1. Adrian Lange & John M. Herbert Department of Chemistry Ohio State University Molecular Spectroscopy Symposium, 6/21/07 Spurious charge-transfer contamination in large-scale TDDFT calculations: A public service announcement

2. JMH group Leif Jacobson Dr. Chris Williams Adrian Lange Shoumik Chatterjee

3. The long-range CT problem in TDDFT *Except those who don’t Everyone* knows that TDDFT woefully underestimates long- range CT excitation energies But just what, precisely, constitutes “long range” ?

4. The long-range CT problem in TDDFT *Except those who don’t e–e– ~1.0/R ~0.5/R ~0.2/R CIS (100% HF exchange) LDA (0% HF exchange) BHLYP (50% HF exchange) B3LYP (20% HF exchange) R / Å [ E(R) – E(4.0Å) ] / eV ~1.0/R ~0.5/R ~0.2/R R / Å Ex. energy / eV Long-range intermolecular CT Dreuw et al., JCP (2003) Everyone* knows that TDDFT woefully underestimates long- range CT excitation energies But just what, precisely, constitutes “long range” ?

5. The long-range CT problem in TDDFT Long-range intramolecular CT Everyone* knows that TDDFT woefully underestimates long- range CT excitation energies But just what, precisely, constitutes “long range” ? Dreuw & Head-Gordon JACS (2004) Magyar & Tretiak JCTC (2007) ET

6. Okay, so long-range ET is out of bounds But perhaps the theory is otherwise okay. After all, it works great* for small, gas-phase molecules. * Typically 0.2–0.3 eV accuracy, for the lowest few valence-type excitations potential energy / eV N1–H bond length / Å uracil singlet excited states

7. Okay, so long-range ET is out of bounds But perhaps the theory is otherwise okay. After all, it works great* for small, gas-phase molecules. Unfortunately, no. Spurious CT states have been observed for acetone/formamide in liquid water and clusters: – Bernasconi, Sprik, Hutter (JPC-B 2003; CPL 2004) – CPMD – Besley (CPL 2004) – Q-Chem – Neugebauer, Gritsenko, Baerends (JCP 2005) – ADF * Typically 0.2–0.3 eV accuracy, for the lowest few valence-type excitations

8. Okay, so long-range ET is out of bounds But perhaps the theory is otherwise okay. After all, it works great* for small, gas-phase molecules. Unfortunately, no. Spurious CT states have been observed for acetone/formamide in liquid water and clusters. Bernasconi, Sprik, Hutter CPL (2004) BUT... popular hybrid functionals like B3LYP and PBE0 push these states up by ~1 eV, above the lowest valence bands. Intensity  / eV BLYP B3LYP PBE0 n*n* CT

9. Okay, so long-range ET is out of bounds But perhaps the theory is otherwise okay. After all, it works great* for small, gas-phase molecules. Unfortunately, no. Spurious CT states have been observed for acetone/formamide in liquid water and clusters. Bernasconi, Sprik, Hutter CPL (2004) BUT... popular hybrid functionals like B3LYP and PBE0 push these states up by ~1 eV, above the lowest valence bands. How robust are these hybrids?

10. A sequence of uracil–water clusters R = 1.5 Å N water = 0 R = 2.0 Å N water = 4 R = 3.0 Å N water = 15 R = 2.5 Å N water = 7 R = 3.5 Å N water = 18 R = 4.0 Å N water = 25 R = 4.5 Å N water = 37 Extracted from a single MD snapshot (T=298 K,  =1.0 g/cm 3 )

11. TD-PBE0 results vs. cluster size Ex. energies below 6 eV40th excitation energy QM region: PBE0/6-31+G*

12. TD-PBE0 results vs. cluster size QM region: PBE0/6-31+G* MM region: TIP3P charges out to 20 Å Ex. energies below 6 eV40th excitation energy

13. Typical CT excited states 4.5 eV 5.6 eV 4.3 eV 4.5 eV blue = detachment density purple = attachment density

14. TDDFT/TDA working equations Solve eigenvalue eqn. Ax =  x for excitation energies , where x = (x ia ) is a vector of occupied (|i>) to virtual (|a>) excitation amplitudes If |i> and |a> are spatially distant, then [Dreuw et al. JCP (2003)] TIP3P charges stabilize water lone pairs on the edge of the cluster, pushing water-to-uracil CT excitations to higher energy A ia,jb = (  a –  i )  ij  ab – c HF (ij|ab)

15. QM/MM: Pure vs. hybrid functionals QM cluster radius / Å Ex. energies below 6 eV 40th excitation energy QM cluster radius / Å B3LYP (c HF = 0.2) behaves much the same as PBE0 (c HF = 0.25) (c HF = 0) (c HF =0.25) (c HF =0)

16. QM/MM electronic absorption spectra B3LYPPBE0Size of QM region R = 1.5 Å (uracil only) R = 2.5 Å (“microhydrated”) R = 4.5 Å (full solvation shell) 40 excited states req’d to reach 6.8 eV

17. Spurious intensity stealing Excitation energies (  i ) and oscillator strengths (ƒ i ) from QM/MM blue = detachment density purple = attachment density

18. Spurious intensity stealing --------------------------------------------------- TDDFT/TDA Excitation Energies --------------------------------------------------- Excited state 1: excitation energy (eV) = 4.2793 Total energy for state 1: -2322.842189693738 Multiplicity: Singlet Trans. Mom.: -0.0640 X -0.0680 Y -0.0578 Z Strength : 0.0013 D(154) --> V( 2) amplitude = 0.9738 Excited state 7: excitation energy (eV) = 5.0455 Total energy for state 7: -2322.814030524368 Multiplicity: Singlet Trans. Mom.: 0.1804 X -0.6829 Y -0.1392 Z Strength : 0.0641 D(151) --> V( 2) amplitude = 0.3653 D(152) --> V( 1) amplitude = 0.3714 D(152) --> V( 2) amplitude = 0.7596 Excited state 8: excitation energy (eV) = 5.0531 Total energy for state 8: -2322.813753118397 Multiplicity: Singlet Trans. Mom.: -0.1953 X 0.4639 Y -0.0488 Z Strength : 0.0317 D(151) --> V( 1) amplitude = -0.5382 D(151) --> V( 2) amplitude = -0.2623 D(152) --> V( 1) amplitude = 0.6666 D(152) --> V( 2) amplitude = -0.2499...... Excited state 9: excitation energy (eV) = 5.0822 Total energy for state 9: -2322.812684185100 Multiplicity: Singlet Trans. Mom.: 0.0763 X -0.3729 Y -0.0894 Z Strength : 0.0190 D(147) --> V( 2) amplitude = 0.3182 D(149) --> V( 2) amplitude = -0.2723 D(151) --> V( 2) amplitude = 0.6347 D(152) --> V( 2) amplitude = -0.5571 Excited state 10: excitation energy (eV) = 5.1582 Total energy for state 10: -2322.809891225142 Multiplicity: Singlet Trans. Mom.: -0.1382 X 0.3452 Y 0.0368 Z Strength : 0.0176 D(140) --> V( 2) amplitude = 0.2390 D(147) --> V( 2) amplitude = 0.5259 D(149) --> V( 2) amplitude = -0.4967 D(151) --> V( 2) amplitude = -0.4595 Excited state 11: excitation energy (eV) = 5.2025 Total energy for state 11: -2322.808260470356 Multiplicity: Singlet Trans. Mom.: -0.0665 X -0.2645 Y -0.2073 Z Strength : 0.0150 D(151) --> V( 2) amplitude = 0.2279 D(154) --> V( 5) amplitude = 0.8782 D(154) --> V( 6) amplitude = -0.3163......

19. Another small system with long-range problems black = attachment density TD-PBE0/6-31+G* calculations on a gas-phase GC base pair

20. Summary: Long-range CT in TDDFT “Long range” is any time (squares of) orbitals do not overlap. Uracil–(H 2 O) 4 is large enough.

21. Summary: Long-range CT in TDDFT “Long range” is any time (squares of) orbitals do not overlap. Uracil–(H 2 O) 4 is large enough. Spurious states impose a major memory bottleneck: N words ~ 2 N O N V N roots N iter / 1.5

22. Summary: Long-range CT in TDDFT “Long range” is any time (squares of) orbitals do not overlap. Uracil–(H 2 O) 4 is large enough. Spurious states impose a major memory bottleneck: N words ~ 2 N O N V N roots N iter / 1.5 QM/MM absorption spectra look okay below 6 eV, even with > 120 QM atoms, but watch out for spurious intensity stealing.

23. Summary: Long-range CT in TDDFT “Long range” is any time (squares of) orbitals do not overlap. Uracil–(H 2 O) 4 is large enough. Spurious states impose a major memory bottleneck: N words ~ 2 N O N V N roots N iter / 1.5 QM/MM absorption spectra look okay below 6 eV, even with > 120 QM atoms, but watch out for spurious intensity stealing. This is a work in progress. Long-range K, subspace truncation, asymptotic correction, etc., are required to make TDDFT a robust method.

24. Long list of spurious charge-transfer states Gaussian user Just because it came from B3LYP doesn’t make it right... Thanks:

26. Spectra from gas-phase clusters Absorption spectrum (Gas-phase QM region) Absorption spectrum (QM/MM) Density of states (Gas-phase QM region) 40 excited states to reach 5.4 eV microhydrated full solvation shell