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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
