Journal of Mathematical Chemistry

, Volume 49, Issue 1, pp 296–324 | Cite as

Reconciling alternate methods for the determination of charge distributions: a probabilistic approach to high-dimensional least-squares approximations

Original Paper


We propose extensions and improvements of the statistical analysis of distributed multipoles (SADM) algorithm put forth by Chipot et al. (Mol Phys 94: 881–895, 1998) for the derivation of distributed atomic multipoles from the quantum-mechanical electrostatic potential. The method is mathematically extended to general least-squares problems and provides an alternative approximation method in cases where the original least-squares problem is computationally not tractable, either because of its ill-posedness or its high-dimensionality. The solution is approximated employing a Monte Carlo method that takes the average of a random variable defined as the solutions of random small least-squares problems drawn as subsystems of the original problem. The conditions that ensure convergence and consistency of the method are discussed, along with an analysis of the computational cost in specific instances.


Least-squares approximation Monte Carlo methods High-dimensional problems 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Chipot C., Ángyán J.G., Millot C.: Statistical analysis of distributed multipoles derived from the molecular electrostatic potential. Mol. Phys. 94, 881–895 (1998)CrossRefGoogle Scholar
  2. 2.
    Cornell W.D., Chipot C.: Alternative Approaches to Charge Distribution Calculations. In: Schleyer, P.V.R., Allinger, N.L., Clark, T., Gasteiger, J., Kollman, P.A., Schaefer, H.F. III, Schreiner, P.R. (eds) Encyclopedia of Computational Chemistry, Vol. 1., pp. 258–263. Wiley and Sons, Chichester (1998)Google Scholar
  3. 3.
    MacKerell A.D. Jr, Bashford D., Bellott M., Dunbrack R.L. Jr, Evanseck J.D., Field M.J., Fischer S., Gao J., Guo H., Ha S., Joseph-McCarthy D., Kuchnir L., Kuczera K., Lau F.T.K., Mattos C., Michnick S., Ngo T., Nguyen D.T., Prodhom B., Reiher W.E. III, Roux B., Schlenkrich M., Smith J.C., Stote R., Straub J., Watanabe M., Wiórkiewicz-Kuczera J., Yin D., Karplus M.: All-atom empirical potential for molecular modeling and dynamics studies of proteins. J. Phys. Chem. B 102, 3586–3616 (1998)CrossRefGoogle Scholar
  4. 4.
    Cornell W.D., Cieplak P., Bayly C.I., Gould I.R., Merz K.M. Jr, Ferguson D.M., Spellmeyer D.C., Fox T., Caldwell J.C., Kollman P.A.: A second generation force field for the simulation of proteins, nucleic acids, and organic molecules. J. Am. Chem. Soc 117, 5179–5197 (1995)CrossRefGoogle Scholar
  5. 5.
    Cox S.R., Williams D.E.: Representation of the molecular electrostatic potential by a net atomic charge model. J. Comput. Chem. 2, 304–323 (1981)CrossRefGoogle Scholar
  6. 6.
    Chirlian L.E., Francl M.M.: Atomic charges derived from electrostatic potentials: a detailed study. J. Comput. Chem. 8, 894–905 (1987)CrossRefGoogle Scholar
  7. 7.
    Bayly C.I., Cieplak P., Cornell W.D., Kollman P.A.: A well-behaved electrostatic potential based method using charge restraints for deriving atomic charges: the resp model. J. Phys. Chem. 97, 10269–10280 (1993)CrossRefGoogle Scholar
  8. 8.
    Francl M.M., Carey C., Chirlian L.E., Gange D.M.: Charges fit to the electrostatic potentials. ii. can atomic charges be unambiguously fit to electrostatic potentials?. J. Comput. Chem. 17, 367–383 (1996)CrossRefGoogle Scholar
  9. 9.
    Meyer Y., Xu H.: Wavelet analysis and chirps. Appl. Comput. Harmon. Anal. 4(4), 366–379 (1997)CrossRefGoogle Scholar
  10. 10.
    Heller E.J.: Time dependent approach to semiclassical dynamics. J. Chem. Phys. 62, 1544–1555 (1975)CrossRefGoogle Scholar
  11. 11.
    Bjorck A.: Numerical Methods for Least Squares Problems. SIAM, Philadelphia, PA (1996)Google Scholar
  12. 12.
    Anderson T.W.: An Introduction to Multivariate Statistical Analysis, 2nd Edition, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. Wiley, New York (1984)Google Scholar
  13. 13.
    V.V. Petrov, Sums of independent random variables, Springer, New York, 1975, translated from the Russian by A.A. Brown, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 82.Google Scholar
  14. 14.
    Edelman A.: Eigenvalues and condition numbers of random matrices. SIAM J. Matrix Anal. Appl. 9(4), 543–560 (1988)CrossRefGoogle Scholar
  15. 15.
    Edelman A.: The distribution and moments of the smallest eigenvalue of a random matrix of Wishart type. Linear Algebra Appl. 159, 55–80 (1991)CrossRefGoogle Scholar
  16. 16.
    Chen Z., Dongarra J.J.: Condition numbers of Gaussian random matrices. SIAM J. Matrix Anal. Appl. 27(3), 603–620 (2005) (electronic)CrossRefGoogle Scholar
  17. 17.
    M. Rudelson, R. Vershynin, The smallest singular value of a random rectangular matrix, preprint (2008).Google Scholar
  18. 18.
    M.J. Frisch, G.W. Trucks, H.B. Schlegel, G.E. Scuseria, M.A. Robb, J.R. Cheeseman, J.A. Montgomery, Jr., T. Vreven, K.N. Kudin, J.C. Burant, J.M. Millam, S.S. Iyengar, J. Tomasi, V. Barone, B. Mennucci, M. Cossi, G. Scalmani, N. Rega, G.A. Petersson, H. Nakatsuji, M. Hada, M. Ehara, K. Toyota, R. Fukuda, J. Hasegawa, M. Ishida, T. Nakajima, Y. Honda, O. Kitao, H. Nakai, M. Klene, X. Li, J.E. Knox, H.P. Hratchian, J.B. Cross, C. Adamo, J. Jaramillo, R. Gomperts, R.E. Stratmann, O. Yazyev, A.J. Austin, R. Cammi, C. Pomelli, J.W. Ochterski, P.Y. Ayala, K. Morokuma, G.A. Voth, P. Salvador, J.J. Dannenberg, V.G. Zakrzewski, S. Dapprich, A.D. Daniels, M.C. Strain, O. Farkas, D.K. Malick, A.D. Rabuck, K. Raghavachari, J.B. Foresman, J.V. Ortiz, Q. Cui, A.G. Baboul, S. Clifford, J. Cioslowski, B.B. Stefanov, G. Liu, A. Liashenko, P. Piskorz, I. Komaromi, R.L. Martin, D.J. Fox, T. Keith, M.A. Al-Laham, C.Y. Peng, A. Nanayakkara, M. Challacombe, P.M.W. Gill, B. Johnson, W. Chen, M.W. Wong, C. Gonzalez, J.A. Pople, Gaussian 03 Revision C.02, Gaussian Inc., Wallingford, CT (2004).Google Scholar
  19. 19.
    Ángyán J.G., Chipot C., Dehez F., Hättig C., Jansen G., Millot C.: Opep: a tool for the optimal partitioning of electric properties. J. Comput. Chem. 24, 997–1008 (2003)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.TOSCA project-teamINRIA Sophia Antipolis-MéditerranéeSophia Antipolis CedexFrance
  2. 2.Theoretical and Computational Biophysics Group, Beckman Institute for Advanced Science and EngineeringUniversity of Illinois at Urbana-ChampaignUrbanaUSA
  3. 3.INRIA & École Normale Supérieure de Cachan BretagneBruzFrance

Personalised recommendations