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qmckl/org/qmckl_examples.org

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#+TITLE: Code examples
#+SETUPFILE: ../tools/theme.setup
#+INCLUDE: ../tools/lib.org
In this section, we provide hands-on examples to demonstrate the capabilities
of the QMCkl library. We furnish code samples in C, Fortran, and Python,
serving as exhaustive tutorials for effectively leveraging QMCkl.
For simplicity, we assume that the wave function parameters are stored in a
[[https://github.com/TREX-CoE/trexio][TREXIO]] file.
* Overlap matrix in the MO basis
The focal point of this example is the numerical evaluation of the overlap
matrix in the MO basis. Utilizing QMCkl, we approximate the MOs at
discrete grid points to compute the overlap matrix \( S_{ij} \) as follows:
\[
S_{ij} = \int \phi_i(\mathbf{r})\, \phi_j(\mathbf{r}) \text{d}\mathbf{r} \approx
\sum_k \phi_i(\mathbf{r}_k)\, \phi_j(\mathbf{r}_k) \delta\mathbf{r}
\]
The code starts by reading a wave function from a TREXIO file. This is
accomplished using the ~qmckl_trexio_read~ function, which populates a
~qmckl_context~ with the necessary wave function parameters. The context
serves as the primary interface for interacting with the QMCkl library,
encapsulating the state and configurations of the system.
Subsequently, the code retrieves various attributes such as the number of
nuclei ~nucl_num~ and coordinates ~nucl_coord~.
These attributes are essential for setting up the integration grid.
The core of the example lies in the numerical computation of the overlap
matrix. To achieve this, the code employs a regular grid in three-dimensional
space, and the grid points are then populated into the ~qmckl_context~ using
the ~qmckl_set_point~ function.
The MO values at these grid points are computed using the
~qmckl_get_mo_basis_mo_value~ function. These values are then used to
calculate the overlap matrix through a matrix multiplication operation
facilitated by the ~qmckl_dgemm~ function.
The code is also instrumented to measure the execution time for the MO
value computation, providing an empirical assessment of the computational
efficiency. Error handling is robustly implemented at each stage to ensure the
reliability of the simulation.
In summary, this example serves as a holistic guide for leveraging the QMCkl
library, demonstrating its ease of use. It provides a concrete starting point
for researchers and developers interested in integrating QMCkl into their QMC
code.
** Python
:PROPERTIES:
:header-args: :tangle mo_ortho.py
:END:
In this example, we will compute numerically the overlap
between the molecular orbitals:
\[
S_{ij} = \int \phi_i(\mathbf{r}) \phi_j(\mathbf{r})
\text{d}\mathbf{r} \sim \sum_{k=1}^{N} \phi_i(\mathbf{r}_k)
\phi_j(\mathbf{r}_k) \delta \mathbf{r}
\]
\[
S_{ij} = \langle \phi_i | \phi_j \rangle
\sim \sum_{k=1}^{N} \langle \phi_i | \mathbf{r}_k \rangle
\langle \mathbf{r}_k | \phi_j \rangle
\]
#+begin_src python :exports code
import numpy as np
import qmckl
#+end_src
#+RESULTS:
First, we create a context for the QMCkl calculation, and load the
wave function stored in =h2o_5z.h5= inside it. It is a Hartree-Fock
determinant for the water molecule in the cc-pV5Z basis set.
#+begin_src python :exports code
trexio_filename = "..//share/qmckl/test_data/h2o_5z.h5"
context = qmckl.context_create()
qmckl.trexio_read(context, trexio_filename)
#+end_src
#+RESULTS:
: None
We now define the grid points $\mathbf{r}_k$ as a regular grid around the
molecule.
We fetch the nuclear coordinates from the context,
#+begin_src python :exports code
nucl_num = qmckl.get_nucleus_num(context)
nucl_charge = qmckl.get_nucleus_charge(context, nucl_num)
nucl_coord = qmckl.get_nucleus_coord(context, 'N', nucl_num*3)
nucl_coord = np.reshape(nucl_coord, (3, nucl_num))
for i in range(nucl_num):
print("%d %+f %+f %+f"%(int(nucl_charge[i]),
nucl_coord[i,0],
nucl_coord[i,1],
nucl_coord[i,2]) )
#+end_src
#+begin_example
8 +0.000000 +0.000000 +0.000000
1 -1.430429 +0.000000 -1.107157
1 +1.430429 +0.000000 -1.107157
#+end_example
and compute the coordinates of the grid points:
#+begin_src python :exports code
nx = ( 120, 120, 120 )
shift = np.array([5.,5.,5.])
point_num = nx[0] * nx[1] * nx[2]
rmin = np.array( list([ np.min(nucl_coord[:,a]) for a in range(3) ]) )
rmax = np.array( list([ np.max(nucl_coord[:,a]) for a in range(3) ]) )
linspace = [ None for i in range(3) ]
step = [ None for i in range(3) ]
for a in range(3):
linspace[a], step[a] = np.linspace(rmin[a]-shift[a],
rmax[a]+shift[a],
num=nx[a],
retstep=True)
dr = step[0] * step[1] * step[2]
#+end_src
#+RESULTS:
Now the grid is ready, we can create the list of grid points
$\mathbf{r}_k$ on which the MOs $\phi_i$ will be evaluated, and
transfer them to the QMCkl context:
#+begin_src python :exports code
point = []
for x in linspace[0]:
for y in linspace[1]:
for z in linspace[2]:
point += [ [x, y, z] ]
point = np.array(point)
point_num = len(point)
qmckl.set_point(context, 'N', point_num, np.reshape(point, (point_num*3)))
#+end_src
#+RESULTS:
: None
Then, we evaluate all the MOs at the grid points (and time the execution),
and thus obtain the matrix $M_{ki} = \langle \mathbf{r}_k | \phi_i \rangle =
\phi_i(\mathbf{r}_k)$.
#+begin_src python :exports code
import time
mo_num = qmckl.get_mo_basis_mo_num(context)
before = time.time()
mo_value = qmckl.get_mo_basis_mo_value(context, point_num*mo_num)
after = time.time()
mo_value = np.reshape( mo_value, (point_num, mo_num) ).T # Transpose to get mo_num x point_num
print("Number of MOs: ", mo_num)
print("Number of grid points: ", point_num)
print("Execution time : ", (after - before), "seconds")
#+end_src
#+begin_example
Number of MOs: 201
Number of grid points: 1728000
Execution time : 5.577778577804565 seconds
#+end_example
and finally we compute the overlap between all the MOs as
$M.M^\dagger$.
#+begin_src python :exports code
overlap = mo_value @ mo_value.T * dr
print (overlap)
#+end_src
#+begin_example
[[ 9.88693941e-01 2.34719693e-03 -1.50518232e-08 ... 3.12084178e-09
-5.81064929e-10 3.70130091e-02]
[ 2.34719693e-03 9.99509628e-01 3.18930040e-09 ... -2.46888958e-10
-1.06064273e-09 -7.65567973e-03]
[-1.50518232e-08 3.18930040e-09 9.99995073e-01 ... -5.84882580e-06
-1.21598117e-06 4.59036468e-08]
...
[ 3.12084178e-09 -2.46888958e-10 -5.84882580e-06 ... 1.00019107e+00
-2.03342837e-04 -1.36954855e-08]
[-5.81064929e-10 -1.06064273e-09 -1.21598117e-06 ... -2.03342837e-04
9.99262427e-01 1.18264754e-09]
[ 3.70130091e-02 -7.65567973e-03 4.59036468e-08 ... -1.36954855e-08
1.18264754e-09 8.97215950e-01]]
#+end_example
** C
In this example, electron-nucleus cusp fitting is added.
:PROPERTIES:
:header-args: :tangle mo_ortho.c
:END:
In this example, we will compute numerically the overlap
between the molecular orbitals:
\[
S_{ij} = \int \phi_i(\mathbf{r}) \phi_j(\mathbf{r})
\text{d}\mathbf{r} \sim \sum_{k=1}^{N} \phi_i(\mathbf{r}_k)
\phi_j(\mathbf{r}_k) \delta \mathbf{r}
\]
\[
S_{ij} = \langle \phi_i | \phi_j \rangle
\sim \sum_{k=1}^{N} \langle \phi_i | \mathbf{r}_k \rangle
\langle \mathbf{r}_k | \phi_j \rangle
\]
We apply the cusp fitting procedure, so the MOs might deviate
slightly from orthonormality.
#+begin_src c :exports code
#include <qmckl.h>
#include <stdio.h>
#include <string.h>
#include <sys/time.h>
int main(int argc, char** argv)
{
const char* trexio_filename = "..//share/qmckl/test_data/h2o_5z.h5";
qmckl_exit_code rc = QMCKL_SUCCESS;
#+end_src
First, we create a context for the QMCkl calculation, and load the
wave function stored in =h2o_5z.h5= inside it. It is a Hartree-Fock
determinant for the water molecule in the cc-pV5Z basis set.
#+begin_src c :exports code
qmckl_context context = qmckl_context_create();
rc = qmckl_trexio_read(context, trexio_filename, strlen(trexio_filename));
if (rc != QMCKL_SUCCESS) {
fprintf(stderr, "Error reading TREXIO file:\n%s\n", qmckl_string_of_error(rc));
exit(1);
}
#+end_src
We impose the electron-nucleus cusp fitting to occur when the
electrons are close to the nuclei. The critical distance
is 0.5 atomic units for hydrogens and 0.1 for the oxygen.
To identify which atom is an oxygen and which are hydrogens, we
fetch the nuclear charges from the context.
#+begin_src c :exports code
int64_t nucl_num;
rc = qmckl_get_nucleus_num(context, &nucl_num);
if (rc != QMCKL_SUCCESS) {
fprintf(stderr, "Error getting nucl_num:\n%s\n", qmckl_string_of_error(rc));
exit(1);
}
double nucl_charge[nucl_num];
rc = qmckl_get_nucleus_charge(context, &(nucl_charge[0]), nucl_num);
if (rc != QMCKL_SUCCESS) {
fprintf(stderr, "Error getting nucl_charge:\n%s\n", qmckl_string_of_error(rc));
exit(1);
}
double r_cusp[nucl_num];
for (size_t i=0 ; i<nucl_num ; ++i) {
switch ((int) nucl_charge[i]) {
case 1:
r_cusp[i] = 0.5;
break;
case 8:
r_cusp[i] = 0.1;
break;
}
}
rc = qmckl_set_mo_basis_r_cusp(context, &(r_cusp[0]), nucl_num);
if (rc != QMCKL_SUCCESS) {
fprintf(stderr, "Error setting r_cusp:\n%s\n", qmckl_string_of_error(rc));
exit(1);
}
#+end_src
We now define the grid points $\mathbf{r}_k$ as a regular grid around the
molecule.
We fetch the nuclear coordinates from the context,
#+begin_src c :exports code
double nucl_coord[nucl_num][3];
rc = qmckl_get_nucleus_coord(context, 'N', &(nucl_coord[0][0]), nucl_num*3);
if (rc != QMCKL_SUCCESS) {
fprintf(stderr, "Error getting nucl_coord:\n%s\n", qmckl_string_of_error(rc));
exit(1);
}
for (size_t i=0 ; i<nucl_num ; ++i) {
printf("%d %+f %+f %+f\n",
(int32_t) nucl_charge[i],
nucl_coord[i][0],
nucl_coord[i][1],
nucl_coord[i][2]);
}
#+end_src
#+begin_example
8 +0.000000 +0.000000 +0.000000
1 -1.430429 +0.000000 -1.107157
1 +1.430429 +0.000000 -1.107157
#+end_example
and compute the coordinates of the grid points:
#+begin_src c :exports code
size_t nx[3] = { 120, 120, 120 };
double shift[3] = {5.,5.,5.};
int64_t point_num = nx[0] * nx[1] * nx[2];
double rmin[3] = { nucl_coord[0][0], nucl_coord[0][1], nucl_coord[0][2] } ;
double rmax[3] = { nucl_coord[0][0], nucl_coord[0][1], nucl_coord[0][2] } ;
for (size_t i=0 ; i<nucl_num ; ++i) {
for (int j=0 ; j<3 ; ++j) {
rmin[j] = nucl_coord[i][j] < rmin[j] ? nucl_coord[i][j] : rmin[j];
rmax[j] = nucl_coord[i][j] > rmax[j] ? nucl_coord[i][j] : rmax[j];
}
}
rmin[0] -= shift[0]; rmin[1] -= shift[1]; rmin[2] -= shift[2];
rmax[0] += shift[0]; rmax[1] += shift[1]; rmax[2] += shift[2];
double step[3];
double* linspace[3];
for (int i=0 ; i<3 ; ++i) {
linspace[i] = (double*) calloc( sizeof(double), nx[i] );
if (linspace[i] == NULL) {
fprintf(stderr, "Allocation failed (linspace)\n");
exit(1);
}
step[i] = (rmax[i] - rmin[i]) / ((double) (nx[i]-1));
for (size_t j=0 ; j<nx[i] ; ++j) {
linspace[i][j] = rmin[i] + j*step[i];
}
}
double dr = step[0] * step[1] * step[2];
#+end_src
Now the grid is ready, we can create the list of grid points
$\mathbf{r}_k$ on which the MOs $\phi_i$ will be evaluated, and
transfer them to the QMCkl context:
#+begin_src c :exports code
double* point = (double*) calloc(sizeof(double), 3*point_num);
if (point == NULL) {
fprintf(stderr, "Allocation failed (point)\n");
exit(1);
}
size_t m = 0;
for (size_t i=0 ; i<nx[0] ; ++i) {
for (size_t j=0 ; j<nx[1] ; ++j) {
for (size_t k=0 ; k<nx[2] ; ++k) {
point[m] = linspace[0][i];
m++;
point[m] = linspace[1][j];
m++;
point[m] = linspace[2][k];
m++;
}
}
}
rc = qmckl_set_point(context, 'N', point_num, point, (point_num*3));
if (rc != QMCKL_SUCCESS) {
fprintf(stderr, "Error setting points:\n%s\n", qmckl_string_of_error(rc));
exit(1);
}
#+end_src
#+RESULTS:
: None
Then, we evaluate all the MOs at the grid points (and time the execution),
and thus obtain the matrix $M_{ki} = \langle \mathbf{r}_k | \phi_i
\rangle = \phi_i(\mathbf{r}_k)$.
#+begin_src c :exports code
int64_t mo_num;
rc = qmckl_get_mo_basis_mo_num(context, &mo_num);
long before, after;
struct timeval timecheck;
double* mo_value = (double*) calloc(sizeof(double), point_num*mo_num);
if (mo_value == NULL) {
fprintf(stderr, "Allocation failed (mo_value)\n");
exit(1);
}
gettimeofday(&timecheck, NULL);
before = (long)timecheck.tv_sec * 1000 + (long)timecheck.tv_usec / 1000;
rc = qmckl_get_mo_basis_mo_value(context, mo_value, point_num*mo_num);
if (rc != QMCKL_SUCCESS) {
fprintf(stderr, "Error getting mo_value)\n");
exit(1);
}
gettimeofday(&timecheck, NULL);
after = (long)timecheck.tv_sec * 1000 + (long)timecheck.tv_usec / 1000;
printf("Number of MOs: %ld\n", mo_num);
printf("Number of grid points: %ld\n", point_num);
printf("Execution time : %f seconds\n", (after - before)*1.e-3);
#+end_src
#+begin_example
Number of MOs: 201
Number of grid points: 1728000
Execution time : 5.608000 seconds
#+end_example
and finally we compute the overlap between all the MOs as
$M.M^\dagger$.
#+begin_src c :exports code
double* overlap = (double*) malloc (mo_num*mo_num*sizeof(double));
rc = qmckl_dgemm(context, 'N', 'T', mo_num, mo_num, point_num, dr,
mo_value, mo_num, mo_value, mo_num, 0.0,
overlap, mo_num);
for (size_t i=0 ; i<mo_num ; ++i) {
printf("%4ld", i);
for (size_t j=0 ; j<mo_num ; ++j) {
printf(" %f", overlap[i*mo_num+j]);
}
printf("\n");
}
}
#+end_src
#+begin_example
0 0.988765 0.002336 0.000000 -0.000734 0.000000 0.000530 0.000000 0.000446 0.000000 -0.000000 -0.000511 -0.000000 -0.000267 0.000000 0.000000 0.001007 0.000000 0.000168 -0.000000 -0.000000 -0.000670 -0.000000 0.000000 -0.000251 -0.000261 -0.000000 -0.000000 -0.000000 -0.000397 -0.000000 -0.000810 0.000000 0.000231 -0.000000 -0.000000 0.000000 -0.000000
...
200 0.039017 -0.013066 -0.000000 -0.001935 -0.000000 -0.003829 -0.000000 0.000996 -0.000000 0.000000 -0.003733 0.000000 0.000065 -0.000000 -0.000000 -0.002220 -0.000000 -0.001961 0.000000 0.000000 -0.004182 0.000000 -0.000000 -0.000165 -0.002445 0.000000 -0.000000 0.000000 0.001985 0.000000 0.001685 -0.000000 -0.002899 0.000000 0.000000 0.000000 -0.000000 0.002591 0.000000 -0.000000 0.000000 0.002385 0.000000 -0.011086 0.000000 -0.003885 0.000000 -0.000000 0.003602 -0.000000 0.000000 -0.003241 0.000000 0.000000 0.002613 -0.007344 -0.000000 -0.000000 0.000000 0.000017 0.000000 0.000000 0.000000 -0.008719 0.000000 -0.001358 -0.003233 0.000000 -0.000000 -0.000000 -0.000000 0.000000 0.003778 0.000000 0.000000 -0.000000 0.000000 0.000000 -0.001190 0.000000 0.000000 -0.000000 0.005578 -0.000000 -0.001502 0.003899 -0.000000 -0.000000 0.000000 -0.003291 -0.001775 -0.000000 -0.002374 0.000000 -0.000000 -0.000000 -0.000000 -0.001290 -0.000000 0.002178 0.000000 0.000000 0.000000 -0.001252 0.000000 -0.000000 -0.000926 0.000000 -0.000000 -0.013130 -0.000000 0.012124 0.000000 -0.000000 -0.000000 -0.000000 0.000000 0.025194 0.000343 -0.000000 0.000000 -0.000000 -0.004421 0.000000 0.000000 -0.000599 -0.000000 0.005289 0.000000 -0.000000 0.012826 -0.000000 0.000000 0.006190 0.000000 0.000000 -0.000000 0.000000 -0.000321 0.000000 -0.000000 -0.000000 0.000000 -0.000000 0.001499 -0.006629 0.000000 0.000000 0.000000 -0.000000 0.008737 -0.000000 0.006880 0.000000 -0.001851 -0.000000 -0.000000 0.000000 -0.007464 0.000000 0.010298 -0.000000 -0.000000 -0.000000 -0.000000 -0.000000 0.000000 0.000540 0.000000 -0.006616 -0.000000 0.000000 -0.002927 -0.000000 0.000000 0.010352 0.000000 -0.003103 -0.000000 -0.007640 -0.000000 -0.000000 0.005302 0.000000 0.000000 -0.000000 -0.000000 -0.010181 0.000000 -0.001108 0.000000 0.000000 -0.000000 0.000000 0.000000 -0.000998 -0.009754 0.013562 0.000000 -0.000000 0.887510
#+end_example
* Fortran
** Checking errors
All QMCkl functions return an error code. A convenient way to handle
errors is to write an error-checking function that displays the
error in text format and exits the program.
#+NAME: qmckl_check_error
#+begin_src f90
subroutine qmckl_check_error(rc, message)
use qmckl
implicit none
integer(qmckl_exit_code), intent(in) :: rc
character(len=*) , intent(in) :: message
character(len=128) :: str_buffer
if (rc /= QMCKL_SUCCESS) then
print *, message
call qmckl_string_of_error(rc, str_buffer)
print *, str_buffer
call exit(rc)
end if
end subroutine qmckl_check_error
#+end_src
** Computing an atomic orbital on a grid
:PROPERTIES:
:header-args: :tangle ao_grid.f90
:END:
The following program, in Fortran, computes the values of an atomic
orbital on a regular 3-dimensional grid. The 100^3 grid points are
automatically defined, such that the molecule fits in a box with 5
atomic units in the borders.
This program uses the ~qmckl_check_error~ function defined above.
To use this program, run
#+begin_src bash :tangle no :exports code
$ ao_grid <trexio_file> <AO_id> <point_num>
#+end_src
#+begin_src f90 :noweb yes
<<qmckl_check_error>>
program ao_grid
use qmckl
implicit none
integer(qmckl_context) :: qmckl_ctx ! QMCkl context
integer(qmckl_exit_code) :: rc ! Exit code of QMCkl functions
character(len=128) :: trexio_filename
character(len=128) :: str_buffer
integer :: ao_id
integer :: point_num_x
integer(c_int64_t) :: nucl_num
double precision, allocatable :: nucl_coord(:,:)
integer(c_int64_t) :: point_num
integer(c_int64_t) :: ao_num
integer(c_int64_t) :: ipoint, i, j, k
double precision :: x, y, z, dr(3)
double precision :: rmin(3), rmax(3)
double precision, allocatable :: points(:,:)
double precision, allocatable :: ao_vgl(:,:,:)
#+end_src
Start by fetching the command-line arguments:
#+begin_src f90
if (iargc() /= 3) then
print *, 'Syntax: ao_grid <trexio_file> <AO_id> <point_num>'
call exit(-1)
end if
call getarg(1, trexio_filename)
call getarg(2, str_buffer)
read(str_buffer, *) ao_id
call getarg(3, str_buffer)
read(str_buffer, *) point_num_x
if (point_num_x < 0 .or. point_num_x > 300) then
print *, 'Error: 0 < point_num < 300'
call exit(-1)
end if
#+end_src
Create the QMCkl context and initialize it with the wave function
present in the TREXIO file:
#+begin_src f90
qmckl_ctx = qmckl_context_create()
rc = qmckl_trexio_read(qmckl_ctx, trexio_filename, 1_8*len(trim(trexio_filename)))
call qmckl_check_error(rc, 'Read TREXIO')
#+end_src
We need to check that ~ao_id~ is in the range, so we get the total
number of AOs from QMCkl:
#+begin_src f90
rc = qmckl_get_ao_basis_ao_num(qmckl_ctx, ao_num)
call qmckl_check_error(rc, 'Getting ao_num')
if (ao_id < 0 .or. ao_id > ao_num) then
print *, 'Error: 0 < ao_id < ', ao_num
call exit(-1)
end if
#+end_src
Now we will compute the limits of the box in which the molecule fits.
For that, we first need to ask QMCkl the coordinates of nuclei.
#+begin_src f90
rc = qmckl_get_nucleus_num(qmckl_ctx, nucl_num)
call qmckl_check_error(rc, 'Get nucleus num')
allocate( nucl_coord(3, nucl_num) )
rc = qmckl_get_nucleus_coord(qmckl_ctx, 'N', nucl_coord, 3_8*nucl_num)
call qmckl_check_error(rc, 'Get nucleus coord')
#+end_src
We now compute the coordinates of opposite points of the box, and
the distance between points along the 3 directions:
#+begin_src f90
rmin(1) = minval( nucl_coord(1,:) ) - 5.d0
rmin(2) = minval( nucl_coord(2,:) ) - 5.d0
rmin(3) = minval( nucl_coord(3,:) ) - 5.d0
rmax(1) = maxval( nucl_coord(1,:) ) + 5.d0
rmax(2) = maxval( nucl_coord(2,:) ) + 5.d0
rmax(3) = maxval( nucl_coord(3,:) ) + 5.d0
dr(1:3) = (rmax(1:3) - rmin(1:3)) / dble(point_num_x-1)
#+end_src
We now produce the list of point coordinates where the AO will be
evaluated:
#+begin_src f90
point_num = point_num_x**3
allocate( points(point_num, 3) )
ipoint=0
z = rmin(3)
do k=1,point_num_x
y = rmin(2)
do j=1,point_num_x
x = rmin(1)
do i=1,point_num_x
ipoint = ipoint+1
points(ipoint,1) = x
points(ipoint,2) = y
points(ipoint,3) = z
x = x + dr(1)
end do
y = y + dr(2)
end do
z = z + dr(3)
end do
#+end_src
We give the points to QMCkl:
#+begin_src f90
rc = qmckl_set_point(qmckl_ctx, 'T', point_num, points, size(points)*1_8 )
call qmckl_check_error(rc, 'Setting points')
#+end_src
We allocate the space required to retrieve the values, gradients and
Laplacian of all AOs, and ask to retrieve the values of the
AOs computed at the point positions.
#+begin_src f90
allocate( ao_vgl(ao_num, 5, point_num) )
rc = qmckl_get_ao_basis_ao_vgl(qmckl_ctx, ao_vgl, ao_num*5_8*point_num)
call qmckl_check_error(rc, 'Setting points')
#+end_src
We finally print the value and Laplacian of the AO:
#+begin_src f90
do ipoint=1, point_num
print '(3(F10.6,X),2(E20.10,X))', points(ipoint, 1:3), ao_vgl(ao_id,1,ipoint), ao_vgl(ao_id,5,ipoint)
end do
#+end_src
#+begin_src f90
deallocate( nucl_coord, points, ao_vgl )
end program ao_grid
#+end_src