Real-World Predictions from Ab Initio Molecular Dynamics Simulations - Multiscale Molecular Methods in Applied Chemistry - page 127

accurately, because in large and condensed phase systems the chemistry often relies
solely on the intermolecular forces of such type.
Several articles on corrections of van der Waals interactions applying the
seamless approach in density functional theory have appeared in the literature
[ 47 - 50 ]. In an early article of a series, Dion et al. developed and applied a van
der Waals density functional in order to treat situations for which nonlocal, long-
ranged interactions, such as van der Waals (vdW) forces, were influential. The
authors suggested the following form:

Z d 3 rd 3 r 0 n

1
2

E nl
c

r 0 Þ r 0 Þ

¼ ð r Þfð r ; n ð (36)

(r, r 0 ) is some given, general
function depending on r r 0 and the densities n in the vicinity of r and r 0 .This
truly non-linear functional has been applied successfully to many systems and
recently a second generation further improved version has been proposed [ 51 ].
R

for the nonlocal correlation energy part in which f

othlisberger and coworkers proposed to add an effective atom centered nonlo-
cal term to the exchange-correlation potential in order to cure the lack of London
dispersion forces in standard density functional theory [ 52 , 53 ]. In particular, the
authors constructed an effective potential consisting of optimized nonlocal terms
dependent on higher angular momentum for all atoms in the system. They modeled
van der Waals forces by an atom-electron interaction, mediated by appropriate
nonlocal effective core potential (ECP) projectors, which were obtained from an
optimization scheme. R

€

othlisberger and coworkers stated that this scheme has some
advantages over empirical pair potential corrections: “

€

First, the improved
electronic properties (dipole moment, quadrupole moment, and polarizability)
indicate that, due to the non-locality of the ECP projectors, the valence
wavefunctions reproduce more of the characteristics of dispersion interactions
than a simple additive atom-atom based correction. Second, properly calibrated
and transferable atomic dispersion calibrated ECPs no longer need any artificial a
priori assignment of interacting groups or atoms” [ 52 ].
Among the approaches discussed here, the most simple and straightforward and
thus the most practical approach is that followed by Stefan Grimme [ 54 , 55 ].
Grimme defined the dispersion corrected total energy E MF-D as

...

E MF D ¼ E MF j E disp ; (37)

where E MF is the Hartree-Fock or DFT mean-field energy and E disp is an empirical
dispersion correction expressed as

s 6 X

N at 1

X

N at

C i 6
R ij

E disp ¼ f dmp R ð R ij Þ: (38)

i¼ 1 j¼iþ 1