Data Types | |
| interface | cfp_9gmic |
| interface | cfp_9gmit |
| interface | cfp_9lgic |
| interface | cfp_9lgit |
| interface | cfp_9lgmc |
| interface | cfp_asyik |
| interface | cfp_asyjy |
| interface | cfp_besi |
| interface | cfp_besj |
| interface | cfp_binom |
| interface | cfp_csevl |
| interface | cfp_eval_poly_horner |
| interface | cfp_gamic |
| interface | cfp_gamlm |
| interface | cfp_gamma_fun |
| interface | cfp_gamma_slatec |
| interface | cfp_initds |
| interface | cfp_jairy |
| interface | cfp_lgams |
| interface | cfp_lngam |
| interface | cfp_lnrel |
Functions/Subroutines | |
| elemental integer function, public | ipair (i) |
| This function is used to index an ordered pair of values (i,j), where i .ge. j, i .ge. 1. The index of the ordered pair is: ipair(i) + j. This process of indexing can be used in a nested way to index quartets of integers, say (i,j,k,l). We compute ij = ipair(i)+j and kl = ipair(k)+l. The index of the quartet (i,j,k,l) is: ipair(ij)+kl. We assume that i.ge.j, k.ge.l, ij.ge.kl. This nesting (triangularization) is used heavilly to index the 2-particle symmetric integrals. More... | |
| subroutine, public | unpack_pq (pq, n, p, q) |
| Assuming \( pq = p + (q-1)*n \) this routine returns p,q given pq,n. More... | |
| subroutine, public | cfp_nlm (INorm, L) |
| subroutine, public | cfp_resh (SH, X, Y, Z, L) |
| subroutine, public | cfp_solh (SH, x, y, z, L) |
| subroutine, public | cfp_zhar (SH, X, Y, Z, L) |
| real(kind=ep1) function, dimension(1:mmax+1), public | boys_function_quad (T, mmax) |
| Quadruple precision version of boys_function. See boys_function for details on the method of evaluation. More... | |
| elemental integer function | dfact (n) |
| Computes the double factorial \((n)!!\) of the integer number n. n must be .ge. -1. More... | |
| real(kind=cfp) function | cfp_eval_poly_horner_single (n, x, a) |
| This function evaluates a polynomial of degree \(n\) at point \(x\) using the Horner form for polynomials. More... | |
| subroutine, public | cfp_eval_poly_horner_many (n, x, n_x, a, res) |
| This subroutine evaluates a set of polynomials of degree \(n\) at a number of points \(x\) using the Horner form for polynomials. More... | |
| subroutine, public | cfp_sph_to_cart_mapping (l, m, c, i_exp, j_exp, k_exp) |
| This routine constructs for a given real solid harmonic L,M the list of cartesian functions which build up this real solid harmonic. On output the array c(:) contains the coefficients with which the individual cartesians contribute to the solid harmonic. Exponents of the cartesians are then given in the arrays i_exp,j_exp,k_exp. More... | |
| subroutine, public | cfp_sph_shell_to_cart_shell (l, c) |
| This routine constructs for a given real solid harmonic L the matrix of coefficients for all M of the transformation from the cartesian harmonics (~x^i*y^j*z^k, i+j+k=L). On output the linear array c(:) emulates 2D matrix with 2*L+1 rows and (L+1)*(L+2)/2 columns. The rows of c correspond to the M values: -L,-L+1,...,0,1,...,L. The coefficients in each row are ordered so that the n-th coefficient in the given row of c corresponds to the coefficient for the cartesian harmonic with the shell canonical index n. More... | |
| subroutine, public | cfp_sph_shell_to_cart_lshells (l, nz, c_nz, nz_can) |
| This routine constructs for all real solid harmonic with L .le. l the matrix of coefficients for all M of the transformation from the cartesian harmonics (~x^i*y^j*z^k, i+j+k=L). On output the linear array c(:) emulates 2D matrix with 2*L+1 rows and (L+1)*(L+2)/2 columns. The rows of c correspond to the M values: -L,-L+1,...,0,1,...,L. The coefficients in each row are ordered so that the n-th coefficient in the given row of c corresponds to the coefficient for the cartesian harmonic with the shell canonical index n. More... | |
Function/Subroutine Documentation
◆ boys_function_quad()
| real(kind=ep1) function, dimension(1:mmax+1), public special_functions_gbl::boys_function_quad | ( | real(kind=ep1), intent(in) | T, |
| integer, intent(in) | mmax | ||
| ) |
Quadruple precision version of boys_function. See boys_function for details on the method of evaluation.
◆ cfp_eval_poly_horner_many()
| subroutine, public special_functions_gbl::cfp_eval_poly_horner_many | ( | integer, intent(in) | n, |
| real(kind=cfp), dimension(n_x), intent(in) | x, | ||
| integer, intent(in) | n_x, | ||
| real(kind=cfp), dimension(n_x,n+1), intent(in) | a, | ||
| real(kind=cfp), dimension(n_x), intent(out) | res | ||
| ) |
This subroutine evaluates a set of polynomials of degree \(n\) at a number of points \(x\) using the Horner form for polynomials.
- Parameters
-
[in] n Order of the polynomials. [in] x Array of points at which to evaluate. [in] n_x The number of points (dimension of x). [in] a The array of the coefficients a(1:n_x,1:n+1): different for each point x. [out] res The polynomials evaluated at points x.
◆ cfp_eval_poly_horner_single()
| real(kind=cfp) function special_functions_gbl::cfp_eval_poly_horner_single | ( | integer, intent(in) | n, |
| real(kind=cfp), intent(in) | x, | ||
| real(kind=cfp), dimension(1:n+1), intent(in) | a | ||
| ) |
This function evaluates a polynomial of degree \(n\) at point \(x\) using the Horner form for polynomials.
- Parameters
-
[in] n Order of the polynomial [in] x The point at which to evaluate. [in] a The array of the coefficients a(1:n+1): \(a_{0},a_{1},\dots,a_{n}\).
◆ cfp_nlm()
| subroutine, public special_functions_gbl::cfp_nlm | ( | real(kind=cfp), dimension(-l:l,0:l), intent(out) | INorm, |
| integer, intent(in) | L | ||
| ) |
- Purpose:
Calculate the full set of inverse normalization factors Ilm for the real unnormalized solid spherical harmonics with l,m .le. L using recursion. The unnormalized solid spherical harmonics are defined in Rico, Lopez, et al. Int. J. Q. Chem.,2012,DOI:10.1002/qua.24356.
\[ I_{lm} = \frac{1}{N_{lm}} = \sqrt{\frac{2\pi(1+\delta_{m,0})}{2l+1}\frac{(l+|m|)!}{(l-|m|)!}}. \]
- Todo:
- {add error checking for case l > size(INorm(,:))}
Nlm is the normalization factor for the real unnormalized solid spherical harmonics zlm.
- Parameters
-
[out] INorm Real array (-L:L,0:L) containing the values of the inverse normalization factors Ilm for all l .le. L and the corresponding values of m.
[in] L Integer angular momentum of the last Ilm to be generated.
◆ cfp_resh()
| subroutine, public special_functions_gbl::cfp_resh | ( | real(kind=cfp), dimension(-l:l,0:l), intent(out) | SH, |
| real(kind=cfp), intent(in) | X, | ||
| real(kind=cfp), intent(in) | Y, | ||
| real(kind=cfp), intent(in) | Z, | ||
| integer, intent(in) | L | ||
| ) |
- Purpose:
Calculate the full set of real normalized spherical harmonics with l,m .le. L using recursion. See Helgaker, p.210, p.215-218 (Section 6.4) for details. The real spherical harmonics are related to the solid ones using the formula:
\[ X_{lm}(\Omega) = S_{lm}(x/r,y/r,z/r)\sqrt{\frac{2l+1}{4\pi}}. \]
We define the normalization factor:\[ n_{lm} = \sqrt{\frac{2l+1}{4\pi}} \]
and then use Helgaker's method to calculate the 'scaled' solid harmonics:\[ X_{lm}(\Omega) = {\overline{S}_{lm}}(x/r,y/r,z/r)=n_{lm}S_{lm}(x/r,y/r,z/r). \]
The real spherical harmonics defined in this way are equivalent to the real spherical harmonics defined using the standard formula:
\[ X_{lm}(\Omega) = \begin{cases} {\sqrt{2}}(-1)^{m}{\cal{R}}\left(Y_{l\vert m\vert}(\Omega)\right), m>0 \\ {\sqrt{2}}(-1)^{m}{\cal{I}}\left(Y_{l\vert m\vert}(\Omega)\right), m<0 \\ Y_{l0}(\Omega), m=0, \end{cases} \]
where the complex spherical harmonics $Y_{lm}(\Omega)$ satisfy the Condon-Shortley phase convention. Note that the convention of Homeier and Steinborn is not to include the factor (-1)**m in the definition of the real spherical harmonics. However, here we follow the std. definition and include the factor (-1)**m in the definition of the real spherical harmonics.
- Todo:
- {add error checking for case L > size(SH(,:))}
- Parameters
-
[out] SH Real array (-L:L,0:L) containing the values of the spherical harmonics for all l .le. L and the corresponding values of m. According to my tests the results are in vast majority of cases accurate to full REAL(kind=cfp). In only a few cases the precision in the last 1 or 2 digits is lost.
[in] X,Y,Z Real input coordinates for which the real harmonics will be evaluated. The radius vector r={X,Y,Z} need not be normalized to 1 since this is done automatically during the calculation.[in] L Integer angular momentum of the last spherical harmonic to be generated. For L = 0; SH = 1/sqrt(4*pi)
◆ cfp_solh()
| subroutine, public special_functions_gbl::cfp_solh | ( | real(kind=cfp), dimension(-l:l,0:l), intent(out) | SH, |
| real(kind=cfp), intent(in) | x, | ||
| real(kind=cfp), intent(in) | y, | ||
| real(kind=cfp), intent(in) | z, | ||
| integer, intent(in) | L | ||
| ) |
- Purpose:
Calculate the full set of real normalized solid spherical harmonics with l,m .le. L using recursion. See Helgaker, p.218 (Section 6.4) for details. For l=m=0: Slm=1.
- Todo:
- {add error checking for case L > size(SH(,:))}
- Parameters
-
[out] SH Real array (-L:L,0:L) containing the values of the solid spherical harmonics for all l .le. L and the corresponding values of m.
[in] X,Y,Z real input coordinates at which the solid harmonics will be evaluated.
[in] L Integer angular momentum of the last spherical harmonic to be generated.
◆ cfp_sph_shell_to_cart_lshells()
| subroutine, public special_functions_gbl::cfp_sph_shell_to_cart_lshells | ( | integer, intent(in) | l, |
| integer, dimension(:), intent(out) | nz, | ||
| real(kind=cfp), dimension(:), intent(out) | c_nz, | ||
| integer, dimension(:), intent(out) | nz_can | ||
| ) |
This routine constructs for all real solid harmonic with L .le. l the matrix of coefficients for all M of the transformation from the cartesian harmonics (~x^i*y^j*z^k, i+j+k=L). On output the linear array c(:) emulates 2D matrix with 2*L+1 rows and (L+1)*(L+2)/2 columns. The rows of c correspond to the M values: -L,-L+1,...,0,1,...,L.
The coefficients in each row are ordered so that the n-th coefficient in the given row of c corresponds to the coefficient for the cartesian harmonic with the shell canonical index n.
◆ cfp_sph_shell_to_cart_shell()
| subroutine, public special_functions_gbl::cfp_sph_shell_to_cart_shell | ( | integer, intent(in) | l, |
| real(kind=cfp), dimension(:), intent(out) | c | ||
| ) |
This routine constructs for a given real solid harmonic L the matrix of coefficients for all M of the transformation from the cartesian harmonics (~x^i*y^j*z^k, i+j+k=L). On output the linear array c(:) emulates 2D matrix with 2*L+1 rows and (L+1)*(L+2)/2 columns. The rows of c correspond to the M values: -L,-L+1,...,0,1,...,L.
The coefficients in each row are ordered so that the n-th coefficient in the given row of c corresponds to the coefficient for the cartesian harmonic with the shell canonical index n.
◆ cfp_sph_to_cart_mapping()
| subroutine, public special_functions_gbl::cfp_sph_to_cart_mapping | ( | integer, intent(in) | l, |
| integer, intent(in) | m, | ||
| real(kind=cfp), dimension(:), intent(out), allocatable | c, | ||
| integer, dimension(:), intent(out), allocatable | i_exp, | ||
| integer, dimension(:), intent(out), allocatable | j_exp, | ||
| integer, dimension(:), intent(out), allocatable | k_exp | ||
| ) |
This routine constructs for a given real solid harmonic L,M the list of cartesian functions which build up this real solid harmonic. On output the array c(:) contains the coefficients with which the individual cartesians contribute to the solid harmonic. Exponents of the cartesians are then given in the arrays i_exp,j_exp,k_exp.
- Todo:
- the construction of the coefficient list (the summations) should be included along the lines of sph_shell_to_cart_shell
◆ cfp_zhar()
| subroutine, public special_functions_gbl::cfp_zhar | ( | real(kind=cfp), dimension(-l:l,0:l), intent(out) | SH, |
| real(kind=cfp), intent(in) | X, | ||
| real(kind=cfp), intent(in) | Y, | ||
| real(kind=cfp), intent(in) | Z, | ||
| integer, intent(in) | L | ||
| ) |
- Purpose:
Calculate the full set of real unnormalized solid spherical harmonics with l,m .le. L using recursion. See Helgaker, p.210, p.218 (Section 6.4) for details. The unnormalized solid spherical harmonics are defined in Rico, Lopez, et al. Int. J. Q. Chem.,2012,DOI:10.1002/qua.24356. The mehod of calculation is to calculate the real solid spherical harmonics and then renormalize them to obtain the surface spherical harmonics using the formula:
\[ z_{lm}(x,y,z) = S_{lm}(x,y,z)\sqrt{\frac{1+\delta_{m,0}}{2}\frac{(l+|m|)!}{(l-|m|)!}}. \]
- Todo:
- {rewrite the algorithm to calculate zlm directly using Helgaker's algorithm for the scaled solid harmonics rather than by calculating Slm first and then renormalizing}
- Todo:
- {add error checking for case L > size(SH(,:))}
- Parameters
-
[out] SH Real array (-L:L,0:L) containing the values of the real unnormalized solidspherical harmonics for all l .le. L and the corresponding values of m.
[in] X,Y,Z Real input coordinates for which the harmonics will be evaluated.
[in] L Integer angular momentum of the last harmonic to be generated.
◆ dfact()
| elemental integer function special_functions_gbl::dfact | ( | integer, intent(in) | n | ) |
Computes the double factorial \((n)!!\) of the integer number n. n must be .ge. -1.
◆ ipair()
| elemental integer function, public special_functions_gbl::ipair | ( | integer, intent(in) | i | ) |
This function is used to index an ordered pair of values (i,j), where i .ge. j, i .ge. 1. The index of the ordered pair is: ipair(i) + j. This process of indexing can be used in a nested way to index quartets of integers, say (i,j,k,l). We compute ij = ipair(i)+j and kl = ipair(k)+l. The index of the quartet (i,j,k,l) is: ipair(ij)+kl. We assume that i.ge.j, k.ge.l, ij.ge.kl. This nesting (triangularization) is used heavilly to index the 2-particle symmetric integrals.
◆ unpack_pq()
| subroutine, public special_functions_gbl::unpack_pq | ( | integer, intent(in) | pq, |
| integer, intent(in) | n, | ||
| integer, intent(out) | p, | ||
| integer, intent(out) | q | ||
| ) |
Assuming \( pq = p + (q-1)*n \) this routine returns p,q given pq,n.
