Quantum chemistry without wave functions › ... › reu › REU_deprince_6_18_19.pdfQuantum chemistry without wave functions (and I don't mean density functional theory) Eugene DePrince

Quantum chemistry without wave functions

Eugene DePrince Department of Chemistry and Biochemistry

Florida State University

June 18th, 2019

Quantum chemistry without wave functions (and I don't mean density functional theory)

Eugene DePrince Department of Chemistry and Biochemistry

Florida State University

June 18th, 2019

electronic structure in complex systems

Class 1: non-dynamical (or strong / static / multireference) electron correlation

Class 2: strong fields, multiple pulses: beyond linear response with real-time methods

- - -- - --

+++ ++ ++E(t)

singlet fission structure/spectra of heavy-atom complexes

spin-state transitions and spin-crossover complexes

JACS, 136, 8050 (2014)

Nat. Commun, 6, 6827 (2015)

“The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble.

Paul A. M. Dirac, 1929



the Schrödinger equation: H = E �i~@ @t

= H or



It therefore becomes desirable that approximate practical methods of applying quantum mechanics should be developed, which can lead to an explanation of the main features of complex atomic systems without too much computation.”

the Schrödinger equation: H = E �i~@ @t

= H or

dynamical electron correlationHartree-Fock MP2 CCSD CCSD(T) full CI

size

of t

he o

ne-e

lect

ron

basi

s se

t

what physics is included

how well is “space” described

cost of “approximate practical methods”

MP2: 2nd-order perturbation theoryCCSD: coupled cluster with single and double excitationsCCSD(T): CCSD + perturbative triple excitationsfull CI: full configuration interaction … wave function includes all possible electronic configurations


size

of t

he o

ne-e

lect

ron

basi

s se

tcost of “approximate practical methods”


two-body correlations


size

of t

he o

ne-e

lect

ron

basi

s se




three-body correlations


size

of t

he o

ne-e

lect

ron

basi

s se





N-body correlations


size

of t

he o

ne-e

lect

ron

basi

s se



X truth!



N-body correlations


size

of t

he o

ne-e

lect

ron

basi

s se



X truth!

feasible computationstwo-body correlations


N-body correlations


size

of t

he o

ne-e

lect

ron

basi

s se



X truth!

feasible computations


size

of t

he o

ne-e

lect

ron

basi

s se



X truth!feasible computationshow do we move this line?


size

of t

he o

ne-e

lect

ron

basi

s se


MP2: 2nd-order perturbation theoryCCSD: coupled cluster with single and double excitationsCCSD(T): CCSD + perturbative triple excitationsfull CI: full configuration interaction … include all possible electronic configurations


1.mild approximations


size

of t

he o

ne-e

lect

ron

basi

s se






size

of t

he o

ne-e

lect

ron

basi

s se



- local correlation approximations

X truth!feasible computations


size

of t

he o

ne-e

lect

ron

basi

s se




how do we move this line?


X truth!feasible computations


size

of t

he o

ne-e

lect

ron

basi

s se




how do we move this line?



1.mild approximations2. improved algorithms and hardware


size

of t

he o

ne-e

lect

ron

basi

s se




1.mild approximations2. improved algorithms and hardware


size

of t

he o

ne-e

lect

ron

basi

s se



vs

GPU CCSD: ~3x acceleration

AED, Hammond, JCTC, 7, 12871295 (2011)AED, Kennedy, Sumpter, Sherrill, Mol. Phys. 112, 844-852 (2014)

Mullinax, Koulias, Gidofalvi, Epifanovsky, AED, in prep.

GPU v2RDM-CASSCF: ~3x acceleration


1.mild approximations2. improved algorithms and hardware3.completely different representations of the electronic structure


size

of t

he o

ne-e

lect

ron

basi

s se



static / strong / non-dynamical / multireference electron correlation

nondynamical correlation

σg

σu

in bonding region, wave function iswell-described by a single electronicconfiguration

−1.2

−1.1

−1.0

−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

ener

gy (E

h)

H−H distance (Å)

Hartree−FockRestrictedE


example: H2 dissociation in a minimal basis


σg σu

−1.2

−1.1

−1.0

−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

ener

gy (E

h)

H−H distance (Å)

Hartree−FockRestrictedE

at dissociation, the MOs become nearly degenerate, the wave function can no longer be described by a single RHF determinant


example: H2 dissociation in a minimal basis


σg σuRestricted

example: H2 dissociation in a minimal basis E

−1.2

−1.1

−1.0

−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

ener

gy (E

h)

H−H distance (Å)

Hartree−FockFull CI

qualitatively correct wave function includes equal contributions from two electronic configurations

(1, 2) =1p2(�2

g � �2u)

nondynamical correlationstatic / strong / non-dynamical / multireference electron correlation

at dissociation, the MOs become nearly degenerate, the wave function can no longer be described by a single RHF determinant

−109.1

−109.0

−108.9

−108.8

−108.7

−108.6

−108.5

−108.4

1.0 1.5 2.0 2.5 3.0 3.5

ener

gy (E

h)

N−N distance (Å)

To describe N2 dissociation, we must ask: which orbitals are important?

complete active space self consistent field method (CASSCF)

conventional methods for nondynamical correlation

−109.1

−109.0

−108.9

−108.8

−108.7

−108.6

−108.5

−108.4

1.0 1.5 2.0 2.5 3.0 3.5

ener

gy (E

h)

N−N distance (Å)


0.0

0.5

1.0

1.5

2.0

1.0 1.5 2.0 2.5 3.0 3.5

occu

patio

n

N−N bond length (Å)

HONO (πu)LUNO (πg)

HONO−1 (σg)LUNO+1 (σu)

π/π*, σ/σ*6 electrons, 6 orbitals



−109.1

−109.0

−108.9

−108.8

−108.7

−108.6

−108.5

−108.4

1.0 1.5 2.0 2.5 3.0 3.5

ener

gy (E

h)

N−N distance (Å)


π/π*, σ/σ*6 electrons, 6 orbitals

core

active: (6e,6o)

virtual



core

virtual


active: (6e,6o)


core

virtual


active: (6e,6o)

| CIi = (1 + C1 + C2 + C3 + . . . + CN )| 0i

active space often by configuration interaction (CI) - full CI in active space: “CAS”

simultaneous optimization of orbitals - mixing between active/virtual and active/core spaces: “SCF”



1

100000

1x1010

1x1015

1x1020

1x1025

1x1030

0 5 10 15 20 25 30 35 40 45 50

k

(ke,ko) active space

num

ber

of v

aria

bles

100

105

1010

1015

1020

1025

1030

beyond (18e,18o) intractable

core

virtual

active: (6e,6o)

- CI



bonding in actinide complexeswhat orbitals are important?


1

100000

1x1010

1x1015

1x1020

1x1025

1x1030

0 5 10 15 20 25 30 35 40 45 50

k


num

ber

of v

aria

bles

100

105

1010

1015

1020

1025

1030


metal: 5f, 6d, 7s, 7pligands (O,N): 2p

25+ electrons, 25+ total orbitals

- CI





1

100000

1x1010

1x1015

1x1020

1x1025

1x1030

0 5 10 15 20 25 30 35 40 45 50

k


100

105

1010

1015

1020

1025

1030




- CI

num

ber

of v

aria

bles

DMRG-drivena

Firm: Q-Chem, Inc. Topic: A14A-T013 Proposal: A2-6031

[19] S. Burer and R. D. Monteiro, Mathematical Programming 95, 329 (2003).[20] J. Povh, F. Rendl, A. Wiegele, Computing 78, 277 (2006).[21] J. Malick, J. Povh, F. Rendl, and A. Wiegele, SIAM J. Optim. 20, 336 (2009).[22] D. A. Mazziotti, Phys. Rev. Lett. 106, 083001 (2011).[23] Y. Shao, L. Fusti-Molnar, Y. Jung, J. Kussmann, C. Ochsenfeld, S. T. Brown, A. T. B. Gilbert, L.

V. Slipchenko,S. V. Levchenko, D. P. O’Neill, R. A. Distasio Jr., R. C. Lochan, T. Wang, G. J. O.Beran, N. A. Besley, J. M., Herbert, C. Y. Lin, T. Van Voorhis, S. H. Chien, A. Sodt, R. P. Steele,V. A. Rassolov, P. E. Maslen, P. P. Korambath, R. D. Adamson, B. Austin, J. Baker, E. F. C. Byrd,H. Dachsel, R. J. Doerksen, A. Dreuw, B. D. Dunietz, A. D. Dutoi, T. R. Furlani, S. R. Gwaltney, A.Heyden, S. Hirata, C.-P. Hsu, G. Kedziora, R. Z. Khalliulin, P. Klunzinger, A. M. Lee, M. S. Lee, W.Liang, I. Lotan, N. Nair, B. Peters, E. I. Proynov, P. A. Pieniazek, Y. M. Rhee, J. Ritchie, E. Rosta,C. D. Sherrill, A. C. Simmonett, J. E. Subotnik, H. L. Woodcock III, W. Zhang, A. T. Bell, A. K.Chakraborty, D. M. Chipman, F. J. Keil, A. Warshel, W. J. Hehre, H. F. Schaefer III, J. Kong, A. I.Krylov, P. M. W. Gill, M. Head-Gordon, Phys. Chem. Chem. Phys. 8, 3172 - 3191 (2006).

[24] E. Epifanovsky, M. Wormit, T. Ku, A. Landau, D. Zuev, K. Khistyaev, P. Manohar, I. Kaliman,A. Dreuw, and A. I. Krylov, J. Comput. Chem. 34, 2293 (2013).

[25] E. Solomonik, D. Matthews, J. Hammond, and J. Demmel, Cyclops Tensor Framework: reducing com-munication and eliminating load imbalance in massively parallel contractions, Tech. Rep. UCB/EECS-2012-210 (EECS Department, University of California, Berkeley, 2012).

[26] M. Nakata, B. J. Braams, K. Fujisawa, M. Fukuda, J. K. Percus, M. Yamashita, and Z. Zhao, J. Chem.Phys. 128, 164113 (2008).

[27] Turney, J. M.; Simmonett, A. C.; Parrish, R. M.; Hohenstein, E. G.; Evangelista, F. A.; Fermann, J.T.; Mintz, B. J.; Burns, L. A.; Wilke, J. J.; Abrams, M. L.; Russ, N. J.; Leininger, M. L.; Janssen, C.L.; Seidl, E. T.; Allen, W. D.; Schaefer, H. F.; King, R. A.; Valeev, E. F.; Sherrill, C. D.; Crawford, T.D. “Psi4: an open-source ab initio electronic structure program”, WIREs Comput. Mol. Sci. 2, 556-565(2012).

[28] A. J. Coleman, Rev. Mod. Phys. 35, 668 (1963).[29] R. M. Erdahl, Rep. Math. Phys. 15, 147 (1979).[30] G. Gidofalvi and D. A. Mazziotti, Phys. Rev. A 72, 052505 (2005).[31] S. R. White, Phys. Rev. Lett. 69, 2863 (1992); S. R. White, Phys. Rev. B 48, 10345 (1993); S. R.

White and R. L. Martin, J. Chem. Phys. 110, 4127 (1999); G. K.-L. Chan and M. Head-Gordon, J.Chem. Phys. 116, 4462 (2002); G. K.-L. Chan and M. Head-Gordon, J. Chem. Phys. 118, 8551 (2003);G. K.-L. Chan, J. Chem. Phys. 120, 3172 (2004).

[32] D. Ghosh, J. Hachmann, T. Yanai, and G. K.-L. Chan, J. Chem. Phys. 128, 144117 (2008).[33] T. Yanai, Y. Kurashige, D. Ghosh, and G. K.-L. Chan, Int. J. Quantum Chem. 109, 2178 (2009).[34] T. Yanai, Y. Kurashige, E. Neuscamman, and G. K.-L. Chan, J. Chem. Phys. 132, 024105 (2010).[35] L. S. Blackford, J. Choi, A. Cleary, E. D’Azevedo, J. Demmel, I. Dhillon, J. Dongarra, S. Hammarling,

G. Henry, A. Petitet, K. Stanley, D. Walker, and R. C. Whaley, ScaLAPACK Users’ Guide (Societyfor Industrial and Applied Mathematics, Philadelphia, PA, 1997).

[36] P. Alpatov, G. Baker, C. Edwards, J. Gunnels, G. Morrow, J. Overfelt, and R. V. D. Geijn, “Plapack:Parallel linear algebra libraries design overview,”.

[37] J. Poulson, B. Marker, R. A. van de Geijn, J. R. Hammond, and N. A. Romero, ACM Trans. Math.Softw. 39, 13:1 (2013).

[38] T. Auckenthaler, H.-J. Bungartz, T. Huckle, L. Krmer, B. Lang, and P. Willems, J. Comput. Sci. 2,272 (2011), social Computational Systems.

[39] H. van Aggelen, B. Verstichel, P. Bultinck, D. Van Neck, and P. W. Ayers, J. Chem. Phys. 136, 014110(2012).

[40] E. Perez-Romero, L. M. Tel, and C. Valdemoro, Int. J. Quantum Chem. 61, 55 (1997).[41] J. Fosso-Tande, D. R. Nascimento, and A. E. DePrince III, submitted (2015).[42] J. Hachmann, J. J. Dorando, M. Avils, and G. K.-L. Chan, J. Chem. Phys. 127, 134309 (2007).[43] J. R. Hammond and D. A. Mazziotti, Phys. Rev. A 71, 062503 (2005).[44] D. A. Mazziotti, Phys. Rev. A 72, 032510 (2005).[45] G. Gidofalvi and D. A. Mazziotti, The Journal of Chemical Physics 125, 144102 (2006).[46] P. Pulay, Chem. Phys. Lett. 73, 393 (1980).

variational 2-RDM (v2RDM) drivenb


E. Epifanovsky, D. Zuev, X. Feng, K. Khistyaev, Y. Shao, and A. I. Krylov, “General imple-mentation of the resolution-of-the-identity and Cholesky representations of electron repulsionintegrals within coupled-cluster and equation-of-motion methods: Theory and benchmarks,”J. Chem. Phys. 139, 134105 (2013).

E. Epifanovsky, M. Wormit, T. Kus, A. Landau, D. Zuev, K. Khistyaev, P. Manohar, I. Kali-man, A. Dreuw, and A.I. Krylov “New implementation of high-level correlated methodsusing a general block-tensor library for high-performance electronic structure calculations,”J. Comput. Chem. 34, 2293 (2013).

VIII. FACILITIES/EQUIPMENT

At the time of submission of this proposal, the DePrince Group at Florida State University has5 modern multi-core Linux servers used for code development and benchmarking. One server isequipped with an NVIDIA Tesla K40c (Kepler) GPU. We have also purchased 160 computer coresat the FSU Research and Computing Center. Q-Chem, Inc. owns six modern multi-core Linuxservers and several other Mac OS, Windows, AIX and Sun Solaris computer servers to support in-house code development and platform porting. The proposed Phase II research e↵ort is limited totheoretical and computational work, and the facilities meet all environmental laws and regulations.

IX. CONSULTANTS

No consultants will be employed during Phase II of this STTR.

X. REFERENCES

[1] C. Garrod, M. V. Mihailovic, and M. Rosina, J. Math. Phys. 16, 868 (1975); M. V. Mihailovic and M.Rosina, Nucl. Phys. A 237, 221 (1975); M. Rosina and C. Garrod, J. Comput. Phys. 18, 300 (1975);R. M. Erdahl, C. Garrod, B. Golli, and M. Rosina, J. Math. Phys. 20, 1366 (1979); R. M. Erdahl, Rep.Math. Phys. 15, 147 (1979).

[2] R. M. Erdahl, Int. J. Quantum Chem. 13, 697 (1978).[3] M. Nakata, H. Nakatsuji, M. Ehara, M. Fukuda, K. Nakata, and K. Fujisawa, J. Chem. Phys. 114,

8282 (2001).[4] D. A. Mazziotti and R. M. Erdahl, Phys. Rev. A 63, 042113 (2001).[5] D. A. Mazziotti, Phys. Rev. A 65, 062511 (2002).[6] D. A. Mazziotti, Phys. Rev. A 74, 032501 (2006).[7] Z. Zhao, B. J. Braams, M. Fukuda, M. L. Overton, and J. K. Percus, J. Chem. Phys. 120, 2095 (2004).[8] M. Fukuda, B. J. Braams, M. Nakata, M. L. Overton, J. K. Percus, M. Yamashita, and Z. Zhao, Math.

Program. 109, 553 (2007).[9] E. Cances, G. Stoltz, and M. Lewin, J. Chem. Phys. 125, 064101 (2006).[10] B. Verstichel, H. van Aggelen, D. Van Neck, P. W. Ayers, and P. Bultinck, Phys. Rev. A 80, 032508

(2009).[11] G. Gidofalvi and D. A. Mazziotti, J. Chem. Phys. 129, 134108 (2008).[12] K. Pelzer, L. Greenman, G. Gidofalvi, and D. A. Mazziotti, J. Phys. Chem. A 115, 5632 (2011).[13] L. Greenman and D. A. Mazziotti, J. Chem. Phys. 133, 164110 (2010).[14] A. V. Sinitskiy, L. Greenman, and D. A. Mazziotti, J. Chem. Phys. 133, 014104 (2010).[15] L. Vandenberghe and S. Boyd, SIAM Rev. 38, 49 (1996).[16] C. Garrod and J. K. Percus, J. Math. Phys. 5, 1756 (1964).[17] D. A. Mazziotti, Phys. Rev. Lett. 93, 213001 (2004).[18] D. A. Mazziotti, ESAIM: Mathematical Modelling and Numerical Analysis 41, 249 (2007).

large-active space CASSCF requires a polynomially scaling approach such as

CASSCF

or

Fosso-Tande, Nguyen, Gidofalvi, AED, JCTC. 12, 2260-2271 (2016).

[a] [b]




conventional methods for nondynamical correlationcomplete active space self consistent field method (CASSCF)

1

100000

1x1010

1x1015

1x1020

1x1025

1x1030

0 5 10 15 20 25 30 35 40 45 50

k


100

105

1010

1015

1020

1025

1030




- 2-RDM

1

100000

1x1010

1x1015

1x1020

1x1025

1x1030

0 5 10 15 20 25 30 35 40 45 50

- CI

num

ber

of v

aria

bles

DMRG-drivena


[19] S. Burer and R. D. Monteiro, Mathematical Programming 95, 329 (2003).[20] J. Povh, F. Rendl, A. Wiegele, Computing 78, 277 (2006).[21] J. Malick, J. Povh, F. Rendl, and A. Wiegele, SIAM J. Optim. 20, 336 (2009).[22] D. A. Mazziotti, Phys. Rev. Lett. 106, 083001 (2011).[23] Y. Shao, L. Fusti-Molnar, Y. Jung, J. Kussmann, C. Ochsenfeld, S. T. Brown, A. T. B. Gilbert, L.

V. Slipchenko,S. V. Levchenko, D. P. O’Neill, R. A. Distasio Jr., R. C. Lochan, T. Wang, G. J. O.Beran, N. A. Besley, J. M., Herbert, C. Y. Lin, T. Van Voorhis, S. H. Chien, A. Sodt, R. P. Steele,V. A. Rassolov, P. E. Maslen, P. P. Korambath, R. D. Adamson, B. Austin, J. Baker, E. F. C. Byrd,H. Dachsel, R. J. Doerksen, A. Dreuw, B. D. Dunietz, A. D. Dutoi, T. R. Furlani, S. R. Gwaltney, A.Heyden, S. Hirata, C.-P. Hsu, G. Kedziora, R. Z. Khalliulin, P. Klunzinger, A. M. Lee, M. S. Lee, W.Liang, I. Lotan, N. Nair, B. Peters, E. I. Proynov, P. A. Pieniazek, Y. M. Rhee, J. Ritchie, E. Rosta,C. D. Sherrill, A. C. Simmonett, J. E. Subotnik, H. L. Woodcock III, W. Zhang, A. T. Bell, A. K.Chakraborty, D. M. Chipman, F. J. Keil, A. Warshel, W. J. Hehre, H. F. Schaefer III, J. Kong, A. I.Krylov, P. M. W. Gill, M. Head-Gordon, Phys. Chem. Chem. Phys. 8, 3172 - 3191 (2006).

[24] E. Epifanovsky, M. Wormit, T. Ku, A. Landau, D. Zuev, K. Khistyaev, P. Manohar, I. Kaliman,A. Dreuw, and A. I. Krylov, J. Comput. Chem. 34, 2293 (2013).

[25] E. Solomonik, D. Matthews, J. Hammond, and J. Demmel, Cyclops Tensor Framework: reducing com-munication and eliminating load imbalance in massively parallel contractions, Tech. Rep. UCB/EECS-2012-210 (EECS Department, University of California, Berkeley, 2012).

[26] M. Nakata, B. J. Braams, K. Fujisawa, M. Fukuda, J. K. Percus, M. Yamashita, and Z. Zhao, J. Chem.Phys. 128, 164113 (2008).

[27] Turney, J. M.; Simmonett, A. C.; Parrish, R. M.; Hohenstein, E. G.; Evangelista, F. A.; Fermann, J.T.; Mintz, B. J.; Burns, L. A.; Wilke, J. J.; Abrams, M. L.; Russ, N. J.; Leininger, M. L.; Janssen, C.L.; Seidl, E. T.; Allen, W. D.; Schaefer, H. F.; King, R. A.; Valeev, E. F.; Sherrill, C. D.; Crawford, T.D. “Psi4: an open-source ab initio electronic structure program”, WIREs Comput. Mol. Sci. 2, 556-565(2012).

[28] A. J. Coleman, Rev. Mod. Phys. 35, 668 (1963).[29] R. M. Erdahl, Rep. Math. Phys. 15, 147 (1979).[30] G. Gidofalvi and D. A. Mazziotti, Phys. Rev. A 72, 052505 (2005).[31] S. R. White, Phys. Rev. Lett. 69, 2863 (1992); S. R. White, Phys. Rev. B 48, 10345 (1993); S. R.

White and R. L. Martin, J. Chem. Phys. 110, 4127 (1999); G. K.-L. Chan and M. Head-Gordon, J.Chem. Phys. 116, 4462 (2002); G. K.-L. Chan and M. Head-Gordon, J. Chem. Phys. 118, 8551 (2003);G. K.-L. Chan, J. Chem. Phys. 120, 3172 (2004).

[32] D. Ghosh, J. Hachmann, T. Yanai, and G. K.-L. Chan, J. Chem. Phys. 128, 144117 (2008).[33] T. Yanai, Y. Kurashige, D. Ghosh, and G. K.-L. Chan, Int. J. Quantum Chem. 109, 2178 (2009).[34] T. Yanai, Y. Kurashige, E. Neuscamman, and G. K.-L. Chan, J. Chem. Phys. 132, 024105 (2010).[35] L. S. Blackford, J. Choi, A. Cleary, E. D’Azevedo, J. Demmel, I. Dhillon, J. Dongarra, S. Hammarling,

G. Henry, A. Petitet, K. Stanley, D. Walker, and R. C. Whaley, ScaLAPACK Users’ Guide (Societyfor Industrial and Applied Mathematics, Philadelphia, PA, 1997).

[36] P. Alpatov, G. Baker, C. Edwards, J. Gunnels, G. Morrow, J. Overfelt, and R. V. D. Geijn, “Plapack:Parallel linear algebra libraries design overview,”.

[37] J. Poulson, B. Marker, R. A. van de Geijn, J. R. Hammond, and N. A. Romero, ACM Trans. Math.Softw. 39, 13:1 (2013).

[38] T. Auckenthaler, H.-J. Bungartz, T. Huckle, L. Krmer, B. Lang, and P. Willems, J. Comput. Sci. 2,272 (2011), social Computational Systems.

[39] H. van Aggelen, B. Verstichel, P. Bultinck, D. Van Neck, and P. W. Ayers, J. Chem. Phys. 136, 014110(2012).

[40] E. Perez-Romero, L. M. Tel, and C. Valdemoro, Int. J. Quantum Chem. 61, 55 (1997).[41] J. Fosso-Tande, D. R. Nascimento, and A. E. DePrince III, submitted (2015).[42] J. Hachmann, J. J. Dorando, M. Avils, and G. K.-L. Chan, J. Chem. Phys. 127, 134309 (2007).[43] J. R. Hammond and D. A. Mazziotti, Phys. Rev. A 71, 062503 (2005).[44] D. A. Mazziotti, Phys. Rev. A 72, 032510 (2005).[45] G. Gidofalvi and D. A. Mazziotti, The Journal of Chemical Physics 125, 144102 (2006).[46] P. Pulay, Chem. Phys. Lett. 73, 393 (1980).

variational 2-RDM (v2RDM) drivenb

large-active space CASSCF requires a polynomially scaling approach such as

CASSCF

or

[a] [b]



E. Epifanovsky, D. Zuev, X. Feng, K. Khistyaev, Y. Shao, and A. I. Krylov, “General imple-mentation of the resolution-of-the-identity and Cholesky representations of electron repulsionintegrals within coupled-cluster and equation-of-motion methods: Theory and benchmarks,”J. Chem. Phys. 139, 134105 (2013).

E. Epifanovsky, M. Wormit, T. Kus, A. Landau, D. Zuev, K. Khistyaev, P. Manohar, I. Kali-man, A. Dreuw, and A.I. Krylov “New implementation of high-level correlated methodsusing a general block-tensor library for high-performance electronic structure calculations,”J. Comput. Chem. 34, 2293 (2013).

VIII. FACILITIES/EQUIPMENT

At the time of submission of this proposal, the DePrince Group at Florida State University has5 modern multi-core Linux servers used for code development and benchmarking. One server isequipped with an NVIDIA Tesla K40c (Kepler) GPU. We have also purchased 160 computer coresat the FSU Research and Computing Center. Q-Chem, Inc. owns six modern multi-core Linuxservers and several other Mac OS, Windows, AIX and Sun Solaris computer servers to support in-house code development and platform porting. The proposed Phase II research e↵ort is limited totheoretical and computational work, and the facilities meet all environmental laws and regulations.

IX. CONSULTANTS

No consultants will be employed during Phase II of this STTR.

X. REFERENCES

[1] C. Garrod, M. V. Mihailovic, and M. Rosina, J. Math. Phys. 16, 868 (1975); M. V. Mihailovic and M.Rosina, Nucl. Phys. A 237, 221 (1975); M. Rosina and C. Garrod, J. Comput. Phys. 18, 300 (1975);R. M. Erdahl, C. Garrod, B. Golli, and M. Rosina, J. Math. Phys. 20, 1366 (1979); R. M. Erdahl, Rep.Math. Phys. 15, 147 (1979).

[2] R. M. Erdahl, Int. J. Quantum Chem. 13, 697 (1978).[3] M. Nakata, H. Nakatsuji, M. Ehara, M. Fukuda, K. Nakata, and K. Fujisawa, J. Chem. Phys. 114,

8282 (2001).[4] D. A. Mazziotti and R. M. Erdahl, Phys. Rev. A 63, 042113 (2001).[5] D. A. Mazziotti, Phys. Rev. A 65, 062511 (2002).[6] D. A. Mazziotti, Phys. Rev. A 74, 032501 (2006).[7] Z. Zhao, B. J. Braams, M. Fukuda, M. L. Overton, and J. K. Percus, J. Chem. Phys. 120, 2095 (2004).[8] M. Fukuda, B. J. Braams, M. Nakata, M. L. Overton, J. K. Percus, M. Yamashita, and Z. Zhao, Math.

Program. 109, 553 (2007).[9] E. Cances, G. Stoltz, and M. Lewin, J. Chem. Phys. 125, 064101 (2006).[10] B. Verstichel, H. van Aggelen, D. Van Neck, P. W. Ayers, and P. Bultinck, Phys. Rev. A 80, 032508

(2009).[11] G. Gidofalvi and D. A. Mazziotti, J. Chem. Phys. 129, 134108 (2008).[12] K. Pelzer, L. Greenman, G. Gidofalvi, and D. A. Mazziotti, J. Phys. Chem. A 115, 5632 (2011).[13] L. Greenman and D. A. Mazziotti, J. Chem. Phys. 133, 164110 (2010).[14] A. V. Sinitskiy, L. Greenman, and D. A. Mazziotti, J. Chem. Phys. 133, 014104 (2010).[15] L. Vandenberghe and S. Boyd, SIAM Rev. 38, 49 (1996).[16] C. Garrod and J. K. Percus, J. Math. Phys. 5, 1756 (1964).[17] D. A. Mazziotti, Phys. Rev. Lett. 93, 213001 (2004).[18] D. A. Mazziotti, ESAIM: Mathematical Modelling and Numerical Analysis 41, 249 (2007).

Fosso-Tande, Nguyen, Gidofalvi, AED, JCTC. 12, 2260-2271 (2016).

[b]

variational 2-RDM (v2RDM) methods

E =1

2

X

pqrs

(pr|qs) 2Dpqrs +

X

pq

hpq1Dp

q ,

P. O. Lowdin Phys. Rev. 97, 1474-1489 (1955).

ground state electronic energy can be written exactly in terms of two-electron reduced-density matrix (2-RDM, 2D) and one-electron RDM (1-RDM, 1D)


*k: number of basis functions

E =1

2

X

pqrs

(pr|qs) 2Dpqrs +

X

pq

hpq1Dp

q ,


H H

prob

abili

ty

x

H2 (2e,2o) / cc-pVDZ

1-RDM → probability density O(k2) elements*




E =1

2

X

pqrs

(pr|qs) 2Dpqrs +

X

pq

hpq1Dp

q ,


H H

prob

abili

ty

x

H2 (2e,2o) / cc-pVDZ 2-RDM → pair probability densityO(k4) elements



H Hpr

obab

ility

x


Since 2D and 1D are much more compact objects than the N-electron wave function, why not invoke the variational principle and minimize E with respect to elements of 2D and 1D?


E =1

2

X

pqrs

(pr|qs) 2Dpqrs +

X

pq

hpq1Dp

q ,

P. O. Lowdin Phys. Rev. 97, 1474-1489 (1955).J. E. Mayer Phys. Rev. 100, 1579-1586 (1955).

H H

prob

abili

ty

x

H2 (2e,2o) / cc-pVDZ 2-RDM → pair probability densityO(k4) elements



H Hpr

obab

ility

x

2D/1D should satisfy a few properties to guarantee they correspond to an antisymmetric N-electron wave function (N-representable) A. J. Coleman, Rev. Mod. Phys. 35, 668-686 (1963).

an ensemble N-representable 1-RDM:

1D(x1|x0

1) = N

Zdx2dx3...dxN

X

k

wk k(x1, x2, x3, ..., xN ) ⇤k(x

01, x2, x3, ..., xN ).

complete ensemble N-representability conditions for the 1-RDM are known (and are simple)



0 �i 1X

i

�i = N

natural spin-orbital occupation numbers (eigenvalues of 1-RDM) must lie between 0 and 1


1D(x1|x0

1) = N

Zdx2dx3...dxN

X

k

wk k(x1, x2, x3, ..., xN ) ⇤k(x

01, x2, x3, ..., xN ).





1D(x1|x0

1) = N

Zdx2dx3...dxN

X

k

wk k(x1, x2, x3, ..., xN ) ⇤k(x

01, x2, x3, ..., xN ).


Tr(1D) = N1Dij +

1Qji = �ij

and

equivalently, the eigenvalues of the 1-RDM and one hole RDM (1Q) must be nonnegative


1D ⌫ 01Q ⌫ 0 (probability of not finding an electron)



2D(x1, x2|x0

1, x02) =

✓N

2

◆Zdx3...dxN

X

k

wk k(x1, x2, x3, ..., xN ) ⇤k(x

01, x

02, x3, ..., xN ).

necessary ensemble N-representability conditions for the 2-RDM include

1. hermiticity2. fixed trace3. antisymmetry with respect to particle exchange

4. nonnegativity of the eigenvalues of 2D (positive semidefiniteness)

2Dpqrs = �2Dqp

rs = �2Dpqsr = 2Dqp

sr


1. hermiticity2. fixed trace3. antisymmetry with respect to particle exchange

4. nonnegativity of the eigenvalues of 2D (positive semidefiniteness)



2D(x1, x2|x0

1, x02) =

✓N

2

◆Zdx3...dxN

X

k

wk k(x1, x2, x3, ..., xN ) ⇤k(x

01, x

02, x3, ..., xN ).

necessary ensemble N-representability conditions for the 2-RDM include

2Dpqrs = �2Dqp

rs = �2Dpqsr = 2Dqp

sr

So, the variational optimization of the ground-state 2-RDM is a semindefinite optimization problem

7

1554354.

[1] P.-O. Lowdin, Phys. Rev. 97, 1474 (1955).[2] A. J. Coleman, Rev. Mod. Phys. 35, 668 (1963).[3] D. A. Mazziotti and R. M. Erdahl, Phys. Rev. A 63,

042113 (2001).[4] C. Garrod and J. K. Percus, J. Math. Phys. 5, 1756

(1964).[5] R. Erdahl, Rep. Math. Phys. 15, 147 (1979).[6] M. Nakata, H. Nakatsuji, M. Ehara, M. Fukuda,

K. Nakata, and K. Fujisawa, J. Chem. Phys. 114, 8282(2001).

[7] D. A. Mazziotti, Phys. Rev. A 65, 062511 (2002).[8] D. A. Mazziotti, Phys. Rev. A 74, 032501 (2006).[9] Z. Zhao, B. J. Braams, M. Fukuda, M. L. Overton, and

J. K. Percus, J. Chem. Phys. 120, 2095 (2004).[10] M. Fukuda, B. J. Braams, M. Nakata, M. L. Overton,

J. K. Percus, M. Yamashita, and Z. Zhao, Math. Prog.109, 553 (2007).

[11] E. Cances, G. Stoltz, and M. Lewin, J. Chem. Phys. 125,064101 (2006).

[12] B. Verstichel, H. van Aggelen, D. Van Neck, P. W. Ayers,and P. Bultinck, Phys. Rev. A 80, 032508 (2009).

[13] B. Verstichel, H. van Aggelen, D. V. Neck, P. Bultinck,and S. D. Baerdemacker, Comput. Phys. Commun. 182,1235 (2011).

[14] J. Fosso-Tande, D. R. Nascimento, and A. E. DePrinceIII, Mol. Phys. 114, 423 (2016).

[15] J. Fosso-Tande, T.-S. Nguyen, G. Gidofalvi, and A. E.DePrince III, J. Chem. Theory Comput. 12, 2260 (2016).

[16] M. Bouten, P. van Leuven, M. Mihailovi, and M. Rosina,Nucl. Phys. A 221, 173 (1974), ISSN 0375-9474.

[17] M. Rosina, Int. J. Quantum Chem. 13, 737 (1978).[18] C. Valdemoro, D. R. Alcoba, O. B. Ona, L. M. Tel, and

E. Perez-Romero, Journal of Mathematical Chemistry50, 492 (2012).

[19] L. Greenman and D. A. Mazziotti, J. Chem. Phys. 128,

114109 (2008).[20] D. A. Mazziotti, Phys. Rev. A 68, 052501 (2003).[21] K. Chatterjee and K. Pernal, J. Chem. Phys. 137, 204109

(2012).[22] H. van Aggelen, B. Verstichel, G. Acke, M. Degroote,

P. Bultinck, P. W. Ayers, and D. V. Neck, Comp. andTheor. Chem. 1003, 50 (2013).

[23] R. E. Borland and K. Dennis, J. Phys. B: At. Mol. Phys.5, 7 (1972).

[24] M. Altunbulak and A. Klyachko, Commun. Math. Phys.282, 287 (2008).

[25] C. Schilling, D. Gross, and M. Christandl, Phys. Rev.Lett. 110, 040404 (2013).

[26] C. L. Benavides-Riveros, J. M. Gracia-Bondıa, andM. Springborg, Phys. Rev. A 88, 022508 (2013).

[27] R. Chakraborty and D. A. Mazziotti, Phys. Rev. A 89,042505 (2014).

[28] C. Schilling, Phys. Rev. A 91, 022105 (2015).[29] I. Theophilou, N. N. Lathiotakis, M. A. L. Marques, and

N. Helbig, J. Chem. Phys. 142, 154108 (2015).[30] T. L. Gilbert, Phys. Rev. B 12, 2111 (1975).[31] E. Perez-Romero, L. M. Tel, and C. Valdemoro, Int. J.

Quantum Chem. 61, 55 (1997).[32] G. Gidofalvi and D. A. Mazziotti, Phys. Rev. A 72,

052505 (2005).[33] J. Povh, F. Rendl, A. Wiegele, Computing 78, 277

(2006).[34] J. Malick, J. Povh, F. Rendl, and A. Wiegele, SIAM J.

Optim. 20, 336 (2009).[35] D. A. Mazziotti, Phys. Rev. Lett. 106, 083001 (2011).[36] J. M. Turney, A. C. Simmonett, R. M. Parrish, E. G. Ho-

henstein, F. A. Evangelista, J. T. Fermann, B. J. Mintz,L. A. Burns, J. J. Wilke, M. L. Abrams, et al., WIRESComput. Mol. Sci. 2, 556 (2012).

7

1554354.

[1] P.-O. Lowdin, Phys. Rev. 97, 1474 (1955).[2] A. J. Coleman, Rev. Mod. Phys. 35, 668 (1963).[3] D. A. Mazziotti and R. M. Erdahl, Phys. Rev. A 63,

042113 (2001).[4] C. Garrod and J. K. Percus, J. Math. Phys. 5, 1756

(1964).[5] R. Erdahl, Rep. Math. Phys. 15, 147 (1979).[6] M. Nakata, H. Nakatsuji, M. Ehara, M. Fukuda,

K. Nakata, and K. Fujisawa, J. Chem. Phys. 114, 8282(2001).

[7] D. A. Mazziotti, Phys. Rev. A 65, 062511 (2002).[8] D. A. Mazziotti, Phys. Rev. A 74, 032501 (2006).[9] Z. Zhao, B. J. Braams, M. Fukuda, M. L. Overton, and

J. K. Percus, J. Chem. Phys. 120, 2095 (2004).[10] M. Fukuda, B. J. Braams, M. Nakata, M. L. Overton,

J. K. Percus, M. Yamashita, and Z. Zhao, Math. Prog.109, 553 (2007).

[11] E. Cances, G. Stoltz, and M. Lewin, J. Chem. Phys. 125,064101 (2006).

[12] B. Verstichel, H. van Aggelen, D. Van Neck, P. W. Ayers,and P. Bultinck, Phys. Rev. A 80, 032508 (2009).

[13] B. Verstichel, H. van Aggelen, D. V. Neck, P. Bultinck,and S. D. Baerdemacker, Comput. Phys. Commun. 182,1235 (2011).

[14] J. Fosso-Tande, D. R. Nascimento, and A. E. DePrinceIII, Mol. Phys. 114, 423 (2016).

[15] J. Fosso-Tande, T.-S. Nguyen, G. Gidofalvi, and A. E.DePrince III, J. Chem. Theory Comput. 12, 2260 (2016).

[16] M. Bouten, P. van Leuven, M. Mihailovi, and M. Rosina,Nucl. Phys. A 221, 173 (1974), ISSN 0375-9474.

[17] M. Rosina, Int. J. Quantum Chem. 13, 737 (1978).[18] C. Valdemoro, D. R. Alcoba, O. B. Ona, L. M. Tel, and

E. Perez-Romero, Journal of Mathematical Chemistry50, 492 (2012).

[19] L. Greenman and D. A. Mazziotti, J. Chem. Phys. 128,

114109 (2008).[20] D. A. Mazziotti, Phys. Rev. A 68, 052501 (2003).[21] K. Chatterjee and K. Pernal, J. Chem. Phys. 137, 204109

(2012).[22] H. van Aggelen, B. Verstichel, G. Acke, M. Degroote,

P. Bultinck, P. W. Ayers, and D. V. Neck, Comp. andTheor. Chem. 1003, 50 (2013).

[23] R. E. Borland and K. Dennis, J. Phys. B: At. Mol. Phys.5, 7 (1972).

[24] M. Altunbulak and A. Klyachko, Commun. Math. Phys.282, 287 (2008).

[25] C. Schilling, D. Gross, and M. Christandl, Phys. Rev.Lett. 110, 040404 (2013).

[26] C. L. Benavides-Riveros, J. M. Gracia-Bondıa, andM. Springborg, Phys. Rev. A 88, 022508 (2013).

[27] R. Chakraborty and D. A. Mazziotti, Phys. Rev. A 89,042505 (2014).

[28] C. Schilling, Phys. Rev. A 91, 022105 (2015).[29] I. Theophilou, N. N. Lathiotakis, M. A. L. Marques, and

N. Helbig, J. Chem. Phys. 142, 154108 (2015).[30] T. L. Gilbert, Phys. Rev. B 12, 2111 (1975).[31] E. Perez-Romero, L. M. Tel, and C. Valdemoro, Int. J.

Quantum Chem. 61, 55 (1997).[32] G. Gidofalvi and D. A. Mazziotti, Phys. Rev. A 72,

052505 (2005).[33] J. Povh, F. Rendl, A. Wiegele, Computing 78, 277

(2006).[34] J. Malick, J. Povh, F. Rendl, and A. Wiegele, SIAM J.

Optim. 20, 336 (2009).[35] D. A. Mazziotti, Phys. Rev. Lett. 106, 083001 (2011).[36] J. M. Turney, A. C. Simmonett, R. M. Parrish, E. G. Ho-

henstein, F. A. Evangelista, J. T. Fermann, B. J. Mintz,L. A. Burns, J. J. Wilke, M. L. Abrams, et al., WIRESComput. Mol. Sci. 2, 556 (2012).


−24.90

−24.85

−24.80

−24.75

−24.70

−24.65

−24.60

0.5 1.0 1.5 2.0 2.5 3.0 3.5

ener

gy (E

h)

B−H distance (Å)

Full CIBH / STO-6G

2D ⌫ 0


2-RDM

−29.00

−28.00

−27.00

−26.00

−25.00

−24.00

0.5 1.0 1.5 2.0 2.5 3.0 3.5

ener

gy (E

h)

B−H distance (Å)

Full CI2−RDM (D)v2RDM (P)

BH / STO-6G

2D ⌫ 0


2-RDM

−24.90

−24.85

−24.80

−24.75

−24.70

−24.65

−24.60

0.5 1.0 1.5 2.0 2.5 3.0 3.5

ener

gy (E

h)

B−H distance (Å)

Full CIBH / STO-6G

2D ⌫ 02Q ⌫ 0


2-RDM

two-hole RDM

−24.90

−24.85

−24.80

−24.75

−24.70

−24.65

−24.60

0.5 1.0 1.5 2.0 2.5 3.0 3.5

ener

gy (E

h)

B−H distance (Å)

Full CI

−24.90

−24.85

−24.80

−24.75

−24.70

−24.65

−24.60

0.5 1.0 1.5 2.0 2.5 3.0 3.5

ener

gy (E

h)

B−H distance (Å)

Full CI2−RDM (DQ)v2RDM (PQ)

BH / STO-6G

2D ⌫ 02Q ⌫ 0


2-RDM

two-hole RDM

C. Garrod and J. K. Percus, J. Math. Phys. 5, 1756 (1964).

−24.90

−24.85

−24.80

−24.75

−24.70

−24.65

−24.60

0.5 1.0 1.5 2.0 2.5 3.0 3.5

ener

gy (E

h)

B−H distance (Å)

Full CI2−RDM (DQ)

2−RDM (DQG)v2RDM (PQ)

v2RDM (PQG)

BH / STO-6G

2D ⌫ 02Q ⌫ 02G ⌫ 0


2-RDM

two-hole RDM

particle-hole RDM

1-positivity: 2-positivity (PQG): partial 3-positivity (T1/T2)

1-positivity < 2-positivity < 3-positivity < N-positivity = Full CI

R. M. Erdahl, Int. J. Quantum Chem. 13, 697 (1978).

D. A. Mazziotti, Phys. Rev. A 74, 032501 (2006).

Z. Zhao, B. J. Braams, M. Fukuda, M. L. Overton, and J. K. Percus, J. Chem. Phys. 120,2095 (2004).

1D, 1Q ⌫ 02D, 2Q, 2G ⌫ 0 T1,T2 ⌫ 0

O(k9)

O(k4)

O(k6) k = number of basis functions

systematically improvable N-representability

C. Garrod and J. K. Percus, J. Math. Phys. 5, 1756 (1964).

variational 2-RDM-driven CASSCF

−109.2

−109.1

−109.0

−108.9

−108.8

−108.7

−108.6

1.0 1.5 2.0 2.5 3.0 3.5

ener

gy (E

h)


CI

N2 / cc-pVQZ / (6e,6o) active space

Fosso-Tande, Nguyen, Gidofalvi, AED, J. Chem. Theory Comput. 12, 2260-2271 (2016).

core

active: (6e,6o)

virtual



−109.2

−109.1

−109.0

−108.9

−108.8

−108.7

−108.6

1.0 1.5 2.0 2.5 3.0 3.5

ener

gy (E

h)


CIPQG


core

active: (6e,6o)

virtual

- v2RDM can replace CI for describing active space


−0.020

−0.015

−0.010

−0.005

0.000

1.0 1.5 2.0 2.5 3.0 3.5

erro

r (E h

)N−N bond length (Å)



−109.2

−109.1

−109.0

−108.9

−108.8

−108.7

−108.6

1.0 1.5 2.0 2.5 3.0 3.5

ener

gy (E

h)


CIPQG


- two-particle N-representability conditions (PQG) provide qualitative accuracy

- three-particle conditions (T1T2) provide quantitative accuracy




−109.2

−109.1

−109.0

−108.9

−108.8

−108.7

−108.6

1.0 1.5 2.0 2.5 3.0 3.5

ener

gy (E

h)


CIPQG

PQG+T1T2

−0.020

−0.015

−0.010

−0.005

0.000

1.0 1.5 2.0 2.5 3.0 3.5

erro

r (E h

)N−N bond length (Å)

- two-particle N-representability conditions (PQG) provide qualitative accuracy


huge active spaces!

0

10

20

30

40

50

60

70

2 3 4 5 6 7 8 9 10 11 12

sing

let-t

riple

t gap

(kca

l mol

-1)

k-acene

PQG (cc-pVDZ)PQG (cc-pVTZ)

DMRG (DZ)experiment

huge active spaces!

- v2RDM-CASSCF with two-particle conditions (PQG) can treat HUGE active spaces

dodecacene: (50e,50o) CASSCF, 1892 orbitals (cc-pVTZ)

conventional (CI-driven) codes are useless beyond 4-acene: (18,18) active space

- performance of PQG is comparable to other state-of-the-art CASSCF-like methods like density-matrix renormalization group (DMRG) … for this problem … other cases require T2

to cross this line,abandon CI-CASSCF

CASSCF (18e,18o)

DMRG results from: J. Hachmann, J.J. Dorando, M. Avils, and G.K.-L. Chan, J. Chem. Phys. 127, 134309 (2007)


0

10

20

30

40

50

60

70

2 3 4 5 6 7 8 9 10 11 12

sing

let-t

riple

t gap

(kca

l mol

-1)

k-acene


DMRG (DZ)experiment

performance?

dodecacene: (50e,50o) CASSCF, 840 orbitals (cc-pVDZ)



CASSCF (18e,18o)

0

10

20

30

40

50

60

70

2 3 4 5 6 7 8 9 10 11 12

sing

let-t

riple

t gap

(kca

l mol

-1)

k-acene


DMRG (DZ)experiment

performance?

dodecacene: (50e,50o) CASSCF, 840 orbitals (cc-pVDZ)



CASSCF (18e,18o)CPU: core i7-6830k (6 cores) 3.2 hours

3.7x faster!!Mullinax, Koulias, Maradzike, Gidofalvi, Epifanovsky, AED, under revision.

GPU: NVIDIA Quadro GP100 0.9 hours

signatures of polyradical character in polyacenes

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

0 5 10 15 20 25 30 35 40 45 50

occu

patio

n

natural orbital number

(a) 2−acene3−acene4−acene5−acene6−acene7−acene8−acene9−acene

10−acene11−acene12−acene

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

1 2 3 4 5 6 7 8 9 10 11 12

occu

patio

n

k−acene

(b)b1ub2gb3gau

- natural orbital occupation numbers convey the onset of polyradical behavior in larger acenes

- linear polyacenes have a history of conflicting predictions from quantum chemical methods (most of which is resolved)



Mullinax, Gidofalvi, Epifanovsky and AED, J. Chem. Theory Comput. 15, 276-289 (2019).

- geometric signatures of polyradical character include(i) larger bond length alternation (BLA) toward ends of molecule than in the middle



- geometric signatures of polyradical character include(i) larger bond length alternation (BLA) toward ends of molecule than in the middle (ii) increasingly similar structures for the lowest-energy singlet and triplet

0.000

0.010

0.020

0.030

0.040

0.050

2 3 4 5 6 7 8 9 10 11 12

RM

S di

ffere

nce

in C−C

bon

d le

ngth

s (Å

)

k−acene

PQGRB3LYPUB3LYP



0.000

0.010

0.020

0.030

0.040

0.050

2 3 4 5 6 7 8 9 10 11 12

RM

S di

ffere

nce

in C−C

bon

d le

ngth

s (Å

)

k−acene

PQGRB3LYPUB3LYP

- geometric signatures of polyradical character include(i) larger bond length alternation (BLA) toward ends of molecule than in the middle (ii) increasingly similar structures for the lowest-energy singlet and triplet

several recent studies suggest that even large CASSCF gets this system wrong …

polyradical character is overestimated unless the σ network is also correlated

moving beyond the ground state

obvious problem: v2RDM methods can treat only ground states

next goal: v2RDM-CASSCF description of electronic excited states

extended random phase approximation (ERPA)equations include only one-electron excitations from the corre-lated ground state, such that they formulate a generalized eigen-value problem in terms of the first-order density matrix (1DM)and second-order density matrix (2DM). We can thus apply thismethod to calculate approximate excitation energies from 2DM’sobtained from the variational 2DM (v2DM) method [13–15]. Thisidea forms an alternative to direct variational optimization of theexcited state 2DM, which would require non-trivial excited-stateN-representability conditions. We shall call this method the ‘Ex-tended RPA’ (ERPA) method, in accordance with the work of Pernal[8]. The ERPA method differs from similar methods previously ap-plied to the variationally optimized 2DM’s [12,13] because it doesnot require an approximation to the 3DM.

More specifically, this paper examines the main motivation forusing ERPA over RPA – the inclusion of correlation effects – and itspossible limitations due to two approximations:

! The input 2DM’s in general are not exact, may represent anensemble of states and may not be N-representable.! The ERPA method truncates the description of the excited states

by including only one-electron excitations from the correlatedground state.

Section 4 first of all demonstrates the effect of correlation,ensemble nature, and N-representability errors on the ERPA spec-trum by comparing the ERPA results from uncorrelated and varia-tionally optimized 2DM’s with those from exact 2DM’s; andsecondly demonstrates the effect of the approximations made inthe ERPA method itself by comparing the ERPA spectra to exactand experimentally determined spectra. Section 2 first presentsthe Equations of Motion in the present context and Section 3 com-ments on the details of our calculations.

2. The extended RPA method: a method to extract excitationenergies from ground-state 2DM

The original framework for the Equations of Motion for the RPAwas proposed by Rowe in the 1960s [2]. An exact excited statejWni ¼ bQ ynjW0i satisfies

bH; bQ ynh i

jW0i ¼ ðEn $ E0ÞbQ ynjW0i

Choosing the bra to be an arbitrary state bAyjW0i leads to a set ofequations

W0bA bH; bQ ynh i!!!

!!!W0

D E¼ ðEn $ E0Þ W0

bA bQ yn!!!

!!!W0

D E

Since bQ yn ¼ jWnihW0j, its Hermitian conjugate annihilates the groundstate

bQ njW0i ¼ 0

so the above equations can be cast in a more symmetrical, equiva-lent, form

W0bA; ½H; bQ yn'h i!!!

!!!W0


bA; bQ ynh i!!!

!!!W0

D E

These are the Equations of Motion, which are in principle exact.However, in order to keep calculations tractable, the excitationoperators bQ yn can be approximated by one-body operators.

The usual RPA then makes an additional approximation: itevaluates the expectation values that occur in the EOM on anuncorrelated Hartree–Fock ground state W0. For such a state,the only terms in the excitation operator that lead to non-vanishing equations are hole-particle and particle-hole excitations,bQ yn ¼

PNh¼1PK

p¼Nþ1Xphaypah $ Yphayhap with N the number of electronsand K the single particle basis dimension. For a correlated ground

state, however, the form of the excitation operators can be general-ized to include all one-electron excitations, bQ yn ¼

PKij¼1cn

ijayj ai. Such

a generalized excitation operator leads to the ‘extended RPA’(ERPA) equationsX

ij

cnij W0 aykal; H;ayj ai

h ih i!!!!!!W0

D E¼ ðEn$E0Þ

X

ij

cnij W0 aykal;a

yj ai

h i!!!!!!W0

D E

ð1Þ

For an uncorrelated Hartree–Fock ground state, these equa-tions reduce to the usual RPA equations. For a correlated groundstate, they depend on the 1DM and 2DM. Even for the exactwave function they depend only on the 1DM and 2DM, whichis a consequence of the one-electron description of the excitationoperator. This generalized eigenvalue problem can thus besolved to obtain approximate excitation energies !n = En $ E0

and the corresponding expansion coefficients of the excitedstates, cn

ij.Like the RPA excitation energies, the ERPA excitation energies

come in pairs (!n, $ !n) due to the special symmetry of the commu-tator matrices. The double commutator matrix on the left-handside and the single-commutator matrix on the right hand side ofEq. (1) are symmetric

W0 aykal; H; ayj ai

h ih i!!!!!!W0

D E¼ W0 ayi aj; H; ayl ak

h ih i!!!!!!W0

D E

W0 aykal; ayj ai

h i!!!!!!W0

D E¼ W0 ayi aj; a

yl ak

" #!! !!W0$ % ð2Þ

and have the additional symmetry

W0 aykal; H; ayj ai

h ih i!!!!!!W0

D E¼ W0 ayl ak; H; ayi aj

h ih i!!!!!!W0

D E

W0 aykal; ayj ai

h i!!!!!!W0

D E¼ $ W0 ayl ak; a

yi aj

" #!! !!W0$ % ð3Þ

These symmetry properties lead to pairs of eigenvalues (!n, $!n) inthe ERPA spectrum.

When the ground state W0 is the exact ground state, thedouble commutator matrix is also positive semi-definite: sinceany state of the form

Pijcija

yj aijW0i has higher energy than the

ground stateX

ijkl

c)kl W0 aykal; H; ayj ai

h ih i!!!!!!W0

D Ecij ¼

X

ijkl

c)kl W0 aykalðH $ E0Þayj ai

!!!!!!W0

D Ecij ð4Þ

þX

ijkl

c)kl W0 ayj aiðH $ E0Þaykal

!!!!!!W0

D Ecij

P 0 ð5Þ

3. Computational details

We used GAMESS [16] to carry out the full diagonalization ofthe CI matrix to obtain the FCI ground state and its exact excitationenergies and subsequently calculated the FCI ground-state 2DMwith our own routines. We computed the variationally optimizedsecond-order density matrices by means of a logarithmic barriermethod [17]. In the variational optimization, we constrained the2DM to have the correct normalization and to satisfy the 2-positiv-ity conditions [17] but did not impose explicit spin conditions [18].The approximate excitation energies for these 2DM’s then followfrom the generalized eigenvalue problem (1). We solved this gen-eralized eigenvalue problem using LAPACK’s dggev routine [19],which yields the generalized eigenvalues as ratio’s a/b. Singulari-ties in the metric matrix on the right hand side of Eq. (1) may giverise to eigenvalues a/0, but these are do not correspond to physicalexcitation energies.

All calculations use the Cartesian cc-pVDZ basis set as listed onthe EMSL Basis Set Exchange [20,21], except for the calculations onthe noble gases, which use the Cartesian aug-cc-pVTZ basis

H. van Aggelen et al. / Computational and Theoretical Chemistry 1003 (2013) 50–54 51

| ni =X

ij

cnij a†j ai| 0i

depends on 2-RDM depends on 1-RDM








bH; bQ ynh i




!!!W0


bA bQ yn!!!

!!!W0

D E


bQ njW0i ¼ 0



!!!W0


bA; bQ ynh i!!!

!!!W0

D E



PNh¼1PK



PKij¼1cn

ijayj ai. Such


ij


h ih i!!!!!!W0

D E¼ ðEn$E0Þ

X

ij

cnij W0 aykal;a

yj ai

h i!!!!!!W0

D E

ð1Þ





W0 aykal; H; ayj ai

h ih i!!!!!!W0


h ih i!!!!!!W0

D E

W0 aykal; ayj ai

h i!!!!!!W0

D E¼ W0 ayi aj; a

yl ak

" #!! !!W0$ % ð2Þ


W0 aykal; H; ayj ai

h ih i!!!!!!W0


h ih i!!!!!!W0

D E

W0 aykal; ayj ai

h i!!!!!!W0


yi aj

" #!! !!W0$ % ð3Þ



Pijcija


ground stateX

ijkl


h ih i!!!!!!W0

D Ecij ¼

X

ijkl


!!!!!!W0

D Ecij ð4Þ

þX

ijkl


!!!!!!W0

D Ecij

P 0 ð5Þ





| ni =X

ij

cnij a†j ai| 0i


previous efforts to combine v2RDM and ERPA were remarkably disappointingvan Aggelen et al., CTC, 1003, 50-54 (2013).








bH; bQ ynh i




!!!W0


bA bQ yn!!!

!!!W0

D E


bQ njW0i ¼ 0



!!!W0


bA; bQ ynh i!!!

!!!W0

D E



PNh¼1PK



PKij¼1cn

ijayj ai. Such


ij


h ih i!!!!!!W0

D E¼ ðEn$E0Þ

X

ij

cnij W0 aykal;a

yj ai

h i!!!!!!W0

D E

ð1Þ





W0 aykal; H; ayj ai

h ih i!!!!!!W0


h ih i!!!!!!W0

D E

W0 aykal; ayj ai

h i!!!!!!W0

D E¼ W0 ayi aj; a

yl ak

" #!! !!W0$ % ð2Þ


W0 aykal; H; ayj ai

h ih i!!!!!!W0


h ih i!!!!!!W0

D E

W0 aykal; ayj ai

h i!!!!!!W0


yi aj

" #!! !!W0$ % ð3Þ



Pijcija


ground stateX

ijkl


h ih i!!!!!!W0

D Ecij ¼

X

ijkl


!!!!!!W0

D Ecij ð4Þ

þX

ijkl


!!!!!!W0

D Ecij

P 0 ð5Þ





| ni =X

ij

cnij a†j ai| 0i


previous efforts to combine v2RDM and ERPA were remarkably disappointing

HeLiBeBCNOFNe

0.00-0.01-0.01-0.40-0.85-0.88-1.61-1.59-0.49

0.000.010.002.677.213.396.87

11.31-0.05

errors in ground-state (v2RDM) and excitation(ERPA) energies (eV), relative to full CI:

ground state excitation energy

van Aggelen et al., CTC, 1003, 50-54 (2013).








bH; bQ ynh i




!!!W0


bA bQ yn!!!

!!!W0

D E


bQ njW0i ¼ 0



!!!W0


bA; bQ ynh i!!!

!!!W0

D E



PNh¼1PK



PKij¼1cn

ijayj ai. Such


ij


h ih i!!!!!!W0

D E¼ ðEn$E0Þ

X

ij

cnij W0 aykal;a

yj ai

h i!!!!!!W0

D E

ð1Þ





W0 aykal; H; ayj ai

h ih i!!!!!!W0


h ih i!!!!!!W0

D E

W0 aykal; ayj ai

h i!!!!!!W0

D E¼ W0 ayi aj; a

yl ak

" #!! !!W0$ % ð2Þ


W0 aykal; H; ayj ai

h ih i!!!!!!W0


h ih i!!!!!!W0

D E

W0 aykal; ayj ai

h i!!!!!!W0


yi aj

" #!! !!W0$ % ð3Þ



Pijcija


ground stateX

ijkl


h ih i!!!!!!W0

D Ecij ¼

X

ijkl


!!!!!!W0

D Ecij ð4Þ

þX

ijkl


!!!!!!W0

D Ecij

P 0 ð5Þ





| ni =X

ij

cnij a†j ai| 0i


previous efforts to combine v2RDM and ERPA were remarkably disappointing

HeLiBeBCNOFNe

0.00-0.01-0.01-0.40-0.85-0.88-1.61-1.59-0.49

0.000.010.002.677.213.396.87

11.31-0.05

errors in ground-state (v2RDM) and excitation(ERPA) energies (eV), relative to full CI:

ground state excitation energy

for open-shell atoms, errors in excitation energies are 10 x larger than errors in the ground state!

what is happening???

van Aggelen et al., CTC, 1003, 50-54 (2013).

2s2 ! 2s12p1

2s1 ! 2p1

2s22p1 ! 2s12p2

2s22p2 ! 2s12p3

Li

Be

B

C

full CIPQG

2.0 2.5 3.0 3.5 4.0 4.5energy (eV)

failures of v2RDM / ERPA

consider low-lying excitations in several second row atoms / STO-3G basis set

2s2 ! 2s12p1

2s1 ! 2p1

2s22p1 ! 2s12p2

2s22p2 ! 2s12p3

Li

Be

B

C

full CIPQG

2.0 2.5 3.0 3.5 4.0 4.5energy (eV)



Li

Be

B

C

full CIPQG

2.0 2.5 3.0 3.5 4.0 4.5energy (eV)

v2RDM ground state density represents and ensemble of these states



ensemble N-representability conditions

- complete ensemble-state N-representability conditions for the 1-RDM

0 �i 1 A.J. Coleman, Rev. Mod. Phys. 35, 668-686 (1963).

X

i

�i = N

natural spin-orbital occupation numbers (eigenvalues of 1-RDM) must lie between 0 and 1

pure-state N-representability conditions

- pure-state conditions: “generalized Pauli constraints”


known empirically: 3 electrons in 6 orbitals:

R.E. Borland and K. Dennis, J. Phys. B: At. Mol. Phys. 5, 7 (1972)

�1 + �6 = 1

�2 + �5 = 1

�3 + �4 = 1

�4 � �5 � �6 0

�i � �i+1

1972 - 2008 … this is the only set of pure-state conditions known



2008: M. Altunbulak and A. Klyachko, Commun. Math. Phys. 282, 287 (2008).


�1 + �6 = 1

�2 + �5 = 1

�3 + �4 = 1

�4 � �5 � �6 0

3 electrons 6 spin orbitals4 constraints

The Pauli Principle Revisited 319

Table 3. N -representability inequalities for system ∧4H8

Inequalities v ∈ S8 w ∈ S70 cvw(a)

λ1 ≤ 1 (1) (1) 1λ5 − λ6 − λ7 − λ8 ≤ 0 (1 5 4 3 2) 1λ1 − λ2 − λ7 − λ8 ≤ 0 (2 3 4 5 6) 1λ1 − λ3 − λ6 − λ8 ≤ 0 (3 4 5 7 6) 1λ1 − λ4 − λ6 − λ7 ≤ 0 (4 5 8 7 6) (1 2 3 4 5) 1λ1 − λ4 − λ5 − λ8 ≤ 0 (4 6)(5 7) 1λ3 − λ4 − λ7 − λ8 ≤ 0 (1 3 2)(4 5 6) 1λ2 − λ4 − λ6 − λ8 ≤ 0 (1 2)(4 5 7 6) 1λ2 + λ3 + λ5 − λ8 ≤ 2 (1 2 3 5 4) 1λ1 + λ3 + λ6 − λ8 ≤ 2 (2 3 6 5 4) 1λ1 + λ2 + λ7 − λ8 ≤ 2 (3 7 6 5 4) 1λ1 + λ2 + λ3 − λ4 ≤ 2 (4 5 6 7 8) (1 2 3 4 5) 1λ1 + λ4 + λ5 − λ8 ≤ 2 (2 4)(3 5) 1λ1 + λ2 + λ5 − λ6 ≤ 2 (3 5 4)(6 7 8) 1λ1 + λ3 + λ5 − λ7 ≤ 2 (2 3 5 4)(7 8) 1



λ2 + λ3 + λ4 + λ5 ≤ 2 (1 2 3 4 5) 1λ1 + λ2 + λ4 + λ7 ≤ 2 (3 4 7 6 5) (1 2 3 4 5) 1λ1 + λ3 + λ4 + λ6 ≤ 2 (2 3 4 6 5) 1λ1 + λ2 + λ5 + λ6 ≤ 2 (3 5)(4 6) 1λ1 + λ2 − λ3 ≤ 1 (3 4 5 6 7 8) 1λ2 + λ5 − λ7 ≤ 1 (1 2 5 4 3)(7 8) 1λ1 + λ6 − λ7 ≤ 1 (2 6 5 4 3)(7 8) (1 2 3 4 5 6) 1λ2 + λ4 − λ6 ≤ 1 (1 2 4 3)(6 7 8) 1λ1 + λ4 − λ5 ≤ 1 (2 4 3)(5 6 7 8) 1λ3 + λ4 − λ7 ≤ 1 (1 3)(2 4)(7 8) 1λ1 + λ8 ≤ 1 (2 8 7 6 5 4 3) (1 2 3 4 5 6 7) 1λ2 − λ3 − λ6 − λ7 ≤ 0 (1 2)(3 4 5 8 7 6) 1λ4 − λ5 − λ6 − λ7 ≤ 0 (1 4 3 2)(5 8 7 6) (1 2 3 4 5 6 7) 1λ1 − λ3 − λ5 − λ7 ≤ 0 (3 4 6)(5 8 7) 1λ2 + λ3 + 2λ4 − λ5 − λ7 + λ8 ≤ 2 (1 4 8 7 5) 1λ1 + λ3 + 2λ4 − λ5 − λ6 + λ8 ≤ 2 (1 4 8 6 7 5 2) (1 2 3 . . . 10 11) 1λ1 + 2λ2 − λ3 + λ4 − λ5 + λ8 ≤ 2 (1 2)(3 4 8 5 6 7) 1λ1 + 2λ2 − λ3 + λ5 − λ6 + λ8 ≤ 2 (1 2)(3 5 4 8 6 7) 1λ1 + λ2 − 2λ3 − λ4 − λ5 ≤ 0 (3 6 4 7 5 8) (1 2 3 . . . 11 12) 1λ1 − λ2 − λ3 + λ6 − 2λ7 ≤ 0 (2 6)(3 4 5 8 7) 1λ1 − λ3 − λ4 − λ5 + λ8 ≤ 0 (2 8 5 7 4 6 3) (1 2 3 . . . 12 13) 1λ1 − λ2 − λ3 − λ7 + λ8 ≤ 0 (2 8 7 3 4 5 6) 12λ1 − λ2 + λ4 − 2λ5 − λ6 + λ8 ≤ 1 (2 4 3 8 5 7 6) 1λ3 + 2λ4 − 2λ5 − λ6 − λ7 + λ8 ≤ 1 (1 4)(2 3 8 5) 12λ1 − λ2 − λ4 + λ6 − 2λ7 + λ8 ≤ 1 (2 6)(3 8 7 4) (1 2 3 . . . 12 13) 12λ1 + λ2 − 2λ3 − λ4 − λ6 + λ8 ≤ 1 (3 8)(4 5 7 6) 1λ1 + 2λ2 − 2λ3 − λ5 − λ6 + λ8 ≤ 1 (1 2)(3 8)(5 7 6) 12λ1 − 2λ2 − λ3 − λ4 + λ6 − 3λ7 + λ8 ≤ 0 (2 6 4 5 3 8 7) 1−λ1 + λ3 + 2λ4 − 3λ5 − 2λ6 − λ7 + λ8 ≤ 0 (1 4 2 3 8 5)(6 7) (1 2 3 . . . 14 15) 12λ1 + λ2 − 3λ3 − 2λ4 − λ5 − λ6 + λ8 ≤ 0 (3 8)(4 7) 1λ1 + 2λ2 − 3λ3 − λ4 − 2λ5 − λ6 + λ8 ≤ 0 (1 2)(3 8)(4 7 5) 1

Adding this vertex gives a polytope P where all facets are covered by Theorem 2. ThusP is the genuine moment polytope for ∧3H8 given by 31 independent inequalities listedin Table 4.


The Pauli Principle Revisited 319



λ1 ≤ 1 (1) (1) 1λ5 − λ6 − λ7 − λ8 ≤ 0 (1 5 4 3 2) 1λ1 − λ2 − λ7 − λ8 ≤ 0 (2 3 4 5 6) 1λ1 − λ3 − λ6 − λ8 ≤ 0 (3 4 5 7 6) 1λ1 − λ4 − λ6 − λ7 ≤ 0 (4 5 8 7 6) (1 2 3 4 5) 1λ1 − λ4 − λ5 − λ8 ≤ 0 (4 6)(5 7) 1λ3 − λ4 − λ7 − λ8 ≤ 0 (1 3 2)(4 5 6) 1λ2 − λ4 − λ6 − λ8 ≤ 0 (1 2)(4 5 7 6) 1λ2 + λ3 + λ5 − λ8 ≤ 2 (1 2 3 5 4) 1λ1 + λ3 + λ6 − λ8 ≤ 2 (2 3 6 5 4) 1λ1 + λ2 + λ7 − λ8 ≤ 2 (3 7 6 5 4) 1λ1 + λ2 + λ3 − λ4 ≤ 2 (4 5 6 7 8) (1 2 3 4 5) 1λ1 + λ4 + λ5 − λ8 ≤ 2 (2 4)(3 5) 1λ1 + λ2 + λ5 − λ6 ≤ 2 (3 5 4)(6 7 8) 1λ1 + λ3 + λ5 − λ7 ≤ 2 (2 3 5 4)(7 8) 1



λ2 + λ3 + λ4 + λ5 ≤ 2 (1 2 3 4 5) 1λ1 + λ2 + λ4 + λ7 ≤ 2 (3 4 7 6 5) (1 2 3 4 5) 1λ1 + λ3 + λ4 + λ6 ≤ 2 (2 3 4 6 5) 1λ1 + λ2 + λ5 + λ6 ≤ 2 (3 5)(4 6) 1λ1 + λ2 − λ3 ≤ 1 (3 4 5 6 7 8) 1λ2 + λ5 − λ7 ≤ 1 (1 2 5 4 3)(7 8) 1λ1 + λ6 − λ7 ≤ 1 (2 6 5 4 3)(7 8) (1 2 3 4 5 6) 1λ2 + λ4 − λ6 ≤ 1 (1 2 4 3)(6 7 8) 1λ1 + λ4 − λ5 ≤ 1 (2 4 3)(5 6 7 8) 1λ3 + λ4 − λ7 ≤ 1 (1 3)(2 4)(7 8) 1λ1 + λ8 ≤ 1 (2 8 7 6 5 4 3) (1 2 3 4 5 6 7) 1λ2 − λ3 − λ6 − λ7 ≤ 0 (1 2)(3 4 5 8 7 6) 1λ4 − λ5 − λ6 − λ7 ≤ 0 (1 4 3 2)(5 8 7 6) (1 2 3 4 5 6 7) 1λ1 − λ3 − λ5 − λ7 ≤ 0 (3 4 6)(5 8 7) 1λ2 + λ3 + 2λ4 − λ5 − λ7 + λ8 ≤ 2 (1 4 8 7 5) 1λ1 + λ3 + 2λ4 − λ5 − λ6 + λ8 ≤ 2 (1 4 8 6 7 5 2) (1 2 3 . . . 10 11) 1λ1 + 2λ2 − λ3 + λ4 − λ5 + λ8 ≤ 2 (1 2)(3 4 8 5 6 7) 1λ1 + 2λ2 − λ3 + λ5 − λ6 + λ8 ≤ 2 (1 2)(3 5 4 8 6 7) 1λ1 + λ2 − 2λ3 − λ4 − λ5 ≤ 0 (3 6 4 7 5 8) (1 2 3 . . . 11 12) 1λ1 − λ2 − λ3 + λ6 − 2λ7 ≤ 0 (2 6)(3 4 5 8 7) 1λ1 − λ3 − λ4 − λ5 + λ8 ≤ 0 (2 8 5 7 4 6 3) (1 2 3 . . . 12 13) 1λ1 − λ2 − λ3 − λ7 + λ8 ≤ 0 (2 8 7 3 4 5 6) 12λ1 − λ2 + λ4 − 2λ5 − λ6 + λ8 ≤ 1 (2 4 3 8 5 7 6) 1λ3 + 2λ4 − 2λ5 − λ6 − λ7 + λ8 ≤ 1 (1 4)(2 3 8 5) 12λ1 − λ2 − λ4 + λ6 − 2λ7 + λ8 ≤ 1 (2 6)(3 8 7 4) (1 2 3 . . . 12 13) 12λ1 + λ2 − 2λ3 − λ4 − λ6 + λ8 ≤ 1 (3 8)(4 5 7 6) 1λ1 + 2λ2 − 2λ3 − λ5 − λ6 + λ8 ≤ 1 (1 2)(3 8)(5 7 6) 12λ1 − 2λ2 − λ3 − λ4 + λ6 − 3λ7 + λ8 ≤ 0 (2 6 4 5 3 8 7) 1−λ1 + λ3 + 2λ4 − 3λ5 − 2λ6 − λ7 + λ8 ≤ 0 (1 4 2 3 8 5)(6 7) (1 2 3 . . . 14 15) 12λ1 + λ2 − 3λ3 − 2λ4 − λ5 − λ6 + λ8 ≤ 0 (3 8)(4 7) 1λ1 + 2λ2 − 3λ3 − λ4 − 2λ5 − λ6 + λ8 ≤ 0 (1 2)(3 8)(4 7 5) 1

Adding this vertex gives a polytope P where all facets are covered by Theorem 2. ThusP is the genuine moment polytope for ∧3H8 given by 31 independent inequalities listedin Table 4.




- number and complexity of constraints increases dramatically with number of electrons and size of basis- not at all obvious which constraints are important or what they even mean, physically - Altunbulak & Klyachko tabulated all constraints for systems with up to 10 orbitals

(a)

Be-

BC+N2+O3+F4+Ne5+

0 1 2 3 4 5 6 7

(b)

1 20 40 60 80 100 120 140 160constraint number

Be-

BC+N2+O3+F4+Ne5+

pure-state N-representability conditions- 5 electrons in 10 spin orbitals [ v2RDM-CASSCF (5e,5o) / cc-pVQZ ] … 160 constraints

- enforcing PQG (ensemble state) constraints

GPC errors (electrons)

AED, J. Chem. Phys. 145, 164109 (2016).

(a)

Be-

BC+N2+O3+F4+Ne5+

0 1 2 3 4 5 6 7

(b)


Be-

BC+N2+O3+F4+Ne5+



- many constraints severely violated


AED, J. Chem. Phys. 145, 164109 (2016).

(a)

Be-

BC+N2+O3+F4+Ne5+

0 1 2 3 4 5 6 7

(b)


Be-

BC+N2+O3+F4+Ne5+




- clearly the ground-state 2-RDM does not represent a pure state

- 5 electrons in 10 spin orbitals [ v2RDM-CASSCF (5e,5o) / cc-pVQZ ] … 160 constraints


AED, J. Chem. Phys. 145, 164109 (2016).

(a)

Be-

BC+N2+O3+F4+Ne5+

0 1 2 3 4 5 6 7

(b)


Be-

BC+N2+O3+F4+Ne5+





Hartree-Fock guess for 1-, 2-RDM



AED, J. Chem. Phys. 145, 164109 (2016).

(a)

Be-

BC+N2+O3+F4+Ne5+

0 1 2 3 4 5 6 7

(b)


Be-

BC+N2+O3+F4+Ne5+



- seeding computation with a different guess yields totally different densities … with exactly the same energy


random guess for 1-, 2-RDM





AED, J. Chem. Phys. 145, 164109 (2016).

(a)

Be-

BC+N2+O3+F4+Ne5+

0 1 2 3 4 5 6 7

(b)


Be-

BC+N2+O3+F4+Ne5+



- seeding computation with a different guess yields totally different densities … with exactly the same energy


random guess for 1-, 2-RDM





AED, J. Chem. Phys. 145, 164109 (2016).

1D(x1|x0

1) = N

Zdx2dx3...dxN

X

k

wk k(x1, x2, x3, ..., xN ) ⇤k(x

01, x2, x3, ..., xN ).


- 2P to 4P excitations from state-averaged (SA) CASSCF averaging 6 states

excitation energies (eV)

Be-

BC+

N2+

O3+

F4+

Ne5+

1.352.994.696.418.139.85

11.65

SA-CASSCF

maximum error


AED, J. Chem. Phys. 145, 164109 (2016).




Be-

BC+

N2+

O3+

F4+

Ne5+

1.413.114.896.748.56

10.4512.39

0.82

1.352.994.696.418.139.85

11.65

SA-CASSCF

ERPAensemble

maximum error


Hartree-Fock guess

AED, J. Chem. Phys. 145, 164109 (2016).




Be-

BC+

N2+

O3+

F4+

Ne5+

1.413.114.896.748.56

10.4512.39

0.82

1.623.415.237.058.87

10.7112.54

0.98

1.352.994.696.418.139.85

11.65

SA-CASSCF

ERPAensemble

maximum error


Hartree-Fock guessrandom guess

AED, J. Chem. Phys. 145, 164109 (2016).




Be-

BC+

N2+

O3+

F4+

Ne5+

1.413.114.896.748.56

10.4512.39

0.82

1.623.415.237.058.87

10.7112.54

0.98

1.352.994.696.418.139.85

11.65

SA-CASSCF

ERPAensemble

maximum error



all hope is lost!

AED, J. Chem. Phys. 145, 164109 (2016).


[-3,2,2,2,2,2,2,-3,-3,-3]<=5,

[3,3,3,-2,-2,-2,-2,-2,-2,3]<=5,

[3,3,5,-3,-1,-3,-1,1,-5,1]<=7,[3,3,5,1,-5,-3,-1,-3,-1,1]<=7,[-1,1,3,5,-1,1,3,-5,-3,-3]<=7,[3,3,5,-3,-1,1,-5,-3,-1,1]<=7,[-1,5,-1,1,3,1,3,-5,-3,-3]<=7,[-1,5,-1,1,1,3,3,-3,-5,-3]<=7,[-1,1,3,1,3,5,-1,-5,-3,-3]<=7,[3,5,3,1,-5,-3,-3,-1,-1,1]<=7,[-1,1,1,3,3,5,-1,-3,-5,-3]<=7,[3,5,3,-3,-3,-1,-1,1,-5,1]<=7,

[1,3,1,5,-1,5,-1,-5,-5,-3]<=9,[1,5,-1,5,-1,3,1,-5,-5,-3]<=9,[3,5,-1,5,-3,1,1,-5,-5,-1]<=9,[3,5,1,5,-5,1,-1,-5,-3,-1]<=9,[1,5,-1,3,1,5,-1,-5,-5,-3]<=9,[3,5,5,1,-5,-1,-3,1,-5,-1]<=9,[3,5,5,1,-5,1,-5,-1,-3,-1]<=9,[3,5,1,5,-5,-1,1,-3,-5,-1]<=9,[1,3,5,-1,1,5,-5,-1,-3,-5]<=9,[1,5,3,-1,1,5,-5,-1,-5,-3]<=9,[1,3,5,1,-1,5,-5,-1,-5,-3]<=9,[3,5,-1,1,1,5,-3,-5,-5,-1]<=9,[3,5,1,1,-1,5,-5,-5,-3,-1]<=9,[3,5,5,-1,-3,1,-5,1,-5,-1]<=9,[1,5,5,-1,-1,3,-5,1,-5,-3]<=9,[5,3,1,5,-5,-1,1,-5,-3,-1]<=9,[1,5,5,3,-5,-1,-1,1,-5,-3]<=9,[1,3,5,5,-5,1,-1,-1,-5,-3]<=9,[3,5,1,-1,1,5,-5,-3,-5,-1]<=9,[1,5,3,5,-5,-1,1,-1,-5,-3]<=9,[5,3,5,-3,-1,1,-5,1,-5,-1]<=9,[1,5,-1,5,-1,1,3,-5,-3,-5]<=9,[5,3,-1,1,1,5,-5,-3,-5,-1]<=9,[1,5,3,5,-5,-1,-1,1,-3,-5]<=9,[1,3,5,5,-5,-1,1,-1,-3,-5]<=9,[5,3,1,-1,1,5,-5,-5,-3,-1]<=9,[5,1,3,5,-5,1,-3,-1,-5,-1]<=9,[1,5,1,3,-1,5,-5,-3,-1,-5]<=9,[5,1,3,5,-5,1,-1,-5,-3,-1]<=9,[1,3,5,1,-1,5,-5,-3,-1,-5]<=9,[1,5,3,5,-5,1,-3,-1,-1,-5]<=9,[5,1,1,3,-1,5,-5,-3,-5,-1]<=9,[5,1,3,1,-1,5,-5,-5,-3,-1]<=9,[1,3,5,5,-5,1,-1,-3,-1,-5]<=9,

[5,3,5,1,-5,-3,-1,1,-5,-1]<=9,[1,5,-1,1,3,5,-1,-5,-3,-5]<=9,[1,1,3,5,-1,5,-1,-5,-3,-5]<=9,[5,3,5,1,-5,1,-5,-3,-1,-1]<=9,[1,1,3,5,-1,5,-3,-1,-5,-5]<=9,[5,5,1,3,-5,1,-5,-3,-1,-1]<=9,

[5,7,9,1,-9,-7,-5,-3,-1,3]<=13,[5,7,9,-7,-5,-3,-1,1,-9,3]<=13,

[-3,9,-1,1,3,5,7,-9,-7,-5]<=13,[-3,1,3,5,7,9,-1,-9,-7,-5]<=13,

[4,9,-1,9,-6,-1,4,-11,-6,-1]<=15,[-1,4,9,-1,4,9,-6,-1,-11,-6]<=15,[9,9,4,-6,-6,-1,-1,4,-11,-1]<=15,[-1,4,9,9,-6,-1,-1,4,-6,-11]<=15,[9,-1,4,4,-1,9,-11,-6,-6,-1]<=15,[-1,4,9,9,-6,-1,4,-1,-11,-6]<=15,[-1,9,4,9,-6,-1,-1,4,-11,-6]<=15,[9,4,-1,-1,4,9,-11,-6,-6,-1]<=15,[9,9,4,4,-11,-6,-6,-1,-1,-1]<=15,[-1,4,4,9,-1,9,-6,-1,-11,-6]<=15,[4,9,-1,4,-1,9,-11,-6,-6,-1]<=15,[4,9,-1,-1,4,9,-6,-11,-6,-1]<=15,[-1,9,-1,4,4,9,-1,-11,-6,-6]<=15,[-1,4,4,9,-1,9,-1,-11,-6,-6]<=15,[-1,9,-1,9,-1,4,4,-11,-6,-6]<=15,

[1,11,-4,1,1,6,6,-4,-9,-9]<=15,[1,6,11,-4,1,6,-9,1,-9,-4]<=15,[6,11,1,6,-9,-4,1,-9,-4,1]<=15,[11,6,-4,1,1,6,-9,-9,-4,1]<=15,[1,6,6,11,-9,1,-4,-4,1,-9]<=15,[6,11,1,-4,1,6,-9,-9,-4,1]<=15,[6,11,-4,1,1,6,-9,-4,-9,1]<=15,[1,6,6,11,-9,-4,1,1,-4,-9]<=15,[6,11,1,6,-9,1,-9,-4,-4,1]<=15,[1,6,6,11,-9,1,-4,1,-9,-4]<=15,[1,6,11,6,-9,-4,1,1,-9,-4]<=15,[6,6,11,-4,-4,1,-9,1,-9,1]<=15,[1,1,1,6,6,11,-4,-4,-9,-9]<=15,[6,6,11,1,-9,1,-9,-4,-4,1]<=15,[6,6,11,1,-9,-4,-4,1,-9,1]<=15,

[-3,7,7,7,-3,7,-3,-3,-3,-13]<=15,[7,-3,7,7,-3,7,-3,-3,-13,-3]<=15,[7,7,-3,-3,7,7,-3,-3,-13,-3]<=15,[7,7,7,-3,-3,7,-13,-3,-3,-3]<=15,

[7,7,-3,7,-3,-3,7,-3,-13,-3]<=15,[7,7,-3,7,-3,7,-3,-13,-3,-3]<=15,[7,7,7,7,-13,-3,-3,-3,-3,-3]<=15,[7,7,7,-3,-3,-3,-3,7,-13,-3]<=15,

[3,13,3,3,-7,3,-7,-7,3,-7]<=15,[13,3,3,3,-7,3,-7,-7,-7,3]<=15,[3,13,3,3,-7,-7,3,3,-7,-7]<=15,[3,13,3,-7,3,3,-7,3,-7,-7]<=15,[3,3,3,3,3,13,-7,-7,-7,-7]<=15,[3,3,13,3,-7,3,-7,3,-7,-7]<=15,[3,13,-7,3,3,3,3,-7,-7,-7]<=15,[3,3,3,13,-7,3,3,-7,-7,-7]<=15,

[-2,8,3,8,3,13,-2,-12,-12,-7]<=20,[-2,13,-2,8,3,8,3,-12,-12,-7]<=20,[-2,8,3,13,-2,8,3,-12,-12,-7]<=20,[-2,13,-2,3,8,3,8,-7,-12,-12]<=20,[-2,3,8,8,3,13,-7,-2,-12,-12]<=20,[-2,8,3,3,8,13,-2,-12,-7,-12]<=20,[-2,8,3,13,-2,3,8,-12,-7,-12]<=20,[-2,3,8,3,8,13,-2,-7,-12,-12]<=20,[-2,3,8,13,-2,3,8,-7,-12,-12]<=20,[-2,3,8,13,-2,8,3,-12,-7,-12]<=20,[-2,13,-2,8,3,3,8,-12,-7,-12]<=20,[-2,13,-2,3,8,8,3,-12,-7,-12]<=20,[-2,8,8,13,-7,-2,3,3,-12,-12]<=20,[-2,3,8,8,3,13,-2,-12,-7,-12]<=20,[8,8,-2,3,3,13,-12,-12,-7,-2]<=20,

[12,7,12,-8,-3,-3,-8,2,-13,2]<=20,[12,7,12,-3,-8,-8,-3,2,-13,2]<=20,[12,12,7,-8,-3,-8,-3,2,-13,2]<=20,[12,12,2,7,-13,-3,-8,-8,-3,2]<=20,[12,7,12,2,-13,-8,-3,-3,-8,2]<=20,[12,7,12,-8,-3,2,-13,-3,-8,2]<=20,[12,12,7,2,-13,-8,-3,-8,-3,2]<=20,[12,12,7,-8,-3,2,-13,-8,-3,2]<=20,[12,7,12,-3,-8,2,-13,-8,-3,2]<=20,[12,12,-3,-3,2,7,-13,-8,-8,2]<=20,[12,7,12,2,-13,-3,-8,-8,-3,2]<=20,[7,12,12,-3,-8,2,-13,-3,-8,2]<=20,[7,12,12,2,-13,-3,-8,-3,-8,2]<=20,[7,12,12,-3,-8,-3,-8,2,-13,2]<=20,[2,7,12,12,-13,-3,-3,2,-8,-8]<=20,

[3,8,13,18,-17,-7,-2,3,-12,-7]<=25,[13,18,-7,-2,3,8,-17,-12,-7,3]<=25,[13,18,8,-12,-7,-7,-2,3,-17,3]<=25,

[13,18,8,3,-17,-12,-7,-7,-2,3]<=25,[13,18,3,8,-17,-7,-12,-7,-2,3]<=25,[7,12,-3,2,7,17,-18,-13,-8,-3]<=25,[-3,7,12,17,-8,-3,2,7,-18,-13]<=25,[-3,2,7,12,7,17,-8,-3,-18,-13]<=25,[-3,17,-3,2,7,7,12,-8,-18,-13]<=25,[-3,2,7,7,12,17,-3,-8,-18,-13]<=25,

[17,17,7,-3,-13,-13,-13,-3,-3,7]<=25,[17,17,7,-13,-13,-3,-3,-3,-13,7]<=25,

[-7,13,3,3,3,13,13,-7,-17,-17]<=25,[-7,3,3,13,13,13,3,-7,-17,-17]<=25,

[-3,27,-8,2,7,12,17,-23,-18,-13]<=35,[-3,2,7,12,17,27,-8,-23,-18,-13]<=35,

[13,18,23,-17,-12,-7,-2,8,-27,3]<=35,[13,18,23,8,-27,-17,-12,-7,-2,3]<=35,

[-7,3,13,13,23,33,-7,-27,-27,-17]<=45,[-7,33,-7,3,13,13,23,-27,-27,-17]<=45,

[17,27,27,7,-33,-23,-13,-13,-3,7]<=45,[17,27,27,-23,-13,-13,-3,7,-33,7]<=45

];

all hope is NOT lost!

- Altunbulak & Klyachko tabulated these constraints M. Altunbulak and A. Klyachko, Commun. Math. Phys. 282, 287 (2008).

- can recast generalized Pauli constraints as linear equality constraints within v2RDM SDP optimization

- resulting optimization requires *far* more iterations to converge … BUT … the 1-RDM is then pure-state N-representable

PQG has 7389 constraints




Be-

BC+

N2+

O3+

F4+

Ne5+

1.413.114.896.748.56

10.4512.39

0.82

1.623.415.237.058.87

10.7112.54

0.98

1.383.044.746.458.189.90

11.62

0.06

1.352.994.696.418.139.85

11.65

SA-CASSCF

ERPA

ensemble pure

maximum error



AED, J. Chem. Phys. 145, 164109 (2016).


- 3P to 5S excitations from state-averaged (SA) CASSCF averaging 4 states


Be2-

B-

CN+

O2+

F3+

Ne4+

0.531.733.154.776.518.33

10.24

0.93

0.491.632.964.466.057.679.30

SA-CASSCF

ERPAensemble

maximum error


Hartree-Fock guess

AED, J. Chem. Phys. 145, 164109 (2016).




Be2-

B-

CN+

O2+

F3+

Ne4+

0.531.733.154.776.518.33

10.24

0.93

0.812.223.805.497.249.04

10.86

1.56

0.491.632.964.466.057.679.30

SA-CASSCF

ERPAensemble

maximum error



AED, J. Chem. Phys. 145, 164109 (2016).




Be2-

B-

CN+

O2+

F3+

Ne4+

0.531.733.154.776.518.33

10.24

0.93

0.812.223.805.497.249.04

10.86

1.56

0.491.632.964.466.057.679.30

SA-CASSCF

maximum error



ERPA

pure

0.491.632.964.466.057.679.30

0.00

ensemble

AED, J. Chem. Phys. 145, 164109 (2016).




Be2-

B-

CN+

O2+

F3+

Ne4+

0.531.733.154.776.518.33

10.24

0.93

0.812.223.805.497.249.04

10.86

1.56

0.491.632.964.466.057.679.30

SA-CASSCF

maximum error



ERPA

pure

0.491.632.964.466.057.679.30

0.00

ensemble

AED, J. Chem. Phys. 145, 164109 (2016).

Current/future directions

Summarysophisticated electronic structure methods are necessary to capture strong correlation effects, but conventional approaches become impractical for large systems

v2RDM-driven CASSCF can treat systems with active spaces as large as (50e,50o), and analytic energy gradients are readily available.

however, treating excited states can be challenging

-

-

-

-

-

standard ERPA often is inadequate; inclusion of higher excitations into excited-state wave functions is necessary

including spin-orbit coupling in two-component v2RDM-CASSCF for relativistic effects

- strategies for dynamical correlation effects (e.g. ERPA, some DFT-inspired approaches)

Acknowledgements

Graduate StudentsElvis MaradzikeBrandon CooperSina MostafanejadNam Vu

Undergraduate StudentsDaniel Gibney

PostdocsRun “Rain” Li

Former Group MembersJacob Fosso-TandeWayne MullinaxDaniel NascimentoJess HaneyLauren Koulias

Collaborators:Jay Foley (William Paterson)Greg Gidofalvi (Gonzaga)Evgeny Epifanovsky (Q-Chem, Inc.)

NSF (CHE-1554354, ACI-1663636)ARO STTR Phase I/II with Q-Chem (69478CHST2)DOE EFRC (DE-SC0016568)

$$$

@DePrinceFSU

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Quantum chemistry without wave functions › ... › reu › REU_deprince_6_18_19.pdfQuantum chemistry without wave functions (and I don't mean density functional theory) Eugene DePrince