The dispersion interaction between quantum mechanics and ... · atoms or molecules with large, diffuse electron clouds, such as π clouds. The general formula for the dispersion interaction

The dispersion interaction between quantum mechanics and effectivefragment potential moleculesQuentin A Smith Klaus Ruedenberg Mark S Gordon and Lyudmila V Slipchenko Citation J Chem Phys 136 244107 (2012) doi 10106314729535 View online httpdxdoiorg10106314729535 View Table of Contents httpjcpaiporgresource1JCPSA6v136i24 Published by the American Institute of Physics Additional information on J Chem PhysJournal Homepage httpjcpaiporg Journal Information httpjcpaiporgaboutabout_the_journal Top downloads httpjcpaiporgfeaturesmost_downloaded Information for Authors httpjcpaiporgauthors

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THE JOURNAL OF CHEMICAL PHYSICS 136 244107 (2012)

The dispersion interaction between quantum mechanics and effectivefragment potential molecules

Quentin A Smith1 Klaus Ruedenberg1 Mark S Gordon1a)

and Lyudmila V Slipchenko2a)

1Department of Chemistry and Ames Laboratory Iowa State University Ames Iowa 50011 USA2Department of Chemistry Purdue University West Lafayette Indiana 47907 USA

(Received 6 April 2012 accepted 31 May 2012 published online 26 June 2012)

A method for calculating the dispersion energy between molecules modeled with the general effec-tive fragment potential (EFP2) method and those modeled using a full quantum mechanics (QM)method eg Hartree-Fock (HF) or second-order perturbation theory is presented C6 dispersioncoefficients are calculated for pairs of orbitals using dynamic polarizabilities from the EFP2 por-tion and dipole integrals and orbital energies from the QM portion of the system Dividing by thesixth power of the distance between localized molecular orbital centroids yields the first term in thecommonly employed London series expansion A C8 term is estimated from the C6 term to achievecloser agreement with symmetry adapted perturbation theory values Two damping functions forthe dispersion energy are evaluated By using terms that are already computed during an ordinaryHF or EFP2 calculation the new method enables accurate and extremely rapid evaluation of thedispersion interaction between EFP2 and QM molecules copy 2012 American Institute of Physics[httpdxdoiorg10106314729535]

I INTRODUCTION

The dispersion interaction is an attractive force betweenatoms and molecules that is caused by the interactions ofinduced multipoles Dispersion arises from the correlatedmovement of electrons an instantaneous multipole on onemolecule may induce a multipole on another molecule1ndash4

While dispersion is generally a weak intermolecular forceit is the major attractive force between neutral atoms andmolecules that lack permanent multipole moments For ex-ample dispersion is singularly responsible for the attractionbetween atoms of the noble gases5 allowing for their con-densation at low temperatures Because dispersion dependson the atomic or molecular dynamic (frequency dependent)polarizability the dispersion energy is significant betweenatoms or molecules with large diffuse electron clouds suchas π clouds

The general formula for the dispersion interaction be-tween two quantum mechanical (QM) molecules comes fromthe second-order term in intermolecular perturbation theory1

Based on this functional form a formula6 for the dispersioninteraction between two molecules modeled with the gen-eral effective fragment potential (EFP2) (Refs 7ndash9) methodwas previously derived and implemented in the generalatomic and molecular electronic structure system (GAMESS)package10 However a formula for the dispersion energyin mixed systemsmdashthose in which one molecule is mod-eled with EFP2 and another with a full ab initio (AI) QMmethod eg Hartree-Fock (HF) or second-order perturba-tion theory (MP2) (Ref 11)mdashhas been lacking This paperpresents a derivation and implementation of the EFP2-AI

a)Authors to whom correspondence should be addressed Electronicaddresses mgordoniastateedu and lslipchenkopurdueedu

dispersion interaction as well as a comparison of resultingEFP2-AI dispersion energy calculations with those of EFP2-EFP2 and symmetry adapted perturbation theory (SAPT)12

The EFP2-AI dispersion term is important in order to describesolute-solvent interactions where the solute may be modeledwith an ab initio method and the solvent with EFP2 Disper-sion is the primary means of solute-solvent interactions withnonpolar solvents eg cyclohexane

The effective fragment potential (EFP) method is a dis-crete solvent method in GAMESS The original EFP methodEFP1 involves fitted parameters explicitly designed for waterwhile the general method EFP2 contains no fitted param-eters and may be applied to any molecule Parameters nec-essary to describe the fragment potential are generated in aMAKEFP calculation for a given isolated molecule The re-sulting potential may be used in future calculations on anynumber of molecules of the specified type with the same in-ternal geometry Potentials can be used to compute Coulombexchange-repulsion polarization charge transfer and disper-sion energy components in the interaction between two EFP2fragments9 With the current development of the EFP2-AIdispersion term all interaction energy terms except chargetransfer may be calculated between an EFP2 fragment and amolecule modeled with a full ab initio method

The EFP2 Coulomb term is calculated via the distributedmultipolar analysis (DMA) method of Stone1 carried outthrough octopole moments The DMA a classical point-wise model cannot account for the overlap of charge den-sities between molecules that occurs at short distances Assuch charge penetration effects are modeled by a distance-dependent cutoff function13 Polarization or induction isthe interaction between an induced dipole on one frag-ment and the electric field of another In the EFP2 method

0021-96062012136(24)24410712$3000 copy 2012 American Institute of Physics136 244107-1


244107-2 Smith et al J Chem Phys 136 244107 (2012)

induction is expressed in terms of localized molecular or-bital (LMO) polarizabilities with polarizability points at eachbond and lone pair The induced dipoles are iterated to self-consistency The exchange-repulsion term derived from firstprinciples1 14 is expressed as a function of intermolecu-lar overlap integrals15 16 Charge transfer is an interactionbetween the occupied valence orbitals of one molecule andthe virtual orbitals of another The implementation of EFP2-EFP2 charge transfer is described in Ref 17 EFP2-AI chargetransfer is currently under development The EFP2-EFP2dispersion term discussed further in Secs II and III isa function of frequency-dependent dynamic polarizabilitiessummed over the entire (imaginary) frequency range6 Exten-sive descriptions of how the potentials are generated and usedto compute intermolecular interaction energies may be foundelsewhere6ndash9 13 15ndash18

Previous work in deriving interaction terms for discreteQMMM methods (that is the interaction between quantummechanical and classical molecular mechanical systems) in-cludes the direct reaction field approach of van Duijnen andco-workers19 Distributed approaches for both multipoles14

and polarizabilities20 have been proposed by Stone1 Claverieand Rein21 have discussed a distributed model for intermolec-ular interactions involving localized orbitals and Karlstroumlmand co-workers22 have used a localized orbital approach forconstructing intermolecular potentials with the non-empiricalmolecular orbital method (NEMO) The EFP2-AI dispersionterm proposed in the present work involves distributed polar-izabilities on the EFP part (which are calculated for localizedorbitals) and localized orbital energies and dipole integrals onthe AI part of the system

The organization of this paper is as follows Section IIpresents a derivation of the EFP2-AI dispersion energyterm beginning from the dispersion energy equation in itsmost general form from intermolecular perturbation theorySection III describes the implementation of the EFP2-AI dis-persion code in GAMESS Computational details concerningthe systems used to test the code appear in Sec IV Section Vcompares dispersion energy values obtained from the EFP2-AI dispersion code with those of EFP2-EFP2 and SAPT Con-clusions are presented in Sec VI

II THEORY

A Dispersion interaction

The general formula for the dispersion interaction be-tween two closed-shell nondegenerate ground state moleculescan be derived with Rayleigh-Schroumldinger perturbation theory(RSPT) following the method of Stone1 While RSPT doesnot account for intermolecular antisymmetry effects such asexchange-dispersion that predominate at short intermonomerdistances it adequately accounts for mid- to long-range ef-fects like the dispersion interaction that is of interest here22 23

The unperturbed Hamiltonian is given by the sum of theHamiltonians for the individual molecules A and B

H 0 =

HA +

HB (1)

The perturbation operator embodies all electrostatic in-teractions between the molecules It can be represented in nu-

merous ways the simplest of which is

V =sumiisinA

sumjisinB

qiqj

Rij

(2)

where qi is the charge on particle i (electron or nucleus) onmolecule A and Rij is the distance between i and j A moreconvenient representation is the multipolar expansion

V = T ABqAqB +xyzsum

a

T ABa

(qAμB

a minus μAa qB

)

minusxyzsumab

T ABab μA

a μBb + (3)

where qA is the net charge on molecule A μBa is the ath di-

rectional (x y z) component of the dipole moment on B etcThe quantities T Ta Tab etc are electrostatic tensors of rankindicated by the number of subscripts T correspondsto charge-charge interactions Ta corresponds to charge-dipole interactions Tab contains dipole-dipole and charge-quadrupole interactions Tabc contains dipole-quadrupole andcharge-octopole interactions and so on The formulae of thefirst three tensors appear below where Ra

ij denotes the ath di-rectional component of the distance between i and j

T ij = 1

Rij

(4)

T ija = nablaa

1

R= minus

Raij

R3ij

(5)

Tij

ab = nablaanablab

1

R= 3 Ra

ijRbij minus R2

ij δab

R5ij

(6)

When the perturbation expansion is carried out thezeroth-order term gives the sum of the ground state energiesof A and B and the first-order term gives the Coulomb (elec-trostatic) interaction energy The second-order energy term inthe perturbation expansion is given by

W primeprime0 = minus

summn

〈0A0B |

V |mn〉〈mn|

V |0A0B〉Wmn minus W0A0B

(7)

where m and n are states of molecules A and B respectively0A represents the ground state of molecule A and Wmn = Em

+ En is the energy of the system in state |mn〉 In Eq (7)either m or n may be zero but not both Formally since anti-symmetry is not imposed on the intermolecular wavefunctionthis expression is valid for exact eigenstates of the isolatedHamiltonians23

This second-order perturbative expression Eq (7) en-compasses both the induction (polarization) and dispersionenergies Induction arises from the interactions between per-manent multipoles on one molecule and induced multipoleson the other As such its representation in Eq (7) occurs inthose terms in the summation in which either m or n is equalto 0 ie one of the interacting molecules is in its ground statewhile the other is in an (induced) excited state The remainderof the terms in the summation those in which both m and n



refer to excited states correspond to the dispersion energy

Edisp = minussumm=0n=0

〈0A0B |

V |mn〉〈mn|

V |0A0B〉EA

m + EBn minus EA

0 minus EB0

(8)

The multipolar expansion of Eq (3) may then be sub-stituted for the perturbation operator in Eq (8) Since the

charges q in Eq (3) are scalar values integrals involving qwill be of the form for example 〈0A|qA|m〉 = qA〈0A|m〉these integrals are zero because the ground and excited statesare orthogonal to each other Therefore the multipole ex-pansion when used in this context properly begins with thedipole-dipole term The dispersion energy corresponding tothe dipole-dipole interaction labeled E

disp

6 is given by

Edisp

6 = minussumm=0n=0

〈0A0B | sumxyz

ab T ABab

μ

A

a

μ

B

b |mn〉〈mn| sumxyz

cd T ABcd

μ

A

c

μ

B

d |0A0B〉EA

m + EBn minus EA

0 minus EB0

(9)

The dipole operators apply only to the states of their respective molecules (m on A or n on B) and tensors Tab do not dependon the states Assuming that the states are separable ie |mn〉 = |m〉|n〉 Eq (9) may then be simplified to

Edisp

6 = minusxyzsumabcd

T ABab T AB

cd

summ=0n=0

1

EAm0 + EB

n0

〈0A|μ

A

a |m〉〈m|μ

A

c |0A〉〈0B |μ

B

b |n〉〈n|μ

B

d |0B〉 (10)

where Em0 = Em minus E0 The energy factors do not permit the easy separation of Eq (10) into portions relating to molecule A andportions relating to molecule B The Casimir-Polder identity1 24

1

A + B= 2

π

int infin

0

AB

(A2 + ω2)(B2 + ω2)dω (11)

may be applied to the denominator of Eq (10) to obtain

Edisp

6 = minus2macr

π

xyzsumabcd

T ABab T AB

cd

int infin

0dω

summ=0n=0

ωAm0 〈0A|

μA

a |m〉〈m|μ

A

c |0A〉macr((

ωAm0

)2 + ω2) ωB

n0〈0B |μ

B

b |n〉〈n|μ

B

d |0B〉macr((

ωBn0

)2 + ω2) (12)

after the energies in Eq (10) are expressed in terms of frequencies eg EAm0 = macrωA

m0From time-dependent (TD) perturbation theory it can be shown that the response of a molecule to an oscillating elec-

tric field is an oscillating dipole moment If a field Fb with frequency ω is given by Fbeminusiωt then the dipole moment is μa

= αab(ω)Fbeminusiωt The components of the frequency-dependent dynamic polarizability tensor α are given by1

αab (ω) =summ

ωm0〈0| μa |m〉〈m|

μb |0〉 + 〈0| μb |m〉〈m|

μa |0〉macr

(ω2

m0 minus ω2) = 2

summ

ωm0 〈0| μa |m〉〈m|

μb |0〉macr

(ω2

m0 minus ω2) (13)

α describes the propagation of a density fluctuation within amolecule2 Expressing the factor ω2 in Eq (13) as ω2 = minus(minusω2) = minus(iω)2 Eq (12) can be recast in terms of the dynamicpolarizability tensor at an imaginary frequency The conceptof imaginary frequencies is purely a mathematical constructwith no actual physical meaning1 Regardless it can be usedto construct functions that are well-behaved and that decreasemonotonically to 0 as ω rarr infin

B EFP-EFP dispersion

If the conversion into terms of dynamic polarizability ten-sors is performed for both molecule A and molecule B as inthe case of EFP2-EFP2 interactions the resulting expression

is

Edisp

6 = minus macr2π

xyzsumabcd

T ABab T AB

cd

int infin

0αA

ac (iω) αBbd (iω) dω

(14)The dynamic polarizability tensors in Eq (14) are im-

plicitly calculated at a single point on each molecule How-ever there are known issues with convergence when using thisapproach to be effective it may require higher terms in themultipolar expansion7 (ie quadrupole polarizabilities mightneed to be considered) A distributed polarizability modelin which polarizability tensors are calculated at multiple ex-pansion points has the advantage of more successful conver-gence and possibly giving a more realistic portrayal of the re-sponse of a molecule to the nonuniform fields arising from



the interacting molecules7 To improve numerical accuracywithout introducing additional complication the form ofEq (14) is used to construct an approximate distributed po-larizability model

Edisp

6 = minus macr2π

sumkisinA

sumjisinB

xyzsumabcd

Tkj

ab Tkj

cd

int infin

0αk

ac (iω) αj

bd (iω) dω

(15)In Eq (15) k and j are LMOs on molecules A and B

respectively and the approximation

αBbd (iω) rarr

sumjisinB

αj

bd (iω) (16)

is used for the distributed polarizabilitiesThe EFP2-EFP2 dispersion energy expression in Eq (15)

is anisotropic as the directional component summation isover all possible x y z pairs In most computational meth-ods an isotropic expressionmdashone that does not include xy xzand other off-diagonal termsmdashis preferred The off-diagonalterms of the frequency-dependent polarizability tensor typi-cally do not contribute greatly to the dispersion energy Thisis because in the case of the EFP method the polarizabil-ity tensor is constructed using the principal orientation fora given molecule the axes are chosen such that the diago-nal components (xx yy and zz) are always dominant Thusomitting the off-diagonal terms from the calculation signif-icantly reduces the total number of terms that must be cal-culated in Eq (15) without serious loss of accuracy Addi-tionally because the majority of the C6 values reported in theliterature are isotropic6 introducing an isotropic expressionfor the EFP2-EFP2 dispersion coefficient allows straightfor-ward comparison with other computational and theoreticalmethods Therefore the following approximation is made

Edisp

6 asymp minus macr2π

sumkisinA

sumjisinB

xyzsumabcd

δacδbdTkj

ab Tkj

cd

timesint infin

0αk

ac (iω) αj

bd (iω) dω

= minus macr2π

sumkisinA

sumjisinB

xyzsumab

Tkj

ab Tkj

ab

int infin

0αk

aa (iω) αj

bb (iω) dω

(17)

A further simplification may be achieved by introducingthe isotropic dynamic polarizability αj (iω) for LMO j andfrequency ω as 13 of the trace of the polarizability tensor at agiven imaginary frequency and employing the spherical atomapproximation

αjxx(iω) asymp αj

yy(iω) asymp αjzz(iω) = αj (iω) (18)

Taking into account thatxyzsumab

Tkj

ab Tkj

ab = 6

R6kj

(19)

the isotropic dispersion energy expression in atomic unitsbecomes

Edisp

6 = minus 3

π

sumkisinA

sumjisinB

1

R6kj

int infin

0αk (iω) αj (iω) dω (20)

Equation (20) is the final expression for the dipole-dipoleterm of the dispersion interaction energy between two EFP2molecules where k and j are expansion points for distributedpolarizabilities on EFP2 molecules A and B respectively andRkj is the distance between these expansion points The in-tegral over the imaginary frequency range is evaluated via a12-point Gauss-Legendre numerical quadrature6 The calcu-lation of the dynamic polarizability tensors α at the necessary(predetermined) frequencies is part of the MAKEFP proce-dure which generates the fragment potential parameters be-fore the calculation of interaction energy components begins

C EFP-AI dispersion

In constructing an expression for the EFP2-AI dispersioninteraction it would be appealing to obtain an equation anal-ogous to Eq (20) However the procedure used to obtain thedynamic polarizability tensors α in a MAKEFP calculation istoo time consuming to be appropriate for the AI molecule es-pecially since many energy and gradient evaluations may beneeded in the calculation Following the approach of Amosand co-workers3 4 values for α associated with each polariz-able point are computed via the time-dependent analog of thecoupled perturbed Hartree-Fock (CPHF)25 equations (Detailsmay be found in Refs 3 4 6 and 25) The time require-ment for this procedure is acceptable in the case of EFP2-EFP2 because the MAKEFP calculation occurs prior to andas a separate step from the EFP2-EFP2 dispersion calcula-tion Constructing the α tensors for the ab initio moleculewould necessarily occur during the calculation of the EFP2-AI interaction thus increasing the run time significantly Forexample if a Hartree-Fock calculation on phenol with the6-311++G(3df2p) basis set (equivalent to 375 basis func-tions) takes approximately 6 min on a given computer aCPHF calculation on the same chemical system requires 10min and a time-dependent Hartree-Fock (TDHF) calcula-tion requires 12 min To avoid these time-consuming calcu-lations the EFP2-AI dispersion derivation must diverge fromthe EFP2-EFP2 derivation at Eq (12) Only the portion ofEq (12) relating to the EFP2 part (molecule B) is recast interms of α giving

EEFPminusAI6 = minus macr

π

xyzsumabcd

T ABab T AB

cd

int infin

0dω

summ=0

ωAm0 〈0A|

μA

a |m〉〈m| μ

A

c |0A〉macr((

ωAm0

)2 + ω2) αB

bd (iω)

= minus 1

π

xyzsumabcd

T ABab T AB

cd

summ=0

〈0A| μ

A

a |m〉〈m| μ

A

c |0A〉int infin

0dω

ωAm0(

ωAm0

)2 + ω2αB

bd (iω)

(21)



To convert from a sum-over-states approach to an orbital-based approach let i correspond to occupied canonical molecularorbitals (MOs) and r to virtual canonical MOs on the AI molecule This changes the dipole integrals in Eq (21) to

summ=0

〈0A| μ

A

a |m〉〈m| μ

A

c |0A〉 rarroccsumk

virsumr

〈k| μ

A

a |r〉〈r| μ

A

c |k〉 (22)

(see Appendix A) The integral over the imaginary frequency range in Eq (21) becomes

int infin

0dω

ωAm0(

ωAm0

)2 + ω2αB

bd (iω) rarrint infin

0dω

ωArk(

ωArk

)2 + ω2αB

bd (iω) where ωrk = ωr minus ωk (23)

(see Appendix B) Invoking Eqs (22) and (23) to change from sum-over-states to the orbital approximation and Eq (16) tochange to distributed polarizability tensors on fragment B Eq (21) becomes

EEFPminusAI6 = minus 1

π

sumjisinB

xyzsumabcd

occsumk

virsumr

Tkj

ab Tkj

cd 〈k| μ

A

a |r〉〈r| μ

A

c |k〉int infin

0dω

ωArk(

ωArk

)2 + ω2α

j

bd (iω) (24)

The isotropic approximation is now employed First off-diagonal directional component terms (which are assumed to be negli-gible) are eliminated

EEFPminusAI6 asymp minus 1

π

sumjisinB

xyzsumabcd

occsumk

virsumr

δacδbdTkj

ab Tkj

cd 〈k| μ

A

a |r〉〈r| μ

A

c |k〉int infin

0dω

ωArk(

ωArk

)2 + ω2α

j

bd (iω)

= minus 1

π

sumjisinB

xyzsumab

occsumk

virsumr

Tkj

ab Tkj

ab 〈k| μ

A

a |r〉〈r| μ

A

a |k〉int infin

0dω

ωArk(

ωArk

)2 + ω2α

j

bb (iω) (25)

The product Tkj

ab Tkj

ab in Eq (25) can be expressed in terms of Rminus6kj as in Eq (19) where Rkj is the distance between the

centroids of orbitals k (on the AI molecule) and j (on the EFP2 fragment) Similar to the direction-independent quantity αj (iω)from Eq (18) the Cartesian component-averaged dipole integrals given by 〈k| μ |r〉〈r| μ |k〉 = 1

3

sumxyza 〈k| μA

a |r〉〈r| μAa |k〉

are introduced The resulting fully isotropic EFP2-AI dispersion equation is


π

sumjisinB

occsumk

virsumr

1

R6kj

〈k| μ |r〉〈r|

μ |k〉int infin

0dω

ωArk(

ωArk

)2 + ω2αj (iω) (26)

The dispersion expansion is commonly expressed in the formof the series

Edisp = C6

R6+ C8

R8+ C10

R10+ (27)

where C6R6 corresponds to the instantaneous dipole-induceddipole interaction C8R8 primarily to the instantaneousdipole-induced quadrupole and instantaneous quadrupole-induced dipole interaction etc (Odd-order terms are almostalways neglected although they are strictly equal to zero onlyfor atoms and for molecules with inversion centers1) Theisotropic EFP2-AI distributed dipole-dipole dispersion termcan be expressed in the form

EEFPminusAI6 = minus

sumkisinA

sumjisinB

Ckj

6

R6kj

(28)

where by comparison with Eq (26)

Ckj

6 = 6

π

virsumr

〈k| μ |r〉〈r|

μ |k〉int infin

0dω

ωArk(

ωArk

)2 + ω2αj (iω)

(29)

III LMO FORMULATION AND IMPLEMENTATION

While Eqs (26) and (29) are formulated in the canoni-cal MO basis in the AI region previous studies with the EFPmethod suggest that formulation in terms of LMOs providesfaster convergence and more accurate results6 7 15 There-fore in the following distributed LMO-based C6 coefficientsand EFP-AI dispersion energy are derived In analogy withEq (28) the LMO reformulation of the EFP2-AI dispersionenergy term is proposed to have the form


sumisinA

sumvisinB

Cv6

R6v

(30)



where |〉 and |v〉 refer to LMOs on the AI and EFP moleculesrespectively Rv is the distance between the centroids of theseLMOs The C6 coefficient between the two LMOs averagedover the directional components Cv

6 is

Cv6 = 1

3middot 6

π

xyzsuma

Cv6a = 2

π

xyzsuma

Cv6a (31)

where the directional components are

Cv6a =

virsumr

〈| μa |r〉sumprime

〈r| μa

∣∣primerang valencesumk

T primek

timesint infin

0dω

ωArk(

ωArk

)2 + ω2αv(iω) (32)

In Eq (32) k sums over the orbitals in the valence spaceand

T primek = LkLkprime (33)

The matrix Lk obtained by performing Boyslocalization26 on the valence MOs is the orthogonaltransformation that expresses the localized AI orbitals |〉 interms of the canonical AI orbitals |k〉 according to

|〉 =sum

k

|k〉Lk |〉 = localized |k〉 = canonical

(34)For convenience the integral over imaginary frequencies

in Eq (32) which includes canonical MO terms ωrk in the AIregion is kept in the MO basis Substituting Eq (34) for andprime into Eq (32) and taking into account the orthogonality ofLk yields

Cv6a =

valencesumk

valencesumkprime

Lk

[virsumr

〈k| μa |r〉〈r|μa|kprime〉

timesint infin

0dω

ωArkprime(

ωArkprime

)2 + ω2αv(iω)

]Lkprime (35)

The LMO-distributed quantities Cv6 obtained by this

method are not identical to the MO-distributed Ckj

6 ofEq (29) but they are equivalent through the transformationto the LMO basis To ensure fast execution of the code onlythe valence orbitals (omitting core orbitals) are used in sum-mations in Eq (35) Equation (35) is implemented in theGAMESS code for EFP-AI dispersion

The distributed LMO-based C6 coefficient in Eq (35) iscomprised of dipole integrals over AI orbitals and an integralover the imaginary frequency range The imaginary frequencyintegral for EFP2 LMO v with AI occupied orbital k and vir-tual orbital r is given by

I vkr =int infin

0dω

ωArk(

ωArk

)2 + ω2αv (iω) where ωrk = ωr minusωk

(36)This integral is computed using a 12-point Gauss-

Legendre quadrature in analogy with the EFP2-EFP2 disper-sion expression1 To convert this integral to a range that is

conducive to the use of a Gauss-Legendre numerical quadra-ture the same transformation of variables that is used in theEFP2-EFP2 dispersion procedure6 may be applied

ω = ν0(1 + t)

(1 minus t) dω = 2ν0dt

(1 minus t)2 (37)

Equation (37) converts the range of integration to be fromminus1 to +1 By using Gauss-Legendre abscissas for t thisprocedure also determines the values of the (imaginary) fre-quencies at which the polarizability tensors must be calcu-lated The substitution shown in Eq (37) was determined inRef 27 where the optimal value of ν0 was found to be 03The imaginary frequency integral in Eq (36) becomes

I vkr =12sum

n=1

wn

2ν0

(1 minus tn)2

ωrk

ω2rk + ω2

n

αv (iωn)

where ωn = ν0(1 + tn)

(1 minus tn) (38)

Here tn and wn are Gauss-Legendre abscissas and weightsand αv (iωn) is 13 of the trace of the polarizability tensor forLMO v at frequency ωn (Eq (18)) Since the MO energies andthe factors ωrk in Eq (38) differ by macr which is 1 in atomicunits the energy difference εrk between virtual MO r and oc-cupied MO k is used in place of the frequency difference ωrk

I vkr =12sum

n=1

wn

2ν0

(1 minus tn)2

εrk

ε2rk + ω2

n

αv (iωn)

where εrk = εr minus εk (39)

Upon computation of the integral I vkr in Eq (39) for agiven EFP2 LMO v and AI MOs k and r I vkr is multipliedby the canonical MO-basis dipole integrals obtained by thestandard transformation

μAocctimesvir MOsa = OCCT middot μAAOs

a middot V IR (40)

where μAAOsa is the matrix of dipole integrals with elements

〈χ1 |μ

A

a |χ2〉 over all pairs of AOs χ1 and χ2 OCC con-sists of the Nvalence = Noccupied ndash Ncore (number of valenceorbitals) columns of the MO coefficient matrix and VIRconsists of the remaining Ntotal ndash Noccupied virtual columnsThis produces a matrix μAocctimesvir MOs

a with dimensions Nvalence

times (Ntotal ndash Noccupied) corresponding to the dipole integrals be-tween valence (k) times virtual (r) canonical MOs its elements

are 〈k| μ

A

a |r〉 The product of the imaginary frequency inte-gral I vkr with the dipole integrals is transformed to the LMObasis using the Boys transformation matrix Lk as shown inEq (35) Finally each Cv

6 coefficient is multiplied by Rminus6v

calculated between the centroids of ab initio LMO and EFP2LMO v Summing over all and v gives the total dipole-dipolecontribution to the EFP2-AI dispersion energy Eq (30)

Two further approximations must be invoked in order toobtain the total dispersion energy from the dipole-dipole dis-persion term First numerous theoretical studies suggest thatin order to compare with experimental or high-level ab initiocalculations a damping function must be used with the dis-persion energy term28ndash35 The damping function Fv

6 is em-ployed to account for both exchange-dispersion and charge



penetration effects which predominate at short intermolecu-lar separations Additionally as with EFP2-EFP2 dispersion6

the total EFP2-AI dispersion energy expansion in Eq (27)is truncated after the C8R8 term which is approximated as13 of the C6R6 term This approximation was determined inRef 6 by comparison of EFP-EFP dispersion energies trun-cated at the C6R6 term with the total SAPT dispersion en-ergies for a selection of dimers Thus the final form for theEFP2-AI dispersion energy becomes

EEFPminusAIdisp = minus4

3

sumisinA

sumvisinB

F v6 Cv

6

R6v

(41)

As was done for the EFP2-EFP2 dispersion interaction6

both Tang-Toennies (TT)6 36 37 and overlap-based37 dampingfunctions have been explored The sixth-order Tang-Toenniesdamping function has the form

FT T6 (Rv) = 1 minus exp (minusβRv)

6sumn=1

(βRv)n

n (42)

where β is a damping parameter chosen to be 15 based on astudy of several small dimers at their equilibrium distances6

The overlap-based damping function is given by37

Foverlap

6 (Sv) = 1 minus S2v(1 minus 2 ln |Sv| + 2(ln |Sv|)2)

(43)where Sv is the overlap integral calculated between LMOs|〉 and |v〉 Dispersion energies obtained using each type ofdamping are presented in Sec V

IV COMPUTATIONAL DETAILS

EFP2-EFP2 and EFP2-AI dispersion calculations wereperformed with the GAMESS10 software package SAPT12

interaction energies were calculated using SAPT-2006The dimers chosen for this studymdashargon methane H2

HF water ammonia methanol (MeOH) dichloromethaneand the sandwich and T-shaped benzene dimersmdashare iden-tical to those used in a previous study of EFP2 dampingfunctions37 (Fig 1) Equilibrium geometries of CH4 H2MeOH and NH3 are taken from Ref 6 The CH2Cl2 dimergeometry is taken from Ref 38 The HF dimer geometryfrom Ref 39 was obtained with coupled cluster with singlesdoubles and perturbative triples40 [CCSD(T)] in the completebasis set limit The Ar dimer geometry from Ref 41 wasobtained with CCSD(T)aug-cc-pVQZ42 43 The geometry ofthe H2O monomer was obtained in Ref 37 by optimizing withCCSD(T)cc-pVTZ (Ref 43) this was followed by a con-strained optimization with MP26-311++G(3df2p)44ndash46

to obtain the dimer geometry37 The sandwich andT-shaped benzene dimer geometries are taken fromRef 47

A Comparison of C6 coefficients

The total EFP2-EFP2 and EFP2-AI C6 coefficients equalto the sum over all LMO pairs of the distributed C6 valueswere obtained for all dimers The 6-311++G(3df3p) basis

FIG 1 Dimers examined in the present study Equilibrium geometriesare shown for (a) argon (b) H2 dimer (c) HF (d) water (e) ammonia(f) methane (g) methanol (MeOH) and (h) dichloromethane Dimer geome-tries for (i) sandwich benzene dimer and (j) T-shaped benzene dimer are con-strained structures

set44ndash46 was used to generate the EFP2 potentials and to per-form the AI Hartree-Fock calculations for all monomer typesexcept benzene For benzene the 6-311++G(3df2p) basisset44ndash46 was used Experimental C6 values from Refs 1 and27 are used for comparison

B Comparison of potential energy curves

As in Ref 37 potential energy curves were generatedfor all dimers except the benzene dimers by varying inter-monomer distances in increments of 02 Aring from ndash08 Aring to+08 Aring with respect to the equilibrium distance while keep-ing the internal monomer geometries fixed For the benzenesandwich dimer the distance between the benzene rings wasvaried from 33 Aring to 60 Aring For the T-shaped benzene dimerthe ring center to ring center distance was varied from 47 Aringto 69 Aring The 6-311++G(3df2p) basis set was used to gen-erate the EFP2 parameters for EFP2-EFP2 and EFP2-AI cal-culations on the Ar H2 H2O CH4 NH3 HF and sandwichand T-shaped benzene dimers The 6-311++G(2d2p) basis44

was used with the MeOH dimers For CH2Cl2 dimers the6-31+G(d) basis48 was used In EFP2-AI calculations thesame basis used for the EFP2 monomer was also used onthe AI molecule For all dimer types other than the ben-zene dimers second-level SAPT calculations were carriedout using the same basis set employed for the EFP2-EFP2and EFP2-AI calculations SAPT data for the benzene dimerscome from Ref 47 in which the aug-cc-pVDZ basis42 wasused The SAPT reference values reported in Sec V arethe sum of the SAPT second-order dispersion and exchange-dispersion values for a given dimer geometry

V RESULTS AND DISCUSSION

The neon dimer provides a concrete example of the pro-cess for determining the C6 coefficients for an EFP2-AI sys-tem Having four valence orbitals (lone pairs) a neon atomis modeled with EFP2 as four α polarizability tensors on the



TABLE I EFP2-EFP2 and EFP2-AI isotropic C6 coefficients (au)ab

AI (left)-EFP2 (right) C6 EFP2 (left)-AI (right) C6 EFP2-EFP2 C6 Experimentalc

Ar 674 (+48) (Identical) 606 (minus58) 643d

H2 102 (minus157) 102 (minus157) 114 (minus58) 121e

HF 170 (minus105) 154 (minus189) 153 (minus195) 190e

H2O 430 (minus53) 406 (minus106) 393 (minus134) 454e

NH3 812 (minus70) 821 (minus60) 781 (minus105) 873e

CH4 1203 (minus72) (Identical) 1204 (minus71) 1296e

MeOH 1977 (minus111) 1972 (minus113) 1958 (minus119) 2222e

CH2Cl2 8430 8430 7558 NAC6H6 ()f 2087 (+211) 2087 (+211) 1805 (+48) 1723d

aCalculated using the 6-311++G(3df3p) basis set for all dimers except the benzene dimer () which was calculated with the 6-311++G(3df2p) basis set for EFP2-EFP2 andEFP2-AI the percent error compared to the experimental C6 value (if available) is shown in parentheses ldquo(Identical)rdquo indicates that there is no difference between AI (left) and AI(right)bFor nonsymmetrical EFP-AI dimers ldquoleftrdquo and ldquorightrdquo correspond to the monomer on the left or right in the dimers pictured in Fig 1 in hydrogen bonded species ldquoleftrdquo correspondsto the hydrogen bond donor and ldquorightrdquo to the hydrogen bond acceptorcExperimental values for footnotes (d) and (e)dFrom Ref 27eFrom Ref 1fEFP2-AI C6 values for sandwich and T-shaped benzene dimers are identical to four significant figures

orbital centroids The EFP2-EFP2 treatment of dispersion en-tails integrals between each α on fragment A with every α onfragment B giving 16 C

pj

6 integrals The EFP2-AI approachoutlined above produces four matrices (one for each αj on theEFP2 part) of dimension 4 times 4 (indexed by the AI valenceLMOs) The diagonal elements of these four matrices give thedistributed C

pj

6 values between the valence orbital p on the AImolecule and the EFP2 expansion point j For comparison thedistributed EFP2-EFP2 C

pj

6 coefficients for the neon dimercalculated with the 6-311++G(3d) basis set are all approx-imately 0286984097 differing from one another beginningin the ninth decimal place The distributed EFP2-AI coeffi-cients with both the AI and the EFP2 part calculated usingthe 6-311++G(3d) basis set are all approximately 02857differing from one another after the third decimal place

Unlike the EFP2-EFP2 C6 coefficient the EFP2-AI C6

coefficient varies somewhat with intermonomer separationThis is because the AI orbitals respond to the perturba-tive presence of the EFP2 fragment The EFP2 Coulombexchange-repulsion and induction energies are iterated intothe AI Hartree-Fock calculation (the dispersion energy itselfis not iterated but added as a correction to the final Hartree-Fock energy) In contrast EFP2 LMOs are held fixed TheEFP2-EFP2 C6 coefficient is calculated from dynamic po-larizabilities that are determined entirely from a MAKEFPcalculation on the individual monomers Since these valuesdo not change in the construction of the dimer the result-ing C6 value is completely independent of distance The dis-tance dependence of the EFP2-AI C6 coefficient is generallynot significant except at very close intermonomer separationswhere it becomes very large The use of a distance- or overlap-dependent damping function helps to mitigate this instabilityof C6 as the intermonomer separation approaches zero

A comparison of the EFP2-AI and EFP2-EFP2 total C6

coefficients (the sum of the Cv6 values over every LMO

|〉and |v〉) appears in Table I For EFP2-AI the reportedC6 value was calculated at the equilibrium separation foreach dimer except the benzene dimer There are two possible

ways to model the nonsymmetric dimers depending on whichmonomer is AI and which is EFP2 Both possibilities are re-ported in Table I where ldquoleftrdquo and ldquorightrdquo refer to the relativepositions of the monomers as they appear in Fig 1 In hydro-gen bonded dimers the ldquoleftrdquo monomer is the hydrogen bonddonor and the ldquorightrdquo monomer is the hydrogen bond accep-tor The calculated EFP2-AI benzene dimer C6 values are thesame to the given number of significant figures for the sand-wich and T-shaped configurations Compared to experimen-tal values1 27 EFP2-EFP2 C6 coefficients are underestimated(by 58 to 195) for all dimer types except the benzenedimer for which C6 is slightly (48) overestimated For alldimer types other than argon and benzene the EFP2-AI C6

coefficients are similarly underestimated compared to the ex-perimental values (53 to 189) The EFP2-AI C6 valuefor the argon dimer is slightly overestimated (48) and thatfor the benzene dimer is rather overestimated compared to theexperimental value (211)

The benzene molecule is a rare case in which the Hartree-Fock method overestimates the static polarizability6 Typi-cally HF underestimates this quantity6 In Ref 6 the staticpolarizability for benzene determined with CPHF was foundto be 13 greater than the experimental value with the largestbasis set examined [6-311G(3df3pd) in that study6] TheTDHF dynamic polarizability was shown6 to exhibit a clearbasis set dependence as well Since the C6 coefficient is afunction of the polarizability and the static polarizability (atzero frequency) is the leading term in the integration overthe imaginary frequency range it is unsurprising that both theEFP2-EFP2 and EFP2-AI benzene dimer C6 values are over-estimated (Table I) with the large 6-311++G(3df2p) basisset used

For all dimers in Table I other than benzene (and ex-cluding CH2Cl2 for which experimental data are not avail-able) the average absolute error in the EFP2-EFP2 C6 valueis 106 compared to experiment The average absolute er-ror in C6 over all unique EFP2-AI dimers other than ben-zene is 103 The largest EFP2-AI error (besides that for



FIG 2 Dimer dispersion energies (kcalmol) as a function of displacement (in Aring) from the equilibrium separation (denoted 0 Aring) The reference SAPT values(red lines) are the sums of dispersion and exchange-dispersion energies at each dimer geometry Dispersion energies plotted for both EFP2-EFP2 and EFP2-AIwere calculated with each of the two possible damping functions Tang-Toennies (dark green for EFP2-EFP2 light green for EFP2-AI) and overlap-baseddamping (dark blue for EFP2-EFP2 light blue for EFP2-AI)

benzene) occurs with the HF dimer when the hydrogen bondacceptor monomer is modeled with Hartree-Fock and the hy-drogen bond donor monomer with EFP2 the magnitude ofthis error is 189 When the EFP2 and ab initio monomersare switched so that the hydrogen bond acceptor is mod-eled with EFP2 and the hydrogen bond donor with Hartree-Fock this error decreases to 105 Similarly for the waterdimer the calculated C6 coefficient differs from the experi-mental value by 106 when Hartree-Fock is used to modelthe hydrogen bond acceptor but by only 53 when Hartree-Fock is used to model the hydrogen bond donor Howeverin the methanol dimer the errors are very similar regardless

of which monomer is modeled with EFP2 and which withHartree-Fock (both are sim11) For the ammonia dimer thetrend reverses there is a difference of 70 between calcu-lated and experimental C6 values when the hydrogen bonddonor is modeled with Hartree-Fock but the difference de-creases slightly to 60 when the hydrogen bond acceptor ismodeled with Hartree-Fock For other nonsymmetric dimers(H2 CH2Cl2 T-shaped benzene) the C6 value does not de-pend on which monomer is modeled with EFP2 and which ismodeled with Hartree-Fock

The difference between EFP2-EFP2 and EFP2-AI C6 co-efficients has two origins The first origin arises from the



TABLE II Effect on EFP2-AI dispersion energy (kcalmol) for selectednonsymmetrical dimers when EFP2 and ab initio monomers are switched

AI (left)-EFP (right) EFP (left)-AI (right)

H2 minus010 minus010HF minus111 minus101H2O minus178 minus171NH3 minus156 minus170MeOH minus050 minus050CH2Cl2 minus434 minus444

Equilibrium dimer geometries 6-311++G(3df3p) basis set used for both EFP2 andab inito (Hartree-Fock) monomers ldquoLeftrdquo and ldquorightrdquo correspond to the monomer onthe left or right in the dimers pictured in Fig 1 (In hydrogen bonded species ldquoleftrdquocorresponds to the hydrogen bond donor and ldquorightrdquo to the hydrogen bond acceptor)

mathematical formulae used with the respective methods TheEFP2-EFP2 C6 coefficient is a function of polarizability ten-sors calculated via the time-dependent Hartree-Fock equa-tions whereas the EFP2-AI approach relies on applying theorbital approximation to a sum-over-states formula The sec-ond difference between EFP2-EFP2 and EFP2-AI C6 valuesresults from the use of perturbed orbitals in EFP2-AI as dis-cussed above EFP2 orbitals are held fixed but the perturbingfield of the EFP2 fragment is taken into account in construct-ing the orbitals on the ab initio monomer these orbitals are inturn used in the calculation of C6 Since monomers subject toa stronger electrostatic field will have correspondingly moreperturbed orbitals this aspect of the EFP2-AI C6 calculationis distance dependent As a result of these two differences theEFP2-AI method is not necessarily invariant to the choice ofEFP2 and AI monomers (as noted above) However despitethe differences in C6 values the difference in the final disper-sion energy appears to be small on the order of a tenth of akcalmol for the nonsymmetrical dimers examined (Table II)

The total dipole-dipole term of the dispersion energy isgiven by dividing each of the distributed Cv

6 values by R6v

where R is the distance between the centroids of LMOs |〉and |v〉 then summing over all and v Given the general sim-ilarity of EFP2-EFP2 and EFP2-AI C6 values (Table I) it isreasonable to expect that EFP2-AI damping functions analo-gous to those of EFP2-EFP2 (Ref 37) may be used Addition-ally the approximation6 utilized with EFP2-EFP2 to account

for higher order terms in the dispersion expansion of Eq (27)appears to apply equally well for the EFP2-AI dispersionthe expansion in Eq (27) is truncated after C8R8 which isthen approximated6 as 13 of the total dipole-dipole disper-sion term C6R6 With the use of a damping function and theapproximation for C8R8 EFP2-AI dispersion energy valuescompare well with those predicted by SAPT (Figs 2 and 3)

Figures 2 and 3 compare EFP2-EFP2 and EFP2-AI dis-persion energies each with the two possible damping func-tions and SAPT values Distances in Fig 2 for the dimersshown in Figs 1(a)ndash1(h) are given as displacements with re-spect to the equilibrium intermonomer separation (denotedby 0 Aring) Distances in Fig 3 for the sandwich and T-shapedbenzene dimers (Figs 1(i) and 1(j)) are the distances be-tween ring centers The SAPT values (red line) are the sumof SAPT dispersion and exchange-dispersion values as theEFP2-EFP2 and EFP2-AI methods do not explicitly modelexchange-dispersion as a quantity separate from the total dis-persion energy For all dimers examined EFP2-EFP2 Tang-Toennies damped values (dark green line) and EFP2-AI TTdamped values (light green line) are very similar to one an-other as are EFP2-EFP2 overlap-based damped values (darkblue line) and EFP2-AI overlap-based damped values (lightblue line) The difference between the two damping functionsin Fig 2 is more significant for hydrogen-bonded dimers (HFH2O MeOH) than for dimers bound primarily by dispersiveforces (Ar H2 CH4 benzene dimers in Fig 3) This is dueto the Rv or overlap dependence of the respective dampingfunctions hydrogen-bonded dimers tend to have shorter equi-librium intermonomer separations compared to dispersion-bound dimers The Tang-Toennies damping function tends toproduce dispersion energies that are too weak compared toSAPT values Thus due to the superior overall agreement ofoverlap-based damping with the SAPT values (Figs 2 and 3)as well as the fact that overlap-based damping is free of arbi-trary parameters that must otherwise be fitted in some man-ner this is the preferred damping function for use with bothEFP2-EFP2 (Ref 37) and EFP2-AI dispersion

The speed of the EFP2-AI dispersion calculation makesthis method appealing for calculations involving large num-bers of molecules modeled with EFP2 as in solute-solventinteractions For all of the dimers examined calculation of

FIG 3 Sandwich (i) and T-shaped (j) benzene dimer dispersion energies (kcalmol) as a function of distance between ring centers (Aring) The reference SAPTvalues (red lines) are the sums of dispersion and exchange-dispersion energies at each dimer geometry Dispersion energies plotted for both EFP2-EFP2 andEFP2-AI were calculated with each of two possible damping functions Tang-Toennies (dark green for EFP2-EFP2 light green for EFP2-AI) and overlap-baseddamping (dark blue for EFP2-EFP2 light blue for EFP2-AI)



the EFP2-AI dispersion term requires a fraction of a secondEven for the largest system examined (benzene dimer 330basis functions on the AI monomer no symmetry imposed)Hartree-Fock convergence requires sim100 min on a givencomputer while the dispersion energy calculation includingorbital localization requires less than a second The computa-tional cost scales linearly with the number of EFP2 fragments(more precisely with the total number of EFP2 LMOs)

VI CONCLUSION

EFP2-AI distributed C6 coefficients can be obtained bymultiplying AI dipole integrals with an integral that con-sists of the EFP2 dynamic polarizability tensor and a func-tion of AI orbital energies The product is transformed fromthe canonical to the localized molecular orbital basis using aBoys localization26 matrix This gives a distributed C6 valuefor each pair of LMOs |〉 on the ab initio molecule and |v〉on the EFP2 fragment For all systems examined this methodyields values similar to the distributed C6 coefficients in theEFP2-EFP2 dispersion interaction The total EFP2-AI dipole-dipole dispersion interaction energies are obtained by multi-plying the distributed Cv

6 values by Rminus6v where Rv is the

distance between the centroids of LMOs |〉 and |v〉 As withthe EFP2-EFP2 dispersion the dispersion expansion is trun-cated after C8R8 the latter term is approximated as 13 of theC6R6 contribution

This EFP2-AI method is shown to yield good agree-ment with both experimentally determined C6 coefficientsand theoretically determined (SAPT) dispersion energies fora variety of dimers examined The EFP2-AI dispersion ener-gies also agree well with those predicted by the EFP2-EFP2method Compared to experimental values the average ab-solute error in the C6 coefficient is 103 for dimers otherthan benzene The C6 coefficient is underestimated for mostdimers although it is overestimated by more than 20 inbenzene dimers This is a result of the observation that thecoupled perturbed and time-dependent Hartree-Fock meth-ods overestimate the benzene static and dynamic polariz-ability tensors which are used in the calculation of the C6

coefficientTwo damping functions an overlap-based damping func-

tion and a Tang-Toennies damping function are evaluatedfor use with the EFP2-AI dispersion energy calculation Adamping function is necessary because the multipole expan-sion is not guaranteed to converge at short intermonomerseparations As previously recommended for EFP2-EFP2dispersion37 the parameter-free overlap-based damping func-tion is found to give better overall agreement with SAPTresults

ACKNOWLEDGMENTS

This work was supported in part by a grant from the USAir Force Office of Scientific Research (US AFOSR) (QASMSG) by a National Science Foundation (NSF) MultiscaleModeling grant (MSG LVS) by a National Science Foun-dation Petascale Applications grant (QAS MSG) by a

National Science Foundation CAREER grant (LVS) andby the Division of Chemical Sciences Office of Basic En-ergy Sciences US Department of Energy (DOE) under Con-tract No DE-AC02-07CH11358 with Iowa State Universitythrough the Ames Laboratory (KR) The authors are grate-ful for many helpful discussions with Dr Ivana Adamovic

APPENDIX A CONVERSION FROM A SUM OVERSTATES TO AN ORBITAL BASED APPROACH

summ=0

〈0A| μ

A

a |m〉〈m| μ

A


virsumr

〈k| μa |r〉〈r|

μc |k〉

(A1)

Each of the (singly) excited states m in Eq (A1) differsfrom the ground state 0A by one spin orbital

From Ref 49Let |K〉 = | mn 〉 and |L〉 = | pn 〉 ie |K〉 and

|L〉 are Slater determinants differing by one spin orbital (de-noted m and p for each determinant respectively)

Also let the sum of one-electron operators be ϑ1

= sumNi=1 h(i)

Then

〈K| ϑ1 |L〉 = 〈m| h |p〉

Let excited state m = | r 〉 differ from the groundstate 0A = | k 〉 in a single spin orbital (k rarr r) (Notethat assuming single excitations does not result in any loss ofgenerality Slater determinants with doubly triply and higherexcited states give integrals 〈K|ϑ1|L〉 equal to zero) Thenaccording to the above

〈0A| μa |m〉 = 〈k| μa |r〉

Since the sum on the LHS of Eq (A1) goes over all ex-cited states (all possible values of spin orbital k in ground state0A are individually ldquoexcitedrdquo to all possible values of virtualspin orbitals r in excited state m) sums over all occupied andall virtual orbitals are included

summ=0

〈0A| μa |m〉 =occsumk

virsumr

〈k| μa |r〉

Doing this for both integrals on the LHS of Eq (A1) givesthe RHS as indicated

APPENDIX B INTEGRAL OVER IMAGINARYFREQUENCY RANGE

int infin

0dω

ωAm0(

ωAm0

)2+ω2αB

bd (iω)rarrint infin

0dω

ωArk(

ωArk

)2+ω2αB

bd (iω)

where ωrk = ωr minus ωk (B1)

The quantity ωm0 was previously defined by EAm0

= macrωAm0 where Em0 = Em ndash E0 is the excited state energy

(so ω is a frequency multiplied by 2π )



Equation (B1) can be justified by a generalization ofKoopmanrsquos theorem The difference between the ground andexcited states is the subtraction of an electron from an oc-cupied orbital (with energy εi or associated frequency times 2π

ωi) and the addition of an electron to a virtual orbital (withenergy εk or associated frequency times 2π ωk) Invoking thefrozen orbital approximation (neglecting the effect of cor-relation with changing numbers of electrons) subtracting anelectron from an occupied orbital changes the Hartree-Fockenergy EHF by ndashεk adding an electron to a previously unoc-cupied orbital changes the HF energy by +εr Thus

ωm0 = ωm minus ω0 = 1

macr(Em minus E0)

= 1

macr([EHF + εr minus εk] minus EHF ) = 1

macr(εr minus εk)

= ωr minus ωk = ωrk

1A J Stone The Theory of Intermolecular Forces (Clarendon Oxford1996)

2R McWeeny Methods of Molecular Quantum Mechanics 2nd ed(Academic London 1989)

3R D Amos N C Handy P J Knowles J E Rice and A J StoneJ Phys Chem 89 2186 (1985)

4A G Ioannou S M Colwell and R D Amos Chem Phys Lett 278 278(1997)

5R Eisenschitz and F Z London Physik 60 491 (1930) F Z LondonPhysik 63 245 (1930) F Z London Phys Chem 33 8 (1937)

6I Adamovic and M S Gordon Mol Phys 103 379 (2005)7P N Day J H Jensen M S Gordon S P Webb W J Stevens M KraussD Garmer H Basch and D Cohen J Chem Phys 105 1968 (1996)

8M S Gordon M A Freitag P Bandyopadhyay J H Jensen V Kairysand W J Stevens J Phys Chem A 105 293 (2001)

9M S Gordon L V Slipchenko H Li and J H Jensen Annu Rep CompChem 3 177 (2007)

10M W Schmidt K K Baldridge J A Boatz S T Elbert M S Gordon JH Jensen S Koseki N Matsunaga K A Nguyen S J Su T L WindusM Dupuis and J A Montgomery J Comput Chem 14 1347 (1993) MS Gordon and M W Schmidt in Theory and Applications of Computa-tional Chemistry edited by C E Dykstra G Frenking K S Kim and GE Scuseria (Elsevier 2005)

11C Moller and S Plesset Phys Rev 46 618 (1934)12B Jeziorski R Moszynski and K Szalewicz Chem Rev 94 1887 (1994)13M A Freitag M S Gordon J H Jensen and W J Stevens J Chem

Phys 112 7300 (2000)14A J Stone Chem Phys Lett 83 233 (1981)15J H Jensen and M S Gordon Mol Phys 89 1313 (1996)

16D D Kemp J M Rintelman M S Gordon and J H Jensen TheorChem Acc 125 481 (2010)

17H Li M S Gordon and J H Jensen J Chem Phys 124 214108 (2006)18D Ghosh D Kosenkov V Vanovschi C F Williams J M Herbert

M S Gordon M W Schmidt L V Slipchenko and A I Krylov J PhysChem A 114 12739 (2010)

19B T Thole and P Th van Duijnen Chem Phys 71 211 (1982) P Th vanDuijnen and A H de Vries Int J Quantum Chem 60 1111 (1996)

20A J Stone Mol Phys 56 1065 (1985)21P Claverie and R Rein Int J Quantum Chem 3 537 (1969)22O Engkvist P O Aringstrand and G Karlstroumlm Chem Rev 100 4087

(2000)23P Claverie Int J Quantum Chem 5 273 (1971)24H B G Casimir and D Polder Phys Rev 73 360 (1948)25Y Yamaguchi J D Goddard Y Osamura and H F Schaefer A New Di-

mension to Quantum Chemistry Analytic Derivative Methods in Ab InitioMolecular Electronic Structure Theory (Oxford New YorkOxford 1994)

26S F Boys Rev Mod Phys 32 306 (1960) C Edmiston and K Rueden-berg ibid 35 457 (1963)

27E K Gross C A Ullrich and U J Gossmann NATO ASI Ser SerB 337 149 (1995)

28J T Ajit J Chem Phys 89 2092 (1988)29D Cvetko A Lausi A Morgante F Tommasini P Cortona and M G

Dondi J Chem Phys 100 2052 (1994)30A G Donchev Phys Rev B 74 235401 (2006)31G Ihm W C Milton T Flavio and G Scoles J Chem Phys 87 3995

(1987)32S H Patil K T Tang and J P Toennies J Chem Phys 116 8118 (2002)33M Wilson P A Madden N C Pyper and J H Harding J Chem Phys

104 8068 (1996)34A J Stone and A J Misquitta Int Rev Phys Chem 26 193 (2007)35R J Wheatley and W Meath J Mol Phys 80 25 (1993)36K T Tang and J P Toennies J Chem Phys 80 3726 (1984)37L V Slipchenko and M S Gordon Mol Phys 107 999 (2009)38L V Slipchenko and M S Gordon J Comput Chem 28 276 (2007)39K A Peterson and T H Dunning J Chem Phys 102 2032 (1995)40K Raghavachari G W Trucks J A Pople and M Head-Gordon Chem

Phys Lett 157 479 (1989)41D E Woon J Chem Phys 100 2838 (1994)42T H Dunning J Chem Phys 157 479 (1989)43R A Kendall T H Dunning and R J Harrison J Chem Phys 96 6796

(1992)44T Clark J Chandrasekhar G W Spitznagel and P V Schleyer J Comput

Chem 4 294 (1983)45M J Frisch J A Pople and J S Binkley J Chem Phys 80 3265 (1984)46R Krishnan J S Binkley R Seeger and J A Pople J Chem Phys 72

650 (1980)47M O Sinnokrot and C D Sherrill J Phys Chem A 108 10200 (2004)48M M Francl W J Pietro W J Hehre J S Binkley M S Gordon D J

Defrees and J A Pople J Chem Phys 77 3654 (1982)49A Szabo and N S Ostlund Modern Quantum Chemistry (Dover Mineola

NY 1996) pp 68ndash70


THE JOURNAL OF CHEMICAL PHYSICS 136 244107 (2012)

The dispersion interaction between quantum mechanics and effectivefragment potential molecules

Quentin A Smith1 Klaus Ruedenberg1 Mark S Gordon1a)

and Lyudmila V Slipchenko2a)

1Department of Chemistry and Ames Laboratory Iowa State University Ames Iowa 50011 USA2Department of Chemistry Purdue University West Lafayette Indiana 47907 USA

(Received 6 April 2012 accepted 31 May 2012 published online 26 June 2012)

A method for calculating the dispersion energy between molecules modeled with the general effec-tive fragment potential (EFP2) method and those modeled using a full quantum mechanics (QM)method eg Hartree-Fock (HF) or second-order perturbation theory is presented C6 dispersioncoefficients are calculated for pairs of orbitals using dynamic polarizabilities from the EFP2 por-tion and dipole integrals and orbital energies from the QM portion of the system Dividing by thesixth power of the distance between localized molecular orbital centroids yields the first term in thecommonly employed London series expansion A C8 term is estimated from the C6 term to achievecloser agreement with symmetry adapted perturbation theory values Two damping functions forthe dispersion energy are evaluated By using terms that are already computed during an ordinaryHF or EFP2 calculation the new method enables accurate and extremely rapid evaluation of thedispersion interaction between EFP2 and QM molecules copy 2012 American Institute of Physics[httpdxdoiorg10106314729535]

I INTRODUCTION

The dispersion interaction is an attractive force betweenatoms and molecules that is caused by the interactions ofinduced multipoles Dispersion arises from the correlatedmovement of electrons an instantaneous multipole on onemolecule may induce a multipole on another molecule1ndash4

While dispersion is generally a weak intermolecular forceit is the major attractive force between neutral atoms andmolecules that lack permanent multipole moments For ex-ample dispersion is singularly responsible for the attractionbetween atoms of the noble gases5 allowing for their con-densation at low temperatures Because dispersion dependson the atomic or molecular dynamic (frequency dependent)polarizability the dispersion energy is significant betweenatoms or molecules with large diffuse electron clouds suchas π clouds

The general formula for the dispersion interaction be-tween two quantum mechanical (QM) molecules comes fromthe second-order term in intermolecular perturbation theory1

Based on this functional form a formula6 for the dispersioninteraction between two molecules modeled with the gen-eral effective fragment potential (EFP2) (Refs 7ndash9) methodwas previously derived and implemented in the generalatomic and molecular electronic structure system (GAMESS)package10 However a formula for the dispersion energyin mixed systemsmdashthose in which one molecule is mod-eled with EFP2 and another with a full ab initio (AI) QMmethod eg Hartree-Fock (HF) or second-order perturba-tion theory (MP2) (Ref 11)mdashhas been lacking This paperpresents a derivation and implementation of the EFP2-AI

a)Authors to whom correspondence should be addressed Electronicaddresses mgordoniastateedu and lslipchenkopurdueedu

dispersion interaction as well as a comparison of resultingEFP2-AI dispersion energy calculations with those of EFP2-EFP2 and symmetry adapted perturbation theory (SAPT)12

The EFP2-AI dispersion term is important in order to describesolute-solvent interactions where the solute may be modeledwith an ab initio method and the solvent with EFP2 Disper-sion is the primary means of solute-solvent interactions withnonpolar solvents eg cyclohexane

The effective fragment potential (EFP) method is a dis-crete solvent method in GAMESS The original EFP methodEFP1 involves fitted parameters explicitly designed for waterwhile the general method EFP2 contains no fitted param-eters and may be applied to any molecule Parameters nec-essary to describe the fragment potential are generated in aMAKEFP calculation for a given isolated molecule The re-sulting potential may be used in future calculations on anynumber of molecules of the specified type with the same in-ternal geometry Potentials can be used to compute Coulombexchange-repulsion polarization charge transfer and disper-sion energy components in the interaction between two EFP2fragments9 With the current development of the EFP2-AIdispersion term all interaction energy terms except chargetransfer may be calculated between an EFP2 fragment and amolecule modeled with a full ab initio method

The EFP2 Coulomb term is calculated via the distributedmultipolar analysis (DMA) method of Stone1 carried outthrough octopole moments The DMA a classical point-wise model cannot account for the overlap of charge den-sities between molecules that occurs at short distances Assuch charge penetration effects are modeled by a distance-dependent cutoff function13 Polarization or induction isthe interaction between an induced dipole on one frag-ment and the electric field of another In the EFP2 method

0021-96062012136(24)24410712$3000 copy 2012 American Institute of Physics136 244107-1







II THEORY




H 0 =

HA +

HB (1)



V =sumiisinA

sumjisinB

qiqj

Rij

(2)



a

T ABa

(qAμB

a minus μAa qB

)

minusxyzsumab

T ABab μA

a μBb + (3)




T ij = 1

Rij

(4)

T ija = nablaa

1

R= minus

Raij

R3ij

(5)

Tij

ab = nablaanablab

1

R= 3 Ra

ijRbij minus R2

ij δab

R5ij

(6)



summn

〈0A0B |

V |mn〉〈mn|


(7)








〈0A0B |

V |mn〉〈mn|

V |0A0B〉EA

m + EBn minus EA

0 minus EB0

(8)



disp

6 is given by

Edisp

6 = minussumm=0n=0

〈0A0B | sumxyz

ab T ABab

μ

A

a

μ

B


cd T ABcd

μ

A

c

μ

B

d |0A0B〉EA

m + EBn minus EA

0 minus EB0

(9)


Edisp

6 = minusxyzsumabcd

T ABab T AB

cd

summ=0n=0

1

EAm0 + EB

n0

〈0A|μ

A

a |m〉〈m|μ

A

c |0A〉〈0B |μ

B

b |n〉〈n|μ

B

d |0B〉 (10)


1

A + B= 2

π

int infin

0

AB

(A2 + ω2)(B2 + ω2)dω (11)


Edisp

6 = minus2macr

π

xyzsumabcd

T ABab T AB

cd

int infin

0dω

summ=0n=0

ωAm0 〈0A|

μA

a |m〉〈m|μ

A

c |0A〉macr((

ωAm0

)2 + ω2) ωB

n0〈0B |μ

B

b |n〉〈n|μ

B

d |0B〉macr((

ωBn0

)2 + ω2) (12)





αab (ω) =summ

ωm0〈0| μa |m〉〈m|

μb |0〉 + 〈0| μb |m〉〈m|

μa |0〉macr

(ω2

m0 minus ω2) = 2

summ

ωm0 〈0| μa |m〉〈m|

μb |0〉macr

(ω2

m0 minus ω2) (13)




is

Edisp

6 = minus macr2π

xyzsumabcd

T ABab T AB

cd

int infin

0αA







Edisp

6 = minus macr2π

sumkisinA

sumjisinB

xyzsumabcd

Tkj

ab Tkj

cd

int infin

0αk

ac (iω) αj

bd (iω) dω



αBbd (iω) rarr

sumjisinB

αj

bd (iω) (16)



Edisp


sumkisinA

sumjisinB

xyzsumabcd

δacδbdTkj

ab Tkj

cd

timesint infin

0αk

ac (iω) αj

bd (iω) dω

= minus macr2π

sumkisinA

sumjisinB

xyzsumab

Tkj

ab Tkj

ab

int infin

0αk

aa (iω) αj

bb (iω) dω

(17)





Tkj

ab Tkj

ab = 6

R6kj

(19)


Edisp

6 = minus 3

π

sumkisinA

sumjisinB

1

R6kj

int infin



C EFP-AI dispersion



π

xyzsumabcd

T ABab T AB

cd

int infin

0dω

summ=0

ωAm0 〈0A|

μA

a |m〉〈m| μ

A

c |0A〉macr((

ωAm0

)2 + ω2) αB

bd (iω)

= minus 1

π

xyzsumabcd

T ABab T AB

cd

summ=0

〈0A| μ

A

a |m〉〈m| μ

A

c |0A〉int infin

0dω

ωAm0(

ωAm0

)2 + ω2αB

bd (iω)

(21)




summ=0

〈0A| μ

A

a |m〉〈m| μ

A


virsumr

〈k| μ

A

a |r〉〈r| μ

A

c |k〉 (22)


int infin

0dω

ωAm0(

ωAm0

)2 + ω2αB


0dω

ωArk(

ωArk

)2 + ω2αB




π

sumjisinB

xyzsumabcd

occsumk

virsumr

Tkj

ab Tkj

cd 〈k| μ

A

a |r〉〈r| μ

A

c |k〉int infin

0dω

ωArk(

ωArk

)2 + ω2α

j

bd (iω) (24)



π

sumjisinB

xyzsumabcd

occsumk

virsumr

δacδbdTkj

ab Tkj

cd 〈k| μ

A

a |r〉〈r| μ

A

c |k〉int infin

0dω

ωArk(

ωArk

)2 + ω2α

j

bd (iω)

= minus 1

π

sumjisinB

xyzsumab

occsumk

virsumr

Tkj

ab Tkj

ab 〈k| μ

A

a |r〉〈r| μ

A

a |k〉int infin

0dω

ωArk(

ωArk

)2 + ω2α

j

bb (iω) (25)

The product Tkj

ab Tkj



3

sumxyza 〈k| μA




π

sumjisinB

occsumk

virsumr

1

R6kj

〈k| μ |r〉〈r|

μ |k〉int infin

0dω

ωArk(

ωArk

)2 + ω2αj (iω) (26)


Edisp = C6

R6+ C8

R8+ C10

R10+ (27)



sumkisinA

sumjisinB

Ckj

6

R6kj

(28)


Ckj

6 = 6

π

virsumr

〈k| μ |r〉〈r|

μ |k〉int infin

0dω

ωArk(

ωArk

)2 + ω2αj (iω)

(29)




sumisinA

sumvisinB

Cv6

R6v

(30)




6 is

Cv6 = 1

3middot 6

π

xyzsuma

Cv6a = 2

π

xyzsuma

Cv6a (31)


Cv6a =

virsumr


〈r| μa


T primek

timesint infin

0dω

ωArk(

ωArk

)2 + ω2αv(iω) (32)




|〉 =sum

k




Cv6a =

valencesumk

valencesumkprime

Lk

[virsumr


timesint infin

0dω

ωArkprime(

ωArkprime

)2 + ω2αv(iω)

]Lkprime (35)





I vkr =int infin

0dω

ωArk(

ωArk





ω = ν0(1 + t)


(1 minus t)2 (37)


I vkr =12sum

n=1

wn

2ν0

(1 minus tn)2

ωrk

ω2rk + ω2

n

αv (iωn)


(1 minus tn) (38)


I vkr =12sum

n=1

wn

2ν0

(1 minus tn)2

εrk

ε2rk + ω2

n

αv (iωn)




a middot V IR (40)


〈χ1 |μ

A




are 〈k| μ

A











3

sumisinA

sumvisinB

F v6 Cv

6

R6v

(41)




6sumn=1

(βRv)n

n (42)



Foverlap




























CH2Cl2 8430 8430 7558 NAC6H6 ()f 2087 (+211) 2087 (+211) 1805 (+48) 1723d



pj


pj


pj

























6 values by R6v









VI CONCLUSION







ACKNOWLEDGMENTS




summ=0

〈0A| μ

A

a |m〉〈m| μ

A


virsumr

〈k| μa |r〉〈r|

μc |k〉

(A1)





= sumNi=1 h(i)

Then

〈K| ϑ1 |L〉 = 〈m| h |p〉


〈0A| μa |m〉 = 〈k| μa |r〉


summ=0


virsumr

〈k| μa |r〉



int infin

0dω

ωAm0(

ωAm0

)2+ω2αB


0dω

ωArk(

ωArk

)2+ω2αB

bd (iω)










macr(Em minus E0)

= 1


macr(εr minus εk)






































II THEORY




H 0 =

HA +

HB (1)



V =sumiisinA

sumjisinB

qiqj

Rij

(2)



a

T ABa

(qAμB

a minus μAa qB

)

minusxyzsumab

T ABab μA

a μBb + (3)




T ij = 1

Rij

(4)

T ija = nablaa

1

R= minus

Raij

R3ij

(5)

Tij

ab = nablaanablab

1

R= 3 Ra

ijRbij minus R2

ij δab

R5ij

(6)



summn

〈0A0B |

V |mn〉〈mn|


(7)








〈0A0B |

V |mn〉〈mn|

V |0A0B〉EA

m + EBn minus EA

0 minus EB0

(8)



disp

6 is given by

Edisp

6 = minussumm=0n=0

〈0A0B | sumxyz

ab T ABab

μ

A

a

μ

B


cd T ABcd

μ

A

c

μ

B

d |0A0B〉EA

m + EBn minus EA

0 minus EB0

(9)


Edisp

6 = minusxyzsumabcd

T ABab T AB

cd

summ=0n=0

1

EAm0 + EB

n0

〈0A|μ

A

a |m〉〈m|μ

A

c |0A〉〈0B |μ

B

b |n〉〈n|μ

B

d |0B〉 (10)


1

A + B= 2

π

int infin

0

AB

(A2 + ω2)(B2 + ω2)dω (11)


Edisp

6 = minus2macr

π

xyzsumabcd

T ABab T AB

cd

int infin

0dω

summ=0n=0

ωAm0 〈0A|

μA

a |m〉〈m|μ

A

c |0A〉macr((

ωAm0

)2 + ω2) ωB

n0〈0B |μ

B

b |n〉〈n|μ

B

d |0B〉macr((

ωBn0

)2 + ω2) (12)





αab (ω) =summ

ωm0〈0| μa |m〉〈m|

μb |0〉 + 〈0| μb |m〉〈m|

μa |0〉macr

(ω2

m0 minus ω2) = 2

summ

ωm0 〈0| μa |m〉〈m|

μb |0〉macr

(ω2

m0 minus ω2) (13)




is

Edisp

6 = minus macr2π

xyzsumabcd

T ABab T AB

cd

int infin

0αA







Edisp

6 = minus macr2π

sumkisinA

sumjisinB

xyzsumabcd

Tkj

ab Tkj

cd

int infin

0αk

ac (iω) αj

bd (iω) dω



αBbd (iω) rarr

sumjisinB

αj

bd (iω) (16)



Edisp


sumkisinA

sumjisinB

xyzsumabcd

δacδbdTkj

ab Tkj

cd

timesint infin

0αk

ac (iω) αj

bd (iω) dω

= minus macr2π

sumkisinA

sumjisinB

xyzsumab

Tkj

ab Tkj

ab

int infin

0αk

aa (iω) αj

bb (iω) dω

(17)





Tkj

ab Tkj

ab = 6

R6kj

(19)


Edisp

6 = minus 3

π

sumkisinA

sumjisinB

1

R6kj

int infin



C EFP-AI dispersion



π

xyzsumabcd

T ABab T AB

cd

int infin

0dω

summ=0

ωAm0 〈0A|

μA

a |m〉〈m| μ

A

c |0A〉macr((

ωAm0

)2 + ω2) αB

bd (iω)

= minus 1

π

xyzsumabcd

T ABab T AB

cd

summ=0

〈0A| μ

A

a |m〉〈m| μ

A

c |0A〉int infin

0dω

ωAm0(

ωAm0

)2 + ω2αB

bd (iω)

(21)




summ=0

〈0A| μ

A

a |m〉〈m| μ

A


virsumr

〈k| μ

A

a |r〉〈r| μ

A

c |k〉 (22)


int infin

0dω

ωAm0(

ωAm0

)2 + ω2αB


0dω

ωArk(

ωArk

)2 + ω2αB




π

sumjisinB

xyzsumabcd

occsumk

virsumr

Tkj

ab Tkj

cd 〈k| μ

A

a |r〉〈r| μ

A

c |k〉int infin

0dω

ωArk(

ωArk

)2 + ω2α

j

bd (iω) (24)



π

sumjisinB

xyzsumabcd

occsumk

virsumr

δacδbdTkj

ab Tkj

cd 〈k| μ

A

a |r〉〈r| μ

A

c |k〉int infin

0dω

ωArk(

ωArk

)2 + ω2α

j

bd (iω)

= minus 1

π

sumjisinB

xyzsumab

occsumk

virsumr

Tkj

ab Tkj

ab 〈k| μ

A

a |r〉〈r| μ

A

a |k〉int infin

0dω

ωArk(

ωArk

)2 + ω2α

j

bb (iω) (25)

The product Tkj

ab Tkj



3

sumxyza 〈k| μA




π

sumjisinB

occsumk

virsumr

1

R6kj

〈k| μ |r〉〈r|

μ |k〉int infin

0dω

ωArk(

ωArk

)2 + ω2αj (iω) (26)


Edisp = C6

R6+ C8

R8+ C10

R10+ (27)



sumkisinA

sumjisinB

Ckj

6

R6kj

(28)


Ckj

6 = 6

π

virsumr

〈k| μ |r〉〈r|

μ |k〉int infin

0dω

ωArk(

ωArk

)2 + ω2αj (iω)

(29)




sumisinA

sumvisinB

Cv6

R6v

(30)




6 is

Cv6 = 1

3middot 6

π

xyzsuma

Cv6a = 2

π

xyzsuma

Cv6a (31)


Cv6a =

virsumr


〈r| μa


T primek

timesint infin

0dω

ωArk(

ωArk

)2 + ω2αv(iω) (32)




|〉 =sum

k




Cv6a =

valencesumk

valencesumkprime

Lk

[virsumr


timesint infin

0dω

ωArkprime(

ωArkprime

)2 + ω2αv(iω)

]Lkprime (35)





I vkr =int infin

0dω

ωArk(

ωArk





ω = ν0(1 + t)


(1 minus t)2 (37)


I vkr =12sum

n=1

wn

2ν0

(1 minus tn)2

ωrk

ω2rk + ω2

n

αv (iωn)


(1 minus tn) (38)


I vkr =12sum

n=1

wn

2ν0

(1 minus tn)2

εrk

ε2rk + ω2

n

αv (iωn)




a middot V IR (40)


〈χ1 |μ

A




are 〈k| μ

A











3

sumisinA

sumvisinB

F v6 Cv

6

R6v

(41)




6sumn=1

(βRv)n

n (42)



Foverlap




























CH2Cl2 8430 8430 7558 NAC6H6 ()f 2087 (+211) 2087 (+211) 1805 (+48) 1723d



pj


pj


pj

























6 values by R6v









VI CONCLUSION







ACKNOWLEDGMENTS




summ=0

〈0A| μ

A

a |m〉〈m| μ

A


virsumr

〈k| μa |r〉〈r|

μc |k〉

(A1)





= sumNi=1 h(i)

Then

〈K| ϑ1 |L〉 = 〈m| h |p〉


〈0A| μa |m〉 = 〈k| μa |r〉


summ=0


virsumr

〈k| μa |r〉



int infin

0dω

ωAm0(

ωAm0

)2+ω2αB


0dω

ωArk(

ωArk

)2+ω2αB

bd (iω)










macr(Em minus E0)

= 1


macr(εr minus εk)




































〈0A0B |

V |mn〉〈mn|

V |0A0B〉EA

m + EBn minus EA

0 minus EB0

(8)



disp

6 is given by

Edisp

6 = minussumm=0n=0

〈0A0B | sumxyz

ab T ABab

μ

A

a

μ

B


cd T ABcd

μ

A

c

μ

B

d |0A0B〉EA

m + EBn minus EA

0 minus EB0

(9)


Edisp

6 = minusxyzsumabcd

T ABab T AB

cd

summ=0n=0

1

EAm0 + EB

n0

〈0A|μ

A

a |m〉〈m|μ

A

c |0A〉〈0B |μ

B

b |n〉〈n|μ

B

d |0B〉 (10)


1

A + B= 2

π

int infin

0

AB

(A2 + ω2)(B2 + ω2)dω (11)


Edisp

6 = minus2macr

π

xyzsumabcd

T ABab T AB

cd

int infin

0dω

summ=0n=0

ωAm0 〈0A|

μA

a |m〉〈m|μ

A

c |0A〉macr((

ωAm0

)2 + ω2) ωB

n0〈0B |μ

B

b |n〉〈n|μ

B

d |0B〉macr((

ωBn0

)2 + ω2) (12)





αab (ω) =summ

ωm0〈0| μa |m〉〈m|

μb |0〉 + 〈0| μb |m〉〈m|

μa |0〉macr

(ω2

m0 minus ω2) = 2

summ

ωm0 〈0| μa |m〉〈m|

μb |0〉macr

(ω2

m0 minus ω2) (13)




is

Edisp

6 = minus macr2π

xyzsumabcd

T ABab T AB

cd

int infin

0αA







Edisp

6 = minus macr2π

sumkisinA

sumjisinB

xyzsumabcd

Tkj

ab Tkj

cd

int infin

0αk

ac (iω) αj

bd (iω) dω



αBbd (iω) rarr

sumjisinB

αj

bd (iω) (16)



Edisp


sumkisinA

sumjisinB

xyzsumabcd

δacδbdTkj

ab Tkj

cd

timesint infin

0αk

ac (iω) αj

bd (iω) dω

= minus macr2π

sumkisinA

sumjisinB

xyzsumab

Tkj

ab Tkj

ab

int infin

0αk

aa (iω) αj

bb (iω) dω

(17)





Tkj

ab Tkj

ab = 6

R6kj

(19)


Edisp

6 = minus 3

π

sumkisinA

sumjisinB

1

R6kj

int infin



C EFP-AI dispersion



π

xyzsumabcd

T ABab T AB

cd

int infin

0dω

summ=0

ωAm0 〈0A|

μA

a |m〉〈m| μ

A

c |0A〉macr((

ωAm0

)2 + ω2) αB

bd (iω)

= minus 1

π

xyzsumabcd

T ABab T AB

cd

summ=0

〈0A| μ

A

a |m〉〈m| μ

A

c |0A〉int infin

0dω

ωAm0(

ωAm0

)2 + ω2αB

bd (iω)

(21)




summ=0

〈0A| μ

A

a |m〉〈m| μ

A


virsumr

〈k| μ

A

a |r〉〈r| μ

A

c |k〉 (22)


int infin

0dω

ωAm0(

ωAm0

)2 + ω2αB


0dω

ωArk(

ωArk

)2 + ω2αB




π

sumjisinB

xyzsumabcd

occsumk

virsumr

Tkj

ab Tkj

cd 〈k| μ

A

a |r〉〈r| μ

A

c |k〉int infin

0dω

ωArk(

ωArk

)2 + ω2α

j

bd (iω) (24)



π

sumjisinB

xyzsumabcd

occsumk

virsumr

δacδbdTkj

ab Tkj

cd 〈k| μ

A

a |r〉〈r| μ

A

c |k〉int infin

0dω

ωArk(

ωArk

)2 + ω2α

j

bd (iω)

= minus 1

π

sumjisinB

xyzsumab

occsumk

virsumr

Tkj

ab Tkj

ab 〈k| μ

A

a |r〉〈r| μ

A

a |k〉int infin

0dω

ωArk(

ωArk

)2 + ω2α

j

bb (iω) (25)

The product Tkj

ab Tkj



3

sumxyza 〈k| μA




π

sumjisinB

occsumk

virsumr

1

R6kj

〈k| μ |r〉〈r|

μ |k〉int infin

0dω

ωArk(

ωArk

)2 + ω2αj (iω) (26)


Edisp = C6

R6+ C8

R8+ C10

R10+ (27)



sumkisinA

sumjisinB

Ckj

6

R6kj

(28)


Ckj

6 = 6

π

virsumr

〈k| μ |r〉〈r|

μ |k〉int infin

0dω

ωArk(

ωArk

)2 + ω2αj (iω)

(29)




sumisinA

sumvisinB

Cv6

R6v

(30)




6 is

Cv6 = 1

3middot 6

π

xyzsuma

Cv6a = 2

π

xyzsuma

Cv6a (31)


Cv6a =

virsumr


〈r| μa


T primek

timesint infin

0dω

ωArk(

ωArk

)2 + ω2αv(iω) (32)




|〉 =sum

k




Cv6a =

valencesumk

valencesumkprime

Lk

[virsumr


timesint infin

0dω

ωArkprime(

ωArkprime

)2 + ω2αv(iω)

]Lkprime (35)





I vkr =int infin

0dω

ωArk(

ωArk





ω = ν0(1 + t)


(1 minus t)2 (37)


I vkr =12sum

n=1

wn

2ν0

(1 minus tn)2

ωrk

ω2rk + ω2

n

αv (iωn)


(1 minus tn) (38)


I vkr =12sum

n=1

wn

2ν0

(1 minus tn)2

εrk

ε2rk + ω2

n

αv (iωn)




a middot V IR (40)


〈χ1 |μ

A




are 〈k| μ

A











3

sumisinA

sumvisinB

F v6 Cv

6

R6v

(41)




6sumn=1

(βRv)n

n (42)



Foverlap




























CH2Cl2 8430 8430 7558 NAC6H6 ()f 2087 (+211) 2087 (+211) 1805 (+48) 1723d



pj


pj


pj

























6 values by R6v









VI CONCLUSION







ACKNOWLEDGMENTS




summ=0

〈0A| μ

A

a |m〉〈m| μ

A


virsumr

〈k| μa |r〉〈r|

μc |k〉

(A1)





= sumNi=1 h(i)

Then

〈K| ϑ1 |L〉 = 〈m| h |p〉


〈0A| μa |m〉 = 〈k| μa |r〉


summ=0


virsumr

〈k| μa |r〉



int infin

0dω

ωAm0(

ωAm0

)2+ω2αB


0dω

ωArk(

ωArk

)2+ω2αB

bd (iω)










macr(Em minus E0)

= 1


macr(εr minus εk)



































Edisp

6 = minus macr2π

sumkisinA

sumjisinB

xyzsumabcd

Tkj

ab Tkj

cd

int infin

0αk

ac (iω) αj

bd (iω) dω



αBbd (iω) rarr

sumjisinB

αj

bd (iω) (16)



Edisp


sumkisinA

sumjisinB

xyzsumabcd

δacδbdTkj

ab Tkj

cd

timesint infin

0αk

ac (iω) αj

bd (iω) dω

= minus macr2π

sumkisinA

sumjisinB

xyzsumab

Tkj

ab Tkj

ab

int infin

0αk

aa (iω) αj

bb (iω) dω

(17)





Tkj

ab Tkj

ab = 6

R6kj

(19)


Edisp

6 = minus 3

π

sumkisinA

sumjisinB

1

R6kj

int infin



C EFP-AI dispersion



π

xyzsumabcd

T ABab T AB

cd

int infin

0dω

summ=0

ωAm0 〈0A|

μA

a |m〉〈m| μ

A

c |0A〉macr((

ωAm0

)2 + ω2) αB

bd (iω)

= minus 1

π

xyzsumabcd

T ABab T AB

cd

summ=0

〈0A| μ

A

a |m〉〈m| μ

A

c |0A〉int infin

0dω

ωAm0(

ωAm0

)2 + ω2αB

bd (iω)

(21)




summ=0

〈0A| μ

A

a |m〉〈m| μ

A


virsumr

〈k| μ

A

a |r〉〈r| μ

A

c |k〉 (22)


int infin

0dω

ωAm0(

ωAm0

)2 + ω2αB


0dω

ωArk(

ωArk

)2 + ω2αB




π

sumjisinB

xyzsumabcd

occsumk

virsumr

Tkj

ab Tkj

cd 〈k| μ

A

a |r〉〈r| μ

A

c |k〉int infin

0dω

ωArk(

ωArk

)2 + ω2α

j

bd (iω) (24)



π

sumjisinB

xyzsumabcd

occsumk

virsumr

δacδbdTkj

ab Tkj

cd 〈k| μ

A

a |r〉〈r| μ

A

c |k〉int infin

0dω

ωArk(

ωArk

)2 + ω2α

j

bd (iω)

= minus 1

π

sumjisinB

xyzsumab

occsumk

virsumr

Tkj

ab Tkj

ab 〈k| μ

A

a |r〉〈r| μ

A

a |k〉int infin

0dω

ωArk(

ωArk

)2 + ω2α

j

bb (iω) (25)

The product Tkj

ab Tkj



3

sumxyza 〈k| μA




π

sumjisinB

occsumk

virsumr

1

R6kj

〈k| μ |r〉〈r|

μ |k〉int infin

0dω

ωArk(

ωArk

)2 + ω2αj (iω) (26)


Edisp = C6

R6+ C8

R8+ C10

R10+ (27)



sumkisinA

sumjisinB

Ckj

6

R6kj

(28)


Ckj

6 = 6

π

virsumr

〈k| μ |r〉〈r|

μ |k〉int infin

0dω

ωArk(

ωArk

)2 + ω2αj (iω)

(29)




sumisinA

sumvisinB

Cv6

R6v

(30)




6 is

Cv6 = 1

3middot 6

π

xyzsuma

Cv6a = 2

π

xyzsuma

Cv6a (31)


Cv6a =

virsumr


〈r| μa


T primek

timesint infin

0dω

ωArk(

ωArk

)2 + ω2αv(iω) (32)




|〉 =sum

k




Cv6a =

valencesumk

valencesumkprime

Lk

[virsumr


timesint infin

0dω

ωArkprime(

ωArkprime

)2 + ω2αv(iω)

]Lkprime (35)





I vkr =int infin

0dω

ωArk(

ωArk





ω = ν0(1 + t)


(1 minus t)2 (37)


I vkr =12sum

n=1

wn

2ν0

(1 minus tn)2

ωrk

ω2rk + ω2

n

αv (iωn)


(1 minus tn) (38)


I vkr =12sum

n=1

wn

2ν0

(1 minus tn)2

εrk

ε2rk + ω2

n

αv (iωn)




a middot V IR (40)


〈χ1 |μ

A




are 〈k| μ

A











3

sumisinA

sumvisinB

F v6 Cv

6

R6v

(41)




6sumn=1

(βRv)n

n (42)



Foverlap




























CH2Cl2 8430 8430 7558 NAC6H6 ()f 2087 (+211) 2087 (+211) 1805 (+48) 1723d



pj


pj


pj

























6 values by R6v









VI CONCLUSION







ACKNOWLEDGMENTS




summ=0

〈0A| μ

A

a |m〉〈m| μ

A


virsumr

〈k| μa |r〉〈r|

μc |k〉

(A1)





= sumNi=1 h(i)

Then

〈K| ϑ1 |L〉 = 〈m| h |p〉


〈0A| μa |m〉 = 〈k| μa |r〉


summ=0


virsumr

〈k| μa |r〉



int infin

0dω

ωAm0(

ωAm0

)2+ω2αB


0dω

ωArk(

ωArk

)2+ω2αB

bd (iω)










macr(Em minus E0)

= 1


macr(εr minus εk)



































summ=0

〈0A| μ

A

a |m〉〈m| μ

A


virsumr

〈k| μ

A

a |r〉〈r| μ

A

c |k〉 (22)


int infin

0dω

ωAm0(

ωAm0

)2 + ω2αB


0dω

ωArk(

ωArk

)2 + ω2αB




π

sumjisinB

xyzsumabcd

occsumk

virsumr

Tkj

ab Tkj

cd 〈k| μ

A

a |r〉〈r| μ

A

c |k〉int infin

0dω

ωArk(

ωArk

)2 + ω2α

j

bd (iω) (24)



π

sumjisinB

xyzsumabcd

occsumk

virsumr

δacδbdTkj

ab Tkj

cd 〈k| μ

A

a |r〉〈r| μ

A

c |k〉int infin

0dω

ωArk(

ωArk

)2 + ω2α

j

bd (iω)

= minus 1

π

sumjisinB

xyzsumab

occsumk

virsumr

Tkj

ab Tkj

ab 〈k| μ

A

a |r〉〈r| μ

A

a |k〉int infin

0dω

ωArk(

ωArk

)2 + ω2α

j

bb (iω) (25)

The product Tkj

ab Tkj



3

sumxyza 〈k| μA




π

sumjisinB

occsumk

virsumr

1

R6kj

〈k| μ |r〉〈r|

μ |k〉int infin

0dω

ωArk(

ωArk

)2 + ω2αj (iω) (26)


Edisp = C6

R6+ C8

R8+ C10

R10+ (27)



sumkisinA

sumjisinB

Ckj

6

R6kj

(28)


Ckj

6 = 6

π

virsumr

〈k| μ |r〉〈r|

μ |k〉int infin

0dω

ωArk(

ωArk

)2 + ω2αj (iω)

(29)




sumisinA

sumvisinB

Cv6

R6v

(30)




6 is

Cv6 = 1

3middot 6

π

xyzsuma

Cv6a = 2

π

xyzsuma

Cv6a (31)


Cv6a =

virsumr


〈r| μa


T primek

timesint infin

0dω

ωArk(

ωArk

)2 + ω2αv(iω) (32)




|〉 =sum

k




Cv6a =

valencesumk

valencesumkprime

Lk

[virsumr


timesint infin

0dω

ωArkprime(

ωArkprime

)2 + ω2αv(iω)

]Lkprime (35)





I vkr =int infin

0dω

ωArk(

ωArk





ω = ν0(1 + t)


(1 minus t)2 (37)


I vkr =12sum

n=1

wn

2ν0

(1 minus tn)2

ωrk

ω2rk + ω2

n

αv (iωn)


(1 minus tn) (38)


I vkr =12sum

n=1

wn

2ν0

(1 minus tn)2

εrk

ε2rk + ω2

n

αv (iωn)




a middot V IR (40)


〈χ1 |μ

A




are 〈k| μ

A











3

sumisinA

sumvisinB

F v6 Cv

6

R6v

(41)




6sumn=1

(βRv)n

n (42)



Foverlap




























CH2Cl2 8430 8430 7558 NAC6H6 ()f 2087 (+211) 2087 (+211) 1805 (+48) 1723d



pj


pj


pj

























6 values by R6v









VI CONCLUSION







ACKNOWLEDGMENTS




summ=0

〈0A| μ

A

a |m〉〈m| μ

A


virsumr

〈k| μa |r〉〈r|

μc |k〉

(A1)





= sumNi=1 h(i)

Then

〈K| ϑ1 |L〉 = 〈m| h |p〉


〈0A| μa |m〉 = 〈k| μa |r〉


summ=0


virsumr

〈k| μa |r〉



int infin

0dω

ωAm0(

ωAm0

)2+ω2αB


0dω

ωArk(

ωArk

)2+ω2αB

bd (iω)










macr(Em minus E0)

= 1


macr(εr minus εk)



































6 is

Cv6 = 1

3middot 6

π

xyzsuma

Cv6a = 2

π

xyzsuma

Cv6a (31)


Cv6a =

virsumr


〈r| μa


T primek

timesint infin

0dω

ωArk(

ωArk

)2 + ω2αv(iω) (32)




|〉 =sum

k




Cv6a =

valencesumk

valencesumkprime

Lk

[virsumr


timesint infin

0dω

ωArkprime(

ωArkprime

)2 + ω2αv(iω)

]Lkprime (35)





I vkr =int infin

0dω

ωArk(

ωArk





ω = ν0(1 + t)


(1 minus t)2 (37)


I vkr =12sum

n=1

wn

2ν0

(1 minus tn)2

ωrk

ω2rk + ω2

n

αv (iωn)


(1 minus tn) (38)


I vkr =12sum

n=1

wn

2ν0

(1 minus tn)2

εrk

ε2rk + ω2

n

αv (iωn)




a middot V IR (40)


〈χ1 |μ

A




are 〈k| μ

A











3

sumisinA

sumvisinB

F v6 Cv

6

R6v

(41)




6sumn=1

(βRv)n

n (42)



Foverlap




























CH2Cl2 8430 8430 7558 NAC6H6 ()f 2087 (+211) 2087 (+211) 1805 (+48) 1723d



pj


pj


pj

























6 values by R6v









VI CONCLUSION







ACKNOWLEDGMENTS




summ=0

〈0A| μ

A

a |m〉〈m| μ

A


virsumr

〈k| μa |r〉〈r|

μc |k〉

(A1)





= sumNi=1 h(i)

Then

〈K| ϑ1 |L〉 = 〈m| h |p〉


〈0A| μa |m〉 = 〈k| μa |r〉


summ=0


virsumr

〈k| μa |r〉



int infin

0dω

ωAm0(

ωAm0

)2+ω2αB


0dω

ωArk(

ωArk

)2+ω2αB

bd (iω)










macr(Em minus E0)

= 1


macr(εr minus εk)





































3

sumisinA

sumvisinB

F v6 Cv

6

R6v

(41)




6sumn=1

(βRv)n

n (42)



Foverlap




























CH2Cl2 8430 8430 7558 NAC6H6 ()f 2087 (+211) 2087 (+211) 1805 (+48) 1723d



pj


pj


pj

























6 values by R6v









VI CONCLUSION







ACKNOWLEDGMENTS




summ=0

〈0A| μ

A

a |m〉〈m| μ

A


virsumr

〈k| μa |r〉〈r|

μc |k〉

(A1)





= sumNi=1 h(i)

Then

〈K| ϑ1 |L〉 = 〈m| h |p〉


〈0A| μa |m〉 = 〈k| μa |r〉


summ=0


virsumr

〈k| μa |r〉



int infin

0dω

ωAm0(

ωAm0

)2+ω2αB


0dω

ωArk(

ωArk

)2+ω2αB

bd (iω)










macr(Em minus E0)

= 1


macr(εr minus εk)











































CH2Cl2 8430 8430 7558 NAC6H6 ()f 2087 (+211) 2087 (+211) 1805 (+48) 1723d



pj


pj


pj

























6 values by R6v









VI CONCLUSION







ACKNOWLEDGMENTS




summ=0

〈0A| μ

A

a |m〉〈m| μ

A


virsumr

〈k| μa |r〉〈r|

μc |k〉

(A1)





= sumNi=1 h(i)

Then

〈K| ϑ1 |L〉 = 〈m| h |p〉


〈0A| μa |m〉 = 〈k| μa |r〉


summ=0


virsumr

〈k| μa |r〉



int infin

0dω

ωAm0(

ωAm0

)2+ω2αB


0dω

ωArk(

ωArk

)2+ω2αB

bd (iω)










macr(Em minus E0)

= 1


macr(εr minus εk)














































6 values by R6v









VI CONCLUSION







ACKNOWLEDGMENTS




summ=0

〈0A| μ

A

a |m〉〈m| μ

A


virsumr

〈k| μa |r〉〈r|

μc |k〉

(A1)





= sumNi=1 h(i)

Then

〈K| ϑ1 |L〉 = 〈m| h |p〉


〈0A| μa |m〉 = 〈k| μa |r〉


summ=0


virsumr

〈k| μa |r〉



int infin

0dω

ωAm0(

ωAm0

)2+ω2αB


0dω

ωArk(

ωArk

)2+ω2αB

bd (iω)










macr(Em minus E0)

= 1


macr(εr minus εk)








































6 values by R6v









VI CONCLUSION







ACKNOWLEDGMENTS




summ=0

〈0A| μ

A

a |m〉〈m| μ

A


virsumr

〈k| μa |r〉〈r|

μc |k〉

(A1)





= sumNi=1 h(i)

Then

〈K| ϑ1 |L〉 = 〈m| h |p〉


〈0A| μa |m〉 = 〈k| μa |r〉


summ=0


virsumr

〈k| μa |r〉



int infin

0dω

ωAm0(

ωAm0

)2+ω2αB


0dω

ωArk(

ωArk

)2+ω2αB

bd (iω)










macr(Em minus E0)

= 1


macr(εr minus εk)



































VI CONCLUSION







ACKNOWLEDGMENTS




summ=0

〈0A| μ

A

a |m〉〈m| μ

A


virsumr

〈k| μa |r〉〈r|

μc |k〉

(A1)





= sumNi=1 h(i)

Then

〈K| ϑ1 |L〉 = 〈m| h |p〉


〈0A| μa |m〉 = 〈k| μa |r〉


summ=0


virsumr

〈k| μa |r〉



int infin

0dω

ωAm0(

ωAm0

)2+ω2αB


0dω

ωArk(

ωArk

)2+ω2αB

bd (iω)










macr(Em minus E0)

= 1


macr(εr minus εk)





































macr(Em minus E0)

= 1


macr(εr minus εk)

































Documents

The dispersion interaction between quantum mechanics and ... · atoms or molecules with large, diffuse electron clouds, such as π clouds. The general formula for the dispersion interaction