Vibrational Spectroscopy of Fluorescent Proteins: A Tool to Investigate the Structure of the Chromophore and Its Environment - Fluorescent Proteins: From Understanding to Design - page 140

and E xc [ ] is the exchange and correlation energy functional. This is the only

unknown part of the system, where most of the many-body effects are hidden.
Several different approximations are commonly used for it, which are variants and
evolutions of the local density approximation; this assumes that a system with
a general nonuniform electron density

r

r

(r) is locally coincident with a uniform

electron gas at the same density

ð

E xc ½ r¼

rð Þ e xc ð rð ÞÞ ;

r r dr

where e xc (

r

) is the exchange and correlation energy density (single variable)
function of a uniform electron gas evaluated for instance through exact sum rules
and Monte Carlo simulations [ 22 - 24 ]. In the last 20 years, tens of different energy
functionals treating the exchange and correlation at different levels were proposed
and their performances evaluated on different systems [ 25 ]. Some functionals of the
last generation include a part of explicit exchange through a Fock-like approach.
This improves the accuracy, especially for the calculations of properties involving
excited states, or of the ground state properties at the very fine level, but at the
expenses of a much larger computational cost.
A further simplification involves the type of basis set - localized orbitals or plane
waves - used for the expansion of the wavefunctions and the strictly related
question of the number of explicit electrons considered. In most cases, the core
electrons are basically inert during the system dynamics: the frozen core approxi-
mation allows treating much larger systems considering only the valence electrons.
This is quite easily implementable within the localized bases schemes because
the core orbitals are expanded using only single or few bases functions whose
coefficients are simply kept fixed during dynamics. The expansion in plane waves,
conversely, brings great save in computational cost due to the possibility of using
the fast Fourier transforms, but implies the use of pseudopotentials for the nuclei
to mimic the effect of their core frozen electrons. Again, in the latest years a large
number of different pseudopotentials with different levels of accuracy and transfer-
ability were optimized and tested on different systems [ 26 ].

3.1 Calculation of the Vibrational Frequencies

The KS energy returns the ground state energy when the KS equation system is self-
consistently resolved. Considering its implicit dependence on the nuclear
coordinates R, this means that E e, 0 (R) E KS [

](R), i.e., the KS energy takes the
part of an effective potential energy for the nuclei. The vibrational frequencies and
vibrational modes are by definition the eigenvalues and eigenvectors of the
dynamical matrix, that is the matrix of second derivatives of E e, 0 (R) with respect
to R, appearing in the second order expansion of the energy with respect to the
equilibrium configuration of the nuclei:

¼

r