A quantum Monte Carlo study of noncovalent interactions

Kenta Hongo

Quantum Monte Carlo in the Apuan Alps VII @ TTI, Vallico Sotto, Tuscany, Italy, 08/03/2012

Assistant Professor,School of Information Science, JAIST

(Prof. Ryo Maezono’s group)

in collaboration with: A. Aspuru-Guzik, M.A. Watson, R.S. Sánchez-Carrera, T. Iitaka (for molecular crystals) R. Maezono, Nguyen Thanh Cuong (for DNA stacking)

Outline

• Motivation of our studies

– Noncovalent interactions

• DMC and DFT studies of

– Polymorphism in para-diiodobenzene (DIB)

– Base-pair-step stacking in B-DNA (AT-AT)

Noncovalent interactions:Challenges for theoretical methods

Noncovalent systems are good challenges to theoretical methods.

Week interactions between complicated molecules

containing hundreds number of electrons…

accuracy:

computational cost:

The energy scale of the interaction is quite small (order of 1 mHa)

Noncovalent bond: hydrogen / dispersion-dominated complexes/mixture of them

Accurate quantum chemistry methods (eg. CCSD(T)) are not routinely applicable

On the other hand, commonly-used DFTs mostly fail to describe noncovalent interactions.

A few QMC studies have been performed so far

eg. For the S22 benchmark: Martin Korth, et. al., J. Phys. Chem. A, 112, 2104, (2008).

Molecular crystals

Many applications to…pharmaceutical materials, semiconductors, superconductors, etc.

DNTTR. S. Sánchez-Carrera, et al,

J. Phys. Chem. C 114, 2334 (2010).

AspirinP. Veshweshwar, et al,

J.Am. Chem. Soc. 127, 16802 (2005)

PiceneR. Mitsuhashi, et al,Nature 464, 76 (2010).

Polymorphism in molecular crystals

• The existence of more than one form of a compound– Each polymorph has different physical and chemical properties

Fluorinated 5,11-bis(triethylsilylethynyl)antradihiophene(diF TES ADT)

O. D. Jurchescu, et al, Phys. Rev. B 80, 085201 (2009).

e.g. effects of polymorphism on charge transport in organic semiconductors

para-diiodobenzene (p-DIB)

Room-temperature hole mobility = 12 cm2/(V s)

π-conjugated organic semiconductors as (opt)electronic devices

-Typical room-temperature charge-carrier mobility ≈ 1 cm2/(V s)

L. M. Schwartz and J. F. Horning, Mol. Cryst. 2, 379 (1967).R. S. Sánchez-Carrera, et al. Chem. Mater. 20, 5832 (2008).

para-diiodobenzene:

Polymorphism in p-DIB

known to exhibit polymorphism (from experiment):

α-phase (Pbca) β-phase (Pccn)

Transition from α- to β-phase occurs at 326 K(The α-phase is more stable than the β-phase at zero temperature)

Previous study of DIB • A. Brillante, et al., JACS 127, 3038 (2005)

DFT calculation: Exchange-correlation functional = BLYP functionalPseudopotential = Troullier-Martins

The α-phase is less stable by ≈0.002 eV/atom than the β-phase at 0 K. Contrary to experimental results!

Good challenge for DMC!

Collaborators & HARIKEN project

The first application of QMC to molecular crystal systems:K. Hongo, M. A. Watson, R. S. Sánchez-Carrera, T. Iitaka, A. Aspruru-Guzik,

J. Phys. Chem. Lett. 1, 1789 (2010).

Prof. Alán Aspuru-Guzik(AG group)

Dr. Mark A. Watson(Princeton University)

Dr. Roel S. Sánchez-Carrera(Former member)

Dr. Toshiaki Iitaka(RIKEN in Japan)

+

Harvard RIKEN (hurricane)

=

Computational details: QMC• DMC calc. using QMCPACK

• Pseudo potential = Trail-Needs type (CASINO library)

– The total number of electrons = 168

• Trial wavefunction = Slater-Jastrow type

– Slater determinant: LDA (cut off energy = 70 Ry)

– Jastrow factor: one- & two-body terms (6 & 4 parameters)

• 1x1x1 simulation cell with a finite size correction

– KZK scheme (Kwee, Zhang, Krakauer, PRL, 100, 126404, 2008)

• computational conditions of DMC

– # of walkers = 16384, # of MC steps = 3.2×107

Relative stability between the two phases

ΔE = E(α) – E(β) (<0: Experiment)

LDA

PW91 PBE B3LYP

-123

29 25

68

-48+/-24

B3LYP+D

-155

DMC

• The DMC result is consistent with experiment, i.e., a negative ΔE.

• GGAs and B3LYP predict a positive ΔE.• LDA and B3LYP+D correctly predict the relative

stability, but strongly overestimate the magnitude of ΔE, compared to DMC.– It is well-known that LDA frequently overbinds.– The Grimme semiempirical dispersion correction is

important.

Biological systems

• Noncovalent bonding: Hydrogen bond, dispersion, and their mixture…

Adenine (A) Guanine (G) Cytosine (C) Thymine (T)

B-DNA

Present study: Base-pair-step stacking in

B-DNAAdenine

Adenine

Thymine

Thymine

d: distance between 2 layers

We applied DMC and DFT’s to this system…

Computational details• DMC calculations using CASINO

• Burkatzki-Fillippi-Dolg PPs with associated basis sets

– VDZ and VTZ basis sets

• Trial wavefunction = Slater-Jastrow type

– LDA, PBE, B3LYP, HF trial nodes (generated by Gaussian 09)

– Jastrow factor: one-, two-, and three-body terms (up to 8the poly.)

• computational conditions of DMC

– # of walkers = 1280, # of MC steps = 105

• DFT calculations using Gaussian 09 – recently developed various functionals examined – Hybrid, DFT+D, LC-DFT…

DMC potential energy curves

Potential energy curves

DFT and QC potential energy curves

DMC potential energy curves

Basis set dependence

Stacking energy at a reference geometry

reference energy = CCSD(T)/CBS (Sponer et. al., Chem. Euro. J. 12, 2854, (2006)

Summary

• We applied DMC and DFT to the polymorphism of DIB.

The DMC result is consistent with the experiment.

The DFT results are not conclusive: The PBE, PW91, and B3LYP fail to give the correct sign of ΔE, while the LDA and B3LYP+D do, but strongly overestimate |ΔE|.

• The base-pair-step stacking interaction in B-DNA (AT-AT stacking) were studied using DMC, HF, MP2, and DFT.

The DMC result is in good agreement with the reference result.

Commonly-used DFT methods fail to describe the stacking, but some new functionals show good performances.

Thank you !

Electronic structure calculations

Molecular orbital (MO) methods (HF, CI, CC, etc.)

Ab-initio calculations Density functional theory (DFT)LDA, PW91, BLYP, B3LYP, etcSolid state

physics; a variational problem of minimizing the energy with respect to the electron density

The Schrodinger equation for many-electron systems

Quantum chemistry;

a variational problem of minimizing the energy with respect to the many-electron wavefunction

Quantum Monte Carlo (QMC)methods

Quantum Monte Carlo methods

Variational Monte Carlo (VMC);

; a choice of trial wavefunction

Accuracy of results strongly depends on the trial wavefuntion adopted.

physical quantity : an expectation value of an operator

Monte Carlo sampling according to 3N dimensional distribution

Diffusion Monte Carlo (DMC); ; an imaginary time-dependent trial

wavefunction that converges to the exact wavefunction after a long enough interval

Purification of a trial wavefunction Iterative operation of the imaginary-time evolution operator

Present study• DIB is a challenging benchmark system for ab initio methods.

– Reliable reference calculations are necessary– But for periodic systems....

highly accurate quantum chemistry methods are not applicable due to their computational cost.

• Diffusion Monte Carlo (DMC) is one of the most promising candidates for treating such systems from the viewpoint of accuracy and computational cost.

• Purpose of our study– To investigate the relative stability of two polymorphs of DIB using DMC and several DFT

methods.

–LDA(SVWN), PW91, PBE, B3LYP–B3LYP+D: Semiempirical dispersion correction to DFT

Stacking energy at a reference geometry

Stac

king

ene

rgy

(kca

l/m

ol)

Example: H2 molecule

VMC-1.17203(1)

CISD (=Full CI)-1.1723

-1.17447 ExactJ. Chem. Phys. 49, 404 (1965).

< Ground state Energy of H2 > (6-311++G** were used in all the calculations)

-1.1330HF

Conventional methods

Trial wfn.

(excited state)

Trial wfn.

= (Ground state)

= (Ground state)

(excited state)

Signal(=target) Noise

DMC-1.17447(4)

The present method

Current topics in electronic structure calculations

A variety of materials:e.g. molecular crystals, etc.

Computational aspects:Accurate quantum chemical calculations can NOT be applied toSOLIDS!Although Full CI and QMC give results of almost the same accuracy, Full CI scales as O(N!), while QMC scales as O(N3 )-O(N4)

Conventional exchange-correlation potentials (LDA, GGA, Hybrid…)

sometimes give different predictions…

universally reliable reference calculations

QMC algorithm is intrinsically parallel

Suitable for recent massive parallel computers

The basic idea of DMC

Trial wfn.

= (Ground state)

(excited state)

Trial wfn.

(excited state)

= (Ground state)

Signal(=target) Noise

VMC energy

Imaginary time steps

DMC energy

Example taken from NiO calculation.

Total Energy (a.u.)

“Purification” of the trial wavefuntion

Computational details: QMC

• DMC calc. using QMCPACK • Pseudo potential = Trail-Needs type

– Taken from CASINO pseudo potential library

– The total number of electrons = 168

• Trial wavefunction = Slater-Jastrow type– Slater determinant: LDA (cut off energy = 70 Ry)

– Jastrow factor: one- and two-body terms

• 6 parameters for the one-body and 4 parameters for the two-body

• Optimized by minimizing the variance of local energy in VMC

• computational conditions of DMC – # of walkers = 16384, # of MC steps = 3.2×107

Computational details: DFT

• All-electron DFT calculations using the Crystal 09 code– The total number of (core) electrons = 584 (416)

• Basis set: – 6-31 G for the hydrogen and carbon atoms– 3-21 G for the iodine atom

• K-point sampling = 1x3x4 (1-2 meV accuracy) • Exchange-correlation functionals

– LDA(SVWN), PW91, PBE, B3LYP– B3LYP+D: Semiempirical dispersion correction to DFT

Grimme Scheme: J. Comput. Chem. 27, 1787 (2006).

• Vibration frequencies within the harmonic approximation

J. Comput. Chem. 25, 888, 1873 (2004).

DFT geometry optimizations:

Lattice constants and volumes

• GGAs incorrectly predict that the α phase has a larger volume than the β one.

• LDA and B3LYP+D give the correct ordering and the most compact structures.

• B3LYP strongly overestimates the cell volume for both phases.

• The deviation from experiment is more than 10% (not satisfactory).

　 　 α-phase 　 　 　 　 β-phase 　 　　 　 a b c V %ΔV 　 a b c V %ΔVSVWN 16.289 6.708 5.956 651 -15 16.387 6.783 5.966 663 -15 PW91 18.721 8.993 5.473 921 20 17.506 8.175 6.091 872 11 PBE 18.748 8.589 5.555 894 16 18.498 8.116 5.815 873 11 B3LYP 18.620 9.733 5.756 1043 36 18.502 9.948 5.831 1073 37 B3LYP+D 16.610 6.731 6.046 676 -12 16.966 6.791 5.998 691 -12 Experiment 　 17.000 7.323 6.168 768 0 　 17.092 7.461 6.154 785 0

Length and volume are in units of Å and Å3, respectively.

Relative energies using the optimized

geometries

• All five functionals predict the α phase to be more stable than the β phase at the optimized geometries.

LDA PW91 PBE B3LYP B3LYP+D

-128(-123)

-165 (29)

-12(25)

-16(68)

-163(-155)Numbers in the parenthesis are ΔE using the experimental geometries

Importance of vibration analysis

in polymorphism of molecular crystals

α-glycine γ-glycine

Lattice energydifference= 1.28 kJ/mol

ΔHexp= 0.27

Vibrational ΔH= 1.01 kJ/mol

Figure: Diagram showing that a correct computation of the lattice energy difference for α- and γ-glycine must have γ more stable than α-glycine by ca. 1.28 kJ/mol.

glycine case: S. A. Rvera, et al. Cryst. Growth Des. 8, 3905 (2008).

The zero-point energy (ZPE) is of the same order of magnitude as the lattice energy difference between the polymorphs. In some cases, the ZPE can influence the relative stability.

Phonon calculations at the Γ point:

Zero-point energy　 ΔEL ΔE0

SVWN -73 9

B3LYP -6 4

B3LYP+D -153 -1

Vibration frequencies calculated within the harmonic approximationusing the Crystal 09 code

All energies are in units of meV.

EL: Electronic energy; E0: Zero-point energy; ΔE := E (α) – E(β)

Refs.: J. Comput. Chem. 25, 888 (2004); 25, 1873 (2004)

Phonon calculations at the Γ point:

Temperature dependence

G = U + PV – TS; U = EL + E0 + ETG: Gibbs free energy; P: Pressure; V: Volume; T: Temperature; S: Entropy; U: Internal energy; EL: Electronic energy; E0: Zero-point energy; ET: Thermal contribution to the vibration energy

P = 0.101325 MPa (Experimental transition temperature: Tpt = 326 K)

[meV]

[Kelvin]

= G(α)-

G(β)

Tpt=471 KTpt=78 K

Tpt=1005 K

Finite-size (FS) effects in QMC

• QMC calculations for periodic systems:– Super cell calculations and their extrapolation to infinite size

– Twist-averaged boundary conditions (for metallic systems)

– Kwee, Zhang, and Krakauer scheme

QMC FS corrections are estimated from modified LDA calculations with finite-size functionals. Phys. Rev. Lett. 100, 126404 (2008).

Phys. Rev. E 64, 016702 (2001).

Phys. Rev. B 53, 1814 (1996).

EFS(α) EFS(β) ΔEFS =EFS(α)-EFS(β)

6206 6156 50

All energies are in units of meV.

The present DMC FS correction = 50 meV

Basis set dependence in DFT energiesin progress…

• Commonly used basis sets in CRYSTAL09: Standard Gaussian (Pople) basis sets

– STO-nG n=2-6 (H-Xe), 3-21G (H-Xe), 6-21G (H-Ar)

– plus polarization and diffuse function extensions

• Standard basis sets are optimized for molecular systems.

– Are they appropriate for molecular crystal systems?

– A number of ab initio studies on molecular crystals have employed standard basis sets.

eg. α-quartz: 6-21G* and 6-31G* for Si and O, respectively [J. Comput. Chem. 25, 888 (2004)]

– No investigation of the basis set dependence because of its computational cost.

• The quality of basis sets significantly affects the final results in some cases.

– It is crucial to investigate the basis set dependence.

Basis sets to be considered:

• Standard Gaussian basis sets– 6-31G & 3-21G (# of AOs = 664)– 6-311G (# of AOs = 816)

• MDT: (# of AOs = 784)– from Dr. Mike D. Towler’s web site:

http://www.tcm.phy.cam.ac.uk/~mdt26/crystal.html– H: (7s1p)/[3s1p], C: (9s3p1d)/[3s2p1d], I: (29s20p10d)/[8s7p3d] (# of AOs =

784)

– Those exponents were optimized with respect to the HF total energy of a typical solid.

• MDT+d: (# of AOs = 824)– add an extra diffuse function to the I atom – The exponent in the d function is optimized with respect to the DFT total

energy.

• opt-MDT: – optimize the exponent in the outermost function centered on each atom, i.e., a

p function with the smallest exponent for H, d functions with the smallest exponents for C and I.

• opt-MDT+d:– optimize exponents in the extra d functions and the outermost functions only.

Energetics at the experimental geometry:

The SVWN total energy of the α phase

• The MDT group gives a better energy than standard Gaussian basis sets.• Optimization of exponents with respect to a solid total energy is very

important; Standard sets are optimal only for molecules, NOT SOLIDS.

243.92

8.33

0.450.31 0.31 0.28

Energetics at the experimental geometry:

The B3LYP total energy of the α phase

• Results similar to SVWN were obtained except for 6-311G, i.e., the B3LYP energy with 6-311G did not converge.

• The exponent optimization also plays an important role in the SCF convergence.

243.99

NA

0.360.24 0.23 0.21

6-311G

Energetics at the experimental geometries:

relative stability ΔE = E(α) – E(β)

• The energy difference depends on the basis set.• The MDT group lowers the energy difference, compared to 6-311G.• An extra d function on the I atom lowers the energy difference. • For some cases in PW91 and B3LYP, the sign of the energy

difference turns to be negative.

　 SVWN PW91 B3LYP

6-31G + 3-21G -122 93 68

6-311G -132 88 NA

MDT -123 25 50

MDT+d -232 -66 -29

opt-MDT -166 20 54

opt-MDT+d -168 -6 14

All energies are in units of meV.

What is a criterion for the “optimal” basis set? NOT CLEAR TO ME….

Summary and future work• We investigated the relative stability of two polymorphs of DIB using the DMC and DFT methods.

• The DMC result correctly predicts that the α phase is more stable than the β phase at zero temperature.

• The DFT results were inconsistent and inconclusive.– A proper treatment of electron correlation is important.

• More systematic study of basis set dependence• Pseudopotential DFT calculations• Phonon dispersion calculations

Polymorphism in molecular crystals

• Computational studies of polymorphism in molecular crystals– mostly relies on molecular mechanics (MM) because of computational costs

– ab-initio studies are also useful for verifying the reliability of MM

• Polymorphism: the existence of more than one form of a compound– Each polymorph has different physical and chemical properties

– Important in, e.g., pharmaceutical industry and so on.

- at zero temperature: ground state energy calculations by DFT - at finite temperature: Car-Parrinello MD simulation based on DFT

For example: (JACS 127, 3038 (2005) )

Phonon calculations, etc. with empirical potentialse.g., S. L. Price, Adv. Drug Delivery Rev. 56, 301 (2004),

Problems in ab-initio studies of Polymorphism in

molecular crystals

• Accuracy of theoretical methods:– The energy difference between polymorphs is quite small (- order of 1 mH)

– Noncovalent bond: hydrogen bond/ dispersion-dominated complexes

• Computational costs:– Unit cell contains a number of complicated molecules

– Periodicity of molecular crystal structuresAccurate quantum chemistry methods such as CI and CC are NOT applicable to periodic systems… Mostly, DFT

Many DFT exchange-correlation functionals fail…

Electron correlation effects play an important role in describing such chemical bondings!

DMC is a good benchmark for molecular crystals