formulations of the closed-shell interactions in …pyykko/no291.pdfformulations of the closed-shell...
TRANSCRIPT
Formulations of the closed-shell interactions in endohedral
systems
Cong Wanga, Michal Strakab, and Pekka Pyykkoa∗
aDepartment of Chemistry, University of Helsinki,POB 55 (A. I. Virtasen aukio 1), 00014 Helsinki, Finland.
E-mail: [email protected] Institute of Organic Chemistry and Biochemistry,
Academy of Sciences of the Czech Republic,Flemingovo nam. 2, 16610 Prague, Czech Republic.
April 13, 2010
Abstract
An attempt is made to express the interaction energy in an endohedral A@Bsystem starting from a one-center (r<)l/(r>)l+1 expansion. Electrostatic, induction,and dispersion contributions are obtained from Rayleigh-Schrodinger perturbationtheory. New electric polarizabilities with r−l−1 radial integrals are calculated for l =0, 1 and 2 for the outer system B. For a ’breakable’ B, they can be related to theusual London formula. The new polarizabilities are used to successfully estimatethe Born-type charge solvation energy and to roughly estimate the lowest-order,l = 1 dispersion term. The latter, London-type expression is now also derived froma Casimir-Polder-type argument. It is applied on A = He-Xe, Zn-Hg, and severalmolecules with B = C60 and the results are compared against MP2 and SCS-MP2supramolecular calculations. The l = 2 dispersion terms are smaller than the l = 1ones.
1 Introduction
While the intermolecular closed-shell interactions have undergone careful scrutiny, much less isknown of the case where one subsystem A is inside the other subsystem, B (see Fig. 1). Anexample is the endohedral system Ng@C60 where Ng is a noble-gas atom. This includes theactual theoretical formulation of the problem.Experimentally, a recent compilation [1] lists existing A@B compounds with A = He, Ar-Xe,H2, N2, H2O, and NH3 among the B = C60 species theoretically considered by us. We are notaware of any experimental data on their heats of formation. A straightforward method wouldbe bomb calorimetry in suitable cases. The rotational levels of ortho-H2 in C60 were observedvia specific heats by Kohama et al. [2]. Earlier quantum chemical values exist on the energies
1
of encapsulation of Ne, H2 and N2 [3], and H2 [4]. For theoretical and experimental data on thereactivities of noble-gas endohedral fullerenes Ng@C60 and Ng2@C60 (Ng=He-Xe), see Osunaet al. [5].The theoretical methods for the intermolecular interactions typically consist of two approaches.[6, 7] One is a direct supramolecular calculation and an energy decomposition based on thiscomputational result, e.g. a Morokuma analysis. [8] Another is a perturbation expansion in-volving the monomers. The second approach is conceptually simpler and, by definition, BasisSet Superposition Error (BSSE)-free. One choice for the perturbation is to use the Coulombinteraction 1/rij directly, for instance through the SAPT [9] method; another starts from amultipole expansion. Although a multipole expansion is less accurate than the full Coulombinteraction, it has the advantage that the interaction energy can be written as a function ofthe properties of the monomers. Such functional forms can also be used in molecular dynamicssimulations. [10]
For an endohedral system A@B, if the inside subsystem A is located at the center of aninternally connected (non-breakable) outside subsystem B, the intermolecular distance R willbe zero. In this case, the traditional two-center multipole expansion, used for deriving London’sformula for dispersion interaction,
Edisp ≈ −3
2
IAIB
IA + IB
αAαB
R6, (1)
will not be valid since R = 0. [11, 12]
In this article, we use a different, one-center expansion for 1/rij to rewrite the Hamiltonianof endohedral intermolecular interactions in terms of multipole operators of the monomers. ARayleigh-Schrodinger perturbation treatment (RSPT) is then carried out for this Hamiltonian.The electrostatic, induction, and dispersion terms are derived from that perturbation expan-sion. It is well-known that the exchange interaction is missing in the RSPT of intermolecularinteractions. [9] Therefore this type of interaction will not be included in our formalism. Theresults are compared with MP2, SAPT, and SCS-MP2 [13] supramolecular calculations. Theconnection with the London formula for a separable outer subsystem is discussed.
2 General Formulation
2.1 The interaction Hamiltonian for an endohedral system
The intermolecular interaction is governed by the interparticle Hamiltonian
Hint =∑
a∈A
∑
b∈B
qaqb
rab
, (2)
Here the summations run over both electrons and nuclei of the two subsystems, A and B. Theqa and qb are the charges of the particles a and b, and rab is the interparticle distance. Thegeometry is illustrated in Fig. 1. The off-center case in Fig. 2 will be considered later. As4πǫ0 = 1, the equations will be valid in atomic units or in Gauss-cgs.
We use the expansion
1
rab
=∞∑
l=0
4π
2l + 1
l∑
m=−l
rla
rl+1b
Y ∗lm(Ωa) Ylm(Ωb). (3)
2
Here it is assumed that ra < rb, ∀a ∈ A, b ∈ B meaning that the electrons and nuclei ofthe inside subsystem, A, are always located inside the outer shell, B. In contrast to nuclearelectric quadrupole interactions, where the nucleus is far smaller than the electronic system,the present A is smaller, but of the same order of magnitude as B. Therefore the convergencewill be a concern. The convergence of the
∑
l R−l expansion was discussed by Ahlrichs [14],
who concluded that the series is only semi-convergent.
The interaction Hamiltonian (2) can be written as
Hint =∞∑
l=0
l∑
m=−l
QA(G)†lm F
B(G)lm (4)
=∞∑
l=0
l∑
m=−l
(−1)mQA(G)l,−m F
B(G)lm (5)
where QA(G)lm is the multipole moment operator for A and the quantities F
B(G)lm characterize B.
Both of them are defined in spherical coordinates. The superscript (G) stands for the ’globalcoordinates’ of the complex, common for A and B, as suggested by Stone. [15]
QA(G)lm ≡
√
4π
2l + 1
∑
a∈A
qarlaYlm(Ωa) (6)
FB(G)lm ≡
√
4π
2l + 1
∑
b∈B
qbr−l−1b Ylm(Ωb), (7)
The relation to the multipole quantities in Cartesian coordinates is shown in Tables 1 and 2.
If we write the contributions from nuclei and electrons separately, the expressions for QA(G)lm
and FB(G)lm become
QA(G)lm ≡
√
4π
2l + 1
NA∑
I=1
ZIRlIYlm(ΩI) −
nA∑
i=1
rliYlm(Ωi)
, (8)
FB(G)lm ≡
√
4π
2l + 1
NB∑
J=1
ZJR−l−1J Ylm(ΩJ) −
nB∑
j=1
r−l−1j Ylm(Ωj)
, (9)
here NA and NB are the numbers of nuclei, and nA and nB are the numbers of electrons,belonging to monomers A and B, respectively. ZI and ZJ are nuclear charges, RI and RJ arenuclear coordinates, and ri and rj are electron coordinates. I and i belong to monomer A,while J and j belong to monomer B.If we consider the internal rotations of the monomers A and B, it may be convenient to adoptthe local coordinates. A rotation through the Euler angles (α, β, γ) can be described by theWigner matrix
QA(L)l,−kA
≡l∑
m=−l
QA(G)l,−mD
(l)−m,−kA
(ξA) (10)
FB(L)lkB
≡l∑
m=−l
FB(G)lm D
(l)mkB
(ξB). (11)
3
Here the ξA and ξB stand for the Euler angles for local rotation for monomers A and B respec-tively. The transformation from the local to the global coordinates is
QA(G)l,−m =
∑
kA
QA(L)l,−kA
D(l)−kA,−m(ξ−1
A ) (12)
FB(G)lm =
∑
kB
FB(L)lkB
D(l)kBm(ξ−1
B ). (13)
Inserting eqns (12) and (13) into (5), we obtain
Hint =∞∑
l=0
l∑
m,kA,kB=−l
(−1)mQA(L)l,−kA
FB(L)l,kB
D(l)−kA,−m(ξ−1
A )D(l)kBm(ξ−1
B ) (14)
=∞∑
l=0
l∑
m,kA,kB=−l
(−1)m(−1)−kA+mQA(L)l,−kA
FB(L)l,kB
D(l)kAm(ξ−1
A )∗D(l)kBm(ξ−1
B ) (15)
=∞∑
l=0
l∑
m,kA,kB=−l
(−1)kAQA(L)l,−kA
FB(L)l,kB
D(l)mkA
(ξA)D(l)kBm(ξ−1
B ) (16)
=∞∑
l=0
l∑
kA,kB=−l
(−1)kAQA(L)l,−kA
FB(L)l,kB
D(l)kBkA
(ξ−1B ξA) (17)
The second, third, and fourth steps employ the properties of Wigner matrices: D(j)m′m(ξ) =
(−1)m′−mD(j)−m′,−m(ξ)∗ [16], D
(j)m′m(ξ) = D
(j)mm′(ξ−1)∗, and
∑
m′ D(j)m′′m′(ξA)D
(j)m′m(ξB) = D
(j)m′′m(ξAξB)
[17] respectively. Eqn (17) may be interpreted as the intermolecular interaction including therelative angular dependence, ξ−1
B ξA. If the local axes of monomer A and B coincide, ξA = ξB, eqn
(17) becomes identical with the global coordinate expression, eqn (5), since D(l)kBkA
(0) = δkBkA.
We then proceed to a RSPT for the Hamiltonian (17). For simplicity, the subscript (L) forlocal coordinates is now omitted.
2.2 First-order Perturbation: Electrostatics
The first-order perturbation correction to the interaction energy becomes the product of ex-pectation values for A and B
E(1) = 〈0A0B|Hint|0A0B〉
=∞∑
l=0
l∑
kA,kB=−l
(−1)kA〈0A|QAl,−kA
|0A〉 〈0B|F Bl,kB
|0B〉D(l)kBkA
(ξ−1B ξA) (18)
=∞∑
l=0
l∑
kA,kB=−l
(−1)kAQAl,−kA
F Bl,kB
D(l)kBkA
(ξ−1B ξA) (19)
where |0A〉 and |0B〉 are the ground-states of A and B, respectively. The quantities QAl,−kA
and F Bl,kB
are the multipole moments defined as the expectation values of the correspondingoperators for the states |0A〉 and |0B〉, respectively. This eqn (19) describes the electrostaticmultipole interaction between the inside and outside subsystems.
2.3 Second-order Perturbation
2.3.1 Induction Interaction
The induction interaction describes the polarization of the electron distribution of one subsys-tem by the static field of the other subsystem. In second-order RSPT, this effect comes from
4
the excited states for one and only one subsystem, i.e. |0AnB〉 or |nA0B〉. Then the interactionenergy is
E(2)ind = −
∑
nB
′⟨
0A0B
∣∣∣Hint
∣∣∣ 0AnB
⟩ ⟨
0AnB
∣∣∣Hint
∣∣∣ 0A0B
⟩
/ (EnB− E0B
)
−∑
nA
′⟨
0A0B
∣∣∣Hint
∣∣∣ nA0B
⟩ ⟨
nA0B
∣∣∣Hint
∣∣∣ 0A0B
⟩
/ (EnA− E0A
) (20)
= −∑
nB
′
∣∣∣∣∣∣
∞∑
l=0
l∑
kA,kB=−l
(−1)kA〈0A|QAl,−kA
|0A〉〈0B|F BlkB
|nB〉D(l)kBkA
(ξ−1B ξA)
∣∣∣∣∣∣
2
/ (EnB− E0B
)
−∑
nA
′
∣∣∣∣∣∣
∞∑
l=0
l∑
kA,kB=−l
(−1)kA〈0A|QAl,−kA
|nA〉〈0B|F BlkB
|0B〉D(l)kBkA
(ξ−1B ξA)
∣∣∣∣∣∣
2
/ (EnA− E0A
) . (21)
Here the prime stands for ”excited states only”. The first term on the right-hand-side (RHS)of eqn (21) describes the induction from the multipoles of the inner monomer A on the outershell B and the second term of the RHS stands for the induction from the multipoles of B onthe inner monomer A.
Introducing the static ’regular’ and ’irregular’ polarizabilities as
αAlkAl′k′
A
≡∑
nA
′
⟨
0A
∣∣∣QA
lkA
∣∣∣ nA
⟩ ⟨
nA
∣∣∣QA
l′k′
A
∣∣∣ 0A
⟩
+⟨
0A
∣∣∣QA
l′k′
A
∣∣∣ nA
⟩ ⟨
nA
∣∣∣QA
lkA
∣∣∣ 0A
⟩
EnA− E0A
(22)
α†BlkBl′k′
B
≡∑
nB
′
⟨
0B
∣∣∣F B
lkB
∣∣∣ nB
⟩ ⟨
nB
∣∣∣F B
l′k′
B
∣∣∣ 0B
⟩
+⟨
0B
∣∣∣F B
l′k′
B
∣∣∣ nB
⟩ ⟨
nB
∣∣∣F B
lkB
∣∣∣ 0B
⟩
EnB− E0B
, (23)
respectively, eqn (21) can be written as
E(2)ind = −1
2
∞∑
l,l′=0
l∑
kA,kB=−l
l′∑
k′
A,k′
B=−l′
(−1)kA+k′
A
[
QAl,−kA
QAl′,−k′
A
α†BlkB l′k′
B
+ αAl,−kA,l′,−k′
A
F Bl,kB
F Bl′,k′
B
]
D(l)kBkA
(ξ−1B ξA)D
(l′)k′
Bk′
A
(ξ−1B ξA.) (24)
2.4 Dispersion Interaction
2.4.1 Formalism
The dispersion interaction comes from the simultaneous charge fluctuation of the two interactingmonomers. In second-order RSPT this effect arises from the doubly excited states |nAnB〉. Theinteraction Hamiltonian (5) gives
E(2)disp = −
∑
nA,nB
′⟨
0A0B
∣∣∣Hint
∣∣∣ nAnB
⟩ ⟨
nAnB
∣∣∣Hint
∣∣∣ 0A0B
⟩
/ (EnA− E0A
+ EnB− E0B
) (25)
= −∑
nA,nB
′
∣∣∣∣∣∣
∞∑
l=1
l∑
kAkB=−l
(−1)kA〈0A|QAl,−kA
|nA〉〈0B|F BlkB
|nB〉D(l)kBkA
(ξ−1B ξA)
∣∣∣∣∣∣
2
/ (EnA− E0A
+ EnB− E0B
) . (26)
Making use of the integral identity [18]
1
a + b=
2
π
∫ ∞
0
a
a2 + z2
b
b2 + z2dz, a > 0, b > 0, (27)
5
and replacing the quantities a and b by EnA− E0A
and EnB− E0B
respectively, eqn (26) istransformed into
E(2)disp = −2
π
∫ ∞
0
∑
nA,nB
′
∣∣∣∣∣∣
∞∑
l=1
l∑
kAkB=−l
(−1)kA〈0A|QAl,−kA
|nA〉〈0B|F BlkB
|nB〉D(l)kBkA
(ξ−1B ξA)
∣∣∣∣∣∣
2
ωnA0A
ω2nA0A
+ ω2
ωnB0B
ω2nB0B
+ ω2dω. (28)
Here the transition frequencies are defined as
h ωnA0A≡ EnA
− E0A(29)
h ωnB0B≡ EnB
− E0B, (30)
meaning that h = 1 from now on. Introducing the dynamical polarizabilities
αAlkAl′k′
A
(ω) ≡∑
nA
′[⟨
0A
∣∣∣QA
lkA
∣∣∣ nA
⟩ ⟨
nA
∣∣∣QA
l′k′
A
∣∣∣ 0A
⟩
+⟨
0A
∣∣∣QA
l′k′
A
∣∣∣ nA
⟩ ⟨
nA
∣∣∣QA
lkA
∣∣∣ 0A
⟩] ωnA0A
ω2nA0A
− ω2
(31)
α†BlkBl′k′
B
(ω) ≡∑
nB
′[⟨
0B
∣∣∣F B
lkB
∣∣∣ nB
⟩ ⟨
nB
∣∣∣F B
l′k′
B
∣∣∣ 0B
⟩
+⟨
0B
∣∣∣F B
l′k′
B
∣∣∣ nB
⟩ ⟨
nB
∣∣∣F B
lkB
∣∣∣ 0B
⟩] ωnB0B
ω2nB0B
− ω2,
(32)
eqn (28) can be written as
E(2)disp = − 1
2π
∞∑
l,l′=1
l∑
kA,kB=−l
l′∑
k′
A,k′
B=−l′
(−1)kA+k′
A
[ ∫ ∞
0αA
l,−kA,l′,−k′
A
(iω) α†BlkBl′k′
B
(iω) dω]
D(l)kBkA
(ξ−1B ξA)D
(l′)k′
Bk′
A
(ξ−1B ξA).
(33)Eqn (33) is the Casimir-Polder type expression for dispersion interaction in the endohedralsystem.
The Casimir-Polder expression may be approximated as [19, 20]
αAlkAl′k′
A
(iω) ≈αA
lkAl′k′
A
1 + (ω/ωI,A)2 (34)
α†BlkBl′k′
B
(iω) ≈α†B
lkBl′k′
B
1 + (ω/ωI,B)2 , (35)
where the α and α† are the regular and irregular static polarizabilities defined in eqn (22) and(23) respectively. The ωI,A and ωI,B are constants interpreted as effective transition frequencies.Then eqn (33) can be integrated analytically yielding
E(2)disp ≈ −1
4
ωI,AωI,B
ωI,A + ωI,B
∞∑
l,l′=1
l∑
kA,kB=−l
l′∑
k′
A,k′
B=−l′
(−1)kA+k′
A
[
αAl,−kA,l′,−k′
A
α†BlkBl′k′
B
]
D(l)kBkA
(ξ−1B ξA)D
(l′)k′
Bk′
A
(ξ−1B ξA)
(36)Choosing ωI,A and ωI,B as the ionization potentials of A and B respectively, we arrive at thegeneralized London-type formula for the dispersion interaction for endohedral system [12]
E(2)disp ≈ −1
4
IAIB
IA + IB
∞∑
l,l′=1
l∑
kA,kB=−l
l′∑
k′
A,k′
B=−l′
(−1)kA+k′
A
[
αAl,−kA,l′,−k′
A
α†BlkBl′k′
B
]
D(l)kBkA
(ξ−1B ξA)D
(l′)k′
Bk′
A
(ξ−1B ξA).
(37)
6
3 Special cases
After obtaining the general results for electrostatic (19), induction (24), and dispersion interac-tions, (33) and (37), we now derive some expressions for particular systems. The special casesinclude both symmetries and explicit lower-order multipole formulae.
3.1 A ∈ O(3)
The highest symmetry of a closed-shell chemical species is the group O(3). The hydrogen atombelongs to O(4), but is not a closed-shell case.
For a non-relativistic closed-shell atom as the inner subsystem A, its simultaneous eigenstateH, L2, Lz = |0A, 0A0A〉. From electrostatics, eqn (19) becomes
E(1) =∞∑
l=0
l∑
kA,kB=−l
(−1)kA〈0A, 0A0A|QAl,−kA
|0A, 0A0A〉 〈0B|F Bl,kB
|0B〉D(l)kBkA
(ξ−1B ξA). (38)
Expressing the matrix element 〈0A, 0A0A|QAl,−kA
|0A, 0A0A〉 by the Wigner-Eckart theorem [21]we have
〈0A, 0A0A|Ql,−kA|0A, 0A0A〉 = 〈0Al; 0A,−kA|0Al; 0A0A〉
〈0A, 0A‖Ql‖0A, 0A〉√2l + 1
, (39)
where the 〈0Al; 0A,−kA|0Al; 0A0A〉 is the Clebsch-Gordon coefficient between the coupled anduncoupled angular momentum bases |j1j2; jjz〉 and |j1j2; j1zj2z〉, respectively. The result van-ishes unless
0 − kA = 0 (40)
l = 0. (41)
Consequently, only the monopole of A will contribute to eqn (19). For a neutral atom A, therewill be no electrostatic interaction under the non-overlap assumption. For charged A, the resultbecomes
E(1) = qA〈0B|F B00|0B〉D(0)
00 (ξ−1B ξA)
= qA
⟨
0B
∣∣∣∣∣∣
NB∑
J=1
ZJ
RJ
−nB∑
j=1
1
rj
∣∣∣∣∣∣
0B
⟩
= qAVB, qA =∑
a∈A
qa, (42)
where VB is the electrostatic potential of B at the origin. The Wigner matrix is 1. Eqn (42)coincides with the classical electrostatic interaction for a point charge in an external potential.Some particular systems, such as H+@C60 or Li+@C60, do not have the cation at the center ofC60. [22] and for them this result is not valid. Off-centre systems are discussed below.
For the induction, the first term of eqn (21) has a similar structure with the electrostaticinteraction and becomes
E(2)ind,A→B = −1
2q2Aα†B
0000 (43)
This result is similar to the Born equation for charge solvation. [23] In addition, eqns (42) and(43) can also describe the leading term of a charged molecule, e.g. H3O+, at the center of the
7
outer shell B. The second term of eqn (21) describes the polarization of A by a multipole ofthe outer shell B, and can be expanded as
E(2)ind,B→A = −
∑
nA
′
∞∑
l,l′=0
l∑
kA,kB=−l
l′∑
k′
A,k′
B=−l′
(−1)kA+k′
A〈0A|QAl,−kA
|nA〉〈nA|QAl′,−k′
A
|0A〉F Bl,kB
F Bl′,k′
B
D(l)kBkA
(ξ−1B ξA)D
(l′)k′
Bk′
A
(ξ−1B ξA)/ (EnA
− E0A) . (44)
Using the Wigner-Eckart theorem for the monomer A,
〈0A|QAl,−kA
|nA〉 = 〈0A, 0A0A|QAl,−kA
|nA, lAmA〉
= 〈lAl; mA,−kA|lAl; 0A0A〉〈0A0A‖QA
l ‖nAlA〉√2l + 1
. (45)
It vanishes unless
mA − kA = 0 (46)
lA = l. (47)
Also the term
〈nA|QAl′,−k′
A
|0A〉 = 〈0A|QA†
l′,−k′
A
|nA〉∗
= (−1)k′
A〈0A, 0A0A|QAl′,k′
A
|nA, lAmA〉∗
= (−1)k′
A〈lAl′; mA, k′A|lAl′; 0A0A〉∗
〈0A0A‖QAl′ ‖nAl′A〉∗√
2l′ + 1, (48)
vanishes unless
mA + k′A = 0 (49)
lA = l′. (50)
Combining these conditions, we obtain kA = −k′A and l = l′. Therefore, all cross terms of
different angular momenta, l 6= l′ in eqn (44), will vanish. Furthermore, the Clebsch-Gordoncoefficients give [24]
〈lAl; mA,−kA|lAl; 0A0A〉 = 〈lAl′; mA, k′A|lAl′; 0A0A〉 = 〈ll; mA,−mA|ll; 00〉 = (−1)l−mA /
√2l + 1.
(51)
There will be no kA dependence of the quantity
(−1)kA〈0A|QAl,−kA
|nA〉〈nA|QAl′,−k′
A
|0A〉. (52)
We write eqn (44) as
E(2)ind,B→A = −
∑
nA
′
∞∑
l=1
l∑
kA,kB,k′
B=−l
(−1)kA(−1)kA〈0A|QAl,−kA
|nA〉〈nA|QAl,kA
|0A〉F Bl,kB
F Bl,k′
B
D(l)kBkA
(ξ−1B ξA)D
(l)k′
B,−kA
(ξ−1B ξA)/ (EnA
− E0A)
= −∑
nA
′
∞∑
l=1
l∑
kA,kB,k′
B=−l
(−1)kA〈0A|QAl0|nA〉〈nA|QA
l0|0A〉F Bl,kB
F Bl,k′
B
D(l)kBkA
(ξ−1B ξA)D
(l)k′
B,−kA
(ξ−1B ξA)
8
/ (EnA− E0A
)
= −1
2
∞∑
l=1
l∑
kA,kB,k′
B=−l
(−1)kAαAl0l0F
Bl,kB
F Bl,k′
B
D(l)kBkA
(ξ−1B ξA)D
(l)k′
B,−kA
(ξ−1B ξA)
= −1
2
∞∑
l=1
l∑
kA,kB,k′
B=−l
(−1)kAαAl F B
l,kBF B
l,k′
B
D(l)kBkA
(ξ−1B ξA)D
(l)k′
B,−kA
(ξ−1B ξA)
= −1
2
∞∑
l=1
l∑
kA,kB,k′
B=−l
(−1)kA+k′
B−kAαA
l F Bl,kB
F Bl,k′
B
D(l)kBkA
(ξ−1B ξA)D
(l)−k′
B,kA
(ξ−1B ξA)∗
= −1
2
∞∑
l=1
l∑
kA,kB,k′
B=−l
(−1)k′
B αAl F B
l,kBF B
l,k′
B
D(l)kBkA
(ξ−1B ξA)D
(l)kA,−k′
B
(ξ−1A ξB)
= −1
2
∞∑
l=1
l∑
kB,k′
B=−l
(−1)k′
B αAl F B
l,kBF B
l,k′
B
D(l)kB ,−k′
B
(0)
= −1
2
∞∑
l=1
l∑
kB,k′
B=−l
(−1)k′
B αAl F B
l,kBF B
l,k′
B
δk′
B,−kB
= −1
2
∞∑
l=1
l∑
kB,=−l
(−1)kB αAl F B
l,kBF B
l,−kB. (53)
Because of the orthogonality 〈0A|qA|nA〉 = qA〈0A|nA〉 = 0, the lowest-order angular momentumfor induction B → A is l = 1. The second step takes k = 0 for the quantity (52). The fourthstep introduces the average polarizability αA, defined as
αAl =
1
2l + 1
l∑
m=−l
(−1)mαAlml,−m. (54)
The fifth, sixth, and seventh steps employ the properties of Wigner matrices as in eqn (17).Therefore, the expression for the polarization of the inside atom A by the outer system B, eqn(53), has no angular dependence. This confirms the intuitive picture that, once a local axis forthe B is set up, the induction energy will not depend on the local axes of A.
The dispersion interaction contains the same polarizabilities as the induction interaction B →A. Therefore the previous symmetry analysis will still be valid. The eqns (33) and (37) become
E(2)disp = − 1
2π
∞∑
l=1
l∑
kB=−l
(−1)kB
[ ∫ ∞
0αA
l (iω) α†BlkBl,−kB
(iω) dω]
(55)
= − 1
2π
∞∑
l=1
(2l + 1)∫ ∞
0αA
l (iω) α†Bl (iω) dω (56)
≈ −1
4
IAIB
IA + IB
∞∑
l=1
l∑
kB=−l
(−1)kB αAl α†B
lkBl,−kB(57)
= −1
4
IAIB
IA + IB
∞∑
l=1
(2l + 1)αAl α†B
l . (58)
Here we introduce the average irregular polarizabilities
α†Bl (iω) ≡ 1
2l + 1
l∑
kB=−l
(−1)kBα†BlkB l,−kB
(iω), (59)
9
α†Bl ≡ 1
2l + 1
l∑
kB=−l
(−1)kBα†BlkB l,−kB
. (60)
It may be convenient to express the irregular polarizabilities in Cartesian format. In anal-ogy with the Buckingham notation, [20] we introduce the Cartesian dipole-dipole, dipole-quadrupole, and quadrupole-quadrupole irregular polarizabilities as follows:
α†Bαβ ≡
∑
nB
′ 〈0B |µα/r3|nB〉 〈nB |µβ/r3| 0B〉 + 〈0B |µβ/r3|nB〉 〈nB |µα/r3| 0B〉
EnB− E0B
(61)
A†Bα,βγ ≡
∑
nB
′〈0B |µα/r3|nB〉
⟨
nB
∣∣∣Θβγ/r
5∣∣∣ 0B
⟩
+⟨
0B
∣∣∣Θβγ/r
5∣∣∣ nB
⟩
〈nB |µα/r3| 0B〉EnB
− E0B
(62)
C†Bαβ,γδ ≡ 1
3
∑
nB
′
⟨
0B
∣∣∣Θαβ/r5
∣∣∣ nB
⟩ ⟨
nB
∣∣∣Θγδ/r
5∣∣∣ 0B
⟩
+⟨
0B
∣∣∣Θγδ/r
5∣∣∣ nB
⟩ ⟨
nB
∣∣∣Θαβ/r5
∣∣∣ 0B
⟩
EnB− E0B
.
(63)
For the sake of simplicity, the summation over all electrons,∑
j, is included implicitly in
eqns (61) - (63). The notations µα/r3 and Θαβ/r5 should be understood as∑
j qjrjα/r3j and
12
∑
j qj(3rjαrjβ − r2j δαβ)/r5
j respectively.
The corresponding averaged quantities are
α†B ≡ 1
3
∑
β
α†Bββ (64)
C†B ≡ 1
5
∑
αβ
C†Bαβ,αβ. (65)
Notice that the average quantities in spherical basis, eqn (60), only equal those in a Cartesianbasis at dipole-dipole level, eqn (64). As for the case of regular polarizabilities, [20, 25, 26] forspherical, linear, or Td-point-group-symmetry monomers B,
α†B2 = 2C†B. (66)
Eqn (66) also holds for Ih point-group symmetry, as can be seen from eqn (77).
As a result, the lowest-order dipole-dipole dispersion interaction can be written as
E(2)disp,DD = − 1
2π
∫ ∞
0αA(iω)
[
α†Bxx (iω) + α†B
yy (iω) + α†Bzz (iω)
]
dω (67)
= − 3
2π
∫ ∞
0αA(iω)α†B(iω)dω (68)
≈ −3
4
IAIB
IA + IB
αA α†B. (69)
The next-order, quadrupole-quadrupole dispersion interaction becomes
E(2)disp,QQ = − 1
2π
∫ ∞
02CA(iω)
[
3C†Bzzzz(iω) + C†B
xxxx(iω) + C†Byyyy(iω) − 2C†B
xxyy(iω) + 4C†Bxyxy(iω) + 4C†B
xzxz(iω)
10
+4C†Byzyz(iω)
]
dω (70)
= − 5
2π
∫ ∞
0αA
2 (iω)α†B2 (iω)dω (71)
≈ −5
4
IAIB
IA + IB
αA2 α†B
2 . (72)
3.2 B ∈ Ih
For a molecule at the centre of C60, the symmetry of B is Ih, the highest finite point group. Theexpectation value 〈0B|Flm(B)|0B〉 will vanish for 0 < l < 6. [27] Notice the angular similarityof the Flm with the corresponding multipole moments, Qlm. Then the electrostatic interactionfrom eqn (19) will be small. The induction from the multipoles of B on the inside monomer A,i. e. the second term on the RHS of eqn (21) is expected to be negligible. We can approximateeqn (21) by
E(2)ind ≈ −
∑
nA
′
∣∣∣∣∣∣
∞∑
l=0
l∑
kA,kB=−l
(−1)kA〈0A|QAl,−kA
|nA〉〈0B|F Bl,kB
|0B〉D(l)kBkA
(ξ−1B ξA)
∣∣∣∣∣∣
2
/ (EnA− E0A
)
(73)
= −1
2
∞∑
l,l′=0
l∑
kA,kB=−l
l′∑
k′
A,k′
B=−l′
(−1)kA+k′
A
[
QAl,−kA
QAl′,−k′
A
α†BlkB l′k′
B
]
D(l)kBkA
(ξ−1B ξA)D
(l′)k′
Bk′
A
(ξ−1B ξA).
(74)
The Ih symmetry can further simplify the structure of the irregular polarizability α†BlkBl′k′
B
. From
the definition (23), the numerator contains 〈0B|F BlkB
|nB〉〈nB|F Bl′k′
B
|0B〉+〈0B|F Bl′k′
B
|nB〉〈nB|F BlkB
|0B〉.Denoting the irreducible representation of the irregular polarizability operator as Γ(l) and not-ing that the ground state is typically Ag, the 〈0B|F B
lkBproduct gives Ag × Γ(l) = Γ(l), requiring
that the |nB〉 state belong to Γ(l) for a non-vanishing matrix element. The product 〈nB|F BlkB
will belong to Γ(l) × Γ(l′) [28], that has to span Ag for a non-vanishing irregular polarizability.Based on this analysis, we present the low-order results in Table 3. The non-vanishing terms,ordered along increasing angular momentum in eqn (74), will be the dipole-dipole, quadrupole-quadrupole, octopole-octopole, dipole-32-pole, quadrupole-hexadecapole, · · · ones. Amongthese quantities, there is only one independent variable for a dipole-dipole or quadrupole-quadrupole polarizability. We chose α†B
1010 = α†Bzz and α†B
2020 = 3C†Bzzzz as that variable. It can be
shown that the non-zero components of the irregular dipole-dipole polarizability obey
α†Bxx = α†B
yy = α†Bzz (75)
α†B1010 = −α†B
111,−1 = −α†B1,−111 (76)
since the x2 − y2, 2z2 − x2 − y2, xy, xz, and yz belong to the irreducible representation Hg .The non-zero irregular quadrupole-quadrupole polarizability obeys
C†Bxxxx = C†B
yyyy = C†Bzzzz = −1/2C†B
xxyy = −1/2C†Bxxzz = −1/2C†B
yyzz = 4/3C†Bxyxy = 4/3C†B
xzxz = 4/3C†Byzyz
(77)
α†B2020 = −α†B
212,−1 = −α†B2,−121 = α†B
222,−2 = α†B2,−222. (78)
We verify (77) numerically.
11
We then derive the explicit expressions for the dipole-dipole and quadrupole-quadrupole induc-tion and dispersion interactions. From eqn (74) we obtain
E(2)ind,DD = −1
2
1∑
kA,k′
A,kB,k′
B=−1
(−1)kA+k′
AQA1,−kA
QA1,−k′
A
α†B1kB1k′
B
D(1)kBkA
(ξ−1B ξA)D
(1)k′
Bk′
A
(ξ−1B ξA)
= −1
2
1∑
kA,k′
A,kB=−1
(−1)kA+k′
AQA1,−kA
QA1,−k′
A
α†B1kB1,−kB
D(1)kBkA
(ξ−1B ξA)D
(1)−kB,k′
A
(ξ−1B ξA)
= −1
2
1∑
kA,k′
A,kB=−1
(−1)kA+k′
A+kBQA
1,−kAQA
1,−k′
A
α†B1010D
(1)kBkA
(ξ−1B ξA)D
(1)−kB ,k′
A
(ξ−1B ξA)
= −1
2
1∑
kA,k′
A,kB=−1
(−1)kA+k′
A+kBQA
1,−kAQA
1,−k′
A
α†B1 D
(1)kBkA
(ξ−1B ξA)D
(1)−kB,k′
A
(ξ−1B ξA)
= −1
2
1∑
kA,k′
A,kB=−1
(−1)kA+k′
A+kB−kB−k′
AQA1,−kA
QA1,−k′
A
α†B1 D
(1)kBkA
(ξ−1B ξA)D
(1)kB,−k′
A
(ξ−1B ξA)∗
= −1
2
1∑
kA,k′
A,kB=−1
(−1)kAQA1,−kA
QA1,−k′
A
α†B1 D
(1)kBkA
(ξ−1B ξA)D
(1)−k′
A,kB
(ξ−1A ξB)
= −1
2
1∑
kA,k′
A=−1
(−1)kAQA1,−kA
QA1,−k′
A
α†B1 δ−k′
A,kA
= −1
2
1∑
kA=−1
(−1)kAQA1,kA
QA1,−kA
α†B1
= −1
2
[
(µAx )2 + (µA
y )2 + (µAz )2
]
α†B1
= −1
2µ2
A α†B1 . (79)
The second, third, and fourth steps use the symmetry properties of the dipole-dipole irregularpolarizability, eqn (76). As a result, the final expression, eqn (79) has no angular dependence.This is the lowest-order induction interaction for a dipolar molecule, e.g. H2O, inside C60.
The next-order contribution is the quadrupole-quadrupole induction. It arises if the insidemolecule has a non-zero quadrupole moment, as e.g. H2 or N2 do. If A has no dipole moment,the induction contribution starts from the following term
E(2)ind,QQ = −1
2
2∑
kA,k′
A,kB,k′
B=−2
(−1)kA+k′
AQA2,−kA
QA2,−k′
A
α†B2kB2k′
B
D(2)kBkA
(ξ−1B ξA)D
(2)k′
Bk′
A
(ξ−1B ξA)
= −1
2
2∑
kA=−2
(−1)kAQA2,kA
QA2,−kA
α†B2
= −1
2
[1
3(QA
xx − QAyy)2 + (QA
zz)2
]
α†B2 . (80)
(81)
The symmetry properties of the quadrupole-quadrupole polarizability, eqn (78), are used in thesecond step to eliminate the angular dependence. In the principal axis system, at the thirdstep, all off-diagonal elements of the quadrupole moment of A vanish. If the inside monomer
12
A has a rotation axis, Cn(n ≥ 3), the quadrupole moment has the additional properties Qxx =Qyy = −1/2Qzz and eqn (81) can be simplified as
E(2)ind,QQ = −1
2(QA
zz)2α†B
2 . (82)
From eqn (33), the dipole-dipole dispersion is
E(2)disp,DD = − 1
2π
1∑
kA,k′
A,kB,k′
B=−1
(−1)kA+k′
A
[ ∫ ∞
0αA
1,−kA,1,−k′
A
(iω) α†B1kB1k′
B
(iω) dω]
D(1)kBkA
(ξ−1B ξA)D
(1)k′
Bk′
A
(ξ−1B ξA)
= − 1
2π
1∑
kA=−1
(−1)kA
[ ∫ ∞
0αA
1,kA,1,−kA(iω) α†B
1 (iω) dω]
= − 3
2π
∫ ∞
0αA
1 (iω) α†B1 (iω) dω (83)
The first step here is analogous with the derivation of eqn (79). The second step uses thedefinition of the average polarizability, eqn (54). Making a London-type approximation, theeqn (83) becomes
E(2)disp,DD ≈ −3
4
IAIB
IA + IB
αA1 α†B
1 (84)
From eqn (33), the quadrupole-quadrupole dispersion is
E(2)disp,QQ = − 1
2π
2∑
kA,k′
A,kB,k′
B=−2
(−1)kA+k′
A
[ ∫ ∞
0αA
2,−kA,2,−k′
A
(iω) α†B2kB2k′
B
(iω) dω]
D(2)kBkA
(ξ−1B ξA)D
(2)k′
Bk′
A
(ξ−1B ξA)
= − 1
2π
2∑
kA=−2
(−1)kA
∫ ∞
0αA
2,kA,2,−kA(iω) α†B
2 (iω) dω
= − 5
2π
∫ ∞
0αA
2 (iω) α†B2 (iω) dω (85)
≈ −5
4
IAIB
IA + IBαA
2 α†B2 (86)
The first step is similar to the quadrupole-quadrupole induction interaction, eqn (81).
There is no angular dependence of dipole-dipole and quadrupole-quadrupole induction anddispersion interactions, eqn (79), (81), (84), and (86). These results support the experimental [2]and computational observations that H2 rotates almost freely inside C60. [4, 29]
4 Connection to London’s formula for breakable systems
B
For a system like He@C60, the α†B1 of the C60 shell can not be separated into individual con-
tributions, although it might be possible to model it in various ways. In contrast, if the outershell consists of isolated pieces, like noble-gas atoms arranged on a sphere, one expects this tobe possible. The simplest case is the dimer Ng2, where one of the atoms is thought to be the’center’ A and the other one is thought to form the ’shell’, B. Another simple case would bea linear Ng3 model, , where the end atoms together form B and the central atom forms A.The figures for these model systems are given as Supplementary Material. We shall show thisconnection and compare the results with numerical calculations.
13
For the He2 model system, the interaction energy from the London formula is [11]
E(2)disp,DD ≈ −3
4IA
[
αA1
]2R−6. (87)
In the new formula [12], eqn (69), if we regard one helium atom as the subsystem A, and theother as the subsystem B, the interaction energy is
E(2)disp,DD ≈ −3
8IAαA
1 α†B1 . (88)
In addition, since the outer-shell belongs to C∞v point group, it has a non-zero irregular dipole,F1. Assuming that it lies along z axis, from eqn (53) we obtain the dipole-dipole inductionenergy as
E(2)ind,DD = −1
2αA
1 (F B10)2, (89)
In case of the lowest-order dispersion interaction, the term (89) will only arise when the outershell B has an irregular dipole moment. At the dipole-dipole level, the one-center, eqn (88)-(89),and two-center, eqn (87), expansions are set equal
−3
4IA
[
αA1
]2R−6 = −3
8IAαA
1 α†B1 − 1
2αA
1 (F B10)
2. (90)
The LHS is the traditional London dispersion. The RHS contains the present dispersion ex-pression. We also formally kept the negligible induction-type term of the new formalism, seeTable 7. Cancelling then the common factors IA and αA
1 and adjusting the prefactor, we getthe connection equations
α†B1 = 2 αA
1 R−6 − 4
3
(F B10)2
IA. (91)
In the He3 linear model system (for a picture, see Supplementary Material) we neglect theinteraction between the two helium atoms at the ends of the system, and only consider thedipole-dipole interaction energy between the two pairs of nearest-neighbor atoms. From Lon-don’s formula we have
E(2)disp,DD ≈ −2 × 3
4IA
[
αA1
]2R−6
= −3
2IA
[
αA1
]2R−6. (92)
In this linear He3 system, eqn (88) still keeps its original form. Due to the D∞h symmetryof B, the lowest-order irregular multipole would be the quadrupole, which is not taken intoaccount. The helium atoms being equal, the ionization energy for the outside subsystem B, IB
is approximately equal to that of the inside subsystem A, or IA.
Therefore, in this He3 model, eqn (88) is simplified to
E(2)disp,DD ≈ −1
8IAαA
1 α†B1 . (93)
Setting the two energies equal
−3
2IA
[
αA1
]2R−6 = −3
8IAαA
1 α†B1 , (94)
14
cancelling the common factors IA and αA1 and adjusting the prefactor, we get the connection
equations for He3:α†B
1 = 4αA1 R−6, (95)
Comparing eqn (91) and eqn (95), we see that the contributions of the end atoms are additive.
In general, for n pairwise individual two-body dispersion interactions at a common R, if A isan atom, the connection equation is
α†B1 = 2nαA
1 R−6, (96)
provided that the term (89) is negligible.As there are many computational approaches, estimating the total dispersion energy from asum of pairwise London-like interactions, such a formula will permit expressing the effect of allneighbors, lumped together, on a given atom or group.If the polarizability contributions from all volume elements could simply be integrated to anirregular polarizability, α†B
1 , the concept could be generalized to arbitrary spherical shells,nanotubes, or holes in a three-dimensional substance, such as fat. It, however, turns out thatat least the regular polarizabilities, α1, do not always scale in this way. For fullerenes the α1
scales as the volume, not as the surface area [30]. For nanotubes, the axial polarizability isfound to scale as R/E2
g , where Eg is the HOMO-LUMO gap, while the perpendicular componentscales as R2 [31].If such an integration were possible, and the thin outer shell would have a polarizability densityρ = dα/da = nα/4πR2, where da is a surface element, we would get for a quasispherical B,
α†B1 = 8πρR−4. (97)
For a nanotube of radius R and the same density ρ,
α†B1 =
3
2π2ρR−4. (98)
For the same R, the sphere/tube ratio becomes 16/3π. For the derivation, see SupplementaryMaterial.
5 Off-center effect
It is known that, at equilibrium, Li+@C60 does not have the cation at the center of C60. [22]More generally, the interaction energies for various off-center positions are needed for moleculardynamics simulations. [10] Therefore we consider the case shown in Fig. 2.Here ∆r is the off-center deviation. We assume the z-axis to be a symmetry axis. Accordingto Fig. 2,
r′j =√
r2j − 2 rj cosθj ∆r + (∆r) 2 (99)
First we consider the off-center effect on a point-charge induction interaction. Via the geometricrelationship (99), we perform a series expansion for the off-center distance ∆r up to secondorder, assuming ∆r < rj, giving
1
r′j=
1
rj+
cosθj
r2j
∆r +3 cos2θj − 1
2r3j
(∆r) 2 + O[
(∆r)3]
=1
rj+
zj
r3j
∆r +3z2
j − r2j
2r5j
(∆r) 2 + O[
(∆r)3]
. (100)
15
Inserting eqn (100) into the Born-like equation (43) and noticing that the cross term 〈0B|1/rj|nB〉〈nB|zj∆r/r2
j |0B〉∆r vanishes due to the mirror symmetry of C60, we have
E(2)ind = −1
2q2Aα†B′
0000
α†B′
0000 = α†B0000 + (α†B
1010 + 2α†B0020) (∆r) 2 + O
[
(∆r)4]
.
(101)
This shows that, when the inner charged subsystem approaches the shell B, the absolute valueof the induction energy increases as (∆r) 2. In addition, when A approaches the shell, its Paulirepulsion with B increases. Adding a Buckingham potential for the repulsive part, we have ageneral form of the point-charge endohedral off-center interaction energy
E(2) ≈ A + B (∆r) 2 + Ce−α(∆r−R). (102)
Here A ≡ −12q2A α†B
0000, B ≡ −12q2Aα†B
1010 − q2Aα†B
0020, C and α are the parameters for the Paulirepulsion, R is the radius of the outer-shell.
We then consider the dipole-dipole induction and dispersion interactions. Since the off-centereffect is on the α†B
1 , we will consider the term r′jk/r
′3j . If the off-center deviation occurs along
the z axis, we have x′ = x, y′ = y, and z′ = z − ∆r. In addition
1
r′3j=
1
r3j
+3zj
r5j
∆r +15z2
j − 3r2j
2r7j
(∆r) 2 + O[
(∆r)3]
, (103)
whence
α†B′
xx = α†Bxx + 12C†B
xzxz∆r2 + 20∑
nB
′ 〈0B|xj/r3j |nB〉〈nB|xj Qzz(B)/r7
j |0B〉EnB
− E0B
(∆r)2
+ 4∑
nB
′ 〈0B|xj/r3j |nB〉〈nB|xj/r
5j |0B〉
EnB− E0B
(∆r)2 + 2∑
nB
′ |〈0B|1/r3j |nB〉|2
EnB− E0B
(∆r)2 + O[
(∆r)4]
(104)
α†B′
yy = α†B′
xx (105)
α†B′
zz = α†B′
xx − 6∑
nB
′ 〈0B|z2j /r
5j |nB〉〈nB|1/r3
j |0B〉EnB
− E0B
(∆r)2 − 6∑
nB
′ 〈0B|1/r3j |nB〉〈nB|z2
j /r5j |0B〉
EnB− E0B
(∆r)2
−6∑
nB
′ 〈0B|zj/r3j |nB〉〈nB|zj/r
5j |0B〉
EnB− E0B
(∆r)2 − 6∑
nB
′ 〈0B|zj/r5j |nB〉〈nB|zj/r
3j |0B〉
EnB− E0B
(∆r)2
+O[
(∆r)4]
. (106)
Therefore, the off-center dipole-dipole induction and dispersion interactions have the samefunctional form, A + B (∆r)2. After adding a Buckingham potential for the repulsive part, theresult is
E(2) ≈ A + B (∆r)2 + Ce−α(∆r−R). (107)
If ∆r ≪ R, a polynomial expansion is possible, as used in the study of the endohedral potentialfor Xe inside C60. [10]
Notice also that inside a cage one can end up with covalent bonding, especially for (monoatomicor composite) transition metal systems, such as Sc3C2@C80. [32] In such cases the currentapproximation will fail.
16
6 Numerical results
6.1 Errata
To begin with, we correct the following errors in our first paper [12] :(1) The factor 3/4 in eqn (12) of that paper (the present eqn (84)) was missing from all numericalresults, including the figures.(2) The unit for α−2 in Table 1 was in 10−3 a.u.(3) The RI-MP2/def2-TZVPP interaction energy for Ne@C60 should be -14.1 kJ mol−1.
6.2 Computational Methods
The supramolecular calculations are performed at MP2 and SCS-MP2 levels using a resolution-of-identity (RI) approximation [33–36] and the Turbomole program package, version 6.0. [37] Ina benchmark test on the Xe· · ·C6H6 interaction, the SCS-MP2 yields a comparable result witha CCSD(T) benchmark [10]. We hence adopt the SCS-MP2 method to model the higher-ordercorrelation beyond MP2. However, for He· · ·C6H6, SCS-MP2 is worse than MP2 comparedwith CCSD(T). [38] Therefore the accuracy of SCS-MP2 is calibrated against CCSD(T) levelusing the MOLPRO program package, version 2006. [39] The SAPT calculations are performedby the MOLPRO program package, version 2006. [39]
The Ahlrichs basis sets, def-SV, def-SVP, [40] def-TZVP, [34] def2-SVP, def2-TZVPP, anddef2-QZVPP, [41] the Dunning correlation-consistent basis sets cc-pVXZ, (X=D,T,Q,5,6) aug-cc-pVXZ, (X=D,T,Q,5,6), and d-aug-cc-pVXZ (X=D,T,Q,5) [42–46], and the Sadlej pVTZbasis set [47] are employed. For Xe, a 26-VE ECP is used. [46] For Cd and Hg, 20-VE ECP areused. [48] The RI-MP2 basis-set limit is estimated by a two-point extrapolation scheme. [49]Since no rigorous derivation is yet known for this scheme, [50] its validity is tested on A · · ·C6H6
model systems. The counterpoise correction [51] for the basis-set superposition error (BSSE) isdone in all supramolecular calculations. The core electrons are frozen in all post-Hartree-Fockcalculations.The ’irregular polarizabilities’ α†
0, α†1 and α†
2 of the C60 and Hen systems were evaluated bythe Dalton program package, version 2.0 [52] at Hartree-Fock, BLYP, and B3LYP [53–56]levels. The response calculations were done using the NUCPOT, NELFLD and EFGCARintegrals. (The factor ’1/3’ in front of the EFGCAR integral in the Dalton manual is missingfrom the program.) As the properties in question have not been studied previously, their basisset convergence and method dependence were checked when affordable. For the basis sets, seeabove. The thresholds and convergence parameters in the Dalton program were carefully testedto avoid numerical noise.The normal, dipole polarizability α1 is also evaluated at Hartree-Fock and B3LYP levels forcomparison. All calculations were done in ’gas phase’, without any solvent effects.
6.3 Calibration
In this section we first calibrate the supramolecular calculations, including the convergence fordifferent basis sets and the accuracy of the SCS-MP2 approach. The reason is that SCS-MP2offers an affordable level to explore the correlation effect beyond MP2. It includes an empiricalcorrection for different spin-components. It is hence desirable to compare the SCS-MP2 resultswith CCSD(T), to check the accuracy, at least for the adopted model system A · · · C6H6.
17
Moreover, we try to examine the basis-set effects on the new polarizabilities α†0, α†
1, and α†2.
We consider the model system He3 and the real-world C60.
6.3.1 MP2 Basis-set limit for A · · · C6H6 (A=He,Xe) complexes
In order to analyze the convergence behavior and the validity of the basis-set extrapolation, weconsider the A · · · C6H6 model complex with A =He or Xe which is above the center of benzenering. The distance from A to the ring midpoint is fixed at 4 A. For benzene, the experimentalgeometry [57] is used. A figure for the structure is in the supplement material.
The results are given in Table 4. Although the def2- series basis sets lack diffuse functions, anextrapolation from def2-TZVPP and def2-QZVPP levels compensates for this limitation, andprovides a pragmatic way to estimate the MP2 basis set limit. Besides, the aug-cc-pVTZ basisset also gives a fairly accurate value, comparing with the basis-set limit. We expect that thespin-component scaling treatment will not change the convergence behavior with respect to thebasis set.
6.3.2 Performance of SCS-MP2
We chose the aug-cc-pVTZ and aug-cc-pVQZ basis sets for comparing the SCS-MP2 approachagainst CCSD(T) on the A · · · C6H6 model complexes at 4 A intermolecular distance. Accordingto the computational results in Table 5, the SCS-MP2 does not improve the accuracy of MP2for the two lightest elements helium and neon, but does so in the other cases A = Ar-Xe andZn-Hg. It was shown earlier for A · · · C6H6 complexes with A = H2O, NH3, and CH4 that MP2gives better results than SCS-MP2, as compared with CCSD(T). As a remedy, different scalingparameters were suggested. [58]
6.3.3 Numerical results on the He2 and He3 systems
The He2 and He3 systems are the simplest prototype of intermolecular interaction. These sys-tems can provide a numerical test for the connection between eqn (69) and London’s formula.They also can be used to illustrate the basis-set effects on α†
1. The structures of the He2 andHe3 systems are given in the supplement material.
In the He3 symmetric linear system, the middle helium atom is chosen as the inner subsystemA, while the two end helium atoms form the outer subsystem B. The z axis is chosen along thethree nuclei. The zz components of α†
1 for subsystem B, calculated with different basis sets,are presented in Table 6.
According to the data in Table 6, the def-TZVP basis set can only give about 1/4 of the valueof α†
1, compared with the much larger aug-cc-pV6Z basis set, while the aug-cc-pVDZ basis setyields rather converged results for this system.
The eqs (91) and (95) form the connection between eqn (69) and London’s approach. Tables 7and 8 present a numerical test for these connections. According to these data, the connectionequation is indeed valid. The additional term from eqn (89)-(91) is negligible.
In addition, we compare the London, α†1-based, MP2, and SAPT calculations in Tables 9, 10,
11, and 12 respectively. All these approaches qualitatively agree with each other. The London-like formula (69) remains 12% difference with the dispersion interaction from SAPT at 7 A.
18
This may regarded as the intrinsic error of this simple formula. At 2 A, the deviation enlargesto 36% which indicates the limitation of multipole expansion.
6.3.4 Basis-set effect of α†0, α†
1, and α†2 for C60.
Here we attempt to test the basis set effect using C60 as example. Since this system is muchlarger than He2 and He3, the calibration is limited to def-SV, def-SVP, def-TZVP, and aug-cc-pVTZ basis sets. The reason to adopt the Sadlej basis set is that it is optimized for electricproperties, i.e. the dipole moment and the normal polarizability α1. The calculated results arein Table 13. Now the new polarizabilities α†
0, α†1, and α†
2 do not depend on diffuse functions,in contrast to the helium systems.
6.4 Point charges inside C60
For point charges, q, at the center of C60, the energy as function of q is listed in Table 14and Fig. 3. The line is calculated from eqn (42) + (43) , using a calculated α†
0 of 0.0923 a.u.at the BLYP/Sadlej-pVTZ level. The result suggests that the interaction energies are indeedproportional to a linear and a quadratic term (q and q2, respectively), as given by eqns (42)and (43). Note that the solvation energy for a charge at the center of B is independent ofits sign. Cations and anions should have the same value, if other things are constant. Note,furthermore, that also a neutral ’B’ can give a non-vanishing VB at the origin.
6.5 Atoms inside C60
This is our perhaps most interesting key result, and it turns out to be a half-success. Themonomer data used are given in Table 15. In Table 16 we report the RI-MP2 and RI-SCS-MP2basis-set-limit values, estimated from an extrapolation from def2-TZVPP and def2-QZVPP inTable 16. These results are compared with the London-type formula, eqn (37) in Table 16 andFig. 4. As an order-of-magnitude estimate, the new expression is potentially useful. It should,however, be noticed that eqn (69) is derived as an estimate for the attractive, l = 1 dispersioncontribution, while the quantum chemical (QC) reference values include all contributions, no-tably the Pauli repulsion and the remaining part of the interelectron interaction (beyond thel = 1 terms of the expansion (3). The QC calculation also makes no effective-excitation-energyapproximation of the London type. If the omitted terms would be small, or cancel, eqn (69)would be ’right for the right reason’. It is seen, however, that the HF-level interaction energy,used as an estimate for the Pauli repulsion, is not small. Moreover, the calculated l = 2 dis-persion from the present approach, based on eqn (72), is smaller, but not always much smaller,than the l = 1 term. Therefore our new endohedral London-like formula may only be useful asa rough semiempirical estimate.As to earlier work, Lo et al. [59] used a dynamical screening approach and obtained for Ar@C60
an interaction energy of ca. 12 kJ mol−1, which is clearly below both our calculated and esti-mated values. For Ne@C60 Slanina et al. [3] obtain about -7.7 kJ mol−1 at MP2/6-31g**+BSSElevel, only one-third of our extrapolated MP2 value. For Xe@C60, an SCS-MP2 interaction en-ergy of ca -63 kJ/mol was reported in our previous work. [10] It was also noticed that the MP2interaction energy in Xe@C60 converged extremely slowly with their basis-set size.
19
6.6 Molecules inside C60
For A@C60 systems, calculated by us, the total structure was first optimized at the RI-MP2/def2-SVP level. In their optimized structures, the inside monomer always preferred thecenter of C60, as opposed to the walls. The largest deviation appears for the LiF molecule,where the center-of-mass of LiF is 7.6 pm away from the center of C60, only 2% of the radiusof the outer shell.The A@B interaction energies for molecules as the inside system A are presented in Table 17.For the new formulae, monomer data from Table 15 are used. For H2 the ’Disp’ estimate isabout half of the calculated MP2 values, which in turn are comparable with the SAPT value ofKorona et al. [4]. The best MP2 value of Slanina et al. [3] of -29 kJ mol−1 is close to the Hes-selmann [4] one of -30.7. For N2 our new formula is comparable with SCS-MP2 but about halfof MP2. Both values are close to the 6-311g** ones of -36 and -65 kJ mol−1 by Slanina et al. [3].
The ’Steric repulsion’ entry in Table 17 is simply the Hartree-Fock-level interaction, minusthe estimated multipolar induction. Our largest steric repulsion term of +85.8 kJ mol−1 forCH4@C60 is comparable with the HF/6-31G value of 73.6 kJ mol−1 by Shameema et al. [60] andsomewhat larger than the DFT values by Charkin et al. [61] (62.4 kJ mol−1, B3LYP/6-31G*)or Rehaman et al. [62] (49.4 kJ mol−1, BP86/SVP). Despite of this effective compression, theeffect on the C-H distance is 0.2 pm or less.Among the lowest-order non-vanishing contributions, dispersion is the dominant intermolecu-lar interaction. The simple formalism gets a roughly comparable value to a supramolecularcalculation.For dipolar molecules, the origin for the quadrupole moment was taken as the center-of-mass[63].
6.7 Off-center effect
Here we calculated the single-point energies at RI-MP2/def2-TZVPP level for He@C60 where thehelium atom is located at various off-center positions. The Pauli repulsion energy is regarded asthe interaction energy at Hartree-Fock level, the dispersion energy is subtracted as EMP2−EHF.From the data in the Table 18 and the fitting curve in Fig. 5, the Pauli repulsion presents aexponential growth towards the outer shell, the absolute value of dispersion interaction alsogoes as (∆r)2 where ∆r is the off-center distance, which has the samefunctional form as eqn(107).
7 Summary
We have shown how to express the interaction energy in endohedral A@B systems via a one-center expansion. This allows us, in principle, to relate this interaction energy to propertiesof the monomers A and B. The electrostatic, induction, and dispersion contributions werederived using Rayleigh-Schrodinger perturbation theory. The dispersion term was also derivedfrom a Casimir-Polder-type argument. New ’irregular’ electric polarizabilities, α†
l , with negativepowers of the radius operator, r, were defined and calculated for the outer subsystem B. For’breakable’ B, they can also be expressed in terms of the polarizabilities of the pieces. Thiswas done for He2 and He3 model systems. The results from the new approach were comparedwith supramolecular calculations. It was found that:(1) The α†
0-based, Born-like formula for ’solvation’ of a charged species in the center of theouter system B performs adequately.
20
(2) The α†1-based, London-like formula for the basically dispersion-induced interaction energy
is a useful rough estimate.(3) The l = 2 dispersion term is smaller, but not always much smaller than the l = 1 one.Therefore the present approach may benefit from further testing.
8 Acknowledgements
We thank Juha Vaara and Bin Gao for technical help with the Dalton program package. Weare grateful to Reinhart Ahlrichs, George Maroulis and Paul Wormer for correspondence aswell as Dage Sundholm for discussions. The referee report greatly helped to improve thepaper, and introduced the term ’irregular multipole operator’. CW and PP belong to theFinnish Center of Excellence in Computational Molecular Science (CMS). CW is supportedby the Magnus Ehrnrooth Foundation. Computational resources were partially provided bythe Center for Scientific Computing (CSC), Espoo, Finland. MS was supported by the CzechScience Foundation (grant No. 203/09/2037) and by the European Reintegration Grant (No.230955) within the 7th European Community Framework Program.
References
[1] T. B. Lee and M. L. McKee, J. Am. Chem. Soc., 2008, 130, 17610–17619.
[2] Y. Kohama, T. Rachi, J. Jing, Z.-F. Li, J. Tang, R. Kumashiro, S. Izumisawa, H. Kawaji,T. Atake, H. Sawa, Y. Murata, K. Komatsu, and K. Tanigaki, Phys. Rev. Lett., 2009, 103,073001.
[3] Z. Slanina, P. Pulay, and S. Nagase, J. Chem. Theory Comput., 2006, 2, 782–785.
[4] T. Korona, A. Hesselmann, and H. Dodziuk, J. Chem. Theory Comput., 2009, 5, 1585–1596.
[5] S. Osuna, M. Swart, and M. Sola, Chem. Eur. J., 2009, 15, 13111–13213.
[6] I. G. Kaplan, Intermolecular Interactions: Physical Picture, Computational Methods andModel Potentials, Wiley, Chichester, 2006.
[7] A. J. Stone, The Intermolecular Forces, Oxford University Press, Oxford, 1996.
[8] K. Morokuma, J. Chem. Phys., 1971, 55, 1236–1244.
[9] B. Jeziorski, R. Moszynski, and K. Szalewicz, Chem. Rev., 1994, 94, 1887–1930.
[10] M. Straka, P. Lantto, and J. Vaara, J. Phys. Chem. A, 2008, 112, 2658–2668.
[11] F. London, Z. Physik, 1930, 63, 245–279.
[12] P. Pyykko, C. Wang, M. Straka, and J. Vaara, Phys. Chem. Chem. Phys., 2007, 9, 2954–2958.
[13] S. Grimme, J. Chem. Phys., 2003, 118, 9095–9102.
[14] R. Ahlrichs, Theo. Chem. Acc., 1976, 41, 7–15.
21
[15] Reference [7], p. 41.
[16] M. E. Rose, Elementary Theory of Angular Momentum, John-Wiley & Sons, New York,1957, p. 73.
[17] J. J. Sakurai, Modern Quantum Mechanics, Addison-Wesley, New York, revised ed., 1994,chapter 5, p. 194.
[18] H. B. G. Casimir and D. Polder, Phys. Rev., 1948, 73, 360–372.
[19] A. Jiemchooroj, P. Norman, and B. E. Sernelius, J. Chem. Phys., 2005, 123, 124312–1–124312–6.
[20] A. D. Buckingham, Adv. Chem. Phys., 1967, 12, 107–143.
[21] J. J. Sakurai, Modern Quantum Mechanics, Addison-Wesley, New York, revised ed., 1994,chapter 5, p. 285.
[22] C. N. Ramachandrana, D. Royb, and N. Sathyamurthy, Chem. Phys. Lett., 2008, 461,87–92.
[23] P. W. Atkins and J. de Paula, Physical Chemistry, Oxford University Press, Oxford, 8thed., 2006, p. 102.
[24] Reference [6], p. 88.
[25] G. Maroulis and A. J. Thakkar, Chem. Phys. Lett., 1989, 156, 87–90.
[26] H. Hettema, P. E. S. Wormer, and A. J. Thakkar, Molec. Phys., 1993, 80, 533–548.
[27] Reference [6], p. 34.
[28] Reference [7], p. 30.
[29] R. J. Cross, J. Phys. Chem. A, 2001, 105, 6943–6944.
[30] G. F. Bertsch, A. Bulgac, D. Tomanek, and Y. Wang, Phys. Rev. Lett., 1991, 67, 2690–2693.
[31] L. Benedict, S. G. Louie, and M. L. Cohen, Phys. Rev. B, 1995, 52, 8541–8549.
[32] S. Taubert, M. Straka, T. O. Pennanen, D. Sundholm, and J. Vaara, Phys. Chem. Chem.Phys., 2008, 10, 7158–7168.
[33] F. Weigend and M. Haser, Theor. Chem. Acc., 1997, 97, 331–340.
[34] A. Schafer, C. Huber, and R. Ahlrichs, J. Chem. Phys., 1994, 100, 5829–5835.
[35] F. Weigend, A. Kohn, and C. Hattig, J. Chem. Phys., 2002, 116, 3175–3183.
[36] C. Hattig, Phys. Chem. Chem. Phys., 2005, 7, 59–66.
[37] R. Ahlrichs, M. Bar, M. Haser, H. Horn, and C. Kolmel, Chem. Phys. Lett., 1989, 162,165–169.
[38] M. Straka, Unpublished results.
22
[39] H. J. Werner, P. J. Knowles, R. Lindh, F. R. Manby, M. Schutz, P. Celani, T. Korona,G. Rauhut, R. D. Amos, A. Bernhardsson, A. Berning, D. L. Cooper, M. J. O. Deegan, A. J.Dobbyn, F. Eckert, C. Hampel, G. Hetzer, A. W. Lloyd, S. J. McNicholas, W. Meyer, M. E.Mura, A. Nicklass, P. Palmieri, R. Pitzer, U. Schumann, H. Stoll, A. J. Stone, R. Tarroni,and T. Thorsteinsson, MOLPRO, version 2006.1, a package of ab initio programs,, 2006.
[40] A. Schafer, H. Horn, and R. Ahlrichs, J. Chem. Phys., 1992, 97, 2571–2577.
[41] F. Weigend and R. Ahlrichs, Phys. Chem. Chem. Phys., 2005, 7, 3297–3305.
[42] T. H. Dunning, J. Chem. Phys., 1989, 90, 1007–1023.
[43] D. E. Woon and T. H. Dunning, J. Chem. Phys., 1994, 100, 2975–2988.
[44] D. E. Woon and T. H. Dunning, Mol. Phys., 1999, 99, 529–547.
[45] A. K. Wilson, T. van Mourik, and T. H. Dunning, J. Mol. Struct. (Theochem), 1997, 388,339–349.
[46] K. A. Peterson, D. Figgen, E. Goll, H. Stoll, and M. Dolg, J. Chem. Phys., 2003, 119,11113–11123.
[47] A. J. Sadlej, Collec. Czech. Chem. Commun., 1998, 53, 1995–2016.
[48] D. Andrae, U. Haussermann, M. Dolg, H. Stoll, and H. Preuss, Theor. Chim. Acta, 1990,77, 123–141.
[49] A. Halkier, T. Helgaker, P. Jørgensen, W. Klopper, H. Koch, J. Olsen, and A. K. Wilson,Chem. Phys. Lett., 1998, 286, 243–252.
[50] W. Kutzelnigg, Phys. Chem. Chem. Phys., 2008, 10, 3460–3468.
[51] S. F. Boys and F. Bernardi, Mol. Phys., 1970, 19, 553–566.
[52] T. Helgaker, H. J. A. Jensen, P. Jørgensen, J. Olsen, K. Ruud, H. Agren, A. A. Auer,K. L. Bak, V. Bakken, O. Christiansen, S. Coriani, P. Dahle, E. K. Dalskov, T. Enevold-sen, B. Fernandez, C. Hattig, K. Hald, A. Halkier, H. Heiberg, H. Hettema, D. Jonsson,S. Kirpekar, R. Kobayashi, H. Koch, K. V. Mikkelsen, P. Norman, M. J. Packer, T. B.Pedersen, T. A. Ruden, A. Sanchez, T. Saue, S. P. A. Sauer, B. Schimmelpfennig, K. O.Sylvester-Hvid, P. R. Taylor, and O. Vahtras, DALTON, a molecular electronic structureprogram, Release 2 0, http://www.kjemi.uio.no/software/dalton/dalton.html, 2005.
[53] A. D. Becke, Phys. Rev. A, 2000, 38, 3098–3100.
[54] S. H. Vosko, L. Wilk, and M. Nusair, Can. J. Phys., 1980, 58, 1200–1210.
[55] C. Lee, W. Yang, and R. G. Parr, Phys. Rev. B, 1988, 37, 785–789.
[56] P. J. Stephens, F. J. Devlin, C. F. Chabalowski, and M. J. Frisch, J. Phys. Chem., 1994,98, 11623–11627.
[57] D. R. Lide, CRC Handbook of Chemistry and Physics, CRC Press, Boca Raton, FL, 80thed., 1999.
[58] J. G. Hill and J. A. Platts, J. Chem. Theory Comput., 2007, 3, 80–85.
23
[59] S. Lo, A. V. Korol, and A. V. Solov’yov, Phys. Rev. A, 2009, 79, 063201–1–063201–10.
[60] O. Shameema, C. N. Ramachandran, and N. Sathyamurthy, J. Phys. Chem. A, 2006, 110,2–4.
[61] O. P. Charkin, N. M. Klimenko, D. O. Charkin, and A. M. Mebel’, Zh. Neorg. Khim.,2004, 49, 792–803.
[62] A. Rehaman, L. Gagliardi, and P. Pyykko, Int. J. Quantum Chem., 2007, 107, 1162–1169.
[63] Reference [6], p. 29.
[64] Reference [6], p. 27.
[65] Reference [6], p. 32.
[66] A. Gelessus, W. Thiel, and W. Weber, J. Chem. Educ., 1995, 72, 505–508.
[67] S. L. Altmann and P. Herzig, Point-Group Theory Tables, Clarendon Press, Oxford, 1994,p. 693.
[68] G. Cha lasinski and M. M. Szczesniak, Mol. Phys., 1988, 63, 205–224.
[69] D. Kolb, W. R. Johnson, and P. Shorer, Phys. Rev. A, 1982, 26, 19–31.
[70] A. Hesselmann, G. Jansen, and M. Schutz, J. Chem. Phys., 2005, 122, 014103–01–014103–17.
[71] S. Kar and Y. K. Ho, Phys. Rev. A, 2009, 80, 062511–1–062511–5.
[72] A. Nicklass, M. Dolg, H. Stoll, and H. Preuss, J. Chem. Phys., 1995, 102, 8942–8952.
[73] D. Goebel, U. Hohm, and G. Maroulis, Phys. Rev. A, 1996, 54, 1973–1978.
[74] A. D. Buckingham and J. E. Cordle, Mol. Phys., 1974, 28, 1037–1047.
[75] D. M. Bishop and J. Pipin, Int. J. Quantum Chem., 1993, 45, 349–361.
[76] C. Graham, D. A. Imrie, and R. E. Raab, Mol. Phys., 1998, 93, 49–56.
[77] G. Maroulis, J. Chem. Phys., 2002, 118, 2673–2687.
[78] K. H. Hellwege and A. M. Hellwege, Molecular Constants from Microwave, MolecularBeam, and Electron Spin Resonance, Springer-Verlag, Berlin, 1974.
[79] K. Hald, F. Pawlowski, P. Jørgensen, and C. Hattig, J. Chem. Phys., 2003, 118, 1292–1300.
[80] G. Maroulis, J. Mol. Struct. (Theochem), 2003, 633, 177–197.
[81] P. E. S. Wormer and H. Hettema, J. Chem. Phys., 1992, 97, 5592–5606.
[82] G. Maroulis, Chem. Phys. Lett., 1994, 226, 420–426.
[83] J. Berkowitz, H. A. Tasman, and W. A. Chupka, J. Chem. Phys., 1962, 36, 2170–2179.
[84] D. E. Stogryn and A. P. Stogryn, Mol. Phys., 1966, 11, 371–393.
24
[85] O. P. Andrade, O. A. V. Amaral, T. L. Fonseca, and M. A. Castro, Molec. Phys., 2002,100, 1975–1979.
[86] O. V. Boltalina, I. N. Ioffe, L. N. Sidorov, G. Seifert, and K. Vietze, J. Am. Chem. Soc.,2000, 122, 9745–9749.
[87] P. J.Mohr, B. N. Taylor, and D. B. Newell, Rev. Mod. Phys., 2006, 80, 633–730.
25
Tables and Figures
Table 1: Multipole-moment operators Q [64] and F in Cartesian coordinates.
Case l Notation Name Cartesian expression
Qlm 0 q Charge∑
a qa
1 µk Dipole moment∑
a qarak
2 Qkl Quadrupole 12
∑
a qa (3rakral − r2aδkl)
3 Ωklm Octupole 12
∑
a qa [5rakralram − r2a (rakδlm + ralδkm + ramδkl)]
Flm 0∑
b qb/rb
1∑
b qbrbk/r3b
2 12
∑
b qb (3rbkrbl − r2bδkl) /r5
b
3 12
∑
b qb [5rbkrblrbm − r2b (rbkδlm + rblδkm + rbmδkl)] /r
7b
Table 2: Conversion of multipole moment operators Q [65] and F between spherical and Carte-sian coordinates.
Spherical Cartesian
Q00 = q
Q10 = µz
Q1±1 = ∓√
12
(µx ± iµy)
Q20 = Qzz
Q2±1 = ∓√
23
(
Qxz ± iQyz
)
Q2±2 =√
16
(
Qxx − Qyy ± 2iQxy
)
F00 =∑
b qb/rb
F10 =∑
b qbzb/r3b
F20 = 12
∑
b qb(3z2b − r2
b )/r5b
26
Table 3: Symmetry properties of irregular multipoles in the Ih point group. Here Γ(l) is definedas the representation, spanned by the multipoles Ylm, ∀m. For the direct product between thesame angular momentum, the symmetric product is used as in the square bracket.
Multipole Irrep.Γ(1) T1u
Γ(2) Hg
Γ(3) T2u + Gu
Γ(4) Gg + Hg
Γ(5) T1u + T2u + Hu [66]
[Γ(1) × Γ(1)] Ag + Hg
Γ(1) × Γ(2) T1u + T2u + Gu + Hu
[Γ(2) × Γ(2)] Ag + Gg + 2Hg
Γ(1) × Γ(3) T2g + 2Gg + 2Hg
Γ(1) × Γ(4) T1u + 2T2u + 2Gu + 2Hu
Γ(2) × Γ(3) 2T1u + 2T2u + 2Gu + 3Hu [67][Γ(3) × Γ(3)] 2Ag + T1g + 2Gg + 3Hg
Γ(1) × Γ(5) Ag + 2T1g + T2g + 2Gg + 3Hg
Γ(2) × Γ(4) Ag + 2T1g + 2T2g + 3Gg + 4Hg
Table 4: RI-MP2 interaction energies (kJ mol−1) for A · · ·C6H6 (A=He,Xe) complexes usingdifferent basis sets. The extrapolation procedure is explained in section 6.3.1.
Basis Set AHe Xe
def2-SVP -0.13 -5.6def2-TZVPP -0.25 -7.7def2-QZVPP -0.39 -9.0Extrapolation (TQ) -0.47 -9.9
cc-pVDZ -0.08 -2.3cc-pVTZ -0.23 -6.8cc-pVQZ -0.36 -8.6cc-pV5Z -0.44 -9.3cc-pV6Z -0.48 -
aug-cc-pVDZ -0.45 -7.0aug-cc-pVTZ -0.50 -8.9aug-cc-pVQZ -0.51 -9.5aug-cc-pV5Z -0.52 -9.7
d-aug-cc-pVDZ -0.46 -7.3a
d-aug-cc-pVTZ -0.51 -9.2a
d-aug-cc-pVQZ -0.50 -9.6a
d-aug-cc-pV5Z - -9.7a
a d-aug-cc-pVXZ for C and H, aug-cc-pVXZ for Xe.
27
Table 5: Comparison of MP2 and SCS-MP2 interaction energies (kJ mol−1) with CCSD(T)ones for A · · ·C6H6 complexes using aug-cc-pVTZ and aug-cc-pVQZ basis sets.
Method AHe Ne Ar Kr Xe Zn Cd Hg
MP2/aug-cc-pVTZ -0.50 -0.98 -4.3 -6.1 -8.6 -9.8 -12.4 -12.8MP2/aug-cc-pVQZ -0.51 -1.04 -4.4 -6.3 -9.0 -10.1 -12.8 -13.1
SCS-MP2/aug-cc-pVTZ -0.37 -0.72 -3.1 -4.4 -6.0 -7.2 -9.1 -9.5SCS-MP2/aug-cc-pVQZ -0.38 -0.77 -3.2 -4.6 -6.4 -7.4 -9.4 -9.8
CCSD(T)/aug-cc-pVTZ -0.49 -0.94 -3.4 -4.6 -6.3 -6.6 -7.9 -7.8CCSD(T)/aug-cc-pVQZ -0.50 -1.00 -3.5 -4.8 -6.6 -6.8 -8.1 -8.1
Table 6: α†xx, α†
yy, α†zz, and α† for a ’B’ consisting of the outer two helium atoms of the
symmetric He3 system at RHe−He = 5 A at B3LYP level.
Basis set def-TZVP aug-cc-pVDZ aug-cc-pVTZ aug-cc-pVQZ aug-cc-pV6Z
α†xx = α†
yy /10−5 a.u. 0.124 0.400 0.428 0.440 0.449
α†zz/10−5 a.u. 0.497 1.60 1.70 1.74 1.77
α†/10−5 a.u. 0.248 0.80 0.852 0.873 0.889
Table 7: The connection between the London-type endohedral eqn (91) and London’s approachfor the He2 system. The aug-cc-pVTZ basis set is used in calculation.
R/A 2αA1 R−6 /10−3 a.u. −4(F B
10)2/3IA /10−3 a.u. α†B1 /10−3 a.u.
Exp HF B3LYP HF B3LYP HF B3LYP2.0 0.95 0.90 1.01 -0.00006 -0.0012 1.02 1.07
3.0 0.0833 0.0793 0.0887 0a 0a 0.0853 0.0943
4.0 0.0148 0.0141 0.0158 0a 0a 0.0147 0.0165
5.0 0.00390 0.00370 0.00413 0a 0a 0.00380 0.00427
6.0 0.00130 0.00124 0.00139 0a 0a 0.00126 0.00142
7.0 0.000517 0.000490 0.000550 0a 0a 0.000497 0.000560
a absolute value less than 10−9 a.u.
28
Table 8: The connection between eqn (95) and London’s approach in He3 system. The aug-cc-pVTZ basis set is used in calculation.
R/A 4αA1 R−6 /10−3 a.u. α†B
1 /10−3 a.u.Exp HF B3LYP HF B3LYP
2.0 1.90 1.81 2.02 2.06 2.14
3.0 0.167 0.159 0.178 0.171 0.189
4.0 0.0297 0.0283 0.0316 0.0295 0.0329
5.0 0.00777 0.00740 0.0083 0.00760 0.00853
6.0 0.00260 0.00248 0.00278 0.00253 0.00283
7.0 0.00103 0.00098 0.00110 0.000997 0.00112
Table 9: The comparison between different methods for the interaction energy (in kJ mol−1) ofthe He2 system. The counterpoise correction for BSSE is performed during the supramolecularHF and MP2 calculations. The energy difference MP2 - HF is used to estimate the dispersioninteraction. Since MP2 contains dispersion interaction at uncoupled-Hartree-Fock level, [68]both Hartree-Fock regular and irregular polarizabilities are used. The aug-cc-pVTZ basis setis adopted.
R/A HF MP2 MP2-HF Londona eqn (69)b
2.0 5.99 4.94 -1.05 -1.07 -1.263.0 0.0643 -0.0429 -0.107 -0.0942 -0.1054.0 0.02594 -0.0174 -0.0180 -0.0167 -0.01815.0 0.05469 -0.02466 -0.02467 -0.02439 -0.024686.0 0.07578 -0.02167 -0.02167 -0.02148 -0.021567.0 0.07184 -0.03786 -0.03786 -0.03583 -0.03614
0.0n594 ≡ 0.0 · · ·0︸ ︷︷ ︸
n
594
a α1,HF = 1.32 a.u. [69]b α†B
1,HF
29
Table 10: The comparison between different methods for the interaction energy (in kJ mol−1)of the He3 system. The counterpoise correction for BSSE is performed in the supramolecularHF and MP2 calculations. The energy difference MP2 - HF is used to estimate the dispersioninteraction. The R denotes the distance between the two nearest helium atoms. The aug-cc-pVTZ basis set is used.
R/A HF MP2 MP2-HF Londona eqn (69)a
2.0 12.0 9.88 -2.12 -2.14 -2.533.0 0.129 -0.0879 -0.217 -0.188 -0.2114.0 0.02119 -0.0349 -0.0361 -0.0335 -0.03635.0 0.05935 -0.02929 -0.02930 -0.02877 -0.029376.0 0.06102 -0.02326 -0.02326 -0.02294 -0.023117.0 0.07236 -0.02147 -0.02147 -0.02116 -0.02123
a α1,HF = 1.32 a.u. [69]b α†B
1,HF
Table 11: The comparison between SAPT and the α†B1 -based eqn (69) for the interaction energy
(in kJ mol−1) of He2. In the SAPT calculation, the monomer is calculated at Hartree-Fock level.The aug-cc-pVTZ basis set is used.
R/A E(1)elst E
(1)exch E
(2)ind E
(2)ex−ind E
(2)disp E
(2)ex−disp δ(HF)a E b
tot eqn (69)c
2.0 -1.28 7.80 -0.369 0.327 -1.86 0.194 -0.496 4.32 -1.263.0 -0.0112 0.0788 -0.00173 0.00142 -0.140 0.02310 -0.02291 -0.0730 -0.1054.0 -0.04844 0.03698 -0.05899 0.05683 -0.0219 0.04319 -0.04176 -0.0213 -0.01815.0 -0.06971 0.05575 -0.07693 0.07240 -0.02547 0.06220 -0.07697 -0.02546 -0.024686.0 -0.07217 0.07572 0d 0d -0.02179 0d 0d -0.02179 -0.021567.0 -0.08396 0d 0d 0d -0.03700 0d 0d -0.03700 -0.03614
a δ(HF) ≡ Eint(HF) − E(1)elst − E
(1)exch − E
(2)ind − E
(2)ex−ind, where the Eint(HF) is the interaction
energy from supramolecular Hartree-Fock calculation. The quantity δ(HF) indicates higher-order induction and exchange effects. [70]b Etot ≡ E
(1)elst + E
(1)exch + E
(2)ind + E
(2)ex−ind + E
(2)disp + E
(2)ex−disp + δ(HF).
c α†B1,HF
d absolute value less than 10−9 kJ mol−1.
30
Table 12: The comparison between SAPT and the α†B1 -based eqn (69) for the interaction energy
(in kJ mol−1) of He3 system. The R denotes to the distance of the two nearest helium atoms.In the SAPT calculation, the monomer is calculated at Hartree-Fock level. The aug-cc-pVTZbasis set is used.
R/A E(1)elst E
(1)exch E
(2)ind E
(2)ex−ind E
(2)disp E
(2)ex−disp δ(HF)a E b
tot eqn (69)c
2.0 -2.56 15.6 -0.735 0.641 -3.74 0.398 -0.958 8.66 -2.533.0 -0.0224 0.158 -0.02345 0.02284 -0.282 0.02630 -0.02582 -0.147 -0.2114.0 -0.03169 0.02140 -0.04179 0.04136 -0.0438 0.04639 -0.04351 -0.0425 -0.03635.0 -0.05196 0.04115 -0.06140 0.07478 -0.0109 0.06441 -0.06139 -0.0109 -0.029376.0 -0.07435 0.06114 -0.08182 0b -0.02358 0.08948 0b -0.02358 -0.023117.0 -0.08792 -0.08196 0b 0b -0.02140 0b 0b -0.02140 -0.02123
a α†B1,HF
b absolute value less than 10−9 kJ mol−1.
Table 13: The new polarizabilities α†0, α†
1, and α†2 for C60 calculated using a BLYP functional
with different basis sets.
Basis set def-SV def-SVP def-TZVP Sadlej-pVTZ aug-cc-pVDZ
α†0/10−3 a.u. 76.2 77.0 90.0 92.3
α†1/10−3 a.u. 7.00 6.88 7.94 7.88 7.72
α†2/10−3 a.u. 0.288 0.287 0.348 0.367 0.347
31
Table 14: Interaction energies (kJ mol−1) between a charge q at the midpoint and C60. RI-MP2/def2-TZVPP and the point-charge electrostatic and induction, eqns (42) and (43) com-pared with quantum chemical, RI-MP2/def2-TZVPP values. The α†B
0 and electrostatic poten-tial V are obtained at the BLYP/Sadlej-pVTZ and RI-MP2/def2-TZVPP levels respectively.The energy for neutral C60 is set to zero.
q/a.u. Eqn. (42) Eqn. (43) Eqn. (42) + (43) RI-MP2-5 -382.7 -3029.3 -3412.0 -3154.7-4 -306.2 -1938.7 -2244.9 -2102.4-3 -229.6 -1090.5 -1319.7 -1060.1-2 -153.1 -484.7 -636.0 -616.8-1 -76.5 -121.2 -197.7 -194.80 0.0 0.0 0.0 0.01 76.5 -121.2 -44.7 -48.32 153.1 -484.7 -331.6 -365.53 229.6 -1090.5 -860.9 -992.74 306.2 -1938.7 -1632.5 -1946.65 382.7 -3029.3 -2646.6 -3429.2
32
Table 15: Data for monomers. I is the experimental ionization potential, µ the experimentaldipole moment, and α1 the experimental average dipole polarizability. [57] The α†B
0 , α†B1 , and
α†B2 are calculated at the BLYP/Sadlej-pVTZ level.
System A I/eV µ /Debye Qzz /Debye A α1 /a.u. α2 /a.u.He 24.59 0 0 1.38 2.45 [71]a
Ne 21.56 0 0 2.68 7.52 [72]b
Ar 15.76 0 0 11.1 52.25 [72]b
Kr 14.00 0 0 16.8 97.39 [72]b
Xe 12.13 0 0 27.3 209.85 [72]b
Zn 9.39 0 0 37.7 324.8 [73]Cd 8.99 0 0 46.9Hg 10.44 0 0 33.8H2 15.43 0 0.46(2) [74] 5.43 16.71 [75]a
N2 15.58 0 -1.394 [76] 11.74 83.26 [77]d
HF 16.04 1.826 2.339 [78] 5.57 [79] 21.22 [80]d
H2O 12.62 1.855 -0.130c [78] 9.79 45.18 [81]e
NH3 10.07 1.472 -2.320 [78] 19.0 82.72 [81]e
CH4 12.61 0 0 17.50 120.9 [82]d
LiF 11.3 [83] 6.327 -5.0 [84] 9.88 [85]d
System B I/eV α†0/10−3 a.u. α†
1/10−3 a.u. α†2/10−3 a.u.
C60 7.57 [86] 92.3 7.88 [12] 0.367
a Sum-over-states with explicitly correlated wavefunction.b Calculated value at CCSD(T) level with a relativistic pesudopotential.c The xx and yy components are not identical, Qxx = −2.500 and Qyy = 2.630.d Finite field difference at CCSD(T) level.e TD-MP2 calculation.Conversion factors: 1 a.u. of energy = 27.2114 eV. 1 a.u. of dipole moment = 2.54175 Debye.1 a.u. of quadrupole moment = 1.34503 Debye A. 1 a.u. of length = 0.5291772 A. [87]
33
Table 16: RI-MP2 and RI-SCS-MP2 interaction energies (kJ mol−1) for the A@C60 com-plexes using different basis sets, the London-type dipole-dipole dispersion, eqn (69), and thequadrupole-quadrupole dispersion, eqn (72). The α†B
1,BLYP/Sadlej−pVTZ and α†B2,BLYP/Sadlej−pVTZ are
used.
AHe Ne Ar Kr Xe Zn Cd Hg
RI-MP2/def2-TZVPP [12] -7.5 -14.1 -71.2 -95.7 -124.5 -125.8 -162.9 -192.5RI-MP2/def2-QZVPP -9.5 -19.0 -77.9 -105.7 -139.2 -141.3 -180.4 -204.5RI-MP2/extrapolation -11.0 -22.6 -81.6 -112.2 -148.5 -151.5 -191.4 -211.9
RI-SCS-MP2/def2-TZVPP -5.1 -9.2 -45.7 -57.4 -63.3 -75.0 -96.6 -124.5RI-SCS-MP2/def2-QZVPP -6.7 -13.0 -51.2 -65.7 -76.1 -88.3 -111.5 -134.6RI-SCS-MP2/extrapolation -7.8 -15.7 -54.1 -71.1 -84.0 -96.8 -120.7 -140.6
Steric repulsiona 2.5 6.3 37.6 69.4 140.5 113.2 146.6 119.7Disp,DD, eqn (69) -4.6 -8.5 -32.3 -46.9 -72.5 -92.0 -120.1 -89.6Disp,QQ, eqn (72) -0.6 -1.9 -11.8 -21.2 -43.3 -60.2Sum of Ster and Disp -2.7 -4.1 -6.5 1.3 24.7 -39.0
a HF/def2-QZVPP results obtained from RI-MP2 calculations.
34
Table 17: Calculated intermolecular interactions from simple formulae, induction eqn (79) anddispersion interaction eqn (69), against RI-MP2 and RI-SCS-MP2 supramolecular calculationswith the two-point extrapolation from def2-TZVPP and def2-QZVPP basis set. The geometryis optimized at RI-MP2/def2-SVP level. The steric repulsion is estimated as the interactionenergy at HF/def2-QZVPP level substracting the induction from eqn (79) and (81). Energiesare in kJ mol−1. The H2 and N2 Hartree-Fock-level energies were not given in the references.The α†B
1,BLYP/Sadlej−pVTZ, and α†B2,BLYP/Sadlej−pVTZ are used.
Method AH2 N2 HF H2O NH3 CH4 LiF
RI-MP2 -30.7a -65.0b -49.5 -71.8 -89.4 -89.4 -211.9RI-SCS-MP2 -21.6a -35.9b -35.2 -50.9 -59.4 -50.1 -140.6
Ind,SAPT -5.0c
Disp,SAPT -36.1c
Steric repulsion - - 28.8 58.7 58.1 85.8 84.1Ind,DD, eqn (79) 0 0 -5.5 -5.9 -3.6 0 -66.6Ind,QQ, eqn (81) -0.1 -0.9 -1.5 -3.9 -1.4 0 -6.7Disp,DD, eqn (69) -15.6 -34.1 -16.3 -26.4 -46.7 -47.2 -26.0Disp,QQ, eqn(86) -3.8 -18.8 -2.4 -9.5 -15.8 -25.3 -Sum of Ster, Ind, and Disp - - 3.1 13.0 -9.4 13.3 -
a From ref. [4]. Both molecules fixed at experimental geometry, def-TZVPP basis set withcounterpoise correction, no RI approximation.b From ref. [3]. Both molecules fixed at experimental geometry, 6-311G(2d, 2p) basis set withcounterpoise correction, no RI approximation.c From ref. [4]. Both molecules fixed at experimental geometry. Energy obtained fromDF-DFT-SAPT calculation with a PBE functional and def-TZVPP basis set.
Table 18: Off-center effect for He@C60. ’Int’ stands for the interaction energy obtained fromMP2/def2-TZVPP calculation directly. Energies in kJ mol−1.
∆r/A Pauli Disp Int0.0 2.5 -9.9 -7.50.5 4.4 -11.4 -6.91.0 15.7 -16.0 -0.31.5 68.4 -26.1 42.42.0 306.7 -51.8 254.92.5 1167.4 -135.5 1031.9
35
ra
rb
rab
A
B
Figure 1: The geometry for an endohedral complex. The ra and rb correspond to the coordinatesof the inside subsystem A and outside subsystem B respectively. The rab distance determinesthe interparticle Coulomb interaction.
rij
ri
rj
'
'
'
rjθ
∆r
O'
O
z
j
Figure 2: The geometry for an off-center endohedral system. O and O’ are the centers of theinside system, A, at center and off-center positions, respectively.
36
-6000
-5000
-4000
-3000
-2000
-1000
0
-6 -4 -2 0 2 4 6
E /k
J m
ol-1
q/e
Charge q at C60 midpoint
QCEstimates
Figure 3: Interaction energies for point charges inside C60. The points are RI-MP2/def2-TZVPP values. The line is from the theoretical expression eqns (42) + (43) with a calculatedC60 V and α†
0 of 0.0292 and 0.0923 a.u. at RI-MP2/def2-TZVPP and BLYP/Sadlej-pVTZlevels respectively.
1
10
100
1000
1 10 100 1000
−E
(es
timat
ed)
/kJ
mol
−1
−E (best) /kJ mol−1
HeNeArKrXeZnCdHg
CH4
H2
N2
HFH O2
NH3
LiF
Figure 4: A comparison of the endohedral interaction energy, estimated from the London-likeformula (84) and the ’best estimate’ from a supramolecular calculation (MP2 for He, Ne andfor most molecules; SCS-MP2 for Ar-Xe, Zn-Hg, N2 and LiF). For a dipolar inner part ’A’, thedipolar induction term (79) is added to the London estimate.
37
−200
0
200
400
600
800
1000
1200
1400
0 0.5 1 1.5 2 2.5 3
E /
kJ m
ol−
1
∆ r /Å
PauliDisp.Int.
Figure 5: Interaction energies for He@C60 with different off-center positions at RI-MP2/def2-TZVPP level. The dashed lines are A + B (∆r)2 fit for dispersion, a Ce−α(∆r−R) fit for Paulirepulsion, and the sum of A + B(∆r)2 with Ce−α(∆r−R) for the interaction energy as suggestedin eqn (107). For ∆r > 2A, terms of higher order than O [(∆r)2] may be necessary.
38