new approaches for ab initio calculations of molecules ... · à soc in general not a perturbation...

|

Stefan KnechtETH Zürich, Laboratorium für Physikalische Chemie, Schweiz

www.reiher.ethz.ch/the-group/people/[email protected]

1

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

|

… summarized in our most recent review on this topic:

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

2

|

▪ Prof. Johannes Neugebauer (U Münster)

▪ Thomas Dresselhaus (U Münster)▪ Prof. Hans Jørgen Aa. Jensen

(SDU Odense)▪ Prof. Jochen Autschbach

(U at Buffalo)

Acknowledgment

ETH Zürich§ Prof. Markus Reiher§ Sebastian Keller§ Christopher Stein§ Dr. Yingjin Ma§ Andrea Muolo§ Stefano Battaglia§ Dr. Erik Hedegaard§ Dr. Arseny Kovyrshin

3

|

§ Understanding the spectroscopy and small-molecule activation (CO, CO2, N2, …) by U-complexes

§ Rationalizing the magnetic properties of f-element complexes

5

What we are aiming for

S. C. Bart et al., JACS, 130, 12356 (2008)

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

|

§ Relativity (spin-orbit coupling) and electron correlation à to be treated on equal footingà 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) requireà multiconfigurational methodsà large active spaces

6

General remarks

| 7

What about DFT?

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

| 8

Wave function heaven: systematic improvability

“FCI heaven”

© original figure by T. Fleig

|

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

§ … construct CI coefficients from correlations among orbitals

9

From traditional to new wave function parametrizations

à tensor construction of expansion coefficients:DMRG

| i =P

i1,i2...iN

Ci1,i2...iN |i1i ⌦ |i2i ⌦ · · ·⌦ |iN i

| i =P

i1,i2...iN

Ci1,i2...iN |i1i ⌦ |i2i ⌦ · · ·⌦ |iN i

|

QCMAQUISFirst 2nd generation quantum-

chemical DMRG code

ü MPS/MPO formalismü Convergence acceleration techniquesü Works with 1-,2-,4-component

Hamiltoniansü Parallelized: MPI/OMPü Interfaces: Molcas, Dalton, Dirac,

Bagel

DMRG-SCF: orbital optimization

DMRG + dynamical correlation

DMRG + environment effects

Analytic gradients for DMRG Spin-orbit coupling/relativity

State-interaction: SOC, NACE, …

FDE-DMRGT. Dresselhaus et al., JCP, 142, 044111 (2015)

S. Battaglia, S. Keller, S. Knecht, A. Muolo, M. Reiher, in preparation

DMRG-srDFTE. Hedegård et al, JCP, 142, 224108 (2015)

NEVPT2: S. Knecht, S. Keller, C. Angeli, M. Reiher, in progress

CASPT2: S. Knecht, S. F. Keller, T. Shiozaki, M. Reiher, in progress

state-specific: Y. Ma, S. Knecht, M. Reiher, in preparation

state-average: Y. Ma, S. Knecht, S. Keller, R. Lindh, M. Reiher, in preparation

10S. Knecht, S. Keller, J. Autschbach,M. Reiher, in preparation

2nd order: Y. Ma, S. Knecht, S. Keller, M. Reiher, in preparation

QCMaquisS. Keller et al, JCP, 143, 244118 (2015)S. Keller and M. Reiher, JCP, 144, 134101(2016)www.reiher.ethz.ch/software/maquis.html

| 11

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

matrix

matrix product

rank-3 tensor

aiai-1

σi/ni physical index

virtual index

FCI tensor representation

MPS representation

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

| 12

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

FCI tensor representation

MPS representation

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

| FCIi =P{nk}

Cn1n2...nL |n1n2 . . . nLi

| FCIi =P

{nk}{aj}An1

1,a1An2

a1,a2. . . AnL�1

aL�2,aL�1AnLaL�1,1 |n1n2 . . . nLi

à optimization and reduction of dimensionality to of A matrices

|

§ 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

… …active subsystem environmentactive orbitals

nl nl+1

|

§ To minimize L we take

§ Find lowest eigenvalue + eigenvector of the effective H

which can be written as14


|

§ 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)

§ Reshape U into § Multiply S with V+ and reshape into§ Move on to sites l+1 and l+2 until end of lattice is reached;

then inverse direction 15


… …active subsystem environmentactive orbitals

nl nl+1

Anl+1al,al+1

Anlal�1,al

|

§ DMRG algorithm: protocol for the iterative improvement of the A matrices

§ Scaling determined by the number of renormalizedbasis 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:

16

MPS optimization: DMRG - summaryS. Keller and M. Reiher, Chimia, 68, 200-203 (2014)

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

|

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

rotations

CASCI§ Variational§ Size consistent§ FCI for a CAS§ Factorial scaling§ Linearly parametrized wave

function§ Invariant wrt orbital rotations

17

Properties of DMRG

|

§ 1-component Hamiltonian§ Spin is a good quantum number§ real algebra§ Local site dimension d == 4

§ 2-/4-component Hamiltonian§ Spin is not a good quantum numberà (Kramers) pairs of spinors:

§ Quaternion symmetry scheme for D2h* and sub-double groups:Real, complex or quaternion algebra

§ Local site dimension d == 2

18

non-relativistic DMRG ßà relativistic DMRG

{ p, ̄p}, ̄p = K̂ p, K̂ = �i⌃yK̂0

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

|

§ 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

A 2nd generation relativistic QC-DMRG codeS. Battaglia, S. Keller, S. Knecht, A. Muolo, M. Reiher, in preparation

|


chemical DMRG code



Bagel






FDE-DMRGT. Dresselhaus et al., JCP, 142, 044111 (2015)


DMRG-srDFTE. Hedegård et al, JCP, 142, 224108 (2015)

NEVPT2: S. Knecht, S. Keller, C. Angeli, M. Reiher, in progress

CASPT2: S. Knecht, S. F. Keller, T. Shiozaki, M. Reiher, in progress

state-specific: Y. Ma, S. Knecht, M. Reiher, in preparation

state-average: Y. Ma, S. Knecht, S. Keller, R. Lindh, M. Reiher, in preparation

20S. Knecht, S. Keller, J. Autschbach,M. Reiher, in preparation

2nd order: Y. Ma, S. Knecht, S. Keller, M. Reiher, in preparation


| 21

QC-DMRG at work..

Applications of relativistic DMRG

|

§ 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)

22

Relativistic DMRG with variational SOC: spectroscopic constants of TlH

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

| 23

Spectroscopic constants of TlH

Method Re (Å) we (cm-1) xewe (cm-1)4c-DMRG[512]-CI(extrapolated)

1.873 1411 27

4c-CISD 1.856 1462 234c-CISDTQ 1.871 1405 204c-CCSD 1.871 1405 194c-CCSD(T) 1.873 1400 244c-CCSDT 1.873 1398 224c-CCSDTQ 1.873 1397 22experiment1 1.872 1390.7 22.7

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

|

§ 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 variational SOC:Spin and magnetic properties of a [Dy]-complex


|

§ 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

Relativistic DMRG with two-step SOC:Magnetic properties of neptunyl

S. Knecht, S. Keller, J. Autschbach, M. Reiher, in preparation

| 26

QC-DMRG at work..

Going beyond DMRG(-SCF): dynamical electron 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

27

DMRG(-SCF) + dynamical correlation

secondary space

inactive space

|

§ 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

28

Going beyond DMRG-SCF:dynamical electron correlation

E. D. Hedegård, S. Knecht, J. S. Kielberg, H. J. Aa. Jensen, M. Reiher, J. Chem. Phys., 142, 224108 (2015)

|

CASPT2- H(0):Fock-type zeroth-order

Hamiltonian+ Size-extensive+ Multi-state formulation possible- Prone to intruder states- Requires 3-particle (and partial

4-particle) reduced density matrix within the active orbital space

NEVPT2+ H(0): Dyall Hamiltonian

(full HCAS-CI for active orbitals, Fock-type for core/secondary orbitals)

+ Size-extensive+ No intruder-state problem+ Multi-state formulation possible- Requires 4-particle reduced

density matrix within the active orbital space

29

Dynamical correlation through MRPT2: CASPT2 or NEVPT2?

|

§ 3-particle (transition) and 4-particle reduced density matrices (RDMs) need to be calculated accurately

§ Compressed MPS wave function for RDM calculation

§ Cumulant approximation of higher–order n-RDMs à loss of N-representability of the n-RDM (possible)!

30

Higher-order n-particle reduced density matrices

S. Guo, M. A. Watson, W. Hu, Q. Sun, G. K.-L. Chan, J. Chem. Theory Comput., doi:10.1021/acs.jctc.5b01225

Y. Kurashige, J. Chalupsky, T. N. Lan, T. Yanai, J. Chem. Phys. 141, 174111 (2014)

|

§ CH2 (bent C2v structure): X3B1 - a1A1 splitting is a benchmark test for correlation methods

§ cc-pVTZ basis, m=1024 for all DMRG-SCF calculations

31

Singlet-triplet gap of methyleneS. Knecht, E. D. Hedegaard, S. Keller, A. Kovyrshin, Y. Ma, A. Muolo, C. Stein, M. Reiher, arXiv: 1512.09267

| 32

QC-DMRG at work..

DMRG in DFT-embedding

|

§ 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

…

active subsystem

environment

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

| 34

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

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

chain of HCN molecules

active subsystem

| 35

QC-DMRG at work..

State-specific DMRG(-SCF) gradients

|

§ Resonance Raman (RR) spectroscopy is a selective spectroscopic techniqueà 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

§ 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. Ma, S. Knecht, M. Reiher, in preparation

J. Neugebauer and B. A. Hess, JCP, 120, 11564 (2004)

|

§ Within the short-time approximation the relative intensity ijfor 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:resonance Raman spectra of uracil

Y. Ma, S. Knecht, M. Reiher, in preparation

|

§ Comparison of DMRG-SCFRR spectra with other ab initio approachesaug-cc-pVTZ basis, m=1000 in all DMRG calculations

38

State-specific DMRG(-SCF) gradients:resonance Raman spectra of uracil C5-C6 C4-O10

be(C5-H11)/be(C6-H12)

|

§ 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

State-specific DMRG(-SCF) gradients:resonance Raman spectra of uracil C5-C6 C4-O10

be(C5-H11)/be(C6-H12)

|

Outlook

2


chemical DMRG code



Bagel







SOET, … ensemble DFT

see talk by B. Senjean see talk by M. Alam

| 41

Thank you for your kind attention

new approaches for ab initio calculations of molecules ... · à soc in general not a perturbation...

Documents