DIIS
Encyclopedia
DIIS also known as Pulay mixing, is an extrapolation
technique. DIIS was developed by Peter Pulay
in the field of computational quantum chemistry
with the intent to accelerate and stabilize the convergence of the Hartree–Fock self consistent field method.
At a given iteration, the approach constructs a linear combination
of approximate error vectors from previous iterations. The coefficients of the linear combination are determined so to best approximate, in a least squares
sense, the null vector
. The newly determined coefficients are then used to extrapolate the function variable for the next iteration.
The DIIS method seeks to minimize the norm of em+1 under the constraint that the coefficients sum to one. This is done by a Lagrange multiplier technique. Introducing an undetermined multiplier λ, a Lagrangian is constructed as
Equating the derivatives of L, with respect to the coefficients and the multiplier, equal to zero, leads to m + 1 linear equation
s to be solved for the m coefficients. The coefficients are then used to update the function variable as
Extrapolation
In mathematics, extrapolation is the process of constructing new data points. It is similar to the process of interpolation, which constructs new points between known points, but the results of extrapolations are often less meaningful, and are subject to greater uncertainty. It may also mean...
technique. DIIS was developed by Peter Pulay
Peter Pulay
Peter Pulay is a theoretical chemist. He is the Roger B. Bost Distinguished Professor of Chemistry in the Department of Chemistry and Biochemistry at the University of Arkansas, U.S....
in the field of computational quantum chemistry
Quantum chemistry
Quantum chemistry is a branch of chemistry whose primary focus is the application of quantum mechanics in physical models and experiments of chemical systems...
with the intent to accelerate and stabilize the convergence of the Hartree–Fock self consistent field method.
At a given iteration, the approach constructs a linear combination
Linear combination
In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results...
of approximate error vectors from previous iterations. The coefficients of the linear combination are determined so to best approximate, in a least squares
Least squares
The method of least squares is a standard approach to the approximate solution of overdetermined systems, i.e., sets of equations in which there are more equations than unknowns. "Least squares" means that the overall solution minimizes the sum of the squares of the errors made in solving every...
sense, the null vector
Null vector
Null vector can refer to:* Null vector * A causal structure in Minkowski space...
. The newly determined coefficients are then used to extrapolate the function variable for the next iteration.
Details
At each iteration, an approximate error vector, ei, corresponding to the variable value, pi is determined. After sufficient iterations, a linear combination of m previous error vectors is constructedThe DIIS method seeks to minimize the norm of em+1 under the constraint that the coefficients sum to one. This is done by a Lagrange multiplier technique. Introducing an undetermined multiplier λ, a Lagrangian is constructed as
Equating the derivatives of L, with respect to the coefficients and the multiplier, equal to zero, leads to m + 1 linear equation
Linear equation
A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable....
s to be solved for the m coefficients. The coefficients are then used to update the function variable as