Generalized Computational Time Correlation Function Approach

J. Phys. Chem. C 2007, 111, 8749-8756

8749

Generalized Computational Time Correlation Function Approach: Quantifying Quadrupole Contributions to Vibrationally Resonant Second-Order Interface-Specific Optical Spectroscopies† Christine Neipert and Brian Space* Department of Chemistry, UniVersity of South Florida, 4202 East Fowler AVenue, CHE205, Tampa, Florida 33620-5250

Alfred B. Roney Department of Chemistry and Biochemistry, UniVersity of the Sciences Philadelphia, Philadelphia, PennsylVania 19104-4495 ReceiVed: October 23, 2006; In Final Form: December 20, 2006

Second-order optical measurements are interface specific (vanishing in isotropic media) in the dipole approximation. Given this approximation, there has been debate, and it is of much interest, to determine the quantitative contribution of bulk media to the second-order optical spectra. Simple estimates and extant experiments have clearly demonstrated that quadrupole contributions can be on the order of dipole contributions for some liquids. However, no definitive set of criteria exists to determine when this will be the case. To this end, a computationally tractable time correlation function formalism is developed that goes beyond the dipole approximation, accounting for dipole, dipole-quadrupole, and pure quadrupole contributions. This theory generalizes an earlier description by both avoiding the rotating wave approximation and including higher order quadrupole contributions. It is, therefore, capable of also describing sum frequency vibrational spectroscopy (SFVS) spectra at low and intermediate frequencies where equally interesting phenomena occur. Further, to date, no implementation models have been proposed to calculate quadrupolar contributions to SFVS signals. Thus, such an approach is presented that is appropriate for use in conjunction with molecular dynamics methods and makes calculating quadrupolar contributions for realistic interfaces possible.

1. Introduction The chemistry that occurs at interfaces is fundamentally important in numerous chemical, biological, and ecological processes. Additionally, the chemical behavior in the interfacial regions of a system is often distinctly different from that in the bulk, and largely governs the dynamics of many reactions. The advent of modern second-order (three-wave mixing) spectroscopies, such as second harmonic/sum frequency/difference frequency generated (SHG/SFG/DFG) spectroscopies, has provided useful tools in probing the molecular structure and motions at interfaces.1-21 Under the usual dipole approximation, three-wave mixing spectroscopies vanish in isotropic media due to the inversion symmetry of such systems.1,22,23 Interfaces serve to break this symmetry and produce a second-order polarization signal. Beyond the dipole approximation, the bulk of a system can contribute coherently to second-order optical measurements through quadrupole (and higher order) effects. While quadrupole contributions can be several orders of magnitude smaller than dipole effects,24 the relative number of absorbers in the bulk versus interfacial regions is large, making the collective quadrupole contribution significant in some systems. Note that the quadrupole contributions from the bulk originate in a region that is roughly the wavelength of light used, while the interfacial contribution is limited to a few molecular layers. The dynamics †

Part of the special issue “Kenneth B. Eisenthal Festschrift”. * Corresponding author. E-mail: [email protected].

between these two regions can be significantly different. Further, simple model calculations by others have clearly demonstrated that quadrupole contributions can be on the order of dipole contributions.25 Also, sum frequency vibrational spectroscopy (SFVS) is being used to examine more complex systems such as crystals, nanoparticles, and nanoarrays, where quadrupole terms may well be important. There is also clear experimental evidence in the literature that the quadrupolar contributions can be important for certain liquids.34,35 However, no definitive criteria exist to determine when they are important. There have been a multitude of careful and interesting interfacial studies of liquid systems, both experimentally5,6,14,17-19,26-31 and theoretically,1,2,7-11,32 using SFVS, a vibrationally resonant version of SFG. These studies have typically assumed the dipole approximation to be adequate in either interpreting or calculating the SFVS signal. Quadrupole contributions have usually, a priori, been assumed to be negligible because they are minimized by taking experimental optical measurements in the reflected (as opposed to transmission) geometry.33 Shen et al. have shown that there are no established general physical criteria that determine when quadrupole contributions can be neglected, and that they need to be investigated on a case by case basis.14,34,35 Further, while experimental determinations (assessing the differences in the signal in the transmission and reflection geometry) can assess the relative importance of the bulk signal, it is not possible to completely separate the bulk and surface contributions.35 Also, the importance of bulk contributions is highly dependent on the particular polarization condition that

10.1021/jp066934c CCC: $37.00 © 2007 American Chemical Society Published on Web 03/20/2007

8750 J. Phys. Chem. C, Vol. 111, No. 25, 2007 is probed.34,35 Thus, in the goal of accurately interpreting threewave mixing spectra, it is to the benefit of both the experimental and theoretical communities to have a general molecularly detailed technique by which quadrupole contributions can be quantified. Therefore, this paper presents a molecularly detailed time correlation function (TCF) approach for calculating bulk quadrupole contributions to the SFVS spectra that is generally valid, extending a previous theory that invoked the rotating wave approximation, making the resulting expressions valid at only high frequencies.1,36,37 The present work also presents TCF expressions for other potentially relevant quadrupole terms that have been previously unreported as TCFs. Further, lower frequency spectra probing intermolecular dynamics, while equally as interesting as intramolecular phenomena, are usually inaccessible experimentally due to a lack of sufficient infrared laser intensity at lower frequencies. However, technological advances in both lasers and detection devices continues to increase the range of frequencies that can be probed in SFVS experiments.31,38-41 Comparison with extant and emerging experiments thus requires a computationally amenable SFVS theory that is capable of describing spectral signatures at low, intermediate, and high frequencies. This is especially relevant because recent theoretical investigations1,8 have demonstrated the importance of probing these regions of lower frequency by identifying novel interfacial vibrational species. For example, at interfaces, hindered translational and rotational modes exhibit well-defined resonant lineshapes; this is in contrast to bulk intermolecular spectra, which are relatively unstructured.42,43 These low-frequency modes may serve as a sensitive measure of interfacial environments in analogy with the use of high frequency, local mode oscillators as interfacial probes. Quadrupole effects have been demonstrated to be important at higher frequencies, and it is expected that they will contribute significantly at these lower frequencies as well. This paper also describes a practical, computationally tractable implementation of the theory, including permanent and induced quadrupole effects. Prior studies7,8,44-49 by us have used a semiclassical approach to calculate the lineshapes of several important vibrationally resonant spectroscopies, including SFVS. In this methodology, classical molecular dynamics (MD) and a spectroscopic model are used to accurately capture the dynamics, the electric moments (dipole, polarizability, etc.), and the derivatives of the electric moments of the system. In general, the response of a system to a given type of spectroscopy is governed by a TCF that is a function of specific system electric moments. Using the combination of MD and a suitable spectroscopic model, these TCFs can be computed, quantum corrected, and Fourier transformed, resulting in the resonant spectral lineshapes. This methodology has been demonstrated to be highly effective in understanding the condensed-phase spectroscopy of water, other liquids, and interfaces.1,7,8,44-51 Thus, given the prior success at modeling other condensed-phase vibrational spectroscopies, the purpose of this paper is to establish appropriate TCF expressions and spectroscopic models to assess the importance of bulk quadrupole effects. This paper concludes with a discussion of proposed applications of the theory. 2. Second-Order Susceptibility: Dipole and Quadrupole Contributions The Nth-order polarization signal a given spectroscopy measures is proportional to the Nth-order susceptibility of a system.1,22,23 Therefore, developing a TCF theory of quadrupole

Neipert et al. contributions to the SFVS spectra requires starting with a general second-order susceptibility expression that incorporates both dipole and quadrupole contributions. The derivation of both the first- and second-order general dipole-quadrupole susceptibility expressions are explicitly presented in appendices A and B to establish the detailed methodology and present polarizability expressions that are later used to simplify the resulting secondorder expressions in the case of SFVS. Letting χ denote a susceptibility, E(ω) denote an applied field of frequency ω, and r denote a coordinate, the total secondorder polarization in terms of tensor components, including both dipole and quadrupole contributions, can be expressed as (2),D P(2) + ∇jP(2),Q i ) Pi ij

(2.1)

In eqs 2.1-2.3, the Einstein summation notation is implied.

∂Ej(ωq) (2)Dq1 ) χ(2)D El(ωp) + P(2),D i ijk Ej(ωq)Ek(ωp) + χijkl ∂rk ∂Ek(ωp) ∂Ej(ωq) ∂El(ωp) q2 q3 χ(2)D + χ(2)D (2.2) ijkl Ej(ωq) ijklm ∂rl ∂rk ∂rm ∂Ek(ωq) + ∂rl ∂Ek(ωq) ∂Em(ωp) ∂El(ωp) q2 q3 χ(2)Q + χ(2)Q (2.3) ijklm Ek(ωq) ijklmn ∂rm ∂rl ∂rn

(2)Qq1 ) χ(2)Q P(2),Q ij ijkl Ek(ωq)El(ωp) + χijklm Em(ωp)

is the total second-order polarization. P(2),D and Here, P(2) i i (2),Q Pij are the dipole and quadrupole moment contributions to the total second-order polarization, respectively. P(2),D contains i terms that collectively contribute to P(2) linearly, and P(2),Q i ij (2) contains terms that collectively contribute to Pi through the . gradient of P(2),Q ij In the dipole approximation, only the first term in eq 2.2 is obtained. Hence, all other terms are inherently quadrupole in origin. Note, the final term in eq 2.2 and the final three terms in eq 2.3 have generally been neglected in the literature when quadrupole contributions have been discussed because they are higher order contributions in the sense that they involve multiple gradients;36,52 they can be important, especially when considering larger systems such as metallic systems, suspended nanoparticles, or colloids, where field gradients would be significant.53,54 The following general expressions for the susceptibilities in eqs 2.2 and 2.3 are given in a form that suppresses the required intrinsic permutation symmetry for brevity; to derive the TCF expressions that follow, the full susceptibility tensors, including intrinsic permutation symmetry, must be considered. (For a discussion of intrinsic permutation symmetry, see, e.g., the work of Boyd.22) Note that µiab (qiab) is a dipole (quadrupole) matrix element between states a and b with a polarization component of i; ωab ≡ ωa - ωb, γab ) γba. γ is a phenomenological damping factor that controls the line width, and is naturally incorporated into the dynamics of a system. (Note that results in Appendices A and B are used to develop the polarization terms that follow.) The first set of terms result from the first term in eq 2.2 and, in contrast to all other expressions below, involve no field gradients:

Quantifying Quadrupole Contributions to SFVS Signals

χ(2)D ijk

)

N

∑ grV

p2

i k µgV µVr µjrg

F(0) VV

(ωs + ωgV + iγgV)(ωp + ωrV + iγrV) i k µrV µjgr µVg

(ωs - ωgV + iγgV)(ωp - ωrV + iγrV) k j µgV µirg µVr

(ωs - ωgr + iγgr)(ωp + ωrV + iγrV)

J. Phys. Chem. C, Vol. 111, No. 25, 2007 8751

+

χ(2)Q ijkl

)

N

∑

p2 grV

ij l qgV µVr µkrg

F(0) VV

(ωs + ωgV + iγgV)(ωp + ωrV + iγrV) ij l µrV µkgr qVg

-

(ωs - ωgV + iγgV)(ωp - ωrV + iγrV) l k µgV qijrg µVr

-


k j µVr µirg µgV

(2.4)

The next set of terms result from the remaining terms in eq 2.2:

-

)

p2

∑ grV

l i µgV µVr qjkrg

F(0) VV

(ωs + ωgV + iγgV)(ωp + ωrV + iγrV) i l µrV qjkgr µVg

(ωs - ωgV + iγgV)(ωp - ωrV + iγrV) l jk qgV µirg µVr


q1 χ(2)Q ijklm

)

N

∑

p2 grV

F(0) VV

ij m kl qgV µVr qrg

(ωs + ωgV + iγgV)(ωp + ωrV + iγrV)

m kl qgV qijrg µVr


-

q2 ) χ(2)D ijkl

N 2

p

F(0) ∑ VV (ω grV

s

kl qrV

(ωs - ωgV + iγgV)(ωp - ωrV + iγrV) µirg

kl qVr

j µgV


(2.5)

q2 χ(2)Q ijklm

)

N

∑

p2 grV

F(0) VV

(ωs + ωgV + iγgV)(ωp + ωrV + iγrV) ij lm k qrV µgr qVg

lm k µgV qijrg qVr

+


q3 χ(2)Q ijklmn

(2.6)

)

N

∑

p2 grV

F(0) VV

)

p2

∑ grV

i lm jk µgV qVr qrg

(ωs + ωgV + iγgV)(ωp + ωrV + iγrV) i lm jk qrV qgr µVg

(ωs - ωgV + iγgV)(ωp - ωrV + iγrV) lm jk qgV µirg qVr


+

(ωs + ωgV + iγgV)(ωp + ωrV + iγrV) ij mn kl qrV qgr qVg

mn kl qgV qijrg qVr


-

+

-

-

(ωs - ωgr + iγgr)(ωp - ωgV + iγgV) -

(ωs - ωgr + iγgr)(ωp - ωgV + iγgV)

(2.10)

mn kl qVr qijrg qgV

lm jk qVr µirg qgV

The last set of terms result from eq 2.3:

-

ij mn kl qgV qVr qrg

(ωs - ωgV + iγgV)(ωp - ωrV + iγrV) F(0) VV

+

-



N

(2.9)

lm k µVr qijrg qgV

-

kl j µVr µirg qgV

q3 χ(2)D ijklm

-

ij lm k qgV qVr µrg

(ωs - ωgV + iγgV)(ωp - ωrV + iγrV)

µjrg

µjgr

-


-

+ ωgV + iγgV)(ωp + ωrV + iγrV) i µVg

+

m kl qVr qijrg µgV

(ωs - ωgr + iγgr)(ωp - ωgV + iγgV) kl qVr

(ωs - ωgV + iγgV)(ωp - ωrV + iγrV)

+

l jk qVr µirg µgV

i µgV

(2.8)


ij m kl µrV qgr qVg

N

-

l k µVr qijrg µgV


q1 χ(2)D ijkl

+

(2.7)

(2.11)

2.1. TCF Expressions for the Quadrupolar Susceptibilities. The general second-order susceptibilities presented above, including the intrinsically permutated terms, represent a starting point for deriving unique quantum mechanical TCFs that describe the response of a particular type of spectroscopy. (Quantum mechanical TCFs can be effectively approximated using classical MD supplemented by a suitable spectroscopic model, and can be quantum corrected using an appropriate reference system.)1,7,8,45,50,55,56 Henceforth, focus will be exclu-

8752 J. Phys. Chem. C, Vol. 111, No. 25, 2007

Neipert et al.

sively on the resonant portion of the various second-order susceptibilities. This is because they provide the dominant, and most informative, contribution to the resonant SFVS spectral lineshape. In this derivation, we are guided by the fact that all time domain response functions must be purely real.37,56 To recast the second-order susceptibility tensors in terms of correlation functions describing SFVS, the dipole (R), dipolequadrupole (Φ,Φ ˜ ), and quadrupole (Υ) polarizabilities, as defined by the first-order solution to the density matrix (see eqs A.13-A.16 in Appendix A), will be used. These are defined as

R (ω) ) ab

1

∑

FVV

p Vn

1

Φ ˜ (ω) ) abc

{

ωnV - ω - iγnV

∑FVV

{

∑FVV

{

p Vn

1

Φ (ω) ) abc

p Vn

Υ

abef

1

(ω) )

a b µVn µnV

bc a µVn qnV

ωnV - ω - iγnV

ab c qVn µnV

ωnV - ω - iγnV

{

∑FVV

p Vn

+

+

ωnV - ω - iγnV

}

a bc qVn µnV

ωnV + ω + iγnV (2.13)

}

a ab µVn qnV

ωnV + ω + iγnV (2.14)

+

}

ab ef qVn qnV

ωnV + ω + iγnV (2.15)

(2)Dq3 (2)Q First, the resonant components of χ(2)D ijk , χijklm , χijkl , and are considered. In pursuit of developing computationally amenable TCF expressions, the following assumptions are made: (1) γrV + γgV ≈ γgr, (2) 1/ωs ≈ 1/ωp, and (3) (where applicable) field derivative terms are symmetric in the sense that (∂Ea/∂rb) ) (∂Eb/∂ra). Note that, for SFVS, the first approximation amounts to equating the frequency of the sum and visible fields, and both the first and second approximations are required to derive the well-known1,9 SFVS TCF that describes the system response in the dipole approximation. Letting χ(2),RES denote only the resonant portion of the susceptibility q3 χ(2)Q ijklmn

,

RES ) χ(2)D ijk

1 p

i ) p

∑ Vgr

∫0

∞

F(0) VV

dt e

{

j ik µrV RVr (ωs)

(ωq - ωgV + iγgV)

(iωqt)

+

ik j (ωs)µVr RrV

}

ωq + ωgV + iγgV (2.16)

〈R (t)µ (0)〉 ik

j

i p

∫0∞ dt e(iω t) 〈µj(0)Rik(t)〉 q

∫0∞ dt e(iω t) 〈Φ˜ ilm(t)qjk(0)〉 i ∞ ∫ dt e(iω t) 〈qjk(0)Φ˜ ilm(t)〉 p 0

i p

q

q

) χ(2)Q,RES ijkl

∫0∞ dt e(iω t) 〈Φijk(t)µl(0)〉 i ∞ ∫ dt e(iω t) 〈µl(0)Φijk(t)〉 p 0

i p

q

q3,RES ) χ(2)Q ijklmn

(2.17)

In deriving eq 2.17 from eq 2.16, the integral identity -i∫∞0 dt eit(a+ib) ) 1/(a + ib) is used to replace the denominator in eq 2.16, the definition of the Heisenberg representation of a timedependent operator is applied to obtain R(t), and the necessary sums over states are performed. Equations 2.18-2.20 represent a generalization (avoiding the rotating wave approximation) and extension (including higher order quadrupolar terms) of an earlier TCF theory.36 They are derived in an analogous fashion to, that is, making the same approximations as, eq 2.17.

(2.18)

q

∫0∞ dt e(iω t) 〈Υijmn(t)qkl(0)〉 i ∞ ∫ dt e(iω t) 〈qkl(0)Υijmn(t)〉 p 0

i p

(2.19)

q

q

ωnV + ω + iγnV (2.12)

+

ab ef qVn qnV

}

b a µVn µnV

q3,RES χ(2)D ) ijklm

(2.20)

Additional simplification of eqs 2.17-2.20 is possible by writing the complex correlation functions, generally denoted by C(t), describing the resonant susceptibilities in terms of their real (CR(t)) and imaginary (CI(t)) parts: C(t) ) CR(t) + iCI(t). When expanded out in such a manner, the resonant frequencydependent susceptibilities can be written as a half-sided transform over their time-dependent imaginary component.1,45 This satisfies the necessary requirement that a time-dependent response function is purely real.56,57 The imaginary component of a correlation function cannot be directly calculated via classical MD; it is only the classical limit of the real part of the complex correlation function that can be calculated directly. However, all two-point, one-time, TCFs have an analytical detailed balance relationship between their real and imaginary parts in the frequency domain given by CR(ω) ) cotanh(βpω/2)CI(ω). Substitution of this relationship into the various resonant susceptibility expressions establishes a definitive quantum classical correspondence, and provides a direct route for calculating the microscopic susceptibilities and spectra for systems of interest. (2)Dq2 (2)Qq1 (2)Qq2 q1 χ(2)D ijkl , χijkl , χijklm , and χijklm can also each be written in terms of a half-sided transform of the difference of two TCFs that are complex conjugates. The rewrite of these four terms requires a slightly different, but relatively similar, set of approximations: ωs ( ωgr ≈ ωs ( ωgV, which is reasonable under typical thermal experimental conditions, and field derivative terms are assumed to be symmetric in the sense that (∂Ea/∂rb) ) (∂Eb/∂ra). Note that, in addition to the above method, (2)Dq3 (2)Q (2)Qq3 the resonant portion of χ(2)D ijk , χijklm , χijkl , and χijklmn can also be written in terms of TCFs using this set of approximations. In this case, the resulting expressions for their resonant susceptibilities are still described by eqs 2.17-2.20. The (2)Dq2 (2)Qq1 (2)Qq2 q1 resonant susceptibility for χ(2)D ijkl , χijkl , χijklm , and χijklm are given by q1,RES ) χ(2)D ijkl

i p

q2,RES χ(2)Q ) ijklm

q

(2.21)

∫0∞ dt eiw t{〈 Ril(t)qjk(0)〉 - 〈qjk(0)Ril(t)〉}

i p

q2,RES χ(2)D ) ijkl

q1,RES χ(2)Q ) ijklm

∫0∞ dt eiw t{〈Φ˜ ikl(t)µj(0)〉 - 〈µj(0)Φ˜ ikl(t)〉}

i p

q

(2.22)

∫0∞ dt eiw t{〈Υijlm(t)µk(0)〉 - 〈µk(0)Υijlm(t)〉} q

(2.23)

∫0

i p

∞

dt eiwqt{〈Φijm(t)qkl(0)〉 - 〈qkl(0)Φijm(t)〉} (2.24)

Thus, TCF formulas are established that are capable of describing the quadrupole contributions to the resonant spectral

Quantifying Quadrupole Contributions to SFVS Signals lineshape at all frequencies. Additionally, this is the first time the quadrupole contribution from the TCF expressions in eqs 2.18, 2.23, and 2.24 have been presented in any capacity. Next, a novel microscopic polarizability model that is compatible with MD simulations is presented to permit the calculation of the above TCFs. 3. Calculation via a Charge-Interaction Model For an isotropic system, SFVS spectra are surface specific in the dipole approximation. In this case, the spectra are proportional to the Fourier transform of a single correlation function: the cross-correlation function between the total system dipole and the polarizability. Previous studies1,7-9,21,32,45-58 have established an effective semiclassical methodology for calculating the total system dipole and polarizability, including induced contributions, using trajectories generated from classical MD simulations. In order to calculate the time-dependent spectroscopic observables inherent in the usual SFVS TCF, a spectroscopic model that supplements the MD force fields must be established. The permanent dipole, polarizability, and derivatives for each species present in the MD simulation are parametrized as a function of molecular geometry via detailed electronic structure calculations or experiment. To account for induced dipoles and polarizabilities arising from interatomic interactions, a point atomic polarizability model of the Thole-Applequist interaction model (referred to here as PAPA) type is used.59,60 The PAPA model’s accuracy arises from its natural incorporation of the dipole and polarizability parameters obtained from electronic structure calculations or experimental measurements and the explicit incorporation of condensed-phase interactions between the polarizable sites. This post-MD calculation of the total (intrinsic + induced) dipole and polarizability is referred to as our spectroscopic model when calculating normal SFVS spectra. In addition to the total dipole and polarizability, calculation of the quadrupole-origin TCFs (that account for contributions from both the isotropic and anisotropic regions of the system to the SFVS signal) require the total quadrupole, dipolequadrupole, and quadrupole polarizabilities to be known. All three quantities can be determined by a novel generalization of the current spectroscopic model. The remainder of this section will systematically outline the equations that need to be computationally implemented within a spectroscopic model to obtain these quantities. Note, the results presented in this section are all in terms of Cartesian tensor components (superscript Greek indices) for a single atom (Roman indices). The various molecular polarizabilities are a sum of all their atomic polarizabilities, repeated Greek indices are to be summed over, subscript “o” denotes is the multipole interpermanent/intrinsic moments, and TRβ...ζ ij action tensor given by ∇R∇β..∇ζ(1/rij), where rij is a vector between atom “i” and atom “j”.61,62 ERi (ERβ i ) is a field (field gradient) including both local and external field contributions, R Rβ whereas Ei,o (Ei,o ) denotes an external field (field gradient). Within an extended PAPA model, the induced dipole is given by eq 3.25. Beyond the dipole approximation, the individual induced dipole moments also have a quadrupole contribution. This contribution is represented by the third term of eq 3.25.63 In the common dipole approximation expression, only the first two terms on the right-hand side are obtained.

µRi ) RERi ) Ri,oERo + Ri,o

Ri,o

qβγ ∑j TRβij µβj - 3 ∑j TRβγ ij j

(3.25)

J. Phys. Chem. C, Vol. 111, No. 25, 2007 8753 From eq 3.25, a supermatrix detailing the effective polarizability, 62 RRβ ij , between every atom pair in the system can be solved. The induced quadrupole is given by Rβγ γ qRβ Ei + ΥRβγζ Eγζ i ) Φi i i

(3.26)

Notice that the induced quadrupole includes contributions from both the dipole-quadrupole (Φ) and pure quadrupole (Υ) polarizabilities. The induced dipole-quadrupole and pure quadrupole moments have been demonstrated to be important in, for example, ice.64 The induced dipole-quadrupole contribution to the quadrupole and dipole-quadrupole polarizability can be calculated in terms of eqs 3.27 and 3.28.62,65 Alternatively, the numerical derivative of the total quadrupole with respect to the field can be taken to obtain the dipole-quadrupole polarizability. The necessary modification of eq 3.28 to obtain Φ ˜ Rβγ (see eqs 2.14 and 2.13) is straightforward.

ΦRβγ i

Eγi

)

ΦRβγ i

γ Ei,o

ΦRβγ i

∑j

Tγζ ij

µζj

ΦRβγ i

-

3

ζη ∑j Tγζη ij qj

3 3 R βγ 1 R Rγ ΦRβγ ) rβi RRγ ij i + ri Ri - ri Ri δRβ 2 2 2

(3.27)

(3.28)

The induced quadrupole due to the quadrupole polarizability and quadrupole polarizability are given by eqs 3.29 and 3.30, respectively.65,66 Alternatively, the numerical derivative of the total quadrupole with respect to the field gradient can be taken to obtain the pure quadrupole polarizability. Rβγζ γζ Eγζ Ei,o + ΥRβγζ ΥRβγζ i i ) Υi i

δ ∑j Tγζδ ij µj -

ΥRβγζ i 3

qδπ ∑j Tγζδπ ij j

(3.29)

2 γ ζRβ 2ΥRβγζ ) rRi Φβγζ + rβi ΦRγζ - ri Φγζ + i i i i δRβ + ri Φi 3 2 3 R γ βζ 3 R ζ βγ - ri ΦRβ rζi ΦγRβ i i δγζ - ri ri Ri - ri ri Ri + 3 2 2 3 β γ ζR 3 β ζ Rγ R β ri ri δγζRi - ri ri Ri - ri ri Ri + rβi ri δγζRR i + 2 2 2 π ζ γ π (3.30) ri rγi δRβRζ i + ri ri δRβRi - ri ri δRβδγβRi 3 In eq 3.29, the multipole interaction tensors Tγζδ and Tγζδπ ij ij are the third and fourth tensors, respectively, for every atom interaction pair ij. These can be computationally expensive to calculate. However, the computational expense can be significantly reduced by (1) exploiting the non-uniqueness of many of the elements present in the multipole interaction tensor for a given rank and (2) using a recursive relationship to generate the N rank interaction tensor from the N - 1 rank tensor. In Appendix C, this recursive relationship is derived. Additionally, symmetries are present in the dipole-quadrupole polarizability: ΦRβγ (pure-quadrupole polarizability ΥRβγζ) is symmetric in R and β (R and β, γ and ζ).65 4. Conclusion A molecularly detailed TCF theory describing quadrupole contributions to second-order optical spectra has been presented. Several of the derived quadrupole origin TCFs are generaliza-


Neipert et al.

tions from an existing theory, while others were previously unreported. A novel methodology has also been developed such that the induced quadrupole and higher order polarizabilities can be calculated using a classical many-body interaction model. Thus, a practical formalism has been established to incorporate bulk quadrupolar contributions to SFVS TCF calculations. Implementing the proposed theory is the subject of ongoing investigation. Acknowledgment. The research at USF was supported by an NSF grant (No. CHE-0312834) and a grant from the Petroleum Research Foundation to B.S. The authors would like to acknowledge the use of the services provided by the Research Oriented Computing center at USF. The authors also thank the Space (Basic and Applied Research) Foundation for partial support. C.N. acknowledges the Latino Graduate Fellowship at USF for partial support.

-it(ωnm-iγnm)

∫-∞t dt′[H′(t′), F(0)]nm eit′(ω

nm-iγnm)

(A.1)

In eq A.1, ωab is the transition frequency between two states a and b, which are eigenstates of the unperturbed Hamiltonian ωab ≡ ωa - ωb. γ is a damping factor that is determined by the dynamics of the system and gives rise to the observed spectral lineshape. The perturbed Hamiltonian is general at this point. To include quadrupole effects, we now specify the perturbed Hamiltonian (q is the quadrupole operator and ∇E ˜ (t) is the field gradient):

H′(t) ) -µ‚E ˜ (t) - q‚∇E ˜ (t)

(A.2)

Inserting eq A.2 into the commutator in eq A.1 gives

[H′(t), F(0)]nm )

∑p E(ωp)e-iω t p

(A.5)

Substituting eqs A.4 and A.5 into eq A.1, F(1) nm(t) can be written as

i -it(ωnm-iγnm) (0) (Fmm F(1) nm(t) ) e p F(0) nn )µnm‚ i

∑p E(ωp) ∫-∞ dt′ e-it′(ω -ω t

p

(0) e-it(ωnm-iγnm)(F(0) mm - Fnn )qnm‚

nm+iγnm)

+

∑p ∇E(ωp)∫-∞dt′ t

e-it′(ωp-ωnm+iγnm) (A.6)

Perturbation expressions for the dipole, the quadrupole, and their mixed contribution to the linear polarizability are required to derive a theory to account for quadrupole contributions to SFVS lineshapes. This appendix presents a derivation of these polarizabilities using density matrix formalism. This presentation goes beyond the dipole approximation (presented previously in the literature)22,23,67 to include both dipole and quadrupole terms that also contribute to the polarization.36 Starting from the interaction Hamiltonian, H′(t), and defining the relevant applied field, E ˜ (t), the first-order term in the density matrix expansion, F(1)(t), can be expressed in terms of the zerothorder term of the density matrix, F(0):22

(-ip )e

E ˜ (t) )

p

Appendix A. Linear Susceptibility: Dipole and Quadrupole Contributions

F(1) nm(t) )

that their thermal excitation does not produce a coherent superposition between states. To isolate the temporal dependence, the highly oscillatory applied field E ˜ (t) can be written as

t Analytic integration (∫-∞ dt′ eit′(a-ib) ) -i(a - bi)-1eit(a-ib)) of eq A.6 yields

1 (0) (0) F(1) nm(t) ) (Fmm - Fnn ) p

(0) [H′(t), F(0)]nm ) (F(0) ˜ (t) + (F(0) nn µnm - µnmFmm)‚E nn qnm -

˜ (t) (A.4) qnmF(0) mm)‚∇E Equation A.3 details the commutator between the perturbed Hamiltonian and the zeroth-order density matrix after projecting with a complete set of states V. Equation A.3 simplifies to eq A.4 when the off-diagonal elements of F(0)(t) can be assumed to be small in comparison to the diagonal elements, and allows the sum over V to be performed. Physically, this implies that occupied states in the system must start in a population, and

[

µnm‚E(ωp)

+ ωnm - ωp - iγnm qnm‚∇E(ωp) ωnm - ωp - iγnm

]

(A.7)

Equation A.7 is the perturbative solution to the first-order power series expansion of the density matrix equation of motion. It is from this term that an expression for the linear susceptibility, χ(1), including both dipole and quadrupole contributions, can be obtained. To generate such an expression, the expectation values of the dipole and quadrupole moments are evaluated:

〈µ˜ (t)〉 )

-iω t F(1) ∑ nmµnm ) ∑〈µ(ωp)〉 e nm p

(A.8)

〈q˜ (t)〉 )

-iω t F(1) ∑ nmqnm ) ∑ 〈q(ωp)〉 e nm p

(A.9)

p

p

When the summations are performed in eqs A.8 and A.9, four terms collectively result. Only one of the four terms is obtained in the electric dipole approximation. Hence, the three other terms are of quadrupole origin. Where N is the atomic number density, the total polarization, P(ωp), is defined by

P(ωp) ) N〈µ(ωp)〉 + N〈q(ωp)〉

(0) (0) ‚E ˜ (t) - qnVFVm ‚∇E ˜ (t) + ∑V {-µnVFVm (0) (0) µVm‚E ˜ (t) + FnV qVm ‚∇E ˜ (t)} (A.3) FnV

∑p

e-itωp

(A.10)

Noting the definition of the total polarization, in terms of Cartesian tensor components, the linear polarization using the derived expression for F(1)(t) is given by

(ωp) ) P(1),D i

(1)D ∑j χ(1)D ij (ωp)Ej(ωp) + ∑ χijk jk

∂Ej(ωp)

qu

(ωp)

χ(1)Q ∑ ijk (ωp) jk di

(ωp) ) P(1),Q ij

+

∂rk ∂Ej(ωp) (A.11) ∂rk

∂Ek(ωp)

χ(1)Q ∑ ijkl (ωp) kl

(A.12) ∂rl

Quantifying Quadrupole Contributions to SFVS Signals

J. Phys. Chem. C, Vol. 111, No. 25, 2007 8755

Here, we have defined the four resulting susceptibility tensors as χ. P(1),D includes terms that collectively contribute linearly i to the total linear polarization, while P(1),Q contributes to the i linear polarization via its gradient. χ(1)D is the susceptibility in ij the standard electric dipole approximation. Both χ(1)D and ij qu χ(1)D result from the expectation value of the dipole moment ij di in eq A.8, whereas χ(1)Q and χ(1)Q result from the expectation ij ij value of the induced quadrupole moment in eq A.10. The expressions for the derived susceptibilities are defined as follows:

χ(1)D ij (ωp) )

N

)

(0) (F(0) ∑ mm - Fnn ) ω nm

N

∑

(F(0) mm

-

p nm

ωnm - ωp - iγnm

∑(F(0)mm - F(0)nn )ω

qijmn

N

∑(F(0)mm - F(0)nn )ω

p nm

nm

(A.14)

( )

qklnm

- ωp - iγnm

(A.16)

∫-∞t dt′[H′(t′), F(1)(t′)]nmeit′(ω

nm-iγnm)

(B.1)

(1) ]‚E ˜ (t) + ∑V [FnV(1)µVm - µnVFVm (1) ]‚∇E ˜ (t) ∑V [FnV(1)qVm - qnVFVm

)

1

∑

(0) F(0) nn - FVV

∑

(0) F(0) VV - Fmm

- ωnV + iγnV)(ωp + ωq - ωnm + iγnm)

×

{[µnV‚E(ωp)][µVm‚E(ωq)] + [µnV‚E(ωp)][qVm‚∇E(ωq)] × [qnV‚∇E(ωp)][µVm‚E(ωq)] + [qnV‚∇E(ωp)][qVm‚∇E(ωq)]} 1

∑ Vpq (ω

(0) e-it(ωp+ωq)(F(0) VV - Fmm) p

- ωVm + iγVm)(ωp + ωq - ωnm + iγnm)

×

{[µVm‚E(ωp)][µnV‚E(ωq)] + [µVm‚E(ωp)][qnV‚∇E(ωq)] ×

(B.3) The expectation value of F(2) with both the dipole and quadrupole moments results in an expression for the secondorder polarization in terms of the second-order susceptibility. Appendix C. Multipole Interaction Tensor and Its Symmetries The general multipole interaction tensor, T, is defined in eq C.1. Here, the Greek/Roman index convention, described and utilized in Section 3, will be used. Additionally, the Roman subscripts, ij, will merely be implied on each variable that is also dependent on a Greek index.

) ∇R∇β∇γ...∇ζTij ) ∇RTβγ...ζ ) ∇R∇βTγ...ζ ) ... TRβγ...ζ ij ij ij (C.1) As the rank of the multipole interaction tensor is increased, the more computationally demanding its calculation becomes. There are, however, inherent symmetries in the form of this tensor that reduce the computational overhead; the vast majority Rβγ of tensor elements are non-unique. Second (TRβ ij ), third (Tij ), Rβγζ and fourth (Tij ) ranked interaction tensors have only 6, 10, and 16 unique elements (out of 9, 27, and 81 total elements, respectively). Additionally, there exists a simple recursive relationship for calculating higher ranked multipole interaction tensors from lower ranked ones. This is illustrated below. Consider the zeroth order interaction tensor, T, where f0(r) is some arbitrary function or r:

T) {[µnV‚E(ωp) +

p Vpq ωp - ωnV + iγnV qnV‚∇E(ωp)][µVm‚E(ωq) + qVm‚∇E(ωq)]} 1

p

(A.15)

In solving for F(2), note that the definition of the perturbed Hamiltonian remains the same (eq A.2), and the additional applied field has been written in the expanded notation of eq A.5 and is denoted using the subscript q. Evaluation of the commutator in eq B.1 yields

[H′(t), F(1)(t)]nm )

∑ Vpq (ω

(0) e-it(ωp+ωq)(F(0) nn - FVV )

[qVm‚∇E(ωp)][µnV‚E(ωq)] + [qVm‚∇E(ωp)][qnV‚∇E(ωq)]}

This appendix presents a derivation of the second-order density matrix that is used in the derivations presented in the text. As illustrated in Appendix A, eq A.7 can be used to derive expressions for the linear susceptibility, including both dipole and quadrupole interactions. Equation A.7 is also a starting point for deriving the second-order density matrix term, F(2)(t), which will facilitate the derivation of second-order susceptibility expressions. Proceeding along the same lines as the linear case gives

-i -it(ωnm-iγnm) e p

p2

p2

Appendix B. Derivation of the Second-Order Density Matrix: Dipole and Quadrupole Contributions

F(2) nm(t) )

1

µknm

nm - ωp - iγnm

p nm χ(1)Q ijkl (ωp) )

qijmn

N

di χ(1)Q ijk (ωp) )

(A.13)

µimn qjknm

F(0) nn )

F(2) nm(t) )

µjnm

nm - ωp - iγnm

p qu (ωp) χ(1)D ijk

µimn

subsequently performing the necessary integration yields

{[µVm‚E(ωp) +

p Vpq ωp - ωVm + iγVm qVm‚∇E(ωp)][µnV‚E(ωq) + qnV‚∇E(ωq)]} (B.2) Inserting the expanded commutator in eq B.2 into eq B.1, and

f0(r) r

(C.2)

1 1 TR ) ∇RT ) ∇Rf0(r) + f0(r)∇R r r

()

)

rR rR rR rR f ′(r) - 3 f0(r) ) 3 {f ′0(r) - rf0(r)} ≡ - 3 f1(r) 2 0 r r r r (C.3)

The definition of f1(r) allows the second ranked tensor to be computed as


TRβ ) ∇βTR )

{f (r) - 31rf ′ (r)} - r

δRβ f (r) ≡ 3 1

3rRrβ 5

r

1

1

δRβ 3rRrβ f (r) - 3 f1(r) (C.4) 5 2 r r Progressing to the third ranked tensor,

{

}

rRrβrγ 1 TRβγ ) ∇γTRβ ) -15 7 f2(r) - rf ′2(r) + 5 r δβγrR + δRγrβ + δRβrγ rRrβrγ 3 f2(r) ≡ -15 7 f3(r) + 5 r r δβγrR + δRγrβ + δRβrγ 3 f2(r) (C.5) r5 Higher order recursive relationships can be derived in an analogous fashion. Implementation of these relationships is critical for optimum computational speed within a spectroscopic model. References and Notes (1) Perry, A.; Neipert, C.; Space, B.; Moore, P. Chem. ReV. 2006, 106, 1234-1258. (2) Kuo, I. W.; Mundy, C. J. Science 2004, 303, 658-660. (3) Ma, G.; Allen, H. Langmuir 2006, 22, 5341-5349. (4) Loch, C.; Ahn, D.; Chen, Z. J. Phys. Chem. B 2006, 110, 914918. (5) Raymond, E. A.; Richmond, G. L. J. Phys. Chem. B 2004, 108, 5051-5059. (6) Shultz, M. J.; Schnitzer, C.; Simonelli, D.; Baldelli, S. Int. ReV. Phys. Chem. 2000, 19, 123-153. (7) Perry, A.; Ahlborn, H.; Moore, P.; Space, B. J. Chem. Phys. 2003, 118, 8411-8419. (8) Perry, A.; Neipert, C.; Ridley, C.; Green, T.; Moore, P.; Space, B. A Theoretical Description of the Polarization Dependence of the Sum Frequency Generation Spectroscopy of the Water/Vapor Interface. J. Chem. Phys. 2005, 125, 144705. (9) Morita, A.; Hynes, J. T. J. Phys. Chem. B 2002, 106, 673-685. (10) Vassilev, P.; Hartnig, C.; Koper, M. T.; Frechard, F.; van Santen, R. A. J. Chem. Phys. 2001, 115, 9815-9820. (11) Benjamin, I. Phys. ReV. Lett. 1994, 73, 2083-2086. (12) Shen, Y. IEEE J. Sel. Top. Quantum Electron. 2000, 6 (6), 13751379. (13) Shen, Y. Pure Appl. Chem. 2001, 73, 1589-1598. (14) Wei, X.; Shen, Y. R. Phys. ReV. Lett. 2001, 86, 4799-4802. (15) Corn, R. M.; Higgins, D. A. Chem. ReV. 1994, 94, 107-125. (16) Eisenthal, K. Chem. ReV. 1996, 96, 1343-1360. (17) Bordenyuk, A. N.; Benderskii, A. V. J. Chem. Phys. 2005, 122, 134713(1)-134713(11). (18) Bonn, M.; Ueba, H.; Wolf, M. J. Phys.: Condens. Matter 2005, 17, S201-S220. (19) Vidal, F.; Tadjeddine, A. Rep. Prog. Phys. 2005, 68, 1095-1127. (20) Lu, R.; Gan, W.; Hua, Wu, B.; Chen, H.; Fei, Wang, H. J. Phys. Chem. B 2004, 108, 7297-7306. (21) Brown, E. C.; Mucha, M.; Jungwirth, P.; Tobias, D. J. J. Phys. Chem. B 2005, 109, 7934-7940. (22) Boyd, R. W. Nonlinear Optics; Academic Press: London, 2003. (23) Shen, Y. Principles of Nonlinear Optics; Wiley: New York, 1984. (24) Adkins, P.; Friedman, R. Molecular Quantum Mechanics, 3rd ed.; Oxford University Press: New York and Oxford, 2001. (25) Shen, Y. Appl. Phys. B 1999, 68, 295-300. (26) Wang, J.; Clarke, M.; Chen, Z. Anal. Chem. 2004, 76, 2159-2167. (27) Tarbuck, T.; Richmond, G. J. Phys. Chem. B 2005, 109, 2086820877. (28) Voges, A.; Al-Abadleh, H.; Musorrafiti, M.; Bertin, P.; Nguyen, S.; Geiger, F. J. Phys. Chem. B 2004, 108, 18675-18682.

Neipert et al. (29) Ma, G.; Allen, H. Langmuir 2006, 22, 5341-5349. (30) Vidal, F.; Busson, B.; Tadjeddine, A.; Peremans, A. J. Chem. Phys. 2003, 119, 12492-12498. (31) Braun, R.; Casson, B.; Bain, C.; van der Ham, E.; Vrehen, Q.; Eliel, E.; Briggs, A.; Davies, P. J. Chem. Phys. 1999, 110, 4634-4640. (32) Morita, A. J. Phys. Chem. B 2006, 110, 3158-3163. (33) Peterson, P.; Saykally, R. Annu. ReV. Phys. Chem. 2006, 57, 333364. (34) Wei, X.; Hong, S.-C.; Lvovsky, A. I.; Held, H.; Shen, Y. R. J. Phys. Chem. B 2000, 104, 3349-3354. (35) Held, H.; Lvovsky, A. I.; Wei, X.; Shen, Y. R. Phys. ReV. B 2002, 66, 2051101-2051107. (36) Morita, A. Chem. Phys. Lett. 2004, 398, 361-366. (37) Mukamel, S. Principles of Nonlinear Optical Spectroscopy; Oxford University Press: Oxford, 1995. (38) Pluchery, O.; Tadjeddine, A. J. Electroanal. Chem. 2001, 500, 379387. (39) Hore, D. K.; Hamamoto, M. Y.; Richmond, G. J. Chem. Phys. 2004, 121, 12589-12594. (40) Hore, D.; King, J.; Moore, F.; Alavi, D.; Hamamoto, M.; Richmond, G. L. Appl. Spectrosc. 2004, 58, 1377-1384. (41) Chen, Z. University of Michigan, Ann Arbor, MI. Private communication, 2005. (42) Ahlborn, H.; Space, B.; Moore, P. B. J. Chem. Phys. 2000, 112, 8083-8088. (43) Castner, E. W., Jr.; Chang, Y. J.; Chu, Y. C.; Walrafen, G. E. J. Chem. Phys. 1995, 102, 653. (44) DeVane, R.; Space, B.; Perry, A.; Neipert, C.; Ridley, C.; Keyes, T. J. Chem. Phys. 2004, 121, 3688-3701. (45) Perry, A.; Neipert, C.; Ridley, C.; Space, B. Phys. ReV. E 2005, 71, 050601(1)-050601(4). (46) Moore, P.; Ahlborn, H.; Space, B. A Combined Time Correlation Function and Instantaneous Normal Mode Investigation of Liquid State Vibrational Spectroscopy. In Liquid Dynamics Experiment, Simulation and Theory; Fayer, M. D., Fourkas, J. T., Eds.; ACS Symposium Series; American Chemical Society: Washington, DC, 2002. (47) Ahlborn, H.; Ji, X.; Space, B.; Moore, P. B. J. Chem. Phys. 2000, 112, 8083-8088. (48) Ji, X.; Ahlborn, H.; Space, B.; Moore, P.; Zhou, Y.; Constantine, S.; Ziegler, L. D. J. Chem. Phys. 2000, 112, 4186-4192. (49) Ahlborn, H.; Ji, X.; Space, B.; Moore, P. B. J. Chem. Phys. 1999, 111, 10622-10632. (50) DeVane, R.; Ridley, C.; Keyes, T.; Space, B. J. Chem. Phys. 2003, 119, 6073-6082. (51) DeVane, R.; Ridley, C.; Perry, A.; Neipert, C.; Keyes, T.; Space, B. J. Chem. Phys. 2004, 121, 3688-3701. (52) Wei, X.; Hong, S.-C.; Lvovsky, A. I.; Held, H.; Shen, Y. R. J. Phys. Chem. B 2000, 104, 3349-3354. (53) Eisenthal, K. Chem. ReV. 2006, 106, 1462-1477. (54) Hao, E.; Schatz, G.; Johnson, R.; Hupp, J. J. Chem. Phys. 2006, 117, 5961-5966. (55) DeVane, R.; Ridley, C.; Keyes, T.; Space, B. Phys. ReV. E 2004, 70, 50101. (56) van Zon, R.; Schofield, J. Phys. ReV. E 2002, 65, 011107. (57) van Zon, R.; Schofield, J. Phys. ReV. E 2002, 65, 011106. (58) Morita, A.; Hynes, J. T. Chem. Phys. 2000, 258, 371-390. (59) Thole, B. Chem. Phys. 1981, 59, 341-350. (60) Neipert, C.; Space, B. Constructing a Time Correlation Function Approach to Calculating Static Field Enhanced Third Order Optical Effects at Interfaces. J. Chem. Phys. 2006, 125, 224706. (61) Jensen, L. Modelling of Optical Response Properties: Application to Nanostructures. Thesis, University of Gorningen, The Netherlands, 2004. (62) Applequist, J. Acc. Chem. Res. 1977, 10, 79-85. (63) Yourshaw, I.; Zhao, Y.; Neumark, D. J. Chem. Phys. 1996, 105, 351. (64) Batista, E.; Xantheas, S.; Jonsson, H. J. Chem. Phys. 1998, 109, 4546-4551. (65) Buckingham, A. Permanent and Induced Molecular Moments and Long-Ranged Intermolecular Forces. In AdVances in Chemical Physics; Hirschfelder, J., Ed.; Interscience Publishers: New York, 1967; Vol. 12. (66) Lorenzi, A. D.; Santis, A. D.; Frattini, R.; Sampoli, M. Phys. ReV. A 1986, 33, 3900. (67) Butcher, P.; Cotter, D. The Elements of Nonlinear Optics; Cambridge University Press: Cambridge, 1990.

Generalized Computational Time Correlation Function Approach

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