View source for ACFDT/RPA calculations

The adiabatic-connection-fluctuation-dissipation theorem (ACFDT) can be used to derive the random-phase approximation (RPA).
The ACFDT-RPA ground-state energy <math>E_{\mathrm{RPA}}</math> is the sum of the ACFDT-RPA correlation energy <math>E_{c}</math> and the Hartree-Fock (HF) energy <math>E_{\mathrm{EXX}}</math> evaluated non self-consistently using Kohn-Sham orbitals computed within density-functional theory (DFT):

<math>E_{\mathrm{RPA}}=E_{\mathrm{c}}+E_{\mathrm{EXX}}</math>.

Note that, here <math>E_{\mathrm{EXX}}</math> includes also the Hartree energy, the kinetic energy, as well as the Ewald energy of the ions, whereas often in literature <math>E_{\mathrm{EXX}}</math> refers only to the exact-exchange (EXX) energy evaluated using DFT orbitals.

If {{TAG|ALGO}}=RPA is set in the {{TAG|INCAR}} file, VASP  calculates the correlation energy within RPA. To this end, VASP first calculates the independent-particle response function, using the unoccupied states given in the {{TAG|WAVECAR}} file, and then determines the correlation energy using the plasmon-fluctuation equation:

<math>E_{c} = \frac{1}{2 \pi} \int_{0}^{\infty}[ \mathrm{Tr} \, \ln(1-\chi( {\rm i} \omega) V) + \chi( {\rm i} \omega) V ]\, d \omega</math>.

More information about the theory behind the RPA is found [[RPA/ACFDT: Correlation energy in the Random Phase Approximation|here]]. 

== General recipe to calculate ACFDT-RPA total energies ==

The ACFDT-RPA total energies can be computed in a four-step procedure or in an all-in-one approach. The all-in-one approach is available as of VASP.6 and can be done in one single step.
Both procedures comprise a self-consistent DFT calculation, diagonalizing the KS Hamiltonian, and the RPA calculation itself. While the all-in-one approach is very convenient, there are several caveats discussed below.

=== Four-step procedure to calculate ACFDT-RPA total energy ===

This four-step procedure is the standard procedure in VASP.5, and is also available in VASP.6 by setting {{TAG|NBANDS}} in the RPA step. 

'''Step 1:''' a standard DFT run. 
:All occupied orbitals (and as usual in VASP, a few unoccupied orbitals) of the DFT Hamiltonian are calculated. For instance,
 {{TAG|EDIFF}} = 1E-8 # required for convergence of RPA total energies
 {{TAG|ISMEAR}} = 0 ; {{TAG|SIGMA}} = 0.05
{{NB|warning| The convergence of DFT eigenenergies influences the convergence of the RPA correlation energy.|:}}
{{NB|tip|Use the Perdew-Burke-Ernzerhof (PBE) pseudopotentials with a small {{TAG|EDIFF}} and avoid {{TAG|ISMEAR}}>0.|:}}
:The DFT run can be done with your favorite setup, but we recommend attaining very high precision (small {{TAG|EDIFF}} tag) and to use a small smearing width ({{TAG|SIGMA}} tag), and to avoid higher-order Methfessel-Paxton smearing (see also {{TAG|ISMEAR}}). That is because Methfessel-Paxton smearing can lead to negative one-electron occupancies, which can result in unphysical correlation energies. We suggest using [[GGA|PBE]] orbitals as input for the ACFDT-RPA run, but other choices, e.g., local-density-approximation (LDA) or [[:Category:Hybrid_functionals|hybrids]], are possible as well. For hybrid functionals, we suggest carefully considering the caveats mentioned in Ref. {{cite|paier:2008}}. Specifically, the RPA dielectric matrix yields too weak screening for hybrid functionals, which potentially deteriorates RPA results.
'''Step 2:''' Compute the HF energy <math>E_{\mathrm{EXX}}</math> using the DFT orbitals. That is, e.g.,
 {{TAG|ALGO}}  = EIGENVAL ; {{TAG|NELM}} = 1
 {{TAG|LWAVE}} = .FALSE.                ! {{FILE|WAVECAR}} not written     
 {{TAG|LHFCALC}} = .TRUE. ; {{TAG|AEXX}} = 1.0  ! {{TAG|ALDAC}} = 0.0 
 {{TAG|ISMEAR}} = 0 ; {{TAG|SIGMA}} = 0.05
:Here, {{TAG|LWAVE}} {{=}} .FALSE. avoids an accidental update of the {{FILE|WAVECAR}} file. Also, VASP automatically sets {{TAG|ALDAC}} {{=}} 1-{{TAG|AEXX}}, i.e., for {{TAG|AEXX}} {{=}} 1.0, {{TAG|ALDAC}} {{=}} 0.0.
{{NB|tip| For insulators and semiconductors with a sizable gap, faster k-point convergence of the HF energy can be obtained by setting {{TAG|HFRCUT}}{{=}}-1, although this slows convergence for metals.|:}}
'''Step 3:''' Compute a sufficient number of unoccupied states.  
:In the {{TAG|OUTCAR}} file of Step 1, search for <code>maximum number of plane-waves:</code> and run VASP again with the following {{TAG|INCAR}} file to determine the unoccupied states by exact diagonalization of the KS Hamiltonian. The KS Hamiltonian must be the same as in step 1, which might be DFT or hybrid.
 {{TAG|NBANDS}} = maximum number of plane-waves (times 2 for vasp_gam)
 {{TAG|ALGO}} = Exact    ! exact diagonalization
 {{TAG|EDIFF}} = 1E-8 # required for convergence of RPA total energies
 {{TAG|LOPTICS}} = .TRUE.
 {{TAG|ISMEAR}} = 0 ; {{TAG|SIGMA}} = 0.05
:For calculations using <code>vasp_gam</code>, that is the gamma-point only version of VASP, {{TAG|NBANDS}} must be set to twice the <code>maximum number of plane-waves:</code> found in the {{TAG|OUTCAR}} file of Step 1. For metals, we recommend to avoid setting {{TAG|LOPTICS}}=.TRUE., since this slows down k-point convergence {{cite|p:2010}}.  
'''Step 4:''' Calculate the ACFDT-RPA correlation energy.
 {{TAG|NBANDS}} =  maximum number of plane-waves
 {{TAG|ALGO}} = ACFDT
 {{TAG|NOMEGA}} = 12     ! default
 {{TAG|ISMEAR}} = 0 ; {{TAG|SIGMA}} = 0.05 
:The number of imaginary-frequency and imaginary-time-grid points {{TAG|NOMEGA}} is usually set to 8-24. For large gap insulators {{TAG|NOMEGA}} = 8 will suffice, while for semiconductors 10-12 suffices.
{{NB|important| For VASP 6.X, {{TAG|NBANDS}} must be set in this step.|:}}

=== All-in-one approach to calculate ACFDT-RPA total energy ===

VASP will read the {{FILE|WAVECAR}} file, perform a self-consistent DFT calculation by iterating until convergence is reached, diagonalize the DFT Hamiltonian in the basis set spanned by all KS orbitals, and finally proceed with the RPA calculation. The all-in-one approach is only available as of VASP.6, and it is particularly convenient for the RPAR and GWR algorithms. An example {{FILE|INCAR}} file is

 {{TAG|ALGO}} = RPAR
 
 {{TAG|EDIFF}} = 1E-8  # required for convergence of RPA total energies
 {{TAG|ISMEAR}} = 0 ; {{TAG|SIGMA}} = 0.05
  
 {{TAG|NOMEGA}} = 8-24    ! for large gap insulators {{TAG|NOMEGA}} = 8 will suffice, for semiconductors 10-12 suffices
    
 {{TAG|LOPTICS}} = .TRUE. ! optional 
 {{TAG|LPEAD}} = .TRUE.   ! required for 6.1 and 6.2 if {{TAG|LOPTICS}} = .TRUE.
{{NB|warning|The tag {{TAG|NBANDS}} must not be set in the {{FILE|INCAR}} file.|:}} 
{{NB|warning|The convergence of DFT eigenenergies influences the convergence of the RPA correlation energy.|:}}
{{NB|tip|Use the Perdew-Burke-Ernzerhof (PBE) pseudopotentials with a small {{TAG|EDIFF}} and avoid {{TAG|ISMEAR}}>0.|:}}
The head of the dielectric function can be calculated only for insulators by setting the {{TAG|LOPTICS}} tag and only the Perturbation-Expression-After-Discretization (PEAD) method is supported, see {{TAG|LPEAD}}. Furthermore, {{TAG|EDIFF}} should be small to calculate the DFT orbitals with high precision. Although it is not strictly required, we recommend starting the calculation from a well-converged DFT {{TAG|WAVECAR}} file.

There are several caveats to the all-in-one approach: 

* There is no general support for ACFDT/RPA on top of hybrid functionals. In VASP <math>\leq</math> 6.2, only LDA and gradient corrected functionals are supported. As of VASP 6.2, the all-in-one approach can be used for Hartree-Fock-type calculations, as well as hybrid functionals without range separation. However, range-separated hybrid functionals (including HSE, {{TAG|LMODELHF}}) yield erroneous results or potentially even crash. Please compare the total energies in a standard DFT/hybrid functional calculation against the energies obtained for the mean-field step in the all-in-one approach: The energies must be exactly the same. If they differ, fall back to the four-step procedure that is explained above.
* To select the new all-in-one approach, it is important "''not to set'' {{TAG|NBANDS}}" in the RPA step. If {{TAG|NBANDS}} is set in the {{FILE|INCAR}} file, VASP.6 proceeds with reading the {{TAG|WAVECAR}} file found in the directory (if not present, random orbitals are used!), and then calculates the correlation energy using these orbitals and one-electron energies.
* The basis set used in the diagonalization is strictly given by all plane waves. That is, there is no option to reduce the basis set size as one can by setting {{TAG|NBANDS}} in Step 3 and Step 4 of the four-step procedure. 
* The exchange energy is only calculated if the low-scaling RPA is selected. 
* As of VASP 6.3, the head of the dielectric function (G=0 component) can be calculated by setting {{TAG|LOPTICS}} = .TRUE. 
* In the all-in-one procedure, VASP automatically sets {{TAG|LMAXFOCKAE}} and {{TAG|NMAXFOCKAE}}. This changes the energies for hybrid functionals and Hartree-Fock slightly. Hence, you need to set {{TAG|LMAXFOCKAE}} {{=}} 4 also in Step 1 of the four-step procedure to obtain the same results as in the all-in-one procedure.

=== Output of ACFDT-RPA total energy ===
The ACFDT-RPA total energy is calculated for 8 different energy cutoffs and a linear regression is used to extrapolate the results to the infinite cutoff limit, see [[ACFDT/RPA calculations#basis_cutoff|section below]]. A successful RPA calculation writes the following lines into the {{FILE|OUTCAR}} file:

       cutoff energy     smooth cutoff   RPA   correlation   Hartree contr. to MP2
 ---------------------------------------------------------------------------------
             316.767           316.767      -17.5265976349      -26.2640927215
             301.683           301.683      -17.3846505665      -26.0990489039
             287.317           287.317      -17.2429031341      -25.9344769084
             273.635           273.635      -17.0686574017      -25.7325162480
             260.605           260.605      -16.8914915810      -25.5277026697
             248.195           248.195      -16.7202601717      -25.3302982602
             236.376           236.376      -16.5559849344      -25.1415392478
             225.120           225.120      -16.3635400223      -24.9210737434
   linear regression    
   converged value                          -19.2585393615      -28.2627347266

Here, the third and fourth columns correspond to the correlation energy (for that specific energy cutoff) in the RPA and the direct second-order Møller–Plesset (MP2) approximation, i.e., the second-order term in RPA. The corresponding results of the linear regression are found in the line starting with <code>converged value</code>. 
<span id="ACFDTR/RPAR">

== Low-scaling ACFDT/RPA algorithm ==
Virtually the same tags and procedures apply to the low-scaling RPA algorithm implemented in VASP 6 {{cite|kaltak:prb:2014}}. However, {{TAG|ALGO}}=ACFDT or {{TAG|ALGO}}=RPA needs to be replaced by either {{TAG|ALGO}}=ACFDTR or {{TAG|ALGO}}=RPAR. 

With this setting VASP calculates the independent-particle polarizability <math> \chi(i\omega) </math> using Green's functions <math>G(i\tau)</math> on the imaginary time axis <math>i\tau</math> by the contraction formula {{cite|rojas:prl:1995}}

<math>
 \chi(i\tau_m) = -G(i\tau_m)G(-i\tau_m)
</math>

Subsequently, a compressed Fourier transformation on the imaginary axes <math>\tau_m,\omega_n</math> yields 

<math>
 \chi(i\omega_n) = \sum_{m=1}^{N} \gamma_{nm} \chi(i\omega_m)
</math>

The remaining step is the evaluation of the correlation energy and is the same as described above. 

Crucial to this approach is the accuracy of the Fourier transformation from <math>\chi(i\tau)\to \chi(i\omega)</math>, which in general depends on two factors: First, the grid order <math>N</math> that can be set by {{TAG|NOMEGA}} in the {{TAG|INCAR}} file. Here, similar choices as for the {{TAG|ALGO}}{{=}}ACFDT are recommended. Second, the grid points <math>\tau_m,\omega_n</math> and Fourier matrix <math>\gamma_{nm}</math> have to be optimized for the same interval as spanned by all possible transition energies in the polarizability. The minimum (maximum) transition energy can be set with the {{TAG|OMEGAMIN}} ({{TAG|OMEGATL}}) tag and should be smaller (larger) than the bandgap, i.e., the maximum transition energy, of the previous DFT calculation. VASP determines these values automatically and writes it in the {{TAG|OUTCAR}} after the lines 

 Response functions by GG contraction: 
These values should be checked for consistency. Furthermore, we recommend inspecting the grid and transformation errors by looking for the following lines in the {{TAG|OUTCAR}} file

 nu_ 1=  0.1561030E+00 ERR=   0.6327933E-05 finished after   1 steps    
 nu_ 2= ...
 Maximum error of frequency grid:  0.3369591E-06

Every frequency point will have a similar line as shown above for the first point. The value after <code>ERR=</code> corresponds to the maximum Fourier-transformation error and should be of similar order as the maximum integration error of the frequency grid.
=== Output of low-scaling ACFDT/RPA ===
Selecting the low-scaling RPA algorithm, VASP computes the total energy in the RPA and writes the following output
  -----------------------------------------------------
  HF-free energy      FHF    =       -25.11173505 eV
  HF+RPA corr. energy TOTEN  =       -36.96463791 eV
  HF+E_corr(extrapolated)    =       -37.70506951 eV
The line <code>HF+RPA corr. energy TOTEN</code> contains the total energy calculated with the largest cutoff {{TAG|ENCUTGW}}. 
The line <code>HF+E_corr(extrapolated)</code> contains the total energy with the extrapolated value for the RPA correlation energy.
=== Singles contribution to the correlation energy ===
The low-scaling RPA algorithm also allows for the determination of the so-called singles contribution,{{cite|klimes:jcp:143}}{{cite|ren:prb:88}} to the total energy represented by following diagrams:

[[File:SinglesDiagrams.png|320px]]

In most textbooks{{cite|mattuck:2012}} this contribution is assumed to be zero. Ren et al.{{cite|ren:prb:88}} pointed out that this is true only in the Hartree-Fock basis set. The singles contribution is non-vanishing for other basis sets. 

The singles contribution is calculated for {{TAG|LSINGLES}}=.TRUE.. In this case following additional lines can be found in OUTCAR:

 HF single shot energy change        -1.23182672
 renormalized HF singles             -1.23310555

Here, the first line contains the value of the singles as proposed by Klimeš et al.,{{cite|klimes:jcp:143}} while the second line contains the singles contribution of Ren et al.{{cite|ren:prb:88}}. In most cases we found that the two values are exceedingly close to each other, which can be understood in the way how the propagator is renormalized.{{cite|klimes:jcp:143}}
<span id="RPAFORCES">
=== Optional: RPA Forces ===
Optionally, RPA forces can be calculated by adding following line to the {{FILE|INCAR}} file:
  {{TAG|LRPAFORCE}} = .TRUE. 
For RPA forces the change in the one-electron density is required.{{cite|ramberger:prl:118}}
This is automatically performed with the linear-response routine within VASP. 
After a successful run, the following block of data is found in the {{FILE|OUTCAR}} file.

 POSITION                                       TOTAL RPA FORCE (eV/Angst)
 -----------------------------------------------------------------------------------
      0.17542     -0.22348      0.17542        -0.292069      7.581315     -0.292069
      1.12850      1.31044      1.12850         0.304683     -7.605527      0.304683
 -----------------------------------------------------------------------------------
    total drift:                                0.012614     -0.024212      0.012614
 
 SUGGESTED UPDATED POSCAR (direct coordinates)  step
 -----------------------------------------------------------------------------------
  -0.00958461  -0.00958461   0.13485779         0.04179056   0.04179056   0.00283088
   0.25787833   0.25787833   0.22191754        -0.04337198  -0.04337198   0.00431513
{{NB|warning|The RPA forces computed for metallic systems are currently not supported.}}

The suggested updated positions are also written to the CONTCAR file (if {{TAG|NSW}}=0). The updated positions are obtained by multiplying the RPA-forces with the parameter {{TAG|DAMP_NEWTON}} and the inverse of the DFT-Hessian (which is also calculated during the RPA force calculations) and adding the resultant vector to the current positions. {{TAG|DAMP_NEWTON}} currently defaults to 0.8, but the user might want to change this to 1.0 (no damping), or smaller values if instabilities are observed. The CONTCAR file can be copied to the POSCAR file, and a few such RPA calculations should recover the groundstate structure. Alternatively, standard relaxations ({{TAG|IBRION}} =1-4) or even MD's can be performed.  
</span>
=== Memory bottleneck and Parallelization ===

The cubic scaling space-time RPA algorithm requires considerably more memory than the corresponding quartic-scaling implementations, two Green's functions <math>G({\bf r,r'},i\tau_n)</math> have to be stored in real-space. To reduce the memory overhead, VASP exploits Fast Fourier Transformations (FFT) to avoid storage of the matrices on the (larger) real space grid, on the one hand. The precision of the FFT can be selected with {{TAG|PRECFOCK}}, where usually the values ''Fast'' sufficient.

On the other hand, the code avoids storage of redundant information, i.e., both the Green's function and polarizability matrices are distributed as well as the individual imaginary grid points. The distribution of the imaginary grid points can be set by hand with the {{TAG|NTAUPAR}} and {{TAG|NOMEGAPAR}} tags, which splits the imaginary grid points {{TAG|NOMEGA}} into {{TAG|NTAUPAR}} time and {{TAG|NOMEGAPAR}} groups. For this purpose both tags have to be divisors of {{TAG|NOMEGA}}. 

The default values are usually reasonable choices provided the tag {{TAG|MAXMEM}} is set correctly and we strongly recommend to set {{TAG|MAXMEM}} instead of {{TAG|NTAUPAR}}. 
{{NB|important|As of version 6.2, {{TAGBL|MAXMEM}} is estimated automatically (if not set) from the "MemAvailable" entry of the Linux kernel in "/proc/meminfo".}}

The required storage for a low-scaling RPA or GW calculation depends mostly on {{TAG|NTAUPAR}}, the number of MPI groups that share same imaginary time points. A rough estimate for the required bytes is given by 

 (NGX*NGY*NGZ)*(NGX_S*NGY_S*NGZ_S) / ( NCPU  / {{TAGBL|NTAUPAR}} ) * 16

where "NCPU" is the number of MPI ranks used for the job,"NGX,NGY,NGZ" denotes the number of FFT grid points for the exact exchange and  "NGX_S,NGY_S,NGZ_S" the number of FFT grid points for the supercell. Note, both grids are written to the {{TAGBL|OUTCAR}} file after the lines

 FFT grid for exact exchange (Hartree Fock)
 FFT grid for supercell:

The smaller {{TAG|NTAUPAR}} is set, the less memory per node the job requires to finish successfully. 

The approximate memory requirement is calculated in advance and printed to screen and {{TAGBL|OUTCAR}} as follows:

 min. memory requirement per mpi rank 1234 MB, per node 9872 MB
</span>

== Some Issues Particular to ACFDT-RPA Calculations on Metals ==

For metals, the RPA groundstate energy converges the fastest with respect to k-points, if the exchange (Eq. (12) in reference {{cite|harl:2010}}) and correlation energy are calculated on the same k-point grid, {{TAG|HFRCUT}} is not set, and the long-wavelength contributions from the polarizability are not considered  (see reference {{cite|harl:2010}}).

To evaluate Eq. (12), a correction energy for <math>E_{\mathrm{EXX}}</math> related to partial occupancies has to be added to the RPA groundstate energy:{{cite|harl:2010}}

<math>E_{\mathrm{RPA}}=E_{\mathrm{c}}+E_{\mathrm{EXX}}+E_{\mathrm{HFc}}</math>.

In vasp.5.4.1, this value is calculated for any HF type calculation (step 2) and can be found in the {{TAG|OUTCAR}} file after the total energy (in the line starting with ''exchange ACFDT corr.  ='').

To neglect the long-wavelength contributions, simply set {{TAG|LOPTICS}}=''.FALSE.'' in the {{TAG|ALGO}}=''Exact'' step (third step), and remove the {{TAG|WAVEDER}} files in the directory.

<span id="basis_cutoff">

== Possible tests and known issues ==
=== Basis set convergence === 
{{:Basis set convergence of RPA-ACFDT calculations}}
Convergence with respect to the number of plane waves can be rather slow, and we recommend to test the calculations carefully. Specifically, the calculations should be performed at the default energy cutoff {{TAG|ENCUT}}, and at an increased cutoff (ideally the default energy cutoff <math>\times 1.3</math>). Another issue is that energy volume-curves are sometimes not particularly smooth. In that case, the best strategy is to set
 {{TAGBL|ENCUT}} = 1.3 times default energy cutoff
 {{TAGBL|ENCUTGWSOFT}} = 0.5 times default energy cutoff

where the default energy cutoff is the usual one (maximum ENMAX in POTCAR files). The frequency integration also needs to be checked carefully, in particular for small gap systems (some symmetry broken atoms) convergence can be rather slow, since the one-electron band gap can be very small, requiring a very small minimum <math>\omega</math> in the frequency integration.

=== K-point convergence: Spline interpolation === 
It is possible to use spline interpolation to improve upon k-point integration errors by rewriting the 
RPA correlation energy as an integral 
<math>
E_c^{{\rm  RPA}}= \sum_{{\mathbf{q}}\in \mathbf{BZ} }\sum_{{\mathbf{G}}} S({\bf q+G})
</math>
over the electronic structure factor:
<math>
S({\bf q}+{\bf G}) =\int {\rm d}\omega 
\left\{(\mathrm{ln}[1-\tilde\chi^0({\mathbf{q}},\mathrm{i}\omega)V({\mathbf{q}})])_{{\mathbf{G,G}}}  +V_{{\mathbf{G,G}}}({\mathbf{q}})\tilde\chi^0({\mathbf{q}},{\mathrm{i}}\omega) \right\}
</math>

If {{TAG|ESF_SPLINE}}=T is set, the code stores the electronic structure factor on a coarse grid defined by {{FILE|KPOINTS}} and performs a tricubic spline interpolation towards finer q-grids iteratively. After each iteration the RPA correlation energy is evaluated and compared to the result of previous interpolation. 
If the difference in energy is smaller than {{TAG|ESF_CONV}} within {{TAG|ESF_NINTER}} interpolation steps, the code consideres the q-point integration to be converged and reports the result to {{FILE|OUTCAR}} in the following format

      cutoff energy     smooth cutoff   RPA   correlation   Hartree contr. to MP2  RPA spline-interp.
 -----------------------------------------------------------------------------------------------------
            166.667           133.333      -12.9738715106      -19.7255874374      -13.4968000908
            158.730           126.984      -12.8840657072      -19.6294580403      -13.4017404001
            151.172           120.937      -12.7775593388      -19.5151822998      -13.3005326847
            143.973           115.178      -12.6604147404      -19.3892142669      -13.1868498210
            137.117           109.694      -12.5530911576      -19.2733151174      -13.0861120393
            130.588           104.470      -12.4659186304      -19.1786165194      -12.9778587892
            124.369            99.495      -12.3690601643      -19.0725742983      -12.8709666989
            118.447            94.758      -12.2461267475      -18.9372318755      -12.7590723870
  linear regression    
  converged value                          -14.0340307585      -20.8751715586      -14.5828037654

The last column contains the result from the spline interpolation for the selected energy cutoffs reported in the first column. 
{{NB|warning|Delete {{FILE|WAVEDER}} for this method.}}
Note, this method is incompatible with k-p perturbation theory, where the 
largest q-point integration error <math>
\lim_{\bf q\to 0} \tilde\chi^0_{{\bf G G}'}({\bf q},{\rm i}\omega) \cdot {\bf V}_{\bf G G'}({\bf q})
</math> is added explicitly to the RPA integral. 
This long-wave contribution is stored in {{FILE|WAVEDER}} and vasp assumes you want to add this term if the file is present in the working directory. 

Also, the long-wave limit is ill-defined for metallic systems, in contrast to the spline interpolation method.  
{{NB|mind|{{TAG|ESF_SPLINES}} can be used for metals.}}

For insulators, the q-point integration error is similar for both methods, i.e. adding the long-wave limit or using tricubic spline interpolation. 
</span>

== Related Tags and Sections ==

* {{TAG|ALGO}} for response functions and ''ACFDT'' calculations
* {{TAG|NOMEGA}}, {{TAG|NOMEGAR}} number of frequency points
* {{TAG|LHFCALC}}, switches on HF calculations
* {{TAG|LOPTICS}}, required in the DFT step to store head and wings 
* {{TAG|ENCUTGW}}, to set cutoff for response functions
* {{TAG|ENCUTGWSOFT}}
* {{TAG|PRECFOCK}} controls the FFT grids in HF, GW, RPA calculations
* {{TAG|NTAUPAR}} controls the number of imaginary time groups in space-time GW and RPA calculations
* {{TAG|NOMEGAPAR}} controls the number of imaginary frequency groups in space-time GW and RPA calculations
* {{TAG|MAXMEM}} sets the available memory per MPI rank on each node
* {{TAG|LFINITE_TEMPERATURE}} switches on Matsubara (finite temperature) formalism

== References ==
<references/>

----

[[Category:Many-body perturbation theory]][[Category:ACFDT]][[Category:Howto]][[Category:Low-scaling GW and RPA]]