TREX/qmckl

mirror of https://github.com/TREX-CoE/qmckl.git synced 2024-12-22 20:36:01 +01:00

Anthony Scemama 2587a046de Fixed #48

2021-10-14 21:40:14 +02:00

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Atomic Orbitals

Context
Radial part
Polynomial part
- General functions for Powers of $x-X_i$
- General functions for Value, Gradient and Laplacian of a polynomial
Combining radial and polynomial parts
- Get
- Provide
- Compute
- Test

The atomic basis set is defined as a list of shells. Each shell $s$ is centered on a nucleus $A$, possesses a given angular momentum $l$ and a radial function $R_s$. The radial function is a linear combination of \emph{primitive} functions that can be of type Slater ($p=1$) or Gaussian ($p=2$):

\[ R_s(\mathbf{r}) = \mathcal{N}_s |\mathbf{r}-\mathbf{R}_A|^{n_s} \sum_{k=1}^{N_{\text{prim}}} a_{ks}\, f_{ks} \exp \left( - \gamma_{ks} | \mathbf{r}-\mathbf{R}_A | ^p \right). \]

In the case of Gaussian functions, $n_s$ is always zero. The normalization factor $\mathcal{N}_s$ ensures that all the functions of the shell are normalized (integrate) to unity. Usually, basis sets are given a combination of normalized primitives, so the normalization coefficients of the primitives, $f_{ks}$, need also to be provided.

Atomic orbitals (AOs) are defined as

\[ \chi_i (\mathbf{r}) = \mathcal{M}_i\, P_{\eta(i)}(\mathbf{r})\, R_{\theta(i)} (\mathbf{r}) \]

where $\theta(i)$ returns the shell on which the AO is expanded, and $\eta(i)$ denotes which angular function is chosen and $P$ are the generating functions of the given angular momentum $\eta(i)$. Here, the parameter $\mathcal{M}_i$ is an extra parameter which allows the normalization of the different functions of the same shell to be different, as in GAMESS for example.

In this section we describe first how the basis set is stored in the context, and then we present the kernels used to compute the values, gradients and Laplacian of the atomic basis functions.

Context

The following arrays are stored in the context:

`type`		Gaussian (`'G'`) or Slater (`'S'`)
`shell_num`		Number of shells
`prim_num`		Total number of primitives
`nucleus_index`	`[nucl_num]`	Index of the first shell of each nucleus
`nucleus_shell_num`	`[nucl_num]`	Number of shells per nucleus
`shell_ang_mom`	`[shell_num]`	Angular momentum of each shell
`shell_prim_num`	`[shell_num]`	Number of primitives in each shell
`shell_prim_index`	`[shell_num]`	Address of the first primitive of each shell in the `EXPONENT` array
`shell_factor`	`[shell_num]`	Normalization factor for each shell
`exponent`	`[prim_num]`	Array of exponents
`coefficient`	`[prim_num]`	Array of coefficients
`prim_factor`	`[prim_num]`	Normalization factors of the primtives
`ao_num`		Number of AOs
`ao_cartesian`		If true, use polynomials. Otherwise, use spherical harmonics
`ao_factor`	`[ao_num]`	Normalization factor of the AO
`ao_shell`	`[ao_num]`	For each AO, specify to which shell it belongs

Computed data:

`coefficient_normalized`	`[prim_num]`	Normalized primitive coefficients
`nucleus_prim_index`	`[nucl_num]`	Index of the first primitive for each nucleus
`nucleus_max_ang_mom`	`[nucl_num]`	Maximum angular momentum for each nucleus
`nucleus_range`	`[nucl_num]`	Distance beyond which all the AOs are zero
`primitive_vgl`	`[5][elec_num][prim_num]`	Value, gradients, Laplacian of the primitives at electron positions
`primitive_vgl_date`	`uint64_t`	Late modification date of Value, gradients, Laplacian of the primitives at electron positions
`shell_vgl`	`[5][elec_num][shell_num]`	Value, gradients, Laplacian of the primitives at electron positions
`shell_vgl_date`	`uint64_t`	Late modification date of Value, gradients, Laplacian of the AOs at electron positions
`ao_vgl`	`[5][elec_num][ao_num]`	Value, gradients, Laplacian of the primitives at electron positions
`ao_vgl_date`	`uint64_t`	Late modification date of Value, gradients, Laplacian of the AOs at electron positions
`nucl_shell_index`	`[nucl_num]`	Index of the first shell for each nucleus
`exponent_sorted`	`[prim_num]`	Array of exponents for sorted primitives
`coeff_norm_sorted`	`[prim_num]`	Array of normalized coefficients for sorted primitives
`prim_factor_sorted`	`[prim_num]`	Normalization factors of the sorted primtives

For H_2 with the following basis set,

HYDROGEN
S   5
1         3.387000E+01           6.068000E-03
2         5.095000E+00           4.530800E-02
3         1.159000E+00           2.028220E-01
4         3.258000E-01           5.039030E-01
5         1.027000E-01           3.834210E-01
S   1
1         3.258000E-01           1.000000E+00
S   1
1         1.027000E-01           1.000000E+00
P   1
1         1.407000E+00           1.000000E+00
P   1
1         3.880000E-01           1.000000E+00
D   1
1         1.057000E+00           1.0000000

we have:

type = 'G'
shell_num = 12
prim_num = 20
ao_num = 38
nucleus_index = [0 , 6]
shell_ang_mom = [0, 0, 0, 1, 1, 2, 0, 0, 0, 1, 1, 2]
shell_factor  = [ 1.,  1.,  1.,  1.,  1.,  1.,  1.,  1.,  1.,  1.,  1.,  1.]
shell_prim_num = [5, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1]
shell_prim_index = [0 , 5 , 6 , 7 , 8 , 9 , 10, 15, 16, 17, 18, 19]
exponent = [ 33.87, 5.095, 1.159, 0.3258, 0.1027, 0.3258, 0.1027, 1.407,
0.388, 1.057, 33.87, 5.095, 1.159, 0.3258, 0.1027, 0.3258, 0.1027, 1.407,
0.388, 1.057]
coefficient = [ 0.006068, 0.045308, 0.202822, 0.503903, 0.383421, 1.0, 1.0,
1.0, 1.0, 1.0, 0.006068, 0.045308, 0.202822, 0.503903, 0.383421, 1.0, 1.0,
1.0, 1.0, 1.0]
prim_factor = [ 1.0006253235944540e+01, 2.4169531573445120e+00, 7.9610924849766440e-01
3.0734305383061117e-01, 1.2929684417481876e-01, 3.0734305383061117e-01,
1.2929684417481876e-01, 2.1842769845268308e+00, 4.3649547399719840e-01,
1.8135965626177861e+00, 1.0006253235944540e+01, 2.4169531573445120e+00,
7.9610924849766440e-01, 3.0734305383061117e-01, 1.2929684417481876e-01,
3.0734305383061117e-01, 1.2929684417481876e-01, 2.1842769845268308e+00,
4.3649547399719840e-01, 1.8135965626177861e+00 ]

Data structure

typedef struct qmckl_ao_basis_struct {
int64_t   shell_num;
int64_t   prim_num;
int64_t   ao_num;
int64_t * nucleus_index;
int64_t * nucleus_shell_num;
int32_t * shell_ang_mom;
int64_t * shell_prim_num;
int64_t * shell_prim_index;
double  * shell_factor;
double  * exponent;
double  * coefficient;
double  * prim_factor;
double  * ao_factor;

int64_t * nucleus_prim_index;
double  * coefficient_normalized;
int32_t * nucleus_max_ang_mom;
double  * nucleus_range;
double  * primitive_vgl;
int64_t   primitive_vgl_date;
double  * shell_vgl;
int64_t   shell_vgl_date;
double  * ao_vgl;
int64_t   ao_vgl_date;

int32_t   uninitialized;
bool      provided;
bool      ao_cartesian;
char      type;
} qmckl_ao_basis_struct;

The uninitialized integer contains one bit set to one for each initialization function which has not been called. It becomes equal to zero after all initialization functions have been called. The struct is then initialized and provided == true. Some values are initialized by default, and are not concerned by this mechanism.

qmckl_exit_code qmckl_init_ao_basis(qmckl_context context);

qmckl_exit_code qmckl_init_ao_basis(qmckl_context context) {

if (qmckl_context_check(context) == QMCKL_NULL_CONTEXT) {
 return qmckl_failwith( context,
                        QMCKL_INVALID_CONTEXT,
                        "qmckl_init_ao_basis",
                        NULL);
}

qmckl_context_struct* const ctx = (qmckl_context_struct* const) context;
assert (ctx != NULL);

ctx->ao_basis.uninitialized = (1 << 14) - 1;

/* Default values */
ctx->ao_basis.ao_cartesian = true;

return QMCKL_SUCCESS;
}

Access functions

In the following functions, when an array is passed as an argument the size of the array should be also passed to check that the array is large enough to accept the data.

#+NAME:post

When all the data for the AOs have been provided, the following function returns true.

bool qmckl_ao_basis_provided (const qmckl_context context);

Initialization functions

To set the basis set, all the following functions need to be called.

qmckl_exit_code  qmckl_set_ao_basis_type             (qmckl_context context, const char      t);
qmckl_exit_code  qmckl_set_ao_basis_shell_num        (qmckl_context context, const int64_t   shell_num);
qmckl_exit_code  qmckl_set_ao_basis_prim_num         (qmckl_context context, const int64_t   prim_num);
qmckl_exit_code  qmckl_set_ao_basis_ao_num           (qmckl_context context, const int64_t   ao_num);
qmckl_exit_code  qmckl_set_ao_basis_nucleus_index    (qmckl_context context, const int64_t * nucleus_index);
qmckl_exit_code  qmckl_set_ao_basis_nucleus_shell_num(qmckl_context context, const int64_t * nucleus_shell_num);
qmckl_exit_code  qmckl_set_ao_basis_shell_ang_mom    (qmckl_context context, const int32_t * shell_ang_mom);
qmckl_exit_code  qmckl_set_ao_basis_shell_prim_num   (qmckl_context context, const int64_t * shell_prim_num);
qmckl_exit_code  qmckl_set_ao_basis_shell_prim_index (qmckl_context context, const int64_t * shell_prim_index);
qmckl_exit_code  qmckl_set_ao_basis_shell_factor     (qmckl_context context, const double  * shell_factor);
qmckl_exit_code  qmckl_set_ao_basis_exponent         (qmckl_context context, const double  * exponent);
qmckl_exit_code  qmckl_set_ao_basis_coefficient      (qmckl_context context, const double  * coefficient);
qmckl_exit_code  qmckl_set_ao_basis_prim_factor      (qmckl_context context, const double  * prim_factor);
qmckl_exit_code  qmckl_set_ao_basis_ao_factor        (qmckl_context context, const double  * ao_factor);
qmckl_exit_code  qmckl_set_ao_basis_cartesian        (qmckl_context context, const bool      cartesian);

#+NAME:pre2

#+NAME:post2

When the basis set is completely entered, other data structures are computed to accelerate the calculations. The primitives within each contraction are sorted in ascending order of their exponents, such that as soon as a primitive is zero all the following functions vanish. Also, it is possible to compute a nuclear radius beyond which all the primitives are zero up to the numerical accuracy defined in the context.

Fortran interfaces

Radial part

TODO Helper functions to accelerate calculations

General functions for Gaussian basis functions

qmckl_ao_gaussian_vgl computes the values, gradients and Laplacians at a given point of n Gaussian functions centered at the same point:

\[ v_i = \exp(-a_i |X-R|^2) \] \[ \nabla_x v_i = -2 a_i (X_x - R_x) v_i \] \[ \nabla_y v_i = -2 a_i (X_y - R_y) v_i \] \[ \nabla_z v_i = -2 a_i (X_z - R_z) v_i \] \[ \Delta v_i = a_i (4 |X-R|^2 a_i - 6) v_i \]

`context`	input	Global state
`X(3)`	input	Array containing the coordinates of the points
`R(3)`	input	Array containing the x,y,z coordinates of the center
`n`	input	Number of computed Gaussians
`A(n)`	input	Exponents of the Gaussians
`VGL(ldv,5)`	output	Value, gradients and Laplacian of the Gaussians
`ldv`	input	Leading dimension of array `VGL`

Requirements

context is not 0
n > 0
ldv >= 5
A(i) > 0 for all i
X is allocated with at least $3 \times 8$ bytes
R is allocated with at least $3 \times 8$ bytes
A is allocated with at least $n \times 8$ bytes
VGL is allocated with at least $n \times 5 \times 8$ bytes

qmckl_exit_code
qmckl_ao_gaussian_vgl(const qmckl_context context,
                 const double *X,
                 const double *R,
                 const int64_t *n,
                 const int64_t *A,
                 const double *VGL,
                 const int64_t ldv);

integer function qmckl_ao_gaussian_vgl_f(context, X, R, n, A, VGL, ldv) result(info)
use qmckl
implicit none
integer*8 , intent(in)  :: context
real*8    , intent(in)  :: X(3), R(3)
integer*8 , intent(in)  :: n
real*8    , intent(in)  :: A(n)
real*8    , intent(out) :: VGL(ldv,5)
integer*8 , intent(in)  :: ldv

integer*8         :: i,j
real*8            :: Y(3), r2, t, u, v

info = QMCKL_SUCCESS

if (context == QMCKL_NULL_CONTEXT) then
info = QMCKL_INVALID_CONTEXT
return
endif

if (n <= 0) then
info = QMCKL_INVALID_ARG_4
return
endif

if (ldv < n) then
info = QMCKL_INVALID_ARG_7
return
endif


do i=1,3
Y(i) = X(i) - R(i)
end do
r2 = Y(1)*Y(1) + Y(2)*Y(2) + Y(3)*Y(3)

do i=1,n
VGL(i,1) = dexp(-A(i) * r2)
end do

do i=1,n
VGL(i,5) = A(i) * VGL(i,1)
end do

t = -2.d0 * ( X(1) - R(1) )
u = -2.d0 * ( X(2) - R(2) )
v = -2.d0 * ( X(3) - R(3) )

do i=1,n
VGL(i,2) = t * VGL(i,5)
VGL(i,3) = u * VGL(i,5)
VGL(i,4) = v * VGL(i,5)
end do

t = 4.d0 * r2
do i=1,n
VGL(i,5) = (t * A(i) - 6.d0) *  VGL(i,5)
end do

end function qmckl_ao_gaussian_vgl_f

integer(c_int32_t) function test_qmckl_ao_gaussian_vgl(context) bind(C)
use qmckl
implicit none

integer(c_int64_t), intent(in), value :: context

integer*8                     :: n, ldv, j, i
double precision              :: X(3), R(3), Y(3), r2
double precision, allocatable :: VGL(:,:), A(:)
double precision              :: epsilon

epsilon = qmckl_get_numprec_epsilon(context)

X = (/ 1.1 , 2.2 ,  3.3 /)
R = (/ 0.1 , 1.2 , -2.3 /)
Y(:) = X(:) - R(:)
r2 = Y(1)**2 + Y(2)**2 + Y(3)**2

n = 10;
ldv = 100;

allocate (A(n), VGL(ldv,5))
do i=1,n
  A(i) = 0.0013 * dble(ishft(1,i))
end do


test_qmckl_ao_gaussian_vgl = &
    qmckl_ao_gaussian_vgl(context, X, R, n, A, VGL, ldv)
if (test_qmckl_ao_gaussian_vgl /= 0) return

test_qmckl_ao_gaussian_vgl = -1

do i=1,n
  test_qmckl_ao_gaussian_vgl = -11
  if (dabs(1.d0 - VGL(i,1) / (&
       dexp(-A(i) * r2) &
       )) > epsilon ) return

  test_qmckl_ao_gaussian_vgl = -12
  if (dabs(1.d0 - VGL(i,2) / (&
       -2.d0 * A(i) * Y(1) * dexp(-A(i) * r2) &
       )) > epsilon ) return

  test_qmckl_ao_gaussian_vgl = -13
  if (dabs(1.d0 - VGL(i,3) / (&
       -2.d0 * A(i) * Y(2) * dexp(-A(i) * r2) &
       )) > epsilon ) return

  test_qmckl_ao_gaussian_vgl = -14
  if (dabs(1.d0 - VGL(i,4) / (&
       -2.d0 * A(i) * Y(3) * dexp(-A(i) * r2) &
       )) > epsilon ) return

  test_qmckl_ao_gaussian_vgl = -15
  if (dabs(1.d0 - VGL(i,5) / (&
       A(i) * (4.d0*r2*A(i) - 6.d0) * dexp(-A(i) * r2) &
       )) > epsilon ) return
end do

test_qmckl_ao_gaussian_vgl = 0

deallocate(VGL)
end function test_qmckl_ao_gaussian_vgl

TODO General functions for Slater basis functions

TODO General functions for Radial functions on a grid

Computation of primitives

Get

qmckl_exit_code qmckl_get_ao_basis_primitive_vgl(qmckl_context context, double* const primitive_vgl);

Provide

Compute

qmckl_context	context	in	Global state
int64_t	prim_num	in	Number of primitives
int64_t	elec_num	in	Number of electrons
int64_t	nucl_num	in	Number of nuclei
int64_t	nucleus_prim_index[nucl_num]	in	Index of the 1st primitive of each nucleus
double	elec_coord[3][elec_num]	in	Electron coordinates
double	nucl_coord[3][elec_num]	in	Nuclear coordinates
double	expo[prim_num]	in	Exponents of the primitives
double	primitive_vgl[5][elec_num][prim_num]	out	Value, gradients and Laplacian of the primitives

integer function qmckl_compute_ao_basis_primitive_gaussian_vgl_f(context, &
 prim_num, elec_num, nucl_num, &
 nucleus_prim_index, elec_coord, nucl_coord, expo, primitive_vgl) &
 result(info)
use qmckl
implicit none
integer(qmckl_context), intent(in)  :: context
integer*8             , intent(in)  :: prim_num
integer*8             , intent(in)  :: nucl_num
integer*8             , intent(in)  :: elec_num
integer*8             , intent(in)  :: nucleus_prim_index(nucl_num+1)
double precision      , intent(in)  :: elec_coord(elec_num,3)
double precision      , intent(in)  :: nucl_coord(nucl_num,3)
double precision      , intent(in)  :: expo(prim_num)
double precision      , intent(out) :: primitive_vgl(prim_num,elec_num,5)

integer*8 :: inucl, iprim, ielec
double precision :: x, y, z, two_a, ar2, r2, v, cutoff

info = QMCKL_SUCCESS

! Don't compute exponentials when the result will be almost zero.
cutoff = -dlog(1.d-15)

do inucl=1,nucl_num
 ! C is zero-based, so shift bounds by one
 do iprim = nucleus_prim_index(inucl)+1, nucleus_prim_index(inucl+1)
    do ielec = 1, elec_num
       x = elec_coord(ielec,1) - nucl_coord(inucl,1)
       y = elec_coord(ielec,2) - nucl_coord(inucl,2)
       z = elec_coord(ielec,3) - nucl_coord(inucl,3)

       r2 = x*x + y*y + z*z
       ar2 = expo(iprim)*r2
       if (ar2 > cutoff) cycle

       v = dexp(-ar2)
       two_a = -2.d0 * expo(iprim) * v

       primitive_vgl(iprim, ielec, 1) = v
       primitive_vgl(iprim, ielec, 2) = two_a * x
       primitive_vgl(iprim, ielec, 3) = two_a * y
       primitive_vgl(iprim, ielec, 4) = two_a * z
       primitive_vgl(iprim, ielec, 5) = two_a * (3.d0 - 2.d0*ar2)

    end do
 end do
end do

end function qmckl_compute_ao_basis_primitive_gaussian_vgl_f

Test

Ideas for improvement

// j : electrons
// l : primitives

k=0;
for (j=0 ; j<elec_num ; ++j) {
for (i=0 ; i<nucl_num ; ++i) {

r2 = nucl_elec_dist[i][j];

if (r2 < nucl_radius2[i]) {

  for (l=0 ; l<prim_num ; ++l) {
    tmp[k].i   = i;
    tmp[k].j   = j;
    tmp[k].m   = m;
    tmp[k].ar2 = -expo[l] *r2;
    ++k;
  }
}
}
}

// sort(tmp) in increasing ar2;
// Identify first ar2 above numerical accuracy threshold
// Compute vectorized exponentials on significant values

Computation of shells

Get

qmckl_exit_code qmckl_get_ao_basis_shell_vgl(qmckl_context context, double* const shell_vgl);

Provide

Compute

`qmckl_context`	`context`	in	Global state
`int64_t`	`prim_num`	in	Number of primitives
`int64_t`	`shell_num`	in	Number of shells
`int64_t`	`elec_num`	in	Number of electrons
`int64_t`	`nucl_num`	in	Number of nuclei
`int64_t`	`nucleus_shell_num[nucl_num]`	in	Number of shells for each nucleus
`int64_t`	`nucleus_index[nucl_num]`	in	Index of the 1st shell of each nucleus
`int64_t`	`shell_prim_index[shell_num]`	in	Index of the 1st primitive of each shell
`int64_t`	`shell_prim_num[shell_num]`	in	Number of primitives per shell
`double`	`elec_coord[3][elec_num]`	in	Electron coordinates
`double`	`nucl_coord[3][elec_num]`	in	Nuclear coordinates
`double`	`expo[prim_num]`	in	Exponents of the primitives
`double`	`coef_normalized[prim_num]`	in	Coefficients of the primitives
`double`	`shell_vgl[5][elec_num][shell_num]`	out	Value, gradients and Laplacian of the shells

integer function qmckl_compute_ao_basis_shell_gaussian_vgl_f(context, &
 prim_num, shell_num, elec_num, nucl_num, &
 nucleus_shell_num, nucleus_index, shell_prim_index, shell_prim_num, &
 elec_coord, nucl_coord, expo, coef_normalized, shell_vgl) &
 result(info)
use qmckl
implicit none
integer(qmckl_context), intent(in)  :: context
integer*8             , intent(in)  :: prim_num
integer*8             , intent(in)  :: shell_num
integer*8             , intent(in)  :: nucl_num
integer*8             , intent(in)  :: elec_num
integer*8             , intent(in)  :: nucleus_shell_num(nucl_num)
integer*8             , intent(in)  :: nucleus_index(nucl_num)
integer*8             , intent(in)  :: shell_prim_index(shell_num)
integer*8             , intent(in)  :: shell_prim_num(shell_num)
double precision      , intent(in)  :: elec_coord(elec_num,3)
double precision      , intent(in)  :: nucl_coord(nucl_num,3)
double precision      , intent(in)  :: expo(prim_num)
double precision      , intent(in)  :: coef_normalized(prim_num)
double precision      , intent(out) :: shell_vgl(shell_num,elec_num,5)

integer*8 :: inucl, iprim, ielec, ishell
double precision :: x, y, z, two_a, ar2, r2, v, cutoff

info = QMCKL_SUCCESS

! Don't compute exponentials when the result will be almost zero.
! TODO : Use numerical precision here
cutoff = -dlog(1.d-15)

do inucl=1,nucl_num

 do ielec = 1, elec_num

    x = elec_coord(ielec,1) - nucl_coord(inucl,1)
    y = elec_coord(ielec,2) - nucl_coord(inucl,2)
    z = elec_coord(ielec,3) - nucl_coord(inucl,3)

    r2 = x*x + y*y + z*z

    do ishell=nucleus_index(inucl)+1, nucleus_index(inucl)+nucleus_shell_num(inucl)
       ! C is zero-based, so shift bounds by one

       shell_vgl(ishell, ielec, 1) = 0.d0
       shell_vgl(ishell, ielec, 2) = 0.d0
       shell_vgl(ishell, ielec, 3) = 0.d0
       shell_vgl(ishell, ielec, 4) = 0.d0
       shell_vgl(ishell, ielec, 5) = 0.d0

       do iprim = shell_prim_index(ishell)+1, shell_prim_index(ishell)+shell_prim_num(ishell)

          ar2 = expo(iprim)*r2
          if (ar2 > cutoff) then
             cycle
          end if

          v = coef_normalized(iprim) * dexp(-ar2)
          two_a = -2.d0 * expo(iprim) * v

          shell_vgl(ishell, ielec, 1) = &
               shell_vgl(ishell, ielec, 1) + v

          shell_vgl(ishell, ielec, 2) = &
               shell_vgl(ishell, ielec, 2) + two_a * x

          shell_vgl(ishell, ielec, 3) = &
               shell_vgl(ishell, ielec, 3) + two_a * y

          shell_vgl(ishell, ielec, 4) = &
               shell_vgl(ishell, ielec, 4) + two_a * z

          shell_vgl(ishell, ielec, 5) = &
               shell_vgl(ishell, ielec, 5) + two_a * (3.d0 - 2.d0*ar2)

       end do

    end do
 end do

end do

end function qmckl_compute_ao_basis_shell_gaussian_vgl_f

qmckl_exit_code qmckl_compute_ao_basis_shell_gaussian_vgl (
      const qmckl_context context,
      const int64_t prim_num,
      const int64_t shell_num,
      const int64_t elec_num,
      const int64_t nucl_num,
      const int64_t* nucleus_shell_num,
      const int64_t* nucleus_index,
      const int64_t* shell_prim_index,
      const int64_t* shell_prim_num,
      const double* elec_coord,
      const double* nucl_coord,
      const double* expo,
      const double* coef_normalized,
      double* const shell_vgl );

Test

Polynomial part

Going from the atomic basis set to AOs implies a systematic construction of all the angular functions of each shell. We consider two cases for the angular functions: the real-valued spherical harmonics, and the polynomials in Cartesian coordinates. In the case of spherical harmonics, the AOs are ordered in increasing magnetic quantum number ($-l \le m \le l$), and in the case of polynomials we choose the canonical ordering, i.e

\begin{eqnarray} p & : & p_x, p_y, p_z \nonumber \\ d & : & d_{xx}, d_{xy}, d_{xz}, d_{yy}, d_{yz}, d_{zz} \nonumber \\ f & : & f_{xxx}, f_{xxy}, f_{xxz}, f_{xyy}, f_{xyz}, f_{xzz}, f_{yyy}, f_{yyz}, f_{yzz}, f_{zzz} \nonumber \\ {\rm etc.} \nonumber \end{eqnarray}

General functions for Powers of $x-X_i$

The qmckl_ao_power function computes all the powers of the n input data up to the given maximum value given in input for each of the $n$ points:

\[ P_{ik} = X_i^k \]

qmckl_context	context	in	Global state
int64_t	n	in	Number of values
double	X[n]	in	Array containing the input values
int32_t	LMAX[n]	in	Array containing the maximum power for each value
double	P[n][ldp]	out	Array containing all the powers of `X`
int64_t	ldp	in	Leading dimension of array `P`

Requirements

context is not QMCKL_NULL_CONTEXT
n > 0
X is allocated with at least $n \times 8$ bytes
LMAX is allocated with at least $n \times 4$ bytes
P is allocated with at least $n \times \max_i \text{LMAX}_i \times 8$ bytes
LDP >= $\max_i$ LMAX[i]

C Header

qmckl_exit_code qmckl_ao_power (
      const qmckl_context context,
      const int64_t n,
      const double* X,
      const int32_t* LMAX,
      double* const P,
      const int64_t ldp );

Source

integer function qmckl_ao_power_f(context, n, X, LMAX, P, ldp) result(info)
use qmckl
implicit none
integer*8 , intent(in)  :: context
integer*8 , intent(in)  :: n
real*8    , intent(in)  :: X(n)
integer   , intent(in)  :: LMAX(n)
real*8    , intent(out) :: P(ldp,n)
integer*8 , intent(in)  :: ldp

integer*8  :: i,k

info = QMCKL_SUCCESS

if (context == QMCKL_NULL_CONTEXT) then
 info = QMCKL_INVALID_CONTEXT
 return
endif

if (n <= ldp) then
 info = QMCKL_INVALID_ARG_2
 return
endif

k = MAXVAL(LMAX)
if (LDP < k) then
 info = QMCKL_INVALID_ARG_6
 return
endif

if (k <= 0) then
 info = QMCKL_INVALID_ARG_4
 return
endif

do i=1,n
 P(1,i) = X(i)
 do k=2,LMAX(i)
    P(k,i) = P(k-1,i) * X(i)
 end do
end do

end function qmckl_ao_power_f

C interface

Fortran interface

Test

integer(c_int32_t) function test_qmckl_ao_power(context) bind(C)
use qmckl
implicit none

integer(qmckl_context), intent(in), value :: context

integer*8                     :: n, LDP
integer, allocatable          :: LMAX(:)
double precision, allocatable :: X(:), P(:,:)
integer*8                     :: i,j
double precision              :: epsilon

epsilon = qmckl_get_numprec_epsilon(context)

n = 100;
LDP = 10;

allocate(X(n), P(LDP,n), LMAX(n))

do j=1,n
 X(j) = -5.d0 + 0.1d0 * dble(j)
 LMAX(j) = 1 + int(mod(j, 5),4)
end do

test_qmckl_ao_power = qmckl_ao_power(context, n, X, LMAX, P, LDP)
if (test_qmckl_ao_power /= QMCKL_SUCCESS) return

test_qmckl_ao_power = QMCKL_FAILURE

do j=1,n
 do i=1,LMAX(j)
    if ( X(j)**i == 0.d0 ) then
       if ( P(i,j) /= 0.d0) return
    else
       if ( dabs(1.d0 - P(i,j) / (X(j)**i)) > epsilon ) return
    end if
 end do
end do

test_qmckl_ao_power = QMCKL_SUCCESS
deallocate(X,P,LMAX)
end function test_qmckl_ao_power

General functions for Value, Gradient and Laplacian of a polynomial

A polynomial is centered on a nucleus $\mathbf{R}_i$

\[ P_l(\mathbf{r},\mathbf{R}_i) = (x-X_i)^a (y-Y_i)^b (z-Z_i)^c \]

The gradients with respect to electron coordinates are

\begin{eqnarray*} \frac{\partial }{\partial x} P_l\left(\mathbf{r},\mathbf{R}_i \right) & = & a (x-X_i)^{a-1} (y-Y_i)^b (z-Z_i)^c \\ \frac{\partial }{\partial y} P_l\left(\mathbf{r},\mathbf{R}_i \right) & = & b (x-X_i)^a (y-Y_i)^{b-1} (z-Z_i)^c \\ \frac{\partial }{\partial z} P_l\left(\mathbf{r},\mathbf{R}_i \right) & = & c (x-X_i)^a (y-Y_i)^b (z-Z_i)^{c-1} \\ \end{eqnarray*}

and the Laplacian is

\begin{eqnarray*} \left( \frac{\partial }{\partial x^2} + \frac{\partial }{\partial y^2} + \frac{\partial }{\partial z^2} \right) P_l \left(\mathbf{r},\mathbf{R}_i \right) & = & a(a-1) (x-X_i)^{a-2} (y-Y_i)^b (z-Z_i)^c + \\ && b(b-1) (x-X_i)^a (y-Y_i)^{b-1} (z-Z_i)^c + \\ && c(c-1) (x-X_i)^a (y-Y_i)^b (z-Z_i)^{c-1}. \end{eqnarray*}

qmckl_ao_polynomial_vgl computes the values, gradients and Laplacians at a given point in space, of all polynomials with an angular momentum up to lmax.

qmckl_context	context	in	Global state
double	X[3]	in	Array containing the coordinates of the points
double	R[3]	in	Array containing the x,y,z coordinates of the center
int32_t	lmax	in	Maximum angular momentum
int64_t	n	inout	Number of computed polynomials
int32_t	L[n][ldl]	out	Contains a,b,c for all `n` results
int64_t	ldl	in	Leading dimension of `L`
double	VGL[n][ldv]	out	Value, gradients and Laplacian of the polynomials
int64_t	ldv	in	Leading dimension of array `VGL`

Requirements

context is not QMCKL_NULL_CONTEXT
n > 0
lmax >= 0
ldl >= 3
ldv >= 5
X is allocated with at least $3 \times 8$ bytes
R is allocated with at least $3 \times 8$ bytes
n >= (lmax+1)(lmax+2)(lmax+3)/6
L is allocated with at least $3 \times n \times 4$ bytes
VGL is allocated with at least $5 \times n \times 8$ bytes
On output, n should be equal to (lmax+1)(lmax+2)(lmax+3)/6
On output, the powers are given in the following order (l=a+b+c):
- Increasing values of l
- Within a given value of l, alphabetical order of the string made by a*"x" + b*"y" + c*"z" (in Python notation). For example, with a=0, b=2 and c=1 the string is "yyz"

C Header

qmckl_exit_code qmckl_ao_polynomial_vgl (
      const qmckl_context context,
      const double* X,
      const double* R,
      const int32_t lmax,
      int64_t* n,
      int32_t* const L,
      const int64_t ldl,
      double* const VGL,
      const int64_t ldv );

Source

integer function qmckl_ao_polynomial_vgl_f(context, X, R, lmax, n, L, ldl, VGL, ldv) result(info)
use qmckl
implicit none
integer*8 , intent(in)  :: context
real*8    , intent(in)  :: X(3), R(3)
integer   , intent(in)  :: lmax
integer*8 , intent(out) :: n
integer   , intent(out) :: L(ldl,(lmax+1)*(lmax+2)*(lmax+3)/6)
integer*8 , intent(in)  :: ldl
real*8    , intent(out) :: VGL(ldv,(lmax+1)*(lmax+2)*(lmax+3)/6)
integer*8 , intent(in)  :: ldv

integer*8         :: i,j
integer           :: a,b,c,d
real*8            :: Y(3)
integer           :: lmax_array(3)
real*8            :: pows(-2:lmax,3)
double precision  :: xy, yz, xz
double precision  :: da, db, dc, dd

info = 0

if (context == QMCKL_NULL_CONTEXT) then
 info = QMCKL_INVALID_CONTEXT
 return
endif

if (lmax < 0) then
 info = QMCKL_INVALID_ARG_4
 return
endif

if (ldl < 3) then
 info = QMCKL_INVALID_ARG_7
 return
endif

if (ldv < 5) then
 info = QMCKL_INVALID_ARG_9
 return
endif


do i=1,3
 Y(i) = X(i) - R(i)
end do

lmax_array(1:3) = lmax
if (lmax == 0) then
 VGL(1,1) = 1.d0
 vgL(2:5,1) = 0.d0
 l(1:3,1) = 0
 n=1
else if (lmax > 0) then
 pows(-2:0,1:3) = 1.d0
 do i=1,lmax
    pows(i,1) = pows(i-1,1) * Y(1)
    pows(i,2) = pows(i-1,2) * Y(2)
    pows(i,3) = pows(i-1,3) * Y(3)
 end do

 VGL(1:5,1:4) = 0.d0
 l  (1:3,1:4) = 0

 VGL(1  ,1  ) = 1.d0
 vgl(1:5,2:4) = 0.d0

 l  (1,2) = 1
 vgl(1,2) = pows(1,1)
 vgL(2,2) = 1.d0

 l  (2,3) = 1
 vgl(1,3) = pows(1,2)
 vgL(3,3) = 1.d0

 l  (3,4) = 1
 vgl(1,4) = pows(1,3)
 vgL(4,4) = 1.d0

 n=4
endif

! l>=2
dd = 2.d0
do d=2,lmax
 da = dd
 do a=d,0,-1
    db = dd-da
    do b=d-a,0,-1
       c  = d  - a  - b
       dc = dd - da - db
       n = n+1

       l(1,n) = a
       l(2,n) = b
       l(3,n) = c

       xy = pows(a,1) * pows(b,2)
       yz = pows(b,2) * pows(c,3)
       xz = pows(a,1) * pows(c,3)

       vgl(1,n) = xy * pows(c,3)

       xy = dc * xy
       xz = db * xz
       yz = da * yz

       vgl(2,n) = pows(a-1,1) * yz
       vgl(3,n) = pows(b-1,2) * xz
       vgl(4,n) = pows(c-1,3) * xy

       vgl(5,n) = &
            (da-1.d0) * pows(a-2,1) * yz + &
            (db-1.d0) * pows(b-2,2) * xz + &
            (dc-1.d0) * pows(c-2,3) * xy

       db = db - 1.d0
    end do
    da = da - 1.d0
 end do
 dd = dd + 1.d0
end do

info = QMCKL_SUCCESS

end function qmckl_ao_polynomial_vgl_f

C interface

Fortran interface

Test

integer(c_int32_t) function test_qmckl_ao_polynomial_vgl(context) bind(C)
use qmckl
implicit none

integer(c_int64_t), intent(in), value :: context

integer                       :: lmax, d, i
integer, allocatable          :: L(:,:)
integer*8                     :: n, ldl, ldv, j
double precision              :: X(3), R(3), Y(3)
double precision, allocatable :: VGL(:,:)
double precision              :: w
double precision              :: epsilon

epsilon = qmckl_get_numprec_epsilon(context)

X = (/ 1.1 , 2.2 ,  3.3 /)
R = (/ 0.1 , 1.2 , -2.3 /)
Y(:) = X(:) - R(:)

lmax = 4;
ldl = 3;
ldv = 100;

d = (lmax+1)*(lmax+2)*(lmax+3)/6

allocate (L(ldl,d), VGL(ldv,d))

test_qmckl_ao_polynomial_vgl = &
   qmckl_ao_polynomial_vgl(context, X, R, lmax, n, L, ldl, VGL, ldv)

if (test_qmckl_ao_polynomial_vgl /= QMCKL_SUCCESS) return
if (n /= d) return

do j=1,n
 test_qmckl_ao_polynomial_vgl = QMCKL_FAILURE
 do i=1,3
    if (L(i,j) < 0) return
 end do
 test_qmckl_ao_polynomial_vgl = QMCKL_FAILURE
 if (dabs(1.d0 - VGL(1,j) / (&
      Y(1)**L(1,j) * Y(2)**L(2,j) * Y(3)**L(3,j)  &
      )) > epsilon ) return

 test_qmckl_ao_polynomial_vgl = QMCKL_FAILURE
 if (L(1,j) < 1) then
    if (VGL(2,j) /= 0.d0) return
 else
    if (dabs(1.d0 - VGL(2,j) / (&
         L(1,j) * Y(1)**(L(1,j)-1) * Y(2)**L(2,j) * Y(3)**L(3,j) &
         )) > epsilon ) return
 end if

 test_qmckl_ao_polynomial_vgl = QMCKL_FAILURE
 if (L(2,j) < 1) then
    if (VGL(3,j) /= 0.d0) return
 else
    if (dabs(1.d0 - VGL(3,j) / (&
         L(2,j) * Y(1)**L(1,j) * Y(2)**(L(2,j)-1) * Y(3)**L(3,j) &
         )) > epsilon ) return
 end if

 test_qmckl_ao_polynomial_vgl = QMCKL_FAILURE
 if (L(3,j) < 1) then
    if (VGL(4,j) /= 0.d0) return
 else
    if (dabs(1.d0 - VGL(4,j) / (&
         L(3,j) * Y(1)**L(1,j) * Y(2)**L(2,j) * Y(3)**(L(3,j)-1) &
         )) > epsilon ) return
 end if

 test_qmckl_ao_polynomial_vgl = QMCKL_FAILURE
 w = 0.d0
 if (L(1,j) > 1) then
    w = w + L(1,j) * (L(1,j)-1) * Y(1)**(L(1,j)-2) * Y(2)**L(2,j) * Y(3)**L(3,j)
 end if
 if (L(2,j) > 1) then
    w = w + L(2,j) * (L(2,j)-1) * Y(1)**L(1,j) * Y(2)**(L(2,j)-2) * Y(3)**L(3,j)
 end if
 if (L(3,j) > 1) then
    w = w + L(3,j) * (L(3,j)-1) * Y(1)**L(1,j) * Y(2)**L(2,j) * Y(3)**(L(3,j)-2)
 end if
 if (dabs(1.d0 - VGL(5,j) / w) > epsilon ) return
end do

test_qmckl_ao_polynomial_vgl = QMCKL_SUCCESS

deallocate(L,VGL)
end function test_qmckl_ao_polynomial_vgl

int  test_qmckl_ao_polynomial_vgl(qmckl_context context);
assert(0 == test_qmckl_ao_polynomial_vgl(context));

Combining radial and polynomial parts

Get

qmckl_exit_code qmckl_get_ao_vgl(qmckl_context context, double* const ao_vgl);

Provide

Compute

`qmckl_context`	`context`	in	Global state
`int64_t`	`ao_num`	in	Number of AOs
`int64_t`	`shell_num`	in	Number of shells
`int64_t`	`elec_num`	in	Number of electrons
`int64_t`	`nucl_num`	in	Number of nuclei
`double`	`elec_coord[3][elec_num]`	in	Electron coordinates
`double`	`nucl_coord[3][nucl_num]`	in	Nuclear coordinates
`int64_t`	`nucleus_index[nucl_num]`	in	Index of the 1st shell of each nucleus
`int64_t`	`nucleus_shell_num[nucl_num]`	in	Number of shells per nucleus
`double`	`nucleus_range[nucl_num]`	in	Range beyond which all is zero
`int32_t`	`nucleus_max_ang_mom[nucl_num]`	in	Maximum angular momentum per nucleus
`int32_t`	`shell_ang_mom[shell_num]`	in	Angular momentum of each shell
`double`	`ao_factor[ao_num]`	in	Normalization factor of the AOs
`double`	`shell_vgl[5][elec_num][shell_num]`	in	Value, gradients and Laplacian of the shells
`double`	`ao_vgl[5][elec_num][ao_num]`	out	Value, gradients and Laplacian of the AOs

integer function qmckl_compute_ao_vgl_f(context, &
 ao_num, shell_num, elec_num, nucl_num, &
 elec_coord, nucl_coord, nucleus_index, nucleus_shell_num, &
 nucleus_range, nucleus_max_ang_mom, shell_ang_mom, &
 ao_factor, shell_vgl, ao_vgl) &
 result(info)
use qmckl
implicit none
integer(qmckl_context), intent(in)  :: context
integer*8             , intent(in)  :: ao_num
integer*8             , intent(in)  :: shell_num
integer*8             , intent(in)  :: elec_num
integer*8             , intent(in)  :: nucl_num
double precision      , intent(in)  :: elec_coord(elec_num,3)
double precision      , intent(in)  :: nucl_coord(nucl_num,3)
integer*8             , intent(in)  :: nucleus_index(nucl_num)
integer*8             , intent(in)  :: nucleus_shell_num(nucl_num)
double precision      , intent(in)  :: nucleus_range(nucl_num)
integer               , intent(in)  :: nucleus_max_ang_mom(nucl_num)
integer               , intent(in)  :: shell_ang_mom(shell_num)
double precision      , intent(in)  :: ao_factor(ao_num)
double precision      , intent(in)  :: shell_vgl(shell_num,elec_num,5)
double precision      , intent(out) :: ao_vgl(ao_num,elec_num,5)

double precision  :: e_coord(3), n_coord(3)
integer*8         :: n_poly
integer           :: l, il, k
integer*8         :: ielec, inucl, ishell
integer           :: lstart(0:20)
double precision  :: x, y, z, r2
double precision  :: cutoff
integer, external :: qmckl_ao_polynomial_vgl_f

double precision, allocatable  :: poly_vgl(:,:)
integer         , allocatable  :: powers(:,:)

allocate(poly_vgl(5,ao_num), powers(3,ao_num))

! Pre-computed data
do l=0,20
 lstart(l) = l*(l+1)*(l+2)/6 +1
end do

info = QMCKL_SUCCESS

! Don't compute polynomials when the radial part is zero.
! TODO : Use numerical precision here
cutoff = -dlog(1.d-15)

do ielec = 1, elec_num
 e_coord(1) = elec_coord(ielec,1)
 e_coord(2) = elec_coord(ielec,2)
 e_coord(3) = elec_coord(ielec,3)
 k=1
 do inucl=1,nucl_num
    n_coord(1) = nucl_coord(inucl,1)
    n_coord(2) = nucl_coord(inucl,2)
    n_coord(3) = nucl_coord(inucl,3)

    ! Test if the electron is in the range of the nucleus
    x = e_coord(1) - n_coord(1)
    y = e_coord(2) - n_coord(2)
    z = e_coord(3) - n_coord(3)

    r2 = x*x + z*z + z*z

    if (r2 > cutoff*nucleus_range(inucl)) then
       cycle
    end if

    ! Compute polynomials
    info = qmckl_ao_polynomial_vgl_f(context, e_coord, n_coord, &
         nucleus_max_ang_mom(inucl), n_poly, powers, 3_8, &
         poly_vgl, 5_8)

    ! Loop over shells
    do ishell = nucleus_index(inucl)+1, nucleus_index(inucl)+nucleus_shell_num(inucl)
       l = shell_ang_mom(ishell)
       do il = lstart(l), lstart(l+1)-1
          ! Value
          ao_vgl(k,ielec,1) = &
               poly_vgl(1,il) * shell_vgl(ishell,ielec,1) * ao_factor(k)

          ! Grad_x
          ao_vgl(k,ielec,2) = ( &
               poly_vgl(2,il) * shell_vgl(ishell,ielec,1) + &
               poly_vgl(1,il) * shell_vgl(ishell,ielec,2) &
               ) * ao_factor(k)

          ! Grad_y
          ao_vgl(k,ielec,3) = ( &
               poly_vgl(3,il) * shell_vgl(ishell,ielec,1) + &
               poly_vgl(1,il) * shell_vgl(ishell,ielec,3) &
               ) * ao_factor(k)

          ! Grad_z
          ao_vgl(k,ielec,4) = ( &
               poly_vgl(4,il) * shell_vgl(ishell,ielec,1) + &
               poly_vgl(1,il) * shell_vgl(ishell,ielec,4) &
               ) * ao_factor(k)

          ! Lapl_z
          ao_vgl(k,ielec,5) = ( &
               poly_vgl(5,il) * shell_vgl(ishell,ielec,1) + &
               poly_vgl(1,il) * shell_vgl(ishell,ielec,5) + &
               2.d0 * ( &
               poly_vgl(2,il) * shell_vgl(ishell,ielec,2) + &
               poly_vgl(3,il) * shell_vgl(ishell,ielec,3) + &
               poly_vgl(4,il) * shell_vgl(ishell,ielec,4) ) &
               ) * ao_factor(k)

          k = k+1
       end do
    end do
 end do
end do

deallocate(poly_vgl, powers)
end function qmckl_compute_ao_vgl_f

qmckl_exit_code qmckl_compute_ao_vgl (
      const qmckl_context context,
      const int64_t ao_num,
      const int64_t shell_num,
      const int64_t elec_num,
      const int64_t nucl_num,
      const double* elec_coord,
      const double* nucl_coord,
      const int64_t* nucleus_index,
      const int64_t* nucleus_shell_num,
      const double* nucleus_range,
      const int32_t* nucleus_max_ang_mom,
      const int32_t* shell_ang_mom,
      const double* ao_factor,
      const double* shell_vgl,
      double* const ao_vgl );

142 KiB Raw Blame History

Atomic Orbitals

Context

Data structure

Access functions

Initialization functions

Fortran interfaces

Radial part

TODO Helper functions to accelerate calculations

General functions for Gaussian basis functions

TODO General functions for Slater basis functions

TODO General functions for Radial functions on a grid

Computation of primitives

Get

Provide

Compute

Test

Ideas for improvement

Computation of shells

Get

Provide

Compute

Test

Polynomial part

General functions for Powers of $x-X_i$

Requirements

C Header

Source

C interface

Fortran interface

Test

General functions for Value, Gradient and Laplacian of a polynomial

Requirements

C Header

Source

C interface

Fortran interface

Test

Combining radial and polynomial parts

Get

Provide

Compute

Test

142 KiB

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