108
3 Ab initio Electronic Structure for Few-Body Systems
3.5.2 Quantitative Problems
1. 2D Monte Carlo: Develop a Monte Carlo code to calculate the numerical value
of π. To do this consider a square box centered at the origin with a dimension of
2 units. A circle of radius r = 1 also centered at the origin will fit just inside the
box. The area of the box and the circle are 4 and π, respectively. For each iteration
use a random number generator to obtain x and y values in the range −1 to 1. If
r
2
= x
2
+ y
2
< 1 the point is within the circle otherwise it is inside the box but
outside the circle. Print the ratio of the number of point inside the circle divided
by the number of points outside the circle after 1,000, 10,000, 100,000, 1,000,000
iterations. This ratio should converge to π/4 which is just the ratio of the areas.
The Monte Carlo is not very efficient for doing iterated integrals with less than
7–10 dimensions so don’t expect a rapid convergence. See the next problem to
understand why the Monte Carlo method is so useful in many areas of science.
2. Multidimensional Monte Carlo: Now modify your code to calculate the 10dimensional integral:
I =
1
−1
dx 1
1
−1
dx 2
1
−1
dx 3 · · ·
1
−1
dx 10
x
2
1 + x
2
1 + x
2
1 + x
2
1 · · · x
2
10
(3.58)
If you attempt to perform this integral using the trapezoidal or similar griding
techniques with 10 point for each coordinate you will need 10
10 points! This
clearly demonstrates why a Monte Carlo method is efficient for integrals with
many dimension as is done in the quantum Monte Carlo method.
3. He Atom: Use two basis functions e
−2r and e
−4r in the Hartree–Fock method
to determine the approximate ground state energy of the He atom. Can you find
exponential parameters better than −2 and −4?
4. He Atom - again: Use the numerical Hartree–Fock code supplied in the additional material to calculate the ground state energy of He. This code will give
the numerically accurate answer for the Hartree–Fock ground state energies.
Carefully explain why this answer is not the experimental value which is also
extremely close to an accurate theoretical calculation. You might want to try
Robert D. Cowans atomic structure code https://www.tcd.ie/Physics/people/
Cormac.McGuinness/Cowan/ to get additional accuracy, information, and physical properties about He and other atoms.
5. Molecular Orbitals: Using the Gamess molecular structure code calculate the
collinear ground state potential energy curves for H 2 and H F.
6. Molecular Orbitals: Using the Gamess molecular structure code calculate the
ground state potential energy curves for H 3 and Li 3 at several internuclear distances. How do you answers compare with other theoretical results?
7. Least squares fitting: Use the linear least squares procedure to determine the C 6
and C 8 coefficients in the long range van der Waals expansion for the Li 2 dimer.
First set all of the weights equal to 1 and then increase the weights for larger r
values (Table 3.3).
3 Ab initio Electronic Structure for Few-Body Systems
3.5.2 Quantitative Problems
1. 2D Monte Carlo: Develop a Monte Carlo code to calculate the numerical value
of π. To do this consider a square box centered at the origin with a dimension of
2 units. A circle of radius r = 1 also centered at the origin will fit just inside the
box. The area of the box and the circle are 4 and π, respectively. For each iteration
use a random number generator to obtain x and y values in the range −1 to 1. If
r
2
= x
2
+ y
2
< 1 the point is within the circle otherwise it is inside the box but
outside the circle. Print the ratio of the number of point inside the circle divided
by the number of points outside the circle after 1,000, 10,000, 100,000, 1,000,000
iterations. This ratio should converge to π/4 which is just the ratio of the areas.
The Monte Carlo is not very efficient for doing iterated integrals with less than
7–10 dimensions so don’t expect a rapid convergence. See the next problem to
understand why the Monte Carlo method is so useful in many areas of science.
2. Multidimensional Monte Carlo: Now modify your code to calculate the 10dimensional integral:
I =
1
−1
dx 1
1
−1
dx 2
1
−1
dx 3 · · ·
1
−1
dx 10
x
2
1 + x
2
1 + x
2
1 + x
2
1 · · · x
2
10
(3.58)
If you attempt to perform this integral using the trapezoidal or similar griding
techniques with 10 point for each coordinate you will need 10
10 points! This
clearly demonstrates why a Monte Carlo method is efficient for integrals with
many dimension as is done in the quantum Monte Carlo method.
3. He Atom: Use two basis functions e
−2r and e
−4r in the Hartree–Fock method
to determine the approximate ground state energy of the He atom. Can you find
exponential parameters better than −2 and −4?
4. He Atom - again: Use the numerical Hartree–Fock code supplied in the additional material to calculate the ground state energy of He. This code will give
the numerically accurate answer for the Hartree–Fock ground state energies.
Carefully explain why this answer is not the experimental value which is also
extremely close to an accurate theoretical calculation. You might want to try
Robert D. Cowans atomic structure code https://www.tcd.ie/Physics/people/
Cormac.McGuinness/Cowan/ to get additional accuracy, information, and physical properties about He and other atoms.
5. Molecular Orbitals: Using the Gamess molecular structure code calculate the
collinear ground state potential energy curves for H 2 and H F.
6. Molecular Orbitals: Using the Gamess molecular structure code calculate the
ground state potential energy curves for H 3 and Li 3 at several internuclear distances. How do you answers compare with other theoretical results?
7. Least squares fitting: Use the linear least squares procedure to determine the C 6
and C 8 coefficients in the long range van der Waals expansion for the Li 2 dimer.
First set all of the weights equal to 1 and then increase the weights for larger r
values (Table 3.3).