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1
Potential energy surfaces and predicted infrared spectra for van der Waals complexes
Daiqian Xie
Department of chemistry, Nanjing University, China
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