Recent developments for the quantum

chemical investigation of molecular

systems with high structural complexity

Stephan IrleWPI-Institute of Transformative Bio-Molecules &

Department of Chemistry, Graduate School of Science

Nagoya University

Nagoya, Japan

APCTCC-6

Gyeongju, Korea, July 10-13, 2013

Group Theme: Quantum Chemistry of Complex Systemshttp://qc.chem.nagoya-u.ac.jp

Quantum Chemistry

Statistical Mechanics

Molecular Dynamics

Method

Development

DFTB, RISM, GA

, Stochastic

Search …

Y i = cn

i jn

n

å

Theoretical

Spectroscopy

(UV/Vis, IR,

Raman)

Nanomaterials

Self-assembly,

reactions, properties

2

Biosystems

Reactions and

ligand-protein

interaction

Solution Chemistry

Solvation,

electron

transfer

AD

AD

A-

D+

Density-Functional Tight-Binding (DFTB)

Tight Binding (extended-Huckel-like) method with parameters from DFT

E (NCC-)DFTB = niei

i

valenceorbitals

å +1

2EAB

rep

A¹B

atoms

å

E (SCC-)DFTB = E (NCC-)DFTB +1

2g ABDqA

A,B

atoms

å DqB

ES( pin-polarized )DFTB = E (SCC-)DFTB +1

2pAl pAl 'WAll '

l 'ÎA

ålÎA

åA

atoms

å

Marcus Elstner

Christof Köhler

Helmut

EschrigGotthard

SeifertThomas

Frauenheim

4

Method Development: Fast QM

Fast QM Method: Approximate DFT

Eschrig, Seifert (1980‟s):

• 2-center approximation

• Minimum basis set

No integrals, DFTB is roughly 1000 times faster than DFT!

Foulke, Haydock (1989):

Introduction

5

ReviewDFTB

Ref.: Oliviera, Seifert, Heine, Duarte, J. Braz. Chem. Soc. 20,

1193-1205 (2009)

...open access

Thomas

HeineHelio

Duarte

6

ReviewDFTB

Density Functional Theory (DFT)

E réëùû= n

iy

i-

1

2Ñ2 + v

ext

r( ) +

rr '( )

r -r '

ò d 3r ' yi

i=1

M

å

+ Exc

réëùû-

1

2

rr( ) r

r '( )

r -r '

d 3ròò d 3r '+1

2

ZaZ

bR

a-R

ba ,b=1a¹b

N

å

= nie

i

i=1

M

å + Erep

at convergence:

Various criteria for convergence possible:

• Electron density

• Potential

• Orbitals

• Energy

• Combinations of above quantities

Walter Kohn/John A. Pople 1998

Self-consistent-charge density-functional

tight-binding (SCC-DFTB)M. Elstner et al., Phys. Rev. B 58 7260 (1998)

E r0 + dr[ ] = ni fi H r0[ ] fi

i

valenceorbitals

å

1

+ ni fi H r0[ ] fi

i

coreorbitals

å

2

+

+ Exc r0[ ]3

-

1

2r0VH r0[ ]

R3

ò

4

- r0Vxc r0[ ]R

3

ò

5

+ Enucl

6

+

+1

2drVH dr[ ]

R3

ò

7

+1

2

¶2Exc

¶dr2

r0

dr2

R3

òò

8

+o 3( )

Second order Taylor-expansion of DFT energy in terms of reference density

r0 and charge fluctuation r (rr0 + r) yields:

Density-functional tight-binding (DFTB) method is derived from terms 1-6

SCC-DFTB method is derived from terms 1-8

Phys. Rev. B, 39, 12520 (1989)

Foulkes + Haydock Ansatz

ReviewDFTB

7

DFTB and SCC-DFTB methods

where

ni and i — occupation and orbital energy ot the ith Kohn-Sham

eigenstate

Erep — distance-dependent diatomic repulsive potentials

qA — induced charge on atom A

AB — distance-dependent charge-charge interaction functional;

obtained from chemical hardness (IP – EA)

EDFTB = niei

i

valenceorbitals

å

term 1

+1

2Erep

AB

A¹B

atoms

å

terms 2-6

ESCC-DFTB = niei

i

valenceorbitals

å

term 1

+1

2g ABDqADqB

A,B

atoms

å

terms 7-8

+1

2Erep

AB

A¹B

atoms

å

terms 2-6

ReviewDFTB

8

DFTB method

Repulsive diatomic potentials replace usual nuclear repulsion

energy

Reference density r0 is constructed from atomic densities

Kohn-Sham eigenstates i are expanded in Slater basis of valence

pseudoatomic orbitals i

The DFTB energy is obtained by solving a generalized DFTB

eigenvalue problem with H0 computed by atomic and diatomic DFT

 

r0 = r0

A

A

atoms

å

 

fi = cmicm

m

AO

å

 

H0C = SCe with Smn = cm cn

Hmn0 = cm

ˆ H r0

M ,r0

N[ ] cn

ReviewDFTB

9

10

Additional induced-charges term allows for a proper description

of charge-transfer phenomena

Induced charge qA on atom A is determined from Mulliken

population analysis

Kohn-Sham eigenenergies are obtained from a generalized,

self-consistent SCC-DFTB eigenvalue problem

SCC-DFTB method (I)

ReviewDFTB

14

Traditional DFTB concept: Hamiltonian matrix elements are approximated to

two-center terms. The same types of approximations are done to Erep.

From Elstner et al., PRB 1998

0

0

(Density superposition)

(Potential superposition)

eff eff A B

eff eff A eff B

V V

V V V

r r r

r r r

A B D

C

A

B

DC

Situation I Situation II

Both approximations are justified by the screening argument: Far away, neutral atoms

have no Coulomb contribution.

Approximations in the DFTB Hamiltonian

ReviewDFTB

15

LCAO ansatz of wave function

Rri

i c

secular equations

0

SHc i

i

variational

principle

pseudoatomic orbital

Example: X4: Atom 1 – 4 are the same atom & have only s shell

1

4

2

3

r12

r23

r14

r34

r13

r24

How to construct?

two-center approximation

nearest neighbor off-diagonal

elements only

(minimum basis set)

Hamiltonian Overlap

pre-computed parameter

•Reference Hamiltonian H0

•Overlap integral Sμν

SCC-DFTB HamiltonianDFTB

16

LCAO ansatz of wave function

Rri

i c

secular equations

0

SHc i

i

variational

principle

pseudoatomic orbital

H11

H22

H33

H44

Atom 1 – 4 are the same atom & have only s shell

Diagonal term

Orbital energy of

neutral free atom

(DFT calculation)

1

4

2

3

r12

r23

r14

r34

r13

r24

Hamiltonian Overlap

qH2

1

Charge-charge

interaction function

Induced

charge

SCC-DFTB HamiltonianDFTB

17

LCAO ansatz of wave function

Rri

i c

secular equations

0

SHc i

i

variational

principle

pseudoatomic orbital

H11

H22

H33

H41 H44

Atom 1 – 4 are the same atom & have only s shell

1

4

2

3

r12

r23

r14

r34

r13

r24

r14

Two-center integral

qSHH2

10

Charge-charge

interaction function

Induced

charge

Hamiltonian Overlap

Lookup tabulated H0

and S at distance r

SCC-DFTB HamiltonianDFTB

r14

18

LCAO ansatz of wave function

Rri

i c

secular equations

0

SHc i

i

variational

principle

pseudoatomic orbital

H11

H22

H33

H41 H43 H44

Atom 1 – 4 are the same atom & have only s shell

1

4

2

3

r12

r23

r34

r13

r24

r34

Two-center integral

qSHH2

10

Charge-charge

interaction function

Induced

charge

Hamiltonian Overlap

Repeat for all off-diagonal terms

Lookup tabulated H0

and S at distance r

SCC-DFTB HamiltonianDFTB

DFTB repulsive potential Erep

Which molecular systems to include?

Development of

(semi-)automatic

fitting:•Knaup, J. et al.,

JPCA, 111, 5637,

(2007)

•Gaus, M. et al.,

JPCA, 113, 11866,

(2009)

•Bodrog Z. et al.,

JCTC, 7, 2654, (2011)

19

Repulsive PotentialsDFTB

21/25

New Confining Potentials

Wa

Conventional potential

r0

Woods-Saxon potential

k

R

rrV

0

)(

R0 = 2.7, k=2

)}(exp{1)(

0rra

WrV

r0 = 3.0, a = 3.0, W = 3.0

Typically, electron

density contracts under

covalent bond formation.

In standard ab initio

methods, this problem

can be remedied by

including more basis

functions.

DFTB uses minimal

valence basis set: the

confining potential is

adopted to mimic

contraction

• •+

• •

1s

σ1s

H H

H2 Δρ = ρ – Σa ρa

H2 difference density1s

Henryk Witek

New Electronic Parameters DFTB Parameterization

21

Band structure for Se (FCC)

Brillouin zone22

New Electronic Parameters DFTB Parameterization

Particle swarm optimization (PSO)

New Electronic Parameters DFTB Parameterization

23

1) Particles (=candidate of a solution) are randomly placed initially in a target space.

2) – 3) Position and velocity of particles are updated based on the exchange of

information between particles and particles try to find the best solution.

4) Particles converges to the place which gives the best solution after a number of

iterations.

•

•

•

••

•

• •••

•

•••••

• •••

• ••••••••

•••••••••••

particle

1)

4)

2)

3)

Particle Swarm Optimization DFTB Parameterization

24

Each particle has

randomly generated

parameter sets (r0, a, W)

within some region

Generating one-center

quantities (atomic orbitals,

densities, etc.)

“onecent”

Computing two-center

overlap and Hamiltonian

integrals for wide range

of interatomic distances

“twocent”

“DFTB+”

Calculating DFTB band

structure

Update the parameter

sets of each particle

Memorizing the best fitness

value and parameter sets

Evaluating “fitness value”

(Difference DFTB – DFT band

structure using specified fitness

points) “VASP”

DFTB Parameterization

orbital

a [2.5, 3.5]

W [0.1, 0.5]

r0 [3.5, 6.5]

density

a [2.5, 3.5]

W [0.5, 2.0]

r0 [6.0, 10.0]

Particle Swarm Optimization

25

Example: Be, HCP crystal structure

DFTB Parameterization

Total density of states (left) and band structure (right) of

Be (hcp) crystral structure

2.286

3.584

•Experimental

lattice constants

•Fermi energy is

shifted to 0 eV

26

Electronic Parameters

DFTB ParameterizationTransferability of optimum parameter sets

for different structures

Artificial crystal structures can be reproduced well

e.g. : Si, parameters were optimized with bcc only

W (orb) 3.33938

a (orb) 4.52314

r (orb) 4.22512

W (dens) 1.68162

a (dens) 2.55174

r (dens) 9.96376

εs -0.39735

εp -0.14998

εd 0.21210

3s23p23d0

bcc 3.081

fcc 3.868

scl 2.532

diamond 5.431

Parameter sets:

Lattice constants:bcc fcc

scl diamond

Expt.

27

Correlation of r(orb) vs. atomic diameter

Atomic Number Z

Ato

mic

dia

mete

r [a

.u.]

Empirically measured radii

(Slater, J. C., J. Chem. Phys.,

41, 3199-3204, (1964).)

Calculated radii with minimal-

basis set SCF functions

(Clementi, E. et al., J. Chem.

Phys., 47, 1300-1307, (1967).)

Expected value using relativistic

Dirac-Fock calculations

(Desclaux, J. P., Atomic Data

and Nuclear Data Tables, 12,

311-406, (1973).)

This work r(orb)

In particular for main group elements, there seems to be a

correlation between r(orb) and atomic diameter.

28

DFTB ParameterizationElectronic Parameters

Rocksalt (space group No. 225)

•NaCl

•MgO

•MoC

•AgCl

…

•CsCl

•FeAl

…

B2 (space group No. 221)

Zincblende (space group No. 216)

•SiC

•CuCl

•ZnS

•GaAs

…

Others

•Wurtzite (BeO, AlO, ZnO, GaN, …)

•Hexagonal (BN, WC)

•Rhombohedral (ABCABC stacking

sequence, BN)

more than 100 pairs tested

29

DFTB ParameterizationBinary Systems

element name

Ga, As hyb-0-2

B, N matsci-0-2

Reference of

previous work :

•d7s1 is used in

POTCAR (DFT)

Further improvement can be performed for specific purpose

but this preliminary sets will work as good starting points 30

DFTB ParameterizationBinary Systems

31

•space group No. 229

•1 lattice constant (a)

Transferability checked (single point calculation)

Reference system in PSO

Experimental lattice constants

available

a

31

DFTB ParameterizationBCC elements

Prof. Henryk

WitekDr. Yoshifumi

Nishimura

Chien-Pin

Chou

32

DFTB ParameterizationErepfit

Gaus, M. et al., JPCA, 113, 11866, (2009)

Prof. Henryk

WitekDr. Yoshifumi

Nishimura

Chien-Pin

Chou

x 1,000,000

Chemistry sans thought

Dr. Matt Addicoat,

(JSPS Postdoc)

Guessing competition:

• What kind of conformations can a

molecule of four different atoms, A, B,

C, D adopt?

Guessing competition:

12 12 6

12 12 2

Guessing competition:

Guessing competition:

• ABCD has a 6 possible structures with

a total of 56 permutations

• ABCDE has 15 possible structures with

a total of 577 permutations

• Genetic algorithm

• Simulated annealing

• Monte-Carlo

• Basin hopping

Automated approaches

Wishlist

• No assumed knowledge / limiting

parameters

• Ensemble of structures

• Broad applicability

• level of theory

• computational chemistry "backend"

• Least amount of human work possible

Kick

• The original Kick (M. Saunders) took a

geometry (input file) and perturbed it

• The Schaefer version generated

random

co-ordinates within a box of pre-set

size

• Adelaide (Addicoat/Metha) Version

works on the same principle as

Schaefer version

• Adds the capability to recognise

“fragments”

Kick

• A fragment is supplied as cartesian co-

ordinates which are rotated by a

random angle (Φ,ϑ,Ψ) before being

"kicked"

• Geometry optimisation, parsing and

resubmission of unique geometries

(frequency, higher level E) automatic.

(x,y,z)

(0,0,0)

-1

Zeise‟s anion

+ 3 +

- or -

Zeise‟s anion

+ 3 + 2

+ 4

- or -

Zeise‟s anion

Zeise‟s anion

• Pt + 3Cl + C2H4

• 20 jobs, 6 minima

• Pt + 3Cl + 2C + 4H

• 2500 jobs to identify global minimum

• Energies up to 12 eV from lowest energy minimum

• Includes many dissociated minima

• 9000 jobs to locate all possible substitutions of

ethylene

1.499

eV

Zeise‟s anion

3.370

eV

3.437

eV

3.553

eV

5.368

eV

... CrazyLego

(made in Nagoya)

J. Comp. Chem. Early View (2013).

DOI: 10.1002/jcc.23420

CrazyLego• Rather than a box (x,y,z), define a

radius (r)

• Translation (x,y,z) and orientation

(φ,θ,ψ) of new fragment are still

chosen randomly

• Can „backtrack‟ and place new

fragment near old fragment

• The „key atom‟ in each fragment is its

centroid

• A fragment location is rejected if it

violates minimum distance constraint

(0,0,0)

r

(0,0,0)

(0,0,0)

([dmim][NO3])2

([emim][NO3])2

M06-2X vs DFTB3-D ([dmim][NO3])2

M06-2X vs DFTB3-D ([emim][NO3])2

M06-2X vs DFTB3-D ([bmim][NO3])2

Global DFTB3-D min. of [emim][NO3]7

(emim+/NO3-)7

The structures of IL clusters are structurally interesting.

J. Comp. Chem. Early View (2013).

DOI: 10.1002/jcc.23420

Summary

• DFTB can be used for pre-

scanning configuration space of

complex systems

• MD of complex systems is

possible on nanosecond

timescale

Acknowledgments

62

From left to right, front row: Tae (Chiang Mai U), Arifin, Akao, Meow (Ubon Ratchathani U), Kato M, Shibata, Usui; back row: Siva, Tim, Nishimoto, SI, Yokogawa, Baba, Anupriya, Hiro, Noguchi

Dr. Matt Addicoat,

(JSPS Postdoc)Dr. Yoshifumi

Nishimura

Thank you for your attention!