Ab initio molecular dynamics: Theory and Implementation

4.2.3 Imposing Pressure: BarostatsKeeping the pressure constant is a desirable feature for many applications of moleculardynamics. The concept of barostats and thus constant–pressure molecular dynamicswas introduced in the framework of extended system dynamics by Hans Andersen14 , see e.g. Refs. 12,270,585,217 for introductions. This method was devised toallow for isotropic fluctuations in the volume of the supercell. A powerful extensionconsists in also allowing for changes of the shape of the supercell to occur as a resultof applying external pressure 459,460,461,678 , including the possibility of non–isotropicexternal stress 460 ; the additional fictitious degrees of freedom in the Parrinello–Rahman approach 459,460,461 are the lattice vectors of the supercell, whereas thestrain tensor is the dynamical variable in the Wentzcovitch approach 678 . Thesevariable–cell approaches make it possible to study dynamically structural phasetransitions in solids at finite temperatures. With the birth of ab initio moleculardynamics both approaches were combined starting out with isotropic volumefluctuations 94 à la Andersen 14 and followed by Born–Oppenheimer 681,682 andCar–Parrinello 201,202,55,56 variable–cell techniques.The basic idea to allow for changes in the cell shape consists in constructingan extended Lagrangian where the primitive Bravais lattice vectors a 1 , a 2 and a 3of the simulation cell are additional dynamical variables similar to the thermostatdegree of freedom ξ, see Eq. (268). Using the 3 × 3 matrix h = [a 1 ,a 2 ,a 3 ] (whichfully defines the cell with volume Ω) the real–space position R I of a particle in thisoriginal cell can be expressed asR I = hS I (274)where S I is a scaled coordinate with components S I,u ∈ [0, 1] that defines theposition of the Ith particle in a unit cube (i.e. Ω unit = 1) which is the scaledcell 459,460 , see Sect. 3.1 for some definitions. The resulting metric tensor G = h t hconverts distances measured in scaled coordinates to distances as given by theoriginal coordinates according to Eq. (106) and periodic boundary conditions areapplied using Eq. (107).In the case of ab initio molecular dynamics the orbitals have to be expressedsuitably in the scaled coordinates s = h −1 r. The normalized original orbitals φ i (r)as defined in the unscaled cell h are transformed according tosatisfying∫Ωφ i (r) = 1 √Ωφ i (s) (275)∫dr φ ⋆ i (r)φ i (r) =Ω unitso that the resulting charge density is given byds φ ⋆ i (s)φ i (s) (276)n(r) = 1 n (s) . (277)Ωin the scaled cell, i.e. the unit cube. Importantly, the scaled fields φ i (s) and thustheir charge density n(s) do not depend on the dynamical variables associated tothe cell degrees of freedom and thus can be varied independently from the cell; the97