ao_basis

This module describes the atomic orbitals basis set.

An AO \(\chi\) centered on nucleus A is represented as:

\[\chi_i({\bf r}) = (x-X_A)^a (y-Y_A)^b (z-Z_A)^c \sum_k c_{ki} e^{-\gamma_{ki} |{\bf r} - {\bf R}_A|^2}\]

The AO coefficients are normalized as:

\[{\tilde c}_{ki} = \frac{c_{ki}}{ \int \left( (x-X_A)^a (y-Y_A)^b (z-Z_A)^c e^{-\gamma_{ki} |{\bf r} - {\bf R}_A|^2} \right)^2 dr}\]

Warning

ao_coef contains the AO coefficients given in input. These do not include the normalization constant of the AO. The ao_coef_normalized provider includes this normalization factor.

The AOs are also sorted by increasing exponent to accelerate the calculation of the two electron integrals.

EZFIO parameters

ao_basis

Name of the AO basis set

ao_num

Number of AOs

ao_prim_num

Number of primitives per AO

ao_prim_num_max

Maximum number of primitives

Default: =maxval(ao_basis.ao_prim_num)

ao_nucl

Index of the nucleus on which the AO is centered

ao_power

Powers of x, y and z for each AO

ao_coef

Primitive coefficients, read from input. Those should not be used directly, as the MOs are expressed on the basis of normalized AOs.

ao_expo

Exponents for each primitive of each AO

ao_md5

MD5 key, specific of the AO basis

ao_cartesian

If true, use AOs in Cartesian coordinates (6d,10f,…)

Default: false

Providers

ao_coef_normalization_factor

File : ao_basis/aos.irp.f

double precision, allocatable   :: ao_coef_normalized   (ao_num,ao_prim_num_max)
double precision, allocatable   :: ao_coef_normalization_factor (ao_num)

Coefficients including the AO normalization

Needs:

ao_coef ao_expo ao_normalized

ao_num ao_power ao_prim_num

ao_prim_num_max primitives_normalized

Needed by:

ao_coef_normalization_libint_factor

ao_coef_normalized_ordered

ao_coef_normalization_libint_factor

File : ao_basis/aos.irp.f

double precision, allocatable   :: ao_coef_normalization_libint_factor  (ao_num)

AO normalization for interfacing with libint

Needs:

ao_coef_normalized ao_expo

ao_l ao_num

ao_prim_num

ao_coef_normalized

File : ao_basis/aos.irp.f

double precision, allocatable   :: ao_coef_normalized   (ao_num,ao_prim_num_max)
double precision, allocatable   :: ao_coef_normalization_factor (ao_num)

Coefficients including the AO normalization

Needs:

ao_coef ao_expo ao_normalized

ao_num ao_power ao_prim_num

ao_prim_num_max primitives_normalized

Needed by:

ao_coef_normalization_libint_factor

ao_coef_normalized_ordered

ao_coef_normalized_ordered

File : ao_basis/aos.irp.f

double precision, allocatable   :: ao_coef_normalized_ordered   (ao_num,ao_prim_num_max)
double precision, allocatable   :: ao_expo_ordered      (ao_num,ao_prim_num_max)

Sorted primitives to accelerate 4 index MO transformation

Needs:

ao_coef_normalized ao_expo

ao_num ao_prim_num

ao_prim_num_max

Needed by:

ao_coef_normalized_ordered_transp

ao_expo_ordered_transp

ao_coef_normalized_ordered_transp

File : ao_basis/aos.irp.f

double precision, allocatable   :: ao_coef_normalized_ordered_transp    (ao_prim_num_max,ao_num)

Transposed ao_coef_normalized_ordered

Needs:

ao_coef_normalized_ordered

ao_num

ao_prim_num_max

Needed by:

ao_coef_normalized_ordered_transp_per_nucl ao_deriv2_x ao_deriv_1_x ao_dipole_x ao_integrals_n_e

ao_integrals_n_e_per_atom ao_overlap ao_overlap_abs ao_pseudo_integrals_local

ao_pseudo_integrals_non_local ao_spread_x ao_two_e_integral_schwartz ao_two_e_integrals_in_map

ao_coef_normalized_ordered_transp_per_nucl

File : ao_basis/aos_transp.irp.f

double precision, allocatable   :: ao_coef_normalized_ordered_transp_per_nucl   (ao_prim_num_max,N_AOs_max,nucl_num)

Needs:

ao_coef_normalized_ordered_transp ao_prim_num

ao_prim_num_max nucl_aos_transposed

nucl_n_aos nucl_num

ao_expo_ordered

File : ao_basis/aos.irp.f

double precision, allocatable   :: ao_coef_normalized_ordered   (ao_num,ao_prim_num_max)
double precision, allocatable   :: ao_expo_ordered      (ao_num,ao_prim_num_max)

Sorted primitives to accelerate 4 index MO transformation

Needs:

ao_coef_normalized ao_expo

ao_num ao_prim_num

ao_prim_num_max

Needed by:

ao_coef_normalized_ordered_transp

ao_expo_ordered_transp

ao_expo_ordered_transp

File : ao_basis/aos.irp.f

double precision, allocatable   :: ao_expo_ordered_transp       (ao_prim_num_max,ao_num)

Transposed ao_expo_ordered

Needs:

ao_coef_normalized_ordered

ao_num

ao_prim_num_max

Needed by:

ao_deriv2_x ao_deriv_1_x ao_dipole_x ao_expo_ordered_transp_per_nucl ao_integrals_n_e

ao_integrals_n_e_per_atom ao_overlap ao_overlap_abs ao_pseudo_integrals_local

ao_pseudo_integrals_non_local ao_spread_x ao_two_e_integral_schwartz ao_two_e_integrals_in_map

ao_expo_ordered_transp_per_nucl

File : ao_basis/aos_transp.irp.f

double precision, allocatable   :: ao_expo_ordered_transp_per_nucl      (ao_prim_num_max,N_AOs_max,nucl_num)

Needs:

ao_expo_ordered_transp ao_prim_num

ao_prim_num_max nucl_aos_transposed

nucl_n_aos nucl_num

ao_l

File : ao_basis/aos.irp.f

integer, allocatable    :: ao_l (ao_num)
integer :: ao_l_max
character*(128), allocatable    :: ao_l_char    (ao_num)

\(l\) value of the AO: :math`a+b+c` in \(x^a y^b z^c\)

Needs:

ao_num

ao_power

l_to_character

Needed by:

ao_cart_to_sphe_coef ao_coef_normalization_libint_factor

ao_l_char_space

nucl_list_shell_aos

ao_l_char

File : ao_basis/aos.irp.f

integer, allocatable    :: ao_l (ao_num)
integer :: ao_l_max
character*(128), allocatable    :: ao_l_char    (ao_num)

\(l\) value of the AO: :math`a+b+c` in \(x^a y^b z^c\)

Needs:

ao_num

ao_power

l_to_character

Needed by:

ao_cart_to_sphe_coef ao_coef_normalization_libint_factor

ao_l_char_space

nucl_list_shell_aos

ao_l_char_space

File : ao_basis/aos.irp.f

character*(4), allocatable      :: ao_l_char_space      (ao_num)

Converts an l value to a string

Needs:

ao_l

ao_num

ao_power

ao_l_max

File : ao_basis/aos.irp.f

integer, allocatable    :: ao_l (ao_num)
integer :: ao_l_max
character*(128), allocatable    :: ao_l_char    (ao_num)

\(l\) value of the AO: :math`a+b+c` in \(x^a y^b z^c\)

Needs:

ao_num

ao_power

l_to_character

Needed by:

ao_cart_to_sphe_coef ao_coef_normalization_libint_factor

ao_l_char_space

nucl_list_shell_aos

ao_power_ordered_transp_per_nucl

File : ao_basis/aos_transp.irp.f

integer, allocatable    :: ao_power_ordered_transp_per_nucl     (3,N_AOs_max,nucl_num)

Needs:

ao_power nucl_aos_transposed

nucl_n_aos

nucl_num

ao_prim_num_max

File : ao_basis/aos.irp.f

integer :: ao_prim_num_max

Max number of primitives.

Needs:

ao_prim_num

Needed by:

ao_coef ao_coef_normalized ao_coef_normalized_ordered

ao_coef_normalized_ordered_transp ao_coef_normalized_ordered_transp_per_nucl ao_expo

ao_expo_ordered_transp ao_expo_ordered_transp_per_nucl

cart_to_sphe_0

File : ao_basis/spherical_to_cartesian.irp.f

double precision, allocatable   :: cart_to_sphe_0       (1,1)

Spherical -> Cartesian Transformation matrix for l=0

cart_to_sphe_1

File : ao_basis/spherical_to_cartesian.irp.f

double precision, allocatable   :: cart_to_sphe_1       (3,3)

Spherical -> Cartesian Transformation matrix for l=1

Needed by:

ao_cart_to_sphe_coef

cart_to_sphe_2

File : ao_basis/spherical_to_cartesian.irp.f

double precision, allocatable   :: cart_to_sphe_2       (6,5)

Spherical -> Cartesian Transformation matrix for l=2

Needed by:

ao_cart_to_sphe_coef

cart_to_sphe_3

File : ao_basis/spherical_to_cartesian.irp.f

double precision, allocatable   :: cart_to_sphe_3       (10,7)

Spherical -> Cartesian Transformation matrix for l=3

Needed by:

ao_cart_to_sphe_coef

cart_to_sphe_4

File : ao_basis/spherical_to_cartesian.irp.f

double precision, allocatable   :: cart_to_sphe_4       (15,9)

Spherical -> Cartesian Transformation matrix for l=4

Needed by:

ao_cart_to_sphe_coef

cart_to_sphe_5

File : ao_basis/spherical_to_cartesian.irp.f

double precision, allocatable   :: cart_to_sphe_5       (21,11)

Spherical -> Cartesian Transformation matrix for l=5

Needed by:

ao_cart_to_sphe_coef

cart_to_sphe_6

File : ao_basis/spherical_to_cartesian.irp.f

double precision, allocatable   :: cart_to_sphe_6       (28,13)

Spherical -> Cartesian Transformation matrix for l=6

Needed by:

ao_cart_to_sphe_coef

cart_to_sphe_7

File : ao_basis/spherical_to_cartesian.irp.f

double precision, allocatable   :: cart_to_sphe_7       (36,15)

Spherical -> Cartesian Transformation matrix for l=7

Needed by:

ao_cart_to_sphe_coef

cart_to_sphe_8

File : ao_basis/spherical_to_cartesian.irp.f

double precision, allocatable   :: cart_to_sphe_8       (45,17)

Spherical -> Cartesian Transformation matrix for l=8

Needed by:

ao_cart_to_sphe_coef

cart_to_sphe_9

File : ao_basis/spherical_to_cartesian.irp.f

double precision, allocatable   :: cart_to_sphe_9       (55,19)

Spherical -> Cartesian Transformation matrix for l=9

Needed by:

ao_cart_to_sphe_coef

l_to_character

File : ao_basis/aos.irp.f

character*(128), allocatable    :: l_to_character       (0:7)

Character corresponding to the “l” value of an AO

Needed by:

ao_l

n_aos_max

File : ao_basis/aos.irp.f

integer, allocatable    :: nucl_n_aos   (nucl_num)
integer :: n_aos_max

Number of AOs per atom

Needs:

ao_nucl

ao_num

nucl_num

Needed by:

ao_coef_normalized_ordered_transp_per_nucl ao_expo_ordered_transp_per_nucl

ao_power_ordered_transp_per_nucl nucl_aos

nucl_aos_transposed nucl_list_shell_aos

n_pt_max_i_x

File : ao_basis/dimensions_integrals.irp.f

integer :: n_pt_max_integrals
integer :: n_pt_max_i_x

Number of points used in the numerical integrations.

Needs:

ao_power

Needed by:

ao_integrals_n_e ao_integrals_n_e_per_atom

ao_two_e_integral_schwartz ao_two_e_integrals_in_map

gauleg_t2

n_pt_max_integrals

File : ao_basis/dimensions_integrals.irp.f

integer :: n_pt_max_integrals
integer :: n_pt_max_i_x

Number of points used in the numerical integrations.

Needs:

ao_power

Needed by:

ao_integrals_n_e ao_integrals_n_e_per_atom

ao_two_e_integral_schwartz ao_two_e_integrals_in_map

gauleg_t2

nucl_aos

File : ao_basis/aos.irp.f

integer, allocatable    :: nucl_aos     (nucl_num,N_AOs_max)

List of AOs centered on each atom

Needs:

ao_nucl ao_num

nucl_n_aos

nucl_num

Needed by:

nucl_aos_transposed

nucl_list_shell_aos

nucl_aos_transposed

File : ao_basis/aos_transp.irp.f

integer, allocatable    :: nucl_aos_transposed  (N_AOs_max,nucl_num)

List of AOs attached on each atom

Needs:

ao_nucl ao_num

nucl_aos nucl_n_aos

nucl_num

Needed by:

ao_coef_normalized_ordered_transp_per_nucl

ao_expo_ordered_transp_per_nucl

ao_power_ordered_transp_per_nucl

nucl_list_shell_aos

File : ao_basis/aos.irp.f

integer, allocatable    :: nucl_list_shell_aos  (nucl_num,N_AOs_max)
integer, allocatable    :: nucl_num_shell_aos   (nucl_num)

Index of the shell type AOs and of the corresponding AOs By convention, for p,d,f and g AOs, we take the index of the AO with the the corresponding power in the x axis

Needs:

ao_l ao_power

nucl_aos nucl_n_aos

nucl_num

nucl_n_aos

File : ao_basis/aos.irp.f

integer, allocatable    :: nucl_n_aos   (nucl_num)
integer :: n_aos_max

Number of AOs per atom

Needs:

ao_nucl

ao_num

nucl_num

Needed by:

ao_coef_normalized_ordered_transp_per_nucl ao_expo_ordered_transp_per_nucl

ao_power_ordered_transp_per_nucl nucl_aos

nucl_aos_transposed nucl_list_shell_aos

nucl_num_shell_aos

File : ao_basis/aos.irp.f

integer, allocatable    :: nucl_list_shell_aos  (nucl_num,N_AOs_max)
integer, allocatable    :: nucl_num_shell_aos   (nucl_num)

Index of the shell type AOs and of the corresponding AOs By convention, for p,d,f and g AOs, we take the index of the AO with the the corresponding power in the x axis

Needs:

ao_l ao_power

nucl_aos nucl_n_aos

nucl_num

Subroutines / functions

ao_power_index:

File : ao_basis/aos.irp.f

integer function ao_power_index(nx,ny,nz)

Unique index given to a triplet of powers:

\(\frac{1}{2} (l-n_x) (l-n_x+1) + n_z + 1\)

ao_value:

File : ao_basis/aos_in_r.irp.f

double precision function ao_value(i,r)

Returns the value of the i-th ao at point $textbf{r}$

Needs:

ao_coef_normalized_ordered_transp ao_expo_ordered_transp

ao_nucl ao_power

ao_prim_num nucl_coord

give_all_aos_and_grad_and_lapl_at_r:

File : ao_basis/aos_in_r.irp.f

subroutine give_all_aos_and_grad_and_lapl_at_r(r,aos_array,aos_grad_array,aos_lapl_array)

input : r(1) ==> r(1) = x, r(2) = y, r(3) = z

output :

• aos_array(i) = ao(i) evaluated at $textbf{r}$

• aos_grad_array(1,i) = $nabla_x$ of the ao(i) evaluated at $textbf{r}$

Needs:

ao_coef_normalized_ordered_transp_per_nucl ao_expo_ordered_transp_per_nucl ao_num

ao_power_ordered_transp_per_nucl ao_prim_num nucl_aos_transposed

nucl_coord nucl_n_aos nucl_num

Called by:

give_all_mos_and_grad_and_lapl_at_r()

give_all_aos_and_grad_at_r:

File : ao_basis/aos_in_r.irp.f

subroutine give_all_aos_and_grad_at_r(r,aos_array,aos_grad_array)

input : r(1) ==> r(1) = x, r(2) = y, r(3) = z

output :

• aos_array(i) = ao(i) evaluated at ro

• aos_grad_array(1,i) = gradient X of the ao(i) evaluated at $textbf{r}$

Needs:

ao_coef_normalized_ordered_transp_per_nucl ao_expo_ordered_transp_per_nucl ao_num

ao_power_ordered_transp_per_nucl ao_prim_num nucl_aos_transposed

nucl_coord nucl_n_aos nucl_num

Called by:

give_all_mos_and_grad_at_r()

give_all_aos_at_r:

File : ao_basis/aos_in_r.irp.f

subroutine give_all_aos_at_r(r,aos_array)

input : r == r(1) = x and so on

output : aos_array(i) = aos(i) evaluated in $textbf{r}$

Needs:

ao_coef_normalized_ordered_transp_per_nucl ao_expo_ordered_transp_per_nucl ao_num

ao_power_ordered_transp_per_nucl ao_prim_num nucl_aos_transposed

nucl_coord nucl_n_aos nucl_num

Called by:

give_all_mos_at_r()

primitive_value:

File : ao_basis/aos_in_r.irp.f

double precision function primitive_value(i,j,r)

Returns the value of the j-th primitive of the i-th AO at point $textbf{r} without the coefficient

Needs:

ao_expo_ordered_transp ao_nucl

ao_power

nucl_coord