Developments in Electronic Structure Theory.

Transcription

1 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

2 A short presentation

3 A short presentation (1) What I do : Quantum Mechanics Wavefunction theories (WFT) Density Functional theory (DFT) Range Separation methods (RS) Main research interest Weak interactions / London dispersion forces / VdW interactions Random Phase Approximation (RPA) - formalism (new approximations, spin-unrestricted generalization,...) - local orbitals (designed new localized orbitals, selection of excitation) - gradients of the energy (forces, dipoles, optimization of geometry,...) -... Numerous collaborations High Harmonics Generation spectra (Electron dynamics) Quantum Monte Carlo methods (QMC)

4 A short presentation (2) Notable Developments Developments in MOLPRO Commercial code Versatile (HF,DFT,CI,CC,MCSCF, MRCI,MRPT,MRCC, FCIQMC,DMRG, Gradients and Properties) lines of code In charge of the RPA code Developments in CHAMP Free code VMC, DMC Optimization of wavefunctions,... Fast calculation of determinants CHAMP@Cornell Developped personal codes, from scratch Calculations on grid, parallel codes Technical side of the work Management of codes (SVN/GIT/BugZilla) Cleaning-up Optimizing pre-existing codes (use of memory,...) HPC Interfaces between codes

5 Context of my main research interest Electronic structure calculations. post-dft calculations in the context of Range Separation. In the WFT part, for the treatment of London dispersion forces, we use the Random Phase Approximation.

6 Range Separation : Motivation [Savin, Rec.dev. (1996)] Get the accurate answer for the right reason at a reasonable cost. WFT (the wavefunction Ψ(x N ) is the key quantity ; determinants) rather costly (size of Hilbert space) improvement of the description is systematic (add more determinants) DFT (the density n(r) is the key quantity) cheap methods ; applicable to big systems ; impressive successes need to approximate E xc [n] ; not systematical Range Separation (a rigourous combination of both approaches) WFT suffers at short range from the e-e coalescence. DFAs are (semi)local approximation, best at short-range. The idea : split the e-e interaction into long- and short-range, treat the long-range with WFT, the short-range with DFT L = 1 L = 2 L = 3 L = 4 L = θ (degree)

7 Range Separation : Realisation [Savin, Rec.dev. (1996)] Split of the Coulomb interaction ; 1 r = v lr ee(r) + v sr ee(r) ˆV lr ee long-range e-e interaction E sr Hxc[n] short-range density functional In a variational way : E exact = min Ψ rigourous mix of WFT and DFT methods. { } Ψ ˆT + ˆV ne + ˆV ee Ψ lr + EHxc[n sr Ψ ] 1/r vee(r) lr vee(r) sr e.g. MCSCF+DFT, CI+DFT,...-> for the static (strong) correlation. [Fromager,Toulouse,Jensen JCP 2007] [Sharkas,Savin,Jensen,Toulouse JCP 2012] With a perturbative treatment, from a mono-determinental reference (RSH) : { } E exact = min Φ ˆT + ˆV ne + ˆV ee Φ lr + EHxc[n sr Φ ] + Ec lr Φ e.g. with RPA -> for long range dynamical correlation (vdw dispersion). [Ángyán,Gerber,Savin,Toulouse PRA 2005] [Toulouse,Gerber,Jansen,Savin,Ángyán PRL 2009]

8 Range Separation : Examples [Savin, Rec.dev. (1996)] Ar 2, aug-cc-pv5z basis, µ = 0.5 bohr Interaction energy (mhartree) Accurate PBE (sr)pbe Internuclear distance (bohr) RSH reference contains no dispersion.

9 Range Separation : Examples [Savin, Rec.dev. (1996)] Ar 2, aug-cc-pv5z basis, µ = 0.5 bohr Interaction energy (mhartree) Accurate drpa (lr)drpa RPAx (lr)rpax Internuclear distance (bohr) RSH reference contains no dispersion. In RPA, it is important to add exchange effects for dispersion interactions.

10 Random Phase Approximation : Origins [Bohm,Pines PR (1951)] The original intention is to decouple by a canonical transformation the particle and field variables in the description of the Uniform Electron Gaz (UEG). The physics : mix of long-range organized behavior (collective plasma oscillations) and of short-range screened explicit interactions. This emerges naturally by the canonical transformation, and involves a cut-off distance, over which long-range behavior correctly describes the system, and under which it no longer captures the main physics : the short-range interaction is then predominant. Hence, in RPA, the energy is the sum of oscillator energies and of a short-range correction. This seems ideal in a range separation context. What is random in RPA? Only the particles in phase with the oscillating field contribute to the oscillations the other particles, having a random phase, are neglected.

11 Random Phase Approximation : formalisms and versions [Ángyán,Liu,Toulouse,Jansen JCTC (2011)] AC-FDT equation [BM,Rocca,Jansen,Ángyán JCTC (submitted)] Ec RPA = 1 1 dω ( [ ]) dα 2 0 2π Tr V ee Π RPA α (iω) Π 0 (iω) (Π RPA α ) 1 = Π 1 0 f Hx,α Versions (include/exclude exchange) This yields different, non-equivalent direct or exchange RPA versions of RPA single-bar or double-bar Formulations (analytical/numerical integrals) density-matrix formulation E RPA c = 1 2 dα tr (P c,α V) dielectric-matrix formulation E RPA dω c = 1 2 tr (log(ɛ(iω)) ɛ(iω) + 1) 2π plasmon formula E RPA c = 1 2 Ricatti equations and rccd E RPA c ω RPA ω TDA = 1 2 tr(bt) +Approximations (in each formulations, for each versions...) SOSEX, SO2

12 Random Phase Approximation : formalisms and versions [Ángyán,Liu,Toulouse,Jansen JCTC (2011)] AC-FDT equation [BM,Rocca,Jansen,Ángyán JCTC (submitted)] Ec RPA = 1 1 dω ( [ ]) dα 2 0 2π Tr V ee Π RPA α (iω) Π 0 (iω) (Π RPA α ) 1 = Π 1 0 f Hx,α Versions (include/exclude exchange) This yields different, non-equivalent direct or exchange RPA versions of RPA single-bar or double-bar Formulations (analytical/numerical integrals) density-matrix formulation E RPA c = 1 2 dα tr (P c,α V) dielectric-matrix formulation E RPA dω c = 1 2 tr (log(ɛ(iω)) ɛ(iω) + 1) 2π plasmon formula E RPA c = 1 2 Ricatti equations and rccd E RPA c ω RPA ω TDA = 1 2 tr(bt) +Approximations (in each formulations, for each versions...) SOSEX, SO2

13 Random Phase Approximation : formalisms and versions [Ángyán,Liu,Toulouse,Jansen JCTC (2011)] AC-FDT equation [BM,Rocca,Jansen,Ángyán JCTC (submitted)] Ec RPA = 1 1 dω ( [ ]) dα 2 0 2π Tr V ee Π RPA α (iω) Π 0 (iω) (Π RPA α ) 1 = Π 1 0 f Hx,α Versions (include/exclude exchange) This yields different, non-equivalent direct or exchange RPA versions of RPA single-bar or double-bar Formulations (analytical/numerical integrals) density-matrix formulation E RPA c = 1 2 dα tr (P c,α V) dielectric-matrix formulation E RPA dω c = 1 2 tr (log(ɛ(iω)) ɛ(iω) + 1) 2π plasmon formula E RPA c = 1 2 Ricatti equations and rccd E RPA c ω RPA ω TDA = 1 2 tr(bt) +Approximations (in each formulations, for each versions...) SOSEX, SO2

14 Random Phase Approximation : Feynman diagrams drpa-i RPAx-II E drpa-i c = 1 2 tr ab ij lr (T lr drccd) ia,jb E RPAx-II c = 1 4 tr ab ij lr (T lr rccdx) ia,jb [McLachlan,Ball RMP 1964] [a lot of diagrams] SOSEX E SOSEX c = 1 2 tr ab ij lr (T lr drccd) ia,jb [Grüneis,et. al. JCP 2009] RPAx-SO2 Ec RPAx-SO2 = 1 2 tr ab ij lr (TrCCDx) lr ia,jb [Szabo,Ostlund JCP 1977] [Toulouse et. al. JCP 2011] Not that good for weak interactions. It is our method of choice.

15 New developments Basis Set Convergence study and Three-point extrapolation scheme. [Franck,BM,Luppi,Toulouse JCP (2015)] Spin-Unrestricted generalization and Calculations on DBH24/08 and AE49. [BM,Reinhardt,Ángyán,Toulouse JCP (2015)] Development and Implementation of the Gradients of RSH+RPA methods. [BM,Szalay,Ángyán JCTC (2014)] Development of Localized Virtual Orbitals and Application for RPA calculations. [BM,Ángyán TCA (2015)]

16 Basis Set Convergence [Franck,BM,Luppi,Toulouse JCP (2015)] Partial wave expansion of the WF [Gori-Giorgi,Savin PRA (2006)] Ψ(r 12) Ψ(0) = 1 + r Ψ µ (r 12) Ψ µ (0) = 1 + µr π +... = c l P l (cosθ) with c l l 2 = c l P l (cosθ) with c l e αl L = 1 L = 2 L = 3 L = 4 L = θ (degree) r 2 θ He r L = 1 L = 2 L = 3 L = θ (degree) Principal number expansion (wrt one-particule basis) Ψ(r 1, r 2) = c i φ i (r 1) φ i (r 2) HF VDZ VTZ VQZ V5Z V6Z Hylleraas r 2 θ He r RSH VDZ VTZ VQZ V5Z V6Z θ (degree) θ (degree)

17 Spin-Unrestricted RPA [BM,Reinhardt,Ángyán,Toulouse JCP (2015)] Implemented most of the variants with a nospinflip/spinflip block structure Normal distributions of errors (kcal/mol) of calculations on AE49. Mean errors of post-rsh calculation are better. Post-RSH calculations dist. of errors have sharper distributions HF+MP RSH+MP Similar results were obtained on the DBH24/08 dataset HF+dRPA-I RSH+dRPA-I We argue that RPAx-SO2 is a good method for a wide range of applications HF+RPAx-SO2 RSH+RPAx-SO2 >cc-pvqz (post-rsh : srpbe, µ = 0.5)

18 Analytical Gradients : Lagrangian formalism énergie énergie énergie paramètre paramètre paramètre lagrangien Given an energy E[κ, V(κ), T(κ)] you have rules for V : E/ V = 0 you have rules for T : R[T] paramètre = 0 lagrangien de dκ = E κ + E V E T + V κ T κ For non-variational methods E[V] [BM,Szalay,Ángyán JCTC (2014)] VV R[T] = = 0 0 TT E[T] ( (a) (a) (b) (b) in E ) κ there is h(κ), (µν σρ) (κ) and S (κ) The solution is to work with an alternative object that is variational : L[κ, V, T, λ] = E[κ, V, T] + tr(λr[t]) énergie L V = E énergie V = 0 énergie paramètre L λ = paramètre paramètre R[T] = 0 lagrangien (a) E[V] R[T] = 0 L T = E ( T + tr λ R ) V T (b) = 0 T E[T] { Φ ˆT + ˆV ne + ˆV lr The RSH+RPA energy is : E = min Hxc[n Φ ] Φ The equations were derived (sr and lr terms) and implemented in MOLPRO. ee Φ + E sr (c) } E[T] L[T] T + E lr,rpa c

19 Analytical Gradients of RSH+RPA Bond Lengths of simple molecules > aug-cc-pvqz r (Å) H2/H H HF/F H H2O/O H HOF/O H RHF+dRPA-I LDA+dRPA-I RHF+SOSEX LDA+SOSEX HNO/N H C2H2/C H HCN/C H C2H4/C H CH4/C H N2/N N HNC/N H NH3/N H N2H2/N H CH2O/C H CH2/C H CO/C O HCN/C N CO2/C O HNC/C N C2H2/C C CH2O/C O HNO/O N N2H2/N N C2H4/C C F2/F F Mean Absolute Errors Bond Lengths of simple molecules RHF+dRPA-I RHF+SOSEX LDA+dRPA-I LDA+SOSEX Interaction Energies HOF/O F Intermonomer distances MP MP LDA+MP LDA+MP LDA+dRPA-I 0.35 LDA+dRPA-I [BM,Szalay,Ángyán JCTC (2014)] Interaction Energies > aug-cc-pvtz ( E) (kcal.mol 1 ) C2H4...F2 NH3...F2 MP2 LDA+MP2 LDA+dRPA-I C2H2...ClF HCN...ClF NH3...Cl2 Intermonomer distances > aug-cc-pvtz R (Å) C2H4...F2 NH3...F2 C2H2...ClF HCN...ClF NH3...Cl2 H2O...ClF H2O...ClF NH3...ClF MP2 LDA+MP2 LDA+dRPA-I NH3...ClF

20 Projected Oscillator Orbitals [BM,A ngya n TCA (2015)] The objective is to construct a set of localized virtual orbitals by multiplying the set of occupied Localized Molecular Orbitals (LMOs) with solid spherical harmonic functions. The orthogonality to the occupied space is ensured by projection. Dipolar oscillator orbital [Foster, Boys RMP (1960)] iα i = (1 P ) (r α Dαi ) ii {z } {z } {z} projector C-H bond in CH2 =O : harmonic LMO where Dαi = hi r α ii

21 Projected Oscillator Orbitals : Key features Dipolar oscillator orbital i α = (ˆ1 ˆP)(ˆr α D i α) i where D i α = i ˆr α i [BM,Ángyán TCA (2015)] This formulation is best employed with Boys LMOs, see : i α = ˆr α i m m ˆr α i The POOs are non-orthogonal, with overlap : S ij αβ = i ˆr αˆr β j i ˆr α m m ˆr β j The fock matrix element are found to be : f ii αβ = 1 2 δ αβ (fim m ˆr αˆr β i + i ˆr αˆr β m f mi ) i ˆr α m f mn n ˆr β i The two-elec integrals are written with a multipole expansion of the lr interaction : i α j kl β = i α ˆr γ k L ij γδ j ˆr δ l β +... A nice result greatly simplifies the equations : i α ˆr β j = i ˆr αˆr β j i ˆr α m m ˆr β j = S ij αβ

22 Projected Oscillator Orbitals : RPA and C 6 [BM,Ángyán TCA (2015)] The working equations are the local RPA Ricatti equations (i.e. local rccd) with the local excitation approximation and spherical average approximation. RPA correlation energy occ Ec RPA,lr 4 = 9 si s j tr ( L ij T ij) ij where s i = S ii αα C 6 coefficients occ E c (2),lr = = ij occ ij 4 s i s j 9 i + j tr ( L ij L ij) 8 s i s j F µ damp (Dij ) 3 i + j D ij 6 where i = f ii f i /s i Error in the molecular C6 coefficients (%) H2 HF H2O N2 CO LDA PBE RHF RSHLDA NH3 CH4 CO2 H2CO N2O C2H2 CH3OH C2H4 CH3NH2 C2H6 CH3CHO CH3OCH3 C3H6 HCl HBr H2S SO2 SiH4 Cl2 COS CS2 > aug-cc-pvtz (RSH : µ = 0.5) MA%E LDA 59.8 PBE 56.7 RHF 15.2 RSHLDA 11.8 CCl4

23 Other Works Efficient calculations of determinants in QMC using the Sherman-Morrison-Woodbury formula. Fractional occupation number calculations and Instabilities in the RPA problem. [BM,Toulouse (in prep)] Electron dynamics for High Harmonics Generation spectra [Coccia,BM,Labeye,Caillat,Taieb,Toulouse,Luppi (submitted)] In real space : relationship between the response functions, exchange holes and localized orbitals. [BM,Ángyán CTC (2015)] Collaborators : János Ángyán, Roland Assaraf, Odile Franck, Georg Jansen, Eleonora Luppi, Peter Reinhardt, Dario Rocca, Julien Toulouse, Cyrus Umrigar.