A STUDY OF INTERPOLATING MOVING LEAST-SQUARES METHOD
ON FITTING POTENTIAL ENERGY SURFACE
_______________________________________
A Dissertation
presented to
the Faculty of the Graduate School
at the University of Missouri-Columbia
_______________________________________________________
in Partial Fulfillment
of the Requirements for the Degree
Doctor of Philosophy
_____________________________________________________
by
YI SHI, B.S.,
Dr. Donald L. Thompson, Dissertation Supervisor
JULY 2016
The undersigned, appointed by the dean of the Graduate School, have examined the dissertation
entitled
A STUDY OF INTERPOLATING MOVING LEAST-SQUARES METHOD
ON FITTING POTENTIAL ENERGY SURFACE
presented by Yi Shi,
a candidate for the degree of doctor of philosophy,
and hereby certify that, in their opinion, it is worthy of acceptance.
Professor Donald L. Thompson
Professor John E. Adams
Professor Shijie Chen
Professor Thomas D. Sewell
To My Parents
ACKNOWLEDGMENTS
I would like first to thank my advisor Professor Donald Thompson. Professor Thompson
gave me lots of guidance and help since I entered the university as a PhD graduate student. I
could not have reached this far without the support and guidance from Professor Thompson. I
would also like to thank my committee members, Professor John E. Adams, Professor Shijie
Chen and Professor Thomas D. Sewell as they provided me much help during my study here. I
also wish to thank my friends, colleagues, teachers, students and university staff who helped me
over the years. Especially I thank for Dr. Damien Bachellerie, Dr. Luis Rivera, Dr. Markus
Froehlich and Dr. John Clay who gave me much useful help in my research and helped the
revision of the dissertation.
ii
Table of Contents
PAGE
ACKNOWLEDGMENTS ................................................................................................................. ii
LIST OF TABLES ............................................................................................................................ v
LIST OF FIGURES .......................................................................................................................... vi
ABSTRACT ..................................................................................................................................... ix
1. INTRODUCTION ............................................................................................................................. 1
2. IMLS ON H2O2 ANALYTICAL SURFACE .................................................................................. 18
2.1 Introduction ................................................................................................................................ 18
2.2 Method and procedure ................................................................................................................ 19
2.3 Results and Discussion ............................................................................................................... 24
2.4 Summary and Conclusion........................................................................................................... 27
3. DYNAMICS-DRIVEN IMLS FITTING ........................................................................................ 34
3.1 Introduction ................................................................................................................................ 34
3.2 Methods ...................................................................................................................................... 36
3.3 Results and discussion ................................................................................................................ 37
3.4 Summary and conclusion ........................................................................................................... 40
4. APPLICATION OF IMLS FITTING FROM ANALYTICAL POTENTIAL ENERGY
SURFACE TO DFT POTENTIAL ENERGY SURFACE ON CO2 + O REACTION .................. 48
4.1 Dynamics-driven IMLS fitting on CO2 + O B3LYP/3-21G PES ............................................... 48
4.1.1 Introduction ...................................................................................................................... 48
4.1.2 Methods............................................................................................................................ 49
4.1.3 Results and discussion ..................................................................................................... 51
4.1.4 Conclusion and future work ............................................................................................. 55
4.2 IMLS fitting on CO2 + O B3LYP/3-21G PES of specified range .............................................. 56
4.2.1 Introduction ...................................................................................................................... 56
4.2.2 Methods............................................................................................................................ 56
4.2.3 Results and discussion ..................................................................................................... 59
4.2.4 Conclusion and future work ............................................................................................. 62
iii
4.3 An algorithm for configurations below cutoff energy in ab initio or DFT potential energy
surface......................................................................................................................................... 65
4.3.1 Introduction ...................................................................................................................... 65
4.3.2 Methods............................................................................................................................ 65
4.3.3 Results and discussion ..................................................................................................... 67
4.3.4 Conclusion and future work ............................................................................................. 71
5. FUTURE ......................................................................................................................................... 93
BIBLIOGRAPHY ........................................................................................................................... 95
VITA ............................................................................................................................................... 98
iv
LIST OF TABLES
Table
Page
4.1 The range of points below 70 kcal/mol of r1, r2 and r. ............................................................ 73
4.2 The range of r, α, θ and φ based on r1 and r2. The units are Å or °. The small φ angles
appeared when α is larger than 170°. The empty cell indicates that there are points
below 70 kcal/mol in the range. .............................................................................................74
4.3 The range of r based on the range of r1 and r2. ....................................................................... 75
4.4 The comparison of a. points below 70 kcal/mol calculated in the dense grid and b.
points below 60 kcal/mol sought by the algorithm for r1 = 1.04 Å, r2 = 1.125 Å, r = 1.8
Å and α = 147.8° and 155.6°. The values of r2, r and α of b are close to those of a but
not the same. .......................................................................................................................... 76
4.5 The comparison of a. points below 70 kcal/mol calculated in the dense grid and b.
points below 60 kcal/mol sought by the algorithm for r1 = 1.0 Å and r2 = 1.125 Å and
r1 = 1.04 Å and r2 = 1.125 Å. The values of θ are those for most α values. The values
of φ are those for most α and θ values. The values of r2 and r of b are close but not the
same. ...................................................................................................................................... 77
4.6 The comparison of a. points below 70 kcal/mol calculated in the dense grid and b.
points below 60 kcal/mol sought by the algorithm for r1 = 1.08 Å and r2 = 1.08 Å. The
values of φ are those for most α and θ values. The values of r2 and r of b are close but
not the same. .......................................................................................................................... 78
4.7 The comparison of a. points below 70 kcal/mol calculated in the dense grid and b.
points below 60 kcal/mol sought by the algorithm for r1 = 1.08 Å and r2 = 1.125 Å.
The values of φ are those for most α and θ values. The values of r2 and r of b are close
but not the same. .................................................................................................................... 79
4.8 The comparison of a. points below 70 kcal/mol calculated in the dense grid and b.
points below 60 kcal/mol sought by the algorithm for r1 = 1.125 Å and r2 = 1.125 Å.
The values of φ are those for most α and θ values. The values of r2 and r of b are close
but not the same. .................................................................................................................... 80
4.9 The CPU time of the works in current study .......................................................................... 81
v
LIST OF FIGURES
Figure
Page
2.1 The H2O2 valence coordinates and ordered bond distance coordinates. ................................. 29
2.2 The RMS errors of (a) energy and (b) gradients with respect to the number of data
2
points. The red is IMLS with weight function
weight function
e-x
x4 +10-14
e-x/d(i)
(x/d(i)2 )4 + 10-14
. The blue is IMLS with
. The brown is IMLS with weight function
2
2.3 The weight functions (a)
e-x/d(i)
with d(i)2 = 10; (b)
-14
(x/d(i)2 )4 + 10
1
(x + 10-4 )2
e-x
x4 +10-14
; (c)
. .......................30
1
(x + 10-4 )2
. The
function are Eq. (10), Eq. (11), Eq. (12) with d2(i) = 10 where x2 in equations is x in the
plots. ........................................................................................................................................ 31
2.4 The RMS errors of (a) energy and (b) gradients with respect to number of data points
with weight function
e-x/d(i)
2
(x/d(i)2 )4 + 10-14
....................................................................................... 32
2.5 Potential energy contour and surface of rOO (2 – 7 Å) and τ (0 - 180°) of the H2O2 with
r1,2 (1.8175 bohr) θ1 (99.76 °), θ2 (80.24 °). (a) H2O2 analytical surface contour. (b)
IMLS contour. (c) the surface of H2O2 analytical surface. ..................................................... 33
3.1 The order of internuclear distance in HONO. ......................................................................... 42
3.2 The total number of PES evaluations vs ab initio point calculated in the trajectories. .......... 43
3.3 cis – trans HONO isomerization reaction. .............................................................................. 44
3.4 The total energy of the HONO isomerization trajectory. The direct dynamics trajectory
(red); the trajectory from IMLS fitted PES with 685 data points (green); the trajectory
from IMLS fitted PES with 2187 data points (blue). .............................................................. 45
3.5 The total angular momentum of components x, y and z for the trajectory computed
from the direct dynamics trajectory, IMLS PES with 685 ab initio points and IMLS
PES with 2817 ab initio points. .............................................................................................. 46
3.6 The isomerization rate from 100 trajectories from (a) the direct dynamics trajectories;
(b) IMLS PES using 685 data points; (c) IMLS PES using 1638 data points. The rates
are ka = 0.59 ps-1, kb = 0.68 ps-1, kc = 0.51 ps-1. ...................................................................... 47
vi
4.1 Spherical coordinates and the order of internuclear coordinates. ........................................... 82
4.2 The total number of PES evaluations in the sampling trajectories with respect to the
number of DFT calculations. .................................................................................................. 83
4.3 Nonreactive scattering and atom exchange reactions. ............................................................ 84
4.4 Opacity functions for CO2 + O atom exchange reaction at Ecoll = 57.7 kcal/mol. The red
line is from B3LYP/3-21G trajectories. The brown line is the error bar for opacity
functions from B3LYP/3-21G trajectories. The blue line is from trajectories from IMLS
potential. The black line is the error bar for the opacity functions from the IMLS PES.. ...... 85
4.5 Opacity functions for CO2 + O atom exchange reaction at Ecoll = 23.1 kcal/mol. The red
line is from B3LYP/3-21G trajectories. The brown line is the error bar for opacity
functions from B3LYP/3-21G trajectories. The blue line is from trajectories from IMLS
potential. The black line is the error bar for the opacity functions from the IMLS PES. ....... 86
4.6 Opacity functions for CO2 + O atom exchange reaction at Ecoll = 34.6 kcal/mol. The red
line is from B3LYP/3-21G trajectories. The brown line is the error bar for opacity
functions from B3LYP/3-21G trajectories. The blue line is from trajectories from IMLS
potential. The black line is the error bar for the opacity functions from the IMLS PES.. ...... 87
4.7 Opacity functions of direct dynamics trajectories of B3LYP/3-21G for CO2 + O atom
exchange reaction at Ecoll = 23.1, 34.6 and 57.7 kcal/mol. The red line is for Ecoll = 57.7
kcal/mol. The brown line is the error bar for Ecoll = 57.7 kcal/mol. The blue line is from
trajectories for Ecoll = 34.6 kcal/mol. The black line is the error bar for Ecoll = 34.6
kcal/mol. The red line is from trajectories for Ecoll = 23.1 kcal/mol. The brown line is
the error bar for Ecoll = 23.1 kcal/mol. .................................................................................... 88
4.8 The total energy of CO2 + O trajectory from direct dynamics and IMLS PES with 992
data points. .............................................................................................................................. 89
4.9 The RMS errors with respect to the number of data points. The energy is in unit of
kcal/mol. The gradients is in unit of kcal/(mol*Å). The bottom lines are RMS errors of
energy. The top lines are RMS errors of gradients.. ............................................................... 90
4.10 The chart of the algorithm for the search of points below cutoff energy. The starting
point for the search is r1,3 1.12 Å, r2,3 1.09 Å, r3 1.82 Å, α3 160°, θ3 60°, φ3 260°. A new
value from ii is given to i after a circle of i. This rule applies to iii, iv, v, vi. ........................ 91
4.11 The number of total points below 60 kcal/mol vs the number of data points selected
from the points below 60 kcal/mol in the search starting from r1 1.12 Å, r2 1.09 Å, r
vii
1.82 Å, α 160°, θ 60°, φ 260° with dr1 0.01 Å, dr2 0.01 Å, dr 0.01 Å, dα 10°, dθ 10°, dφ
20°. .......................................................................................................................................... 92
viii
ABSTRACT
Interpolating moving least-squares method (IMLS) is a highly accurate fitting method. There
are a lot of studies on how to use IMLS to fit potential energy surfaces (PESs). However, there
are still only few IMLS-fitted PESs for 4-atom systems based on data calculated at electronic
structure method. The purpose of the current study is to find more efficient ways to use IMLS to
fit ab initio or DFT PES of 4-atom. Three methods have been investigated to fit the PES of CO2
+ O using points computed at B3LYP/3-21G level. Dawes et al. developed the dynamics-driven
IMLS method by fitting HONO at the HF/cc-pVDZ level. [R. Dawes, A. F. Wagner and D. L.
Thompson, J. Phys. Chem. A, 113, 4709 (2009).] In current study, the dynamics-driven IMLS
method is used to fit the CO2 + O PES for collision energy below 57.7 kcal/mol for 992 data
points. There is good agreement between opacity functions at three collision energy below 57.7
kcal/mol calculated using classical trajectories with the IMLS potential and those from direct
dynamics trajectories. Since ab initio or DFT calculation is only performed for the data point
calculation, the method can be used to fit PES of highly accurate theoretical method only by
changing the ab initio or DFT method. In the second method, grids of specified range of
CO2 + O PES are calculated. A new range is determined to include points below 70 kcal/mol.
The RMS error of energy and gradients are 0.19 kcal/mol and 13 kcal/(mol Å) respectively with
528 data points selected from the range by random points. In the third method, an algorithm is
developed to search for all the points below the cutoff energy 60 kcal/mol. The results show that
the algorithm successfully finds the points below the cutoff energy. The algorithm is used to fit
the points of energy below 60 kcal/mol. The points below 60 kcal/mol are fitted using 1287 DFT
points. However, there are many DFT calculations required in the last two methods besides the
ix
data point calculation. Much more work is needed to develop these two methods in order to fit
PES of highly accurate theoretical method.
x
CHAPTER 1
INTRODUCTION
The rules of the molecule behaviors can be fully understood with accurate potential energy
surface (PES) of the system. Accurate fitting of electronic structure theory data points is of prime
interest to theoretical chemists. Several research groups have made a lot of effort to develop
practical and accurate methods. Though PES of highly accurate theoretical method for system
with more than four atoms have been fitted, the latest developments of fitting PES of highly
accurate theoretical method is still mainly at the size of 4-atom molecule concerning the
accuracy of the PES and the cost of the calculation of the data points for the fitted PES. This is
due that the number of data points needed for fitting a PES increases rapidly as the size of the
molecule increases. Therefore, fitting a PES of more than four atoms can be expected if there is a
major advancement of the power of the calculation of computer technology. This chapter
summarizes the different PES fitting methods and their progress.
Analytical functions were used in early potential energy surface fitting. Bentley 1 fit the
interaction of Ne with a rigid H2O molecule. The fit was performed based on 106 points using an
analytical function involving Legendre polynomials. The energy of the points on excited states
was obtained using configuration interaction calculation. They fit the surface to interpret the
experiments involving Ne* with H2O. Schinke et al.2 fitted the configuration interaction (CI)
surface for H3+ system using analytical functions including inverse powers of the Ri. The
analytical functions was fit to 650 energy values calculated using complete CI calculations in
space by 5s and 3p-type Gaussian basis functions. The average deviation from the CI energies
was only 0.16 kcal/mol over a range of 35 eV. They calculated the angle dependence of the
1
probabilities for the vibrational transitions and the differential cross sections using this potential.
These results were in good agreements with experimental data. Polanyi and Schreiber 3 used an
London-Eyring-Polanyi-Sato (LEPS) functional form to fit the F + H2 surface. Truhlar and
Horowitz4 fit the H3 surface using analytical functions including London potential using 267 CI
points calculated by Liu and Siegbahn.5 Vance and Gallup6 developed analytical functions as a
sum of terms of atomic energy, two-body and three-body terms for three-atom surface as
analytical functions and tested using the analytical functions by fitting 36 CI data points of H3.
Many efficient methods have been developed for potential energy surface fitting from then
on. Spline functions have been used for potential surface fitting, especially cubic splines. A onedimensional spline, S(x), is a particular set of polynomial functions of degree m defined xi ≤ x ≤
xi+1, i = 1, …, n - 1. The spline functions should pass through all the data points f(x1), f(x2), …
f(xn). All the derivatives up to the (m - 1)st should be continuous at xi, i = 1, 2, …, n. The cubic
spline has the form
𝑆𝑖 (𝑥) = 𝑆 ′′ (𝑥𝑖 )
𝑥2
𝑆 ′′ (𝑥𝑖+1 ) − 𝑆 ′′ (𝑥𝑖 ) (𝑥 − 𝑥𝑖 )3
+(
) + 𝛼𝑖 𝑥 + 𝛽𝑖 ,
2
𝑥𝑖+1 − 𝑥𝑖
6
(1.1)
where i = 1, 2, ⋯, n.
The cubic splines are used piecewise as shown in Eq. (1.1). There are n – 1 cubic splines
connecting (xi, xi+1) with i = 1, 2, …, n – 1. The coefficients in Eq. (1.1) are second derivatives
S’’(xi), S’’(xi+1), αi, βi. The coefficients can be obtained by requiring the functions pass through
all the data points as Eq. (1.2)
𝑆𝑖 (𝑥𝑖 ) = 𝑓(𝑥𝑖 ), 𝑖 = 1, 2, ⋯ 𝑛.
(1.2)
and the first derivatives be continuous at the data points as Eq. (1.3) and
2
′
(𝑥𝑖 ) = 𝑆𝑖′ (𝑥𝑖 ), 𝑖 = 2, 3, ⋯ , 𝑛 − 1.
𝑆𝑖−1
(1.3)
setting up end conditions; for example, specifying the first or second derivatives at x1 and xn. The
high-dimensional S(x) can be calculated with the similar procedure. McLaughlin and Thompson7
used spline interpolating polynomials to fit the reaction HeH+ + H2 → He + H3+. Three degrees
of freedom that were fitted in the reaction were the internuclear distance of H2, the HeH+
internuclear separation and the center of mass of H2 and the hydrogen atom of HeH+. They fit
300 C2v ab initio points. They fit a 10×10 grid on two degrees of freedom using 2-dimensional
cubic splines. They used three-point spline fit for the H2 distance. The majority of points for the
10×10 grid are below 5 a.u.. The remainder of the points is at 50 a.u. for two coordinates. They
calculated the classical trajectories using the spline-fitted PES to study the vibrational energy
partitioning in the reaction products. Sathyamurthy and Raff 8 used cubic splines to fit
one-dimensional
potentials,
the
Morse
two-dimensional
analytical
potential
potential
for
and
collinear
the
D-Cl-H
Lennard-Jones
and
potential,
three-dimensional
diatomics-in-molecules (DIM) potential for He-H2+ and valence bond potential for D-Cl-H. It
was shown that the one-dimensional cubic spline fit was very accurate while 2-D and 3-D fit
were substantially less accurate. The trajectory from a (15×15×15) cubic spline fit would not
give a point-to-point match to trajectories computed using the analytical PES. However, the total
reaction cross sections, energy partitioning distribution, and spatial scattering distributions from
the quasiclassical trajectories using the spline fit PES are in good agreement with those
computed using the analytical potentials. Sathyamurthy et al. 9 did time-dependent quantum
mechanical (QM) scattering calculations using the analytical potential and (15×15) splinefitted
PES. The analytical PES was analytical functions parameterized as a hypothetical analog of
H2 + H exchange reaction by Kellerhals et al10.. They found excellent agreement of the results
3
computed using the two potentials. The error of the time-dependent QM calculations was less
than quasiclassical trajectory (QCT) calculations from the splinefitted potential. Sathyamurthy et
al.11 did a 2D cubic spline fit of ab initio data to understand that there was no enhancement of
reaction probability by selective partitioning of the energy into initial H2+ from a DIM fit to ab
initio data for He + H2+ → HeH+ + H reaction which was in contrast with experiment. A
significant enhancement was found from the QCT computed using the splinefitted potential.
They found it was the small difference between the DIM potential and the splinefitted potential
that contributed to the difference of reaction probability from the two potentials. They further
found out that the translation-vibration energy transfer and the reaction probability depended on
the inner wall of the surface by directing another study on the DIM surface and splinefitted
surface. 12 Sathyamurthy and Raff 13 used 3-D cubic splines to fit the interaction between
CO2 + H2. The fit was done on a (13×9×9) grid in three coordinates R, distance between C and
the center of H2, θ and φ, angles between CO2 and the vector C to the center of H2, between H2
and the vector and included 1053 points. An analytical potential of CO2 and a Morse potential
for H2 were used. They studied vibrational energy transfer and the deexcitation probability of
CO2*(0, 0, 1, 0) + H2 → CO2(0, 0, 0, 0) + H2 using the fitted potential. The deexcitation
probability and the computed isotope ratios from QCT are in a good agreement with
experimental data. Wu et al.14 fit the potential energy surface of H + H2O → OH + H2 reaction.
The system was specified as HaOHbHc. They used cubic spline functions to fit three variables,
O-Ha
distance,
Ha-Hb
distance
and
OHbHc
angle.
A
grid
(14×18×21)
of
(7o, 9e)-CAS+1+2/aug-pvdz points, 5292 points in total were fitted. Analytical functions, like
the Morse function, power functions have been used to represent other coordinates of the
4
molecule. The cross sections, rate coefficients, product energy, angular and alignment results
from the QCT calculations are in good agreement with experiments.
Braams and Bowman
15
developed a method for global surface fitting based on
permutationally invariant polynomial basis. They used an analytical equation Eq. (1.4) to
represent the potential energy surface. Equation (1.4) was used for a four-atom molecule. The
specific representation of the polynomial basis for molecules consisting of other number of
atoms can be derived from this equation.
𝑀
𝑓
𝑎 𝑏 𝑐 𝑑 𝑒
𝑉 = ∑ 𝐶𝑎𝑏𝑐𝑑𝑒𝑓 [𝑦12
𝑦13 𝑦14 𝑦23 𝑦24 𝑦34 ] ;
(𝑚 = 𝑎 + 𝑏 + 𝑐 + 𝑑 + 𝑒 + 𝑓),
(1.4)
𝑚=0
The variable yij are internuclear distances or the Morse variables, exp(-rij/a) and rij is the
internuclear distance in the Morse variables. The summation over all powers of yij in one
monomial should not be larger than M. The M could be determined based on the system. The
monomial y12ay13by14cy23dy24ey34f was symmetrized. The monomial would be symmetrized
differently for molecules with different symmetries. The property of the symmetrized monomial
was that there would be no change for Eq. (1.4) after permutation of identical atoms in the
molecule. A database of electronic energies was calculated. The data points were calculated from
trajectories or in the vicinity of the stationary points. A least squares fit was used to fit the
scattered data points using Eq. (1.4). A simple weighting was used on the data. The weighting is
varied for different systems. The expansion coefficients in Eq. (1.4) could then be obtained from
the least squares fit. The permutation invariant polynomial method has been used to fit global
potential energy surfaces for different systems by Joel Bowman’s group and other groups.
Conforti et al. 16 fit global potential energy surface for H2O(1A1) + O(3P) collisions at
5
CASSCF(10, 8)+MP2/TZV level. The three lowest triplet electronic states were state averaged.
Therefore, the fitted potential took the consideration of the three electronic states. Approximately
110 000 CASSCF(10, 8)+MP2/TZV points were calculated for the potential. The M = 7
polynomial including 500 terms was used to fit the database. The root mean square deviation
(RMSD) was less than 3 kcal/mol for energy below 50 kcal/mol and less than 10 kcal/mol below
150 kcal/mol. Both hydrogen abstraction and elimination reactions were included in the fitted
potential energy surface. They performed QCT calculations from the fitted PESs. The cross
sections for HO + HO products and H + OOH products were both lower than B3LYP direct
dynamics cross sections from previous study. The H + OOH results were in good agreements
with molecular beam experiments for the relative cross section. Li et al. 17 fit ~48 000
ROHF-UCCSD(T)-F12b/AVTZ points for HO + CO → CO2 + H reaction using permutation
invariant polynomial. The fitted PES is improved from the original PES of the system by making
some necessary adjustments. The fitted PES is named as CCSD-2/d PES as the original PES is
CCSD-1/d. The RMSD of the fitted surface CCSD-2/d was 1.09 kcal/mol. The rate coefficients
from the calculated total capture and integral cross sections from the QCT calculations of
CCSD-2/d were in reasonably good agreement with experimental values in both low and high
pressure limits. There were still significant differences between the experimental translational
and angular distributions and the calculated results from the QCT method. The calculated
product angular distributions also failed to reproduce the observed forward bias. Homayoon et
al.18 fit the global potential energy surface of CH3NO2 using 114 000 at UB3LYP/6-311+G(d,p)
and CASSCF(8,8)/6-311+G(d,p) levels. CASSCF(8,8)/6-311+G(d,p) is a supplemental method
to UB3LYP/6-311+G(d,p). It is used to some asymptotic region of CH3O + NO. The weighted
root-mean square error is 0.56 kcal/mol. They studied the decomposition of nitromethane from
6
the QCT calculations from the fitted potential. They observed the roaming pathway to form
CH3O + NO. Fu et al.19 fit the two low-lying doubly states (D0 and D1) of the global potential
energy surface of NO3. The database consisted of 90 000 ab initio points at (MS)-CAS(17e,
13o)PT2/aug-cc-pVTZ levels. The weighted RMS errors of fitted D0 and D1 surfaces are 2.39
kJ/mol and 2.59 kJ/mol respectively for energies up to 370 kJ/mol relative to the global
minimum of D0 surface. Their QCT calculations from the fitted surface supported the roaming
pathway of NO3 photodissociation.
Ischtwan and Collins 20 raised a method to use modified Shepard method to fit potential
energy surface as Eq. (1.5).
1
𝑉[𝝆(𝐑); 𝑖] = 𝑉[𝐑(𝑖)] + [𝝆 − 𝝆(𝑖)]𝑇 𝐆𝜌 (𝑖) + 2 [𝝆 − 𝝆(𝑖)]𝑇 𝐅𝜌 (𝑖)[𝝆 − 𝝆(𝑖)] + ⋯,
(1.5)
The point R is near the point R(i). The R is the interatomic distances. The ρ is the inverse of the
interatomic distances R. The G(i) and F(i) are the gradients and matrix of second derivatives at
R(i), respectively. The energy of R is calculated based on the energy V, first and second
derivatives G(i) and F(i) of point R(i).
𝑁𝑑
𝑉(𝐑) = ∑ 𝑤𝑖 (𝐑)𝑉[𝝆(𝐑); 𝑖]
(1.6)
𝑖=1
The Eq. (1.6) is a weighted sum of all terms by substituting R(i) in Eq. (1.5) using the Nd data
points. The wi(R) is the weight function with respect to R(i). The Nd data points will be selected
in the reaction path. (1). A large number of classical trajectories will be calculated using Eq. (1.6)
based on Nd data points. (2). Weights will be assigned to the NT points in the trajectories. The
weights of NT trajectory points will be large if they are close to the NT points and away from the
7
Nd data points. Points with largest weights will be added to the Nd data points. Weights will be
assigned again to the left trajectory points with the new data points. (3). Points with largest
weights will be again added to the data points. Step (2) and (3) will continue until Nextra data
points are added. Then step (1) to (3) will repeat. The procedure to add data points will continue
until an observable trajectory average converges. Jordan et al.21 used this method to fit OH + H2
→ H2O + H reaction at UHF/6-311++G(d,p) level. There were 457 data points with energy and
derivatives selected for the fitting. The probabilities of reaction OH + H2 → H2O + H and
fragmentation converged with 457 data points. The fragmentation indicated to the fragmentation
of the reactants OH or H2. Castillo et al.22 fit the H + N2O → OH + N2 reaction. The fit used
1100 data points including energy, first and second derivatives at QCISD(T)/cc-pVTZ levels.
The total cross sections at low collision energy and the OH state-resolved differential cross
sections from the QCT calculations using the fitted potential are in a good agreement with
experiment data. Zhou et al.23 fit H + CH4 → H2 + CH3 reaction. They used QCT to select 30
000 UCCSD(T)/aug-cc-pVTZ energies and UCCSD(T)/6-311++G(3df,2dp) derivatives and
Hessians for the fit. Satisfactory agreement was achieved between the previous PES and the
present PES on the total reaction probabilities and integral cross sections for H + CH4 →
H2 + CH3 reaction. Collins et al. 24 fit the diabatic potential energy matrix (DPEM) for three
lowest electronic states of OH3. The DPEM which couples nuclear motion in different adiabatic
states could be used to study chemical processes involving more than one electronic state and
nonadiabatic dynamics. The energy, derivatives and hessians of 2983 points were calculated at
MRCI/6-311++G(2df,2pd) levels. The DPEM of these points were then constructed from the
energy, derivatives and hessians. The difference of data point selection comparing to their
previous fitting study was that the surface-hoping trajectories were used in the fitting. The
8
geometry and energy of asymptotes from the DPEM agreed well with experimental data. They
used the DPEM to study the possible conical intersections. Good agreements were observed
between classical simulations using this DPEM and cross molecular beam experiments.
Hollebeek et al.25 systematically discussed reproducing kernel Hilbert space (RKHS) method
for constructing potential energy surface. The f(x) is a function of a multivariate variable x.
There exists a reproducing kernel Q(x, x’) with the following properties:
𝑓(𝐱) = 〈𝑓(𝐱 ′ ), 𝑄(𝐱, 𝐱 ′ )〉′
(1.7)
𝑄(𝐱, 𝐲) = 〈𝑄(𝐱, 𝐱 ′ ), 𝑄(𝐲, 𝐱 ′ )〉′
(1.8)
and
This is reproducing property of the function f(x). The prime (‘) indicates that the inner product
(<,>’) is performed over x’. The interpolation problem using RKHS is reduced to solve the linear
inverse problem:
𝑓(𝐱 𝒊 ) = 〈𝑓(𝐱 ′ ), 𝑄(𝐱 𝒊 , 𝐱 ′ )〉′
𝑖 = 1, ⋯ , 𝑀.
(1.9)
The f(xi) (i = 1, …, M) on the left hand side of Eq. (1.9) is M data points. The problem is to find
the function f(x) on the right hand side of Eq. (1.9). They listed different types of reproducing
kernel Q(x, x’). These Q(x, x’) has specific function forms. The f(x) can be solved by
substituting these Q(x, x’) into Eq. (1.9). They showed that the RKHS could be used not only
with a D-dimensional grid of data points but also scattered data points. They showed three
examples of potential energy surface fitting using RKHS. The first two are two fit of
two-dimensional interaction He-CO (57 MP4 data) and Ar-OH (87 the coupled electron pair
approximation data) using the RKHS method. The third one is the fit of three two-body terms
9
and one three-body term in many-body expansion of O(D1) + H2 reactive system. Both the
classical and QM calculations from the RKHS potential were in a qualitative agreement with
experimental results. Ho and Rabitz26 presented a new potential energy surface fitting method,
the RKHS-HDMR method. The high dimensional model representation (HDMR) of an arbitrary
M-dimensional function, as shown in Eq. (1.10).
𝑀
𝐼𝑀 = ∏(𝑃𝑚 + (𝐼1 − 𝑃𝑚 ))
𝑚=1
𝑀
𝑀
𝑀
𝑀
= ∏ 𝑃𝑚 + ∑ (𝐼1 − 𝑃𝑚 ) ∏ 𝑃𝑛 + ∑ ∑ (𝐼1 − 𝑃𝑚 ) (𝐼1 − 𝑃𝑛 ) ∏ 𝑃𝑙 + ⋯
𝑚=1
𝑚=1
𝑛≠𝑚
𝑚=1 𝑛=𝑚+1
𝑀
𝑙≠𝑚,𝑛
𝑀
+ ∑ 𝑃𝑚 ∏(𝐼1 − 𝑃𝑛 ) + ∏(𝐼1 − 𝑃𝑚 ) ,
𝑚=1
𝑛≠𝑚
(1.10)
𝑚=1
The IM, I1 are M-dimensional and 1-D identity operator. The Pm (m = 1, 2, …, M) are 1D
interpolation operators defined as Eq. (1.11).
𝐾𝑚
𝜇
𝑃𝑚 𝑓(𝐱) = ∑ 𝑞𝑚 (𝑥𝑚 , 𝑥𝑚 )𝜆𝜇𝑚 (𝐱/𝑥𝑚 ),
𝑚 = 1,2, ⋯ , 𝑀.
(1.11)
𝜇=1
The f(x) can be written as Eq. (1.12) for regular distribution data.
𝐾1 𝐾2
𝐾𝑀
𝑀
𝜇
𝑓(𝑥1 , 𝑥2 , ⋯ , 𝑥𝑀 ) = ∑ ∑ ⋯ ∑ 𝜆𝜇1 ,𝜇2,⋯,𝜇𝑀 ∏ 𝑞𝑚 (𝑥𝑚𝑚 , 𝑥𝑚 )
𝜇1 𝜇 2
𝜇𝑀
(1.12)
𝑚=1
qm(xmμm, xm) is one-dimensional reproducing kernel. xmμm are the data points, xm is a variable.
λμ1,μ2,…,μM are the coefficients. The qm and λμm in Eq. (1.11) can be understood the same way as
those in Eq. (1.12). The Eq. (1.11) satisfies the interpolation conditions Eq. (1.13).
10
𝜈
𝜈 = 𝑓(𝐱/𝑥𝑚 , 𝑥𝑚 ),
𝑃𝑚 𝑓(𝐱)|𝑥𝑚=𝑥𝑚
𝜈 = 1, 2, ⋯ , 𝐾𝑚 .
(1.13)
The x/xm is (x1, x2, …, xm-1, xm+1, …, xM). The Eq. (1.10) can be further written into Eq. (1.14)
through orthogonal decomposition.
𝑀
𝐼𝑀 =
𝐿∗𝑥
+∑
𝑀
𝐿∗𝐱/𝑥𝑖1
𝑖1=1
+
𝑀
+∑
𝑀
𝐿∗𝐱/𝑥𝑖1 /𝑥𝑖2
∑
𝑖1=1 𝑖2=𝑖1+1
𝑀
+∑
∑
𝑀
∑ 𝐿∗𝐱/𝑥𝑖1 /𝑥𝑖2 /𝑥𝑖3 + ⋯
𝑖1=1 𝑖2=𝑖1+1 𝑖3=𝑖2+1
𝐿∗𝐱/𝑥𝑖1 /𝑥𝑖2 /⋯/𝑥𝑖𝛼 + ⋯ + 𝐿∗𝐱/𝑥𝑖1 /𝑥𝑖2 /⋯/𝑥𝑖𝑀
∑
(1.14)
1≤𝑖1<𝑖2<⋯<𝑖𝛼≤𝑀
where
𝐿∗𝐱 = 𝐿𝐱 ,
𝐿∗𝐱/𝑥𝑖1 = 𝐿𝐱/𝑥𝑖1 − 𝐿∗𝐱 ,
𝐿∗𝐱/𝑥𝑖1 /𝑥𝑖2 = 𝐿𝐱/𝑥𝑖1 /𝑥𝑖2 − {𝐿∗𝐱/𝑥𝑖1 + 𝐿∗𝐱/𝑥𝑖2 + 𝐿∗𝐱 },
𝐿∗𝐱/𝑥𝑖1 /𝑥𝑖2 /⋯/𝑥𝑖𝛼 = 𝐿𝐱/𝑥𝑖1 /⋯/𝑥𝑖𝛼 −
𝐿∗𝐱/𝑥𝑖
∑
0≤𝑘≤𝛼−1
1≤𝜅1 <𝜅2 <⋯<𝜅𝑘 ≤𝛼
𝜅1 /⋯/𝑥𝑖𝜅𝑘
,
𝛼 = 3, ⋯ , 𝑀.
(1.15)
𝑀
𝐿𝐱 = ∏ 𝑃𝑚 ,
𝑚=1
𝐿𝐱/𝑥𝑖1 /𝑥𝑖2 /⋯/𝑥𝑖𝛼 =
∏
𝑃𝑚 .
(1.16)
1≤𝑚≠𝑖1 ,𝑖2 ,⋯,𝑖𝛼 ≤𝑀
The IM(0) is Lx* = Π Pm (m =1, …, M) in Eq. (1.10). The IM(α) includes terms until Σ L*x/xi1/xi2/…/xiα.
All the terms after in Eq. (1.14) are truncated. Both IM(0) and IM(α) are approximation of IM. The
successive decomposition techniques are used for the HDMR schemes. As in a 3-atom potential,
11
the truncated second-order projector I3(2) (P11 + P21 + P31 - P11P21 – P21P31 – P31P11 + P11P21P31)
can be used to the potential f(x1, x2, x3). The P11, P21, P31f(x) will leave 2D function. The
truncated first-order projector I2(1) ( Pj2 + Pk2 – Pj2Pk2) can be used to the 2D functions to reduce
them to 1D functions. Further I1(0) is used to 1D functions to reduce them to analytical
representations with coefficients that can be obtained using the grid data points. They used
two-level HDMR decomposition scheme and full grid tensor-product RKHS scheme to fit 980
and 1748 ab initio points of C(1D) + H2, respectively. The two-level HDMR decomposition
scheme used I2(1) to reduce the 3-D function to 1-D function. It then used P1(2), P2(2), P3(2) on the
1-D functions. The RMS errors were 0.25 and 0.5 kcal/mol for the two schemes. The example
showed that the HDMR scheme can fit data points not necessarily in a full grid while it achieved
an accuracy close to the tensor-product scheme.
Neural networks have also been used to fit potential energy surfaces. Blank et al.27 creaated a
method using neural network to fit global potential energy surface. A two-layer, feed-forward
neural network was used to illustrate the method for surface fitting. There are an input layer, a
hidden layer, an output layer and a bias node connected to the hidden layer and output layer. The
input layer has the coordinates of the potential. The yi will be calculated first as the weighted
sum as Eq. (1.17) at the node i in the hidden layer.
𝑠
𝑦𝑖 = 𝑤𝑖0 + ∑ 𝑤𝑖𝑗 𝑎𝑗 .
(1.17)
𝑗=1
The value wi0 is a constant to node i from the bias node, aj are the coordinates of the potential to
node i and wij is weight associated with aj at node i. There are s coordinates of the potential. The
g(yi) will be calculated using a sigmoidal function Eq. (1.15) at the node i in the hidden layer.
12
𝑔(𝑦𝑖 ) = (1 + exp(−𝑦𝑖 ))−1 .
(1.15)
The transfer function in the single node in the output layer is a linear transformation of the
summed g(yi) from the nodes in the hidden layer and another input from the bias node to obtain
the energy. A training algorithm will be performed on the neural network. The weights in the
neural network will be optimized through the training algorithm. So the neural network will
predict with least error the training data. The optimized neural network can be used as a fitted
potential energy surface. They used the neural network method to fit two degrees of freedom and
three degrees of freedom cross sections of an empirical model of CO/Ni(111). They calculated
the surface diffusion rates using the fitted two-degree-of-freedom model. They also fit 750 DFT
data of 12 degees-of-freedom H2/Si(100) potentials. Brown et al.28 developed a neural network
method by using Bayesian inference. They used probability of the correct weights,
P({wij}|{Vk},Hi) given the data and network architecture. The probability is evaluated via the
Bayes’ theorem:
𝑃({𝑤𝑖𝑗 }|{𝑉𝑘 }, 𝐻𝑖 ) =
𝑃({𝑉𝑘 }|{𝑤𝑖𝑗 }, 𝐻𝑖 )𝑃({𝑤𝑖𝑗 }|𝐻𝑖 )
.
𝑃({𝑉𝑘 }|𝐻𝑖 )
(1.16)
The terms on the right hand side can be straightforwardly evaluated. The best {wij} can be
obtained by maximizing the probability. The weights will be used for the neural network
architecture. They used the method to fit four-dimensional (HF)2 analytical potential and DZF
SCF level data of HF-HCl potential. The results of diffusion Monte Carlo of the two neural
network potentials are in good agreement with the analytical surface for (HF)2 and the
experiments for HF-HCl respectively. Gassner et al. 29 used neural network to fit three-body
interaction energies in H2O-Al3+-H2O including 15 interatomic distances. Analytical functions
were used for the two-body interaction. Both H2O molecules are held rigid in the energy
13
calculation and simulation. The training data included 13400 points at HF method. They
performed a simulation of 200 molecules and one Al3+ using the neural potential. They studied
the distance and orientation of the water molecules in different shells with respect to Al3+ and
compared to three-body analytical potential and two-body pair potential without three-body
interaction. Prudente and Neto30 used the neural network to fit the X2Π and 22Π state of HCl+ ion
and its dipole moment function from X2Π state to 22Π state based on 31 ab initio points. They
calculated the photo dissociation cross section for transition from X2Π to 22Π state. Prudente et
al. 31 also used neural network to fit Dykstra-Swope’s 32 and Meyer-Botschwina-Burton’s 33 ab
initio points of H3+. They calculated the vibrational levels of H3+ using the neural network
potential. Lorenz et al.34 used the neural network to fit six-degree-of-freedom H2/K(2×2)/Pd(100)
DFT potential energy surface. They used 619 points as the training data and 40 points as the test
data. The errors of training data and test data are well below 0.1 eV. The absorption probability
from their simulation using the neural network potential agreed well with that from their
independent fit analytical potential of the H2/K(2×2)/Pd(100) potential. Raff et al.35 developed a
neural network method to generate fitted potential energy surface for molecular dynamics and
Monte Carlo studies. They used trajectories from empirical potential to sample several thousand
ab initio data. The neural network was used to fit the data to run trajectories. An algorithm was
used to add more data points based on current density of data points from the trajectory until the
neural network converged to experimental potential and force field. They used this method to fit
12-dimensional potential for vinyl bromide (CH2CHBr) dissociation. An ensemble of
nonreactive trajectories and dissociation trajectories sampled 1400 MP4(SDQ) points. The neural
network potential reproduced ab initio results very well. They also used the method to fit
hypersurface of five-atom silicon cluster that occurs in the simulation of cutting experiment. The
14
Tersoff potential was used for the early cutting MD simulations containing 1675 atoms. The
simulation produced 18 000 configurations of Si5 in the front of the tool, in the chip and within a
few unit cell distances beneath the tool. Another 10 000 five-atom Si5 configurations near
equilibrium were selected. These points were calculated at B3LYP6-31G** level. Another
thousand configurations were added to the data from neural network molecular dynamics
calculations as the final data points for the neural network. There were no more configurations
not sampled well from new molecular dynamics calculations after the addition of the extra
thousand new data points.
Maisuradze and Thompson36 first used interpolating moving least-squares method (IMLS) to
fit a potential energy surface. They fitted 1-D, 2-D and 3-D of the analytical potential energy
surface of the reaction HN2 → N2 + H by Koizumi, Schatz and Walch.37 Maisuradze et al.38
systematically studied the parameters in the weight function, the degree of IMLS fit, the number
and placement of points by fitting the Morse potential and 1-D curve of HN2 → N2 + H potential
energy surface. Later Maisuradze et al.39 first used IMLS to fit a 6-D analytical H2O2 potential
energy surface by Kuhn et al..40 They were able to study the effects of the weight function
parameters, the degree and partial degree of IMLS, the sampling of the data points, the number
of data points and the optimal automatic data point selection by fitting the potential energy
surface. They were able to fit the potential energy surface with RMS error less than 1 kcal/mol
using 1350 data points. Kawano et al.41 studied two cutoff methods, fixed radius cutoff method
and density adaptive cutoff method by fitting the analytical H2O2 potential energy surface. A
cutoff value will be set up to select only data points within the cutoff value for the fitting. The
computational time with the cutoff methods was one order of magnitude lower. Guo et al.42
studied IMLS focusing on classical dynamical calculations. They found only 152 data points and
15
their symmetrical points were needed for the H2O2 dissociation rate. Later Guo et al. 43 used
IMLS to fit H2CN unimolecular reaction at CCSD(T)/aug-cc-pvtz level. The rate of H2CN
dissociation converged with as little as 330 data points. Guo et al.44 further developed the IMLS
method called local IMLS method. The coefficients of previous points fitted by IMLS will be
stored. The energy of further points can be calculated as shown in Eq. (1.17) and (1.18).
𝑀
𝐿𝑍𝑘 𝑉(𝑍) = ∑ 𝑎𝑖 (𝑍𝑘 )𝑏𝑖 (𝑍),
(1.17)
𝑖=1
𝑁
𝑉𝑓𝑖𝑡𝑡𝑒𝑑 (𝑍) = ∑ 𝑤𝑖 (𝑍)𝐿𝑍𝑖 𝑉(𝑍)
(1.18)
𝑖=1
The ai(Zk) are the coefficients of point Zk. The coefficients of Zk are used to points Z that are
close to Zk as shown in Eq. (1.17). The equation (1.17) calculates the energy of Z using the
coefficients of Zi. The wi(Z) in Eq. (1.18) is the normalized weight function. The energy of a new
point Z can be calculated as the weighted sum of terms by using coefficients from all data points
in local IMLS as shown in Eq. (1.18). They found that the local IMLS was at the same accuracy
as standard IMLS. Dawes et al. 45 used IMLS using low-density ab initio Hessian values to
provide high-density Hessian values for modified Shepard or L-IMLS to fit global potential
energy surface. They used this method to fit 1D Morse oscillator, 3D HCN and 6D H2O2
analytical potential energy surface. The potentials fitted to the analytical potentials from the
method were very accurate. Dawes et al.46 developed an efficient way to select data points for
IMLS. They calculated the negative square difference of two successive order IMLS and added
new data points at the minimum of the negative square difference. They were able to achieve
accuracy of RMS error 0.2 kcal/mol with 686 data points using energy and gradients with this
16
new data selection strategy. Dawes et al.47 also used trajectories to sample data points for IMLS
fitting. They used the method to fit the cis – trans HONO isomerization at HF/cc-pVDZ level.
There were about 430 data points needed when the error tolerance of 1 kcal/mol is used to run
the trajectory to sample data points. Dawes et al.48 fit two 3-atom potential energy surfaces, CH2
at Davidson-corrected MRCI calculations with double-, triple- and quadruple-ζ basis sets
extrapolated to complete basis set (CBS) limit and HCN at CCSD(T)/CBS based on
aug-cc-pCVDZ, aug-cc-pCVTZ and aug-cc-pCVQZ. Less than 500 ab initio points with energy
and gradients were needed for both potentials. They calculated the vibrational levels of the two
molecules using the fitted potential energy surface. There were significantly better results from
the CH2 fitted potential than previous calculated results. The RMS error of vibrational levels
from the HCN fitted potential with some corrections was 3.2 cm-1 comparing to the experiments.
Although much progress is made in the IMLS PES fitting, there are still only few electronic
structure theory points of PES of 4-atom fitted using IMLS comparing to the progress of fitting
ab initio or DFT PES of 4-atom by other fitting methods. The current study is not to further
increase the accuracy of IMLS for fitting PES based on the basis functions, the data selection or
the cutoff strategy. The current study focuses on finding ways that can make IMLS methods fit
highly accurate ab initio data points of 4-atom PES. The next two chapters discuss using
previous IMLS methods on fitting 4-atom PES, either analytical or ab initio. The fourth chapter
discusses several ways to use IMLS methods developed in previous studies to fit the
B3LYP/3-21G PES of CO2 + O reaction. The last chapter briefly talks about the future of IMLS
fitting on PES.
17
CHAPTER 2
IMLS ON H2O2 ANALYTICAL SURFACE
2.1 Introduction
Maisuradze and Thompson36 first used IMLS to fit a PES for the global function of the
reaction HN2 → N2 + H by Koizumi, Schatz and Walch.37 They fit 1-D, 2-D and the complete
3-D of the surface. More IMLS studies have been done by Donald L. Thompson’s group using
the H2O2 analytical surface by Kuhn et al..40 Maisuradze et al.39 first fitted this six-dimensional
potential energy surface and studied the weight functions, the degree of the basis set and the
automatic selection of data points. The surface is fit over the range of 100 kcal/mol. They were
successful to fit the surface with 1350 data points with an accuracy of less than 1 kcal/mol. Later
Kawano et al.41 studied the cutoff strategies by fitting this surface. However, the cutoff strategy
is not used in the current study. Guo et al.42 first studied the IMLS fitted surface on the
perspective of classical dynamics of H2O2. They used the IMLS fitted surface to calculate the
dissociation rate of the reaction H2O2 → 2 OH. They found that a fairly good dissociation rate
can be calculated with as low as 152 nonsymmetrical data points comparing to the rate from the
analytical surface. It means that a reasonable dynamic result can be obtained using a fitted
surface with lower accuracy for this reaction system. Besides their work on the study of the
dynamics of H2O2 surface, Guo et al.44 developed a newer version of IMLS, L-IMLS. They fit
the H2O2 analytical surface to demonstrate the efficacy of the new IMLS method. The most
recent work on IMLS was done by Dawes et al..45 They selected a HCN analytical PES and the
H2O2 PES for 3-D and 6-D IMLS fitting respectively. Since the IMLS method is an accurate but
expensive relative to some common fitting method, they studied on using IMLS to provide high
18
density IMLS points from a small density of ab initio points for other cheap fitting methods.
They used the modified-Shepard method to fit the IMLS points. They first used
high-dimensional model representation (HDMR) style polynomials as the basis set for the IMLS
in this work. Dawes et al.46 presented a more robust data point selection scheme for the IMLS
fitting in the later work. Instead of data points selected from a series of random points, they
calculated the data points based on finding the minimum of a function. The function is the
negative squared difference of the two successive order IMLS. They were able to achieve highly
accurate surface with many fewer data points compared to randomly selecting points.
In this chapter, we will present the fit using IMLS on this six-dimensional analytical potential
energy surface. Both the results on this and next chapter will be preparation for the IMLS fit on
ab initio surface. The specifics of the how the surface is fit are given in section 2.2. The
discussion of the fit is given in section 2.3.
2.2 Method and procedure
Here is a description of one-dimensional IMLS method. There are N + 1 data points, xi {i = 0,
1, ⋯ N} with values fi {i = 0, 1, ⋯N}.
The mth order polynomials are used as Eq. (2.1).
𝑚
𝑝(𝑥) = ∑ 𝑎𝑖 𝑥 𝑖
(2.1)
𝑖=0
The sum of square deviation is given in Eq. (2.2).
𝑁
𝐸𝑥 (𝑝) = ∑[𝑝(𝑥𝑗 ) − 𝑓𝑖 ]
𝑖=0
19
2
(2.2)
The best least squares fit can be obtained with a polynomial that minimizes Eq. (2.2). In the
weighted least squares, a weight is added in each difference term in Eq. (2.2). This gives:
𝑁
𝐸𝑥 (𝑝) = ∑ 𝑤𝑖 (𝑥)[𝑝(𝑥𝑗 ) − 𝑓𝑖 ]
2
(2.3)
𝑖=0
where wi(x) is a positive weight function with features: wi(x) → ∞ when x → 𝑥𝑖 and wi(x) → 0
when x → ∞. The two most used weight functions are Eq. (2.4) and Eq. (2.5).
𝑤𝑖 (𝑥) =
1
,
(𝑥 − 𝑥𝑖 )2𝑛 + 𝜀
(2.4)
𝑤𝑖 (𝑥) =
𝑒𝑥𝑝[−(𝑥 − 𝑥𝑖 )2 ]
.
(𝑥 − 𝑥𝑖 )2𝑛 + 𝜀
(2.5)
The n is a small positive integer. This can be adjusted based on the dimensions of the fit and
whether gradients or Hessians are used in the fitting.
Setting ∂Ex(p)/∂ai = 0, i = 1, 2, ⋯ m, gives the normal equations:
[∑ wi (x)xi 0 ]a0 + ⋯ + [∑ wi (x)xi m ]am = ∑ wi (x)fi
[∑ wi (x)xi1 ]a0 + ⋯ + [∑ wi (x)xi m+1 ]am = ∑ wi (x)xi fi
⋮
[∑ wi (x)xi m ]a0 + ⋯ + [∑ wi (x)xi 2m ]am = ∑ wi (x)xim fi .
(2.6)
The matrix-vector form of Eq. (2.6) is:
BT ∙ W ∙ B ∙ a = BT ∙ W ∙ f,
(2.7)
20
where B is a (N + 1) × (m + 1) matrix, BT is the transpose of B, W is a (N + 1) × (N + 1)
diagonal matrix, and a and f are column vectors; explicitly, they are:
B=
1 x0
1 x1
[1
xN
⋯
⋯
⋮
⋯
x0m
x1m
,
m
xN
]
f0
f
f = [ 1 ],
⋮
fN
a0
a1
a = [ ⋮ ],
am
W = dia[w0(x), w1(x), ⋯, wN(x)].
(2.8)
Either by solving the normal equations Eq. (2.6) or its matrix-vector form Eq. (2.7), a set of
coefficients ai {i = 0, 1, ⋯ , m} can be obtained. The condition that N is not less than m must be
satisfied to solve the equations. When m = 0 or 1, Eq. (2.6) and (2.7) has analytical solutions.
When m is higher, numerical methods must be used to solve the normal equations. Every time a
new x needs to be evaluated, a new set of normal equations will be set up. A new set of
coefficients ai {i = 0, 1, ⋯ , m} is obtained as the solutions of the equations. With the new
coefficients, the value at the new x, f(x) = p(x) = Σaixi can be evaluated with the polynomial
representations. There are variables other than x for higher-dimensional IMLS. But the way it
works is the same as discussed for one-dimensional IMLS.
The higher-dimensional model representation (HDMR) style polynomials is the formula
Eq. (2.9).
𝐷
2
𝐶𝐷
𝑉 𝐻𝐷𝑀𝑅 (𝑞1 , 𝑞2 , ⋯ , 𝑞𝐷 ) = 𝑣0 + ∑ 𝑣𝑖 (𝑞𝑖 ) + ∑ 𝑣𝑖𝑗 (𝑞𝑖 , 𝑞𝑗 ) + ⋯.
𝑖=1
(2.9)
𝑖<𝑗=1
where D is the dimension of the basis set. The first term is a constant, the second is onecoordinate terms, the third is two-coordinate terms and so on. For example, for a basis function is
21
denoted as (6, 5, 4, 4), it comprises the one-, two-, three- and four-coordinate terms truncated at
6th, 5th, 4th, and 4th orders, respectively.
The range of analytical H2O2 PES fitted by IMLS are O-H (1.5 – 8 bohr), O-O (2 – 7 bohr),
O-O-H angle (0.6 – 179.4°), and a dihedral angle H-O-O-H (0.6 – 179.4°). The valence
coordinates is shown in Figure 2.1. The cutoff energy is selected at 100 kcal/mol. There are 54
seed points selected from a sparse grid to start IMLS. Both energy and gradients of each point
are used for the fit. The high-dimensional model representation (HDMR) style polynomial is
used as the basis set. Both fixed basis set and dynamic basis set are used. The dynamics basis set
is used since there are not enough data points for the large basis at the beginning of the fitting.
The size of the basis set is (6, 3) with 82 terms in the fixed basis set. The one-order lower basis
set is (5, 2) with 46 terms. The starting basis set is (6, 3) in dynamic basis set. It increases to
(10, 7, 5, 4) as the total number of energy and gradients is four times greater than the number of
terms in the basis. There are 591 terms in (10, 7, 5, 4). Three weight functions are tested in the fit
as Eq. (2.10), (2.11) and (2.12).
𝑤(𝑥) =
𝑒
(
𝑥 2
−(
)
𝑑(𝑖)
,
𝑥 8
) + 10−14
𝑑(𝑖)
(2.10)
2
𝑒 −𝑥
𝑤(𝑥) = 8
,
𝑥 + 10−14
(2.11)
1
.
(𝑥 2 + 10−4 )4
(2.12)
𝑤(𝑥) =
The variable x in the weight functions is the distance between the evaluation point and data
points. The internuclear distance is used as the coordinates for fitting as shown in Figure 2.1.
22
Since there are two H- and O- atoms in the molecule, the shorter O-H is the first coordinate. The
other O-H is the second coordinate. This prevents that the same data points can be represented by
two different internuclear distances. The O-O, two O-O-H angles and dihedral angle are third to
sixth coordinates. A thousand random points below 100 kcal/mol energy are used to select data
points. Five points with largest difference of two successive order IMLS are selected. Another
1000 random points below 100 kcal/mol energy are used to calculate the RMS errors once a set
of fifty data points are added. The final fitting included 3054 points.
The minimum H2O2 is r1 (1.8175 bohr), r2 (1.8175 bohr), r3 (2.7445 bohr), θ1 (99.76°), θ2
(80.24°) and τ (114.32°). The energy of the minimum is -19039.23 cm-1 (-54.435 kcal/mol). All
the energy of the points is relative to the minimum of H2O2. The RMS error of energy is
calculated as Eq. (2.13).
∑1000
(𝐸𝐼𝑀𝐿𝑆,𝑖 − 𝐸𝑃𝐸𝑆,𝑖 )2
1
√
(𝑒𝑛𝑒𝑟𝑔𝑦)
𝑅𝑀𝑆 𝑒𝑟𝑟𝑜𝑟
=
.
1000
(2.13)
The EIMLS, i and EPES, i are IMLS energy and analytical energy from 1000 random points.
The gradients are calculated using finite difference method as Eq. (2.14).
𝜕𝐸 ∆𝐸
(𝑖 = 1, 2 ⋯ 6).
=
𝜕𝑟𝑖 ∆𝑟𝑖
(2.14)
The internuclear distance ri is defined in Figure 2.1. The gradients RMS error is calculated as Eq.
(2.15).
𝑅𝑀𝑆 𝑒𝑟𝑟𝑜𝑟 (𝑔𝑟𝑎𝑑𝑖𝑒𝑛𝑡𝑠) =
√
∑1000
∑61(
1
𝜕𝐸
𝜕𝑟𝑗
−
𝐼𝑀𝐿𝑆,𝑖
6000
23
𝜕𝐸
𝜕𝑟𝑗
)2
𝑃𝐸𝑆,𝑖
.
(2.15)
The (∂E/∂rj)IMLS,i and (∂E/∂rj)PES,i are gradients relative to jth internuclear distance of ith random
point of 1000 random points from the IMLS and the analytical PES.
2.3 Results and Discussion
Here is a brief description of the analytical H2O2 PES. There are four different surfaces in
Kuhn et al.’s study.40 The surfaces are fitted on 8145 DFT and 5310 CASPT2 points with
analytical equations. The surface represents the ground state PES up to 40000 cm-1 (114.36
kcal/mol) above the equilibrium geometry with 100 – 107 cm-1 (0.29 – 0.31 kcal/mol) standard
deviation comparing to the points from electronic structure calculations. The surface includes
only the dissociation channel H2O2 → 2OH. However, the dissociation products HO2 and H are
included in the surface. The fitted surface only includes the H2O2 → 2OH channel, H2O2
stationary points, HO2 + O products and their surrounding points. They used associated Legendre
polynomials Pjm(θi) for the angular potential term. The PCPSDE surface is used as the analytical
surface here. The PCPSDE surface is credited as the best surface by the authors among the four
different potentials. The cutoff energy and the range of the coordinates are given in section 2.2.
The RMS errors of energy and gradients with different weight functions are shown in Fig.
2.2. The reason to show the accuracy of gradients is that the energy only gives the shape of the
potential energy surface. It is gradients that are used for the dynamics calculation of the molecule.
The RMS errors of energy and gradients decrease very fast with the addition of data points.
There are 3054 data points. The final RMS errors of energy and gradients are 0.18 kcal/mol and
1.3 kcal/(mol*bohr).
The three different weight functions are Eq. (2.10), (2.11) and (2.12). The variable x is the
distance between the evaluation point and the data point. The weight of the evaluation point
24
should be calculated for all the data points in Eq. (2.3). The d(i) is the distance of the Rth closest
data points to the ith data point when the weight of the evaluation point at the ith data point is
calculated. Therefore, the d(i) is a value based on the density of other data points near i th data
point. The value of d(i) adjusts the weight. The distance of 20th closest data point is used in the
fit. The plot of three weight functions is shown in Figure 2.3. The x is the square of the distance
between the data point and the evaluation point which is different from the x in equations, the
distance between the evaluation point and the data point. The Figure 2.3 (a) shows the weight
function Eq. (2.10) as d(i)2 = 10. Since the d(i) is the distance between a data point and its 20th
closest data point, the parameter d(i) adjusts the weight function Eq. (2.10) for different data
point. If the d(i) is large, the weight function decreases slower with the increase of x. The weight
function Eq. (2.12) decrease very fast comparing to weight functions Eq. (2.10) and Eq. (2.11). It
means that only the weights of points closest to the evaluation point are large. Unlike Eq. (2.12),
weight functions Eq. (2.10) and (2.11) have a flat area before the drastic fall. Therefore, points in
this flat area are assigned the largest weights. The peak at zero in Eq. (2.12) means that only
points at the position of evaluation point are assigned the maximum weight. Once the points are
off the evaluation point, the weight decreases very fast.
Even at 3054 data points, the weights from Eq. (2.12) of the data points are mostly smaller
than 1. The weight functions of Eq. (2.10), Eq. (2.11), and Eq. (2.12) for almost all the
evaluation points are in the tail areas. Since the characteristic difference of the three weight
functions are insignificant in this area after the fast decrease of the weights, the choice of the
three weight functions does not affect the accuracy of the fitting in this area. No evaluation
points from the 1000 random points have been found with weight larger than 1000 at any of the
data points for Eq. (2.10), Eq. (2.11) and Eq. (2.12) with as many as 3054 data points from the
25
IMLS using fixed basis set (6, 3). The average of d(i)2 for the data points are calculated as
Eq. (2.16).
𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑜𝑓 𝑑(𝑖)2 𝑜𝑓 𝑡ℎ𝑒 𝑑𝑎𝑡𝑎 𝑝𝑜𝑖𝑛𝑡𝑠 =
2
∑𝑁
1 𝑑(𝑖)
.
𝑁
(2.16)
The average d(i)2 is 10.95 at the beginning of the fitting with 54 data points. The average
decreases to 0.91 with 2254 data points. Therefore, the weight function Eq. (2.10) is close to Fig.
2.3 (a) at the beginning. It gradually changes to the weight function Eq. (2.11) as Fig. 2.3 (b)
with more data points. This is the average d2(i) of the all the data points. The weight function is
varied for each single data point. Both the RMS errors of energy and gradients are almost the
same using the three different weight functions. It shows that the choice of weight function is not
an important factor that affects the fitting. This result is in agreement with previous discussion of
the weight functions.
The RMS errors of energy and gradients with respect to number of data points from IMLS
fitting with (6, 3) and with dynamic basis set are shown in Fig. 2.4. The dynamic basis set is
started at (6, 3) including 82 terms with 54 data points. The basis set increases to (7, 4) including
133 terms at 59 data points. It then increases to (8, 5, 3) including 219 terms at 94 data points. It
then increases to (9, 6, 4) including 360 terms at 154 data points. It finally increases to
(10, 7, 5, 4) including 591 terms at 254 data points. The RMS errors from dynamic basis set with
(10, 7, 5, 4) are much smaller than those from the fixed basis set (6, 3). The larger basis set
greatly improves the accuracy. The RMS error of energy from the large basis set is below 0.2
kcal/mol while that is around 0.5 kcal/mol for the small basis set.
26
A contour plot of both the analytical surface and IMLS surface are shown in Figure 2.5 (a)
and (b). The contour plot is the scan of the O-O coordinate (2 – 7 bohr) and the dihedral (0.5 –
179.5°) with other coordinates at equilibrium values. The contour is created from 51*51 points.
The contour describes the dissociation of H2O2 to 2 OH with the rotation of two O-H at the same
time. As the O-O distance is large enough, the total energy is independent of the O-O distance
and the dihedral. The rotation of the dihedral affects the energy in the shorter O-O distance. This
contributes to the well in the contour on the top left area which corresponds to the minimum of
H2O2. The contour from IMLS surface is quite the same as the contour from the analytical
surface in the lower energy area (100 kcal/mol). The cutoff energy for the IMLS fitting is 100
kcal/mol. Even in the left side of the contour with as high as 150 kcal/mol energy, the IMLS
surface still agrees well with the analytical surface.
2.4 Summary and Conclusion
The analytical surface of H2O2 is fitted using IMLS. The range of the fitting is O-H (1.5 –
8 bohr), O-O (2 – 7 bohr), O-O-H angle (0.6 – 179.4°), dihedral H-O-O-H (0.6 – 179.4°) with
cutoff energy 100 kcal/mol. There are 54 seed points. The difference of two successive order
IMLS is calculated from 1000 random points. Five points with the largest difference from the
random points are added to the data points. The RMS errors of energy and gradients of IMLS
PES with 3054 data points are 0.18 kcal/mol and 1.2 kcal/(mol*bohr) respectively. There are
three weight function used in the fitting. The RMS errors are similar using the three weight
functions. However, the RMS errors are much larger for the small basis set. Therefore, the basis
set has an important effect on the RMS errors of the IMLS fitted PES.
Besides the RMS errors are calculated for the IMLS fitted PES. A contour is calculated from
both the fitted PES and the analytical PES. There is an excellent agreement between the two
27
contours. Even the region of the IMLS contour above 100 kcal/mol is similar to that of the
contour from the analytical PES. Dawes et al. fitted the H2O2 analytical PES.46 They fitted the
PES with the RMS error of 0.2 kcal/mol using 686 data points of energy and gradients and with
the RMS error of 0.1 kcal/mol using 1446 data points. In current study, the RMS error of energy
is below 0.2 kcal/mol at 2754 data points for the first time. Therefore, Dawes et al. fitted the PES
much more efficiently than the current study. Their efficiency of the fitting can be contributed to
their use of L-IMLS and a new data selection strategy used in their fitting. They found the
minimum of the function of negative square difference of two successive order IMLS. They
added new data points where the minimum of the function is. However, comparing to the earlier
and the other fitting of the PES using IMLS, they fitted the PES using different sampling method
for both data selection and RMS error estimation. Their fitting error is 1 kcal/mol with 1350 data
points. In the current study, the RMS error is smaller than 1 kcal/mol using 554 data points for
the first time. This is due to several factors. They mainly include the use of both energy and
gradients of the data points and the use of large size HDMR basis set in current fitting.
28
6
θ1
r1
r
5
θ2
τ
r2
1
2
4
3
Figure 2.1 The H2O2 valence coordinates and ordered bond distance coordinates.
29
Energy (kcal/mol)
(a)
Gradients kcal/(mol*bohr)
(b)
Number of data points
Figure 2.2 The RMS errors of (a) energy and (b) gradients with respect to the number of data
𝒙
points. The red is IMLS with weight function
𝟐
𝒆−𝒙
𝟐
−(
)
𝒆 𝒅(𝒊)
(
𝒙 𝟖
) + 𝟏𝟎−𝟏𝟒
𝒅(𝒊)
. The blue is IMLS with weight
𝟏
function 𝒙𝟖 + 𝟏𝟎−𝟏𝟒 . The brown is IMLS with weight function (𝒙𝟐 +𝟏𝟎−𝟒 )𝟐 .
30
(a)
(b)
2
(c)
4
x
−(
Figure 2.3 The weight functions (a)
𝒙
𝒆 𝒅(𝒊)𝟐
)
𝒙 𝟒
((
) +𝟏𝟎−𝟏𝟒 )
𝒅(𝒊)𝟐
𝒆−𝒙
𝟏
with d(i)2 = 10; (b) (𝒙𝟒 +𝟏𝟎−𝟏𝟒 ); (c) (𝒙+𝟏𝟎−𝟒 )𝟐 .
The function are Eq. (10), Eq. (11), Eq. (12) with d2(i) = 10 where x2 in equations is x in the plots.
31
(a)
Gradients kcal/(mol*bohr)
Energy (kcal/mol)
(10, 7, 5, 4)
(6, 3)
(b)
Number of data points
Figure 2.4 The RMS errors of energy and gradients with respect to number of data points with
𝒙
weight function
𝟐
−(
)
𝒆 𝒅(𝒊)
(
𝒙 𝟖
) +𝟏𝟎−𝟏𝟒
𝒅(𝒊)
for different basis.
32
τ/
(a)
(b)
rOO/Å
kcal/mol
(c)
Figure 2.5 Potential energy contour and surface of rOO (2 – 7 Å) and τ (0 - 180°) of the H2O2
with r1,2 (1.8175 bohr) θ1 (99.76°), θ2 (80.24°). (a) H2O2 analytical surface contour. (b) IMLS
contour. (c) the surface of H2O2 analytical surface.
33
CHAPTER 3
DYNAMICS-DRIVEN IMLS FITTING
3.1 Introduction
Dawes et al. described the method to grow PES on the exploration of a set of trajectories
using IMLS.47 They called the method dynamics-driven IMLS fitting, the method studied in the
chapter. The trajectories are started using direct dynamics method. The initial points from the
trajectory are stored until there are enough data points to perform IMLS fit. Two successive
order IMLS are calculated. If the difference of the two successive order IMLS is larger than the
tolerance value, the point is calculated using ab initio or DFT method. The point is added to the
data points. The procedure of addition of data points continues until there are enough data points
for the fitted PES. Dawes et al. used this approach to fit cis – trans isomerization of HONO at
HF/cc-pVDZ level as illustration. They studied different error tolerance to fit the PES for the
isomerization reaction. The smaller the error tolerance, the more accurate of the PES is. At the
same time, more ab initio points are used for the PES. Only hundreds of ab initio points are
needed to accurately predict the isomerization rate using this approach. Therefore, highly
accurate PES is not necessary for the rate calculation. In the same paper, they also studied the
acceleration of direct dynamics calculation using IMLS, called IMLS-accelerated direct
dynamics. The difference between the method and the dynamics-driven IMLS fitting is much
smaller error tolerance should be used in order to satisfy the convergence of the trajectory in the
method. They studied the method also on HONO isomerization. They studied the speedup and
convergence of the accelerated trajectories using different error tolerance values. They also
found the deviation between the direct dynamics trajectory and IMLS-accelerated direct
dynamics trajectory is not much during the short time length, 150 fs in their study.
34
Much earlier work on PES fitting using trajectories was GROW method by Ischtwan and
Collins.20 The initial ab initio points from the minimum energy path are used for modified
Shepard method to calculate trajectories. Ab initio points are selected based on the similarity
between points from exploratory trajectories and previous ab initio points and also between these
points and other points in the exploratory trajectory. They calculated a large number of points
from the trajectories. These points are assigned weights using Eq. (3.1). The NT is the number of
points calculated from the trajectory.
𝑁𝑇
𝑣𝑗 [𝐑(𝑛)]
1
ℎ[𝐑(𝑗)] =
∑ { 𝑁𝑑
}
𝑁𝑇 − 1
∑
𝑣𝑘 [𝐑(𝑛)]
𝑛=1
𝑛≠𝑗
(3.1)
𝑘=1
vj is the weights of the point to other trajectory points excluding itself. The vk is the weights of
the point to the data points. Therefore, the trajectory point close to the trajectory points and far
from the data points has a large weight h[R(j)]. The points with largest weights are added to Nd
data points. The Nd increases. And the NT decreases. The procedure to assign weights to NT
trajectory points and add them to Nd data points continues until Nextra data points are added.
Another number of trajectories are calculated using Nd + Nextra data points. A new NT trajectory
points are calculated to add another Nextra points as described above. Enough data points are
calculated for the potential energy surface by following this procedure. They demonstrated the
method by fitting the abstraction reaction 3NH + H2 → NH2 + H on the MCSCF method. They
first calculated the energy, gradients and second derivative matrix of points on the intrinsic
reaction path. Then, they calculated the trajectory with these data points using modified Shepard
method. They added Nextra three times based on the procedure described above after three
iterations. Finally, they compared the improvement of the accuracy of the PES after each
35
iteration by comparing the energy, gradients and Hessian matrix of the potential with ab initio
values.
The dynamics-driven IMLS fitting approach has lots of similarities comparing to GROW
method. However, this method is developed based on IMLS fitting. Therefore, the difference
between the method and the GROW is due that the IMLS method is developed in order to best
use IMLS to fit the potential energy surface.
3.2 Methods
The ab initio method for the trajectories is HF/cc-pVDZ level. As for the dynamics-driven
IMLS method, the energy and gradients of the first 12 points in the first trajectory are calculated.
Two successive order IMLS fitting are started with these ab initio seed points. If the difference
of the two IMLS fits is smaller than the error tolerance ɛ, the analytical IMLS gradients are used
in the trajectory. Otherwise, ab initio points are calculated. These ab initio points are saved as
data points. Both error tolerance value ε of 1 kcal/mol and 0.5 kcal/mol are used in the
trajectories. The high-dimensional model representation (HDMR) style polynomial basis set is
used. The final basis set is (10, 7, 5, 4). The one-order lower basis set is (9, 6, 4). Smaller basis
set is used in the earlier trajectories when the number of ab initio points is not enough to support
larger basis. The length of the trajectories is set to 10 ps with a step size of 0.1 fs and the velocity
Verlet integrator. The initial condition for the trajectory is 8 quanta added to the H-O-N bending
mode and zero point energy in other normal modes. The total energy comparing to that of the cis
– HONO minimum is about 49 kcal/mol. The initial configuration is a random cis – HONO
configuration. The trajectory is terminated if the energy drift is greater than 10 kcal/mol or if the
isomerization to trans – HONO happens or if it is beyond 10 ps. There are 685 data points
36
selected from 325 trajectories with error tolerance ε of 1 kcal/mol. There are 2817 data points
selected from 785 trajectories with error tolerance ε of 0.5 kcal/mol.
3.3 Results and discussion
The total number of PES evaluations with respect to the number of ab initio points in the
trajectories is shown in Figure 3.2. The slope increases greatly with the increase of the number of
ab initio points. The slope is almost vertical in later trajectories. The slope is vertical about 700
data points for ε of 1 kcal/mol. The calculation of a new ab initio point is very rare in the later
trajectories. However, the slope is vertical at about 2200 data points for ε of 0.5 kcal/mol.
Enough ab initio points are calculated in the cis – trans HONO isomerization as shown in the
Figure 3.2. This is the same as results in Dawes et al.’s paper.47 Dawes et al. showed that the
slope of the total number of PES evaluations vs the number of ab initio evaluations is close to
vertical with about 430 ab initio points for the isomerization with ε of 1 kcal/mol and 800 ab
initio points with ε of 0.5 kcal/mol.47 Some earlier trajectories are terminated due to energy drift
or isomerization before the full length of the trajectories. The total PES evaluations is about
7 × 106 and 1 × 107 for ε of 1 kcal/mol and 0.5 kcal/mol respectively. The mean error of
dynamics-driven IMLS fitting should be one order lower than the error tolerance. Therefore, the
mean error for the cis – trans isomerization should be 0.1 kcal/mol with 685 ab initio points and
0.05 kcal/mol with 2187 ab initio points based on the .
The RMS errors are calculated using trajectories with the initial condition as discussed in
method section. Trajectories are calculated from the IMLS fitted surface using 685 ab initio and
2187 ab initio points. The step size, dt, is 0.1 fs. The error is calculated by comparing the IMLS
value to ab initio value every 10 time steps in the trajectory. Two trajectories from 685 ab initio
points are calculated at 6.61 ps and 0.55 ps. The errors are calculated from 6613 and 550 ab
37
initio points. The RMS errors of energy are 0.12 kcal/mol and 0.18 kcal/mol respectively. The
RMS errors of gradients are 2.3 kcal/(mol*bohr) and 3.6 kcal/(mol*bohr) respectively. Therefore,
the RMS errors of the IMLS PES are varied using different trajectories. Another two trajectories
from IMLS fitted PES with 2187 ab initio points are at 3.318 and 1.852 ps. The initial conditions
are the same as those of trajectories from IMLS PES with 685 ab initio points. The errors are
calculated from 3318 and 1852 ab initio points. The RMS errors of energy for 2187 ab initio
points are 0.074 kcal/mol and 0.081 kcal/mol respectively. The RMS errors of gradients are 1.5
kcal/(mol*bohr) and 1.7 kcal/(mol*bohr) respectively. Both RMS errors are a little higher than
one order lower of ɛ. However, the mean error is smaller than the RMS error. Therefore, it is in
favor of the statement that the mean error of the fitted PES is one order lower than the error
tolerance value. The RMS error from IMLS with 2187 ab initio points is larger than one half of
the errors from IMLS PES with 685 ab initio points. But they are smaller than the RMS errors
from IMLS PES with 685 ab initio points.
The total energy of HONO trajectory from the direct dynamics trajectory, IMLS fitted PES
with 685 data points and with 2187 data points is shown in Figure 3.4. The initial conditions for
the three trajectories are the same. The time length is 2.5 ps. As shown in Figure 3.4, the total
energy convergence is 1 – 2 kcal/mol for trajectories from IMLS fitted PES with 685 and 2187
ab initio points. The total energy convergence for the direct dynamics trajectory is 0.01 kcal/mol
which is the accuracy of the velocity Verlet integrator. The starting total energy of the trajectory
from IMLS PES with 1638 ab initio points is different from that of the direct dynamics trajectory
and the trajectory from IMLS PES with 685 ab initio points. Since the excitation energy is
calculated from the frequency of the H-O-N bending mode, the difference of the total energy
comes from the difference of the frequency of the mode. The wave number of the cis-HONO is
38
1510 cm-1 from the PES with 1638 ab initio points while it is 1522.70 cm-1 from IMLS with 685
ab initio points and 1522.61 cm-1 from the ab initio calculation. The frequency of H-O-N
bending mode is still oscillating as large as 1583 cm-1, as low as 1502 cm-1 as the number of the
ab initio points increases to 1980. The frequency is 1521.02 cm-1 from PES with 2186 ab initio
points. Therefore, the cis – HONO configurations are still not sampled as well as the PES with
685 ab initio points for ε of 1 kcal/mol. This is related to that the slope for ε of 0.5 kcal/mol is
still not as vertical as that for ε of 1 kcal/mol even at 2187 ab initio points. The frequency can be
further improved after adding more data points for ε of 0.5 kcal/mol. Therefore, the frequency
and total energy computed from IMLS PES with 685 ab initio points are more accurate. The
RMS errors of IMLS PES with 2187 ab initio points are smaller. How the IMLS PES is
improved by using smaller ε? More work is needed to study this.
The total angular momentum of x, y and z for the direct dynamics trajectory, trajectories
from the two IMLS PES are shown in Figure 3.5. As the angular momentum is removed from the
initial configuration, the total angular momentum is close to zero. The angular momentum
convergence is better for the IMLS trajectories than the direct dynamics trajectory.
The isomerization rate is calculated using one hundred trajectories with the initial condition
as discussed in the method section. The total time for the trajectory is set 3 ps with a step size of
0.1 fs and the velocity Verlet integrator. The cis – trans HONO isomerization is defined as
HONO stays in the trans configuration for more than three periods of tortional mode. The
starting time of the trans – HONO is counted as the dihedral τ larger than 150°. If the dihedral
becomes smaller than 120° after it is 150°, the time of the trans – HONO is counted from zero
again. The total isomerized trajectories at the time t, Nt is calculated. The Nt is calculated at a
step size dt of 0.1 ps. The –ln(Nt/N0) is calculated with respect to the time. The least squares
39
method is used to fit the value –ln(Nt/N0) during the 3 ps. The least squares fit to these points are
shown in Figure 3.5. The points in the ballistic area are not included in the fit. The isomerization
rates for direct dynamics trajectories, IMLS PES with 685 data points and IMLS PES with 1638
data points are kdirect of 0.59 ps-1, kIMLS,685 of 0.68 ps-1, kIMLS,1638 of 0.51 ps-1. Therefore, there is a
good agreement for the isomerization rate between the ab initio PES and the IMLS PES. The
total energy convergence of trajectories for direct dynamics trajectories is at order of
0.01 kcal/mol. These for IMLS PES with 685 data points and IMLS PES with 1638 data points
are at order of 1 kcal/mol.
3.4 Summary and conclusion
The dynamics-driven IMLS fitting approach is used to fit cis – trans isomerization of HONO
at HF/cc-pVDZ level. Eight quanta excitation energy is added to H-O-N bending normal mode
as the initial condition. The error tolerance values of trajectories ε are 1 kcal/mol and
0.5 kcal/mol. About 700 ab initio points and 1700 ab initio points are calculated from the
trajectories for the isomerization of cis – trans HONO with ε of 1 kcal/mol and 0.5 kcal/mol
respectively. The real mean error of the fitted surface is one order lower than the error tolerance.
It should be 0.1 kcal/mol and 0.05 kcal/mol for the two error tolerance. Trajectories are used to
calculate the RMS errors of the two fitted PES. The RMS error is calculated in favor of the
statement that the mean error is one order lower than the error tolerance value. There is a good
agreement of isomerization rate between HF/cc-pVDZ PES and IMLS PES. Therefore, both
IMLS PES are fitted good enough for isomerization rate calculation. However, Dawes et al.47
fitted the same PES using IMLS with less data points, especially for ε of 0.5 kcal/mol, only about
800 data points. Dawes et al. used the L-IMLS instead of the IMLS method. Besides, they used a
slightly larger basis set (10, 7, 5, 5) comparing to the basis (10, 7, 5, 4) in the current study.
40
Other factors include the initial conditions of the trajectories and the definition of an
isomerization. Besides, the frequency of the H-O-N mode from the PES with ɛ of 0.5 kcal/mol is
not correct with 1980 ab initio points. More ab initio points need be calculated to see whether the
frequency of the mode can be correct.
As in Chapter 1, the RMS error of the fitted PES is calculated. The contours of the fitted PES
and the analytical PES are compared. In this chapter, the RMS error of the fitted PES is
calculated. The ensemble averages of trajectories from the ab initio PES and the IMLS PES are
compared. The ensemble average is selected as the isomerization rate in the study. Both the static
property as the RMS error, the contour from the fitted PES and the dynamic property, the
isomerization rate from the trajectories can be calculated to estimate the accuracy of the fitted
PES comparing to the real PES. Comparing to the static property, it is more difficult to reach an
agreement on the dynamic property of the PES. As the total energy from the fitted PES should be
conserved, the fitted PES should be accurate enough in order to conserve the total energy of a
trajectory.
41
r3
r1
r2
r5
r6
r4
Figure 3.1 The order of internuclear distance in HONO.
42
The total number of PES evaluations
ε = 1 kcal/mol
ε = 0.5 kcal/mol
0
The number of ab initio evaluations
Figure 3.2 The total number of PES evaluations vs ab initio point calculated in the trajectories.
43
cis – trans HONO isomerization
Figure 3.3 cis – trans HONO isomerization reaction.
44
Total energy (kcal/mol)
Direct
IMLS 1 kcal/mol
IMLS 0.5 kcal/mol
t (ps)
Figure 3.4 The total energy of the HONO isomerization trajectory. The direct dynamics
trajectory (red); the trajectory from IMLS fitted PES with 685 data points (green); the trajectory
from IMLS fitted PES with 2187 data points (blue).
45
component
z
y,
x,
Angular momentum of
(0.020*Å2*atomic mass/fs)
t (ps)
Direct
IMLS ε = 1 kcal/mol
IMLS ε = 0.5 kcal/mol
Figure 3.5 The total angular momentum of components x, y and z for the trajectory computed
from the direct dynamics trajectory, IMLS PES with 685 ab initio points and IMLS PES with
2817 ab initio points.
46
0.589*x – 0.161
R2 = 0.99
Direct
0.68*x – 0.26
R2 = 0.97
0.511*x – 0.126
R2 = 0.99
Direct
-ln(Nt/N0)
Direct
(a)
(b)
(c)
t (ps)
Figure 3.6 The isomerization rate from 100 trajectories from (a) the direct dynamics trajectories;
(b) IMLS PES using 685 data points; (c) IMLS PES using 1638 data points. The rates are k a =
0.59 ps-1, kb = 0.68 ps-1, kc = 0.51 ps-1.
47
CHAPTER 4
APPLICATION OF IMLS FITTING FROM ANALYTICAL POTENTIAL
ENERGY SURFACE TO DFT POTENTIAL ENERGY SURFACE ON
CO2 + O REACTION
4.1 Dynamics-driven IMLS fitting on CO2 + O B3LYP/3-21G PES
4.1.1 Introduction
Yeung et al.49 did cross-molecular-beam experiments on O(3P) + CO2 collisions. Products
included nonreactive scattering, isotope exchange and O-atom abstraction were observed in the
experiments. They measured the laboratory-frame time-of-fight (TOF) distribution and angular
distribution of the products in all three channels. The resulting total center of mass (c.m.)
translational energy distribution and angular distribution were derived from the experiment data.
Theoretically, they studied the minimum and saddle points of the potential energy surface using
CCSD(T)/aug-cc-pVTZ, B3LYP/6-311G(d), BMK/6-311G(d) electronic structure methods. W4
theory calculations were performed for stationary points. W4 theory represents a layered
extrapolation to the relativistic FCI basis set limit. They further performed direct dynamics
trajectories at B3LYP/6-311G(d) and BMK/6-311G(d) levels at different collision energies. The
theoretical c.m. angular and translational energy distributions for the products in the three
channels were calculated from the trajectories. The theoretical results at collision energy close to
the experiments were compared to experiments. There was good agreement between theoretical
and experimental results in both nonreactive scattering and isotope exchange channels. They
48
studied the dynamics of the reaction based on the experimental and theoretical results. They
found: (1).The theoretical product yield for isotope exchange reaction was overestimated as the
maximum impact parameter in direct dynamics trajectories 5a0 was below the collision diameter
in the experiments. (2).There was large energy transfer in the nonreactive scattering. There were
nonreactive scattering that a CO3 complex was formed. The O leaving the complex was the same
O that collided with CO2. (3).There would be a short-lived CO3 complex formed in the isotope
exchange reaction. (4).They thought that the O atom would attack the C atom first to form a
complex with CO2 in the oxygen abstraction reaction. The O2 was formed as the departing O
atom attached to another O atom.
Since there is no fitted potential energy surface for the reaction, the system is selected as the
PES of 4-atom for the study of IMLS fitting on ab initio or DFT PES.
4.1.2 Methods
The dynamics-driven IMLS method is used to fit the CO2 + O reaction for Ecoll below
57.7 kcal/mol. The reference energy of CO3 minimum is used. Yeung et al.49 studied the reaction
with collision energy as high as 149.9 kcal/mol using direct dynamics trajectories. There are
much more configurations in trajectories as the collision energy is higher. Many more
trajectories are needed to sample those configurations. Therefore, a small collision energy upper
limit is used for the fitting. The low electronic structure method B3LYP/3-21G is selected as the
cost of the calculations is cheap. The initial condition is started at reactant side CO2 + O. A
random phase of CO2 is selected from a CO2 trajectory with ZPE energy in each normal mode.
The initial orientation of CO2 and impact parameter b are sampled randomly. The maximum
impact parameter bmax is 2.6 Å. The O atom is 5 Å from the center of mass of CO2. The velocity
49
vector of O atom is parallel to the x-axis. The velocity of the O atom is calculated from the
collision energy as Eq. (4.1).
𝐸𝑐𝑜𝑙𝑙 =
1 2
𝜇𝑣
2 𝑟𝑒𝑙
(4.1)
where μ is the reduced mass of CO2 and O which is mco2mo/(mco2 + mo), vrel is the relative
velocity of O atom comparing to CO2. The vo is calculated by treating the vco2 = 0. The collision
energy of these trajectories Ecoll is 57.7 kcal/mol. The initial 12 points from the first trajectory are
calculated from the DFT method as seed points for the IMLS fitting. The IMLS is calculated
using previous data points. The high-dimensional model representation (HDMR) style
polynomial is used as the basis set. The initial basis is (6, 3). The basis increases to (10, 7, 5, 4)
as the data points increase. The IMLS with an order lower basis (9, 6, 4) is also calculated. If the
difference of the IMLS between the two basis is larger than the error tolerance value, ɛ, of
1 kcal/mol, the DFT calculation is performed at that point. The DFT point is added to the data
points. The DFT gradients are used to propagate the trajectory. Otherwise, the IMLS fitted
gradients are used in the trajectory. The internuclear distances are used as the fitting coordinates.
The order of the internuclear distances is given in Figure 4.1. The internuclear distances of all
data points are stored in the way that the shortest C-O distance is the first coordinate and the
second shortest C-O distance is the second coordinate. Since all three O atoms are the same, this
prevents selecting the same configuration as the data point. The integration algorithm is the
velocity Verlet method with a time step size 0.1 fs. The trajectory is terminated when any C-O
bond length is larger than 5 Å after 80 fs or the total energy is more than 10 kcal/mol off the
initial total energy. A certain number of trajectories are calculated starting from CO3 complex
before trajectories using the CO2 + O initial condition. Enough data points from CO3
50
configurations need be calculated before running trajectories starting from CO2 + O. The
nonreactive trajectories only sample CO2 + O configurations. There are much more nonreactive
trajectories than atom exchange trajectories at this collision energy. If trajectories are started at
first from CO2 + O, the trajectories are terminated once they enter the CO3 complex due to
convergence problems of total energy. It causes that the CO3 complex configurations are not
sampled enough by the trajectories. Therefore, the sampling trajectories are started first from the
CO3 complex. The total energy of CO3 trajectories is 30 and 50 kcal/mol with CO3 minimum as
the reference energy. The random phase CO3 points are selected from direct dynamics
trajectories in which the time length of CO3 is long enough. There are 40 sampling trajectories
started from CO3 with total energy 30 kcal/mol and 60 trajectories from CO3 with total energy
50 kcal/mol. Two hundred more trajectories with the CO2 + O initial condition are calculated
after those trajectories sampling CO3 configurations. There are final 992 data points selected
from these trajectories. There are very few trajectories terminated due to convergence problems
in the later trajectories. Besides, many trajectories are able to be calculated without even a single
DFT calculation. These indicate the fitted PES is good enough for trajectory calculation.
4.1.3 Results and discussion
The PES evaluations with respect to the DFT calculations for data point sampling are shown
in Figure 4.2. The slope increases as the increase of the DFT points. The slope is not as vertical
as that of dynamics-driven IMLS fitting of cis – trans isomerization of HONO. The difference
between the two reactions is that the trajectories of CO2 + O is usually less than 0.2 ps while
HONO isomerization trajectories are much longer. Therefore, the total PES evaluations from the
300 trajectories is 5 × 105, less than one tenth of the total PES evaluations 7 × 106. Besides, the
configurations included in cis – trans HONO isomerization are less than those in CO2 + O
51
reaction. Much more PES evaluations for the collision reaction are needed to obtain the vertical
slope for CO2 + O reactions. Although the slope is not vertical in Figure 4.2, enough data points
are calculated for the study of the reaction.
The opacity functions for isotope exchange reaction and nonreactive scattering at E coll of
57.7 kcal/mol from B3LYP/3-21G direct dynamics trajectories and IMLS trajectories are shown
in Figure 4.4. The opacity functions are functions that give the probability of a reaction at a
given impact parameter. It is calculated from Nreact,b/Ntotal,b. The Nreact,b and Ntotal,b are the number
of reaction trajectory and number of total trajectory for given impact parameter b. The b range is
divided into ten ranges. These b ranges start from 0 to 2.45 Å with 0.25 Å as the increment.
Trajectories falling into each range are counted as trajectories with b value, the right bound of
the range. The x-axis is impact parameter of the collision reaction. The y-axis is the probability
of a reaction. Both the opacity reactions from DFT calculations and IMLS fitted potential are
calculated from 400 trajectories. The opacity functions are average of two sets of 200 trajectories.
The error bars are the standard deviation between the two sets of trajectories. The initial
conditions of the trajectories from IMLS potential started at CO2 + O are the same as those of the
direct dynamics trajectories. Since it is either an atom exchange reaction or a nonreactive
scattering for a collision at this Ecoll, the sum of the probability of atom exchange reactions and
nonreactive scattering is 1 at a given impact parameter. There are 992 data points calculated from
sampling trajectories for the IMLS potential. There is a good agreement between opacity
functions from direct dynamics trajectories and classical trajectories from IMLS potential. The
opacity functions for atom exchange reaction from the IMLS potential is slightly smaller than
those from direct dynamics trajectories. Some atom exchange trajectories are predicted to be
nonreactive scattering trajectories by the IMLS potential. There are oscillations in the opacity
52
functions. A large set of trajectories can be calculated to improve the oscillations in the opacity
functions. Or these oscillations reflect the possibility of the reaction due to the physics of the
reaction. There are some trajectories with total energy convergence problems from the IMLS
PES. These total energy convergence is caused by the inaccuracy introduced from the IMLS
forces. The accumulation of the inaccuracy leads to the total energy convergence problem. There
are 16 trajectories from the IMLS potential terminated due to the convergence problem. These 16
direct dynamics trajectories using the same initial conditions are 15 atom exchange reaction
trajectories and 1 nonreactive scattering trajectory. The opacity functions at other two collision
energy Ecoll of 23.1 and 34.6 kcal/mol from direct dynamics trajectories and classical trajectories
from IMLS potential are shown in Figure 4.5 and 4.6. Both the opacity functions at the two
different collision energy are calculated from 400 trajectories. The opacity functions are average
from two set of 200 trajectories. The error bars are the standard deviation between the two sets of
trajectories. There is a good agreement between opacity functions from direct dynamics
trajectories and classical trajectories from the IMLS potential at the two collision energies. There
are eight trajectories having convergence problems at an Ecoll of 23.1 kcal/mol. Six of them are
nonreactive scattering trajectories in direct dynamics trajectories. Two of them are atom
exchange reactions in direct dynamics trajectories. There are three classical trajectories from
IMLS potential having convergence problems at an Ecoll of 34.6 kcal/mol. Two of them are atom
exchange reactions in direct dynamics trajectories. One is nonreactive scattering trajectory in
direct dynamics trajectory. However, the time length of the nonreactive scattering trajectory is
0.323 ps. It indicates that the CO3 complex is formed in the nonreactive scattering. The
convergence problems at three different collision energy shows that the CO3 complex regions are
not fitted well in the potential. More trajectories started from CO3 need be calculated to select
53
data points in the CO3 configurations. More trajectories of isotope exchange reaction have
convergence problems at high collision energy, Ecoll of 34.6 and 57.7 kcal/mol. More trajectories
of nonreactive scattering have convergence problems at an Ecoll of 23.1 kcal/mol. The opacity
functions from direct dynamics trajectories of B3LYP/3-21G at Ecoll of 23.1, 34.6 and
57.7 kcal/mol are shown in Figure 4.6. The opacity functions for isotope exchange reaction
increase as the Ecoll increases for b ≤ 1.5 Å. There are no isotope exchange reactions for b ≥
1.75 Å.
Compared to the fitting of PES of HONO isomerization, the main difference is that the
trajectories from PES of CO2 + O is very short, 0.2 – 0.3 ps comparing to 2 – 3 ps for trajectories
from PES of HONO isomerization. Therefore, the PES of current reaction should be easier to fit
as the longer the trajectory the more accurate of the fitted PES needs be to meet the convergence
of total energy. Besides, there are fragments in the PES of CO2 + O reaction. This does not make
the fitting more difficult as it only includes more configurations in the PES. Besides, the
collision energy of the CO2 + O trajectory is 57.7 kcal/mol with the reference energy of CO2 + O.
The total energy of HONO isomerization is about 49 kcal/mol with reference energy of cis –
HONO minimum. The total energy for the two reactions is comparable.
The total energy of the CO2 + O trajectory from the direct dynamics trajectory and IMLS
fitted PES with 992 DFT points is shown in Figure 4.8. The large total energy drift at the
beginning and at the end of the trajectory is due that the PES is multireference for CO2 + O
configuration. The C-O distance is large at the beginning and the end of the trajectory. Therefore,
B3LYP/3-21G method has some problems in the long C-O range of the PES. The total energy
drift for the trajectory from IMLS fitted PES with 992 DFT points is less than 1 kcal/mol during
the 0.172 ps time length.
54
4.1.4 Conclusion and future work
The dynamics-driven IMLS method has been used to fit the PES of CO2 + O at
B3LYP/3-21G with Ecoll below 57.7 kcal/mol. The IMLS PES is fitted with 992 data points.
There is an excellent agreement between the opacity functions from IMLS PES and from direct
dynamics trajectories. However, there are few trajectories with convergence problems for the
IMLS PES. These results show that more data points are needed in the CO3 region. The current
PES is a type of AB2 + C of PES of four atoms. It enriches the pool of the PES fitted by the
dynamics-driven IMLS. This is the second PES fitted using this method. The first PES is HONO
isomerization at HF/cc-pVDZ. Some treatment has to be done differently when using trajectories
to sample the data points for the PES comparing to the fitting of HONO isomerization. The
trajectories with initial configuration CO3 need be used first for data selection.
The dynamics-driven IMLS fitting can be used to next fit PES of CO2 + O at B3LYP/3-21G
with Ecoll below 149.9 kcal/mol. Many more data points are needed for the fitting of the PES at
the high Ecoll. Much more trajectories are needed to select enough data points for the fitting. The
ab initio or DFT calculations is only performed for data points in the trajectories. Therefore,
other method can be used the same way as B3LYP/3-21G in the method. The difference of
computational time needed for the fitting using different methods only comes from the data point
calculation. The PES of CO2 + O can be fitted using the dynamics-driven IMLS using the
coupled cluster method, CCSD(T)/aug-cc-pVTZ.
55
4.2 IMLS fitting on CO2 + O B3LYP/3-21G PES of specified range
4.2.1 Introduction
As in the previous section, good agreement is observed between direct dynamics trajectories
and trajectories from the fitted potential using dynamics-driven IMLS method. However, those
configurations that are not frequented enough in trajectories can be only sampled by running a
large number of trajectories. The fitted potential from the method based on trajectory will miss
these configurations if not enough trajectories are calculated. The exact configurations in the
trajectories are not known. For example, if a random configuration is given, it is hard to know
whether the configuration is included in the potential or not. The H2O2 analytical PES has been
fitted for the specified range using IMLS in Chapter 2. In the fitting, random points are used to
select data points for IMLS. There is a problem to use this method to fit a real ab initio or DFT
PES. Even for a very low level ab initio or DFT method, the cost is much more expensive than
the analytical PES. Therefore, if there is only a small number of points below the cutoff energy
in the specified range for the fitting, it is not practical to use this method to select data points. An
arbitrary range cannot be used for the data selection using random point to fit an ab initio or DFT
PES. Therefore, a range should be found including many points below cutoff energy. What
configurations are below cutoff energy in the specified range of the PES should be known to find
the range including many points below cutoff energy. This section studies the method that fits
the specified range of the PES below a cutoff energy.
4.2.2 Methods
The range for the fitting is determined using spherical coordinates. The coordinates are given
in Figure 4.1. The r1, r2 and α are the first and second C-O distance and the angle between r1 and
r2. The r, θ and φ spherical coordinates determine the third O atom. The range of PES is r1 (1.0 –
56
1.125 Å), r2 (1.0 – 1.125 Å), r (1.2 – 1.9 Å), α (80 – 179°), θ (0 – 89°), φ (0 – 360°). This range
is selected based on the range for each coordinate in trajectories of the collision reaction in
previous section. The r1, r2 and r should increase to (1.0 – 1.3 Å), (1.0 – 1.3 Å), (1.2 – 5 Å)
respectively to include all points in the trajectories of the CO2 + O reaction. However, the
smaller range is selected to be fitted first. Based on the results of the smaller range of the PES,
the complete range can be fitted later. A uniform grid is calculated to find the points below cutoff
energy 70 kcal/mol in the range of the PES. The grid values of r1, r2 are 1.0 Å, 1.04 Å, 1.08 Å
and 1.125 Å for the r1 and r2 range. Only points of r1 ≤ r2 are calculated. The step size for dr, dα,
dθ and dφ are 0.1 Å, 9.9°, 29.7° and 36° for r, α, θ and φ range respectively. The internuclear
distance of the molecule is used as the fitting coordinates. The points with the same internuclear
distance as points in r1 ≥ r2 can be found in the range of r1 ≤ r2. Therefore, fitting the PES of r1 ≤
r2 is equivalent to fitting of PES with both r1 ≤ r2 and r1 ≥ r2. A second grid of PES of smaller
range and smaller step size is calculated based on this first grid. Points below 70 kcal/mol can be
found more accurately in a denser grid. The grid values of r1 and r2 are the same as the first grid.
The step size of dr, dα, dθ and dφ are 0.1 Å, 9.9°, 9.9° and 12° respectively. The step size of dθ
and dφ are smaller than that of the first grid. The range of r for second grid is shown in Table 4.1.
The α range is dependent on the r1, r2 and r. Even the largest α range in the new grid (100, 179°)
is smaller than the α range (80, 179°) of the first grid. The specific α range for each r1, r2 and r is
not shown here.
The θ is the angle between the third C-O bond and the vector perpendicular to the OCO plane.
The OCO plane refers to the first two O and C atoms. Based on the first grid of the PES, the
general description of the PES in the range is given below. If θ is small, the energy of CO3 does
not change much by rotating the third C-O. This is changing φ while fixing r1, r2, r, α and θ. If θ
57
is large, the energy of CO3 is highly dependent on φ. The energy can increase to as high as more
than 1000 kcal/mol when the third C-O bond is on top of either one of the other two C-O bond
by changing φ. There is too much repulsion in these configurations. The energy only decreases to
as low as 60 or 70 kcal/mol when the third C-O bond is far away from the top of the two C-O.
The projection of their third C-O bond is close to the angular bisector of the complementary
angle of the O-C-O angle for these low energy points. The r1 and r2 affects the energy more than
r, α and θ. The energy is small at large r1, r2, r and α.
The range of points below 70 kcal/mol found in the second grid is shown in Table 4.2. This
however is an approximate range as the φ range should be different for different r, α and θ values.
However, the same φ range applies to all the values in r, α and θ ranges. Besides, the range of r1
and r2 is calculated based on a single value r1 and r2 from the range. The values are shown below
the r1 and r2 ranges in the parentheses in the table. Based on this range below 70 kcal/mol in
Table 4.2, a new range is determined to include as many points in this range as possible for the
fitting. The range of the PES is r1 (1 – 1.125 Å), r2 (1 – 1.125 Å), α (120 – 175°), θ (30 – 80°)
and φ (240 – 276°). The r range is different for different r1 and r2. Table 4.3 gives the r range at
different r1 and r2 range. There are 28 seed points with energy ≤ 70 kcal/mol from a sparse grid.
Fifty random points from the range are used to select extra data points. The square difference of
the two successive order IMLS are calculated for the 50 points. The five points with the largest
square difference from every 50 points are added to the data points. The basis is final at
(10, 7, 5, 4) with enough data points. Fifty random points are used to calculate the RMS error of
energy and gradients once every 50 new data points are added. The procedure of data point
selection is repeated. Finally, there are 528 data points calculated for the PES.
58
4.2.3 Results and discussion
As described in method section 4.2.2, the points below 70 kcal/mol are summarized in Table
4.2 based on the second grid of the PES. As the energy of PES indicated, the range of r, α, θ and
φ are highly dependent on the values of r1 and r2 for the points ≤ 70 kcal/mol. The range of r, α, θ
and φ are given at the different r1 and r2 ranges. Only points with r1 ≤ r2 are considered. Therefore,
the six cells at the left bottom of the table for points of r1 ≥ r2 are empty. Besides, some cells at
the top right of the table are empty. There are no points below 70 kcal/mol found in these ranges.
Finally, there are six r1 and r2 ranges with points below cutoff energy.
There are some general trends for the distribution of points below 70 kcal/mol. (1).The
ranges of other coordinates of the points below 70 kcal/mol are large as r1 and r2 values are large.
(2).In each six r1 and r2 ranges, the ranges of α, θ and φ are large as r values are large. The third
C-O bond is less repulsed by two other C-O bonds at large r value. (3).The θ and φ ranges are
large as α values are large. The two C-O bond is less repulsed at large α values. (4).The θ value
is the angle between the third C-O and the vector perpendicular to the OCO plane. The CO3 is
closer to a planar geometry if the θ value is large. The CO3 is far from planar geometry if the θ
value goes to zero. There is more repulsion between the third C-O bond and two other C-O
bonds in the close planar geometry. Therefore, the φ range is large as θ values are small. The
range of the φ value is the range of the rotation of the third C-O bond. Besides, there are not
much configurations at the small θ value since the rotation of the third C-O bond includes small
number of configurations. Therefore, the φ range in Table 4.2 is determined based on the φ range
of the largest θ value in the grid.
These general trends can be applied to the points below the cutoff energy. How these trends
affect the determination of the range for the fitting is illustrated using the r1 (1.06 – 1.1 Å), r2
59
(1.06 – 1.1 Å) range, one of the six ranges. (1).First of all, the range is large as r1 and r2 values
are large in the six ranges. (2).The α range is large for large r in (1.5 – 1.9 Å). The α range (140 –
179°) is determined based on the α range for different r values. (3).The θ range is large for large r
and large α values in r (1.5 – 1.9 Å) and α (140 – 179°). The θ (30 – 89°) is determined based on
different r and α values. (4).The φ range is large for large r, α and small θ. The φ range for the
largest θ value for r = 1.9 Å are (216 – 288°), (228 – 300°), (240 – 300°). The φ range is different
for different α values. There are three φ ranges for r = 1.9 Å. The φ range for r = 1.5 Å is (240 –
264°). There are no points below 70 kcal/mol for large α values when r = 1.5 Å. Based on the
analysis of φ range for different r and α values at largest θ value, the φ range is determined as
(240 – 276°) for all r and α values in the range. Therefore, there are points below 70 kcal/mol left
off and points above 70 kcal/mol included in the determined new range as comparing to the φ
range for r = 1.5 Å and 1.9 Å case. Besides, there is small φ range starting to appear for r =
1.8 Å and α = 169°. The φ range of small values is large for large r and α values. There is no
such φ range for small r and α values. The same analysis is used to determine the range for the
rest of r1 and r2 values.
The RMS errors of the energy and gradients with respect to the number of data points are
shown in Figure 4.9. There are some noticeable fluctuations since only 50 random points are
used to select new data points and to calculate the RMS error. There are less fluctuations of the
RMS errors with large number of random points to select data points. There are five data points
added from every 50 random points. The RMS errors are calculated after 50 new data points are
added. The number of data points that are added before each RMS error is calculated also affects
the fluctuation of the RMS error. The RMS errors are 0.19 kcal/mol and 13 kcal/(mol*Å) for
energy and gradients with final 528 data points. As there are many points below cutoff energy in
60
the new range, the cost of fitting the range is relatively cheap. The RMS errors in red are from
fitting the range given in method section. The range is selected to avoid geometries close to
planar configurations. The RMS errors in blue are those from fitting the range without avoiding
selecting such geometries by increasing range α from (120 – 175°) to (120 – 179°) and θ from
(30 – 80°) to (30 – 89°). Both the RMS errors of energy and gradients are much larger than those
from the range without points close to planar geometries. The final RMS errors of energy and
gradients are 0.38 kcal/mol and 30 kcal/(mol*Å) with 528 data points. The large RMS errors are
due that the gradients with respect to internuclear distances for the data points close to planar
geometries are incorrect. In order to calculate the gradients with respect to the internuclear
distance, ∂E/∂q, the gradients with respect to the Cartesian coordinates, ∂E/∂x need be converted
to the gradients with respect to internuclear distances, ∂E/∂q. The derivatives of Cartesian
coordinates with respect to internuclear distances, ∂x/∂q need be calculated in order to calculate
∂E/∂q = (∂E/∂x) (∂x/∂q). The derivatives of internuclear distances with respect to Cartesian
coordinates, ∂q/∂x are easy to calculate. Singular value decomposition is used to calculate the
inverse matrix of derivatives of internuclear distances with respect to Cartesian coordinates
(∂q/∂x)-1. The derivatives of internuclear distances with respect to Cartesian coordinates for 4atom systems ∂q/∂x is 12×6 matrix M as Eq. (4.2).
𝜕𝑞1
𝜕𝑥1
𝜕𝑞1
𝑀 = 𝜕𝑥2
⋮
𝜕𝑞1
[𝜕𝑥12
𝜕𝑞2
𝜕𝑥1
𝜕𝑞2
𝜕𝑥2
⋮
𝜕𝑞2
𝜕𝑥12
𝜕𝑞6
𝜕𝑥1
𝜕𝑞6
⋯
𝜕𝑥2
⋱
⋮
𝜕𝑞6
⋯
𝜕𝑥12 ]
⋯
61
(4.2)
The matrix can be decomposed by singular value decomposition as M = UΣV*. The V* is the
conjugate transpose matrix of V. The inverse matrix is calculated as M-1 = M* = VΣ-1U*. Since
Σ is a diagonal matrix of non-negative real numbers, Σ-1 is calculated as the reciprocals of all the
diagonal real numbers in Σ. For example, the derivatives of internuclear distances with respect to
the Cartesian coordinates M of a geometry from the spherical coordinates (r1 = 1.018 Å, r2 =
1.116 Å, r = 1.9 Å, α = 147.1°, θ = 90°, φ = 248.6°) can be decomposed into UΣV*. It is a planar
geometry as θ = 90°. The diagonal nonnegative numbers in Σ are 2, 1.76, 1.70, 1.06, 0.94 and
7.5×10-12. The near zero value 7.5×10-12 is the result of the planar geometry. Further calculation
of Σ-1 introduces a huge number from the near zero value. This causes the large unrealistic
gradients due to introduction of the huge number from the calculation of Σ-1 using Σ. The range
of gradients to Cartesian coordinates ∂E/∂x is below 103 kcal/(mol Å). But the gradients to
internuclear distance ∂E/∂q using the singular value decomposition are at the order of 109
kcal/(mol Å) for this planar geometry. Gradients of a point not planar but close to planar, e.g. θ =
85° are reasonable numbers. But the gradients are not correct through the conversion from ∂E/∂x
to ∂E/∂q. Since incorrect gradients are introduced in the data points, this affects the accuracy of
the evaluation of all the gradients of those random points for error estimation. Besides, a less
important factor is in the calculation of RMS errors. Since the gradients of the random points
close planar geometry are incorrect, it is impossible to predict whether the IMLS gradients are
correct at these random points. These problems cause the higher RMS errors for PES with range
α (120 – 179°) and θ (30 – 89°) comparing to with range α (120 – 175°) and θ (30 – 80°).
4.2.4 Conclusion and future work
The initial range for the fitting is selected as r1 (1.0 – 1.125 Å), r2 (1.0 – 1.125 Å), r (1.2 –
1.9 Å), α (80 – 179°), θ (0 – 89°) and φ (0 – 360°). As discussed above, it is not possible to fit
62
the range of the PES even at a low level theoretical method. Grids of the range of the PES are
calculated to know the shape of the PES. A new range for the fitting, r1 (1.0 – 1.125 Å), r2 (1.0 –
1.125 Å), α (120 – 175°), θ (30 – 80°) and φ (240 – 276°) is determined to include points below
70 kcal/mol based on the grids. The r range is different for different r1 and r2 values. The r range
is shown in Table 4.3. The new range can be fitted using the random point strategy discussed in
Chapter 2. The RMS error of the energy and gradients with 528 data points are 0.19 kcal/mol and
13 kcal/(mol*Å), respectively. The RMS errors are much larger if points close to planar
geometry are included for the fitting. Therefore, an accurate PES can be fitted with a small
number of data points using the method. However, some points below the cutoff energy are left
off. Besides, there are some points above the cutoff energy included in this new range. Therefore,
it is necessary to show what points below cutoff energy are missed and what points above cutoff
energy are included in the current range of PES. This can be easily done by comparing the
current range and the points below 70 kcal/mol in the grid of the PES. The fitting is performed at
a low level DFT method. A fitted PES of a low level theoretical method is not very useful in the
study of the system when compared to the recent developments made in the field of theoretical
chemistry. These are preliminary results for fitting a highly accurate PES. Even though the PES
is fitted at low level theoretical method, the results show the number of data points needed for
the PES of a specified range. However, there are points below cutoff energy left in the new range
and the results cannot show the number of data points needed for all the points below cutoff
energy in the range. Future work is needed to find out the number of data points for fitting all the
points below cutoff energy. The PES of low level theoretical method agrees with highly accurate
PES of more expensive theoretical method qualitatively, but not quantitatively. The results at
low theoretical level give an idea of the number of data points needed for the fitting of highly
63
accurate PES. There are many ab initio or DFT calculations needed in the grids of the PES.
Besides, ab initio or DFT calculations are still needed to find out some random points below
cutoff energy during data point selection. Therefore, a large number of ab initio or DFT
calculations is needed in this method. Ab initio or DFT calculations can only be used to calculate
data points to fit PES at expensive theoretical method. Therefore, much more work is needed to
use the current method to fit PES at highly accurate theoretical method. One way is to fit the low
ab initio or DFT PES first using current method. The low theoretical level data points are then
replaced by expensive theoretical data points. This can be an easy and straightforward way to use
current method to fit PES at expensive theoretical method. However, there can be better ways to
use the current method to fit PES at highly accurate theoretical method.
It is not necessary to know what configurations are included in the fitting for dynamicsdriven IMLS. The configurations are included in the trajectories to sample the data points. Grids
of PES are needed to find a range to include mainly the points below cutoff energy. However,
the range determined in the study still misses many points below the cutoff energy. It also
includes many points above the cutoff energy. It is necessary to find out what points below
cutoff energy are missed and what points above cutoff energy are included for the range. In order
to include all the points below cutoff energy, new range should be included besides the current
range of the PES. Only increasing the range of each coordinate not only include more points
below cutoff energy, but also include more points above cutoff energy. Discrete ranges need be
used to include all the points below the cutoff energy. The discrete ranges can be determined
based on the grids of the PES. Further study is necessary on how to find out these discrete ranges
to include most configurations below cutoff energy while not include too many configurations
above cutoff energy. Since the current range does not include all the points below cutoff energy,
64
the number of data points needed is only for the current range, not for all the points below cutoff
energy. It is only possible to know the number of data points needed for an accurate PES by
fitting the range of the PES including all the points below cutoff energy.
4.3 An algorithm for configurations below cutoff energy in ab initio or DFT potential
energy surface
4.3.1 Introduction
The previous section finds the range of the PES for points below cutoff energy based on the
grids of the PES. The range determined does not include all the points below cutoff energy.
Several discrete ranges should be determined to mainly include the points below the cutoff
energy by comparing to the grids of the PES. However, it is difficult to find all the points below
cutoff energy using these discrete ranges. This section studies an algorithm that can
automatically find all the points below cutoff energy. If an algorithm can find all the points
below cutoff energy, these points can then be fitted. Therefore, the PES below the cutoff energy
can be fitted without calculating the grids of the PES. The algorithm may be an easier way to fit
the points below cutoff energy in a range of the PES.
4.3.2 Methods
A configuration below cutoff energy is found through random points in the range. The
specific value of the spherical coordinates of the configuration is r1 1.12 Å, r2 1.09 Å, r 1.82 Å, α
160°, θ 60° and φ 260°. The search for configurations below cutoff energy 60 kcal/mol in the ab
initio or DFT potential energy surface is started from this configuration. The search is performed
at the PES of CO2 + O at B3LYP/3-21G level. The range using the spherical coordinates is the
same as that in section 4.2, r1 (1.0 – 1.125 Å), r2 (1.0 – 1.125 Å), r (1.2 – 1.9 Å), α (100 – 179°),
65
θ (0 – 89°) and φ (0 – 360°). The only difference is that α range in section 3.3 is (80 – 179°), a
little larger than the α range here. However, there are not many points below 60 kcal/mol in α
(80 – 100°). The cutoff energy 60 kcal/mol is selected close to Ecoll = 57.7 kcal/mol used in
dynamics-driven IMLS fitting. The step size for r1, r2, r, α, θ and φ are dr1, dr2, dr, dα, dθ and dφ
for minor change of the configuration from the previous configuration. They are 0.02 Å, 0.02 Å,
0.1 Å, 5°, 5° and 10°, respectively. (1).The φ1 value is the angular bisector of the complementary
angle of α, π + α/2. The search of points below cutoff energy is started by increasing dφ from φ 1
until the energy is above the cutoff energy or the φ is larger than the upper φ bound. The search
continues by decreasing dφ from φ1 - dφ until the energy is above the cutoff energy or the φ is
below the lower φ bound. Once a circle of (1) is complete, then (2) proceeds. (2).The θ increases
by dθ from the θ value of the configuration. The (1) scheme is performed for every new θ value.
The starting φ is the same φ1 for every new θ value. The θ increases to the upper θ bound value
or when there are no points below cutoff energy at that θ value. The θ next decreases to the lower
θ bound or the value when there are no points below cutoff energy for that θ value. Once a circle
of (2) is complete, the (3) proceeds. (3).The α increases by dα. The α increases to the upper
bound α value 179°. The (1), (2) scheme are performed for every new α. The starting value for θ
is the maximum θ value of the previous circle of (2) for all the circles of (2), but the first circle of
(2). The starting θ value is the θ value of the initial configuration in the first circle of (2). The
starting φ is the φ1 value defined as in (1). Since the α is a new value, the φ1 is different
comparing to the φ1 for the old α value. It is α + (2π – α)/2. Once the α increases to the maximum
value, it decreases by dα. It decreases to the lower bound α value 100°. Once the circle of (3) is
complete, the (4) proceeds. (4).The r value increases by dr to the upper bound r value. The (1), (2)
and (3) scheme are performed for every new r value. The starting α is the value of the beginning
66
configuration. The starting θ and φ are the same as in (3). The r next decreases to lower bound r
value by dr. Once the circle of (4) is complete, the (5) proceeds. (5).The r2 value increases to the
upper bound r2 value by dr2. The (1), (2), (3) and (4) scheme are performed for every new r2
value. The starting r value is the r value of the initial configuration. The starting values of α, θ
and φ are the same as in (5). The r2 value decreases to the lower bound r2 value by dr2. Once the
circle of r2 is complete, the (5) starts. (6).The r1 value increases to the upper bound r1 value by dr1.
The starting value of r2 is the r2 value of the initial configuration. The starting values of r, α, θ
and φ are the same as in (5). The (1), (2), (3), (4) and (5) schemes are performed for every new r1
value. The r1 next decreases to r1 lower bound value by dr1. Figure 4.10 shows an illustrated
example of the procedure. It shows the starting values, hierarchy and the termination requirement
of each circle (1), (2), (3), (4), (5) and (6). The red and blue signs indicate once a circle is
complete a new value is given to the current level from a circle of one level higher. A new circle
at this level is started with this new value. The points below the cutoff energy are found through
the procedure. Since there are no termination steps used for the moving of r1, r2, r and α, there are
some points above the cutoff energy calculated in the algorithm.
4.3.3 Results and discussion
The percentage of points below 60 kcal/mol comparing to the total number of points is 71%.
There are 46941 points below 60 kcal/mol. There are 19248 points above 60 kcal/mol. The
algorithm only stops further moving θ of the configuration by dθ when there are no points below
cutoff energy at that θ. The grids of the PES shows that there are no points below cutoff energy
once out of the range of θ for points below cutoff energy either by further decreasing the lower
bound or increasing the upper bound of the range. Therefore, it does not search further in the θ
range as all the points are above cutoff energy if the search continues in θ. A new α is given to
67
start another circle of changing θ. It stops moving φ by dφ once the energy is above the cutoff
energy. It samples all the points below cutoff energy in the φ range without searching out of the
φ range. A new θ starts another circle of changing φ. However, there is no such step to stop
moving of dr1, dr2, dr and dα when the search is in the range of points above cutoff energy. No
such rule on these coordinates makes the algorithm simple. Besides, there are not many points
calculated even the search enters r1, r2, r and α range above cutoff energy as the termination rules
of θ and φ makes very few calculations under these r1, r2, r and α values. There are some points
above cutoff energy calculated in this way.
The points below 60 kcal/mol from the algorithm are compared to the regions below 70
kcal/mol from a dense grid calculated at the same DFT level. The dense grid is the second grid in
previous section. The dense grid is calculated at r1 = 1.0 Å, 1.04 Å, 1.08 Å, 1.125 Å, r2 = 1.0 Å,
1.04 Å, 1.08 Å, 1.125 Å, with step size dr = 0.1 Å, dα = 10°, dθ = 10° and dφ = 12°. The ranges
of r and α are dependent on r1 and r2. The φ and θ ranges are (0, 89°) and (0, 360°), respectively.
It is the second grid calculated in section 4.2. The cutoff energy 70 kcal/mol is used in the dense
grid as there are still not many points below 60 kcal/mol in the dense grid. The comparison
between the regions in the dense grid and from the algorithm is not at the same cutoff energy.
The regions in the dense grid cannot be completely matched with those from the algorithm as the
cutoff energy is different. However, it is adequate to see whether the algorithm finds all the
points below cutoff energy by comparing it to the dense grid. The Tables 4.4 – 4.8 show the
points below 70 kcal/mol in the dense grid and points below 60 kcal/mol sought by the algorithm.
The “a” is the points from the dense grid. The “b” is the points sought by the algorithm. The
points at r1 = 1.04 Å, r2 = 1.08 Å, r = 1.8 Å, α = 147.8° and 155.6° of “a” and r1 = 1.04 Å, r2 =
1.09 Å, r = 1.82 Å, α = 150° and 155° of “b” are shown in Table 4.4. The algorithm finds the
68
points that are calculated from the dense grid. The θ range of “b” is smaller than that of “a”. The
cutoff energy used in the algorithm is smaller than that in the dense grid. The comparison of the
points sought by the algorithm and calculated from the dense grid at other r1, r2 values, that are
(1.0, 1.125 Å), (1.04, 1.08 Å), (1.04, 1.125 Å), (1.08, 1.08 Å), (1.08, 1.125 Å) and (1.125, 1.125
Å) are shown in Tables 4.4 – 8. Table 4.4 shows the φ range for the different r, α and θ values.
The φ range is different for different r, α and θ values for points below cutoff energy. The φ
range is for all the values for α and θ range in Table 4.5 – 8. Therefore, the φ range is determined
based on the points below cutoff energy of different α and θ values.
The regions of “a” below 70 kcal/mol are (1).r1 1.0 Å, r2 1.125 Å, r 1.9 Å, α 156 – 179°, θ 30
– 89°, φ 252 – 276°; (2).r1 1.04 Å, r2 1.08 Å, r 1.8 – 1.9 Å, α 150 – 179°, θ 20 – 89°, φ 240 – 276°
and r1 1.04 Å, r2 1.08 Å, r 1.8 – 1.9 Å, α 171 – 179°, θ 0 – 60°, φ 84 – 96°; (3).r1 1.04 Å, r2 1.125
Å, r 1.7 – 1.9 Å, α 140 – 179°, θ 0 – 89°, φ 240 – 288° and r1 1.04 Å, r2 1.125 Å, r 1.6 – 1.9 Å, α
159 – 179°, θ 0 – 89°, φ 84 – 96°; (4).r1 1.08 Å, r2 1.08 Å, r 1.5 – 1.9 Å, α 140 – 179°, θ 0 – 89°,
φ 240 – 288° and r1 1.08 Å, r2 1.08 Å, r 1.7 – 1.9 Å, α 169 – 179°, θ 0 – 89°, φ 72 – 96°; (5).r1
1.08 Å, r2 1.125 Å, r 1.4 – 1.9 Å, α 130 – 179°, θ 0 – 89°, φ 228 – 276° and r1 1.08 Å, r2 1.125 Å,
r 1.6 – 1.9 Å, α 159 – 179°, θ 0 – 89°, φ 72 – 96°; (6).r1 1.125 Å, r2 1.125 Å, r 1.3 – 1.9 Å, α 120
– 179°, θ 0 – 89°, φ 228 – 288° and r1 1.125 Å, r2 1.125 Å, r 1.6 – 1.9 Å, α 159 – 179°, θ 0 – 89°,
φ 72 – 96°. The regions of “b” found by the algorithm are the following. These regions are
labelled according to the number of regions of “a” they are corresponding to. Therefore, they are
(2).r1 1.04 Å, r2 1.09 Å, r 1.82, α 150 – 175°, θ 30 – 85°, φ 240 – 280°; (3).r1 1.04 Å, r2 1.1 Å, r
1.72 – 1.82 Å, α 150 – 175°, θ 30 – 85°, φ 240 – 260°; (4).r1 1.08 Å, r2 1.09 Å, r 1.52 – 1.82 Å, α
140 – 175°, θ 10 – 85°, φ 240 – 280°; (5).r1 1.08 Å, r2 1.1 Å, r 1.52 – 1.82 Å, α 130 – 175°, θ 0 –
85°, φ 230 – 280°; (6).r1 1.1 Å, r2 1.1 Å, r 1.42 – 1.82 Å, α 125 – 175°, θ 0 – 85°, φ 240 – 280°.
69
Most regions below 70 kcal/mol of “a” are found corresponding to the regions in the algorithm.
But the range of these corresponding regions of “b” is smaller. The cutoff energy is 60 kcal/mol
for the algorithm as 70 kcal/mol in the dense grid. Due to the step size dr, dα and dθ, the
maximum r, α and θ of b are 1.82 Å, 175° and 85° in the search respectively. Therefore, no
points with r ≥ 1.82 Å, α of 175° and θ of 85° in regions of b can be compared to those points in
a. The points from regions of “b” are compared to those from regions of “a” with closest r1, r2
and r values due the grid values of each coordinate. Another group of points in regions of “a” are
missed in regions of “b”. They are the points with φ between 84 – 96° in regions of a (2), (3), (4),
(5) and (6). The search should go through a range of φ above cutoff energy before finding the
points in the small φ range. The rule to stop moving φ does not allow the search to go through
the range. Another starting point with the small φ value can be used to search these points with
small φ values. However, as most points are in the regions with large φ values, the majority of
the points are found without using another starting point. Besides, there are much more these
points with small φ values at large α values. As the points are at large α values, the geometry of
these points are close to the points with large φ values. The O-C-O is close to a line at large α
values. The points with small φ is close to points with large φ due to symmetry. Another group
of points in regions of “a” are missed in regions of “b”. They are points with r = 1.6 Å in region
(3) of “a”, r = 1.4 Å in region (5) of “a” and r = 1.3 Å in region (6) of “a”. The points are not
below 60 kcal/mol but below 70 kcal/mol at these small r values in regions of “b”. Therefore, it
is the same as the smaller α, θ and φ range in “b” due to the smaller cutoff energy.
The algorithm can find most points below cutoff energy. A fitting is performed based on the
algorithm. There are two search in the fitting. The first search is used to select data points. The
second search is to calculate the RMS errors. The first search is started at r1 of 1.12 Å, r2 of 1.09
70
Å, r of 1.82 Å, α of 160°, θ of 60°, φ of 260°. The step size is dr1 of 0.01 Å, dr2 of 0.01 Å, dr of
0.05 Å, dα of 10°, dθ of 10°, dφ of 20°. The procedure used in the dynamics-driven fitting is
used in the fitting. The first 12 points in the search are used as seed points. The fitting is started
at (6, 3) basis. The basis finally increases to (10, 7, 5, 4) with enough data points. During the
search, the difference of two successive order IMLS is calculated. If the difference is larger than
1 kcal/mol, the point is added as data point. There are 1287 data points calculated after the search.
The number of total points below cutoff energy comparing to the number of data points during
the search is shown in Figure 4.11. Another search is started to calculate the RMS error of the
fitting. The search is started at r1 of 1.1 Å, r2 of 1.06 Å, r of 1.88 Å, α of 154°, θ of 14°, φ of 226°.
The step size is dr1 of 0.02 Å, dr2 of 0.02 Å, dr of 0.01 Å, α of 10°, θ of 10°, φ of 20°. The RMS
error is calculated at every 10 points. The final RMS error of energy is 0.32 kcal/mol from 1338
data points. The RMS error of gradients is 18 kcal/(mol Å). As all the grid values of r1, r2 and r in
the search for the RMS error calculation are included in these values in the search for data points,
there is a bias on the RMS error. The discrete values of α, θ and φ between the two search are
alternate. Therefore, there is no bias for the angle values of the points to calculate the RMS error.
4.3.4 Conclusion and future work
The algorithm successfully finds the regions below cutoff energy. There are some points of
small φ values missed in the search comparing to the grids of the PES. These points of small φ
value can be found using another starting point with small φ value. However, most points are in
the large φ range. Further, an accurate fitting is performed on the regions using the algorithm.
There are 1287 data points needed in the fitting. The estimated RMS error is 0.32 kcal/mol.
Therefore, the fitting method based on the algorithm is another way to fit a range of PES below
cutoff energy. Comparing to the method in section 4.2, there is no need to calculate the grids of
71
the PES. Besides, there is no need to determine a range that covers all the points below cutoff
energy. The algorithm already searches all the points below the cutoff energy. However, the ab
initio or DFT calculations are needed for all the points in the algorithm whether these points are
above or below the cutoff energy. The same as in section 4.2, the data points from the algorithm
using low theoretical level method can be replaced by high theoretical level data points. The data
selection in the method is not the best comparing to the method in section 4.2. The use of data
selection in the method is due to the use of the algorithm.
The step size of the search can affect the data selection. The effects have not been studied.
The algorithm cannot go through all the possible points below cutoff energy due to the discrete
step size. Future study is needed to study the step size effect on the accuracy of the fitting. The
step size effect of the search to estimate RMS error also needs be studied. The final goal of the
study is to develop the method to fit PES of highly accurate theoretical method.
The computational costs for each of the work in the current study are shown in Table 4.9.
72
Table 4.1 The range of points below 70 kcal/mol of r1, r2 and r.
r1/Å
r2/Å
r/Å
1.0
1.125
1.9
1.04
1.08
1.8 – 1.9
1.04
1.125
1.6 – 1.9
1.08
1.08
1.5 – 1.9
1.08
1.125
1.4 – 1.9
1.125
1.125
1.3 – 1.9
73
Table 4.2 The range of r, α, θ and φ based on r1 and r2. The units are Å or °. The small φ angles
appeared when α is larger than 170°. The empty cell indicates that there are points below 70
kcal/mol in the range.
r2
r1
1.0 – 1.02 (1.0)
1.02 –
(1.04)
1.0 – 1.02 (1.0)
1.02
–
(1.04)
r 1.8 – 1.9 Å
α 150 – 179°
θ 20 – 89°
φ 84 – 96° and
240 – 276°
r 1.5 – 1.9 Å
α 140 – 179°
θ 30 – 89°
φ 72 – 96° and
240 – 276°
1.06
1.06 – 1.1 (1.08)
1.1 –
(1.125)
1.06 1.06 – 1.1 (1.08)
1.125
74
1.1
–
1.125
(1.125)
r 1.9 – 1.9 Å
α 150 - 179°
θ 40 – 89°
φ 84 – 96° and
240 – 264°
r 1.6 – 1.9 Å
α 140 – 179°
θ 20 – 89°
φ 72 – 96° and
240 – 276°
r 1.4 – 1.9 Å
α 120 – 179°
θ 30 – 89°
φ 72 – 96° and
240 – 288°
r 1.3 – 1.9 Å
α 110 – 179°
θ 20 – 89°
φ 72 – 108° and
240 – 276°
Table 4.3 The range of r based on the range of r1 and r2.
r2
1.0 – 1.02
1.02 – 1.06
1.06 – 1.1
r1
1.0 – 1.02
1.1 – 1.125
1.9 – 1.9
1.02 – 1.06
1.8 – 1.9
1.6 – 1.9
1.06 – 1.1
1.5 – 1.9
1.4 – 1.9
1.1 – 1.125
1.3 – 1.9
75
Table 4.4 The comparison of a. points below 70 kcal/mol calculated in the dense grid and b.
points below 60 kcal/mol sought by the algorithm for r1 = 1.04 Å, r2 = 1.125 Å, r = 1.8 Å and α =
147.8° and 155.6°. The values of r2, r and α of b are close to those of a but not the same.
a
b
r1
r2
r
α/°
θ/°
φ/°
1.04
1.08
1.8
147.8
59.3
240 – 252
69.2
228 – 264
79.11
228 – 264
89
228 – 264
75
235 – 245
80
235 – 245
85
235 – 245
19.7
240 – 264
29.7
228 – 276
39.6
228 – 276
49.4
228 – 276
59.3
228 – 276
69.2
240 – 276
79.1
240 – 276
89
240 – 276
45
237 – 247
50
237 – 257
55
237 – 257
60
237 – 267
65
237 – 267
70
237 – 247
75
237 – 267
80
237 – 267
85
237 – 267
1.04
1.09
1.82
150
a
155.6
b
155
76
Table 4.5 The comparison of a. points below 70 kcal/mol calculated in the dense grid and b.
points below 60 kcal/mol sought by the algorithm for r1 = 1.0 Å and r2 = 1.125 Å and r1 = 1.04 Å
and r2 = 1.125 Å. The values of θ are those for most α values. The values of φ are those for most
α and θ values. The values of r2 and r of b are close but not the same.
r1
r2
r
α/°
a
1.0
1.125
1.9
155.8 – 179 29.7 – 89
252 – 276
a
1.04
1.08
1.9
147.8 – 179 0 – 89
240 – 288
171.2 – 179 0 – 89
72 – 96
147.8 – 179 19.7 – 89
240 – 276
171.2 – 179 0 – 49.4
84 – 96
150 – 175
240 – 280
1.8
b
1.04
1.09
1.82
θ/°
30 – 85
φ/°
No points for both a and b below 1.8 Å for r1 = 1.04 Å and r2 = 1.08 Å.
a
1.04
1.125
139.8
– 39.6 – 89
169.2
139.8 – 179 19.8 – 89
1.6
1.7
179
1.1
84 – 96
240 – 288
169.2 – 179 0 – 89
84 – 96
139.8 – 179 0 – 89
228 – 288
159.4 – 179 0 – 89
72 – 84
1.72
150 – 175
50 – 85
240 – 260
1.82
145 – 175
0 – 85
250 – 280
1.9
1.04
240 – 276
139.8 – 179 0 – 89
1.8
b
0 – 89
240 – 264
No points for a below 1.6 Å for r1 = 1.04 Å and r2 = 1.125 Å. No points for b below 1.72 Å.
77
Table 4.6 The comparison of a. points below 70 kcal/mol calculated in the dense grid and b.
points below 60 kcal/mol sought by the algorithm for r1 = 1.08 Å and r2 = 1.08 Å. The values of
φ are those for most α and θ values. The values of r2 and r of b are close but not the same.
a
1.08
1.08
1.5
139.8
159.4
– 69.2 – 89
240 – 264
1.6
139.8
169.2
– 29.7 – 89
240 – 276
1.7
139.8 – 179
0 – 89
240 – 288
179
0 – 89
84 – 96
139.8 – 179
0 – 89
240 – 288
169.2 – 179
0 – 89
72 – 96
139.8 – 179
0 – 89
228 – 288
169.2 – 179
0 – 89
72 – 108
1.52
155
75 – 85
267.5
1.62
140 – 170
50 – 85
240 – 270
1.72
135 – 175
10 – 85
240 – 280
1.82
135 – 175
0 – 85
230 – 280
1.8
1.9
b
1.08
1.09
No points for both a and b below 1.5 Å for r1 = 1.08 Å and r2 = 1.09 Å.
78
Table 4.7 The comparison of a. points below 70 kcal/mol calculated in the dense grid and b.
points below 60 kcal/mol sought by the algorithm for r1 = 1.08 Å and r2 = 1.125 Å. The values of
φ are those for most α and θ values. The values of r2 and r of b are close but not the same.
a
r1
r2
r
α
θ
φ
1.08
1.125
1.4
120 – 159.3
49.4 – 89
240 – 264
1.5
119.9 –
169.1
120 – 179
39.6 – 89
240 – 276
19.8 – 89
240 – 276
179
0 – 89
84 – 96
129.8 – 179
0 – 89
228 – 276
179
0 – 89
72 – 108
129.8 – 179
0 – 89
228 – 288
159.3 – 179
0 – 89
72 – 96
129.8 – 179
0 – 89
228 – 300
149.5 – 179
0 – 89
60 – 108
1.52
130 – 165
50 – 85
230 – 260
1.62
130 – 175
30 – 85
240 – 280
1.72
130 – 175
0 – 85
230 – 280
1.82
135 – 175
0 – 85
230 – 290
1.6
1.7
1.8
1.9
b
1.08
1.1
No points for a below 1.4 Å at r1 = 1.08 Å and r2 = 1.125 Å. No points for b below 1.5 Å at r1 =
1.08 Å and r2 = 1.1 Å.
79
Table 4.8 The comparison of a. points below 70 kcal/mol calculated in the dense grid and b.
points below 60 kcal/mol sought by the algorithm for r1 = 1.125 Å and r2 = 1.125 Å. The values
of φ are those for most α and θ values. The values of r2 and r of b are close but not the same.
a
1.125
1.125
1.3
115 – 155
49.4 – 89
228 – 264
1.4
240 – 276
1.5
109.9 –
29.7 – 89
169.1
109.9 – 179 9.9 – 89
1.6
109.9 – 179 0 – 89
228 – 288
179
228 – 288
169.1 – 179 0 – 89
72 – 96
119.8 – 179 0 – 89
216 – 288
159.3 – 179 0 – 89
72 – 108
119.8 – 179 0 – 89
216 – 300
149.4 – 179 0 – 89
60 - 96
1.42
120 – 160
55 – 89
240 – 260
1.52
120 – 170
40 – 85
240 – 280
1.62
125 – 175
10 – 85
240 – 280
1.72
130 – 175
0 – 85
230 – 290
1.82
130 – 175
0 – 85
230 – 290
1.8
1.9
1.1
1.1
84 – 96
119.8 – 179 0 – 89
1.7
b
0 – 89
228 – 276
No points for a below 1.3 Å at r1 = 1.125 Å and r2 = 1.125 Å. No points for b below 1.4 Å at r1 =
1.1 Å and r2 = 1.1 Å.
80
Table 4.9 The CPU time of the works in current study
The IMLS study
The CPU time
Chapter 2
The IMLS fitting of H2O2 to select 3054 data
points and calculate the RMS errors
Chapter 3
The dynamics-driven IMLS method to select
685 data points for cis – trans HONO
isomerization
Chapter 3
The calculation of 100 trajectories using IMLS
PES with 685 data points
Chapter 4
The dynamics-driven IMLS method to select
992 data points for CO2 + O PES
Chapter 4
The calculation of 120 trajectories using IMLS
PES with 992 data points
Chapter 4
The second method to select 528 data points
and calculate the RMS errors
Chapter 4
The algorithm to select 1287 data points in the
third method
15d 5h – 61d 8 h
16d 15h
12d 15h
3d 11 h
2d 1h
6h
2d 5h
81
θ
r2
r
r1
5
α
3
1
6
2
4
Figure 4.1 Spherical coordinates and the order of internuclear coordinates.
82
PES evaluations
DFT calculations
Figure 4.2 The total number of PES evaluations in the sampling trajectories with respect to the
number of DFT calculations.
83
Nonreactive scattering
Atom exchange reaction
Figure 4.3 Nonreactive scattering and atom exchange reactions.
Oxygen atom abstraction
84
P(b)
Direct
Direct Error bar
IMLS
IMLS Error bar
b (Å)
Figure 4.4 Opacity functions for CO2 + O atom exchange reaction at Ecoll = 57.7 kcal/mol. The
red line is from B3LYP/3-21G trajectories. The brown line is the error bar for opacity functions
from B3LYP/3-21G trajectories. The blue line is from trajectories from IMLS potential. The
black line is the error bar for the opacity functions from the IMLS PES.
85
P(b)
Direct
Direct Error bar
IMLS
IMLS Error bar
b (Å)
Figure 4.5 Opacity functions for CO2 + O atom exchange reaction at Ecoll = 23.1 kcal/mol. The
red line is from B3LYP/3-21G trajectories. The brown line is the error bar for opacity functions
from B3LYP/3-21G trajectories. The blue line is from trajectories from IMLS potential. The
black line is the error bar for the opacity functions from the IMLS PES.
86
P(b)
Direct
Direct Error bar
IMLS
IMLS Error bar
b (Å)
Figure 4.6 Opacity functions for CO2 + O atom exchange reaction at Ecoll = 34.6 kcal/mol. The
red line is from B3LYP/3-21G trajectories. The brown line is the error bar for opacity functions
from B3LYP/3-21G trajectories. The blue line is from trajectories from IMLS potential. The
black line is the error bar for the opacity functions from the IMLS PES.
87
P(b)
57.7 kcal/mol
57.7 kcal/mol Error bar
34.6 kcal/mol
34.6 kcal/mol Error bar
23.1 kcal/mol
23.1 kcal/mol Error bar
b (Å)
Figure 4.7 Opacity functions of direct dynamics trajectories of B3LYP/3-21G for CO2 + O atom
exchange reaction at Ecoll = 23.1, 34.6 and 57.7 kcal/mol. The red line is for Ecoll = 57.7 kcal/mol.
The brown line is the error bar for Ecoll = 57.7 kcal/mol. The blue line is from trajectories for Ecoll
= 34.6 kcal/mol. The black line is the error bar for Ecoll = 34.6 kcal/mol. The red line is from
trajectories for Ecoll = 23.1 kcal/mol. The brown line is the error bar for Ecoll = 23.1 kcal/mol.
88
Total energy/(kcal/mol)
direct
IMLS
t/ps
Figure 4.8 The total energy of CO2 + O trajectory from direct dynamics and IMLS with 992 data
points.
89
(Gradienst ) (kcal/mol* Å)
(Energy) (kcal/mol)
α (120 – 175°), θ (30 – 80°)
α (120 – 179°), θ (30 – 89°)
Number of data points
Figure 4.9 The RMS errors with respect to the number of data points. The energy is in unit of
kcal/mol. The gradients is in unit of kcal/(mol*Å). The bottom lines are RMS errors of energy.
The top lines are RMS errors of gradients.
90
vi
r1,starting = r1,3
r1 ↑dr1 until r1 ˃ r1,upper
r1 - dr1 ↓dr until r1 ˂ r1,lower
v
r1
r2,starting = r2,3
r2 ↑dr2 until r2 ˃ r2,upper
r2 - dr2 ↓dr until r2 ˂ r2,lower
iv
r1, r2
rstarting = r3
r ↑dr until r ˃ rupper
r - dr ↓dr until r ˂ rlower
iii
r1, r2, r
αstarting = α3
α ↑dα until α ˃ αupper
α - dα ↓dα until α ˂ αlower
ii
r1, r2, r, α
θstarting = θ3 (for the first circle) or θmax for the last α
θ ↑dθ until θ ˃ θupper or no points at θ ˂ Ecutoff
θ - dθ ↓dθ until θ ˂ θlower or no points at θ ˂ Ecutoff
i
r1, r2, r, α, θ
φstarting = φ1 = (2π – α)/2 + α
φ1 ↑dφ until φ ˃ φupper or E ˃ Ecutoff
φ1 - dφ↓dφ until φ ˂ φlower or E ˃ Ecutoff
Figure 4.10 The chart of the algorithm for the search of points below cutoff energy. The starting
point for the search is r1,3 1.12 Å, r2,3 1.09 Å, r3 1.82 Å, α3 160°, θ3 60°, φ3 260°. A new value
from ii is given to i after a circle of i. This rule applies to iii, iv, v, vi.
91
The number of total points
The number of data points
Figure 4.11 The number of total points below 60 kcal/mol vs the number of data points selected
from the points below 60 kcal/mol in the search starting from r1 1.12 Å, r2 1.09 Å, r 1.82 Å, α
160°, θ 60°, φ 260° with dr1 0.01 Å, dr2 0.01 Å, dr 0.01 Å, dα 10°, dθ 10°, dφ 20°.
92
CHAPTER 5
FUTURE
Lots of studies have been done on IMLS to fit the potential energy surface. It is a very
accurate and efficient fitting method. Both analytical potential energy surface and high level ab
initio potential energy surface have been fitted using this method. PES of system as large as
4-atom molecule has been fitted. The main purpose of the current study is to apply these methods
to fit electronic structure theory data points from PES of a four-atom molecule, since there are
not many electronic structure theory data points from PES of 4-atom system fitted by IMLS so
far. The dynamics-driven IMLS fitting has been successfully used to fit the B3LYP/3-21G PES
of CO2 + O with Ecoll below 57.7 kcal/mol. Only 992 B3LYP/3-21G points are needed to
accurately fit the PES of CO2 + O with Ecoll below 57.7 kcal/mol. The second method is based on
knowing the shape of PES of a specified range before the fitting. Some work has been done on
using this method. Besides, an algorithm has been developed to search all the points below the
cutoff energy in the PES. Fitting is performed on the points below the cutoff energy found in the
algorithm. The future work of three methods has been discussed in chapter 4. It is important to
apply the IMLS methods to real high level ab initio PES. Current results on fitting a low level
PES have shown it is promising to fit high level PES of 4-atom systems. Further adjustments or
more work or complete new methods based on current study are still needed in order to fit the
PES of 4-atom at the high level ab initio method.
On the other hand, there are many studies on how to efficiently use IMLS method. The
weight functions, data selection, cutoff value to include data points to evaluate a single point and
the basis set have been studied previously. Among these studies, a new IMLS method is
93
developed, L-IMLS. Coefficients of data points evaluated by IMLS are stored. New evaluation
point can be calculated using weighted expansion based on the coefficients stored in these data
points. If there are other ways to use less data points for the IMLS fitting, this can help fitting the
PES using IMLS. Finding more efficient way for data selection should be a practical and useful
way to improve the fitting based on current progress of IMLS fitting. Therefore, other possible
studies can be directed to optimize the data selection for IMLS fitting. As can be seen in the
results of current study, there are many data points needed in the fitting of four-atom system PES.
The number of data points should increase exponentially if increasing the number of atoms in the
molecule or requiring more accurate PES, smaller RMS error. Therefore, the number of data
points needed for IMLS to achieve these purposes can decrease if the more efficient and accurate
IMLS fitting method is used. Since the cost of higher level electronic structure method is very
expensive, the computational time can be saved a lot by decreasing the number of data points for
the fitting. Besides, current results support the accurate fitting of PES of 4-atom of high level
theoretical method using IMLS. It usually takes hundreds or thousands of data points using
IMLS to fit an accurate PES of 4-atom. These are in agreement with other fitting methods.
Currently, these fitting methods are used to fit the PES of 4-atom. As mentioned in chapter 1,
there are still not many studies of the fitting of the PES of more than four atoms of high level
theoretical method. This is due to the large number of data point needed for PES of more than
four atoms. Therefore, fitting these PES really needs the major advancement of computer
technology.
94
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97
VITA
Yi Shi was born November 6, 1984, in Zhuzhou city, Hunan Province, China. He
attended elementary and middle school in Hunan Province. He received his B. S. in
Chemistry at Capital Normal University in Beijing, China in 2007. He is currently a PhD
candidate in Department of Chemistry at the University of Missouri, Columbia.
98