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 Developments in Electronic Structure Theory Get the accurate answer for the right reason at a reasonable cost. Accuracy & Cost DFT (the density n(r) is the key quantity) cheap methods ; applicable to big systems unable to systematically improve the results Range Separation WFT (the wavefunction Ψ(x N ) is the key quantity) rather costly improvement of the description is systematic QMC ( stochastic method) good method in terms of accuracy and scaling costly ; ideal for parallelisation WFT as input for QMC

3 DFT+WFT 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.

4 Range Separation [Savin, Rec.dev. (1996)] WFT suffers at short range from the description of the e-e coalescence. DFAs are essentially (semi)local approximation and are best at short-range. The idea is to split the e-e interaction into long- and short- range parts: 1 r = v lr ee(r) + v sr ee(r) 1/r vee(r) lr vee(r) sr versus... and as a result to rigourously mix WFT and DFT methods : in a variational way: { } E = min Ψ ˆT + ˆV ne + Ŵee lr Ψ + EHxc[n sr Ψ ] Ψ e.g. MCSCF+DFT, CI+DFT,... -> for the static correlation. with a perturbative treatment, from a mono-determinental reference (RSH): { } E = min Φ ˆT + ˆV ne + Ŵee Φ lr + EHxc[n sr Φ ] + Ec lr Φ e.g. with RPA -> for long range dynamical correlation.

5 Random Phase Approximation: Origins [Bohm,Pines PR 82 (1951)] The original intention is to decouple by a canonical transformation the particle and field variables in the description of the UEG. Go from Ĥ = Ĥ part. + Ĥ field + Ĥ part./field to Ĥ RPA = Ĥ part. + Ĥ oscillations + Ĥ sr part./part The physics: mix of long-range organized behavior (collective plasma oscillations) and of short-range screened interactions. 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. Only the particles in phase with the oscillating field contribute to the oscillations the other particles, having a random phase, are neglected.

6 Random Phase Approximation: formalisms and flavors ACFDT equation [Langreth,Perdew PRB 15 (1977)] Ec ACFDT = 1 1 dω dα 2 2π Tr ( [ W ee Π RPA α (iω) Π (iω) ]) (Π RPA α ) 1 = Π 1 f Hx,α Flavors (include/exclude exchange) [Ángyán,Liu,Toulouse,Jansen JCTC 7 (211)] direct or exchange RPA single-bar or double-bar This yields the flavors of RPA called: drpa-i, drpa-ii, RPAx-I, RPAx-II Formulations (analytical or numerical integrals) density-matrix formulation E = 1 2 dα tr (P α W) dielectric-matrix formulation E = 1 dω 2 tr (log(ɛ(iω)) + 1 ɛ(iω)) 2π plasmon formula E = 1 2 ωrpa ω TDA Ricatti equations and rccd E = 1 2 tr(bt) +Approximations (in each formulations, for each flavors...) SOSEX, RPAx-SO2

7 Random Phase Approximation: formalisms and flavors ACFDT equation [Langreth,Perdew PRB 15 (1977)] Ec ACFDT = 1 1 dω dα 2 2π Tr ( [ W ee Π RPA α (iω) Π (iω) ]) (Π RPA α ) 1 = Π 1 f Hx,α Flavors (include/exclude exchange) [Ángyán,Liu,Toulouse,Jansen JCTC 7 (211)] direct or exchange RPA single-bar or double-bar This yields the flavors of RPA called: drpa-i, drpa-ii, RPAx-I, RPAx-II Formulations (analytical or numerical integrals) density-matrix formulation E = 1 2 dα tr (P α W) dielectric-matrix formulation E = 1 dω 2 tr (log(ɛ(iω)) + 1 ɛ(iω)) 2π plasmon formula E = 1 2 ωrpa ω TDA Ricatti equations and rccd E = 1 2 tr(bt) +Approximations (in each formulations, for each flavors...) SOSEX, RPAx-SO2

8 Random Phase Approximation: formalisms and flavors ACFDT equation [Langreth,Perdew PRB 15 (1977)] Ec ACFDT = 1 1 dω dα 2 2π Tr ( [ W ee Π RPA α (iω) Π (iω) ]) (Π RPA α ) 1 = Π 1 f Hx,α Flavors (include/exclude exchange) [Ángyán,Liu,Toulouse,Jansen JCTC 7 (211)] direct or exchange RPA single-bar or double-bar This yields the flavors of RPA called: drpa-i, drpa-ii, RPAx-I, RPAx-II Formulations (analytical or numerical integrals) density-matrix formulation E = 1 2 dα tr (P α W) dielectric-matrix formulation E = 1 dω 2 tr (log(ɛ(iω)) + 1 ɛ(iω)) 2π plasmon formula E = 1 2 ωrpa ω TDA Ricatti equations and rccd E = 1 2 tr(bt) +Approximations (in each formulations, for each flavors...) SOSEX, RPAx-SO2

9 Full-range DFT + RPA DFT calculation followed by an RPA calculation with KS orbitals and energies. { } E DFT = min Φ ˆT + ˆV ne Φ + E Hxc [n Φ ] Φ E = Φ ˆT + ˆV ne Φ + EHx HF [Φ] + Ec RPA DFT (LDA/GGA/...) do not describe correctly the long-range dispersion forces, while RPA does. Still problems with: short-range correlation energies (far too negative) strong dependence on basis size simple van der Waals dimers Interaction energy (mhartree) Ne 2 Accurate drpa-i SOSEX RPAx-II RPAx-SO2 Interaction energy (mhartree) Ar 2 Accurate drpa-i SOSEX RPAx-II RPAx-SO Internuclear distance (bohr) Internuclear distance (bohr) >PBE, aug-cc-pv6z [Toulouse,Zhu,Savin,Jansen,Ángyán JCP 135 (211)]

10 Range-Separated Hybrid + RPA Hybrid DFT with exact HF exchange and RPA correlation, both at long-range. { } E = min Φ ˆT + ˆV ne + Ŵee Φ lr + EHxc[n sr Φ ] + Ec lr,rpa Φ Range separation greatly improves RPA (see srpbe+rpax-so2). Basis dependence is reduced. Interaction energy (mhartree) Ne 2 Accurate drpa-i SOSEX RPAx-II RPAx-SO2 Interaction energy (mhartree) Ar 2 Accurate drpa-i SOSEX RPAx-II RPAx-SO Internuclear distance (bohr) Internuclear distance (bohr) >srpbe, aug-cc-pv6z, µ =.5 [Toulouse,Zhu,Savin,Jansen,Ángyán JCP 135 (211)]

11 Range-Separated Hybrid + RPA >srpbe, aug-cc-pvdz, µ =.5 [Toulouse,Zhu,Savin,Jansen,Ángyán JCP 135 (211)] Hybrid DFT with exact HF exchange and RPA correlation, both at long-range. { } E = min Φ ˆT + ˆV ne + Ŵee Φ lr + EHxc[n sr Φ ] + Ec lr,rpa Φ Range separation greatly improves RPA (see srpbe+rpax-so2). Basis dependence is reduced. for ex.: performance on the S22 dataset is very good for srpbe+rpax-so2. % of error on interaction energy H bonded dispersion dispersion +multipoles MP2 drpa-i SOSEX RPAx-II RPAx-SO2 system in S22 set

12 New developments DFT+WFT Basis Set Convergence study and Three-point extrapolation scheme. Spin-Unrestricted generalization and Calculations on DBH24/8 and AE49. Development and Implementation of the gradients of RSH+RPA methods. In general Development of Localized Virtual Orbitals and Application for RPA calculations. WFT+QMC Implementation of the calculation of weights of VB structures in QMC.

13 Basis Set Convergence [Franck,BM,Luppi,Toulouse JCP 142 (215)] Wavefunction basis convergence [Gori-Giorgi,Savin PRA 73 (26)] Both partial wave and principal number expansions lead to exponential convergence of the wavefunction r 2 θ He r 1 Ψ (a.u.) HF VDZ VTZ VQZ V5Z V6Z Hylleraas θ (degree) Extrapolation formula for total energies We proposed a three-point extrapolation scheme for total RSH+correlation energies (fitting E X = E + B exp( βx )). E = E XYZ = E 2 Y E X E Z 2E Y E X E Z Ψ lr,µ (a.u.) RSH VDZ VTZ VQZ V5Z V6Z θ (degree) >errors (mhartree) wrt. cc-pv6z He Ne N 2 H 2O E D E T E Q E E DTQ E TQ

14 Spin-Unrestricted RPA [BM,Reinhardt,Ángyán,Toulouse JCP 142 (215)] Implemented most of the variants with a nospinflip/spinflip block structure Normal distributions of errors (kcal/mol) of calculations on AE HF+MP2 RSH+MP2 Mean errors of post-rsh calculation are better Post-RSH calculations dist. of errors have sharper distributions HF+dRPA-I RSH+dRPA-I Similar results were obtained on the DBH24/8 dataset 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 =.5)

15 Analytical Gradients: Lagrangian formalism [BM,Szalay,Ángyán JCTC 1 (214)] énergie énergie énergie paramètre paramètre paramètre lagrangien Given an energy E[κ, V(κ), T(κ)] you have rules for V: E/ V = you have rules for T: R[T] paramètre = lagrangien de dκ = E κ + E V E T + V κ T κ For non-variational methods E[V] VV R[T] = = 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 = énergie paramètre L λ = paramètre paramètre R[T] = lagrangien E[V] R[T] = L T = E ( T + tr λ R ) V T (a) (b) { } For RSH+RPA the energy is: E = min Φ ˆT + ˆV ne + Ŵee Φ lr + EHxc[n sr Φ ] + Ec lr,rpa Φ The equations were derived (sr and lr terms) and implemented in MOLPRO. (based on RSH+MP2 of [Chabbal, Stoll, Werner, Leininger, MP 18 (21)]) = T E[T] (c) E[T] L[T] T

16 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 -.8 RHF+dRPA-I LDA+dRPA-I RHF+SOSEX LDA+SOSEX HCN/C H C2H4/C H CH4/C H N2/N N CH2O/C H CH2/C H HNC/N H NH3/N H N2H2/N H HNO/N H C2H2/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.16 LDA+dRPA-I.13 RHF+SOSEX.21 LDA+SOSEX.14 Interaction Energies HOF/O F Intermonomer distances MP2.76 MP2.75 LDA+MP2.63 LDA+MP2.53 LDA+dRPA-I.35 LDA+dRPA-I.23 [BM,Szalay,Ángyán JCTC 1 (214)] 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

17 Projected Oscillator Orbitals [BM,A ngya n, TCA (submitted)] 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 32 (196)] iα i = (1 P ) (r α Dαi ) ii {z } {z } {z} projector C-H bond in CH2 O: harmonic where Dαi = hi r α ii LMO Lone pair in CH2 O:

18 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 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 [BM,Ángyán, TCA (submitted)] 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 simplifies the equations: i α ˆr β j = i ˆr αˆr β j i ˆr α m m ˆr β j = S ij αβ

19 Projected Oscillator Orbitals: RPA and C 6 [BM,Ángyán, TCA (submitted)] The working equations are the local RPA Ricatti equations (i.e. local rccd) with the local excitation approximation (M ij k αl β M ij. i αj β = M ij αβ ) and spherical average approximation (M ii 1 αβ δ αβ 3 M ii. αα = 1 δαβ 3 mi ). RPA correlation energy occ Ec RPA,lr 4 = 9 si s j tr ( L ij T ij) ij where R ij = = 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 =.5) MA%E LDA 59.8 PBE 56.7 RHF 15.2 RSHLDA 11.8 CCl4

20 VB-QMC: Motivation VB-SCF (VB structures = Lewis structures) Good dissociation energies Interpretative tools (weight of structures,... ) Non-orthogonal orbitals -> costly Introduce dynamical correlation by breathing (L-BO-, S-BO-, SD-BOVB) -> costly VB-QMC in the hope of treating big systems Ψ = e J ( ) Static correlation taken care of by VB. Non-orthogonality of orbitals is not an issue for QMC. Dynamical correlation hopefully taken care of by Jastrow. But: we wish to conserve the interpretative tools, i.e. the weights (and the calculation of separated structure). WFT as input for QMC

21 VB-QMC: weights of VB structures We wish to transfert the weights of structures to VB-QMC. j Chirgwin-Coulson weights w i = c i c j S ij and Löwdin weights w c i c j S i = ( c j (S 1 2 ) ji ) 2 j ij ( ij c j (S 1 2 ) ji ) 2 i j In both cases, S ij needs to be computed, via S ij = dr Ψ i(r) Ψ j (R) Ψ (R) Ψ (R) Ψ2 (R) Li2 VB-SCF VB-QMC S-BOVB (jc) (jco) LiH VB-SCF VB-QMC S-BOVB (jc) (jco) VB-QMC = VB-SCF Li 2 : mainly covalent character ionic too important in BOVB (structures are mixing) F VB-SCF VB-QMC S-BOVB (jc) (jco) FH VB-SCF VB-QMC S-BOVB (jc) (jco) F 2 : Jastrow very similar to BO no mixing of structures It is unclear if the VB weights are the referent ones (when there is a too preponderant structure, mixing occurs).

22 Other Works All (and more) is implemented in the Quantum Chemistry Package MOLPRO Fractional occupation number calculations and Instabilities in the RPA problem. Efficient calculations of determinants in QMC using the Sherman-Morrison-Woodbury formula. Collaborators at the UPMC... Odile Franck Eleanora Luppi Peter Reinhardt Julien Toulouse... and at the Université de Lorraine: János Ángyán Dario Rocca