Date post: | 11-Jul-2015 |
Category: |
Technology |
Upload: | stephan-irle |
View: | 1,470 times |
Download: | 6 times |
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!