Matt Welborn in Tom Miller’s group in Caltech recently published a paper on improved subsystem selection in projector-based WF-in-DFT embedding. Congratulations Matt!

# Category Archives: Research

# Unsöld functional – a new hybrid functional for correlation

There are about four or five distinct ingredients used in modern density functionals:

- integrals of functions of the density and its gradients (GGAs, etc)
- inclusion of the kinetic energy density in such expressions (meta-GGAs)
- a fraction of exact Hartree–Fock exchange (or a length-scale-screened variant)
- a contribution from a wavefunction-based correlation method like MP2 or RPA

Now there is another:

Tim Wiles and F R Manby, ‘Wavefunction-like Correlation Model for Use in Hybrid Density Functionals’, *J. Chem. Theory Comput.*, Article ASAP, DOI: 10.1021/acs.jctc.8b00337.

In this paper we introduce the UW12 correlation functional:

- an explicit functional of occupied orbitals (i.e. a functional of the 1-RDM)
- an F12-like model of dynamic electron correlation
- can be self-consistently optimized in a straightforward way
- most of the complicated stuff handled through familiar DFT quadrature
- no double-summation over virtual orbitals, so fast basis-set convergence

This functional can be used as a viable (and hybrid) alternative to use of MP2 in double-hybrid functionals, as we demonstrate in this paper. Our *hybrid* XCH-BLYP-UW12 functional has only one parameter, and is competitive with more highly parameterised *double hybrids* for reaction barrier test sets.

If you don’t have access and want to read the paper, just email Fred.

Congratulations to Tim for publishing this work!

# How Markovian is exciton dynamics in purple bacteria?

Felix’s work on Markovianity in photosynthetic energy transport has been published: Felix Vaughan, N Linden and F R Manby, J. Chem. Phys. **146**, 124113 (2017); http://dx.doi.org/10.1063/1.4978568.

Modelling the interaction between excitons and the surrounding environment is a non-trivial problem. Many interesting insights about excitonic energy transfer have made use of a Markovian or “memoryless” approximation to this interaction. In this paper we assess the applicability of this approximation by employing a new metric of non-Markovianity.

We find that for smooth spectral densities the Markovian approximation works well provided that a precise change to the system Hamiltonian is made, which for the dimer system studied corresponds to an increase in the coupling strength between chromophores. We also find that discrete vibrational modes resonant with the eigenstates of the Hamiltonian induce the greatest degree of non-markovianity. Ultimately we conclude that to model exciton dynamics coupled to realistic spectral densities a Markovian approximation is not suitable.

# Delocalization errors in WF-in-DFT embedding are large, but fixable

Delocalization error in approximate DFT clearly manifests itself in homodimer cation systems (like H_{2}^{+} or (H_{2}O)_{2}^{+}), with GGA functionals typically leading to large energy errors and qualitatively incorrect structures. It also causes problems in a variety of other chemically important contexts.

We have found that the delocalization error in densities can cause major errors in WF-in-DFT embedding – these errors are not particular to the projector-based scheme we use,but simple expose a limitation of partitioning systems based on the electron density when that electron density is qualitatively flawed.

Following work from Kieron Burke, we have found the simple expedient of using Hartree-Fock densities in WF-in-DFT calculations really improves reliability in cases where there is a serious delocalization error, and doesn’t cause major problems (in the examples we have studied) when there is not a big delocalization error.

You can read about this work in a paper that has just appeared online: Pennifold *et al.*, ‘Correcting density-driven errors in projection-based embedding’, *J. Chem. Phys. ***146**, 084113 (2017);** **DOI: 10.1063/1.4974929.

# Pump-probe experiments on photosynthetic light-harvesting complexes are not easy to interpret

Clem’s paper on the interpretation of pump-probe experiments on the purple-bacteria light-harvesting complex LHII is now in print in J Phys Chem B.

Through careful and extensive calculations involving molecular dynamics, time-dependent density functional theory, and quantum dynamics we have shown that the interpretation of anisotropy decay rates in terms of strength of coupling to a dissipative bath is not so easily justified. The reason is that static (or inhomogeneous) disorder itself produces anisotropy decay at about the experimentally observed rate.

The paper also contains an epic, paper-length appendix on how to compute such quantities for the circularly degenerate oscillator model.

Congratulations Clem!

C. Stross, M. W. Van der Kamp, T. A. A. Oliver, J. N. Harvey, N. Linden and F. R. Manby, “How Static Disorder Mimics Decoherence in Anisotropy Pump–Probe Experiments on Purple-Bacteria Light Harvesting Complexes”, J. Phys. Chem. B, **120**, 11449-11463 (2016), DOI: 10.1021/acs.jpcb.6b09916

# Quantum Drude oscillators without the dipole approximation

Interactions between Drude oscillators (two opposite charges, one fixed, one mobile, interacting by a spring) are almost always modelled in the dipole approximation. This is because the Hamiltonian for describing Drude oscillators in this way is quadratic in the creation/annihilation operators (or equivalently, is quadratic in position and momentum) and can therefore be diagonalized exactly through a normal-mode transformation.

Second-order perturbation theory for the interaction of two such oscillators produces the expected –*C*_{6} *R*^{–6} dispersion interaction. At short distances this diverges, and the exact interaction energy ceases to be defined at all.

Therefore people use screening functions to fix up the short range. How much are the screening functions fixing up the divergence, and how much are they just compensating for the use of the dipole interaction Hamiltonian? Now we can find out, because Mainak has calculated the (almost) exact binding energy between Drude oscillators with the full Coulomb interaction.

Quantum mechanics of Drude oscillators with full Coulomb interaction

M. Sadhukhan and F. R. Manby, Phys. Rev. B **94**, 115106 (2016) 10.1103/PhysRevB.94.115106

# CCSD(T)-in-DFT for enzymology

Our paper *A Projector-Embedding Approach for Multiscale Coupled-Cluster Calculations Applied to Citrate Synthase *has appeared online in JCTC, and was one of the most highly downloaded papers soon after publication.

The great thing about this is that by tuning in CCSD(T) for just a handful of reacting atoms, almost all of the dependence on the approximate exchange-correlation functional is eliminated.