Abstract
The purpose is to fit an accurate smooth function of the many-body expansion type to a multidimensional large data set using a basis-set type method. By adopting a combined-hyperbolic-inverse-power-representation for the basis, the novel approach is tested in detail for the ground electronic state of tri-hydrogen and hydroperoxyl systems, assuming that their potential energy surfaces are single-sheeted representable. It is also shown that the method can be easily applicable to potential energy curves by considering as prototypes molecular oxygen and the hydroxyl radical.
This work is financed by FEDER through “Programa Operacional Factores de Competitividade – COMPETE” and national funds under the auspices of Fundação para a Ciência e a Tecnologia, Portugal (Project Nos. PTDC/QUI-QUI/099744/2008 and PTDC/AAC-AMB/099737/2008).
I. INTRODUCTION
II. METHOD
A. Analytic representation of n-body energies
B. Basis set expansion
III. THE TWO-BODY CASE
IV. THREE-BODY SYSTEMS
A. Trihydrogen
B. The hydroperoxyl radical
V. CONCLUDING REMARKS
Key Topics
- Polynomials
- 16.0
- Ab initio calculations
- 12.0
- Molecular beam epitaxy
- 5.0
- Potential energy surfaces
- 5.0
- Ground states
- 4.0
Figures
Potential energy curve for the electronic ground state of OH as obtained by fitting MRCI+Q/AVQZ energies with the EHFACE2U (dashed line) and CHIPR (solid line) formalisms. The errors for the latter are indicated in the bottom panel, while the short-range repulsive region is shown in the insert of the top panel.
Potential energy curve for the electronic ground state of OH as obtained by fitting MRCI+Q/AVQZ energies with the EHFACE2U (dashed line) and CHIPR (solid line) formalisms. The errors for the latter are indicated in the bottom panel, while the short-range repulsive region is shown in the insert of the top panel.
As in Figure 1 but for . For better visibility, the error bars referring to the highly repulsive points have been truncated but, as shown from the insert, the percent errors are still rather small.
As in Figure 1 but for . For better visibility, the error bars referring to the highly repulsive points have been truncated but, as shown from the insert, the percent errors are still rather small.
HH contracted basis from a fit to energies for equilateral triangular geometries of trihydrogen. Points were chosen from the range 0.8 ⩽ R/a_{0} ⩽ 6.0 or reproduction of the 10 pivotal ones (solid dots) democratically selected over the same range. The arrows indicate the distributed origins that have been optimally defined for each fit. Shown in gray is the fully optimized basis b8 suitably scaled by 0.5.
HH contracted basis from a fit to energies for equilateral triangular geometries of trihydrogen. Points were chosen from the range 0.8 ⩽ R/a_{0} ⩽ 6.0 or reproduction of the 10 pivotal ones (solid dots) democratically selected over the same range. The arrows indicate the distributed origins that have been optimally defined for each fit. Shown in gray is the fully optimized basis b8 suitably scaled by 0.5.
Rmsd for 9477 points covering a uniformly dense region defined by 0.8 ⩽ R _{1}/a_{0} ⩽ 6.0 (similarly for R _{2}) and included angles , spaced by and , respectively. All fits were done with the basis b8.
Rmsd for 9477 points covering a uniformly dense region defined by 0.8 ⩽ R _{1}/a_{0} ⩽ 6.0 (similarly for R _{2}) and included angles , spaced by and , respectively. All fits were done with the basis b8.
Lowest adiabatic potential energy surface for collinear geometries of as fitted to 9477 DMBE points with a polynomial expansion up to power 20 in the basis b8. See text.
Lowest adiabatic potential energy surface for collinear geometries of as fitted to 9477 DMBE points with a polynomial expansion up to power 20 in the basis b8. See text.
Fits to 9477 (top panel) and 420 (bottom) points for with a polynomial expansion up to power 20 in basis b8.
Fits to 9477 (top panel) and 420 (bottom) points for with a polynomial expansion up to power 20 in basis b8.
O–H and O–O contracted basis as obtained from a fit to energies for 1D cuts along the PES keeping the angle fixed at the equilibrium value: in black for O–OH stretching, in gray for H–OO. For visibility, the optimized basis have been multiplied by suitable constant scaling factors.
O–H and O–O contracted basis as obtained from a fit to energies for 1D cuts along the PES keeping the angle fixed at the equilibrium value: in black for O–OH stretching, in gray for H–OO. For visibility, the optimized basis have been multiplied by suitable constant scaling factors.
Contour plot for bond stretching in O–O–H fixed at its equilibrium included angle. Contours are equally spaced by , starting at .
Contour plot for bond stretching in O–O–H fixed at its equilibrium included angle. Contours are equally spaced by , starting at .
Contour plot for an O atom moving around OH fixed at its equilibrium geometry and lying along the X axis with the center of the bond fixed at the origin. Contours are equally spaced by , starting at . In dash are contours equally spaced by , starting at .
Contour plot for an O atom moving around OH fixed at its equilibrium geometry and lying along the X axis with the center of the bond fixed at the origin. Contours are equally spaced by , starting at . In dash are contours equally spaced by , starting at .
Isotropic components of the O–OH and interactions as predicted with the diatomic molecule fixed at its equilibrium geometry (reference energy taken at asymptote).
Isotropic components of the O–OH and interactions as predicted with the diatomic molecule fixed at its equilibrium geometry (reference energy taken at asymptote).
Tables
Some non-zero coefficients employed in the expansion of Eq. (3) .
Some non-zero coefficients employed in the expansion of Eq. (3) .
CHIPR curves ^{ a } for ground state OH and O_{2}.
CHIPR curves ^{ a } for ground state OH and O_{2}.
H–H contracted basis ^{ a } for H_{3}.
H–H contracted basis ^{ a } for H_{3}.
Saddle point attributes ^{ a } for the exchange reaction.
Saddle point attributes ^{ a } for the exchange reaction.
O–H and O–O contracted basis ^{ a } for HO_{2}.
O–H and O–O contracted basis ^{ a } for HO_{2}.
Accumulated unweighted rmsd (in ) of the CHIPR fits ^{ a } to DMBE IV for ground-state HO_{2}.
Accumulated unweighted rmsd (in ) of the CHIPR fits ^{ a } to DMBE IV for ground-state HO_{2}.
Attributes of major stationary points of HO_{2} potential energy surface. ^{ a } Unless indicated otherwise, energies are relative to the covalent minimum.
Attributes of major stationary points of HO_{2} potential energy surface. ^{ a } Unless indicated otherwise, energies are relative to the covalent minimum.
A summary of rate constants ^{ a } for the reaction. ^{ b }
A summary of rate constants ^{ a } for the reaction. ^{ b }
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