Progress Towards the Accurate Calculation of Anharmonic Vibrational States of Fluxional Molecules...

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

What Are the Challenges???

Harmonic analysis pathologically fails

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-

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

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

The General Scheme

I

The General SchemeMonte Carlo SampleConfiguration Space

Monte Carlo Sample Configuration Space

Use importance sampling Monte Carlo

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

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

Sampling function based on harmonic ground state

Monte Carlo Sample Configuration Space

R

Rela

tive

Prob

abili

ty

Sampling function based on harmonic ground state

Monte Carlo Sample Configuration Space

R

Rela

tive

Prob

abili

ty

Sampling function based on harmonic ground state

Monte Carlo Sample Configuration Space

R

Rela

tive

Prob

abili

ty

Sampling function based on harmonic ground state

Monte Carlo Sample Configuration Space

R

Rela

tive

Prob

abili

ty

Sampling function based on harmonic ground state

Monte Carlo Sample Configuration Space

R

Rela

tive

Prob

abili

ty

I

I

The General Scheme

Construct Initial Basis

Monte Carlo SampleConfiguration Space

I

I

IBuild and DiagonalizeHamiltonian Matrix

The General Scheme

Construct Initial Basis

Monte Carlo SampleConfiguration Space

I

I

I IBuild and DiagonalizeHamiltonian Matrix

Assign Eigenstates &Build New Basis

The General Scheme

Construct Initial Basis

Monte Carlo SampleConfiguration Space

I

I

I IBuild and DiagonalizeHamiltonian Matrix

Assign Eigenstates &Build New Basis

The General Scheme

Construct Initial Basis

Monte Carlo SampleConfiguration Space

The Evolving Basis

The Evolving BasisHarmonic oscillator basis

€

H 1( )

The Evolving BasisHarmonic oscillator basis

€

H 1( )

€

˜ Ψ 1( ){ }, E 1( )

{ }

The Evolving BasisHarmonic oscillator basis

€

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

€

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

€

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

€

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

€

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

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

∑

€

H 1( )

€

˜ Ψ 1( ){ }, E 1( )

{ }

€

Ψ 2( ){ }

MxM Matrix

Construction of 2nd Iteration Basis

€

H 1( )

€

˜ Ψ 1( ){ }, E 1( )

{ }

€

Ψ 2( ){ }

MxM Matrix

Suppose active space only contains ground state

Construction of 2nd Iteration Basis

€

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

€

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

€

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

€

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

I

I

I IBuild and DiagonalizeHamiltonian Matrix

Assign Eigenstates &Build New Basis

The General Scheme

Construct Initial Basis

Monte Carlo SampleConfiguration Space

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

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

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

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

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

The Problem of Assignment

€

H j( )

€

˜ Ψ j( ){ }, E j( )

{ }

€

Ψ j +1( ){ }

Eigenstates used to

construct new basis

The Problem of Assignment

€

H j( )

€

˜ Ψ j( ){ }, E j( )

{ }

€

Ψ j +1( ){ }

Eigenstates used to

construct new basis

DVR

€

1,1

€

0,2

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

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

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

The Hunt for an Effective Metric

€

Ψm,nj( ) Basis function Un-assigned eigenstate

€

φkj( )

€

Ψm,nj( ) Basis function Un-assigned eigenstate

€

φkj( )

€

φkj( ) Ψm,n

j( )2

The Hunt for an Effective Metric

€

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

€

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

€

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

The Hunt for an Effective Metric

Perform calculation using an assignment scheme based on

€

φkj( ) Ψm,n

j( )2

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

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

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

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

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.

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

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

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

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

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

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

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

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

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

AcknowledgementsDr. Anne B. McCoy

Dr. Sara E. Ray

Bethany A. Wellen

Andrew S. Petit

Annie L. Lesiak

Dr. Charlotte E.

Hinkle

Dr. Samantha Horvath

The Initial Basis (2D Example)

€

Ψ0,01( ) ∝ e

−α 1r 2

2 e−

α 2R 2

2

The Initial Basis (2D Example)

€

Ψ0,01( ) ∝ e

−α 1r 2

2 e−

α 2R 2

2

Basis functions change each iteration

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

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

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

€

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 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 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)

€

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 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( )