📚 Numerical Methods – Master Projects

📘 Overview

This repository contains numerical methods projects completed during my Master’s program. The projects involve scientific programming in Fortran and Python, focusing on:

🔹 Project 1: Interpolation / Extrapolation
    • Potential of O₂
    • Potential of the C₂H₅ radical

🔹 Project 2: Integration & Differentiation
    • Average value
    • Numerical derivatives of VO₂(R)
    • Integration

🔹 Project 3: Model Fitting
    • Spectroscopic Hamiltonian analysis

🧠 Developed using Fortran for high-performance numerical tasks and Python for data fitting and visualization.

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Project I: Interpolation/Extrapolation (Fortran)

📌Exercise 1: Potential of O₂

We consider the internuclear potential of the O₂ molecule:

$$V(R) = -D \left\{1 + c_1(R - R^*) + c_2(R - R^*)^2 + c_3(R - R^*)^3 \right\} e^{-\alpha(R - R^*)}$$

Where:

• D = 3.8886 eV
• R* = 2.2818 a₀
• α = 3.3522498 a₀⁻¹
• c₁ = 3.6445906 a₀⁻¹
• c₂ = 3.9281238 a₀⁻²
• c₃ = 2.0986689 a₀⁻³

This is a model adjusted to experimental spectroscopic data.

We want to interpolate the potential over the interval [1.8, 7]a₀ using these points:

$$\{ R_q \} = \{1.8, 2.1, 2.4, 2.8, 3.3, 4.0, 5.0, 7.0\}$$

🧩Interpolation Methods:

• Lagrange Polynomial ( L(R) )
• Polynomial in powers of ( 1/R )
• Rational Polynomial (RatInt.f)
• Cubic Splines (Spline.f)

Plot the interpolation error:

$$|V(R_i) - V_{\text{Int}}(R_i)|$$

on a logarithmic scale for R in [1.8, 10], and compare the behavior of the interpolation and extrapolation methods.

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📌Exercise 2: Potential of the C₂H₅ Radical

We use the model potential:

$$V(\phi) = e^{-\alpha \cos(6\phi)}, \quad \alpha = 0.2$$

This potential has the same symmetry properties as the exact one.

🔧Tasks:

• Build an interpolation scheme V̄(φ) using Minv.f.
• Determine the minimum number of interpolation points {φ_q} such that the standard deviation:

$$\phi_{q+\frac{1}{2}} = \frac{1}{2}(\phi_{q+1} + \phi_q)$$ $$\sigma = \sqrt{ \frac{1}{N - 1} \sum_{q=1}^{N-1} \left[ V(\phi_{q+1/2}) - \bar{V}(\phi_{q+1/2}) \right]^2 } < 10^{-5}$$

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Project II: Integration – Differentiation (Fortran)

📌Exercise 1: Average Value

We want to compute the average distance of the electron to the nucleus in the hydrogen atom’s 3s orbital:

$$\langle r \rangle_{3s} = \langle \phi_{3s} | r | \phi_{3s} \rangle = 4\pi \int_0^\infty \phi_{3s}^2(\vec{r}) r^3 dr$$

With:

$$\phi_{3s}(\vec{r}) = \frac{1}{\sqrt{4\pi}} \cdot \frac{2}{81\sqrt{3}} \left(\frac{Z}{a_0}\right)^{3/2} (27 - 18\rho + 2\rho^2) e^{-\rho/3}, \quad \rho = \frac{Zr}{a_0}$$

🧩Methods:

• Gauss-Laguerre Quadrature (Laguerre_Quad.c)
• Monte Carlo Method (with same RNG seed)
• (Optional) Monte Carlo with Importance Sampling

Compare with the analytical value:

$$\langle r \rangle_{ns} = \frac{n^2 a_0}{Z} \left(1 + \frac{1}{2}\left(1 - \frac{\ell(\ell + 1)}{n^2} \right)\right)$$

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📌Exercise 2: Numerical Derivatives of the VO₂ Potential

Reuse the O₂ potential from Project I and compute its first and second derivatives at:

$$R = R^*$$

Study the effect of:

• Number of points used
• Spacing of points

Compare with the analytical derivatives.

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📌Exercise 3: Laser Pulse Integration

Consider a laser pulse defined by the function:

$$A(t) = A_0 \sin(\omega t) e^{-\alpha(t - T)^2}$$

Where:

• ω = 2
• T = 5π
• α = 0.05

🔧Tasks:

• Compute the fluence Φ:

$$\Phi = \int_0^{2T} A^2(t) dt$$

🔧Using:

• Trapezoidal method
• Simpson's rule

Increase the number of integration points until the relative convergence reaches 10⁻⁸.

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Project III: Model Fitting (Python)

📌Exercise 1: Study of a Spectroscopic Hamiltonian

We consider the vibrational spectroscopic Hamiltonian:

$$E_{\text{sp}}(n) = T_0 + \sum_{\alpha=1}^{6} \omega_\alpha \left(n_\alpha + \frac{1}{2} \right) + \sum_{\alpha \leq \beta} x_{\alpha\beta} \left(n_\alpha + \frac{1}{2} \right)\left(n_\beta + \frac{1}{2} \right) + \sum_{\alpha \leq \beta \leq \gamma} y_{\alpha\beta\gamma} \left(n_\alpha + \frac{1}{2} \right)\left(n_\beta + \frac{1}{2} \right)\left(n_\gamma + \frac{1}{2} \right)$$

Adjust the parameters T₀, {ω_α}, {x_αβ}, and {y_αβγ} using the least squares method. These parameters can be determined based on the list {E_exp(n)} (see the file HFCO_exp.dat), which contains the first 150 calculated vibrational levels of the HFCO molecule.

🔧Tasks:

• Fit Only ω_α and x_αβ and use a regular matrix inversion method to perform the least squares fit:

numpy.linalg.inv

• Include y_αβγ terms. Some of these parameters cannot be determined. if the state (n₁ = 1, n₂ = 1, n₃ = 1, n₄ = n₅ = n₆ = 0) does not appear in the list, then y₁₂₃ cannot be fitted.

• Test with Singular Value Decomposition:

numpy.linalg.svd

Print:

• The fitted parameters
• The residuals $E_{\text{exp}}(n) - E_{\text{sp}}(n)$ for all levels

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License

The package is licensed under the MIT License.