Hartree-Fock theory plays a central role in Quantum Chemistry. It serves on the one hand as a basis for electron correlation approaches while on the other it is also essential for density functional theory. The evaluation of the HF exchange term is a rather time consuming operation for larger molecules and as a result for a wide range of chemically or biologically interesting systems. This motivates the development of various efficient approaches to exchange evaluation. The Chain of Spheres exchange (COSX) algorithm, one of the most efficient methods available for this purpose today, has been developed in the institute since 2009. The algorithm has favorable scaling and contraction properties which makes it especially suitable for larger systems. In combination with the resolution of identity for the Coulomb term (RI-J) it enables large scale HF and DFT calculations.
The possibility of robust fitting in the COSX context is of a more theoretical interest which has also been investigated during the last few years. Several other ideas are also being considered including various projection based approaches, linear scaling RI approaches and range separation.
To realize the full potential of any approximation to the SCF energy, it is inevitable to obtain the corresponding gradients and the Hessian. These can then be used in geometry optimizations and to calculate various properties. Gradients are determined in an approximate manner for the COSX algorithm, while for Hessian calculation its main advantage is in the CP-SCF step.
Basis set incompleteness affects quantum chemical methods in various ways. Since practical methods correspond to the expansion of the wave function in a truncated basis, it becomes important to find a good way to approximate the basis set limit. Basis set extrapolation and explicitly correlated (F12) methods are two different ways to address the same question. At the MP2 level, the additional cost of F12 methods is comparatively high. Nevertheless, our investigations show that combined with various approximate SCF approaches the F12 is fair competitor to extrapolation methods with respect to efficiency as well as accuracy.
Incompleteness effects are also connected to numerical errors, in fact the incompleteness of basis sets and the incompleteness of numerical grids are closely related. There is a conceptual similarity between complementary auxiliary basis (CABS) functions of F12 approaches and the so-called dealiasing functions of seminumerical methods, which are used to cancel numerical errors. We argue that the CABS sets, or other basis sets developed based on similar principles can generalize the notion of grid specific dealiasing functions. This procedure is also a generalization of the overlap fitting method in which the dealiasing set is identical to the atomic orbital basis.
The COSX method can also be used beneficially in correlation methods. In these cases the SCF density is replaced by an effective density in which the amplitudes of various excited determinants are contracted with molecular orbital coeffcients. Thus, the evaluation of the so called singles Fock term in the LPNO-CCSD method, and the external exchange term in the SCS-MP3 method can both be significantly accelerated without significant loss of accuracy. In the former case the contraction involves the singles, in the latter case the doubles amplitudes, but in both cases the approximated terms are form the bottleneck of the corresponding method.
[1] G. Tasi, R. Izsak, G. Matisz, A. G. Csaszar, M. Kallay, B. Ruscic and J. F. Stanton, The origin of systematic error in the standard enthalpies of formation of hydrocarbons computed via atomization schemes, ChemPhysChem, 2006, 7, 1664.
[2] R. Izsák, M. Szőri, P. J. Knowles and B. Viskolcz, High Accuracy ab Initio Calculations on Reactions of OH with 1-Alkenes. The Case of Propane, J. Chem. Theory Comput., 2009, 5, 2313.
[3] R. Izsák and F. Neese, An overlap fitted chain of spheres exchange method, J. Chem. Phys., 2011, 135, 144105.
[4] M.-E. Pandelia, D. Bykov, R. Izsák, P. Infossi, M.-T. Giudici-Orticoni, E. Bill, F. Neese and W. Lubitz, Electronic structure of the unique [4Fe-3S] cluster in O2-tolerant hydrogenases characterized by 57Fe Mössbauer and EPR spectroscopy, PNAS, 2012, 110, 483.
[5] D. G. Liakos, R. Izsák, F. Neese and E. F. Valeev, What is the most efficient way to reach the canonical MP2 basis set limit?, Mol. Phys., 2013, 111, 2653.
[6] R. Izsák, F. Neese and W. Klopper, Robust fitting techniques in the chain of spheres approximation to the Fock exchange: The role of the complementary space, J. Chem. Phys., 2013, 139, 094111.
[7] D. Bykov, T. Petrenko, R. Izsák, S. Kossmann, U. Becker, E. Valeev, F. Neese, Efficient implementation of the analytic second derivatives of Hartree–Fock and hybrid DFT energies: a detailed analysis of different approximations, Mol. Phys., 2015, 113, 1961.
[8] A. K. Dutta, F. Neese, R. Izsák, Speeding up Equation of Motion Coupled Cluster Theory with the Chain of Spheres Approximation, J. Chem. Phys., 2016, 144, 034102.
[9] E. Rebolini, R. Izsák, S. S. Reine, T. Helgaker, T. B. Pedersen, Comparison of Three Efficient Approximate Exact-Exchange Algorithms: The Chain-of-Spheres Algorithm, Pair-Atomic Resolution-of-the-Identity Method, and Auxiliary Density Matrix Method, J. Chem. Theory Comput., 2016, 12, 3514.
[10] A. K. Dutta, F. Neese, R. Izsák, Towards a pair natural orbital coupled cluster method for excited states, J. Chem. Phys., 2016, 145, 034102.
Full publicationlist
Equation of Motion (EOM) Coupled Cluster theory is a standard way of obtaining reliable excitation energies and properties of excited states. The COSX approximation is particularly well suited to accelerate the most expensive term involved in the EOM equations with very good accuracy, see Figure 1. We were able to calculate the first two excited states of the trans retinal molecule (539 basis functions) in less then two days on eight cores only, which is the first such EOM calculation reported in the literature for such a large non-symmetric system.
Ionization operators can also be obtained within the EOM framework. The resulting equations are special cases of the general EOM equations for ionization potentials (IP) and electron attachments (EA). The advantages of the COSX approximation are even more significant in the EA case than for regular EOM; calculations up to a hundred atoms can now be carried out using our code. The accurate calculation of IP and EA values can be a valuable contribution to redox potential calculations, such as those involved in Photosystem II.
In order to speed up EOM calculations to an even larger extent, we currently use a strategy which employs the domain based pair natural orbital (DLPNO) framework for the ground state coupled cluster calculation in combination with the similarity transformed equation of motion (STEOM) method for the excited state part. The DLPNOs are back transformed for the purposes for the excited state calculation, while the excitation manifold is truncated to the singles space, while the effect of the doubles is still taken care of using a similarity transformation. The latter is parametrized in terms of IP and EA eigenstates, which also benefit from the COSX approximation. This way, we have been able to calculate excitation energies for molecules of 80-90 atoms (e.g., Chlorophyll A, more than a 1000 basis functions), thus providing an alternative for time dependent density functional theory for systems which are out of reach for traditional EOM calculations, see Figure 2.
Dr. Berraud-Pache, Romain
+49 (0)208 306 - 2189
berraudpache((atsign))kofo.mpg.de
Dr. Demoulin, Baptiste Francis Francois
+49(0)208/306-2189
bdemoulin((atsign))kofo.mpg.de
Dr. Huntington, Lee
+49 (0)208 306 - 3591
Dr. Izsák, Róbert
+49 (0)208/306-2164
robert.izsak((atsign))kofo.mpg.de
Lechner, Marvin
+49 (0)208 306 - 2189
marvin.lechner((atsign))kofo.mpg.de
Dr. Sen, Avijit
+49 (0)208 306 - 2163
avijit.sen((atsign))kofo.mpg.de