progress towards the accurate calculation of anharmonic vibrational states of fluxional molecules...
TRANSCRIPT
![Page 1: Progress Towards the Accurate Calculation of Anharmonic Vibrational States of Fluxional Molecules and Clusters Without a Potential Energy Surface Andrew](https://reader031.vdocuments.net/reader031/viewer/2022013004/56649e855503460f94b873fa/html5/thumbnails/1.jpg)
Progress Towards the Accurate Calculation of Anharmonic Vibrational States of Fluxional
Molecules and Clusters Without a Potential Energy Surface
Andrew S. Petit and Anne B. McCoyThe Ohio State University
|0,0> = 0 cm-1
rOHb (Å)
0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
RO
F (
Å)
2.0
2.2
2.4
2.6
2.8
3.0
3.2
|1,0> = 1533 cm-1
rOHb (Å)
0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
RO
F (
Å)
2.0
2.2
2.4
2.6
2.8
3.0
3.2
|2,0> = 2934 cm-1
rOHb (Å)
0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0R
OF (
Å)
2.0
2.2
2.4
2.6
2.8
3.0
3.2
|3,0> = 4469 cm-1
rOHb (Å)
0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0R
OF (
Å)
2.0
2.2
2.4
2.6
2.8
3.0
3.2
![Page 2: Progress Towards the Accurate Calculation of Anharmonic Vibrational States of Fluxional Molecules and Clusters Without a Potential Energy Surface Andrew](https://reader031.vdocuments.net/reader031/viewer/2022013004/56649e855503460f94b873fa/html5/thumbnails/2.jpg)
What Are the Challenges???
Harmonic analysis pathologically fails
![Page 3: Progress Towards the Accurate Calculation of Anharmonic Vibrational States of Fluxional Molecules and Clusters Without a Potential Energy Surface Andrew](https://reader031.vdocuments.net/reader031/viewer/2022013004/56649e855503460f94b873fa/html5/thumbnails/3.jpg)
What Are the Challenges???
Harmonic analysis pathologically fails
Photon energy
0 500 1000 1500 2000
Sig
nal
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
Experimentharmonic
shared protonstretch
H3O2-
![Page 4: Progress Towards the Accurate Calculation of Anharmonic Vibrational States of Fluxional Molecules and Clusters Without a Potential Energy Surface Andrew](https://reader031.vdocuments.net/reader031/viewer/2022013004/56649e855503460f94b873fa/html5/thumbnails/4.jpg)
What Are the Challenges???
Harmonic analysis pathologically fails
An accurate and general treatment of these highly fluxional systems requires
either a global PES or a grid-based sampling of configuration space
![Page 5: Progress Towards the Accurate Calculation of Anharmonic Vibrational States of Fluxional Molecules and Clusters Without a Potential Energy Surface Andrew](https://reader031.vdocuments.net/reader031/viewer/2022013004/56649e855503460f94b873fa/html5/thumbnails/5.jpg)
What Are the Challenges???
Harmonic analysis pathologically fails
An accurate and general treatment of these highly fluxional systems requires
either a global PES or a grid-based sampling of configuration space
Large basis set expansions required to fully capture the anharmoncity of
these systems
![Page 6: Progress Towards the Accurate Calculation of Anharmonic Vibrational States of Fluxional Molecules and Clusters Without a Potential Energy Surface Andrew](https://reader031.vdocuments.net/reader031/viewer/2022013004/56649e855503460f94b873fa/html5/thumbnails/6.jpg)
The General Scheme
![Page 7: Progress Towards the Accurate Calculation of Anharmonic Vibrational States of Fluxional Molecules and Clusters Without a Potential Energy Surface Andrew](https://reader031.vdocuments.net/reader031/viewer/2022013004/56649e855503460f94b873fa/html5/thumbnails/7.jpg)
I
The General SchemeMonte Carlo SampleConfiguration Space
![Page 8: Progress Towards the Accurate Calculation of Anharmonic Vibrational States of Fluxional Molecules and Clusters Without a Potential Energy Surface Andrew](https://reader031.vdocuments.net/reader031/viewer/2022013004/56649e855503460f94b873fa/html5/thumbnails/8.jpg)
Monte Carlo Sample Configuration Space
Use importance sampling Monte Carlo
![Page 9: Progress Towards the Accurate Calculation of Anharmonic Vibrational States of Fluxional Molecules and Clusters Without a Potential Energy Surface Andrew](https://reader031.vdocuments.net/reader031/viewer/2022013004/56649e855503460f94b873fa/html5/thumbnails/9.jpg)
Use importance sampling Monte Carlo
€
fr x ( )dV =
fr x ( )
gr x ( )
gr x ( )
V
∫V
∫ dV ≈V
N
fr x i( )
gr x i( )i=1
N
∑
Monte Carlo Sample Configuration Space
![Page 10: Progress Towards the Accurate Calculation of Anharmonic Vibrational States of Fluxional Molecules and Clusters Without a Potential Energy Surface Andrew](https://reader031.vdocuments.net/reader031/viewer/2022013004/56649e855503460f94b873fa/html5/thumbnails/10.jpg)
Use importance sampling Monte Carlo
Normalized probability distribution that is peaked, ideally, where the function of interest, f, is peaked.
€
fr x ( )dV =
fr x ( )
gr x ( )
gr x ( )
V
∫V
∫ dV ≈V
N
fr x i( )
gr x i( )i=1
N
∑
Monte Carlo Sample Configuration Space
![Page 11: Progress Towards the Accurate Calculation of Anharmonic Vibrational States of Fluxional Molecules and Clusters Without a Potential Energy Surface Andrew](https://reader031.vdocuments.net/reader031/viewer/2022013004/56649e855503460f94b873fa/html5/thumbnails/11.jpg)
Sampling function based on harmonic ground state
Monte Carlo Sample Configuration Space
R
Rela
tive
Prob
abili
ty
![Page 12: Progress Towards the Accurate Calculation of Anharmonic Vibrational States of Fluxional Molecules and Clusters Without a Potential Energy Surface Andrew](https://reader031.vdocuments.net/reader031/viewer/2022013004/56649e855503460f94b873fa/html5/thumbnails/12.jpg)
Sampling function based on harmonic ground state
Monte Carlo Sample Configuration Space
R
Rela
tive
Prob
abili
ty
![Page 13: Progress Towards the Accurate Calculation of Anharmonic Vibrational States of Fluxional Molecules and Clusters Without a Potential Energy Surface Andrew](https://reader031.vdocuments.net/reader031/viewer/2022013004/56649e855503460f94b873fa/html5/thumbnails/13.jpg)
Sampling function based on harmonic ground state
Monte Carlo Sample Configuration Space
R
Rela
tive
Prob
abili
ty
![Page 14: Progress Towards the Accurate Calculation of Anharmonic Vibrational States of Fluxional Molecules and Clusters Without a Potential Energy Surface Andrew](https://reader031.vdocuments.net/reader031/viewer/2022013004/56649e855503460f94b873fa/html5/thumbnails/14.jpg)
Sampling function based on harmonic ground state
Monte Carlo Sample Configuration Space
R
Rela
tive
Prob
abili
ty
![Page 15: Progress Towards the Accurate Calculation of Anharmonic Vibrational States of Fluxional Molecules and Clusters Without a Potential Energy Surface Andrew](https://reader031.vdocuments.net/reader031/viewer/2022013004/56649e855503460f94b873fa/html5/thumbnails/15.jpg)
Sampling function based on harmonic ground state
Monte Carlo Sample Configuration Space
R
Rela
tive
Prob
abili
ty
![Page 16: Progress Towards the Accurate Calculation of Anharmonic Vibrational States of Fluxional Molecules and Clusters Without a Potential Energy Surface Andrew](https://reader031.vdocuments.net/reader031/viewer/2022013004/56649e855503460f94b873fa/html5/thumbnails/16.jpg)
I
I
The General Scheme
Construct Initial Basis
Monte Carlo SampleConfiguration Space
![Page 17: Progress Towards the Accurate Calculation of Anharmonic Vibrational States of Fluxional Molecules and Clusters Without a Potential Energy Surface Andrew](https://reader031.vdocuments.net/reader031/viewer/2022013004/56649e855503460f94b873fa/html5/thumbnails/17.jpg)
I
I
IBuild and DiagonalizeHamiltonian Matrix
The General Scheme
Construct Initial Basis
Monte Carlo SampleConfiguration Space
![Page 18: Progress Towards the Accurate Calculation of Anharmonic Vibrational States of Fluxional Molecules and Clusters Without a Potential Energy Surface Andrew](https://reader031.vdocuments.net/reader031/viewer/2022013004/56649e855503460f94b873fa/html5/thumbnails/18.jpg)
I
I
I IBuild and DiagonalizeHamiltonian Matrix
Assign Eigenstates &Build New Basis
The General Scheme
Construct Initial Basis
Monte Carlo SampleConfiguration Space
![Page 19: Progress Towards the Accurate Calculation of Anharmonic Vibrational States of Fluxional Molecules and Clusters Without a Potential Energy Surface Andrew](https://reader031.vdocuments.net/reader031/viewer/2022013004/56649e855503460f94b873fa/html5/thumbnails/19.jpg)
I
I
I IBuild and DiagonalizeHamiltonian Matrix
Assign Eigenstates &Build New Basis
The General Scheme
Construct Initial Basis
Monte Carlo SampleConfiguration Space
![Page 20: Progress Towards the Accurate Calculation of Anharmonic Vibrational States of Fluxional Molecules and Clusters Without a Potential Energy Surface Andrew](https://reader031.vdocuments.net/reader031/viewer/2022013004/56649e855503460f94b873fa/html5/thumbnails/20.jpg)
The Evolving Basis
![Page 21: Progress Towards the Accurate Calculation of Anharmonic Vibrational States of Fluxional Molecules and Clusters Without a Potential Energy Surface Andrew](https://reader031.vdocuments.net/reader031/viewer/2022013004/56649e855503460f94b873fa/html5/thumbnails/21.jpg)
The Evolving BasisHarmonic oscillator basis
![Page 22: Progress Towards the Accurate Calculation of Anharmonic Vibrational States of Fluxional Molecules and Clusters Without a Potential Energy Surface Andrew](https://reader031.vdocuments.net/reader031/viewer/2022013004/56649e855503460f94b873fa/html5/thumbnails/22.jpg)
€
H 1( )
The Evolving BasisHarmonic oscillator basis
![Page 23: Progress Towards the Accurate Calculation of Anharmonic Vibrational States of Fluxional Molecules and Clusters Without a Potential Energy Surface Andrew](https://reader031.vdocuments.net/reader031/viewer/2022013004/56649e855503460f94b873fa/html5/thumbnails/23.jpg)
€
H 1( )
€
˜ Ψ 1( ){ }, E 1( )
{ }
The Evolving BasisHarmonic oscillator basis
![Page 24: Progress Towards the Accurate Calculation of Anharmonic Vibrational States of Fluxional Molecules and Clusters Without a Potential Energy Surface Andrew](https://reader031.vdocuments.net/reader031/viewer/2022013004/56649e855503460f94b873fa/html5/thumbnails/24.jpg)
€
H 1( )
€
˜ Ψ 1( ){ }, E 1( )
{ }
Eigenstates used toconstruct new basis
Assignment via:
€
max ˜ Ψ k1( ) ˆ r Ψm−1
1( )2 ⎛
⎝ ⎜
⎞ ⎠ ⎟⇒ ˜ Ψ m
1( )
The Evolving BasisHarmonic oscillator basis
![Page 25: Progress Towards the Accurate Calculation of Anharmonic Vibrational States of Fluxional Molecules and Clusters Without a Potential Energy Surface Andrew](https://reader031.vdocuments.net/reader031/viewer/2022013004/56649e855503460f94b873fa/html5/thumbnails/25.jpg)
€
H 1( )
€
˜ Ψ 1( ){ }, E 1( )
{ }
Eigenstates used toconstruct new basis
Assignment via:
€
max ˜ Ψ k1( ) ˆ r Ψm−1
1( )2 ⎛
⎝ ⎜
⎞ ⎠ ⎟⇒ ˜ Ψ m
1( )
€
Ψ 2( ){ }
The Evolving BasisHarmonic oscillator basis
![Page 26: Progress Towards the Accurate Calculation of Anharmonic Vibrational States of Fluxional Molecules and Clusters Without a Potential Energy Surface Andrew](https://reader031.vdocuments.net/reader031/viewer/2022013004/56649e855503460f94b873fa/html5/thumbnails/26.jpg)
€
H 1( )
€
˜ Ψ 1( ){ }, E 1( )
{ }
Eigenstates used toconstruct new basis
Assignment via:
€
max ˜ Ψ k1( ) ˆ r Ψm−1
1( )2 ⎛
⎝ ⎜
⎞ ⎠ ⎟⇒ ˜ Ψ m
1( )
€
Ψ 2( ){ }
Define an active space:
€
Ψm(2)
{ }active
= ˜ Ψ m(1)
{ }active
The Evolving BasisHarmonic oscillator basis
![Page 27: Progress Towards the Accurate Calculation of Anharmonic Vibrational States of Fluxional Molecules and Clusters Without a Potential Energy Surface Andrew](https://reader031.vdocuments.net/reader031/viewer/2022013004/56649e855503460f94b873fa/html5/thumbnails/27.jpg)
€
H 1( )
€
˜ Ψ 1( ){ }, E 1( )
{ }
Eigenstates used toconstruct new basis
Assignment via:
€
max ˜ Ψ k1( ) ˆ r Ψm−1
1( )2 ⎛
⎝ ⎜
⎞ ⎠ ⎟⇒ ˜ Ψ m
1( )
€
Ψ 2( ){ }
Define an active space:
€
Ψm(2)
{ }active
= ˜ Ψ m(1)
{ }active
€
Ψ m∉active2( ) = κ m,m−k
2( ) rkΨm−k2( )
k=1
m
∑
The Evolving BasisHarmonic oscillator basis
![Page 28: Progress Towards the Accurate Calculation of Anharmonic Vibrational States of Fluxional Molecules and Clusters Without a Potential Energy Surface Andrew](https://reader031.vdocuments.net/reader031/viewer/2022013004/56649e855503460f94b873fa/html5/thumbnails/28.jpg)
€
H 1( )
€
˜ Ψ 1( ){ }, E 1( )
{ }
Eigenstates used toconstruct new basis
Assignment via:
€
max ˜ Ψ k1( ) ˆ r Ψm−1
1( )2 ⎛
⎝ ⎜
⎞ ⎠ ⎟⇒ ˜ Ψ m
1( )
€
Ψ 2( ){ }
Define an active space:
€
Ψm(2)
{ }active
= ˜ Ψ m(1)
{ }active
€
Ψ m∉active2( ) = κ m,m−k
2( ) rkΨm−k2( )
k=1
m
∑
The Evolving BasisHarmonic oscillator basis
![Page 29: Progress Towards the Accurate Calculation of Anharmonic Vibrational States of Fluxional Molecules and Clusters Without a Potential Energy Surface Andrew](https://reader031.vdocuments.net/reader031/viewer/2022013004/56649e855503460f94b873fa/html5/thumbnails/29.jpg)
The Evolving Basis
€
H j( )
€
˜ Ψ j( ){ }, E j( )
{ }
Eigenstates used toconstruct new basis
Assignment via:
€
max ˜ Ψ kj( ) ˆ r Ψm−1
j( )2 ⎛
⎝ ⎜
⎞ ⎠ ⎟⇒ ˜ Ψ m
j( )
€
Ψ j +1( ){ }
Define an active space:
€
Ψm( j +1)
{ }active
= ˜ Ψ m( j )
{ }active
€
Ψ m∉activej +1( ) = κ m,m−k
j +1( ) rkΨm−kj +1( )
k=1
m
∑
![Page 30: Progress Towards the Accurate Calculation of Anharmonic Vibrational States of Fluxional Molecules and Clusters Without a Potential Energy Surface Andrew](https://reader031.vdocuments.net/reader031/viewer/2022013004/56649e855503460f94b873fa/html5/thumbnails/30.jpg)
€
H 1( )
€
˜ Ψ 1( ){ }, E 1( )
{ }
€
Ψ 2( ){ }
MxM Matrix
Construction of 2nd Iteration Basis
![Page 31: Progress Towards the Accurate Calculation of Anharmonic Vibrational States of Fluxional Molecules and Clusters Without a Potential Energy Surface Andrew](https://reader031.vdocuments.net/reader031/viewer/2022013004/56649e855503460f94b873fa/html5/thumbnails/31.jpg)
€
H 1( )
€
˜ Ψ 1( ){ }, E 1( )
{ }
€
Ψ 2( ){ }
MxM Matrix
Suppose active space only contains ground state
Construction of 2nd Iteration Basis
![Page 32: Progress Towards the Accurate Calculation of Anharmonic Vibrational States of Fluxional Molecules and Clusters Without a Potential Energy Surface Andrew](https://reader031.vdocuments.net/reader031/viewer/2022013004/56649e855503460f94b873fa/html5/thumbnails/32.jpg)
€
H 1( )
€
˜ Ψ 1( ){ }, E 1( )
{ }
€
Ψ 2( ){ }
MxM Matrix
Suppose active space only contains ground state
€
Ψ 11( ) = ra0
1( )e−
αr 2
2
Basis functions shown prior to orthonormalization
Construction of 2nd Iteration Basis
![Page 33: Progress Towards the Accurate Calculation of Anharmonic Vibrational States of Fluxional Molecules and Clusters Without a Potential Energy Surface Andrew](https://reader031.vdocuments.net/reader031/viewer/2022013004/56649e855503460f94b873fa/html5/thumbnails/33.jpg)
€
H 1( )
€
˜ Ψ 1( ){ }, E 1( )
{ }
€
Ψ 2( ){ }
MxM Matrix
Suppose active space only contains ground state
€
Ψ 11( ) = ra0
1( )e−
αr 2
2
Basis functions shown prior to orthonormalization
Construction of 2nd Iteration Basis
€
Ψ 02( ) = a0
2( ) + a12( )r + a2
2( )r2 +K + aM −12( ) rM −1
( )e−
αr 2
2
![Page 34: Progress Towards the Accurate Calculation of Anharmonic Vibrational States of Fluxional Molecules and Clusters Without a Potential Energy Surface Andrew](https://reader031.vdocuments.net/reader031/viewer/2022013004/56649e855503460f94b873fa/html5/thumbnails/34.jpg)
€
H 1( )
€
˜ Ψ 1( ){ }, E 1( )
{ }
€
Ψ 2( ){ }
MxM Matrix
Suppose active space only contains ground state
€
Ψ 11( ) = ra0
1( )e−
αr 2
2
Basis functions shown prior to orthonormalization
Construction of 2nd Iteration Basis
€
Ψ 02( ) = a0
2( ) + a12( )r + a2
2( )r2 +K + aM −12( ) rM −1
( )e−
αr 2
2
€
Ψ 12( ) = rΨ 0
2( ) = a02( )r + a1
2( )r2 + a22( )r3 +K + aM −1
2( ) rM( )e
−αr 2
2
![Page 35: Progress Towards the Accurate Calculation of Anharmonic Vibrational States of Fluxional Molecules and Clusters Without a Potential Energy Surface Andrew](https://reader031.vdocuments.net/reader031/viewer/2022013004/56649e855503460f94b873fa/html5/thumbnails/35.jpg)
€
H 1( )
€
˜ Ψ 1( ){ }, E 1( )
{ }
€
Ψ 2( ){ }
MxM Matrix
Suppose active space only contains ground state
Basis functions shown prior to orthonormalization
Construction of 2nd Iteration Basis
€
Ψ 02( ) = a0
2( ) + a12( )r + a2
2( )r2 +K + aM −12( ) rM −1
( )e−
αr 2
2
€
Ψ 12( ) = rΨ 0
2( ) = a02( )r + a1
2( )r2 + a22( )r3 +K + aM −1
2( ) rM( )e
−αr 2
2
Effective size of basis grows each iteration
![Page 36: Progress Towards the Accurate Calculation of Anharmonic Vibrational States of Fluxional Molecules and Clusters Without a Potential Energy Surface Andrew](https://reader031.vdocuments.net/reader031/viewer/2022013004/56649e855503460f94b873fa/html5/thumbnails/36.jpg)
I
I
I IBuild and DiagonalizeHamiltonian Matrix
Assign Eigenstates &Build New Basis
The General Scheme
Construct Initial Basis
Monte Carlo SampleConfiguration Space
![Page 37: Progress Towards the Accurate Calculation of Anharmonic Vibrational States of Fluxional Molecules and Clusters Without a Potential Energy Surface Andrew](https://reader031.vdocuments.net/reader031/viewer/2022013004/56649e855503460f94b873fa/html5/thumbnails/37.jpg)
Testing the Method
Δr (Å)
Ener
gy (c
m-1
)
E0=237.5 cm-1
E1=637.5 cm-1
E2=937.5 cm-1
E3=1137.5 cm-1
E4=1237.5 cm-1
![Page 38: Progress Towards the Accurate Calculation of Anharmonic Vibrational States of Fluxional Molecules and Clusters Without a Potential Energy Surface Andrew](https://reader031.vdocuments.net/reader031/viewer/2022013004/56649e855503460f94b873fa/html5/thumbnails/38.jpg)
Testing the MethodE0=237.5 cm-1
E1=637.5 cm-1
E2=937.5 cm-1
E3=1137.5 cm-1
E4=1237.5 cm-1
![Page 39: Progress Towards the Accurate Calculation of Anharmonic Vibrational States of Fluxional Molecules and Clusters Without a Potential Energy Surface Andrew](https://reader031.vdocuments.net/reader031/viewer/2022013004/56649e855503460f94b873fa/html5/thumbnails/39.jpg)
Testing the MethodE0=237.5 cm-1
E1=637.5 cm-1
E2=937.5 cm-1
E3=1137.5 cm-1
E4=1237.5 cm-1
![Page 40: Progress Towards the Accurate Calculation of Anharmonic Vibrational States of Fluxional Molecules and Clusters Without a Potential Energy Surface Andrew](https://reader031.vdocuments.net/reader031/viewer/2022013004/56649e855503460f94b873fa/html5/thumbnails/40.jpg)
Testing the MethodE0=237.5 cm-1
E1=637.5 cm-1
E2=937.5 cm-1
E3=1137.5 cm-1
E4=1237.5 cm-1
![Page 41: Progress Towards the Accurate Calculation of Anharmonic Vibrational States of Fluxional Molecules and Clusters Without a Potential Energy Surface Andrew](https://reader031.vdocuments.net/reader031/viewer/2022013004/56649e855503460f94b873fa/html5/thumbnails/41.jpg)
Moving on to 2D
Horvath et al. J. Phys. Chem. A 2008, 112, 12337-12344.
(Å)
(Å)
€
rOF
€
rOHb2D PES and G matrix elements constructed by S. Horvath from a bicubic spline interpolation of a grid of points on which MP2/aug-cc-pVTZ calculations were performed.
€
rOHband
€
rOF .
Highly anharmonic system with a strong coupling observed between
Our goal is to accurately capture all states with up to 3 total quanta of excitation.
rOHb (Å)
0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4
RO
F (
Å)
2.0
2.2
2.4
2.6
2.8
3.0
3.2
0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 11000 12000 13000 14000 15000
![Page 42: Progress Towards the Accurate Calculation of Anharmonic Vibrational States of Fluxional Molecules and Clusters Without a Potential Energy Surface Andrew](https://reader031.vdocuments.net/reader031/viewer/2022013004/56649e855503460f94b873fa/html5/thumbnails/42.jpg)
The Problem of Assignment
€
H j( )
€
˜ Ψ j( ){ }, E j( )
{ }
€
Ψ j +1( ){ }
Eigenstates used to
construct new basis
![Page 43: Progress Towards the Accurate Calculation of Anharmonic Vibrational States of Fluxional Molecules and Clusters Without a Potential Energy Surface Andrew](https://reader031.vdocuments.net/reader031/viewer/2022013004/56649e855503460f94b873fa/html5/thumbnails/43.jpg)
The Problem of Assignment
€
H j( )
€
˜ Ψ j( ){ }, E j( )
{ }
€
Ψ j +1( ){ }
Eigenstates used to
construct new basis
DVR
€
1,1
€
0,2
![Page 44: Progress Towards the Accurate Calculation of Anharmonic Vibrational States of Fluxional Molecules and Clusters Without a Potential Energy Surface Andrew](https://reader031.vdocuments.net/reader031/viewer/2022013004/56649e855503460f94b873fa/html5/thumbnails/44.jpg)
The Problem of Assignment
€
H j( )
€
˜ Ψ j( ){ }, E j( )
{ }
€
Ψ j +1( ){ }
Eigenstates used to
construct new basis
DVR Iteration 2
€
1,1
€
0,2
![Page 45: Progress Towards the Accurate Calculation of Anharmonic Vibrational States of Fluxional Molecules and Clusters Without a Potential Energy Surface Andrew](https://reader031.vdocuments.net/reader031/viewer/2022013004/56649e855503460f94b873fa/html5/thumbnails/45.jpg)
The Problem of Assignment
€
H j( )
€
˜ Ψ j( ){ }, E j( )
{ }
€
Ψ j +1( ){ }
Eigenstates used to
construct new basis
DVR Iteration 2 Iteration 3
€
1,1
€
0,2
![Page 46: Progress Towards the Accurate Calculation of Anharmonic Vibrational States of Fluxional Molecules and Clusters Without a Potential Energy Surface Andrew](https://reader031.vdocuments.net/reader031/viewer/2022013004/56649e855503460f94b873fa/html5/thumbnails/46.jpg)
The Problem of Assignment
DVR Iteration 2 Iteration 3
€
1,1
€
0,2
Identity of problem states dependent on parameters of algorithm AND exact distribution of Monte Carlo sampling points
![Page 47: Progress Towards the Accurate Calculation of Anharmonic Vibrational States of Fluxional Molecules and Clusters Without a Potential Energy Surface Andrew](https://reader031.vdocuments.net/reader031/viewer/2022013004/56649e855503460f94b873fa/html5/thumbnails/47.jpg)
The Hunt for an Effective Metric
€
Ψm,nj( ) Basis function Un-assigned eigenstate
€
φkj( )
![Page 48: Progress Towards the Accurate Calculation of Anharmonic Vibrational States of Fluxional Molecules and Clusters Without a Potential Energy Surface Andrew](https://reader031.vdocuments.net/reader031/viewer/2022013004/56649e855503460f94b873fa/html5/thumbnails/48.jpg)
€
Ψm,nj( ) Basis function Un-assigned eigenstate
€
φkj( )
€
φkj( ) Ψm,n
j( )2
The Hunt for an Effective Metric
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€
Ψm,nj( ) Basis function Un-assigned eigenstate
€
φkj( )
€
φkj( ) Ψm,n
j( )2
€
φkj( ) ˆ r OF Ψm−1,n
j( )2
+ φkj( ) ˆ r OHb
Ψm,n−1j( )
2
The Hunt for an Effective Metric
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€
Ψm,nj( ) Basis function Un-assigned eigenstate
€
φkj( )
€
φkj( ) Ψm,n
j( )2
€
φkj( ) ˆ r OF Ψm−1,n
j( )2
+ φkj( ) ˆ r OHb
Ψm,n−1j( )
2
Compare moments of the two sets of wavefunctions:
€
rOF − rOF k( )2
k
rOF − rOF k( ) rOHb− rOHb k( )
rOHb− rOHb k( )
2
k
⎧
⎨
⎪ ⎪ ⎪
⎩
⎪ ⎪ ⎪
⎫
⎬
⎪ ⎪ ⎪
⎭
⎪ ⎪ ⎪
The Hunt for an Effective Metric
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€
Fk,l;m,nj( ) =
1
1+Ek,l
j( ) − Em,nj( )
Hk,l;m,nj( )
⎛
⎝ ⎜
⎞
⎠ ⎟
2
Gruebele et al. Int. Rev. Phys. Chem 1998, 17, 91-145.
The Hunt for an Effective Metric
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The Hunt for an Effective Metric
Perform calculation using an assignment scheme based on
€
φkj( ) Ψm,n
j( )2
![Page 53: Progress Towards the Accurate Calculation of Anharmonic Vibrational States of Fluxional Molecules and Clusters Without a Potential Energy Surface Andrew](https://reader031.vdocuments.net/reader031/viewer/2022013004/56649e855503460f94b873fa/html5/thumbnails/53.jpg)
Gruebele et al. Int. Rev. Phys. Chem 1998, 17, 91-145.
The Hunt for an Effective Metric
Perform calculation using an assignment scheme based on
€
φkj( ) Ψm,n
j( )2
Significantly contaminated eigenstates are rebuilt
![Page 54: Progress Towards the Accurate Calculation of Anharmonic Vibrational States of Fluxional Molecules and Clusters Without a Potential Energy Surface Andrew](https://reader031.vdocuments.net/reader031/viewer/2022013004/56649e855503460f94b873fa/html5/thumbnails/54.jpg)
The Hunt for an Effective Metric
Perform calculation using an assignment scheme based on
€
φkj( ) Ψm,n
j( )2
Significantly contaminated eigenstates are rebuilt
(Å)
(Å)
DVR
![Page 55: Progress Towards the Accurate Calculation of Anharmonic Vibrational States of Fluxional Molecules and Clusters Without a Potential Energy Surface Andrew](https://reader031.vdocuments.net/reader031/viewer/2022013004/56649e855503460f94b873fa/html5/thumbnails/55.jpg)
The Hunt for an Effective Metric
Perform calculation using an assignment scheme based on
€
φkj( ) Ψm,n
j( )2
Significantly contaminated eigenstates are rebuilt
Iteration23
(Å)
(Å)
DVR
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The Hunt for an Effective Metric
Perform calculation using an assignment scheme based on
€
φkj( ) Ψm,n
j( )2
Significantly contaminated eigenstates are rebuilt
Iteration23 24 25 26 30 100
(Å)
(Å)
DVR
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The Hunt for an Effective Metric
Iteration23 24 25 26 30 100
(Å)
(Å)
DVR
Contamination occurs under the radar of overlaps, transition moments, moment distributions, etc.
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The Hunt for an Effective Metric
Iteration23 24 25 26 30 100
(Å)
(Å)
DVR
Contamination occurs under the radar of overlaps, transition moments, moment distributions, etc.
€
Fk,l;m,nj( ) =
1
1+Ek,l
j( ) − Em,nj( )
Hk,l;m,nj( )
⎛
⎝ ⎜
⎞
⎠ ⎟
2
![Page 59: Progress Towards the Accurate Calculation of Anharmonic Vibrational States of Fluxional Molecules and Clusters Without a Potential Energy Surface Andrew](https://reader031.vdocuments.net/reader031/viewer/2022013004/56649e855503460f94b873fa/html5/thumbnails/59.jpg)
The Hunt for an Effective Metric
Iteration Number
F 0,2;
m,n
Largest F0,2;m,n
2nd Largest F0,2;m,n
€
F0,2;m,nj( ) =
1
1+E0,2
j( ) − Em,nj( )
H0,2;m,nj( )
⎛
⎝ ⎜
⎞
⎠ ⎟
2
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The Hunt for an Effective Metric
Iteration Number
F 0,2;
m,n
Largest F0,2;m,n
2nd Largest F0,2;m,n (Å)
(Å)
24 25 26DVR
€
F0,2;m,nj( ) =
1
1+E0,2
j( ) − Em,nj( )
H0,2;m,nj( )
⎛
⎝ ⎜
⎞
⎠ ⎟
2
![Page 61: Progress Towards the Accurate Calculation of Anharmonic Vibrational States of Fluxional Molecules and Clusters Without a Potential Energy Surface Andrew](https://reader031.vdocuments.net/reader031/viewer/2022013004/56649e855503460f94b873fa/html5/thumbnails/61.jpg)
Developed a methodology that uses importance sampling Monte Carlo and and evolving basis to intelligently sample
configuration space and minimize size of basis required
Conclusions and Future Work
![Page 62: Progress Towards the Accurate Calculation of Anharmonic Vibrational States of Fluxional Molecules and Clusters Without a Potential Energy Surface Andrew](https://reader031.vdocuments.net/reader031/viewer/2022013004/56649e855503460f94b873fa/html5/thumbnails/62.jpg)
Developed a methodology that uses importance sampling Monte Carlo and and evolving basis to intelligently sample
configuration space and minimize size of basis required
The method has been shown to be able to accurately describe a highly anharmonic Morse oscillator
Conclusions and Future Work
![Page 63: Progress Towards the Accurate Calculation of Anharmonic Vibrational States of Fluxional Molecules and Clusters Without a Potential Energy Surface Andrew](https://reader031.vdocuments.net/reader031/viewer/2022013004/56649e855503460f94b873fa/html5/thumbnails/63.jpg)
Developed a methodology that uses importance sampling Monte Carlo and and evolving basis to intelligently sample
configuration space and minimize size of basis required
The method has been shown to be able to accurately describe a highly anharmonic Morse oscillator
Promising developments in tackling the problem of multi-dimensional eigenstate assigment
Conclusions and Future Work
![Page 64: Progress Towards the Accurate Calculation of Anharmonic Vibrational States of Fluxional Molecules and Clusters Without a Potential Energy Surface Andrew](https://reader031.vdocuments.net/reader031/viewer/2022013004/56649e855503460f94b873fa/html5/thumbnails/64.jpg)
Complete the optimization of the multi-dimensional algorithm’s treatment of spurious resonances
Developed a methodology that uses importance sampling Monte Carlo and and evolving basis to intelligently sample
configuration space and minimize size of basis required
The method has been shown to be able to accurately describe a highly anharmonic Morse oscillator
Promising developments in tackling the problem of multi-dimensional eigenstate assigment
Conclusions and Future Work
![Page 65: Progress Towards the Accurate Calculation of Anharmonic Vibrational States of Fluxional Molecules and Clusters Without a Potential Energy Surface Andrew](https://reader031.vdocuments.net/reader031/viewer/2022013004/56649e855503460f94b873fa/html5/thumbnails/65.jpg)
Complete the optimization of the multi-dimensional algorithm’s treatment of spurious resonances
Developed a methodology that uses importance sampling Monte Carlo and and evolving basis to intelligently sample
configuration space and minimize size of basis required
The method has been shown to be able to accurately describe a highly anharmonic Morse oscillator
Promising developments in tackling the problem of multi-dimensional eigenstate assigment
Conclusions and Future Work
Further multi-dimensional benchmarking
![Page 66: Progress Towards the Accurate Calculation of Anharmonic Vibrational States of Fluxional Molecules and Clusters Without a Potential Energy Surface Andrew](https://reader031.vdocuments.net/reader031/viewer/2022013004/56649e855503460f94b873fa/html5/thumbnails/66.jpg)
Complete the optimization of the multi-dimensional algorithm’s treatment of spurious resonances
Couple algorithm with ab initio electronic structure calculations
Developed a methodology that uses importance sampling Monte Carlo and and evolving basis to intelligently sample
configuration space and minimize size of basis required
The method has been shown to be able to accurately describe a highly anharmonic Morse oscillator
Promising developments in tackling the problem of multi-dimensional eigenstate assigment
Conclusions and Future Work
Further multi-dimensional benchmarking
![Page 67: Progress Towards the Accurate Calculation of Anharmonic Vibrational States of Fluxional Molecules and Clusters Without a Potential Energy Surface Andrew](https://reader031.vdocuments.net/reader031/viewer/2022013004/56649e855503460f94b873fa/html5/thumbnails/67.jpg)
AcknowledgementsDr. Anne B. McCoy
Dr. Sara E. Ray
Bethany A. Wellen
Andrew S. Petit
Annie L. Lesiak
Dr. Charlotte E.
Hinkle
Dr. Samantha Horvath
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The Initial Basis (2D Example)
€
Ψ0,01( ) ∝ e
−α 1r 2
2 e−
α 2R 2
2
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The Initial Basis (2D Example)
€
Ψ0,01( ) ∝ e
−α 1r 2
2 e−
α 2R 2
2
Basis functions change each iteration
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The Initial Basis (2D Example)
€
Ψ0,01( ) ∝ e
−α 1r 2
2 e−
α 2R 2
2
€
Ψ m,n1( ) = κ m,n;m− j ,n
1( ) r jΨm− j,n1( ) +
j=1
m
∑ κ m,n;m,n− j1( ) R jΨm,n− j
1( )
j=1
n
∑
Basis functions change each iteration
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The Initial Basis (2D Example)
€
Ψ0,01( ) ∝ e
−α 1r 2
2 e−
α 2R 2
2
€
Ψ m,n1( ) = κ m,n;m− j ,n
1( ) r jΨm− j,n1( ) +
j=1
m
∑ κ m,n;m,n− j1( ) R jΨm,n− j
1( )
j=1
n
∑
Basis functions change each iteration
Phase factors
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The Initial Basis (2D Example)
€
Ψ0,01( ) ∝ e
−α 1r 2
2 e−
α 2R 2
2
€
Ψ m,n1( ) = κ m,n;m− j ,n
1( ) r jΨm− j,n1( ) +
j=1
m
∑ κ m,n;m,n− j1( ) R jΨm,n− j
1( )
j=1
n
∑
Basis functions change each iteration
Phase factors
Will be made orthogonal to all previously built states and normalized
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€
Ψ m,n1( ) = κ m,n;m− j ,n
1( ) r jΨm− j,n1( ) +
j=1
m
∑ κ m,n;m,n− j1( ) R jΨm,n− j
1( )
j=1
n
∑
€
Ψ0,0
Ψ1,0 Ψ0,1
Ψ2,0 Ψ1,1 Ψ0,2
Ψ3,0 Ψ2,1 Ψ1,2 Ψ0,3
The Initial Basis (2D Example)
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€
Ψ m,n1( ) = κ m,n;m− j ,n
1( ) r jΨm− j,n1( ) +
j=1
m
∑ κ m,n;m,n− j1( ) R jΨm,n− j
1( )
j=1
n
∑
€
Ψ0,0
Ψ1,0 Ψ0,1
Ψ2,0 Ψ1,1 Ψ0,2
Ψ3,0 Ψ2,1 Ψ1,2 Ψ0,3
The Initial Basis (2D Example)
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€
Ψ m,n1( ) = κ m,n;m− j ,n
1( ) r jΨm− j,n1( ) +
j=1
m
∑ κ m,n;m,n− j1( ) R jΨm,n− j
1( )
j=1
n
∑
€
Ψ0,0
Ψ1,0 Ψ0,1
Ψ2,0 Ψ1,1 Ψ0,2
Ψ3,0 Ψ2,1 Ψ1,2 Ψ0,3
€
Ψ 3,01( ) = κ 3,0;2,0
1( ) rΨ2,01( ) + κ 3,0;1,0
1( ) r2 Ψ1,01( ) + κ 3,0;0,0
1( ) r3 Ψ0,01( )
The Initial Basis (2D Example)
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€
Ψ m,n1( ) = κ m,n;m− j ,n
1( ) r jΨm− j,n1( ) +
j=1
m
∑ κ m,n;m,n− j1( ) R jΨm,n− j
1( )
j=1
n
∑
€
Ψ0,0
Ψ1,0 Ψ0,1
Ψ2,0 Ψ1,1 Ψ0,2
Ψ3,0 Ψ2,1 Ψ1,2 Ψ0,3
The Initial Basis (2D Example)
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€
Ψ m,n1( ) = κ m,n;m− j ,n
1( ) r jΨm− j,n1( ) +
j=1
m
∑ κ m,n;m,n− j1( ) R jΨm,n− j
1( )
j=1
n
∑
€
Ψ0,0
Ψ1,0 Ψ0,1
Ψ2,0 Ψ1,1 Ψ0,2
Ψ3,0 Ψ2,1 Ψ1,2 Ψ0,3
The Initial Basis (2D Example)
€
Ψ 2,11( ) = κ 2,1;2,0
1( ) RΨ2,01( ) + κ 2,1;1,1
1( ) rΨ1,11( ) + κ 2,1;0,1
1( ) r2 Ψ0,11( )