PhD Defence Part 1

Development and Application of GVVPT2 Gradients and

Nonadiabatic Coupling Terms

Daniel P. TheisUniversity of North Dakota

Chemistry DepartmentGrand Forks, ND

2

• Electronic Structure Theory and the GVVPT2 Method.

Outline

◦ Importance of each property.

◦ Challenges for evaluating those properties.

◦ Benchmark calculations.

• N2O2 Dissociation.

• Analytic Expressions for the GVVPT2 Molecular Gradients, Nonadiabatic Coupling Terms, and State-Specific Electric Dipole Moments.

3

Electronic Structure Theory

);()();(),(ˆ xrxxrxr eleleleNe EVT

);(

)(

xr

xel

elE

)(

)(

)(

)(

TG

TS

TH

Tk

F

F

F

Rxn

Electronic structure theory studies the behavior of a chemical system by determining the electronic energy and electronic wave function that influences the system.

Ene

rgy

xOO

)S(O2)(O 4122

g

In order to generate reliable results the electronic structure method needs to account for the static and dynamic correlation that influences the chemical system.

4

Ene

rgy

xOO

)S(O2)(O 4122

g

Electronic Structure Theory: Properties of a Reliable Method

+

18

17

16

15

14

13

22

21 0

807

04

23

22

21

22

21

)(O 122

g )S(O2 4

Electronic Structure Theory: Static Correlation

5

In order to generate reliable results the electronic structure method needs to account for the static and dynamic correlation that influences the chemical system.

Electronic Structure Theory: Dynamic Correlation

6

True Dynamic Correlation

SCF Approximation of that Correlation

)(x

)(x

)(xelE

SCF Procedure

)(xSTART

In order to generate reliable results the electronic structure method needs to account for the static and dynamic correlation that influences the chemical system.

Time Required to Perform the Calculation

Time Required to Perform the

Calculation

GVVPT2

7

No Static Correlation

Includes Static Correlation

No Dynamic Correlation

HF MCSCF

Includes Dynamic Correlation

DFT (Time ≈ HF)PT

CC, CI

MRPTMRCC, MRCI

Electronic Structure Theory: The GVVPT2 Method

In order to generate reliable results the electronic structure method needs to account for the static and dynamic correlation that influences the chemical system.

0.00

1.00

2.00

3.00

4.00

5.00

6.00

1.00 2.00 3.00 4.00 5.00 6.00

xLiH (Å)

Rel

ativ

e E

nerg

y (e

V)

GVVPT2 potential energy surfaces for the X 1Sg

+and A 1Sg+ states of LiH

1. GVVPT2 takes into account static and dynamic correlation effects.

2. GVVPT2 can determine accurate electronic energies for systems with low lying, nearly degenerate electronic states.

The Benefits of the GVVPT2 Method

8

1. GVVPT2 takes into account static and dynamic correlation effects.

2. GVVPT2 can determine accurate electronic energies for systems with low lying, nearly degenerate electronic states.

3. GVVPT2 potential energy surfaces are contentious, differentiable functions of geometry, that ensure the evaluation of molecular gradients.

The Benefits of the GVVPT2 Method

9

GVVPT2 potential energy surface of Mn2 (X 1Sg

+)

3.0 4.0 5.0 6.00.00

0.02

0.04

0.06

0.08

xMnMn (Å)

Rel

ativ

e E

nerg

y (e

V)

cc-pVTZCBS, B=1

CBS, B=1.63

cc-pVQZ

3.0 4.0 5.0

-0.27

0.00

0.27

xMnMn (Å)

Ene

rgy

+ 6

2606

.90

(eV

)

MCQDPT potential energy surface of Mn2 (X 1Sg

+)

10

The Importance of Electronic State Properties

)(

)(

)(

)(

TG

TS

TH

Tk

F

F

F

Rxn

The determination of macroscopic date often requires the evaluation of properties of the electronic states

Ene

rgy

xOO

)S(O2)(O 4122

g

)(xμel

elel

el

dx

d

dx

dE

x1

x2

11

Molecular Gradients

xxx xxg

)()( elel Exx

x

a

el

dx

dE )(Analytic molecular gradients lead to the efficient determination of:

• Transition States

• Minimum energy paths

OO O

O

O

O

x1 x1

x2x2

q q

• Minima (Possible reaction intermediates)

12

Molecular Gradients

◦ Harmonic frequencies and normal modes of vibration

OO O

OO O

OO O

• Second derivatives (Hessians)

• Transition States

• Minimum energy paths

◦ Approximations of H(T), S(T), etc. b

belab

ela

ba

el

ab

x

xgxg

dxdx

EdH

2

)()(

)()(

,,

2

xx

xx

xx

xxx xxg

)()( elel Exx

x

a

el

dx

dE )(Analytic molecular gradients lead to the efficient determination of:

• Minima (Possible reaction intermediates)

13

)(

)()(

21

21

x

xxκ

x

x

elel

elelel

Nonadiabatic Coupling TermsE

nerg

y

xAB

AB*

AB

AB → AB* → A + Bhv

Nonadiabatic coupling terms determine the likelihood that a radiationless electronic transition will occur during a chemical reaction.

14

Nonadiabatic coupling terms determine the likelihood that a radiationless electronic transition will occur during a chemical reaction.

)(

)()(

21

21

x

xxκ

x

x

elel

elelel

Nonadiabatic Coupling Terms

0.00

1.00

2.00

3.00

4.00

5.00

6.00

1.00 2.00 3.00 4.00 5.00 6.00

xLiH (Å)

Rel

ativ

e E

nerg

y (e

V)

GVVPT2 potential energy surfaces for the X 1Sg

+and A 1Sg+ states of LiH Types of Reactions this will

Affect:• Charge Transfer Reactions• Photochemical Reactions

15

Electronic Dipole Moments

Electronic dipole moments are used:

• To calculate vibrational excitation strengths.

• To evaluating the energies of intermolecular reactions.

16

Electronic Dipole Moments

Electronic dipole moments are used:

• In some implicit solvation models.

NH3 + CH3Cl → CH3NH3+ + Cl-

No Solvation

Implicit Solvation (H2O)

CH3NH3+ + Cl-

NH3 + CH3Cl

Ene

rgy

Reaction Coordinate

17

Electronic Dipole Moments

0EE Eμ

)(,elel

LR E0E

E

a

el

dE

dE )(

Electronic dipole moments are used:

• In some implicit solvation models.elelel

HF μμ ˆ,el

ael

NH3 + CH3Cl → CH3NH3+ + Cl-

No Solvation

Implicit Solvation (H2O)

CH3NH3+ + Cl-

NH3 + CH3Cl

Ene

rgy

Reaction Coordinate

18

Developing Computer Codes to Determine Those Properties: Challenges that were Addressed

Helgaker, T.; Jørgensen, P. Theor. Chim. Acta. 1989, 75, 111.

Challenge Solution

Complicated formulas define EL

GVV and |YLGVV(1)

First develop codes for the GVVPT2 dipole moments and the MRCISD gradients and nonadiabatic coupling terms.

)()()()( )2()2(

21)2( xZxZxHxH MMMMMM

effMM

)()()()()(2)( )1()2( xCxXxHxCxCIxZ PMQPMQPMMPMMM

)()()()1( xxx m qII

qI HDXe

)(

)(tanh)(

x

xx

m

mm I

II

e

e

e E

ED

e

eeeLq

qIIII HE

m

xxxx mmm22

41

21 )()()()(

)()()()( )2(2 xAxHxAx M

effMMM

GVVE )()()()( xAxHxAx TTTT

MRCISDE

19

Developing Computer Codes to Determine Those Properties: Challenges that were Addressed

Helgaker, T.; Jørgensen, P. Theor. Chim. Acta. 1989, 75, 111.

Challenge Solution

The presence of electronic structure parameters in EL

GVV and |YL

GVV(1)

By using a Lagrangian based approach to derive the analytical formulas it was not necessary to evaluate the derivatives of the electronic structure parameters.

m a

m

m

GVV

m I a

mI

mI

GVV

a

GVV

a

GVV

x

A

A

E

x

C

C

E

x

E

dx

dE 2222 )(x

)()()()( )2()2(

21)2( xZxZxHxH MMMMMM

effMM

)()()()()(2)( )1()2( xCxXxHxCxCIxZ PMQPMQPMMPMMM

)()()()( )2(2 xAxHxAx M

effMMM

GVVE

20

Developing Computer Codes to Determine Those Properties: Challenges that were Addressed

Helgaker, T.; Jørgensen, P. Theor. Chim. Acta. 1989, 75, 111.

Challenge Solution

|YLGVV(1) is not orthogonal to the

other 1st order GVVPT2 wave functions.

Replace |YLGVV(1) by |YL

GVV(2) when deriving an expression for the nonadiabatic coupling terms. Once the expression is obtained eliminate all the 3rd and 4th order terms.

)()1()0()( nGVVGVVGVVnGVV

);(0)1()1( nOGVVGVV

)4(0)2()2( OGVVGVV

3n

21

Developing Computer Codes to Determine Those Properties: Challenges that were Addressed

Helgaker, T.; Jørgensen, P. Theor. Chim. Acta. 1989, 75, 111.

Challenge Solution

Unstable numerical algorithms This problem still needs to be resolved.

Numerical Algorithms are procedures that are used to perform mathematical calculations or operations.

Example: Tan(q)

22

Developing Computer Codes to Determine Those Properties: Progress Summary

Helgaker, T.; Jørgensen, P. Theor. Chim. Acta. 1989, 75, 111.

Property Analytic Formulas

Written Code

Debugged Code

Benchmark Tests

GVVPT2 Molecular Gradients Done Done Done Done

GVVPT2 Nonadiabatic Coupling Terms

Done Done Incomplete Incomplete

GVVPT2 Dipole Moments Done Done Done Done

MRCISD Molecular Gradients Done Done Done Done

MRCISD Nonadiabatic Coupling Terms

Done Done Incomplete Incomplete

23

GVVPT2 Electric Dipole Moments for the X 1Sg

+ State of LiH

1 2 3 4 5 6 7 80

1

2

3

4

5

6

7

8

z (D

ebye

)

RLiH

(Ang.)

• Roos Aug. TZ Basis Set

Technical Details: LiH (C∞v)

• 1:1 SA-MCSCF MOs

• (2:10)-CAS + 1 Core Orb.

24

GVVPT2 Electric Dipole Moments for the A 1Sg

+ State of LiH

• Roos Aug. TZ Basis Set

Technical Details: LiH (C∞v)

• 1:1 SA-MCSCF MOs

• (2:10)-CAS + 1 Core Orb.

2 3 4 5 6 7 8

-6

-4

-2

0

2

4

6

8

10

12

z (D

ebye

)

RLiH

(Ang.)

25

Molecule Analytical Values Deviation from Numerical Values

Geometry Description X Y Z X Y Z

H2O O 4.517477 -0.086969 0.000000 -3.22×10-6 9.00×10-8 0.00

Asym. Str. (0.5 Å) H1 -4.482958 -0.020627 0.000000 3.33×10-6 -8.90×10-7 0.00

H2 -0.034519 0.107596 0.000000 -3.45×10-6 -8.40×10-7 0.00

LiH (X 1S+) H 0.000000 0.000000 -0.014113 0.00 0.00 0.00

Avoided Crossing Li 0.000000 0.000000 0.014113 0.00 0.00 0.00

LiH (A 1S+) H 0.000000 0.000000 -0.004441 0.00 0.00 0.00

Avoided Crossing Li 0.000000 0.000000 0.004441 0.00 0.00 0.00

• cc-pVTZ Basis Set

Technical Details: H2O (Cs – Broken Sym.)

• RCO = 1.205 Å, RCH = 1.611 Å, and RCH = 1.111 Å.

• (8:6)-CAS + 1 Core Orb.

• HCH = 116.1o and OCH = 121.9o1 2

• Roos Aug. TZ Basis Set

Technical Details: LiH (C∞v)

• 9:1 SA-MCSCF MOs

• (2:10)-CAS + 1 Core Orb.

• RLiH = 3.400 Å

Analytical GVVPT2 Gradients for H2O and LiH

26

Molecule Analytical Values Deviation from Numerical Values

Geometry Description X Y Z X Y Z

H2O O 4.521281 -0.105445 0.000000 -3.14×10-6 0.00 0.00

Asym. Str. (0.5 Å) H1 -4.483541 -0.017956 0.000000 3.15×10-6 -1.00×10-8 0.00

H2 -0.037740 0.123401 0.000000 0.00 1.00×10-8 0.00

LiH (X 1S+) H 0.000000 0.000000 -0.014491 0.00 0.00 0.00Avoided Crossing Li 0.000000 0.000000 0.014491 0.00 0.00 0.00

LiH (A 1S+) H 0.000000 0.000000 -0.004457 0.00 0.00 0.00Avoided Crossing Li 0.000000 0.000000 0.004457 0.00 0.00 0.00

• cc-pVTZ Basis Set

Technical Details: H2O (Cs – Broken Sym.)

• RCO = 1.205 Å, RCH = 1.611 Å, and RCH = 1.111 Å.

• (8:6)-CAS + 1 Core Orb.

• HCH = 116.1o and OCH = 121.9o1 2

• Roos Aug. TZ Basis Set

Technical Details: LiH (C∞v)

• 9:1 SA-MCSCF MOs

• (2:10)-CAS + 1 Core Orb.

• RLiH = 3.400 Å

Analytical MRCISD Gradients for H2O and LiH

Method Geometry

Geometry Description R1 R1

GVVPT2 (C2v) 1.382 Å 1.382 Å 85.2o

GVVPT2 (Cs) 1.383 Å 1.381 Å 85.2o

MRCISD (C2v) 1.391 Å 1.391 Å 85.4o

GVVPT2 and MRCISD minimum energy geometries along the conical intersection seam between the first two 1A1 states of O3

• aug(sp)-cc-pVDZ Basis Set

Technical Details: H2CO (Cs – Broken Sym.)

• (12:7)-CAS

27

• Geometry optimizations, gradients calculations, and frequency calculations verify that the GVVPT2 method accurately describes the chemically important regions of most potential energy surfaces.

Conclusions

28

• The GVVPT2 gradients are continuous across potential energy surfaces, including regions of avoided crossings.

• Analytic formulas for GVVPT2 electric dipole moments, molecular gradients, and nonadiabatic coupling terms have been developed which scale at approximately 2-3 times the speed of the GVVPT2 energy.

• Computational implementation of GVVPT2 electric dipole moments and analytic gradients show excellent agreement with finite difference calculations.

• MRCISD and GVVPT2 predictions for the minimum energy geometries along the conical intersection seam between the first two 1A1 states of O3 are in close agreement with one another.

N

O

N

O

N

O

N

O+

• Experimental Geometry:

◦ C2v Symmetry◦ RNN = 2.2630 Å; RNO = 1.1515 Å; ONN = 97.17o

• Electronic State: 1A1

• Bonding:

◦ ED = 710 40 cm-1

◦ Bonding occurs through the NO p* orbitals

Nitric Oxide Dimer – N2O2

29

30

Ene

rgy

(eV

)

RNN (a.u.) RNN (a.u.)

Marouani, S. et al. J. Phys. Chem. A. 2010, 114, 3025.

The Excited States of N2O2

MRCI CASSCF

310.00 eV

4.77 eV

5.45 eV

Focus of the Study

(8 2 0)

(7 3 0)

(8 1 1)

(6 4 0)

(7 2 1)

(8 0 2)

(6 3 1)

NO (X 2P) + NO (A 2S+) ~ (F)20(p2p)8(p2p)1(R3s)1*

NO (X 2P) + NO (a 4P) ~ (F)20(p2p)7(p2p)3(R3s)0*

NO (X 2P) + NO (X 2P) ~ (F)20(p2p)8(p2p)2(R3s)0*

(F)20 = 2×(s1s)2(s1s)2(s2s)2(s2s)2* *

N N

O ON N

O O

N N

O ON N

O O

p2p Orbitals

N N

O ON N

O O

N N

O ON N

O O

*p2p Orbitals

N N

O O

N N

O O

R3s Orbitals

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

2.263 2.763 3.263 3.763 4.263 4.763 5.263 5.763

Ene

rgy

(eV

)Potential Energy Surfaces for the Electronic Singlet and

Triplet States of the Lowest Dissociation Limit

Vertical Excitation Energies of (NO)2.

State GVVPT2 (eV) MRCISD (eV)

1 1A1 0.00 0.00

1 3B1 0.27 0.27

1 1B1 0.29 0.39

2 1A1 0.53 0.51

1 3B2 0.64 0.59

1 3A2 0.41 0.61

1 1A2 0.54 0.62

2 3B2 0.98 0.87

32

RNN (Å)

1 1A1 2 1A1 1 1A2 1 1B1 1 3A2 1 3B1 1 3B2 2 3B2

East, A. L. L.. J. Chem. Phys. 1998, 109, 2185.

6.4

6.5

6.6

6.7

6.8

6.9

7.0

7.1

7.2

2.263 2.763 3.263 3.763 4.263 4.763 5.263 5.763

RNN (Å)

Ene

rgy

(eV

)

MRCISD calculation of the Energy at the Dissociation Limit:

◦ The adiabatic (X 2P) → (a 4P) excitation energy (4.79 eV) closely agreed with the experimentally measured value (4.78 eV).

◦ The minimum energy geometry (1.151 Å) closely agreed with the experimentally measured bond length (1.152 Å).

Potential Energy Surfaces for the Triplet Electronic States of the Second Lowest Dissociation Limit

331 3A1 2 3A1 2 3A2 3 3A2 2 3B1 3 3B1 3 3B2 4 3B2

Potential Energy Surfaces for the 1A1 and 3A1 States

34

Potential Energy Surfaces for the 1A2 and 3A2 States

35

Potential Energy Surfaces for the 1B1 and 3B1 States

36

Potential Energy Surfaces for the 3B2 States

37

• Photodissociation studies of N2O2 suggested the existence of “dark states”, that undergo nonadiabatic transitions.

Interpretation and Conclusions

38

• GVVPT2 is capable of generating accurate potential energy surfaces of the NO + NO dissociation limits of N2O2.

• From those calculations several areas of potentially strong nonadiabatic coupling were identified.

• Many of those states have B2 symmetry and involve a excitation energy of 5 – 6 eV. These results are consistent with photofragment measurements which predict that the 244 – 190 nm UV bands involve B2 electronic states.

Dr. Mark R. HoffmannDr. Yuriy G. Khait

Patrick TamukangRashel MokambeJason HicksErik Timmian

Jennifer TheisJeremy and Kate Casper

National Science Foundation

Acknowledgements

39