A finite basis representation Lanczos calculation of the bend energy levels of methane

2003 ◽  
Vol 118 (15) ◽  
pp. 6946-6956 ◽  
Author(s):  
Xiao-Gang Wang ◽  
Tucker Carrington
Keyword(s):  
Energy Levels ◽  
Finite Basis ◽  
Finite Basis Representation

scholarly journals USING LEBEDEV GRIDS, SINE SPHERICAL HARMONICS, AND MONOMER CONTRACTED BASIS FUNCTIONS TO CALCULATE BENDING ENERGY LEVELS OF HF TRIMER

2003 ◽  
Vol 02 (04) ◽  
pp. 599-608 ◽  
Author(s):  
XIAO-GANG WANG ◽  
TUCKER CARRINGTON
Keyword(s):  
Spherical Harmonics ◽  
Energy Levels ◽  
Finite Basis ◽  
Chem Phys ◽  
Previous Method ◽  
Basis Functions ◽  
Bending Energy ◽  
Quadrature Scheme ◽  
Finite Basis Representation ◽  
Quadrature Grid

We calculate energy levels of a six-dimensional bending Hamiltonian for HF trimer using a finite basis representation (FBR) in conjunction with the Lanczos eigensolver. We improve on our previous method [J. Chem. Phys.115, 9781 (2001)] using three techniques: (1) Lebedev's quadrature scheme is used to reduce the size of quadrature grid by a factor of 3.4. (2) Since the barrier separating the two equivalent versions of HF trimer is high and wide, it is a good approximation to confine the bending motion to one well by using sine spherical harmonics basis functions (this reduces the size of the basis by a factor of 8). (3) The sine spherical harmonic basis is contracted for each monomer to generate a very efficient basis. It is shown that the best approach is to combine all the three techniques.

scholarly journals Nonparametric regression with adaptive truncation via a convex hierarchical penalty

Biometrika ◽  
2018 ◽  
Vol 106 (1) ◽  
pp. 87-107
Author(s):  
Asad Haris ◽  
Ali Shojaie ◽  
Noah Simon
Keyword(s):  
Nonparametric Regression ◽  
Empirical Studies ◽  
Finite Basis ◽  
Additive Models ◽  
Computationally Efficient ◽  
Penalized Estimation ◽  
Minimax Rates ◽  
Minimax Rate ◽  
Finite Basis Representation ◽  
Different Levels

SUMMARY We consider the problem of nonparametric regression with a potentially large number of covariates. We propose a convex, penalized estimation framework that is particularly well suited to high-dimensional sparse additive models and combines the appealing features of finite basis representation and smoothing penalties. In the case of additive models, a finite basis representation provides a parsimonious representation for fitted functions but is not adaptive when component functions possess different levels of complexity. In contrast, a smoothing spline-type penalty on the component functions is adaptive but does not provide a parsimonious representation. Our proposal simultaneously achieves parsimony and adaptivity in a computationally efficient way. We demonstrate these properties through empirical studies and show that our estimator converges at the minimax rate for functions within a hierarchical class. We further establish minimax rates for a large class of sparse additive models. We also develop an efficient algorithm that scales similarly to the lasso with the number of covariates and sample size.

Vibrational Energy Levels via Finite-Basis Calculations Using a Quasi-Analytic Form of the Kinetic Energy

2011 ◽  
Vol 7 (5) ◽  
pp. 1428-1442 ◽  
Author(s):  
Juana Vázquez ◽  
Michael E. Harding ◽  
John F. Stanton ◽  
Jürgen Gauss
Keyword(s):  
Kinetic Energy ◽  
Vibrational Energy ◽  
Energy Levels ◽  
Analytic Form ◽  
Finite Basis ◽  
Vibrational Energy Levels

VIBRATIONALLY AVERAGED POTENTIAL ENERGY SURFACES AND PREDICTED INFRARED SPECTRA OF THE He–18O13C18O AND He–16O13C16O COMPLEXES

2008 ◽  
Vol 07 (04) ◽  
pp. 707-717 ◽  
Author(s):  
YALI CUI ◽  
HONG RAN ◽  
DAIQIAN XIE
Keyword(s):  
Potential Energy ◽  
Normal Mode ◽  
Potential Energy Surfaces ◽  
Energy Levels ◽  
Finite Basis ◽  
Lanczos Algorithm ◽  
Stretching Vibration ◽  
Representation Method ◽  
Energy Surfaces ◽  
Variable Representation

Vibrationally averaged potential energy surfaces for isotopic He–CO 2 complexes ( He –18 O 13 C 18 O and He –16 O 13 C 16 O ) are presented. Based on the latest ab initio potential of He –16 O 12 C 16 O (Ran H, Xie D, J Chem Phys128:124323, 2008.) including the Q3 normal mode for the v3 antisymmetric stretching vibration of the CO 2 molecule, the averaged potentials for both He –18 O 13 C 18 O and He –16 O 13 C 16 O are obtained by integrating the potential energy surfaces over the Q3 normal mode. The averaged potentials have T-shaped global minima and two equivalent linear local minima. The radial discrete variable representation/angular finite basis representation method and Lanczos algorithm are employed to calculate the related rovibrational energy levels. The calculated band origin shifts of He –18 O 13 C 18 O and He –16 O 13 C 16 O are 0.1066 and 0.0914 cm-1, respectively, which agree very well with the observed values of 0.1123 and 0.0929 cm-1. The calculated rovibrational transitions of He –18 O 13 C 18 O and He –16 O 13 C 16 O are also in very good agreement with the available experimental spectra.

A Perturbation Theory for the Population of Atomic Energy Levels in Non-Lte Plasmas

1988 ◽  
Vol 102 ◽  
pp. 343-347
Author(s):  
M. Klapisch
Keyword(s):  
Perturbation Theory ◽  
Small Parameter ◽  
Classification Scheme ◽  
Sum Rules ◽  
Energy Levels ◽  
Natural Classification ◽  
Quantum Numbers ◽  
Formal Expansion ◽  
Atomic Energy Levels ◽  
As Effects

AbstractA formal expansion of the CRM in powers of a small parameter is presented. The terms of the expansion are products of matrices. Inverses are interpreted as effects of cascades.It will be shown that this allows for the separation of the different contributions to the populations, thus providing a natural classification scheme for processes involving atoms in plasmas. Sum rules can be formulated, allowing the population of the levels, in some simple cases, to be related in a transparent way to the quantum numbers.

Low-lying energy levels of the FeH molecule

1999 ◽  
Vol 97 (1) ◽  
pp. 93-103 ◽  
Author(s):  
DANIEL F. HULLAH, RICHARD F. BARROW, JOHN
Keyword(s):  
Energy Levels

Do Atomic Spectra Determine Energy Levels?

1995 ◽  
Vol 5 (8) ◽  
pp. 949-961 ◽  
Author(s):  
C. Billionnet
Keyword(s):  
Energy Levels ◽  
Atomic Spectra
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