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DMRG in Quantum Chemistry:
From its relation to traditional methods to n-orbital
density matrices and beyond
Markus Reiher
Laboratorium für Physikalische Chemie, ETH Zürich, Switzerland
http://www.reiher.ethz.ch
April 2016
DMRG in Quantum Chemistry Markus Reiher 1 / 77
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Outline
DMRG in Quantum Chemistry Markus Reiher 2 / 77
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Non-Relativistic Many-Electron Hamiltonian
many-electron Hamiltonian in position space (Hartree atomic units)
Hel =
N∑i
(−1
2∇2i −
∑I
ZIriI
)+
N∑i
-
Standard Procedure: Construction of Many-Electron Basis
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 n
orbitals from n one-electron basis functions.
From the N orbitals with the lowest energy, the Hartree–Fock (HF)
Slater determinant is constructed.
The other determinants (configurations) are obtained by subsequent
substitution of orbitals in the HF Slater determinant Φ0:
Φ0 → {Φai } → {Φabij } → {Φabcijk} → ... (3)
Determinants are classified by number of ’excitations’
(= substitutions in HF reference determinant).
DMRG in Quantum Chemistry Markus Reiher 4 / 77
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Standard Full Configuration Interaction (FCI)
Including all possible excited Slater determinants for a finite or infinite
one-electron basis set leads to the so-called full CI approach.
Number of Slater determinants nSD for N spin orbitals chosen from a
set of L spin orbitals (colloquial: N electrons in L spin orbitals):
nSD =
(L
N
)=
L!
N !(L−N)! (4)
Example: There are ≈ 1012 different possibilities to distribute 21electrons in 43 spin orbitals.
CI calculations employing a complete set of Slater determinants are
only feasible for small basis sets.
DMRG in Quantum Chemistry Markus Reiher 5 / 77
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Truncated CI Wave Functions
Standard recipe to avoid the factorial scaling of the many-electron
basis-set size: truncate basis! Note: basis is pre-defined!
Assumption: Substitution hierarchy is a useful measure to generate a
systematically improvable basis set.
CIS: all singly-(S)-excited determinants are included:
ΨCISel = C0Φ0 +∑(ai)
C(ai)Φai (5)
CISD: all singly- and doubly-(D)-excited determinants are included:
ΨCISDel = C0Φ0 +∑(ai)
C(ai)Φai +
∑(ai)(bj)
C(ai,bj)Φabij (6)
C0, C(ai), C(ai,bj) ∈ {CI} (7)DMRG in Quantum Chemistry Markus Reiher 6 / 77
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Determination of the CI Expansion Coefficients CI
ECIel =
〈ΨCIel
∣∣Hel ∣∣ΨCIel 〉〈ΨCIel
∣∣ΨCIel 〉 =∑
I,J C∗ICJ 〈ΦI |Hel |ΦJ〉∑
I,J C∗ICJ 〈ΦI |ΦJ〉
(8)
CI determined from variational principle:
Calculate all derivatives ∂ECIel / ∂C∗K and set them equal to zero, which
yields the CI eigenvalue problem:
H ·C = Eel ·C (9)
Essential: H is constructed from matrix elements 〈ΦI |Hel |ΦJ〉 inpre-defined determinantal basis {ΦI}
DMRG in Quantum Chemistry Markus Reiher 7 / 77
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Standard ’Technical’ Trick: Second Quantization
Operators and wave functions are expressed in terms of creation and
annihilation operators to implement the Slater–Condon rules for the
evaluation of matrix elements 〈ΦI |Hel |ΦJ〉 directly into the formalism.
Hel in second quantization (i, j, k, l are spin orbital indices):
⇒ Hel =∑ij
〈φi|h(i) |φj〉 c†i cj
+1
2
∑ijkl
〈φi(1) 〈φk(2)| g(1, 2) |φl(2)〉φj(1)〉c†i c†jckcl (10)
CI wave function in second quantization:
ΨFCIel = C0Φ0 +∑(ai)
C(ai)c†aciΦ0 +
∑(ai)(bj)
C(ai,bj)c†bcjc
†aciΦ0 · · · (11)
DMRG in Quantum Chemistry Markus Reiher 8 / 77
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CI Energy in Second Quantization
ECIel =D
ΨCIel
˛̨̨Hel
˛̨̨ΨCIel
E(12)
=
LXij
nSDXIJ
C∗ICJ tIJij| {z }
γij
〈φi(1)|h(1) |φj(1)〉| {z }≡hij
+LXijkl
nSDXIJ
C∗ICJTIJijkl| {z }
Γijkl
〈φi(1) 〈φk(2)| g(1, 2) |φl(2)〉φj(1)〉| {z }gijkl
(13)
=LXij
γijhij +LXijkl
Γijklgijkl (14)
tIJij or TIJijkl are matrix elements of determinantal basis functions over
pairs or quadruples of elementary operators c† and c.
γij are Γijkl are density matrix elements (compare: RDMFT and 2RDM
methods).
DMRG in Quantum Chemistry Markus Reiher 9 / 77
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Is there a better way to construct the finite-dimensional
determinantal basis set in order to avoid the factorial scaling?
DMRG in Quantum Chemistry Markus Reiher 10 / 77
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Coupled-Cluster — An Advanced CI-type Wave FunctionAnsatz:
ΨCCel = exp (T ) Φ0 (15)
Excitation operator:
T = T1 + T2 + T3 + · · · (16)
where
Tα =X
ab · · ·| {z }α times
ij · · ·| {z }α times
cluster-amplitudesz}|{tab···ij··· · · · c†bcjc
†aci| {z }
α pairs c†c
⇒ T1 =Xai
tai c†aci (17)
Notation:
CCS (T = T1), CCSD (T = T1 + T2), CCSDT (T = T1 + T2 + T3) ,· · ·Coupled-cluster improves on truncated CI, because certain (disconnected)
higher excited configurations (e.g., tai c†acit
bcjkc†cckc
†bcj) are included.
DMRG in Quantum Chemistry Markus Reiher 11 / 77
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However, all of this is too expensive and often too limited
(MRCC would be nice, but requires 5e-RDMs ...)
DFT is the workhorse in transition metal chemistry
(structures(!), energies, spectroscopic signatures)
in critical cases, its accuracy is difficult to assess
... illustrated by an example
DMRG in Quantum Chemistry Markus Reiher 12 / 77
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The Spin State Problem in Quantum ChemistryTo predict the correct energy gaps between states of different spin S
is crucial in chemistry (e.g., two-state reactivity); in kJ/mol
Present-day DFT has difficulties to cope with this problem
E.g., the results strongly depend on exact exchange admixture c3
Example: the thermal spin-crossover complex [Fe(phen)2(NCS)2]
N
N
N
N
Fe
N
NCS
CS
0 0.05 0.1 0.15 0.2
c3 parameter
−75
−25
25
75
∆ E
a/b / k
J/m
ol
S=0/S=2 splitting
S=0/S=1 splitting
M. Reiher, Inorg. Chem. 41 (2002) 6928;
M. Reiher, Faraday Discus. 135 (2007) 97
DFT ∆EeLS/HS
B3LYP(c3=0.00) 82.6
BP86/RI 66.0
BLYP 48.1
B3LYP(c3=0.12) 11.0
B3LYP(c3=0.13) 5.5
B3LYP(c3=0.14) −0.3B3LYP(c3=0.15) −6.1B3LYP −33.4HF −407.5DMRG in Quantum Chemistry Markus Reiher 13 / 77
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Ab initio Methods for benchmarks needed
Well-established for open-shell transition metal complexes:
CASSCF/CASPT2 and MRCI
(incl. RASSCF/RASPT2)
DMRG in Quantum Chemistry Markus Reiher 14 / 77
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Orbital subspaces in CAS approaches
inactive orbitals ~ 50 - 100
active orbitals CASSCF: up to 18
DMRG-SCF: up to 100
virtual orbitals ~ 102 - 103
orbitals are subdivided in doubly occupied
inactive orbitals, active orbitals and
unoccupied virtual orbitals
huge virtual orbital subspace is not
considered in CAS approaches (but should
be (dynamic correlation)!)
Exponential scaling of many-particle basis
set size permits exact diagonalization up to
only CAS(18,18)
DMRG in Quantum Chemistry Markus Reiher 15 / 77
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Ab initio Methods for benchmarks needed
Well-established for open-shell transition metal complexes:
CASSCF/CASPT2 and MRCI
(incl. RASSCF/RASPT2)
... but what if active orbital space is too large?
Promising new approaches?
DMRG in Quantum Chemistry Markus Reiher 16 / 77
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Parameterization of the Wave Function
|Ψ〉 =∑
i1i2···iN
Ci1i2···iN |i1〉 ⊗ |i2〉 ⊗ · · · ⊗ |iN 〉
Restricted sum over basis states with a certain excitation pattern
→ yields a pre-defined (!) many-particle basis setConfiguration Interaction ansatz
|CI〉 =(
1 +∑µ
Cµτ̂µ
)|HF〉
Coupled Cluster ansatz
|CC〉 =(∏
µ
[1 + tµτ̂µ])|HF〉
numerous specialized selection/restriction protocols
DMRG in Quantum Chemistry Markus Reiher 17 / 77
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Instead of standard CI-type calculations by
diagonalization/projection
|Ψ〉 =∑
i1i2···iN
Ci1i2···iN |i1〉 ⊗ |i2〉 ⊗ · · · ⊗ |iN 〉
construct CI coefficients from correlations among orbitals
|Ψ〉 =∑
i1i2···iN
Ci1i2···iN |i1〉 ⊗ |i2〉 ⊗ · · · ⊗ |iN 〉
DMRG in Quantum Chemistry Markus Reiher 18 / 77
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Density Matrix Renormalization Group: Principles
New wavefunction parametrization:
|ψ〉 =∑σ
cσ|σ〉 −→ |ψ〉 =∑σ
Mσ1 Mσ2 · · · MσL |σ〉
complete active space (CAS) partitioning and iterative sweeping:
In essence: dimension reduction by least-squares fitting
DMRG in Quantum Chemistry Markus Reiher 19 / 77
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Reviews on DMRG in Quantum Chemistry
1 Ö Legeza, R. M. Noack, J. Sólyom and L. Tincani, Applications of Quantum Information in the Density-Matrix
Renormalization Group, Lect. Notes Phys., 739,653-664 (2008)
2 G. K.-L. Chan, J. J. Dorando, D. Ghosh, J. Hachmann, E. Neuscamman, H. Wang, T. Yanai, An Introduction to the
Density Matrix Renormalization Group Ansatz in Quantum Chemistry, Prog. Theor. Chem. and Phys., 18, 49 (2008)
3 D. Zgid and G. K.-L. Chan, The Density Matrix Renormalisation Group in Quantum Chemistry, Ann. Rep. Comp.
Chem., 5, 149, (2009)
4 K. H. Marti, M. Reiher, The Density Matrix Renormalization Group Algorithm in Quantum Chemistry, Z. Phys. Chem.,
224, 583-599 (2010)
5 G. K.-L. Chan and S. Sharma, The density matrix renormalization group in quantum chemistry, Ann. Rev. Phys. Chem.,
62, 465 (2011)
6 K. H. Marti, M. Reiher, New Electron Correlation Theories for Transition Metal Chemistry, Phys. Chem. Chem. Phys.,
13, 6750-6759 (2011)
7 Y. Kurashige, Multireference electron correlation methods with density matrix renormalisation group reference functions,
Mol. Phys. 112, 1485-1494 (2014)
8 S. Wouters and D. Van Neck, The density matrix renormalization group for ab initio quantum chemistry, Eur. Phys. J.
D 68, 272 (2014)
9 T. Yanai, Y. Kurashige, W. Mizukami, J. Chalupský, T. N. Lan, M. Saitow, Density matrix renormalization group for ab
initio Calculations and associated dynamic correlation methods,Int. J. Quantum Chem. 115, 283-299 (2015)
10 S. Szalay, M. Pfeffer, V. Murg, G. Barcza, F. Verstraete, R. Schneider, Ö. Legeza, Tensor product methods and
entanglement optimization for ab initio quantum chemistry, arXiv:1412.5829 (2015)
DMRG in Quantum Chemistry Markus Reiher 20 / 77
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DMRG Parameters
Active space selection, as in MCSCF, but need additional parameters:
Number of renormalized basis states m
Orbital ordering
Choice of initial environment guess
These parameters do depend on each other.
S. Keller, M. Reiher, Chimia, 2014, 68, 200–203
DMRG in Quantum Chemistry Markus Reiher 21 / 77
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Two variants of DMRG:
Traditional DMRG
|ψ〉 =∑LR CLR|σL〉 ⊗ |σR〉
coefficients valid for one
bipartition into L and R
(need basis
transformations)
Excited states only with
state averaging
faster for ground states?
MPS DMRG (2nd generation)
|ψ〉 =∑σM
σ1 Mσ2 · · · MσL |σ〉
coefficients valid for the whole
system
Easy and efficient
implementation of
〈φ|ψ〉〈ψ|a†i Ĥaj |ψ〉3- and 4-body RDMs
excited states
DMRG in Quantum Chemistry Markus Reiher 22 / 77
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Matrix product states (MPS) and Matrix product
operators (MPO)
|ψ〉 =∑σ
cσ|σ〉 → |ψ〉 =∑σ
∑a1,...,aL−1
Mσ11a1 Mσ2a1a2 · · · MσLaL−11 |σ〉
Ŵ =∑σ,σ′
wσσ′ |σ〉〈σ′| →
Ŵ =∑σσ′
∑b1,...,bL−1
Wσ1σ′11b1
· · ·W σlσ′lbl−1bl · · ·WσLσ
′L
bL−11|σ〉〈σ′|
S. Keller, M. Dolfi, M. Troyer, M. Reiher, J. Chem. Phys. 143, 244118 (2015)
DMRG in Quantum Chemistry Markus Reiher 23 / 77
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Matrix product operators (MPO)
Ĥ =L∑ij σ
tij ĉ†iσ ĉjσ +
1
2
L∑ijklσ σ′
Vijklĉ†iσ ĉ†kσ′ ĉlσ′ ĉjσ,
DMRG in Quantum Chemistry Markus Reiher 24 / 77
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MPO compression
Consider the terms c†1c†2c5c7 + c
†1c†3c6c7
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MPS-MPO operations: expectation values
DMRG in Quantum Chemistry Markus Reiher 26 / 77
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Spin symmetry adaptation: MPS and MPO put together
F[a′]i′j′(l + 1) =
∑aijkss′
Wigner-9j[jsj′
aka′is′i′
]M
[s′]†
i′i W[k]ss
′
aa′ F[a]ij (l)M
[s]j′j
S. Keller, M. Reiher, J. Chem. Phys. 144, 134101 (2016)
DMRG in Quantum Chemistry Markus Reiher 27 / 77
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The QCMaquis DMRG program
Input: tij and Vijkl integrals, target state quantum numbers
The system (atom, molecule, ...) is encoded in tij
Co-developed with the ALPS MPS project (Troyer).
Integrated into the Molcas program package.
Source code available from www.reiher.ethz.ch/software
Parallelized by MPI and threading
C++ / C++11, 60k lines core program + 50k lines associated libs
+ standard libs (boost, etc.)
S. Keller, M. Dolfi, M. Troyer, M. Reiher, J. Chem. Phys. 143, 244118 (2015)
DMRG in Quantum Chemistry Markus Reiher 28 / 77
www.reiher.ethz.ch/software
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DMRG in Quantum Chemistry Markus Reiher 29 / 77
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A well working example: FCI for N2 (14e, 28o)
DMRG in Quantum Chemistry Markus Reiher 30 / 77
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A well working example: FCI for N2 (14e, 28o)
DMRG in Quantum Chemistry Markus Reiher 31 / 77
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Does DMRG Work for Compact Molecules?
Original ’opinion’ in the DMRG community:
Works only for pseudo-one-dimensional, non-compact systems!
Test for a mononuclear transition metal system CAS(10,10): CoH
m Esinglet/Eh Etriplet/Eh ∆E/kJmol−1
64 −1381.952 054 −1381.995 106 113.0376 −1381.952 063 −1381.995 109 113.0291 −1381.952 070 −1381.995 110 113.00109 −1381.952 073 −1381.995 110 112.99CAS(10,10) −1381.952 074 −1381.995 110 112.99CASPT2(10,10) −1382.189 527 −1382.241 333 130.57DFT/BP86 −1383.504 019 −1383.585 212 213.1DFT/B3LYP −1383.202 267 −1383.279 574 203.0
K. Marti, I. Malkin Ondik, G. Moritz, M. Reiher, J. Chem. Phys 128 (2008) 014104
DMRG in Quantum Chemistry Markus Reiher 32 / 77
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Example: Analysis of Spin Density Distributions
with DMRG
DMRG in Quantum Chemistry Markus Reiher 33 / 77
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Noninnocent Iron Nitrosyl Complexes
(a) Fe(salen)(NO) conformation a
(b) Fe(salen)(NO) conformation b
(c) Fe(porphyrin)(NO)
transition metal nitrosyl complexes
have a complicated electronic
structure
qualitatively different spin
densities reported by Conradie
& GhoshJ. Conradie, A. Ghosh, J. Phys. Chem. B 2007, 111,
12621.
systematic comparison of DFT
spin densities with CASSCF:K. Boguslawski, C. R. Jacob, M. Reiher, J. Chem.
Theory Comput. 2011, 7, 2740;
see also work by K. Pierloot et al.
DMRG in Quantum Chemistry Markus Reiher 34 / 77
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DFT Spin Densities: A Case Study
(a) OLYP (b) OPBE (c) BP86 (d) BLYP
(e) TPSS (f) TPSSh (g) M06-L (h) B3LYP
Only for high-spin complexes similar spin densities are obtained
⇒ [Fe(NO)]2+ moiety determines the spin density
DMRG in Quantum Chemistry Markus Reiher 35 / 77
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The Model System for Accurate Reference Calculations
zyx
Fe
N
O
�
dpc dpc
dpcdpc
Structure:
Four point charges of −0.5 e model a square-planarligand field (ddp = 1.131 Å)
⇒ Similar differences in DFT spin densities as present forlarger iron nitrosyl complexes
Advantage of the small system size:
Standard correlation methods (CASSCF, . . .) can be efficiently
employed
Study convergence of the spin density w.r.t. the size of the active
orbital space
K. Boguslawski, C. R. Jacob, M. Reiher, J. Chem. Theory Comput. 2011, 7, 2740.
DMRG in Quantum Chemistry Markus Reiher 36 / 77
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DFT Spin Densities
(a) OLYP (b) OPBE (c) BP86 (d) BLYP
(e) TPSS (f) TPSSh (g) M06-L (h) B3LYP
Spin density isosurface plots Spin density difference plots w.r.t.
OLYP
⇒ Similar differences as found for the large iron nitrosyl complexes
DMRG in Quantum Chemistry Markus Reiher 37 / 77
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Reference Spin Densities
from Standard Electron Correlation Methods
DMRG in Quantum Chemistry Markus Reiher 38 / 77
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Reference Spin Densities from CASSCF Calculations
Defining the active orbital space:
Minimal active space: Fe 3d- and both NO π∗-orbitals ⇒ CAS(7,7)
Consider also both NO π-orbitals ⇒ CAS(11,9)
Additional shell of Fe d-orbitals (double-shell orbitals) is gradually
included ⇒ CAS(11,11) to CAS(11,14)
Include one ligand σ-orbital and the antibonding σ∗-orbital upon the
CAS(11,11) ⇒ CAS(13,13)
⇒ Analyze convergence of spin density w.r.t. the dimension of the activeorbital space
DMRG in Quantum Chemistry Markus Reiher 39 / 77
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CASSCF Results: Oscillating Spin Densities
CAS(7,7) CAS(11,9) CAS(11,11) CAS(11,12) CAS(11,14)
CAS(11,13) CAS(13,13) CAS(13,14) CAS(13,15) CAS(13,16)
CAS(11,14): spin
density; all others:
spin density
differences
stable CAS with all
important orbitals is
difficult to obtain
⇒ Reference spindensities for very
large CAS required
DMRG in Quantum Chemistry Markus Reiher 40 / 77
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DMRG Spin Densities — Measures of Convergence
Qualitative convergence measure: spin density difference plots
Quantitative convergence measure:
∆abs =
∫|ρs1(r)− ρs2(r)|dr < 0.005 (18)
∆sq =
√∫|ρs1(r)− ρs2(r)|2dr < 0.001 (19)
Quantitative convergence measure: quantum fidelity Fm1,m2
Fm1,m2 = |〈Ψ(m1)|Ψ(m2)〉|2 (20)
⇒ Reconstructed CI expansion of DMRG wave function can be used!K. Boguslawski, K. H. Marti, M. Reiher, J. Chem. Phys. 2011, 134, 224101.
DMRG in Quantum Chemistry Markus Reiher 41 / 77
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DMRG Spin Densities for Large Active Spaces∆abs and ∆sq for DMRG(13,y)[m] calculations w.r.t. DMRG(13,29)[2048] reference
Method ∆abs ∆sq Method ∆abs ∆sq
DMRG(13,20)[128] 0.030642 0.008660 DMRG(13,29)[128] 0.032171 0.010677
DMRG(13,20)[256] 0.020088 0.004930 DMRG(13,29)[256] 0.026005 0.006790
DMRG(13,20)[512] 0.016415 0.003564 DMRG(13,29)[512] 0.010826 0.003406
DMRG(13,20)[1024] 0.015028 0.003162 DMRG(13,29)[1024] 0.003381 0.000975
DMRG(13,20)[2048] 0.014528 0.003028
DMRG(13,20;128) DMRG(13,20;256) DMRG(13,20;512) DMRG(13,20;1024) DMRG(13,20;2048)
DMRG(13,29;128) DMRG(13,29;256) DMRG(13,29;512) DMRG(13,29;1024) DMRG(13,29;2048)
DMRG in Quantum Chemistry Markus Reiher 42 / 77
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Importance of Empty Ligand Orbitals
Table: Some important Slater determinants with large CI weights from DMRG(13,29)[m]Upper part: Slater determinants containing an occupied dx2−y2 -double-shell orbital (marked in
bold face). Bottom part: Configurations with occupied ligand orbitals (marked in bold face). 2:
doubly occupied; a: α-electron; b: β-electron; 0: empty.
CI weight
Slater determinant m = 128 m = 1024
b2b222a0a0000000 0000000 a 00000 0.003 252 0.003 991
bb2222aa00000000 0000000 a 00000 −0.003 226 −0.003 611222220ab00000000 0000000 a 00000 −0.002 762 −0.003 328ba2222ab00000000 0000000 a 00000 0.002 573 0.003 022
b2a222a0b0000000 0000000 a 00000 −0.002 487 −0.003 017202222ab00000000 0000000 a 00000 0.002 405 0.002 716
b222a2a0b0000000 0000000 0 0000a 0.010 360 0.011 558
22b2a2a0a0000000 0000000 0 b0000 0.009 849 0.011 366
22b2a2a0b0000000 0000000 0 a0000 −0.009 532 −0.011 457b2222aab00000000 0000000 0 0000a −0.009 490 −0.010 991a2222baa00000000 0000000 0 0000b −0.009 014 −0.010 017b2b222a0a0000000 0000000 0 0a000 0.008 820 0.010 327
DMRG in Quantum Chemistry Markus Reiher 43 / 77
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Assessment of CASSCF Spin Densities (Differences)
CAS(11,11) CAS(11,12) CAS(11,13) CAS(11,14)
CAS(13,13) CAS(13,14) CAS(13,15) CAS(13,16)
⇒ CASSCF spin densities oscillate around DMRG reference distribution
DMRG in Quantum Chemistry Markus Reiher 44 / 77
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DFT–DMRG Spin Densities Differences
OLYP OPBE BP86 BLYP
TPSS TPSSh M06-L B3LYP
K. Boguslawski, K. H. Marti, Ö. Legeza, M. Reiher, J. Chem. Theory Comput. 8 2012 1970
DMRG in Quantum Chemistry Markus Reiher 45 / 77
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Analyzing DMRG and correlated wave functions
with concepts from quantum information theory
DMRG in Quantum Chemistry Markus Reiher 46 / 77
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Entanglement Measures for Embedded Subsystems
See pioneering work by Ö. Legeza within QC-DMRG
Consider one or two orbitals embedded in a CAS
Measure for entanglement of states on orbital i with those defined on
environment:
Ö. Legeza, J. Sólyom, Phys. Rev. B 2003, 68, 195116.
von-Neumann-type single-orbital entropy
s(1)i = −4∑
α=1
ωα,i lnωα,i (21)
(ωα,i is an eigenvalue of the 1o-RDM of spatial orbital i — states
defined on all other orbitals of the CAS have been traced out)
DMRG in Quantum Chemistry Markus Reiher 47 / 77
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Entanglement Measures for Embedded Subsystems
Measure for entanglement of states living on orbital i and orbital j
with those in the environment:
von-Neumann-type two-orbital entropy
s(2)ij = −16∑α=1
ωα,ij lnωα,ij (22)
(ωα,ij is an eigenvalue of the 2o-RDM of two spatial orbitals i and j
— states defined on all other orbitals of the CAS have been traced
out)
Note the difference between ne-RDMs and no-RDMs !
DMRG in Quantum Chemistry Markus Reiher 48 / 77
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Entanglement Measures for Embedded Subsystems
s(2)ij contains also the ’on-site’ entropies for the two orbitals
⇒ Subtract these contributions to obtain the ’inter-orbital entropy’:J. Rissler, R.M. Noack, S.R. White, Chem. Phys. 2006, 323, 519.
Mutual information
Iij ∝ s(2)ij − s(1)i − s(1)j (23)
Successfully applied to optimize orbital ordering and to enhance
DMRG convergence by Ö. Legeza
DMRG in Quantum Chemistry Markus Reiher 49 / 77
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How To Read the (New) Mutual Information Plots
DMRG in Quantum Chemistry Markus Reiher 50 / 77
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Entanglement and Orbitals — Single Orbital Entropy
0 5 10 15 20 25 300
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Orbital index
s(1
)
K. Boguslawski, P. Tecmer, Ö. Legeza, M. Reiher,
J. Phys. Chem. Lett. 2012, 3, 3129.
Three groups of orbitals
⇒ large single orbital entropy⇒ medium single orbital entropy⇒ tiny single orbital entropyConfigurations belonging to the
third block have small CI ⇒important for dynamic
correlation
Note: more structure than in
spectrum of 1e-RDM
zyx
Fe
N
O
�
dpc dpc
dpcdpc
4 point charges in xy-plane at dpc = 1.133 Å
Natural orbital basis: CAS(11,14)SCF/cc-pVTZ
DMRG(13,29) with DBSS (mmin = 128,mmax = 1024)
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Entanglement and Orbitals — Mutual Information
1
3
5
7
9
11
1315
17
19
21
23
25
27
29
2
4
6
8
10
12
1416
18
20
22
24
26
28
A
100
10−1
10−2
10−3
K. Boguslawski, P. Tecmer, Ö. Legeza, M. Reiher,
J. Phys. Chem. Lett. 2012, 3, 3129.
Three groups of orbitals
⇒ high entanglement⇒ medium entanglement⇒ weak entanglementStrong entanglement for pairs:
(d, π∗)⇐⇒ (d, π∗)∗π ⇐⇒ π∗
σMetal ⇐⇒ σLigand⇒ Important for static correlation (⇐⇒
chemical intuition of constructing a
CAS)
zyx
Fe
N
O
�
dpc dpc
dpcdpc
4 point charges in xy-plane at dpc = 1.133 Å
Natural orbital basis: CAS(11,14)SCF/cc-pVTZ
DMRG(13,29) with DBSS (mmin = 128,mmax = 1024)
DMRG in Quantum Chemistry Markus Reiher 52 / 77
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Entanglement Measures can Monitor
Bond Breaking/Formation Processes: Dinitrogen
K. Boguslawski, P. Tecmer, G. Barcza, O. Legeza, M. Reiher, J. Chem. Theory Comput. 9 2013
2959–2973 [arxiv: 1303.7207]
DMRG in Quantum Chemistry Markus Reiher 53 / 77
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Bond Breaking in Dinitrogen at 1.12 Ångström
Part of entanglement is already encoded in molecular orbitals changing with the structure !
Atomic-like non-orthogonal basis fcts. exhibit large entanglement measures among each other.
DMRG in Quantum Chemistry Markus Reiher 54 / 77
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Bond Breaking in Dinitrogen at 1.69 Ångström
Part of entanglement is already encoded in molecular orbitals changing with the structure !
Atomic-like non-orthogonal basis fcts. exhibit large entanglement measures among each other.
DMRG in Quantum Chemistry Markus Reiher 55 / 77
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Bond Breaking in Dinitrogen at 2.22 Ångström
Part of entanglement is already encoded in molecular orbitals changing with the structure !
Atomic-like non-orthogonal basis fcts. exhibit large entanglement measures among each other.
DMRG in Quantum Chemistry Markus Reiher 56 / 77
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Bond Breaking in Dinitrogen at 3.18 Ångström
Part of entanglement is already encoded in molecular orbitals changing with the structure !
Atomic-like non-orthogonal basis fcts. exhibit large entanglement measures among each other.
DMRG in Quantum Chemistry Markus Reiher 57 / 77
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Artifacts of Small Active Space CalculationsComparison of DMRG(11,9)[220] and DMRG(11,11)[790] to
DMRG(13,29)[128,1024,10−5] for [Fe(NO)]2+
1
3
5
7
9
2
4
6
8
A
100
10−1
10−2
10−3
1
3
5
7
9
11
2
4
6
8
10
A
100
10−1
10−2
10−3
1
3
5
7
9
11
1315
17
19
21
23
25
27
29
2
4
6
8
10
12
1416
18
20
22
24
26
28
A
100
10−1
10−2
10−3
Iij (and s(1)i) overestimated for
small active spaces
⇒ Entanglement of orbitals too large⇒ Missing dynamic correlation is
captured in an artificial way
DMRG in Quantum Chemistry Markus Reiher 58 / 77
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How to choose an active space?
DMRG in Quantum Chemistry Markus Reiher 59 / 77
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How to choose an active space? — Opinions
“... the choice of the active space actually used in more complex
systems is highly subjective and can lead to serious problems.“
R. D. Bach in The Chemistry of Peroxides, Z. Rappoport (Ed.), John Wiley & Sons, (2006) 4.
”CAS-based methods are another alternative, although the selection
of the active space is a tremendous challenge.“
Y. Shao, L. Molnar, Y. Jung, J. Kussmann, C. Ochsenfeld et al., Phys. Chem. Chem. Phys., 8, (2006) 3172–3191.
“Choosing the “correct” active space for a specific application is by
no means trivial; often the practitioner must “experiment” with
different choices in order to assess adequacy and convergence
behavior. While every chemical system poses its own challenges,
certain rules of thumb apply.”
P. Å. Malmqvist, K. Pierloot, A. R. M. Shahi, C. J. Cramer, L. Gagliardi, J. Chem. Phys., 128, (2008) 204109.
DMRG in Quantum Chemistry Markus Reiher 60 / 77
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How to choose an active space? — Recipes
B. O. Roos (1989)The Complete Active Space Self-Consistent Field Method and Its Applications in
Electronic Structure Calculations; in Ab Initio Methods in Quantum Chemistry,
K.P. Lawley (Ed.), John Wiley & Sons Ltd., 399–446.
- most active orbitals should appear paired (one highly occupied and
one corresponding almost empty orbital)
- conjugated and aromatic bonds should be included in the CAS
- both the bonding and antibonding orbitals of a bond that is broken
have to be included
- orbitals describing C-H bonds are not to be included in active space
DMRG in Quantum Chemistry Markus Reiher 61 / 77
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How to choose an active space? — Recipes
J. M. Bofill, P. Pulay (1989)J. Chem. Phys. 90, 3637–3646.
- unrestricted Hartree-Fock natural orbitals (UNOs) are used as
starting orbitals for CASSCF calculations
- orbitals with occupation numbers between 0.02 and 1.98 are
selected for the active space
- the idea has been explored further by comparison to DMRG results:
S. Keller, K. Boguslawski, T. Janowski, M. Reiher, P. Pulay (2015)J. Chem. Phys. 142, 244104.
DMRG in Quantum Chemistry Markus Reiher 62 / 77
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Can one choose the CAS in an automated way?
DMRG in Quantum Chemistry Markus Reiher 63 / 77
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Exploit Two Advantages of DMRG for the CAS choice
DMRG results obtained after (micro)iterations, qualitative structure
of wave function converges quickly
The CAS is not limited by exponentially scaling basis size (the scaling
depends on the no. of orbitals, not on the no. of electrons; 100
orbitals are about the limit)
DMRG in Quantum Chemistry Markus Reiher 64 / 77
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Example: Transition Structure of Ozone
0 5 10 15 20 25 30DMRG microiteration steps
-224.40
-224.38
-224.35
-224.33
-224.30
-224.28
ener
gy
/ H
artr
ee
HFCASCIDMRG
O3
transition state energy
15 20 25 30iteration
-224.3845
-224.3840
-224.3835
-224.3830
ener
gy
/ H
artr
ee
0 5 10 15 20 25 30DMRG microiteration steps
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
abs.
CI
coef
fici
ent
SD1: 111001000000110110SD2: 110110000000111001SD3: 111100000000111100SD4: 111100000000111001SD5: 111010000000110101SD6: 101101000000111100
O3
transition state CI coefficients
G. Moritz, M. Reiher, J. Chem. Phys. 126 (2007) 244109
(see this reference also for a ’first-generation’ DMRG flow chart)
DMRG in Quantum Chemistry Markus Reiher 65 / 77
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Example: Transition Structure of Ozone
0 5 10 15 20 25 30DMRG microiteration steps
-224.40
-224.38
-224.35
-224.33
-224.30
-224.28
ener
gy
/ H
artr
ee
HFCASCIDMRG
O3
transition state energy
15 20 25 30iteration
-224.3845
-224.3840
-224.3835
-224.3830
ener
gy
/ H
artr
ee
0 5 10 15 20 25 30DMRG microiteration steps
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
abs.
CI
coef
fici
ent
SD1: 111001000000110110SD2: 110110000000111001SD3: 111100000000111100SD4: 111100000000111001SD5: 111010000000110101SD6: 101101000000111100
O3
transition state CI coefficients
G. Moritz, M. Reiher, J. Chem. Phys. 126 (2007) 244109
(see this reference also for a ’first-generation’ DMRG flow chart)
DMRG in Quantum Chemistry Markus Reiher 66 / 77
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Example: Transition Structure of Ozone
0 5 10 15 20 25 30DMRG microiteration steps
-224.40
-224.38
-224.35
-224.33
-224.30
-224.28
ener
gy
/ H
artr
ee
HFCASCIDMRG
O3
transition state energy
15 20 25 30iteration
-224.3845
-224.3840
-224.3835
-224.3830
ener
gy
/ H
artr
ee
0 5 10 15 20 25 30DMRG microiteration steps
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
abs.
CI
coef
fici
ent
SD1: 111001000000110110SD2: 110110000000111001SD3: 111100000000111100SD4: 111100000000111001SD5: 111010000000110101SD6: 101101000000111100
O3
transition state CI coefficients
G. Moritz, M. Reiher, J. Chem. Phys. 126 (2007) 244109
(see this reference also for a ’first-generation’ DMRG flow chart)
DMRG in Quantum Chemistry Markus Reiher 67 / 77
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Exploit Two Advantages of DMRG for the CAS choice
DMRG is iterative
DMRG can handle large CAS sizes
=⇒ ... towards an automated CAS determination
DMRG in Quantum Chemistry Markus Reiher 68 / 77
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Partially converged entanglement information is sufficient
E = -2658.144420 EH E = -2665.877397 EH E = -2665.878045 EH
E = -1449.266845 EH E = -1449.290301 EH E = -1449.302426 EH
HF orbitals, m = 2004 sweepsHF guess
HF orbitals, m = 2004 sweepsHF guess
HF orbitals, m = 5008 sweeps
CI-DEAS guess
HF orbitals, m = 200016 sweeps
CI-DEAS guess
HF orbitals, m = 5008 sweeps
CI−DEAS guess
HF orbitals, m = 200016 sweeps
CI−DEAS guess
sufficiently accurate entanglement information can be obtained from partially
converged calculations C. J. Stein, M. Reiher, J. Chem. Theory Comput, 2016, 12, 1760–1771.
DMRG in Quantum Chemistry Markus Reiher 69 / 77
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Complete Automation of CAS Selection
generate initial orbitals recommended: CASSCF with small CAS or DMRG-SCF
initial DMRG calculation with a large CAS around the Fermi level
recommended settings: CI-DEAS guess, m = 500, 8 sweeps
max. si (1) > 0.14 ?
single-configuration case
generate threshold diagrams
calculate entanglement measures
plateaus?Can orbitals with
si (1) < 1-2 % max. si (1) be excluded? include orbitals kept in the first,
clearly identifiable plateau at low thresholds
converge calculation with
CASSCF or DMRG-SCF
consistency test: Do entanglement measures of the final calculation agree qualitatively with those of the initial calculation?YES
YES
NO
NO
Flowchart for the Automated Selection of Active Orbital Spaces
select CAS anyway
YES
Is it feasible to converge a calculation with this CAS size?
YES
NO
consistency test:
increase initial CAS
NO
STOP: unfeasible for DMRG
C. J. Stein, M. Reiher, J. Chem. Theory Comput, 2016, 12, 1760–1771.
DMRG in Quantum Chemistry Markus Reiher 70 / 77
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Same CAS selection for different types of orbitals
MnO4-
HF orbitals
CAS(10,10)-SCF orbitals
split-localized orbitals
DMRG(38,25)[500]-SCF orbitals
the same CAS is automatically selected for all orbital bases
considered C. J. Stein, M. Reiher, J. Chem. Theory Comput, 2016, 12, 1760–1771.DMRG in Quantum Chemistry Markus Reiher 71 / 77
-
Subtile correlation effects are automatically captured
DMRG-SCF orbitalsm = 1000, 20 sweeps
HF guess
CrF63- CrF6
correlation effects attributed to the covalency of the bonds are automatically
accounted for
C. J. Stein, M. Reiher, J. Chem. Theory Comput, 2016, 12, 1760–1771.
DMRG in Quantum Chemistry Markus Reiher 72 / 77
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The Cu2O2+2 torture track
·−1
bis(µ-oxo) peroxo
F
CR-CCSD(TQ)LCAS(16,14)-SCFDMRG(48,36)[500]-SCFDMRG(32,28)[1000]-SCF#DMRG(48,36)[500]-SCF
automated approach gives qualitatively correct wave
function C. J. Stein, M. Reiher, J. Chem. Theory Comput, 2016, 12, 1760–1771.
discrepancy to coupled-cluster result can be explained by (still) missing
dynamical correlation C. J. Cramer et al., JPC A (2006), 110, 1991.
DMRG in Quantum Chemistry Markus Reiher 73 / 77
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Regularizing effect of short-range DFT: Water
E. D. Hedeg̊ard, S. Knecht, J. S. Kielberg, H. J. A. Jensen, and M. Reiher, J. Chem. Phys. 142 2015 224108
DMRG in Quantum Chemistry Markus Reiher 74 / 77
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DMRG–srDFT for the WCCR10 test set
N
NCu N
N
N
NCu N
N
7Å
Calculated dissociation energies in kJ/mol
Method De (kJ/mol) D0 (kJ/mol)
DMRG[2000](30,22) 173.5 165.1
DMRG[2000](20,18) 169.9 161.5
DMRG[2000](10,10) 132.8 124.3
DMRG[2000](30,22)-srPBE 225.1 216.6
DMRG[2000](20,18)-srPBE 227.9 219.4
DMRG[2000](10,10)-srPBE 216.5 208.0
PBE 240.2 231.8
PBE (full complex/def2-TZVP) 257.5 249.0
PBE (full complex/def2-QZVPP from WCCR10) 247.5 239.0
Exp. (from WCCR10) 226.7 218.2
E. D. Hedeg̊ard, S. Knecht, J. S. Kielberg, H. J. A. Jensen, and M. Reiher, J. Chem. Phys. 142 2015 224108;WCCR10: T. Weymuth, E. P. A. Couzijn, P. Chen, M. Reiher, J. Chem. Theory Comput. 10 2014 3092
DMRG in Quantum Chemistry Markus Reiher 75 / 77
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Conclusion
new and efficient second-generation DMRG program for full
Schrödinger–Coulomb (Dirac–Gaunt/Breit) Hamiltonian available:
QCMaquis
HF-, split-localized- and CASSCF-orbitals allow to select an active
space that is in line with the literature suggestions
the whole CAS evaluation procedure is fast
... and can be automatized
dynamic correlation introduced through short-range DFT regularizes
the CAS
DMRG in Quantum Chemistry Markus Reiher 76 / 77
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Acknowledgments
Christopher Stein, Sebastian Keller
Stefan Knecht, Arseny Kovyrshin
Yingjin Ma, Erik Hedeg̊ard
Andrea Muolo, Pawel Tecmer
Katharina Boguslawski
... as well as the rest
of the group −→
2nd editionFinancial Support: ETH, SNF, FCI
Thank you for your kind
attention!
DMRG in Quantum Chemistry Markus Reiher 77 / 77
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