Adiabatic Connection Formula Research Articles

Generalized adiabatic connection in density functional theory

A generalized adiabatic connection is developed for density functional theory. The method extends the well-known adiabatic connection formula and provides a general link between the Kohn–Sham and the physical system. When the complimentary error function is used as a special case, the expression for the exchange-correlation functional does not have the 1/r12 Coulomb component. The exact contributions from the physical system and the noninteracting system are established: The physical system limit has a dominant contribution, while the noninteracting system limit has no contribution.

Novel Density Functional Methodology for the Computation of Accurate Electronic and Thermodynamic Properties of Molecular Systems and Improved Long-Range Behavior

A novel general purpose density functional methodology for the computation of accurate electronic and thermodynamic properties of molecules and improved long-range behavior is reported. Assuming the separability of the exchange (Ex) and correlation (Ec) contributions to the total exchange-correlation energy functional (Exc), the Ex term consists of a hybrid mixture of 37.5% Hartree−Fock exchange and the appropriate local spin density exchange using the adiabatic connection formula. We demonstrated that Ex and its corresponding potential Vx [=dEx/dρ(r)] have the proper asymptotic limits at r = 0 and r → ∞. Ec consists of the Vosko, Wilk, and Nusair formula for the free-electron gas correlation energy and a generalized gradient approximation term with one adjustable parameter. Vc [=dEc/dρ(r)] was shown to obey the r → ∞ limit of the corresponding potential derived from exact atomic exchange-correlation computations; namely, Vc is proportional to r-4. Most importantly, we demonstrated that, at r values where dispersion forces are operating, Vc is proportional to 1/rn (n = 4, 6, 8, ...). The reported method was denoted by K2-BVWN because it used two adjustable parameters in its formulation. The K2-BVWN scheme scales as N3, where N is the number of basis functions, compared to ∼N7 for Gaussian-2 (G2) ab initio theory and related methods, ∼N5 for Barone's mPW1,3PW, and ∼N4 for Becke's three-parameter density functional approaches. The K2-BVWN/6-311g(d) model predicted the structures of numerous molecular systems with remarkable accuracy. The results of K2-BVWN/6-311+g(df), K2-BVWN/6-311+g(2df), and K2-BVWN/6-311+g(3df) computations on Li through Ar atoms showed that the calculated energies at all three levels of theory are comparable to within 0.1 kcal/mol, thus demonstrating the fast convergence of atomic energies as the size of basis sets increased. Accordingly, the thermochemical properties of molecular systems could, in principle, be calculated to increasing levels of accuracy depending on the size of the basis sets used, in accordance with the usual practice in Hartree−Fock theory. In a data set comprised of 350 atomic and molecular systems, which included the G2 data set, we demonstrated that the K2-BVWN/6-311+g(2df) level of theory is a reliable model for the computation of room-temperature heats of formation, ionization potentials, and electron and proton affinities of normal valent compounds with average errors of 1.4 kcal/mol, 0.07, 0.07, and 0.05 eV, respectively. The outliers in calculated heats of formation consisted mainly of hypervalent compounds, nonhydrides containing multiple chlorine atoms, H2O, and HF. Enthalpies of formation of these outliers were computed using the K2-BVWN/6-311+g(3df,p) level of theory. Further refinement of calculated heats of formation of outliers was achieved through the use of atom equivalent corrections instead of increasing the size of basis sets beyond 6-311+g(3df,p). Interestingly, the only atoms that required a correction for the latter step were oxygen (0.6 kcal/mol), aluminum (0.60 kcal/mol), silicon (0.60 kcal/mol), phosphorus (0.60 kcal/mol), sulfur (0.60 kcal/mol), and chlorine (0.60 kcal/mol). Comparison of the results obtained from the K2-BVWN method and corresponding ones from G2/6-311+g(3df,2p) ab initio theory, HFS-BVWN/6-311+g(3df,p), bond additivity correction BAC-G2/6-311++g(3df,2pd), G2(MP2)/6-311+g(3df,2p), G2(MP2, SVP)/6-311+g(3df,2p), B3LYP/6-311+g(3df,2p), and mPW1,3PW/6-311++g(3df,3pd) demonstrated that G2 ab initio theory and the K2-BVWN density functional scheme have comparable average errors in computed heats of formation, ionization potentials, and electron and proton affinities of molecules and are superior to all other approaches. Furthermore, we showed that the interaction energies of nine noble gas dimers were remarkably well reproduced using K2-BVWN/3-21+g(d,p) and K2-BVWN/3-21g levels of theory. Moreover, binding energies of the hydrogen-bonded isoelectronic systems (H2O)2 and (HF)2 and the interaction energy of the charge-transfer complex formed between Cl2 and ethylene were also reliably predicted to within less than 0.5 kcal/mol from corresponding experimental values. The G2 data set complemented by the reported molecular systems investigated in this work was recommended as a critical test for evaluating novel ab initio and density functional methodologies. The K2-BVWN method has been implemented in the Gaussian series of programs.

Closed-form expression relating the second-order component of the density functional theory correlation energy to its functional derivative

For helping to improve approximations to the density-functional exchange-correlation energy, Exc[n], and its functional derivative, the difference between the second-order component of the correlation energy, Ec(2)[n], and the integral ∫dr vc(2)([n];r)n(r), involving its functional derivative, vc(2)([n];r), is given in terms of only the occupied Kohn–Sham orbitals and the exchange potential. The quantity 2Ec(2)[n] is especially significant because it is the initial slope in the adiabatic connection formula for Exc[n]. The analytic expression for 2Ec(2)[n]−∫dr vc(2)([n];r)n(r) is obtained for any spherically symmetric two-electron test density. Numerical examples are presented.

Exchange−Correlation Energy Density from Virial Theorem

The virial of the exchange potential in density functional theory yields the exchange energy, but the virial of the correlation potential does not yield the correlation energy. Via the adiabatic connection formula, we define a hypercorrelated potential whose virial is exactly the correlation energy. This exchange−correlation energy density is uniquely determined by the exchange−correlation energy functiional. We calculate the virial energy density both exactly and within several popular functionals, LDA, PBE, and BLYP, on several atoms. The well-known differences between the potentials generated by these functionals is reflected in this energy density. We speculate on how accurately the correlation energy can be estimated from knowledge of the exact density, and on the construction of an energy density hybrid.

Magnetic coupling in ionic solids studied by density functional theory

Magnetic coupling in ionic solids is studied using a density functional theory, DFT, approach applied to suitable cluster models representing KNiF3, K2NiF4, and La2CuO4. A mapping between eigenstates of the exact nonrelativistic and spin model Hamiltonians allows us to obtain the magnetic coupling constant J and to compare the DFT values with either experiment or previous theoretical studies based on the use of accurate wave functions. In the present work different correlation and exchange functionals are explored. Numerical results show that it is possible to reach very good agreement with experiment. Surprisingly, it is shown that the difficulty of the local spin density approximation in describing the antiferromagnetic behavior of these compounds lies not in the correlation but in the exchange part of the density functional. Hybrid functionals, which include a component of the full, nonlocal, “exact” exchange interaction yield qualitatively and semiquantitatively correct magnetic interactions. The origin of this behavior is discussed from the point of view of the adiabatic connection formula.

Molecular energies and properties from density functional theory: Exploring basis set dependence of Kohn?Sham equation using several density functionals

The performance of four commonly used density functionals (VWN, BLYP, BP91, and Becke's original three-parameter approximation to the adiabatic connection formula, referred to herein as the adiabatic connection method or ACM) was studied with a series of six Gaussian-type atomic basis sets [DZP, 6–31G**, DZVP, TZVP, TZ2P, and uncontracted aug-cc-pVTZ (UCC)]. The geometries and dipole moments of over 100 first-row and second-row molecules and reaction energies of over 300 chemical reactions involving such molecules were computed using each of the four density functionals in combination with each of the six basis sets. The results were compared to experimentally determined values. Based on overall mean absolute theory versus experiment errors, it was found that ACM is the best choice for predictions of both energies of reaction [overall mean absolute theory versus experiment error (MATvEE) of 4.7 kcal/mol with our most complete (UCC) basis set] and molecular geometries (overall MATvEE of 0.92 pm for bond distances and 0.88° for bond angles with the UCC basis set). For routine calculations with moderate basis sets (those of double-ζ type: DZP, 6–31G**, and DZVP) the DZVP basis set was, on average, the best choice. There were, however, examples of reactions where significantly larger basis sets were required to achieve reasonable accuracy (errors ≤ 5 kcal/mol). For dipole moments, ACM, BP91, and BLYP performed comparably (overall MATvEE of 0.071, 0.067, and 0.059 debye, respectively, with the UCC basis set). Basis sets that include additional polarization functions and diffuse functions were found to be important for accurate density functional theory predictions of dipole moments. © 1997 by John Wiley & Sons, Inc.

Local energy and chemical potential equations and the exchange-correlation potential

Local energy and chemical potential equations are considered in some detail in relation to low-order density matrices. Some asymptotic properties can be extracted in exact form. The spatial derivative of the chemical potential equation referred to above yields the external force, defined as the (negative of the) gradient of the potential energy of the nuclear framework. This quantity, by utilizing the differential virial theorem, can be expressed as a sum of three terms: (i) a Laplacian contribution known explicitly in terms of the ground-state electron density; (ii) a kinetic part derivable from the "near-diagonal" behaviour of the first-order density matrix; and (iii) a term from electron–electron interactions, that involves the electronic pair correlation function. Following the work of Holas and March, this allows the exchange-correlation potential of density functional theory to be expressed in terms of low-order density matrices. Finally, scaling of electron–electron interactions is briefly considered, as well as the adiabatic connection formula in density functional theory. Such scaling arguments lead to a kinetic correction to the Harbola–Sahni form of the exchange-only potential. Key words: external force, first-order density matrix, electronic pair function.

Coordinate scaling and adiabatic connection formula for ensembles of fractionally occupied excited states

AbstractThe coordinate scaling for the density matrix of ensembles of fractional occupied states of the density functional theory is explored. The ground‐state adiabatic connection formula is extended to the ensemble exchange‐correlation energy. © 1995 John Wiley & Sons, Inc.