278
E.F. Sheka
15.3 Discussion and Conclusive Remarks
The odd electrons of benzenoid units and the correlation of these electrons having
different spins are the main building stones of the molecular theory of sp 2 nanocarbons. In contrast to the theory of aromaticity, the molecular theory accepts that the
odd electrons with different spins occupy different places in the space so that the
configuration interaction (CI) becomes the central point of the theory. Consequently,
a multi-determinant presentation of the wave function of the system of the weakly
interacting odd electrons is absolutely mandatory on the way of the theory implementation at the computational level. However, the efficacy of the available CI computational techniques is quite restricted in regards large polyatomic systems, which
does not allow performing extensive computational experiments. On the other hand,
the modern computational science of sp 2 nanocarbons, in general, and graphene, in
particular, is, actually, the field of such experiments due to its steadily grown importance caused by prevailing computations over other empirical technique, which is
evidently the case of graphene. Facing the problem, computationists have addressed
standard single-determinant software albeit not often being aware of how correct
are the obtained results. The current paper attempts to present the molecular theory
of graphene in terms of the single-determinant computational schemes as well as to
analyze the reliability of the obtained results.
The open-shell presentation of the wave functions is the first step towards the
multi-determinant computational schemes so that naturally one has to address this
form of the function presentation. Unrestricted Hartree-Fock (UHF) and density
functional techniques (UDFT) are to be the basic grounds for the techniques used.
In spite of a partial suiting of both approaches to the CI ones, both UHF and UDFT
schemes provide spin-contaminated solutions with the relevant energies that exceed
the pure-spin ones. Much higher energies and, thus, much less reliability correspond
to the standard computational HF and DFT schemes in the restricted closed-shell
approach. Nevertheless, a predominant majority of the DFT computations related
to graphene have been performed in this approximation, which greatly impugns the
reliability of the results obtained.
In the case of the unrestricted approach, the situation is better but this does not
remove the issue about the result reliability. On the example of the application of
the UHF-based theory to graphene, were obtained answers to most of the questions.
These answers lead the foundation of the current paper. It should be noted that getting them has required the performance of system computational experiments in the
majority of cases.
Before passing to the answers, one should pay attention to the fact that the inner
features of the unrestricted computational schemes open the possibility in issuing
three criteria that can distinguish electrons systems by the electrons correlation.
These criteria are presented by the following quantities: (1) the energy misalignment
E RU ≥ 0; (2) the total number of effectively unpaired electrons N D = 0; and
(3) the squared spin misalignment ˆ
S 2 ≥ 0. A detailed description of the values is
given in the relevant Section. When all the quantities are zero, the electrons are noncorrelated (that is the case of the benzene molecule), and the relevant sp 2 systems
E.F. Sheka
15.3 Discussion and Conclusive Remarks
The odd electrons of benzenoid units and the correlation of these electrons having
different spins are the main building stones of the molecular theory of sp 2 nanocarbons. In contrast to the theory of aromaticity, the molecular theory accepts that the
odd electrons with different spins occupy different places in the space so that the
configuration interaction (CI) becomes the central point of the theory. Consequently,
a multi-determinant presentation of the wave function of the system of the weakly
interacting odd electrons is absolutely mandatory on the way of the theory implementation at the computational level. However, the efficacy of the available CI computational techniques is quite restricted in regards large polyatomic systems, which
does not allow performing extensive computational experiments. On the other hand,
the modern computational science of sp 2 nanocarbons, in general, and graphene, in
particular, is, actually, the field of such experiments due to its steadily grown importance caused by prevailing computations over other empirical technique, which is
evidently the case of graphene. Facing the problem, computationists have addressed
standard single-determinant software albeit not often being aware of how correct
are the obtained results. The current paper attempts to present the molecular theory
of graphene in terms of the single-determinant computational schemes as well as to
analyze the reliability of the obtained results.
The open-shell presentation of the wave functions is the first step towards the
multi-determinant computational schemes so that naturally one has to address this
form of the function presentation. Unrestricted Hartree-Fock (UHF) and density
functional techniques (UDFT) are to be the basic grounds for the techniques used.
In spite of a partial suiting of both approaches to the CI ones, both UHF and UDFT
schemes provide spin-contaminated solutions with the relevant energies that exceed
the pure-spin ones. Much higher energies and, thus, much less reliability correspond
to the standard computational HF and DFT schemes in the restricted closed-shell
approach. Nevertheless, a predominant majority of the DFT computations related
to graphene have been performed in this approximation, which greatly impugns the
reliability of the results obtained.
In the case of the unrestricted approach, the situation is better but this does not
remove the issue about the result reliability. On the example of the application of
the UHF-based theory to graphene, were obtained answers to most of the questions.
These answers lead the foundation of the current paper. It should be noted that getting them has required the performance of system computational experiments in the
majority of cases.
Before passing to the answers, one should pay attention to the fact that the inner
features of the unrestricted computational schemes open the possibility in issuing
three criteria that can distinguish electrons systems by the electrons correlation.
These criteria are presented by the following quantities: (1) the energy misalignment
E RU ≥ 0; (2) the total number of effectively unpaired electrons N D = 0; and
(3) the squared spin misalignment ˆ
S 2 ≥ 0. A detailed description of the values is
given in the relevant Section. When all the quantities are zero, the electrons are noncorrelated (that is the case of the benzene molecule), and the relevant sp 2 systems