1. Development and Application of
GVVPT2 Gradients and
Nonadiabatic Coupling Terms
Daniel P. Theis
University of North Dakota
Chemistry Department
Grand Forks, ND

2. 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. 3
Electronic Structure Theory
  );()();(),(ˆ xrxxrxr elelel
eNe EVT  
);(
)(
xr
x
el
el
E



)(
)(
)(
)(
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.
Energy
xOO
)S(O2)(O 412
2

g

4. 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
Energy
xOO
)S(O2)(O 412
2

g
Electronic Structure Theory:
Properties of a Reliable Method

5. +
1
8
1
7
1
6
1
5
1
4
1
3
2
2
2
1 0
8
0
7
0
4
2
3
2
2
2
1
2
2
2
1 
)(O 12
2

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.

6. Electronic Structure Theory:
Dynamic Correlation
6
True Dynamic
Correlation
SCF Approximation
of that Correlation
)(x
)(x
)(xel
E
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.

7. 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
MRPT
MRCC, 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.

8. 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)
RelativeEnergy(eV)
GVVPT2 potential energy surfaces for
the X 1g
+and A 1g
+ 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

9. 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 1g
+)
3.0 4.0 5.0 6.0
0.00
0.02
0.04
0.06
0.08
xMnMn (Å)
RelativeEnergy(eV)
3.0 4.0 5.0
-0.27
0.00
0.27
xMnMn (Å)
Energy+62606.90(eV)
MCQDPT potential energy
surface of Mn2 (X 1g
+)

10. 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
Energy
xOO
)S(O2)(O 412
2

g

)(xμel
el
el
el
dx
d
dx
dE







11. x1
x2
11
Molecular Gradients
  xxx xxg

 )()( elel
E
xx
x



a
el
dx
dE )(Analytic molecular gradients lead to the efficient
determination of:
• Transition States
• Minimum energy paths
O
O O
O
O
O
x1 x1
x2
x2
q q
• Minima (Possible reaction intermediates)

12. 12
Molecular Gradients
◦ Harmonic frequencies and normal modes
of vibration
O
O O
O
O O
O
O O
• Second derivatives (Hessians)
• Transition States
• Minimum energy paths
◦ Approximations of H(T), S(T), etc. b
b
el
ab
el
a
ba
el
ab
x
xgxg
dxdx
Ed
H








2
)()(
)(
)(
,,
2
xx
x
x
xx
  xxx xxg

 )()( elel
E
xx
x



a
el
dx
dE )(Analytic molecular gradients lead to the efficient
determination of:
• Minima (Possible reaction intermediates)

13. 13
 
  )(
)()(
2
1
2
1
x
xxκ
x
x
elel
elelel




Nonadiabatic Coupling Terms
Energy
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. 14
Nonadiabatic coupling terms determine the likelihood
that a radiationless electronic transition will occur
during a chemical reaction.
 
  )(
)()(
2
1
2
1
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)
RelativeEnergy(eV)
GVVPT2 potential energy surfaces for
the X 1g
+and A 1g
+ states of LiH Types of Reactions this will
Affect:
• Charge Transfer Reactions
• Photochemical Reactions

15. 15
Electronic Dipole Moments
Electronic dipole moments are used:
• To calculate vibrational excitation strengths.
• To evaluating the energies of intermolecular reactions.

16. 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
Energy
Reaction Coordinate

17. 17
Electronic Dipole Moments
  0EE Eμ

 )(,
elel
LR E
0E
E




a
el
dE
dE )(
Electronic dipole moments are used:
• In some implicit solvation models.
elelel
HF   μμ ˆ,
el
a
el
  ˆ
NH3 + CH3Cl → CH3NH3
 + Cl
No Solvation
Implicit Solvation
(H2O)
CH3NH3
 + Cl
NH3 + CH3Cl
Energy
Reaction Coordinate

18. 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
E
GVV and |
GVV(1)
First develop codes for the GVVPT2 dipole moments and
the MRCISD gradients and nonadiabatic coupling terms.
  
 )()()()( )2()2(
2
1)2(
xZxZxHxH MMMMMM
eff
MM
  )()()()()(2)( )1()2(
xCxXxHxCxCIxZ PMQPMQPMMPMMM 
)()()()1(
xxx m qI
I
qI HDX e

 
)(
)(tanh
)(
x
x
x
m
m
m I
I
I
e
e
e
E
E
D



   

e
eee
Lq
qI
III
HE
m
xxxx mmm
22
4
1
2
1
)()()()( 
)()()()( )2(2
xAxHxAx 

  M
eff
MMM
GVV
E )()()()( xAxHxAx 

  TTTT
MRCISD
E

19. 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 E
GVV
and |
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(
2
1)2(
xZxZxHxH MMMMMM
eff
MM
  )()()()()(2)( )1()2(
xCxXxHxCxCIxZ PMQPMQPMMPMMM 

)()()()( )2(2
xAxHxAx 

  M
eff
MMM
GVV
E

20. 20
Developing Computer Codes to Determine Those Properties:
Challenges that were Addressed
Helgaker, T.; Jørgensen, P. Theor. Chim. Acta. 1989, 75, 111.
Challenge Solution
|
GVV(1) is not orthogonal to
the other 1st order GVVPT2
wave functions.
Replace |
GVV(1) by |
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
 
3n

21. 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. 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. 23
GVVPT2 Electric Dipole Moments for the
X 1g
+ State of LiH
1 2 3 4 5 6 7 8
0
1
2
3
4
5
6
7
8
z
(Debye)
RLiH
(Ang.)
• Roos Aug. TZ Basis Set
Technical Details: LiH (C∞v)
• 1:1 SA-MCSCF MOs
• (2:10)-CAS + 1 Core Orb.

24. 24
GVVPT2 Electric Dipole Moments for the
A 1g
+ 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
(Debye)
RLiH
(Ang.)

25. 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 1+) 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 1+) 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 = 121.9o
1 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. 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 1+) H 0.000000 0.000000 -0.014491 0.00 0.00 0.00
Avoided Crossing Li 0.000000 0.000000 0.014491 0.00 0.00 0.00
LiH (A 1+) H 0.000000 0.000000 -0.004457 0.00 0.00 0.00
Avoided 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 = 121.9o
1 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

27. 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

28. • 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.

29. 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. 30
Energy(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

31. 31
0.00 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 2+) ~ (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 O
N N
O O
N N
O O
N N
O O
p2p Orbitals
N N
O O
N N
O O
N N
O O
N N
O O
*p2p Orbitals
N N
O O
N N
O O
R3s Orbitals

32. 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
Energy(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.

33. 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)
Energy(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

34. Potential Energy Surfaces for the 1A1 and 3A1 States
34

35. Potential Energy Surfaces for the 1A2 and 3A2 States
35

36. Potential Energy Surfaces for the 1B1 and 3B1 States
36

37. Potential Energy Surfaces for the 3B2 States
37

38. • 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.

39. Dr. Mark R. Hoffmann
Dr. Yuriy G. Khait
Patrick Tamukang
Rashel Mokambe
Jason Hicks
Erik Timmian
Jennifer Theis
Jeremy and Kate Casper
National Science Foundation
Acknowledgements
39