Following earlier studies 67101112, each atom i(i = 1, 2, ..., n) is considered as a three-dimensional (3D) harmonic quantum oscillator, where an electron with mass m_i oscillates harmonically with a frequency ω_i, if unaffected by other atoms. Atom i is fixed in the 3D point r_i; it has an instantaneous dipole moment ex_i, where e is the elementary charge and x_i the 3D vector of instantaneous displacement (whose equilibrium value is zero). Thus, a classic Hamiltonian for the system of coupled oscillators is

H = ∑ i m i 2 ( d x i d t ) 2 + ∑ i m i ω i 2 2 x i 2 + ∑ i ∑ j > i e 2 θ i j | r i j | 3 x i x j ; MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGibascqGH9aqpdaaeqbqaamaalaaabaGaemyBa02aaSbaaSqaaiabdMgaPbqabaaakeaacqaIYaGmaaWaaeWaaeaadaWcaaqaaiabdsgaKjabbIha4naaBaaaleaacqWGPbqAaeqaaaGcbaGaemizaqMaemiDaqhaaaGaayjkaiaawMcaamaaCaaaleqabaGaeGOmaidaaaqaaiabdMgaPbqab0GaeyyeIuoakiabgUcaRmaaqafabaWaaSaaaeaacqWGTbqBdaWgaaWcbaGaemyAaKgabeaaiiGakiab=L8a3naaDaaaleaacqWGPbqAaeaacqaIYaGmaaaakeaacqaIYaGmaaGaeeiEaG3aa0baaSqaaiabdMgaPbqaaiabbkdaYaaaaeaacqWGPbqAaeqaniabggHiLdGccqGHRaWkdaaeqbqaamaaqafabaGaemyzau2aaWbaaSqabeaacqaIYaGmaaGcdaWcaaqaaGGabiab+H7aXnaaBaaaleaacqWGPbqAcqWGQbGAaeqaaaGcbaGaeiiFaWNaeeOCai3aaSbaaSqaaiabdMgaPjabdQgaQbqabaGccqGG8baFdaahaaWcbeqaaiabiodaZaaaaaGccqqG4baEdaWgaaWcbaGaemyAaKgabeaakiabbIha4naaBaaaleaacqWGQbGAaeqaaaqaaiabdQgaQjabg6da+iabdMgaPbqab0GaeyyeIuoaaSqaaiabdMgaPbqab0GaeyyeIuoakiabcUda7aaa@71A9@

here θ_ij/|r_ij|³ = (δ - 3n_ij ⊗ n_ij)/|r_ij|³ is the usual dipole-dipole interaction tensor (where δ is the 3D unit matrix, n_ij ⊗ n_ij the tensor product of vectors n_ij = r_ij/|r_ij|, and r_ij = r_i - r_j; i ≠ j). As is usually done (see Refs. 6101112), relatively week quadruple-dipole and so on interactions of oscillators are ignored in addition to possible inharmoniousness of the oscillators.

Using the substitution yi=ximi MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqqG5bqEdaWgaaWcbaGaemyAaKgabeaakiabg2da9iabbIha4naaBaaaleaacqWGPbqAaeqaaOWaaOaaaeaacqWGTbqBdaWgaaWcbaGaemyAaKgabeaaaeqaaaaa@36BE@12, a conventional quantum mechanical Hamiltonian operator for coupled oscillators is obtained from [1]:

H ^ = − ℏ 2 2 ∑ i ∂ 2 ∂ y i 2 + ∑ i ω i 2 2 y i 2 + ∑ i ∑ j > i e 2 m i m j θ i j | r i j | 3 y i y j ; MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGibasgaqcaiabg2da9iabgkHiTmaalaaabaGaeS4dHG2aaWbaaSqabeaacqaIYaGmaaaakeaacqaIYaGmaaWaaabuaeaadaWcaaqaaiabgkGi2oaaCaaaleqabaGaeeOmaidaaaGcbaGaeyOaIyRaeeyEaK3aa0baaSqaaiabdMgaPbqaaiabikdaYaaaaaaabaGaemyAaKgabeqdcqGHris5aOGaey4kaSYaaabuaeaadaWcaaqaaGGaciab=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@733E@

here ℏ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWIpecAaaa@2DD5@ is the reduced Planck constant. By the introduction of a 3n-dimensional vector Y = [y₁, ..., y_n] composed of n 3D vectors y_i, and a 3n × 3n matrix B = ||b_ij|| composed of n × n 3D blocks bij[bij=ωi2δ if i=j, bij=(e2/mimj)(θij/|rij|3) if i≠j] MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaieqacqWFIbGydaWgaaWcbaGaemyAaKMaemOAaOgabeaakiabcUfaBjab=jgaInaaBaaaleaacqWGPbqAcqWGQbGAaeqaaOGaeyypa0dcciGae4xYdC3aa0baaSqaaiabdMgaPbqaaiabikdaYaaaiiqakiab9r7aKjabbccaGiabbMgaPjabbAgaMjabbccaGiabdMgaPjabg2da9iabdQgaQjabcYcaSiabbccaGiab=jgaInaaBaaaleaacqWGPbqAcqWGQbGAaeqaaOGaeyypa0JaeiikaGYaaSGbaeaacqWGLbqzdaahaaWcbeqaaiabikdaYaaaaOqaamaakaaabaGaemyBa02aaSbaaSqaaiabdMgaPbqabaGccqWGTbqBdaWgaaWcbaGaemOAaOgabeaaaeqaaaaakiabcMcaPiabcIcaOiab9H7aXTWaaSbaaeaacqWGPbqAcqWGQbGAaeqaaOGaei4la8IaeiiFaWNaeeOCai3aaSbaaSqaaiabdMgaPjabdQgaQbqabaGccqGG8baFdaahaaWcbeqaaiabiodaZaaakiabcMcaPiabbccaGiabbMgaPjabbAgaMjabbccaGiabdMgaPjabgcMi5kabdQgaQjabc2faDbaa@7036@, we obtain a simple Hamiltonian

H ^ = − ℏ 2 2 d 2 d Y 2 + 1 2 Y B Y MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacuWGibasgaqcaiabg2da9iabgkHiTmaalaaabaGaeS4dHG2aaWbaaSqabeaacqaIYaGmaaaakeaacqaIYaGmaaWaaSaaaeaacqWGKbazdaahaaWcbeqaaiabbkdaYaaaaOqaaiabdsgaKjabbMfaznaaCaaaleqabaGaeeOmaidaaaaakiabgUcaRmaalaaabaGaeGymaedabaGaeGOmaidaaiabbMfazjabhkeacjabbMfazbaa@3FA2@

for oscillation in a 3n dimensional potential well determined by the 3n × 3n matrix B. The 3n eigenvalues (Ω₁)², ..., (Ω_3n)² of matrix B are squared frequencies of 3n independent one-dimensional oscillations along its eigenvectors. If dipole-dipole interactions are absent (i.e., all θ_ij ≡ 0, where 0 is the zero 3D matrix), Ω₁ = Ω₂ = Ω₃ = ω₁, ..., Ω_3n-2 = Ω_3n-1 = Ω_3n = Ω_n. If dipole-dipole interactions are small, Ω₁ > 0, ..., Ω_3n > 0 are eigenvalues of the positively determined matrix B^1/2 (while in the case of too large dipole-dipole interactions the system becomes unstable, and at least one Ω² becomes negative). The frequencies Ω > 0 of the stable system of oscillators determine the ground state energy of this system 12: the system's energy is W=∑s=13nℏ2Ωs=ℏ2Sp[B1/2] MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGxbWvcqGH9aqpdaaeWbqaamaalaaabaGaeS4dHGgabaGaeGOmaidaaiabl2==UnaaBaaaleaacqWGZbWCaeqaaaqaaiabdohaZjabg2da9iabigdaXaqaaiabiodaZiabd6gaUbqdcqGHris5aOGaeyypa0ZaaSaaaeaacqWIpecAaeaacqaIYaGmaaGaee4uamLaeeiCaa3aamWaaeaacqWHcbGqdaahaaWcbeqaaiabigdaXiabc+caViabikdaYaaaaOGaay5waiaaw2faaaaa@4970@ in the presence of dipole-dipole interactions, while the energy of the same but non-interacting oscillators is W0=ℏ2∑i=1n3ωi MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGxbWvdaWgaaWcbaGaeeimaadabeaakiabg2da9maalaaabaGaeS4dHGgabaGaeGOmaidaamaaqahabaGaeG4mamdcciGae8xYdC3aaSbaaSqaaiabdMgaPbqabaaabaGaemyAaKMaeyypa0JaeGymaedabaGaemOBa4ganiabggHiLdaaaa@3D6D@. Thus, the energy of VdW dispersion forces is

Δ W = W − W 0 = ℏ 2 ( Sp [ B 1 / 2 ] − ∑ i = 1 n 3 ω i ) . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqqHuoarcqWGxbWvcqGH9aqpcqWGxbWvcqGHsislcqWGxbWvdaWgaaWcbaGaeGimaadabeaakiabg2da9maalaaabaGaeS4dHGgabaGaeGOmaidaamaabmaabaGaee4uamLaeeiCaa3aamWaaeaacqWHcbGqdaahaaWcbeqaaiabigdaXiabc+caViabikdaYaaaaOGaay5waiaaw2faaiabgkHiTmaaqahabaGaeG4mamdcciGae8xYdC3aaSbaaSqaaiabdMgaPbqabaaabaGaemyAaKMaeyypa0JaeGymaedabaGaemOBa4ganiabggHiLdaakiaawIcacaGLPaaacqGGUaGlaaa@4F35@

In general, one can compute all 3n eigenvalues (Ω_i)² of 3n × 3n matrix B(and thus eigenvalues Ω_i of B^1/2 and their sum Sp [B^1/2]) in a time proportional to (3n)³. Computationally, this solves a problem of exact calculation of VdW dispersion forces for any system of polarizable dipoles.

However, to get a physical understanding of the main terms contributing to these forces, one has to consider the main terms of Sp[B^1/2].

Matrix B can be presented as

B = B 0 + Δ B = [ ω 1 2 δ 0 ... 0 0 ω 2 2 δ ... 0 ... ... ... ... 0 0 ... ω n 2 δ ] + [ 0 b 12 ... b 1 n b 21 0 ... b 2 n ... ... ... ... b n 1 b n 2 ... 0 ] , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWHcbGqcqGH9aqpcqWHcbGqdaWgaaWcbaGaeGimaadabeaakiabgUcaRiabfs5aejabhkeacjabg2da9maadmaabaqbaeqabqabaaaaaeaaiiGacqWFjpWDdaqhaaWcbaGaeGymaedabaGaeeOmaidaaGGabOGae4hTdqgabaGaeCimaadabaGaeiOla4IaeiOla4IaeiOla4cabaGaeCimaadabaGaeCimaadabaGae8xYdC3aa0baaSqaaiabikdaYaqaaiabbkdaYaaakiab+r7aKbqaaiabc6caUiabc6caUiabc6caUaqaaiabhcdaWaqaaiabc6caUiabc6caUiabc6caUaqaaiabc6caUiabc6caUiabc6caUaqaaiabc6caUiabc6caUiabc6caUaqaaiabc6caUiabc6caUiabc6caUaqaaiabhcdaWaqaaiabhcdaWaqaaiabc6caUiabc6caUiabc6caUaqaaiab=L8a3naaDaaaleaacqWGUbGBaeaacqqGYaGmaaGccqGF0oazaaaacaGLBbGaayzxaaGaey4kaSYaamWaaeaafaqabeabeaaaaaqaaiabhcdaWaqaaGqabiab9jgaInaaBaaaleaacqaIXaqmcqaIYaGmaeqaaaGcbaGaeiOla4IaeiOla4IaeiOla4cabaGae0Nyai2aaSbaaSqaaiabigdaXiabd6gaUbqabaaakeaacqqFIbGydaWgaaWcbaGaeGOmaiJaeGymaedabeaaaOqaaiabhcdaWaqaaiabc6caUiabc6caUiabc6caUaqaaiab9jgaInaaBaaaleaacqaIYaGmcqWGUbGBaeqaaaGcbaGaeiOla4IaeiOla4IaeiOla4cabaGaeiOla4IaeiOla4IaeiOla4cabaGaeiOla4IaeiOla4IaeiOla4cabaGaeiOla4IaeiOla4IaeiOla4cabaGae0Nyai2aaSbaaSqaaiabd6gaUjabigdaXaqabaaakeaacqqFIbGydaWgaaWcbaGaemOBa4MaeGOmaidabeaaaOqaaiabc6caUiabc6caUiabc6caUaqaaiabhcdaWaaaaiaawUfacaGLDbaacqGGSaalaaa@9007@

where matrix B₀ corresponds to uncoupled oscillators and ΔB to weak coupling of the oscillators. Now let us consider an auxiliary matrix Bλ = B₀ + λΔB (where λ is a small multiplier), and present its square root, Bλ1/2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWHcbGqdaqhaaWcbaacciGae83UdWgabaGaeGymaeJaei4la8IaeGOmaidaaaaa@326D@, as a series Bλ1/2=B01/2+λZ1+λ2Z2+... MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWHcbGqdaqhaaWcbaacciGae83UdWgabaGaeGymaeJaei4la8IaeGOmaidaaOGaeyypa0JaeCOqai0aa0baaSqaaiabicdaWaqaaiabigdaXiabc+caViabikdaYaaakiabgUcaRiab=T7aSjabhQfaAnaaBaaaleaacqaIXaqmaeqaaOGaey4kaSIae83UdW2aaWbaaSqabeaacqaIYaGmaaGccqWHAbGwdaWgaaWcbaGaeGOmaidabeaakiabgUcaRiabc6caUiabc6caUiabc6caUaaa@4724@, where B01/2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWHcbGqdaqhaaWcbaGaeGimaadabaGaeGymaeJaei4la8IaeGOmaidaaaaa@31A0@ is a diagonal matrix with 3D blocks (B01/2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWHcbGqdaqhaaWcbaGaeGimaadabaGaeGymaeJaei4la8IaeGOmaidaaaaa@31A0@)_ij = ω_iδ if i = j, and (B01/2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWHcbGqdaqhaaWcbaGaeGimaadabaGaeGymaeJaei4la8IaeGOmaidaaaaa@31A0@)_ij = 0 if i≠j, and matrices Z₁, Z₂,... have to be calculated.

Since Bλ1/2Bλ1/2=B0+λ[B01/2Z1+Z1B01/2]+λ2[B01/2Z2+Z1Z1+Z2B01/2]+... MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWHcbGqdaqhaaWcbaacciGae83UdWgabaGaeGymaeJaei4la8IaeGOmaidaaOGaeCOqai0aa0baaSqaaiab=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@6F4F@ and Bλ1/2Bλ1/2=B0+λΔB MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWHcbGqdaqhaaWcbaacciGae83UdWgabaGaeGymaeJaei4la8IaeGOmaidaaOGaeCOqai0aa0baaSqaaiab=T7aSbqaaiabigdaXiabc+caViabikdaYaaakiabg2da9iabhkeacnaaBaaaleaacqaIWaamaeqaaOGaey4kaScccaGae43UdWMaeuiLdqKaeCOqaieaaa@4082@,

B 0 1 / 2 Z 1 + Z 1 B 0 1 / 2 = Δ B B 0 1 / 2 Z 2 + Z 2 B 0 1 / 2 = − Z 1 Z 1 ⋯ B 0 1 / 2 Z m + Z m B 0 1 / 2 = − ( Z 1 Z m − 1 + ... + Z m − 1 Z 1 ) ;  and so on . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@8D9D@

This system can be solved recursively: equation MZ + ZM = Y (where matrices consist of equal-size blocks (M)_ij, (Z)_ij, (Y)_ij, i, j = 1, ..., n) unambiguously determines Z for any Y when M = ||ω_iδ_ijδ|| is a positively determined diagonal matrix: (MZ+ZM)ij=∑p[ωiδipδ⋅(Z)pj+(Z)ip⋅ωpδpjδ]=[ωi+ωj](Z)ij MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqGGOaakcqWHnbqtcqWHAbGwcqGHRaWkcqWHAbGwcqWHnbqtcqGGPaqkdaWgaaWcbaGaemyAaKMaemOAaOgabeaakiabg2da9maaqafabaWaamWaaeaaiiGacqWFjpWDdaWgaaWcbaGaemyAaKgabeaakiab=r7aKnaaBaaaleaacqWGPbqAcqWGWbaCaeqaaGGabOGae4hTdqMaeyyXICTaeiikaGIaeCOwaOLaeiykaKYaaSbaaSqaaiabdchaWjabdQgaQbqabaGccqGHRaWkcqGGOaakcqWHAbGwcqGGPaqkdaWgaaWcbaGaemyAaKMaemiCaahabeaakiabgwSixlab=L8a3naaBaaaleaacqWGWbaCaeqaaOGae8hTdq2aaSbaaSqaaiabdchaWjabdQgaQbqabaGccqGF0oazaiaawUfacaGLDbaaaSqaaiabdchaWbqab0GaeyyeIuoakiabg2da9iabbUfaBjab=L8a3naaBaaaleaacqWGPbqAaeqaaOGaey4kaSIae8xYdC3aaSbaaSqaaiabdQgaQbqabaGccqGGDbqxcqGGOaakcqWHAbGwcqGGPaqkdaWgaaWcbaGaemyAaKMaemOAaOgabeaaaaa@731B@, and from [ω_i + ω_j](Z)_ij = (Y)_ij = (Y)_ij we have (Z)_ij = (Y)_ij/(ω_i + ω_j).

Denoting 1/(ω_i + ω_j) as μ_ij and bij/(ωi+ωj)={e2/[mimj(ωi+ωj)]}⋅(θij/|rij|3) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaajugWbiabhkgaIPWaaSbaaSqaaiabdMgaPjabdQgaQbqabaGccqGGVaWlcqqGOaakiiGacqWFjpWDdaWgaaWcbaGaemyAaKgabeaakiabgUcaRiab=L8a3naaBaaaleaacqWGQbGAaeqaaOGaeiykaKIaeyypa0ZaaiWaaeaadaWcgaqaaiabdwgaLnaaCaaaleqabaGaeGOmaidaaaGcbaGaei4waS1aaOaaaeaacqWGTbqBdaWgaaWcbaGaemyAaKgabeaakiabd2gaTnaaBaaaleaacqWGQbGAaeqaaaqabaGccqqGOaakcqWFjpWDdaWgaaWcbaGaemyAaKgabeaakiabgUcaRiab=L8a3naaBaaaleaacqWGQbGAaeqaaOGaeiykaKIaeiyxa0faaaGaay5Eaiaaw2haaiabgwSixpaabmaabaacceqcLbCacqGF4oqClmaaBaaabaGaemyAaKMaemOAaOgabeaakiabc+caViabcYha8jabbkhaYnaaBaaaleaacqWGPbqAcqWGQbGAaeqaaOGaeiiFaW3aaWbaaSqabeaacqaIZaWmaaaakiaawIcacaGLPaaaaaa@66FC@ as h_ij, we have

( Z 1 ) i j = ( Δ B ) i j / ( ω i + ω j ) = h i j ; ( Z 2 ) i j = − ( Z 1 Z 1 ) i j / ( ω i + ω j ) = − ∑ k h i k h k j ⋅ μ i j ; ( Z 3 ) i j = − ( Z 1 Z 2 + Z 2 Z 1 ) i j / ( ω i + ω j ) = + ∑ k ∑ p h i k h k p h p j ⋅ ( μ i p + μ k j ) μ i j ; ( Z 4 ) i j = − ∑ k ∑ p ∑ q h i k h k p h p q h q j ⋅ ( μ i p μ i q + μ i q μ k q + μ i p μ p j + μ k j μ k q + μ k j μ p j ) μ i j ;  and so on . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaafaqaaeabbaaaaeaacqGGOaakcqWHAbGwdaWgaaWcbaGaeGymaedabeaakiabcMcaPmaaBaaaleaacqWGPbqAcqWGQbGAaeqaaOGaeyypa0JaeiikaGIaeuiLdqKaeCOqaiKaeiykaKYaaSbaaSqaaiabdMgaPjabdQgaQbqabaGccqGGVaWldaqadaqaaGGaciab=L8a3naaBaaaleaacqWGPbqAaeqaaOGaey4kaSIae8xYdC3aaSbaaSqaaiabdQgaQbqabaaakiaawIcacaGLPaaacqGH9aqpieqacqGFObaAdaWgaaWcbaGaemyAaKMaemOAaOgabeaakiabcUda7aqaaiabcIcaOiabhQfaAnaaBaaaleaacqaIYaGmaeqaaOGaeiykaKYaaSbaaSqaaiabdMgaPjabdQgaQbqabaGccqGH9aqpcqGHsislcqGGOaakcqWHAbGwdaWgaaWcbaGaeGymaedabeaakiabhQfaAnaaBaaaleaacqaIXaqmaeqaaOGaeiykaKYaaSbaaSqaaiabdMgaPjabdQgaQbqabaGccqGGVaWlcqGGOaakcqWFjpWDdaWgaaWcbaGaemyAaKgabeaakiabgUcaRiab=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L8a3naaBaaaleaacqWGQbGAaeqaaOGaeiykaKIaeyypa0Jaey4kaSYaaabuaeaadaaeqbqaaiab+HgaOnaaBaaaleaacqWGPbqAcqWGRbWAaeqaaOGae4hAaG2aaSbaaSqaaiabdUgaRjabdchaWbqabaGccqGFObaAdaWgaaWcbaGaemiCaaNaemOAaOgabeaaaeaacqWGWbaCaeqaniabggHiLdGccqGHflY1cqGGOaakcqWF8oqBdaWgaaWcbaGaemyAaKMaemiCaahabeaakiabgUcaRiab=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X7aTnaaBaaaleaacqWGPbqAcqWGXbqCaeqaaOGaey4kaSIae8hVd02aaSbaaSqaaiabdMgaPjabdghaXbqabaGccqWF8oqBdaWgaaWcbaGaem4AaSMaemyCaehabeaakiabgUcaRiab=X7aTnaaBaaaleaacqWGPbqAcqWGWbaCaeqaaOGae8hVd02aaSbaaSqaaiabdchaWjabdQgaQbqabaGccqGHRaWkcqWF8oqBdaWgaaWcbaGaem4AaSMaemOAaOgabeaakiab=X7aTnaaBaaaleaacqWGRbWAcqWGXbqCaeqaaOGaey4kaSIae8hVd02aaSbaaSqaaiabdUgaRjabdQgaQbqabaGccqWF8oqBdaWgaaWcbaGaemiCaaNaemOAaOgabeaakiabcMcaPiab=X7aTnaaBaaaleaacqWGPbqAcqWGQbGAaeqaaOGaei4oaSJaeeiiaaIaeeyyaeMaeeOBa4MaeeizaqMaeeiiaaIaee4CamNaee4Ba8MaeeiiaaIaee4Ba8MaeeOBa4MaeiOla4caaaaa@381E@

Matrices Z₁, Z₂, Z₃,... do not depend on λ. Series λZ1+λ2Z2+λ3Z3+... MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWF7oaBcqWHAbGwdaWgaaWcbaGaeGymaedabeaakiabgUcaRiab=T7aSnaaCaaaleqabaGaeGOmaidaaOGaeCOwaO1aaSbaaSqaaiabikdaYaqabaGccqGHRaWkcqWF7oaBdaahaaWcbeqaaiabiodaZaaakiabhQfaAnaaBaaaleaacqaIZaWmaeqaaOGaey4kaSIaeiOla4IaeiOla4IaeiOla4caaa@40A6@ converges at λ = 1, if the terms b_ij of the matrix ΔB (and consequently h_ij) are sufficiently small. Thus, the energy of dispersion forces

Δ W = ℏ 2 ( Sp [ B 1 / 2 ] − Sp [ B 0 1 / 2 ] ) = ℏ 2 Sp [ Z 2 ] + ℏ 2 Sp [ Z 3 ] + ℏ 2 Sp [ Z 4 ] + ... , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqqHuoarcqWGxbWvcqGH9aqpdaWcaaqaaiabl+qiObqaaiabikdaYaaadaqadaqaaiabbofatjabbchaWnaadmaabaGaeCOqai0aaWbaaSqabeaacqaIXaqmcqGGVaWlcqaIYaGmaaaakiaawUfacaGLDbaacqGHsislcqqGtbWucqqGWbaCdaWadaqaaiabhkeacnaaDaaaleaacqaIWaamaeaacqaIXaqmcqGGVaWlcqaIYaGmaaaakiaawUfacaGLDbaaaiaawIcacaGLPaaacqGH9aqpdaWcaaqaaiabl+qiObqaaiabikdaYaaacqqGtbWucqqGWbaCdaWadaqaaiabhQfaAnaaBaaaleaacqaIYaGmaeqaaaGccaGLBbGaayzxaaqcLbwacqGHRaWkkmaalaaabaGaeS4dHGgabaGaeGOmaidaaiabbofatjabbchaWnaadmaabaGaeCOwaO1aaSbaaSqaaiabiodaZaqabaaakiaawUfacaGLDbaajugybiabgUcaROWaaSaaaeaacqWIpecAaeaacqaIYaGmaaGaee4uamLaeeiCaa3aamWaaeaacqWHAbGwdaWgaaWcbaGaeGinaqdabeaaaOGaay5waiaaw2faaKqzGfGaey4kaSIaeiOla4IaeiOla4IaeiOla4IaeiilaWcaaa@6C17@

can be presented as a sum of pairwise, triple, quadruple, etc. interactions (the term Sp[Z1]=∑iSp[hii] MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqqGtbWucqqGWbaCcqqGBbWwcqWHAbGwdaWgaaWcbaGaeGymaedabeaakiabc2faDjabg2da9maaqafabaGaee4uamLaeeiCaaNaee4waSfcbeGae8hAaG2aaSbaaSqaaiabdMgaPjabdMgaPbqabaGccqGGDbqxaSqaaiabdMgaPbqab0GaeyyeIuoaaaa@4208@ is equal to 0, since all diagonal blocks h_ii ≡ 0).

The term

ℏ 2 Sp [ Z 2 ] = ℏ 2 ∑ i Sp[ − μ i i ∑ k h i k h k i ] = ∑ i ∑ < k { − ℏ 2 ⋅ ω i + ω k 2 ω i ω k Sp[ h i k h k i ] } MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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L8a3naaBaaaleaacqWGPbqAaeqaaOGae8xYdC3aaSbaaSqaaiabdUgaRbqabaaaaOGaee4uamLaeeiCaaNaee4waSLae4hAaG2aaSbaaSqaaiabdMgaPjabdUgaRbqabaGccqGFObaAdaWgaaWcbaGaem4AaSMaemyAaKgabeaakiabb2faDbGaay5Eaiaaw2haaaWcbaGaeyipaWJaem4AaSgabeqdcqGHris5aaWcbaGaemyAaKgabeqdcqGHris5aaaa@80B7@

is a sum of conventional London's energies ΔW_ik 10111213 of pairwise interactions: since Sp[θ_ik θ_ki] = 6,

Δ W i k = − 3 ℏ ( ω i + ω k ) 2 ⋅ ( γ i k ) 2 , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqqHuoarcqWGxbWvdaWgaaWcbaGaemyAaKMaem4AaSgabeaakiabg2da9iabgkHiTmaalaaabaGaeG4mamJaeS4dHGMaeiikaGccciGae8xYdC3aaSbaaSqaaiabdMgaPbqabaGccqGHRaWkcqWFjpWDdaWgaaWcbaGaem4AaSgabeaakiabcMcaPaqaaiabbkdaYaaacqGHflY1daqadaqaaiab=n7aNnaaBaaaleaacqWGPbqAcqWGRbWAaeqaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacqaIYaGmaaGccqGGSaalaaa@4B08@

where

γ i k = ω i ω k ω i + ω k ⋅ ( α i α k | r i k | 3 ) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFZoWzdaWgaaWcbaGaemyAaKMaem4AaSgabeaakiabg2da9maalaaabaWaaOaaaeaacqWFjpWDdaWgaaWcbaGaemyAaKgabeaakiab=L8a3naaBaaaleaacqWGRbWAaeqaaaqabaaakeaacqWFjpWDdaWgaaWcbaGaemyAaKgabeaakiabgUcaRiab=L8a3naaBaaaleaacqWGRbWAaeqaaaaakiabgwSixpaabmaabaWaaSaaaeaadaGcaaqaaiab=f7aHnaaBaaaleaacqWGPbqAaeqaaOGae8xSde2aaSbaaSqaaiabdUgaRbqabaaabeaaaOqaaiabcYha8jabbkhaYnaaBaaaleaacqWGPbqAcqWGRbWAaeqaaOGaeiiFaW3aaWbaaSqabeaacqaIZaWmaaaaaaGccaGLOaGaayzkaaaaaa@5391@

is a convenient dimensionless parameter for the interaction of oscillators i and k, and α_i = e²/(m_iωi2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFjpWDdaqhaaWcbaGaemyAaKgabaGaeGOmaidaaaaa@30FA@) is the electronic polarizability of atom i.

The term

ℏ 2 Sp [ Z 3 ] = ∑ i ∑ < k    ∑ < p { ℏ 2 ⋅ ω i + ω k + ω p ω i ω k ω p ⋅ Sp[ h i k h k p h p i ] } MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWcaaqaaiabl+qiObqaaiabikdaYaaacqqGtbWucqqGWbaCdaWadaqaaiabhQfaAnaaBaaaleaacqaIZaWmaeqaaaGccaGLBbGaayzxaaGaeyypa0ZaaabuaeaadaaeqbqaamaaqafabaWaaiWaaeaadaWcaaqaaiabl+qiObqaaiabikdaYaaacqGHflY1daWcaaqaaGGaciab=L8a3naaBaaaleaacqWGPbqAaeqaaOGaey4kaSIae8xYdC3aaSbaaSqaaiabdUgaRbqabaGccqGHRaWkcqWFjpWDdaWgaaWcbaGaemiCaahabeaaaOqaaiab=L8a3naaBaaaleaacqWGPbqAaeqaaOGae8xYdC3aaSbaaSqaaiabdUgaRbqabaGccqWFjpWDdaWgaaWcbaGaemiCaahabeaaaaGccqGHflY1cqqGtbWucqqGWbaCcqqGBbWwieqacqGFObaAdaWgaaWcbaGaemyAaKMaem4AaSgabeaakiab+HgaOnaaBaaaleaacqWGRbWAcqWGWbaCaeqaaOGae4hAaG2aaSbaaSqaaiabdchaWjabdMgaPbqabaGccqqGDbqxaiaawUhacaGL9baaaSqaaiabgYda8iabdchaWbqab0GaeyyeIuoaaSqaaiabgYda8iabdUgaRjabbccaGiabbccaGaqab0GaeyyeIuoaaSqaaiabdMgaPbqab0GaeyyeIuoaaaa@75FA@

is the sum of the energies of triple interactions, ΔW_ikp, which depends on the geometry of the triangles (Fig 1a) formed by the atoms involved 681113. Because Sp[θ_ikθ_kpθ_pi] = 3+9cosφ_k·cosφ_p·cosφ_i,

Δ W i k p = 3 ℏ ( ω i + ω k + ω p ) 2 ⋅ γ i k γ k p γ p i ⋅ ( 1 + 3 cos ⁡ ϕ i cos ⁡ ϕ k cos ⁡ ϕ p ) ,   w h e r e   i ≠ k ≠ p ≠ i . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqqHuoarcqWGxbWvdaWgaaWcbaGaemyAaKMaem4AaSMaemiCaahabeaakiabg2da9maalaaabaGaeG4mamJaeS4dHGMaeiikaGccciGae8xYdC3aaSbaaSqaaiabdMgaPbqabaGccqGHRaWkcqWFjpWDdaWgaaWcbaGaem4AaSgabeaakiabgUcaRiab=L8a3naaBaaaleaacqWGWbaCaeqaaOGaeiykaKcabaGaeGOmaidaaiabgwSixlab=n7aNnaaBaaaleaacqWGPbqAcqWGRbWAaeqaaOGae83SdC2aaSbaaSqaaiabdUgaRjabdchaWbqabaGccqWFZoWzdaWgaaWcbaGaemiCaaNaemyAaKgabeaakiabgwSixlabcIcaOiabigdaXiabgUcaRiabiodaZiGbcogaJjabc+gaVjabcohaZjab=v9aQnaaBaaaleaacqWGPbqAaeqaaOGagi4yamMaei4Ba8Maei4CamNae8x1dO2aaSbaaSqaaiabdUgaRbqabaGccyGGJbWycqGGVbWBcqGGZbWCcqWFvpGAdaWgaaWcbaGaemiCaahabeaakiabcMcaPiabcYcaSiabbccaGmXvP5wqSXMqHnxAJn0BKvguHDwzZbqegqvATv2CG4uz3bIuV1wyUbaceaGaa43Daiaa+HgacaGFLbGaa4NCaiaa+vgacqqGGaaicqWGPbqAcqGHGjsUcqWGRbWAcqGHGjsUcqWGWbaCcqGHGjsUcqWGPbqAcqGGUaGlaaa@92B9@

Thus, a triple interaction can be both repulsive and attractive: attractive, when atoms i, k, p stay nearly along a line; repulsive, when these atoms form an acute or right-angled triangle 6.

The quadruple interactions (contributing to the energy term ℏ2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWcaaqaaiabl+qiObqaaiabikdaYaaaaaa@2ED7@Sp[Z₄] can consist of not only four different atoms, but also three or two. Four-, three- and two-atom interactions contain terms like Sp[h_ikh_kph_pqh_qi], Sp[h_ikh_kih_iph_pi], Sp[h_ikh_kih_ikh_ki], respectively (where different indices denote different particles). After simple but voluminous calculations it can be shown that

ℏ 2 Sp [ Z 4 ] = ∑   i ∑ < k 2 Γ i k i k + ∑   i ∑ < k ∑ < p ( 4 Γ i k i p + 4 Γ k i k p + 4 Γ p i p k ) + ∑   i ∑ < k ∑ < p ∑ < q ( 8 Γ i k p q + 8 Γ i k q p + 8 Γ i p k q ) , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@A0DF@

where Γikpq=ℏ16(ωi+ωk+ωp+ωq+ωkωqωi+ωp+ωiωpωk+ωq)⋅γikγkpγpqγqi⋅(-Sp[θikθkpθpqθqi]) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqqHtoWrdaWgaaWcbaGaemyAaKMaem4AaSMaemiCaaNaemyCaehabeaakiabg2da9maalaaabaGaeS4dHGgabaGaeeymaeJaeeOnaydaamaabmaabaacciGae8xYdC3aaSbaaSqaaiabdMgaPbqabaGccqGHRaWkcqWFjpWDdaWgaaWcbaGaem4AaSgabeaakiabgUcaRiab=L8a3naaBaaaleaacqWGWbaCaeqaaOGaey4kaSIae8xYdC3aaSbaaSqaaiabdghaXbqabaGccqGHRaWkdaWcaaqaaiab=L8a3naaBaaaleaacqWGRbWAaeqaaOGae8xYdC3aaSbaaSqaaiabdghaXbqabaaakeaacqWFjpWDdaWgaaWcbaGaemyAaKgabeaakiabgUcaRiab=L8a3naaBaaaleaacqWGWbaCaeqaaaaakiabgUcaRmaalaaabaGae8xYdC3aaSbaaSqaaiabdMgaPbqabaGccqWFjpWDdaWgaaWcbaGaemiCaahabeaaaOqaaiab=L8a3naaBaaaleaacqWGRbWAaeqaaOGaey4kaSIae8xYdC3aaSbaaSqaaiabdghaXbqabaaaaaGccaGLOaGaayzkaaGaeyyXICTae83SdC2aaSbaaSqaaiabdMgaPjabdUgaRbqabaGccqWFZoWzdaWgaaWcbaGaem4AaSMaemiCaahabeaakiab=n7aNnaaBaaaleaacqWGWbaCcqWGXbqCaeqaaOGae83SdC2aaSbaaSqaaiabdghaXjabdMgaPbqabaGccqGHflY1daqadaqaaiabb2caTiabbofatjabbchaWjabbUfaBHGabiab+H7aXnaaBaaaleaacqWGPbqAcqWGRbWAaeqaaOGae4hUde3aaSbaaSqaaiabdUgaRjabdchaWbqabaGccqGF4oqCdaWgaaWcbaGaemiCaaNaemyCaehabeaakiab+H7aXnaaBaaaleaacqWGXbqCcqWGPbqAaeqaaOGaeeyxa0facaGLOaGaayzkaaaaaa@99B3@.

In two-atom quadruple terms Γ_ikik, -Sp[θ_ikθ_kiθ_ikθ_ki] = -18. In three-atom quadruple terms like Γ_ikip,

-Sp[θ_ik θ_ki θ_ik θ_pi] = 9 - 9cos²φ_i

is always negative (Fig. 1b), i.e., neighbor k of atom i increases attraction of i to p, and neighbor p of i increases attraction of i to k. In four-atom quadruple terms Γ_ikpq, the value of -Sp[θ_ikθ_kpθ_kpθ_qi] varies from -18 to +9 depending on the mutual arrangement all four atoms.

Consider the system of oscillators described above, where dipole moments ex_a, ex_b of atoms a, b are now fixed. The potential energy of this system is

U = x a g a b x b + X ( a b ) G a ( a b ) x a + X ( a b ) G b ( a b ) x b + 1 2 X ( a b ) A ( a b ) X ( a b ) . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaeHbuLwBLnhiov2DGi1BTfMBaGabciaa=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@7B32@

Here g_ab = e²θ_ab/|r_ab|³, X^(ab) is a 3·(n-2)-dimensional vector composed of n-2 3D vectors x_i (i ≠ a, b), Ga(ab) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaieqacqWFhbWrdaqhaaWcbaGaemyyaegabaGaeiikaGIaemyyaeMaemOyaiMaeiykaKcaaaaa@338B@, Gb(ab) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaieqacqWFhbWrdaqhaaWcbaGaemOyaigabaGaeiikaGIaemyyaeMaemOyaiMaeiykaKcaaaaa@338D@ are 3 × 3·(n-2) matrices composed of n-2 3D matrices g_ia or g_ib, respectively (i ≠ a, b), and the 3·(n-2) × 3·(n-2) matrix A^(ab) = ||a_ij|| is composed of (n-2) × (n-2) 3D blocks a_ij [i ≠ a, b, j ≠ a, b, a_ij = m_iωi2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFjpWDdaqhaaWcbaGaemyAaKgabaGaeGOmaidaaaaa@30FA@δ if i = j, a_ij = g_ij if i ≠ j]. The system's energy minimum is determined by equations ddX(ab)U=0 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWcaaqaaiabdsgaKbqaaiabdsgaKjabbIfaynaaCaaaleqabaGaeeikaGIaeeyyaeMaeeOyaiMaeeykaKcaaaaakiabdwfavjabg2da9iabicdaWaaa@3837@, so that X(ab)=−[A(ab)]-1Ga(ab)xa−[A(ab)]-1Gb(ab)xb MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqqGybawdaahaaWcbeqaaiabbIcaOiabdggaHjabdkgaIjabbMcaPaaakiabg2da9iabgkHiTiabcUfaBHqabiab=feabnaaCaaaleqabaGaeeikaGIaemyyaeMaemOyaiMaeeykaKcaaOGaeiyxa01aaWbaaSqabeaacqqGTaqlcqqGXaqmaaGccqWFhbWrdaqhaaWcbaGaemyyaegabaGaeiikaGIaemyyaeMaemOyaiMaeiykaKcaaOGaeeiEaG3aaSbaaSqaaiabdggaHbqabaGccqGHsislcqGGBbWwcqWFbbqqdaahaaWcbeqaaiabbIcaOiabdggaHjabdkgaIjabbMcaPaaakiabc2faDnaaCaaaleqabaGaeeyla0IaeeymaedaaOGae83raC0aa0baaSqaaiabdkgaIbqaaiabcIcaOiabdggaHjabdkgaIjabcMcaPaaakiabbIha4naaBaaaleaacqWGIbGyaeqaaaaa@5D02@, and the energy at minimum is

U min ⁡ = { − 1 2 x a G a ( a b ) [ A ( a b ) ] -1 G a ( a b ) x a − 1 2 x b G b ( a b ) [ A ( a b ) ] -1 G b ( a b ) x b } − x a G a ( a b ) [ A ( a b ) ] -1 G b ( a b ) x b + x a g a b x b . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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DgaNnaaBaaaleaacqWGHbqycqWGIbGyaeqaaOGaeeiEaG3aaSbaaSqaaiabdkgaIbqabaGccqGGUaGlaaa@A19F@

The terms in {} are not interesting to us at this stage, they correspond to the interaction of separate fixed dipoles a and b with a polarizable environment. The term −xaGa(ab)[A(ab)]-1Gb(ab)xb MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqGHsislcqqG4baEdaWgaaWcbaGaemyyaegabeaakiabhEeahnaaDaaaleaacqWGHbqyaeaacqGGOaakcqWGHbqycqWGIbGycqGGPaqkaaGccqGGBbWwcqWHbbqqdaahaaWcbeqaaiabbIcaOiabdggaHjabdkgaIjabbMcaPaaakiabc2faDnaaCaaaleqabaGaeeyla0IaeeymaedaaOGaeC4raC0aa0baaSqaaiabdkgaIbqaaiabcIcaOiabdggaHjabdkgaIjabcMcaPaaakiabbIha4naaBaaaleaacqWGIbGyaeqaaaaa@4B60@ corresponds to the interaction of dipoles a and b via a polarizable environment; and the last term corresponds to the direct interaction of dipoles a and b. Thus, the polarizable medium-modified energy of electrostatic interaction of fixed dipoles a and b is

U a -medium- b ( 1 ) = x a ( g a b − G a ( a b ) [ A ( a b ) ] -1 G b ( a b ) )  x b MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBamXvP5wqSXMqHnxAJn0BKvguHDwzZbqegyvzYrwyUfgarqqtubsr4rNCHbGeaGqiA8vkIkVAFgIELiFeLkFeLk=iY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqVepeea0=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@7713@

when the "medium" consists of oscillators described at a microscopic level.

On the other hand, we have a classic "macroscopic" equation for the medium-modified energy of interaction of the same two dipoles. The modification is described by permittivity, ε:

U a b ( ε ) = x a [ 1 ε ] g a b x b . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBamXvP5wqSXMqHnxAJn0BKvguHDwzZbqegyvzYrwyUfgarqqtubsr4rNCHbGeaGqiA8vkIkVAFgIELiFeLkFeLk=iY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfeaY=biLkVcLq=JHqVepeea0=as0db9vqpepesP0xe9Fve9Fve9GapdbaqaaeGacaGaaiaabeqaamqadiabaaGcbaqegqvATv2CG4uz3bIuV1wyUbacfiGaa8xvamaaDaaaleaacqWGHbqycqWGIbGyaeaaryqqYLwySbacgaGaa4hkaGGaciab9v7aLjaa+LcaaaaccaGccqaF9aqpcqqG4baEdaWgaaWcbaGaemyyaegabeaakmaadmaabaWaaSaaaeaacqaIXaqmaeaacqqF1oqzaaaacaGLBbGaayzxaaacdeGaaS3zamaaBaaaleaacqWGHbqycqWGIbGyaeqaaOGaeeiEaG3aaSbaaSqaaiabdkgaIbqabaGccqGGUaGlaaa@5B56@

The value 1ε MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWcaaqaaiabigdaXaqaaGGaciab=v7aLbaaaaa@2F5A@ is a scalar in a uniform macroscopic medium; but [1ε]ab=δ−Ga(ab)[A(ab)]-1Gb(ab)gab−1 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWadaqaamaalaaabaGaeGymaedabaacciGae8xTdugaaaGaay5waiaaw2faamaaBaaaleaacqWGHbqycqWGIbGyaeqaaOGaeyypa0dcceGae4hTdqMaeyOeI0IaeC4raC0aa0baaSqaaiabdggaHbqaaiabcIcaOiabdggaHjabdkgaIjabcMcaPaaakiabcUfaBjabhgeabnaaCaaaleqabaGaeeikaGIaemyyaeMaemOyaiMaeeykaKcaaOGaeiyxa01aaWbaaSqabeaacqqGTaqlcqqGXaqmaaGccqWHhbWrdaqhaaWcbaGaemOyaigabaGaeiikaGIaemyyaeMaemOyaiMaeiykaKcaaGqabOGae03zaC2aa0baaSqaaiabdggaHjabdkgaIbqaaiabgkHiTiabigdaXaaaaaa@5593@ is a 3 × 3 tensor that depends on the 3D positions of dipoles a and b when in a non-uniform "microscopic" medium. This can be seen from comparison of Equations [18] and [19], which describe the same energy U⁽¹⁾_a-medium-b = U^(ε)_ab in "micro-" and "macroscopic" terms. However, the effective value of medium-induced modification of dipole-dipole interaction strength can still be defined by the equation

1 ε a b 2 =   Sp [ [ 1 ε ] a b g a b [ 1 ε ] b a g b a ] / Sp [ g a b g b a ] . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWcaaqaaiabigdaXaqaaGGaciab=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@5E3C@

Defined thus, the scalar ε_ab coincides with the conventional permittivity in a uniform macroscopic medium. When the "medium" consists of oscillators described at a microscopic level, ε_ab corresponds to the dielectric permittivity for the interaction of dipoles a and b via the polarizable environment.

From equations [18] – [20], εab−2= Sp[(gab−Ga(ab)[A(ab)]-1Gb(ab))(gba−Gb(ab)[A(ab)]-1Ga(ab))] / Sp[gabgba] MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@84B9@. To estimate this value with accuracy up to triple interactions (i.e. up to the third powers of small g matrices), it remains only to find the main (g-independent) term of [A^(ab)]^-1, since products like Ga(ab)Gb(ab) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaieqacqWFhbWrdaqhaaWcbaGaemyyaegabaGaeiikaGIaemyyaeMaemOyaiMaeiykaKcaaOGae83raC0aa0baaSqaaiabdkgaIbqaaiabcIcaOiabdggaHjabdkgaIjabcMcaPaaaaaa@3A6C@ already contain second powers of g. From expansion similar to that presented in equation [5], one obtains

1 ε a b 2 ≈ 1 − ∑ p ≠ a , b 2 m p ω p 2 ⋅ Sp[ g a b g b p g p a ] Sp[ g a b g b a ] = 1 − ∑ p ≠ a , b 2 α p |r a b | 3 |r a p | 3 |r p b | 3 ⋅ Sp[ θ a b θ b p θ p a ] Sp[ θ a b θ b a ] .

If the sum is taken over the oscillating electrons only (while the nuclei positions are fixed), ε_ab is an electronic component of dielectric permittivity.

Equations [10] – [14] show that energies of pairwise, triple and quadruple interactions are proportional to ~ℏ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWIpecAaaa@2DD5@ω·γ_ikγ_ki, ~ℏ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWIpecAaaa@2DD5@ω·γ_ikγ_kpγ_pi and ~ℏ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWIpecAaaa@2DD5@ω·γ_ikγ_kpγ_pqγ_qi, respectively, where ℏ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWIpecAaaa@2DD5@ω is an oscillator excitation energy. The γ values are small for non-bonded contacts. Indeed, ωiωkωi+ωk inγik=ωiωkωi+ωk⋅(αiαk|rik|3) MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaadaWcaaqaamaakaaabaacciGae8xYdC3aaSbaaSqaaiabdMgaPbqabaGccqWFjpWDdaWgaaWcbaGaem4AaSgabeaaaeqaaaGcbaGae8xYdC3aaSbaaSqaaiabdMgaPbqabaGccqGHRaWkcqWFjpWDdaWgaaWcbaGaem4AaSgabeaaaaGccqqGGaaiieaacqGFPbqAcqGFUbGBtCvAUfeBSjuyZL2yd9gzLbvyNv2CaeHbuLwBLnhiov2DGi1BTfMBaGabaiaa9bcacqWFZoWzdaWgaaWcbaGaemyAaKMaem4AaSgabeaakiabg2da9maalaaabaWaaOaaaeaacqWFjpWDdaWgaaWcbaGaemyAaKgabeaakiab=L8a3naaBaaaleaacqWGRbWAaeqaaaqabaaakeaacqWFjpWDdaWgaaWcbaGaemyAaKgabeaakiabgUcaRiab=L8a3naaBaaaleaacqWGRbWAaeqaaaaakiabgwSixpaabmaabaWaaSaaaeaadaGcaaqaaiab=f7aHnaaBaaaleaacqWGPbqAaeqaaOGae8xSde2aaSbaaSqaaiabdUgaRbqabaaabeaaaOqaaiabcYha8jabbkhaYnaaBaaaleaacqWGPbqAcqWGRbWAaeqaaOGaeiiFaW3aaWbaaSqabeaacqaIZaWmaaaaaaGccaGLOaGaayzkaaaaaa@7357@ is normally close to 1/2, because ω_i and ω_k are of equal order of magnitude 13, and the α values are about 1 ų for atoms typical in proteins (0.40 ų for H atoms, 0.69 – 0.85 ų for O atoms, 0.90–0.97 ų for N atoms, 1.03 – 1.32 ų for C atoms 13). Thus, the dimension less value γ is about 0.02 for the closest non-bonded contact of atoms (|r| ≈ 2.2–3.4 Å for atoms H, O, N, C 13).

The very small value of γ at "non-bonded" distances means that triple, quadruple, etc., non-bonded atom-atom contacts are very weak compared to those that are pairwise.

However, the situation changes when the γ factor involves two atoms connected by a covalent bond. The atom-atom distance in this case is at least two times smaller than that for the closest non-bonded contact. Thus, the γ factor for covalently bonded atoms is about tenfold (see Eq. [11]) larger than the γ factor for the closest non-bonded contact of atoms. In this case, the energy of high-order interactions can approach that of a pairwise interaction.

Taking an average estimate of γ = 0.15 for bonded atoms (that corresponds to a typical covalent bond length of 1.5 Å between chemically equal atoms with polarizability of 1 ų), we see (Fig. 2) a significant change in the VdW interaction energy for the cost of triple and quadruple interactions of three atoms (when a pair of these atoms is covalently bonded) and, to a lesser extent, for the cost of four-atom interactions (when two pairs of atoms are covalently bonded). It is useful to mention here that one study cited herein 14 has already discussed the general idea that ''bond-bond'' interactions are more appropriate for conformational analysis than ''atom-atom'' interactions.

In the case of the collinear arrangement of atoms and bonds the attraction can grow by 70% (compare values in columns "configuration and its energy" and "pairwise" in Fig. 2), while in the case of the orthogonal arrangement of atoms and bonds, the attraction can decrease by 6%. The decrease is less than the increase because, as mentioned above, the quadruple interaction is mainly attractive. The main part of this attraction is due to the three-atom quadruple interaction. This can be divided in two parts: Equation [15] can be presented as Sp[θ_ikθ_kiθ_ipθ_pi] = 12 + (9cos²φ_i - 3), where the first (larger) term is an orientation-independent constant (actually, it simply increases the VdW attraction of any atom involved in covalent bonding to all other atoms, see Fig. 1b; in available force fields 12345 this term is implicitly taken into account by ascribing and fitting different pairwise attraction forces to atoms in different covalent states). The second, the orientation-dependent part equals zero if averaged in 3D space over all possible atom-to-bond orientations (see examples in Fig. 2); this part is rather similar to that for the interaction of a separate atom with a remote covalent bond (cf. Fig.1b to Fig. 1a).

The orientation-dependent terms, which mainly arise from triple interactions, can increase or decrease VdW energy by ~±20–40% at extremes. This energy effect is significant, being comparable to that caused by the replacement of an N atom (α_N ≈ 0.95 ų) by a C (α_C ≈ 1.2 ų) or O (α_O ≈ 0.75 ų) atom in a pairwise VdW interaction (see equations [10,11] at ω_N≈ω_C≈ω_O). Moreover, the orientation effect is much stronger than the effect of replacing the pairwise energies of two C-N non-bonded contacts with the sum of pairwise energies of C-C and N-N contacts (α_Cα_C + α_Nα_N - 2α_Cα_N ≈ 5% × α_Cα_N). Note that the contact replacements are the main VdW effects currently used to distinguish ''correct'' protein folds from those that are ''wrong''.

Thus, we indeed see that any conformational analysis should by no means ignore the cumulative effect of the coupling of multi-atom VdW interactions with covalent bonds. Figure 3 shows that the energy difference between two structures of equal compactness may change twice due to this effect.

It is necessary to note that at short distances, which are the most important for VdW interactions, one can expect effects of inharmonicity, higher multipole interactions, etc. 1112. These effects are avoided in the "harmonic oscillator" model used in this work. Nevertheless, they exist in reality, and their existence makes the expressions obtained for many-body interactions approximate to the same extent as the conventional London's expression is for pairwise interactions. In particular, the value of γ for each covalent bond may be treated as a parameter, the value of which should be obtained from a fit of the experimental data in the same way as the London's pairwise interaction energy 111314. The necessity of fitting the experimental data is underlined by a relatively slow convergence of both the orientation-dependent and orientation-independent series in Figure 2.

VdW forces and the electronic part of dielectric permittivity are evidently connected: both originate from dipole-dipole interactions. The relationship between VdW forces and the macroscopic dielectric permittivity of the medium is well studied in the continuum approximation 111416. However, this approach is useful and relatively simple when applied to large uniform bodies, like drops or layers. This work concerns interactions of small fragments composed of various atoms, and the frequency-dependent dielectric permeability, taken as a macroscopic characteristic, seems not to be an appropriate tool to investigate this case. Here we will consider an electronic part of the microscopic dielectric permittivity created by and acting at interacting harmonic oscillators, with the goal of demonstrating that multi-atom interaction creates a kind of "VdW permittivity" for pairwise VdW forces in addition to elucidating a connection between VdW forces and the electronic part of microscopic dielectric permittivity.

The role of dielectric permittivity in VdW forces is ambiguous. On the one hand, the VdW interaction of two atoms is proportional to the square of the electrostatic interaction between their fluctuating electronic polarizations. Thus, one might expect the pairwise VdW interaction to be inversely proportional to the square of the electronic part of the dielectric permittivity. On the other hand, the medium's electrons (which create the electronic component of dielectric permittivity) are involved in VdW interactions of "their own" atoms, and it is not clear if they are "free" enough to influence the VdW interactions of the other atoms in such a strong manner.

Following equations [8] – [10], [12], [13], one can present the total VdW energy ΔW as

∑ i ∑ < k Δ W i k + ∑ i ∑ < k    ∑ < p Δ W i k p + ... = ∑ i ∑ < k Δ W i k { 1 − ∑ p ≠ i , j ( ω p ) 2 ( ω i + ω p ) ( ω p + ω k ) ⋅ 2 α p |r i k | 3 |r i p | 3 |r p k | 3 ⋅ Sp[ θ i k θ k p θ p i ] Sp[ θ i k θ k i ] + ... } , MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=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@D0C2@

and see that Φik≈{1−∑p≠i,j(ωp)2(ωi+ωp)(ωp+ωk)⋅2αp|rik|3|rip|3|rpk|3⋅Sp[θikθkpθpi]Sp[θikθki]+...}−1 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFMoGrdaWgaaWcbaGaemyAaKMaem4AaSgabeaakiabgIKi7oaacmaabaGaeGymaeJaeyOeI0YaaabuaeaadaWcaaqaaiabcIcaOiab=L8a3naaBaaaleaacqWGWbaCaeqaaOGaeiykaKYaaWbaaSqabeaacqaIYaGmaaaakeaacqGGOaakcqWFjpWDdaWgaaWcbaGaemyAaKgabeaakiabgUcaRiab=L8a3naaBaaaleaacqWGWbaCaeqaaOGaeiykaKIaeiikaGIae8xYdC3aaSbaaSqaaiabdchaWbqabaGccqGHRaWkcqWFjpWDdaWgaaWcbaGaem4AaSgabeaakiabcMcaPaaacqGHflY1daWcaaqaaiabikdaYiab=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@A23B@ plays the role of "microscopic permittivity" for the pairwise VdW interactions of atoms i, k. This "permittivity" depends on the distance between atoms i and k, and the space distribution of the other atoms with oscillating electrons around them. The value Φ_ik looks similar to, but does not coincide with the square of the electronic part of dielectric permittivity εab2 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWF1oqzdaqhaaWcbaGaemyyaeMaemOyaigabaGaeGOmaidaaaaa@3211@ given by equation [21]. Thus the naive idea that VdW interaction may be inversely proportional to the square of the electronic part of the dielectric permittivity turned out to be wrong. However, the values Φ_ik and ε_ik are related if all atoms have equal electron oscillation frequencies ω. Since the value ε_ik of the electronic component of the dielectric permittivity is not far from unity 13, we see in this case that

Φ i k ≈ ε i k . MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaiiGacqWFMoGrdaWgaaWcbaGaemyAaKMaem4AaSgabeaakiabgIKi7oaakaaabaGae8xTdu2aaSbaaSqaaiabdMgaPjabdUgaRbqabaaabeaakiabc6caUaaa@3853@

Thus, on average, MB effects decrease the pairwise VdW energy roughly in proportion to the square root of the electronic part of the microscopic dielectric permittivity for the dipole-dipole interaction via a polarizable atomic environment.

The main effect is caused by the electronic permittivity, which is pertinent to small, atomic distances, where the main VdW interaction takes place. One can expect that the electronic component of the microscopic dielectric permittivity for interacting dipoles at these distances is significantly smaller than the electronic dielectric permittivity at macroscopic distances: the decrease in VdW energy caused by three-atom interaction in liquid argon 7 is approximately five-fold less than that expected the from its macroscopic permittivity.