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Transforming the Vibrational Hamiltonian of a Polyatomic Molecule Using Van Vleck Perturbation Theory Andreana Rosnik Polik Group, Hope College Midwest Undergraduate Computational Chemistry Consortium Conference February 2012

Transcript of Transforming the Vibrational Hamiltonian of a …discus/muccc/muccc18/MUCCC18-Rosnik.pdfTransforming...

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Transforming the Vibrational Hamiltonian of a Polyatomic Molecule Using Van Vleck Perturbation Theory

Andreana Rosnik Polik Group, Hope College Midwest Undergraduate Computational Chemistry Consortium Conference February 2012

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Motivation ◦  Scientific models can provide us with a

quantitative understanding of the natural world ◦ Comprehensible, accurate models are

essential to advancing science and understanding the universe ◦ This presentation will introduce a model for

chemical bonding and molecular vibrations

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Potential Energy Surfaces

•  Potential Energy Surfaces (PES) are chemists’ models for reactivity

•  Provide information about transition states, reaction pathways, and relative energies of products and reactants

•  Mathematical construct –  A PES is a potential energy function of the

form V(q1, q2,…q3N-6 ) –  qi are molecular internal coordinates (e.g.

bond lengths and angles)

•  How do you find a PES if you can’t measure it directly?

–  Experiment: Measure vibrational levels

–  Theory: Compute V(q1, q2,…q3N-6 )

•  Goal: Connect experiment and theory by calculating vibrational levels from V

J. Harvey, University of Bristol, http://www.chm.bris.ac.uk/pt/harvey/msci_pract/back_qm.html

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Molecular Motion �  Molecules translate, rotate, and vibrate

�  Only vibrations will be considered

–  Vibrations involve changes in the internal coordinates of atoms, whereas rotations and translations do not

–  Therefore vibrations are the coordinates of PES

�  Molecules will be treated as anharmonic oscillators

◦  The harmonic oscillator gives the zero-order model

�  There are 3N-6 degrees of vibrational freedom, where N is the number of atoms in the molecule

Comparison in one dimension of harmonic (parabolic curve) and anharmonic (semi-parabolic curve) oscillators and their vibrational energy levels. L. Kieran, Deakin University, and W. Coleman, Wellesley College. http://jchemed.chem.wisc.edu/JCEDLib/WebWare/collection/ reviewed/JCE2005p1263_2WW/index.html

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Connecting Theory and Experiment

...!41

!31

21 2 +++= ∑∑∑

klmnnmlkklmn

klmmlkklmk

kk qqqqhcqqqhcqhcV φφω

• The potential energy function V that will be used is

Harmonic Oscillator

First-Order Anharmonicities

Second-Order Anharmonicities

• Theory computes the ω’s and Φ’s, the frequencies and force constants, respectively

• In order to relate the ω’s and Φ’s to the vibrational levels mathematically, the Schrödinger equation must be solved…

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Schrödinger Equation

• Schrödinger’s Expression:

)()()(2 2

22

xExxVdxd

mΨ=Ψ⎥

⎤⎢⎣

⎡+−

• Heisenberg’s Expression:

⎟⎟⎟⎟⎟

⎜⎜⎜⎜⎜

=

⎟⎟⎟⎟⎟

⎜⎜⎜⎜⎜

⎥⎥⎥⎥

⎢⎢⎢⎢

3

2

1

3

2

1

ccc

Eccc

HHHHH

cc

bbba

abaa

• Mathematically Equivalent:

–  Eigenfunction is a linear combination of basis functions

• The matrix method is implemented in this derivation because it is easier to

implement on computers

Ψ=Ψ EH

...)( 321 +++=Ψ cba cccx ψψψ)(xΨ

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Diagonalization of the Hamiltonian �  In the matrix method, solving the Schrödinger equation

is equivalent to diagonalizing the Hamiltonian

�  T is a unitary matrix whose rows are the coefficients of the basis functions of H

�  Λ is a diagonal matrix whose diagonal corresponds to the energy eigenvalues of H

⎥⎥⎥⎥

⎢⎢⎢⎢

=Λ=−

00000000

00

3

2

1

1

EE

E

THT

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Approximate Solutions �  The Schrödinger equation is not exactly soluble for two

main reasons: ◦  The vibrational Hamiltonian is infinite-dimensional ◦  There are both large and small interactions between

harmonic oscillator basis functions �  Large interactions involve resonances between

vibrational modes �  Small interactions between states are

anharmonicities �  Thus, a means of approximately solving the Schrödinger

equation must be used…

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Solving the Schrödinger Equation via Higher Order Van Vleck Perturbation Theory

...~~~~ˆ 21 +ʹ′ʹ′+ʹ′+==− HHHHTHT o λλ

• Define the Hamiltonian as an infinite expansion and truncate after a certain order ...ˆ 2 +ʹ′ʹ′+ʹ′+= HHHH o λλ

•  λ is an ordering parameter (and is usually set to one for simplicity) • Second-order Van Vleck perturbation theory will be used here • Define the transformation matrix

...2

1 22

+−+== SSieT Si λλλ

• Carry out the transformation

( )2221

kkk

k qphc +∑ω ∑klm

mlkklm qqqhc φ!31 ∑

klmnnmlkklmn qqqqhc φ

!41

Harmonic Oscillator

First-Order Anharmonicity

Second-Order Anharmonicity

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Why Use VVPT?

� Van Vleck perturbation theory treats both large and small interactions ◦  First order PT does not treat interactions

between states ◦  Second order PT only treats small

interactions ◦ VVPT treats large and small interactions

differently, thereby accounting for the complexities of molecules ◦  Selection rules make infinite sums finite

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Solving the Schrödinger Equation via Perturbation Theory

�  First order PT accounts only for corrections to a state itself

�  Second order PT accounts for small interactions between states

�  Van Vleck PT accounts for both small and large interactions between states

aaoaaa HHH ʹ′+=~

∑ −

ʹ′ʹ′+ʹ′ʹ′+ʹ′+=

γ γ

γγoo

a

aaaaaa

oaaa EE

HHHHHH~

∑⎥⎥⎦

⎢⎢⎣

ʹ′ʹ′+

ʹ′ʹ′+ʹ′ʹ′+ʹ′+=

γ γ

γγ

γ

γγoo

b

baoo

a

baabab

oab EE

HHEEHH

HHHH21~

Other states

Other states

Other states

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Context of Derivation �  Two attempts have been made in the literature to use Van Vleck perturbation theory

to describe large interactions in a general fashion, but...

◦  …Both are in error: One author provided an incorrect derivation, and the other did not provide one

◦  Both papers contain expressions fraught with typographical errors

�  Since there are incorrect equations in the literature, scientists have access to several inconsistent equations describing the same quantities

�  My work has involved…

�  thoroughly investigating underlying assumptions

�  developing a complete, accurate expression containing all terms as well as a simplified one, in which neglected terms are identified

�  Van Vleck perturbation theory has been used in a case-by-case manner to arrive at a result

�  I seek a general method to calculate vibrational energy levels from a potential energy surface

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WARNING: Take two ibuprofen tablets before proceeding.

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On-Diagonal Matrix Elements

∑ −

ʹ′ʹ′+ʹ′ʹ′+ʹ′+=

γ γ

γγoo

a

aaaaaa

o

EEHH

HHHH~

( ) ( )

( )

⎟⎠

⎞⎜⎝

⎛ +⎟⎠

⎞⎜⎝

⎛ +−⎟⎠

⎞⎜⎝

⎛ +⎟⎠

⎞⎜⎝

⎛ +−

⎟⎠

⎞⎜⎝

⎛ +⎟⎠

⎞⎜⎝

⎛ +−⎟⎠

⎞⎜⎝

⎛ +⎟⎠

⎞⎜⎝

⎛ +−−−

⎟⎠

⎞⎜⎝

⎛ +⎟⎠

⎞⎜⎝

⎛ +−−−⎟⎠

⎞⎜⎝

⎛ +⎟⎠

⎞⎜⎝

⎛ +−−−

∑∑

∑∑

∑∑

≠≠

≠≠≠

≠≠≠≠

21

211

41

21

21

)4(2

21

211

41

21

21

2

21

21

221

21

2

22

2

2222

2222

2222

lklk k

kllkkklklk

k

lk

kkl

lkm

llmkkmlk

mlmlkkmlk klm

klm

mkmkllmlk klm

klmlkklmm

mlk klm

klm

vvhcvvhc

vvhcvvN

hc

vvN

hcvvN

hc

ωφφ

ωωωφ

ωφφωωωω

φ

ωωωωφ

ωωωωφ

mlkmlk klm

klmkkk

klk

lkkl

lk Nhchchc ωωω

φφ

ωωω

φ ∑∑∑≠≠≠

−−−

+4576

7)4(64

3 22

222

∑∑≠

⎟⎠

⎞⎜⎝

⎛ +⎟⎠

⎞⎜⎝

⎛ ++⎥⎥⎦

⎢⎢⎣

⎡+⎟

⎞⎜⎝

⎛ ++lk

lkkkllk

kkkkk hchc21v

21v

41

41

21v

161 2

φφ⎟⎠

⎞⎜⎝

⎛ += ∑ 21vk

kkhc ω

( ) ⎟⎠

⎞⎜⎝

⎛ +⎟⎠

⎞⎜⎝

⎛ +⎟⎟⎠

⎞⎜⎜⎝

⎛++ ∑∑ 2

1v21v

2lk

l

k

k

l

klklB

ωω

ωω

ζ α

αα

22

2

22

222

211

485

21

)4()38(

161

⎟⎠

⎞⎜⎝

⎛ +−⎟⎠

⎞⎜⎝

⎛ +−

−− ∑∑ k

kkkkkk

lkl

lkkkl

k

vhcvhcω

φωωωωω

φ

where ))()()(( mlkmlkmlkmlkklmN ωωωωωωωωωωωω −++−−−++=

• Red circles identify harmonic frequencies • Orange circles identify force constants

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Second-Order Off-Diagonal Matrix Elements for a 2-2 Resonance

[ ]∑ ++++=++++klm

nmlkmnkl

nmlknmlk nnnnK

nnnnHnnnn 2/1; )1)(1)(1)(1(4

1,1,,~,,1,1

( ) ⎥⎦

⎤⎢⎣

⎡ ++−++−−−+

=

∑ 2/1ln

;

))(())(())((2nmlk

mlnklmknnlmkkmnmlkmnkl

klmnmnkl

B

K

ωωωωωωωωζζωωωωζζωωωωζζ

φαααααα

αα

⎥⎦

⎤⎢⎣

+−−+

+−−+

+++

++⋅− ∑

rnmrlkrnmrlkrmnrklr ωωωωωωωωωωωωφφ

111141

⎥⎦

⎤⎢⎣

++−+

++−+

+−+

+−⋅− ∑

rnlrmkrnlrmkrrkmr ωωωωωωωωωωωω

φφ1111

41

ln

⎥⎦

⎤⎢⎣

++−+

++−+

+−+

+−⋅− ∑

rmlrnkrmlrnkrlmrknr ωωωωωωωωωωωωφφ

111141

where

• Red circles identify denominators where resonances could occur, i.e. where the denominator could be zero • Orange circles identify force constants

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Notes on Equations

� Evaluated using harmonic oscillator matrix elements

� Describe vibrational energy levels of polyatomic molecules

� Written in non-restrictive summations ◦ Non-restrictive summations involve summing

over all possible combinations of vibrational modes ◦ Restrictive summations include only one of

each combination of modes

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Comparing Theory and Experiment

� The vibrational energy levels of the theoretical PES can be compared to spectroscopically measured vibrational energy levels

�  Force constants ωk, Φklm, and Φklmn were calculated by finite differences using the aug-cc-pVTQ basis set and CCSD(T) method

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Comparison- H2CO Value VPT2

(2nd-Order PT) VPT2+K (VVPT)

Experiment

ω1 2933.7 2933.7 2927.8

… … … …

ω6 1268.0 1268.0 1277.2

x11 -32.0 -32.0 -27.3

… … … …

x66 -2.0 -2.0 -2.7

K26,5 - 146.5 149.2

K36,5 - -187.0 -130.4

K11,55 - -141.9 -145.3

σ   139.4 20.9 -

% Error   1.4 0.2 -

• VVPT yields more accurate results than second-order perturbation theory for molecules with large interactions (resonances)

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Progress Report �  Some difficulties have been encountered in using non-

restrictive summations in the on-diagonal expressions ◦  Each step in the derivation is being coded into a

computer program in order to validate each step and to analyze the equations’ behavior

�  These programs will be applied to several molecules to determine their vibrational energy levels

�  Once the equations have been finalized and some sample applications are developed, a manuscript will be published (It is currently in the works)

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Future Work ◦  Publish corrected equations with comprehensive derivations ◦  Apply computer programs to several molecules ◦  Compare results to results of other methods (e.g. second-order

perturbation theory) ◦  Define new computational technique for vibrational states

(VPT2+K) ◦  Work with Gaussian and WebMO developers to integrate

VPT2+K into software

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Acknowledgements �  Dr. Polik �  Howard Dobbs �  John D., Kent K., and former Polik group members �  Hope College Chemistry Department �  Dean of Natural and Applied Sciences �  National Science Foundation