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Los Alamos, 30.07.2008
The dynamics of angular degrees of freedom: new basis set and grid
representations of Hamiltonian operatorsClemens Clemens WoywodWoywod
UniversityUniversity ofof TromsTromsøø
woywodwoywod@@ch.tum.dech.tum.de
Florian RuppFlorian Rupp
TU MuenchenTU Muenchen
rupp@ma.tum.derupp@ma.tum.de
Los Alamos, 30.07.2008
OutlineOutline
Part I: Recipe & ingredients for a vibrational calculation
Part II: A new basis set for angular motion & comparison with Legendre functions
Part III: Overview of localized & delocalized representations,construction of localized mixed basis functions,applications
Part IV: Conclusions
Los Alamos, 30.07.2008
Recipe & ingredients for aRecipe & ingredients for avibrational calculationvibrational calculation
Part Part II
Los Alamos, 30.07.2008
Vibrational spectrum of HVibrational spectrum of H22OO
⎪⎩
⎪⎨
⎧−=⎟⎟
⎠
⎞⎜⎜⎝
⎛−−
= 1,...,2,1,0,)(
)(cos)( Nnab
axnxnπσν
0,2≠
−= n
abσ
How to compute very accurately the How to compute very accurately the vibrationalvibrational levels of Hlevels of H22O in the O in the electronic ground state? electronic ground state?
Compute potential energy surface on dense grid in (r1, r2, α) – space
Make decision: definition of potential energy operator V(r1, r2, α) directly on grid or via analytical model function
Select basis functions for description of vibrational wave functions. If V(r1, r2, α) is defined on discrete set of points basis functions are still needed for representation of T operator
Popular basis functions for radial degrees of freedom: Popular basis functions for radial degrees of freedom:
Nnab
axnab
xn ,...,2,1,)(
)(sin2)( =⎟⎟⎠
⎞⎜⎜⎝
⎛−−
−=
πµ
0,1=
−= n
abσ
See e.g. Colbert & Miller, JCP See e.g. Colbert & Miller, JCP 9696, 1982 (1992), 1982 (1992)
Los Alamos, 30.07.2008
Vibrational basis setsVibrational basis sets
⎪⎩
⎪⎨
⎧= ∑∫
=
)()(1
N
kk
b
a
xfwdxxfnfor
Nabw ν,−
=
Why are the Why are the µµnn(x(x) and ) and ννnn(x(x)) functions popular? functions popular?
They yield analytic expressions for <µm|T|µn> and <νm|T|νn> for finite and infinite definition intervals [a,b]
They are associated with an equidistant quadrature grid (relation to Chebychev)
The quadrature rule is of Gaussian accuracy (discrete orthogonality)
nforN
abw µ,1+
−=
Which basis sets are appropriate for bending motion? Which basis sets are appropriate for bending motion?
The bending kinetic energy operator is:
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
+∂∂
−=∧
xx
xcT bendbend )cot(2
2
Θ=
21
bendc
Los Alamos, 30.07.2008
Legendre basis for bending motionLegendre basis for bending motion
))3cos(5)cos(3(25))(cos( 522 xxxP +=σ
In this form, Tbend is hermitian on [0,π] with respect to volume element sin(x) dx
µn(x) and νn(x) functions perform badly as basis functions for Tbend
The standard basis functions for Tbend are derived from Legendre polynomials Pl(x):
21))(cos(00 =xPσ
)cos(23))(cos(11 xxP =σ ))4cos(35)2cos(209(
29))(cos( 1333 xxxP ++=σ
Pl(cos(x)) are the eigenfunctions of Tbend → diagonal analytic representation
Pl(cos(x)) are associated with quadrature rule of Gaussian accuracy
Grid point density increases moderately towards interval limits
PPll(cos(x(cos(x)) are suitable bending basis functions for harmonic type )) are suitable bending basis functions for harmonic type potential functions potential functions →→ performance good because the density of excited performance good because the density of excited state wave functions accumulates at interval borders state wave functions accumulates at interval borders
Los Alamos, 30.07.2008
Normalized Legendre functions Pl(cos(x)) 0
3
12
4
5
Los Alamos, 30.07.2008
A new basis set for angular A new basis set for angular motion & comparison with motion & comparison with
Legendre functionsLegendre functions
Part Part IIII
Los Alamos, 30.07.2008
The The ηηn n (x) angular basis functions(x) angular basis functions
is of Gaussian accuracyis of Gaussian accuracy
Nnx
nxxn ,...,2,1,)sin()sin(2)( ==
πη
Can we formulate basis functions for bending motion that are Can we formulate basis functions for bending motion that are analoganalog to the to the µµnn(x(x) and ) and ννnn(x(x)) functions? functions?
How about:
⎟⎠⎞
⎜⎝⎛
++=
1sin
1 Nk
Nwk
ππ
Properties of Properties of ηηnn(x(x) functions :) functions :
they are orthonormal on [0,π] wrt to volume element sin(x) dx
the matrix elements <ηm|Tbend |ηn> have simple analytic solutions
they are related to an equidistant quadrature grid
the quadrature rule
)()sin()(10∑∫=
=N
kkk xfwdxxxf
π
Los Alamos, 30.07.2008
ηηnn(x(x) functions) functions1
4
23
5
6
Los Alamos, 30.07.2008
Definition of model HamiltonianDefinition of model Hamiltonian
relatively harmonic potential relatively harmonic potential →→ well suited for well suited for LegendreLegendre basisbasis
VVariationalariational BBasis asis RRepresentionepresention (VBR) for H(VBR) for H
dxxwxOxxOxb
anmnm )()()()()( ∫ ⎥⎦
⎤⎢⎣⎡=
∧∧
ϕϕϕϕ
We compare the performance of We compare the performance of ηηnn(x(x) and ) and PPll(cos(x(cos(x)) basis functions)) basis functions
Model system: pure bending motion of HModel system: pure bending motion of H22OO
( ) ( ) ( ) ( )443
32
210 )cos()cos()cos()cos( xcxcxcxccTH bend +++++=∧∧
For a true VBR, all matrix elements must be evaluated exactly
Los Alamos, 30.07.2008
Legendre VBR for H2O bending
Los Alamos, 30.07.2008
ηηnn(x(x)) VBR for H2O bending
Los Alamos, 30.07.2008
How to improve performance of How to improve performance of ηηnn(x(x) ?) ?Obviously, a basis formed exclusively by Obviously, a basis formed exclusively by ηηnn(x(x) functions is incomplete) functions is incomplete
Can we use the complementary functions?Can we use the complementary functions?
))sin(exp()( xrxsr −=
Can we supplement the Can we supplement the ηηnn(x(x) functions? For example: ) functions? For example:
1...,2,1,0,)sin()cos(
−= Nnx
nxnσ
))sin(exp()( xtxst −=
{ })())(cos(,))(1)(( xsxPxsx tnrm −η
Switching functions (1-sr(x)), st(x) keep basis orthogonal
Evaluation of matrix elements tedious
Los Alamos, 30.07.2008
Supplementation of Supplementation of ηηnn(x(x) functions) functionsIs direct basis extension an option? For example:Is direct basis extension an option? For example:
( ) ( ) HUXUAUXU TT =
We construct representation of H in this mixed basis → symmetric matrix ALoewdin (symmetric) orthogonalization of mixed basis:- Diagonalization of overlap matrix yields eigenvector matrix U and the matrix
X=diag(1/√ε1, 1/√ε2, … , 1/√εN)- Orthogonalization of mixed basis according to:
⎭⎬⎫
⎩⎨⎧
))(cos(,...)),(cos()),(cos(),(,...),(),(
1100
21
xPxPxPxxx
MM
N
σσσηηη
H is the desired representation of H in orthonormal mixed basis
Los Alamos, 30.07.2008
Mixed basis (+ P0,P1) VBR H2O bending
Los Alamos, 30.07.2008
Overview of localized & Overview of localized & delocalized representations,delocalized representations,
construction of localized mixed construction of localized mixed basis functions,basis functions,
applicationsapplications
Part Part IIIIII
Los Alamos, 30.07.2008
Overview: representationsOverview: representationsWe differentiate between infinitely localized, nearly localized We differentiate between infinitely localized, nearly localized and and delocalized basis functionsdelocalized basis functions
Discrete representations are only possible for local operatorsDiscrete representations are only possible for local operators
DDiscrete iscrete VVariable ariable RRepresentation (DVR) of local operator O is a epresentation (DVR) of local operator O is a matrix diagonal over the grid points matrix diagonal over the grid points xxkk..
If the grid is related to orthogonal basis functions If the grid is related to orthogonal basis functions ϕϕmm(x(x) through a ) through a quadraturequadrature rule of the form:rule of the form:
then we can define a then we can define a FFinite inite BBasis asis RRepresentation (FBR): epresentation (FBR):
)()()(1∑∫=
=N
kkk
b
a
xfwdxxwxf
k
N
kknkm
b
anm wxOxdxxwxOx ∑∫
=
∧∧
⎥⎦⎤
⎢⎣⎡≈⎥⎦
⎤⎢⎣⎡
1)()()()()( ϕϕϕϕ
Los Alamos, 30.07.2008
VBR, FBR, DVR, NDVRVBR, FBR, DVR, NDVRFBR matrix is an approximation to the VBR matrixFBR matrix is an approximation to the VBR matrix
FBR and DVR are equivalent:FBR and DVR are equivalent:
In analogy we can define:In analogy we can define:
NDVR matrix is an approximation to the DVR matrix:NDVR matrix is an approximation to the DVR matrix:
NDVR basis functions are approximately localized at grid points xk.
The term The term ““DVR calculationDVR calculation”” is not exact:is not exact:
DVR results are not variational!
How to perform How to perform ““DVR calculationDVR calculation”” for mixed basis functions for mixed basis functions QQnn(x(x)?)?
Definition of grid points through zeroes of QN+M+1(x) (from ηN+1(x) and PM(cos(x)))→ explicit derivation of orthogonal mixed basis
Establishment of quadrature rule for mixed basis → derivation of Λ matrix
tDVRFBR OO ΛΛ=
tNDVRVBR OO ΛΛ=
VTH DVRNDVRDVR +=
Los Alamos, 30.07.2008
Qm(x) VBR for m=1,2,3,4 from ηηnn(x(x) (n=1) (n=1--8) , 8) , PPll(cos(x(cos(x)) (l=1)) (l=1--4)4)1
234
12
3
4
Los Alamos, 30.07.2008
Qm(x) NDVR localized at xk, k=1,2,3,6 from ηηnn(x(x) (n=1) (n=1--30) , 30) , PPll(cos(x(cos(x)) (l=1)) (l=1--2)2)
Los Alamos, 30.07.2008
V1 = 300 + 3000 cos(x) + 8000 (cos(x))2 – 3000 (cos(x))3 – 2000 (cos(x))4
cbend = 10.0
Los Alamos, 30.07.2008
Legendre VBR for V1
Los Alamos, 30.07.2008
Qm(x) VBR for V1 from ηηnn(x(x) (n=1) (n=1--NN--2) , 2) , PPll(cos(x(cos(x)) (l=1)) (l=1--2)2)
Los Alamos, 30.07.2008
Legendre VBR for V1
Los Alamos, 30.07.2008
Qm(x) VBR for V1 from ηηnn(x(x) (n=1) (n=1--NN--2) , 2) , PPll(cos(x(cos(x)) (l=1)) (l=1--2)2)
Los Alamos, 30.07.2008
Legendre DVR for V1
Los Alamos, 30.07.2008
Qm(x) DVR for V1 from ηηnn(x(x) (n=1) (n=1--NN--2) , 2) , PPll(cos(x(cos(x)) (l=1)) (l=1--2)2)
Los Alamos, 30.07.2008
Legendre DVR for V1
Los Alamos, 30.07.2008
Qm(x) DVR for V1 from ηηnn(x(x) (n=1) (n=1--NN--2) , 2) , PPll(cos(x(cos(x)) (l=1)) (l=1--2)2)
Los Alamos, 30.07.2008
V2 = 420 - 4000 (cos(x))2 + 10000 (cos(x))4
cbend = 10.0
Los Alamos, 30.07.2008
Legendre VBR for V2
Los Alamos, 30.07.2008
Qm(x) VBR for V2 from ηηnn(x(x) (n=1) (n=1--NN--2) , 2) , PPll(cos(x(cos(x)) (l=1)) (l=1--2)2)
Los Alamos, 30.07.2008
Legendre VBR for V2
Los Alamos, 30.07.2008
Qm(x) VBR for V2 from ηηnn(x(x) (n=1) (n=1--NN--2) , 2) , PPll(cos(x(cos(x)) (l=1)) (l=1--2)2)
Los Alamos, 30.07.2008
Legendre DVR for V2
Los Alamos, 30.07.2008
Qm(x) DVR for V2 from ηηnn(x(x) (n=1) (n=1--NN--2) , 2) , PPll(cos(x(cos(x)) (l=1)) (l=1--2)2)
Los Alamos, 30.07.2008
Legendre DVR for V2
Los Alamos, 30.07.2008
Qm(x) DVR for V2 from ηηnn(x(x) (n=1) (n=1--NN--2) , 2) , PPll(cos(x(cos(x)) (l=1)) (l=1--2)2)
Los Alamos, 30.07.2008
ConclusionsConclusions
Part Part IVIV
Los Alamos, 30.07.2008
LegendreLegendre functions form for many applications a good basis set for functions form for many applications a good basis set for bending degrees of freedombending degrees of freedom
However, they offer a limited flexibility in particular for the However, they offer a limited flexibility in particular for the description of description of states with larger density close to the states with larger density close to the centercenter of the intervalof the interval
The combination of The combination of ηηnn(x(x) and ) and PPll(cos(x(cos(x)))) functions appears to be an functions appears to be an interesting alternative to the pure interesting alternative to the pure LegendreLegendre basisbasis
For mixed basis VBR calculations:For mixed basis VBR calculations:
all matrix elements can be evaluated analytically
computationally efficient because of Loewdin orthogonalization
more homogeneous accuracy distribution for different eigenstates
For mixed basis DVR calculations:For mixed basis DVR calculations:
explicit orthogonalization complicated → but needs to be performed only once since kinetic energy operator is always the same
quadrature rule for mixed basis set has been derived
accuracy similar to Legendre DVR can be reached, but further improvement necessary
Los Alamos, 30.07.2008
AcknowledgmentsAcknowledgments
Cooperation:Cooperation:U Kassel: Dietmar Kolb
Financial Support:Financial Support:CTCC at the University of Tromsø
And to you:And to you:
TUSEN TAKK!
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