1

Potential energy surfaces and predicted infrared spectra for van der Waals complexes

Daiqian Xie

Department of chemistry, Nanjing University, China

dqxie@itcc.nju.edu.cn

2009.11

5th International Meeting on Mathematical Methods for ab initio Quantum Chemistry, Nice

2

Outline

• Introduction

• Rg－(linear molecule) vdW complexes

• H2－(linear molecule) vdW complexes

• Path integral Monte Carlo Study for HeN-N2O

• Summary

3

Van der Waals interactions

Collections of molecules

Stability of molecular complexes

Biochemical reactions

The spectroscopy provides very useful information for the intermolecular potential energy surface (IPES) and dynamics.

Theoretical study of the ro-vibrational spectraprovide complete description of the spectra.

Introduction

4

Rigid monomer molecule

Separate the inter- and intra- molecular vibrations and fix the geometry of the linear molecule.

The rovibrational Hamiltonian:

( ) ( )

( ) ( )[ ] ( ) ( )[ ] ( )θθµµ

θ

µθθθ

θθµµθ

,ˆˆˆˆ2

ˆˆˆˆˆ2cot

ˆ2ˆ2

1sin

ˆsin

sin1

222ˆ,ˆ,ˆ,,ˆ

22

2222

22

2

2

2

22

RVJiJJiJR

JiJJiJJR

JJR

JIRR

JJJRH

yxyxyxyxz

zz

zyx

+−−+⋅∂∂

+−++⋅+

−+

+

∂∂

∂∂−

++

∂∂

−=

V(R, θ) : potential energy of intermolecular interactionµ: reduced mass of the vdW moleculeI: moment of inertia of the linear molecule

θ

R

r0BA

Rg

5

This strategy is usually good for microwave spectra.

But not enough for infrared spectra, which involves excitation of the intermolecule vibrations.

For example, for the infrared spectra of He-N2O,

Band origin shift: Calculated :-0.019 cm -1, Observed: 0.253 cm -1

Y. Zhou and D.Q. Xie, J. Chem. Phys. 120, 8574(2004)

An improved way is to include the dependence of intramolecular modes

6

Q normal mode

θ1

θ2

ϕ

R

Q

2 2 2 22 2 1 2

1 1 1 22 2 21 2 1

ˆ ˆ ˆˆ1 1 ( )ˆˆ ( , , , , )2 2 2 2Q

H B V R QR Q I R

ϕ θ θµ µ µ

∂ ∂ − −= − − + + + +

∂ ∂j J j jj

The observed infrared spectra usually involve excitationof only one intermolecule vibrations.

( Linear molecule)

D. Q. Xie*, H. Ran, Y. Zhou, International Reviews in Physical Chemistry, 26（3）， 487 （2007）

7

Rg－(linear molecule) vdW complexes

θ

R

rBA

Rg

Q

( ) ( ) ( )

( ) ( ) ( )

2 2 2

2 2 2 21 2 1

2 22 2

1 1

21

ˆ1 1 1 1 1ˆ ( ) sin2 2 2 2 sin sin

1 cotˆ ˆ ˆ ˆ ˆ ˆ ˆ22 2

1 ˆ ˆ ˆ ˆ , ,2

z

Q

z x y x y z

x y x y

JHR Q I R

J J J iJ J iJ JR R

J iJ J iJ V R QR

θµ µ µ θ θ θ θ

θµ µ

θµ θ

∂ ∂ ∂ ∂= − − + + − + ∂ ∂ ∂ ∂

+ − + + + − ⋅

∂ + ⋅ + − − + ∂

(in the Body-Fixed frame)

Rare-gas atom

(XZ plane)

8

The three-dimensional PES can be divided as:

Vmon(Q): PES for isolated linear molecule.∆V(R, θ, Q) : IPES at fixed Q

mon( , , ) ( ) ( , , )V R Q V Q V R Qθ θ= + ∆

1 2 1 2

1 2

, , ,, , ,

( , , , , , ) ( ) ( ) (cos ) ( , , )JMp nJp K Jpn j K v v v v j KMj K v v

R Q c R Q P Cθ α β γ ψ ψ θ α β γΨ = ∑

1/ 2 * *0( , , ) [2(1 )] [ ( 1) ], 0,1

Jp J J K p JKM K MK M KC D D pα β γ δ

− + +−= + + − =

2 2 2

2

22

1 ( ) ( ) ( )2 mon v v v

d V Q Q E QdQ

ψ ψµ

− + =

Total wavefunction (in FBR):FBR: finite basis representation

(total parity is given by (-1)J+p)

9

The oneThe one--dimensional energy curve for the dimensional energy curve for the QQ33 coordinate of COcoordinate of CO22

2

2

CO 3 3 323

1 ( ) ( ) ( )2 v v v

d V Q Q E QM dQ

ψ ψ − + =

3 C O 1 C O 2( ) / 2Q r r= −

Q3

-.4 -.2 0.0 .2 .4

V

V without scaleV with scale

E0obs.1 = 2349.149cm-1

E0cal.= 2418.759cm-1 (without scale)E0cal.= 2349.1483cm-1 (scaled)

Scale factor = 0.971449

1. J. Mol. Spec. 76, 430 (1979)

For example of Rg-CO2 complex:

10

Then the PODVR grid points for Q can be determined by(PODVR: potential optimized discrete variable representation)

mn m nX xψ ψ=

2( , ) ( ) ( , , ) ( ) ( , , )v v v kv kk

V R Q V R Q Q dQ T V R Qθ ψ θ ψ θ∞

−∞= ∆ = ∆∑∫

Vibrationally averaged intermolecular PES：

Only a few PODVR grid points for Q are sufficient !

The FBR can be conveniently transformed to the DVR. Any local functions are assumed to be diagonal in the DVR.

11

The matrix elements of the Hamiltonian in the DVR are given by

( ){

( ) ( ) }

2 2

2 2

2 22 2

1 12 2

0 1 0 1

ˆ

1 12 2

1 1 1ˆ 1 22 2

ˆ ˆ1 1

q

K K q q K K

K K q q K KQ

K JK K K K JK K K

q K H qK

q qR Q

j J J KR I R

β β α α β β

α α β βα α

α β αβ

α α δ δ δ δ δ δµ µ

β β δ δ δ δ δµ µ

δ β β δ δ β β δ δ

′ ′ ′ ′ ′ ′1 2

′ ′ ′ ′ ′1 1

+ −′ ′ ′ ′+ + − −

′ ′ ′ ′ =

∂ ∂′ ′− ⋅ ⋅ ⋅ + − ⋅ ⋅ ⋅ +∂ ∂

′ + ⋅ ⋅ ⋅ + + − ⋅ ⋅

′ ′− + Λ ⋅ ⋅ − + Λ ⋅ ⋅ ⋅j j

( ), ,( , , )

q q

Ka q K K q qV R Q

α α

β α α β β

δ

θ δ δ δ δ

′

′ ′

⋅

+ ⋅ ⋅ ⋅ ⋅

The IPES can be constructed at a few PODVR points of Q, so that the possible interpolation error could be avoided.

( ) ( )11 ±−+=± KKJJJKΛ

12

Five PODVR grid points are enough0.6050, 0.7332, 0.8607, 1.0000, 1.1694 Å

KrH H3.38 År

How many PODVR points are enough for the r coordinate?

The vibrational energy levels (in cm-1) of H2 at a fixed Kr-H distance of 3.38Å.

v 6 DVR points 5 DVR points 4 DVR points 0 0.0 0.0 0.0 1 4157.6683 4157.6618 4157.6379 2 8079.4942 8079.4714 8079.4976 3 11769.4622 11769.5329 11768.7091

For Example:

13

Lanczos algorithm: (Lanczos 1950)

• Three term recursion:

where

( )[ ] kkkkkk βψβψαψ /2111 −−−− −−= H

kkk ψψα HT=

( ) 2111 −−−− −−= kkkkk ψβψαβ H

−

=

14

• Reduction of H:

KNNNKN

K

KK ×××× =

= QHQT T2221

00

00

α

αββα

KKationdiagonaliz

KKLanczos

NN ××× → → DTH

15

• Diagonalization (QL or inverse iteration):

– Lanczos eigenvectors:

– Lanczos eigenstates:

≡

)(

)(2

)(1

)(

KKi

Ki

Ki

Ki

z

zz

z

kK

ki

K

k

Ki z ψφ

)(

1

)( ∑==

)()()()( Ki

Ki

Ki

K E zzT =

16

• Convergence:

• Scaling laws:

– Memory ∝ N, CPU ∝ N 2 .– Capable of handling problems with N up to 108.

1)()()()(

++= KK

KiKK

iK

iK

i zE ψβφφH

If , converges!0)( →KKiz )(Kiφ

17

• Finite precision arithmetic– Loss of orthonormality among Lanczos states:

– Unnormalized Lanczos eigenstates

Observation: sum of norms of converged copies in a cluster = # of copies:

– Arithmetic average

∑ ≠=′

′′

K

kkkk

Kik

Kki

Ki

Ki zz

,

)()()()( 1ψψφφ

,kkkk ′′ ≠ δψψ 0>>′− kk

22 ( )

1

M Km n m i

iMχ φ χ φ

== ∑

( ) ( )K Ki i

iMφ φ =∑

H. Guo, R. Chen, D.Q. Xie, J. Theo. Comp. Chem. 1,173 (2002)

18

Calculation of the transition intensity

The line intensity of a transition at a temperature T :

2/ /

, ,

( )[ ]Jpn J p nE kT E kT JMp J M pJpn J p n J p n Jpn n A nM M A X Y Z

I E E e e µ′ ′ ′− − ′ ′ ′′ ′ ′ ′ ′ ′ ′→′ ′ ′ ′=

∝ − − Ψ Ψ∑∑ ∑

component of the dipole moment along A axis of the space-fixed (SF) frame

zyixyix µµµµµµµµ =−=−+−=+ 0),(21

1),(21

1

11

1

( , , )g h ghh

Dµ µ α β γ=−

′ = ∑

The calculated dipole moment in BF frame is rewritted as:

Transfomation between BF and SF frames:

(Wigner rotation function

19

The transition intensity:

'

/ /'

' '12

0 0 ' '

' '

' '

(2 1)(2 1)( )[ ]

1 1[(1 )(1 )] ( 1) ( 1)

1 1( 1) ( 1)

Jpn J p nE kT E kTJpn J p n J p n Jpn

K J pK K

K hK

K J p K J p J p K

I J J E E e e

J J J JK K h K K h

J J J JK K h K K h

δ δ

′ ′ ′− −′ ′ ′ ′ ′ ′→

− +′

′ ′ ′ ′ ′ ′+ + + + + + +

∝ + + − −

+ + − + − −

+ − + − − − −

∑∑ ∑2

Jnp J n pK h K dµ τ

′ ′ ′′

Ψ Ψ∫

The rovibrational wave function can be rewritten as:

( , , , , , ) ( , , ) ( , , )JMp Jnp Jpn K KMK

R Q R Q Cθ α β γ θ α β γΨ = Ψ∑

(Winger 3-j symbol )

20

Consider the Q3 normal mode for the antisymmetric vibrational stretching of N2O molecule

Q3=8.834062(r1-r1e)-4.056573(r2-r2e)

N N Or1 r2

Y. Zhou and D.Q. Xie, J. Chem. Phys. 124, 144317(2006)

He-N2O: rovibraional spectra in the v3 region of N2O

21

• CCSD(T) method with aug-cc-pVTZ basis set

• Bond function (3s 3p 2d1f 1g) (for 3s and 3p, α=0.9,0.3,0.1; for 2d, α=0.6,0.2, for f, g,α=0.3)

• The supermolecular approach

• Full CP was used to correct BSSE

• 5 PODVR points for Q3 coordinate:-2.0202, -0.9586, 0.0, 0.9586, 2.0202and 250 geometries for each IPES

))()(()(int BABBAABAvdW VVVV χχχχχχ +++−+=

22Contour plot of the intermolecular potential energy of He–N2O.

Contours are labeled in cm-1.

R(a

o)

4

5

6

7

8

9

10

-7

-10-13

-5-8-11

θ

30 60 90 120 150 180

-5

-8-11

Q3= -2.0202 Q3= 0

Q3= 2.0202

θ

0 30 60 90 120 150 180

R(a

o)

4

5

6

7

8

9

10

Q3= 0.9586

-4-7

-10

23

Vibrational energy levels (in cm-1) for 4He-N2O and 3He-N2O

Ground state state 4He-N2O 3He-N2O 4He-N2O 3He-N2O

0 0 -21.4252 -18.2289 -21.2548 -18.0738 0 1 -9.3540 -7.5059 -9.3467 -7.4957 0 2 -7.2367 -5.4854 -7.2031 -5.4566 0 3 -4.1228 -1.5844 -4.0833 -1.5431 1 0 -2.2008 -2.1451

3ν

Frequency of v3 band of N2O: 2223.7567cm-1

Band shift in 4He: 0.2532cm-1 (obs), 0.1704cm-1 (cal)in 3He: 0.2170cm-1 (obs), 0.1551cm-1 (cal)

24

J. Tang and A. R. W. Mckellar, J. Chem. Phys. 117, 2586 (2002).

3He-N2O

Transition frequency/cm-12223.4 2223.6 2223.8 2224.0 2224.2 2224.4

Rel

ativ

e in

tens

ity

0.0

.5

1.0

211-220101-110

221-220

331-330330-331

220-221

110-101220-211

441-440

440-441

111-110

110-111

Transition frequency/cm-12223.6 2223.8 2224.0 2224.2 2224.4

Rel

ativ

e in

tens

ity

0.0

.5

1.0

211-220

101-110

111-110221-220

441-440 440-441330-331

220-221

110-111

220-211

110-101

331-330

4He-N2O

The calculated line intensities of 3He-N2O and 4He-N2O at a rotatotaional tempreture of 2K.

25

He-CO2:Assignment of the hot band in the infrared spectra

1. M. J. Weida, J. M. Sperhac, and D. J. Nesbitt, J. Chem. Phys. 101, 8351 (1994)2. Y. Xu and W. Jager, Journal of Molecular Structure 599, 211 (2001) 3. K. Nauta and R. E. Miller, J. Chem. Phys. 115 (22), 10254 (2001)4. J. Tang and A. R. W. Mckellar, J. Chem. Phys. 121, 181 (2004) 5. A. R. W. Mckellar, J. Chem. Phys. 125, 114310 (2006 )

1. G. S. Yan, M. H. Yang, and D. Q. Xie, J. Chem. Phys. 109, 10284 (1998)2. T. Korona, R. Moszynski, F. Thibault, J.-M. Launay, B. Bussery-Honvault,

J. Boissoles, and P. E. S. Wormer, J. Chem. Phys. 115, 3074 (2001)

3. F. Negri, F. Ancliotto, G. Mistura, and F. Toigo, J. Chem. Phys. 111, 6439 (1999)

Experimental researches:

Theoretical researches:

26

Weida, et al., J. Chem. Phys. 101, 8351 (1994)

O1 C O2θ

R

Q3

He

27

Korona, et al., J. Chem. Phys. 115, 3074 (2001)

28

NewNew abab initioinitio potential calculation:potential calculation:

•• JacobiJacobi coordinatecoordinate (R, θ, Q3) ;;

•• about about 700 symmetry unique points;symmetry unique points;

• aVQZ basis set plus basis set plus bond functions;;

•• CCSD(T) levellevel

Five PODVR grid points, -0.21715, -0.10325, 0.0, 0.10325, and 0.21715 a0,

for the Q3 coordinate

O1 C O2θ

R

Q3

He

29

Energy levels (cm-1) of bound vibrational states

R (B

ohr)

6

8

10

12

14

θ (degree)30 60 90 120 150

30 60 90 120 150

6

8

10

12

14

30 60 90 120 150

θ (degree) θ (degree)

R (B

ohr)

(0, 0) (0, 1) (0, 2)

(0, 3) (0, 4)

-1.235-1.268(0, 4)

-3.974-3.954(0, 3)

-7.599-7.593(0, 2)

-8.700-8.687(0, 1)

-16.875-16.979(0, 0)

v4 stateGround state(ns, nb)

No bound states with the stretching excitation

8.292 vs 9±2 cm-1

30

524←4133.00.0542202←2112.50.02543±22351.040116221←1109.9-0.1251221←1104.60.02744±22350.797815322←2118.7-0.1955423←3122.7-0.00375±22350.544414423←3126.3-0.0948313←21210.0.01546±22350.531113423←4140.60.156110←2111.00.03855±22350.186912524←4310.90.1457212←2210.1-0.0196±22350.153611440←4310.2-0.1242616←6431.00.04793±22349.736710423←3300.00.1059422←5050.9-0.04225±22349.52989220←1112.4-0.3215551←4040.1-0.02237±22348.94348220←2111.8-0.2586220←2112.3-0.02573±22348.52637321←2125.4-0.1282202←3136.90.03865±22348.44576422←3313.9-0.0433101←2127.10.04185±22348.39665422←4130.6-0.095422←3131.5-0.00412±22348.31794202←1114.20.147422←5151.6-0.06634±22348.26193321←3123.0-0.0797321←3121.20.00263±22348.00222101←1104.80.1106111←2020.20.00963±22347.93051

assignmentIν(obs.-cal.)assignmentIν(obs.-cal.)Iν(cm-1)

Ref.Our workObs.

Ref. : J. Chem. Phys. 115(2001)3074

New Assignment of ‘hot band’ transitions:

31

ν(cm-1)2347.0 2347.5 2348.0 2348.5

Rel

ativ

e In

tens

ity

0.0

.5

1.0cal.obs.

5 15-

6 06

3 31-

4 22

4 13-

5 24

4 14-

5 05

3 12-

4 23

1 11-

2 20

2 11-

3 22

3 13-

4 04

4 14-

4 23

1 10-

2 21

2 12-

3 03

5 14-

5 23

(413

-422

+ 3

13-3

22)

4 13-

4 22

3 13-

3 22

1 11-

2 02

4 32-

4 41

2 12-

2 21

Transitions from v3=0 to v3=1 state of CO2 with the complex at the van der Waals ground state at T=4.1 K.

Calculated shift of band origin: 0.101 cm-1 ; Observed value: 0.094 cm-1

Journal of Chemical Physics, 128, 124323(2008)

32

H2－(linear molecule) vdW complexes

θ1

θ2

ϕ

R

Q

H H

2 2 2 22 2 1 2

1 1 1 22 2 21 2 1

ˆ ˆ ˆˆ1 1 ( )ˆˆ ( , , , , )2 2 2 2Q

H B V R QR Q I R

ϕ θ θµ µ µ

∂ ∂ − −= − − + + + +

∂ ∂j J j jj

33

Radial DVR/Angular FBRSin-DVR for R coordinate

PODVR for Q coordinate

The basis set for angular coordinate (θ1,θ2,ϕ):

1 2

1 2

121 2 0 0 1 , 2

1 , 2

, , , ; (2 2 ) [ ( , ) ( )

( 1) ( , ) ( )]

JK m MK j m j K m

J P JM K j m j m K

j j m K JMP D Y

D Y

δ δ θ φ θ

θ φ θ

−−

+− − −

= + Θ

+ − Θ

2( ) ( ) ( , ) ( ) ( , , )v v v kv kk

V R Q V R Q Q dQ T V R Qψ ψ∞

−∞= ∆ = ∆∑∫

1 0 1 0( ) ( ) ( , ) ( ) ( , )v v k k kk

d R Q d R Q Q dQ T R Q Tα α αψ ψ µ∞

= =−∞= = ∑∫

Vibrationally averaged intermolecular PES：

Transition diploe moments：

34

In the angular FBR, the angular kinetic energy matrix elements:

21 2 1 2 1 2

21 2 1 2 1 2 1 2 1 2

21 1 2 2 1 1 2 2

ˆ ˆˆ, , , ; ( ) , , , ;ˆ ˆ ˆ ˆ ˆ ˆˆ ˆ ˆ, , , ; ( ) ( ) ( ) , , , ;

[ ( 1) 2 2 ( ) ( 1) ( 1)] ( ) ( ) ( , ) ( , )

(1 ( ,0) ( ,0))

j j m K JMp j j m K JMp

j j m K JMp J j j m K JMp

J J K m K m j j j j j j j j K K m m

K m

δ δ δ δ

δ δ

′ ′ ′ ′ − −

′ ′ ′ ′= − + − + + +

′ ′ ′ ′= + − + − + + + +

′ ′− +

2

J j j

J j j j j J j j

1

1

2

2

12 1 1 2 2

12 1 1 2 2

12 1 1 2 2

12

( ) ( ) ( , 1) ( , 1)

(1 ( ,0) ( ,0)) ( ) ( ) ( , 1) ( , 1)

(1 ( ,0) ( ,0)) ( ) ( ) ( , 1) ( , )

(1 ( ,0) ( ,0))

JK j m

JK j m

JK j K m

JK j K m

j j j j K K m m

K m j j j j K K m m

K m j j j j K K m m

K m

δ δ δ δ

δ δ δ δ δ δ

δ δ δ δ δ δ

δ δ

− −

+ +

− −−

+ +−

′ ′ ′ ′Λ Λ − −

′ ′ ′ ′− + Λ Λ + +

′ ′ ′ ′ ′− + Λ Λ −

− + Λ Λ

1 2

1 2

1 1 2 2

12 1 1 2 2

12 1 1 2 2

( ) ( ) ( , 1) ( , )

(1 ( ,0) ( ,0)) ( ) ( ) ( , ) ( , 1)

(1 ( ,0) ( ,0)) ( ) ( ) ( , ) ( , 1)

j m j K m

j m j K m

j j j j K K m m

K m j j j j K K m m

K m j j j j K K m m

δ δ δ δ

δ δ δ δ δ δ

δ δ δ δ δ δ

+ −−

− +−

′ ′ ′ ′+

′ ′ ′ ′+ + Λ Λ +

′ ′ ′ ′ ′+ + Λ Λ −

21 2 1 1 2 1 1 1 1 2 2

21 2 2 1 2 2 2 1 1 2 2

ˆ, , , ; , , , ; ( 1) ( ) ( ) ( , ) ( , )ˆ, , , ; , , , ; ( 1) ( ) ( ) ( , ) ( , )

j j m K JMp j j m K JMp j j j j j j K K m m

j j m K JMp j j m K JMp j j j j j j K K m m

δ δ δ δ

δ δ δ δ

′ ′ ′ ′ ′ ′ ′ ′= +

′ ′ ′ ′ ′ ′ ′ ′= +

j

j

35

Fitting of the potential energy surface

1 2 1 2 1 2 1 21 2 1 2

1 2 1 2 1 2( , , , , ) ( , ) ( , , ) exp ( , ) ( , , )l l l l l l l l l l l ll l l l l l

V R Q g R Q A d R Q Aϕ θ θ θ θ ϕ θ θ ϕ

∆ =

∑ ∑

(For a set value of R and Q)

1 2 1 21 2

1 2 1 1 , 2 2( , , ) ( , ) ( , )0

ll l l l m l m

m l

l l lA Y Y

m mθ θ ϕ θ ϕ θ ϕ

<−

=− <

= −

∑

with

21 ϕϕϕ −= ),min( 21 lll =<

(both l1 and must be even )21 lll ++

where

36

Fitting of the dipole moment surface

1 2 1 21 2

1 2 ; ; 1 2( , , , , ) ( , ) ( , , )l l l l l ll l l

R Q g R Q Aα α αµ ϕ θ θ θ θ ϕ= ∑ , α= x, y, or z .

The angular form of µz is the same as that for the potential energy.

1 2 1 2

1 2; 1 2 1 1 2( , , ) ( ) ( ) cos( )1 1l l l x l m l mm

l l lA m

m mθ θ ϕ θ θ ϕ− +

= Θ Θ − + −

∑

1 2 1 2

1 2; 1 2 1 1 2( , , ) ( ) ( ) sin( )1 1l l l y l m l mm

l l lA m

m mθ θ ϕ θ θ ϕ− +

= Θ Θ − + −

∑

37

Calculation of the transition intensity

'

/ /

'*

'

'*

'

'*

'

(2 1)(2 1)( )[ ]

1( 1)

1( 1)

1( 1)

Jpn J p nE kT E kTJpn J p n J p n Jpn

K Jnp J n pK h K

K hK

J p Jnp J n pK h K

J p JnpK h K

I J J E E e e

J Jd

K K h

J Jd

K K h

J JK K h

µ τ

µ τ

µ

′ ′ ′− −′ ′ ′ ′ ′ ′→

′ ′ ′′+ +

′ ′ ′+′− +

′ ′+′+ −

′∝ + + − −

− Ψ Ψ −

+ − Ψ Ψ

+ − Ψ Ψ − −

∑ ∑ ∑ ∫

∫

2'

*'

1( 1)

J n p

J p J p Jnp J n pK h K

d

J Jd

K K h

τ

µ τ

′ ′ ′

′ ′ ′ ′ ′+ + +′− −

+ − Ψ Ψ −

∫

∫

*1 2 1 2

*1 2

( , , , , , , , ) ( , , , , ) ( , , )

( 1) ( , , , , ) ( , , )

JMp Jnp Jn K MK

K

J p Jnp JK M K

R Q R Q D

R Q D

ϕ θ θ α β γ ϕ θ θ α β γ

ϕ θ θ α β γ

+

+− −

Ψ = Ψ +

− Ψ

∑

38

H2: possibly the most abundant molecule in the universe

pH2: indistinguishable boson

superfluid behavior

Nuclear spin state: I = 0 for pH2, I = 0, 2 for oD2I = 1 for oH2 and pD2

At low temperature: jH = 0 for pH2 and oD2jH = 1 for oH2 and pD2

H2-N2O: rovibraional spectra in the v3 region of N2O

Y. Zhou, H. Ran, and D.Q. Xie, J. Chem. Phys., 125, 174310(2006)

39

• CCSD(T) method•• aug-cc-pVTZ basis set for H atom

cc-pVTZ plus diffuse functions (1s1p1d) for Oand N atoms

• Bond function (3s 3p 2d1f) (for 3s and 3p, α=0.9,0.3,0.1; for 2d, α=0.6,0.2, for f, α=0.3)

• 4 PODVR points for Q3 coordinate:-1.6507, -0.5246, 0.5246, 1.6507and about 5000 geometries in totalR: [4.5a0, 15.0a0] with 12 pointsθ1, θ2, ϕ in increments of 30o

40

θ 2

0

30

60

90

120

150

180

θ10 30 60 90 120 150 180

ϕ

0

30

60

90

120

150

180

θ1

30 60 90 120 150 180

(a)

(c)

(b)

Contour plots of potential energy surface for N2O-H2 at R = 5.43 a0 for Q3 = -0.5246. (a) ϕ = 0o (b) ϕ = 90o (c) θ2 = 93.6o.The contour spacing is 30 cm-1

H H

N N O

θ1

θ2

R

Q3

ϕ

41

Pure vibrational energy levels (in cm-1) for first ten bound states of four species of N2O−hydrogen complexes.

n N2O−pH2 N2O−oH2 N2O−oD2 N2O−pD2

1 -64.7179 31.4662 -84.6173 -48.1361

2 -35.0508 63.2391 -52.9458 -14.6583

3 -27.6110 77.4146 -40.6190 2.9199

4 -23.3189 81.7838 -36.4570 8.8001

5 -18.1627 86.5550 -30.2769 12.9811

6 -11.0445 88.6383 -28.0395 14.7727

7 -6.7396 91.2522 -23.2572 16.6810

8 -0.9296 96.5342 -17.8034 21.0470

9 101.2074 -13.6489 24.1322

10 105.8023 -8.5608 29.8419

The ground state energies for oH2 and pD2 are 118.48675cm-1 and 59.78042 cm-1

Vmin = -242.43 cm-1

42

Band origin shift:

N2O-pH2: 0.2219 cm-1(cal); 0.2261 cm-1(obs)

N2O-oH2: 0.4236 cm-1(cal); 0.6238 cm-1(obs)

N2O-oD2: 0.3585 cm-1(cal); 0.4534 cm-1(obs)

N2O-pD2: 0.5437 cm-1(cal); 0.7900 cm-1(obs)

43

θ230 60 90 120 150 180

θ2

0 30 60 90 120 150 180

R (a

0)

56789

101112

n = 2 n = 3

The contour plots of probability density integrated over two angular variables θ1 and ϕ for some vibration states of N2O−pH2 at Q3 = -0.5246.

44

θ230 60 90 120 150 180

θ20 30 60 90 120 150 180

R (a

0)

5

6

7

8

9

10 n = 2 n = 3

The contour plots of probability density integrated over two angular variables θ1 and ϕ for some vibration states of N2O−oD2 at Q3 = -0.5246.

45

The calculated line intensities at a rotational temperature of T = 1.5K. The transition frequencies are relative to the band origin.

a J. Tang and A. R. W. McKellar, JCP 117, 8308 (2002)

Transition frequency/cm-1.4 .6 .8 1.0 1.2 1.4 1.6 1.8 2.0

Rel

ativ

e in

tens

ity

0.0

.5

1.0

N2O-pH2

Transition frequency/cm-1.6 .8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4

Rel

ativ

e in

tens

ity

0.0

.5

1.0 this workexperimental

101-000

211-110313-212

303-202

322-221

321-220

312-211

212-111

202-101

N2O-oH2101-000

212-111

202-101

211-110 313-212

303-202

46Transition frequency/cm-1.4 .6 .8 1.0 1.2 1.4 1.6

Rel

ativ

e in

tens

ity

0.0

.2

.4

.6

.8

1.0

1.2

Transition frequency/cm-1.2 .4 .6 .8 1.0 1.2 1.4 1.6 1.8

Rel

ativ

e in

tens

ity

0.0

.2

.4

.6

.8

1.0101-000

211-110

313-212303-202

211-212

221-110

111-000

212-111

212-101202-111

202-101

110-101

101-000

212-111

202-101

212-101

211-110

313-212

303-202

N2O-orthoD2

N2O-paraD2

111-000

The calculated line intensities for N2O-D2at a rotational temperature of T = 1.5K. The transition frequencies are relative to the band origin.

47

Spectroscopic studies offer insights into dynamical and structural information.

From vdW complexes to clusters：

Nano-dropletClustersDimer

48

In 4He, sharp rotational lines, absent Q branch→ free linear rotorIn 3He, broad line → conventional rotation diffusion in viscous solution

S. Grebenev et al., Science, 278, 2083 (1998)

Superfluidity Within a Small Helium-4 Cluster

OCS in4He droplet

OCS in3He droplet

OCS inmixed

4He-3He droplet

49

High resolution infrared spectra of CO2 solvated with helium atomsJ.Tang and A.R.W.McKellar, J. Chem. Phys. 121, 181 (2004)

Turnaround at critical point ⇒decoupling from environment, i.ethe rotor is no longer dragged by 4He.

Due to microscopic superfludity

50

For quantum solvents such as 4He and para-H2, microscopic superfluidity and transition from vdWcomplexes to quantum solvation have been established.

Theoretical description requires both QM approach to the finite temperature dynamics and accurate intermolecular PES

51

Path integral Monte Carlo Study for HeN-N2O

Hamiltonian for a 4He cluster doped with a N2O molecule:

2 2N O He He N Oˆ ˆ ˆ

NH H H Vν ν ν= + +

2 2

2

N O N ON O

ˆ2

H Bm

ν ν= +2

20p L

2 3

1N O 2 QB Iν

ν νϕ ϕ−=

( )2

He1 He

ˆ2

Ni

i ji i j

H um= <

= + −∑ ∑p r r

2 2He N O He-N O1

N

N

i

V Vν ν=

= ∑

{pi ; ri }: the momenta and positions of the helium atoms{p0; r0}: the momenta and position of the N2O molecule

V=0,1

52

ˆˆ ( ) He βρ β −=

{ }ˆTr ( )Z ρ β=

1/ Bk Tβ =

Any physical observable can be evaluated as

{ }0( , | 0 )i i= >q r r

{ }1ˆ ˆ ˆTr ( )1 ˆ ˆ( )

O OZ

d d d d OZ

βρ β

ρ β

=

′ ′ ′ ′ ′ ′= ∫ q q Ω Ω qΩ q Ω qΩ qΩ

(Density operator)

(Partition function)

imaginary time

53

In the path integral picture with a finite number （K) of discretization of the imaginary time, the density matrix can be written as

rot trN O2

ˆ

ˆ ˆ ( )1 1

2 1

k

H

K KT H

k k k k k kk k

e

d d e e

β

τ τ

−

− −+ +

= =

′ ′ =

∏ ∏∫ ∫ ΩqΩ qΩ

q Ω Ω Ω q q

/ Kτ β=

1 1( ) ( )′ ′=q Ω qΩ

1 1( ) ( )K K+ + =q Ω qΩ

( )N O2 1

0

2 1 (cos )4

B l ll

l

l e pυτ γπ

∞− +

=

+∑( ) ( )

( ) ( )

1

2 21 1

3 / 2 3 / 20.5

/ 4 / 4

1 14 4

k k

k k N k ki i i

NV R V R

R R r r

e

e e

τ

τ λτ

π τ πλτ+

+ +

− +

− − Λ − −

Λ

∑×

The Bosonic exchange effect was incorporated into PIMC byIncluding permutation sampling.

54

Distance distribution of the i th He atom,

01

1 Ki ik k

k

RK β=

= −∑ r r

cos iiiR β

θ ⋅= n R

2PIMCE a bβ

τ= +

Orientation estimator:

The “raw” PIMC data were extrapolated to the limit of ,0τ →

55

2 2( ) ( 1) ( 1)eff effE J B J J D J J= + − +

( ) ( ) { }10 Tr H H Hn n e e ne nZ

β τ ττ − −〈 • 〉 = •

1 0E Eν νβ β

ν = =∆ = −

Vibrational band origin shifit:

Effective rotational and centrifugal distortion constants:

(Orientational correlation function)

[ ][ ]{ })ττττ

ββττ

33

0

22

)1(4)1(2exp)1()42exp(

)1()1(exp)42exp(1

+++−++−

×+++−++−= ∑

>

JDJBJDJBJJ

JDJJBJDBZ J

56

The formula for analytic potential energy surface:

( )[ ] ( )[ ]∑ ∑= =

−

+

×+×=12

6 02 12

)(cos,exp),(),(max

i

lL

l

lii

i

lPC

RRDfRBRARV θθθθθ

Short-range term Long-range term

∑ ∑= =

−

+=

4

0

5

0

20 ]12

)(cos)exp(),(i

l

l

li

i

lPgRRdRA θθis short range linear term:),( θRA

12)(cos)(

12)(cos),(

4

0

4

1 +−+

+= ∑∑

== lPbR

lPdRB l

l

ll

l

l θθθis short range exponential term:),( θRB

is dumping factor which is defined as( )[ ]RDfi θ !/1)( 0 kxexfn

kkx

n ∑ =−−=

12)(cos)(

4

0 += ∑

= lPbD l

l

l θθwhere

)(cosθlP is Legendra polynomial

Refitting for He-N2O PES

57

Contour plot in Jacobi coordinates of the He−N2O PES

V=-62.27 cm-1

V=-32.88cm-1

N N Oθ

R

He

58

Effective rotational temperature, T=0.37K

J. Chem. Phys., 121, 181, 2004.

Rotational constant

B=0.4190 cm-1 , for ground state of N2O

0.4156 cm-1 , for ground state of N2O

PIMC simulation parameters

59

• Accept ratio0.3-0.6, to enhance the efficiency of PIMC sampling.

• Number of time slices of transitional freedom:N=256, 384,512.

• Number of time slices of rotation freedom:N= 32, 64, 128

• Sampling stepsMore than 2,000,000 sampling steps to reach quasi-ergodicity

60

0.230-21.973-22.20364512

0.224-21.615-21.839128512

0.235-22.090-22.326128384

0.241-22.443-22.68464384

0.251-23.546-23.79764256

0.186-21.295-21.481Rot-vib Energy Calculation

0.206-20.904-21.110Extrapolation

0.225-21.719-21.94464640

RotationTranslationΔv(cm-1)Ev=1(cm-1)Ev=0(cm-1)

Number of time slices

( )0→τ

Benchmark calculation: He-N2O dimer of different time slices

61

4HeN-N2O

T=0.37K

Angular distribution of helium atoms around the N2O probe moleculein the 4HeN−N2O complexes. The density is normalized.

62

N=5 N=6

N=7 N=8

4He density ρ(r, θ) and averaged positions of the 4He atomsaround the N2O molecule. The ‘bond’ only shows the configurations of rings.

63

Vibrational band origin shift for the 4HeN−N2O clusters

64

Effective rotational constant Beff

4HeN-N2OT=0.37K

65

Summary

1. It now becomes possible to include the explicit dependence of the intramolecular degrees of freedom in the studies of the rovibrational spectra of the complex.

2. With the explicit involvement of one intramolecular vibrationalcoordinate that is related to the transition in the infrared spectra, full prediction of the infrared spectra including the shift of band origin can be achieved.

3. The vibrationally averaged potentials are essential to simulate the spectroscopic properties of the related vdW clusters.

66

Further developments :

. complexes with a non-linear molecule

. complexes with an open-shell linear molecule.

67

Acknowledgments

Collaborators:

Prof. Hua Guo, University of New Mexico, USAProf. P.N. Roy, University of Waterloo, Canada

Graduate Students: Hong Ran, Zheng Li, Lecheng Wang, Yanzi Zhou

Funding：NSFC; MOE; MOST

68