Sum rules for variational wavefunctions.
Sum rule properties of finite set of wave functions obtained from application of variational principles to basis functions linear combinations
Dalgarno, A.↗
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Extending GPU-accelerated Gaussian integrals in the TeraChem software package to f type orbitals: Implementation and applications
Here, the increasing availability of graphics processing units (GPUs) for scientific computing has prompted interest in accelerating quantum chemical calculations through their use. However, the complexity of integral kernels for high angular momentum basis functions often limits the utility of GPU implementations with large basis sets or for metal containing systems. In this work, we report the implementation of f function support in the GPU-accelerated TeraChem software package through the development of efficient kernels for the evaluation of Hamiltonian integrals. The high efficiency of the resulting code is demonstrated through density functional theory (DFT) calculations on increasingly large organic molecules and transition metal complexes, as well as coupled cluster singles and doubles calculations on water clusters. Preliminary investigations into Ni(I) catalysis with DFT and the photochemistry of MnH(CH 3 ) with complete active space self-consistent field are also carried out. Overall, our GPU-accelerated software appears to be well-suited for fast simulation of large transition metal containing systems, as well as organic molecules.
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
Excitation of helium Rydberg states and doubly excited resonances in strong extreme ultraviolet fields: Full-dimensional quantum dynamics using exponentially tempered Gaussian basis sets
Recently optimized exponentially tempered Gaussian basis sets [P. R. Kapralova-Zdanska and J. Smydke, J. Chem. Phys. 138, 024105 (2013)] are employed in quantitative simulations of helium absorption cross-sections and two-photon excitation yields of doubly excited resonances. Linearly polarized half-infinite and Gaussian laser pulses at wavelengths 38–58 nm and large intensities up to 100 TW/cm{sup 2} are considered. The emphasis is laid on convergence of the results with respect to the quality of the Gaussian basis sets (typically limited by a number of partial waves, density, and spatial extent of the basis functions) as well as to the quality of the basis set of field-free states (typically limited by the maximum rotational quantum number and maximum excitation of the lower electron). Particular attention is paid to stability of the results with respect to varying complex scaling parameter. Moreover, the study of the dynamics is preceded by a thorough check of helium energies and oscillator strengths as they are obtained with the exponentially tempered Gaussian basis sets, being also compared with yet unpublished emission wavelengths measured in electric discharge experiments.
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
Identifying attributes of CO 2 leakage zones in shallow aquifers using a parametric level set method: Original Research Article: Identifying attributes of CO 2 leakage zones in shallow aquifers using a parametric level set method
Leakage through abandoned wells and geologic faults poses the greatest risk to CO 2 storage permanence. For shallow aquifers, secondary CO 2 plumes emanating from the leak zones may go undetected for a sustained period of time and has the greatest potential to cause large-scale and long-term environmental impacts. Identification of the attributes of leak zones, including their shape, location, and strength, is required for environmental risk assessment. This study applies a parametric level set (PaLS) method to characterize leak zones. Level set methods are appealing for tracking topological changes and recovering unknown shapes of objects. However, level set evolution using the conventional level set methods is computationally challenging. In PaLS, the level set function is approximated by using a weighted sum of basis functions and the level set evolution problem is replaced by an optimization problem. The efficacy of PaLS is demonstrated by reconstructing attributes of a CO 2 leak zone in a carbonate aquifer. Our results show that PaLS is a robust source identification method that can recover the approximate leak zone locations in the presence of measurement errors, model parameter uncertainty, and inaccurate guesses of source flux strengths. Accurate delineation of source geometry is achievable when a relatively dense observation network is used. The PaLS-based, source recovery framework introduced in this work is generic and can be adapted for any reactive transport model by simply switching the pre- and post-processing routines.
54 ENVIRONMENTAL SCIENCES↗
Non-oscillatory spectral Fourier methods for shock wave calculations
A non-oscillatory spectral Fourier method is presented for the solution of hyperbolic partial differential equations. The method is based on adding a nonsmooth function to the trigonometric polynomials which are the usual basis functions for the Fourier method. The high accuracy away from the shock is enhanced by using filters. Numerical results confirm that no oscillations develop in the solution. Also, the accuracy of the spectral solution of the inviscid Burgers equation is shown to be higher than a fixed order.
Cai, Wei↗
Essentially nonoscillatory spectral Fourier methods for shock wave calculations
An essentially nonoscillatory spectral Fourier method for the solution of hyperbolic partial differential equations is presented. The method is based on adding a nonsmooth function to the trigonometric polynomials which are the usual basis functions for the Fourier method. The high accuracy away from the shock is enhanced by using filters. Numerical results confirm that essentially no oscillations develop in the solution.
Cai, Wei↗
Time-dependent treatment of scattering. II - Novel integral equation approach to quantum wave packets
The novel wave-packet propagation scheme presented is based on the time-dependent form of the Lippman-Schwinger integral equation and does not require extensive matrix inversions, thereby facilitating application to systems in which some degrees of freedom express the potential in a basis expansion. The matrix to be inverted is a function of the kinetic energy operator, and is accordingly diagonal in a Bessel function basis set. Transition amplitudes for various orbital angular momentum quantum numbers are obtainable via either Fourier transform of the amplitude density from the time to the energy domain, or the direct analysis of the scattered wave packet.
Sharafeddin, Omar A.↗
A Meshless Local Petrov-Galerkin Method for Euler-Bernoulli Beam Problems
An accurate and yet simple Meshless Local Petrov-Galerkin (MLPG) formulation for analyzing beam problems is presented. In the formulation, simple weight functions are chosen as test functions. The use of these functions shows that the weak form can be integrated with conventional Gaussian integration. The MLPG method was evaluated by applying the formulation to a variety of patch test and thin beam problems. The formulation successfully reproduced exact solutions to machine accuracy when test functions with C2 continuity and an appropriate order of basis functions are used.
Raju, I. S.↗
Spatial Statistical Data Fusion (SSDF)
As remote sensing for scientific purposes has transitioned from an experimental technology to an operational one, the selection of instruments has become more coordinated, so that the scientific community can exploit complementary measurements. However, tech nological and scientific heterogeneity across devices means that the statistical characteristics of the data they collect are different. The challenge addressed here is how to combine heterogeneous remote sensing data sets in a way that yields optimal statistical estimates of the underlying geophysical field, and provides rigorous uncertainty measures for those estimates. Different remote sensing data sets may have different spatial resolutions, different measurement error biases and variances, and other disparate characteristics. A state-of-the-art spatial statistical model was used to relate the true, but not directly observed, geophysical field to noisy, spatial aggregates observed by remote sensing instruments. The spatial covariances of the true field and the covariances of the true field with the observations were modeled. The observations are spatial averages of the true field values, over pixels, with different measurement noise superimposed. A kriging framework is used to infer optimal (minimum mean squared error and unbiased) estimates of the true field at point locations from pixel-level, noisy observations. A key feature of the spatial statistical model is the spatial mixed effects model that underlies it. The approach models the spatial covariance function of the underlying field using linear combinations of basis functions of fixed size. Approaches based on kriging require the inversion of very large spatial covariance matrices, and this is usually done by making simplifying assumptions about spatial covariance structure that simply do not hold for geophysical variables. In contrast, this method does not require these assumptions, and is also computationally much faster. This method is fundamentally different than other approaches to data fusion for remote sensing data because it is inferential rather than merely descriptive. All approaches combine data in a way that minimizes some specified loss function. Most of these are more or less ad hoc criteria based on what looks good to the eye, or some criteria that relate only to the data at hand.
Braverman, Amy J.↗
Adaptive Sparse Grid Construction in a Context of Local Anisotropy and Multiple Hierarchical Parents
We consider general strategy for hierarchical multidimensional interpolation based on sparse grids, where the interpolation nodes and locally supported basis functions are constructed from tensors of a one dimensional hierarchical rule. We consider four different hierarchies that are tailored towards general functions, high or low order polynomial approximation, or functions that satisfy homogeneous boundary conditions. The main advantage of the locally supported basis is the ability to choose a set of functions based on the observed behavior of the target function. We present an alternative to the classical surplus refinement techniques, where we exploit local anisotropy and refine using functions with not strictly decreasing support. The more flexible refinement strategy improves stability and reduces the total number of expensive simulations, resulting in significant computational saving. We demonstrate the advantages of the different hierarchies and refinement techniques by application to series of simple functions as well as a system of ordinary differential equations given by the Kermack-McKendrick SIR model.
Stoyanov, Miroslav K.↗
Using a pruned, nondirect product basis in conjunction with the multi-configuration time-dependent Hartree (MCTDH) method
In this paper, we propose a pruned, nondirect product multi-configuration time dependent Hartree (MCTDH) method for solving the Schrödinger equation. MCTDH uses optimized 1D basis functions, called single particle functions, but the size of the standard direct product MCTDH basis scales exponentially with D, the number of coordinates. We compare the pruned approach to standard MCTDH calculations for basis sizes small enough that the latter are possible and demonstrate that pruning the basis reduces the CPU cost of computing vibrational energy levels of acetonitrile (D = 12) by more than two orders of magnitude. Using the pruned method, it is possible to do calculations with larger bases, for which the cost of standard MCTDH calculations is prohibitive. Pruning the basis complicates the evaluation of matrix-vector products. In this paper, they are done term by term for a sum-of-products Hamiltonian. When no attempt is made to exploit the fact that matrices representing some of the factors of a term are identity matrices, one needs only to carefully constrain indices. In this paper, we develop new ideas that make it possible to further reduce the CPU time by exploiting identity matrices.
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
Kinetic fluid moments closure for a magnetized plasma with collisions
A novel method aimed at a kinetic moments closure for a magnetized plasma with collisions is proposed. The velocity distribution function for each species is expanded in 8 Gaussian Radial Basis Functions (GRBFs) which are essentially shifted Maxwellians at eight representative 3D-velocity points of drift. The vector of 8 fluid moments (for particle density, 3 particle fluxes, total energy density, and 3 energy fluxes) has an 8x8 analytic linear matrix relation to the vector of 8 GRBF density weighs in 3D-real space. The 8 fluid moments with sources for each species are advanced in time while the 8 GRBF weighs are determined from the 8x8 inverse matrix. The two closure moments (for the stress tensor and the energy weighted stress tensor) are linearly determined from the GRBF weighs. Most importantly the velocity moments of the nonlinear Coulomb Fokker-Planck collision operator [Rosenbluth et al, Phys. Rev 107, 1957] are evaluated from the GRBF weights. Generalization from 8 to 12, 16, 20 .., in an energy weighted moment hierarchy is straightforward. The electric field follows from a generalized vorticity (quasi-neutral charge conservation) equation. A strong drift ordering approximation can be applied to eliminate any spuriously unstable high frequency cyclotron motions. A novel weak drift ordering two-time step scheme avoids the vorticity equation by following ion cyclotron motion in time to get the electric field with ion gyroaveraging. Inclusion of low-beta magnetic perturbations is straightforward.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
Theoretical study of NMR, infrared and Raman spectra on triple-decker phthalocyanines
Electronic structures and magnetic properties of multi-decker phthalocyanines were studied by theoretical calculation. Electronic structures, excited processes at multi-states, isotropic chemical shifts of {sup 13}C, {sup 14}N and {sup 1}H-nuclear magnetic resonance (NMR), principle V-tensor in electronic field gradient (EFG) tensor and asymmetry parameters (η), vibration mode in infrared (IR) and Raman spectra of triple-decker phthalocyanines were calculated by density functional theory (DFT) and time-dependent DFT using B3LYP as basis function. Electron density distribution was delocalized on the phthalocyanine rings with electron static potential. Considerable separation of chemical shifts in {sup 13}C, {sup 14}N and {sup 1}H-NMR was originated from nuclear spin interaction between nitrogen and carbon atoms, nuclear quadrupole interaction based on EFG and η of central metal under crystal field. Calculated optical absorption at multi-excited process was derived from overlapping π-orbital on the phthalocyanine rings. The vibration modes in IR and Raman spectra were based on in-plane deformation and stretching vibrations of metal-ligand coordination bond on the deformed structure.
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗
Practical method for highly accurate large-scale surface calculations
An accurate and efficient film linearized muffin-tin orbital (FLMTO) technique for surface electronic-structure calculations is presented which uses only 60-70 basis functions, as opposed to the 300 functions used in the linear augmented plane-wave method. Calculations for three different (3d and 4d) transition-metal films resulted in high quality results for five-layer slabs of Cu(001), Fe(001), and Ru(001), in addition to good results for the work functions and projected density of states. By retaining the LMTO small basis size, computer time and memory are reduced, making practical the study of systems with a larger number of atoms in the two-dimensional unit cell.
Fernando, G. W.↗
Parameter dimension of turbulence-induced phase errors and its effects on estimation in phase diversity
The method of phase diversity has been used in the context of incoherent imaging to estimate jointly an object that is being imaged and phase aberrations induced by atmospheric turbulence. The method requires a parametric model for the phase-aberration function. Typically, the parameters are coefficients to a finite set of basis functions. Care must be taken in selecting a parameterization that properly balances accuracy in the representation of the phase-aberration function with stability in the estimates. It is well known that over parameterization can result in unstable estimates. Thus a certain amount of model mismatch is often desirable. We derive expressions that quantify the bias and variance in object and aberration estimates as a function of parameter dimension.
Thelen, Brian J.↗
A quasi-meshfree method for constructing boundary-aware reproducing bases on geometrically complex domains using manifold geodesics
In many applications, physical domains are geometrically complex making it challenging to perform coarse-scale approximation. A defeaturing process is often used to simplify the domain in preparation for approximation and analysis at the coarse scale. In this paper, a methodology is presented for constructing a coarse-scale reproducing basis on geometrically complex domains given an initial fine-scale mesh of the fully featured domain. The initial fine-scale mesh can be of poor quality and extremely refined. The construction of the basis functions begins with a coarse-scale covering of the domain and generation of weighting functions with local support. Manifold geodesics are used to define distances within the local support for general applicability to non-convex domains. Conventional moving least squares is used to construct the coarse-scale reproducing basis. Applications in quasi-interpolation and linear elasticity are presented.
97 MATHEMATICS AND COMPUTING↗
Prediction of {sup 2}D Rydberg energy levels of {sup 6}Li and {sup 7}Li based on very accurate quantum mechanical calculations performed with explicitly correlated Gaussian functions
Very accurate variational nonrelativistic finite-nuclear-mass calculations employing all-electron explicitly correlated Gaussian basis functions are carried out for six Rydberg {sup 2}D states (1s{sup 2}nd, n= 6, Horizontal-Ellipsis , 11) of the {sup 7}Li and {sup 6}Li isotopes. The exponential parameters of the Gaussian functions are optimized using the variational method with the aid of the analytical energy gradient determined with respect to these parameters. The experimental results for the lower states (n= 3, Horizontal-Ellipsis , 6) and the calculated results for the higher states (n= 7, Horizontal-Ellipsis , 11) fitted with quantum-defect-like formulas are used to predict the energies of {sup 2}D 1s{sup 2}nd states for {sup 7}Li and {sup 6}Li with n up to 30.
74 ATOMIC AND MOLECULAR PHYSICS↗