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Discrete Variable Representation (DVR) [1], [5]

A wavefunction $ \psi $ is represented as a linear combination of given interpolation functions $ \phi _{n} $ by

$\displaystyle \psi (x)=\sum _{n=0}^{N-1}a_{n}\phi _{n}(x)$ (1)

such that for given interpolation points $ x_{i}=x_{0},\ldots ,x_{N} $

$\displaystyle \psi (x_{i})=\sum _{n=0}^{N-1}a_{n}\phi _{n}(x_{i}).$

If the $ \phi _{n} $ satisfy the orthonormality relationin the arguments

$\displaystyle \sum ^{N-1}_{n=0}\phi _{n}^{*}(x_{i})\phi _{n}(x_{j})=\delta _{ij}$ (2)

then this interpolation expression, i.e. the matrix $ \phi _{n}(x_{j}) $ can be directly inverted to give

$\displaystyle a_{n}=\sum _{j=0}^{N-1}\psi (x_{j})\phi _{n}^{*}(x_{j})$ (3)

for then

$\displaystyle \psi (x_{i})=\sum _{n=0}^{N-1}\sum _{j=0}^{N-1}\psi (x_{j})\phi _{n}^{*}(x_{j})\phi _{n}(x_{i})=\sum _{j=0}^{N-1}\psi (x_{j})\delta _{ij}.$

If the $ \phi _{\mathsf{n}} $ furthermore satisfy the orthonormality relation in their order

$\displaystyle \left\langle \phi _{n}\vert \phi _{m}\right\rangle =\int \phi _{n}^{*}(x)\phi _{m}(x)dx=\delta _{mn}$ (4)

or, more generally,

$\displaystyle \int w(x)\phi _{n}^{*}(x)\phi _{m}(x)dx=\delta _{mn}$

then, choosing the sampling points $ x_{j} $ as the zeros of the N-th polynomial $ \phi _{N} $, it follows from Gaussian integration theory that their discrete representations also fulfill the orthonormality relation

$\displaystyle \sum ^{N-1}_{j=0}w_{j}\phi _{n}^{*}(x_{j})\phi _{m}(x_{j})=\delta _{mn}$ (5)

with constant point weights $ w_{j} $ because their degrees are all smaller than $ N $ (see appendix 4.1). The coefficients become

$\displaystyle a_{n}=\sum _{j=0}^{N-1}w_{j}\psi (x_{j})\phi _{n}^{*}(x_{j}).$

In this case, the scalar product formula simplifies to

$\displaystyle \left\langle \psi \vert \chi \right\rangle =\sum _{n}\frac{1}{w_{n}}a_{n}^{*}b_{n}=\sum _{j=0}^{N-1}\frac{1}{w_{j}}\psi ^{*}(x_{j})\chi (x_{j})$

with $ b_{j} $ the coefficients of the representation of $ \chi $.

Substitution of the coefficients (3) in the representation (1) results in

$\displaystyle \psi (x)$ $\displaystyle =$ $\displaystyle \sum _{i=0}^{N-1}\sum _{j=0}^{N-1}w_{j}\phi _{i}^{*}(x_{j})\psi (x_{j})\phi _{i}(x)$  
  $\displaystyle =$ $\displaystyle \sum _{j=0}^{N-1}\psi _{j}\xi _{j}(x)$  

with orthonormal coordinate eigenfunctions $ \xi $$ _{j(x)} $ and the sampling values $ \psi $$ _{j} $:
$\displaystyle \xi _{j}(x)$ $\displaystyle =$ $\displaystyle \sqrt{w_{j}}\sum _{i=0}^{N-1}\phi _{i}^{*}(x_{j})\phi _{i}(x),$  
$\displaystyle \psi _{j}$ $\displaystyle =$ $\displaystyle \sqrt{w_{j}}\psi (x_{j}).$  

Then the derivative operators can be written as
$\displaystyle \frac{d^{n}\psi (x)}{dx^{n}}$ $\displaystyle =$ $\displaystyle \sum _{j=0}^{N-1}\psi _{j}\frac{d^{n}\xi _{j}(x)}{dx^{n}},$  
$\displaystyle \frac{d^{n}\xi _{j}(x)}{dx^{n}}$ $\displaystyle =$ $\displaystyle \sqrt{w_{j}}\sum _{i=0}^{N-1}\phi _{i}^{*}(x_{j})\frac{d^{n}\phi _{i}(x)}{dx^{n}}.$  


next up previous
Next: Comparison Up: Evaluation of [1] Previous: Global representation of and
Andreas Markmann 2003-10-22