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Title: Accelerating molecular property calculations with nonorthonormal Krylov space methods

Abstract

Here, we formulate Krylov space methods for large eigenvalue problems and linear equation systems that take advantage of decreasing residual norms to reduce the cost of matrix-vector multiplication. The residuals are used as subspace basis without prior orthonormalization, which leads to generalized eigenvalue problems or linear equation systems on the Krylov space. These nonorthonormal Krylov space (nKs) algorithms are favorable for large matrices with irregular sparsity patterns whose elements are computed on the fly, because fewer operations are necessary as the residual norm decreases as compared to the conventional method, while errors in the desired eigenpairs and solution vectors remain small. We consider real symmetric and symplectic eigenvalue problems as well as linear equation systems and Sylvester equations as they appear in configuration interaction and response theory. The nKs method can be implemented in existing electronic structure codes with minor modifications and yields speed-ups of 1.2-1.8 in typical time-dependent Hartree-Fock and density functional applications without accuracy loss. The algorithm can compute entire linear subspaces simultaneously which benefits electronic spectra and force constant calculations requiring many eigenpairs or solution vectors. The nKs approach is related to difference density methods in electronic ground state calculations, and particularly efficient for integral direct computationsmore » of exchange-type contractions. By combination with resolution-of-the-identity methods for Coulomb contractions, three- to fivefold speed-ups of hybrid time-dependent density functional excited state and response calculations are achieved.« less

Authors:
 [1];  [1];  [1];  [1]
  1. Univ. of California, Irvine, CA (United States)
Publication Date:
Research Org.:
Univ. of California, Irvine, CA (United States)
Sponsoring Org.:
USDOE Office of Science (SC), Basic Energy Sciences (BES) (SC-22)
OSTI Identifier:
1250890
Alternate Identifier(s):
OSTI ID: 1250585
Grant/Contract Number:  
SC0008694
Resource Type:
Accepted Manuscript
Journal Name:
Journal of Chemical Physics
Additional Journal Information:
Journal Volume: 144; Journal Issue: 17; Journal ID: ISSN 0021-9606
Publisher:
American Institute of Physics (AIP)
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; 74 ATOMIC AND MOLECULAR PHYSICS; eigenvalues; subspaces; linear equations; excitation energies; ground states

Citation Formats

Furche, Filipp, Krull, Brandon T., Nguyen, Brian D., and Kwon, Jake. Accelerating molecular property calculations with nonorthonormal Krylov space methods. United States: N. p., 2016. Web. doi:10.1063/1.4947245.
Furche, Filipp, Krull, Brandon T., Nguyen, Brian D., & Kwon, Jake. Accelerating molecular property calculations with nonorthonormal Krylov space methods. United States. doi:10.1063/1.4947245.
Furche, Filipp, Krull, Brandon T., Nguyen, Brian D., and Kwon, Jake. Tue . "Accelerating molecular property calculations with nonorthonormal Krylov space methods". United States. doi:10.1063/1.4947245. https://www.osti.gov/servlets/purl/1250890.
@article{osti_1250890,
title = {Accelerating molecular property calculations with nonorthonormal Krylov space methods},
author = {Furche, Filipp and Krull, Brandon T. and Nguyen, Brian D. and Kwon, Jake},
abstractNote = {Here, we formulate Krylov space methods for large eigenvalue problems and linear equation systems that take advantage of decreasing residual norms to reduce the cost of matrix-vector multiplication. The residuals are used as subspace basis without prior orthonormalization, which leads to generalized eigenvalue problems or linear equation systems on the Krylov space. These nonorthonormal Krylov space (nKs) algorithms are favorable for large matrices with irregular sparsity patterns whose elements are computed on the fly, because fewer operations are necessary as the residual norm decreases as compared to the conventional method, while errors in the desired eigenpairs and solution vectors remain small. We consider real symmetric and symplectic eigenvalue problems as well as linear equation systems and Sylvester equations as they appear in configuration interaction and response theory. The nKs method can be implemented in existing electronic structure codes with minor modifications and yields speed-ups of 1.2-1.8 in typical time-dependent Hartree-Fock and density functional applications without accuracy loss. The algorithm can compute entire linear subspaces simultaneously which benefits electronic spectra and force constant calculations requiring many eigenpairs or solution vectors. The nKs approach is related to difference density methods in electronic ground state calculations, and particularly efficient for integral direct computations of exchange-type contractions. By combination with resolution-of-the-identity methods for Coulomb contractions, three- to fivefold speed-ups of hybrid time-dependent density functional excited state and response calculations are achieved.},
doi = {10.1063/1.4947245},
journal = {Journal of Chemical Physics},
number = 17,
volume = 144,
place = {United States},
year = {2016},
month = {5}
}

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Works referenced in this record:

Recent Advances in Wave Function-Based Methods of Molecular-Property Calculations
journal, December 2011

  • Helgaker, Trygve; Coriani, Sonia; Jørgensen, Poul
  • Chemical Reviews, Vol. 112, Issue 1
  • DOI: 10.1021/cr2002239

Linear and nonlinear response functions for an exact state and for an MCSCF state
journal, April 1985

  • Olsen, Jeppe; Jo/rgensen, Poul
  • The Journal of Chemical Physics, Vol. 82, Issue 7
  • DOI: 10.1063/1.448223

Response functions from Fourier component variational perturbation theory applied to a time-averaged quasienergy
journal, January 1998


A perspective on nonresonant and resonant electronic response theory for time-dependent molecular properties
journal, January 2011

  • Norman, Patrick
  • Physical Chemistry Chemical Physics, Vol. 13, Issue 46
  • DOI: 10.1039/c1cp21951k

The iterative calculation of a few of the lowest eigenvalues and corresponding eigenvectors of large real-symmetric matrices
journal, January 1975


A new method for large-scale Cl calculations
journal, August 1972


Large-scaleRPA calculations of chiroptical properties of organic molecules: Program RPAC
journal, February 1983

  • Bouman, Thomas D.; Hansen, Aage E.; Voigt, Bj�rn
  • International Journal of Quantum Chemistry, Vol. 23, Issue 2
  • DOI: 10.1002/qua.560230230

Direct atomic‐orbital‐based time‐dependent Hartree–Fock calculations of frequency‐dependent polarizabilities
journal, February 1992

  • Feyereisen, Martin; Nichols, Jeff; Oddershede, Jens
  • The Journal of Chemical Physics, Vol. 96, Issue 4
  • DOI: 10.1063/1.461995

Solution of the large matrix equations which occur in response theory
journal, February 1988

  • Olsen, Jeppe; Jensen, Hans Jørgen Aa; Jørgensen, Poul
  • Journal of Computational Physics, Vol. 74, Issue 2
  • DOI: 10.1016/0021-9991(88)90081-2

Principles for a direct SCF approach to LICAO - MO ab - initio calculations : Direct
journal, September 1982

  • Almlöf, J.; Faegri, K.; Korsell, K.
  • Journal of Computational Chemistry, Vol. 3, Issue 3
  • DOI: 10.1002/jcc.540030314

Improvements on the direct SCF method: Improved Direct SCF Method
journal, January 1989

  • Häser, Marco; Ahlrichs, Reinhart
  • Journal of Computational Chemistry, Vol. 10, Issue 1
  • DOI: 10.1002/jcc.540100111

Linear scaling computation of the Fock matrix
journal, April 1997

  • Challacombe, Matt; Schwegler, Eric
  • The Journal of Chemical Physics, Vol. 106, Issue 13
  • DOI: 10.1063/1.473575

Towards an order-
journal, January 1998

  • Fonseca Guerra, C.; Snijders, J. G.; te Velde, G.
  • Theoretical Chemistry Accounts, Vol. 99, Issue 6
  • DOI: 10.1007/s002140050021

Treatment of electronic excitations within the adiabatic approximation of time dependent density functional theory
journal, July 1996


An efficient implementation of time-dependent density-functional theory for the calculation of excitation energies of large molecules
journal, November 1998

  • Stratmann, R. Eric; Scuseria, Gustavo E.; Frisch, Michael J.
  • The Journal of Chemical Physics, Vol. 109, Issue 19
  • DOI: 10.1063/1.477483

Implementation of time-dependent density functional response equations
journal, May 1999

  • van Gisbergen, S. J. A.; Snijders, J. G.; Baerends, E. J.
  • Computer Physics Communications, Vol. 118, Issue 2-3
  • DOI: 10.1016/S0010-4655(99)00187-3

First-order one-electron properties in the integral-direct coupled cluster singles and doubles model
journal, July 1997

  • Halkier, Asger; Koch, Henrik; Christiansen, Ove
  • The Journal of Chemical Physics, Vol. 107, Issue 3
  • DOI: 10.1063/1.474384

Calculation of excitation energies within time-dependent density functional theory using auxiliary basis set expansions
journal, January 1997


Analytical time-dependent density functional derivative methods within the RI-J approximation, an approach to excited states of large molecules
journal, February 2005

  • Rappoport, Dmitrij; Furche, Filipp
  • The Journal of Chemical Physics, Vol. 122, Issue 6
  • DOI: 10.1063/1.1844492

Efficient linear-scaling calculation of response properties: Density matrix-based Laplace-transformed coupled-perturbed self-consistent field theory
journal, June 2008

  • Beer, Matthias; Ochsenfeld, Christian
  • The Journal of Chemical Physics, Vol. 128, Issue 22
  • DOI: 10.1063/1.2940731

Linear-scaling implementation of molecular response theory in self-consistent field electronic-structure theory
journal, April 2007

  • Coriani, Sonia; Høst, Stinne; Jansík, Branislav
  • The Journal of Chemical Physics, Vol. 126, Issue 15
  • DOI: 10.1063/1.2715568

Molecular-orbital-free algorithm for excited states in time-dependent perturbation theory
journal, August 2008

  • Lucero, Melissa J.; Niklasson, Anders M. N.; Tretiak, Sergei
  • The Journal of Chemical Physics, Vol. 129, Issue 6
  • DOI: 10.1063/1.2965535

Representation independent algorithms for molecular response calculations in time-dependent self-consistent field theories
journal, February 2009

  • Tretiak, Sergei; Isborn, Christine M.; Niklasson, Anders M. N.
  • The Journal of Chemical Physics, Vol. 130, Issue 5
  • DOI: 10.1063/1.3068658

Time-dependent Hartree–Fock frequency-dependent polarizability calculation applied to divide-and-conquer electronic structure method
journal, January 2010


Efficient time-dependent density functional theory approximations for hybrid density functionals: Analytical gradients and parallelization
journal, February 2011

  • Petrenko, Taras; Kossmann, Simone; Neese, Frank
  • The Journal of Chemical Physics, Vol. 134, Issue 5
  • DOI: 10.1063/1.3533441

Pseudospectral time-dependent density functional theory
journal, March 2008

  • Ko, Chaehyuk; Malick, David K.; Braden, Dale A.
  • The Journal of Chemical Physics, Vol. 128, Issue 10
  • DOI: 10.1063/1.2834222

Efficient Semi-numerical Implementation of Global and Local Hybrid Functionals for Time-Dependent Density Functional Theory
journal, August 2015

  • Maier, Toni M.; Bahmann, Hilke; Kaupp, Martin
  • Journal of Chemical Theory and Computation, Vol. 11, Issue 9
  • DOI: 10.1021/acs.jctc.5b00624

An unconventional scf method for calculations on large molecules
journal, June 1986

  • Cremer, Dieter; Gauss, J�Rgen
  • Journal of Computational Chemistry, Vol. 7, Issue 3
  • DOI: 10.1002/jcc.540070305

Linear and sublinear scaling formation of Hartree–Fock-type exchange matrices
journal, August 1998

  • Ochsenfeld, Christian; White, Christopher A.; Head-Gordon, Martin
  • The Journal of Chemical Physics, Vol. 109, Issue 5
  • DOI: 10.1063/1.476741

Linear scaling computation of the Fock matrix. II. Rigorous bounds on exchange integrals and incremental Fock build
journal, June 1997

  • Schwegler, Eric; Challacombe, Matt; Head-Gordon, Martin
  • The Journal of Chemical Physics, Vol. 106, Issue 23
  • DOI: 10.1063/1.473833

Super-matrix methods
journal, May 1989


Long-range charge-transfer excited states in time-dependent density functional theory require non-local exchange
journal, August 2003

  • Dreuw, Andreas; Weisman, Jennifer L.; Head-Gordon, Martin
  • The Journal of Chemical Physics, Vol. 119, Issue 6
  • DOI: 10.1063/1.1590951

TD-DFT Performance for the Visible Absorption Spectra of Organic Dyes:  Conventional versus Long-Range Hybrids
journal, November 2007

  • Jacquemin, Denis; Perpète, Eric A.; Scuseria, Gustavo E.
  • Journal of Chemical Theory and Computation, Vol. 4, Issue 1
  • DOI: 10.1021/ct700187z

Assessing Excited State Methods by Adiabatic Excitation Energies
journal, July 2011

  • Send, Robert; Kühn, Michael; Furche, Filipp
  • Journal of Chemical Theory and Computation, Vol. 7, Issue 8
  • DOI: 10.1021/ct200272b

Ab initio non-adiabatic molecular dynamics
journal, January 2013

  • Tapavicza, Enrico; Bellchambers, Gregory D.; Vincent, Jordan C.
  • Physical Chemistry Chemical Physics, Vol. 15, Issue 42
  • DOI: 10.1039/c3cp51514a

Time-dependent density functional theory within the Tamm–Dancoff approximation
journal, December 1999


Toward a systematic molecular orbital theory for excited states
journal, January 1992

  • Foresman, James B.; Head-Gordon, Martin; Pople, John A.
  • The Journal of Physical Chemistry, Vol. 96, Issue 1
  • DOI: 10.1021/j100180a030

The Davidson Method
journal, January 1994

  • Crouzeix, M.; Philippe, B.; Sadkane, M.
  • SIAM Journal on Scientific Computing, Vol. 15, Issue 1
  • DOI: 10.1137/0915004

Derivative studies in hartree-fock and møller-plesset theories
journal, March 1979

  • Pople, J. A.; Krishnan, R.; Schlegel, H. B.
  • International Journal of Quantum Chemistry, Vol. 16, Issue S13
  • DOI: 10.1002/qua.560160825

The determination of hyperpolarisabilities using density functional theory
journal, July 1993

  • Colwell, Susan M.; Murray, Christopher W.; Handy, Nicholas C.
  • Chemical Physics Letters, Vol. 210, Issue 1-3
  • DOI: 10.1016/0009-2614(93)89131-Z

An efficient implementation of second analytical derivatives for density functional methods
journal, August 2002


Time-Dependent Hartree—Fock Theory for Molecules
journal, July 1964


A multiconfigurational time-dependent hartree-fock approach
journal, August 1979


Transition moments and dynamic polarizabilities in a second order polarization propagator approach
journal, December 1980

  • Nielsen, Egon S.; Jo/rgensen, Poul; Oddershede, Jens
  • The Journal of Chemical Physics, Vol. 73, Issue 12
  • DOI: 10.1063/1.440119

Gauge invariant coupled cluster response theory
journal, May 1999

  • Pedersen, Thomas Bondo; Koch, Henrik; Hättig, Christof
  • The Journal of Chemical Physics, Vol. 110, Issue 17
  • DOI: 10.1063/1.478742

On the density matrix based approach to time-dependent density functional response theory
journal, April 2001

  • Furche, Filipp
  • The Journal of Chemical Physics, Vol. 114, Issue 14
  • DOI: 10.1063/1.1353585

Rationale for mixing exact exchange with density functional approximations
journal, December 1996

  • Perdew, John P.; Ernzerhof, Matthias; Burke, Kieron
  • The Journal of Chemical Physics, Vol. 105, Issue 22, p. 9982-9985
  • DOI: 10.1063/1.472933

Balanced basis sets of split valence, triple zeta valence and quadruple zeta valence quality for H to Rn: Design and assessment of accuracy
journal, January 2005

  • Weigend, Florian; Ahlrichs, Reinhart
  • Physical Chemistry Chemical Physics, Vol. 7, Issue 18, p. 3297-3305
  • DOI: 10.1039/b508541a

Property-optimized Gaussian basis sets for molecular response calculations
journal, October 2010

  • Rappoport, Dmitrij; Furche, Filipp
  • The Journal of Chemical Physics, Vol. 133, Issue 13
  • DOI: 10.1063/1.3484283

Efficient molecular numerical integration schemes
journal, January 1995

  • Treutler, Oliver; Ahlrichs, Reinhart
  • The Journal of Chemical Physics, Vol. 102, Issue 1
  • DOI: 10.1063/1.469408

An iteration method for the solution of the eigenvalue problem of linear differential and integral operators
journal, October 1950

  • Lanczos, C.
  • Journal of Research of the National Bureau of Standards, Vol. 45, Issue 4
  • DOI: 10.6028/jres.045.026

Methods of conjugate gradients for solving linear systems
journal, December 1952

  • Hestenes, M. R.; Stiefel, E.
  • Journal of Research of the National Bureau of Standards, Vol. 49, Issue 6
  • DOI: 10.6028/jres.049.044

A space-saving modification of Davidson's eigenvector algorithm
journal, November 1990

  • van Lenthe, Johan H.; Pulay, Peter
  • Journal of Computational Chemistry, Vol. 11, Issue 10
  • DOI: 10.1002/jcc.540111008

Krylov-space algorithms for time-dependent Hartree–Fock and density functional computations
journal, July 2000

  • Chernyak, Vladimir; Schulz, Michael F.; Mukamel, Shaul
  • The Journal of Chemical Physics, Vol. 113, Issue 1
  • DOI: 10.1063/1.481770

Dynamic Thick Restarting of the Davidson, and the Implicitly Restarted Arnoldi Methods
journal, January 1998

  • Stathopoulos, Andreas; Saad, Yousef; Wu, Kesheng
  • SIAM Journal on Scientific Computing, Vol. 19, Issue 1
  • DOI: 10.1137/S1064827596304162

The effect of non-optimal bases on the convergence of Krylov subspace methods
journal, May 2005


On the Efficiency of Algorithms for Solving Hartree–Fock and Kohn–Sham Response Equations
journal, April 2011

  • Kauczor, Joanna; Jørgensen, Poul; Norman, Patrick
  • Journal of Chemical Theory and Computation, Vol. 7, Issue 6
  • DOI: 10.1021/ct100729t

Computational Variants of the Lanczos Method for the Eigenproblem
journal, January 1972


Harnessing the meta-generalized gradient approximation for time-dependent density functional theory
journal, October 2012

  • Bates, Jefferson E.; Furche, Filipp
  • The Journal of Chemical Physics, Vol. 137, Issue 16
  • DOI: 10.1063/1.4759080

Density‐functional thermochemistry. III. The role of exact exchange
journal, April 1993

  • Becke, Axel D.
  • The Journal of Chemical Physics, Vol. 98, Issue 7, p. 5648-5652
  • DOI: 10.1063/1.464913

Time-dependent density functional theory investigation of the absorption, fluorescence, and phosphorescence spectra of solvated coumarins
journal, October 2006

  • Jacquemin, Denis; Perpète, Eric A.; Scalmani, Giovanni
  • The Journal of Chemical Physics, Vol. 125, Issue 16
  • DOI: 10.1063/1.2361290

Thioindigo Dyes:  Highly Accurate Visible Spectra with TD-DFT
journal, February 2006

  • Jacquemin, Denis; Preat, Julien; Wathelet, Valérie
  • Journal of the American Chemical Society, Vol. 128, Issue 6
  • DOI: 10.1021/ja056676h

Role of Donor−Acceptor Strengths and Separation on the Two-Photon Absorption Response of Cytotoxic Dyes:  A TD-DFT Study
journal, August 2005

  • Badaeva, Ekaterina A.; Timofeeva, Tatiana V.; Masunov, Artëm
  • The Journal of Physical Chemistry A, Vol. 109, Issue 32
  • DOI: 10.1021/jp0521510

Degenerate two-photon-absorption spectral studies of highly two-photon active organic chromophores
journal, March 2004

  • He, Guang S.; Lin, Tzu-Chau; Dai, Jianming
  • The Journal of Chemical Physics, Vol. 120, Issue 11
  • DOI: 10.1063/1.1647537

Protein-Induced Color Shift of Carotenoids in β-Crustacyanin
journal, July 2015

  • Gamiz-Hernandez, Ana P.; Angelova, Iva Neycheva; Send, Robert
  • Angewandte Chemie International Edition, Vol. 54, Issue 39
  • DOI: 10.1002/anie.201501609

A ruthenium(II) polypyridyl complex for direct imaging of DNA structure in living cells
journal, October 2009

  • Gill, Martin R.; Garcia-Lara, Jorge; Foster, Simon J.
  • Nature Chemistry, Vol. 1, Issue 8
  • DOI: 10.1038/nchem.406

On the dimerization of chlorophyll in photosystem II
journal, January 2011

  • Nilsson Lill, Sten O.
  • Physical Chemistry Chemical Physics, Vol. 13, Issue 35
  • DOI: 10.1039/c1cp21390c

Energy Transfer Followed by Electron Transfer in a Supramolecular Triad Composed of Boron Dipyrrin, Zinc Porphyrin, and Fullerene:  A Model for the Photosynthetic Antenna-Reaction Center Complex
journal, June 2004

  • D'Souza, Francis; Smith, Phillip M.; Zandler, Melvin E.
  • Journal of the American Chemical Society, Vol. 126, Issue 25
  • DOI: 10.1021/ja030647u

Cadmium selenide semiconductor nanocrystals: a theoretical study
journal, May 1998


Excited-State Structure of Oligothiophene Dendrimers: Computational and Experimental Study
journal, December 2010

  • Badaeva, Ekaterina; Harpham, Michael R.; Guda, Ramakrishna
  • The Journal of Physical Chemistry B, Vol. 114, Issue 48
  • DOI: 10.1021/jp109624d

Turbomole
journal, July 2013

  • Furche, Filipp; Ahlrichs, Reinhart; Hättig, Christof
  • Wiley Interdisciplinary Reviews: Computational Molecular Science, Vol. 4, Issue 2
  • DOI: 10.1002/wcms.1162

A new mixing of Hartree–Fock and local density‐functional theories
journal, January 1993

  • Becke, Axel D.
  • The Journal of Chemical Physics, Vol. 98, Issue 2
  • DOI: 10.1063/1.464304

Gaussian basis sets of quadruple zeta valence quality for atoms H–Kr
journal, December 2003

  • Weigend, Florian; Furche, Filipp; Ahlrichs, Reinhart
  • The Journal of Chemical Physics, Vol. 119, Issue 24
  • DOI: 10.1063/1.1627293

Accurate Coulomb-fitting basis sets for H to Rn
journal, January 2006

  • Weigend, Florian
  • Physical Chemistry Chemical Physics, Vol. 8, Issue 9
  • DOI: 10.1039/b515623h

Real-time time-dependent density functional theory approach for frequency-dependent nonlinear optical response in photonic molecules
journal, October 2007

  • Takimoto, Y.; Vila, F. D.; Rehr, J. J.
  • The Journal of Chemical Physics, Vol. 127, Issue 15
  • DOI: 10.1063/1.2790014

Modeling Fast Electron Dynamics with Real-Time Time-Dependent Density Functional Theory: Application to Small Molecules and Chromophores
journal, April 2011

  • Lopata, Kenneth; Govind, Niranjan
  • Journal of Chemical Theory and Computation, Vol. 7, Issue 5
  • DOI: 10.1021/ct200137z

A Jacobi–Davidson Iteration Method for Linear Eigenvalue Problems
journal, April 1996

  • G. Sleijpen, Gerard L.; Van der Vorst, Henk A.
  • SIAM Journal on Matrix Analysis and Applications, Vol. 17, Issue 2
  • DOI: 10.1137/S0895479894270427

Energy-Specific Linear Response TDHF/TDDFT for Calculating High-Energy Excited States
journal, October 2011

  • Liang, Wenkel; Fischer, Sean A.; Frisch, Michael J.
  • Journal of Chemical Theory and Computation, Vol. 7, Issue 11
  • DOI: 10.1021/ct200485x

Nonlinear response theory with relaxation: The first-order hyperpolarizability
journal, November 2005

  • Norman, Patrick; Bishop, David M.; Jensen, Hans Jørgen Aa.
  • The Journal of Chemical Physics, Vol. 123, Issue 19
  • DOI: 10.1063/1.2107627

Finite lifetime effects on the polarizability within time-dependent density-functional theory
journal, June 2005

  • Jensen, L.; Autschbach, J.; Schatz, G. C.
  • The Journal of Chemical Physics, Vol. 122, Issue 22
  • DOI: 10.1063/1.1929740

Turbo charging time-dependent density-functional theory with Lanczos chains
journal, April 2008

  • Rocca, Dario; Gebauer, Ralph; Saad, Yousef
  • The Journal of Chemical Physics, Vol. 128, Issue 15
  • DOI: 10.1063/1.2899649

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