qmckl/org/qmckl_determinant.org

Slater Determinant

• Context
  • Data structure
  • Access functions
  • Initialization functions
  • Fortran Interfaces
  • Test
• Computation
  • Determinant matrix
    • Get
    • Provide
    • Compute alpha
    • Compute beta
    • Test
  • Inverse of Determinant matrix
    • Get
    • Provide
    • Compute alpha
    • Compute beta
    • Test

The slater deteminant is required for the calculation of the wavefunction, gradient, and derivatives. These quantities will be used to calculate the local Energy (\[E_L\]).

ψ(x) = det|ϕ₁(x₁)…ϕᵢ(yᵢ)…ϕₙ(xₙ)|

The above slater-matrix is also required and is denoted by Dᵢⱼ(x) such that:

ψ(x) = det|Dᵢⱼ(x)|

We also require the inverse of the slater-matrix which is denoted by D⁻¹ᵢⱼ(x). Using this notation, the acceptance probability which is proportional to ψ(y)/ψ(x) can be calculated as follows:

ψ(yᵢ)/ψ(xᵢ) = ∑ⱼDᵢⱼ(y)D⁻¹ⱼᵢ(x)

Concerning the gradient and laplacian, in fact what is actually calculated is the ratio of the gradient/laplacian and the determinant of the slater matrix:

∇ψ(x)/ψ(x)

and

∇²ψ(x)/ψ(x)

This avoids the unnecessary multiplication and division of by the determinant ψ(x).

Context

The following arrays are stored in the context:

type	char	α ('A') or β ('B') determinant
walk_num	int64_t	Number of walkers
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][walk_num][alpha_num]	Index of MOs for each walker
mo_index_beta	mo_index[det_num_beta][walk_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][walk_num]	The α slater matrix for each determinant of each walker.
det_value_alpha_date	int64_t	Date of The α slater matrix for each determinant of each walker.
det_value_beta	[det_num_beta][walk_num]	The β slater matrix for each determinant of each walker.
det_value_beta_date	int64_t	Date of The β slater matrix for each determinant of each walker.
det_adj_matrix_alpha	[det_num_alpha][walk_num][alpha_num][alpha_num]	Adjoint of the α slater matrix for each determinant of each walker.
det_adj_matrix_alpha_date	int64_t	Date of the Adjoint of the α slater matrix for each determinant of each walker.
det_adj_matrix_beta	[det_num_beta][walk_num][beta_num][beta_num]	Adjoint of the β slater matrix for each determinant of each walker.
det_adj_matrix_beta_date	int64_t	Date of the Adjoint of the β slater matrix for each determinant of each walker.
det_vgl_alpha	[5][det_num_alpha][walk_num][alpha_num][alpha_num]	Value, gradients, Laplacian of Dᵅᵢⱼ(x) at electron positions
det_vgl_alpha_date	int64_t	Late modification date of Value, gradients, Laplacian of the MOs at electron positions
det_vgl_beta	[5][det_num_beta][walk_num][beta_num][beta_num]	Value, gradients, Laplacian of Dᵝᵢⱼ(x) at electron positions
det_vgl_beta_date	int64_t	Late modification date of Value, gradients, Laplacian of the MOs at electron positions
det_inv_matrix_alpha	[det_num_alpha][walk_num][alpha_num][alpha_num]	Inverse of the α electron slater matrix for each determinant of each walker.
det_inv_matrix_alpha_date	int64_t	Date for the Inverse of the α electron slater matrix for each determinant of each walker.
det_inv_matrix_beta	[det_num_beta][walk_num][beta_num][beta_num]	Inverse of the β electron slater matrix for each determinant of each walker.
det_inv_matrix_beta_date	int64_t	Date for the Inverse of the β electron slater matrix for each determinant of each walker.

Data structure

typedef struct qmckl_determinant_struct {
char      type;
int64_t  walk_num;  
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;
int64_t   det_value_alpha_date;
int64_t   det_vgl_alpha_date;
int64_t   det_adj_matrix_alpha_date;
int64_t   det_inv_matrix_alpha_date;
int64_t   det_value_beta_date;
int64_t   det_vgl_beta_date;
int64_t   det_adj_matrix_beta_date;
int64_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.

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);

#+NAME:post

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_walk_num         (const qmckl_context context, const int64_t walk_num);
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);
qmckl_exit_code  qmckl_set_determinant_mo_index_beta    (const qmckl_context context, const int64_t* mo_index_beta);

#+NAME:pre2

#+NAME:post2

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

Fortran Interfaces

Test

Computation

Determinant matrix

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);

Provide

Compute alpha

qmckl_context	context	in	Global state
int64_t	det_num_alpha	in	Number of determinants
int64_t	walk_num	in	Number of walkers
int64_t	alpha_num	in	Number of electrons
int64_t	beta_num	in	Number of electrons
int64_t	elec_num	in	Number of electrons
int64_t	mo_index_alpha[det_num_alpha][walk_num][alpha_num]	in	MO indices for electrons
int64_t	mo_num	in	Number of MOs
double	mo_vgl[5][walk_num][elec_num][mo_num]	in	Value, gradients and Laplacian of the MOs
double	det_vgl_alpha[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, walk_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, iwalk, 1)

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

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

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

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

Compute beta

qmckl_context	context	in	Global state
int64_t	det_num_beta	in	Number of determinants
int64_t	walk_num	in	Number of walkers
int64_t	alpha_num	in	Number of electrons
int64_t	beta_num	in	Number of electrons
int64_t	elec_num	in	Number of electrons
int64_t	mo_index_beta[det_num_beta][walk_num][beta_num]	in	Number of electrons
int64_t	mo_num	in	Number of MOs
double	mo_vgl[5][walk_num][elec_num][mo_num]	in	Value, gradients and Laplacian of the MOs
double	det_vgl_beta[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, walk_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, iwalk, 1)

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

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

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

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

Test

Inverse of Determinant matrix

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);

Provide

Compute alpha

qmckl_context	context	in	Global state
int64_t	det_num_alpha	in	Number of determinants
int64_t	walk_num	in	Number of walkers
int64_t	alpha_num	in	Number of electrons
int64_t	mo_num	in	Number of MOs
double	det_vgl_alpha[det_num_alpha][walk_num][5][alpha_num][mo_num]	in	determinant matrix Value, gradients and Laplacian of the MOs
double	det_value_alpha[det_num_alpha][walk_num]	out	value of determinant matrix
double	det_adj_matrix_alpha[det_num_alpha][walk_num][alpha_num][alpha_num]	out	adjoint of determinant matrix
double	det_inv_matrix_alpha[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, mo_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
integer*8, intent(in)             :: mo_num
double precision, intent(in)      :: det_vgl_alpha(mo_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

allocate(matA(mo_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

if (mo_num <= 0) then
 info = QMCKL_INVALID_ARG_5
 return
endif

LDA = alpha_num
do idet = 1, det_num_alpha
do iwalk = 1, walk_num
! Value
matA = det_vgl_alpha(1:mo_num, 1:alpha_num, 1, iwalk, idet)
res = qmckl_invert(context, alpha_num, alpha_num, LDA, matA, det_l)
det_adj_matrix_alpha(1:mo_num, 1:alpha_num, iwalk, idet) = matA
det_inv_matrix_alpha(1:mo_num, 1:alpha_num, iwalk, idet) = matA/det_l
det_value_alpha(iwalk, idet) = det_l
end do
end do

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 int64_t mo_num,
const double* det_vgl_alpha,
double* const det_value_alpha,
double* const det_adj_matrix_alpha,
double* const det_inv_matrix_alpha );

Compute beta

qmckl_context	context	in	Global state
int64_t	det_num_beta	in	Number of determinants
int64_t	walk_num	in	Number of walkers
int64_t	beta_num	in	Number of electrons
int64_t	mo_num	in	Number of MOs
double	det_vgl_beta[det_num_beta][walk_num][5][beta_num][mo_num]	in	determinant matrix Value, gradients and Laplacian of the MOs
double	det_value_beta[det_num_beta][walk_num]	out	value of determinant matrix
double	det_adj_matrix_beta[det_num_beta][walk_num][beta_num][beta_num]	out	adjoint of determinant matrix
double	det_inv_matrix_beta[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, mo_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
integer*8, intent(in)             :: mo_num
double precision, intent(in)      :: det_vgl_beta(mo_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(mo_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

if (mo_num <= 0) then
 info = QMCKL_INVALID_ARG_5
 return
endif

LDA = beta_num
do idet = 1, det_num_beta
do iwalk = 1, walk_num
! Value
matA = det_vgl_beta(1:mo_num, 1:beta_num, 1, iwalk, idet)
res = qmckl_invert(context, beta_num, beta_num, LDA, matA, det_l)
det_adj_matrix_beta(1:mo_num, 1:beta_num, iwalk, idet) = matA
det_inv_matrix_beta(1:mo_num, 1:beta_num, iwalk, idet) = matA/det_l
det_value_beta(iwalk, idet) = det_l
end do
end do

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 int64_t mo_num,
const double* det_vgl_beta,
double* const det_value_beta,
double* const det_adj_matrix_beta,
double* const det_inv_matrix_beta );

Test