APPROACHING EXACT QUANTUM CHEMISTRYBY STOCHASTIC …€¦ · and Arnab Chakraborty Department of...

APPROACHING EXACT QUANTUM CHEMISTRY BY STOCHASTIC WAVE FUNCTION SAMPLING AND

DETERMINISTIC COUPLED-CLUSTER COMPUTATIONS

Office of Basic Energy SciencesChemical Sciences, Geosciences

& Biosciences Division

Piotr Piecuch, J. Emiliano Deustua, Jun Shen, Ilias Magoulas, Stephen H. Yuwono, and Arnab Chakraborty

Department of Chemistry and Department of Physics & Astronomy, Michigan State University, East Lansing, Michigan 48824, USA

Quantum International Frontiers, 2nd EditionShanghai, China, November 18-22, 2019

MANY THANKS TO PROFESSOR MAŁGORZATA BICZYSKO FOR INVITATION AND WARM HOSPITALITY

ˆ ( ; ) ( ) ( ; )eH Eµ µ µΨ = ΨX R R X R

ˆ( ( )ˆ ,) ˆ i j

ei i j i

iH z v= = = +

= +∑ ∑ ∑x x x

1ˆ( )2

, ˆ ) 1( ,M

= − ∇ − =∑ xx x

THE ELECTRONIC SCHRÖDINGER EQUATION

pbiradical

Define a basis set of one-electron functions, e.g., LCAO-type molecular spin-orbitals obtained by solving mean-field (e.g., Hartree-Fock) equations

Construct all possible Slater determinants that can be formed from these one-electron states

{ }, i dim dim

( ), 1, ,dimr

V r Vϕ

= ∞ < ∞

≡ =x

Exact case : n practi ce :

( ) ( )1( , , )

! ( ) ( )N

r r NN

ϕ ϕΦ =

x xx x

SOLVING THE ELECTRONIC SCHRÖDINGER EQUATION

The exact wave function can be written as a linear combination of all Slater determinants

Determine the coefficients c and the energies Eμ by solving the matrix eigenvalue problem:

This procedure, referred to as the full configuration interaction approach (FCI), yields the exact solution within a given one-electron basis set

( , , ) ( , , )

( , , )

N r r r r Nr r

I I NI

Ψ = Φ

x x x x

Eµ µµ=HC C

where the matrix elements of the Hamiltonian are

1 1 1ˆ ˆ( , , ) ( , , )KL K L N K N L NH H d d H∗= Φ Φ = Φ Φ∫ x x x x x x

SOLVING THE ELECTRONIC SCHRÖDINGER EQUATION

THE PROBLEM WITH FCI

The high dimensionality of the FCI eigenvalue problem makes this approach inapplicable to systems with more than a few electrons and realistic basis sets

Alternative approaches are needed in order to study the majority of chemical problems of interest

Dimensions of the FCI spaces for many-electron systems

SINGLE-REFERENCE COUPLED-CLUSTER (CC) THEORY(F. Coester, 1958; F. Coester and H. Kümmel, 1960; J. Čížek, 1966,1969; J. Čížek and J. Paldus, 1971)

1p-1h, singles (S) 2p-2h, doubles (D) 3p-3h, triples (T)

← iterative N6

← iterative N8

← iterative N10

SINGLE-REFERENCE COUPLED-CLUSTER (CC) THEORY(F. Coester, 1958; F. Coester and H. Kümmel, 1960; J. Čížek, 1966,1969; J. Čížek and J. Paldus, 1971)

1p-1h, singles (S) 2p-2h, doubles (D) 3p-3h, triples (T)

CPU time scaling with the system size

exact theory (full CI), approximationsA Am N m N= ⇒ < ⇒

(J. Čížek, 1966)

ARGUMENTS IN FAVOR OF THE CC THEORY

Size-extensivity of the resulting approximations (no loss of accuracy occurs when the system is made larger)

Linked cluster (diagram) theorem (Brueckner, 1955; Goldstone, 1957)

Connected cluster theorem (Hubbard, 1957; Hugenholtz, 1957)

Separability or size consistency if the reference state separates correctly

Fastest convergence toward the exact, full CI, limit

Taken from R.J. Bartlett and M. Musiał, Rev. Mod.

Phys., 2007

MBPT(2)

MBPT(4)MBPT(6)

CCSDTQ

CISDTQ

1.0 1.5 2.0

Relative bond length (ROH/ROH,eq.)

Taken from N.P. Bauman, J. Shen, and P. Piecuch, Mol. Phys., 2017

1.0 1.5 2.0

H2OARGUMENTS IN FAVOR OF THE CC THEORY

1.0 1.5 2.0

CCSDTQ

H2OARGUMENTS IN FAVOR OF THE CC THEORY

CI expansion CC expansion

CIS = CI(1p-1h)

CISD = CI(2p-2h)

CISDT = CI(3p-3h)

CISDTQ = CI(4p-4h)

EXCITED STATES: EQUATION-OF-MOTION CC (EOMCC) THEORY, SYMMETRY-ADAPTED-CLUSTER CONFIGURATION INTERACTION

APPROACH (SAC-CI), AND RESPONSE CC METHODS(H. Monkhorst, 1977; D. Mukherjee and P.K. Mukherjee, 1979; H. Nakatsuji and K. Hirao, 1978; K. Emrich, 1981; M.

Takahashi and J. Paldus; 1986; J. Geertsen, M. Rittby, and R.J. Bartlett, 1989)

Example: EOMCC(K. Emrich, 1981; J. Geertsen, M. Rittby, and R.J. Bartlett, 1989; J.F. Stanton and R.J. Bartlett, 1993)

Basic approximation: EOMCCSD

Higher-order methods: EOMCCSDT, EOMCCSDTQ, etc.

Example: EOMCC(K. Emrich, 1981; J. Geertsen, M. Rittby, and R.J. Bartlett, 1989; J.F. Stanton and R.J. Bartlett, 1993)

State EOMCCSD EOMCCSDT EOMCCSDTQ Experiment

a 4Σ– 0.95 0.66 0.65 0.74

A 2Δ 3.33 3.02 3.00 2.88

B 2Σ– 4.41 3.27 3.27 3.23

C 2Σ+ 5.29 4.07 4.04 3.94

Adiabatic excitation energies of the CH radical (in eV) [S. Hirata, 2004]

Vertical excitation energies of C2 (in eV) [K. Kowalski and P. Piecuch, 2002; S. Hirata, 2004]

errors relative to full CI

State EOMCCSD EOMCCSDT EOMCCSDTQ Experiment

a 4Σ– 0.95 0.66 0.65 0.74

A 2Δ 3.33 3.02 3.00 2.88

B 2Σ– 4.41 3.27 3.27 3.23

C 2Σ+ 5.29 4.07 4.04 3.94

Adiabatic excitation energies of the CH radical (in eV) [S. Hirata, 2004]

> 1 eV errors

Vertical excitation energies of C2 (in eV) [K. Kowalski and P. Piecuch, 2002; S. Hirata, 2004]

errors relative to full CI

1.5321.091

105.8o

120.8o

φH1-C1-C2-C3 = 0.0o

φH5-C3-C2-C1 = 0.0o

1.4951.081H1

φH1-C1-C2-C3 = 0.0o

1.081116.1o

121.6o

φH5-C3-C2-C1 = 0.0o

CCSD(T)Vib mode [1/cm]

1 411.2i2 157.1i3 168.14 237.65 336.6

%Av Err 89.3

MRCI Vib mode [1/cm]

1 139.2i2 61.13 277.94 361.35 372.5

… … … …

Trimethylene biradical

CASSCF: 330.0i, 314.5iMCQDPT2: 131.7i

KEY CHALLENGE: How to incorporate Tn and Rn components with n > 2, needed to achieve a quantitative description, without running into prohibitive computational costs of CCSDT, CCSDTQ, and similar schemes?TRADITIONAL SOLUTION: Noniterative perturbative corrections of the CCSD(T) type, iterative CCSDT-n and similar approximations, and their linear-response CCSDR3, CC3, etc. counterparts (replace iterative N8 and N10 steps of CCSDT and CCSDTQ by iterative N6 plus noniterative N7 (N9) or iterative N7 or N9 operations)

BETTER SOLUTION: CC(P;Q) MOMENT EXPANSIONS[J. Shen and P. Piecuch, Chem. Phys., 2012; J. Chem. Phys., 2012; original moment expansions: P. Piecuch and K.

Kowalski, 2000-2001; biorthogonal moment expansions: P. Piecuch and M. Włoch, 2005-2006]

( )( ) ( ; )PP Q EE P Qµµ µδ+ = +

CC (µ = 0) or EOMCC (µ > 0)energy obtained in the P space H(P)

( )( ) ( ; )PP Q EE P Qµµ µδ+ = +

Correction due to correlation effects captured by the Q space H(Q)

( )( ) ( ; )PP Q EE P Qµµ µδ+ = +

moments of CC/EOMCC equations

( )( ) ( ; )PP Q EE P Qµµ µδ+ = +

Examples:

( )( ) ( ; )PP Q EE P Qµµ µδ+ = +

Examples: P space: singly and doubly excited determinants (CCSD) Q space: triply excited determinants

( )( ) ( ; )PP Q EE P Qµµ µδ+ = +

CR-CC(2,3)

( )( ) ( ; )PP Q EE P Qµµ µδ+ = +

CR-CC(2,3) P space: singly and doubly excited determinants (CCSD) Q space: triply and quadruply excited determinants

( )( ) ( ; )PP Q EE P Qµµ µδ+ = +

CR-CC(2,3) P space: singly and doubly excited determinants (CCSD) Q space: triply and quadruply excited determinants

CR-CC(2,4)

(CR-CC(2,3)) (CCSD)0 0 0 0

,00, ,(2,3), (2,3) ) (2)(2 ijk

abcabc

ijki j k a b c

E E δ δ< < < <

= + = ∑ M

1 2 1 2 1 2

(CCSD) (CCSD)0, 0

(CCSD)

(CCSD) (CCSD)0, 0

(CCSD)0,

0,(2) / (2

ijk abca

T T T T T TC

ijk abc abca

abc abc ijki

jk ijk a

bc bc ijk ijk

k H e He He

H D HL

− − + += =

= Φ Φ − Φ Φ

Example: CR-CC(2,3), robust noniterative triples correction to CCSDiterative no

2nu4 (N6) + noniterative no

3nu4 (N7); CCTYP=CR-CCL in GAMESS

F2/cc-pVTZ

(CR-CC(2,3)) (CCSD)0 0 0 0

,00, ,(2,3), (2,3) ) (2)(2 ijk

abcabc

ijki j k a b c

E E δ δ< < < <

= + = ∑ M

1 2 1 2 1 2

(CCSD) (CCSD)0, 0

(CCSD)

(CCSD) (CCSD)0, 0

(CCSD)0,

0,(2) / (2

ijk abca

T T T T T TC

ijk abc abca

abc abc ijki

jk ijk a

bc bc ijk ijk

k H e He He

H D HL

− − + += =

= Φ Φ − Φ Φ

F2/cc-pVTZCCSDT

(CR-CC(2,3)) (CCSD)0 0 0 0

,00, ,(2,3), (2,3) ) (2)(2 ijk

abcabc

ijki j k a b c

E E δ δ< < < <

= + = ∑ M

1 2 1 2 1 2

(CCSD) (CCSD)0, 0

(CCSD)

(CCSD) (CCSD)0, 0

(CCSD)0,

0,(2) / (2

ijk abca

T T T T T TC

ijk abc abca

abc abc ijki

jk ijk a

bc bc ijk ijk

k H e He He

H D HL

− − + += =

= Φ Φ − Φ Φ

F2/cc-pVTZCCSDT

(CR-CC(2,3)) (CCSD)0 0 0 0

,00, ,(2,3), (2,3) ) (2)(2 ijk

abcabc

ijki j k a b c

E E δ δ< < < <

= + = ∑ M

1 2 1 2 1 2

(CCSD) (CCSD)0, 0

(CCSD)

(CCSD) (CCSD)0, 0

(CCSD)0,

0,(2) / (2

ijk abca

T T T T T TC

ijk abc abca

abc abc ijki

jk ijk a

bc bc ijk ijk

k H e He He

H D HL

− − + += =

= Φ Φ − Φ Φ

F2/cc-pVTZCCSDT

CCSD(T)

(CR-CC(2,3)) (CCSD)0 0 0 0

,00, ,(2,3), (2,3) ) (2)(2 ijk

abcabc

ijki j k a b c

E E δ δ< < < <

= + = ∑ M

1 2 1 2 1 2

(CCSD) (CCSD)0, 0

(CCSD)

(CCSD) (CCSD)0, 0

(CCSD)0,

0,(2) / (2

ijk abca

T T T T T TC

ijk abc abca

abc abc ijki

jk ijk a

bc bc ijk ijk

k H e He He

H D HL

− − + += =

= Φ Φ − Φ Φ

F2/cc-pVTZ

CR-CC(2,3)

CCSD(T)

(CR-CC(2,3)) (CCSD)0 0 0 0

,00, ,(2,3), (2,3) ) (2)(2 ijk

abcabc

ijki j k a b c

E E δ δ< < < <

= + = ∑ M

1 2 1 2 1 2

(CCSD) (CCSD)0, 0

(CCSD)

(CCSD) (CCSD)0, 0

(CCSD)0,

0,(2) / (2

ijk abca

T T T T T TC

ijk abc abca

abc abc ijki

jk ijk a

bc bc ijk ijk

k H e He He

H D HL

− − + += =

= Φ Φ − Φ Φ

FunctionalBDE (with ZPE),

kcal/mol, 6-31G(d) / 6-311++G(d,p)

BHandHLYP (50% HF) 1.2 / -2.2

MPW1K (42.8 % HF) 8.9 / 6.0

M06-2X (54 % HF) 13.5 / 9.2

MPW1PW91 (25 % HF) 18.1 / 16.1

B3LYP (20 % HF) 17.8 / 15.9

B3LYP+D3 (20 % HF) 21.2 / 24.7

M06 (27 % HF) 27.4 / 26.2

TPSSh (10 % HF) 24.5 / 23.0

ωB97X-D (22 % HF) 26.8 / 24.8

BLYP 25.7 / 24.8

TPSS 29.1 / 28.1

MPWPW91 30.3 / 29.7

M06-L 31.3 / 29.8

BP86 30.6 / 30.0

BP86+D3 35.2 / 39.7

B97-D 35.1 / 34.8

CIM-CR-CC(2,3)/CCSD 39.8 / 37.8

Experiment 37 ± 3, 36 ± 4

LARGE SYSTEMS (LOCAL CC): CIM-CR-CC(2,3) STUDY OF Co-C BOND DISSOCIATON IN METHYLCOBALAMIN

[P.M. Kozłowski, M. Kumar, P. Piecuch, W. Li, N.P. Bauman, J.A. Hansen, P. Lodowski, and M. Jaworska,

J. Chem. Theory Comput. 8, 1870 (2012)]

Exp.CIM-CR-CC(2,3)

Structural model: 58 atoms; 234 electrons.

[In GAMESS: CIMTYP = GSECIM]

FunctionalBDE (with ZPE),

kcal/mol, 6-31G(d) / 6-311++G(d,p)

BHandHLYP (50% HF) 1.2 / -2.2

MPW1K (42.8 % HF) 8.9 / 6.0

M06-2X (54 % HF) 13.5 / 9.2

MPW1PW91 (25 % HF) 18.1 / 16.1

B3LYP (20 % HF) 17.8 / 15.9

B3LYP+D3 (20 % HF) 21.2 / 24.7

M06 (27 % HF) 27.4 / 26.2

TPSSh (10 % HF) 24.5 / 23.0

ωB97X-D (22 % HF) 26.8 / 24.8

BLYP 25.7 / 24.8

TPSS 29.1 / 28.1

MPWPW91 30.3 / 29.7

M06-L 31.3 / 29.8

BP86 30.6 / 30.0

BP86+D3 35.2 / 39.7

B97-D 35.1 / 34.8

CIM-CR-CC(2,3)/CCSD 39.8 / 37.8

Experiment 37 ± 3, 36 ± 4

CASSCF(11,10), CASPT2(11,10) 15.1, 53.8

LARGE SYSTEMS (LOCAL CC): CIM-CR-CC(2,3) STUDY OF Co-C BOND DISSOCIATON IN METHYLCOBALAMIN

[P.M. Kozłowski, M. Kumar, P. Piecuch, W. Li, N.P. Bauman, J.A. Hansen, P. Lodowski, and M. Jaworska,

J. Chem. Theory Comput. 8, 1870 (2012)]

Exp.CIM-CR-CC(2,3)

Structural model: 58 atoms; 234 electrons.

[In GAMESS: CIMTYP = GSECIM]

an aromatic system with first excited-state dominated by aπ π* electronic transition

Vertical excitation energies of cis-7-hydroxyquinoline in various molecular environments

Theory: G. Fradelos, J.J. Lutz, P. Piecuch, T.A. Wesolowski, and M. Włoch, J. Chem. Theory Comput., 2011. Experiment: S. Leutwyler et al., J. Phys. Chem. A, 2008

G. Fradelos, J.J. Lutz, P. Piecuch, and T.A. Wesolowski, in preparation

Environment Exp. EOMCCSD Error δ-CR-EOMCC(2,3) Errorbare 30830 35046 4216 30711 -119

an aromatic system with first excited-state dominated by aπ π* electronic transition

(Values in cm-1 obtained using 6-311+G(d)/6-31+G(d) basis set; 350 cm-1 ≈ 1 kcal/mol )

Environment Exp. EOMCCSD Error δ-CR-EOMCC(2,3) Errorbare 30830 35046 4216 30711 -119NH3 29925 34263 4338 29922 -3

Environment Exp. EOMCCSD Error δ-CR-EOMCC(2,3) Errorbare 30830 35046 4216 30711 -119NH3 29925 34263 4338 29922 -3H2O 30240 34500 4260 30199 -41

Environment Exp. EOMCCSD Error δ-CR-EOMCC(2,3) Errorbare 30830 35046 4216 30711 -119NH3 29925 34263 4338 29922 -3H2O 30240 34500 4260 30199 -41H2O + H2O 29193 33699 4506 29378 185

Environment Exp. EOMCCSD Error δ-CR-EOMCC(2,3) Errorbare 30830 35046 4216 30711 -119NH3 29925 34263 4338 29922 -3H2O 30240 34500 4260 30199 -41H2O + H2O 29193 33699 4506 29378 185NH3 + H2O + H2O 28340 33218 4878 28863 523

Reactant TS Barrier Height (kcal/mol)

CCSD 26.827 47.979 20.9

CCSD(T) 1.123 14.198 15.8CR-CC(2,3) 0.848 14.636 16.3

CCSDT -154.244157 -154.232002 7.6

Various CC energies (in millihartree) relative to full CCSDT (in hartree), cc-pVDZ basis set

AUTOMERIZATION OF CYCLOBUTADIENE

[J. Shen and P. Piecuch, J. Chem. Phys., 2012]

CCSD 26.827 47.979 20.9

CCSD(T) 1.123 14.198 15.8CR-CC(2,3) 0.848 14.636 16.3

CCSDT -154.244157 -154.232002 7.6

[J. Shen and P. Piecuch, J. Chem. Phys., 2012]

T1 and T2decoupled from T3

Core ( , , ,...)i j k

Active Unoccupied ( , , ,...)A B C

Virtual ( , , ,...)a b c

Active Occupied ( , , ,...)I J K

Unoccupied ( , , ,...)a b c

Occupied ( , , ,...)i j k

CAPTURING THE COUPLING OF LOWER- AND HIGHER-ORDER CLUSTERS: ACTIVE-SPACE CC APROACHES (CCSDt, CCSDtq, etc.)

[Key concepts: Oliphant and Adamowicz, 1991; Piecuch, Oliphant, and Adamowicz, 1993; Piecuch and Adamowicz, 1994; Piecuch, Kucharski, and Bartlett, 1999; Kowalski and Piecuch, 2000-2001; Gour, Piecuch, and Włoch, 2005-2006;

Shen, Ajala, and Piecuch, 2013-2017; cf., also, CASCC work by Adamowicz et al.]

𝑇𝑇 CCSDt = 𝑇𝑇1 + 𝑇𝑇2 + 𝑡𝑡3 , 𝑇𝑇 CCSDtq = 𝑇𝑇1 + 𝑇𝑇2 + 𝑡𝑡3 + 𝑡𝑡4, etc.

𝑡𝑡3 = �𝐈𝐈>𝑗𝑗>𝑘𝑘𝑎𝑎>𝑏𝑏>𝐂𝐂

𝑡𝑡𝑎𝑎𝑏𝑏𝐂𝐂𝐈𝐈𝑗𝑗𝑘𝑘 𝐸𝐸𝐈𝐈𝑗𝑗𝑘𝑘𝑎𝑎𝑏𝑏𝐂𝐂, 𝑡𝑡4 = �

𝐈𝐈>𝐉𝐉>𝑘𝑘>𝑙𝑙𝑎𝑎>𝑏𝑏>𝐂𝐂>𝐃𝐃

𝑡𝑡𝑎𝑎𝑏𝑏𝐂𝐂𝐃𝐃𝐈𝐈𝐉𝐉𝑘𝑘𝑙𝑙 𝐸𝐸𝐈𝐈𝐉𝐉𝑘𝑘𝑙𝑙𝑎𝑎𝑏𝑏𝐂𝐂𝐃𝐃

Method CPU Time ScalingCCSDt/EOMCCSDt NoNuno

CCSDtq/EOMCCSDtq No2Nu

2no2nu

CCSD 26.827 47.979 20.9

CCSD(T) 1.123 14.198 15.8CR-CC(2,3) 0.848 14.636 16.3

CCSDt 20.786 20.274 7.3

CCSDT -154.244157 -154.232002 7.6

[J. Shen and P. Piecuch, J. Chem. Phys, 2012]

CCSD 26.827 47.979 20.9

CCSD(T) 1.123 14.198 15.8CR-CC(2,3) 0.848 14.636 16.3

CCSDt 20.786 20.274 7.3

CCSDT -154.244157 -154.232002 7.6

[J. Shen and P. Piecuch, J. Chem. Phys, 2012]

t3 misses some dynamical

correlations

( )( ) ( ; )PP Q EE P Qµµ µδ+ = +

Examples:

( )( ) ( ; )PP Q EE P Qµµ µδ+ = +

Examples: P space: singles, doubles, and a subset of triples defined via active orbitals,

as in CCSDt Q space: remaining triples not captured by CCSDt

( )( ) ( ; )PP Q EE P Qµµ µδ+ = +

CC(t;3)

( )( ) ( ; )PP Q EE P Qµµ µδ+ = +

CC(t;3) P space: singles, doubles, and a subset of triples and quadruples defined

via active orbitals, as in CCSDtq Q space: remaining triples and quadruples not captured by CCSDtq

( )( ) ( ; )PP Q EE P Qµµ µδ+ = +

CC(t;3) P space: singles, doubles, and a subset of triples and quadruples defined

via active orbitals, as in CCSDtq Q space: remaining triples and quadruples not captured by CCSDtq

CC(t,q;3,4)

Reactant TSCCSDT -154.244157 -154.232002CCSD 26.827 47.979

CCSD(T) 1.123 14.198CR-CC(2,3) 0.848 14.636

CCSDt(I) 20.786 20.274CCSD(T)-h(I) -0.371 -4.548CC(t;3)(I) -0.137 0.071

Various CC energies (in millihartree) relative to full CCSDT (in hartree), cc-pVDZ

Barrier heights (in kcal/mol) →

[J. Shen and P. Piecuch, J. Chem. Phys. 136, 144104 (2012)]

CCSD(T) 1.123 14.198CR-CC(2,3) 0.848 14.636

-22.2-25.2

con_TS dis_TS g-but gt_TS t-butThe Conrotatoryand DisrotatoryIsomerization Pathways of Bicyclo[1.1.0]-butane to Butadiene (enthalpies at 0 K in kcal/mol)

-22.2-25.2

58.6c A. R. Berner and A. Lüchow, J. Phys. Chem. A 114, 13222 (2010)

con_TS dis_TS g-but gt_TS t-but

OMR3-DMCc 40.4(5) 58.6(5) -25.2(5) -22.2(5) -27.9(5)

The Conrotatoryand DisrotatoryIsomerization Pathways of Bicyclo[1.1.0]-butane to Butadiene (enthalpies at 0 K in kcal/mol)

-22.2-25.2

a A. Kinal and P. Piecuch, J. Phys. Chem. A 111, 734 (2007)

OMR3-DMCc 40.4(5) 58.6(5) -25.2(5) -22.2(5) -27.9(5)

CCSD(T)a 40.4 21.8 -25.1 -22.3 -28.0

-22.2-25.2

a A. Kinal and P. Piecuch, J. Phys. Chem. A 111, 734 (2007)

OMR3-DMCc 40.4(5) 58.6(5) -25.2(5) -22.2(5) -27.9(5)

CR-CC(2,3)a 41.1 66.1 -24.9 -22.1 -27.9

CCSD(T)a 40.4 21.8 -25.1 -22.3 -28.0

-22.2-25.2

a A. Kinal and P. Piecuch, J. Phys. Chem. A 111, 734 (2007)b J. Shen and P. Piecuch, J. Chem. Phys. 136, 144104 (2012)

OMR3-DMCc 40.4(5) 58.6(5) -25.2(5) -22.2(5) -27.9(5)

CR-CC(2,3)a 41.1 66.1 -24.9 -22.1 -27.9

CCSD(T)a 40.4 21.8 -25.1 -22.3 -28.0

CCSDtb 40.1 59.0 -27.2 -25.3 -31.1

-22.2-25.2

a A. Kinal and P. Piecuch, J. Phys. Chem. A 111, 734 (2007)b J. Shen and P. Piecuch, J. Chem. Phys. 136, 144104 (2012)

CC(t;3)b 40.2 60.1 -25.3 -22.6 -28.3

OMR3-DMCc 40.4(5) 58.6(5) -25.2(5) -22.2(5) -27.9(5)

CR-CC(2,3)a 41.1 66.1 -24.9 -22.1 -27.9

CCSD(T)a 40.4 21.8 -25.1 -22.3 -28.0

CCSDtb 40.1 59.0 -27.2 -25.3 -31.1

Magnesium Dimer[S.H. Yuwono, I. Magoulas, J. Shen, and P. Piecuch, Mol. Phys. 117, 1486 (2019)]

(CC(t;3)/AwCQZ)

(CCSDT/AwCQZ

(CCSD(T)/AwCQZ) 1e

(CR-CC )

(2,3)/AwCQZ

e i ent) -e

366.7 cm

1.4 cm

3.4 cmD

=CCSDT

CCSD(T)CR-CC(2,3)

CC(t;3)

Magnesium Dimer[S.H. Yuwono, I. Magoulas, J. Shen, and P. Piecuch, Mol. Phys. 117, 1486 (2019)]

(CC(t;3)/AwCQZ)

(CCSDT/AwCQZ

(CCSD(T)/AwCQZ) 1e

(CR-CC )

(2,3)/AwCQZ

e i ent) -e

366.7 cm

1.4 cm

3.4 cmD

=CCSDT

CCSD(T)CR-CC(2,3)

CC(t;3)

The CC(t;3), CC(t,q;3), CC(t,q;3,4), etc. hierarchy works well, but it requires choosing user- and system-dependent active orbitals to select the dominant Tn

and Rn components with n > 2 prior to the determination of CC(P;Q) corrections, i.e., it is not a black-box methodology.

and Rn components with n > 2 prior to the determination of CC(P;Q) corrections, i.e., it is not a black-box methodology.QUESTIONS:

and Rn components with n > 2 prior to the determination of CC(P;Q) corrections, i.e., it is not a black-box methodology.QUESTIONS: Is there an automated way of determining P spaces reflecting on the nature

of states being calculated, while using corrections δµ(P;Q) to capture the remaining correlations of interest?

Can this be done such that the resulting electronic energies rapidly converge to their high-level (CCSDT, CCSDTQ, etc.) parents, even when higher–than–two-body clusters become large, at the small fraction of the computational effort and with an ease of a black-box computation?

Both questions have positive answers if we fuseDETERMINISTIC CC(P;Q) METHODOLOGY

withSTOCHASTIC CI AND CC MONTE CARLO.

[J.E. Deustua, J. Shen, and P. Piecuch, Phys. Rev. Lett. 119, 223003 (2017)]

CI QUANTUM MONTE CARLO (CIQMC)

CC MONTE CARLO (CCMC)

0 0 0( )

forlim ( ) lim for

c S Ee S E

τ ττ − −

→∞ →∞

Ψ =Ψ = Φ = ∞ > <

0 0( ) ( ) ( )K KK

c cτ τ τΨ = Φ + Φ∑

0 0( )If , lim =0 and we obtain ( ) ( )K

KL L KL

cS E H c E cτ

ττ→∞

∂→ ∞ = ∞

∂ ∑

( ) ( ) ( ) ( )KKK K KL L

c H S c H cτ τ ττ ≠

∂= − − −

∂ ∑

CIQMC (FCIQMC, CISDT-MC, CISDTQ-MC, etc.)

( )( ) ( ) ( )K K KL L

L Kc c H cτ τ τ τ τ

+ ∆ = −∆ ∑( ) [1 ( ) ] ( )K KK Kc H S cτ τ τ τ+ ∆ = − − ∆

birth and death spawning

WALKER POPULATION DYNAMICS

,( ) ~ , 1K K K Kc N s sαα α

τ δ= = ±∑

( )(( ) ( )( ))

KK K KL LK

LH cHc S c ττ

∂−=

∂− − ∑

1. Place a certain number of walkers on a reference determinant (or determinants) and set S at some value above E0.2. In every time step, attempt

i. spawning: spawn walkers at different determinants.ii. birth or death: create or destroy walkers at a given determinant.iii. annihilation: eliminate pairs of oppositely signed walkers at a given

determinant.3. Once a critical (or sufficiently large) number of walkers is reached, start applying energy shifts in S to stabilize walker population and reach convergence.

,( ) ~ , 1K K K Kc N s sαα α

τ δ= = ±∑

( )(( ) ( )( ))

KK K KL LK

LH cHc S c ττ

∂−=

∂− − ∑

( ) [1 ( ) ] ( )K KK Kc H S cτ τ τ τ+ ∆ = − − ∆( )

( ) ( ) ( )K K KL LL K

c c H cτ τ τ τ τ≠

+ ∆ = −∆ ∑

,( ) ~ , 1K K K Kc N s sαα α

τ δ= = ±∑

( )(( ) ( )( ))

KK K KL LK

LH cHc S c ττ

∂−=

∂− − ∑

( ) [1 ( ) ] ( )K KK Kc H S cτ τ τ τ+ ∆ = − − ∆( )

( ) ( ) ( )K K KL LL K

+ ∆ = −∆ ∑

,( ) ~ , 1K K K Kc N s sαα α

τ δ= = ±∑

( )(( ) ( )( ))

KK K KL LK

LH cHc S c ττ

∂−=

∂− − ∑

( ) [1 ( ) ] ( )K KK Kc H S cτ τ τ τ+ ∆ = − − ∆( )

( ) ( ) ( )K K KL LL K

+ ∆ = −∆ ∑

,( ) ~ , 1K K K Kc N s sαα α

τ δ= = ±∑

( )(( ) ( )( ))

KK K KL LK

LH cHc S c ττ

∂−=

∂− − ∑

( ) [1 ( ) ] ( )K KK Kc H S cτ τ τ τ+ ∆ = − − ∆( )

( ) ( ) ( )K K KL LL K

+ ∆ = −∆ ∑

,( ) ~ , 1K K K Kc N s sαα α

τ δ= = ±∑

( )(( ) ( )( ))

KK K KL LK

LH cHc S c ττ

∂−=

∂− − ∑

( ) [1 ( ) ] ( )K KK Kc H S cτ τ τ τ+ ∆ = − − ∆( )

( ) ( ) ( )K K KL LL K

+ ∆ = −∆ ∑

,( ) ~ , 1K K K Kc N s sαα α

τ δ= = ±∑

( )(( ) ( )( ))

KK K KL LK

LH cHc S c ττ

∂−=

∂− − ∑

( ) [1 ( ) ] ( )K KK Kc H S cτ τ τ τ+ ∆ = − − ∆( )

( ) ( ) ( )K K KL LL K

+ ∆ = −∆ ∑

,( ) ~ , 1K K K Kc N s sαα α

τ δ= = ±∑

( )(( ) ( )( ))

KK K KL LK

LH cHc S c ττ

∂−=

∂− − ∑

( ) [1 ( ) ] ( )K KK Kc H S cτ τ τ τ+ ∆ = − − ∆( )

( ) ( ) ( )K K KL LL K

+ ∆ = −∆ ∑

,( ) ~ , 1K K K Kc N s sαα α

τ δ= = ±∑

( )(( ) ( )( ))

KK K KL LK

LH cHc S c ττ

∂−=

∂− − ∑

( ) [1 ( ) ] ( )K KK Kc H S cτ τ τ τ+ ∆ = − − ∆( )

( ) ( ) ( )K K KL LL K

+ ∆ = −∆ ∑

,( ) ~ , 1K K K Kc N s sαα α

τ δ= = ±∑

( )(( ) ( )( ))

KK K KL LK

LH cHc S c ττ

∂−=

∂− − ∑

( ) [1 ( ) ] ( )K KK Kc H S cτ τ τ τ+ ∆ = − − ∆( )

( ) ( ) ( )K K KL LL K

+ ∆ = −∆ ∑

,( ) ~ , 1K K K Kc N s sαα α

τ δ= = ±∑

( )(( ) ( )( ))

KK K KL LK

LH cHc S c ττ

∂−=

∂− − ∑

( ) [1 ( ) ] ( )K KK Kc H S cτ τ τ τ+ ∆ = − − ∆( )

( ) ( ) ( )K K KL LL K

+ ∆ = −∆ ∑

,( ) ~ , 1K K K Kc N s sαα α

τ δ= = ±∑

( )(( ) ( )( ))

KK K KL LK

LH cHc S c ττ

∂−=

∂− − ∑

( ) [1 ( ) ] ( )K KK Kc H S cτ τ τ τ+ ∆ = − − ∆( )

( ) ( ) ( )K K KL LL K

+ ∆ = −∆ ∑

,( ) ~ , 1K K K Kc N s sαα α

τ δ= = ±∑

( )(( ) ( )( ))

KK K KL LK

LH cHc S c ττ

∂−=

∂− − ∑

( ) [1 ( ) ] ( )K KK Kc H S cτ τ τ τ+ ∆ = − − ∆( )

( ) ( ) ( )K K KL LL K

+ ∆ = −∆ ∑

,( ) ~ , 1K K K Kc N s sαα α

τ δ= = ±∑

( )(( ) ( )( ))

KK K KL LK

LH cHc S c ττ

∂−=

∂− − ∑

( ) [1 ( ) ] ( )K KK Kc H S cτ τ τ τ+ ∆ = − − ∆( )

( ) ( ) ( )K K KL LL K

+ ∆ = −∆ ∑

,( ) ~ , 1K K K Kc N s sαα α

τ δ= = ±∑

( )(( ) ( )( ))

KK K KL LK

LH cHc S c ττ

∂−=

∂− − ∑

( ) [1 ( ) ] ( )K KK Kc H S cτ τ τ τ+ ∆ = − − ∆( )

( ) ( ) ( )K K KL LL K

+ ∆ = −∆ ∑

,( ) ~ , 1K K K Kc N s sαα α

τ δ= = ±∑

( )(( ) ( )( ))

KK K KL LK

LH cHc S c ττ

∂−=

∂− − ∑

( ) [1 ( ) ] ( )K KK Kc H S cτ τ τ τ+ ∆ = − − ∆( )

( ) ( ) ( )K K KL LL K

+ ∆ = −∆ ∑

,( ) ~ , 1K K K Kc N s sαα α

τ δ= = ±∑

( )(( ) ( )( ))

KK K KL LK

LH cHc S c ττ

∂−=

∂− − ∑

( ) [1 ( ) ] ( )K KK Kc H S cτ τ τ τ+ ∆ = − − ∆( )

( ) ( ) ( )K K KL LL K

+ ∆ = −∆ ∑

,( ) ~ , 1K K K Kc N s sαα α

τ δ= = ±∑

( )(( ) ( )( ))

KK K KL LK

LH cHc S c ττ

∂−=

∂− − ∑

( ) [1 ( ) ] ( )K KK Kc H S cτ τ τ τ+ ∆ = − − ∆( )

( ) ( ) ( )K K KL LL K

+ ∆ = −∆ ∑

,( ) ~ , 1K K K Kc N s sαα α

τ δ= = ±∑

( )(( ) ( )( ))

KK K KL LK

LH cHc S c ττ

∂−=

∂− − ∑

( ) [1 ( ) ] ( )K KK Kc H S cτ τ τ τ+ ∆ = − − ∆( )

( ) ( ) ( )K K KL LL K

+ ∆ = −∆ ∑

,( ) ~ , 1K K K Kc N s sαα α

τ δ= = ±∑

( )(( ) ( )( ))

KK K KL LK

LH cHc S c ττ

∂−=

∂− − ∑

( ) [1 ( ) ] ( )K KK Kc H S cτ τ τ τ+ ∆ = − − ∆( )

( ) ( ) ( )K K KL LL K

+ ∆ = −∆ ∑

,( ) ~ , 1K K K Kc N s sαα α

τ δ= = ±∑

( )(( ) ( )( ))

KK K KL LK

LH cHc S c ττ

∂−=

∂− − ∑

( ) [1 ( ) ] ( )K KK Kc H S cτ τ τ τ+ ∆ = − − ∆( )

( ) ( ) ( )K K KL LL K

+ ∆ = −∆ ∑

,( ) ~ , 1K K K Kc N s sαα α

τ δ= = ±∑

( )(( ) ( )( ))

KK K KL LK

LH cHc S c ττ

∂−=

∂− − ∑

( ) [1 ( ) ] ( )K KK Kc H S cτ τ τ τ+ ∆ = − − ∆( )

( ) ( ) ( )K K KL LL K

+ ∆ = −∆ ∑

,( ) ~ , 1K K K Kc N s sαα α

τ δ= = ±∑

( )(( ) ( )( ))

KK K KL LK

LH cHc S c ττ

∂−=

∂− − ∑

( ) [1 ( ) ] ( )K KK Kc H S cτ τ τ τ+ ∆ = − − ∆( )

( ) ( ) ( )K K KL LL K

+ ∆ = −∆ ∑

,( ) ~ , 1K K K Kc N s sαα α

τ δ= = ±∑

( )(( ) ( )( ))

KK K KL LK

LH cHc S c ττ

∂−=

∂− − ∑

( ) [1 ( ) ] ( )K KK Kc H S cτ τ τ τ+ ∆ = − − ∆( )

( ) ( ) ( )K K KL LL K

+ ∆ = −∆ ∑

,( ) ~ , 1K K K Kc N s sαα α

τ δ= = ±∑

( )(( ) ( )( ))

KK K KL LK

LH cHc S c ττ

∂−=

∂− − ∑

( ) [1 ( ) ] ( )K KK Kc H S cτ τ τ τ+ ∆ = − − ∆( )

( ) ( ) ( )K K KL LL K

+ ∆ = −∆ ∑

,( ) ~ , 1K K K Kc N s sαα α

τ δ= = ±∑

( )(( ) ( )( ))

KK K KL LK

LH cHc S c ττ

∂−=

∂− − ∑

( ) [1 ( ) ] ( )K KK Kc H S cτ τ τ τ+ ∆ = − − ∆( )

( ) ( ) ( )K K KL LL K

+ ∆ = −∆ ∑

,( ) ~ , 1K K K Kc N s sαα α

τ δ= = ±∑

( )(( ) ( )( ))

KK K KL LK

LH cHc S c ττ

∂−=

∂− − ∑

In CCMC, instead of sampling determinants by walkers, one samples the space of excitation amplitudes (amplitudes of “excitors”) by excitor particles (“excips”).

CCMC (CCSDT-MC, CCSDTQ-MC, etc.)

( ) [1 ( ) ] ( )K KK Kc H S cτ τ τ τ+ ∆ = − − ∆( )

( ) ( ) ( )K K KL LL K

+ ∆ = −∆ ∑

To accelerate convergence, one can use the initiator CIQMC (i-CIQMC) and CCMC (i-CCMC) approaches, where only those determinants or excitorsthat acquire a walker/excip population exceeding a preset value na are allowed to spawn new walkers onto empty determinants/excitors. One can start i-CIQMC and i-CCMC simulations by placing a certain, sufficiently large, number of walkers/excips on the reference determinant (in our case, the RHF state).

Developing a Stochastic CC(P;Q) Approach

Developing a Stochastic CC(P;Q) Approach1. Start a CIQMC (e.g., i-CIQMC) or CCMC (e.g., i-CCMC) propagation by placing a certain number of walkers or excips on the reference determinant.

2. Extract a list of the most important determinants or cluster amplitude types relevant to the CC theory of interest (triples for CCSDT; triples and quadruples for CCSDTQ, etc.) from the CIQMC or CCMC propagation at a given time τ to define the P space for CC(P) calculations as follows: if the target approach is CCSDT, the P space is defined as all singles, all doubles,

and a subset of triples having at least nP (e.g., one) positive or negative walkers/excips on them.

if the target approach is CCSDTQ, the P space is defined as all singles, all doubles, and a subset of triples and quadruples having at least nP (e.g., one) positive or negative walkers/excips on them, etc.

F2 /cc-pVDZ, R = 2 Re, i-FCIQMC

3. Solve the CC(P) equations. if the target approach is CCSDT, use if the target approach is CCSDTQ, use

)PT T T T= + +(MC) (MC)

1PT T T TT= + + +

RECOVERING CCSDT ENERGETICS FOR F2/cc-pVDZMONTE CARLO APPROACH = i-FCIQMC (∆τ = 0.0001 a.u., na = 3)

FCIQMC

R = Re

R = 5ReR = 2Re

R = 1.5Re

FCIQMC

R = Re

R = 5ReR = 2Re

R = 1.5Re

)PT T T T= + +(MC) (MC)

1PT T T TT= + + +

4. Correct the CC(P) energy for the remaining triples (if the target approach is CCSDT), triples and quadruples (if the target approach is CCSDTQ), etc. using the non-iterative CC(P;Q) correction δ(P;Q).

)PT T T T= + +(MC) (MC)

1PT T T TT= + + +

FCIQMC

CC(P;Q)EN

R = Re

R = 5ReR = 2Re

R = 1.5Re

RECOVERING CCSDT ENERGETICS FOR F2/cc-pVDZMONTE CARLO APPROACH = i-CISDTQ-MC (∆τ = 0.0001 a.u., na = 3)

CISDTQ-MC

CC(P;Q)EN

R = Re

R = 5ReR = 2Re

R = 1.5Re

[J.E. Deustua, J. Shen, and P. Piecuch, in preparation]

RECOVERING CCSDT ENERGETICS FOR F2/cc-pVDZMONTE CARLO APPROACH = i-CISDT-MC (∆τ = 0.0001 a.u., na = 3)

CISDT-MC

CC(P;Q)EN

R = Re

R = 5ReR = 2Re

R = 1.5Re

RECOVERING CCSDT ENERGETICS FOR F2/cc-pVDZMONTE CARLO APPROACH = i-CCSDT-MC (∆τ = 0.0001 a.u., na = 3)

CCSDT-MC

CC(P;Q)EN

R = Re

R = 5ReR = 2Re

R = 1.5Re

MC Iter. % ofTriples in P

CC(P) (mEh)

CC(P;Q)MP(mEh)

CC(P;Q)EN(mEh)

Wall Time (s)

MC CC(P;Q) Total

0 0 45.638 CCSD

6.357 CCSD(2)T

1.862CR-CC(2,3)

10,000 4 12.199 1.887 0.915 3 2 5

20,000 10 4.127 0.596 0.279 10 5 15

30,000 21 0.802 0.067 -0.009 28 13 41

40,000 35 0.456 0.036 -0.007 66 31 97

∞ 100 -199.058201 Eh 208

R = 2 Re

Errors relative to CCSDT

Errors relative to CCSDTCCSD: 45.638 mEhCCSD(T): −23.596 mEh

CC(P) (mEh)

CC(P;Q)MP(mEh)

CC(P;Q)EN(mEh)

Wall Time (s)

MC CC(P;Q) Total

0 0 45.638CCSD

6.357 CCSD(2)T

1.862 CR-CC(2,3)

10,000 3 12.687 2.069 0.978 3 2 5

20,000 9 3.672 0.583 0.280 9 3 12

30,000 17 1.393 0.154 0.030 17 8 25

40,000 28 0.627 0.053 -0.005 32 16 48

∞ 100 -199.058201 Eh 208

R = 2 Re

RECOVERING CCSDT ENERGETICS FOR F2/cc-pVDZMONTE CARLO APPROACH = i-CISDT-MC (∆τ = 0.0001 a.u., na = 3)

RECOVERING CCSDT ENERGETICS FOR LARGER BASIS SETSMONTE CARLO APPROACH = i-FCIQMC (∆τ = 0.0001 a.u., na = 3)

F2/cc-pVTZ , R = 2Re F2/aug-cc-pVTZ , R = 2Re

FCIQMC

CC(P;Q)EN

CC(P) (mEh)

CC(P;Q)MP(mEh)

CC(P;Q)EN(mEh)

Speedup rel. to CCSDT

0 0 65.036CCSD

9.808 CCSD(2)T

5.595 CR-CC(2,3)

30,000 4 8.065 0.858 0.454 90

40,000 10 4.408 0.330 0.093 30

50,000 23 2.208 0.125 0.002 10

∞ 100 -199.253022 Eh 1

R = 2 Re

RECOVERING CCSDT ENERGETICS FOR F2/aug-cc-pVTZMONTE CARLO APPROACH = i-FCIQMC (∆τ = 0.0001 a.u., na = 3)

RECOVERING CCSDT ENERGETICS FOR AUTOMERIZATION OF CYCLOBUTADIENE/cc-pVDZ

MONTE CARLO APPROACH = i-FCIQMC/i-CISDT-MC (∆τ = 0.0001 a.u., na = 3)

[J.E. Deustua, J. Shen, and P. Piecuch, Phys. Rev. Lett. 119, 223003 (2017); in preparation]

MC Iter. % of Triplesin P space

CC(P;Q)MP(kcal/mol

CC(P;Q)EN(kcal/mol

Total Wall Time (hrs)

MC CC(P;Q) Total

0 0/0 9.6 CCSD(2)T

8.7CR-CC(2,3)

0/0 0.4/0.4 0.4/0.4

40,000 15-16/12-13 1.5/2.8 0.9/2.0 1.0/0.3 1.9/1.4 2.9/1.7

50,000 31-31/24-26 0.5/0.6 0.2/0.3 3.1/0.7 5.9/4.3 9.0/5.0

60,000 52-52/41-42 0.1/0.2 0.0/0.1 11.6/1.4 13.6/9.8 25.2/11.2

∞ 100 7.6 kcal/mol 41.05

Errors relative to CCSDTCCSD: 13.3 kcal/molCCSD(T): 8.2 kcal/mol

MONTE CARLO APPROACH = i-FCIQMC/i-CISDT-MC (∆τ = 0.0001 a.u., na = 3)

MC APPROACH = i-FCIQMC, i-CISDTQ-MC, i-CISDT-MC (∆τ = 0.0001 a.u., na = 3)

CC(P;Q)EN

MC = FCIQMC MC = CISDTQ-MC MC = CISDT-MC

MC Iter. % of Triples/Quads CC(P) (mEh) CC(P;Q)MP (mEh) CC(P;Q)EN (mEh)

0 0/0 15.582 CCSD -28.302 CCSD(2)T -35.823 CR-CC(2,3)

10,000 3/1 10.165 -2.390 -3.945

20,000 5/1 4.282 -0.084 -0.403

40,000 13/3 0.969 0.267 0.199

80,000 36/16 0.030 0.022 0.021

∞ 100/100 -75.916679 Eh

RECOVERING CCSDTQ ENERGETICS FOR H2O/cc-pVDZMONTE CARLO APPROACH = i-FCIQMC (∆τ = 0.0001 a.u., na = 3)

Errors relative to CCSDTQ

CCSD 10.849 mEh CCSDT −40.126 mEhCCSD(T) −90.512 mEh CCSDTQ −4.733 mEh

J.E. Deustua, J. Shen, and P. Piecuch, in preparation

Errors relative to FCI:

FCIQMC

CC(P;Q)EN

Triples

R = Re R = 3Re

R = 3Re

WHAT IS NEXT?Higher-order CC methods, excited states, corrections toward FCI, and more … .

[J.E. Deustua, S.H. Yuwono, J. Shen, and P. Piecuch, J. Chem. Phys. 150, 111101 (2019)]EXCITED STATES

Total electronic energies of ground and excited states of CH+ resulting from EOMCC(P) calculations (errors relative to full EOMCCSDT in millihartree): R = Re

[J.E. Deustua, S.H. Yuwono, J. Shen, and P. Piecuch, J. Chem. Phys. 150, 111101 (2019); J.E. Deustua, S.H. Yuwono, A. Chakraborty, J. Shen, and P. Piecuch, in preparation]

EXCITED STATES

Total electronic energies of excited states of CH+ resulting from EOMCC(P) and CC(P;Q) calculations (errors relative to full EOMCCSDT in millihartree)

EXCITED STATES

[J.E. Deustua, S.H. Yuwono, A. Chakraborty, J. Shen, and P. Piecuch, in preparation]

Excited State EOMCCSD EOMCCSDT ExperimentA 2Δu 4.291 3.945 3.761B 2Σu

+ 7.123 4.597 4.315

Adiabatic excitation energies (eV) of the two low-lying excited states of CNC

EXCITED STATES

[J.E. Deustua, S.H. Yuwono, A. Chakraborty, J. Shen, and P. Piecuch, in preparation]

Excited State EOMCCSD EOMCCSDT ExperimentA 2Δu 4.291 3.945 3.761B 2Σu

+ 7.123 4.597 4.315

Adiabatic excitation energies (eV) of the two low-lying excited states of CNC

Total electronic energies of excited states of CNC resulting from EOMCC(P) and CC(P;Q) calculations (errors relative to full EOMCCSDT in millihartree)

HOW ABOUT ATTACKING FCI (EXACT THEORY!) ITSELF?[J.E. Deustua, I. Magoulas, J. Shen, and P. Piecuch, J. Chem. Phys. 149, 151101 (2018)]

2 31 2 1 1 2 1[ (1 (1/ 2) (1/ 6) )] 0a

i N CH T T T TT TΦ + + + + + + Φ =3T

2 31 2 1 1 2 1

2 2 41 2 1 2 1

[ (1 (1/ 2) (1/ 6)

(1/ 2) (1/ 2) (1/ 24) )] 0

abij N

H T T T TT T

T T T T T

Φ + + + + + +

+ + + + + Φ =

20 1 2 1[ ( (1/ 2) )]N CE H H T T T= Φ Φ + Φ + + Φ

2 31 2 1 1 2 1[ (1 (1/ 2) (1/ 6) )] 0a

i N CH T T T TT TΦ + + + + + + Φ =3T

2 31 2 1 1 2 1

2 2 41 2 1 2 1

[ (1 (1/ 2) (1/ 6)

(1/ 2) (1/ 2) (1/ 24) )] 0

abij N

H T T T TT T

T T T T T

Φ + + + + + +

+ + + + + Φ =

In standard CCSD, we neglect equations corresponding to projections on higher-than doubly excited determinants and terms containing T3 and T4.

20 1 2 1[ ( (1/ 2) )]N CE H H T T T= Φ Φ + Φ + + Φ

2 31 2 1 1 2 1[ (1 (1/ 2) (1/ 6) )] 0a

i N CH T T T TT TΦ + + + + + + Φ =3T

2 31 2 1 1 2 1

2 2 41 2 1 2 1

[ (1 (1/ 2) (1/ 6)

(1/ 2) (1/ 2) (1/ 24) )] 0

abij N

H T T T TT T

T T T T T

Φ + + + + + +

+ + + + + Φ =

20 1 2 1[ ( (1/ 2) )]N CE H H T T T= Φ Φ + Φ + + Φ

2 31 2 1 1 2 1[ (1 (1/ 2) (1/ 6) )] 0a

i N CH T T T TT TΦ + + + + + + Φ =3T

2 31 2 1 1 2 1

2 2 41 2 1 2 1

[ (1 (1/ 2) (1/ 6)

(1/ 2) (1/ 2) (1/ 24) )] 0

abij N

H T T T TT T

T T T T T

Φ + + + + + +

+ + + + + Φ =

OBSERVATION: T3, T4 extracted from Full CI E0 becomes exact !!!

20 1 2 1[ ( (1/ 2) )]N CE H H T T T= Φ Φ + Φ + + Φ

2 31 2 1 1 2 1[ (1 (1/ 2) (1/ 6) )] 0a

i N CH T T T TT TΦ + + + + + + Φ =3T

2 31 2 1 1 2 1

2 2 41 2 1 2 1

[ (1 (1/ 2) (1/ 6)

(1/ 2) (1/ 2) (1/ 24) )] 0

abij N

H T T T TT T

T T T T T

Φ + + + + + +

+ + + + + Φ =

Historically, used to develop externally corrected CCSD-type methods, where one extracts T3 and T4 from a non-CC wave function (for example, PUHF, as in ACP methods of Paldus, Čížek, and Piecuch, or MRCI, as in RMRCC of Paldus and Li).

OBSERVATION: T3, T4 extracted from Full CI E0 becomes exact !!!

20 1 2 1[ ( (1/ 2) )]N CE H H T T T= Φ Φ + Φ + + Φ

CI (FCIQMC) expansion CC expansion

1. Start a FCIQMC propagation by placing a certain number of walkers on the reference determinant.

2. Extract C1, C2, C3, and C4 from a FCIQMC wave function at a given time τ using a formula(MC)

0( ) ( ) , where ( ) / .n n K K K KK

C C c E c N Nτ τ τ≡ = =∑

2. Extract C1, C2, C3, and C4 from a FCIQMC wave function at a given time τ using a formula

3. Perform cluster analysis of the FCIQMC wave function at time τ to extract

(MC)0( ) ( ) , where ( ) / .n n K K K K

KC C c E c N Nτ τ τ≡ = =∑

HOW ABOUT ATTACKING FCI (EXACT THEORY!) ITSELF?

(MC) (MC)3 4and( ) ( ) .T Tτ τ

[J.E. Deustua, I. Magoulas, J. Shen, and P. Piecuch, J. Chem. Phys. 149, 151101 (2018)]

4. Solve CCSD equations corrected for terms containing to determine

(MC)0( ) ( ) , where ( ) / .n n K K K K

(MC) (MC)3 4an) )d( (T Tτ τ= =3 4T T

(MC) (MC)3 4and( ) ( ) .T Tτ τ

1 2 0and and energy ( ).T T E τ

5. Check convergence by repeating steps 2−4 at some later FCIQMC propagation time τ’ > τ.

(MC)0( ) ( ) , where ( ) / .n n K K K K

(FCI)0 0It is guaranteed that im ( ) .l E E

→∞=

(MC) (MC)3 4an) )d( (T Tτ τ= =3 4T T

(MC) (MC)3 4and( ) ( ) .T Tτ τ

5. Check convergence by repeating steps 2−4 at some later FCIQMC propagation time τ’ > τ.

(MC)0( ) ( ) , where ( ) / .n n K K K K

(FCI)0 0It is guaranteed that im ( ) .l E E

→∞=

(MC) (MC)3 4an) )d( (T Tτ τ= =3 4T T

CLUSTER-ANALYSIS-DRIVEN FCIQMC = CAD-FCIQMC

(MC) (MC)3 4and( ) ( ) .T Tτ τ

R = Re

R = 1.5Re

R = 2.0Re

MC Iters

R = Re R = 1.5 Re R = 2 ReCAD-FCIQMC i-FCIQMC CAD-FCIQMC i-FCIQMC CAD-FCIQMC i-FCIQMC

0 3.744 (CCSD) 217.821 (RHF) 10.043 (CCSD) 269.961 (RHF) 22.032 (CCSD) 363.954 (RHF)10,000 0.611 8.381 3.335 45.802 12.351 119.89620,000 -0.073 1.596 2.597 18.345 8.485 72.65030,000 -0.175 0.586 0.473 15.937 4.794 49.20340,000 -0.211 -2.217 0.873 4.855 0.138 44.62750,000 -0.440 1.456 1.310 3.247 -1.693 31.44860,000 -0.046 1.911 0.501 0.588 -0.225 19.66070,000 -0.235 0.302 0.685 0.811 -0.377 11.33380,000 0.189 -0.686 0.063 -0.128 -0.425 12.61190,000 -0.177 -0.981 -0.171 0.014 -1.657 10.089

100,000 -0.036 0.139 -0.302 1.956 -0.816 5.680110,000 0.129 -0.710 -0.189 1.088 -0.580 4.797120,000 -0.035 0.597 0.020 -0.241 -0.555 4.041130,000 0.086 -0.503 -0.020 -0.720 -1.166 3.107140,000 0.098 0.080 -0.084 0.497 -0.666 1.981150,000 -0.055 0.308 -0.156 0.990 -0.620 1.630160,000 0.078 -0.400 -0.059 -1.002 -0.434 1.328

∞ -76.241860 (FCI) -76.072348 (FCI) -75.951665 (FCI)

R = Re

R = 1.5Re

R = 2.0Re

MC Iters

R = Re R = 1.5 Re R = 2 ReCAD-FCIQMC i-FCIQMC CAD-FCIQMC i-FCIQMC CAD-FCIQMC i-FCIQMC

0 3.744 (CCSD) 217.821 (RHF) 10.043 (CCSD) 269.961 (RHF) 22.032 (CCSD) 363.954 (RHF)10,000 0.611 8.381 3.335 45.802 12.351 119.89620,000 -0.073 1.596 2.597 18.345 8.485 72.65030,000 -0.175 0.586 0.473 15.937 4.794 49.20340,000 -0.211 -2.217 0.873 4.855 0.138 44.62750,000 -0.440 1.456 1.310 3.247 -1.693 31.44860,000 -0.046 1.911 0.501 0.588 -0.225 19.66070,000 -0.235 0.302 0.685 0.811 -0.377 11.33380,000 0.189 -0.686 0.063 -0.128 -0.425 12.61190,000 -0.177 -0.981 -0.171 0.014 -1.657 10.089

100,000 -0.036 0.139 -0.302 1.956 -0.816 5.680110,000 0.129 -0.710 -0.189 1.088 -0.580 4.797120,000 -0.035 0.597 0.020 -0.241 -0.555 4.041130,000 0.086 -0.503 -0.020 -0.720 -1.166 3.107140,000 0.098 0.080 -0.084 0.497 -0.666 1.981150,000 -0.055 0.308 -0.156 0.990 -0.620 1.630160,000 0.078 -0.400 -0.059 -1.002 -0.434 1.328

∞ -76.241860 (FCI) -76.072348 (FCI) -75.951665 (FCI)

R = Re

R = 1.5Re

R = 2.0Re

6[ ] [ 0;FCI sho ]rt -FCIQMC run CCSD[ ]ie eθ θ⇒ ≈ +N N N

CPU time savings (in terms of system size ):N

Storage (in terms of orbital (M) and electron (N) nos):4~ list of dets and nos of walkersNM M⇒ +

[I. Magoulas, J.E. Deustua, J. Shen, and P. Piecuch, to be submitted in the coming days]

“THE FINAL FRONTIER”: STRONGLY CORRELATED REGIME (as in, e.g., the Mott metal−insulator transitions)

WE CAN EXTEND CAD-FCIQMC TO SYSTEMS WITH LARGER

NUMBERS OF STRONGLY CORRELATED ELECTRONS,

CONVERGING FCI ENERGIES OUT OF THE EARLY STAGES OF

FCIQMC PROPAGATIONS COMBINED WITH INEXPENSIVE CCSD-

LIKE, POLYNOMIAL STEPS, BUT TO LEARN MORE ABOUT IT, YOU

WILL HAVE TO JOIN US AT THE “QUANTUM INTERNATIONAL

FRONTIERS 2019” CONFERENCE IN SHANGHAI IN NOVEMBER.

Benzene/cc-pVDZ:

RHF MO basis

30 correlated electrons (6 core MOs frozen)

114 basis functions

IF YOU JOIN US AT THE “QUANTUM INTERNATIONAL

FRONTIERS 2019” CONFERENCE IN SHANGHAI, YOU

WILL ALSO WITNESS A MAJOR BREAKTROUGH IN

ELECTRONIC STRUCTURE THEORY. WE WILL SHOW

THAT WE CAN USE CAD-FCIQMC TO OBTAIN WELL-

CONVERGED FCI ENERGIES FOR MANY-ELECTRON

SYSTEMS CHARACTERIZED BY HILBERT SPACE

DIMENSIONALITIES THAT ARE ORDERS OF MAGNITUDE

BEYOND THE REACH OF CONVENTIONAL HAMILTONIAN

DIAGONALIZATIONS.

EXAMPLE: BENZENE MOLECULE IN A cc-pVDZ BASIS.

FCI DIMENSION: 7.67 × 1035 (5.56 × 1034 CSFs).

Method Total Energy/Hartree

CCSD −231.5…

CCSDT −231.5…

CCSDT(2)Q = CR-CC(3,4)A −231.5…

CCSDTQ −231.5… (J. Gauss)

i-FCIQMC (1B) −231.5… ± 0.0… (A. Alavi)

CAD-FCIQMC ext. energy (1B) −231.5…

CAD-FCIQMC[1-5] (1B) −231.5…

CAD-FCIQMC[1,(3+4)/2] (1B) −231.5…

i-FCIQMC (2B) −231.5… ± 0.0… (A. Alavi)

CAD-FCIQMC ext. energy (2B) −231.5…

CAD-FCIQMC[1-5] (2B) −231.5…

CAD-FCIQMC[1,(3+4)/2] (2B) −231.5…

Sz = 0 Determinants

Singly excited 2,790

Doubly excited 2,844,405

Triply excited 1,371,327,160

Quadruply excited 374,473,981,350

All (Full CI) 7.67 × 1035 (5.56 × 1034 CSFs)

Benzene/cc-pVDZ:

RHF MO basis

30 correlated electrons (6 core MOs frozen)

114 basis functions

−231.5…

Search for Chem 580

THANK YOU

“Algebraic and Diagrammatic Methods for Many-Fermion Systems”https://pages.wustl.edu/ppiecuch/course-videos

Search for Chem 580 in YouTube

Emiliano Deustua (2014-present)

Ilias Magoulas (2015-present)

Dr. Jun Shen (2010-present)

Stephen Yuwono (2017-present)

Arnab Chakraborty (2018-present)

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