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

Similar documents
Resolution of identity approach for the Kohn-Sham correlation energy within the exact-exchange random-phase approximation

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

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

Exact exchange (spin current) density-functional methods for magnetic properties, spin-orbit effects, and excited states

Extending Kohn-Sham density-functional theory

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

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

TDDFT in Chemistry and Biochemistry III

Optimized Effective Potential method for non-collinear Spin-DFT: view to spin-dynamics

Combining density-functional theory and many-body methods

Dispersion energies from the random-phase approximation with range separation

Introduction to density-functional theory. Emmanuel Fromager

Random-phase approximation correlation methods for molecules and solids

Orbital dependent correlation potentials in ab initio density functional theory

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

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

Convergence behavior of RPA renormalized many-body perturbation theory

The calculation of the universal density functional by Lieb maximization

Pseudopotentials for hybrid density functionals and SCAN

An Approximate DFT Method: The Density-Functional Tight-Binding (DFTB) Method

Combining density-functional theory and many-body methods

Exchange-Correlation Functional

Computational Methods. Chem 561

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

All electron optimized effective potential method for solids

Introduction to Density Functional Theory

OVERVIEW OF QUANTUM CHEMISTRY METHODS

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

Key concepts in Density Functional Theory (I) Silvana Botti

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

Density Functional Theory: from theory to Applications

Post Hartree-Fock: MP2 and RPA in CP2K

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

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

Orbital functionals derived from variational functionals of the Green function

Non-covalent force fields computed ab initio

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

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

Advanced Electronic Structure Theory Density functional theory. Dr Fred Manby

XYZ of ground-state DFT

Density Functional Theory - II part

Spring College on Computational Nanoscience May Variational Principles, the Hellmann-Feynman Theorem, Density Functional Theor

The adiabatic connection

Ab-initio studies of the adiabatic connection in density-functional theory

Teoría del Funcional de la Densidad (Density Functional Theory)

Molecular Magnetism. Magnetic Resonance Parameters. Trygve Helgaker

Density Func,onal Theory (Chapter 6, Jensen)

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

Electron Correlation - Methods beyond Hartree-Fock

Density Functional Theory

Key concepts in Density Functional Theory

Electronic Structure: Density Functional Theory

Density functional theory in the solid state

Computational Chemistry I

Linear-response excitations. Silvana Botti

The potential of Potential Functional Theory

Non-collinear OEP for solids: SDFT vs CSDFT

The electronic structure of materials 2 - DFT

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

Module 6 1. Density functional theory

Molecular Magnetic Properties

Spring College on Computational Nanoscience

1 Density functional theory (DFT)

From Quantum Mechanics to Materials Design

The Self Interaction Correction revisited

DFT calculations of NMR indirect spin spin coupling constants


Introduction to DFTB. Marcus Elstner. July 28, 2006

Intermolecular Forces in Density Functional Theory

Electric properties of molecules

Kohn Sham density functional theory [1 3] is. Role of the Exchange Correlation Energy: Nature s Glue STEFAN KURTH, JOHN P. PERDEW.

Modified Becke-Johnson (mbj) exchange potential

Electronic Structure Calculations and Density Functional Theory

QUANTUM CHEMISTRY FOR TRANSITION METALS

Orbital Density Dependent Functionals

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

The frequency-dependent Sternheimer equation in TDDFT

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

Orbitals and orbital energies in DFT and TDDFT. E. J. Baerends Vrije Universiteit, Amsterdam

Short Course on Density Functional Theory and Applications VII. Hybrid, Range-Separated, and One-shot Functionals

Introduction to Computational Chemistry

Rabi oscillations within TDDFT: the example of the 2 site Hubbard model

Periodic Trends in Properties of Homonuclear

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

High-level Quantum Chemistry Methods and Benchmark Datasets for Molecules

Quantum Mechanical Simulations

ABC of ground-state DFT

Density Functional Theory

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

Electronic structure theory: Fundamentals to frontiers. 1. Hartree-Fock theory

Yingwei Wang Computational Quantum Chemistry 1 Hartree energy 2. 2 Many-body system 2. 3 Born-Oppenheimer approximation 2

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

7/29/2014. Electronic Structure. Electrons in Momentum Space. Electron Density Matrices FKF FKF. Ulrich Wedig

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

DFT: Exchange-Correlation

Key concepts in Density Functional Theory (II) Silvana Botti

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

Dispersion Interactions from the Exchange-Hole Dipole Moment

Density matrix functional theory vis-á-vis density functional theory

Transcription:

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 Theoretische Chemie Universität Erlangen Nürnberg J. Gebhardt, G. Gebhardt, T. Gimon, W. Hieringer, C. Neiß, I. Nikiforidis, H. Schulz, E. Trushin, K.-G. Warnick, T. Wölfle A. Görling (University Erlangen Nürnberg) Seattle 213 1 / 36

Overview 1 Introduction Orbital-dependent functionals 2 Exact-exchange (EXX) Kohn-Sham methods 3 Direct RPA and EXXRPA correlation energy Adiabatic-connection fluctuation-dissipation theorem for KS correlation energy EXXRPA methods Direct RPA methods Selfconsistent direct RPA Correlation potentials Total energies, reaction energies Bond dissociation, static correlation Weak interactions, Van-der-Waals bonds 4 Concluding Remarks 5 Literature A. Görling (University Erlangen Nürnberg) Seattle 213 2 / 36

Motivation Orbital-dependent functionals Orbital- and eigenvalue-dependent exchange-correlation functionals in Kohn-Sham methods in order to improve accuracy get rid of Coulomb self-interactions and self-interactions in general (qualitatively correct KS spectra, improved input data for TDDFT) enable description of static correlation (bond formation and breaking), i.e., multiconfiguration cases enable description of Van der Waals interactions and noncovalent, mean interactions A. Görling (University Erlangen Nürnberg) Seattle 213 3 / 36

Kohn-Sham formalism Electronic Schrödinger equation [ ˆT + ˆV ee + v]ψ = E Ψ Kohn-Sham equation [ ˆT + ˆv s ]Φ = E,s Φ [ 1 2 2 + v s (r)] φ i (r) = ɛ i φ i (r) v s = v + v H + v xc Relation Kohn-Sham to real electronic system HK ˆT + v s Φ ρ Ψ ˆT + ˆVee + ˆv E = T s + U[ρ ] + E xc [ρ ] + dr v(r) ρ(r) HK v xc (r) = δe xc δρ(r) A. Görling (University Erlangen Nürnberg) Seattle 213 4 / 36

Orbital-dependent functionals I Historic development of DFT Ground state energy of an electronic system E = T s + U + E x + E c + dr v nuc (r)ρ(r) Thomas-Fermi-Dirac E = T s [ρ]+ U[ρ] +E x [ρ] + E c [ρ]+ dr v nuc (r)ρ(r) δe/δρ(r) = µ Conventional Kohn-Sham E = T s [{φ i }]+ U[ρ] +E x [ρ] + E c [ρ]+ dr v nuc (r)ρ(r) [ ˆT + ˆv nuc + ˆv H + ˆv x + ˆv c ]φ i = ε i φ i Kohn-Sham with orbital dependent functionals E = T s [{φ i }]+ U[ρ]+E x [{φ i }] +E c [{φ i }]+ dr v nuc (r)ρ(r) [ ˆT + ˆv nuc + ˆv H + ˆv x + ˆv c ]φ i = ε i φ i A. Görling (University Erlangen Nürnberg) Seattle 213 5 / 36

Orbital-dependent functionals II Conventional Kohn-Sham methods E = T s [{φ i }] + U[ρ ] + E x [ρ ] + E c [ρ ] + dr v nuc (r) ρ (r) [ ˆT + ˆvnuc + ˆv H [ρ ] + ˆv x [ρ ] + ˆv c [ρ ]] φ i = ε i φ i KS methods with orbital- and eigenvalue-dependent xc-functionals E = T s [{φ i }] + U[ρ ] + E x [{φ i }] + E c [{ε s }, {φ s }] + dr v nuc (r) ρ(r) [ ˆT + ˆvnuc + ˆv H [ρ ] + ˆv x [{φ i }] + ˆv c [{ε s }, {φ s }]] φ i = ε i φ i A. Görling (University Erlangen Nürnberg) Seattle 213 6 / 36

Exact treatment of KS exchange Exchange energy occ. E x = dr dr φ i(r )φ j (r )φ j (r)φ i (r) r r i,j Exchange potential v x ( r) = δe x[{φ i }] δρ( r) dr χ (r, r ) v x (r ) = t x (r) occ. KS response function χ (r, r ) = 4 occ. t x (r) = 4 i unocc. a i unocc. a φ i (r)φ a (r) a ˆv NL x i ɛ i ɛ a φ i (r)φ a (r)φ a (r )φ i (r ) ε i ε a Plane wave methods for solid numerically stable Gaussian basis set methods for molecules numerically demanding A. Görling (University Erlangen Nürnberg) Seattle 213 7 / 36

Exact-exchange Gaussian basis set KS method Auxiliary basis set: Electrostatic potential of Gaussian functions v x (r) = v x,k f k (r) with f k (r) = dr g k (r )/ r r k Incorporation of exact conditions to treat asymptotic of v x (r) dr ρ x (r) = 1 with ρ x (r) = v x,k g k (r) k φ HOMO v x φ HOMO = φ HOMO ˆv NL x φ HOMO Balancing of auxiliary and orbital basis sets (uncontracted orbital basis sets required) JCP 127, 5412 (27) By preprocessing of auxiliary basis set and regularization techniques, EXX calculations with standard contracted orbital basis sets possible (aug-cc-pcvxz, x = 3, 4, 5) A. Görling (University Erlangen Nürnberg) Seattle 213 8 / 36

EXX vs. GGA (PBE) orbitals of methane 2 t2u 2.934 ev 2 t2u +.533 ev 3 a1g.396 ev 1 t2u 9.448 ev 2 a1g 17.54 ev -5 3 a1g 4.438 ev -1-15 1 t2u 14.724 ev -2 2 a1g 22.437 ev contour value.32 A. Go rling (University Erlangen Nu rnberg) Seattle 213 9 / 36

EXX vs. GGA (PBE) orbitals of methane 2 t2u 2.934 ev 2 t2u +.533 ev 3 a1g.396 ev 1 t2u 9.448 ev 2 a1g 17.54 ev -5 3 a1g 4.438 ev -1-15 1 t2u 14.724 ev -2 2 a1g 22.437 ev contour value.13 A. Go rling (University Erlangen Nu rnberg) Seattle 213 1 / 36

Band gaps of semiconductors FLAPW vs. PP EXX band gaps EXX+VWNc Exp. FLAPW a PP b Si Γ Γ 3.21 3.26 3.4 Γ L 2.28 2.35 2.4 Γ X 1.44 1.5 SiC Γ Γ 7.24 7.37 Γ L 6.21 6.3 Γ X 2.44 2.52 2.42 Ge Γ Γ 1.21 1.28 1. Γ L.94 1.1.7 Γ X 1.28 1.34 1.3 GeAs Γ Γ 1.74 1.82 1.63 Γ L 1.86 1.93 Γ X 2.12 2.15 2.18 C Γ Γ 6.26 6.28 7.3 Γ L 9.16 9.18 Γ X 5.33 5.43 a PRB 83, 4515 (211) b PRB 59, 131 (1997) A. Görling (University Erlangen Nürnberg) Seattle 213 11 / 36

Summary exact-exchange Kohn-Sham methods Problem of Coulomb self-interaction is solved Physically meaningful orbital and eigenvalue spectra (Improved input data for time-dependent DFT) Similar total energies of exact-exchange-only Kohn-Sham and HF methods but completely different orbital and eigenvalue spectra A. Görling (University Erlangen Nürnberg) Seattle 213 12 / 36

Adiabatic-connection fluctuation-dissipation theorem for DFT correlation energy I E c = 1 2π 1 dα drdr 1 r r dω [ ] χ α (r, r, iω) χ (r, r, iω) Integration of response functions along complex frequencies 1 2π dω dr dr g(r, r ) χ α (r, r, iω) = [ = dr dr g(r, r ) ρ α 2 (r, r ) 1 ] 2 ρ(r)ρ(r ) + ρ(r)δ(r r ) dω a a 2 + ω 2 = π 2 later on g(r, r ) = 1 r r χ α (r, r, iω) = 2 n E n E (E n E ) 2 + ω 2 Ψα ˆρ(r) Ψ α n Ψ α n ˆρ(r ) Ψ α V c (α) = Ψ (α) ˆV ee Ψ (α) Φ ˆV ee Φ A. Görling (University Erlangen Nürnberg) Seattle 213 13 / 36

Adiabatic-connection fluctuation-dissipation theorem for DFT correlation energy II E c = 1 2π 1 dα drdr 1 r r dω Integration along adiabatic connection E c = 1 [ ] χ α (r, r, iω) χ (r, r, iω) = 1 dα V c (α) with V c (α) = Ψ (α) ˆV ee Ψ (α) Φ ˆV ee Φ E c = Ψ (α) ˆT + α ˆV ee Ψ (α) Φ ˆT + α ˆV ee Φ dα V c (α) Ψ (α) = min Ψ ρ Ψ ˆT + α ˆV ee Ψ From Hellmann-Feynman theorem follows de c (α) dα = V c (α) A. Görling (University Erlangen Nürnberg) Seattle 213 14 / 36

Response functions occ χ (r, r, iω) = 4 KS response function χ (r, r, iω) i unocc a ɛ ai ɛ 2 ai + ω2 ϕ i(r)ϕ a (r)ϕ a (r )ϕ i (r ) Basics time-dependent DFT ρ (1) = X α=1 v (1) = X v (1) s = X [v (1) + F Hxc X α=1 v (1) ] v s = v + v H + v xc v s (1) = v (1) pert + v (1) H + v(1) xc v (1) Hxc = δv Hxc δρ ρ(1) = F Hxc ρ (1) = F Hxc X α=1 v (1) Response functions χ α (r, r, iω) from EXX-TDDFT X α = X + α X F Hx X α X α = [1 αx F Hx ] 1 X X α = X [X αh Hx ] 1 X with X F Hx X = H Hx A. Görling (University Erlangen Nürnberg) Seattle 213 15 / 36

Exact frequency-dependent exchange kernel OEP-like equation for sum f Hx (ω, r, r ) of Coulomb and EXX kernel dr dr X (r, r, ω) f Hx (ω, r, r ) X (r, r, ω) = h Hx (ω, r, r ) with h Hx (ω, r, r ) = 1 4 Y (r)λ(ω) [A + B + ] λ(ω)y(r ) + ω 2 1 4 Y (r)λ(ω)ɛ 1 [A + B + ] ɛ 1 λ(ω)y(r) + a ˆv NL x ˆv x j Y ia(r)λ ia(ω) φ i(r)φ j(r) + ɛ a ɛ j i j a + b ˆv NL x ˆv x i Y ia(r)λ ia(ω) φ a(r)φ b (r) + ɛ b ɛ i a b i A ia,jb = 2(ai jb) (ab ji) ia,jb = δ ij ϕ a ˆv NL x B ia,jb = 2(ai bj) (aj bi) ˆv x ϕ b δ ab ϕ i ˆv NL x ˆv x ϕ j λ ia,jb = δ ia,jb 4ɛ ia ɛ 2 ia +ω2 ɛ ia,jb = δ ia,jb ɛ ia = δ ia,jb (ɛ a ɛ i) Y ia(r) = φ i(r)φ a(a) RPA: h H (r, r, ω) = Y (r)λ(ω) C λ(ω)y(r) with C ia,jb = (ia jb) A. Görling (University Erlangen Nürnberg) Seattle 213 16 / 36

E c = 1 2π dω 1 dα RI-EXXRPA method I drdr 1 [ ] χ α(r, r, iω) χ (r, r, iω) r r Representation of χ α(iω), h x(iω) in RI basis set with respect to Coulomb norm with X α = [1 αx F Hx ] 1 X X α = X [X αh Hx ] 1 X E c = 1 2π dω 1 (using F Hx = X 1 H Hx X 1 ) ] dα T r [S 1 X [X α H Hx ] 1 X S 1 X X (iω) = D λ(iω)d with D ia,h = (ϕ iϕ a f h ) Coul and λ ia,jb = δ ia,jb 4ɛ ia ɛ 2 ia + ω2 and H(iω) = 1 4 DT λ(iω) [A+B+ ] λ(iω)d + (iω) 2 1 4 DT λ(iω) ɛ 1 [A+B+ ] ɛ 1 λ(iω)d + W 1(iω) + W T 1 (iω) + W 2(iω) + W T 2 (iω) A. Görling (University Erlangen Nürnberg) Seattle 213 17 / 36

RI-EXXRPA method II E c = 1 2π dω 1 ] dα T r [S 1 X [X α H Hx ] 1 X S 1 X Orthonormalize RI basis set, i.e. make S = E, and use X = ( X ) 2 1 ( X) 1 2 carry out analytic integration over coupling constant and E c = 1 2π ( ) ] dω T r [( X (iω)) 1 2 U(iω) τ 1 (iω) ln[ 1 + τ (iω) ]+1 U(iω) T ( X (iω)) 1 2 with ( X (iω)) 1 2 H(iω)( X(iω)) 1 2 = U(iω) τ (iω) U (iω) Complete exchange kernel can be treated (RI-EXXRPA+) N 5 scaling A. Görling (University Erlangen Nürnberg) Seattle 213 18 / 36

E c = 1 2π dω 1 RI2-dRPA method RI2-dRPA ] dα T r [S 1 X [X α H H ] 1 X S 1 X For drpa, with second RI approximation and S = E, H H simplifies to H H = X X With spectral representation X (iω) = V(iω) σ(iω) V (iω) E c = 1 2π dω Tr [ln[1 + σ(iω)] σ(iω)] Only KS response function X (iω) required N 4 scaling A. Görling (University Erlangen Nürnberg) Seattle 213 19 / 36

Self-consistent direct RPA Optimized-effective-potential equation for vc drpa dr χ (r, r ) vc drpa (r ) = t drpa c (r) with t drpa c (r) = s dr δedrpa c δφ s (r ) δφ s (r ) δv s (r) + s δe drpa c δɛ s δɛ s δv s (r) v drpa c is represented in auxiliary basis set X v drpa c = t drpa c A. Görling (University Erlangen Nürnberg) Seattle 213 2 / 36

drpa correlation potentials I.2 v drpa c for neon atoms.15.1 v c drpa (a.u.).5 -.5 -.1 Exact -.15 SC-dRPA PW92 LDA -.2 PBE.1.1 1 1 r(a.u.) drpa correlation potential qualitatively correct in contrast to LDA and GGA correlation potential A. Görling (University Erlangen Nürnberg) Seattle 213 21 / 36

drpa correlation potentials II v drpa c for C 2 H 2 v[a.u.] -.2 -.4 -.6 -.8-1 -1.2-1.4 drpa v c v x -5-4 -3-2 -1 1 2 3 4 5 r[a.u.] A. Görling (University Erlangen Nürnberg) Seattle 213 22 / 36

Results EXXRPA correlation methods I Deviations of total energies from CCSD(T) energies, E = E Method E CCSD(T) Orbital basis: aug-cc-pcvtz RI basis (for E c and v xc): aug-cc-pvtz/mp2-fit Energy difference to CCSD(T) [Hartree].6.5.4.3.2.1 -.1 CCSD MP2 EXXRPA@EXX EXXRPA@dRPA NH 2 CONH 2 HCOOCH 3 HCONH 2 H 2 CCO CH 3 CHO C 2 H 4 O HCOOH HNCO HCHO C 2 H 5 OH CH 3 OH C 2 H 6 C 2 H 4 C 2 H 2 CH 4 NH 3 CO 2 CO H 2 O 2 H 2 O H 2 RMS [Hartree].34.33.32.31.3.1.9.8.7.6.5.4.3.2.1 CCSD MP2 B3LYP EXXRPA+@dRPA EXXRPA+@EXX drpa@drpa drpa@exx A. Görling (University Erlangen Nürnberg) Seattle 213 23 / 36

Results EXXRPA correlation methods II Deviations of total energies from CCSD(T) energies, E = E Method E CCSD(T) Orbital basis: aug-cc-pcvqz RI basis (for E c and v xc): aug-cc-pvqz/mp2-fit Energy difference to CCSD(T) [Hartree].6.5.4.3.2.1 -.1 CCSD MP2 EXXRPA@EXX EXXRPA@dRPA NH 2 CONH 2 HCOOCH 3 HCONH 2 H 2 CCO CH 3 CHO C 2 H 4 O HCOOH HNCO HCHO C 2 H 5 OH CH 3 OH C 2 H 6 C 2 H 4 C 2 H 2 CH 4 NH 3 CO 2 CO H 2 O 2 H 2 O H 2.37.36.35.34.33.1.9.8.7.6.5.4.3.2.1 RMS [Hartree] CCSD MP2 B3LYP EXXRPA+@SC-RPA EXXRPA+@EXX drpa@drpa drpa@exx A. Görling (University Erlangen Nürnberg) Seattle 213 24 / 36

Results EXXRPA correlation methods III Deviations of reaction energies from CCSD(T) reaction energies Orbital basis: aug-cc-pcvtz RI basis (for E x and v xc): aug-cc-pvtz/mp2-fit 3.5 3 2.5 RMS [kcal/mol] 2 1.5 1.5 drpa@drpa drpa@exx B3LYP MP2 CCSD 1 kcal/mol =.43 ev EXXRPA+@dRPA EXXRPA+@EXX A. Görling (University Erlangen Nürnberg) Seattle 213 25 / 36

Results EXXRPA correlation methods IV Deviations of reaction energies from CCSD(T) reaction energies Orbital basis: aug-cc-pcvqz RI basis (for E x and v xc): aug-cc-pvqz/mp2-fit 3.5 3 2.5 RMS [kcal/mol] 2 1.5 1.5 drpa@drpa drpa@exx B3LYP MP2 CCSD 1 kcal/mol =.43 ev EXXRPA+@dRPA EXXRPA+@EXX A. Görling (University Erlangen Nürnberg) Seattle 213 26 / 36

Dissociation of H 2 Orbital basis: aug-cc-pcvqz RI basis (for E c and v xc ): aug-cc-pvqz/mp2-fit -.75 -.8 EXXRPA SC-EXXRPA FCI HF -.85 -.9 E [a.u.] -.95-1 -1.5-1.1-1.15-1.2 2 3 4 5 6 7 8 9 1 r [a.u.] Dissociation limit of other molecules (CO, N 2, etc.) is also treated correctly A. Görling (University Erlangen Nürnberg) Seattle 213 27 / 36

Coupling constant integrand -.5 HF-RPA 1 E c = dα V c (α) -.5 HF-RPA V c (α) [a.u.] -.1 -.15 -.2 -.25 -.5 r=1.4 a r=6. a r=1. a r=2. a r=3. a EXX-RPA V c HL (α) [a.u.] -.1 -.15 -.2 -.25 -.5 r=1.4 a r=6. a r=1. a r=2. a r=3. a EXX-RPA V c (α) [a.u.] -.1 -.15 -.2 -.25 V c HL (α) [a.u.] -.1 -.15 -.2 -.25 (ia ia).2.4.6.8 1 coupling strength V c (α).2.4.6.8 1 coupling strength Vc HL (α) A. Görling (University Erlangen Nürnberg) Seattle 213 28 / 36

Dissociation of N 2-18 -18.2 HF EXX CCSD RI-EXX-RPA Orbital basis: avtz RI basis: avtz Energy [hartree] -18.4-18.6-18.8-19 -19.2-19.4 1.5 2 2.5 3 3.5 4 4.5 5 5.5 r [bohr] Special treatment of singularity in ω-integrand ( τ 1 ln [ 1 + τ (iω) ] + 1 ) A. Görling (University Erlangen Nürnberg) Seattle 213 29 / 36

Twisting of ethene Orbital basis: avqz -77.8 RI basis: avqz -77.9-78 Energy [Hartree] -78.1-78.2-78.3-78.4-78.5-78.6 2 4 6 8 1 12 14 16 18 Torsion angle [degree] HF EXX CCSD B3LYP RI-EXXRPA+ MCSCF A. Görling (University Erlangen Nürnberg) Seattle 213 3 / 36

RI-EXXRPA for noncovalent bonds Interaction energies in kcal/mol Complex BasisCCSD(T) drpa drpa EXXRPA+ EXXRPA+ MP2 CCSD a @EXX @DRPA @EXX @drpa (NH 3 ) 2 4 2.52 2.57 2.83 2.99 3.11 3 3. 2.36 2.41 2.84 2.89 3. 2.75 CBS 3.15 2.62 2.64 2.81 3.2 3.18 2.88 (H 2 O) 2 4 4.25 4.33 4.91 5.7 5.2 3 4.83 3.95 4.3 4.81 4.92 4.72 4.6 CBS 5.7 4.4 4.46 4.92 5.9 5.21 4.74 (HCONH 2 ) 2 4 14.32 14.67 16.1 16.42 15.57 3 15.37 13.65 13.95 15.56 15.9 15.8 14.65 CBS 16.11 14.75 15.6 16.28 16.67 15.86 15.28 (HCOOH) 2 4 16.41 16.98 18.79 19.34 18.23 3 17.86 15.51 16.3 18.28 18.73 17.61 17.8 CBS 18.81 16.97 17.49 19.6 19.6 18.61 17.92 (C 2 H 4 ) 2 4 1.8 1.5 1.55 1.42 1.55 3 1.38.99.99 1.44 1.33 1.47 1.4 CBS 1.48 1.16 1.5 1.64 1.44 1.6 1.14 C 2 H 4 C 2 H 2 4 1.31 1.21 1.42 1.56 1.64 3 1.44 1.23 1.16 1.34 1.5 1.59 1.26 CBS 1.5 1.37 1.22 1.49 1.58 1.67 1.31 (CH 4 ) 2 4.35.36.44.44.48 3.5.31.32.41.41.46.38 CBS.53.37.35.44.44.53.41 RMS 3 1.19.97.19.39.13.45 RMS CBS.93.73.19.37.15.51 a CBS obtained via F12-CCSD 1 kcal/mol =.43 ev A. Görling (University Erlangen Nürnberg) Seattle 213 31 / 36

S22 test set of dimer binding energies E [kcal/mol] 4 2-2 -4-6 -8-1 Hydrogen Standard benchmark for non-covalent interactions E = E Binding Method EBinding CCSD(T) Mixed -12 drpa@drpa EXXRPA+@dRPA Dispersion -14 MP2 PBE B3LYP -16 1 2 3 4 5 6 7 8 9 1 11 12 13 14 15 16 17 18 19 2 21 22 1 NH 3 (C 2h ) 2 (H 2 O) 2 (C s) 3 (HCOOH) 2 (C 2h ) 4 (CHONH 2 ) 2 5 Uracil-Uracil (C 2h ) 6 2-Pyridoxine-2-Aminopyridine 7 AT (WC) 8 (CH 4 ) 2 (D 3d ) 9 (C 2 H 4 ) 2 (D 2d ) 1 Bz CH 4 (C 3) 11 Bz-Bz (C 2h ) 12 Pyrazine-Pyrazine (C s) 13 Uracil-Uracil (C 2) 14 Indole-Bz (stacked) 15 AT (stacked) 16 C 2 H 4 C 2 H 2 (C 2v) 17 Bz H 2 O (C s) 18 Bz NH 3 (C s) 19 Bz HCN (C s) 2 Bz-Bz (C 2v) 21 Indole-Bz (T-shaped) 22 Phenole-Phenole A. Görling (University Erlangen Nürnberg) Seattle 213 32 / 36

Directional non-covalent interactions in Ar Br2 I CCSD(T) EXXRPA + @drpa 1 kcal/mol =.43 ev A. Go rling (University Erlangen Nu rnberg) Seattle 213 33 / 36

Directional non-covalent interactions in Ar Br2 II CCSD(T) B3LYP 1 kcal/mol =.43 ev A. Go rling (University Erlangen Nu rnberg) Seattle 213 34 / 36

Summary EXXRPA correlation functional combines accuracy at equilibrium geometries with a correct description of dissociation (static correlation) and a highly accurate treatment of VdW interactions EXXRPA correlation functional is self-interaction free Self-consistent drpa yields qualitatively reasonable correlation potentials in contrast to conventional KS methods Promising starting point for further developments, e.g., completely self-consistent EXXRPA method or inclusion of correlation in kernel Orbital-dependent functionals open up fascinating new possibilities in DFT A. Görling (University Erlangen Nürnberg) Seattle 213 35 / 36

Literature Phys. Rev. Lett. 79, 289 (1997) Phys. Rev. B 81, 155119 (21) EXX for solids Phys. Rev. B 83, 4515 (211) EXX for molecules Phys. Rev. Lett. 83, 5459 (1999) J. Chem. Phys. 128, 1414 (28) Phys. Rev. A 8, 1257 (29) Phys. Rev. Lett. 12, 2333 (29) Mol. Phys. 18, 359 (21) Mol. Phys. 19, 2473 (211) Review TDEXX Int. J. Quantum. Chem. 11, 222 (21) J. Chem. Phys. 134, 3412 (211) EXX-RPA Phys. Rev. Lett. 16, 931 (211) J. Chem. Phys. 136, 13412 (212) A. Görling (University Erlangen Nürnberg) Seattle 213 36 / 36

2024 © sciencedocbox.com Privacy Policy | Terms of Service | Feedback