5 Introduction to Quantum Vibrational Spectroscopy
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effects into account. However, there are limitations in the applicability of MP theory,
which led to the development of more advanced approaches.
Unlike the HF formalism, configuration interaction (CI) theory utilizes multiple
Slater determinants to construct configuration state functions (CSF), which are then
linearly combined to describe the wavefunction of the quantum system. The first
term in the expansion of the CI wavefunction is equivalent to the HF ground-state
wavefunction, while the higher terms capture the effects of the correlated motion
of the electrons. In the CI formalism, the wavefunction is a combination of the
HF reference states plus all possible excited states. This is reflected by mixing the
ground CSFs and the excited CSFs. If all possibilities of orbital occupations are
included (full-CI, FCI), an exact solution to the electron correlation problem can
be achieved. Unfortunately, the number of excited configurations is enormously
large, and in practice, the number of CI terms representing the electronic excitations needs to be truncated. The abbreviations for truncated CI variants reflect the
excitation levels treated; ‘S’ for single excitations, ‘D’ for double, ‘T’ for triple,
‘Q’ for quadruple. This leads to CI single and double excitations (CISD), CI single,
double, and triple excitations (CISDT), etc. From the point of view of quantum theory,
CI is the most complete approach to describe the electronic structure of molecular
systems. However, this corresponds only to FCI, that is, the case in which all orbital
occupations possible for the quantum system are treated. The FCI method is useful
for validation and benchmarking purposes of lower-level quantum methods, where
its extensive computational cost remains manageable. In practical terms, unless FCI
conditions are achieved, the application of truncated variants is often linked to considerable inaccuracies. Truncated CI methods capture a rapidly decreasing amount of
the ‘exact’ correlation energy with an increase of the system size, which limits their
usefulness in treating larger molecules and heavy atoms. Multiconfigurational selfconsistent field (MCSCF) is an analogous approach that additionally applies a similar
CI-like concept also to derive the one-electron functions that are subsequently used
to construct CSFs.
Coupled-cluster theory (CC) expands the molecular orbitals obtained at HF level
using an exponential cluster operator (acting as the excitation operator) and constructs
a multi-electron wavefunction that includes electron correlation. The CC formalism
may be considered as an alternative to CI, which produces an equivalent combination of one-electron functions to yield the multi-electron wavefunction. However,
unlike linear combination assumed in the latter, the exponential expansion used in
the former grants its size-extensivity resulting in an improved limiting behavior of
the CC correlation energy upon truncation to a given excitation level (e.g., CCSD).
Similar practical limitations as those found in CI apply here as well. The number
of treated electronic excitations needs to be limited in order to make the method
applicable in terms of the associated computational demand. This leads to variants,
abbreviated analogous to CI variants, e.g., CCSD, CCSDT, etc. Unfortunately, in
practical use, the CCSD level yields moderately correct results considering its cost,
while the more accurate CCSDT proves to be too expensive for most applications.
For this reason, CCSD(T) was introduced as a variant approximating the triple excitations via perturbation theory. Note, however, that when truncated at the same level,
91
effects into account. However, there are limitations in the applicability of MP theory,
which led to the development of more advanced approaches.
Unlike the HF formalism, configuration interaction (CI) theory utilizes multiple
Slater determinants to construct configuration state functions (CSF), which are then
linearly combined to describe the wavefunction of the quantum system. The first
term in the expansion of the CI wavefunction is equivalent to the HF ground-state
wavefunction, while the higher terms capture the effects of the correlated motion
of the electrons. In the CI formalism, the wavefunction is a combination of the
HF reference states plus all possible excited states. This is reflected by mixing the
ground CSFs and the excited CSFs. If all possibilities of orbital occupations are
included (full-CI, FCI), an exact solution to the electron correlation problem can
be achieved. Unfortunately, the number of excited configurations is enormously
large, and in practice, the number of CI terms representing the electronic excitations needs to be truncated. The abbreviations for truncated CI variants reflect the
excitation levels treated; ‘S’ for single excitations, ‘D’ for double, ‘T’ for triple,
‘Q’ for quadruple. This leads to CI single and double excitations (CISD), CI single,
double, and triple excitations (CISDT), etc. From the point of view of quantum theory,
CI is the most complete approach to describe the electronic structure of molecular
systems. However, this corresponds only to FCI, that is, the case in which all orbital
occupations possible for the quantum system are treated. The FCI method is useful
for validation and benchmarking purposes of lower-level quantum methods, where
its extensive computational cost remains manageable. In practical terms, unless FCI
conditions are achieved, the application of truncated variants is often linked to considerable inaccuracies. Truncated CI methods capture a rapidly decreasing amount of
the ‘exact’ correlation energy with an increase of the system size, which limits their
usefulness in treating larger molecules and heavy atoms. Multiconfigurational selfconsistent field (MCSCF) is an analogous approach that additionally applies a similar
CI-like concept also to derive the one-electron functions that are subsequently used
to construct CSFs.
Coupled-cluster theory (CC) expands the molecular orbitals obtained at HF level
using an exponential cluster operator (acting as the excitation operator) and constructs
a multi-electron wavefunction that includes electron correlation. The CC formalism
may be considered as an alternative to CI, which produces an equivalent combination of one-electron functions to yield the multi-electron wavefunction. However,
unlike linear combination assumed in the latter, the exponential expansion used in
the former grants its size-extensivity resulting in an improved limiting behavior of
the CC correlation energy upon truncation to a given excitation level (e.g., CCSD).
Similar practical limitations as those found in CI apply here as well. The number
of treated electronic excitations needs to be limited in order to make the method
applicable in terms of the associated computational demand. This leads to variants,
abbreviated analogous to CI variants, e.g., CCSD, CCSDT, etc. Unfortunately, in
practical use, the CCSD level yields moderately correct results considering its cost,
while the more accurate CCSDT proves to be too expensive for most applications.
For this reason, CCSD(T) was introduced as a variant approximating the triple excitations via perturbation theory. Note, however, that when truncated at the same level,