A quantum Monte Carlo study of
noncovalent interactions
Kenta Hongo
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)
Quantum Monte Carlo in the Apuan Alps VII @ TTI, Vallico Sotto, Tuscany, Italy, 08/03/2012
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
accuracy:
The energy scale of the interaction is quite small (order of 1 mHa)
Noncovalent bond: hydrogen / dispersion-dominated complexes/mixture of them
computational cost:
Week interactions between complicated molecules
containing hundreds number of electrons…
 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).
Noncovalent systems are good challenges to theoretical methods.
Molecular crystals
Many applications to…
pharmaceutical materials, semiconductors, superconductors, etc.
Aspirin
DNTT
Picene
P. Veshweshwar, et al,
J.Am. Chem. Soc. 127, 16802 (2005)
R. S. Sánchez-Carrera, et al,
J. Phys. Chem. C 114, 2334 (2010).
R. 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
e.g. effects of polymorphism on charge transport in organic semiconductors
Fluorinated 5,11-bis(triethylsilylethynyl)antradihiophene
(diF TES ADT)
O. D. Jurchescu, et al, Phys. Rev. B 80, 085201 (2009).
para-diiodobenzene (p-DIB)
π-conjugated organic semiconductors as (opt)electronic devices
-Typical room-temperature charge-carrier mobility ≈ 1 cm2/(V s)
para-diiodobenzene:
Room-temperature hole mobility = 12 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).
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 functional
Pseudopotential = 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
Prof. Alán Aspuru-Guzik
(AG group)
Dr. Mark A. Watson Dr. Roel S. Sánchez-Carrera
(Princeton University)
(Former member)
=
+
Harvard
Dr. Toshiaki Iitaka
(RIKEN in Japan)
RIKEN
(hurricane)
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).
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
100
ΔE (meV/unit cell)
68
29
0
25
B3LYP+D
LDA
PW91
PBE
DMC
ΔE = E(α) – E(β)
(<0: Experiment)
B3LYP
-48+/-24
-100
-123
-155
-200
• 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
B-DNA
Adenine (A)
Guanine (G)
Cytosine (C)
Thymine (T)
• Noncovalent bonding: Hydrogen bond, dispersion, and their mixture…
Present study:
Base-pair-step stacking in B-DNA
Adenine
Thymine
d: distance between 2 layers
Adenine
Thymine
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
-330.50
DMC/LDA
DMC/B3LYP
-330.52
DMC/PBE
DMC/HF
-330.54
-330.56
-330.58
-330.60
-330.62
2.5
3.5
4.5
5.5
6.5
7.5
Potential energy curves
DNA ATAT stacking/LDA&DMC
DNA ATAT stacking/PBE&DMC
15
DMC(TZ)
PBE(TZ)
30
10
Energy difference (kcal/mol)
Energy difference (kcal/mol)
DMC
LDA
5
0
-5
-10
20
10
0
-10
-20
-15
2
3
4
5
6
7
8
9
10
2
11
3
4
5
6
7
8
9
Distance (angstrom)
Distance (angstrom)
DNA ATAT stacking/B3LYP&DMC
DNA ATAT stacking/HF&DMC
10
11
50
DMC(TZ)
B3LYP(TZ)
Energy difference (kcal/mol)
Energy difference (kcal/mol)
40
30
20
10
0
-10
-20
2
3
4
5
6
7
Distance (angstrom)
8
9
DMC(TZ)
HF(TZ)
30
10
11
20
10
0
-10
-20
2
3
4
5
6
7
Distance (angstrom)
8
9
10
11
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
The Schrodinger equation for many-electron systems
Ab-initio calculations
Solid state physics;
Density functional theory (DFT)
LDA, PW91, BLYP, B3LYP, etc
dE [n( r ) ]
=0
dn
Quantum chemistry;
a variational problem of minimizing the
energy with respect to the electron density
dE [ Y( r1 , ,rN ) ]
=0
dY
a variational problem of minimizing the energy
with respect to the many-electron wavefunction
Molecular orbital (MO) methods (HF, CI, CC, etc.)
Quantum Monte Carlo (QMC)methods
ò dR × F ( R) × Oˆ F(R)
ò dR × F (R)F(R)
*
Oˆ =
*
ò dR × F
=
2
[ ()
( )] =
× F -1 R × Oˆ F R
ò dR × F
2
( )
( )
F -1 R × Oˆ F R
Quantum Monte Carlo methods
physical quantity : an expectation value of an operator
Oˆ =
( ) ( )
ò dR × F (R)F(R)
*
d
R
×
F
R × Oˆ F R
ò
*
=
[ ()
( )] =
2
d
R
×
F
× F -1 R × Oˆ F R
ò
ò dR × F
2
( )
( )
F -1 R × Oˆ F R
Monte Carlo sampling according to 3N dimensional distribution
Variational Monte Carlo (VMC);
( )
( )
F R = YT R
F
; a choice of trial wavefunction
Accuracy of results strongly depends on the trial wavefuntion adopted.
Diffusion Monte Carlo (DMC);
( )
F( R,0) = Y ( R)
F = F R,t
; an imaginary time-dependent trial wavefunction that converges to
the exact wavefunction after a long enough interval 
T
F = exp[ -tH ] × Yinit
Iterative operation of the imaginary-time evolution operator
Purification of a trial wavefunction
2
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 (kcal/mol)
-5
-15
-20
-25
PBE0
B3LYP
TPSS
BLYP
PW91
PBE
reference
DMC
MP2
ωB97X-D
HF
ωB97
B97-D
LC-wPBE
CAM-B3LYP
M06-2X
M06
M05-2X
M05
B97-1
mPW1PW91
-10
SVWN
Stacking energy at a reference geometry
15
10
5
0
Example: H2 molecule
< Ground state Energy of H2 > (6-311++G** were used in all the calculations)
-1.1330
HF
Conventional
methods
-1.17203(1)
VMC
-1.1723
CISD (=Full CI)
The present method
-1.17447(4)
-1.17447
DMC
Exact
J. Chem. Phys. 49, 404 (1965).
Trial wfn. (t ) = (Ground state) ×exp(-iE0t ) +å(excited state) ×exp(-iE t )
n
Signal(=target)
n
Noise
Trial wfn. (t ) = (Ground state) ×exp[-( E0 - ET )t ] +å (excited state) ×exp[-( En - ET )t ]
n
Current topics in electronic structure calculations
A variety of materials:
e.g. molecular crystals, etc.
Conventional exchange-correlation potentials (LDA, GGA, Hybrid…)
sometimes give different predictions…
universally reliable reference calculations
Computational aspects:
Accurate quantum chemical calculations can NOT be applied to SOLIDS!
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)
QMC algorithm is intrinsically parallel
Suitable for recent massive parallel computers
The basic idea of DMC
Trial wfn. (t ) = (Ground state) ×exp(-iE0t ) +å(excited state) ×exp(-iE t )
n
Signal(=target)
n
Noise
Trial wfn. (t ) = (Ground state) ×exp[-( E0 - ET )t ] +å (excited state) ×exp[-( En - ET )t ]
n
“Purification” of the trial wavefuntion
Total Energy (a.u.)
VMC energy
Example taken from NiO calculation.
DMC energy
Imaginary time steps
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
SVWN
PW91
PBE
B3LYP
B3LYP+D
Experiment
a
16.289
18.721
18.748
18.620
16.610
17.000
α-phase
b
c
V
%ΔV
6.708
5.956 651
-15
8.993
5.473 921
20
8.589
5.555 894
16
9.733
5.756 1043
36
6.731
6.046 676
-12
7.323
6.168 768
0
a
16.387
17.506
18.498
18.502
16.966
17.092
β-phase
b
c
V %ΔV
6.783
5.966 663 -15
8.175
6.091 872
11
8.116
5.815 873
11
9.948
5.831 1073
37
6.791
5.998 691 -12
7.461
6.154 785
0
Length and volume are in units of Å and Å3, respectively.
• 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).
Relative energies
using the optimized geometries
ΔE (meV/unit cell)
0
LDA
PW91
-50
PBE
B3LYP
-12
(25)
-16
(68)
B3LYP+D
-100
-150
-128
(-123)
-165
(29)
-163
(-155)
-200
Numbers in the parenthesis are ΔE using the experimental geometries
• All five functionals predict the α phase to be more stable than the
β phase at the optimized geometries.
Importance of vibration analysis
in polymorphism of molecular crystals
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.
glycine case:
S. A. Rvera, et al. Cryst. Growth Des. 8, 3905 (2008).
ΔHexp= 0.27
Lattice energy
difference
= 1.28 kJ/mol
α-glycine
γ-glycine
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.
Phonon calculations at the Γ point:
Zero-point energy
ΔEL
ΔE0
SVWN
-73
9
B3LYP
-6
4
-153
-1
B3LYP+D
All energies are in units of meV.
EL: Electronic energy; E0: Zero-point energy; ΔE := E (α) – E(β)
Vibration frequencies calculated within the harmonic approximation
using the Crystal 09 code
Refs.: J. Comput. Chem. 25, 888 (2004); 25, 1873 (2004)
Phonon calculations at the Γ point:
Temperature dependence
P = 0.101325 MPa (Experimental transition temperature: Tpt = 326 K)
200
Tpt=78 K
100
= G(α)-G(β)
ΔG [meV]
150
Tpt=471 K
50
0
-50
0
1000
500
Tpt=1005 K
1500
LDA
B3LYP
B3LYP+D
-100
-150
-200
T [Kelvin]
G = U + PV – TS; U = EL + E0 + ET
G: 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
Finite-size (FS) effects in QMC
• QMC calculations for periodic systems:
– Super cell calculations and their extrapolation to infinite size
Phys. Rev. B 53, 1814 (1996).
– Twist-averaged boundary conditions (for metallic systems)
Phys. Rev. E 64, 016702 (2001).
– Kwee, Zhang, and Krakauer scheme
Phys. Rev. Lett. 100, 126404 (2008).
 QMC FS corrections are estimated from modified LDA calculations
with finite-size functionals.
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 energies
in 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
1000.0
243.92
6-31G + 3-21G
E(α)+56233
(hartree)
100.0
10.0
8.33
6-311G
1.0
0.45
0.31
0.31
0.28
MDT+d
opt-MDT
opt-MDT+d
MDT
0.1
•
•
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.
Energetics at the experimental geometry:
The B3LYP total energy of the α phase
1000.0
243.99
6-31G + 3-21G
E(α)+56286
(mhartree)
100.0
10.0
NA
6-311G
1.0
0.36
0.24
0.23
0.21
opt-MDT
opt-MDT+d
MDT
MDT+d
0.1
•
•
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.
Energetics at the experimental geometries:
relative stability ΔE = E(α) – E(β)
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.
•
•
•
•
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.
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
• 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.
• Computational studies of polymorphism in molecular crystals
– mostly relies on molecular mechanics (MM) because of computational costs
Phonon calculations, etc. with empirical potentials
e.g., S. L. Price, Adv. Drug Delivery Rev. 56, 301 (2004),
– ab-initio studies are also useful for verifying the reliability of MM
For example: (JACS 127, 3038 (2005) )
- at zero temperature: ground state energy calculations by DFT
- at finite temperature: Car-Parrinello MD simulation based on DFT
Problems in ab-initio studies of
Polymorphism in molecular crystals
• Computational costs:
– Unit cell contains a number of complicated molecules
– Periodicity of molecular crystal structures
Accurate quantum chemistry methods such as CI and CC
are NOT applicable to periodic systems… Mostly, DFT
• Accuracy of theoretical methods:
– The energy difference between polymorphs is quite small (- order of 1 mH)
– Noncovalent bond: hydrogen bond/ dispersion-dominated complexes
Electron correlation effects play an important
role in describing such chemical bondings!
Many DFT exchange-correlation functionals fail…
DMC is a good benchmark for molecular crystals