int. symp. molecular spectroscopy ohio state univ., 2005 the ground state four dimensional morphed...

Int. Symp. Molecular SpectroscopyOhio State Univ., 2005

The Ground State Four Dimensional Morphed Potentials of HBr and HI Dimers

Collaborator: J. W. Bevan, TAMU

Funding: National Science Foundation

Robert R. LuccheseDepartment of Chemistry

Texas A&M University


Morphing of Intermolecular Potentials

• Compute a full potential energy surface (PES) using a quantum chemistry model

• Morph potential to obtain best possible agreement with experiment

Vmorphed R,θ1,θ2 ,φ( ) =S1 θ1,θ2 ,φ( )Vab initio Rmorph θ1,θ2 ,φ( ),θ1,θ2 ,φ( )

Rmorph θ1,θ2 ,φ( ) =S2 θ1,θ2 ,φ( ) R−RF( ) + 1+S3 θ1,θ2 ,φ( )⎡⎣ ⎤⎦RF


Morphing Functions

Sα θ1,θ2 ,φ( ) = Cα,iFλα,iθ1,θ2 ,φ( )

i∑

λ = lx ,n, ′θ1, ′θ2 , ′φ( )

Fλ θ1,θ2 ,φ( ) = Nλ I l1 ,l2 ,m θ1,θ2 ,φ( ) I l1 ,l2 ,m ′θ1, ′θ2 , ′φ( )m=−min l1 ,l2( )

min l1 ,l2( )

∑l2=0

lx

∑l1=0

lx

∑⎡

⎣⎢⎢

⎤

⎦⎥⎥

n

Il1 ,l2 ,l θ1,θ2 ,φ( ) = l1,m, l2 ,−ml,0 Yl1 ,m θ1,φ( )Yl2 ,−m θ2 ,0( )m∑

The Fλ are defined so that they approach Dirac delta functions located at′θ1,′θ2,′φ( ) aslx increases with n = 1.


Interpolation of PES

• The Vab initio must be interpolated onto a fine grid

• Interpolation is done using reproducing kernel Hilbert space (RKHS) fitting functions of Ho and Rabitz

• Interpolate the transformed function for correct behavior at large and small R

q R,Ri( ) =114

1R>

7 1−79R<

R>

⎛

⎝⎜⎞

⎠⎟

V R,Ω( )=logVab initio R,Ω( )−Vlower

−Vlower

⎛

⎝⎜⎞

⎠⎟


Regularized Non-Linear Least Squares

• Function to be minimized

• The C0α,i

correspond to no morphing• The regularization parameter reduces the linear

dependence among the parameters Cα,i

• One obtains the best fit of the experiment that is also as close as possible to the original ab initio potential

F Cα,i ,( ) =Okexpt−Ok

calc Cα,i( )σk

⎡

⎣⎢⎢

⎤

⎦⎥⎥

2

k=1

M

∑ +2 Cα,i −Cα,i0

( )2

α,i∑


Regularized Non-Linear Least Squares

• The quality of the fit is characterized by the root-mean-square deviation from the experiment

χ ( ) =1

M

Okexpt −Ok

calc Cα ,i( )

σ k

⎡

⎣⎢⎢

⎤

⎦⎥⎥

2

k=1

M

∑⎧

⎨⎪

⎩⎪

⎫

⎬⎪

⎭⎪

1 2

• (∞) gives the quality of the ab initio potential

• (0) gives the quality of an unconstrained fit


Types of Experimental Data Used in Morphing

• Rotation constants (B0), value of R0

• Distortion constants (DJ), curvature in R direction

• , curvature in θ direction

• Dθ , coupling between R and θ directions

• Intermolecular bending and stretching vibrational transition frequencies

• D and H isotopes• Second virial coefficients, well depth

P2 cosθ( )


(HX)2 Interaction Potential

• Two identical isomers• Tunneling splitting is very sensitive to the shape of

the barrier• Potential is a function of four intermolecular

coordinates


Data Used in (HBr)2 Fit, (v5, K)

Observable Isotopomer Fit Exp. σk

B(0,0) 10-2 cm-1 H79Br:H81Br 2.458 2.459a 0.003

B(1,0) 10-2 cm-1 H79Br:H81Br 2.423 2.425a 0.003

B(0,0) 10-2 cm-1 H79Br:D81Br 2.449 2.444b 0.003

B(0,1) 10-2 cm-1 H79Br:H81Br 2.458 2.459c 0.003

B(0,2) 10-2 cm-1 H79Br:H81Br 2.457 2.458c 0.003

B(1,1) 10-2 cm-1 H79Br:H81Br 2.424 2.424c 0.003

D(0,0) 10-8 cm-1 H79Br:H81Br 4.07 4.09a 0.01

D(1,0)10-8 cm-1 H79Br:H81Br 3.60 3.60a 0.01

D(0,0) 10-8 cm-1 H79Br:D81Br 3.99 3.97b 0.01


More Data Used in (HBr)2 Fit, (v5, K)


P2(cos ) (H79Br)(0,0) H79Br:H81Br 0.186 0.190 0.001

P2(cos ) (H81Br)(0,0) H79Br:H81Br 0.185 0.190 0.001

P2(cos ) (H79Br)(1,0) H79Br:H81Br 0.202 0.198 0.001

P2(cos ) (H81Br)(1,0) H79Br:H81Br 0.203 0.199 0.001

P2(cos ) (H79Br)(0,0) H79Br:D81Br -0.242 -0.237 0.001

P2(cos ) (D81Br)(0,0) H79Br:D81Br 0.664 0.668 0.001

5 cm-1 H79Br:H81Br 15.03 15.03 0.01

A(5=0) cm-1 H79Br:H81Br 9.27 9.32 0.01


Still More Data Used in (HBr)2 Fit, (v5, K)


B(T=231.9 K) 10-4 m3 mol-1 H79Br:H79Br -3.144 -3.160 0.016

B(T=333.4 K) 10-4 m3 mol-1 H79Br:H79Br -1.438 -1.434 0.007

B(T=444.5 K) 10-4 m3 mol-1 H79Br:H79Br -0.769 -0.768 0.004

G 2.73


Morphed (HBr)2 PES, θ2 vs θ1

Low barrier between the two equivalent structures


Morphed (HBr)2 PES, R vs

Near the equilibrium structure


Morphed (HBr)2 PES, R vs

At the top of the barrier


(HBr)2 Wave Functions

E = 15.03 cm–1

v5 =0 v5 =1


Features of the Morphed (HBr)2 PES

-680

-660

-640

-620

-600

-580

-560

7560453015

θ1 ( )deg

ab initio 3 parameter 7 parameter

4.10

4.08

4.06

4.04

4.02

4.00

3.987560453015

θ1 ( )deg

ab initio 3 parameter 7 parameter


(HBr)2 Potential

• Ab initio potential computed using a large TZV(3d,3f) basis set with MP2 and BSSE

• The use of the log(V) interpolation with RKHS fitting functions is dramatically better that a direct fit of V.

• χ()/ χ(10)~15.4, with 7 fitting parameters and 20 experimental observations.


Data Used in (HI)2 Fit, χ = 2.91

Model Exp. Uncer.

B(v4=0,v5=0,K=0) 10-2 cm-1 1.271 1.262 0.001

B(v4=1,v5=0,K=0) 10-2 cm-1 1.255 1.255 0.001

B(v4=0,v5=1,K=0) 10-2 cm-1 1.232 1.237 0.001

B(v4=0,v5=0,K=1) 10-2 cm-1 1.272 1.279 0.001

D(v4=0,v5=0,K=0) 10-8 cm-1 1.42 1.34 0.13

D(v4=1,v5=0,K=0) 10-8 cm-1 1.29 1.25 0.02

D(v4=0,v5=1,K=0) 10-8 cm-1 1.07 0.95 0.18

D(v4=0,v5=1,K=1) 10-8 cm-1 1.45 1.40 0.06

(B-C) (v4=0,v5=1,K=1) 10-4 cm-1 0.369 0.413 0.034

P2(cos ) (v4=0,v5=0,K=0) 0.212 0.213 0.002

P2(cos ) (v4=0,v5=1,K=0) 0.203 0.206 0.002

E(v4=1,v5=0,K=0)-E (v4=0,v5=0,K=1) cm-1 13.92 13.92 0.01

E(v4=0,v5=1,K=0)-E (v4=0,v5=0,K=0) cm-1 17.08 17.08 0.01


Morphed (HI)2 PES, θ2 vs θ1

θ2

θ1


(HI)2 Wave Functions

v5 =0 v5 =1E = 17.08 cm–1


(HI)2 Potential

• Ab initio potential computed using a aug-cc-pvtz basis set with CCSD(T) and BSSE and an ECP for I.

• χ = 2.91, with 6 fitting parameters and 13 experimental observations.

• The state seems to be very weakly tunneling in the geared motion, with a significant probability at the symmetric geometry.

• Further refinements of the potential are in progress.


Conclusions

• Potential morphing can lead to accurate representations in the regions of the potential which have been experimentally interrogated.

• Morphed potentials have estimated errors that increase only slowly away from the experimental region.

• Morphed potentials have yielded accurate predictions of unmeasured spectroscopic constants.

• Goal is to develop general morphing parameters that can be used to adjust a given level of ab initio theory for accurate predictions of intermolecular interactions.

int. symp. molecular spectroscopy ohio state univ., 2005 the ground state four dimensional morphed...

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