Code examples
Table of Contents
1 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.
1.1 Python
In this example, we will compute numerically the overlap between the molecular orbitals:
\[ S_{ij} = \int \phi_i(\mathbf{r}) \phi_j(\mathbf{r}) \text{d}\mathbf{r} \sim \sum_{k=1}^{N} \phi_i(\mathbf{r}_k) \phi_j(\mathbf{r}_k) \delta \mathbf{r} \] \[ S_{ij} = \langle \phi_i | \phi_j \rangle \sim \sum_{k=1}^{N} \langle \phi_i | \mathbf{r}_k \rangle \langle \mathbf{r}_k | \phi_j \rangle \]
import numpy as np import qmckl
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.
trexio_filename = "..//share/qmckl/test_data/h2o_5z.h5" context = qmckl.context_create() qmckl.trexio_read(context, trexio_filename)
We now define the grid points \(\mathbf{r}_k\) as a regular grid around the molecule.
We fetch the nuclear coordinates from the context,
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]) )
8 +0.000000 +0.000000 +0.000000 1 -1.430429 +0.000000 -1.107157 1 +1.430429 +0.000000 -1.107157
and compute the coordinates of the grid points:
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]
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:
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)))
Then, we evaluate all the MOs at the grid points (and time the execution), and thus obtain the matrix \(M_{ki} = \langle \mathbf{r}_k | \phi_i \rangle = \phi_i(\mathbf{r}_k)\).
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")
Number of MOs: 201 Number of grid points: 1728000 Execution time : 5.577778577804565 seconds
and finally we compute the overlap between all the MOs as \(M.M^\dagger\).
overlap = mo_value @ mo_value.T * dr print (overlap)
[[ 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]]
1.2 C
In this example, electron-nucleus cusp fitting is added.
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.
#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;
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.
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); }
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.
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); }
We now define the grid points \(\mathbf{r}_k\) as a regular grid around the molecule. We fetch the nuclear coordinates from the context,
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]); }
8 +0.000000 +0.000000 +0.000000 1 -1.430429 +0.000000 -1.107157 1 +1.430429 +0.000000 -1.107157
and compute the coordinates of the grid points:
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];
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:
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); }
Then, we evaluate all the MOs at the grid points (and time the execution), and thus obtain the matrix \(M_{ki} = \langle \mathbf{r}_k | \phi_i \rangle = \phi_i(\mathbf{r}_k)\).
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);
Number of MOs: 201 Number of grid points: 1728000 Execution time : 5.608000 seconds
and finally we compute the overlap between all the MOs as \(M.M^\dagger\).
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"); } }
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
2 Fortran
2.1 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.
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
2.2 Computing an atomic orbital on a grid
The following program, in Fortran, computes the values of an atomic orbital on a regular 3-dimensional grid. The 1003 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
$ ao_grid <trexio_file> <AO_id> <point_num>
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 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(:,:,:)
Start by fetching the command-line arguments:
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
Create the QMCkl context and initialize it with the wave function present in the TREXIO file:
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')
We need to check that ao_id is in the range, so we get the total
number of AOs from QMCkl:
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
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.
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')
We now compute the coordinates of opposite points of the box, and the distance between points along the 3 directions:
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)
We now produce the list of point coordinates where the AO will be evaluated:
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
We give the points to QMCkl:
rc = qmckl_set_point(qmckl_ctx, 'T', point_num, points, size(points)*1_8 ) call qmckl_check_error(rc, 'Setting points')
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.
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')
We finally print the value and Laplacian of the AO:
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
deallocate( nucl_coord, points, ao_vgl ) end program ao_grid