Vibrational Spectroscopy of Fluorescent Proteins: A Tool to Investigate the Structure of the Chromophore and Its Environment - Fluorescent Proteins: From Understanding to Design - page 139

parametrically on R. The nuclear equation is generally treated at the classical level
or quantized within the harmonic approximation, while the electron equation can be
solved with a variety of different QM methods. A widely used one is the Density
Functional Theory (DFT; for a review on DFT, see [ 21 ]), which is a good com-
promise between the size of the system (up to ~100 atoms when considering the
chromophore and the other atoms in its pocket in the protein) and accuracy.
The DFT is based on the one-to-one correspondence between the multivariable
ground state electron wave-function and two single variable functions, i.e., the
external field (e.g., that of the nuclei) and the ground state density r (r). Conse-
quently, the uniqueness of a density functional describing the ground state energy
and a minimal principle for it can be derived, which would in principle allow to
find the exact ground state electron density (and the wavefunction). In practice,
however, the DFT is commonly used within the Kohn-Sham (KS) scheme, which
exploits the expansion of the electron wavefunctions in single-electron orbitals

f i ,

leading to the very simple form for the electron density

X

N

2

rð r Þ¼ jf i ð r Þj

i

€

that, combined with the minimal principle, allows rewriting the electron Schr odinger

equation in a set of separated equations for electrons

X
j L ij f j ð

1
2 r

Þþ d

E xc ½r
d

2

þ v ð Þþ V H ð f i ð Þ¼ Þ

r r r r

n ð r Þ

i.e., the single electron Schr odinger equation in an effective potential, called the

Kohn-Sham potential, which is the functional derivative of the KS energy (total
electron energy) with respect to the density. In this way, the N electron problem is
remapped in a set of N equivalent single electron equations in an effective external
potential depending on the electron density. Even if this set of equations has
to be solved self-consistently, there is a great save in computational cost.

€

L ij
are Lagrange multipliers that account for the orbitals orthonormality and the
Kohn- Sham energy is

ð dr v ð dr V H ð

1
2

E KS ½r¼ T s ½rþ ð r Þrð r Þþ r Þrð r Þþ E xc ½r;

where v (r) is the external potential (implicitly dependent on the nuclear
coordinates), V H (r) is the Hartree potential (the coulomb part), T s the kinetic part

f i

X

N

1
2 r

2

T s ½r¼ f i

i