Chemistry Reference
In-Depth Information
MD simulations can be of considerable interest for simulating biological systems since most living organisms work
at about 310 K. The addition of QM methods to MD, within the DFT framework, i.e first principles MD was
originally developed in 1985 in the Car-Parrinello molecular dynamics (CPMD) approach, in which MD simulations
can be made with the potential energy of the systems being computed at the DFT level of theory [468].
The scope of DFT formalism in particular and QM methods in general has expanded considerable, with the advent
of QM/MM schemes, which can take into account tens of thousands of atoms enabling the study of a protein active
site at the QM level whereas the rest of the protein and the solvent is treated by means of classical MM force fields.
The implementation of the CPMD and other modification of the QM/MM mixed codes provides a very interesting
and stimulating way of investigating at the QM level, macromolecular systems of pharmacological interest.
In a QM/MM approach the computational effort is concentrated in the part of the system where it is most needed.
The effects of the surroundings are taken into account with a more expedient model. The intricacies of QM/MM lies
in finding an appropriate treatment for the coupling between QM and MM regions. Electrostatic, van der Waals and
bonded interactions can be included between the QM and MM regions. Hydrogen link atoms are often introduced to
saturate the shell of QM-atoms covalently bonded to MM-atoms. The QM region can be treated at levels of theory
spanning from semiempirical to ab initio and DFT Hamiltonians [467].
For a Hamiltonian with M nuclei and N electrons, in the Born-Oppenheimer approximation, the electronic motion
can be decoupled since the mass of the nuclei is much larger than the electron mass. The nuclei moves in a potential
given by electronic ground state energy and the electrons can be relaxed to the ground state for a given ionic
configuration. The total energy is the expectation value of the Hamiltonian and the many-electron wavefunction is in
3N dimensional space. The ground state wavefunction has the lowest energy and obeys the conservation laws and
the symmetries of the particles.
Early approaches to solve the Schrodinger equations concentrated in improving the wavefunctions using Slater
determinants in an HF approach. However, correlation is not included and configuration interaction (CI) had to be
considered whereas the wavefunction is a linear combination of Slater determinants. The growth is factorial with the
increase of the number of electrons and difficulties lies in exact solutions of these equations.
Ab initio
methods aim
at the solution of the Schrodinger equation, which cannot be solved exactly for polyelectronic systems resorting to
approximations.
Hartre-Fock (HF) is one of the simplest approximations whereas the probability of finding an electron is assumed
independent of the probability of finding other electrons. Each electron feels the average potential of all the other
electrons. This method only includes the quantum mechanical exchange term but does not include electron-
correlation which may be computed in various ways. In the Moller-Plesset (MP) many perturbation theory it is
treated as a perturbation and corrections can be made at any order of energy and wave function, of which MP is one
the most common. The HF approximation can also be extended to several electron configurations leading to the so-
called multiconfigurational methods, which although very computationally demanding can reach a very high level of
accuracy.
For a molecule, solid or atom, i.e an N-electron system, the time-independent Schrodinger equation depends only on
the potential yielding minimization or direct solution strategies. Effectively, we can obtain the ground state energy
by minimizing the expectation value of the energy using the variational method. However, it was noted that it should
be possible to use the density as key variable, as in condensed matter physics, providing thus a viable, versatile
alternative. Systems differ only by their potential energy.
A prescription can be provided, which deals with the kinetic energy and the electron-electron interaction, mapping
the many-body problem into a single-body problem. Although the charge density and electrostatic potential were
not reproduced in the early 1920s, it was recognized that the kinetic energy density in a uniform electron gas
approximation could lead to ground state minimization of the energy functional via constraints. With the
developments of Green and correlation functions, the chain of correlation is broken and only in 1964 did Hohenberg
and Kohn show that all properties of the many-body system could be determined and are a function of the ground
state density
