Recent Developments and Applications of the DFTB

Method

Keiji MorokumaCherry L. Emerson Center for Scientific Computation

and Department of Chemistry, Emory University, Atlanta, GA

(from 9/06, also Fukui Institute for Fundamental Chemistry, Kyoto University, Kyoto, Japan)

A Symposium on DFTB: Theory and Applications 232st ACS National Meeting

San Francisco, September 10-14, 2006

closed

Emory GroupDr. Guishan Zheng (Gaussian, analytical

functions, TM, nano)

Prof. Henryk Witek (Hessian,TM)

Dr. Stephan Irle (nano,TM)

Zhi Wang (nano)

Benjamin Finck (nano)

Dr. Petia Bobadova-Parvanova (TM)

Dr. Djamaladdin G. Musaev (TM)

Prof. Rajeev Prabhakar (TM)

Gaussian, Inc.Dr. Michael J. Frisch

Dr. Thom Vreven

Paderborn GroupProf. Marcus Elstner (Braunschweig)

Dr. Christof Köhler (Bremen)

Prof. Thomas Frauenheim (Bremen)

SupportNational Science Foundation

Air Force Office of Scientific Research (DURIP)

Mitsubishi Chemical Corporation

ACS, Petroleum Research Funds

Gaussian, Inc.

Pacific Northwest National Laboratory, EMSL Grand Challenge

Acknowledgment

I. Analytical Hessian in DFTBWitek, Irle and KM, J. Chem. Phys. 121, 5163 (2004).

Witek, KM and Stradomska, J. Chem. Phys., 121, 5171 (2004). Witek and KM, J. Comp. Chem. 25, 1858 (2004).

Malolepsza, Witek and KM, Chem. Phys. Lett. 412 237 (2005) .Witek, Stradomska and KM, J. Theo. Comp. Chem. 4, 639 (2005).Witek, Zheng, Irle, de Jong and KM, J. Chem. Phys. In revision.

Analytical Hessian

€

GabDFTB =

∂ 2E rep

∂a∂b+ 2 ni Umi

b cμmcν i∂Hμν

0

∂a−ε i∂Sμν∂a

⎡

⎣ ⎢

⎤

⎦ ⎥

μν

AO

∑im

MO

∑ +

+ ni cμicν i∂ 2Hμν

0

∂a∂b−ε i∂ 2Sμν∂a∂b

−∂ε i∂b

∂Sμν∂a

⎡

⎣ ⎢

⎤

⎦ ⎥

μν

AO

∑i

MO

∑

€

GabSCC-DFTB =

∂ 2E rep

∂a∂b+ 2 ni Umi

b cμmcν i∂Hμν

0

∂a+

1

2γMK + γNK( )ΔqK

K

atoms

∑ −ε i ⎛

⎝ ⎜

⎞

⎠ ⎟∂Sμν∂a

⎡

⎣ ⎢

⎤

⎦ ⎥

μν

AO

∑im

MO

∑ +

+ ni cμicν i∂ 2Hμν

0

∂a∂b+

1

2γMK + γNK( )ΔqK

K

atoms

∑ −ε i ⎛

⎝ ⎜

⎞

⎠ ⎟∂ 2Sμν∂a∂b

⎡

⎣ ⎢

⎤

⎦ ⎥

μν

AO

∑i

MO

∑ −

− ni cμicν i∂ε i∂b

−1

2

∂γMK∂b

+∂γNK∂b

⎛

⎝ ⎜

⎞

⎠ ⎟ΔqK −

1

2γMK + γNK( )

∂ΔqK∂bK

atoms

∑K

atoms

∑ ⎡

⎣ ⎢

⎤

⎦ ⎥

μν

AO

∑i

MO

∑∂Sμν∂a

+

+∂ 2γAK∂a∂b

ΔqAΔqKK

atoms

∑ +∂γAK∂a

∂ΔqA∂b

ΔqKK

atoms

∑ +∂γAK∂a

ΔqA∂ΔqK∂bK

atoms

∑

Accuracy of DFTB frequenciesTesting set of 66 molecules, 1304 distinct vibrational modes (cm-1)

mean absolute deviation

standard deviation

maximal absolute deviation

scaling

factor

SCC-DFTB 56 82 529 0.9933

DFTB 60 87 536 0.9917

AM1 69 95 670 0.9566

PM3 74 102 918 0.9762

HF/cc-pVDZ 30 49 348 0.9102

BLYP/cc-pVDZ 34 47 235 1.0043

B3LYP/cc-pVDZ 29 42 246 0.9704

IR and Raman Spectra of C70

Dispersion Contribution

SCC-DFTB-D (SCC-DFTB with dispersion) is based on a London-type dispersion energy between atoms :

London dispersion interaction energy is only valid in case of non-interacting charge densities. Therefore, damping function f(R) has to be introduced for small interatomic distances:

N=7, M=7, d=3.0, R0=3.8Å 1st row, 4.8Å 2nd row elements

M. Elstner et al., JCP 114, 5167 (2001))

undamped

damped

We implemented dispersion contributions to Hessian

II. Development of analytical functions

for DFTB parameters

Guishan Zheng, COMP 342, Wednesday, 4.45pm

Motivation and Form

Original two-center parameters (overlap, Hamiltonian and repulsion) given on several hundred 1D grid points

Functions are smooth and accurate for high order energy derivative

Much smaller parameter data base Functional form used:

€

f (R) = Cii=1

10

∑ exp(−αβ iR)

An example of fitting: Overlap integral

An example of fitting:Core-core repulsion

RMS and maximum deviation between fitted values and original grid values of repulsion curves. The unit used is Hartree.

RMS E [a.u.] Max E [a.u.]hh_r 3.25E-05 2.44E-04hc_r 1.83E-04 1.19E-03hn_r 8.78E-05 7.27E-04ho_r 1.55E-04 1.27E-03ch_r 1.83E-04 1.19E-03cc_r 6.97E-04 4.76E-03cn_r 2.41E-04 1.19E-03co_r 1.17E-04 6.08E-04nh_r 8.78E-05 7.27E-04nc_r 2.41E-04 1.19E-03nn_r 4.71E-04 1.71E-03no_r 6.57E-04 4.07E-03oh_r 1.55E-04 1.27E-03oc_r 1.17E-04 6.08E-04on_r 6.57E-04 4.07E-03oo_r 1.35E-04 3.38E-04

Test optimization calculations for 264 molecules

• All 264 molecules consist of H, C, N and O atoms. • Geometry optimization starting from the same point

using the original parameter and the fitted function forms• The optimized geometries are superposed in order to

compare how close they are.• The average geometry superposition deviation of 264

molecules between the original numerical grids and the new analytical functions are 0.007Å!

• Total energies are slightly different (should not mix two total energies), but energy differences (bond energies, etc.) are very close.

III. Development of DFTB parametersfor first-row transition metal elements (Sc, Ti, Fe, Co, Ni)

Guishan Zheng, Henryk Witek, Petia Bobadova-Parvanova, Stephan Irle, Djamaladdin G. Musaev, Rajeev Prabhakar, Keiji Morokuma,

Marcus Elstner, Christof Köhler and Thomas Frauenheim, J. Chem. Theo. Comp. to be submitted

Guishan Zheng, COMP 342, Wednesday, 4:45pm

-----

Aslo Henryk Witek, COMP 339, Wednesday, 3:00pm

Motivation

• Extend the applicability of DFTB to problems containing transition metal elements. No reliable semiempirical method for TM. (Cu, Zn: Elstner, Cui, et al. 2003. Au: Koskinen, Seifert,

2006).

• Candidate for low-level QM method in the ONIOM(QM:QM) ot (QM:QM:MM) method

Parameterization for First-Row Transition Metal Elements

• Only 3d, 4s and 4p orbitals are taken into account.

• Spin-polarized DFTB scheme (SDFTB) needs to be considered.

Self-consistency of and spin densities on each shell of atoms is important.

• Analytical functions used for all two-center parameters

• Parameters determined for M-M and M-X with M = Sc, Ti, Fe, Co, and Ni, and X = H, C, N, and O.

Comparison between DFTB and B3LYP/SDD results

H C N OTi 0.06 0.06 0.01 0.02Fe 0.04 0.10 0.06 0.05Co 0.03 0.12 0.03 0.03Ni 0.04 0.16 0.08 0.02

Absolute mean bond length difference (Å) between DFTB and B3LYP/SDD results

Absolute mean bond angle difference (degree) between DFTB and

B3LYP/SDD results

H C N OTi 7.2 6.7 3.2 4.8Fe 10.1 2.8 15.6 7.8Co 2.1 5.4 2.9 3.8Ni 3.6 1.7 16.0 9.4

Energy differences of different spin states

SDFTB and DFT (B3LYP/SDD+6-31G(d)) relative energies of different spin states Re lat ive Energ ies (kca l/mol) Compound Multi-

plicities DFT SDFTB Fe -H

FeH 4 2 43.7 33.5 -10.2 FeH 2 3 1 22.0 11.7 -10.3

Fe -C Fe(CH 3)2 3 1 33.1 19.7 -13.4

Fe(C 2H4)+1 4 2 41.6 32.4 -9.2 FeCp +1 5 3 13.8 16.7 2.9

Fe -N Fe(NH 2)2 5 1 33.3 12.4 -20.9 Fe (NH)2 5 3 8.2 27.6 19.4

Fe( 1-N2) 4 2 25.5 37.1 11.6 Fe -O, Fe -Fe

FeO 5 1 10.6 0.6 -10.0 Fe 2O2 3 1 40.4 6.0 -34.4 FeO 2 3 1 26.4 8.0 -18.4

Fe(O 2)+1 4 2 47.1 14.0 -33.1 Fe 2O4 3 1 7.0 0.7 -6.3

Conclusions1. SDFTB analytical parameters for M-M and M-X (M=Sc,

Ti, Fe, CO, Ni; X=H, C, N, O) have been determined.

2. SDFTB optimized geometries for M-containing compounds agree well with B3LYP results, except for very weak bonding cases.

3. SDFTB energetic orders qualitatively agree with B3LYP in most cases. Quantitatively cases exist with over- and underestimation of as large as 50 kcal/mol. Use with care. More tests needed.

4. A good candidate as a low-level QM method in ONIOM. Calibration in progress.

5. These analytical parameters will be available for download very soon.

IV. Implementation of the DFTB method

into Gaussian03

COMP 342, Wednesday, 4.45pm: An efficient implementation of Density-Functional based Tight-Binding method (DFTB) in Gaussian 03 program:

The calculation of energies, gradients, vibrational frequencies and IR spectrums Guishan Zheng, Michael Frisch and Keiji Morokuma

Motivation / Implementation

• Extend the applicability of DFTB to wider range of problems, e.g. TS.

• Take advantage of existing Gaussian functionalities, e.g. SCF convergence techniques, partial geometry optimization

• A smooth combination with ONIOM method• Numerically efficient and stable implementation for

dealing with large molecular systems, including Gradient, Hessian, IR and Raman intensities.

• SDFTB (both restricted and unrestricted) formalism with explicit on-site charge interaction and “analytical” parameter functions.

V. Applications to Nano Structure and Dynamics

COMP 126, Monday 425pm: Quantum chemical molecular dynamics study of catalyst-free SWNT growth from SiC-derived carbon

Zhi Wang, Stephan Irle, Guishan Zheng, K. Morokuma, Michiko Kusunoki

COMP 159, Tuesday 230pm: DFTB-based QM/MD simulations of nanostructure formation processes far from thermodynamic equilibrium

Stephan Irle, Zhi Wang, Guishan Zheng, Keiji Morokuma

COMP 396, Thursday 1105am: The use of ONIOM in computational nanomaterials research

Stephan Irle, Zhi Wang, Keiji Morokuma

PHYS 269, Tuesday 120pm, QM/MD simulations of carbon nanotube and fullerene growth and dynamics

Stephan Irle, Guishan Zheng, Zhi Wang, Keiji Morokuma

Va. Dynamics of formation of fullerenes and carbon nanotubes

0.0 ps 0.1 ps 1.6 ps 8.5 ps 14.5 ps

40.2 ps 56.8 ps 81.1 ps 94.7 ps 104.1 ps

158.1 ps 320.1 ps 320.4 ps 360.0 ps 361.5 ps

2000 K 2000 K 2000 K 2000 K 2000 K

2000 K 2000 K 3000 K 3000 K 3000 K

3000 K 3000 K 3000 K 3000 K

+10 C2 +20 C2

+10 C2 +10 C2

“Shrinking Hot Giant” Road of Fullerene FormationJ. Phys. Chem. B 110, 14531(2006).

Sc-entrapment in S3 trajectory at 2000 K(Finck. Irle, Morokuma: unpublished)

0 ps 0.7 ps 1.3 ps

12.1 ps 27.8 ps 36.5 ps

Nanotube growth from graphene sheets from the C face of SiC (JCP, 2006; unpublished)

42ps 48ps: cap form

48ps: remove Si

96ps

42ps: Add one layer SiC

60ps78ps

108ps

Vb. The Origin of Linear Relationship

between CH2/NH/O-SWNT Reaction

Energies and Sidewall Curvature:

Armchair Nanotubes

G. Zheng, Z. Wang, S. Irle, and K. Morokuma, J. Am. Chem. Soc. in press.

Linear relationship has long been recognized experimentally and

computationally between the electrophile reaction energy and the

sidewall curvature of SWCNT.

What is the origin?

• DFTB: SCC-DFTB, geometry optimizations by Gaussian’s ‘external’ keyword/script

• DFT: B3LYP/6-31G(d)

• System: 15 Å (n,n) SWNT+O/CH2/NH DFTB: n=2~13 DFT: n=2~6

• Studied Thermodynamic stabilities of exo- and endo-Adducts

1.41.61.82.02.22.4-100-90-80-70-60-50-40-30-20-10X=CH2

Ca-Ca distance [A]

Δ [ / ]E Kcal mol

(6,6)SWNT (8,8)SWNT (10,10)SWNT graphite

1.4 1.6 1.8 2.0 2.2 2.4-100-90-80-70-60-50-40-30-20-10X=NH

1.4 1.6 1.8 2.0 2.2 2.4-100-90-80-70-60-50-40-30-20-10X=O

Exo-addition

Endo-addition

1.41.61.82.02.22.4-50-40-30-20-100102030405060X=CH2

Δ [ / ]E Kcal mol

(6,6)SWNT (8,8)SWNT graphite

1.4 1.6 1.8 2.0 2.2 2.4-50-40-30-20-100102030405060X=NH

Ca-Ca distance [A]1.4 1.6 1.8 2.0 2.2 2.4-50-40-30-20-100102030405060X=O

Closed vs. Open Form(DFTB)

open

closed

closed

0.000.030.060.090.120.150.18-120

-110-100-90-80-70-60-50-40-30-20-100

0.000.030.060.090.120.150.18-120-110-100-90-80-70-60-50-40-30-20-100

0.000.030.060.090.120.150.18-120-110-100-90-80-70-60-50-40-30-20-100

(4,4)

(13,13)(13,13)

Δ( / )E kcal mol

(4,4)

(4,4)

(13,13)(13,13)

(13,13)

(6,6)(8,8)

(4,4)(13,13) (10,10)

1/d(A-1)

(4,4)

(4,4)

(13,13)(13,13)

(13,13)

exo(l) exo(s) endoX=OX=NHX=CH2

DFTB Stabilization Energy : Exo/Endo 1/d Plots

openclosedclosed

Relationship of energy components vs 1/d

A: ΔE = DEF + INT for CH2 adducts on the 15 Å SWNT with DFTB

B: INT = ES + EX + ORB for CH2 adducts on the 5 Å SWNT with

HF/STO-3G.

exoendo

Closed only

C CX

C CX

C CXexo(l) exo(s) endo

Scheme 1. Schematic depiction of the different bonding types in exohedrally and

endohedrally functionalized SWNTs.

Comparison of Orbital Interaction between Exo vs. Endo Adducts

a1 b1

endo

exo

Scheme 2. Schematic depiction of major SWNT→CH2 (donative a1) and CH2→CNT

back- (donative b1) interactions for exohedral and endohedral reactions of .SWNTs

Vc. Handedness by DFTB-D

Vs.

Suenaga et al. PRL’05 Tashiro et al. JACS’06

K. Suenaga, H. Shinohara, S. Iijima: Determination of Handedness in Optical Active Chiral DWNTs, PRL 95, 187406 (2005)

Models:(14,3)@(17,10) (3,14)@(17,10) (14,3)@(10,17) (3,14)@(10,17)

tiltedmodels

simulatedHRTEM of

tiltedmodels

HRTEM ofreal tube

“Fringe counting: this is a (14,3)@(17,10) DWNT”

DWNT with same LL/RR handedness are predominantly found!

NCC-DFTB-D: 15 Å tubes(14,3)SWNT: C252H34 (17,10)SWNT: C374H54

GDVE02+ opt=loose external DFTBD full geometry optimizations

Total stoichiometry: C626H88

SWNT NCC-DFTB-D [ha] DWNT Δ ( ) [ ]E init eVΔ ( ) [ ]E opt eV17_10 -668.332824 a -15.10 -21.9710_17 -668.332824 b -14.88 -21.9614_3 -449.100762 c -14.82 -21.963_14 -449.100762 d -15.07 -21.97

Absolutely no predominance found!

SWNT L NCC-DFTB-D [ha] DWNT DE(init) [eV]DE(opt) [eV]17_10 15.0 -668.332824 a -15.10 -21.9710_17 15.0 -668.332824 b -14.88 -21.9614_3 15.0 -449.100762 c -14.82 -21.963_14 15.0 -449.100762 d -15.07 -21.9717_10 30.0 -1258.486464 a -43.19 -43.2910_17 30.0 b -42.80 -43.4214_3 30.0 -837.675932 c3_14 30.0 d9_7 15.0 -385.995271 a 246.84 -18.4220_5 15.0 -642.431432 b 246.83 -18.42

Still no predominance found!

Our presumption holds: Chirality preference may be caused at DWNT nucleation stage

What about defects? I-V defects are underway.

Longer tube, different chirality:

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.

Using SCC-DFTB-D w/Co instead of Rh

+y -y0.0 kcal/mol +2.9 kcal/mol

Found two different isomers corresponding to preference for a certain chirality over another.

Conclusion

DFTB is a very useful approximate method for geometries and energies of large molecular systems.

Parameters for transition metal elements (Sc, Ti, Fe, Co and Ni) have been determined.

More general method of determination of inter-element parameters is needed.

An excellent candidate for low-level QM in ONIOM(QM:QM) and ONIM(QM:QM:MM) calculations.

DFTB energy, gradient and Hessian will be available in the Gaussian code very soon.