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        "#**Computacional Chemistry - PL 3**\n",
        "\n",
        "**Unitary (orthogonal) matrices and transformations**\n",
        "\n",
        "\\\\\n",
        "\n",
        "---\n",
        "Supporting Bibliography:\n",
        "\n",
        "Attila Szabo and Neil S. Ostlund, Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory, Dover Publications Inc., New York, 1996,  **Capítulo 1, Secção 1.1**\n",
        "\n",
        "https://www.geeksforgeeks.org/applications-of-eigenvalues-and-eigenvectors/\n",
        "---\n",
        "\n",
        "\\\\\n",
        "\n",
        "**1. Unitrary Transformations**\n",
        "\n",
        "We verified that\n",
        "\n",
        "$\\boldsymbol{\\Omega} = \\textbf{U}^{\\dagger}\\textbf{OU} =\n",
        "\\begin{bmatrix}\n",
        "cos \\theta & sin \\theta   \\\\\n",
        "sin \\theta & -cos \\theta\n",
        "\\end{bmatrix}\n",
        "\\begin{bmatrix}\n",
        "O_{11} & O_{12}   \\\\\n",
        "O_{21} & O_{22}  \n",
        "\\end{bmatrix}\n",
        "\\begin{bmatrix}\n",
        "cos \\theta & sin \\theta   \\\\\n",
        "sin  \\theta & -cos \\theta\n",
        "\\end{bmatrix} =\n",
        "$\n",
        "\n",
        "\\\\\n",
        "$\\begin{bmatrix}\n",
        "O_{11} cos^{2} \\theta + O_{22} sin^{2} \\theta + O_{12} sin 2\\theta & \\frac{1}{2}(O_{11}-O_{22}) sin 2\\theta - O_{12} cos 2\\theta\\\\\n",
        "\\frac{1}{2}(O_{11}-O_{22}) sin 2\\theta - O_{12} cos 2\\theta & O_{11} sin^{2} \\theta + O_{22} cos^{2} \\theta - O_{12} sin 2\\theta  \n",
        "\\end{bmatrix}\n",
        "$\n",
        "\n",
        "\\\\\n",
        "**Problem 1**\n",
        "\n",
        "Show that for $\\boldsymbol{\\Omega}$ to be diagonal, the angle $\\theta$ must be given by\n",
        "\n",
        "\\\\\n",
        "$\\theta_{0} = \\frac{1}{2}tan^{-1}\\frac{2O_{12}}{O_{11}-O_{22}} = \\frac{1}{2} arctan\\frac{2O_{12}}{O_{11}-O_{22}}$\n",
        "\n",
        "\\\\\n",
        "**Problem 2**\n",
        "\n",
        "Write a Python program that calculates the value of $\\theta_{0}$ given the values ​​of the elements of the matrix **O** ($O_{11}, O_{12}, O_{22}$).\n",
        "\n",
        "Note1: Define an arbitrary Hermitean matrix **O** ($2\\times2$)\n",
        "\n",
        "Note2: Don't forget to import the numpy library\n",
        "\n",
        "Note3: the `np.arctan()` function can be used to calculate $tan^{-1}$ or $arctan$\n",
        "\n",
        "Note4: the returned values ​​are in radians. To convert the values ​​to degrees you can use `np.degrees()`"
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        "# Place your code here and make any necessary modifications\n",
        "#\n",
        "#\n",
        "#"
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      "source": [
        "**Problem 3**\n",
        "\n",
        "What are the eigenvalues ​​of the matrix **O**?\n",
        "\n",
        "Recall that $\\boldsymbol{\\Omega} = \\textbf{U}^{\\dagger}\\textbf{OU} =\n",
        "\\begin{bmatrix}\n",
        "\\omega_1 & 0 \\\\\n",
        "0 & \\omega_2\n",
        "\\end{bmatrix}$\n",
        "\n",
        "\\\\\n",
        "**Problem 4**\n",
        "\n",
        "Write a program that calculates the eigenvalues ​​of the matrix **O** defined above"
      ],
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      "source": [
        "# Place your code here and make any necessary modifications\n",
        "#\n",
        "#\n",
        "#"
      ],
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      "source": [
        "**Problem 5**\n",
        "\n",
        "Note that you can directly use the value of $\\theta_{0}$ and the product $\\boldsymbol{\\Omega} = \\textbf{U}^{\\dagger}\\textbf{OU} =\n",
        "\\begin{bmatrix}\n",
        "\\omega_1 & 0 \\\\\n",
        "0 & \\omega_2\n",
        "\\end{bmatrix}$ to obtain the eigenvalues. Use the definition of the matrix $\\textbf{U} = \\begin{bmatrix}\n",
        "cos \\theta & sin \\theta \\\\\n",
        "sin \\theta & -cos \\theta\n",
        "\\end{bmatrix}$ to write a program that computes the eigenvalues.\n",
        "\n",
        "Note 1: the function `numpy.matrix.H()` returns the transposed complex conjugate matrix. If the matrix is ​​real, we can use `np.transpose()` which returns the transposed matrix. The `.T` attribute can be used in the same way, i.e. `A.T` gives us the transpose of A.\n",
        "\n",
        "Note 2: The `np.diag(matrix)` function returns the diagonal elements of a matrix\n",
        "\n",
        "Note 3: the `@` operator can be used as a shorthand for `np.matmul` (`np.matmul(A,B)` is equal to `A @ B`)\n",
        "\n",
        "Note 4. Trigonometric functions have the general form np.function, e.g. `np.cos()`"
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        "# Place your code here and make any necessary modifications\n",
        "#\n",
        "#"
      ],
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      "source": [
        "**Problem 6**\n",
        "\n",
        "Show that *tr*(**AB**) = *tr*(**BA**). Use this result to show that the trace of a matrix is ​​invariant under a unitary transformation, i.e., if $\\boldsymbol{\\Omega} = \\textbf{U}^{\\dagger}\\textbf{OU}$, then *tr*$\\boldsymbol{\\Omega}$ = *tr*$\\textbf{O}$\n",
        "\n",
        "\\\\\n",
        "**2. Secular Determinant**\n",
        "\n",
        "As we saw in the class, we can also use the Secular Determinant method to solve an eigenvalue equation and, therefore, diagonalize a matrix:\n",
        "\n",
        "$\\textbf{O}\\textbf{c} = \\omega \\textbf{c}$ pode reescrever-se como  $(\\textbf{O}-\\omega\\textbf{1})\\textbf{c} = 0$\n",
        "and has solutions\n",
        "$|\\textbf{O}-\\omega \\textbf{1}| = 0$\n",
        "\n",
        "We discovered that for a matrix ($2\\times2$) the eigenvalues ​​are given by:\n",
        "\n",
        "\\begin{equation}\n",
        "\\begin{split}\n",
        "\\omega_{1} = \\frac{1}{2} [O_{11}+O_{22} -[(O_{11}-O_{22})^{2}+4O_{12}O_{21}]^{\\frac{1}{2}}]  \\\\\n",
        "\\omega_{2} = \\frac{1}{2} [O_{11}+O_{22} + [(O_{11}-O_{22})^{2}+4O_{12}O_{21}]^{\\frac{1}{2}}]\n",
        "\\end{split}\n",
        "\\end{equation}\n",
        "\n",
        "and therefore, the eigenvectors are:\n",
        "\n",
        "\\begin{equation}\n",
        "\\begin{split}\n",
        "O_{11}c_{1}^{i} + O_{12}c_{2}^{i} = \\omega_{i} c_{1}^{i} \\\\\n",
        "O_{21}c_{1}^{i} + O_{22}c_{2}^{i} = \\omega_{i} c_{2}^{i} \\\\\n",
        "\\end{split}\n",
        "\\space i = 1,2 ;  \\text{  } (c_{1}^{i})^{2}+(c_{2}^{i})^{2} = 1\n",
        "\\end{equation}\n",
        "\n",
        "Remember also that we can write\n",
        "\n",
        "\\begin{equation}\n",
        "\\begin{pmatrix}\n",
        "O_{11} & O_{12} \\\\\n",
        "O_{12} & O_{22}\n",
        "\\end{pmatrix}\n",
        "\\begin{pmatrix}\n",
        "c_1^1 \\\\\n",
        "c_2^1\n",
        "\\end{pmatrix}\n",
        "= \\omega_1\n",
        "\\begin{pmatrix}\n",
        "c_1^1 \\\\\n",
        "c_2^1\n",
        "\\end{pmatrix}\n",
        "\\quad \\text{e} \\quad\n",
        "\\begin{pmatrix}\n",
        "O_{11} & O_{12} \\\\\n",
        "O_{12} & O_{22}\n",
        "\\end{pmatrix}\n",
        "\\begin{pmatrix}\n",
        "c_1^2 \\\\\n",
        "c_2^2\n",
        "\\end{pmatrix}\n",
        "= \\omega_2\n",
        "\\begin{pmatrix}\n",
        "c_1^2 \\\\\n",
        "c_2^2\n",
        "\\end{pmatrix}\n",
        "\\end{equation}\n",
        "\n",
        "**Problem 7**\n",
        "\n",
        "Calculate the eigenvalues ​​and eigenvectors of the symmetric matrix $\\textbf{O}$ ($2 \\times 2$) using the secular determinant method\n",
        "\n",
        "\\begin{equation*}\n",
        "\\textbf{O} = \\begin{bmatrix}\n",
        "O_{11} & O_{12}   \\\\\n",
        "O_{21} & O_{22}  \n",
        "\\end{bmatrix}\n",
        "=\n",
        "\\begin{bmatrix}\n",
        "a & b   \\\\\n",
        "b & a  \n",
        "\\end{bmatrix}\n",
        "\\end{equation*}\n",
        "\n",
        "\n",
        "**Problem 8**\n",
        "\n",
        "Consider the following matrices\n",
        "\n",
        "\\begin{equation*}\n",
        "\\boldsymbol{A} = \\begin{bmatrix}\n",
        "3 & 1 \\\\\n",
        "1 & 3 \\\\\n",
        "\\end{bmatrix}\n",
        "\\text{e}\\space\n",
        "\\boldsymbol{B} = \\begin{bmatrix}\n",
        "3 & 1 \\\\\n",
        "1 & 2 \\\\\n",
        "\\end{bmatrix}\n",
        "\\end{equation*}\n",
        "\n",
        "Check if they are Hermitian.\n",
        "\n",
        "**Problem 9**\n",
        "\n",
        "Write the secular determinant for each of the matrices and obtain the eigenvalues.\n",
        "\n",
        "**Problem 10**\n",
        "\n",
        "Using the eigenvalues ​​from the previous exercise, also obtain the eigenvectors. Verify that they are orthogonal.\n",
        "\n",
        "**Exercício 11**\n",
        "\n",
        "Write a Python program that calculates eigenvalues ​​of matrices $2\\times2$ and apply the program to matrices **A** and **B**.\n",
        "\n",
        "Tips: After declaring the matrix, start by defining the coefficients $a, b, c$ of the quadratic equation $ax²+bx+c$ of the determinants $|\\textbf{A}-ω\\textbf{1}|$ and $|\\textbf{B}-ω\\textbf{1}|$ in terms of the matrix elements.\n",
        "\n",
        "Remember that for a matrix $\\textbf{O} = \\begin{bmatrix} O_{11} & O_{12} \\\\ O_{21} & O_{22} \\end{bmatrix}$\n",
        "\n",
        "\\\\\n",
        "$|\\textbf{O}-ω\\textbf{1}| = \\begin{vmatrix}\n",
        "O_{11} - ω & O_{12}\\\\\n",
        "O_{21} & O_{22}-ω\n",
        "\\end{vmatrix} = (O_{11} - ω)(O_{22}-ω) - (O_{21}×O_{12})$\n",
        "\n",
        "\\\\\n",
        "$= O_{11}O_{22}-O_{11}ω-ωO_{22}+ω² -(O_{21}×O_{12})$\n",
        "\n",
        "\\\\\n",
        "$= ω² -(O_{11}+O_{22})ω + (O_{11}O_{22})-(O_{21}×O_{12})$\n",
        "\n",
        "\\\\\n",
        "having $a, b, c$, we can find the roots of the quadratic equation $ax²+bx+c$ using the function `np.roots([a, b, c])`"
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        "#\n",
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        "*** Assessment Exercise (Home)**\n",
        "\n",
        "As expected, there are libraries that perform this task more easily and allow you to efficiently diagonalize/find the eigenvalues ​​of matrices larger than $2\\times2$ .\n",
        "\n",
        "The Numpy library has the function `linalg.eig(a)` which, given a square matrix, returns the values ​​and eigenvectors of that matrix.\n",
        "\n",
        "Syntax\n",
        "\n",
        " ```\n",
        " # Importing the library\n",
        " import numpy as np\n",
        "\n",
        " # Stores 2 matrices, W and v, with eigenvalues ​​and eigenvectors of the matrix m\n",
        " w, v = np.linalg.eig(m)\n",
        " ```\n",
        "\n",
        " Write a program that calculates eigenvalues ​​and eigenvectors of square matrices using this function. You can use the previously defined matrices **A** and **B**."
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        "# Coloque aqui o seu código e faça as modificações necessárias\n",
        "#\n",
        "#"
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