1. Application of DFTB in molecular electronics
Jeffrey R Reimers, Gemma C. Solomon, Zheng-Li Cai, Noel S. Hush,
School of Chemistry, The University of Sydney, Australia
Alessio Gagliardi, Thomas Frauenheim,
Department of Theoretical Physics, Paderborn University, Germany,
Theoretical Physics Department, University of Bremen, Germany
Alessandro Pecchia, and Aldo Di Carlo
Department of Electronic Engineering, University of Rome "Tor Vergata", Italy

2. Summary What is Molecular Electronics
The “gDFTB” method for molecular electronics applications
Why use DFTB ?
Problems with standard DFT
Does DTFB offer any intrinsic advantages ?
Is DFTB accurate enough ?
Use of gDFTB in interpreting experiment
Implementing Symmetry in DFTB
Nature of molecular conduction channels

3. Molecular Electronics: Measuring single molecule conduction
Wang et al.
PRB 68 (2003)
Nanopore
Kushmerick et al. PRL 89 (2002)
Cross-wire
STM Break Junction
B. Xu & N. J. Tao Science (2003) 301, 1221
Scanning Probe
Cui et al.
Science 294
(2001) 571
Electromigration
H. S. J. van der Zant et al.
Faraday Discuss. (2006)
131, 347
Nanocluster
Mechanical Break Junction
Dadosh et al. Nature
436 (2005) 677
Reichert et al. PRL

4. Single-Molecule Conductivity
L ELECTRODE
R ELECTRODE
MOLECULE

5. Single-Molecule Conductivity
L ELECTRODE
R ELECTRODE
MOLECULE
Molecular Orbitals
Fermi energy

6. Single-Molecule Conductivity
L ELECTRODE
R ELECTRODE
MOLECULE
Molecular Orbitals eV V I

7. Finding a true molecular signature: Inelastic Electron Tunnelling Spectroscopy (IETS)
h/e V
Elastic V
h/e
dI/dV
Inelastic
h/e V
d2I/dV2
h/e V

8. Application to molecules
J. GKushmerick, J. Lazorcik, C. H. Patterson & R. Shashidhar
Nano Lett. (2004) 4(4) 639
W. Wang, T. Lee, I. Kretzschmar & M. Reed Nano Lett. (2004) 4(4) 643

9. Shot noise measurements
Garcia et al. Phys. Rev. B (2004) 69,
Thygesen & Jacobsen
Phys. Rev. Lett. (2005) 94,
Djukic & Van Ruitenbeek Nano Lett. (2006) 6(4), 789
Smit et al. Nature (2002) 419, 906

10. “gDFTB” Method for Calculating the Current
Non-Equilibrium Green’s Function (NEGF) formalism
Implementation developed at Tor Vergata
Reduces to Landauer Formalism in some instances (eg., coherent current but not for IETS)
DFTB implementation developed at Paderborn / Dresden
called “gDFTB”
calculates the system Hamiltionain H for electrode-molecule-electrode system
requires an optimized geometry
requires vibrational analysis for IETS
See Poster COMP 300 by Gagliardi et al.

11. Diagonal blocks are the energies of each part
Partitioning the Electrode-Molecule-Electrode Hamiltonian Operator for the System Energy L M R
Diagonal blocks are the energies of each part
Mujica, Kemp, Ratner, J. Chem. Phys. 101 (1994) 6849.

12. Off- Diagonal blocks are the interaction energies
Partitioning the Electrode-Molecule-Electrode Hamiltonian Operator for the System Energy L M R
Off- Diagonal blocks are the interaction energies
Mujica, Kemp, Ratner, J. Chem. Phys. 101 (1994) 6849.

13. Landauer Formalism
Mujica, Kemp, Ratner, J. Chem. Phys. 101 (1994) 6849.

14. Why DFTB? General Serious Failures of DFT
Dispersion
Covalent bond breakage
Partial electron removal/addition (long range electron-transfer processes)
Extended  conjugation
ALL RELEVANT
TO PHOTONICS AND
MOLECULAR ELECTRONICS !
Can DFTB do better ???
Reimers, Cai, Bilić, Hush, Ann. N.Y. Acad. Sci (2003) 235.

15. DFT Failure (1): Dispersion error leads to poor adsorption energies
Molecule
Surface
Observed
PW91 Calculated
NH3
Au(111)
7.5-10 8
Benzene 9 2
Cu(111) 14 1
Cu(110) 23 6
kcal/mol
kcal/mol
DFT calculations for benzene on a Cu13 model cluster for (110) = 19 kcal/mol
CASPT2 dispersion energy error for DFT = 15 kcal/mol
Bilić, Reimers, Hush & Hafner J. Chem. Phys. 116 (2002) 8981
Bilić, Reimers, Hoft, Ford & Hush J. Theor. Comput. Chem (2006).

16. DFT Failure (2): Covalent Bond Breakage
H2: Source of long-range correlation
Single bonds break properly if  and  electrons have different orbitals
Cai & Reimers J. Chem. Phys. 112 (2000) 527

17. Time-Dependent DFT (TDDFT) collapses for excited states
The triplet instability has a profound effect for TDDFT and its analogue RPA (use H0 + H1 + H2 )
CIS is OK (uses H0 + H1 )
Cai & Reimers J. Chem. Phys. 112 (2000) 527

18. Electrode cluster – molecule – Electrode cluster MODEL SYSTEM
Application to the weak electrode-electrode through molecule bonds that drive single-molecule conductivity experiments
Electrode cluster – molecule – Electrode cluster
MODEL SYSTEM
Typical pair of weakly coupled orbitals
Actually there are 2 such pairs !
Solomon, Reimers and Hush J. Chem. Phys. 112 (2000) 527

19. Fermi Level of system is OPEN SHELL
Solomon, Reimers and Hush J. Chem. Phys. 112 (2000) 527

20. Closed-Shell treatments lead to split orbitals
Closed-Shell GGA density functionals have incorrect asymptotes but maintain double degeneracy … results in additional weak conduction channels … is useful
Closed-shell hybrid density fucntionals gives asymptotically very poor result … perceived as strong coupling, resultant currents x 100 too high … useless
Open-shell calculation gives asymptotically correct answer
Solomon, Reimers and Hush J. Chem. Phys. 112 (2000) 527

21. DFT Failure (3): Partial Electron Removal
All modern functionals have an incorrect asymptotic potential
Should be Is
Vx as a function of nuclear - electron distance r for the H atom
Taken from Tozer et al. J. Chem. Phys. 112 (2000) P3507

22. DFT band lineup error for phenylthiol (RSH) on gold(111)
RSH RS• RS– Adsorbate Gold (111)
Obs PW PW PW Bridge FCC PW Obs.
Band-gap error 5.6 eV
Band lineup error 3.4 eV
Bilić, Reimers and Hush J. Chem. Phys. 122 (2005)

23. DFT Failure (4): Conjugated  Systems
Examples …. overestimation of metallic-like properties
Collapse of band-gap in oligoporphyrin molecular wires
Appearance of charge-transfer bands in porphyrins and chlorophylls
Loss of band gap in polyacetylene, very high NLO properties

24. Oligoporphyrins
Sendt, Johnston, Hough, Crossley, Hush & Reimers J. Am. Chem. Soc. 124 (2002) 9299
Cai, Sendt & Reimers J. Chem. Phys. 117 (2002) 5543

25. Can DFTB be better ? Dispersion – yes, via empirical corrections
Covalent bond breakage – yes, no singlet/triplet thus no triplet instability !
Partial electron removal/addition (long range electron-transfer processes) ???
Extended  conjugation ???

26. SCC-DFTB errors for properties of 63 Mg complexes
Property
B3LYP
SCC-DFTB
AM1
PM3
MNDO-d
PM5
Bond length / Å
Ave
.00
.02
-.03
-.10
-.05
-.06 
.06
.11
.18
.10
.09
Bond Ang. / ° -1 -2 -9 1 4 14 25 15 3
IP / eV
.20
-.13
.27
.33
.23
.22
.34
.66
.51
.65
.44
.50
Hf / kcal mol-1
149 -4 -6 7
246 19 18 23
Incr. Ligand 8 16 6
Binding / kcal mol-1 21 17
Deprotonation -7 2
-17
Energy / kcal mol-1 10 11 9 27 22
Comp. to either experiment or else CBS or else QCISD
Cai, Lopez, Reimers, Cui, Elstner in prep.

27. SCC-DFTB geometries of thiols on Au(111)
Alkane chain
S head group
p(5  5) Au surface cell
optimized geometry has S on a top site
DFT calculations predict either FCC or bridge-distorted FCC site
experiments indicate top site but may involve Au adatom instead
Solomon,Gagliardi, Pecchia, Frauenheim, Di Carlo, Reimers, Hush J.C.P. (2006) 124,

28. Observed and gDFTB-calculated IETS
Reed’s experiment
Calculations match and enhance experimental assignment
W. Wang, T. Lee, I. Kretzschmar & M. Reed Nano Lett. (2004) 4(4)
Binding site
Solomon,Gagliardi, Pecchia, Frauenheim, Di Carlo, Reimers, Hush J.C.P. (2006) 124,

29. Effect of the binding site on CH intensity
Wang, Lee, Kretzschmar & Reed Nano Lett. (2004) 4 643
Opt structure Calculated IETS
lower energy
higher energy
J. Kushmerick, Lazorcik, Patterson & Shashidhar Nano Lett. (2004) 4 639
Solomon,Gagliardi, Pecchia, Frauenheim, Di Carlo, Reimers, Hush J.C.P. (2006) 124,

30. Importance of molecular symmetry
Vibrations are characterized by their symmetry.
What are the selection rules for IETS?
What is the nature of the conduction channels through the molecule?
How many are there?
What is the role of the junction region?
What is the role of the molecule and its molecular orbitals

31. Implementing symmetry in SCC-DFTB
Find all atoms related by the Albelian symmetry operators C2 (two-fold rotation),  (reflection plane), and i (inversion)
Construct the transformation S that forces all atomic orbitals (AO), Cartesian tensor components, etc., to be eigenfunctions of these operators
Transform the Kohn-Sham matrix H, force vector, Hessian matrix of second derivatives, etc. from AO basis and Cartesian coordinates into symmetry adapted representations:
H = ST H S
Diagonalize H to get symmetry-adapted molecular orbitals C
Back transformation to get molecular orbitals in AO basis
C = S C
Solomon,Gagliardi, Pecchia, Frauenheim, Di Carlo, Reimers, Hush J.C.P. (2006) submitted

32. Numerical Advantages
Numerical error is removed (numbers that should be zero ARE zero)
Force optimization of transition states and saddle points
Block diagonalization gives speedup (eg,  4 for C2v)
eg. Say that H has transforms according to the C2v point group
symmetry operators C2z, xz, yz, and E
irreducible representations a1, a2, b1, and b2 a1 a2 b1 b2
a1 a b1 b2
H =

33. What is the point group in gDFTB calculations?
Symmetry of entire system H is C2h (operators are C2z, xy, and i)
Symmetry of molecular component HM is C2h
Symmetry of individual molecule- electrode couplings JL and JR is Cs only
gDFTB equations use JL and JR explicitly hence there a new quantity is needed, the
MOLECULAR CONDUCANCE POINT GROUP
Solomon,Gagliardi, Pecchia, Frauenheim, Di Carlo, Reimers, Hush J.C.P. (2006) submitted

34. Determining the Molecular Conductance Point Group
Eg., for chemisorbed 1,4-benzenedithiol S- C6H4-S
All symmetry operators that enforce end-to-end symmetry are lost
All other symmetry operators are retained
In this case, D2h  C2v
Solomon,Gagliardi, Pecchia, Frauenheim, Di Carlo, Reimers, Hush J.C.P. (2006) submitted

35. Conduction split into symmetry channels
Total transmission
A2 component
Ef = Fermi energy of Au, controls low-voltage conductivity … its B1 !
Solomon,Gagliardi, Pecchia, Frauenheim, Di Carlo, Reimers, Hush J.C.P. (2006) submitted

36. The transmission through each symmetry block can then be partitioned in other ways:
1. Büttiker eigenchannels (shot noise)
2. Junction eignchannels coupled by the molecule
3. Interference between Molecular Conductance Orbitals coupled through the junction
Solomon,Gagliardi, Pecchia, Frauenheim, Di Carlo, Reimers, Hush Nano Letts (2006) in press

37. Harnessing the power of DFTB
Au atoms per electrode:
Black- 3
Red- 25
Solomon,Gagliardi, Pecchia, Frauenheim, Di Carlo, Reimers, Hush J.C.P. (2006) submitted

38. Conclusions
gDFTB formalism provides powerful application areas to molecules coupled to solid-state devices
implementation of symmetry into SCC-DFTB code
provides faster and more stable central algorithm
provides key information for understanding molecular systems
must be careful to use DFTB only for suitable properties
initial applications in molecular electronics encouraging
conduction channels
IETS vibrational spectroscopy
basic behaviour of method not yet fully characterized
ready for testing on large systems

39. The end

40. ACS Abstract Application of DFTB in molecular electronics
Jeffrey R Reimers, Gemma C. Solomon, Zheng-Li Cai, Noel S. Hush, Alessio Gagliardi, Thomas Frauenheim, Alessandro Pecchia5, and Aldo Di Carlo, (1) School of Chemistry, The University of Sydney, Sydney, 2006, Australia, (2) School of Molecular and Microbial Biosciences, The University of Sydney, Sydney, 2006, Australia, (3) Theoretical Physics Department, University of Bremen, Germany, Vogeliusweg , Paderborn, , Germany, (4) Bremen Center for Computational Materials Science, Bremen University, Bibliothekstrasse 1, Bremen, 28359, Germany, (5) Department of Electronic Engeneering, University of Rome "Tor Vergata", Rome, Italy
Molecular electronics involves the passing of current between two electrodes through a single conducting molecule. Calculations in this area require not only the ability to handle large systems including metal-electrode fragments but also require accurate positioning of molecular and metallic energy bands and must treat occupied and virtual orbitals on an equivalent footing. Each of these requirements presents difficulties for standard DFT calculations, making DFTB an attractive alternative proposition. We present enhancements to the SCC-DFTB program that allow it to diagnose and utilize molecular symmetry, increasing computational speed and accuracy whilst providing important information concerning molecular orbitals and molecular vibrations. Optimized geometries are then obtained for molecules sandwiched between gold electrodes, leading to Green's-function based calculations of steady-state through-molecule electrical conductivity and incoherent inelastic tunnelling spectroscopy (IETS) arising from electrical current activation of molecular vibrational modes.

41. When the junction symmetry is less than that of the Molecular Conductance Point Group
Black- 3 Au, exact
Green- 3 Au, using higher symmetry
Red- 25 Au, exact
Solomon,Gagliardi, Pecchia, Frauenheim, Di Carlo, Reimers, Hush J.C.P. (2006) submitted