Kinetic Energy Operator in curvilinear coordinates: numerical approach

La Grande Motte, February 2008

Kinetic Energy Operator in curvilinear coordinates: numerical approach

A. NautsY. JustumM. Desouter-Lecomte

L. Bomble (PhD)

Spectroscopy, floppy systems

control quantum gates


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: -The torsion of a chemical fragment (ex: Methyl) can be described by one coordinates: a dihedral angle, .-....

-All 3N curvilinear coordinates will be noted: q=[qi]

O C

H2H3

H1H


ˆ T q (q,q ) 2

2(q) 1

qi (q)G ij(q) q j

ij (q)

2

2G ij

2

qiq j 2

2G ij ln

q j G ij

q jj

qi

i

ij (q)

d (q)dq1dqn

Kinetic energy operator in curvilinear coordinates

)det()(J

qx

qxg jiij

1

gq

gG

where

and

Contravariant conponents of metric tensor

f2ij(q)

f1i(q)

extrapotential term (function of J and )


Where is the problem?

With few degrees of freedom, analytical expressions are easily determined:up to 3-4 atom systemsor with (quasi) orthogonal coordinates: Jacobi

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.

ij ii

jj

ij

jij

2

ji

2ij

2

qqG

qJlnG

2qqG

2),(

qqT

f2ij(q) f1

i(q)

Ex: Numerical

integrations


Example:For the 1D-Kinetic energy operator of methanol:

One active (dynamical) coordinate, , and 11 frozen coordinates

O C

H2H3

H1H

G()=A/B

A = -6 RCH2 (MCH3 MOH RCO

2 - 2 MH RCO (3 MOH RCH Cos(aHCO) + MCH3 RCOH Cos(aHOC)) + MH (3 MCHO RCH

2 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 RCO

2 - 12 MH MO RCH RCO Cos(aHCO) + 6 MH MCO RCH

2 Cos(aHCO)2 + 3 MH MCH3O RCH2

Sin(aHCO)2) Sin(aHOC)2

B = 6 MH RCH2 RCOH

2 Sin(aHCO)2 (2 MCH3 MO RCO2 -12 MH MO RCH RCO Cos(aHCO) + 6 MH

MCO RCH2 Cos(aHCO)2 + 3 MH MCH3O RCH

2 Sin(aHCO)2) Sin(aHOC)2


Where is the problem?

With few degrees of freedom, analytical expressions are easily determined:up to 3-4 atom systemsor with (quasi) orthogonal coordinates: Jacobi

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.

ij ii

jj

ij

jij

2

ji

2ij

2

qqG

qJlnG

2qqG

2),(

qqT

f2ij(q) f1

i(q)

Ex: Numerical

integrations


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 procedures1-2 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


IIIbJacobian and its derivatives

(Numerical) Calculation of TI

Mass-weighted cartesian coordinates in terms of the

curvilinear ones

II g matrix and its derivatives

IVKinetic energy

operator

IIIaG matrix and its derivatives

x(q)

xq i

2xq iq j

gij x

q i

x

q j

xq i

xq j

g ij

qk 2x

qkq ixq j x

qi2x

qkq j

J g

ln Jqk 1

2g 1 g

qk

ˆ T e (q, q )

or

f2ij(q) and f1

i (q)

G g 1

Gqk G

gqk G

ˆ T e(q,q ) 2

2Gij 2

qiq jij

2

2Gij ln J

q j G ij

q jj

q i

i


Cartesian coordinates in BF

x(Qdyn )

xQact

i

2xQact

i Qactj

O1O2 O1 RH1 O1 R1 O2 a1H1 O2 R2 O1 a2 H1 phi

Z-matrix

O1 O2

H1 H2

R

R1 R2a1 a2phi

V1

V2 V3

Cartesian construction with the vectors in any order.

=> Polyspherical, Jacobi vectors....

Analytical expression

bunch of vectors

Qzmat

Qzmat

Qacti

2Qzmat

Qacti Qact

j

Especially developed for MCTDH


Further transformations: Qzmat => Qdyn

Qzmati

Qdynk

1 if Qzmati Qdyn

k

0 if Qzmati Qdyn

k

Qzmat : Coordinates associated with the Z-matrix or the bunch of vectorsQdyn : Coordinates used in the dynamic (active, inactive)

Transformations

Identity Linear combinations Polar transformations

Qzmati C(i,m)Qdyn

m

m

Qzmati

Qdynk C(i, k)

Qzmat Qdyn Qzmati 1 Qdyn

k1 cos(Q dynk2 )

Qzmati 2 Qdyn

k1 sin(Qdynk2 )

Qzmati 1

Qdynk1

cos(Qdynk2 )

Qzmati 1

Qdynk 2

Qdynk1 sin(Q dyn

k2 )


Rigid constraints: Qinact = CteFlexible constraints: Qinact = Qinact(Qact)

Rigid or flexible constraints

Q inacti

Qactk 0

Q inacti

Qactk has to be calculated

Qdyn is split in active and inactive coordinates.The inactive coordinates are not used in the dynamics.

level(cm-1)

rigidconstraint

(1d)[1]

adiabaticconstraint

(1d)[1]

HADA(1+2) [2]

(0 0 0) 725,4 721,9 3508,95(0 2 0) 2167,9 2167,2 4952,48(0 4 0) 3586,0 3590,2 6383,93[0 0 0] 4366,3 4396,0 7318,47

max diff 0,3 < 0,1 < 0,01

Test : H-CN (Jacobi)

Spectrum of HCN/CNH (diagonalization) :

[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

Spectrum of HCN/CNH (WP)[3] :

Z-matrix :

CX C (1-)RN X R C 180,0H X r C N 0,0

Normalization : (x,R,r)=1 with x=cos()

constants=MC/MCN

H-CN/CN-H spectrum : floppy states

0

500

1000

1500

7000 12000 17000 22000 27000 32000 37000Frequency cm -1

Arb

itrar

y un

itsk = 0k = 20k = 25k = 30

adiabatic constraint(1d : )

C X

H

(1-)R

NR

r


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, 2006WP on 1,3-dibromopropane by R. Brogaard in Denmark


Tnum and MCTDH

-In Tnum, the elements of the G tensor can be known only on the full dimensionality grid!

=> 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

Incompatibility between Tnum and MCTDH!

Can be easily used with CDVR!!


Taylor expansion of G around Q0

-The calculation of the derivatives of G (up to the second order) is already implemented in Tnum.

This form can be used with MCTDH

G(Q) G(Q0) G(Q)Q i

Q0

(Q i Q0i )

i

12

2G(Q)Q iQ j

Q0

(Q i Q0i )(Q j Q0

j )i, j

This approach is well known:

1. R. Wallace, CP, 11 p189 1975: H2O, CH of benzene (Zero order)2. E. L Sibert III, W. P. Reinhardt, J. T. Hynes, JCP, 81, p1115 1984: CH in benzene (first order)3. L. Halonen, T. Carrington Jr JCP 88 p4171 1988: H2X (third order)4. L Lespade, S. Robin D. Cavagnat, JPC, 97, p6134, 1993: Cyclohexene (second order)....


Taylor expansion of G around Q0

The Taylor expansion may give non-hermitian KEO !

Ex: AB-C with Jacobi coordinates :

The Taylor expansion (2d order)

T(x,R,r) 12

1MR2

1mr2

x

(1 x2) x

T(x,R,r)

R R0

MR03

( 2x0)

x(x x0)

x

2d order in x and R.

Non-hermitian with Legendre polynomial

Always Hermitian with the sine and the HO basis-sets.

Example: H2O in valence coordinates

&geom zmat=T nat=3 /16.1. 11. 1 2

1 1 1 &niveau nrho=1 read_nameQ=t /r1OH 1.r2OH 1.a 1.6

Z-matrix

reference geometry, Q0

-------------------------------------------------HAMILTONIAN-SECTIONmodes | r1OH | r2OH | a------------------------------------------------# Zero order part: -1/2*G^ij(Qref)------------------------------------------------- -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....

12

1M

1m

defined constraints(here no constraint)

Number of terms for (H2O)2H+ (D2d): 29,92,279


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 propagationTime independent methodsClassical Trajectories (Hamilton)

- Hard to find a basis set well adapted to T (with a diagonal representation).


Curvilinear coordinates: Z-matrix

atom 2 : distance d2 from

atom 1atom 1:at origin

atom 3 :distance d3 from atom 2

and angle a3 between atoms 1, 2 and 3At1 At2

At3

d3

d2

3

x 1zmat

000

zmat

x 1zmat

Q k

000

zmat

x 2zmat 00

d 2

zmat

x 2

zmat

Q k

00d2

Q k

zmat

x 3zmat

x 2zmat d 3

sin( 3 )0

cos( 3 )

zmat

x 3

zmat

Q k x 2

zmat

Q k d 3

Q k

sin( 3 )0

cos( 3 )

zmat

d 3

3

Q k

cos( 3 )0

sin( 3 )

zmat


Curvilinear coordinates: Z-matrix

atom 2 : distance d2 from

atom 1atom 1:at origin

atom 3 :distance d3 from atom 2

and angle a3 between atoms 1, 2 and 3At1 At2

At3

d3

d2

3

x 1zmat

000

zmat

x 1zmat

Q k

000

zmat

zmat

k

zmat2

zmat

zmat2

000

Qx

.200

x

zmat

zmat

k

zmat2

k

zmat3

zmat

zmat2

zmat3

8660.00

5.0.0*.2

5.00

8660.0.1

Qx

Qx

5.00

8660.0.2xx

d2=2.d3=2.a3=120°

Qk= d3

3

2

dd

3

d3

d3

d3

Kinetic Energy Operator in curvilinear coordinates: numerical approach

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