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

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

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

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

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

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

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

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

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

for then

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

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

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

or, more generally,

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

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

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

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

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

In this case, the scalar product formula simplifies to

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

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

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

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

$$= \sum _{j=0}^{N-1}\psi _{j}\xi _{j}(x)$$

with orthonormal coordinate eigenfunctions $\xi$$_{j(x)}$ and the sampling values $\psi$$_{j}$:

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

$$\psi _{j} = \sqrt{w_{j}}\psi (x_{j}).$$

Then the derivative operators can be written as

$$\frac{d^{n}\psi (x)}{dx^{n}} = \sum _{j=0}^{N-1}\psi _{j}\frac{d^{n}\xi _{j}(x)}{dx^{n}},$$

$$\frac{d^{n}\xi _{j}(x)}{dx^{n}} = \sqrt{w_{j}}\sum _{i=0}^{N-1}\phi _{i}^{*}(x_{j})\frac{d^{n}\phi _{i}(x)}{dx^{n}}.$$