@@ -13,7 +13,7 @@ In the Fleur [[input]] file there is a flag for writing out the necessary data f

* ``gw=0``: No output files are generated (default; the same as omitting ``gw`` and ``numbands`` altogether).

* ``gw=1``: Some basic parameters are written to several files. Otherwise Fleur runs as usual.

* ``gw=2``: All output files necessary for Spex are created and Fleur stops after one iteration (without updating the potential). In this case you should also set ``pot8=T`` and the maximal number of bands ``N``. The latter overrides the default ``neigd`` value in [[fl7para]].

* ``gw=3``: Self-consistent cycle for QSGW. Same as ``gw=1`` but adds {$ \Sigma^{\mathrm{QSGW}}_{\mathrm{xc}}(\mathbf{r},\mathbf{r}')-v_{\mathrm{xc}}(\mathbf{r}) $} to the Hamiltonian in each iteration.

* ``gw=3``: Self-consistent cycle for QSGW. Same as ``gw=1`` but adds :math:`{ \Sigma^{\mathrm{QSGW}}_{\mathrm{xc}}(\mathbf{r},\mathbf{r}')-v_{\mathrm{xc}}(\mathbf{r}) }` to the Hamiltonian in each iteration.

The output files are

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@@ -59,26 +59,26 @@ As an indication for such a case the overlap of core states with basis and wave

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(:OBSOLETE: !!! Overcompleteness of the interstitial plane waves

The set of interstitial plane waves (IPW) (i.e., plane waves where the MT spheres are cut out) is overcomplete in the sense that a function (defined only in the interstitial region) has a nonunique representation in this set: one can always add the Fourier coefficients of a function, which is restricted to the MT spheres and zero outside. This leads to the fact that the IPW coefficients of the wave functions might not go quickly to zero for large G vectors, especially for a large '''G''' cutoff in the DFT calculation and large MT radii. In the ''GW'' calculation this gives rise to bad convergence with respect to the parameter ``GCUT`` ('''G''' cutoff in Spex) as well as jumps in the band structure. In this case Spex can exploit the nonuniqueness of the coefficients to make them approach zero for large '''G''' vectors as quickly as possible. Use the keyword ``MINCPW`` for this.:)

The set of interstitial plane waves (IPW) (i.e., plane waves where the MT spheres are cut out) is overcomplete in the sense that a function (defined only in the interstitial region) has a nonunique representation in this set: one can always add the Fourier coefficients of a function, which is restricted to the MT spheres and zero outside. This leads to the fact that the IPW coefficients of the wave functions might not go quickly to zero for large G vectors, especially for a large :math:`G` cutoff in the DFT calculation and large MT radii. In the ``GW`` calculation this gives rise to bad convergence with respect to the parameter ``GCUT`` (:math:`G` cutoff in Spex) as well as jumps in the band structure. In this case Spex can exploit the nonuniqueness of the coefficients to make them approach zero for large :math:`G` vectors as quickly as possible. Use the keyword ``MINCPW`` for this.:)

Setting up ``spex.inp``

-----------------------

Mixed Basis (section ``MBASIS``)

""""""""""""""""""""""""""""""""""""

The bare Coulomb scattering {$ v(\mathbf r,\mathbf r') $} couples two electrons whose incoming and outgoing waves give rise to wave-function products (occupied/unoccupied). The mixed basis is optimized for representing wave-function products and derived from the FLAPW basis set. It is a combined set of interstitial plane waves (IPWs) and MT functions and is defined in section ``MBASIS`` of the input file ``spex.inp`` with the keywords

The bare Coulomb scattering :math:`{ v(\mathbf r,\mathbf r') }` couples two electrons whose incoming and outgoing waves give rise to wave-function products (occupied/unoccupied). The mixed basis is optimized for representing wave-function products and derived from the FLAPW basis set. It is a combined set of interstitial plane waves (IPWs) and MT functions and is defined in section ``MBASIS`` of the input file ``spex.inp`` with the keywords

* ``GCUT``: '''G''' cutoff for the interstitial plane waves,

* ``GCUT``: :math:`G` cutoff for the interstitial plane waves,

* ``LCUT``: L cutoffs (one for every atom type) for the MT functions,

* ``SELECT``: determines the pairs of radial functions from whose products the MT functions are constructed. For every atom type two l cutoffs ``l1`` and ``l2`` must be specified for the functions {$u_l(r)$}. Optionally l cutoffs ``l3`` and ``l4`` can also be defined for {$\dot u_l(r)$}, which are neglected by default. The local orbitals, on the other hand, are always used in the construction of the products by default, but can also be chosen individually by series of zeros (neglect) and ones (select), e.g., ``10100``, ``lo1`` and ``lo2`` for occupied and unoccupied states. (Instead of ``0000111``, one can also use ``4031``.) Examples are --- ``2;3`` --- ``2,2;3,2`` --- ``2,2,10100;3,2,1110`` --- ``2,,10100;3,2,1110``

* ``SELECT``: determines the pairs of radial functions from whose products the MT functions are constructed. For every atom type two l cutoffs ``l1`` and ``l2`` must be specified for the functions :math:`{u_l(r)}`. Optionally l cutoffs ``l3`` and ``l4`` can also be defined for :math:`{\dot u_l(r)}`, which are neglected by default. The local orbitals, on the other hand, are always used in the construction of the products by default, but can also be chosen individually by series of zeros (neglect) and ones (select), e.g., ``10100``, ``lo1`` and ``lo2`` for occupied and unoccupied states. (Instead of ``0000111``, one can also use ``4031``.) Examples are --- ``2;3`` --- ``2,2;3,2`` --- ``2,2,10100;3,2,1110`` --- ``2,,10100;3,2,1110``

The so-defined set of product MT functions is usually highly linearly dependent. Therefore, in a subsequent optimization step one diagonalizes the overlap matrix and retains only those eigenfunctions with eigenvalues exceeding a tolerance value which is defined by the keyword ``TOL``.

* ``OPTIMIZE``: Truncation of the polarization and related matrices to ``n``x``n`` by ``MB=n`` where ``n`` is an integer number between 1 and the size of the full mixed basis. Alternatively, one can use ``PW=n`` where plane waves are used instead of the mixed basis (or better: projections of plane waves onto the mixed basis) in which case the Coulomb matrix is analytically known. In both cases (``MB`` and ``PW``), you can also specify the minimum Coulomb eigenvalue instead of ``n`` as a negative real number, e.g. ``MB=-1.0`` . Alternatively, you can specify a "pseudo" reciprocal cutoff radius as a positive real number, e.g. ``MB=4.5``, which is more suitable for convergence tests. This cutoff value {$K$} is related to the minimum Coulomb eigenvalue {$c$} by {$K= \sqrt{4\pi/c}$}. The options ``MB`` and ``PW`` can speed up the calculation considerably.

* ``OPTIMIZE``: Truncation of the polarization and related matrices to ``n``x``n`` by ``MB=n`` where ``n`` is an integer number between 1 and the size of the full mixed basis. Alternatively, one can use ``PW=n`` where plane waves are used instead of the mixed basis (or better: projections of plane waves onto the mixed basis) in which case the Coulomb matrix is analytically known. In both cases (``MB`` and ``PW``), you can also specify the minimum Coulomb eigenvalue instead of ``n`` as a negative real number, e.g. ``MB=-1.0`` . Alternatively, you can specify a "pseudo" reciprocal cutoff radius as a positive real number, e.g. ``MB=4.5``, which is more suitable for convergence tests. This cutoff value :math:`{K}` is related to the minimum Coulomb eigenvalue :math:`{c}` by :math:`{K= \sqrt{4\pi/c}}`. The options ``MB`` and ``PW`` can speed up the calculation considerably.

If any of the keywords (or the whole section) is not defined, Spex guesses reasonable values according to the FLAPW basis parameters.

Coulomb Matrix (section ``COULOMB``)

""""""""""""""""""""""""""""""""""""

In many-body perturbation theory the perturbing force is the Coulomb interaction. Thus its representation in the mixed basis is one of the basic ingredients in the implementation. The IPWs are plane waves {$\exp[i(\mathbf{k+G})\mathbf{r}]$} from which the MT spheres are cut away. In a MT sphere a plane wave is given by the Rayleigh expansion over l quantum numbers. In a practical implementation we must introduce an l cutoff here (keyword ``LEXP``). Multipole interactions between the periodic MT functions and their periodic images are calculated in an Ewald summation involving a real and a Fourier-space sum. The cutoff radii of these are determined during the program run. There is a scaling parameter (``SCALE``) which must be adjusted if one of the sums take much longer to evaluate than the other one. In most calculations, the default values work reliably, though.

In many-body perturbation theory the perturbing force is the Coulomb interaction. Thus its representation in the mixed basis is one of the basic ingredients in the implementation. The IPWs are plane waves :math:`{\exp[i(\mathbf{k+G})\mathbf{r}]}` from which the MT spheres are cut away. In a MT sphere a plane wave is given by the Rayleigh expansion over l quantum numbers. In a practical implementation we must introduce an l cutoff here (keyword ``LEXP``). Multipole interactions between the periodic MT functions and their periodic images are calculated in an Ewald summation involving a real and a Fourier-space sum. The cutoff radii of these are determined during the program run. There is a scaling parameter (``SCALE``) which must be adjusted if one of the sums take much longer to evaluate than the other one. In most calculations, the default values work reliably, though.

The most important entry in ``spex.inp`` is the job definition. It defines what Spex should do. Each job begins with one of the keywords ``HF, QP, SUSCEP, SUSCEPR, DIELEC`` or ``SCREEN`` followed by additional parameters:

* ``HF``: Hartree-Fock correction, e.g., ``HF A:(1-3,5)`` where A is a k-point label or index and ``1-3,5`` are band indices. A spin index (up or down) can be specified by ``A:u(...)`` and ``A:d(...)``. More than one k-point definition is possible in one job, e.g., ``HF A:(1-3,5) 12:(1,2,6)``. Normally, only diagonal elements are calculated, i.e., {$ \langle \phi_{n\mathbf{k}} | \Sigma_{\mathrm{x}} | \phi_{n'\mathbf{k}} \rangle $} with {$ n=n' $}. Full calculations of the self-energy matrix (i.e., {$ n \neq n' $}) are possible by adding ``FULL`` (e.g., ``HF FULL A:(1-10)``). This allows a subsequent self-consistent HF calculation.

* ``HF``: Hartree-Fock correction, e.g., ``HF A:(1-3,5)`` where A is a k-point label or index and ``1-3,5`` are band indices. A spin index (up or down) can be specified by ``A:u(...)`` and ``A:d(...)``. More than one k-point definition is possible in one job, e.g., ``HF A:(1-3,5) 12:(1,2,6)``. Normally, only diagonal elements are calculated, i.e., :math:`{ \langle \phi_{n\mathbf{k}} | \Sigma_{\mathrm{x}} | \phi_{n'\mathbf{k}} \rangle }` with :math:`{ n=n' }`. Full calculations of the self-energy matrix (i.e., :math:`{ n \neq n' }`) are possible by adding ``FULL`` (e.g., ``HF FULL A:(1-10)``). This allows a subsequent self-consistent HF calculation.

* ``GW``: Quasiparticle correction with the ''GW'' approximation. Spin, k points and bands are defined as in ``HF``. A full self-energy calculation is also possible with ``GW FULL``.

* ``SUSCEP``: Calculation of the susceptibility, e.g., ``SUSCEP A:{0..2,0.01}`` where A is a k-point label or index and a frequency mesh from 0 to 2 Ha in 0.02 Ha steps is specified. A more general frequency mesh can also be specified as a list of real or complex frequencies in a file. Then simply write the file name into the brackets, e.g., ``{foo}``. A spin index (``uu, dd, ud, du`` or ``+``) can be specified with ``A:ud{...}``. The spin index ``+`` denotes addition of ``uu`` and ``dd`` and is the default. Furthermore you can add ``MB`` or ``PW`` (see above) and ``OUT=n`` (separated with commas) to the job definition. With ``OUT=n`` an ``n``x``n`` matrix is written instead of only the head element. You can also write the full matrix to a binary file with ``OUT=BIN``. An expression like ``A:u{0..2,0.01},MB=100,OUT=4`` defines one job which can be followed by additional expressions each defining one job.

* ``SUSCEPR``: Calculation of the renormalized susceptibility. Spin, k points and frequencies are defined as in ``SUSCEP``. Additionally you can specify a TDDFT kernel as in ``SUSCEPR A:{0..2,0.01},KERN=ALDA,MB=100``. As spin labels only ``ud``, ``du`` and ``+`` are allowed where the latter is the default. The former spin indices require a TDDFT kernel.

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@@ -102,7 +102,7 @@ Output files

The main results of the Spex calculation is written to standard output ([[SpexOutputFile|output example]]). You should pipe the output to a file (e.g. ``spex > spex.out``). The spectra are written to separate files:

* ``suscep``: Susceptibility head element.

* ``suscep.XXX.YYY``: Susceptibility ``XXX/YYY`` matrix element (key: ``OUT=NNN`` and ``XXX,YYY`` {$\le$} ``NNN``).

* ``suscep.XXX.YYY``: Susceptibility ``XXX/YYY`` matrix element (key: ``OUT=NNN`` and ``XXX,YYY`` :math:`{\le}` ``NNN``).

* ``suscep.bin``: Binary output of the susceptibility matrix (key: ``OUT=BIN``).

* ``suscepr``: Renormalized susceptibility head element.

@@ -118,7 +118,7 @@ If more than one job is defined, the job number is appended to the file names (b

There are a number of additional files which can be written during the program run (``WRITE INFO``) and which tell you more about intermediate results. Some of them are numbered, such that files are never overwritten. From time to time you should remove them from disk, because the highest file count is ``NNN=999``, and they clog up directory listings.

* ``spex.sf.NNN``: If projections ``MB=...`` or ``PW=...`` are used, these files contain the spectral function on the mesh given by ``FSPEC`` for any '''k''' point projected on {$exp(ikr)$}. Column 1: Frequency, column 2: value. The files are useful to check whether the mesh is dense enough and whether the spectral function approaches zero quickly. If it does not, ``GCUT`` might be underconverged or you must use ``MINCPW``.

* ``spex.sf.NNN``: If projections ``MB=...`` or ``PW=...`` are used, these files contain the spectral function on the mesh given by ``FSPEC`` for any '''k''' point projected on :math:`{exp(ikr)}`. Column 1: Frequency, column 2: value. The files are useful to check whether the mesh is dense enough and whether the spectral function approaches zero quickly. If it does not, ``GCUT`` might be underconverged or you must use ``MINCPW``.

* ``spex.sew.NNN``: Correlation self-energy on the imaginary frequency axis (only for ``CONTINUE``). Column 1: imaginary frequency, column 2: fitted value (real part), column 3: fitted value (imaginary part), column 4: numerical value (real part), column 5: numerical value (imaginary part).

* ``spex.sec.NNN``: Correlation self-energy on the real frequency axis. Column 1: real frequency, column 2: (fitted) value (real part), column 3: (fitted) value (imaginary part).

* ``spex.head``: Head element of screened interaction on imaginary axis.

The polarization function gives the linear change in the electronic density of a non-interacting system with respect to changes in the effective potential. It is, thus, a fundamental quantity in the calculation of screening properties of a many-electron systems. For example, the dielectric function, instrumental in the calculation of spectroscopic quantities (e.g. EELS) and the screened interaction needed in ``GW``, is related to the polarization matrix through :math:`{\varepsilon(\mathbf{k},\omega)=1-P(\mathbf{k},\omega)v(\mathbf{k})}`, here given in matrix notation. The corresponding explicit formula for matrix elements of ``P`` in the mixed product basis is

We have implicitly defined the spectral function ``S`` in the last equation, an explicit expression for which is basically given by the formula in the middle with the :math:`{1/(\omega...)}` replaced by :math:`{\delta(\omega...)}`. (Technically, the :math:`{M_{\mathbf{k}\mu}(\mathbf{r})}` form the eigenbasis of the Coulomb matrix, so they are linear combinations of the mixed product basis functions.)