New Approaches for Ab Initio Calculations of Molecules with Strong Electron Correlation

New Approaches for ab initio Calculations of Molecules with Strong Electron Correlation

Stefan Knecht ETH Zürich, Laboratorium für Physikalische Chemie, Schweiz www.reiher.ethz.ch/the-group/people/person-detail.html?persid=193620 [email protected]

| 1 … summarized in our most recent review on this topic:

http://arxiv.org/abs/1512.09267 … to appear soon in Chimia

| 2 Acknowledgment

ETH Zürich ▪ Prof. Johannes Neugebauer § Prof. Markus Reiher (U Münster) § Sebastian Keller ▪ Thomas Dresselhaus (U Münster) § Christopher Stein ▪ Prof. Hans Jørgen Aa. Jensen § Dr. Yingjin Ma (SDU Odense) § Andrea Muolo ▪ Prof. Jochen Autschbach (U at Buffalo) § Stefano Battaglia § Dr. Erik Hedegaard § Dr. Arseny Kovyrshin

| 3 | 4 What we are aiming for

§ Understanding the spectroscopy § Rationalizing the magnetic and small-molecule activation properties of f-element

(CO, CO2, N2, …) by U-complexes complexes

I. Castro-Rodriguez and K. Meyer, Chem. Commun. 13, 1353 (2006) EPR inactive EPR active

S. C. Bart et al., JACS, 130, 12356 (2008) | 5 General remarks

§ Relativity (spin-orbit coupling) and electron correlation à to be treated on equal footing à SOC in general not a perturbation for f-elements § Large number of open- and near-degenerate electronic shells (like in f-elements or 3d-metals) require à multiconfigurational methods à large active spaces

| 6 What about DFT?

à DFT is likely to fail for 3d- and f-element complexes à little progress for 3d-complexes albeit intense research in past decades à not much is known for f-elements à difficult to find a systematic improvability for DFT

| 7 Wave function heaven: systematic improvability

“FCI heaven”

| 8 From traditional to new wave function parametrizations

§ Instead of standard CI-type calculations by diagonalization/projection …

= Ci1,i2...iN i1 i2 iN | i i1,i2...iN | i⌦| i⌦···⌦ | i P § … construct CI coefficients from correlations among orbitals

= Ci1,i2...iN i1 i2 iN | i i1,i2...iN | i⌦| i⌦···⌦ | i P à tensor construction of expansion coefficients: DMRG | 9 state-average: Y. Ma, S. Knecht, S. Battaglia, S. Keller, S. Knecht, A. Muolo, NEVPT2: S. Knecht, S. Keller, S. Keller, R. Lindh, M. Reiher, in preparation M. Reiher, in preparation C. Angeli, M. Reiher, in progress

Analytic gradients for DMRG Spin-orbit coupling/relativity DMRG + dynamical correlation

state-specific: Y. Ma, S. Knecht, DMRG-srDFT M. Reiher, in preparation E. Hedegård et al, JCP, 142, 224108 (2015) CASPT2: S. Knecht, S. F. Keller, QCMAQUIS T. Shiozaki, M. Reiher, in progress First 2nd generation quantum- chemical DMRG code

ü MPS/MPO formalism ü Convergence acceleration techniques ü Works with 1-,2-,4-component Hamiltonians ü Parallelized: MPI/OMP ü Interfaces: Molcas, Dalton, Dirac, Bagel QCMaquis S. Keller et al, JCP, 143, 244118 (2015) S. Keller and M. Reiher, JCP, 144, 134101(2016) www.reiher.ethz.ch/software/maquis.html

FDE-DMRG 2nd order: Y. Ma, S. Knecht, T. Dresselhaus et al., JCP, 142, 044111 (2015) S. Keller, M. Reiher, in preparation

DMRG + environment effects State-interaction: SOC, NACE, … DMRG-SCF: orbital optimization

S. Knecht, S. Keller, J. Autschbach, | 10 M. Reiher, in preparation Matrix product states (MPS) representation: SVD of FCI tensor

matrix FCI tensor representation

matrix product

σi/ni physical index singular value decomposition M(m x n) = U(m x m) s(m x n) V(n x n)* ai-1 ai virtual index rank-3 tensor MPS representation

| 11 Matrix product states (MPS) representation: SVD of FCI tensor

FCI tensor representation

n1n2...nL FCI = C n1n2 ...nL | i nk | i {P}

singular value decomposition M(m x n) = U(m x m) s(m x n) V(n x n)*

MPS representation

n1 nL 1 nL n2 FCI = A1,a1 Aa1,a2 ...AaL 2,aL 1 AaL 1,1 n1n2 ...nL | i n a | i { k}{ j } P | 12 à optimization and reduction of dimensionality to of A matrices MPS optimization: DMRG

… … active subsystem active orbitals environment nl nl+1 § Search for which minimizes

§ Introduce Lagrangian multiplier λ and solve

§ by optimizing the entries of (two sites combined) at a time while keeping all others fixed

| 13 MPS optimization: DMRG

§ To minimize L we take

§ Find lowest eigenvalue + eigenvector of the effective H

which can be written as

| 14 MPS optimization: DMRG

… … active subsystem active orbitals environment nl nl+1

§ Perform SVD of v reshaped into = U S V+ § Truncate sum of singular values to the largest m values (ßà eigenvalues of reduced density matrix) nl § Reshape U into Aal 1,al n + l+1 § Multiply S with V and reshape into Aal,al+1 § Move on to sites l+1 and l+2 until end of lattice is reached; then inverse direction | 15 MPS optimization: DMRG - summary

S. Keller and M. Reiher, Chimia, 68, 200-203 (2014) § DMRG algorithm: protocol for the iterative improvement of the A matrices § Scaling determined by the number of renormalized basis states m § Orbital ordering is crucial but can be optimized § Sum of discarded singular values can be used as a quality measure and/or for extrapolation of energies § Start guess for environment is important, recipes: Ö. Legeza and J. Solyom, Phys. Rev. B, 68, 195116 (2003) Y. Ma, S. Keller, C. Stein, S. Knecht, R. Lindh, M. Reiher, in preparation | 16 Properties of DMRG

DMRG CASCI § Variational § Variational § Size consistent § Size consistent § (approximate) FCI for a CAS § FCI for a CAS § Polynomial scaling (~L4 m3) § Factorial scaling § MPS wave function § Linearly parametrized wave § For large m invariant wrt orbital function rotations § Invariant wrt orbital rotations

| 17 non-relativistic DMRG ßà relativistic DMRG

§ 1-component Hamiltonian § 2-/4-component Hamiltonian § Spin is a good quantum number § Spin is not a good quantum number § real algebra à (Kramers) pairs of spinors: § Local site dimension d == 4 , ¯ , ¯ = Kˆ , Kˆ = i⌃ Kˆ { p p} p p y 0 § Quaternion symmetry scheme for

D2h* and sub-double groups: Real, complex or quaternion algebra § Local site dimension d == 2

and

S. Knecht, Ö. Legeza, and M. Reiher, J. Chem. Phys., 140, 041101 (2014) S. Battaglia, S. Keller, S. Knecht, A. Muolo, M. Reiher, in preparation | 18 A 2nd generation relativistic QC-DMRG code S. Battaglia, S. Keller, S. Knecht, A. Muolo, M. Reiher, in preparation § MPS representation of wave function

§ MPO representation of Hamiltonian

§ Full support for linear symmetry, D2h* and sub-double groups exploiting an operator string classification (ΔMk)

| 19 state-average: Y. Ma, S. Knecht, S. Battaglia, S. Keller, S. Knecht, A. Muolo, NEVPT2: S. Knecht, S. Keller, S. Keller, R. Lindh, M. Reiher, in preparation M. Reiher, in preparation C. Angeli, M. Reiher, in progress

Analytic gradients for DMRG Spin-orbit coupling/relativity DMRG + dynamical correlation

FDE-DMRG 2nd order: Y. Ma, S. Knecht, T. Dresselhaus et al., JCP, 142, 044111 (2015) S. Keller, M. Reiher, in preparation

DMRG + environment effects State-interaction: SOC, NACE, … DMRG-SCF: orbital optimization

S. Knecht, S. Keller, J. Autschbach, | 20 M. Reiher, in preparation QC-DMRG at work..

Applications of relativistic DMRG

| 21 Relativistic DMRG with variational SOC: spectroscopic constants of TlH

§ Comparison with relativistic CI and CC methods § 4-component Dirac-Coulomb Hamiltonian § MO integrals through interface to Dirac § Dyall.cv3z basis set § Active space: all MP2 NOs with occupations between 1.98 and 0.0001 à DMRG-CI with a CAS(14,94)

S. Knecht, Ö. Legeza, and M. Reiher, J. Chem. Phys., 140, 041101 (2014) S. Battaglia, S. Keller, S. Knecht, A. Muolo, M. Reiher, in preparation

| 22 Spectroscopic constants of TlH

-1 -1 Method Re (Å) we (cm ) xewe (cm ) 4c-DMRG[512]-CI 1.873 1411 27 (extrapolated) 4c-CISD 1.856 1462 23 4c-CISDTQ 1.871 1405 20 4c-CCSD 1.871 1405 19 4c-CCSD(T) 1.873 1400 24 4c-CCSDT 1.873 1398 22 4c-CCSDTQ 1.873 1397 22 experiment1 1.872 1390.7 22.7

1R.-D. Urban et al., Chem. Phys. Lett., 158, 443 (1989)

| 23 Relativistic DMRG with variational SOC: Spin and magnetic properties of a [Dy]-complex

S. Battaglia, S. Keller, S. Knecht, A. Muolo, M. Reiher, in preparation § Lanthanide complexes are ubiquitous as contrast agents in magnetic resonance imaging (signal enhancing)

§ Symmetrized [Dy]-DOTA complex (C4*-double group) § Study convergence of (approximate) spin and magnetization density wrt the active orbital space: from CAS(9in7) to CAS(47,57) § cc-pVTZ basis sets § All-electron X2C Hamiltonian

| 24 Relativistic DMRG with two-step SOC: Magnetic properties of neptunyl S. Knecht, S. Keller, J. Autschbach, M. Reiher, in preparation § Study magnetic properties of a prototypical 5f1 molecule § Comparison with CASSCF/CASPT2-SO reference data § MPS-state-interaction approach (à “RASSI” in CI world) ANO-RCC-VTZ basis set, no symmetry, Cholesky decomposition for two-electron integrals, m = 256 for all DMRG-SCF calculations

| 25 QC-DMRG at work..

Going beyond DMRG(-SCF): dynamical electron correlation

| 26 DMRG(-SCF) + dynamical correlation

§ DMRG(-SCF) is an efficient approach to take into account static correlation effects § Dynamical correlation is missing to a large extent § Requires excitations from the inactive/active to the active/secondary orbital space

secondary space

inactive space | 27 Going beyond DMRG-SCF: dynamical electron correlation

§ DMRG can in principle be combined with any post-HF approach (“diagonalize-the-perturb” ansatz): § (Internally-contracted) multi-reference CI § Multi-reference CC § Multi-reference perturbation theory to 2nd order (MRPT2): CASPT2, NEVPT2, … àCASSCF/CASPT2 is one of the most successful and versatile approaches in quantum chemistry (landmark CASPT2 paper has more than 1200 citations!) § Conceptually different approach: short-range DFT-long-range DMRG E. D. Hedegård, S. Knecht, J. S. Kielberg, H. J. Aa. Jensen, M. Reiher, J. Chem. Phys., 142, 224108 (2015)

| 28 Dynamical correlation through MRPT2: CASPT2 or NEVPT2?

CASPT2 NEVPT2 - H(0):Fock-type zeroth-order + H(0): Dyall Hamiltonian Hamiltonian (full HCAS-CI for active orbitals, + Size-extensive Fock-type for core/secondary + Multi-state formulation possible orbitals) + - Prone to intruder states Size-extensive + - Requires 3-particle (and partial No intruder-state problem 4-particle) reduced density + Multi-state formulation possible matrix within the active orbital - Requires 4-particle reduced space density matrix within the active orbital space

| 29 Higher-order n-particle reduced density matrices

§ 3-particle (transition) and 4-particle reduced density matrices (RDMs) need to be calculated accurately § Compressed MPS wave function for RDM calculation

S. Guo, M. A. Watson, W. Hu, Q. Sun, G. K.-L. Chan, J. Chem. Theory Comput., doi:10.1021/acs.jctc.5b01225 § Cumulant approximation of higher–order n-RDMs à loss of N-representability of the n-RDM (possible)!

| 30 Y. Kurashige, J. Chalupsky, T. N. Lan, T. Yanai, J. Chem. Phys. 141, 174111 (2014) Singlet-triplet gap of methylene

S. Knecht, E. D. Hedegaard, S. Keller, A. Kovyrshin, Y. Ma, A. Muolo, C. Stein, M. Reiher, arXiv: 1512.09267 3 1 § CH2 (bent C2v structure): X B1 - a A1 splitting is a benchmark test for correlation methods § cc-pVTZ basis, m=1024 for all DMRG-SCF calculations

| 31 QC-DMRG at work..

DMRG in DFT-embedding

| 32 DMRG embedded in a quantum environment: DMRG in DFT-embedding environment

…

active subsystem

T. Dresselhaus, J. Neugebauer, S. Knecht, S. Keller, Y. Ma, M. Reiher, J. Chem. Phys., 142, 044111 (2015)

§ frozen density embedding (FDE) scheme is used with a freeze-and-thaw strategy for a self-consistent polarization of the orbital-optimized wave function and the environmental densities

| 33 DMRG embedded in a quantum environment: DMRG in DFT-embedding chain of HCN molecules

active subsystem

T. Dresselhaus, J. Neugebauer, S. Knecht, S. Keller, Y. Ma, M. Reiher, J. Chem. Phys., 142, 044111 (2015)

| 34 QC-DMRG at work..

State-specific DMRG(-SCF) gradients

| 35 State-specific DMRG(-SCF) gradients: resonance Raman spectra of uracil Y. M a , S . Knecht, M. Reiher, in preparation § Resonance Raman (RR) spectroscopy is a selective spectroscopic technique à resonance conditions act like a filter such that only certain peaks are selectively enhanced in the vibrational spectrum § Benchmark data for RR spectrum of uracil of the bright

excited state (S2) in gas phase available J. Neugebauer and B. A. Hess, JCP, 120, 11564 (2004) § RR in the short-time approximation requires only knowledge about the excited-state gradient at the minimum structure

| 36 State-specific DMRG(-SCF) gradients: resonance Raman spectra of uracil Y. M a , S . Knecht, M. Reiher, in preparation

§ Within the short-time approximation the relative intensity ij for the j-th mode is given by

: harmonic vibrational frequency : normal mode displacement § Independent mode displaced harmonic oscillator model: obtain intensity from partial derivative of the excited-state electronic energy wrt normal mode qj at qj=0:

| 37 State-specific DMRG(-SCF) gradients: be(C5-H11)/ resonance Raman spectra of uracil be(C6-H12) C5-C6 C4-O10

§ Comparison of DMRG-SCF RR spectra with other ab initio approaches aug-cc-pVTZ basis, m=1000 in all DMRG calculations

| 38 State-specific DMRG(-SCF) gradients: be(C5-H11)/ resonance Raman spectra of uracil be(C6-H12) C5-C6 C4-O10

§ Rydberg-type orbitals play a non-negligible role for the intensities in the RR spectrum § Unbalanced treatment of static and dynamical correlation can lead to unphysical intensities

§ Entanglement analysis of S2: considerable σ bonding and valence π/π∗ orbital entanglement

| 39 Outlook

Analytic gradients for DMRG Spin-orbit coupling/relativity DMRG + dynamical correlation

QCMAQUIS First 2nd generation quantum- chemical DMRG code

ü MPS/MPO formalism ü Convergence acceleration techniques ü Works with 1-,2-,4-component SOET, … Hamiltonians ensemble DFT ü Parallelized: MPI/OMP see talk by B. Senjean see talk by M. Alam ü Interfaces: Molcas, Dalton, Dirac, Bagel QCMaquis S. Keller et al, JCP, 143, 244118 (2015) S. Keller and M. Reiher, JCP, 144, 134101(2016) www.reiher.ethz.ch/software/maquis.html

DMRG + environment effects State-interaction: SOC, NACE, … DMRG-SCF: orbital optimization| 2 Thank you for your kind attention

| 41