Car-Parrinello method
Encyclopedia
The Car–Parrinello method is a type of ab initio
(first principles) molecular dynamics
, usually employing periodic boundary conditions, planewave basis set
s, and density functional theory
, proposed by Roberto Car
and Michele Parrinello
in 1985, who were subsequently awarded the Dirac Medal
by ICTP in 2009.
In contrast to Born–Oppenheimer molecular dynamics
wherein the nuclear (ions) degree of freedom are propagated using ionic forces which are calculated at each iteration by approximately solving the electronic problem with conventional matrix diagonalization methods, the Car–Parrinello method explicitly introduces the electronic degrees of freedom as (fictitious) dynamical variables, writing an extended Lagrangian
for the system which leads to a system of coupled equations of motion for both ions and electrons. In this way an explicit electronic minimization at each time step, as done in Born-Oppenheimer MD, is not needed: after an initial standard electronic minimization, the fictitious dynamics of the electrons keeps them on the electronic ground state
corresponding to each new ionic configuration visited along the dynamics, thus yielding accurate ionic forces. In order to maintain this adiabaticity condition, it is necessary that the fictitious mass of the electrons is chosen small enough to avoid a significant energy transfer from the ionic to the electronic degrees of freedom. This small fictitious mass in turn requires that the equations of motion are integrated using a smaller time step than the one (1–10 fs) commonly used in Born–Oppenheimer molecular dynamics.
where E[{ψi},{RI}] is the Kohn–Sham energy density functional, which outputs energy values when given Kohn–Sham orbitals and nuclear positions.
where δij is the Kronecker delta.
where Λij is a Lagrangian multiplier matrix to comply with the orthonormality constraint.
Ab initio
ab initio is a Latin term used in English, meaning from the beginning.ab initio may also refer to:* Ab Initio , a leading ETL Tool Software Company in the field of Data Warehousing.* ab initio quantum chemistry methods...
(first principles) molecular dynamics
Molecular dynamics
Molecular dynamics is a computer simulation of physical movements of atoms and molecules. The atoms and molecules are allowed to interact for a period of time, giving a view of the motion of the atoms...
, usually employing periodic boundary conditions, planewave basis set
Basis set (chemistry)
A basis set in chemistry is a set of functions used to create the molecular orbitals, which are expanded as a linear combination of such functions with the weights or coefficients to be determined. Usually these functions are atomic orbitals, in that they are centered on atoms. Otherwise, the...
s, and density functional theory
Density functional theory
Density functional theory is a quantum mechanical modelling method used in physics and chemistry to investigate the electronic structure of many-body systems, in particular atoms, molecules, and the condensed phases. With this theory, the properties of a many-electron system can be determined by...
, proposed by Roberto Car
Roberto Car
Roberto Car is an Italian physicist, who works on simulation of molecular dynamics phenomena.-Life:Car studied physics and attained a doctorate in 1971 in nuclear technology at the Politecnico di Milano...
and Michele Parrinello
Michele Parrinello
Michele Parrinello is an Italian physicist particularly known for his work in molecular dynamics...
in 1985, who were subsequently awarded the Dirac Medal
Dirac Prize
The Dirac Prize is the name of four prominent awards in the field of theoretical physics, computational chemistry, and mathematics, awarded by different organizations, named in honour of Professor Paul Dirac, one of the great theoretical physicists of the 20th Century.- The Dirac Medal and Lecture...
by ICTP in 2009.
In contrast to Born–Oppenheimer molecular dynamics
Molecular dynamics
Molecular dynamics is a computer simulation of physical movements of atoms and molecules. The atoms and molecules are allowed to interact for a period of time, giving a view of the motion of the atoms...
wherein the nuclear (ions) degree of freedom are propagated using ionic forces which are calculated at each iteration by approximately solving the electronic problem with conventional matrix diagonalization methods, the Car–Parrinello method explicitly introduces the electronic degrees of freedom as (fictitious) dynamical variables, writing an extended Lagrangian
Lagrangian
The Lagrangian, L, of a dynamical system is a function that summarizes the dynamics of the system. It is named after Joseph Louis Lagrange. The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics by Irish mathematician William Rowan Hamilton known as...
for the system which leads to a system of coupled equations of motion for both ions and electrons. In this way an explicit electronic minimization at each time step, as done in Born-Oppenheimer MD, is not needed: after an initial standard electronic minimization, the fictitious dynamics of the electrons keeps them on the electronic ground state
Ground state
The ground state of a quantum mechanical system is its lowest-energy state; the energy of the ground state is known as the zero-point energy of the system. An excited state is any state with energy greater than the ground state...
corresponding to each new ionic configuration visited along the dynamics, thus yielding accurate ionic forces. In order to maintain this adiabaticity condition, it is necessary that the fictitious mass of the electrons is chosen small enough to avoid a significant energy transfer from the ionic to the electronic degrees of freedom. This small fictitious mass in turn requires that the equations of motion are integrated using a smaller time step than the one (1–10 fs) commonly used in Born–Oppenheimer molecular dynamics.
Lagrangian
where E[{ψi},{RI}] is the Kohn–Sham energy density functional, which outputs energy values when given Kohn–Sham orbitals and nuclear positions.
Orthogonality constraint
where δij is the Kronecker delta.
Equations of motion
The equations of motion are obtained by finding the stationary point of the Lagrangian under variations of ψi and RI, with the orthogonality constraint.where Λij is a Lagrangian multiplier matrix to comply with the orthonormality constraint.
Born–Oppenheimer limit
In the formal limit where μ → 0, the equations of motion approach Born–Oppenheimer molecular dynamics.See also
- Computational chemistryComputational chemistryComputational chemistry is a branch of chemistry that uses principles of computer science to assist in solving chemical problems. It uses the results of theoretical chemistry, incorporated into efficient computer programs, to calculate the structures and properties of molecules and solids...
- Car–Parrinello Molecular Dynamics
- List of quantum chemistry and solid state physics software