Stores radial Dirac spinor: F_nk = (f, g)
\[
\psi_{n\kappa m} = \frac{1}{r}
\begin{pmatrix}
f_{n\kappa}(r)\,\Omega_{\kappa m}\\
g_{n\kappa}(r)\,\Omega_{-\kappa m}
\end{pmatrix},
\quad
F_{n\kappa} =
\begin{pmatrix}
f_{n\kappa}(r)\\
g_{n\kappa}(r)
\end{pmatrix}
\]
- Construction.
- Takes in n and k=kappa values + grid
- A shared pointer to the Grid is stored (to avoid many copies of Grid)
- Operator Overloads
- Intuitive operator overloads are provided (Fa, Fb are DiracSpinors):
- v * Fa, where v is a double or vector works the obvious way
- Fa * Fb = <Fa|Fb>
- Fa == Fb returns true if {na,ka}=={nb,kb}
- Fa > Fb : first compares n, and then kappa (via kappa_index)
- Fa +/- Fb : Adds/subtracts the two spinors (and updates p0/pinf)
- You can make copies: auto Fnew = Fa
- And you can re-assign: Fb = Fa (copies all, including n and kappa)
- n has no setter; kappa only via set_new_kappa(). Avoids angular errors.
- non-const accessors (e.g., en(), f(), max_pt()) return mutable references to variables
- Continuum orbitals
- Note
- Continuum (and other unlabelled) orbitals have n = 0. Since they are identified only by kappa, all n = 0 orbitals with the same kappa compare equal (operator==), and their nk_index() coincides with that of a bound state (see nk_index). They must not be mixed with each other, or with bound states, in anything that identifies orbitals by {n, kappa}. This might need to be updated later.
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| | DiracSpinor (int in_n, int in_kappa, std::shared_ptr< const Grid > in_rgrid, bool negative_energy=false) |
| | Constructor: Requires n (PQN), kappa (Dirac QN), and grid.
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| int | n () const |
| | Principal quantum number, n.
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| int | kappa () const |
| | Dirac quantum number, kappa.
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| int | l () const |
| | Orbital angular momentum Q number.
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| double | jjp1 () const |
| | j(j+1)
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| int | twoj () const |
| | 2j (twice the total angular momentum)
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| int | twojp1 () const |
| | 2j+1
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| int | parity () const |
| | (-1)^l, returns +/- 1
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| std::size_t | k_index () const |
| | kappa index (see Angular::kappa_to_kindex)
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| Index | nk_index () const |
| | (n,kappa) index
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| std::string | symbol (bool gnuplot=false) const |
| | Single-electron term symbol (e.g., 6s_1/2). Gnuplot=true => 6s_{1/2}.
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| std::string | shortSymbol () const |
| | e.g., 6p_1/2 => 6p-, 6p_3/2 => 6p+
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| bool & | exotic () |
| | Checks if spinor is for "exotic" lepton, or regular electron.
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| bool | exotic () const |
| | Checks if spinor is for "exotic" lepton, or regular electron.
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| bool | negativeEnergyStateQ () const |
| | Returns true if this is a negative-energy (Dirac sea) state.
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| void | set_new_kappa (int new_kappa) |
| | Changes 'kappa' angular quantum number. Use with caution!
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| const Grid & | grid () const |
| | Returns a const reference to the radial grid.
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| std::shared_ptr< const Grid > | grid_sptr () const |
| | Returns copy of shared_ptr to grid [shared resource].
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| double | en () const |
| | Single-particle energy, not including rest energy.
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double & | en () |
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| const std::vector< double > & | f () const |
| | Upper (large) radial component function, f.
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std::vector< double > & | f () |
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| double | f (std::size_t i) const |
| | Upper (large) radial component, f(r_i)
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double & | f (std::size_t i) |
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| const std::vector< double > & | g () const |
| | Lower (small) radial component function, g.
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std::vector< double > & | g () |
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| double | g (std::size_t i) const |
| | Lower (small) radial component, g(r_i)
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double & | g (std::size_t i) |
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| auto | min_pt () const |
| | First non-zero point (index for f[i])
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auto & | min_pt () |
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| auto | max_pt () const |
| | Effective size(); index after last non-zero point (index for f[i])
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auto & | max_pt () |
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| double | r0 () const |
| | r0 = r[min_pt] (in atomic units)
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| double | rinf () const |
| | rinf = r[max_pt - 1], the last point of the spinor (au); 0 if max_pt = 0
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| auto | its () const |
| | Number of iterations until energy convergence (for latest routine only)
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auto & | its () |
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| auto | occ_frac () const |
| | Occupation fraction. =1 for closed shells. =1/(2j+1) for valence.
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auto & | occ_frac () |
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| auto | eps () const |
| | Fractional energy convergence: eps = |(en'-en)/en|.
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auto & | eps () |
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| const DiracSpinor & | scale (const double factor) |
| | Scales DiracSpinor (f and g) by constant.
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| const DiracSpinor & | scale (const std::vector< double > &v) |
| | Scales DiracSpinor (f and g) by function of r.
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| void | normalise (double norm_to=1.0) |
| | By default normalises to 1, but can normalise to other number.
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| void | zero_boundaries () |
| | Forces f(r) and g(r) to be zero outside of [p0,pinf)
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| double | norm () const |
| | Returns the norm, defined: Norm = Sqrt[<a|a>].
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| double | norm2 () const |
| | Returns the norm^2, defined: Norm2 = <a|a>
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| int | num_nodes (double rel_cut=1.0e-3) const |
| | Number of nodes (sign changes) in f(r).
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| std::pair< double, double > | r0pinfratio () const |
| | Returns [f[p0]/f_max , f[pinf-1]/f_max] - for tests; {0,0} if f = 0.
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| std::vector< double > | rho () const |
| | Radial density: rho(r) = (2j+1) * x_occ * |F^2|(r)
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| int | num_electrons () const |
| | Number of occupied electrons: (2j+1)*occ_frac, rounded to an integer.
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| DiracSpinor & | operator+= (const DiracSpinor &rhs) |
| | Adds rhs to this, in place (should generally have same kappa, but forced)
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| DiracSpinor & | operator-= (const DiracSpinor &rhs) |
| | Subtracts rhs from this, in place; should generally have same kappa (see +=)
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| DiracSpinor & | add_scaled (double x, const DiracSpinor &rhs) |
| | Adds x*rhs to this, in place (no temporary); should generally have same kappa.
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| DiracSpinor & | operator*= (const double x) |
| | Scales this by scalar x, in place.
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| DiracSpinor & | operator*= (const std::vector< double > &v) |
| | Multiplies this by function v(r), pointwise, in place.
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| void | orthog (const DiracSpinor &rhs) |
| | A more correct way to force orthogonality than normal.
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| static bool | comp_l (const DiracSpinor &lhs, const DiracSpinor &rhs) |
| | Comparator (for sorting): by l.
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| static bool | comp_j (const DiracSpinor &lhs, const DiracSpinor &rhs) |
| | Comparator (for sorting): by 2j.
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| static bool | comp_ki (const DiracSpinor &lhs, const DiracSpinor &rhs) |
| | Comparator (for sorting): by kappa_index.
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| static bool | comp_n (const DiracSpinor &lhs, const DiracSpinor &rhs) |
| | Comparator (for sorting): by n.
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| static bool | comp_en (const DiracSpinor &lhs, const DiracSpinor &rhs) |
| | Comparator (for sorting): by energy.
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| static std::pair< double, std::string > | check_ortho (const std::vector< DiracSpinor > &a, const std::vector< DiracSpinor > &b) |
| | Returns worst |<a|b>| (or ||<a|b>|-1| for a=b) {val, state_names}.
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| static void | orthonormaliseOrbitals (std::vector< DiracSpinor > &orbs, int num_its=1) |
| | (approximately) OrthoNormalises a set of any orbitals
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| static DiracSpinor | orthogonaliseWrt (const DiracSpinor &Fin, const std::vector< DiracSpinor > &orbs) |
| | Forces Fin to be orthogonal to orbs.
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| static DiracSpinor | orthonormaliseWrt (const DiracSpinor &Fin, const std::vector< DiracSpinor > &orbs) |
| | As orthogonaliseWrt(), but normalises Fin afterwards.
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| static std::string | shortSymbol (int n, int kappa) |
| | e.g., 6p_1/2 => 6p-, 6p_3/2 => 6p+
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| static std::string | state_config (const std::vector< DiracSpinor > &orbs, Branch branch=Branch::both) |
| | Returns formatted states string (e.g., '7sp5d') given list of orbs.
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| static std::string | config_string (const std::vector< int > &n_per_l) |
| | Formats a list of numbers into the compressed states-string notation.
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| static DiracSpinor | exactHlike (int n, int k, std::shared_ptr< const Grid > rgrid, double zeff, double alpha=PhysConst::alpha, double mass=1.0) |
| | Constructs an exact H-like (pointlike Coulomb) DiracSpinor.
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| static DiracSpinor | exactHlike_cntm (double en, int k, std::shared_ptr< const Grid > rgrid, double zeff, double alpha=PhysConst::alpha, double mass=1.0) |
| | Constructs an exact (Dirac-Coulomb) continuum DiracSpinor.
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| static std::vector< DiracSpinor > | HlikeBasis (int max_l, int num_ns, std::shared_ptr< const Grid > rgrid, double zeff, double alpha=PhysConst::alpha, double mass=1.0) |
| | Constructs a basis of H-like (pointlike) DiracSpinors.
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| static std::vector< DiracSpinor > | HlikeBasis (const std::string &basis_string, std::shared_ptr< const Grid > rgrid, double zeff, double alpha=PhysConst::alpha, double mass=1.0) |
| | Constructs a basis of H-like (pointlike) DiracSpinors.
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| static const DiracSpinor * | find (int n, int k, const std::vector< DiracSpinor > &orbs) |
| | Searches for {n,k} in list of orbitals, returns pointer (may be null)
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| static int | max_tj (const std::vector< DiracSpinor > &orbs) |
| | Returns maximum (2j) found in {orbs}.
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| static int | max_l (const std::vector< DiracSpinor > &orbs) |
| | Returns maximum l found in {orbs}.
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| static int | max_n (const std::vector< DiracSpinor > &orbs) |
| | Returns maximum n found in {orbs}.
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| static int | max_n (const std::vector< DiracSpinor > &orbs, int kappa) |
| | Returns maximum n found in {orbs}, for given kappa.
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| static std::size_t | max_kindex (const std::vector< DiracSpinor > &orbs) |
| | Returns maximum kappa_index found in {orbs}.
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| static double | max_En (const std::vector< DiracSpinor > &orbs) |
| | Returns maximum Energy found in {orbs}.
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| static double | min_En (const std::vector< DiracSpinor > &orbs) |
| | Returns minimum Energy found in {orbs}.
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| static std::pair< std::vector< DiracSpinor >, std::vector< DiracSpinor > > | split_by_energy (const std::vector< DiracSpinor > &orbitals, double Fermi_energy, int n_min_core=1, int n_max_excited=9999, bool include_negative_energy=true) |
| | Splits orbitals into two groups (i.e., core, excited) by energy.
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| static std::pair< std::vector< DiracSpinor >, std::vector< DiracSpinor > > | split_by_core (const std::vector< DiracSpinor > &orbitals, const std::vector< DiracSpinor > &core, int n_min_core=1) |
| | Splits orbitals into two groups (i.e., core, excited)
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| static std::vector< DiracSpinor > | subset (const std::vector< DiracSpinor > &basis, const std::string &subset_string, bool exclude_negative_energy=true) |
| | Takes a subset of a basis (by copy), according to subset_string.
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| double | operator* (const DiracSpinor &Fa, const DiracSpinor &Fb) |
| | Returns radial integral (Fa,Fb) = Int(fa*fb + ga*gb)
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| DiracSpinor | operator+ (DiracSpinor lhs, const DiracSpinor &rhs) |
| | Returns lhs + rhs; should generally have same kappa (see +=)
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| DiracSpinor | operator- (DiracSpinor lhs, const DiracSpinor &rhs) |
| | Returns lhs - rhs; should generally have same kappa (see +=)
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| DiracSpinor | operator* (DiracSpinor Fa, const double x) |
| | Returns Fa scaled by scalar x.
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| DiracSpinor | operator* (const double x, DiracSpinor Fa) |
| | Returns Fa scaled by scalar x.
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| DiracSpinor | operator* (const std::vector< double > &v, DiracSpinor Fa) |
| | Returns Fa multiplied by function v(r), pointwise.
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| bool | operator== (const DiracSpinor &lhs, const DiracSpinor &rhs) |
| | Equality: true if {n,kappa} match.
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| bool | operator!= (const DiracSpinor &lhs, const DiracSpinor &rhs) |
| | Inequality: true if {n,kappa} differ.
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| bool | operator< (const DiracSpinor &lhs, const DiracSpinor &rhs) |
| | Ordering: compares n, then kappa (via kappa_index)
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| bool | operator> (const DiracSpinor &lhs, const DiracSpinor &rhs) |
| | Ordering: compares n, then kappa (via kappa_index)
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| bool | operator<= (const DiracSpinor &lhs, const DiracSpinor &rhs) |
| | Ordering: compares n, then kappa (via kappa_index)
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| bool | operator>= (const DiracSpinor &lhs, const DiracSpinor &rhs) |
| | Ordering: compares n, then kappa (via kappa_index)
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| std::ostream & | operator<< (std::ostream &ostr, const DiracSpinor &Fa) |
| | Writes short symbol to ostream.
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1.9.8