The Adiabatic Connection Method

Another possibility for improving the energies obtained by DFT methods would be to combine them with ab initio results. There would be very gross approximations, but this could be another way to learn more about the unknown functionals. Let s calculate the unknown integral found in eq. (27). Using the trapezoidal formula for definite integrals, we obtain, 

This can be calculated exactly provided that Hartree-Fock, correlated, and density functional calculations are carried out. Since the idea is to improve results at the minimum computational cost, the following approximations can be made in obtaining I, which is the most difficult part of the total energy to calculate. (The others, the noninteracting kinetic energy Ts and the nuclear-electron interaction are trivially obtained.) 

If we assume that p = p °, we can also assume that the direct terms are 

Notice that there are no implications about the similarity or dissimilarity of Fs with respect to Pm. Under these conditions, the integral I becomes, 

Density functional techniques are available for the calculation of the molecular and electronic structures of ground state systems. With the functionals available today, these compete with the best ab initio methods. This article focuses on the theoretical aspects associated with the Kohn Sham density functional procedure. While there is much room for improvement, the Kohn-Sham exchange-correlation functional offers a great opportunity for theoretical development without returning to the uniform electron gas approximation. Theoretical work in those areas will contribute significantly to the development of new, highly precise density functional methods. 

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Burke, K., Emzerhof, M., Perdew, J. P., 1997, The Adiabatic Connection Method A Non-Empirical Hybrid , Chem. Phys. Lett., 265, 115. 

Andzelm, J. W., D. T. Nguyen, R. Eggenberger, D. R. Salahub, and A. T. Hagler. 1995. Applications of the Adiabatic Connection Method to Conformational Equilibria and Reactions Involving Formic Acid. Computers and Chemistry 19, 145. 

The quantity 2Ec(2)[n] is particularly important because it is the initial slope in the adiabatic connection method (coupling-constant formula) for Ec[n] [10,14-18]. In fact, quite good use of the evaluation of 2Ec(2)[n] has recently been made by Ernzerhof [19] and by Perdew, Burke, and Ernzerhof [20-23] in their modeling of... 

A Quantum Monte Carlo Approach to the Adiabatic Connection Method... 

A fourth form of gradient correction has recently been proposed by Becke [107], based the adiabatic connection method (ACM). It uses a linear combination of the HF, LDA and B88 exchange contributions, together with the... 

Furthermore, recent work has shown that the approaches rooted in the adiabatic connection method (ACM), in which a fraction of HF exchange is mixed together with DF exchange, provide even better results [47-59]. Some questions are open about the empirical parameters which rule the ratio between HF and DF... 

Adamo, C., Barone, V., 1998a, Implementation and Validation of the Lacks-Gordon Exchange Functional in Conventional Density Functional and Adiabatic Connection Methods , J. Comput. Chem., 19, 418. 

Adamo, C., di Matteo, A., Barone, V., 1999, From Classical Density Functionals to Adiabatic Connection Methods. The State of the Art , Adv. Quantum Chem., 36, 45. 

Adamo, C. and Barone, V. 1998. Exchange Functionals with Improved Long-range Behavior and Adiabatic Connection Methods Without Adjustable Parameters The zwPW and zwPWlPW Models , J. Chem. Phys., 108, 664. 

Adamo C. Barone V. Exchange functionals with improved long-range behavior and adiabatic connection methods without adjustable parameters the mPW and mPW 1PW models. J. Chem. Phys. 1998, 108, 664—675. 

The results from the nonlocal and hybrid DFT calculations show trends similar to those observed for the MO calculations. The computed geometries are very similar to each other the calculated bond lengths for the breaking bond range from 2.14 A, calculated by the hybrid adiabatic connection method (ACM) [24], to 2.16 A, calculated by the nonlocal BLYP [25, 26] functional. A comparison of the results from the nonlocal Becke-Perdew [27] calculations using DZVP and a TZVP basis sets shows that the geometries are influenced to a very small extent by basis set effects [12]. The activation energies calculated by the different nonlocal methods are too low by 1-3 kcal/mol, whereas the hybrid DFT methods overestimate Ea by approximately the same amount. Both the nonlocal and the hybrid DFT methods tend to overestimate the heat of reaction by up to 7 kcal/mol, calculated by the ACM/6-31G method. 

Geometry errors for 108 molecules were reported by Scheiner et al. [55], comparing several ab initio and DFT methods. They found that Becke s original three-parameter function (which they denote ACM, for adiabatic connection method B3LYP was developed as a modification of this [49]), with a 6-31G -type and with the 6-3IG ... 

From Classical Density Functionals to Adiabatic Connection Methods. The State of the Art. 

The first-principle method is being developed for systems with long-range dispersion forces. There are two ways to include dispersion forces in first-principle calculations. A semiempirical van der Waals interaction can be taken into account in ab initio calculations. It is realized by using the Lenard-Jones potential of the form (11.26). The second approach is based on the adiabatic connection fluctuation-dissipation theorem. This theory includes seamless long-range dispersion forces... 

The B3LYP hybrid functional (Becke 1993), the first hybrid functional, is the most frequently used functional (or method) in all functionals (or all theories) in quantum chemistry calculations. This functional uses three parameters as the mixing ratios to form the adiabatic connections between the Hartree-Fock exchange integral and the LDA exchange functional and between the LYP-GGA correlation functional and the LDA correlation functional, and to combine with the attenuated GGA term of the B88 exchange functional. 

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