Some recent developments in

practical quantum chemistry

calculations

MARTIN HEAD-GORDON

Department of Chemistry,

University of California, Berkeley, and,

Chemical Sciences Division,

Lawrence Berkeley National Laboratory

Outline

1. ωB97 density functionals

2. ALMO-EDA for intermolecular interaction

3. SOS-MP2 and O2 for ground states

4. SOS-CIS(D), SOS-CIS(D0) for excited states

Outline

Three new density functionals

• 100% long-range exact exchange & dispersion.

• ωB97, ωB97X, ωB97X-D

• Dr. Jeng-Da Chai

• J.D. Chai, MHG, JCP 128, 084106 (2008)

• PCCP 44, 6615 (2008), CPL 467, 176 (2008).

2 new long-range corrected B97 functionals

(Jeng-Da Chai)

• B97: 100% long-range exact exchange

• B97X: also a fraction of short-range exact exchange

• For short-range GGA, use the short-range LSDA (Gill)

with new GGA parameters for the enhancement factor

EXCB97 EX

LRHF EXSRB97 ECss

B97 ECosB97

EXCB97X EX

LRHF cXEXSRHF EX

SRB97 ECssB97 ECos

B97

EXSRB97 dr x

SRLSDA cix (

x s2

1 x s2

)i

i0

m

Empirical atom-atom dispersion (-D) corrections

• Additional non-local correlation energy contribution:

• C6i are atomic C6 factors; f damps at short-range

• Greatly improves dispersion-dominated interactions:• R. Ahlrichs, R. Penco, G. Scoles, Chem. Phys. 19, 119 (1977)

• Q. Wu and W.T. Yang, J. Chem. Phys. 116, 515 (2002)

• S. Grimme, J. Comput. Chem. 25, 1463 (2004)

• S. Grimme, J. Comput. Chem. 27, 1787 (2006)

• We add this correction to wB97X & fully reoptimize

Edisp s6

C6

ij

Rij6

i j

atoms

fdamp Rij C6

ij C6

iC6

j fdampa 1 a(Rij / Rr )

12 1

Test set performance (Jeng-Da Chai)

Ar2+ dissociation (Jeng-Da Chai)

HF

CCSD(T) B97

B97X

B97-1B3LYP

BLYP

Outline

Analyzing intermolecular interactions

• ALMO EDA/CTA: a new, well-defined method

• Energies, charge flow, donor-acceptor orbitals

• Dr. Rustam Khaliullin, Prof. Alex Bell (UCB)

• Khalliulin et al, JPC A 111, 8753 (2007);

• JCP 128, 184112 (2008); Chem. Eur. J. 15, 851 (2009).

Absolutely localized molecular orbitals (ALMO’s)

• For molecular clusters and liquids, molecular fragments

are well-defined.

• Define absolutely localized molecular orbitals as:

• Transformation T is molecule-blocked

• ALMO’s are non-orthogonal

• No charge transfer between fragments

• No borrowing your neighbor’s basis functions for your own

selfish variational purposes!

xi x Tx,xi

x

T

TA 0 0

0 TB 0

0 0 TC

Energetics for water dimer (aug-cc-pVDZ basis)

A single step correction for charge transfer

• ALMO-SCF performed first…

• Converged Fock matrix has occupied-virtual coupling

• So perform a single full diagonalization

• Energy change is defined as linear in the density change

E Tr P - p F

Tr XVO

FOV

diag F p P

F F p

min e

SCFp

R.Z Khaliullin, MHG, A.T.Bell, J. Chem Phys. 124, 204105 (2006)

Improved energetics for the water dimer

Interaction energy analysis: an incomplete history

• K. Kitaura and K. Morokuma, Int. J. Quantum Chem. 10, 325 (1976). “Morokuma decomposition”

• P.S. Bagus, K. Hermann, C.W. Bauschlicher, J. Chem. Phys. 80, 4378 (1984) “CSOV analysis”

• E.D. Glendening, A. Streitwieser, J. Chem. Phys. 100, 2900 (1994).

• “natural orbital decomposition”

• Y.R. Mo, J.L. Gao, S.D. Peyerimhoff, J. Chem. Phys. 112, 5530 (2000).

• “BLW EDA”

• Alternative development is symmetry-adapated perturbation theory (SAPT). See e.g. B. Jeziorski, R. Moszynski & K. Szalewicz, Chem. Rev. 94, 1887 (1994); A.J. Misquitta, B. Jeziorski & K. Szalewicz, Phys. Rev. Lett. 91 (2003)

Analysis of binding energies

• Geometric distortion

• Non-interacting fragment densities p(0)

• Frozen electrostatics: Interaction energy using p(0)

• Coupling of permanent moments & exchange repulsion

• Polarization: treat by ALMO-SCF (new)

• Induction effects treated fully self-consistently

• Donor-acceptor interactions: charge-transfer correction (new).

• Can be decomposed into forward/back donation

• Higher-order charge transfer: full SCF

• Not decomposable

E Tr P - p F

Tr XVO

FOV

R.Z Khaliullin, R.Lochan, E.Cobar, A.T.Bell, MHG, J. Phys. Chem A 111, 8753 (2007)

Complementary occupied-virtual orbital pairs

• XVO is decomposed into (x,y) intermolecular pairs:

• The principal orbitals for charge transfer are obtained by

singular value decomposition of

R.Z Khaliullin, A.T.Bell, MHG, J. Chem. Phys. 128, 184112 (2008)

E Tr XVO

FOV

Tr XVO

x , y F

OV

y ,x

y

x

X

VO

x , y

X

VO

x, y V

xx(x, y)U

y

 

Principal virtual

(acceptor) orbitals

Principal occupied

(donor) orbitals

Weight of each

donor-acceptor

pair

Forward donation in (CO)5W-CO

(Rustam Khaliullin)

• E(LM) = 101 kJ/mol

• Q(LM) = 0.04 e

• 1st complementary orbital pair:

• 98% of E

• 97% of Q

• Bold: donor orbital

• Faint: acceptor orbital

R.Z Khaliullin, A.T.Bell, MHG, J. Chem. Phys. 128, 184112 (2008)

Back-donation in (CO)5W-CO

(Rustam Khaliullin)

• E(ML) = 142 kJ/mol

• Q(ML) = 0.25 e

• 1st complementary orbital pair:

• 50% of E

• 50% of Q

• Bold: donor orbital

• Faint: acceptor orbital

R.Z Khaliullin, A.T.Bell, MHG, J. Chem. Phys. 128, 184112 (2008)

Form of the donor & acceptor orbitals

Donor orbital does not

rotate with rotation of

the donor molecule!

Outline

Scaled opposite spin (SOS) MP2 and O2

• 4th order scaling, removes systematic MP2 error

• Dr. Yousung Jung, Dr. Rohini Lochan

• JCP 121, 9793 (2004); PCCP 8, 2831 (2006);

JCP 126, 164101 (2007); JCTC 3, 988 (2007);

Wavefunction-based levels of theory

• Uncorrelated (mean-field) level.

• Hartree-Fock (HF) / Single excitation CI (CIS)

• Cheap (~M2), self-interaction free, and quite inaccurate

• Second order pair correlation treatment.

• MP2 for ground state. M5 cost, fair accuracy.

• CIS(D) and related methods are excited state analogs

• Self-consistent pair correlation treatment (& beyond!)

• Coupled cluster theory for ground state

• Linear response/equation-of-motion coupled cluster

• Systematic framework for high accuracy (high cost: M6)

Improved MP2: scaled opposite spin (SOS) MP2

• A 1-parameter empirical modification of MP2

• Motivated by Grimme’s empirical 2-parameter SCS-MP2 method

• (Statistically) better chemistry:

• Scaled (by 1.3) to correct systematic deficiencies of MP2

• Cheaper cost:

• SOS-MP2 has only 4th order computational cost scaling!

ESOSMP2 EHF cEOS(2)

EMP2 EHF EOS(2) ESS

(2)

Y. Jung, R. Lochan, A.D. Dutoi, and MHG, J. Chem. Phys. 121, 9793-9802 (2004).

Atomization energies (Rohini Lochan)

• Independently assess atomization energies…

• 148 molecules from the G2 database

• Compare against CCSD(T) (cc-pVTZ basis)

• SOS-MP2 is significantly better (and cheaper) than MP2

MP2 SOS-MP2

RMS dev 14.9 6.2

Max dev 38.1 21.1

SOS-MP2 energy timings

• t/seconds (Opteron 2 GHz). cc-pVDZ basis.

• Tight cutoffs (12/9)

• SOSMP2 is 4th order scaling.

• Faster than RI-MP2 for large systems

HF MP2 RIMP2 SOSMP2

(alanine)4 (3D) 2340 3087 245 282

(alanine)8 (3D) 15729 47480 5392 3394

(alanine)16 (3D) 71178 ---- 145600 48630

Incremental time

For MP2 energy

SOS-MP2 analytical gradient timings

• t/seconds. 6-31G** basis. Tight cutoffs (12/9)

• SOS-MP2 is faster for large systems.

• Advantage increases with size

HF MP2 RIMP2 SOSMP2

(alanine)4 1704 9160 4470 4141

(alanine)8 9596 121473 46953 30853

(alanine)16 46305 ---- 781151 227251

Total time

for force

What to do about large radicals?

• Hartree-Fock theory can suffer large spin contamination

• Which causes MP2 (and SOS-MP2) to perform poorly

• Cure? Optimize orbitals with scaled opposite spin

second order correlation (O2 method):

EO2 E0 c EOS(2)

S2

UHF 2.08 S2

exact 0.75

S2

MP2 1.92

R.C. Lochan and MHG, J. Chem. Phys. 126, 164101 (2007)

Performance of the O2 method

• 148 atomization energies (kcal/mol): choose c=1.2

• 12 bond lengths (doublet diatomic radicals) (Å)

• 12 harmonic frequencies (doublet radicals) (cm-1)

1.1 1.2 1.3

RMS dev 9.9 5.8 18.5

HF MP2 O2(1.2)

RMS dev 0.043 0.030 0.014

HF MP2 O2(1.2)

RMS dev 292 380 142

Large unsaturated radicals

Method

öS 2 BE (kcal/mol)

Phenalenyl radicalb

HF 2.0764

MP2 1.9238

O2 (1.2)c 0.7634

Phenalenyl dimer

HF 3.1991 30.97

MP2 2.9198 -5.96

O2 (1.2)c 0.0000 -21.56

R.C. Lochan and MHG, J. Chem. Phys. 126, 164101 (2007)

Outline

Scaled opposite spin (SOS) CIS(D), CIS(D0)

• Excited state analog of ground state MP2

• 4th order scaling, better than CIS(D) error

• Dr. Young Min Rhee, Dr. David Casanova

• JCP 111, 5314 (2007); JCP 128, 164106 (2008);

• JCTC (in press)

Excited state CIS(D): Spin-component scaling?

• CIS(D) energy can also be split into spin components…

• And thus we can define an opposite spin-scaled method:

• 2 parameters are involved -- one for each of the 2 terms

• “spectator” pair correlations are scaled as the ground state

• “active” pair correlations will be scaled differently, by cU

• 4th order scaling of computational cost

ECIS(D) ECIS EOSCIS(D) ESS

CIS(D)

ESOSCIS(D) ECIS cUEOSactive cTEOS

spectator

Examples: CIS vs CIS(D) vs experiment

• 00 excitations. aug-cc-pVTZ. HF/CIS structures/ZPE

Same examples: CIS(D) vs SOS-CIS(D)

• 00 excitations. aug-cc-pVTZ. HF/CIS structures/ZPE

Statistics: 43 adiabatic excitation energies

• Significant improvement over CIS(D): scale factor is 1.4

SOS-CIS(D) Rydbergn**

CIS(D):

MAE=0.30eV

SOS-CIS(D):

MAE=0.13eV

Computational costs for 10 lowest excited states

• aug-cc-pVTZ basis (except for ZnBC-BC: 6-31G*)

• Timings on 1CPU of a 2.0 GHz Opteron processor.

CPU time (min) Molecule

No. of

basis CIS(D)b RI-CIS(D) SOS-CIS(D)

Acrolein (C3H4O) 276c 65 3 4

Hexatriene (C6H8) 460c 942 23 25

Styrene (C8H8) 552c 4944 57 55

Azulene (C10H8) 644c - 126 109

Anthracene (C14H10) 874c - 504 352

Pyrene (C16H10) 966c - 809 528

ZnBC-BC (C46H36N8Zn) 918d - 8372 3010

Conical intersections between excited states

Non-degenerate:

SOS-CIS(D) fails

Quasi-degenerate:

SOS-CIS(D0) is OK

D. Casanova, Y.M. Rhee and M. Head-Gordon, J. Chem. Phys. 128, 164106 (2008).

Further development of the SOS-CIS(D0) method.

• Theory and implementation of the analytical gradient

• O(M4) cost

• Approaches CC2 accuracy

• Young Min Rhee, MHG,

• JCTC (in press)

• Young Min Rhee, MHG

• (in preparation)

Text

gradient CPU times

(Grateful) Acknowledgements

• All members of my group -- past and present

• For the topics covered here:

• Jeng-Da Chai, Rustam Khaliullin, Yousung Jung, Rohini

Lochan, Young Min Rhee, David Casanova

• All Q-Chem employees

• Particularly Jing Kong and Yihan Shao.

• Academic collaborators

• Q-Chem Customers

Funding acknowledgements:

U.S. Department of Energy

BES, SciDAC, TMS

National Institutes of Health

National Science Foundation

MHG is a part-owner of Q-Chem

Dual basis RI-MP2 gradient

Rob Distasio, Ryan Steele

6-311G(3df,3pd) 6-311G(d,p)

• Significant (4 times) speedup

• Insignificant loss of accuracy

• Less time than large basis SCF alone!

Alanine tetrapeptide timings (hours) on a 2 GHz Opteron computer

Rob Distasio,

Ryan Steele,

MHG,

Mol. Phys.

105, 2731

(2007)

A (quick) look at the CIS(D) method

• Perturbative doubles/triples correlation correction to CIS

• Double excitations for pairs “active” in CIS transition

• Amplitudes are excited state analogs of MP2 amplitudes

• Triple excitations for “spectator” electron pairs

• Amplitudes are disconnected products, involving ground state (MP1) doubles, T2, and CIS amplitudes.

ECIS(D) ECIS CIS |V |U20 CIS |V |T2U10

U20 1

4bijabij

ab

ijab

1

4

ij

ab |V |U10

a b i j ij

ab

ijab

.

Spin-component scaling for CIS(D)

• With some algebra, it can be shown that scaled opposite

spin (SOS)-CIS(D) can be evaluated with M4 effort.

• One can also scale the two spin-components separately,

giving “spin-component scaled” (SCS)-CIS(D)

• SCS-CIS(D) has 2 new excited state scaling parameters,

and 2 parameters fixed by ground state SCS-MP2

• Computational cost is M5 as for CIS(D) itself…

• But the chemistry might be better….

Charge-transfer states

• Correct behavior

• (self-interaction free)

-1

0

1

2

3

4

5

-2 -1 0 1 2 3 4

R (Angstrom)

En

erg

y (

eV

)

N N

N N

H

H

N N

N N

Zn

N N

N N

H

H

N N

N N

Zn

R

TD-B3LYP

CIS(D) family

Perturbation theory for excited states

• CIS problem is zero order:

• Expand Hamiltonian and cluster amplitudes in powers of

the Moller-Plesset fluctuation potential:

• CIS(D) is 2nd order non-degenerate perturbation theory:

ASS(0)bi

(0) i

(0)bi(0)

A

ASS

(0)0 0 L

0 DDD

(0)0

0 0 DTT(0)

M O

0 ASD

(1)AST

(1) L

ADS

(1)ADD

(1)ADT

(1)

0 ATD(1)

ATT(1)

M O

ASS

(2)ASD

(2)0 L

ADS

(2)ADD

(2)ADT

(2)

ATS(1)

ATD(2)

ATT(2)

M O

...

i

(2) bi(0)†ASS

(2)bi

(0) bi(0)†ASD

(1)DDD

(0) i

(0) 1

ADS

(1)bi

(0)

Quasi-degenerate perturbation theory

• Treat all other single excitations as quasidegenerate… so

we must rediagonalize the SS block with correlation:

• Treat energy-dependent response matrix by (rapidly

convergent) binomial expansion which we truncate…

• Truncation at zero order in gives CIS(D0)…

ASS(02) ASS

(0) ASS(2) ASD(1)

DDD(0)

1

ADS

(1)

DDD(0)

1

DDD(0)

1

1 1 DDD

(0) 1

1 2 ...

DDD(0)

1

ASS(0) ASS

(2) ASD(1) DDD

(0) 1

ADS

(1) bS bSD. Casanova, Y.M. Rhee and M. Head-Gordon, J. Chem. Phys. 128, 164106 (2008).