Lab Home | Phone | Search
Center for Nonlinear Studies  Center for Nonlinear Studies
 Home 
 People 
 Current 
 Affiliates 
 Alumni 
 Visitors 
 Students 
 Research 
 ICAM-LANL 
 Quantum 
 Publications 
 Publications 
 2007 
 2006 
 2005 
 2004 
 2003 
 2002 
 2001 
 2000 
 <1999 
 Conferences 
 Workshops 
 Sponsorship 
 Talks 
 Colloquia 
 Colloquia Archive 
 Seminars 
 Postdoc Seminars Archive 
 Quantum Lunch 
 CMS Colloquia 
 Q-Mat Seminars 
 Q-Mat Seminars Archive 
 Archive 
 Kac Lectures 
 Dist. Quant. Lecture 
 Ulam Scholar 
 Colloquia 
 
 Jobs 
 Students 
 Summer Research 
 Student Application 
 Visitors 
 Description 
 Past Visitors 
 Services 
 General 
 PD Travel Request 
 
 History of CNLS 
 
 Maps, Directions 
 CNLS Office 
 T-Division 
 LANL 
 
Monday, November 04, 2013
3:00 PM - 4:00 PM
CNLS Conference Room (TA-3, Bldg 1690)

Colloquium

Nonlinear approximations in quantum chemistry: some results and challenges

Greg Beylkin
University of Colorado

A typical approach to solving partial differential or integral equations is to approximate their solution via a fixed (perhaps, an adaptive) basis and then find the coefficients of the representation. Nonlinear approximations express solutions via linear combinations of functions chosen from a set much larger than a basis. The problem then is to find an optimal (or near optimal) representation of the solution in this form while satisfying the underlying equations.

Nonlinear approximations to find solutions of quantum chemistry problems (within one-particle theories, e.g., Hartree-Fock) has its origins in seminal papers of Boys, Longstaff and Singer (1960). The authors used linear combinations of Gaussians and optimized their exponents and coefficients via Newton's method. Unfortunately, such approach turned out to be practical only for very small molecules. Consequently, construction of spatial orbitals has been performed off-line and the resulting sets of functions used as a fixed basis. More recent multiresolution approaches (e.g., via multiwavelets), use dynamically refined selection of basis functions to represent spatial orbitals. However, since the basis functions do not resemble the spatial orbitals, such multiresolution methods require a relatively large number of parameters in resulting representations.

Another example of nonlinear approximations are separated representations of multivariate functions. In a paper with M. Mohlenkamp and F. Prez (2008), we used such representations to solve the multiparticle Schrödinger equation. Since the wavefunction is antisymmetric, it is natural to approximate it as a sum of Slater determinants. Many current methods do so, but they impose additional structural constraints on the determinants, such as orthogonality between orbitals or an excitation pattern. Our method has no such constraints and instead, for a specified accuracy, we are looking for the best separated approximation to the wavefunction.

Using new algorithms for optimizing nonlinear approximations via exponentials (or Gaussians), T. Haut and I recently developed an adaptive method for one-particle theories. Compared to previous multiresolution approaches, we do not use bases and, as a result, significantly reduce the number of independent parameters needed to represent solutions. We are planning to use these functional forms within a separated representation to accelerate the algorithm for solving the Schrödinger equation (whose current implementation using multiwavelet bases is too slow to be competitive with one-particle methods).

The talk will describe relevant nonlinear approximations, algorithms for constructing them, current results and, more generally, challenges in solving multivariate problems.

Host: Terry Haut