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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

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