1. Automated construction of Potential Energy
Surfaces for van der Waals systems
Ernesto Quintas Sánchez
Richard Dawes
Missouri University of Science and Technology
Chemistry Department

2. Xx Born - Oppenheimer approximation  PES
The Potential Energy Surface
system wave function
nuclear coordinates
electronic coordinates
Xx Born - Oppenheimer approximation  PES
clamped-nuclei Schrödinger equation
electronic wave function
nuclear wave function
Once the level of theory on the electronic structure calculations is established,
AUTOFIT codes are designed to assist in the development of accurate PESs ready to be used to predict and understand the molecular spectroscopy and dynamics of …

3. Xx Born - Oppenheimer approximation  PES
The Potential Energy Surface
system wave function
nuclear coordinates
electronic coordinates
Xx Born - Oppenheimer approximation  PES
clamped-nuclei Schrödinger equation
electronic wave function
Once the level of theory on the electronic structure calculations is established,
AUTOFIT codes are designed to assist in the development of accurate PESs ready to be used to predict and understand the molecular spectroscopy and dynamics of …

4. Xx Born - Oppenheimer approximation  PES
The Potential Energy Surface
system wave function
nuclear coordinates
electronic coordinates
Xx Born - Oppenheimer approximation  PES
clamped-nuclei Schrödinger equation
electronic wave function
single–point interaction energies
electronic structure method (MP2, CCSD(T), …)
basis set (AVDZ, AVTZ, …)
program (MOLPRO, CFOUR, …)
fitting process
Once the level of theory on the electronic structure calculations is established,
AUTOFIT codes are designed to assist in the development of accurate PESs ready to be used to predict and understand the molecular spectroscopy and dynamics of …
𝑉 𝑓𝑖𝑡 𝑅 = 𝑖=1 𝑀 𝑪 𝒊 𝐵 𝑖 ({ 𝑅 })

5. Xx Born - Oppenheimer approximation  PES
The Potential Energy Surface
system wave function
nuclear coordinates
electronic coordinates
Xx Born - Oppenheimer approximation  PES
clamped-nuclei Schrödinger equation
electronic wave function
single–point interaction energies
electronic structure method (MP2, CCSD(T), …)
basis set (AVDZ, AVTZ, …)
program (MOLPRO, CFOUR, …)
fitting process
Once the level of theory on the electronic structure calculations is established,
AUTOFIT codes are designed to assist in the development of accurate PESs ready to be used to predict and understand the molecular spectroscopy and dynamics of …
𝑉 𝑓𝑖𝑡 𝑅 = 𝑖=1 𝑀 𝑪 𝒊 𝐵 𝑖 ({ 𝑅 })

6. Xx Born - Oppenheimer approximation  PES
The Potential Energy Surface
system wave function
nuclear coordinates
electronic coordinates
Xx Born - Oppenheimer approximation  PES
clamped-nuclei Schrödinger equation
electronic wave function
single–point interaction energies
electronic structure method (MP2, CCSD(T), …)
basis set (AVDZ, AVTZ, …)
program (MOLPRO, CFOUR, …)
fitting process
Once the level of theory on the electronic structure calculations is established,
AUTOFIT codes are designed to assist in the development of accurate PESs ready to be used to predict and understand the molecular spectroscopy and dynamics of …
𝑉 𝑓𝑖𝑡 𝑅 = 𝑖=1 𝑀 𝑪 𝒊 𝐵 𝑖 ({ 𝑅 })
grid of points
fitting strategy and numerical methods
evaluate the error
persistence, experience, intuition
computational resources

7. Xx Born - Oppenheimer approximation  PES
The Potential Energy Surface
system wave function
nuclear coordinates
electronic coordinates
Xx Born - Oppenheimer approximation  PES
clamped-nuclei Schrödinger equation
electronic wave function
single–point interaction energies
electronic structure method (MP2, CCSD(T), …)
basis set (AVDZ, AVTZ, …)
program (MOLPRO, CFOUR, …)
fitting process
Once the level of theory on the electronic structure calculations is established,
AUTOFIT codes are designed to assist in the development of accurate PESs ready to be used to predict and understand the molecular spectroscopy and dynamics of …
𝑉 𝑓𝑖𝑡 𝑅 = 𝑖=1 𝑀 𝑪 𝒊 𝐵 𝑖 ({ 𝑅 })
grid of points
fitting strategy and numerical methods
evaluate the error
persistence, experience, intuition
computational resources
vdW systems composed of two rigid fragments

8. Automated construction of a PES
AUTOFIT package
AUTOFIT - PES
AUTOFIT - PLOT
summary (relevant info: basis, parameters, errors…)
evaluate the PES
1D or 2D cuts of the PES
2D cut R-optimized
rigid2D code: linear molecule + atom
rigid3D code: fragment + atom
rigid4D code: two linear molecules
rigid5D code: fragment + linear molecule
rigid6D code: two fragments

9. Automated construction of a PES
AUTOFIT package
AUTOFIT - PES
AUTOFIT - PLOT
summary (relevant info: basis, parameters, errors…)
evaluate the PES
1D or 2D cuts of the PES
2D cut R-optimized
rigid2D code: linear molecule + atom
rigid3D code: fragment + atom
rigid4D code: two linear molecules
rigid5D code: fragment + linear molecule
rigid6D code: two fragments

10. Automated construction of a PES
AUTOFIT package
AUTOFIT - PES
AUTOFIT - PLOT
summary (relevant info: basis, parameters, errors…)
evaluate the PES
1D or 2D cuts of the PES
2D cut R-optimized
rigid2D code: linear molecule + atom
rigid3D code: fragment + atom
rigid4D code: two linear molecules
rigid5D code: fragment + linear molecule
rigid6D code: two fragments
CO2 - dimer

11. Automated construction of a PES
AUTOFIT package
AUTOFIT - PES
AUTOFIT - PLOT
summary (relevant info: basis, parameters, errors…)
evaluate the PES
1D or 2D cuts of the PES
2D cut R-optimized
rigid2D code: linear molecule + atom
rigid3D code: fragment + atom
rigid4D code: two linear molecules
rigid5D code: fragment + linear molecule
rigid6D code: two fragments
CO2 - dimer
HC2NC – H2

12. Automated construction of a PES
AUTOFIT package
AUTOFIT - PES
AUTOFIT - PLOT
summary (relevant info: basis, parameters, errors…)
evaluate the PES
1D or 2D cuts of the PES
2D cut R-optimized
rigid2D code: linear molecule + atom
rigid3D code: fragment + atom
rigid4D code: two linear molecules
rigid5D code: fragment + linear molecule
rigid6D code: two fragments
CO – O2
extended angles
CO2 - dimer
HC2NC – H2

13. basic logic steps in the procedure
AUTOFIT-PES: How does it work..?
each pass through the loop adds another generation of data to refine the PES
The figure illustrate the basic logic steps in the procedure
basic logic steps in the procedure

14. AUTOFIT-PES: How does it work..?
# GENERAL INFORMATION:
CO-N ! System LABEL. Maximum 15 char. It must be at least one empty space after the LABEL
! [xunitE] PES units, Energy: 1=hartree, 2=kcal/mol, 3=cm-1, 4=meV
! [restart] 1=yes, 0=no
...
# FRAGMENTS INFORMATION:
! [natom1] number of atoms in fragment 1
O ! element label for atom 1, e.g. 'H1‘
# CODE CONTROL:
1d ! [acc](kcal/mol) accuracy target -> ( 1d-3 kcal/mol = 0.35 cm-1 = meV )
6d ! [E_range](kcal/mol) energy range of interest (above asymptote)
! [maxpoints] maximum number of high level ab intio data points

15. AUTOFIT-PES: How does it work..?
# GENERAL INFORMATION:
...
! [xseed] seed for the random number generator algorithm. 0 = Sobol sequence
# CODE CONTROL:
! [numpoints] number of high-level points in the seed-grid
internal coordinates

16. AUTOFIT-PES: How does it work..?
# CODE CONTROL:
...
! [code_flag] 1 = Gaussian, 2 = Molpro, 3 = CFOUR
! [ab_flag] 1 = single point energies; 2 = also gradients

17. AUTOFIT-PES: How does it work..?
# CODE CONTROL:
...
! [code_flag] 1 = Gaussian, 2 = Molpro, 3 = CFOUR
! [ab_flag] 1 = single point energies; 2 = also gradients
*** title
memory,750,m
GTHRESH,energy=1d-8
geomtyp=xyz
basis=cvqz-f12
{hf;maxit,1000}
{ccsd(t)-f12b,scale_trip=1,gem_beta=1.5d0;core}
molpro_energy =energy
---
sample input “header” (Molpro)

18. AUTOFIT-PES: How does it work..?
# FRAGMENTS INFORMATION:
O ! element label for atom 1, e.g. 'H1‘
C ! element label for atom 2, e.g. 'H2'
N ! ...
N ! ...
d0 ! mass of atom 1 (atomic units)
12.d ! mass of atom 2
d0 ! ...
0d ! atom 1 fragment 1 (atoms placed along z-axis), Cartesian coordinates (Angstroms)
0d ! ...
0d ! ...
0d ! atom 2 fragment 1 0d ! ...
d0 ! ...
0d ! atom 1 fragment 2 (atoms placed along z-axis), Cartesian coordinates (Angstroms)
0d ! ...
0d ! ...
...
*** title
memory,750,m
GTHRESH,energy=1d-8
geomtyp=xyz
basis=cvqz-f12
{hf;maxit,1000}
{ccsd(t)-f12b,scale_trip=1,gem_beta=1.5d0;core}
molpro_energy =energy
---
sample input “header” (Molpro)

19. local interpolating moving least squares (L-IMLS)
AUTOFIT-PES: How does it work..?
local interpolating moving least squares (L-IMLS)
𝑣 𝑗 𝑅, 𝜃 1 , 𝜃 2 ,𝜑 = 𝑘=1 𝑲 𝑙 1 =0 𝑳𝟏 𝑙 2 =0 𝑳𝟐 𝑚=0 min⁡(𝑳𝟏,𝑳𝟐) 𝑪 𝒌, 𝒍 𝟏 , 𝒍 𝟐 ,𝒎 𝒆 𝑘𝛼𝑅 𝑷 𝑙 1 𝑚 ( 𝜃 1 ) 𝑷 𝑙 2 𝑚 ( 𝜃 2 ) 𝐜𝐨𝐬⁡(𝑚𝜑)

20. AUTOFIT-PES: How does it work..?
The core of the automated procedure is the determination of new data locations by using two successive degrees of the fitting basis
determination of new data locations using two successive degrees of the fitting basis

21. AUTOFIT-PES: How does it work..?
The core of the automated procedure is the determination of new data locations by using two successive degrees of the fitting basis
𝑽 𝟐𝒏𝒅 𝐑 =𝛼+𝛽𝑹+𝛾 𝑹 𝟐
𝑽 𝟑𝒓𝒅 𝐑 =𝛼+𝛽𝑹+𝛾 𝑹 𝟐 +𝛿 𝑹 𝟑
determination of new data locations using two successive degrees of the fitting basis

22. AUTOFIT-PES: How does it work..?
The core of the automated procedure is the determination of new data locations by using two successive degrees of the fitting basis
𝑽 𝟐𝒏𝒅 𝐑 =𝛼+𝛽𝑹+𝛾 𝑹 𝟐
𝑽 𝟑𝒓𝒅 𝐑 =𝛼+𝛽𝑹+𝛾 𝑹 𝟐 +𝛿 𝑹 𝟑
determination of new data locations using two successive degrees of the fitting basis

23. AUTOFIT-PES: How does it work..?
The core of the automated procedure is the determination of new data locations by using two successive degrees of the fitting basis
𝑽 𝟐𝒏𝒅 𝐑 =𝛼+𝛽𝑹+𝛾 𝑹 𝟐
𝑽 𝟑𝒓𝒅 𝐑 =𝛼+𝛽𝑹+𝛾 𝑹 𝟐 +𝛿 𝑹 𝟑
determination of new data locations using two successive degrees of the fitting basis

24. AUTOFIT-PES: How does it work..?
The core of the automated procedure is the determination of new data locations by using two successive degrees of the fitting basis
𝑽 𝟐𝒏𝒅 𝐑 =𝛼+𝛽𝑹+𝛾 𝑹 𝟐
𝑽 𝟑𝒓𝒅 𝐑 =𝛼+𝛽𝑹+𝛾 𝑹 𝟐 +𝛿 𝑹 𝟑
determination of new data locations using two successive degrees of the fitting basis

25. AUTOFIT-PES: How does it work..?
# CODE CONTROL:
...
! [K] maximum power of R = exp(αr)
! [L1] maximum value of L1 in angular basis
! [L2] maximum value of L2 in angular basis
! [L] maximum value of L=L1+L2 in angular basis
𝑣 𝑗 𝑅, 𝜃 1 , 𝜃 2 ,𝜑 = 𝑘=1 𝑲 𝑙 1 =0 𝑳𝟏 𝑙 2 =0 𝑳𝟐 𝑚=0 min⁡(𝑳𝟏,𝑳𝟐) 𝑪 𝑘, 𝑙 1 , 𝑙 2 ,𝑚 𝒆 𝑘𝛼𝑅 𝑷 𝑙 1 𝑚 ( 𝜃 1 ) 𝑷 𝑙 2 𝑚 ( 𝜃 2 ) 𝐜𝐨𝐬⁡(𝑚𝜑)
high degree (K,L1,L2,L) = (6,6,6,6)  basis size: 301
lower degree (K,L1,L2,L) = (5,5,5,5)  basis size: 171
determination of new data locations using two successive degrees of the fitting basis

26. AUTOFIT-PES: How does it work..?

27. AUTOFIT-PES: How does it work..?
# CODE CONTROL:
...
3d ! [acc] (kcal/mol) accuracy target  ( 1d-3 kcal/mol = 0.35 cm-1 = meV )

28. Example: CO2 dimer GENERAL.dat ...
*** START AUTOMATIC DATA POINT GENERATION ***
accuracy target: 1.00E-03 kcal/mol (0.35 cm-1)
LOOP RMS MEAN DEV # points
E E
E E
E E
E E
E E
E E
E E
E E
E E
E E
E E
E E
E E
E E

29. Example: CO2 dimer GENERAL.dat ... Summary of errors: RMS1 = Global
RMS2 = below asymptote
RMS3 = 0.3 kcal/mol (~100 cm-1) above global minimum
Ri Rf <|V(R)|> RMS % RMS % RMS %
E E E E
E E E E
E E E E
E E E E
E E E E
E E E E

30. AUTOFIT-PES: How does it work..?
# GENERAL INFORMATION:
...
! [restart] 1 = yes, 0 = no
# CODE CONTROL:
3d ! [acc](kcal/mol) accuracy target -> ( 1d-3 kcal/mol = 0.35 cm-1 = meV )
the final PES can be improved
(globally or in any specific region of the configuration space)
up to high accuracy with very small remaining errors compared to the underlying reference data,
up to any desired accuracy …
the ultimate goal ,
since,
apart from the intrinsic limitations of the chosen reference electronic structure method

31. AUTOFIT-PES: How does it work..?
# GENERAL INFORMATION:
...
! [restart] 1 = yes, 0 = no
# CODE CONTROL:
3d ! [acc](kcal/mol) accuracy target -> ( 1d-3 kcal/mol = 0.35 cm-1 = meV )
! [focus] 0 = no, 1 = yes; if "focus=0", all energy range is considered
3.0d ! [increment] (kcal/mol) considered energies = asymptotic energy - "increment"; if "focus=1"
! [focus_onR] 0 = no, 1 = yes; if "focus_onR=0", all R-range is considered
9.0d ! [minR] minimum R considered; if "focus_onR=1" (after low- and seed-grid are computed)
15.0d ! [maxR] maximum R considered; if "focus_onR=1" (after low- and seed-grid are computed)
the methodology does not contain any approximations…
the final PES can be improved
(globally or in any specific region of the configuration space)
up to high accuracy with very small remaining errors compared to the underlying reference data,
up to any desired accuracy …
the ultimate goal ,
since,
apart from the intrinsic limitations of the chosen reference electronic structure method
… the goal is to obtain a PES that represents the ab initio data with such high-fidelity, that when used in dynamics or spectroscopic studies the results directly reflect the underlying level of the electronic structure calculations.

32. purely ab initio spectroscopy
Examples
CO2 - CS2
CO dimer
Experiment
Calculated
0.0000a
0.00
0.8772c
1.18
7.0768m
6.85 k
11.73
J=1, A+
2.654b
2.60
2.934d
3.06
J=1, A-
3.859e
3.48
JKa
Experiment
Calculated 00
41.880
41.918 11
44.441 22
44.329
44.277 33
48.573 44
46.039
45.942
Predictive quality :
0.29 cm-1 RMS
for 68 levels
Calc. is for pure intermolecular vibrations, while Exp. is for intermolecular + CO2 2 vibration
CCSD(T)-F12b/VTZ-F12
CCSD(T)-F12b(AE)/CBS
purely ab initio spectroscopy

33. first version will be released soon
SUMMARY
AUTOFIT package
rigid2D code: linear molecule + atom
rigid3D code: molecule + atom
rigid4D code: (vdW) two linear molecules
rigid5D code: molecule + linear molecule
rigid6D code: two molecules
AUTOFIT - PES
AUTOFIT - PLOT
summary (basis, parameters, errors…)
evaluate the PES
1D or 2D cuts of the PES
2D cut R-optimized
enable non-experts in PES fitting methods to bridge electronic structure calculations with spectroscopic and dynamics research
no human intervention
no system-specific modifications
designed to run in parallel
the final PES can be improved
first version will be released soon …
The goal is to enable a broad community of non-
experts in PES fitting methods to bridge electronic structure calculations with spectroscopic and dynamics research, making systematic studies of vdW systems much more a routine.

34. ACKNOWLEDGEMENTS Prof. Richard Dawes Dr. Steve Ndengué
Dr. Andrew Powell
Sangeeta Sur
Bradley Welch
CHE
DE-SC