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_factor provider includes this normalization factor.

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

Complex Gaussian-Type Orbitals (cGTOs)

Complex Gaussian-Type Orbitals (cGTOs) are also supported:

\[\chi_i(\mathbf{r}) = x^a y^b z^c \sum_k c_{ki} \left( e^{-\alpha_{ki} \mathbf{r}^2 - \imath \mathbf{k}_{ki} \cdot \mathbf{r} - \imath \phi_{ki}} + \text{C.C.} \right)\]

where:

$\alpha \in \mathbb{C}$ and $\Re(\alpha) > 0$ (specified by ao_expo and ao_expo_im),
$\mathbf{k} = (k_x, k_y, k_z) \in \mathbb{R}^3$ (specified by ao_expo_pw),
$\phi = \phi_x + \phi_y + \phi_z \in \mathbb{R}$ (specified by ao_expo_phase).

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

ao_normalized

Use normalized basis functions

Default: true

primitives_normalized

Use normalized primitive functions

Default: true

use_cgtos

If true, use cgtos for AO integrals

Default: False

ao_expo_im: imag part for Exponents for each primitive of each cGTOs AO

ao_expo_pw: plane wave part for each primitive GTOs AO

ao_expo_phase: phase shift for each primitive GTOs AO

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_normalized_ordered

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_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_integrals_pt_chrg
ao_overlap
ao_overlap_abs
ao_pseudo_integrals_local
ao_pseudo_integrals_non_local
ao_spread_x

ao_two_e_integral_alpha
ao_two_e_integral_erf_schwartz
ao_two_e_integral_schwartz
ao_two_e_integrals_erf_in_map
ao_two_e_integrals_in_map
cholesky_ao_num

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_integrals_pt_chrg
ao_overlap
ao_overlap_abs
ao_pseudo_integrals_local
ao_pseudo_integrals_non_local
ao_spread_x

ao_two_e_integral_alpha
ao_two_e_integral_erf_schwartz
ao_two_e_integral_schwartz
ao_two_e_integrals_erf_in_map
ao_two_e_integrals_in_map
cholesky_ao_num

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_first_of_shell

File : ao_basis/aos.irp.f

integer, allocatable    :: ao_first_of_shell    (shell_num)

Index of the shell to which the AO corresponds

Needs:

shell_ang_mom

shell_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_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_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_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_cgtos_norm_ord_transp
ao_coef_norm_cgtos
ao_coef_norm_cgtos_ord
ao_coef_normalized

ao_coef_normalized_ordered
ao_coef_normalized_ordered_transp
ao_coef_normalized_ordered_transp_per_nucl
ao_expo
ao_expo_cgtos_ord_transp

ao_expo_im
ao_expo_ordered_transp
ao_expo_ordered_transp_per_nucl
ao_expo_phase
ao_expo_pw

ao_shell

File : ao_basis/aos.irp.f

integer, allocatable    :: ao_shell     (ao_num)

Index of the shell to which the AO corresponds

Needs:

ao_num

shell_ang_mom

shell_num

cart_to_sphe_0