- Generated on for Multidip by
1.14.0
|
Multidip 1.0
Multi-photon matrix elements
|
Special integrals needed by MULTIDIP. More...
Functions/Subroutines | |
| complex(wp) function | nested_exp_integ (z, a, c, n, m, s, k) |
| Multi-dimensional triangular integral of exponentials and powers. | |
| complex(wp) function | nested_coul_integ (z, a, c, n, m, s, l, k) |
| Multi-dimensional triangular integral of Coulomb-Hankel functions and powers. | |
| complex(wp) function | nested_cgreen_integ (z, a, r0, c, n, sa, sb, m, l, k) |
| Multi-dimensional integral of Coulomb-Hankel and Coulomb-Green functions and powers. | |
Special integrals needed by MULTIDIP.
Module containing routines for calculation of the multi-dimensional dipole integrals used in outer correction of transition dipole elements in MULTIDIP.
| complex(wp) function multidip_integ::nested_cgreen_integ | ( | real(wp), intent(in) | z, |
| real(wp), intent(in) | a, | ||
| real(wp), intent(in) | r0, | ||
| real(wp), intent(in) | c, | ||
| integer, intent(in) | n, | ||
| integer, intent(in) | sa, | ||
| integer, intent(in) | sb, | ||
| integer, dimension(:), intent(in) | m, | ||
| integer, dimension(:), intent(in) | l, | ||
| complex(wp), dimension(:), intent(in) | k ) |
Multi-dimensional integral of Coulomb-Hankel and Coulomb-Green functions and powers.
Evaluate the many-dimensional integral
\[ \int\limits_a^{+\infty} \dots \int\limits_a^{+\infty} H_N(r_N) \dots g_3(r_3, r_2) r_2^{m_2} g_2(r_2, r_1) r_1^{m_1} H_1(r_1) \mathrm{d}r_1 \dots \]
using the asymptotic form of Coulomb-Hankel functions. The arrays passed to this function need to be ordered from the right-most integral to the right. This function iterates over all possible orderings of \( (r_1, r_2, \dots, r_N) \) and for each of these it splits the integral into (hyper-)triangular integrals and integrates those using nested_coul_integ.
For small values of "a" it may increase the accuracy if the leading interval (a,r0) (or more specifically the set Q = (a,+∞)^N \ (b,+∞)^N) is integrated numerically. For such use, provide r0 > a. Otherwise set r0 to zero.
When the global paramter coulomb_check is on, the integral is not attempted if any of the Coulomb functions in the integrand is not sufficiently well approximated by the asymptotic form at the R-matrix radius (or at the asymptotic radius if given). In such a case, the function returns the NaN constant.
The one-dimensional case with c = 0, m = 0, l1 = l2 and real momenta is treated in a special way using a closed-form formula from the article: H. F. Arnoldus, T. F. George: Analytical evaluation of elastic Coulomb integrals, J. Math. Phys. 33 (1992) 578–583.
The (adapted) formula reads:
\[ \lim_{c \rightarrow 0+} \int\limits_a^{+\infty} X_l(k_2; r) Y_l(k_1; r) \exp(-cr) dr = \frac{X_l'(k_2; a) Y_l(k_1; a) - X_l(k_2; a) Y_l'(k_1; a)}{k_2^2 - k_1^2} \,, \]
where \( X_l \) and \( Y_l \) stand for arbitrary Coulomb functions and primes denote derivatives with respect to the radius.
| Z | Residual ion charge |
| a | Lower bound of the integration |
| r0 | Optional radius from which to apply asymptotic integrals |
| c | Damping factor (additional exp(-c*r) added to all r^m functions) |
| N | Dimension (number of integration variables) |
| sa | Sign of the right-most Coulomb-Hankel function |
| sb | Sign of the left-most Coulomb-Hankel function |
| m | Array of integer powers of length N |
| l | Array of angular momenta of length N + 1 |
| k | Array of linear momenta of length N + 1 |
Definition at line 355 of file multidip_integ.f90.
| complex(wp) function multidip_integ::nested_coul_integ | ( | real(wp), intent(in) | z, |
| real(wp), intent(in) | a, | ||
| real(wp), intent(in) | c, | ||
| integer, intent(in) | n, | ||
| integer, dimension(1:n), intent(in) | m, | ||
| integer, dimension(1:2*n), intent(in) | s, | ||
| integer, dimension(1:2*n), intent(in) | l, | ||
| complex(wp), dimension(1:2*n), intent(in) | k ) |
Multi-dimensional triangular integral of Coulomb-Hankel functions and powers.
Evaluate the nested many-dimensional triangular integral
\[ \int\limits_a^{+\infty} \dots \int\limits_{r_3}^{+\infty} H_4(r_2) r_2^{m_2} H_3(r_2) \int\limits_{r_2}^{+\infty} H_2(r_1) r_1^{m_1} H_1(r_1) \mathrm{d}r_1 \mathrm{d}r_2 \mathrm{d}r_3 \dots \]
using the asymptotic form of Coulomb-Hankel functions. The arrays passed to this function need to be ordered from the inner-most (right-most) integral outward. For each term, use nested_exp_integ.
| Z | Residual ion charge |
| a | Lower bound |
| c | Damping factor (additional exp(-c*r) added to all r^m functions) |
| N | Dimension (number of integration variables) |
| m | Array of integer powers of length N |
| s | Array of integer signs (+1 or -1) in exponents of length 2*N |
| l | Array of angular momenta of length 2*N |
| k | Array of linear momenta of length 2*N |
Definition at line 227 of file multidip_integ.f90.
| complex(wp) function multidip_integ::nested_exp_integ | ( | real(wp), intent(in) | z, |
| real(wp), intent(in) | a, | ||
| real(wp), intent(in) | c, | ||
| integer, intent(in) | n, | ||
| integer, dimension(0:n-1), intent(in) | m, | ||
| integer, dimension(0:2*n-1), intent(in) | s, | ||
| complex(wp), dimension(0:2*n-1), intent(in) | k ) |
Multi-dimensional triangular integral of exponentials and powers.
Evaluate the nested many-dimensional triangular integral
\[ \int\limits_a^{+\infty} \dots \int\limits_{r_3}^{+\infty} \mathrm{e}^{\mathrm{i}s_4 \theta_4(r_2)} r_2^{m_2} \mathrm{e}^{\mathrm{i}s_3 \theta_3(r_2)} \int\limits_{r_2}^{+\infty} \mathrm{e}^{\mathrm{i}s_2 \theta_2(r_1)} r_1^{m_1} \mathrm{e}^{\mathrm{i}s_1 \theta_1(r_1)} \mathrm{d}r_1 \mathrm{d}r_2 \mathrm{d}r_3 \dots \]
where
\[ \theta_1(r_1) = k_1 r_1 + \log(2 k_1 r_1)/k_1 - \pi l_1/2 + \sigma_{l_1}(k_1) \]
is the asymptotic phase of a Coulomb function. The arrays passed to this function need to be ordered from the inner-most (right-most) integral outward. The function result does not contain the overall phase and damp factor, which needs to be added manually.
| Z | Residual ion charge |
| a | Lower bound |
| c | Damping factor (additional exp(-c*r) added to all r^m functions) |
| N | Dimension (number of integration variables) |
| m | Array of integer powers of length N |
| s | Array of integer signs (+1 or -1) in exponents of length 2*N |
| k | Array of linear momenta of length 2*N |
Definition at line 68 of file multidip_integ.f90.
1.14.0