Gradients conjugate density matrices

Millam, J. M., Scuseria, G. E., 1997, Linear Scaling Conjugate Gradient Density Matrix Search as an Alternative to Diagonalization for First Principles Electronic Structure Calculations , J. Chem. Phys., 106, 5569. 

Daniels AD, Milliam JM, Scuseria GE. Semiempirical methods with conjugate gradient density matrix search to replace diagonalization for molecular systems containing thousands of atoms. J Chem Phys 1997 107 425 431. 

Application of the Millam-Scuseria conjugate-gradient density-matrix search method (Section 15.5) to semiempirical methods aUowed single-point AMI energy calculations to be done on a 19995-atom polymer of glydne and a 6304-atom RNA molecule [A. D. Daniels, J. M. Millam, and G. E. Scuseria, /. Chem. Phys., 107,425 (1997)]. 

Direct minimization techniques. The variational principle indicates that we want to minimize the energy as a function of the MO coefficients or the corresponding density matrix elements, as given by eq. (3.54). In this formulation, the problem is no different from other types of non-linear optimizations, and the same types of technique, such as steepest descent, conjugated gradient or Newton-Raphson methods can be used (see Chapter 12 for details). 

Finally, the minimization of the total energy with respect to the density-matrix expansion coefficients Ki i can be done through a conjugate-gradient technique, which scales essentially as N instead of by solving a matrix eigenvalue problem (which would scale as tP). 

Optimization of the Atomic-orbital Density Matrix Using the Conjugate-gradient Method with a Multilevel Preconditioner. 

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