Performance of numerical atom-centered basis sets in the ground-state correlated calculations of...

Performance of Numerical Atom-Centered Basis Sets inthe Ground-State Correlated Calculations of NoncovalentInteractions: Water and Methane Dimer Cases

Maxim Zakharov

Numerical atom-centered basis sets (orbitals) (NAO) are known

for their compactness and rapid convergence in the Hartree–

Fock and density-functional theory (DFT) molecular electronic-

structure calculations. To date, not much is known about the

performance of the numerical sets against the well-studied

Gaussian-type bases in correlated calculations. In this study,

one instance of NAO [Blum et al., The Fritz Haber Institute ab

initio Molecular Simulations Package (FHI-aims), 2009] was

thoroughly examined in comparison to the correlation-

consistent basis sets in the ground-state correlated

calculations on the hydrogen-bonded water and dispersion-

dominated methane dimers. It was shown that these NAO

demonstrate improved, comparing to the unaugmented

correlation-consistent based, convergence of interaction

energies in correlated calculations. However, the present

version of NAO constructed in the DFT calculations on

covalently-bound diatomics exhibits enormous basis-set

superposition error (BSSE)—even with the largest bases.

Moreover, these basis sets are essentially unable to capture

diffuse character of the wave function, necessary for example,

for the complete convergence of correlated interaction

energies of the weakly-bound complexes. The problem is

usually treated by addition of the external Gaussian diffuse

functions to the NAO part, what indeed allows to obtain

accurate results. However, the operation increases BSSE with

the resulting hybrid basis sets even further and breaks down

the initial concept of NAO (i.e., improved compactness) due to

the significant increase in their size. These findings clearly

point at the need in the alternative strategies for the

construction of sufficiently-delocalized and BSSE-balanced

purely-numerical bases adapted for correlated calculations,

possible ones were outlined here. For comparison with the

considered NAOs, a complementary study on the convergence

properties of the correlation-consistent basis sets, with a

special emphasis on BSSE, was also performed. Some of its

conclusions may represent independent interest. VC 2013 Wiley

Periodicals, Inc.

DOI: 10.1002/qua.24407

Introduction

Basis-set effects constitute a notorious source of error in

quantum-mechanical computations. Most of the molecular

(quantum chemical) electronic-structure computations

operate with atom-centered basis sets for the representation

of the electronic wave function. This property uses the

natural localization of electrons within molecules and

hence provides rather rapid convergence (in comparison to

e.g., fully-delocalized plane waves) of energies and molecular

properties with respect to the basis-set size.

Among atom-centered bases, the most common in use are the

analytical Gaussian-type orbitals (GTO) and Slater-type orbitals

(STO). They represent the radial part of molecular orbital as a linear

combination of Gaussian or Slater functions, centered in specific

positions of space (usually, but not necessarily, on the nuclei within

the molecule). The coefficients in the linear combinations are opti-

mized during the variational self-consistent field (SCF) procedure.*

The advantage of STO is that they correspond to the radial

solutions of the Schr€odinger equation for the hydrogen

atom and, therefore, possess correct asymptotic at zero and

large distances, hence providing better (as compared to the

pure GTO) convergence. The advantage of GTO is that the

two-electron integrals represented by Gaussian functions

can be easily computed analytically. This is in contrast to

STO, which call for the numerical evaluation of the two-elec-

tron integrals. In practice, each Slater function of the basis

set expansion can be represented by the linear combina-

tions of Gaussians (the so-called contracted Gaussians) with

the fixed, during the SCF procedure, coefficients. This opera-

tion combines best of each basis type, and it is commonly

implemented and works fine in most of the real-world

situations.

Another alternative to the above-introduced analytical basis

sets in molecular calculations are the numerical atomic orbitals

(NAO):

wiðr; RÞ ¼ uiðjr � RjÞYlmðXÞ=jr � Rj (1)

M. Zakharov

IDEA Group, Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy

E-mail: [email protected]

VC 2013 Wiley Periodicals, Inc.

*If, in addition, the spatial positions of the exponents are optimized, then one

deals with the floating basis functions. They are, however, not common in

electronic structure calculations nowadays due to of the greater number of

variational parameters.

International Journal of Quantum Chemistry 2013, DOI: 10.1002/qua.24407 1

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here, r and R are the coordinates of an electron and the basis-

set center, ui(|r � R|) is the radial shape, and Ylm(X) are the

spherical harmonics. The radial shape is numerically tabulated

during the basis-set construction. This gives additional flexibil-

ity in representation of the radial part of the atomic orbitals

(AO) and, in theory, may provide better convergence of ener-

gies and molecular properties, as compared to the analytical

bases. So far, several versions of NAO are implemented at the

level of density-functional theory (DFT),[1] including local-den-

sity approximation (LDA)[2] and generalized-gradient approxi-

mation (GGA),[3] Hartree–Fock (HF),[4,5] and correlated levels:

Refs. [6–16], Refs. [15–22], and Refs. [15,16,23], respectively.

This study will deal with the numerical basis sets implemented

in the FHI-aims software package (the so-called Tier (n) basis

sets).[15,16] The compactness of the numerical basis sets can be

particularly advantageous in correlated calculations which are

more sensitive to the basis-set size due to the higher (in com-

parison to SCF) scaling of computational cost with respect to

the basis size. The price to be paid for the presumably better

convergence at correlated level is, however, the necessity to

numerically evaluate two-electron integrals (like in the case of

STO). This requires sophisticated techniques employing auxil-

iary basis sets[24–27] already at the HF step, to take care of this

issue. Another serious disadvantage of NAO is the necessity to

compute numerical derivatives of AO with respect to the nu-

clear coordinates, to calculate harmonic frequencies at the SCF,

or energy gradients at correlated levels. In general, compared

to GTO, NAO are more labor-intensive for the implementation

of correlated methods and gradient techniques.

There are two major, closely related sources of basis set error

in electronic structure calculations with atom-centered bases

(both numerical and analytical): basis-set incompleteness error

(BSIE) and basis-set superposition error (BSSE).[28,29] BSIE arises

when a basis set does not describe physical situation properly,

that is, contains insufficient (for the particular system) number

of AO to represent each molecular orbital, or does not properly

describe delocalized character of the wave function (i.e., does

not include necessary diffuse functionsy). For a given finite basis

set, BSIE leads to too high energies with respect to the com-

plete basis set (CBS). BSSE is a peculiarity of the atom-centered

basis sets (both analytical and numerical) which stems from the

mutual augmentation of the monomer’s basis sets within the

complex. As the consequence, energies obtained with finite

bases without BSSE correction are below the BSSE-corrected

ones and, moreover, are often unphysically below the CBS limit.

The wave function becomes BSSE-contaminated what can

affect, for example, electronic densities. Thus, BSSE is the prop-

erty of atom-centred basis sets which leads to the artificial

decrease of the complex’ energy. It is well-understood now,

that mathematically, it is the multicenter one- and two-electron

integrals expressed in the AO basis are responsible for BSSE.[30]

Because BSSE corrupts virtual orbitals as well, correlated and

excited-state methods are much more sensitive to BSSE than

SCF. Contribution of the intermolecular BSSE to the total bind-

ing energy of weakly-bound complexes at their equilibrium

geometries is usually considerable even with large basis sets

(this will be demonstrated in section Results). However, a num-

ber of notorious examples demonstrate importance of intramo-

lecular BSSE, for example, in the artificial distortion of ben-

zene,[31–33] planar arenes,[34–38] and other nonrigid molecules,[39]

observed at correlated level of theory (the correct planar shapes

were recovered after the BSSE correction[40]). A recent computa-

tional study[41] observed strong BSSE effect in some biopoly-

mers at the DFT level with double-zeta quality basis sets, what

led to erroneous ordering of the principal isomers. Finally, no-

ticeable intramolecular BSSE was recently found in interaction

energies of fluoride dimer at the CCSD(T) level with augmented

correlation-consistent basis sets.[42]

Two major ways to treat BSSE exist: (1) the chemical Hamil-

tonian approach (CHA)[30,43,44] and, (2) the counterpoise

correction (CP).[28,45] Historical overview of the BSSE correc-

tion developments can be found in Ref. [46]. In CHA, the

terms responsible for BSSE are removed from the SCF

equations directly.z CHA-SCF thus produces BSSE-free SCF

reference. It opened the way for the development of a priori

BSSE-free correlated methods on CHA.[47,48] Very encouraging

results on the closed- and open-shell hydrogen-bonded

systems and van der Waals complexes were obtained with

the BSSE-free second-order Møller–Plesset perturbation

theory (MP2) on the CHA reference.[47,48] The major disad-

vantage of CHA is that the Fock matrix in the SCF equations

is non-Hermitian anymore. Although it does not affect

computational performance of the methodology, it brings

additional complexity for the analytical formulation and

implementation of correlated methods and gradient techni-

ques on CHA. Furthermore, at the moment, the procedure is

not well-formulated for the case of covalently-bound frag-

ments (i.e., it cannot be used transparently for the removal of

intramolecular BSSE; Istvan Mayer, private communication).

CP is a conceptually simple procedure of the BSSE correction.

The idea of CP is to add the ‘‘ghost’’ orbitals, borrowed from

the entire complex, to each monomer within the complex.

Technically, these orbitals are obtained by setting nuclear

charges of other fragments within the complex to zero and

omitting their electrons. BSSE per fragment is then computed

as the difference between the energies of the monomer in its

own basis (the so-called monomer-centered basis set) and in

the basis of ghost orbitals (the so-called dimer-centered basis

set). The disadvantages of CP are well-known. CP is an a

posteriori approach—it corrects for BSSE at the energy level

but not at the wave function level. Wave function-specific

properties, for example, electron densities can be affected by

BSSE, even if PES is corrected by means of CP.[49] Application

of the CP correction to open-shell, or charged complexes is

yDiffuse functions are STO or GTO with small coefficients (\0.1) at the expo-

nents what provides their slow decay at large electron-nuclear distances.

zThese are the multicenter one- and two-electron integrals. In CHA, the two-

center integrals at most are retained. The multicenter character of the wave

function is captured by the multicenter overlap integrals. Avoiding the multi-

center integrals thus can be an additional factor of speed-up of CHA-SCF, com-

pared to the conventional SCF.

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2 International Journal of Quantum Chemistry 2013, DOI: 10.1002/qua.24407 WWW.CHEMISTRYVIEWS.ORG

often problematic, if possible.x As both of these situations are

normally present in molecules, application of CP to molecules

to diminish BSSE at the intramolecular level (for example, in

atomic CP) is complicated. Even in such pathological cases,

CP correction is often possible but is very tedious.[40] Finally,

computational cost of the CP-correction scales linearly with

respect to the number of fragments n in the complex: the

procedure requires (n þ 1) energy calculations in the com-

posite basis set. This is in strong contrast to CHA in which

computational cost is essentially independent on the number

of fragments. It is also worth to mention that there are local-

orbital (LO) correlated methods[50–52] which were designed to

provide better computational scaling. The nice side effect of

the LO methods is the reduction of BSSE in correlation

energy. In principle, one can consider the combination of

CHA HF and LO correlation energies to reduce BSSE in total

energy, because it is easier to formulate/implement CHA at

the HF than at correlated levels.

In spite of the promising results demonstrated with numerical

basis sets in the earlier ground-state HF[17–22] and DFT calcula-

tions,[13,16] not much is known about the performance of the

numerical sets in correlation calculations, owing to the fact of

comparatively late emergence of the technology. In particular,

not much is known about the performance of NAO against the

well-studied analytical GTO-based sets. This aspect is of para-

mount importance for the further adoption of the technology

for the following reason. As it was mentioned above, computa-

tion of the two-electron integrals over the nonintegrable AO

(like NAO or STO) is more expensive than in the case of the

Gaussian ones. That means an additional prefactor in computa-

tional time of their evaluation (plus, a number of other technical

difficulties). Thus, to justify the real advantage of NAO vs. GTO

in correlated calculations, one shall clearly show that the above

disadvantages are compensated by significantly smaller basis-

set size and hence, by better scaling of computational cost (i.e.,

smaller number of the involved two-electron integrals). At the

moment, there are two studies on correlated methods com-

bined with NAO, giving some insights at their performance in

comparison to more conventional atom-centered basis

sets.[23,53] The first study[23] is a great step forward in the evolu-

tion of the numerical atom-centered sets, which for the first

time demonstrates a feasible and computationally-efficient

NAO-based implementation of correlated methods [random

phase approximation (RPA) on the DFT and HF references, as

well as common MP2], thanks to the employed numerical tech-

niques for the acceleration of calculation of the two-electron

numerical integrals. Both studies contain extensive benchmarks

on the performance of NAO in application to well-established

reference data sets representing both covalent and noncovalent

interactions, as well as comparisons of NAO vs. several correla-

tion-consistent basis sets of Dunning and coworkers[54–56] in

application to selected complexes. However, neither study ana-

lyzed the whole range of the basis-set effects (including the

role of diffuse and core-valence functions, frozen-core approxi-

mation, and BSSE) with the full range of cardinal numbers for

both NAO and correlation-consistent basis sets.{ The present

investigation was dedicated to clarify these aspects (in particular

in relation to the weakly-bound complexes) and hence comple-

ments the previous NAO studies. A special attention has been

paid to the effect of BSSE with the Tier (n) sets, which was al-

ready recognized to be large.[23]

As it is mentioned above, for comparison with the Tier (n) sets,

the author has chosen the well-defined analytical correlation con-

sistent basis sets. Although there are number of alternative ana-

lytical Gaussian-based basis sets such as the Pople’s sets,[57] the

atomic natural orbitals (ANO) basis sets,[58,59] the polarization-con-

sistent basis sets,[60–64] and the completeness-optimized sets[65,66]

which are either more compact, or less prone to BSSE, the correla-

tion-consistent ones were chosen due to their ubiquity in the

benchmark studies of various systems. Knowing the differences

(in size and quality) between the correlation-consistent and other

Gaussian-based basis sets, one always can estimate the differen-

ces between the latter and the numerical ones. For the reference

and comparison purpose, a rather comprehensive study on the

convergence and BSSE properties of the correlation-consistent

basis sets was undertaken. That allowed to thoroughly clarify the

role of diffuse and core-valence functions[56] in the convergence

of interaction energies of both complexes. Some of the findings

of this complimentary study, in the author’s opinion, may repre-

sent independent (from the initial NAO agenda) interest.

Because BSSE is most readily observable on the characteris-

tic energy scales/equilibrium distances of the weakly-bound

complexes,** the author has chosen for the benchmarks two

extensively studied complexes from the S22 database[67]yy—

the water and methane dimers. The first one represents purely

hydrogen-bonded system, the second—the dispersion-domi-

nated one. Because all two-electron integrals over the correla-

tion-consistent basis sets in the FHI-aims code are computed

numerically (in the same fashion as the integrals over NAO),

the calculations of potential-energy curves with the largest

xThe ghost-orbital calculations for open-shell fragment shall describe the

same spin state as for the fragment in its own basis set. Because the symmetry

(and spin) of the wave function of the monomers can be changed in the com-

posite ghost-orbital basis, this condition is not always automatically fulfilled.

This is because the minimum-energy state in the composite basis set would

not be necessarily of the same multiplicity as it is in the basis of the monomer.

Consequently, the SCF procedure can be difficult to converge to the same

electronic spin state. This is especially the case for the complexes having sev-

eral electronic states of the same symmetry. In the case of charged, for

instance, protonated complexes the charge can significantly shift electron

density within the complex. Herein, the assignment of charge to the frag-

ments (proton and the rest) to apply CP becomes ambiguous.

{For example, most of the ‘‘converged’’ results[23] were obtained with the largest

numerical Tier 4 set, augmented with the diffuse functions of the correlation-

consistent aug-cc-pV5Z basis. Unfortunately, these data neither provide magni-

tudes of the BSSE contribution to the interaction/atomization energies obtained

with that hybrid set, nor the basis set size per each complex, what would be

very informative for the direct comparison of these hybrid bases with for exam-

ple, the reference aug-cc-pV5Z set.**This is because BSSE magnitude grows slower with the decrease of the dis-

tance between the basis set centers, comparing to the electrostatic interac-

tions (see any figure in the supporting information).yyThe S22 database contains only noncovalently-bound complexes (hydro-

gen-bonded, dispersion-dominated, and the mixed ones).




correlation-consistent (as well as numerical) basis set for the

whole database appeared to be quite demanding computa-

tionally. It is known that the S22 complexes with the similar

type of bonding (i.e., the H-bonded, mixed, and dispersion-

dominated ones) typically exhibit similar BSSE behavior (see

e.g., Table 6 of Ref. [68]).zz Moreover, the results of this study,

concerning the convergence and BSSE properties of the Tier

(n) basis set, are consistent with the results of aforementioned

Ref. [23], observed in several other systems. Therefore, the

author is confident that the limitation of the investigation by

the above two complexes does not affect generality of its con-

clusions regarding to the convergence and BSSE properties of

the Tier (n) sets.

In this work, the two correlated methods were chosen for

the benchmark of the performance of NAO: the RPA[69] using

Kohn–Sham orbitals and orbital energies produced with the

GGA PBE density functional[70] (this combination is referred to

as RPA@PBE[71]) as well as common MP2, all implemented in

the FHI-aims code.[15] The choice of RPA@PBE for the bench-

marks of basis set is somewhat unusual and needs some clari-

fication. The first (major) reason for this choice is straightfor-

ward: at the moment, RPA (based on orbitals obtained with HF

or DFT) and MP2 are the only two ground-state correlated

methodologies (except DFT) which are implemented in FHI-

aims (i.e., in any NAO code) and it is perfectly logical to clarify

how well these numerical bases, combined with available cor-

related methods, behave as compared to more conventional

Gaussian-based sets. The second reason is the following: there

is a serious interest in the physical and chemical communities

in the development of ‘‘post-RPA’’ methods [like RPA based on

the DFT rather than HF orbitals,[71] or RPA with ‘‘single excita-

tions’’ (RPAþSE@PBE)[72])]capable to treat weak interactions

(see Ref. [73] for an overview) and much of the effort comes

from the community around the NAO-based FHI-aims code.

Note that RPA on top of Kohn–Sham (rather than HF) orbi-

tals/orbital energies was chosen here because it is well-

known that RPA with the HF orbitals behaves poor in the

case of weak interactions. Since the previous work[23] recog-

nized somewhat large BSSE with all RPA-based methods/Tier

(n) basis sets, it motivated the author to clarify BSSE behav-

ior of RPA (combined with the numerical bases) more thor-

oughly. Although the accuracy of RPA@PBE and RPAþSE

@PBE methods in the case of water and methane dimers

appears to be imperfect (at least with the PBE orbitals/or-

bital energies) as compared to the reference CCSD(T) data,xx

all the results related to the performance of NAO vs. corre-

lation-consistent prove to be consistent with the MP2

calculations.

The study is organized as follows. In section Results, analy-

ses of BSSE behavior and convergence of the BSSE-corrected

interaction energies of the water dimer (section Basis-set

superposition error and basis set convergence with correla-

tion-consistent basis sets) and methane dimer (section Basis-

set superposition error and basis set convergence with Tier (n)

sets), computed with the correlation-consistent and numerical

Tier (n) basis sets at correlated (RPA@PBE and MP2) and SCF

(DFT and HF) levels, are presented. The results of section

Results will be discussed in section Discussion and summarized

in section Conclusions.

Because CHA is not available in standard quantum chemical

software, the common CP correction was used throughout the

study. Its application for the elimination of BSSE at intermolec-

ular level is feasible for all complexes of the S22 set.

Results

BSSE and basis set convergence with correlation-consistent

basis sets

BSSE-convergence. As the first step, for comparison with the

numerical Tier (n) basis sets, BSSE effect in correlated

RPA@PBE calculations with the two cc-pVDZ and cc-pV5Z[54]

unaugmented correlation-consistent polarized valence basis

sets was analyzed. Both bases do not possess extra diffuse

functions and have no core-valence functions. Although this

type of analysis has been performed previously for the con-

sidered complexes in many accurate studies, and the results

are in principle predictable, this step is necessary to provide

quantitative framework for the direct comparison of the cor-

relation-consistent bases versus the unaugmented Tier (n)

ones at the RPA level.

In course of the study, it will be found that inclusion of dif-

fuse functions from the large correlation-consistent basis sets

zzIt is of course the consequence of the two facts: (1) all S22 complexes contain

the second-row elements at most, and, (2) the equilibrium distances of the

complexes with the similar type of bonding are close.xxFor example, RPA@PBE with converged aug-cc-pV5Z set underestimates the

reference CCSD(T) interaction energies[68]) of water and methane dimers by

about 14 and 26%, respectively, whereas the RPA@PBE including the ‘‘single

excitations’’ (RPAþSE@PBE) overestimates them by 12 and 15% (all the interac-

tion energies are BSSE-corrected). In both cases, it is worse than MP2 which

interaction energy is within 2% of the reference CCSD(T) interaction energy

for water dimer and is underestimated by about 7% for the methane dimer);

however, on average (RPAþSE)@PBE is a significant improvement over

RPA@PBE for the S22 set. Note, that total RPA@PBE energy (the sum of PBE SCF

and RPA correlation energy) is completely missing the exact HF exchange. It

would also be informative to mention that the observed discrepancies

between the CCSD(T) and MP2 results often can be cured by using spin-com-

ponent scaling versions of MP2 (see e.g., Ref. [63]) but this discussion goes

beyond the present study.

xxFor example, RPA@PBE with converged aug-cc-pV5Z set underestimates the

reference CCSD(T) interaction energies[68]) of water and methane dimers by

about 14 and 26%, respectively, whereas the RPA@PBE including the ‘‘single

excitations’’ (RPAþSE@PBE) overestimates them by 12 and 15% (all the interac-

tion energies are BSSE-corrected). In both cases, it is worse than MP2 which

interaction energy is within 2% of the reference CCSD(T) interaction energy

for water dimer and is underestimated by about 7% for the methane dimer);

however, on average (RPAþSE)@PBE is a significant improvement over

RPA@PBE for the S22 set. Note, that total RPA@PBE energy (the sum of PBE SCF

and RPA correlation energy) is completely missing the exact HF exchange. It

would also be informative to mention that the observed discrepancies

between the CCSD(T) and MP2 results often can be cured by using spin-com-

ponent scaling versions of MP2 (see e.g., Ref. [63]) but this discussion goes

beyond the present study.



is of importance for the convergence of interaction energies at

correlated level with both NAO and correlation-consistent

sets.{{ Therefore, the BSSE influence on the interaction ener-

gies obtained with the smallest correlation-consistent cc-pVDZ

basis augmented with the diffuse functions of the aug-cc-

pV5Z set was investigated (to best of our knowledge, this

path has not been followed in the literature). This combined

set will be designated as cc-pVDZþdf5Z. For direct comparison

with NAO, the author analyzed BSSE effect at the RPA level

with the correlation-consistent basis sets augmented with their

native (i.e., specifically optimized) diffuse functions aug-cc-

pV(n)Z, n ¼ 2–5 (number n is often is termed in the literature

as ‘‘cardinal number’’).[55] Finally, the role of core polarization

effects in the convergence of interaction energies and BSSE

was clarified in calculations with the correlation-consistent

polarized core-valence aug-cc-pCV5Z basis set[56]—an exten-

sion of the valence aug-cc-pV5Z set which possesses

additional polarization functions on core orbitals. Note that

core-electron correlation effects can be captured only if the

basis set contains core-valence functions. It shall be noted that

further increase of the basis set size of correlation-consistent

sets did not lead to the noticeable change in the BSSE-

corrected interaction energies of either complex, hence

aug-cc-pCV5Z basis can be considered as nearly converged. In

particular, the water dimer interaction energies are within 2%

agreement with those from the benchmark MP2 study on water

dimer,[74] using a larger uncontracted ANO-type basis set.

To get an idea about the magnitude and spatial distribution

of BSSE, BSSE-curves as well as BSSE-uncorrected RPA@PBE

interaction energy curves of the water and methane dimers,

computed along their S22 pathways,[68]*** are provided with

all employed correlation-consistent basis sets (Supporting in-

formation, Figs. S1 and S4). For the reference, BSSE-corrected

interaction energy curves produced with the largest aug-cc-

pCV5Z set are displayed at the same graphs.

Because CP-corrected and uncorrected energy curves have

different positions of the minima in most considered cases

(and so do the BSSE values in these points), in Tables 1 and 3

(as well as in Tables 2 and 4 dedicated to NAO), all results are

given for the both minima of the CP-corrected and uncor-

rected curves. For the unambiguity of the analysis on the con-

vergence of BSSE and CP-corrected interaction energies with

respect to the relevant cardinal numbers, their values are also

computed in the lowest energy point of the RPA@PBE/aug-cc-

pCV5Z energy curve.

Finally, to compare BSSE performance of the RPA vs. MP2

methods, single-point MP2 energies and BSSE, computed in

the lowest energy point of the reference RPA@PBE/aug-cc-

pCV5Z energy curve with the largest unaugmented (cc-pV5Z)

and augmented (aug-cc-pV5Z, aug-cc-pCV5Z) basis sets in the

frozen-core and full-electron (FE) modes, respectively, are given

(Tables 1 and 3).

The following BSSE properties are observed in the present

data. For the largest considered correlation-consistent aug-cc-

pCV5Z basis set comprising diffuse and core-valence functions,

BSSE contribution to the total RPA interaction energy of water

and methane dimer in the FE calculations is still significant—

about 4.8 and 11.8%, respectively. At the MP2 level, the situa-

tion is better with the BSSE contributions of �2.5 and �6.9%

of the converged RPA energy (and with �2.8 and �5.6% of

the MP2 interaction energy computed in the same points;

Tables 1 and 3). This demonstrates that BSSE in the RPA-based

methods converges slower than in the MP2 case (in agree-

ment with Ref. [23]). This probably would require sextuple-fquality bases to provide comparable BSSE-convergence in the

BSSE-uncorrected calculations.

Using the aug-cc-pV5Z basis set in the frozen-core MP2 cal-

culations produces slightly stronger BSSE than aug-cc-pCV5Z

in the full electron mode (�2.9 vs. �2.5%) for the water

dimer and slightly weaker one in the case of methane dimer

(�6.5 vs. �6.9%). Obviously, there is no advantage in using

basis sets containing core-valence functions in the FE calcula-

tions vs. the frozen-core calculations, as a remedy against

BSSE in the ground state. This is especially true for the

considered complexes because additional core polarization

functions contribute to the BSSE-corrected ground-stateyyy

interaction energies only slightly (e.g., within 0.02 and 0.22%

of the reference aug-cc-pCV5Z RPA interaction energy for the

water and methane dimers, respectively), whereas being

considerably larger (19 and 12% for aug-cc-pCV5Z vs. aug-

cc-pV5Z for water and methane dimers, respectively). Note

these ratios will be even greater for the unaugmented basis

sets (because augmentation by the diffuse functions increase

the basis set size considerably). The author expects the same

situation for other complexes comprising second-row

elements.

As one can see, the core-valence basis functions decrease

BSSE incurring from the core orbitals. This nice side effect is

advantageous in those cases when the core-orbital contribu-

tion to the interaction energy is significant and the FE

calculations (using the core-valence functions in the basis

set) are necessary, whereas application of CP to the

complex (fragment counterpoise) or to the atoms (atomic

counterpoise) is difficult/impossible either computationally

(due to the linear scaling of the computational cost) or

conceptually (e.g., in the case of molecules, ionic, and free-

radical complexes with delocalized charge/spin densities).

However, one shall be aware that even in this case BSSE

due to the valence orbitals is significant (what is shown in

the present calculations) and needs to be corrected. This as-

pect is particularly important in relation to the Tier (n) sets,

owing their greater overall BSSE (both core- and valence-or-

bital one; see the next subsection).{{Augmentation of the numerical Tier 4 basis set by the diffuse functions of

the aug-cc-pV5Z set is often used in the FHI-aims community to provide nearly

complete convergence of the former.***S22 pathways are obtained by discrete displacements of the monomers

within the complex with respect to their center of mass, without their subse-

quent geometry reoptimization in the displaced positions.

yyyIn general, however, one cannot expect that the effect of core polarization

functions on PES of molecular complexes in selected excited states (involving

core excitations) is negligible.




Table

1.

BB

SE-u

nco

rre

cte

d(E

NOCP

int

),B

SSE-

corr

ect

ed

(ECP

int

)in

tera

ctio

ne

ne

rgie

s,an

dB

SSE

(d)

(in

kcal

mo

l21)

of

me

than

ed

ime

rco

mp

ute

dw

ith

the

corr

ela

tio

n-c

on

sist

en

tb

asis

sets

atth

eR

PA@

PB

Eth

eo

ryle

vel

inth

ere

leva

nt

po

ints

of

the

inte

ract

ion

en

erg

ycu

rve

s.

cc-p

VD

Zcc

-pV

5Z

cc-V

DZþ

df5

Zau

g-c

c-p

VD

Zau

g-c

c-p

VT

Zau

g-c

c-p

VQ

Zau

g-c

c-p

V5

Zau

g-c

c-p

CV

5Z

BS

size

48

40

22

20

82

18

43

34

57

46

82

RPA

(FU

LL)

RNOCP

min

0.0

0.0

�0

.20

.00

.00

.00

.00

.0

ENOCP

int

(RNOCP

min

)�

6.1

39

�4

.56

7�

9.8

29

�4

.77

6�

5.5

45

�4

.72

5�

4.7

47

�4

.44

1

ECP

int

(RNOCP

min

)�

2.5

59

�4

.08

1�

2.4

95

�3

.35

8�

3.8

64

�4

.12

1�

4.2

06

�4

.20

7

d(RNOCP

min

)3

.58

0.4

86

7.3

34

1.4

18

1.6

81

0.6

04

0.5

40

0.2

34

RCP

min

0.2

0.1

0.1

0.2

0.1

0.1

0.1

0.1

ENOCP

int

(RCP

min

)�

5.8

63

�4

.51

4�

8.7

61

�4

.60

9�

5.4

54

�4

.66

9�

4.6

85

�4

.41

5

ECP

int

(RCP

min

)�

3.0

20

�4

.10

3�

3.8

71

�3

.55

9�

3.9

40

�4

.14

7�

4.2

15

�4

.21

5

d(RCP

min

)2

.84

30

.41

14

.89

01

.05

01

.51

40

.52

10

.46

90

.20

1

ENOCP

int

(RCP�REF

min

)�

6.0

85

�4

.51

4�

8.7

61

�4

.76

8�

5.4

54

�4

.66

9�

4.6

85

�4

.41

5

ECP

int

(RCP�REF

min

)�

2.9

05

�4

.10

3�

3.8

71

�3

.55

5�

3.9

40

�4

.14

7�

4.2

15

�4

.21

5

d(RCP�REF

min

)/%

(EREF

int

(RPA

))3

.17

9/7

5.4

4%

0.4

11

/9.75

%4

.89

0/1

16

.01

%1

.21

3/2

8.7

8%

1.5

14

/35

.92

%0

.52

1/1

2.3

6%

0.4

69

/11.15

%0

.20

1/4.77

%

RPA

(FC

)*

d(RCP�REF

min

)/%

(EREF

int

(RPA

))–

0.3

53

/8.37

%–

––

–0

.22

9/5.45

%

MP

2(F

ULL

)

ENOCP

int

(RCP�REF

min

)–

�5

.11

6–

––

–�

5.1

06

�4

.96

0

ECP

int

(RCP�REF

min

)–

�4

.80

6–

––

–�

4.8

53

�4

.85

5

d(RCP�REF

min

)/%

(EREF

int

(RPA

))/%

(EREF

int

(MP

2))

–0

.30

9/7.33

%/6

.37

%–

––

–0

.25

3/5.99

%/5

.20

%0

.10

5/2.48

%/2

.16

%

MP

2(F

C)

ENOCP

int

(RCP�REF

min

)–

�5

.04

0–

––

–�

4.9

46

–

ECP

int

(RCP�REF

min

)–

�4

.77

5–

––

–�

4.8

22

–

d(RCP�REF

min

)/%

(EREF

int

(RPA

))/%

(EREF

int

(MP

2))

–0

.26

5/6.29

%/5

.46

%–

––

–0

.12

4/2.93

%/2

.55

%–

Su

ffix

‘‘df5

Z’’

de

sig

nat

es

aug

me

nta

tio

nb

yth

ed

iffu

sefu

nct

ion

so

fth

eau

g-c

c-p

V5

Zb

asi

s.RNOCP

min

andRCP

min

corr

esp

on

dto

the

low

est

en

erg

yp

oin

tso

fth

eB

SS

E-u

nco

rre

cte

dan

dB

SSE

-co

rre

cte

din

tera

ctio

n

en

erg

ycu

rve

so

bta

ine

dw

ith

ea

chb

asi

s.RCP�REF

min

refe

rsto

the

low

est

en

erg

yp

oin

to

fth

eB

SSE

-co

rre

cte

dR

PA@

PB

Ein

tera

ctio

ne

ne

rgy

curv

ep

rod

uce

dw

ith

the

refe

ren

ceau

g-c

c-p

CV

5Z

ba

sis.

BS

size

isth

e

nu

mb

er

of

the

ba

sis

fun

ctio

ns

for

the

com

ple

x.A

llth

ed

ista

nce

sar

ed

efi

ne

das

the

dis

pla

cem

en

t(i

nA

ng

str€ o

ms)

wit

hre

spe

ctto

the

cen

ter

of

ma

sso

fth

ere

fere

nce

stru

ctu

refr

om

the

S22

da

tab

ase

,

acco

rdin

gto

Re

f.[6

2].

Full-

ele

ctro

n(F

ULL

)an

dfr

oze

n-c

ore

(FC

)M

P2

en

erg

ies

are

com

pu

ted

wit

hth

ela

rge

stco

nsi

de

red

corr

ela

tio

n-c

on

sist

en

tse

tsin

theRCP�REF

min

po

int.

Th

eth

ird

valu

e(%EREF

int

(MP

2))

inth

e

d(RCP�REF

min

)fi

eld

sd

esi

gn

ate

sth

efr

acti

on

of

BS

SEre

lati

veto

the

CP

-co

rre

cte

dM

P2

inte

ract

ion

en

erg

yco

mp

ute

din

the

refe

ren

celo

we

st-e

ne

rgy

po

int

of

the

RPA

@P

BE/

aug

-cc-

pC

V5

Zcu

rve

.Fr

oze

n-c

ore

RPA

valu

es

RPA

(FC

)is

ane

stim

ate

de

rive

db

ym

ult

iply

ing

the

rati

ob

etw

ee

nth

efr

oze

n-c

ore

vs.

full-

ele

ctro

nM

P2

valu

es

and

the

full-

ele

ctro

nR

PA@

PB

Ee

ne

rgy.



Table

2.

BB

SE-u

nco

rre

cte

d(E

NOCP

int

),B

SSE-

corr

ect

ed

(ECP

int

)in

tera

ctio

ne

ne

rgie

san

dB

SSE

(d)

(in

kcal

mo

l21)

of

me

than

ed

ime

rco

mp

ute

dw

ith

the

Tie

r(n

)b

asis

sets

atth

eR

PA@

PB

Eth

eo

ryle

vel

inth

e

rele

van

tp

oin

tso

fth

ein

tera

ctio

ne

ne

rgy

curv

es.

Tie

r1

Tie

r2

Tie

r3

Tie

r4

Tie

r1þ

df5

ZT

ier

2þ

df5

ZT

ier

3þ

df5

ZT

ier

4þ

df5

Z

BS

size

48

13

82

34

28

42

20

31

04

06

45

6

RPA

(FU

LL)

RNOCP

min

0.0

�0

.2�

0.1

�0

.1�

0.2

�0

.2�

0.2

�0

.1

ENOCP

int

(RNOCP

min

)�

6.4

14

�1

1.7

22

�1

1.9

86

�9

.94

4�

11

.10

6�

10

.78

0�

11

.33

2�

8.2

63

ECP

int

(RNOCP

min

)�

3.0

42

�2

.97

6�

3.7

75

�3

.84

3�

2.9

41

�3

.20

1�

3.2

51

�3

.93

4

d(RNOCP

min

)3

.37

28

.74

78

.21

16

.10

18

.16

57

.57

98

.08

04

.33

0

RCP

min

0.2

0.1

0.1

0.1

0.1

0.1

0.1

0.1

ENOCP

int

(RCP

min

)�

6.0

97

�1

0.7

78

�1

1.2

45

�9

.37

3�

9.9

75

�9

.59

5�

10

.23

5�

7.7

69

ECP

int

(RCP

min

)�

3.3

36

�4

.11

0�

4.1

08

�4

.15

3�

4.0

88

�4

.19

8�

4.2

11

�4

.21

5

d(RCP

min

)2

.76

16

.66

87

.13

75

.22

05

.88

65

.39

76

.02

53

.55

3

ENOCP

int

(RCP�REF

min

)�

6.3

31

�1

0.7

78

*�

11

.24

5*

�9

.37

3*

�9

.97

5*

�9

.59

5*

�1

0.2

35

*�

7.7

69

*

ECP

int

(RCP�REF

min

)�

3.2

86

�4

.11

0�

4.1

08

�4

.15

3�

4.0

88

�4

.19

8�

4.2

11

�4

.21

5

d(RCP�REF

min

)/%

(EREF

int

(RPA

))3

.04

5/7

2.2

4%

6.6

68

/15

8.4

8%

7.1

37

/16

9.3

2%

5.2

20

/123.84

%5

.88

6/1

39

.64

%5

.39

7/1

28

.04

%6

.02

5/1

42

.94

%3

.55

3/84.29

%

RPA

(FC

)*

d(RCP�REF

min

)/%

(EREF

int

(RPA

))–

––

1.5

05

/35

.71

%–

––

1.2

98

/30.96

%

MP

2(F

ULL

)

ENOCP

int

(RCP�REF

min

)–

––

�7

.42

5–

––

�6

.42

2

ECP

int

(RCP�REF

min

)–

––

�4

.81

6–

––

�4

.85

0

d(RCP�REF

min

)/%

(EREF

int

(RPA

))/%

(EREF

int

(MP

2))

––

–2

.60

9/61.91

%/5

3.7

5%

––

–1

.57

2/37.30

%/3

2.3

8%

MP

2(F

C)

ENOCP

int

(RCP�REF

min

)–

––

�5

.53

4–

––

�5

.39

3

ECP

int

(RCP�REF

min

)–

––

�4

.78

2–

––

�4

.81

6

D( R

CP�REF

min

)/%

(EREF

int

(RPA

))/%

(EREF

int

(MP

2))

––

–0

.75

2/17.85

%/1

5.4

9%

––

–0

.57

7/13.70

%/1

1.8

9%

Suff

ix‘‘d

f5Z

’’d

esi

gn

ate

sau

gm

en

tati

on

by

the

dif

fuse

fun

ctio

ns

of

the

aug

-cc-

pV

5Z

bas

is.RNOCP

min

andRCP

min

corr

esp

on

dto

the

low

est

en

erg

yp

oin

tso

fth

eB

SSE-

un

corr

ect

ed

and

BSS

E-co

rre

cte

din

tera

ctio

n

en

erg

ycu

rve

so

bta

ine

dw

ith

eac

hb

asis

.RCP�REF

min

refe

rsto

the

low

est

en

erg

yp

oin

to

fth

eB

SSE-

corr

ect

ed

RPA

@P

BE

inte

ract

ion

en

erg

ycu

rve

pro

du

ced

wit

hth

ere

fere

nce

aug

-cc-

pC

V5

Zb

asis

.B

Ssi

zeis

the

nu

mb

er

of

the

bas

isfu

nct

ion

sfo

rth

eco

mp

lex.

All

the

dis

tan

ces

are

de

fin

ed

asth

ed

isp

lace

me

nt

(in

An

gst

r€ om

s)w

ith

resp

ect

toth

ece

nte

ro

fm

ass

of

the

refe

ren

cest

ruct

ure

fro

mth

eS2

2d

atab

ase

,ac

cord

ing

toR

ef.

[62

].Fu

ll-e

lect

ron

and

fro

zen

-co

reM

P2

en

erg

ies

are

com

pu

ted

wit

hth

ela

rge

stT

ier

(n)

sets

inth

eRCP�REF

min

po

int.

Th

eth

ird

valu

e(%EREF

int

(MP

2))

inth

ed(RCP�REF

min

)fi

eld

sd

esi

gn

ate

sth

efr

acti

on

of

BSS

E

rela

tive

toth

eC

P-c

orr

ect

ed

MP

2in

tera

ctio

ne

ne

rgy

com

pu

ted

inth

ere

fere

nce

low

est

-en

erg

yp

oin

to

fth

eR

PA@

PB

E/au

g-c

c-p

CV

5Z

curv

e.Fr

oze

n-c

ore

RPA

valu

es

RPA

(FC

)is

ane

stim

ate

de

rive

db

ym

ult

iply

ing

the

rati

ob

etw

ee

nth

efr

oze

n-c

ore

vs.

full-

ele

ctro

nM

P2

valu

es

and

the

full-

ele

ctro

nR

PA@

PB

Ee

ne

rgy.




Table

3.

BB

SE-u

nco

rre

cte

d(E

NOCP

int

),B

SSE-

corr

ect

ed

(ECP

int

)in

tera

ctio

ne

ne

rgie

san

dB

SSE

(d)

(in

kcal

mo

l21)

of

me

than

ed

ime

rco

mp

ute

dw

ith

the

corr

ela

tio

n-c

on

sist

en

tb

asis

sets

atth

eR

PA@

PB

Eth

eo

ryle

vel

inth

ere

leva

nt

po

ints

of

the

inte

ract

ion

en

erg

ycu

rve

s.

cc-p

VD

Zcc

-pV

5Z

cc-V

DZþ

df5

ZA

ug

-cc-

pV

DZ

Au

g-c

c-p

VT

ZA

ug

-cc-

pV

QZ

Au

g-c

c-p

V5

ZA

ug

-cc-

pC

V5

Z

BS

size

68

62

23

40

11

82

76

52

88

94

10

02

RPA

(FU

LL)

RNOCP

min

0.1

0.0

<�

0.4

�0

.2�

0.2

0.0

0.0

0.0

ENOCP

int

(RNOCP

min

)�

0.2

00

�0

.43

1�

3.6

19

�1

.16

7�

1.1

15

�0

.53

4�

0.5

02

�0

.43

7

ECP

int

(RNOCP

min

)0

.00

4�

0.3

54

>0

.01

5�

0.1

51

�0

.27

8�

0.3

83

�0

.39

1�

0.3

92

d(RNOCP

min

)0

.20

40

.07

7>

3.6

35

1.0

16

0.8

37

0.1

51

0.1

11

0.0

45

RCP

min

0.7

0.0

0.0

0.1

0.0

0.0

0.0

0.0

ENOCP

int

(RCP

min

)�

0.1

29

�0

.43

1�

2.6

54

�1

.04

2�

1.0

83

�0

.53

4�

0.5

02

�0

.43

7

ECP

int

(RCP

min

)�

0.0

75

�0

.35

4�

0.3

84

�0

.31

0�

0.3

56

�0

.38

3�

0.3

91

�0

.39

2

d(RCP

min

)0

.05

50

.07

72

.27

00

.73

20

.72

70

.15

10

.11

10

.04

5

ENOCP

int

(RCP�REF

min

)�

0.1

89

�0

.43

1�

2.6

54

�1

.11

1�

1.0

83

�0

.53

4�

0.5

02

�0

.43

7

ECP

int

(RCP�REF

min

)0

.06

7�

0.3

54

�0

.38

4�

0.2

92

�0

.35

6�

0.3

83

�0

.39

1�

0.3

92

d(RCP�REF

min

)/%

(EREF

int

(RPA

))0

.25

6/6

5.3

0%

0.0

77

/19.64

%2

.27

0/5

79

.38

%0

.81

9/2

08

.98

%0

.72

7/1

85

.49

%0

.15

1/3

8.5

1%

0.1

11

/28.40

%0

.04

5/11.59

%

RPA

(FC

)*

d(RCP�REF

min

)/%

(RCP�REF

min

(RPA

))–

0.0

44

/11.15

%–

––

–0

.04

6/11.83

%–

MP

2(F

ULL

)

ENOCP

int

(RCP�REF

min

)–

�0

.49

8–

––

–�

0.5

43

�0

.51

1

ECP

int

(RCP�REF

min

)–

�0

.46

1–

––

–�

0.4

84

�0

.48

4

d(RCP�REF

min

)/%

(EREF

int

(RPA

))%

(EREF

int

(MP

2))

–0

.03

7/9.56

%/7

.64

%–

––

–0

.06

0/15.21

%/1

2.3

2%

0.0

27

/6.88

%/5

.57

%

MP

2(F

C)

ENOCP

int

(RCP�REF

min

)–

�0

.48

0–

––

–�

0.5

08

–

ECP

int

(RCP�REF

min

)–

1�

0.4

59

––

––

�0

.48

2–

d(RCP�REF

min

)/%

(EREF

int

(RPA

))/%

(EREF

int

(MP

2))

–0

.02

1/5.32

%/4

.31

%–

––

–0

.02

5/6.50

%/5

.26

%–

Suff

ix‘‘d

f5Z

’’d

esi

gn

ate

sau

gm

en

tati

on

by

the

dif

fuse

fun

ctio

ns

of

the

aug

-cc-

pV

5Z

bas

is.RNOCP

min

andRCP

min

corr

esp

on

dto

the

low

est

en

erg

yp

oin

tso

fth

eB

SSE-

un

corr

ect

ed

and

BSS

E-co

rre

cte

din

tera

ctio

n

en

erg

ycu

rve

so

bta

ine

dw

ith

eac

hb

asis

.RCP�REF

min

refe

rsto

the

low

est

en

erg

yp

oin

to

fth

eB

SSE-

corr

ect

ed

RPA

@P

BE

inte

ract

ion

en

erg

ycu

rve

pro

du

ced

wit

hth

ere

fere

nce

aug

-cc-

pC

V5

Zb

asis

.B

Ssi

zeis

the

nu

mb

er

of

the

bas

isfu

nct

ion

sfo

rth

eco

mp

lex.

All

the

dis

tan

ces

are

de

fin

ed

asth

ed

isp

lace

me

nt

(in

An

gst

r€ om

s)w

ith

resp

ect

toth

ece

nte

ro

fm

ass

of

the

refe

ren

cest

ruct

ure

fro

mth

eS2

2d

atab

ase

,ac

cord

ing

toR

ef.

[62

].Fu

ll-e

lect

ron

and

fro

zen

-co

reM

P2

en

erg

ies

are

com

pu

ted

wit

hth

ela

rge

stco

nsi

de

red

corr

ela

tio

n-c

on

sist

en

tse

tsin

theRCP�REF

min

po

int.

Th

eth

ird

valu

e(%EREF

int

(MP

2))

inth

ed(RCP�REF

min

)fi

eld

s

de

sig

nat

es

the

frac

tio

no

fB

SSE

rela

tive

toth

eC

P-c

orr

ect

ed

MP

2in

tera

ctio

ne

ne

rgy

com

pu

ted

inth

ere

fere

nce

low

est

-en

erg

yp

oin

to

fth

eR

PA@

PB

E/au

g-c

c-p

CV

5Z

curv

e.Fr

oze

n-c

ore

RPA

valu

es

RPA

(FC

)is

an

est

imat

ed

eri

ved

by

mu

ltip

lyin

gth

era

tio

be

twe

en

the

fro

zen

-co

revs

.fu

ll-e

lect

ron

MP

2va

lue

san

dth

efu

ll-e

lect

ron

RPA

@P

BE

en

erg

y.



Table

4.

BB

SE-u

nco

rre

cte

d(E

NOCP

int

),B

SSE-

corr

ect

ed

(ECP

int

)in

tera

ctio

ne

ne

rgie

s,an

dB

SSE

(d)

(in

kcal

mo

l21)

of

me

than

ed

ime

rco

mp

ute

dw

ith

the

Tie

r(n

)b

asis

sets

atth

eR

PA@

PB

Eth

eo

ryle

vel

inth

e

rele

van

tp

oin

tso

fth

ein

tera

ctio

ne

ne

rgy

curv

es.

Tie

r1

Tie

r2

Tie

r3

Tie

r4

Tie

r1þ

df5

ZT

ier

2þ

df5

ZT

ier

3þ

df5

ZT

ier

4þ

df5

Z

BS

size

68

19

83

58

40

83

40

47

06

30

68

0

RPA

(FU

LL)

RNOCP

min

�0

.1�

0.2

<�

0.4

<�

0.4

<�

0.4

<�

0.4

<�

0.4

0.0

ENOCP

int

(RNOCP

min

)�

1.1

87

�1

.19

6�

1.5

97

�1

.38

7�

4.5

53

�5

.88

0�

2.9

74

�0

.69

8

ECP

int

(RNOCP

min

)�

0.0

03

�0

.11

1>

0.0

40

>0

.02

0�

0.0

35

�0

.05

1�

0.0

61

�0

.39

2

d(RNOCP

min

)1

.18

51

.08

4>

1.6

37

>1

.40

7>

4.5

17

>5

.82

9>

2.9

14

1.0

38

9

RCP

min

0.4

0.1

0.0

0.0

0.0

0.0

0.0

0.0

ENOCP

int

(RCP

min

)-0

.87

3-1

.09

1-1

.35

4-1

.15

5-3

.48

4-4

.74

5-2

.45

2-0

.69

8

ECP

int

(RCP

min

)�

0.1

75

�0

.24

5�

0.3

45

�0

.35

6�

0.3

97

�0

.39

5�

0.3

95

�0

.39

2

D(R

CP

min

)0

.69

80

.84

61

.00

80

.79

93

.08

74

.35

02

.05

71

.03

89

ENOCP

int

(RCP�REF

min

)�

1.1

59

�1

.15

2�

1.3

54

�1

.15

5�

3.4

84

�4

.74

5�

2.4

52

�0

.69

8

ECP

int

(RCP�REF

min

)�

0.0

88

�0

.23

1�

0.3

45

�0

.35

6�

0.3

97

�0

.39

5�

0.3

95

�0

.39

2

d(RCP�REF

min

)/%

(EREF

int

(RPA

))1

.07

1/2

73

.21

%0

.92

1/2

34

.95

%1

.00

8/2

56

.14

%0

.79

9/203.83

%3

.08

7/7

87

.50

%4

.35

0/1

10

9.6

9%

2.0

57

/52

4.7

4%

1.0

38

9/265.03

%

RPA

(FC

)*

d(RCP�REF

min

)/%

(EREF

int

)–

––

0.1

51

/38.52

%–

––

0.2

56

/65.36

%

MP

2(F

ULL

)

ENOCP

int

(RCP�EF

min

)–

––

�0

.96

6–

––

�1

.00

1

ECP

int

(RCP�REF

min

)–

––

�0

.45

9–

––

�0

.48

2

d(RCP�REF

min

)%/(EREF

int

(RPA

))/%

(EREF

int

(MP

2))

––

0.5

08

/129.64

%/1

04

.99

%–

––

0.5

19

/132.61

%/1

07

.40

%

MP

2(F

C)

ENOCP

int

(RCP�EF

min

)–

––

�0

.55

3–

––

�0

.60

8

ECP

int

(RCP�REF

min

)–

––

–0

.45

6–

––

�0

.48

0

d(RCP�REF

min

)%/(EREF

int

(RPA

))/%

(EREF

int

(MP

2))

––

0.0

96

/24.60

%/1

9.9

2%

––

0.1

28

/32.70

%/2

6.4

8%

Suff

ix‘‘d

f5Z

’’d

esi

gn

ate

sau

gm

en

tati

on

by

the

dif

fuse

fun

ctio

ns

of

the

aug

-cc-

pV

5Z

bas

is.RNOCP

min

andRCP

min

corr

esp

on

dto

the

low

est

en

erg

yp

oin

tso

fth

eB

SSE-

un

corr

ect

ed

and

BSS

E-co

rre

cte

din

tera

ctio

n

en

erg

ycu

rve

so

bta

ine

dw

ith

eac

hb

asis

.RCP�REF

min

refe

rsto

the

low

est

en

erg

yp

oin

to

fth

eB

SSE-

corr

ect

ed

RPA

@P

BE

inte

ract

ion

en

erg

ycu

rve

pro

du

ced

wit

hth

ere

fere

nce

aug

-cc-

pC

V5

Zb

asis

.B

Ssi

zeis

the

nu

mb

er

of

the

bas

isfu

nct

ion

sfo

rth

eco

mp

lex.

All

the

dis

tan

ces

are

de

fin

ed

asth

ed

isp

lace

me

nt

(in

An

gst

r€ om

s)w

ith

resp

ect

toth

ece

nte

ro

fm

ass

of

the

refe

ren

cest

ruct

ure

fro

mth

eS2

2d

atab

ase

,ac

cord

ing

toR

ef.

[62

].Fu

ll-e

lect

ron

and

fro

zen

-co

reM

P2

en

erg

ies

are

com

pu

ted

wit

hth

ela

rge

stT

ier

(n)

sets

inth

eRCP�REF

min

po

int.

Th

eth

ird

valu

e(%EREF

int

(MP

2))

inth

ed(RCP�REF

min

)fi

eld

sd

esi

gn

ate

sth

efr

acti

on

of

BSS

E

rela

tive

toth

eC

P-c

orr

ect

ed

MP

2in

tera

ctio

ne

ne

rgy

com

pu

ted

inth

ere

fere

nce

low

est

-en

erg

yp

oin

to

fth

eR

PA@

PB

E/au

g-c

c-p

CV

5Z

curv

e.Fr

oze

n-c

ore

RPA

valu

es

RPA

(FC

)is

ane

stim

ate

de

rive

db

ym

ult

iply

ing

the

rati

ob

etw

ee

nth

efr

oze

n-c

ore

vs.

full-

ele

ctro

nM

P2

valu

es

and

the

full-

ele

ctro

nR

PA@

PB

Ee

ne

rgy.




Using the FE regime with correlation-consistent bases with-

out additional core-valence functions produces about two

times greater BSSE for both complexes at both theory levels

with the aug-cc-pV5Z vs. aug-cc-pCV5Z basis set. It is a well-

known effect of BSSE in the FE calculations. What is more

interesting is that BSSE in the FE calculations with the basis

sets without core-valence basis functions converges to zero

nonmonotonically with respect to the cardinal number (Figs. 5

and 6). This ‘‘nonanalytical’’ behavior of BSSE transfers to the

BSSE-uncorrected energy surfaces and makes complete basis-

set interpolations on the first cardinal numbers meaningless.

In contrast, in the frozen-core correlated calculations of stabili-

zation (interaction) energies of weakly-bound complexes with

correlation-consistent basis sets, BSSE itself usually converges

monotonically to zero with respect to the cardinal number

(see e.g., Ref. [75]). In spite of that, the dependence of interac-

tion energies of the BSSE-uncorrected interaction/binding

energies is often nonmonotonic, as compared to the CP-cor-

rected results—even with the frozen-core approximation

employed (see e.g., Ref. [42], Fig. 1). The author shall stress

that the latter property is of paramount importance because

monotonic energy curves require fewer cardinal points to

obtain CBS limit, whereas nonmonotonic dependence makes

the extrapolation ambiguous (depending on the selection of

the cardinal points) and in general requires larger basis sets

(see e.g., Ref. [76], Fig. 3).

Another, often overlooked, property of the BSSE-uncorrected

calculations usually manifested in the case of weakly-bound

complexes is that the binding energy as a function of cardinal

number converges to the CBS limit from below (irrespectively

to the frozen-core approximation) what is utterly unphysical.

Contrarily, BSSE correction restores correct ‘‘variational’’ de-

pendence with respect to the cardinal number and, of course,

makes this dependence monotonic (Figs. 5 and 6 and the

references above). This in principle makes possible to use a

wider range of the basis sets (rather than the correlation-con-

sistent ones) for the complete basis-set limit interpolations.

Finally, it is worth to mention that in the case of both com-

plexes, BSSE-corrected interaction energies not only monotoni-

cally increase (in absolute value) with increase of the cardinal

number but are also closer to the converged interaction ener-

gies in nearly the whole range of the cardinal numbers (Figs. 5

and 6). This is often the case; however, some systems may ex-

hibit opposite behavior due to the favorable error compensa-

tion of BSSE and BSIE.[42]

In summary, using FE mode with the basis sets containing

no additional core-valence basis functions in the ground-state

calculations of PES is not practical from the computational

efficiency point of view, because the unpolarized core basis

functions do not contribute to the stabilization energies (at

both SCF and correlated levels) while their employment in the

correlation energies calculations increases basis set size.

Furthermore, unpolarized core basis functions produce

pathological BSSE behavior in the BSSE-uncorrected correlated

calculations (coming from the correlation energy) which needs

to be corrected even with the largest numerical sets. As the

next subsection will show, this finding has direct implications

for the present Tier (n) basis sets because the latter possess

no specifically-optimized core valence functions and are

supposed to be actively used with the ‘‘post-RPA’’ methods

(such as RPA@PBE) exhibiting about two times greater BSSE

than MP2.

Basis-set convergence in the BSSE-corrected calculations. For

each considered correlation-consistent basis set, its CP-cor-

rected RPA@PBE interaction energy, computed in its lowest

energy point, will be compared to the reference aug-cc-pCV5Z

interaction energy in its lowest energy point (0.1 and 0.1 A

displacements with respect to[67] for water and methane

dimers, respectively). It will be the test for the absolute values

of the interaction energies obtained with each basis set. In

addition, the positions of the minimal energy points of each

interaction energy curve will be compared with that of the ref-

erence curve. It will give us an estimate for the quality of the

interaction energy curves obtained with the different bases.

CP-corrected RPA@PBE interaction energies and lowest energy

positions for water dimer and methane dimers with all basis

sets are again given in Table 1 (water dimer), Table 3 (methane

dimer), whereas CP-corrected curves produced with each basis

Figure 1. CP-corrected interaction energy curves of water dimer computed

at the RPA@PBE level of theory with the cc-pVDZ, cc-pV5Z (brown curves),

aug-cc-pVDZ, aug-cc-pVTZ, aug-cc-pVQZ, aug-cc-pV5Z (black curves) as

well as with the aug-cc-pCV5Z (light grey curves), and cc-pVDZþdf5Z (blue

curves) basis sets. B) Is a zoomed version of A). [Color figure can be viewed

in the online issue, which is available at wileyonlinelibrary.com.]



set type are presented in Figure 1 (water dimer) and Figure 3

(methane dimer).

Single-point MP2 interaction energies, computed with

the largest numerical correlation-consistent unaugmented

(cc-pV5Z), augmented (aug-cc-pV5Z) and augmented with the

diffuse and polarization function (aug-cc-pCV5Z) basis sets in

the lowest energy points of the RPA@PBE/aug-cc-pCV5Z

interaction energy curves of both complexes, are given in

Tables 1 and 3.

Let us discuss the results. In the case of water dimer, the

effect of basis-set size as well as diffuse functions on the BSSE-

corrected potential energy curves of water dimer is relatively

small: already cc-pVDZ basis set produces potential energy

curves which match the quality of the aug-cc-pV5Z curves, in

agreement with the author’s previous experience.[77] The larg-

est considered basis sets with diffuse functions (aug-cc-pV5Z)

and without them (cc-pV5Z) converge to different values with

the difference of about 2.7% with RPA and 1% with MP2 meth-

ods. The fact that medium-size basis sets describe pure hydro-

gen-bonded systems at correlated level reasonably well and

do not require massive sets of diffuse functions is the conse-

quence of the fact that pure hydrogen-bonded systems are

largely driven by electrostatics, while dynamic correlations play

a moderate role. This is corroborated by the already good

results for the water dimer obtained in the DFT GGA calcula-

tions (Table 5). Recently, a practical strategy to construct

approximated correlation-consistent-based basis sets was pro-

posed.[78,79] It starts from a fully augmented aug-cc-pV(n)Z ba-

sis and gradually removes extra diffuse functions with respect

to their cardinal number from the heavy atoms until minimal

augmentation (i.e., corresponding the diffuse functions of aug-

Figure 3. CP-corrected interaction energy curves of methane dimer com-

puted at the RPA@PBE level of theory with the Tier 1–4 (brown curves) as

well as with the Tier 1–4-basis sets augmented with the diff-aug-cc-pV5Z

set (blue curves). Aug-cc-pV5Z (black) and aug-cc-pCV5Z (light gray) inter-

action energy curves are given for reference. B) And C) are zoomed ver-

sions of A). [Color figure can be viewed in the online issue, which is

available at wileyonlinelibrary.com.]

Figure 2. CP-corrected interaction energy curves of water dimer computed

at the RPA@PBE level of theory with the Tier 1–4 (brown curves) as well as

with the Tier 1–4 basis sets augmented with the diff-aug-cc-pV5Z set (blue

curves). Aug-cc-pV5Z (black) and aug-cc-pCV5Z (light gray) interaction

energy curves are given for reference. B) And C) are zoomed versions of A).

[Color figure can be viewed in the online issue, which is available at

wileyonlinelibrary.com.]




cc-pDZ basis), while keeping H and He completely unaug-

mented. This gives rise to the so-called partially-augmented

basis sets. Alternatively, minimally-augmented sets can be

obtained, for example, by combining unaugmented cc-pV(n)Z

with the diffuse functions of the Pople’s basis sets[57] (which

are of sp-type) set on the heavy atoms.[80] It was shown that

in many applications (reaction barriers, electron affinities,

ionization potentials, and atomization energies), partially, or

even minimally-augmented bases perform better than the

fully-augmented ones with the smaller cardinal number, while

usually being more compact. Although the author of this

study cannot judge explicitly on the convergence and BSSE-

performance of these partially-augmented sets in calculations

of water and methane dimer as the presented PES curves

were given BSSE uncorrected,[80]zzz he supposes this strategy

can be more favorable in the case of pure H-bonded systems

than the augmentation of small unaugmented bases with the

diffuse functions of larger augmented ones (such ascc-

pVDZþdiff-aug-cc-pV5Z), owing to the above arguments. Note,

however, that this argument applies only to the neutral H-

bonded systems: for the anionic systems (for which diffuse

functions are known to be more important), the situation can

be considerably different. In addition, the author’s[77] and other

researcher’s experience (e.g., Ref. [81]) indicates that diffuse

functions are particularly important in correlated calculations

for the description of the hydrogens directly involved in

Figure 4. CP-corrected interaction energy curves of methane dimer com-

puted at the RPA@PBE level of theory with the Tier 1–4 (brown curves) as

well as with the Tier 1–4 basis sets augmented with the diff-aug-cc-pV5Z

set (blue curves). Aug-cc-pV5Z (black) and aug-cc-pCV5Z (light gray) inter-

action energy curves are given for reference. B) And C) are zoomed ver-

sions of A). [Color figure can be viewed in the online issue, which is

available at wileyonlinelibrary.com.]

Figure 5. BSSE energies (dotted lines), CP-uncorrected (dashed lines), and

CP-corrected (solid lines) interaction energies of the water dimer computed

at the level of FE RPA@PBE in the lowest energy point of the aug-cc-pV5Z

curve as a function of cardinal number of the aug-cc-pV(n)Z (n ¼ 2–5) A)

and Tier (n) (n ¼ 1–4) B) basis sets.

zzzThe depth of interaction energy curves of six noncovalently-bound com-

plexes computed with aug-cc-pV(n)Z (n ¼ 2-4) unphysically decreases with

the increase of cardinal number what is clearly due to the effect of BSSE (Fig.

1–6 of Ref. [80]).



hydrogen-bonding, so at least minimal augmentation of hydro-

gens is often necessary.

In the case of methane dimer, the situation is opposite. The

native diffuse functions of the aug-cc-pVDZ basis sets are neces-

sary for the qualitatively-correct description of the BSSE-cor-

rected potential energy surfaces as well as absolute values of

interaction energy. Moreover, the difference in the interaction

energies of the methane dimer obtained with the largest cc-

pV5Z and aug-cc-pV5Z sets constitutes about 10% at the

RPA@PBE and 5% at the MP2 levels. These observations suggest

that diffuse functions in general, and the diffuse functions cor-

responding to higher cardinal numbers in particular, play very

important role in the description of dispersion-dominated com-

plexes. Following this hypothesis, the following combined basis

was constructed: the smallest correlation-consistent cc-pVDZ

augmented with the diffuse functions of the aug-cc-pV5Z set.

This operation readily brought the cc-pVDZ RPA interaction

energy to 98% of the aug-cc-pCV5Z value. It is better than the

interaction energy of the cc-pV5Z set and is on pair with aug-

cc-pVQZ (90 and 97.9% of the aug-cc-pCV5Z energy at the RPA

level), while being substantially more compact than the latter

two ones (by 83 and 55%, respectively). In terms of computa-

tional time, calculation of the total RPAþSE@PBE correlated

energy for the methane dimer with the cc-pVDZþdf5Z basis is

about 2.3 times faster on the same computer platform than

with aug-cc-pVQZ (of identical accuracy), most of this time

comes from the evaluation of the three-center integrals (neces-

sary for the resolution of identity approximation). With the nu-

merical Tier (n) basis sets, the effect of the augmentation is sim-

ilar. Because the removal of diffuse functions from the diffuse

part in the combined cc-pVDZþdf5Z basis will certainly

decrease interaction energy of the complex below the aug-cc-

pVQZ values, it is unlikely that there is much potential for fur-

ther reduction of the basis set size. However, there can be fur-

ther reduction of the basis set size by using more compact

instances of GTO for the unaugmented ‘‘roots.’’ In view of the

particular success of this combined basis set in calculation of

interaction energy of dispersion-dominated methane dimer, it

would be interesting to benchmark the efficiency of this strat-

egy in a wider class of dispersion-dominated and van der Waals

complexes (as well in application to the anionic complexes, for

the calculation of electric and optical molecular properties etc.).

If these expectations confirm, it can be a good argument for

more universal use of this strategy. In fact, BSSE-corrected

potential energy curves of the pure hydrogen-bonded com-

plexes are already rather well-described by the double-zeta

quality bases. The situation improves even further with the

combined cc-pVDZþdf5Z basis set. Although at the RPA level it

recovers ‘‘only’’ 92% of the converged BSSE-corrected interaction

energy of water dimer, the lowest energy point produced with

this basis falls into the same group as those produces with the

largest bases. Despite this strategy is not that efficient for the

description of pure H-bonded systems (presumably in compari-

son to the aforementioned partially-augmented basis sets), it

can be a good compromise for the calculation of potential-

energy surfaces of mixed systems including both H-bonded and

dispersion-dominated (van der Waals) parts.

In spite of the apparent computational advantage in calcula-

tion of interaction energy of the methane dimer (and presum-

ably other similar dispersion-dominated complexes), a straight

augmentation of the cc-pVDZ basis by the diffuse functions of

the aug-cc-pV5Z set has a negative side effect: the combined

basis sets suffer from enormous BSSE (116 and 579.4% of the

reference interaction energies for water and methane dimers

at the FE RPA level, respectively)xxx which leads to unaccept-

able quality of the uncorrected potential energy surfaces. One

may guess that strong BSSE was the reason why this promis-

ing property has been largely overseen so far (the author has

not find examples in the literature where this direction was

explored). One shall remember that in the original procedure

by Dunning et al., the ‘‘native’’ diffuse functions corresponding

to a particular unaugmented ‘‘root’’ basis set were optimized

in the highly-correlated multireference calculations on anions

and atoms, to obtain converged electron affinities (comparable

to the available experimental values). Herein, the exponents of

the ‘‘root’’ remained untouched during the optimization.[55]

Figure 6. BSSE energies (dotted lines), CP-uncorrected (dashed lines), and

CP-corrected (solid lines) interaction energies of the methane dimer com-

puted at the level of FE RPA@PBE computed in the lowest energy point of

the aug-cc-pV5Z curve as a function of cardinal number of the aug-cc-

pV(n)Z (n ¼ 2–5) A) and Tier (n) (n ¼ 1–4) B) basis sets.

xxxAt the MP2 level, this ratio would be about two times smaller for each com-

plex, according to the differences between the RPA and MP2 BSSE, observed

with other basis sets and, of course, it would be even smaller (by a factor of

several) in the frozen-core calcualtion.




Most probably, unbalanced BSSE in this case stems from the

fact that the exponents of the aug-cc-pV5Z diffuse functions

were not optimized specifically with the cc-pVDZ ‘‘root.’’

Unfortunately, this procedure is not applicable in general for

the optimization of an arbitrary diffuse part. In fact, the

double-f bases contain at most d angular momenta, their aug-

mentation by the diffuse functions with higher angular

momenta and subsequent optimization in the Dunning’s fash-

ion will most probably return diffuse exponents to the regular

(nondiffuse) values. This argumentation calls for the alternative

procedures for the optimization of diffuse functions with high

angular momenta. Let us make some guesses about it. Because

interaction energy of the methane dimer with the combined

cc-pVDZþdiff-aug-cc-pV5Z basis set is lower than that of cc-

pV5Z, one may think about optimization of the external diffuse

functions in calculations of small dispersion-dominated dimers,

rather than in the atomic/anionic calculations. Such a procedure

is more tedious and moreover is not universal than optimization

in atomistic calculations, because the diffuse functions of a

given atom may strongly depend on the type of surrounding

atoms constituting the weakly-bound complex. At least one

rather successful example of this strategy is known to

the author. The recently developed N07D and N07T{{{ basis

sets[82–84] exhibit good performance not only in the calculations

of hyperfine coupling constants—their initial target, but also in

a variety of molecular properties (structural, magnetic, and elec-

tric)[83,85,86] for a fraction of computational cost of their correla-

tion-consistent counterparts. This success was in part stipulated

by addition of extra p- and d-type diffuse functions to the Pop-

le’s basis and their optimization in the CP-corrected MP2 calcu-

lations of selected noncovalently-bound molecular dimers. How-

ever, it is an open question whether this procedure can be

easily applied for optimization of the external diffuse functions

of higher angular momenta (than those presented in the

unaugmented part). For sure, there would be no problem in

using this composite (unoptimized) basis sets in the CHA-based

calculations.

BSSE and basis set convergence with Tier (n) sets

BSSE-convergence. In the FHI-aims notation, Tier 1–4 NAO sets

approximately correspond to the double- to quintuple-f quality

analytical sets.[15,16] RPA@PBE BSSE-curves, plus BSSE uncor-

rected interaction energy curves of water, and methane dimers

computed with the full range of the FHI-aims NAO basis sets—

from Tier 1–4 are depicted in Supporting Information, Figures

S2 and S5. BSSE-corrected curves produced with the largest Tier

4 set are given there as well. As in the case of Gaussian sets,

the author investigated BSSE effect on the interaction energy

curves obtained with the hybrid NAO-Gaussian basis sets: Tier

Table 5. BBSE-uncorrected (ENOCPint ), BSSE-corrected (ENOCPint ) interaction energies, and BSSE (d) (in kcal mol21) of water and methane dimers computed

with correlation-consistent and Tier (n) basis sets at the DFT/PBE theory level in the lowest energy points of the reference RPA@PBE/aug-cc-pCV5Z

interaction energy curves (0.1 and 0.0 A displacements w.r.t. Ref. [62] for water and methane dimers, respectively).

Water dimer Methane dimer

ENOCPint ECPint d/(EREFint RPA)%/EREFint HF)%/ ENOCPint ECPint d/(EREFint RPA)%/EREFint HF)%/

PBE

cc-pVDZ �8.433 �4.962 3.470/82.32%/- �0.281 �0.069 0.213/54.25%/-

cc-pV5Z �5.138 �4.841 0.297/7.05%/- �0.096 �0.094 0.002/0.57%/-

cc-pVDZþdf5Z �6.355 �4.817 1.538/36.50%/- �0.469 �0.072 0.397/101.29%/-

aug-cc-pVDZ �4.993 �4.794 0.200/4.74%/- �0.406 �0.102 0.304/77.69%/-

aug-cc-pVTZ �4.853 �4.804 0.049/1.16%/- �0.123 �0.101 0.022/5.66%/-

aug-cc-pVQZ �4.873 �4.838 0.035/0.83%/- �0.100 �0.095 0.005/1.07%/-

aug-cc-pV5Z �4.858 �4.841 0.017/0.40%/- �0.100 �0.096 0.004/0.90%/-

aug-cc-pCV5Z �4.854 �4.841 0.013/0.31%/- �0.099 �0.096 0.004/0.91%/-

Tier 1 �5.496 �5.163 0.033/0.78%/- �0.091 �0.078 0.013/3.25%/-

Tier 2 �4.942 �4.866 0.076/1.80%/- �0.097 �0.092 0.005/1.26%/-

Tier 3 �4.873 �4.835 0.038/0.90%/- �0.100 �0.096 0.005/1.17%/-

Tier 4 �4.858 �4.842 0.016/0.38%/- �0.104 �0.096 0.008/2.01%/-

Tier 1þdf5Z �4.964 �4.864 0.100/2.37%/- �0.129 �0.098 0.031/7.88%/-

Tier 2þdf5Z �4.821 �4.841 0.019/0.45%/- �0.093 �0.095 0.002/0.56%/-

Tier 3þdf5Z �4.851 �4.842 0.010/0.21%/- �0.099 �0.096 0.003/0.85%/-

Tier 4þdf5Z �4.819 �4.840 0.021/0.50%/- �0.103 �0.096 0.007/1.70%/-

HF

cc-pV5Z �3.8470 �3.757 0.091/2.15%/2.42% 0.364 0.366 0.002/0.45%/0.49%

aug-cc-pV5Z �3.764 �3.751 0.0130/0.31%/0.35% 0.364 0.366 0.002/0.47%/0.50%

aug-cc-pCV5Z �3.763 �3.750 0.0130/0.30%/0.34% 0.362 0.366 0.004/1.10%/1.18%

Tier 4 �3.941 �3.751 0.190/4.50%/5.06% 0.341 0.366 0.025/6.33%/6.77%

Tier 4þdf5Z �3.829 �3.751 0.078/1.86%/2.09% 0.355 0.366 0.011/2.92%/3.12%

The suffix ‘‘df5Z’’ designates augmentation by the diffuse functions of the-aug-cc-pV5Z basis. HF energies of the complexes are computed with the

largest considered correlation-consistent and Tier (n) basis sets. The second (EREFint RPA) and the third (EREFint HF) values in the BSSE fields designates

the fraction of BSSE relative to the reference RPA@PBE/aug-cc-pCV5Z interaction energy and the HF energy, respectively, computed in the lowest energy

points of the RPA interaction energy curves.

{{{D07 and T07 refer to double- and triple-f, respectively. It is interesting to

note that these basis sets have two versions specifically optimized with B3LYP

and PBE0 density functionals, respectively.



1–4, augmented with the diffuse functions of the aug-cc-pV5Z

set (Supporting Information, Figs. S3 and S6). This combination

will be referred to as Tier (n) þ df5Z (n ¼ 1–4).

For certainty, BSSE values of each basis set type (NAO—both

pure and hybrid and the correlation-consistent Gaussians) were

compared with the counterpoise-corrected RPA interaction energy

of the maximal aug-cc-pCV5Z basis set. This comparison was

made in the lowest energy point of the RPA interaction energy

curves of water and methane dimers produced with the aug-cc-

pVC5Z basis. To evaluate the quality of both BBSE-uncorrected

and corrected interaction energy curves obtained with used basis

sets, the positions of their lowest energy points, as well as the

BSSE values and interaction energies (CP-corrected and uncor-

rected) computed in these points**** are also presented. All

correlated results produced with the numerical sets are contained

in Table 2 (water dimer) and Table 4 (methane dimer).

For comparison of the BBSE effects in the RPA against DFT,

HF as well as MP2 methods, CP-corrected and uncorrected

interaction energies and BSSE values, computed with numeri-

cal (as well as correlation-consistent) basis sets in the lowest

energy points of the RPA@PBE/aug-cc-pCV5Z interaction

energy curves of each complex (with all available basis sets for

DFT/PBE and with the largest ones for MP2 and HF), are also

provided. The DFT and HF results are presented in Table 5.

All RPA calculations were performed in the FE mode. As the

calculations with the large correlation-consistent aug-cc-pCV5Z

basis set (contains core-valence functions) showed (see the pre-

vious subsection) the contribution of the core orbitals into the

BSSE-corrected interaction, energies of the S22-like (second-row)

complexes appears to be tiny. Moreover, using core orbitals in

correlated calculations strongly increases BSSE (compared to the

frozen-core calculations with the same basis sets) and BSSE cor-

rection needs to be done. Because the present version of the

Tier (n) sets does not possess core-valence functions, FE mode in

the ground-state correlated calculations with NAO is in principle

not justified from the computational cost perspective for any

complex (does not matter how heavy the atoms and what the

fraction of core electrons is). Nevertheless, in this study, the

author stayed in the FE mode for the compatibility reasons,

because most of the previous calculations with FHI-aims are per-

formed in this mode.[23,71–73] The author may guess that the rea-

son of this is that RPA and ‘‘post-RPA’’ (as well as GW-based)

methods implemented in FHI-aims are dedicated for both

ground and excited-state calculations. The latter, in general, may

require the FE calculations (if some core excitations are involved

in an excited state). Still, the above studies do not present com-

parison of the excited-state properties in the FE vs. the frozen-

core regimes and it is even unclear whether the core orbital

effects can be captured effectively without the core-valence

functions in the basis. It would be interesting to see such studies

in future. In this work, the effect of the core orbitals on BSSE of

the two weakly-bound complexes was studied with the largest

ones Tier 4 and Tier (4)þdf5Z bases in the FE and frozen-

core MP2 calculations. The results were compared with the

results obtained with the largest cc-pV5Z, aug-cc-pV5Z, and

aug-cc-pCV5Z correlation consistent sets. All calculations were

performed in the lowest energy point of the RPA@PBE/aug-cc-

pVC5Z energy curve. Because the orbital energies of the 1s orbi-

tals of oxygen and nitrogen are an order of magnitude greater

than the energies of the next 2s orbitals (also not involved in the

covalent bonding), the former introduce large part of BSSE and

hence were excluded from the frozen-core correlation energy

calculations. As it was mentioned in the introduction, the com-

parison of the RPA BSSE behavior represents independent inter-

est due to its presumably greater (vs. MP2) BSSE susceptibility.[23]

Here, the frozen-core BSSE RPA values were estimated from the

ratio between the frozen-core and FE MP2 BSSE values.

The results show that although BSSE performance of Tier (n)

in the pure DFT calculations is excellent (Table 5), this is no

longer the case in correlated calculations (but also in the HF

ones, to lesser extent). For instance, in the FE mode BSSE of

the largest Tier 4 (and Tier 4þdiff-aug-cc-pV5Z) basis sets is

about an order of magnitude greater than that of the largest

correlation-consistent cc-pV5Z and aug-cc-pV5Z sets and is

about three to four times greater in the frozen-core mode, as

compared to the same sets. In terms of the absolute values,

BSSE of the Tier (n) sets in the frozen-core mode does not

reach controlled convergence even with the largest Tier 4 sets

and constitutes about 35% at the RPA level (18% at the MP2

level) of the reference RPA interaction energy for the water

and 40% at RPA (25% at MP2) for the methane dimers, respec-

tively (Tables 2 and 4). Such huge BSSE magnitudes can be

noticeable even at the intramolecular level.yyyy It is also inter-

esting to note that HF BSSE of the Tier 4 is about two times

(4.5 vs. 2.15% of interaction energy) greater for the dimer and

more than 10 times for the methane dimer (6.33 vs. 0.45%), as

compared to BSSE with cc-pV5Z. These findings indicate two

things. (1) Basis sets obtained in the multireference calcula-

tions are more HF-adapted (in particular are better fit to repre-

sent the nonlocal HF exchange) than the DFT-optimized ones,

what is not surprising. (2) Particularly large BSSE at correlated

level with the largest Tier (n) set in the case of methane can

be considered as a strong sign of the basis set incompleteness

at characteristic distances of the weakly bound complexes

(what is again not surprising owing to the procedure of their

construction). This will be corroborated in the next subsection.

The described BSSE issues were in fact the reason of the

recent effort to construct less BSSE-sensitive version of Tier (n)

sets.[53] It was achieved by diminishing the core-orbital contri-

bution to the correlated part of BSSE in the FE calculations by

****The results show that it is indeed sufficient to capture essential trends in

the BSSE influence on the interaction energy curves even on the fixed S22 geo-

metries without reoptimization of the fragments. The accuracy of such a com-

parison is about 0.1 A (the smallest displacement step of the S22 pathway).

yyyySee for example, Figure 11 of Ref. [23], it demonstrates substantial (and

even diverging up to Tier 3, which is probably the result of the FE calculations)

BSSE for the N2 molecule at the MP2 and RPA@HF levels with the Tier (n) (n ¼1–4) sets. Another example, the hybrid Tier (n)þdf5Z basis sets (which are

found to be more BSSE-sensitive) exhibit sight over convergence of the inter-

action energies of methane dimer (Fig. 4), what is possibly due to the intramo-

lecular BSSE. Note that for the pure (unaugmented) NAO as well as the

correlation-consistent ones (which both are less BSSE-sensitive) no overcon-

vergence was observed.




introducing additional core-valence basis functions (‘‘core

polarization functions’’ in the FHI-aims terminology). The latter

were optimized in the MP2 calculations on the covalently-

bound homonuclear dimers (H2, C2, N2, and O2). The effect of

the significant reduction of the core orbital BSSE in correlated

calculations with correlation-consistent sets by improving the

core orbital basis was shown in section Basis-set superposition

error and basis set convergence with correlation-consistent ba-

sis sets. As expected, the operation decreased the core orbital

part of BSSE of the resulting Tier (n) significantly. Couple of

the remarks regarding this work shall be made. (1) As it was

discussed above (section Basis-set superposition error and ba-

sis set convergence with correlation-consistent basis sets), the

influence of the core valence orbitals on the interaction ener-

gies of the second-row weakly-bound complexes appears to

be negligible. This is why the frozen core calculations in this

case is a much more computationally-efficient way to elimi-

nate core-orbital contribution to BSSE than adding the core-va-

lence functions (and hence increasing the bases set size). (2)

As the present study demonstrates the core orbitals is not the

only origin of large BSSE of the Tier (n) sets at correlated level.

The frozen-core MP2 calculations (Tables 1–4) show that BSSE

with the Tier (4) basis is two (water dimer) to five (methane

dimer) times greater than with the corresponding cc-pV5Z ba-

sis. This aspect was omitted by the authors because they com-

pared Tier (n) sets augmented with the extra core-valence

functions against the correlation-consistent sets without core-

valence basis functions in the FE calculations.zzzz It can be

clearly seen from the comparison of aug-cc-pCV5Z vs. aug-cc-

pV5Z that addition of core valence functions decrease BSSE in

correlated full electron calculations with correlation-consistent

basis sets by a factor of two (Tables 1 and 3). Still large BSSE

in the resulting numerical sets suggested the authors to com-

bine these sets with the atomic CP. Although atomic CP is not

proportionally more expensive with respect to the number of

atoms at the SCF level (because of the smaller number of

occupied orbitals in the ghost-orbital calculations of atoms),[87]

it quickly becomes unfeasible at correlated level for the large

systems (because of the high number of virtual orbitals in the

ghost orbital basis), not to mention conceptual difficulties to

apply CP at the intramolecular level in some cases.

In summary, at correlated level, CP correction is mandatory

(atomic CP is preferable, if possible) with the present version

of the Tier (n) sets at the whole range of the cardinal num-

bers. Under these conditions, the bases provide the accuracy

similar or better to the corresponding unaugmented correla-

tion-consistent sets. However, one shall always remember

that due to the linear scaling of its computational cost, at

correlated level, CP correction is practically limited by the sys-

tems comprising several fragments. It is important to men-

tion that using the present Tier (n) sets with the BSSE-free

CHA-based correlated methods can hypothetically fix the

problem of unbalanced BSSE. However, implementation of

CHA with NAO (and other non-Gaussian basis sets) may meet

additional technical difficulties due to the use of the auxiliary

basis sets.

Basis-set convergence in the BSSE-corrected calculations. CP-cor-

rected RPA@PBE interaction energies of all numerical bases,

computed at their lowest energy points are compared to the

reference aug-cc-pCV5Z interaction energy in its lowest energy

point. As usually, it would be a benchmark for the conver-

gence of absolute values of the interaction energies obtained

with each basis set in a specific point of PES. To estimate the

quality of the interaction energy curves obtained with the dif-

ferent bases, the positions of their minimal energy points are

compared to that of the reference curve.

CP-corrected RPA@PBE interaction energies and the lowest

energy points computed with all basis sets are again given in

Table 2 (water dimer) and Table 4 (methane dimer). Single-

point MP2 interaction energies, computed with the largest nu-

merical (Tier 4) and hybrid (Tier 4þdf5Z) basis sets in the low-

est energy points of the RPA@PBE/ aug-cc-pCV5Z interaction

energy curves of both complexes, are also presented in the

same tables. CP-corrected interaction energy curves, produced

with each basis set type, are presented in Figure 2 (water

dimer) and Figure 4 (methane dimer).

The results of this part confirm, that comparing to the corre-

spondingxxxx correlation-consistent sets, numerical Tier (n) basis

sets appears to be about 10–35% more compact (depending

on the complex and whether original NAO were augmented

by Gaussian diffuse functions, or not) in the BSSE-corrected

correlated calculations. However, one shall remember that the

improved convergence of the hybrid Tier (n)þdf5Z sets is

achieved by the augmentation with the large sets of the Gaus-

sian-type diffuse functions. The pure numerical Tier (n) sets are

unable to reproduce delocalized diffuse character of the wave

function necessary for the convergence of interaction energies

of the two weakly-bound complexes. For example, at the RPA

level, Tier 4 set is missing about 1.5 and 9% of the reference

CP-corrected interaction energies of water and methane

dimers, respectively (and about 1 and 7% at the MP2 level).

This property is impossible to correct by any manipulations

with the radial part of the Eq. (1) to behave more diffuse-like-

{{{ and it was only cured by addition of the external Gaus-

sian diff-aug-cc-pV5Z diffuse functions. That, however, leads to

even greater overall BSSE (which is already huge for the pure

Tier (n) sets), what was shown in this study. Note that this

operation breaks the initial concept of NAO (i.e., compactness)

due to the augmentation with a large portion of diffuse Gaus-

sians: in both cases, the hybrid Tier 4 augmented by the diff-

aug-cc-pV5Z functions is nearly twice as large then Tier 4 itself.

It is known that the diffuse functions (in particular of d-type)

zzzzMoreover, in these comparions, the diffuse-unaugmented (pure numerical)

version of the Tier (n) sets was compared against the diffuse-augmented (aug-

cc-pV(n)Z) correlation consistent bases which are significantly larger than cc-

pV(n)Z.

xxxxThat is, unaugmented Tier (n) vs. unaugmented correlation-consistent

ones, and the hybrid Tier (n)þdf5Z sets vs. the augmented correlation-consist-

ent ones.{{{{It can be done by increasing the so-called cutting potential in the FHI-

aims software.



are important for the description of electric properties of the

electronegative atoms, as well as some geometrical parameters

of the complexes involving multiple bonds.[88–90] It was also

recently shown that aforementioned Pople’s-based N07T basis

sets possessing additional diffuse functions provide quantita-

tive description of the rotational strength of some small chiral

organic molecules.[86] It indicates that there can be implica-

tions for the present purely-numerical Tier (n) basis sets in the

prediction of spectroscopic properties requiring substantial

delocalization of the wave function. Again, the reason of this

convergence behavior can be attributed to the procedure of

the construction of these numerical sets. The present Tier (n)

bases were optimized in the DFT LDA calculations on the

homonuclear covalently-bound complexes (H2, O2, N2, etc.),

starting from the trial radial function shape (hydrogen-like,

atom-like, and cation-like functions), to provide converged

energies of the dimers.[16] Obviously, such a procedure is con-

ceptually inapt to capture the long-range diffuse character of

the wave function, necessary for the adequate descriptions of

noncovalent interactions. Most probably, the tightness of the

DFT-optimized Tier (n) sets (in particular, in the valence region)

is also the reason of their poor BSSE performance in correlated

calculations. In addition, the present DFT-based procedure is

of course unable to generate the core-valence basis functions

of the Tier (n) bases. Besides of the situations when core-or-

bital contributions are important for the producing accurate

ground-state PES, this may have serious implications for the

description of the specific excited states using transitions from

the core electrons. As it was discussed in the BSSE part of this

section, there was rather successful effort to construct addi-

tional core polarization functions for the Tier (n) sets for the

several second row atoms as a means to decrease BSSE.[53] To

be feasible, the procedure indeed used a correlated methodol-

ogy (MP2 for the moment), what corroborates the conclusions

of this part.

BSSE and role of integration cut-off in correlated calculations

with Tier (n) sets

In the end of this section, it would be interesting to mention

another interesting (but rather technical) aspect of BSSE in cor-

related calculations with the considered numerical atom-cen-

tered basis sets. It was found that at correlated level, BSSE of

the hybrid Tier (n)þdf5Z sets is particularly sensitive to the

cut-off parameter for the numerical evaluation of the one- and

two-electron integrals required for correlated (and HF) calcula-

tions: the so-called radial multiplier (RM) in the FHI-aims termi-

nology. On the other hand, no pathologies were found at the

DFT level with small integration cutoff, owing its already negli-

gible BSSE. Full-electron RPA BSSE-uncorrected interaction

energy curves of water and methane dimers produced with

the largest Tier 4þdf5Z and small integration cut-off (RM ¼ 2)

demonstrate not only stronger overbinding, as compared to

the case of tight cut-off (RM ¼ 6), but also wrong asymptotic

behavior at large separations—the interaction energies do not

converge to zero up to 10 A (Supporting Information, Figs. S3

and S6). The same effect was observed with other Tier

(n)þdf5Z sets. Furthermore, BSSE-uncorrected interaction curve

of methane dimer exhibits discontinuities. These pathologies

can be exclusively attributed to the dependence of BSSE on

the integration cut-off. It is corroborated by the analyses of

relevant BSSE curves: indeed, correction for BSSE eliminates

these pathologies and produces CP-corrected interaction

curves identical for both RM ¼ 2 and RM ¼ 6. The smaller cut-

off would accelerate calculation of the necessary integrals by a

factor of several. However, as CP correction is often difficult/

impossible to apply at the intramolecular level, the author

would not recommend to use this property routinely. For the

pure numerical sets, the dependence of BSSE on the integra-

tion cut-off is much weaker (Supporting Information, Figs. S2

and S5). Finally, doing the frozen-core calculations instead of

the FE ones may also decrease the requirements to the inte-

gration cut-off due to the significant decrease of BSSE.

Discussion

In the BSSE-corrected correlated calculations on the water and

methane dimers, the numerical Tier (n) basis sets appear to be

about 10–35% more compact (depending on the system and

whether they are augmented by the auxiliary diffuse function

or not) than the corresponding correlation-consistent ones of

the similar accuracy. It shall be remembered, that although

‘‘universal’’ (in the sense that the procedure can systematically

produce almost any type of the basis functions for any atom),

the correlation-consistent basis sets optimized in the atomic

calculations is not the most compact instance of the Gaussian

sets. For example, polarization-consistent basis sets of Jen-

sen[60] were obtained by minimizing the SCF energies of the

covalently-bound dimers, similarly to the Tier (n) basis sets.

Again, similarly to the NAO case, the resulting basis sets

appear to be more compact in the SCF calculations of atom-

ization energies by up to several dozen per cents, comparing

to the correlation-consistent ones.[64]

Despite the favorable convergence behavior demonstrated

in this study by the numerical Tier (n) bases (vs. the unaug-

mented correlation-consistent ones) in the correlated BSSE-cor-

rected calculations on the water and methane dimers, a num-

ber of issues have been revealed. In particular, enormous BSSE

at all range of the cardinal numbers, the lack of diffuse and

core-valence polarization functions. Let us make some assump-

tions about the possible reasons of these difficulties. One

apparent reason can be attributed to the procedure of the

construction of the Tier (n) basis sets—specifically in the DFT/

LDA calculations on the covalently-bound homonuclear

dimers, rather than in the correlated calculations.[16] Because

at the DFT level BSSE vanishes rapidly with increase of the ba-

sis set size (especially in the nonhybrid case), such an optimi-

zation shall result in the basis sets more balanced in the DFT

calculations. Indeed, at the DFT/PBE (GGA) level, BSSE perform-

ance of the Tier (n) basis set exceeds BSSE performance of the

correlation-consistent basis sets in the case of water dimer

(however, it is not the case for the methane dimer; Table 5).

The latter fact, by the way, can be a direct indication of the

overlocalization of the spatial part of the DFT-based numerical




bases and, hence, the basis set incompleteness at larger dis-

tances. Thus, one obviously shall not expect balanced basis

sets at one theory level, if it was constructed at another one.

In line with this argumentation, the similar pathological BSSE

behavior exhibits composite cc-pVDZþdf5Z set which diffuse

part was optimized with the cc-pV5Z (rather than cc-pVDZ)

‘‘root’’ basis (see the discussion in section Basis-set superposi-

tion error and basis set convergence with correlation-consist-

ent basis sets). Going back to the Tier (n) case, their overlocali-

zation is most probably the reason of the increased BSSE

sensitivity in the wave function calculations. To illustrate this

point, one can imagine completely delocalized plane waves

orbitals, which are BSSE free. This argument is corroborated by

the fact that addition of the external diffuse functions of the

aug-cc-pV5Z to the Tier (4) basis decreases HF BSSE by a factor

of two for both water and methane dimer (Table 5). However,

this operation is in general impractical in the correlated case,

because it increases BSSE in correlation energies due to the

enormous amount of additional virtual orbitals generated by

the diffuse functions (Table 2 and 4). Therefore, it would be

desirable to increase the intrinsic delocalization of the spatial

orbitals of the numerical basis sets, rather than adding auxil-

iary diffuse functions (especially those optimized at different

theory levels). There is a technical way to do it by increasing

the so-called confining radius (which is responsible for the

span of the radial part of the basis function).[16] As it was men-

tioned in the previous section, doing so a posteriori for the

original LDA-optimized Tier (n) basis sets does not cause a no-

table influence on the correlated interaction energies of both

complexes (what would be expected, owing to the fact that

LDA is a local theory). In this connection, it would be interest-

ing to mention the recent work[53] on the optimization of

core-valence functions of the Tier (n) sets again. Although it

was successful in optimizing the core-valence orbitals of the

Tier (n) sets in the MP2 calculations (at least as the means to

decrease core-orbital contribution to BSSE), the procedure did

not lead to any change in the basis set coefficients of the va-

lence orbitals of the MP2-optimized sets, as compared to the

LDA-optimized ones. As the result, the problem of large BSSE

at correlated level coming from the valence orbitals of the Tier

(n) sets remained unsolved. The author of this study therefore

suggests, that to solve the problem of the overlocalization of

the Tier (n) sets systemically (what is related to both BSSE and

convergence issues), one shall optimize not only the basis set

coefficients but, simultaneously, the confining radii during the

basis sets generation in correlated calculations. Furthermore,

the optimization of the basis set coefficients simultaneously

with the confining radii (in correlated calculations) can

hypothetically be the main advantage of the NAO vs. the

Gaussian-type functions, because it could increase the diffu-

siveness of the spatial orbitals without introducing additional

diffuse functions (and hence increasing the basis set size). This

hypothesis needs to be verified for further progress of the

technology.***** Because the increase of the spatial span of the

radial part of the numerical bases leads to the increase of the

computational cost for the numerical evaluation of the two-

electron integrals at the SCF and correlated levels, it would be

desirable to have several types of NAO: those adapted for the

simulations of the covalently bound complexes; those for the

simulations on weak interactions (with the diffuse-like orbitals)

and, finally, those with additional core-valance functions. It

would be analogs to the Gaussian world with the regular,

diffuse, and core-valence functions.

There are several major strategies to construct atom-cen-

tered basis sets: (1) in the calculations on atoms and atomic

anions (when combined with multireference correlated meth-

ods, it allows to produce almost any type of basis functions;

e.g., Ref. [54–56]; (2) in the calculations on the covalently-

bound dimers (it allows to produce efficient valence-type basis

functionsyyyyy; e.g., Refs. [60,65]; (3) in the correlated calcula-

tions on the weakly-bound dimers (this path can be followed

to optimize diffuse-type functions, although it is not univer-

sal);[84] and (4) in the properties (dipole moments and polariz-

abilities) calculations[91–93] (this path may be also explored to

generate external diffuse functions). More complete analysis of

the strategies to construct Gaussian-type basis sets can be

found, for example, in reviews.[94,95] It is an open question

whether it is possible to cover all the observed issues of the

Tier (n) sets staying only at the level of the covalently-bound

(single-reference) diatomic calculations to produce BSSE-bal-

anced, correlation-adapted, purely-numerical basis sets beyond

the second row. One thing is clear: either way (whether atom-

istic or diatomic calculations are used for the basis set con-

struction) requires correlation methodologies which go well-

beyond those currently available in FHI-aims (i.e., MP2, RPA,

and ‘‘post-RPA’’ methods).zzzzz

Finally, it shall be stressed that the observed difficulties of

the numerical sets in correlated calculations are not related to

their numerical nature itself and most probably will be rele-

vant to any atom-centered basis sets constructed in the low-

level SCF-based procedures.

Conclusions

This study confirms improved, in comparison to the unaug-

mented correlation-consistent basis sets, convergence of the

considered numerical atom-centered Tier (n) sets in the ground-

*****The easiest way to check this hypothesis is to optimize Tier (n) basis sets in

calculations on the noble gas atoms and their dimers at the MP2 level. It is fea-

sible at present in FHI-aims. Herein, several strategies are possible. For exam-

ple: (1) optimizing the core and valence orbitals in the atomic calculations and

auxiliary numerical diffuse functions (and core-valence ones, if necessary) in

the dimer calculations at their minima; or, (2) optimizing basis functions of all

types (including confining radii) in calculations on the noble gas dimers.

yyyyyWhen applied to the diatomic anions, the procedure in principle allows to

obtain diffuse functions (with reservation that these diffuse functions are

obtained from the DFT calculations)[62].zzzzzFor example, the above discussed procedure of optimization of the core

polarization functions for H, C, N, and O in the MP2 calculations on their dia-

tomics[53] cannot be extended to an arbitrary element of the periodic table

simply because MP-n theory completely fails to describe for example, metals

and their compounds. The same limitation is of course relevant to the proper-

ties-optimized sets (e.g., Ref. [91]). This particular problem can be solved only

by going to the higher levels of theory like the coupled clusters theory and

may in general require multireference calculations to take to account static

correlations in some open-shell atoms and anions.



state correlated BSSE-corrected calculations of potential energy

surfaces of the water and methane dimers—the two important

instances of the noncovalently-bound complexes.

However, the present DFT/LDA-optimized version of NAO

cannot be recommended per se for use in correlated calcu-

lations for the following reasons: (1) due to the enormous

magnitude and slow convergence of BSSE especially pro-

nounced at the ‘‘post-RPA’’ theory levels, requiring CP even

with the largest basis sets (sometimes—atomic CP), (2) due

to the need in external Gaussian diffuse functions necessary

for the complete convergence of the interaction energies of

the noncovalently-bound complexes and, finally, (3) due to

the lack of polarized core-valence functions. To reveal the

full potential of NAO in correlated calculations, these issues

shall be addressed carefully. That would clearly require the

update of the existing DFT-based construction procedures

using highly correlated methods for the basis sets construc-

tion. Possible strategies for the optimizations have been dis-

cussed throughout the work. A recent, rather successful

attempt to generate core-valence numerical basis functions

for several atoms in correlated calculations[53] corroborates

the above conclusions. Most likely, the observed difficulties

will be valid for any atom-centered basis sets optimized in

the HF/DFT calculations.

Second, the work may inspire additional investigations to

answer the question whether improved convergence of the

combined basis sets (unaugmented part with smaller cardi-

nal numbers plus the diffuse part with larger ones) demon-

strated in the calculations of dispersion dominated methane

dimer will be valid for a wider class of weakly-bound

complexes.

Finally, and ones again, the study demonstrates paramount

importance of the BSSE effect in correlated studies of weak

intermolecular interactions with both numerical and correla-

tion-consistent atom-centered sets what, in principle, calls for

a wider adaption of the a priory BSSE-free CHA. That would

allow one to make calculations with the conventional atom-

centered basis sets more reliable and take full advantage of

using more exotic instances of them.

Acknowledgments

The author thanks the theory group of the Fritz Haber Insti-

tute and Garching Supercomputer Center for the access to the

FHI-aims software and provided computational time. Prof. Kirk

Peterson, Prof. Frank Jensen, and Dr. Mariana Rossi Carvalho

are gratefully acknowledged for the helpful comments and

insightful discussions. Prof. Vincenzo Barone is acknowledged

for the introduction to the N07 basis sets construction

procedure.

Keywords: basis set convergence � basis set superposition

error � correlation-consistent basis sets � diffuse func-

tions � hydrogen bonding � methane dimmer � numerical

atom-centered basis sets � random phase approximation � water

dimmer � noncovalent interactions

How to cite this article: M. Zakharov, Int. J. Quantum Chem.

2013, DOI: 10.1002/qua.24407

Additional Supporting Information may be found in the

online version of this article.

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Received: 24 October 2012Revised: 28 December 2012Accepted: 18 January 2013Published online on Wiley Online Library



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