (pq) J-1 pq J
14
Other way to find 2Tc J-1 tpq J g pq
Let use the expression of the Laplacian in
spherical coordinates 2Tc -h/2p D,
This expression can be re-expressed by

2Tc (tpq) g pq
15
Quantum Expression of Tc for J0 in Wilson
Normalization dtWilson dq1 dq2 dq3n-6
This normalization can be helpful to calculate
some integrals. (jEuclide) ÂEuclide fEuclide
dtEuclide (jWilson) ÂWilson fWilson
dtWilson (jEuclide) ÂEuclide fEuclide J dq1
dq3n-6 (jWilson) ÂWilson fWilson dq1
dq3n-6 (J0.5jEu) (J0.5 ÂEuJ-0.5) (J0.5
fEu) dtWilson (jWilson) ÂWilson fWilson
dtWilson (jWilson) ÂWilson
fWilson 2TcW J0.5 TcEuJ-0.5 J0.5 J-1 tpq J
g pq J-0.5 J-0.5tpqJ g pq J-0.5
2 TcWilson J-0.5 tpq J g pq J-0.5
16
2 TcWilson J-0.5 tpq J g pq J-0.5
OR 2TcEuclide J-1 tpq J g 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
17
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
18
Other formulation Pickett expression JCP,
56,1715 (1972)
Starting from 2 TcWilson J-0.5 tpq 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)
19
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
20
2- Polyspherical Parametrization
The N-atom system is parametrized by (N-1)
vectors described by their Spherical Coordinates
((Ri,?i, ?i), i1,...,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 gt
adapted to the description of large amplitude
motion OR 2- the momenta
conjugated with the polyspherical
coordinates ((Ri,?i, ?i),i1,...,N-1) gt
adapted to the description of highly excited
semi-rigid systems

21
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)

22
(No Transcript)
23
(No Transcript)
24
(No Transcript)
25
ORTHOGONAL SET OF VECTORS
H
H
C
O
O
C
BF Gz
F
H
Jacobi Vectors
Radau Vectors
Polyspherical Coordinates R3, R2, R1, ?1, ??
et ???out-of-plane dihedral angle)
26
Non Orthogonal Set of Vectors
H
O
C
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
27
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
28
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

29
Choix du Body Fixed
The (Gz)BF is chosen parallel to RN-1 LN-1 is
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 Momenta Li (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)

30
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
31
with
One finds that
32
The problem of no-commutation are such that
33
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
34
(No Transcript)
35
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)
36
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
37
Definition of Curvilinear Normal
Coordinates,Qi, In terms of polyspherical
coordinates qj
Normal Modes Defined in terms of Polyspherical
coordinates
Polyspherical Coordinates
Pq is substituted by (tL) PQ in 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))
38
JACOBI-WILSON STRATEGY
Application to HFCO et H2CO Up to10000 cm-1
39
Improvement of the zero-order basis set
On can take into account to the diagonal
anharmonicity
40
H Matrix calculation
semi-analytical estimation of its action
pseudo spectral scheme used
Spectral Representation
Grid Representation
41
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

42
LANCZOS METHOD

• Iterative Construction of the Krylov subspace
  generated by un, nO,N
• 1- Initialization A first guess vector u0 is
  chosen
• 2- Propagation The following vector un1 is
  calculated
• bn1 un1 (H an) un
  bn un-1
• with
• an ltunHungt
• bn1 ltun1Hungt

43
LANCZOS FEATURES

• Lanczos Method

• Avoid the determination of the full H matrix.
• The convergence is slower when the state density
  increases

44
DIAGONALISATION DE H
Méthode de Lanczos
Diagonalisation directe
45
One has to open some window energy

• Spectral Transform .
• Lanczos applied to G(Eref-H)-1. or
  exp(-a(H-Eref)2)

46
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
47
Prediagonalization step in order to reduce the
off-diagonal terms
The working basis set Banh is 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 ui
,Ei is used during the Davidson scheme
H
48
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
49
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

50
Selectivity of the energy transfer in
HFCO whose out-of-plane mode is excited
6 modes 2981 cm-1 CH
stretch 1837cm-1 C0 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,
n14,,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
Eexc14000-20000cm-1?
51
CONVERGENCE OF THE DAVIDSON SCHEME

• État 60
• État 120
• État 180

Error On the Energy (cm-1)
Lanczos
Nombre d itérations Davidson

• Between 1 and 60 Davidson iterations are
  required to calculate each state.
• . DE is not a correct indicator of the
  convergence

52
Convergence criterion

• IIq(M)II constitue an excellent
• Indicator of the convergence
• And the accuracy of the eigenenergy and
  eigenstate.

• I IIq(M)IImax lt 10 cm-1
• Pour DE lt 0.01 cm-1
• IIq(M)IImax lt 50 cm-1
• Pour DE lt 0.5 cm-1

53
Numerical Cost
Nombre d actions de H
Energie d excitation (cm-1)
54
Determination of an Active Space specificaly
built to study a given state
Example state 10n6gt in HFCO (State n 1774, in
a 100,000 state primitive basis set Caracterized
by vmax(in plane)8 and Emax32,000cm-1) We
begin with a Davidson calculation performed on
10n6gt(0) in a 7,000 state basis set defined by
vmax(in plane)4 and Emax24,000cm-1 The
Davidson scheme in this small basis set converges
and provides an estimation of the eigenstate
studied 10n6gt(1) The largest
v1,,v6gtcontributions of this estimated
eigenstate 10n6gt(1) are retained in a small
(368 states)  active space  Po used in the
calculation in the large (100,000 states) basis
set
55
Application to the calculation of highly Excited
overtones in HFCO State 10n6 State n1700
Similar coefficients obtained for
different overtones The nature of the
coupling is identical for these overtones
CH stretch
The CH stretch is the more coupled mode
HCO bend
56
Main features of this new 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 100 000
  states)
• Calculation of the eigenstates and eigenenergies
• The accuracy of the results can be controled
  (with qM)
• Low memory cost
• Faster and more efficient than Lanczos
• Very easy to use because it depends only on one
  parameter e
• It is adapted to calculate a series of coupled
  states

57
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

58

• 3 MCTDH Method (Time Dependant method)
• 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)
59
The MCTDH Approach

• Primitive Basis set v1,,v9gt,vi0,1,,vimax
  
• The MCTDH Active Space is generated
• by the configurations
• ji1(1)(Q1,t).. ji9(2)(Q9 ,t)
  ij0,1,,ijmax j1,,9
• (time-dependant functions which are adapted to
  the location of the wave-packet describing the
  system)
• It is efficient if ijmaxlt vimax

60
Time Dependent Coefficient and Functions to
estimate

Time dependent coefficient to optimize
Time Dependent 3D functions to optimize

Density Matrix
Mean Field Hamiltonian
Projection on the Space generated by Functions
jj(k) (Qk)
61
Application to the calculation of highly excited
overtones in HFCO (nn6gt) Spectrum obtained with
filtered Diagonalization
62
Fraction of Energy in the different Normal Modes
10n6
20n6
The CH stretch is an energy reservoir