Table of Contents

Kohn-Sham random-phase approximation

This chapter describes three different programs that are related to Kohn-Sham based RPA correlation methods. The first one is the density fitting RPA program of Heßelmann et al. described in section Density fitting RPA programs. The second one is the RPATDDFT program by Toulouse et al., see section Random-phase approximation (RPATDDFT) program. And finally section Self consistent RPA programs presents a set of programs which can be used to perform self consistent EXX and RPA calculations developed by Trushkin and Görling.

All of the different codes are capable to perform standard RPA correlation energy calculations, but have distinct features which separate them from each other. More details about each program's capabilities can be found in the respective subsections.

Density fitting RPA programs

Electron correlation energies within the random-phase approximation can be calculated by the programs DIRPA, RPAX2 and ACFDT that are subdirectives of the driver command KSRPA. These methods should be used in conjunction with Kohn-Sham reference determinants, i.e., orbitals and orbital energies from a preceeding DFT calculation should be supplied. An alterntive implementation of various RPA electron correlation methods by Julien Toulouse and co-workers is described in the subsection Random-phase approximation (RPATDDFT) program. The second alternative implementation of RPA is based on the adiabatic-connection fluctuation-dissipation and is described in the subsection RIRPA program. This implementation additionally allows calculations with σ-functionals, which at negligible computational cost significantly improves over RPA in the calculation of various chemical properties.

The self-consistent random phase approximation method (J. Chem. Phys. 139, 084113 (2013), Phys. Rev. Lett. 134, 016402 (2025)) is implemented in the SCRPA program. The program uses the optimized effective potential method to construct the exact exchange potential and the random phase approximation correlation potential. The SCEXX program offers the self-consistent exact exchange optimized effective potential method (J. Chem. Phys. 127, 054102 (2007), J. Chem. Phys. 155, 054109 (2021)) that neglects correlation.

A typical input to calculate the RPAX2 correlation energy is given by: 

basis={
set,orbital; default,<basis>
set,mp2fit;  default,<basis>/mp2fit}

ks,pbe
{ksrpa; rpax2,orb=2100.2}

All methods are implemented using density-fitting of the two-electron repulsion integrals, see Refs. [2,4]. Because of this, auxiliary basis sets for fitting occupied-virtual orbital pairs have to be given, see section auxiliary basis sets for density fitting or resolution of the identity. No point-group symmetry can be used for all methods described in this section.

References:

RPA:
$[1]$ F. Furche, Phys. Rev. B 64, 195120 (2001).
RPAX2:
$[2]$ A. Heßelmann, Phys. Rev. A 85, 012517 (2012).
$[3]$ A. Heßelmann, Top. Curr. Chem. 365 97 (2015)
ACFDT(ALDA):
$[4]$ A. Heßelmann and A. Görling, J. Chem. Theory Comput. 9, 4382 (2013).

DIRPA program

The direct RPA program (implemented with the algorithm described in [1]) has the following options:

Note that in case of MODE=1 or MODE=2 it is recommended to have the wavefunction (wfu) directory located on a scratch partition. E.g., add the command line option -W /scratch/$USER/wfu to the Molpro command.

RPAX2 program

The RPAX2 method is an extension to the RPA and accounts for higher order particle-hole pair exchange contributions [2,3]. The RPAX2 program has the same options as the DIRPA program, see section DIRPA program. The following list shows a few additional ones that can be used:

Spin-unrestricted calculations can be done using the URPAX2 program. In this case the orbitals from a preceeding unrestricted Kohn-Sham calculation have to be passed to the program (via the ORB key).

ACFDT program

The ACFDT (adiabatic connection fluctuation-dissipation theorem) method is an alternative approach to derive the RPA. If used in conjunction with local adiabatic exchange-correlation kernels, the method can also describe electron-electron interaction contributions beyond the RPA. Currently, the ALDA xc-kernel can be used in the program (ACFDT(ALDA) method), see also Ref. [4]. The ACFDT program has the following options:

The ACFDT(ALDA) method is ill-defined for short electron-electron distances, see Ref. [4] and F. Furche and T. Van Voorhis, J. Chem. Phys. 122, 164106 (2005). Because of this, the method does not have a defined basis set limit and the ACFDT program should not generally be used to calculate ACFDT(ALDA) correlation energies. Instead, the ACFDT2 should be utilised which implements the hybrid approach as described in Ref. [4] and which has, in addition to the ones given above, the following options:

For applying the correction as described in Ref. [4], the vaules of SCAL and MU have to be set to the values SCAL=0.6 and MU=2d0.

The ACFDT3 program implements an approximation to the ACFDT(ALDA) method assuming that the xc-kernel matrix depends linearly on the coupling strength (which is true for the exchange contribution but not, in general, for the correlation contribution to the kernel). Within this approximation the coupling-strength integration can be done analytically leading to a performance improvement over the ACFDT and ACFDT2 programs. The options for ACFDT3 are identical to the ones given above for ACFDT and ACFDT2.

Random-phase approximation (RPATDDFT) program

The random-phase approximation program (rpatddft) can be used to calculate RPA correlation energies after a SCF calculation. Additionnally, it can be used to calculate dynamic dipole polarizabilities, C$_6$ dispersion coefficients, and excitation energies. The program currently works without point-group symmetry.

List of the main keywords:

Calculation of RPA correlation energies [1] (see options below)

Calculation of dynamic dipole polarizabilities and C$_6$ dispersion coefficients [10] (see options below)

Calculation of excitation energies [11] (see options below)

as well as contextual options (see later for an explanation on this):

Number of quadrature points for the Gauss-Legendre numerical integration along the adiabatic connection for RPA calculations (default is 7). If LAMBDA and WEIGHT are given, assumes a one-point quadrature with given abscissa and weight.

Options for the numerical integration over the frequency variable of RPA calculations. METHFREQ governs the type of quadrature used (0(default) is Gauss-Chebyshev, 1 and 2 are Gauss-Legendre, 3 is Clenshaw-Curtis) and NFREQ governs the number of quadrature points (default is 16).

and global options, shared by all commands inside the rpatddft block:

Specify the exchange and correlation kernel for EXCIT (if only one argument is given, it is understood to be the exchange-correlation kernel).

Calculation of RPA correlation energies ECORR, <list of methods>
If no method is given, a SO2-RCCD calculation will be done (see below).

There are two main RPA variants [1]: dRPA (direct RPA, without the inclusion of an Hartree-Fock exchange kernel in the response function) and RPAx (with the Hartree-Fock exchange kernel included in the response function).

There are four main formulations in which the RPA equations can be derived. The adiabatic-connection fluctuation-dissipation theorem (ACFDT) equation involves integrations both over frequency and coupling constant: an analytical integration along the frequency variable followed by a quadrature on the coupling constant yields the adiabatic connection formulation (AC) [1], while an analytic integration on the coupling constant followed by a numerical integration over the frequency yields the dielectric formulation (DIEL) [2]. Two other formulations are obtained when the two integrations are carried out analytically: the plasmon formula (PLASMON) [1] and the ring coupled cluster doubles formulation (RCCD) [3]. These four formulations are not in general equivalent.

Most variants+formulations can readily by used in a spin-unrestricted context [6]. This is implemented in the code and does not need any further input from the user: the RPA program recognizes the spin-unrestricted character of a SCF calculation that was done beforehand and acts accordingly.

Gradients of most of the RCCD-formulation RPA energies are available, both without range-separation with RHF orbitals and with range-separation with RSH orbitals [9]. The calculations are triggered by the presence of the keyword FORCE or OPTG after the energy-related section (see examples at the end of the section).

The user can test the RPA program using make rpatddfttest, which proposes a variety of tests for RPA correlation energy calculations.

The keywords for the methods are constructed on the model:

<variant>-<formulation>-<alternative>

For the AC formulation, the available methods are:

For the DIEL formulation, the available methods are:

For the RCCD formulation, the available methods are:

For the PLASMON formulation, the available methods are:

Note that to all these keywords are associated energy variables defined as :

ECORR_VARIANT_FORMULATION_ALTERNATIVE

(see the examples below).

Example of a dRPA-I calculation using the PBE functional: 

{rks,pbe,orbital,2101.2}
{rhf,start=2101.2,maxit=0}
{rpatddft;
 orb,2101.2;
 ecorr,DRPAI-AC
}
e=ECORR_DRPAI_AC

Example of a range-separated RPAx-I calculation using the short-range PBE exchange-correlation functional and the range-separated parameter mu=0.5: 

{int;erf,0.5}
{rks,exerfpbe,ecerfpbe;rangehybrid;orbital,2101.2}
{rpatddft;
 orb,2101.2;
 ecorr,RPAXI-AC
}

Example of several RPA calculations in the same run: 

{rhf,orbital,2101.2}
{rpatddft;
 orb,2101.2;
 ecorr,DRPAI-AC,RPAXII-RCCD,DRPAI-DIEL
}
e1=ECORR_DRPAI_AC
e2=ECORR_RPAXII_RCCD
e3=ECORR_DRPAI_DIEL

(this way, the calculations are done with the same transformed integrals, i.e. without redoing the integral transformation).

Example of a dRPA-I gradient calculation: 

{rks,pbe,orbital,2101.2}
{rpatddft;
 orb,2101.2;
 ecorr,DRPAI-RCCD
}
force

Example of a geometry optimization at the LDA+dRPA-I level: 

{int;erf,0.5}
{rks,exerf,ecerf;rangehybrid;orbital,2101.2}
{rpatddft;
 orb,2101.2;
 ecorr,DRPAI-RCCD
}
optg

Calculation of properties, excitation energies and oscillator strengths
EXCIT, METHOD=<method> The EXCIT calculations output shows the excitation energies in ua, eV and nm, the oscillator strengths in length and velocity gauge, as well as the major excitations involved in each mode. The methods available are:

The exchange density functionals (FUNCX) available are:

The correlation density functionals (FUNCC) available are:

Example of a range-separated time-dependent density-functional theory calculation using the short-range LDA exchange-correlation functional and the range-separated parameter mu=0.5: 

mu=0.5
{int;erf,mu;save}
{rks,exerf,ecerf;rangehybrid;orbital,2101.2}
{int}
{setmu,mu}
{rpatddft;
 orb,2101.2;
 excit,method=rs-tddft;
 dftkernel,funcx=ldaxerf,funcc=ldacerf
}

Example of a TDHF-TDA calculation with writing of several files for interfacing with a real-time propagation code (see Ref. [14]): 

{rpatddft;
integral,1;
writefile,transmom.dat,energies.dat,virtual.dat,transmom_v.dat,amplitudes.dat,transmom_a.dat;
orb,2330.2;
excit,method=tdhf;
NOSPINBLOCK;
tda}

The files are: transmom.dat: transition moments in position form; energies.dat: total energies of the states; virtual.dat: virtual positive-energy orbitals; transmom_v.dat: transition moments in velocity form; amplitudes.dat: coefficients of excited-state wave functions over single-excited determinants; transmom_a.dat: transition moments in acceleration form

References
$[1]$ J. G. Ángyán, R.-F. Liu, J. Toulouse, and G. Jansen, J. Chem. Theory Comput. 7, 3116 (2011).
$[2]$ G. Jansen, B. Mussard, D. Rocca, J. G. Ángyán (in prep).
$[3]$ J. Toulouse, W. Zhu, A. Savin, G. Jansen, and J. G. Ángyán, J. Chem. Phys. 135, 084119 (2011).
$[4]$ F. Furche, Phys. Rev. B 64, 195120 (2001).
$[5]$ J. Toulouse, I. C. Gerber, G. Jansen, A. Savin, and J. G. Ángyán, Phys. Rev. Lett. 102, 096404 (2009).
$[6]$ B. Mussard, P. Reinhardt, J. G. Ángyán, and J. Toulouse, J. Chem. Phys. (submitted).
$[7]$ Heßelmann, A., Görling, A., Phys. Rev. Lett. 106, 093001 (2011).
$[8]$ Heßelmann, A., Phys. Rev. A 85, 012517 (2012).
$[9]$ B. Mussard, P. G. Szalay, J. G. Ángyán, J. Chem. Theory Comput. 10, 1968 (2014).
$[10]$ J. Toulouse, E. Rebolini, T. Gould, J. F. Dobson, P. Seal, J. G. Ángyán, J. Chem. Phys. 138, 194106 (2013).
$[11]$ E. Rebolini, A. Savin, J. Toulouse, Mol. Phys. 111, 1219 (2013).
$[12]$ J. Toulouse, A. Savin, and H.-J. Flad, Int. J. Quantum Chem. 100, 1047 (2004).
$[13]$ S. Paziani, S. Moroni, P. Gori-Giorgi, and G. B. Bachelet, Phys. Rev. B 73, 155111 (2006).
$[14]$ E. Coccia, B. Mussard, M. Lebeye, J. Caillat, R. Taieb, J. Toulouse, and E. Luppi, Int. J. Quant. Chem. 116, 1120 (2016).

Self consistent RPA programs

RIRPA program

The RIRPA and URIRPA programs allow non-self-consistent spin-restricted and spin-unrestricted resolution of identity (RI) random phase approximation (RPA) [1-3] and σ-functional [4-6] calculations. These methods should be used in conjunction with conventional Kohn-Sham (KS) density functional theory (DFT) calculations, i.e. data from a preceding KS DFT calculation should be provided. Conventional KS DFT means calculations with LDA, GGA and hybrid exchange-correlation functionals.

Bibilography:
RPA:
[1] F. Furche, Phys. Rev. B, 195120 (2001)
[2] A. Heßelmann and A. Görling, Mol. Phys. 109, 2473 (2011)
[3] X. Ren, P. Rinke, C. Joas, and M. Scheffler, J. Mater. Sci. 47, 7447 (2012)
σ-functionals:
[4] E. Trushin, A. Thierbach, A. Görling, J. Chem. Phys. 154, 014104 (2021)
[5] S. Fauser, E. Trushin, C. Neiss, A. Görling, J. Chem. Phys. 155, 134111 (2021)
[6] C. Neiss, S. Fauser, A. Görling, J. Chem. Phys. 158, 044107 (2023)
Other papers cited in the documentation:
[7] E. Trushin, A. Görling, J. Chem. Phys. 155, 054109 (2021)

We kindly ask you to cite the original publications of the corresponding methods in the publications that result from these programs.

Example input file for spin-restricted calculations for the CO molecule:

examples/co_rirpa.inp
gthresh,twoint=1d-20,energy=1d-10,orbital=1d-8 ! tighter thresholds are recommended
gdirect ! integral-direct mode

basis={
default,aug-cc-pwCVQZ ! orbital basis
set,ri; default,aug-cc-pwCVQZ/mp2fit ! RI basis
}

symmetry,nosym ! RIRPA does not use symmetry

angstrom
geometry={
2

C 0.000000 0.000000 -0.646514
O 0.000000 0.000000 0.484886
}

df-ks,pbe,df_basis=aug-cc-pwCV5Z/mp2fit,maxit=200 ! DFT calculation
{cfit,basis_coul=aug-cc-pwCV5Z/mp2fit,basis_exch=aug-cc-pwCV5Z/mp2fit}

acfd;rirpa ! RPA/sigma-functional calculation; one can use alternatively: ksrpa;rirpa

As well as an example of spin-unrestricted calculation for the NH2 molecule:

examples/nh2_urirpa.inp
gthresh,twoint=1d-20,energy=1d-10,orbital=1d-8 ! tighter thresholds are recommended
gdirect ! integral-direct mode

basis={
default,aug-cc-pwCVQZ ! orbital basis
set,ri; default,aug-cc-pwCVQZ/mp2fit ! RI basis
}

symmetry,nosym ! URIRPA does not use symmetry

angstrom
geometry={
3

N 0.000000 0.000000 0.142235
H 0.000000 0.800646 -0.497821
H 0.000000 -0.800646 -0.497821
}

spin=1

df-uks,pbe,df_basis=aug-cc-pwCV5Z/mp2fit,maxit=200 ! DFT calculation
{cfit,basis_coul=aug-cc-pwCV5Z/mp2fit,basis_exch=aug-cc-pwCV5Z/mp2fit}
 
acfd;urirpa ! RPA/sigma-functional calculation; one can use alternatively: ksrpa;urirpa

The following options are available for the RIRPA and URIRPA programs:

SCEXX program

The SCEXX and USCEXX programs allow self-consistent exact-exchange calculations for closed-shell and open-shell systems.

Bibilography:
[1] A. Heßelmann, A.W. Götz, F. Della Sala, A. Görling J. Chem. Phys. 127, 054102 (2007)
[2] E. Trushin, A. Görling, J. Chem. Phys. 155, 054109 (2021)
[3] E. Trushin, A. Görling, J. Chem. Phys. 159, 244109 (2023)
[4] P. Bleiziffer, A. Heßelmann, A. Görling J. Chem. Phys. 139, 084113 (2013)

NOTE: We have tutorial which provides practical Hands-on examples about the use of SCEXX and USCEXX programs and post-processing of results of calculations. This tutorial is a good supplement to this documentation. Link to Tutorial

Important: Read carefully the information below about the selection of basis sets and the thr_fai_oep parameter:
To obtain numerically stable exchange potentials, one must either use regularization techniques to carefully handle the small eigenvalues of the response matrix or to use auxiliary basis sets that are balanced to the orbital basis set. The latter can be done by manually constructing specific orbital and auxiliary basis sets that are sufficiently balanced. This has been possible for a number of atoms and molecules with quite large orbital basis sets [1], but does not qualify as a general applicable routine approach. The SCEXX and USCEXX programs therefore contain a new preprocessing scheme for auxiliary basis sets that effectively removes linear combinations of auxiliary basis functions that couple poorly to products of occupied times unoccupied Kohn-Sham orbitals and enable the construction of numerically stable exchange potentials with standard basis sets [2].

The preprocessing step to remove linear combinations of auxiliary basis functions that couple poorly to products of occupied times unoccupied Kohn-Sham orbitals is implemented according to Sec II5 of Ref. [2]. It involves the threshold thr_fai_oep, which determines how many linear combinations of auxiliary basis functions are removed and varies with respect to the size of the orbital basis set used. In Ref. [2], this scheme was tested using Dunning correlation consistent basis sets and recommended thresholds are

Orbital basis set thr_fai_oep
aug-cc-pwCVTZ 5e-2
aug-cc-pwCVQZ 1.7e-2
aug-cc-pwCV5Z 5e-3

These thresholds are expected to work also for orbital basis sets without augmentation cc-pwCVXZ (X=T,Q,5) and without core-polarization functions aug-cc-pVXZ (X=T,Q,5). As an auxiliary basis set (OEP), the aug-cc-pVXZ/mp2fit (X=T,Q,5) family of basis sets works best. In particular, according to Ref. [2], it is recommended to use aug-cc-pVDZ/mp2fit auxiliary basis sets for atoms up to neon and aug-cc-pVTZ/mp2fit auxiliary basis sets for heavier atoms.

The code allows to perform calculations symmetrized in ordinary, spin or both ordinary and spin space according to Ref. [3], see parameters space_sym and spin_sym below.

Below is an example input file for spin-restricted calculations for the CO molecule. Note that the input record from a preceding calculation is mandatory for initialization of orbitals and eigenvalues as starting point for EXX calculation, whereas it can come from HF or DFT calculations with maxit=0.

examples/co_scexx.inp
gdirect ! integral-direct mode

basis={
default,aug-cc-pwCVQZ ! orbital basis
set,oep;default,aug-cc-pVDZ/mp2fit ! OEP basis
set,dfit;default,aug-cc-pwCV5Z/mp2fit ! density fitting basis
}

symmetry,nosym ! SCEXX does not use symmetry

angstrom
geometry={
2

C        0.000000    0.000000   -0.646514
O        0.000000    0.000000    0.484886
}

df-hf,maxit=0,df_basis=dfit ! HF calculation with 0 iteration
{cfit,basis_coul=dfit,basis_exch=dfit}

acfd;scexx,thr_fai_oep=1.7d-2 ! SCEXX calculation

As well as an example of a spin-unrestricted calculation for the BeF molecule:

examples/bef_uscexx.inp
gdirect ! integral-direct mode

basis={
default,aug-cc-pwCVQZ ! orbital basis
set,oep;default,aug-cc-pVDZ/mp2fit ! OEP basis
set,dfit;default,aug-cc-pwCV5Z/mp2fit ! density fitting basis
}

symmetry,nosym ! USCEXX does not use symmetry

angstrom
geometry={
2

Be    0.0000000    0.0000000   -0.6823625
F     0.0000000    0.0000000    0.6823625

}

charge=0
spin=1

df-uhf,maxit=0,df_basis=dfit ! HF calculation with 0 iterations
{cfit,basis_coul=dfit,basis_exch=dfit}

acfd;uscexx,thr_fai_oep=1.7d-2 ! USCEXX calculation

The following options are available for the `SCEXX` and `USCEXX` programs:

The following parameters are only relevant for the USCEXX program:

Pitfalls:

Since the local exchange potential is important in self-consistent exact-exchange calculations, we provide an illustration of how to plot it, including separate contributions. Let us assume that we have performed calculations for CO with the following options: 

acfd;scexx,thr_fai_oep=1.7d-2,plot_z=1

At the end there is the file vx-final.z with reference $v_x^{ref}$, rest $v_x^{rest}$ and full $v_x$ exchange potentials. The potentials can be plotted using Python and matplotlib as follows: 

import numpy as np
import matplotlib.pyplot as plt
 
def load_potential(potfile):
    """Read data from potential-file.
 
    Args:
      potfile: Path to file with potential data.
 
    Returns:
      x: x coordinate.
      y: y coordinate.
      z: z coordinate.
      coord: coordinate on path.
      vref: reference potential.
      vrest: rest potential.
      v: full potential.
    """
    x, y, z, coord = list(), list(), list(), list()
    vref, vrest, v = list(), list(), list()
    for line in open(potfile):
        aux = line.split()
        x.append(float(aux[1]))
        y.append(float(aux[2]))
        z.append(float(aux[3]))
        coord.append(float(aux[4]))
        vref.append(float(aux[6]))
        vrest.append(float(aux[7]))
        v.append(float(aux[8]))
    x, y, z, coord = np.array(x), np.array(y), np.array(z), np.array(coord)
    vref, vrest, v = np.array(vref), np.array(vrest), np.array(v)
    return x, y, z, coord, vref, vrest, v
 
_, _, _, coord, vxref, vxrest, vx = load_potential('vx-final.z')
 
plt.plot(coord, vxref, color='orangered', label='$v_{x}^{ref}$')
plt.plot(coord, vxrest, color='dodgerblue', label='$v_{x}^{rest}$')
plt.plot(coord, vx, color='orange', label='$v_x$')
 
plt.ylabel('Potential (Hartree)', fontsize=16)
plt.xlabel('r (Bohr)', fontsize=16)
plt.xlim(-5, 5)
plt.legend(frameon=False, fontsize=14)
plt.tight_layout()
plt.show()

In the similar way, for spin-unrestricted calculations with USCEXX, one ends up with two files vxa-final.z and vxb-final.z with data for $\alpha$ and $\beta$ spin channels, respectively. For BeF one obtains: 

import numpy as np
import matplotlib.pyplot as plt
 
def load_potential(filename):
    """Use function from example with CO."""
    pass
 
_, _, _, coord, _, _, vxa = load_potential('vxa-final.z')
_, _, _, coord, _, _, vxb = load_potential('vxb-final.z')
 
plt.plot(coord, vxa, color='orangered', label=r'$v_{x}^\alpha$')
plt.plot(coord, vxb, color='dodgerblue', label=r'$v_{x}^\beta$')
 
plt.ylabel('Potential (Hartree)', fontsize=16)
plt.xlabel('r (Bohr)', fontsize=16)
plt.xlim(-5, 5)
plt.legend(frameon=False, fontsize=14)
plt.tight_layout()
plt.show()

SCRPA program

The SCRPA and USCRPA programs allow spin-restricted and spin-unrestricted self-consistent random phase approximation calculations.

Bibilography:
[1] A. Heßelmann, A.W. Götz, F. Della Sala, A. Görling J. Chem. Phys. 127, 054102 (2007)
[2] E. Trushin, A. Görling, J. Chem. Phys. 155, 054109 (2021)
[3] P. Bleiziffer, A. Heßelmann, A. Görling J. Chem. Phys. 139, 084113 (2013)
[4] E. Trushin, S. Fauser, A. Mölkner, J. Erhard, A. Görling Phys. Rev. Lett. 134, 016402 (2025)

NOTE: We have tutorial which provides practical Hands-on examples about the use of SCRPA and USCRPA programs and post-processing of results of calculations. This tutorial is a good supplement to this documentation. Link to Tutorial

Important: Read carefully the information below about the selection of basis sets and the thr_fai_oep parameter:
To obtain numerically stable potentials using optimized effective potential method, one must either use regularization techniques to carefully handle the small eigenvalues of the response matrix or to use auxiliary basis sets that are balanced to the orbital basis set. The latter can be done by manually constructing specific orbital and auxiliary basis sets that are sufficiently balanced. This has been possible for a number of atoms and molecules with quite large orbital basis sets [1], but does not qualify as a general applicable routine approach. The SCRPA and USCRPA programs therefore contain a new preprocessing scheme for auxiliary basis sets that effectively removes linear combinations of auxiliary basis functions that couple poorly to products of occupied times unoccupied Kohn-Sham orbitals and enable the construction of numerically stable exchange potentials with standard basis sets [2].

The preprocessing step to remove linear combinations of auxiliary basis functions that couple poorly to products of occupied times unoccupied Kohn-Sham orbitals is implemented according to Sec II5 of Ref. [2]. It involves the threshold thr_fai_oep, which determines how many linear combinations of auxiliary basis functions are removed and varies with respect to the size of the orbital basis set used. In Ref. [2], this scheme was tested using Dunning correlation consistent basis sets and recommended thresholds are

Orbital basis set thr_fai_oep
aug-cc-pwCVTZ 5e-2
aug-cc-pwCVQZ 1.7e-2
aug-cc-pwCV5Z 5e-3

These thresholds are expected to work also for orbital basis sets without augmentation cc-pwCVXZ (X = T, Q, 5) and without core-polarization functions aug-cc-pVXZ (X = T, Q, 5). As an auxiliary basis set (OEP), the aug-cc-pVXZ/mp2fit (X = T, Q, 5) family of basis sets works best. In particular, according to Ref. [2], it is recommended to use aug-cc-pVDZ/mp2fit auxiliary basis sets for atoms up to neon and aug-cc-pVTZ/mp2fit auxiliary basis sets for heavier atoms. As resolution of identity (RI) basis sets aug-cc-pwCVQZ/mp2fit or aug-cc-pwCV5Z/mp2fit basis sets are suitable.

Below is an example input file for spin-restricted calculations for the hygrogen molecule. Note that the input record from a preceding calculation is mandatory for initialization of orbitals and eigenvalues as starting point for RPA calculation, whereas it can come from HF or DFT calculations with maxit=0.

examples/h2_scrpa.inp
gdirect ! integral-direct mode

basis={
default,aug-cc-pwCVTZ ! orbital basis
set,ri;default,aug-cc-pwCVQZ/mp2fit ! RI BASIS
set,oep;default,aug-cc-pVDZ/mp2fit ! OEP basis
set,dfit;default,aug-cc-pwCV5Z/mp2fit ! density fitting basis
}

symmetry,nosym ! SCRPA does not use symmetry

angstrom
geometry={
2

H 0.0 0.0 0.370946
H 0.0 0.0 -0.370946
}

df-hf,maxit=0,df_basis=dfit ! HF calculation with 0 iteration
{cfit,basis_coul=dfit,basis_exch=dfit}

acfd;scrpa,thr_fai_oep=5d-2,plot_z=1 ! SCRPA calculation

As well as an example of a spin-unrestricted calculation for the lithium molecule:

examples/li_uscrpa.inp
gdirect ! integral-direct mode

basis={
default,aug-cc-pwCVTZ ! orbital basis
set,ri;default,aug-cc-pwCVQZ/mp2fit ! RI BASIS
set,oep;default,aug-cc-pVDZ/mp2fit ! OEP basis
set,dfit;default,aug-cc-pwCV5Z/mp2fit ! density fitting basis
}

symmetry,nosym ! USCRPA does not use symmetry

angstrom
geometry={
1

Li 0.0 0.0 0.0
}

spin=1

df-uhf,maxit=0,df_basis=dfit ! HF calculation with 0 iteration
{cfit,basis_coul=dfit,basis_exch=dfit}

acfd;uscrpa,thr_fai_oep=5d-2,plot_z=1 ! USCRPA calculation

The following options are available for the `SCRPA` and `USCRPA` programs:

The following parameters are only relevant for the `USCRPA` code:

Pitfalls:

Since the local exchange and correlation potentials are important in self-consistent RPA calculations, we provide an illustration of how to plot these potentials. Let us assume that we have performed calculations for hydrogen molecule with the following options: 

acfd;scrpa,thr_fai_oep=1.7d-2,plot_z=1

At the end one has the files vx-final.z with reference $v_x^{ref}$, rest $v_x^{rest}$ and full $v_x$ exchange potentials and vc-final.z with correlation potential. The potentials can be plotted using Python and matplotlib as follows: 

import numpy as np
import matplotlib.pyplot as plt
 
def load_potential(potfile):
    """Read data from potential-file.
 
    Args:
      potfile: Path to file with potential data.
 
    Returns:
      x: x coordinate.
      y: y coordinate.
      z: z coordinate.
      coord: coordinate on path.
      vref: reference potential.
      vrest: rest potential.
      v: full potential.
    """
    x, y, z, coord = list(), list(), list(), list()
    vref, vrest, v = list(), list(), list()
    for line in open(potfile):
        aux = line.split()
        x.append(float(aux[1]))
        y.append(float(aux[2]))
        z.append(float(aux[3]))
        coord.append(float(aux[4]))
        vref.append(float(aux[6]))
        vrest.append(float(aux[7]))
        v.append(float(aux[8]))
    x, y, z, coord = np.array(x), np.array(y), np.array(z), np.array(coord)
    vref, vrest, v = np.array(vref), np.array(vrest), np.array(v)
    return x, y, z, coord, vref, vrest, v
 
_, _, _, coord, vxref, vxrest, vx = load_potential('vx-final.z')
_, _, _, coord, _, _, vc = load_potential('vc-final.z')
 
plt.plot(coord, vx, color='dodgerblue', label='$v_x$')
plt.plot(coord, vc, color='orangered', label='$v_c$')
plt.plot(coord, vx+vc, color='orange', label='$v_{xc}$')
 
plt.ylabel('Potential (Hartree)', fontsize=16)
plt.xlabel('r (Bohr)', fontsize=16)
plt.xlim(-5, 5)
plt.legend(frameon=False, fontsize=14)
plt.tight_layout()
plt.show()

In the similar way, for spin-unrestricted calculations with USCRPA, one ends up with four files vxa-final.z, vxb-final.z, vca-final.z, vcb-final.z with data for $\alpha$ and $\beta$ spin channels. For lithium atom, one can plot, e.g., exchange-correlation potentials for $\alpha$ and $\beta$ spins: 

import numpy as np
import matplotlib.pyplot as plt
 
def load_potential(potfile):
    """Use function from example with CO."""
    pass
 
_, _, _, coord, _, _, vxa = load_potential('vxa-final.z')
_, _, _, coord, _, _, vca = load_potential('vca-final.z')
 
_, _, _, coord, _, _, vxb = load_potential('vxb-final.z')
_, _, _, coord, _, _, vcb = load_potential('vcb-final.z')
 
plt.plot(coord, vxa+vca, color='dodgerblue', label=r'$v_{xc,\alpha}$')
plt.plot(coord, vxb+vcb, color='orangered', label=r'$v_{xc,\beta}$')
 
plt.ylabel('Potential (Hartree)', fontsize=16)
plt.xlabel('r (Bohr)', fontsize=16)
plt.xlim(-5, 5)
plt.legend(frameon=False, fontsize=14)
plt.tight_layout()
plt.show()