Orthogonalization Procedures and the Localization of Wannier Functions

Carlson, B. C.; Keller, Joseph M.

doi:10.1103/physrev.105.102

Cited by 248 publications

(112 citation statements)

References 2 publications

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“…According to the Carlson-Keller theorem, 37 the orthogonal orbitals thus constructed most closely resemble the initial nonorthogonal set. In this sense, they are expected to retain most of the localized character of R a , especially for compact atomic orbital sets.…”

Section: B Virtual Orbitalsmentioning

confidence: 99%

Fast noniterative orbital localization for large molecules

Aquilante

Pedersen

Merás³

et al. 2006

The Journal of Chemical Physics

164

192

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We use Cholesky decomposition of the density matrix in atomic orbital basis to define a new set of occupied molecular orbital coefficients. Analysis of the resulting orbitals ͑"Cholesky molecular orbitals"͒ demonstrates their localized character inherited from the sparsity of the density matrix.Comparison with the results of traditional iterative localization schemes shows minor differences with respect to a number of suitable measures of locality, particularly the scaling with system size of orbital pair domains used in local correlation methods. The Cholesky procedure for generating orthonormal localized orbitals is noniterative and may be made linear scaling. Although our present implementation scales cubically, the algorithm is significantly faster than any of the conventional localization schemes. In addition, since this approach does not require starting orbitals, it will be useful in local correlation treatments on top of diagonalization-free Hartree-Fock optimization algorithms.

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Section: B Virtual Orbitalsmentioning

confidence: 99%

Fast noniterative orbital localization for large molecules

Aquilante

Pedersen

Merás³

et al. 2006

The Journal of Chemical Physics

164

192

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“…In the present scheme, this problem is addressed by using the weighted modification of the S −1/2 orthogonalization derived by Carlson and Keller, 55 who proved the following theorem. If the f n form a set of non-orthogonal orbitals and the g n are the orthogonal orbitals that minimize the weighted average of mean square deviations, Manifestly, w runs from 1 to 100 if the respective ξ A n run from 0 to ξ A max .…”

Section: Weighted Inter-atomic Orthogonalizationmentioning

confidence: 99%

A comprehensive analysis of molecule-intrinsic quasi-atomic, bonding, and correlating orbitals. I. Hartree-Fock wave functions

West

Schmidt

Gordon

et al. 2013

The Journal of Chemical Physics

203

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Through a basis-set-independent web of localizing orbital-transformations, the electronic wave function of a molecule is expressed in terms of a set of orbitals that reveal the atomic structure and the bonding pattern of a molecule. The analysis is based on resolving the valence orbital space in terms of an internal space, which has minimal basis set dimensions, and an external space. In the internal space, oriented quasi-atomic orbitals and split-localized molecular orbitals are determined by new, fast localization methods. The density matrix between the oriented quasi-atomic orbitals as well as the locations of the split-localized orbitals exhibit atomic populations and inter-atomic bonding patterns. A correlation-adapted quasi-atomic basis is determined in the external orbital space. The general formulations are specified in detail for Hartree-Fock wave functions. Applications to specific molecules exemplify the general scheme. Through a basis-set-independent web of localizing orbital-transformations, the electronic wave function of a molecule is expressed in terms of a set of orbitals that reveal the atomic structure and the bonding pattern of a molecule. The analysis is based on resolving the valence orbital space in terms of an internal space, which has minimal basis set dimensions, and an external space. In the internal space, oriented quasi-atomic orbitals and split-localized molecular orbitals are determined by new, fast localization methods. The density matrix between the oriented quasi-atomic orbitals as well as the locations of the split-localized orbitals exhibit atomic populations and interatomic bonding patterns. A correlation-adapted quasi-atomic basis is determined in the external orbital space. The general formulations are specified in detail for Hartree-Fock wave functions. Applications to specific molecules exemplify the general scheme. © 2013 AIP Publishing LLC.

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“…The optimal orthonormalization scheme here is L€ owdin's symmetric orthonormalization, [203] as it changesĈV as little as possible. [204] The above-described algorithm leads for Dt!0 to Uðt1DtÞ!UðtÞ. However, for finite Dt, one has to take some care.…”

Section: Appendix A: Propagation Using Substepsmentioning

confidence: 99%

A general method to describe intersystem crossing dynamics in trajectory surface hopping

Mai

Marquetand

González

2015

Int J of Quantum Chemistry

269

406

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Intersystem crossing is a radiationless process that can take place in a molecule irradiated by UV‐Vis light, thereby playing an important role in many environmental, biological and technological processes. This paper reviews different methods to describe intersystem crossing dynamics, paying attention to semiclassical trajectory theories, which are especially interesting because they can be applied to large systems with many degrees of freedom. In particular, a general trajectory surface hopping methodology recently developed by the authors, which is able to include nonadiabatic and spin‐orbit couplings in excited‐state dynamics simulations, is explained in detail. This method, termed SHARC, can in principle include any arbitrary coupling, what makes it generally applicable to photophysical and photochemical problems, also those including explicit laser fields. A step‐by‐step derivation of the main equations of motion employed in surface hopping based on the fewest‐switches method of Tully, adapted for the inclusion of spin‐orbit interactions, is provided. Special emphasis is put on describing the different possible choices of the electronic bases in which spin‐orbit can be included in surface hopping, highlighting the advantages and inconsistencies of the different approaches. © 2015 Wiley Periodicals, Inc.

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Orthogonalization Procedures and the Localization of Wannier Functions

Cited by 248 publications

References 2 publications

Fast noniterative orbital localization for large molecules

Fast noniterative orbital localization for large molecules

A comprehensive analysis of molecule-intrinsic quasi-atomic, bonding, and correlating orbitals. I. Hartree-Fock wave functions

A general method to describe intersystem crossing dynamics in trajectory surface hopping

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