Detailed Description

Exact relativistic hydrogen-like (Coulomb) wavefunctions.

In the form

\[ \psi_{n\kappa m}(\vb{r}) = \frac{1}{r} \begin{pmatrix} f_{n\kappa}(r)\,\Omega_{\kappa m}(\hat n) \\ i\,g_{n\kappa}(r)\,\Omega_{-\kappa,m}(\hat n) \end{pmatrix}. \]

From: H. A. Bethe and E. E. Salpeter, Quantum Mechanics of One-and Two-Electron Atoms (Plenum, New York, 1977).

Note: Uses some numerically unstable functions, including Gamma functions and confluent hypergeometric functions. So, for some inputs, may be numerically unstable. For reasonable inputs (i.e., Zeff=5, up to n=~10), good to at least parts in 10^12

The optional electron mass parameter m defaults to 1 (atomic units). The full relativistic energy is E = m*c^2 + enk = m/alpha^2 + enk.

Typedefs
using	DiracQN = qip::StrongType< DiracTypes::DiracQN, int >

using	AlphaFS = qip::StrongType< DiracTypes::AlphaFS, double >

using	Zeff = qip::StrongType< DiracTypes::Zeff, double >

using	PrincipalQN = qip::StrongType< DiracTypes::PrincipalQN, double >

using	RaB = qip::StrongType< DiracTypes::RaB, double >

Enumerations
enum class	DiracTypes { DiracQN , AlphaFS , Zeff , PrincipalQN , RaB }

Functions
double	enk (PrincipalQN n, DiracQN k, Zeff z, AlphaFS a, double m=1.0)
	Energy, without rest mass. `m` is the electron mass (default 1 a.u.)

double	Enk (PrincipalQN n, DiracQN k, Zeff z, AlphaFS a, double m=1.0)
	Full energy: Enk = m/alpha^2 + enk. `m` is the electron mass (default 1 a.u.)

double	gamma (DiracQN k, Zeff z, AlphaFS a)
	Relativistic factor gamma = Sqrt[k^2 - (aZ)^2].

double	f (RaB r, PrincipalQN n, DiracQN k, Zeff z, AlphaFS a, double m=1.0)
	Upper (large) radial component. `m` is the electron mass (default 1 a.u.)

double	g (RaB r, PrincipalQN n, DiracQN k, Zeff z, AlphaFS a, double m=1.0)
	Lower (small) radial component. `m` is the electron mass (default 1 a.u.)

double	gfratio (double r, int k, double z, double a, double e, double m=1.0)
	Ratio g/f at r, for given energy `e` (without rest mass) and mass `m`.

Function Documentation

◆ enk()

double DiracHydrogen::enk	(	PrincipalQN	n,
		DiracQN	k,
		Zeff	z,
		AlphaFS	a,
		double	m
	)

Energy, without rest mass. m is the electron mass (default 1 a.u.)

◆ Enk()

double DiracHydrogen::Enk	(	PrincipalQN	n,
		DiracQN	k,
		Zeff	z,
		AlphaFS	a,
		double	m
	)

Full energy: Enk = m/alpha^2 + enk. m is the electron mass (default 1 a.u.)

◆ gamma()

double DiracHydrogen::gamma	(	DiracQN	k,
		Zeff	z,
		AlphaFS	a
	)

Relativistic factor gamma = Sqrt[k^2 - (aZ)^2].

◆ f()

double DiracHydrogen::f	(	RaB	r,
		PrincipalQN	n,
		DiracQN	k,
		Zeff	z,
		AlphaFS	a,
		double	m
	)

Upper (large) radial component. m is the electron mass (default 1 a.u.)

◆ g()

double DiracHydrogen::g	(	RaB	r,
		PrincipalQN	n,
		DiracQN	k,
		Zeff	z,
		AlphaFS	a,
		double	m
	)

Lower (small) radial component. m is the electron mass (default 1 a.u.)

◆ gfratio()

double DiracHydrogen::gfratio	(	double	r,
		int	k,
		double	z,
		double	a,
		double	e,
		double	m
	)

Ratio g/f at r, for given energy e (without rest mass) and mass m.