84
3 Ab initio Electronic Structure for Few-Body Systems
more than one nucleus we shall omit the nucleus-nucleus interaction ˆ
V nn term that
is included in the nuclear part.
Accordingly, here we deal only with the terms associated with the interaction of
the electrons among themselves and with the nuclei. This means that the additional
complexity of the theoretical treatment depends not only on the inclusion of more
particles (with the consequent inapplicability of the pure central field) but also on the
fermionic nature of the electrons that induces a change of sign of the wavefunction
following the change of assignment of the electrons to the related wavefunctions. As
a matter of fact, in the remainder of the section, we shall discuss the details of the
theoretical treatment of multi-electron and the multi-atom features of the potential
energy surfaces governing reactive chemical processes and with their implications
on the motion of the nuclei.
It is, in fact, important to stress out here that while the detail of the ab initio
techniques provided in the book is instrumental to the understanding of the utilization
of their outcomes in dynamical studies it would have been impossible to confine the
discussion of the subject to a mere provision of some key literature references and
to the listing of the most popular computer programs (see for example [1, 14–17]).
Accordingly, the discussion given in this book of the most popular ab initio techniques
will be directed toward the motivation of either the direct utilization of their outcomes
for dynamical calculations or the construction of appropriate functional forms for
fitting high level of theory potential energy values.
For this reason, we anticipate from the next chapter that here we deal only with
the pure electronic wavefunction component (r) of the total wavefunction of
the molecular (electrons + nuclei) system described there due to the adoption of the
Born–Oppenheimer approximation (that separates the nuclear subproblem from the
electronic one). We also mention that here we use r as the electronic coordinate (of
dimension 3 for a single electron and 3K for K electrons).
3.1.2 Quantum Monte Carlo
The first method to mention, thanks also to the fact that most reactions occur on
the fundamental potential energy surface, is the quantum Monte Carlo (QMC) [18,
19] one. QMC calculations can produce reliable numerical solutions of the quantum
many-body problem by the direct numerical integration of the many-electron multidimensional Schrödinger equation based on repeated random sampling. Its Metropolis–
Hastings algorithm is also a method for obtaining a sequence of random samples from
a probability distribution for which direct sampling is difficult.
1
1 The Metropolis–Hastings algorithm works by generating a sequence of sample values in such a
way that, as more and more sample values are produced, the distribution of values more closely
approximates the desired distribution, P(x). These sample values are produced iteratively, with
the distribution of the next sample being dependent only on the current one. Specifically, at each
iteration, the algorithm picks a candidate for the next sample value based on the current one. Then,
with some probability, the candidate is either accepted (in which case the candidate value is used in
3 Ab initio Electronic Structure for Few-Body Systems
more than one nucleus we shall omit the nucleus-nucleus interaction ˆ
V nn term that
is included in the nuclear part.
Accordingly, here we deal only with the terms associated with the interaction of
the electrons among themselves and with the nuclei. This means that the additional
complexity of the theoretical treatment depends not only on the inclusion of more
particles (with the consequent inapplicability of the pure central field) but also on the
fermionic nature of the electrons that induces a change of sign of the wavefunction
following the change of assignment of the electrons to the related wavefunctions. As
a matter of fact, in the remainder of the section, we shall discuss the details of the
theoretical treatment of multi-electron and the multi-atom features of the potential
energy surfaces governing reactive chemical processes and with their implications
on the motion of the nuclei.
It is, in fact, important to stress out here that while the detail of the ab initio
techniques provided in the book is instrumental to the understanding of the utilization
of their outcomes in dynamical studies it would have been impossible to confine the
discussion of the subject to a mere provision of some key literature references and
to the listing of the most popular computer programs (see for example [1, 14–17]).
Accordingly, the discussion given in this book of the most popular ab initio techniques
will be directed toward the motivation of either the direct utilization of their outcomes
for dynamical calculations or the construction of appropriate functional forms for
fitting high level of theory potential energy values.
For this reason, we anticipate from the next chapter that here we deal only with
the pure electronic wavefunction component (r) of the total wavefunction of
the molecular (electrons + nuclei) system described there due to the adoption of the
Born–Oppenheimer approximation (that separates the nuclear subproblem from the
electronic one). We also mention that here we use r as the electronic coordinate (of
dimension 3 for a single electron and 3K for K electrons).
3.1.2 Quantum Monte Carlo
The first method to mention, thanks also to the fact that most reactions occur on
the fundamental potential energy surface, is the quantum Monte Carlo (QMC) [18,
19] one. QMC calculations can produce reliable numerical solutions of the quantum
many-body problem by the direct numerical integration of the many-electron multidimensional Schrödinger equation based on repeated random sampling. Its Metropolis–
Hastings algorithm is also a method for obtaining a sequence of random samples from
a probability distribution for which direct sampling is difficult.
1
1 The Metropolis–Hastings algorithm works by generating a sequence of sample values in such a
way that, as more and more sample values are produced, the distribution of values more closely
approximates the desired distribution, P(x). These sample values are produced iteratively, with
the distribution of the next sample being dependent only on the current one. Specifically, at each
iteration, the algorithm picks a candidate for the next sample value based on the current one. Then,
with some probability, the candidate is either accepted (in which case the candidate value is used in