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Updated ECP documentation
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@ -107,54 +107,40 @@ arrays are 0-based. Hence, we introduce the ~index~ type which is an
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#+end_src
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#+end_src
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:end:
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:end:
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* TODO Effective core potentials (ecp group)
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* Effective core potentials (ecp group)
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An effective core potential (ECP) $V_A^{\text{pp}}$ replacing the
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core electrons of atom $A$ is the sum of a local component
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$V_A^{\text{l}}$ and a non-local component $V_A^{\text{nl}}$.
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The local component is given by
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An effective core potential (ECP) $V_A^{\text{ECP}}$ replacing the
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core electrons of atom $A$ is expressed as
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\[
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\[
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\hat{V}_A^{\text{l}}(r) = -\frac{Z_A^{\text{eff}}}{r} +
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V_A^{\text{ECP}} =
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\frac{Z_A^{\text{eff}}}{r}\, \exp\left( -\alpha_A\, r^2\right) +
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V_{A \ell_{\max}} +
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Z_{\text{eff}}\, \alpha_A\, r\, \exp\left( -\beta_A\, r^2\right) +
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\sum_{\ell=0}^{\ell_{\max} -1}
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\gamma_A \exp\left( -\delta_A\, r^2\right),
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\sum_{m=-\ell}^{\ell} | Y_{\ell m} \rangle \left[
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V_{A \ell} - V_{A \ell_{\max}} \right] \langle Y_{\ell m} |
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\]
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\]
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and the component obtained after localizing the non-local operator is
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The functions $V_{A\ell}$ are parameterized as:
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\[
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\[
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\hat{V}_A^{\text{nl}}(r) =
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V_{A \ell}(\mathbf{r}) = \sum_{q=1}^{N_{q \ell}} \beta_{A q \ell}\, |\mathbf{r}-\mathbf{R}_A}|^{n_{A q \ell}}\, e^{-\alpha_{A q \ell} |\mathbf{r}-\mathbf{R}_A|^2}}
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\zeta_A\, \exp\left( -\eta_A\, r^2\right) |0\rangle \langle 0| +
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\mu_A \, \exp\left( -\nu_A \, r^2\right) |1\rangle \langle 1|
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\]
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\]
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where $r=|\mathbf{r-R}_A|$ is the distance to the nucleus on which the
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See http://dx.doi.org/10.1063/1.4984046 for more info.
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potential is centered, $Z_A^{\text{eff}}$ is the effective charge
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due to the removed electrons, $|0\rangle \langle 0|$
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and $|1\rangle \langle 1|$ are projections over zero and one principal angular
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momenta, respectively (generalization to higher angular momenta is
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straightforward), and all the parameters labeled by Greek
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letters are parameters.
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- $\hat{V}_\text{ecp,l} = \sum_A \hat{V}_A^{\text{l}}$ : local component
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- $\hat{V}_\text{ecp,nl} = \sum_A \hat{V}_A^{\text{nl}}$ : non-local component
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#+NAME: ecp
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#+NAME: ecp
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| Variable | Type | Dimensions | Description |
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| Variable | Type | Dimensions | Description |
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|-----------------------+---------+------------------------------------------+----------------------------|
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|-----------------------+---------+------------------------------------------+----------------------------------------------------------------------------------------------|
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| ~lmax_plus_1~ | ~int~ | ~(nucleus.num)~ | $l_{\max} + 1$ |
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| ~lmax_plus_1~ | ~int~ | ~(nucleus.num)~ | $\ell_{\max} + 1$, one higher than the maximum angular momentum in the removed core orbitals |
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| ~z_core~ | ~float~ | ~(nucleus.num)~ | Charges to remove |
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| ~z_core~ | ~float~ | ~(nucleus.num)~ | Charges to remove |
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| ~local_n~ | ~int~ | ~(nucleus.num)~ | Number of local function |
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| ~local_n~ | ~int~ | ~(nucleus.num)~ | Number of local functions $N_{q \ell}$ |
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| ~local_num_n_max~ | ~int~ | | Maximum value of ~local_n~ |
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| ~local_num_n_max~ | ~int~ | | Maximum value of ~local_n~, used for dimensioning arrays |
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| ~local_exponent~ | ~float~ | ~(ecp.local_num_n_max, nucleus.num)~ | |
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| ~local_exponent~ | ~float~ | ~(ecp.local_num_n_max, nucleus.num)~ | $\alpha_{A q \ell_{\max}}}$ |
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| ~local_coef~ | ~float~ | ~(ecp.local_num_n_max, nucleus.num)~ | |
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| ~local_coef~ | ~float~ | ~(ecp.local_num_n_max, nucleus.num)~ | $\beta_{A q \ell_{\max}}}$ |
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| ~local_power~ | ~int~ | ~(ecp.local_num_n_max, nucleus.num)~ | |
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| ~local_power~ | ~int~ | ~(ecp.local_num_n_max, nucleus.num)~ | $n_{A q \ell_{\max}}$ |
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| ~non_local_n~ | ~int~ | ~(nucleus.num)~ | |
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| ~non_local_n~ | ~int~ | ~(nucleus.num)~ | $N_{q \ell_{\max}}$ |
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| ~non_local_num_n_max~ | ~int~ | | |
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| ~non_local_num_n_max~ | ~int~ | | Maximum value of ~non_local_n~, used for dimensioning arrays |
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| ~non_local_exponent~ | ~float~ | ~(ecp.non_local_num_n_max, nucleus.num)~ | |
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| ~non_local_exponent~ | ~float~ | ~(ecp.non_local_num_n_max, nucleus.num)~ | $\alpha_{A q \ell}$ |
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| ~non_local_coef~ | ~float~ | ~(ecp.non_local_num_n_max, nucleus.num)~ | |
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| ~non_local_coef~ | ~float~ | ~(ecp.non_local_num_n_max, nucleus.num)~ | $\beta_{A q \ell}$ |
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| ~non_local_power~ | ~int~ | ~(ecp.non_local_num_n_max, nucleus.num)~ | |
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| ~non_local_power~ | ~int~ | ~(ecp.non_local_num_n_max, nucleus.num)~ | $n_{A q \ell}$ |
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#+CALL: json(data=ecp, title="ecp")
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#+CALL: json(data=ecp, title="ecp")
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@ -462,11 +448,11 @@ prim_factor =
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#+NAME: mo
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#+NAME: mo
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| Variable | Type | Dimensions | Description |
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| Variable | Type | Dimensions | Description |
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|---------------+---------+--------------------+------------------------------------------|
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|---------------+---------+--------------------+-----------------------------------------------------------------------|
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| ~type~ | ~str~ | | String identify the set of MOs |
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| ~type~ | ~str~ | | Free text to identify the set of MOs (HF, Natural, Local, CASSCF, /etc/) |
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| ~num~ | ~int~ | | Number of MOs |
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| ~num~ | ~int~ | | Number of MOs |
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| ~coefficient~ | ~float~ | ~(ao.num, mo.num)~ | MO coefficients |
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| ~coefficient~ | ~float~ | ~(ao.num, mo.num)~ | MO coefficients |
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| ~class~ | ~str~ | ~(mo.num)~ | Core, Inactive, Active, Virtual, Deleted |
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| ~class~ | ~str~ | ~(mo.num)~ | Choose among: Core, Inactive, Active, Virtual, Deleted |
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| ~symmetry~ | ~str~ | ~(mo.num)~ | Symmetry in the point group |
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| ~symmetry~ | ~str~ | ~(mo.num)~ | Symmetry in the point group |
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| ~occupation~ | ~float~ | ~(mo.num)~ | Occupation number |
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| ~occupation~ | ~float~ | ~(mo.num)~ | Occupation number |
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