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Function space requirements for the single-electron functions within the multiparticle Schrödinger equation

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10.1063/1.4811396

### Abstract

Our previously described method to approximate the many-electron wavefunction in the multiparticle Schrödinger equation reduces this problem to operations on many single-electron functions. In this work, we analyze these operations to determine which function spaces are appropriate for various intermediate functions in order to bound the output. This knowledge then allows us to choose the function spaces in which to control the error of a numerical method for single-electron functions. We find that an efficient choice is to maintain the single-electron functions in L 2 ∩ L 4, the product of these functions in L 1 ∩ L 2, the Poisson kernel applied to the product in L 4, a function times the Poisson kernel applied to the product in L 2, and the nuclear potential times a function in L 4/3. Due to the integral operator formulation, we do not require differentiability.

© 2013 AIP Publishing LLC

Received 11 September 2012
Accepted 04 June 2013
Published online 24 June 2013

Acknowledgments: This material is based upon work supported by the National Science Foundation under Grant No. DMS-0545895.

Article outline:

I. INTRODUCTION

A. Sketch of the algorithm

B. Operations on single-electron functions

II. PRELIMINARY ANALYSIS

A. Function space basics

B. Energy estimate and update

C. Analysis of the Gaussian convolutions in the Green function

1. Convolution and Gaussians

2. Integral representation of the Green function

3. Operator norms

4. Ensuring accuracy in the Green function

III. OPERATION DIAGRAM ANALYSIS

A. Function space inequalities and calculations

B. Analysis using *L* ^{2}

1. Product

2. Nuclear potential

3. Poisson convolution

C. Analysis using *L* ^{[2, u]} for 3 < *u*

1. Product

2. Nuclear potential

3. Poisson convolution

4. Gaussian convolution

D. Analysis using *H* ^{1}

1. Background results

2. Product

3. Nuclear potential

4. Poisson convolution

5. Gaussian convolution

IV. CHOOSING THE FUNCTION SPACES USING THE CORE ORBITAL

A. The core orbital

B. Truncation near the singularities

1. Flattening the core orbital

2. Truncating the nuclear potential

3. Truncating the poisson kernel

4. Truncating the Green function

C. Comparison of bounds

1. Sharpness

2. Intermediate objects

3. Minimal output size

V. IMPLEMENTATION AND TESTING

A. Bound collection

B. Truncation radius collection

C. Vector and matrix amplification

D. Antisymmetric inner products using the singular value decomposition (SVD)

E. Validation tests

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2013-06-24

2014-04-19

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