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
Int. Symp. Molecular SpectroscopyOhio State Univ., 2005
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
Int. Symp. Molecular SpectroscopyOhio State Univ., 2005
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.
Int. Symp. Molecular SpectroscopyOhio State Univ., 2005
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
⎛
⎝⎜⎞
⎠⎟
Int. Symp. Molecular SpectroscopyOhio State Univ., 2005
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∑
Int. Symp. Molecular SpectroscopyOhio State Univ., 2005
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
Int. Symp. Molecular SpectroscopyOhio State Univ., 2005
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( )
Int. Symp. Molecular SpectroscopyOhio State Univ., 2005
(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
Int. Symp. Molecular SpectroscopyOhio State Univ., 2005
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
Int. Symp. Molecular SpectroscopyOhio State Univ., 2005
More Data Used in (HBr)2 Fit, (v5, K)
Observable Isotopomer Fit Exp. σk
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
Int. Symp. Molecular SpectroscopyOhio State Univ., 2005
Still More Data Used in (HBr)2 Fit, (v5, K)
Observable Isotopomer Fit Exp. σk
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
Int. Symp. Molecular SpectroscopyOhio State Univ., 2005
Morphed (HBr)2 PES, θ2 vs θ1
Low barrier between the two equivalent structures
Int. Symp. Molecular SpectroscopyOhio State Univ., 2005
Morphed (HBr)2 PES, R vs
Near the equilibrium structure
Int. Symp. Molecular SpectroscopyOhio State Univ., 2005
Morphed (HBr)2 PES, R vs
At the top of the barrier
Int. Symp. Molecular SpectroscopyOhio State Univ., 2005
(HBr)2 Wave Functions
E = 15.03 cm–1
v5 =0 v5 =1
Int. Symp. Molecular SpectroscopyOhio State Univ., 2005
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
Int. Symp. Molecular SpectroscopyOhio State Univ., 2005
(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.
Int. Symp. Molecular SpectroscopyOhio State Univ., 2005
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
Int. Symp. Molecular SpectroscopyOhio State Univ., 2005
Morphed (HI)2 PES, θ2 vs θ1
θ2
θ1
Int. Symp. Molecular SpectroscopyOhio State Univ., 2005
(HI)2 Wave Functions
v5 =0 v5 =1E = 17.08 cm–1
Int. Symp. Molecular SpectroscopyOhio State Univ., 2005
(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.
Int. Symp. Molecular SpectroscopyOhio State Univ., 2005
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.