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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
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ˆ ( ; ) ( ) ( ; )eH Eµ µ µΨ = ΨX R R X R
1 1 1
ˆ( ( )ˆ ,) ˆ i j
N N N
ei i j i
iH z v= = = +
= +∑ ∑ ∑x x x
2
1
1ˆ( )2
, ˆ ) 1( ,M
i jij
Ai i
A Ai
vZr
zR=
= − ∇ − =∑ xx x
THE ELECTRONIC SCHRÖDINGER EQUATION
+
a
ac c
b
pbiradical
conc
erted
bc
ab
cp
bp
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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 V
V r Vϕ
= ∞ < ∞
≡ =x
Exact case : n practi ce :
1 1
1
1
1
1
( ) ( )1( , , )
! ( ) ( )N
N N
r r N
r r N
r r NN
ϕ ϕ
ϕ ϕΦ =
x x
x xx x
SOLVING THE ELECTRONIC SCHRÖDINGER EQUATION
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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
1 1
1
1 1
1
( , , ) ( , , )
( , , )
N N
N
N r r r r Nr r
I I NI
c
c
µµ
µ
< <
Ψ = Φ
= Φ
∑
∑
x x 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
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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
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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)
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← 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= ⇒ < ⇒
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kp-kh
(J. Čížek, 1966)
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ARGUMENTS IN FAVOR OF THE CC THEORY
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Size-extensivity of the resulting approximations (no loss of accuracy occurs when the system is made larger)
ARGUMENTS IN FAVOR OF THE CC THEORY
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Size-extensivity of the resulting approximations (no loss of accuracy occurs when the system is made larger)
ARGUMENTS IN FAVOR OF THE CC THEORY
Linked cluster (diagram) theorem (Brueckner, 1955; Goldstone, 1957)
Connected cluster theorem (Hubbard, 1957; Hugenholtz, 1957)
MBPT
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Size-extensivity of the resulting approximations (no loss of accuracy occurs when the system is made larger)
Separability or size consistency if the reference state separates correctly
ARGUMENTS IN FAVOR OF THE CC THEORY
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Size-extensivity of the resulting approximations (no loss of accuracy occurs when the system is made larger)
Separability or size consistency if the reference state separates correctly
ARGUMENTS IN FAVOR OF THE CC THEORY
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Size-extensivity of the resulting approximations (no loss of accuracy occurs when the system is made larger)
Separability or size consistency if the reference state separates correctly
Fastest convergence toward the exact, full CI, limit
ARGUMENTS IN FAVOR OF THE CC THEORY
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Size-extensivity of the resulting approximations (no loss of accuracy occurs when the system is made larger)
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)
CISDT
CCSDTQ
CISD
CISDTQ
CCSD
CCSDT
ARGUMENTS IN FAVOR OF THE CC THEORY
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-10.0
-8.0
-6.0
-4.0
-2.0
0.0
2.0
4.0
6.0
8.0
10.0
1.0 1.5 2.0
Erro
r rel
ativ
e to
full
CI(m
illih
artr
ee)
Relative bond length (ROH/ROH,eq.)
CCSD
H2O
Taken from N.P. Bauman, J. Shen, and P. Piecuch, Mol. Phys., 2017
ARGUMENTS IN FAVOR OF THE CC THEORY
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-10.0
-8.0
-6.0
-4.0
-2.0
0.0
2.0
4.0
6.0
8.0
10.0
1.0 1.5 2.0
Erro
r rel
ativ
e to
full
CI(m
illih
artr
ee)
Relative bond length (ROH/ROH,eq.)
CCSD
CCSDT
H2OARGUMENTS IN FAVOR OF THE CC THEORY
Taken from N.P. Bauman, J. Shen, and P. Piecuch, Mol. Phys., 2017
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-10.0
-8.0
-6.0
-4.0
-2.0
0.0
2.0
4.0
6.0
8.0
10.0
1.0 1.5 2.0
Erro
r rel
ativ
e to
full
CI(m
illih
artr
ee)
Relative bond length (ROH/ROH,eq.)
CCSD
CCSDT
CCSDTQ
H2OARGUMENTS IN FAVOR OF THE CC THEORY
Taken from N.P. Bauman, J. Shen, and P. Piecuch, Mol. Phys., 2017
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Size-extensivity of the resulting approximations (no loss of accuracy occurs when the system is made larger)
Separability or size consistency if the reference state separates correctly
Fastest convergence toward the exact, full CI, limit
ARGUMENTS 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)
CCSDT
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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)
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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)
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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)
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)
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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
(eV)
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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
(eV)
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1.5321.091
105.8o
H1
H4
H2 H5
H3
C1
C2
C3
1.088
1.102
120.8o
φH1-C1-C2-C3 = 0.0o
φH5-C3-C2-C1 = 0.0o
1.4951.081H1
H4
H2
H6
H5
H3
C1
C2
C3
φH1-C1-C2-C3 = 0.0o
1.081116.1o
121.6o
1.100
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)
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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+ = +
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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]
CC (µ = 0) or EOMCC (µ > 0)energy obtained in the P space H(P)
( )( ) ( ; )PP Q EE P Qµµ µδ+ = +
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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]
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)
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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]
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)
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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]
moments of CC/EOMCC equations
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)
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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)
Examples:
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]
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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)
Examples: P space: singly and doubly excited determinants (CCSD) Q space: triply excited determinants
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]
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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)
Examples: P space: singly and doubly excited determinants (CCSD) Q space: triply excited determinants
CR-CC(2,3)
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]
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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)
Examples: P space: singly and doubly excited determinants (CCSD) Q space: triply excited determinants
CR-CC(2,3) P space: singly and doubly excited determinants (CCSD) Q space: triply and quadruply excited determinants
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]
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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)
Examples: P space: singly and doubly excited determinants (CCSD) Q space: triply excited determinants
CR-CC(2,3) P space: singly and doubly excited determinants (CCSD) Q space: triply and quadruply excited determinants
CR-CC(2,4)
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]
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(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
, ( )
, (2)
(2)
)
ijk abca
T T T T T TC
ijk abc abca
abc abc ijki
bc
jk ijk a
ij
bc bc ijk ijk
k H e He He
D E
H
H D HL
− − + += =
=
= Φ
= Φ Φ − Φ Φ
Φ
M
Example: CR-CC(2,3), robust noniterative triples correction to CCSDiterative no
2nu4 (N6) + noniterative no
3nu4 (N7); CCTYP=CR-CCL in GAMESS
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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
, ( )
, (2)
(2)
)
ijk abca
T T T T T TC
ijk abc abca
abc abc ijki
bc
jk ijk a
ij
bc bc ijk ijk
k H e He He
D E
H
H D HL
− − + += =
=
= Φ
= Φ Φ − Φ Φ
Φ
M
Example: CR-CC(2,3), robust noniterative triples correction to CCSDiterative no
2nu4 (N6) + noniterative no
3nu4 (N7); CCTYP=CR-CCL in GAMESS
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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
, ( )
, (2)
(2)
)
ijk abca
T T T T T TC
ijk abc abca
abc abc ijki
bc
jk ijk a
ij
bc bc ijk ijk
k H e He He
D E
H
H D HL
− − + += =
=
= Φ
= Φ Φ − Φ Φ
Φ
M
Example: CR-CC(2,3), robust noniterative triples correction to CCSDiterative no
2nu4 (N6) + noniterative no
3nu4 (N7); CCTYP=CR-CCL in GAMESS
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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
, ( )
, (2)
(2)
)
ijk abca
T T T T T TC
ijk abc abca
abc abc ijki
bc
jk ijk a
ij
bc bc ijk ijk
k H e He He
D E
H
H D HL
− − + += =
=
= Φ
= Φ Φ − Φ Φ
Φ
M
Example: CR-CC(2,3), robust noniterative triples correction to CCSDiterative no
2nu4 (N6) + noniterative no
3nu4 (N7); CCTYP=CR-CCL in GAMESS
CCSD
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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
, ( )
, (2)
(2)
)
ijk abca
T T T T T TC
ijk abc abca
abc abc ijki
bc
jk ijk a
ij
bc bc ijk ijk
k H e He He
D E
H
H D HL
− − + += =
=
= Φ
= Φ Φ − Φ Φ
Φ
M
Example: CR-CC(2,3), robust noniterative triples correction to CCSDiterative no
2nu4 (N6) + noniterative no
3nu4 (N7); CCTYP=CR-CCL in GAMESS
CCSD
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F2/cc-pVTZ
CR-CC(2,3)
CCSDT
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
, ( )
, (2)
(2)
)
ijk abca
T T T T T TC
ijk abc abca
abc abc ijki
bc
jk ijk a
ij
bc bc ijk ijk
k H e He He
D E
H
H D HL
− − + += =
=
= Φ
= Φ Φ − Φ Φ
Φ
M
Example: CR-CC(2,3), robust noniterative triples correction to CCSDiterative no
2nu4 (N6) + noniterative no
3nu4 (N7); CCTYP=CR-CCL in GAMESS
CCSD
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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)]
DFT
Exp.CIM-CR-CC(2,3)
Structural model: 58 atoms; 234 electrons.
[In GAMESS: CIMTYP = GSECIM]
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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)]
DFT
Exp.CIM-CR-CC(2,3)
Structural model: 58 atoms; 234 electrons.
[In GAMESS: CIMTYP = GSECIM]
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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
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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
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
(Values in cm-1 obtained using 6-311+G(d)/6-31+G(d) basis set; 350 cm-1 ≈ 1 kcal/mol )
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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 -119NH3 29925 34263 4338 29922 -3
Vertical excitation energies of cis-7-hydroxyquinoline in various molecular environments
(Values in cm-1 obtained using 6-311+G(d)/6-31+G(d) basis set; 350 cm-1 ≈ 1 kcal/mol )
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
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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 -119NH3 29925 34263 4338 29922 -3H2O 30240 34500 4260 30199 -41
Vertical excitation energies of cis-7-hydroxyquinoline in various molecular environments
(Values in cm-1 obtained using 6-311+G(d)/6-31+G(d) basis set; 350 cm-1 ≈ 1 kcal/mol )
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
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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 -119NH3 29925 34263 4338 29922 -3H2O 30240 34500 4260 30199 -41H2O + H2O 29193 33699 4506 29378 185
Vertical excitation energies of cis-7-hydroxyquinoline in various molecular environments
(Values in cm-1 obtained using 6-311+G(d)/6-31+G(d) basis set; 350 cm-1 ≈ 1 kcal/mol )
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
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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
Vertical excitation energies of cis-7-hydroxyquinoline in various molecular environments
(Values in cm-1 obtained using 6-311+G(d)/6-31+G(d) basis set; 350 cm-1 ≈ 1 kcal/mol )
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
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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
TS
AUTOMERIZATION OF CYCLOBUTADIENE
[J. Shen and P. Piecuch, J. Chem. Phys., 2012]
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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
TS
AUTOMERIZATION OF CYCLOBUTADIENE
[J. Shen and P. Piecuch, J. Chem. Phys., 2012]
T1 and T2decoupled from T3
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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
2nu4
CCSDtq/EOMCCSDtq No2Nu
2no2nu
4
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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 20.786 20.274 7.3
CCSDT -154.244157 -154.232002 7.6
Various CC energies (in millihartree) relative to full CCSDT (in hartree), cc-pVDZ basis set
TS
AUTOMERIZATION OF CYCLOBUTADIENE
[J. Shen and P. Piecuch, J. Chem. Phys, 2012]
T1 and T2decoupled from T3
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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 20.786 20.274 7.3
CCSDT -154.244157 -154.232002 7.6
Various CC energies (in millihartree) relative to full CCSDT (in hartree), cc-pVDZ basis set
TS
AUTOMERIZATION OF CYCLOBUTADIENE
[J. Shen and P. Piecuch, J. Chem. Phys, 2012]
T1 and T2decoupled from T3
t3 misses some dynamical
correlations
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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)
Examples:
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]
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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)
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
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]
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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)
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
CC(t;3)
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]
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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)
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
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
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]
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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)
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
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)
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]
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AUTOMERIZATION OF CYCLOBUTADIENE
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
TS
10.6
Barrier heights (in kcal/mol) →
[J. Shen and P. Piecuch, J. Chem. Phys. 136, 144104 (2012)]
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AUTOMERIZATION OF CYCLOBUTADIENE
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
TS
10.6
Barrier heights (in kcal/mol) →
[J. Shen and P. Piecuch, J. Chem. Phys. 136, 144104 (2012)]
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AUTOMERIZATION OF CYCLOBUTADIENE
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
TS
10.6
Barrier heights (in kcal/mol) →
[J. Shen and P. Piecuch, J. Chem. Phys. 136, 144104 (2012)]
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AUTOMERIZATION OF CYCLOBUTADIENE
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
TS
10.6
Barrier heights (in kcal/mol) →
[J. Shen and P. Piecuch, J. Chem. Phys. 136, 144104 (2012)]
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-27.9
40.4
-22.2-25.2
58.6
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)
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-27.9
40.4
-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)
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-27.9
40.4
-22.2-25.2
58.6c A. R. Berner and A. Lüchow, J. Phys. Chem. A 114, 13222 (2010)
a A. Kinal and P. Piecuch, J. Phys. Chem. A 111, 734 (2007)
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)
CCSD(T)a 40.4 21.8 -25.1 -22.3 -28.0
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-27.9
40.4
-22.2-25.2
58.6c A. R. Berner and A. Lüchow, J. Phys. Chem. A 114, 13222 (2010)
a A. Kinal and P. Piecuch, J. Phys. Chem. A 111, 734 (2007)
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)
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
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-27.9
40.4
-22.2-25.2
58.6c A. R. Berner and A. Lüchow, J. Phys. Chem. A 114, 13222 (2010)
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)
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)
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
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-27.9
40.4
-22.2-25.2
58.6c A. R. Berner and A. Lüchow, J. Phys. Chem. A 114, 13222 (2010)
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)
con_TS dis_TS g-but gt_TS t-but
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)
The Conrotatoryand DisrotatoryIsomerization Pathways of Bicyclo[1.1.0]-butane to Butadiene (enthalpies at 0 K in kcal/mol)
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
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Magnesium Dimer[S.H. Yuwono, I. Magoulas, J. Shen, and P. Piecuch, Mol. Phys. 117, 1486 (2019)]
(CC(t;3)/AwCQZ)
1
(CCSDT/AwCQZ
(CCSD(T)/AwCQZ) 1e
(ex m
(CR-CC )
p
(2,3)/AwCQZ
er
-1e
1
) -1
e
e i ent) -e
409.6
4
= 39
366.7 cm
30.3
c
c
1
m
4
m
1.4 cm
3.4 cmD
D
D
D
D
−
−
=
=
=
=CCSDT
CCSD(T)CR-CC(2,3)
CC(t;3)
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Magnesium Dimer[S.H. Yuwono, I. Magoulas, J. Shen, and P. Piecuch, Mol. Phys. 117, 1486 (2019)]
(CC(t;3)/AwCQZ)
1
(CCSDT/AwCQZ
(CCSD(T)/AwCQZ) 1e
(ex m
(CR-CC )
p
(2,3)/AwCQZ
er
-1e
1
) -1
e
e i ent) -e
409.6
4
= 39
366.7 cm
30.3
c
c
1
m
4
m
1.4 cm
3.4 cmD
D
D
D
D
−
−
=
=
=
=CCSDT
CCSD(T)CR-CC(2,3)
CC(t;3)
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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.
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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.QUESTIONS:
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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.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?
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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.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?
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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.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)]
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CI QUANTUM MONTE CARLO (CIQMC)
CC MONTE CARLO (CCMC)
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0 0 0( )
0 0
0
forlim ( ) lim for
0 for
H S
c S Ee S E
S E
τ
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
L K
c H S c H cτ τ ττ ≠
∂= − − −
∂ ∑
CIQMC (FCIQMC, CISDT-MC, CISDTQ-MC, etc.)
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( )( ) ( ) ( )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
α
τ δ= = ±∑
( )(( ) ( )( ))
KK K KL LK
K
LH cHc S c ττ
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.
CIQMC (FCIQMC, CISDT-MC, CISDTQ-MC, etc.)
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birth and death spawning
WALKER POPULATION DYNAMICS
,( ) ~ , 1K K K Kc N s sαα α
α
τ δ= = ±∑
( )(( ) ( )( ))
KK K KL LK
K
LH cHc S c ττ
ττ
≠
∂−=
∂− − ∑
CIQMC (FCIQMC, CISDT-MC, CISDTQ-MC, etc.)
( ) [1 ( ) ] ( )K KK Kc H S cτ τ τ τ+ ∆ = − − ∆( )
( ) ( ) ( )K K KL LL K
c c H cτ τ τ τ τ≠
+ ∆ = −∆ ∑
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birth and death spawning
WALKER POPULATION DYNAMICS
,( ) ~ , 1K K K Kc N s sαα α
α
τ δ= = ±∑
( )(( ) ( )( ))
KK K KL LK
K
LH cHc S c ττ
ττ
≠
∂−=
∂− − ∑
CIQMC (FCIQMC, CISDT-MC, CISDTQ-MC, etc.)
( ) [1 ( ) ] ( )K KK Kc H S cτ τ τ τ+ ∆ = − − ∆( )
( ) ( ) ( )K K KL LL K
c c H cτ τ τ τ τ≠
+ ∆ = −∆ ∑
![Page 82: APPROACHING EXACT QUANTUM CHEMISTRYBY STOCHASTIC …€¦ · and Arnab Chakraborty Department of Chemistry and Department of Physics & Astronomy, Michigan State University, East Lansing,](https://reader034.vdocuments.net/reader034/viewer/2022043004/5f8939eeeb3d1800a4714023/html5/thumbnails/82.jpg)
birth and death spawning
WALKER POPULATION DYNAMICS
,( ) ~ , 1K K K Kc N s sαα α
α
τ δ= = ±∑
( )(( ) ( )( ))
KK K KL LK
K
LH cHc S c ττ
ττ
≠
∂−=
∂− − ∑
CIQMC (FCIQMC, CISDT-MC, CISDTQ-MC, etc.)
( ) [1 ( ) ] ( )K KK Kc H S cτ τ τ τ+ ∆ = − − ∆( )
( ) ( ) ( )K K KL LL K
c c H cτ τ τ τ τ≠
+ ∆ = −∆ ∑
![Page 83: APPROACHING EXACT QUANTUM CHEMISTRYBY STOCHASTIC …€¦ · and Arnab Chakraborty Department of Chemistry and Department of Physics & Astronomy, Michigan State University, East Lansing,](https://reader034.vdocuments.net/reader034/viewer/2022043004/5f8939eeeb3d1800a4714023/html5/thumbnails/83.jpg)
birth and death spawning
WALKER POPULATION DYNAMICS
,( ) ~ , 1K K K Kc N s sαα α
α
τ δ= = ±∑
( )(( ) ( )( ))
KK K KL LK
K
LH cHc S c ττ
ττ
≠
∂−=
∂− − ∑
CIQMC (FCIQMC, CISDT-MC, CISDTQ-MC, etc.)
( ) [1 ( ) ] ( )K KK Kc H S cτ τ τ τ+ ∆ = − − ∆( )
( ) ( ) ( )K K KL LL K
c c H cτ τ τ τ τ≠
+ ∆ = −∆ ∑
![Page 84: APPROACHING EXACT QUANTUM CHEMISTRYBY STOCHASTIC …€¦ · and Arnab Chakraborty Department of Chemistry and Department of Physics & Astronomy, Michigan State University, East Lansing,](https://reader034.vdocuments.net/reader034/viewer/2022043004/5f8939eeeb3d1800a4714023/html5/thumbnails/84.jpg)
birth and death spawning
WALKER POPULATION DYNAMICS
,( ) ~ , 1K K K Kc N s sαα α
α
τ δ= = ±∑
( )(( ) ( )( ))
KK K KL LK
K
LH cHc S c ττ
ττ
≠
∂−=
∂− − ∑
CIQMC (FCIQMC, CISDT-MC, CISDTQ-MC, etc.)
( ) [1 ( ) ] ( )K KK Kc H S cτ τ τ τ+ ∆ = − − ∆( )
( ) ( ) ( )K K KL LL K
c c H cτ τ τ τ τ≠
+ ∆ = −∆ ∑
![Page 85: APPROACHING EXACT QUANTUM CHEMISTRYBY STOCHASTIC …€¦ · and Arnab Chakraborty Department of Chemistry and Department of Physics & Astronomy, Michigan State University, East Lansing,](https://reader034.vdocuments.net/reader034/viewer/2022043004/5f8939eeeb3d1800a4714023/html5/thumbnails/85.jpg)
birth and death spawning
WALKER POPULATION DYNAMICS
,( ) ~ , 1K K K Kc N s sαα α
α
τ δ= = ±∑
( )(( ) ( )( ))
KK K KL LK
K
LH cHc S c ττ
ττ
≠
∂−=
∂− − ∑
CIQMC (FCIQMC, CISDT-MC, CISDTQ-MC, etc.)
( ) [1 ( ) ] ( )K KK Kc H S cτ τ τ τ+ ∆ = − − ∆( )
( ) ( ) ( )K K KL LL K
c c H cτ τ τ τ τ≠
+ ∆ = −∆ ∑
![Page 86: APPROACHING EXACT QUANTUM CHEMISTRYBY STOCHASTIC …€¦ · and Arnab Chakraborty Department of Chemistry and Department of Physics & Astronomy, Michigan State University, East Lansing,](https://reader034.vdocuments.net/reader034/viewer/2022043004/5f8939eeeb3d1800a4714023/html5/thumbnails/86.jpg)
birth and death spawning
WALKER POPULATION DYNAMICS
,( ) ~ , 1K K K Kc N s sαα α
α
τ δ= = ±∑
( )(( ) ( )( ))
KK K KL LK
K
LH cHc S c ττ
ττ
≠
∂−=
∂− − ∑
CIQMC (FCIQMC, CISDT-MC, CISDTQ-MC, etc.)
( ) [1 ( ) ] ( )K KK Kc H S cτ τ τ τ+ ∆ = − − ∆( )
( ) ( ) ( )K K KL LL K
c c H cτ τ τ τ τ≠
+ ∆ = −∆ ∑
![Page 87: APPROACHING EXACT QUANTUM CHEMISTRYBY STOCHASTIC …€¦ · and Arnab Chakraborty Department of Chemistry and Department of Physics & Astronomy, Michigan State University, East Lansing,](https://reader034.vdocuments.net/reader034/viewer/2022043004/5f8939eeeb3d1800a4714023/html5/thumbnails/87.jpg)
birth and death spawning
WALKER POPULATION DYNAMICS
,( ) ~ , 1K K K Kc N s sαα α
α
τ δ= = ±∑
( )(( ) ( )( ))
KK K KL LK
K
LH cHc S c ττ
ττ
≠
∂−=
∂− − ∑
CIQMC (FCIQMC, CISDT-MC, CISDTQ-MC, etc.)
( ) [1 ( ) ] ( )K KK Kc H S cτ τ τ τ+ ∆ = − − ∆( )
( ) ( ) ( )K K KL LL K
c c H cτ τ τ τ τ≠
+ ∆ = −∆ ∑
![Page 88: APPROACHING EXACT QUANTUM CHEMISTRYBY STOCHASTIC …€¦ · and Arnab Chakraborty Department of Chemistry and Department of Physics & Astronomy, Michigan State University, East Lansing,](https://reader034.vdocuments.net/reader034/viewer/2022043004/5f8939eeeb3d1800a4714023/html5/thumbnails/88.jpg)
birth and death spawning
WALKER POPULATION DYNAMICS
,( ) ~ , 1K K K Kc N s sαα α
α
τ δ= = ±∑
( )(( ) ( )( ))
KK K KL LK
K
LH cHc S c ττ
ττ
≠
∂−=
∂− − ∑
CIQMC (FCIQMC, CISDT-MC, CISDTQ-MC, etc.)
( ) [1 ( ) ] ( )K KK Kc H S cτ τ τ τ+ ∆ = − − ∆( )
( ) ( ) ( )K K KL LL K
c c H cτ τ τ τ τ≠
+ ∆ = −∆ ∑
![Page 89: APPROACHING EXACT QUANTUM CHEMISTRYBY STOCHASTIC …€¦ · and Arnab Chakraborty Department of Chemistry and Department of Physics & Astronomy, Michigan State University, East Lansing,](https://reader034.vdocuments.net/reader034/viewer/2022043004/5f8939eeeb3d1800a4714023/html5/thumbnails/89.jpg)
birth and death spawning
WALKER POPULATION DYNAMICS
,( ) ~ , 1K K K Kc N s sαα α
α
τ δ= = ±∑
( )(( ) ( )( ))
KK K KL LK
K
LH cHc S c ττ
ττ
≠
∂−=
∂− − ∑
CIQMC (FCIQMC, CISDT-MC, CISDTQ-MC, etc.)
( ) [1 ( ) ] ( )K KK Kc H S cτ τ τ τ+ ∆ = − − ∆( )
( ) ( ) ( )K K KL LL K
c c H cτ τ τ τ τ≠
+ ∆ = −∆ ∑
![Page 90: APPROACHING EXACT QUANTUM CHEMISTRYBY STOCHASTIC …€¦ · and Arnab Chakraborty Department of Chemistry and Department of Physics & Astronomy, Michigan State University, East Lansing,](https://reader034.vdocuments.net/reader034/viewer/2022043004/5f8939eeeb3d1800a4714023/html5/thumbnails/90.jpg)
birth and death spawning
WALKER POPULATION DYNAMICS
,( ) ~ , 1K K K Kc N s sαα α
α
τ δ= = ±∑
( )(( ) ( )( ))
KK K KL LK
K
LH cHc S c ττ
ττ
≠
∂−=
∂− − ∑
CIQMC (FCIQMC, CISDT-MC, CISDTQ-MC, etc.)
( ) [1 ( ) ] ( )K KK Kc H S cτ τ τ τ+ ∆ = − − ∆( )
( ) ( ) ( )K K KL LL K
c c H cτ τ τ τ τ≠
+ ∆ = −∆ ∑
![Page 91: APPROACHING EXACT QUANTUM CHEMISTRYBY STOCHASTIC …€¦ · and Arnab Chakraborty Department of Chemistry and Department of Physics & Astronomy, Michigan State University, East Lansing,](https://reader034.vdocuments.net/reader034/viewer/2022043004/5f8939eeeb3d1800a4714023/html5/thumbnails/91.jpg)
birth and death spawning
WALKER POPULATION DYNAMICS
,( ) ~ , 1K K K Kc N s sαα α
α
τ δ= = ±∑
( )(( ) ( )( ))
KK K KL LK
K
LH cHc S c ττ
ττ
≠
∂−=
∂− − ∑
CIQMC (FCIQMC, CISDT-MC, CISDTQ-MC, etc.)
( ) [1 ( ) ] ( )K KK Kc H S cτ τ τ τ+ ∆ = − − ∆( )
( ) ( ) ( )K K KL LL K
c c H cτ τ τ τ τ≠
+ ∆ = −∆ ∑
![Page 92: APPROACHING EXACT QUANTUM CHEMISTRYBY STOCHASTIC …€¦ · and Arnab Chakraborty Department of Chemistry and Department of Physics & Astronomy, Michigan State University, East Lansing,](https://reader034.vdocuments.net/reader034/viewer/2022043004/5f8939eeeb3d1800a4714023/html5/thumbnails/92.jpg)
birth and death spawning
WALKER POPULATION DYNAMICS
,( ) ~ , 1K K K Kc N s sαα α
α
τ δ= = ±∑
( )(( ) ( )( ))
KK K KL LK
K
LH cHc S c ττ
ττ
≠
∂−=
∂− − ∑
CIQMC (FCIQMC, CISDT-MC, CISDTQ-MC, etc.)
( ) [1 ( ) ] ( )K KK Kc H S cτ τ τ τ+ ∆ = − − ∆( )
( ) ( ) ( )K K KL LL K
c c H cτ τ τ τ τ≠
+ ∆ = −∆ ∑
![Page 93: APPROACHING EXACT QUANTUM CHEMISTRYBY STOCHASTIC …€¦ · and Arnab Chakraborty Department of Chemistry and Department of Physics & Astronomy, Michigan State University, East Lansing,](https://reader034.vdocuments.net/reader034/viewer/2022043004/5f8939eeeb3d1800a4714023/html5/thumbnails/93.jpg)
birth and death spawning
WALKER POPULATION DYNAMICS
,( ) ~ , 1K K K Kc N s sαα α
α
τ δ= = ±∑
( )(( ) ( )( ))
KK K KL LK
K
LH cHc S c ττ
ττ
≠
∂−=
∂− − ∑
CIQMC (FCIQMC, CISDT-MC, CISDTQ-MC, etc.)
( ) [1 ( ) ] ( )K KK Kc H S cτ τ τ τ+ ∆ = − − ∆( )
( ) ( ) ( )K K KL LL K
c c H cτ τ τ τ τ≠
+ ∆ = −∆ ∑
![Page 94: APPROACHING EXACT QUANTUM CHEMISTRYBY STOCHASTIC …€¦ · and Arnab Chakraborty Department of Chemistry and Department of Physics & Astronomy, Michigan State University, East Lansing,](https://reader034.vdocuments.net/reader034/viewer/2022043004/5f8939eeeb3d1800a4714023/html5/thumbnails/94.jpg)
birth and death spawning
WALKER POPULATION DYNAMICS
,( ) ~ , 1K K K Kc N s sαα α
α
τ δ= = ±∑
( )(( ) ( )( ))
KK K KL LK
K
LH cHc S c ττ
ττ
≠
∂−=
∂− − ∑
CIQMC (FCIQMC, CISDT-MC, CISDTQ-MC, etc.)
( ) [1 ( ) ] ( )K KK Kc H S cτ τ τ τ+ ∆ = − − ∆( )
( ) ( ) ( )K K KL LL K
c c H cτ τ τ τ τ≠
+ ∆ = −∆ ∑
![Page 95: APPROACHING EXACT QUANTUM CHEMISTRYBY STOCHASTIC …€¦ · and Arnab Chakraborty Department of Chemistry and Department of Physics & Astronomy, Michigan State University, East Lansing,](https://reader034.vdocuments.net/reader034/viewer/2022043004/5f8939eeeb3d1800a4714023/html5/thumbnails/95.jpg)
birth and death spawning
WALKER POPULATION DYNAMICS
,( ) ~ , 1K K K Kc N s sαα α
α
τ δ= = ±∑
( )(( ) ( )( ))
KK K KL LK
K
LH cHc S c ττ
ττ
≠
∂−=
∂− − ∑
CIQMC (FCIQMC, CISDT-MC, CISDTQ-MC, etc.)
( ) [1 ( ) ] ( )K KK Kc H S cτ τ τ τ+ ∆ = − − ∆( )
( ) ( ) ( )K K KL LL K
c c H cτ τ τ τ τ≠
+ ∆ = −∆ ∑
![Page 96: APPROACHING EXACT QUANTUM CHEMISTRYBY STOCHASTIC …€¦ · and Arnab Chakraborty Department of Chemistry and Department of Physics & Astronomy, Michigan State University, East Lansing,](https://reader034.vdocuments.net/reader034/viewer/2022043004/5f8939eeeb3d1800a4714023/html5/thumbnails/96.jpg)
birth and death spawning
WALKER POPULATION DYNAMICS
,( ) ~ , 1K K K Kc N s sαα α
α
τ δ= = ±∑
( )(( ) ( )( ))
KK K KL LK
K
LH cHc S c ττ
ττ
≠
∂−=
∂− − ∑
CIQMC (FCIQMC, CISDT-MC, CISDTQ-MC, etc.)
( ) [1 ( ) ] ( )K KK Kc H S cτ τ τ τ+ ∆ = − − ∆( )
( ) ( ) ( )K K KL LL K
c c H cτ τ τ τ τ≠
+ ∆ = −∆ ∑
![Page 97: APPROACHING EXACT QUANTUM CHEMISTRYBY STOCHASTIC …€¦ · and Arnab Chakraborty Department of Chemistry and Department of Physics & Astronomy, Michigan State University, East Lansing,](https://reader034.vdocuments.net/reader034/viewer/2022043004/5f8939eeeb3d1800a4714023/html5/thumbnails/97.jpg)
birth and death spawning
WALKER POPULATION DYNAMICS
,( ) ~ , 1K K K Kc N s sαα α
α
τ δ= = ±∑
( )(( ) ( )( ))
KK K KL LK
K
LH cHc S c ττ
ττ
≠
∂−=
∂− − ∑
CIQMC (FCIQMC, CISDT-MC, CISDTQ-MC, etc.)
( ) [1 ( ) ] ( )K KK Kc H S cτ τ τ τ+ ∆ = − − ∆( )
( ) ( ) ( )K K KL LL K
c c H cτ τ τ τ τ≠
+ ∆ = −∆ ∑
![Page 98: APPROACHING EXACT QUANTUM CHEMISTRYBY STOCHASTIC …€¦ · and Arnab Chakraborty Department of Chemistry and Department of Physics & Astronomy, Michigan State University, East Lansing,](https://reader034.vdocuments.net/reader034/viewer/2022043004/5f8939eeeb3d1800a4714023/html5/thumbnails/98.jpg)
birth and death spawning
WALKER POPULATION DYNAMICS
,( ) ~ , 1K K K Kc N s sαα α
α
τ δ= = ±∑
( )(( ) ( )( ))
KK K KL LK
K
LH cHc S c ττ
ττ
≠
∂−=
∂− − ∑
CIQMC (FCIQMC, CISDT-MC, CISDTQ-MC, etc.)
( ) [1 ( ) ] ( )K KK Kc H S cτ τ τ τ+ ∆ = − − ∆( )
( ) ( ) ( )K K KL LL K
c c H cτ τ τ τ τ≠
+ ∆ = −∆ ∑
![Page 99: APPROACHING EXACT QUANTUM CHEMISTRYBY STOCHASTIC …€¦ · and Arnab Chakraborty Department of Chemistry and Department of Physics & Astronomy, Michigan State University, East Lansing,](https://reader034.vdocuments.net/reader034/viewer/2022043004/5f8939eeeb3d1800a4714023/html5/thumbnails/99.jpg)
birth and death spawning
WALKER POPULATION DYNAMICS
,( ) ~ , 1K K K Kc N s sαα α
α
τ δ= = ±∑
( )(( ) ( )( ))
KK K KL LK
K
LH cHc S c ττ
ττ
≠
∂−=
∂− − ∑
CIQMC (FCIQMC, CISDT-MC, CISDTQ-MC, etc.)
( ) [1 ( ) ] ( )K KK Kc H S cτ τ τ τ+ ∆ = − − ∆( )
( ) ( ) ( )K K KL LL K
c c H cτ τ τ τ τ≠
+ ∆ = −∆ ∑
![Page 100: APPROACHING EXACT QUANTUM CHEMISTRYBY STOCHASTIC …€¦ · and Arnab Chakraborty Department of Chemistry and Department of Physics & Astronomy, Michigan State University, East Lansing,](https://reader034.vdocuments.net/reader034/viewer/2022043004/5f8939eeeb3d1800a4714023/html5/thumbnails/100.jpg)
birth and death spawning
WALKER POPULATION DYNAMICS
,( ) ~ , 1K K K Kc N s sαα α
α
τ δ= = ±∑
( )(( ) ( )( ))
KK K KL LK
K
LH cHc S c ττ
ττ
≠
∂−=
∂− − ∑
CIQMC (FCIQMC, CISDT-MC, CISDTQ-MC, etc.)
( ) [1 ( ) ] ( )K KK Kc H S cτ τ τ τ+ ∆ = − − ∆( )
( ) ( ) ( )K K KL LL K
c c H cτ τ τ τ τ≠
+ ∆ = −∆ ∑
![Page 101: APPROACHING EXACT QUANTUM CHEMISTRYBY STOCHASTIC …€¦ · and Arnab Chakraborty Department of Chemistry and Department of Physics & Astronomy, Michigan State University, East Lansing,](https://reader034.vdocuments.net/reader034/viewer/2022043004/5f8939eeeb3d1800a4714023/html5/thumbnails/101.jpg)
birth and death spawning
WALKER POPULATION DYNAMICS
,( ) ~ , 1K K K Kc N s sαα α
α
τ δ= = ±∑
( )(( ) ( )( ))
KK K KL LK
K
LH cHc S c ττ
ττ
≠
∂−=
∂− − ∑
CIQMC (FCIQMC, CISDT-MC, CISDTQ-MC, etc.)
( ) [1 ( ) ] ( )K KK Kc H S cτ τ τ τ+ ∆ = − − ∆( )
( ) ( ) ( )K K KL LL K
c c H cτ τ τ τ τ≠
+ ∆ = −∆ ∑
![Page 102: APPROACHING EXACT QUANTUM CHEMISTRYBY STOCHASTIC …€¦ · and Arnab Chakraborty Department of Chemistry and Department of Physics & Astronomy, Michigan State University, East Lansing,](https://reader034.vdocuments.net/reader034/viewer/2022043004/5f8939eeeb3d1800a4714023/html5/thumbnails/102.jpg)
birth and death spawning
WALKER POPULATION DYNAMICS
,( ) ~ , 1K K K Kc N s sαα α
α
τ δ= = ±∑
( )(( ) ( )( ))
KK K KL LK
K
LH cHc S c ττ
ττ
≠
∂−=
∂− − ∑
CIQMC (FCIQMC, CISDT-MC, CISDTQ-MC, etc.)
( ) [1 ( ) ] ( )K KK Kc H S cτ τ τ τ+ ∆ = − − ∆( )
( ) ( ) ( )K K KL LL K
c c H cτ τ τ τ τ≠
+ ∆ = −∆ ∑
![Page 103: APPROACHING EXACT QUANTUM CHEMISTRYBY STOCHASTIC …€¦ · and Arnab Chakraborty Department of Chemistry and Department of Physics & Astronomy, Michigan State University, East Lansing,](https://reader034.vdocuments.net/reader034/viewer/2022043004/5f8939eeeb3d1800a4714023/html5/thumbnails/103.jpg)
birth and death spawning
WALKER POPULATION DYNAMICS
,( ) ~ , 1K K K Kc N s sαα α
α
τ δ= = ±∑
( )(( ) ( )( ))
KK K KL LK
K
LH cHc S c ττ
ττ
≠
∂−=
∂− − ∑
CIQMC (FCIQMC, CISDT-MC, CISDTQ-MC, etc.)
( ) [1 ( ) ] ( )K KK Kc H S cτ τ τ τ+ ∆ = − − ∆( )
( ) ( ) ( )K K KL LL K
c c H cτ τ τ τ τ≠
+ ∆ = −∆ ∑
![Page 104: APPROACHING EXACT QUANTUM CHEMISTRYBY STOCHASTIC …€¦ · and Arnab Chakraborty Department of Chemistry and Department of Physics & Astronomy, Michigan State University, East Lansing,](https://reader034.vdocuments.net/reader034/viewer/2022043004/5f8939eeeb3d1800a4714023/html5/thumbnails/104.jpg)
birth and death spawning
WALKER POPULATION DYNAMICS
,( ) ~ , 1K K K Kc N s sαα α
α
τ δ= = ±∑
( )(( ) ( )( ))
KK K KL LK
K
LH cHc S c ττ
ττ
≠
∂−=
∂− − ∑
CIQMC (FCIQMC, CISDT-MC, CISDTQ-MC, etc.)
( ) [1 ( ) ] ( )K KK Kc H S cτ τ τ τ+ ∆ = − − ∆( )
( ) ( ) ( )K K KL LL K
c c H cτ τ τ τ τ≠
+ ∆ = −∆ ∑
![Page 105: APPROACHING EXACT QUANTUM CHEMISTRYBY STOCHASTIC …€¦ · and Arnab Chakraborty Department of Chemistry and Department of Physics & Astronomy, Michigan State University, East Lansing,](https://reader034.vdocuments.net/reader034/viewer/2022043004/5f8939eeeb3d1800a4714023/html5/thumbnails/105.jpg)
birth and death spawning
WALKER POPULATION DYNAMICS
,( ) ~ , 1K K K Kc N s sαα α
α
τ δ= = ±∑
( )(( ) ( )( ))
KK K KL LK
K
LH cHc S c ττ
ττ
≠
∂−=
∂− − ∑
CIQMC (FCIQMC, CISDT-MC, CISDTQ-MC, etc.)
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
c c H cτ τ τ τ τ≠
+ ∆ = −∆ ∑
![Page 106: APPROACHING EXACT QUANTUM CHEMISTRYBY STOCHASTIC …€¦ · and Arnab Chakraborty Department of Chemistry and Department of Physics & Astronomy, Michigan State University, East Lansing,](https://reader034.vdocuments.net/reader034/viewer/2022043004/5f8939eeeb3d1800a4714023/html5/thumbnails/106.jpg)
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).
![Page 107: APPROACHING EXACT QUANTUM CHEMISTRYBY STOCHASTIC …€¦ · and Arnab Chakraborty Department of Chemistry and Department of Physics & Astronomy, Michigan State University, East Lansing,](https://reader034.vdocuments.net/reader034/viewer/2022043004/5f8939eeeb3d1800a4714023/html5/thumbnails/107.jpg)
Developing a Stochastic CC(P;Q) Approach
[J.E. Deustua, J. Shen, and P. Piecuch, Phys. Rev. Lett. 119, 223003 (2017)]
![Page 108: APPROACHING EXACT QUANTUM CHEMISTRYBY STOCHASTIC …€¦ · and Arnab Chakraborty Department of Chemistry and Department of Physics & Astronomy, Michigan State University, East Lansing,](https://reader034.vdocuments.net/reader034/viewer/2022043004/5f8939eeeb3d1800a4714023/html5/thumbnails/108.jpg)
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.
[J.E. Deustua, J. Shen, and P. Piecuch, Phys. Rev. Lett. 119, 223003 (2017)]
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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.
[J.E. Deustua, J. Shen, and P. Piecuch, Phys. Rev. Lett. 119, 223003 (2017)]
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[J.E. Deustua, J. Shen, and P. Piecuch, Phys. Rev. Lett. 119, 223003 (2017)]
Developing a Stochastic CC(P;Q) Approach
F2 /cc-pVDZ, R = 2 Re, i-FCIQMC
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[J.E. Deustua, J. Shen, and P. Piecuch, Phys. Rev. Lett. 119, 223003 (2017)]
Developing a Stochastic CC(P;Q) Approach
F2 /cc-pVDZ, R = 2 Re, i-FCIQMC
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[J.E. Deustua, J. Shen, and P. Piecuch, Phys. Rev. Lett. 119, 223003 (2017)]
Developing a Stochastic CC(P;Q) Approach
F2 /cc-pVDZ, R = 2 Re, i-FCIQMC
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[J.E. Deustua, J. Shen, and P. Piecuch, Phys. Rev. Lett. 119, 223003 (2017)]
Developing a Stochastic CC(P;Q) Approach
F2 /cc-pVDZ, R = 2 Re, i-FCIQMC
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[J.E. Deustua, J. Shen, and P. Piecuch, Phys. Rev. Lett. 119, 223003 (2017)]
Developing a Stochastic CC(P;Q) Approach
F2 /cc-pVDZ, R = 2 Re, i-FCIQMC
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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.
[J.E. Deustua, J. Shen, and P. Piecuch, Phys. Rev. Lett. 119, 223003 (2017)]
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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.
3. Solve the CC(P) equations. if the target approach is CCSDT, use if the target approach is CCSDTQ, use
etc.
( )1
(MC32
)PT T T T= + +(MC) (MC)
3(
2 4)
1PT T T TT= + + +
[J.E. Deustua, J. Shen, and P. Piecuch, Phys. Rev. Lett. 119, 223003 (2017)]
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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
[J.E. Deustua, J. Shen, and P. Piecuch, Phys. Rev. Lett. 119, 223003 (2017)]
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RECOVERING CCSDT ENERGETICS FOR F2/cc-pVDZMONTE CARLO APPROACH = i-FCIQMC (∆τ = 0.0001 a.u., na = 3)
FCIQMC
CC(P)
R = Re
R = 5ReR = 2Re
R = 1.5Re
[J.E. Deustua, J. Shen, and P. Piecuch, Phys. Rev. Lett. 119, 223003 (2017)]
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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.
3. Solve the CC(P) equations. if the target approach is CCSDT, use if the target approach is CCSDTQ, use
etc.
( )1
(MC32
)PT T T T= + +(MC) (MC)
3(
2 4)
1PT T T TT= + + +
[J.E. Deustua, J. Shen, and P. Piecuch, Phys. Rev. Lett. 119, 223003 (2017)]
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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.
3. Solve the CC(P) equations. if the target approach is CCSDT, use if the target approach is CCSDTQ, use
etc.
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).
( )1
(MC32
)PT T T T= + +(MC) (MC)
3(
2 4)
1PT T T TT= + + +
[J.E. Deustua, J. Shen, and P. Piecuch, Phys. Rev. Lett. 119, 223003 (2017)]
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RECOVERING CCSDT ENERGETICS FOR F2/cc-pVDZMONTE CARLO APPROACH = i-FCIQMC (∆τ = 0.0001 a.u., na = 3)
FCIQMC
CC(P)
CC(P;Q)EN
R = Re
R = 5ReR = 2Re
R = 1.5Re
[J.E. Deustua, J. Shen, and P. Piecuch, Phys. Rev. Lett. 119, 223003 (2017)]
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RECOVERING CCSDT ENERGETICS FOR F2/cc-pVDZMONTE CARLO APPROACH = i-CISDTQ-MC (∆τ = 0.0001 a.u., na = 3)
CISDTQ-MC
CC(P)
CC(P;Q)EN
R = Re
R = 5ReR = 2Re
R = 1.5Re
[J.E. Deustua, J. Shen, and P. Piecuch, in preparation]
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RECOVERING CCSDT ENERGETICS FOR F2/cc-pVDZMONTE CARLO APPROACH = i-CISDT-MC (∆τ = 0.0001 a.u., na = 3)
CISDT-MC
CC(P)
CC(P;Q)EN
R = Re
R = 5ReR = 2Re
R = 1.5Re
[J.E. Deustua, J. Shen, and P. Piecuch, in preparation]
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RECOVERING CCSDT ENERGETICS FOR F2/cc-pVDZMONTE CARLO APPROACH = i-CCSDT-MC (∆τ = 0.0001 a.u., na = 3)
CCSDT-MC
CC(P)
CC(P;Q)EN
R = Re
R = 5ReR = 2Re
R = 1.5Re
[J.E. Deustua, J. Shen, and P. Piecuch, Phys. Rev. Lett. 119, 223003 (2017)]
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MC Iter. % ofTriples in P
space
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)
0 2 2
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
RECOVERING CCSDT ENERGETICS FOR F2/cc-pVDZMONTE CARLO APPROACH = i-FCIQMC (∆τ = 0.0001 a.u., na = 3)
Errors relative to CCSDT
Errors relative to CCSDTCCSD: 45.638 mEhCCSD(T): −23.596 mEh
[J.E. Deustua, J. Shen, and P. Piecuch, Phys. Rev. Lett. 119, 223003 (2017)]
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MC Iter. % ofTriples in P
space
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)
0 2 2
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)
Errors relative to CCSDTCCSD: 45.638 mEhCCSD(T): −23.596 mEh
Errors relative to CCSDT
[J.E. Deustua, J. Shen, and P. Piecuch, in preparation]
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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)
CC(P;Q)EN
[J.E. Deustua, J. Shen, and P. Piecuch, Phys. Rev. Lett. 119, 223003 (2017)]
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MC Iter. % ofTriples in P
space
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)
~300
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)
Errors relative to CCSDTCCSD: 65.036 mEhCCSD(T): −27.209 mEh
Errors relative to CCSDT
[J.E. Deustua, J. Shen, and P. Piecuch, Phys. Rev. Lett. 119, 223003 (2017)]
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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]
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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
Errors relative to CCSDT
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]
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RECOVERING CCSDT ENERGETICS FOR AUTOMERIZATION OF CYCLOBUTADIENE/cc-pVDZ
MC APPROACH = i-FCIQMC, i-CISDTQ-MC, 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
CC(P)
CC(P;Q)EN
MC = FCIQMC MC = CISDTQ-MC MC = CISDT-MC
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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)
CC(P;Q)EN
Triples
Quads
R = Re R = 3Re
R = 3Re
R R
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WHAT IS NEXT?Higher-order CC methods, excited states, corrections toward FCI, and more … .
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[J.E. Deustua, S.H. Yuwono, J. Shen, and P. Piecuch, J. Chem. Phys. 150, 111101 (2019)]EXCITED STATES
WHAT IS NEXT?Higher-order CC methods, excited states, corrections toward FCI, and more … .
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[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
WHAT IS NEXT?Higher-order CC methods, excited states, corrections toward FCI, and more … .
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[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
WHAT IS NEXT?Higher-order CC methods, excited states, corrections toward FCI, and more … .
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[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
WHAT IS NEXT?Higher-order CC methods, excited states, corrections toward FCI, and more … .
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[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)
WHAT IS NEXT?Higher-order CC methods, excited states, corrections toward FCI, and more … .
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[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)
WHAT IS NEXT?Higher-order CC methods, excited states, corrections toward FCI, and more … .
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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)
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); J.E. Deustua, S.H. Yuwono, A. Chakraborty, J. Shen, and P. Piecuch, in preparation]
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EXCITED STATES
WHAT IS NEXT?Higher-order CC methods, excited states, corrections toward FCI, and more … .
[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
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EXCITED STATES
WHAT IS NEXT?Higher-order CC methods, excited states, corrections toward FCI, and more … .
[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)
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HOW ABOUT ATTACKING FCI (EXACT THEORY!) ITSELF?[J.E. Deustua, I. Magoulas, J. Shen, and P. Piecuch, J. Chem. Phys. 149, 151101 (2018)]
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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
C
H T T T TT T
T T T T T
Φ + + + + + +
+ + + + + Φ =
3
4 3
T
T T
(…)
20 1 2 1[ ( (1/ 2) )]N CE H H T T T= Φ Φ + Φ + + Φ
HOW ABOUT ATTACKING FCI (EXACT THEORY!) ITSELF?[J.E. Deustua, I. Magoulas, J. Shen, and P. Piecuch, J. Chem. Phys. 149, 151101 (2018)]
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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
C
H T T T TT T
T T T T T
Φ + + + + + +
+ + + + + Φ =
3
4 3
T
T T
(…)
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
HOW ABOUT ATTACKING FCI (EXACT THEORY!) ITSELF?[J.E. Deustua, I. Magoulas, J. Shen, and P. Piecuch, J. Chem. Phys. 149, 151101 (2018)]
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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
C
H T T T TT T
T T T T T
Φ + + + + + +
+ + + + + Φ =
3
4 3
T
T T
(…)
20 1 2 1[ ( (1/ 2) )]N CE H H T T T= Φ Φ + Φ + + Φ
HOW ABOUT ATTACKING FCI (EXACT THEORY!) ITSELF?[J.E. Deustua, I. Magoulas, J. Shen, and P. Piecuch, J. Chem. Phys. 149, 151101 (2018)]
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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
C
H T T T TT T
T T T T T
Φ + + + + + +
+ + + + + Φ =
3
4 3
T
T T
(…)
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
HOW ABOUT ATTACKING FCI (EXACT THEORY!) ITSELF?[J.E. Deustua, I. Magoulas, J. Shen, and P. Piecuch, J. Chem. Phys. 149, 151101 (2018)]
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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
C
H T T T TT T
T T T T T
Φ + + + + + +
+ + + + + Φ =
3
4 3
T
T T
(…)
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
HOW ABOUT ATTACKING FCI (EXACT THEORY!) ITSELF?[J.E. Deustua, I. Magoulas, J. Shen, and P. Piecuch, J. Chem. Phys. 149, 151101 (2018)]
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CI (FCIQMC) expansion CC expansion
HOW ABOUT ATTACKING FCI (EXACT THEORY!) ITSELF?[J.E. Deustua, I. Magoulas, J. Shen, and P. Piecuch, J. Chem. Phys. 149, 151101 (2018)]
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CI (FCIQMC) expansion CC expansion
1. Start a FCIQMC propagation by placing a certain number of walkers on the reference determinant.
HOW ABOUT ATTACKING FCI (EXACT THEORY!) ITSELF?[J.E. Deustua, I. Magoulas, J. Shen, and P. Piecuch, J. Chem. Phys. 149, 151101 (2018)]
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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≡ = =∑
HOW ABOUT ATTACKING FCI (EXACT THEORY!) ITSELF?[J.E. Deustua, I. Magoulas, J. Shen, and P. Piecuch, J. Chem. Phys. 149, 151101 (2018)]
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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
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)]
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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
3. Perform cluster analysis of the FCIQMC wave function at time τ to extract
4. Solve CCSD equations corrected for terms containing to determine
(MC)0( ) ( ) , where ( ) / .n n K K K K
KC C c E c N Nτ τ τ≡ = =∑
(MC) (MC)3 4an) )d( (T Tτ τ= =3 4T T
HOW ABOUT ATTACKING FCI (EXACT THEORY!) ITSELF?
(MC) (MC)3 4and( ) ( ) .T Tτ τ
1 2 0and and energy ( ).T T E τ
[J.E. Deustua, I. Magoulas, J. Shen, and P. Piecuch, J. Chem. Phys. 149, 151101 (2018)]
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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
3. Perform cluster analysis of the FCIQMC wave function at time τ to extract
4. Solve CCSD equations corrected for terms containing to determine
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
KC C c E c N Nτ τ τ≡ = =∑
(FCI)0 0It is guaranteed that im ( ) .l E E
ττ
→∞=
(MC) (MC)3 4an) )d( (T Tτ τ= =3 4T T
HOW ABOUT ATTACKING FCI (EXACT THEORY!) ITSELF?
(MC) (MC)3 4and( ) ( ) .T Tτ τ
1 2 0and and energy ( ).T T E τ
[J.E. Deustua, I. Magoulas, J. Shen, and P. Piecuch, J. Chem. Phys. 149, 151101 (2018)]
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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
3. Perform cluster analysis of the FCIQMC wave function at time τ to extract
4. Solve CCSD equations corrected for terms containing to determine
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
KC C c E c N Nτ τ τ≡ = =∑
(FCI)0 0It is guaranteed that im ( ) .l E E
ττ
→∞=
(MC) (MC)3 4an) )d( (T Tτ τ= =3 4T T
CLUSTER-ANALYSIS-DRIVEN FCIQMC = CAD-FCIQMC
HOW ABOUT ATTACKING FCI (EXACT THEORY!) ITSELF?
(MC) (MC)3 4and( ) ( ) .T Tτ τ
1 2 0and and energy ( ).T T E τ
[J.E. Deustua, I. Magoulas, J. Shen, and P. Piecuch, J. Chem. Phys. 149, 151101 (2018)]
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R R
HOW ABOUT ATTACKING FCI (EXACT THEORY!) ITSELF?[J.E. Deustua, I. Magoulas, J. Shen, and P. Piecuch, J. Chem. Phys. 149, 151101 (2018)]
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R R
R = Re
R = 1.5Re
R = 2.0Re
HOW ABOUT ATTACKING FCI (EXACT THEORY!) ITSELF?[J.E. Deustua, I. Magoulas, J. Shen, and P. Piecuch, J. Chem. Phys. 149, 151101 (2018)]
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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 R
R = Re
R = 1.5Re
R = 2.0Re
HOW ABOUT ATTACKING FCI (EXACT THEORY!) ITSELF?[J.E. Deustua, I. Magoulas, J. Shen, and P. Piecuch, J. Chem. Phys. 149, 151101 (2018)]
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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
HOW ABOUT ATTACKING FCI (EXACT THEORY!) ITSELF?[J.E. Deustua, I. Magoulas, J. Shen, and P. Piecuch, J. Chem. Phys. 149, 151101 (2018)]
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⇒ +
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HOW ABOUT ATTACKING FCI (EXACT THEORY!) ITSELF?
[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.
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Benzene/cc-pVDZ:
RHF MO basis
30 correlated electrons (6 core MOs frozen)
114 basis functions
HOW ABOUT ATTACKING FCI (EXACT THEORY!) ITSELF?
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).
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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…
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Search for Chem 580
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“Algebraic and Diagrammatic Methods for Many-Fermion Systems”https://pages.wustl.edu/ppiecuch/course-videos
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Emiliano Deustua (2014-present)
Ilias Magoulas (2015-present)
Dr. Jun Shen (2010-present)
Stephen Yuwono (2017-present)
Arnab Chakraborty (2018-present)