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DIIS (direct inversion in the iterative subspace or direct inversion of the iterative subspace), 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.[1][2]

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.


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 constructed

\mathbf e_{m+1}=\sum_i^m\ c_i\mathbf e_i.

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

L&=\|e_{m+1}\|^2-\lambda\left(\sum_i\ c_i-1\right),\\
&=\sum_{ij}c_jB_{ji}c_i-\lambda\left(\sum_i\ c_i-1\right),\ \mathrm{where}\ B_{ij}=\langle\mathbf e_j|\mathbf e_i\rangle.

Equating the derivatives of L, with respect to the coefficients and the multiplier, equal to zero, leads to m + 1 linear equations to be solved for the m coefficients. The coefficients are then used to update the function variable as

\mathbf p_{m+1}=\sum_i^m c_i\mathbf p_i.


  1. ^ Pulay, Péter (1980). "Convergence acceleration of iterative sequences. the case of SCF iteration". Chemical Physics Letters 73 (2): 393–398. doi:10.1016/0009-2614(80)80396-4. 
  2. ^ Pulay, Péter (1982). "Improved SCF Convergence Acceleration". Journal of Computational Chemistry 3 (4): 556–560. doi:10.1002/jcc.540030413. 


  • Garza, Alejandro J.; Scuseria, Gustavo E. (2012). "Comparison of self-consistent field convergence acceleration techniques". Journal of Chemical Physics 173 (5): 054110. doi:10.1063/1.4740249. 
  • Rohwedder, Thorsten; Schneider, Reinhold (2011). "An analysis for the DIIS acceleration method used in quantum chemistry calculations". Journal of Mathematical Chemistry 49 (9): 1889. doi:10.1007/s10910-011-9863-y. 

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