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Development of Modern Density Functional Methods: Combining the Correlation Factor Model and the Local
Hybrid Approach

Final Report Summary - CFMLHA (Development of Modern Density Functional Methods: Combining the Correlation Factor Model and the Local<br/>Hybrid Approach)

Background, Motivation, and Objectives
The goal of the present project is to design new approximations for the exchange-correlation (XC) energy of Kohn-Sham density functional theory (DFT). This computational methodology has revolutionized theoretical chemistry and physics ever since it was awarded a Nobel prize in 1996. Due to its widespread use, new advances in the DFT methods have far-reaching consequences. Many fields are affected including chemistry, biochemistry, physics, pharmacology, medicinal chemistry, materials science, and catalysis.

Since the accuracy of DFT calculations exclusively depends on the available approximations to the unknown exchange-correlation energy, the development of new approximations has received an immense interest in the modern quantum chemistry. Most of the illnesses of the contemporary exchange-correlation functionals are attributed to the use of semilocal exchange (X). This project aims to compensate for the lack of state-of-the-art functionals for chemistry that build on nonlocal exactly computed exchange instead. Especially a nonempirical construction of such functionals has so far remained a serious problem that we aim to solve in the present work.

Two methodologies, the correlation factor (CF) model and the local hybrid (LH) approach, have been considered as a means to reach this objective. Both methods constitute promising potential remedies for the illnesses of global hybrids, which are typical examples of XC functionals employing exact exchange. Unlike global hybrids, the CF model and the LH approach can be designed in such a way that they recover exact exchange in regions of a system with vanishing or negligible correlation. The correlation factor approach approximates the XC energy through models to the so-called XC hole expressed as a product of an exchange hole and a correlation factor. The model has been little explored so far and potentially suffered from an increasing complexity and a limited flexibility. Local hybrids, in turn, are plagued by the empirical character of the underlying local mixing function. This project provides an answer to the question whether these disadvantages can be overcome on the way to a successful design of the much sought-after nonempirical exact-exchange-based functionals.

The work carried out to achieve the project's objectives
To develop the target functionals from the correlation factor model, we designed both constituting components of the CF model. Formulas for the exchange hole and the correlation factor were proposed. We examined their mathematical properties and developed the underlying theory and methodology for the evaluation of the unknown parameters involved using the constraint-satisfaction approach. Building on the equations elaborated, the subsequent research activities focused on the software implementation, testing, and performance validation.

In order to benefit from the exact treatment of exchange and to avoid the complexity of the exact exchange hole at the same time, we based the CF model on a model exchange hole reproducing the exact exchange energy per electron. This model hole is multiplied by a five-parameter CF inspired by the ansatz of Bahmann and Ernzerhof (Bahmann, H.; Ernzerhof, M.; J. Chem. Phys. 128, 234104 (2008)). Various physical conditions constrain the parameters in the ansatz for CF. The local spin density approximation determines the depth of the XC hole for zero interelectronic distances as well as its cusp for small interelectronic distances. The XC hole is also constrained to contain one electron while the range and oscillatory character of the hole are determined from semilocal approximations. The physics of these approximations is well-understood and we make use of it to control the behavior of the correlation factor in the core and density tail regions, which are of special interest to us. In these regions, correlation vanishes or becomes negligible compared to exchange and the functionals constructed here have to correctly reproduce this behavior.

Four different correlation factor models were designed. The first model, called CFXN, builds on a normalized exchange hole model. It performs extremely well for atoms but does not provide a sufficient accuracy for the atomization energies of molecules. To improve upon CFXN, the CFX model was developed, in which the correlation factor multiplies a not normalized exchange hole. The normalization is replaced by the exact curvature, which significantly improves the atomization energies as well as reaction barriers. To reflect the complementary benefits of the exact normalization and curvature conditions, we designed an empirical hybrid, CFXhyb. In this model, the normalized and not normalized exchange hole models are mixed in a linear fashion before the correlation factor is applied. Finally, we introduced gradient corrections to the on-top value and cusp of the exchange-correlation hole in CFX, which led to a computationally simplified CFXs model.

The exchange hole model as well as the correlation factor were implemented in Mathematica and thereafter in the modified version of Gaussian, a commercially available suite of quantum chemistry programs. Using the computer code, we evaluated all the correlation factor models, CFXN, CFX, CFXhyb, and CFXs, and validated their performance against the unrestricted Hartree-Fock, semilocal and hybrid density functional approximations. The evaluation was carried out for total and correlation energies (atoms), atomization energies (molecules) and barrier heights (hydrogen-transfer reactions). Visualisation of the computed X and XC holes and the CF then confirmed that the desired outcome in terms of the hole properties and the CF behavior in the regions of interest has been achieved.

Importantly, we could demonstrate that the correlation factor model is flexible enough to construct exact-exchange-based functionals directly thereof, i.e. without the parallel comparative development of local hybrids. We did undertake initial steps to design an empirical-parameter-free local hybrid scheme and proposed a potential ansatz that could serve as a local hybrid formulation of our CFX model. Yet, given the success and great promise the correlation factor models have shown in the design of nonempirical functionals building on exact exchange, we preferred to focus an getting an in-depth knowledge of the correlation CF approach. Its simplifications and refinements will in the future better facilitate the design of equivalent exact-exchange-based XC functionals from the local hybrid scheme.

The main results (overview)
Building on the correlation factor model, we have designed novel density functional approximations that stand out in the contemporary landscape of exchange-correlation functionals due to a unique combination of crucial physical properties. The use of 100% exact exchange eliminates the error due to approximating the exchange energy but raises a challenge for the construction of a compatible approximation to the correlation energy. Rising to this challenge, we have developed several prototypes of such functionals that largely obviate empirical parameters, yet display a very promising performance for thermochemistry and kinetics. Importantly, they also offer the profound advantage of correctly describing the core and density tail regions of atoms and molecules, thereby curing the well-known illnesses of the widely used global hybrid functionals.

Conclusions, Potential impact and use
The construction of functionals with the described properties is a rare achievement and solves a long-standing challenge in DFT that defies theorists for decades already. We have proposed avenues for further improvements of the designed functionals that will, together with our computational algorithm and software implementation, pave the way for the first-principles construction of exchange-correlation functionals based on full exact exchange. The growing insight into the empirical-parameter-free correlation factor model has opened the door for the reformulation of the local hybrid ansatz and elimination of its underdetermined parameters.