Combining density-functional theory and many-body methods

Size: px
Start display at page:

Download "Combining density-functional theory and many-body methods"

Transcription

1 Combining density-functional theory and many-body methods Julien Toulouse Université Pierre & Marie Curie and CNRS, Paris, France Vrije Universiteit Amsterdam, Netherlands November 2017

2 Outline 2/23 1 A brief overview of DFT/many-body hybrids 2 Range-separated hybrids for the ground state 3 Range-separated hybrids for excited states

3 Outline 3/23 1 A brief overview of DFT/many-body hybrids 2 Range-separated hybrids for the ground state 3 Range-separated hybrids for excited states

4 Overview of DFT/many-body hybrids 1 Kohn-Sham DFT (1965): { } E = min Φ ˆT + ˆV ne Φ +E Hxc [n Φ ] Φ where Φ is a single determinant Semilocal density-functional approximations (DFAs) for E xc [n]: LDA, GGAs, meta-ggas = work well in many cases but still limitations: self-interaction error, strong/static correlation, van der Waals dispersion interactions 2 Hybrid approximations (Becke 1993): E xc = a x Ex HF [Φ]+(1 a x )Ex DFA [n]+ec DFA [n] with Hartree-Fock (HF) exchange: Ex HF [Φ] = Φ Ŵ ee Φ E H [n] one empirical parameter: a x 0.25 = reduces self-interaction error 4/23

5 Overview of DFT/many-body hybrids 5/23 3 Double-hybrid approximations (Grimme 2006): E xc = a x Ex HF [Φ]+(1 a x )Ex DFA [n]+(1 a c )Ec DFA [n]+a c Ec MP2 with second-order Møller-Plesset (MP2) perturbative correlation: E MP2 c occ = i<j unocc a<b Φ ab ij Ŵ ee Φ 2 ε a +ε b ε i ε j two empirical parameters: a x 0.5 and a c 0.25 = further reduces self-interaction error, partially account for van der Waals dispersion, but fails for strongly correlated systems

6 Overview of DFT/many-body hybrids 6/23 4 General DFT/many-body hybrid scheme (Sharkas, Toulouse, Savin 2011) { } E = min Ψ ˆT + ˆV ne +λŵ ee Ψ +ĒHxc[n λ Ψ ] Ψ with one empirical parameter λ and the complement density functional: Ē λ Hx[n] = (1 λ)e Hx [n] and Ē λ c [n] = E c [n] λ 2 E c [n 1/λ ] (1 λ 2 )E c [n] Single determinant: Ψ Φ = a hybrid approximation: E xc = λex HF [Φ]+(1 λ)ex DFA [n]+(1 λ 2 )Ec DFA [n] Second-order perturbation: one-parameter double-hybrid approximation: E xc = λex HF [Φ]+(1 λ)ex DFA [n]+(1 λ 2 )Ec DFA [n]+λ 2 Ec MP2 Non-perturbative approaches: Ψ = n c nφ n = for strong correlation (Sharkas, Savin, Jensen, Toulouse, JCP, 2012)

7 Leininger, Stoll, Werner, Savin, CPL, 1997; Fromager, Toulouse, Jensen, JCP, /23 1. Overview of many-body hybrids 2. Range-separated hybrids for the ground state 3. Range-separated hybrids for excited states Overview of DFT/many-body hybrids 5 Range-separated DFT/many-body hybrid scheme (Savin 1996) { } E = min Ψ ˆT + ˆV ne +Ŵee Ψ +E lr Hxc[n sr Ψ ] Ψ with a long-range interaction Ŵee lr = erf(µr ij ) i<j r ij a short-range density functional EHxc sr [n] and one empirical parameter µ controlling the range of the separation Single determinant: Ψ Φ = a lrhf+srdft hybrid: E xc = E lr,hf x [Φ]+Ex sr,dfa [n]+ec sr,dfa similar to the LC scheme (Hirao et al. 2001) Many-body perturbation theory = lrmp2/lrrpa+srdft hybrids: E xc = E lr,hf x [n] [Φ]+Ex sr,dfa [n]+ec sr,dfa [n]+ec lr,mp2/rpa Ángyán, Gerber, Savin, Toulouse, PRA, 2005; Toulouse, Gerber, Jansen, Savin, Ángyán, PRL, 2009; Janesko, Henderson, Scuseria, JCP, 2009 Non-perturbative approaches: Ψ = n c nφ n

8 Outline 8/23 1 A brief overview of DFT/many-body hybrids 2 Range-separated hybrids for the ground state 3 Range-separated hybrids for excited states

9 Range-separated hybrids: lrhf+srdft 9/23 E lrhf+srdft xc = E lr,hf x Ángyán, Gerber, Savin, Toulouse, PRA, 2005 [Φ]+E sr,dfa x [n]+ec sr,dfa Short-range exchange and correlation DFAs have been developed, e.g.: short-range LDA: Ex/c srlda [n] = n(r) ǫ sr,unif x/c (n(r)) dr Toulouse, Savin, Flad, IJQC, 2004; Paziani, Moroni, Gori-Giorgi, Bachelet, PRB, 2006 [n] short-range PBE: E srpbe x/c [n] = n(r) ǫ srpbe x/c (n(r), n(r)) dr Toulouse, Colonna, Savin, JCP, 2005; Goll, Werner, Stoll, Leininger, Gori-Giorgi, Savin, CP, 2006

10 lrhf+srdft: reduction of the self-interaction error Dissociation of H + 2 molecule: H+ 2 H H +0.5 Binding energy (kcal/mol) exact lrhf+srpbe PBE0 PBE Internuclear distance (Angstrom) = removal of the large self-interaction error coming from the long-range part of the PBE exchange Mussard, Toulouse, MP, /23

11 Adding long-range correlation: lrmp2/lrrpa+srdft Long-range MP2: occ = E lr,mp2 c unocc i<j a<b Ángyán, Gerber, Savin, Toulouse, PRA, 2005 Φ ab ij Ŵ lr ee Φ 2 ε a +ε b ε i ε j = + Long-range direct RPA (drpa) = sum of all direct ring diagrams E lr,drpa c = + + Toulouse, Gerber, Jansen, Savin, Ángyán, PRL, 2009; Janesko, Henderson, Scuseria, JCP, 2009 Long-range RPA with exchange (RPAx-SO2) = sum of all direct + some exchange ring diagrams E lr,rpax-so2 c = Toulouse, Zhu, Savin, Jansen, Ángyán, JCP, /23

12 Fast basis convergence of long-range perturbation theory Total energy of N 2 calculated with Gaussian basis sets (µ = 0.5 bohr 1, srpbe functional, Dunning basis cc-pvxz): Total energy (hartree) MP2 lrmp2+srpbe VDZ VTZ VQZ V5Z V6Z Basis sets of increasing sizes = Exponential basis convergence of lrmp2+srpbe Franck, Mussard, Luppi, Toulouse, JCP, /23

13 Test of lrmp2/lrrpa+srdft on atomization energies A G2 subset of 49 atomization energies of small molecules (µ = 0.5 bohr 1, srpbe functional, cc-pvqz, spin unrestricted): Standard deviation (kcal/mol) MP2 drpa RPAx-SO2 lrmp2+srpbe lrdrpa+srpbe lrrpax-so2+srpbe Mean error (kcal/mol) = Range separation decreases the standard deviation for all methods = All range-separated methods give a mean error of 5 kcal/mol Mussard, Reinhardt, Ángyán, Toulouse, JCP, /23

14 Test of lrmp2/lrrpa+srdft on reaction barrier heights DBH24/08 set: 24 barrier heights of reactions with small molecules (µ = 0.5 bohr 1, srpbe functional, aug-cc-pvqz, spin unrestricted): Standard deviation (kcal/mol) Mean error (kcal/mol) MP2 drpa RPAx-SO2 lrmp2+srpbe lrdrpa+srpbe lrrpax-so2+srpbe = Range separation improves all methods = All range-separated methods give a mean error 1 kcal/mol Mussard, Reinhardt, Ángyán, Toulouse, JCP, /23

15 Toulouse, Zhu, Savin, Jansen, Ángyán, JCP, /23 1. Overview of many-body hybrids 2. Range-separated hybrids for the ground state 3. Range-separated hybrids for excited states Test of lrmp2/lrrpa+srdft on weak interactions S22 set: 22 equilibrium interaction energies of weakly-interacting molecular systems from water dimer to DNA base pairs (µ = 0.5 bohr 1, srpbe functional, aug-cc-pvdz): % of error on interaction energy H bonded dispersion dispersion +multipoles lrmp2+srpbe lrdrpa+srpbe lrrpax SO2+srPBE system in S22 set = lrrpax-so2+srpbe/avdz gives a mean absolute error of 4%

16 Comparitive test for lrrpa+srdft on weak interactions Test set of 10 molecular dimers (from 6 to 32 atoms) with 8 configurations sampling the repulsive wall, minimum, and asymptotic regions (µ = 0.5 bohr 1, srpbe functional, aug-cc-pvtz): Mean absolute error (kcal/mol) LC-ωPBE-D B3LYP-D lrrpax-so2+srpbe 0.29 M M vdw-df = lrrpax-so2+srpbe is globally less accurate that LC-ωPBE-D3 and B3LYP-D3: the short-range xc functional still needs to be improved Taylor et al., JCP, /23

17 Toward self-consistent DFT/many-body hybrids Fully self-consistent MP2+DFT hybrid using the OEP method E xc = λex HF [{φ i }]+(1 λ)ex B [n]+(1 λ 2 )Ec LYP [n]+λ 2 Ec MP2 [{φ i,ε i }] local exchange-correlation potential including the MP2 term 0 CO molecule -0.5 v xc (hartree) -1 improves accuracy of EA -1.5 BLYP (λ=0) MP2+DFT (λ=0.65) MP2 (λ=1) accurate reference r (bohr) Śmiga, Franck, Mussard, Buksztel, Grabowski, Luppi, Toulouse, JCP, /23

18 Outline 18/23 1 A brief overview of DFT/many-body hybrids 2 Range-separated hybrids for the ground state 3 Range-separated hybrids for excited states

19 Time-dependent range-separated hybrids Linear-response TDDFT equation χ 1 (ω) = χ 1 0 (ω) f Hxc(ω) = excitation energies, linear-response properties Range separation for exchange kernel is now standard: f xc = f lr,hf x +f sr,dfa x +f DFA c Tawada, Tsuneda, Yanagisawa, Yanai, Hirao, JCP, 2004 Here, range separation for both exchange and correlation kernels: f xc = f lr,hf x +f sr,dfa x +f sr,dfa c +fc lr,(2) (ω) Rebolini, Savin, Toulouse, MP, 2013; Rebolini, Toulouse, JCP, 2016 Other similar schemes: Pernal, JCP, 2012; Fromager, Knecht, Jensen, JCP, 2013; Hedegård, Heiden, Knecht, Fromager, Jensen, JCP, /23

20 Second-order correlation self-energy and kernel δ δg 0 (1,2) δ δg 0 (3,4) E MP2 c = + Σ (2) c (2,1)= Ξ (2) c (2,4;1,3)= 2 1 2,3 1, , , /23

21 In an orbital basis 21/23 Effective long-range second-order correlation kernel (lrbse2): f lr,(2) occ c,ia,jb (ω) = k<l unocc c<d Φ a i Ŵlr ee Φ cd kl Φcd kl Ŵlr ee Φ b j ω (ε c +ε d ε k ε l ) = The correlation kernel brings the effect of the double excitations Calculation of excitation energies in two steps: 1 lrtdhf+srtdlda calculation in the TDA: A X 0 = ω 0 X 0 2 perturbative addition of lrbse2 kernel: ω = ω 0 +X 0 flr,(2) c (ω 0 ) X 0 Zhang, Steinmann, Yang, JCP, 2013 Rebolini, Toulouse, JCP, 2016

22 Test of lrbse2+srtddft on excitation energies 56 singlet and triplet excitation energies of 4 small molecules N 2, CO, H 2 CO, C 2 H 4 (µ = 0.35 bohr 1, srlda functional, TDA, Sadlej+ basis): 0.5 Standard deviation (ev) Valence TDLDA Valence lrtdhf+srtdlda Valence lrbse2+srtdlda 0.1 Rydberg TDLDA Rydberg lrtdhf+srtdlda Rydberg lrbse2+srtdlda Mean error (ev) = lrbse2+srtdlda provides a slight overall improvement over lrtdhf+srtdlda Rebolini, Toulouse, JCP, /23

23 Summary and Acknowledgments 23/23 DFT/many-body hybrid methods based on a decomposition of the e-e interaction into long-range and short-range parts lrhf+srdft reduces the self-interaction error lrmp2/lrrpa+srdft has a fast basis convergence lrmp2/lrrpa+srdft accounts for van der Waals dispersion interactions self-consistent MP2+DFT using the OEP method lrbse2+srtddft for excitation energies: frequency-dependent second-order long-range correlation kernel bringing the effect of the double excitations Acknowledgments J. Ángyán, A. Buksztel, F. Colonna, H.-J. Flad, O. Franck, E. Fromager, I. Gerber, I. Grabowski, G. Jansen, H. J. Aa. Jensen, E. Luppi, B. Mussard, E. Rebolini, P. Reinhardt, A. Savin, K. Sharkas, S. Śmiga, K. Szalewicz, D. Taylor, W. Zhu

Combining density-functional theory and many-body methods

Combining density-functional theory and many-body methods Combining density-functional theory and many-body methods Julien Toulouse Université Pierre & Marie Curie and CNRS, Paris, France Albuquerque New Mexico, USA June 2016 Outline 2/23 1 A brief overview of

More information

Extending Kohn-Sham density-functional theory

Extending Kohn-Sham density-functional theory Extending Kohn-Sham density-functional theory Julien Toulouse Université Pierre & Marie Curie and CNRS, Paris, France Email: julien.toulouse@upmc.fr Web page: www.lct.jussieu.fr/pagesperso/toulouse/ January

More information

Multideterminant density-functional theory for static correlation. Julien Toulouse Université Pierre & Marie Curie and CNRS, Paris, France

Multideterminant density-functional theory for static correlation. Julien Toulouse Université Pierre & Marie Curie and CNRS, Paris, France Multideterminant density-functional theory for static correlation Julien Toulouse Université Pierre & Marie Curie and CNRS, Paris, France Basel, Switzerland September 2015 Kohn-Sham DFT and static correlation

More information

Combining many-body methods and density-functional theory. Julien Toulouse Université Pierre & Marie Curie and CNRS, Paris, France

Combining many-body methods and density-functional theory. Julien Toulouse Université Pierre & Marie Curie and CNRS, Paris, France Combining many-body methods and density-funtional theory Julien Toulouse Université Pierre & Marie Curie and CNRS, Paris, Frane September 2014 2/1 Hybrids ombining many-body methods and DFT Goal: improve

More information

Range-separated density-functional theory with long-range random phase approximation

Range-separated density-functional theory with long-range random phase approximation Range-separated density-functional theory with long-range random phase approximation Julien Toulouse 1 Wuming Zhu 1, Andreas Savin 1, János Ángyán2 1 Laboratoire de Chimie Théorique, UPMC Univ Paris 6

More information

Adiabatic-connection fluctuation-dissipation density-functional theory based on range separation

Adiabatic-connection fluctuation-dissipation density-functional theory based on range separation Adiabatic-connection fluctuation-dissipation density-functional theory based on range separation Julien Toulouse 1 I. Gerber 2, G. Jansen 3, A. Savin 1, W. Zhu 1, J. Ángyán 4 1 Laboratoire de Chimie Théorique,

More information

Dispersion energies from the random-phase approximation with range separation

Dispersion energies from the random-phase approximation with range separation 1/34 Dispersion energies from the random-phase approximation with range separation Julien Toulouse Université Pierre & Marie Curie and CNRS, Paris, France Web page: www.lct.jussieu.fr/pagesperso/toulouse/

More information

DFT basée sur le théorème de fluctuation-dissipation avec séparation de portée pour les interactions de van der Waals

DFT basée sur le théorème de fluctuation-dissipation avec séparation de portée pour les interactions de van der Waals DFT basée sur le théorème de fluctuation-dissipation avec séparation de portée pour les interactions de van der Waals Julien Toulouse 1 Iann Gerber 2, Georg Jansen 3, Andreas Savin 1, János Ángyán 4 1

More information

Théorie de la fonctionnnelle de la densité avec séparation de portée pour les forces de van der Waals

Théorie de la fonctionnnelle de la densité avec séparation de portée pour les forces de van der Waals Théorie de la fonctionnnelle de la densité avec séparation de portée pour les forces de van der Waals Julien Toulouse 1 Iann Gerber 2, Georg Jansen 3, Andreas Savin 1, János Ángyán 4 1 Laboratoire de Chimie

More information

Long-range/short-range energy decomposition in density functional theory

Long-range/short-range energy decomposition in density functional theory p. 1/2 Long-range/short-range energy decomposition in density functional theory Julien Toulouse François Colonna, Andreas Savin Laboratoire de Chimie Théorique, Université Pierre et Marie Curie, Paris

More information

Fractional-charge and Fractional-spin errors in Range-Separated Density-Functional Theory with Random Phase Approximation.

Fractional-charge and Fractional-spin errors in Range-Separated Density-Functional Theory with Random Phase Approximation. Fractional-charge and Fractional-spin errors in Range-Separated Density-Functional Theory with Random Phase Approximation. Bastien Mussard, Julien Toulouse Laboratoire de Chimie Théorique Institut du Calcul

More information

Assessment of range-separated time-dependent density-functional theory for calculating C 6 dispersion coefficients

Assessment of range-separated time-dependent density-functional theory for calculating C 6 dispersion coefficients 1/10 Assessment of range-separated time-dependent density-functional theory for calculating C 6 dispersion coefficients Julien Toulouse 1,2, Elisa Rebolini 1, Tim Gould 3, John F. Dobson 3, Prasenjit Seal

More information

Electronic structure theory: Fundamentals to frontiers. 2. Density functional theory

Electronic structure theory: Fundamentals to frontiers. 2. Density functional theory Electronic structure theory: Fundamentals to frontiers. 2. Density functional theory MARTIN HEAD-GORDON, Department of Chemistry, University of California, and Chemical Sciences Division, Lawrence Berkeley

More information

Developments in Electronic Structure Theory.

Developments in Electronic Structure Theory. Developments in Electronic Structure Theory. Bastien Mussard Laboratoire de Chimie Théorique Institut du Calcul et de la Simulation Sorbonne Universités, Université Pierre et Marie Curie bastien.mussard@upmc.fr

More information

Developments in Electronic Structure Theory.

Developments in Electronic Structure Theory. Developments in Electronic Structure Theory. Bastien Mussard Laboratoire de Chimie Théorique Institut du Calcul et de la Simulation Sorbonne Universités, Université Pierre et Marie Curie bastien.mussard@upmc.fr

More information

Adiabatic connections in DFT

Adiabatic connections in DFT Adiabatic connections in DFT Andreas Savin Torino, LCC2004, September 9, 2004 Multi reference DFT Kohn Sham: single reference Variable reference space capability to approach the exact result Overview The

More information

Range-separated hybrids

Range-separated hybrids Range-separated hybrids A. Savin MSSC2009 (Torino) Hybrids Wave function (Y) + density functional (n) hybrid E pluribus unum ("Out of many one") : Seal of the USA Overview Components (Y, n) Hybrids (Y

More information

Density-functional theory at noninteger electron numbers

Density-functional theory at noninteger electron numbers Density-functional theory at noninteger electron numbers Tim Gould 1, Julien Toulouse 2 1 Griffith University, Brisbane, Australia 2 Université Pierre & Marie Curie and CRS, Paris, France July 215 Introduction

More information

JULIEN TOULOUSE, PAOLA GORI-GIORGI, ANDREAS SAVIN

JULIEN TOULOUSE, PAOLA GORI-GIORGI, ANDREAS SAVIN Scaling Relations, Virial Theorem, and Energy Densities for Long-Range and Short-Range Density Functionals JULIEN TOULOUSE, PAOLA GORI-GIORGI, ANDREAS SAVIN Laboratoire de Chimie Théorique, CNRS et Université

More information

A multiconfigurational hybrid density-functional theory

A multiconfigurational hybrid density-functional theory THE JOURNAL OF CHEMICAL PHYSICS 137, 044104 (01) A multiconfigurational hybrid density-functional theory Kamal Sharkas, 1,,a) Andreas Savin, 1 Hans Jørgen Aa. Jensen, 3 and Julien Toulouse 1,b) 1 Laboratoire

More information

Introduction to density-functional theory. Emmanuel Fromager

Introduction to density-functional theory. Emmanuel Fromager Institut de Chimie, Strasbourg, France Page 1 Emmanuel Fromager Institut de Chimie de Strasbourg - Laboratoire de Chimie Quantique - Université de Strasbourg /CNRS M2 lecture, Strasbourg, France. Institut

More information

Advanced Quantum Chemistry III: Part 3. Haruyuki Nakano. Kyushu University

Advanced Quantum Chemistry III: Part 3. Haruyuki Nakano. Kyushu University Advanced Quantum Chemistry III: Part 3 Haruyuki Nakano Kyushu University 2013 Winter Term 1. Hartree-Fock theory Density Functional Theory 2. Hohenberg-Kohn theorem 3. Kohn-Sham method 4. Exchange-correlation

More information

Quantum Monte Carlo wave functions and their optimization for quantum chemistry

Quantum Monte Carlo wave functions and their optimization for quantum chemistry Quantum Monte Carlo wave functions and their optimization for quantum chemistry Julien Toulouse Université Pierre & Marie Curie and CNRS, Paris, France CEA Saclay, SPhN Orme des Merisiers April 2015 Outline

More information

arxiv: v1 [physics.chem-ph] 24 Jan 2019

arxiv: v1 [physics.chem-ph] 24 Jan 2019 Range-separated multideterminant density-functional theory with a short-range correlation functional of the on-top pair density Anthony Ferté, Emmanuel Giner, and Julien Toulouse Laboratoire de Chimie

More information

Dispersion Interactions from the Exchange-Hole Dipole Moment

Dispersion Interactions from the Exchange-Hole Dipole Moment Dispersion Interactions from the Exchange-Hole Dipole Moment Erin R. Johnson and Alberto Otero-de-la-Roza Chemistry and Chemical Biology, University of California, Merced E. R. Johnson (UC Merced) Dispersion

More information

OVERVIEW OF QUANTUM CHEMISTRY METHODS

OVERVIEW OF QUANTUM CHEMISTRY METHODS OVERVIEW OF QUANTUM CHEMISTRY METHODS Outline I Generalities Correlation, basis sets Spin II Wavefunction methods Hartree-Fock Configuration interaction Coupled cluster Perturbative methods III Density

More information

van der Waals forces in density functional theory: Perturbational long-range electron-interaction corrections

van der Waals forces in density functional theory: Perturbational long-range electron-interaction corrections van der Waals forces in density functional theory: Perturbational long-range electron-interaction corrections János G. Ángyán and Iann C. Gerber Laboratoire de Cristallographie et de Modélisation des Matériaux

More information

Teoría del Funcional de la Densidad (Density Functional Theory)

Teoría del Funcional de la Densidad (Density Functional Theory) Teoría del Funcional de la Densidad (Density Functional Theory) Motivation: limitations of the standard approach based on the wave function. The electronic density n(r) as the key variable: Functionals

More information

A new generation of density-functional methods based on the adiabatic-connection fluctuation-dissipation theorem

A new generation of density-functional methods based on the adiabatic-connection fluctuation-dissipation theorem A new generation of density-functional methods based on the adiabatic-connection fluctuation-dissipation theorem Andreas Görling, Patrick Bleiziffer, Daniel Schmidtel, and Andreas Heßelmann Lehrstuhl für

More information

Pseudo-Hermitian eigenvalue equations in linear-response electronic-structure theory

Pseudo-Hermitian eigenvalue equations in linear-response electronic-structure theory 1/11 Pseudo-Hermitian eigenvalue equations in linear-response electronic-structure theory Julien Toulouse Université Pierre & Marie Curie and CNRS, 4 place Jussieu, Paris, France Web page: www.lct.jussieu.fr/pagesperso/toulouse/

More information

One-electron Hamiltonians: HF & DFT

One-electron Hamiltonians: HF & DFT MSSC06 Ab initio Modelling in Solid State Chemistry Torino, 4 9/09/06 One-electron Hamiltonians: HF & DFT Bartolomeo Civalleri Department of Chemistry IS Centre of xcellence University of Torino bartolomeo.civalleri@unito.it

More information

Role of van der Waals Interactions in Physics, Chemistry, and Biology

Role of van der Waals Interactions in Physics, Chemistry, and Biology Role of van der Waals Interactions in Physics, Chemistry, and Biology How can we describe vdw forces in materials accurately? Failure of DFT Approximations for (Long-Range) Van der Waals Interactions 1

More information

RESEARCH ARTICLE. Excited states from range-separated density-functional perturbation theory

RESEARCH ARTICLE. Excited states from range-separated density-functional perturbation theory To appear in Molecular Physics Vol. 00, No. 00, Month 200x, 1 15 RESEARCH ARTICLE Excited states from range-separated density-functional perturbation theory Elisa Rebolini 1,2,4, Julien Toulouse 1,2, Andrew

More information

Model Hamiltonians in Density Functional Theory

Model Hamiltonians in Density Functional Theory Centre de Recherches Mathématiques CRM Proceedings and Lecture Notes Volume 41, 2006 Model Hamiltonians in Density Functional Theory Paola Gori-Giorgi, Julien Toulouse, and Andreas Savin Abstract. The

More information

arxiv: v3 [physics.chem-ph] 16 Feb 2016

arxiv: v3 [physics.chem-ph] 16 Feb 2016 Range-separated time-dependent density-functional theory with a frequency-dependent second-order Bethe-Salpeter correlation kernel Elisa Rebolini and Julien Toulouse Sorbonne Universités, UPMC Univ Paris

More information

Importing ab-initio theory into DFT: Some applications of the Lieb variation principle

Importing ab-initio theory into DFT: Some applications of the Lieb variation principle Importing ab-initio theory into DFT: Some applications of the Lieb variation principle Trygve Helgaker, Andy Teale, and Sonia Coriani Centre for Theoretical and Computational Chemistry (CTCC), Department

More information

arxiv: v2 [physics.chem-ph] 14 Jan 2013

arxiv: v2 [physics.chem-ph] 14 Jan 2013 Multi-configuration time-dependent density-functional theory based on range separation Emmanuel Fromager a, Stefan Knecht b and Hans Jørgen Aa. Jensen b a Laboratoire de Chimie Quantique, arxiv:1211.4829v2

More information

Computational Methods. Chem 561

Computational Methods. Chem 561 Computational Methods Chem 561 Lecture Outline 1. Ab initio methods a) HF SCF b) Post-HF methods 2. Density Functional Theory 3. Semiempirical methods 4. Molecular Mechanics Computational Chemistry " Computational

More information

CLIMBING THE LADDER OF DENSITY FUNCTIONAL APPROXIMATIONS JOHN P. PERDEW DEPARTMENT OF PHYSICS TEMPLE UNIVERSITY PHILADELPHIA, PA 19122

CLIMBING THE LADDER OF DENSITY FUNCTIONAL APPROXIMATIONS JOHN P. PERDEW DEPARTMENT OF PHYSICS TEMPLE UNIVERSITY PHILADELPHIA, PA 19122 CLIMBING THE LADDER OF DENSITY FUNCTIONAL APPROXIMATIONS JOHN P. PERDEW DEPARTMENT OF PHYSICS TEMPLE UNIVERSITY PHILADELPHIA, PA 191 THANKS TO MANY COLLABORATORS, INCLUDING SY VOSKO DAVID LANGRETH ALEX

More information

XYZ of ground-state DFT

XYZ of ground-state DFT XYZ of ground-state DFT Kieron Burke and Lucas Wagner Departments of Physics and of Chemistry, University of California, Irvine, CA 92697, USA January 5-9th, 2014 Kieron (UC Irvine) XYZ of ground-state

More information

RPA step by step. Bastien Mussard, János G. Ángyán, Sébastien Lebègue CRM 2, FacultÃl des Sciences UniversitÃl Henry PoincarÃl, Nancy

RPA step by step. Bastien Mussard, János G. Ángyán, Sébastien Lebègue CRM 2, FacultÃl des Sciences UniversitÃl Henry PoincarÃl, Nancy RPA step by step Bastien Mussard, János G. Ángyán, Sébastien Lebègue CRM, FacultÃl des Sciences UniversitÃl Henry PoincarÃl, Nancy Eötvös University Faculty of Science Budapest, Hungaria 0 mars 01 bastien.mussard@uhp-nancy.fr

More information

Intermediate DFT. Kieron Burke and Lucas Wagner. Departments of Physics and of Chemistry, University of California, Irvine, CA 92697, USA

Intermediate DFT. Kieron Burke and Lucas Wagner. Departments of Physics and of Chemistry, University of California, Irvine, CA 92697, USA Intermediate DFT Kieron Burke and Lucas Wagner Departments of Physics and of Chemistry, University of California, Irvine, CA 92697, USA October 10-19th, 2012 Kieron (UC Irvine) Intermediate DFT Lausanne12

More information

Lecture 8: Introduction to Density Functional Theory

Lecture 8: Introduction to Density Functional Theory Lecture 8: Introduction to Density Functional Theory Marie Curie Tutorial Series: Modeling Biomolecules December 6-11, 2004 Mark Tuckerman Dept. of Chemistry and Courant Institute of Mathematical Science

More information

Multi-configuration time-dependent density-functional theory based on range separation

Multi-configuration time-dependent density-functional theory based on range separation Multi-configuration time-dependent density-functional theory based on range separation Emmanuel Fromager, Stefan Knecht, and Hans Jørgen Aa. Jensen Citation: J. Chem. Phys. 138, 8411 213); doi: 1.163/1.4792199

More information

Optimization of quantum Monte Carlo wave functions by energy minimization

Optimization of quantum Monte Carlo wave functions by energy minimization Optimization of quantum Monte Carlo wave functions by energy minimization Julien Toulouse, Roland Assaraf, Cyrus J. Umrigar Laboratoire de Chimie Théorique, Université Pierre et Marie Curie and CNRS, Paris,

More information

Multi-reference Density Functional Theory. COLUMBUS Workshop Argonne National Laboratory 15 August 2005

Multi-reference Density Functional Theory. COLUMBUS Workshop Argonne National Laboratory 15 August 2005 Multi-reference Density Functional Theory COLUMBUS Workshop Argonne National Laboratory 15 August 2005 Capt Eric V. Beck Air Force Institute of Technology Department of Engineering Physics 2950 Hobson

More information

Introduction to Computational Chemistry: Theory

Introduction to Computational Chemistry: Theory Introduction to Computational Chemistry: Theory Dr Andrew Gilbert Rm 118, Craig Building, RSC andrew.gilbert@anu.edu.au 3023 Course Lectures Introduction Hartree Fock Theory Basis Sets Lecture 1 1 Introduction

More information

Computational Chemistry I

Computational Chemistry I Computational Chemistry I Text book Cramer: Essentials of Quantum Chemistry, Wiley (2 ed.) Chapter 3. Post Hartree-Fock methods (Cramer: chapter 7) There are many ways to improve the HF method. Most of

More information

Comparison of exchange-correlation functionals: from LDA to GGA and beyond

Comparison of exchange-correlation functionals: from LDA to GGA and beyond Comparison of ehange-correlation functionals: from LDA to GGA and beyond Martin Fuchs Fritz-Haber-Institut der MPG, Berlin, Germany Density-Functional Theory Calculations for Modeling Materials and Bio-Molecular

More information

Density functional theory in the solid state

Density functional theory in the solid state Density functional theory in the solid state Ari P Seitsonen IMPMC, CNRS & Universités 6 et 7 Paris, IPGP Department of Applied Physics, Helsinki University of Technology Physikalisch-Chemisches Institut

More information

Accurate van der Waals interactions from ground state electron density

Accurate van der Waals interactions from ground state electron density Accurate van der Waals interactions from ground state electron density Alexandre Tkatchenko Theory Department, Fritz Haber Institut der MPG Berlin, Germany tkatchen@fhi berlin.mpg.de Haber Institute EXCITCM09,

More information

The calculation of the universal density functional by Lieb maximization

The calculation of the universal density functional by Lieb maximization The calculation of the universal density functional by Lieb maximization Trygve Helgaker, Andy Teale, and Sonia Coriani Centre for Theoretical and Computational Chemistry (CTCC), Department of Chemistry,

More information

Non-covalent force fields computed ab initio

Non-covalent force fields computed ab initio Non-covalent force fields computed ab initio Supermolecule calculations Symmetry-adapted perturbation theory (SAPT) Supermolecule calculations Requirements: E = E AB E A E B. Include electron correlation,

More information

Density Functional Theory - II part

Density Functional Theory - II part Density Functional Theory - II part antonino.polimeno@unipd.it Overview From theory to practice Implementation Functionals Local functionals Gradient Others From theory to practice From now on, if not

More information

1 Density functional theory (DFT)

1 Density functional theory (DFT) 1 Density functional theory (DFT) 1.1 Introduction Density functional theory is an alternative to ab initio methods for solving the nonrelativistic, time-independent Schrödinger equation H Φ = E Φ. The

More information

NWChem: Hartree-Fock, Density Functional Theory, Time-Dependent Density Functional Theory

NWChem: Hartree-Fock, Density Functional Theory, Time-Dependent Density Functional Theory NWChem: Hartree-Fock, Density Functional Theory, Time-Depent Density Functional Theory Hartree-Fock! Functionality! Input! Wavefunctions! Initial MO vectors! Direct and semidirect algorithms! Convergence,

More information

Dispersion Interactions in Density-Functional Theory

Dispersion Interactions in Density-Functional Theory Dispersion Interactions in Density-Functional Theory Erin R. Johnson and Alberto Otero-de-la-Roza Chemistry and Chemical Biology, University of California, Merced E. R. Johnson (UC Merced) Dispersion from

More information

Introduction to Computational Quantum Chemistry: Theory

Introduction to Computational Quantum Chemistry: Theory Introduction to Computational Quantum Chemistry: Theory Dr Andrew Gilbert Rm 118, Craig Building, RSC 3108 Course Lectures 2007 Introduction Hartree Fock Theory Configuration Interaction Lectures 1 Introduction

More information

Exchange Correlation Functional Investigation of RT-TDDFT on a Sodium Chloride. Dimer. Philip Straughn

Exchange Correlation Functional Investigation of RT-TDDFT on a Sodium Chloride. Dimer. Philip Straughn Exchange Correlation Functional Investigation of RT-TDDFT on a Sodium Chloride Dimer Philip Straughn Abstract Charge transfer between Na and Cl ions is an important problem in physical chemistry. However,

More information

Orbital dependent correlation potentials in ab initio density functional theory

Orbital dependent correlation potentials in ab initio density functional theory Orbital dependent correlation potentials in ab initio density functional theory noniterative - one step - calculations Ireneusz Grabowski Institute of Physics Nicolaus Copernicus University Toruń, Poland

More information

Session 1. Introduction to Computational Chemistry. Computational (chemistry education) and/or (Computational chemistry) education

Session 1. Introduction to Computational Chemistry. Computational (chemistry education) and/or (Computational chemistry) education Session 1 Introduction to Computational Chemistry 1 Introduction to Computational Chemistry Computational (chemistry education) and/or (Computational chemistry) education First one: Use computational tools

More information

Tutorial I: IQ MOL and Basic DFT and MP2 Calculations 1 / 30

Tutorial I: IQ MOL and Basic DFT and MP2 Calculations 1 / 30 Tutorial I: IQ MOL and Basic DFT and MP2 Calculations Q-Chem User Workshop, Denver March 21, 2015 1 / 30 2 / 30 Introduction to IQMOL DFT and MP2 Calculations 3 / 30 IQMOL and Q-CHEM IQMOL is an open-source

More information

Methods for van der Waals Interactions

Methods for van der Waals Interactions Methods for van der Waals Interactions Alexandre Tkatchenko Theory Department, Fritz Haber Institut der MPG Berlin, Germany tkatchen@fhi berlin.mpg.de Haber Institute FHI DFT and Beyond Workshop, Jul.

More information

Excitation energies from Görling Levy perturbation theory along the range-separated adiabatic connection

Excitation energies from Görling Levy perturbation theory along the range-separated adiabatic connection Excitation energies from Görling Levy perturbation theory along the range-separated adiabatic connection Elisa Rebolini 1, Andrew M. Teale 3,4,5, Trygve Helgaer 4,5, Andreas Savin 2, and Julien Toulouse

More information

Molecular Mechanics: The Ab Initio Foundation

Molecular Mechanics: The Ab Initio Foundation Molecular Mechanics: The Ab Initio Foundation Ju Li GEM4 Summer School 2006 Cell and Molecular Mechanics in BioMedicine August 7 18, 2006, MIT, Cambridge, MA, USA 2 Outline Why are electrons quantum? Born-Oppenheimer

More information

Prediction of spectroscopic parameters for bio-organic and bio-inorganic intermediates in complex systems

Prediction of spectroscopic parameters for bio-organic and bio-inorganic intermediates in complex systems Prediction of spectroscopic parameters for bio-organic and bio-inorganic intermediates in complex systems Erik Donovan Hedegård Department of Physics, Chemistry and Pharmacy University of Southern Denmark

More information

DFT calculations of NMR indirect spin spin coupling constants

DFT calculations of NMR indirect spin spin coupling constants DFT calculations of NMR indirect spin spin coupling constants Dalton program system Program capabilities Density functional theory Kohn Sham theory LDA, GGA and hybrid theories Indirect NMR spin spin coupling

More information

Density Func,onal Theory (Chapter 6, Jensen)

Density Func,onal Theory (Chapter 6, Jensen) Chem 580: DFT Density Func,onal Theory (Chapter 6, Jensen) Hohenberg- Kohn Theorem (Phys. Rev., 136,B864 (1964)): For molecules with a non degenerate ground state, the ground state molecular energy and

More information

Density Functional Theory

Density Functional Theory Chemistry 380.37 Fall 2015 Dr. Jean M. Standard October 28, 2015 Density Functional Theory What is a Functional? A functional is a general mathematical quantity that represents a rule to convert a function

More information

High-level Quantum Chemistry Methods and Benchmark Datasets for Molecules

High-level Quantum Chemistry Methods and Benchmark Datasets for Molecules High-level Quantum Chemistry Methods and Benchmark Datasets for Molecules Markus Schneider Fritz Haber Institute of the MPS, Berlin, Germany École Polytechnique Fédérale de Lausanne, Switzerland دانشگاه

More information

Linear response time-dependent density functional theory

Linear response time-dependent density functional theory Linear response time-dependent density functional theory Emmanuel Fromager Laboratoire de Chimie Quantique, Université de Strasbourg, France fromagere@unistra.fr Emmanuel Fromager (UdS) Cours RFCT, Strasbourg,

More information

Random-phase approximation and beyond for materials: concepts, practice, and future perspectives. Xinguo Ren

Random-phase approximation and beyond for materials: concepts, practice, and future perspectives. Xinguo Ren Random-phase approximation and beyond for materials: concepts, practice, and future perspectives Xinguo Ren University of Science and Technology of China, Hefei USTC-FHI workshop on frontiers of Advanced

More information

Orbital Density Dependent Functionals

Orbital Density Dependent Functionals Orbital Density Dependent Functionals S. Kluepfel1, P. Kluepfel1, Hildur Guðmundsdóttir1 and Hannes Jónsson1,2 1. Univ. of Iceland; 2. Aalto University Outline: Problems with GGA approximation (PBE, RPBE,...)

More information

Jack Simons, Henry Eyring Scientist and Professor Chemistry Department University of Utah

Jack Simons, Henry Eyring Scientist and Professor Chemistry Department University of Utah 1. Born-Oppenheimer approx.- energy surfaces 2. Mean-field (Hartree-Fock) theory- orbitals 3. Pros and cons of HF- RHF, UHF 4. Beyond HF- why? 5. First, one usually does HF-how? 6. Basis sets and notations

More information

GEM4 Summer School OpenCourseWare

GEM4 Summer School OpenCourseWare GEM4 Summer School OpenCourseWare http://gem4.educommons.net/ http://www.gem4.org/ Lecture: Molecular Mechanics by Ju Li. Given August 9, 2006 during the GEM4 session at MIT in Cambridge, MA. Please use

More information

The adiabatic connection

The adiabatic connection The adiabatic connection Trygve Helgaker, Andy Teale, and Sonia Coriani Centre for Theoretical and Computational Chemistry (CTCC), Department of Chemistry, University of Oslo, Norway Dipartimento di Scienze

More information

Lawrence Berkeley National Laboratory Lawrence Berkeley National Laboratory

Lawrence Berkeley National Laboratory Lawrence Berkeley National Laboratory Lawrence Berkeley National Laboratory Lawrence Berkeley National Laboratory Title Long-Range Corrected Hybrid Density Functionals with Damped Atom-Atom Dispersion Corrections Permalink https://escholarship.org/uc/item/8559x299

More information

Molecular Magnetism. Magnetic Resonance Parameters. Trygve Helgaker

Molecular Magnetism. Magnetic Resonance Parameters. Trygve Helgaker Molecular Magnetism Magnetic Resonance Parameters Trygve Helgaker Centre for Theoretical and Computational Chemistry (CTCC), Department of Chemistry, University of Oslo, Norway Laboratoire de Chimie Théorique,

More information

Lecture 5: More about one- Final words about the Hartree-Fock theory. First step above it by the Møller-Plesset perturbation theory.

Lecture 5: More about one- Final words about the Hartree-Fock theory. First step above it by the Møller-Plesset perturbation theory. Lecture 5: More about one- determinant wave functions Final words about the Hartree-Fock theory. First step above it by the Møller-Plesset perturbation theory. Items from Lecture 4 Could the Koopmans theorem

More information

Self-Consistent Implementation of Self-Interaction Corrected DFT and of the Exact Exchange Functionals in Plane-Wave DFT

Self-Consistent Implementation of Self-Interaction Corrected DFT and of the Exact Exchange Functionals in Plane-Wave DFT Self-Consistent Implementation of Self-Interaction Corrected DFT and of the Exact Exchange Functionals in Plane-Wave DFT Kiril Tsemekhman (a), Eric Bylaska (b), Hannes Jonsson (a,c) (a) Department of Chemistry,

More information

Electronic band structure, sx-lda, Hybrid DFT, LDA+U and all that. Keith Refson STFC Rutherford Appleton Laboratory

Electronic band structure, sx-lda, Hybrid DFT, LDA+U and all that. Keith Refson STFC Rutherford Appleton Laboratory Electronic band structure, sx-lda, Hybrid DFT, LDA+U and all that Keith Refson STFC Rutherford Appleton Laboratory LDA/GGA DFT is good but... Naive LDA/GGA calculation severely underestimates band-gaps.

More information

FONCTIONNELLES DE DENSITÉ AVEC SÉPARATION DE PORTÉE: AU CIEL DFT SANS ÉCHELLE DE JACOB

FONCTIONNELLES DE DENSITÉ AVEC SÉPARATION DE PORTÉE: AU CIEL DFT SANS ÉCHELLE DE JACOB FONCTIONNELLES DE DENSITÉ AVEC SÉPARATION DE PORTÉE: AU CIEL DFT SANS ÉCHELLE DE JACOB, Iann Gerber, Paola Gori-Giorgi, Julien Toulouse, Andreas Savin Équipe de Modélisation Quantique et Cristallographique

More information

Recent advances in quantum Monte Carlo for quantum chemistry: optimization of wave functions and calculation of observables

Recent advances in quantum Monte Carlo for quantum chemistry: optimization of wave functions and calculation of observables Recent advances in quantum Monte Carlo for quantum chemistry: optimization of wave functions and calculation of observables Julien Toulouse 1, Cyrus J. Umrigar 2, Roland Assaraf 1 1 Laboratoire de Chimie

More information

Electronic Structure Calculations and Density Functional Theory

Electronic Structure Calculations and Density Functional Theory Electronic Structure Calculations and Density Functional Theory Rodolphe Vuilleumier Pôle de chimie théorique Département de chimie de l ENS CNRS Ecole normale supérieure UPMC Formation ModPhyChem Lyon,

More information

Study of Ozone in Tribhuvan University, Kathmandu, Nepal. Prof. S. Gurung Central Department of Physics, Tribhuvan University, Kathmandu, Nepal

Study of Ozone in Tribhuvan University, Kathmandu, Nepal. Prof. S. Gurung Central Department of Physics, Tribhuvan University, Kathmandu, Nepal Study of Ozone in Tribhuvan University, Kathmandu, Nepal Prof. S. Gurung Central Department of Physics, Tribhuvan University, Kathmandu, Nepal 1 Country of the Mt Everest 2 View of the Mt Everest 3 4 5

More information

3/23/2010 More basics of DFT Kieron Burke and friends UC Irvine Physics and Chemistry References for ground-state DFT ABC of DFT, by KB and Rudy Magyar, http://dft.uci.edu A Primer in Density Functional

More information

Electric properties of molecules

Electric properties of molecules Electric properties of molecules For a molecule in a uniform electric fielde the Hamiltonian has the form: Ĥ(E) = Ĥ + E ˆµ x where we assume that the field is directed along the x axis and ˆµ x is the

More information

Post Hartree-Fock: MP2 and RPA in CP2K

Post Hartree-Fock: MP2 and RPA in CP2K Post Hartree-Fock: MP2 and RPA in CP2K A tutorial Jan Wilhelm jan.wilhelm@chem.uzh.ch 4 September 2015 Further reading MP2 and RPA by Mauro Del Ben, Jürg Hutter, Joost VandeVondele Del Ben, M; Hutter,

More information

TDDFT in Chemistry and Biochemistry III

TDDFT in Chemistry and Biochemistry III TDDFT in Chemistry and Biochemistry III Dmitrij Rappoport Department of Chemistry and Chemical Biology Harvard University TDDFT Winter School Benasque, January 2010 Dmitrij Rappoport (Harvard U.) TDDFT

More information

Many electrons: Density functional theory Part II. Bedřich Velický VI.

Many electrons: Density functional theory Part II. Bedřich Velický VI. Many electrons: Density functional theory Part II. Bedřich Velický velicky@karlov.mff.cuni.cz VI. NEVF 514 Surface Physics Winter Term 013-014 Troja 1 st November 013 This class is the second devoted to

More information

Analytical Gradients for the Range-Separated Random Phase Approximation Correlation Energies Using a Lagrangian Framework

Analytical Gradients for the Range-Separated Random Phase Approximation Correlation Energies Using a Lagrangian Framework Analytical Gradients for the Range-Separated Random Phase Approximation Correlation Energies Using a Lagrangian Framework Bastien Mussard a, János G. Ángyán a, Péter G. Szalay b a CRM 2, Université de

More information

Quantum Chemical and Dynamical Tools for Solving Photochemical Problems

Quantum Chemical and Dynamical Tools for Solving Photochemical Problems 2.165430 3.413060 3.889592 9 H 3.413060 2.165430 1.099610 2.165430 3.413060 10 H 3.889592 3.413060 2.165430 1.099610 2.165430 11 H 3.413060 3.889592 3.413060 2.165430 1.099610 12 H 2.165430 3.413060 3.889592

More information

Excitation energies from density-functional theory some failures and successes. Trygve Helgaker

Excitation energies from density-functional theory some failures and successes. Trygve Helgaker 1 Excitation energies from density-functional theory some failures and successes Trygve Helgaker Centre for Theoretical and Computational Chemistry, University of Oslo, Norway Ola Lutnæs, Simen Reine,

More information

Introduction to multiconfigurational quantum chemistry. Emmanuel Fromager

Introduction to multiconfigurational quantum chemistry. Emmanuel Fromager Institut de Chimie, Strasbourg, France Page 1 Emmanuel Fromager Institut de Chimie de Strasbourg - Laboratoire de Chimie Quantique - Université de Strasbourg /CNRS M2 lecture, Strasbourg, France. Notations

More information

MO Calculation for a Diatomic Molecule. /4 0 ) i=1 j>i (1/r ij )

MO Calculation for a Diatomic Molecule. /4 0 ) i=1 j>i (1/r ij ) MO Calculation for a Diatomic Molecule Introduction The properties of any molecular system can in principle be found by looking at the solutions to the corresponding time independent Schrodinger equation

More information

Key concepts in Density Functional Theory (I) Silvana Botti

Key concepts in Density Functional Theory (I) Silvana Botti From the many body problem to the Kohn-Sham scheme European Theoretical Spectroscopy Facility (ETSF) CNRS - Laboratoire des Solides Irradiés Ecole Polytechnique, Palaiseau - France Temporary Address: Centre

More information

Molecular Magnetic Properties

Molecular Magnetic Properties Molecular Magnetic Properties Trygve Helgaker Hylleraas Centre, Department of Chemistry, University of Oslo, Norway and Centre for Advanced Study at the Norwegian Academy of Science and Letters, Oslo,

More information

Electronic quantum effect on hydrogen bond geometry in. water dimer

Electronic quantum effect on hydrogen bond geometry in. water dimer Electronic quantum effect on hydrogen bond geometry in water dimer Danhui Li 1,2, Zhiyuan Zhang 1,2 Wanrun Jiang 1,2 Depeng Zhang 1,2 Yu Zhu 1,2 and Zhigang Wang 1,2* 1 Institute of Atomic and Molecular

More information

5/27/2012. Role of van der Waals Interactions in Physics, Chemistry, and Biology

5/27/2012. Role of van der Waals Interactions in Physics, Chemistry, and Biology Role of van der Waals Interactions in Physics, Chemistry, and Biology 1 Role of van der Waals Interactions in Physics, Chemistry, and Biology Karin Jacobs: Take van der Waals forces into account in theory,

More information