Computational Physics - Department of Physics

16.12 Exercises 535

to make modules or classes of trial wave functions, both many-body wave functions and single-

particle wave functions and the quantum numbers involved, such as spin, value ofnxandny

quantum numbers.

The new item you need to pay attention to is the calculation ofthe Slater Determinant. This

is an additional complication to your VMC calculations. If we stick to harmonic oscillator like

wave functions, the trial wave function for say anN= 6 electron quantum dot can be written

as

ψT(r 1 ,r 2 ,...,r 6 ) =Det(φ 1 (r 1 ),φ 2 (r 2 ),...,φ 6 (r 6 ))

6

∏

i<j

exp

(

ari j

( 1 +βri j)

)

, (16.57)

whereDetis a Slater determinant and the single-particle wave functions are the harmonic

oscillator wave functions for thenx= 0 , 1 andny= 0 , 1 orbitals. For theN= 12 quantum dot,

the trial wave function can take the form

ψT(r 1 ,r 2 ,...,r 12 ) =Det(φ 1 (r 1 ),φ 2 (r 2 ),...,φ 12 (r 12 ))

12

∏

i<j

exp

(

ari j

2 ( 1 +βri j)

)

, (16.58)

In this case you need to include thenx= 2 andny= 2 wave functions as well. Observe that

ri=

√

ri^2 x+r^2 iy. Use the Hermite polynomials defined in the appendix.

(1f) Write a function which sets up the Slater determinant handle larger systems as well. Find
the Hermite polynomials which are needed fornx= 0 , 1 , 2 and obviouslynyas well. Compute
the ground state energies of quantum dots forN= 6 andN= 12 electrons, following the
same set up as in exercise 1e) forω= 0. 01 ,ω= 0. 28 andω= 1. 0. The calculations should
include parallelization, blocking, importance sampling and energy minimization using the
conjugate gradient approach. To test your Slater determinant code, you should reproduce
the unperturbed single-particle energies when the electron-electron repulsion is switched
off. Convince yourself that the unperturbed ground state energies forN= 6 is 10 ωand for
N= 12 we obtain 28 ω. What is the expected total spin of the ground states?
1g) With the optimal parameters for the ground state wave function, compute again the onebody
density. Discuss your results and compare the results with those obtained with a pure har-
monic oscillator wave functions. Run a Monte Carlo calculations without the Jastrow factor
as well and compute the same quantities. How important are the correlations induced by the
Jastrow factor? Compute also the expectation value of the kinetic energy and potential energy
usingω= 0. 01 ,ω= 0. 28 andω= 1. 0. Comment your results.

Additional material on Hermite polynomials

The Hermite polynomials are the solutions of the following differential equation

d^2 H(x)

dx^2

− 2 xdH(x)

dx

+ (λ− 1 )H(x) = 0. (16.59)

The first few polynomials are

H 0 (x) = 1 ,

H 1 (x) = 2 x,

H 2 (x) = 4 x^2 − 2 ,

H 3 (x) = 8 x^3 − 12 x,

and