DISPERSION CORRECTION

Dispersion correction treatment in DFT

DFT, approximations must be made for how electrons interact with each other.

Standard XC functionals include:

  • Local density approximation (LDA)
  • Generalized gradient approximation (GGA) functionals
  • Hybrid XC functionals

Standard XC functionals do not describe dispersion because:

  1. instantaneous density fluctuations are not considered
  2. they are “short-sighted” in that they consider only local properties to calculate the XC energy

Ground-Binding with incorrect asymptotics

The ground method does not describe the long range asymptotics and give incorrect shapes of binding curves and underestimate the binding of well separated molecules.

The result with LDA for dispersion bonded systems have limited and inconsistent accuracy and the asymptotic form of the interaction is incorrect.

Step one-Simple C6 corrections (DFT-D)

The basic requirement for DFT-based dispersion scheme: it yields reasonable −1/r6 asymptotic behavior for the interaction of particles in the gas phase, where r is the distance between the particles.

Approach: add an additional energy term which accounts for the missing long range attraction.

Four shortcomings:

  • the C6/r^6 dependence represents only the leading term of the correction and neglects both many-body dispersion effects and faster decaying terms such as the C8/r^8 or C10/r^10
  • It is not clear where one should obtain theC6 coefficients. The reliance on experimental data (ionization potentials and polarizabilities) limits the set of elements that can be treated to those present in typical organic molecules.
  • C6 coefficients are kept constant during the calculation, and so effects of different chemical states of the atom or the influence of its environment are neglected.
  • C6/r^6 function diverges for small separations (small r) and this divergence must be removed.

With the simple correction schemes the dispersion correction diverges at short inter-atomic separations and so must be “damped”. The damping function f(rAB, A, B) is equal to one for large r and decreases Edisp to zero or to a constant for small r.

Issues with damping function:

  • The shape of the underlying binding curve is sensitive to the XC functional used and so the damping functions must be adjusted so as to be compatible with each exchange-correlation or exchange functional.
  • This fitting is also sensitive to the definition of atomic size and must be done carefully since the damping function can actually affect the binding energies even more than the asymptotic C6 coefficients.
  • The fitting also effectively includes the effects of C8/r^8 or C10/r^10 and higher contributions.

Step two – Environment-dependent corrections

The simple “DFT-D” schemes: the dispersion coefficients are predetermined and constant quantities. The errors introduced by this approximation can be large.

The unifying concept:

The dispersion coefficient of an atom in a molecule depends on the effective volume of the atom. When the atom is “squeezed”, its electron cloud becomes less polarizable leading to a decrease of the C6 coefficients.

Three step 2 methods:

  • DFT-D3 of Grimme

Capture the environmental dependence of the C6 coefficients by considering the number of neighbors each atom has.

  • vdW(TS) of Tkatchenko and Scheffler

Relies on reference atomic polarizabilities and reference atomic C6 coefficients to calculate the dispersion energy.

During the calculation on the system of interest the electron density of a molecule is divided between the individual atoms and for each atom its corresponding density is compared to the density of a free atom.

  • BJ by Becke-Johnson

Based on the fact that around an electron at r1 there will be a region of electron density depletion, the so-called XC hole. This creates asymmetric electron density and thus non-zero dipole and higher-order electrostatic moments, which causes polarization in other atoms to an extent given by their polarizability.

C6 coefficients are altered through two effects:

  1. The polarizabilities of atoms in molecules are scaled from their reference atom values according to their effective atomic volumes.
  2. The dipole moments respond to the chemical environment through changes of the exchange hole, although this effect seems to be difficult to quantify precisely.

Step three – Long-range density functionals

Approaches that do not rely on external input parameters but rather obtain the dispersion interaction directly from the electron density.

Termed non-local correlation functionals since they add non-local (i.e., long range) correlations to local or semi-local correlation functionals.

Dispersion corrected DFT theory

Commonly used DFs do not describe the long-range dispersion interactions correctly. All semilocal DFs and conventional hybrid functionals asymptotically cannot provide the correct -C6/R^6 dependence of the dispersion interaction energy on the interatomic (molecular) distance R.

The failure of standard DFs comes from its inability to describe instantaneous electron correlations. In  more precise picture, electromagnetic zero-point energy fluctuations in the vacuum lead to ‘virtual’ excitations to allowed atomic or molecular electronic states. The corresponding densities interact electrostatically. They are not represented by conventional (hybrid) functionals that only consider electron exchange but do not employ virtual orbitals.

The computationally most efficient basic approaches to account for London dispersion effects in DFT includes:

  • Nonlocal vdW-DFs
  • ‘pure’ [semilocal (hybrid] DFs, which are highly parameterized forms of standard meta hybrid approximations (e.g., the M0XX family of functionals)
  • DFT-D methods (atom pairwise sum over  -C6/R^6 potentials)
  • Dispersion-correcting atom-centered one-electron potentials (1ePOT, called DCACP or in local variants LAP or DCP)

vdW-DF and Related Methods

(vdW-DF(2004), vdW-DF (2010), VV09, AND VV10)

Currently most widely used form, a nonempirical way to compute the dispersion energy. A supermolecular calculation of the total energy of the complex and the fragments is performed to obtain the interaction energy.

Approximation:

Total exchange-correlation energy Exc = Ex^LDA/GGA + Ec^LDA/GGA+Ec^NL

LDA (local density approximation) or GGA (semilocal) type are used for the short-ranged parts;

Ec^NL represents the nonlocal term describing the dispersion energy

Nonlocal vs Local (examination on the terminology)

In the DFT community, the dispersion energy is understood as an inherently nonlocal property. It must be described by a kernel which depends on two electron coordinates simultaneously.

In a WF picture, long-range dispersion has no nonlocal component.

Physically, dispersion is the Coulomb interaction between (local, fragment centered) transition densities. These can be plotted like ‘normal’ densities and have no ‘mysterious’ nonlocal character (except that virtual orbitals are needed for their construction).

Summary

  • Typically, Ec^NL is computed non-self-consistently, i.e., it is simply an add-on to the self-consistent filed (SCF)-DF energy similar as in DFT-D.
  • vdW-DF works better with short-range components that are basically repulsive such as Hartree-Fock.
  • Dispersion effects are naturally included via the charge density so that charge-transfer dependence of dispersion is automatically included in a physically sound manner. If performed self-consistently, the correction in turn also changes the density.
  • Whether double-counting effects of correlation at short range are present in the mentioned vdW-DFs is currently unknown.

Reference

Grimme, S., Density functional theory with London dispersion corrections. Wiley Interdisciplinary Reviews: Computational Molecular Science 2011, 1 (2), 211-228.