qmckl/org/qmckl_determinant.org

Slater Determinant

• Context
  • Data structure
  • Access functions
  • Initialization functions
  • Fortran Interfaces
  • Test
• Computation
  • Determinant matrix
    • Get
    • Provide
    • Compute
    • Test
  • Inverse of Determinant matrix
    • Get
    • Provide
    • Compute
    • 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	int64_t	Number of determinants per walker
fermi_num	int64_t	Number of number of fermions
mo_index_list	mo_index[walk_num][det_num]	Index of MOs for each walker

Computed data:

det_value_list	[walk_num][det_num]	The slater matrix for each determinant of each walker.
det_value_list_date	int64_t	The slater matrix for each determinant of each walker.
det_adj_matrix_list	[walk_num][det_num][fermi_num][fermi_num]	Adjoint of the slater matrix for each determinant of each walker.
det_adj_matrix_list_date	int64_t	Adjoint of the slater matrix for each determinant of each walker.
det_vgl	[5][walk_num][det_num][fermi_num][fermi_num]	Value, gradients, Laplacian of Dᵢⱼ(x) at electron positions
det_vgl_date	int64_t	Late modification date of Value, gradients, Laplacian of the MOs at electron positions
det_inv_matrix_list	[walk_num][det_num][fermi_num][fermi_num]	Inverse of the slater matrix for each determinant of each walker.
det_inv_matrix_list_date	int64_t	Inverse of the 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;  
int64_t  fermi_num;  
int64_t* mo_index_list;

double  * det_matrix_list;
double  * det_adj_matrix_list;
double  * det_inv_matrix_list;
double  * det_vgl;
int64_t   det_matrix_list_date;
int64_t   det_adj_matrix_list_date;
int64_t   det_inv_matrix_list_date;
int64_t   det_vgl_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          (const qmckl_context context, const int64_t det_num);
qmckl_exit_code  qmckl_set_determinant_fermi_num        (const qmckl_context context, const int64_t fermi_num);
qmckl_exit_code  qmckl_set_determinant_mo_index_list    (const qmckl_context context, const int64_t* mo_index_list);

#+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_context	context	in	Global state
double	det_vgl[5][walk_num][det_num][fermi_num][fermi_num]	out	Value, gradients and Laplacian of the MOs

qmckl_exit_code qmckl_get_det_vgl(qmckl_context context, double* const det_vgl);

Provide

Compute

qmckl_context	context	in	Global state
int64_t	walk_num	in	Number of walkers
int64_t	fermi_num	in	Number of electrons
int64_t	mo_index_list[fermi_num]	in	Number of electrons
int64_t	mo_num	in	Number of MOs
double	mo_vgl[5][walk_num][fermi_num][mo_num]	in	Value, gradients and Laplacian of the MOs
double	det_vgl[5][walk_num][fermi_num][fermi_num]	out	Value, gradients and Laplacian of the Det

integer function qmckl_compute_det_vgl_f(context, &
 walk_num, fermi_num, mo_index_list, mo_num, mo_vgl, det_vgl) &
 result(info)
use qmckl
implicit none
integer(qmckl_context)  , intent(in)  :: context
integer*8, intent(in)             :: walk_num
integer*8, intent(in)             :: fermi_num
integer*8, intent(in)             :: mo_num
integer*8, intent(in)             :: mo_index_list(fermi_num)
double precision, intent(in)      :: mo_vgl(mo_num, fermi_num, walk_num, 5)
double precision, intent(inout)   :: det_vgl(fermi_num, fermi_num, walk_num, 5)
integer*8 :: 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 (fermi_num <= 0) then
 info = QMCKL_INVALID_ARG_3
 return
endif

do iwalk = 1, walk_num
do ielec = 1, fermi_num
  do imo = 1, fermi_num
    mo_id = mo_index_list(imo)
    ! Value
    det_vgl(imo, ielec, iwalk, 1) = mo_vgl(mo_id, ielec, iwalk, 1)

    ! Grad_x
    det_vgl(imo, ielec, iwalk, 1) = mo_vgl(mo_id, ielec, iwalk, 2)

    ! Grad_y
    det_vgl(imo, ielec, iwalk, 1) = mo_vgl(mo_id, ielec, iwalk, 3)

    ! Grad_z
    det_vgl(imo, ielec, iwalk, 1) = mo_vgl(mo_id, ielec, iwalk, 4)

    ! Lap
    det_vgl(imo, ielec, iwalk, 1) = mo_vgl(mo_id, ielec, iwalk, 5)
  end do
end do
end do

end function qmckl_compute_det_vgl_f

qmckl_exit_code qmckl_compute_det_vgl (
const qmckl_context context,
const int64_t walk_num,
const int64_t fermi_num,
const int64_t* mo_index_list,
const int64_t mo_num,
const double* mo_vgl,
double* const det_vgl );

Test

Inverse of Determinant matrix

Get

qmckl_context	context	in	Global state
double	det_inv_matrix_list[5][walk_num][det_num]	out	Value, gradients and Laplacian of the MOs

qmckl_exit_code qmckl_get_det_inv_matrix(qmckl_context context, double* const det_inv_matrix);

qmckl_exit_code qmckl_compute_get_det_inv_matrix (
const qmckl_context context,
double* const det_inv_matrix_list );

Provide

Compute

qmckl_context	context	in	Global state
int64_t	walk_num	in	Number of walkers
int64_t	fermi_num	in	Number of electrons
int64_t	mo_index_list[fermi_num]	in	Number of electrons
int64_t	mo_num	in	Number of MOs
double	mo_vgl[5][walk_num][fermi_num][mo_num]	in	Value, gradients and Laplacian of the MOs
double	det_inv_matrix_list[5][walk_num][fermi_num][fermi_num]	out	Value, gradients and Laplacian of the Det

integer function qmckl_compute_det_inv_matrix_f(context, &
 walk_num, fermi_num, mo_index_list, mo_num, mo_vgl, det_inv_matrix_list) &
 result(info)
use qmckl
implicit none
integer(qmckl_context)  , intent(in)  :: context
integer*8, intent(in)             :: walk_num
integer*8, intent(in)             :: fermi_num
integer*8, intent(in)             :: mo_num
integer*8, intent(in)             :: mo_index_list(fermi_num)
double precision, intent(in)      :: mo_vgl(mo_num, fermi_num, walk_num, 5)
double precision, intent(inout)   :: det_inv_matrix_list(fermi_num, fermi_num, walk_num, 5)
integer*8 :: 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 (fermi_num <= 0) then
 info = QMCKL_INVALID_ARG_3
 return
endif

do iwalk = 1, walk_num
do ielec = 1, fermi_num
  do imo = 1, fermi_num
    mo_id = mo_index_list(imo)
    ! Value
    det_inv_matrix_list(imo, ielec, iwalk, 1) = mo_vgl(mo_id, ielec, iwalk, 1)
  end do
end do
end do

end function qmckl_compute_det_inv_matrix_f

qmckl_exit_code qmckl_compute_det_inv_matrix (
const qmckl_context context,
const int64_t walk_num,
const int64_t fermi_num,
const int64_t* mo_index_list,
const int64_t mo_num,
const double* mo_vgl,
double* const det_inv_matrix_list );

Test