TREX/trexio
1
0
Fork 0
mirror of https://github.com/TREX-CoE/trexio.git synced 2024-12-22 20:35:44 +01:00
Code Issues Projects Releases Wiki Activity

Fixed documentation

Browse Source
This commit is contained in:
Anthony Scemama 2023-02-28 09:44:09 +01:00
parent d63bc67892
commit 2465293581
1 changed files with 116 additions and 94 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

160
trex.org
View File

@ -144,9 +144,19 @@
** Electron (electron group)
The chemical system consists of nuclei and electrons, where the
nuclei are considered as fixed point charges with Cartesian
coordinates. The wave function is stored in the spin-free
formalism, and therefore, it is necessary for the user to
explicitly store the number of electrons with spin up
($N_\uparrow$) and spin down ($N_\downarrow$). These numbers
correspond to the normalization of the spin-up and spin-down
single-particle reduced density matrices.
We consider wave functions expressed in the spin-free formalism, where
the number of \uparrow and \downarrow electrons is fixed.
#+NAME:electron
| Variable | Type | Dimensions | Description |
|----------+-------+------------+-------------------------------------|
@ -208,9 +218,9 @@
*** Gaussian and Slater-type orbitals
We consider here basis functions centered on nuclei. Hence, we enable
the possibility to define /dummy atoms/ to place basis functions in
random positions.
We consider here basis functions centered on nuclei. Hence, it is
possibile to define /dummy atoms/ to place basis functions in
arbitrary positions.
The atomic basis set is defined as a list of shells. Each shell $s$ is
centered on a center $A$, possesses a given angular momentum $l$ and a
@ -553,50 +563,42 @@ power = [
* Orbitals
** Atomic orbitals (ao group)
Going from the atomic basis set to AOs implies a systematic
construction of all the angular functions of each shell. We
consider two cases for the angular functions: the real-valued
spherical harmonics, and the polynomials in Cartesian coordinates.
In the case of real spherical harmonics, the AOs are ordered as
$0, +1, -1, +2, -2, \dots, +m, -m$ (see [[https://en.wikipedia.org/wiki/Table_of_spherical_harmonics#Real_spherical_harmonics][Wikipedia]]).
In the case of polynomials we
impose the canonical (or alphabetical) ordering), i.e
\begin{eqnarray}
p & : & p_x, p_y, p_z \nonumber \\
d & : & d_{xx}, d_{xy}, d_{xz}, d_{yy}, d_{yz}, d_{zz} \nonumber \\
f & : & f_{xxx}, f_{xxy}, f_{xxz}, f_{xyy}, f_{xyz}, f_{xzz}, f_{yyy}, f_{yyz}, f_{yzz}, …f_{zzz} \nonumber \\
{\rm etc.} \nonumber
\end{eqnarray}
Note that there is no exception for $p$ orbitals in spherical
coordinates: the ordering is $0,+1,-1$ which corresponds $p_z, p_x, p_y$.
AOs are defined as
\[
\chi_i (\mathbf{r}) = \mathcal{N}_i\, P_{\eta(i)}(\mathbf{r})\, R_{s(i)} (\mathbf{r})
\chi_i (\mathbf{r}) = \mathcal{N}_i'\, P_{\eta(i)}(\mathbf{r})\, R_{s(i)} (\mathbf{r})
\]
where $i$ is the atomic orbital index,
$P$ encodes for either the
polynomials or the spherical harmonics, $s(i)$ returns the
shell on which the AO is expanded, and $\eta(i)$ denotes which
angular function is chosen.
$\mathcal{N}_i$ is a normalization factor that enables the
possibility to have different normalization coefficients within a
shell, as in the GAMESS convention where
$\mathcal{N}_{x^2} \ne \mathcal{N}_{xy}$ because
\[ \left[ \iiint \left(x-X_A \right)^2 R_{s(i)}
(\mathbf{r}) dx\, dy\, dz \right]^{-1/2} \ne
\left[ \iiint \left( x-X_A \right) \left( y-Y_A \right) R_{s(i)}
(\mathbf{r}) dx\, dy\, dz \right]^{-1/2}. \]
where $i$ is the atomic orbital index, $P$ refers to either
polynomials or spherical harmonics, and $s(i)$ specifies the shell
on which the AO is expanded.
$\eta(i)$ denotes the chosen angular function. The AOs can be
expressed using real spherical harmonics or polynomials in Cartesian
coordinates. In the case of real spherical harmonics, the AOs are
ordered as $0, +1, -1, +2, -2, \dots, + m, -m$ (see [[https://en.wikipedia.org/wiki/Table_of_spherical_harmonics#Real_spherical_harmonics][Wikipedia]]). In
the case of polynomials, the canonical (or alphabetical) ordering is
used,
| $p$ | $p_x, p_y, p_z$ |
| $d$ | $d_{xx}, d_{xy}, d_{xz}, d_{yy}, d_{yz}, d_{zz}$ |
| $f$ | $f_{xxx}, f_{xxy}, f_{xxz}, f_{xyy}, f_{xyz}$, |
| | $f_{xzz}, f_{yyy}, f_{yyz}, f_{yzz}, f_{zzz}$ |
| $\vdots$ | |
Note that for \(p\) orbitals in spherical coordinates, the ordering
is $0,+1,-1$ which corresponds to $p_z, p_x, p_y$.
$\mathcal{N}_i'$ is a normalization factor that allows for different
normalization coefficients within a single shell, as in the GAMESS
convention where each individual function is unit-normalized.
Using GAMESS convention, the normalization factor of the shell
$\mathcal{N}_d$ in the ~basis~ group is appropriate for instance
for the $d_z^2$ function (i.e.
$\mathcal{N}_{d}\equiv\mathcal{N}_{z^2}$) but not for the $d_{xy}$
AO, so the correction factor $\mathcal{N}_i'$ for $d_{xy}$ in the
~ao~ groups is the ratio $\frac{\mathcal{N}_{xy}}{\mathcal{N}_{z^2}}$.
In such a case, one should set the normalization of the shell (in
the [[Basis set (basis group)][Basis set]] section) to $\mathcal{N}_{z^2}$, which is the
normalization factor of the atomic orbitals in spherical coordinates.
The normalization factor of the $xy$ function which should be
introduced here should be $\frac{\mathcal{N}_{xy}}{\mathcal{N}_{z^2}}$.
#+NAME: ao
| Variable | Type | Dimensions | Description |
@ -679,9 +681,9 @@ power = [
\[ \langle p q \vert \hat{O} \vert r s \rangle \] in physicists
notation, where $p,q,r,s$ are indices over atomic orbitals.
# TODO: Physicist / Chemist functions
# Functions are provided to get the indices in physicists or chemists
# notation.
# TODO: Physicist / Chemist functions
# Functions are provided to get the indices in physicists or chemists
# notation.
- \[ \hat{W}_{\text{ee}} = \sum_{i=2}^{N_\text{elec}} \sum_{j=1}^{i-1} \frac{1}{\vert \mathbf{r}_i - \mathbf{r}_j \vert} \] : electron-electron repulsive potential operator.
- \[ \hat{W}^{lr}_{\text{ee}} = \sum_{i=2}^{N_\text{elec}}
@ -1085,12 +1087,12 @@ power = [
* Correlation factors
** Jastrow factor (jastrow group)
The Jastrow factor is an $N$-electron function to which the CI
expansion is multiplied: $\Psi = \Phi \times \exp(J)$,
The Jastrow factor is an $N$-electron function which multiplies the CI
expansion: $\Psi = \Phi \times \exp(J)$,
In the following, we use the notations $r_{ij} = |\mathbf{r}_i - \mathbf{r}_j|$ and
$R_{i\alpha} = |\mathbf{r}_i - \mathbf{R}_\alpha|$, where indices
$i$ and $j$ correspond to electrons and $\alpha$ to nuclei.
$i$ and $j$ refer to electrons and $\alpha$ to nuclei.
Parameters for multiple forms of Jastrow factors can be saved in
TREXIO files, and are described in the following sections. These
@ -1112,20 +1114,27 @@ power = [
$J_{\text{eN}}$ contains electron-nucleus terms:
\[
J_{\text{eN}}(\mathbf{r},\mathbf{R}) = \sum_{i=1}^{N_\text{elec}} \sum_{\alpha=1}^{N_\text{nucl}}
\frac{a_{1,\alpha}\, g_\alpha(R_{i\alpha})}{1+a_{2,\alpha}\, g_\alpha(R_{i\alpha})} +
\sum_{p=2}^{N_\text{ord}^a} a_{p+1,\alpha}\, [g_\alpha(R_{i\alpha})]^p - J_{eN}^\infty
J_{\text{eN}}(\mathbf{r},\mathbf{R}) = \sum_{i=1}^{N_\text{elec}} \sum_{\alpha=1}^{N_\text{nucl}}\left[
\frac{a_{1,\alpha}\, f_\alpha(R_{i\alpha})}{1+a_{2,\alpha}\,
f_\alpha(R_{i\alpha})} + \sum_{p=2}^{N_\text{ord}^a} a_{p+1,\alpha}\, [f_\alpha(R_{i\alpha})]^p - J_{\text{eN}}^\infty
\right]
\]
$J_{\text{ee}}$ contains electron-electron terms:
\[
J_{\text{ee}}(\mathbf{r}) =
\sum_{i=1}^{N_\text{elec}} \sum_{j=1}^{i-1}
\frac{b_1\, f(r_{ij})}{1+b_2\, f(r_{ij})} +
\sum_{p=2}^{N_\text{ord}^b} a_{p+1}\, [f(r_{ij})]^p - J_{ee}^\infty
\left[
\frac{\frac{1}{2}\big(1 + \delta^{\uparrow\downarrow}_{ij}\big)\,b_1\, f_{\text{ee}}(r_{ij})}{1+b_2\, f_{\text{ee}}(r_{ij})} +
\sum_{p=2}^{N_\text{ord}^b} b_{p+1}\, [f_{\text{ee}}(r_{ij})]^p - J_{\text{ee},ij}^\infty
\right]
\]
and $J_{\text{eeN}}$ contains electron-electron-Nucleus terms:
$\delta^{\uparrow\downarrow}_{ij}$ is zero when the electrons $i$ and
$j$ have the same spin, and one otherwise.
$J_{\text{eeN}}$ contains electron-electron-Nucleus terms:
\[
J_{\text{eeN}}(\mathbf{r},\mathbf{R}) =
@ -1135,28 +1144,41 @@ power = [
\sum_{p=2}^{N_{\text{ord}}}
\sum_{k=0}^{p-1}
\sum_{l=0}^{p-k-2\delta_{k,0}}
c_{lkp\alpha} \left[ f({r}_{ij}) \right]^k
c_{lkp\alpha} \left[ g_{\text{ee}}({r}_{ij}) \right]^k \nonumber \\
\left[ \left[ g_\alpha({R}_{i\alpha}) \right]^l + \left[ g_\alpha({R}_{j\alpha}) \right]^l \right]
\left[ g_\alpha({R}_{i\,\alpha}) \, g_\alpha({R}_{j\alpha}) \right]^{(p-k-l)/2}
\left[ g_\alpha({R}_{i\,\alpha}) \,
g_\alpha({R}_{j\alpha}) \right]^{(p-k-l)/2}
\]
$c_{lkp\alpha}$ are non-zero only when $p-k-l$ is even.
The terms $J_{\text{ee}}^\infty$ and $J_{\text{eN}}^\infty$ are shifts to ensure that
$J_{\text{ee}}$ and $J_{\text{eN}}$ have an asymptotic value of zero.
The terms $J_{\text{ee},ij}^\infty$ and $J_{\text{eN}}^\infty$ are shifts to ensure that
$J_{\text{eN}}$ and $J_{\text{ee}}$ have an asymptotic value of zero:
\[
J_{\text{eN}}^{\infty} =
\frac{a_{1,\alpha}\, \kappa_\alpha^{-1}}{1+a_{2,\alpha}\,
\kappa_\alpha^{-1}} + \sum_{p=2}^{N_\text{ord}^a} a_{p+1,\alpha}\, \kappa_\alpha^{-p}
\]
\[
J_{\text{ee},ij}^{\infty} =
\frac{\frac{1}{2}\big(1 + \delta^{\uparrow\downarrow}_{ij}\big)\,b_1\,
\kappa_{\text{ee}}^{-1}}{1+b_2\, \kappa_{\text{ee}}^{-1}} +
\sum_{p=2}^{N_\text{ord}^b} b_{p+1}\, \kappa_{\text{ee}}^{-p}
\]
$f$ and $g$ are scaling function defined as
\[
f(r) = \frac{1-e^{-\kappa\, r}}{\kappa} \text{ and }
g_\alpha(r) = e^{-\kappa_\alpha\, r}.
f_\alpha(r) = \frac{1-e^{-\kappa_\alpha\, r}}{\kappa_\alpha} \text{ and }
g_\alpha(r) = e^{-\kappa_\alpha\, r},
\]
*** Mu
[[https://aip.scitation.org/doi/10.1063/5.0044683][Mu-Jastrow]] is based on a one-parameter correlation factor that has
been introduced in the context of transcorrelated methods. This
correlation factor imposes the electron-electron cusp and it is
correlation factor imposes the electron-electron cusp, and it is
built such that the leading order in $1/r_{12}$ of the effective
two-electron potential reproduces the long-range interaction of the
range-separated density functional theory. Its analytical
@ -1167,15 +1189,15 @@ power = [
J_{\text{eN}}(\mathbf{r}, \mathbf{R})
\].
The electron-electron cusp is incorporated in the three-body term.
The electron-electron cusp is incorporated in the three-body term
\[
J_\text{eeN} (\mathbf{r}, \mathbf{R}) =
\sum_{i=1}^{N_\text{elec}} \sum_{j=1}^{i-1} \, u\left(\mu, r_{ij}\right) \,
\Pi_{\alpha=1}^{N_{\text{nucl}}} \, E_\alpha({R}_{i\alpha}) \, E_\alpha({R}_{j\alpha})
\Pi_{\alpha=1}^{N_{\text{nucl}}} \, E_\alpha({R}_{i\alpha}) \, E_\alpha({R}_{j\alpha}),
\]
$u$ is an electron-electron function given by the symmetric function
where ww$u$ is an electron-electron function
\[
u\left(\mu, r\right) = \frac{r}{2} \, \left[ 1 - \text{erf}(\mu\, r) \right] - \frac{1}{2 \, \mu \, \sqrt{\pi}} \exp \left[ -(\mu \, r)^2 \right].
@ -1186,7 +1208,7 @@ power = [
electrons.
An envelope function has been introduced to cancel out the Jastrow
effects between two-electrons when they are both close to a nucleus
effects between two-electrons when at least one is close to a nucleus
(to perform a frozen-core calculation). The envelope function is
given by
@ -1223,11 +1245,11 @@ power = [
| Variable | Type | Dimensions | Description |
|---------------+----------+---------------------+-----------------------------------------------------------------|
| ~type~ | ~string~ | | Type of Jastrow factor: ~CHAMP~ or ~Mu~ |
| ~ee_num~ | ~dim~ | | Number of Electron-electron parameters |
| ~en_num~ | ~dim~ | | Number of Electron-nucleus parameters |
| ~ee_num~ | ~dim~ | | Number of Electron-electron parameters |
| ~een_num~ | ~dim~ | | Number of Electron-electron-nucleus parameters |
| ~ee~ | ~float~ | ~(jastrow.ee_num)~ | Electron-electron parameters |
| ~en~ | ~float~ | ~(jastrow.en_num)~ | Electron-nucleus parameters |
| ~ee~ | ~float~ | ~(jastrow.ee_num)~ | Electron-electron parameters |
| ~een~ | ~float~ | ~(jastrow.een_num)~ | Electron-electron-nucleus parameters |
| ~en_nucleus~ | ~index~ | ~(jastrow.en_num)~ | Nucleus relative to the eN parameter |
| ~een_nucleus~ | ~index~ | ~(jastrow.een_num)~ | Nucleus relative to the eeN parameter |
@ -1241,11 +1263,11 @@ power = [
#+begin_src python :tangle trex.json
"jastrow": {
"type" : [ "string", [] ]
, "ee_num" : [ "dim" , [] ]
, "en_num" : [ "dim" , [] ]
, "ee_num" : [ "dim" , [] ]
, "een_num" : [ "dim" , [] ]
, "ee" : [ "float" , [ "jastrow.ee_num" ] ]
, "en" : [ "float" , [ "jastrow.en_num" ] ]
, "ee" : [ "float" , [ "jastrow.ee_num" ] ]
, "een" : [ "float" , [ "jastrow.een_num" ] ]
, "en_nucleus" : [ "index" , [ "jastrow.en_num" ] ]
, "een_nucleus" : [ "index" , [ "jastrow.een_num" ] ]