Handbook of Solvents - George Wypych - ChemTech - Ventech!

420 Jacopo Tomasi, Benedetta Mennucci, Chiara Cappelli
Quantum mechanics (QM) is universally acknowledged as the appropriate theory to
treat material systems at the level of phenomena of interest to chemistry. Therefore quantum
mechanics is the legitimate theory level at which to treat liquids.
We recall here the opening statement of a famous book on quantum chemistry: 1 “In so
far as quantum mechanics is correct, chemical questions are problems in applied mathematics.”
Actually, the mathematics to apply to liquid systems is a hard nut to crack. Fortunately
a sequence of many, but reasonable, approximations can be introduced. We shall consider
and exploit them in the following section of this chapter.
It is important to stress one point before beginning. The naïve picture we have summarized
can be recovered without much difficulty in the quantum formulation. This will be a
semiclassical model: the identity of the constituting particle is preserved; their motions, as
well as those of nuclei within each molecule, are treated as in classical mechanics: the quantum
methods add the necessary details to describe interactions. This semiclassical quantum
description can be extended to treat problems going beyond the possibilities of purely classical
models, as, for example, to describe chemical reactions and chemical equilibria in solution.
The limits between the classical and quantum parts of the model are quite flexible,
and one may shift them in favor of the quantum part of the model to treat some specific phenomenon,
or in favor of the classical part, to make the description of larger classes of phenomena
faster.
We shall enter into more details later. It is sufficient here to underline that the choice of
using a quantum approach as reference is not a caprice of theoreticians: it makes descriptions
(and predictions) safer and, eventually, simpler.
We shall start with the introduction of some basic simplifications in the quantum
model.
8.2 BASIC SIMPLIFICATIONS OF THE QUANTUM MODEL
The quantum mechanical description of a material system is obtained as solution of the pertinent
Schrödinger equations.
The first Schrödinger equation is the famous equation everybody knows:
()() Ψ Ψ()
Hx x = E x
[8.1]
The function Ψ(x) describes the “state of the system” (there are many states for each system).
It explicitly depends on a set of variables, collectively called x, that are the coordinates
of the particles constituting the system (electrons and nuclei). E is a number, obtained by
solving the equation, which corresponds to the energy of the system in that state. H(x),
called Hamiltonian, technically is an operator (i.e., a mathematical construct acting on the
function placed at its right to give another function). Eq. [8.1] is an eigenvalue function:
among the infinite number of possible functions depending on the variables x, only a few
have the notable property of giving, when H(x) is applied to them, exactly the same function,
multiplied by a number. To solve eq. [8.1] means to find such functions.
The first step in the sequence of operations necessary to reach a description of properties
of the system is to give an explicit formulation of the Hamiltonian. This is not a difficult
task; often its formulation is immediate. The problems of “applied mathematics” are related
to the solution of the equation, not to the formulation of H.
The second Schrödinger equation adds more details. It reads: