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q-posev 2021-09-16 16:41:46 +02:00
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@ -107,54 +107,43 @@ arrays are 0-based. Hence, we introduce the ~index~ type which is an
#+end_src
:end:
* TODO Effective core potentials (ecp group)
An effective core potential (ECP) $V_A^{\text{pp}}$ replacing the
core electrons of atom $A$ is the sum of a local component
$V_A^{\text{l}}$ and a non-local component $V_A^{\text{nl}}$.
The local component is given by
* Effective core potentials (ecp group)
An effective core potential (ECP) $V_A^{\text{ECP}}$ replacing the
core electrons of atom $A$ is expressed as
\[
\hat{V}_A^{\text{l}}(r) = -\frac{Z_A^{\text{eff}}}{r} +
\frac{Z_A^{\text{eff}}}{r}\, \exp\left( -\alpha_A\, r^2\right) +
Z_{\text{eff}}\, \alpha_A\, r\, \exp\left( -\beta_A\, r^2\right) +
\gamma_A \exp\left( -\delta_A\, r^2\right),
V_A^{\text{ECP}} =
V_{A \ell_{\max}} +
\sum_{\ell=0}^{\ell_{\max} -1}
\sum_{m=-\ell}^{\ell} | Y_{\ell m} \rangle \left[
V_{A \ell} - V_{A \ell_{\max}} \right] \langle Y_{\ell m} |
\]
and the component obtained after localizing the non-local operator is
The functions $V_{A\ell}$ are parameterized as:
\[
\hat{V}_A^{\text{nl}}(r) =
\zeta_A\, \exp\left( -\eta_A\, r^2\right) |0\rangle \langle 0| +
\mu_A \, \exp\left( -\nu_A \, r^2\right) |1\rangle \langle 1|
V_{A \ell}(\mathbf{r}) =
\sum_{q=1}^{N_{q \ell}}
\beta_{A q \ell}\, |\mathbf{r}-\mathbf{R}_{A}|^{n_{A q \ell}}\,
e^{-\alpha_{A q \ell} |\mathbf{r}-\mathbf{R}_{A}|^2 }
\]
where $r=|\mathbf{r-R}_A|$ is the distance to the nucleus on which the
potential is centered, $Z_A^{\text{eff}}$ is the effective charge
due to the removed electrons, $|0\rangle \langle 0|$
and $|1\rangle \langle 1|$ are projections over zero and one principal angular
momenta, respectively (generalization to higher angular momenta is
straightforward), and all the parameters labeled by Greek
letters are parameters.
- $\hat{V}_\text{ecp,l} = \sum_A \hat{V}_A^{\text{l}}$ : local component
- $\hat{V}_\text{ecp,nl} = \sum_A \hat{V}_A^{\text{nl}}$ : non-local component
See http://dx.doi.org/10.1063/1.4984046 for more info.
#+NAME: ecp
| Variable | Type | Dimensions | Description |
|-----------------------+---------+------------------------------------------+----------------------------|
| ~lmax_plus_1~ | ~int~ | ~(nucleus.num)~ | $l_{\max} + 1$ |
| ~z_core~ | ~float~ | ~(nucleus.num)~ | Charges to remove |
| ~local_n~ | ~int~ | ~(nucleus.num)~ | Number of local function |
| ~local_num_n_max~ | ~int~ | | Maximum value of ~local_n~ |
| ~local_exponent~ | ~float~ | ~(ecp.local_num_n_max, nucleus.num)~ | |
| ~local_coef~ | ~float~ | ~(ecp.local_num_n_max, nucleus.num)~ | |
| ~local_power~ | ~int~ | ~(ecp.local_num_n_max, nucleus.num)~ | |
| ~non_local_n~ | ~int~ | ~(nucleus.num)~ | |
| ~non_local_num_n_max~ | ~int~ | | |
| ~non_local_exponent~ | ~float~ | ~(ecp.non_local_num_n_max, nucleus.num)~ | |
| ~non_local_coef~ | ~float~ | ~(ecp.non_local_num_n_max, nucleus.num)~ | |
| ~non_local_power~ | ~int~ | ~(ecp.non_local_num_n_max, nucleus.num)~ | |
| Variable | Type | Dimensions | Description |
|-----------------------+---------+------------------------------------------+----------------------------------------------------------------------------------------------|
| ~lmax_plus_1~ | ~int~ | ~(nucleus.num)~ | $\ell_{\max} + 1$, one higher than the maximum angular momentum in the removed core orbitals |
| ~z_core~ | ~float~ | ~(nucleus.num)~ | Charges to remove |
| ~local_n~ | ~int~ | ~(nucleus.num)~ | Number of local functions $N_{q \ell}$ |
| ~local_num_n_max~ | ~int~ | | Maximum value of ~local_n~, used for dimensioning arrays |
| ~local_exponent~ | ~float~ | ~(ecp.local_num_n_max, nucleus.num)~ | $\alpha_{A q \ell_{\max}}$ |
| ~local_coef~ | ~float~ | ~(ecp.local_num_n_max, nucleus.num)~ | $\beta_{A q \ell_{\max}}$ |
| ~local_power~ | ~int~ | ~(ecp.local_num_n_max, nucleus.num)~ | $n_{A q \ell_{\max}}$ |
| ~non_local_n~ | ~int~ | ~(nucleus.num)~ | $N_{q \ell_{\max}}$ |
| ~non_local_num_n_max~ | ~int~ | | Maximum value of ~non_local_n~, used for dimensioning arrays |
| ~non_local_exponent~ | ~float~ | ~(ecp.non_local_num_n_max, nucleus.num)~ | $\alpha_{A q \ell}$ |
| ~non_local_coef~ | ~float~ | ~(ecp.non_local_num_n_max, nucleus.num)~ | $\beta_{A q \ell}$ |
| ~non_local_power~ | ~int~ | ~(ecp.non_local_num_n_max, nucleus.num)~ | $n_{A q \ell}$ |
#+CALL: json(data=ecp, title="ecp")
@ -461,14 +450,14 @@ prim_factor =
* Molecular orbitals (mo group)
#+NAME: mo
| Variable | Type | Dimensions | Description |
|---------------+---------+--------------------+------------------------------------------|
| ~type~ | ~str~ | | String identify the set of MOs |
| ~num~ | ~int~ | | Number of MOs |
| ~coefficient~ | ~float~ | ~(ao.num, mo.num)~ | MO coefficients |
| ~class~ | ~str~ | ~(mo.num)~ | Core, Inactive, Active, Virtual, Deleted |
| ~symmetry~ | ~str~ | ~(mo.num)~ | Symmetry in the point group |
| ~occupation~ | ~float~ | ~(mo.num)~ | Occupation number |
| Variable | Type | Dimensions | Description |
|---------------+---------+--------------------+--------------------------------------------------------------------------|
| ~type~ | ~str~ | | Free text to identify the set of MOs (HF, Natural, Local, CASSCF, /etc/) |
| ~num~ | ~int~ | | Number of MOs |
| ~coefficient~ | ~float~ | ~(ao.num, mo.num)~ | MO coefficients |
| ~class~ | ~str~ | ~(mo.num)~ | Choose among: Core, Inactive, Active, Virtual, Deleted |
| ~symmetry~ | ~str~ | ~(mo.num)~ | Symmetry in the point group |
| ~occupation~ | ~float~ | ~(mo.num)~ | Occupation number |
#+CALL: json(data=mo, title="mo")