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Fixed documentation
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@ -144,8 +144,18 @@
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** Electron (electron group)
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We consider wave functions expressed in the spin-free formalism, where
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the number of \uparrow and \downarrow electrons is fixed.
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The chemical system consists of nuclei and electrons, where the
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nuclei are considered as fixed point charges with Cartesian
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coordinates. The wave function is stored in the spin-free
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formalism, and therefore, it is necessary for the user to
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explicitly store the number of electrons with spin up
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($N_\uparrow$) and spin down ($N_\downarrow$). These numbers
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correspond to the normalization of the spin-up and spin-down
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single-particle reduced density matrices.
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We consider wave functions expressed in the spin-free formalism, where
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the number of \uparrow and \downarrow electrons is fixed.
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#+NAME:electron
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| Variable | Type | Dimensions | Description |
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@ -208,9 +218,9 @@
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*** Gaussian and Slater-type orbitals
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We consider here basis functions centered on nuclei. Hence, we enable
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the possibility to define /dummy atoms/ to place basis functions in
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random positions.
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We consider here basis functions centered on nuclei. Hence, it is
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possibile to define /dummy atoms/ to place basis functions in
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arbitrary positions.
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The atomic basis set is defined as a list of shells. Each shell $s$ is
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centered on a center $A$, possesses a given angular momentum $l$ and a
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@ -553,50 +563,42 @@ power = [
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* Orbitals
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** Atomic orbitals (ao group)
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Going from the atomic basis set to AOs implies a systematic
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construction of all the angular functions of each shell. We
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consider two cases for the angular functions: the real-valued
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spherical harmonics, and the polynomials in Cartesian coordinates.
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In the case of real spherical harmonics, the AOs are ordered as
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$0, +1, -1, +2, -2, \dots, +m, -m$ (see [[https://en.wikipedia.org/wiki/Table_of_spherical_harmonics#Real_spherical_harmonics][Wikipedia]]).
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In the case of polynomials we
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impose the canonical (or alphabetical) ordering), i.e
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\begin{eqnarray}
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p & : & p_x, p_y, p_z \nonumber \\
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d & : & d_{xx}, d_{xy}, d_{xz}, d_{yy}, d_{yz}, d_{zz} \nonumber \\
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f & : & f_{xxx}, f_{xxy}, f_{xxz}, f_{xyy}, f_{xyz}, f_{xzz}, f_{yyy}, f_{yyz}, f_{yzz}, …f_{zzz} \nonumber \\
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{\rm etc.} \nonumber
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\end{eqnarray}
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Note that there is no exception for $p$ orbitals in spherical
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coordinates: the ordering is $0,+1,-1$ which corresponds $p_z, p_x, p_y$.
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AOs are defined as
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\[
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\chi_i (\mathbf{r}) = \mathcal{N}_i\, P_{\eta(i)}(\mathbf{r})\, R_{s(i)} (\mathbf{r})
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\chi_i (\mathbf{r}) = \mathcal{N}_i'\, P_{\eta(i)}(\mathbf{r})\, R_{s(i)} (\mathbf{r})
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\]
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where $i$ is the atomic orbital index,
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$P$ encodes for either the
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polynomials or the spherical harmonics, $s(i)$ returns the
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shell on which the AO is expanded, and $\eta(i)$ denotes which
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angular function is chosen.
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$\mathcal{N}_i$ is a normalization factor that enables the
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possibility to have different normalization coefficients within a
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shell, as in the GAMESS convention where
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$\mathcal{N}_{x^2} \ne \mathcal{N}_{xy}$ because
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\[ \left[ \iiint \left(x-X_A \right)^2 R_{s(i)}
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(\mathbf{r}) dx\, dy\, dz \right]^{-1/2} \ne
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\left[ \iiint \left( x-X_A \right) \left( y-Y_A \right) R_{s(i)}
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(\mathbf{r}) dx\, dy\, dz \right]^{-1/2}. \]
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where $i$ is the atomic orbital index, $P$ refers to either
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polynomials or spherical harmonics, and $s(i)$ specifies the shell
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on which the AO is expanded.
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$\eta(i)$ denotes the chosen angular function. The AOs can be
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expressed using real spherical harmonics or polynomials in Cartesian
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coordinates. In the case of real spherical harmonics, the AOs are
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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
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the case of polynomials, the canonical (or alphabetical) ordering is
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used,
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| $p$ | $p_x, p_y, p_z$ |
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| $d$ | $d_{xx}, d_{xy}, d_{xz}, d_{yy}, d_{yz}, d_{zz}$ |
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| $f$ | $f_{xxx}, f_{xxy}, f_{xxz}, f_{xyy}, f_{xyz}$, |
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| | $f_{xzz}, f_{yyy}, f_{yyz}, f_{yzz}, f_{zzz}$ |
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| $\vdots$ | |
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Note that for \(p\) orbitals in spherical coordinates, the ordering
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is $0,+1,-1$ which corresponds to $p_z, p_x, p_y$.
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$\mathcal{N}_i'$ is a normalization factor that allows for different
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normalization coefficients within a single shell, as in the GAMESS
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convention where each individual function is unit-normalized.
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Using GAMESS convention, the normalization factor of the shell
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$\mathcal{N}_d$ in the ~basis~ group is appropriate for instance
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for the $d_z^2$ function (i.e.
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$\mathcal{N}_{d}\equiv\mathcal{N}_{z^2}$) but not for the $d_{xy}$
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AO, so the correction factor $\mathcal{N}_i'$ for $d_{xy}$ in the
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~ao~ groups is the ratio $\frac{\mathcal{N}_{xy}}{\mathcal{N}_{z^2}}$.
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In such a case, one should set the normalization of the shell (in
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the [[Basis set (basis group)][Basis set]] section) to $\mathcal{N}_{z^2}$, which is the
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normalization factor of the atomic orbitals in spherical coordinates.
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The normalization factor of the $xy$ function which should be
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introduced here should be $\frac{\mathcal{N}_{xy}}{\mathcal{N}_{z^2}}$.
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#+NAME: ao
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| Variable | Type | Dimensions | Description |
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@ -637,18 +639,18 @@ power = [
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over atomic orbitals.
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#+NAME: ao_1e_int
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| Variable | Type | Dimensions | Description |
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|-----------------------+---------+--------------------+--------------------------------------------------------------------------|
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| ~overlap~ | ~float~ | ~(ao.num, ao.num)~ | $\langle p \vert q \rangle$ |
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| ~kinetic~ | ~float~ | ~(ao.num, ao.num)~ | $\langle p \vert \hat{T}_e \vert q \rangle$ |
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| ~potential_n_e~ | ~float~ | ~(ao.num, ao.num)~ | $\langle p \vert \hat{V}_{\text{ne}} \vert q \rangle$ |
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| ~ecp~ | ~float~ | ~(ao.num, ao.num)~ | $\langle p \vert \hat{V}_{\text{ecp}} \vert q \rangle$ |
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| ~core_hamiltonian~ | ~float~ | ~(ao.num, ao.num)~ | $\langle p \vert \hat{h} \vert q \rangle$ |
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| ~overlap_im~ | ~float~ | ~(ao.num, ao.num)~ | $\langle p \vert q \rangle$ (imaginary part) |
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| ~kinetic_im~ | ~float~ | ~(ao.num, ao.num)~ | $\langle p \vert \hat{T}_e \vert q \rangle$ (imaginary part) |
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| ~potential_n_e_im~ | ~float~ | ~(ao.num, ao.num)~ | $\langle p \vert \hat{V}_{\text{ne}} \vert q \rangle$ (imaginary part) |
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| ~ecp_im~ | ~float~ | ~(ao.num, ao.num)~ | $\langle p \vert \hat{V}_{\text{ECP}} \vert q \rangle$ (imaginary part) |
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| ~core_hamiltonian_im~ | ~float~ | ~(ao.num, ao.num)~ | $\langle p \vert \hat{h} \vert q \rangle$ (imaginary part) |
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| Variable | Type | Dimensions | Description |
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|-----------------------+---------+--------------------+--------------------------------------------------------------------------|
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| ~overlap~ | ~float~ | ~(ao.num, ao.num)~ | $\langle p \vert q \rangle$ |
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| ~kinetic~ | ~float~ | ~(ao.num, ao.num)~ | $\langle p \vert \hat{T}_e \vert q \rangle$ |
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| ~potential_n_e~ | ~float~ | ~(ao.num, ao.num)~ | $\langle p \vert \hat{V}_{\text{ne}} \vert q \rangle$ |
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| ~ecp~ | ~float~ | ~(ao.num, ao.num)~ | $\langle p \vert \hat{V}_{\text{ecp}} \vert q \rangle$ |
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| ~core_hamiltonian~ | ~float~ | ~(ao.num, ao.num)~ | $\langle p \vert \hat{h} \vert q \rangle$ |
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| ~overlap_im~ | ~float~ | ~(ao.num, ao.num)~ | $\langle p \vert q \rangle$ (imaginary part) |
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| ~kinetic_im~ | ~float~ | ~(ao.num, ao.num)~ | $\langle p \vert \hat{T}_e \vert q \rangle$ (imaginary part) |
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| ~potential_n_e_im~ | ~float~ | ~(ao.num, ao.num)~ | $\langle p \vert \hat{V}_{\text{ne}} \vert q \rangle$ (imaginary part) |
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| ~ecp_im~ | ~float~ | ~(ao.num, ao.num)~ | $\langle p \vert \hat{V}_{\text{ECP}} \vert q \rangle$ (imaginary part) |
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| ~core_hamiltonian_im~ | ~float~ | ~(ao.num, ao.num)~ | $\langle p \vert \hat{h} \vert q \rangle$ (imaginary part) |
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#+CALL: json(data=ao_1e_int, title="ao_1e_int")
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@ -679,9 +681,9 @@ power = [
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\[ \langle p q \vert \hat{O} \vert r s \rangle \] in physicists
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notation, where $p,q,r,s$ are indices over atomic orbitals.
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# TODO: Physicist / Chemist functions
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# Functions are provided to get the indices in physicists or chemists
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# notation.
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# TODO: Physicist / Chemist functions
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# Functions are provided to get the indices in physicists or chemists
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# notation.
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- \[ \hat{W}_{\text{ee}} = \sum_{i=2}^{N_\text{elec}} \sum_{j=1}^{i-1} \frac{1}{\vert \mathbf{r}_i - \mathbf{r}_j \vert} \] : electron-electron repulsive potential operator.
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- \[ \hat{W}^{lr}_{\text{ee}} = \sum_{i=2}^{N_\text{elec}}
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@ -695,14 +697,14 @@ power = [
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\]
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#+NAME: ao_2e_int
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| Variable | Type | Dimensions | Description |
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|-----------------------+----------------+---------------------------------------------------+-----------------------------------------------|
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| ~eri~ | ~float sparse~ | ~(ao.num, ao.num, ao.num, ao.num)~ | Electron repulsion integrals |
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| ~eri_lr~ | ~float sparse~ | ~(ao.num, ao.num, ao.num, ao.num)~ | Long-range electron repulsion integrals |
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| ~eri_cholesky_num~ | ~dim~ | | Number of Cholesky vectors for ERI |
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| ~eri_cholesky~ | ~float sparse~ | ~(ao.num, ao.num, ao_2e_int.eri_cholesky_num)~ | Cholesky decomposition of the ERI |
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| ~eri_lr_cholesky_num~ | ~dim~ | | Number of Cholesky vectors for long range ERI |
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| ~eri_lr_cholesky~ | ~float sparse~ | ~(ao.num, ao.num, ao_2e_int.eri_lr_cholesky_num)~ | Cholesky decomposition of the long range ERI |
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| Variable | Type | Dimensions | Description |
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|-----------------------+----------------+---------------------------------------------------+-----------------------------------------------|
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| ~eri~ | ~float sparse~ | ~(ao.num, ao.num, ao.num, ao.num)~ | Electron repulsion integrals |
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| ~eri_lr~ | ~float sparse~ | ~(ao.num, ao.num, ao.num, ao.num)~ | Long-range electron repulsion integrals |
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| ~eri_cholesky_num~ | ~dim~ | | Number of Cholesky vectors for ERI |
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| ~eri_cholesky~ | ~float sparse~ | ~(ao.num, ao.num, ao_2e_int.eri_cholesky_num)~ | Cholesky decomposition of the ERI |
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| ~eri_lr_cholesky_num~ | ~dim~ | | Number of Cholesky vectors for long range ERI |
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| ~eri_lr_cholesky~ | ~float sparse~ | ~(ao.num, ao.num, ao_2e_int.eri_lr_cholesky_num)~ | Cholesky decomposition of the long range ERI |
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#+CALL: json(data=ao_2e_int, title="ao_2e_int")
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@ -1085,12 +1087,12 @@ power = [
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* Correlation factors
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** Jastrow factor (jastrow group)
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The Jastrow factor is an $N$-electron function to which the CI
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expansion is multiplied: $\Psi = \Phi \times \exp(J)$,
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The Jastrow factor is an $N$-electron function which multiplies the CI
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expansion: $\Psi = \Phi \times \exp(J)$,
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In the following, we use the notations $r_{ij} = |\mathbf{r}_i - \mathbf{r}_j|$ and
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$R_{i\alpha} = |\mathbf{r}_i - \mathbf{R}_\alpha|$, where indices
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$i$ and $j$ correspond to electrons and $\alpha$ to nuclei.
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$i$ and $j$ refer to electrons and $\alpha$ to nuclei.
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Parameters for multiple forms of Jastrow factors can be saved in
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TREXIO files, and are described in the following sections. These
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@ -1109,54 +1111,74 @@ power = [
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\]
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$J_{\text{eN}}$ contains electron-nucleus terms:
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$J_{\text{eN}}$ contains electron-nucleus terms:
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\[
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J_{\text{eN}}(\mathbf{r},\mathbf{R}) = \sum_{i=1}^{N_\text{elec}} \sum_{\alpha=1}^{N_\text{nucl}}
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\frac{a_{1,\alpha}\, g_\alpha(R_{i\alpha})}{1+a_{2,\alpha}\, g_\alpha(R_{i\alpha})} +
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\sum_{p=2}^{N_\text{ord}^a} a_{p+1,\alpha}\, [g_\alpha(R_{i\alpha})]^p - J_{eN}^\infty
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J_{\text{eN}}(\mathbf{r},\mathbf{R}) = \sum_{i=1}^{N_\text{elec}} \sum_{\alpha=1}^{N_\text{nucl}}\left[
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\frac{a_{1,\alpha}\, f_\alpha(R_{i\alpha})}{1+a_{2,\alpha}\,
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f_\alpha(R_{i\alpha})} + \sum_{p=2}^{N_\text{ord}^a} a_{p+1,\alpha}\, [f_\alpha(R_{i\alpha})]^p - J_{\text{eN}}^\infty
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\right]
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\]
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$J_{\text{ee}}$ contains electron-electron terms:
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\[
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J_{\text{ee}}(\mathbf{r}) =
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\sum_{i=1}^{N_\text{elec}} \sum_{j=1}^{i-1}
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\frac{b_1\, f(r_{ij})}{1+b_2\, f(r_{ij})} +
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\sum_{p=2}^{N_\text{ord}^b} a_{p+1}\, [f(r_{ij})]^p - J_{ee}^\infty
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\left[
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\frac{\frac{1}{2}\big(1 + \delta^{\uparrow\downarrow}_{ij}\big)\,b_1\, f_{\text{ee}}(r_{ij})}{1+b_2\, f_{\text{ee}}(r_{ij})} +
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\sum_{p=2}^{N_\text{ord}^b} b_{p+1}\, [f_{\text{ee}}(r_{ij})]^p - J_{\text{ee},ij}^\infty
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\right]
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\]
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$\delta^{\uparrow\downarrow}_{ij}$ is zero when the electrons $i$ and
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$j$ have the same spin, and one otherwise.
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and $J_{\text{eeN}}$ contains electron-electron-Nucleus terms:
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$J_{\text{eeN}}$ contains electron-electron-Nucleus terms:
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\[
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J_{\text{eeN}}(\mathbf{r},\mathbf{R}) =
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J_{\text{eeN}}(\mathbf{r},\mathbf{R}) =
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\sum_{\alpha=1}^{N_{\text{nucl}}}
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\sum_{i=1}^{N_{\text{elec}}}
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\sum_{j=1}^{i-1}
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\sum_{p=2}^{N_{\text{ord}}}
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\sum_{k=0}^{p-1}
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\sum_{l=0}^{p-k-2\delta_{k,0}}
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c_{lkp\alpha} \left[ f({r}_{ij}) \right]^k
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\left[ \left[ g_\alpha({R}_{i\alpha}) \right]^l + \left[ g_\alpha({R}_{j\alpha}) \right]^l \right]
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\left[ g_\alpha({R}_{i\,\alpha}) \, g_\alpha({R}_{j\alpha}) \right]^{(p-k-l)/2}
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c_{lkp\alpha} \left[ g_{\text{ee}}({r}_{ij}) \right]^k \nonumber \\
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\left[ \left[ g_\alpha({R}_{i\alpha}) \right]^l + \left[ g_\alpha({R}_{j\alpha}) \right]^l \right]
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\left[ g_\alpha({R}_{i\,\alpha}) \,
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g_\alpha({R}_{j\alpha}) \right]^{(p-k-l)/2}
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\]
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$c_{lkp\alpha}$ are non-zero only when $p-k-l$ is even.
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The terms $J_{\text{ee}}^\infty$ and $J_{\text{eN}}^\infty$ are shifts to ensure that
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$J_{\text{ee}}$ and $J_{\text{eN}}$ have an asymptotic value of zero.
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The terms $J_{\text{ee},ij}^\infty$ and $J_{\text{eN}}^\infty$ are shifts to ensure that
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$J_{\text{eN}}$ and $J_{\text{ee}}$ have an asymptotic value of zero:
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\[
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J_{\text{eN}}^{\infty} =
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\frac{a_{1,\alpha}\, \kappa_\alpha^{-1}}{1+a_{2,\alpha}\,
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\kappa_\alpha^{-1}} + \sum_{p=2}^{N_\text{ord}^a} a_{p+1,\alpha}\, \kappa_\alpha^{-p}
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\]
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\[
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J_{\text{ee},ij}^{\infty} =
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\frac{\frac{1}{2}\big(1 + \delta^{\uparrow\downarrow}_{ij}\big)\,b_1\,
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\kappa_{\text{ee}}^{-1}}{1+b_2\, \kappa_{\text{ee}}^{-1}} +
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\sum_{p=2}^{N_\text{ord}^b} b_{p+1}\, \kappa_{\text{ee}}^{-p}
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\]
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$f$ and $g$ are scaling function defined as
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\[
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f(r) = \frac{1-e^{-\kappa\, r}}{\kappa} \text{ and }
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g_\alpha(r) = e^{-\kappa_\alpha\, r}.
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f_\alpha(r) = \frac{1-e^{-\kappa_\alpha\, r}}{\kappa_\alpha} \text{ and }
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g_\alpha(r) = e^{-\kappa_\alpha\, r},
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\]
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*** Mu
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[[https://aip.scitation.org/doi/10.1063/5.0044683][Mu-Jastrow]] is based on a one-parameter correlation factor that has
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been introduced in the context of transcorrelated methods. This
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correlation factor imposes the electron-electron cusp and it is
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correlation factor imposes the electron-electron cusp, and it is
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built such that the leading order in $1/r_{12}$ of the effective
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two-electron potential reproduces the long-range interaction of the
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range-separated density functional theory. Its analytical
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@ -1167,15 +1189,15 @@ power = [
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J_{\text{eN}}(\mathbf{r}, \mathbf{R})
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\].
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The electron-electron cusp is incorporated in the three-body term.
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The electron-electron cusp is incorporated in the three-body term
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\[
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J_\text{eeN} (\mathbf{r}, \mathbf{R}) =
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\sum_{i=1}^{N_\text{elec}} \sum_{j=1}^{i-1} \, u\left(\mu, r_{ij}\right) \,
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\Pi_{\alpha=1}^{N_{\text{nucl}}} \, E_\alpha({R}_{i\alpha}) \, E_\alpha({R}_{j\alpha})
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\Pi_{\alpha=1}^{N_{\text{nucl}}} \, E_\alpha({R}_{i\alpha}) \, E_\alpha({R}_{j\alpha}),
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\]
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$u$ is an electron-electron function given by the symmetric function
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where ww$u$ is an electron-electron function
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\[
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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].
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@ -1186,7 +1208,7 @@ power = [
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electrons.
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An envelope function has been introduced to cancel out the Jastrow
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effects between two-electrons when they are both close to a nucleus
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effects between two-electrons when at least one is close to a nucleus
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(to perform a frozen-core calculation). The envelope function is
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given by
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@ -1223,11 +1245,11 @@ power = [
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| Variable | Type | Dimensions | Description |
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|---------------+----------+---------------------+-----------------------------------------------------------------|
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| ~type~ | ~string~ | | Type of Jastrow factor: ~CHAMP~ or ~Mu~ |
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| ~ee_num~ | ~dim~ | | Number of Electron-electron parameters |
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| ~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" ] ]
|
||||
|
||||
Loading…
Reference in New Issue
Block a user