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

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 …

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 …

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 })

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 })

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

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

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

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

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

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

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

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

AUTOFIT-PES: How does it work..? # GENERAL INFORMATION: -------------------------------------- CO-N2 ! System LABEL. Maximum 15 char. It must be at least one empty space after the LABEL 2 ! [xunitE] PES units, Energy: 1=hartree, 2=kcal/mol, 3=cm-1, 4=meV 0 ! [restart] 1=yes, 0=no ... # FRAGMENTS INFORMATION: ------------------------------------------ 2 ! [natom1] number of atoms in fragment 1 O1 ! element label for atom 1, e.g. 'H1‘ # CODE CONTROL: -------------------------- 1d-3 ! [acc](kcal/mol) accuracy target -> ( 1d-3 kcal/mol = 0.35 cm-1 = 0.043 meV ) 6d0 ! [E_range](kcal/mol) energy range of interest (above asymptote) 60000 ! [maxpoints] maximum number of high level ab intio data points

AUTOFIT-PES: How does it work..? # GENERAL INFORMATION: ------------------------------------- ... 0 ! [xseed] seed for the random number generator algorithm. 0 = Sobol sequence # CODE CONTROL: -------------------------- 500 ! [numpoints] number of high-level points in the seed-grid internal coordinates

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

AUTOFIT-PES: How does it work..? # CODE CONTROL: -------------------------- ... 2 ! [code_flag] 1 = Gaussian, 2 = Molpro, 3 = CFOUR 1 ! [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)

AUTOFIT-PES: How does it work..? # FRAGMENTS INFORMATION: ------------------------------------------ O1 ! element label for atom 1, e.g. 'H1‘ C1 ! element label for atom 2, e.g. 'H2' N1 ! ... N2 ! ... 15.9949146221d0 ! mass of atom 1 (atomic units) 12.d0 ! mass of atom 2 14.0030740052d0 ! ... 0d0 ! atom 1 fragment 1 (atoms placed along z-axis), Cartesian coordinates (Angstroms) 0d0 ! ... 0d0 ! ... 0d0 ! atom 2 fragment 1 0d0 ! ... 1.12820589486d0 ! ... 0d0 ! atom 1 fragment 2 (atoms placed along z-axis), Cartesian coordinates (Angstroms) 0d0 ! ... 0d0 ! ... ... *** 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)

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)

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

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

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

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

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

AUTOFIT-PES: How does it work..? # CODE CONTROL: ---------------- ... 6 ! [K] maximum power of R = exp(αr) 6 ! [L1] maximum value of L1 in angular basis 6 ! [L2] maximum value of L2 in angular basis 6 ! [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

AUTOFIT-PES: How does it work..?

AUTOFIT-PES: How does it work..? # CODE CONTROL: ---------------- ... 3d-3 ! [acc] (kcal/mol) accuracy target  ( 1d-3 kcal/mol = 0.35 cm-1 = 0.043 meV )

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 0 3.130934E-02 5.079335E-03 198 1 2.580193E-02 5.037868E-03 210 2 1.168961E-02 2.888773E-03 222 3 1.153931E-02 2.873410E-03 234 4 9.758163E-03 2.420913E-03 246 15 3.538952E-03 8.636190E-04 690 16 3.316861E-03 8.113605E-04 702 17 3.029152E-03 7.713979E-04 714 18 2.603664E-03 6.573480E-04 726 25 1.189262E-03 3.053100E-04 966 26 1.075628E-03 2.759447E-04 978 27 1.062809E-03 2.842090E-04 990 28 1.051966E-03 2.755224E-04 1002 9.644447E-04 2.562896E-04 1014

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)|> RMS1 % RMS2 % RMS3 % 3.00 3.60 1.45763E+00 6.293E-03 0.43 3.992E-03 0.27 4.062E-03 0.28 3.60 4.20 1.00336E+00 4.558E-03 0.45 3.302E-03 0.33 4.465E-03 0.45 4.20 4.80 5.95141E-01 2.920E-03 0.49 2.343E-03 0.39 6.389E-03 1.07 7.20 7.80 4.13953E-02 1.229E-04 0.30 1.229E-04 0.30 0.000E+00 0.00 7.80 8.40 2.38776E-02 8.173E-05 0.34 8.173E-05 0.34 0.000E+00 0.00 8.40 9.00 1.29081E-02 8.655E-05 0.67 8.655E-05 0.67 0.000E+00 0.00

AUTOFIT-PES: How does it work..? # GENERAL INFORMATION: -------------------------------------- ... 0 ! [restart] 1 = yes, 0 = no # CODE CONTROL: -------------------------- 3d-3 ! [acc](kcal/mol) accuracy target -> ( 1d-3 kcal/mol = 0.35 cm-1 = 0.043 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

AUTOFIT-PES: How does it work..? # GENERAL INFORMATION: -------------------------------------- ... 0 ! [restart] 1 = yes, 0 = no # CODE CONTROL: -------------------------- 3d-3 ! [acc](kcal/mol) accuracy target -> ( 1d-3 kcal/mol = 0.35 cm-1 = 0.043 meV ) 1 ! [focus] 0 = no, 1 = yes; if "focus=0", all energy range is considered 3.0d0 ! [increment] (kcal/mol) considered energies = asymptotic energy - "increment"; if "focus=1" 0 ! [focus_onR] 0 = no, 1 = yes; if "focus_onR=0", all R-range is considered 9.0d0 ! [minR] minimum R considered; if "focus_onR=1" (after low- and seed-grid are computed) 15.0d0 ! [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.

purely ab initio spectroscopy Examples CO2 - CS2 CO dimer Experiment Calculated 0.0000a 0.00 0.8772c 1.18 7.0768m 6.85 11.8743k 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

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 http://web.mst.edu/~dawesr/index.html 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.

ACKNOWLEDGEMENTS Prof. Richard Dawes Dr. Steve Ndengué Dr. Andrew Powell Sangeeta Sur Bradley Welch CHE-1566246 DE-SC0010616