Local hybrids:

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CDFT 2007 A Coastal Voyage of Current Density
Functional Theory
Local hybrids a new generation of the
exchange-correlation functionals for precise
thermochemistry, kinetics, and response
properties
Alexei V. Arbuznikov, Martin Kaupp, Hilke
Bahmann Institute of Inorganic Chemistry,
University of Würzburg, Germany
Collaborations Alexander Rodenberg
Tromsø-Trondheim, Norway, September 21, 2007
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Outline

• Introduction Explicit density functionals vs.
  orbital-dependent ones
• - features of implementation
• - principal classes of functionals (Jacobs
  ladder, etc.)

• Closer look on the traditional (global) hybrid
  functionals

• Local hybrid functionals
• - theory
• - performance for thermochemistry and reaction
  barriers
• - self-consistent implementation (localized
  local hybrid
• potentials), and performance for response
  properties
• (nuclear shielding constants)

• Links of local hybrids to other
  orbital-dependent functionals
• Beckes real-space model of non-dynamical
  correlation (B05),
• and ways to incorporate current-density
  dependence therein

• Conclusions and outlook

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Density Functional Theory (DFT), Kohn-Sham
approach (KS)

• Energy functional
  

no current dependency yet ?
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Jacobs ladder of density functional
approximations to the exchange-correlation energy
(J. P. Perdew and K. Schmidt, in Density
Functional Theory and its Application to
Materials, V. Van Doren, C. Van Alsenoy, and P.
Geerlings, Eds., AIP, Melville, New York, 2001.)
some other modern functionals that hardly fit
to Jacobs ladder
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Exchange-correlation functionals (another point
of view)

• Explicit density functionals
• - LSDA (?) S-VWN
• - GGA (??) B88-P86, PW91, PBE
• - Laplacian-dependent meta-GGA (?2?)
  Proynov-Salahub-95

2. Occupied-orbital dependent functionals -
?-dependent meta-GGA PKZB, TPSS - global
hybrids B3LYP, PBE0PBE - local hybrids (to be
considered in more details below) - explicit
models for non-dynamical correlation
Becke-05 - range-separated functionals
CAM-B3LYP LC-?PBE - self-interaction
corrections SIC-PZ - many current-density
dependent functionals
3. General nonlocal functionals (virtual-orbital
dependent) - Görling-Levy, 1994 -
Engel-Dreizler, 1999 - Bartlett et el.,
2005 - B2-PLYP (Grimme et el., 2006)
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Hybrids balance between self-interaction
elimination and inclusion of non-dynamical
correlation
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The Next Step in DFT Local Hybrid Functionals
An example admixture of exact exchange within
the CO molecule
Local hybrid functional J. Jaramillo, G. E.
Scuseria, M. Ernzerhof J. Chem. Phys. 118, 1068
(2003).
Earlier hints F. G. Cruz, K.-C. Lam, K. Burke,
J. Chem. Phys. 102, 4911 (1998)
J. P. Perdew, K. Schmidt, The Jacobs
ladder paper (2001).
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Local hybrid functionals

• good reaction barriers (better than B3LYP),

• But! Poor thermochemistry
• (atomization energies are underestimated by ca.
  15-20 kcal/mol ! )

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Other kinds of LMFs

• scaled down ?t ?W /? g ? t, ? lt 1
• H. Bahmann, A. Rodenberg, A. V.
  Arbuznikov, M. Kaupp J. Chem. Phys. 126, 011103
  (2007).

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A properly chosen combination of t- and s-LMFs
may satisfy all these constraints!
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Local vs. global hybrids thermochemical tests
(G2-1 set 55 molecules)
a0 (S-VWN) MAE 34.5 kcal mol-1 (overbinding)
a1 (EXX-VWN) MAE 28.5 kcal mol-1 (underbinding)
A thermochemically competitive local hybrid
functional without GGA part! H. Bahmann, A.
Rodenberg, A. Arbuznikov, M. Kaupp J. Chem. Phys.
126, 011103 (2007).
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Development of mean absolute errors (in kcal
? mol-1) for atomization energies upon
enlargening the test set B3LYP vs. local hybrids
! Only one adjustable parameter in our local
hybrid vs. three parameters in B3LYP !
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Computational kinetics Classical reaction
barriers with local hybrid functionals
Test sets HTBH38, NHTBH38 (76 barriers)
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Computational kinetics Classical reaction
barriers with local hybrid functionals
DFT estimates, kcal / mol
Test sets HTBH38, NHTBH38 (76 barriers)
Best classical barrier estimates, kcal/mol
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What happens as soon as the bond is stretched?
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• Intermediate summary about local hybrid
  functionals
• provide excellent thermochemistry (extremely
  stable wrt the extension
• of the test set up to G3)
• simultaneously improve reaction barriers

What to do if we want other properties?!
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Construction of localized hybrid
exchange-correlation potentials
Global hybrids

• LHF/CEDA approximation
• F. Della Sala, A. Görling,
• J. Chem. Phys. 115, 5718 (2001)
• O.V. Gritsenko, E. J. Baerends,
• Phys Rev. A 64, 042506 (2001)

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Nuclear shieldings for main-group compounds
(IGLO-IV basis set, GIAO) Linear regression
analysis for comparison between theory and
experiment A. V. Arbuznikov, M. Kaupp, Chem.
Phys. Lett. 386, 8 (2004)
22 molecules - 32 shieldings C2H2, C2H4, CH4,
CH3F, CHF3, CO, CO2, F2, H2CO, H2O, H2S,
HCl, HCN, HF, N2, N2O, NH3, O3, P2H2,
PH3, PN, SO2
See also W. Hieringer, F. Della Sala, A.
Görling, Chem. Phys. Lett. 383, 115 (2004) A.
M. Teale, D. J. Tozer, Chem. Phys. Lett. 383,
109 (2004)
Results for electronic g-tensors of 3d transition
metal complexes are also very encouraging ! A.
V. Arbuznikov, M. Kaupp, Chem. Phys. Lett. 391,
16 (2004)
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Self-consistent implementation of local hybrid
functionals provides the first localized
local-hybrid potentials!
traditional hybrid functionals and potentials
Now we can do property calculations (e.g.
NMR/EPR)! A. V. Arbuznikov, M. Kaupp, H. Bahmann
J. Chem. Phys. 124, 204102 (2006).
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Self-consistent implementation of local hybrid
functionals
Two-step procedure
1. Constructing functional derivatives with
respect to the orbitals (FDOs)
2. Transforming FDOs into localized local-hybrid
potentials using OEP-based techniques (in our
case LHF/CEDA approach)
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Performance of LLH potentials for nuclear
shieldings of small main group compounds

• Thermochemically optimized

• Optimized for nuclear shieldings

Local hybrid functionals are more universal than
global hybrid functionals.
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Local hybrid functionals links to other
approaches
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Implementation

• Code MAG - ReSpect ("Relativistic
  Spectroscopy") - since 1998
• by V. G. Malkin, O. L. Malkina, R.
  Reviakine, A. V. Arbuznikov, M. Kaupp,
• B. Schimmelpfennig, I. Malkin, M. Repiský, S.
  Komorovský, P. Hrobarik, E. Malkin,
• T. Helgaker, and K. Ruud, version 2.1, 2006.
• relativistic two-component scheme (Douglas-Kroll
  transformation)
• traditional and new exotic xc functionals and
  potentials OEP
• various NMR and EPR parameters
• etc.

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Conclusions

• General usefulness of the orbital-dependent
  functionals higher accuracy than that
• of explicit density functionals (but, higher
  computational cost)

• Novel class of exchange-correlation functionals
  - local hybrid functionals (position-dependent
  exact exchange admixture) - has been developed
  simultaneously good description of atomization
  energies, reaction barriers, and (via localized
  local hybrid potentials) NMR chemical shifts

Outlook

• Improvement of the computational efficiency

• Better understanding, why do local hybrids
  perform so well (adiabatic connection?)

• Search for still better local mixing functions

• Self-consistent implementation of Beckes
  real-space model for NDC (programming only) -
  without and with jp- dependence

• Switching from LHF/CEDA to the numerically
  stable OEP according to
• A. He?elmann, A. W. Götz, F. Della Sala,
  and A. Görling, J. Chem. Phys. 127, 054102 (2007)

• Extension for solids?

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Acknowledgements
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Thank you for your attention!