PAW-XML

PAW-XML is a data format for atomic datasets necessary for doing Projector Augmented-Wave calculations.

Current version of the PAW XML specification: 0.7

Codes implementing the PAW XML format: gpaw, abinit, atompaw

Introduction

This page contains information about the PAW-XML data format for the atomic datasets necessary for doing Projector Augmented-Wave calculations¹. We use the term dataset instead of pseudo-potential because the PAW method is not a pseudo-potential method.

An example XML file for nitrogen PAW dataset using LDA can be seen here: n.lda.xml

Note: Hartree atomic units are used in the XML file ($\hbar = m = e = 1$).

What defines a dataset?

The following quantities define a minimum PAW dataset (the notation from Ref. ² is used here):

The following quantities can be optionally provided:

Specification of the elements

An element looks like this:

<name> ... </name>

or for an empty element:

<name/>

Tip: An XML-tutorial can be found here.

The header

The first two lines should look like this:

<?xml version="1.0"?>
<paw_dataset version="0.7">

The first line must be present in all XML files. Everything else is put inside an element with name paw_dataset, and this element has an attribute called version.

A comment

It is recommended to put a comment giving the units and a link to this web page:

<!-- Nitrogen dataset for the Projector Augmented Wave method. -->
<!-- Units: Hartree and Bohr radii.                            -->
<!-- http://www.where.org/paw_dataset.html                     -->

The atom element

<atom symbol="N" Z="7" core="2" valence="5"/>

The atom element has attributes symbol, Z, core and valence (chemical symbol, atomic number, number of core electrons and number of valence electrons).

Exchange-correlation

The xc_functional element defines the exchange-correlation functional used for generating the dataset. It has the two attributes type and name.

The type attribute can be LDA, GGA, MGGA or HYB.

The name attribute designates the exchange-correlation functional and can be specified in the following ways:

• Taking the names from the Libxc library. The correlation and exchange names are stripped from their XC_ part and combined with a +-sign. Here is an example for an LDA functional:

  <xc_functional type="LDA", name="LDA_X+LDA_C_PW"/>
  

  and this is what PBE will look like:

  <xc_functional type="GGA", name="GGA_X_PBE+GGA_C_PBE"/>
  

• Using one of the following pre-defined aliases:

  type	name	Libxc equivalent	Reference
  LDA	PW	LDA_X+LDA_C_PW	LDA exchange; Perdew, Wang, PRB 45, 13244 (1992)
  GGA	PW91	GGA_X_PW91+GGA_C_PW91	Perdew et al PRB 46, 6671 (1992)
  GGA	PBE	GGA_X_PBE+GGA_C_PBE	Perdew, Burke, Ernzerhof, PRL 77, 3865 (1996)
  GGA	RPBE	GGA_X_RPBE+GGA_C_PBE	Hammer, Hansen, Nørskov, PRB 59, 7413 (1999)
  GGA	revPBE	GGA_X_PBE_R+GGA_C_PBE	Zhang, Yang, PRL 80, 890 (1998)
  GGA	PBEsol	GGA_X_PBE_SOL+GGA_C_PBE_SOL	Perdew et al, PRL 100, 136406 (2008)
  GGA	AM05	GGA_X_AM05+GGA_C_AM05	Armiento, Mattsson, PRB 72, 085108 (2005)
  GGA	BLYP	GGA_X_B88+GGA_C_LYP	Becke, PRA 38, 3098 (1988); Lee, Yang, Parr, PRB 37, 785 (1988)

  Examples:

  <xc_functional type="LDA", name="PW"/>
  

  <xc_functional type="GGA", name="PBE"/>
  

Generator

<generator type="scalar-relativistic" name="MyGenerator-2.0">
  Frozen core: [He]
</generator>

This element contains character data describing in words how the dataset was generated. The type attribute must be one of: non-relativistic, scalar-relativistic or relativistic.

Energies

<ae_energy kinetic="53.777460" xc="-6.127751"
           electrostatic="-101.690410" total="-54.040701"/>
<core_energy kinetic="43.529213"/>

The kinetic energy of the core electrons, $E^{kin}_c$, is used in the PAW method. The other energies are convenient to have for testing purposes and can also be useful for checking the quality of the underlying atomic calculation.

Valence states

<valence_states>
  <state n="2" l="0" f="2"  rc="1.10" e="-0.6766" id="N-2s"/>
  <state n="2" l="1" f="3"  rc="1.10" e="-0.2660" id="N-2p"/>
  <state       l="0"        rc="1.10" e=" 0.3234" id="N-s1"/>
  <state       l="1"        rc="1.10" e=" 0.7340" id="N-p1"/>
  <state       l="2"        rc="1.10" e=" 0.0000" id="N-d1"/>
</valence_states>

The valence_states element contains several state elements, defined by a unique id as well as l and n quantum numbers. For each of them it is also required to provide the energy e, the occupation f and the matching radius of the partial waves rc.

The number of state elements determines the size of the partial wave basis. It is equal to the number of radial functions (radial parts of the $\phi_i$, $\tilde{\phi}_i$ and $\tilde{p}_i$) and is noted $n_{waves}$ in the rest of this document.

For this dataset, the first two lines describe bound eigenstates with occupation numbers and principal quantum numbers. Notice, that the three additional unbound states should have no f and n attributes. In this way, we know that only the first two bound states (with f and n attributes) should be used for constructing an initial guess for the wave functions.

Radial grids

There can be one or more definitions of radial grids.

Example:

<radial_grid eq="r=d*i" d="0.1" istart="0" iend="9" id="g1">
  <values>
    0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
  </values>
  <derivatives>
    0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1
  </derivatives>
</radial_grid>

This defines one radial grid as $r_i = di$ where $i$ runs from 0 to 9. Inside the <radial_grid> element we have the 10 values of $r_i$ followed by the 10 values of the derivatives $dr_i/di$.

All functions (densities, potentials, ...) that use this grid are given as 10 numbers defining the radial part of the function. The radial part of the function must be multiplied by a spherical harmonics: $f_{\ell m}(\mathbf{r}) = f_\ell(r) Y_{\ell m}(\theta, \phi)$.

Each radial grid has a unique id:

<radial_grid eq="r=d*i" d="0.01" istart="0" iend="99" id="lin">
<radial_grid eq="r=a*exp(d*i)" a="1.056e-4" d="0.05" istart="0" iend="249" id="log">

and each numerical function must refer to one of these ids:

<function grid="lin">
  ... ...
</function>

In this example, the function element should contain 100 numbers (i = 0, ..., 99). Each number must be separated by a character or by one or more 's or 's (no commas). For numbers with scientific notation, use this format: 1.23456e-5 or 1.23456E-5 and not 1.23456D-5.

A program can read the values for $r_i$ and $dr_i/di$ from the file or evaluate them from the eq and associated parameter attributes. There are currently six types of radial grids:

The istart and iend attributes indicating the range of i should always be present.

Although it is possible to define as many radial grids as desired, it is recommended to minimize the number of grids in the dataset.

Shape function for the compensation charge

The general formulation of the compensation charge uses an expansion over the partial waves ij and the spherical harmonics:

$$\sum_{\ell m} C_{\ell m \ell_i m_i \ell_j m_j} \hat{Q}^{\ell}{i j}(r) Y{\ell m}(\theta, \phi),$$

where $C_{\ell m \ell_i m_i \ell_j m_j}$ is a Gaunt coefficient.

The standard expression ¹ for the shape function $\hat{Q}^{\ell}_{ij}(\mathbf{r})$ is a product of the multipole moment $Q^{\ell}_{i j}$ and a shape function $g_\ell(r)$:

$$\hat{Q}^{\ell}_{i j}(r) = Q^{\ell}_{i j} g_\ell(r)$$

Several formulations ¹³ define $g_\ell(r) \propto r^\ell k(r)$, where $k(r)$ is an $l$-independent shape function:

Example:

<shape_function type="gauss" rc="3.478505426185e-01">

Another formulation ⁴ defines directly $g_\ell(r)$:

For bessel the four parameters ($\alpha_1^\ell$, $q_1^\ell$, $\alpha_2^\ell$, $q_2^\ell$) must be determined from rc for each value of $l$ as described in Ref. ⁴.

Example:

<shape_function type="bessel" rc="3.478505426185e-01">

There is also a more general formulation where $\hat{Q}^{\ell}_{i j}(r)$ is given in a numerical form. Several shape functions can be set (with the <shape_function> tag), depending on $l$ and/or combinations of partial waves (specified using the optional state1 and state2 attributes). See for instance section II.C of Ref. ⁵.

Example 1, defining numerically $g_\ell(r)$ in $\hat{Q}^{\ell}_{ij}(r)=Q^{\ell}_{i j} g_\ell(r)$:

<shape_function type="numeric" l=0 grid="g1">
  ... ...
</shape_function>

Example 2, defining directly $\hat{Q}^{\ell}_{i j}(r)$ for states $i$=N-2s and $j$=N-2p, and $l$=0:

<shape_function type="numeric" l=0 state1="N-2s" state2="N-2p" grid="g1">
  ... ...
</shape_function>

Radial functions

Continuing, we have now reached the all-electron (resp. pseudo core, pseudo valence) density:

<ae_core_density grid="g1">
  6.801207147443e+02 6.801207147443e+02 6.665042896724e+02
  ... ...
</ae_core_density>

<pseudo_core_density rc="1.1" grid="g1">
  ... ...
</pseudo_core_density>

<pseudo_valence_density rc="1.1" grid="g1">
  ... ...
</pseudo_valence_density>

The numbers inside the ae_core_density (resp. pseudo_core_density, pseudo_valence_density) element defines the radial part of $n_c(\mathbf{r})$ (resp. $\tilde{n}_c(\mathbf{r})$, $\tilde{n}_v(\mathbf{r})$). The radial part must be multiplied by $Y_{00} = (4\pi)^{-1/2}$ to get the full density ($Y_{00}n_c(\mathbf{r})$ should integrate to the number of core electrons). The pseudo core density and the pseudo valence density are defined similarly and also have a rc attribute specifying the matching radius.

The ae_partial_wave, pseudo_partial_wave and projector_function elements contain the radial parts of the $\phi_i(\mathbf{r})$, $\tilde{\phi}_i(\mathbf{r})$ and $\tilde{p}_i(\mathbf{r})$ functions for the states listed in the valence_states element above (five states in the nitrogen example). All functions must have an attribute state="..." referring to one of the states listed in the valence_states element:

<ae_partial_wave state="N-2s" grid="g1">
  -8.178800366898029e+00 -8.178246914143839e+00 -8.177654917302689e+00
  ... ...
</ae_partial_wave>

<pseudo_partial_wave state="N-2s" grid="g1">
  ... ...
</pseudo_partial_wave>

<projector_function state="N-2s" grid="g1">
  ... ...
</projector_function>

<ae_partial_wave state="N-2p" grid="g1">
  ... ...
</ae_partial_wave>

Remember that the radial part of these functions must be multiplied by a spherical harmonics:

$$\phi_i(\mathbf{r}) = \phi_i(r) Y_{\ell_i m_i}(\theta, \phi)$$

Zero potential

The zero potential, $\bar{v}$ (see section VI.D of Ref. ¹) is defined similarly to the densities; the radial part must be multiplied by $Y_{00} = (4\pi)^{-1/2}$ to get the full potential. The zero_potential element has a rc attribute specifying the cut-off radius of $\bar{v}(\mathbf{r})$:

<zero_potential rc="1.1" grid="g1">
  ... ...
</zero_potential>

The Kresse-Joubert formulation

The Kresse-Joubert formulation of the PAW method ⁴ is very similar to the original formulation of Blöchl ¹. However, the Kresse-Joubert formulation does not use $\bar{v}$ directly, but indirectly through the local ionic pseudopotential, $v_H[\tilde{n}_{Zc}]$. Therefore, the following transformation is necessary:

$$v1_H[\tilde{n}_{Zc}] = v_H[\tilde{n}_c + (N_c - Z - \tilde{N}_c) g_{00} Y_{00}] + \bar{v} + v_{xc}[\tilde{n}_v + \tilde{n}_c] - v_{xc}[\tilde{n}_v + \tilde{n}_c + (N_v - \tilde{N}_v - \tilde{N}_c) g_{00} Y_{00}],$$

where $N_c$ is the number of core electrons, $N_v$ is the number of valence electrons, $\tilde{N}_c$ is the number of electrons contained in the pseudo core density and $\tilde{N}_v$ is the number of electrons contained in the pseudo valence density.

The Hartree potential from the density $n$ is defined as:

$$v_H[n](r_1) = 4\pi \int_0^\infty r_2^2 dr_2 \frac{n(r_2)}{r_>},$$

where $r_>$ is the larger of $r_1$ and $r_2$.

Note: In the Kresse-Joubert formulation, the symbol $\tilde{n}$ is used for what we here call $\tilde{n}_v$ and in the Blöchl formulation, we have $\tilde{n} = \tilde{n}_c + \tilde{n}_v$.

It is also possible to add an element kresse_joubert_local_ionic_pseudopotential that contains the $v_H[\tilde{n}_{Zc}](r)$ function directly, so that no conversion is necessary:

<kresse_joubert_local_ionic_pseudopotential rc="1.3" grid="log">
  ... ...
</kresse_joubert_local_ionic_pseudopotential>

The kresse_joubert_local_ionic_pseudopotential element has a rc attribute specifying the matching radius. This matching radius corresponds to the maximum of all the matching radii used in the formalism.

Kinetic energy differences

<kinetic_energy_differences>
  1.744042161013e+00 0.000000000000e+00 2.730637956456e+00
  ... ...
<kinetic_energy_differences>

This element contains the symmetric $\Delta E^\text{kin}_{ij}$ matrix:

$$\Delta E^\text{kin}_{ij} = \langle \phi_i | \hat{T} | \phi_j \rangle - \langle \tilde{\phi}_i | \hat{T} | \tilde{\phi}_j \rangle$$

where $\hat{T}$ is the kinetic energy operator used by the generator.

With $n_{waves}$ partial waves (see $n_{waves}$ definition), we have a $n_{waves} \times n_{waves}$ matrix listed as $n_{waves}^2$ numbers.

Meta-GGA

Datasets for use with MGGA functionals must also include information on the core kinetic energy density and pseudo core kinetic energy density ; the latters are defined with these two elements:

<ae_core_kinetic_energy_density grid="g1"> 
  ... ...
</ae_core_kinetic_energy_density> 
<pseudo_core_kinetic_energy_density rc="1.1" grid="g1">
  ... ...
</pseudo_core_kinetic_energy_density>

These densities are defined similarly to the core and valence densities (see above). The pseudo_core_kinetic_energy_density element has a rc attribute specifying its matching radius.

Exact exchange integrals

The core-core contribution to the exact exchange energy $X^{\text{core-core}}$ and the symmetric core-valence PAW-correction matrix $X_{ij}^{\text{core-valence}}$ are given as:

$$X^{\text{core-core}} = -\frac{1}{4}\sum_{cc'} \iint d\mathbf{r} d\mathbf{r}' \frac{\phi_c(\mathbf{r})\phi_{c'}(\mathbf{r}) \phi_c(\mathbf{r}')\phi_{c'}(\mathbf{r}')} {|\mathbf{r}-\mathbf{r}'|}$$

$$X_{ij}^{\text{core-valence}} = -\frac{1}{2}\sum_c \iint d\mathbf{r} d\mathbf{r}' \frac{\phi_i(\mathbf{r})\phi_c(\mathbf{r}) \phi_j(\mathbf{r}')\phi_c(\mathbf{r}')} {|\mathbf{r}-\mathbf{r}'|}$$

The $X_{ij}^{\text{core-valence}}$ coefficients depend only on pairs of the radial basis functions $\phi_i(r)$ and can be evaluated by summing over radial integrals times $3-j$ symbols according to:

$$X_{ij}^{\text{core-valence}} = -\delta_{l_i l_j} \delta_{m_i m_j} \sum_{c L} \frac{N_c}{2} {\begin{pmatrix}l_c & L & l_i \\ 0 & 0 & 0\end{pmatrix}}^2 \int r^2 dr \int {r'}^2 d{r'} \frac{r^{L}_{<}}{r^{L+1}_{>}} \phi_i(r) \phi_c(r) \phi_j(r') \phi_c(r'),$$

where $N_{c}$ is the number of core electrons corresponding to $l_{c}$, namely $N_c=2(2l_c+1)$, $r_{>}$ (resp. $r_{<}$) is the larger (resp. smaller) of $r$ and $r'$.

$X^{\text{core-core}}$ can be specified in the core attribute of the <exact_exchange> element.

With $n_{waves}$ valence states (see $n_{waves}$ definition), $X_{ij}^{\text{core-valence}}$ is a $n_{waves} \times n_{waves}$ matrix. It can be specified as $n_{waves}^2$ numbers inside the <exact_exchange> element:

<exact_exchange core="...">
  ... ...
</exact_exchange>

Optional elements

<paw_radius rc="2.3456781234">

Although not necessary, it may be helpful to provide the following item(s) in the dataset:

• Radius of the PAW augmentation region paw_radius

  This radius defines the region (around the atom) outside which all pseudo quantities are equal to the all-electron ones. It is equal to the maximum of all the cut-off and matching radii. Note that -- for better lisibility -- the paw_radius element should be provided in the header of the file.

End of the dataset

</paw_dataset>

How to use the datasets

Most likely, the radial functions will be needed on some other type of radial grid than the one used in the dataset. The idea is that one should read in the radial functions and then transform them to the radial grids used by the specific implementation. After the transformation, some sort of normalization may be necessary.

Plotting the radial functions

The first 10-20 lines of the XML-datasets, should be pretty much human readable, and should give an overview of what kind of dataset it is and how it was generated. The remaining part of the file contains numerical data for all the radial functions. To get an overview of these functions, you can extract that data with the pawxml.py program and then pass it on to your favorite plotting tool.

Note: The pawxml.py program is very primitive and is only included in order to demonstrates how to parse XML using SAX from a Python program. Parsing XML from Fortran or C code with SAX should be similar.

Usage

It works like this:

$ pawxml.py [options] dataset[.gz]

Examples:

[~]$ pawxml.py -x pseudo_core_density N.LDA | xmgrace -
[~]$ pawxml.py -x ae_partial_wave -s N2p N.LDA > N.ae.2p 
[~]$ pawxml.py -x pseudo_partial_wave -s N2p N.LDA > N.ps.2p 
[~]$ xmgrace N.??.2p

References