High-level Quantum Chemistry Methods and Benchmark ... · High-level Quantum Chemistry Methods and...

High-level Quantum Chemistry Methodsand Benchmark Datasets for Molecules

Markus SchneiderFritz Haber Institute of the MPS, Berlin, Germany

École Polytechnique Fédérale de Lausanne, Switzerland

May 11th, 2016 - اصفهانصنعتیدانشگاه

May 11th, 2016 - اصفهانصنعتیدانشگاه 1/48

Outline1 Introduction2 Levels of Theory3 Wavefunction-based methods

Hartree-FockConfiguration interaction (CI)Coupled-Cluster (CC)DLPNO-CCSD(T)2nd order Møller-Plesset (MP2)Higher order Møller-Plesset

4 Practical considerationsCounterpoise correctionExtrapolation schemes and basis sets

5 Benchmark datasets6 Summary


Introduction

I. Introduction

(often) very basic task:Calculate properties for a system!system:

solids, surfaces, molecules, …properties:

potential energy, free energy, lattice constant, atomic distances,UV spectra, IR spectra, …

example: conformational search of cation-peptide systems in gas-phase


Introduction

which quantum chemistry computer program should I use?there are many!

https://en.wikipedia.org/wiki/List_of_quantum_chemistry_and_solid-state_physics_softwareMay 11th, 2016 - اصفهانصنعتیدانشگاه 4/48

https://en.wikipedia.org/wiki/List_of_quantum_chemistry_and_solid-state_physics_software

Introduction

define your problem and evaluate choice of software!license (GPL, academic, commercial, …)support, manual, …basis set (numeric atomic orbitals (NAO), plane waves (PW),

Gaussian type orbitals (GTO), …)method (semi-empirical, DFT, post-Hartree-Fock, …)

often compromise between accuracy and computational costs

one should verify accuracy of method of choice!

→ verify against experiment

→ verify against theory (higher-level method)


Introduction

verification against experimentbut be careful, example: AcFA6-Na+ at 4K

PBE0+MBD (tight) PBE0+MBD (tight) + ZPE (PBE+MBD)

0.000

0.050

0.100

0.150

0.200

Rela

tive

ener

gy [e

V]

0.0

1.0

2.0

3.0

4.0

5.0

[kca

l/mol

]

theory predicts one distinctive global minimumBUT: conformer experimentally excluded

Who’s wrong? Experiment or theory?May 11th, 2016 - اصفهانصنعتیدانشگاه 6/48

Levels of Theory

II. Levels of Theory - Overview

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puta

tiona

lcos

tsFull CI

Wavefunction-based methods(CCSD(T), MP2, RPA, …)

DFT(xc-functionals: LDA, GGA, hybrids, …)

Semi-empirical methods(AM1, PM3, tight-binding, …)

Empirical methods (force fields)

accu

racy

(?)


Levels of Theory

Timings

qualitative example: Phenylalanine + Ca2+

timings depend on basis set, integration grid, implementation, …May 11th, 2016 - اصفهانصنعتیدانشگاه 8/48

Levels of Theory

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tsFull CI





accu

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Levels of Theory

Empirical Force Field Potential Energy Function

Epot = Ebond + Eangle + Etorsion + ECoulomb + EvdW

bonded terms:Ebond =

∑i<j Kb

ij(rij − r0ij)2

…

non-bonded terms:ECoulomb =

∑i<j

qiqj4πε0rij

EvdW =∑

i<j

[Aijr12ij

− Bijr6ij

]

e.g. OPLS, CHARMM22


Parameters are derivedfrom specific data sets(with experimental- orQM-derived properties)by a fitting algorithm.

Cations are alwaysnon-bonded!

Levels of Theory

example: Histidine + Zn2+ + H2Oforce field global minimum DFT global minimum

Hirshfeld charges of the Zn2+ cation over structure data set

0

10

20

30

40

50

60

70

0.5 0.6 0.7 0.8

co

un

t

Hirshfeld charge


⇒ opposes the idea offixed-point charges

⇒ (one solution:polarizable force fields

→ buffered or scaled by factor;e.g. AMOEBA

Levels of Theory

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accu

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Levels of Theory

Density-functional theory (DFT)

→ this workshop ,→ talks by Sergey Levchenko (“DFT in Practice”) and

Weitao Yang (“DFT and the Exchange-Correlation Functional”)→ practical session 1 last tuesday

solve Kohn-Sham equations self-consistently[− ℏ2

2m∇2 + V(r)]ϕi(r) = εi ϕi(r)

with ϕi(r) … non-interacting Kohn-Sham orbitals that fulfill∑N1 |ϕi(r)|2 = n(r)

V = Vext(r) +∫ e2n(r ′)

|r−r ′| d3r ′ + Vxc[n(r)]density-functional theory (DFT) is an exact method butdensity-functional approximation (DFA) is not


Levels of Theory

local-density approximation (LDA):ELDA

xc =∫εxc(n) n(r) d3r ′

generalized gradient approximation (GGA):EGGA

xc =∫εxc(n, ∇n) n(r) d3r ′

hybrid functionals include Hartree-Fock exact exchange functional EHFx

EHFx = −1

2∑

i,j∫ ∫

ϕ∗i (r)ϕ∗

j (r) 1|r−r ′|ϕi(r ′)ϕj(r ′) d3r d3r ′

e.g. PBE0: EPBE0x = αEHF

x + (1 − α)EPBEx + EPBE

c

→ higher computational costs ??= higher accuracy


Wavefunction-based methods

III. Wavefunction-based methods

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Wavefunction-based methods

Quick recap

n-electron system described by Schrödinger equationin Born-Oppenheimer approximation(fixed nuclei / classical treatment separately from electrons):

HBOψ = Eψ

HBO =n∑

i=1

−12∆i −

N∑j=1

Zj|ri − Rj|

+ 12

n∑j=i

1|ri − rj|

analytical solution impossible ⇒ use numerical technique


Wavefunction-based methods Hartree-Fock

Hartree-Fock

n quasi-independent electrons describedby a (antisymmetric) Slater determinant

ψ(r1, r2, . . . , rn) =

∣∣∣∣∣∣∣∣∣ϕ1(r1) ϕ2(r1) . . . ϕn(r1)ϕ1(r2) ϕ2(r2) . . . ϕn(r2). . . . . . . . . . . .

ϕ1(rn) ϕ2(rn) . . . ϕn(rn)

∣∣∣∣∣∣∣∣∣variational principle (ground state energy E0 ≤ EHF = ⟨HHF⟩)→ optimize orbitals w.r.t. EHFwith EHF =

∑ni=1

(−1

2∆i −∑N

j=1Zj

|ri−Rj|

)+ 1

2∑n

i=1∑n

j=1(Jij − Kij)Jij … Coulomb term → Coulomb repulsion between electronsKij … exchange term → no classical counterpart;

→ corrects for self-interaction Jii of electrons

self-consistent solution yields EHF, |ψHF⟩May 11th, 2016 - اصفهانصنعتیدانشگاه 18/48

Wavefunction-based methods Hartree-Fock

Hartree-Fock method is size consistent ,Hartree-Fock method is a mean field approximation /instead, electrons are correlatedremaining correlation error: Ecorr = Eexact − EHF

exact energy = Hartree-Fock + correlation energy= “mean field” + “instantaneous e- - e- interaction”

correlation energy Ecorr is ∼ 1% of Eexact

→ chemically significant

how to calculate Ecorr?→ use HF as starting point

→ apply post-HF wave-function based method on topMay 11th, 2016 - اصفهانصنعتیدانشگاه 19/48

Wavefunction-based methods Configuration interaction (CI)

Configuration interaction (CI)

post-Hartree–Fock linear variational method

instead of one Slater determinant→ linear combination of Slater determinants

⇒ |Ψ>= c0|ΨHF> +∑

i,a cai |Ψa

i > +∑occ.

i<j∑unocc.

a<b cabij |Ψab

ij > + · · ·ΨHF … Slater determinantΨa

i … determinant with one occupied orbital replaced by virtual orbitalΨab

ij … determinant with two occupied orbitals replacedby two virtual ones

→ take into account all possible Slater determinants obtained by excitingall possible electrons to all possible virtual orbitals ⇒ Full CI



truncation of expansion:CIS … configuration interaction “singles”CID … configuration interaction “doubles”CISD … configuration interaction “singles and doubles”…

↿ ⇃↿ ↿ ⇂ ↿

↿ ⇂ ⇃ ↿ ⇂ ↿↿ ⇂ ↿ ⇂ ↿ ↿ ⇂ ⇃↿ ⇂ ↿ ⇂ ↿ ⇂ ↿ ⇂ ↿ ⇂HF “singles” “singles” “doubles” “doubles”



problem: truncated CI not size-consistent

slow convergence with excitations→ many configurations must be taken into account

in order to approximate exact energy

Full CI + infinite basis set = exact solution of Schrödinger equation⇒ Complete-CI

method scales ∼ exponentiallly→ suitable only for very small systems


Wavefunction-based methods Coupled-Cluster (CC)

Coupled-Cluster Method

again, replace electronic wave functions in HF Slater determinant byvirtual orbitals, so-called excitations

|ΨCC>= eT|ΨHF>= eT1+T2+T3+...|ΨHF>

T1 =∑i,a

tai a+

a ai, T2 = 14

∑i,j,a,b

tabij a+

a a+b aiaj, T3 = . . .

i, j … occupied (internal) HF orbitalsa, b … canonical unoccupied (external; virtual) orbitalst...... … singles and doubles wavefunction amplitudes (⇒ CCSD), to be determineda+

..., a... … creation and destruction operators

total energy E =<ΨCC|HBO|ΨCC>=<ΨHF|e−THBOeT|ΨHF>

= EHF + Ecorr =<ΨHF|HBO|ΨHF>+ . . .



do the math…

singles and doubles residuals determine wavefunction amplitudes t......

Rai =<Ψa

i |e−T H eT|ΨHF>= 0

Rabij =<Ψab

ij |e−T H eT|ΨHF>= 0

|Ψai >= a+

a ai |ΨHF>, |Ψabij >= a+

a a+b aiaj |ΨHF>

▶ set of non-linear equations▶ iterative solution gives wavefunction amplitudes ta

i , tabij

currently being implemented in FHI-aimsMay 11th, 2016 - اصفهانصنعتیدانشگاه 24/48


Configuration Interaction (CI) vs Coupled-cluster (CC)

Configuration Interaction (CI): |ΨCI>= (1 + T1 + T2 + . . .)|ΨHF>

Coupled-cluster (CC): |ΨCC>= eT1+T2+...|ΨHF>

example: consider only “doubles”CI: |ΨCID>= (1 + T2)|ΨHF>

CC: |ΨCCD>= eT2 |ΨHF>=(1 + T2 + T2

22! + . . .

)|ΨHF>

T2 … “doubles” excitation operator (one electron pair)T 2

2 ⇒ simultaneous “doubles” excitationof two non-interacting electron pairs

⇒ CI is not size consistent⇒ CC is size consistent



CCSD gives results often above chemical accuracy (∼ 1 kcal/mol) /CCSDT scales with O(N8) /

▶ instead CCSD(T), i.e. perturbative triples correction, based onwavefunction amplitudes obtained from CCSD; scales O(N7)

▶ CCSD(T) often very accurate▶ CCSD(T) = “gold standard of quantum chemistry”

sufficiently large basis set required→ still restricted to rather small systems


Wavefunction-based methods DLPNO-CCSD(T)

DLPNO-CCSD(T)

how to improve computational costs of CCSD(T)?→ C. Riplinger, F. Neese, J. Chem. Phys., 138, 034106 (2013).

localization → truncation method and/or tail estimationEcorr =

∑ individual pair correlation energies


“strong pair approximation”


external space (virtual orbitals) typically one order of magnitude largerthan internal space (occupied orbitals) ⇒ bottleneck

▶ truncation scheme, based on the occupation numbervirtual orbital representation is not fixed→ transform between different virtual orbital sets, e.g.

tai =

∑r

Qarii tr

i , tabij =

∑r,s

Qarij trs

ij Qbsij

i, j … occupied (internal) HF orbitalsa, b … canonical virtual (external) orbitalsr, s … virtual (external) orbitals w.r.t. new virtual basis

domain approximation:restrict the virtual space to a subset (domain) of the orbitals, ∀ ij

tai =

∑r∈[i]

Qarii tr

i , tabij =

∑r,s∈[i,j]

Qarij trs

ij Qbsij



new virtual basis of orbitals?

canonical Molecular Orbitals (MOs)?no, too “chaotic” space, not local

Projected Atomic Orbitals (PAOs)?yes, but PAOs are maximally localized

→ accurate description of external space?Pair Natural Orbitals (PNOs)!

Yes!constructed from so-called virtual pair densitylocalized in same region of space as internal electron pair, but alsocertain amount of delocalizationexcitations are only allowed into respective local domains

L(ocal)PNO approch = “strong pair approximation” + PNOs



LPNO method expands PNOs in terms of virtual MOs⇒ PNOs are local, but expansion is not⇒ scales with O(N5)

instead expand PNOs in terms of PAOs⇒ D(omain based)LPNO-CCSD(T)⇒ scales almost linearly


implemented in ORCA

Wavefunction-based methods 2nd order Møller-Plesset (MP2)

2nd order Møller-Plesset perturbation theory (MP2)

perturbative treatment: H = H0 + λVH0 … unperturbated Hamiltonian which solution is known→ here: Hartree-FockλV … small perturbation → here: correlation potential

EMP2 =∑

i,j,a,b <ϕi(1)ϕj(2)| 1r12

|ϕa(1)ϕb(2)>

×2<ϕa(1)ϕb(2)| 1

r12|ϕi(1)ϕj(2)>−<ϕa(1)ϕb(2)| 1

r12|ϕj(1)ϕi(2)>

ϵi+ϵj−ϵa−ϵb

ϕi,j … occupied orbitalsϕa,b … unoccupied (virtual) orbitalsϵi,j,a,b … corresponding orbital energies

→ problematic for systems with small HOMO-LUMO gapscales with O(N5)

implemented in FHI-aimsMay 11th, 2016 - اصفهانصنعتیدانشگاه 31/48

Wavefunction-based methods Higher order Møller-Plesset

Higher order Møller-Plesset perturbation theory

performance of MP3 very often worse than MP2obviously computational costs increase with higher order

Møller-Plesset theory diverges for higher nOlsen et al.; J. Chem. Phys., 2000, 112, pp 9736-9748


Practical considerations Counterpoise correction

IV. Practical considerations - Counterpoise correction

in weakly bound clusters→ artificial strengthening of the intermolecular interaction→ basis set superposition error (BSSE)

monomer A approaches monomer B→ monomer A “borrows” basis function from monomer B, and vice versa

→ dimer artificially stabilizedproblem: inconsistent treatment at short (“extra” basis functionsavailable) and long (“extra” basis functions not available) distancescomplete basis-set limit: BSSE → 0

⇒ always apply counterpoise correction (CP)S. F. Boys, F. Bernardi; Mol. Phys., 1970, 19, 553-566


Practical considerations Counterpoise correction

uncorrected interaction energy ∆Eint(AB):∆Eint(AB) = EAB(AB) − EA(A) − EB(B)

subscripts denote basisparentheses denote system(here: assuming rigid conformers; often reasonable approximation)

estimation of basis set superposition error (BSSE):EBSSE(A) = EAB(A) − EA(A)EBSSE(B) = EAB(B) − EA(B)EAB(A) < EA(A) ⇒ EBSSE(A) < 0 (error is stabilizing)EAB(B) < EB(B) ⇒ EBSSE(B) < 0 (error is stabilizing)e.g. calculate EAB(A)→ consider only basis of monomer B, not atom (electrons and nuclear

charges) → “ghost atoms”

⇒ ∆ECPint (AB) = EAB(AB) − EAB(A) − EAB(B)


Practical considerations Extrapolation schemes and basis sets

Extrapolation schemes and basis sets

problem for wave-function based methods: require large basis setslow convergence with basis set sizebut computational costs are high

→ use extrapolation schemecalculate for smaller basis sets → extrapolate to basis-set completeness



systematic basis sets required→ designed to converge systematically to the complete-basis-set

(CBS) limit using empirical extrapolation techniques

in contrast to e.g. DFTmoderately sized basis sets enough to reach complete basis set limit

e.g. numeric atom-centered basis functions in FHI-aimsminimal, tier1, tier2, tier3

e.g. Pople basis sets6-31G*, 6-311G*, …widely used (HF, DFT, …)mostly used for light elementsnot too systematic



systematic basis sets requirede.g. correlation-consistent basis sets (Dunning et al.)(J. Chem. Phys, 1989, 90 (2), pp 1007–1023)

denoted cc-pVNZ (correlation-consistent polarized valence N zeta)cc-pVDZ: double-zetacc-pVTZ: triple-zetacc-pVQZ: quadruple-zetacc-pV5Z: quintuple-zetacc-pCVNZ: including core-polarization(e.g. for cations where semi-core states treatment is important)aug-cc-pVNZ: augmented versions with added diffuse functions

systematic, popular, performance well knownuse for MP2, coupled-cluster; not for HF, DFT



e.g. Ahlrichs/Karlsruhe basis sets(Phys. Chem. Chem. Phys., 2005, 7, pp 3297-3305)

def2-SVPdef2-TZV: valence triple-zetadef2-TZVP: valence triple-zeta plus polarizationdef2-TZVPP: valence triple-zeta plus polarization (doubly polarized)def2-QZVPP: valence quadruple-zeta plus polarization (doublypolarized)…

also efficient for HF, DFTuse doubly polarized versions (PP) for MP2, coupled-cluster, …

in FHI-aims: NAO-VCC-nZ(Numeric Atom-centered Orbitals -

Valence Correlation Consistent n-Zeta)(New J. Phys., 2013, 15, 123033)

use for MP2, RPA, GW, …constructed following Dunning’s“correlation-consistent” recipe


Igor Ying Zhang


Extrapolation schemes

simple two-point extrapolation scheme by Halkier et al.(Chem. Phys. Lett., 1998, 286, 243)

ECBS = E[n1]n31 − E[n2]n3

2n3

1 − n32

n1, n2 … basis set cardinal numbersn = 3: triple-zetan = 4: quadruple-zeta…

extrapolation scheme used for correlation energy,sometimes also used for total energy



more precise: different basis set behavior forcorrelation energy Ecorr and SCF energy ESCF

→ use different extrapolation schemes for Ecorr and ESCF, e.g.:

SCF extrapolation scheme by Kanton and Martin(Theor. Chem. Acc., 2006, 115, pp 330–333)

EnSCF = ECBS

SCF + Ae−α√

n

A, α, CBS-extrapolated energy ECBSSCF

→ 3 parameters to be determined from least-square fitting→ need at least 3 different basis set calculations, e.g. n = T/Q/5

correlation energy extrapolation scheme by Truhlar(Chem. Phys. Lett., 1998, 294, pp 45–48)

Encorr = ECBS

corr + Bn−β

B, β, CBS-extrapolated energy ECBScorr

→ 3 parameters to be determined from least-square fitting→ need at least 3 different basis set calculations, e.g. n = T/Q/5


Benchmark datasets

V. Benchmark datasets

purpose: provide highly accurate QM calculations of molecularstructures, energies, and properties

used as benchmarks for testingused for parameterization of computational methods

collection of datasets: database→ e.g. Benchmark Energy and Geometry DataBase (BEGDB)

(Řezáč et al. Collect. Czech. Chem. Commun., 2008, 73,pp 1261-1270)

http://www.begdb.com/

→ e.g. NIST Computational Chemistry Comparison and BenchmarkDataBase (CCCBDB)http://cccbdb.nist.gov/


http://www.begdb.com/

http://cccbdb.nist.gov/

Benchmark datasets

S22: Noncovalent Complexes

(Jurecka et al.; Phys. Chem. Chem. Phys., 2006, 8, pp 1985-1993)widely popularset of 22 small (< 30 atoms) complexes containing only C, N, O and H,and single, double and triple bondsmix of hydrogen-bonded and dispersion-bonded complexestypical noncovalent interactions representedgeometry relaxation using counterpoise-corrected gradient optimizationseveral methods appliedsmaller complexes optimized with CCSD(T)using cc-pVTZ/cc-pVQZ without counterpoise correction


Benchmark datasets

S66: A Well-balanced Database of Benchmark InteractionEnergies Relevant to Biomolecular Structure

(Řezáč et al.; J. Chem. Theory Comput., 2011, 7 (8), pp 2427-2438)

66 noncovalent complexes23 hydrogen-bond dominated complexes23 dispersion-dominated complexes20 complexes with mixed electrostatic/dispersion interaction

MP2/CBS calculations in aug-cc-pVTZ/aug-cc-pVQZ+ CCSD(T) correction in aug-cc-pVDZ


Benchmark datasets

X40: Noncovalent Interactions of Halogenated Molecules

(Řezáč et al.; J. Chem. Theory Comput., 2012, 8 (11), pp 4285-4292)

40 noncovalent complexes of organic halides, halohydrides,and halogen moleculesvariety of interaction types

composite CCSD(T)/CBS scheme appliedtriple-zeta basis sets on all atoms but hydrogen


Benchmark datasets

L7: Large Noncovalent Complexes

(Sedlak et al.; J. Chem. Theory Comput., 2013, 9 (8), pp 3364-3374)

seven large complexes (number of atoms: 48-112)

MP2/CBS binding energies + ∆QCISD(T)/6-31G*(0.25) correctionMP2/CBS binding energies + ∆QCISD(T)/aug-cc-pVDZ correctionMP2/CBS binding energies + ∆CCSD(T)/6-31G**(0.25,0.15)


Benchmark datasets

and many more…


Summary

VI. Summary

wave-function based methods often very accuratebut computationally expensive (large basis set required)

CI, CCSD(T), DLPNO-CCSD(T), MP2, …

numerically challengingcounterpoise correction must be applied!almost always: extrapolation scheme→ use appropriate basis sets (cc-…, def2-…)

verify your method used for production Bbut: don’t get lost becoming too accurate


Summary

VI. Summary

wave-function based methods often very accuratebut computationally expensive (large basis set required)

CI, CCSD(T), DLPNO-CCSD(T), MP2, …

numerically challengingcounterpoise correction must be applied!almost always: extrapolation scheme→ use appropriate basis sets (cc-…, def2-…)

verify your method used for production Bbut: don’t get lost becoming too accurate

ممنونخیلیMay 11th, 2016 - اصفهانصنعتیدانشگاه 48/48

High-level Quantum Chemistry Methods and Benchmark ... · High-level Quantum Chemistry Methods and...

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