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 ci|Φi〉. 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.