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



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-



An accurate and general treatment of these highly fluxional systems requires

either a global PES or a grid-based sampling of configuration space



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


€

fr x ( )dV =

fr x ( )

gr x ( )

gr x ( )

V

∫V

∫ dV ≈V

N

fr x i( )

gr x i( )i=1

N

∑



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

∑


Sampling function based on harmonic ground state


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



I

I

I IBuild and DiagonalizeHamiltonian Matrix

Assign Eigenstates &Build New Basis

The General Scheme



The Evolving Basis

The Evolving BasisHarmonic oscillator basis

€

H 1( )


€

H 1( )

€

˜ Ψ 1( ){ }, E 1( )

{ }


€

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( )


€

H 1( )

€

˜ Ψ 1( ){ }, E 1( )

{ }


Assignment via:

€


1( )2 ⎛

⎝ ⎜

⎞ ⎠ ⎟⇒ ˜ Ψ m

1( )

€

Ψ 2( ){ }


€

H 1( )

€

˜ Ψ 1( ){ }, E 1( )

{ }


Assignment via:

€


1( )2 ⎛

⎝ ⎜

⎞ ⎠ ⎟⇒ ˜ Ψ m

1( )

€

Ψ 2( ){ }

Define an active space:

€

Ψm(2)

{ }active

= ˜ Ψ m(1)

{ }active


€

H 1( )

€

˜ Ψ 1( ){ }, E 1( )

{ }


Assignment via:

€


1( )2 ⎛

⎝ ⎜

⎞ ⎠ ⎟⇒ ˜ Ψ m

1( )

€

Ψ 2( ){ }


€

Ψm(2)

{ }active

= ˜ Ψ m(1)

{ }active

€

Ψ m∉active2( ) = κ m,m−k

2( ) rkΨm−k2( )

k=1

m

∑


The Evolving Basis

€

H j( )

€

˜ Ψ j( ){ }, E j( )

{ }


Assignment via:

€

max ˜ Ψ kj( ) ˆ r Ψm−1

j( )2 ⎛

⎝ ⎜

⎞ ⎠ ⎟⇒ ˜ Ψ m

j( )

€

Ψ j +1( ){ }


€

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


€

H 1( )

€

˜ Ψ 1( ){ }, E 1( )

{ }

€

Ψ 2( ){ }

MxM Matrix


€

Ψ 11( ) = ra0

1( )e−

αr 2

2

Basis functions shown prior to orthonormalization


€

H 1( )

€

˜ Ψ 1( ){ }, E 1( )

{ }

€

Ψ 2( ){ }

MxM Matrix


€

Ψ 11( ) = ra0

1( )e−

αr 2

2



€

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


€

Ψ 11( ) = ra0

1( )e−

αr 2

2



€

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




€

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



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

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


€

H j( )

€

˜ Ψ j( ){ }, E j( )

{ }

€

Ψ j +1( ){ }

Eigenstates used to

construct new basis

DVR

€

1,1

€

0,2


€

H j( )

€

˜ Ψ j( ){ }, E j( )

{ }

€

Ψ j +1( ){ }

Eigenstates used to

construct new basis

DVR Iteration 2

€

1,1

€

0,2


€

H j( )

€

˜ Ψ j( ){ }, E j( )

{ }

€

Ψ j +1( ){ }

Eigenstates used to

construct new basis

DVR Iteration 2 Iteration 3

€

1,1

€

0,2


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( )

€


€

φkj( )

€

φkj( ) Ψm,n

j( )2


€


€

φkj( )

€

φkj( ) Ψm,n

j( )2

€

φkj( ) ˆ r OF Ψm−1,n

j( )2

+ φkj( ) ˆ r OHb

Ψm,n−1j( )

2


€


€

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

⎧

⎨

⎪ ⎪ ⎪

⎩

⎪ ⎪ ⎪

⎫

⎬

⎪ ⎪ ⎪

⎭

⎪ ⎪ ⎪


€

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.



Perform calculation using an assignment scheme based on

€

φkj( ) Ψm,n

j( )2

Gruebele et al. Int. Rev. Phys. Chem 1998, 17, 91-145.



€

φkj( ) Ψm,n

j( )2

Significantly contaminated eigenstates are rebuilt



€

φkj( ) Ψm,n

j( )2


(Å)

(Å)

DVR



€

φkj( ) Ψm,n

j( )2


Iteration23

(Å)

(Å)

DVR



€

φkj( ) Ψm,n

j( )2


Iteration23 24 25 26 30 100

(Å)

(Å)

DVR


Iteration23 24 25 26 30 100

(Å)

(Å)

DVR

Contamination occurs under the radar of overlaps, transition moments, moment distributions, etc.


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


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


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



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


Complete the optimization of the multi-dimensional algorithm’s treatment of spurious resonances












Further multi-dimensional benchmarking


Couple algorithm with ab initio electronic structure calculations






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


€

Ψ0,01( ) ∝ e

−α 1r 2

2 e−

α 2R 2

2

Basis functions change each iteration


€

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

∑



€

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


1( )

j=1

n

∑


Phase factors


€

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


1( )

j=1

n

∑


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


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 m,n1( ) = κ m,n;m− j ,n

1( ) r jΨm− j,n1( ) +

j=1

m


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( )


€

Ψ m,n1( ) = κ m,n;m− j ,n

1( ) r jΨm− j,n1( ) +

j=1

m


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 m,n1( ) = κ m,n;m− j ,n

1( ) r jΨm− j,n1( ) +

j=1

m


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 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( )

progress towards the accurate calculation of anharmonic vibrational states of fluxional molecules...

Documents

new basisassignment

harmonic analysis

general treatment

active space

fluxional systems

global pes

mxm matrixconstruction

potential energy surfaceandrew