====== XML specification for atomic PAW datasets ======
//Current version of the specification: **0.7**// \\ //Codes implementing the PAW XML format: [[https://wiki.fysik.dtu.dk/gpaw|gpaw]], [[http://www.abinit.org|abinit]], [[http://users.wfu.edu/natalie/papers/pwpaw/man.html|atompaw]]// \\ \\
===== Introduction =====
This page contains information about the PAW-XML data format for the atomic datasets necessary for doing Projector Augmented-Wave calculations [(BLOCHL1994)]. 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: {{atomicdata:specs:n.lda.xml| n.lda.xml}}.
Note: Hartree atomic units are used in the XML file (ℏ = m = e = 1).
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===== What defines a dataset? =====
The following quantities define a minimum PAW dataset (the notation from Ref. [(BLOCHL2003)] is used here): ^Quantity ^Description ^ |Z |Atomic number | |$E_\text{XC}[n]$ |Exchange-correlation functional | |$E^\text{kin}_c$ |Kinetic energy of the core electrons | |$g_{\ell m}(\mathbf{r})$ |Shape function for compensation charge| |$n_c(r)$ |All-electron core density | |$\tilde{n}_c(r)$ |Pseudo electron core density | |$\tilde{n}_v(r)$ |Pseudo electron valence density | |$\bar{v}(r)$ |Zero potential | |$\phi_i(\mathbf{r})$ |All-electron partial waves | |$\tilde{\phi}_i(\mathbf{r})$|Pseudo partial waves | |$\tilde{p}_i(\mathbf{r})$ |Projector functions | |$\Delta E^\text{kin}_{ij}$ |Kinetic energy differences | \\ The following quantities can be optionally provided: ^Quantity ^Description ^ |$r_{PAW}$ |Radius of the PAW augmentation region (max. of matching radii)| |$v_H[\tilde{n}_{Zc}](r)$ |Kresse-Joubert local ionic pseudopotential | |$\tilde{Q}_{ij}(\mathbf{r})$|State-dependent shape function for compensation charge | |$\tau_c(r)$ |Core kinetic energy density | |$\tilde{\tau}_c(r)$ |Pseudo core kinetic energy density | |$X^{\text{core-core}}$ |Core-core contribution to exact exchange | |$X_{ij}^{\text{core-val}}$ |Core-valence exact-exchange correction matrix | \\ \\
===== Specification of the elements =====
An element looks like this: ... or for an empty element:
Tip: An XML-tutorial can be found here.
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===== The header =====
The first two lines should look like this: 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: \\ \\
===== The ''atom'' element =====
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 [[http://www.tddft.org/programs/octopus/wiki/index.php/Libxc:manual#Available_functionals|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: and this is what PBE will look like:
  • 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:
    • \\ \\
    ===== Generator =====
    Frozen core: [He] 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 =====
    The kinetic energy of the core electrons, $E^\text{kin}_c$, 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 =====
    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 [[paw-xml#Radial functions|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: 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 This defines one radial grid as $r_i = di$ where $i$ runs from 0 to 9. Inside the ''%%%%'' 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: and each numerical function must refer to one of these ids: ... ... 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: ^''%%eq%%'' ^parameters ^ |''%%r=d*i%%'' |''%%d%%'' | |''%%r=a*exp(d*i)%%'' |''%%a%%'' and ''%%d%%''| |''%%r=a*(exp(d*i)-1)%%'' |''%%a%%'' and ''%%d%%''| |''%%r=a*i/(1-b*i)%%'' |''%%a%%'' and ''%%b%%''| |''%%r=a*i/(n-i)%%'' |''%%a%%'' and ''%%n%%''| |''%%r=(i/n+a)^5/a-a^4%%''|''%%a%%'' and ''%%n%%''| The ''%%istart%%'' and ''%%iend%%'' attributes indicating the range of $i$ should always be present. Although it is possible to define as 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 [(BLOCHL1994)] for the //shape function// $\hat{Q}^{\ell}_{i j}(\mathbf{r})$ is a product of the multipole moment $Q^{\ell}_{i j}$ and a shape function $g_l(r)$: $$\hat{Q}^{\ell}_{i j}(r) = Q^{\ell}_{i j} g_\ell(r),$$ Several formulations [(BLOCHL1994)] [(HOLZWARTH2001)] define $g_\ell(r) \propto r^\ell k(r)$, where $k(r)$ is an $l$-independent shape function: ^''%%type%%'' ^parameters ^$k(r)$ ^ |''%%gauss%%''|''%%rc%%'' |$\exp(-(r/r_c)^2)$ | |''%%sinc%%'' |''%%rc%%'' |$[\sin(\pi r/r_c)/(\pi r/r_c)]^2$| |''%%exp%%'' |''%%rc%%'' and ''%%lamb%%''|$\exp(-(r/r_c)^\lambda) $ | Example: Another formulation [(KRESSE1999)] defines directly $g_l(r)$: ^''%%type%%'' ^parameters ^$g_l(r)$ ^ |''%%bessel%%''|''%%rc%%'' |$\sum_{i=1}^2 \alpha_i^\ell j_\ell(q_i^\ell r)$| For ''%%bessel%%'' the four parameters ($\alpha_1^l$, $q_1^l$, $\alpha_2^l$ and $q_2^l$) must be determined from ''%%rc%%'' for each value of $l$ as described in [(KRESSE1999)]. Example: 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 ''%%%%'' 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 [(LAASONEN1993)]. Example 1, defining numerically $g_l(r)$ in $\hat{Q}^{\ell}_{i j}(r)=Q^{\ell}_{i j} g_\ell(r)$: ... ... ... Example 2, defining directly $\hat{Q}^{\ell}_{i j}(r)$ for states $i$=''%%N-2s%%'' and $j$=''%%N-2p%%'', and $l$=0: ... ... ... \\ \\
    ===== Radial functions =====
    Continuing, we have now reached the //all-electron// (resp. //pseudo core//, //pseudo valence//) density: 6.801207147443e+02 6.801207147443e+02 6.665042896724e+02 ... ... ... ... ... ... 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: -8.178800366898029e+00 -8.178246914143839e+00 -8.177654917302689e+00 ... ... ... ... ... ... ... ... 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 [(BLOCHL1994)]) 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 $\bar{v}(\mathbf{r})$: ... ... \\ \\
    ===== The Kresse-Joubert formulation =====
    The Kresse-Joubert formulation of the PAW method [(KRESSE1999)] is very similar to the original formulation of Blöchl [(BLOCHL1994)] . 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: $$v_H[\tilde{n}_{Zc}] = v_H[\tilde{n}_c + (N_c - Z - \tilde{N}_c) g_{00} Y_{00}] + \bar{v} + v_{xc}[\tilde{n}_v + \tilde{n}_c] - v_{xc}[\tilde{n}_v + \tilde{n}_c + (N_v - \tilde{N}_v - \tilde{N}_c) g_{00} Y_{00}]$$ 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: ... ... 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 =====
    1.744042161013e+00 0.000000000000e+00 2.730637956456e+00 ... ... 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 [[paw-xml#Valence states|$n_{waves}$ 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: ... ... ... ... 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 ''%%%%'' element.\\ With $n_{waves}$ partial waves (see [[paw-xml#Valence states|$n_{waves}$ 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 ''%%%%'' element: ... ... \\ \\
    ===== Optional elements =====
    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 =====
    \\ \\
    ===== 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.
    #!/usr/bin/env python from __future__ import print_function import os import sys import xml.sax from optparse import OptionParser from math import exp, log class Reader(xml.sax.handler.ContentHandler): def __init__(self, list, name, state): self.list = list self.name = name self.state = state self.grid = {} xml.sax.handler.ContentHandler.__init__(self) def read(self, filename): if filename.endswith('.gz'): import gzip source = gzip.open(filename) else: source = open(filename) xml.sax.parse(source, self) def startElement(self, name, attrs): if name == 'state' and self.list: print(attrs['id'], file=sys.stderr) elif name == 'radial_grid': istart = int(attrs['istart']) iend = int(attrs['iend']) ii = range(istart, iend + 1) id = attrs['id'] eq = attrs['eq'] if eq == 'r=a*exp(d*i)': a = float(attrs['a']) d = float(attrs['d']) self.grid[id] = [a * exp(d * i) for i in ii] elif eq == 'r=a*i/(n-i)': a = float(attrs['a']) n = float(attrs['n']) self.grid[id] = [a * i / (n - i) for i in ii] elif eq == 'r=a*(exp(d*i)-1)': a = float(attrs['a']) d = float(attrs['d']) self.grid[id] = [a * (exp(d * i) - 1.0) for i in ii] elif eq == 'r=d*i': d = float(attrs['d']) self.grid[id] = [d * i for i in ii] elif eq == 'r=(i/n+a)^5/a-a^4': a = float(attrs['a']) n = float(attrs['n']) self.grid[id] = [(i / n + a)**5 / a - a**4 for i in ii] else: raise RuntimeError('Unknown grid type: ' + attrs['eq']) elif name == self.name and attrs.get('state', None) == self.state: self.r = self.grid[attrs['grid']] self.data = '' else: self.data = None def characters(self, data): if self.data is not None: self.data += data def endElement(self, name): if self.data is not None: for r, x in zip(self.r, self.data.split()): print(r, x) op = OptionParser(usage='%prog [options] setup[.gz]', version='%prog 0.2') op.add_option('-x', '--extract', help='Function to extract.', metavar='') op.add_option('-s', '--state', help='Select valence state.', metavar='') op.add_option('-l', '--list', action='store_true', help='List valence states.') options, args = op.parse_args() if len(args) != 1: op.error('incorrect number of arguments') reader = Reader(options.list, options.extract, options.state) reader.read(args[0]) Usage: It works like this: $ pawxml.py [options] dataset[.gz] Options: |''%%--version%%'' |Show program's version number and exit.| |''%%-h, --help%%'' |Show this help message and exit. | |''%%-x , --extract=%%'' |Function to extract. | |''%%-s, --state=%%''|Select valence state. | |''%%-l, --list%%'' |List valence states | 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 =====
    [(BLOCHL1994>[[http://journals.aps.org/prb/abstract/10.1103/PhysRevB.50.17953 | P. E. Blöchl, Projector Augmented-Wave method, Phys. Rev. B 50, 17953-19979 (1994) ]] )] [(BLOCHL2003>[[http://www.ias.ac.in/matersci/bmsjan2003/Bms3867.pdf | P. E. Blöchl, C. J. Forst and J. Schimpl, Projector Augmented-Wave method: Ab initio molecular dynamics with full wave functions, Bulletin of Materials Science 26, 33-41 (2003) ]] )] [(KRESSE1999>[[http://journals.aps.org/prb/abstract/10.1103/PhysRevB.59.1758 | G. Kresse and D. Joubert, Form ultrasoft pseudopotentials to the projector augmented-wave method, Phys. Rev. B 59, 1758-1775 (1999) ]] )] [(HOLZWARTH2001>[[http://www.sciencedirect.com/science/article/pii/S0010465500002447 | N. A. W. Holzwarth, A. R. Tackett, and G. E. Matthews, A Projector Augmented Wave (PAW) code for electronics structure calculations: Part I atompaw for generating atom-centered functions, Computer Physics Communications 135, 329-347 (2001) ]] )] [(LAASONEN1993>[[http://journals.aps.org/prb/abstract/10.1103/PhysRevB.47.10142 | K. Laasonen, A. Pasquarello, R. Car, C. Lee and D. Vanderbilt, Car-Parrinello molecular dynamics with Vanderbilt ultrasoft pseudopotentials, Phys. Rev. B 47, 10142-10153 (1993) ]] )] refnote-id : 1 reference-format : [] reference-base : text multi-ref-id : note note-id-format : [] note-id-base : text back-ref-base : text back-ref-format : 1 back-ref-caret : merge notes-separator : none ~~REFNOTES~~