Cite this: Phys. Chem. Chem. Phys., 2016, 18, 3011
The van der Waals interactions in rare-gas dimers: the role of interparticle interactions
Yu-Ting Chen,aKerwin Huiaand Jeng-Da Chai*ab
We investigate the potential energy curves of rare-gas dimers with various ranges and strengths of interparticle interactions (nuclear–electron, electron–electron, and nuclear–nuclear interactions). Our investigation is based on the highly accurate coupled-cluster theory associated with those interparticle interactions. For comparison, the performances of the corresponding Hartree–Fock theory, second- order Møller–Plesset perturbation theory, and density functional theory are also investigated. Our results reveal that when the interparticle interactions retain the long-range Coulomb tails, the nature of van der Waals interactions in the rare-gas dimers remains similar. By contrast, when the interparticle interactions are sufficiently short-range, the conventional van der Waals interactions in the rare-gas dimers completely disappear, yielding purely repulsive potential energy curves.
I. Introduction
The van der Waals (vdW) interactions1–5 are omnipresent in materials and biological systems. These interactions are of fundamental importance in numerous fields, involving molecular and condensed matter physics, supramolecular chemistry, structural biology, surface science, and nanoscience. While vdW interactions are individually weak (e.g., compared to covalent bonds or electrostatic interactions between permanent charges, dipoles, etc.), they are collectively important in the determination of the structure, stability, and function of a vast variety of systems, such as the interaction between graphene layers, the self-assembly of functional nanomaterials, the structure of biomacromolecules (e.g., DNA, RNA, and proteins), and the molecular recognition of proteins.6
In particular, the potential energy curve of a rare-gas dimer is predominantly determined by the interplay between the exchange-repulsion energy at short internuclear distances and the attractive vdW interaction at large internuclear distances, exhibiting a potential minimum (the vdW minimum) at an intermediate internuclear distance. The exchange-repulsion energy arises from the overlap of the electron densities of the two atoms. On the other hand, the vdW interaction, also known as London dispersion interaction or induced dipole–induced dipole interaction, arises from the Coulomb correlation of electron density fluctuations in the two well-separated atoms.
The potential energy curve can be conveniently approximated
by the Lennard-Jones (LJ) potential4 VLJðRÞ ¼ 4e s
R
! "12
$ s R
! "6
# $
; (1)
where R is the internuclear distance, s is the distance at which the potential is zero, and $e is the minimum of the potential, which is reached at R = 21/6s. Here the term R$12 models the exchange-repulsion energy, dominant at short internuclear distances, while the term R$6models the attractive vdW inter- action, dominant at large internuclear distances. Whereas the attractive term is physically based, the repulsive term has no theoretical justification (i.e., chosen for computational efficiency).
Note that the exchange-repulsion energy should decay almost exponentially with the internuclear distance. Nevertheless, due to its computational simplicity, the LJ potential is widely used in computer simulations even though more accurate potentials exist.
However, the R$6dependence of the vdW interaction may not be applied to macroscopic systems like colloids and bio- logical membranes. In these systems, the vdW interaction between two objects immersed in a medium is strongly influenced by the dielectric properties of the objects and the medium.
Accordingly, the resulting vdW interaction can be very different from the conventional R$6expression,2,7,8and can be completely repulsive under certain conditions.2,9,10 Several fascinating phenomena have been discovered in these non-R$6macroscopic vdW systems.9–12
Is it possible to create non-R$6 vdW interactions between rare-gas atoms in vacuum? Conceptually, the types of inter- particle interactions (nuclear–electron, electron–electron, and nuclear–nuclear interactions), traditionally given by the Coulomb interactions, should play a fundamental role in determining the properties of atoms and molecules. Hence, we expect that non-R$6
aDepartment of Physics, National Taiwan University, Taipei 10617, Taiwan.
E-mail: [email protected]
bCenter for Theoretical Sciences and Center for Quantum Science and Engineering, National Taiwan University, Taipei 10617, Taiwan
Received 17th October 2015, Accepted 10th December 2015 DOI: 10.1039/c5cp06317e
www.rsc.org/pccp
PCCP
PAPER
vdW interactions can appear by tuning the effective interparticle interactions of rare-gas atoms in vacuum. As a proof of concept, in this work, we address how the nature of vdW interactions in rare-gas dimers (i.e., the simplest vdW systems) changes with varying interparticle interactions, using the highly accurate coupled-cluster theory associated with those interparticle inter- actions. The rest of this paper is organized as follows. In Section II, we describe our model systems and computational details. We compare the results obtained from the coupled-cluster theory with those obtained from different computational methods, and give our comments on the connection between this study and a popular scheme in density functional theory in Section III. Our conclusions are given in Section IV.
II. Model systems and computational details
For a system consisting of M nuclei and N electrons in the Born–Oppenheimer approximation (as the nuclei are much heavier than the electrons), the electronic Hamiltonian4
He¼ $ !h2 2me
XN
i¼1
ri2$ e2 4pe0
XN
i¼1
XM
A¼1
ZAf rð ÞiA
þ e2 4pe0
XN
i¼1
XN
j4 i
f r% &ij (2)
is the sum of the kinetic energy of electrons, the nuclear–
electron attraction energy, and the electron–electron repulsion energy, respectively. Here ZAis the atomic number of nucleus A, meis the mass of an electron, $e is the charge of an electron, riA= |ri$ RA| is the distance between electron i and nucleus A, rij= |ri$ rj| is the distance between electrons i and j, and f (r) is the interparticle interaction operator with r being the inter- particle distance. The electronic Schro¨dinger equation
HeCe= EeCe (3)
is solved for the electronic energy Eeand the electronic wave- function Ce, which describes the motion of the electrons for fixed nuclear positions. The total energy
Etotal¼ Eeþ e2 4pe0
XM
A¼1
XM
B4 A
ZAZBf Rð ABÞ (4)
is obtained by adding the nuclear–nuclear repulsion energy to the electronic energy, where RAB = |RA $ RB| is the distance between nuclei A and B. One can obtain Etotalas a function of the nuclear positions, commonly known as the potential energy curve (or surface).
Traditionally, f (r) is given by the Coulomb interaction 1/r.
However, in this work, we consider two types of f (r): erf(or)/r and erfc(or)/r, which are generated by splitting the Coulomb interaction into two components.13,14 The former (the erf interaction) retains the long-range Coulomb tail without the singularity at r = 0, while the latter (the erfc interaction) is a short-range interaction with a singularity at r = 0. Physically, 1/o specifies the distance beyond which erf(or)/r approaches
1/r and the distance beyond which erfc(or)/r becomes insigni- ficant (see Fig. 1). Similar to the Coulomb case,15,16the nuclear- attraction and two-electron repulsion integrals modified for the erf and erfc interaction operators can be evaluated analytically over Gaussian basis functions,16–18facilitating an efficient evalua- tion of the integrals needed for solving eqn (3) and the equations associated with related approximate methods (see below). In principle, other types of f (r) can also be adopted.17,19–22
Similar to the Coulomb case, solving eqn (3) for a given f (r) is, however, extremely difficult even for the ground-state energy and wavefunction of a very small system, due to the prohibi- tively expensive computational cost. Practically, one searches for approximate solutions to eqn (3), obtained from ab initio wavefunction methods,4,13,14,23such as Hartree–Fock (HF) theory, second-order Møller–Plesset perturbation theory (MP2), coupled- cluster theory with iterative singles and doubles (CCSD), and CCSD with a perturbative treatment of triple substitutions (CCSD(T)). Among them, the CCSD(T) method with a sufficiently Fig. 1 Interparticle interaction as a function of interparticle distance (in atomic units).
Fig. 2 Potential energy curves of the He–He dimer associated with the long-range interparticle interactions erf(or)/r, calculated using the corresponding CCSD(T). The o = N case is equivalent to the Coulomb interaction 1/r.
large basis set is generally expected to provide highly accurate results for a variety of small- to medium-sized systems.
Alternatively, Kohn–Sham density functional theory (KS-DFT),24 a popular method for the study of the ground-state properties of large systems, can also be employed. Similar to the Coulomb case, density functional approximations (DFAs), such as the local density approximation (LDA) and generalized-gradient approximations (GGAs), to the exchange–correlation (XC) energy functional for a given f (r) are needed in the corres- ponding KS-DFT.25,26 Here, the LDA exchange energy func- tional for the erf interaction is obtained by subtracting the LDA exchange energy functional for the erfc interaction27from
the LDA exchange energy functional for the Coulomb inter- action,28whereas the LDA correlation energy functional for the erfc interaction is obtained by subtracting the LDA correlation energy functional for the erf interaction29from the LDA corre- lation energy functional for the Coulomb interaction.30 In addition, as the Perdew–Burke–Ernzerhof (PBE) XC energy functional (i.e., a popular GGA) for the Coulomb interaction31 and its variant for the erfc interaction32 are both available, their difference gives the PBE XC energy functional for the erf interaction.
To illustrate how the nature of vdW interactions in rare-gas dimers changes with varying interparticle interactions, we calculate
Fig. 3 Potential energy curves of the He–He dimer associated with the long-range interparticle interactions erf(or)/r, calculated using the corresponding CCSD(T), CCSD, MP2, HF, PBE, and LDA. The o = N case is equivalent to the Coulomb interaction 1/r.
the potential energy curves of the He–He dimer associated with the interparticle interactions erf(or)/r (o = N, 10.00, 2.00, 1.70, 1.40, and 1.10 bohr$1) and erfc(or)/r (o = 0.00, 0.10, 0.20, 0.25, 0.30, and 0.40 bohr$1), using the corresponding CCSD(T), CCSD, MP2, HF, and KS-DFT employing the PBE and LDA XC energy functionals for the associated interactions.4,13,14,23
All calculations are performed using a development version of Q-Chem 4.0.33The results are computed using a large aug-cc- pVQZ basis set34 with a high-quality EML (250, 590) grid, consisting of 250 Euler–Maclaurin radial grid points35 and 590 Lebedev angular grid points.36 The counterpoise correc- tion37is employed to reduce the basis set superposition error (BSSE).
III. Results and discussion
The potential energy curves of the He–He dimer associated with the long-range interparticle interactions erf(or)/r, calculated using the corresponding CCSD(T), are presented in Fig. 2.
Similar to the Coulomb case (i.e., the o = N case of the erf interaction), all the potential energy curves resemble the LJ potentials. For a smaller o, the strength of the erf interaction is weaker. Consequently, the electrons are more loosely bound to the nucleus, and the atoms are more polarizable, yielding larger values of s and e, respectively, (see eqn (1)).4Owing to the long-range nature of the erf interaction, the attractive vdW interaction is shown to have the [erf(oR)]2R$6asymptote (essentially retaining the R$6 asymptote of conventional vdW interactions) at sufficiently large internuclear distances R, based on the second- order perturbation theory (see the Appendix).
In comparison with the highly accurate CCSD(T) results, the He–He potential energy curves associated with the erf inter- actions, calculated using the corresponding CCSD, MP2, HF, PBE, and LDA are presented in Fig. 3. As shown, CCSD performs similarly to CCSD(T), and slightly outperforms MP2. Besides,
CCSD(T), CCSD, and MP2 exhibit the correct R$6vdW asymptotes.
By contrast, due to the lack of electron correlation, HF completely fails to describe the attractive vdW interactions, yielding purely repulsive potential energy curves for all the o values studied.
Within the framework of KS-DFT, PBE consistently outperforms LDA. However, in view of the large errors associated with the vdW minima and the incorrect vdW asymptotes (decaying much faster than R$6),38,39 LDA, PBE, and possibly other semilocal density functionals40 cannot accurately describe long-range vdW inter- actions,41wherein a fully nonlocal XC energy functional should be essential.25,26,38
On the other hand, the potential energy curves of the He–He dimer associated with the short-range interparticle interactions erfc(or)/r, calculated using the corresponding CCSD(T), are shown in Fig. 4. In contrast to the Coulomb case (i.e., the o = 0 case of the erfc interaction), the potential energy curves show strong o-dependence. It resembles the LJ potential only for a vanishingly small o, displays a metastable state for an intermediate o (around 0.25 bohr$1), and becomes purely repulsive for a o larger than 0.30 bohr$1(see the Appendix).
For comparison, the He–He potential energy curves asso- ciated with the erfc interactions, calculated using the corres- ponding CCSD, MP2, HF, PBE, and LDA are shown in Fig. 5.
With the increase of o, the potential energy curves obtained from all the computational methods become very similar. As would be expected on physical grounds, semilocal density functionals can be surprisingly accurate for short-range XC effects.40 PBE is shown to consistently perform better than LDA. Besides, due to the dominance of exchange-repulsion energy for a sufficiently large o, even HF theory can be reliably accurate.
Similar to the Coulomb case, the overall trends of LDA and PBE are opposite to those of HF and MP2, implying that a combination of the HF exchange, MP2 correlation, and DFAs (e.g., LDA or GGAs) in KS-DFT (i.e., hybrid DFT42 or double- hybrid DFT43) may achieve a more favorable balance between cost and performance than CCSD(T) for the vdW interactions in large rare-gas dimers under the erf and erfc interactions.
In addition, we calculate the potential energy curves of the He–Ne and Ne–Ne dimers associated with the erf and erfc interactions, using the corresponding CCSD(T), as shown in Fig. 6–9. For the Coulomb case, the values of s for the He–Ne and Ne–Ne dimers are larger than that for the He–He dimer.
Nevertheless, similar trends are also found for the potential energy curves of He–Ne, Ne–Ne, and possibly other rare-gas dimers.
To test the transferability of the above observed trends for other vdW systems, we calculate the potential energy curves for the lowest triplet states of H2 (a simple vdW system)44 associated with the erf and erfc interactions, using the corres- ponding CCSD (i.e., an exact theory for any two-electron system).
As shown in Fig. 10 and 11, the major features of the potential energy curves remain very similar to those found for rare-gas dimers.
Here we comment on the connection between this study and long-range corrected (LC) hybrid functionals for systems with Fig. 4 Potential energy curves of the He–He dimer associated with the
short-range interparticle interactions erfc(or)/r, calculated using the corresponding CCSD(T). The o = 0 case is equivalent to the Coulomb interaction 1/r.
Coulomb interactions.45–56These functionals model the short- range interaction (e.g., the erfc interaction) by a DFA in KS-DFT and the complementary long-range interaction (e.g., the erf interaction) by HF exchange or a fully nonlocal (i.e., orbital- dependent) XC energy component from ab initio wavefunction methods. In Fig. 3 and 5, compared to the highly accurate CCSD(T) results, LDA and PBE perform reasonably well for sufficiently short-range interparticle interactions, whereas they perform poorly for long-range interparticle interactions. Accord- ingly, our findings are also in support of the key feature of the LC hybrid functionals for systems with Coulomb interactions, which have recently been found to provide supreme performance for a very wide range of applications,57,58 especially for problems
related to the asymptote of the XC potential,59–66self-interaction errors,67,68fundamental gaps,69–82and charge-transfer excitations.83–89 Besides, empirical atom–atom dispersion potentials51,55,56,90–92or MP2 correlation energy43,53,93–95 can be added to the KS-DFT energy in order to improve the description of noncovalent inter- actions (e.g., vdW interactions). Alternatively, KS-DFT may also be combined with symmetry-adapted perturbation theory (SAPT)96–104 to yield accurate results for noncovalent interactions.105–111 In addition, to appropriately describe strong static correlation, it could be essential to develop a combined LC hybrid scheme with random phase approximations (RPAs)25,112–114for small- to medium-sized systems or with thermally-assisted-occupation density functional theory (TAO-DFT)115–117for large-sized systems.
Fig. 5 Potential energy curves of the He–He dimer associated with the short-range interparticle interactions erfc(or)/r, calculated using the corresponding CCSD(T), CCSD, MP2, HF, PBE, and LDA. The o = 0 case is equivalent to the Coulomb interaction 1/r.
IV. Conclusions
In conclusion, we have developed a comprehensive under- standing of the physics involved in controlling the vdW inter- actions in rare-gas dimers. Specifically, we have examined the potential energy curves of the rare-gas dimers associated with a variety of interparticle interactions, using the highly accurate CCSD(T) method as well as other computational methods. The long-range interparticle interactions are shown to be essential for retaining the main features of conventional vdW interactions, which cannot be appropriately described by LDA, PBE, and possibly other semilocal density functionals in KS-DFT, but can be accurately described by MP2, CCSD, and possibly other fully nonlocal XC energy components from ab initio wavefunction methods. On the other hand, the nature of vdW interactions is shown to change drastically with the short-range interparticle interactions, wherein LDA, PBE, and possibly other semilocal density functionals in KS-DFT perform reasonably well for
Fig. 8 Potential energy curves of the Ne–Ne dimer associated with the long- range interparticle interactions erf(or)/r, calculated using the corresponding CCSD(T). The o = N case is equivalent to the Coulomb interaction 1/r.
Fig. 6 Potential energy curves of the He–Ne dimer associated with the long- range interparticle interactions erf(or)/r, calculated using the corresponding CCSD(T). The o = N case is equivalent to the Coulomb interaction 1/r.
Fig. 7 Potential energy curves of the He–Ne dimer associated with the short- range interparticle interactions erfc(or)/r, calculated using the corresponding CCSD(T). The o = 0 case is equivalent to the Coulomb interaction 1/r.
Fig. 9 Potential energy curves of the Ne–Ne dimer associated with the short-range interparticle interactions erfc(or)/r, calculated using the corres- ponding CCSD(T). The o = 0 case is equivalent to the Coulomb interaction 1/r.
Fig. 10 Potential energy curves for the lowest triplet states of H2associated with the long-range interparticle interactions erf(or)/r, calculated using the corres- ponding CCSD. The o = N case is equivalent to the Coulomb interaction 1/r.
sufficiently short-range interparticle interactions (e.g., erfc(or)/r with o = 0.30 bohr$1 or larger). Therefore, our findings also support the main feature of the LC hybrid functionals for systems with Coulomb interactions. Although only the vdW interactions in rare-gas dimers and the triplet H2molecule are studied and discussed in this work, our conclusion may remain appropriate for other vdW-dominated systems.
Appendix: asymptote of the interaction energy curve between two well-
separated rare-gas atoms associated with the long-range (erf) interparticle interactions
Similar to the derivation for the Coulomb case (e.g., see Chapter 3 of ref. 5), we derive an analytical expression for the asymptote of the interaction energy curve between two well-separated rare- gas atoms associated with the long-range interparticle inter- action operator f (r): erf(or)/r (the erf interaction), based on the second-order perturbation theory.118 Since in the Coulomb case, a rare-gas atom has no permanent multipole moments in its nondegenerate ground state,3 presumably this remains correct for the erf interaction with a sufficiently large o or for the erfc interaction [f (r): erfc(or)/r] with a sufficiently small o.
Also note that the finite speed of propagation of electromag- netic signals is not taken into account in our derivation.5For brevity, the Einstein summation convention119is adopted here.
Based on this convention, when an index variable appears twice in a term, it implies a summation of that term over all possible values of the index.
Consider a rare-gas atom A, composed of a nucleus situated at ra=0 and NA electrons situated at ra (a = 1, 2,. . .,NA) with respect to the nucleus of A. The electric potential at a point r,
due to the charge distribution, is
VAðrÞ ¼ 1 4pe0
XNA
a¼0
eAaf rðj $ rajÞ; (5)
where eAa=0= NAe is the nuclear charge of A and eAa= $e (a = 1, 2. . .,NA) is the charge of an electron. The Taylor series expan- sion of VA(r) around the nucleus of A gives
VAðrÞ ¼ 1 4pe0
X
a
eAafðrÞ $X
a
eAariarifðrÞ þ1 2!
X
a
eAariarjarirjfðrÞ þ . . .
" #
¼ 1 4pe0
eAtot$ mAiriþ QAijrirjþ . . .
h i
fðrÞ; (6)
where the first term is from an electric monopole eAtot¼P
a
eAa, the second term is from an electric dipole, whose ith Cartesian component is mAi ¼P
a eAaria, the third term is from an electric quadrupole source, QAij ¼ 1
2!
P
a
eAariarja, and so on.
Consider a second rare-gas atom B, composed of a nucleus situated at rb=0and NBelectrons situated at rb(b = 1,2,. . .,NB) with respect to the nucleus of B. Let R be the separation distance vector pointing from the nucleus of A towards the nucleus of B. The interaction energy between atoms A and B is
UAB¼XNB
b¼0
eBbVA%Rþ rb&
; (7)
where eBb=0= NBe is the nuclear charge of B, and eBb= $e (b = 1, 2,. . .,NB) is the charge of an electron. The Taylor series expan- sion of VA(R + rb) around the nucleus of B gives
VA%Rþ rb&
¼ VAðRÞ þ ribriVAðRÞ þ1
2!ribrjbrirjVAðRÞ þ . . .
¼ 1 þ ribriþ1
2!ribrjbrirjþ . . .
# $
VAðRÞ:
(8) Substituting eqn (6) and (8) into eqn (7) produces
UAB¼ X
b
eBb 1þ ribriþ1
2!ribrjbrirjþ . . .
# $
VAðRÞ
¼ eh Btotþ mBiriþ QBijrirjþ . . .i 1 4pe0
& eh Atot$ mAiriþ QAijrirjþ . . .i fðRÞ
¼ 1
4pe0
eAtoteBtotþ e% AtotmBiri$ eBtotmAiri&
$ mAimBjrirj h
þ e! AtotQBijrirjþ eBtotQAijrirj"
$ m! AiQBjkrirjrk$ mBiQAjkrirjrk"
þ QAijQBklrirjrkrlþ . . .i fðRÞ:
(9) Fig. 11 Potential energy curves for the lowest triplet states of H2asso-
ciated with the short-range interparticle interactions erfc(or)/r, calculated using the corresponding CCSD. The o = 0 case is equivalent to the Coulomb interaction 1/r.
Here eBtot¼P
b
eBb, mBi ¼P
b
eBbrib, QBij¼ 1 2!
P
b
eBbribrjb, and so on.
Since atoms A and B are both neutral, eAtot= eBtot= 0. Accordingly, UAB ¼ 1
4pe0 $mAimBjrirj$ m! AiQBjk$ mBiQAjk"
rirjrk h
þQAijQBklrirjrkrlþ . . .i fðRÞ
(10)
can be expressed as a sum of dipole–dipole (dd), dipole–quadrupole (dq), quadrupole–quadrupole (qq), and other contributions.
To evaluate the interaction energy between ground-state rare-gas atoms A and B, the classical interaction energy given by eqn (10) should be first converted into a quantum mechanical operator.
Perturbation theory may then be adopted to obtain the various perturbation contributions to the interaction energy at large R.
Let the Hamiltonian of an isolated rare-gas atom X (X = A, B) be HX. The Schro¨dinger equation
HXcXn= EXncXn (11) is solved for the nth excited-state energy EXnand wavefunction cXn, where the n = 0 case refers to the ground state. Accordingly, the full Hamiltonian of rare-gas atoms A and B can be expressed as
H = HA+ HB+ UAB. (12) The interaction energy between ground-state rare-gas atoms A and B can be calculated as
DEint= E0$ (EA0+ EB0), (13) where E0is the ground-state energy of H.
To circumvent the need for solving the Schro¨dinger equa- tion with Hamiltonian H, E0may be expressed in terms of {EAn, cAn;EBn, cBn}, based on perturbation theory.5,118Since atoms A and B are well-separated, an appropriate unperturbed Hamiltonian is the sum of the Hamiltonians of the isolated atoms A and B,
H0= HA+ HB. (14)
Consequently,
H = H0+ UAB, (15)
where UABgiven by eqn (10) is the perturbation.
A. Zeroth-order theory
H0C(0)n = E(0)n C(0)n . At large R, the effects of electron exchange are insignificant. Accordingly, for the nth excited state, C(0)n = cArcBs
and E(0)n = EAr + EBs, where the isolated atoms A and B are described by quantum numbers r and s, respectively. For the ground state, C(0)0 = cA0cB0and E(0)0 = EA0+ EB0. Correspondingly, DEint= E0$ (EA0+ EB0) E (E(0)0 ) $ (EA0+ EB0) = (EA0+ EB0) $ (EA0+ EB0) = 0.
Therefore, to obtain a nonvanishing DEint, it is necessary to go beyond the zeroth-order theory.
B. First-order theory
E(1)n = hC(0)n |UAB|C(0)n i. Since in the Coulomb case, the isolated rare-gas atom X (X = A, B) has no permanent multipole moments in its nondegenerate ground state,3presumably this
holds true for the erf interaction with a sufficiently large o or for the erfc interaction with a sufficiently small o. Accordingly, the dipole terms are vanished hcX0|mXi|cX0i = 0, the quadrupole terms are vanished hcX0|QXij|cX0i = 0, and so on. Therefore, the first-order correction to the ground-state energy is
E0ð1Þ ¼ Cð0Þ0 jUABjCð0Þ0
D E
¼ Cð0Þ0
D ''' 1
4pe0 $mAimBjrirj$ m! AiQBjk$ mBiQAjk"
rirjrk h
þ QAijQBklrirjrkrlþ . . .i
fðRÞ Cð0Þ0
''
' E
¼ $ 1
4pe0 rirjfðRÞ Cð0Þ0 mAimBj
''
' '''Cð0Þ0
D E
h
þ rirjrkfðRÞ Cð0Þ0 '''!mAiQBjk$ mBiQAjk"'''Cð0Þ0
D E
$ rirjrkrlfðRÞ Cð0Þ0 '''QAijQBkl'''Cð0Þ0
D E
þ . . .i
¼ $ 1
4pe0 rirjfðRÞ cA0cB0 mAimBj
''
' '''cA0cB0
D E
h
þ rirjrkfðRÞ cA0cB0'''!mAiQBjk$ mBiQAjk"'''cA0cB0
D E
$rirjrkrlfðRÞ cA0cB0'''QAijQBkl'''cA0cB0
D E
þ . . .i
¼ $ 1
4pe0 rirjfðRÞD cA0
'' 'mAi
'' 'cA0
E cB0 mBj
'' ' '''cB0
D E
h
þ rirjrkfðRÞ D cA0
'' 'mAi
'' 'cA0
E cB0'''QBjk'''cB0
D E
!
$D cB0
'' 'mBi
'' 'cB0
E cA0'''QAjk'''cA0
D E"
$rirjrkrlfðRÞ cA0'''QAij'''cA0
D ED
cB0
'' 'QBkl
'' 'cB0
Eþ . . .i
¼ 0:
(16) Accordingly, DEint= E0$ (EA0+ EB0) E (E(0)0 + E(1)0 ) $ (EA0+ EB0) = E(1)0 = 0. Therefore, to obtain a nonvanishing DEint, it is also necessary to go beyond the first-order theory.
C. Second-order theory
Enð2Þ¼ $ P
man
Cð0Þm jUABjCð0Þn
( )
'' ''2
Emð0Þ$ Enð0Þ . The second-order correction to the ground-state energy is
E0ð2Þ¼ $X
ma0
Cð0Þm jUABjCð0Þ0
D E
''
' '''2
Eð0Þm $ E0ð0Þ
; (17)
which is always nonpositive.
From eqn (10), if only the dipole–dipole contribution is retained, we have
UABdd ¼ $ 1
4pe0mAimBjrirjfðRÞ: (18) Accordingly, the second-order correction to the ground-state energy due to the dipole–dipole contribution is
E0ð2Þ;dd ¼ $X
ma0
Cð0Þm ''UABdd' 'Cð0Þ0
D E
''
' '''2
Emð0Þ$ E0ð0Þ
¼ $X
ma0
Cð0Þm $ 1 4pe0
* +
mAimBjrirjfðRÞ ''
''
'' ''Cð0Þ0
, -
'' ''
'' ''
2
Emð0Þ$ E0ð0Þ
¼ $ 1
4pe0
ð Þ2 X
ra0
X
sa0
cArcBs'''mAimBjrirjfðRÞ'''cA0cB0
D E
''
' '''2
ErAþ EsB$ E0A$ E0B
¼ $ 1
4pe0
ð Þ2 X
ra0
X
sa0
cArcBs'''mAimBjrirjfðRÞ'''cA0cB0
D E
cA0cB0'''mAi0mBj0ri0rj0fðRÞ'''cArcBs
D E
EAr þ EsB$ E0A$ EB0
¼ $ 1
4pe0
ð Þ2.rirjfðRÞ/
ri0rj0fðRÞ
. /
&X
ra0
X
sa0
cArcBs mAimBj
''
' '''cA0cB0
D E
cA0cB0 mAi0mBj0
''
' '''cArcBs
D E
ErAþ EsB$ E0A$ E0B
¼ $ 1
4pe0
ð Þ2.rirjfðRÞ/
ri0rj0fðRÞ
. /
&X
ra0
X
sa0
DcAr
'' 'mAi
'' 'cA0
E cBs mBj
'' ' '''cB0
D ED
cA0
'' 'mAi0
'' 'cAr
E cB0 mBj0
'' ' '''cBs
D E
ErAþ EsB$ E0A$ E0B
:
(19)
In eqn (19), the terms (r = 0, s a 0) and (r a 0, s = 0) are excluded in the summation, due to the vanishing dipole terms, i.e., hcX0|mXi|cX0i = 0 (X = A, B).
' For the erf interaction, f ðRÞ ¼erfðoRÞ
R .
rirjfðRÞ ¼ rirjerfðoRÞ R ¼ ri 1
RrjerfðoRÞ þ erfðoRÞrj1 R
# $
¼ ri 1 R
@erfðoRÞ
@R R^j$erfðoRÞ R2 R^j
# $
¼ ri 1 R2
@erfðoRÞ
@R Rj$erfðoRÞ R3 Rj
# $
¼ @
@R 1 R2
@erfðoRÞ
@R
# $
R^iRjþ 1 R2
@erfðoRÞ
@R dij$ @
@R
erfðoRÞ R3
# $
R^iRj$erfðoRÞ R3 dij
¼ R @
@R 1 R2
@erfðoRÞ
@R $erfðoRÞ R3
# $
0 1
R^iR^jþ 1 R2
@erfðoRÞ
@R $erfðoRÞ R3
# $
dij
¼ $4o ffiffiffip
p 1
R2e$o2R2$4o3 ffiffiffip
p e$o2R2þ 3
R3erfðoRÞ $ 1 R2
2o ffiffiffip p e$o2R2
# $
R^iR^j
þ 2o ffiffiffip
p 1
R2e$o2R2$erfðoRÞ R3
# $
dij:
(20)
Since e$o2R2decays faster than polynomials when R is large, rirjfðRÞ ( $erfðoRÞ
R3 %dij$ 3 ^RiR^j&at large R. Accordingly,
E0ð2Þ;dd( $ 1 4pe0
ð Þ2
½erfðoRÞ*2
R6 %dij$ 3 ^RiR^j&
di0j0$ 3 ^Ri0R^j0
% &
&X
ra0
X
sa0
DcAr
'' 'mAi
'' 'cA0
E cBs mBj
'' ' '''cB0
D ED
cA0
'' 'mAi0
'' 'cAr
E cB0 mBj0
'' ' '''cBs
D E
ErAþ EBs $ EA0 $ E0B
: (21) Similar to the Coulomb case (e.g., see Chapter 3 of ref. 5), we adopt the rotational average of cAr mAi
'' ' 'cA0
( )
cA0 mAi0
'' ' 'cAr
( )
¼ 1
3dii0 cAr''mA' 'cA0
( )
'' ''2, and the rotational average of cBs mBj
'' ' '''cB0
D E
cB0 mBj0
'' ' '''cBs
D E
¼1 3djj0
'' 'D
cBs
''
'mB'''cB0E'''2, where mA¼P
a
eAara and mB¼P
b
eBbrb. Also, note that
X3
i¼1
X3
j¼1
X3
i0¼1
X3
j0¼1
dij$ 3 ^RiR^j
% &
di0j0$ 3 ^Ri0R^j0
% &
dii0djj0 ¼X3
i¼1
X3
j¼1
dij$ 3 ^RiR^j
% &2
¼ 6: (22)
Therefore, from eqn (21),
Eð2Þ;dd0 ( $ 1 24p2e02
½erfðoRÞ*2 R6
X
ra0
X
sa0
cAr''mA' 'cA0
( )
'' ''2 cBs''mB' 'cB0
( )
'' ''2 ErAþ EsB$ E0A$ E0B
: (23) From eqn (10), retaining also the dipole–quadrupole, quadrupole–
quadrupole, and other contributions will produce additional terms in eqn (17), involving rirjrkf (R), rirjrkrlf (R), and so on. For the erf interaction, it can be shown that rirjf (R) decays more slowly than rirjrkf (R), and rirjrkf (R) decays more slowly than rirjrkrlf (R), and so on. Accordingly, E(2)0 E E(2),dd0 at large R.
Therefore, in the second-order theory, the interaction energy between rare-gas atoms A and B at large R is
DEint ¼ E0$ E0Aþ E0B
% &
( E0ð0Þþ Eð1Þ0 þ E0ð2Þ
! "
$ E0Aþ EB0
% &
¼ E0ð2Þ( E0ð2Þ;dd( $ 1 24p2e02
½erfðoRÞ*2 R6
&X
ra0
X
sa0
cAr''mA' 'cA0
( )
'' ''2 cBs''mB' 'cB0
( )
'' ''2 ErAþ EsB$ E0A$ E0B
;
(24) which has the [erf(oR)]2R$6asymptote.
' For the erfc interaction, f ðRÞ ¼erfcðoRÞ
R .
In the second-order theory, the interaction energy between rare-gas atoms A and B, DEint¼ E0$ E0Aþ E0B
% &
( E0ð2Þ¼
$ P
ma0
Cð0Þm jUABjCð0Þ0
D E
''
' '''2
Emð0Þ$ Eð0Þ0
, is always nonpositive. Therefore, it is necessary to go beyond the second-order theory to describe the repulsive interaction energy at large R (as discussed in our paper), which is, however, beyond the scope of our discussion here.
Acknowledgements
This work was supported by the Ministry of Science and Tech- nology of Taiwan (Grant No. MOST104-2628-M-002-011-MY3), National Taiwan University (Grant No. NTU-CDP-104R7818), the Center for Quantum Science and Engineering at NTU (Subproject No. NTU-ERP-104R891401 and NTU-ERP-104R891403), and the National Center for Theoretical Sciences of Taiwan. We would like to thank Prof. Peter Gill (ANU) and Su-Kuan Chu (NTU) for useful discussions. Yu-Ting Chen would like to give special thanks to her family.
References
1 A. J. Stone, The Theory of Intermolecular Forces, Clarendon Press, Oxford, 1996.
2 V. A. Parsegian, Van der Waals Forces: A Handbook for Biologists, Chemists, Engineers, and Physicists, Cambridge University Press, New York, 2006.
3 K. Lucas, Molecular Models for Fluids, Cambridge University Press, New York, 2007.
4 F. Jensen, Introduction to Computational Chemistry, Wiley, New York, 2007.
5 A. Salam, Molecular Quantum Electrodynamics: Long-Range Intermolecular Interactions, Wiley, Hoboken, NJ, 2010.
6 Y. S. Lee, Self-Assembly and Nanotechnology: A Force Balance Approach, Wiley, New York, 2008.
7 H. C. Hamaker, Physica, 1937, 4, 1058.
8 E. M. Lifshitz, Dokl. Akad. Nauk SSSR, 1954, 97, 643; E. M.
Lifshitz, Dokl. Akad. Nauk SSSR, 1955, 100, 879; E. M.
Lifshitz, Zh. Eksp. Teor. Fiz., 1955, 29, 94, [J. Exp. Theor.
Phys., 1956, 2, 73].
9 S.-W. Lee and W. M. Sigmund, Colloids Surf., A, 2002, 204, 43.
10 A. Milling, P. Mulvaney and I. Larson, J. Colloid Interface Sci., 1996, 180, 460.
11 B. W. Ninham and V. A. Parsegian, J. Chem. Phys., 1970, 53, 3398.
12 L. Bergstro¨m, Adv. Colloid Interface Sci., 1997, 70, 125.
13 R. D. Adamson, J. P. Dombroski and P. M. W. Gill, Chem.
Phys. Lett., 1996, 254, 329.
14 R. D. Adamson and P. M. W. Gill, THEOCHEM, 1997, 398, 45.
15 S. F. Boys, Proc. R. Soc. London, Ser. A, 1950, 200, 542.
16 P. M. W. Gill, Adv. Quantum Chem., 1994, 25, 141.
17 P. M. W. Gill and R. D. Adamson, Chem. Phys. Lett., 1996, 261, 105.
18 T. R. Adams, R. D. Adamson and P. M. W. Gill, J. Chem.
Phys., 1997, 107, 124.
19 J. Toulouse, F. Colonna and A. Savin, Phys. Rev. A: At., Mol., Opt. Phys., 2004, 70, 062505.
20 A. D. Dutoi and M. Head-Gordon, J. Phys. Chem. A, 2008, 112, 2110.
21 J.-D. Chai and M. Head-Gordon, Chem. Phys. Lett., 2008, 467, 176.
22 J. A. Parkhill, J.-D. Chai, A. D. Dutoi and M. Head-Gordon, Chem. Phys. Lett., 2009, 478, 283.
23 P. M. W. Gill, 2013, private communication.
24 P. Hohenberg and W. Kohn, Phys. Rev., 1964, 136, B864;
W. Kohn and L. J. Sham, Phys. Rev., 1965, 140, A1133.
25 S. Ku¨mmel and L. Kronik, Rev. Mod. Phys., 2008, 80, 3.
26 A. J. Cohen, P. Mori-Sa´nchez and W. Yang, Chem. Rev., 2011, 112, 289.
27 P. M. W. Gill, R. D. Adamson and J. A. Pople, Mol. Phys., 1996, 88, 1005.
28 P. A. M. Dirac, Proc. Cambridge Philos. Soc., 1930, 26, 376.
29 S. Paziani, S. Moroni, P. Gori-Giorgi and G. B. Bachelet, Phys. Rev. B: Condens. Matter Mater. Phys., 2006, 73, 155111.
30 J. P. Perdew and Y. Wang, Phys. Rev. B: Condens. Matter Mater. Phys., 1992, 45, 13244.
31 J. P. Perdew, K. Burke and M. Ernzerhof, Phys. Rev. Lett., 1996, 77, 3865.
32 E. Goll, H.-J. Werner and H. Stoll, Phys. Chem. Chem. Phys., 2005, 7, 3917; E. Goll, H.-J. Werner, H. Stoll, T. Leininger, P. Gori-Giorgi and A. Savin, Chem. Phys., 2006, 329, 276;
E. Goll, H.-J. Werner and H. Stoll, Chem. Phys., 2008, 346, 257.
33 Y. Shao, Z. Gan, E. Epifanovsky, A. T. B. Gilbert, M. Wormit, J. Kussmann, A. W. Lange, A. Behn, J. Deng, X. Feng, D. Ghosh, M. Goldey, P. R. Horn, L. D. Jacobson, I. Kaliman, R. Z. Khaliullin, T. Kus´, A. Landau, J. Liu, E. I. Proynov, Y. M. Rhee, R. M. Richard, M. A. Rohrdanz, R. P. Steele, E. J. Sundstrom, H. L. Woodcock III, P. M.
Zimmerman, D. Zuev, B. Albrecht, E. Alguire, B. Austin, G. J. O. Beran, Y. A. Bernard, E. Berquist, K. Brandhorst, K. B. Bravaya, S. T. Brown, D. Casanova, C.-M. Chang, Y. Chen, S. H. Chien, K. D. Closser, D. L. Crittenden, M. Diedenhofen, R. A. DiStasio Jr, H. Do, A. D. Dutoi, R. G. Edgar, S. Fatehi, L. Fusti-Molnar, A. Ghysels, A. Golubeva-Zadorozhnaya, J. Gomes, M. W. D. Hanson- Heine, P. H. P. Harbach, A. W. Hauser, E. G. Hohenstein, Z. C. Holden, T.-C. Jagau, H. Ji, B. Kaduk, K. Khistyaev, J. Kim, J. Kim, R. A. King, P. Klunzinger, D. Kosenkov, T. Kowalczyk, C. M. Krauter, K. U. Lao, A. Laurent, K. V.
Lawler, S. V. Levchenko, C. Y. Lin, F. Liu, E. Livshits, R. C. Lochan, A. Luenser, P. Manohar, S. F. Manzer, S.-P. Mao, N. Mardirossian, A. V. Marenich, S. A. Maurer, N. J. Mayhall, E. Neuscamman, C. M. Oana, R. Olivares- Amaya, D. P. O’Neill, J. A. Parkhill, T. M. Perrine, R. Peverati, A. Prociuk, D. R. Rehn, E. Rosta, N. J. Russ, S. M. Sharada, S. Sharma, D. W. Small, A. Sodt, T. Stein, D. Stu¨ck, Y.-C. Su, A. J. W. Thom, T. Tsuchimochi, V. Vanovschi, L. Vogt, O. Vydrov, T. Wang, M. A. Watson, J. Wenzel, A. White, C. F. Williams, J. Yang, S. Yeganeh, S. R. Yost, Z.-Q. You, I. Y. Zhang, X. Zhang, Y. Zhao, B. R. Brooks, G. K. L. Chan, D. M. Chipman, C. J.
Cramer, W. A. Goddard III, M. S. Gordon, W. J. Hehre, A. Klamt, H. F. Schaefer III, M. W. Schmidt, C. D. Sherrill, D. G. Truhlar, A. Warshel, X. Xu, A. Aspuru-Guzik, R. Baer, A. T. Bell, N. A. Besley, J.-D. Chai, A. Dreuw, B. D. Dunietz, T. R. Furlani, S. R. Gwaltney, C.-P. Hsu, Y. Jung, J. Kong, D. S. Lambrecht, W. Z. Liang, C. Ochsenfeld, V. A.
Rassolov, L. V. Slipchenko, J. E. Subotnik, T. Van Voorhis, J. M. Herbert, A. I. Krylov, P. M. W. Gill and M. Head- Gordon, Mol. Phys., 2015, 113, 184.
34 T. H. Dunning Jr., J. Chem. Phys., 1989, 90, 1007; D. E.
Woon and T. H. Dunning Jr., J. Chem. Phys., 1994, 100, 2975.
35 C. W. Murray, N. C. Handy and G. J. Laming, Mol. Phys., 1993, 78, 997.
36 V. I. Lebedev and D. N. Laikov, Dokl. Math., 1999, 59, 477.
37 S. F. Boys and F. Bernardi, Mol. Phys., 1970, 19, 553.
38 M. Dion, H. Rydberg, E. Schro¨der, D. C. Langreth and B. I. Lundqvist, Phys. Rev. Lett., 2004, 92, 246401.
39 A. Ruzsinszky, J. P. Perdew and G. I. Csonka, J. Phys.
Chem. A, 2005, 109, 11015.
40 J. P. Perdew, A. Ruzsinszky, G. I. Csonka, L. A. Constantin and J. Sun, Phys. Rev. Lett., 2009, 103, 026403.
41 J. F. Dobson, K. McLennan, A. Rubio, J. Wang, T. Gould, H. M. Le and B. P. Dinte, Aust. J. Chem., 2001, 54, 513.
42 A. D. Becke, J. Chem. Phys., 1993, 98, 5648.
43 S. Grimme, J. Chem. Phys., 2006, 124, 034108.
44 O. Gritsenko and E. J. Baerends, J. Chem. Phys., 2006, 124, 054115.
45 A. Savin, in Recent Developments and Applications of Modern Density Functional Theory, ed. J. M. Seminario, Elsevier, Amsterdam, 1996, pp. 327–357.
46 H. Iikura, T. Tsuneda, T. Yanai and K. Hirao, J. Chem.
Phys., 2001, 115, 3540.
47 T. Yanai, D. P. Tew and N. C. Handy, Chem. Phys. Lett., 2004, 393, 51.
48 O. A. Vydrov, J. Heyd, A. V. Krukau and G. E. Scuseria, J. Chem. Phys., 2006, 125, 074106.
49 E. Livshits and R. Baer, Phys. Chem. Chem. Phys., 2007, 9, 2932.
50 J.-D. Chai and M. Head-Gordon, J. Chem. Phys., 2008, 128, 084106.
51 J.-D. Chai and M. Head-Gordon, Phys. Chem. Chem. Phys., 2008, 10, 6615.
52 M. A. Rohrdanz and J. M. Herbert, J. Chem. Phys., 2008, 129, 034107.
53 J.-D. Chai and M. Head-Gordon, J. Chem. Phys., 2009, 131, 174105.
54 R. Peverati and D. G. Truhlar, J. Phys. Chem. Lett., 2011, 2, 2810.
55 Y.-S. Lin, C.-W. Tsai, G.-D. Li and J.-D. Chai, J. Chem. Phys., 2012, 136, 154109.
56 Y.-S. Lin, G.-D. Li, S.-P. Mao and J.-D. Chai, J. Chem. Theory Comput., 2013, 9, 263.
57 C.-W. Tsai, Y.-C. Su, G.-D. Li and J.-D. Chai, Phys. Chem.
Chem. Phys., 2013, 15, 8352.
58 J.-C. Lee, J.-D. Chai and S.-T. Lin, RSC Adv., 2015, 5, 101370.
59 M. Levy, J. P. Perdew and V. Sahni, Phys. Rev. A: At., Mol., Opt. Phys., 1984, 30, 2745.
60 C.-O. Almbladh and U. von Barth, Phys. Rev. B: Condens.
Matter Mater. Phys., 1985, 31, 3231.
61 R. van Leeuwen and E. J. Baerends, Phys. Rev. A: At., Mol., Opt. Phys., 1994, 49, 2421.
62 A. Go¨rling, Phys. Rev. Lett., 1999, 83, 5459.
63 P. W. Ayers and M. Levy, J. Chem. Phys., 2001, 115, 4438.
64 A. P. Gaiduk and V. N. Staroverov, J. Chem. Phys., 2009, 131, 044107.
65 A. P. Gaiduk, D. S. Firaha and V. N. Staroverov, Phys. Rev.
Lett., 2012, 108, 253005.
66 C.-R. Pan, P.-T. Fang and J.-D. Chai, Phys. Rev. A: At., Mol., Opt. Phys., 2013, 87, 052510.
67 J. P. Perdew and A. Zunger, Phys. Rev. B: Condens. Matter Mater. Phys., 1981, 23, 5048.
68 T. Bally and G. N. Sastry, J. Phys. Chem. A, 1997, 101, 7923.
69 J. P. Perdew and M. Levy, Phys. Rev. Lett., 1983, 51, 1884.
70 L. J. Sham and M. Schlu¨ter, Phys. Rev. Lett., 1983, 51, 1888.
71 L. J. Sham and M. Schlu¨ter, Phys. Rev. B: Condens. Matter Mater. Phys., 1985, 32, 3883.
72 G. K.-L. Chan, J. Chem. Phys., 1999, 110, 4710.
73 M. Gru¨ning, A. Marini and A. Rubio, J. Chem. Phys., 2006, 124, 154108.
74 E. Sagvolden and J. P. Perdew, Phys. Rev. A: At., Mol., Opt.
Phys., 2008, 77, 012517.
75 A. J. Cohen, P. Mori-Sa´nchez and W. Yang, Phys. Rev. B:
Condens. Matter Mater. Phys., 2008, 77, 115123.
76 P. Mori-Sa´nchez, A. J. Cohen and W. Yang, Phys. Rev. Lett., 2008, 100, 146401.
77 P. Mori-Sa´nchez, A. J. Cohen and W. Yang, Phys. Rev. Lett., 2009, 102, 066403.
78 T. Tsuneda, J.-W. Song, S. Suzuki and K. Hirao, J. Chem.
Phys., 2010, 133, 174101.
79 T. Stein, H. Eisenberg, L. Kronik and R. Baer, Phys. Rev.
Lett., 2010, 105, 266802.
80 X. Andrade and A. Aspuru-Guzik, Phys. Rev. Lett., 2011, 107, 183002.
81 W. Yang, A. J. Cohen and P. Mori-Sa´nchez, J. Chem. Phys., 2012, 136, 204111.
82 J.-D. Chai and P.-T. Chen, Phys. Rev. Lett., 2013, 110, 033002.
83 A. Dreuw, J. L. Weisman and M. Head-Gordon, J. Chem.
Phys., 2003, 119, 2943.
84 D. J. Tozer, J. Chem. Phys., 2003, 119, 12697.
85 O. Gritsenko and E. J. Baerends, J. Chem. Phys., 2004, 121, 655.
86 A. Dreuw and M. Head-Gordon, J. Am. Chem. Soc., 2004, 126, 4007.
87 A. Dreuw and M. Head-Gordon, Chem. Rev., 2005, 105, 4009.
88 M. Hellgren and E. K. U. Gross, Phys. Rev. A: At., Mol., Opt.
Phys., 2012, 85, 022514.
89 W.-T. Peng and J.-D. Chai, Phys. Chem. Chem. Phys., 2014, 16, 21564.
90 S. Grimme, J. Comput. Chem., 2004, 25, 1463.
91 S. Grimme, J. Comput. Chem., 2006, 27, 1787.
92 S. Grimme, J. Antony, S. Ehrlich and H. Krieg, J. Chem.
Phys., 2010, 132, 154104.
93 E. Bre´mond and C. Adamo, J. Chem. Phys., 2011, 135, 024106.
94 J.-D. Chai and S.-P. Mao, Chem. Phys. Lett., 2012, 538, 121.
95 K. Hui and J.-D. Chai, e-print arXiv:1510.00381.
96 P. Jankowski, B. Jeziorski, S. Rybak and K. Szalewicz, J. Chem. Phys., 1990, 92, 7441.
97 H. L. Williams, K. Szalewicz, B. Jeziorski, R. Moszynski and S. Rybak, J. Chem. Phys., 1993, 98, 1279.
98 B. Jeziorski, R. Moszynski and K. Szalewicz, Chem. Rev., 1994, 94, 1887.
99 T. Korona, H. L. Williams, R. Bukowski, B. Jeziorski and K. Szalewicz, J. Chem. Phys., 1997, 106, 5109.
100 E. M. Mas, K. Szalewicz, R. Bukowski and B. Jeziorski, J. Chem. Phys., 1997, 107, 4207.
101 R. Bukowski, J. Sadlej, B. Jeziorski, P. Jankowski, K. Szalewicz, S. A. Kucharski, H. L. Williams and B. M. Rice, J. Chem. Phys., 1999, 110, 3785.
102 M. Jeziorska, P. Jankowski, K. Szalewicz and B. Jeziorski, J. Chem. Phys., 2000, 113, 2957.
103 M. Jeziorska, W. Cencek, K. Patkowski, B. Jeziorski and K. Szalewicz, J. Chem. Phys., 2007, 127, 124303.
104 K. U. Lao and J. M. Herbert, J. Phys. Chem. A, 2012, 116, 3042.
105 A. Heßelmann and G. Jansen, Phys. Chem. Chem. Phys., 2003, 5, 5010.
106 A. J. Misquitta, R. Podeszwa, B. Jeziorski and K. Szalewicz, J. Chem. Phys., 2005, 123, 214103.
107 A. Tekin and G. Jansen, Phys. Chem. Chem. Phys., 2007, 9, 1680.
108 K. Yu, J. G. McDaniel and J. R. Schmidt, J. Phys. Chem. B, 2011, 115, 10054.
109 D. E. Taylor, F. Rob, B. M. Rice, R. Podeszwa and K. Szalewicz, Phys. Chem. Chem. Phys., 2011, 13, 16629.
110 K. U. Lao and J. M. Herbert, J. Chem. Phys., 2014, 140, 044108.
111 G. Jansen, WIREs Comput. Mol. Sci., 2014, 4, 127.
112 J. P. Perdew, A. Ruzsinszky, L. A. Constantin, J. Sun and G. I. Csonka, J. Chem. Theory Comput., 2009, 5, 902.
113 D. Peng, S. N. Steinmann, H. van Aggelen and W. Yang, J. Chem. Phys., 2013, 139, 104112.
114 H. van Aggelen, Y. Yang and W. Yang, Phys. Rev. A: At., Mol., Opt. Phys., 2013, 88, 030501(R).
115 J.-D. Chai, J. Chem. Phys., 2012, 136, 154104.
116 J.-D. Chai, J. Chem. Phys., 2014, 140, 18A521.
117 C.-S. Wu and J.-D. Chai, J. Chem. Theory Comput., 2015, 11, 2003.
118 D. J. Griffiths, Introduction to Quantum Mechanics, Prentice Hall, Upper Saddle River, NJ, 2nd edn, 2004.
119 See, for example, http://mathworld.wolfram.com/Einstein Summation.html.
