Adiabatic-connection Theorem Research Articles

Recent Advances in Cartesian-Grid DFT in Atoms and Molecules.

In the past several decades, density functional theory (DFT) has evolved as a leading player across a dazzling variety of fields, from organic chemistry to condensed matter physics. The simple conceptual framework and computational elegance are the underlying driver for this. This article reviews some of the recent developments that have taken place in our laboratory in the past 5 years. Efforts are made to validate a viable alternative for DFT calculations for small to medium systems through a Cartesian coordinate grid- (CCG-) based pseudopotential Kohn–Sham (KS) DFT framework using LCAO-MO ansatz. In order to legitimize its suitability and efficacy, at first, electric response properties, such as dipole moment ( μ ), static dipole polarizability ( α ), and first hyperpolarizability ( β ), are calculated. Next, we present a purely numerical approach in CCG for proficient computation of exact exchange density contribution in certain types of orbital-dependent density functionals. A Fourier convolution theorem combined with a range-separated Coulomb interaction kernel is invoked. This takes motivation from a semi-numerical algorithm, where the rate-deciding factor is the evaluation of electrostatic potential. Its success further leads to a systematic self-consistent approach from first principles, which is desirable in the development of optimally tuned range-separated hybrid and hyper functionals. Next, we discuss a simple, alternative time-independent DFT procedure, for computation of single-particle excitation energies, by means of “adiabatic connection theorem” and virial theorem. Optical gaps in organic chromophores, dyes, linear/non-linear PAHs, and charge transfer complexes are faithfully reproduced. In short, CCG-DFT is shown to be a successful route for various practical applications in electronic systems.

Excitation energies through Becke’s exciton model within a Cartesian-grid KS DFT

Photon-induced electronic excitations are ubiquitously observed in organic chromophore. In this context, we present a simple, alternative time-independent DFT procedure, for the computation of single-particle excitation energies, in particular, the lower bound excited singlet states, which are of primary interest in photochemistry. This takes inspiration from recently developed Becke’s exciton model, where a key step constitutes the accurate evaluation of correlated singlet–triplet splitting energy. It introduces a non-empirical model, from both “adiabatic connection theorem” and “virial theorem” to analyze the role of 2e\(^-\) integral in such calculations. The latter quantity is efficiently mapped onto a real grid and computed accurately using a purely numerical strategy. Illustrative calculations are performed on 10 \(\pi \)-electron organic chromophores within a Cartesian grid implementation of pseudopotential Kohn–Sham (KS) DFT, developed in our laboratory, taking SBKJC-type basis functions within B3LYP approximation. The triplet and singlet excitation energies corresponding to first singly excited configuration are found to be in excellent agreement with TD-B3LYP calculations. Further, we perform the same for a set of larger molecular systems using the asymptotically corrected LC-BLYP, in addition to B3LYP. A systematic comparison with theoretical best estimates demonstrates the viability and suitability of current approach in determining optical gaps, combining predictive accuracy with moderate computational cost.

Electron correlation in chemical bonds

We study the correlation energy in the H2 molecule as a function of bond distance and the electron coupling parameter that modifies the strength of electron–electron repulsion. For the unscaled electron repulsion we find an unusual complexity of the correlation energy dependence on the bond distance. This complexity remains if the electron repulsion is scaled down. We pay particular attention to the range of bond distances near the H2 equilibrium geometry. We confirm that good performance of the modern exchange–correlation functionals in molecular geometry optimization is due to the cancellation of errors between the exchange and correlation components. These components have different functional dependence on the electron coupling parameter, so the cancellation of errors no longer occurs when the electron repulsion is scaled down. This observation is important if the adiabatic connection theorem is used to construct the exchange–correlation functionals. It is also relevant for the performance of exchange–correlation functionals to study ionic interactions. The performance of the correlation operator in the molecular environment is also investigated and discussed.

Toward reliable adiabatic connection models free from adjustable parameters

Methods rooted in the adiabatic connection theorem offer a significant improvement over standard density functional approximations. Among them, the most successful model is the so-called B3LYP functional, which includes three empirical parameters optimized by a best fit to atomization energies. We introduce a related functional, B1LYP, in which the ratio between Hartree-Fock and density functional exchange is determined a priori from purely theoretical considerations and no further parameters are present. The numerical results obtained on a standard molecular data set and on some ‘delicate’ chemical systems indicate that this model has essentially the same performance as the B3LYP model, but within a more satisfactory theoretical framework.