Ritz method

The Ritz method is a variational method named after Walter Ritz, in which the ansatz function is a linear combination of N known basis functions $\left\lbrace \Psi _{i}\right\rbrace$, parametrized by unknown coefficients:

$\Phi =\sum _{i=1}^{N}c_{i}\Psi _{i}.$

With a known hamiltonian, we can write its expected value as

$\varepsilon ={\frac {\left\langle \sum _{i=1}^{N}c_{i}\Psi _{i}|{\hat {H}}|\sum _{i=1}^{N}c_{i}\Psi _{i}\right\rangle }{\left\langle \sum _{i=1}^{N}c_{i}\Psi _{i}|\sum _{i=1}^{N}c_{i}\Psi _{i}\right\rangle }}={\frac {\sum _{i=1}^{N}\sum _{j=1}^{N}c_{i}^{*}c_{j}H_{ij}}{\sum _{i=1}^{N}\sum _{j=1}^{N}c_{i}^{*}c_{j}S_{ij}}}\equiv {\frac {A}{B}}$.

The basis functions are usually not orthogonal, so that the overlap matrix S is has nonzero diagonal elements. Either $\left\lbrace c_{i}\right\rbrace$ or $\left\lbrace c_{i}^{*}\right\rbrace$ (the conjugation of the first) can be used to minimze the expectation value. For instance, by making the partial derivatives of $\varepsilon$ over $\left\lbrace c_{i}^{*}\right\rbrace$ zero, the following equality is obtained for every k = 1,2,...,N:

${\frac {\partial \varepsilon }{\partial c_{k}^{*}}}={\frac {\sum _{j=1}^{N}c_{j}(H_{kj}-\varepsilon S_{kj})}{\mathbf {B} }}=0$,

which leads to a set of N secular equations:

$\sum _{j=1}^{N}c_{j}\left(H_{kj}-\varepsilon S_{kj}\right)=0\;\;\;\;\;\;\;\;for\;\;\;k=1,2,...,N$.

In the above equations, energy $\varepsilon$ and the coefficients $\left\lbrace c_{j}\right\rbrace$ are unknown. With respect to c, this is a homogeneous set of linear equations, which has a solution when the determinant of the coefficients to these unknowns is zero:

$det\left(H_{kj}-\varepsilon S_{kj}\right)=0$,

which in turn is true only for N values of $\varepsilon$. Furthermore, since the hamiltonian is a hermitian operator. matrix H is also hermitian and the values of $\varepsilon _{i}$ will be real. The lowest value among $\varepsilon _{i}$ (i=1,2,..,N), $\varepsilon _{0}$, will be the best approximation to the ground state for the basis functions used. The remaining N-1 energies are estimates of excited state energies. An approximation for the wave function of state i can be obtained by finding the coefficients $\left\lbrace c_{j}\right\rbrace$ from the corresponding secular equation.