Best N-term approximation in electronic structure calculations. II. Jastrow factors
ESAIM: Modélisation mathématique et analyse numérique, Tome 41 (2007) no. 2, pp. 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 (2) near electron-electron and electron-nuclear cusps. Based on Nitsche’s characterization of best N-term approximation spaces A q α (H 1 ), we prove that (2) A q α (H 1 ) for q>1 and α=1 q-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 : 10.1051/m2an:2007016
Classification : 41A50, 41A63, 65Z05, 81V70
Mots clés : best N-term approximation, wavelets, electron correlations, Jastrow factor
Flad, Heinz-Jürgen  ; Hackbusch, Wolfgang  ; Schneider, Reinhold 1

1 Institut für Informatik Christian-Albrechts-Universität zu Kiel, Christian-Albrechts-Platz 4, 24098 Kiel, Germany. ; Christian-Albrechts-Universität Kiel, Christian-Albrechts-Platz 4, 24098 Kiel, Germany.
@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: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {261--279},
     publisher = {EDP-Sciences},
     volume = {41},
     number = {2},
     year = {2007},
     doi = {10.1051/m2an:2007016},
     mrnumber = {2339628},
     zbl = {1135.81029},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an:2007016/}
}
TY  - JOUR
AU  - Flad, Heinz-Jürgen
AU  - Hackbusch, Wolfgang
AU  - Schneider, Reinhold
TI  - Best $N$-term approximation in electronic structure calculations. II. Jastrow factors
JO  - ESAIM: Modélisation mathématique et analyse numérique
PY  - 2007
SP  - 261
EP  - 279
VL  - 41
IS  - 2
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an:2007016/
DO  - 10.1051/m2an:2007016
LA  - en
ID  - M2AN_2007__41_2_261_0
ER  - 
%0 Journal Article
%A Flad, Heinz-Jürgen
%A Hackbusch, Wolfgang
%A Schneider, Reinhold
%T Best $N$-term approximation in electronic structure calculations. II. Jastrow factors
%J ESAIM: Modélisation mathématique et analyse numérique
%D 2007
%P 261-279
%V 41
%N 2
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an:2007016/
%R 10.1051/m2an:2007016
%G en
%F 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: Modélisation mathématique et analyse numérique, Tome 41 (2007) no. 2, pp. 261-279. doi : 10.1051/m2an:2007016. http://www.numdam.org/articles/10.1051/m2an:2007016/

[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 | Zbl

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

[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 | Zbl

[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

[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

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

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

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

[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

[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

[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

[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

[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

[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

[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

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

[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 | Zbl

[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

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

Cité par Sources :