Integrals and derivatives for correlated Gaussian functions using matrix differential calculus

Kinghorn, Donald B.

doi:10.1002/(sici)1097-461x(1996)57:2<141::aid-qua1>3.0.co;2-y

Cited by 33 publications

(21 citation statements)

References 13 publications

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“…Several approaches have been proposed to to deal with high computational demands [39,41,42,[59][60][61][62][63][64]. In this work we have used an approach that combines a stochastic selection of the parameters [41,42,59] with a direct optimization that uses the analytic energy gradient [39,40,64]. In the present calculations the basis set was grown from zero to 4000 functions.…”

Section: Computational Detailsmentioning

confidence: 99%

Ground-state energy and relativistic corrections for positronium hydride

Bubin

Varga

2011

Phys. Rev. A

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Variational calculations of the ground state of positronium hydride (HPs) are reported, including various expectation values, electron-positron annihilation rates, and leading relativistic corrections to the total and dissociation energies. The calculations have been performed using a basis set of 4000 thoroughly optimized explicitly correlated Gaussian basis functions. The relative accuracy of the variational energy upper bound is estimated to be of the order of 2×10 −10 , which is a significant improvement over previous nonrelativistic results.

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Section: Computational Detailsmentioning

confidence: 99%

Ground-state energy and relativistic corrections for positronium hydride

Bubin

Varga

2011

Phys. Rev. A

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“…In our atomic calculations the optimization is performed with the procedure that utilizes analytical derivatives of the energy with respect to the Gaussian exponential parameters. [13][14][15] The derivatives form the gradient vector. Its elements depend on the derivatives of the Hamiltonian and overlap matrix elements.…”

Section: 2mentioning

confidence: 99%

Isotope shifts of the three lowest S1 states of the B+ ion calculated with a finite-nuclear-mass approach and with relativistic and quantum electrodynamics corrections

Bubin

Komasa

Stanke

et al. 2010

The Journal of Chemical Physics

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We present very accurate quantum mechanical calculations of the three lowest S-states ͓1s 2 2s 2 ͑ 1 S 0 ͒, 1s 2 2p 2 ͑ 1 S 0 ͒, and 1s 2 2s3s͑ 1 S 0 ͔͒ of the two stable isotopes of the boron ion, 10 B + and 11 B + . At the nonrelativistic level the calculations have been performed with the Hamiltonian that explicitly includes the finite mass of the nucleus as it was obtained by a rigorous separation of the center-of-mass motion from the laboratory frame Hamiltonian. The spatial part of the nonrelativistic wave function for each state was expanded in terms of 10 000 all-electron explicitly correlated Gaussian functions. The nonlinear parameters of the Gaussians were variationally optimized using a procedure involving the analytical energy gradient determined with respect to the nonlinear parameters. The nonrelativistic wave functions of the three states were subsequently used to calculate the leading ␣ 2 relativistic corrections ͑␣ is the fine structure constant; ␣ =1/ c, where c is the speed of light͒ and the ␣ 3 quantum electrodynamics ͑QED͒ correction. We also estimated the ␣ 4 QED correction by calculating its dominant component. A comparison of the experimental transition frequencies with the frequencies obtained based on the energies calculated in this work shows an excellent agreement. The discrepancy is smaller than 0.4 cm −1 .

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“…where I is the identity matrix and vec denotes the tensor stacking operator [31]. Formally, the same expression can be obtained by computing the derivative of eq.…”

Section: Approximate Dynamic Mapsmentioning

confidence: 99%

Action Recognition with Dynamic Image Networks

Bilen

Fernando

Gavves

et al. 2018

IEEE Trans. Pattern Anal. Mach. Intell.

196

164

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Abstract-We introduce the concept of dynamic image, a novel compact representation of videos useful for video analysis, particularly in combination with convolutional neural networks (CNNs). A dynamic image encodes temporal data such as RGB or optical flow videos by using the concept of 'rank pooling'. The idea is to learn a ranking machine that captures the temporal evolution of the data and to use the parameters of the latter as a representation. When a linear ranking machine is used, the resulting representation is in the form of an image, which we call dynamic because it summarizes the video dynamics in addition of appearance. This is a powerful idea because it allows to convert any video to an image so that existing CNN models pre-trained for the analysis of still images can be immediately extended to videos. We also present an efficient and effective approximate rank pooling operator, accelerating standard rank pooling algorithms by orders of magnitude, and formulate that as a CNN layer. This new layer allows generalizing dynamic images to dynamic feature maps. We demonstrate the power of the new representations on standard benchmarks in action recognition achieving state-of-the-art performance.

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Integrals and derivatives for correlated Gaussian functions using matrix differential calculus

Cited by 33 publications

References 13 publications

Ground-state energy and relativistic corrections for positronium hydride

Ground-state energy and relativistic corrections for positronium hydride

Isotope shifts of the three lowest S1 states of the B+ ion calculated with a finite-nuclear-mass approach and with relativistic and quantum electrodynamics corrections

Action Recognition with Dynamic Image Networks

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