Some recent developments in practical quantum chemistry ... · Some recent developments in...
Transcript of Some recent developments in practical quantum chemistry ... · Some recent developments in...
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).