Extensive benchmarking calculations are presented to assess the accuracy of the standard approximate coupled cluster singles and doubles method (CC2) in studying * excited states properties of model protein chains containing a phenylalanine residue, namely capped peptides, whose ground state conformers adopt the prototypical secondary structural features of proteins. First, the dependence with the basis-set of the CC2 excitation energies, CC2 geometry optimizations and amide A region frequencies of the lowest * excited state in a reference system, the N-acetyl-phenylalaninylamide, are investigated and the results are compared with experimental data. Second, at the best level of theory determined, the CC2/aug(N,O,)-cc-pVDZ//CC2/cc-

In Kohn-Sham electronic structure computations, wave functions have singularities at nuclear positions. Because of these singularities, plane-wave expansions give a poor approximation of the eigenfunctions. In conjunction with the use of pseudo-potentials, the PAW (projector augmented-wave) method circumvents this issue by replacing the original eigenvalue problem by a new one with the same eigenvalues but smoother eigenvectors. Here a slightly different method, called VPAW (variational PAW), is proposed and analyzed. This new method allows for a better convergence with respect to the number of plane-waves. Some numerical tests on an idealized case corroborate this efficiency. This work has been recently announced in [3]. arXiv:1711.06529v1 [math.NA] 17 Nov 2017 1. the number N I of PAW functions used to build T I , 2. a cut-off r c radius which will set the acting domain of T I , more precisely:

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This paper deals with a general class of algorithms for the solution of fixed-point problems that we refer to as Anderson–Pulay acceleration. This family includes the DIIS technique and its variant sometimes called commutator-DIIS, both introduced by Pulay in the 1980s to accelerate the convergence of self-consistent field procedures in quantum chemistry, as well as the related Anderson acceleration which dates back to the 1960s, and the wealth of techniques they have inspired. Such methods aim at accelerating the convergence of any fixed-point iteration method by combining several iterates in order to generate the next one at each step. This extrapolation process is characterised by its depth, i.e. the number of previous iterates stored, which is a crucial parameter for the efficiency of the method. It is generally fixed to an empirical value. In the present work, we consider two parameter-driven mechanisms to let the depth vary along the iterations. In the first one, the depth grows until a certain nondegeneracy condition is no longer satisfied; then the stored iterates (save for the last one) are discarded and the method ``restarts’’. In the second one, we adapt the depth continuously by eliminating at each step some of the oldest, less relevant, iterates. In an abstract and general setting, we prove under natural assumptions the local convergence and acceleration of these two adaptive Anderson–Pulay methods, and we show that one can theoretically achieve a superlinear convergence rate with each of them. We then investigate their behaviour in quantum chemistry calculations. These numerical experiments show that both adaptive variants exhibit a faster convergence than a standard fixed-depth scheme, and require on average less computational effort per iteration. This study is complemented by a review of known facts on the DIIS, in particular its link with the Anderson acceleration and some multisecant-type quasi-Newton methods.

This paper deals with a general class of algorithms for the solution of fixed-point problems that we refer to as Anderson-Pulay acceleration. This family includes the DIIS technique and its variant sometimes called commutator-DIIS, both introduced by Pulay in the 1980s to accelerate the convergence of self-consistent field procedures in quantum chemistry, as well as the related Anderson acceleration, which dates back to the 1960s, and the wealth of methods it inspired. Such methods aim at accelerating the convergence of any fixed-point iteration method by combining several previous iterates in order to generate the next one at each step. The size of the set of stored iterates is characterised by its depth, which is a crucial parameter for the efficiency of the process. It is generally fixed to an empirical value in most applications.In the present work, we consider two parameter-driven mechanisms to let the depth vary along the iterations. One way to do so is to let the set grow until the stored iterates (save for the last one) are discarded and the method "restarts". Another way is to "adapt" the depth by eliminating some of the older, less relevant, iterates at each step. In an abstract and general setting, we prove under natural assumptions the local convergence and acceleration of these two types of Anderson-Pulay acceleration methods and demonstrate how to theoretically achieve a superlinear convergence rate. We then investigate their behaviour in calculations with the Hartree-Fock method and the Kohn-Sham model of density functional theory. These numerical experiments show that the restarted and adaptive-depth variants exhibit a faster convergence than that of a standard fixed-depth scheme, and require on average less computational effort per iteration. This study is complemented by a review of known facts on the DIIS, in particular its link with the Anderson acceleration and some multisecant-type quasi-Newton methods.

In this article, a numerical analysis of the projector augmented-wave (PAW) method is presented, restricted to the case of dimension one with Dirac potentials modeling the nuclei in a periodic setting. The PAW method is widely used in electronic ab initio calculations, in conjunction with pseudopotentials. It consists in replacing the original electronic Hamiltonian H by a pseudo-Hamiltonian H P AW via the PAW transformation acting in balls around each nuclei. Formally, the new eigenvalue problem has the same eigenvalues as H and smoother eigenfunctions. In practice, the pseudo-Hamiltonian H P AW has to be truncated, introducing an error that is rarely analyzed. In this paper, error estimates on the lowest PAW eigenvalue are proved for the one-dimensional periodic Schrödinger operator with double Dirac potentials.