Linear variational method#
\[
\phi = \sum_i c_i f_i
\]
\(c_i\) -Variational coefficients
\(f_i\) Trial functions
Example: \(\phi(x) = c_1 sin (2x)+c_2 sin(2x)\)
Minimizing energy by varying linear coefficients#
\[
\mid \phi \rangle = \sum_i c_i \mid f_i \rangle
\]
\[
E(c_1,c_2,...c_N) = \frac{\langle \phi \mid \hat{H} \mid \phi \rangle}{\langle\phi \mid \phi \rangle}
\]
\[
E(c_1,c_2,...c_N) = \frac{\sum_i \sum_j c_i c_j\langle f_i \mid \hat{H} \mid f_j \rangle}{\sum_i \sum_j c_i c_j\langle f_i \mid f_j \rangle} = \frac{\sum_i \sum_j H_{ij}}{\sum_i \sum_j c_i c_jS_{ij}}
\]
It’s a linear algebra problem#
\[\mid \phi\rangle = c_1\mid f_1\rangle+ c_1\mid f_2\rangle\]
\[E(c_1,c_2) = \langle \phi \mid \hat{H} \mid \phi \rangle \]
Minimizing \(E(c_1,c_2)\) with respect to (\(c_1\),\(c_2\)) gives rise 2 linear equations.
\[
c_1(H_{11}-ES_{11})+c_2(H_{11}-ES_{12}) = 0 \newline
c_1(H_{21}-ES_{21})+c_2(H_{21}-ES_{22}) = 0
\]
The set of linear equations has nontrivial solution only when determinant of matrix elements is zero:
\[\begin{split}
\begin{vmatrix}
H_{11}-ES_{11} & H_{12}-ES_{12} \\
H_{21}-ES_{21} & H_{22}-ES_{22} \\
\end{vmatrix} = 0
\end{split}\]
Determinant for N trial functions#
Variational minimization of coeffients of N trial functions \((c_1, c_2,...c_N)\) leads to an N by N determinant:
\[
\mid \phi \rangle = \sum_i c_i \mid f_i \rangle
\]
\[\begin{split}
\begin{vmatrix}
H_{11}-ES_{11} & H_{12}-ES_{12} & \dots & H_{1N}-ES_{1N} \\
H_{21}-ES_{21} & H_{22}-ES_{22} & \dots & H_{2N}-ES_{2N} \\
H_{31}-ES_{31} & H_{32}-ES_{32} & \dots & H_{3N}-ES_{3N} \\
\dots & \dots & \dots & \dots \\
H_{N1}-ES_{11} & H_{N2}-ES_{N2} & \dots & H_{NN}-ES_{NN}
\end{vmatrix} = 0
\end{split}\]