Small Atomic Orbital Basis Set First-Principles Quantum Chemical Methods for Large Molecular and Periodic Systems: A Critical Analysis of Error Sources

Sure, R.; Brandenburg, J. G.; Grimme, S. ChemistryOpen, EarlyView, DOI: 10.1002/open.201500192 (CC by-nc-nd)
Contributed by Grant Hill

The use of density functional theory (DFT) calculations to produce insights into the chemistry unveiled by experiment is widespread due to its relative ease-of-use and ability to explain many chemical phenomena of interest. In particular, the B3LYP hybrid functional [1] and the Pople-type 6-31G* basis set [2] are incredibly popular, with some referring to this as Default Favourite Theory (a play on DFT).[3] A recent review by Grimme and co-workers sets out a case against using this B3LYP/6-31G* model chemistry by careful examination of errors.

The review contends that the use of small basis sets such as 6-31G* leads to relatively large errors due to both basis set superposition error (BSSE) and basis set incompleteness error (BSIE). It's not entirely clear how to separate these two terms and as a result the review mostly focuses on the BSSE element, with some emphasis on intramolecular BSSE in addition to the more familiar intermolecular BSSE. The question of why B3LYP/6-31G* still performs well in a number of cases is then examined in terms of a fortuitous cancellation of errors between BSSE and London forces (dispersion energy). It is demonstrated that this cancellation cannot be relied upon in all cases and a convincing case is made for choosing different basis sets and methods. While the review mostly focuses on intermolecular interactions, there is some generalisation to other problems of interest.

A number of alternative methods for including dispersion in DFT calculations are reviewed, and final recommendations are made that can easily be incorporated into the workflow of a non-specialist, without a significant increase in computational cost. This includes the use of the def2-SVP basis set of Weigend, Ahlrichs and co-workers.[4] This review should make for interesting reading for anyone routinely using DFT methods in conjunction with double-zeta basis sets.

References:
[1]a) Stephens, P. J.; Devlin, F. J.; Chablowski, C. F.; Frisch, M. J. J. Phys. Chem. 1994, 98, 11623. b) Becke, A. D. J. Chem. Phys. 1993, 98, 5648.
[2] Hehre, W. J.; Ditchfield, R.; Pople, J. A. J. Chem. Phys. 1972, 56, 2257.
[3] I first heard this at the Computational Molecular Science conference in 2008, but I can't recall the originator. The late Nick Handy responded by suggesting that DFT could instead be "Damn Fine Theory".
[4] See Weigend, F.; Ahlrichs, R. Phys. Chem. Chem. Phys. 2005, 7, 3297 and references therein. These basis sets are available to download from the EMSL basis set exchange in formats suitable for most electronic structure packages.

Intrinsic Atomic Orbitals: An Unbiased Bridge between Quantum Theory and Chemical Concepts

Gerald Knizia J. Chem. Theory Comput. doi:10.1021/ct400687b (Article ASAP)
Contributed by J. Grant Hill.

Readers of Computational Chemistry Highlights are already aware of the power of quantum chemistry in determining/predicting physical observables, but familiar chemical concepts (including the covalent bond!) are, at best, loosely defined in quantum mechanics. This is neatly summarised in Knizia's paper:

"quantum chemistry methods can determine benzene's heat of formation with an accuracy of <2 kJ/mol; but, strictly speaking, they can neither determine the partial charges on benzene's carbon atoms, nor can they show that benzene has twelve localized sigma-bonds and a delocalized pi-system."

 A number of techniques including the quantum theory of atoms in molecules [1], natural bond orbitals [2] and modern valence bond theory [3], to name just a few, have all sought to connect quantum mechanics with concepts known to all chemists, including partial charges, electronegativity, Lewis structures etc. All have been successful to some degree, but often include some assumptions or have complicated programs that require specialist knowledge to run and interpret.

This work by Knizia defines a new method of intrinsic atom orbitals (IAOs) that can exactly express the occupied MOs of an accurate wave function. This is then combined with an orbital localization procedure to construct bond orbitals (IBOs). It is demonstrated that this method is insensitive to basis set size (unlike Mulliken charges), correctly predicts differences in electronegativities, produces the "correct" bond orbitals for some nontrivial cases and can calculate oxidation states of transition-metal complexes (building upon the work of Sit et al.[4]).

This initial implementation of the method shows great promise as an additional tool in the translation of quantum mechanical results into the chemical vernacular, and it will be interesting to observe how it may be applied. My own attraction to the method is due to it combining simplicity of approach with the natural emergence of Lewis structures in the absence of any bias/assumption.

Footnote: The IAO/IBO method will be included in the next public release of the MOLPRO program package, and a sample python implementation will be made available on the author's website (it had not yet been uploaded at the time of writing). Algorithm details are included in an appendix.

References

[1] R. F. W. Bader. A quantum theory of molecular structure and its applications. Chem. Rev. 1991, 91, 893.

[2] E. D. Glendening, C. R. Landis and F. Weinhold. Natural Bond Orbital Methods. WIREs Comput. Mol. Sci. 2012, 2, 1.

[3] J. Gerratt, D. L. Cooper, P. B. Karadakov and M. Raimondi. Modern valence bond theory. Chem. Soc. Rev. 1997, 26, 87; S. Shaik and P. C. Hiberty. A primer on qualitative valence bond theory - a theory coming of age. WIREs Comput. Mol. Sci. 2011, 1, 18.

[4] P. H.-L. Sit, R. Car, M. H. Cohen and A. Selloni. Simple, Unambiguous Theoretical Approach to Oxidation State Determination via First-Principles Calculation. Inorg. Chem. 2011, 50, 10259.

Empirical correction of nondynamical correlation energy for density functionals

Wanyi Jiang, Chris C. Jeffrey, and Angela K. Wilson J. Phys. Chem. A 2012, 116, 9969 (Paywall)
Contributed by Grant Hill.

The inability of common density functionals to correctly account for dispersion interactions has been addressed in a number of ways, but the most popular method is to add an empirical dispersion correction to the DFT energy [1]. In this paper, Wilson and co-workers propose an empirical correction for nondynamic correlation that operates in a superficially similar way.

If one defines nondynamic correlation as the significant contribution of several electronic configurations to the total energy of a system, it can be seen that this type of correlation becomes important for a number of chemically relevant situations, including the breaking of covalent bonds. Some of the methods typically used to recover nondynamic correlation include CASSCF and MRCI, with the common theme that they quickly become expensive in terms of computational cost, and that a degree of expertise is required in the choice of which orbitals and electrons to include in the active space of nondynamic correlation. The method proposed attempts to bypass these difficulties by carrying out a standard DFT calculation, then adding a correction for nondynamic correlation (with an empirical scale factor) via a CASCI calculation including a small set of orbitals in the active space. The authors suggest that this choice of orbitals can be automated, producing a computationally efficient black-box method.

The initial results indicate that the method performs well for the torsion of ethylene and automerization of cyclobutadiene, yet when investigating barrier heights it seems that the best results are produced when only the transition state is empirically corrected. The results presented suggest that the method is worthy of further investigation, and I for one would be very interested to see if how it performs for spin-state splittings of transition metal complexes [2].

References
[1] See S. Grimme, J. Antony, S. Ehrlich, and H. Krieg J. Chem. Phys. 2010, 132, 154104 and references therein.
[2] For a perspective on spin-state splittings in bio-inorganic systems see M. Swart Int. J. Quantum Chem. 2012, DOI: 10.1002/qua.24255

A Paramagnetic Bonding Mechanism for Diatomics in Strong Magnetic Fields

Kai K. Lange, E. I. Tellgren, M. R. Hoffmann and T. Helgaker Science 2012, 337, 327 (Paywall)

A new bonding mechanism
As chemists we are familiar with two types of strong bonds occurring between atoms, covalent and ionic. This paper shows that when very strong magnetic fields (of the order of 10⁵ T) are applied, a third bonding mechanism arrises. Helgaker and co-workers term this perpendicular paramagnetic bonding.


Binding triplet H₂
Whilst previous Hartree-Fock calculations have shown that the lowest triplet state of H₂ becomes bound in strong magnetic fields, this investigation uses the recently developed LONDON code to demonstrate the same phenomena at the Full-CI level. Similar calculations (again, with a very strong magnetic field) on the triplet state of He₂ show a considerable strengthening of the interaction between the constituent atoms.

The nature of this bonding
By examining the behaviour of the molecular orbitals under different orientations relative to the external magnetic field, the authors note a stabilisation of antibonding orbitals in the perpendicular orientation, leading to a new type of bonding interaction. Although the potential of a new chemical bonding mechanism is undoubtably exciting, the magnetic fields required are beyond those that can be currently generated in a lab. However, such fields are present on some stellar objects and the findings of this paper are likely to aid in the spectroscopy of such bodies.

Accurate ab Initio Spin Densities

Katharina Boguslawski, Konrad H. Marti, Örs Legeza and Markus Reiher Journal of Chemical Theory and Computation 2012, 8, 1970 (Paywall)

The spin density problem
Obtaining accurate spin density distributions and predicting the correct ground state from a number of close lying states of different spin is a challenging problem in quantum chemistry, particularly when transition metal systems are considered. This paper highlights how density-matrix renomarlization group¹ (DMRG) based methods can be used to calculate spin density distributions for molecules that are too large to be treated by complete-active-space self-consistent-field (CASSCF) methods. What I found particularly exciting is the prospect of using DMRG results in the benchmarking and development of new density functionals.

Benchmarking DMRG
The electronic structure of a simple model system of iron nitrosyl is manipulated by surrounding it with point-charges (simulating ligands), adjusting the position of these charges means the system becomes either single- or multi-reference. The results convincingly show how DMRG can converge to a CASSCF(7,7) reference spin density by controlling the number of reference active-system states in the DMRG. The DMRG calculations were carried out using the Reiher group's Qc-Dmrg-ETH code.

Spin densities for large active spaces and comparison to DFT
A problem arrises in CASSCF when the number of active electrons and orbitals required to correctly describe the system becomes too large (this article suggests 18 electrons in 18 orbitals as a practical limit). The paper goes on to show that much larger active spaces are possible with DMRG and, more importantly, that the spin density converges very quickly both with respect to active space and the number of active-system states. The resulting spin densities are then compared to a number of common density functionals, none of which accurately reproduces the DMRG result. Whilst DMRG calculations are not yet commonplace in the computational chemistry community, this paper convinces me that they will have an important role to play in solving the spin density problem.

References
(1) For an introduction to DMRG see: Chan, G. K.-L.; Dorando, J. J.; Ghosh, D.; Hachmann. J.; Neuscamman, E.; Wang, H.;Yanai, T. An Introduction to the Density Matrix Renormalization Group Ansatz in Quantum Chemistry. In Frontiers in Quantum Systems in Chemistry and Physics, 1st ed.; Wilson, S., Grout, P. J.; Maruani, J., Delgado-Barrio, G., Piecuch, P., Eds.; Springer: Dordrecht, The Netherlands, 2008; Vol. 18, pp 49-65. arXiv:0711.1398v1.