Ab initio molecular dynamics: Theory and Implementation

The eigenvalues of A when multiplied by P are precisely the normal mode frequencies{λ} (s) . Since the transformation is unitary, its Jacobian is unity. Finally, it isconvenient to define a set of normal mode massesM (s)I= λ (s) M I (326)that vary along the imaginary time axis s = 1, . . ., P, where λ (1) = 0 for thecentroid mode u (1)I .Based on these transformations the Lagrangian corresponding to the ab initiopath integral expressed in normal modes is obtained 644L AIPI = 1 PP∑s=1{∑iΛ (s)ij+ ∑ ij{P∑ ∑ N+s=1I=1µ(〈〈 ˙φ(s) iφ (s)i∣∣∣φ (s)j1(2 M ′(s)I〉 [ ( )} ] (s)(s) ˙φ i − E {φ i } (s) ,{R I u (1)I , . . .,u(P) I〉− δ ij) }) 2 ∑N˙u (s) 1I −I=12 M(s) IωP2(u (s)I) 2}, (327)where the masses M ′(s)Iwill be defined later, see Eq. (338). As indicated, theelectronic energy E (s) is always evaluated in practice in terms of the “primitive”path variables {R I } (s) in Cartesian space. The necessary transformation to switchforth and back between “primitive” and normal mode variables is easily performedas given by the relations Eq. (325).The chief advantage of the normal mode representation Eq. (325) for the presentpurpose is that the lowest–order normal mode u (1)Iu (1)I= R c I = 1 P∑R (s′ )I(328)Pturns out to be identical to the centroid R c Iof the path that represents the Ithnucleus. The centroid force can also be obtained from the matrix U accordingto 644∂E∂u (1)I= 1 PP∑s ′ =1s ′ =1∂E (s′ )∂R (s′ )Isince U 1s = U † s1 = 1/√ P and the remaining normal mode forces are given by∂E∂u (s)I= 1 √P P ∑s ′ =1∂E (s′ )U ss ′∂R (s′ )I(329)for s = 2, . . ., P (330)in terms of the “primitive” forces −∂E (s) /∂R (s)I. Here, E on the left–hand–sidewith no superscript (s) refers to the average electronic energy E = (1/P) ∑ Ps=1 E(s)from which the forces have to be derived. Thus, the force Eq. (329) acting on eachcentroid variable u (1)I, I = 1, . . .N, is exactly the force averaged over imaginarytime s = 1, . . ., P, i.e. the centroid force on the Ith nucleus as already given in115