Computational Physics

290 Quantum molecular dynamics

parts of the pseudopotential, and the electrostatic energy. The expressions are:

∇RnElocal=−

∑

K

iKVlocal,n(K)e−iK·Rnn∗(K) (9.90)

∇RnEnonlocal=

∑

j

fj

∑

l,mεn

[(Fjlmn )∗hnlm∇RnFjlmn +∇Rn(Fnlm)∗hnlmFnlm]; (9.91)

∇RnEES=−

∑

K= 000

iK

n∗tot

K^2

nncore(K)e−iK·Rn+∇RnEovrl, (9.92)

where

∇RnFlmn =−

1

√



∑

K

iKe−iK·Rnc∗j(K)Ylm(Kˆ)plm(K), (9.93)

and

∇RnEovrl=

∑′

n′

∑

L







ZnZn′

|Rn−Rn′−L|^3

erfc







|Rn−Rn′−L|

√

2 (ξn^2 +ξn^2 ′)







+

2

√

π

1

√

ξn^2 +ξn^2 ′

ZnZn′

|Rn−Rn′−L|^2

exp







|Rn−Rn′−L|

√

2 (ξn^2 +ξn^2 ′)













×(Rn−Rn′−L). (9.94)
The full implementation of the Car–Parrinello is quite cumbersome. For a hydro-
gen dimer, you should find a result similar to that ofFigure 9.4. Note that this
frequency depends on the nuclear mass. Similar results should be found for a silicon
dimer.

Exercises

9.1 [C] The Car–Parrinello method can be used to find the minimum of any variational
energy functional. We use it in this problem for finding the ground state of a particle
in a one-dimensional, infinitely deep potential well. This problem was treated in
Chapter 3,Section 3.2.1. We useNvariational basis functions of the form
χr(x)=xr(x−a)(x+a), r=0, 1, 2,...,N− 1
from which a variational stateψ(x)is built as
ψ(x)=

∑

r

Crχr(x).

The Euler–Lagrange equations for the Lagrangian with potential

E[Cr]=

∑

rs

CrCs〈χr|H|χs〉=

∑

rs

CrCsHrs