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Stefan KnechtETH Zürich, Laboratorium für Physikalische Chemie, Schweiz

www.reiher.ethz.ch/the-group/people/person-detail.html?persid=193620stefan.knecht@phys.chem.ethz.ch

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Towards Large-Scale Quantum Chemistry with Second-Generation Density Matrix Renormalization Group - QCMaquis

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Review of our QCMaquis toolbox based on the density matrix renormalization group approach

Chimia, 70, 244-251 (2016)http://arxiv.org/abs/1512.09267

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§ Entirely written in C++, hosted on Gitlab§ OpenMP parallelization available§ Ambient library (http://ambientcxx.org) developed at

ETH/CSCS for MPI parallelization (in progress)§ Boost§ ALPS library (Algorithms and Libraries for Physics

Simulations, http://alps.comp-phys.org)§ Linear algebra libraries: dgemm, daxpy, SVD, …

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Library requirements for QCMaquis

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Acknowledgment

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

(now Toulouse, France)§ Dr. Erik Hedegaard

(now Uppsala, Sweden)§ Dr. Arseny Kovyrshin

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ETH Zürich§ Group of Prof. M. Troyer

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

§ Rationalizing the magnetic properties of f-element complexes

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Motivation: Large-Scale Quantum Chemistry in terms of molecular size

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

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

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§ Relativity (spin-orbit coupling [SOC]) and electron correlation à to be treated on equal footingà SOC in general not a perturbation for f-elements

§ Large number of near-degenerate electronic shells requireà multi-configurational methodsà large active orbital spaces

§ Wave function methods preferential (solution of the electronic Schrödinger equation

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General remarks

Hel {RI}el ({ri}) = Eel({RI}) {RI}

el ({ri})

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How to approximate Ψel?

§ Construct many-electron (determinantal) basis set {ΦI} from a given (finite) one-electron (orbital) basis set φi

§ From the solution of the Roothaan–Hall equations, one obtains as norbitals from n one-electron basis functions:

§ From the N orbitals with the lowest energy, the Hartree–Fock (HF) Slater determinant Φ0 is constructed

§ The other determinants (many-particle basis states) are obtained by subsequent substitution of orbitals in the HF Slater determinant Φ0:

FC = SC✏

{�I} ! �ai ,�

bj , . . . ! �ab

ij ,�acik , . . . ! �abc

ijk ,�abdijl , . . .

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

§ … construct CI coefficients from correlations among orbitals

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From traditional to new wave function parameterizations

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

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QCMAQUISFirst 2nd generation quantum-

chemical DMRG code

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

Hamiltoniansü Parallelized: OMP/(MPI)ü 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

12S. 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

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

renormalized states) invariant wrtorbital rotations

§ Up to 100 active orbitals possible: CAS(x,100)

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

function§ Invariant wrt orbital rotations§ Computational limit: CAS(18,18)

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Properties of DMRG

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

renormalized states) invariant wrtorbital rotations

§ Up to 100 active orbitals possible: CAS(x,100)

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

function§ Invariant wrt orbital rotations§ Computational limit: CAS(18,18)

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Properties of DMRG

ü Ö. Legeza et al., Lect. Notes Phys., 739, 653 (2008) ü K. Marti and M. Reiher, Z. Phys. Chem. 224, 583 (2010)ü U. Schollwöck, Ann. Phys., 326, 96 (2011)ü G. K.-L. Chan, WIREs, 2, 907 (2012) ü Y. Kurashige, Mol. Phys., 112, 1485 (2013)ü S. Wouters and D. van Neck, Eur. Phys. J. D, 68, 272 (2014)ü S. Szalay et al., Int. J. Quant. Chem. 115, 1342 (2015)

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Matrix product states (MPS) representation:Singular value decomposition of the 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)*

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Matrix product states (MPS) representation:Singular value decomposition of the 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

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§ Quantum chemical Hamiltonian in second quantization

§ Consider occupation-number-vector basis states and§ The coefficients of a general operator

… may be encoded in matrix-product form

… combining both, the operator reads

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MPS Structure of Operators: MPOs

|�i |�0iw��0

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§ Simplify by contraction over the local site indices

… which then yields

§ Motivation for this: Entries of the resulting matrices are the elementary creation and annihilation operators acting on a single site (=orbital)! à 4x4 matrix

§ MPS concept has been transferred to operators (MPOs)à First 2nd generation QC-DMRG:

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MPS Structure of Operators: MPOs

S. Keller, M. Dolfi, M. Troyer, M. Reiher, J. Chem. Phys. 143, 244118 (2015)

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§ Quantum chemical Hamiltonian in second quantization

§ Represent Hamiltonian terms in a tree-like data structure by forking them from a trivial branch of identity operators. In- and outgoing branches on site i translate into tensor indices bi−1 and bi.

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Compact MPO constructionS. Keller, M. Dolfi, M. Troyer, M. Reiher, J. Chem. Phys. 143, 244118 (2015)

M. Dolfi, B. Bauer, S. Keller, A. Kosenkov, T. Ewart, A. Kantian, T. Giamarchi, M. Troyer, Comp. Phys. Commun 185, 3420 (2014)

Uncompressed MPO network Compressed MPO network

L5 scaling L4 scaling

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§ 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

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MPS optimization: DMRG

… …active subsystem environmentactive orbitals

nl nl+1

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§ To minimize L we take

§ Find lowest EV+ EVC of the effective H (dimension:16 m2)

which can be written as18

MPS optimization: DMRG

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§ 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 19

MPS optimization: DMRG

… …active subsystem environmentactive orbitals

nl nl+1

Anl+1al,al+1

Anlal�1,al

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

§ Scaling determined by the number of renormalizedbasis states m: current limits around m=10 000

§ 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:

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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

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QC-DMRG at work..

Going beyond DMRG(-SCF): dynamical electron correlation

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§ 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

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DMRG(-SCF) + dynamical correlation

secondary space

inactive space

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§ 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

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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)

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

Hamiltonian+ (Nearly) 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 of size L(~ L2n elements with n=3,4)

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 of size L(~ L2n elements with n=4)

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Dynamical correlation through MRPT2: CASPT2 or NEVPT2?

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§ 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)!

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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)

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§ CAAR::DIRAC project (OLCF):§ Optimized MPI implementation of

QC-DMRG algorithm based on the Ambient library (http://ambientcxx.org) developed at ETH/CSCS

§ GPU-accelerated calculation of higher-order n-particle reduced density matrix elements (for n=4, ca. L8 elements are needed)

à Requires optimized re-usage of pre-contracted matrix-matrix products in order to minimize data exchange between CPU and GPU

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Outlook: Towards Large-Scale Quantum Chemistry in an HPC sense

h |c†i c†j . . . cmcn . . . |�i

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Thank you for your kind attention