qmckl/org/qmckl_slater_determinant.org

Slater Determinant

• Context

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)|

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	uint64_t	Number of walkers
det_num	uint64_t	Number of determinants per walker
fermi_num	uint64_t	Number of number of fermions
mo_index_list	mo_index[walk_num][det_num]	Index of MOs for each walker

Computed data:

det_matrix_list	[walk_num][det_num][mo_num][fermi_num]	The slater matrix for each determinant of each walker.
det_vgl	[5][walk_num][det_num]	Value, gradients, Laplacian of the MOs at electron positions
det_vgl_date	uint64_t	Late modification date of Value, gradients, Laplacian of the MOs at electron positions