Intermolecular vibronic intensity borrowing of solvent from solute through
dipole-dipole and dipole-dipole-quadrupole interactions
Rintaro Shimada and Hiro-o Hamaguchi
Citation: The Journal of Chemical Physics 140, 204506 (2014); doi: 10.1063/1.4879058 View online: http://dx.doi.org/10.1063/1.4879058
View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/140/20?ver=pdfcov Published by the AIP Publishing
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Molecular near-field antenna effect in resonance hyper-Raman scattering:
Intermolecular vibronic intensity borrowing of solvent from solute through
dipole-dipole and dipole-quadrupole interactions
Rintaro Shimadaa)and Hiro-o Hamaguchia),b)
Department of Applied Chemistry and Institute of Molecular Science, National Chiao Tung University, 1001 University Road, Hsinchu 30010, Taiwan
(Received 26 February 2014; accepted 8 May 2014; published online 28 May 2014)
We quantitatively interpret the recently discovered intriguing phenomenon related to resonance Hyper-Raman (HR) scattering. In resonance HR spectra of all-trans-β-carotene (β-carotene) in so-lution, vibrations of proximate solvent molecules are observed concomitantly with the solute β-carotene HR bands. It has been shown that these solvent bands are subject to marked intensity en-hancements by more than 5 orders of magnitude under the presence of β-carotene. We have called this phenomenon the molecular-near field effect. Resonance HR spectra of β-carotene in benzene, deuterated benzene, cyclohexane, and deuterated cyclohexane have been measured precisely for a quantitative analysis of this effect. The assignments of the observed peaks are made by referring to the infrared, Raman, and HR spectra of neat solvents. It has been revealed that infrared active and some Raman active vibrations are active in the HR molecular near-field effect. The observed spectra in the form of difference spectra (between benzene/deuterated benzene and cyclohexane/deuterated cyclohexane) are quantitatively analyzed on the basis of the extended vibronic theory of resonance HR scattering. The theory incorporates the coupling of excited electronic states of β-carotene with the vibrations of a proximate solvent molecule through solute–solvent dipole–dipole and dipole– quadrupole interactions. It is shown that the infrared active modes arise from the dipole–dipole in-teraction, whereas Raman active modes from the dipole–quadrupole interaction. It is also shown that vibrations that give strongly polarized Raman bands are weak in the HR molecular near-field effect. The observed solvent HR spectra are simulated with the help of quantum chemical calculations for various orientations and distances of a solvent molecule with respect to the solute. The observed spectra are best simulated with random orientations of the solvent molecule at an intermolecular distance of 10 Å. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4879058]
I. INTRODUCTION
Hyper-Raman (HR) scattering is one of the nonlinear analogues of Raman scattering that provides molecular vibra-tional spectra with unique selection and polarization rules.1–11 Recently, we discovered and reported a new phenomenon in resonance HR scattering of all-trans-β-carotene (hereafter ab-breviated as β-carotene) in organic solvents.12Certain solvent HR bands were found to gain intensities by more than 5 orders of magnitude in the presence of β-carotene. No solvent HR bands were observed from neat solvents without β-carotene under exactly the same experimental conditions. This phe-nomenon was interpreted in terms of intermolecular vibronic coupling in which solute (β-carotene) electronic states inter-act with vibrations of neighboring solvent molecules.13 We called it the molecular near-field effect12in analogy with near-field Raman scattering like Surface Enhanced Raman Scatter-ing (SERS) and Tip Enhanced Raman ScatterScatter-ing (TERS), in
a)A part of this research was performed while R. Shimada and H. Hamaguchi
were at Department of Chemistry, the University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan.
b)Author to whom correspondence should be addressed. Electronic mail:
hhama@nctu.edu.tw
which molecules in the vicinity of a photo-excited small metal structure are subject to giant Raman intensity enhancements ascribed to the local excitations of electrons.
We proposed an extended vibronic theory incorpo-rating solute/solvent intermolecular vibronic coupling and accounted for the observed HR intensity enhancement as in-termolecular vibronic intensity borrowing.13,14 However, the origin of solute/solvent vibronic coupling was not discussed in these previous papers with coupling constants introduced in a purely phenomenological manner. The selection rules as well as orientation dependence of the effect were not elucidated either.
In the present paper, we develop our vibronic theory to discuss in further details of the intermolecular vibronic in-teraction and elucidate the physical origin of the molecular near-field effect in resonance HR scattering. High signal to noise ratio (S/N) experimental data for benzene and cyclo-hexane in the forms of difference spectra (C6H6–C6D6 for
benzene and C6H12–C6D12 for cyclohexane) are used in
or-der to eliminate the background signals due to HR signal of solute as well as two-photon excited fluorescence. Thanks to the high symmetry of benzene and cyclohexane, clear selec-tion rules are derived experimentally by comparing the ob-served HR spectra with the already well-established band
0021-9606/2014/140(20)/204506/12/$30.00 140, 204506-1 © 2014 AIP Publishing LLC
assignments.15–18 These selection rules are well accounted for in terms of solute/solvent electrostatic multipole tions, namely, dipole–dipole and dipole–quadrupole interac-tions. The orientation dependence of the enhanced solvent HR spectra is then theoretically calculated. The calculation shows that the observed solvent HR spectra are best reproduced by assuming random orientations of the solute molecules around β-carotene. The mechanism of resonance HR molecular near-field effect is thus clearly identified as vibronic intensity bor-rowing due to the solute/solvent dipole–dipole and dipole– quadrupole interactions.
II. THEORY
In previous papers,13,14 we extended the vibronic the-ory of resonance HR scattering10,11,19–22to incorporate an in-termolecular vibronic coupling term in a phenomenological manner. Now the theory is further developed to a more rig-orous form in which solvent molecular states are explicitly taken into account.
A. Intermolecular vibronic coupling
Consider a bimolecular system where a solute (labeled by A) and a solvent molecule (labeled by B) are weakly cou-pled. Here, we assume that the solute has multiple low lying excited electronic states that may be in resonance with the incident/scattered electromagnetic field, whereas the solvent (ordinary organic solvent) is completely off from electronic resonances.
In the absence of overlap of the molecular electronic wave functions, a zero-order electronic state of the bimolec-ular system can be expressed as a product of the solute and solvent electronic states. Suppose we have a zero-order total excited electronic state|N](0) = |n]A|g]B in which the solute is in the excited state n and the solvent is in the ground state g. Here,|]A and|]B denote ket vectors in the pure electronic spaces of the solute and the solvent, respectively. By applying the perturbation theory, the first-order total electronic state|N] is approximated as |N] ≈ |n]A|g]B+ ng=et |e]A|t]B × [t|B[e|AH|n]A|g] B εA n + εgB −εA e + εtB , (1)
where His the intermolecular interaction Hamiltonian whose physical nature is to be discussed later. εAn, ε
A e, ε B g, and ε B t are the eigen energies of the corresponding pure electronic states denoted by the indices. When the solute possesses several low lying electronic states e whose energies are close to one an-other, the relation|εA
n − ε A e| |ε B g − ε B
t | holds for any sol-vent states t which are t= g. Thus, contributions from solvent excited states (t= g) are neglected in the summation. Since we are interested in how vibrations of the solvent perturb the solute electronic states, the interaction Hamiltonian His ex-panded into a Taylor series of the solvent normal coordinates
QB b to yield |N] = |n]A|g]B+ n=e |e]A|g]B[g| B[e|AH0|n]A|g]B εA n − εeA + b n=e |e]A|g]B h b en εA n − εeA QBb + · · ·, (2) where hben= [g|B[e|A ∂H ∂QB b 0 |n]A|g]B. (3) The first term is the zero-order term, and the second term represents a correction to the zero-order term where electronic states of the solute are coupled due to the interaction with nearby solvent at a fixed geometry. A similar term (without neglecting the contribution from the excited electronic states of solvent) has been used by Myers and Birge23to account for the solvation effect of β-carotene in solution on the oscilla-tor strength of molecular electronic transition. The third term is the term that we focus in the present paper, which repre-sents the mixing of solute electronic states due to the vibration of the nearby solvent, namely, solute/solvent intermolecular vibronic coupling. A similar term has recently been derived to estimate the magnitude of non-Condon intermolecular cou-pling in the dimer units of cyanobacterial light harvesting pro-tein, C-phycocyanin.24
By assuming the electrostatic interaction as the origin of intermolecular vibronic coupling, the interaction Hamilto-nian in Eq.(3)may be expanded into a multipole interaction series.25,26For the interaction between neutral molecules, we have H= H(μμ)+ H(μ)+ · · · = αα Tαα(μμ) μAαμ B α+ αβ Tαβ(μ)μAαβB+ AβμBα+ · · · , (4) where μ and are the dipole and traceless quadrupole mo-ment operators, respectively, of the solute and the solvent (μA, Aor μB, B, respectively). The subscripts α, α(= 1z, 1x, 1y) and β(= 20, 21c, 21s, 22c, 22s) denote the tensorial components of the multipole moments as they are defined by the regular spherical tensor notation27,28 in their respective molecular fixed frame of axis. The regular spherical tensor notation is connected to the Cartesian tensor notation by the following relationship:25 μ1z= μz, μ1x = μx, μ1y = μy, (5a) 20= zz, 21c= 2 √ 3xz, 21s= 2 √ 3yz, (5b) 22c= 1 √ 3(xx− yy), 22s= 2 √ 3xy.
The orientation factors Tαα(μμ) and Tαβ(μ) are the func-tions of orientafunc-tions and posifunc-tions of molecule A and B of which explicit expressions are given in Appendix A. Note that the orientation factor between multipoles of ith and jth
rank is proportional to R−i−j−1, where R is the distance be-tween the coupled molecules. The contributions from higher-order multipole interactions may be neglected at a certain in-termolecular distance or longer. In Eq. (4), only the terms with i + j ≤ 3 or up to R−4 dependence is included. The first term is the dipole–dipole interaction and the second term gives the two dipole–quadrupole interactions where the ori-gins of dipole and quadrupole moments are permuted among a solute-solvent pair.
By substituting the first term of Eq. (4) into Eq. (3), we obtain an expression for intermolecular vibronic coupling with dipole–dipole interaction,
hben(μμ)= αα Tαα(μμ) [e|AμAα|n] A ∂[g|BμB α|g] B ∂QB b 0 . (6)
Here, the interaction term is expressed as a product of three quantities, namely, the orientation factor, the electronic transi-tion dipole moment of solute, and the dipole moment deriva-tive (also known as the vibrational transition dipole moment) of solvent in the ground state. The electronic and vibrational contributions to intermolecular vibronic coupling are thus segregated into the pure coordinate spaces of the solute and the solvent. The two transition dipole moments determine the electronic and vibrational properties of intermolecular vi-bronic coupling. The coupled electronic states of the solute must have the symmetries between which electronic dipole transition is allowed. The vibrational modes of solvent should possess non-vanishing vibrational transition dipole moments, in other words, they must be infrared active vibrations. Above all, the relative orientation of the interacting transition dipoles determines the overall magnitude of the coupling constant. We here note that although the orientation factor does not de-pend on normal coordinates, orientation dede-pendence of hben(μμ) does depend on normal coordinates because the direction of vibrational transition dipoles may differ by mode to mode even within a molecule. In short, each vibration may have its own favorable direction.
Similarly, by substituting the second term of Eq.(4)into Eq.(3), the intermolecular vibronic coupling through dipole– quadrupole interaction is formulated
hben(μ)= αβ Tαβ(μ) [e|AμAα|n]A ∂[g|BB β|g]B ∂QBb 0 + [e|AA β|n] A ∂[g|BμB α|g]B ∂QBb 0 . (7)
The first term inside the square bracket consists of an elec-tronic transition dipole moment of solute and a quadrupole moment derivative of solvent in the ground state. Through this term, the electronic states of solute are coupled by the vi-brations of solvent that changes the ground state quadrupole moment. By considering symmetry, those vibrations indeed coincide with the Raman active vibrations with exception of isotropic totally symmetric modes. This similarity of selection rule arises from the fact that both quadrupole moment and po-larizability are expressed by symmetric second rank tensors, and the exception arises from the additional property of the quadrupole moment tensor that it has no trace component.
In the second term inside the square bracket of Eq.(7), the dipole moment derivative of solvent appears again. There-fore, in addition to the dipole–dipole term, infrared active modes of solvent may promote coupling of solute electronic states through this term as well. The counterpart of the in-teraction is the electronic transition quadrupole moment of solute, which introduces a different electronic selection rule in the intermolecular vibronic coupling from that from the dipole–dipole coupling term.
By the substitution of higher-order multipole interaction terms, such as a dipole–octapole term, it is possible to show that even both IR and Raman inactive but HR active vibra-tions, which were conventionally classified as “dark” modes, may be intermolecular vibronic coupling active. However, the magnitude of such higher order terms diminishes rapidly due to the R−i−j−1dependence of the orientation factor and may not be detectable. In fact, it will be shown in Sec.IVthat the contribution from the multipole interaction terms higher than dipole–quadrupole interaction can be neglected in the present case of the β-carotene/solvent system.
B. Resonance HR scattering
The vibronic theory of resonance HR scattering is ob-tained by following the derivation given in our previous report.14We omit the detailed derivation in the present paper and only to recite the final result from the reference. Accord-ing to the formulation made in the present paper, two minor changes are introduced: (1) The intermolecular vibronic cou-pling constant is replaced with the one given by Eq.(3); (2) solvent degree of freedom is added in the electronic coordi-nate space.
From our previous study on the excitation profiles of the enhanced solvent bands, the enhanced HR bands of solvent under current experimental condition are shown to be origi-nated from the 1-0 resonance first order vibronic term of the hyperpolarizability tensor elements.14The relevant term with the present notation is as follows:
(IHR)if ∝ |(B1)if|2, (8) (B1)if = mu,v b F(ω0)(εn− εe)−1(Mλ)gnhbne(Mμ)em(Mν)mg × (f |v )v QBb u(u|i ), (9) where (Mλ)gn≡ [g|B[g|ARλ|n]A|g]B, (10) and F(ω0)= {(εmu− εgi− ¯ω0)(εnv− εgi− 2¯ω0− inv,gi)}−1. (11) Here, ω0and are the angular frequency of the incident
elec-tromagnetic field and the damping constant, respectively. | ) denotes ket vectors in the pure vibrational spaces. Indices m, e, n, and g refer to the adiabatic total electronic states in the excited states|m]A|g]B,|e]A|g]B, and|n]A|g]B and that in the
ground state |g]A|g]B, respectively, and f, v, u, and i desig-nate the adiabatic vibrational states which span over the com-bined spaces of solute and solvent nuclear degrees of freedom. Equation(9)represents HR transitions involving solvent nor-mal modes in the presence of the molecular near-field effect where optically allowed two-photon transition (Mμ)em(Mν)mg (possibly 3Ag ← 1Ag or 2Ag ← 1Ag) and optically allowed resonant one-photon transition (Mλ)gn(1Bu→ 1Ag) in the so-lute electronic manifold are coupled through intermolecular vibronic interaction hb
ne. Substituting Eq.(6)into Eq.(9), we obtain hyperpolarizability element arising from the dipole– dipole interaction (B1)if = mu,v b F(ω0)(εn− εe)−1 × α (Mλ)gn[n|AμAα|e] A (Mμ)em(Mν)mg × α Tαα(μμ) ∂[g|BμBα|g]B ∂QB b 0 × (f |v )v QBb u(u|i ). (12) Similarly, substituting Eq.(7)into Eq.(9)gives the one aris-ing from the dipole–quadrupole interactions
(B1)if = mu,v b F(ω0)(εn− εe)−1 × α (Mλ)gn[n|AμAα|e] A (Mμ)em(Mν)mg × β Tαβ(μ) ∂[g|BBβ|g]B ∂QB b 0 + β (Mλ)gn[n|AAβ|e] A (Mμ)em(Mν)mg × α Tαβ(μ) ∂[g|BμB α|g] B ∂QB b 0 × (f |v )(v| QB b |u)(u |i ). (13) Because of the explicit incorporation of vibrational tran-sition dipole as well as quadrupole moments in the expression of hyperpolarizability, not only the infrared active but also the Raman active but HR inactive vibrations of solvent are ex-pected to be enhanced through the intermolecular vibronic coupling. Therefore, examining the symmetry species of the enhanced bands in detail will be a crucial test for the validity of our theory.
III. EXPERIMENTAL A. Apparatus
A picosecond cw mode-locked Ti:sapphire oscillator (Spectra Physics, Tsunami) was used as the light source. The center wavelength, the repetition rate, and the typical pulse duration were 810 nm, 82 MHz, and 3 ps, respectively. The output of the oscillator was attenuated by a neutral density
fil-ter and focused into a sample solution contained in a quartz fluorescence cuvette by an achromatic lens. The 90◦scattered HR light was collected by a camera lens and dispersed by a polychromator (Horiba Jobin Yvon, iHR-320), and detected by a liquid nitrogen cooled charge coupled device detector (Roper Scientific, Spec-10 2KB-EV/LN). Scattered incident radiation was eliminated by a dichroic filter placed in front of the entrance slit of the spectrometer. Excitation power at the sample point was 300 mW. We note that the quadratic re-sponse of the HR signals has been confirmed with the excita-tion power up to 300 mW. For each sample soluexcita-tion, averag-ing over 40 spectra with 180 s exposure was done followed by subtraction by a neat solvent spectrum acquired in the same manner. A fraction of the incident beam was separated and in-troduced into a β-barium borate crystal for second harmonic generation. The intensity of the fundamental and the second harmonic signal were simultaneously monitored during HR measurements and used for making correction for any fluctua-tion of pulse characteristics during exposure. Wavelength de-pendence of instrumental sensitivity was corrected. Because of the large chromatic aberration of the camera lens used for collecting HR scattering light, the spectral range suitable for quantitative discussion is limited only within the Raman shift from 300 cm−1to 1800 cm−1.
B. Samples
All-trans-β-carotene, benzene-h6(high-performance
liq-uid chromatography grade), cyclohexane-h12 (spectroscopic
grade), and tetrahydrofuran (high-performance liquid chro-matography grade) were purchased from Wako Chemi-cal Corp. and used as received. All-deuterated benzene-d6
(D:99.5%) and cyclohexane-d12 (D:99.5%) were also
com-mercially obtained from Cambridge Isotope Laboratories, Inc., and used as received. Set volume of a concentrated stock solution of β-carotene in tetrahydrofuran was trans-ferred with a micropipette to a volumetric flask. The sol-vent, tetrahydrofuran, was evaporated completely before re-maining β-carotene was redissolved into desired solvent. The final concentration of the sample solution was 1.1 × 10−4 M for both benzene-h6 and benzene-d6 solution, and 1.4
× 10−4M for both cyclohexane-h
12 and cyclohexane-d12
so-lutions. The sample preparation was carried out under deep red light in order to prevent the sample from undergoing photoisomerization.
C. Density functional calculation
All of the density functional theory calculations were per-formed using the Gaussian 03W program suite.29 The op-timized geometry and the normal mode of vibrations were determined analytically using the B3LYP/6-3111G(d,p) level of theory computations. All obtained vibrational frequencies were scaled by a factor of 0.98 to achieve better agreement with the observed frequencies. The ground state permanent dipole and traceless quadrupole moments were obtained from the single point calculations at optimized geometry under the same basis sets as those used in the vibrational frequency
calculations. The normal coordinate derivatives of dipoles (quadrupoles) were estimated from the change in the ground state permanent dipole (quadrupole) moments derived by the single point calculation under distorted molecular structures along the relevant normal coordinate.
IV. RESULTS AND DISCUSSION A. Difference spectra
Resonance HR spectra of β-carotene in benzene-h6 (a;
red), in benzene-d6 (a; blue), and their difference spectrum
(b) are shown in Figure1. The vertical axis of the difference spectrum is taken so that bands observed in the hydrogenated solution appear as positive peaks and those in the deuterated solution as negative. In the difference spectrum, numbers of positive and negative peaks are observed. HR bands of solute and the two-photon fluorescence background should be absent in the difference spectrum, since these signals are insensitive to the deuteration of solvent and thus are canceled out by sub-traction. Therefore, all the bands in the difference spectrum are solely ascribed to the enhanced HR signals of solvents. Note that non-resonance HR bands from bulk solvents, which are roughly an order of magnitude smaller in intensity than the enhanced bands, have no effect on the difference spectra because they had been pre-subtracted from the solution phase spectra as described in Sec.III.
As is predicted by our vibronic theory incorporating dipole–dipole and dipole–quadrupole interactions, many of the infrared as well as Raman active vibrations of solvent are found in the difference spectrum. The most prominent band observed in the difference spectrum at 679 cm−1for positive (and at 501 cm−1for negative direction) is assigned to an in-frared active vibration of benzene, which, in fact, is the most intense IR band in the finger print region. All the other in-frared active vibrations 1481 and 1039 cm−1 (and 1334 and 814 cm−1 for -d6) are also found in the difference spectrum
with the relative intensities similar to those in the infrared
ab-3.0 2.5 2.0 1.5 1.0 0.5 0.0 Intensity 1600 1200 800 400 Wavenumber / cm-1 -0.5 0.0 0.5 intensity (a) (b) 679 501 1481 1334 1556 814 1039 1178 865 946 672 576 609 355 854 1579
FIG. 1. Resonance HR spectra of all-trans-β-carotene (a) in benzene-h6(red)
and in benzene-d6(blue), and their difference spectrum (b). Dotted line is an
eye guide showing the baseline.
16 12 8 4 0 Intensity 1600 1200 800 400 Wavenumber / cm-1 -1.0 -0.5 0.0 Intensity (a) (b) 686 939 1123 1081 985 912 795 1217 722 859 1267 1158 1031 1451 902 1347
FIG. 2. Resonance HR spectra of all-trans-β-carotene (a) in cyclohexane-h12
(red) and in cyclohexane-d12(blue), and their difference spectrum (b). Dotted
line is the eye guide showing the baseline.
sorption spectrum. In addition to the above-mentioned three IR active vibrations, numbers of peaks are found and as-signed to Raman active vibrations of the solvent. A simi-lar and more pronounced observation is made in the case of cyclohexane(-h12/-d12) solutions [Figs.2(a)and2(b)]. The
as-signments of the observed enhanced peaks in the two differ-ence spectra are made by referring to infrared, Raman, and hyper-Raman spectra of corresponding neat solvents. Results are tabulated in TablesIandII. The symmetry species of each band is taken from the literature.15–18Note that the mutual ex-clusion rule holds between infrared active and Raman active vibrations (as well as between Raman active and HR active vibrations) due to the presence of inversion symmetry in ben-zene and cyclohexane. Therefore, there is little ambiguity in these assignments of the observed solvent vibrations.
Here, we note that the relative intensities of the Raman active vibration in the difference HR spectrum are consider-ably different from those observed in the Raman spectrum; totally symmetric vibrations are generally prominent in the Raman spectrum but this is not the case with the HR differ-ence spectrum. This tendency can be qualitatively explained by considering the traceless property of the quadrupole mo-ment tensor. The well-known totally symmetric breathing mode of benzene, which gives a highly intense Raman band at 993 cm−1(943 cm−1for -d6), is almost completely absent in
the difference spectrum. Similar trend is found for the cyclo-hexane case where the strong Raman active totally symmetric mode at 802 cm−1(723 cm−1for -d12) shows only weak
in-tensities in the difference spectrum. These totally symmetric Raman bands are highly polarized and derive their intensi-ties mostly from the isotropic part of the polarizability ten-sor. Generally speaking, these totally symmetric vibrations have the same symmetry with the isotropic part of a symmet-ric second rank tensor. Since quadrupole moment tensor is a symmetric second-rank tensor and does not have the isotropic part, those vibrations have little effect on the quadrupole mo-ment derivatives that determine the activity in the intermolec-ular vibronic coupling.
TABLE I. Observed and reference vibrational frequencies of vibrational modes of benzene(-h6, -d6), and their assignments.
Observed frequency Reference frequency
(enhanced bands) (cm−1) (neat solvent) (cm−1) Assignmenta Sym. speciesa Activityb -h6 -d6 -h6c -d6d
1579 1556 1586 1558 Ring stretch e2g R
1481 1334 1479 1330 Ring stretch+ deform e1u IR HR
1348 1056 CH/CD bend a2g 1310 1285 Ring stretch b2u HR 1178 865 1177 868 CH/CD bend e2g R 1149 824a CH/CD bend b2u HR 1039 814 1036 812 CH/CD bend e1u IR HR 1011 970 Ring deform b1u HR 993 838 CH/CD bend b2g 946 993 945 Ring stretch a1g R 973 795a CH/CD bend e 2u HR 854 672 850 664 CH/CD bend e1g R 702 600 Ring deform b2g 679 501 676 500 CH/CD bend a2u IR HR 609 576 606 578 Ring deform e2g R 355 403 352a Ring deform e 2u HR aReference15.
bIR, R, and HR refer to IR, Raman, and HR active vibrations, respectively. cReference17.
dReference16unless otherwise noted.
TABLE II. Observed and reference vibrational frequencies of vibrational modes of cyclohexane(-h12, -d12), and their assignments.
Observed frequency Reference frequency
(enhanced bands) (cm−1) (neat solvent) (cm−1) Assignmentsa Sym. speciesa Activityb
-h12 -d12 -h12c -d12c 1451 1123 1466 1117 CH2/CD2 scis a1g R 1451 1081 1458 1069 CH2/CD2 scis eu IR HR 1451 1081 1458 1090 CH2/CD2 scis a2u IR HR 1451 1081 1445 1072 CH2/CD2 scis eg R 1348 864a CH2/CD2 twist a1u HR 1347 1347 1163 CH2/CD2 wag eu IR HR 1347 795 1347 794 CH2/CD2 wag eg R 1320 1126a CH2/CD2 wag a 2g 1267 939 1268 937 CH2/CD2 twist eg R 1267 985 1261 991 CH2/CD2 twist eu IR HR 1158 1158 1012 CH2/CD2 rock a1g R 1107 842a CH2/CD2 wag a1u HR 1091 1187a CC stretch+ CC torsion a 1u HR 1057 778a CH2/CD2 twist a2g 1031 912 1039 916 CH2/CD2 rock a2u IR HR 1031 1217 1029 1212 CC stretch eg R 902 686 906 686 CH2/CD2 rock eu IR HR 859 722 863 719 CC stretch eu IR HR 722 802 723 CC stretch a1g R 785 633 CH2/CD2 rock eg R 522 393 CCC deform a2u IR HR 423 372 CCC deform+ CC torsion eg R 384 298 CCC deform+ CC torsion a1g R 238 203a CCC deform+ CC torsion e u IR HR aReference15.
bIR, R, and HR refer to IR, Raman, and HR active vibrations, respectively. cReference18unless otherwise noted.
Finally, the contribution of higher-order multipole interaction terms, such as a dipole-octapole term, is dis-cussed. Benzene possesses five “dark” normal modes that are IR/Raman inactive but HR active in the wavenumber region lower than 1600 cm−1. None of those bands, except for the weak one at 355 cm−1 in benzene-d6, are observed in the
difference spectrum. The absence of dark modes indicates negligible contribution from the higher-order multipole interaction, most probably due to the higher order inverse dependence on R.
B. Orientation analysis
In Sec.IV A, the selection rules of the enhanced solvent bands have been discussed. Next, we try to quantitatively an-alyze the observed enhanced solvent band intensity based on Eqs.(12)and(13). Analysis has been conducted in two steps. At first, a HR spectral pattern from a single solute-solvent pair at a selected geometry is calculated to investigate the ori-entational properties of the intermolecular vibronic coupling. Then, the theory is extended to cover a polymolecular system where a solute interacts with a number of solvent molecules.
Two assumptions are made in order to reduce the com-putational complexities. First, the initial state is assumed to be the vibrational ground state of the ground electronic state. Second, harmonic vibrational wave functions are assumed. Furthermore, in the case of solute with an inversion symme-try, as in the present β-carotene case, the second term in the square bracket in Eq.(13)vanishes because quadrupole tran-sition between one-photon and two-photon allowed excited states are forbidden,
(Mλ)gn[n|AAβ|e] A
(Mμ)em(Mν)mg = 0. (14) As a result, the following interaction Hamiltonian is chosen:
hbne = α Tzα(μμ)[n|AμAz|e]A ∂[g|BμB α|g] B ∂QB b 0 + β Tzβ(μ)[n|AμA z|e] A ∂[g|BB β|g]B ∂QBb 0 . (15)
Here, the direction of solute local frame of axis is taken so that transition dipole moment μA
ne is oriented parallel to the molecular fixed z-axis. Substituting Eq.(15)into Eq.(9)then into Eq.(8), HR intensity of a mode QB
b is given by IHR(Qb)∝ |G(ω0)|2 α Tzα(μμ) ∂[g|BμBα|g]B ∂QB b 0 + β Tzβ(μ) ∂[g|BB β|g]B ∂QB b 0 ⎤ ⎦ 2 × |(1b|1b) 1b QBb 0b |2, (16) G(ω0)= m F(ω0)(εn− εe)−1(Mλ)gn[n|AμAz|e] A × (Mμ)em(Mν)mg. (17) 1 0 Normalized Intensity 1500 1000 500 Wavenumber / cm-1
FIG. 3. Vibrational frequency dependence of |G(ω0)|2 (dotted line),
|(1b|Qb|0b)|2(dashed line), and the product|G(ω0)|2|(1b|Qb|0b)|2(bold solid
line). Curves are normalized at 1000 cm−1.
A HR spectral pattern from a single solute-solvent pair at a given geometry can be calculated from Eq. (16) since numerical evaluation of the normal coordinate dependence of each quantity in the equation is possible. Under the har-monic oscillator approximation, the vibrational matrix ele-ment|(1b|Qb|0b)|2yields a factor proportional to ω−1b , where ωb is the frequency of the relevant vibration. A resonance electronic transition term G(ω0) also exhibits normal
coordi-nate dependence because each vibrational modes is in reso-nance to different vibronic states (1-0 resoreso-nance mechanism, see Ref.14for detail). ωbdependence of G(ω0) can be
calcu-lated based on the parameters determined by excitation pro-file measurements.14The ω
bdependence of these two factors are plotted in Fig.3, in which ωb dependence are shown to cancel with each other accidentally in the current region of interest to give almost ωb independent plot for the product |G(ω0)|2|(1b|Qb|0b)|2. The normal coordinate derivatives of dipoles and quadrupoles in Cartesian tensor formulation are estimated using the Gaussian 03W program suite.
Figure 4 shows calculated HR spectral patterns of the enhanced solvent vibrations of benzene with a fixed
x z y (a) (b) ×64 (c) ×729 0 Intensity 0 Intensity 0 Intensity 1600 1200 800 400 Wavenumber / cm-1
FIG. 4. Calculated HR spectral patterns of the enhanced solvent vibrations of benzene at intermolecular distances of 5 Å (a), 10 Å (b), and 15 Å (c). The corresponding relative geometries of the solvent with respect to solute are shown on the right side of each spectrum. Red and blue colored bars indicate the intensity of HR bands arisen from the dipole–dipole and dipole– quadrupole interactions, respectively. The solid gray curve shows the simu-lated overall difference spectra with artificial bandwidth of 20 cm−1. Spectra in (b) and (c) are magnified by the factors designated in each graph.
geometry (see figure) at three different distances, 5, 10, and 15 Å. The bars indicate the intensities of individual normal modes, where red and blue colors correspond to bands arising from the dipole–dipole and dipole–quadrupole interactions, respectively. For a clear comparison with the observed spec-tra, the intensities of benzene-h6 are plotted in the positive
directions and those of benzene-d6are shown in the negative
directions. The solid gray curve indicates the simulated over-all difference spectra with artificial bandwidth of 20 cm−1on each band. The simulated spectral patterns show comparable contributions from the dipole–dipole (red bar) and dipole– quadrupole induced (blue bar) HR bands, when the inter-molecular distance R is 5 Å. When R is 15 Å, however, the red bar dominates the spectral pattern. This R dependence is not surprising since intensity of dipole–dipole induced vibrations follow R−6 dependence while dipole–quadrupole induced ones follow R−8. It is interesting to obtain an R value close to the size of the first solvent shell (∼10 Å) in order to repro-duce the observed relative intensity of bands arising from dif-ferent multipole interactions. The results imply that the mag-nitude of quadrupole derivative of the solvent is, in fact, large enough to be able to give rise to the intermolecular vibronic coupling thorough interaction with the transition dipole of the solute.
The simulated spectral patterns also show drastic depen-dence on the solvent orientation with respect to the solute. Figure5shows calculated HR spectral patterns at a fixed dis-tance (10 Å) with different orientations. The intensity of a particular vibrational mode increases when the molecular ori-entation is set to its favorable direction. This situation is best illustrated in Figs. 5(a)–5(c), where red bars (dipole–dipole interaction) show two distinct spectral patterns (a) and (b), (c). Since the interaction between two dipoles is maximized when they are aligned in the parallel geometry but vanishes when they are perpendicular to each other, the vibrational modes whose transition dipole moments pointing parallel to the z-axis of solute local axis are favored. It can be eas-ily seen that only the out of plane vibration (679/501 cm−1) gain large enhancement for the configuration (a), whereas, in contrast, only the in-plane vibrations (1481/1334, 1039/814 cm−1) become prominent for the configurations (b) and (c). The blue bars (dipole–quadrupole interaction) follow differ-ent and more complicated oridiffer-entation dependence. As a re-sult, the enhanced bands exhibit characteristic spectral pat-terns on the relative orientation of the interacting molecule pairs.
Finally, we briefly discuss the effect of intermolecular distance R on the simulated HR spectral profiles. In contrast to the angular parameters which changes the spectral pat-terns greatly, R mainly governs the overall intensity as the intensity of dipole–dipole (red-bar) and quadrupole–dipole induced vibrations (blue bar) are proportional to R−6 and R−8, respectively. When solvent is located at the apex of so-lute, where R∼ 17 Å (Van der Waals radius of β-carotene (long axis) is estimated to be 14 Å23 and that of benzene is 3 Å,30,31 respectively), HR intensity is about 1–2 orders of magnitude smaller than when it is located alongside, where R ∼ 8 Å (Van der Waals radius of β-carotene (short axis) is es-timated to be 5 Å23 and that of benzene 3 Å, respectively),
x z y (a) (b) ×5 (c) ×5 0 intensity 1600 1200 800 400 Wavenumber / cm-1 (d) 0 Intensity 0 Intensity 0 Intensity
FIG. 5. Calculated HR spectral patterns of the enhanced solvent vibrations of benzene (a)–(c) and the observed difference spectrum in benzene solution (d). For the calculated spectra, a benzene molecule is placed with its ring plane parallel to xy (a), xz (b), and yz (c) plane of the solute local frame of axis. The corresponding relative geometries of the solvent with respect to solute are shown on the right side of each spectrum. Red and blue colored bars indicate the intensity of HR bands arisen from the dipole–dipole and dipole–quadrupole interactions, respectively. The solid gray curve shows the simulated overall difference spectra with artificial bandwidth of 20 cm−1. Spectra in (b) and (c) are magnified by the factors designated in each graph. The intermolecular distance is set to 10 Å.
as expected from the doubled distance (2−6 = 1/64, 2−8 = 1/256).
So far, all the discussion has been based on a single solute-solvent pair. In order to account for the experimen-tally observed HR spectra, the theory must be extended to cover a polymolecular system where a solute can interact with more than one solvent molecule, since observed HR signal is generated from solutes that are surrounded by many solvent molecules. This is easily achieved by simply treating the in-teraction in N molecular system as a collection of N pairwise interactions because electrostatic interaction is strictly pair-wise additive. In other words, one hyper-Raman transition ac-companies only one vibrational transition of the solvent. The ergodicity assures that the rotational and translational motion of the molecules during the HR measurement is taken into ac-count by taking ensemble average of the orientation factor in Eq.(16)over all the possible configurations of solute-solvent pairs when deriving the HR intensity
IHR(Qb)∝ |G(ω0)|2 ωb ⎡ ⎣ α Tzα(μμ) 2 ∂[g|BμB α|g]B ∂QB b 0 2 + β Tzβ(μ) 2 ∂[g|BBβ|g]B ∂QB b 0 2⎤ ⎦ . (18)
Here, the overline ( ¯ ) symbol denotes taking the ensemble average of the quantity under the bar. Because of the cen-trosymmetry of the solvent, benzene/cyclohexane, the dipole derivative, and quadrupole derivative are mutually exclusive, i.e., only either, if any, of the terms can have nonzero value for a normal mode QB
b. Therefore, the cross term is omitted when expanding the squared square bracket in Eq.(16).
In evaluating the ensemble average of the solvent config-uration in the vicinity of solute, we adopt an approximation to further simplify the calculation. Although there are many sol-vent molecules surrounding the solute, the contribution from the solvent molecules with longer R may be neglected because the high order (6 or 8) inverse dependence of HR signal on R. Only the solvents in the first solvation shell at representative intermolecular distance R0is considered. Then, the following
two extreme cases about the solvent orientation were com-pared: (1) A solute and the closest solvent molecules have a specific preferential orientation; (2) the solute and solvent molecule have no preferential orientation and can freely ro-tate around its center of inertia.
Simulated HR spectra for the above case 1 is the same as those shown in Figs.4and5. Therefore, we can immediately conclude that the simulated HR spectra do not resemble the observed HR spectra, indicating unlikely presence of a spe-cific preferential orientation in the solution.
In order to evaluate the case 2 where molecules can freely rotate around, i.e., random orientation, each orienta-tion parameters were taken the rotaorienta-tional average over the relative orientation of solvent fixed local frame of axes. By taking the rotational average over solvent orientation (AppendixB) Tzα(μμ) 2= 1 3R −6 0 [3cos2θ+ 1] ≈ 2 3R −6 0 , (19a) Tzβ(μ) 2 =3 5R −8 0 [2cos2θ+ 1] ≈ R0−8, (19b)
where θ is the polar angle in spherical coordinate designat-ing the location of the solvent molecules on solute local co-ordinate system. Assuming the free rotation of solute gives the orientation average cos2θ= 1 / 3, yielding the last step in
Eqs.(19a)and(19b). HR intensity (Eq.(18)) becomes
IHR(Qb)∝ 2 3R −6 0 |G(ω0)|2 ωb α ∂[g|BμB α|g] B ∂QB b 0 2 + R−8 0 |G(ω0)|2 ωb β ∂[g|BB β|g] B ∂QBb 0 2 . (20)
The HR signals from the same multipole interaction mech-anism now have fixed spectral patterns because directional-ity of multipoles is averaged out. The intensdirectional-ity of each vi-brational mode becomes proportional to the summed squared magnitude of the corresponding dipole/quadrupole transition moments, leaving R0 as the only parameter to determine the
whole HR spectral profile.
Figure 6 shows the comparison between the observed enhanced HR bands of benzene and the simulated difference spectrum after taking the rotational average of the solvent
0 Intensity 1600 1200 800 400 Wavenumber / cm-1 0 intensity (a) (b)
FIG. 6. Comparison of the observed (a) and simulated (b) difference HR spectra measured in benzene-h6/-d6 solutions. Simulated curves are
calcu-lated with the representative intermolecular distance R0of 10 Å.
orientation. Figure 7 shows a similar comparison for the cyclohexane solution. Excellent agreement between the simulated HR spectra and the observed difference spectra in both solutions are achieved. Most of the characteristic features including the absence of strong totally symmetric Raman active vibrations mentioned earlier are reproduced in the simulated spectra. The result suggests that the presence of solute does not interfere solvent from freely rotating. The intermolecular distance to the nearest neighboring solvent are estimated to be roughly 10 Å from the simulation, which is in good agreement with the sum of van der Waals radii of the solute and solvents, indicating the possibility of efficient detection of the solvent molecules in the first solvation shell.
So far, the experiments were conducted only for non-polar solvents, because of the experimental difficulty aris-ing from very poor solubility of β-carotene in polar solvents. However, we would like to emphasize that the theory should be equally applicable to the polar solvents, because the inter-action Hamiltonian is based on the normal coordinate deriva-tives of the permanent dipoles/quadrupoles of the solvent but not on the permanent dipoles/quadrupoles themselves. Since the polar solvents are likely to form solvent structure due to permanent dipole–dipole interactions, specific solute–solvent orientation may exist. It would be of great interest, although
0 Intensity 1600 1200 800 400 Wavenumber / cm-1 0 Intensity (a) (b)
FIG. 7. Comparison of the observed (a) and simulated (b) difference HR spectra measured in cyclohexane-h12/-d12solutions. Simulated curves are
calculated with the representative intermolecular distance R0of 10 Å.
challenging, to study the present molecular near-field effect in polar solvents.
C. General discussion
The present theory clearly indicates that the resonance HR molecular near-field effect is more rigorously interpreted as a photophysical response of a bimolecular system to an incident electromagnetic field. It can also be considered, however, as a resonant optical process of molecule A (β-carotene) that incorporates pairwise intermolecular interac-tion with nearby molecule B (benzene and cyclohexane). The transient electronic excitation of molecule A in this optical process may be regarded as a quasi-light source that acts on and detect molecule B in the near-field. The latter view is essentially the same as those discussed by Andrews and his co-workers32,33 under the framework of quantum electrody-namics with virtual photon concept. It also better fits to the existing concept of near-field optics and spectroscopy. Here, we discuss two other analogous optical processes that may also be understood on the same basis.
Surface-Enhanced Raman/Hyper-Raman scattering (SEHRS)34–37 and TERS38–41 have apparent strong resem-blance to the HR molecular near-field effect. They both require a probe (β-carotene in the present case and nano-structured metal in SERS). Incident laser wavelength must be tuned into the absorption of the probe. Observed enhanced Raman/HR spectra do not seem to follow the selection rules of conventional Raman scattering. When we first reported the HR signal enhancement by β-carotene in solution,12,13 this resemblance lead us to introduce the term “molecular near-field effect,” though at that time the origin of intensity enhancement was unclear. However, the present study has proved that its intensity enhancement mechanism is quite different from what is thought for SERS. The most widely accepted mechanism of SERS assumes the adsorbate Raman intensity enhancement by amplified electric field in the near-field of metal probe that is produced by the excitation of surface plasmon-polaritons.34,42,43In this electromagnetic mechanism, the interaction between probe and the adsorbate is not really essential for the Raman process. The field generated by the probe acts as if an external light source to the adsorbed molecule and hence it is possible to regard the whole process as a stepwise two independent optical processes; the excitation of surface plasmon-polaritons of the probe occurs first and Raman scattering of the adsorbate follows. On the other hand, in the HR molecular near-field effect, the transient electronic excitation acts as if a near-field antenna to let surrounding solvent’s nuclear motion intrude into the resonance HR process of the probe molecule. The intermolecular interaction is embedded in the HR process, and it is not possible to further break down the whole process into two respective optical processes of the probe and the solvent. We note the possibility that, contrary to what is presently accepted, SERS/SEHRS and TERS should also be viewed as inseparable optical processes that arise from the molecular near-field effect.
Interaction between the two transition moments of nearby molecules during an optical process is also
well-known as Fluorescence Resonance Energy Transfer, FRET.44,45 In FRET, a photo excited donor molecule inter-acts with a nearby acceptor molecule in the ground state via a transition dipole-transition dipole interaction to create an excited state acceptor and a ground state donor, resulting in fluorescence emission from the acceptor molecule. Similar to the HR molecular near-field effect, intensity (or lifetime) of FRET signal shows large dependence on the relative ge-ometries of interacting molecules owing to the large orienta-tional dependence of the dipole–dipole interaction. Because of this property, FRET is known as a molecular ruler. The FRET process is considered as non-radiative energy transfer between donor and acceptor molecule during a fluorescence process.
There is a distinction to be discussed although the ba-sic intermolecular interactions are similar for FRET and the molecular near-field effect. In FRET, electronic transitions take place between the real states of the molecules, hence the interacting transition dipole moments must be in resonance with each other, in other words, must connect states with the same amount of energy gap to satisfy energy conservation. This condition limits the FRET process to occur only among specific pairs of donor and acceptor molecules for which the emission spectrum of the former overlaps with the absorption of the latter. In the HR molecular near-field effect, the inter-molecular energy transfer is not real, since there will be no net exchange of energy between the donor (molecule A) and the acceptor (molecule B). Although multipole interactions take place between these molecules, the energy for vibrational ex-citation of molecule B is provided via the intermediate states of molecule A by the electromagnetic field as the energy dif-ference between the incident and scattered photon energies. Therefore, interactions can take place with a variety of vibra-tional transition dipole/quadrupole moments of the solvent ir-respective of their transition energies. There is no restriction of the solvent molecule to be detected in the HR molecular near-field effect.
V. CONCLUSIONS
With the excellent reproduction of the observed spectra by the simulated spectra based on Eq.(20), we consider that the molecular level mechanism and the origin of the mysteri-ous phenomenon, the “molecular near-field effect,” has been fully elucidated. The origin of the effect is identified as an electromagnetic field generated by transient electronic exci-tation produced during the HR optical process. The excita-tion acts as if a near-field antenna to let surrounding solvent’s nuclear motion intrude into the resonance HR process of β-carotene by way of dipole–dipole and dipole–quadrupole in-teractions. The intensity enhancement is well accounted for under the framework of resonance HR scattering where the solute “lend” its electronic resonance to the nearby solvent through the intermolecular vibronic coupling. The “molecular near-field effect” is more specifically termed as the “molecu-lar near-field antenna effect.”
The present study has shed light on a new type of optical process that specifically incorporates intermolecular interac-tion in the intermediate state. The selecinterac-tion rule in this optical
process is determined not only by the interactions with the incident electromagnetic field but also by the near-field mul-tipolar intermolecular interactions. This class of optical pro-cesses may open up new possibilities of spectroscopy, molec-ular near-field spectroscopy, which provides direct observa-tion methods of molecules lying in the vicinity of a molecu-lar probe. We note the possibility that the SERS/SEHRS and TERS mechanism can also be explained on the same basis of molecular near-field antenna effect. With the use of Eq.(20), it is now possible to extract and discuss geometric information of solvating molecules around a solute with high molecular specificity characteristic of vibrational spectroscopy. It will be a powerful tool for investigating nanoscale intermolecular interactions in complex molecular systems.
ACKNOWLEDGMENTS
Part of this research has been supported by the National Science Council of Taiwan (NSCT) Grant No. NSC102-2113-M-009-003 and “Aiming to the Top University Project” of Ministry of Education in Taiwan. The authors are grateful to Professor S.-H. Lin for his insightful comments on inter-molecular interactions.
APPENDIX A: DEFINITION OF ORIENTATION FACTORS
According to Ref.25, the orientation factors in the spher-ical tensor formulation are defined as follows:
Tαα(μμ) = R−3 3rαAr B α+ cαα , (A1) Tα(μ)20 = R−41 2 15rzB2rαA+ 6rzBcαz− 3rαA , (A2a) Tα(μ)21c = R−4√3rxBcαz+ cαxrzB+ 5r B xr B zr A α , (A2b) Tα(μ)21s = R−4√3ryBcαz+ cαyrzB+ 5r B yr B zr A α , (A2c) Tα(μ)22c = R−4 √ 3 2 5rxB2−ryB2rαA+ 2rxBcαx− 2ryBcαy , (A2d) Tα(μ)22s = R−4√35rxBr B yr A α + r B xcαy+ ryBcαx , (A2e) where rαA= eAα· eAB, rαB = e B α· eBA and rαB= e B α· eBA (A3) eAα and e B
α (α, α = x, y, z) are the unit vectors defining the Cartesian local coordinate systems for the molecule A and B, respectively (Figure8). eABand eBA(= −eAB) are unit vectors in the direction connecting the origins of local coordinates from A to B and from B to A, respectively. rA
α and rαB are projections of eAB(eBAin the case of latter) on local frame of axes of molecule A and B, respectively. cααcorresponds to an
FIG. 8. The coordinate system of molecular frame of axes.
element of rotation matrix which rotates eAα into e B
α. We can write the following relationship:
eBα= i ciαeAi (A4) and thus, rαB= eBα· eBA= − i ciαeAi · eAB= − i ciαriA. (A5)
APPENDIX B: ORIENTATION AVERAGE OF ORIENTATION FACTORS
We consider a problem of taking rotational average over the relative orientation of molecule B (solvent) fixed local frame of axes. Overline notation ( ¯ ) for the quantity after the rotational average is introduced as follows:
f = 1 8π2 π 0 2π 0 2π 0 f( , , ) sin d dd, (B1)
where Euler angles , , and express the rotation of the molecule B fixed local frame of axes with respect to the molecule A fixed coordinate system.
1. Dipole–dipole interaction
By substituting Eq. (A5) into (A1), we obtain the ex-pression for orientation factor of dipole–dipole interaction in terms of rA α and cαα alone, Tαα(μμ) = R−3 −3rA α i ciαriA+ cαα . (B2)
Taking squared modulus, we get Tαα(μμ) 2= R−6 ⎡ ⎣9 rA α 2 i,j ciαcj αriAr A j − 6rA α i ciαcααriA+ cααcαα . (B3)
Taking the orientation average of both sides of the equa-tion yields Tαα(μμ) 2 = R−6 ⎡ ⎣9 rA α 2 i,j ciαcj α riArjA − 6rA α i ciαcαα riA+ cααcαα
= R−6 ⎡ ⎣9 rA α 2 i,j δij 3 r A i r A j − 6r A α i δiα 3 r A i + 1 3 ⎤ ⎦ = R−6 3rαA2 i riA2− 2rαA2+1 3 . (B4)
Notice that rαAare independent of Euler angles , , and , thus, are invariant under rotational average. When deriv-ing the second line, the followderiv-ing property of the rotational averages of direction cosine is used46
ciαcj α = δij
3 , (B5)
where δij is the Kronecker delta function. Substituting the equality i riA2= 1, (B6) Eq.(B1)is simplified to Tαα(μμ) 2 = R−6rA α 2 +1 3 . (B7)
Defining polar θ and azimuthal φ angles in the spherical coordinate designating the location of the solvent molecules on solute local coordinate system,
rxA= sinθcosφ, ryA= sinθsinφ, and rzA= cosθ. (B8) The z-component of the orientation factor is derived,
Tzα(μμ) 2 = R−6cos2θ+1 3 . (B9)
2. Dipole–quadrupole orientation factor
Orientation average of the orientation factors of dipole– quadrupole interaction can be calculated in the same manner as dipole–dipole interaction. All five factors results in the ex-act same expression,
Tzβ(μ) 2=3 5R −82rA z 2 + 1=3 5R −8[2 cos2θ+ 1]. (B10) In the above derivation, the following relationships are used in addition to Eq.(B5):46
ciαcj αckαclα = δijδkl+ δikδj l+ δilδj k 15 , (B11) ciαcj αckβclβ = 4δijδkl− δikδj l− δilδj k 30 . (B12)
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