diff --git a/trex.org b/trex.org index 80d146f..b7c5723 100644 --- a/trex.org +++ b/trex.org @@ -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")