Slater Determinant

1. Context
2. Computation
- 2.1. Determinant matrix
- 2.2. Inverse of Determinant matrix

1 Context

The following arrays are stored in the context:

`type`	`char`	α (`'A'`) or β (`'B'`) determinant
`det_num_alpha`	`int64_t`	Number of determinants per walker
`det_num_beta`	`int64_t`	Number of determinants per walker
`mo_index_alpha`	`mo_index[det_num_alpha][walker.num][alpha_num]`	Index of MOs for each walker
`mo_index_beta`	`mo_index[det_num_beta][walker.num][beta_num]`	Index of MOs for each walker

Computed data:

`up_num`	`int64_t`	Number of number of α electrons
`donwn_num`	`int64_t`	Number of number of β electrons
`det_value_alpha`	`[det_num_alpha][walker.num]`	The α slater matrix for each determinant of each walker.
`det_value_alpha_date`	`uint64_t`	Date of The α slater matrix for each determinant of each walker.
`det_value_beta`	`[det_num_beta][walker.num]`	The β slater matrix for each determinant of each walker.
`det_value_beta_date`	`uint64_t`	Date of The β slater matrix for each determinant of each walker.
`det_adj_matrix_alpha`	`[det_num_alpha][walker.num][alpha_num][alpha_num]`	Adjoint of the α slater matrix for each determinant of each walker.
`det_adj_matrix_alpha_date`	`uint64_t`	Date of the Adjoint of the α slater matrix for each determinant of each walker.
`det_adj_matrix_beta`	`[det_num_beta][walker.num][beta_num][beta_num]`	Adjoint of the β slater matrix for each determinant of each walker.
`det_adj_matrix_beta_date`	`uint64_t`	Date of the Adjoint of the β slater matrix for each determinant of each walker.
`det_vgl_alpha`	`[5][det_num_alpha][walker.num][alpha_num][alpha_num]`	Value, gradients, Laplacian of Dᵅᵢⱼ(x) at electron positions
`det_vgl_alpha_date`	`uint64_t`	Late modification date of Value, gradients, Laplacian of the MOs at electron positions
`det_vgl_beta`	`[5][det_num_beta][walker.num][beta_num][beta_num]`	Value, gradients, Laplacian of Dᵝᵢⱼ(x) at electron positions
`det_vgl_beta_date`	`uint64_t`	Late modification date of Value, gradients, Laplacian of the MOs at electron positions
`det_inv_matrix_alpha`	`[det_num_alpha][walker.num][alpha_num][alpha_num]`	Inverse of the α electron slater matrix for each determinant of each walker.
`det_inv_matrix_alpha_date`	`uint64_t`	Date for the Inverse of the α electron slater matrix for each determinant of each walker.
`det_inv_matrix_beta`	`[det_num_beta][walker.num][beta_num][beta_num]`	Inverse of the β electron slater matrix for each determinant of each walker.
`det_inv_matrix_beta_date`	`uint64_t`	Date for the Inverse of the β electron slater matrix for each determinant of each walker.

1.1 Data structure

typedef struct qmckl_determinant_struct {
  char     type;
  int64_t  det_num_alpha;
  int64_t  det_num_beta ;
  int64_t  up_num;
  int64_t  down_num;
  int64_t* mo_index_alpha;
  int64_t* mo_index_beta;

  double  * det_value_alpha;
  double  * det_value_beta;
  double  * det_vgl_alpha;
  double  * det_adj_matrix_alpha;
  double  * det_inv_matrix_alpha;
  double  * det_vgl_beta;
  double  * det_adj_matrix_beta;
  double  * det_inv_matrix_beta;
  uint64_t  det_value_alpha_date;
  uint64_t  det_vgl_alpha_date;
  uint64_t  det_adj_matrix_alpha_date;
  uint64_t  det_inv_matrix_alpha_date;
  uint64_t  det_value_beta_date;
  uint64_t  det_vgl_beta_date;
  uint64_t  det_adj_matrix_beta_date;
  uint64_t  det_inv_matrix_beta_date;

  int32_t   uninitialized;
  bool      provided;
} qmckl_determinant_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_determinant(qmckl_context context);

qmckl_exit_code qmckl_init_determinant(qmckl_context context) {

  if (qmckl_context_check(context) == QMCKL_NULL_CONTEXT) {
    return false;
  }

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

  ctx->det.uninitialized = (1 << 5) - 1;

  return QMCKL_SUCCESS;
}

1.2 Access functions

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

bool      qmckl_determinant_provided           (const qmckl_context context);

1.3 Initialization functions

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

qmckl_exit_code  qmckl_set_determinant_type             (const qmckl_context context, const char t);
qmckl_exit_code  qmckl_set_determinant_det_num_alpha    (const qmckl_context context, const int64_t det_num_alpha);
qmckl_exit_code  qmckl_set_determinant_det_num_beta     (const qmckl_context context, const int64_t det_num_beta);
qmckl_exit_code  qmckl_set_determinant_mo_index_alpha   (const qmckl_context context, const int64_t* mo_index_alpha, const int64_t size_max);
qmckl_exit_code  qmckl_set_determinant_mo_index_beta    (const qmckl_context context, const int64_t* mo_index_beta, const int64_t size_max);

When the basis set is completely entered, other data structures are computed to accelerate the calculations.

1.4 Fortran Interfaces

1.5 Test

2 Computation

2.1 Determinant matrix

2.1.1 Get

qmckl_exit_code qmckl_get_det_vgl_alpha(qmckl_context context, double* const det_vgl_alpha);
qmckl_exit_code qmckl_get_det_vgl_beta(qmckl_context context, double* const det_vgl_beta);

2.1.2 Provide

2.1.3 Compute alpha

Variable	Type	In/Out	Description
`context`	`qmckl_context`	in	Global state
`det_num_alpha`	`int64_t`	in	Number of determinants
`walk_num`	`int64_t`	in	Number of walkers
`alpha_num`	`int64_t`	in	Number of electrons
`beta_num`	`int64_t`	in	Number of electrons
`elec_num`	`int64_t`	in	Number of electrons
`mo_index_alpha`	`int64_t[det_num_alpha][walk_num][alpha_num]`	in	MO indices for electrons
`mo_num`	`int64_t`	in	Number of MOs
`mo_vgl`	`double[5][elec_num][mo_num]`	in	Value, gradients and Laplacian of the MOs
`det_vgl_alpha`	`double[det_num_alpha][walk_num][5][alpha_num][alpha_num]`	out	Value, gradients and Laplacian of the Det

integer function qmckl_compute_det_vgl_alpha_f(context, &
     det_num_alpha, walk_num, alpha_num, beta_num, elec_num, &
     mo_index_alpha, mo_num, mo_vgl, det_vgl_alpha) &
     result(info)
  use qmckl
  implicit none
  integer(qmckl_context)  , intent(in)  :: context
  integer*8, intent(in)             :: det_num_alpha
  integer*8, intent(in)             :: walk_num
  integer*8, intent(in)             :: alpha_num
  integer*8, intent(in)             :: beta_num
  integer*8, intent(in)             :: elec_num
  integer*8, intent(in)             :: mo_num
  integer*8, intent(in)             :: mo_index_alpha(alpha_num, walk_num, det_num_alpha)
  double precision, intent(in)      :: mo_vgl(mo_num, elec_num, 5)
  double precision, intent(inout)   :: det_vgl_alpha(alpha_num, alpha_num, 5, walk_num, det_num_alpha)
  integer*8 :: idet, iwalk, ielec, mo_id, imo

  info = QMCKL_SUCCESS

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

  if (walk_num <= 0) then
     info = QMCKL_INVALID_ARG_2
     return
  endif

  if (alpha_num <= 0) then
     info = QMCKL_INVALID_ARG_3
     return
  endif

  do idet  = 1, det_num_alpha
  do iwalk = 1, walk_num
    do ielec = 1, alpha_num
      do imo = 1, alpha_num
        mo_id = mo_index_alpha(imo,iwalk,idet)
        ! Value
        det_vgl_alpha(imo, ielec, 1, iwalk, idet) = mo_vgl(mo_id, ielec, 1)

        ! Grad_x
        det_vgl_alpha(imo, ielec, 2, iwalk, idet) = mo_vgl(mo_id, ielec, 2)

        ! Grad_y
        det_vgl_alpha(imo, ielec, 3, iwalk, idet) = mo_vgl(mo_id, ielec, 3)

        ! Grad_z
        det_vgl_alpha(imo, ielec, 4, iwalk, idet) = mo_vgl(mo_id, ielec, 4)

        ! Lap
        det_vgl_alpha(imo, ielec, 5, iwalk, idet) = mo_vgl(mo_id, ielec, 5)
      end do
    end do
  end do
  end do

end function qmckl_compute_det_vgl_alpha_f

qmckl_exit_code qmckl_compute_det_vgl_alpha (
      const qmckl_context context,
      const int64_t det_num_alpha,
      const int64_t walk_num,
      const int64_t alpha_num,
      const int64_t beta_num,
      const int64_t elec_num,
      const int64_t* mo_index_alpha,
      const int64_t mo_num,
      const double* mo_vgl,
      double* const det_vgl_alpha );

2.1.4 Compute beta

Variable	Type	In/Out	Description
`context`	`qmckl_context`	in	Global state
`det_num_beta`	`int64_t`	in	Number of determinants
`walk_num`	`int64_t`	in	Number of walkers
`alpha_num`	`int64_t`	in	Number of electrons
`beta_num`	`int64_t`	in	Number of electrons
`elec_num`	`int64_t`	in	Number of electrons
`mo_index_beta`	`int64_t[det_num_beta][walk_num][beta_num]`	in	Number of electrons
`mo_num`	`int64_t`	in	Number of MOs
`mo_vgl`	`double[5][elec_num][mo_num]`	in	Value, gradients and Laplacian of the MOs
`det_vgl_beta`	`double[det_num_beta][walk_num][5][beta_num][beta_num]`	out	Value, gradients and Laplacian of the Det

integer function qmckl_compute_det_vgl_beta_f(context, &
     det_num_beta, walk_num, alpha_num, beta_num, elec_num, &
     mo_index_beta, mo_num, mo_vgl, det_vgl_beta) &
     result(info)
  use qmckl
  implicit none
  integer(qmckl_context)  , intent(in)  :: context
  integer*8, intent(in)             :: det_num_beta
  integer*8, intent(in)             :: walk_num
  integer*8, intent(in)             :: alpha_num
  integer*8, intent(in)             :: beta_num
  integer*8, intent(in)             :: elec_num
  integer*8, intent(in)             :: mo_num
  integer*8, intent(in)             :: mo_index_beta(beta_num,walk_num,det_num_beta)
  double precision, intent(in)      :: mo_vgl(mo_num, elec_num, 5)
  double precision, intent(inout)   :: det_vgl_beta(beta_num, beta_num, 5, walk_num, det_num_beta)
  integer*8 :: idet, iwalk, ielec, mo_id, imo

  info = QMCKL_SUCCESS

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

  if (walk_num <= 0) then
     info = QMCKL_INVALID_ARG_2
     return
  endif

  if (beta_num <= 0) then
     info = QMCKL_INVALID_ARG_3
     return
  endif

  do idet = 1, det_num_beta
  do iwalk = 1, walk_num
    do ielec = 1, beta_num
      do imo = 1, beta_num
        mo_id = mo_index_beta(imo, iwalk, idet)
        ! Value
        det_vgl_beta(imo, ielec, 1, iwalk, idet) = mo_vgl(mo_id, alpha_num + ielec, 1)

        ! Grad_x
        det_vgl_beta(imo, ielec, 2, iwalk, idet) = mo_vgl(mo_id, alpha_num + ielec, 2)

        ! Grad_y
        det_vgl_beta(imo, ielec, 3, iwalk, idet) = mo_vgl(mo_id, alpha_num + ielec, 3)

        ! Grad_z
        det_vgl_beta(imo, ielec, 4, iwalk, idet) = mo_vgl(mo_id, alpha_num + ielec, 4)

        ! Lap
        det_vgl_beta(imo, ielec, 5, iwalk, idet) = mo_vgl(mo_id, alpha_num + ielec, 5)
      end do
    end do
  end do
  end do

end function qmckl_compute_det_vgl_beta_f

qmckl_exit_code qmckl_compute_det_vgl_beta (
      const qmckl_context context,
      const int64_t det_num_beta,
      const int64_t walk_num,
      const int64_t alpha_num,
      const int64_t beta_num,
      const int64_t elec_num,
      const int64_t* mo_index_beta,
      const int64_t mo_num,
      const double* mo_vgl,
      double* const det_vgl_beta );

2.1.5 Test

2.2 Inverse of Determinant matrix

2.2.1 Get

qmckl_exit_code qmckl_get_det_inv_matrix_alpha(qmckl_context context, double* const det_inv_matrix_alpha);
qmckl_exit_code qmckl_get_det_inv_matrix_beta(qmckl_context context, double* const det_inv_matrix_beta);
qmckl_exit_code qmckl_get_det_adj_matrix_alpha(qmckl_context context, double* const det_adj_matrix_alpha);
qmckl_exit_code qmckl_get_det_adj_matrix_beta(qmckl_context context, double* const det_adj_matrix_beta);
qmckl_exit_code qmckl_get_det_alpha(qmckl_context context, double* const det_adj_matrix_alpha);
qmckl_exit_code qmckl_get_det_beta(qmckl_context context, double* const det_adj_matrix_beta);

2.2.2 Provide

2.2.3 Compute alpha

Variable	Type	In/Out	Description
`context`	`qmckl_context`	in	Global state
`det_num_alpha`	`int64_t`	in	Number of determinants
`walk_num`	`int64_t`	in	Number of walkers
`alpha_num`	`int64_t`	in	Number of electrons
`det_vgl_alpha`	`double[det_num_alpha][walk_num][5][alpha_num][alpha_num]`	in	determinant matrix Value, gradients and Laplacian of the MOs
`det_value_alpha`	`double[det_num_alpha][walk_num]`	out	value of determinant matrix
`det_adj_matrix_alpha`	`double[det_num_alpha][walk_num][alpha_num][alpha_num]`	out	adjoint of determinant matrix
`det_inv_matrix_alpha`	`double[det_num_alpha][walk_num][alpha_num][alpha_num]`	out	inverse of determinant matrix

integer function qmckl_compute_det_inv_matrix_alpha_f(context, &
     det_num_alpha, walk_num, alpha_num, det_vgl_alpha, det_value_alpha, det_adj_matrix_alpha, det_inv_matrix_alpha) &
     result(info)
  use qmckl
  implicit none
  integer(qmckl_context)  , intent(in)  :: context
  integer*8, intent(in)             :: det_num_alpha
  integer*8, intent(in)             :: walk_num
  integer*8, intent(in)             :: alpha_num
  double precision, intent(in)      :: det_vgl_alpha(alpha_num, alpha_num, 5, walk_num, det_num_alpha)
  double precision, intent(inout)   :: det_value_alpha(walk_num, det_num_alpha)
  double precision, intent(inout)   :: det_adj_matrix_alpha(alpha_num, alpha_num, walk_num, det_num_alpha)
  double precision, intent(inout)   :: det_inv_matrix_alpha(alpha_num, alpha_num, walk_num, det_num_alpha)
  double precision,dimension(:,:),allocatable  :: matA
  double precision                  :: det_l
  integer*8 :: idet, iwalk, ielec, mo_id, imo, LDA, res, i, j

  allocate(matA(alpha_num, alpha_num))

  info = QMCKL_SUCCESS

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

  if (det_num_alpha <= 0) then
     info = QMCKL_INVALID_ARG_2
     return
  endif

  if (walk_num <= 0) then
     info = QMCKL_INVALID_ARG_3
     return
  endif

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

  LDA = alpha_num
  do idet = 1, det_num_alpha
     do iwalk = 1, walk_num
        ! Value
        matA(1:alpha_num,1:alpha_num) = &
             det_vgl_alpha(1:alpha_num, 1:alpha_num, 1, iwalk, idet)

        res = qmckl_adjugate(context,                 &
             alpha_num, matA, LDA,                    &
             det_adj_matrix_alpha(1:alpha_num, 1:alpha_num, iwalk, idet), &
             int(size(det_adj_matrix_alpha,1),8),     &
             det_l)

        det_inv_matrix_alpha(1:alpha_num, 1:alpha_num, iwalk, idet) = &
             (1.d0/det_l) * &
             det_adj_matrix_alpha(1:alpha_num, 1:alpha_num, iwalk, idet)

        det_value_alpha(iwalk, idet) = det_l
     end do
  end do

  deallocate(matA)
end function qmckl_compute_det_inv_matrix_alpha_f

qmckl_exit_code qmckl_compute_det_inv_matrix_alpha (
      const qmckl_context context,
      const int64_t det_num_alpha,
      const int64_t walk_num,
      const int64_t alpha_num,
      const double* det_vgl_alpha,
      double* const det_value_alpha,
      double* const det_adj_matrix_alpha,
      double* const det_inv_matrix_alpha );

2.2.4 Compute beta

Variable	Type	In/Out	Description
`context`	`qmckl_context`	in	Global state
`det_num_beta`	`int64_t`	in	Number of determinants
`walk_num`	`int64_t`	in	Number of walkers
`beta_num`	`int64_t`	in	Number of electrons
`det_vgl_beta`	`double[det_num_beta][walk_num][5][beta_num][beta_num]`	in	determinant matrix Value, gradients and Laplacian of the MOs
`det_value_beta`	`double[det_num_beta][walk_num]`	out	value of determinant matrix
`det_adj_matrix_beta`	`double[det_num_beta][walk_num][beta_num][beta_num]`	out	adjoint of determinant matrix
`det_inv_matrix_beta`	`double[det_num_beta][walk_num][beta_num][beta_num]`	out	inverse of determinant matrix

integer function qmckl_compute_det_inv_matrix_beta_f(context, &
     det_num_beta, walk_num, beta_num, det_vgl_beta, det_value_beta, det_adj_matrix_beta, det_inv_matrix_beta) &
     result(info)
  use qmckl
  implicit none
  integer(qmckl_context)  , intent(in)  :: context
  integer*8, intent(in)             :: det_num_beta
  integer*8, intent(in)             :: walk_num
  integer*8, intent(in)             :: beta_num
  double precision, intent(in)      :: det_vgl_beta(beta_num, beta_num, 5, walk_num, det_num_beta)
  double precision, intent(inout)   :: det_value_beta(walk_num, det_num_beta)
  double precision, intent(inout)   :: det_adj_matrix_beta(beta_num, beta_num, walk_num, det_num_beta)
  double precision, intent(inout)   :: det_inv_matrix_beta(beta_num, beta_num, walk_num, det_num_beta)
  double precision,dimension(:,:),allocatable  :: matA
  double precision                  :: det_l
  integer*8 :: idet, iwalk, ielec, mo_id, imo, LDA, res

  allocate(matA(beta_num, beta_num))

  info = QMCKL_SUCCESS

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

  if (det_num_beta <= 0) then
     info = QMCKL_INVALID_ARG_2
     return
  endif

  if (walk_num <= 0) then
     info = QMCKL_INVALID_ARG_3
     return
  endif

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

  LDA = beta_num
  do idet = 1, det_num_beta
     do iwalk = 1, walk_num
        ! Value
        matA(1:beta_num,1:beta_num) = &
             det_vgl_beta(1:beta_num, 1:beta_num, 1, iwalk, idet)

        res = qmckl_adjugate(context,                 &
             beta_num, matA, LDA,                    &
             det_adj_matrix_beta(1, 1, iwalk, idet), &
             int(size(det_adj_matrix_beta,1),8),     &
             det_l)

        det_inv_matrix_beta(1:beta_num, 1:beta_num, iwalk, idet) = &
             (1.d0/det_l) * &
             det_adj_matrix_beta(1:beta_num, 1:beta_num, iwalk, idet)

        det_value_beta(iwalk, idet) = det_l
     end do
  end do


  deallocate(matA)
end function qmckl_compute_det_inv_matrix_beta_f

qmckl_exit_code qmckl_compute_det_inv_matrix_beta (
      const qmckl_context context,
      const int64_t det_num_beta,
      const int64_t walk_num,
      const int64_t beta_num,
      const double* det_vgl_beta,
      double* const det_value_beta,
      double* const det_adj_matrix_beta,
      double* const det_inv_matrix_beta );