Best $N$-term approximation in electronic structure calculations. II. Jastrow factors
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 41 (2007) no. 2, p. 261-279

We present a novel application of best $N$-term approximation theory in the framework of electronic structure calculations. The paper focusses on the description of electron correlations within a Jastrow-type ansatz for the wavefunction. As a starting point we discuss certain natural assumptions on the asymptotic behaviour of two-particle correlation functions ${ℱ}^{\left(2\right)}$ near electron-electron and electron-nuclear cusps. Based on Nitsche’s characterization of best $N$-term approximation spaces ${A}_{q}^{\alpha }\left({H}^{1}\right)$, we prove that ${ℱ}^{\left(2\right)}\in {A}_{q}^{\alpha }\left({H}^{1}\right)$ for $q>1$ and $\alpha =\frac{1}{q}-\frac{1}{2}$ with respect to a certain class of anisotropic wavelet tensor product bases. Computational arguments are given in favour of this specific class compared to other possible tensor product bases. Finally, we compare the approximation properties of wavelet bases with standard gaussian-type basis sets frequently used in quantum chemistry.

DOI : https://doi.org/10.1051/m2an:2007016
Classification:  41A50,  41A63,  65Z05,  81V70
Keywords: best N-term approximation, wavelets, electron correlations, Jastrow factor
@article{M2AN_2007__41_2_261_0,
author = {Flad, Heinz-J\"urgen and Hackbusch, Wolfgang and Schneider, Reinhold},
title = {Best $N$-term approximation in electronic structure calculations. II. Jastrow factors},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {41},
number = {2},
year = {2007},
pages = {261-279},
doi = {10.1051/m2an:2007016},
zbl = {1135.81029},
mrnumber = {2339628},
language = {en},
url = {http://www.numdam.org/item/M2AN_2007__41_2_261_0}
}

Flad, Heinz-Jürgen; Hackbusch, Wolfgang; Schneider, Reinhold. Best $N$-term approximation in electronic structure calculations. II. Jastrow factors. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 41 (2007) no. 2, pp. 261-279. doi : 10.1051/m2an:2007016. http://www.numdam.org/item/M2AN_2007__41_2_261_0/

[1] S. Agmon, Lectures on exponential decay of solutions of second-order elliptic equations: Bounds on eigenfunctions of N-body Schrödinger operators, Mathematical Notes 29. Princeton University Press (1982). | MR 745286 | Zbl 0503.35001

[2] H.-J. Bungartz and M. Griebel, Sparse grids. Acta Numer. 13 (2004) 147-269. | Zbl 1122.65405

[3] C.E. Campbell, E. Krotscheck and T. Pang, Electron correlations in atomic systems. Phys. Rep. 223 (1992) 1-42.

[4] D. Ceperley, Ground state of the fermion one-component plasma: A Monte Carlo study in two and three dimensions. Phys. Rev. B 18 (1978) 3126-3138.

[5] J.W. Clark, Variational theory of nuclear matter, in Progress in Nuclear and Particle Physics, Vol. 2, D.H. Wilkinson Ed., Pergamon, Oxford (1979) 89-199.

[6] E.T. Copson, Asymptotic Expansions. Cambridge University Press, Cambridge (1967). | MR 168979 | Zbl 1096.41001

[7] W. Dahmen, S. Prößdorf and R. Schneider, Wavelet approximation methods for pseudodifferential equations. II: Matrix compression and fast solution. Adv. Comp. Maths. 1 (1993) 259-335. | Zbl 0826.65093

[8] W. Dahmen, S. Prößdorf and R. Schneider, Wavelet approximation methods for pseudodifferential equations. I: Stability and convergence. Math. Z. 215 (1994) 583-620. | Zbl 0794.65082

[9] R.A. Devore, Nonlinear approximation. Acta Numer. 7 (1998) 51-150. | Zbl 0931.65007

[10] R.A. Devore, B. Jawerth and V. Popov, Compression of wavelet decompositions. Amer. J. Math. 114 (1992) 737-785. | Zbl 0764.41024

[11] R.A. Devore, S.V. Konyagin and V.N. Temlyakov, Hyperbolic wavelet approximation. Constr. Approx. 14 (1998) 1-26. | Zbl 0895.41016

[12] N.D. Drummond, M.D. Towler and R.J. Needs, Jastrow correlation factor for atoms, molecules, and solids. Phys. Rev. B 70 (2004) 235119.

[13] H.-J. Flad and A. Savin, Transfer of electron correlation from the electron gas to inhomogeneous systems via Jastrow factors. Phys. Rev. A. 50 (1994) 3742-3746.

[14] H.-J. Flad and A. Savin, A new Jastrow factor for atoms and molecules, using two-electron systems as a guiding principle. J. Chem. Phys. 103 (1995) 691-697.

[15] H.-J. Flad, W. Hackbusch, D. Kolb and R. Schneider, Wavelet approximation of correlated wavefunctions. I. Basics. J. Chem. Phys. 116 (2002) 9641-9657.

[16] H.-J. Flad, W. Hackbusch, H. Luo and D. Kolb, Diagrammatic multiresolution analysis for electron correlations. Phys. Rev. B 71 (2005) 125115.

[17] H.-J. Flad, W. Hackbusch and R. Schneider, Best $N$-term approximation in electronic structure calculations. I. One-electron reduced density matrix. ESAIM: M2AN 40 (2006) 49-61. | Numdam | Zbl 1100.81050

[18] S. Fournais, M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof and T. Ostergaard Sorensen, Sharp regularity results for Coulombic many-electron wave functions. Commun. Math. Phys. 255 (2005) 183-227. | Zbl 1075.35063

[19] D.E. Freund, B.D. Huxtable and J.D. Morgan Iii, Variational calculations on the helium isoelectronic sequence. Phys. Rev. A 29 (1984) 980-982.

[20] P. Fulde, Electron Correlations in Molecules and Solids, 2nd edition. Springer, Berlin (1993).

[21] P. Fulde, Ground-state wave functions and energies of solids. Int. J. Quant. Chem. 76 (2000) 385-395.

[22] J. Garcke and M. Griebel, On the computation of the eigenproblems of hydrogen and helium in strong magnetic and electric fields with the sparse grid combination technique. J. Comp. Phys. 165 (2000) 694-716. | Zbl 0979.65101

[23] R. Gaudoin, M. Nekovee, W.M.C. Foulkes, R.J. Needs and G. Rajagopal, Inhomogeneous random-phase approximation and many-electron trial wave functions. Phys. Rev. B 63 (2001) 115115.

[24] W. Hackbusch, B.N. Khoromskij and E. Tyrtyshnikov, Hierarchical Kronecker tensor-product approximation. J. Numer. Math. 13 (2005) 119-156. | Zbl 1081.65035

[25] A. Halkier, T. Helgaker, P. Jørgensen, W. Klopper, H. Koch, J. Olsen and A.K. Wilson, Basis-set convergence in correlated calculations on Ne 286 (1998) 243-252.

[26] T. Helgaker, W. Klopper, H. Koch and J. Noga, Basis-set convergence of correlated calculations on water. J. Chem. Phys. 106 (1997) 9639-9646.

[27] T. Helgaker, P. Jørgensen and J. Olsen, Molecular Electronic-Structure Theory. Wiley, New York (1999).

[28] R.N. Hill, Rates of convergence and error estimation formulas for the Rayleigh-Ritz variational method. J. Chem. Phys. 83 (1985) 1173-1196.

[29] M. Hoffmann-Ostenhof and R. Seiler, Cusp conditions for eigenfunctions of n-electron systems. Phys. Rev. A 23 (1981) 21-23.

[30] M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof and H. Stremnitzer, Local properties of Coulombic wave functions. Commun. Math. Phys. 163 (1994) 185-215. | Zbl 0812.35105

[31] C.-J. Huang, C.J. Umrigar and M.P. Nightingale, Accuracy of electronic wave functions in quantum Monte Carlo: The effect of high-order correlations. J. Chem. Phys. 107 (1997) 3007-3013.

[32] T. Kato, On the eigenfunctions of many-particle systems in quantum mechanics. Commun. Pure Appl. Math. 10 (1957) 151-177. | Zbl 0077.20904

[33] E. Krotscheck, Variations on the electron gas. Ann. Phys. (N.Y.) 155 (1984) 1-55.

[34] E. Krotscheck, Theory of inhomogeneous quantum systems. III. Variational wave functions for Fermi fluids. Phys. Rev. B 31 (1985) 4267-4278.

[35] E. Krotscheck, W. Kohn and G.-X. Qian, Theory of inhomogeneous quantum systems. IV. Variational calculations of metal surfaces. Phys. Rev. B 32 (1985) 5693-5712.

[36] W. Kutzelnigg, ${r}_{12}$-Dependent terms in the wave function as closed sums of partial wave amplitudes for large l. Theoret. Chim. Acta 68 (1985) 445-469.

[37] W. Kutzelnigg and J.D. Morgan Iii, Rates of convergence of the partial-wave expansions of atomic correlation energies. J. Chem. Phys. 96 (1992) 4484-4508.

[38] H. Luo, D. Kolb, H.-J. Flad, W. Hackbusch and T. Koprucki, Wavelet approximation of correlated wavefunctions. II. Hyperbolic wavelets and adaptive approximation schemes. J. Chem. Phys. 117 (2002) 3625-3638.

[39] H. Luo, D. Kolb, H.-J. Flad and W. Hackbusch, Perturbative calculation of Jastrow factors. Phys. Rev. B. 75 (2007) 125111.

[40] P.-A. Nitsche, Sparse approximation of singularity functions. Constr. Approx. 21 (2005) 63-81. | Zbl 1073.65118

[41] P.-A. Nitsche, Best N-term approximation spaces for tensor product wavelet bases. Constr. Approx. 24 (2006) 49-70. | Zbl 1101.41021

[42] T. Pang, C.E. Campbell and E. Krotscheck, Local structure of electron correlations in atomic systems. Chem. Phys. Lett. 163 (1989) 537-541.

[43] K.E. Schmidt and J.W. Moskowitz, Correlated Monte Carlo wave functions for the atoms He through Ne. J. Chem. Phys. 93 (1990) 4172-4178.

[44] E.M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton University Press (1993). | MR 1232192 | Zbl 0821.42001

[45] G. Stollhoff, The local ansatz extended. J. Chem. Phys. 105 (1996) 227-234.

[46] G. Stollhoff and P. Fulde, On the computation of electronic correlation energies within the local approach. J. Chem. Phys. 73 (1980) 4548-4561.

[47] J.D. Talman, Linked-cluster expansion for Jastrow-type wave functions and its application to the electron-gas problem. Phys. Rev. A 10 (1974) 1333-1344.

[48] J.D. Talman, Variational calculation for the electron gas at intermediate densities. Phys. Rev. A 13 (1976) 1200-1208.

[49] C.J. Umrigar, K.G. Wilson and J.W. Wilkins, Optimized trial wave functions for quantum Monte Carlo calculations. Phys. Rev. Lett. 60 (1988) 1719-1722.

[50] A.J. Williamson, S.D. Kenny, G. Rajagopal, A.J. James, R.J. Needs, L.M. Fraser, W.M.C. Foulkes and P. Maccallum, Optimized wavefunctions for quantum Monte Carlo studies of atoms and solids. Phys. Rev. B 53 (1996) 9640-9648.

[51] H. Yserentant, On the regularity of the electronic Schrödinger equation in Hilbert spaces of mixed derivatives. Numer. Math. 98 (2004) 731-759. | Zbl 1062.35100

[52] H. Yserentant, Sparse grid spaces for the numerical solution of the electronic Schrödinger equation. Numer. Math. 101 (2005) 381-389. | Zbl 1084.65125