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logos Determinant based Separable MRPT2 E. Giner1, Jean Paul Malrieu1 1Laboratoire de Physique et Chimie Quantiques / IRSAMC, Toulouse, France Tuesday 14 of January E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos Current works, motivations .. PhD supervisors : M. Caffarel and A. Scemama Past investment : Fixed Node Diffusion Monte Carlo (FN-DMC) In principle : exact energy but ... Needs function in input with good nodal properties My PhD : Do correlated wf based method provide better nodes ? With selected CI wf (CIPSI see later) ⇒ systematic improvment of the nodes ⇒ dramatic improvment of the FN-DMC energy ! E. G, Anthony Scemama, Michel Caffarel. Canadian Journal of Chemistry, 1-26, 2013 The talk : 1 Recall on SR and MR-PT2 performances and defects 2 Proposal of a strictly separable MRPT2 E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos Current works, motivations .. PhD supervisors : M. Caffarel and A. Scemama Past investment : Fixed Node Diffusion Monte Carlo (FN-DMC) In principle : exact energy but ... Needs function in input with good nodal properties My PhD : Do correlated wf based method provide better nodes ? With selected CI wf (CIPSI see later) ⇒ systematic improvment of the nodes ⇒ dramatic improvment of the FN-DMC energy ! E. G, Anthony Scemama, Michel Caffarel. Canadian Journal of Chemistry, 1-26, 2013 The talk : 1 Recall on SR and MR-PT2 performances and defects 2 Proposal of a strictly separable MRPT2 E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos Current works, motivations .. PhD supervisors : M. Caffarel and A. Scemama Past investment : Fixed Node Diffusion Monte Carlo (FN-DMC) In principle : exact energy but ... Needs function in input with good nodal properties My PhD : Do correlated wf based method provide better nodes ? With selected CI wf (CIPSI see later) ⇒ systematic improvment of the nodes ⇒ dramatic improvment of the FN-DMC energy ! E. G, Anthony Scemama, Michel Caffarel. Canadian Journal of Chemistry, 1-26, 2013 The talk : 1 Recall on SR and MR-PT2 performances and defects 2 Proposal of a strictly separable MRPT2 E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos Current works, motivations .. PhD supervisors : M. Caffarel and A. Scemama Past investment : Fixed Node Diffusion Monte Carlo (FN-DMC) In principle : exact energy but ... Needs function in input with good nodal properties My PhD : Do correlated wf based method provide better nodes ? With selected CI wf (CIPSI see later) ⇒ systematic improvment of the nodes ⇒ dramatic improvment of the FN-DMC energy ! E. G, Anthony Scemama, Michel Caffarel. Canadian Journal of Chemistry, 1-26, 2013 The talk : 1 Recall on SR and MR-PT2 performances and defects 2 Proposal of a strictly separable MRPT2 E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos Current works, motivations .. PhD supervisors : M. Caffarel and A. Scemama Past investment : Fixed Node Diffusion Monte Carlo (FN-DMC) In principle : exact energy but ... Needs function in input with good nodal properties My PhD : Do correlated wf based method provide better nodes ? With selected CI wf (CIPSI see later) ⇒ systematic improvment of the nodes ⇒ dramatic improvment of the FN-DMC energy ! E. G, Anthony Scemama, Michel Caffarel. Canadian Journal of Chemistry, 1-26, 2013 The talk : 1 Recall on SR and MR-PT2 performances and defects 2 Proposal of a strictly separable MRPT2 E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos Current works, motivations .. PhD supervisors : M. Caffarel and A. Scemama Past investment : Fixed Node Diffusion Monte Carlo (FN-DMC) In principle : exact energy but ... Needs function in input with good nodal properties My PhD : Do correlated wf based method provide better nodes ? With selected CI wf (CIPSI see later) ⇒ systematic improvment of the nodes ⇒ dramatic improvment of the FN-DMC energy ! E. G, Anthony Scemama, Michel Caffarel. Canadian Journal of Chemistry, 1-26, 2013 The talk : 1 Recall on SR and MR-PT2 performances and defects 2 Proposal of a strictly separable MRPT2 E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos Current works, motivations .. PhD supervisors : M. Caffarel and A. Scemama Past investment : Fixed Node Diffusion Monte Carlo (FN-DMC) In principle : exact energy but ... Needs function in input with good nodal properties My PhD : Do correlated wf based method provide better nodes ? With selected CI wf (CIPSI see later) ⇒ systematic improvment of the nodes ⇒ dramatic improvment of the FN-DMC energy ! E. G, Anthony Scemama, Michel Caffarel. Canadian Journal of Chemistry, 1-26, 2013 The talk : 1 Recall on SR and MR-PT2 performances and defects 2 Proposal of a strictly separable MRPT2 E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos Current works, motivations .. PhD supervisors : M. Caffarel and A. Scemama Past investment : Fixed Node Diffusion Monte Carlo (FN-DMC) In principle : exact energy but ... Needs function in input with good nodal properties My PhD : Do correlated wf based method provide better nodes ? With selected CI wf (CIPSI see later) ⇒ systematic improvment of the nodes ⇒ dramatic improvment of the FN-DMC energy ! E. G, Anthony Scemama, Michel Caffarel. Canadian Journal of Chemistry, 1-26, 2013 The talk : 1 Recall on SR and MR-PT2 performances and defects 2 Proposal of a strictly separable MRPT2 E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos Current works, motivations .. PhD supervisors : M. Caffarel and A. Scemama Past investment : Fixed Node Diffusion Monte Carlo (FN-DMC) In principle : exact energy but ... Needs function in input with good nodal properties My PhD : Do correlated wf based method provide better nodes ? With selected CI wf (CIPSI see later) ⇒ systematic improvment of the nodes ⇒ dramatic improvment of the FN-DMC energy ! E. G, Anthony Scemama, Michel Caffarel. Canadian Journal of Chemistry, 1-26, 2013 The talk : 1 Recall on SR and MR-PT2 performances and defects 2 Proposal of a strictly separable MRPT2 E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos Current works, motivations .. PhD supervisors : M. Caffarel and A. Scemama Past investment : Fixed Node Diffusion Monte Carlo (FN-DMC) In principle : exact energy but ... Needs function in input with good nodal properties My PhD : Do correlated wf based method provide better nodes ? With selected CI wf (CIPSI see later) ⇒ systematic improvment of the nodes ⇒ dramatic improvment of the FN-DMC energy ! E. G, Anthony Scemama, Michel Caffarel. Canadian Journal of Chemistry, 1-26, 2013 The talk : 1 Recall on SR and MR-PT2 performances and defects 2 Proposal of a strictly separable MRPT2 E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos Recall : Rayleigh–Schrödinger Perturbation Theory (RS-PT) Partitioning of the Hamiltonian : H = H0 + V H0|ψn >= E0 n |ψn > We distinguish the fundamental state |ψ0 > from the excited functions |α > Expression of the second order perturbative energy for the ground state : E(2) = E0 + α <α|H|ψ0>2 E0 0 −E0 α E(2) = E0 + α < α|H|ψ0 > cα Only difference : H0 ⇒ denominators E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos Recall : Single Reference Perturbation Theory (SR-PT) SR ⇔ |ψ0 >= |HF > Two main choices for H0 in Quantum Chemistry : Moller Plesset H0 (MP) Epstein Nesbet H0 (EN) Main difference : MP is mono electronic : H0(MP) = F = Norb n=1 ia† i ai E0 0 − E0 α = i + j − k − l EN is bielectronic (diagonal part of the Hamiltonian) : H0(EN) = Ndet n=1 |Dn >< Dn|H|Dn >< Dn| E0 0 − E0 α =< HF|H|HF > − < α|H|α > E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos Recall : Single Reference Perturbation Theory (SR-PT) SR ⇔ |ψ0 >= |HF > Two main choices for H0 in Quantum Chemistry : Moller Plesset H0 (MP) Epstein Nesbet H0 (EN) Main difference : MP is mono electronic : H0(MP) = F = Norb n=1 ia† i ai E0 0 − E0 α = i + j − k − l EN is bielectronic (diagonal part of the Hamiltonian) : H0(EN) = Ndet n=1 |Dn >< Dn|H|Dn >< Dn| E0 0 − E0 α =< HF|H|HF > − < α|H|α > E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos Recall : Single Reference Perturbation Theory (SR-PT) SR ⇔ |ψ0 >= |HF > Two main choices for H0 in Quantum Chemistry : Moller Plesset H0 (MP) Epstein Nesbet H0 (EN) Main difference : MP is mono electronic : H0(MP) = F = Norb n=1 ia† i ai E0 0 − E0 α = i + j − k − l EN is bielectronic (diagonal part of the Hamiltonian) : H0(EN) = Ndet n=1 |Dn >< Dn|H|Dn >< Dn| E0 0 − E0 α =< HF|H|HF > − < α|H|α > E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos Recall : Single Reference Perturbation Theory (SR-PT) SR ⇔ |ψ0 >= |HF > Two main choices for H0 in Quantum Chemistry : Moller Plesset H0 (MP) Epstein Nesbet H0 (EN) Main difference : MP is mono electronic : H0(MP) = F = Norb n=1 ia† i ai E0 0 − E0 α = i + j − k − l EN is bielectronic (diagonal part of the Hamiltonian) : H0(EN) = Ndet n=1 |Dn >< Dn|H|Dn >< Dn| E0 0 − E0 α =< HF|H|HF > − < α|H|α > E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos Recall : Single Reference Perturbation Theory (SR-PT) SR ⇔ |ψ0 >= |HF > Two main choices for H0 in Quantum Chemistry : Moller Plesset H0 (MP) Epstein Nesbet H0 (EN) Main difference : MP is mono electronic : H0(MP) = F = Norb n=1 ia† i ai E0 0 − E0 α = i + j − k − l EN is bielectronic (diagonal part of the Hamiltonian) : H0(EN) = Ndet n=1 |Dn >< Dn|H|Dn >< Dn| E0 0 − E0 α =< HF|H|HF > − < α|H|α > E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos Separability of such approaches MP : Strictly separable for closed shell systems EN : Separable with localized orbitals ⇒ Explained by J.P Malrieu and F. Spiegelman in 1979 E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos Multi Reference Perturbation Theory (MRPT) : first considerations MRPT ⇔ |ψ0 >= N i=1 ci|Di > Two different classes of theory : contracted and uncontracted perturbers... why ? Consider the action of ˆ Vijkl = a† i a† j akal on |ψ0 > ⇒ generates a set of determinants {Dα}. Two options Consider all the Dα as different perturbers Build contracted functions from the set {Dα} E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos Contracted MRPT2 : CASPT2 and NEVPT2 In both method the perturbers are contracted functions. Main differences : H0 is bielectronic for the active space in NEVPT2 No intruder states in NEVPT2 Separability is ensured in NEVPT2, not in CASPT2 E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos Example of uncontracted MRPT2 : the CIPSI algorithm Some remarks about a FCI space : Exponentially large size but very few importants reference determinants Selecting by class of excitation (CISD,CISDTQ etc ...) introduces too many non contributing determinants Why not try to select them by their energy contribution ? ⇒ Selection by perturbation Most importants determinants treated variationally, the others perturbatively Ref : B. Huron, J. P. Malrieu, and P. Rancurel J. Chem. Phys. 58, 5745 (1973) E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos Example of uncontracted MRPT2 : the CIPSI algorithm Some remarks about a FCI space : Exponentially large size but very few importants reference determinants Selecting by class of excitation (CISD,CISDTQ etc ...) introduces too many non contributing determinants Why not try to select them by their energy contribution ? ⇒ Selection by perturbation Most importants determinants treated variationally, the others perturbatively Ref : B. Huron, J. P. Malrieu, and P. Rancurel J. Chem. Phys. 58, 5745 (1973) E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos Example of uncontracted MRPT2 : the CIPSI algorithm Some remarks about a FCI space : Exponentially large size but very few importants reference determinants Selecting by class of excitation (CISD,CISDTQ etc ...) introduces too many non contributing determinants Why not try to select them by their energy contribution ? ⇒ Selection by perturbation Most importants determinants treated variationally, the others perturbatively Ref : B. Huron, J. P. Malrieu, and P. Rancurel J. Chem. Phys. 58, 5745 (1973) E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos Example of uncontracted MRPT2 : the CIPSI algorithm Some remarks about a FCI space : Exponentially large size but very few importants reference determinants Selecting by class of excitation (CISD,CISDTQ etc ...) introduces too many non contributing determinants Why not try to select them by their energy contribution ? ⇒ Selection by perturbation Most importants determinants treated variationally, the others perturbatively Ref : B. Huron, J. P. Malrieu, and P. Rancurel J. Chem. Phys. 58, 5745 (1973) E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos Example of uncontracted MRPT2 : the CIPSI algorithm Some remarks about a FCI space : Exponentially large size but very few importants reference determinants Selecting by class of excitation (CISD,CISDTQ etc ...) introduces too many non contributing determinants Why not try to select them by their energy contribution ? ⇒ Selection by perturbation Most importants determinants treated variationally, the others perturbatively Ref : B. Huron, J. P. Malrieu, and P. Rancurel J. Chem. Phys. 58, 5745 (1973) E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos Example of uncontracted MRPT2 : the CIPSI algorithm Some remarks about a FCI space : Exponentially large size but very few importants reference determinants Selecting by class of excitation (CISD,CISDTQ etc ...) introduces too many non contributing determinants Why not try to select them by their energy contribution ? ⇒ Selection by perturbation Most importants determinants treated variationally, the others perturbatively Ref : B. Huron, J. P. Malrieu, and P. Rancurel J. Chem. Phys. 58, 5745 (1973) E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos The CIPSI energy CIPSI calculation : general selected CI wf (no particular class is privileged) Two energetic contribution : Variational energy : EVar = <ψ0|H|ψ0> <ψ0|ψ0> The perturbative energy EPT2 = Dα 2 E0 0 −E0 α Note that here the Dα are treated separately ⇔ externally uncontracted MRPT2 E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos The CIPSI perturbative energy As for SRPT, two main choices for H0 : One body : MP with an averaged Fock operator E0 0 − E0 Dα = ¯i + ¯j − ¯ k − ¯l Two body : EN with the Diagonal part of H E0 0 − E0 Dα =< ψ0|H|ψ0 > − < Dα|H|Dα > In all cases not strictly separable properties. Unbalanced treatment between |Dα > and the |ψ0 > ⇒ E0 0 − E0 Dα too large ⇒ not strictly separable We implemented the EN. Bad formal properties ... But good numerical aspects ! E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos Numerical results : CIPSI calculations on the Oxygen ground state in cc-pVXZ -75.05 -75 -74.95 -74.9 -74.85 -74.8 -74.75 1 10 100 1000 10000 100000 Energy/Hartree Number of determinants in the CIPSI wf E Var DZ E(CIPSI) DZ E FCI DZ E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos Numerical results : CIPSI calculations on the Oxygen ground state in cc-pVXZ -75.05 -75 -74.95 -74.9 -74.85 -74.8 -74.75 1 10 100 1000 10000 100000 Energy/Hartree Number of determinants in the CIPSI wf E Var DZ E(CIPSI) DZ E FCI DZ E var TZ E(CIPSI) TZ E FCI TZ E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos Numerical results : CIPSI calculations on the Oxygen ground state in cc-pVXZ -75.05 -75 -74.95 -74.9 -74.85 -74.8 -74.75 1 10 100 1000 10000 100000 Energy/Hartree Number of determinants in the CIPSI wf E Var DZ E(CIPSI) DZ E FCI DZ E var TZ E(CIPSI) TZ E FCI TZ E var QZ E(CIPSI) QZ E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos Numerical results : CIPSI calculations on the F2 breaking bond -199.12 -199.1 -199.08 -199.06 -199.04 -199.02 -199 -198.98 -198.96 1 1.5 2 2.5 3 3.5 4 Energy (Hartree) Inter atomic distance (Angstrom) Convergence of the CIPSI energy 500 dets 1k dets 5k dets 10k dets 50k dets 2*F FCI E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos CIPSI summary CIPSI advantages : Converges fast to the FCI wf Automatic construction CIPSI disadvantages : Quality not uniform as a function of the geometry (i.e MR situtation) Energy not size consistant The greater is your wf, the more expensive is your EPT2 We want to have a good MRPT ! E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos HMZ proposition main idea HMZ = J.L. Heully,J.P. Malrieu and A. Zaitsevski When you have a CASSCF wf : |I >∈ CASSCF has been stabilized by interactions in the CASSCF T† IJ |I >= |J >⇒< I|H|J > Most of these interactions can be repeated to the |Dα > T† IJ |Dα >= |Dβ >⇒< Dα|H|Dβ > The interactions in the active space are most the same in the outer space ! < I|H|J >=< Dα|H|Dβ > in most cases Why not repeating the interactions to |Dα > ? Correct the unbalanced treatment between |ψ0 > and |Dα > ⇒ taking into account a part of the |Dα >, |Dβ > interactions ! E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos HMZ proposition main idea HMZ = J.L. Heully,J.P. Malrieu and A. Zaitsevski When you have a CASSCF wf : |I >∈ CASSCF has been stabilized by interactions in the CASSCF T† IJ |I >= |J >⇒< I|H|J > Most of these interactions can be repeated to the |Dα > T† IJ |Dα >= |Dβ >⇒< Dα|H|Dβ > The interactions in the active space are most the same in the outer space ! < I|H|J >=< Dα|H|Dβ > in most cases Why not repeating the interactions to |Dα > ? Correct the unbalanced treatment between |ψ0 > and |Dα > ⇒ taking into account a part of the |Dα >, |Dβ > interactions ! E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos HMZ proposition main idea HMZ = J.L. Heully,J.P. Malrieu and A. Zaitsevski When you have a CASSCF wf : |I >∈ CASSCF has been stabilized by interactions in the CASSCF T† IJ |I >= |J >⇒< I|H|J > Most of these interactions can be repeated to the |Dα > T† IJ |Dα >= |Dβ >⇒< Dα|H|Dβ > The interactions in the active space are most the same in the outer space ! < I|H|J >=< Dα|H|Dβ > in most cases Why not repeating the interactions to |Dα > ? Correct the unbalanced treatment between |ψ0 > and |Dα > ⇒ taking into account a part of the |Dα >, |Dβ > interactions ! E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos HMZ proposition main idea HMZ = J.L. Heully,J.P. Malrieu and A. Zaitsevski When you have a CASSCF wf : |I >∈ CASSCF has been stabilized by interactions in the CASSCF T† IJ |I >= |J >⇒< I|H|J > Most of these interactions can be repeated to the |Dα > T† IJ |Dα >= |Dβ >⇒< Dα|H|Dβ > The interactions in the active space are most the same in the outer space ! < I|H|J >=< Dα|H|Dβ > in most cases Why not repeating the interactions to |Dα > ? Correct the unbalanced treatment between |ψ0 > and |Dα > ⇒ taking into account a part of the |Dα >, |Dβ > interactions ! E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos HMZ proposition main idea HMZ = J.L. Heully,J.P. Malrieu and A. Zaitsevski When you have a CASSCF wf : |I >∈ CASSCF has been stabilized by interactions in the CASSCF T† IJ |I >= |J >⇒< I|H|J > Most of these interactions can be repeated to the |Dα > T† IJ |Dα >= |Dβ >⇒< Dα|H|Dβ > The interactions in the active space are most the same in the outer space ! < I|H|J >=< Dα|H|Dβ > in most cases Why not repeating the interactions to |Dα > ? Correct the unbalanced treatment between |ψ0 > and |Dα > ⇒ taking into account a part of the |Dα >, |Dβ > interactions ! E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos HMZ proposition main idea HMZ = J.L. Heully,J.P. Malrieu and A. Zaitsevski When you have a CASSCF wf : |I >∈ CASSCF has been stabilized by interactions in the CASSCF T† IJ |I >= |J >⇒< I|H|J > Most of these interactions can be repeated to the |Dα > T† IJ |Dα >= |Dβ >⇒< Dα|H|Dβ > The interactions in the active space are most the same in the outer space ! < I|H|J >=< Dα|H|Dβ > in most cases Why not repeating the interactions to |Dα > ? Correct the unbalanced treatment between |ψ0 > and |Dα > ⇒ taking into account a part of the |Dα >, |Dβ > interactions ! E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos The HMZ proposition Intermediate Effective Hamiltonian theory (IEHT) key idea. ∆IJ = Dα HIαHJα E0 0 −E0 α Reference wf is a CASSCF wf ⇒ add a term in the denominator to ensure separability. ∆HMZ IJ = Dα HIαHJα E0 0 −(E0 α +BIα) BIα = 1 cI J∈M T† IJ |Dα>=0 HIJcJ E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos The HMZ proposition Intermediate Effective Hamiltonian theory (IEHT) key idea. ∆IJ = Dα HIαHJα E0 0 −E0 α Reference wf is a CASSCF wf ⇒ add a term in the denominator to ensure separability. ∆HMZ IJ = Dα HIαHJα E0 0 −(E0 α +BIα) BIα = 1 cI J∈M T† IJ |Dα>=0 HIJcJ E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos The HMZ proposition Intermediate Effective Hamiltonian theory (IEHT) key idea. ∆IJ = Dα HIαHJα E0 0 −E0 α Reference wf is a CASSCF wf ⇒ add a term in the denominator to ensure separability. ∆HMZ IJ = Dα HIαHJα E0 0 −(E0 α +BIα) BIα = 1 cI J∈M T† IJ |Dα>=0 HIJcJ E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos The HMZ proposition Intermediate Effective Hamiltonian theory (IEHT) key idea. ∆IJ = Dα HIαHJα E0 0 −E0 α Reference wf is a CASSCF wf ⇒ add a term in the denominator to ensure separability. ∆HMZ IJ = Dα HIαHJα E0 0 −(E0 α +BIα) BIα = 1 cI J∈M T† IJ |Dα>=0 HIJcJ E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos HMZ main idea Repeat the excitations of the active space in the outer space |I > has been stabilized by interactions with |J > : T† IJ BIα = repeating all these excitations to |Dα > ⇒ takes into account the interactions between |Dα > and the |Dβ >= T† IJ |Dα > ⇒ takes into account a piece of the 2nd order on the wave function Ex : |Dα >= pure inactive double excitation ⇒ strictly separable ! E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos HMZ main idea Repeat the excitations of the active space in the outer space |I > has been stabilized by interactions with |J > : T† IJ BIα = repeating all these excitations to |Dα > ⇒ takes into account the interactions between |Dα > and the |Dβ >= T† IJ |Dα > ⇒ takes into account a piece of the 2nd order on the wave function Ex : |Dα >= pure inactive double excitation ⇒ strictly separable ! E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos HMZ main idea Repeat the excitations of the active space in the outer space |I > has been stabilized by interactions with |J > : T† IJ BIα = repeating all these excitations to |Dα > ⇒ takes into account the interactions between |Dα > and the |Dβ >= T† IJ |Dα > ⇒ takes into account a piece of the 2nd order on the wave function Ex : |Dα >= pure inactive double excitation ⇒ strictly separable ! E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos HMZ main idea Repeat the excitations of the active space in the outer space |I > has been stabilized by interactions with |J > : T† IJ BIα = repeating all these excitations to |Dα > ⇒ takes into account the interactions between |Dα > and the |Dβ >= T† IJ |Dα > ⇒ takes into account a piece of the 2nd order on the wave function Ex : |Dα >= pure inactive double excitation ⇒ strictly separable ! E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos HMZ main idea Repeat the excitations of the active space in the outer space |I > has been stabilized by interactions with |J > : T† IJ BIα = repeating all these excitations to |Dα > ⇒ takes into account the interactions between |Dα > and the |Dβ >= T† IJ |Dα > ⇒ takes into account a piece of the 2nd order on the wave function Ex : |Dα >= pure inactive double excitation ⇒ strictly separable ! E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos HMZ as a MRPT2 HMZ historically thought for IEHT with a MP H0 We decided to use this idea to make a MRPT2 EN zeroth order Hamiltonian Work with local orbitals New expression for the cα : cα = I cI E0−(Hαα+BIα) BIα = 1 cI J∈M T† IJ |Dα>=0 HIJcJ EPT2 = α < Dα|H|ψ0 > cα E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos HMZ as a MRPT2 HMZ historically thought for IEHT with a MP H0 We decided to use this idea to make a MRPT2 EN zeroth order Hamiltonian Work with local orbitals New expression for the cα : cα = I cI E0−(Hαα+BIα) BIα = 1 cI J∈M T† IJ |Dα>=0 HIJcJ EPT2 = α < Dα|H|ψ0 > cα E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos HMZ as a MRPT2 HMZ historically thought for IEHT with a MP H0 We decided to use this idea to make a MRPT2 EN zeroth order Hamiltonian Work with local orbitals New expression for the cα : cα = I cI E0−(Hαα+BIα) BIα = 1 cI J∈M T† IJ |Dα>=0 HIJcJ EPT2 = α < Dα|H|ψ0 > cα E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos The HMZ PT2 : main ideas To resume the HMZ PT2 key ideas : MR in the construction of cα cα = I cItIα A bit of 2nd order in |ψ > is taken Hαα + BIα SR in the determination of the BIα E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos Comparison with EN -199.13 -199.125 -199.12 -199.115 -199.11 -199.105 1.4 1.6 1.8 2 2.2 2.4 2.6 Energy (Hartree) Inter Fluorine distance (Angstrom) Study of the Separability of the F2+He with EN EN (F2+He) EN F2 + EN He E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos Our HMZ-PT2 implementation : first numerical results of separability -199.16 -199.15 -199.14 -199.13 -199.12 -199.11 1.2 1.4 1.6 1.8 2 2.2 Energy (Hartree) Inter fluorine distance (Angstrom) Study of the Separability of the F2+He with HMZ PT2 (F2+He) HMZ PT2 F2 HMZ PT2 + He HMZ PT2 E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos Our HMZ-PT2 implementation : first numerical results -199.2 -199.15 -199.1 -199.05 -199 -198.95 -198.9 -198.85 -198.8 -198.75 1 1.5 2 2.5 3 3.5 4 Energy (Hartree) Inter atomic distance (Angstrom) HMZ PT2 vs VP PT2, comparaison to near FCI PES (cc-pVDZ) VP PT2 HMZ PT2 Estimated FCI 2* F(HMZ PT2) E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos Method R ∆E(mH) k CAS 1.531 22.1 0.430 CAS+EN 1.529 18.7 0.482 CAS+HMZ 1.412 55.8 1.006 Estimated FCI 1.461 45.3 0.773 Exact NR 1.412 62.0 1.121 TABLE: Basis cc-pVDZ : F2 spectroscopic properties for CASSCF, CASSCF+EN PT2, CASSCF+HMZ PT2 and estimated FCI (CIPSI with 50k determiannts) E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos Our ms degenerated MRPT2 Problem with HMZ and EN : Don’t give S2 eigenfunctions .. By taking restrictions regarding the exchange integrals on the BIα and H0 ⇒ We get a HMZ+S2 like PT2 that gives eigenfunctions of S2 Ex : the quartet fundamental of N Method ms = 3 2 ms = 1 2 E1 2 − E3 2 EN -0.09837593 -0.10280613 0.00443 HMZ -0.09920624 -0.10280613 0.00360 HMZ+S2 -0.100864901592813 -0.100864901592843 3.010−14 E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos Our ms degenerated MRPT2 Problem with HMZ and EN : Don’t give S2 eigenfunctions .. By taking restrictions regarding the exchange integrals on the BIα and H0 ⇒ We get a HMZ+S2 like PT2 that gives eigenfunctions of S2 Ex : the quartet fundamental of N Method ms = 3 2 ms = 1 2 E1 2 − E3 2 EN -0.09837593 -0.10280613 0.00443 HMZ -0.09920624 -0.10280613 0.00360 HMZ+S2 -0.100864901592813 -0.100864901592843 3.010−14 E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos Our ms degenerated MRPT2 Problem with HMZ and EN : Don’t give S2 eigenfunctions .. By taking restrictions regarding the exchange integrals on the BIα and H0 ⇒ We get a HMZ+S2 like PT2 that gives eigenfunctions of S2 Ex : the quartet fundamental of N Method ms = 3 2 ms = 1 2 E1 2 − E3 2 EN -0.09837593 -0.10280613 0.00443 HMZ -0.09920624 -0.10280613 0.00360 HMZ+S2 -0.100864901592813 -0.100864901592843 3.010−14 E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos Conclusion and perspectives We show that : Our HMZ MRPT2 is strictly separable for open shell system Modifying the H0 leads to an ms invariant version Work in development : Need more test ! (N2 for instance) Think about optimization (split the excitation class) Future work : Dress the Hamiltonian matrix with the HMZ PT2 (internally uncontracted) ⇒ change the coefficients of the wf ⇒ good for the nodes and FN-DMC Generalization to an MR in a CAS is possible ⇒ Avoid the full CASSCF wf for the MRPT2 E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos Thank you for your attention E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos Dressed Hamiltonian : key idea If the FCI was known, writing the eigenvalue equation for a given |DI > : (HII − E0)cI + J∈M CJHIJ + α∈O CαHIα = 0 ⇔ (HII + 1 cI α∈O CαHIα − E0)cI + J∈M CJHIJ = 0 ⇔ (HII + ∆II − E0)cI + J∈M CJHIJ = 0 ∆II : dressing of HII by the effect of O : ∆II = 1 cI α∈O CαHIα Very general : can also define some ∆IJ, ∆JI ⇒ must guess the cα Ex : Coupled Cluster, SC2 ... E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos The CIPSI-like algorithm (1) Starting with : |ψ(0) >= N(0) i=1 ci|Di > do k = 0, Niterations 1 look at all the determinants |Dα > for which < Dα|H|ψ(k) >= 0 (connected) 2 for each |Dα > compute the perturbative energy contribution α 3 Sort all the |Dα > by energy contribution 4 Select the n most important ones 5 Diagonalize H in the new set of determinants : N(k+1) = N(k) + n 6 you have a new reference wf : |ψ(k+1) >= N(k+1) i=1 ci|Di > end do E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos Numerical result : the Oxygen ground state in cc-pVDZ basis set -74.94 -74.92 -74.9 -74.88 -74.86 -74.84 -74.82 -74.8 -74.78 0 200 400 600 800 1000 1200 1400 1600 1800 2000 Energy/Hartree Number of determinants Convergence of the Energy for the oxygen atom (cc-pVDZ) E(Variational) E(CIPSI) E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos The HMZ derivation (1) HMZ key idea : introduction of BIα How to get to the formulae ? |α > has several parents : cα|α >= I cIαT† Iα |I > Each parents contributes ∝ its coefficient : cIα = cItIα The tIα is given by the RS PT : tIα = HIα E0−Hαα The |α > interacts with all the |β >= T† Iα |J > This interaction dresses the Hαα : Hαα = Hαα + ∆I αα ∆I αα ≡ 1 cα β,T† Iα |J>=0 Hαβcβ The term ∆I αα looks like the BIα from HMZ ... E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos The HMZ derivation (2) ∆I αα ≡ 1 cα β,T† Iα |J>=0 Hαβcβ cβ cα = J tJβcJ I tIαcI Approximation for cα, cβ in ∆I αα : unique parents |I > is locally the lone parent of |α > |α >= T† Iα |I >, cα = tIαcI |β >= T† Iα |J >= T† Jβ |J > has locally only |J > for parent cβ = tJβcJ because T† Iα = T† Jβ , so are the amplitudes : tIα = tJβ, ⇒ cβ = tIαcJ So that : cβ cα = tIαcJ tIαcI = cJ cI E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos The HMZ derivation (3) Then, what about the Hαβ ? Most are already known ! |α >= T† Iα |I > and |β >= T† Iα |J > |J >= T† IJ |I > ⇒|I > and |J > differ by the same electrons/orbitals than |α > and |β > If T† IJ is a double excitation then Hαβ = HIJ ⇒ assume the same thing for the T† IJ that are single excitations ! ⇒∆I αα = 1 cI J,T† Iα |J>=0 HIJcJ = BIα E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos Our HMZ-PT2 implementation : first numerical results -199.2 -199.1 -199 -198.9 -198.8 -198.7 -198.6 -198.5 -198.4 1 1.5 2 2.5 3 3.5 4 Energy (Hartree) Inter atomic distance (Angstrom) HMZ PT2 vs VP PT2, comparaison to near FCI PES (cc-pVDZ) VP PT2 HMZ PT2 CASSCF Estimated FCI E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos Our HMZ-PT2 implementation : Analysis by excitation class -199 -198.95 -198.9 -198.85 -198.8 -198.75 -198.7 -198.65 -198.6 -198.55 -198.5 1 1.5 2 2.5 3 3.5 4 Energy (Hartree) Inter atomic distance (Angstrom) HMZ and VP treatments of 2h,2p and 1h,1p excitations in F2 HMZ 2h 2p VP 2h 2p HMZ 1h 1p VP 1h 1p E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos HMZ Example |Dα > = pure inactive double excitation on a |DI > in a CASSCF wave function |Dα > has only one parent |I > All the excitations T† IJ are repeatable on |Dα > BIα = E0 0 − E0 I = correlation energy for |DI > Denominator becomes E0 I − E0 α Real energy of excitation Strictly separable cα = cI HIα HII−Hαα E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos HMZ Example |Dα > = pure inactive double excitation on a |DI > in a CASSCF wave function |Dα > has only one parent |I > All the excitations T† IJ are repeatable on |Dα > BIα = E0 0 − E0 I = correlation energy for |DI > Denominator becomes E0 I − E0 α Real energy of excitation Strictly separable cα = cI HIα HII−Hαα E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos HMZ Example |Dα > = pure inactive double excitation on a |DI > in a CASSCF wave function |Dα > has only one parent |I > All the excitations T† IJ are repeatable on |Dα > BIα = E0 0 − E0 I = correlation energy for |DI > Denominator becomes E0 I − E0 α Real energy of excitation Strictly separable cα = cI HIα HII−Hαα E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos HMZ Example |Dα > = pure inactive double excitation on a |DI > in a CASSCF wave function |Dα > has only one parent |I > All the excitations T† IJ are repeatable on |Dα > BIα = E0 0 − E0 I = correlation energy for |DI > Denominator becomes E0 I − E0 α Real energy of excitation Strictly separable cα = cI HIα HII−Hαα E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos HMZ Example |Dα > = pure inactive double excitation on a |DI > in a CASSCF wave function |Dα > has only one parent |I > All the excitations T† IJ are repeatable on |Dα > BIα = E0 0 − E0 I = correlation energy for |DI > Denominator becomes E0 I − E0 α Real energy of excitation Strictly separable cα = cI HIα HII−Hαα E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos HMZ Example |Dα > = pure inactive double excitation on a |DI > in a CASSCF wave function |Dα > has only one parent |I > All the excitations T† IJ are repeatable on |Dα > BIα = E0 0 − E0 I = correlation energy for |DI > Denominator becomes E0 I − E0 α Real energy of excitation Strictly separable cα = cI HIα HII−Hαα E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos HMZ Example |Dα > = pure inactive double excitation on a |DI > in a CASSCF wave function |Dα > has only one parent |I > All the excitations T† IJ are repeatable on |Dα > BIα = E0 0 − E0 I = correlation energy for |DI > Denominator becomes E0 I − E0 α Real energy of excitation Strictly separable cα = cI HIα HII−Hαα E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos HMZ Example |Dα > = pure inactive double excitation on a |DI > in a CASSCF wave function |Dα > has only one parent |I > All the excitations T† IJ are repeatable on |Dα > BIα = E0 0 − E0 I = correlation energy for |DI > Denominator becomes E0 I − E0 α Real energy of excitation Strictly separable cα = cI HIα HII−Hαα E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos F2 Triplet : HMZ for ms = 1, ms = 0 -199.2 -199.1 -199 -198.9 -198.8 -198.7 -198.6 -198.5 -198.4 -198.3 -198.2 -198.1 1 2 3 4 5 6 7 8 Energy (Hartree) Inter atomic distance (Angstrom) F2 triplet HMZ HMZ Sz=1 HMZ Sz=0 E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos F2 Triplet : HMZ+S2 for ms = 1, ms = 0 -199.12 -199.1 -199.08 -199.06 -199.04 -199.02 -199 -198.98 -198.96 -198.94 1.5 2 2.5 3 3.5 4 Energy (Hartree) Inter atomic distance (Angstrom) F2 triplet HMZ + S2 HMZ Sz=1 HMZ Sz=0 E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos Our mz degenerated MRPT2 : triplet fo F2 1e-13 1e-12 1e-11 1e-10 1e-09 1e-08 1e-07 1e-06 1e-05 0.0001 0.001 0.01 1.5 2 2.5 3 3.5 4 Energy (Hartree) Inter atomic distance Difference between Sz=0 and Sz=1 PT2 calculations for HMZ and HMZ+S2 perturbations HMZ (Sz=0 - Sz=1) HMZ + S2 (Sz=0 - Sz=1) E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos Separability of such approaches MP (with canonical orbitals) : Strictly separable for closed shell systems 2 ×MP2 for Ne = -0.371046562 Hartree MP2 for Ne2 (32 Angstrom)= -0.371046562 Hartree EN : Depends on the locality of the orbitals ⇒ Explained by J.P Malrieu and F. Spiegelman in 1979 2 ×EN for Ne = -0.4641138 Hartree EN for Ne2 (delocalized orbitals) = -0.4156731 Hartree EN for Ne2 (localized orbitals) = -0.4641138 Hartree EN is potentially separable for closed shell system ! E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos Separability of such approaches MP (with canonical orbitals) : Strictly separable for closed shell systems 2 ×MP2 for Ne = -0.371046562 Hartree MP2 for Ne2 (32 Angstrom)= -0.371046562 Hartree EN : Depends on the locality of the orbitals ⇒ Explained by J.P Malrieu and F. Spiegelman in 1979 2 ×EN for Ne = -0.4641138 Hartree EN for Ne2 (delocalized orbitals) = -0.4156731 Hartree EN for Ne2 (localized orbitals) = -0.4641138 Hartree EN is potentially separable for closed shell system ! E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos Separability of such approaches MP (with canonical orbitals) : Strictly separable for closed shell systems 2 ×MP2 for Ne = -0.371046562 Hartree MP2 for Ne2 (32 Angstrom)= -0.371046562 Hartree EN : Depends on the locality of the orbitals ⇒ Explained by J.P Malrieu and F. Spiegelman in 1979 2 ×EN for Ne = -0.4641138 Hartree EN for Ne2 (delocalized orbitals) = -0.4156731 Hartree EN for Ne2 (localized orbitals) = -0.4641138 Hartree EN is potentially separable for closed shell system ! E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos Separability of such approaches MP (with canonical orbitals) : Strictly separable for closed shell systems 2 ×MP2 for Ne = -0.371046562 Hartree MP2 for Ne2 (32 Angstrom)= -0.371046562 Hartree EN : Depends on the locality of the orbitals ⇒ Explained by J.P Malrieu and F. Spiegelman in 1979 2 ×EN for Ne = -0.4641138 Hartree EN for Ne2 (delocalized orbitals) = -0.4156731 Hartree EN for Ne2 (localized orbitals) = -0.4641138 Hartree EN is potentially separable for closed shell system ! E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos Separability of such approaches MP (with canonical orbitals) : Strictly separable for closed shell systems 2 ×MP2 for Ne = -0.371046562 Hartree MP2 for Ne2 (32 Angstrom)= -0.371046562 Hartree EN : Depends on the locality of the orbitals ⇒ Explained by J.P Malrieu and F. Spiegelman in 1979 2 ×EN for Ne = -0.4641138 Hartree EN for Ne2 (delocalized orbitals) = -0.4156731 Hartree EN for Ne2 (localized orbitals) = -0.4641138 Hartree EN is potentially separable for closed shell system ! E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos Separability of such approaches MP (with canonical orbitals) : Strictly separable for closed shell systems 2 ×MP2 for Ne = -0.371046562 Hartree MP2 for Ne2 (32 Angstrom)= -0.371046562 Hartree EN : Depends on the locality of the orbitals ⇒ Explained by J.P Malrieu and F. Spiegelman in 1979 2 ×EN for Ne = -0.4641138 Hartree EN for Ne2 (delocalized orbitals) = -0.4156731 Hartree EN for Ne2 (localized orbitals) = -0.4641138 Hartree EN is potentially separable for closed shell system ! E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos Separability of such approaches MP (with canonical orbitals) : Strictly separable for closed shell systems 2 ×MP2 for Ne = -0.371046562 Hartree MP2 for Ne2 (32 Angstrom)= -0.371046562 Hartree EN : Depends on the locality of the orbitals ⇒ Explained by J.P Malrieu and F. Spiegelman in 1979 2 ×EN for Ne = -0.4641138 Hartree EN for Ne2 (delocalized orbitals) = -0.4156731 Hartree EN for Ne2 (localized orbitals) = -0.4641138 Hartree EN is potentially separable for closed shell system ! E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos Separability of such approaches MP (with canonical orbitals) : Strictly separable for closed shell systems 2 ×MP2 for Ne = -0.371046562 Hartree MP2 for Ne2 (32 Angstrom)= -0.371046562 Hartree EN : Depends on the locality of the orbitals ⇒ Explained by J.P Malrieu and F. Spiegelman in 1979 2 ×EN for Ne = -0.4641138 Hartree EN for Ne2 (delocalized orbitals) = -0.4156731 Hartree EN for Ne2 (localized orbitals) = -0.4641138 Hartree EN is potentially separable for closed shell system ! E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2

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logos Separability of such approaches MP (with canonical orbitals) : Strictly separable for closed shell systems 2 ×MP2 for Ne = -0.371046562 Hartree MP2 for Ne2 (32 Angstrom)= -0.371046562 Hartree EN : Depends on the locality of the orbitals ⇒ Explained by J.P Malrieu and F. Spiegelman in 1979 2 ×EN for Ne = -0.4641138 Hartree EN for Ne2 (delocalized orbitals) = -0.4156731 Hartree EN for Ne2 (localized orbitals) = -0.4641138 Hartree EN is potentially separable for closed shell system ! E. Giner, Jean Paul Malrieu Determinant based Separable MRPT2