2.1. Theoretical backgrounds of computational quantum chemistry
2.1.7. Post-Hartree-Fock methods
Post-Hartree-Fock methods are advanced wave function approaches used in quantum chemistry to improve upon the inherent limitations of the HF method.
While HF provides a good starting point for describing electronic structure, it neglects important electron correlation effects which are crucial for accurately
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predicting molecular properties. Post-Hartree-Fock methods thus aim to incorporate electron correlation more rigorously in the wave functions leading to more accurate results.
There are several widely used post-Hartree-Fock methods, including configuration interaction (CI), many-body perturbation theory (MBPT), multi- configuration (MC), coupled cluster (CC) and multi-reference CI (MRCI)... These methods include electron correlation contributions taking the HF wavefunctions as reference for generating electronic configurations via electron excitations.
Accordingly, electronic configurations include the singly excited (called the singles), doubly excited (doubles), triply excited (triples)… wavefunctions. The most commonly used CC methods are the CCSD (coupled cluster including singles and doubles), CCSD(T) (CCSD with a perturbative triples correction), and CCSDT (CC with all possible singles, doubles and triple excitations). CCSD(T) is often referred to as a "gold standard" method in computational quantum chemistry in view of its high accuracy results and computational demands, as compared to higher levels such as the CCSDT, CCSDTQ… methods. Such a designation stems from its high level of accuracy and reliability in treating electron correlation effects. With the test of time, CCSD(T) has become a benchmark against other methods that are evaluated and compared.
The CCSD(T) method is known for its remarkable accuracy in predicting molecular energetics and properties, including total atomization energies, bond dissociation energies, reaction energies, and other parameters related to a potential energy surface. However, it is important to note that CCSD(T) is typically limited to moderate-sized molecules due to its high computational cost, and its applicability to larger systems (over 20 atoms) is expected to be challenging, even with access to high-performance computers (HPC).
The choice of basis set in CCSD(T) calculations also requires a careful consideration. In the case of large molecules, performing CCSD(T) calculations can be computationally expensive, leading to a temptation of selecting excessively small
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basis sets that may compromise the accuracy of the results. It does not make sense running a high level post-HF computation using a small basis set.
When the size of the molecule or the level of electron correlation requires a sophisticated treatment, the complete basis set (CBS) extrapolation is a method used to determine the total energy of a chemical species as if it is computed with an infinitely large basis set. It involves performing calculations using a series of basis sets of increasing size and then extrapolating the results to the limit of an infinite basis set. The CBS extrapolation helps to reduce the basis set dependence of the computed results and provides a more accurate value of the system's electronic energy, which is important in obtaining reliable and accurate theoretical results.
Performing CCSD(T) calculations using a series of two or three basis sets, namely aug-cc-pVDZ, aug-cc-pVTZ, and aug-cc-pVQZ (in which cc stands for correlation consistent) is a common practice in quantum chemistry. The CBS energy can be estimated from the equation:
𝐸(𝑥) = 𝐸𝐶𝐵𝑆+ 𝐵𝑒−(𝑥−1)+ 𝐶𝑒−(𝑥−1)2 (2.11) where 𝑥 = 2, 3, and 4 for the aug-cc-pVDZ, aug-cc-pVTZ, and aug-cc-pVQZ basis sets, respectively [68]. Alternatively, for a simpler approach (although it requires more RAM resources during calculations), the CBS values can be readily extrapolated using a pre-established keyword in the ORCA program [69].
In addition to its computational cost, the CC method has limitations in accurately describing spin-contaminated structures, which are typically characterized by T1 values greater than 0.02 for closed electron systems and 0.04 for open electron systems [70]. In other words, because the CC method is based on HF reference, it is mainly good for systems that can be characterized by a single reference. To address this challenge and obtain a more precise electronic structure description of such molecules, alternative methods that consider multi-reference methods become necessary. Two common multireference approaches that can handle spin-contaminated systems are the multi-configurational (MC) method and
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the Multi-reference Configuration Interaction (MRCI). In particular, the Complete Active Space Self-Consistent Field (CASSCF) which is a specific version of the MC method, can be carried out by several non-commercial software such as the GAMES, ORCA, MOLCAS… The CASSCF wave function is in turn used as reference for a following treatment by 2d-order perturbation theory leading to the CASPT2 method. For its part, the MRCI method used the MC wave functions as references for a subsequent expansion of the configuration interaction whose convergence is variationally determined. By definition, these methods allow us to treat with confidence systems having a multi-reference character and imperatively the excited states. Nevertheless, the main limitation of these methods is that they usually require a huge amount of computing time. For their routine use, an access to superior supercomputing resources is necessary!