Excited States Variational Principles and Quantum Monte Carlo

Excited States, Variational Principles, and Quantum Monte Carlo Eric Neuscamman June 13, 2018

motivation charge transfer core spectroscopy double excitations

the menu aperitif: fun with QMC main course: excited states with QMC dessert: mean field theory

real-space Gutzwiller energy +12 (Hartrees) -0. 6 RHF -0. 8 -1. 0 -1. 2 -1. 4 MRCI+Q -1. 6 -1. 8 1. 0 1. 5 2. 0 2. 5 3. 0 C-C distance (Angstroms) Van Der Goetz & Neuscamman, JCTC 2017 3. 5 4. 0 Van Der Goetz & Neuscamman, unpublished

real-space Gutzwiller energy +12 (Hartrees) -0. 6 RHF -0. 8 -1. 0 +Jastrows -1. 2 -1. 4 MRCI+Q -1. 6 -1. 8 1. 0 1. 5 2. 0 2. 5 3. 0 C-C distance (Angstroms) Van Der Goetz & Neuscamman, JCTC 2017 3. 5 4. 0 Van Der Goetz & Neuscamman, unpublished

real-space Gutzwiller energy +12 (Hartrees) -0. 6 RHF -0. 8 -1. 0 +Jastrows -1. 2 -1. 4 +DMC -1. 6 -1. 8 1. 0 1. 5 2. 0 2. 5 3. 0 C-C distance (Angstroms) Van Der Goetz & Neuscamman, JCTC 2017 3. 5 MRCI+Q 4. 0 Van Der Goetz & Neuscamman, unpublished

the menu aperitif: fun with QMC main course: excited states with QMC dessert: mean field theory

excited state variational principles energy not stationary not unique size consistency challenges global minimum: Choi, Lebeda, Messmer, CPL 1970 Van Voorhis et al, JCP 2017

excited state variational principles size consistency challenges energy not stationary not unique global minimum: Choi, Lebeda, Messmer, CPL 1970 Van Voorhis et al, JCP 2017 QMC Zhao & Neuscamman, ar. Xiv 1804. 09663 Shea & Neuscamman, JCTC 2017 Blunt & Neuscamman, JCP 2017 Robinson, Flores, Neuscamman, JCP 2017 Neuscamman, JCP 2016 Zhao & Neuscamman, JCTC 2016, 2017

fairness in thioformaldehyde H C S : H + Jastrow + state-specific orb. opt. excited state VMC ex p . d e g r 2 D e T S v C cc-p. VDZ n I+P C o c IPS M n cc-p. VTZ O u C E aug-cc-p. VTZ 2. 0 2. 1 2. 2 2. 3 Judge & King, Can J Phys 1975 2. 4 CASPT 2/MRCI Pineda Flores & Neuscamman, in preparation

? #$%!? ^*#? @ in cc-p. VDZ 3. 8 e. V state-averaged CASSCF: 2. 7 e. V to 5. 9 e. V (yikes!) state-specific CASSCF: 4. 4 e. V [ ] state-specific CASPT 2: 3. 5 e. V Pineda Flores & Neuscamman, in preparation

optical gaps 14 Li. F VMC gap (e. V) 12 10 8 6 Zn O 4 Si 2 0 Li. H C 0 2 4 6 8 Experimental gap (e. V) 10 12 14 Zhao & Neuscamman, ar. Xiv 1804. 09663

Zinc Oxide 6. 0 G 0 W 0 HF 5. 5 5. 0 Gap (e. V) 4. 5 ent Experim 4. 0 3. 5 3. 0 2. 5 2. 0 E 0 G 0 W 0 PB E G W 0 HS 0 E G 0 W 0 PB Bechstedt et al, PRB 2009 Pulci et al, PRB 2010 Kresse et al, PRB 2007 (2 papers) Zhao & Neuscamman, ar. Xiv 1804. 09663

Zinc Oxide 6. 0 G 0 W 0 HF 5. 5 5. 0 Gap (e. V) 4. 5 ent Experim GW HSE 4. 0 3. 5 3. 0 2. 5 2. 0 E 0 W PBE G G 0 W 0 PB E G W 0 HS 0 E G 0 W 0 PB Bechstedt et al, PRB 2009 Pulci et al, PRB 2010 Kresse et al, PRB 2007 (2 papers) Zhao & Neuscamman, ar. Xiv 1804. 09663

Zinc Oxide 6. 0 G 0 W 0 HF 5. 5 Gap (e. V) VMC-PBE 0 5. 0 4. 5 VMC-LDA A VMC LD ent Experim GW HSE 4. 0 VMC PBE 0 3. 5 E 0 W PBE G G 0 W 0 PB E G W 0 HS 3. 0 2. 5 2. 0 0 E G 0 W 0 PB VMC-LDA for Li. F Bechstedt et al, PRB 2009 Pulci et al, PRB 2010 Kresse et al, PRB 2007 (2 papers) Zhao & Neuscamman, ar. Xiv 1804. 09663

the menu aperitif: fun with QMC main course: excited states with QMC dessert: mean field theory

the quantum chemistry cookbook FCI-QMC Hartree Fock MP 2 CCSD ground state mean field theory

the quantum chemistry cookbook varational principle minimally correlated ansatz Hartree Fock stationary energy size consistent Fock-build cost orbitals consistent with their own mean field

ingredients varational principle minimally correlated ansatz Hartree Fock Lagrangian stationary energy size consistent Fock-build cost orbitals consistent with their own mean field

pie in the sky varational principle minimally correlated ansatz MP 2 Lagrangian CCSD a mean field platform for ground excited states Shea & Neuscamman, ar. Xiv 1806. 00853

the ansatz mean-field: minimal correlation, relaxed orbitals Shea & Neuscamman, ar. Xiv 1806. 00853

behavioral economics selects state guarantees stationary energy Shea & Neuscamman, ar. Xiv 1806. 00853

automatic differentiation repeat one more time… Fock build cost dot product Tensor. Flow Shea & Neuscamman, ar. Xiv 1806. 00853

water singlet excitations, cc-p. VDZ error vs EOM-CCSD (e. V) 1. 25 1. 00 0. 75 CIS 0. 50 0. 25 0. 00 -0. 25 -0. 50 -0. 75 -1. 00 CIS ESMF CIS(D)Shea & Neuscamman, ar. Xiv 1806. 00853 ESMP 2

water singlet excitations, cc-p. VDZ error vs EOM-CCSD (e. V) 1. 25 1. 00 0. 75 CIS 0. 50 0. 25 0. 00 -0. 25 -0. 50 -0. 75 -1. 00 ESMF CIS(D)Shea & Neuscamman, ar. Xiv 1806. 00853 ESMP 2

water singlet excitations, cc-p. VDZ error vs EOM-CCSD (e. V) 1. 25 1. 00 0. 75 CIS 0. 50 0. 25 0. 00 -0. 25 -0. 50 -0. 75 -1. 00 CIS(D) ESMF CIS(D)Shea & Neuscamman, ar. Xiv 1806. 00853 ESMP 2

water singlet excitations, cc-p. VDZ 1. 00 MP 0. 75 ES CIS 2 error vs EOM-CCSD (e. V) 1. 25 0. 50 0. 25 0. 00 -0. 25 -0. 50 -0. 75 -1. 00 CIS(D) ESMF CIS(D)Shea & Neuscamman, ar. Xiv 1806. 00853 ESMP 2

CH 2 O singlet excitations, cc-p. VDZ error vs EOM-CCSD (e. V) 2. 0 1. 5 CIS 1. 0 0. 5 0. 0 -0. 5 -1. 0 -1. 5 CIS ESMF CIS(D)Shea & Neuscamman, ar. Xiv 1806. 00853 ESMP 2

CH 2 O singlet excitations, cc-p. VDZ error vs EOM-CCSD (e. V) 2. 0 1. 5 CIS 1. 0 0. 5 0. 0 -0. 5 -1. 0 -1. 5 ESMF CIS(D)Shea & Neuscamman, ar. Xiv 1806. 00853 ESMP 2

CH 2 O singlet excitations, cc-p. VDZ error vs EOM-CCSD (e. V) 2. 0 1. 5 CIS(D) 1. 0 CIS 0. 5 0. 0 -0. 5 -1. 0 -1. 5 ESMF CIS(D)Shea & Neuscamman, ar. Xiv 1806. 00853 ESMP 2

CH 2 O singlet excitations, cc-p. VDZ error vs EOM-CCSD (e. V) 2. 0 1. 5 CIS(D) 1. 0 0. 5 0. 0 -0. 5 -1. 0 -1. 5 ESMP 2 ESMF CIS(D)Shea & Neuscamman, ar. Xiv 1806. 00853 ESMP 2

error vs EOM-CCSD (e. V) 2. 0 1. 5 CIS 1. 0 0. 5 0. 0 -0. 5 -1. 0 -1. 5 CIS ESMF CIS(D)Shea & Neuscamman, ar. Xiv 1806. 00853 ESMP 2

error vs EOM-CCSD (e. V) 2. 0 1. 5 CIS 1. 0 0. 5 0. 0 -0. 5 -1. 0 -1. 5 ESMF CIS(D)Shea & Neuscamman, ar. Xiv 1806. 00853 ESMP 2

error vs EOM-CCSD (e. V) 2. 0 1. 5 CIS 1. 0 0. 5 0. 0 -0. 5 -1. 0 -1. 5 ESMF CIS(D)Shea & Neuscamman, ar. Xiv 1806. 00853 ESMP 2

error vs EOM-CCSD (e. V) 2. 0 1. 5 ESMP 2 1. 0 CIS 0. 5 0. 0 -0. 5 -1. 0 -1. 5 ESMF CIS(D)Shea & Neuscamman, ar. Xiv 1806. 00853 ESMP 2

error vs EOM-CCSD (e. V) 2. 0 1. 5 ESMP 2 1. 0 CIS 0. 5 0. 0 -0. 5 within 0. 1 e. V of MRCI+Q -1. 0 -1. 5 ESMF CIS(D)Shea & Neuscamman, ar. Xiv 1806. 00853 ESMP 2

P 2 ES M -C M EO CI S(D ) CS D comparison to EOM(2, 3) mean unsigned error (e. V) 0. 52 0. 11 0. 06 max unsigned error (e. V) 1. 48 0. 22 0. 10

future directions Monte Carlo oo-MSJ in solids core & CT excitations generalized backflow better optimizers vectorization mean field theory DIIS, NR, etc. production code CC theory CASSCF s. CI & DMRG is there a Kohn-Sham analogue hiding nearby?

acknowledgements - Gas Phase Chemical Physics - Early Career Research Program