Functions
template<class Real >
Real	Normal_Hermite_Ftns (Real r, unsigned int n, Real PI)
template<class Real >
Real	Potential_QAGI (Real t, void *params)
template<class Real >
int	Hermite_Potential (Hermite_data< Real > &data, TNT::Matrix< Real > &P)
template<class Real >
void	Hermite_Kinetic (TNT::Matrix< Real > &K)
template<class Real >
void	Hermite_Moment (int ell, TNT::Matrix< Real > &P)

Function Documentation

template<class Real >

Real Normal_Hermite_Ftns	(	Real	r,
		unsigned int	n,
		Real	PI
	)

The non-normalized Hermite functions $ h_n $ have a numerically stable implemenation via the three-term recursion

$\begin{eqnarray*} h_0(r) & = & \exp(-r^2/2), \\ h_{n+1}(r) & = & 2r h_n(r) - 2n h_{n-1}(r), \qquad n=0,1,\ldots. \end{eqnarray*}$

The normalization factors are computed, likewise recursively, from the relationships

$\begin{eqnarray*} \int h_0^2~dr & = & \sqrt{\pi}, \\ \int h_{n+1}^2~dr & = & 2(n+1) \int h_n^2~dr, \qquad n=0,1,\ldots. \end{eqnarray*}$

See [1] for more details.

The value $\pi$ (of the template parameter type) is passed as a parameter since it is required to compute the normalization factors.

template<class Real >

Real Potential_QAGI	(	Real	t,
		void *	params
	)

An integral on the real line is transformed to an integral over the interval $ (0, 1] $ via the following variable transformation:

$\int_{-\infty}^\infty F(t)~dt = \int_0^1 \left[ F\left(\frac{1-t}{t}\right) + F\left(\frac{t-1}{t}\right) \right] \frac{1}{t^2}~dt.$

Potential_QAGI defines the integrand

$F(t) = \phi_m(t) \phi_n(t) f(t),$

where the function $ f $ as well as the indices $ m, n $ are passed in the structure Hermite_data as parameters.

template<class Real >

int Hermite_Potential	(	Hermite_data< Real > &	data,
		TNT::Matrix< Real > &	P
	)

The coefficients of the "potential" energy matrix,

$P_{mn} = P_{mn} + \int \phi_m \phi_n V~dr,$

are computed via numerical quadrature.The integrand $ V, $ together with quadrature parameters, are passed in the structure Hermite_data.

template<class Real >

void Hermite_Kinetic ( TNT::Matrix< Real > & K )

The coefficients of the "kinetic" energy matrix,

$K_{mn} = K_{mn} + \int \phi_m' \phi_n'~dr$

can be computed explicitly:

$\begin{eqnarray*} K_{mm} & = & (2m + 3)/2, \\ K_{m,m+2} & = & - ((m+1)(m+2))^{1/2} / 2. \end{eqnarray*}$

template<class Real >

void Hermite_Moment	(	int	ell,
		TNT::Matrix< Real > &	P
	)

Assembles the projection matrix $ P $ of the polynomial $V(r) = r^\ell$ onto the basis of Hermite polynomials. The coefficients

$P_{m,m},\; P_{m,m+1}, \ldots, P_{m,m+\ell}, \qquad m=0,1,\ldots,$

coincide with the coefficients $ c_k $ in the expansion

$r^\ell H_m(r) = \cdots + c_m H_m(r) + \cdots + c_{m+\ell} H_{m+\ell}(r).$

This is a linear system of equations in $c_0, \ldots, c_{m+\ell}$ whose coefficients are those of the $H_0, \ldots, H_{m+\ell}$ and which can be solved via back-substitution.

Hermite approximation

Functions

Function Documentation