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![20
Self-consistent-charge density-functional
tight-binding (SCC-DFTB)
M. Elstner et al., Phys. Rev. B 58 7260 (1998)
E r[ ] = ni fi
ˆH r0[ ] fi
i
valence
orbitals
å
1
+ ni fi
ˆH r0[ ] fi
i
core
orbitals
å
2
+ Exc r0[ ]
3
-
1
2
r0VH r0[ ]
R3
ò
4
-
- r0Vxc r0[ ]
R3
ò
5
+ Enucl
6
+
1
2
r1VH r1[ ]
R3
ò
7
+
1
2
d2
Exc
dr1
2
r0
r1
2
R3
òò
8
+ o 2( )
Approximate density functional theory (DFT) method!
Second order-expansion of DFT energy in terms of reference density 0 and
charge fluctuation 1 ( 0 + 1) yields:
Density-functional tight-binding (DFTB) method is derived from terms 1-6
Self-consistent-charge density-functional tight-binding (SCC-DFTB)
method is derived from terms 1-8
o(3)
Lecture II DFTB](https://image.slidesharecdn.com/dftb-talk-3427-130702051611-phpapp02/85/Density-Functional-Tight-Binding-DFTB-as-fast-approximate-DFT-method-An-introduction-20-320.jpg)
![22
DFTB method
Repulsive diatomic potentials replace usual nuclear repulsion
energy
Reference density 0 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
H0
C = SCe with Smn = cm cn
Hmn
0
= cm
ˆH r0
M
,r0
N
[ ] cn
Lecture II DFTB](https://image.slidesharecdn.com/dftb-talk-3427-130702051611-phpapp02/85/Density-Functional-Tight-Binding-DFTB-as-fast-approximate-DFT-method-An-introduction-22-320.jpg)
![35
Spin-polarized DFTB (SDFTB)
Lecture II DFTB
for systems with different and spin densities, we have
total density = +
magnetization density S = -
2nd-order expansion of DFT energy at ( 0,0) yields
E r,rS
[ ]= ni fi
ˆH r0[ ] fi
i
valence
orbitals
å
1
+ ni fi
ˆH r0[ ] fi
i
core
orbitals
å
2
+ Exc r0[ ]
3
-
1
2
r0VH r0[ ]
R3
ò
4
-
- r0Vxc r0[ ]
R3
ò
5
+ Enucl
6
+
1
2
r1VH r1[ ]
R3
ò
7
+
1
2
d2
Exc
dr1
2
r0 ,0( )
r1
2
R3
òò
8
+
1
2
d2
Exc
drS
( )
2
r0 ,0( )
rS
( )
2
R3
òò
9
+ o 2( )
The Spin-Polarized SCC-DFTB (SDFTB) method is derived from terms 1-9
o(3)](https://image.slidesharecdn.com/dftb-talk-3427-130702051611-phpapp02/85/Density-Functional-Tight-Binding-DFTB-as-fast-approximate-DFT-method-An-introduction-35-320.jpg)
![0 1 2 3 4 5
0
20
40
60
80
40
0 1 2 3 4 5
0
20
40
60
80
0 1 2 3 4 5
0
20
40
60
80
0 1 2 3 4 5
0
20
40
60
80
0 1 2 3 4 5
0
20
40
60
80
Time[ps] Time[ps]Time[ps]
Te= 0K
Te= 1500K
Te= 10kK
Te=0 K always yields SCC convergence problemSCC iterations(time)
Maximum iteration number is 70
0 1 2 3 4 5
0
20
40
60
80
0 1 2 3 4 5
0
20
40
60
80
0 2 4
0
20
40
60
80
0 1 2 3 4 5
0
20
40
60
80
(A) H10C60 Fe38 (B) Fe13C10 (C) Fe6C2
kbTe(10kK) ~0.87 eV
~half-width of 3d band in Fe38
DFTBLecture II](https://image.slidesharecdn.com/dftb-talk-3427-130702051611-phpapp02/85/Density-Functional-Tight-Binding-DFTB-as-fast-approximate-DFT-method-An-introduction-40-320.jpg)
![2). DFTB band structure fitting
•Optimization of parameter sets for Woods-Saxon confining potential (orbital
and density) and unoccupied orbital energies
•Fixed orbital energies for electron occupied orbitals
•Valence orbitals : [1s] for 1st row
[2s, 2p] for 2nd row
[ns, np, md] for 3rd – 6th row
(n ≥ 3, m = n-1 for group 1-12, m = n for group 13-18)
•Fitting points : valence bands + conduction bands (depending on the system,
at least including up to ~+5 eV with respect to Fermi level)
Electronic Parameters DFTB Parameterization
1). DFT band structure calculations
•VASP 4.6
•One atom per unit cell
•PAW (projector augmented wave) method
•32 x 32 x 32 Monkhorst-Pack k-point sampling
•cutoff = 400 eV
•Fermi level is shifted to 0 eV
49](https://image.slidesharecdn.com/dftb-talk-3427-130702051611-phpapp02/85/Density-Functional-Tight-Binding-DFTB-as-fast-approximate-DFT-method-An-introduction-49-320.jpg)
![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
53](https://image.slidesharecdn.com/dftb-talk-3427-130702051611-phpapp02/85/Density-Functional-Tight-Binding-DFTB-as-fast-approximate-DFT-method-An-introduction-53-320.jpg)
![Correlation of r(orb) vs. atomic diameter
Atomic Number Z
Atomicdiameter[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.
65
DFTB ParameterizationElectronic Parameters](https://image.slidesharecdn.com/dftb-talk-3427-130702051611-phpapp02/85/Density-Functional-Tight-Binding-DFTB-as-fast-approximate-DFT-method-An-introduction-65-320.jpg)
![74
E ~
1/R6
damping f(R ) = [1-exp(-3(rij/Rij,vdW
)7)]3
Rij,vdW = 3.8 Å (for 1st row), 4.8 A (for 2nd row)
DFTB+Dispersion
Elstner et al., JCP 114, 5149 (2001)
DFTB-D choice of C6
parameters:
- generally hybridization
dependent (i.e. not
simply atomic values)
- use “empirical” values
for parameters to match
BSSE-corrected MP2
interaction energies
R [Å]
E [ha]
ExtensionsLecture II](https://image.slidesharecdn.com/dftb-talk-3427-130702051611-phpapp02/85/Density-Functional-Tight-Binding-DFTB-as-fast-approximate-DFT-method-An-introduction-74-320.jpg)
![80
Performance
Accuracy of DFTB Geometries and
Energies for Fullerene Isomers
Fullerene Isomers Geometries and Energies vs B3LYP/6-31G(d)
G. Zheng, SI, M.
Elstner, K.
Morokuma, Chem.
Phys. Lett. 412, 210
(2005)
RMS [Å] NCC-DFTB SCC-DFTB AM1 PM3
C20-C36 0.025 0.019 0.035 0.030
C60-C86 0.014 0.014 0.016 0.015
R2
(lin. reg.) NCC-DFTB SCC-DFTB AM1 PM3
C20-C36 0.88 0.93 0.77 0.73
C60-C86 0.97 0.98 0.86 0.84
Geometries
Energetics
•102 Fullerene Isomers
• small cage non-IPR C20-C36 (35), large cage IPR C70-C86 (67)
R2: Energy linear regression between E(Method) and E(B3LYP)
Lecture II](https://image.slidesharecdn.com/dftb-talk-3427-130702051611-phpapp02/85/Density-Functional-Tight-Binding-DFTB-as-fast-approximate-DFT-method-An-introduction-80-320.jpg)
![95
A B C
DFT:PW91[1] -6.24 -5.63 -1.82
SCC-DFTB[2] -5.17 -4.68 -1.86
Adhesion energies (eV/atom)
A B C
PW91: An ultrasoft pseudopotential with a plane-wave cutoff of 290 eV for the single metal and the
projector augmented wave method with a plane-wave cutoff of 400 eV for the metal cluster
Fe-Fe and Fe-C DFTB parameters from: G. Zheng et al., J. Chem. Theor. Comput. 3, 1349 (2007)
[1] Phys. Rev. B 75, 115419 (2007) [2] Fermi broadening=0.13 eV
H10C60Fe H10C60Fe
H10C60Fe55
Fe55
icosahedron
H-terminated (5,5) armchair SWNT + Fe atom/cluster
PerformanceLecture II](https://image.slidesharecdn.com/dftb-talk-3427-130702051611-phpapp02/85/Density-Functional-Tight-Binding-DFTB-as-fast-approximate-DFT-method-An-introduction-95-320.jpg)
Uploaded byStephan Irle
Density-Functional Tight-Binding (DFTB) as fast approximate DFT method - An introduction
The document provides a comprehensive overview of the density-functional tight-binding (DFTB) method, a fast approximate DFT method used in computational materials science. It discusses various DFTB approaches, implementations, and extensions, along with associated resources and literature references. The document highlights the formulation, application, and significance of DFTB and its self-consistent-charge variant in simulating complex material properties.
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Density-Functional Tight-Binding (DFTB) as fast approximate DFT method - An introduction
- 1. All lectures onlineat: http://qc.chem.nagoya-u.ac.jp/presentations.html
- 2. 2 Density-Functional Tight-Binding (DFTB)as fast approximate DFT method Helmut Eschrig Gotthard Seifert Thomas Frauenheim Marcus Elstner Lecture II: Introduction to the Density-Functional Tight-Binding (DFTB) Method
- 3. 3 Density-Functional Tight-Binding 1. Tight-Binding 2.Density-Functional Tight-Binding (DFTB) 3. Bond Breaking in DFTB 4. Extensions 5. Performance and Applications
- 4.
- 5. 55 Lecture II 1.Tight-Binding Resources 1. http://www.dftb.org 2. DFTB Porezag, D., T. Frauenheim, T. Köhler, G. Seifert, and R. Kaschner, Construction of tight-binding-like potentials on the basis of density-functional theory: application to carbon. Phys. Rev. B, 1995. 51: p. 12947-12957. 3. DFTB Seifert, G., D. Porezag, and T. Frauenheim, Calculations of molecules, clusters, and solids with a simplified LCAO-DFT-LDA scheme. Int. J. Quantum Chem., 1996. 58: p. 185-192. 4. SCC-DFTB Elstner, M., D. Porezag, G. Jungnickel, J. Elsner, M. Haugk, T. Frauenheim, S. Suhai, and G. Seifert, Self-consistent-charge density-functional tight- binding method for simulations of complex materials properties. Phys. Rev. B, 1998. 58: p. 7260-7268. 5. SCC-DFTB-D Elstner, M., P. Hobza, T. Frauenheim, S. Suhai, and E. Kaxiras, Hydrogen bonding and stacking interactions of nucleic acid base pairs: A density-functional-theory based treatment. J. Chem. Phys,, 2001. 114: p. 5149-5155. 6. SDFTB Kohler, C., G. Seifert, U. Gerstmann, M. Elstner, H. Overhof, and T. Frauenheim, Approximate density-functional calculations of spin densities in large molecular systems and complex solids. Phys. Chem. Chem. Phys., 2001. 3: p. 5109- 5114. 7. DFTB3 Gaus, M.; Cui, C.; Elstner, M. DFTB3: Extension of the Self-Consistent-Charge Density-Functional Tight-Binding Method (SCC-DFTB). J. Chem. Theory Comput., 2011. 7: p. 931-948.
- 6. 6 Standalone fast andefficient DFTB implementation with several useful extensions of the original DFTB method. It is developed at the Bremen Center for Computational Materials Science (Prof. Frauenheim, Balint Aradi) and is the successor of the old Paderborn DFTB and Dylax codes. Free for non-commercial use. DFTB+ as part of Accelrys' Materials Studio package, providing a user friendly graphical interface and the possibility to combine DFTB with other higher or lower level methods. DFTB integrated in the ab initio DFT code deMon DFTB in the Gaussian code Amber is a package of molecular simulation programs distributed by UCSF, developed mainly for biomolecular simulations. The current version of Amber includes QM/MM support, whereby part of the system can be treated quantum mechanically, and DFTB is among the quantum mechanical methods available. Amber also has a stand-alone (pure QM) implementation. CHARMm (Chemistry at HARvard Macromolecular Mechanics) DFTB integrated in the Amsterdam Density Functional (ADF) program suite. DFTB+ DFTB+/Accelrys deMon GAUSSIAN G09 AMBER CHARMm ADF Implementations Lecture II 1. Tight-Binding
- 7. 7 • Tight binding(TB) approaches work on the principle of treating electronic wavefunction of a system as a superposition of atom-like wavefunction (known to chemists as LCAO approach) • Valence electrons are tightly bound to the cores (not allowed to delocalize beyond the confines of a minimal LCAO basis) • Semi-empirical tight-binding (SETB): Hamiltonian Matrix elements are approximated by analytical functions (no need to compute integrals) • TB energy for N electrons, M atoms system: • This separation of one-electron energies and interatomic distance- dependent potential vj,k constitutes the TB method Tight-Binding Lecture II 1. Tight-Binding
- 8. 8 are eigenvalues ofa Schrodinger-like equation • solved variationally using atom-like (minimum, single-zeta) AO basis set, leading to a secular equation: where H and S are Hamiltonian and overlap matrices in the basis of the AO functions. In orthogonal TB, S = 1 (overlap between atoms is neglected) • H and S are constructed using nearest-neighbor relationships; typically only nearest-neighbor interactions are considered: Similarity to extended Hückel method Tight-Binding Lecture II 1. Tight-Binding
- 9. 9 • Based onapproximation by W. Wolfsberg and L. J. Helmholz (1952) H Ci = i S Ci • H – Hamiltonian matrix constructed using nearest neighbor relationships • Ci – column vector of the i-th molecular orbital coefficients • i – orbital energy • S – overlap matrix • H - choose as a constant – valence shell ionization potentials • H = K S (H + H )/2 • K – Wolfsberg Helmholz constant, typically 1.75 Extended Huckel (EHT) Method Lecture II 1. Tight-Binding
- 10. Categories of TBapproaches 10 Source: http://beam.acclab.helsinki.fi/~akrashen/esctmp.html Lecture II 1. Tight-Binding
- 11. Slater-Koster (SK) Approximation(I) 11 Source: http://beam.acclab.helsinki.fi/~akrashen/esctmp.html Lecture II 1. Tight-Binding
- 12. SK Approximation (II) 12 Source:http://beam.acclab.helsinki.fi/~akrashen/esctmp.html Lecture II 1. Tight-Binding
- 13. SK Approximation (III) 13 Source:http://beam.acclab.helsinki.fi/~akrashen/esctmp.html Lecture II 1. Tight-Binding
- 14. SK Approximation (IV) 14 Source:http://beam.acclab.helsinki.fi/~akrashen/esctmp.html Lecture II 1. Tight-Binding
- 15.
- 16. 16 Density-Functional Tight-Binding 2. Density-FunctionalTight-Binding (DFTB)
- 17. 17 DFTB MethodQuick reviewI Taken from Oliviera, Seifert, Heine, Duarte, J. Braz. Chem. Soc. 20, 1193-1205 (2009) ...open access Thomas Heine Helio Duarte
- 18. 18 DFTB MethodQuick reviewII Density Functional Theory (DFT) 2 3 1 3 3 , 1 1 '1 ' 2 ' '1 1 ' 2 ' 2 M i i ext i i N xc M i i rep i r E n v r d r r r Z Zr r E d rd r r r R R n E at convergence: Various criteria for convergence possible: • Electron density • Potential • Orbitals • Energy • Combinations of above quantities Walter Kohn/John A. Pople 1998
- 19. 19 DFTB MethodQuick reviewIII Phys. Rev. B, 39, 12520 (1989) Foulkes + Haydock Ansatz
- 20. 20 Self-consistent-charge density-functional tight-binding (SCC-DFTB) M.Elstner et al., Phys. Rev. B 58 7260 (1998) E r[ ] = ni fi ˆH r0[ ] fi i valence orbitals å 1 + ni fi ˆH r0[ ] fi i core orbitals å 2 + Exc r0[ ] 3 - 1 2 r0VH r0[ ] R3 ò 4 - - r0Vxc r0[ ] R3 ò 5 + Enucl 6 + 1 2 r1VH r1[ ] R3 ò 7 + 1 2 d2 Exc dr1 2 r0 r1 2 R3 òò 8 + o 2( ) Approximate density functional theory (DFT) method! Second order-expansion of DFT energy in terms of reference density 0 and charge fluctuation 1 ( 0 + 1) yields: Density-functional tight-binding (DFTB) method is derived from terms 1-6 Self-consistent-charge density-functional tight-binding (SCC-DFTB) method is derived from terms 1-8 o(3) Lecture II DFTB
- 21. 21 DFTB and SCC-DFTBmethods 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) Lecture II DFTB
- 22. 22 DFTB method Repulsivediatomic potentials replace usual nuclear repulsion energy Reference density 0 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 H0 C = SCe with Smn = cm cn Hmn 0 = cm ˆH r0 M ,r0 N [ ] cn Lecture II DFTB
- 23. 23 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 A B D C A B D C 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 Lecture II DFTB
- 24. SCC-DFTB matrix elements 24 LCAOansatz of wave function Rr i i c secular equations 0SHc i i variational principle pseudoatomic orbital 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 Hamiltonian Overlap pre-computed parameter •Reference Hamiltonian H0 •Overlap integral Sμν
- 25. SCC-DFTB matrix elements 25 LCAOansatz of wave function Rr i i c secular equations 0SHc 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 qH 2 1 Charge-charge interaction function Induced charge
- 26. SCC-DFTB matrix elements 26 LCAOansatz of wave function Rr i i c secular equations 0SHc 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 qSHH 2 10 Charge-charge interaction function Induced charge Hamiltonian Overlap Lookup tabulated H0 and S at distance r
- 27. r14 SCC-DFTB matrix elements 27 LCAOansatz of wave function Rr i i c secular equations 0SHc 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 qSHH 2 10 Charge-charge interaction function Induced charge Hamiltonian Overlap Repeat until building off-diagonal term Lookup tabulated H0 and S at distance r
- 28.
- 29. DFTB repulsive potentialErep Which molecular systems to include? Development of (semi- )automatic fitting: •Knaup, J. et al., JPCA, 111, 56 37, (2007) •Gaus, M. et al., JPCA, 113, 11 866, (2009) •Bodrog Z. et al., JCTC, 7, 2654, (2011) 29 Lecture II DFTB
- 30. 30 Additional induced-chargesterm 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) Lecture II DFTB
- 31. 31 SCC-DFTB method (II) Basicassumptions: •Only transfer of net charge between atoms •Size and shape of atom (in molecule) unchanged Only second-order terms (terms 7-8 on slide 16): Lecture II DFTB
- 32.
- 33. 33 SCC-DFTB method (IV) Severalpossible formulations for : Mataga-Nishimoto < Klopmann-Ohno < DFTB Klopmann-Ohno: Lecture II DFTB
- 34. 34 Gradient for theDFTB methods The DFTB force formula The SCC-DFTB force formula computational effort: energy calculation 90% gradient calculation 10% Fa = - ni cmicni ¶Hmn 0 ¶a -ei ¶Smn ¶a é ë ê ù û ú mn AO å i MO å - ¶Erep ¶a Lecture II DFTB
- 35. 35 Spin-polarized DFTB (SDFTB) LectureII DFTB for systems with different and spin densities, we have total density = + magnetization density S = - 2nd-order expansion of DFT energy at ( 0,0) yields E r,rS [ ]= ni fi ˆH r0[ ] fi i valence orbitals å 1 + ni fi ˆH r0[ ] fi i core orbitals å 2 + Exc r0[ ] 3 - 1 2 r0VH r0[ ] R3 ò 4 - - r0Vxc r0[ ] R3 ò 5 + Enucl 6 + 1 2 r1VH r1[ ] R3 ò 7 + 1 2 d2 Exc dr1 2 r0 ,0( ) r1 2 R3 òò 8 + 1 2 d2 Exc drS ( ) 2 r0 ,0( ) rS ( ) 2 R3 òò 9 + o 2( ) The Spin-Polarized SCC-DFTB (SDFTB) method is derived from terms 1-9 o(3)
- 36. where pA l— spin population of shell l on atom A WA ll’ — spin-population interaction functional Spin populations pA l and induced charges qA are obtained from Mulliken population analysis 36 Spin-polarized DFTB (SDFTB) Lecture II DFTB
- 37. 37 Spin-polarized DFTB (SDFTB) LectureII DFTB Kohn-Sham energies are obtained by solving generalized, self-consistent SDFTB eigenvalue problems where H- C- = SC- e- H¯ C¯ = SC¯ e¯ M,N,K: indexing specific atoms
- 38. 38 SCC-DFTB w/fractional orbitaloccupation numbers 1 2 2 tot i i rep i E f E q q 0vi i v c H S Fractional occupation numbers fi of Kohn-Sham eigenstates replace integer ni TB-eigenvalue equation Lecture II DFTB E 2fi 0 1 2 Finite temperature approach (Mermin free energy EMermin) 1 exp / 1 i i B e f k T 2 ln 1 ln 1e B i i i i i S k f f f f Te: electronic temperature Se: electronic entropy 0 1 2 N rep i i i i i EH H S F f c c q q SR R R R 0 1i f Atomic force M. Weinert, J. W. Davenport, Phys. Rev. B 45, 13709 (1992) EMermin = Etot - TeSe
- 39. 39 Fermi-Dirac distribution function:Energy derivative for Mermin Free Energy M. Weinert, J. W. Davenport, Phys. Rev. B 45, 13709 (1992) elect TS HF pulay charge TS i i i i i i i i i i i e i i i i F F F F F f x f T f x x f S f x x x elect HF pulay charge i i i i i i i i i F F F F f f f x x x Correction term arising from Fermi distribution function cancels out Lecture II DFTB
- 40. 0 1 23 4 5 0 20 40 60 80 40 0 1 2 3 4 5 0 20 40 60 80 0 1 2 3 4 5 0 20 40 60 80 0 1 2 3 4 5 0 20 40 60 80 0 1 2 3 4 5 0 20 40 60 80 Time[ps] Time[ps]Time[ps] Te= 0K Te= 1500K Te= 10kK Te=0 K always yields SCC convergence problemSCC iterations(time) Maximum iteration number is 70 0 1 2 3 4 5 0 20 40 60 80 0 1 2 3 4 5 0 20 40 60 80 0 2 4 0 20 40 60 80 0 1 2 3 4 5 0 20 40 60 80 (A) H10C60 Fe38 (B) Fe13C10 (C) Fe6C2 kbTe(10kK) ~0.87 eV ~half-width of 3d band in Fe38 DFTBLecture II
- 41. 41 Density-Functional Tight-Binding 3. BondBreaking in DFTB
- 42. Bond breakingLecture II SCC-DFTBand SDFTB Dissociation of H2 +
- 43. Bond breakingLecture II SCC-DFTBand SDFTB Dissociation of H2 +
- 44. Bond breakingLecture II SCC-DFTBand SDFTB Dissociation of H2 + (correct) (wrong)
- 45. Bond breakingLecture II SCC-DFTBand SDFTB Dissociation of H2 M. Lundberg, Y. Nishimoto, SI, Int. J. Quant. Chem. 112, 1701 (2012)
- 46.
- 47. 47 Extensions M=Sc, Ti, Fe,Co, Ni X=M,CHON d-elements with X partners: Geometries very good - “ballpark” energies only Recommendation: Use ONIOM(QM:QM), esp. ONIOM(DFT:DFTB) Lecture II
- 48. 48/25 New Confining Potentials Wa Conventionalpotential r0 Woods-Saxon potential k R r rV 0 )( R0 = 2.7, k=2 )}(exp{1 )( 0 rra W rV 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 density 1s Henryk Witek Electronic Parameters DFTB Parameterization 48
- 49. 2). DFTB bandstructure fitting •Optimization of parameter sets for Woods-Saxon confining potential (orbital and density) and unoccupied orbital energies •Fixed orbital energies for electron occupied orbitals •Valence orbitals : [1s] for 1st row [2s, 2p] for 2nd row [ns, np, md] for 3rd – 6th row (n ≥ 3, m = n-1 for group 1-12, m = n for group 13-18) •Fitting points : valence bands + conduction bands (depending on the system, at least including up to ~+5 eV with respect to Fermi level) Electronic Parameters DFTB Parameterization 1). DFT band structure calculations •VASP 4.6 •One atom per unit cell •PAW (projector augmented wave) method •32 x 32 x 32 Monkhorst-Pack k-point sampling •cutoff = 400 eV •Fermi level is shifted to 0 eV 49
- 50. Band structure forSe (FCC) Brillouin zone 50 Electronic Parameters DFTB Parameterization
- 51. Particle swarm optimization(PSO) Electronic Parameters DFTB Parameterization 51
- 52. 1) Particles (=candidateof 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 52
- 53. Each particle has randomlygenerated 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 53
- 54. Example: Be, HCPcrystal 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 54 Electronic Parameters
- 55. 55 Band structure fittingfor BCC crystal structures •space group No. 229 •1 lattice constant (a) Transferability checked (single point calculation) Reference system in PSO Experimental lattice constants available No POTCAR file for Z ≥ 84 in VASPa 55
- 56. 56 Band structure fittingfor FCC crystal structures Reference system in PSO Experimental lattice constants available •space group No. 225 •1 lattice constant (a) a Transferability checked (single point calculation) 56
- 57. 57 Band structure fittingfor SCL crystal structures Reference system in PSO Experimental lattice constants available •space group No. 221 •1 lattice constant (a) a Transferability checked (single point calculation) 57
- 58. 58 Band structure fittingfor HCP crystal structures Reference system in PSO Experimental lattice constants available •space group No. 194 •2 lattice constants (a, c) c a Transferability checked (single point calculation) 58
- 59. 59 Band structure fittingfor Diamond crystal structures Reference system in PSO Experimental lattice constants available •space group No. 227 •1 lattice constant (a) a Transferability checked (single point calculation) 59
- 60. DFTB ParameterizationTransferability ofoptimum 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. 60
- 61. Influence of virtualorbital energy (3d) to Al (fcc) band structure OPT The bands of upper part are shifted up constantly as orb (3d) becomes larger61
- 62. Influence of W(orb)to Al (fcc) band structure OPT The bands of upper part go lower as W(orb) becomes larger 62
- 63. Influence of a(orb)to Al (fcc) band structure OPT Too small a(orb) gives the worse band structure 63
- 64. Influence of r(orb)to Al (fcc) band structure OPT r(orb) strongly influences DFTB band structure 64
- 65. Correlation of r(orb)vs. atomic diameter Atomic Number Z Atomicdiameter[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. 65 DFTB ParameterizationElectronic Parameters
- 66. Straightforward application tobinary crystal structures 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 66
- 67. Selected examples forbinary crystal structures 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 67
- 68. 68 Extensions Gab DFTB = ¶2 Erep ¶a¶b + 2 niUmi b cmmcni ¶Hmn 0 ¶a -ei ¶Smn ¶a é ë ê ù û ú mn AO å im MO å + + ni cmicni ¶2 Hmn 0 ¶a¶b -ei ¶2 Smn ¶a¶b - ¶ei ¶b ¶Smn ¶a é ë ê ù û ú mn AO å i MO å ( ) ( ) ( ) ååå å å å å å å å å å å D D+D D +DD+ +ú û ù ê ë é D +-D÷ ø ö ç è æ +-- - ú ú û ù ê ê ë é ÷ ø ö ç è æ -D+++ + ú ú û ù ê ê ë é ÷ ø ö ç è æ -D+++= atomsatomsatoms 2 MO AO atoms atoms MO AO 2atoms02 MO AO atoms0 rep 2 DFTB-SCC 2 1 2 1 2 1 2 1 2 K K A AK K K AAK K KA AK i K K K NKMKK NKMKi iii i i K KNKMKiii im i K KNKMKim b miiab b q q a q b q a qq ba a S b q q bbb ccn ba S q ba H ccn a S q a H ccUn ba E G ¶ ¶ ¶ ¶g ¶ ¶ ¶ ¶g ¶¶ g¶ ¶ ¶ ¶ ¶ gg ¶ ¶g ¶ ¶g ¶ ¶e ¶¶ ¶ egg ¶¶ ¶ ¶ ¶ egg ¶ ¶ ¶¶ ¶ mn mn nm mn mnmn nm mn mnmn nm Analytical Hessian for DFTB and SCC-DFTB H. A. Witek, S. Irle, K. Morokuma, J. Chem. Phys. 121, 5163 (2005) Lecture II
- 69. 69 Extensions To determineresponse of molecular orbitals to nuclear perturbation, one has to solve a set of iterative coupled- perturbed SCC-DFTB equations Coupled Perturbed SCC-DFTB Equations Lecture II
- 70.
- 71. 71 How to treatinterlayer interactions in graphite cheaply? Addition of empirical London dispersion term! • Ahlrichs et al., Chem. Phys. 19, 119 (1977): HFD (Hartree-Fock Dispersion), Ci parameters from expt. • Mooij et al., J. Phys. Chem. A 103, 9872 (1999): PES with London dispersion, C6 parameters from MP2 calculations • Elstner et al., J. Chem. Phys. 114, 5149 (2001): SCC-DFTB-D (dispersion- augmented SCC-DFTB), C6 parameters via Halgren,J. Am. Chem. Soc. 114, 7872 (1992) from expt. • Yang et al., J. Chem. Phys. 116, 515 (2002): DFT+vdW (DFT including van der Waals interactions), C6 parameters from expt. • Grimme, J. Comput. Chem. 25, 1463 (2004): DFT-D, C6 parameters from expt. • … several more plus many reviews and benchmarks Etot = EQM - f (R ) C6 /R6 (+1/R8 etc. terms) f (R ): damping function Fritz London 1/R6: 1922 ExtensionsLecture II
- 72.
- 73. 73 DFTB+Dispersion 4 7 , 66 3 3 , , ,2 2 , , 6 1 ( , ) * ( , ), , cubic mean rule 3 ( , 1 exp 3.0* ,damping funct ) ( ion ) * 2 , ij ij vd dispersio w tot DFTB dispersion ij i vdw j vdw ij vdw i vdw n i j j vdw i j E E E i j C i j r R R R E i j f r R R C f R i j a a a < æ öæ öæ ö ç ÷ç ÷= - - ç ÷ç ÷ç ÷ç ÷è = + = - + = èè = ø øø + å ( ) ( ) 1/ 21/ 2 / / , : atomic polarizability parameer , : Slater Kirkwood effective number of electrons i i j j i j i j N N N N a a a + Elstner et al., JCP 114, 5149 (2001) Lecture IX Extensions
- 74. 74 E ~ 1/R6 damping f(R) = [1-exp(-3(rij/Rij,vdW )7)]3 Rij,vdW = 3.8 Å (for 1st row), 4.8 A (for 2nd row) DFTB+Dispersion Elstner et al., JCP 114, 5149 (2001) DFTB-D choice of C6 parameters: - generally hybridization dependent (i.e. not simply atomic values) - use “empirical” values for parameters to match BSSE-corrected MP2 interaction energies R [Å] E [ha] ExtensionsLecture II
- 75.
- 76. Principles of GRRM ADD Anharmonicdownward distortion S. Maeda and K. Ohno, J. Phys. Chem. A, 2005, 109, 5742. 76 UDC DDC One of the most powerful and reliable methods to find reaction pathways. Extensions GRRM with DFTB Lecture II
- 77. Example: Formaldehyde (RB3LYP/6-31G(d)) 4EQs 9TSs 3DDCs 9UDCs DDCat DFTB DFTB B3LYP Match EQ 4 4 4 TS 9 9 6 ~ 8 DDC 4 3 3 UDC 7 9 7 : Reproduced by DFTB : Found TS at DFTB 77 ExtensionsLecture II
- 78. 78 Density-Functional Tight-Binding 5. Performanceand Applications
- 79. Performance for smallorganic molecules (mean absolut deviations) • Reaction energies: ~ 5 kcal/mole • Bond-lenghts: ~ 0.014 A° • Bond angles: ~ 2° •Vib. Frequencies: ~6-7 % 79 Performance SCC-DFTB: general comparison with experiment Lecture II
- 80. 80 Performance Accuracy of DFTBGeometries and Energies for Fullerene Isomers Fullerene Isomers Geometries and Energies vs B3LYP/6-31G(d) G. Zheng, SI, M. Elstner, K. Morokuma, Chem. Phys. Lett. 412, 210 (2005) RMS [Å] NCC-DFTB SCC-DFTB AM1 PM3 C20-C36 0.025 0.019 0.035 0.030 C60-C86 0.014 0.014 0.016 0.015 R2 (lin. reg.) NCC-DFTB SCC-DFTB AM1 PM3 C20-C36 0.88 0.93 0.77 0.73 C60-C86 0.97 0.98 0.86 0.84 Geometries Energetics •102 Fullerene Isomers • small cage non-IPR C20-C36 (35), large cage IPR C70-C86 (67) R2: Energy linear regression between E(Method) and E(B3LYP) Lecture II
- 81. What about non-cagecarbon cluster structures? Some C28 isomers as example (from Scuseria, CPL 301, 98 (1999)) PerformanceLecture II
- 82. Performance What about non-cagecarbon cluster structures? Some C28 isomers as example TACC’04 Symp. Proceedings AM1 and PM3 are performing very bad! DFTB includes effects of polarization functions through parameterization Wrong minimum structures! Lecture II
- 83. 83 Performance Accuracy of DFTBGeometries Table 3. The relative energies (in eV) of 11 different isomeric singlet structures of C28 calculated with B3LYP/6-31G(d), and DFTB. Structure a B3LYP/6-31G(d) DFTB B3LYP/6-31G(d) // DFTB buckyD2 0.00 0.00 0.00 ring 3.32 8.10 3.43 c24-6 3.17 3.56 3.66 2+2r14 5.08 9.66 5.22 2+2r16 6.01 10.25 6.13 c20-6o 5.41 5.52 5.96 c20-6 m 5.57 5.62 6.09 2 + 4 7.97 10.28 8.52 central7 5.86 6.07 6.47 8 + 8 7.43 9.43 7.41 4 + 4 9.91 14.27 10.20 R2 b 0.75 0.99 a Structures illustrated below, with the labels taken from “Portmann, S.; Galbraith, J. M.; Schaefer, H. F.; Scuseria, G. E.; Lüthi, H. P. Chem. Phys. Lett. 1999, 301, 98- 104.” b Squared correlation coefficients R2 in the linear regression analysis with B3LYP/6-31G(d) energies. Lecture II
- 84. 84 Performance Accuracy of DFTBfrequencies 84 calculated frequencies scaled frequencies Testing set of 66 molecules, 1304 distinct vibrational modes Lecture II
- 85. 85 Performance Accuracy of DFTBfrequencies Testing set of 66 molecules, 1304 distinct vibrational modes mean absolute deviation standard deviation maximal absolute deviation scaling factor SCC-DFTB 56 cm-1 82 cm-1 529 cm-1 0.9933 DFTB 60 cm-1 87 cm-1 536 cm-1 0.9917 AM1 69 cm-1 95 cm-1 670 cm-1 0.9566 PM3 74 cm-1 102 cm-1 918 cm-1 0.9762 HF/cc-pVDZ 30 cm-1 49 cm-1 348 cm-1 0.9102 BLYP/cc-pVDZ 34 cm-1 47 cm-1 235 cm-1 1.0043 B3LYP/cc-pVDZ 29 cm-1 42 cm-1 246 cm-1 0.9704 Lecture II
- 86. triphenylene Energy, frequencies, and Ramanand IR intensities calculations DFT BLYP/cc-pVDZ Linux 2.4GHz machine 32 hours SCC-DFTB Linux 333MHz machine 24 seconds Performance Efficiency of DFTB frequencies Lecture II
- 87. Performance Improvement of DFTBERep H. A. Witek, et al, J. Theor. Comp. Chem. 4, 639 (2005) and others Lecture II
- 88. 88 H. A. Witek,SI, G. Zheng, W. A. de Jong, K. Morokuma, J. Chem. Phys. 125, 214706 (2005) Performance C28 D2 Harmonic IR and Raman Spectra of C28 Lecture II
- 89. 89 H. A. Witek,SI, G. Zheng, W. A. de Jong, K. Morokuma, J. Chem. Phys. 125, 214706 (2005) Performance Harmonic IR and Raman Spectra of C60 C60 D2h~Ih Lecture II
- 90. 90 H. A. Witek,SI, G. Zheng, W. A. de Jong, K. Morokuma, J. Chem. Phys. 125, 214706 (2005) Performance Harmonic IR and Raman Spectra of C70 C70 D5h Lecture II
- 91. 91 Performance SCC-DFTB: Systematic comparisonwith other methods by Walter Thiel et al. Thiel and coworkers, J. Phys. Chem. A 111, 5751 (2007) (reference: exptl. H0) Lecture II
- 92. 92 Performance Thiel and coworkers,J. Phys. Chem. A 111, 5751 (2007) SCC-DFTB: Systematic comparison with other methods by Walter Thiel et al. Lecture II
- 93. 93 Performance But: NCC- andSCC-DFTB for radicals: own comparison with B3LYP/6-311G** Lecture II
- 94. 94 Performance NCC- and SCC-DFTBfor radicals: own comparison NCC-DFTB B3LYP/6-311G(d,p) SCC-DFTB B3LYP/6-311G(d,p) Lecture II
- 95. 95 A B C DFT:PW91[1]-6.24 -5.63 -1.82 SCC-DFTB[2] -5.17 -4.68 -1.86 Adhesion energies (eV/atom) A B C PW91: An ultrasoft pseudopotential with a plane-wave cutoff of 290 eV for the single metal and the projector augmented wave method with a plane-wave cutoff of 400 eV for the metal cluster Fe-Fe and Fe-C DFTB parameters from: G. Zheng et al., J. Chem. Theor. Comput. 3, 1349 (2007) [1] Phys. Rev. B 75, 115419 (2007) [2] Fermi broadening=0.13 eV H10C60Fe H10C60Fe H10C60Fe55 Fe55 icosahedron H-terminated (5,5) armchair SWNT + Fe atom/cluster PerformanceLecture II
- 96. 96 Acknowledgements • Marcus Elstner •Jan Knaup • Alexey Krashenninikov • Yasuhito Ohta • Thomas Heine • Keiji Morokuma • Marcus Lundberg • Yoshio Nishimoto • Some others Acknowledgements Lecture II