Handbook of Solvents - George Wypych - ChemTech - Ventech!

422 Jacopo Tomasi, Benedetta Mennucci, Chiara Cappelli
2) If it is small but not negligible, it may be used to correct the solutions
given by equations [8.6] with the use of appropriate mathematical tools:
the factorized formulation with corrections continues to be simpler than
the original one.
3) There are cases in which this factorization is profitably used, in spite of
the fact that interactions between particles of A and B are of the same
magnitude of interactions within both A and B.
We shall now briefly introduce two factorizations of the last type that are of paramount
importance in molecular quantum chemistry, then consider other factorizations of direct interest
for the description of liquid systems.
The first factorization we are introducing regards the electronic and nuclear coordinates
(called r and R respectively) of the same system: H(x) = H e (r)+H n (R). It is clear that
the electrostatic interaction between two electrons (and between two nuclei) within a given
molecule is of the same magnitude as that concerning a couple electron-nucleus. In spite of
this, the factorization is performed, with a tremendous effect on the evolution of quantum
chemistry. Its physical justification rests on the very different masses nuclei and electrons
have, and therefore on their velocities. It is often called Born-Oppenheimer (BO) approximation
and in the following we shall use this acronym. The BO approximation makes possible
the introduction of the concept of potential energy surface (PES), a very useful model to
describe the motion of nuclei (often in the semiclassical approximation) taking into account
the interactions with the quantum description of the electrons.
Some more details can be useful in the following. To apply this factorization it is imperative
to follow a given order. First, to solve the electronic equation with H e (R) at a fixed
geometry of nuclei: the output is a wave function and an energy both parametrically dependent
on the nuclear geometry R given as input: Ψ e (r;R) and E e (R) (actually there will be a set
of electronic states, each with its energy). Second, to repeat the same calculations at different
nuclear geometries R’, many times, until a sufficiently detailed description of the function
E e (R) is reached. E e (R) is the PES we have mentioned: it may be used to define the
potential operator within the nuclear Hamiltonian H n (R) and then to compute vibrational
and rotational states (another factorization is here introduced) or used in a semiclassical way
to study the effect of nuclear motions.
The second factorization we are introducing regards the electronic part of the system
(after the BO factorization). The main procedure in use leads to factorization into many separate
parts, each regarding one electron only:
elec
() ( )
e
H r = ∑ hi ri
i
Here again the coupling terms are of the same order of magnitude as the interactions
left within each one-electron Hamiltonian (that explicitly regards the electrostatic interaction
of the electron with all the nuclei of the systems, placed at fixed positions). The trick allowing
this factorization consists in introducing within each one-electron Hamiltonian (the
symmetry of electrons makes them all equal) an averaged description of the couplings,
based on the yet unknown wave functions of the other electrons. The calculation proceeds
iteratively: starting from a first guess of the averaged interaction, the description is progressively
refined using intermediate values of the one-electron wave functions. The final out-