Car-Parrinello method

Car-Parrinello Molecular Dynamics or CPMD refers to either the molecular dynamics method, also known as the Car-Parrinello method, or the software package in which the method was implemented.

The CPMD method is related to the more widespread Born-Oppenheimer Molecular Dynamics Method (BOMD) in that quantum mechanical effects of the electrons are taken into account in the calculation of energies and forces for classical nuclear movements. BOMD deals with the electronic structure in the context of the time-independent Schrödinger equation. CPMD explicitly considers the electronic structure as an active degree of freedom via dynamic variables. The CPMD software package contains a parallelized plane-wave / pseudopotential implementation of the CPMD method based on density functional theory.

Car – Parrinello method

The Car-Parrinello method is a molecular dynamics method that is usually used in combination with periodic boundary conditions, plane-wave basis theorems and density functional theory. The method was proposed in 1985 by Roberto Car and Michele Parrinello, who were awarded the Dirac Medal by the ICTP in 2009.

With the Born-Oppenheimer molecular dynamics method, the electronic structure of a specific core position is calculated. This results in correction terms for the Born-Oppenheimer approximation that couple electron and nuclear movements. The core movements in the BOMD consist of classical ionic contributions and the correction terms.

In contrast, the CPMD method uses a Lagrange operator for the core movements, which explicitly contains the electronic degrees of freedom as fictitious, dynamic variables. This leads to coupled equations of motion for nuclei and electrons. An optimization of the electrons in every time step, as required in the BOMD, is avoided with the CPMD: After an initial optimization of the electronic structure, the electronic structure is kept at the basic state level of the respective core configuration by the fictitious dynamic variables.

In order to adhere to this adiabaticity condition, the fictitious electron mass is chosen so small that no significant energy transfer takes place from the nuclei to the electrons. However, this fictitious small electron mass means that the equations of motion can only be integrated over significantly smaller time intervals than the 1–10 fs usual for BOMD.

General approach

In the CPMD, the electrons close to the nucleus are usually approximated by pseudopotentials and the wave function of the valence electrons by plane-wave basis sets. The electron density of the ground state is calculated self-consistently for a fixed core configuration using density functional theory. The forces acting on the cores are calculated using this density and from this the next core configuration.

Fictional dynamics

CPMD is an approximation to the BOMD method. In the BOMD, the wave function of the electrons or the electron density is calculated in each journal. CPMD uses a fictitious dynamic to approximate the electronic ground state without recalculating the ground state in each step. These fictitious dynamics use a fictitious electron mass (400–800 au are more common) to ensure that little energy is transferred from the nuclei to the electrons (adiabaticity condition). An increase in the fictitious electron mass would lead to the system being able to leave the approximate BOMD ground state.

Lagrange functional

{\ displaystyle {\ mathcal {L}} = {\ frac {1} {2}} \ left (\ sum _ {I} ^ {\ mathrm {cores}} \ M_ {I} {\ dot {\ vec { R}}} _ {I} ^ {2} + \ mu \ sum _ {i} ^ {\ mathrm {orbitals}} \ int \ mathrm {d} ^ {3} {\ vec {r}} \, | {\ dot {\ psi}} _ {i} ({\ vec {r}}, t) | ^ {2} \ right) -E \ left [\ {\ psi _ {i} \}, \ {{ \ vec {R}} _ {I} \} \ right],}

where the Kohn-Sham energy density is functional , which assigns an energy to given Kohn-Sham orbitals and core positions. ${\ displaystyle E [\ {\ psi _ {i} \}, \ {{\ vec {R}} _ {I} \}]}$

Orthogonality condition

{\ displaystyle \ int \ mathrm {d} ^ {3} {\ vec {r}} \ \ psi _ {i} ^ {*} ({\ vec {r}}, t) \ psi _ {j} ( {\ vec {r}}, t) = \ delta _ {ij},}

where is the Kronecker delta . ${\ displaystyle \ delta _ {ij}}$

Equations of motion

The equations of motion are obtained by minimizing the Lagrange functional while varying and maintaining the orthogonality condition, ${\ displaystyle \ psi _ {i}}$ ${\ displaystyle {\ vec {R}} _ {I}}$

{\ displaystyle {\ begin {aligned} M_ {I} {\ ddot {\ vec {R}}} _ {I} & = - {\ vec {\ nabla}}} _ {I} E \ left [\ {\ psi _ {i} \}, \ {{\ vec {R}} _ {I} \} \ right] \\ mu {\ ddot {\ psi}} _ {i} ({\ vec {r}} , t) & = - {\ frac {\ delta E} {\ delta \ psi _ {i} ^ {*} ({\ vec {r}}, t)}} + \ sum _ {j} \ Lambda _ {ij} \ psi _ {j} ({\ vec {r}}, t), \ end {aligned}}}

where the Lagrange multipliers are used to ensure compliance with the orthogonality condition. ${\ displaystyle \ Lambda _ {ij}}$

Born – Oppenheimer limit

In the borderline case , the equations of motion correspond to those of the BOMD. ${\ displaystyle \ mu \ to 0}$

Individual evidence

^ Unified Approach for Molecular Dynamics and Density-Functional Theory . tape 55 , no. 22 , 1995, p. 2471-2474 , doi : 10.1103 / PhysRevLett.55.2471 .

[1] Unified Approach for Molecular Dynamics and Density-Functional Theory . tape 55 , no. 22 , 1995, p. 2471-2474 , doi : 10.1103 / PhysRevLett.55.2471 .