Calculating dipole and quadrupole polarizabilities relevant to surface enhanced Raman spectroscopy

Spectrochimica Acta Part A 55 (1999) 625 – 638

Calculating dipole and quadrupole polarizabilities relevant to surface enhanced Raman spectroscopy Gary S. Kedziora, George C. Schatz * Department of Chemistry, Northwestern Uni6ersity, E6anston, IL 60208 -3113, USA Received 22 May 1998; accepted 16 July 1998

Abstract We have used ab initio electronic structure calculations to calculate the frequency dependent dipole – dipole polarizability (a), quadrupole–dipole polarizability (A), quadrupole – quadrupole polarizability (C), (electric) dipole– magnetic dipole polarizability (G), and their normal coordinate derivatives for CO. The polarizability derivatives are of relevance to the interpretation of surface enhanced Raman (SER) spectra. Basis set convergence of these spectroscopic properties is studied, along with the effect of including electron correlation at the second-order polarization propagator (SOPPA) level, and the variation of the results with excitation frequency. The largest basis set SOPPA results that we have generated appear to be converged to within 20% or better for most of the properties we have studied, however in a few cases the convergence is much poorer. The most difficult property to converge involves the off-diagonal component of the derivatives of the quadrupole – dipole polarizability tensor. Our results show that the ratio of the largest components of A to the largest components of a are on the order of one atomic unit in size, and a similar statement can be made concerning the corresponding ratios of the normal coordinate derivatives. This means that the ratio of field derivatives to field strengths will also have to be on the order of one atomic unit in order for A and C to contribute comparably to a in determining SERS intensities. © 1999 Elsevier Science B.V. All rights reserved. Keywords: Dipole; Quadrupole; Raman spectroscopy

1. Introduction Surface enhanced Raman spectroscopy (SERS) is an important spectroscopic tool for studying molecules adsorbed on noble metal surfaces, especially silver [1]. Generally, the spectra are very similar in appearance to normal Raman spectra of * Corresponding author. Tel.: +1-847-4915657; fax: +1847-4917713; e-mail: [email protected].

the same molecules in solution, but the intensity per molecule is enhanced by 106 or more, which makes it possible to observe submonolayer coverages and in special cases individual molecules. It is generally agreed that a major mechanism for the observed enhancement is the enhanced electromagnetic fields that exist in the vicinity of rough noble metal surfaces or metal nanoparticles at frequencies where the particles exhibit plasmon resonances [1]. There have been several calcula-

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tions of these fields for noble metal nanoparticles using electrodynamics methods and assuming bulk dielectric constants [2 – 4], and the resulting SERS enhancements (averaged over the particle surface) are typically 104 or larger (depending on particle shape and size), thus accounting for most of the observed enhancement. Another aspect of this electromagnetic mechanism is that the only property of the molecule that is important in determining the SERS spectrum is its dipole polarizability tensor a and normal coordinate derivative a%. Except for producing the enhanced fields, the surface plays a relatively passive role, causing (1) the local electric field to be perpendicular to the surface and (2) the molecule to orient in a specific way relative to this surface normal. Estimates of SERS spectra based on these assumptions and using a and a% from zero frequency Hartree–Fock calculations give results for molecules like pyridine and trans-1,2-bis(4 pyridyl) ethylene that match up reasonably well with measured spectra [5,6]. Despite the apparent success of the dipole polarizability-based field enhancement model, there are aspects of the experimental results which suggest that other factors may be involved. Since the enhanced fields are typically found near highly curved surfaces (radius of curvature:1 –10 nm), enhanced fields must necessarily be associated with enhanced field derivatives. However the importance of the field derivatives in SERS is uncertain. In 1981, Moskovits and coworkers [7] proposed that enhanced field derivatives might lead to enhanced SERS via the quadrupole – dipole term in the expression for the induced dipole moment [8]. If mk is the kth component of the induced moment, then the relation between this and the multipole polarizabilities is given by 1 1 (E mk = akmEm + Ak.mn n + Gk,mH: m +… 3 (xm v

(1)

In this expression, akm is an element of the dipole–dipole polarizability tensor (which is normally the only term that is considered), Ak,mn is an element of the dipole – quadrupole polarizability tensor, and Gk,m is an element of the (electric–) dipole–magnetic – dipole polarizability. In Eq. (1)

and all subsequent equations, the Einstein summation convention is followed. Typical Raman intensities are proportional to dakm /dQ 2, where Q is the vibrational normal coordinate for the molecule being studied. However if the field derivative term (En /(xm is large enough, then the second term in Eq. (1) can become important, leading to changes in Raman scattering spectra. The magnetic field, H, is related to the field derivative by Maxwell’s equation, 1 9× E= H: c Therefore, in situations where the field derivative is expected to be important, the magnetic field may also give a substantial contribution to the Raman spectrum. A consequence of large field derivative effects would be to change the selection rules for Raman scattering, leading to the appearance of Ramanforbidden peaks in SERS spectra. Such peaks have in fact been observed in the spectra of benzene [9,10] but it is still uncertain if this is the correct explanation of these experiments, or whether these forbidden lines are due to lowering of molecular symmetry due to adsorption induced distortion of the molecule. The importance of the A term was studied recently by Campion et al. [11] for benzene adsorbed onto flat silver surfaces. They found that the ratio of the forbidden a2u to allowed a1g modes of benzene was independent of excitation wavelength, which is at odds with the expected result from the field derivative mechanism. However they noted that large field derivatives (such that the ratio of E to (E/(x is on the order of a ˚ ) should exist near the silver surfaces (based few A on jellium calculations), so they concluded that the absence of a field derivative contribution must be due to the fact that the molecules are not located at positions where the field derivative is large. The Campion experiments refer to flat surfaces, so field derivative effects associated with surface curvature would be missing. However typical SERS experiments are done with surfaces that exhibit significant curvature on a 1–10 nm scale, thereby providing another mechanism whereby the field derivative term can contribute.

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One impediment to developing a quantitative understanding of the field derivative effect is the fact that values of (Ak,mn /(Q have never been determined for molecules of interest in SERS. Although values of Ak,mn have occasionally been reported from theory [12,13], methods for measuring or determining (Ak,mn /(Q for use in the interpretation of Raman spectra have not generally been available. (One exception here arises in Raman optical activity measurements [14], but the components of A available from these measurements are different from those needed for SERS.) Recently, a new electronic structure code, DALTON, has been developed [15] which provides frequency dependent values of a and A directly from Hartree – Fock or better calculations, thereby making it possible to study field derivative effects in SERS at a more quantitative level than in the past. This code also determines the quadrupole-quadrupole polarizability tensor C, which also can contribute to Raman intensities when field derivatives are large enough. In this paper we use DALTON to calculate dipole – dipole, dipole – quadrupole, quadrupole– quadrupole, and electric –dipole– magnetic–dipole polarizabilities and normal coordinate derivatives for CO, which is one of the simplest molecules that has been the subject of SERS experiments [16]. Since the calculation of these polarizability properties is new to this paper, part of our work will be concerned with establishing the quality of the results, including convergence with respect to basis set and influence of electron correlation. We will also study the frequency dependence of the results, as this will be of use in establishing the accuracy of earlier theoretical studies of SERS spectra, including one study of benzene [5] that considered only zero-frequency information. In a future paper [17], we will combine the spectroscopic information presented here with electrodynamics calculations of fields in the vicinity of metal particles to make estimates of enhancement factors that are relevant to a variety of SERS measurements. We begin the paper (Section 2) by summarizing the basic formulas and conventions concern-

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ing a, A, C, and G. Section 3 covers the details of the computations. Section 4 presents convergence studies and frequency dependence of the results for each of the CO polarizabilities and derivatives at the time-dependent Hartree Fock and SOPPA levels of theory. Section 5 summarizes our conclusions.

2. Theory We begin by writing down the interaction Hamiltonian that determines Raman intensities when field strengths, field derivatives, and magnetic field strengths are significant. Our theory follows from the work of Barron and Buckingham [18] and of Buckingham [8]. The interaction Hamiltonian for a molecule in a field with angular frequency v expanded through terms that scale linearly with inverse wavelength is given as Hint = − mkEk (v)−

1 (Ek (v) − mkHk (v), 3 (xl

(2)

where we have used Buckingham’s conventions[8] for the multipole moments. The dipolemoment vector is m, U is the quadrupole-moment tensor, and m is the magnetic–moment vector. We derive the Raman intensity expression using standard time-dependent perturbation theory, restricting our consideration to planewave fields; the treatment of more general fields will be presented later [17]. If the fields are expressed as plane waves with wavevector k and polarization o for the field of frequency v and with k(s) and o (s) for frequency v (s), then the resulting expression of the Raman scattering cross section is:

)

ds i (s) (s) (s) 8 o (s) n omanm − (o k omkn + o m k n ok )Ak,mn dV 3 (s) − i(o (s) n (k×e)m − (k× e)m )Gn,m

)

2 1 (s) + o (s) k k l omknCkl,mn 3

(3)

In this equation we leave out the magnetic–dipole–magnetic–dipole polarizability and the magnetic–dipole–electric–quadrupole polarizability

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terms that would result from including all terms in the interaction Hamiltonian Eq. (2). Note that one consequence of the gauge condition on the electromagnetic wave is that omkm =0. This means that there is a lack of uniqueness in some of the elements of A, with only the sums in Eq. (3) having physical meaning. A similar statement applies to C. In past work, several different conventions for choosing these arbitrary elements have been developed, as reviewed by Dykstra et al. [19]. It should be understood that the polarizability tensors in Eq. (3) are really transition averages over the vibrational wavefunctions. These are expanded in normal coordinates, as originally developed by Placzek [20] (see also [18]). For example, the expansion for the dipole – dipole polarizability is n% anm n = (amn )edn%n +% j

 

(amn n% Qj n+ … (Qj e

(4) Note also, that to arrive at Eq. (3) approximations are made that require the photon energy be small compared to the first electronic excitation energy and require the vibrational transition energy be small compared to the electronic excitation energy.

3. Computational methods We studied the basis set convergence properties of the polarizabilities using the augmented correlation-consistent polarized-valence basis set series, aug-cc-pVXZ, where X=D,T and Q [21]. The carbon and oxygen contraction schemes for the three basis sets is (10s5p2d)/[4s3p2d] for augcc-pVDZ, (11s6p3d2f)/[5s4p3d2f] for aug-ccpVTZ, and (13s7p4d3f2g)/[6s5p4d3f2g] for aug-cc-pVQZ. These basis sets were designed for calculations of electron affinities using correlated wavefunctions, and for our purposes provides a balance of diffuse functions and polarization functions. In the discussion that follows and in the tables and figures we will use the abbreviated respective designations ADZ, ATZ, and AQZ for these basis sets.

Frequency-dependent and static dipole-dipole, dipole–quadrupole, quadrupole–quadrupole, and (electric-)dipole –magnetic dipole polarizabilities were calculated using the random phase approximation (RPA) and the second-order polarization propagator approximation (SOPPA) [22]. RPA is equivalent to time-dependent Hartree–Fock or CPHF in the limit of zero frequency. SOPPA adds correlation to the polarizabilities through second order, which includes only single and double excitations relative to the HF reference, but it is not the same as time-dependant MP2 (TDMP2) theory. For a discussion of the differences between TDMP2 and SOPPA see reference [23]. For each of the basis sets in the series, the bond length of CO was first optimized at the SCF level. Polarizability derivatives with respect to the bond length were calculated using the forward-difference approximation with a step size of 1 × 10 − 4 Bohr. Tests with the ADZ basis showed that the forward-difference approximation agrees with the central-difference derivative approximation to three significant figures or more in some cases. To ensure adequate precision in the finite difference results, which in our case is a modest two significant figures, we used tight thresholds in all the calculations. The threshold for neglecting integrals was 10 − 15 a.u.; the threshold for the gradient of the SCF density was 10 − 9 a.u.; and the threshold for the length of the residual in solving the linear equations in the RPA and SOPPA calculations was 10 − 7 a.u.. We encountered no convergence problems with the larger basis sets using these thresholds. The polarizabilities were calculated at the ex˚ perimental equilibrium bond length (1.128323 A [24]) in addition to the SCF bond length. All of the polarizabilities from the literature with which we compare in this paper were calculated at the experimental bond length. For many molecules, the experimental geometry may not be available. Therefore, it is interesting to get an estimate the effect of the bond length difference on the polarizabilities and their derivatives to give us an idea of the magnitude of the errors we might expect. All calculations were performed with the singleprocessor version of the DALTON electronic structure program [15] on an IBM SP2.

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Table 1 Average polarizability, polarizability anisotropy, and their derivatives with respect to bond lengtha

Methodc SCF/RPA

SCF/SOPPA

EXP/SOPPA

Photon

a¯

Da

Energyb

0.00

1.24

2.47

0.00

1.24

2.47

0.00

1.24

2.47

0.00

1.24

2.47

Basis ADZ ATZ AQZ ADZ ATZ AQZ ADZ ATZ AOZ

11.99 12.08 12.08 13.08 13.05 12.98 13.32 13.37 13.32

12.34 12.43 12.43 13.19 13.16 13.08 13.42 13.48 13.43

12.23 12.32 12.32 13.50 13.48 13.40 13.75 13.81 13.76

3.30 2.93 2.86 4.62 4.05 3.94 5.04 4.60 4.53

3.30 2.92 2.86 4.64 4.06 3.95 5.07 4.63 4.55

3.29 2.90 2.84 4.70 4.10 3.98 5.16 4.70 4.62

4.77 4.92 4.97 6.62 6.65 6.63 6.97 7.13 7.14

4.83 4.98 5.03 6.72 6.75 6.73 7.07 7.24 7.26

5.05 5.19 5.25 7.03 7.06 7.04 7.43 7.61 7.62

6.86 6.92 7.09 12.09 12.30 12.94 12.89 13.12 13.87

6.68 6.73 6.90 11.59 11.78 12.37 12.64 12.87 13.59

6.58 6.63 6.79 11.38 11.56 12.14 12.50 12.73 13.44

da¯ dQ

dDa dQ

a

The value of each property is presented for various methods and basis sets and at three different photon energies. Photon energy in eV. c The field before the slash indicates the geometry used. The field after the slash indicates the method used to calculate the ˚ , ATZ 1.104011 A ˚ , and AQZ 1.101985 A ˚ . The experimental geometry is polarizability. The SCF geometries are ADZ 1.110319 A ˚ [24]. 1.128323 A b

4. Results

4.1. Dipole– dipole polarizabilities Dipole–dipole polarizabilities of CO have been studied by many authors. It is common to find them expressed in terms of the average polarizability and the polarizability anisotropy. For a diatomic the average polarizability is a =(axx + ayy + azz )/3, and the polarizability anisotropy is Da= azz −axx, where z is taken along the molecular axis. In Table 1 we present the average polarizability and the polarizability anisotropy and their derivatives at three different photon energies: 0.00, 1.24 and 2.47 eV. These photon energies correspond to a static field, a 1000 nm laser, and a 500 nm laser, respectively. At each photon energy for each property, the properties are presented from calculations with the three basis sets and various methods. The use of correlation induces the largest change on the values of the dipole – dipole properties, and the largest change is on the derivatives. For example, the derivative of Da with respect to the bond length increases by up to about 80% from RPA to SOPPA, while Da increases by about 40%. The average polarizability increases

by about 10%, while the derivative of a increases by about 40%. The bond length increases by 1.6, 2.2 and 2.3% from the SCF ADZ, ATZ, and AQZ geometry to the experimental geometry, respectively. The corresponding changes in a are about the same, but the increase of Da ranges from about 9–16%. The geometry changes in the derivative of a range from ca. 5 to 8%, and the geometry changes in the Da derivatives range from ca. 7 to nearly 11%. Basis set changes effect the a results by less than 1% and the Da values by about 10–15%. The corresponding changes for the derivatives are a few percent. Frequency changes the properties by a few percent, but the effect is more pronounced for the derivatives. Recent accurate studies of the dipole–dipole polarizability of CO include the basis set convergence and correlation convergence study of Peterson and Dunning [25] on the static a and Da. They used finite differences [26] to compute static dipole–dipole polarizabilities. Their SCF a values (12.11, 12.31 and 12.34 a.u. for ADZ, ATZ, and AQZ) agree well with our SCF/RPA values and their MP2 values (13.00 a.u. ADZ, 13.17 a.u. ATZ, and 13.16 a.u. AQZ) agree well with our SOPPA results. Peterson and Dunning report 3.47 a.u. ADZ, 3.27 a.u. ATZ, and 3.21 a.u. AQZ for

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SCF Da values and 3.96, 3.70 and 3.64 a.u. for MP2 Da values. Their SCF results are larger than ours by a few tenths of an a.u., most of which is due to the geometry difference. We have performed static SCF calculations of Da at the experimental bond length and get 3.54 a.u. ADZ, 3.24 a.u. ATZ, and 3.20 a.u. AQZ. Our EXP/ SOPPA Da values are larger than Peterson and Dunning’s MP2 results by more than one a.u. Ha¨ttig and Hess [27] published a study of dipole–dipole polarizabilities of CO, including higher-order multipole polarizabilities, with respect to basis set size at the RPA and TDMP2 levels of theory. They did not, however, include polarizability derivatives in their study. The largest basis set used by Ha¨ttig and Hess has the contraction scheme (17s11p8d5f3g)/[9s8p6d4f3g], which is larger than the aug-cc-pVQZ (AQZ) basis set we used. Their static polarizabilities are 12.34 for a and 3.20 for Da at the RPA level and 13.18 for a and 3.63 for Da at the TDMP2 level of theory. Rozyczko et al. [28] have calculated both static and frequency-dependent dipole – dipole polarizabilities at the RPA and EOM-CCSD levels of theory with a (14s10p4d)/[7s5p2d] basis set. Our EXP/SOPPA average polarizability results are very similar to their CI-like EOM-CCSD results, but our polarizability anisotropy is significantly larger. They get 13.28 a.u. for a and 4.32 a.u. for Da The change in the polarizabilities with frequency in their study and ours is nearly the same. The quadratic EOM-CCSD polarizabilities are somewhat smaller than the CI-like EOMCCSD polarizabilities. The static polarizabilities at this level of theory are a = 13.07 a.u. and Da = 4.17 a.u. Maroulis [29] calculated the static dipole– dipole polarizabilities using a finite-field method with a (14s9p4d3f)/[9s6p4d3f] basis set optimized for polarizability calculations at various levels of single-reference correlation theory. His average polarizability at the SCF, MP2 and CCSD(T) levels of theory are 12.31, 13.16 and 13.08 a.u., respectively. The SCF, MP2, and CCSD(T) values for the anisotropy are 3.21, 3.65 and 3.61 a.u., respectively.

The experimental results quoted in the Rozyczko et al. paper are 13.08 a.u. [30] for a and 3.59 a.u. for Da [31]. Ha¨ttig and Hess quote dipole oscillator strength distribution data polarizabilities [32] which have an estimated error to be less than 1%. These values are a= 13.08 a.u. and Da = 3.57 a.u. Our SOPPA results overshoot the experimental average polarizability by a couple tenths of an atomic unit but overshoot the experimental anisotropy by a significant amount (1–1.5 a.u.). There are fewer papers that present dipole– dipole polarizability derivatives in the literature. Theoretical papers include [33] and [34]. Ja¨pelt et al. have calculated the derivatives at infinite ˚ with a small wavelength, 4880, 4358 and 3511 A STO basis using RPA. They report 4.86 a.u. for ˚ the static da/dQ and 5.18 a.u. for the 4880 A result, respectively, and for the anisotropy derivative they report 6.45 and 6.66 a.u. for the ˚ , respectively. The static and static and 4880 A ˚ 4880 A results are most comparable to our ADZ ˚ ). Sunil RPA results at 0.00 and 2.47 eV (5000 A and Jordan use the finite-field method to calculate the static polarizability derivatives using several standard correlation methods and an (11s7p4d)/[6s5p4d] basis set. The SCF derivatives are 5.16 a.u. for the average polarizability derivative and 6.92 a.u. for the anisotropy derivative. The MP2 results are da/dQ = 5.92 and dDa/ dQ= 9.33 a.u. More correlation tends to reduce these values. For example, the highest correlated level they used was CCD + ST for which the polarizabilty and anisotropy derivatives are 5.45 and 8.49 a.u., respectively. Maroulis [29] calculated the derivatives with an (11s7p4d2f)/ [6s5p4d2f] basis. His average polarizability derivatives at the SCF, MP2, and CCSD(T) levels of theory are 5.17, 5.96 and 5.52 a.u., respectively. The corresponding values for the anisotropy derivatives are 6.82, 9.21 and 8.28 a.u. Our SOPPA derivatives are larger than the experimental values [24] which are 5.54 and 9.7 a.u. for the polarizability derivative and anisotropy derivative, respectively.

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4.2. Dipole– quadrupole polarizabilities The dipole – quadrupole polarizability derivatives are quite sensitive to the basis set and require correlation, while the dipole – quadrupole polarizabilities themselves appear to be relatively insensitive to basis set and correlation. Table 2 displays the dipole– quadrupole polarizabilities and their derivatives. There is roughly a 20% increase in the magnitude of Az,zz between the SCF/RPA results and the SCF/SOPPA results, but the derivative of Az,zz changes drastically and even changes sign. There is a large change in dAz,zz /dQ between the double-zeta result and the triple-zeta result, which seems to indicate that f functions are necessary for adequate convergence of the dipole – quadrupole polarizability derivatives. The small changes in bond length (compare the SCF/SOPPA and EXP/SOPPA results in Table 2) produce relatively large changes in the dipole– quadrupole polarizability derivatives but only small changes in the polarizabilities. The magnitude of the Az,zz derivative changes between 16% and 33% going from SCF/SOPPA to EXP/ SOPPA and the magnitude of the Ay,yz derivative changes by about 5% using the larger AQZ basis set. With the smaller basis sets the relative difference is rather large, but the absolute difference is comparable to the absolute difference using the AQZ basis set. Photon frequency has a larger effect on Ay,yz than does basis set or correlation. Fig. 1 demonstrates that there are well-behaved trends in the Ay,yz values with respect to basis set and frequency, for a given method, but this is not the case for dAy,yz /dQ. Fig. 2 shows that the ADZ RPA results are ordered inconsistently with respect to the others. Also, the dAy,yz /dQ values do not appear to be near convergence with respect the basis set, since there remains large relative change between the ATZ results and the AQZ results. Recent calculations of the static dipole– quadrupole polarizabilities of CO include those of Ha¨ttig and Hess [27] and Maroulis [35]. Using the basis set discussed above in Section 3, Ha¨ttig and Hess report Az,zz = − 11.39 a.u. from a RPA calculation and Az,zz = − 13.93 a.u. from a TDMP2 calculation. For Ay,yz they get −13.77 and

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− 15.14 a.u. for RPA and TDMP2 respectively. Maroulis [35] using the same basis as he did in [29] described above, calculated the static dipole– quadrupole polarizabilities at the SCF, MP2, MP3, and the approximate MP4 levels, DQ-MP4 and SDQ-MP4. The Az,zz results are − 11.46, −14.24, −13.25 and − 13.92 a.u. at the SCF, MP2, MP3, and SDQ-MP4 levels, and the Ay,yz results are − 13.91, −15.42, − 14.64 and − 15.01 a.u. at the respective levels of theory. We know of no calculations on the dipole–quadrupole or quadrupole–quadrupole polarizabilities in the literature that include triple excitations, which are expected to be important for highly accurate results.

4.3. Quadrupole–quadrupole polarizabilities When looking at trends in the quadrupole– quadrupole polarizabilities and their deriviatives, shown in Tables 3 and 4, we find nothing unusual as we did in the dipole–quadrupole polarizability case. The quadrupole–quadrupole properties change little with frequency in the frequency range we have considered. The largest relative frequency-dependent change is in Cyz,yz and its derivative, although the magnitude of the change is comparable to the other quadrupole–quadrupole properties. Inclusion of correlation at the SOPPA level changes the quadrupole–quadrupole polarizabilities from 5 to ca. 9% and the derivatives about 15–18% for dCzz,zz /dQ and a couple percent for the other derivatives. The 1 –2% changes between the SCF and experimental bond lengths result in comparable relative changes for the SOPPA quadrupole–quadrupole polarizabilties; the derivative changes are somewhat larger at 4–9%. The most striking observation of the quadrupole–quadrupole data is the convergence of Cyy,yy with respect to the basis set. There is a large change in Cyy,yy between the ATZ results and the AQZ results indicating that g functions may not have high enough angular momentum to demonstrate adequate convergence. At the RPA level of theory, Ha¨ttig and Hess get for static quadrupole–quadrupole polarizabilities Czz,zz = 42.45, Cyz,yz = 35.30 and Cyy,yy = 23.79 a.u., and at the static coupled-perturbed

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Fig. 1. The y,yz component of A plotted versus basis set for various methods and photon energies. Each line is defined by a three field description with each field separated by a slash. The first field indicates the photon energy in eV; the second field indicates the geometry used; and the final field indicates the method used to calculate the polarizability.

MP2 level of theory they get Czz,zz = 47.07, Cyz,yz =38.13 and Cyy,yy =26.19 a.u. Maroulis [35] reports Czz,zz =42.29, Cyz,yz =32.24 and Cyy,yy = 23.52 a.u. for SCF; Czz,zz =47.40, Cyz,yz = 38.44 and Cyy,yy = 26.20 a.u. for MP2; and Czz,zz = 46.25, Cyz,yz =37.61 and Cyy,yy =25.51 a.u. for SDQ-MP4. Our static RPA values are lower than the others because ours were evaluated at the SCF

geometries, which are lower than the experimental geometry. The static SCF Czz,zz result is 42.37 a.u. using the AQZ basis at the experimental geometry; also Cyz,yz = 35.13 and Cyy,yy = 23.15 a.u. at the experimental geometry calculated with RPA and AQZ basis. All of our SOPPA static quadrupole–quadrupole polarizabilities are lower than the other values from correlated calculations.

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Fig. 2. The y,yz component of the derivative of A plotted versus basis set for various methods and photon energies. Each line is defined by a three field description with each field separated by a slash. The first field indicates the photon energy in eV; the second field indicates the geometry used; and the final field indicates the method used to calculate the polarizability.

4.4. Dipole– magnetic – dipole polarizabilities Our dipole – magnetic – dipole polarizabilities and corresponding derivatives are presented in Table 5. These calculated properties change very little with respect to the basis set, the addition of electron correlation, or the small changes in bond length. There is, however, a substantial variation with photon frequency. Also, the use of the

gauge-invariant London orbitals [14] has little effect on our results. This may be expected since we evaluated the property only at the center of mass. However, there may be some additional errors introduced in the derivative with respect to bond length since we used finite differences to calculate the derivative. We were unable to find dipole–magnetic–dipole polarizability calculations on CO similar to ours in the literature.

Basis ADZ ATZ AQZ ADZ ATZ AQZ ADZ ATZ AOZ −11.90 −11.53 −11.46 −14.36 −13.68 −13.48 −14.55 −13.88 −13.69

0.00

Energyb

−12.00 −11.62 −11.55 −14.49 −13.81 −13.60 −14.68 −14.00 −13.81

1.24

−12.29 −11.92 −11.85 −14.89 −14.20 −13.98 −15.09 −14.40 −14.19

2.47

−14.32 −14.03 −13.92 −14.68 −14.31 −14.15 −14.65 −14.24 −14.05

0.00

Ay,yz

−14.56 −14.27 −14.16 −14.93 −14.57 −14.41 −14.91 −14.50 −14.32

1.24

−15.34 −15.05 −14.93 −15.76 −15.42 −15.26 −15.76 −15.37 −15.19

2.47

0.42 2.02 2.07 −5.31 −3.69 −3.59 −6.16 −4.72 −4.72

0.00

dAz,zz dQ

0.44 2.06 2.11 −5.35 −3.72 −3.62 −6.22 −4.77 −4.77

1.24

0.53 2.19 2.25 −5.48 −3.81 −3.70 −6.40 −4.92 −4.92

2.47

0.58 1.70 2.14 0.74 1.67 2.04 0.68 1.58 2.00

0.00

dQ

dAy,yz

−0.24 1.54 1.99 0.63 1.55 1.92 0.56 1.46 1.88

1.24

0.41 0.95 1.42 0.19 1.09 1.47 0.11 0.97 1.39

2.47

a Each property is calculated with three basis sets for each method. Properties calculated with the RPA method use the SCF geometry from each basis set. Polarizabilities calculated with the SOPPA method use the SCF geometries and the experimental geometry. b Photon energy in eV. c The field before the slash indicates the geometry used. The field after the slash indicates the method used to calculate the polarizability. The SCF geometries are ˚ , ATZ 1.104011 A ˚ , and AQZ 1.101985 A ˚ . The experimental geometry is 1.128323 A ˚ [24]. ADZ 1.110319 A

EXP/SOPPA

SCF/SOPPA

Methodc SCF/RPA

Az,zz

Photon

Table 2 Dipole-quadrupole polarizabilities and their derivatives calculated at three different photon energiesa

634 G.S. Kedziora, G.C. Schatz / Spectrochimica Acta Part A 55 (1999) 625–638

G.S. Kedziora, G.C. Schatz / Spectrochimica Acta Part A 55 (1999) 625–638

635

Table 3 Quadrupole-quadrupole polarizabilities calculated at three different photon energiesa

Methodc SCF/RPA

SCF/SOPPA

EXP/SOPPA

Photon

Czz,zz

Energyb

0.00

1.24

2.47

0.00

1.24

2.47

0.00

1.24

2.47

Basis ADZ ATZ AQZ ADZ ATZ AQZ ADZ ATZ AQZ

39.92 41.54 41.68 43.28 44.19 43.96 43.78 44.93 44.76

40.13 41.78 41.92 43.54 44.47 44.23 44.05 45.21 45.03

40.80 42.50 42.65 44.37 45.32 45.07 44.88 46.06 45.88

32.26 33.81 34.38 34.01 35.37 35.71 34.52 36.04 36.47

32.63 34.19 34.77 34.44 35.81 36.16 34.97 36.50 36.94

33.82 35.41 36.00 35.83 37.23 37.59 36.42 38.01 38.46

18.64 21.88 22.84 19.95 23.14 23.99 20.09 23.38 24.29

18.74 22.00 22.97 20.06 23.27 24.12 20.21 23.52 24.42

19.04 22.36 23.34 20.41 23.67 24.54 20.56 23.92 24.85

Cyz,yz

Cyy,yy

a

Each property is calculated using three basis sets. Polarizabilities calculated using the RPA method use the SCF geometry from each basis set. Polarizabilities calculated with the SOPPA method use the SCF geometries and the experimental geometry. The polarizabilities are given in atomic units. b Photon energy in eV. c The field before the slash indicates the geometry used. The field after the slash indicates the method used to calculate the ˚ , ATZ 1.104011 A ˚ , and AQZ 1.101985 A ˚ . The experimental geometry is polarizability. The SCF geometries are ADZ 1.110319 A ˚ [24]. 1.128323 A

4.5. Implications of the polarizability results We plan to defer a detailed study of the physical implications of the results we have presented to a future publication. However, it is of interest to do some rough estimates here, both for gas phase Raman and SERS. One can use Eq. (3) to make estimates of gas phase Raman intensities. We see that the A and G terms are multiplied by the wavevector k, which is proportional to v/c, and the C term is multiplied by (v/c)2. For static fields, k is zero; the magnitude of k for a 500 nm planewave is 6.6 ×10 − 4 a.u., and the magnitude of k for a 1000 nm planewave is 3.3×10 − 4 a.u. Clearly, as is well known, these terms have a negligible contribution to gas-phase Raman intensities when the derivatives with respect to the dipole – dipole polarizability elements are nonzero. To make an estimate of the SERS intensities, we need an expression analogous to Eq. (3) but for molecules that experience fields close to the surfaces of metal particles, rather than planewave fields. This will be considered elsewhere [17], but here we note that the contribution of the a term relative to the A term will be determined by the ratio of the derivatives of a to A (with appropriate

sums over tensor components inserted) multiplied by the ratio of the field strength to field derivative. Since a and A terms are of comparable magnitude in atomic units, the corresponding ratio of the field derivative strengths to field derivatives will also have to be on the order of one atomic unit of distance to make the terms comparable. This means that the distance scale over which the field must change significantly must be one bohr. If instead the relevant ratio of field strengths to field derivatives is 1 nm, then we expect that the A term contribution will be smaller than the a term contribution by over an order of magnitude. A similar argument applies to the ratio of the C and G terms relative to the A term. Note also that if the fields are varying rapidly over space, then the Placzek approximation (Eq. (4)) will no longer be valid. We would then also expect that the use of the gauge invariant forms of the multipole polarizabilties [8] would be essential.

5. Conclusion In this paper we have calculated a few of the higher multipole contributions to the Raman in-

G.S. Kedziora, G.C. Schatz / Spectrochimica Acta Part A 55 (1999) 625–638

636

Table 4 Quadrupole–quadrupole polarizability derivatives calculated at three different photon energiesa Photon

dCzz,zz

dCyz,yz dQ

dQ

Methodc SCF/RPA

SCF/SOPPA

EXP/SOPPA

dCyy,yy dQ

Energyb

0.00

1.24

2.47

0.00

1.24

2.47

0.00

1.24

2.47

Basis ADZ ATZ AQZ ADZ ATZ AQZ ADZ ATZ AQZ

12.24 13.45 13.48 14.44 15.59 15.54 15.06 16.48 16.58

12.26 13.47 13.50 14.47 15.63 15.57 15.10 16.52 16.63

12.31 13.53 13.56 14.57 15.74 15.67 15.23 16.67 16.78

14.57 14.02 14.67 14.82 14.33 14.92 15.15 14.87 15.54

15.04 14.45 15.11 15.33 14.83 15.42 15.70 15.42 16.10

16.64 15.96 16.61 17.11 16.60 17.19 17.63 17.40 18.09

4.03 5.07 5.84 3.98 5.05 5.81 4.26 5.46 6.32

4.08 5.12 5.88 4.01 5.07 5.83 4.30 5.50 6.35

4.24 5.26 6.02 4.12 5.15 5.90 4.42 5.60 6.45

a Each property is calculated using three basis sets. Polarizabilities derivatives calculated with the RPA method use the SCF geometry from each basis set. Polarizabilities derivatives calculated with the SOPPA method use the SCF geometries and the experimental geometry. b Photon energy in eV. c The field before the slash indicates the geometry used. The field after the slash indicates the method used to calculate the ˚ , ATZ 1.104011 A ˚ , and AQZ 1.101985 A ˚ . The experimental geometry is polarizability. The SCF geometries are ADZ 1.110319 A ˚ [24]. 1.128323 A

tensities of CO for the first time to our knowledge. We have examined their convergence with respect to the basis set and investigated the importance of electron correlation through the use of a secondorder correlation treatment (SOPPA) for static fields and two time-dependent fields that span the range of the common wavelengths used in SERS. We find that higher-order properties generally require larger basis sets, and are more sensitive to the inclusion of correlation. This is especially true of the off-diagonal A terms, which are not converged at the largest basis sets and SOPPA level. Small changes in the bond length can have an amplified effect on some of the polarizabilities and their derivatives. Most notable in this regard are Da and the derivatives of the A tensor. A 2% change in the bond length produces a 15% change in Da and a 33% change in dAz,zz /dQ. The AQZ/SOPPA numbers we have calculated should prove useful for determining gas-phase Raman and SERS intensities of the CO molecule. This same technology may be extended to molecules like benzene that provide better tests of the field-derivative mechanism of SERS. However, our dipole-quadrupole polarizability results for

CO demonstrate that large basis sets and inclusion of correlation can be crucial to obtaining results that can be used for interpretation of SERS experiments. Obtaining adequate multipole polarizabilities of larger molecules may require prohibitively large computational resources. The results that we have obtained so far indicate that the higher-order multipole properties are generally comparable in magnitude to the lowest-order ones when expressed in atomic units. This means that the ratio of field strengths to field derivatives will have to be on the order of one atomic unit of length to make the higher-order terms comparable to the lower-order terms. In future work we will make more detailed estimates of the contributions of these terms for surface features that are more commonly found in SERS.

Acknowledgements We thank A. Polubotko for several discussions that stimulated us to do these calculations. We also thank R.P. Van Duyne for useful comments.

G.S. Kedziora, G.C. Schatz / Spectrochimica Acta Part A 55 (1999) 625–638

637

Table 5 Magnetic polarizability tensor and derivative calculated at two photon energiesa Photon

dGx,y

Gx,y

dQ b

Energy

1.24

2.47

1.24

2.47

Methodc SCF/RPA

Basis ADZ ATZ AQZ

−0.74 −0.72 −0.71

−1.57 −1.53 −1.53

−0.74 −0.70 −0.69

−1.71 −1.61 −1.59

SCF/RPA/Lon.

ADZ ATZ AQZ

−0.73 −0.72 −0.71

−1.56 −1.53 −1.53

−0.72 −0.69 −0.69

−1.67 −1.60 −1.59

SCF/SOPPA

ADZ ATZ AQZ

−0.71 −0.72 −0.72

−1.53 −1.54 −1.54

−0.50 −0.51 −0.52

−1.20 −1.23 −1.24

EXP/SOPPA

ADZ ATZ AQZ

−0.73 −0.74 −0.74

−1.57 −1.60 −1.60

−0.50 −0.51 −0.52

−1.21 −1.25 −1.26

a Each property is calculated using three basis sets. Properties calculated using the RPA method use the SCF bond length from each basis set. Properties calculated with the SOPPA method use the SCF bond lengths and the experimental bond length. b Photon energy in eV. c The field before the slash indicates the geometry used. The field after the slash indicates the method used to calculate the ˚ , 1.104011 A ˚ , and 1.101985 A ˚ AQZ. The experimental geometry is 1.128323 A ˚ polarizability. The SCF geometries are 1.110319 A [24].

Acknowledgements We thank A. Polubotko for several discussions that stimulated us to do these calculations. We also thank R.P. Van Duyne for useful comments. This research was supported by ARO Grant DAAG55-97-1-0133 and by PRF Grant 29507AC6,5.

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