Chapter 2
Molecular States
Abstract This chapter will introduce the quantum mechanical equation of motion,
i.e., the time-dependent Schrödinger equation. We will show how the separation of
variables can be exploited to partition the molecular wavefunction in translational,
rotational, vibrational, and electronic components, with special emphasis on the
Born–Oppenheimer approximation and its breakdown. We shall then provide an
overview of the electronic structure and reactivity of excited states commonly found
in organic molecules.
Keywords Molecular states · Born · Oppenheimer approximation
Electric excited states · Photochemical reactions
2.1 The Time-Dependent Schrödinger Equation
In the first sections of this chapter we develop the quantum mechanical theory of
molecular states. The basic physical principles can be found in any textbook of
quantum mechanics, for instance in Merzbacher [1] or Sakurai and Napolitano [2].
For more chemical approaches, see Atkins and Friedman [3] or Levine [4] among
others. Good introductions to excited states and the theory of photochemistry are
Michl and Bonaˇ ci´ c-Koutecký [5] and Klessinger and Michl [6].
In nonrelativistic quantum mechanics the time evolution of a physical system is
given by the time-dependent Schrödinger equation (TDSE):
i
dΨ (x, t)
dt
= ˆ
H Ψ (x, t)
(2.1)
where ˆ
H is the Hamiltonian operator. The physical state of the system is described
by the wavefunction Ψ , which depends on time t and on the collection of spatial and
spin coordinates x of all the particles belonging to the system.
© Springer International Publishing AG, part of Springer Nature 2018
M. Persico and G. Granucci, Photochemistry, Theoretical Chemistry
and Computational Modelling, https://doi.org/10.1007/978-3-319-89972-5_2
25
Molecular States
Abstract This chapter will introduce the quantum mechanical equation of motion,
i.e., the time-dependent Schrödinger equation. We will show how the separation of
variables can be exploited to partition the molecular wavefunction in translational,
rotational, vibrational, and electronic components, with special emphasis on the
Born–Oppenheimer approximation and its breakdown. We shall then provide an
overview of the electronic structure and reactivity of excited states commonly found
in organic molecules.
Keywords Molecular states · Born · Oppenheimer approximation
Electric excited states · Photochemical reactions
2.1 The Time-Dependent Schrödinger Equation
In the first sections of this chapter we develop the quantum mechanical theory of
molecular states. The basic physical principles can be found in any textbook of
quantum mechanics, for instance in Merzbacher [1] or Sakurai and Napolitano [2].
For more chemical approaches, see Atkins and Friedman [3] or Levine [4] among
others. Good introductions to excited states and the theory of photochemistry are
Michl and Bonaˇ ci´ c-Koutecký [5] and Klessinger and Michl [6].
In nonrelativistic quantum mechanics the time evolution of a physical system is
given by the time-dependent Schrödinger equation (TDSE):
i
dΨ (x, t)
dt
= ˆ
H Ψ (x, t)
(2.1)
where ˆ
H is the Hamiltonian operator. The physical state of the system is described
by the wavefunction Ψ , which depends on time t and on the collection of spatial and
spin coordinates x of all the particles belonging to the system.
© Springer International Publishing AG, part of Springer Nature 2018
M. Persico and G. Granucci, Photochemistry, Theoretical Chemistry
and Computational Modelling, https://doi.org/10.1007/978-3-319-89972-5_2
25