Florence J. Lin- Overall rotation due to internal motion in the three-body problem: Applications in molecular dissociation and collisions

Overall rotation due to internal motion in the three-body problem: Applications in molecular dissociation and collisions

Florence J. LinUniversity of Southern CaliforniaDepartment of Mathematics, KAP 1083620 S. Vermont AvenueLos Angeles, CA 90089-2532

California American Physical Society 2007Lawrence Berkeley National LaboratoryBerkeley, CAOctober 27, 2007

F. J. Lin, California APS 2007, LBNL, Berkeley, CA, October 27, 2007; Copyright 2007 F. J. Lin

2

Abstract

As a result of the conservation of total rotational angular momentum in an N-body system, an internal motion with nonzero orbital angular momentum produces a net overall rotation of a generalized Eckart frame of a polyatomic molecular system in the center-of-mass frame, regardless of whether or not the total rotational angular momentum vanishes. Examples appear in a net rotation of 20 degrees in the recoil angle of an atom departing from a dissociating triatomic molecule and in a net overall rotation of 42 degrees in 100,000 reduced time units in the computational dynamics of a protein. While the diatomic molecule in atom-diatomic molecule scattering has previously been treated as a point mass, this approach describes the contribution to the deflection angle of the atom due to rotation of the diatomic target molecule. When an N-body system returns to its original shape over the time interval, the net rotation due to internal motion is an example of a classical geometric phase. Other applications appear in the dissociation of polyatomic molecules, in the separation of overall rotation and internal motions of N-body systems, and in the dynamics of molecular rotors and machines.References:F. J. Lin, Hamiltonian dynamics of atom-diatomic molecule complexes and collisions, Discrete and Continuous Dynamical Systems, Supplement, vol. 2007, 655 – 666 (2007).J. E. Marsden, R. Montgomery, and T. Ratiu, Reduction, symmetry, and phases in mechanics, Memoirs of the American Mathematical Society, vol. 88, no. 436, (American Mathematical Society, Providence, RI, 1990).


3

Introduction

Traditionally, molecular motions have been separated into translation, rotation, and internal motions [15]. Symplectic reduction [11-14] provides a systematic method to obtain reduced phase spaces and has been applied to describe N-body molecular dynamics [7-9]. The first reduction by translational symmetry parameterized by the total linear momentum leads to dynamics in the center-of-mass frame; a second reduction by rotational symmetry parameterized by the orbital angular momentum leads to dynamics in the internal frame. While the total rotational angular momentum is conserved, the orbital angular momentum is not conserved and couples the dynamics of overall rotation with the internal motions. The net overall rotation in the center-of-mass frame due to the internal dynamics follows from [9] (i) theconservation of the total rotational angular momentum in the center-of-mass frame and, equivalently, from (ii) Hamilton’s equations in the center-of-mass frame. This net rotation can be described in terms of various internal coordinates and is mass-dependent. After describing various observations of this net rotation [3, 5, 16] and describing the net rotation in terms of Jacobi coordinates, the net rotation is described in terms of Eckart generalized coordinates for a generalized Eckart frame as a reduced “internal” phase space [9]. When the total rotational angular momentum vanishes, then the coordinates of overall rotation and internal motions are separated when the internal angular momentum vanishes.


4

Outline

(1) The dynamics in the “internal” reduced phase space can be described in various internal coordinates

(2) The internal dynamics are coupled to the overall rotation in the center-of-mass frame; i.e., net overall rotation due to internal motions at zero total angular momentum has been observed

(a) in a differential geometric study of a triatomic molecule,(b) in the net rotation of the recoil angle of the departing O atom in the dissociation of NO2,(c) in the net overall rotation in a computer simulation of a three-helix bundle protein.

(3) In Jacobi coordinates, the net overall rotation due to internal motions(a) can be derived from the conservation of total rotational angular momentum,(b) can be derived from Hamilton’s equations in the center-of-mass frame, using reduction theory.

(4) In generalized Eckart coordinates, the net overall rotation due to internal motions can be derived from the conservation of total rotational angular momentum.

(5) Further applications(a) The net rotation of the recoil angle of a departing atom in a triatomic dissociation at nonzero

total angular momentum is due to internal motions of the diatomic fragment.(b) The deflection angle of the scattered atom in atom-diatomic molecule scattering includes a

contribution due to the internal motion, e.g., rotation, of the diatomic molecule.(6) Geometric formulation

(a) The geometric phase is the holonomy of a molecular connection.(b) The geometric phase is expressed in terms of a molecular gauge potential.

(7) Summary


5

Various internal coordinates can be used to describe the dynamics of three-body to N-body molecular systems:

Guichardet coordinates

(Tachibana and Iwai, 1986)

Jacobi coordinates

Internuclear distances

Bond angles, bond lengths, dihedral angles, …

(Zhou, Cook, and Karplus, 2000)


6

Background: Observations of net rotation at zero total rotational angular momentum

(a) Net overall rotation in a differential geometric study of triatomic molecular dynamics (Guichardet, 1984)At zero total angular momentum a purely vibrational motion, i.e., an internal motion, can take a molecule with a specified initial shape to a final configuration with the same shape but differing from the initial configuration by a net rotation

(Tacbibana and Iwai, 1986)

∑∫=∆k

kk dqaθ

( )( )( )

( ) ( )( )2121

23

22322

21311

2321123

33222

3211

2 )(

)(/)(/

)(/

qqmmqqmmmqmmmqD

qDqmmqmmaqDqmmma

qDqmma

+++++=

++−=+=

=


7


(b) Net rotation of the recoil angle of the O atom in the photodissociation of NO2 (Demyanenko, Dribinski, Reisleret al., 1999)At zero total angular momentum when the (remaining) NO fragment rotates by an angle θ, the direction of the recoil velocity vector R of the departing O atom rotates by an angle θR with net rotation ∆θR:

Jacobi coordinates

m1 = O, m2 = N, m3 = 0

o2022

2

≈+

=∆ ∫C

i

dRmr

mrR

γ

γ

θµ

θ

( )321

321

21

21 , mmmmmm

mmmmm

+++

=+

= µ


8


(c) Net overall rotation in the dynamics of a model three-helix bundle protein (Zhou, Cook, and Karplus, 2000)At zero total angular momentum, the flexible protein molecule rotates by 42 degrees in 105 reduced time steps due to internal motions, i.e.,

(Zhou, Cook, and Karplus, 2000)

degrees 42=∆Θ


9

Jacobi coordinates for three-body systems, e.g., NO2 or an atom-diatomic molecule complex

m1 = O, m2 = N, m3 = Oor

m1 = Cl, m2 = H, m3 = Ar

R

θr θR

m1

m3

m2

θ

r


10

The net rotation ∆θR of a generalized Eckart frame due to internal motions corresponds to the orbital angular momentum:(Lin, 2007)

The total rotational angular momentum is conserved.The total rotational angular momentum is the sum of two terms (Delves,1960), i.e.,

which leads to a differential equation for the angle θR of rotation of a generalized Eckartframe.

Integration of the angular velocity of rotation yields the net rotation of a generalized Eckart frame.

The net rotation of a generalized Eckart frame

has dynamic and geometric terms

where (a) the first term vanishes when the total angular momentum vanishes and (b) the second counter-rotary term is in the opposite direction to the rotation of the diatomic molecule in the rotating frame.

( ) θθµθθµ &&&&l 22222 mrmrRmrRj J RrRtot ++=+=+=

Rr θθθ −=( )

321

321

21

21 , mmmmmm

mmmmm

+++

=+

= µ

( ) ( )geomRdynRR θθθ ∆+∆≡∆

( )

( ) θµ

θ

µθ

dmrR

mr

dtmrR

J

geomR

t

t

totdynR

f

i

∫

∫

+−=∆

+=∆

22

2

22


11

The Hamiltonian for three-body dynamics in Jacobi coordinates (Lin, 2007)

The Hamiltonian in the lab frame is

The reduced Hamiltonian in the center-of-mass frame is

with amended potential given by

with

The reduced Hamiltonian in terms of “internal” coordinates and θR is

( )cm,12312321321321 rrrrpppppprrr −−+++= ,,2

12

121),,,,,( 2

3

2

2

2

1

θVmmm

H

( ) ( )RrVm

H ,,21

21,,, 2 θ

µ linlin pRrRrp ppppRr ++=

( ) ( ) ( )321

2

2,,,,

mmmRrVRrV

+++= lin

p

plin

θθ

Rr θθθ −≡

( ) ( )

( ) ( ) ( )( ) ( )RrV

mrRpJ

mrRpJp

mrpp

mppppRrH

RrVR

pmrpp

mpppppRrH

tottotRrRr

RrRrRr

Rr

Rr

,,2222

,,,,,

,,2222

,,,,,,,

22

2

222

222

2

2

2

222

θµµµ

θ

θµµ

θθ

θθθθθ

θθθθ

linlin

linlin

pp

pp

++

−+

+−

+++=

++++=


12

The coupling of overall rotation and internal motions in Hamilton’s equations is given by dθR/dt (Lin, 2007)

The twice-reduced Hamiltonian is

The twice-amended potential is

The twice-reduced Hamilton’s equations for “internal” motions on a reduced phase space are

µ

θ θ

R

r

pR

mrp

mpr

=

=

=

&

&

&

2

( ) ( )( )( )[ ]

( )

( ) ( )( )( )[ ]2222

22223

2

/,,

,,

/,,

mrmrRpJpJRr

RVp

RrVp

mrmrRpJpJRr

rV

mrpp

tottotR

tottotr

+

+−−

∂∂

−=

∂∂

−=

+

+−−

∂∂

−−=

µθ

θθ

µθ

θθ

θ

θθθ

&

&

&

( ) ( )

( )RrmrR

pJpmrpp

mppppRrH

pJ

totRrRr

tot

ptotJ

,,V 222

,,,,,

,

222

222

,

θµµ

θ

θ

θ

θθθθ

−++

−+++=

−

lin

lin

p

p

( ) ( ) ( )( )( )22

2321

2

,

2

2,,,,

mrRpJ

mmmRrVRrV

tot

pJtot

+−

+

+++=−

µ

θθ

θ

θ

linp

plin

( )RrVp

mrRmrJ

R

totR

,,

22

2

θθ

µθθ

θ ∂∂

=

+−

=

&

&&

The net overall rotation ∆θR is coupled to the internal motions by


13

Applications: (a) Net rotation of recoil angle, ∆θR, in triatomicphotodissociation and (b) scattering angle χ in atom-diatomic molecule collisions(a) Net rotation ∆θR of the recoil angle

in triatomic photodissociation with internal motionsThe magnitude |∆θR| of the net rotation of the recoil angle for zero total angular momentum with internal motions (Demyanenko, Dribinski, Reisler, et al.,1999) is

The net rotation of the recoil angle for arbitrary total angular momentum including internal motions (Lin, 2007) is

(Demyanenko, Dribinski, Reisler, et al., 1999)

θµ

θθ

θ

dmrR

mrC

i

R ∫ +−=∆ 22

2

∫∫ +−

+=∆

C

i

C

i

dmrR

mrdtmrR

Jt

t

totR

θ

θ

θµµ

θ 22

2

22


14

Applications: (a) Net rotation of recoil angle, ∆θR, in triatomicphotodissociation and (b) scattering angle χ in atom-diatomic molecule collisions

(Weston and Schwarz, 1972)

(b) Scattering angle χ in atom-diatomic molecule collisionsThe atom-diatomic molecule scattering angle χ omitting internal motions (Cross and Herschbach, 1965) is

The scattering angle χ including internal motions in atom-diatomic molecule scattering (Lin, 2007) is

( )

dtR

J

t

tot

R

∫∞

−=

−=

min

2

min

2

2

µπχ

θπχ

( )

( )

∫∫∞∞

++

+−=

θ

θ

θµµ

πχminmin

22

2

22 22tt

tot dmrR

mrdtmrR

J


15

Applications in N-body dynamics in Eckart generalized coordinates: Net overall rotation of a three-helix bundle protein(Zhou, Cook, and Karplus, 2000; Lin, 2007)

∑+Θ=λ

λλ qBAJ tot &&

The total rotational angular momentum is

( ) ( )geomdyn ∆Θ+∆Θ≡∆Θ

( )

( ) ∑ ∫

∫

−=∆Θ

=∆Θ

λλ

λ dqA

B

dtA

J

geom

t

t

totdyn

f

i

∑+Θ=λ

λλ dq

ABdmolA

ABA λ

λ =

The net overall rotation due to internal motions is

The molecular connection is

The molecular gauge potential is

degrees 42

motions internal todueprotein bundlehelix- threeflexible a ofrotation Overall iii)(

degrees 20

NO ofon dissociati in thefragment NO theofrotation the todue atom O

departing theof angle recoil theofRotation ii)(

study geometric aldifferenti ain molecule triatomica ofrotation Overall i)(

:momentumangular totalzeroat rotation net of Examples

22

2

2

3

1

=−=∆Θ

≈+

−

=∆

∑ ∫

∫

∑ ∫=

λλ

λ

θ

θ

θµµ

µ

θ

dqA

B

dRr

r

dqa

C

i NO

NO

kkk


16

Geometric formulation: The net overall rotation ∆θR in terms of a geometric phase (Marsden, Montgomery, and Ratiu, 1990; Lin, 2007)

The coordinates of the “internal”reduced phase space are expressed in Jacobi coordinates by r, θ, R.For a closed loop in configuration space, the contributions to the net overall rotation are

the dynamic phase (∆ θR)dyn

the geometric phase (∆θR)geom

(Marsden, Montgomery,and, Ratiu, 1990)

For a closed loop in configuration space, the geometric contribution to the net rotation of a generalized Eckart frame

is expressible in terms of a molecular gauge potential (a ratio of moments of inertia, in this case)

and as the holonomy of a molecular connection

( ) ∫∫ −=+

−=∆ θθµ

θ θ dAdmrR

mrgeomR 22

2

22

2

mrRmrA

+=

µθ

θθθµ

θ θ dAddmrR

mrd RRmol +=+

+= 22

2

A


17

Summary: Net rotation in the N-body dynamics in Eckartgeneralized coordinates and in Jacobi coordinates (Lin, 2007)

( ) ( )geomRdynRR θθθ ∆+∆≡∆

Special case: In Jacobi coordinates, the net rotation ∆θR of a generalized Eckartframe in the center-of-mass frame due to internal motions is

( ) ( )geomdyn ∆Θ+∆Θ≡∆Θ

( )

( ) ∑ ∫

∫

−=∆Θ

=∆Θ

λλ

λ dqA

B

dtA

J

geom

t

t

totdyn

f

i

General case: In Eckartgeneralized coordinates, the net rotation ∆Θ of a generalized Eckart frame in the center-of-mass frame due to internal motions is

( )

( ) θµ

θ

µθ

dmrR

mr

dtmrR

J

geomR

t

t

totdynR

f

i

∫

∫

+−=∆

+=∆

22

2

22


18

Summary: Physical explanation and examples

Physical explanation: Net rotation is due to nonzero orbital or internal angular momentum, respectively:

Examples:Net overall rotation in a three-helix bundle protein (including nonzero total angular momentum and internal motions): 42 degrees

Examples (cont’d.):Net rotation of the recoil angle of a departing atom in photodissociation(including nonzero total rotational angular momentum): 20 degrees

Deflection angle of an atom scattering off a diatomic molecule (including internal motions)

AcknowledgementWiSE Program, Office of the Provost, University of Southern California

∫∫ +−

+=∆

C

i

C

i

dmrR

mrdtmrR

Jt

t

totR

θ

θ

θµµ

θ 22

2

22

∑ ∫∫ −=∆Θλ

λλ dq

ABdt

AJf

i

t

t

tot

( )

( )

∫∫∞∞

++

+−=

θ

θ

θµµ

πχminmin

22

2

22 22tt

tot dmrR

mrdtmrR

J

⎪⎪

⎩

⎪⎪

⎨

⎧

=⎟⎠

⎞⎜⎝

⎛−

≠⎟⎠

⎞⎜⎝

⎛−

=∆Θ

∫ ∑

∫ ∑−

=

−

=

0,/

0,/

32

1

32

1

tot

t

t

N

tot

t

t

N

tot

JAdtqB

JAdtqBJ

f

i

f

i

λλλ

λλλ

&

&


19

References

1. R. J. Cross and D. R. Herschbach, Classical scattering of an atom from a diatomic rigid rotor, J. Chem. Phys. 43, 3530 – 3540 (1965).

2. L. M. Delves, Tertiary and general order collisions, Nucl. Phys. 9, 391 – 399 (1958/59); 20, 275 – 308 (1960).3. A. V. Demyanenko, V. Dribinski, H. Reisler, H. Meyer, and C. X. W. Qian, Quantum-product state-dependent

anisotropies in photoinitiated unimolecular decomposition, J. Chem. Phys. 111, 7383 -- 7396 (1999).4. C. Eckart, Some studies concerning rotating axes and polyatomic molecules, Phys. Rev. 47, 552 – 558 (1935).5. A. Guichardet, On rotation and vibration motions of molecules, Ann. Inst. Henri Poincaré, Phys. Théor. 40, 329

– 342 (1984).6. J. Jellinek and D. H. Li, Separation of the energy of overall rotation in any N-body system, Phys. Rev. Lett. 62,

241 -- 244 (1989).7. F. J. Lin and J. E. Marsden, Symplectic reduction and topology for applications in classical molecular

dynamics, J. Math. Phys. 33, 1281 – 1294 (1992).8. F. J. Lin, Symplectic reduction, geometric phase, and internal dynamics in three-body molecular dynamics,

Physics Letters A 234, 291 – 300 (1997).9. F. J. Lin, Hamiltonian dynamics of atom-diatomic molecule complexes and collisions, Discrete and Continuous

Dynamical Systems, Supplement 2007, 655 – 666 (2007).10. J. E. Marsden, R. Montgomery, and T. Ratiu, Reduction, symmetry, and phases in mechanics, Mem. Amer.

Math. Soc., Vol. 88, No. 436 (American Mathematical Society, Providence, RI, 1990).11. J. E. Marsden and T. Ratiu, Introduction to Mechanics and Symmetry, (Springer-Verlag, New York, 1994).12. J. Marsden and A. Weinstein, Reduction of symplectic manifolds with symmetry, Rep. Math. Phys. 5, 121 –

130 (1974).13. K. R. Meyer, Symmetries and integrals in mechanics, in M. M. Peixoto, ed., Dynamical Systems, (Academic,

New York, 1973), pp. 259 – 272.14. K. R. Meyer and G. R. Hall, Introduction to Hamiltonian Dynamical Systems and the N-body Problem,

(Springer-Verlag, New York, 1992).15. E. B. Wilson, Jr., J. C. Decius, and P. C. Cross, Molecular Vibrations: The Theory of Infrared and Raman

Vibrational Spectra, (Dover, New York, 1980), republication of McGraw-Hill edition of 1955.16. Y. Zhou, M. Cook, and M. Karplus, Proteins at zero-total angular momentum: The importance of long-range

correlations, Biophys. J. 79, 2902 – 2908 (2000).

Documents

Florence J. Lin- Overall rotation due to internal motion in the three-body problem: Applications in molecular dissociation and collisions