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 leftlbracePsi_i ight brace, parametrized by unknown coefficients:

Related Topics:
Variational method - Walter Ritz

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: Phi = sum_{i=1}^N c_i Psi_i.

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

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: arepsilon = rac{leftlangle sum_{i=1}^N c_iPsi_i|hat{H}|sum_{i=1}^Nc_iPsi_i ight angle}{leftlangle sum_{i=1}^N c_iPsi_i|sum_{i=1}^Nc_iPsi_i ight angle} = rac{sum_{i=1}^Nsum_{j=1}^Nc_i^*c_jH_{ij}}{sum_{i=1}^Nsum_{j=1}^Nc_i^*c_jS_{ij}} equiv rac{A}{B} .

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The basis functions are usually not orthogonal, so that the overlap matrix S is has nonzero diagonal elements. Either leftlbrace c_i ight brace or leftlbrace c_i^* ight brace (the conjugation of the first) can be used to minimze the expectation value. For instance, by making the partial derivatives of arepsilon over leftlbrace c_i^* ight brace zero, the following equality is obtained for every k = 1,2,...,N:

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: rac{partialarepsilon}{partial c_k^*} = rac{sum_{k=1}^Nc_i(H_{kj}-arepsilon S_{kj})}{mathbf{B}} = 0 ,

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which leads to a set of N secular equations:

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:sum_{j=1}^N c_j left( H_{kj} - arepsilon S_{kj} ight) = 0 ;;;;;;;; for ;;; k = 1,2,...,N.

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In the above equations, energy arepsilon and the coefficients leftlbrace c_j ight brace 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:

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:det left( H_{kj} - arepsilon S_{kj} ight) = 0,

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which in turn is true only for N values of arepsilon. Furthermore, since the hamiltonian is a hermitian operator. matrix H is also hermitian and the values of arepsilon_i will be real. The lowest value among arepsilon_i (i=1,2,..,N), arepsilon_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 leftlbrace c_j ight brace from the corresponding secular equation.

Related Topics:
Hermitian operator - Hermitian

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