3.1 Ab initio electronic structure methods
Foundations of electronic structure methods are based on the variational theorem in quantum mechanics that states that the exact wave function of the ground state of a given Hamiltonian alone is the solution of the variational minimization of the expectation value of the Hamiltonian: minimize 〈Ψ|H|Ψ〉 subject to 〈Ψ|Ψ〉=1 yields H|Ψ〉=E|Ψ〉. In this manner, the variational theorem solves the time-independent Schrodinger equation, which conforms to the quantum master equation, as noted above. As a practical implementation, one can arrive at very close approximations to the exact groundstate solution by expanding the wavefunction in terms of finite basis sets: |Ψ〉=Σi cii〉. Lynchpin methods that enable the implementation of the variational calculation for many-electron systems, which are further subject to the constraints of Pauli’s exclusion principle, are Hartree-Fock methods and methods based on electronic density functional theory.
3.1.1 The Hartree-Fock (HF) approximation : HF corresponds to the conventional single-electron picture of the electronic structure where the distribution of the N electrons is given simply by the product of one-electron distributions. Hartree-Fock theory, by assuming a determinant form for the wavefunction, imposes the property of antisymmetry; nevertheless, the form neglects correlation between electrons. The electrons are subject to an average non-local potential arising from the other electrons, which can lead to an inadequate description of the electronic structure. Although qualitatively correct in many materials and compounds, Hartree-Fock theory can be insufficient to make accurate quantitative predictions. These predictions can be improved using higher-order perturbation theory-based methods [19].
3.1.2 Density functional theory (DFT): DFT is a formally exact theory [20]. It is distinct from quantum chemical methods in that it is a non-interacting theory and does not yield a correlated N-body wavefunction. DFT has come to prominence over the last decade as a method capable of very accurate results at a low computational cost. In practice, approximations are required to implement the theory and the accuracy is context-dependent. The Hohenberg-Kohn theorem states that if N interacting electrons move in a potential external Vext(r), the groundstate electron density n0(r) minimizes the energy functional E[n(r)]. The practical utility of DFT is in constructing the energy functional by augmenting a free electron gas reference energy functional (which is precisely known) with a parameterized form of energy terms that account for exchange and electron correlation (determined based on more accurate techniques such as quantum Monte Carlo or QMC methods, see below). Variational techniques similar to those utilized in HF methods can then be employed to obtain the ground state solution in DFT.
Softwares for quantum chemical calculations are available under open source or commercial licenses that make it easy to model molecular systems using electronic structure methods:
(https://en.wikipedia.org/wiki/List_of_quantum_chemistry_and_solid-state_physics_software). They have been the driving force to parametrize the force fields of classical simulations such as those in (Eq. 11) below.