Kinetic Energy Operator in curvilinear coordinates:
numerical approach
A. Nauts
Y. Justum
M. Desouter-Lecomte
Spectroscopy, floppy
systems
L. Bomble (PhD)
control
quantum gates
La Grande Motte, February 2008
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.
H
-....
f
O
H1
C
-All 3N curvilinear coordinates will be noted: q=[qi]
H3
La Grande Motte, February 2008
H2
Kinetic energy operator in curvilinear coordinates
Contravariant conponents of metric
tensor
Tˆq (q,q ) = 
extrapotential term (function
of J and )


ij
(q)
(q)G (q) j  (q)

i
2 ij
q
q
2
1
ij 
 2
 2 ij 2
ln

G

ij

=  G  i j  

G

 (q)

j
j 
i

q
q q
q q
ij  2
i  2
j
f2ij (q)
d = (q)dq
1
where
and
G=g
dq
n
1
f1i (q)
x  x 
g ij =  i
j
 q q
J(q) = det( g )
La Grande Motte, February 2008
Where is the problem?
 2
  2 ij   2
T (q,  q ) =    G  i j    
2
2
ij 
i 
 q q
f2ij (q)
 ln J G ij  
j G q j  q j  qi

ij
f1i (q)
With few degrees of freedom, analytical expressions are easily determined:
up to 3-4 atom systems
or with (quasi) orthogonal coordinates: Jacobi
Ex:
Numerical
integrations
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.
La Grande Motte, February 2008
Example:
For the 1D-Kinetic energy operator of methanol:
One active (dynamical) coordinate, f, and 11 frozen
coordinates
H
H1
f
O
C
H3
H2
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
Where is the problem?
 2
  2 ij   2
T (q,  q ) =    G  i j    
2
2
ij 
i 
 q q
f2ij (q)
 ln J G ij  
j G q j  q j  qi

ij
f1i (q)
With few degrees of freedom, analytical expressions are easily determined:
up to 3-4 atom systems
or with (quasi) orthogonal coordinates: Jacobi
Ex:
Numerical
integrations
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.
La Grande Motte, February 2008
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
1-2 active coordinate(s) (inversion of ammonia[2], ring puckering[3,4], torsion[5])
6 active 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
ˆ (q, ) =  
T
e
q
ij
 2
2

G
i
j   
2
q q
2
i 
2
ij
 ln J G ij 


G


i
q j
q j 
j
 q
ij
(Numerical) Calculation of T
IIIa
G matrix and
its derivatives
I
Mass-weighted cartesian
coordinates in terms of the
curvilinear ones
G = g 1
x(q)
x
q i
 x
q i q j
2
IV
Kinetic energy
operator
x x 
x x
gij =  i
=
j
q i q j
 q q
g ij
 2 x x
x  2 x
=

q k
q k q i q j
qi q k q j
II
g matrix and
its derivatives
G
g
= G k G
k
q
q
Tˆe (q, q )
or
f2ij (q) and f1i (q)
J=
g
ln J 1 1  g
= g
q k
2
q k
La Grande Motte, February 2008
IIIb
Jacobian and
its derivatives
Cartesian coordinates in BF
H1
a1
R1
O1
phi
R
H2
a2
O2
R2
O1
O2 O1
H1 O1
H1 O2
Z-matrix
R
R1 O2 a1
R2 O1 a2 H1 phi
Qzmat
Qzmat
Q iact
2Qzmat
j
Q iact Q act
x(Qdyn )
V2
V3
Cartesian construction with the
vectors in any order.
x
Q iact
=> Polyspherical, Jacobi vectors....
V1
2 x
j
Q iact Q act
Especially developed
for MCTDH
bunch of vectors
Analytical expression
La Grande Motte, February 2008

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
Qzmat = Qdyn
1 if Qizmat = Qkdyn
Qizmat 
= 
k
i
k
Qdyn 
0 if Qzmat  Qdyn
Linear combinations
Qizmat =  C(i,m)Qmdyn
m
Q izmat
= C(i, k)
Q kdyn
La Grande Motte, February 2008
Polar transformations
i1
k1
k2

Qzmat = Qdyn cos(Q dyn )
 i
k1
k2
2

Qzmat = Qdyn sin( Qdyn )
Q i 1
k2
 zmat
= cos(Q dyn
)
k1
Q dyn

1
Q izmat
k1
2
sin( Q kdyn
)
 k 2 = Q dyn

Q dyn
Rigid or flexible constraints
Qdyn is split in active and inactive coordinates.
The inactive coordinates are not used in the dynamics.
Rigid constraints:
Flexible constraints:
Q iinact
=0
k
Q act
Qinact = Cte
Qinact = Qinact(Qact)
Q iinact 
has to be calculated
k
Q act
La Grande Motte, February 2008
Test : H-CN (Jacobi)
H

C
Z-matrix :
r
(1-m)R
m=MC/MCN
X
mR
N
Spectrum of HCN/CNH (diagonalization) :
level
(cm-1)
rig id
constraint
(1d)[1]
(0 0 0)
725,4
(0 2 0)
2167,9
(0 4 0)
3586,0
[0 0 0]
4366,3
max diff
0,3
constants
adiabatic
constraint
(1d)[1]
721,9
2167,2
3590,2
4396,0
< 0,1
HADA
(1+2) [2]
3508,95
4952,48
6383,93
7318,47
< 0,01
[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
C
X C (1-m)R
N X
m R C 180,0
H X
r C

N
0,0
Normalization : (x,R,r)=1 with x=cos()
Spectrum of HCN/CNH (WP)[3] :
adiabatic
constraint
(1d : )
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
Tnum and MCTDH
-In Tnum, the elements of the G tensor can be
known only on the full dimensionality grid!
Incompatibility
between Tnum
and MCTDH!
=> 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
What can be done?
-Use a fitting procedure
-Taylor expansion of G
La Grande Motte, February 2008
Can be easily
used with
CDVR!!
Taylor expansion of G around Q0
G(Q)
1 2G(Q)
i
i
i
i
j
j
G(Q) = G(Q0 )  
(Q

Q
)

(Q

Q
)(Q

Q

0
0
0) 
i
i
j
Q Q0
2 i,j Q Q Q
i
0
-The calculation of the derivatives of G (up to the
second order) is already implemented in Tnum.
This form can
be used with
MCTDH
This approach is well known:
1.
2.
3.
4.
....
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
Taylor expansion of G around Q0
The Taylor expansion may give non-hermitian KEO !
Ex: AB-C with Jacobi coordinates :
1  1
1  
2 
T(x,R,r) =   2  2  (1 x ) 
2 MR
mr x
x


The Taylor expansion (2d order)
T(x,R,r) = 
R  R 


0
(x  x 0 ) 

3 (2x 0 )
x
x
 MR 0 
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
Example: H2O in valence coordinates
&geom zmat=T nat=3 /
16.
Z-matrix
1. 1
1. 1 2
defined constraints
(here no constraint)
111
&niveau nrho=1 read_nameQ=t /
r1OH 1.
reference
r2OH 1.
geometry, Q0
a 1.6
Number of terms for (H2O)2H+
(D2d): 29,92,279
------------------------------------------------HAMILTONIAN-SECTION
modes | r1OH | r2OH | a
-----------------------------------------------1  1 1 
# Zero order part: -1/2*G^ij(Qref)
   
2 M m 
-------------------------------------------------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
....
Advantages/drawback of Tnum
+ 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
Curvilinear coordinates: Z-matrix
3
At1
d2
At3
atom 3 :
distance d3 from atom 2
and angle a3 between atoms 1, 2
and 3
d3
At2
atom 1:
at origin
atom 2 :
distance d2 from
atom 1
x zmat
= x zmat
3
2
 0 
 
=  0 
 
d 2 
zmat
zmat
x1
x
Q k
zmat
1
0 zmat
 
= 0 
0 
 
x zmat
2
0 zmat
 
= 0 

0 

x zmat
2
Q k
x 3zmat x zmat
= 2k
k
Q
Q


 0 


=  0 
d 2 
k 
LaGrande
Q  Motte, February 2008
zmat
 sin( 3 ) zmat


 d 3  0

 cos( )

3 
 sin( 3 ) zmat
d 3 


0
 
k 
Q 

 cos( 3 )
cos( 3 )zmat
3 

d3
0


Q k 

sin( 3 ) 
Curvilinear coordinates: Z-matrix
3
At1
d2
At3
atom 3 :
distance d3 from atom 2
and angle a3 between atoms 1, 2
and 3
d3
At2
atom 1:
at origin
atom 2 :
distance d2 from
atom 1
x zmat
= x zmat
3
2
 0 
 
=  0 
 
d 2 
zmat
zmat
x1
x
Q k
zmat
1
0 zmat
 
= 0 
0 
 
x zmat
2
0 zmat
 
= 0 

0 

x zmat
2
Q k
x 3zmat x zmat
= 2k
k
Q
Q


 0 


=  0 
d 2 
k 
LaGrande
Q  Motte, February 2008
zmat
 sin( 3 ) zmat


 d 3  0

 cos( )

3 
 sin( 3 ) zmat
d 3 


0
 
k 
Q 

 cos( 3 )
cos( 3 )zmat
3 

d3
0


Q k 

sin( 3 ) 
Curvilinear coordinates: Z-matrix
3
At1
d2
At2
atom 1:
at origin
At3
d3
d2=2.
d3=2.
a3=120°
atom 3 :
distance d3 from atom 2
and angle a3 between atoms 1, 2
and 3
Qk= d3
atom 2 :
distance d2 from
atom 1

x 3zmat
0.8660
 zmat
= x 2  2. 0 
 0.5 
d 3
d 3
zmat
x1
x
Q k
zmat
1
0 zmat
 
= 0 
0 
 
0 zmat
 
= 0 

0 


x zmat
2
0
=  0 
2.
zmat
zmat
0 

x zmat
2
= 0
k
Q d 2
0 Motte, February 2008
La
dGrande
3
zmat
0.8660


x 3zmat x zmat
= 2 k  1. 0 
k
Q
Q
 0.5 
  0 .5 
2. * 0. 0 
 3
0.8660
d 3
zmat
zmat
