Long-range correction

In this chapter, various types of corrections for incorporating physical effects neglected in exchange-correlation functionals are discussed. Since the corrections reviewed in this chapter have clear physical meanings in common with each other, they are distinctly different from empirical terms that are added simply to enhance the ability to reproduce a particular physical property. It is, therefore, possible to determine whether the corresponding physical effect is included in an exchange-correlation functional in advance and to carry out a correction for the functional if it has not aiready been included. 

Tsuneda, Density Functional Theory in Quantum Chemistry, DOI 10.1007/978-4-431-54825-6 6, Springer Japan 2014 

In the above formulation, the only parameter /z naturally depends on the exchange functional corrected. For example, it has been confirmed that the /r value of the B88 and PBE exchange functionals is optimized as /r = 0.47 for electronic ground states at around equilibrium geometries (Song et al. 2007a) and as /r = 0.33 for others (likuraet al. 2001). 

02N(C2H2)bNH2 with respect to the number of units n. The DFT (LC-BOP, BOP, and B3LYP) results were obtained by the coupled-pertuibed Kohn-Sham method (see Sect. 4.7), the HF result was given by the coupled-perturbed Hartree-Fock method, and the ab initio results were provided by the finite-field method (see Sect. 4.7). The aug-cc-pVDZ basis functions tire used. See Kamiya et al. (2005) 

The CAM-B3LYP functional (Yanai et al. 2004) is a long-range corrected hybrid functional using 

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Calculating nonbonded interactions only to a certain distance imparts an error in the calculation. If the cutoff radius is fairly large, this error will be very minimal due to the small amount of interaction at long distances. This is why many bulk-liquid simulations incorporate 1000 molecules or more. As the cutoff radius is decreased, the associated error increases. In some simulations, a long-range correction is included in order to compensate for this error. 

Since only one molecule is added to (or removed from) the system, U is simply the interaction of the added (or removed) molecule with the remaining ones. If one attempts to add a new molecule, N is the number of molecules after addition, otherwise it is the number of molecules prior to removal. If a cutoff for the interaction potential is employed, long-range corrections to must be taken into account because of the density change of /As. Analytic expressions for these corrections can be found in the appendix of Ref. 33. 

In the majority of cases the force associated with the MM interactions is composed of a Coulombic term (typically a long-range correction is applied), non-Coulombic forces (Lennard-Jones 6-12 type potentials are the most commonly used formulation), and intramolecular force field contributions. The QM/MM coupling is composed of the Coulombic interactions with all core (Ni) and layer (N2) atoms plus non-Coulombic forces with all atoms in the layer region (N2). As the latter contributions correspond to the coupling terms in the core and layer regions, no violation of momentum conservation occurs. 

The principal effect of the j, / corrections amounts more or less to a rescaling of the B°°10(R) transition elements. Specifically, the long-range correction (XL — 23) looks much like.a scaled-down version of B 10(R), and the overlap correction (XL = 01) is more or less a scaled-down version of B l0(R), so that to a first approximation, scaling factors could be used to model these corrections. 

Likura H, Tsuneda T, Yanai T, Hirao K (2001) A long-range correction scheme for generalized-gradient-approximation exchange functionals, J Chem Phys, 115 3540—3544... 

An approach to address the proper r decay of the exchange-correlation potential is the LC scheme, where LC indicates a long-range correction. HF exchange is only applied to long-range electron correlation, with a functional defined as... 

Chai, J.-D. Head-Goidon, M. Systematic optmiizatioii of long-range corrected hybrid density functionals, J. Chem. Phys. 2008,128,084106-084115. 

Song, J.-W. Tsuneda, T Sato, T Hirao, K. Calculations of alkane energies using long-range corrected DFT combined with intramolecular van der Waals correlation, Org. Lett. 2010,12, 1440-1443. 

It is also worth noting that B3LYP and related functionals provide very poor reaction energetics for the aldol and Mannich reactions. The more modem functionals, such as M06-2x and the long-range-corrected functionals CAM-B3LYP and LC-(oPBE, perform much better, with mean errors of less than 2 kcalmol" ... 

Note that the last term in Eq. [77] involves a derivative of the potential with respect to the explicit volume dependence. Such an explicit dependence comes about when employing, for example, long range corrections to Lennard-Jones potentials " or when Ewald summation is used to obtain energies and forces from electrostatic interactions. Equations [76] have the conserved energy... 

The results outlined above for the RPM and the CHD fluid clearly show a discrepancy between MFFSS analyses (which are consistent with Ising-like criticality) and calculations of Cy in NVT MC simulations. To investigate this discrepancy, we have studied a class of fluids defined by the potential in Eq. (5) with a = 6 (which should give rise to Ising-like criticality since a > 4.7) and a = 4 and a = 3.1 (which are classical since a < 4.5). All simulations reported in this section were carried out in cubic simulation cells of side L. The potentials were truncated at L/2, and a long-range correction was calculated... 

Rowley, L.A., Nicholson, D., and Parsonage, N.G. (1978). Long-range corrections to grand canonical ensemble Monte Carlo calculations for adsorption systems. J. Comput. Phys., 26, 66—79. 

Rohrdanz MA, Martins KM, Herbert JM (2009) A long-range-corrected density functional that performs weU for both ground-state properties and time-dependent density functional theory excitation energies, including charge-transfer excited states. J Chem Phys 130 054112... 

As mentioned above, various correction schemes have been developed up to the present. However, there is room for further improvement in conventional correction schemes. Conventional hybrid functionals give poor excitation energies in TDDFT calculations as mentioned later. Asymptotic and SICs have little (or worse) effect on reproducibilities of molecular chemical properties. Recently, it has been proved that a long-range correction for exchange functionals obviously brings solutions to various... 

DPT problems that have never been solved by other functionals or corrections. In later sections, we will briefly review the background of the long-range correction scheme and will reveal the applicabilities of this scheme. 

Long-range correction schemes for exchange functionals... 

It is easily confirmed that Eq. (67) reproduces the original GGA exchange functional for fjb = 0. Parameter jx is determined to optimize bond distances of homonuclear diatomic molecules up to the third period as /r = 0.33. This scheme is called long-range correction (EC) scheme . The applicabilities of the EC scheme will be discussed in the later section. 

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