If :math:`{N}` is the number of LAPW basis functions, one would naively expect the number of product functions to be roughly :math:`{N^2}`. In the case of the interstitial plane waves, this is not so, since, with a cutoff :math:`{g_\mathrm{max}}`, the maximum momentum of the product would be :math:`{2g_\mathrm{max}}`, leading to :math:`{8N}` as the number of product plane waves. Fortunately, it turns out that the basis size can be much smaller in practice. Therefore, we introduce a reciprocal cutoff radius :math:`{G_\mathrm{max}}` for the interstitial plane waves and find that, instead of :math:`{G_\mathrm{max}=2g_\mathrm{max}}`, good convergence is achieved already with :math:`{G_\mathrm{max}=0.75g_\mathrm{max}}`, the default value. The parameter :math:`{G_\mathrm{max}}` can be set to a different value with the keyword ``GCUT`` in the section ``MBASIS`` of the input file.
If :math:`{N}` is the number of LAPW basis functions, one would naively expect the number of product functions to be roughly :math:`{N^2}`. In the case of the interstitial plane waves, this is not so, since, with a cutoff :math:`{g_\mathrm{max}}`, the maximum momentum of the product would be :math:`{2g_\mathrm{max}}`, leading to :math:`{8N}` as the number of product plane waves. Fortunately, it turns out that the basis size can be much smaller in practice. Therefore, we introduce a reciprocal cutoff radius :math:`{G_\mathrm{max}}` for the interstitial plane waves and find that, instead of :math:`{G_\mathrm{max}=2g_\mathrm{max}}`, good convergence is achieved already with :math:`{G_\mathrm{max}=0.75g_\mathrm{max}}`, the default value. The parameter :math:`{G_\mathrm{max}}` can be set to a different value with the keyword ``GCUT`` in the section ``MBASIS`` of the input file.
(*) The set of MT products selected by ``SELECT`` can still be highly linearly dependent. Therefore, in a subsequent optimization step one diagonalizes the MT overlap matrix and retains only those eigenfunctions whose eigenvalues exceed a predefined tolerance value. This tolerance is 0.0001 by default and can be changed with the keyword ``TOL`` in the input file.
(*) The set of MT products selected by ``SELECT`` can still be highly linearly dependent. Therefore, in a subsequent optimization step one diagonalizes the MT overlap matrix and retains only those eigenfunctions whose eigenvalues exceed a predefined tolerance value. This tolerance is 0.0001 by default and can be changed with the keyword ``TOL`` in the input file.
The mixed product basis can still be quite large. In the calculation of the screened interaction, each matrix element, when represented in the basis of Coulomb eigenfunctions, is multiplied by :math:`{\sqrt{v_\mu v_\nu}}` with the Coulomb eigenvalues :math:`{\{v_\mu\}}`. This gives an opportunity for reducing the basis-set size further by introducing a Coulomb cutoff :math:`{v_\mathrm{min}}`. The reduced basis set is then used for the polarization function, the dielectric function, and the screened interaction. The parameter :math:`{v_\mathrm{min}}` can be specified after the keyword ``OPTIMIZE MB`` in three ways: first, as a "pseudo" reciprocal cutoff radius :math:`{\sqrt{4\pi/v_\mathrm{min}}}` (which derives from the plane-wave Coulomb eigenvalues :math:`{v_\mathbf{G}=4\pi/G^2}), second, directly as the parameter :math:`{v_\mathrm{min}}` by using a negative real number, and, finally, as the number of basis functions that should be retained when given as an integer. The so-defined basis functions are mathematically close to plane waves. For testing purposes, one can also enforce the usage of plane waves (or rather projections onto plane waves) with the keyword ``OPTIMIZE PW``, in which case the Coulomb matrix is known analytically. No optimization of the basis is applied, if ``OPTIMIZE`` is omitted.
The mixed product basis can still be quite large. In the calculation of the screened interaction, each matrix element, when represented in the basis of Coulomb eigenfunctions, is multiplied by :math:`{\sqrt{v_\mu v_\nu}}` with the Coulomb eigenvalues :math:`{\{v_\mu\}}`. This gives an opportunity for reducing the basis-set size further by introducing a Coulomb cutoff :math:`{v_\mathrm{min}}`. The reduced basis set is then used for the polarization function, the dielectric function, and the screened interaction. The parameter :math:`{v_\mathrm{min}}` can be specified after the keyword ``OPTIMIZE MB`` in three ways: first, as a "pseudo" reciprocal cutoff radius :math:`{\sqrt{4\pi/v_\mathrm{min}}}` (which derives from the plane-wave Coulomb eigenvalues :math:`{v_\mathbf{G}=4\pi/G^2}`), second, directly as the parameter :math:`{v_\mathrm{min}}` by using a negative real number, and, finally, as the number of basis functions that should be retained when given as an integer. The so-defined basis functions are mathematically close to plane waves. For testing purposes, one can also enforce the usage of plane waves (or rather projections onto plane waves) with the keyword ``OPTIMIZE PW``, in which case the Coulomb matrix is known analytically. No optimization of the basis is applied, if ``OPTIMIZE`` is omitted.
In summary, there are a number of parameters that influence the accuracy of the basis set. Whenever a new physical system is investigated, it is recommendable to converge the basis set for that system. The parameters to consider in this respect are ``GCUT``, ``LCUT``, ``SELECT``, and ``OPTIMIZE``.
In summary, there are a number of parameters that influence the accuracy of the basis set. Whenever a new physical system is investigated, it is recommendable to converge the basis set for that system. The parameters to consider in this respect are ``GCUT``, ``LCUT``, ``SELECT``, and ``OPTIMIZE``.
...
@@ -124,7 +131,7 @@ FFT (WFPROD)
...
@@ -124,7 +131,7 @@ FFT (WFPROD)
--------------
--------------
When the interaction potential is represented in the mixed product basis, the coupling to the single-particle states involve projections of the form
When the interaction potential is represented in the mixed product basis, the coupling to the single-particle states involve projections of the form
The calculation of these projections can be quite expensive. Therefore, there are a number of keywords that can be used for acceleration. Most of them are, by now, somewhat obsolete. An important keyword, though, is ``FFT`` in the section ``WFPROD`` of the input file. When used, the interstitial terms are evaluated using Fast Fourier Transformations (FFTs), i.e., by transforming into real space (where the convolutions turn into products), instead of by explicit convolutions in reciprocal space. For small systems the latter is faster, but for large systems it is recommendable to use FFTs because of a better scaling with system size. A run with FFTs can be made to yield results identical to the explicit summation. This requires an FFT reciprocal cutoff radius of :math:`{2G_\mathrm{max}+g_\mathrm{max}}`, which can be achieved by setting ``FFT EXACT``, but such a calculation is quite costly. It is, therefore, advisable to use smaller cutoff radii, thereby sacrificing a bit of accuracy but speeding up the computations a lot. If given without an argument, Spex will use 2/3 of the above ''exact'' cutoff. One can also specify a cutoff by a real-valued argument explicitly, good compromises between accuracy and speed are values between 6 and 8 Bohr'^-1^'.
The calculation of these projections can be quite expensive. Therefore, there are a number of keywords that can be used for acceleration. Most of them are, by now, somewhat obsolete. An important keyword, though, is ``FFT`` in the section ``WFPROD`` of the input file. When used, the interstitial terms are evaluated using Fast Fourier Transformations (FFTs), i.e., by transforming into real space (where the convolutions turn into products), instead of by explicit convolutions in reciprocal space. For small systems the latter is faster, but for large systems it is recommendable to use FFTs because of a better scaling with system size. A run with FFTs can be made to yield results identical to the explicit summation. This requires an FFT reciprocal cutoff radius of :math:`{2G_\mathrm{max}+g_\mathrm{max}}`, which can be achieved by setting ``FFT EXACT``, but such a calculation is quite costly. It is, therefore, advisable to use smaller cutoff radii, thereby sacrificing a bit of accuracy but speeding up the computations a lot. If given without an argument, Spex will use 2/3 of the above *exact* cutoff. One can also specify a cutoff by a real-valued argument explicitly, good compromises between accuracy and speed are values between 6 and 8 Bohr'^-1^'.
@@ -41,13 +41,14 @@ Each Spex run needs a job definition, which defines what Spex should do, e.g., `
...
@@ -41,13 +41,14 @@ Each Spex run needs a job definition, which defines what Spex should do, e.g., `
Details of these jobs are explained in subsequent sections. The job definition must not be omitted but may be empty: ``JOB``, in which case Spex will just read the wavefunctions and energies, perform some checks and some elemental calculations (e.g., Wannier interpolation), and stop. In principle, Spex supports multiple jobs such as ``JOB GW 1:(1-5) DIELEC 1:{0:1,0.01}``. This feature is, however, seldom used and is not guaranteed to work correctly in all versions.
Details of these jobs are explained in subsequent sections. The job definition must not be omitted but may be empty: ``JOB``, in which case Spex will just read the wavefunctions and energies, perform some checks and some elemental calculations (e.g., Wannier interpolation), and stop. In principle, Spex supports multiple jobs such as ``JOB GW 1:(1-5) DIELEC 1:{0:1,0.01}``. This feature is, however, seldom used and is not guaranteed to work correctly in all versions.
(*) The LAPW method relies on a partitioning of space into MT spheres and the interstitial region. The basis functions are defined differently in the two regions, interstitial plane waves in the latter and numerical functions in the spheres with radial parts :math:`{u(r)}, {\dot{u}(r)=\partial u(r)/\partial\epsilon}`, :math:`{u^\mathrm{LO}(r)}` and spherical harmonics :math:`{Y_{lm}(\hat{\mathbf{r}})}` The plane waves and the angular part of the MT functions can be converged straightforwardly with the reciprocal cutoff radius :math:`{g_\mathrm{max}}` and the maximal l quantum number :math:`{l_\mathrm{max}}`, respectively, whereas the radial part of the MT functions is not converged as easily. The standard LAPW basis is restricted to the functions :math:`{u}` and :math:`{\dot{u}}`. Local orbitals :math:`{u^\mathrm{LO}}` can be used to extend the basis set, to enable the description of semicore and high-lying conduction states. The accuracy of the radial MT basis can be analyzed with the keyword MTACCUR e1 e2 which gives the MT representation error [Phys. Rev. B 83, 081101] in the energy range between e1 and e2. (If unspecified, e1 and e2 are chosen automatically.) The results are written to the output files spex.mt.t where t is the atom type index, or spex.mt.s.t with the spin index s(=1 or 2) for spin-polarized calculations. The files contain sets of data for all l quantum numbers, which can be plotted separately with gnuplot (e.g., plot "spex.mt.1" i 3 for :math:`{l=3}`
(*) The LAPW method relies on a partitioning of space into MT spheres and the interstitial region. The basis functions are defined differently in the two regions, interstitial plane waves in the latter and numerical functions in the spheres with radial parts :math:`{u(r)}, {\dot{u}(r)=\partial u(r)/\partial\epsilon}`, :math:`{u^\mathrm{LO}(r)}` and spherical harmonics :math:`{Y_{lm}(\hat{\mathbf{r}})}` The plane waves and the angular part of the MT functions can be converged straightforwardly with the reciprocal cutoff radius :math:`{g_\mathrm{max}}` and the maximal l quantum number :math:`{l_\mathrm{max}}`, respectively, whereas the radial part of the MT functions is not converged as easily. The standard LAPW basis is restricted to the functions :math:`{u}` and :math:`{\dot{u}}`. Local orbitals :math:`{u^\mathrm{LO}}` can be used to extend the basis set, to enable the description of semicore and high-lying conduction states. The accuracy of the radial MT basis can be analyzed with the keyword MTACCUR e1 e2 which gives the MT representation error [Phys. Rev. B 83, 081101] in the energy range between e1 and e2. (If unspecified, e1 and e2 are chosen automatically.) The results are written to the output files spex.mt.t where t is the atom type index, or spex.mt.s.t with the spin index s(=1 or 2) for spin-polarized calculations. The files contain sets of data for all l quantum numbers, which can be plotted separately with gnuplot (e.g., plot "spex.mt.1" i 3 for :math:`{l=3}`)
| Examples | ``MTACCUR -1 2`` | Calculate MT representation error between -1 and 2 Hartree |
| Examples | ``MTACCUR -1 2`` | Calculate MT representation error between -1 and 2 Hartree |
...
@@ -116,19 +118,19 @@ BANDINFO
...
@@ -116,19 +118,19 @@ BANDINFO
--------
--------
(*) In some cases, it may be necessary to replace the energy eigenvalues, provided by the mean-field (DFT) code, by energies (e.g., GW quasiparticle energies) obtained in a previous Spex calculation, for example, to determine the GW Fermi energy or to perform energy-only self-consistent calculations. This can be achieved with the keyword ``ENERGY file``, where ``file`` contains the new energies in eV. The format of ``file`` corresponds to the output of the ``spex.extr`` utility: ``spex.extr g spex.out > file``. It must be made sure that ``file`` contains energy values for the whole irreducible Brillouin zone. Band energies not contained in ``file`` will be adjusted so that the energies are in accending order (provided that there is at least one energy value for the particular k point).
(*) In some cases, it may be necessary to replace the energy eigenvalues, provided by the mean-field (DFT) code, by energies (e.g., GW quasiparticle energies) obtained in a previous Spex calculation, for example, to determine the GW Fermi energy or to perform energy-only self-consistent calculations. This can be achieved with the keyword ``ENERGY file``, where ``file`` contains the new energies in eV. The format of ``file`` corresponds to the output of the ``spex.extr`` utility: ``spex.extr g spex.out > file``. It must be made sure that ``file`` contains energy values for the whole irreducible Brillouin zone. Band energies not contained in ``file`` will be adjusted so that the energies are in accending order (provided that there is at least one energy value for the particular k point).
(*) This keyword modifies the exchange splitting of a collinear magnetic system, i.e., it shifts spin-up and spin-down energies relative to to each other so as to increase or decrease the exchange splitting. With ``DELTAEX x``, the spin-up (spin-down) energies are lowered (elevated) by x/2. The parameter x can be used to enforce the Goldstone condition in spin-wave calculations [Phys. Rev. B 94, 064433 (2016)]
(*) This keyword modifies the exchange splitting of a collinear magnetic system, i.e., it shifts spin-up and spin-down energies relative to to each other so as to increase or decrease the exchange splitting. With ``DELTAEX x``, the spin-up (spin-down) energies are lowered (elevated) by x/2. The parameter x can be used to enforce the Goldstone condition in spin-wave calculations [Phys. Rev. B 94, 064433 (2016)]
(*) If specified, Spex only reads the LAPW basis set from the input data, provided by the mean-field (DFT) code, but performs the diagonalization of the Hamiltonian at the k points itself. This calculation effectively replaces the second run of the DFT code. In this sense, the name of the keyword is a bit misleading, as the calculation is non-iterative. The keyword ``ITERATE`` is mostly intended for testing and debugging. It is not available for executables compiled with ``-DLOAD`` (configured with ``--enable-load``).
(*) If specified, Spex only reads the LAPW basis set from the input data, provided by the mean-field (DFT) code, but performs the diagonalization of the Hamiltonian at the k points itself. This calculation effectively replaces the second run of the DFT code. In this sense, the name of the keyword is a bit misleading, as the calculation is non-iterative. The keyword ``ITERATE`` is mostly intended for testing and debugging. It is not available for executables compiled with ``-DLOAD`` (configured with ``--enable-load``).
@@ -13,7 +13,7 @@ It needs input from a converged DFT calculation, which can be generated by Fleur
...
@@ -13,7 +13,7 @@ It needs input from a converged DFT calculation, which can be generated by Fleur
If you use SPEX for your research, please cite the following work:
If you use SPEX for your research, please cite the following work:
.. highlights:: Christoph Friedrich, Stefan Blügel, Arno Schindlmayr, "Efficient implementation of the GW approximation within the all-electron FLAPW method", Phys. Rev. B 81, 125102 (2010).
.. highlights:: Christoph Friedrich, Stefan Blügel, Arno Schindlmayr, "Efficient implementation of the GW approximation within the all-electron FLAPW method", *Phys. Rev. B 81, 125102 (2010)*.
where :math:`{v^\mathrm{ext}}`, :math:`{v^\mathrm{H}}`, :math:`{\Sigma^\mathrm{xc}}`, :math:`{\psi_{\mathbf{k}n}}`, :math:`{E_{\mathbf{k}n})}` are the external and Hartree potential, the ``GW`` self-energy, and the quasiparticle wavefunction and energy, respectively. Taking the difference :math:`{\Sigma^\mathrm{xc}-v^\mathrm{xc}}` as a small perturbation, we can write the quasiparticle energy as a correction on the mean-field eigenvalue
where :math:`{v^\mathrm{ext}}`, :math:`{v^\mathrm{H}}`, :math:`{\Sigma^\mathrm{xc}}`, :math:`{\psi_{\mathbf{k}n}}`, :math:`{E_{\mathbf{k}n})}` are the external and Hartree potential, the ``GW`` self-energy, and the quasiparticle wavefunction and energy, respectively. Taking the difference :math:`{\Sigma^\mathrm{xc}-v^\mathrm{xc}}` as a small perturbation, we can write the quasiparticle energy as a correction on the mean-field eigenvalue
with the single-particle wavefunction :math:`{\phi_{\mathbf{k}n}}` and the frequency-independent potential :math:`{v^{\mathrm{xc}}}`, which in the case of a KS solution would correspond to the local exchange-correlation potential; the nonlocal Hartree-Fock exchange potential and the ''hermitianized'' self-energy of QSGW (see below) are other examples. :math:`{Z_{\mathbf{k}n}=[1-\partial\Sigma^{\mathrm{xc}}/\partial\omega(\epsilon_{\mathbf{k}n})]^{-1}}` is called the renormalization factor. The two expressions on the right-hand side correspond to the "linearized" and "direct" (iterative) solutions given in the output. The direct solution takes into account the non-linearity of the quasiparticle equation and is thus considered the more accurate result. However, there is usually only little difference between the two values.
with the single-particle wavefunction :math:`{\phi_{\mathbf{k}n}}` and the frequency-independent potential :math:`{v^{\mathrm{xc}}}`, which in the case of a KS solution would correspond to the local exchange-correlation potential; the nonlocal Hartree-Fock exchange potential and the *hermitianized* self-energy of QSGW (see below) are other examples. :math:`{Z_{\mathbf{k}n}=[1-\partial\Sigma^{\mathrm{xc}}/\partial\omega(\epsilon_{\mathbf{k}n})]^{-1}}` is called the renormalization factor. The two expressions on the right-hand side correspond to the "linearized" and "direct" (iterative) solutions given in the output. The direct solution takes into account the non-linearity of the quasiparticle equation and is thus considered the more accurate result. However, there is usually only little difference between the two values.
Up to this point, the job syntax for Hartree Fock (``JOB HF``), PBE0 (``JOB PBE0``), screened exchange (``JOB SX``), COHSEX (``JOB COSX``), and ''GT'' (``JOB GT`` and ``JOB GWT``) calculations is identical to the one of ``GW`` calculations, e.g., ``JOB HF FULL X:(1-10)``. Except the latter (``GT``), all of these methods are mean-field approaches, so one only gets one single-particle energy (instead of a ''linearized'' and a ''direct'' solution) for each band.
Up to this point, the job syntax for Hartree Fock (``JOB HF``), PBE0 (``JOB PBE0``), screened exchange (``JOB SX``), COHSEX (``JOB COSX``), and ''GT'' (``JOB GT`` and ``JOB GWT``) calculations is identical to the one of ``GW`` calculations, e.g., ``JOB HF FULL X:(1-10)``. Except the latter (``GT``), all of these methods are mean-field approaches, so one only gets one single-particle energy (instead of a ''linearized'' and a ''direct'' solution) for each band.
.. _spectral:
.. _spectral:
...
@@ -77,11 +77,13 @@ SPECTRAL (SENERGY)
...
@@ -77,11 +77,13 @@ SPECTRAL (SENERGY)
(*) It should be pointed out that the quasiparticle energies given in the output rely on the quasiparticle approximation. The more fundamental equation, as it were, is the Dyson equation
(*) It should be pointed out that the quasiparticle energies given in the output rely on the quasiparticle approximation. The more fundamental equation, as it were, is the Dyson equation
which links the interacting Green function {$G$} to the non-interacting KS one {$G_0$} and which, in principle, requires the self-energy to be known on the complete :math:`{\omega}` axis. The spectral function measured in photoelectron spectroscopy is directly related to the Green function by
which links the interacting Green function {$G$} to the non-interacting KS one :math:`{G_0}` and which, in principle, requires the self-energy to be known on the complete :math:`{\omega}` axis. The spectral function measured in photoelectron spectroscopy is directly related to the Green function by
where the trace (tr) is over the eigenstates and :math:`{\epsilon_\mathrm{F}}` is the Fermi energy. The spectral function can be evaluated using the line
where the trace (tr) is over the eigenstates and :math:`{\epsilon_\mathrm{F}}` is the Fermi energy. The spectral function can be evaluated using the line
...
@@ -91,25 +93,25 @@ in the section ``SENERGY`` of the input file. This option does not require the `
...
@@ -91,25 +93,25 @@ in the section ``SENERGY`` of the input file. This option does not require the `
the keyword ``SPECTRAL`` in the section ``SENERGY``, in the examples below from -10 eV to 1 eV in steps of 0.01 eV. If there is no explicit mesh, Spex chooses one automatically. The spectral function is then written to the file "spectral", one block of data for each k point given in the job definition. There is another optional parameter given at the end of the line (second example below), which can be used to bound the imaginary part of the self-energy and, thus, the quasiparticle peak widths from below by this value (unset if not given). This can be helpful in the case of very sharp quasiparticle peaks that are otherwise hard to catch with the frequency mesh.
the keyword ``SPECTRAL`` in the section ``SENERGY``, in the examples below from -10 eV to 1 eV in steps of 0.01 eV. If there is no explicit mesh, Spex chooses one automatically. The spectral function is then written to the file "spectral", one block of data for each k point given in the job definition. There is another optional parameter given at the end of the line (second example below), which can be used to bound the imaginary part of the self-energy and, thus, the quasiparticle peak widths from below by this value (unset if not given). This can be helpful in the case of very sharp quasiparticle peaks that are otherwise hard to catch with the frequency mesh.
Alhough ``GW`` calculations can be readily performed with the default settings, the user should be familiar to some degree with the details of the computation and how he/she can influence each step of the program run. Also note that the default settings might work well for one physical system but be unsuitable for another.
Alhough ``GW`` calculations can be readily performed with the default settings, the user should be familiar to some degree with the details of the computation and how he/she can influence each step of the program run. Also note that the default settings might work well for one physical system but be unsuitable for another.
can be understood as a scattering potential that contains the exact exchange potential and correlation effects through the inclusion of ''W'', the dynamically screened interaction, which incorporates the screening of the many-electron system into an effective dynamical potential, obtained from the dielectric function :math:`{\varepsilon}` through
can be understood as a scattering potential that contains the exact exchange potential and correlation effects through the inclusion of ''W'', the dynamically screened interaction, which incorporates the screening of the many-electron system into an effective dynamical potential, obtained from the dielectric function :math:`{\varepsilon}` through
This integral equation turns into a matrix equation
This integral equation turns into a matrix equation
...
@@ -117,13 +119,13 @@ This integral equation turns into a matrix equation
...
@@ -117,13 +119,13 @@ This integral equation turns into a matrix equation
if the quantities :math:`{W}`, :math:`{\varepsilon}`, and :math:`{v}` ; are represented in the :ref:`mbp`, which thus has to be converged properly. The dielectric function, in turn, describes the change of the internal potential through screening processes and is related to the polarization matrix by :math:`{\varepsilon(\mathbf{k},\omega)=1-P(\mathbf{k},\omega)v(\mathbf{k})}` in matrix notation. The polarization function is one of the main quantities in the ``GW`` scheme. Its evaluation is described in the section :ref:`polar`.
if the quantities :math:`{W}`, :math:`{\varepsilon}`, and :math:`{v}` ; are represented in the :ref:`mbp`, which thus has to be converged properly. The dielectric function, in turn, describes the change of the internal potential through screening processes and is related to the polarization matrix by :math:`{\varepsilon(\mathbf{k},\omega)=1-P(\mathbf{k},\omega)v(\mathbf{k})}` in matrix notation. The polarization function is one of the main quantities in the ``GW`` scheme. Its evaluation is described in the section :ref:`polar`.
The self-energy can be written as the sum of two terms, the first of which is the exact non-local exchange potential of Hartree-Fock theory, the remainder can be interpreted as a correlation self-energy and has the mathematical form of the [[#Eq:Selfene|self-energy]] with :math:{W(\omega)} replaced by :math:`{W^\mathrm{c}(\omega)=W(\omega)-v}`. The frequency integration is carried out analytically for the exchange part (by summing over the residues). The correlation part is more complex to evaluate because of the frequency dependence of the interaction. (Fast electrons experience a different potential than slow electrons.)
The self-energy can be written as the sum of two terms, the first of which is the exact non-local exchange potential of Hartree-Fock theory, the remainder can be interpreted as a correlation self-energy and has the mathematical form of the [[#Eq:Selfene|self-energy]] with :math:`{W(\omega)}` replaced by :math:`{W^\mathrm{c}(\omega)=W(\omega)-v}`. The frequency integration is carried out analytically for the exchange part (by summing over the residues). The correlation part is more complex to evaluate because of the frequency dependence of the interaction. (Fast electrons experience a different potential than slow electrons.)
There are several ways to represent the self-energy as a function of frequency. The default method is ``analytic continuation``, in which the screened interaction and the self-energy are evaluated on a mesh of purely imaginary frequencies. The self-energy is then analytically continued to the complete complex frequency plane (:math:`{\Sigma^\mathrm{xc}(i\omega)\rightarrow\Sigma^\mathrm{xc}(z)}`, :math:`{\omega\in\cal{R}}`, :math:`{z\in\cal{C}}`). This has several advantages over the usage of the real frequency axis. First, :math:`{W(\omega)}` is a hermitian (or real-symmetric) matrix if :math:`{\omega}` is purely imaginary. Second, ``W`` and :math:`{\Sigma^{\mathrm{xc}}}` show a lot of structure along the real axis, whereas they are much smoother on the imaginary axis, thereby making it easier to sample and interpolate these functions. Third, after the analytic continuation the self-energy is known, in principle, as an analytic function on the complete complex plane. And fourth, the method requires only few parameters and is, therefore, easy to handle. The main disadvantage lies is the badly controlled extrapolation of the Pade approximants, which can sometimes produce "outlier values", with a potential adverse effect on the accuracy and reliability of the method. Therefore, there is a more sophisticated but also more complex method called ``contour integration``, in which the frequency convolution is performed explicitly, yielding the self-energy directly for selected frequencies on the real axis. In this method, we also mostly integrate along the imaginary frequency axis.
There are several ways to represent the self-energy as a function of frequency. The default method is ``analytic continuation``, in which the screened interaction and the self-energy are evaluated on a mesh of purely imaginary frequencies. The self-energy is then analytically continued to the complete complex frequency plane (:math:`{\Sigma^\mathrm{xc}(i\omega)\rightarrow\Sigma^\mathrm{xc}(z)}`, :math:`{\omega\in\cal{R}}`, :math:`{z\in\cal{C}}`). This has several advantages over the usage of the real frequency axis. First, :math:`{W(\omega)}` is a hermitian (or real-symmetric) matrix if :math:`{\omega}` is purely imaginary. Second, ``W`` and :math:`{\Sigma^{\mathrm{xc}}}` show a lot of structure along the real axis, whereas they are much smoother on the imaginary axis, thereby making it easier to sample and interpolate these functions. Third, after the analytic continuation the self-energy is known, in principle, as an analytic function on the complete complex plane. And fourth, the method requires only few parameters and is, therefore, easy to handle. The main disadvantage lies is the badly controlled extrapolation of the Pade approximants, which can sometimes produce "outlier values", with a potential adverse effect on the accuracy and reliability of the method. Therefore, there is a more sophisticated but also more complex method called ``contour integration``, in which the frequency convolution is performed explicitly, yielding the self-energy directly for selected frequencies on the real axis. In this method, we also mostly integrate along the imaginary frequency axis.
MESH (SENERGY)
MESH (SENERGY)
--------------
--------------
All methods employ a mesh of purely imaginary frequencies, which extends from 0 to some maximal :math:`{i\omega_\mathrm{max}}`. The number of mesh points :math:{N} and the maximal frequency must be provided as parameters, for example ``MESH 10 10.0``, which is the default. The mesh is defined by :math:`{i\omega_n=i\omega_\mathrm{max}f_n/f_N}` with :math:`{f_n=\{(N-1)/[0.9(n-1)]-1\}^{-1}}`, :math:`{n=1,2,...}`. It is fine for small :math:`{\omega}`, where the quantities have a lot of structure, and coarse for large :math:{\omega}. Sometimes it is helpful to make the mesh even finer for small :math:`{\omega}`. This is possible by specifying, for example, ``10+3``, which would yield three, two, and one extra equidistant frequencies in the ranges [:math:`{\omega_1,\omega_2}`], [:math:`{\omega_2,\omega_3}`], and [:math:`{\omega_3,\omega_4}`], respectively. If the second argument is defined negative (:math:`{-\omega_\mathrm{max}}`), then :math:`{f_n=\{N/(n-1)-1\}^{-1}}`. The latter definition is rarely used. One can also employ a user-defined mesh provided in a file (one frequency per line, comments ``#...`` are allowed).
All methods employ a mesh of purely imaginary frequencies, which extends from 0 to some maximal :math:`{i\omega_\mathrm{max}}`. The number of mesh points :math:`{N}` and the maximal frequency must be provided as parameters, for example ``MESH 10 10.0``, which is the default. The mesh is defined by :math:`{i\omega_n=i\omega_\mathrm{max}f_n/f_N}` with :math:`{f_n=\{(N-1)/[0.9(n-1)]-1\}^{-1}}`, :math:`{n=1,2,...}`. It is fine for small :math:`{\omega}`, where the quantities have a lot of structure, and coarse for large :math:`{\omega}`. Sometimes it is helpful to make the mesh even finer for small :math:`{\omega}`. This is possible by specifying, for example, ``10+3``, which would yield three, two, and one extra equidistant frequencies in the ranges [:math:`{\omega_1,\omega_2}`], [:math:`{\omega_2,\omega_3}`], and [:math:`{\omega_3,\omega_4}`], respectively. If the second argument is defined negative (:math:`{-\omega_\mathrm{max}}`), then :math:`{f_n=\{N/(n-1)-1\}^{-1}}`. The latter definition is rarely used. One can also employ a user-defined mesh provided in a file (one frequency per line, comments ``#...`` are allowed).
| Examples | ``MESH 12 15.0`` | Use a mesh containing twelve frequencies for [0,i15] htr |
| Examples | ``MESH 12 15.0`` | Use a mesh containing twelve frequencies for [0,i15] htr |
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@@ -140,19 +142,19 @@ CONTINUE (SENERGY)
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@@ -140,19 +142,19 @@ CONTINUE (SENERGY)
-------------------
-------------------
The keyword ``CONTINUE`` in the section ``SENERGY`` chooses the analytic continuation method. It can have optional parameters. If given without parameters (or if not specified at all), Pade approximants are used. An integer number, such as ``CONTINUE 2``, lets Spex perform a fit to an n-pole function (here, n=2) with the Newton method. The latter approach is somewhat obsolete by now and recommended only for test calculations. The argument can have additional flags +c*. Any combination is possible. They are explained in the examples.
The keyword ``CONTINUE`` in the section ``SENERGY`` chooses the analytic continuation method. It can have optional parameters. If given without parameters (or if not specified at all), Pade approximants are used. An integer number, such as ``CONTINUE 2``, lets Spex perform a fit to an n-pole function (here, n=2) with the Newton method. The latter approach is somewhat obsolete by now and recommended only for test calculations. The argument can have additional flags +c*. Any combination is possible. They are explained in the examples.
The second method is ``contour integration``, in which the frequency integration is performed explicitly, however not along the real frequency axis but on a deformed integration contour that avoids the real frequency axis as well as possible. This integration contour starts from :math:`{-\infty}`, describes an infinite quarter circle to :math:`{-i\infty}`, then runs along the imaginary frequency axis to :math:`{i\infty}`, and finishes, again with an infinite quarter circle, to :math:`{\infty}`. The two quarter circles do not contribute to the integral (because the integrand behaves as :math:`{\propto \omega^{-2}}`). Furthermore, depending on the frequency argument :math:`{\omega}` of the self-energy, one has to add a few residues coming from the poles of the Green function in the interval [:math:`{0,\omega-\epsilon_\mathrm{F}}`] if :math:`{\omega>\epsilon_\mathrm{F}}` and [:math:`{\epsilon_\mathrm{F}-\omega,0}`] otherwise, which requires the correlation screened interaction :math:`{W^\mathrm{c}(\omega)}` to be evaluated on this interval of the real axis. As a consequence, the calculations are more demanding in terms of computational complexity and cost (time and memory). Also, contour integration requires additional input parameters and is therefore somewhat more difficult to apply. However, the results are more accurate. In particular, they are not affected by the "ill-definedness" of the analytic continuation.
The second method is ``contour integration``, in which the frequency integration is performed explicitly, however not along the real frequency axis but on a deformed integration contour that avoids the real frequency axis as well as possible. This integration contour starts from :math:`{-\infty}`, describes an infinite quarter circle to :math:`{-i\infty}`, then runs along the imaginary frequency axis to :math:`{i\infty}`, and finishes, again with an infinite quarter circle, to :math:`{\infty}`. The two quarter circles do not contribute to the integral (because the integrand behaves as :math:`{\propto \omega^{-2}}`). Furthermore, depending on the frequency argument :math:`{\omega}` of the self-energy, one has to add a few residues coming from the poles of the Green function in the interval [:math:`{0,\omega-\epsilon_\mathrm{F}}`] if :math:`{\omega>\epsilon_\mathrm{F}}` and [:math:`{\epsilon_\mathrm{F}-\omega,0}`] otherwise, which requires the correlation screened interaction :math:`{W^\mathrm{c}(\omega)}` to be evaluated on this interval of the real axis. As a consequence, the calculations are more demanding in terms of computational complexity and cost (time and memory). Also, contour integration requires additional input parameters and is therefore somewhat more difficult to apply. However, the results are more accurate. In particular, they are not affected by the "ill-definedness" of the analytic continuation.
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@@ -161,18 +163,20 @@ CONTOUR (SENERGY)
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@@ -161,18 +163,20 @@ CONTOUR (SENERGY)
The corresponding keyword is called ``CONTOUR`` and belongs to the section ``SENERGY``. Obviously, ``CONTOUR`` and ``CONTINUE`` must not be given at the same time. The keyword ``CONTOUR`` expects two arguments. The first defines the frequencies :math:`{\omega}`, for which the self-energy :math:`{\Sigma^\mathrm{xc}(\omega)}` is to be evaluated. At least, two frequencies are needed to approximate the self-energy as a linear function in :math:`{\omega}` and, thus, to calculate the ''linearized'' solution of the quasiparticle equation (see above). For this, a single value suffices (example ``0.01``), with which the self-energy for a state :math:`{\mathbf{k}n}` is evaluated at two frequencies (:math:`{\epsilon_{\mathbf{k}n}-0.01}` and :math:`{\epsilon_{\mathbf{k}n}+0.01}`). The more accurate ''direct'' solution is only available if we specify a range of frequencies for the self-energy instead of a single number. This is possible by an argument such as :math:`{-0.1:0.15,0.01}`. Here, the range of values is relative to :math:`{\epsilon_{\mathbf{k}n}}`. Note that the range is flipped (to :math:`{-0.15:0.1,0.01}` in the example) for occupied states :math:`{\mathbf{k}n}` to reflect the fact that occupied and unoccupied states tend to shift in opposite directions by the renormalization. One can also specify an absolute frequency mesh by [:math:`{...}`], i.e., relative to the Fermi energy. This is mandatory for ``FULL`` calculations. It is sometimes a bit inconvenient to determine suitable values for the upper and lower bound of [:math:`{...}`].
The corresponding keyword is called ``CONTOUR`` and belongs to the section ``SENERGY``. Obviously, ``CONTOUR`` and ``CONTINUE`` must not be given at the same time. The keyword ``CONTOUR`` expects two arguments. The first defines the frequencies :math:`{\omega}`, for which the self-energy :math:`{\Sigma^\mathrm{xc}(\omega)}` is to be evaluated. At least, two frequencies are needed to approximate the self-energy as a linear function in :math:`{\omega}` and, thus, to calculate the ''linearized'' solution of the quasiparticle equation (see above). For this, a single value suffices (example ``0.01``), with which the self-energy for a state :math:`{\mathbf{k}n}` is evaluated at two frequencies (:math:`{\epsilon_{\mathbf{k}n}-0.01}` and :math:`{\epsilon_{\mathbf{k}n}+0.01}`). The more accurate ''direct'' solution is only available if we specify a range of frequencies for the self-energy instead of a single number. This is possible by an argument such as :math:`{-0.1:0.15,0.01}`. Here, the range of values is relative to :math:`{\epsilon_{\mathbf{k}n}}`. Note that the range is flipped (to :math:`{-0.15:0.1,0.01}` in the example) for occupied states :math:`{\mathbf{k}n}` to reflect the fact that occupied and unoccupied states tend to shift in opposite directions by the renormalization. One can also specify an absolute frequency mesh by [:math:`{...}`], i.e., relative to the Fermi energy. This is mandatory for ``FULL`` calculations. It is sometimes a bit inconvenient to determine suitable values for the upper and lower bound of [:math:`{...}`].
Therefore, Spex allows the usage of wildcards for one of the boundaries or both (see below). The second argument to ``CONTOUR`` gives the increment of the (equidistant) real-frequency mesh for :math:`{W(\omega')}`. The lower and upper boundaries of this mesh are predetermined already by the first argument. As a third method, Spex also allows to omit the second argument altogether. Then, it uses a ''hybrid'' method where the screened interaction is analytically continued from the imaginary (where it has to be known for the integral along this axis, see above) to the real axis, thereby obviating the need of calculating and storing ''W'' on a real-frequency mesh. The disadvantage is that the badly controlled Pade extrapolation introduces an element of randomness (also see keyword ``SMOOTH`` below). Our experience so far is that the ''hybrid'' method is in-between the two other methods in terms of both computational cost and numerical accuracy.
Therefore, Spex allows the usage of wildcards for one of the boundaries or both (see below). The second argument to ``CONTOUR`` gives the increment of the (equidistant) real-frequency mesh for :math:`{W(\omega')}`. The lower and upper boundaries of this mesh are predetermined already by the first argument. As a third method, Spex also allows to omit the second argument altogether. Then, it uses a ''hybrid'' method where the screened interaction is analytically continued from the imaginary (where it has to be known for the integral along this axis, see above) to the real axis, thereby obviating the need of calculating and storing ''W'' on a real-frequency mesh. The disadvantage is that the badly controlled Pade extrapolation introduces an element of randomness (also see keyword ``SMOOTH`` below). Our experience so far is that the ''hybrid'' method is in-between the two other methods in terms of both computational cost and numerical accuracy.
(*) Independently of whether :math:`{\Sigma^\mathrm{xc}(i\omega_n)}` (``CONTINUE``) or :math:`{\Sigma^\mathrm{xc}(\omega_n)}` (``CONTOUR``) is evaluated, an important step in the calculation of the self-energy is to perform the frequency convolution :math:`{\int_{-\infty}^\infty G(z+i\omega')W(i\omega')d\omega'}` with :math:`{z=i\omega}` or :math:`{z=\omega}`. For this frequency integration, we interpolate ``W`` and then perform the convolution with the Green function analytically. The keyword ``FREQINT`` determines how the interpolation should be done. It can take two values, ``SPLINE`` and ``PADE`` for spline [after the transformation :math:`{\omega\rightarrow \omega/(1+\omega)}`] and Pade interpolation. The default is ``SPLINE``. In the case of ``GT`` calculations, there is a similar frequency integration with the ``T`` matrix replacing ``W``. There, the default is ``PADE``.
(*) Independently of whether :math:`{\Sigma^\mathrm{xc}(i\omega_n)}` (``CONTINUE``) or :math:`{\Sigma^\mathrm{xc}(\omega_n)}` (``CONTOUR``) is evaluated, an important step in the calculation of the self-energy is to perform the frequency convolution :math:`{\int_{-\infty}^\infty G(z+i\omega')W(i\omega')d\omega'}` with :math:`{z=i\omega}` or :math:`{z=\omega}`. For this frequency integration, we interpolate ``W`` and then perform the convolution with the Green function analytically. The keyword ``FREQINT`` determines how the interpolation should be done. It can take two values, ``SPLINE`` and ``PADE`` for spline [after the transformation :math:`{\omega\rightarrow \omega/(1+\omega)}`] and Pade interpolation. The default is ``SPLINE``. In the case of ``GT`` calculations, there is a similar frequency integration with the ``T`` matrix replacing ``W``. There, the default is ``PADE``.
Spex can reuse data from a previous ``GW`` run that has finished successfully, crashed, or has been stopped by the user. A ``GW`` calculation consists mainly of a loop over the irreducible k points. For each k point, Spex (a) calculates the matrix ``W``(``'k``') and (b) updates the self-energy matrix (or expectation values) :math:`{\Sigma^{xc}}` with the contribution of the current k point (and its symmetry-equivalent k points). After completion of step (b), the current self-energy is always written to the (binary) files "spex.sigx" and "spex.sigc". If the ``RESTART`` option is specified (independently of its argument), Spex also writes the matrix ``W(k)`` (in addition to some other data) to the (HDF5) file "spex.cor" unless it is already present in that file. If it is present, the corresponding data is read instead of being calculated. In this way, the keyword ``RESTART`` enables reusage of the calculated ``W(k)`` from a previous run. The matrix ``W(k)`` does not depend on the k points and band indices defined after ``JOB``. So, these parameters can be changed before a run with ``RESTART``, in which the ``W`` data is then reused. For example, band structures can be calculated efficiently in this way (see below). Especially for long calculations, it is recommended to use the ``RESTART`` option. Spex can also restart a calculation using self-energy data contained in "spex.sigx" and "spex.sigc". To this end, an argument is added: ``RESTART 2``. Spex then starts with the k point, at which the calculation was interrupted before. In contrast to "spex.cor", the files "spex.sigx" and "spex.sigc" do depend on the job definition, which must therefore ``not`` be changed before a run with ``RESTART 2``. However, there are few parameters (see below) that may be changed before a rerun with ``RESTART 2``. These concern details of solving the quasiparticle equation, which follows after completion of the self-energy calculation. The following logical table gives an overview.
Spex can reuse data from a previous ``GW`` run that has finished successfully, crashed, or has been stopped by the user. A ``GW`` calculation consists mainly of a loop over the irreducible k points. For each k point, Spex (a) calculates the matrix ``W``(``'k``') and (b) updates the self-energy matrix (or expectation values) :math:`{\Sigma^{xc}}` with the contribution of the current k point (and its symmetry-equivalent k points). After completion of step (b), the current self-energy is always written to the (binary) files "spex.sigx" and "spex.sigc". If the ``RESTART`` option is specified (independently of its argument), Spex also writes the matrix ``W(k)`` (in addition to some other data) to the (HDF5) file "spex.cor" unless it is already present in that file. If it is present, the corresponding data is read instead of being calculated. In this way, the keyword ``RESTART`` enables reusage of the calculated ``W(k)`` from a previous run. The matrix ``W(k)`` does not depend on the k points and band indices defined after ``JOB``. So, these parameters can be changed before a run with ``RESTART``, in which the ``W`` data is then reused. For example, band structures can be calculated efficiently in this way (see below). Especially for long calculations, it is recommended to use the ``RESTART`` option. Spex can also restart a calculation using self-energy data contained in "spex.sigx" and "spex.sigc". To this end, an argument is added: ``RESTART 2``. Spex then starts with the k point, at which the calculation was interrupted before. In contrast to "spex.cor", the files "spex.sigx" and "spex.sigc" do depend on the job definition, which must therefore ``not`` be changed before a run with ``RESTART 2``. However, there are few parameters (see below) that may be changed before a rerun with ``RESTART 2``. These concern details of solving the quasiparticle equation, which follows after completion of the self-energy calculation. The following logical table gives an overview.
The different rules for "spex.cor" and "spex.sigx/c" are motivated by the facts that (a) the file "spex.cor" is much bigger than "spex.sigx/c" (so, writing of "spex.cor" to harddisc should not be the default), (b) the files "spex.sigx/c" include the updated self-energy (requiring more computation than for ``W``, thus representing "more valuable" data).
The different rules for "spex.cor" and "spex.sigx/c" are motivated by the facts that (a) the file "spex.cor" is much bigger than "spex.sigx/c" (so, writing of "spex.cor" to harddisc should not be the default), (b) the files "spex.sigx/c" include the updated self-energy (requiring more computation than for ``W``, thus representing "more valuable" data).
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@@ -344,15 +350,17 @@ One can also go beyond the perturbative solution of the [[#Eq:qpeq|quasiparticle
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@@ -344,15 +350,17 @@ One can also go beyond the perturbative solution of the [[#Eq:qpeq|quasiparticle
[[#QSGW]]If the job definition contains ``FULL`` and [[#IBZlabel|``IBZ``]], the full ``GW`` self-energy matrix is evaluated for the whole IBZ, which enables self-consistent calculations in the framework of the ``quasiparticle self-consistent GW`` (QSGW) approach. In this approach, one creates a mean-field system from the ``GW`` self-energy whose single-particle energies are as close as possible to the quasiparticle energies. This mean-field system is subsequently solved to self-consistency in a DFT code. The resulting solution can then serve as a starting point for a new one-shot ``GW`` calculation, which constitutes the second iteration and so on until the quasiparticle energies are converged. The construction of the mean-field system is, to some extent, arbitrary. We use the following definition, which is slightly modified from the original work [PRL 93, 126406]:
[[#QSGW]]If the job definition contains ``FULL`` and [[#IBZlabel|``IBZ``]], the full ``GW`` self-energy matrix is evaluated for the whole IBZ, which enables self-consistent calculations in the framework of the ``quasiparticle self-consistent GW`` (QSGW) approach. In this approach, one creates a mean-field system from the ``GW`` self-energy whose single-particle energies are as close as possible to the quasiparticle energies. This mean-field system is subsequently solved to self-consistency in a DFT code. The resulting solution can then serve as a starting point for a new one-shot ``GW`` calculation, which constitutes the second iteration and so on until the quasiparticle energies are converged. The construction of the mean-field system is, to some extent, arbitrary. We use the following definition, which is slightly modified from the original work [PRL 93, 126406]:
for off-diagonal elements. The ``hermitianized`` QSGW operator is then obtained from :math:`{\Sigma^\mathrm{xc,H}=(A+A^\dagger)/2}`. The difference to the original definition is the inclusion of the renormalization factor to better reproduce the ``GW`` quasiparticle energies. The ``hermitianized`` matrix, or rather the difference :math:`{\Sigma^{\mathrm{xc,H}}-v^\mathrm{xc}}`, is written to the file "spex.qsgw", which is later read by the DFT code. In Fleur, the following steps are required:
for off-diagonal elements. The *hermitianized* QSGW operator is then obtained from :math:`{\Sigma^\mathrm{xc,H}=(A+A^\dagger)/2}`. The difference to the original definition is the inclusion of the renormalization factor to better reproduce the ``GW`` quasiparticle energies. The *hermitianized* matrix, or rather the difference :math:`{\Sigma^{\mathrm{xc,H}}-v^\mathrm{xc}}`, is written to the file "spex.qsgw", which is later read by the DFT code. In Fleur, the following steps are required:
* ``rm fleur.qsgw`` - remove any previous version of the hermitianized matrix.
* ``rm fleur.qsgw`` - remove any previous version of the *hermitianized* matrix.
* ``rm broyd*`` - remove Broyden information about previous iterations because this information is inconsistent with the new Hamiltonian (the SCF calculation does not converge otherwise).
* ``rm broyd*`` - remove Broyden information about previous iterations because this information is inconsistent with the new Hamiltonian (the SCF calculation does not converge otherwise).
* Set ``gw=3`` in the Fleur input file.
* Set ``gw=3`` in the Fleur input file.
* Run Fleur.
* Run Fleur.
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@@ -389,7 +397,9 @@ Polarization function
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@@ -389,7 +397,9 @@ Polarization function
=====================
=====================
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
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.)
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.)
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@@ -425,12 +435,13 @@ The most important keyword in the calculation of the polarization function is ``
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@@ -425,12 +435,13 @@ The most important keyword in the calculation of the polarization function is ``
| | | | and an accumulated stretching factor of 30 at 5 htr. |
| | | | and an accumulated stretching factor of 30 at 5 htr. |
(*) In the limit :math:`{k\rightarrow 0}`, the projections in the numerator of [[#Eq:P|``P``]] approach linearly to zero. However, when calculating the dielectric function, one has to multiply with :math:`{\sqrt{4\pi}/k}` (square root of Coulomb matrix) in this limit. So, the first order of :math:`{\langle e^{i\mathbf{kr}} \phi_{\mathbf{q}n} | \phi_{\mathbf{k+q}n'} \rangle}` (corresponding to :math:`{\mu=1}`) in ``'k``' becomes relevant. Using k·p perturbation theory, one can show that the linear term is proportional to :math:`{(\epsilon_{\mathbf{q}n'}-\epsilon_{\mathbf{q}n})^{-1}}`. This can lead to numerical problems if the two energies are very close to each other. Therefore, when treating the Γ point (k=0), Spex multiplies the linear term with this energy difference, resulting in smooth and non-divergent values, and takes the energy difference into account by replacing :math:`{S(\omega)\rightarrow S(\omega)/\omega}` in the [[#Eq:P|frequency integration]], thereby avoiding the numerical difficulties. (As an alternative, the energy differences can also be incorporated into the integration weights, which is arguably even more stable numerically, see option ``INT`` below.) By default, Spex does that only for k=0. The behavior can be changed with the keyword ``MULTDIFF`` in the section ``SUSCEP``.
(*) In the limit :math:`{k\rightarrow 0}`, the projections in the numerator of [[#Eq:P|``P``]] approach linearly to zero. However, when calculating the dielectric function, one has to multiply with :math:`{\sqrt{4\pi}/k}` (square root of Coulomb matrix) in this limit. So, the first order of :math:`{\langle e^{i\mathbf{kr}} \phi_{\mathbf{q}n} | \phi_{\mathbf{k+q}n'} \rangle}` (corresponding to :math:`{\mu=1}`) in :math:`k` becomes relevant. Using k·p perturbation theory, one can show that the linear term is proportional to :math:`{(\epsilon_{\mathbf{q}n'}-\epsilon_{\mathbf{q}n})^{-1}}`. This can lead to numerical problems if the two energies are very close to each other. Therefore, when treating the Γ point (k=0), Spex multiplies the linear term with this energy difference, resulting in smooth and non-divergent values, and takes the energy difference into account by replacing :math:`{S(\omega)\rightarrow S(\omega)/\omega}` in the [[#Eq:P|frequency integration]], thereby avoiding the numerical difficulties. (As an alternative, the energy differences can also be incorporated into the integration weights, which is arguably even more stable numerically, see option ``INT`` below.) By default, Spex does that only for k=0. The behavior can be changed with the keyword ``MULTDIFF`` in the section ``SUSCEP``.