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SIRIUS 7.5.0
Electronic structure library and applications
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Finds a solution to radial Schrodinger, Koelling-Harmon or Dirac equation. More...
#include <radial_solver.hpp>
Inherited by sirius::Bound_state, and sirius::Enu_finder.
Public Member Functions | |
| Radial_solver (int zn__, std::vector< double > const &v__, Radial_grid< double > const &radial_grid__) | |
| std::tuple< int, std::vector< double >, std::vector< double >, std::vector< double >, std::vector< double > > | solve (relativity_t rel__, int dme__, int l__, int k__, double enu__) const |
| int | solve (relativity_t rel__, int dme__, int l__, double enu__, std::vector< double > &p__, std::vector< double > &rdudr__, std::array< double, 2 > &uderiv__) const |
| Integrates the radial equation for a given energy and finds the m-th energy derivative of the radial solution. More... | |
| int | num_points () const |
| int | zn () const |
| double | radial_grid (int i__) const |
| Radial_grid< double > const & | radial_grid () const |
Protected Member Functions | |
| template<relativity_t rel, bool prevent_overflow> | |
| int | integrate_forward_rk4 (double enu__, int l__, int k__, Spline< double > const &chi_p__, Spline< double > const &chi_q__, std::vector< double > &p__, std::vector< double > &dpdr__, std::vector< double > &q__, std::vector< double > &dqdr__) const |
| Integrate system of two first-order differential equations forward starting from the origin. More... | |
Protected Attributes | |
| int | zn_ |
| Positive charge of the nucleus. More... | |
| Radial_grid< double > const & | radial_grid_ |
| Radial grid. More... | |
| Spline< double > | ve_ |
| Electronic part of potential. More... | |
Finds a solution to radial Schrodinger, Koelling-Harmon or Dirac equation.
Non-relativistic radial Schrodinger equation:
\[ -\frac{1}{2}p''(r) + V_{eff}(r)p(r) = Ep(r) \]
where \( V_{eff}(r) = V(r) + \frac{\ell (\ell+1)}{2r^2} \), \( p(r) = u(r)r \) and \( u(r) \) is the radial wave-function. Energy derivatives of radial solutions obey the slightly different equation:
\[ -\frac{1}{2}\dot{p}''(r) + V_{eff}(r)\dot{p}(r) = E\dot{p}(r) + p(r) \]
\[ -\frac{1}{2}\ddot{p}''(r) + V_{eff}(r)\ddot{p}(r) = E\ddot{p}(r) + 2\dot{p}(r) \]
and so on. We can generalize the radial Schrodinger equation like this:
\[ -\frac{1}{2}p''(r) + \big(V_{eff}(r) - E\big) p(r) = \chi(r) \]
where now \( p(r) \) represents m-th energy derivative of the radial solution and \( \chi(r) = m \frac{\partial^{m-1} p(r)} {\partial^{m-1}E} \)
Let's now decouple second-order differential equation into a system of two first-order euquations. From \( p(r) = u(r)r \) we have
\[ p'(r) = u'(r)r + u''(r) = 2q(r) + \frac{p(r)}{r} \]
where we have introduced a new variable \( q(r) = \frac{u'(r) r}{2} \). Differentiating \( p'(r) \) again we arrive to the following equation for \( q'(r) \):
\[ p''(r) = 2q'(r) + \frac{p'(r)}{r} - \frac{p(r)}{r^2} = 2q'(r) + \frac{2q(r)}{r} + \frac{p(r)}{r^2} - \frac{p(r)}{r^2} \]
\[ q'(r) = \frac{1}{2}p''(r) - \frac{q(r)}{r} = \big(V_{eff}(r) - E\big) p(r) - \frac{q(r)}{r} - \chi(r) \]
Final expression for a linear system of differential equations for m-th energy derivative is:
\begin{eqnarray*} p'(r) &=& 2q(r) + \frac{p(r)}{r} \\ q'(r) &=& \big(V_{eff}(r) - E\big) p(r) - \frac{q(r)}{r} - \chi(r) \end{eqnarray*}
Scalar-relativistic equations look similar. For m = 0 (no energy derivative) we have:
\begin{eqnarray*} p'(r) &=& 2Mq(r) + \frac{p(r)}{r} \\ q'(r) &=& \big(V(r) - E + \frac{\ell(\ell+1)}{2Mr^2}\big) p(r) - \frac{q(r)}{r} \end{eqnarray*}
where \( M = 1 + \frac{\alpha^2}{2}\big(E-V(r)\big) \). Because M depends on energy, the m-th energy derivatives of the scalar-relativistic solution take a slightly different form. For m=1 we have:
\begin{eqnarray*} \dot{p}'(r) &=& 2M\dot{q}(r) + \frac{\dot{p}(r)}{r} + \alpha^2 q(r) \\ \dot{q}'(r) &=& \big(V(r) - E + \frac{\ell(\ell+1)}{2Mr^2}\big) \dot{p}(r) - \frac{\dot{q}(r)}{r} - \big(1 + \frac{\ell(\ell+1)\alpha^2}{4M^2r^2}\big)p(r) \end{eqnarray*}
For m=2:
\begin{eqnarray*} \ddot{p}'(r) &=& 2M\ddot{q}(r) + \frac{\ddot{p}(r)}{r} + 2 \alpha^2 \dot{q}(r) \\ \ddot{q}'(r) &=& \big(V(r) - E + \frac{\ell(\ell+1)}{2Mr^2}\big) \ddot{p}(r) - \frac{\ddot{q}(r)}{r} - 2 \big(1 + \frac{\ell(\ell+1)\alpha^2}{4M^2r^2}\big)\dot{p}(r) + \frac{\ell(\ell+1)\alpha^4}{4M^3r^2} p(r) \end{eqnarray*}
Derivation of IORA.
First thing to do is to expand the \( 1/M \) to the first order in \( E \):
\[ \frac{1}{M} = \frac{1}{1-\frac{\alpha^2}{2} V(r)} - \frac{\alpha^2 E}{2 (1-\frac{\alpha^2}{2} V(r))^2} + O(E^2) = \frac{1}{M_0} - \frac{\alpha^2}{2} \frac{1}{M_0^2} E \]
From this expansion we can derive the expression for \( M \):
\[ M = \Big(\frac{1}{M}\Big)^{-1} = \frac{M_0}{1 - \frac{\alpha^2}{2}\frac{E}{M_0}} \]
Now we insert this expressions into the scalar-relativistic equations:
\[ \left\{ \begin{array}{ll} \displaystyle p'(r) = 2 \frac{M_0}{1 - \frac{\alpha^2}{2}\frac{E}{M_0}} q(r) + \frac{p(r)}{r} \\ \displaystyle q'(r) = \Big(V(r) - E + \frac{\ell(\ell+1)}{2r^2} \big(\frac{1}{M_0} - \frac{\alpha^2}{2} \frac{E}{M_0^2} \big) \Big) p(r) - \frac{q(r)}{r} = \Big(V(r) - E + \frac{\ell(\ell+1)}{2 M_0 r^2} - \frac{\alpha^2}{2} \frac{\ell(\ell+1) E}{2 M_0^2 r^2} \Big) p(r) - \frac{q(r)}{r} \end{array} \right. \]
The first energy derivatives are then:
\[ \left\{ \begin{array}{ll} \displaystyle \dot{p}'(r) = 2 \frac{M_0}{1 - \frac{\alpha^2}{2}\frac{E}{M_0}} \dot{q}(r) + \frac{\dot{p}(r)}{r} + \frac{\alpha^2}{(1 - \frac{\alpha^2 E}{2 M_0})^2} q(r) \\ \displaystyle \dot{q}'(r) = \Big(V(r) - E + \frac{\ell(\ell+1)}{2 M_0 r^2} - \frac{\alpha^2}{2} \frac{\ell(\ell+1)}{2 M_0^2 r^2} E \Big) \dot{p}(r) - \frac{\dot{q}(r)}{r} - \Big(1 + \frac{\alpha^2}{2} \frac{\ell(\ell+1)}{2 M_0^2 r^2} \Big) p(r) \end{array} \right. \]
The second energy derivatives are:
\[ \left\{ \begin{array}{ll} \displaystyle \ddot{p}'(r) = 2 \frac{M_0}{1 - \frac{\alpha^2}{2}\frac{E}{M_0}} \ddot{q}(r) + \frac{\ddot{p}(r)}{r} + \frac{2 \alpha^2}{(1 - \frac{\alpha^2 E}{2 M_0})^2} \dot{q}(r) + \frac{\alpha^4}{2 M_0 (1 - \frac{\alpha^2 E}{2 M_0})^3} q(r) \\ \displaystyle \ddot{q}'(r) = \Big(V(r) - E + \frac{\ell(\ell+1)}{2 M_0 r^2} - \frac{\alpha^2}{2} \frac{\ell(\ell+1)}{2 M_0^2 r^2} E \Big) \ddot{p}(r) - \frac{\ddot{q}(r)}{r} - 2 \Big(1 + \frac{\alpha^2}{2} \frac{\ell(\ell+1)}{2 M_0^2 r^2} \Big) \dot{p}(r) \end{array} \right. \]
Definition at line 135 of file radial_solver.hpp.
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Definition at line 639 of file radial_solver.hpp.
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Integrate system of two first-order differential equations forward starting from the origin.
Use Runge-Kutta 4th order method
Definition at line 150 of file radial_solver.hpp.
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Definition at line 652 of file radial_solver.hpp.
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Integrates the radial equation for a given energy and finds the m-th energy derivative of the radial solution.
| [in] | rel | Type of relativity |
| [in] | dme | Order of energy derivative. |
| [in] | l | Oribtal quantum number. |
| [in] | enu | Integration energy. |
| [out] | p | \( p(r) = ru(r) \) radial function. |
| [out] | rdudr | \( ru'(r) \). |
| [out] | uderiv | \( u'(R) \) and \( u''(R) \). |
Returns \( p(r) = ru(r) \), \( ru'(r)\), \( u'(R) \) and \( u''(R) \).
Surface derivatives:
From
\[ u(r) = \frac{p(r)}{r} \]
we have
\[ u'(r) = \frac{p'(r)}{r} - \frac{p(r)}{r^2} = \frac{1}{r}\big(p'(r) - \frac{p(r)}{r}\big) \]
\[ u''(r) = \frac{p''(r)}{r} - \frac{p'(r)}{r^2} - \frac{p'(r)}{r^2} + 2 \frac{p(r)}{r^3} = \frac{1}{r}\big(p''(r) - 2 \frac{p'(r)}{r} + 2 \frac{p(r)}{r^2}\big) \]
Definition at line 780 of file radial_solver.hpp.
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Definition at line 811 of file radial_solver.hpp.
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Definition at line 816 of file radial_solver.hpp.
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Definition at line 821 of file radial_solver.hpp.
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Definition at line 826 of file radial_solver.hpp.
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Positive charge of the nucleus.
Definition at line 139 of file radial_solver.hpp.
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Radial grid.
Definition at line 142 of file radial_solver.hpp.
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Electronic part of potential.
Definition at line 145 of file radial_solver.hpp.
1.9.3